Content-Type: multipart/mixed; boundary="-------------1102081820642" This is a multi-part message in MIME format. ---------------1102081820642 Content-Type: text/plain; name="11-17.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="11-17.keywords" Hamiltonian Systems, splitting of separatrices, exponentially small phenomena, Melnikov method, complex matching ---------------1102081820642 Content-Type: application/postscript; name="splitting13.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="splitting13.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.98 Copyright 2009 Radical Eye Software %%Title: splitting13.dvi %%CreationDate: Tue Feb 8 19:13:06 2011 %%Pages: 115 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMBX9 CMR9 CMBX12 CMR10 CMMI10 CMEX10 CMR7 %%+ CMMI7 CMSY10 CMSY7 CMR5 MSBM10 CMTI10 ZapfChancery-MediumItalic %%+ CMBX10 CMBXTI10 MSBM7 CMMI5 CMSY5 CMEX7 CMR8 CMMI12 CMMI8 CMTT10 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: /usr/bin/dvips -o splitting13.ps splitting13.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2011.02.08:1913 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: psfrag.pro 0 0 %% %% This is file `psfrag.pro', %% generated with the docstrip utility. %% %% The original source files were: %% %% psfrag.dtx (with options: `filepro') %% %% Copyright (c) 1996 Craig Barratt, Michael C. Grant, and David Carlisle. %% All rights reserved. %% %% This file is part of the PSfrag package. %% userdict begin /PSfragLib 90 dict def /PSfragDict 6 dict def /PSfrag { PSfragLib begin load exec end } bind def end PSfragLib begin /RO /readonly load def /CP /currentpoint load def /CM /currentmatrix load def /B { bind RO def } bind def /X { exch def } B /MD { { X } forall } B /OE { end exec PSfragLib begin } B /S false def /tstr 8 string def /islev2 { languagelevel } stopped { false } { 2 ge } ifelse def [ /sM /tM /srcM /dstM /dM /idM /srcFM /dstFM ] { matrix def } forall sM currentmatrix RO pop dM defaultmatrix RO idM invertmatrix RO pop srcFM identmatrix pop /Hide { gsave { CP } stopped not newpath clip { moveto } if } B /Unhide { { CP } stopped not grestore { moveto } if } B /setrepl islev2 {{ /glob currentglobal def true setglobal array astore globaldict exch /PSfrags exch put glob setglobal }} {{ array astore /PSfrags X }} ifelse B /getrepl islev2 {{ globaldict /PSfrags get aload length }} {{ PSfrags aload length }} ifelse B /convert { /src X src length string /c 0 def src length { dup c src c get dup 32 lt { pop 32 } if put /c c 1 add def } repeat } B /Begin { /saver save def srcFM exch 3 exch put 0 ne /debugMode X 0 setrepl dup /S exch dict def { S 3 1 roll exch convert exch put } repeat srcM CM dup invertmatrix pop mark { currentdict { end } stopped { pop exit } if } loop PSfragDict counttomark { begin } repeat pop } B /End { mark { currentdict end dup PSfragDict eq { pop exit } if } loop counttomark { begin } repeat pop getrepl saver restore 7 idiv dup /S exch dict def { 6 array astore /mtrx X tstr cvs /K X S K [ S K known { S K get aload pop } if mtrx ] put } repeat } B /Place { tstr cvs /K X S K known { bind /proc X tM CM pop CP /cY X /cX X 0 0 transform idtransform neg /aY X neg /aX X S K get dup length /maxiter X /iter 1 def { iter maxiter ne { /saver save def } if tM setmatrix aX aY translate [ exch aload pop idtransform ] concat cX neg cY neg translate cX cY moveto /proc load OE iter maxiter ne { saver restore /iter iter 1 add def } if } forall /noXY { CP /cY X /cX X } stopped def tM setmatrix noXY { newpath } { cX cY moveto } ifelse } { Hide OE Unhide } ifelse } B /normalize { 2 index dup mul 2 index dup mul add sqrt div dup 4 -1 roll exch mul 3 1 roll mul } B /replace { aload pop MD CP /bY X /lX X gsave sM setmatrix str stringwidth abs exch abs add dup 0 eq { pop } { 360 exch div dup scale } ifelse lX neg bY neg translate newpath lX bY moveto str { /ch X ( ) dup 0 ch put false charpath ch Kproc } forall flattenpath pathbbox [ /uY /uX /lY /lX ] MD CP grestore moveto currentfont /FontMatrix get dstFM copy dup 0 get 0 lt { uX lX /uX X /lX X } if 3 get 0 lt { uY lY /uY X /lY X } if /cX uX lX add 0.5 mul def /cY uY lY add 0.5 mul def debugMode { gsave 0 setgray 1 setlinewidth lX lY moveto lX uY lineto uX uY lineto uX lY lineto closepath lX bY moveto uX bY lineto lX cY moveto uX cY lineto cX lY moveto cX uY lineto stroke grestore } if dstFM dup invertmatrix dstM CM srcM 2 { dstM concatmatrix } repeat pop getrepl /temp X S str convert get { aload pop [ /rot /scl /loc /K ] MD /aX cX def /aY cY def loc { dup 66 eq { /aY bY def } { % B dup 98 eq { /aY lY def } { % b dup 108 eq { /aX lX def } { % l dup 114 eq { /aX uX def } { % r dup 116 eq { /aY uY def } % t if } ifelse } ifelse } ifelse } ifelse pop } forall K srcFM rot tM rotate dstM 2 { tM concatmatrix } repeat aload pop pop pop 2 { scl normalize 4 2 roll } repeat aX aY transform /temp temp 7 add def } forall temp setrepl } B /Rif { S 3 index convert known { pop replace } { exch pop OE } ifelse } B /XA { bind [ /Kproc /str } B /XC { ] 2 array astore def } B /xs { pop } XA XC /xks { /kern load OE } XA /kern XC /xas { pop ax ay rmoveto } XA /ay /ax XC /xws { c eq { cx cy rmoveto } if } XA /c /cy /cx XC /xaws { ax ay rmoveto c eq { cx cy rmoveto } if } XA /ay /ax /c /cy /cx XC /raws { xaws { awidthshow } Rif } B /rws { xws { widthshow } Rif } B /rks { xks { kshow } Rif } B /ras { xas { ashow } Rif } B /rs { xs { show } Rif } B /rrs { getrepl dup 2 add -1 roll //restore exec setrepl } B PSfragDict begin islev2 not { /restore { /rrs PSfrag } B } if /show { /rs PSfrag } B /kshow { /rks PSfrag } B /ashow { /ras PSfrag } B /widthshow { /rws PSfrag } B /awidthshow { /raws PSfrag } B end PSfragDict RO pop end %%EndProcSet %%BeginProcSet: 8r.enc 0 0 % File 8r.enc TeX Base 1 Encoding Revision 2.0 2002-10-30 % % @@psencodingfile@{ % author = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry, % W. Schmidt, P. Lehman", % version = "2.0", % date = "27nov06", % filename = "8r.enc", % email = "tex-fonts@@tug.org", % docstring = "This is the encoding vector for Type1 and TrueType % fonts to be used with TeX. This file is part of the % PSNFSS bundle, version 9" % @} % % The idea is to have all the characters normally included in Type 1 fonts % available for typesetting. This is effectively the characters in Adobe % Standard encoding, ISO Latin 1, Windows ANSI including the euro symbol, % MacRoman, and some extra characters from Lucida. % % Character code assignments were made as follows: % % (1) the Windows ANSI characters are almost all in their Windows ANSI % positions, because some Windows users cannot easily reencode the % fonts, and it makes no difference on other systems. The only Windows % ANSI characters not available are those that make no sense for % typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen % (173). quotesingle and grave are moved just because it's such an % irritation not having them in TeX positions. % % (2) Remaining characters are assigned arbitrarily to the lower part % of the range, avoiding 0, 10 and 13 in case we meet dumb software. % % (3) Y&Y Lucida Bright includes some extra text characters; in the % hopes that other PostScript fonts, perhaps created for public % consumption, will include them, they are included starting at 0x12. % These are /dotlessj /ff /ffi /ffl. % % (4) hyphen appears twice for compatibility with both ASCII and Windows. % % (5) /Euro was assigned to 128, as in Windows ANSI % % (6) Missing characters from MacRoman encoding incorporated as follows: % % PostScript MacRoman TeXBase1 % -------------- -------------- -------------- % /notequal 173 0x16 % /infinity 176 0x17 % /lessequal 178 0x18 % /greaterequal 179 0x19 % /partialdiff 182 0x1A % /summation 183 0x1B % /product 184 0x1C % /pi 185 0x1D % /integral 186 0x81 % /Omega 189 0x8D % /radical 195 0x8E % /approxequal 197 0x8F % /Delta 198 0x9D % /lozenge 215 0x9E % /TeXBase1Encoding [ % 0x00 /.notdef /dotaccent /fi /fl /fraction /hungarumlaut /Lslash /lslash /ogonek /ring /.notdef /breve /minus /.notdef /Zcaron /zcaron % 0x10 /caron /dotlessi /dotlessj /ff /ffi /ffl /notequal /infinity /lessequal /greaterequal /partialdiff /summation /product /pi /grave /quotesingle % 0x20 /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash % 0x30 /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question % 0x40 /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O % 0x50 /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore % 0x60 /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o % 0x70 /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef % 0x80 /Euro /integral /quotesinglbase /florin /quotedblbase /ellipsis /dagger /daggerdbl /circumflex /perthousand /Scaron /guilsinglleft /OE /Omega /radical /approxequal % 0x90 /.notdef /.notdef /.notdef /quotedblleft /quotedblright /bullet /endash /emdash /tilde /trademark /scaron /guilsinglright /oe /Delta /lozenge /Ydieresis % 0xA0 /.notdef /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen /registered /macron % 0xB0 /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown % 0xC0 /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis % 0xD0 /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls % 0xE0 /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis % 0xF0 /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]FontType 0 ne{/Metrics exch def dict begin Encoding{exch dup type/integertype ne{ pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def}ifelse}forall Metrics/Metrics currentdict end def}{{1 index type /nametype eq{exit}if exch pop}loop}ifelse[2 index currentdict end definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{dup sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[ exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}if} forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def}def end %%EndProcSet %%BeginProcSet: special.pro 0 0 %! TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N /vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N /rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N /@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{ /hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B /@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{ /urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known {userdict/md get type/dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{ itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack} if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{ noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{ Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale }if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState save N userdict maxlength dict begin/magscale true def normalscale currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts /psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR/showpage{}N/erasepage{}N/setpagedevice{pop}N/copypage{}N/p 3 def @MacSetUp}N/doclip{psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N /@beginspecial{SDict begin/SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N} N/@setspecial{CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR}{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N /setpagedevice{pop}N/copypage{}N newpath}N/@endspecial{count ocount sub{ pop}repeat countdictstack dcount sub{end}repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N/@fedspecial{end}B/li{lineto}B /rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X/yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet %%BeginProcSet: color.pro 0 0 %! TeXDict begin/setcmykcolor where{pop}{/setcmykcolor{dup 10 eq{pop setrgbcolor}{1 sub 4 1 roll 3{3 index add neg dup 0 lt{pop 0}if 3 1 roll }repeat setrgbcolor pop}ifelse}B}ifelse/TeXcolorcmyk{setcmykcolor}def /TeXcolorrgb{setrgbcolor}def/TeXcolorgrey{setgray}def/TeXcolorgray{ setgray}def/TeXcolorhsb{sethsbcolor}def/currentcmykcolor where{pop}{ /currentcmykcolor{currentrgbcolor 10}B}ifelse/DC{exch dup userdict exch known{pop pop}{X}ifelse}B/GreenYellow{0.15 0 0.69 0 setcmykcolor}DC /Yellow{0 0 1 0 setcmykcolor}DC/Goldenrod{0 0.10 0.84 0 setcmykcolor}DC /Dandelion{0 0.29 0.84 0 setcmykcolor}DC/Apricot{0 0.32 0.52 0 setcmykcolor}DC/Peach{0 0.50 0.70 0 setcmykcolor}DC/Melon{0 0.46 0.50 0 setcmykcolor}DC/YellowOrange{0 0.42 1 0 setcmykcolor}DC/Orange{0 0.61 0.87 0 setcmykcolor}DC/BurntOrange{0 0.51 1 0 setcmykcolor}DC /Bittersweet{0 0.75 1 0.24 setcmykcolor}DC/RedOrange{0 0.77 0.87 0 setcmykcolor}DC/Mahogany{0 0.85 0.87 0.35 setcmykcolor}DC/Maroon{0 0.87 0.68 0.32 setcmykcolor}DC/BrickRed{0 0.89 0.94 0.28 setcmykcolor}DC/Red{ 0 1 1 0 setcmykcolor}DC/OrangeRed{0 1 0.50 0 setcmykcolor}DC/RubineRed{ 0 1 0.13 0 setcmykcolor}DC/WildStrawberry{0 0.96 0.39 0 setcmykcolor}DC /Salmon{0 0.53 0.38 0 setcmykcolor}DC/CarnationPink{0 0.63 0 0 setcmykcolor}DC/Magenta{0 1 0 0 setcmykcolor}DC/VioletRed{0 0.81 0 0 setcmykcolor}DC/Rhodamine{0 0.82 0 0 setcmykcolor}DC/Mulberry{0.34 0.90 0 0.02 setcmykcolor}DC/RedViolet{0.07 0.90 0 0.34 setcmykcolor}DC /Fuchsia{0.47 0.91 0 0.08 setcmykcolor}DC/Lavender{0 0.48 0 0 setcmykcolor}DC/Thistle{0.12 0.59 0 0 setcmykcolor}DC/Orchid{0.32 0.64 0 0 setcmykcolor}DC/DarkOrchid{0.40 0.80 0.20 0 setcmykcolor}DC/Purple{ 0.45 0.86 0 0 setcmykcolor}DC/Plum{0.50 1 0 0 setcmykcolor}DC/Violet{ 0.79 0.88 0 0 setcmykcolor}DC/RoyalPurple{0.75 0.90 0 0 setcmykcolor}DC /BlueViolet{0.86 0.91 0 0.04 setcmykcolor}DC/Periwinkle{0.57 0.55 0 0 setcmykcolor}DC/CadetBlue{0.62 0.57 0.23 0 setcmykcolor}DC /CornflowerBlue{0.65 0.13 0 0 setcmykcolor}DC/MidnightBlue{0.98 0.13 0 0.43 setcmykcolor}DC/NavyBlue{0.94 0.54 0 0 setcmykcolor}DC/RoyalBlue{1 0.50 0 0 setcmykcolor}DC/Blue{1 1 0 0 setcmykcolor}DC/Cerulean{0.94 0.11 0 0 setcmykcolor}DC/Cyan{1 0 0 0 setcmykcolor}DC/ProcessBlue{0.96 0 0 0 setcmykcolor}DC/SkyBlue{0.62 0 0.12 0 setcmykcolor}DC/Turquoise{0.85 0 0.20 0 setcmykcolor}DC/TealBlue{0.86 0 0.34 0.02 setcmykcolor}DC /Aquamarine{0.82 0 0.30 0 setcmykcolor}DC/BlueGreen{0.85 0 0.33 0 setcmykcolor}DC/Emerald{1 0 0.50 0 setcmykcolor}DC/JungleGreen{0.99 0 0.52 0 setcmykcolor}DC/SeaGreen{0.69 0 0.50 0 setcmykcolor}DC/Green{1 0 1 0 setcmykcolor}DC/ForestGreen{0.91 0 0.88 0.12 setcmykcolor}DC /PineGreen{0.92 0 0.59 0.25 setcmykcolor}DC/LimeGreen{0.50 0 1 0 setcmykcolor}DC/YellowGreen{0.44 0 0.74 0 setcmykcolor}DC/SpringGreen{ 0.26 0 0.76 0 setcmykcolor}DC/OliveGreen{0.64 0 0.95 0.40 setcmykcolor} DC/RawSienna{0 0.72 1 0.45 setcmykcolor}DC/Sepia{0 0.83 1 0.70 setcmykcolor}DC/Brown{0 0.81 1 0.60 setcmykcolor}DC/Tan{0.14 0.42 0.56 0 setcmykcolor}DC/Gray{0 0 0 0.50 setcmykcolor}DC/Black{0 0 0 1 setcmykcolor}DC/White{0 0 0 0 setcmykcolor}DC end %%EndProcSet %%BeginFont: CMTT10 %!PS-AdobeFont-1.0: CMTT10 003.002 %%Title: CMTT10 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMTT10. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMTT10 known{/CMTT10 findfont dup/UniqueID known{dup /UniqueID get 5000832 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMTT10 def /FontBBox {-4 -233 537 696 }readonly def /UniqueID 5000832 def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMTT10.) readonly def /FullName (CMTT10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 53 /five put dup 55 /seven put dup 57 /nine put dup 58 /colon put dup 68 /D put dup 84 /T put dup 88 /X put dup 97 /a put dup 99 /c put dup 100 /d put dup 104 /h put dup 112 /p put dup 116 /t put dup 119 /w put dup 120 /x put readonly def currentdict end currentfile eexec D9D66F633B846AB284BCF8B0411B772DE5CE3DD325E55798292D7BD972BD75FA 0E079529AF9C82DF72F64195C9C210DCE34528F540DA1FFD7BEBB9B40787BA93 51BBFB7CFC5F9152D1E5BB0AD8D016C6CFA4EB41B3C51D091C2D5440E67CFD71 7C56816B03B901BF4A25A07175380E50A213F877C44778B3C5AADBCC86D6E551 E6AF364B0BFCAAD22D8D558C5C81A7D425A1629DD5182206742D1D082A12F078 0FD4F5F6D3129FCFFF1F4A912B0A7DEC8D33A57B5AE0328EF9D57ADDAC543273 C01924195A181D03F5054A93B71E5065F8D92FE23794DDF2E5ECEBA191DB82B3 7A69521B0C4D40495B5D9CE7A3AF33D17EE69979B82B715BAD8A5904C5DE0260 6C15950CCF6E188A0CDF841EB68E5A2F88253E382140F87C87E55C9EA93B8C89 14A36CDF630D6BE7CD36DBDCE22B21778E8648B97B7EC6742EB5114BDF0454B0 0EA7B1FE236C84C0E5308C871F67B973892890557AA12E00B2C20C71F516C397 3F3BBD14A1D0149CA064391056E45E9470FC7F6F556ABC82653B3C8049AB5CF4 BA83C8F2158C236B2FFD4208846013BAF4165E8BB8D334C8FF2E8D74AF5DAB2F D44788869B08399421AAA900ECC6A2D594641C121660D4B5F512938994C18DD0 FCD9B008F68F0351D21ED735B2740CB1E0C1CCD25EB548C35B844601D98828DB 556F71D07E081A593FF12DAF83676492A0FFE16E95717A07082B43A966C1EE8F 8A59E1255E1705C43A23CF29A5E4A6547C93F1680A870EE7BAD8CF74D838CD5E F806911D8FE4262ED8E7F5BC58B92C9C6D74F8AD45FBB021EC7E97393018B9DB B1B84E7B243ADB05ADD3F1DB3692ADC5D47FEC7DF93080669E63281F1576B673 125EDF08016664BE73364F65389F7C3B66623AD1754ECBEF9E5CE6948D933787 A5674279ACB2EBECD3B4E6361419AB32028A27670C9F3E18B746A10B00AF6D77 4EC00E3BE521C02A99AE5BAA98F793EB1228952BE67934B91472E01AF7B816BC 56D7F19F631A1927846D800C107B1E9CBFF9D2DD513B4A8CE2E0DFD77B1ED178 E43FA7052765E9FAF89989D490D8FEF6C536EC0D4AE27A74F474B98DA9E6B92F 15E063DB260571979A5DE2423920CE1F59F56EB11E00E3BB9D466A8263E1E385 2014BEFDA8D1EA3EDA04BE32AEE6CD15C5C010A1DF7F705A2C0C18E87C8DCCE9 05D9163181CBA56C0FAC8C06A2990554C8E759D076B01BBEADE3B5FB8B551390 6C8E4A2A1C6E7D9C708614626F3770C0AB7DD2027469C77975C27576065862AD 04E5E50CEBE907E3E991FA0C627302C0E207B4D5992BEBAB5853AD1C0D271728 C76F40A79392ACCA7358F948AC65DC823CFDA59E1FF69CEBB6B7EC3CF21669E4 70D999508F9C49E2D9F8818CA53C977D93E15FBBBAF75B1E84F0BA62BCC4BAFA 4EEC82D804C8A8C0210F3E5E258BB1F6921AF02BA9861BAD5C3D5FC8CEFABA8A A607E547B802096F7AEB09FBA99C83C9A494B94408DD607CA6561A6E6660C473 62CF8D35F31D052F6C6C8138A8E1430CBA7EA6973D6D510C1A06B3FBD79D9364 240C1A00272DA44B89A9FE8D5BF36DC1B5EBB4A78ADBE9C5EDB485F093D9517D 69E1AC9A8E6C9D7C324E3797CFEAD9A18E82E03F69B2CED7D5DDCD1A218BF2E2 ED2293AE999FE2A4B5213A10083EE0407BCF8007670B8C737EAB30311C868D84 121149ACB4A27F3ED6C0C181C98AAAF51B105F264B5672D7F745131ABAB5BEA4 0C9B43C0DD9116D6DC61F90BE72018F290D26D5E9D341055CAF09C9F45333CDB D45B7954271767F638EEC499F7B53C2CC5774EA7A7F024C4CABFB93D9CB1856A 0C671A4ECA7C62EA5242648A84E7F3AFB9547A0AFC29593CFCE6D8B873A78157 D337CABD291431C0A2CE1F37E0CD7340567AC206FF98E4B5A6410F70F750451C 550EFB54AA259A1B236CA9CB730D2CEF125EC65D959441F7CC9768F777B44844 CC9842A307C72B740680ACBBF6AA35FA7A94825069BF7696ED81A371A9E5475A 9D997F2DFAD339AADF797F7E03E654234455AC3D17702A420EE0A597BA31BDE4 FEB8DBA7C61D311CC90441A620164DC22DC2D373973EF84CC553453AB1B3337F 7B39983B8DFFB3A9425F119B45C1CD37A76F905777B3154CA6200792F1759D06 E017890F4041A385F2238E3C48B6C8EE6F5258463FDBFF7AC762F6C4363926D6 50F004D473B7B7F73CA686B559C2885F1AA761653C727A77D73431E9D110E76A 2E55C68CD50F43997C9B2FC4710F8C8540909829E215678E63BB8363C4B8AF05 9986102BB36580D9CA95CD216B7C321822CB41B2E0422CD077F3B55E0246FDB2 44D5976F67296B5B0BE4B06F6E43535C21164E6C5089C3E9BA2D6B30888C57DE 49DC8D9D46C0D5EDC47ACF2C03B72DE3B69512508539019B759280BABEA12BC9 385308A0395C4CD33182A10A5A229743379C2075D82D8BFCE4A66E1AA087A091 8F5372684FA5037D1B92D50CD9CB4F50AD4F8EE7D51F1C9E63C721CB5B9BD011 6F0A8DD4FDCD2B008F223A1036D90F0F3B252487DE7898F9AFBB3A9D9CD49E0C EF4ADAD5155A98D2125ED5A3D3907F67301649519419F33CD942E8DDEAC1BDA0 E90C431B198F646766A8FA9F8D1561B57E126EF604838C0C1966655CF31FB7EB C8CCC434FC1C96046D38203E1791EC824A3D7AED85C029288D4608CA7668A2BE 484C99639F121845B22EEFCE0A3B808261921AA042AE19E641769E91277BEC29 4594082CCB3058F90FAC4A700A8A827ACA00FCF574ABC8EB7DBCECD97F2B22C0 0AA19E8739B81AF8C6F621D69B8E6F29BAE233FBA655A0AF5BDFD7F5C6B9167C 6BC7AB693D45EF2AD999F5DA3CEFA39BA48A17EE6D9F2C4DAB91AE3F0044DC3F 5D5506CE4675AA928B0092D6F173644F91295216D8BBB14CDDE0AD524A4D545C 1B5E284A3BF0396664081CFB4F186A84A0D24D61E82F4767C1E55A0642720CF3 909FA1AB8EAB78030B59BEA067DEDBD2F1D0340E790AB2777DB18248521934A8 BB38A58B7F633DEA4291B0D5D13E9A882C974697CC6D3B49E030C94EA29B5506 CC29C44D01B4751B453A46A9F6BF3BF135AE87A4CE232AF57B66578310DE41E0 2A6AC422117F1963C4D7CC306BD25A6E724E51921779F22F029733122E23E2F0 CB340008813ABB104380C80A492B3FC6D0BB07CB8D8409E9576891EF6E5C9D08 EB8320DFA31BAFFBD336D0C2BBC3D3B2D30368B9860768FC080D30569C7F7811 0EBEDA2962476113625EEB555490B8CE4C5F99D74ED10F738C61854CFF8B41C6 9402E56BE8856144A1A05D0B05F4CB7EF728B2F4F5A439F18C3B68CEFA41E59A D8308ADC92EC1289DC84CF48D2CDEFF509A145BF945E1E00D552D329EBD2A7C4 21D58082CC8FA790E981F4AC8EAB99950678FD3A7DA3DF13778681B208DD71A0 7C3CBD0664B37C9EDC6B601D79A2C51FB54DAEE849F93209793849104E722D3F 52DFAF7047EEEDDFE744787A5801E4AC2C3D58EC5DDC15FCEE03990C53B0C57A FC54F125A04C8E4A0ADAA725808C587E7DAFB9F784FA2875689979D316DC22BD AA36B306A1ABCF907B63C6476737B746099973CAEA8C1E2C5C41F27E0F7DE8D7 F0D942E34E92F43FE902653D4D2EBB6F3B9F7928B1550A82AF234D45D028F429 067652BD3D391BF423AE72B9CB1E8D91E898161BE3A7849D456A861A2046711E E934DC59442AE7D81661CE8EF727D8D7DDC0270E937E40F896AEAE6171661431 C1025C53172F9D366834BA0054FBFD84503FBAE328B6FDEA180F8EA35B1DA937 5CC3B8F00C206908C2FFFFA6A7AC6915D15EA44BDCF29E2BFCFD4A849535F19B 0D307C696BE8205C7D84B9C77F02EF27D911056EDBB4080E4D3ED72788666CAD CD91B0ECE27A177DB23320A7FA9C31408B4D02D2A4B1CC6DDE1A6CAC3D8EC1EC 2226EC98E51046D1EC26FA20EE62D24747D83CF4941DCE5CCEEC0DBE387149CD E05B19FFCAFC0D117F9A3E60DCD4C815228D98EF95EB559AD0ACC0D50FFDF714 56C3C812EA5ADBB013BBD956A7C4CC0ED7D3E25D5C9AF5E626F18297F75D4957 F5B0B33379114B903FE98BCF35C3FF76FEE1D9AEB711F2962276531F7380EE3F E368720E0292A170A15C5539B1FC7BB954EE2624B504CB8C805B8D31AC38307F 0513606F09211AE64DAC447693B2A0AD15E9A64C34F5A911ECD0ABCA90E9791D 67C6BD202B0858EF96E7722305B8AC02B01AB1706CC6AE875A8DDD15EE349046 EAA65005E7866B506EDFB7A5A2AFD5C9E9DCC821A79EE9C1EA2C7BBA32A40BC7 CEC26DB1AC473C8C3960ACEC581B37D6569E8C8C42950BAB7930B65E1570E3F8 9A7FA719F1DCFDA45A3BF2AAB32C9A93BA3552608A61C623DE59BCB346E87EF5 9CF025A87803161221C5C1C6F6B3403712C76E9D755C7BD68D7F2DC03C14CDF0 C1BBED1D648B905B4B17037B7263C1EA7A7F06FAAC4E09E08483A8D714C19861 327CD9C32DDF850302DD6DDE24912D00C22ECDF3CDFB18FA831A41A7488EC203 F564CFE30D506F0829A96D35A7E09C3DCD107D589B627A15B55C5D6649126BEC 60B88C55ECCBB4E680265D9EAB4CE22965D3B1AF759B01ACB0D0E6C92B6B4EFD A81E6A648708979487FC591CF09631310D46891423F4EC159A73E30D8DD147A4 B0EACF6D45D18CD16CEB8176F03ABCB41F2234747B9733C8FAF34AE5D43D3BA5 0CE0FACFC9B087F84FB6C68678BC6E76022B1526D6E5B3A48EC1A110BD75F45F 1C4DC6D39F254976453F57DF873B7D635C80C42026DE020E5BAFE0DA0D54D1E1 DC634D2621BA184347E5252F645A6A1DB7657C48124186F0E4C644077457C24D 55753C651A9A7B6349867641464B515B821349C795A645420508673B93750D0C 7A3B33EB1F09782033742AE8F3A23FC02284E6C03818FADD1731361542E3FA3E 75B8D52B668C3E18A4AE967D0FC3157083D952AFB8144D549E69EAAC51C279C5 E5D88A0D9D53013DFFB4352A1598FF84DCDE6FA32FC377306B9B92C0F96EE149 8CD55E7B2445B86CCA7A547FA732D52D59025129FD8C6333AC0DF4F0CFF6287E F2036D5DBBB3B91B92F12FEBE0B61A313A4DB5A9CF0BB3DDB781A56FEBFFACCB 8CB9D1D3DBDBC4CB6AAE6769E470582403CB920630221B68BCB625CD4605FA8F D3D5B7A1A28D15E44B38E92E906C138E72C15B86F64C38E23BF0440052A8C914 54397F49DBED99D0AF7CEA3B0A05FF37C2D7EAE1412567E6776333237C31E3C0 49949EC8BFD6E0F6446CE2D4DCD2C1524A288818CC5D159BF8463A847AE4A2B9 CC8C58F822804B81B13BF4F2DEB6229C4F51F093075581791D02C36A13B855A0 34900AA7CD4F1A797652656FE3A8425A38F421C4CC0ACA1CDD44FA6B31219276 1CDE1CD63D6A58CE705CB56CCA1260F9B86E989019071563A9B4C274A87558CA 6EF1660D574EDA276801F0057740E2C3B80D253D697736484D892CE1AB128B8A DECD69712F5E70E895FBAA927E8194D792A04AB6CE205E04E38A433BBB793FB4 E8BBC4279D58A223C6673D909D6AFECD246E66A52F4CB35E5931D24C828489BD 4ECAF621A220D8ECF702BEB01C4FC7510197D3F6D15321EC87175ADBA6434ECD 2B5A306E91375CAD22CD94301763E4A8B981472890422C5488FCD523C9CB17DC ED22FBF12D5F7525D0D6BCFE8CE85B0DFB1D6F989C267FFBA0A996D309E4A934 3DB54A9D29C88B9D55D7300DA3D46419256C5A07A2A529A8DE8BD1727281F5FE 97033D861E0531B14E811378EC1AF1CC7EE9BA2B07D935843D3053F673979F8C FAFD59D555B56CE338F606747238B22BD62C42BB7238FEA335678D474A643570 A9E7B4970E8C541CE9DBC7BF70ED7BA33639D6744A18379455029E934C95E2EF 639C4848CE9A0879B51649FAB023A71782444B451F92A34CB8A124270CCF86D4 D18EEF5C1D2B2A29012613851C49F50702D63BACF95EE2AB4D72B375E0A62615 E0991E130A67ECBA9E05329B740708F1CB148724C3A6E5E3AEC1F88EBCA398D2 1CA8827C977D72734310233176D1AE26C55CF2CEACA62223315C28FCF6305C7E A22414D4739A059F552F1F9372CCCA5FED4F9AC987942848EB498900269511F3 F408CBEA0659B954F5F1B18AE4FB270213646F9B28AE4439D2BA2D3E0AAAA780 5E530E4EFC8A060EB979E12191044509DA0C14397AFF949E12DC970658D5EAF5 4EA963F5BC1407A32F3837CA6A24B7F3D60EB8E6222B702E25ED903F9D21AE50 664A095009BDEAF4B78DAF94E5A55D48366CABF07791A1684B2F54EA69070844 4F031AF8DF416C2D3679F8BA038B0DC9DD0400CA6B34667BCBBC07E62C1668A8 35A8C57C9048A7227E672E89681B54D662079A189A9E96A3CA96D8DD10189B04 1DA49BA2729F1CA585B1BD5C467295285D52E47CA904235A1A3E48EFAE9EB6F6 01374125CE89D53C276858668CF45D2F092DDCAA52418E0BB94C2B8266B4D88A 5D911507BB1DDA3D8F6E7C14A91CA11AE799EC42E993098E18CADA70BD2A1D82 2C39326C6E3F9E84CD9758B9AE43D79BF99E6A0CD713E95B3D9B7DB90D127DE0 DAFEBF850CAAACBD860B5DEF2082F1ADA64B44B193C4A1417BE221FDCA36456C BE5934C8CE3ED55AE3A11697C2D682B7D0F72D48976451D205783BE25DBD2507 39C14FFB4BB828DFD187104F38A7F11D5F0698C11E8C1D4F107CACE573FDC4B1 C56FDAE47024D6FD16A2FEABB434CA320300FC4B6C1B6CA08F76C60B7C08A665 99F404DBA8A2A1EB18EF6750E4EC186E31561A3F080BA6562967546715859481 7BA782940F5C5D06626D6F6A412CA7C13820EC7C1DF23E15E5829F698CF617BE D940523E4EE4ADECEC48C24297DBAD528BA1DCE7AC335A1D15D55415B108EFC8 6D45030D27B3EA63B2B4CD771DBE66AE0218ABB1153D4B7482289D1313CEF184 5C960B1E3C3C953912CC6F4521D1E15636C1545EEE457EFB87B88C9E43CC2F38 6BC4BC96969F4FF28ABB06F4454C01CEF1B6DC538F1E832FC1666D977E5A881B F72F1B4C7DD4BE167A5535F1163A0706F9A0B26400178DF8A128FB5EBE6A7B81 E478AD183EC06622B591337B9F1872AAEA356F4FC67EE767B34CB5A4D90702D9 39FB846947F4096FB3DCF16EC81455164783BA0B5D723060DAFF411B68307E81 7BEA1D9A47A5AA3D648E618C83C60F060029E6EC4D46B045FA7415BAB2AD0AA5 ED9C729C24136F6AF61E6409C0B5CA760B16225641E268A68CFB8260BBEAFC77 6626EBD97195E77CAB425CFB0096D805D9EE699E41680D095AE9FA10122A7882 2F00F495C9EB2102DF0D3E61833BC0A2E468C5CF7AB430FDB7C0BE3DF2C0D230 1580BAA25D65F599378D873165482A1FBB224AEA89C6BCCFBDBA42AE1C5DCF41 06969F585CD3B737D1388D6359F5468D88FCD2279BDB270F6A858FB7D2ABDEFE 5EE8FB79FA437F8F50237B92C307B73B0DCB808D07A9C3255CB9B3B17039CE5A 288103D05D132863FB522A02CEE3839EF9AF7F07D99732F0B8B384745369FB3E 7901166478F4A16076A1504C5E98D17408494E270BBF4470ED12B4332422679F 759F1D93984D7E506D16950DB6C2682FE1379EFFA6F6C95DD71F6E55BE3EF6AF E0CB25388EEB436E6527806FC75484133F6E561DEB979D5C1FFEFDAF2A6D964E 03BAE0BD593C2992AD84569C81050F7A793C5263E50C2F50B98C4CC703EAE17A 6AEDAACE312DAFAF5278D125B6EFC5587484F61DAFF46B87B7C9B1EEDECA4859 314A9A9E2248467DE1E54D90DD671660B9040B3E0DD982260822177EFD757266 74A16C83A7FB168016A320D3DF3BD7726F1F4EC90EE5DFE810C96B099FD4368D 906AE4699049EFD37E8EF058D4B97BF71106445AADD4FC6E90615A0066823A36 673B8DE32322BBE861AE251226B4385AB28702831270DBD25D666FBB0AD7B96E A44E891EA1EAF0F87013AFC982E33D67A28E96E0C9CB99B9E4192536830D9901 931A8CAFA41289633B20BA3BD7AA3414B6DA8D57CCF2FBE39920CC06361F075B CC40335DB9A0071CFF77F6B7BB47F3100DBDC9C4A58C2B81EC99E8E966AF3390 E3FBCC28BA1D79961C8A1584266454DF772FBA99664D74D4A89FC82FFEDFCFE1 4C9E4A04291E803D142E37E7ACA66AB279378F2F192FFB2B5BBAD18B95F03136 2CB594A3D6D3F8576B90A6C4DAD6D6C8EE07AF682F925F01D0B26CBA347C03BE F3B0585CF4539FDC66915E22117078CC94D621F31DCB3E021998A5D6EE94CA4B E214D07517283D56973D8E4367392BF6C1150DEBF459D141AE0941C1C8C5CFBE E735D796E365A1B0F60BB4CF2801EAFE4889EE5F338D3C4885368281B3C95CCE 251C28A90D318A8A0384439B38D63B94757252062EA44E88509FDD2E75FAAB71 7329622828B2785C1A8B26351BC7448C1719C88FE99BCB73F7DEA427FBFCDF4F 00EE079B0C712F7D2C8DF98D4830A9D8C8B70A8C5D54BC1DF3171135278BED55 DA1CF0E696B12935EB59B606AA3C0CC50C7A259AF32F4C81D9B39470D9993CBC 8644403D2833B34AF40511654F96BCE0064860F39849DC65622EDFA8E6EDE9A4 7DA62658B43CBCBC5E8F53D4FAE6BD0253C0C81E6C53CF5C3A9E3FCCD9AE773C FFEC1AEB6A3E1BF76F90C6748E41DEDD0B2CA1D43717DC554F6A3B9BF2392D24 BA65BA823265CB453323E8EFEDDBC5D2ADFB60F744206A028BB73B27F0672208 1B1E35E982901815AD345642C5AFB06E6C5B432D12D11D0E9012C6B763746FC0 A2B42143BC174C525BD3FE52BF99C9C301B7E591CDEF451A4B2DDAC6A751FE77 66E98BA573A3EEA5967F58691C47B09CEBE352F9A9DA4B5DD9ED6E3482283ACB 506A2D6812D4676F3EB6FAD213BF986954219C25778AA79FCA0480EDC4FAABBA C858E073AF9E730725247C5C189C065B947E837C8CEDF0FFEB7DD80B2A7B7650 5850C70CBCE9A590E3A30EBF2C1ED1B1BFBA16499036AB8F879E3B8186EE2454 E318F5CC571A951409B0E69BB7A24213F763461ED764B530D2B46DD8E4FBB600 3E4164792700A912941A52336F854107B6333A96AD20CBD1742254B2B54A972B 3583047D43D202EC047CC23B82B9A4EBBD75E0C33F398E59546F733274CF2E7E 02ABC4E14A1E65E65A8ED13E0A65E4E64ADA9DB1DDD531844FD7D591A617EC74 03730DC7FDC3F6298C4728CBACB1BC9737DDDB8D941425DE145DA043A710A148 B9BB3EC722CEB29C19EF05608D01D55454CC24DE67F68DFCFFD671EEB94CC075 B815C6A45549EC23DB85923EBE614678C070E5C8D318C9D04C98BE7418101EB5 78946D805BC78580A7EF7997467C254D74D5092245B0BD81EBED69E888FBA1E2 544DE0BB152A0901F7BCE0A3EC5E54CF9F5595B56A90B89A6B9ED602DB9FF8DB 7AFEA58FE04476A1973304A8C76D1382E662077D02FD81BE3E03DEFA2D97148C 6B5C232FC5BFC9DE4665A311DDC3B5F1C1CD64BB9C55B200F32EAB40360422F5 9F50625FAE9DD08A8BB3C39D282B7AC5306671526361ED2C379A528BA81DFF14 211B6E5DD8C731904052C7D8502433AB3AF59AAB9EEA10C1CB629C10F0F24DE6 B78ACEC019FB7BF22A3B979E719F5DE598C58FF8074C8E8681F6A1F401570490 9BF052B23B8F7A145C59174B2AE897B4599F155B8D3825FA54222D3F56DD91A4 09F5C00077A7B5F0C4B0FC8E4B8175703FEC53549A80FA0B8B14AD07A99EB2B2 A974E898F62BFE90F132A37E2DCA4F43E13DB577913213A61C9C3BD9336AEB3C BE770DEF99EB9336AFD6F672B517A89E498488371938C573B40D709C838D7083 194DC83C7784B7734022650FBDC39FE0CA07748E5F6AFC8EE6E2EE98763933D8 14901FDEB3C6B2B20CF4AFE5BDA993616660D1BC6A8E77C180BB5757F2A22D01 7A5A32FCFB655CC893A8ABF5E1A2260C9BE38530230CDE952ACF0B77A4AE35ED 74B5E419A5734DB4452DFF68C724446933E8623806AE21AE841E5C923473A908 C80DBFD1B8F061C79FEE265DBB5D9145EB2BDAFB2043F674EA7339210E894FA9 4FF78C6BDCD3AF55834D4A05A92CE22E0F6746130013ED0F1628BFF1109DA71D 85A35F40871DD6EDDB2FE6C82BE91C762DF8E6D8F1E3DECF77A703341F4D587E A30C5ED0923B060EA6EEA5B735D0D62E41842C07B032A43C5D42182A2087CC39 A4D9B855AA0BC6B2582BC133908392F13CDA64C769B2B01695D2A12BAF043C91 8C6792D3B1BA80063E3556AD68D0B2AE350504DB78CBD73D1561064DA0D11E8B 2EB840E1818F6F1CA6D5D91E52FFDD417F70C6497495901E69E4E11B9A365A71 6EA21BF29FE0A023A3DFFDE181B90C51D705519A69CA824B117CD99334626049 A9AD892D4D31764965BD6147146667E32C964A18B5E01E5BF051784EBD5E4169 02FC22B7ABDC397E50AD6499394EE2DB30DC19E0102171113C43E1AAB211D466 E1C6234FE0FFF1ADFAE148D7BF05C9CABE06778A0E2C6A56E93F96FF799BE362 DC32ED3D871C435AAFA50BE0BD343D09A2F5BFE27752D26A1E8E8C305EF102FA 6C77B8D2301785D96B78B8B9EADEB9C7C3C667E24918676AD61A2451E14B42AB 75B3939F8C03CDC48A7525AEA53A636BE67964972A449835E233632EDE4C96C7 D3176D49C9E4CC33CD6CFF25615175A3A05C380AE8A154B66363BAB6B22645C6 122B80B49FB98E367ACB90AE4AC512F7C6FA027343F17500247A68846FCD3DE2 A37EBB0670047408CCA826F8995FD8C8DD48515A205BCFFA4807C5B21DDDE424 A9D5A46135985AB1F8A8AE4E95E3D9D2739253665056B037580A3AE4688EDB13 36E984761E273E922B05F0711E60E750C7E0E898D641FCD7CACF9B29E9B43366 76A3DB0457E66EC804C9A0FF7EFC222A9115A65AAF61D183052616C66FF633A5 8C0089CDBCC1E679A5881169D33E5C017ECCE81A3B397E6E1A8E7BD1EBCFD495 2593AC3E2A62EFCA163BBF333B338252C4645C16E1B29452424C0ED3865FC300 9328FBBBA96C4EABE1AE9E2A2AD54173071522DDBF92615F331E01A6473486A0 A53AE37F3A14E55B2D16A3F269B9FDDD26302C255228DDD757E1078ABCBA62D9 8CCDB59FC8CFF572054BF356BDA2BB153FF4CBC37B5A5A68E8D9B115AE6C70AF 1FAE23B7A5D89788524D47F998ACE6765DF1B6EBB459181702E23F987EABF169 71F273F5941F2F6E437C878DF8466CF3F157E1CCF863B9EA68DC4CF11115560C E328712A5366994A7526D73287CE3540F26BE9CB29BF9E4FA63E3315E0C493E8 2D0F37323C0250355B5737AB9AD13971CA80C44842EBB97183BE9AA0886862CC 4A14EDB62DD4F2DD90973171D622BA4F2A942676E1ED161362C1127A86A0C520 6A9871563076C7F05B95713536B934D49A1738024BCD5280E6EC074A297101D8 76802FB7D1933AE9646143FFB85D78805A2143782EAA3C19C973B24EFE54471B 1A10DA6DEFE0C755BDDB3DD830A5A66FAD7700C60A12770EE53DFA665AD93217 3CA820F27696AB81E10861E6D913950A30C0F8EEFBA57A0CE25310C912BE3F31 540C1562AE76794DF23E5D1BFA017D2F365F94CB2DCB92206331071EC5578660 17265939BB5A679D14EBEC64747497DD0A1F2E3E884CD70E870CD9B4AEDF306C CF2727CA7883CCD49D02496B6814361415F1FA43F16F4EABA7FF05E0E0217A58 AFDEE7549D937E51B3F7498EC031BF887D4C020C30FBCE70BD8BBC702D71AB8A 5DF73F5F1A425DE44630858BD3DDD46872166E30BABB976CEE9EF8B5CD07BF53 9D35C11867664B6C106DEC1BBD868FB17430AF88A3F851321104CB17604844BE B1CC74106378A01EF9C6C01F8B9BD341CFACB4D45E0467F5A1ADBA1DE47A1E06 967499D9050935413D0077115CEB632AEE05A4A453BE43597064310DF7A776E6 1467530C18E58F24C89A408BDB34C67E15EEC36C0092A5022199BE19905EFEE4 9D8A4162A84511AF9AF715EDD353AAA66445A6D9FD185775CE486C614F02BEFD 4F8908F9F85E0F3F2DFB538E8E7CB80DE5BD20E848BA45FBD6DA8078F7575758 3C3520E2AF1C1859891606226C1786B2719C6B2D3672F1863B389C72DE5A3AA7 1D292686D7B1F3590135FAD1A37A28D6B01963E6A63B2A79F4C2830C34385F4C D5347496D98FB8452010C150F1C54303B93CB6D396851B9C63622AC2068070E3 DDCAFEE674CE5D77B715B6EA70AF25F8C08073AA79146E483C45918643BC8AD0 EEE7F6C630F5019262069A3B988D54FB51095AA609DCFC2BA4468296A7132A3D 86A4604C6CEE9CC02F4BC3F472FAB5C318DF4869ABA4123DC405927940D4B2B1 D21FAEC43CDC0671E91564A9B16DA8AD39BC1E31BFDEED83B1187137F7E2F775 4D110CAD205766C41D8C1EA68DFB01FEBD1B2295D9531D06075599A5DA427FD2 527ECE3944CD84D31E512E28CB163850B43975C60CD13D202DA617A8AA1DBA2B 6AD36FF68F20554CF2E0ABB993C3FD893FDCEA0B4B65B4BD340DCE13E726088F CD89CFD8C1432FA83E889BA23D377E450FE19553544E81719F93E2C5197EBF2E 09F0791D26AC57695B093764060F41EF4D4455C5DF2B9DA6D1EEC826832F1440 B57955153D122DCBA28CE6F8C14E6ABF34DB3648516E5A142F081A161A79D840 E5D2AB5D8B81A08DF81B972AC693BD9EA8D79CD2315A6215D4E16932B1C53EA3 3548FF922564CC140A843C93AAE230E8B9891029E1336A673B262F5C7278764A 969938795986A9DAA13961500C0F433335BE6182088FF94EDB3AA0E27809A5DF F80EF14E93CDA47B7B6E58CF75CD9D450E658537C8D6F53D7A65CA0F12000B5C 3AE27F6A9F890F22040EFBA72AFEB08C23E5B5AF604BB13859C2373C66549DCA 845C775FDE05FF445176901B14F13751ED30AE9E72FF23CEE8B0E1846EC08A76 EA5C204C3D0C92E882D692FB304B01D1DCB042664C65FDF643F61F4673D0695B 4CF66237F834C21E44EEB6ABDB615CBCC138EC5A72E38D0F18D02033EA327D1A FDD141B75227E31EC7EF426C5BE1ABDDDC64A67EE4F4C905021900A968721A90 CA0AD2C701CAB4E0DA5195F3C3CE002ED84AB231B88E711D4C3DA6B11DE823CD 4E3A389E4597A0557C2944D76560C3C85CB9F3B8A8C033AA75F60E1CA5AF5257 57AAC5D349E5AB2A1FA0F2B9BA58CE09D0DDB1D08471513DEB8E8B9313E61865 DD43F55508E55F0411CF8E96A548438E58CF261506FA8D03C1E74B06DE0A3A8C 463FB8E4B113AC39360ED8C35E867AAB150BC699D28854124B117B2F353DE8E3 1035B5D4D9B910C777FCB5D173ABF12F9DA0D45F9DDA76AA6B97EC09A5B5ED35 2D050DD9109AF8F2FE021A2CC49C247F3E9BDE8B14C6313564B8BCAAFC3E4CA3 F7DFF5664236037364B149F4BC314767F0736196D115EA618242AFE398C3B088 E544BC5511ABDAEEAA6802233F9FCED700A7A58D1DC6ED2C68AC60440F050983 2837C94EDE4E9E24E88D9238911A5460CFA0338DF1F5EEB27F3D102BE34C8983 728956E2626E93402BE0A01DE6045995D64D05AE3453390BFA7B1B7A293556FE 31EB720A940E72C99D234DC5E79CFF902D012893650017903E7BDBC84A6FB7E2 3395D692A7E77E04161EAF3F0A54B4448591F2070093AA83B0883321E55DBB90 2DFCA2E3E3A6D37B30F5F413FC0780F55CA48F9A6B76D5CDE0830034E312F068 04203569C795FF5A96AE1AF9F99E2E1D73859107B409C8B36E27434E30CD0737 03DC4255CF12F05633C26CD54DC541A56F938716FEFF97D568811D722ADF74D5 C5FEC6837F432FBD02C44B337F82253DBE14109AA8838F66FEA3A8BA54BB5319 6479D714F3DBCD3A881BC49A5A903AC984D537A435EBC938BBD745F7911E4F1A 937B8C07EC25BE586F1CCF8DFF6629A56C3119984D8B3C305772136E91D34A48 9B4618C5031BA79A2AC0284440D786F13E706C372132165D7A42E2636CFE9949 360EC40BCBB721A0935CD2FF8AE6539E6F2D8A929093A8E723216CD97B3EBB2E 397629AEED29B7757E41F14E1F38937BF59E9FFA2E6F6B7769731DE55717E9D3 C922B14B1260EB7C73CDEC7F30FC45A7A540CF2085B1D59F042557720F7C7B25 E27EE2E97E88B32FDEE329046C2694DBBAF882F8D58DEF428933DC8DF333DD83 0A1F1D8EE749AC06C6E5E70FECA70DA75867DFA2597DA115F005DEC644FFABED 902CA83EDEE230F0F914479CF17165FD399D3B0DB9195368529266BFA158DF7B 557C4AAA538AF19A9F7DE415FB97A60457DBDAA36F07319DA3EC809E6AEEFB6A C51F11D544AF547740269656FFDF018C3724B271A7FAC8FD5B4A5B8547161ECF 5BF184CEC813DE7AA6A8001DB42ED43D6E0D847ACA1CBDDD800EEC056A60EB46 E8FEF149113019FA2BA6701FF8FE3A3D435518AF8426F6277CAE62BC5D0301B9 9D3F669E5F94541D69E3C1EE929CE12CFA8AA0976C08CC36EFDDDB16413A7056 3341E66EEB0C3964 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark {restore}if %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.0: CMR12 003.002 %%Title: CMR12 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMR12. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMR12 known{/CMR12 findfont dup/UniqueID known{dup /UniqueID get 5000794 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMR12 def /FontBBox {-34 -251 988 750 }readonly def /UniqueID 5000794 def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMR12.) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 19 /acute put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 56 /eight put dup 61 /equal put dup 66 /B put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 77 /M put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 121 /y put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.0: CMMI12 003.002 %%Title: CMMI12 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMMI12. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMMI12 known{/CMMI12 findfont dup/UniqueID known{dup /UniqueID get 5087386 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMMI12 def /FontBBox {-31 -250 1026 750 }readonly def /UniqueID 5087386 def /PaintType 0 def /FontInfo 10 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMMI12.) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /ascent 750 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 22 /mu put dup 24 /xi put dup 34 /epsilon put dup 59 /comma put dup 60 /less put dup 64 /partialdiff put dup 67 /C put dup 84 /T put dup 96 /lscript put dup 97 /a put dup 105 /i put dup 112 /p put dup 114 /r put dup 117 /u put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.0: CMSY10 003.002 %%Title: CMSY10 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMSY10. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMSY10 known{/CMSY10 findfont dup/UniqueID known{dup /UniqueID get 5096651 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMSY10 def /FontBBox {-29 -960 1116 775 }readonly def /UniqueID 5096651 def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMSY10.) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 7 /minusplus put dup 14 /openbullet put dup 15 /bullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 28 /lessmuch put dup 33 /arrowright put dup 39 /similarequal put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 59 /emptyset put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 112 /radical put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.0: CMEX10 003.002 %%Title: CMEX10 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMEX10. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMEX10 known{/CMEX10 findfont dup/UniqueID known{dup /UniqueID get 5092766 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMEX10 def /FontBBox {-24 -2960 1454 772 }readonly def /UniqueID 5092766 def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMEX10.) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 10 /angbracketleftbig put dup 11 /angbracketrightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 40 /braceleftBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 68 /angbracketleftBig put dup 69 /angbracketrightBig put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 98 /hatwide put dup 99 /hatwider put dup 101 /tildewide put dup 102 /tildewider put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.0: CMMI8 003.002 %%Title: CMMI8 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMMI8. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMMI8 known{/CMMI8 findfont dup/UniqueID known{dup /UniqueID get 5087383 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMMI8 def /FontBBox {-24 -250 1110 750 }readonly def /UniqueID 5087383 def /PaintType 0 def /FontInfo 10 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMMI8.) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /ascent 750 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 59 /comma put dup 115 /s put dup 117 /u put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont %%BeginFont: CMBXTI10 %!PS-AdobeFont-1.0: CMBXTI10 003.002 %%Title: CMBXTI10 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMBXTI10. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMBXTI10 known{/CMBXTI10 findfont dup/UniqueID known{dup /UniqueID get 5000771 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMBXTI10 def /FontBBox {-29 -250 1274 754 }readonly def /UniqueID 5000771 def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMBXTI10.) readonly def /FullName (CMBXTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 72 /H put dup 80 /P put dup 97 /a put dup 98 /b put dup 100 /d put dup 101 /e put dup 103 /g put dup 105 /i put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.0: CMR8 003.002 %%Title: CMR8 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMR8. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMR8 known{/CMR8 findfont dup/UniqueID known{dup /UniqueID get 5000791 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMR8 def /FontBBox {-36 -250 1070 750 }readonly def /UniqueID 5000791 def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMR8.) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /zero put dup 49 /one put dup 50 /two put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont %%BeginFont: CMEX7 %!PS-AdobeFont-1.0: CMEX7 003.002 %%Title: CMEX7 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMEX7. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMEX7 known{/CMEX7 findfont dup/UniqueID known{dup /UniqueID get 5092763 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMEX7 def /FontBBox {-12 -2951 1627 770 }readonly def /UniqueID 5092763 def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMEX7.) readonly def /FullName (CMEX7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 16 /parenleftBig put dup 17 /parenrightBig put dup 98 /hatwide put dup 101 /tildewide put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.0: CMSY5 003.002 %%Title: CMSY5 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMSY5. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMSY5 known{/CMSY5 findfont dup/UniqueID known{dup /UniqueID get 5096646 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMSY5 def /FontBBox {21 -944 1448 791 }readonly def /UniqueID 5096646 def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMSY5.) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 48 /prime put dup 49 /infinity put dup 106 /bar put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.0: CMMI5 003.002 %%Title: CMMI5 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMMI5. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMMI5 known{/CMMI5 findfont dup/UniqueID known{dup /UniqueID get 5087380 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMMI5 def /FontBBox {37 -250 1349 750 }readonly def /UniqueID 5087380 def /PaintType 0 def /FontInfo 10 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMMI5.) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /ascent 750 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 13 /gamma put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 26 /rho put dup 27 /sigma put dup 34 /epsilon put dup 59 /comma put dup 61 /slash put dup 96 /lscript put dup 97 /a put dup 99 /c put dup 100 /d put dup 107 /k put dup 109 /m put dup 110 /n put dup 114 /r put dup 115 /s put dup 117 /u put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.0: MSBM7 003.002 %%Title: MSBM7 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name MSBM7. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/MSBM7 known{/MSBM7 findfont dup/UniqueID known{dup /UniqueID get 5032014 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /MSBM7 def /FontBBox {-45 -504 2613 1004 }readonly def /UniqueID 5032014 def /PaintType 0 def /FontInfo 7 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name MSBM7.) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 84 /T put dup 90 /Z put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.0: CMBX10 003.002 %%Title: CMBX10 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMBX10. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMBX10 known{/CMBX10 findfont dup/UniqueID known{dup /UniqueID get 5000768 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMBX10 def /FontBBox {-56 -250 1164 750 }readonly def /UniqueID 5000768 def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMBX10.) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 66 /B put dup 67 /C put dup 69 /E put dup 72 /H put dup 74 /J put dup 76 /L put dup 77 /M put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 123 /endash put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.0: CMBX12 003.002 %%Title: CMBX12 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMBX12. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMBX12 known{/CMBX12 findfont dup/UniqueID known{dup /UniqueID get 5000769 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMBX12 def /FontBBox {-53 -251 1139 750 }readonly def /UniqueID 5000769 def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMBX12.) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.0: CMTI10 003.002 %%Title: CMTI10 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMTI10. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMTI10 known{/CMTI10 findfont dup/UniqueID known{dup /UniqueID get 5000828 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMTI10 def /FontBBox {-35 -250 1124 750 }readonly def /UniqueID 5000828 def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMTI10.) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 18 /grave put dup 19 /acute put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 87 /W put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.0: MSBM10 003.002 %%Title: MSBM10 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name MSBM10. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/MSBM10 known{/MSBM10 findfont dup/UniqueID known{dup /UniqueID get 5031982 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /MSBM10 def /FontBBox {-55 -420 2343 920 }readonly def /UniqueID 5031982 def /PaintType 0 def /FontInfo 7 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name MSBM10.) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put dup 84 /T put dup 90 /Z put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.0: CMR5 003.002 %%Title: CMR5 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMR5. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMR5 known{/CMR5 findfont dup/UniqueID known{dup /UniqueID get 5000788 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMR5 def /FontBBox {-10 -250 1304 750 }readonly def /UniqueID 5000788 def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMR5.) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 105 /i put dup 108 /l put dup 110 /n put dup 111 /o put dup 116 /t put dup 117 /u put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.0: CMSY7 003.002 %%Title: CMSY7 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMSY7. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMSY7 known{/CMSY7 findfont dup/UniqueID known{dup /UniqueID get 5096648 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMSY7 def /FontBBox {-15 -951 1251 782 }readonly def /UniqueID 5096648 def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMSY7.) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 20 /lessequal put dup 21 /greaterequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 67 /C put dup 68 /D put dup 82 /R put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 110 /backslash put dup 112 /radical put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.0: CMMI7 003.002 %%Title: CMMI7 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMMI7. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMMI7 known{/CMMI7 findfont dup/UniqueID known{dup /UniqueID get 5087382 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMMI7 def /FontBBox {-1 -250 1171 750 }readonly def /UniqueID 5087382 def /PaintType 0 def /FontInfo 10 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMMI7.) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /ascent 750 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 67 /C put dup 68 /D put dup 73 /I put dup 77 /M put dup 78 /N put dup 82 /R put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.0: CMR7 003.002 %%Title: CMR7 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMR7. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMR7 known{/CMR7 findfont dup/UniqueID known{dup /UniqueID get 5000790 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMR7 def /FontBBox {-27 -250 1122 750 }readonly def /UniqueID 5000790 def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMR7.) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 22 /macron put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 61 /equal put dup 73 /I put dup 82 /R put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 101 /e put dup 103 /g put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 116 /t put dup 117 /u put dup 120 /x put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.0: CMMI10 003.002 %%Title: CMMI10 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMMI10. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMMI10 known{/CMMI10 findfont dup/UniqueID known{dup /UniqueID get 5087385 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMMI10 def /FontBBox {-32 -250 1048 750 }readonly def /UniqueID 5087385 def /PaintType 0 def /FontInfo 10 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMMI10.) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /ascent 750 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.0: CMR10 003.002 %%Title: CMR10 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMR10. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMR10 known{/CMR10 findfont dup/UniqueID known{dup /UniqueID get 5000793 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMR10 def /FontBBox {-40 -250 1009 750 }readonly def /UniqueID 5000793 def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMR10.) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 16 /dotlessi put dup 18 /grave put dup 19 /acute put dup 21 /breve put dup 22 /macron put dup 33 /exclam put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 127 /dieresis put readonly def currentdict end currentfile eexec D9D66F633B846AB284BCF8B0411B772DE5CE3DD325E55798292D7BD972BD75FA 0E079529AF9C82DF72F64195C9C210DCE34528F540DA1FFD7BEBB9B40787BA93 51BBFB7CFC5F9152D1E5BB0AD8D016C6CFA4EB41B3C51D091C2D5440E67CFD71 7C56816B03B901BF4A25A07175380E50A213F877C44778B3C5AADBCC86D6E551 E6AF364B0BFCAAD22D8D558C5C81A7D425A1629DD5182206742D1D082A12F078 0FD4F5F6D3129FCFFF1F4A912B0A7DEC8D33A57B5AE0328EF9D57ADDAC543273 C01924195A181D03F5054A93B71E5065F8D92FE23794D2DB9B8591E5F01442D8 569672CF86B91C3F79C5DDC97C190EE0082814A5B5A2A5E77C790F087E729079 24A5AC880DDED58334DD5E8DC6A0B2BD4F04B17334A74BF8FF5D88B7B678A04A 2255C050CB39A389106B0C672A1912AFA86A49EFD02E61E6509E50EE35E67944 8FC63D91C3D2794B49A0C2993832BC4CDC8F7BD7575AD61BCDF42E2E421AA93E 3FF9E4FAD980256D8B377043A07FC75D6169338028692CCA8CD1FE92FD60AD26 D57B7519B80A8F8DCE9CEE5CDF720AF268D3C14099498A843D76E3B6C0328F24 D36EFE7F5C4E5B5C612786200C8DE3A41EE5F1FFAF4097653CFCDC8F4FD32E0B 03EDB3E413283B9EFB0AC33B055617005BC9B0057FD68C52D1B0E67F0C571685 767F2AA85ADE4E0104A1C777733D5E318A22A9944336E5B98D965E50D31F357A 8B6EA5A0EA98E1B027CE68C2EDB149EDDD04ED74A1B3D206D471A0C11C11449B DE190BBFEBC08C9E1B7513B43DA3134D6B11A2516E6E86B67F68C970A320D05E 94FEC57FB347606DF89989C33482BD09D011C55AA920319E7B26A205D3D0F004 22466F09C0482A164CFB27EF6ED2B040ECCC3DCAF345B5A73676F193D43123B7 72FD6CFC5E37930E61EBD5A6307E4DE70194E6384EC0D79DB6AD86D3B319A31C 8B0589D0FE28241D8ACE280D0530EE99C80723E560BB72AE9D53F4713181F491 344B06D3027BA4E9E94D4305BE1D817197C54C8FF56CD6964165F6448ECC8A8A 64B48B4F0FD69299A137589E2491A283509B21A3A5772F75B7602A9F60AE559B 07A58436D04222C73EAEA72DE9A5A441F88D27C11F4F91255EFE280E91A4ACAC 1E98A4E5E6C57B9AE86FD218C3CD8F24A4104156A80F13821384E529783C52C8 78B94AB3A0096090867ED32E8A30980E737922037F75F062BD83BF4F5929BC51 CC22AEE2DBBAAA001CFFBFF41D258424FAD888FFF1BEAB796A44E3126159E120 7E4025C676CF94888A1971AEF8B6764B3AF4A92D36FAF6FC56FD049710EE3782 BC2CD84FE2473F133BE03C1346B875463F126DCAB15C7A9BCC9A727D23611462 4E8D2BFD2466600285D79518712B8681ABCD69608E6AA9578F7BD771EC36E01A 5A17BC17E375020ECA59B43790ABEB9DF5F4FBBEF807E5699EFEAC563E1ACC5D EFA336E75DE6D8248E9381BB110884FDC89C2F9A41EBBC9A8A1F98E6A41F68BE EE30E25CA148C1EFF42DFF8C214A6537AB11F260B8C329A4947B5FC8DC9C5622 4DF7BF4FBFB00380D47BABB03BC30627AA74103E553F55278F538EDD8C1E64CE 0F1398CA0AB5A86630139B4A7E8FC02804CAFF3830114640AE50D2FDA3B561B5 C63AD7EE3347804CBB40FB1E77A6C89735DD870351C3A1811591AB493251B904 314F65791963C0412377C1D02362C5E9655F1C3D4803CD379A8EF24C48218C2E DF1165840462BF37DDE1B8D5FF09FA2C3B261E2F1A65ECFBE5D4EAD43B52C029 EEB3948CB8A252CBAF545C8FA1C31E920E23A12DD7222CEF2D2A513BD758EA13 DA33BF5FBF1D734653EB83DA2D374A5B9A0CE316F24EE375D6DF6BDA49954C2E DB25A88821193636119D469BA66E5DAA9C92520FD4F84426A4E54273FA469084 7517817A6EE3E21176D333825E88046F50B3CF6938AF9BA79A2F51398239EB91 1A2D07F7FCD948427FF62F40FF95E39FE1A1AA8451411563FD5388472251C155 69BDE9283B41900B21EB1190D06E6B13B7794FED020D2C1BDD205AE77B084BCE EF628249398B496DE85B406FC2E1939EF00DFC84C07E26CF72EC401BAAE756E5 7F6673216E7560D1C2A723CB405EE5CA474A07F61B81F8836482F73DC9516D67 CE0CB770EAD755B6B356198B4B97EBB29C63456953270CCC8D5650C1D006E69D 38DE2DFEAB27DAD50A817F0D645D30AF5B75A7B53CBD3D2B8D87BD0A7E525AF3 22F7ADDFCE31716914C2318260C2E2B4664893921B68C5A93334A361D94A759C 0D7B146D6FD94F0442D672BDA0F6432E18F3C5DFA37ADA378D95B75F413C9ED1 BB5C606A3EC7DFB3F796F59B0478C13FD1900381EFE0BB5242D5B5D34D03AF1D 4BDC93EAF8020E26CA23C8B0E7DDEBBC6762A557067A4CE05A524188A8F02E2F 3625DA38DFCF381727887F5646A3995A8A38A5FB1E5D5EBB395FDD0B7C8E71AD B48EEDB62AB2CE99D121435EFBBFCEEA69AE9ED8238B60CC7288DE33C766CDFE 15B767B4AE2E6CE0965E77272AC9F86023DA620548CFAC85BC751C44218A29C9 849F1C2DCBDFAD895B54E51A569952ED50F82DC8A19F367E7E44643854EFD6B3 FCAEB04E55E4661C82D31E2932611748480EF61FB2FBFB0CFB940BEA81AFCD84 4C6A6332D7A600170E38A8EAFCD4F93DC153C43175434C86BC747348FAC61B76 1FEC9027C1A193E55C80F1F20B5317AA0A05AAA36AE235F6E49F06E570FEE798 84857D7552EA92EF3EFAD52DE39C2F8F43C59E3A957B7B926FC95FC4B60186DF 7F3523EE2AB74E294C8C4BCD8B4975E84849E0FBDA6C0B0F24A636DFA578B122 CF97BC5089E21E9F5298D1C9F30CB8BAFF6A3A11BB4D9A0A5CF2B18D055C44CA 4FD4D8FE1AF3630907DE7E585AA811F9CD11FB2C8FC791851D651009FA5DF20B 3C33FD2FF848A9E3F5652BD294965A332DD3F246C91B0ADA34017FF2451D1394 F9C3C95AAC6EC8062BE98E8914D51DA6A164AD13938693D446044859D03A949D F9AC5DF4A000CDA98BB516D762CB9F6D44B5268FD0C26E88BC4A760C0F75A140 DEBDECA4F511128B7D2805872160C55236F0A0FA7637FF0D4E94AC079CD3C8A7 D03A5A56F26B0438B577C46011A10532FEBCAD14FBD6032E224F45691A726886 56F305231EB2FCDF59C8BBFCB5DBD2D093A0E84D62AC93A2312CA69295E937C4 8DBA1802B85F54B5E7E6D6216A918F911FF705D3B5CF055F1D873B96283A0B53 59344D910CD396D883F6F7836BA65FAB4393A773A8F6BC298069E5BA38210EED 49C9D920F718E3FCE692527DC7CCE6963BF744F2C91BC5952564196D60574E86 87A0FAB21F2DB2BD5A51D7FBD8FC19946D24E5A228462C4772F978E650ADCE3B 8D66B9C21279C531CA1C3A8ECE3420BB65837287A7222CC3673A2A5F8BBFDB60 C719CD073EF9A23675198462C7C87B24CC92D6AEE5C25AC63855CC3281494342 D28F3D2FDE0C183486769A4FD5B0143193D31FCB2C2A14E487BBD96D0BADBB64 D1B56021C363A795BF10E2DB448261C363A54A4AC1182B470C457AA82DF3F5D1 F4B329806141EBD53CAE309319B94133D7EBDC2D0453A905ADD207364371E178 0A95C2686E3B34C4A978BFC0EE968C39ABA00889BC5149162C2B54483D44FD3B 5CFF41F611C7E03B94945F414560E874D7CF27FFD0630890D7D7EA66CBD15448 229059E1C436BB33D69552B5367AB5D53591C4678D0C704DD3EA23F5D9E8A7AC 17D003C19E333E726FFFA2961F33C70F429085F7BFE3E2510F59B78F58B19CB4 01B48E184BAD9020FECCE3AF52048A056981DAEA02AE78197E65855DDB170616 F54278395D9EA50DC83761AE759F9CDEF9E1948E7002414FC05286ED793E6662 3347F2A9AF8917493D7305B92CF93E8E9185F70015F5594084298A6C2F9FD3C0 689F262AC9FEDC9B89577ECDE92F08D3142209FBCE7B5C0A840CC767BCA56C20 4E4E545E2BE4D21C53855CEE4CD0AB35D1A604C0FFFF77DBAE4289752276559F A05FEE65F45ECAF44E95E23FAB6052195C7948AF0B1126482D4E02D72BF8AB03 DE0F1A632F7672AD9DDE70EDC82AA993678A82BEAD0BC2649C4707FD8509810D 364B5C6FE0E10772E95288C622C2F06C634F4DF8C7FD1432BC9310D5F24FEE3F 7AB324863D6DABAA1576E70643CA79EF4D7DF4105093D66CEE0F3B87D2164A7F 26EA05F5C4645B22D3E1BFD2219657712C168FD90DE801FB0F32759E80DEC1E1 43CEEB19FED12D757205043FC98FEC62D6A8D8B97BC083B4A0E985AF7850D6FD 8716B9957C1C35A0675BC53DF672C425C79F43FDABAEE7D63F092CF271C9A9D7 C41F40C4189510987887942E60A412B3EEC84C9A6E1AC7D54D528F5604B72C08 94B7882621A5BF1F325B92FF96B80878CC550D1AE4D8196E41CB1251856609A5 C4D3BD05A922D0D45E039D9450DEF8490A3E924E41434194910BF60BA1B08BE1 B41824345627745541A4F1703E956328F6227D11C74946B38CFB096139979E56 4E723B889B44C6D78673868C89912F8B4F0B4B485F1587A637B630F92E6072D5 7F3B44EA6FD96BBD4FC28A6C1D90805E3BE3E42A7BC9C880762966C55BC04E01 204D083AE976FAE6F37C94F27E68F8C0F28D52B17F6C0FD7C9150701FD78F8CE B8E8DC9260E3974005EB5CA728171F482D765016C94D4ADFE4A42EF42212BC56 7E4EEEE8B0D2A7856CD4E44F55C0BAB762F92CB8D64C17022D4BF3A47C12F5E6 279FC23101FEE93753653CE8CEDC3B75C9CCB29BF1D4554C6120DE8EE750FCBB E38B5D915206974962E320362E59B3F21B3AB1875703191043D03284D4467346 CFF2F98CEB4845B73ED8E003E0DC94251B73E13A9B51A3F1430BCF6A21EB9B7A 65E17FA411F53BE6432F1506232B8159E008FA257F884A4A01AC53BE91754D78 BF14A5B0FBFB9C31BF4908355F8A762052968DF526D118708CCB0B7CB5BEE285 6DAB6CD2E3934178E60BECB11AAB5478623CF6C50C92F8BB5D1A583609028FA7 B8A53B791BDC9EF76A124F3F7641857E4BEA0837CB36176EC9A522EA7F41B8D3 63C37D1145367BD300F17B54522A834BBB74DE12BF9EB26ACE6F24A046D58F89 4D4B7DF74875F1A0C1C9D97BE0849593D7B398EB4B00BEBC8C8D1497B6EF831A A35380FFB7F1AFA4D888AA52C9482E8B1755CC209905F98F40D95B44D4DCBCB6 67423D1BC2F3560FF0A8B4F0CAC352A4EE2C1D946E45AAEC8A6AD40303F3382C DF0756BFA3B1ED64C169E56ED1C760F2FF0E24DC5C9F41306EF8D2628153D30A 5DCB0791126BEFD4947D7EF08301FE015F2B0008DFFCBF9F2D4D859FD43EC7D9 C5BE237E9BF6665B7B1BEBB362F0C0C3A8D86010B9C97FA741C97C2E0513386C 9C26C235B14DD2A58BFDAC7B5F63DB4DA6D5D37D0098175A9071590E1DF66A3D B8173A047C29D7D35557F06132CC920B5460B8AFC11D23D09A4E45D089F5EB51 963FA1A6256E359D485107FD143B2BF21FDE9DA5744BC2615E86C31C89470CF0 D06C6397D9FCCB316EA9989430240759D2C4945D941F159FC02327F34B042BAB B5C3A47C78E8C1A6FBCD396B1A51CC4B020B8AD401841EDABACECDB482D6EC5B 72D2BFEB4556720FADD49D07307C8B22ACB7E310CA4151A85C71EEF70E8D15DE B3B00F26E0E166C14647A65ADA228A3D1C89025BE059306565DB1B1EFC37D358 8C1EB024254AFD049BA977BD4C2C605050E17940A89D0D4C5D963E792320F5DB 3706682E03D25D9E02487247819551465092CC22B6B56E93F3AB528038FEC3F0 668F866707A19B0463BE706EC729D2EE1653AAC7E29BD25BFB3241D4792F5152 ED415B4E7FA92C2EE5A22E27E8B75542C492E56D811C192E95542A6FE0BFE5A5 69273C2ABED4300D491B92D2AECDD278404CB84B1BB1BD7AFEC858215837D118 C0E928BE7E07CFEEB51A6D21375B772B8248C994564014015232A0DA4BEA1754 3274F407FED0837A236371F1A32056240F2015B1E7F4B2CA72C6B58610A66F13 407CFFBA5E0A2893C1F572D50F51286E9133B5A84239C9493B0574E77D281D01 11D00683354A000C9700EAFBC1FD104EA19DFCB87470190E7E2CE26E3A6FD0FF 2620B87B82AC8686B6206B530F17E9348BC7D04B948348802CE53A312443DB87 4DBBA5313A6A2A8DAB8A1CC9A594FF8C299281C0A261C8CB2226B732FBEEDE40 2C6ACC74A1A61379E2E1CD5548CD908268A32FA83D8504C442EA0E183ADBF7FF 9FD09C037AB03516ECCA93FF048235BD11A25DB07F164512A079C5392AC7F889 CE96AE5C8D9580BCAFCC087C35E76EED1A671E87C12E3045E15A687134736DF8 DA984772AFD189D68571A2ED7256F1E204230E41D3D9DD876F938951714A3973 0CA9310489F8E807C1C7A4E51AEA5BC030610A5D7263FF7E0F9FDE3E5E37A362 5B919000BD94D978583B942EB79CF2BEAC33FEBC9A67272EB10865BA8FB75FD7 9D280AB59F91B96C16C982DE848D76D8FA8620DFD7C80B7DEAE7264350D6FB3A EF04794DA3305844A7CF718F6D1A4A3AFF6826173A076A1372ABFC54ED3AC6C2 09C9287FC830556CA694E21CA5342ECA7B10C90AFC4783D841D7B1E34FA3DB7A 2B706F3E21B0FBAB23E7257962FC3BC309CEA2C7239A9D6B44CC96825115ABD2 AF9A2566D2F3382C01569FBDB94C8D664A5DA0F7DC3DD140CA77C743D7BC1420 324ECF9E4780280EB119885E96A6C619CE3C0C8E1E264E2DEB137E5DC8149786 486D65667ECF47B1A1E20E9E6E4FC8323E0BC8E61BDD3BCDFC6575C69C03E31A EFFC290472CBBD049DE3F840AEE37A2486034240F80E75D8A79E0762377DF660 52B12EAA16D678990B11A9BFBC03C1D4FCDA9FD4FFBB3E88352438102F10B7C5 9F04C013B6575B5E948FAB58EA691984A0E54E6B9F3F505FFFEF74D06FA1CDF3 4B8A95904C8A2763AA8AF5B71D00F5DE09DC1CDF87A08B6D181453063E14C12D B7BB3775A6E2A901636273D9EEB833EA8CF20FD83AE899E28DADE10EEEC20BD7 BD93085A4B1AC80AC1AE8280C14767F1A487BD066007A0D050317BD081131A14 6EA0898ED59E46DA7B6254BDCCBC660686E2EDA0E77A705A653733BB5C5497D0 B130359F866CF293FB6EF0C2AC5BAA2DB0DED045E2DED3A2612D078333260359 16CF0CCB272D34767EA069E0F0B0D42327A18529D72E890EDA6195C2688438ED E9ACDBEED41E81CA8EB5E43C2B09CE266EFCA03F2D7FF57F12B06F9E54FCC6A6 546676F6FFC5B8B7D3F0982B6FF0D21D949309F0C0B175CC1D0976F8C55C6AED 6E821C39041E22D91AB30922F2B2EC2746BC7DAB484991542FBC82D87B487507 559AB466F73EE23C2D3194DC5CE4C9AE66D3164613AC5CBB3DB501B64DA7C91B C7ED2EE9027FC0906820B35D4F2CF66C4F9CE4A884B7C07155BCA884ECA5EB3A ABB83F84DB1F5639599DC7D3F51241AB5D95C3BCB7AB1EC90B4BC989F74FB354 04B2D7366A34D335A47B8C00C05CB423482BF6C7970A95545424A08AFF9A035B 7F83F52B65A9799CE76E303B85664B624C65E9CA58184C7BE2BB9D9C86A4DE5A 8165EE3DA2E652B5022EE7893896BABD88931DE1D538F615787645DF5ACBBA0B A8E5B899A37321AA7D4B283AC9234978C2DD81813A1EE5DB6EC170DAC1B6EF02 94892635B498765C07A38D2E9DB0B7581B11056C28278F89B0E60998379C07EB C0EAEDC32AA69B8B836F92A61AFD35688315B2C3F860632FC13E4BDFB63214BC 41CC6859EAB3AC3034449213CAB99FA1D216563419CD6D6CE4E1B56F33E6C654 7AA9DCB5B05FC068DF02AC32408C8010AD004F6CCA9887830927F8CBCD49CDB5 18CAC1EAFF815FF2F6F527F936948201565003022C6C7390B4E3C2B219FB4F76 9F12BD25CA7B3B61D1A2F8DFEE795D04D5428B42FB66E0C254AF7B7A10CEF7FD E5ADA5E217BE24851180E9A1700FBA66C7D2B0D7BFDE4F4EED1D24B821A40947 5620363657F6D048E651A689822CF815E72FC8AE9D835BE31D1DD8B54C9A717F 4DC319B4B59AE073936EA40B070524C7E71D5A7B64436DA107749746B516E29F E3BBCB8F8C473E706670E11E5B221716F315FF097CD1841D0069FA69EA1898FF 9F9EC2518C77806A19730C97F54BEAD604548D553D4A6EDB247853225E24E7E9 89D71F6BC94DB986467E755CCC99069B313F5745B02B4BB608A39F0A0A732B87 7EA2DED68219754BF1FBCA350327572D769C962EF9242132D93A5C8E9725D8D3 AAAEC15ED0F362471AA58488620156F3474FA59CA080EA96FE995D2B3DEEADF3 3141D157481C66507725ACA5953CBBE1ACEE7E3F02C72C6552D15EB3D612730E 61A06A43575568DC3CF3844BABF04CA767E2995196097015E0C4F622C4356B6B F41DBAFD797A4B9D7AC22332C552043EF98913D0D9B50CA6B7CDAF903BC5C04F D20A952BA5CC35B646ACD0A287C956B98C450051AF6AAF79DF37F8954473F8F6 652BF03AE2AE82B99D820CF93F5FC0BA17EBD7AF90313E70594EB5C354023BFA 07912408F1757319C7288E99872B907D5AB583B082EEED8AB079C63E38B07D11 6744856E689A479CB3A8BC081F33CB06755926204981DC0A45B3ACC18F6865BB EE2C50DB43B62E3630FC1D9B1FFB3BFFAA6D0A20C0381ADF48E4D916BEE85BA2 BB40F538F55C11D50F882B73913840B45161262BC8B0012694C3EF26452F9B77 2CD7C7AD6BFEEAFE31C8A721C2D46AA00C10681BA9970D09F1E10DDC250E2AC3 9A160EC8C9654FCEB36AC2B586E978D54744FC8A0E963D8EF6E228ADD22D093B B889C940206F504F14DD921D909BE06EC9BACBC23EB9E9D137FBC983570FFD2E CC5D2EB5D2A4A8604A4AD418B800EDC6B89809E00091A2315EA8C071AB631CCD 04C3EA77541646B7F9D70B7EA564D92A65C4DED8966517660EFB9F77E172CD8F 1B5FA83270898C3709DE964B0C4A899418435F92C13E94F5C1C5B9D7DF3E8793 5F9E83CE1430E3EF275C26835080FD2AC78DE88BF65298A8EB4181F215181356 A09245577F997AB5F23F31DB7E081C7385F461B083E96E3F08890BCEFBFBB6FC E597C3F22E0F0F7B6A93C55879A8EA022411E0C1864E987D2F00CE0427E4E8DE D92F9621E3F7751A3CE5CF1EC306A78C0D1CC6781A016A285C0D195E17B59A63 C593D2DC78B6071636FFBF968B8E7970871E4B60DF79B9BF341528C4F0C5008F 191FABC891AD46A68212B20FCC87E25AE386DFA935CC1563EA96F1339B6CC859 CEA88CCE5DD0993F19E3902D9683ADA47980535FB5597CF57C046C48E0D0140A C40AF806F16E94FA0C046C0F32AB5BE121A7157A42654E7C1FCBA31379FDA9F7 BFA8A0D0384D3B29B9DEA1500AFBE314EF78B201E5F679AA38F12A87D1E1512F F71D12EF75F3DC48BD8BA01A6D091F0C5B2B961F4371E588F79C825EBC124C1C E334BD4714EDEBB709068CE0669738B49AF675692317C3E530C6AE57CA21B59A 0B542AC77FF404DEEF116796DFAF46DD8C9290CDCA4A16FFE88A0D60ACD6CB47 440014F30DA7C27C199D99A1E6D89D0A0C31CD1AD4891668DFF5FFFAE992F10E C7B88872E9AC808E0663929D199AB424136E03E284D8F44DF9235246B9D22C9D 79BBFCC936171A58B943DF0EA5ADFE2818992B46DF97AE05915E078FD6AAF8B7 1047AD85A1CB13CAF70D095264B8AF9A3FFE6A603B10EFC6A20DD56A9947DC94 06DE0A6AC5B655C71870FA1BA99322E51D46A52D3E7AEF8189C9F1DE58AC9833 7B325CC59375D0F61488AD740909A855DC069F260A465080F5570CE7D41CBADA 47C1AB404D32CBE9E1B8220A27C8003ADE08A63AB714709CE4312179218E4924 6CDEBDCAE80C39AC4063789B6A407EA9C19D12688DF86B62D63BC05E9D65ACD6 CD5DBF04B896DA28B466D584426F531F4A3F9C4A717A0C5ED8C8CB4191E950DB CCD311A20B14B417714501AEB2F6F8510242AF91BEDB6D70B1A52B91FB3BBC8F A8F9731E745F2BA8ED8B00CFD8A93B2BB05DC2DFFB58CC51647721D4F7F3FB37 F36472F6B02B8A82EA0CB44DC84256134A55480B519A066B8716B4C8899DAF9D D5663AFF2DE36A22493F26251275EE9EDBA4407FF5654A263BC7E03509B2FAEB E1A7FFB356CAE2A1DD97D6D5A602B7D3EC0929A6DC9FC0BA64146B4292A93D6D 63AC99AC051FD258535F8DF458C6DC4A4088D6AC9CBD937D09BB576AED3A5E22 89E8E72D7E8260E15F0BCCCEF9F4F6CC8A4CD1B956046D818FBA9029512D981A 6F5AA8D1692A8B8E569E86E4C0DB55745D40451FB32DBB822BC495B23EAB87B3 394DDE8E0A8EF71FAC42D53468D1CB1D7B550A47C7F3722A2E41B4F607D85114 3F8C7747EE66624D0A302B3AEFF5D18F34B617BE92386F530FA9025A9C103CE9 71961454C5BF4219F49EE8C789A9BC7F63500C8A735DF316A6854BB5A5B214C0 2A57BB9900E406B2FC13BE715ECE4AACBB0756D1622C478F8FDA34582D839A1A C0532A0B9A2164C4B260EEE16CEFB7FA65451A4B2DDAC6A751F90DE96D25579B A36AC6ECDC5F35C78896CD75094B94AF7A64FDC21B0DB5CD98CE841305B6AA01 BA0CEEF83D4B8D7191ABA51FCA4AE38E374DE01977C13C3AA747764F9F899A23 09593F7929450057E27BFA25D69CFAE73DD23DF51F656B9088034D4F8E832480 DB24BE6820F1402424CE91258B1C760B7B267FACD82DAB23D9CE5BB9930C5940 3A25CDF21A397AEFFA35EAB1C90347EF3961693799541B6E9E6D0C721175311D 5C8EC1832B6E83C2CB3E4716F48CE05507447E290E5C9AA8696AF68926C8FAFD 2392C817B8F774002CBAF73D683CA449BB310095986B388BAB9EF6573E2BB4C9 1AB8A75DB1D127B4AE7A00F6321E82794BABD4D8814BD8CF2016EE3FC828DDB6 948DCAF3920E05C59E54F7323939DD8007ECC3CBC8498CF1AED76F99B5E8695D 829948C67FB70470D7559CFFC18F6D29EEB340597C0CFC94CECCFBDEAED491A6 4634DADCFCF79D47E4AEF5AEA1F2ABAB63ADB2AE14453D129262B15105DEA220 E42F72DD544A34124186A61701B1DE66C5F0C41B72068216B1119FE60022543E 2F760EA2F572CB6F0F16A6805C6D9831ABE3B365E99F45D68F5EAD5F515417B6 8604134075A2A572E39D30A211A875A4730D95B8D5FF033724D3144AE565D9A9 F8C81F1A4ECF9604F81C8C228422A4AA3B41F6A25B8B90F75A1073FC8F97CCFB F005DC64D338BD682120F16D0C5C5A84995673C6C2C851C9FF10A36CC5DEB9E5 C95C84F5F08DF1178333898EA1A5DD235D36BBD540E7BC26D70DCF4BC023002F 0AB73AB67BB075D7520D7CE72E528EC29E12687F88E2B63C78F552D411F741CB 7D0210DD76081F1BD2A720E559FEFBC8E25D0C67E537C0BD61BF6335C8F3DC3B FACCDAC376BDCB2AD4954BB62EBD275540EB32AFE40016800A3DBB1533D46BCB C2176B8BD6BC6579118CA3AF966211D4FE891B918DC0C4B55C68AD1333FEAB0A 17A30891CE8A1E83A0E83999C524553F137AF0D768D241D576C1D83C1A9473DD F625CCE234026ECB11F4B42556896BE79999EBC0B99FE53DDE44AFE136F10701 73D7AF2FFFDFF2789DDA2D6E73094BE25FE79636D4373D503552C0B76FC0B998 B282DEA1B0DB4E681C77096213DB6DAF0AB501C1974271664D8D4DF4C8557369 3C138D4F2B7C46FFFCF9F2CF710F8AF667C2130B69C698112D4DAA7AA3979638 0636E4212746E1048C74BB17A917622B6A3693799B61E31791C41E33A1F66F93 1FAA6BE313BA8AE794C103D1FF63DA904CBA135CB68AA71040D5DC54FC4056E7 7CA2BB35988541C9EF9412A4836F88CE91742DF1FA5E07485678BCF79134040A F068B7730A2D17494BD3185DE8522327B23A0FDF2CC388891FD16FFDD3605A5A 6B41BE612088E4F901836E02ACFBA635A73EC6FFA67BBAF6E4F58DD25A64D91A DBC40B2113C7B19A19D7E7F9F762420CE6FB005A43A086EADEE7C88F69C7D7BB B59CF68223289A512C0FE0C7A590D6C8174243C3F43F01F216E2009FFE57F63B B8EFB55CA62AE40EBA96E5C8D8E1742B24F83F621717A771A7A615AA2373FE5D 01EAD4897E0763CCA342215C705A459F97350F28CF05B933F92DD601F46245B0 AEF9E2ACE1359FE8207FFDABC084BC2F4180D5D08C1ED8EED0F8511DD4F70966 026EB2834670BC9A18AEE2B564F06763A6444A311CED273DA6075200DD492E14 C0634BE0D7AEC5FDC53E14B766CA83DD6C682D1B44EE1C72727E7C46D379522F B0B70E7CEA2AC13FFD22C7593EF5658A9CA196A2C614D5FDA066A17F3ACB2D4E B65F64C5E6D912D1AEAFA9E71FC2F49C28CE53D3DB4CAC6182BFA3C55AC786BB 5451BF3D795855D89616022829F44A2BF7FB1BDB96A42FA469C5D9896635A2FF 30348287C9C3FA00C1E2C3EBBE992B310863F654C7562A22773A5148DF78B9EF 4FA417C12C3208B9FF934A964E3E1F71B1784B138A29EED33CA7285FC1E08BDC C9744AA41195B48B8568FD9FD11EDC99325102F07E886560C3F2BDC00B20180D 686B2FD132918DA9B4C088FDE08D86152CBDA2F514C1C074FC68FC2C2115F03C 1C527953013978882A080E73FDE400C7E7A7E51A12BF0C52093E834FC1F5285E 4DACC1354F66C48C133487780B61B170EF9CF3734F7E6FA9448B818E9AF1898C 86DCA52875B5E59A5E4D6149FBE37DE8CDEF76B797462F06354D5E9C0C62AAA2 3C0A90883FE0FFAD489ACA8E670D6C13F27F8E3707C8E73CEE480C14DBDC27CA 20BFF22CE3A67C3C26F506807B2BA11B04533108AFC15E3F6108D2AE14C91803 9BAF2E1950694B57141705BA8F730AEDAC1E76732C327078CEB42D46DA7A6AE5 B4445A5512217DFB39CF13DE34E2E485639851B72A409AE6DABB2D6C734B3DBC DFBA2451DA1DAA8115CA9B3D71B9C6B07B09618AA4C4F673A91F6451F6CA0AF3 C9D75F66C83E885A16701DA48C76A2102905B8CCB2F6B1E95BF1742AE207A13E DAE2DEEFCC048288432B6AE8FBEEB0F238F7A38405ECF07F5B524A6C07661FE6 95F7042FA9540CF3313A96E3500C24141E83377BB8A2C201DAB48D982F5DCC96 0DE56555DBEED09E09E37422D89505C204E257C19A307204620B46CD95343EA3 7BCF5808E562A1A01F9CC62DCAE4EE399FF9148A791304DD3DE38FA3A30CF829 C9CF4D1A5F32B6E63E8C9D8B7A6D2F656C8215D594C6B911EBC33F0E2BE968A1 1E61AF16CE3CF8608E204AAFDA797024C5E968F97E0A01BA46429729B07B9CB8 0A5088EA0AB2065FDAC24CD1506FC85D8A7C0932CEBAC1D750997E17EC65B3CD BEAE3998631477A19FE5E16BCA1E79F442873FF17704C4D368228B0C3F0935C6 2A6DE71AD9A1A09A16B8BD08181CC180203CD1753BCB8FE15E53C2775860ABBA 6F5E56F3753A3AC9539C7FA816E4AB5C71AACCF529F6693147CEFD02F3F6970F 63063204D78AECC1A2173D43912D5EBA8BA0A53B6F47A5892CF4A1D9970DE44C A0B245FE5C0E330AD4BF1DEE72D814B6A515E06C03971F8B160781C880BE39A1 5079470D32C7EF3E936BBF9D717B587EC83211533AEA3689E85EE7103E59F1A5 9C6D8D84A18F324F221B1AFB68130939E5E308EC3A56B613FD4F9AD5D3CC90B9 86A7D15BC5CFFAF11E24F7CB73AABFF94853E15C8DD0D46A2EBDB067F03D53C8 13AD401A609B8200010B28D99C6A05FEA64367E2F31D08D13C1B273E3E0BF12E 658416989AA61DDB74B70C20E2FA7B7767CB27989290E8C78B4209863936759E 038B0CCBEA65B8CDC911B51E62EE11277088A55F285A7A49B3B5BA9A70889559 317B536B5A37FB75F2543F74E7777851AA44975855AB55E84F8AC147AF870C58 7E92B20112679501348FE44CB10800C87F277EFC5C115DDB30D92FBBDCC49A5B D03B13A20E5D87E3BF0229327B7908A349F43F99A556431AD2CC40A625E36CC2 09BF9FC825E4A05E2EF054760D113E8810DD3FF42D29C5036522B5263E6C1A57 DA640F4C5017CD1663CC91ADC5EB6A15CDE8D5998245DC727E3EDAA0D93044CD C5FDC4FF193140D05F1A809A08DD906D7D2B81AE1C1B67CEA004304E5FB0FF6C 1E4FE4CE19EF100A9642E94601F4A6BEF746EF4D4F9C909CF7000B4FC82D3DAF 00660A6B7D413228DEE6B7280A0328B89AA88CD331FD0BCD5D9E771787D4F657 47DA164DEC7E92BA7D96F7911217C660E367E1C4FBD92DF4540E60E95F9DF44A AE199FB203F9D29AA107EA9146DB82657BD67CE63611725968C6C2DA456556F5 DC56F31C0353FB7B9B979945D1D3B65DC86198AE3B2E01B06796432BF0106498 6DDA53A77F7627C77966AA33CF0CECD8F0A9275770F9B15820781EF174FF78E6 18EEF59199DF29B51285099EB64501C68DFB83D1DBD31C15B4210F1BAD4849E9 AC4A3695D453C843D6AB47F1454B1728512939F5B0D020A634116920A116BCE2 F3452A0373B9A6612BDE20AAF7CFABEB2756BF38B009D75EA2B1B4E72CAB512D FA7AD56112F27F9B4CDC7D097771EFCEE623F2B614FFDDC0F71566E4E9C1E755 EA2AD854255A7AD6245B5E4D16539912025C297A369AEFA2AC2B40EC3CFAC728 A4F578827E94DDA52E3D005D9801221953E20C81756157888131DCB90CEA2045 116575BFED687F69C15CB5B1B3052E4707FFC96FE6F3280BFBCA3048FCD51E42 C24DB3565225809410DF3791F75453A567C09B46E3B8DD58BCF1A827A742E66B E777C2ED0DF4FA27D5E6EF0B30856480A5712A939C4F2A211992CA12C53E4A43 3F372C54383E9D7CF2256FC14BA2EDB168F624374E0A493E084ADFA1773704BB CC05DD6C21EC663A471E1EB2D7935AE7F9CAB51EF2A99651A3CA4132E5019AFB F890C15CB3BCF2F1D8452A4BC37992E986DEE708F843EEB491B85F186CED51EA 83471F12CC04C923FACD1C295FC8CE60392FC0C1E785065DCDC869E111D11E4D C42C4EAEF40CDACFEAF9E9D4988B3D5E264D2CC76F2FE513DE1C06B7DB3614C3 827FF2AE5A3887F6991CAFA10C6DEB196EA826300130518CDE0B9A56D905906F C1FF4832830B7FBF4D93A6BA0961397C83165370EB5306FA7F6FE18F3FDA92C1 0DB2CDFA23D7D85FD9CC6C59E9E75F167693A095872AD6CE9261B7ECD70D1881 47F62D79AB089F0AB29B80F2193D6C44F41E0A37951CF0D56C15E48B77086768 AB9739ACFBCE15EBD3D150F581DFE9090EA47B74FB9106B69875B6145104CB32 939712BBCF728D140E8C521B4C513F3C7EC8C10B565A9AA6C35C62A016384EBF FACD52FF7F8441C91CB8A3826AFC138BBFE683F24989D6BD3215186D202779EC 086D5CAF4AA3CEBBE7DD32A91C1FE8800C69FCC6F8A3FDC89DA54ECC52B1E18D C6ED4DA239F05DA2230CA36EEAB56CC3CF2F0BC9D4044B32B2D2F4A99FCFF529 8D3A4B3B154E1D323E06EAE90ABB9E9399239BB300026AD1C9A6A935EDE1052A 2F0B8EFFB5BE931018C54D3AB6FECCF3B14E06469F51E292DA7FD3BC800206BB F627017681C8C62CA9318C0ED4BA1FD6F2E48B4AD85BBE086FCA7631E3888B9D D7C6720CAA219215C0089FE85178B4BA28CF84BCD43A3FCC1CAF820F4D44936C CC1D40F78A28EC64E33D4A9CF1C0816D7423826048CFDCDF6A01B2E9DBDA9B74 8AB8FAAE1C1E1DE11F04D18D13466D01A2C8D537CD89F8264D2B48C193841DC3 2410A7AB99292C9EE2B7DD4E700A783A9DEB532A60BF96ADF960F29FA65CDD11 4ECC8F07941F0A4E71B7271A8A9C6C4FCD041708F6CE74EB48DB56C864BD42CE CCEB18FDBF3CBFA8E3BCFA3F3DA2379F12412F17927E4144A35AE2409DED3AFA 15C31ABF0F5789ABEAEBEECD14C6367F73C72743DDEDB6CEA540CAFBE76E7BE4 DA66F228DC1808EC39E0CE147342FF9836B51DB87B4FFE3398F893B2018DD9AA 6E38941A1B7BE399594A1C79E15A62449A97DBCACFBF885FCBA1CD43BE29D11F 5CD9055F477C84E0735F86F68471B107FD9DE3C0E37B7A83E1CCE872F6BE25D2 0A2586363BDC69B64B4D643D05B2C601CBA81C5872DB99B5D9573BADDA057E54 27F43F592E2C74B22774545255ECA1FA08598108EDC1E7064852B8ACC9F5D761 AED330E8276A4061715DF2262E9BF34294805EF67DF53EEE4581C38A7AC245C4 61CD3C550E01D3E851A007225A2D30E370D2B6A3DC32AB74C3D9C27B0446BAF9 931D5144D9E550A548F617A76B955CD3B8E5259F6C79FA53C1E2FC67871C9344 7BF902FD4DB30FA616D397ECE826E791D0C59D833C036FF64B4DD2E16F38F824 FB517E0D8AC47CDAE4B8FF7B1430F1565281BE2B78537A0C5F0779D711122B50 B91A2130166BB56535A47FDB97B5F45BF1FE5BF9D551E118F2270025D0C900B4 6E519E690702575F36005E903B38A28C75D422E3D72820675BDF5C94C61C0EF4 75D3D684CE605A5AA4E87A418322FF5CF130AB500DC4F4EDDDBD1E67AF62E431 1E3DCD80DF204ABD61CCBB510EEBFB5D6DBC9E800174968DDEEFEB4223F0EEED 521EE60440DA57FC600FF80056F28837A11A27A5048C49B32B49D632E403F900 DEFDDBE7C8B20FD508B03073FBC576EE099C8A99F8092F79265AEED198FFB83B AFB9B731196103BFC6D44C2BD451800CB8E2FBD2537DCF65C196497739274EE2 7697B6DDA7BB48063B4B2ACB333A5B44883E259D8E4FEBEE3DF11AD40878BF45 01B7B73E96A0AF819B57F212769F659093C95F94FB669A5369677A4D60A4844A BC03942FE1776491DEE5DC2794707DC72E04842AA6834D5ACD1AA31DA9754098 0FE8D93CEC6115929411AB7D7647173E21DFD7188AA138B59EECB77C48993826 BA56B7D5D7672B1AB1C6D1D8372BAA17C09EEC3A78225E1C483502EE742223EF 38923FE531F07C57ACC845E93E944125AF6A8E2527C8D09D8A29528E45C47F09 39A39C44B67B1DB0AE614392539ABC52D96E4177D99233C9264E4F8CCD8AA72F B54023FE7CCA69FA753B6DB48D931E1CD673A90FE048D270819A07255A8E48A6 8D3B0388865A9A702692849B91E15AE341958B04C7DDAC2C9CECDFD870ACA8C2 50A931A8E30DDCB85F171E941318C6048EA6B77E4EBC1B8C195B8DF41507AC62 501E95EFB165CED97C5A2E21CB75FEA6380742E8CBE5A9EC79FE790CC005F8D0 91CE9699745E75075712516793D650D35374BDAEA2C401D641AD34358F3572C6 13CCC6B7A98D06392274CB9284BD56D7C927F2AD7CABC4A1C78EFFFEABE726CC C7B3D0ABB0486B0B488E01A34350CF4CE8B630CE2AAD3D5BCC6F424A1DE720F6 C23E9550685AEA45C4F22F96D1C902409E2D391E882910FD5CCECB6C718BB52A 8ADFB12522F848AD873DC653C9A40EDDDF35B5FB8460DC546B09E2042E248A87 4E81A1702F0A9616ACE9B1E4872F513AD2235613998246B5D94A361C8C85EF60 2135ED0CD684470469798A24D14F0F9B8E6AF6031F1EBE422D4AC99293EE485B D042744706322784D38C7342F3EF36AA62DA8CBDCE17A4C40A0D0901A4A15890 097F319907EE6007D2B90EDD8D4433E342DCA08D76024B2E655FF08D84F7C78E 2D149C147C59A32CDC6EEAAA29F239775CC378A1EAF650B526F19F9FB473EC00 CB7962BC6AE0FCA66760484C30507E70C8D3630203FAAE4D5A5C147361BF7A90 09FBD43CA2E50689682FBD670DBD3999144964548DFCAB44A75FCB8033187308 193B0CFD83200A4127BD2DC178DBD1A4EEC3F48716B78690A62917E67B73A440 B598726E9919874A224272A101B647D58760227170A04671D81140ADF645D9E4 9B3A42967A4C82D49DB8E6E2A61CC107FD1BF3C005FC8EEA9D111A2A3C623917 343C89FB990855245A61A812BB7EE226C64101F55499C6EC12C072DEFCB691A4 FE0BA9EE405999421E05AFC6E6781D2AD130177CDF36D1B8B8634BAA642C13A6 AD8D47833AC2E461540644E4601D291DA09D44A3761B4DBFEFE80D6CD99C579B 583A194388D6B9D1D3192C3491489BC28FD9DA779EE6B6C84AA7FE3B79403003 A3486D64F3F154BA9C0FA69786A67E3EA2A767432B91D3C72E5B40667D213A75 90BBD9A453410CF798C7409FDDDFF0F0E35FB67C0E715A06F56BB9B36E3AB56D E3166A388DA2B5621C62FF03F10136DE44D1DD09562B247F936B7D2669B2FB73 8492BFF9EF41D3F87CA174DB75CF89ACFA6C2ECDFE919299665AC9BCF672914A 7DC82F52CCBF6D05E4A2EE338A3EDE19801298EF1B4580FFB305EC331571BA2D EFB18350EE3F71B57861E96BAE397EF51AC598A0220FC5905A70AD4314D75733 A6FFEB7DC0666AA9DA53FDD26C83FC5061E21E8A74EB046124BD29782A28503D 80C9A1E6182DF2610507704FDA03A7A332A43BF762871743689C189D3128269B B2D1592B8314E9962116AF31157A01C55A3F19DB3FE570F194296FE793ED9A1F 84BBDFA401B44ABB1524A71E2C4CF0D60A012AF6C9ACDEEB36E46B6555F6D1CD CDA20FE2A4D6D1A10F7A6E3DBFC39DC51E24C0F6B9DFFB5B89154F81CBEB1465 AB1078B9F352B9F4FB2827B767929D199AB424136E922A0E33C68F7E810163BA 32C3A3C05AA456BED90F2B1485E95C0565FA85CE0D2B5A1937EDCD406C0AB704 8C3CF27DF14BD0E32EA9378F6267EAEEEC7DCD54E960E5E9D3583B00C021D76B D0DC676F4368970A5A055B7A232831C130F82C9B865CC50967A65AE972B425E0 D223027612B7C6290C9D905424AEEFE9CF75D05CAC62BB742CF92365B20E3F2B 7E1FB7B6724451D828A31B94BD628F2BAD92F05B9059D8F9912A73DDCA3C9BF2 31164EE7B11EC2166B573CA0A92545E866E65430B1C529ABD5CD0AC936D66FD2 62C44F3C8249794EE9E7D610E1C0A93C95EBFE98BE2DEC0823FF1ED3B87E1574 BA5B4B61FE35A5E9B08C30E4916FB631F7AA75D62054212F96BE1147A60F07E6 7B8C973B5850A6B1E407C067D49135F4FCB987BDDC8510DF9B579BAF062F3A6A 6081A8C0B506021AE3BFF8CEEF0CF1C7EF84BF08339D2F46BC4F91022EF73455 9912D8254C6CDB22C8037C221D3D7AC317DCA6BB13B29C68248FADF097BC56C4 98842A21E9ED86B704F35773A182D3FE8CAF00D6F7EE91DE0E77C4CAFBA0B5A4 488A60D39E5E2B3D734611C784A734C71D2D17BD4B934592F7E469A68308C783 F559A61A99365BA7A2F66BDE7FC34F88A144C258ABBA9B55752D936387B8F5C4 D756311122A48BBB25DA9C01AE005499EDF26D13A3C1CE03105B8B5FAA573698 8ADD1368E93685625B669AA3B463EF465D5E878237F7EEA0500BC613CE9065AC 8967F1E5A3C61686033221A4136FB2950BEF205FA905CCE8F914B1787872D727 3DD1512F83D15BAC8A322F2ECEE919D708E2501EBB070ABA2624A568F55ACEF0 B040C47CEA84B0633460821B3E507E2D382BAEEE6C0AC765D7084E254603DCF9 375DE5B307C7AB5B1160FF0C2DFFC7E51612BEBE5B79E91FA31B321974203F64 28543C4FA38632C9FC083CF5A6E3F2E4A2BCA2E7AEE3B7130D93E1253995C0FA 07380D53B9BD3165BF495C441D85C6774B901A21698D540A37C2F7FA0B28EC16 CCB559BF21CA1B478564D748AB67DB950333C533DC13F8D1E8194584E00E01B7 2AAB72E3567FE9BBC028111ABC86604DB1602524EB8D6356B5EFCA3BA4970356 D277EC66F550619B2FBEAD915CD6919ADF2E99327A79D4778EE1B9F1E377FD19 289F26D6826DD852F76840A78AE970BAB06CB7AACE8F061D9306C80F53892345 273CFD89A0B324B14E7D22D7C294C2A591F795A5E18228B0D84F64215172B23E 85622148D8EE03CB22E3AE81E7798411D1309E4D50E4ECA5078A0E6DC10DE6DB 3BF2BFA14E068F20C53D97A08D0067208BC09A13C27CBAD11BFCEF0359B7E5D5 E2BAD80E366A16ED86466D7B74797CB91C7D4C63A99114EFD231332AA4CD1D5E 7BBF0745452BC64B2A18C2F8435DAEE50E3453B36BBABF1AF08299DC328F4AB6 3AB7BDCA224261EE60F62663B1181809A7DA78B74D109DEF5B06D7D2603B0930 EC58BA1CA9B6DEC56ECB97899C0D317D182162CADE21843F0BE8A7A697D388F3 F623BD7B6661524AA18B44F30F6A33A093C4C9EDBD8CCF9785AA9AE8DF53A2A2 A3E58DCAAC7820AA0B36585C7E79D8AB747030DB148968CB2DF2FACA8E68C08D 5D72D24725CF5B5D47D57FE02D4FB390EB5AA0AB9B90BBDB9E80D9867C934640 997F33B6C2CEAE875D8303C151FAC4378FF893E2B464B42E2712AFDF1893FCCB 9D00199C6C32BF5657352774DAA1CD0FB86EDC497C2715F77EB8B59315AEEC86 2E8F306A47AC0E75D64E38C4A4E988D1F61E37A9EAD98887F5C53E102669355C 2FB7CB285AF3D328F108918111C4001809CF61F71C822459024DB40652134943 6AFD01F9AE6E1570F39A8B30A2ADDC7CF29381B3D109AD0BF965CCDCBB537D39 375E1243B3FB7AE7D5F6353102FF0D32FB00AC3F3ED9CC12E44DDFBBAA786201 C5D5EB7582924EF090698E9C47D4E04EC425FA65823570A5BB7D370BFCB5F04A EB4B1990835EE4385299E8C8D9634231CC226B97BA369F97B702898EA286FF56 5847756A6E37625FC01908C102A8955F5445C86D41B58FA52EC218145DF05693 CBA4E2018296C21262E49DAE165944F0BE305057C8EB5B21260E0A4B16553994 A45213B894E12536F3337D3BE966A73CA0D3920E0EB565447B4F11E6F9983CF7 3636B2BE03C887AB2E336D6A1BAAD7F34B260E83C27B2C1F5A99771CCB57A94B A48D07B8F06D40B0C768C5C36CD20C5DB197027272A77B1F147C3737D3D128EE 8D8BB0E0000FE6870CD85728CB39C617E162AE9BEF4191AA0BA07D8E485954AD 4ECBD923BFDD910EFA7850AEBACA10D55E564DAC820E700B0880A45F1D1E496C FAE927518FB33869EF27408EF0DCD4CD89724C5ADF709C71DB7CF2DE1362C708 485D1D61035477B753149BE8A973E31C4403BD5D7B8EB51B9F62DCBE669868C0 C7A02931A19D32206F84B2FAF118B69CFBB4A7389857A7125137DF03CA9AB291 D9D2282229B1DC77BF145D70E784DE2EE3239CEC54C4E657C82338FF1944F9CC 5EE28ED208FED40FFC02A9B2EC2755B0BAC745E270C2892FF8571CFAD624C46E 8272B5439C4E870748A74FAA837AC15B558126A6D8DDD92B59410809299FCC2B 42A3DB3EFD43C82C35A568220C0D572C945958FFDB55D38CE09FCD361ECDE77A 8C103CAA8CDD94DF92C17A25B0AEF7C819D4524D9F72F65FBBED4C3200ADD196 2FB41FE37E8F25AD6C69CD4F03023E4A6390182DEE4D98D0E4179E2B5DA7279C 5EC234BE64988BE02A244B07918A2AE5849A1C704BDC59417CB0B2868C7A7272 D169CD0F76CF262428CCD618D62D27736EC37AA209CAA3A6B02693918FA23B1D 6C7DEDB156FA49E5738B148EFE407D20F5A1FFDEF965BC2B8CF5A5D9B201C704 5D97CA619D4097582A155AB9928BF65CCAFF6C4DBCFD3C470AF90F2C4C44AF02 0B08C83A2649F14A8A5225A0FEC1B615B94776A9E586562D722B9BF7013A82F9 AD62664EF8802ADD4F865DBD1C1F0352948163C1FAB8ABA4C7FF55560917E2D6 130928770BAC576ECC1C2BF51294DE2EB6B4377F6ACFFC4C6FDE54405DAA5277 4593CCB9F92E402B51A445C30A105FD1E1BBD5623A9386E40207B269ABE5CF0C DFC27F74D196F7C127BA73C5312A52966AD451664884997B9B36FF67B0B67ABA B1E4FC184073EFF48599F81819EA69A51DBB7B4559242B5565A8924C67E0E590 3A7EF3D5E6F3AADB6F36A2494FFE3E902B96F321119742CF3B889832C41DA34A A974B02B3D2F1CD1F87A926C150E5A7E6386D842CDC54FE3A910157D0F0ADBEC 7392AF0D08967B4288AFCF0F31D80AF63E516B3FD38AC7B0D9CDB248AE46E27E DA1635B68EBEA6D80C48E7255C609D7AF4B726D50E4E62B1EA1783F42589947C F86DF41D5321D52386D8D28C5CBD467E66DE6AD41E3B0CB23CE275C01C1B7AC4 14EF57D0EB39AF405A83062C138D2608982A2D42AC2323412F8DB912DC3D84D4 B0F8022EB47445CE922F79768909B9C5EA42375BB856547B256C676F7EE7F3B8 69160D213291ABC2B15E500E4842B41748DD0480478AAFC899062440CB42B674 5769F101E964394DFFC1805BE53DB429A578F37C4AA6268921BDA8A13C91C8F1 1C7C56B6FB0E66415DB1FBAB3F784C61FF96B46EBFFBED08121C6C35757464E6 89FC8BDBEF070396F488E6C4D15BA572217B8E0C9B293E756B01207C096085D6 37D8511287DC927EDF7F69F8DF308F1CDD8E9EA12DDD5B3E8C3DDC6E4BF8ED63 48577FEED9DEB081266C688E86E1138A1E1612640DB3020B3CD2D0163FB48BB2 9759133B76AB23F554BFC7AD5EDED9676869B9C2D16956E8FD1FFC5704EC4032 3B77775EDA224B6062F6C7E47AB493F09699730A55B38614B454445C9A4C5E26 5841F65EB7D6424CF316ADFA3B1D6349BE81581F90F31CA82E558E1A65DEA452 DC5F5A2D8EBAEED341B295A660D68CAB32D7687C95ABA7FDDC2E249431B423B8 A0511F095971304768AB4C4384E3DB56573F6F9E9BFC670D023D08DE98064FFD 03F42623FF0B40393DBBE0ECD0CB4300838E6E37D5C24536016A243167E5D212 33452DE44911367064C66BD03E92AD7334522B3904B74E3CBC2BD46E3C02E953 0C137D12A36476DE4A1E8648CFB5D43AA6E7314AD81C5BDA73E1EC2016EECB43 1B8ED0E9F61374E36C11D7D11C5812B0127FB9E5A9F10CB766320BEEA9C3282F 5D2D748704B31C96953AF327E69EB5725F7EAC53D0E07DF930D49F36597855BB 1A97DF995EA05BF2EAE8C26374935069BC523BCF4F6BAA6F87F49EE146AF9981 AA4694DD6D4202A1E5B7D63CCF9F52019FEAFDC25C90F0C6B11CF2FBEB6CC165 9D560BBF3137E8F58382E4FA84D43338F89B623BA2EDF0343A86541031AD6C59 DE8549621A443A6BA12609D56DC18BAD904CBF4B3CC7B942F01E26E90472B4F8 2E5F8B0A4211835DD2EFD06846375670148049E218CEC59C00349DB031BAEE76 DCB2AE8E80E4F5C0E860DFEBA7B5E0B441059B8A96B97C4CBBF2FD409257BF96 93F3D41293E50BE4B020D3D950307324C8D7E39E1FCE31B4E7486D66D5491FB9 477D3A23BE78FF4E5A0E61762C958D4BCFA7005375D336C5559FF521961B3EB8 004EEBF6D163154B6D4BDD939CD8DF85A2C2D1012AF661B20A5CFF5A83AD7580 83D10C8B4E880CF8854311E253EAD5A6633FCBD28A0F62EA1BF57E3A7E2F9419 678A58080EA581DDB9762605DBEED9B1FB10FDAC5B6F3F4D7235723125FDA61B ED77B36C149DE11FD85CE0917C7664E9B1276D40D2E375A70A5AE048B10EA99D BE4C5DE780F8945E71FA6DA35E7C83C8ADF0B1ECA0E5DC368A849529CED7237F D35AB812F42839FC7F6D397BA3F0BC75AF2A86393E4B07AB7782E689512A4452 18C8EE1E7DD4801C8B06BBB0F074CE63746FD5D4906C493980271CC516E8EE29 AFCFD5AC194A25B43BF3284B7FD452096ECBB3A9F1E01D6E3E6DF0D2F3E543F9 D1D1CFCE489976D53F01D8230FC143723058B133E2D6807416F82954D1CE920C C2657B3AA9FAB2F17E0687F912C658574B3F92356BCF515A961B0FA64CF02C32 73A40CEF0F0E47F16E84895D419FAEF6A720664AEF72BD0935BCB84BDFAB1BDC A30238D92ADDFFFA5D44840865C0C47A251515932A3A6560A220B97EE63D72D3 B876F96FFA6DFCA3938158BADF3D2AE3E87804F44DE3A34CB4A7E111BCB06E12 1AA0A1140DEAE247F7783ABE883E5B447ECAE60E879B65B8A08B34C291E2671B 868A2D47E859BDC88A623BD3EDBB68191A007F492A8BAD449BAD5A777AC27EFA D54432B5E36337B34E618F1BF682AAFEF74895B7F4AF1A9C89924CE97491F6C2 380E2872B04FFD364CB45CE295E147FB2D30B188D936604EE0BFCC5FEBAEBC20 AF3313F42D4901FC6245BDF6978B49D02D18E63AC8ADAA66880B72C2BBEB4102 065A79F196279087046952CC1735735F9DEA15E4EFE429371B7BA6BB18732D21 44354983053098F18D0C51315CBAC346FF039C6FBE3D587DB639C9C3525391C5 0622663FBC671AD888E142941462C1794A399F9D9B6924E0B8F993DFF1889DFC 3D6C756C551AF3E9290F75178C736FC306AA1C6FA3412BDD184890CB8C5CD361 46BF42B6F4E012F557450D357FE9F552759E46F2DBA4FA83EDF61D591F6B4999 519672A0A8DD669EC40B6D0DAECDC45BA4EDDA09159F5462F7F111628E9C3479 7F97BF003D5AAADDB00D8405ACE78E8D304D2F347566EA0B0FD0F016C374155C F804DD1BF1118B821D86F0C889AC5E659EA83FB03A2D1BBA89931A42FD93B363 DA64004C7C72AA616CFA84AED915B4036972FF589CFF0EF9CD56DC4CB97610CA FF7772AA825AE14126CCECBCAC80F493872EB5DB4A896FE5E9D0B5A66F0932A2 E1C501D6700B577C400160F3E8D7A6116EBC5DB135E1EC20F2FFAA20B1C1434D C996F12FEDCF7EA747A3BEF432640EF473BC0C826B160A66FF9B3E3AAD6B719C E650E4E8EB513027F86224D004E35AB80FE6CF3CBA9EB43AFB2DC6212CC769D9 6B484DC43AC80314F71C6D47F673F254FE69BEDA3D1FAB1F62F81FAAC1890E12 0B728B44C1C4677B176ED8E4F95AB36483A52A952507F10A8C4F9F559383DCA6 EBA25D9DED7D280D75DBDB2D25427FB569B1FA6BD3A56C5E5770CA6EC183384D 81EBA78527F08AF236027936F232143D604905ECD062F16E0694195377017CD8 6E91A4B77A681F165235A4E7EEC1D341604180129B029040FC1B5EB67D7036D3 6DF2D7E1B1D10E797F648281AD69D84B3400C9D428818D139012B0CC62C72AED 1D1A4765C417E2A4A5C17DE68237553D858260D586448E52E6E258CEAD67FD66 BDDC17F46531781544EB72405E6E4CE219A84C547A13B83A1FB4F74B10318EBC 70EDB935E3B2D8FE2C78ADFBB40CC3391E4F3E1EFC749FAB513205F0A86A0AB1 1EE7951946C9F41F4B161171BBAE601E1E5588BCED4D7BF43FF15F84FB76ED38 4B3F3D387A2150B4571300071EB902D6CBA6297108FF949D1CB0CAC72E79257B 9400848D726AC57E1587DC8DCD84EE353E4B198C2D971909E637E9A118D205F3 EF2667EBBF540984DCAA206B1A90B087A1B24FF319DCBB058A06B55190D8D07C 9C6EBFFCD1059EFA438CC2E147863236ACC8594A35887AC50CDAEF28F83FF3FE 175D39CB192FDB3BA801C564E0C374C547335D08363CBF194002A5A6C7DF14A8 5ADCD41F8184C71CFB6FF1463F2C4A3F009AA2A8E2CB310C8B0114D068A026CC A9EA7531F369389B87BD3F6E4887EB36DB4DA9F7B5046B465C7DFBCD9AF13B29 FDD99E9C5A869DDF3ABA13115515C1D50E1428407E487209FF3FF219C343362E 32DF1CC22BAC0E3465BE2F29AC38F9BB1BD743B07A3038EC83B0330A1F818A36 C96A4BC453FE45524586E242699B07E267E6378613E7ABF9505AAD67C90FBD85 2F5A163C01077A344CC619951050F74E6A7EBD83B3B7114D4CDCEC2625511F8F 69E27D6907D281E49411E2A504733CDB34A174D472ED8A8DBD1B31FE6A1EB6D2 581F1A2AA2C707E34C793AEA0EBBCBCA512F8B6EA111E4E3F84A8BCCCB97FB96 B66833CF6BC79997F52DA7D7E66E937E1E4904B32386DCB5FEA7678336CC3843 458D5D57CCB361B57E57AFA2D685236B6B5E518AA4ECA3A6D8FA027FE65A17E9 F0F0EACE5F0D0BCAFE713888026EFB44DD55E3C4C89467E91B7AF726F92A93F0 801149AC9321FA0BF01D43C72EC33B110D4486C743CC463663E87CE9050A65F4 7CED2B58DF652A82185EC4AB2421BA686462E85D91B3F21B628BB112889DD3B1 6E3C8F12E57F82DBD5715E133AFC1792881100289B43F468814403A5FAE7B3E4 075C211D80CEF180CCDA8BD5C97172B3BA46FC4741B71E7D0A203DC395ECD6A5 EAE607117A665B4FAEC2ABACDBC1BCEAB6B1CD4D3E6CAE79D2D1514A3A14F848 E6B7CCEDFBFF650940D8B9EC84B7641E1C1FAABA80926C16D5699D749BE0D030 D1F7EC0D1A6855119FFCCA75998563FE17713355664CE5318A42FF6C7475FB4D 29F0FD475D14DFD28254422AD260702C4419212A9F4B156986F84C115C85A075 45D328137F9977B1D624B581873DFEE757976458E919AC56C93DDDF16C2A4BC7 4E3D3D3C63809BF7246BB048751DEE1660C948B3DFB2CB100C11F127B5D93CAB 41085B57E6D913E58884F0256E41EB786A7D4FC6E86F808E88B571229AE60092 30820DD4639EDE6AF18493C0692D2D75C709DBAB254E6D6769A3CF5EF1624128 238E2D43F70DC88EF6056A9046AF693B3F5500EBA8C5CAA54C9F71D8B64E0546 38760C1543D11741F394440C5BE1B6B419CEDC21C94FAF3226A3042583CE79E0 25C3D54743A32597D3BE5C50ECE5214C816AB198AF4B76B6DC29CAE36BECB42D 577837918F31736CEC04F96700F8A6554C9055DD0413C80A0C786C8E5D393593 C2ABE7CB39B06D093FF7AD439308DD9FF628CBCAF13C8D6F1E43A3F3917CCAEB D74A2AAD345666922D6B83D8F26DE01A92E733A4FA22385A58AE3D7E25E45702 4C114A914C25200FA2956E859C10B5181247337EA820F7ABE8D113B02DD33557 E45CF6020A6F57C51396FE9C526D354A301666C99E367F2F15FBF16D8DF11451 4549586F71D15DFEF78AE03B4519F4CD5819580E081220EC5F447BB851054544 4F1706D4584BB5B506DCE3858FCDA4F7E04AC306A3B0C1E0CF605B7DF30ADB71 01D68FA98A9F37779F1D425321236C79551479B9893C7A40A9C2F9A6B2FF162B 4D147B39E64C75E04BC31E7D2CF972A26E3CAA007D2359E1D19B5F5A1D23CB86 0E32090CEED9D5CB01A0443FCE29F1376279A3D1FD229600A8FA1D16C10945BE D8674D73B1E7F0178284F61DA31B9E6F4B72A6A3759AAF00ABEF388AE45C08C8 75504891484E991E498AF127E0272122C8E7B510CF07CF86E4823B2FD58C9D3A 819B5A61636264A683F57CE9E7C6BFA87590CACB0C5602CD6D843C7CD9E02AB3 E03BE08AAF33D1E76691DAEA13155F2CFCFD443736AC3278A89EA63950124F69 516F70FE9CFCAE06CE8AE1F08A962A0FBC018D2A4F20B6AA1538B9DFAB7A436E A95D13CB0500106B9E4887FD4E7EB66044BACA62A9A3B219B66F8FDE6FB58BE0 81461491163A10B89930B6249C240B506E4C8B5ED868ADF15E299E5CA2750B47 9D9F80280BA1DB311F816CCA8527465CF58F6A36D37F3ED76E07B0B5409344EB 8AB48DF98DF775DBBE260AF2B4E8E3D8859C774B60D632EF74C498B223176806 5E836F293E43A72952167A7420E73B8E6DE2D34A4C092C33C24F7263FDF21221 AE459E644579C22131A62B24EF7CAAA928743C16665B250B497345F7CA138A8A BC1B07B5BE5D0E8992D2E9210657D6AC9886ABA43B80C9C6B39DCEAA72143AE3 26F0F4B5598BEF3A82E3BDB0B37998B08E452EA86C83BFC01CBB676ACBADC971 42DDA755D53AAD1F6F1D909FD229B5C088436BE5780A39C979303E0E48E10F3C 82784AFE8F47FCA541867FE1E68D66A6DFB5160DB6A19A2B9B69D35BD17B739F 0942BB041055DE1F9C471F75A17ED8CC5C1ED16D94EE5537FD244D04BDF63699 C3C0B596C2F78FB4ED8D03A03A395F8E6AACD1FE4057F386AE5208FB49634384 FAD98EE5F135AF77320C9E196F9F5BB2EACC00FCD6DB85C60851C78DF2BEFB6A 06BE8BAAABCEF29423929DC743B0A9E75CE7498CEDA3EEBC0A77936824A225AC ECB31588601797FB5B6216077A05077906643BEA9EC99ED4C04BCE39B209780F BAE660A58C3CB49B7791429D9B4E09D3A9B0A222B409B1613CB0B9304F2DEBC0 EE452688A48597026F5F595CADE6F1EDAF59DB68A55BA8404947B546A980386F FE29BDE1640EA48FBF089A0477292A86BACAF7442DD2F401AA71A057A50791BE B5BB38161FD1EAE0C977DDBEC529E43B47C0F8D92E052894C03F5B3B68529F3A 69D27CF63A8721D649D827C3BCE69FF658794AA39A41AB674EE38D7702CC45C4 977125CADBD9D6342FF289DD886FDAE6FCA02260C17BD20CFA83D32D0FD7C495 FE250F75E08BB44E621D5E746A10A6B5574622CE7C3479E767AA4BDFDC2F1A00 D49F509044D13AF26FC851EA1C2CA8583D82EA390AFF5E3B1CA78D514943C3C0 F65445AE9CDC179A6C1FCF89B8B76CCF9286634DDFAFEDA1F5E6DC9008A8C2B3 C2A0E9C9C1BB2F2AFE2B11117614F02317FABDEB29007CF1CEDFBECB97D6A656 9CD4736336355CA92DE3CF5478BFAD716FD5BCB2DB37E3AA365B3E7EED305D98 994C4C04023AC413061D0C78AF6256E4E9D2173170FB3E3FB3AB7C59D178B5B0 D38957D96D114B13503C3605CF72EB451FE9A2ED629DF6B69500FDAC089FB93B 9962897D7E438C0A638968AC06EED7AEA8F18CDF9EDDD82FC3A180758B982AD2 30DFD1FFA15765649B47DB8F698D0ACE13889B553895D221FDF7DF9437DE568A B2549817F5FFEB1F578A9C07CE360A97AEBD2B900E940844C8917325C08ED32C 4A639E14771C72A36AA243483FC101BBA46F840030D50F7DEFAAA5E8D5471B7A 2E074D653AB7B2F3E7846426CA0D56B4FE3352BEEC87874C717302B8DCBE8491 056C59862993375ADE9832A9B315D6995FC3CFFA889B59F6D52E02B2CB1C3358 ED1250B9A0943F873C39E8A8015F0CEBA78CFFFA61CFEE4398A6E4698419A2DA 31BEAC9921FA0D06E4F5F20F67039055699409859CA1B0434BB91C8BBECC4DBD F504CAFAFC17899FC3D7BFB7BE3C02231D95B220EBE6A8DCC086444A8879B3E2 5D0F668BF035F0EAB97B4C43029C1A81356FBC5002B476720B5EC6811AEB9066 B821FB26A8C92A68E638E40A5C7E87C7347F913A004581918AC791D0FEFACC53 F8C43B449E279DBB08561BECAEF22E899F6D69F8DF0F6DA4E60EA3B278096521 B36B4005AA69F7A7BC670FD0437F371D381D2E0E2AC39A160EC8C942CCD6833B 28C324ECD33039FC468D97BA6F233D2397E3DCB58D1B42C13E07E5FF3889EF89 947228FB8D9559E73A51AA7D06154085C362AF4DEDEB55B8D4C783F74D1ACCF2 6E50C0242E39D84CA612ED8B2E9051160A6946DB0A29DBCF9003458E3DF47AE3 1C080A6E673A266C6434B9147809485B6AB90009BBD9555F4080D999713FAC4D F653E36643A43A1CD7DB9148DFB3AC2FC38F3A37209C2C0959F15134EDE718B6 7700680D76C4999AD29ECA821C09007DB97CA42BBB39281EA7EAF79C616566B6 1FE58BC44A17FFF520DEBF2A72B62039AD1626408C4FFFB3537B4D8AA53450A5 5CBECEEB36F85A4B95E5BE79C89C31EE3E500C879BC9AD1A25849713E92809DE C845EC2F024719660A840C194ACD83B7DC54353CF9A8BBDFFB5B2CCCA1A64813 03D30014375273D8F311387C3A494BEBDF729FD4F0A8F33C869EA5BDE3CDD44E 8BC368061634B74101D75D0EDA801D1EC5733605BF151F26DBDF2242E255358B 7FDB00D1D659CE703FD9B52BA2576FF24C139E683E4289376019BF92A8D26EEA E783988F9AD8EF0B41ED845D2A5199343CF462571D5065F5166FD7974590FC06 80983106C682E0D9A3BB0F6FB34D68EBA9E6EEDC55DF2667D59B72CC219C9744 A242DEEFEF077AE863C340C6B6E02E9208CFCD56E3EB4A07F7FE90108BC5B7A4 39698DFBC3589118CE239887BEB39D6F7214AB6310C9243BF6270AE824AA2003 12F724AAEAAC31A00376E9A2C94B295789C1939F2224F824EB0CE6C511F917A0 8A3585EAC56C3281D0CDFBEF46A0283003D72BC10B0A0DEE056D8ACCBD0D333E E750C5E0D5ACA4D821DA6CA3D0225828005FF415879AF9A61E85175A9D53C8C9 42B890A5E6ADD5F7EE5A20F13E64977F025D82E6A66335FC6116F356FEF1BE1F A7889428AEF3F0DAF25A55A803B3CC29E95DBE46F2C6095D45D684A40948D86F C4B7724C1A99EECD2DB22EACCE31CC4AEF3B607408A170506129175C72939D70 A0FB53C232D99EC4F43EB17235CD59B6881162A89BB6981727EDEF38CCD7E82A 5B16F4212BA513178DB4932B93E550FB73823625B64B0FFC6A5E174F33D630CB 8AF474C0DC24E43FDB57FB54D40DFB3642B6DCAA7FAEC850A9E35C1ED722C39E 68CE0AC12DCA9F87124C48A3E15A8FB55F18905386916F86C81B16856A75853F ADB29EEC10F02ACC0386488659BFBB652B72D822A7AAE7D04E3385EB826E52B9 0F452D0020D434AD676D1CB90A081748E8EDFBEFD1629B5A27B25A821D7A4331 B498289832BB697862A80B00BCF9E3D3CCD5B582409CEE88858AF035B3649E8A 821DB706979B9F31CBB0CDAB99F138FBC4996D8D801D68F387C8589839F7989B 8FD17476970C1A951622AB8900F4EFD787A2B16565C0279E1DC6F8D424F30121 B63EFB021060E8B1DC9D93E089A7094A6FEA55ED9082705FB9BCCC4CD1D11279 89FFF96BB7AD5D304662ED4CC904981AC1AF7D2E4F3E9056C625004C3DED06BE 98E5E553BB05BF024B9A4466C40D798EEAB045F9CA8786D5DAF9D44AA4ECA3CC 19B3B4D643E1515B8311E53ABC9E1401893D4458C16C6F1DCEA7975459FACC34 C842D34ED7D0456B8DB3EBBD624D2F5CFCA397F4AF808238EDA988A50A3729B7 AE1567DBAE8F6F334FF3664D44DDC2AF6A36EB7C8A7A6409D34DBB9D73CB3878 36DBE7AA3D64EC8F4A786B4A6AD5362421B6F17446114701510A8BAD88D0FE37 E00BC603ECC0759752BC16AB4AEDB4398D3BF64116169708085F6DC8CF4A9CD0 6BD3AA4E5002B1F39FAE1216EB62012DEF2088ADB703D384DD0FF3E40E6A65E5 3003CFC3FDB2B5857E816AD88DDAEF7AE19AE6007B776B0F552FD06E8B65D0E9 A030BAF88AB3B112C1BBE6D7E8BF1D2634FDC2922B801200FE3FEEF3E6BDDF58 576933E2C08F13EA75FC91F7CFCB3B94697E683C0A3214F0266C4197D1FC2326 15D91062779B0375AA20987A9AC1AB2801CFF975791700088AF517DFF70713A6 95EAAF6EED3F27092933E0819A4A08554511BCA0492CDB142DC499497CE0806D 82DD00F9AC7D7980C4E79E21ECD129D4B5CB09804F71D0A2BCE0671AB28096A7 71810C3752DED964DA1921832E376053A449364689B0B7BEC7271C399BEAE1E8 408CDA40DFA385DB85EF5E258A0BB0A71990C11CA8A18FBC7F33DDF80CCAEA35 D37D8412A43707015C11FA7CABB2E9D706D7AAC812067354E0C5AD396F52895C 5852B283FDC5295CFF9F6E92A12226B12A70679EC6A8149BBC765FDBCB7E8842 8310EA6C13A21F2C4FC6853B81A3F277A3F9039F4687389F47DEDDA4A12C9DAB 4B167DD8810B3766DA9AD7F28B39E52AAFBA6A8D90088518C9B1CF5F344DF5D0 E6A37AC9365EF5DE2218DED4D38AB1C68D5A10C231F7978C22707FBA46824007 7180EC0C65DEAD1F8B5BEF63575014DDE2914E6F87723209341E096CF73A7AB0 B5C2A7F740553AE54CB6DD12B835A4F6E6E858BF11B979C3B325E481E1A7294B C2414DF6AD56440127E29A9FF1472882CB0FE8DBD403DFAF2BEADF872428D25F 67B9438DFB7442982EDC49B2AE5C28BBA64B23E7EA4A11F3E8B5DC498254C665 C4BC2CF336CFD5413F2DCBC40E0873993C5199E0053392DD0062BF2D2E981F12 47C0C98C5FB8DD84BCFC85978C2DA7BA5635A964A9CDBF679D8B9374DE87C31E AA3838FEEFAB59635152A3709BC9FEC86C9CCB6BD55AE027C5D614F16CF0A800 708F10BD4B13BE1C01C1DCAABA7857163128C20AC8D10D18F3D2433B0549B153 B810C41C27692E419257A74445D8BD1017E825D49EFA9A74AEDDDCA0676D313A DA416668D8CC99F500131CC1D52E2CB4343B8FCA08AF1548AE59459F208855C3 0B4F8B21B6493410839FA98F335F6F7D4F8009B014EFCB97AE7994A70D81A303 C4186EC8674B77C53F2A5B56581E990F47089A674082C5A590697CEAD545DB27 E894C29A2F5F5D4676920828BB64A79AEEFA572E6FA19773C946F4A7AFBF4056 2909E8EB7069C9E67E96EA2E122A5FEBA6144CE2439C56E8AA0CAD954637B461 E04E1C9E90AC6B9A20EC20DBBE51F38412FB2E542E12EAEC1D1BA1C999937A7D AF6D89D3E547E00F4B2B0B91513CF66C4EA845B7F78E877E555141BDE08AD034 D908BC53F12C6CCA63238C4AE53940345119E39E8E0A15D13444DA5F59F75860 9F25EFE2ECCD000D255CE26EA4D238DA00979D34E0CFF31FDB30F3E27881BBCA DE948CF632FAC74FB900479E24406B90C2DBB558813683268259EFAFC5BCAB97 AFE55FEA52715BAFA858C3BD792232E52A0BA9825C837196A05D7FC151BC4D9A 3A335CCB560DF1FE4B51C2C7787DF0D7E841836BDBFC352241AFCA98B20C6674 D343B5408AAA4D4DEF828BF9E1FAACCE3C7CAB423320314A88C9F0684974BDAA 1FC8A39BE5EA7C0153664BC945FB5450219E2EB1CAA88C701FA4C49906DCA485 916B04F906531ECD256BAD59CB2241D1B926561047DAA39E12E6AA62D8F3591C 32CCACB6801C75263EF9EF386EEB4215AE59007CB0E7400FED401E369A86A8B2 11985B8D4BA72D787C1E2A12659B500F500758C4DD72BC97AFA5128C7C530644 9C53FC3E15BA3ECA5B050EACCF8F5C6B3631F6EF03340F0C0B80C4295419BD1F 51B0952F235EC3647B01470F4795E91012183C46DCD9201F62A36ECB2F264E74 001CB70159F15B931F170BCFF2C0C21159904F23EB8A21F2176CA60701EF7821 18DF2687DA3BDCF9C6D55268719780C0356F50ADFAEFB41161A14C7236139FE2 7143C35E2AE087557071A678A4ADF9C980CBB3C4C3951215BC8569F818CB504E E9C26644094B2157B6060EE54E257CF9DB029051A044CBDD3D863C25253AA08E 37ED164D7A70563FDB2BD7FF069FB2C16986DFC16BE3AFEA2B97AB7C5E18FC4E 779A80B4923BDAFCDB5FC0498B314773C37A5D6888F47EE0C2B32332FDD533CE F5079692DAE12BBE9F6C254D319176D3BA69445645F2122CCFD2E8430992DFC8 0131F57F63000E98D87D032BE7306279B56D611CF42707C848F5261BFAEB23E4 2E4CECF4E69D1BA90B2EC660323AF19F0FA9F91384EBC4F69C921417F2603E85 5E59464A58760C01A3F0B7FA898E1D0432C5DFC8015B3A3D4870188C241024C6 102248F40FD7A2627261BB9B78566CD5552B633C17E898CCD9790EC87078B694 C1C9CB273EFFB72E2AF86B5B624FE821EF7BA1FAC4BE94E822407B249845868D 8A64FD99F17F48909D91CAEE424E8A8684E3AC1484D63EF5AED8FE6195588E73 A1D73D5AB5129DBCFF35C23CD541A067386685DE931C6F44A049120DDA2B8847 6FDB76D11A9303EE8624C16F98B9570B53FDC4C4FC2BA4E91340FFFD1A389FD8 54A750F8C813D201E2F0AD0E10628F150BCF4D17BD93515399BA57F679F76F57 19E9D9EECC393337C630B88D67E143704094AA2AF2D553E369C054FEBFB57EE5 FAD78F3C848851BC6DF488AB9D9CA7844735E405AEC5F352E6C4A23E80E0AF84 8257C4C9DFACD56393813E467262EAE909F6CF3F73631DC08D5263EE631BE20C C16AE6B667C887D3A477B8C08B7C4FED9FBDE626E142DC9377AAFC405D7B990E EFAC2EB714E3CB669D7C78A279AC779C3DCE6706524ABDF3A3FB53D98DA6E3D0 8E4BA092009FFA336DFA13DA730A3292EECCA7039FD2967AA866DDFACE108AF3 4436FCC16F49C288203F00DD116CBEB0E7B2E3A104F4F224851E671ACF696A99 CD164DEF91E4E39991492DCE44FC902C592BC032950124F0C53FCB9295C24675 130DCB5567D000D9DE6F6FB4DAD3C3CCD9CAD2B41AAC596203E242270A2AA9F8 E576947A530F68FF3DBBB5527F61DE011772F96B9C26FD6D4278D724D565623C 1191C9585282A70F01B10379CCC99B15CCA03CFCC0AD897DC5B733ECF490D749 D358EBD66A8CF3D780CDB3205FA261B18E72607DE185D47D2471E68264FA95F7 6A684FB269B8098771EDA1334CD990B5CC94FDFCFCFE88190E55CA21A2E8D5C6 13F34C7B9CB26F57983F4A09EA28F54B643A9BBE9353272845D2298C94149019 3A5FAE86E81622EEA9378C14F46E4F1E0A7BF1ED42790B6F90074FF46D835B72 F92DDE0718115B30BC0AC74039D0489E9C7733C4572498CE478E3D69ED3AE2D6 FEF73EE95DB6D5450F197D60E2DBD4288DF66797FE0364A7BF99D16D166FAEC9 180F73D1B8ECCFCDC9FF19528D9588D9BE357100E1FF5E10E9AB38203B1A3444 ABE34CAD4BBF05CBE8E6E184B1E843F22DA2A1B174E9B54AE5FF4844B7DA0307 01C5BDEB23208F9F198C658E355A9005B78764E191F481631CD2494AFFE79187 F54C6C28BABD15420A528CB7A2AE1DA02646265F170F34A388FDBABA6CCF6E1F BD84163C13A65184397A551B5ACE1CA0EF44EA22E24CBEA44B8FE287BA22DABC 7ED2BA81A850A2D0962E4205375A2240CF2414A137AB620E7911A00161A9B052 924393975CE3E85720C6F9CF5074CFB38FB91D09EBF4C6626C45E384F3651E5B 1B470CC96316A607D70BB09F84F46AA9D1AB7D07119C2D414E4812B23DC3A9AE FCB14D80D4B8895485E15FC1AEC88E1BFF8F2297055BEBA758647CE450DD2E60 EC65BBB922F329C668B8081D8189ED173F83558DE36008647D488BD7BD0148E3 9B7DBCBE419CCECA5535F0E17974EB141AA810FB5E3A3CF230A4B6E8DF27D495 3FCB8995AE6D80034F6D42421102912E0723996D601CA66E094AE4F8BD7949D2 CF1BE45B7A1C6F8CD17252E4EB0C623B2D782B38DFB9B797102E24F7397EF124 633C2669F634AB20C1BE58CA56EBE8863D94C0BCDE0799EEBA3EE9A2BFFDE8B6 CBE9EFA7E9A4C0FD75D518AE 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark {restore}if %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.0: CMR9 003.002 %%Title: CMR9 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMR9. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMR9 known{/CMR9 findfont dup/UniqueID known{dup /UniqueID get 5000792 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMR9 def /FontBBox {-39 -250 1036 750 }readonly def /UniqueID 5000792 def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMR9.) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 45 /hyphen put dup 46 /period put dup 72 /H put dup 73 /I put dup 77 /M put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont %%BeginFont: CMBX9 %!PS-AdobeFont-1.0: CMBX9 003.002 %%Title: CMBX9 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMBX9. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMBX9 known{/CMBX9 findfont dup/UniqueID known{dup /UniqueID get 5000767 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMBX9 def /FontBBox {-58 -250 1195 750 }readonly def /UniqueID 5000767 def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMBX9.) readonly def /FullName (CMBX9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 65 /A put dup 97 /a put dup 98 /b put dup 99 /c put dup 114 /r put dup 115 /s put dup 116 /t put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.0: CMR17 003.002 %%Title: CMR17 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMR17. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMR17 known{/CMR17 findfont dup/UniqueID known{dup /UniqueID get 5000795 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMR17 def /FontBBox {-33 -250 945 749 }readonly def /UniqueID 5000795 def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMR17.) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 69 /E put dup 72 /H put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 120 /x put dup 121 /y put readonly def currentdict end currentfile eexec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cleartomark {restore}if %%EndFont TeXDict begin 39158280 55380996 1000 600 600 (splitting13.dvi) @start /Fa 135[44 44 2[44 3[44 7[44 3[44 44 1[44 8[44 3[44 15[44 9[44 44 1[44 1[44 2[44 44 44 44 44 44 45[{}21 83.022 /CMTT10 rf /Fb 194[91 12[59 48[{}2 119.552 /CMR12 rf /Fc 231[51 24[{}1 119.552 /CMMI12 rf /Fd 179[82 76[{}1 119.552 /CMSY10 rf /Fe 154[66 101[{}1 119.552 /CMEX10 rf /Ff 201[0 32[77 14[77 6[{}3 99.6264 /CMSY10 rf /Fg 138[41 1[33 55[20 59[{}3 66.4176 /CMMI8 rf /Fh 138[56 2[44 1[49 6[33 7[51 41 11[57 16[70 2[51 3[76 27 24[46 11[59 22[{}13 99.6264 /CMMI12 rf /Fi 140[49 50 2[59 65 94 3[35 1[53 1[53 59 1[53 59 97[{}11 99.6264 /CMBXTI10 rf /Fj 205[35 35 35 48[{}3 66.4176 /CMR8 rf /Fk 154[38 2[38 80[40 40 16[{}4 58.1154 /CMEX7 rf /Fl 149[19 56[57 18 44[31 2[45{}5 41.511 /CMSY5 rf /Fm 138[35 1[28 28 3[37 51 1[31 6[31 27 1[32 25 34[31 1[19 24[28 6[34 30 5[34 1[28 31 3[31 33 12[{}20 41.511 /CMMI5 rf /Fn 165[39 5[39 84[{}2 58.1154 /MSBM7 rf /Fo 175[65 7[74 17[49 49 49 49 49 49 2[30 46[{}9 83.022 /CMBXTI10 rf /Fp 132[48 1[50 50 69 50 53 37 38 39 50 53 48 53 80 27 50 1[27 53 48 29 44 53 42 53 46 12[66 53 72 1[65 2[91 57 1[49 1[75 2[63 1[69 68 8[48 48 48 48 48 48 48 48 48 48 1[27 32 32[53 56 11[{}50 83.022 /CMBX10 rf /Fq 133[50 59 59 81 59 62 44 44 46 1[62 56 62 93 31 59 1[31 62 56 34 51 62 50 62 54 12[78 62 2[77 1[88 106 67 2[42 88 2[74 86 81 80 85 6[31 56 56 56 56 56 56 56 56 56 56 1[31 1[31 31[62 65 11[{}52 99.6264 /CMBX12 rf /Fr 144[33 111[{ TeXBase1Encoding ReEncodeFont}1 83.022 /ZapfChancery-MediumItalic rf /Fs 133[34 40 39 55 38 45 28 34 35 38 42 42 47 68 21 38 25 25 42 38 25 38 42 38 38 42 3[25 1[25 51 2[83 2[59 47 61 1[56 64 62 74 52 1[44 32 62 64 54 56 63 59 58 62 6[25 42 42 42 42 42 42 42 42 42 42 1[25 30 25 2[34 34 25 19[42 42 3[73 1[47 51 11[{}70 83.022 /CMTI10 rf /Ft 165[55 5[55 1[60 3[60 10[60 67[{}5 83.022 /MSBM10 rf /Fu 138[31 22 4[28 31 1[17 2[17 48[28 28 28 28 28 28 28 28 28 4[43 1[22 22 40[{}18 41.511 /CMR5 rf /Fv 143[55 1[34 3[20 2[34 34 19[57 13[52 36 12[0 3[45 66 19 14[66 11[52 52 13[52 2[34 52 1[52{}19 58.1154 /CMSY7 rf /Fw 143[69 4[42 23 32 32 42 42 9[55 55 60 55 59 82 51 52 45 50 70 68 58 66 68 1[57 63 56 45 70 49 60 44 64 44 1[66 5[42 4[0 3[55 83 9[65 5[83 4[83 1[65 1[65 2[65 65 2[65 1[42 42 6[65 65 2[42 65 23 65{}53 83.022 /CMSY10 rf /Fx 133[32 34 38 48 33 39 25 31 31 30 34 33 41 59 21 35 27 23 39 32 2[35 30 29 36 28 13[50 3[53 63 3[29 4[54 48 4[52 34 52 20 20 23[31 5[31 38 34 39 1[33 40 39 39 1[32 34 30 1[30 35 38 43 11[{}52 58.1154 /CMMI7 rf /Fy 135[35 2[37 26 4[33 37 55 19 2[19 1[33 1[30 3[33 2[33 19 1[19 8[48 8[24 11[51 3[33 33 33 33 33 33 33 33 33 33 4[51 1[26 26 17[33 22[{}31 58.1154 /CMR7 rf /Fz 143[83 55 55 4[39 39 1[83 46 1[83 46 7[46 1[120 5[39 1[88 10[51 51 2[73 73 1[74 1[74 1[74 1[74 6[73 73 7[67 6[66 66 4[62 62 4[44 44 61 61 50 50 2[46 28 39 39 48 48 4[35 35 38 38{}44 83.022 /CMEX10 rf /FA 133[39 41 47 59 40 48 30 39 37 37 42 1[50 73 25 43 34 29 48 40 41 39 43 36 36 44 35 5[57 48 1[78 48 1[49 51 63 66 53 1[67 81 57 71 1[36 69 65 53 61 69 59 63 62 44 1[65 42 65 23 23 18[54 4[39 52 54 52 49 1[36 47 43 47 36 41 50 48 48 1[39 41 36 1[37 43 47 53 11[{}76 83.022 /CMMI10 rf /FB 128[42 3[42 37 44 44 60 44 46 32 33 33 44 46 42 46 69 23 44 25 23 46 42 25 37 46 37 46 42 1[23 42 23 42 23 1[62 1[85 62 62 60 46 61 1[57 65 62 76 52 65 43 30 62 65 54 57 63 60 59 62 3[65 1[23 23 42 42 42 42 42 42 42 42 42 42 42 23 28 23 65 1[32 32 23 4[42 23 10[42 42 1[42 42 1[23 1[69 46 46 48 2[60 65 60 2[58 65 69 52{}95 83.022 /CMR10 rf /FC 134[71 71 1[71 75 52 53 55 71 75 67 75 112 37 1[41 37 75 1[41 61 75 60 75 65 12[94 1[100 3[105 1[81 2[50 105 2[88 103 2[102 6[37 67 67 67 67 67 67 67 67 67 2[37 45 33[78 11[{}44 119.552 /CMBX12 rf /FD 134[41 41 55 41 43 30 30 30 1[43 38 43 64 21 41 1[21 43 38 23 34 43 34 43 38 9[79 9[70 3[28 58 25[21 26 32[43 12[{}30 74.7198 /CMR9 rf /FE 139[34 35 36 14[39 49 43 31[67 65[{}7 74.7198 /CMBX9 rf /FF 134[51 3[54 38 38 38 2[49 54 81 27 2[27 54 2[43 54 43 54 49 12[70 54 5[89 3[35 1[77 64 66 2[69 4[76 4[49 5[49 49 49 1[27 1[27 2[38 38 20[49 17[81 1[{}35 99.6264 /CMR12 rf /FG 134[70 70 3[51 52 51 1[73 66 73 111 36 2[36 73 66 40 58 73 58 1[66 24[99 2[90 69[{}20 143.462 /CMR17 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%BeginPaperSize: a4 /setpagedevice where { pop << /PageSize [595 842] >> setpagedevice } { /a4 where { pop a4 } if } ifelse %%EndPaperSize end %%EndSetup %%Page: 1 1 TeXDict begin 1 0 bop Black Black Black Black 169 621 a FG(Exp)t(onen)l(tially)43 b(small)f(splitting)g(of)h(separatrices)f (for)h(one)h(and)f(a)g(half)807 803 y(degrees)g(of)g(freedom)g (Hamiltonian)g(systems)388 1044 y FF(Inmaculada)33 b(Baldom\023)-49 b(a,)33 b(Ernest)h(F)-8 b(on)m(tic)m(h,)33 b(Marcel)g(Guardia)f(and)h (T)-8 b(ere)33 b(M.)g(Seara)1601 1240 y(F)-8 b(ebruary)33 b(8,)f(2011)p Black Black 1792 1529 a FE(Abstract)p Black Black 394 1650 a FD(In)20 b(this)i(pap)r(er)f(w)n(e)h(study)e(the)h (problem)g(of)i(exp)r(onen)n(tially)e(small)h(splitting)g(of)h (separatrices)g(of)f(one)f(degree)h(of)278 1741 y(freedom)e(classical)h (Hamiltonian)f(systems)f(with)g(a)g(non-autonomous)g(p)r(erturbation)g (whic)n(h)f(is)i(fast)f(and)g(p)r(erio)r(dic)278 1833 y(in)30 b(time.)45 b(W)-6 b(e)29 b(pro)n(vide)g(the)g(asymptotic)h (form)n(ula)g(for)h(the)d(measure)i(of)h(the)d(splitting)i(for)h(b)r (oth)e(the)g(so-called)278 1924 y(regular)f(and)f(singular)h(cases.)39 b(In)26 b(the)h(latter)g(w)n(e)h(sho)n(w)f(that)g(Melnik)n(o)n(v)g (theory)f(fails)j(to)e(predict)g(correctly)g(the)278 2015 y(\014rst)f(order)g(of)g(the)g(splitting)g(of)g(separatrices.)71 2288 y FC(1)135 b(In)l(tro)t(duction)71 2470 y FB(In)28 b(this)g(pap)r(er)f(w)n(e)g(consider)f(the)i(familiy)g(of)g (Hamiltonian)f(systems)g(of)h(the)g(form)1114 2688 y FA(H)1204 2571 y Fz(\022)1265 2688 y FA(x;)14 b(y)s(;)1444 2632 y(t)p 1440 2669 39 4 v 1440 2745 a(")1489 2688 y FB(;)g FA(")1565 2571 y Fz(\023)1648 2688 y FB(=)23 b FA(H)1805 2700 y Fy(0)1842 2688 y FB(\()p FA(x;)14 b(y)s FB(\))19 b(+)f FA(\026")2225 2654 y Fx(\021)2265 2688 y FA(H)2334 2700 y Fy(1)2386 2571 y Fz(\022)2447 2688 y FA(x;)c(y)s(;)2626 2632 y(t)p 2622 2669 V 2622 2745 a(")2670 2688 y FB(;)g FA(")2746 2571 y Fz(\023)3744 2688 y FB(\(1\))71 2912 y(where)27 b FA(H)380 2924 y Fy(0)417 2912 y FB(\()p FA(x;)14 b(y)s FB(\))28 b(is)g(giv)n(en)f(b)n (y)g(a)g(classical)f(Hamiltonian)1565 3138 y FA(H)1634 3150 y Fy(0)1671 3138 y FB(\()p FA(x;)14 b(y)s FB(\))24 b(=)1985 3082 y FA(y)2029 3052 y Fy(2)p 1985 3119 81 4 v 2004 3195 a FB(2)2094 3138 y(+)18 b FA(V)h FB(\()p FA(x)p FB(\))71 3339 y(and)i FA(H)295 3351 y Fy(1)332 3339 y FB(\()p FA(x;)14 b(y)s(;)g(\034)9 b FB(;)14 b FA(")p FB(\))23 b(is)e(a)g(2)p FA(\031)s FB(-p)r(erio)r(dic)g(time)h (dep)r(enden)n(t)g(Hamiltonian)f(with)h(zero)e(a)n(v)n(erage.)32 b(W)-7 b(e)22 b(assume)f FA(H)7 b FB(\()p FA(x;)14 b(y)s(;)g(\034)9 b FB(;)14 b FA(")p FB(\))71 3439 y(is)21 b(analytic)f(and)g(w)n(e)h (study)g(the)g(problem)f(of)h(the)g(splitting)g(of)g(separatrices.)33 b(The)20 b(parameter)g FA(")g FB(is)h(a)f(small)h(parameter)71 3539 y(but)33 b(this)f(is)g(not)g(the)g(case)f(for)h FA(\026)p FB(,)h(whic)n(h)f(is)g(considered)f(a)h(parameter)e(of)i (order)f(one.)50 b(The)32 b(results)f(in)i(this)f(pap)r(er)71 3638 y(are)c(v)-5 b(alid)29 b(not)f(only)h(for)f FA(\026)h FB(small,)g(but)g(also)f(for)h(\014nite)g(v)-5 b(alues)28 b(of)h FA(\026)p FB(,)h(generically)d(for)h FA(\026)d FB(=)g(1.)40 b(W)-7 b(e)29 b(will)g(see)g(that)g(the)71 3738 y(results)f(are)g(di\013eren)n(t)h(dep)r(ending)g(on)g(the)g (other)f(parameter)f FA(\021)i Fw(\025)24 b FB(0,)29 b(whic)n(h)g(app)r(ears)e(in)i(\(1\))q(,)g(and)f(on)h(the)g(analytic)71 3837 y(prop)r(erties)e(of)g FA(H)7 b FB(.)195 3937 y(When)22 b FA(\026")520 3907 y Fx(\021)581 3937 y FB(is)e(small,)i(the)f (Hamiltonian)g(system)f(asso)r(ciated)g(to)h FA(H)27 b FB(is)21 b(a)f(small)h(p)r(erturbation)f(of)h(the)g(Hamiltonian)71 4037 y(system)27 b(asso)r(ciated)g(to)g FA(H)912 4049 y Fy(0)949 4037 y FB(:)1691 4210 y(_)-37 b FA(x)83 b FB(=)g FA(y)1695 4334 y FB(_)-38 b FA(y)86 b FB(=)d Fw(\000)p FA(V)2086 4300 y Fv(0)2109 4334 y FB(\()p FA(x)p FB(\))p FA(:)p Black 1501 w FB(\(2\))p Black 71 4508 a(W)-7 b(e)34 b(assume)f(that)i(this)f(system)g(has)f(a)h(h)n(yp)r(erb)r(olic)f(or)g (parab)r(olic)g(critical)g(p)r(oin)n(t)h(at)g(the)h(origin)d(with)j (stable)f(and)71 4607 y(unstable)e(manifolds)h(whic)n(h)f(coincide)g (along)g(a)g(separatrix)e(\()p FA(q)2120 4619 y Fy(0)2158 4607 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2349 4619 y Fy(0)2386 4607 y FB(\()p FA(u)p FB(\)\).)52 b(The)33 b(coincidence)f(of)g(the)h(stable)g(and)71 4707 y(unstable)28 b(in)n(v)-5 b(arian)n(t)27 b(manifolds)h(is)g(not)h(a)f(generic)f (phenomenon)h(ev)n(en)g(for)f(Hamiltonian)h(systems)g(of)g(one)g(and)g (half)71 4807 y(degrees)e(of)i(freedom)f(as)g(\(1\).)37 b(Therefore,)26 b(one)h(can)h(exp)r(ect)g(that)f(the)h(homo)r(clinic)g (connection)f(of)34 b(\(2\))28 b(breaks)e(do)n(wn)71 4906 y(when)f(w)n(e)f(add)g(the)h(non-autonomous)e(part)h(to)g(the)h (system.)36 b(Nev)n(ertheless,)24 b(the)h(symplectic)f(structure)g (ensures)g(the)71 5006 y(existence)30 b(of)g(in)n(tersections)f(b)r(et) n(w)n(een)h(the)h(p)r(erturb)r(ed)f(in)n(v)-5 b(arian)n(t)29 b(manifolds.)45 b(The)30 b(main)g(goal)f(of)h(this)h(pap)r(er)f(is)g (to)71 5106 y(pro)n(vide)c(an)i(asymptotic)f(form)n(ula)f(for)h(this)h (splitting.)195 5205 y(Our)c(\014rst)h(observ)-5 b(ation)23 b(is)h(that,)i(b)r(eing)f(the)g(Hamiltonian)f FA(H)32 b FB(fast)24 b(in)h(time,)h(a)n(v)n(eraging)21 b(theory)j([AKN88,)g (LM88)o(])71 5305 y(tells)j(us)g(that,)h(ev)n(en)e(for)h FA(\026")959 5275 y Fx(\021)1022 5305 y FB(=)c Fw(O)r FB(\(1\),)k(the)h(solutions)e(of)h(the)h(Hamiltonian)f(system)f(asso)r (ciated)g(to)h(\(1\))g(are)f(close)h(to)71 5404 y(the)f(solutions)g(of) 32 b(\(2\).)37 b(Therefore,)25 b(the)i(problem)e(of)h(measuring)f(the)i (splitting)f(of)g(separatrices)e(can)i(b)r(e)g(considered,)71 5504 y(in)i(general,)e(for)h FA(\021)f Fw(\025)d FB(0.)p Black 1940 5753 a(1)p Black eop end %%Page: 2 2 TeXDict begin 2 1 bop Black Black 195 272 a FB(Due)32 b(to)f(the)h(2)p FA(\031)s(")p FB(-p)r(erio)r(dicit)n(y)e(of)i(the)f (Hamiltonian)h FA(H)38 b FB(it)31 b(is)h(con)n(v)n(enien)n(t)e(to)h (consider)f(the)i(P)n(oincar)n(\023)-39 b(e)28 b(map)j FA(P)3788 284 y Fx(t)3813 292 y Fu(0)71 372 y FB(de\014ned)24 b(in)g(a)g(P)n(oincar)n(\023)-39 b(e)21 b(section)i(\006)1182 384 y Fx(t)1207 392 y Fu(0)1267 372 y FB(=)g Fw(f)p FB(\()p FA(x;)14 b(y)s(;)g(t)1624 384 y Fy(0)1661 372 y FB(\);)28 b(\()p FA(x;)14 b(y)s FB(\))23 b Fw(2)h Ft(R)2098 342 y Fy(2)2135 372 y Fw(g)p FB(.)35 b(If)25 b FA(\026)e FB(=)f(0,)j(the)f(phase)g(p)r(ortrait)f(of)h FA(P)3384 384 y Fx(t)3409 392 y Fu(0)3470 372 y FB(is)g(giv)n(en)f(b)n(y)71 490 y(the)30 b(lev)n(el)f(curv)n(es)f(of)h(the)h(Hamiltonian)f FA(H)1452 502 y Fy(0)1490 490 y FB(\()p FA(x;)14 b(y)s FB(\))26 b(=)1809 453 y Fx(y)1845 428 y Fu(2)p 1809 471 69 4 v 1827 518 a Fy(2)1907 490 y FB(+)19 b FA(V)g FB(\()p FA(x)p FB(\).)44 b(The)29 b(origin)f(is)i(a)f(h)n(yp)r(erb)r(olic)g(or) f(parab)r(olic)g(\014xed)71 589 y(p)r(oin)n(t)g(of)f FA(P)435 601 y Fx(t)460 609 y Fu(0)525 589 y FB(with)h(stable)f(and)h (unstable)f(curv)n(es)g(giv)n(en)f(b)n(y)i(the)g(homo)r(clinic)f (connection)g(\()p FA(q)3079 601 y Fy(0)3117 589 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)3308 601 y Fy(0)3345 589 y FB(\()p FA(u)p FB(\)\).)195 689 y(In)39 b(the)g(h)n(yp)r(erb)r(olic)f (case,)i(a)e(classical)f(result)i(of)f(a)n(v)n(eraging)e(theory)h ([AKN88)o(,)i(LM88)o(])g(is)f(that,)k(for)c FA(")g FB(small)71 788 y(enough,)33 b(there)e(exists)h(a)g(h)n(yp)r(erb)r(olic)g(\014xed)g (p)r(oin)n(t)g(of)g FA(P)1899 800 y Fx(t)1924 808 y Fu(0)1993 788 y FB(corresp)r(onding)e(to)i(a)g(h)n(yp)r(erb)r(olic)g(p)r(erio)r (dic)g(orbit)f(of)i FA(H)7 b FB(,)71 888 y(and)26 b(their)f(stable)h (and)g(unstable)f(in)n(v)-5 b(arian)n(t)25 b(curv)n(es)g FA(C)1821 858 y Fx(s)1856 888 y FB(\()p FA(t)1918 900 y Fy(0)1956 888 y FB(\))h(and)g FA(C)2239 858 y Fx(u)2282 888 y FB(\()p FA(t)2344 900 y Fy(0)2382 888 y FB(\).)37 b(These)25 b(curv)n(es)g(lie)h(near)f(the)h(unp)r(erturb)r(ed)71 988 y(separatrix.)34 b(In)24 b(the)g(parab)r(olic)f(case)g(our)g (\(standard\))h(h)n(yp)r(otheses)f(will)i(ensure)e(that)h(the)h(origin) e(will)h(still)g(b)r(e)h(a)e(\014xed)71 1087 y(p)r(oin)n(t)28 b(with)g(similar)f(prop)r(erties.)195 1187 y(Generically)k(the)g(curv)n (es)f FA(C)1105 1157 y Fx(s)1141 1187 y FB(\()p FA(t)1203 1199 y Fy(0)1240 1187 y FB(\))i(and)f FA(C)1534 1157 y Fx(u)1578 1187 y FB(\()p FA(t)1640 1199 y Fy(0)1677 1187 y FB(\))h(in)n(tersect)f(giving)f(rise)g(to)h(some)g(homo)r (clinic)g(p)r(oin)n(ts)g FA(z)3516 1199 y Fx(h)3559 1187 y FB(.)48 b(If)31 b(this)71 1287 y(in)n(tersection)22 b(is)g(transv)n(ersal)e(at)j FA(z)1142 1299 y Fx(h)1184 1287 y FB(,)h(the)f(curv)n(es)e(enclose)h(lob)r(es)g(whose)g(area)f Fw(A)i FB(do)r(es)g(not)f(dep)r(end)h(on)g(the)g(homo)r(clinic)71 1386 y(p)r(oin)n(t)29 b(w)n(e)f(ha)n(v)n(e)g(c)n(hosen)g(\(see)g (Figure)g(1\).)41 b(The)28 b(measure)g(of)h(this)g(area)e(in)i(terms)g (of)f FA(")h FB(is)f(the)i(main)e(purp)r(ose)g(of)h(this)71 1486 y(pap)r(er.)64 b(Another)37 b(quan)n(tit)n(y)f(that)h(can)g(b)r(e) g(used)g(at)g(homo)r(clinic)f(p)r(oin)n(ts)h(to)g(measure)f(the)h (transv)n(ersalit)n(y)d(of)j(the)71 1586 y(in)n(tersection)26 b(is)i(the)f(angle)g(b)r(et)n(w)n(een)g(the)h(curv)n(es)e FA(C)1741 1555 y Fx(s)1777 1586 y FB(\()p FA(t)1839 1598 y Fy(0)1876 1586 y FB(\))i(and)f FA(C)2162 1555 y Fx(u)2206 1586 y FB(\()p FA(t)2268 1598 y Fy(0)2306 1586 y FB(\),)g(but)h(this)g (quan)n(tit)n(y)f(dep)r(ends)h(on)f(the)g(c)n(hosen)71 1685 y(homo)r(clinic)h(p)r(oin)n(t.)37 b(An)29 b(in)n(v)-5 b(arian)n(t)26 b(quan)n(tit)n(y)i(related)f(to)g(the)i(angle)e(is)g (the)i(so-called)d Fs(L)l(azutkin)j(invariant)37 b FB(\(see,)28 b(for)71 1785 y(instance)f([GL)-7 b(T91)o(]\).)p Black Black Black 640 3096 a /PSfrag where{pop(A)[[0(Bl)1 0]](x)[[1(Bl)1 0]](y)[[2(Bl)1 0]](z)[[3(Bl)1 0]]4 0 -1/Begin PSfrag}{userdict /PSfrag{pop}put}ifelse 640 3096 a @beginspecial 0 @llx -1 @lly 306 @urx 138 @ury 1440 @rhi @setspecial %%BeginDocument: FigureEight.eps %!PS-Adobe-3.0 EPSF-3.0 %%Creator: cairo 1.11.1 (http://cairographics.org) %%CreationDate: Mon Feb 7 00:17:41 2011 %%Pages: 1 %%DocumentData: Clean7Bit %%LanguageLevel: 2 %%BoundingBox: 0 -1 306 138 %%EndComments %%BeginProlog /cairo_eps_state save def /dict_count countdictstack def /op_count count 1 sub def userdict begin /q { gsave } bind def /Q { grestore } bind def /cm { 6 array astore concat } bind def /w { setlinewidth } bind def /J { setlinecap } bind def /j { setlinejoin } bind def /M { setmiterlimit } bind def /d { setdash } bind def /m { moveto } bind def /l { lineto } bind def /c { curveto } bind def /h { closepath } bind def /re { exch dup neg 3 1 roll 5 3 roll moveto 0 rlineto 0 exch rlineto 0 rlineto closepath } bind def /S { stroke } bind def /f { fill } bind def /f* { eofill } bind def /n { newpath } bind def /W { clip } bind def /W* { eoclip } bind def /BT { } bind def /ET { } bind def /pdfmark where { pop globaldict /?pdfmark /exec load put } { globaldict begin /?pdfmark /pop load def /pdfmark /cleartomark load def end } ifelse /BDC { mark 3 1 roll /BDC pdfmark } bind def /EMC { mark /EMC pdfmark } bind def /cairo_store_point { /cairo_point_y exch def /cairo_point_x exch def } def /Tj { show currentpoint cairo_store_point } bind def /TJ { { dup type /stringtype eq { show } { -0.001 mul 0 cairo_font_matrix dtransform rmoveto } ifelse } forall currentpoint cairo_store_point } bind def /cairo_selectfont { cairo_font_matrix aload pop pop pop 0 0 6 array astore cairo_font exch selectfont cairo_point_x cairo_point_y moveto } bind def /Tf { pop /cairo_font exch def /cairo_font_matrix where { pop cairo_selectfont } if } bind def /Td { matrix translate cairo_font_matrix matrix concatmatrix dup /cairo_font_matrix exch def dup 4 get exch 5 get cairo_store_point /cairo_font where { pop cairo_selectfont } if } bind def /Tm { 2 copy 8 2 roll 6 array astore /cairo_font_matrix exch def cairo_store_point /cairo_font where { pop cairo_selectfont } if } bind def /g { setgray } bind def /rg { setrgbcolor } bind def /d1 { setcachedevice } bind def %%EndProlog 11 dict begin /FontType 42 def /FontName /DejaVuSans def /PaintType 0 def /FontMatrix [ 1 0 0 1 0 0 ] def /FontBBox [ 0 0 0 0 ] def /Encoding 256 array def 0 1 255 { Encoding exch /.notdef put } for Encoding 65 /A put Encoding 120 /x put Encoding 121 /y put Encoding 122 /z put /CharStrings 5 dict dup begin /.notdef 0 def /A 1 def /x 2 def /y 3 def /z 4 def end readonly def /sfnts [ <0001000000090080000300106376742000691d39000005f8000001fe6670676d7134766a0000 07f8000000ab676c7966c8aa1e8e0000009c0000055c68656164f34bbbfa000008a400000036 686865610cb80656000008dc00000024686d747817f1014600000900000000146c6f63610000 0e3400000914000000186d617870047206710000092c00000020707265703b07f1000000094c 0000056800020066fe96046605a400030007001a400c04fb0006fb0108057f0204002fc4d4ec 310010d4ecd4ec301311211125211121660400fc73031bfce5fe96070ef8f272062900020010 0000056805d50002000a00c2404100110100040504021105050401110a030a0011020003030a 0711050406110505040911030a08110a030a4200030795010381090509080706040302010009 050a0b10d4c4173931002f3ce4d4ec1239304b5358071005ed0705ed071005ed0705ed071008 ed071005ed071005ed071008ed5922b2200c01015d40420f010f020f070f080f005800760070 008c000907010802060309041601190256015802500c67016802780176027c03720477077808 87018802800c980299039604175d005d090121013301230321032302bcfeee0225fe7be50239 d288fd5f88d5050efd1903aefa2b017ffe8100000001003b000004790460000b014340460511 060706041103040707060411050401020103110202010b110001000a11090a0101000a110b0a 0708070911080807420a070401040800bf05020a0704010408000208060c10d44bb00a544bb0 0f545b4bb010545b4bb011545b58b90006004038594bb0145458b90006ffc03859c4d4c41117 3931002f3cec321739304b5358071005ed071008ed071008ed071005ed071005ed071008ed07 1008ed071005ed59220140980a04040a1a04150a260a3d04310a55045707580a660a76017a04 7607740a8d04820a99049f049707920a900aa601a904af04a507a30aa00a1c0a03040505090a 0b1a03150515091a0b2903260525092a0b200d3a013903370534073609390b300d4903460545 094a0b400d590056015902590357055606590756085609590b500d6f0d78017f0d9b019407ab 01a407b00dcf0ddf0dff0d2f5d005d09022309012309013309010464fe6b01aad9febafebad9 01b3fe72d9012901290460fddffdc101b8fe48024a0216fe71018f000001003dfe56047f0460 000f018b40430708020911000f0a110b0a00000f0e110f000f0d110c0d00000f0d110e0d0a0b 0a0c110b0b0a420d0b0910000b058703bd0e0bbc100e0d0c0a09060300080f040f0b1010d44b b00a544bb008545b58b9000b004038594bb0145458b9000bffc03859c4c4111739310010e432 f4ec113911391239304b5358071005ed071008ed071008ed071005ed071008ed0705ed173259 220140f0060005080609030d160a170d100d230d350d490a4f0a4e0d5a095a0a6a0a870d800d 930d120a000a09060b050c0b0e0b0f1701150210041005170a140b140c1a0e1a0f2700240124 022004200529082809250a240b240c270d2a0e2a0f201137003501350230043005380a360b36 0c380d390e390f30114100400140024003400440054006400740084209450a470d490e490f40 115400510151025503500450055606550756085709570a550b550c590e590f50116601660268 0a690e690f60117b08780e780f89008a09850b850c890d890e890f9909950b950c9a0e9a0fa4 0ba40cab0eab0fb011cf11df11ff11655d005d050e012b01353332363f01013309013302934e 947c936c4c543321fe3bc3015e015ec368c87a9a488654044efc94036c000000000100580000 03db04600009009d401a081102030203110708074208a900bc03a905080301000401060a10dc 4bb00b544bb00c545b58b90006ffc038594bb0135458b9000600403859c432c411393931002f ecf4ec304b5358071005ed071005ed592201404205021602260247024907050b080f0b18031b 082b08200b36033908300b400140024503400440054308570359085f0b600160026603600460 0562087f0b800baf0b1b5d005d1321150121152135012171036afd4c02b4fc7d02b4fd650460 a8fcdb93a8032500013500b800cb00cb00c100aa009c01a600b800660000007100cb00a002b2 0085007500b800c301cb0189022d00cb00a600f000d300aa008700cb03aa0400014a003300cb 000000d9050200f4015400b4009c01390114013907060400044e04b4045204b804e704cd0037 047304cd04600473013303a2055605a60556053903c5021200c9001f00b801df007300ba03e9 033303bc0444040e00df03cd03aa00e503aa0404000000cb008f00a4007b00b80014016f007f 027b0252008f00c705cd009a009a006f00cb00cd019e01d300f000ba018300d5009803040248 009e01d500c100cb00f600830354027f00000333026600d300c700a400cd008f009a00730400 05d5010a00fe022b00a400b4009c00000062009c0000001d032d05d505d505d505f0007f007b 005400a406b80614072301d300b800cb00a601c301ec069300a000d3035c037103db01850423 04a80448008f0139011401390360008f05d5019a0614072306660179046004600460047b009c 00000277046001aa00e904600762007b00c5007f027b000000b4025205cd006600bc00660077 061000cd013b01850389008f007b0000001d00cd074a042f009c009c0000077d006f0000006f 0335006a006f007b00ae00b2002d0396008f027b00f600830354063705f6008f009c04e10266 008f018d02f600cd03440029006604ee00730000140000960000b707060504030201002c2010 b002254964b040515820c859212d2cb002254964b040515820c859212d2c20100720b00050b0 0d7920b8ffff5058041b0559b0051cb0032508b0042523e120b00050b00d7920b8ffff505804 1b0559b0051cb0032508e12d2c4b505820b0fd454459212d2cb002254560442d2c4b5358b002 25b0022545445921212d2c45442d2cb00225b0022549b00525b005254960b0206368208a108a 233a8a10653a2d000001000000024cccd96494365f0f3cf5001f080000000000c76891d40000 0000c76891d4f7d6fd330d72095500000008000000010000000000010000076dfe1d00000de2 f7d6fa510d7200010000000000000000000000000000000504cd00660579001004bc003b04bc 003d04330058000000000000004400000140000002c4000004900000055c0001000000050354 002b0068000c000200100099000800000415021600080004b8028040fffbfe03fa1403f92503 f83203f79603f60e03f5fe03f4fe03f32503f20e03f19603f02503ef8a4105effe03ee9603ed 9603ecfa03ebfa03eafe03e93a03e84203e7fe03e63203e5e45305e59603e48a4105e45303e3 e22f05e3fa03e22f03e1fe03e0fe03df3203de1403dd9603dcfe03db1203da7d03d9bb03d8fe 03d68a4105d67d03d5d44705d57d03d44703d3d21b05d3fe03d21b03d1fe03d0fe03cffe03ce fe03cd9603cccb1e05ccfe03cb1e03ca3203c9fe03c6851105c61c03c51603c4fe03c3fe03c2 fe03c1fe03c0fe03bffe03befe03bdfe03bcfe03bbfe03ba1103b9862505b9fe03b8b7bb05b8 fe03b7b65d05b7bb03b78004b6b52505b65d40ff03b64004b52503b4fe03b39603b2fe03b1fe 03b0fe03affe03ae6403ad0e03acab2505ac6403abaa1205ab2503aa1203a98a4105a9fa03a8 fe03a7fe03a6fe03a51203a4fe03a3a20e05a33203a20e03a16403a08a4105a096039ffe039e 9d0c059efe039d0c039c9b19059c64039b9a10059b19039a1003990a0398fe0397960d0597fe 03960d03958a410595960394930e05942803930e0392fa039190bb0591fe03908f5d0590bb03 9080048f8e25058f5d038f40048e25038dfe038c8b2e058cfe038b2e038a8625058a41038988 0b05891403880b03878625058764038685110586250385110384fe038382110583fe03821103 81fe0380fe037ffe0340ff7e7d7d057efe037d7d037c64037b5415057b25037afe0379fe0378 0e03770c03760a0375fe0374fa0373fa0372fa0371fa0370fe036ffe036efe036c21036bfe03 6a1142056a530369fe03687d036711420566fe0365fe0364fe0363fe0362fe03613a0360fa03 5e0c035dfe035bfe035afe0359580a0559fa03580a035716190557320356fe03555415055542 0354150353011005531803521403514a130551fe03500b034ffe034e4d10054efe034d10034c fe034b4a13054bfe034a4910054a1303491d0d05491003480d0347fe0346960345960344fe03 43022d0543fa0342bb03414b0340fe033ffe033e3d12053e14033d3c0f053d12033c3b0d053c 40ff0f033b0d033afe0339fe033837140538fa033736100537140336350b05361003350b0334 1e03330d0332310b0532fe03310b03302f0b05300d032f0b032e2d09052e10032d09032c3203 2b2a25052b64032a2912052a25032912032827250528410327250326250b05260f03250b0324 fe0323fe03220f03210110052112032064031ffa031e1d0d051e64031d0d031c1142051cfe03 1bfa031a42031911420519fe031864031716190517fe031601100516190315fe0314fe0313fe 031211420512fe0311022d05114203107d030f64030efe030d0c16050dfe030c0110050c1603 0bfe030a100309fe0308022d0508fe030714030664030401100504fe03401503022d0503fe03 02011005022d0301100300fe0301b80164858d012b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b 2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b 2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b 2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b 2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b002b2b2b2b 2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b 2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b 2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b 2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b 2b1d00> ] def /f-0-0 currentdict end definefont pop %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 0 -1 306 138 %%EndPageSetup q 0 -1 306 139 rectclip q 0 137.9 306 -138 re W n 0 g 0.8 w 0 J 0 j [] 0.0 d 4 M q 1 0 0 -1 0 137.899994 cm 144.398 9.5 m 144.398 137.5 l S Q q 1 0 0 -1 0 137.899994 cm 288.398 73.5 m 0.398 73.5 l S Q q 1 0 0 -1 0 137.899994 cm 144.398 73.5 m 176.398 109.5 236.125 184.445 260.398 73.5 c 261.684 66.074 269.039 45.328 250.863 43.031 c 240.648 41.352 230.855 30.637 230.688 17.5 c S Q q 1 0 0 -1 0 137.899994 cm 144.453 73.332 m 176.453 37.332 236.18 -37.613 260.453 73.332 c 261.738 80.754 269.094 101.5 250.918 103.797 c 240.703 105.48 230.91 116.195 230.738 129.332 c S Q q 1 0 0 -1 0 137.899994 cm 253.266 23.688 m 236.902 29.344 l 239.125 27.223 l S Q q 1 0 0 -1 0 137.899994 cm 236.863 29.121 m 239.488 29.828 l S Q BT 32 0 0 32 254.431763 112.519763 Tm /f-0-0 1 Tf (A)Tj ET q -1 0 0 -1 0 137.899994 cm -144.453 73.668 m -112.453 109.668 -52.727 184.613 -28.453 73.668 c -27.172 66.246 -19.816 45.5 -37.992 43.203 c -48.207 41.52 -58 30.805 -58.168 17.668 c S Q q -1 0 0 -1 0 137.899994 cm -144.398 73.5 m -112.398 37.5 -52.672 -37.445 -28.398 73.5 c -27.117 80.926 -19.762 101.672 -37.938 103.969 c -48.152 105.648 -57.945 116.363 -58.113 129.5 c S Q BT 32 0 0 32 287.828564 56.399994 Tm /f-0-0 1 Tf (x)Tj ET BT 32 0 0 32 145.542871 120.399994 Tm /f-0-0 1 Tf (y)Tj ET BT 32 0 0 32 271.828564 101.6 Tm /f-0-0 1 Tf (z)Tj ET q 1 0 0 -1 0 137.899994 cm 271.688 35.355 m 252.543 41.785 l 255.398 39.355 l S Q q 1 0 0 -1 0 137.899994 cm 252.828 41.785 m 256.113 42.215 l S Q Q Q showpage %%Trailer count op_count sub {pop} repeat countdictstack dict_count sub {end} repeat cairo_eps_state restore %%EOF %%EndDocument @endspecial 640 3096 a /End PSfrag 640 3096 a 640 2607 a /Hide PSfrag 640 2607 a -100 2665 a FB(PSfrag)26 b(replacemen)n(ts)p -100 2694 741 4 v 640 2697 a /Unhide PSfrag 640 2697 a 574 2797 a { 574 2797 a Black Fw(A)p Black 574 2797 a } 0/Place PSfrag 574 2797 a 593 2897 a { 593 2897 a Black FA(x)p Black 593 2897 a } 1/Place PSfrag 593 2897 a 596 2980 a { 596 2980 a Black FA(y)p Black 596 2980 a } 2/Place PSfrag 596 2980 a 558 3084 a { 558 3084 a Black FA(z)597 3096 y Fx(h)p Black 558 3084 a } 3/Place PSfrag 558 3084 a Black 1337 3362 a FB(Figure)h(1:)36 b(Splitting)28 b(of)g(separatrices.)p Black Black 195 3547 a(Classical)22 b(p)r(erturbation)h(theory)f(applied)h(to)g(our)f (problem)h(pro)n(vides)e(the)j(so-called)e(Melnik)n(o)n(v)g(p)r(oten)n (tial)h(\(called)71 3647 y(also)j(sometimes)i(P)n(oincar)n(\023)-39 b(e)24 b(F)-7 b(unction,)28 b(see)f(for)g(instance)h([DG00)o(]\),)g (whic)n(h)g(is)f(giv)n(en)g(b)n(y)1064 3864 y FA(L)14 b FB(\()p FA(t)1197 3876 y Fy(0)1234 3864 y FB(\))23 b(=)1377 3751 y Fz(Z)1460 3771 y Fy(+)p Fv(1)1423 3939 y(\0001)1595 3864 y FA(H)1664 3876 y Fy(1)1715 3797 y Fz(\000)1753 3864 y FA(q)1790 3876 y Fy(0)1827 3864 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2018 3876 y Fy(0)2055 3864 y FB(\()p FA(u)p FB(\))p FA(;)g(")2243 3830 y Fv(\000)p Fy(1)2332 3864 y FB(\()p FA(t)2394 3876 y Fy(0)2450 3864 y FB(+)k FA(u)p FB(\);)c(0)2692 3797 y Fz(\001)2743 3864 y FA(du:)71 4077 y FB(Using)38 b(this)g(function,)j(P)n(oincar)n(\023) -39 b(e,)37 b(and)h(later)f(Melnik)n(o)n(v)g([Mel63)o(],)j(pro)n(v)n (ed)d(that,)j(if)f FA(\026")2968 4047 y Fx(\021)3046 4077 y FB(is)f(small)f(enough,)j(non-)71 4176 y(degenerate)29 b(critical)g(p)r(oin)n(ts)h(of)g FA(L)g FB(giv)n(e)f(rise)h(to)g (transv)n(ersal)d(in)n(tersections)i(b)r(et)n(w)n(een)h(the)h(in)n(v)-5 b(arian)n(t)29 b(curv)n(es)g FA(C)3683 4146 y Fx(s)3719 4176 y FB(\()p FA(t)3781 4188 y Fy(0)3818 4176 y FB(\))71 4276 y(and)c FA(C)295 4246 y Fx(u)339 4276 y FB(\()p FA(t)401 4288 y Fy(0)439 4276 y FB(\),)h(and)g(the)g(area)e(b)r(et)n(w) n(een)i(the)g(lob)r(es)f(is)h(giv)n(en)f(asymptotically)g(b)n(y)g FA(L)p FB(\()p FA(t)2753 4246 y Fy(1)2753 4297 y(0)2790 4276 y FB(\))15 b Fw(\000)f FA(L)p FB(\()p FA(t)3035 4246 y Fy(2)3035 4297 y(0)3072 4276 y FB(\),)27 b(b)r(eing)f FA(t)3406 4246 y Fy(1)3406 4297 y(0)3469 4276 y FB(and)f FA(t)3658 4246 y Fy(2)3658 4297 y(0)3721 4276 y FB(t)n(w)n(o)71 4376 y(consecutiv)n(e)i(critical)h(p)r(oin)n(ts)g(of)g FA(L)p FB(.)39 b(In)28 b(our)g(setting,)h(where)e FA(H)2111 4388 y Fy(0)2149 4376 y FB(\()p FA(x;)14 b(y)s FB(\))29 b(and)f FA(H)2601 4388 y Fy(1)2638 4376 y FB(\()p FA(x;)14 b(y)s(;)g(\034)9 b FB(;)14 b(0\))29 b(will)f(b)r(e)h(either)f (algebraic)71 4475 y(or)f(trigonometric)f(p)r(olynomials)g(in)i(\()p FA(x;)14 b(y)s FB(\),)29 b(the)e(P)n(oincar)n(\023)-39 b(e)25 b(function)j FA(L)f FB(is)h(asymptotically)e(giv)n(en)h(b)n(y:) 1318 4685 y FA(L)14 b FB(\()p FA(t)1451 4697 y Fy(0)1488 4685 y FB(\))24 b Fw(')e FA(")1670 4650 y Fx(\014)1715 4685 y FA(e)1754 4650 y Fv(\000)p Fx(a=")1911 4685 y FA(g)1967 4567 y Fz(\022)2038 4628 y FA(t)2068 4640 y Fy(0)p 2038 4665 68 4 v 2053 4742 a FA(")2116 4567 y Fz(\023)2190 4685 y FA(;)180 b(")23 b Fw(!)g FB(0)1141 b(\(3\))71 4894 y(b)r(eing)41 b FA(g)s FB(\()p FA(s)p FB(\))h(a)f(2)p FA(\031)s FB(-p)r(erio)r(dic)g(function,)k(and)c FA(a)46 b(>)g FB(0,)e(and)d FA(\014)46 b FB(some)41 b(computable)g (constan)n(ts.)77 b(The)42 b(constan)n(t)f FA(a)71 4993 y FB(is)e(indep)r(enden)n(t)h(of)g(the)g(p)r(erturbation:)60 b(it)39 b(turns)h(out)f(that)h(the)f(time)h(parameterization)e(of)h (the)h(unp)r(erturb)r(ed)71 5093 y(separatrix)20 b(has)i(alw)n(a)n(ys)f (singularities)g(in)i(the)g(complex)f(plane)g(\(see)g([F)-7 b(on95)o(,)23 b(BF04)o(]\))g(and)f(the)h(constan)n(t)f FA(a)g FB(is)h(nothing)71 5193 y(but)28 b(the)g(imaginary)e(part)h(of)h (the)g(singularit)n(y)e(closest)h(to)g(the)h(real)f(axis.)195 5292 y(The)34 b(straighforw)n(ard)c(application)j(of)g(Melnik)n(o)n(v)g (metho)r(d)h(to)f(our)g(Hamiltonian)g(pro)n(vides)f(a)h(form)n(ula)f (for)h(the)71 5392 y(area)26 b(whic)n(h)i(reads:)1227 5492 y Fw(A)23 b(')g FA(\026")1493 5457 y Fx(\021)1547 5399 y Fz(\020)1596 5492 y FA(K)6 b(")1712 5457 y Fx(\014)1756 5492 y FA(e)1795 5457 y Fv(\000)p Fx(a=")1971 5492 y FB(+)18 b Fw(O)e FB(\()p FA(\026")2257 5457 y Fx(\021)2297 5492 y FB(\))2330 5399 y Fz(\021)2393 5492 y FA(;)69 b(")23 b Fw(!)g FB(0)1049 b(\(4\))p Black 1940 5753 a(2)p Black eop end %%Page: 3 3 TeXDict begin 3 2 bop Black Black 71 272 a FB(with)35 b FA(K)40 b(>)35 b FB(0,)h(a)e(computable)h(constan)n(t.)57 b(Therefore,)36 b(Melnik)n(o)n(v)d(p)r(oten)n(tial)i(is)f(exp)r(onen)n (tially)h(small)f(in)h FA(")f FB(and)h(a)71 372 y(direct)30 b(application)g(of)g(classical)f(p)r(erturbation)h(theory)g(only)g (ensures)g(the)g(v)-5 b(alidit)n(y)31 b(of)f(suc)n(h)g(an)g(appro)n (ximation)f(if)71 471 y FA(\026)23 b Fw(\034)g FA(e)289 441 y Fv(\000)p Fx(a=")446 471 y FB(.)195 571 y(W)-7 b(e)24 b(w)n(an)n(t)f(to)h(emphasize)f(that)h(the)g(asymptotic)f(size)g (with)i(resp)r(ect)e(to)g FA(")h FB(of)f(the)h(Melnik)n(o)n(v)f(p)r (oten)n(tial)g(is)h(giv)n(en)f(b)n(y)71 671 y(\(3\))g(pro)n(vided)g FA(H)607 683 y Fy(0)644 671 y FB(\()p FA(x;)14 b(y)s FB(\))24 b(and)f FA(H)1086 683 y Fy(1)1123 671 y FB(\()p FA(x;)14 b(y)s(;)g(\034)9 b FB(;)14 b(0\))24 b(are)f(either)g (algebraic)e(or)i(algebraic)e(in)j FA(y)i FB(and)d(trigonometric)f(p)r (olynomi-)71 770 y(als)i(in)g FA(x)p FB(.)37 b(The)24 b(study)h(of)f(the)h(Melnik)n(o)n(v)e(p)r(oten)n(tial)i(for)e(general)g (analytic)h(Hamiltonian)g(systems)g(with)h(fast)g(p)r(erio)r(dic)71 870 y(p)r(erturbations)30 b(has)g(not)g(b)r(een)h(previously)f(done.)46 b(This)30 b(study)h(strongly)e(dep)r(ends)i(on)g(the)g(analyticit)n(y)e (prop)r(erties)71 969 y(of)j(the)h(Hamiltonian)f FA(H)39 b FB(and)32 b(therefore,)g(ev)n(en)g(if)h(the)f(P)n(oincar)n(\023)-39 b(e)29 b(function)k(can)f(b)r(e)h(estimated)f(for)g(some)f(concrete)71 1069 y(systems)26 b([LS80,)h(MP94)n(,)h(SMH91)o(],)g(a)e(general)g (study)h(of)h(this)f(function)h(seems)e(to)h(require)f(more)h(p)r(o)n (w)n(erful)f(analytic)71 1169 y(to)r(ols.)39 b(Ev)n(en)27 b(if)i(to)f(compute)h(the)g(\014rst)f(asymptotic)g(order)f(for)h (general)f(Hamiltonian)h(systems)g(seems)g(no)n(w)n(ada)n(ys)e(a)71 1268 y(problem)j(out)h(of)f(reac)n(h,)g(to)h(obtain)f(\(non-sharp\))g (exp)r(onen)n(tially)g(small)g(upp)r(er)h(b)r(ounds)g(w)n(as)e(already) g(ac)n(hiev)n(ed)h(b)n(y)71 1368 y(Neish)n(tadt)e(in)h([Ne)-9 b(\025)-32 b(\02084)n(])27 b(using)g(a)n(v)n(eraging)e(tec)n(hniques)i (and)g(b)n(y)g([FS90)o(])g(using)g(complex)g(extensions)g(of)g(the)h (in)n(v)-5 b(arian)n(t)71 1468 y(manifolds.)195 1567 y(Once)33 b(w)n(e)g(kno)n(w)g(that)h(the)f(splitting)h(is)f(exp)r(onen) n(tially)g(small,)i(a)d(natural)h(question)g(whic)n(h)g(arises)f(is)i (whether)71 1667 y(the)c(Melnik)n(o)n(v)g(p)r(oten)n(tial)g(giv)n(es)f (the)h(correct)f(asymptotic)h(\014rst)g(order)f(of)h(the)g(splitting.) 45 b(In)31 b(comparison)d(with)j(the)71 1766 y(problem)21 b(of)h(giving)f(exp)r(onen)n(tially)g(small)h(upp)r(er)g(b)r(ounds)g (for)f(the)i(splitting,)g(this)f(problem)f(is)h(m)n(uc)n(h)g(more)f(in) n(tricate.)71 1866 y(The)h(results)f(in)h(this)g(direction)g(strongly)e (dep)r(end)j(on)f(the)g(b)r(eha)n(vior)e(of)i(the)g(homo)r(clinic)g (orbit)g(\()p FA(q)3165 1878 y Fy(0)3202 1866 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)3393 1878 y Fy(0)3430 1866 y FB(\()p FA(u)p FB(\)\))22 b(around)71 1966 y(its)28 b(complex)f(singularities)f(and)i(on)f(the)h(analytical)e(prop)r (erties)h(of)h(the)g(p)r(erturbation.)195 2065 y(As)h(w)n(e)e(ha)n(v)n (e)g(already)g(explained,)h(insp)r(ecting)g(form)n(ula)g(\(3\),)g(one)g (sees)f(that)i(Melnik)n(o)n(v)e(theory)g(w)n(orks)g(pro)n(vided)71 2165 y FA(\026")160 2135 y Fx(\021)223 2165 y FB(=)c Fr(o)t FB(\()p FA(")419 2135 y Fx(\014)464 2165 y FA(e)503 2135 y Fv(\000)p Fx(a=")660 2165 y FB(\).)37 b(Namely)-7 b(,)26 b(one)g(needs)h(the)g(size)f(of)g(the)h(p)r(erturbation)f(to)g (b)r(e)h(exp)r(onen)n(tially)f(small)g(with)h(resp)r(ect)71 2265 y(to)35 b FA(")p FB(.)61 b(This)35 b(is)h(not)g(the)f(natural)g (setting)h(and)f(therefore)g(the)h(\014rst)f(w)n(orks)f(dealing)h(with) h(this)g(problem)f(tried)h(to)71 2364 y(enlarge)24 b(the)i(size)f(of)h (the)g(p)r(erturbation)f FA(\026")1459 2334 y Fx(\021)1499 2364 y FA(H)1568 2376 y Fy(1)1631 2364 y FB(for)g(whic)n(h)g(Melnik)n (o)n(v)g(theory)g(actually)g(measures)f(the)i(splitting.)36 b(In)71 2464 y(fact,)28 b(under)f(certain)g(non-degeneracy)e (conditions,)j(it)g(su\016ces)f(to)g(tak)n(e)g FA(\021)k FB(big)c(enough)g(and)h FA(\026)g FB(of)f(order)f(1.)195 2564 y(In)h(other)f(w)n(ords,)g(all)g(the)h(results)g(v)-5 b(alidating)26 b(the)h(prediction)f(of)h(the)g(Melnik)n(o)n(v)f (approac)n(h)e(require)i(some)g(arti\014-)71 2663 y(cial)c(condition)h (ab)r(out)g(the)g(smallness)f(of)g(the)i(p)r(erturbation.)34 b(The)23 b(reason,)f(roughly)g(sp)r(eaking,)h(is)f(the)i(follo)n(wing.) 34 b(T)-7 b(o)71 2763 y(pro)n(v)n(e)19 b(that)i(Melnik)n(o)n(v)e (theory)h(giv)n(es)f(asymptotically)h(the)h(\014rst)f(order)f(of)i(the) g(splitting)g(one)f(needs)g(to)h(p)r(erform)f(\\com-)71 2862 y(plex)25 b(p)r(erturbation)f(theory".)35 b(Namely)-7 b(,)25 b(one)g(lo)r(oks)e(for)i(complex)f(parameterizations)f FA(Z)2923 2832 y Fx(u;s)2917 2883 y(\026)3017 2862 y FB(\()p FA(u;)14 b(t)3164 2874 y Fy(0)3201 2862 y FB(\))25 b(of)g(the)g(p)r(erturb)r(ed)71 2962 y(in)n(v)-5 b(arian)n(t)27 b(curv)n(es)h FA(C)741 2932 y Fx(u;s)836 2962 y FB(\()p FA(t)898 2974 y Fy(0)935 2962 y FB(\))h(of)g(the)g(P)n(oincar)n(\023) -39 b(e)25 b(map)k FA(P)1816 2974 y Fx(t)1841 2982 y Fu(0)1907 2962 y FB(as)f(a)g(p)r(erturbation)g(of)h(the)g (time-parameterization)d(of)j(the)71 3062 y(unp)r(erturb)r(ed)j (separatrix)e FA(Z)1007 3074 y Fy(0)1044 3062 y FB(\()p FA(u)p FB(\))g(=)f(\()p FA(q)1349 3074 y Fy(0)1387 3062 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1578 3074 y Fy(0)1615 3062 y FB(\()p FA(u)p FB(\)\).)50 b(One)31 b(needs)h(to)f(pro)n(v)n(e)g (that)h(b)r(oth)g(p)r(erturb)r(ed)g(parameteriza-)71 3161 y(tions)24 b(are)g(close)g(enough)g(to)g(the)h(unp)r(erturb)r(ed)h (one)e(not)h(only)f(for)g(real)g(v)-5 b(alues)24 b(of)h(time)g FA(u)p FB(,)g(as)f(classical)f(p)r(erturbation)71 3261 y(theory)35 b(requires,)h(but)g(also)f(in)h(some)f(complex)g(domains.) 61 b(The)36 b(main)f(no)n(v)n(elt)n(y)g(in)h(the)g(pro)r(ofs)e(of)i (exp)r(onen)n(tially)71 3361 y(small)28 b(splitting,)i(that)f(w)n(as)e (disco)n(v)n(ered)g(b)n(y)i(Lazutkin)f(in)h(his)g(pioneer)f(pap)r(er)g ([Laz84)o(],)h(is)f(that)h(the)h(p)r(erturb)r(ed)f(and)71 3460 y(unp)r(erturb)r(ed)e(manifolds,)f(as)g(w)n(ell)g(as)f(the)i (solutions)f(of)g(the)h(v)-5 b(ariational)24 b(equations)i(along)f (them,)i(need)f(to)h(b)r(e)f(close)71 3560 y(enough)h(when)h(one)g (considers)f(complex)g(times)h(in)h(a)e(domain)h(whic)n(h)f(con)n (tains)g(a)h(real)f(in)n(terv)-5 b(al)27 b(and)h(whic)n(h)g(reac)n(hes) 71 3659 y(a)d(neigh)n(b)r(orho)r(o)r(d)g(of)h(order)e FA(")i FB(of)g(the)g(singularities)f(of)h(the)g(unp)r(erturb)r(ed)g (homo)r(clinic)g(orbit.)36 b(Clearly)-7 b(,)25 b(when)h(time)h(is)71 3759 y(real,)21 b(the)g(homo)r(clinic)f(orbit)g(is)g(a)g(b)r(ounded)h (solution)f(and)g(it)h(is)f(easy)f(to)h(see)g(that)h(the)g(p)r(erturb)r (ed)f(in)n(v)-5 b(arian)n(t)20 b(manifolds)71 3859 y(are)30 b(close)g(to)h(it.)47 b(Ho)n(w)n(ev)n(er,)29 b(when)i(w)n(e)g(reac)n(h) e(a)i(neigh)n(b)r(orho)r(o)r(d)f(of)g(its)h(singularities,)g(the)g (homo)r(clinic)g(itself)g(blo)n(ws)71 3958 y(up,)25 b(and)e(it)h(is)f (not)h(clear)f(that)g(the)h(p)r(erturb)r(ed)g(in)n(v)-5 b(arian)n(t)23 b(manifolds)g(are)f(close)h(to)h(it)g(an)n(ymore.)33 b(Of)24 b(course)e(assuming)71 4058 y(arti\014cially)i(that)h(the)g(p)r (erturbation)f(is)h(small)f(enough)g(\(adding)h(extra)f(p)r(o)n(w)n (ers)f(of)i FA(\021)j FB(in)d(the)g(p)r(erturbativ)n(e)f(term\))h(one) 71 4158 y(can)k(see)g(that)h(the)f(manifolds)h(are)e(close)h(to)g(the)h (unp)r(erturb)r(ed)g(homo)r(clinic)f(in)h(a)f(complex)g(domain)g(whic)n (h)g(reac)n(hes)71 4257 y(a)36 b(neigh)n(b)r(orho)r(o)r(d)g(of)g(size)g FA(")h FB(of)f(the)h(singularities)f(of)g(the)h(unp)r(erturb)r(ed)h (homo)r(clinic)e(tra)5 b(jectory)-7 b(.)62 b(Consequen)n(tly)71 4357 y(the)34 b(Melnik)n(o)n(v)e(approac)n(h,)h(that)h(is)f(based)g(on) g(the)h(fact)g(that)g(the)f(p)r(erturb)r(ed)h(manifolds)f(are)g(w)n (ell)g(appro)n(ximated)71 4456 y(b)n(y)h(the)h(unp)r(erturb)r(ed)g (homo)r(clinic,)i(still)e(w)n(orks.)56 b(This)35 b(w)n(as)e(the)i (approac)n(h)e(used)h(in)h([DS97,)g(Gel97a)n(,)g(BF04)o(])g(for)71 4556 y FA(\021)f(>)c(`)p FB(.)51 b(The)32 b(constan)n(t)f FA(`)h FB(w)n(as)f(called)h(the)h Fs(or)l(der)42 b FB(of)32 b(the)h(p)r(erturbation)e FA(H)2520 4568 y Fy(1)2558 4556 y FB(.)50 b(Roughly)32 b(sp)r(eaking,)g(it)h(is)f(the)h(order)71 4656 y(of)28 b(the)g(singularities)e(of)i(the)g(unp)r(erturb)r(ed)g (homo)r(clinic)g(tra)5 b(jectory)25 b(\()p FA(q)2363 4668 y Fy(0)2401 4656 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2592 4668 y Fy(0)2629 4656 y FB(\()p FA(u)p FB(\)\))28 b(closest)f(to)h(the)g(real)f(axis)f(of)i(the)71 4755 y(function)g FA(f)9 b FB(\()p FA(u)p FB(\))23 b(=)f FA(H)737 4767 y Fy(1)775 4755 y FB(\()p FA(q)844 4767 y Fy(0)881 4755 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1072 4767 y Fy(0)1109 4755 y FB(\()p FA(u)p FB(\))p FA(;)g(t=")p FB(;)g(0\),)27 b(for)g(an)n(y)g FA(t)c Fw(2)g Ft(R)p FB(.)195 4855 y(In)29 b(the)f(aforemen)n(tioned)f(w)n(orks,)g(the)i (condition)f FA(\021)f(>)d(`)j FB(ensures)h(that)g(the)h(p)r(erturb)r (ed)f(parameterizations)e FA(Z)3756 4825 y Fx(u:s)3750 4875 y(\026)71 4955 y FB(are)k(close)f(to)i(the)g(parameterization)e (of)h(the)h(unp)r(erturb)r(ed)g(separatrix)e FA(Z)2487 4967 y Fy(0)2555 4955 y FB(ev)n(en)h(at)g(a)h(distance)f(of)h(order)e FA(")h FB(of)h(the)71 5054 y(singularities)26 b(of)i FA(Z)685 5066 y Fy(0)750 5054 y FB(closest)e(to)i(the)g(real)f(axis.)36 b(Nev)n(ertheless,)27 b(as)g(w)n(e)g(will)h(see)f(in)h(this)g(pap)r (er,)f(the)h(condition)g FA(\021)e(>)d(`)71 5154 y FB(is)33 b(su\016cien)n(t)h(but)g(not)f(necessary)f(to)h(ensure)g(that)h(Melnik) n(o)n(v)e(approac)n(h)g(still)i(predicts)f(correctly)f(the)i(size)f(of) g(the)71 5253 y(splitting.)68 b(What)39 b(is)f(imp)r(ortan)n(t)f(is)h (the)h(relativ)n(e)e(size)g(b)r(et)n(w)n(een)h(the)h(homo)r(clinic)f (orbit)f FA(Z)3106 5265 y Fy(0)3181 5253 y FB(and)h(the)h(di\013erence) 71 5353 y(b)r(et)n(w)n(een)i(the)g(homo)r(clinic)f(and)h(the)g(p)r (erturb)r(ed)g(manifolds,)j(and)d(analogously)d(b)r(et)n(w)n(een)j(the) g(solutions)f(of)h(the)71 5453 y(corresp)r(onding)25 b(v)-5 b(ariational)25 b(equations.)36 b(In)27 b(other)g(w)n(ords,)e (as)i(the)g(parameterizations)e(of)i(the)g(in)n(v)-5 b(arian)n(t)26 b(manifolds)p Black 1940 5753 a(3)p Black eop end %%Page: 4 4 TeXDict begin 4 3 bop Black Black 71 272 a FB(can)36 b(b)r(e)h(written)g(as)f FA(Z)828 242 y Fx(u;s)822 293 y(\026)961 272 y FB(=)h FA(Z)1120 284 y Fy(0)1182 272 y FB(+)24 b(\()p FA(Z)1366 242 y Fx(u;s)1360 293 y(\026)1485 272 y Fw(\000)g FA(Z)1631 284 y Fy(0)1668 272 y FB(\),)39 b(the)f(Melnik)n(o)n(v)d(metho)r(d)i(giv)n(es)f(the)h(correct)e (asymptotic)i(term)71 372 y(for)e(the)i(size)e(of)h(the)h(splitting)f (pro)n(vided)f(the)h(homo)r(clinic)g FA(Z)2098 384 y Fy(0)2171 372 y FB(is)g(bigger)f(than)h(the)g(di\013erence)g FA(Z)3317 342 y Fx(u;s)3311 392 y(\026)3435 372 y Fw(\000)24 b FA(Z)3581 384 y Fy(0)3618 372 y FB(.)62 b(Call)71 471 y FA(r)39 b FB(to)d(the)g(order)f(of)h(the)h(singularities)d(of)j FA(p)1505 483 y Fy(0)1542 471 y FB(\()p FA(u)p FB(\))f(closest)f(to)h (the)h(real)e(axis.)61 b(Then,)39 b(the)d(size)g(of)g FA(p)3333 483 y Fy(0)3370 471 y FB(\()p FA(u)p FB(\))h(at)f(p)r(oin)n (ts)71 571 y FA(u)d FB(whic)n(h)g(are)f FA(")p FB(-close)g(to)h(the)g (singularities)f(is)h Fw(O)r FB(\()p FA(")1761 541 y Fv(\000)p Fx(r)1850 571 y FB(\).)54 b(Lo)r(oking)32 b(at)h(the)h (relativ)n(e)e(size)g(of)h(grad)o FA(H)3334 583 y Fy(0)3371 571 y FB(\()p FA(q)3440 583 y Fy(0)3478 571 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)3669 583 y Fy(0)3706 571 y FB(\()p FA(u)p FB(\)\))71 671 y(and)33 b FA(\026")327 640 y Fx(\021)368 671 y FB(grad)n FA(H)598 683 y Fy(1)636 671 y FB(\()p FA(q)705 683 y Fy(0)742 671 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)933 683 y Fy(0)970 671 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(;)14 b FA(")p FB(\),)36 b(one)e(can)f(guess)g(that)h(the)g(\014rst)g(one)f(is)h(strictly)f (bigger)g(than)h(the)g(second)f(if)71 770 y FA(\021)22 b Fw(\000)d FB(\()p FA(`)h Fw(\000)f FA(r)r FB(\))26 b FA(>)f Fw(\000)p FA(r)r FB(.)41 b(This)29 b(new)g(approac)n(h)e(allo) n(ws)h(us)h(to)g(pro)n(v)n(e)e(in)i(this)g(pap)r(er)g(that)g FA(Z)2885 782 y Fy(0)2922 770 y FB(\()p FA(u)p FB(\))g(is)g(strictly)g (bigger)e(than)71 870 y FA(Z)134 840 y Fx(u;s)128 890 y(\026)228 870 y FB(\()p FA(u;)14 b(t)375 882 y Fy(0)412 870 y FB(\))21 b Fw(\000)g FA(Z)608 882 y Fy(0)645 870 y FB(\()p FA(u)p FB(\))31 b(pro)n(vided)g FA(\021)h(>)d(`)21 b Fw(\000)f FB(2)p FA(r)r FB(,)33 b(ev)n(en)e(if)h FA(u)f FB(is)g(at)h(a)f(distance)g FA(")g FB(of)h(the)f(singularit)n(y)-7 b(.)47 b(F)-7 b(or)31 b FA(`)e Fw(\025)g FB(2)p FA(r)r FB(,)k(the)71 969 y(condition)25 b(for)g(the)h(parameterizations)d(and) i(for)g(the)g(solutions)g(of)g(the)h(v)-5 b(ariational)24 b(equations)g(to)i(b)r(e)f(close)g(coincide)71 1069 y(and)i(is)h(giv)n (en)f FA(\021)f(>)c(\021)731 1039 y Fv(\003)793 1069 y FB(=)h FA(`)18 b Fw(\000)g FB(2)p FA(r)r FB(.)195 1169 y(F)-7 b(or)39 b FA(`)j(<)g FB(2)p FA(r)g FB(w)n(e)d(will)h(not)f(reac) n(h)f(v)-5 b(alues)39 b(of)g FA(\021)k FB(suc)n(h)c(that)g FA(`)26 b Fw(\000)g FB(2)p FA(r)45 b(<)d(\021)k(<)c FB(0.)72 b(In)39 b(fact,)k FA(\021)f FB(will)e(reac)n(h)e(\014rst)71 1268 y(the)33 b(\\natural")d(limit)j FA(\021)i FB(=)30 b(0,)j(where)f(the)h(partial)e(deriv)-5 b(ativ)n(es)32 b(of)g FA(H)2321 1280 y Fy(0)2358 1268 y FB(\()p FA(x;)14 b(y)s FB(\))33 b(and)f FA(H)2818 1280 y Fy(1)2856 1268 y FB(\()p FA(x;)14 b(y)s(;)g(\034)9 b FB(;)14 b FA(")p FB(\))33 b(ev)-5 b(aluated)32 b(on)g(the)71 1368 y(unp)r(erturb)r(ed)26 b(homo)r(clinic)g(are)f(not)i(close)e(ev)n(en)g(for)h(real)f(v)-5 b(alues.)36 b(Ev)n(en)25 b(if)i(for)e(concrete)h(examples)f([GOS10)o(,) h(Gel00)o(])71 1468 y(one)33 b(can)f(pro)n(v)n(e)g(the)h(existence)g (of)g(in)n(v)-5 b(arian)n(t)33 b(manifolds)f(and)h(compute)h(the)f (size)g(of)g(their)g(splitting)h(for)f(negativ)n(e)71 1567 y(v)-5 b(alues)21 b(of)h FA(\021)s FB(,)h(in)f(this)g(pap)r(er)f (w)n(e)g(deal)g(with)h(general)f(Hamiltonians)g(and)g FA(\021)26 b Fw(\025)d FB(0.)35 b(This)21 b(implies)h(that)g(the)g(unp) r(erturb)r(ed)71 1667 y(system)i(and)g(the)h(p)r(erturbation)e(can)h (ha)n(v)n(e)f(the)i(same)e(size)h(\(see)g(Section)h(1.1)e(for)h(some)f (problems)h(whic)n(h)g(corresp)r(ond)71 1766 y(to)h(this)g(natural)f (limit)i FA(\021)g FB(=)d(0\).)35 b(When)26 b FA(\021)g FB(=)d(0,)i(one)f(can)h(apply)g(classical)e(a)n(v)n(eraging)f(theory)i (to)h(see)f(that)i(w)n(e)e(are)g(still)71 1866 y(in)i(a)f(p)r (erturbativ)n(e)g(setting)h(and)f(the)h(p)r(erturb)r(ed)h(real)d(in)n (v)-5 b(arian)n(t)25 b(manifolds)g(are)g FA(\026")p FB(-close)f(to)i (the)g(real)f(unp)r(erturb)r(ed)71 1966 y(separatrix.)44 b(Nev)n(ertheless,)30 b(as)g(w)n(e)g(will)h(see)f(in)h(this)f(pap)r (er,)h(the)g(solutions)f(of)g(the)h(v)-5 b(ariational)29 b(equations)h(are)f(not)71 2065 y(close)c(enough)h(in)g(this)g(case.)36 b(This)26 b(implies)g(that,)h(as)e(is)h(stated)g(in)h(Theorem)e(2.5,)h (Melnik)n(o)n(v)e(form)n(ula)i(do)r(es)f(not)h(giv)n(e)71 2165 y(the)i(correct)e(\014rst)h(asymptotic)h(term)f(of)h(the)g (splitting)g(generically)-7 b(.)195 2265 y(In)24 b(conclusion,)g (\(under)g(certain)f(non-degeneracy)e(conditions\))j(Melnik)n(o)n(v)e (theory)h(giv)n(es)f(the)j(correct)d(prediction)71 2364 y(pro)n(vided)1511 2464 y FA(\021)k(>)d(\021)1710 2430 y Fv(\003)1771 2464 y FB(=)g(max)o Fw(f)p FA(`)18 b Fw(\000)g FB(2)p FA(r)n(;)c FB(0)p Fw(g)p FA(:)195 2613 y FB(The)28 b(so)f(called)h(\\singular")e(case)h(o)r(ccurs)f(when)j(the)f (di\013erence)g FA(Z)2330 2583 y Fx(u;s)2324 2634 y(\026)2424 2613 y FB(\()p FA(u;)14 b(t)2571 2625 y Fy(0)2608 2613 y FB(\))k Fw(\000)h FA(Z)2799 2625 y Fy(0)2836 2613 y FB(\()p FA(u)p FB(\))28 b(has)f(the)h(same)f(size)h(as)f(the)71 2713 y(unp)r(erturb)r(ed)h(homo)r(clinic)g FA(Z)1026 2725 y Fy(0)1063 2713 y FB(\()p FA(u)p FB(\))g(when)g FA(u)g FB(reac)n(hes)e(a)h(neigh)n(b)r(orho)r(o)r(d)g(at)h(a)f (distance)h FA(")g FB(of)g(the)g(singularities)f(of)g FA(Z)3790 2725 y Fy(0)3827 2713 y FB(.)71 2813 y(Consequen)n(tly)-7 b(,)28 b(the)h(in)n(v)-5 b(arian)n(t)27 b(manifolds)i(are)e(not)i(w)n (ell)f(appro)n(ximated)f(b)n(y)i(the)g(unp)r(erturb)r(ed)g(homo)r (clinic)f(in)h(this)71 2912 y(complex)g(region.)40 b(Let)30 b(us)f(note)g(that)h(this)f(singular)f(case)h(can)g(only)f(happ)r(en)i (if)g FA(`)25 b Fw(\025)h FB(2)p FA(r)31 b FB(and)e FA(\021)g FB(=)d FA(\021)3307 2882 y Fv(\003)3345 2912 y FB(.)42 b(In)30 b(this)f(case,)71 3012 y(w)n(e)f(need)h(to)g(obtain)f(a)g (di\013eren)n(t)h(appro)n(ximation)e(of)h(the)i(manifolds)e(in)h(this)g (region)e(of)i(the)g(complex)f(plane.)40 b(Close)71 3111 y(to)29 b(the)g(singularities)e(of)i(the)g(homo)r(clinic)g(orbit,)g(an) f(equation)h(for)f(the)h(leading)f(term)h(is)g(obtained)f(and)h(it)g (is)g(called)71 3211 y(the)f Fs(inner)i(e)l(quation)p FB(.)37 b(This)27 b(is)h(a)f(non-in)n(tegrable)f(equation)h(without)h (parameters,)e(whic)n(h)h(needs)h(a)f(deep)h(study)-7 b(.)195 3311 y(Summarizing,)34 b(on)f(the)h(one)f(hand,)i(the)e(in)n(v) -5 b(arian)n(t)32 b(manifolds)h(are)g(w)n(ell)g(appro)n(ximated)e(b)n (y)i(the)h(unp)r(erturb)r(ed)71 3410 y(homo)r(clinic)27 b(in)h(a)f(complex)h(region)e(con)n(taing)g(the)i(real)f(line.)37 b(On)28 b(the)g(other)f(hand,)g(the)h(inner)g(equation)f(pro)n(vides)f (a)71 3510 y(go)r(o)r(d)j(appro)n(ximation)f(near)h(the)i (singularities)e Fw(\006)p FA(ia)p FB(.)43 b(Finally)-7 b(,)30 b(matc)n(hing)g(tec)n(hniques)f(are)g(required)g(to)h(matc)n(h)g (the)71 3610 y(di\013eren)n(t)d(appro)n(ximations)e(obtained)h(for)h (the)g(in)n(v)-5 b(arian)n(t)26 b(manifolds.)37 b(Roughly)26 b(sp)r(eaking,)g(the)i(di\013erence)f(b)r(et)n(w)n(een)71 3709 y(the)36 b(solutions)f(of)h(the)g(inner)f(equation)g(replaces)g (the)h(Melnik)n(o)n(v)e(p)r(oten)n(tial)i(in)g(the)g(asymptotic)f(form) n(ula)g(for)g(the)71 3809 y(splitting.)195 3908 y(The)f(\014rst)e (author)h(who)f(dealt)h(with)h(this)f(singular)f(case)g(w)n(as)h (Lazutkin)f(in)i([Laz84)n(,)f(Laz03)o(].)53 b(He)34 b(studied)f(the)71 4008 y(splitting)24 b(of)g(separatrices)e(of)i(the)g(Chirik)n(o)n(v)f (standard)g(map,)i(and)f(ga)n(v)n(e)e(the)i(main)g(idea)g(that)g (inspired)g(all)g(the)g(w)n(orks)71 4108 y(in)k(the)g(sub)5 b(ject:)37 b(as)27 b(w)n(e)g(explained)h(ab)r(o)n(v)n(e,)e(one)i(needs) f(to)h(deal)f(with)h(suitable)g(complex)f(parameterizations)e(of)j(the) 71 4207 y(in)n(v)-5 b(arian)n(t)32 b(manifolds.)53 b(These)33 b(parameterizations)e(are)h(analytic)g(in)i(a)f(complex)f(strip,)j (whose)d(size)h(is)g(limited)h(b)n(y)71 4307 y(the)h(singularities)f (of)h(the)g(unp)r(erturb)r(ed)h(homo)r(clinic)f(orbit.)58 b(A)35 b(complete)g(pro)r(of)g(w)n(as)f(published)h(y)n(ears)e(later)i (b)n(y)71 4407 y(Gelfreic)n(h)28 b(in)i([Gel99)o(].)40 b(A)30 b(fundamen)n(tal)f(to)r(ol)f(in)h(Lazutkin's)g(w)n(ork)e(is)i (the)g(use)g(of)f(\\\015o)n(w)g(b)r(o)n(x)h(co)r(ordinates",)e(called) 71 4506 y(\\straigh)n(tening)21 b(the)j(\015o)n(w")f(in)g([Gel00],)h (around)e(one)i(of)f(the)h(manifolds.)35 b(In)24 b(this)g(w)n(a)n(y)-7 b(,)23 b(one)g(obtains)g(a)g(p)r(erio)r(dic)g(func-)71 4606 y(tion)h(whose)g(v)-5 b(alues)25 b(are)e(related)h(with)h(the)g (distance)f(b)r(et)n(w)n(een)h(the)g(manifolds)f(and)h(whose)f(zeros)f (corresp)r(ond)g(to)h(the)71 4705 y(in)n(tersections)i(b)r(et)n(w)n (een)h(them.)37 b(Consequen)n(tly)-7 b(,)26 b(the)i(result)e(ab)r(out)h (exp)r(onen)n(tially)g(small)f(splitting)i(is)f(deriv)n(ed)f(from)71 4805 y(some)j(prop)r(erties)f(of)h(analytic)g(p)r(erio)r(dic)g (functions)h(b)r(ounded)g(in)f(complex)g(strips)g(\(see,)h(for)f (instance,)g(Prop)r(osition)71 4905 y(2.7)e(in)g([DS97]\).)195 5004 y(In)35 b(the)f(pap)r(ers)g([Sau01)n(,)h(LMS03)o(])f(the)h (authors)e(in)n(tro)r(duced)g(a)h(di\013eren)n(t)g(approac)n(h)e(that)j (a)n(v)n(oided)d(the)j(\\\015o)n(w-)71 5104 y(b)r(o)n(x)29 b(co)r(ordinates")g(of)h(Lazutkin's)f(metho)r(d.)45 b(The)30 b(authors)f(w)n(ork)n(ed)f(with)j(the)f(original)f(v)-5 b(ariables)29 b(of)h(the)g(problem)71 5204 y(and)25 b(w)n(ere)f(able)g (to)h(measure)f(the)h(distance)g(b)r(et)n(w)n(een)g(the)g(manifolds)g (without)g(using)g(\\\015o)n(w)f(b)r(o)n(x)g(co)r(ordinates".)34 b(The)71 5303 y(idea)26 b(w)n(as)f(the)i(follo)n(wing:)35 b(b)r(eing)26 b(b)r(oth)h(manifolds)f(giv)n(en)g(b)n(y)g(the)g(graphs)f (of)i(suitable)f(functions)g(that)h(are)e(solutions)71 5403 y(of)31 b(the)g(same)f(equation,)h(their)g(di\013erence)g (satis\014es)f(a)h(linear)f(equation)g(and)h(is)f(b)r(ounded)i(in)f (some)f(complex)h(strip.)p Black 1940 5753 a(4)p Black eop end %%Page: 5 5 TeXDict begin 5 4 bop Black Black 71 272 a FB(Studying)28 b(the)g(prop)r(erties)f(of)h(b)r(ounded)g(solutions)f(of)g(this)i (linear)e(equation,)g(where)g(p)r(erio)r(dicit)n(y)h(also)e(pla)n(ys)h (a)g(role,)71 372 y(one)g(obtains)g(exp)r(onen)n(tially)g(small)g (results.)195 471 y(The)32 b(metho)r(d)g(in)g([Sau01)o(,)g(LMS03)o(])g (uses)f(the)h(fact)g(that,)h(in)f(the)g(considered)e(systems,)i(the)g (manifolds)g(can)f(b)r(e)71 571 y(written)24 b(as)g(graphs)e(of)j(the)f (gradien)n(t)f(of)h(generating)f(functions)h(in)h(suitable)f(domains.) 35 b(These)23 b(generating)g(functions)71 671 y(are)j(solutions)g(of)h (the)g(Hamilton-Jacobi)f(equation)g(asso)r(ciated)g(to)h(system)f (\(1\))q(.)36 b(Solving)27 b(these)g(partial)f(di\013eren)n(tial)71 770 y(equations)j(one)h(can)f(obtain)h(parameterizations)e(of)i(the)g (global)f(manifolds.)44 b(Ev)n(en)30 b(if)g(in)h(this)f(pap)r(er)g(w)n (e)f(deal)h(with)71 870 y(general)38 b(Hamiltonian)h(systems)g(and)g (then)h(the)g(in)n(v)-5 b(arian)n(t)38 b(manifolds)h(ma)n(y)f(not)i(b)r (e)g(graphs)e(globally)-7 b(,)41 b(w)n(e)e(ha)n(v)n(e)71 969 y(adapted)23 b(the)g(metho)r(d)g(in)h([Sau01)n(,)g(LMS03)o(].)35 b(W)-7 b(e)24 b(w)n(ork)d(with)j(the)f(Hamilton-Jacobi)f(equation)g(in) h(suitable)g(domains)71 1069 y(where)k(the)h(manifolds)f(are)g(giv)n (en)g(b)n(y)g(graphs)f(and)i(then)g(w)n(e)f(measure)g(the)g(splitting)h (there.)195 1169 y(W)-7 b(e)28 b(w)n(an)n(t)e(to)g(emphasize)h(that,)g (as)f(far)h(as)f(the)h(authors)f(kno)n(w,)g(there)h(are)e(no)i(general) e(results)i(dealing)f(with)h(the)71 1268 y(singular)33 b(case.)55 b(The)34 b(previous)f(results)g(in)i(the)f(singular)f(case)g (\(see)h([Gel00)o(,)g(T)-7 b(re97)o(,)34 b(Oli06)n(,)h(GOS10)n(]\))g (only)f(dealt)71 1368 y(with)28 b(particular)e(examples.)195 1468 y(In)g(this)h(pap)r(er)e(w)n(e)h(giv)n(e)e(results)i(that)g(con)n (tain)f(the)h(so-called)f(regular)f(case)h FA(\021)h(>)d(\021)2840 1437 y Fv(\003)2904 1468 y FB(\(see)j(Section)g(2.1\),)f(in)i(whic)n(h) 71 1567 y(the)32 b(Melnik)n(o)n(v)f(form)n(ula)f(predicts)i(correctly)e (the)i(di\013erence)g(b)r(et)n(w)n(een)f(the)i(manifolds,)f(but)g(w)n (e)g(also)e(consider)h(the)71 1667 y(so-called)h(singular)h(case)g FA(\021)k FB(=)c FA(\021)1137 1637 y Fv(\003)1176 1667 y FB(,)i(in)g(whic)n(h)f(the)g(Melnik)n(o)n(v)f(form)n(ula)g(do)r(es)g (not)h(predict)g(correctly)f(an)n(ymore)f(the)71 1766 y(di\013erence)27 b(b)r(et)n(w)n(een)h(the)g(p)r(erturb)r(ed)g (manifolds)f(and)h(one)f(has)g(to)g(consider)g(an)g(alternativ)n(e)g (form)n(ula.)195 1866 y(Studying)f(the)f(phenomenon)g(of)g(splitting)g (in)h(general)d(Hamiltonian)i(systems)g(w)n(e)f(ha)n(v)n(e)g(found)i (examples)e(where)71 1966 y(the)g(Melnik)n(o)n(v)e(theory)g(do)r(es)i (not)f(predict)g(correctly)f(the)i(form)n(ula)e(for)h(the)h(area)e(of)h (the)h(lob)r(es)f(\(4\))h(in)f(sev)n(eral)f(asp)r(ects.)71 2065 y(In)k(section)g(2.2.4)f(w)n(e)h(will)h(see)f(mo)r(dels)g(where)g (the)h(constan)n(t)e FA(K)32 b FB(is)27 b(not)f(correctly)f(giv)n(en)g (b)n(y)h(the)h(Melnik)n(o)n(v)e(form)n(ula,)71 2165 y(as)30 b(it)h(happ)r(ens)g(in)g(some)f(mo)r(dels)h(studied)g(b)r(efore)f ([Gel00)o(,)h(T)-7 b(re97)o(,)31 b(Oli06)n(,)g(GOS10)o(].)46 b(But)31 b(w)n(e)g(will)g(also)e(encoun)n(ter)71 2265 y(examples)e(where)g(the)h(Melnik)n(o)n(v)e(form)n(ula)h(\(4\))h (neither)f(predicts)h(this)g(constan)n(t)e(nor)h(the)h(correct)e(p)r(o) n(w)n(er)h FA(\014)32 b FB(in)c(\(4\).)195 2364 y(W)-7 b(e)32 b(ha)n(v)n(e)e(seen)h(that)g(the)g(b)r(eha)n(vior)f(of)h(the)h (splitting)f(is)g(extremely)g(sensitiv)n(e)f(on)h(the)g(sign)g(of)g FA(`)20 b Fw(\000)h FB(2)p FA(r)33 b FB(and)e(the)71 2464 y(v)-5 b(alue)26 b(of)h FA(\021)s FB(.)37 b(In)26 b(consequence,)g(one)g(has)g(to)h(obtain)f(the)h(di\013eren)n(t)f (\014rst)h(asymptotic)f(orders)e(separately)-7 b(,)26 b(taking)g(in)n(to)71 2564 y(accoun)n(t)h(the)h(prop)r(erties)e(of)i (eac)n(h)f(case.)36 b(W)-7 b(e)28 b(summarize)e(the)i(main)g (particularities)e(of)i(eac)n(h)f(case:)p Black 195 2730 a Fw(\017)p Black 41 w FA(\021)h(>)d(\021)481 2699 y Fv(\003)544 2730 y FB(=)f(max\()p FA(`)19 b Fw(\000)f FB(2)p FA(r)n(;)c FB(0\):)39 b(under)28 b(certain)g(non-degeneracy)e (conditions,)j(the)g(Melnik)n(o)n(v)e(form)n(ula)h(\(4\))h(giv)n(es)278 2829 y(the)f(correct)f(\014rst)g(order,)f(that)i(is,)g(the)g(correct)e (constan)n(ts)g FA(K)6 b FB(,)28 b FA(\014)k FB(and)27 b FA(a)p FB(.)p Black 195 2995 a Fw(\017)p Black 41 w FA(`)19 b Fw(\000)g FB(2)p FA(r)27 b(<)e FB(0)k(and)f FA(\021)h FB(=)24 b(0:)39 b(there)29 b(app)r(ears)e(a)i(correcting)e (term)i(whic)n(h)f(replaces)g FA(K)34 b FB(in)29 b(the)h(Melnik)n(o)n (v)d(form)n(ula.)278 3095 y(This)21 b(term)g(can)f(b)r(e)h(obtained)g (through)f(classical)f(p)r(erturbation)h(theory)g(tec)n(hniques.)34 b(Since)21 b(it)g(nev)n(er)f(v)-5 b(anishes,)278 3194 y(the)31 b(\014rst)e(asymptotic)g(order)g(is)g(non-degenerate)f(if)i (and)g(only)f(if)h(neither)g(is)g(the)g(Melnik)n(o)n(v)e(p)r(oten)n (tial.)43 b(Note)278 3294 y(that)28 b(in)g(this)g(case,)f(for)g(real)g (v)-5 b(alues)27 b(of)g(the)h(v)-5 b(ariables)27 b FA(H)34 b FB(is)28 b(not)f(a)g(p)r(erturbation)g(of)h FA(H)3088 3306 y Fy(0)3125 3294 y FB(.)p Black 195 3460 a Fw(\017)p Black 41 w FA(`)20 b Fw(\000)f FB(2)p FA(r)28 b(>)e FB(0)j(and)g FA(\021)g FB(=)d FA(\021)1054 3430 y Fv(\003)1118 3460 y FB(=)g FA(`)19 b Fw(\000)g FB(2)p FA(r)r FB(:)41 b(as)29 b(in)g(the)h(previous)e(case,)h(the)h(Melnik)n(o)n(v)e(form)n(ula)h (\(4\))g(fails)g(to)g(predict)278 3560 y(correctly)37 b(the)h(constan)n(t)f FA(K)6 b FB(.)67 b(Nev)n(ertheless,)39 b(the)f(correcting)e(term)i(has)f(a)h(signi\014can)n(tly)f(di\013eren)n (t)h(origin,)278 3659 y(since)c(it)g(comes)f(from)g(the)i(study)e(of)h (the)g(aforemen)n(tioned)f Fs(inner)i(e)l(quation)p FB(.)56 b(Let)33 b(us)h(note)g(that)g(the)g(range)278 3759 y FA(\021)27 b Fw(2)c FB([0)p FA(;)14 b(`)k Fw(\000)g FB(2)p FA(r)r FB(\))28 b(remains)f(op)r(en.)p Black 195 3925 a Fw(\017)p Black 41 w FA(`)20 b Fw(\000)g FB(2)p FA(r)30 b FB(=)d(0)j(and)g FA(\021)h FB(=)c(0:)42 b(as)30 b(in)g(the)h (previous)e(case,)h(w)n(e)g(need)h(to)f(consider)f(an)h Fs(inner)j(e)l(quation)k FB(to)30 b(obtain)g(a)278 4025 y(candidate)c(for)f(the)h(\014rst)g(asymptotic)f(order.)35 b(This)26 b(candidate)f(di\013ers)h(from)f(the)h(Melnik)n(o)n(v)f(form) n(ula)g(b)n(y)g(b)r(oth)278 4124 y(the)30 b(constan)n(t)f FA(K)35 b FB(and)29 b(the)h(exp)r(onen)n(t)f FA(\014)t FB(.)42 b(Nev)n(ertheless,)29 b(to)h(obtain)f(the)g(true)h(\014rst)f (order,)f(one)h(has)g(to)g(mak)n(e)278 4224 y(still)22 b(one)f(more)g(mo)r(di\014cation)h(to)f(the)h(form)n(ula)f(giv)n(en)f (b)n(y)i(the)g(inner)f(equation,)h(whic)n(h)g(implies)g(a)f(c)n(hange)f (in)i(the)278 4324 y(constan)n(t)28 b(pro)n(vided)f(b)n(y)g(the)i (inner)f(equation.)37 b(Note,)29 b(that)f(the)h(c)n(hange)d(in)j(the)f (exp)r(onen)n(t)g FA(\014)33 b FB(is)27 b(a)h(substan)n(tial)278 4423 y(qualitativ)n(e)34 b(c)n(hange)g(in)h(the)g(b)r(eha)n(vior)e(of)i (the)g(splitting.)59 b(Ev)n(en)34 b(if)h(this)g(fact)g(w)n(as)e (already)h(p)r(oin)n(ted)g(out)h(in)278 4523 y([Bal06)o(],)27 b(the)h(presen)n(t)e(pap)r(er,)h(as)f(far)h(as)f(the)h(authors)f(kno)n (w,)h(is)g(the)g(\014rst)g(w)n(ork)e(that)j(rigorously)c(pro)n(v)n(es)h (that)278 4622 y(this)j(phenomenon)g(actually)f(happ)r(ens.)195 4788 y(The)40 b(structure)f(of)h(this)g(pap)r(er)f(go)r(es)g(as)g (follo)n(ws.)72 b(First)39 b(in)h(Section)g(2)f(w)n(e)g(in)n(tro)r (duce)h(some)f(notation,)j(the)71 4888 y(h)n(yp)r(otheses)33 b(and)h(w)n(e)f(state)h(the)g(main)g(results.)55 b(In)34 b(Section)g(3)f(w)n(e)g(giv)n(e)g(some)h(heuristic)f(ideas)g(of)h(the)g (pro)r(of)g(and)71 4988 y(w)n(e)c(compare)f(our)h(metho)r(ds)g(to)h (those)f(of)g(some)g(of)g(the)h(aforemen)n(tioned)e(previous)h (results.)44 b(Section)31 b(4)f(is)g(dev)n(oted)71 5087 y(to)e(describ)r(e)g(the)h(pro)r(of)f(of)g(the)h(main)g(theorem.)39 b(T)-7 b(o)28 b(mak)n(e)f(this)i(section)f(more)g(readable,)f(the)i (pro)r(of)f(of)g(the)h(partial)71 5187 y(results)e(obtained)g(in)h (this)g(section)f(are)g(deferred)g(to)g(the)h(follo)n(wing)f(sections,) g(that)h(is,)f(Sections)h(5-9.)p Black 1940 5753 a(5)p Black eop end %%Page: 6 6 TeXDict begin 6 5 bop Black Black 71 272 a Fq(1.1)112 b(Motiv)-6 b(ation)38 b(of)g(the)f(problem)71 425 y FB(Throughout)26 b(the)i(pap)r(er)f(w)n(e)g(will)h(use)f(the)h(classical)e(notation)h FA(H)2193 437 y Fy(0)2249 425 y FB(+)17 b FA(\026")2420 395 y Fx(\021)2461 425 y FA(H)2530 437 y Fy(1)2594 425 y FB(whic)n(h)28 b(encoun)n(ters)e(the)i(singular)e(case)71 525 y FA(\021)34 b FB(=)c FA(\021)285 495 y Fv(\003)323 525 y FB(,)k(where)d(the)i(v)-5 b(alue)32 b(of)g FA(\021)1134 495 y Fv(\003)1204 525 y FB(dep)r(ends)h(on)f(the)g(prop)r(erties)f(of) h FA(H)2360 537 y Fy(0)2430 525 y FB(and)g FA(H)2665 537 y Fy(1)2702 525 y FB(.)51 b(F)-7 b(or)31 b(general)g(p)r (erturbations)g FA(H)3813 537 y Fy(1)71 638 y FB(of)i(general)e (classical)h(Hamiltonian)h(systems)f FA(H)1652 650 y Fy(0)1689 638 y FB(\()p FA(x;)14 b(y)s FB(\))33 b(=)2021 601 y Fx(y)2057 576 y Fu(2)p 2021 619 69 4 v 2038 667 a Fy(2)2121 638 y FB(+)21 b FA(V)e FB(\()p FA(x)p FB(\))34 b(this)g(w)n(ork)d(\014nishes)i(the)g(general)f(problem,)71 738 y(initiated)c(and)f(partially)f(solv)n(ed)g(in)i([DS97)o(,)g (Gel97a)n(,)g(BF04)o(,)f(BF05)o(])h(for)e FA(\021)h(>)22 b(`)p FB(,)27 b(of)h(the)f(splitting)h(of)f(separatrices)e(in)71 838 y(the)j(singular)e(and)i(regular)d(cases)i FA(\021)f Fw(\025)d FA(\021)1379 807 y Fv(\003)1417 838 y FB(.)195 937 y(Nev)n(ertheless)31 b(w)n(e)h(w)n(an)n(t)f(to)h(stress)f(that)h (it)g(is)g(of)f(particular)g(in)n(terest)g(the)i(non-p)r(erturbativ)n (e)d(case)h FA(\026)f FB(=)g(1)i(and)71 1037 y FA(\021)26 b FB(=)d(0.)36 b(F)-7 b(or)27 b(instance,)h(one)f(can)g(consider)g(mo)r (dels)g(of)h(the)g(form)1413 1268 y FA(H)1503 1151 y Fz(\022)1564 1268 y FA(x;)14 b(y)s(;)1743 1212 y(t)p 1739 1249 39 4 v 1739 1325 a(")1788 1151 y Fz(\023)1872 1268 y FB(=)1970 1212 y FA(y)2014 1182 y Fy(2)p 1970 1249 81 4 v 1989 1325 a FB(2)2079 1268 y(+)2172 1247 y Fz(e)2162 1268 y FA(V)2243 1151 y Fz(\022)2304 1268 y FA(x;)2402 1212 y(t)p 2398 1249 39 4 v 2398 1325 a(")2447 1151 y Fz(\023)3744 1268 y FB(\(5\))71 1522 y(taking)32 b FA(V)19 b FB(\()p FA(x)p FB(\))33 b(=)670 1489 y Fy(1)p 649 1503 75 4 v 649 1551 a(2)p Fx(\031)747 1455 y Fz(R)802 1476 y Fy(2)p Fx(\031)786 1551 y Fy(0)905 1501 y Fz(e)894 1522 y FA(V)19 b FB(\()p FA(x;)14 b(\034)9 b FB(\))p FA(d\034)45 b FB(and)32 b FA(H)1513 1534 y Fy(1)1564 1522 y FB(\()q FA(x;)14 b(y)s(;)g(\034)9 b FB(\))32 b(=)1978 1501 y Fz(e)1968 1522 y FA(V)18 b FB(\()p FA(x;)c(\034)9 b FB(\))24 b Fw(\000)e FA(V)c FB(\()p FA(x)p FB(\).)54 b(In)33 b(this)g(case,)h(under)e(certain)g(generic)71 1622 y(h)n(yp)r(otheses)26 b(ab)r(out)g FA(V)19 b FB(,)27 b(whic)n(h)g(are)f(sp)r(eci\014ed)g(in)h(Section)g(2.1,)f(our)g(result) g(in)h(Theorem)f(2.6)g(pro)n(vides)f(a)h(form)n(ula)g(for)71 1721 y(the)i(splitting)g(ev)n(en)f(for)g FA(\021)902 1691 y Fv(\003)963 1721 y FB(=)c(0.)195 1821 y(Hamiltonian)29 b(\(5\))f(is)g(a)g(particular)f(case)h(of)g(p)r(erio)r(dic)g(time)h (dep)r(enden)n(t)g(Hamiltonians)f FA(H)7 b FB(\()p FA(x;)14 b(y)s(;)g(t=")p FB(\).)38 b(After)29 b(the)71 1920 y(c)n(hange)g(of)h (time)g FA(t)d FB(=)g FA("\034)9 b FB(,)31 b(these)f(systems)g(b)r (ecome)g(time)g(dep)r(enden)n(t)h(Hamiltonian)f(systems)f(with)i(slo)n (w)e(dynamics)71 2020 y(and)e(classical)f(a)n(v)n(eraging)f(theory)i (can)g(b)r(e)h(applied)g(to)f(them.)38 b(So)27 b(it)h(is)f(natural)g (to)h(split)1254 2248 y FA(H)1344 2131 y Fz(\022)1405 2248 y FA(x;)14 b(y)s(;)1584 2192 y(t)p 1580 2229 39 4 v 1580 2305 a(")1629 2131 y Fz(\023)1713 2248 y FB(=)23 b FA(H)1870 2260 y Fy(0)1907 2248 y FB(\()p FA(x;)14 b(y)s FB(\))19 b(+)f FA(H)2270 2260 y Fy(1)2321 2131 y Fz(\022)2382 2248 y FA(x;)c(y)s(;)2561 2192 y(t)p 2557 2229 V 2557 2305 a(")2606 2131 y Fz(\023)71 2502 y FB(with)28 b FA(H)329 2514 y Fy(0)366 2502 y FB(\()p FA(x;)14 b(y)s FB(\))24 b(=)700 2469 y Fy(1)p 679 2483 75 4 v 679 2531 a(2)p Fx(\031)777 2435 y Fz(R)833 2456 y Fy(2)p Fx(\031)816 2531 y Fy(0)925 2502 y FA(H)c FB(\()q FA(x;)14 b(y)s(;)g(\034)9 b FB(\))28 b FA(d\034)38 b FB(and)27 b FA(H)1664 2514 y Fy(1)1715 2502 y FB(\()p FA(x;)14 b(y)s(;)g(\034)9 b FB(\))24 b(=)f FA(H)e FB(\()p FA(x;)14 b(y)s(;)g(\034)9 b FB(\))19 b Fw(\000)f FA(H)2636 2514 y Fy(0)2674 2502 y FB(\()p FA(x;)c(y)s FB(\).)195 2602 y(Av)n(eraging)30 b(theory)h(tells)h(us)g(that,)h(ev)n(en)f(if)g(in)g(this)g(mo)r(del)g FA(\021)i FB(=)29 b(0,)k(the)f(solutions)f(of)h(the)g(whole)g (Hamiltonian)71 2701 y(system)26 b(asso)r(ciated)f(to)h FA(H)33 b FB(can)26 b(b)r(e)h(appro)n(ximated,)e(up)h(to)g(long)g (times,)h(b)n(y)f(the)g(solutions)g(of)g FA(H)3156 2713 y Fy(0)3193 2701 y FB(.)37 b(Moreo)n(v)n(er,)24 b(in)i(the)71 2801 y(case)31 b(that)h FA(H)504 2813 y Fy(0)541 2801 y FB(\()p FA(x;)14 b(y)s FB(\))33 b(has)e(a)h(h)n(yp)r(erb)r(olic)f (critical)h(p)r(oin)n(t)g(the)g(stable)g(and)f(unstable)h(in)n(v)-5 b(arian)n(t)31 b(manifolds)h(of)g FA(H)38 b FB(are)71 2900 y FA(")p FB(-close)26 b(to)i(the)h(stable)e(and)h(unstable)g (manifolds)g(of)g FA(H)1858 2912 y Fy(0)1895 2900 y FB(.)38 b(It)29 b(is)f(then)g(natural)f(to)h(study)g(if)h(the)f(homo)r(clinic)g (orbit)g(of)71 3000 y(the)c(Hamiltonian)f(system)g(\(2\))g(splits)h (and)f(to)g(giv)n(e)f(an)i(asymptotic)e(form)n(ula,)i(for)e FA(")i FB(small)f(enough,)g(of)g(this)h(splitting.)71 3100 y(Our)31 b(result)g(in)h(Theorem)e(2.6)h(giv)n(es)g(the)g (splitting)h(of)g(these)f(manifolds)h(in)g(this)f(case)g(for)g(the)h(v) -5 b(alue)31 b FA(\021)3442 3070 y Fv(\003)3511 3100 y FB(=)e(0)i(for)g(a)71 3199 y(wide)d(class)e(of)i(Hamiltonians.)195 3299 y(One)40 b(also)f(encoun)n(ters)g(the)h(singular)f(case)g(for)g FA(\021)1841 3269 y Fv(\003)1923 3299 y FB(=)k(0,)g(when)d(one)f (studies)h(the)h(splitting)f(of)g(separatrices)71 3399 y(phenomenon)32 b(in)g(a)g(resonance)f(of)h(one)g(and)g(a)g(half)g (degrees)f(of)h(freedom)g(Hamiltonian)g(systems)g(whic)n(h)g(are)f (close)71 3498 y(to)d(completely)h(in)n(tegrable)f(ones)g(\(in)h(the)g (sense)f(of)h(Liouville-Arnold\).)39 b(This)29 b(setting)g(do)r(es)f (not)h(\014t)g(exactly)f(in)h(our)71 3598 y(h)n(yp)r(otheses)35 b(but,)k(as)c(w)n(e)h(will)g(see)f(in)i(a)e(forthcoming)g(pap)r(er,)j (the)e(metho)r(ds)h(used)e(in)i(this)f(pap)r(er)f(can)h(b)r(e)g(easily) 71 3697 y(adapted)27 b(to)h(that)f(case)g(\(see)h(section)f(2.3\).)71 3930 y Fq(1.2)112 b(Historical)37 b(remarks)i(and)f(related)g(problems) 71 4083 y FB(Historically)-7 b(,)36 b(the)g(results)f(ab)r(out)h(exp)r (onen)n(tially)f(small)g(splitting)h(of)f(separatrices)f(can)h(b)r(e)h (classi\014ed)e(in)n(to)i(three)71 4183 y(groups:)f(upp)r(er)28 b(b)r(ounds,)g(v)-5 b(alidation)27 b(of)h(the)g(Melnik)n(o)n(v)e (approac)n(h)g(and)h(asymptotics)g(for)g(the)h(singular)e(case.)195 4282 y(Some)h(results,)g(dealing)f(with)i(quite)f(general)f(systems,)h (whic)n(h)g(obtain)f(exp)r(onen)n(tially)h(small)g(upp)r(er)g(b)r (ounds)g(for)71 4382 y(the)22 b(splitting)g(for)g(Hamiltonian)g (systems.)34 b(Neish)n(tadt)22 b(in)g([Ne)-9 b(\025)-32 b(\02084)o(])22 b(ga)n(v)n(e)e(exp)r(onen)n(tially)h(small)h(upp)r(er)g (b)r(ounds)g(for)f(the)71 4482 y(splitting)e(for)g(t)n(w)n(o)g(degrees) f(of)h(freedom)g(Hamiltonian)g(systems.)33 b(F)-7 b(or)19 b(second)g(order)e(equations)i(with)h(a)f(rapidly)f(forced)71 4581 y(p)r(erio)r(dic)24 b(term,)g(sev)n(eral)e(authors)h(ga)n(v)n(e)f (sharp)h(exp)r(onen)n(tially)g(small)h(upp)r(er)g(b)r(ounds)g(in)g([F) -7 b(on93)o(,)24 b(F)-7 b(on95)o(,)24 b(FS96])g(and,)71 4681 y(for)31 b(the)g(higher)g(dimensional)g(case,)g(the)h(pap)r(ers)f ([Sau01)n(,)h(Sim94)o(])g(ga)n(v)n(e)d(\(non-sharp\))i(exp)r(onen)n (tially)f(small)h(upp)r(er)71 4781 y(b)r(ounds.)195 4880 y(The)36 b(P)n(oincar)n(\023)-39 b(e)32 b(map)j(of)g(a)g (non-autonomous)f(Hamiltonian)h(in)g(the)h(plane)f(is)g(a)g(particular) f(case)g(of)i(a)f(planar)71 4980 y(area)25 b(preserving)f(map.)37 b(F)-7 b(or)26 b(the)g(Hamiltonian)g(\(1\))h(the)g(P)n(oincar)n(\023) -39 b(e)23 b(map)j FA(P)38 b FB(is)27 b(a)f(near)f(the)i(iden)n(tit)n (y)f(area)f(preserving)71 5079 y(map.)36 b(Rigorous)24 b(upp)r(er)i(b)r(ounds)g(for)f(the)i(splitting)f(of)f(area)g (preserving)f(maps)h(close)g(to)h(the)g(iden)n(tit)n(y)g(w)n(ere)f(giv) n(en)g(in)71 5179 y([FS90)o(].)195 5279 y(The)37 b(second)g(group)f(of) h(results)f(is)h(concerned)f(with)i(the)f(question)g(ab)r(out)g(the)g (v)-5 b(alidit)n(y)37 b(of)g(the)g(asymptotics)71 5378 y(pro)n(vided)25 b(b)n(y)g(the)i(Melnik)n(o)n(v)d(theory)-7 b(.)36 b(Sev)n(eral)25 b(authors)f(in)j(the)f(last)f(15)h(y)n(ears)e (ha)n(v)n(e)g(tried)i(to)g(ensure)f(the)i(v)-5 b(alidit)n(y)26 b(of)71 5478 y(the)k(form)n(ula)f(pro)n(vided)g(b)n(y)h(the)g(Melnik)n (o)n(v)f(p)r(oten)n(tial)h(\(3\))g(to)g(compute)g(the)g(asymptotic)g (form)n(ula)f(for)g(the)i(area)d Fw(A)p FB(.)p Black 1940 5753 a(6)p Black eop end %%Page: 7 7 TeXDict begin 7 6 bop Black Black 71 272 a FB(The)26 b(results)g(in)h(this)f(direction)g(strongly)f(dep)r(end)i(on)f(the)h (b)r(eha)n(vior)e(of)h(the)h(homo)r(clinic)f(orbit)g(around)f(its)i (complex)71 372 y(singularities)f(and)h(on)g(the)g(analytical)f(prop)r (erties)g(of)h(the)h(p)r(erturbation.)36 b(F)-7 b(or)26 b(this)i(reason,)d(the)j(existing)f(results)f(in)71 471 y(this)i(direction)f(mostly)g(deal)g(with)i(sp)r(eci\014c)e(examples.) 195 571 y(The)h(most)f(studied)h(example)f(in)g(the)h(literature)f(has) g(b)r(een)g(the)h(rapidly)f(p)r(erturb)r(ed)h(p)r(endulum)g(with)g(a)f (p)r(ertur-)71 671 y(bation)g(only)g(dep)r(ending)h(on)g(time,)1584 802 y(\177)-47 b FA(x)23 b FB(=)g(sin)14 b FA(x)19 b FB(+)f FA(\026")2091 768 y Fx(\021)2145 802 y FB(sin)2275 746 y FA(t)p 2271 783 39 4 v 2271 859 a(")2319 802 y(;)71 985 y FB(whic)n(h)35 b(in)g(our)f(notation)g(corresp)r(onds)f(to)i FA(H)1555 997 y Fy(0)1592 985 y FB(\()p FA(x;)14 b(y)s FB(\))36 b(=)e FA(y)1963 955 y Fy(2)2000 985 y FA(=)p FB(2)23 b(+)f(cos)13 b FA(x)24 b Fw(\000)f FB(1)34 b(and)h FA(H)2792 997 y Fy(1)2829 985 y FB(\()p FA(x;)14 b(t=")p FB(\))36 b(=)e Fw(\000)p FA(x)14 b FB(sin\()p FA(t=")p FB(\).)58 b(The)71 1085 y(\014rst)27 b(result)g(concerning)f(this)i (system)f(w)n(as)f(obtained)i(b)n(y)f(Holmes,)g(Marsden)f(and)i(Sc)n (heurle)e(in)i([HMS88])f(\(follo)n(w)n(ed)71 1184 y(b)n(y)e([Sc)n(h89)o (,)g(Ang93)o(]\),)h(where)f(they)h(con\014rmed)e(the)i(prediction)f(of) g(the)h(Melnik)n(o)n(v)e(p)r(oten)n(tial)h(establishing)g(exp)r(onen-) 71 1284 y(tially)32 b(small)g(upp)r(er)h(and)f(lo)n(w)n(er)e(b)r(ounds) j(for)f(the)g(area)f Fw(A)i FB(pro)n(vided)e FA(\021)j Fw(\025)d FB(8,)i(whic)n(h)f(coincide)g(with)h(the)g(Melnik)n(o)n(v)71 1383 y(prediction.)j(Later)25 b(the)i(w)n(ork)d([EKS93)n(])j(v)-5 b(alidated)25 b(the)i(same)e(result)h(for)g FA(\021)g Fw(\025)d FB(3.)35 b(Delshams)26 b(and)g(Seara)f(established)71 1483 y(rigourosly)34 b(the)i(result)g(in)h([DS92])f(for)g FA(\021)41 b(>)c FB(0)f(and)g(an)g(analogous)e(result)i(for)g FA(\021)41 b(>)c FB(5)f(w)n(as)g(obtained)g(b)n(y)g(Gelfre-)71 1583 y(ic)n(h)d(in)h([Gel94)o(].)55 b(The)34 b(latter)f(t)n(w)n(o)g (pap)r(ers)g(used)h(a)f(di\013eren)n(t)g(approac)n(h)f(inspired)h(b)n (y)h([GL)-7 b(T91)o(].)55 b(F)-7 b(or)33 b(a)g(simpli\014ed)71 1682 y(p)r(erturbation)27 b(in)h([Sau95)o(])f(an)h(alternativ)n(e)e (pro)r(of,)h(using)g(P)n(arametric)f(Resurgence,)g(w)n(as)h(done.)195 1782 y(The)39 b(only)f(w)n(orks)f(whic)n(h)h(pro)n(vide)g(\(partial\))g (results)g(for)g(some)g(general)f(Hamiltonian)h(as)g(\(1\))h(taking)f FA(\021)k FB(big)71 1882 y(enough,)35 b(are)f([DS97)o(,)h(Gel97a)n(,)g (BF04)o(,)f(BF05)o(].)58 b(In)34 b([DS97,)g(Gel97a)o(],)i(a)e(pro)r(of) g(for)g(the)g(v)-5 b(alidit)n(y)35 b(of)f(the)h(Melnik)n(o)n(v)71 1981 y(metho)r(d)30 b(for)e(general)g(rapidly)h(p)r(erio)r(dic)g (Hamiltonian)g(p)r(erturbations)f(of)h(a)g(class)f(of)i(second)e(order) g(equations)g(w)n(as)71 2081 y(giv)n(en.)34 b(The)22 b(case)g(of)g(a)g(p)r(erturb)r(ed)g(second)g(order)e(equation)i(with)h (a)e(parab)r(olic)g(p)r(oin)n(t)i(w)n(as)e(studied)h(in)h([BF04)o(,)f (BF05)o(].)195 2180 y(A)28 b(Melnik)n(o)n(v)f(theory)f(for)h(t)n(wist)h (maps)f(can)g(b)r(e)h(found)g(in)g([DR97])f(and)h(some)f(results)g(ab)r (out)g(the)h(v)-5 b(alidit)n(y)28 b(of)f(the)71 2280 y(prediction)g(giv)n(en)g(b)n(y)g(the)h(P)n(oincar)n(\023)-39 b(e)25 b(function)j(for)f(area)f(preserving)g(maps)h(w)n(ere)g(giv)n (en)g(in)g([DR98].)195 2380 y(The)21 b(generalization)e(of)i(the)g (splitting)g(problem)g(to)g(higher)f(dimensional)g(systems)g(has)h(b)r (een)g(ac)n(hiev)n(ed)f(b)n(y)g(sev)n(eral)71 2479 y(authors,)27 b(mainly)h(in)h(the)g(Hamiltonian)f(case.)38 b(See,)28 b(for)g(instance,)g([Eli94)o(,)g(T)-7 b(re94)o(,)28 b(LMS03)o(,)h(DG00) o(])g(and)f(references)71 2579 y(therein.)60 b(Some)35 b(results)g(ab)r(out)h(the)f(v)-5 b(alidit)n(y)36 b(of)f(the)h(Melnik)n (o)n(v)e(metho)r(d)i(for)f(higher)g(dimensional)g(Hamiltonian)71 2679 y(systems)24 b(can)g(b)r(e)h(found)g(in)f([Gal94)o(,)h(CG94)o(,)f (DGJS97,)g(GGM99,)g(Sau01)o(,)h(DGS04].)35 b(Finally)-7 b(,)26 b(in)e(a)g(non)h(Hamiltonian)71 2778 y(setting,)36 b(in)f([BS06)o(])f(the)h(splitting)g(of)f(a)g(hetero)r(clinic)g(orbit)g (for)g(some)g(degenerate)f(unfoldings)h(of)h(the)g(Hopf-zero)71 2878 y(singularit)n(y)26 b(of)i(v)n(ector)e(\014elds)i(in)f Ft(R)1195 2848 y Fy(3)1260 2878 y FB(w)n(as)g(found.)195 2977 y(As)33 b(w)n(e)g(already)e(explained,)j(all)f(the)g(results)f(v) -5 b(alidating)33 b(the)g(prediction)g(of)g(the)g(Melnik)n(o)n(v)f (approac)n(h)f(require)71 3077 y(some)e(arti\014cial)g(condition)g(ab)r (out)h(the)g(smallness)e(of)i(the)g(p)r(erturbation.)42 b(The)30 b(third)g(group)e(of)i(results)f(deals)g(with)71 3177 y(the)h(so)g(called)f(\\singular)f(case")h FA(\021)h FB(=)d FA(\021)1343 3147 y Fv(\003)1411 3177 y FB(for)j(whic)n(h)g(one) f(needs)h(to)g(study)g(the)h Fs(inner)h(e)l(quation)k FB(and)30 b(use)g(matc)n(hing)71 3276 y(tec)n(hniques)d(to)h(relate)f (di\013eren)n(t)g(appro)n(ximations)f(for)h(the)h(in)n(v)-5 b(arian)n(t)26 b(manifolds.)195 3376 y(The)f(\014rst)f(result)g(ab)r (out)h(splitting)f(of)h(separatrices)d(in)j(the)g(singular)e(case)g(w)n (as)h(done)g(b)n(y)g(V.)h(Lazutkin)f(in)h([Laz84)n(])71 3476 y(\(see)36 b(also)f([Laz03)o(]\))h(who)g(ga)n(v)n(e)f(a)h(form)n (ula)f(for)h(the)g(splitting)h(for)f(the)h(Chirik)n(o)n(v)d(standard)h (map)i(\(whic)n(h)f(can)g(b)r(e)71 3575 y(reduced)29 b(to)f(a)h(near)f(the)i(iden)n(tit)n(y)f(map)g(through)f(a)h(simple)g (scaling\).)40 b(In)30 b(our)e(notation,)h(this)g(case)f(corresp)r (onds)g(to)71 3675 y FA(\021)115 3645 y Fv(\003)176 3675 y FB(=)23 b(0,)195 3774 y(After)30 b(Lazutkin's)e(w)n(ork,)g(some)h (authors)f(used)h(his)g(metho)r(d)g(and)g(obtained)g(results)f(for)h (the)g(inner)g(equation)g(of)71 3874 y(sev)n(eral)18 b(sp)r(eci\014c)i(equations.)34 b(In)20 b([GS01)o(])g(there)g(is)g(a)g (rigorous)d(study)j(of)g(the)h(inner)e(equation)h(of)g(the)g(H)n(\023) -39 b(enon)19 b(map)h(using)71 3974 y(Resurgence)f(Theory)g([)810 3953 y(\023)803 3974 y(Eca81a)m(,)1113 3953 y(\023)1105 3974 y(Eca81b)n(],)j(and)e(in)g([BS08)o(])g(the)g(authors)f(studied)i (the)f(inner)g(system)g(asso)r(ciated)e(to)i(the)71 4073 y(Hopf-zero)27 b(singularit)n(y)f(using)i(functional)g(analysis)e(tec)n (hniques.)38 b(The)28 b(corresp)r(onding)e(inner)i(equation)f(for)g (sev)n(eral)71 4173 y(p)r(erio)r(dically)36 b(p)r(erturb)r(ed)h(second) e(order)h(equations)f(w)n(as)h(giv)n(en)g(b)n(y)g(Gelfreic)n(h)g(in)h ([Gel97b)o(])g(and)f(he)h(called)f(them)71 4273 y(Reference)26 b(Systems.)36 b(In)26 b([OSS03)o(])g(there)g(is)g(a)f(rigorous)f (analysis)h(of)g(the)i(inner)f(equation)f(for)h(the)g(Hamilton-Jacobi) 71 4372 y(equation)36 b(asso)r(ciated)f(to)i(a)f(p)r(endulum)i (equation)e(with)h(p)r(erturbation)g(term)f FA(H)2730 4384 y Fy(1)2768 4372 y FB(\()p FA(x;)14 b(t=")p FB(\))38 b(=)g(\(cos)13 b FA(x)25 b Fw(\000)f FB(1\))14 b(sin\()p FA(t=")p FB(\))71 4472 y(b)n(y)29 b(using)g(Resurgence)f(Theory)-7 b(.)41 b(The)29 b(only)g(result)g(whic)n(h)h(deals)e(with)i(the)g (inner)f(equation)g(asso)r(ciated)f(to)h(general)71 4571 y(p)r(olynomial)18 b(Hamiltonian)h(lik)n(e)g(\(1\))g(is)g([Bal06)o(],)i (where)e(this)g(analysis)f(is)h(done)g(using)g(functional)g(analysis)f (tec)n(hniques.)71 4671 y(Finally)-7 b(,)28 b(in)f([MSS10b],)h(the)g (authors)e(study)i(the)g(inner)f(equation)g(of)h(the)g(McMillan)g(Map.) 195 4771 y(Besides)36 b(the)h(w)n(ork)e(of)i(Lazutkin,)h(there)e(are)g (v)n(ery)f(few)i(w)n(orks)e(with)i(rigorous)d(pro)r(ofs)h(in)i(the)g (singular)e(case.)71 4870 y(In)f([Gel00)o(])g(there)g(is)f(a)h (detailed)f(sk)n(etc)n(h)g(of)h(the)g(pro)r(of)g(for)f(the)h(splitting) g(of)g(separatrices)d(for)j(the)g(equation)f(of)h(a)71 4970 y(p)r(endulum)j(with)g(p)r(erturbation)f FA(H)1232 4982 y Fy(1)1270 4970 y FB(\()p FA(x;)14 b(t=")p FB(\))37 b(=)h FA(x)14 b FB(sin\()p FA(t=")p FB(\))36 b(and)h FA(\021)2258 4940 y Fv(\003)2334 4970 y FB(=)g Fw(\000)p FB(2.)63 b(A)36 b(complete)h(rigorous)d(pro)r(of)i(whic)n(h)71 5070 y(also)e(co)n(v)n(er)f(some)h(\\under)g(the)i(limit")f(cases,)g (that)h(is)e FA(\021)39 b(<)c(\021)2106 5039 y Fv(\003)2179 5070 y FB(=)g Fw(\000)p FB(2)f(is)h(done)f(in)i([GOS10)o(].)59 b(Numerical)34 b(results)71 5169 y(ab)r(out)41 b(the)g(splitting)g(for) f(this)h(problem)f(can)h(b)r(e)g(found)g(in)g([BO93)n(,)g(Gel97b)o(].) 77 b(In)41 b([Oli06)o(])g(it)g(w)n(as)e(obtained)i(a)71 5269 y(rigorous)30 b(pro)r(of)i(for)g(the)g(p)r(endulum)i(with)f(p)r (erturbation)f FA(H)2046 5281 y Fy(1)2083 5269 y FB(\()p FA(x;)14 b(t=")p FB(\))31 b(=)g(\(cos)13 b FA(x)22 b Fw(\000)f FB(1\))14 b(sin\()p FA(t=")p FB(\),)34 b(for)d(whic)n(h)i FA(\021)3621 5239 y Fv(\003)3690 5269 y FB(=)e(0.)71 5368 y(T)-7 b(resc)n(hev)27 b(ga)n(v)n(e)g(in)i([T)-7 b(re97)o(])29 b(an)f(asymptotic)h(form)n(ula)e(for)i(the)g(splitting)g (in)g(the)g(case)f(of)h(a)f(p)r(endulum)i(with)f(certain)71 5468 y(p)r(erturbations,)i(for)g(whic)n(h)g FA(\021)1034 5438 y Fv(\003)1101 5468 y FB(=)d(0,)k(using)f(a)f(di\013eren)n(t)i (metho)r(d)f(called)g(Con)n(tin)n(uous)f(Av)n(eraging.)45 b(Concerning)30 b(2-)p Black 1940 5753 a(7)p Black eop end %%Page: 8 8 TeXDict begin 8 7 bop Black Black 71 272 a FB(dimensional)25 b(symplectic)h(maps,)g(the)h(study)f(of)g(the)g(splitting)g(for)g(the)g (H)n(\023)-39 b(enon)25 b(and)h(McMillan)g(maps)f(ha)n(v)n(e)g(recen)n (tly)71 372 y(b)r(een)20 b(completed)f(in)h([BG10)o(])f(and)g([MSS10a)o (])h(resp)r(ectiv)n(ely)-7 b(.)33 b(In)19 b(b)r(oth)h(cases,)g FA(\021)2547 342 y Fv(\003)2609 372 y FB(=)i(0.)34 b(Finally)-7 b(,)21 b(in)f([GG10)o(],)h(com)n(bining)71 471 y(n)n(umerical)27 b(and)g(analytical)g(tec)n(hniques,)g(the)h(authors)e(study)i(the)g (Hamiltonian-Hopf)g(bifurcation.)195 571 y(Another)41 b(w)n(ork)f(dealing)h(with)g(a)g(singular)f(case)h(is)g([Lom00)n(],)k (where)c(the)g(author)g(pro)n(v)n(ed)e(the)j(splitting)g(of)71 671 y(separatrices)25 b(for)h(a)h(certain)f(class)g(of)h(rev)n(ersible) e(systems)i(in)g Ft(R)2116 640 y Fy(4)2153 671 y FB(.)37 b(A)27 b(related)f(problem)h(ab)r(out)g(adiabatic)f(in)n(v)-5 b(arian)n(ts)71 770 y(in)28 b(the)g(harmonic)e(oscillator)g(using)h (matc)n(hing)g(tec)n(hniques)h(and)f(Resurgence)g(Theory)f(w)n(as)h (done)g(in)h([BSSV98)o(].)71 1045 y FC(2)135 b(Notation)46 b(and)f(main)g(results)71 1227 y FB(In)35 b(this)f(section)g(w)n(e)h (presen)n(t)e(the)i(main)g(problem)f(w)n(e)g(consider,)h(the)g(h)n(yp)r (otheses)f(w)n(e)g(assume)g(and)g(the)h(rigorous)71 1326 y(statemen)n(t)27 b(of)h(the)g(main)g(results.)71 1559 y Fq(2.1)112 b(Notation)37 b(and)i(h)m(yp)s(otheses)71 1712 y FB(W)-7 b(e)28 b(consider)e(Hamiltonian)i(systems)f(with)h (Hamiltonian)f(function)i(of)e(the)h(form)1171 1940 y FA(H)1261 1823 y Fz(\022)1322 1940 y FA(x;)14 b(y)s(;)1501 1884 y(t)p 1497 1921 39 4 v 1497 1997 a(")1546 1823 y Fz(\023)1630 1940 y FB(=)23 b FA(H)1787 1952 y Fy(0)1824 1940 y FB(\()p FA(x;)14 b(y)s FB(\))19 b(+)f FA(\026")2207 1905 y Fx(\021)2247 1940 y FA(H)2316 1952 y Fy(1)2367 1823 y Fz(\022)2428 1940 y FA(x;)c(y)s(;)2608 1884 y(t)p 2603 1921 V 2603 1997 a(")2652 1823 y Fz(\023)2727 1940 y FA(;)994 b FB(\(6\))71 2168 y(where)1565 2300 y FA(H)1634 2312 y Fy(0)1671 2300 y FB(\()p FA(x;)14 b(y)s FB(\))24 b(=)1985 2243 y FA(y)2029 2213 y Fy(2)p 1985 2281 81 4 v 2004 2357 a FB(2)2094 2300 y(+)18 b FA(V)h FB(\()p FA(x)p FB(\))1389 b(\(7\))71 2472 y(and)27 b FA(V)47 b FB(is)27 b(either)h(a)f(p)r(olynomial)g(or)g(a)g(trigonometric)f(p)r (olynomial.)36 b(In)28 b(the)g(\014rst)f(case)g(w)n(e)g(assume)g(that) 1413 2732 y FA(H)1482 2744 y Fy(1)1533 2732 y FB(\()q FA(x;)14 b(y)s(;)g(\034)9 b FB(\))24 b(=)1990 2628 y Fx(N)1960 2653 y Fz(X)1919 2831 y Fx(k)q Fy(+)p Fx(l)p Fy(=)p Fx(n)2134 2732 y FA(a)2178 2744 y Fx(k)q(l)2240 2732 y FB(\()p FA(\034)9 b FB(\))p FA(x)2396 2697 y Fx(k)2439 2732 y FA(y)2483 2697 y Fx(l)3744 2732 y FB(\(8\))71 2996 y(and)27 b(in)h(the)g(second)f(one)804 3178 y FA(H)873 3190 y Fy(1)924 3178 y FB(\()p FA(x;)14 b(y)s(;)g(\034)9 b FB(\))24 b(=)f FA(a)p FB(\()p FA(\034)9 b FB(\))p FA(x)20 b FB(+)1729 3099 y Fz(X)1613 3278 y Fx(k)q Fy(=)p Fv(\000)p Fx(N)s(;:::)o(;N)1658 3338 y(l)p Fy(=0)p Fx(;:::)o(;N)1979 3178 y FA(a)2023 3190 y Fx(k)q(l)2085 3178 y FB(\()p FA(\034)9 b FB(\))p FA(e)2233 3144 y Fx(k)q(ix)2336 3178 y FA(y)2380 3144 y Fx(l)2428 3178 y FB(=)2555 3099 y Fz(X)2516 3278 y Fx(i)p Fy(+)p Fx(j)s Fv(\025)p Fx(n)2727 3178 y Fz(b)-45 b FA(a)2772 3190 y Fx(ij)2830 3178 y FB(\()p FA(\034)9 b FB(\))p FA(x)2986 3144 y Fx(i)3016 3178 y FA(y)3060 3144 y Fx(j)3094 3178 y FA(;)627 b FB(\(9\))71 3509 y(where)29 b(the)i(second)e(equalit)n(y)h(de\014nes)g FA(n)g FB(and)f Fz(b)-45 b FA(a)1611 3521 y Fx(ij)1669 3509 y FB(.)45 b(Ev)n(en)29 b(if)i(in)f(the)h(second)e(case)g FA(H)2789 3521 y Fy(1)2857 3509 y FB(can)g(ha)n(v)n(e)g(terms)h(of)g (the)h(form)71 3609 y FA(a)p FB(\()p FA(\034)9 b FB(\))p FA(x)p FB(,)29 b(w)n(e)d(will)h(refer)f(to)h FA(H)961 3621 y Fy(1)1025 3609 y FB(as)f(a)h(trigonometric)e(p)r(olynomial.)36 b(In)27 b(b)r(oth)g(cases)f(w)n(e)g(will)h(refer)f(to)h FA(n)g FB(as)f(the)h(order)f(of)71 3708 y FA(H)140 3720 y Fy(1)177 3708 y FB(.)195 3808 y(The)i(equations)f(asso)r(ciated)f(to) i(the)g(Hamiltonian)f(\(6\))h(are)1257 3920 y Fz(8)1257 3995 y(>)1257 4020 y(>)1257 4045 y(<)1257 4194 y(>)1257 4219 y(>)1257 4244 y(:)1387 4041 y FB(_)-38 b FA(x)24 b FB(=)e FA(y)g FB(+)c FA(\026")1765 4006 y Fx(\021)1805 4041 y FA(@)1849 4053 y Fx(y)1889 4041 y FA(H)1958 4053 y Fy(1)2009 3923 y Fz(\022)2070 4041 y FA(x;)c(y)s(;)2249 3984 y(t)p 2245 4021 39 4 v 2245 4097 a(")2294 3923 y Fz(\023)1387 4240 y FB(_)-38 b FA(y)26 b FB(=)d Fw(\000)p FA(V)1658 4206 y Fv(0)1681 4240 y FB(\()p FA(x)p FB(\))d Fw(\000)e FA(\026")1984 4206 y Fx(\021)2024 4240 y FA(@)2068 4252 y Fx(x)2110 4240 y FA(H)2179 4252 y Fy(1)2230 4123 y Fz(\022)2291 4240 y FA(x;)c(y)s(;)2470 4184 y(t)p 2466 4221 V 2466 4297 a(")2515 4123 y Fz(\023)2590 4240 y FA(:)3703 4140 y FB(\(10\))71 4468 y(F)-7 b(rom)27 b(no)n(w)f(on,)i(w)n (e)f(call)g(unp)r(erturb)r(ed)g(system)h(to)f(the)g(system)h(de\014ned) f(b)n(y)g(the)h(Hamiltonian)f FA(H)3237 4480 y Fy(0)3302 4468 y FB(and)g(w)n(e)g(refer)g(to)71 4567 y FA(H)140 4579 y Fy(1)207 4567 y FB(as)i(the)h(p)r(erturbation.)42 b(Let)30 b(us)f(observ)n(e)f(that)i(the)g(term)g FA(a)p FB(\()p FA(\034)9 b FB(\))p FA(x)31 b FB(in)f(\(9\))g(corresp)r(onds)d (to)j(a)f(term)g(in)h(\(10\))g(whic)n(h)71 4667 y(only)d(dep)r(ends)h (on)f(time.)195 4767 y(W)-7 b(e)28 b(dev)n(ote)f(the)h(rest)f(of)h(the) g(section)f(to)h(state)f(the)h(h)n(yp)r(otheses)f(w)n(e)g(assume)g(on)g FA(H)7 b FB(.)71 4982 y Fp(2.1.1)94 b(Hyp)s(otheses)30 b(on)i(the)f(unp)s(erturb)s(ed)h(system)71 5136 y FB(W)-7 b(e)28 b(assume)f(the)h(follo)n(wing)e(h)n(yp)r(otheses)h(corresp)r (onding)f(to)h(the)h(unp)r(erturb)r(ed)g(system)p Black 71 5302 a Fp(HP1)p Black 41 w FA(H)369 5314 y Fy(0)406 5302 y FB(\()p FA(x;)14 b(y)s FB(\))24 b(=)f FA(y)754 5271 y Fy(2)791 5302 y FA(=)p FB(2)11 b(+)g FA(V)16 b FB(\()p FA(x)p FB(\),)26 b(where)d FA(V)42 b FB(is)24 b(either)f(a)h(p)r(olynomial)f(or)f(a)i(trigonometric)e(p)r(olynomial)h (and)g(satis\014es)278 5401 y(one)28 b(of)f(the)h(follo)n(wing)f (conditions)p Black 1940 5753 a(8)p Black eop end %%Page: 9 9 TeXDict begin 9 8 bop Black Black Black 278 272 a Fp(HP1.1)p Black 41 w FA(H)651 284 y Fy(0)716 272 y FB(has)27 b(a)g(h)n(yp)r(erb)r (olic)g(critical)g(p)r(oin)n(t)h(at)f(\(0)p FA(;)14 b FB(0\))28 b(with)g(eigen)n(v)-5 b(alues)26 b Fw(f)p FA(\025;)14 b Fw(\000)p FA(\025)p Fw(g)27 b FB(with)h FA(\025)c(>)e FB(0,)28 b(and)f(then)1471 508 y FA(V)18 b FB(\()p FA(x)p FB(\))24 b(=)f Fw(\000)1835 452 y FA(\025)1883 422 y Fy(2)p 1835 489 86 4 v 1857 565 a FB(2)1930 508 y FA(x)1977 474 y Fy(2)2033 508 y FB(+)18 b Fw(O)2199 441 y Fz(\000)2237 508 y FA(x)2284 474 y Fy(3)2321 441 y Fz(\001)2498 508 y FB(as)27 b FA(x)c Fw(!)g FB(0)p FA(:)p Black 278 735 a Fp(HP1.2)p Black 41 w FA(H)651 747 y Fy(0)716 735 y FB(has)k(a)g(parab)r(olic)f(critical)h(p)r(oin)n(t)h(at)g(\(0)p FA(;)14 b FB(0\))27 b(and)g(then)1436 918 y FA(V)19 b FB(\()p FA(x)p FB(\))24 b(=)f FA(v)1766 930 y Fx(m)1829 918 y FA(x)1876 883 y Fx(m)1958 918 y FB(+)18 b Fw(O)2123 850 y Fz(\000)2161 918 y FA(x)2208 883 y Fx(m)p Fy(+1)2356 850 y Fz(\001)2532 918 y FB(as)27 b FA(x)d Fw(!)f FB(0)p FA(;)827 b FB(\(11\))461 1100 y(for)27 b(certain)g FA(m)c Fw(2)g Ft(N)p FB(,)28 b FA(m)23 b Fw(\025)g FB(3,)k(whic)n(h)h(is)f (called)g(the)h(order)e(of)i FA(V)47 b FB(and)27 b FA(v)2733 1112 y Fx(m)2819 1100 y Fw(2)d Ft(R)p FB(.)p Black 71 1266 a Fp(HP2)p Black 41 w FB(The)30 b(critical)g(p)r(oin)n(t)g(\(0)p FA(;)14 b FB(0\))30 b(has)f(stable)h(and)g(unstable)g(in)n(v)-5 b(arian)n(t)29 b(manifolds)h(whic)n(h)g(coincide)g(along)f(a)g(sepa-) 278 1366 y(ratrix.)278 1499 y(W)-7 b(e)23 b(denote)g(b)n(y)f(\()p FA(q)858 1511 y Fy(0)896 1499 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1087 1511 y Fy(0)1124 1499 y FB(\()p FA(u)p FB(\)\))23 b(a)f(real-analytic)f(time)i(parametrization)e(of)h(the)h (separatrix)e(with)i(some)f(c)n(hosen)278 1598 y(\(\014xed\))33 b(initial)g(condition.)50 b(It)32 b(is)g(w)n(ell)g(kno)n(wn)g(\(see)g ([F)-7 b(on95)o(])32 b(for)g(the)g(h)n(yp)r(erb)r(olic)g(case)f(and)h ([BF04)o(])g(for)g(the)278 1698 y(parab)r(olic)g(one\))h(that)h(there)f (exists)g FA(\032)f(>)g FB(0)g(suc)n(h)h(that)h(the)f(parametrization)e (\()p FA(q)2927 1710 y Fy(0)2965 1698 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)3156 1710 y Fy(0)3193 1698 y FB(\()p FA(u)p FB(\)\))34 b(is)f(analytic)f(in)278 1798 y(the)c(complex)g (strip)f Fw(fj)p FB(Im)14 b FA(u)p Fw(j)22 b FA(<)h(\032)p Fw(g)p FB(.)278 1931 y(W)-7 b(e)34 b(assume)f(that)h(there)g(exists)f (a)g(real-analytic)f(time)i(parametrization)e(of)i(the)g(separatrix)d (\()p FA(q)3440 1943 y Fy(0)3478 1931 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)3669 1943 y Fy(0)3706 1931 y FB(\()p FA(u)p FB(\)\))278 2030 y(analytic)28 b(on)h Fw(fj)p FB(Im)13 b FA(u)p Fw(j)25 b FA(<)f(a)p Fw(g)k FB(suc)n(h)h(that)g(the)g (only)f(singularities)f(of)i(\()p FA(q)2513 2042 y Fy(0)2550 2030 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2741 2042 y Fy(0)2778 2030 y FB(\()p FA(u)p FB(\)\))30 b(in)f(the)g(lines)f Fw(f)p FB(Im)14 b FA(u)24 b FB(=)g Fw(\006)p FA(a)p Fw(g)278 2130 y FB(are)j Fw(\006)p FA(ia)p FB(.)278 2263 y(More)33 b(precisely)-7 b(,)33 b(Hyp)r(othesis)h Fp(HP2)f FB(implies)g(that)g (one)g(of)g(the)h(t)n(w)n(o)e(follo)n(wing)g(situations)h(is)g (satis\014ed)g(\(see)278 2362 y(the)28 b(remarks)e(in)i(Section)g (2.1.3\):)p Black 278 2528 a Fp(HP2.1)p Black 41 w FB(In)h(the)h(p)r (olynomial)e(case,)h(the)g(singularities)f Fw(\006)p FA(ia)g FB(of)h(the)h(homo)r(clinic)f(orbit)g(are)f(branc)n(hing)g(p)r (oin)n(ts)461 2628 y(\(or)34 b(p)r(oles\))h(of)g(the)g(same)f(order,)h (i.e.)58 b(there)35 b(exists)f(an)g(irreducible)g(rational)g(n)n(um)n (b)r(er)g FA(r)k FB(=)c FA(\013=\014)39 b(>)c FB(1)461 2728 y(\(indep)r(enden)n(t)29 b(of)e(the)h(singularit)n(y\))f(and)g FA(\027)i(>)22 b FB(0)27 b(suc)n(h)h(that)g(\()p FA(q)2469 2740 y Fy(0)2506 2728 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2697 2740 y Fy(0)2734 2728 y FB(\()p FA(u)p FB(\)\))28 b(can)f(b)r(e)h(expressed)f(as)1222 2953 y FA(q)1259 2965 y Fy(0)1296 2953 y FB(\()p FA(u)p FB(\))d(=)e Fw(\000)1864 2896 y FA(C)1923 2908 y Fv(\006)p 1594 2933 656 4 v 1594 3010 a FB(\()p FA(r)f Fw(\000)d FB(1\)\()p FA(u)g Fw(\007)g FA(ia)p FB(\))2127 2986 y Fx(r)r Fv(\000)p Fy(1)2273 2860 y Fz(\020)2322 2953 y FB(1)g(+)g Fw(O)2547 2860 y Fz(\020)2597 2953 y FB(\()p FA(u)g Fw(\007)g FA(ia)p FB(\))2883 2918 y Fy(1)p Fx(=\014)2995 2860 y Fz(\021\021)1217 3176 y FA(p)1259 3188 y Fy(0)1296 3176 y FB(\()p FA(u)p FB(\))24 b(=)1633 3120 y FA(C)1692 3132 y Fv(\006)p 1529 3157 323 4 v 1529 3233 a FB(\()p FA(u)18 b Fw(\007)g FA(ia)p FB(\))1815 3209 y Fx(r)1876 3084 y Fz(\020)1925 3176 y FB(1)g(+)g Fw(O)2151 3084 y Fz(\020)2200 3176 y FB(\()p FA(u)g Fw(\007)g FA(ia)p FB(\))2486 3142 y Fy(1)p Fx(=\014)2598 3084 y Fz(\021\021)3703 3068 y FB(\(12\))461 3408 y(for)39 b FA(u)k Fw(2)h Ft(C)c FB(and)f(either)h Fw(j)p FA(u)26 b Fw(\000)g FA(ia)p Fw(j)43 b FA(<)g(\027)i FB(and)40 b(arg\()p FA(u)26 b Fw(\000)g FA(ia)p FB(\))44 b Fw(2)f FB(\()p Fw(\000)p FB(3)p FA(\031)s(=)p FB(2)p FA(;)14 b(\031)s(=)p FB(2\))38 b(or)h Fw(j)p FA(u)26 b FB(+)h FA(ia)p Fw(j)43 b FA(<)f(\027)k FB(and)461 3507 y(arg\()p FA(u)20 b FB(+)g FA(ia)p FB(\))29 b Fw(2)g FB(\()p Fw(\000)p FA(\031)s(=)p FB(2)p FA(;)14 b FB(3)p FA(\031)s(=)p FB(2\))29 b(resp)r(ectiv)n(ely)-7 b(.)46 b(Let)31 b(us)g(p)r(oin)n(t)g(out)g(that)g(the)h(real-analytic)d(c)n (haracter)g(of)461 3607 y(\()p FA(q)530 3619 y Fy(0)568 3607 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)759 3619 y Fy(0)796 3607 y FB(\()p FA(u)p FB(\)\))28 b(implies)g(that)f FA(C)1488 3619 y Fv(\000)1568 3607 y FB(=)p 1656 3540 66 4 v 23 w FA(C)1721 3619 y Fy(+)1776 3607 y FB(.)p Black 278 3740 a Fp(HP2.2)p Black 41 w FB(In)32 b(the)h(trigonometric)e (case,)h FA(q)1601 3752 y Fy(0)1638 3740 y FB(\()p FA(u)p FB(\))h(has)f(logarithmic)f(singularities)g(at)h Fw(\006)p FA(ia)f FB(of)h(the)h(form)f FA(q)3606 3752 y Fy(0)3643 3740 y FB(\()p FA(u)p FB(\))f Fw(\030)461 3839 y FB(ln\()p FA(u)20 b Fw(\007)f FA(ia)p FB(\))29 b(\(where)g(w)n(e)f(tak)n(e)h (di\013eren)n(t)g(branc)n(hes)f(of)h(the)h(logarithm)e(whether)h(w)n(e) f(are)h(close)f(to)h(+)p FA(ia)f FB(or)461 3939 y Fw(\000)p FA(ia)p FB(:)35 b(w)n(e)24 b(tak)n(e)h(arg)o(\()p FA(u)13 b Fw(\000)g FA(ia)p FB(\))23 b Fw(2)g FB(\()p Fw(\000)p FB(3)p FA(\031)s(=)p FB(2)p FA(;)14 b(\031)s(=)p FB(2\))24 b(and)h(arg)o(\()p FA(u)13 b FB(+)g FA(ia)p FB(\))23 b Fw(2)g FB(\()p Fw(\000)p FA(\031)s(=)p FB(2)p FA(;)14 b FB(3)p FA(\031)s(=)p FB(2\))24 b(resp)r(ectiv)n(ely\).)35 b(In)25 b(this)461 4039 y(case,)i(one)g(can)g(see)g(that)h(there)g (exists)f FA(M)32 b Fw(2)23 b Ft(N)28 b FB(suc)n(h)f(that,)h(if)g FA(u)23 b Fw(2)g Ft(C)p FB(,)28 b Fw(j)p FA(u)18 b Fw(\007)g FA(ia)p Fw(j)k FA(<)h(\027)5 b FB(,)1267 4291 y(cos\()p FA(q)1448 4303 y Fy(0)1485 4291 y FB(\()p FA(u)p FB(\)\))24 b(=)1921 4211 y Fz(b)1904 4232 y FA(C)1969 4202 y Fy(1)1963 4252 y Fv(\006)p 1750 4272 424 4 v 1750 4350 a FB(\()p FA(u)18 b Fw(\007)g FA(ia)p FB(\))2036 4326 y Fy(2)p Fx(=)l(M)2197 4199 y Fz(\020)2247 4291 y FB(1)g(+)g Fw(O)2472 4199 y Fz(\020)2522 4291 y FB(\()p FA(u)g Fw(\007)g FA(ia)p FB(\))2808 4257 y Fy(2)p Fx(=)l(M)2945 4199 y Fz(\021\021)1277 4545 y FB(sin\()p FA(q)1448 4557 y Fy(0)1485 4545 y FB(\()p FA(u)p FB(\)\))24 b(=)1921 4464 y Fz(b)1904 4485 y FA(C)1969 4455 y Fy(2)1963 4505 y Fv(\006)p 1750 4526 V 1750 4603 a FB(\()p FA(u)18 b Fw(\007)g FA(ia)p FB(\))2036 4579 y Fy(2)p Fx(=)l(M)2197 4453 y Fz(\020)2247 4545 y FB(1)g(+)g Fw(O)2472 4453 y Fz(\020)2522 4545 y FB(\()p FA(u)g Fw(\007)g FA(ia)p FB(\))2808 4510 y Fy(2)p Fx(=)l(M)2945 4453 y Fz(\021\021)1438 4770 y FA(p)1480 4782 y Fy(0)1517 4770 y FB(\()p FA(u)p FB(\))24 b(=)1836 4714 y FA(C)1895 4726 y Fv(\006)p 1750 4751 287 4 v 1750 4827 a FB(\()p FA(u)18 b Fw(\007)g FA(ia)p FB(\))2060 4678 y Fz(\020)2110 4770 y FB(1)g(+)g Fw(O)2335 4678 y Fz(\020)2384 4770 y FB(\()p FA(u)h Fw(\007)f FA(ia)p FB(\))2671 4736 y Fy(2)p Fx(=)l(M)2808 4678 y Fz(\021)o(\021)3703 4520 y FB(\(13\))461 5001 y(with)34 b(arg)o(\()p FA(u)22 b Fw(\000)g FA(ia)p FB(\))32 b Fw(2)g FB(\()p Fw(\000)p FB(3)p FA(\031)s(=)p FB(2)p FA(;)14 b(\031)s(=)p FB(2\))31 b(and)i(arg\()p FA(u)22 b FB(+)f FA(ia)p FB(\))32 b Fw(2)h FB(\()p Fw(\000)p FA(\031)s(=)p FB(2)p FA(;)14 b FB(3)p FA(\031)s(=)p FB(2\))31 b(if)j(w)n(e)e(are)g(dealing)h(with)g(the)461 5101 y(singularit)n(y)26 b(+)p FA(ia)h FB(or)g Fw(\000)p FA(ia)f FB(resp)r(ectiv)n(ely)-7 b(.)36 b(W)-7 b(e)28 b(also)f(ha)n(v)n(e)f(that)i FA(C)2521 5113 y Fy(+)2600 5101 y FB(=)p 2687 5034 116 4 v 22 w FA(C)2746 5113 y Fv(\000)2826 5101 y FB(=)23 b Fw(\006)p FA(i)p FB(2)p FA(=)-5 b(M)9 b FB(.)461 5217 y(F)-7 b(or)27 b(con)n(v)n(enience,)f(in)i(the)g(trigonometric)e(case,)h(w)n(e)g(tak)n (e)g(the)h(con)n(v)n(en)n(tion)e FA(r)g FB(=)d(1)k(and)g FA(\014)h FB(=)22 b FA(M)9 b FB(.)p Black 1940 5753 a(9)p Black eop end %%Page: 10 10 TeXDict begin 10 9 bop Black Black 71 272 a Fp(2.1.2)94 b(Hyp)s(otheses)30 b(on)i(the)f(p)s(erturbation)p Black 71 425 a(HP3)p Black 41 w FB(The)g(function)g FA(H)871 437 y Fy(1)908 425 y FB(\()p FA(x;)14 b(y)s(;)g(\034)9 b FB(\))32 b(is)e(real-analytic)e(in)j(\()p FA(x;)14 b(y)s(;)g(\034)9 b FB(\))29 b Fw(2)f Ft(C)2326 395 y Fy(2)2384 425 y Fw(\002)20 b Ft(T)31 b FB(and)f(2)p FA(\031)s FB(-p)r(erio)r(dic)g(in)g FA(\034)9 b FB(.)46 b(F)-7 b(urthermore,)278 525 y(either)27 b(it)g(is)g(a)f(p)r(olynomial)g(of)h (the)g(form)g(\(8\))g(if)g FA(V)19 b FB(\()p FA(x)p FB(\))28 b(is)e(a)h(p)r(olynomial)f(or)g(it)h(is)g(a)f(trigonometric)f(p)r (olynomial)278 625 y(of)j(the)g(form)f(\(9\))h(if)g FA(V)19 b FB(\()p FA(x)p FB(\))29 b(is)e(a)g(trigonometric)f(p)r(olynomial.)37 b(Moreo)n(v)n(er,)24 b(it)29 b(has)e(zero)f(mean)1647 754 y Fz(Z)1730 774 y Fy(2)p Fx(\031)1693 942 y Fy(0)1822 867 y FA(H)1891 879 y Fy(1)1928 867 y FB(\()p FA(x;)14 b(y)s(;)g(\034)9 b FB(\))14 b FA(d\034)34 b FB(=)23 b(0)p FA(:)p Black 71 1129 a Fp(HP4)p Black 41 w FB(Let)28 b(us)f(consider)g(the)h(order)e(of)i FA(H)1403 1141 y Fy(1)1440 1129 y FB(,)g FA(n)f FB(giv)n(en)g(in)h(\(8\))g(or)e(\(9\).) 37 b(W)-7 b(e)28 b(ask)f FA(H)2641 1141 y Fy(1)2706 1129 y FB(to)h(satisfy:)p Black 278 1295 a Fp(HP4.1)p Black 41 w FB(In)g(the)g(h)n(yp)r(erb)r(olic)f(case)g(\()p FA(H)1512 1307 y Fy(0)1577 1295 y FB(satis\014es)g Fp(HP1.1)p FB(\),)g FA(n)22 b Fw(\025)h FB(1.)p Black 278 1428 a Fp(HP4.2)p Black 41 w FB(In)28 b(the)g(parab)r(olic)e(case)h(\()p FA(H)1468 1440 y Fy(0)1533 1428 y FB(satis\014es)g Fp(HP1.2)p FB(\),)g(2)p FA(n)18 b Fw(\000)g FB(2)k Fw(\025)h FA(m)p FB(.)p Black 71 1611 a Fp(Remark)f(2.1.)p Black 32 w Fs(L)l(et)g(us)f(p)l(oint)h(out)f(that,)j(in)e(fact,)j Fo(HP4.1)d Fs(do)l(es)g(not)g(add)h(any)f(extr)l(a)f(hyp)l(othesis)j (on)e(the)g(Hamiltonian,)71 1710 y(sinc)l(e)30 b(it)g(c)l(an)f(always)i (b)l(e)f(taken)g(with)g FA(n)23 b Fw(\025)g FB(1)29 b Fs(\(the)h(c)l(onstant)f(terms)g(in)h FB(\()p FA(x;)14 b(y)s FB(\))30 b Fs(do)h(not)e(play)j(any)e(r)l(ole\).)71 1876 y FB(Let)c(us)g(consider)f(the)i(function)g FA(H)1180 1888 y Fy(1)1217 1876 y FB(\()p FA(q)1286 1888 y Fy(0)1324 1876 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1515 1888 y Fy(0)1552 1876 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))27 b(that)f(is:)36 b FA(H)2167 1888 y Fy(1)2231 1876 y FB(ev)-5 b(aluated)26 b(on)g(the)g(separatrix.)35 b(Then,)26 b(w)n(e)g(de\014ne)71 1976 y FA(`)d FB(to)f(b)r(e)i(the)f (order)f(of)h(the)g(branc)n(hing)f(p)r(oin)n(ts)h Fw(\006)p FA(ia)p FB(,)g(namely)-7 b(,)24 b(the)g(maxim)n(um)e(of)h(the)h(orders) d(of)i(the)h(branc)n(hing)e(p)r(oin)n(ts)71 2075 y(of)30 b(the)h(monomials)f(of)g FA(H)899 2087 y Fy(1)936 2075 y FB(.)46 b(This)31 b(parameter)e(w)n(as)g(already)g(de\014ned)i(in)g ([DS97)o(,)g(BF04)o(].)46 b(Let)30 b(us)h(p)r(oin)n(t)g(out)f(that)h FA(`)71 2175 y FB(can)c(b)r(e)h(simply)g(de\014ned)g(as)753 2352 y FA(`)22 b FB(=)102 b(max)898 2407 y Fx(n)p Fv(\024)p Fx(k)q Fy(+)p Fx(l)p Fv(\024)p Fx(N)1225 2352 y Fw(f)o FA(k)s FB(\()p FA(r)21 b Fw(\000)d FB(1\))h(+)f FA(l)r(r)r FB(;)c FA(a)1808 2364 y Fx(k)q(l)1870 2352 y FB(\()p FA(\034)9 b FB(\))24 b Fw(6\021)f FB(0)p Fw(g)179 b FB(\(p)r(olynomial) 27 b(case\))753 2529 y FA(`)22 b FB(=)170 b(max)898 2586 y Fv(j)p Fx(k)q Fv(j\024)p Fx(N)s(;)27 b Fy(0)p Fv(\024)p Fx(l)p Fv(\024)p Fx(N)1360 2529 y Fw(f)o FB(2)p Fw(j)p FA(k)s Fw(j)p FA(=)-5 b(M)26 b FB(+)19 b FA(l)r FB(;)14 b FA(a)1871 2541 y Fx(k)q(l)1932 2529 y FB(\()p FA(\034)9 b FB(\))24 b Fw(6\021)f FB(0)p Fw(g)179 b FB(\(trigonometric)26 b(case\).)3703 2467 y(\(14\))71 2762 y(Note)34 b(that)h(in)g(the)g (trigonometric)e(case,)j(if)f FA(H)1603 2774 y Fy(1)1640 2762 y FB(\()p FA(x;)14 b(y)s(;)g(\034)9 b FB(\))36 b(=)e FA(a)p FB(\()p FA(\034)9 b FB(\))p FA(x)p FB(,)38 b(then)d FA(H)2575 2774 y Fy(1)2612 2762 y FB(\()p FA(q)2681 2774 y Fy(0)2719 2762 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2910 2774 y Fy(0)2947 2762 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))36 b(has)e(a)g(logarithmic)71 2862 y(singularit)n(y)26 b(\(see)i(Hyp)r(othesis)f Fp(HP2.2)p FB(\).)36 b(In)28 b(this)g(case)f(w)n(e)g(mak)n(e)g(the)h(con)n(v)n(en)n(tion)e FA(`)d FB(=)f(0.)p Black 71 3028 a Fp(HP5)p Black 41 w FB(W)-7 b(e)28 b(assume)f FA(\021)f Fw(\025)d FA(\021)929 2998 y Fv(\003)990 3028 y FB(=)g(max)o Fw(f)p FB(0)p FA(;)14 b(`)j Fw(\000)h FB(2)p FA(r)r Fw(g)p FB(.)195 3194 y(In)29 b(the)g(case)e FA(`)d(<)g FB(2)p FA(r)r FB(,)29 b(w)n(e)f(still)g(ha)n(v)n(e)f(to)i(assume)e(a)h(generic)f (extra)h(h)n(yp)r(othesis.)38 b(By)28 b(the)h(de\014nition)g(of)f FA(`)g FB(in)h(\(14\))o(,)71 3294 y(w)n(e)g(already)g(kno)n(w)g(that)h FA(H)961 3306 y Fy(1)998 3294 y FB(\()p FA(q)1067 3306 y Fy(0)1105 3294 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1296 3306 y Fy(0)1333 3294 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))31 b(has)e(branc)n(hing)g(p)r(oin)n(ts)h(of)f(degree)g FA(`)h FB(at)f FA(u)e FB(=)f Fw(\006)p FA(ia)p FB(.)43 b(Therefore,)29 b(if)i(w)n(e)71 3393 y(consider)26 b(the)i(F)-7 b(ourier)27 b(expansion)1038 3588 y FA(H)1107 3600 y Fy(1)1145 3588 y FB(\()p FA(q)1214 3600 y Fy(0)1251 3588 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1442 3600 y Fy(0)1479 3588 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))24 b(=)1885 3509 y Fz(X)1817 3691 y Fx(k)q Fv(2)p Fn(Z)p Fv(nf)p Fy(0)p Fv(g)2086 3588 y FA(H)2162 3545 y Fy([)p Fx(k)q Fy(])2155 3610 y(1)2240 3588 y FB(\()p FA(q)2309 3600 y Fy(0)2347 3588 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2538 3600 y Fy(0)2575 3588 y FB(\()p FA(u)p FB(\)\))p FA(e)2758 3554 y Fx(ik)q(\034)2860 3588 y FA(;)71 3892 y FB(all)34 b(the)i(F)-7 b(ourier)33 b(co)r(e\016cien)n(ts)i FA(H)1142 3849 y Fy([)p Fx(k)q Fy(])1135 3914 y(1)1220 3892 y FB(\()p FA(q)1289 3904 y Fy(0)1327 3892 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1518 3904 y Fy(0)1555 3892 y FB(\()p FA(u)p FB(\)\))35 b(ha)n(v)n(e)f(at)g FA(u)h FB(=)g Fw(\006)p FA(ia)f FB(a)g(branc)n(hing)g(p)r(oin)n(t)h(of)g (order)e(less)h(than)h(or)71 3992 y(equal)29 b(to)g FA(`)p FB(.)42 b(Next)30 b(h)n(yp)r(othesis,)g(whic)n(h)f(will)h(b)r(e)g(only) f(used)g(in)h(the)g(case)f FA(`)19 b Fw(\000)g FB(2)p FA(r)29 b(<)d FB(0,)j(requires)f(a)h(sligh)n(tly)g(stronger)71 4103 y(condition)j(on)g(the)g(F)-7 b(ourier)31 b(co)r(e\016cien)n(ts)h FA(H)1500 4060 y Fy([)p Fv(\006)p Fy(1])1493 4125 y(1)1627 4103 y FB(\()p FA(q)1696 4115 y Fy(0)1733 4103 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1924 4115 y Fy(0)1961 4103 y FB(\()p FA(u)p FB(\)\),)34 b(namely)-7 b(,)33 b(that)f(these)h(co)r (e\016cien)n(ts)e(also)g(ha)n(v)n(e)g(these)71 4202 y(branc)n(hings)26 b(p)r(oin)n(ts)i FA(u)22 b FB(=)h Fw(\006)p FA(ia)j FB(of)i(order)e FA(`)i FB(and)f(not)h(less.)p Black 71 4380 a Fp(HP6)p Black 41 w FB(If)33 b FA(`)d(<)g FB(2)p FA(r)r FB(,)j(the)g(F)-7 b(ourier)31 b(co)r(e\016cien)n(ts)h FA(H)1626 4337 y Fy([)p Fv(\006)p Fy(1])1619 4402 y(1)1752 4380 y FB(\()p FA(q)1821 4392 y Fy(0)1859 4380 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2050 4392 y Fy(0)2087 4380 y FB(\()p FA(u)p FB(\)\))32 b(ha)n(v)n(e)f(branc)n(hing)g(p)r(oin)n(ts)h(of)g(order)f Fs(exactly)40 b FA(`)32 b FB(at)278 4480 y FA(u)23 b FB(=)g Fw(\006)p FA(ia)p FB(.)36 b(That)27 b(is)h(w)n(e)f(are)g (assuming)f(that)1334 4674 y FA(A)d FB(=)48 b(lim)1507 4726 y Fx(u)p Fv(!)p Fx(ia)1672 4674 y FB(\()p FA(u)18 b Fw(\000)g FA(ia)p FB(\))1958 4639 y Fx(`)1990 4674 y FA(H)2066 4631 y Fy([1])2059 4696 y(1)2141 4674 y FB(\()p FA(q)2210 4686 y Fy(0)2247 4674 y FB(\()p FA(u)p FB(\))p FA(;)c(p)2438 4686 y Fy(0)2475 4674 y FB(\()p FA(u)p FB(\)\))24 b Fw(6)p FB(=)e(0)p FA(:)p Black 71 4917 a Fp(Remark)31 b(2.2.)p Black 40 w Fs(Hyp)l(othesis)g Fo(HP6)f Fs(is)g(generic)g(b)l(e)l(c)l(ause)f(it)h(is)g(e)l(quivalent)g(to)f (assume)g(that)h(some)g(c)l(o)l(e\016cient)g(in)f(the)71 5028 y(L)l(aur)l(ent)g(exp)l(ansions)h(of)g FA(H)968 4985 y Fy([)p Fv(\006)p Fy(1])961 5050 y(1)1095 5028 y FB(\()p FA(q)1164 5040 y Fy(0)1201 5028 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1392 5040 y Fy(0)1429 5028 y FB(\()p FA(u)p FB(\)\))31 b Fs(at)e(the)h(p)l(oints)g FA(u)23 b FB(=)f Fw(\006)p FA(ia)29 b Fs(is)h(non-zer)l(o.)p Black 1919 5753 a FB(10)p Black eop end %%Page: 11 11 TeXDict begin 11 10 bop Black Black 71 272 a Fp(2.1.3)94 b(Some)31 b(remarks)h(ab)s(out)g(the)f(h)m(yp)s(otheses)p Black 195 425 a Fw(\017)p Black 41 w FB(Let)22 b(us)f(p)r(oin)n(t)h (out)f(that)h(the)g(time)g(parametrization)d(of)i(the)h(separatrix)e (has)g(alw)n(a)n(ys)g(singularities)g(for)h(complex)278 525 y(time)28 b(\(see)f([F)-7 b(on95)o(])27 b(for)g(the)h(h)n(yp)r(erb) r(olic)e(case)h(and)g([BF04)o(])g(for)g(the)g(parab)r(olic)f(one\).)37 b(The)27 b(real)f(restriction)g(in)278 625 y Fp(HP2)i FB(is)f(that)h(there)g(exists)f(only)g(one)g(singularit)n(y)f(in)i(the) g(lines)g Fw(f)p FB(Im)13 b FA(u)23 b FB(=)g Fw(\006)p FA(a)p Fw(g)p FB(.)p Black 195 789 a Fw(\017)p Black 41 w FB(The)40 b(conditions)g(satis\014ed)f(in)h Fp(HP2.1)f FB(and)h Fp(HP2.2)f FB(are)f(consequence)h(of)46 b Fp(HP2)p FB(.)74 b(Indeed,)43 b(let)d FA(u)3606 759 y Fv(\003)3684 789 y FB(b)r(e)g(a)278 888 y(singularit)n(y)27 b(of)g(\()p FA(q)853 900 y Fy(0)891 888 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1082 900 y Fy(0)1119 888 y FB(\()p FA(u)p FB(\)\).)37 b(W)-7 b(e)28 b(ha)n(v)n(e)f(that:)p Black 372 1052 a Fp({)p Black 41 w FB(If)33 b FA(V)51 b FB(is)33 b(a)f(p)r(olynomial,)h(let)g FA(M)41 b FB(b)r(e)33 b(its)g(degree.)50 b(Then)33 b FA(u)2330 1022 y Fv(\003)2400 1052 y FB(is)g(a)f(branc)n (hing)f(p)r(oin)n(ts)i(\(or)f(p)r(ole\))g(of)h(order)461 1152 y(2)p FA(=)p FB(\()p FA(M)22 b Fw(\000)14 b FB(2\).)35 b(That)25 b(is,)h(if)f FA(u)g FB(b)r(elongs)g(to)g(a)g(neigh)n(b)r (orho)r(o)r(d)f(of)h FA(u)2468 1122 y Fv(\003)2505 1152 y FB(,)h(then)g(\()p FA(q)2810 1164 y Fy(0)2848 1152 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)3039 1164 y Fy(0)3076 1152 y FB(\()p FA(u)p FB(\)\))25 b(can)g(b)r(e)h(expressed)461 1252 y(as)1151 1457 y FA(q)1188 1469 y Fy(0)1225 1457 y FB(\()p FA(u)p FB(\))d(=)g Fw(\000)1651 1400 y FA(C)6 b FB(\()p FA(M)27 b Fw(\000)18 b FB(2\))p 1523 1437 619 4 v 1523 1515 a(2\()p FA(u)f Fw(\000)h FA(u)1793 1491 y Fv(\003)1831 1515 y FB(\))1863 1491 y Fy(2)p Fx(=)p Fy(\()p Fx(M)6 b Fv(\000)p Fy(2\))2165 1364 y Fz(\020)2214 1457 y FB(1)18 b(+)g Fw(O)2440 1364 y Fz(\020)2489 1457 y FB(\()p FA(u)g Fw(\000)g FA(u)2718 1422 y Fv(\003)2756 1457 y FB(\))2788 1422 y Fy(2)p Fx(=)p Fy(\()p Fx(M)6 b Fv(\000)p Fy(2\))3066 1364 y Fz(\021\021)1146 1682 y FA(p)1188 1694 y Fy(0)1225 1682 y FB(\()p FA(u)p FB(\))23 b(=)1730 1626 y FA(C)p 1458 1663 610 4 v 1458 1740 a FB(\()p FA(u)18 b Fw(\000)g FA(u)1687 1716 y Fv(\003)1725 1740 y FB(\))1757 1716 y Fx(M)s(=)p Fy(\()p Fx(M)6 b Fv(\000)p Fy(2\))2092 1590 y Fz(\020)2141 1682 y FB(1)18 b(+)g Fw(O)2366 1590 y Fz(\020)2416 1682 y FB(\()p FA(u)g Fw(\000)g FA(u)2645 1648 y Fv(\003)2683 1682 y FB(\))2715 1648 y Fy(2)p Fx(=)p Fy(\()p Fx(M)6 b Fv(\000)p Fy(2\))2993 1590 y Fz(\021)o(\021)461 1910 y FB(for)27 b(some)g(constan)n(t)g FA(C)i Fw(6)p FB(=)23 b(0.)36 b(This)28 b(fact)g(is)f(pro)n(v)n(ed)f (in)i([BF04)o(].)p Black 372 2041 a Fp({)p Black 41 w FB(If)34 b FA(V)53 b FB(is)33 b(a)h(trigonometric)e(p)r(olynomial,)i (let)h(us)e(call)g FA(M)43 b FB(to)33 b(its)h(degree.)54 b(Then,)36 b(for)d FA(u)g FB(b)r(elonging)g(to)h(a)461 2140 y(neigh)n(b)r(orho)r(o)r(d)26 b(of)i FA(u)1125 2110 y Fv(\003)1163 2140 y FB(,)f(\()p FA(q)1282 2152 y Fy(0)1320 2140 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1511 2152 y Fy(0)1548 2140 y FB(\()p FA(u)p FB(\)\))28 b(are)f(of)g(the)h(form)1315 2325 y FA(q)1352 2337 y Fy(0)1390 2325 y FB(\()p FA(u)p FB(\))23 b(=)g FA(C)d FB(log)1813 2258 y Fz(\000)1869 2325 y Fw(\000)e FA(i)p FB(\()p FA(u)g Fw(\000)g FA(u)2210 2291 y Fv(\003)2248 2325 y FB(\))2280 2258 y Fz(\001)2337 2325 y FB(+)g Fw(O)2488 2258 y Fz(\000)2526 2325 y FB(\()p FA(u)g Fw(\000)g FA(u)2755 2291 y Fv(\003)2793 2325 y FB(\))2825 2291 y Fy(2)p Fx(=)l(M)2963 2258 y Fz(\001)1311 2501 y FA(p)1353 2513 y Fy(0)1390 2501 y FB(\()p FA(u)p FB(\))23 b(=)1740 2445 y FA(C)p 1623 2482 300 4 v 1623 2558 a FB(\()p FA(u)18 b Fw(\000)g FA(u)1852 2534 y Fv(\003)1890 2558 y FB(\))1950 2501 y(+)g Fw(O)2101 2434 y Fz(\000)2140 2501 y FB(\()p FA(u)g Fw(\000)g FA(u)2369 2467 y Fv(\003)2407 2501 y FB(\))2439 2467 y Fy(2)p Fx(=)l(M)2576 2434 y Fz(\001)461 2728 y FB(with)34 b(the)f(constan)n(t)f FA(C)39 b FB(=)31 b Fw(\006)p FA(i)p FB(2)p FA(=)-5 b(M)41 b FB(dep)r(ending)33 b(on)g(Im)14 b FA(q)2308 2740 y Fy(0)2345 2728 y FB(\()p FA(u)p FB(\))32 b Fw(!)g(\0071)h FB(resp)r(ectiv)n(ely) -7 b(.)52 b(Indeed,)35 b(\014rst)e(w)n(e)461 2827 y(note)23 b(that,)i(due)f(to)f(the)h(fact)g(that)f(Re)14 b FA(q)1714 2839 y Fy(0)1751 2827 y FB(\()p FA(u)p FB(\))24 b Fw(2)f FB([0)p FA(;)14 b FB(2)p FA(\031)s FB(],)24 b(the)g(condition)f Fw(j)p FA(q)2788 2839 y Fy(0)2825 2827 y FB(\()p FA(u)p FB(\))p Fw(j)h(!)f FB(+)p Fw(1)g FB(as)g FA(u)f Fw(!)h FA(u)3583 2797 y Fv(\003)3644 2827 y FB(forces)461 2927 y(to)29 b Fw(j)p FB(Im)14 b FA(q)737 2939 y Fy(0)774 2927 y FB(\()p FA(u)p FB(\))p Fw(j)26 b(!)f FB(+)p Fw(1)j FB(as)g FA(u)h FB(go)r(es)f(to)h FA(u)1734 2897 y Fv(\003)1771 2927 y FB(.)41 b(Assume)29 b(that)g(Im)14 b FA(q)2475 2939 y Fy(0)2513 2927 y FB(\()p FA(u)p FB(\))25 b Fw(!)g(\0001)k FB(as)f FA(u)c Fw(!)i FA(u)3267 2897 y Fv(\003)3304 2927 y FB(.)41 b(W)-7 b(e)29 b(note)g(that)461 3027 y(in)k(this)f(case,)h (since)f FA(q)1179 3039 y Fy(0)1216 3027 y FB(\()p FA(u)p FB(\))g(is)g(a)g(real)f(analytic)h(function,)i(then)p 2557 2966 86 4 v 33 w FA(u)2605 3003 y Fv(\003)2675 3027 y FB(is)e(also)f(a)h(singularit)n(y)e(of)i FA(q)3559 3039 y Fy(0)3629 3027 y FB(and)g(it)461 3126 y(satis\014es)e(Im)14 b FA(q)920 3138 y Fy(0)957 3126 y FB(\()p FA(u)p FB(\))29 b Fw(!)f FB(+)p Fw(1)i FB(as)g FA(u)e Fw(!)p 1680 3065 V 29 w FA(u)1728 3102 y Fv(\003)1765 3126 y FB(.)47 b(W)-7 b(e)31 b(p)r(erform)f(the)h(c)n(hange)f(of)g(v)-5 b(ariables)30 b FA(x)e FB(=)g FA(i)14 b FB(log)g FA(w)33 b FB(and)e(w)n(e)461 3226 y(emphasize)c(that,)h(if)g(Im)14 b FA(x)24 b Fw(!)f(\0001)p FB(,)k(then)h FA(w)e Fw(!)d FB(0.)37 b(F)-7 b(rom)27 b(the)h(fact)f(that)1860 3386 y FA(dx)p 1859 3423 91 4 v 1859 3499 a(du)1983 3442 y FB(=)2071 3367 y Fz(p)p 2154 3367 285 4 v 75 x Fw(\000)p FB(2)p FA(V)18 b FB(\()p FA(x)p FB(\))q FA(;)461 3643 y FB(w)n(e)27 b(obtain)h(that)1682 3709 y FA(du)p 1675 3746 105 4 v 1675 3822 a(dw)1813 3766 y FB(=)22 b FA(iw)1990 3731 y Fx(M)s(=)p Fy(2)p Fv(\000)p Fy(1)2213 3766 y FB(\()p FA(c)2281 3778 y Fy(0)2337 3766 y FB(+)c Fw(O)r FB(\()p FA(w)r FB(\)\))461 3935 y(for)27 b(some)h(constan)n(t)f FA(c)1168 3947 y Fy(0)1205 3935 y FB(.)38 b(Henceforth,)27 b(in)n(tegrating)g(b)r(oth)h(sides)g (of)f(the)i(previous)d(di\013eren)n(tial)i(equation,)461 4035 y(w)n(e)d(obtain)h FA(u)14 b Fw(\000)g FA(u)1027 4005 y Fv(\003)1088 4035 y FB(=)23 b FA(iw)1266 4005 y Fx(M)s(=)p Fy(2)1403 4035 y FB(\()p FA(c)1471 4047 y Fy(1)1523 4035 y FB(+)14 b Fw(O)r FB(\()p FA(w)r FB(\)\),)28 b(for)e(some)f(constan)n(t)g FA(c)2579 4047 y Fy(1)2616 4035 y FB(,)h(whic)n(h)g(implies)g(that)g FA(w)f FB(=)3531 3968 y Fz(\000)3584 4035 y Fw(\000)14 b FA(i)p FB(\()p FA(u)g Fw(\000)461 4166 y FA(u)509 4136 y Fv(\003)547 4166 y FB(\))579 4099 y Fz(\001)617 4116 y Fy(2)p Fx(=)l(M)754 4099 y Fz(\000)792 4166 y FA(c)828 4178 y Fy(2)885 4166 y FB(+)20 b Fw(O)1038 4099 y Fz(\000)1076 4166 y FB(\()p FA(u)g Fw(\000)g FA(u)1309 4136 y Fv(\003)1346 4166 y FB(\))1378 4136 y Fy(2)p Fx(=)l(M)1516 4099 y Fz(\001\001)1622 4166 y FB(for)29 b(a)g(suitable)h(constan)n(t)f FA(c)2507 4178 y Fy(2)2544 4166 y FB(.)44 b(and)30 b(the)g(results)f(follo)n(ws)g (going)f(bac)n(k)461 4266 y(to)g(the)g(original)e(v)-5 b(ariables.)p Black 195 4430 a Fw(\017)p Black 41 w FB(In)29 b(fact,)h(let)f(us)g(observ)n(e)e(that)i(the)g(h)n(yp)r(otheses)f (considered)g(ab)r(out)h(the)g(expansions)f(of)g(\()p FA(q)3222 4442 y Fy(0)3260 4430 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)3451 4442 y Fy(0)3488 4430 y FB(\()p FA(u)p FB(\)\))29 b(giv)n(en)278 4529 y(in)j(\(12\))e(and)h(\(13\))g(\()p Fp(HP2.1)f FB(and)h Fp(HP2.2)p FB(\))f(are)g(w)n(eak)n(er)f(than)i (what)g(usually)g(happ)r(ens)g(when)g(the)h(p)r(oten)n(tial)278 4629 y FA(V)47 b FB(is)27 b(a)g(p)r(olynomial)g(or)f(a)h(trigonometric) f(p)r(olynomial)h(as)g(w)n(e)g(ha)n(v)n(e)f(seen)h(previously)-7 b(.)36 b(This)27 b(w)n(eakness)f(comes)278 4729 y(from)f(the)g(fact)g (that)g(the)g(second)g(terms)f(in)h(the)h(expansions)d(are,)i(in)g (fact,)g(of)g(greater)e(order.)35 b(W)-7 b(e)25 b(assume)f(this)278 4828 y(w)n(eak)n(er)k(h)n(yp)r(othesis)h(to)g(sho)n(w)g(that)g(our)g (results)g(could)g(b)r(e)h(applied)g(to)f(more)f(general)h(p)r(oten)n (tials)g(as)f(long)h(as)278 4928 y(Hyp)r(othesis)f Fp(HP2)f FB(is)h(satis\014ed.)p Black 195 5092 a Fw(\017)p Black 41 w FB(F)-7 b(rom)27 b Fp(HP2.1)p FB(,)g(taking)g(in)n(to)g(accoun)n (t)g(that)g(the)h(homo)r(clinic)f(connection)g(is)h(a)f(solution)g(of)g (the)h(unp)r(erturb)r(ed)278 5192 y(Hamiltonian)d(system)g(and)f(iden)n (tifying)h(terms)g(of)f(the)h(same)g(order)e(in)i(\()p FA(u)13 b Fw(\000)g FA(ia)p FB(\),)24 b(one)h(can)f(deduce)h(that)g(in) g(the)278 5291 y(p)r(olynomial)i(case,)g(the)h(degree)f(of)g FA(V)47 b FB(is)27 b(2)p FA(r)r(=)p FB(\()p FA(r)22 b Fw(\000)c FB(1\).)36 b(In)28 b(fact,)g(there)f(exists)h(a)f(constan)n (t)g FA(v)3187 5303 y Fv(1)3280 5291 y Fw(2)d Ft(R)j FB(suc)n(h)h(that)1299 5483 y FA(V)19 b FB(\()p FA(x)p FB(\))24 b(=)e FA(v)1628 5495 y Fv(1)1699 5483 y FA(x)1779 5425 y Fu(2)p Fm(r)p 1756 5434 103 3 v 1756 5467 a(r)q Fl(\000)p Fu(1)1873 5483 y FB(\(1)c(+)g Fr(o)t FB(\(1\)\))195 b(as)54 b FA(x)24 b Fw(!)f(1)p FA(:)873 b FB(\(15\))p Black 1919 5753 a(11)p Black eop end %%Page: 12 12 TeXDict begin 12 11 bop Black Black Black 195 272 a Fw(\017)p Black 41 w FB(Hip)r(othesis)34 b Fp(HP4.2)e FB(is)g(to)h(ensure)g(that) g(the)g(parab)r(olic)f(critical)g(p)r(oin)n(t)h(\(0)p FA(;)14 b FB(0\))33 b(of)g(the)g(unp)r(erturb)r(ed)h(system)278 372 y(p)r(ersists)25 b(when)h(w)n(e)f(add)g(the)h(p)r(erturbation)e (and)i(that)f(it)h(k)n(eeps)e(its)i(parab)r(olic)e(c)n(haracter.)34 b(Therefore)24 b(it)i(is)f(the)278 471 y(natural)30 b(h)n(yp)r(othesis) h(to)f(deal)h(with)g(and)g(it)g(is)g(the)g(same)f(one)g(that)h(w)n(as)f (considered)g(in)h([BF04)o(].)46 b(Namely)-7 b(,)32 b(if)278 571 y(the)e(p)r(erturbation)g(has)f(order)f FA(n)h FB(with)i(2)p FA(n)19 b Fw(\000)g FB(2)26 b FA(<)g(m)p FB(,)k(when)g(the)g(p)r (erturbation)f(is)h(added)f(the)h(system)g(migh)n(t)278 671 y(undergo)j(bifurcations)f(and)h(the)h(in)n(v)-5 b(arian)n(t)32 b(manifolds)h(migh)n(t)g(ev)n(en)g(disapp)r(ear.)53 b(The)33 b(only)g(study)g(done)g(in)278 770 y(one)28 b(of)f(these)h(bifurcation)f(cases)g(can)g(b)r(e)h(found)g(in)g([BF05)o (].)p Black 195 926 a Fw(\017)p Black 41 w FB(The)i(form)f(of)h(the)g (p)r(erturb)r(ed)g(Hamiltonian)f FA(H)1829 938 y Fy(1)1896 926 y FB(considered)g(is)g(more)g(restrictiv)n(e)f(than)i(necessary)-7 b(.)41 b(In)30 b(fact,)278 1026 y(our)d(result)h(can)f(b)r(e)h(applied) f(to)h(an)n(y)f(Hamiltonian)g(of)h(the)g(form)1471 1259 y FA(H)1540 1271 y Fy(1)1577 1259 y FB(\()p FA(x;)14 b(y)s(;)g(\034)9 b FB(;)14 b FA(")p FB(\))24 b(=)2072 1155 y Fx(N)2041 1180 y Fz(X)2039 1356 y Fx(n)p Fy(=0)2178 1259 y FA(")2217 1225 y Fx(n)2262 1259 y FA(H)2338 1225 y Fx(n)2331 1279 y Fy(1)2383 1259 y FB(\()p FA(x;)14 b(y)s(;)g(\034)9 b FB(\))278 1492 y(if)37 b(the)f(functions)g FA(H)956 1462 y Fx(n)949 1512 y Fy(1)1001 1492 y FB(\()p FA(q)1070 1504 y Fy(0)1107 1492 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1298 1504 y Fy(0)1335 1492 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))37 b(ha)n(v)n(e)e(a)g(singularit)n(y)g (of)g(order)g(less)g(or)g(equal)g(than)h FA(`)23 b FB(+)h FA(n)p FB(.)61 b(In)36 b(this)278 1591 y(case,)27 b(the)h(result)f(w)n (ould)g(b)r(e)h(the)f(same)g(but)h(one)f(has)g(to)g(sligh)n(tly)g (adapt)g(the)h(de\014nition)f(of)h(the)g(constan)n(t)e FA(b)h FB(in)278 1691 y(Theorem)g(2.6.)p Black 195 1847 a Fw(\017)p Black 41 w FB(Recall)h(the)g(Hamiltonian)1275 2032 y FA(H)1365 1915 y Fz(\022)1426 2032 y FA(x;)14 b(y)s(;)1605 1976 y(t)p 1601 2013 39 4 v 1601 2089 a(")1649 1915 y Fz(\023)1734 2032 y FB(=)22 b FA(H)1890 2044 y Fy(0)1928 2032 y FB(\()p FA(x;)14 b(y)s FB(\))19 b(+)f FA(\026")2311 1998 y Fx(\021)2351 2032 y FA(H)2420 2044 y Fy(1)2471 1915 y Fz(\022)2532 2032 y FA(x;)c(y)s(;)2711 1976 y(t)p 2707 2013 V 2707 2089 a(")2756 1915 y Fz(\023)2831 2032 y FA(:)278 2234 y FB(Let)37 b(us)e(p)r(oin)n(t)i(out)e(that)i(in)f (the)g(case)f FA(`)24 b Fw(\000)f FB(2)p FA(r)40 b Fw(\024)c FB(0,)i(Hyp)r(othesis)e Fp(HP5)g FB(corresp)r(onds)e(to)h FA(\021)41 b Fw(\025)36 b FB(0,)i(whic)n(h)e(is)278 2334 y(optimal)25 b(in)h(the)f(sense)g(that)g(it)h(includes)f(the)h(case)e (suc)n(h)h(that)g(the)h(p)r(erturbation)f(is)g(of)g(the)g(same)g(order) f(as)g(the)278 2434 y(unp)r(erturb)r(ed)29 b(system.)278 2561 y(The)41 b(case)e FA(`)44 b FB(=)g(2)p FA(r)f FB(is)d(what)h(t)n (ypically)e(happ)r(ens)i(in)f(near)g(in)n(tegrable)f(Hamiltonian)h (systems)g(close)g(to)g(a)278 2661 y(resonance)28 b(and)g(in)h(general) f(p)r(erio)r(dic)g(systems)h(with)g(slo)n(w)f(dynamics,)h(therefore,)f (in)h(this)g(sense)g(Hyp)r(othesis)278 2760 y Fp(HP5)f FB(is)f(optimal)h(in)g(the)g(generic)e(case.)278 2888 y(In)j(the)g(case)f FA(`)19 b Fw(\000)f FB(2)p FA(r)27 b(>)e FB(0)j(one)g(ma)n(y)g(think)h(to)f(also)g(ask)g FA(\021)g Fw(\025)c FB(0.)39 b(Nev)n(ertheless,)28 b(our)g(tec)n (hniques)g(only)h(pro)n(vide)278 2988 y(optimal)f(exp)r(onen)n(tially)f (upp)r(er)h(b)r(ounds)f(if)h FA(\021)22 b Fw(\000)c FA(`)g FB(+)g(2)p FA(r)25 b Fw(\025)e FB(0.)278 3115 y(F)-7 b(or)25 b(lo)n(w)n(er)f(v)-5 b(alues)24 b(of)i FA(\021)s FB(,)g(that)f(is)g(0)e Fw(\024)f FA(\021)27 b(<)22 b(`)14 b Fw(\000)g FB(2)p FA(r)r FB(,)25 b(using)g(similar)f(to)r(ols)h(as)f (the)i(ones)e(presen)n(ted)h(in)g(this)h(pap)r(er,)278 3215 y(one)g(could)f(easily)g(pro)n(v)n(e)g(the)h(existence)f(of)h(the) g(p)r(erturb)r(ed)g(in)n(v)-5 b(arian)n(t)25 b(manifolds)h(and)f (obtain)h(\(non-optimal\))278 3315 y(exp)r(onen)n(tially)31 b(small)f(upp)r(er)h(b)r(ounds)g(for)g(the)g(di\013erence)g(b)r(et)n(w) n(een)g(them.)47 b(This)31 b(case)f(can)h(b)r(e)g(called)f Fs(b)l(elow)278 3414 y(the)k(singular)g(c)l(ase)39 b FB(\(see)31 b([GOS10)o(]\).)50 b(T)-7 b(o)31 b(obtain)h(an)f (asymptotic)h(form)n(ula)e(for)i(the)g(di\013erence)f(b)r(et)n(w)n(een) h(the)278 3514 y(in)n(v)-5 b(arian)n(t)26 b(manifolds)h(in)g(the)g Fs(b)l(elow)j(the)g(singular)f(c)l(ase)34 b FB(is)27 b(a)f(problem)h(whic)n(h)g(remains)f(op)r(en.)37 b(Some)26 b(ideas)h(to)278 3613 y(deal)h(with)g(this)g(case)e(b)n(y)i(using)f(a)n (v)n(eraging)d(theory)j(can)h(b)r(e)g(found)f(in)h([GOS10)o(].)p Black 195 3769 a Fw(\017)p Black 41 w FB(Hyp)r(othesis)h Fp(HP6)f FB(for)g(the)h(case)f FA(`)d(<)f FB(2)p FA(r)31 b FB(has)d(an)g(analogous)f(condition)h(for)g(the)h(general)e(case)h FA(`)c Fw(\025)h FB(2)p FA(r)31 b FB(whic)n(h)278 3869 y(is)i(stated)g(in)g(Theorem)g(2.6.)52 b(Therefore)31 b(w)n(e)i(are)f(not)h(adding)g(an)f(extra)g(h)n(yp)r(othesis)h(for)f (the)i(\\easier")c(case)278 3968 y FA(`)23 b(<)g FB(2)p FA(r)r FB(.)71 4196 y Fq(2.2)112 b(Main)39 b(results)71 4350 y FB(By)28 b(Hyp)r(othesis)h Fp(HP1)p FB(,)g(system)g(\(6\))g (with)g FA(\026)c FB(=)g(0)j(has)g(either)h(a)f(h)n(yp)r(erb)r(olic)h (or)f(parab)r(olic)f(p)r(oin)n(t)i(at)g(the)g(origin.)39 b(In)71 4449 y(the)23 b(second)f(case,)h(Hyp)r(othesis)f Fp(HP4.2)g FB(ensures)g(that)h(the)g(origin)e(is)i(also)e(a)h(critical) g(p)r(oin)n(t)h(of)g(the)g(p)r(erturb)r(ed)g(system)71 4549 y(\()p FA(\026)j Fw(6)p FB(=)f(0\))k(whic)n(h)g(is)g(also)f(parab) r(olic.)40 b(In)29 b(the)h(h)n(yp)r(erb)r(olic)f(case,)f(the)i(next)f (theorem)g(ensures)f(that)h(the)h(unp)r(erturb)r(ed)71 4648 y(h)n(yp)r(erb)r(olic)25 b(critical)f(p)r(oin)n(t)i(of)f(the)h (unp)r(erturb)r(ed)g(system)f(b)r(ecomes)g(a)g(h)n(yp)r(erb)r(olic)g(p) r(erio)r(dic)g(orbit)g(whic)n(h)g(is)g(close)g(to)71 4748 y(the)j(origin.)p Black 71 4893 a Fp(Theorem)h(2.3.)p Black 38 w Fs(L)l(et)f(us)f(assume)h(Hyp)l(otheses)h Fo(HP1.1)p Fs(,)h Fo(HP3)p Fs(,)f Fo(HP4.1)p Fs(.)39 b(T)-6 b(ake)28 b FA(\021)f Fw(\025)22 b FB(0)28 b Fs(and)g(\014x)g (any)g(value)g FA(\026)3635 4905 y Fy(0)3696 4893 y FA(>)22 b FB(0)p Fs(.)71 4993 y(Then,)30 b(ther)l(e)g(exists)f FA(")785 5005 y Fy(0)845 4993 y FA(>)23 b FB(0)29 b Fs(such)g(that)h (for)g(any)g Fw(j)p FA(\026)p Fw(j)23 b FA(<)g(\026)1910 5005 y Fy(0)1976 4993 y Fs(and)30 b FA(")23 b Fw(2)g FB(\(0)p FA(;)14 b(")2427 5005 y Fy(0)2464 4993 y FB(\))p Fs(,)30 b(system)36 b FB(\(6\))29 b Fs(has)i(a)e(hyp)l(erb)l(olic)j(p)l (erio)l(dic)71 5093 y(orbit)e FB(\()p FA(x)348 5105 y Fx(p)387 5093 y FB(\()p FA(t=")p FB(\))p FA(;)14 b(y)640 5105 y Fx(p)678 5093 y FB(\()p FA(t=")p FB(\)\))30 b Fs(which)h(satis\014es)f(that,)g(for)h FA(t)23 b Fw(2)g Ft(R)p Fs(,)1354 5182 y Fz(\014)1354 5232 y(\014)1354 5281 y(\014)1354 5331 y(\014)1382 5302 y FA(x)1429 5314 y Fx(p)1481 5185 y Fz(\022)1557 5246 y FA(t)p 1553 5283 V 1553 5359 a(")1601 5185 y Fz(\023)1662 5182 y(\014)1662 5232 y(\014)1662 5281 y(\014)1662 5331 y(\014)1708 5302 y FB(+)1791 5182 y Fz(\014)1791 5232 y(\014)1791 5281 y(\014)1791 5331 y(\014)1819 5302 y FA(y)1860 5314 y Fx(p)1912 5185 y Fz(\022)1988 5246 y FA(t)p 1983 5283 V 1983 5359 a(")2032 5185 y Fz(\023)2093 5182 y(\014)2093 5232 y(\014)2093 5281 y(\014)2093 5331 y(\014)2144 5302 y Fw(\024)f FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2443 5268 y Fx(\021)r Fy(+1)71 5504 y Fs(for)30 b(a)h(c)l(onstant)e FA(K)f(>)23 b FB(0)29 b Fs(indep)l(endent)h(of)h FA(")e Fs(and)i FA(\026)p Fs(.)p Black 1919 5753 a FB(12)p Black eop end %%Page: 13 13 TeXDict begin 13 12 bop Black Black 195 272 a FB(The)29 b(pro)r(of)f(of)h(this)g(theorem,)f(whic)n(h)h(w)n(as)f(done)g(in)h ([DS97])f(for)h FA(\021)f(>)c(`)p FB(,)29 b(is)f(giv)n(en)g(in)h (Section)g(5.)40 b(An)29 b(alternativ)n(e)71 372 y(pro)r(of)35 b(for)g(v)-5 b(alues)36 b(of)f FA(\021)40 b(>)c Fw(\000)p FB(1)p FA(=)p FB(2)e(without)i(explicit)h(b)r(ounds)e(for)h(the)g(p)r (erio)r(dic)f(orbit)h(can)f(b)r(e)h(found)g(in)g([F)-7 b(on95)o(].)71 471 y(F)g(or)27 b(the)g(case)g(when)g(p)r(erturbation)g (only)g(dep)r(ends)g(on)h(time)f(in)h([F)-7 b(on93)o(])27 b(the)h(existence)f(of)g(the)h(p)r(erio)r(dic)f(orbit)g(with)71 571 y(explicit)h(b)r(ounds)f(w)n(as)g(giv)n(en)g(for)g FA(\021)f(>)d Fw(\000)p FB(2.)195 671 y(T)-7 b(o)37 b(use)f(the)h(same) f(notation)g(in)h(b)r(oth)g(the)h(h)n(yp)r(erb)r(olic)e(and)g(parab)r (olic)g(cases,)h(in)g(the)g(latter)g(one)f(w)n(e)g(de\014ne)71 770 y(\()p FA(x)150 782 y Fx(p)189 770 y FA(;)14 b(y)267 782 y Fx(p)305 770 y FB(\))23 b(=)g(\(0)p FA(;)14 b FB(0\).)195 870 y(The)32 b(next)f(step)g(is)g(to)g(study)h(the)f(stable)g(and)g (unstable)g(in)n(v)-5 b(arian)n(t)30 b(manifolds)h(of)h(the)f(p)r(erio) r(dic)g(orbit)g(\()p FA(x)3640 882 y Fx(p)3679 870 y FA(;)14 b(y)3757 882 y Fx(p)3795 870 y FB(\).)71 969 y(In)26 b(the)g(unp)r(erturb)r(ed)h(case)e(\(that)h(is)g FA(\026)d FB(=)g(0\))j(w)n(e)f(kno)n(w)g(that)i(they)f(coincide)f (along)g(the)h(separatrix)e(\()p FA(q)3354 981 y Fy(0)3392 969 y FA(;)14 b(p)3471 981 y Fy(0)3508 969 y FB(\))26 b(giv)n(en)f(in)71 1069 y Fp(HP2)p FB(.)37 b(When)28 b FA(\026)23 b Fw(6)p FB(=)f(0)28 b(they)f(generically)f(split.)195 1169 y(T)-7 b(o)35 b(measure)f(the)h(splitting)h(of)f(the)g(in)n(v)-5 b(arian)n(t)34 b(manifolds)h(let)g(us)g(consider)f(the)i(2)p FA(\031)s(")p FB(-P)n(oincar)n(\023)-39 b(e)31 b(map)k FA(P)3608 1181 y Fx(t)3633 1189 y Fu(0)3705 1169 y FB(in)g(a)71 1268 y(transv)n(ersal)e(section)j(\006)846 1280 y Fx(t)871 1288 y Fu(0)945 1268 y FB(=)1046 1201 y Fz(\010)1095 1268 y FB(\()p FA(x;)14 b(y)s(;)g(t)1322 1280 y Fy(0)1359 1268 y FB(\);)g(\()p FA(x;)g(y)s FB(\))24 b Fw(2)f Ft(R)1782 1238 y Fy(2)1819 1201 y Fz(\011)1868 1268 y FB(.)62 b(This)36 b(P)n(oincar)n(\023)-39 b(e)33 b(map)j(has)f(a)h(\(h)n(yp)r(erb)r(olic) f(or)g(parab)r(olic\))71 1368 y(\014xed)23 b(p)r(oin)n(t)h(\()p FA(x)559 1380 y Fx(p)598 1368 y FB(\()p FA(t)660 1380 y Fy(0)697 1368 y FA(=")p FB(\))p FA(;)14 b(y)888 1380 y Fx(p)926 1368 y FB(\()p FA(t)988 1380 y Fy(0)1025 1368 y FA(=")p FB(\)\).)36 b(W)-7 b(e)23 b(will)h(see)f(that)g(this)h (\014xed)f(p)r(oin)n(t)g(has)g(stable)g(and)g(unstable)h(in)n(v)-5 b(arian)n(t)22 b(curv)n(es.)195 1468 y(As)37 b FA(P)380 1480 y Fx(t)405 1488 y Fu(0)479 1468 y FB(is)g(an)g(area)e(preserving)g (map,)k(w)n(e)e(measure)e(the)j(splitting)f(giving)f(an)g(asymptotic)h (form)n(ula)f(for)g(the)71 1567 y(area)d(of)i(the)g(lob)r(es)g (generated)e(b)n(y)i(these)g(curv)n(es)e(b)r(et)n(w)n(een)i(t)n(w)n(o)f (transv)n(ersal)f(homo)r(clinic)h(p)r(oin)n(ts.)59 b(Moreo)n(v)n(er,)34 b(b)n(y)71 1667 y(the)i(area)f(preserving)f(c)n(haracter)g(of)i FA(P)1346 1679 y Fx(t)1371 1687 y Fu(0)1408 1667 y FB(,)i(the)e(area)f Fw(A)h FB(of)g(these)g(lob)r(es)g(do)r(es)f(not)h(dep)r(end)h(on)f(the) g(c)n(hoice)f(of)h(the)71 1766 y(homo)r(clinic)30 b(p)r(oin)n(ts.)44 b(Other)29 b(quan)n(tities)h(measuring)e(the)j(splitting,)g(as)e(the)h (distance)g(along)f(a)g(transv)n(ersal)f(section)71 1866 y(to)i(the)i(unp)r(erturb)r(ed)f(separatrix,)f(or)f(the)j(angle)d(b)r (et)n(w)n(een)i(these)g(curv)n(es)e(at)i(an)f(homo)r(clinic)h(p)r(oin)n (t,)h(can)e(b)r(e)h(easily)71 1966 y(deriv)n(ed)c(from)g(our)g(w)n (ork.)195 2065 y(The)40 b(quan)n(titativ)n(e)e(measure)h(of)g(this)h (splitting)g(di\013ers)f(substan)n(tially)f(dep)r(ending)i(whether)f FA(`)26 b Fw(\000)g FB(2)p FA(r)45 b(<)d FB(0)d(or)71 2165 y FA(`)20 b Fw(\000)g FB(2)p FA(r)31 b Fw(\025)e FB(0)h(\(see)h(Hyp)r(othesis)g Fp(HP2)g FB(for)f(the)i(de\014nition)f (of)g FA(r)j FB(and)c(\(14\))h(for)g(the)g(de\014nition)g(of)g FA(`)p FB(\).)47 b(Therefore,)31 b(w)n(e)71 2265 y(split)24 b(these)g(results)g(in)n(to)f(t)n(w)n(o)h(di\013eren)n(t)g(theorems.)35 b(First,)24 b(Theorem)f(2.5)h(deals)f(with)h(the)h(case)e FA(`)11 b Fw(\000)g FB(2)p FA(r)25 b(<)e FB(0)g(and)h(then)71 2364 y(Theorem)g(2.6)g(deals)g(with)i(the)f(case)f FA(`)13 b Fw(\000)g FB(2)p FA(r)25 b Fw(\025)e FB(0.)35 b(W)-7 b(e)26 b(will)f(giv)n(e)f(a)g(complete)h(description)g(of)g(the)g(pro)r (of)f(of)h(these)g(t)n(w)n(o)71 2464 y(theorems)j(in)i(Section)f(4.)41 b(W)-7 b(e)29 b(also)f(refer)h(to)g(Section)g(3)f(for)h(an)g(heuristic) g(idea)f(of)h(the)h(main)f(features)g(of)g(the)g(pro)r(of)71 2564 y(of)e(our)g(main)h(results.)71 2779 y Fp(2.2.1)94 b(Main)31 b(result)g(for)h(the)g(case)g FA(`)23 b(<)f FB(2)p FA(r)71 2933 y FB(As)30 b(w)n(e)g(will)h(see,)g(in)f(this)h (case,)f(the)h(asymptotic)f(co)r(e\016cien)n(t)g(for)g(the)h(area)e(of) h(the)h(lob)r(e)f(b)r(et)n(w)n(een)h(t)n(w)n(o)e(consecutiv)n(e)71 3032 y(homo)r(clinic)24 b(p)r(oin)n(ts)g(is)g(v)n(ery)g(related)f(with) i(the)g(so-called)e(Melnik)n(o)n(v)g(P)n(oten)n(tial.)34 b(F)-7 b(or)24 b(this)h(reason,)e(\014rst)h(of)g(all)h(w)n(e)e(are)71 3132 y(going)j(to)i(obtain)f(an)g(asymptotic)g(form)n(ula)g(for)g(it.) 195 3231 y(The)j(Melnik)n(o)n(v)e(P)n(oten)n(tial)h(\(called)g(also)f (sometimes)i(P)n(oincar)n(\023)-39 b(e)26 b(F)-7 b(unction,)30 b(see)g(for)f(instance)g([DG00)o(]\),)i(is)e(giv)n(en)71 3331 y(b)n(y)923 3484 y FA(L)994 3367 y Fz(\022)1055 3484 y FA(u;)1154 3428 y(t)p 1149 3465 39 4 v 1149 3541 a(")1198 3367 y Fz(\023)1282 3484 y FB(=)1370 3371 y Fz(Z)1453 3391 y Fy(+)p Fv(1)1416 3560 y(\0001)1588 3484 y FA(H)1657 3496 y Fy(1)1708 3417 y Fz(\000)1746 3484 y FA(q)1783 3496 y Fy(0)1820 3484 y FB(\()p FA(u)19 b FB(+)f FA(s)p FB(\))p FA(;)c(p)2152 3496 y Fy(0)2189 3484 y FB(\()p FA(u)k FB(+)g FA(s)p FB(\))p FA(;)c(")2517 3450 y Fv(\000)p Fy(1)2606 3484 y FB(\()p FA(t)19 b FB(+)f FA(s)p FB(\))2841 3417 y Fz(\001)2893 3484 y FA(ds:)705 b FB(\(16\))195 3682 y(Let)28 b(us)g(p)r(oin)n(t)f(out)h(that,)g(b)n(y) f(Hyp)r(othesis)h Fp(HP4)p FB(,)f(this)h(in)n(tegral)f(is)g(uniformly)g (con)n(v)n(ergen)n(t.)35 b(Moreo)n(v)n(er)1530 3865 y FA(L)p FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b(=)g FA(M)9 b FB(\()p FA(\034)28 b Fw(\000)18 b FA(")2200 3831 y Fv(\000)p Fy(1)2289 3865 y FA(u)p FB(\))p FA(;)1311 b FB(\(17\))71 4048 y(where)27 b FA(M)36 b FB(is)28 b(the)g(2)p FA(\031)s FB(-p)r(erio)r(dic)f(function)907 4284 y FA(M)9 b FB(\()p FA(s)p FB(\))24 b(=)1211 4170 y Fz(Z)1294 4191 y Fy(+)p Fv(1)1257 4359 y(\0001)1429 4284 y FA(H)1498 4296 y Fy(1)1549 4216 y Fz(\000)1588 4284 y FA(q)1625 4296 y Fy(0)1662 4284 y FB(\()p FA(r)r FB(\))p FA(;)14 b(p)1844 4296 y Fy(0)1882 4284 y FB(\()p FA(r)r FB(\))p FA(;)g(")2061 4249 y Fv(\000)p Fy(1)2151 4284 y FA(r)21 b FB(+)d FA(s)2331 4216 y Fz(\001)2383 4284 y FA(dr)26 b FB(=)2577 4205 y Fz(X)2577 4383 y Fx(k)q Fv(6)p Fy(=0)2711 4284 y FA(M)2801 4249 y Fy([)p Fx(k)q Fy(])2879 4284 y FA(e)2918 4249 y Fx(ik)q(s)71 4568 y FB(whic)n(h,)h(b)n(y)i Fp(HP3)p FB(,)e(has)g(zero)g(mean.)37 b(Here)27 b FA(M)1550 4538 y Fy([)p Fx(k)q Fy(])1656 4568 y FB(denotes)g(the)h FA(k)s FB(-F)-7 b(ourier)26 b(co)r(e\016cien)n(t)h(of)h FA(M)9 b FB(.)195 4668 y(In)25 b([DS97])f(\(p)r(olar)g(case\))g(and)h ([BF04)o(])g(\(branc)n(hing)e(p)r(oin)n(t)i(case\),)g(it)g(w)n(as)e (seen)h(that)h(Hyp)r(othesis)g Fp(HP6)f FB(\(together)71 4767 y(with)j(Hyp)r(otheses)f Fp(HP3)h FB(and)f Fp(HP4)p FB(\),)h(allo)n(ws)e(us)i(to)f(giv)n(e)g(an)g(asymptotic)g(form)n(ula)g (for)g(the)h(F)-7 b(ourier)25 b(co)r(e\016cien)n(ts)i(of)71 4867 y FA(M)36 b FB(and)28 b(henceforth)f(w)n(e)g(will)h(obtain)f(an)h (asymptotic)f(form)n(ula)f(for)h(the)h(functions)g FA(M)37 b FB(and)27 b FA(L)p FB(.)p Black 71 5033 a Fp(Lemma)k(2.4)g FB(\([DS97,)d(BF04)o(]\))p Fp(.)p Black 41 w Fs(L)l(et)i(us)f(assume)g (Hyp)l(otheses)i Fo(HP2)p Fs(,)g Fo(HP3)p Fs(,)g Fo(HP4)g Fs(and)f Fo(HP6)p Fs(.)39 b(L)l(et)1715 5272 y FA(f)1756 5284 y Fy(0)1816 5272 y FB(=)1914 5215 y FA(Ai)2005 5185 y Fv(\000)p Fx(`)p Fv(\000)p Fy(1)p 1914 5253 260 4 v 1968 5329 a FB(\000\()p FA(`)p FB(\))2183 5272 y FA(;)71 5498 y Fs(wher)l(e)30 b FA(A)g Fs(is)g(the)g(c)l(onstant)f(de\014ne)l (d)h(by)h(Hyp)l(othesis)f Fo(HP6)p Fs(.)40 b(Then:)p Black 1919 5753 a FB(13)p Black eop end %%Page: 14 14 TeXDict begin 14 13 bop Black Black Black 169 272 a Fs(1.)p Black 42 w(The)31 b(\014rst)e(F)-6 b(ourier)30 b(c)l(o)l(e\016cients)h (of)f FA(M)38 b Fs(ar)l(e)31 b(given)f(by:)p 1243 412 165 4 v 1243 489 a FA(M)1333 465 y Fy([1])1430 489 y FB(=)23 b FA(M)1608 455 y Fy([)p Fv(\000)p Fy(1])1757 489 y FB(=)g Fw(\000)1977 433 y FB(1)p 1920 470 156 4 v 1920 546 a FA(")1959 522 y Fx(`)p Fv(\000)p Fy(1)2085 489 y FA(f)2126 501 y Fy(0)2163 489 y FA(e)2202 452 y Fv(\000)2264 419 y Fx(a)p 2264 433 37 4 v 2266 480 a(")2328 397 y Fz(\020)2377 489 y FB(1)18 b(+)g Fw(O)2602 397 y Fz(\020)2652 489 y FA(")2704 430 y Fu(1)p 2701 439 35 3 v 2701 472 a Fm(\014)2750 397 y Fz(\021)o(\021)2863 489 y FA(:)p Black 169 733 a Fs(2.)p Black 42 w(If)31 b Fw(j)p FA(k)s Fw(j)23 b(6)p FB(=)f(1)p Fs(,)1605 882 y FA(M)1695 848 y Fy([)p Fx(k)q Fy(])1796 882 y FB(=)h Fw(O)1966 765 y Fz(\022)2094 826 y FB(1)p 2037 863 156 4 v 2037 939 a FA(")2076 915 y Fx(`)p Fv(\000)p Fy(1)2203 882 y FA(e)2242 848 y Fv(\000j)p Fx(k)q Fv(j)2379 826 y Fm(a)p 2379 835 33 3 v 2381 868 a(")2426 765 y Fz(\023)2501 882 y FA(:)p Black 169 1110 a Fs(3.)p Black 42 w(F)-6 b(or)30 b FA(u)23 b Fw(2)g Ft(R)30 b Fs(and)g FA(t)23 b Fw(2)h Ft(R)p Fs(,)1023 1338 y FA(L)1094 1221 y Fz(\022)1154 1338 y FA(u;)1253 1282 y(t)p 1248 1319 39 4 v 1248 1395 a(")1297 1221 y Fz(\023)1381 1338 y FB(=)f Fw(\000)1601 1282 y FB(2)p 1544 1319 156 4 v 1544 1395 a FA(")1583 1371 y Fx(`)p Fv(\000)p Fy(1)1709 1338 y FB(Re)1835 1221 y Fz(\022)1896 1338 y FA(f)1937 1350 y Fy(0)1974 1338 y FA(e)2013 1291 y Fv(\000)p Fx(i)2088 1227 y Fk(\020)2138 1259 y Fx(u)p Fv(\000)p Fx(t)p 2138 1273 117 4 v 2180 1320 a(")2264 1227 y Fk(\021)2309 1221 y Fz(\023)2384 1338 y FA(e)2423 1301 y Fv(\000)2484 1268 y Fx(a)p 2484 1282 37 4 v 2486 1329 a(")2548 1246 y Fz(\020)2598 1338 y FB(1)18 b(+)g Fw(O)2823 1246 y Fz(\020)2872 1338 y FA(")2924 1279 y Fu(1)p 2921 1288 35 3 v 2921 1321 a Fm(\014)2970 1246 y Fz(\021\021)3083 1338 y FA(;)278 1566 y Fs(wher)l(e)31 b FA(a)e Fs(and)i FA(\014)j Fs(ar)l(e)c(the)g(c)l (onstants)f(de\014ne)l(d)h(in)g(Hyp)l(othesis)g Fo(HP2)p Fs(.)p Black 71 1732 a Fp(Theorem)g(2.5)g FB(\(Main)d(Theorem:)36 b(case)25 b FA(`)17 b Fw(\000)f FB(2)p FA(r)25 b(<)e FB(0\))p Fp(.)p Black 40 w Fs(L)l(et)28 b(us)h(assume)g(Hyp)l(otheses)g Fo(HP1)p Fs(-)p Fo(HP6)i Fs(and)e FA(`)16 b Fw(\000)g FB(2)p FA(r)26 b(<)c FB(0)p Fs(.)71 1832 y(Then,)35 b(given)f FA(\026)587 1844 y Fy(0)653 1832 y FA(>)29 b FB(0)p Fs(,)k(ther)l(e)h (exists)e FA(")1327 1844 y Fy(0)1393 1832 y FA(>)d FB(0)j Fs(such)i(that)f(for)g(any)h FA(\026)29 b Fw(2)g(fj)p FA(\026)p Fw(j)g(\024)g FA(\026)2699 1844 y Fy(0)2736 1832 y Fw(g)k Fs(and)g FA(")c Fw(2)g FB(\(0)p FA(;)14 b(")3277 1844 y Fy(0)3314 1832 y FB(\))33 b Fs(the)g(invariant)71 1931 y(manifolds)e(of)g(the)e(p)l(erio)l(dic)j(orbit)e(given)g(in)g (The)l(or)l(em)g(2.3)h(split)f(and)g(the)g(ar)l(e)l(a)g(of)g(the)g(c)l (orr)l(esp)l(onding)h(lob)l(es)f(for)g(the)71 2031 y(Poinc)l(ar)n(\023) -40 b(e)31 b(map)f(is)h(given)f(by)1069 2259 y Fw(A)23 b FB(=)g(4)p Fw(j)p FA(\026)p Fw(j)p FA(")1423 2224 y Fx(\021)r Fy(+1)p Fv(\000)p Fx(`)1627 2259 y FA(e)1666 2221 y Fv(\000)1727 2188 y Fx(a)p 1727 2202 37 4 v 1729 2250 a(")1791 2142 y Fz(\022)1852 2163 y(\014)1852 2213 y(\014)1852 2263 y(\014)1880 2259 y FA(f)1921 2271 y Fy(0)1958 2259 y FA(e)1997 2224 y Fx(iC)t Fy(\()p Fx(\026;")p Fy(\))2219 2163 y Fz(\014)2219 2213 y(\014)2219 2263 y(\014)2265 2259 y FB(+)18 b Fw(O)2430 2142 y Fz(\022)2572 2203 y FB(1)p 2501 2240 182 4 v 2501 2316 a Fw(j)c FB(ln)g FA(")p Fw(j)2693 2142 y Fz(\023\023)2829 2259 y FA(;)71 2491 y Fs(wher)l(e)30 b FA(C)6 b FB(\()p FA(\026;)14 b(")p FB(\))30 b Fs(is)g(a)f(function)h(de\014ne)l(d)f(in)h Fw(fj)p FA(\026)p Fw(j)23 b(\024)f FA(\026)1756 2503 y Fy(0)1794 2491 y Fw(g)17 b(\002)g FB(\(0)p FA(;)d(")2085 2503 y Fy(0)2122 2491 y FB(\))p Fs(,)30 b(analytic)g(in)g FA(\026)f Fs(and)h(such)g(that)f FA(C)6 b FB(\()p FA(\026;)14 b(")p FB(\))24 b(=)e Fw(O)r FB(\()p FA(\026")3777 2461 y Fx(\021)3818 2491 y FB(\))71 2591 y Fs(uniformly)31 b(in)e FA(")p Fs(,)h(and)h FA(f)848 2603 y Fy(0)914 2591 y Fs(is)f(given)h(in)e(L)l(emma)h(2.4.)71 2807 y Fp(2.2.2)94 b(Main)31 b(result)g(for)h(the)g(case)g FA(`)23 b Fw(\025)f FB(2)p FA(r)71 2960 y FB(The)38 b(case)g FA(`)j Fw(\025)f FB(2)p FA(r)h FB(is)e(essen)n(tially)e(di\013eren)n(t)i(from)f(the)h (previous)e(case)g FA(`)k(<)g FB(2)p FA(r)g FB(in)d(the)h(sense)f(that) h(w)n(e)f(are)f(not)71 3060 y(able)g(to)g(ha)n(v)n(e)g(\\a)f(priori")g (estimates)h(for)g(the)h(asymptotic)f(co)r(e\016cien)n(t)g(of)h(the)f (area)f(of)i(the)g(lob)r(es)f(b)r(et)n(w)n(een)g(t)n(w)n(o)71 3159 y(consecutiv)n(e)d(homo)r(clinic)g(p)r(oin)n(ts.)59 b(Suc)n(h)35 b(asymptotic)f(co)r(e\016cien)n(t)h(dep)r(ends)g(on)g(an)f (unkno)n(wn)g(function)i(\()p FA(f)9 b FB(\()e(^)-49 b FA(\026)p FB(\))35 b(in)71 3259 y(Theorem)h(2.6\))h(whic)n(h)g(comes) f(from)h(the)h(study)f(of)g(the)g(di\013erence)h(b)r(et)n(w)n(een)f (adequate)f(appro)n(ximations)f(of)i(the)71 3358 y(in)n(v)-5 b(arian)n(t)27 b(manifolds)h(near)g(the)g(singularities)g Fw(\006)p FA(ia)p FB(.)38 b(F)-7 b(or)28 b(that)g(reason)f(w)n(e)h (note)g(that,)h(in)g(this)f(case,)g(the)h(analogous)71 3458 y(condition)e(to)h(Hyp)r(othesis)f Fp(HP6)h FB(is)f FA(f)9 b FB(\()e(^)-49 b FA(\026)p FB(\))23 b Fw(6)p FB(=)g(0,)k(whic)n(h)h(is)f(assumed)g(in)h(Theorem)f(2.6.)p Black 71 3624 a Fp(Theorem)j(2.6)g FB(\(Main)d(Theorem:)36 b(case)25 b FA(`)17 b Fw(\000)f FB(2)p FA(r)25 b Fw(\025)e FB(0\))p Fp(.)p Black 40 w Fs(L)l(et)28 b(us)h(assume)g(Hyp)l(otheses)g Fo(HP1)p Fs(-)p Fo(HP5)i Fs(and)e FA(`)16 b Fw(\000)g FB(2)p FA(r)26 b Fw(\025)c FB(0)p Fs(.)71 3724 y(Then,)38 b(ther)l(e)e(exists)f(an)h(entir)l(e)f(analytic)i(function)e FA(f)9 b FB(\()e(^)-49 b FA(\026)p FB(\))36 b Fs(such)g(that,)h(if)g FA(f)9 b FB(\()e(^)-49 b FA(\026)p FB(\))34 b Fw(6)p FB(=)f(0)i Fs(for)43 b FB(^)-49 b FA(\026)34 b Fw(2)g(fj)7 b FB(^)-49 b FA(\026)p Fw(j)33 b FA(<)40 b FB(^)-49 b FA(\026)3472 3736 y Fy(1)3510 3724 y Fw(g)22 b(\\)h Ft(R)35 b Fs(for)71 3823 y(some)j FB(^)-49 b FA(\026)334 3835 y Fy(1)397 3823 y Fw(2)26 b Ft(R)p Fs(,)32 b(ther)l(e)f(exists)g FA(")1071 3835 y Fy(0)1134 3823 y FA(>)25 b FB(0)31 b Fs(such)g(that)g(for)h(any)g FB(\()p FA(";)14 b(\026)p FB(\))25 b Fw(2)h(f)p FB(\()p FA(";)14 b(\026)p FB(\))26 b Fw(2)g Ft(R)2648 3793 y Fy(2)2685 3823 y FB(;)45 b(0)25 b FA(<)g(")h(<)f(")3104 3835 y Fy(0)3141 3823 y FA(;)45 b Fw(j)p FA(\026")3321 3793 y Fx(\021)r Fv(\000)p Fx(`)p Fy(+2)p Fx(r)3557 3823 y Fw(j)26 b FA(<)32 b FB(^)-49 b FA(\026)3746 3835 y Fy(1)3783 3823 y Fw(g)p Fs(,)71 3923 y(the)28 b(invariant)h(manifolds)g(of)g(the)f(p)l(erio)l(dic)i (orbit)f(given)f(in)g(The)l(or)l(em)h(2.3)g(split)g(and)f(the)g(ar)l(e) l(a)g(of)h(the)f(c)l(orr)l(esp)l(onding)71 4023 y(lob)l(es)i(for)h(the) f(Poinc)l(ar)n(\023)-40 b(e)31 b(map)f(is)h(given)f(by)p Black 195 4189 a Fw(\017)p Black 41 w Fs(If)h FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b(>)e FB(0)p Fs(,)1054 4333 y Fw(A)g FB(=)g(4)p Fw(j)p FA(\026)p Fw(j)p FA(")1408 4299 y Fx(\021)r Fy(+1)p Fv(\000)p Fx(`)1611 4333 y FA(e)1650 4296 y Fv(\000)1712 4263 y Fx(a)p 1712 4277 37 4 v 1714 4324 a(")1776 4216 y Fz(\022)1837 4263 y(\014)1837 4313 y(\014)1864 4333 y FA(f)1928 4266 y Fz(\000)1966 4333 y FA(\026")2055 4299 y Fx(\021)r Fy(+2)p Fx(r)r Fv(\000)p Fx(`)2291 4266 y Fz(\001)2329 4263 y(\014)2329 4313 y(\014)2376 4333 y FB(+)18 b Fw(O)2541 4216 y Fz(\022)2757 4277 y FB(1)p 2612 4314 332 4 v 2612 4390 a Fw(j)c FB(ln)g FA(")p Fw(j)2794 4366 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)2953 4216 y Fz(\023\023)p Black 195 4561 a Fw(\017)p Black 41 w Fs(If)31 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b FB(=)e(0)p Fs(,)810 4789 y Fw(A)g FB(=)g(4)p Fw(j)p FA(\026)p Fw(j)p FA(")1164 4755 y Fx(\021)r Fy(+1)p Fv(\000)p Fx(`)1367 4789 y FA(e)1406 4752 y Fv(\000)1468 4719 y Fx(a)p 1468 4733 37 4 v 1470 4780 a(")1514 4752 y Fy(+)p Fx(\026)1605 4727 y Fu(2)1637 4752 y Fx(")1668 4727 y Fu(4)p Fm(\021)1734 4752 y Fy(Im)10 b Fx(b)i Fy(ln)1941 4729 y Fu(1)p 1941 4738 29 3 v 1941 4772 a Fm(")1997 4672 y Fz(\022)2058 4694 y(\014)2058 4743 y(\014)2058 4793 y(\014)2086 4789 y FA(f)22 b FB(\()p FA(\026")2270 4755 y Fx(\021)2311 4789 y FB(\))14 b FA(e)2396 4755 y Fx(iC)t Fy(\()o Fx(\026")2567 4730 y Fm(\021)2605 4755 y Fx(;")p Fy(\))2686 4694 y Fz(\014)2686 4743 y(\014)2686 4793 y(\014)2732 4789 y FB(+)k Fw(O)2897 4672 y Fz(\022)3038 4733 y FB(1)p 2968 4770 182 4 v 2968 4846 a Fw(j)c FB(ln)g FA(")p Fw(j)3160 4672 y Fz(\023\023)3296 4789 y FA(;)278 5022 y Fs(wher)l(e)34 b FA(b)27 b Fw(2)i Ft(C)j Fs(is)h(a)g(c)l(onstant)f(c)l(ompletely)i (determine)l(d)f(by)g(the)g(Hamiltonian)g FA(H)40 b Fs(and)33 b FA(C)6 b FB(\()h(^)-49 b FA(\026;)14 b(")p FB(\))33 b Fs(is)f(a)h(function)278 5121 y(de\014ne)l(d)d(in)g Fw(fj)7 b FB(^)-49 b FA(\026)p Fw(j)23 b(\024)g FA(\026)959 5133 y Fy(1)996 5121 y Fw(g)18 b(\002)g FB(\(0)p FA(;)c(")1289 5133 y Fy(0)1326 5121 y FB(\))p Fs(,)30 b(analytic)h(in)36 b FB(^)-48 b FA(\026)30 b Fs(and)g(such)g(that)f FA(C)6 b FB(\()h(^)-49 b FA(\026)q(;)14 b(")p FB(\))23 b(=)g Fw(O)r FB(\()7 b(^)-49 b FA(\026)p FB(\))30 b Fs(uniformly)h(in)f FA(")p Fs(.)p Black 1919 5753 a FB(14)p Black eop end %%Page: 15 15 TeXDict begin 15 14 bop Black Black 71 272 a Fp(2.2.3)94 b(Some)31 b(commen)m(ts)g(ab)s(out)g(the)h(results)p Black 195 425 a Fw(\017)p Black 41 w FB(The)i(constan)n(t)f FA(b)h FB(app)r(earing)f(in)h(Theorem)f(2.6)g(can)g(b)r(e)h(computed)g (explicitly)g(as)g(it)g(is)g(sho)n(w)n(ed)e(in)i(form)n(ula)278 525 y(\(75\))f(in)g(Prop)r(osition)f(4.15.)51 b(In)33 b(particular,)g FA(b)f FB(=)g(0)g(when)h(the)h(Hamiltonian)e FA(H)2920 537 y Fy(1)2991 525 y FB(in)h(\(8\))g(and)g(\(9\))g(do)r(es)f (not)278 625 y(dep)r(end)g(on)f FA(y)s FB(.)47 b(F)-7 b(or)31 b(this)g(reason,)f(in)i(the)f(previous)f(results)h(obtained)g (in)g(the)g(singular)f(case)g(corresp)r(onding)278 724 y(to)h FA(\021)f FB(=)e FA(`)20 b Fw(\000)f FB(2)p FA(r)30 b FB(=)e(0,)i(see)g([T)-7 b(re97)o(,)30 b(Gel00)o(,)h(Oli06)o(,)f (GOS10)o(],)h(this)g(term)f(do)r(es)g(not)g(app)r(ear.)45 b(The)30 b(app)r(earance)278 824 y(of)j(this)g(logarithmic)f(term)g(in) h(the)h(asymptotic)e(form)n(ula)g(had)g(already)f(b)r(een)j(detected)f (in)g([Bal06)n(].)53 b(Let)33 b(us)278 923 y(also)28 b(p)r(oin)n(t)g(out)g(that)h(an)f(analogous)e(phenomenon)i(happ)r(ens)g (in)g(the)h(analytic)f(unfoldings)g(of)g(the)h(Hopf-zero)278 1023 y(singularit)n(y)e(\(see)g([BS06)o(,)h(BS08)o(]\))g(and)f(in)h(w)n (eak)f(resonances)e(of)j(area)e(preserving)g(maps)h([SV09].)p Black 195 1189 a Fw(\017)p Black 41 w FB(The)22 b(constan)n(t)e FA(C)6 b FB(\()h(^)-49 b FA(\026)q(;)14 b(")p FB(\))21 b(app)r(earing)f(in)i(Theorems)e(2.5)h(and)g(2.6)f(also)h(satis\014es)f FA(C)6 b FB(\()h(^)-49 b FA(\026)q(;)14 b(")p FB(\))23 b(=)f(0)f(if)h(the)g(Hamiltonian)278 1289 y FA(H)347 1301 y Fy(1)416 1289 y FB(in)32 b(\(8\))g(and)g(\(9\))g(do)r(es)f(not)h (dep)r(end)g(on)g FA(y)s FB(.)49 b(Moreo)n(v)n(er,)30 b(it)i(also)f(satis\014es)g(lim)2906 1301 y Fx(")p Fv(!)p Fy(0)3055 1289 y FA(C)6 b FB(\()h(^)-49 b FA(\026;)14 b(")p FB(\))30 b(=)g FA(C)3494 1301 y Fy(0)3531 1289 y FB(\()7 b(^)-49 b FA(\026)q FB(\))32 b(for)f(a)278 1388 y(certain)d(function)h FA(C)941 1400 y Fy(0)979 1388 y FB(\()7 b(^)-49 b FA(\026)p FB(\))28 b(whic)n(h)h(is)f(en)n (tire)g(analytic)f(in)35 b(^)-48 b FA(\026)p FB(.)39 b(W)-7 b(e)29 b(pro)n(v)n(e)d(this)j(fact)f(in)h(Section)f(9.4)f(and)i (w)n(e)e(giv)n(e)278 1488 y(an)h(explicit)g(expression)e(of)h FA(C)1240 1500 y Fy(0)1278 1488 y FB(\()7 b(^)-49 b FA(\026)p FB(\))28 b(in)g(terms)f(of)h(sev)n(eral)e(explicitly)h(computable)h (auxiliary)e(functions.)p Black 195 1654 a Fw(\017)p Black 41 w FB(If)j(one)f(w)n(eak)n(ens)f(Hyp)r(othesis)h Fp(HP3)g FB(to)g(admit)g(Hamiltonian)g(systems)g(with)h Fw(C)2849 1624 y Fy(1)2914 1654 y FB(dep)r(endence)g(on)f FA(\034)9 b FB(,)29 b(one)f(can)278 1753 y(get)g(analogous)d(results)i (to)h(the)g(ones)f(obtained)g(in)h(Theorems)e(2.5)h(and)g(2.6.)p Black 195 1919 a Fw(\017)p Black 41 w Fp(Comparison)33 b(with)g(Melnik)m(o)m(v)p FB(.)41 b(In)30 b(the)f(case)f FA(`)19 b Fw(\000)g FB(2)p FA(r)28 b(<)d FB(0,)k(taking)g(in)n(to)g (accoun)n(t)f(that)h(b)n(y)g(Theorem)f(2.5,)278 2019 y FA(C)6 b FB(\()p FA(\026;)14 b(")p FB(\))29 b(=)f Fw(O)r FB(\()p FA(\026")844 1989 y Fx(\021)885 2019 y FB(\),)k(one)e(has)g (that)h(Melnik)n(o)n(v)f(theory)g(predicts)g(the)h(area)e(of)i(the)g (lob)r(es)f(correctly)g(pro)n(vided)278 2119 y FA(\021)g(>)c FB(0)j(ev)n(en)h(if)g(in)g(this)g(case)e(this)i(theory)f(cannot)h(b)r (e)g(directly)f(applied)h(due)g(to)f(the)h(exp)r(onen)n(tial)f (smallness.)278 2218 y(If)36 b FA(\021)i FB(=)d(0,)i(it)e(do)r(es)g (not)g(predict)f(the)i(area)d(correctly)h(in)h(general.)57 b(Nev)n(ertheless,)36 b(since)f FA(C)42 b Fw(\021)34 b FB(0)h(when)g(the)278 2318 y(p)r(erturbation)24 b(do)r(es)g(not)g (dep)r(end)h(on)f FA(y)s FB(,)h(in)g(this)f(case)g(Melnik)n(o)n(v)f (theory)g(predicts)h(the)h(asymptotic)f(size)g(of)g(the)278 2417 y(area)k(of)i(the)f(lob)r(es)h(ev)n(en)f(if)g FA(\021)h FB(=)25 b(0,)30 b(that)g(is,)f(when)h(the)g(p)r(erturbation)f(has)f (the)i(same)f(size)g(as)g(the)h(in)n(tegrable)278 2517 y(system.)278 2650 y(In)e(the)g(cases)e FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b(>)e FB(0)k(and)g FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b FB(=)e(0,)k(w)n(e)g(kno)n(w)f(that)i(the)g(function)g FA(f)9 b FB(\()e(^)-49 b FA(\026)p FB(\))28 b(app)r(earing)e(in)i (Theorem)e(2.6,)278 2750 y(satis\014es)h(that)h(for)34 b(^)-49 b FA(\026)28 b FB(small)1734 2849 y FA(f)9 b FB(\()e(^)-49 b FA(\026)p FB(\))24 b(=)e FA(f)2050 2861 y Fy(0)2105 2849 y FB(+)c Fw(O)r FB(\()7 b(^)-49 b FA(\026)q FB(\))p FA(;)278 2998 y FB(where)30 b FA(f)562 3010 y Fy(0)627 2998 y Fw(2)e Ft(C)i FB(is)g(a)g(constan)n(t)g(indep)r(enden)n (t)h(of)f FA(\026)p FB(.)46 b(In)30 b([Bal06)o(],)h(it)g(is)f(seen)g (that)g(the)h(constan)n(t)f FA(f)3460 3010 y Fy(0)3527 2998 y FB(coincides)278 3098 y(with)f(the)f(constan)n(t)e(that)i (Melnik)n(o)n(v)f(theory)g(giv)n(es)f(in)i(fron)n(t)f(of)h(the)g(exp)r (onen)n(tial)f(term)g(\(see)h(Lemma)f(2.4\).)278 3231 y(In)j(other)f(w)n(ords,)g(this)g(means)g(that)h(for)f(the)h(case)f FA(`)19 b Fw(\000)g FB(2)p FA(r)29 b(>)c FB(0,)30 b(if)g(one)f(assumes) g(that)g FA(\021)h(>)c(`)19 b Fw(\000)g FB(2)p FA(r)r FB(,)31 b(Melnik)n(o)n(v)278 3330 y(theory)c(also)g(predicts)g(the)h (asymptotic)f(b)r(eha)n(vior)f(of)i(the)g(area)e(of)i(the)g(lob)r(es)f (correctly)-7 b(.)278 3463 y(In)30 b(the)g(case)e FA(`)20 b Fw(\000)f FB(2)p FA(r)28 b FB(=)e(0,)j FA(f)1178 3475 y Fy(0)1245 3463 y FB(also)f(corresp)r(onds)g(to)h(the)h(Melnik)n(o)n (v)e(theory)h(prediction.)42 b(Nev)n(ertheless,)29 b(since)278 3563 y(a)i(logarithmic)f(term)h(app)r(ears)e(in)i(the)h(exp)r(onen)n (tial,)f(in)g(the)g(case)g FA(\021)g FB(=)d(0)j(the)g(Melnik)n(o)n(v)f (prediction)h(is)f(v)-5 b(alid)278 3662 y(pro)n(vided)1798 3797 y Fw(j)p FA(\026)p Fw(j)23 b(\034)2145 3740 y FB(1)p 2033 3777 265 4 v 2033 3794 a Fz(p)p 2116 3794 182 4 v 71 x Fw(j)14 b FB(ln)g FA(")p Fw(j)2308 3797 y FA(:)278 4006 y FB(Of)21 b(course,)g(if)g FA(b)h FB(=)h(0,)e(as)f(happ)r(ens)h (when)f(the)h(p)r(erturbation)f(do)r(es)g(not)h(dep)r(end)g(on)f FA(y)s FB(,)i(the)e(Melnik)n(o)n(v)g(prediction)278 4106 y(is)28 b(v)-5 b(alid)28 b(for)f(an)n(y)f FA(\026)i FB(small)f(and)h (indep)r(enden)n(t)g(of)g FA(")p FB(.)71 4321 y Fp(2.2.4)94 b(Examples)71 4475 y FB(In)28 b(this)g(section)f(w)n(e)g(apply)g (Theorems)g(2.5)g(and)g(2.6)g(to)g(some)g(examples.)36 b(W)-7 b(e)28 b(consider)f(the)h(Du\016ng)g(equation)1499 4705 y FA(H)1568 4717 y Fy(0)1606 4705 y FB(\()p FA(x;)14 b(y)s FB(\))23 b(=)1919 4649 y FA(y)1963 4619 y Fy(2)p 1919 4686 81 4 v 1939 4762 a FB(2)2028 4705 y Fw(\000)2121 4649 y FA(x)2168 4619 y Fy(2)p 2121 4686 85 4 v 2143 4762 a FB(2)2234 4705 y Fw(\000)2327 4649 y FA(x)2374 4619 y Fy(4)p 2327 4686 V 2349 4762 a FB(4)71 4911 y(with)i(di\013eren) n(t)g(p)r(erturbations.)36 b(The)25 b(Du\016ng)g(equation)f(has)h(t)n (w)n(o)f(separatrices)e(forming)j(a)f(\014gure)g(eigh)n(t,)i(whic)n(h)e (are)71 5011 y(parameterized)i(b)n(y)1005 5180 y(\000)1057 5146 y Fv(\006)1113 5180 y FB(\()p FA(u)p FB(\))d(=)f(\()p Fw(\006)p FA(q)1469 5192 y Fy(0)1507 5180 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1698 5192 y Fy(0)1735 5180 y FB(\()p FA(u)p FB(\)\))23 b(=)1990 5038 y Fz( )2055 5180 y Fw(\006)2184 5056 y(p)p 2253 5056 42 4 v 68 x FB(2)p 2130 5161 219 4 v 2130 5237 a(cosh)13 b FA(u)2359 5180 y(;)h Fw(\007)2471 5056 y(p)p 2539 5056 42 4 v 2539 5124 a FB(2)g(sinh)f FA(u)p 2471 5161 334 4 v 2509 5246 a FB(cosh)2666 5209 y Fy(2)2717 5246 y FA(u)2814 5038 y Fz(!)2893 5180 y FA(:)71 5404 y FB(The)27 b(singularities)f(of)h(these)g(separatrices)d (whic)n(h)j(are)f(closer)g(to)h(the)g(real)f(axis)h(are)f FA(u)c FB(=)h Fw(\006)p FA(i\031)s(=)p FB(2)i(and)i FA(r)f FB(=)d(2)j(\(see)h(the)71 5504 y(de\014nition)h(of)f FA(r)k FB(in)d(Hyp)r(othesis)f Fp(HP2)p FB(\).)p Black 1919 5753 a(15)p Black eop end %%Page: 16 16 TeXDict begin 16 15 bop Black Black 195 272 a FB(W)-7 b(e)25 b(consider)f(t)n(w)n(o)f(di\013eren)n(t)i(t)n(yp)r(es)f(of)h(p)r (erturbations)e(and)i(w)n(e)f(study)g(ho)n(w)g(the)h(separatrix)e(\000) 3211 242 y Fy(+)3290 272 y FB(splits.)36 b(The)25 b(\014rst)71 372 y(p)r(erturbation)i(that)h(w)n(e)f(consider)g(is)1259 601 y FA(H)7 b FB(\()p FA(x;)14 b(y)s FB(\))23 b(=)1648 545 y FA(y)1692 515 y Fy(2)p 1648 582 81 4 v 1668 658 a FB(2)1757 601 y Fw(\000)1850 545 y FA(x)1897 515 y Fy(2)p 1850 582 85 4 v 1872 658 a FB(2)1963 601 y Fw(\000)2056 545 y FA(x)2103 515 y Fy(4)p 2056 582 V 2078 658 a FB(4)2169 601 y(+)18 b FA(\026")2341 567 y Fx(\021)2382 601 y FA(x)2429 567 y Fx(n)2488 601 y FB(sin)2618 545 y FA(t)p 2614 582 39 4 v 2614 658 a(")71 810 y FB(for)27 b FA(n)c Fw(2)g Ft(N)28 b FB(and)f FA(\021)g Fw(\025)22 b FB(0.)37 b(Then)27 b(the)h(order)f(of)g(the)h(p)r(erturbation)f(is)h FA(`)22 b FB(=)h FA(n)28 b FB(\(see)f(the)h(de\014nition)g(of)g FA(`)f FB(in)h(\(14\))o(\).)195 909 y(If)f(one)f(applies)g(Melnik)n(o)n (v)f(theory)h(to)g(these)g(Hamiltonian)g(systems,)h(obtains)e(the)i (follo)n(wing)e(prediction)h(for)g(the)71 1009 y(area)g(of)i(the)g(lob) r(es)1205 1144 y Fw(A)c FB(=)e Fw(j)p FA(\026)p Fw(j)p FA(")1517 1109 y Fx(\021)1681 1088 y FB(2)1733 1035 y Fm(n)p 1732 1044 37 3 v 1736 1077 a Fu(2)1779 1057 y Fy(+2)1867 1088 y FA(\031)p 1568 1125 464 4 v 1568 1201 a FB(\()p FA(n)c Fw(\000)g FB(1\)!)c FA(")1901 1177 y Fx(n)p Fv(\000)p Fy(1)2041 1144 y FA(e)2080 1106 y Fv(\000)2153 1073 y Fx(\031)p 2141 1087 65 4 v 2141 1135 a Fy(2)p Fx(")2238 1144 y FB(+)k Fw(O)2403 1076 y Fz(\000)2442 1144 y FA(\026)2492 1109 y Fy(2)2529 1144 y FA(")2568 1109 y Fy(2)p Fx(\021)2641 1076 y Fz(\001)2693 1144 y FA(:)987 b FB(\(18\))71 1336 y(F)-7 b(or)33 b FA(n)g(<)f FB(4,)j(whic)n(h)e(corresp)r(onds)f(to)h FA(`)22 b Fw(\000)g FB(2)p FA(r)35 b(<)e FB(0,)i(one)e(can)g(apply)g(Theorem)g(2.5)f(to)i (see)f(that)h(Melnik)n(o)n(v)e(theory)71 1436 y(predicts)27 b(correctly)f(the)i(area)e(of)i(the)g(lob)r(es)f(for)g(an)n(y)g FA(\021)f Fw(\025)d FB(0.)36 b(Namely)-7 b(,)1092 1668 y Fw(A)24 b FB(=)e Fw(j)p FA(\026)p Fw(j)p FA(")1404 1633 y Fx(\021)1568 1612 y FB(2)1620 1559 y Fm(n)p 1619 1568 37 3 v 1623 1601 a Fu(2)1666 1581 y Fy(+2)1754 1612 y FA(\031)p 1455 1649 464 4 v 1455 1725 a FB(\()p FA(n)c Fw(\000)g FB(1\)!)c FA(")1788 1701 y Fx(n)p Fv(\000)p Fy(1)1928 1668 y FA(e)1967 1630 y Fv(\000)2040 1597 y Fx(\031)p 2028 1611 65 4 v 2028 1659 a Fy(2)p Fx(")2121 1551 y Fz(\022)2182 1668 y FB(1)k(+)g Fw(O)2407 1551 y Fz(\022)2548 1612 y FB(1)p 2478 1649 182 4 v 2478 1725 a Fw(j)c FB(ln)g FA(")p Fw(j)2670 1551 y Fz(\023\023)2806 1668 y FA(:)874 b FB(\(19\))195 1894 y(The)28 b(case)g FA(n)23 b FB(=)g(4)28 b(corresp)r(onds)e(to)i FA(`)23 b FB(=)g(2)p FA(r)r FB(.)39 b(In)28 b(this)g(case,)f(since)h(the)h(p)r (erturbation)e(do)r(es)h(not)g(dep)r(end)g(on)g FA(y)s FB(,)g(w)n(e)71 1993 y(ha)n(v)n(e)e(that)i FA(b)23 b FB(=)f(0)28 b(and)f FA(C)i FB(=)23 b(0.)37 b(Then,)27 b(for)g(an)n(y)g FA(\021)g Fw(\025)22 b FB(0,)27 b(the)h(form)n(ula)f (of)h(the)g(area)e(is)h(giv)n(en)g(b)n(y)1145 2219 y Fw(A)c FB(=)g Fw(j)p FA(\026)p Fw(j)p FA(")1457 2185 y Fx(\021)1507 2163 y FB(4)14 b Fw(j)o FA(f)23 b FB(\()p FA(\026")1770 2133 y Fx(\021)1810 2163 y FB(\))p Fw(j)p 1507 2200 359 4 v 1602 2276 a FA(")1641 2252 y Fx(n)p Fv(\000)p Fy(1)1876 2219 y FA(e)1915 2182 y Fv(\000)1988 2149 y Fx(\031)p 1976 2163 65 4 v 1976 2210 a Fy(2)p Fx(")2068 2102 y Fz(\022)2130 2219 y FB(1)17 b(+)i Fw(O)2355 2102 y Fz(\022)2496 2163 y FB(1)p 2426 2200 182 4 v 2426 2276 a Fw(j)14 b FB(ln)g FA(")p Fw(j)2617 2102 y Fz(\023)q(\023)2754 2219 y FA(;)926 b FB(\(20\))71 2450 y(where)27 b FA(f)9 b FB(\()e(^)-49 b FA(\026)p FB(\))28 b(satis\014es)1505 2605 y FA(f)9 b FB(\()e(^)-49 b FA(\026)p FB(\))24 b(=)1840 2549 y(2)1892 2497 y Fm(n)p 1891 2506 37 3 v 1895 2539 a Fu(2)1942 2549 y FA(\031)s(i)p 1790 2586 281 4 v 1790 2662 a FB(\()p FA(n)19 b Fw(\000)f FB(1\)!)2099 2605 y(+)g Fw(O)e FB(\()7 b(^)-48 b FA(\026)p FB(\))14 b FA(:)1287 b FB(\(21\))71 2798 y(Then,)37 b(as)e(w)n(e)f(ha)n(v)n(e)g(explained)h (in)g(Section)h(2.2.3,)f(Melnik)n(o)n(v)f(theory)h(predicts)f (correctly)g(the)i(area)d(of)i(the)h(lob)r(es)71 2897 y(pro)n(vided)29 b FA(\021)i(>)c FB(0.)45 b(F)-7 b(or)30 b FA(\021)h FB(=)c(0)j(and)g(\014xed)g FA(\026)h FB(indep)r(enden)n(t)g (of)f FA(")p FB(,)h(the)g(\014rst)f(order)f(dep)r(ends)i(on)f(the)h (full)g(jet)g(of)f FA(f)9 b FB(\()e(^)-49 b FA(\026)p FB(\))71 2997 y(and)30 b(then)g(the)g(Melnik)n(o)n(v)f(function)h(do)r (es)g(not)g(predict)f(it)i(correctly)-7 b(.)42 b(Nev)n(ertheless)29 b(b)r(oth)h(form)n(ulas)f(\(19\))g(and)h(\(20\))71 3096 y(giv)n(e)d(a)g(\014rst)g(order)f(term)i(of)f(the)h(same)f(t)n(yp)r(e.) 195 3196 y(The)32 b(case)e FA(n)f(>)g FB(4)i(corresp)r(onds)f(to)h FA(`)e(>)g FB(2)p FA(r)r FB(.)48 b(Theorem)31 b(2.6)f(giv)n(es)h (results)f(for)h FA(\021)24 b FB(+)d(4)f Fw(\000)h FA(n)29 b Fw(\025)g FB(0.)48 b(Applying)31 b(this)71 3296 y(theorem,)c(one)g (has)g(that)999 3533 y Fw(A)c FB(=)g Fw(j)p FA(\026)p Fw(j)p FA(")1311 3499 y Fx(\021)1361 3472 y FB(4)1417 3401 y Fz(\014)1417 3451 y(\014)1444 3472 y FA(f)1508 3405 y Fz(\000)1546 3472 y FA(\026")1635 3442 y Fx(\021)r Fy(+4)p Fv(\000)p Fx(n)1852 3405 y Fz(\001)1890 3401 y(\014)1890 3451 y(\014)p 1361 3514 557 4 v 1555 3590 a FA(")1594 3566 y Fx(n)p Fv(\000)p Fy(1)1928 3533 y FA(e)1967 3496 y Fv(\000)2040 3463 y Fx(\031)p 2028 3477 65 4 v 2028 3524 a Fy(2)p Fx(")2121 3416 y Fz(\022)2182 3533 y FB(1)18 b(+)g Fw(O)2407 3416 y Fz(\022)2613 3477 y FB(1)p 2478 3514 312 4 v 2478 3590 a Fw(j)c FB(ln)g FA(")p Fw(j)2660 3566 y Fx(n)p Fv(\000)p Fy(4)2800 3416 y Fz(\023\023)71 3764 y FB(where)30 b FA(f)9 b FB(\()e(^)-49 b FA(\026)p FB(\))31 b(also)e(satis\014es)h(\(21\).)45 b(Then,)32 b(for)e FA(\021)h(>)d(n)20 b Fw(\000)g FB(4)30 b(Melnik)n(o)n(v)f(theory)h(predicts)h(the)f(area)g(correctly)f(and)h (for)71 3863 y FA(\021)c FB(=)d FA(n)18 b Fw(\000)g FB(4)27 b(and)h FA(\026)f FB(\014xed)h(and)f(indep)r(enden)n(t)i(of)e FA(")h FB(it)g(do)r(es)f(not.)195 3963 y(T)-7 b(o)25 b(see)f(ho)n(w)h(the)g(\014rst)g(asymptotic)f(order)g(of)g(the)i(area)d (of)i(the)g(lob)r(es)g(c)n(hanges)e(when)i(the)h(p)r(erturbation)e(dep) r(ends)71 4063 y(on)j FA(y)k FB(and)c FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b FB(=)e(0,)k(w)n(e)g(consider)g(the)h(follo)n(wing)f(p) r(erturbation)g(of)g(the)h(Du\016ng)g(equation,)996 4292 y FA(H)7 b FB(\()p FA(x;)14 b(y)s FB(\))24 b(=)1385 4236 y FA(y)1429 4206 y Fy(2)p 1385 4273 81 4 v 1405 4349 a FB(2)1495 4292 y Fw(\000)1588 4236 y FA(x)1635 4206 y Fy(2)p 1588 4273 85 4 v 1609 4349 a FB(2)1701 4292 y Fw(\000)1794 4236 y FA(x)1841 4206 y Fy(4)p 1794 4273 V 1815 4349 a FB(4)1907 4292 y(+)18 b FA(\026)2054 4175 y Fz(\022)2115 4292 y FA(x)2162 4258 y Fy(4)2214 4292 y FB(sin)2344 4236 y FA(t)p 2339 4273 39 4 v 2339 4349 a(")2406 4292 y FB(+)g FA(\025x)2584 4258 y Fy(2)2623 4292 y FA(y)e FB(cos)2819 4236 y FA(t)p 2815 4273 V 2815 4349 a(")2864 4175 y Fz(\023)71 4518 y FB(with)28 b FA(\025)23 b Fw(2)h Ft(R)p FB(.)37 b(F)-7 b(or)27 b(this)h(example,)f(Melnik)n(o)n (v)f(theory)h(predicts)h(that)f(the)h(area)e(of)i(the)g(lob)r(es)f(is) 1300 4733 y Fw(A)c FB(=)g Fw(j)p FA(\026)p Fw(j)1596 4677 y FB(4)p FA(\031)p 1583 4714 118 4 v 1583 4790 a FB(3)p FA(")1664 4766 y Fy(3)1710 4733 y Fw(j)p FB(2)18 b(+)1876 4660 y Fw(p)p 1945 4660 42 4 v 73 x FB(2)p FA(\025)p Fw(j)p FA(e)2097 4696 y Fv(\000)2171 4663 y Fx(\031)p 2159 4677 65 4 v 2159 4724 a Fy(2)p Fx(")2256 4733 y FB(+)g Fw(O)2421 4666 y Fz(\000)2459 4733 y FA(\026)2509 4699 y Fy(2)2547 4666 y Fz(\001)2598 4733 y FA(:)71 4942 y FB(Note)32 b(that)g(if)g(one)f(tak)n(es)g FA(\025)g FB(=)e(0,)k Fw(A)f FB(coincides)f(with)h(\(18\))g(with)g FA(n)e FB(=)f(4)j(and)f FA(\021)i FB(=)d(0.)49 b(On)31 b(the)i(other)e(hand,)i(if)f(one)71 5041 y(tak)n(es)26 b FA(\025)e FB(=)f Fw(\000)508 4973 y(p)p 576 4973 42 4 v 576 5041 a FB(2)28 b(the)g(Melnik)n(o)n(v)e(function)i(is)g (degenerate)e(since)h(the)h(\014rst)g(order)e(v)-5 b(anishes.)195 5141 y(F)e(or)30 b(this)h(p)r(erturbation)f FA(`)e FB(=)f(2)p FA(r)33 b FB(and)d(therefore)g(w)n(e)g(can)g(apply)g(Theorem)g(2.6.)45 b(Using)30 b(form)n(ula)g(\(75\))o(,)i(one)e(can)71 5241 y(easily)24 b(see)g(that)h FA(b)e FB(=)f Fw(\000)p FB(4)858 5172 y Fw(p)p 926 5172 V 926 5241 a FB(2)p FA(\025i)p FB(.)36 b(Therefore,)24 b(the)h(true)g(\014rst)f(asymptotic)g(order)g (of)g(the)h(area)e(of)i(the)g(lob)r(es)f(is)h(giv)n(en)f(b)n(y)900 5467 y Fw(A)g FB(=)e Fw(j)p FA(\026)p Fw(j)1201 5411 y FB(4)p 1183 5448 76 4 v 1183 5524 a FA(")1222 5500 y Fy(3)1269 5467 y FA(e)1308 5429 y Fv(\000)1382 5396 y Fx(\031)p 1370 5410 65 4 v 1370 5458 a Fy(2)p Fx(")1444 5429 y Fv(\000)p Fy(4)1529 5381 y Fv(p)p 1584 5381 34 3 v 48 x Fy(2)p Fx(\025\026)1696 5404 y Fu(2)1740 5429 y Fy(ln)1817 5407 y Fu(1)p 1817 5416 29 3 v 1817 5449 a Fm(")1873 5350 y Fz(\022)1934 5371 y(\014)1934 5421 y(\014)1934 5471 y(\014)1962 5467 y FA(f)9 b FB(\()p FA(\026)p FB(\))p FA(e)2165 5432 y Fx(iC)t Fy(\()p Fx(\026;")p Fy(\))2388 5371 y Fz(\014)2388 5421 y(\014)2388 5471 y(\014)2434 5467 y FB(+)18 b Fw(O)2599 5350 y Fz(\022)2740 5411 y FB(1)p 2670 5448 182 4 v 2670 5524 a Fw(j)c FB(ln)g FA(")p Fw(j)2862 5350 y Fz(\023\023)2998 5467 y FA(;)682 b FB(\(22\))p Black 1919 5753 a(16)p Black eop end %%Page: 17 17 TeXDict begin 17 16 bop Black Black 71 272 a FB(where)27 b FA(C)6 b FB(\()p FA(\026;)14 b(")p FB(\))23 b(=)g Fw(O)r FB(\()p FA(\026)p FB(\))29 b(and)e FA(f)9 b FB(\()p FA(\026)p FB(\))28 b(satis\014es)1399 472 y FA(f)9 b FB(\()p FA(\026)p FB(\))23 b(=)1683 416 y FA(\031)s(i)p 1683 453 79 4 v 1702 529 a FB(3)1786 380 y Fz(\020)1836 472 y FB(2)18 b(+)1979 400 y Fw(p)p 2048 400 42 4 v 72 x FB(2)o FA(\025)2137 380 y Fz(\021)2206 472 y FB(+)g Fw(O)e FB(\()p FA(\026)p FB(\))f FA(:)71 660 y FB(One)27 b(can)g(tak)n(e)g FA(\026)c FB(=)g(1)k(and)h(write)f(form)n(ula)g(\(22\))g(as)1056 870 y Fw(A)c FB(=)1366 814 y(4)p 1243 851 288 4 v 1243 938 a FA(")1282 914 y Fy(3)p Fv(\000)p Fy(4)1400 866 y Fv(p)p 1454 866 34 3 v 1454 914 a Fy(2)p Fx(\025)1541 870 y FA(e)1580 832 y Fv(\000)1653 800 y Fx(\031)p 1641 814 65 4 v 1641 861 a Fy(2)p Fx(")1734 753 y Fz(\022)1795 774 y(\014)1795 824 y(\014)1795 874 y(\014)1822 870 y FA(f)9 b FB(\(1\))p FA(e)2017 836 y Fx(iC)t Fy(\(1)p Fx(;")p Fy(\))2232 774 y Fz(\014)2232 824 y(\014)2232 874 y(\014)2278 870 y FB(+)18 b Fw(O)2444 753 y Fz(\022)2585 814 y FB(1)p 2515 851 182 4 v 2515 927 a Fw(j)c FB(ln)g FA(")p Fw(j)2706 753 y Fz(\023\023)2842 870 y FA(:)71 1075 y FB(Therefore,)20 b(the)h(correcting)e(logarithmic)g(term)h (implies)g(a)g(drastic)g(c)n(hange)f(in)h(the)h(degree)e(of)h(the)h(p)r (olynomial)e(term)i(of)71 1175 y(the)k(asymptotics.)35 b(Note)24 b(that)h(one)e(can)h(tak)n(e)g(an)n(y)f FA(\025)h Fw(2)f Ft(R)i FB(and)f(then)h(the)f(p)r(olynomial)g(term)g(can)g(c)n (hange)f(arbitrarily)-7 b(,)71 1274 y(b)r(oth)28 b(increasing)e(or)h (decreasing)f(the)i(degree.)35 b(Finally)-7 b(,)28 b(if)g(one)f(tak)n (es)g FA(\025)d FB(=)e(0,)27 b(one)h(reco)n(v)n(ers)d(form)n(ula)h (\(20\).)71 1503 y Fq(2.3)112 b(Near)35 b(in)m(tegrable)h(Hamiltonian)h (systems)f(of)f FF(1)2363 1463 y Fj(1)p 2363 1480 36 4 v 2363 1537 a(2)2443 1503 y Fq(degrees)h(of)f(freedom)h(close)f(to)g (a)326 1619 y(resonance)71 1772 y FB(The)d(results)g(obtained)g(in)h (this)f(w)n(ork)f(can)h(b)r(e)h(easily)f(adapted)g(to)g(study)g(near)g (in)n(tegrable)f(Hamiltonian)h(systems)71 1872 y(of)k(1)226 1839 y Fy(1)p 226 1853 34 4 v 226 1900 a(2)305 1872 y FB(degrees)f(of)h(freedom)g(close)f(to)h(a)g(resonance.)62 b(Let)36 b(us)g(consider)f(an)h(analytic)g(Hamiltonian)g(system)g(with) 71 1971 y(Hamiltonian)1386 2071 y FA(h)p FB(\()p FA(x;)14 b(I)7 b(;)14 b(\034)9 b FB(\))24 b(=)f FA(h)1867 2083 y Fy(0)1904 2071 y FB(\()p FA(I)7 b FB(\))19 b(+)f FA(\016)s(h)2201 2083 y Fy(1)2238 2071 y FB(\()p FA(x;)c(I)7 b(;)14 b(\034)9 b FB(\))p FA(;)1169 b FB(\(23\))71 2207 y(where)27 b FA(\016)f Fw(\034)d FB(1)k(is)g(a)g(small)h(parameter,)e(\()p FA(x;)14 b(\034)9 b FB(\))24 b Fw(2)g Ft(T)1690 2177 y Fy(2)1727 2207 y FB(,)k FA(I)i Fw(2)23 b Ft(R)28 b FB(and)f FA(h)2219 2219 y Fy(1)2284 2207 y FB(is)g(a)g(trigonometric)f (p)r(olynomial)h(as)g(a)g(function)71 2306 y(of)33 b FA(x)p FB(.)54 b(When)33 b FA(\016)i FB(=)d(0,)i(the)g(Hamiltonian)e (system)h(is)g(completely)g(in)n(tegrable)f(\(in)h(the)h(sense)e(of)h (Liouville-Arnold\))71 2406 y(and)27 b(the)h(phase)f(space)g(is)h (foliated)f(b)n(y)h(in)n(v)-5 b(arian)n(t)26 b(tori)h(with)h(frequency) f FA(!)s FB(\()p FA(I)7 b FB(\))24 b(=)e(\()p FA(@)2744 2418 y Fx(I)2783 2406 y FA(h)2831 2418 y Fy(0)2868 2406 y FB(\()p FA(I)7 b FB(\))p FA(;)14 b FB(1\).)195 2506 y(In)31 b(particular,)g(if)g(for)g(certain)f FA(I)7 b FB(,)32 b(there)f(exists)f FA(k)h Fw(2)e Ft(Z)1962 2476 y Fy(2)2031 2506 y FB(suc)n(h)i(that)g FA(!)s FB(\()p FA(I)7 b FB(\))21 b Fw(\001)f FA(k)32 b FB(=)c(0,)j(the)g(corresp)r (onding)e(torus)i(is)71 2605 y(foliated)g(b)n(y)h(p)r(erio)r(dic)f (orbits.)49 b(When)32 b FA(\016)h(>)c FB(0)i(\(but)i(small)e(enough\),) h(it)g(is)g(a)f(w)n(ell)h(kno)n(wn)e(fact)i(that)g(t)n(ypically)f(this) 71 2705 y(torus,)c(a)g(resonan)n(t)f(torus,)h(breaks)f(do)n(wn.)195 2805 y(Let)i(us)g(consider)e(the)i(simplest)g(setting)g(and)f(let)h(us) g(assume)e(that)1151 3013 y FA(h)1199 3025 y Fy(0)1236 3013 y FB(\()p FA(I)7 b FB(\))23 b(=)1464 2957 y FA(I)1507 2927 y Fy(2)p 1464 2994 81 4 v 1483 3070 a FB(2)1573 3013 y(+)18 b FA(G)p FB(\()p FA(I)7 b FB(\))194 b(with)28 b FA(G)p FB(\()p FA(I)7 b FB(\))24 b(=)f Fw(O)2577 2946 y Fz(\000)2615 3013 y FA(I)2658 2979 y Fy(3)2696 2946 y Fz(\001)2747 3013 y FA(:)71 3201 y FB(Then)31 b FA(I)36 b FB(=)29 b(0)h(corresp)r(onds)g(to)h(the)g(resonan)n(t)f(v)n(ector)g FA(!)s FB(\(0\))f(=)f(\(0)p FA(;)14 b FB(1\).)47 b(T)-7 b(o)31 b(study)g(the)h(dynamics)f(of)g(the)g(p)r(erturb)r(ed)71 3300 y(system)c(around)g(this)h(resonance,)e(one)h(usually)g(p)r (erforms)g(the)h(rescaling)1487 3492 y FA(I)i FB(=)1641 3418 y Fw(p)p 1710 3418 41 4 v 74 x FA(\016)s(y)113 b FB(and)e FA(\034)32 b FB(=)2355 3436 y FA(t)p 2315 3473 110 4 v 2315 3490 a Fw(p)p 2384 3490 41 4 v 71 x FA(\016)71 3716 y FB(and)27 b(tak)n(es)g FA(")c FB(=)594 3645 y Fw(p)p 664 3645 V 664 3716 a FA(\016)30 b FB(as)d(a)g(new)h(parameter.) 35 b(Then,)28 b(one)f(obtains)h(the)f(Hamiltonian)819 3924 y FA(H)7 b FB(\()p FA(x;)14 b(y)s(;)g(t)p FB(\))24 b(=)1275 3868 y FA(y)1319 3838 y Fy(2)p 1275 3905 81 4 v 1295 3981 a FB(2)1385 3924 y(+)1495 3868 y(1)p 1478 3905 76 4 v 1478 3981 a FA(")1517 3957 y Fy(2)1563 3924 y FA(G)p FB(\()p FA("y)s FB(\))19 b(+)f FA(V)h FB(\()p FA(x)p FB(\))h(+)e FA(F)2236 3807 y Fz(\022)2297 3924 y FA(x;)2396 3868 y(t)p 2392 3905 39 4 v 2392 3981 a(")2440 3807 y Fz(\023)2520 3924 y FB(+)g FA(R)2681 3807 y Fz(\022)2742 3924 y FA(x;)c("y)s(;)2960 3868 y(t)p 2955 3905 V 2955 3981 a(")3004 3807 y Fz(\023)3079 3924 y FA(;)71 4129 y FB(where)1248 4328 y FA(V)19 b FB(\()p FA(x)p FB(\))24 b(=)f Fw(h)p FA(h)1618 4340 y Fy(1)1655 4328 y FB(\()p FA(x;)14 b FB(0)p FA(;)g(\034)9 b FB(\))p Fw(i)24 b FB(=)2106 4271 y(1)p 2081 4308 92 4 v 2081 4385 a(2)p FA(\031)2197 4215 y Fz(Z)2280 4235 y Fy(2)p Fx(\031)2243 4403 y Fy(0)2372 4328 y FA(h)2420 4340 y Fy(1)2457 4328 y FB(\()p FA(x;)14 b FB(0)p FA(;)g(\034)9 b FB(\))14 b FA(d\034)1167 4506 y(F)e FB(\()p FA(x;)i(\034)9 b FB(\))25 b(=)e FA(h)1586 4518 y Fy(1)1623 4506 y FB(\()p FA(x;)14 b FB(0)p FA(;)g(\034)9 b FB(\))19 b Fw(\000)f(h)p FA(h)2077 4518 y Fy(1)2115 4506 y FB(\()p FA(x;)c FB(0)p FA(;)g(\034)9 b FB(\))p Fw(i)1089 4631 y FA(R)q FB(\()p FA(x;)14 b(I)7 b(;)14 b(\034)9 b FB(\))24 b(=)f FA(h)1586 4643 y Fy(1)1623 4631 y FB(\()p FA(x;)14 b(I)7 b(;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(h)2047 4643 y Fy(1)2084 4631 y FB(\()p FA(x;)c FB(0)p FA(;)g(\034)9 b FB(\))p FA(;)71 4787 y FB(whic)n(h)27 b(can)h(b)r(e)g(written)f(as)1192 4915 y FA(H)1281 4798 y Fz(\022)1342 4915 y FA(x;)14 b(y)s(;)1522 4859 y(t)p 1517 4896 39 4 v 1517 4972 a(")1566 4798 y Fz(\023)1650 4915 y FB(=)23 b FA(H)1807 4927 y Fy(0)1844 4915 y FB(\()p FA(x;)14 b(y)s FB(\))19 b(+)f FA(\026H)2257 4927 y Fy(1)2308 4798 y Fz(\022)2369 4915 y FA(x;)c(y)s(;)2549 4859 y(t)p 2544 4896 V 2544 4972 a(")2593 4915 y(;)g(")2669 4798 y Fz(\023)71 5096 y FB(with)1247 5289 y FA(H)1316 5301 y Fy(0)1353 5289 y FB(\()p FA(x;)g(y)s FB(\))24 b(=)1667 5233 y FA(y)1711 5202 y Fy(2)p 1667 5270 81 4 v 1686 5346 a FB(2)1776 5289 y(+)18 b FA(V)h FB(\()p FA(x)p FB(\))1093 5489 y FA(H)1162 5501 y Fy(1)1200 5489 y FB(\()p FA(x;)14 b(y)s(;)g(\034)5 b(;)14 b(")p FB(\))23 b(=)g FA(F)12 b FB(\()p FA(x;)i(\034)9 b FB(\))20 b(+)2045 5432 y(1)p 2028 5469 76 4 v 2028 5546 a FA(")2067 5522 y Fy(2)2113 5489 y FA(G)p FB(\()p FA("y)s FB(\))f(+)f FA(R)q FB(\()p FA(x;)c("y)s(;)g(\034)9 b FB(\))p FA(:)p Black 1919 5753 a FB(17)p Black eop end %%Page: 18 18 TeXDict begin 18 17 bop Black Black 71 272 a FB(Here)30 b FA(\026)h FB(is)g(in)g(fact)g(a)f(fak)n(e)g(parameter,)g(since)h(w)n (e)f(are)g(in)n(terested)g(in)h(the)h(case)d FA(\026)g FB(=)f(1.)46 b(This)30 b(system)h(is)g(similar)f(to)71 372 y(the)d(ones)e(considered)g(in)i(this)g(pap)r(er.)36 b(Let)26 b(us)g(p)r(oin)n(t)h(out)f(also)f(that,)i(b)n(y)f (de\014nition,)h FA(")2848 342 y Fv(\000)p Fy(2)2937 372 y FA(G)p FB(\()p FA("y)s FB(\))f(and)h FA(R)q FB(\()p FA(x;)14 b("y)s(;)g(\034)9 b FB(\))26 b(are)71 471 y(of)h(order)g FA(")p FB(.)195 571 y(Let)g(us)g(assume)f(that)h(the)g(Hamiltonian)g FA(H)34 b FB(satis\014es)26 b(Hyp)r(otheses)g Fp(HP1)p FB(-)p Fp(HP4)g FB(and)h(instead)f(of)34 b Fp(HP5)26 b FB(satis\014es)71 671 y(the)e(alternativ)n(e)f(h)n(yp)r(othesis)g (that)h FA(V)19 b FB(,)25 b(whic)n(h)f(is)g(a)f(trigonometric)g(p)r (olynomial,)h(has)f(the)h(same)g(degree)e(as)i FA(h)3549 683 y Fy(1)3610 671 y FB(in)g(\(23\))71 770 y(as)30 b(a)g(function)i (of)e FA(x)p FB(.)47 b(Then,)32 b(using)e(the)h(to)r(ols)f(considered)g (in)h(this)g(pap)r(er,)g(one)g(can)f(giv)n(e)g(an)g(asymptotic)g(form)n (ula)71 870 y(analogous)35 b(to)i(the)g(one)g(giv)n(en)f(in)i(Theorem)e (2.6.)65 b(Let)37 b(us)g(p)r(oin)n(t)h(out)f(that)g(in)h(this)f (setting,)j(ev)n(en)d(if)g(the)h(terms)71 969 y FA(")110 939 y Fv(\000)p Fy(2)199 969 y FA(G)p FB(\()p FA("y)s FB(\))28 b(and)g FA(R)q FB(\()p FA(x;)14 b("y)s(;)g(\034)9 b FB(\))29 b(are)f(of)g(order)f FA(")h FB(and)g(therefore)f(smaller)g (than)i FA(F)12 b FB(\()p FA(x;)i(\034)9 b FB(\),)30 b(the)e(function)h FA(f)9 b FB(\()p FA(\026)p FB(\))29 b(app)r(earing)71 1069 y(in)35 b(Theorem)g(2.6)f(dep)r(ends)h(not)h (only)e(on)h FA(F)47 b FB(but)36 b(also)e(on)h(the)h(full)f(jet)h(in)g FA(y)i FB(of)d FA(G)g FB(and)g FA(R)q FB(.)59 b(The)36 b(reason)d(is)i(that)71 1169 y(these)29 b(terms)f(b)r(ecome)h(of)g(the) g(same)g(order)e(as)i FA(V)19 b FB(\()p FA(x)p FB(\))30 b(and)e FA(F)12 b FB(\()p FA(x;)i(\034)9 b FB(\))31 b(close)d(to)h(the) g(singularities)f(of)h(the)g(unp)r(erturb)r(ed)71 1268 y(separatrix.)40 b(Moreo)n(v)n(er,)28 b(for)h(these)g(systems,)g(the)h (\014rst)g(asymptotic)e(order)h(also)f(has)h(the)h(logarithmic)e(term)h (in)h(the)71 1368 y(exp)r(onen)n(tial)35 b(as)f(it)i(happ)r(ens)g(in)f (Theorem)g(2.6)f(for)h FA(`)23 b Fw(\000)g FB(2)p FA(r)39 b FB(=)c(0.)60 b(W)-7 b(e)36 b(plan)f(to)g(study)h(rigorously)d(these)i (kind)h(of)71 1468 y(systems)27 b(in)h(future)g(w)n(ork.)71 1742 y FC(3)135 b(Heuristic)45 b(ideas)h(of)f(the)h(pro)t(of)71 1924 y FB(The)32 b(rigorous)f(pro)r(ofs)g(of)i(asymptotic)f(form)n (ulas)f(for)h(measuring)g(the)g(splitting)h(of)g(separatrices)d (require)i(a)g(signif-)71 2024 y(ican)n(t)i(amoun)n(t)f(of)h(tec)n (hnicalities.)57 b(F)-7 b(or)33 b(the)i(con)n(v)n(enience)e(of)h(the)g (reader,)h(ev)n(en)f(though)g(in)g(Section)g(4)g(w)n(e)g(giv)n(e)f(a)71 2123 y(precise)g(description)g(of)g(the)h(en)n(tire)f(pro)r(of)g(of)h (Theorems)e(2.5)h(and)g(2.6,)h(w)n(e)g(\014rst)f(dev)n(ote)g(this)h (section)f(to)g(giv)n(e)g(an)71 2223 y(heuristic)d(description)f(of)h (our)f(strategy)f(explaining)i(the)g(main)g(di\013erences)f(resp)r(ect) h(to)g(the)g(ones)f(already)f(used)i(in)71 2323 y(the)i(literature.)48 b(W)-7 b(e)32 b(also)e(explain)h(the)h(main)g(no)n(v)n(elties)e(w)n(e)h (ha)n(v)n(e)f(in)n(tro)r(duced)i(to)f(o)n(v)n(ercome)e(the)j (di\016culties)g(that)71 2422 y(our)27 b(general)f(setting)h(in)n(v)n (olv)n(es.)71 2655 y Fq(3.1)112 b(Measuring)39 b(the)f(splitting)f(b)m (y)h(using)g(generating)g(functions)71 2808 y FB(The)27 b(main)h(idea)f(of)h(the)g(metho)r(d)g(that)g(w)n(e)f(are)f(going)h(to) g(explain,)h(w)n(as)e(in)n(tro)r(duced)h(in)h([LMS03)o(,)g(Sau01)o(],)g (based)f(on)71 2907 y(ideas)32 b(b)n(y)g(P)n(oincar)n(\023)-39 b(e)29 b([P)n(oi99)n(].)51 b(Roughly)32 b(sp)r(eaking,)h(if)g(the)f(in) n(v)-5 b(arian)n(t)32 b(manifolds)g(can)g(b)r(e)h(expressed)e(in)i(a)f (suitable)71 3007 y(w)n(a)n(y)-7 b(,)22 b(then)i(the)f(area)e(of)i(the) g(lob)r(es)f(generated)g(b)n(y)g(the)i(p)r(erturb)r(ed)f(manifolds)f(b) r(et)n(w)n(een)h(t)n(w)n(o)f(consecutiv)n(e)g(homo)r(clinic)71 3107 y(p)r(oin)n(ts)31 b(and)h(also)e(the)i(distance)f(b)r(et)n(w)n (een)h(the)g(manifolds)f(can)g(b)r(e)h(simply)g(computed)g(b)n(y)f(the) h(di\013erence)g(b)r(et)n(w)n(een)71 3206 y(t)n(w)n(o)27 b(functions.)195 3306 y(Let)34 b(us)e(explain)h(this)g(approac)n(h)e (in)j(more)e(detail.)53 b(Assume)33 b(that)g(w)n(e)g(are)f(in)h(the)h (Hamiltonian)e(setting)h(giv)n(en)71 3406 y(in)g(Section)g(2.)52 b(As)33 b(the)g(main)g(goal)e(is)i(to)f(measure)g(the)h(distance)g(of)g (the)g(stable)f(and)h(unstable)g(manifolds)f(of)h(the)71 3505 y(p)r(erio)r(dic)24 b(orbit)f(\()p FA(x)664 3517 y Fx(p)703 3505 y FB(\()p FA(t=")p FB(\))p FA(;)14 b(y)956 3517 y Fx(p)994 3505 y FB(\()p FA(t=")p FB(\)\))24 b(in)g(a)f(P)n (oincar)n(\023)-39 b(e)21 b(section)i(\006)2053 3517 y Fx(t)2078 3525 y Fu(0)2115 3505 y FB(,)h(it)g(is)g(useful)g(to)g (obtain)f(these)h(manifolds)g(as)f(graphs.)71 3605 y(The)28 b(stable)f(and)g(unstable)h(manifolds)f(of)h(the)g(unp)r(erturb)r(ed)g (system)f(can)g(b)r(e)h(expressed)f(as)g(graphs)f(as)1065 3787 y FA(y)g FB(=)c FA(')p FB(\()p FA(x;)14 b(t=")p FB(\))24 b(=)f FA(y)1685 3799 y Fx(p)1723 3787 y FB(\()p FA(t=")p FB(\))18 b(+)g FA(@)2043 3799 y Fx(x)2085 3787 y FA(S)2141 3753 y Fx(s;u)2235 3787 y FB(\()p FA(x)h Fw(\000)f FA(x)2463 3799 y Fx(p)2502 3787 y FB(\()p FA(t=")p FB(\))p FA(;)c(t=")p FB(\))71 3970 y(in)28 b(some)f(complex)g(domains,) g(where)g(the)h(functions)g FA(S)1849 3940 y Fx(s;u)1971 3970 y FB(are)f(called)g(generating)f(functions.)37 b(As)28 b(it)g(is)g(p)r(oin)n(ted)f(out)71 4070 y(in)e([Sau01)o(])g(the)g (generating)e(functions)i FA(S)1407 4040 y Fx(s;u)1501 4070 y FB(\()p FA(q)s(;)14 b(\034)9 b FB(\))26 b(are)e(solutions)g(of)h (the)g(Hamilton-Jacobi)e(equation)h(asso)r(ciated)g(to)71 4169 y(our)j(Hamiltonian)g(system)g(after)h(the)g(c)n(hange)e(of)i(v)-5 b(ariables)1345 4352 y FA(q)26 b FB(=)d FA(x)18 b Fw(\000)g FA(x)1691 4364 y Fx(p)1730 4352 y FB(\()p FA(t=")p FB(\))p FA(;)97 b(p)23 b FB(=)f FA(y)g Fw(\000)c FA(y)2364 4364 y Fx(p)2402 4352 y FB(\()p FA(t=")p FB(\))71 4535 y(and)27 b(the)h(c)n(hange)f(of)g(time)h FA(\034)33 b FB(=)23 b FA(t=")p FB(.)195 4634 y(Then,)32 b(to)f(measure)f(the)h(distance)g (b)r(et)n(w)n(een)g(the)g(stable)g(and)g(the)g(unstable)g(manifolds)f (in)i(a)e(P)n(oincar)n(\023)-39 b(e)28 b(section)71 4734 y(w)n(e)f(just)h(need)g(to)f(compute:)1264 4834 y FA(d)p FB(\()p FA(q)s(;)14 b(t)1446 4846 y Fy(0)1483 4834 y FB(\))24 b(=)e FA(@)1670 4846 y Fx(q)1707 4834 y FA(S)1763 4799 y Fx(u)1806 4834 y FB(\()p FA(q)s(;)14 b(t)1945 4846 y Fy(0)1983 4834 y FA(=")p FB(\))k Fw(\000)g FA(@)2241 4846 y Fx(q)2277 4834 y FA(S)2333 4799 y Fx(s)2369 4834 y FB(\()p FA(q)s(;)c(t)2508 4846 y Fy(0)2545 4834 y FA(=")p FB(\))1045 b(\(24\))71 4983 y(and)27 b(it)h(is)g(standard)e(that)i(the) g(area)e(of)i(the)g(lob)r(es)f(is)h(giv)n(en)f(b)n(y)832 5166 y Fw(A)d FB(=)e FA(S)1065 5131 y Fx(u)1109 5166 y FB(\()p FA(q)1178 5178 y Fy(2)1215 5166 y FA(;)14 b(t)1282 5178 y Fy(0)1319 5166 y FA(=")p FB(\))k Fw(\000)g FA(S)1589 5131 y Fx(s)1624 5166 y FB(\()p FA(q)1693 5178 y Fy(2)1731 5166 y FA(;)c(t)1798 5178 y Fy(0)1835 5166 y FA(=")p FB(\))k Fw(\000)g FB(\()p FA(S)2137 5131 y Fx(u)2181 5166 y FB(\()p FA(q)2250 5178 y Fy(1)2287 5166 y FA(;)c(t)2354 5178 y Fy(0)2391 5166 y FA(=")p FB(\))k Fw(\000)g FA(S)2661 5131 y Fx(s)2696 5166 y FB(\()p FA(q)2765 5178 y Fy(1)2803 5166 y FA(;)c(t)2870 5178 y Fy(0)2907 5166 y FA(=")p FB(\)\))g FA(;)614 b FB(\(25\))71 5348 y(where)36 b FA(q)357 5360 y Fy(1)394 5348 y FB(,)i FA(q)492 5360 y Fy(2)566 5348 y FB(are)d(the)i(co)r(ordinates)e(of)h(t)n(w)n(o)g(consecutiv)n(e) f(homo)r(clinic)h(p)r(oin)n(ts)g(in)h(the)f(section)g(\006)3318 5360 y Fx(t)3343 5368 y Fu(0)3380 5348 y FB(.)63 b(Note)36 b(that,)71 5448 y(thanks)27 b(to)h(the)g(symplectic)f(structure,)g Fw(A)h FB(do)r(es)g(not)f(dep)r(end)h(on)g FA(t)2236 5460 y Fy(0)2273 5448 y FB(.)p Black 1919 5753 a(18)p Black eop end %%Page: 19 19 TeXDict begin 19 18 bop Black Black 195 272 a FB(W)-7 b(e)28 b(p)r(erform)e(the)h(c)n(hange)f(of)g(v)-5 b(ariables)26 b FA(q)g FB(=)c FA(q)1691 284 y Fy(0)1729 272 y FB(\()p FA(u)p FB(\),)27 b(where)f FA(q)2167 284 y Fy(0)2204 272 y FB(\()p FA(u)p FB(\))h(is)g(the)g(\014rst)g(comp)r(onen)n(t)f(of) h(the)g(unp)r(erturb)r(ed)71 372 y(homo)r(clinic)g(orbit.)37 b(In)28 b(this)f(w)n(a)n(y)-7 b(,)27 b(w)n(e)g(w)n(ork)f(with)i(the)g (function)1489 546 y FA(T)1550 512 y Fx(u;s)1643 546 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)f FA(S)2005 512 y Fx(u;s)2099 546 y FB(\()p FA(q)2168 558 y Fy(0)2205 546 y FB(\()p FA(u)p FB(\))p FA(;)14 b(\034)9 b FB(\))71 721 y(that)25 b(is,)g(w)n(e)f(write)g(the)h(p)r(erturb)r(ed)g (manifolds)f(as)g(functions)g(of)h(time)g FA(\034)34 b FB(and)24 b(the)h(\\time)g(o)n(v)n(er)d(the)j(homo)r(clinic)f(orbit") 71 821 y FA(u)p FB(,)f(whic)n(h)g(parameterizes)e(the)i(unp)r(erturb)r (ed)g(homo)r(clinic)f(orbit.)35 b(These)23 b(functions)g(satisfy)f(a)g (new)h(Hamilton-Jacobi)71 920 y(equation,)k(whic)n(h)g(is)h(easier)e (to)i(deal)f(with.)195 1020 y(Let)h(us)g(consider)e(their)i (di\013erence)1417 1195 y(\001\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)e FA(T)1852 1161 y Fx(u)1895 1195 y FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b Fw(\000)f FA(T)2252 1161 y Fx(s)2287 1195 y FB(\()p FA(u;)c(\034)9 b FB(\))p FA(:)195 1370 y FB(The)29 b(\014rst)f(observ)-5 b(ation)26 b(is)j(that,)f(when)h FA(\026)24 b FB(=)f(0,)29 b FA(@)1786 1382 y Fx(u)1829 1370 y FA(T)1890 1339 y Fx(u;s)1984 1370 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)g FA(@)2335 1382 y Fx(u)2378 1370 y FA(T)2427 1382 y Fy(0)2464 1370 y FB(\()p FA(u)p FB(\))g(=)g FA(p)2731 1382 y Fy(0)2768 1370 y FB(\()p FA(u)p FB(\))p FA(@)2924 1382 y Fx(q)2961 1370 y FA(S)3017 1339 y Fx(u;s)3111 1370 y FB(\()p FA(q)3180 1382 y Fy(0)3218 1370 y FB(\()p FA(u)p FB(\))p FA(;)14 b(\034)9 b FB(\))25 b(=)e(\()p FA(p)3631 1382 y Fy(0)3669 1370 y FB(\()p FA(u)p FB(\)\))3813 1339 y Fy(2)71 1469 y FB(whic)n(h)37 b(corresp)r(onds)d(to)j(the)g(parameterization)e(of)i(the)g(unp)r (erturb)r(ed)g(separatrix.)62 b(Then,)40 b(b)n(y)c(analyticit)n(y)g (with)71 1569 y(resp)r(ect)27 b(to)h(the)g(regular)d(parameter)i FA(\026)p FB(,)g(w)n(e)h(ha)n(v)n(e)e(that)i(\001\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)e Fw(O)r FB(\()p FA(\026)p FB(\).)195 1668 y(The)34 b(second)f(observ)-5 b(ation)32 b(is)i(that,)i(as)d(the)h(exp)r(erts)f(in)h(this)g(area)e(kno)n(w,)j (\001\()p FA(u;)14 b(\034)9 b FB(\))34 b(is)g(exp)r(onen)n(tially)f (small)g(in)71 1768 y(the)d(singular)e(parameter)h FA(")p FB(.)43 b(T)-7 b(o)29 b(obtain)h(sharp)f(estimates)g(of)h(\001\()p FA(u;)14 b(\034)9 b FB(\),)31 b(w)n(e)e(need)h(to)g(b)r(ound)g(it,)h (and)e(consequen)n(tly)71 1868 y FA(T)132 1838 y Fx(u)174 1868 y FB(\()p FA(u;)14 b(\034)9 b FB(\))35 b(and)f FA(T)632 1838 y Fx(s)667 1868 y FB(\()p FA(u;)14 b(\034)9 b FB(\),)37 b(in)d(a)g(region)f(of)h(the)h(complex)f(plane)g(that,)i(on)e(one)g (hand,)i(con)n(tains)d(a)h(segmen)n(t)g(of)g(the)71 1967 y(real)26 b(line)i(ha)n(ving)e(t)n(w)n(o)g(v)-5 b(alues)27 b(of)g FA(u)g FB(giving)f(rise)h(to)g(t)n(w)n(o)g(consecutiv)n(e)f (homo)r(clinic)h(p)r(oin)n(ts)g(and,)g(on)g(the)h(other)e(hand,)71 2067 y(in)n(tersects)h(a)g(neigh)n(b)r(orho)r(o)r(d)f(su\016cien)n(tly) i(close)f(to)g(the)h(singularities)f Fw(\006)p FA(ia)f FB(of)i FA(T)2659 2079 y Fy(0)2696 2067 y FB(\()p FA(u)p FB(\).)195 2167 y(Assume)f(that)h(w)n(e)e(can)h(construct)f (parameterizations)f FA(T)2054 2136 y Fx(u;s)2147 2167 y FB(\()p FA(u;)14 b(\034)9 b FB(\))28 b(of)f(the)g(p)r(erturb)r(ed)h (in)n(v)-5 b(arian)n(t)25 b(manifolds)i(sat-)71 2266 y(isfying)h(b)r(oth)h(that)g(they)f(are)g(2)p FA(\031)s FB(-p)r(erio)r(dic)g(with)h(resp)r(ect)f(to)g FA(\034)38 b FB(and)29 b(that)f(they)h(are)f(real-analytic)e(and)i(b)r(ounded)h (in)71 2366 y(some)e(complex)h(domain)f(whic)n(h)h(con)n(tains)f(t)n(w) n(o)g(real)g(v)-5 b(alues)27 b(of)h FA(u)g FB(whic)n(h)g(giv)n(e)f (rise)g(to)h(t)n(w)n(o)f(consecutiv)n(e)g(homo)r(clinic)71 2465 y(p)r(oin)n(ts.)36 b(No)n(w)26 b(w)n(e)h(are)e(going)g(to)i (explain)f(ho)n(w)g(an)g(exp)r(onen)n(tially)g(small)g(upp)r(er)h(b)r (ound)g(of)g(the)g(di\013erence)f(\001)h(can)f(b)r(e)71 2565 y(deriv)n(ed.)35 b(The)26 b(\014rst)f(p)r(oin)n(t)g(is)g(that,)i (b)r(eing)e FA(T)1510 2535 y Fx(u)1578 2565 y FB(and)g FA(T)1798 2535 y Fx(s)1858 2565 y FB(solutions)g(of)g(the)h(same)e (partial)h(di\013eren)n(tial)g(equation)g(\(with)71 2665 y(di\013eren)n(t)k(b)r(oundary)f(conditions\),)h(\001\()p FA(u;)14 b(\034)9 b FB(\))30 b(satis\014es)e(a)h(homogeneous)e(linear)h (partial)h(di\013eren)n(tial)f(equation.)41 b(One)71 2764 y(can)29 b(see)g(that)h(this)g(equation)f(is)h(conjugated)f(to)g (\()p FA("@)1776 2776 y Fx(u)1840 2764 y FB(+)19 b FA(@)1968 2776 y Fx(\034)2010 2764 y FB(\))p FA(Y)f FB(\()p FA(u;)c(\034)9 b FB(\))28 b(=)e(0.)42 b(Let)30 b(us)f(assume)g(for)g(a)h(momen)n(t)f (that)h(\001)71 2864 y(is)j(a)f(solution)h(of)g(this)g(equation.)53 b(In)33 b(fact,)h(in)g(Theorems)e(4.21)f(and)i(4.17,)g(w)n(e)g(will)g (see)g(that)g(this)g(is)g(true)g(after)g(a)71 2964 y(suitable)25 b(c)n(hange)f(of)h(v)-5 b(ariables.)35 b(Then,)26 b(b)r(eing)g(\001)f (2)p FA(\031)s FB(-p)r(erio)r(dic)g(in)g FA(\034)35 b FB(w)n(e)25 b(obtain)g(that)h(\001\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)e(\003\()p FA(u=")14 b Fw(\000)g FA(\034)9 b FB(\),)25 b(where)71 3063 y(\003\()p FA(s)p FB(\))j(is)f(a)g(2)p FA(\031)s FB(-p)r(erio)r(dic)g(function.)38 b(This)27 b(fact)h(implies)g(that)1470 3255 y(\001\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)1845 3176 y Fz(X)1845 3354 y Fx(k)q Fv(2)p Fn(Z)1979 3255 y FB(\003)2037 3267 y Fx(k)2078 3255 y FA(e)2117 3220 y Fv(\000)p Fx(ik)2238 3198 y Fm(u)p 2238 3207 36 3 v 2242 3240 a(")2287 3255 y FA(e)2326 3220 y Fx(ik)q(\034)2428 3255 y FA(:)71 3513 y FB(No)n(w,)j(a)g(b)r (ound)h Fw(j)p FB(\001\()p FA(u;)14 b(\034)9 b FB(\))p Fw(j)24 b(\024)f FA(M)36 b FB(for)27 b Fw(j)p FB(Im)14 b FA(u)p Fw(j)23 b(\024)g FA(a)1635 3483 y Fv(0)1658 3513 y FB(,)k(automatically)g(giv)n(es)1665 3714 y Fw(j)p FB(\003)1746 3726 y Fx(k)1787 3714 y Fw(j)c(\024)g FA(M)9 b(e)2050 3679 y Fv(\000j)p Fx(k)q Fv(j)2187 3657 y Fm(a)2219 3640 y Fl(0)p 2187 3666 55 3 v 2200 3699 a Fm(")71 3910 y FB(whic)n(h)23 b(implies)g(that)h Fw(j)p FB(\001\()p FA(u;)14 b(\034)9 b FB(\))h Fw(\000)g FB(\003)1186 3922 y Fy(0)1223 3910 y Fw(j)23 b(\024)f FB(2)p FA(M)9 b(e)1527 3880 y Fv(\000)1588 3858 y Fm(a)1620 3841 y Fl(0)p 1588 3867 V 1601 3900 a Fm(")1680 3910 y FB(for)23 b(real)f(v)-5 b(alues)23 b(of)g FA(u)p FB(.)35 b(The)23 b(bigger)f(the)i(size)e(of)i (the)f(strip)g(where)g(w)n(e)71 4010 y(can)k(b)r(ound)h Fw(j)p FB(\001\()p FA(u;)14 b(\034)9 b FB(\))p Fw(j)29 b FB(the)f(smaller)e(the)i(exp)r(onen)n(tial)f(that)h(giv)n(es)f(the)h (b)r(ound)g(for)f(real)f(v)-5 b(alues)28 b(of)f FA(u)p FB(.)37 b(Note)27 b(that)h(the)71 4109 y(constan)n(t)h(\003)466 4121 y Fy(0)533 4109 y FB(do)r(es)h(not)g(app)r(ear)f(neither)g(in)i (the)f(form)n(ula)f(of)h(the)g(area)f(\(25\))o(,)i(nor)e(in)h(the)h (form)n(ula)e(of)g(the)i(distance)71 4209 y(\(24\))f(If)h(w)n(e)f(use)g (Melnik)n(o)n(v)g(theory)f(the)i(exp)r(ected)g(exp)r(onen)n(tial)f(exp) r(onen)n(t)g(is)h FA(a)p FB(,)g(where)f Fw(\006)p FA(ai)f FB(are)h(the)h(singularities)71 4309 y(of)c FA(T)214 4321 y Fy(0)251 4309 y FB(.)37 b(Then,)28 b(to)f(obtain)h(sharp)e(b)r (ounds,)i(it)g(w)n(ould)f(b)r(e)h(enough)f(to)h(tak)n(e)f FA(a)2499 4278 y Fv(0)2545 4309 y FB(=)c FA(a)18 b Fw(\000)g FA(")p FB(.)195 4408 y(In)31 b(some)f(cases,)h(the)g(c)n(hange)f(of)g (v)-5 b(ariables)30 b(whic)n(h)g(conjugates)g(the)h(original)e(partial) h(di\013eren)n(tial)h(equation)f(for)71 4508 y(\001\()p FA(u;)14 b(\034)9 b FB(\))28 b(with)h(\()p FA("@)667 4520 y Fx(u)729 4508 y FB(+)18 b FA(@)856 4520 y Fx(\034)897 4508 y FB(\))p FA(Y)h FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)f(0)k(is)g(not)h(close)f(enough)g(to)g(the)h(iden)n(tit)n(y)-7 b(.)38 b(This)27 b(fact)h(implies)g(the)g(app)r(earance)71 4607 y(of)j(the)g(constan)n(t)f FA(C)6 b FB(\()p FA(\026;)14 b(")p FB(\))32 b(and)f(the)g(logarithmic)f(term)h(in)g(the)g (asymptotic)g(form)n(ulas)f(obtained)g(in)h(Theorems)f(2.5)71 4707 y(and)k(2.6.)58 b(This)34 b(c)n(hange)g(of)g(v)-5 b(ariables)34 b(is)g(obtained,)i(essen)n(tially)-7 b(,)36 b(studying)e(the)h(v)-5 b(ariational)34 b(equation)g(along)f(the)71 4807 y(p)r(erturb)r(ed)h(in)n(v)-5 b(arian)n(t)32 b(manifolds.)55 b(Therefore,)34 b(the)g(existence)f(of)h(these)f(terms,)i(whic)n(h)f(w) n(ere)f(not)g(presen)n(t)g(in)h(the)71 4906 y(Melnik)n(o)n(v)k (prediction,)43 b(sho)n(ws)38 b(that,)43 b(to)d(study)g(the)g(exp)r (onen)n(tially)f(small)g(splitting)h(of)f(separatrices,)i(it)f(is)f (not)71 5006 y(enough)34 b(to)g(lo)r(ok)g(for)g(the)h(\014rst)f(order)f (appro)n(ximations)g(of)h(the)h(in)n(v)-5 b(arian)n(t)33 b(manifolds)i(close)e(to)i(the)g(singularities.)71 5106 y(One)30 b(has)f(to)h(lo)r(ok)f(also)g(for)h(the)g(\014rst)g(order)f (of)h(certain)f(solutions)g(of)h(the)h(v)-5 b(ariational)28 b(equation)i(of)g(the)g(p)r(erturb)r(ed)71 5205 y(in)n(v)-5 b(arian)n(t)31 b(manifolds)i(close)f(to)g(the)h(singularities.)51 b(In)33 b(fact,)h(these)f(terms)f(app)r(ear)g(when)h(these)f(certain)g (solutions)71 5305 y(of)j(the)g(v)-5 b(ariational)33 b(equation)h(of)h(the)g(p)r(erturb)r(ed)g(in)n(v)-5 b(arian)n(t)34 b(manifolds)g(close)g(to)h(the)g(singularities)e(are)h(not)h(w)n(ell)71 5404 y(appro)n(ximated)26 b(b)n(y)h(the)h(solutions)f(of)h(the)g(v)-5 b(ariational)26 b(equation)h(of)g(the)h(unp)r(erturb)r(ed)g (separatrix.)195 5504 y(Then,)g(roughly)f(sp)r(eaking)f(one)i(can)f (conclude)g(that)h(Melnik)n(o)n(v)e(theory)h(giv)n(es)g(the)h(correct)e (answ)n(er)g(if:)p Black 1919 5753 a(19)p Black eop end %%Page: 20 20 TeXDict begin 20 19 bop Black Black Black 195 272 a Fw(\017)p Black 41 w FB(The)33 b(p)r(erturb)r(ed)f(in)n(v)-5 b(arian)n(t)31 b(manifolds)h(are)f(w)n(ell)h(appro)n(ximated)e(b)n(y)i(the)h(unp)r (erturb)r(ed)f(separatrix)e(close)i(to)278 372 y(the)c(singularit)n(y) -7 b(.)p Black 195 538 a Fw(\017)p Black 41 w FB(The)29 b(solutions)f(of)h(the)g(v)-5 b(ariational)27 b(equation)h(along)f(the) i(p)r(erturb)r(ed)g(in)n(v)-5 b(arian)n(t)28 b(manifold)g(are)g(w)n (ell)g(appro)n(xi-)278 637 y(mated)g(b)n(y)f(certain)g(solutions)g(of)h (the)g(v)-5 b(ariational)26 b(equation)h(along)f(the)i(unp)r(erturb)r (ed)g(separatrix.)71 803 y(In)33 b(all)f(the)i(other)e(cases,)h(the)g (splitting)g(is)g(giv)n(en)f(b)n(y)g(an)h(alternativ)n(e)e(form)n(ula.) 52 b(This)33 b(fact,)h(is)f(explained)f(in)h(more)71 903 y(detail)27 b(Section)h(3.4.)71 1135 y Fq(3.2)112 b(The)38 b Fi(b)-6 b(o)g(omer)g(ang)41 b(domains)71 1289 y FB(In)36 b(this)g(pap)r(er)g(w)n(e)f(deal)h(with)g(general)f (Hamiltonian)g(whose)g(in)n(v)-5 b(arian)n(t)35 b(manifolds,)j(in)e (general,)h(are)e(not)h(global)71 1388 y(graphs)18 b(o)n(v)n(er)h FA(q)s FB(.)34 b(Therefore,)20 b(the)h(approac)n(h)d(explained)i(in)g (the)g(previous)f(section)h(cannot)f(b)r(e)h(used)g(straigh)n(tforw)n (ardly)-7 b(.)71 1488 y(Nev)n(ertheless,)22 b(w)n(e)f(will)h(see)f (that)i(there)e(are)g(alw)n(a)n(ys)f(regions)g(in)i(the)g(phase)f (space)g(where)g(b)r(oth)i(manifolds)e(are)g(graphs)71 1588 y(and)29 b(w)n(e)f(will)i(use)e(one)h(of)g(these)g(regions)e(to)i (measure)f(the)h(splitting.)42 b(Consequen)n(tly)-7 b(,)28 b(b)r(eing)i(the)f(area)e(of)i(the)h(lob)r(es)71 1687 y(an)d(in)n(v)-5 b(arian)n(t)27 b(quan)n(tit)n(y)-7 b(,)27 b(this)h(will)g(giv)n(e)e(the)i(w)n(an)n(ted)f(result.)195 1787 y(As)i(w)n(e)g(ha)n(v)n(e)f(explained,)h(w)n(e)f(are)g(forced)h (to)f(\014nd)i(parameterizations)c FA(T)2550 1757 y Fx(u;s)2673 1787 y FB(of)j(the)g(in)n(v)-5 b(arian)n(t)28 b(manifolds)h(whic)n(h)71 1886 y(ha)n(v)n(e)21 b(to)h(b)r(e)h(analytic)f(in)h(a)f(common)g (complex)g(domain)g(whic)n(h)g(reac)n(hes)f(p)r(oin)n(ts)i(at)f(a)g (distance)g FA(")g FB(of)h(the)g(singularities.)71 1986 y(Moreo)n(v)n(er)28 b(w)n(e)i(also)g(need)h(to)f(guaran)n(tee)f(that)i (our)f(domain)g(con)n(tains)g(an)h(op)r(en)f(set)h(of)g(real)f(v)-5 b(alues)30 b(of)h FA(u)f FB(\(this)h(will)71 2086 y(b)r(e)g(enough)f (to)h(ensure)f(that)h(the)g(domain)f(con)n(tains)g FA(u)1847 2098 y Fy(1)1914 2086 y FB(and)h FA(u)2127 2098 y Fy(2)2194 2086 y FB(that)g(giv)n(e)f(rise)g(to)h(homo)r(clinic)f(p)r(oin)n(ts)h (since)f(they)71 2185 y(are)c FA(")i FB(close\).)195 2285 y(T)-7 b(o)29 b(this)g(end)g(let)g(us)f(observ)n(e)f(that)i(w)n(e) g(ha)n(v)n(e)e(no)i(hop)r(e)f(to)h(construct)f(parameterizations)f FA(T)3186 2255 y Fx(u;s)3279 2285 y FB(\()p FA(u;)14 b(\034)9 b FB(\))30 b(for)e(v)-5 b(alues)71 2385 y(of)28 b FA(u)h FB(suc)n(h)f(that)h FA(p)654 2397 y Fy(0)691 2385 y FB(\()p FA(u)p FB(\))24 b(=)h(0,)j(at)g(least)h(in)f(a)h (general)e(case.)39 b(In)28 b(fact,)h(the)g(unp)r(erturb)r(ed)g(homo)r (clinic)g(connection)f(can)71 2484 y(b)r(e)33 b(expressed)f(as)g(graph) o Fw(f)p FA(p)f FB(=)1094 2413 y Fz(p)p 1177 2413 278 4 v 71 x Fw(\000)p FB(2)p FA(V)18 b FB(\()p FA(q)s FB(\))q Fw(g)j([)i FB(graph)n Fw(f)p FA(p)32 b FB(=)f Fw(\000)2080 2413 y Fz(p)p 2163 2413 V 71 x Fw(\000)p FB(2)p FA(V)18 b FB(\()p FA(q)s FB(\))q Fw(g)p FB(.)52 b(Then)33 b(if)h FA(p)2904 2496 y Fy(0)2941 2484 y FB(\()p FA(u)3021 2496 y Fy(0)3058 2484 y FB(\))e(=)g(0,)i(for)e(some)h(v)-5 b(alue)71 2584 y FA(u)119 2596 y Fy(0)156 2584 y FB(,)28 b(the)g(unp)r(erturb)r(ed)h(homo)r(clinic)f(connection)f(cannot)h(b)r (e)g(expressed)f(as)g(a)h(graph)f(o)n(v)n(er)f(the)i(base)f(in)i(the)f (original)71 2683 y(v)-5 b(ariables)22 b(\()p FA(q)s(;)14 b(p)p FB(\))23 b(in)h(a)e(neigh)n(b)r(orho)r(o)r(d)g(of)h(\()p FA(q)1450 2695 y Fy(0)1488 2683 y FB(\()p FA(u)1568 2695 y Fy(0)1605 2683 y FB(\))p FA(;)14 b FB(0\).)35 b(This)23 b(fact)h(implies)f(that)g(the)h(Hamilton-Jacobi)e(equation)g(that)71 2783 y FA(T)132 2753 y Fx(u;s)253 2783 y FB(has)27 b(to)h(satisfy)f(is) g(not)h(de\014ned)g(for)f FA(u)22 b FB(=)h FA(u)1612 2795 y Fy(0)1649 2783 y FB(.)195 2883 y(W)-7 b(e)30 b(will)g(alw)n(a)n (ys)e(k)n(eep)h(in)h(mind)h(that)f(w)n(e)f(need)h(to)f(c)n(hec)n(k)g (this)h(condition)g(\()p FA(p)2708 2895 y Fy(0)2745 2883 y FB(\()p FA(u)p FB(\))d Fw(6)p FB(=)f(0\))j(if)i(w)n(e)e(w)n(an)n(t)g (to)h(use)f(the)71 2982 y(parameterizations)c FA(T)806 2952 y Fx(u;s)900 2982 y FB(.)195 3082 y(F)-7 b(or)31 b(this)h(reason)d(w)n(e)i(de\014ne)h(the)g(follo)n(wing)e Fs(b)l(o)l(omer)l(ang)k(domains)39 b FB(\(see)32 b(Figure)e(2\),)j(in)e (whic)n(h)g FA(p)3347 3094 y Fy(0)3384 3082 y FB(\()p FA(u)p FB(\))f Fw(6)p FB(=)f(0,)j(and)71 3182 y(hence)27 b(the)h(functions)g FA(T)863 3151 y Fx(s;u)985 3182 y FB(will)f(b)r(e)h(w)n(ell)g(de\014ned)g(in)g(them.)517 3343 y FA(D)588 3308 y Fx(s)586 3363 y(\024;d)706 3343 y FB(=)14 b Fw(f)o FA(u)23 b Fw(2)g Ft(C)p FB(;)48 b Fw(j)p FB(Im)14 b FA(u)p Fw(j)23 b FA(<)f FB(tan)14 b FA(\014)1604 3355 y Fy(1)1641 3343 y FB(Re)g FA(u)k FB(+)g FA(a)g Fw(\000)g FA(\024";)c Fw(j)p FB(Im)g FA(u)p Fw(j)23 b FA(<)f FB(tan)14 b FA(\014)2669 3355 y Fy(2)2706 3343 y FB(Re)g FA(u)k FB(+)g FA(a)g Fw(\000)g FA(\024";)795 3477 y Fw(j)p FB(Im)c FA(u)p Fw(j)22 b FA(>)h FB(tan)14 b FA(\014)1293 3489 y Fy(2)1330 3477 y FB(Re)g FA(u)k FB(+)g FA(a)g Fw(\000)g FA(d)p Fw(g)517 3602 y FA(D)588 3568 y Fx(u)586 3623 y(\024;d)706 3602 y FB(=)c Fw(f)o FA(u)23 b Fw(2)g Ft(C)p FB(;)48 b Fw(j)p FB(Im)14 b FA(u)p Fw(j)23 b FA(<)f Fw(\000)14 b FB(tan)f FA(\014)1682 3614 y Fy(1)1720 3602 y FB(Re)h FA(u)j FB(+)i FA(a)f Fw(\000)g FA(\024";)c Fw(j)p FB(Im)f FA(u)p Fw(j)23 b FA(<)g FB(tan)13 b FA(\014)2747 3614 y Fy(2)2785 3602 y FB(Re)g FA(u)18 b FB(+)g FA(a)h Fw(\000)f FA(\024";)795 3737 y Fw(j)p FB(Im)c FA(u)p Fw(j)22 b FA(>)h FB(tan)14 b FA(\014)1293 3749 y Fy(2)1330 3737 y FB(Re)g FA(u)k FB(+)g FA(a)g Fw(\000)g FA(d)p Fw(g)789 3861 y([)h(f)p FA(u)j Fw(2)i Ft(C)p FB(;)37 b Fw(j)p FB(Im)14 b FA(u)p Fw(j)23 b FA(<)g Fw(\000)14 b FB(tan)f FA(\014)1751 3873 y Fy(1)1788 3861 y FB(Re)h FA(u)k FB(+)g FA(a)g Fw(\000)g FA(\024";)c Fw(j)p FB(Im)g FA(u)p Fw(j)22 b FA(>)h Fw(\000)14 b FB(tan)f FA(\014)2894 3873 y Fy(2)2931 3861 y FB(Re)h FA(u)k FB(+)g FA(a)h Fw(\000)f FA(d;)795 3986 y FB(Re)c FA(u)22 b(<)h FB(0)p Fw(g)13 b FA(;)3703 3666 y FB(\(26\))71 4169 y(where)27 b FA(\014)358 4181 y Fy(1)418 4169 y Fw(2)d FB(\(0)p FA(;)14 b(\031)s(=)p FB(2\))27 b(is)g(an)n(y)g(\014xed)h(angle.)195 4269 y(T)-7 b(o)32 b(c)n(ho)r(ose)g FA(\014)638 4281 y Fy(2)707 4269 y FB(w)n(e)g(use)g(the)h(follo)n(wing.)50 b(First)33 b(let)f(us)h(p)r(oin)n(t)f(out)g(that)h(the)g(zeros)e(of)h FA(p)3031 4281 y Fy(0)3068 4269 y FB(\()p FA(u)p FB(\))h(are)e (isolated)h(in)g Ft(C)p FB(.)71 4369 y(Moreo)n(v)n(er,)c(close)h(to)h (the)h(singularities)e FA(u)e FB(=)f Fw(\006)p FA(ia)p FB(,)k FA(p)1777 4381 y Fy(0)1814 4369 y FB(\()p FA(u)p FB(\))h(can)e(not)i(v)-5 b(anish.)44 b(Then,)31 b(in)f(order)f(to)h (assure)f(that)h FA(p)3701 4381 y Fy(0)3738 4369 y FB(\()p FA(u)p FB(\))71 4468 y(do)r(es)g(not)g(v)-5 b(anish)30 b(in)g(the)h(whole)f(domains)f FA(D)1554 4438 y Fx(s)1552 4492 y(\024;d)1679 4468 y FB(and)h FA(D)1914 4438 y Fx(u)1912 4492 y(\024;d)2009 4468 y FB(,)h(one)f(has)g(to)g(c)n(ho)r(ose)f(an)h (angle)f FA(\014)3123 4480 y Fy(2)3190 4468 y FB(suc)n(h)h(that)h FA(\014)3610 4480 y Fy(2)3674 4468 y FA(>)c(\014)3813 4480 y Fy(1)71 4568 y FB(has)d(a)h(p)r(ositiv)n(e)g(lo)n(w)n(er)e(b)r (ound)j(indep)r(enden)n(t)g(of)f FA(")g FB(and)g FA(\026)g FB(and)g(suc)n(h)g(that)g(the)g(lines)g Fw(j)p FB(Im)15 b FA(u)p Fw(j)22 b FB(=)h(tan)14 b FA(\014)3261 4580 y Fy(2)3298 4568 y FB(Re)g FA(u)f FB(+)g FA(a)25 b FB(do)f(not)71 4667 y(con)n(tain)29 b(an)n(y)g(zero)f(of)i FA(p)841 4679 y Fy(0)878 4667 y FB(\()p FA(u)p FB(\).)44 b(Then,)30 b(taking)f FA(")e(>)f FB(0)j(and)h FA(d)c(>)h FB(0)i(indep)r(enden)n(t) i(of)e FA(")p FB(,)h(b)r(oth)g(small)g(enough,)g(one)f(can)71 4767 y(guaran)n(tee)d(that)h FA(p)673 4779 y Fy(0)711 4767 y FB(\()p FA(u)p FB(\))g(do)r(es)h(not)f(v)-5 b(anish)28 b(neither)f(in)h FA(D)1891 4737 y Fx(s)1889 4791 y(\024;d)2014 4767 y FB(nor)f(in)g FA(D)2329 4737 y Fx(u)2327 4791 y(\024;d)2424 4767 y FB(.)195 4867 y(W)-7 b(e)20 b(will)f(use)g(these)g Fs(b)l(o)l(omer)l(ang)k(domains)k FB(as)19 b(fundamen)n(tal)g(domains)f (to)h(measure)f(the)i(splitting.)34 b(It)20 b(is)f(imp)r(ortan)n(t)71 4966 y(to)27 b(emphasize)g(that)h(b)r(oth)g FA(D)1016 4936 y Fx(s)1014 4990 y(\024;d)1139 4966 y FB(and)g FA(D)1372 4936 y Fx(u)1370 4990 y(\024;d)1494 4966 y FB(reac)n(h)f(a)g(neigh)n(b) r(orho)r(o)r(d)f(of)i(the)g(singularities)e Fw(\006)p FA(ia)h FB(of)g(size)h FA(")p FB(.)p Black 1919 5753 a(20)p Black eop end %%Page: 21 21 TeXDict begin 21 20 bop Black Black Black Black Black 293 1606 a /PSfrag where{pop(u1)[[0(Bl)1 0]](u2)[[1(Bl)1 0]](b1)[[2(Bl)1 0]](b2)[[3(Bl)1 0]](a1)[[4(Bl)1 0]](a2)[[5(Bl)1 0]](a3)[[6(Bl)1 0]](a4)[[7(Bl)1 0]](D1)[[8(Bl)1 0]](D)[[9(Bl)1 0]](D4)[[10(Bl)1 0]](D3)[[11(Bl)1 0]]12 0 -1/Begin PSfrag}{userdict /PSfrag{pop}put}ifelse 293 1606 a @beginspecial 0 @llx 603 @lly 513 @urx 821 @ury 1700 @rhi @setspecial %%BeginDocument: boomerangs.eps %!PS-Adobe-3.0 EPSF-3.0 %%Creator: 0.46 %%Pages: 1 %%Orientation: Portrait %%BoundingBox: 0 603 513 821 %%HiResBoundingBox: -0.4 603.33928 512.4 820.87415 %%EndComments %%Page: 1 1 0 842 translate 0.8 -0.8 scale 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap gsave [1 0 0 1 0 0] concat 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 550 52.362183 moveto 550 292.36218 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 330 172.36218 moveto 640 172.36218 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 340 172.36218 moveto 630 52.362183 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 340 172.36218 moveto 630 292.36218 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 550.01276 259.1399 moveto 492.18152 172.26678 lineto 550.01276 85.393658 lineto 569.73214 77.272894 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 571.12359 77.03186 moveto 507.87542 172.26678 lineto 571.38362 267.75741 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 90 107.36218 moveto 46.57326 172.5971 lineto 89.8136 237.55202 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 550 259.23718 moveto 569.86607 267.49611 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 498.76282 162.16525 moveto 502.80783 164.63305 503.51325 168.21404 503.56104 172.01424 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 521.23871 161.1551 moveto 504.31866 166.20586 lineto 508.35927 162.16525 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 504.06612 166.20586 moveto 507.85419 167.46855 lineto stroke gsave [1 0 0 -1 520.5014 161.50916] concat gsave /newlatin1font {findfont dup length dict copy dup /Encoding ISOLatin1Encoding put definefont} def /Sans-ISOLatin1 /Sans newlatin1font 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (b2) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 366.93791 160.90256 moveto 368.44692 163.69293 370.38121 165.63272 369.96837 172.26678 curveto stroke gsave [1 0 0 -1 372.99884 167.00409] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (b1) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 2.5 setlinewidth 0 setlinejoin 0 setlinecap newpath 551.71589 76.330616 moveto 548.18036 76.330616 lineto stroke gsave [1 0 0 -1 554.06866 57.361923] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a1) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 2.5 setlinewidth 0 setlinejoin 0 setlinecap newpath 552.02955 268.226 moveto 548.49402 268.226 lineto stroke gsave [1 0 0 -1 557.85712 297.71933] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a2) show grestore grestore gsave [1 0 0 1 0.8928571 -0.3571429] concat 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 572.67857 101.82647 moveto 552.14286 106.29075 lineto 555.71429 102.89789 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 551.42857 106.64789 moveto 556.25 107.71932 lineto stroke grestore gsave [1 0 0 -1 575 107.11218] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a3) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 547.5 83.790752 moveto 542.32143 85.040752 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 520 77.362181 moveto 547.85714 83.790752 lineto 542.67857 79.862181 lineto stroke gsave [1 0 0 -1 480.25 75.290749] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a4) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 471.99378 193.73252 moveto 508.10673 182.11576 lineto 501.03566 181.61069 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 507.34912 181.86323 moveto 503.56104 185.90384 lineto stroke gsave [1 0 0 -1 443.30594 203.32896] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D) show grestore grestore gsave [1 0 0 -1 586.89862 227.06755] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D1) show grestore grestore 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 90.029846 85.646195 moveto 82.70624 96.252797 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 70.016199 267.78932 moveto 82.643106 248.34388 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 70.268737 267.41051 moveto 89.903578 259.26616 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2.5 setlinewidth 0 setlinejoin 0 setlinecap newpath 91.743577 77.241513 moveto 88.208047 77.241513 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 122.32143 155.04075 moveto 118.48777 159.00471 115.20957 163.52413 116.07143 172.18361 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 115.24163 91.433614 moveto 93.813062 105.36218 lineto 97.384493 100.36218 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 94.348773 105.00503 moveto 98.81306 104.29075 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 98.928572 154.32647 moveto 115 163.61218 lineto 112.14285 160.04076 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 55.892857 158.96932 moveto 60.509019 163.16006 61.309551 167.69767 60.714286 172.36218 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 93.035714 153.61218 moveto 62.142857 165.57647 lineto 65.892857 162.00504 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 82.88871 96.647895 moveto 133.96014 172.36218 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 33.035714 147.18361 moveto 50.357143 159.32647 lineto 47.5 154.86218 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 50.357143 159.1479 moveto 46.25 159.1479 lineto stroke gsave [1 0 0 -1 93.607147 150.68359] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (b2) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 62.142857 165.75504 moveto 66.607143 165.75504 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 115 163.79075 moveto 110 163.43361 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 82.76462 96.417927 moveto 32.05232 172.5971 lineto 82.610403 248.5446 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 46.57326 172.5971 moveto 101.64396 255.18595 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 90 52.362183 moveto 90 292.36218 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 310 172.36218 moveto 0 172.36218 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 300 172.36218 moveto 10 52.362183 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 300 172.36218 moveto 10 292.36218 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 90.061413 259.26616 moveto 82.643105 248.62799 lineto stroke gsave [1 0 0 -1 97.696182 291.60379] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a2) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 101.59504 89.878404 moveto 46.57326 172.5971 lineto 89.8136 237.55202 lineto stroke gsave [1 0 0 -1 49.273823 55.019501] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a1) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 82.780475 96.342016 moveto 69.910714 77.094323 lineto stroke gsave [1 0 0 -1 9.6428566 141.68361] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D3) show grestore grestore gsave [1 0 0 -1 135.35715 212.82648] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D4) show grestore grestore gsave [1 0 0 -1 112.6702 93.076469] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a3) show grestore grestore gsave [1 0 0 -1 112.65235 69.607162] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a4) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 114.11664 68.714305 moveto 93.04521 82.642876 lineto 94.830923 77.107162 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 92.876128 82.897896 moveto 97.697557 81.826467 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 69.953064 77.186167 moveto 101.20466 90.065612 lineto 101.26779 90.002478 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 70.142468 267.53678 moveto 82.662002 248.35622 lineto 89.966712 237.35848 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2.5 setlinewidth 0 setlinejoin 0 setlinecap newpath 91.79716 266.7533 moveto 88.26163 266.7533 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 101.6141 254.3002 moveto 73.741136 265.95843 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 561.07143 59.862189 moveto 552.85714 73.433617 lineto 553.92857 66.647903 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 552.85714 73.076474 moveto 558.21428 70.219332 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 68.214285 57.005046 moveto 86.428571 74.862189 lineto 84.285714 68.79076 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 86.428571 74.862189 moveto 80.714285 73.433617 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 103.21429 276.6479 moveto 93.214285 267.36219 lineto 95.357142 272.71933 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 93.571428 267.00504 moveto 99.285714 268.79076 lineto stroke gsave [1 0 0 -1 31.567266 96.252808] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (u1) show grestore grestore gsave [1 0 0 -1 22.728432 255.09929] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (u2) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 568.2108 287.17164 moveto 554.06867 268.98889 lineto 555.5839 276.05996 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 554.06867 268.98889 moveto 560.12958 272.27188 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 51.517779 88.929199 moveto 67.68022 78.322597 lineto 62.124381 79.33275 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 67.68022 78.322597 moveto 65.659915 83.120822 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 44.446712 252.82645 moveto 45.709402 253.07899 67.932758 265.95843 67.932758 265.95843 curveto 64.902301 260.90767 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 68.185296 266.21097 moveto 62.629457 265.45336 lineto stroke grestore showpage %%EOF %%EndDocument @endspecial 293 1606 a /End PSfrag 293 1606 a 293 320 a /Hide PSfrag 293 320 a -447 378 a FB(PSfrag)26 b(replacemen)n(ts)p -447 408 741 4 v 293 411 a /Unhide PSfrag 293 411 a 208 499 a { 208 499 a Black FA(u)256 511 y Fy(1)p Black 208 499 a } 0/Place PSfrag 208 499 a 208 598 a { 208 598 a Black 5 w FB(\026)-47 b FA(u)256 610 y Fy(1)p Black 208 598 a } 1/Place PSfrag 208 598 a 209 694 a { 209 694 a Black FA(\014)256 706 y Fy(1)p Black 209 694 a } 2/Place PSfrag 209 694 a 209 793 a { 209 793 a Black FA(\014)256 805 y Fy(2)p Black 209 793 a } 3/Place PSfrag 209 793 a 221 909 a { 221 909 a Black FA(ia)p Black 221 909 a } 4/Place PSfrag 221 909 a 156 1002 a { 156 1002 a Black Fw(\000)p FA(ia)p Black 156 1002 a } 5/Place PSfrag 156 1002 a 11 1088 a { 11 1088 a Black FA(i)p FB(\()p FA(a)18 b Fw(\000)g FA(d)p FB(\))p Black 11 1088 a } 6/Place PSfrag 11 1088 a -32 1187 a { -32 1187 a Black FA(i)p FB(\()p FA(a)g Fw(\000)g FA(\024")p FB(\))p Black -32 1187 a } 7/Place PSfrag -32 1187 a 71 1276 a { 71 1276 a Black FA(D)142 1246 y Fy(out)p Fx(;s)140 1296 y(\032;\024)p Black 71 1276 a } 8/Place PSfrag 71 1276 a 127 1372 a { 127 1372 a Black FA(D)198 1342 y Fx(s)196 1396 y(\024;d)p Black 127 1372 a } 9/Place PSfrag 127 1372 a 63 1475 a { 63 1475 a Black FA(D)134 1445 y Fy(out)p Fx(;u)132 1495 y(\032;\024)p Black 63 1475 a } 10/Place PSfrag 63 1475 a 127 1572 a { 127 1572 a Black FA(D)198 1541 y Fx(u)196 1595 y(\024;d)p Black 127 1572 a } 11/Place PSfrag 127 1572 a Black 770 1872 a FB(Figure)27 b(2:)37 b(The)27 b Fs(b)l(o)l(omer)l(ang)k(domains) 36 b FA(D)2120 1842 y Fx(u)2118 1896 y(\024;d)2242 1872 y FB(and)28 b FA(D)2475 1842 y Fx(s)2473 1896 y(\024;d)2597 1872 y FB(de\014ned)g(in)g(\(26\).)p Black Black Black 71 2095 a Fp(Remark)k(3.1.)p Black 40 w Fs(L)l(et)e(us)f(observe)j (that)e(the)g(domains)h FA(D)1881 2065 y Fx(u)1879 2119 y(\024;d)2006 2095 y Fs(and)g FA(D)2239 2065 y Fx(s)2237 2119 y(\024;d)2364 2095 y Fs(have)g(di\013er)l(ent)f(shap)l(e.)41 b(We)30 b(wil)t(l)h(give)g(al)t(l)g(the)71 2195 y(pr)l(o)l(ofs)g(in)f (the)g(unstable)f(c)l(ase.)39 b(A)n(l)t(l)30 b(of)h(them)f(ar)l(e)g (analo)l(gous,)h(and)f(even)g(simpler,)h(in)f(the)g(stable)g(one.)195 2340 y FB(T)-7 b(o)39 b(study)g(the)g(di\013erence)f(b)r(et)n(w)n(een)h (the)g(manifolds,)i(w)n(e)e(consider)e(\001\()p FA(u;)14 b(\034)9 b FB(\))43 b(=)e FA(T)2945 2310 y Fx(u)2987 2340 y FB(\()p FA(u;)14 b(\034)9 b FB(\))27 b Fw(\000)e FA(T)3359 2310 y Fx(s)3394 2340 y FB(\()p FA(u;)14 b(\034)9 b FB(\))39 b(in)g(the)71 2439 y(domain)27 b FA(R)429 2451 y Fx(\024;d)549 2439 y FB(=)c FA(D)708 2409 y Fx(s)706 2463 y(\024;d)822 2439 y Fw(\\)18 b FA(D)966 2409 y Fx(u)964 2463 y(\024;d)1089 2439 y FB(whic)n(h)28 b(is)f(de\014ned)h(as)616 2609 y FA(R)679 2621 y Fx(\024;d)800 2609 y FB(=)22 b Fw(f)p FA(u)g Fw(2)i Ft(C)p FB(;)47 b Fw(j)p FB(Im)15 b FA(u)p Fw(j)22 b FA(<)h FB(tan)14 b FA(\014)1707 2621 y Fy(2)1744 2609 y FB(Re)g FA(u)k FB(+)g FA(a)g Fw(\000)g FA(\024";)c Fw(j)p FB(Im)f FA(u)p Fw(j)23 b FA(>)g FB(tan)14 b FA(\014)2772 2621 y Fy(2)2809 2609 y FB(Re)g FA(u)k FB(+)g FA(a)g Fw(\000)g FA(d;)1208 2733 y Fw(j)p FB(Im)d FA(u)p Fw(j)22 b FA(<)h Fw(\000)14 b FB(tan)f FA(\014)1785 2745 y Fy(1)1822 2733 y FB(Re)h FA(u)k FB(+)g FA(a)g Fw(\000)g FA(\024")p Fw(g)13 b FA(:)3703 2672 y FB(\(27\))195 2890 y(W)-7 b(e)28 b(recall)f(that)h FA(p)782 2902 y Fy(0)819 2890 y FB(\()p FA(u)p FB(\))23 b Fw(6)p FB(=)g(0)k(if)h FA(u)23 b Fw(2)g FA(R)1399 2902 y Fx(\024;d)1524 2890 y FB(and)k(hence)h(w)n(e)f(can)g(use)h(the)g(functions)g FA(T)2896 2860 y Fx(s;u)3017 2890 y FB(in)g(this)g(domain.)p Black Black Black 1136 4428 a /PSfrag where{pop(D6)[[0(Bl)1 0]](D5)[[1(Bl)1 0]](b1)[[2(Bl)1 0]](b2)[[3(Bl)1 0]](a1)[[4(Bl)1 0]](a2)[[5(Bl)1 0]](a3)[[6(Bl)1 0]](a4)[[7(Bl)1 0]](D3)[[8(Bl)1 0]](D)[[9(Bl)1 0]](D1)[[10(Bl)1 0]]11 0 -1/Begin PSfrag}{userdict /PSfrag{pop}put}ifelse 1136 4428 a @beginspecial 183 @llx 532 @lly 424 @urx 739 @ury 1700 @rhi @setspecial %%BeginDocument: BoomerangInterseccio.eps %!PS-Adobe-3.0 EPSF-3.0 %%Creator: 0.46 %%Pages: 1 %%Orientation: Portrait %%BoundingBox: 183 532 424 739 %%HiResBoundingBox: 183.02858 532.50579 423.82858 738.35135 %%EndComments %%Page: 1 1 0 842 translate 0.8 -0.8 scale 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap gsave [1 0 0 1 0 0] concat 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 309.49312 179.15962 moveto 329.19666 170.8393 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 318.21429 248.19424 moveto 334.28572 257.47995 lineto 331.42857 253.90853 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 275.17858 252.83709 moveto 279.79474 257.02783 280.59527 261.56544 280.00001 266.22995 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 312.32143 247.47995 moveto 281.42858 259.44424 lineto 285.17858 255.87281 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 288.78157 170.69424 moveto 353.24586 266.22995 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 252.32143 241.05138 moveto 269.64286 253.19424 lineto 266.78572 248.72995 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 269.64286 253.01567 moveto 265.53572 253.01567 lineto stroke gsave [1 0 0 -1 312.89285 244.55136] concat gsave /newlatin1font {findfont dup length dict copy dup /Encoding ISOLatin1Encoding put definefont} def /Sans-ISOLatin1 /Sans newlatin1font 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (b2) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 281.42858 259.62281 moveto 285.89286 259.62281 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 334.28572 257.65852 moveto 329.28572 257.30138 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 309.03898 352.59094 moveto 251.33804 266.46487 lineto 309.1932 179.21427 lineto 321.07142 184.23718 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 265.85898 266.46487 moveto 320.92968 349.05372 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 309.28572 146.22995 moveto 309.28572 386.22995 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 529.28572 266.22995 moveto 231.40755 266.22995 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 519.28572 266.22995 moveto 229.28572 146.22995 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 519.28572 266.22995 moveto 229.28572 386.22995 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 309.34713 353.13393 moveto 301.92883 342.49576 lineto stroke gsave [1 0 0 -1 265.19617 386.18582] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a2) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 320.88076 183.74617 moveto 265.85898 266.46487 lineto 309.09932 331.41979 lineto stroke gsave [1 0 0 -1 284.9881 158.17299] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a1) show grestore grestore gsave [1 0 0 -1 228.92857 235.55138] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D3) show grestore grestore gsave [1 0 0 -1 326.58093 158.83208] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a4) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 328.58093 158.83208 moveto 311.21486 176.10886 lineto 313.35771 170.39457 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 311.35828 175.91745 moveto 316.62614 173.68531 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 289.42819 361.40455 moveto 301.94772 342.22399 lineto 309.25243 331.22625 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2.5 setlinewidth 0 setlinejoin 0 setlinecap newpath 311.08288 360.1925 moveto 307.54735 360.1925 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 320.35714 348.43361 moveto 309.28571 353.07647 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 309.28571 201.29075 moveto 328.92857 171.11218 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 309.53766 353.01758 moveto 329.2412 361.3379 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 309.33025 330.88645 moveto 328.97311 361.06502 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 343.7044 252.06883 moveto 341.56725 254.3655 337.36071 259.09659 335.87572 265.95842 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 266.04893 180.7268 moveto 302.54069 183.88353 lineto 297.48993 181.10561 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 302.28815 183.88353 moveto 297.23739 185.6513 lineto stroke gsave [1 0 0 -1 236.37569 186.91399] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D1) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 348.03572 179.1479 moveto 317.85714 178.61218 lineto 322.14286 176.46932 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 317.67857 178.61218 moveto 321.96429 180.93361 lineto stroke gsave [1 0 0 -1 349.28571 190.93361] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 487.3986 252.82644 moveto 484.64347 255.36669 481.50617 256.95154 481.33769 265.95843 curveto stroke gsave [1 0 0 -1 461.61942 260.42291] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (b1) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 281.60714 376.6479 moveto 306.42857 361.11219 lineto 298.92857 363.07647 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 306.25 361.11219 moveto 301.42857 366.11219 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2.5 setlinewidth 0 setlinejoin 0 setlinecap newpath 310.8749 171.46934 moveto 307.33937 171.46934 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 299.10714 159.50504 moveto 306.78571 168.61219 lineto 305.17857 163.07647 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 306.96428 168.25504 moveto 305.35714 167.8979 301.42857 166.11219 301.42857 166.11219 curveto stroke grestore showpage %%EOF %%EndDocument @endspecial 1136 4428 a /End PSfrag 1136 4428 a 1136 3190 a /Hide PSfrag 1136 3190 a 395 3248 a FB(PSfrag)f(replacemen)n(ts)p 395 3277 741 4 v 1136 3280 a /Unhide PSfrag 1136 3280 a 914 3370 a { 914 3370 a Black 931 3349 a Fz(e)914 3370 y FA(D)985 3330 y Fy(out)o Fx(;s)983 3395 y(\032;d;\024)p Black 914 3370 a } 0/Place PSfrag 914 3370 a 906 3495 a { 906 3495 a Black 923 3474 a Fz(e)906 3495 y FA(D)977 3455 y Fy(out)o Fx(;u)975 3520 y(\032;d;\024)p Black 906 3495 a } 1/Place PSfrag 906 3495 a 1051 3615 a { 1051 3615 a Black FA(\014)1098 3627 y Fy(1)p Black 1051 3615 a } 2/Place PSfrag 1051 3615 a 1051 3715 a { 1051 3715 a Black FA(\014)1098 3727 y Fy(2)p Black 1051 3715 a } 3/Place PSfrag 1051 3715 a 1063 3831 a { 1063 3831 a Black FA(ia)p Black 1063 3831 a } 4/Place PSfrag 1063 3831 a 998 3923 a { 998 3923 a Black Fw(\000)p FA(ia)p Black 998 3923 a } 5/Place PSfrag 998 3923 a 854 4009 a { 854 4009 a Black FA(i)p FB(\()p FA(a)18 b Fw(\000)g FA(d)p FB(\))p Black 854 4009 a } 6/Place PSfrag 854 4009 a 810 4109 a { 810 4109 a Black FA(i)p FB(\()p FA(a)h Fw(\000)f FA(\024")p FB(\))p Black 810 4109 a } 7/Place PSfrag 810 4109 a 975 4205 a { 975 4205 a Black FA(R)1038 4217 y Fx(\024;d)p Black 975 4205 a } 8/Place PSfrag 975 4205 a 970 4294 a { 970 4294 a Black FA(D)1041 4264 y Fx(s)1039 4317 y(\024;d)p Black 970 4294 a } 9/Place PSfrag 970 4294 a 970 4393 a { 970 4393 a Black FA(D)1041 4363 y Fx(u)1039 4417 y(\024;d)p Black 970 4393 a } 10/Place PSfrag 970 4393 a Black 1175 4694 a FB(Figure)27 b(3:)37 b(The)27 b(domain)h FA(R)2068 4706 y Fx(\024;d)2193 4694 y FB(de\014ned)g(in)f (\(27\).)p Black Black 195 4906 a(The)j(domain)g FA(R)729 4918 y Fx(\024;d)826 4906 y FB(,)h(where)f(w)n(e)f(measure)g(the)i (di\013erence)e(b)r(et)n(w)n(een)h(the)h(in)n(v)-5 b(arian)n(t)29 b(manifolds,)h(is)g(considerably)71 5006 y(di\013eren)n(t)h(from)f(the) h(ones)f(used)g(in)h(previous)e(w)n(orks)g(\(see)i(for)f(instance)g ([Sau01)o(]\),)i(where)e(the)h(analogous)d(domains)71 5106 y(lo)r(ok)d(lik)n(e)h(a)f(diamonds.)36 b(In)26 b([Sau01)o(],)h (the)f(author)f(considers)g(systems)h(for)f(whic)n(h)h(the)g(unp)r (erturb)r(ed)h(separatrix)d(is)i(a)71 5205 y(graph)f(globally)f(and)i (then)g(they)h(can)e(w)n(ork)g(in)h(suc)n(h)f(wide)h(domains.)36 b(Nev)n(ertheless,)25 b(w)n(e)h(ha)n(v)n(e)e(to)i(restrict)f(ourselv)n (es)71 5305 y(to)i(this)h(narro)n(w)n(er)d Fs(fundamental)30 b(domain)35 b FA(R)1520 5317 y Fx(\024;d)1645 5305 y FB(to)28 b(ensure)f(that)h FA(p)2228 5317 y Fy(0)2265 5305 y FB(\()p FA(u)p FB(\))23 b Fw(6)p FB(=)g(0)k(on)g(it.)195 5404 y(Once)j(w)n(e)f(ha)n(v)n(e)g(the)h(di\013erence)f(\001)h(in)g FA(R)1507 5416 y Fx(\024;d)1605 5404 y FB(,)g(using)f(the)h(argumen)n (ts)f(exp)r(osed)g(in)h(the)g(previous)f(subsection)g(one)71 5504 y(can)e(obtain)g(exp)r(onen)n(tially)g(small)h(upp)r(er)f(b)r (ounds)h(for)f(\001.)p Black 1919 5753 a(21)p Black eop end %%Page: 22 22 TeXDict begin 22 21 bop Black Black 195 272 a FB(Recall)25 b(that)f(our)g(goal)f(is)i(to)f(giv)n(e)g(an)g(asymptotic)g(form)n(ula) g(for)g(the)h(area)e(of)i(the)g(lob)r(e)f(b)r(et)n(w)n(een)h(t)n(w)n(o) e(consecutiv)n(e)71 372 y(homo)r(clinic)28 b(p)r(oin)n(ts.)39 b(Henceforth,)28 b(once)g(w)n(e)g(found)h(the)f(\014rst)g(asymptotic)g (term)g(of)h(\001,)f(whic)n(h)h(w)n(e)e(call)h(\001)3509 384 y Fy(0)3547 372 y FB(,)g(w)n(e)g(will)71 471 y(use)j(the)g(argumen) n(ts)e(exp)r(osed)i(in)g(the)g(previous)f(section)g(to)h(b)r(ound)g (the)h(di\013erence)e(\001\()p FA(u;)14 b(\034)9 b FB(\))22 b Fw(\000)e FB(\001)3273 483 y Fy(0)3311 471 y FB(\()p FA(u;)14 b(\034)9 b FB(\).)47 b(W)-7 b(e)31 b(will)71 571 y(come)c(bac)n(k)g(to)g(the)h(problem)f(of)h(\014nding)g(\001)1485 583 y Fy(0)1550 571 y FB(in)g(Section)f(3.4.)71 803 y Fq(3.3)112 b(P)m(arameterizations)38 b(of)g(the)f(in)m(v)-6 b(arian)m(t)39 b(manifolds)g(of)f(the)f(p)s(erturb)s(ed)h(system)71 956 y FB(In)h(this)h(section)f(w)n(e)g(are)f(going)g(to)h(explain)g (the)h(strategy)e(w)n(e)h(use)g(for)g(pro)n(ving)f(the)h(existence)g (of)h FA(T)3494 926 y Fx(u;s)3627 956 y FB(in)f(the)71 1056 y(corresp)r(onding)26 b(b)r(o)r(omerang)g(domains)h FA(D)1433 1016 y Fx(u;s)1431 1081 y(\024;d)1528 1056 y FB(.)37 b(In)27 b(fact)h(w)n(e)f(will)h(alw)n(a)n(ys)e(deal)h(with)h FA(@)2808 1068 y Fx(u)2852 1056 y FA(T)2913 1025 y Fx(u;s)3006 1056 y FB(.)195 1155 y(W)-7 b(e)31 b(b)r(egin)g(our)f(construction)g (near)f(the)i(origin)f(\()p FA(q)s(;)14 b(p)p FB(\))28 b(=)f(\(0)p FA(;)14 b FB(0\).)46 b(In)30 b(terms)h(of)f(the)h(new)g(v) -5 b(ariable)29 b FA(u)i FB(this)f(corre-)71 1255 y(sp)r(onds)d(to)h (tak)n(e)f(Re)13 b FA(u)28 b FB(near)e Fw(\0001)i FB(for)f(the)h (unstable)f(in)n(v)-5 b(arian)n(t)27 b(manifold)g(and)h(near)e(+)p Fw(1)i FB(for)f(the)h(stable)f(one.)195 1354 y(Giv)n(en)h FA(\032)479 1366 y Fy(1)539 1354 y Fw(\025)23 b FB(0,)k(w)n(e)g (consider)g(the)h(follo)n(wing)e(domains:)1421 1528 y FA(D)1492 1498 y Fx(u)1490 1548 y Fv(1)p Fx(;\032)1610 1556 y Fu(1)1669 1528 y FB(=)d Fw(f)p FA(u)f Fw(2)h Ft(C)p FB(;)14 b(Re)g FA(u)23 b(<)f Fw(\000)p FA(\032)2422 1540 y Fy(1)2459 1528 y Fw(g)1421 1630 y FA(D)1492 1599 y Fx(s)1490 1650 y Fv(1)p Fx(;\032)1610 1658 y Fu(1)1669 1630 y FB(=)h Fw(f)p FA(u)f Fw(2)h Ft(C)p FB(;)14 b(Re)g FA(u)23 b(>)f(\032)2357 1642 y Fy(1)2395 1630 y Fw(g)p FA(:)3703 1581 y FB(\(28\))71 1811 y(It)32 b(is)g(not)f(di\016cult)i (to)f(pro)n(v)n(e)e(that)i(the)g(constan)n(t)f FA(\032)1760 1823 y Fy(1)1829 1811 y FB(can)g(b)r(e)h(tak)n(en)g(big)f(enough)g(so)g (that)h FA(p)3097 1823 y Fy(0)3134 1811 y FB(\()p FA(u)p FB(\))g(do)r(es)g(not)g(v)-5 b(anish)71 1911 y(in)26 b(these)g(domains.)36 b(Henceforth)25 b(the)i(Hamilton-Jacobi)d(form)n (ulation)h(is)h(allo)n(w)n(ed)e(in)i(these)g(domains)f(\(see)h(\(47\))g (and)71 2010 y(\(48\))o(\).)41 b(The)29 b(\014rst)f(result)h(is)f (Theorem)g(4.3,)g(where)h(w)n(e)f(pro)n(v)n(e)f(the)i(existence)g(of)f FA(@)2713 2022 y Fx(u)2757 2010 y FA(T)2818 1980 y Fx(s;u)2940 2010 y FB(and)h(w)n(e)f(see)g(that)h(b)r(oth)g(are)71 2110 y(w)n(ell)k(appro)n(ximated)e(b)n(y)i FA(@)941 2122 y Fx(u)985 2110 y FA(T)1034 2122 y Fy(0)1104 2110 y FB(in)g FA(D)1277 2080 y Fx(u;s)1275 2131 y Fv(1)p Fx(;\032)1395 2139 y Fu(1)1432 2110 y FB(.)53 b(This)33 b(result)g(giv)n(es)f(the)h (existence)g(of)g(lo)r(cal)g(in)n(v)-5 b(arian)n(t)32 b(manifolds)h(and,)71 2210 y(moreo)n(v)n(er,)25 b(pro)n(vides)h (suitable)i(prop)r(erties)e(of)i(them.)195 2309 y(In)23 b(the)h(case)e(that)h FA(p)821 2321 y Fy(0)858 2309 y FB(\()p FA(u)p FB(\))g Fw(6)p FB(=)g(0)f(the)i(next)f(step)g(is)g(to)f (extend)i FA(@)2122 2321 y Fx(u)2165 2309 y FA(T)2226 2279 y Fx(u;s)2342 2309 y FB(to)f(the)h(so-called)d Fs(outer)k(domains) 31 b FB(\(see)23 b(Figure)71 2409 y(4\))k(de\014ned)h(b)n(y)865 2508 y FA(D)936 2474 y Fy(out)p Fx(;u)934 2529 y(\032;\024)1118 2508 y FB(=)23 b Fw(f)p FA(u)f Fw(2)i Ft(C)p FB(;)14 b Fw(j)p FB(Im)f FA(u)p Fw(j)23 b FA(<)g Fw(\000)14 b FB(tan)f FA(\014)2070 2520 y Fy(1)2107 2508 y FB(Re)h FA(u)k FB(+)g FA(a)g Fw(\000)g FA(\024";)c FB(Re)g FA(u)22 b(>)h Fw(\000)p FA(\032)p Fw(g)865 2610 y FA(D)936 2576 y Fy(out)p Fx(;s)934 2631 y(\032;\024)1110 2610 y FB(=)1198 2543 y Fz(\010)1247 2610 y FA(u)f Fw(2)i Ft(C)p FB(;)14 b Fw(\000)p FA(u)22 b Fw(2)h FA(D)1777 2576 y Fy(out)p Fx(;u)1775 2631 y(\032;\024)1936 2543 y Fz(\011)1998 2610 y FA(;)3703 2561 y FB(\(29\))71 2756 y(where)k FA(\024)c(>)f FB(0,)28 b(whic)n(h)f(migh)n(t)h(dep)r(end)g(on)f FA(")p FB(,)h(is)f(suc)n(h)h(that)g FA(a)18 b Fw(\000)g FA(\024")23 b(>)f FB(0.)37 b(The)27 b(constan)n(t)g FA(\032)h FB(will)g(b)r(e)g (tak)n(en)f FA(\032)c(>)f(\032)3693 2768 y Fy(1)3730 2756 y FB(,)28 b(in)71 2855 y(order)d(to)i(ensure)f(that)h FA(D)896 2825 y Fv(\003)894 2876 y(1)p Fx(;\032)1014 2884 y Fu(1)1067 2855 y Fw(\\)17 b FA(D)1210 2825 y Fy(out)p Fx(;)p Fv(\003)1208 2876 y Fx(\032;\024)1387 2855 y Fw(6)p FB(=)23 b Fw(;)j FB(for)g Fw(\003)d FB(=)f FA(u;)14 b(s)p FB(.)37 b(Since)26 b(w)n(e)h(ha)n(v)n(e)e(already)h(pro)n(v)n(ed)f(the) i(existence)f(of)h(lo)r(cal)71 2955 y(in)n(v)-5 b(arian)n(t)26 b(manifolds)i(de\014ned)g(in)g FA(D)1250 2925 y Fx(u;s)1248 2976 y Fv(1)p Fx(;\032)1368 2984 y Fu(1)1404 2955 y FB(,)g(therefore)f FA(@)1848 2967 y Fx(u)1891 2955 y FA(T)1952 2925 y Fx(u;s)2073 2955 y FB(are)g(de\014ned)h(in)g FA(D)2666 2925 y Fv(\003)2664 2976 y(1)p Fx(;\032)2784 2984 y Fu(1)2839 2955 y Fw(\\)19 b FA(D)2984 2925 y Fy(out)o Fx(;)p Fv(\003)2982 2976 y Fx(\032;\024)3165 2955 y FB(for)27 b Fw(\003)c FB(=)f FA(u;)14 b(s)p FB(.)p Black Black Black 305 4511 a /PSfrag where{pop(d)[[0(Bl)1 0]](a)[[1(Bl)1 0]](b)[[2(Bl)1 0]](D2)[[3(Bl)1 0]](D3)[[4(Bl)1 0]](D)[[5(Bl)1 0]](D1)[[6(Bl)1 0]](r)[[7(Bl)1 0]](rr)[[8(Bl)1 0]](r1)[[9(Bl)1 0]](rr1)[[10(Bl)1 0]](s)[[11(Bl)1 0]](u)[[12(Bl)1 0]](v)[[13(Bl)1 0]]14 0 -1/Begin PSfrag}{userdict /PSfrag{pop}put}ifelse 305 4511 a @beginspecial 11 @llx 619 @lly 483 @urx 821 @ury 1700 @rhi @setspecial %%BeginDocument: outer.eps %!PS-Adobe-3.0 EPSF-3.0 %%Creator: 0.46 %%Pages: 1 %%Orientation: Portrait %%BoundingBox: 11 619 483 821 %%HiResBoundingBox: 11.211606 619.88571 482.13257 820.6857 %%EndComments %%Page: 1 1 0 842 translate 0.8 -0.8 scale 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap gsave [1 0 0 1 0 0] concat 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 217.78572 126.64791 moveto 222.96429 126.64791 lineto stroke gsave [1 0 0 -1 61.058746 165.19264] concat gsave /newlatin1font {findfont dup length dict copy dup /Encoding ISOLatin1Encoding put definefont} def /Sans-ISOLatin1 /Sans newlatin1font 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (r) show grestore grestore gsave [1 0 0 -1 235.82143 121.82648] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (s) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 3 setlinewidth 0 setlinejoin 0 setlinecap newpath 212.07143 187.00505 moveto 217.78571 187.00505 lineto stroke gsave [1 0 0 -1 184 194.36218] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (b) show grestore grestore gsave [1 0 0 -1 111 165.36218] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (rr) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 214.57143 27.005054 moveto 214.57143 277.00505 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 189.92857 117.71934 moveto 181.90492 124.88054 174.82344 133.5361 176.71429 152.00505 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 59.571426 62.005054 moveto 59.571426 242.00505 lineto stroke gsave [1 0 0 -1 193 117.36218] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 234.57143 117.00505 moveto 217.57143 127.00505 lineto 220.82143 122.21934 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 294.57143 152.00505 moveto 19.571426 152.00505 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 269.57143 152.00505 moveto 59.571426 62.005054 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 269.57143 152.00506 moveto 59.571426 242.00505 lineto stroke gsave [1 0 0 -1 51.523216 58.520027] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (u) show grestore grestore gsave [1 0 0 -1 53.523216 253.6013] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (v) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 3 setlinewidth 0 setlinejoin 0 setlinecap newpath 212.07144 116.00505 moveto 217.78572 116.00505 lineto stroke gsave [1 0 0 -1 154.5 140.57648] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (d) show grestore grestore 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 109.57143 27.005054 moveto 109.57143 277.00505 lineto stroke gsave [1 0 0 -1 10.92857 143.07648] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D) show grestore grestore gsave [1 0 0 -1 148.92857 174.29077] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D2) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 398.21428 126.64791 moveto 393.03571 126.64791 lineto stroke gsave [-1 0 0 1 0 0] concat grestore gsave [-1 0 0 1 0 0] concat grestore 0 0 0 setrgbcolor [] 0 setdash 3 setlinewidth 0 setlinejoin 0 setlinecap newpath 403.92857 187.00505 moveto 398.21429 187.00505 lineto stroke gsave [-1 0 0 1 0 0] concat grestore gsave [-1 0 0 1 0 0] concat grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 401.42857 27.005054 moveto 401.42857 277.00505 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 556.42857 62.005054 moveto 556.42857 242.00505 lineto stroke gsave [-1 0 0 1 0 0] concat grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 381.42857 117.00505 moveto 398.42857 127.00505 lineto 395.17857 122.21934 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 321.42857 152.00505 moveto 596.42857 152.00505 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 346.42857 152.00505 moveto 556.42857 62.005054 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 346.42857 152.00506 moveto 556.42857 242.00505 lineto stroke gsave [-1 0 0 1 0 0] concat grestore gsave [-1 0 0 1 0 0] concat grestore 0 0 0 setrgbcolor [] 0 setdash 3 setlinewidth 0 setlinejoin 0 setlinecap newpath 403.92856 116.00505 moveto 398.21428 116.00505 lineto stroke gsave [-1 0 0 1 0 0] concat grestore 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 506.42857 27.005054 moveto 506.42857 277.00505 lineto stroke gsave [-1 0 0 1 0 0] concat grestore gsave [-1 0 0 1 0 0] concat grestore gsave [1 0 0 -1 558.85712 142.36218] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D1) show grestore grestore gsave [1 0 0 -1 538.71429 162.36218] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (r1) show grestore grestore gsave [1 0 0 -1 488.42856 161.6479] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (rr1) show grestore grestore gsave [1 0 0 -1 427.14285 173.07648] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D3) show grestore grestore gsave [1 0 0 -1 405 117.36218] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a) show grestore grestore gsave [1 0 0 -1 318 122.36218] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (s) show grestore grestore gsave [1 0 0 -1 402 199.36218] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (b) show grestore grestore grestore showpage %%EOF %%EndDocument @endspecial 305 4511 a /End PSfrag 305 4511 a 305 3025 a /Hide PSfrag 305 3025 a -435 3083 a FB(PSfrag)26 b(replacemen)n(ts)p -435 3113 741 4 v 305 3116 a /Unhide PSfrag 305 3116 a 221 3199 a { 221 3199 a Black FA(\014)268 3211 y Fy(1)p Black 221 3199 a } 0/Place PSfrag 221 3199 a 232 3315 a { 232 3315 a Black FA(ia)p Black 232 3315 a } 1/Place PSfrag 232 3315 a 168 3408 a { 168 3408 a Black Fw(\000)p FA(ia)p Black 168 3408 a } 2/Place PSfrag 168 3408 a 75 3483 a { 75 3483 a Black FA(D)146 3452 y Fy(out)p Fx(;u)144 3503 y(\032;\024)p Black 75 3483 a } 3/Place PSfrag 75 3483 a 83 3582 a { 83 3582 a Black FA(D)154 3552 y Fy(out)p Fx(;s)152 3603 y(\032;\024)p Black 83 3582 a } 4/Place PSfrag 83 3582 a 79 3682 a { 79 3682 a Black FA(D)150 3652 y Fx(u)148 3702 y Fv(1)p Fx(;\032)268 3710 y Fu(1)p Black 79 3682 a } 5/Place PSfrag 79 3682 a 79 3781 a { 79 3781 a Black FA(D)150 3751 y Fx(s)148 3802 y Fv(1)p Fx(;\032)268 3810 y Fu(1)p Black 79 3781 a } 6/Place PSfrag 79 3781 a 234 3897 a { 234 3897 a Black FB(-)p FA(\032)p Black 234 3897 a } 7/Place PSfrag 234 3897 a 197 3996 a { 197 3996 a Black FB(-)p FA(\032)268 4008 y Fy(1)p Black 197 3996 a } 8/Place PSfrag 197 3996 a 262 4096 a { 262 4096 a Black FA(\032)p Black 262 4096 a } 9/Place PSfrag 262 4096 a 225 4196 a { 225 4196 a Black FA(\032)268 4208 y Fy(1)p Black 225 4196 a } 10/Place PSfrag 225 4196 a -20 4291 a { -20 4291 a Black FA(i)p FB(\()p FA(a)18 b Fw(\000)g FA(\024")p FB(\))p Black -20 4291 a } 11/Place PSfrag -20 4291 a 220 4399 a { 220 4399 a Black FA(u)268 4411 y Fy(1)p Black 220 4399 a } 12/Place PSfrag 220 4399 a 220 4499 a { 220 4499 a Black 220 4452 48 4 v -1 x FA(u)267 4511 y Fy(1)p Black 220 4499 a } 13/Place PSfrag 220 4499 a Black 810 4776 a FB(Figure)27 b(4:)37 b(The)27 b(outer)g(domains)g FA(D)1960 4746 y Fy(out)p Fx(;u)1958 4797 y(\032;\024)2147 4776 y FB(and)g FA(D)2379 4746 y Fy(out)p Fx(;s)2377 4797 y(\032;\024)2558 4776 y FB(de\014ned)h(in)g(\(29\))o(.)p Black Black 195 5006 a(In)i(Theorem)e(4.4)g(it)i(is)f(pro)n(v)n(ed)f(that)h FA(@)1455 5018 y Fx(u)1499 5006 y FA(T)1560 4976 y Fx(u;s)1653 5006 y FB(\()p FA(u;)14 b(\034)9 b FB(\))30 b(can)f(b)r(e)g(extended)h (to)f(the)g Fs(outer)i(domain)37 b FA(D)3340 4972 y Fy(out)p Fx(;)p Fv(\003)3338 5026 y Fx(\032;\024)3494 5006 y FB(,)30 b Fw(\003)25 b FB(=)g FA(u;)14 b(s)p FB(,)71 5106 y(and)27 b(that)h(is)g(w)n(ell)f(appro)n(ximated)f(\(in)i(some)f(norm\))h(b)n(y) f FA(@)1936 5118 y Fx(u)1979 5106 y FA(T)2028 5118 y Fy(0)2065 5106 y FB(\()p FA(u)p FB(\))h(there.)195 5205 y(In)g(the)f(case)f(that)h FA(p)837 5217 y Fy(0)874 5205 y FB(\()p FA(u)p FB(\))h(can)e(v)-5 b(anish)27 b(in)g(the)h(outer)e (domains)h(the)g(pro)r(cedure)f(b)r(ecomes)h(a)f(little)i(tec)n (hnical.)36 b(The)71 5305 y(main)28 b(idea)f(is)h(to)g(use)g (parameterizations)d(of)j(the)h(in)n(v)-5 b(arian)n(t)27 b(manifolds)g(of)h(the)g(form)g(\()p FA(Q)p FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(P)e FB(\()p FA(u;)i(\034)9 b FB(\)\))29 b(to)f(extend)71 5404 y(them)h(to)f(a)h(new)f(domain)g (where)g FA(p)1207 5416 y Fy(0)1244 5404 y FB(\()p FA(u)p FB(\))d Fw(6)p FB(=)f(0)k(do)r(es)h(not)f(v)-5 b(anish)28 b(an)n(ymore)f(and)i(that)g(o)n(v)n(erlaps)d(with)j(the)g Fs(b)l(o)l(omer)l(ang)71 5504 y(domain)37 b FA(D)445 5464 y Fx(u;s)443 5529 y(\024;d)569 5504 y FB(\(see)30 b(Theorem)e(4.6\).)42 b(It)30 b(is)f(imp)r(ortan)n(t)g(to)g(emphasize)g (here)g(that)g(these)h(new)f(domains)g(are)f(still)i(far)p Black 1919 5753 a(22)p Black eop end %%Page: 23 23 TeXDict begin 23 22 bop Black Black 71 272 a FB(a)n(w)n(a)n(y)34 b(from)i(the)g(singularities)f Fw(\006)p FA(ia)h FB(of)g FA(T)1440 284 y Fy(0)1477 272 y FB(\()p FA(u)p FB(\),)i(henceforth)e (these)g(parameterizations)e(are)h(straigh)n(tforw)n(ard)f(\(see)71 372 y(Theorem)e(4.7\).)54 b(Once)33 b(w)n(e)g(ha)n(v)n(e)f(pro)n(v)n (ed)g(the)h(existence)g(of)h(the)f(parameterizations)f(of)h(the)h(in)n (v)-5 b(arian)n(t)32 b(manifolds)71 471 y(for)j(v)-5 b(alues)36 b(of)f FA(u)h FB(far)f(from)h(the)g(singularities)e(but)j (inside)f(the)g Fs(b)l(o)l(omer)l(ang)i(domains)44 b FA(D)2996 431 y Fx(u;s)2994 496 y(\024;d)3091 471 y FB(,)38 b(w)n(e)d(can)h(reco)n(v)n(er)d(the)71 581 y(generating)e(functions)j FA(@)890 593 y Fx(u)933 581 y FA(T)994 551 y Fx(u;s)1088 581 y FB(\()p FA(u;)14 b(\034)9 b FB(\))34 b(and)e(extend)i(them)f(to)g (the)h(whole)e Fs(b)l(o)l(omer)l(ang)k(domains)41 b FA(D)3297 541 y Fx(u;s)3295 606 y(\024;d)3425 581 y FB(in)33 b(Theorem)71 681 y(4.8.)195 780 y(W)-7 b(e)28 b(w)n(an)n(t)f(to)h(emphasize)f(here)g (that)p Black 195 917 a Fw(\017)p Black 41 w FB(W)-7 b(e)28 b(are)f(able)g(to)g(extend)h(the)g(manifolds)f(up)g(to)h(a)f (distance)g(of)g(order)f FA(")i FB(of)f(the)h(singularities)e(in)i(all) f(the)h(cases)278 1017 y(without)g(using)g(an)n(y)f(inner)g(equation)g (ev)n(en)g(in)h(the)g(singular)e(case)h FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b Fw(\025)e FB(0)k(and)h FA(\021)e FB(=)c FA(`)d Fw(\000)f FB(2)p FA(r)r FB(.)p Black 195 1168 a Fw(\017)p Black 41 w FB(The)28 b(outer)f(domain)g FA(D)1032 1138 y Fy(out)p Fx(;)p Fv(\003)1030 1189 y Fx(\032;\024)1214 1168 y FB(con)n(tains)f(the)i Fs(b)l(o)l(omer)l(ang)j(domain)k FA(D)2463 1138 y Fv(\003)2461 1192 y Fx(\024;d)2586 1168 y FB(for)27 b Fw(\003)22 b FB(=)h FA(u;)14 b(s)p FB(.)71 1395 y Fq(3.4)112 b(The)38 b(asymptotic)g(\014rst)f(order)g(of)h FF(\001)71 1548 y FB(Ev)n(en)24 b(though)h(w)n(e)f(ha)n(v)n(e)g(pro)n (v)n(ed)f(the)j(existence)e(of)h(the)h(in)n(v)-5 b(arian)n(t)24 b(manifolds)g(in)i(the)f Fs(b)l(o)l(omer)l(ang)j(domains)p FB(,)f(w)n(e)d(need)71 1647 y(some)35 b(extra)g(information)g(to)h (detect)g(the)g(asymptotic)g(\014rst)f(order)g(of)h(this)g (di\013erence.)61 b(The)36 b(main)g(idea)f(is)h(that)71 1747 y(functions)24 b(whic)n(h)h(are)e(of)h(algebraic)f(order)f(with)j (resp)r(ect)f(to)g FA(")g FB(near)g(the)g(singularities)g Fw(\006)p FA(ia)f FB(are)g(exp)r(onen)n(tially)h(small)71 1847 y(for)33 b(real)f(v)-5 b(alues)33 b(of)h FA(u)p FB(.)54 b(Th)n(us,)35 b(the)e(main)h(p)r(oin)n(t)g(to)f(compute)g(the)h (di\013erence)g(and)f(capture)g(the)h(asymptotic)f(\014rst)71 1946 y(order)d(is)h(to)g(b)r(e)h(able)f(to)h(giv)n(e)e(the)i(main)f (terms)g(of)h(this)f(di\013erence)h(close)e(to)h(the)h(singularities,)f (concretely)-7 b(,)32 b(up)g(to)71 2046 y(distance)26 b(of)g(order)f FA(")h FB(of)h(the)f(singularities.)36 b(F)-7 b(or)25 b(that)i(w)n(e)f(need)g(to)h(giv)n(e)e(b)r(etter)i (appro)n(ximations)d(of)i(the)h(generating)71 2146 y(functions)h FA(T)490 2115 y Fx(u;s)583 2146 y FB(\()p FA(u;)14 b(\034)9 b FB(\))29 b(near)d(of)i(the)g(singularities)e Fw(\006)p FA(ia)h FB(of)h(the)g(homo)r(clinic)f(connection.)195 2245 y(T)-7 b(o)28 b(this)g(end,)f(w)n(e)h(de\014ne)f(the)h(so-called)f Fs(inner)i(domains)36 b FB(\(see)28 b(Figure)f(5\),)g(whic)n(h)h(are)e (de\014ned)i(as)487 2394 y FA(D)558 2360 y Fy(in)o Fx(;)p Fy(+)p Fx(;u)556 2415 y(\024;c)770 2394 y FB(=)13 b Fw(f)p FA(u)23 b Fw(2)g Ft(C)p FB(;)14 b(Im)g FA(u)22 b(>)h Fw(\000)14 b FB(tan)f FA(\014)1666 2406 y Fy(1)1703 2394 y FB(\(Re)i FA(u)j FB(+)g FA(c")2072 2360 y Fx(\015)2114 2394 y FB(\))h(+)f FA(a;)c FB(Im)f FA(u)23 b(<)g Fw(\000)14 b FB(tan)f FA(\014)2859 2406 y Fy(2)2896 2394 y FB(Re)h FA(u)k FB(+)g FA(a)g Fw(\000)g FA(\024";)858 2529 y FB(Im)c FA(u)23 b(<)g Fw(\000)14 b FB(tan)f FA(\014)1389 2541 y Fy(0)1426 2529 y FB(Re)h FA(u)k FB(+)g FA(a)g Fw(\000)g FA(\024")p Fw(g)486 2666 y FA(D)557 2631 y Fy(in)o Fx(;)p Fv(\000)p Fx(;u)555 2686 y(\024;c)770 2666 y FB(=)848 2598 y Fz(\010)897 2666 y FA(u)k Fw(2)i Ft(C)p FB(;)19 b(\026)-47 b FA(u)22 b Fw(2)i FA(D)1363 2631 y Fy(in)o Fx(;)p Fy(+)p Fx(;u)1361 2686 y(\024;c)1552 2598 y Fz(\011)495 2804 y FA(D)566 2770 y Fy(in)o Fx(;)p Fy(+)p Fx(;s)564 2825 y(\024;c)770 2804 y FB(=)848 2737 y Fz(\010)897 2804 y FA(u)e Fw(2)i Ft(C)p FB(;)14 b Fw(\000)5 b FB(\026)-47 b FA(u)22 b Fw(2)h FA(D)1427 2770 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)1425 2825 y(\024;c)1617 2737 y Fz(\011)494 2943 y FA(D)565 2908 y Fy(in)o Fx(;)p Fv(\000)p Fx(;s)563 2963 y(\024;c)770 2943 y FB(=)848 2875 y Fz(\010)897 2943 y FA(u)f Fw(2)i Ft(C)p FB(;)14 b Fw(\000)p FA(u)22 b Fw(2)h FA(D)1427 2908 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)1425 2963 y(\024;c)1617 2875 y Fz(\011)3703 2669 y FB(\(30\))71 3092 y(for)k FA(\024)c(>)f FB(0,)27 b FA(c)c(>)g FB(0)k(and)g FA(\015)h Fw(2)c FB(\(0)p FA(;)14 b FB(1\).)36 b(On)27 b(the)h(other)f(hand,)g FA(\014)1994 3104 y Fy(1)2059 3092 y FB(and)h FA(\014)2268 3104 y Fy(2)2332 3092 y FB(are)f(the)h(angles)e(considered)g(in)i(the)g(de\014nition)71 3191 y(of)e(the)g(b)r(o)r(omerang)f(domains)g(in)h(\(26\))g(and)g FA(\014)1535 3203 y Fy(0)1598 3191 y FB(is)g(an)n(y)f(angle)h (satisfying)f(that)h FA(\014)2644 3203 y Fy(1)2697 3191 y Fw(\000)15 b FA(\014)2824 3203 y Fy(0)2887 3191 y FB(has)25 b(a)h(p)r(ositiv)n(e)g(lo)n(w)n(er)e(b)r(ound)71 3291 y(indep)r(enden)n(t)k(of)g FA(")f FB(and)h FA(\026)p FB(.)37 b(Let)27 b(us)h(observ)n(e)e(that,)i(if)g FA(u)22 b Fw(2)i FA(D)2024 3261 y Fy(in)o Fx(;)p Fv(\006)p Fx(;)p Fv(\003)2022 3312 y Fx(\024;c)2209 3291 y FB(,)k Fw(\003)22 b FB(=)h FA(u;)14 b(s)p FB(,)27 b(then)h Fw(O)r FB(\()p FA(\024")p FB(\))c Fw(\024)e(j)p FA(u)c Fw(\007)g FA(ia)p Fw(j)23 b(\024)g(O)r FB(\()p FA(")3623 3261 y Fx(\015)3666 3291 y FB(\).)p Black Black Black 783 4834 a /PSfrag where{pop(D)[[0(Bl)1 0]](D1)[[1(Bl)1 0]](D3)[[2(Bl)1 0]](D4)[[3(Bl)1 0]](b0)[[4(Bl)1 0]](b1)[[5(Bl)1 0]](b2)[[6(Bl)1 0]](a)[[7(Bl)1 0]](a1)[[8(Bl)1 0]](a2)[[9(Bl)1 0]]10 0 -1/Begin PSfrag}{userdict /PSfrag{pop}put}ifelse 783 4834 a @beginspecial 31 @llx 543 @lly 483 @urx 815 @ury 1700 @rhi @setspecial %%BeginDocument: Inners.eps %!PS-Adobe-3.0 EPSF-3.0 %%Creator: 0.46 %%Pages: 1 %%Orientation: Portrait %%BoundingBox: 31 543 483 815 %%HiResBoundingBox: 31.6 543.5602 482.36754 815 %%EndComments %%Page: 1 1 0 842 translate 0.8 -0.8 scale 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap gsave [1 0 0 1 0 0] concat gsave [1 0 0 -1 0.1197834 424.74529] concat grestore gsave [1 0 0 -1 165 62.362183] concat gsave /newlatin1font {findfont dup length dict copy dup /Encoding ISOLatin1Encoding put definefont} def /Sans-ISOLatin1 /Sans newlatin1font 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a) show grestore grestore gsave [1 0 0 -1 165 372.36218] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a1) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 210.11173 62.538956 moveto 162.12948 85.519926 lineto 168.82174 79.206473 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 162.25575 85.393657 moveto 170.2107 84.130966 lineto stroke gsave [1 0 0 -1 210 62.362183] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a2) show grestore grestore gsave [1 0 0 -1 74.413094 140.23947] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 160 87.362183 moveto 180 137.36218 lineto 95 77.362183 lineto 160 87.362183 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 160 87.362176 moveto 172.74511 119.06481 209.31635 211.92221 209.31635 211.92221 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 254.55844 102.81878 moveto 253.33845 104.91998 251.93763 109.59608 252.79067 112.41524 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 203.79828 198.78328 moveto 198.11961 203.7448 197.69056 208.12304 197.9899 212.42034 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 267.94296 200.29851 moveto 263.39728 205.69373 263.64981 212.41458 263.64981 212.42034 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 160 52.362183 moveto 160 372.36218 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 305 212.36218 moveto 40 212.36218 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 159.85664 123.02184 moveto 285.36809 212.42034 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 97.459416 77.312437 moveto 310 112.36218 lineto 250 112.36218 lineto stroke gsave [1 0 0 -1 0.1197834 424.74529] concat 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 160 87.362183 moveto 180 137.36218 lineto 95 77.362183 lineto 160 87.362183 lineto stroke grestore gsave [1 0 0 -1 0.1197834 424.74529] concat 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 160 87.362176 moveto 172.74511 119.06481 209.31635 211.92221 209.31635 211.92221 curveto stroke grestore 0 0 0 setrgbcolor [] 0 setdash 3 setlinewidth 0 setlinejoin 0 setlinecap newpath 162 57.362183 moveto 158 57.362183 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 3 setlinewidth 0 setlinejoin 0 setlinecap newpath 162 367.36218 moveto 158 367.36218 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 100 122.36218 moveto 145 97.362183 lineto 139.27478 97.768026 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 144.57808 97.515488 moveto 141.80016 101.5561 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 99.626295 305.73318 moveto 147.10346 327.57773 lineto 141.80016 323.03204 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 146.72466 327.32519 moveto 140.53747 327.07265 lineto stroke gsave [1 0 0 -1 68.09964 297.07709] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D1) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 482.45942 87.411929 moveto 462.45942 137.41193 lineto 547.45942 77.411929 lineto 482.45942 87.411929 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 482.45942 87.411922 moveto 469.71431 119.11456 433.14307 211.97196 433.14307 211.97196 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 438.66114 198.83303 moveto 444.33981 203.79455 444.76886 208.17279 444.46952 212.47009 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 374.51646 200.34826 moveto 379.06214 205.74348 378.80961 212.46433 378.80961 212.47009 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 482.45942 52.411929 moveto 482.45942 372.41193 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 337.45942 212.41193 moveto 602.45942 212.41193 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 482.60278 123.07159 moveto 357.09133 212.47009 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 482.33964 337.43286 moveto 462.33964 287.43286 lineto 547.33964 347.43286 lineto 482.33964 337.43286 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 482.33964 337.43286 moveto 469.59453 305.73023 433.02329 212.87283 433.02329 212.87283 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 3 setlinewidth 0 setlinejoin 0 setlinecap newpath 480.45942 57.411929 moveto 484.45942 57.411929 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 3 setlinewidth 0 setlinejoin 0 setlinecap newpath 480.45942 367.41193 moveto 484.45942 367.41193 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 542.45942 122.41193 moveto 497.45942 97.411929 lineto 503.18464 97.817772 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 497.88134 97.565234 moveto 500.65926 101.60585 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 542.83312 305.78293 moveto 495.35596 327.62748 lineto 500.65926 323.08179 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 495.73476 327.37494 moveto 501.92195 327.1224 lineto stroke gsave [1 0 0 -1 541.44177 133.08276] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D3) show grestore grestore gsave [1 0 0 -1 541.44177 307.92038] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D4) show grestore grestore gsave [1 0 0 -1 177.28177 207.87465] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (b2) show grestore grestore gsave [1 0 0 -1 244.70946 208.88481] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (b1) show grestore grestore gsave [1 0 0 -1 228.547 112.1627] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (b0) show grestore grestore grestore showpage %%EOF %%EndDocument @endspecial 783 4834 a /End PSfrag 783 4834 a 783 3708 a /Hide PSfrag 783 3708 a 43 3766 a FB(PSfrag)j(replacemen)n(ts)p 43 3795 741 4 v 783 3798 a /Unhide PSfrag 783 3798 a 523 3876 a { 523 3876 a Black FA(D)594 3846 y Fy(in)o Fx(;)p Fy(+)p Fx(;u)592 3896 y(\024;c)p Black 523 3876 a } 0/Place PSfrag 523 3876 a 522 3985 a { 522 3985 a Black FA(D)593 3955 y Fy(in)o Fx(;)p Fv(\000)p Fx(;u)591 4006 y(\024;c)p Black 522 3985 a } 1/Place PSfrag 522 3985 a 531 4095 a { 531 4095 a Black FA(D)602 4065 y Fy(in)o Fx(;)p Fy(+)p Fx(;s)600 4115 y(\024;c)p Black 531 4095 a } 2/Place PSfrag 531 4095 a 530 4204 a { 530 4204 a Black FA(D)601 4174 y Fy(in)o Fx(;)p Fv(\000)p Fx(;s)599 4225 y(\024;c)p Black 530 4204 a } 3/Place PSfrag 530 4204 a 699 4319 a { 699 4319 a Black FA(\014)746 4331 y Fy(0)p Black 699 4319 a } 4/Place PSfrag 699 4319 a 699 4419 a { 699 4419 a Black FA(\014)746 4431 y Fy(1)p Black 699 4419 a } 5/Place PSfrag 699 4419 a 699 4519 a { 699 4519 a Black FA(\014)746 4531 y Fy(2)p Black 699 4519 a } 6/Place PSfrag 699 4519 a 711 4634 a { 711 4634 a Black FA(ia)p Black 711 4634 a } 7/Place PSfrag 711 4634 a 646 4727 a { 646 4727 a Black Fw(\000)p FA(ia)p Black 646 4727 a } 8/Place PSfrag 646 4727 a 458 4813 a { 458 4813 a Black FA(i)p FB(\()p FA(a)18 b Fw(\000)g FA(\024")p FB(\))p Black 458 4813 a } 9/Place PSfrag 458 4813 a Black 1138 5099 a FB(Figure)27 b(5:)37 b(The)27 b Fs(inner)j(domains)36 b FB(de\014ned)28 b(in)g(\(30\))o(.)p Black Black 195 5305 a(Let)k(us)f(observ)n(e)f (that)i(simply)f(rewriting)g FA(\026)e FB(:=)g FA(\026")1856 5275 y Fx(\021)r Fv(\000)p Fx(\021)1980 5250 y Fl(\003)2019 5305 y FB(,)k(one)e(can)g(include)g(the)h(regular)e(case)g(\()p FA(\021)j(>)c(\021)3532 5275 y Fv(\003)3571 5305 y FB(\))i(in)h(the)71 5404 y(singular)g(one.)52 b(This)33 b(is)g(v)n(ery)f(con)n(v)n(enien)n (t)g(since)g(one)h(can)g(pro)n(v)n(e)e(the)i(results)g(for)f(b)r(oth)h (cases)f(at)h(the)h(same)e(time.)71 5504 y(Therefore,)26 b(from)i(no)n(w)f(on)g(in)h(this)g(section,)f(w)n(e)g(will)h(fo)r(cus)g (on)f(the)h(singular)e(case.)p Black 1919 5753 a(23)p Black eop end %%Page: 24 24 TeXDict begin 24 23 bop Black Black 195 272 a FB(When)34 b(studying)f(the)h(functions)f FA(@)1344 284 y Fx(u)1388 272 y FA(T)1449 242 y Fx(u;s)1575 272 y FB(ev)-5 b(aluated)33 b(in)h(the)f(inner)g(domains,)h(one)f(can)g(distinguish)g(the)h(cases) 71 372 y FA(`)t Fw(\000)t FB(2)p FA(r)25 b(<)d FB(0)f(or)e FA(`)t Fw(\000)t FB(2)p FA(r)25 b Fw(\025)e FB(0.)34 b(The)20 b(di\013erence)h(b)r(et)n(w)n(een)f(these)h(t)n(w)n(o)f (cases,)g(roughly)g(sp)r(eaking,)h(is)f(that,)j(when)d FA(`)t Fw(\000)t FB(2)p FA(r)25 b(<)e FB(0,)71 471 y(the)f(appro)n (ximation)d(of)i(the)h(manifolds)f(in)g(the)h(inner)f(domain)g(is)g (still)h(giv)n(en)e(b)n(y)h(the)h(\014rst)f(order)f(p)r(erturbation)g (theory)71 571 y(as)27 b(is)g(stated)h(in)g(Prop)r(osition)e(4.18.)35 b(In)28 b(the)g(case)f FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b Fw(\025)e FB(0)k(this)h(fact)f(is)h(not)g(true)f(an)n(ymore.)195 671 y(Analyzing)39 b FA(@)643 683 y Fx(u)686 671 y FA(T)747 640 y Fx(u;s)880 671 y FB(close)f(to)g(the)h(singularit)n(y)f FA(ia)p FB(,)j(one)d(can)h(see)f(that,)k(if)e FA(u)25 b Fw(\000)g FA(ia)42 b FB(=)f Fw(O)r FB(\()p FA(")p FB(\),)h(then)e FA(@)3558 683 y Fx(u)3601 671 y FA(T)3662 640 y Fx(u;s)3795 671 y FB(is)71 770 y(of)34 b(order)f Fw(O)r FB(\(1)p FA(=")619 740 y Fy(2)p Fx(r)688 770 y FB(\).)58 b(F)-7 b(or)33 b(this)i(reason)e(w)n(e)h(p)r(erform)g(the)g(c)n(hange)f(of)i (v)-5 b(ariables)33 b FA(u)g FB(=)h FA(ia)23 b FB(+)f FA("z)38 b FB(and)c(w)n(e)g(study)g(the)71 870 y(functions)c FA( )488 840 y Fx(u;s)582 870 y FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))28 b(=)e FA(")929 840 y Fy(2)p Fx(r)r Fv(\000)p Fy(1)1083 870 y FA(T)1144 840 y Fx(u;s)1238 870 y FB(\()p FA(ia)19 b FB(+)h FA("z)t(;)14 b(\034)9 b FB(\).)43 b(The)30 b(\014rst)g(order)e(in)i FA(")f FB(of)h(these)g(functions)g(v)n (eri\014es)f(the)h(so)f(called)71 969 y Fs(inner)h(e)l(quation)k FB(whic)n(h)28 b(do)r(es)g(not)g(dep)r(end)g(on)g FA(")p FB(.)37 b(Their)28 b(solutions)f FA( )2336 930 y Fx(u;s)2333 992 y Fy(0)2430 969 y FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))29 b(w)n(ere)d(studied)j(in)f([Bal06)n(].)38 b(Then,)28 b(in)71 1069 y(Theorem)f(4.16)f(w)n(e)h(pro)n(vide)g(a)g(b)r(ound)h (for)f Fw(j)p FA( )1547 1039 y Fx(u;s)1641 1069 y FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))19 b Fw(\000)f FA( )1989 1029 y Fx(u;s)1986 1091 y Fy(0)2084 1069 y FB(\()p FA(z)t(;)c(\034)9 b FB(\))p Fw(j)p FB(.)37 b(This)28 b(is)f(kno)n(wn)g(as)g Fs(c)l(omplex)j(matching)p FB(.)195 1169 y(It)k(is)g(imp)r(ortan)n(t)f (to)h(emphasize)f(here)g(that)h(w)n(e)f(ha)n(v)n(e)g(not)h(needed)f(to) h(use)f(the)i(inner)e(solutions)g FA( )3459 1129 y Fx(u;s)3456 1191 y Fy(0)3553 1169 y FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))35 b(to)71 1268 y(extend)g(our)f(functions)h FA(T)929 1238 y Fx(u;s)1057 1268 y FB(to)g(the)g(inner)f(domains)g (since)h(w)n(e)f(already)g(knew)g(their)h(existence.)58 b(Henceforth)35 b(to)71 1368 y(b)r(ound)h Fw(j)p FA( )415 1338 y Fx(u;s)509 1368 y FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))24 b Fw(\000)f FA( )867 1328 y Fx(u;s)864 1390 y Fy(0)962 1368 y FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))p Fw(j)36 b FB(w)n(e)f(ha)n(v)n(e)f(exploited)h(the)h(same)f(idea)g(as)f (the)i(one)f(for)g(studying)g(the)h(di\013erence)71 1468 y(\001)25 b(=)f FA(T)315 1437 y Fx(u)376 1468 y Fw(\000)19 b FA(T)521 1437 y Fx(s)555 1468 y FB(.)40 b(Let)28 b(us)h(explain)f(it) h(with)g(more)e(detail.)40 b(As)28 b(w)n(e)g(ha)n(v)n(e)f(explained)i (in)f(Section)h(3.3,)f(w)n(e)g(ha)n(v)n(e)f(already)71 1567 y(pro)n(v)n(ed)34 b(the)i(existence)g(of)g(generating)e(functions) i FA(T)1806 1537 y Fx(u;s)1935 1567 y FB(in)h(the)f(whole)f Fs(b)l(o)l(omer)l(ang)j(domains)p FB(,)h(therefore,)e(w)n(e)f(can)71 1667 y(translate)30 b(their)h(prop)r(erties)f(to)g(the)i(new)f (functions)g FA( )1860 1637 y Fx(u;s)1955 1667 y FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))29 b(=)f FA(")2305 1637 y Fy(2)p Fx(r)r Fv(\000)p Fy(1)2459 1667 y FA(T)2520 1637 y Fx(u;s)2614 1667 y FB(\()p FA(ia)20 b FB(+)g FA("z)t(;)14 b(\034)9 b FB(\).)47 b(No)n(w)31 b(w)n(e)f(consider)g(the)71 1766 y(di\013erence)e(\001)p FA( )569 1736 y Fx(u;s)689 1766 y FB(=)c FA(@)822 1778 y Fx(z)860 1766 y FA( )917 1736 y Fx(u;s)1031 1766 y Fw(\000)18 b FA(@)1158 1778 y Fx(z)1197 1766 y FA( )1251 1778 y Fy(0)1288 1766 y FB(.)40 b(Suc)n(h)29 b(a)f(function)h(\(whic)n(h)g(is)f(kno)n(wn\))g (satis\014es)g(a)g(non-homogeneous)e(linear)71 1866 y(equation)h(whic)n (h)g(can)g(b)r(e)g(\\easily")f(studied.)37 b(Summarizing,)27 b(w)n(e)f(ha)n(v)n(e)g(just)i(obtained)f(an)g(\\a)g(p)r(osteriori")e(b) r(ound)j(of)71 1966 y(\001)p FA( )197 1936 y Fx(u;s)292 1966 y FB(.)35 b(This)24 b(mak)n(es)f(our)g Fs(c)l(omplex)k(matching)32 b FB(considerably)23 b(simpler)h(b)r(ecause)f(w)n(e)h(just)g(need)g(to) g(use)g(Gron)n(w)n(all-lik)n(e)71 2065 y(tec)n(hniques.)195 2165 y(In)35 b(b)r(oth)f(cases)f FA(`)23 b Fw(\000)f FB(2)p FA(r)36 b(<)d FB(0)h(and)g FA(`)22 b Fw(\000)h FB(2)p FA(r)36 b Fw(\025)d FB(0,)j(w)n(e)e(ha)n(v)n(e)e(no)n(w)i (accurate)f(appro)n(ximations)f(for)h FA(T)3416 2135 y Fx(u;s)3544 2165 y FB(near)g(the)71 2265 y(singularities.)44 b(Let)31 b(us)f(call)g(them)h FA(T)1266 2225 y Fx(u;s)1254 2287 y Fy(0)1360 2265 y FB(.)45 b(The)31 b(\014rst)f(order)f (asymptotics)h(for)g(the)h(di\013erence)f(\001)e(=)f FA(T)3359 2234 y Fx(u)3422 2265 y Fw(\000)20 b FA(T)3568 2234 y Fx(s)3633 2265 y FB(comes)71 2364 y(from)36 b FA(T)337 2334 y Fx(u)325 2385 y Fy(0)404 2364 y Fw(\000)23 b FA(T)553 2334 y Fx(s)541 2385 y Fy(0)624 2364 y FB(after)36 b(a)g(c)n(hange)g(of)g(v)-5 b(ariables.)62 b(Recall)36 b(that,)j(as)c(w)n(e)h(ha)n(v)n(e)g(explained)g(in)g(Section)h(3.1,)g (in)g(some)71 2464 y(cases,)30 b(this)g(c)n(hange)f(of)h(v)-5 b(ariables)29 b(implies)h(an)g(additional)g(correcting)e(term)i(in)h FA(T)2740 2434 y Fx(u)2728 2484 y Fy(0)2802 2464 y Fw(\000)20 b FA(T)2948 2434 y Fx(s)2936 2484 y Fy(0)2982 2464 y FB(.)45 b(Finally)-7 b(,)31 b(w)n(e)f(b)r(ound)g(the)71 2564 y(remainder)c(b)n(y)i(using)f(the)h(tec)n(hniques)f(explained)h (in)f(Section)h(3.1.)71 2838 y FC(4)135 b(Description)45 b(of)h(the)f(pro)t(ofs)g(of)g(Theorems)g(2.5)h(and)e(2.6)71 3020 y FB(W)-7 b(e)28 b(dev)n(ote)f(this)h(section)f(to)g(pro)n(v)n(e)f (Theorems)h(2.5)g(and)g(2.6.)71 3252 y Fq(4.1)112 b(Basic)38 b(notations)71 3406 y FB(First,)27 b(w)n(e)h(in)n(tro)r(duce)f(some)g (basic)g(notations)g(whic)n(h)g(will)h(b)r(e)g(used)g(through)f(the)h (pap)r(er.)195 3505 y(W)-7 b(e)28 b(denote)g(b)n(y)f Ft(T)d FB(=)e Ft(R)p FA(=)p FB(\(2)p FA(\031)s Ft(Z)p FB(\))28 b(the)g(real)f(1-dimensional)f(torus)h(and)g(b)n(y)1364 3688 y Ft(T)1419 3700 y Fx(\033)1487 3688 y FB(=)22 b Fw(f)p FA(\034)33 b Fw(2)23 b Ft(C)p FA(=)p FB(\(2)p FA(\031)s Ft(Z)p FB(\);)14 b Fw(j)p FB(Im)g FA(\034)9 b Fw(j)24 b FA(<)f(\033)s Fw(g)14 b FA(;)71 3870 y FB(with)28 b FA(\033)e(>)d FB(0,)k(the)h(torus)f(with)h(a)f(complex)h(strip.)195 3970 y(Giv)n(en)g(a)f(function)h FA(h)23 b FB(:)g FA(D)e Fw(\002)d Ft(T)1175 3982 y Fx(\033)1243 3970 y Fw(!)23 b Ft(C)p FB(,)28 b(where)f FA(D)e Fw(\032)d Ft(C)28 b FB(is)g(an)f(op)r(en)g(set,)h(w)n(e)f(denote)h(its)g(F)-7 b(ourier)26 b(series)h(b)n(y)1528 4165 y FA(h)p FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b(=)1881 4086 y Fz(X)1881 4265 y Fx(k)q Fv(2)p Fn(Z)2015 4165 y FA(h)2063 4130 y Fy([)p Fx(k)q Fy(])2141 4165 y FB(\()p FA(u)p FB(\))p FA(e)2292 4130 y Fx(ik)q(\034)71 4426 y FB(and)k(its)h(a)n(v)n(erage)d (b)n(y)1288 4581 y Fw(h)p FA(h)p Fw(i)p FB(\()p FA(u)p FB(\))f(=)e FA(h)1671 4546 y Fy([0])1746 4581 y FB(\()p FA(u)p FB(\))h(=)2004 4524 y(1)p 1979 4562 92 4 v 1979 4638 a(2)p FA(\031)2094 4468 y Fz(Z)2177 4488 y Fy(2)p Fx(\031)2141 4656 y Fy(0)2269 4581 y FA(h)p FB(\()p FA(u;)14 b(\034)9 b FB(\))14 b FA(d\034)5 b(:)195 4777 y FB(In)28 b(an)n(y)f(Banac)n(h)f(space)h(\()p Fw(X)12 b FA(;)i Fw(k)19 b(\001)f(k)p FB(\),)28 b(w)n(e)f(de\014ne)h(the)g(follo)n(wing) e(balls)1467 4958 y FA(B)t FB(\()p FA(R)q FB(\))d(=)g Fw(f)o FA(x)h Fw(2)f(X)12 b FB(;)i Fw(k)p FA(x)p Fw(k)23 b FA(<)g(R)q Fw(g)p 1467 5028 68 4 v 1467 5095 a FA(B)t FB(\()p FA(R)q FB(\))g(=)g Fw(f)o FA(x)h Fw(2)f(X)12 b FB(;)i Fw(k)p FA(x)p Fw(k)23 b(\024)g FA(R)q Fw(g)13 b FA(:)195 5278 y FB(By)35 b(Hyp)r(othesis)g Fp(HP3)p FB(,)h(the)f(Hamiltonian)g FA(H)41 b FB(in)35 b(\(6\))g(is)g(analytic)f (in)h FA(\034)45 b FB(=)34 b FA(t=")p FB(.)58 b(By)34 b(the)i(compactness)d(of)i Ft(T)p FB(,)71 5378 y(actually)28 b(there)g(exists)g(a)f(constan)n(t)h FA(\033)1283 5390 y Fy(0)1349 5378 y FB(suc)n(h)g(that)h FA(H)35 b FB(is)28 b(con)n(tin)n(uous)f(in)p 2420 5311 56 4 v 29 w Ft(T)2475 5390 y Fx(\033)2513 5398 y Fu(0)2578 5378 y FB(and)i(analytic)e(in)i Ft(T)3210 5390 y Fx(\033)3248 5398 y Fu(0)3285 5378 y FB(.)39 b(F)-7 b(rom)28 b(no)n(w)g(on,)71 5478 y(w)n(e)f(\014x)h(0)22 b FA(<)h(\033)j(<)d(\033)671 5490 y Fy(0)709 5478 y FB(.)p Black 1919 5753 a(24)p Black eop end %%Page: 25 25 TeXDict begin 25 24 bop Black Black 195 272 a FB(Throughout)29 b(the)h(pro)r(of)f(of)g(Theorems)g(2.5)g(and)g(2.6)g(w)n(e)g(will)h (use)f(the)h(analyticit)n(y)f(in)h FA(\026)p FB(.)43 b(W)-7 b(e)30 b(\014x)f(an)h(arbitrary)71 372 y(v)-5 b(alue)36 b FA(\026)344 384 y Fy(0)419 372 y FA(>)h FB(0.)62 b(Ev)n(en)36 b(if)h(w)n(e)f(do)g(not)g(write)g(it)h(explicitly)-7 b(,)38 b(all)e(functions)h(whic)n(h)f(app)r(ear)g(in)g(this)h(pap)r(er) f(will)g(b)r(e)71 471 y(analytic)27 b(in)h FA(\026)23 b Fw(2)g FA(B)t FB(\()p FA(\026)784 483 y Fy(0)822 471 y FB(\).)195 571 y(F)-7 b(rom)28 b(no)n(w)f(on,)g(w)n(e)g(w)n(ork)f (with)i(the)g(fast)g(time)g FA(\034)33 b FB(=)22 b FA(t=")p FB(.)37 b(Then,)27 b(denoting)2639 541 y Fv(0)2685 571 y FB(=)c FA(d=d\034)9 b FB(,)28 b(w)n(e)f(ha)n(v)n(e)g(the)h(system) 1196 686 y Fz(\032)1300 753 y FA(x)1347 723 y Fv(0)1454 753 y FB(=)22 b FA(")14 b FB(\()p FA(y)21 b FB(+)d FA(\026")1860 719 y Fx(\021)1900 753 y FA(@)1944 765 y Fx(y)1984 753 y FA(H)2053 765 y Fy(1)2104 753 y FB(\()q FA(x;)c(y)s(;)g(\034)9 b FB(\))q(\))1304 852 y FA(y)1348 822 y Fv(0)1454 852 y FB(=)22 b Fw(\000)p FA(")14 b FB(\()p FA(V)1757 818 y Fv(0)1781 852 y FB(\()p FA(x)p FB(\))19 b(+)f FA(\026")2083 818 y Fx(\021)2123 852 y FA(@)2167 864 y Fx(x)2209 852 y FA(H)2278 864 y Fy(1)2329 852 y FB(\()q FA(x;)c(y)s(;)g(\034)9 b FB(\))q(\))14 b FA(;)3703 803 y FB(\(31\))195 1031 y(In)26 b(order)e(to)h(mak)n(e)f(Sections)h(4-9)f(more)g(readable,)h(w) n(e)f(will)i(denote)f(b)n(y)g FA(K)30 b FB(an)n(y)25 b(constan)n(t)f(indep)r(enden)n(t)i(of)f FA(\026)h FB(and)71 1131 y FA(")h FB(to)h(state)f(all)g(the)h(b)r(ounds.)71 1363 y Fq(4.2)112 b(The)38 b(p)s(erio)s(dic)g(orbit)71 1517 y FB(In)e(the)g(parab)r(olic)e(case,)j(Hyp)r(othesis)f Fp(HP4.2)e FB(on)i FA(H)1839 1529 y Fy(1)1912 1517 y FB(implies)f(that)h(the)g(origin)f(is)g(still)h(a)g(critical)f(p)r(oin) n(t)g(of)h(the)71 1616 y(p)r(erturb)r(ed)21 b(system)g(\(31\).)35 b(In)21 b(the)h(h)n(yp)r(erb)r(olic)e(case,)i(the)g(next)f(theorem)g (states)f(the)i(existence)f(and)g(useful)g(prop)r(erties)71 1716 y(of)27 b(a)h(h)n(yp)r(erb)r(olic)f(p)r(erio)r(dic)g(orbit)g (close)g(to)h(the)g(origin)e(of)i(the)g(p)r(erturb)r(ed)g(system.)p Black 71 1882 a Fp(Theorem)33 b(4.1.)p Black 40 w Fs(L)l(et)e(us)f (assume)h(Hyp)l(otheses)h Fo(HP1.1)p Fs(,)g Fo(HP3)p Fs(,)g Fo(HP4.1)f Fs(and)g FA(\021)e Fw(\025)24 b FB(0)p Fs(.)42 b(Then,)32 b(ther)l(e)f(exists)f FA(")3657 1894 y Fy(0)3719 1882 y FA(>)25 b FB(0)71 1981 y Fs(such)31 b(that)g(for)h(any)g Fw(j)p FA(\026)p Fw(j)26 b FA(<)f(\026)989 1993 y Fy(0)1058 1981 y Fs(and)31 b FA(")26 b Fw(2)g FB(\(0)p FA(;)14 b(")1516 1993 y Fy(0)1553 1981 y FB(\))p Fs(,)32 b(system)37 b FB(\(31\))31 b Fs(has)h(a)g FB(2)p FA(\031)s Fs(-p)l(erio)l(dic)g(orbit)g FB(\()p FA(x)3036 1993 y Fx(p)3075 1981 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(y)3262 1993 y Fx(p)3302 1981 y FB(\()p FA(\034)9 b FB(\)\))27 b(:)f Ft(T)3574 1993 y Fx(\033)3644 1981 y Fw(!)g Ft(C)3813 1951 y Fy(2)71 2081 y Fs(which)31 b(is)f(r)l(e)l (al-analytic)h(and)g(satis\014es)1336 2264 y FB(sup)1319 2334 y Fx(\034)7 b Fv(2)p Fn(T)1441 2342 y Fm(\033)1493 2264 y FB(\()p Fw(j)p FA(x)1595 2276 y Fx(p)1634 2264 y FB(\()p FA(\034)i FB(\))p Fw(j)20 b FB(+)e Fw(j)p FA(y)1933 2276 y Fx(p)1971 2264 y FB(\()p FA(\034)9 b FB(\))p Fw(j)q FB(\))24 b Fw(\024)e FA(b)2283 2276 y Fy(0)2320 2264 y Fw(j)p FA(\026)p Fw(j)p FA(")2455 2229 y Fx(\021)r Fy(+1)2579 2264 y FA(;)71 2499 y Fs(wher)l(e)30 b FA(b)341 2511 y Fy(0)401 2499 y FA(>)23 b FB(0)29 b Fs(is)h(a)g(c)l(onstant)f (indep)l(endent)i(of)f FA(")g Fs(and)g FA(\026)p Fs(.)195 2665 y FB(This)e(theorem)f(is)h(pro)n(v)n(ed)e(in)h(Section)h(5.)195 2765 y(Once)e(w)n(e)f(kno)n(w)g(the)h(existence)f(of)h(the)g(p)r(erio)r (dic)g(orbit,)f(w)n(e)h(p)r(erform)f(the)h(time)g(dep)r(enden)n(t)h(c)n (hange)d(of)i(v)-5 b(ariables)1635 2876 y Fz(\032)1739 2942 y FA(q)26 b FB(=)d FA(x)c Fw(\000)f FA(x)2086 2954 y Fx(p)2124 2942 y FB(\()p FA(\034)9 b FB(\))1739 3042 y FA(p)23 b FB(=)f FA(y)g Fw(\000)c FA(y)2078 3054 y Fx(p)2116 3042 y FB(\()p FA(\034)9 b FB(\))3703 2993 y(\(32\))71 3244 y(whic)n(h)27 b(transforms)g(system)g(\(31\))g(in)n (to)h(a)f(Hamiltonian)g(system)g(with)i(Hamiltonian)e(function)h FA(")3213 3223 y Fz(b)3194 3244 y FA(H)7 b FB(\()p FA(q)s(;)14 b(p;)g(\034)9 b FB(\):)942 3459 y Fz(b)923 3480 y FA(H)e FB(\()p FA(q)s(;)14 b(p;)g(\034)9 b FB(\))24 b(=)1362 3424 y FA(p)1404 3394 y Fy(2)p 1362 3461 80 4 v 1381 3537 a FB(2)1470 3480 y(+)18 b FA(V)32 b FB(\()q FA(q)21 b FB(+)d FA(x)1854 3492 y Fx(p)1893 3480 y FB(\()p FA(\034)9 b FB(\)\))20 b Fw(\000)e FA(V)33 b FB(\()p FA(x)2297 3492 y Fx(p)2336 3480 y FB(\()p FA(\034)9 b FB(\)\))20 b Fw(\000)e FA(V)2647 3446 y Fv(0)2684 3480 y FB(\()p FA(x)2763 3492 y Fx(p)2802 3480 y FB(\()p FA(\034)9 b FB(\)\))15 b FA(q)1371 3651 y FB(+)j FA(\026")1543 3617 y Fx(\021)1602 3630 y Fz(b)1583 3651 y FA(H)1652 3663 y Fy(1)1689 3651 y FB(\()p FA(q)s(;)c(p;)g(\034)9 b FB(\))3703 3540 y(\(33\))71 3830 y(with)881 3981 y Fz(b)862 4002 y FA(H)931 4014 y Fy(1)968 4002 y FB(\()p FA(q)s(;)14 b(p;)g(\034)9 b FB(\))24 b(=)p FA(H)1391 4014 y Fy(1)1428 4002 y FB(\()p FA(x)1507 4014 y Fx(p)1546 4002 y FB(\()p FA(\034)9 b FB(\))20 b(+)e FA(q)s(;)c(y)1876 4014 y Fx(p)1914 4002 y FB(\()p FA(\034)9 b FB(\))19 b(+)g FA(p;)14 b(\034)9 b FB(\))19 b Fw(\000)f FA(H)2453 4014 y Fy(1)2490 4002 y FB(\()p FA(x)2569 4014 y Fx(p)2608 4002 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(y)2795 4014 y Fx(p)2834 4002 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))1340 4186 y Fw(\000)18 b FA(D)r(H)1563 4198 y Fy(1)1600 4186 y FB(\()p FA(x)1679 4198 y Fx(p)1719 4186 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(y)1906 4198 y Fx(p)1945 4186 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))2182 4069 y Fz(\022)2287 4135 y FA(q)2286 4235 y(p)2370 4069 y Fz(\023)2445 4186 y FA(;)3703 4114 y FB(\(34\))71 4418 y(where)20 b(w)n(e)g(ha)n(v)n(e)g(denoted)g FA(D)r(H)1050 4430 y Fy(1)1111 4418 y FB(=)i(\()p FA(@)1274 4430 y Fx(x)1316 4418 y FA(H)1385 4430 y Fy(1)1423 4418 y FA(;)14 b(@)1504 4430 y Fx(y)1544 4418 y FA(H)1613 4430 y Fy(1)1650 4418 y FB(\).)35 b(W)-7 b(e)21 b(ha)n(v)n(e)e(added)i(the)g(terms)f FA(V)33 b FB(\()p FA(x)2818 4430 y Fx(p)2857 4418 y FB(\()p FA(\034)9 b FB(\)\))22 b(and)f FA(H)3244 4430 y Fy(1)3281 4418 y FB(\()p FA(x)3360 4430 y Fx(p)3399 4418 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(y)3586 4430 y Fx(p)3625 4418 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))71 4531 y(for)27 b(con)n(v)n(enience.)36 b(Note)27 b(that)h(they)g(don't)g(generate)e(an)n(y)h(term)g(in)h(the)g (di\013eren)n(tial)g(equations)e(asso)r(ciated)h(to)3699 4510 y Fz(b)3680 4531 y FA(H)7 b FB(.)195 4631 y(Since)28 b Fw(j)p FB(\()p FA(x)514 4643 y Fx(p)553 4631 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(y)740 4643 y Fx(p)780 4631 y FB(\()p FA(\034)9 b FB(\)\))p Fw(j)24 b FB(=)f Fw(O)1138 4563 y Fz(\000)1176 4631 y FA(\026")1265 4601 y Fx(\021)r Fy(+1)1389 4563 y Fz(\001)1427 4631 y FB(,)1498 4610 y Fz(b)1478 4631 y FA(H)1547 4643 y Fy(1)1612 4631 y FB(can)k(b)r(e)h(split)g(as)1278 4811 y Fz(b)1259 4832 y FA(H)1328 4844 y Fy(1)1365 4832 y FB(\()p FA(q)s(;)14 b(p;)g(\034)9 b FB(\))24 b(=)1761 4811 y Fz(b)1742 4832 y FA(H)1818 4798 y Fy(1)1811 4853 y(1)1855 4832 y FB(\()p FA(q)s(;)14 b(p;)g(\034)9 b FB(\))19 b(+)f FA(")2280 4811 y Fz(b)2261 4832 y FA(H)2337 4798 y Fy(2)2330 4853 y(1)2374 4832 y FB(\()p FA(q)s(;)c(p;)g(\034)9 b FB(\))p FA(;)71 5015 y FB(where)908 5122 y Fz(b)889 5143 y FA(H)965 5109 y Fy(1)958 5164 y(1)1002 5143 y FB(\()p FA(q)s(;)14 b(p;)g(\034)9 b FB(\))24 b(=)f FA(H)1448 5155 y Fy(1)1485 5143 y FB(\()p FA(q)s(;)14 b(p;)g(\034)9 b FB(\))19 b Fw(\000)f FA(H)1921 5155 y Fy(1)1958 5143 y FB(\(0)p FA(;)c FB(0)p FA(;)g(\034)9 b FB(\))19 b Fw(\000)f FA(D)r(H)2467 5155 y Fy(1)2504 5143 y FB(\(0)p FA(;)c FB(0)p FA(;)g(\034)9 b FB(\))2785 5026 y Fz(\022)2889 5093 y FA(q)2888 5192 y(p)2971 5026 y Fz(\023)p Black 1919 5753 a FB(25)p Black eop end %%Page: 26 26 TeXDict begin 26 25 bop Black Black 71 272 a FB(and)250 251 y Fz(b)230 272 y FA(H)306 242 y Fy(2)299 293 y(1)343 272 y FB(\()p FA(q)s(;)14 b(p;)g(\034)9 b FB(\))27 b(is)e(the)h (remaining)e(part.)36 b(In)26 b(fact,)g(w)n(e)f(can)g(giv)n(e)g(a)g (more)g(precise)f(form)n(ula)h(for)3170 251 y Fz(b)3151 272 y FA(H)3227 242 y Fy(1)3220 293 y(1)3289 272 y FB(and)3468 251 y Fz(b)3449 272 y FA(H)3525 242 y Fy(2)3518 293 y(1)3587 272 y FB(in)h(b)r(oth)71 372 y(the)i(p)r(olynomial)f(and)g(the)h (trigonometric)e(cases:)275 521 y Fz(b)256 542 y FA(H)332 507 y Fy(1)325 562 y(1)369 542 y FB(\()p FA(q)s(;)14 b(p;)g(\034)9 b FB(\))23 b(=)828 463 y Fz(X)736 642 y Fy(2)p Fv(\024)p Fx(k)q Fy(+)p Fx(l)p Fv(\024)p Fx(N)1054 542 y FA(a)1098 554 y Fx(k)q(l)1160 542 y FB(\()p FA(\034)9 b FB(\))p FA(q)1309 507 y Fx(k)1352 542 y FA(p)1394 507 y Fx(l)3024 542 y FB(\(p)r(olynomial)27 b(case\))275 760 y Fz(b)256 781 y FA(H)332 747 y Fy(1)325 802 y(1)369 781 y FB(\()p FA(q)s(;)14 b(p;)g(\034)9 b FB(\))23 b(=)852 702 y Fz(X)736 881 y Fx(k)q Fy(=)p Fv(\000)p Fx(N)s(;:::)o(;N)1102 781 y FA(a)1146 793 y Fx(k)q Fy(0)1220 781 y FB(\()p FA(\034)9 b FB(\))1343 714 y Fz(\000)1382 781 y FA(e)1421 747 y Fx(ik)q(q)1536 781 y Fw(\000)18 b FB(1)g Fw(\000)g FA(ik)s(q)1877 714 y Fz(\001)3703 781 y FB(\(35\))741 1021 y(+)940 942 y Fz(X)824 1121 y Fx(k)q Fy(=)p Fv(\000)p Fx(N)s(;:::)n(;N)1190 1021 y FA(a)1234 1033 y Fx(k)q Fy(1)1307 1021 y FB(\()p FA(\034)9 b FB(\))1430 954 y Fz(\000)1470 1021 y FA(e)1509 987 y Fx(ik)q(q)1624 1021 y Fw(\000)18 b FB(1)1749 954 y Fz(\001)1800 1021 y FA(p)g FB(+)2059 942 y Fz(X)1943 1121 y Fx(k)q Fy(=)p Fv(\000)p Fx(N)s(;:::)o(;N)1988 1181 y(l)p Fy(=2)p Fx(;:::)o(;N)2309 1021 y FA(a)2353 1033 y Fx(k)q(l)2415 1021 y FB(\()p FA(\034)9 b FB(\))p FA(e)2563 987 y Fx(ik)q(q)2661 1021 y FA(p)2703 987 y Fx(l)2728 1021 y FA(;)185 b FB(\(trigonometric)26 b(case\))71 1329 y(where)h FA(a)355 1341 y Fx(k)q(l)445 1329 y FB(are)f(the)i(functions)g(de\014ned)g(in)g(\(8\))g(and)f(\(9\)) h(and)f(ha)n(v)n(e)g(zero)f(a)n(v)n(erage,)f(that)j(is)1784 1498 y Fw(h)1836 1477 y Fz(b)1816 1498 y FA(H)1892 1464 y Fy(1)1885 1518 y(1)1929 1498 y Fw(i)c FB(=)e(0)p FA(:)1566 b FB(\(36\))71 1666 y(Let)28 b(us)f(p)r(oin)n(t)h(out)g(that)890 1645 y Fz(b)870 1666 y FA(H)946 1636 y Fy(1)939 1687 y(1)1011 1666 y FB(is)g FA(H)1164 1678 y Fy(1)1229 1666 y FB(subtracting)e(its)i(linear)f(terms)g(in)h(\()p FA(x;)14 b(y)s FB(\),)28 b(and)g(hence)g(it)g(is)f(of)h(order)e FA(n)d FB(=)f(2.)195 1776 y(The)28 b(Hamiltonian)863 1755 y Fz(b)843 1776 y FA(H)919 1746 y Fy(2)912 1797 y(1)984 1776 y FB(is)g(giv)n(en)e(b)n(y:)280 1930 y Fz(b)261 1951 y FA(H)337 1916 y Fy(2)330 1971 y(1)374 1951 y FB(\()p FA(q)s(;)14 b(p;)g(\034)9 b FB(\))24 b(=)876 1872 y Fz(X)741 2051 y Fy(2)p Fv(\024)p Fx(k)q Fy(+)p Fx(l)p Fv(\024)p Fx(N)6 b Fv(\000)p Fy(1)1145 1951 y FA(c)1181 1963 y Fx(k)q(l)1243 1951 y FB(\()p FA(\034)j FB(\))p FA(q)1392 1916 y Fx(k)1434 1951 y FA(p)1476 1916 y Fx(l)3019 1951 y FB(\(p)r(olynomial)27 b(case\))280 2169 y Fz(b)261 2190 y FA(H)337 2156 y Fy(2)330 2211 y(1)374 2190 y FB(\()p FA(q)s(;)14 b(p;)g(\034)9 b FB(\))24 b(=)857 2111 y Fz(X)741 2290 y Fx(k)q Fy(=)p Fv(\000)p Fx(N)s(;:::)o(;N)1107 2190 y FA(c)1143 2202 y Fx(k)q Fy(0)1217 2190 y FB(\()p FA(\034)9 b FB(\))1340 2123 y Fz(\000)1379 2190 y FA(e)1418 2156 y Fx(ik)q(q)1533 2190 y Fw(\000)18 b FB(1)g Fw(\000)g FA(ik)s(q)1874 2123 y Fz(\001)3703 2190 y FB(\(37\))746 2430 y(+)945 2351 y Fz(X)829 2530 y Fx(k)q Fy(=)p Fv(\000)p Fx(N)s(;:::)n(;N)1195 2430 y FA(c)1231 2442 y Fx(k)q Fy(1)1305 2430 y FB(\()p FA(\034)9 b FB(\))1428 2363 y Fz(\000)1467 2430 y FA(e)1506 2396 y Fx(ik)q(q)1621 2430 y Fw(\000)18 b FB(1)1746 2363 y Fz(\001)1797 2430 y FA(p)g FB(+)2057 2351 y Fz(X)1941 2530 y Fx(k)q Fy(=)p Fv(\000)p Fx(N)s(;:::)n(;N)1943 2590 y(l)p Fy(=2)p Fx(;:::)o(;N)6 b Fv(\000)p Fy(1)2307 2430 y FA(c)2343 2442 y Fx(k)q(l)2405 2430 y FB(\()p FA(\034)j FB(\))p FA(e)2553 2396 y Fx(ik)q(q)2650 2430 y FA(p)2692 2396 y Fx(l)2718 2430 y FA(;)190 b FB(\(trigonometric) 26 b(case\))71 2734 y(where)20 b FA(c)340 2746 y Fx(k)q(l)423 2734 y FB(are)f(2)p FA(\031)s FB(-p)r(erio)r(dic)h(functions)h(whic)n (h,)h(in)f(general,)f(do)g(not)h(ha)n(v)n(e)e(zero)h(a)n(v)n(erage.)31 b(As)21 b(w)n(e)f(will)h(see)f(in)h(Corollary)71 2833 y(5.6)27 b(the)h(functions)g FA(c)742 2845 y Fx(k)q(l)831 2833 y FB(are)f(2)p FA(\031)s FB(-p)r(erio)r(dic)g(and)g(satisfy)1641 2991 y Fw(j)p FA(c)1700 3003 y Fx(k)q(l)1762 2991 y FB(\()p FA(\034)9 b FB(\))p Fw(j)24 b(\024)f FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2218 2957 y Fx(\021)2258 2991 y FA(:)1422 b FB(\(38\))195 3150 y(In)19 b(the)g(case)e(that)i(the)g(unp) r(erturb)r(ed)g(Hamiltonian)f(has)g(a)g(parab)r(olic)f(p)r(oin)n(t)i (at)f(the)h(origin,)g(since)g(\()p FA(x)3345 3162 y Fx(p)3384 3150 y FA(;)14 b(y)3462 3162 y Fx(p)3500 3150 y FB(\))23 b(=)g(\(0)p FA(;)14 b FB(0\),)71 3249 y(w)n(e)27 b(ha)n(v)n(e)g(that)g FA(c)600 3261 y Fx(k)q(l)686 3249 y FB(=)22 b(0.)p Black 71 3395 a Fp(Remark)45 b(4.2.)p Black 46 w Fs(The)c(p)l(erio)l(dic)h (orbit)f FB(\()p FA(x)1432 3407 y Fx(p)1471 3395 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(y)1658 3407 y Fx(p)1697 3395 y FB(\()p FA(\034)9 b FB(\)\))p Fs(,)45 b(and)40 b(c)l(onse)l(quently)g(the)g(Hamiltonians)3258 3375 y Fz(b)3239 3395 y FA(H)6 b Fs(,)3402 3375 y Fz(b)3382 3395 y FA(H)3451 3407 y Fy(1)3489 3395 y Fs(,)3576 3375 y Fz(b)3556 3395 y FA(H)3632 3365 y Fy(1)3625 3416 y(1)3669 3395 y Fs(,)3757 3375 y Fz(b)3737 3395 y FA(H)3813 3365 y Fy(2)3806 3416 y(1)71 3495 y Fs(dep)l(end)27 b(on)e(the)h(p)l(ar)l (ameters)g FA(\026)p Fs(,)h FA(")p Fs(.)37 b(We)26 b(wil)t(l)h(not)e (write)h(this)g(dep)l(endenc)l(e)h(explicitly)g(but)e(we)h(wil)t(l)h (emphasize)g(it)f(when)71 3595 y(ne)l(c)l(essary.)39 b(In)29 b(p)l(articular,)j(the)d(Hamiltonian)1624 3574 y Fz(b)1604 3595 y FA(H)1680 3565 y Fy(1)1673 3615 y(1)1747 3595 y Fs(is)h(simply)h(the)f(limit,)h(when)f FA(")23 b Fw(!)g FB(0)p Fs(,)29 b(of)3051 3574 y Fz(b)3032 3595 y FA(H)3108 3565 y Fy(1)3145 3595 y Fs(.)71 3823 y Fq(4.3)112 b(Di\013eren)m(t)38 b(parameterizations)h(of)e(the)h(in)m(v)-6 b(arian)m(t)38 b(manifolds)71 3976 y FB(The)c(next)h(step)g(is)f(to)g (pro)n(v)n(e)f(the)i(existence)f(of)g(the)h(unstable)f(and)h(stable)f (in)n(v)-5 b(arian)n(t)33 b(manifolds)h(of)h(the)g(p)r(erio)r(dic)71 4076 y(orbit)27 b(giv)n(en)g(in)h(Theorem)e(4.1.)195 4175 y(W)-7 b(e)22 b(will)g(consider)f(t)n(w)n(o)f(di\013eren)n(t)i (strategies)e(to)i(\014nd)g(suitable)f(parameterizations)e(of)j(these)f (in)n(v)-5 b(arian)n(t)21 b(manifolds)71 4275 y(dep)r(ending)31 b(on)f(the)h(domain)f(w)n(e)h(are.)44 b(On)31 b(the)g(one)f(hand,)h (when)g(it)g(is)g(p)r(ossible,)g(w)n(e)f(will)h(follo)n(w)f([LMS03)o(,) h(Sau01)n(])71 4375 y(\(see)e(also)f([GOS10)o(]\),)j(and)e(w)n(e)g (will)h(write)f(the)h(in)n(v)-5 b(arian)n(t)28 b(manifolds)h(as)g (graphs)f(of)h(suitable)g(generating)f(functions)71 4474 y(whic)n(h)33 b(are)f(solutions)g(of)h(a)g(Hamilton-Jacobi)e(equation)i (in)g(appropriate)e(v)-5 b(ariables.)52 b(On)33 b(the)g(other)g(hand,)h (when)71 4574 y(this)28 b(is)f(not)h(p)r(ossible,)f(w)n(e)g(will)h (obtain)f(suitable)h(parameterizations)d(of)j(in)n(v)-5 b(arian)n(t)26 b(manifolds.)195 4674 y(T)-7 b(o)28 b(in)n(tro)r(duce)f (the)h(\014rst)f(metho)r(d,)h(let)g(us)g(consider)e(the)i(symplectic)g (c)n(hange)e(of)i(v)-5 b(ariables)26 b(\(see)i([Bal06)n(]\))1687 4774 y Fz(\()1796 4831 y FA(q)e FB(=)d FA(q)1984 4843 y Fy(0)2021 4831 y FB(\()p FA(u)p FB(\))1796 4953 y FA(p)g FB(=)2023 4896 y FA(w)p 1958 4933 192 4 v 1958 5009 a(p)2000 5021 y Fy(0)2037 5009 y FB(\()p FA(u)p FB(\))2159 4953 y FA(;)3703 4916 y FB(\(39\))71 5159 y(where)k(\()p FA(q)380 5171 y Fy(0)418 5159 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)609 5171 y Fy(0)646 5159 y FB(\()p FA(u)p FB(\)\))28 b(is)g(the)g(parameterization)e(of)h(the)h(homo)r(clinic)g(orbit)f(giv) n(en)g(in)h(Hyp)r(othesis)g Fp(HP2)p FB(.)37 b(This)28 b(is)g(a)71 5259 y(w)n(ell)f(de\014ned)h(c)n(hange)f(for)g(an)n(y)g FA(u)22 b Fw(2)i Ft(C)j FB(suc)n(h)g(that)h FA(p)1729 5271 y Fy(0)1766 5259 y FB(\()p FA(u)p FB(\))23 b Fw(6)p FB(=)g(0)k(and)h(leads)f(to)g(a)g(new)h(Hamiltonian)f(giv)n(en)g(b)n(y) 1299 5467 y FA(")p 1338 5400 76 4 v(H)7 b FB(\()p FA(u;)14 b(w)r(;)g(\034)9 b FB(\))24 b(=)f FA(")1876 5446 y Fz(b)1857 5467 y FA(H)1946 5350 y Fz(\022)2007 5467 y FA(q)2044 5479 y Fy(0)2081 5467 y FB(\()p FA(u)p FB(\))p FA(;)2305 5411 y(w)p 2240 5448 192 4 v 2240 5524 a(p)2282 5536 y Fy(0)2319 5524 y FB(\()p FA(u)p FB(\))2441 5467 y FA(;)14 b(\034)2523 5350 y Fz(\023)2599 5467 y FA(;)1081 b FB(\(40\))p Black 1919 5753 a(26)p Black eop end %%Page: 27 27 TeXDict begin 27 26 bop Black Black 71 272 a FB(where)330 251 y Fz(b)311 272 y FA(H)34 b FB(is)28 b(the)g(Hamiltonian)f (de\014ned)h(in)g(\(33\).)195 382 y(Let)d(us)f(recall)g(that)g(when)h FA(\026)e FB(=)g(0,)1323 361 y Fz(b)1303 382 y FA(H)31 b FB(b)r(ecomes)24 b FA(H)1799 394 y Fy(0)1861 382 y FB(de\014ned)h(in)f(\(7\))q(.)35 b(Then,)26 b(the)e(separatrix)f(of)h (the)h(unp)r(erturb)r(ed)71 482 y(system)i(\()p FA(\026)d FB(=)e(0\))28 b(for)p 768 415 76 4 v 27 w FA(H)34 b FB(can)27 b(b)r(e)h(parameterized)f(as)f(a)i(graph)e(as)h FA(w)f FB(=)c FA(p)2399 494 y Fy(0)2436 482 y FB(\()p FA(u)p FB(\))2548 452 y Fy(2)2586 482 y FB(.)195 581 y(If)i(w)n(e)f(w)n(an)n (t)g(to)g(obtain)g(parameterizations)e(of)j(the)g(p)r(erturb)r(ed)f(in) n(v)-5 b(arian)n(t)22 b(manifolds,)j(w)n(e)e(can)g(tak)n(e)f(in)n(to)h (accoun)n(t)71 681 y(the)29 b(w)n(ell)g(kno)n(wn)g(fact)g(that,)h(lo)r (cally)-7 b(,)29 b(they)h(are)e(Lagrangian)f(and)i(can)f(b)r(e)i (obtained)f(as)g(graphs)e(of)j(some)e(functions)71 781 y(whic)n(h)f(are)g(solutions)g(of)g(the)h(Hamilton-Jacobi)f(equation.) 195 880 y(Therefore,)j(w)n(e)g(lo)r(ok)f(for)g(the)i(in)n(v)-5 b(arian)n(t)29 b(manifolds)g(as)h(graphs)e(of)i(generating)f(functions) h(whic)n(h)g(are)f(solutions)71 980 y(of)37 b(the)g(Hamilton-Jacobi)e (equation)h(asso)r(ciated)g(to)g(the)h(Hamiltonian)g FA(")p 2503 913 V(H)6 b FB(.)65 b(That)36 b(is,)j(w)n(e)e(write)f(them) i(as)e FA(w)41 b FB(=)71 1079 y FA(@)115 1091 y Fx(u)158 1079 y FA(T)219 1049 y Fx(u;s)313 1079 y FB(\()p FA(u;)14 b(\034)9 b FB(\),)28 b(where)f(the)h(functions)g FA(T)1360 1049 y Fx(u;s)1481 1079 y FB(satisfy)1320 1257 y FA(@)1364 1269 y Fx(\034)1405 1257 y FA(T)12 b FB(\()p FA(u;)i(\034)9 b FB(\))18 b(+)g FA(")p 1800 1190 V(H)7 b FB(\()p FA(u;)14 b(@)2037 1269 y Fx(u)2080 1257 y FA(T)e FB(\()p FA(u;)i(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))24 b(=)e(0)1101 b(\(41\))71 1435 y(and)27 b(certain)g(limiting)h(prop)r(erties.)195 1534 y(The)c(solutions)e(of)h(this)g(equation)g(giv)n(e)f (parameterizations)f(of)i(the)g(in)n(v)-5 b(arian)n(t)22 b(manifolds,)i(whic)n(h,)g(in)g(the)f(original)71 1634 y(v)-5 b(ariables,)26 b(read)1394 1779 y(\()p FA(q)s(;)14 b(p)p FB(\))23 b(=)1688 1662 y Fz(\022)1749 1779 y FA(q)1786 1791 y Fy(0)1824 1779 y FB(\()p FA(u)p FB(\))p FA(;)1982 1723 y(@)2026 1735 y Fx(u)2070 1723 y FA(T)2131 1693 y Fx(u;s)2224 1723 y FB(\()p FA(u;)14 b(\034)9 b FB(\))p 1982 1760 437 4 v 2105 1836 a FA(p)2147 1848 y Fy(0)2184 1836 y FB(\()p FA(u)p FB(\))2429 1662 y Fz(\023)2504 1779 y FA(:)1176 b FB(\(42\))195 1975 y(Notice)36 b(that)g(in)g(v)-5 b(ariables)34 b(\()p FA(q)s(;)14 b(p)p FB(\))36 b(the)g(condition)g FA(p)1894 1987 y Fy(0)1931 1975 y FB(\()p FA(u)p FB(\))g(=)52 b(_)-39 b FA(q)2217 1987 y Fy(0)2255 1975 y FB(\()p FA(u)p FB(\))36 b Fw(6)p FB(=)g(0)g(ensures)e(that)i(the)g(manifolds)g(can)f (b)r(e)71 2075 y(written)c(as)f(graphs)g(o)n(v)n(er)f(the)i(v)-5 b(ariable)30 b FA(q)k FB(through)d(the)g(functions)g FA(S)2336 2045 y Fx(u;s)2430 2075 y FB(\()p FA(q)s(;)14 b(\034)9 b FB(\))30 b(=)f FA(T)2801 2045 y Fx(u;s)2894 2075 y FB(\()p FA(q)2966 2039 y Fv(\000)p Fy(1)2963 2097 y(0)3056 2075 y FB(\()p FA(q)s FB(\))p FA(;)14 b(\034)9 b FB(\))32 b(wic)n(h)f(v)n(erify)f(the)71 2186 y(classical)c (Hamilton-Jacobi)g(equation)h(asso)r(ciated)f(to)i(the)g(Hamiltonian) 2492 2165 y Fz(b)2472 2186 y FA(H)7 b FB(\()p FA(q)s(;)14 b(p;)g(\034)9 b FB(\).)195 2286 y(When)26 b(this)f(metho)r(d)g(cannot)f (b)r(e)h(used,)g(that)g(is)g(when)g FA(p)1996 2298 y Fy(0)2033 2286 y FB(\()p FA(u)p FB(\))f(can)h(v)-5 b(anish,)25 b(w)n(e)f(lo)r(ok)g(for)g(the)h(in)n(v)-5 b(arian)n(t)24 b(manifolds)71 2385 y(as)j(parameterizations:)1507 2485 y(\()p FA(q)s(;)14 b(p)p FB(\))24 b(=)e(\()p FA(Q)p FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)14 b(P)e FB(\()p FA(v)s(;)i(\034)9 b FB(\)\))1292 b(\(43\))71 2631 y(in)25 b(suc)n(h)g(a)f(w)n(a)n(y)g(that)h(\()p FA(Q)p FB(\()p FA(v)17 b FB(+)c FA(t;)h(\034)22 b FB(+)1240 2599 y Fx(t)p 1237 2613 32 4 v 1237 2660 a(")1278 2631 y FB(\))p FA(;)14 b(P)e FB(\()p FA(v)17 b FB(+)c FA(t;)h(\034)22 b FB(+)1795 2599 y Fx(t)p 1792 2613 V 1792 2660 a(")1833 2631 y FB(\)\))k(are)e (solutions)g(of)h(the)g(di\013eren)n(tial)g(equation)f(asso)r(ciated)g (to)71 2731 y(the)j(Hamiltonian)g(\(33\))o(.)37 b(These)26 b(kind)h(of)g(parameterizations)e(w)n(ere)h(used)g(in)i([DS92)o(,)f (DS97,)g(Gel97a)n(,)g(Gel00)o(,)g(BF04)o(,)71 2831 y(BF05)o(].)195 2930 y(Then,)h(it)g(is)g(straigh)n(tforw)n(ard)c(to)k(see)f(\([Gel97a)o (]\))h(that)g(\()p FA(Q;)14 b(P)e FB(\))28 b(has)f(to)g(satisfy)673 3183 y Fw(L)730 3195 y Fx(")779 3066 y Fz(\022)882 3132 y FA(Q)882 3232 y(P)989 3066 y Fz(\023)1073 3183 y FB(=)1161 3041 y Fz( )1792 3132 y FA(P)j FB(+)18 b FA(\026")2047 3102 y Fx(\021)2087 3132 y FA(@)2131 3144 y Fx(p)2189 3111 y Fz(b)2169 3132 y FA(H)2238 3144 y Fy(1)2276 3132 y FB(\()p FA(Q;)c(P)r(;)g(\034)9 b FB(\))1268 3243 y Fw(\000)14 b FB(\()p FA(V)1446 3213 y Fv(0)1469 3243 y FB(\()p FA(Q)k FB(+)g FA(x)1715 3255 y Fx(p)1754 3243 y FB(\()p FA(\034)9 b FB(\)\))20 b Fw(\000)e FA(V)2065 3213 y Fv(0)2088 3243 y FB(\()p FA(x)2167 3255 y Fx(p)2207 3243 y FB(\()p FA(\034)9 b FB(\)\)\))20 b Fw(\000)e FA(\026")2572 3213 y Fx(\021)2612 3243 y FA(@)2656 3255 y Fx(q)2712 3222 y Fz(b)2693 3243 y FA(H)2762 3255 y Fy(1)2799 3243 y FB(\()p FA(Q;)c(P)r(;)g(\034)9 b FB(\))3146 3041 y Fz(!)3225 3183 y FA(;)455 b FB(\(44\))71 3430 y(where)27 b Fw(L)368 3442 y Fx(")431 3430 y FB(is)h(the)g(op)r(erator)1660 3530 y Fw(L)1717 3542 y Fx(")1776 3530 y FB(=)22 b FA(")1902 3496 y Fv(\000)p Fy(1)1991 3530 y FA(@)2035 3542 y Fx(\034)2095 3530 y FB(+)c FA(@)2222 3542 y Fx(v)3703 3530 y FB(\(45\))71 3687 y(and)252 3666 y Fz(b)232 3687 y FA(H)301 3699 y Fy(1)366 3687 y FB(is)28 b(the)g(Hamiltonian)f(de\014ned)h(in)g(\(34\)) o(.)195 3787 y(Both)39 b(parameterizations)e(\(42\),)42 b(\(44\))d(satisfy)g(that,)j(\014xing)d FA(\034)53 b FB(=)42 b FA(\034)2459 3799 y Fv(\003)2497 3787 y FB(,)g(they)e(giv)n (e)e(parameterizations)f(of)i(the)71 3886 y(in)n(v)-5 b(arian)n(t)40 b(curv)n(es)g(of)h(the)g(\014xed)g(p)r(oin)n(t)h(of)f (the)g(2)p FA(\031)s FB(-P)n(oincar)n(\023)-39 b(e)38 b(map)j(from)g(the)g(section)g FA(\034)55 b FB(=)45 b FA(\034)3249 3856 y Fv(\003)3329 3886 y FB(to)c(the)h(section)71 3986 y FA(\034)33 b FB(=)22 b FA(\034)272 3956 y Fv(\003)330 3986 y FB(+)c(2)p FA(\031)s FB(.)71 4217 y Fq(4.4)112 b(Existence)38 b(of)f(the)h(lo)s(cal)g(in)m(v)-6 b(arian)m(t)38 b(manifolds)71 4371 y FB(In)28 b(this)g(section)f(w)n(e)g(will)h (\014nd)g(the)g(lo)r(cal)f(in)n(v)-5 b(arian)n(t)26 b(manifolds)i(of)f (the)h(origin)f(of)g(the)h(Hamiltonian)g(system)f(\(33\).)195 4470 y(First,)35 b(w)n(e)e(recall)f(the)i(b)r(eha)n(vior)e(of)h(the)h (separatrix)e(\()p FA(q)1992 4482 y Fy(0)2029 4470 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2220 4482 y Fy(0)2257 4470 y FB(\()p FA(u)p FB(\)\))34 b(as)f(Re)13 b FA(u)33 b Fw(!)f(\0061)p FB(,)j(whic)n(h)e(is)g(substan)n(tially)71 4570 y(di\013eren)n(t)28 b(dep)r(ending)g(whether)f(\(0)p FA(;)14 b FB(0\))27 b(is)h(a)f(h)n(yp)r(erb)r(olic)g(or)g(a)g(parab)r (olic)f(p)r(oin)n(t)i(of)f(the)h(unp)r(erturb)r(ed)h(system.)195 4669 y(In)f(the)g(h)n(yp)r(erb)r(olic)f(case,)g(b)n(y)g(Hyp)r(othesis)h Fp(HP1.1)p FB(,)f(close)f(to)i FA(x)23 b FB(=)g(0)k(the)h(p)r(oten)n (tial)g(b)r(eha)n(v)n(es)e(as)1517 4896 y FA(V)19 b FB(\()p FA(x)p FB(\))24 b(=)f Fw(\000)1882 4839 y FA(\025)1930 4809 y Fy(2)p 1881 4877 86 4 v 1904 4953 a FB(2)1977 4896 y FA(x)2024 4861 y Fy(2)2080 4896 y FB(+)18 b Fw(O)r FB(\()p FA(x)2310 4861 y Fy(3)2349 4896 y FB(\))p FA(:)1299 b FB(\(46\))71 5101 y(Therefore,)26 b Fw(f)p FA(\025;)14 b Fw(\000)p FA(\025)p Fw(g)27 b FB(are)g(the)h(eigen)n(v)-5 b(alues)26 b(of)i(the)g(critical)f(p)r(oin)n(t.)37 b(Moreo)n(v)n(er,)25 b(there)j(exist)f(constan)n(ts)g FA(c)3455 5113 y Fv(\006)3534 5101 y Fw(6)p FB(=)c(0)k(suc)n(h)71 5201 y(that)h(as)f(Re)14 b FA(u)22 b Fw(!)h(\0071)k FB(the)h(separatrix)e(b)r(eha)n(v)n(es)h(as) 1374 5378 y FA(q)1411 5390 y Fy(0)1448 5378 y FB(\()p FA(u)p FB(\))c(=)g FA(c)1707 5390 y Fv(\006)1763 5378 y FA(e)1802 5344 y Fv(\006)p Fx(\025u)1955 5378 y FB(+)18 b Fw(O)2120 5311 y Fz(\000)2158 5378 y FA(e)2197 5344 y Fv(\006)p Fy(2)p Fx(\025u)2364 5311 y Fz(\001)2416 5378 y FA(;)1369 5516 y(p)1411 5528 y Fy(0)1448 5516 y FB(\()p FA(u)p FB(\))23 b(=)g Fw(\006)p FA(\025c)1820 5528 y Fv(\006)1876 5516 y FA(e)1915 5481 y Fv(\006)p Fx(\025u)2068 5516 y FB(+)18 b Fw(O)2233 5448 y Fz(\000)2271 5516 y FA(e)2310 5481 y Fv(\006)p Fy(2)p Fx(\025u)2477 5448 y Fz(\001)2529 5516 y FA(:)3703 5446 y FB(\(47\))p Black 1919 5753 a(27)p Black eop end %%Page: 28 28 TeXDict begin 28 27 bop Black Black 71 272 a FB(In)28 b(the)h(parab)r(olic)e(case,)h(using)g(Hyp)r(othesis)g Fp(HP1.2)p FB(,)g(in)h([BF04)o(])f(it)h(is)g(seen)f(that)g(there)h (exists)f(a)g(constan)n(t)f FA(c)3625 284 y Fy(0)3691 272 y FB(suc)n(h)71 372 y(that)h(as)f(Re)14 b FA(u)22 b Fw(!)h(\0071)k FB(the)h(separatrix)e(b)r(eha)n(v)n(es)h(as)1266 588 y FA(q)1303 600 y Fy(0)1341 588 y FB(\()p FA(u)p FB(\))c(=)1635 532 y FA(c)1671 544 y Fy(0)p 1573 569 196 4 v 1573 664 a FA(u)1679 606 y Fu(2)p 1631 615 125 3 v 1631 648 a Fm(m)p Fl(\000)p Fu(2)1798 588 y FB(+)18 b Fw(O)1963 471 y Fz(\022)2058 532 y FB(1)p 2034 569 89 4 v 2034 645 a FA(u)2082 621 y Fx(\027)2133 471 y Fz(\023)2208 588 y FA(;)1262 821 y(p)1304 833 y Fy(0)1341 821 y FB(\()p FA(u)p FB(\))23 b(=)f Fw(\000)1819 765 y FB(2)p FA(c)1897 777 y Fy(0)p 1638 802 477 4 v 1638 888 a FB(\()p FA(m)c Fw(\000)h FB(2\))p FA(u)2012 829 y Fm(m)p 1976 838 125 3 v 1976 872 a(m)p Fl(\000)p Fu(2)2143 821 y FB(+)f Fw(O)2308 704 y Fz(\022)2445 765 y FB(1)p 2379 802 173 4 v 2379 878 a FA(u)2427 854 y Fx(\027)t Fy(+1)2562 704 y Fz(\023)2637 821 y FA(;)3703 709 y FB(\(48\))71 1051 y(where)27 b FA(m)h FB(is)f(the)h(order)e(of)i(the)g(p)r(oten)n (tial)f(\(11\))h(and)f FA(\027)h(>)23 b FB(2)p FA(=)p FB(\()p FA(m)18 b Fw(\000)g FB(2\).)195 1150 y(W)-7 b(e)34 b(lo)r(ok)f(for)f(the)i(parameterizations)d(of)i(the)h(lo)r(cal)f(in)n (v)-5 b(arian)n(t)32 b(manifolds)h(in)h(the)f(domains)g FA(D)3334 1120 y Fx(u;s)3332 1171 y Fv(1)p Fx(;\032)3490 1150 y FB(de\014ned)g(in)71 1250 y(\(28\))o(.)195 1349 y(By)k(\(47\))e(and)g(\(48\),)i(the)f(constan)n(t)f FA(\032)g FB(can)g(b)r(e)h(tak)n(en)f(big)g(enough)g(so)g(that)g FA(p)2761 1361 y Fy(0)2798 1349 y FB(\()p FA(u)p FB(\))h(do)r(es)f(not) g(v)-5 b(anish)36 b(in)f(these)71 1449 y(domains.)60 b(Then,)38 b(as)c(w)n(e)i(explained)f(in)g(Section)h(4.3,)h(w)n(e)e (can)g(lo)r(ok)g(for)g(the)h(in)n(v)-5 b(arian)n(t)34 b(manifolds)h(b)n(y)g(means)h(of)71 1549 y(generating)22 b(functions)h FA(T)887 1519 y Fx(u;s)1004 1549 y FB(de\014ned)h(in)g FA(D)1450 1519 y Fv(\003)1448 1569 y(1)p Fx(;\032)1595 1549 y FB(with)g Fw(\003)f FB(=)f FA(u;)14 b(s)23 b FB(resp)r(ectiv)n (ely)-7 b(,)23 b(whic)n(h)h(are)e(solutions)h(of)g(the)h(Hamilton-)71 1648 y(Jacobi)i(equation)h(\(41\).)37 b(Moreo)n(v)n(er,)25 b(w)n(e)i(imp)r(ose)h(the)f(asymptotic)h(conditions)943 1820 y(lim)844 1873 y Fy(Re)11 b Fx(u)p Fv(!\0001)1170 1820 y FA(p)1212 1784 y Fv(\000)p Fy(1)1212 1842 y(0)1301 1820 y FB(\()p FA(u)p FB(\))19 b Fw(\001)f FA(@)1517 1832 y Fx(u)1561 1820 y FA(T)1622 1785 y Fx(u)1664 1820 y FB(\()p FA(u;)c(\034)9 b FB(\))24 b(=)f(0)82 b(\(for)28 b(the)g(unstable)f(manifold\))626 b(\(49\))951 1995 y(lim)853 2048 y Fy(Re)11 b Fx(u)p Fv(!)p Fy(+)p Fv(1)1178 1995 y FA(p)1220 1959 y Fv(\000)p Fy(1)1220 2017 y(0)1309 1995 y FB(\()p FA(u)p FB(\))19 b Fw(\001)f FA(@)1525 2007 y Fx(u)1569 1995 y FA(T)1630 1960 y Fx(s)1664 1995 y FB(\()p FA(u;)c(\034)9 b FB(\))24 b(=)f(0)82 b(\(for)28 b(the)g(stable)f(manifold\))p FA(:)695 b FB(\(50\))71 2206 y(The)28 b(functions)g FA(T)661 2175 y Fx(u;s)782 2206 y FB(will)h(giv)n(e)d(us)i(parameterizations)e(of)i(the)g(in)n(v) -5 b(arian)n(t)27 b(manifolds)g(of)h(\(0)p FA(;)14 b FB(0\))28 b(of)f(the)i(Hamiltonian)71 2305 y(system)k(with)i (Hamiltonian)e(\(33\))h(of)g(the)g(form)f(\(42\))h(and)g(th)n(us,)h (these)f(asymptotic)f(conditions)h(ensure)f(that)h(the)71 2405 y(in)n(v)-5 b(arian)n(t)26 b(manifolds)i(tend)g(to)f(the)h (critical)f(p)r(oin)n(t)h(\(0)p FA(;)14 b FB(0\))27 b(as)g(Re)14 b FA(u)23 b Fw(!)g(\0061)p FB(.)195 2504 y(W)-7 b(e)28 b(note)g(that)g(when)g FA(\026)23 b FB(=)f(0)27 b(a)h(solution)f(of)34 b(\(41\))27 b(satisfying)g(b)r(oth)h(asymptotic)f(conditions)g(\(49\))g (and)h(\(50\))f(is)1560 2723 y FA(T)1609 2735 y Fy(0)1646 2723 y FB(\()p FA(u)p FB(\))c(=)1869 2610 y Fz(Z)1952 2630 y Fx(u)1915 2798 y Fv(\0001)2051 2723 y FA(p)2093 2688 y Fy(2)2093 2743 y(0)2130 2723 y FB(\()p FA(v)s FB(\))14 b FA(dv)s(;)1343 b FB(\(51\))71 2943 y(whic)n(h)27 b(corresp)r(onds)f(to)i(the)g(the)g(unp)r(erturb)r(ed)g(separatrix.)195 3042 y(The)j(next)g(theorem)g(giv)n(es)e(the)i(existence)g(of)g(the)g (in)n(v)-5 b(arian)n(t)30 b(manifolds)g(in)h(the)g(domains)g FA(D)3219 3012 y Fv(\003)3217 3063 y(1)p Fx(;\032)3372 3042 y FB(with)g Fw(\003)d FB(=)g FA(u;)14 b(s)71 3142 y FB(de\014ned)30 b(in)g(\(28\))o(.)44 b(W)-7 b(e)30 b(state)f(the)h(results)g(for)f(the)h(unstable)f(in)n(v)-5 b(arian)n(t)29 b(manifold.)43 b(The)30 b(stable)f(one)h(has)f (analogous)71 3242 y(prop)r(erties.)p Black 71 3399 a Fp(Theorem)46 b(4.3.)p Black 47 w Fs(L)l(et)41 b(us)g(assume)g(Hyp)l (otheses)h Fo(HP1.1)p Fs(,)j Fo(HP3)p Fs(,)g Fo(HP4)d Fs(and)g(take)g FA(\021)47 b Fw(\025)c FB(0)p Fs(.)73 b(L)l(et)40 b FA(\032)3422 3411 y Fy(1)3503 3399 y FA(>)k FB(0)d Fs(b)l(e)g(a)71 3498 y(r)l(e)l(al)d(numb)l(er)e(big)i(enough)g (such)f(that)h FA(p)1385 3510 y Fy(0)1422 3498 y FB(\()p FA(u)p FB(\))f Fw(6)p FB(=)f(0)h Fs(for)h FA(u)f Fw(2)g FA(D)2139 3468 y Fx(u)2137 3519 y Fv(1)p Fx(;\032)2257 3527 y Fu(1)2293 3498 y Fs(.)62 b(Then,)40 b(ther)l(e)e(exists)f FA(")3120 3510 y Fy(0)3193 3498 y FA(>)g FB(0)g Fs(such)g(that)g(for)71 3598 y FA(")23 b Fw(2)g FB(\(0)p FA(;)14 b(")361 3610 y Fy(0)398 3598 y FB(\))26 b Fs(and)h FA(\026)c Fw(2)h FA(B)t FB(\()p FA(\026)915 3610 y Fy(0)952 3598 y FB(\))p Fs(,)k(the)e(Hamilton-Jac)l(obi)i(e)l(quation)33 b FB(\(41\))26 b Fs(has)h(a)f(unique)g(\(mo)l(dulo)h(an)g(additive)h(c)l(onstant\))71 3698 y(r)l(e)l(al-analytic)j(solution)f(in)g FA(D)1030 3667 y Fx(u)1028 3718 y Fv(1)p Fx(;\032)1148 3726 y Fu(1)1203 3698 y Fw(\002)18 b Ft(T)1341 3710 y Fx(\033)1416 3698 y Fs(satisfying)31 b(the)f(asymptotic)h(c)l(ondition)37 b FB(\(49\))o Fs(.)195 3797 y(Mor)l(e)l(over,)27 b(ther)l(e)d(exists)f (a)h(r)l(e)l(al)f(c)l(onstant)g FA(b)1581 3809 y Fy(1)1641 3797 y FA(>)g FB(0)g Fs(indep)l(endent)h(of)g FA(")f Fs(and)h FA(\026)p Fs(,)h(such)f(that)f(for)i FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)f FA(D)3496 3767 y Fx(u)3494 3818 y Fv(1)p Fx(;\032)3614 3826 y Fu(1)3655 3797 y Fw(\002)5 b Ft(T)3780 3809 y Fx(\033)3825 3797 y Fs(,)1318 3981 y Fw(j)p FA(@)1385 3993 y Fx(u)1429 3981 y FA(T)1490 3946 y Fx(u)1532 3981 y FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b Fw(\000)g FA(@)1873 3993 y Fx(u)1916 3981 y FA(T)1965 3993 y Fy(0)2002 3981 y FB(\()p FA(u)p FB(\))p Fw(j)k(\024)g FA(b)2284 3993 y Fy(1)2320 3981 y Fw(j)p FA(\026)p Fw(j)p FA(")2455 3946 y Fx(\021)r Fy(+1)2580 3981 y FA(:)195 4152 y FB(The)34 b(asymptotic)e(b)r(eha)n (vior)g(of)h(the)h(in)n(v)-5 b(arian)n(t)32 b(manifolds)g(when)i(Re)14 b FA(u)31 b Fw(!)i FB(+)p Fw(1)g FB(is)g(qualitativ)n(ely)f(di\013eren) n(t)h(for)71 4252 y(the)f(h)n(yp)r(erb)r(olic)f(case)g(and)g(the)h (parab)r(olic)f(case.)48 b(F)-7 b(or)31 b(this)h(reason)e(w)n(e)h(pro)n (v)n(e)f(separately)g(Theorem)h(4.3)f(for)h(these)71 4351 y(t)n(w)n(o)c(cases.)35 b(W)-7 b(e)28 b(deal)g(with)g(the)g(h)n (yp)r(erb)r(olic)f(case)g(in)g(Section)h(6.1)f(and)g(with)h(the)g (parab)r(olic)f(case)f(in)i(Section)g(6.2.)195 4551 y(In)h(the)h(rest)e (of)h(the)g(pap)r(er)g(w)n(e)f(will)h(assume)f(the)i(whole)e(set)h(of)g (Hyp)r(otheses)f Fp(HP1)p FB(,)i Fp(HP2)p FB(,)f Fp(HP3)p FB(,)g Fp(HP4)p FB(,)g Fp(HP5)71 4650 y FB(and)e Fp(HP6)p FB(.)71 4881 y Fq(4.5)112 b(The)38 b(global)g(in)m(v)-6 b(arian)m(t)39 b(manifolds)71 5034 y FB(The)24 b(next)g(step)h(is)f(to) g(extend)g(the)h(in)n(v)-5 b(arian)n(t)23 b(manifolds)h(to)g(a)f(wider) h(domain)g(whic)n(h)g(con)n(tains)f(a)h(region)f(close)g(to)h(the)71 5133 y(singularities)29 b Fw(\006)p FA(ia)g FB(of)h(the)h(separatrix)d (\(see)i(Hyp)r(othesis)h Fp(HP2)p FB(\).)45 b(In)30 b(the)h(general)d (case)i(the)g(function)h FA(p)3461 5145 y Fy(0)3498 5133 y FB(\()p FA(u)p FB(\))g(migh)n(t)71 5233 y(v)-5 b(anish)28 b(and)h(therefore,)f(the)g(symplectic)h(c)n(hange)e(\(39\))i(is)f(not)h (w)n(ell)f(de\014ned.)40 b(F)-7 b(or)28 b(this)h(reason)e(one)h(cannot) g(use)g(the)71 5333 y(Hamilton-Jacobi)e(equation)h(\(41\))g(an)n (ymore.)35 b(Instead)28 b(w)n(e)f(lo)r(ok)g(for)g(parameterizations) 1413 5504 y(\()p FA(q)s(;)14 b(p)p FB(\))23 b(=)g(\()p FA(Q)1805 5470 y Fx(u;s)1899 5504 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)14 b(P)2190 5470 y Fx(u;s)2286 5504 y FB(\()p FA(v)s(;)g(\034)9 b FB(\)\))p Black 1919 5753 a(28)p Black eop end %%Page: 29 29 TeXDict begin 29 28 bop Black Black 71 272 a FB(whic)n(h)27 b(are)g(solutions)g(of)g(the)h(partial)f(di\013eren)n(tial)g(equation)g (\(44\).)195 372 y(Nev)n(ertheless,)33 b(there)f(are)f(some)h(cases,)g (as)g(happ)r(ens)g(for)g(the)g(classical)f(p)r(endulum,)k(in)d(whic)n (h)g FA(p)3357 384 y Fy(0)3394 372 y FB(\()p FA(u)p FB(\))h(do)r(es)e (not)71 471 y(v)-5 b(anish)26 b(for)f FA(u)e Fw(2)g Ft(C)p FB(,)k(and)f(then)h(one)e(can)h(use)g(the)h(Hamilton-Jacobi)d(equation) i(in)g(the)h(whole)e(domain,)h(whic)n(h)h(mak)n(es)71 571 y(the)h(pro)r(of)g(of)g(Theorems)f(2.5)g(and)h(2.6)f(remark)-5 b(ably)27 b(simpler.)38 b(Section)28 b(4.5.1)f(is)h(dev)n(oted)f(to)h (this)h(simpler)e(case)h(and)71 671 y(Section)f(4.5.2)g(to)g(the)h (general)e(one.)71 886 y Fp(4.5.1)94 b(The)32 b(global)e(in)m(v)-5 b(arian)m(t)33 b(manifolds)e(in)g(the)h(case)g FA(p)2190 898 y Fy(0)2227 886 y FB(\()p FA(u)p FB(\))23 b Fw(6)p FB(=)g(0)71 1039 y(In)36 b(this)h(section)e(w)n(e)h(extend)g(the)h (parameterizations)d(\(42\))i(of)g(the)g(in)n(v)-5 b(arian)n(t)35 b(manifolds)h(to)g(the)h(outer)e(domains)71 1139 y FA(D)142 1104 y Fy(out)p Fx(;)p Fv(\003)140 1159 y Fx(\032;\024)295 1139 y FB(,)g Fw(\003)d FB(=)g FA(u;)14 b(s)p FB(,)34 b(\(see)g(Figure)e(4\))h(de\014ned)h(b)n(y)g(\(29\),)g(in)g(the)g(case) e(that)h FA(p)2531 1151 y Fy(0)2568 1139 y FB(\()p FA(u)p FB(\))g Fw(6)p FB(=)f(0.)54 b(W)-7 b(e)33 b(emphasize)g(that)g(these)71 1238 y(domains)j(reac)n(h)g(a)h(region)f(whic)n(h)h(is)h(a)e(distance)h (of)h Fw(O)r FB(\()p FA(")p FB(\))g(to)f(the)g(singularities)f FA(u)j FB(=)g Fw(\006)p FA(ia)d FB(of)h(the)h(unp)r(erturb)r(ed)71 1338 y(separatrix.)195 1438 y(The)d(constan)n(t)g FA(\032)g FB(will)g(b)r(e)g(tak)n(en)g FA(\032)g(>)g(\032)1533 1450 y Fy(1)1570 1438 y FB(,)i(where)e FA(\032)1921 1450 y Fy(1)1993 1438 y FB(is)g(the)g(constan)n(t)g(giv)n(en)f(b)n(y)g (Theorem)h(4.3,)h(in)f(order)f(to)71 1537 y(ensure)27 b(that)h FA(D)581 1507 y Fx(u)579 1558 y Fv(1)p Fx(;\032)699 1566 y Fu(1)754 1537 y Fw(\\)19 b FA(D)899 1507 y Fy(out)o Fx(;u)897 1558 y(\032;\024)1081 1537 y Fw(6)p FB(=)j Fw(;)p FB(.)195 1637 y(Since)h(in)g(this)g(section)f(w)n(e)g(are)g (assuming)f(that)i FA(p)1751 1649 y Fy(0)1788 1637 y FB(\()p FA(u)p FB(\))g Fw(6)p FB(=)g(0)f(in)h(the)g(whole)f Fs(outer)j(domain)p FB(,)g(the)e(symplectic)g(c)n(hange)71 1736 y(of)j(v)-5 b(ariables)25 b(\(39\))g(is)h(still)h(w)n(ell)e (de\014ned)i(there.)36 b(Then,)27 b(it)f(is)g(enough)f(to)h(lo)r(ok)g (for)f(the)i(analytic)e(con)n(tin)n(uation)g(of)h(the)71 1836 y(generating)g(functions)i FA(T)896 1806 y Fx(u;s)1017 1836 y FB(obtained)f(in)h(Theorem)f(4.3.)p Black 71 1999 a Fp(Theorem)d(4.4.)p Black 34 w Fs(L)l(et)g FA(\032)834 2011 y Fy(1)895 1999 y Fs(b)l(e)g(the)g(c)l(onstant)f(c)l(onsider)l(e)l (d)j(in)e(The)l(or)l(em)h(4.3)g(and)f(let)g(us)g(c)l(onsider)h FA(\032)3146 2011 y Fy(2)3207 1999 y Fs(such)f(that)g FA(\032)3597 2011 y Fy(2)3657 1999 y FA(>)f(\032)3788 2011 y Fy(1)3825 1999 y Fs(,)71 2099 y FA(\024)119 2111 y Fy(1)189 2099 y FA(>)32 b FB(0)j Fs(big)h(enough)g(and)g FA(")994 2111 y Fy(0)1064 2099 y FA(>)c FB(0)j Fs(smal)t(l)h(enough.)56 b(Then,)37 b(for)f FA(\026)d Fw(2)h FA(B)t FB(\()p FA(\026)2504 2111 y Fy(0)2541 2099 y FB(\))p Fs(,)k FA(")32 b Fw(2)i FB(\(0)p FA(;)14 b(")2946 2111 y Fy(0)2982 2099 y FB(\))p Fs(,)38 b(the)d(function)g FA(T)3613 2069 y Fx(u)3656 2099 y FB(\()p FA(u;)14 b(\034)9 b FB(\))71 2199 y Fs(obtaine)l(d)31 b(in)f(The)l(or)l(em)g(4.3)h(c)l(an)f(b)l(e)g(analytic)l(al)t(ly)i (extende)l(d)e(to)g(the)f(domain)i FA(D)2623 2168 y Fy(out)p Fx(;u)2621 2219 y(\032)2655 2227 y Fu(2)2688 2219 y Fx(;\024)2747 2227 y Fu(1)2802 2199 y Fw(\002)18 b Ft(T)2940 2211 y Fx(\033)2985 2199 y Fs(.)195 2298 y(Mor)l(e)l(over,)27 b(ther)l(e)c(exists)g(a)h(r)l(e)l(al)f(c)l(onstant)g FA(b)1580 2310 y Fy(2)1639 2298 y FA(>)g FB(0)g Fs(indep)l(endent)h(of) g FA(")f Fs(and)g FA(\026)p Fs(,)i(such)f(that)f(for)h FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)f FA(D)3492 2268 y Fy(out)p Fx(;u)3490 2319 y(\032)3524 2327 y Fu(2)3557 2319 y Fx(;\024)3616 2327 y Fu(1)3656 2298 y Fw(\002)t Ft(T)3780 2310 y Fx(\033)3825 2298 y Fs(,)1260 2542 y Fw(j)p FA(@)1327 2554 y Fx(u)1370 2542 y FA(T)1431 2507 y Fx(u)1474 2542 y FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b Fw(\000)f FA(@)1814 2554 y Fx(u)1857 2542 y FA(T)1906 2554 y Fy(0)1943 2542 y FB(\()p FA(u)p FB(\))p Fw(j)23 b(\024)2248 2485 y FA(b)2284 2497 y Fy(2)2320 2485 y Fw(j)p FA(\026)p Fw(j)p FA(")2455 2455 y Fx(\021)r Fy(+1)p 2199 2523 430 4 v 2199 2615 a Fw(j)p FA(u)2270 2591 y Fy(2)2325 2615 y FB(+)18 b FA(a)2452 2591 y Fy(2)2489 2615 y Fw(j)2512 2573 y Fx(`)p Fy(+1)2638 2542 y FA(;)71 2785 y Fs(wher)l(e)30 b FA(T)354 2797 y Fy(0)421 2785 y Fs(is)g(the)g(unp)l(erturb)l(e)l(d)f(sep)l(ar)l(atrix,)h(given)h(in) 36 b FB(\(51\))o Fs(.)195 2949 y FB(The)28 b(pro)r(of)f(of)h(this)g (theorem)f(is)g(giv)n(en)g(in)h(Section)f(7.1.)36 b(The)28 b(results)f(for)g(the)h(stable)f(manifold)h(are)f(analogous.)71 3164 y Fp(4.5.2)94 b(The)32 b(global)e(in)m(v)-5 b(arian)m(t)33 b(manifolds)e(for)h(the)g(general)f(case)71 3317 y FB(W)-7 b(e)26 b(dev)n(ote)e(this)i(section)f(to)g(obtain)g(parameterizations)e (of)i(the)h(global)e(in)n(v)-5 b(arian)n(t)25 b(manifolds)g(for)g(the)g (general)f(case,)71 3417 y(that)31 b(is,)i(considering)c(Hamiltonian)j (systems)e(for)h(whic)n(h)g FA(p)2010 3429 y Fy(0)2047 3417 y FB(\()p FA(u)p FB(\))h(migh)n(t)f(v)-5 b(anish)31 b(in)h(the)f(outer)g(domains)g(de\014ned)g(in)71 3516 y(\(29\))o(.)36 b(In)25 b(this)g(case,)f(w)n(e)g(cannot)g(use)g(the)h (Hamilton-Jacobi)e(equation)h(\(41\))g(in)h(the)f(whole)g(outer)g (domains.)35 b(F)-7 b(or)24 b(this)71 3616 y(reason,)i(w)n(e)h(lo)r(ok) g(\014rst)g(for)g(parameterizations)1413 3795 y(\()p FA(q)s(;)14 b(p)p FB(\))23 b(=)g(\()p FA(Q)1805 3761 y Fx(u;s)1899 3795 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)14 b(P)2190 3761 y Fx(u;s)2286 3795 y FB(\()p FA(v)s(;)g(\034)9 b FB(\)\))71 3974 y(whic)n(h)27 b(are)g(solutions)g(of)g(the)h(partial) f(di\013eren)n(tial)g(equation)g(\(44\).)37 b(Our)27 b(strategy)f(will)i(b)r(e:)p Black 195 4138 a Fw(\017)p Black 41 w FB(T)-7 b(o)28 b(obtain)f(the)h(parameterizations)d(\()p FA(Q)1574 4108 y Fx(u;s)1669 4138 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)14 b(P)1960 4108 y Fx(u;s)2055 4138 y FB(\()p FA(v)s(;)g(\034)9 b FB(\)\))30 b(in)d(a)h(transition)e (domain)i(\(Theorem)f(4.5\).)p Black 195 4302 a Fw(\017)p Black 41 w FB(T)-7 b(o)28 b(extend)f(them)i(up)f(to)f(a)g(region)g (where)g(w)n(e)g(can)g(ensure)g(that)h FA(p)2418 4314 y Fy(0)2455 4302 y FB(\()p FA(u)p FB(\))g(do)r(es)f(not)h(v)-5 b(anish)27 b(\(Theorem)g(4.6\).)p Black 195 4467 a Fw(\017)p Black 41 w FB(T)-7 b(o)29 b(reco)n(v)n(er)c(in)k(this)g(new)f(region)g (the)g(represen)n(tations)f(\(42\))h(through)g(the)h(generating)e (function)i FA(T)3518 4437 y Fx(u;s)3640 4467 y FB(of)f(the)278 4567 y(manifolds,)g(whic)n(h)f(are)g(solution)g(of)h(the)g (Hamilton-Jacobi)e(equation)h(\(41\))g(\(Theorem)g(4.7\).)p Black 195 4731 a Fw(\017)p Black 41 w FB(T)-7 b(o)29 b(extend)h(the)f(generating)f(function)i FA(@)1597 4743 y Fx(u)1640 4731 y FA(T)1701 4701 y Fx(u;s)1795 4731 y FB(\()p FA(u;)14 b(\034)9 b FB(\))30 b(up)f(to)g(a)g(distance)g(of)g (order)f FA(")h FB(of)g(the)g(singularit)n(y)-7 b(,)29 b(as)f(it)278 4831 y(w)n(as)f(done)g(in)h(the)g(easier)e(case)h FA(p)1325 4843 y Fy(0)1362 4831 y FB(\()p FA(u)p FB(\))c Fw(6)p FB(=)g(0)k(in)h(Theorem)f(4.4)g(\(Theorem)g(4.8\).)195 4994 y(First)d(w)n(e)f(are)f(going)g(to)i(construct)f(the)g(t)n(w)n(o)g (dimensional)g(parameterizations)e(of)j(the)f(in)n(v)-5 b(arian)n(t)23 b(manifolds)g(from)71 5094 y(the)37 b(parameterizations) d(of)i(the)h(lo)r(cal)e(in)n(v)-5 b(arian)n(t)36 b(manifolds)g(giv)n (en)f(in)i(Theorem)e(4.3,)j(whic)n(h)e(w)n(ere)g(obtained)g(b)n(y)71 5193 y(using)27 b(the)h(Hamilton-Jacobi)e(equation.)37 b(W)-7 b(e)27 b(lo)r(ok)g(for)g(them)i(in)e(the)h(transition)f(domains) 1583 5373 y FA(I)1626 5339 y Fx(u)1619 5394 y(\032;)6 b Fy(\026)-39 b Fx(\032)1735 5373 y FB(=)23 b FA(D)1894 5333 y Fy(out)p Fx(;u)1892 5391 y(\024;)6 b Fy(\026)-39 b Fx(\032)2071 5373 y Fw(\\)19 b FA(D)2216 5339 y Fx(u)2214 5394 y Fv(1)p Fx(;\032)1583 5514 y FA(I)1626 5479 y Fx(s)1619 5534 y(\032;)6 b Fy(\026)-39 b Fx(\032)1735 5514 y FB(=)23 b FA(D)1894 5474 y Fy(out)p Fx(;s)1892 5532 y(\024;)6 b Fy(\026)-39 b Fx(\032)2063 5514 y Fw(\\)19 b FA(D)2208 5479 y Fx(s)2206 5534 y Fv(1)p Fx(;\032)3703 5443 y FB(\(52\))p Black 1919 5753 a(29)p Black eop end %%Page: 30 30 TeXDict begin 30 29 bop Black Black 71 272 a FB(with)36 b(\026)-49 b FA(\032)24 b(>)f(\032)29 b FB(\(see)f(Figure)g(6\).)38 b(T)-7 b(aking)28 b(in)n(to)g(accoun)n(t)f(the)i(c)n(hange)e(of)h(v)-5 b(ariables)27 b(\(39\),)h(it)h(is)f(natural)g(to)g(lo)r(ok)f(for)h(the) 71 372 y(parameterizations)d(of)j(the)g(in)n(v)-5 b(arian)n(t)26 b(manifolds)i(\()p FA(Q)1806 342 y Fx(u;s)1900 372 y FA(;)14 b(P)2002 342 y Fx(u;s)2096 372 y FB(\))28 b(of)g(the)g(form) 1304 554 y FA(Q)1370 519 y Fx(u;s)1464 554 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f FA(q)1802 566 y Fy(0)1853 554 y FB(\()p FA(v)f FB(+)c Fw(U)2090 519 y Fx(u;s)2185 554 y FB(\()p FA(v)s(;)c(\034)9 b FB(\)\))1304 702 y FA(P)1369 668 y Fx(u;s)1463 702 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)1774 646 y FA(@)1818 658 y Fx(u)1862 646 y FA(T)1923 616 y Fx(u;s)2030 646 y FB(\()p FA(v)d FB(+)c Fw(U)2267 616 y Fx(u;s)2362 646 y FB(\()p FA(v)s(;)c(\034)9 b FB(\)\))p 1774 683 811 4 v 1863 759 a FA(p)1905 771 y Fy(0)1942 759 y FB(\()p FA(v)22 b FB(+)c Fw(U)2179 735 y Fx(u;s)2273 759 y FB(\()p FA(v)s(;)c(\034)9 b FB(\)\))2594 702 y FA(;)3703 652 y FB(\(53\))71 933 y(where)34 b Fw(U)378 903 y Fx(u;s)507 933 y FB(de\014ne)h(a)f(c)n(hange)f(of)h(v)-5 b(ariables)33 b FA(u)h FB(=)h FA(v)26 b FB(+)c Fw(U)1958 903 y Fx(u;s)2053 933 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))36 b(in)f(suc)n(h)f(a)g(w)n(a)n(y)f(that)i(\()p FA(Q)3112 903 y Fx(u;s)3206 933 y FA(;)14 b(P)3308 903 y Fx(u;s)3402 933 y FB(\))35 b(satisfy)f(the)71 1033 y(system)27 b(of)h(equations)f(\(44\))o(.)p Black Black Black 332 2579 a /PSfrag where{pop(b)[[0(Bl)1 0]](D4)[[1(Bl)1 0]](D3)[[2(Bl)1 0]](D6)[[3(Bl)1 0]](D1)[[4(Bl)1 0]](D5)[[5(Bl)1 0]](D2)[[6(Bl)1 0]](r)[[7(Bl)1 0]](r1)[[8(Bl)1 0]](r2)[[9(Bl)1 0]](r3)[[10(Bl)1 0]](s)[[11(Bl)1 0]](u)[[12(Bl)1 0]](v)[[13(Bl)1 0]]14 0 -1/Begin PSfrag}{userdict /PSfrag{pop}put}ifelse 332 2579 a @beginspecial 0 @llx 0 @lly 409 @urx 178 @ury 1700 @rhi @setspecial %%BeginDocument: TransInfty.eps %!PS-Adobe-3.0 EPSF-3.0 %%Creator: cairo 1.11.1 (http://cairographics.org) %%CreationDate: Mon Feb 7 14:39:15 2011 %%Pages: 1 %%DocumentData: Clean7Bit %%LanguageLevel: 2 %%BoundingBox: 0 0 409 178 %%EndComments %%BeginProlog /cairo_eps_state save def /dict_count countdictstack def /op_count count 1 sub def userdict begin /q { gsave } bind def /Q { grestore } bind def /cm { 6 array astore concat } bind def /w { setlinewidth } bind def /J { setlinecap } bind def /j { setlinejoin } bind def /M { setmiterlimit } bind def /d { setdash } bind def /m { moveto } bind def /l { lineto } bind def /c { curveto } bind def /h { closepath } bind def /re { exch dup neg 3 1 roll 5 3 roll moveto 0 rlineto 0 exch rlineto 0 rlineto closepath } bind def /S { stroke } bind def /f { fill } bind def /f* { eofill } bind def /n { newpath } bind def /W { clip } bind def /W* { eoclip } bind def /BT { } bind def /ET { } bind def /pdfmark where { pop globaldict /?pdfmark /exec load put } { globaldict begin /?pdfmark /pop load def /pdfmark /cleartomark load def end } ifelse /BDC { mark 3 1 roll /BDC pdfmark } bind def /EMC { mark /EMC pdfmark } bind def /cairo_store_point { /cairo_point_y exch def /cairo_point_x exch def } def /Tj { show currentpoint cairo_store_point } bind def /TJ { { dup type /stringtype eq { show } { -0.001 mul 0 cairo_font_matrix dtransform rmoveto } ifelse } forall currentpoint cairo_store_point } bind def /cairo_selectfont { cairo_font_matrix aload pop pop pop 0 0 6 array astore cairo_font exch selectfont cairo_point_x cairo_point_y moveto } bind def /Tf { pop /cairo_font exch def /cairo_font_matrix where { pop cairo_selectfont } if } bind def /Td { matrix translate cairo_font_matrix matrix concatmatrix dup /cairo_font_matrix exch def dup 4 get exch 5 get cairo_store_point /cairo_font where { pop cairo_selectfont } if } bind def /Tm { 2 copy 8 2 roll 6 array astore /cairo_font_matrix exch def cairo_store_point /cairo_font where { pop cairo_selectfont } if } bind def /g { setgray } bind def /rg { setrgbcolor } bind def /d1 { setcachedevice } bind def %%EndProlog 11 dict begin /FontType 42 def /FontName /DejaVuSans def /PaintType 0 def /FontMatrix [ 1 0 0 1 0 0 ] def /FontBBox [ 0 0 0 0 ] def /Encoding 256 array def 0 1 255 { Encoding exch /.notdef put } for Encoding 49 /one put Encoding 50 /two put Encoding 51 /three put Encoding 52 /four put Encoding 53 /five put Encoding 54 /six put Encoding 68 /D put Encoding 98 /b put /CharStrings 9 dict dup begin /.notdef 0 def /D 1 def /one 2 def /two 3 def /three 4 def /four 5 def /five 6 def /six 7 def /b 8 def end readonly def /sfnts [ <0001000000090080000300106376742000691d39000006a0000001fe6670676d7134766a0000 08a0000000ab676c79666174166b0000009c0000060468656164f34bbbfa0000094c00000036 686865610cb8065a0000098400000024686d74782e94058d000009a8000000246c6f63610000 1b5c000009cc000000286d61787004760671000009f400000020707265703b07f10000000a14 0000056800020066fe96046605a400030007001a400c04fb0006fb0108057f0204002fc4d4ec 310010d4ecd4ec301311211125211121660400fc73031bfce5fe96070ef8f2720629000200c9 000005b005d500080011002e4015009509810195100802100a0005190d32001c09041210fcec f4ec113939393931002fecf4ec30b2601301015d011133200011100021252120001110002901 0193f40135011ffee1fecbfe42019f01b20196fe68fe50fe61052ffb770118012e012c0117a6 fe97fe80fe7efe960000000100e10000045a05d5000a004040154203a00402a005810700a009 081f061c03001f010b10d44bb00f5458b9000100403859ecc4fcec31002fec32f4ecd4ec304b 5358592201b40f030f04025d3721110535253311211521fe014afe990165ca014afca4aa0473 48b848fad5aa0000000100960000044a05f0001c009a4027191a1b03181c1105040011050504 4210a111940da014910400a00200100a02010a1c171003061d10fc4bb015544bb016545b4bb0 14545b58b90003ffc03859c4d4ecc0c011123931002fec32f4ecf4ec304b5358071005ed0705 ed11173959220140325504560556077a047a05761b87190704000419041a041b051c74007606 751a731b741c82008619821a821b821ca800a81b115d005d25211521353600373e0135342623 220607353e01333204151406070600018902c1fc4c73018d33614da7865fd3787ad458e80114 455b19fef4aaaaaa7701913a6d974977964243cc3132e8c25ca5701dfeeb00000001009cffe3 047305f000280070402e0015130a86091f862013a0150da00993061ca020932391068c15a329 161c13000314191c2620101c03141f09062910fc4bb016544bb014545b58b90009ffc03859c4 c4d4ecf4ec11173939310010ece4f4e4ec10e6ee10ee10ee10ee11123930014009641e611f61 20642104005d011e0115140421222627351e013332363534262b013533323635342623220607 353e01333204151406033f91a3fed0fee85ec76a54c86dbec7b9a5aeb6959ea39853be7273c9 59e6010c8e03251fc490ddf22525c33132968f8495a67770737b2426b42020d1b27cab000002 0064000004a405d50002000d0081401d010d030d0003030d4200030b07a00501038109010c0a 001c0608040c0e10dc4bb00b544bb00d545b58b9000cffc03859d43cc4ec32113931002fe4d4 3cec321239304b5358071004c9071005c9592201402a0b002a0048005900690077008a000716 012b0026012b0336014e014f0c4f0d5601660175017a0385010d5d005d090121033311331523 11231121350306fe0201fe35fed5d5c9fd5e0525fce303cdfc33a8fea00160c300000001009e ffe3046405d5001d005e4023041a071186101d1aa00714a010890d02a000810d8c07a41e171c 010a031c000a10061e10fc014bb016544bb014545b58b90010ffc038594bb00f5458b9001000 403859c4d4ec10c4ee310010e4e4f4ec10e6ee10fec410ee1112393013211521113e01333200 15140021222627351e0133323635342623220607dd0319fda02c582cfa0124fed4feef5ec368 5ac06badcacaad51a15405d5aafe920f0ffeeeeaf1fef52020cb3130b69c9cb6242600000002 008fffe3049605f0000b0024005840241306000d860c00a01606a01c16a510a00c8922911c8c 250c22091c191e131c03211f1b2510fcececf4ece4310010e4f4e4fce410ee10ee10ee111239 304014cb00cb01cd02cd03cd04cb05cb0607a41eb21e025d015d012206151416333236353426 01152e01232202033e0133320015140023200011100021321602a4889f9f88889f9f01094c9b 4cc8d30f3bb26be10105fef0e2fefdfeee0150011b4c9b033bbaa2a1bbbba1a2ba0279b82426 fef2feef575dfeefebe6feea018d0179016201a51e000000000200baffe304a40614000b001c 0038401903b90c0f09b918158c0fb81b971900121247180c06081a461d10fcec3232f4ec3100 2fece4f4c4ec10c6ee30b6601e801ea01e03015d013426232206151416333236013e01333200 111002232226271523113303e5a79292a7a79292a7fd8e3ab17bcc00ffffcc7bb13ab9b9022f cbe7e7cbcbe7e702526461febcfef8fef8febc6164a80614013500b800cb00cb00c100aa009c 01a600b800660000007100cb00a002b20085007500b800c301cb0189022d00cb00a600f000d3 00aa008700cb03aa0400014a003300cb000000d9050200f4015400b4009c0139011401390706 0400044e04b4045204b804e704cd0037047304cd04600473013303a2055605a60556053903c5 021200c9001f00b801df007300ba03e9033303bc0444040e00df03cd03aa00e503aa04040000 00cb008f00a4007b00b80014016f007f027b0252008f00c705cd009a009a006f00cb00cd019e 01d300f000ba018300d5009803040248009e01d500c100cb00f600830354027f000003330266 00d300c700a400cd008f009a0073040005d5010a00fe022b00a400b4009c00000062009c0000 001d032d05d505d505d505f0007f007b005400a406b80614072301d300b800cb00a601c301ec 069300a000d3035c037103db0185042304a80448008f0139011401390360008f05d5019a0614 072306660179046004600460047b009c00000277046001aa00e904600762007b00c5007f027b 000000b4025205cd006600bc00660077061000cd013b01850389008f007b0000001d00cd074a 042f009c009c0000077d006f0000006f0335006a006f007b00ae00b2002d0396008f027b00f6 00830354063705f6008f009c04e10266008f018d02f600cd03440029006604ee007300001400 00960000b707060504030201002c2010b002254964b040515820c859212d2cb002254964b040 515820c859212d2c20100720b00050b00d7920b8ffff5058041b0559b0051cb0032508b00425 23e120b00050b00d7920b8ffff5058041b0559b0051cb0032508e12d2c4b505820b0fd454459 212d2cb002254560442d2c4b5358b00225b0022545445921212d2c45442d2cb00225b0022549 b00525b005254960b0206368208a108a233a8a10653a2d000001000000024ccc7a827b3e5f0f 3cf5001f080000000000c76891d400000000c76891d4f7d6fd330d7209550000000800000001 0000000000010000076dfe1d00000de2f7d6fa510d7200010000000000000000000000000000 000904cd0066062900c9051700e1051700960517009c051700640517009e0517008f051400ba 0000000000000044000000c4000001340000023000000318000003d4000004940000056c0000 06040001000000090354002b0068000c000200100099000800000415021600080004b8028040 fffbfe03fa1403f92503f83203f79603f60e03f5fe03f4fe03f32503f20e03f19603f02503ef 8a4105effe03ee9603ed9603ecfa03ebfa03eafe03e93a03e84203e7fe03e63203e5e45305e5 9603e48a4105e45303e3e22f05e3fa03e22f03e1fe03e0fe03df3203de1403dd9603dcfe03db 1203da7d03d9bb03d8fe03d68a4105d67d03d5d44705d57d03d44703d3d21b05d3fe03d21b03 d1fe03d0fe03cffe03cefe03cd9603cccb1e05ccfe03cb1e03ca3203c9fe03c6851105c61c03 c51603c4fe03c3fe03c2fe03c1fe03c0fe03bffe03befe03bdfe03bcfe03bbfe03ba1103b986 2505b9fe03b8b7bb05b8fe03b7b65d05b7bb03b78004b6b52505b65d40ff03b64004b52503b4 fe03b39603b2fe03b1fe03b0fe03affe03ae6403ad0e03acab2505ac6403abaa1205ab2503aa 1203a98a4105a9fa03a8fe03a7fe03a6fe03a51203a4fe03a3a20e05a33203a20e03a16403a0 8a4105a096039ffe039e9d0c059efe039d0c039c9b19059c64039b9a10059b19039a1003990a 0398fe0397960d0597fe03960d03958a410595960394930e05942803930e0392fa039190bb05 91fe03908f5d0590bb039080048f8e25058f5d038f40048e25038dfe038c8b2e058cfe038b2e 038a8625058a410389880b05891403880b03878625058764038685110586250385110384fe03 8382110583fe0382110381fe0380fe037ffe0340ff7e7d7d057efe037d7d037c64037b541505 7b25037afe0379fe03780e03770c03760a0375fe0374fa0373fa0372fa0371fa0370fe036ffe 036efe036c21036bfe036a1142056a530369fe03687d036711420566fe0365fe0364fe0363fe 0362fe03613a0360fa035e0c035dfe035bfe035afe0359580a0559fa03580a03571619055732 0356fe035554150555420354150353011005531803521403514a130551fe03500b034ffe034e 4d10054efe034d10034cfe034b4a13054bfe034a4910054a1303491d0d05491003480d0347fe 0346960345960344fe0343022d0543fa0342bb03414b0340fe033ffe033e3d12053e14033d3c 0f053d12033c3b0d053c40ff0f033b0d033afe0339fe033837140538fa033736100537140336 350b05361003350b03341e03330d0332310b0532fe03310b03302f0b05300d032f0b032e2d09 052e10032d09032c32032b2a25052b64032a2912052a25032912032827250528410327250326 250b05260f03250b0324fe0323fe03220f03210110052112032064031ffa031e1d0d051e6403 1d0d031c1142051cfe031bfa031a42031911420519fe031864031716190517fe031601100516 190315fe0314fe0313fe031211420512fe0311022d05114203107d030f64030efe030d0c1605 0dfe030c0110050c16030bfe030a100309fe0308022d0508fe030714030664030401100504fe 03401503022d0503fe0302011005022d0301100300fe0301b80164858d012b2b2b2b2b2b2b2b 2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b 2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b 2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b 2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b 2b2b2b2b2b002b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b 2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b 2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b 2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b2b 2b2b2b2b2b2b2b2b2b2b2b1d00> ] def /f-0-0 currentdict end definefont pop %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 0 0 409 178 %%EndPageSetup q 0 0 409 178 rectclip q 0 g 0.8 w 0 J 0 j [ 2.4 0.8] 0 d 4 M q 1 0 0 -1 0 177.08284 cm 72.398 0.684 m 72.398 176.684 l S Q [] 0.0 d q 1 0 0 -1 0 177.08284 cm 136.398 0.684 m 136.398 176.684 l S Q q 1 0 0 -1 0 177.08284 cm 192.398 88.684 m 0.398 88.684 l S Q [ 2.4 0.8] 0 d q 1 0 0 -1 0 177.08284 cm 32.398 16.684 m 176.398 88.684 l 32.398 160.684 l S Q [] 0.0 d q 1 0 0 -1 0 177.08284 cm 260.117 0.398 m 260.117 176.398 l S Q q 1 0 0 -1 0 177.08284 cm 203.566 88.641 m 395.566 88.641 l S Q [ 2.4 0.8] 0 d q 1 0 0 -1 0 177.08284 cm 363.566 16.641 m 219.566 88.641 l 363.566 160.641 l S Q [ 2.4 0.8] 0 d q 1 0 0 -1 0 177.08284 cm 323.602 0.684 m 323.602 176.684 l S Q BT 32 0 0 32 366.318066 150.582129 Tm /f-0-0 1 Tf (D1)Tj ET BT 32 0 0 32 330.679883 45.364646 Tm /f-0-0 1 Tf (D2)Tj ET BT 32 0 0 32 281.465259 94.01359 Tm /f-0-0 1 Tf (D3)Tj ET BT 32 0 0 32 95.920435 62.335214 Tm /f-0-0 1 Tf (D4)Tj ET BT 32 0 0 32 35.957779 99.67045 Tm /f-0-0 1 Tf (D5)Tj ET BT 32 0 0 32 0.319596 148.885077 Tm /f-0-0 1 Tf (D6)Tj ET [] 0.0 d q 1 0 0 -1 0 177.08284 cm 236.602 80.082 m 238.34 82.637 238.848 85.602 239 88.684 c S Q BT 32 0 0 32 238.6 90.799997 Tm /f-0-0 1 Tf (b)Tj ET 1.6 w q 1 0 0 -1 0 177.08284 cm 323.609 36.398 m 323.609 140.484 l S Q q 1 0 0 -1 0 177.08284 cm 363.398 16.484 m 323.602 36.484 l S Q q 1 0 0 -1 0 177.08284 cm 363.398 160.684 m 323.602 140.484 l S Q q 1 0 0 -1 0 177.08284 cm 363.199 16.484 m 363.602 161.082 l S Q q 1 0 0 -1 0 177.08284 cm 72.305 140.629 m 32.562 160.711 l S Q q 1 0 0 -1 0 177.08284 cm 32.422 16.742 m 72.305 36.684 l S Q q 1 0 0 -1 0 177.08284 cm 72.16 36.684 m 72.445 140.484 l S Q q 1 0 0 -1 0 177.08284 cm 32.848 17.168 m 32.848 160.566 l S Q Q Q showpage %%Trailer count op_count sub {pop} repeat countdictstack dict_count sub {end} repeat cairo_eps_state restore %%EOF %%EndDocument @endspecial 332 2579 a /End PSfrag 332 2579 a 332 1066 a /Hide PSfrag 332 1066 a -408 1124 a FB(PSfrag)f(replacemen)n(ts)p -408 1153 741 4 v 332 1157 a /Unhide PSfrag 332 1157 a 248 1240 a { 248 1240 a Black FA(\014)295 1252 y Fy(1)p Black 248 1240 a } 0/Place PSfrag 248 1240 a 102 1340 a { 102 1340 a Black FA(D)173 1300 y Fy(out)p Fx(;u)177 1358 y Fy(\026)-39 b Fx(\032)q(;\024)p Black 102 1340 a } 1/Place PSfrag 102 1340 a 110 1454 a { 110 1454 a Black FA(D)181 1414 y Fy(out)p Fx(;s)185 1472 y Fy(\026)g Fx(\032;\024)p Black 110 1454 a } 2/Place PSfrag 110 1454 a 139 1551 a { 139 1551 a Black FA(D)210 1521 y Fx(u)208 1571 y Fv(1)p Fx(;\032)p Black 139 1551 a } 3/Place PSfrag 139 1551 a 139 1650 a { 139 1650 a Black FA(D)210 1620 y Fx(s)208 1671 y Fv(1)p Fx(;\032)p Black 139 1650 a } 4/Place PSfrag 139 1650 a 203 1750 a { 203 1750 a Black FA(I)246 1720 y Fx(u)239 1770 y(\032;)6 b Fy(\026)-39 b Fx(\032)p Black 203 1750 a } 5/Place PSfrag 203 1750 a 203 1850 a { 203 1850 a Black FA(I)246 1819 y Fx(s)239 1870 y(\032;)6 b Fy(\026)-39 b Fx(\032)p Black 203 1850 a } 6/Place PSfrag 203 1850 a 225 1965 a { 225 1965 a Black Fw(\000)8 b FB(\026)-50 b FA(\032)p Black 225 1965 a } 7/Place PSfrag 225 1965 a 225 2065 a { 225 2065 a Black Fw(\000)p FA(\032)p Black 225 2065 a } 8/Place PSfrag 225 2065 a 289 2164 a { 289 2164 a Black FA(\032)p Black 289 2164 a } 9/Place PSfrag 289 2164 a 289 2264 a { 289 2264 a Black 8 w FB(\026)g FA(\032)p Black 289 2264 a } 10/Place PSfrag 289 2264 a 54 2359 a { 54 2359 a Black FA(i)p FB(\()p FA(a)18 b Fw(\000)g FA(\016)s FB(\))p Black 54 2359 a } 11/Place PSfrag 54 2359 a 247 2467 a { 247 2467 a Black FA(u)295 2479 y Fy(1)p Black 247 2467 a } 12/Place PSfrag 247 2467 a 247 2567 a { 247 2567 a Black 247 2521 48 4 v -1 x FA(u)295 2579 y Fy(1)p Black 247 2567 a } 13/Place PSfrag 247 2567 a Black 826 2844 a FB(Figure)27 b(6:)36 b(The)28 b(transition)f(domains)g FA(I)2110 2814 y Fx(u)2103 2865 y(\032;)6 b Fy(\026)-39 b Fx(\032)2224 2844 y FB(and)27 b FA(I)2428 2814 y Fx(s)2421 2865 y(\032;)6 b Fy(\026)-39 b Fx(\032)2542 2844 y FB(de\014ned)28 b(in)g(\(52\))o(.)p Black Black 195 3076 a(The)i(next)f(theorem)f(ensures)h(that)g(this)g (c)n(hange)f(of)h(v)-5 b(ariables)28 b(exists)h(and)g(it)g(is)g(w)n (ell)g(de\014ned)g(in)h(the)f(transition)71 3175 y(domains)e FA(I)442 3145 y Fv(\003)435 3196 y Fx(\032;)6 b Fy(\026)-39 b Fx(\032)528 3175 y FB(,)28 b Fw(\003)22 b FB(=)h FA(u;)14 b(s)p FB(.)36 b(W)-7 b(e)28 b(state)g(it)g(for)f Fw(U)1531 3145 y Fx(u)1574 3175 y FB(.)37 b(The)28 b(results)f(for)g Fw(U)2256 3145 y Fx(s)2319 3175 y FB(are)g(analogous.)p Black 71 3341 a Fp(Theorem)h(4.5.)p Black 38 w Fs(L)l(et)e FA(\032)844 3353 y Fy(1)908 3341 y Fs(b)l(e)i(the)f(c)l(onstant)f(c)l (onsider)l(e)l(d)j(in)e(The)l(or)l(em)h(4.3)g(and)g(let)f(us)g(c)l (onsider)h FA(\032)3194 3353 y Fy(3)3258 3341 y Fs(and)g FA(\032)3460 3353 y Fy(4)3524 3341 y Fs(such)f(that)71 3441 y FA(\032)114 3453 y Fy(4)184 3441 y FA(>)33 b(\032)325 3453 y Fy(3)395 3441 y FA(>)f(\032)535 3453 y Fy(1)607 3441 y Fs(and)k FA(")813 3453 y Fy(0)885 3441 y Fs(smal)t(l)g(enough)g (\(which)h(might)f(dep)l(end)g(on)f FA(\032)2357 3453 y Fx(i)2385 3441 y Fs(,)i FA(i)32 b FB(=)h(1)p FA(;)14 b FB(2)p FA(;)g FB(3)p Fs(\).)53 b(Then,)38 b(for)e FA(")c Fw(2)i FB(\(0)p FA(;)14 b(")3615 3453 y Fy(0)3651 3441 y FB(\))36 b Fs(and)71 3540 y FA(\026)23 b Fw(2)g FA(B)t FB(\()p FA(\026)371 3552 y Fy(0)409 3540 y FB(\))p Fs(,)31 b(ther)l(e)e(exists)h(a)g(r)l(e)l(al-analytic)h(function)f Fw(U)1862 3510 y Fx(u)1928 3540 y FB(:)23 b FA(I)2017 3510 y Fx(u)2010 3561 y(\032)2044 3569 y Fu(3)2078 3561 y Fx(;\032)2132 3569 y Fu(4)2187 3540 y Fw(\002)18 b Ft(T)2325 3552 y Fx(\033)2393 3540 y Fw(!)23 b Ft(C)30 b Fs(such)g(that)p Black 195 3707 a Fw(\017)p Black 41 w Fs(Ther)l(e)h(exists)e(a)i(c)l(onstant)e FA(b)1184 3719 y Fy(3)1243 3707 y FA(>)23 b FB(0)29 b Fs(indep)l(endent)i(of)f FA(")g Fs(and)g FA(\026)g Fs(such)g(that)f(for)i FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b Fw(2)g FA(I)3096 3676 y Fx(u)3089 3727 y(\032)3123 3735 y Fu(3)3156 3727 y Fx(;\032)3210 3735 y Fu(4)3265 3707 y Fw(\002)18 b Ft(T)3403 3719 y Fx(\033)3448 3707 y Fs(,)1661 3901 y Fw(jU)1744 3867 y Fx(u)1788 3901 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))p Fw(j)25 b(\024)d FA(b)2148 3913 y Fy(3)2185 3901 y Fw(j)p FA(\026)p Fw(j)p FA(")2320 3867 y Fx(\021)r Fy(+1)2444 3901 y FA(:)p Black 195 4117 a Fw(\017)p Black 41 w Fs(If)31 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b Fw(2)f FA(I)700 4087 y Fx(u)693 4138 y(\032)727 4146 y Fu(3)761 4138 y Fx(;\032)815 4146 y Fu(4)870 4117 y Fw(\002)18 b Ft(T)1008 4129 y Fx(\033)1053 4117 y Fs(,)30 b(then)g FA(v)21 b FB(+)d Fw(U)1497 4087 y Fx(u)1541 4117 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))25 b Fw(2)e FA(D)1904 4087 y Fx(u)1902 4138 y Fv(1)p Fx(;\032)2022 4146 y Fu(1)2059 4117 y Fs(.)p Black 195 4283 a Fw(\017)p Black 41 w Fs(The)32 b(p)l(ar)l(ameterizations)g (of)g(the)e(invariant)i(manifolds)g FB(\()p FA(Q)2192 4253 y Fx(u)2236 4283 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)14 b(P)2527 4253 y Fx(u)2571 4283 y FB(\()p FA(v)s(;)g(\034)9 b FB(\)\))32 b Fs(in)37 b FB(\(53\))31 b Fs(satisfy)g(the)g(system)g(of)278 4383 y(e)l(quations)37 b FB(\(44\))29 b Fs(and)i(ther)l(e)f(exists)f(a)h(c)l(onstant)f FA(b)1862 4395 y Fy(4)1922 4383 y FA(>)23 b FB(0)29 b Fs(such)h(that)f(for)i FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b Fw(2)g FA(I)2907 4353 y Fx(u)2900 4403 y(\032)2934 4411 y Fu(3)2967 4403 y Fx(;\032)3021 4411 y Fu(4)3076 4383 y Fw(\002)18 b Ft(T)3214 4395 y Fx(\033)3259 4383 y Fs(,)1519 4578 y Fw(j)p FA(Q)1608 4544 y Fx(u)1651 4578 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))20 b Fw(\000)e FA(q)1980 4590 y Fy(0)2017 4578 y FB(\()p FA(v)s FB(\))p Fw(j)24 b(\024)f FA(b)2295 4590 y Fy(4)2331 4578 y Fw(j)p FA(\026)p Fw(j)p FA(")2466 4544 y Fx(\021)r Fy(+1)1515 4713 y Fw(j)p FA(P)1603 4678 y Fx(u)1646 4713 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(p)1980 4725 y Fy(0)2017 4713 y FB(\()p FA(v)s FB(\))p Fw(j)24 b(\024)f FA(b)2295 4725 y Fy(4)2331 4713 y Fw(j)p FA(\026)p Fw(j)p FA(")2466 4678 y Fx(\021)r Fy(+1)2591 4713 y FA(;)278 4896 y Fs(wher)l(e)31 b FB(\()p FA(q)582 4908 y Fy(0)619 4896 y FA(;)14 b(p)698 4908 y Fy(0)735 4896 y FB(\))30 b Fs(is)g(the)g(p)l(ar)l(ametrization)i(of)e(the)g(unp)l(erturb)l(e)l (d)f(sep)l(ar)l(atrix)h(given)g(in)g(Hyp)l(othesis)h Fo(HP2)p Fs(.)195 5062 y FB(The)d(pro)r(of)f(of)h(this)g(theorem)f(is)g (deferred)g(to)h(section)f(7.2.2.)195 5162 y(Ha)n(ving)22 b(the)h(parameterizations)d(\()p FA(Q)1379 5132 y Fx(u;s)1474 5162 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)14 b(P)1765 5132 y Fx(u;s)1860 5162 y FB(\()p FA(v)s(;)g(\034)9 b FB(\)\))24 b(in)f(the)g(transition)f(domains)g FA(I)3075 5132 y Fv(\003)3068 5182 y Fx(\032)3102 5190 y Fu(3)3135 5182 y Fx(;\032)3189 5190 y Fu(4)3234 5162 y Fw(\002)8 b Ft(T)3362 5174 y Fx(\033)3430 5162 y FB(for)22 b Fw(\003)g FB(=)h FA(u;)14 b(s)p FB(,)71 5261 y(w)n(e)26 b(extend)g(them)h(till)f (w)n(e)g(arriv)n(e)f(to)h(a)f(region)g(where)h(w)n(e)g(can)f(ensure)h (that)g FA(p)2557 5273 y Fy(0)2594 5261 y FB(\()p FA(u)p FB(\))h(do)r(es)f(not)g(v)-5 b(anish)26 b(an)n(ymore.)35 b(This)71 5361 y(region)29 b(consists)h(of)g(a)g(piece)h(of)f(the)h Fs(b)l(o)l(omer)l(ang)i(domains)39 b FB(de\014ned)31 b(in)g(\(26\))f(\(see)g(Figure)g(2\),)h(in)g(whic)n(h)f FA(p)3516 5373 y Fy(0)3553 5361 y FB(\()p FA(u)p FB(\))e Fw(6)p FB(=)g(0,)71 5461 y(and)f(hence)h(the)g(parameterizations)d (\(42\))i(will)h(b)r(e)g(w)n(ell)g(de\014ned)g(in)f(them.)p Black 1919 5753 a(30)p Black eop end %%Page: 31 31 TeXDict begin 31 30 bop Black Black 195 272 a FB(The)28 b(next)h(step)f(is)g(to)g(extend)g(the)g(parameterizations)e(\()p FA(Q)2102 242 y Fx(u;s)2196 272 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)14 b(P)2487 242 y Fx(u;s)2583 272 y FB(\()p FA(v)s(;)g(\034)9 b FB(\)\))30 b(pro)n(vided)d(in)h (Theorem)f(4.5)g(up)71 372 y(to)k(domains)g(whic)n(h)g(in)n(tersect)g (the)h Fs(b)l(o)l(omer)l(ang)i(domains)39 b FA(D)2057 342 y Fx(u)2055 395 y(\024;d)2184 372 y FB(and)31 b FA(D)2420 342 y Fx(s)2418 395 y(\024;d)2546 372 y FB(resp)r(ectiv)n(ely)-7 b(.)48 b(T)-7 b(o)31 b(this)h(end,)g(w)n(e)f(de\014ne)71 471 y(the)d(follo)n(wing)e(domains)920 670 y Fz(e)903 691 y FA(D)974 651 y Fy(out)o Fx(;u)972 716 y(\032;d;\024)1155 691 y FB(=)d FA(D)1314 656 y Fy(out)p Fx(;u)1312 711 y(\032;\024)1491 691 y Fw(\\)1565 574 y Fz(\032)1628 691 y FA(u)f Fw(2)i Ft(C)p FB(;)14 b Fw(j)p FB(Im)f FA(u)p Fw(j)23 b FA(<)g Fw(\000)14 b FB(tan)f FA(\014)2450 703 y Fy(2)2487 691 y FB(Re)h FA(u)k FB(+)g FA(a)g Fw(\000)2903 635 y FA(d)p 2903 672 44 4 v 2904 748 a FB(2)2956 574 y Fz(\033)928 902 y(e)910 923 y FA(D)981 883 y Fy(out)p Fx(;s)979 948 y(\032;d;\024)1155 923 y FB(=)23 b FA(D)1314 889 y Fy(out)p Fx(;s)1312 944 y(\032;\024)1484 923 y Fw(\\)1557 806 y Fz(\032)1620 923 y FA(u)f Fw(2)i Ft(C)p FB(;)14 b Fw(j)p FB(Im)g FA(u)p Fw(j)22 b FA(>)h FB(tan)14 b FA(\014)2364 935 y Fy(2)2401 923 y FB(Re)g FA(u)k FB(+)g FA(a)g Fw(\000)2817 867 y FA(d)p 2817 904 V 2818 980 a FB(2)2870 806 y Fz(\033)2946 923 y FA(;)3703 807 y FB(\(54\))71 1142 y(whic)n(h)27 b(are)g(depicted)h(in)g(Figure)f(7.)195 1242 y(W)-7 b(e)41 b(w)n(an)n(t)f(to)h(emphasize)f(that)h(to)f(extend)h (the)g(parameterizations)d(\()p FA(Q)2623 1212 y Fx(u;s)2717 1242 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)14 b(P)3008 1212 y Fx(u;s)3104 1242 y FB(\()p FA(v)s(;)g(\034)9 b FB(\)\))42 b(to)f(these)f(new)71 1342 y(domains,)25 b(has)h(no)f(tec)n(hnical)h(di\016culties)g(since)g(they)g(are)f(far)g (from)h(the)g(singularities)e FA(u)f FB(=)g Fw(\006)p FA(ia)p FB(.)35 b(Consequen)n(tly)25 b(the)71 1441 y(next)j(theorem)f (is)g(a)g(classical)g(p)r(erturbativ)n(e)f(result.)p Black Black Black 290 2971 a /PSfrag where{pop(D6)[[0(Bl)1 0]](D5)[[1(Bl)1 0]](b1)[[2(Bl)1 0]](b2)[[3(Bl)1 0]](r1)[[4(Bl)1 0]](r2)[[5(Bl)1 0]](a1)[[6(Bl)1 0]](a2)[[7(Bl)1 0]](a3)[[8(Bl)1 0]](a4)[[9(Bl)1 0]](D1)[[10(Bl)1 0]](D)[[11(Bl)1 0]](D4)[[12(Bl)1 0]](D3)[[13(Bl)1 0]]14 0 -1/Begin PSfrag}{userdict /PSfrag{pop}put}ifelse 290 2971 a @beginspecial 7 @llx 564 @lly 587 @urx 810 @ury 1700 @rhi @setspecial %%BeginDocument: OuterParam.eps %!PS-Adobe-3.0 EPSF-3.0 %%Creator: 0.46 %%Pages: 1 %%Orientation: Portrait %%BoundingBox: 7 564 587 810 %%HiResBoundingBox: 7.4832511 564.68339 586.59801 809.93219 %%EndComments %%Page: 1 1 0 842 translate 0.8 -0.8 scale 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap gsave [1 0 0 1 0 0] concat gsave [0.8907956 0 0 0.9694639 117.08825 37.514288] concat 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 48.42644 246.9755 moveto 41.008132 236.33733 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 4.938287 160.30644 moveto 59.830417 242.806 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2.5 setlinewidth 0 setlinejoin 0 setlinecap newpath 510.17246 257.72057 moveto 506.63693 257.72057 lineto stroke gsave [1 0 0 -1 486.39835 264.28964] concat gsave /newlatin1font {findfont dup length dict copy dup /Encoding ISOLatin1Encoding put definefont} def /Sans-ISOLatin1 /Sans newlatin1font 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a2) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 2.5 setlinewidth 0 setlinejoin 0 setlinecap newpath 510.08092 62.417932 moveto 506.54539 62.417932 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 463.34221 153.00592 moveto 465.58305 155.65792 465.88753 155.73919 466.73718 159.17099 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 479.60374 148.86444 moveto 466.89341 153.9152 lineto 470.73355 149.69039 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 325.30294 148.6119 moveto 326.81195 151.40227 328.74624 153.34206 328.3334 159.97612 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 73.365027 151.50009 moveto 68.365027 151.14295 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2.5 setlinewidth 0 setlinejoin 0 setlinecap newpath 50.108604 62.328829 moveto 46.573074 62.328829 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 73.606657 79.142954 moveto 52.178089 93.07152 lineto 55.74952 88.07152 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 48.394873 73.355535 moveto 41.071267 83.962137 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 41.253737 84.357235 moveto 92.325167 160.07152 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 80.686457 142.75009 moveto 76.852797 146.71405 73.574597 151.23347 74.436457 159.89295 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 508.36503 40.071523 moveto 508.36503 280.07152 lineto stroke gsave [1 0 0 -1 545.26367 214.77689] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D6) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 288.36503 160.07152 moveto 676.8239 159.72528 lineto stroke gsave [1 0 0 1 0 0] concat gsave [1 0 0 -1 509.68344 64.520302] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a1) show grestore grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 51.400741 141.32152 moveto 20.507884 153.28581 lineto 24.257884 149.71438 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 20.507884 153.46438 moveto 24.97217 153.46438 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 14.257884 146.67866 moveto 18.874046 150.8694 19.674578 155.40701 19.079313 160.07152 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath -8.599259 134.89295 moveto 8.72217 147.03581 lineto 5.865027 142.57152 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 8.72217 146.85724 moveto 4.615027 146.85724 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 59.781497 77.230601 moveto 4.759716 159.9493 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 41.145502 84.051356 moveto 28.275741 64.803663 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 298.36503 160.07152 moveto 676.8239 5.0005932 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 298.36503 160.07152 moveto 676.8239 314.44997 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 508.37779 246.84924 moveto 450.54655 159.97612 lineto 508.4351 72.047958 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 529.48862 64.7412 moveto 466.24045 159.97612 lineto 529.74865 255.46675 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 41.129647 84.127267 moveto -9.582653 160.30644 lineto 40.97543 236.25394 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 48.36503 40.071523 moveto 48.36503 280.07152 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 268.36503 160.07152 moveto -120.21641 159.72528 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 258.36503 160.07152 moveto -119.83863 3.6007035 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 258.36503 160.07152 moveto -119.4377 317.64752 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 467.0418 153.54681 moveto 472.8345 154.4411 lineto stroke gsave [1 0 0 -1 478.86642 149.21849] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (b2) show grestore grestore gsave [1 0 0 -1 331.36386 154.71342] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (b1) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 478.36503 65.071521 moveto 506.22217 71.500092 lineto 501.0436 67.571521 lineto stroke gsave [1 0 0 -1 413.18253 58.874096] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a4) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 430.35881 181.44186 moveto 466.47176 169.8251 lineto 459.40069 169.32003 lineto stroke gsave [1 0 0 -1 401.67096 191.0383] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D) show grestore grestore 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 28.507495 255.24612 moveto 41.027029 236.06556 lineto 48.331739 225.06782 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 28.381226 255.49866 moveto 41.008133 236.05322 lineto stroke gsave [1 0 0 -1 26.051901 64.518196] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a1) show grestore grestore gsave [1 0 0 -1 49.423901 266.2673] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a2) show grestore grestore gsave [1 0 0 -1 51.972176 138.39293] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (b2) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 57.293599 142.03581 moveto 73.365027 151.32152 lineto 70.507877 147.7501 lineto stroke gsave [1 0 0 -1 -38.727673 127.32995] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D3) show grestore grestore gsave [1 0 0 -1 93.722176 200.53581] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D4) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 52.7138 92.71437 moveto 57.178087 92.00009 lineto stroke gsave [1 0 0 -1 71.035225 80.785812] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a3) show grestore grestore gsave [1 0 0 -1 71.01738 57.316502] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a4) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 72.481666 56.423645 moveto 51.41024 70.352216 lineto 53.195953 64.816502 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 51.241158 70.607236 moveto 56.062587 69.535807 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2.5 setlinewidth 0 setlinejoin 0 setlinecap newpath 50.162187 259.78207 moveto 46.626657 259.78207 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 465.71415 169.57257 moveto 461.92607 173.61318 lineto stroke gsave [1.122592 0 0 1.0314979 -131.44233 -38.69591] concat 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 10.354063 40.946946 moveto 153.21429 105.57647 lineto 160.16226 117.95818 lineto 208.21428 193.07647 lineto 160.30059 266.89932 lineto 151.42857 280.93361 lineto 10.859139 345.25541 lineto 10.354063 345.50795 lineto 10.354063 40.946946 lineto closepath stroke grestore gsave [1.0432221 0 0 0.9585686 0 0] concat gsave [1 0 0 -1 -89.102272 228.42659] concat gsave /Sans-ISOLatin1 findfont 43.043259 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D5) show grestore grestore grestore gsave [1.122592 0 0 1.0314979 -131.44233 -38.69591] concat 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 578.75 280.3979 moveto 526.07143 192.36218 lineto 578.39286 105.69376 lineto 719.99999 42.362183 lineto 719.99999 342.36219 lineto 719.99999 342.36219 lineto 578.75 280.3979 lineto closepath stroke grestore gsave [1.122592 0 0 1.0314979 -131.44233 -38.69591] concat 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 567.45319 106.60686 moveto 562.65497 107.86955 lineto stroke grestore gsave [1.0432221 0 0 0.9585686 0 0] concat gsave [1 0 0 -1 373.04871 151.79845] concat gsave /Sans-ISOLatin1 findfont 43.043259 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D1) show grestore grestore grestore gsave [1.0432221 0 0 0.9585686 0 0] concat gsave [1 0 0 -1 -112.38706 159.7011] concat gsave /Sans-ISOLatin1 findfont 43.043259 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (r1) show grestore grestore grestore gsave [1.0432221 0 0 0.9585686 0 0] concat gsave [1 0 0 -1 626.99304 159.42935] concat gsave /Sans-ISOLatin1 findfont 43.043259 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (r2) show grestore grestore grestore grestore grestore showpage %%EOF %%EndDocument @endspecial 290 2971 a /End PSfrag 290 2971 a 290 1434 a /Hide PSfrag 290 1434 a -450 1492 a FB(PSfrag)g(replacemen)n(ts)p -450 1521 741 4 v 290 1524 a /Unhide PSfrag 290 1524 a 68 1614 a { 68 1614 a Black 85 1593 a Fz(e)68 1614 y FA(D)139 1574 y Fy(out)p Fx(;s)137 1639 y(\032;d;\024)p Black 68 1614 a } 0/Place PSfrag 68 1614 a 60 1739 a { 60 1739 a Black 77 1718 a Fz(e)60 1739 y FA(D)131 1699 y Fy(out)p Fx(;u)129 1764 y(\032;d;\024)p Black 60 1739 a } 1/Place PSfrag 60 1739 a 206 1859 a { 206 1859 a Black FA(\014)253 1871 y Fy(1)p Black 206 1859 a } 2/Place PSfrag 206 1859 a 206 1959 a { 206 1959 a Black FA(\014)253 1971 y Fy(2)p Black 206 1959 a } 3/Place PSfrag 206 1959 a 182 2058 a { 182 2058 a Black Fw(\000)p FA(\032)p Black 182 2058 a } 4/Place PSfrag 182 2058 a 247 2158 a { 247 2158 a Black FA(\032)p Black 247 2158 a } 5/Place PSfrag 247 2158 a 218 2274 a { 218 2274 a Black FA(ia)p Black 218 2274 a } 6/Place PSfrag 218 2274 a 153 2366 a { 153 2366 a Black Fw(\000)p FA(ia)p Black 153 2366 a } 7/Place PSfrag 153 2366 a 8 2452 a { 8 2452 a Black FA(i)p FB(\()p FA(a)18 b Fw(\000)g FA(d)p FB(\))p Black 8 2452 a } 8/Place PSfrag 8 2452 a -35 2552 a { -35 2552 a Black FA(i)p FB(\()p FA(a)g Fw(\000)g FA(\024")p FB(\))p Black -35 2552 a } 9/Place PSfrag -35 2552 a 68 2640 a { 68 2640 a Black FA(D)139 2610 y Fy(out)p Fx(;s)137 2661 y(\032;\024)p Black 68 2640 a } 10/Place PSfrag 68 2640 a 124 2737 a { 124 2737 a Black FA(D)195 2707 y Fx(s)193 2760 y(\024;d)p Black 124 2737 a } 11/Place PSfrag 124 2737 a 60 2840 a { 60 2840 a Black FA(D)131 2809 y Fy(out)p Fx(;u)129 2860 y(\032;\024)p Black 60 2840 a } 12/Place PSfrag 60 2840 a 124 2936 a { 124 2936 a Black FA(D)195 2906 y Fx(u)193 2960 y(\024;d)p Black 124 2936 a } 13/Place PSfrag 124 2936 a Black 919 3237 a FB(Figure)27 b(7:)36 b(The)28 b(domains)1798 3216 y Fz(e)1781 3237 y FA(D)1852 3197 y Fy(out)o Fx(;u)1850 3262 y(\032;d;\024)2038 3237 y FB(and)2217 3216 y Fz(e)2200 3237 y FA(D)2271 3197 y Fy(out)o Fx(;s)2269 3262 y(\032;d;\024)2449 3237 y FB(de\014ned)g(in)g (\(54\).)p Black Black 195 3453 a(The)g(next)g(theorem)f(giv)n(es)f (the)i(results)f(for)g(the)h(unstable)g(manifold.)37 b(F)-7 b(or)27 b(the)h(stable)f(one)g(are)g(analogous.)p Black 71 3612 a Fp(Theorem)34 b(4.6.)p Black 41 w Fs(L)l(et)e FA(\032)859 3624 y Fy(4)928 3612 y Fs(and)g FA(\024)1139 3624 y Fy(1)1208 3612 y Fs(b)l(e)g(the)g(c)l(onstants)f(c)l(onsider)l (e)l(d)i(in)f(The)l(or)l(ems)h(4.5)g(and)f(4.4,)i FA(d)3222 3624 y Fy(0)3287 3612 y FA(>)26 b FB(0)31 b Fs(and)i FA(")3654 3624 y Fy(0)3717 3612 y FA(>)27 b FB(0)71 3712 y Fs(smal)t(l)33 b(enough.)46 b(Then,)34 b(for)f FA(\026)27 b Fw(2)g FA(B)t FB(\()p FA(\026)1305 3724 y Fy(0)1343 3712 y FB(\))32 b Fs(and)h FA(")27 b Fw(2)h FB(\(0)p FA(;)14 b(")1870 3724 y Fy(0)1906 3712 y FB(\))p Fs(,)33 b(ther)l(e)g(exist)e(functions)h FB(\()p FA(Q)2862 3681 y Fx(u)2906 3712 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)14 b(P)3197 3681 y Fx(u)3241 3712 y FB(\()p FA(v)s(;)g(\034)9 b FB(\)\))34 b Fs(de\014ne)l(d)e(in)88 3801 y Fz(e)71 3822 y FA(D)142 3782 y Fy(out)p Fx(;u)140 3847 y(\032)174 3855 y Fu(4)206 3847 y Fx(;d)261 3855 y Fu(0)293 3847 y Fx(;\024)352 3855 y Fu(1)409 3822 y Fw(\002)20 b Ft(T)549 3834 y Fx(\033)627 3822 y Fs(satisfying)35 b(e)l(quation)k FB(\(44\))33 b Fs(and)g(such)g(that)g(they)h(ar)l(e)f(the)g(analytic)h (c)l(ontinuation)f(of)h(the)f(p)l(ar)l(ame-)71 3921 y(terizations)d(of) h(the)f(invariant)g(manifolds)i(obtaine)l(d)f(in)f(The)l(or)l(em)h (4.5.)195 4021 y(Mor)l(e)l(over,)g(ther)l(e)e(exists)f(a)h(c)l(onstant) e FA(b)1453 4033 y Fy(5)1513 4021 y FA(>)c FB(0)28 b Fs(indep)l(endent)h(of)g FA(")g Fs(and)g FA(\026)f Fs(such)h(that)f (for)i FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b Fw(2)3328 4000 y Fz(e)3311 4021 y FA(D)3382 3981 y Fy(out)p Fx(;u)3380 4046 y(\032)3414 4054 y Fu(4)3447 4046 y Fx(;d)3502 4054 y Fu(0)3534 4046 y Fx(;\024)3593 4054 y Fu(1)3645 4021 y Fw(\002)15 b Ft(T)3780 4033 y Fx(\033)3825 4021 y Fs(,)1411 4212 y Fw(j)p FA(Q)1500 4177 y Fx(u)1543 4212 y FB(\()p FA(v)s(;)f(\034)9 b FB(\))20 b Fw(\000)e FA(q)1872 4224 y Fy(0)1909 4212 y FB(\()p FA(v)s FB(\))p Fw(j)24 b(\024)f FA(b)2187 4224 y Fy(5)2224 4212 y Fw(j)p FA(\026)p Fw(j)p FA(")2359 4177 y Fx(\021)r Fy(+1)1411 4313 y Fw(j)p FA(P)1499 4279 y Fx(u)1542 4313 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(p)1876 4325 y Fy(0)1913 4313 y FB(\()p FA(v)s FB(\))p Fw(j)24 b(\024)f FA(b)2191 4325 y Fy(5)2228 4313 y Fw(j)p FA(\026)p Fw(j)p FA(")2363 4279 y Fx(\021)r Fy(+1)2487 4313 y FA(:)195 4484 y FB(The)28 b(pro)r(of)f(of)h(this)g(theorem)f(is)g(giv)n(en)g(in)h(Section)f (7.2.3.)195 4683 y(Theorem)k(4.6)g(pro)n(vides)f(parameterizations)g (of)h(the)h(in)n(v)-5 b(arian)n(t)31 b(manifolds)g(of)h(the)g(form)f (\(43\))g(in)h(the)g(domains)88 4772 y Fz(e)71 4793 y FA(D)142 4753 y Fy(out)p Fx(;u)140 4818 y(\032;d;\024)325 4793 y FB(and)501 4772 y Fz(e)484 4793 y FA(D)555 4753 y Fy(out)o Fx(;s)553 4818 y(\032;d;\024)706 4793 y FB(.)j(In)25 b(particular,)f(they)h(are)e(de\014ned)i(in)g(the)f(follo)n(wing)g (transition)g(domains,)g(whic)n(h)h(are)e(depicted)71 4893 y(in)28 b(Figure)f(8.)1553 5072 y FA(I)1596 5033 y Fy(out)o Fx(;u)1589 5098 y(\024;d)1778 5072 y FB(=)1882 5052 y Fz(e)1865 5072 y FA(D)1936 5033 y Fy(out)p Fx(;u)1934 5098 y(\032;d;\024)2114 5072 y Fw(\\)18 b FA(D)2258 5038 y Fx(u)2256 5093 y(\024;d)1561 5223 y FA(I)1604 5183 y Fy(out)o Fx(;s)1597 5248 y(\024;d)1778 5223 y FB(=)1882 5202 y Fz(e)1865 5223 y FA(D)1936 5183 y Fy(out)p Fx(;s)1934 5248 y(\032;d;\024)2106 5223 y Fw(\\)g FA(D)2250 5189 y Fx(s)2248 5244 y(\024;d)2345 5223 y FA(;)3703 5147 y FB(\(55\))71 5404 y(where,)33 b(b)n(y)f(construction,)g FA(p)1006 5416 y Fy(0)1043 5404 y FB(\()p FA(u)p FB(\))g(do)r(es)g(not) g(v)-5 b(anish.)51 b(Then,)33 b(w)n(e)f(can)g(use)g(these)g(domains)f (as)h(transition)f(domains)71 5504 y(where)g(w)n(e)g(can)g(go)f(bac)n (k)h(to)g(the)h(parameterizations)d(\(42\))i(and)g(where)g(the)h (Hamilton-Jacobi)e(equation)h(\(41\))g(can)p Black 1919 5753 a(31)p Black eop end %%Page: 32 32 TeXDict begin 32 31 bop Black Black 71 272 a FB(b)r(e)28 b(used.)37 b(T)-7 b(o)27 b(obtain)g(them,)h(w)n(e)g(lo)r(ok)f(for)g(c)n (hanges)f(of)h(v)-5 b(ariables)27 b FA(v)f FB(=)d FA(u)18 b FB(+)g Fw(V)2558 242 y Fx(u;s)2652 272 y FB(\()p FA(u;)c(\034)9 b FB(\))28 b(whic)n(h)g(satisfy)1399 455 y FA(Q)1465 420 y Fx(u;s)1559 455 y FB(\()p FA(u)18 b FB(+)g Fw(V)1798 420 y Fx(u;s)1893 455 y FB(\()p FA(u;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))24 b(=)f FA(q)2350 467 y Fy(0)2387 455 y FB(\()p FA(u)p FB(\))p FA(;)1181 b FB(\(56\))71 637 y(where)31 b FA(Q)381 607 y Fx(u;s)508 637 y FB(are)g(the)h (\014rst)g(comp)r(onen)n(ts)g(of)g(the)g(parameterizations)e(obtained)i (in)g(Theorem)g(4.6.)49 b(Once)32 b(w)n(e)g(ha)n(v)n(e)71 737 y(them,)27 b(w)n(e)f(will)g(de\014ne)h(the)f(generating)f (functions)h FA(T)1783 707 y Fx(u;s)1903 737 y FB(whic)n(h)g(giv)n(e)f (the)i(parameterizations)d(\(42\))o(.)37 b(Let)26 b(us)g(observ)n(e)71 837 y(that)f(if)g FA(p)363 849 y Fy(0)400 837 y FB(\()p FA(u)p FB(\))g(do)r(es)f(not)h(v)-5 b(anish)24 b(in)h(the)g(outer)g (domains,)f(the)h(c)n(hanges)f(of)g(v)-5 b(ariables)24 b FA(v)i FB(=)c FA(u)13 b FB(+)g Fw(V)3144 807 y Fx(u;s)3237 837 y FB(\()p FA(u;)h(\034)9 b FB(\))25 b(are)f(de\014ned)71 936 y(in)i(the)g(whole)g(domain)f(and)h(they)g(are)f(the)h(in)n(v)n (erse)f(of)h(the)g(c)n(hanges)e FA(u)f FB(=)g FA(v)18 b FB(+)c Fw(U)2626 906 y Fx(u;s)2721 936 y FB(\()p FA(v)s(;)g(\034)9 b FB(\))27 b(obtained)f(in)g(Theorem)f(4.5.)p Black Black Black 312 2474 a /PSfrag where{pop(D6)[[0(Bl)1 0]](D5)[[1(Bl)1 0]](b1)[[2(Bl)1 0]](b2)[[3(Bl)1 0]](a1)[[4(Bl)1 0]](a2)[[5(Bl)1 0]](a3)[[6(Bl)1 0]](a4)[[7(Bl)1 0]](D1)[[8(Bl)1 0]](D)[[9(Bl)1 0]](D4)[[10(Bl)1 0]](D3)[[11(Bl)1 0]](D8)[[12(Bl)1 0]](D7)[[13(Bl)1 0]]14 0 -1/Begin PSfrag}{userdict /PSfrag{pop}put}ifelse 312 2474 a @beginspecial 7 @llx 565 @lly 577 @urx 810 @ury 1700 @rhi @setspecial %%BeginDocument: TransBoom.eps %!PS-Adobe-3.0 EPSF-3.0 %%Creator: 0.46 %%Pages: 1 %%Orientation: Portrait %%BoundingBox: 7 565 577 810 %%HiResBoundingBox: 7.6282817 565.08339 576.4 809.53219 %%EndComments %%Page: 1 1 0 842 translate 0.8 -0.8 scale 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap gsave [1 0 0 1 0 0] concat 0 0 0 setrgbcolor [2.787893 0.92929768] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 160.22631 276.94812 moveto 153.61811 266.6348 lineto stroke 0 0 0 setrgbcolor [2.787893 0.92929768] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 121.48725 192.92559 moveto 170.38492 272.90594 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2.3232441 setlinewidth 0 setlinejoin 0 setlinecap newpath 571.63819 284.99073 moveto 568.48876 284.99073 lineto stroke gsave [0.9585686 0 0 1.0432221 0 0] concat gsave [1 0 0 -1 593.93805 284.20706] concat gsave /newlatin1font {findfont dup length dict copy dup /Encoding ISOLatin1Encoding put definefont} def /Sans-ISOLatin1 /Sans newlatin1font 37.171906 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a2) show grestore grestore grestore 0 0 0 setrgbcolor [] 0 setdash 2.3232441 setlinewidth 0 setlinejoin 0 setlinecap newpath 571.46609 100.02622 moveto 568.31665 100.02622 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 529.83145 185.848 moveto 531.82758 188.41902 532.09881 188.49781 532.85568 191.82482 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 544.31715 181.83299 moveto 532.99485 186.72952 lineto 536.41563 182.63372 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 406.86668 181.58816 moveto 408.2109 184.29332 409.93395 186.17388 409.5662 192.60536 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 182.44149 184.38816 moveto 177.98752 184.04192 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2.3232441 setlinewidth 0 setlinejoin 0 setlinecap newpath 161.47223 99.829434 moveto 158.3228 99.829434 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 182.65674 114.24052 moveto 163.56826 127.74377 lineto 166.74968 122.89645 lineto stroke 0 0 0 setrgbcolor [2.787893 0.92929768] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 160.19819 108.62983 moveto 153.67435 118.91255 lineto stroke 0 0 0 setrgbcolor [2.787893 0.92929768] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 153.8369 119.29558 moveto 199.3311 192.69785 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 188.96339 175.90535 moveto 185.54838 179.74826 182.62818 184.12968 183.39592 192.52473 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 569.93758 76.362183 moveto 569.93758 309.03352 lineto stroke gsave [0.9585686 0 0 1.0432221 0 0] concat gsave [1 0 0 -1 628.86133 235.55168] concat gsave /Sans-ISOLatin1 findfont 37.171906 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D6) show grestore grestore grestore 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 373.96255 192.69785 moveto 720 192.36218 lineto stroke gsave [0.9585686 0 0 1.0432221 0 0] concat gsave [1 0 0 -1 595.79657 95.918587] concat gsave /Sans-ISOLatin1 findfont 37.171906 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a1) show grestore grestore grestore 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 162.8758 174.5204 moveto 135.35658 186.11935 lineto 138.69707 182.65697 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 135.35658 186.29246 moveto 139.33335 186.29246 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 129.78911 179.71395 moveto 133.90117 183.77672 134.61428 188.17577 134.08402 192.69785 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 109.42807 168.28813 moveto 124.85792 180.0602 lineto 122.31279 175.73223 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 124.85792 179.88708 moveto 121.1993 179.88708 lineto stroke 0 0 0 setrgbcolor [2.787893 0.92929768] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 170.34134 112.38657 moveto 121.32818 192.57936 lineto stroke 0 0 0 setrgbcolor [2.787893 0.92929768] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 153.74048 118.99904 moveto 142.27616 100.3391 lineto stroke 0 0 0 setrgbcolor [2.787893 0.92929768] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 382.87051 192.69785 moveto 577.64031 105.06435 lineto stroke 0 0 0 setrgbcolor [2.787893 0.92929768] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 382.87051 192.69785 moveto 578.50339 280.2633 lineto stroke 0 0 0 setrgbcolor [2.7878931 0.92929769] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 569.94895 276.82571 moveto 518.43313 192.60536 lineto 569.94895 108.38501 lineto stroke 0 0 0 setrgbcolor [2.7878931 0.92929769] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 588.75438 100.27854 moveto 532.41319 192.60536 lineto 588.98602 285.18008 lineto stroke 0 0 0 setrgbcolor [2.787893 0.92929768] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 153.72636 119.07264 moveto 108.55206 192.92559 lineto 153.58898 266.55395 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 160.17161 76.362183 moveto 160.17161 309.03352 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 356.14664 192.69785 moveto 10.000001 192.36218 lineto stroke 0 0 0 setrgbcolor [2.787893 0.92929768] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 347.23868 192.69785 moveto 164.82578 110.56502 lineto stroke 0 0 0 setrgbcolor [2.787893 0.92929768] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 347.23868 192.69785 moveto 165.36957 275.25173 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 533.12703 186.37238 moveto 538.28714 187.23936 lineto stroke gsave [0.9585686 0 0 1.0432221 0 0] concat gsave [1 0 0 -1 567.15851 174.62842] concat gsave /Sans-ISOLatin1 findfont 37.171906 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (b2) show grestore grestore grestore gsave [0.9585686 0 0 1.0432221 0 0] concat gsave [1 0 0 -1 430.08472 179.73485] concat gsave /Sans-ISOLatin1 findfont 37.171906 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (b1) show grestore grestore grestore 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 543.21371 100.59878 moveto 568.02873 106.83105 lineto 563.41568 103.02244 lineto stroke gsave [0.9585686 0 0 1.0432221 0 0] concat gsave [1 0 0 -1 506.11844 94.505852] concat gsave /Sans-ISOLatin1 findfont 37.171906 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a4) show grestore grestore grestore gsave [0.9585686 0 0 1.0432221 0 0] concat gsave [1 0 0 -1 495.42093 213.49147] concat gsave /Sans-ISOLatin1 findfont 37.171906 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D) show grestore grestore grestore 0 0 0 setrgbcolor [2.787893 0.92929768] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 142.4826 284.96619 moveto 153.63495 266.37133 lineto 160.14195 255.70941 lineto stroke 0 0 0 setrgbcolor [2.787893 0.92929768] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 142.37012 285.21102 moveto 153.61811 266.35936 lineto stroke gsave [0.9585686 0 0 1.0432221 0 0] concat gsave [1 0 0 -1 145.28401 95.296227] concat gsave /Sans-ISOLatin1 findfont 37.171906 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a1) show grestore grestore grestore gsave [0.9585686 0 0 1.0432221 0 0] concat gsave [1 0 0 -1 165.95897 280.89441] concat gsave /Sans-ISOLatin1 findfont 37.171906 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a2) show grestore grestore grestore gsave [0.9585686 0 0 1.0432221 0 0] concat gsave [1 0 0 -1 170.44667 164.56825] concat gsave /Sans-ISOLatin1 findfont 37.171906 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (b2) show grestore grestore grestore 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 168.12514 175.21288 moveto 182.44149 184.21504 lineto 179.89636 180.75268 lineto stroke gsave [0.9585686 0 0 1.0432221 0 0] concat gsave [1 0 0 -1 86.159515 154.28745] concat gsave /Sans-ISOLatin1 findfont 37.171906 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D3) show grestore grestore grestore gsave [0.9585686 0 0 1.0432221 0 0] concat gsave [1 0 0 -1 209.24486 222.31749] concat gsave /Sans-ISOLatin1 findfont 37.171906 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D4) show grestore grestore grestore 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 164.04547 127.39752 moveto 168.02224 126.70505 lineto stroke gsave [0.9585686 0 0 1.0432221 0 0] concat gsave [1 0 0 -1 188.16193 111.03409] concat gsave /Sans-ISOLatin1 findfont 37.171906 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a3) show grestore grestore grestore gsave [0.9585686 0 0 1.0432221 0 0] concat gsave [1 0 0 -1 188.14534 89.224113] concat gsave /Sans-ISOLatin1 findfont 37.171906 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (a4) show grestore grestore grestore 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 181.6546 92.214975 moveto 162.88427 105.71822 lineto 164.47497 100.35155 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 162.73365 105.96545 moveto 167.02856 104.92674 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2.3232441 setlinewidth 0 setlinejoin 0 setlinecap newpath 161.51997 285.74689 moveto 158.37053 285.74689 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 503.35417 212.15292 moveto 528.7049 203.41628 lineto 522.40602 202.92663 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 528.47196 203.29773 moveto 523.77173 207.46749 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 10.354063 40.946946 moveto 164.64286 110.57647 lineto 115 192.36218 lineto 160.33167 266.95047 lineto 165.35714 275.21932 lineto 10.859139 345.25541 lineto 10.354063 345.50795 lineto 10.354063 40.946946 lineto closepath stroke gsave [1 0 0 -1 28.285719 235.79051] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D5) show grestore grestore 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 578.75 280.3979 moveto 526.07143 192.36218 lineto 578.39286 104.68361 lineto 720 42.362183 lineto 720 342.36218 lineto 578.75 280.3979 lineto closepath stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 567.45319 106.60686 moveto 562.65497 107.86955 lineto stroke gsave [1 0 0 -1 463.76157 178.58023] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D1) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 153.54319 266.4635 moveto 153.54319 266.4635 lineto 142.17897 285.90894 lineto 149.97648 282.15765 lineto 163.23531 261.45645 lineto 121.40739 192.46983 lineto 163.73017 123.49045 lineto 152.43869 105.14376 lineto 141.92643 100.29341 lineto 153.79572 119.23377 lineto 108.84393 192.46983 lineto 153.54319 266.4635 lineto closepath stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 579.06994 280.85818 moveto 526.28947 192.21729 lineto 578.56487 104.33401 lineto 589.17147 99.788332 lineto 532.60293 192.21729 lineto 532.60293 192.21729 lineto 588.91893 284.89879 lineto 589.17146 285.15133 lineto 579.06994 280.85818 lineto closepath stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 113.13709 261.1602 moveto 143.09442 242.21984 lineto 135.07634 244.52425 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 142.81032 242.44081 moveto 137.06507 248.31232 lineto stroke gsave [1 0 0 -1 75.508904 281.11072] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D7) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 603.56615 120.749 moveto 575.02934 115.4457 lineto 583.11056 120.24392 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 575.02934 115.4457 moveto 584.87832 113.67793 lineto stroke gsave [1 0 0 -1 602.0509 129.58783] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D8) show grestore grestore 0 0 0 setrgbcolor [2.787893 0.92929768] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 152.07432 105.12533 moveto 205.78281 192.2776 lineto stroke 0 0 0 setrgbcolor [2.787893 0.92929768] 0 setdash 0.92929769 setlinewidth 0 setlinejoin 0 setlinecap newpath 151.82375 280.01983 moveto 205.53224 192.86756 lineto stroke grestore showpage %%EOF %%EndDocument @endspecial 312 2474 a /End PSfrag 312 2474 a 312 896 a /Hide PSfrag 312 896 a -428 953 a FB(PSfrag)h(replacemen)n(ts)p -428 983 741 4 v 312 986 a /Unhide PSfrag 312 986 a 90 1075 a { 90 1075 a Black 107 1054 a Fz(e)90 1075 y FA(D)161 1035 y Fy(out)p Fx(;s)159 1100 y(\032;d;\024)p Black 90 1075 a } 0/Place PSfrag 90 1075 a 82 1201 a { 82 1201 a Black 99 1180 a Fz(e)82 1201 y FA(D)153 1161 y Fy(out)p Fx(;u)151 1226 y(\032;d;\024)p Black 82 1201 a } 1/Place PSfrag 82 1201 a 228 1321 a { 228 1321 a Black FA(\014)275 1333 y Fy(1)p Black 228 1321 a } 2/Place PSfrag 228 1321 a 228 1420 a { 228 1420 a Black FA(\014)275 1432 y Fy(2)p Black 228 1420 a } 3/Place PSfrag 228 1420 a 239 1536 a { 239 1536 a Black FA(ia)p Black 239 1536 a } 4/Place PSfrag 239 1536 a 175 1629 a { 175 1629 a Black Fw(\000)p FA(ia)p Black 175 1629 a } 5/Place PSfrag 175 1629 a 30 1715 a { 30 1715 a Black FA(i)p FB(\()p FA(a)18 b Fw(\000)g FA(d)p FB(\))p Black 30 1715 a } 6/Place PSfrag 30 1715 a -13 1814 a { -13 1814 a Black FA(i)p FB(\()p FA(a)g Fw(\000)g FA(\024")p FB(\))p Black -13 1814 a } 7/Place PSfrag -13 1814 a 90 1903 a { 90 1903 a Black FA(D)161 1873 y Fy(out)p Fx(;s)159 1923 y(\032;\024)p Black 90 1903 a } 8/Place PSfrag 90 1903 a 146 1999 a { 146 1999 a Black FA(D)217 1969 y Fx(s)215 2023 y(\024;d)p Black 146 1999 a } 9/Place PSfrag 146 1999 a 82 2102 a { 82 2102 a Black FA(D)153 2072 y Fy(out)p Fx(;u)151 2122 y(\032;\024)p Black 82 2102 a } 10/Place PSfrag 82 2102 a 146 2199 a { 146 2199 a Black FA(D)217 2168 y Fx(u)215 2222 y(\024;d)p Black 146 2199 a } 11/Place PSfrag 146 2199 a 118 2317 a { 118 2317 a Black FA(I)161 2277 y Fy(out)p Fx(;s)154 2342 y(\024;d)p Black 118 2317 a } 12/Place PSfrag 118 2317 a 110 2438 a { 110 2438 a Black FA(I)153 2398 y Fy(out)p Fx(;u)146 2463 y(\024;d)p Black 110 2438 a } 13/Place PSfrag 110 2438 a Black 947 2740 a FB(Figure)27 b(8:)36 b(The)28 b(domains)f FA(I)1852 2700 y Fy(out)o Fx(;u)1845 2765 y(\024;d)2038 2740 y FB(and)h FA(I)2243 2700 y Fy(out)o Fx(;s)2236 2765 y(\024;d)2421 2740 y FB(de\014ned)g(in)g(\(55\).)p Black Black 195 2954 a(W)-7 b(e)20 b(state)f(the)h(next)f(theorem)g(for)g(the)g (unstable)h(manifold)f(in)g(the)h(domain)f FA(I)2619 2915 y Fy(out)p Fx(;u)2612 2980 y(\024;d)2778 2954 y FB(.)34 b(The)19 b(stable)g(case)g(is)g(analogous.)p Black 71 3121 a Fp(Theorem)29 b(4.7.)p Black 39 w Fs(L)l(et)f FA(d)848 3133 y Fy(0)885 3121 y Fs(,)h FA(\024)987 3133 y Fy(1)1024 3121 y Fs(,)g FA(\032)1121 3133 y Fy(4)1186 3121 y Fs(b)l(e)g(the)f(c)l(onstants)f(given)i(in)f(The)l(or)l(em)i (4.6,)g FA(\024)2657 3133 y Fy(2)2717 3121 y FA(>)23 b(\024)2853 3133 y Fy(1)2890 3121 y Fs(,)29 b FA(d)2987 3133 y Fy(1)3047 3121 y FA(<)23 b(d)3178 3133 y Fy(0)3243 3121 y Fs(and)29 b FA(")3442 3133 y Fy(0)3502 3121 y FA(>)23 b FB(0)k Fs(smal)t(l)71 3220 y(enough.)39 b(Then,)31 b(for)f FA(")23 b Fw(2)h FB(\(0)p FA(;)14 b(")1052 3232 y Fy(0)1088 3220 y FB(\))30 b Fs(and)h FA(\026)23 b Fw(2)g FA(B)t FB(\()p FA(\026)1612 3232 y Fy(0)1650 3220 y FB(\))p Fs(,)30 b(and)g(incr)l(e)l(asing)h FA(\024)2338 3232 y Fy(1)2405 3220 y Fs(if)f(ne)l(c)l(essary,)p Black 195 3386 a Fw(\017)p Black 41 w Fs(Ther)l(e)42 b(exists)f(a)h(r)l(e)l (al-analytic)g(function)f Fw(V)1730 3356 y Fx(u)1817 3386 y FB(:)j FA(I)1927 3346 y Fy(out)o Fx(;u)1920 3411 y(\024)1959 3419 y Fu(2)1991 3411 y Fx(;d)2046 3419 y Fu(1)2112 3386 y Fw(\002)26 b Ft(T)2258 3398 y Fx(\033)2347 3386 y Fw(!)44 b Ft(C)d Fs(which)h(satis\014es)48 b FB(\(56\))p Fs(.)73 b(Mor)l(e)l(over,)45 b(if)278 3506 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)g FA(I)618 3467 y Fy(out)o Fx(;u)611 3532 y(\024)650 3540 y Fu(2)682 3532 y Fx(;d)737 3540 y Fu(1)795 3506 y Fw(\002)18 b Ft(T)933 3518 y Fx(\033)978 3506 y Fs(,)30 b(then)g FA(u)18 b FB(+)g Fw(V)1425 3476 y Fx(u)1468 3506 y FB(\()p FA(u;)c(\034)9 b FB(\))24 b Fw(2)f FA(I)1807 3467 y Fy(out)p Fx(;u)1800 3532 y(\024)1839 3540 y Fu(1)1872 3532 y Fx(;d)1927 3540 y Fu(0)1995 3506 y Fs(and)1660 3706 y Fw(j)q(V)1742 3672 y Fx(u)1785 3706 y FB(\()p FA(u;)14 b(\034)9 b FB(\))p Fw(j)23 b(\024)g FA(b)2149 3718 y Fy(6)2186 3706 y Fw(j)p FA(\026)p Fw(j)p FA(")2321 3672 y Fx(\021)r Fy(+1)2445 3706 y FA(:)278 3889 y Fs(with)31 b FA(b)495 3901 y Fy(6)561 3889 y Fs(a)f(c)l(onstant)f(indep)l(endent)i(of)f FA(\026)g Fs(and)g FA(")p Fs(.)p Black 195 4055 a Fw(\017)p Black 41 w Fs(Ther)l(e)h(exists)e(a)i(gener)l(ating)f(function)f FA(T)1606 4024 y Fx(u)1672 4055 y FB(:)23 b FA(I)1761 4015 y Fy(out)p Fx(;u)1754 4080 y(\024)1793 4088 y Fu(2)1826 4080 y Fx(;d)1881 4088 y Fu(1)1938 4055 y Fw(\002)18 b Ft(T)2076 4067 y Fx(\033)2144 4055 y Fw(!)24 b Ft(C)29 b Fs(such)h(that)1359 4237 y FA(@)1403 4249 y Fx(u)1447 4237 y FA(T)1508 4203 y Fx(u)1550 4237 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)e FA(p)1897 4249 y Fy(0)1934 4237 y FB(\()p FA(u)p FB(\))p FA(P)2111 4203 y Fx(u)2155 4237 y FB(\()p FA(u)c FB(+)g Fw(V)2394 4203 y Fx(u)2437 4237 y FB(\()p FA(u;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))p FA(;)278 4420 y Fs(wher)l(e)30 b FA(P)577 4390 y Fx(u)649 4420 y Fs(is)f(the)g(function)g(obtaine)l(d)h(in)f(The)l(or) l(em)h(4.6,)h(and)e(satis\014es)g(e)l(quation)36 b FB(\(41\))p Fs(.)i(Then,)30 b(we)f(have)h(that)278 4520 y FB(\()p FA(q)s(;)14 b(p)p FB(\))24 b(=)e(\()p FA(q)641 4532 y Fy(0)679 4520 y FA(;)14 b(p)758 4532 y Fy(0)795 4520 y FB(\()p FA(u)p FB(\))907 4489 y Fv(\000)p Fy(1)996 4520 y FA(@)1040 4532 y Fx(u)1084 4520 y FA(T)1145 4489 y Fx(u)1187 4520 y FB(\()p FA(u;)g(\034)9 b FB(\)\))29 b Fs(is)g(a)f(p)l(ar)l(ametrization)h(of)g(the)g(unstable)e(invariant)i (manifold)h(of)f(the)g(form)278 4619 y FB(\(42\))p Fs(.)39 b(Mor)l(e)l(over,)31 b(ther)l(e)f(exists)f(a)i(c)l(onstant)e FA(b)1754 4631 y Fy(7)1813 4619 y FA(>)23 b FB(0)29 b Fs(such)h(that,)g(for)h FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)f FA(I)2828 4579 y Fy(out)p Fx(;u)2821 4644 y(\024)2860 4652 y Fu(2)2893 4644 y Fx(;d)2948 4652 y Fu(1)3005 4619 y Fw(\002)18 b Ft(T)3143 4631 y Fx(\033)3188 4619 y Fs(,)1422 4819 y Fw(j)p FA(@)1489 4831 y Fx(u)1533 4819 y FA(T)1594 4784 y Fx(u)1636 4819 y FB(\()p FA(u;)c(\034)9 b FB(\))19 b Fw(\000)f FA(@)1976 4831 y Fx(u)2020 4819 y FA(T)2069 4831 y Fy(0)2106 4819 y FB(\()p FA(u)p FB(\))p Fw(j)23 b(\024)f FA(b)2387 4831 y Fy(7)2424 4819 y Fw(j)p FA(\026)p Fw(j)p FA(")2559 4784 y Fx(\021)r Fy(+1)2684 4819 y FA(:)195 5034 y FB(This)28 b(theorem)f(is)h(pro)n(v)n(ed)e(in)h(Section)h (7.2.4.)195 5134 y(The)21 b(\014nal)f(step)h(is)f(to)h(extend)f(the)h (just)g(obtained)g(parameterizations)d(of)i(the)h(form)f(\(42\))g(to)h (the)g(whole)f Fs(b)l(o)l(omer)l(ang)71 5234 y(domains)37 b FA(D)479 5204 y Fx(u)477 5257 y(\024;d)604 5234 y FB(and)29 b FA(D)838 5204 y Fx(s)836 5257 y(\024;d)963 5234 y FB(de\014ned)h(in)f (\(26\))h(\(see)f(also)f(Figure)h(2\).)42 b(In)30 b(particular)e(the)i (whole)f Fs(b)l(o)l(omer)l(ang)j(domains)71 5333 y FB(con)n(tain)f(p)r (oin)n(ts)g(up)h(to)f(a)h(distance)f FA(\024")g FB(of)g(the)h (singularities)f Fw(\006)p FA(ia)p FB(.)47 b(The)32 b(next)g(theorem)f (giv)n(es)f(the)i(results)f(for)g(the)71 5433 y(unstable)c(manifold.)37 b(The)28 b(stable)f(one)g(satis\014es)g(analogous)f(ones.)p Black 1919 5753 a(32)p Black eop end %%Page: 33 33 TeXDict begin 33 32 bop Black Black Black 71 272 a Fp(Theorem)33 b(4.8.)p Black 41 w Fs(L)l(et)e FA(\024)862 284 y Fy(2)931 272 y Fs(and)g FA(d)1136 284 y Fy(1)1205 272 y Fs(b)l(e)h(the)f(c)l (onstants)f(given)i(in)g(The)l(or)l(em)g(4.7,)i FA(d)2693 284 y Fy(2)2756 272 y FA(<)26 b(d)2890 284 y Fy(1)2927 272 y Fs(,)32 b FA(\024)3032 284 y Fy(3)3095 272 y FA(>)26 b(\024)3234 284 y Fy(2)3302 272 y Fs(big)32 b(enough)g(and)71 372 y FA(")110 384 y Fy(0)177 372 y FA(>)d FB(0)k Fs(smal)t(l)h (enough.)50 b(Then,)35 b(for)g FA(\026)29 b Fw(2)i FA(B)t FB(\()p FA(\026)1594 384 y Fy(0)1631 372 y FB(\))j Fs(and)g FA(")29 b Fw(2)h FB(\(0)p FA(;)14 b(")2165 384 y Fy(0)2202 372 y FB(\))p Fs(,)35 b(the)e(function)h FA(T)2827 342 y Fx(u)2869 372 y FB(\()p FA(u;)14 b(\034)9 b FB(\))34 b Fs(obtaine)l(d)h(in)e(The)l(or)l(em)71 471 y(4.7)e(c)l(an)f(b)l(e)f (analytic)l(al)t(ly)k(extende)l(d)c(to)h(the)g(domain)h FA(D)1848 441 y Fx(u)1846 495 y(\024)1885 503 y Fu(3)1917 495 y Fx(;d)1972 503 y Fu(2)2026 471 y Fw(\002)18 b Ft(T)2164 483 y Fx(\033)2209 471 y Fs(.)195 571 y(Mor)l(e)l(over,)27 b(ther)l(e)c(exists)g(a)g(r)l(e)l(al)h(c)l(onstant)e FA(b)1579 583 y Fy(8)1639 571 y FA(>)h FB(0)g Fs(indep)l(endent)h(of)g FA(")f Fs(and)g FA(\026)p Fs(,)i(such)f(that)f(for)h FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)f FA(D)3492 541 y Fx(u)3490 594 y(\024)3529 602 y Fu(3)3561 594 y Fx(;d)3616 602 y Fu(2)3656 571 y Fw(\002)t Ft(T)3780 583 y Fx(\033)3825 571 y Fs(,)1260 821 y Fw(j)p FA(@)1327 833 y Fx(u)1370 821 y FA(T)1431 787 y Fx(u)1474 821 y FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b Fw(\000)f FA(@)1814 833 y Fx(u)1857 821 y FA(T)1906 833 y Fy(0)1943 821 y FB(\()p FA(u)p FB(\))p Fw(j)23 b(\024)2248 765 y FA(b)2284 777 y Fy(8)2320 765 y Fw(j)p FA(\026)p Fw(j)p FA(")2455 735 y Fx(\021)r Fy(+1)p 2199 802 430 4 v 2199 894 a Fw(j)p FA(u)2270 870 y Fy(2)2325 894 y FB(+)18 b FA(a)2452 870 y Fy(2)2489 894 y Fw(j)2512 852 y Fx(`)p Fy(+1)2638 821 y FA(;)71 1068 y Fs(wher)l(e)30 b FA(T)354 1080 y Fy(0)421 1068 y Fs(is)g(the)g(unp)l(erturb)l(e)l(d)f(sep)l(ar)l(atrix)h(given)g (in)36 b FB(\(51\))p Fs(.)195 1234 y FB(The)28 b(pro)r(of)f(of)h(this)g (theorem)f(is)g(giv)n(en)g(in)h(Section)f(7.2.5.)p Black 71 1400 a Fp(Remark)36 b(4.9.)p Black 42 w Fs(L)l(et)d(us)g(p)l(oint)g (out)g(that)g(these)h(domains)h(satisfy)f FA(D)2305 1370 y Fx(u)2303 1424 y(\024;d)2429 1400 y Fw(\032)29 b FA(D)2594 1370 y Fy(out)p Fx(;u)2592 1421 y(\032;\024)2786 1400 y Fs(and)34 b FA(D)3022 1370 y Fx(s)3020 1424 y(\024;d)3147 1400 y Fw(\032)29 b FA(D)3312 1370 y Fy(out)o Fx(;s)3310 1421 y(\032;\024)3496 1400 y Fs(if)34 b FA(\032)f Fs(is)h(big)71 1500 y(enough.)45 b(Ther)l(efor)l(e,)35 b(in)c(the)h(c)l(ase)h(that)e FA(p)1434 1512 y Fy(0)1471 1500 y FB(\()p FA(u)p FB(\))h Fs(do)l(es)h(not)e(vanish,)j(The)l(or)l(em)f(4.4)g(ensur)l(es)e(that)g (the)h(functions)g FA(T)3757 1470 y Fx(u;s)71 1600 y Fs(ar)l(e)e(alr)l(e)l(ady)h(de\014ne)l(d)f(in)g FA(D)948 1569 y Fx(u)946 1623 y(\024;d)1073 1600 y Fs(and)g FA(D)1305 1569 y Fx(s)1303 1623 y(\024;d)1430 1600 y Fs(r)l(esp)l(e)l(ctively.) 195 1712 y(L)l(et)g(us)f(observe)i(that,)f(if)h FA(")e Fs(is)h(smal)t(l)h(enough,)f FA(D)1773 1682 y Fy(in)p Fx(;)p Fv(\006)p Fx(;s)1771 1732 y(\024;c)1979 1712 y Fw(\032)22 b FA(D)2137 1682 y Fx(s)2135 1735 y(\024;d)2262 1712 y Fs(and)30 b FA(D)2494 1682 y Fy(in)p Fx(;)p Fv(\006)p Fx(;u)2492 1732 y(\024;c)2708 1712 y Fw(\032)22 b FA(D)2866 1682 y Fx(u)2864 1735 y(\024;d)2961 1712 y Fs(.)195 1878 y FB(After)37 b(Theorem)f(4.4)f(and)h(4.8)g(there)g(is)g(no)g (di\013erence)g(b)r(et)n(w)n(een)g(the)h(case)f FA(p)2761 1890 y Fy(0)2798 1878 y FB(\()p FA(u)p FB(\))h Fw(6)p FB(=)g(0,)h(when)f(the)g(in)n(v)-5 b(arian)n(t)71 1978 y(manifolds)33 b(can)f(b)r(e)i(written)f(as)g(graphs)e(globally)-7 b(,)34 b(and)e(the)i(general)e(case)g(when)h FA(p)2793 1990 y Fy(0)2863 1978 y FB(can)g(v)-5 b(anish:)48 b(w)n(e)32 b(ha)n(v)n(e)g(found)71 2077 y Fs(b)l(o)l(omer)l(ang)f(domains)36 b FB(whic)n(h)28 b(in)n(tersect)g(the)g(real)f(line)i(and)f(whic)n(h)g (reac)n(h)e(neigh)n(b)r(orho)r(o)r(ds)h(of)h(size)g FA(\024")g FB(of)g(the)g(singu-)71 2177 y(larities)d(where)g(b)r(oth)h(manifolds)g (can)f(b)r(e)h(written)g(as)f(graphs.)35 b(This)25 b(will)h(b)r(e)h (the)f(starting)e(p)r(oin)n(t)i(in)g(our)f(strategy)g(to)71 2276 y(measure)h(the)i(distance)g(b)r(et)n(w)n(een)f(the)h(in)n(v)-5 b(arian)n(t)27 b(manifolds.)71 2509 y Fq(4.6)112 b(The)38 b(asymptotic)g(\014rst)f(order)g(of)h Fh(@)1840 2524 y Fg(u)1886 2509 y Fh(T)1957 2473 y Fg(u;s)2091 2509 y Fq(close)g(to)f(the)g(singularities)i Ff(\006)p Fh(ia)71 2662 y FB(Theorems)25 b(4.4)h(and)h(4.8)f(are)f(v)-5 b(alid)27 b(for)f FA(\021)h Fw(\025)22 b FB(max)p Fw(f)p FB(0)p FA(;)14 b(`)h Fw(\000)h FB(2)p FA(r)r Fw(g)p FB(.)37 b(W)-7 b(e)27 b(notice)f(that)h(when)g FA(`)c Fw(\024)f FB(2)p FA(r)29 b FB(the)e(result)g(is)f(true)h(for)71 2762 y FA(\021)32 b Fw(\025)d FB(0,)j(but)g(only)e(if)i FA(`)d(<)f FB(2)p FA(r)34 b FB(Theorems)c(4.4)h(and)g(4.8)f(giv)n(e)g (a)h Fs(classic)l(al)42 b FB(p)r(erturbativ)n(e)30 b(result)h(with)h (resp)r(ect)f(to)g(the)71 2861 y(singular)d(parameter)g FA(")p FB(,)i(in)f(the)h(sense)f(that)h(the)f(main)h(term)f(of)g FA(@)2212 2873 y Fx(u)2256 2861 y FA(T)2317 2831 y Fx(u;s)2440 2861 y FB(is)g(giv)n(en)g(b)n(y)g(the)g(unp)r(erturb)r(ed)h(separatrix) 71 2961 y FA(@)115 2973 y Fx(u)158 2961 y FA(T)207 2973 y Fy(0)272 2961 y FB(in)e(the)g(whole)g Fs(outer)37 b FB(domains.)g(This)28 b(fact)g(is)f(not)h(true)g(an)n(ymore)e(in)i(the) h(case)e FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b Fw(\025)d FB(0)28 b(and)f(then)i(w)n(e)e(will)71 3061 y(ha)n(v)n(e)k(to)h(lo)r (ok)f(for)h(di\013eren)n(t)g(appro)n(ximations)e(of)i(the)h(in)n(v)-5 b(arian)n(t)31 b(manifolds)h(close)f(to)h(the)h(singularities)e FA(u)f FB(=)g Fw(\006)p FA(ia)p FB(,)71 3160 y(b)n(y)i(using)f (suitable)h(solutions)f(of)h(the)h(so-called)d Fs(inner)41 b FB(equations.)50 b(Consequen)n(tly)-7 b(,)32 b(the)g(case)g FA(`)e(<)g FB(2)p FA(r)k FB(is)e(easier)f(to)71 3260 y(deal)f(with,)j(b)r(ecause)d(it)h(is)g(a)f(regular)f(case)h(and)h (there)f(is)h(no)g(need)g(of)f(using)h Fs(inner)40 b FB(equations)30 b(to)g(obtain)h(a)f(b)r(etter)71 3359 y(appro)n(ximation)c(of)h FA(@)763 3371 y Fx(u)807 3359 y FA(T)868 3329 y Fx(u;s)989 3359 y FB(near)g(the)g(singularities)g Fw(\006)p FA(ia)g FB(of)g FA(T)2087 3371 y Fy(0)2124 3359 y FB(.)195 3459 y(W)-7 b(e)28 b(separate)f(b)r(oth)h(cases)e FA(`)d(<)f FB(2)p FA(r)30 b FB(and)e FA(`)22 b Fw(\025)h FB(2)p FA(r)30 b FB(in)e(the)g(corresp)r(onding)d(sections)i(b)r(elo)n (w.)71 3675 y Fp(4.6.1)94 b(The)32 b(asymptotic)f(\014rst)h(order)g(of) g FA(@)1654 3687 y Fx(u)1698 3675 y FA(T)1759 3645 y Fx(u;s)1884 3675 y Fp(for)g(the)g(case)g FA(`)23 b(<)f FB(2)p FA(r)71 3828 y FB(In)28 b(this)g(section)f(w)n(e)g(will)h (assume)f(that)h FA(`)22 b(<)h FB(2)p FA(r)30 b FB(and)d(henceforth)h (w)n(e)f(are)f(dealing)h(with)i(v)-5 b(alues)27 b(of)g FA(\021)g Fw(\025)22 b FB(0.)195 3928 y(T)-7 b(o)30 b(obtain)f(the)h (main)g(term)f(of)h FA(@)1276 3940 y Fx(u)1320 3928 y FA(T)1381 3898 y Fx(u;s)1494 3928 y Fw(\000)19 b FA(@)1622 3940 y Fx(u)1666 3928 y FA(T)1715 3940 y Fy(0)1781 3928 y FB(w)n(e)30 b(just)g(need)g(to)f(use)h(classical)e(p)r(erturbation)h (theory)g(ev)n(en)g(in)71 4027 y(the)i Fs(inner)h(domains)38 b FA(D)850 3997 y Fy(in)p Fx(;)p Fv(\006)p Fx(;)p Fv(\003)848 4048 y Fx(\024;c)1035 4027 y FB(,)31 b Fw(\003)c FB(=)g FA(u;)14 b(s)p FB(,)31 b(de\014ned)f(in)h(\(30\))f(\(see)g(Figure)g (5\).)44 b(Let)31 b(us)f(observ)n(e)e(that,)k(if)e FA(u)d Fw(2)h FA(D)3642 3997 y Fy(in)p Fx(;)p Fv(\006)p Fx(;)p Fv(\003)3640 4048 y Fx(\024;c)3827 4027 y FB(,)71 4127 y Fw(\003)22 b FB(=)h FA(u;)14 b(s)p FB(,)27 b(then)h Fw(O)r FB(\()p FA(\024")p FB(\))c Fw(\024)e(j)p FA(u)d Fw(\007)f FA(ia)p Fw(j)k(\024)h(O)r FB(\()p FA(")1434 4097 y Fx(\015)1477 4127 y FB(\).)195 4227 y(The)29 b(next)f(prop)r (osition)f(giv)n(es)g(the)i(\014rst)f(order)e(asymptotic)i(terms)g(of)g FA(@)2527 4239 y Fx(u)2570 4227 y FA(T)2631 4196 y Fx(u;s)2744 4227 y Fw(\000)18 b FA(@)2871 4239 y Fx(u)2915 4227 y FA(T)2964 4239 y Fy(0)3029 4227 y FB(close)27 b(to)h FA(u)c FB(=)f FA(ia)p FB(,)28 b(that)h(is)71 4326 y(in)f FA(D)239 4296 y Fy(in)o Fx(;)p Fy(+)p Fx(;)p Fv(\003)237 4347 y Fx(\024;c)423 4326 y FB(,)g Fw(\003)22 b FB(=)h FA(u;)14 b(s)p FB(.)36 b(The)28 b(study)f(close)g(to)h FA(u)22 b FB(=)h Fw(\000)p FA(ia)k FB(can)g(b)r(e)h(done)f(analogously) -7 b(.)p Black 71 4506 a Fp(Prop)s(osition)29 b(4.10.)p Black 39 w Fs(L)l(et)g(us)f(assume)h FA(`)17 b Fw(\000)g FB(2)p FA(r)25 b(<)e FB(0)28 b Fs(and)i FB(0)22 b FA(<)h(\015)k(<)c FB(min)p Fw(f)p FB(1)p FA(;)2501 4473 y Fx(`)p Fy(+1)p 2498 4487 117 4 v 2498 4534 a Fx(r)r Fy(+1)2625 4506 y Fw(g)29 b Fs(wher)l(e)g FA(\015)34 b Fs(is)29 b(the)h(c)l(onstant)e (involve)l(d)71 4605 y(in)f(the)h(de\014nition)g(of)g(the)g(inner)f (domains)i(in)34 b FB(\(30\))o Fs(.)k(L)l(et)27 b(us)g(c)l(onsider)h (the)g(c)l(onstant)e FA(\024)2854 4617 y Fy(3)2919 4605 y Fs(given)i(by)g(The)l(or)l(em)g(4.8)h(and)71 4705 y FA(c)107 4717 y Fy(1)167 4705 y FA(>)23 b FB(0)29 b Fs(and)h(let)g(us)f (de\014ne)h(the)g(c)l(onstant)750 4888 y FA(\027)796 4853 y Fv(\003)857 4888 y FB(=)23 b(min)14 b Fw(f)o FA(\027)1184 4853 y Fv(\003)1179 4908 y Fy(1)1223 4888 y FA(;)g(\027)1306 4853 y Fv(\003)1301 4908 y Fy(2)1344 4888 y FA(;)g FB(1)k Fw(\000)g FB(max)p Fw(f)p FB(0)p FA(;)c(`)j Fw(\000)h FB(2)p FA(r)i FB(+)e(1)p Fw(g)p FA(;)c(r)n(;)g(`;)g(`)j FB(+)h(1)g Fw(\000)g FB(\()p FA(r)j FB(+)d(1\))p FA(\015)5 b Fw(g)22 b FA(>)h FB(0)p FA(;)71 5070 y Fs(wher)l(e)1447 5231 y FA(\027)1493 5197 y Fv(\003)1488 5252 y Fy(1)1554 5231 y FB(=)g(min)p Fw(f)p FB(\(2)p FA(r)e Fw(\000)d FA(`)p FB(\))p FA(\015)5 b(;)14 b FB(1)p Fw(g)1447 5415 y FA(\027)1493 5381 y Fv(\003)1488 5435 y Fy(2)1554 5415 y FB(=)1642 5298 y Fz(\032)1746 5364 y FA(`)p FB(\(1)k Fw(\000)g FA(\015)5 b FB(\))83 b Fs(if)30 b FA(`)23 b(>)f FB(0)1746 5464 y(1)c Fw(\000)g FA(\015)187 b Fs(if)30 b FA(`)23 b FB(=)f(0)2451 5415 y FA(:)p Black 1919 5753 a FB(33)p Black eop end %%Page: 34 34 TeXDict begin 34 33 bop Black Black 71 272 a Fs(L)l(et)29 b(us)g(also)i(de\014ne)f(the)g(functions)902 493 y Fw(T)968 459 y Fx(u)947 514 y Fy(0)1011 493 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)18 b Fw(\000)g FA(\026")1484 459 y Fx(\021)1538 380 y Fz(Z)1621 401 y Fy(0)1584 569 y Fv(\0001)1720 493 y FA(H)1789 505 y Fy(1)1827 493 y FB(\()p FA(q)1896 505 y Fy(0)1933 493 y FB(\()p FA(u)g FB(+)h FA(t)p FB(\))p FA(;)14 b(p)2256 505 y Fy(0)2293 493 y FB(\()p FA(u)k FB(+)g FA(t)p FB(\))p FA(;)c(\034)28 b FB(+)18 b FA(")2759 459 y Fv(\000)p Fy(1)2848 493 y FA(t)p FB(\))c FA(dt)910 739 y Fw(T)976 705 y Fx(s)955 759 y Fy(0)1011 739 y FB(\()p FA(u;)g(\034)9 b FB(\))24 b(=)18 b Fw(\000)g FA(\026")1484 705 y Fx(\021)1538 626 y Fz(Z)1621 646 y Fy(0)1584 815 y(+)p Fv(1)1719 739 y FA(H)1788 751 y Fy(1)1826 739 y FB(\()p FA(q)1895 751 y Fy(0)1932 739 y FB(\()p FA(u)h FB(+)f FA(t)p FB(\))p FA(;)c(p)2255 751 y Fy(0)2292 739 y FB(\()p FA(u)k FB(+)g FA(t)p FB(\))p FA(;)c(\034)28 b FB(+)18 b FA(")2758 705 y Fv(\000)p Fy(1)2847 739 y FA(t)p FB(\))c FA(dt;)3703 614 y FB(\(57\))71 959 y Fs(wher)l(e)32 b FA(H)376 971 y Fy(1)444 959 y Fs(is)g(the)g(function)f(de\014ne)l(d)g(in)38 b FB(\(8\))32 b Fs(and)40 b FB(\(9\))31 b Fs(and)h FB(\()p FA(q)2072 971 y Fy(0)2110 959 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2301 971 y Fy(0)2338 959 y FB(\()p FA(u)p FB(\)\))31 b Fs(is)h(the)g(p)l(ar)l(ameterization)g(of)h(the)e(unp)l (er-)71 1058 y(turb)l(e)l(d)d(sep)l(ar)l(atrix)h(given)g(in)f(Hyp)l (othesis)i Fo(HP2)p Fs(.)39 b(Then,ther)l(e)29 b(exists)f FA(")2360 1070 y Fy(0)2420 1058 y FA(>)23 b FB(0)28 b Fs(and)h(a)g(c)l(onstant)e FA(b)3174 1070 y Fy(9)3234 1058 y FA(>)c FB(0)28 b Fs(such)g(that)g(for)71 1158 y(any)i FA(")23 b Fw(2)g FB(\(0)p FA(;)14 b(")520 1170 y Fy(0)557 1158 y FB(\))30 b Fs(and)g FA(\026)23 b Fw(2)h FA(B)t FB(\()p FA(\026)1081 1170 y Fy(0)1118 1158 y FB(\))30 b Fs(the)g(fol)t(lowing)i(b)l(ounds)e(ar)l(e)g(satis\014e)l(d.)p Black 195 1311 a Fw(\017)p Black 41 w Fs(If)h FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b Fw(2)h FA(D)733 1281 y Fy(in)o Fx(;)p Fy(+)p Fx(;u)731 1332 y(\024)770 1340 y Fu(3)802 1332 y Fx(;c)852 1340 y Fu(1)941 1311 y Fw(\002)18 b Ft(T)1079 1323 y Fx(\033)1124 1311 y Fs(,)1116 1497 y Fw(j)p FA(@)1183 1509 y Fx(u)1227 1497 y FA(T)1288 1462 y Fx(u)1330 1497 y FB(\()p FA(u;)c(\034)9 b FB(\))19 b Fw(\000)f FA(@)1670 1509 y Fx(u)1714 1497 y FA(T)1763 1509 y Fy(0)1800 1497 y FB(\()p FA(u)p FB(\))g Fw(\000)g FA(@)2057 1509 y Fx(u)2101 1497 y Fw(T)2167 1462 y Fx(u)2146 1517 y Fy(0)2210 1497 y FB(\()p FA(u;)c(\034)9 b FB(\))p Fw(j)24 b(\024)f FA(b)2575 1509 y Fy(9)2612 1497 y Fw(j)p FA(\026)p Fw(j)p FA(")2747 1462 y Fx(\021)r Fv(\000)p Fx(`)p Fy(+)p Fx(\027)2951 1437 y Fl(\003)2990 1497 y FA(:)p Black 195 1693 a Fw(\017)p Black 41 w Fs(If)31 b FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b Fw(2)h FA(D)733 1663 y Fy(in)o Fx(;)p Fy(+)p Fx(;s)731 1714 y(\024)770 1722 y Fu(3)802 1714 y Fx(;c)852 1722 y Fu(1)933 1693 y Fw(\002)18 b Ft(T)1071 1705 y Fx(\033)1116 1693 y Fs(,)1128 1879 y Fw(j)p FA(@)1195 1891 y Fx(u)1239 1879 y FA(T)1300 1844 y Fx(s)1334 1879 y FB(\()p FA(u;)c(\034)9 b FB(\))19 b Fw(\000)f FA(@)1674 1891 y Fx(s)1710 1879 y FA(T)1759 1891 y Fy(0)1796 1879 y FB(\()p FA(u)p FB(\))g Fw(\000)g FA(@)2053 1891 y Fx(u)2097 1879 y Fw(T)2163 1844 y Fx(s)2142 1899 y Fy(0)2199 1879 y FB(\()p FA(u;)c(\034)9 b FB(\))p Fw(j)23 b(\024)g FA(b)2563 1891 y Fy(9)2600 1879 y Fw(j)p FA(\026)p Fw(j)p FA(")2735 1844 y Fx(\021)r Fv(\000)p Fx(`)p Fy(+)p Fx(\027)2939 1819 y Fl(\003)2978 1879 y FA(:)195 2075 y FB(This)28 b(prop)r(osition)f(is)g(pro)n(v)n(ed)f(in)i (Section)f(7.1.)71 2288 y Fp(4.6.2)94 b(The)32 b(\014rst)g(asymptotic)f (order)h(of)g FA(@)1654 2300 y Fx(u)1698 2288 y FA(T)1759 2258 y Fx(u;s)1884 2288 y Fp(for)g(the)g(case)g FA(`)23 b Fw(\025)f FB(2)p FA(r)71 2441 y FB(Theorems)e(4.4)g(and)h(4.8)g(giv)n (e)f(the)h(existence)g(of)g(parameterizations)e(of)i(the)h(in)n(v)-5 b(arian)n(t)20 b(manifolds)h(of)g(the)h(form)e(\(42\))h(in)71 2541 y FA(D)142 2511 y Fx(s)140 2564 y(\024;d)234 2572 y Fu(2)290 2541 y FB(and)f FA(D)515 2511 y Fx(u)513 2564 y(\024;d)607 2572 y Fu(2)663 2541 y FB(for)f FA(")i FB(small)e(enough)h (and)g FA(\024)g FB(big)h(enough.)33 b(Nev)n(ertheless,)21 b(when)g FA(\021)26 b FB(=)d FA(`)t Fw(\000)t FB(2)p FA(r)e FB(the)g(parameterizations)71 2640 y(of)26 b(the)g(p)r(erturb)r (ed)g(in)n(v)-5 b(arian)n(t)24 b(manifolds)i(are)e(not)i(w)n(ell)g (appro)n(ximated)e(b)n(y)h(the)h(unp)r(erturb)r(ed)h(separatrix)c(when) j FA(u)g FB(is)71 2740 y(at)k(a)f(distance)h(of)g(order)e Fw(O)r FB(\()p FA(")p FB(\))j(of)f(the)g(singularities)f FA(u)d FB(=)h Fw(\006)p FA(ia)p FB(.)43 b(F)-7 b(or)29 b(this)i(reason,)e(to)g(obtain)h(the)g(\014rst)g(asymptotic)71 2840 y(order)22 b(of)h(the)g(di\013erence)g(b)r(et)n(w)n(een)h(the)f (manifolds,)h(w)n(e)f(need)g(to)g(lo)r(ok)g(for)f(b)r(etter)i(appro)n (ximations)d FA(T)3318 2810 y Fx(u;s)3435 2840 y FB(in)i(the)h(inner)71 2939 y(domains)30 b(de\014ned)i(in)f(\(30\).)48 b(W)-7 b(e)32 b(obtain)e(them)i(through)f(a)g(singular)e(limit.)49 b(Since)31 b(w)n(e)g(are)f(dealing)h(with)h(the)f(case)71 3039 y FA(\021)26 b Fw(\025)d FA(`)18 b Fw(\000)g FB(2)p FA(r)r FB(,)28 b(the)g(\014rst)f(step)h(is)f(to)h(de\014ne)g(a)f(new)h (parameter)1686 3205 y(^)-48 b FA(\026)23 b FB(=)f FA(\026")1929 3171 y Fx(\021)r Fv(\000)p Fy(\()p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(\))2219 3205 y FA(:)1461 b FB(\(58\))71 3382 y(Then,)28 b(the)g(Hamiltonian)950 3361 y Fz(b)931 3382 y FA(H)34 b FB(reads)942 3576 y Fz(b)923 3597 y FA(H)7 b FB(\()p FA(q)s(;)14 b(p;)g(\034)9 b FB(\))24 b(=)1362 3541 y FA(p)1404 3511 y Fy(2)p 1362 3578 80 4 v 1381 3654 a FB(2)1470 3597 y(+)18 b FA(V)32 b FB(\()q FA(q)21 b FB(+)d FA(x)1854 3609 y Fx(p)1893 3597 y FB(\()p FA(\034)9 b FB(\)\))20 b Fw(\000)e FA(V)33 b FB(\()p FA(x)2297 3609 y Fx(p)2336 3597 y FB(\()p FA(\034)9 b FB(\)\))20 b Fw(\000)e FA(V)2647 3563 y Fv(0)2684 3597 y FB(\()p FA(x)2763 3609 y Fx(p)2802 3597 y FB(\()p FA(\034)9 b FB(\)\))15 b FA(q)1371 3768 y FB(+)24 b(^)-48 b FA(\026")1543 3734 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)1711 3747 y Fz(b)1692 3768 y FA(H)1761 3780 y Fy(1)1798 3768 y FB(\()p FA(q)s(;)14 b(p;)g(\034)9 b FB(\))3703 3657 y(\(59\))71 3954 y(and,)26 b(from)468 3933 y Fz(b)448 3954 y FA(H)7 b FB(,)26 b(one)g(can)f (de\014ne)h(the)g(Hamiltonian)p 1729 3887 76 4 v 26 w FA(H)32 b FB(in)26 b(\(40\))g(using)f(again)g(the)h(c)n(hange)f(\(39\)) o(.)37 b(On)25 b(the)h(other)f(hand,)71 4054 y(from)j(Theorems)g(4.4)g (and)g(4.8,)h(one)f(can)g(obtain)h(b)r(ounds)g(for)f(the)h (parameterizations)d(of)j(the)g(in)n(v)-5 b(arian)n(t)28 b(manifolds)71 4153 y(in)j(terms)f(of)38 b(^)-49 b FA(\026)31 b FB(and)f FA(")p FB(.)47 b(W)-7 b(e)31 b(state)f(them)h(for)g(the)g (unstable)g(manifold.)46 b(The)31 b(stable)f(manifold)h(satis\014es)f (analogous)71 4253 y(b)r(ounds.)p Black 71 4406 a Fp(Corollary)g(4.11.) p Black 38 w Fs(L)l(et)d(us)g(c)l(onsider)h(the)g(c)l(onstants)e FA(\024)1848 4418 y Fy(3)1913 4406 y Fs(and)i FA(d)2115 4418 y Fy(2)2180 4406 y Fs(de\014ne)l(d)f(in)h(The)l(or)l(em)h(4.8.)39 b(Then)28 b(the)f(function)h FA(T)3808 4376 y Fx(u)71 4506 y Fs(obtaine)l(d)j(in)f(The)l(or)l(ems)g(4.4)h(and)g(4.8,)g(which) g(is)g(de\014ne)l(d)f(for)g FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)f FA(D)2449 4475 y Fx(u)2447 4529 y(\024)2486 4537 y Fu(3)2518 4529 y Fx(;d)2573 4537 y Fu(2)2628 4506 y Fw(\002)18 b Ft(T)2766 4518 y Fx(\033)2811 4506 y Fs(,)30 b(satis\014es)1254 4742 y Fw(j)p FA(@)1321 4754 y Fx(u)1364 4742 y FA(T)1425 4708 y Fx(u)1468 4742 y FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b Fw(\000)f FA(@)1808 4754 y Fx(u)1851 4742 y FA(T)1900 4754 y Fy(0)1937 4742 y FB(\()p FA(u)p FB(\))p Fw(j)23 b(\024)2193 4686 y FA(b)2229 4698 y Fy(8)2266 4686 y Fw(j)7 b FB(^)-49 b FA(\026)p Fw(j)p FA(")2401 4656 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p 2193 4723 442 4 v 2199 4815 a Fw(j)p FA(u)2270 4791 y Fy(2)2325 4815 y FB(+)18 b FA(a)2452 4791 y Fy(2)2489 4815 y Fw(j)2512 4773 y Fx(`)p Fy(+1)2644 4742 y FA(;)71 4973 y Fs(wher)l(e)30 b FA(T)354 4985 y Fy(0)421 4973 y Fs(is)g(the)g(unp)l(erturb)l(e)l(d)f (sep)l(ar)l(atrix)h(given)g(in)36 b FB(\(51\))p Fs(.)195 5126 y FB(W)-7 b(e)36 b(w)n(an)n(t)e(to)h(study)g(the)h(in)n(v)-5 b(arian)n(t)34 b(manifolds)h(close)f(to)h(the)g(singularities)f FA(u)h FB(=)g Fw(\006)p FA(ia)p FB(,)i(that)e(is,)i(in)e(the)h(inner)71 5226 y(domains)26 b(de\014ned)h(in)f(\(30\).)37 b(Since)26 b(the)h(study)g(of)g(b)r(oth)f(in)n(v)-5 b(arian)n(t)26 b(manifolds)g(close)g(either)g(to)h FA(u)22 b FB(=)h FA(ia)j FB(or)g FA(u)c FB(=)h Fw(\000)p FA(ia)j FB(is)71 5325 y(analogous,)f(w)n(e)i(only)h(study)f(them)i(in)e(the)h(domain)g FA(D)1829 5295 y Fy(in)o Fx(;)p Fy(+)p Fx(;u)1827 5346 y(\024;c)2018 5325 y FB(.)37 b(Then,)28 b(w)n(e)f(consider)f(the)i(c)n (hange)f(of)g(v)-5 b(ariables)1666 5504 y FA(z)26 b FB(=)d FA(")1858 5470 y Fv(\000)p Fy(1)1946 5504 y FB(\()p FA(u)c Fw(\000)f FA(ia)p FB(\))p FA(:)1447 b FB(\(60\))p Black 1919 5753 a(34)p Black eop end %%Page: 35 35 TeXDict begin 35 34 bop Black Black 71 272 a FB(The)31 b(v)-5 b(ariable)30 b FA(z)k FB(is)d(called)f(the)i Fs(inner)g (variable)p FB(,)j(in)c(con)n(trap)r(osition)e(to)i(the)g Fs(outer)i(variable)39 b FA(u)p FB(.)47 b(W)-7 b(e)31 b(note)g(that,)h(b)n(y)71 372 y(de\014nition)h(of)g FA(T)594 384 y Fy(0)664 372 y FB(in)g(\(51\))f(and)h(using)g(the)g(expansion)f (around)g(the)h(singularities)f(of)g FA(p)2919 384 y Fy(0)2956 372 y FB(\()p FA(u)p FB(\))h(in)h(\(12\))e(and)h(\(13\))o(,)i (w)n(e)71 471 y(ha)n(v)n(e)26 b(that)1191 607 y FA(@)1235 619 y Fx(u)1279 607 y FA(T)1328 619 y Fy(0)1364 607 y FB(\()p FA("z)c FB(+)c FA(ia)p FB(\))23 b(=)1858 547 y FA(C)1923 517 y Fy(2)1917 568 y(+)p 1805 588 221 4 v 1805 664 a FA(")1844 640 y Fy(2)p Fx(r)1913 664 y FA(z)1956 640 y Fy(2)p Fx(r)2049 515 y Fz(\020)2099 607 y FB(1)18 b(+)g Fw(O)2324 515 y Fz(\020)2373 607 y FB(\()p FA("z)t FB(\))2519 573 y Fy(1)p Fx(=\014)2631 515 y Fz(\021)o(\021)71 771 y FB(and,)27 b(using)h(the)g(results)f(of)g(Corollary)e(4.11,)i(w)n (e)g(ha)n(v)n(e)f(that)1047 983 y Fw(j)p FA(@)1114 995 y Fx(u)1157 983 y FA(T)1218 949 y Fx(u;s)1312 983 y FB(\()p FA("z)c FB(+)c FA(ia;)c(\034)9 b FB(\))18 b Fw(\000)g FA(@)1859 995 y Fx(u)1903 983 y FA(T)1952 995 y Fy(0)1989 983 y FB(\()p FA("z)j FB(+)d FA(ia)p FB(\))p Fw(j)23 b(\024)g FA(K)2637 927 y Fw(j)7 b FB(^)-49 b FA(\026)p Fw(j)p 2528 964 313 4 v 2528 1040 a FA(")2567 1016 y Fy(2)p Fx(r)2637 1040 y Fw(j)p FA(z)t Fw(j)2726 1016 y Fx(`)p Fy(+1)2851 983 y FA(:)71 1196 y FB(Hence,)28 b(in)g(order)e(to)h(catc)n(h)g(the)h(terms)g(of)f(the)h(same)f(order)f (in)i FA(")p FB(,)g(w)n(e)f(scale)g(the)h(generating)e(function)i(as) 1271 1365 y FA( )1328 1330 y Fx(u;s)1423 1365 y FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))23 b(=)g FA(")1762 1330 y Fy(2)p Fx(r)r Fv(\000)p Fy(1)1916 1365 y FA(C)1981 1329 y Fv(\000)p Fy(2)1975 1385 y(+)2071 1365 y FA(T)2132 1330 y Fx(u;s)2225 1365 y FB(\()p FA(ia)18 b FB(+)g FA("z)t(;)c(\034)9 b FB(\))p FA(:)1053 b FB(\(61\))195 1533 y(Then,)28 b(the)g (Hamilton-Jacobi)e(equation)h(\(41\))h(reads)1142 1702 y FA(@)1186 1714 y Fx(\034)1228 1702 y FA( )21 b FB(+)d FA(")1425 1668 y Fy(2)p Fx(r)1495 1702 y FA(C)1560 1666 y Fv(\000)p Fy(2)1554 1723 y(+)p 1649 1635 76 4 v 1649 1702 a FA(H)1739 1635 y Fz(\000)1777 1702 y FA(ia)g FB(+)g FA("z)t(;)c(")2109 1668 y Fv(\000)p Fy(2)p Fx(r)2229 1702 y FA(C)2294 1668 y Fy(2)2288 1722 y(+)2344 1702 y FA(@)2388 1714 y Fx(z)2426 1702 y FA( )s(;)g(\034)2565 1635 y Fz(\001)2627 1702 y FB(=)22 b(0)p FA(;)924 b FB(\(62\))71 1882 y(where)p 311 1815 V 27 w FA(H)34 b FB(is)28 b(the)g(Hamiltonian)f (function)h(de\014ned)g(in)g(\(40\).)37 b(The)27 b(corresp)r(onding)f (Hamiltonian)h(is)1138 2050 y Fw(H)q FB(\()p FA(z)t(;)14 b(w)r(;)g(\034)9 b FB(\))24 b(=)e FA(")1646 2016 y Fy(2)p Fx(r)1716 2050 y FA(C)1781 2015 y Fv(\000)p Fy(2)1775 2071 y(+)p 1870 1984 V 1870 2050 a FA(H)1960 1983 y Fz(\000)1998 2050 y FA(ia)c FB(+)g FA("z)t(;)c(")2330 2016 y Fv(\000)p Fy(2)p Fx(r)2450 2050 y FA(C)2515 2016 y Fy(2)2509 2071 y(+)2565 2050 y FA(w)r(;)g(\034)2708 1983 y Fz(\001)2761 2050 y FA(:)919 b FB(\(63\))71 2219 y(W)-7 b(e)28 b(study)g(equation)f (\(62\))g(in)h(the)g(domain)f Fw(D)1558 2189 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)1556 2239 y(\024;c)1766 2219 y Fw(\002)18 b Ft(T)1904 2231 y Fx(\033)1949 2219 y FB(,)28 b(where)1287 2400 y Fw(D)1353 2366 y Fy(in)o Fx(;)p Fy(+)p Fx(;u)1351 2421 y(\024;c)1565 2400 y FB(=)1644 2333 y Fz(\010)1692 2400 y FA(z)e Fw(2)e Ft(C)p FB(;)14 b FA(ia)k FB(+)g FA("z)26 b Fw(2)d FA(D)2360 2366 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)2358 2421 y(\024;c)2549 2333 y Fz(\011)2612 2400 y FA(:)1068 b FB(\(64\))195 2581 y(T)-7 b(o)21 b(study)g(equation)f (\(62\),)i(as)e(a)h(\014rst)g(step)g(it)g(is)g(natural)f(to)h(study)g (it)g(in)g(the)g(limit)h(case)e FA(")j FB(=)g(0.)34 b(In)21 b(the)g(p)r(olynomial)71 2681 y(case)27 b(it)h(reads)596 2894 y FA(@)640 2906 y Fx(\034)681 2894 y FA( )735 2906 y Fy(0)791 2894 y FB(+)884 2838 y(1)p 884 2875 42 4 v 884 2951 a(2)935 2894 y FA(z)978 2860 y Fy(2)p Fx(r)1061 2894 y FB(\()q FA(@)1138 2906 y Fx(z)1176 2894 y FA( )1230 2906 y Fy(0)1267 2894 y FB(\))1299 2852 y Fy(2)1355 2894 y Fw(\000)1504 2838 y FB(1)p 1448 2875 154 4 v 1448 2951 a(2)p FA(z)1533 2927 y Fy(2)p Fx(r)1630 2894 y FB(+)1742 2838 y(^)-49 b FA(\026)p 1723 2875 75 4 v 1723 2951 a(z)1766 2927 y Fx(`)1956 2815 y Fz(X)1821 2997 y Fy(\()p Fx(r)r Fv(\000)p Fy(1\))p Fx(k)q Fy(+)p Fx(r)r(l)p Fy(=)p Fx(`)2247 2834 y FA(C)2312 2798 y Fx(k)q Fy(+)p Fx(l)p Fv(\000)p Fy(2)2306 2855 y(+)p 2235 2875 289 4 v 2235 2951 a FB(\(1)18 b Fw(\000)g FA(r)r FB(\))2481 2927 y Fx(k)2533 2894 y FA(a)2577 2906 y Fx(k)q(l)2639 2894 y FB(\()p FA(\034)9 b FB(\))2762 2827 y Fz(\000)2801 2894 y FA(z)2844 2860 y Fy(2)p Fx(r)2913 2894 y FA(@)2957 2906 y Fx(z)2996 2894 y FA( )3050 2906 y Fy(0)3087 2827 y Fz(\001)3125 2843 y Fx(l)3174 2894 y FB(=)22 b(0)p FA(:)377 b FB(\(65\))71 3159 y(The)28 b(solutions)e(of)i(this)g(equation)f(w)n(ere)f(studied)j (in)e(detail)h(in)g([Bal06)n(],)g(where)f(equation)g(\(65\))g(w)n(as)g (rewritten)g(as)894 3409 y FA(@)938 3421 y Fx(\034)979 3409 y FA( )1033 3421 y Fy(0)1089 3409 y FB(+)1182 3353 y(1)p 1182 3390 42 4 v 1182 3466 a(2)1233 3409 y FA(z)1276 3375 y Fy(2)p Fx(r)1359 3409 y FB(\()q FA(@)1436 3421 y Fx(z)1474 3409 y FA( )1528 3421 y Fy(0)1565 3409 y FB(\))1598 3367 y Fy(2)1653 3409 y Fw(\000)1802 3353 y FB(1)p 1746 3390 154 4 v 1746 3466 a(2)p FA(z)1831 3442 y Fy(2)p Fx(r)1928 3409 y FB(+)2040 3353 y(^)-49 b FA(\026)p 2021 3390 75 4 v 2021 3466 a(z)2064 3442 y Fx(`)2150 3305 y(N)2119 3330 y Fz(X)2127 3509 y Fx(l)p Fy(=0)2253 3409 y FA(A)2315 3421 y Fx(l)2341 3409 y FB(\()p FA(\034)9 b FB(\))2464 3342 y Fz(\000)2503 3409 y FA(z)2546 3375 y Fy(2)p Fx(r)2615 3409 y FA(@)2659 3421 y Fx(z)2697 3409 y FA( )2751 3421 y Fy(0)2789 3342 y Fz(\001)2827 3358 y Fx(l)2875 3409 y FB(=)23 b(0)p FA(;)675 b FB(\(66\))71 3652 y(where)1331 3796 y FA(A)1393 3808 y Fx(l)1418 3796 y FB(\()p FA(\034)9 b FB(\))25 b(=)1774 3718 y Fz(X)1639 3900 y Fy(\()p Fx(r)r Fv(\000)p Fy(1\))p Fx(k)q Fy(+)p Fx(r)r(l)p Fy(=)p Fx(`)2065 3737 y FA(C)2130 3701 y Fx(k)q Fy(+)p Fx(l)p Fv(\000)p Fy(2)2124 3757 y(+)p 2053 3777 289 4 v 2053 3853 a FB(\(1)18 b Fw(\000)g FA(r)r FB(\))2299 3829 y Fx(k)2351 3796 y FA(a)2395 3808 y Fx(k)q(l)2457 3796 y FB(\()p FA(\034)9 b FB(\))p FA(;)1114 b FB(\(67\))71 4034 y(and)27 b FA(a)276 4046 y Fx(k)q(l)365 4034 y FB(are)g(the)g(co)r (e\016cien)n(ts)g(of)g FA(H)1229 4046 y Fy(1)1294 4034 y FB(in)g(\(8\))h(and)f FA(C)1744 4046 y Fy(+)1827 4034 y FB(is)g(giv)n(en)f(in)i Fp(HP2)p FB(.)36 b(This)27 b(equation)g(is)g(in)h(fact)f(the)h(Hamilton-)71 4134 y(Jacobi)e(equation)h(asso)r(ciated)g(to)g(the)h(non-autonomous)e (Hamiltonian)1000 4379 y Fw(H)1070 4391 y Fy(0)1108 4379 y FB(\()p FA(z)t(;)14 b(w)r(;)g(\034)9 b FB(\))24 b(=)1516 4323 y(1)p 1516 4360 42 4 v 1516 4436 a(2)1568 4379 y FA(z)1611 4345 y Fy(2)p Fx(r)1680 4379 y FA(w)1741 4345 y Fy(2)1797 4379 y Fw(\000)1946 4323 y FB(1)p 1890 4360 154 4 v 1890 4436 a(2)p FA(z)1975 4412 y Fy(2)p Fx(r)2072 4379 y FB(+)2184 4323 y(^)-49 b FA(\026)p 2165 4360 75 4 v 2165 4436 a(z)2208 4412 y Fx(`)2294 4275 y(N)2263 4300 y Fz(X)2270 4479 y Fx(l)p Fy(=0)2397 4379 y FA(A)2459 4391 y Fx(l)2485 4379 y FB(\()p FA(\034)9 b FB(\))2608 4312 y Fz(\000)2647 4379 y FA(z)2690 4345 y Fy(2)p Fx(r)2759 4379 y FA(w)2820 4312 y Fz(\001)2859 4328 y Fx(l)2898 4379 y FA(;)782 b FB(\(68\))71 4627 y(whic)n(h)27 b(satis\014es)g(that) h Fw(H)c(!)f(H)1064 4639 y Fy(0)1129 4627 y FB(as)k FA(")c Fw(!)g FB(0,)k(where)g Fw(H)i FB(is)e(the)h(Hamiltonian)g(function)g (de\014ned)g(in)g(\(63\))o(.)195 4727 y(In)g(the)h(trigonometric)d (case,)h(an)g(analogous)f(equation)h(to)h(\(65\))f(is)h(obtained.)37 b(There)27 b(are)g(only)h(t)n(w)n(o)f(di\013erences.)71 4826 y(First,)k(one)f(has)f(to)h(consider)g(the)g(de\014nition)h(of)f FA(`)g FB(giv)n(en)g(in)g(\(14\))g(asso)r(ciated)f(to)h(this)h(t)n(yp)r (e)f(of)g(systems.)45 b(Secondly)-7 b(,)71 4936 y(in)27 b(the)g(trigonometric)e(case,)i(the)g(co)r(e\016cien)n(ts)f(in)h(fron)n (t)g(of)f FA(a)2021 4948 y Fx(k)q(l)2083 4936 y FB(\()p FA(\034)9 b FB(\))29 b(are)c(expressed)h(in)h(terms)g(of)f(the)i(co)r (e\016cien)n(ts)3728 4915 y Fz(b)3712 4936 y FA(C)3777 4906 y Fy(1)3771 4957 y Fv(\006)3827 4936 y FB(,)87 5032 y Fz(b)71 5053 y FA(C)136 5023 y Fy(2)130 5073 y Fv(\006)214 5053 y FB(and)f FA(C)434 5065 y Fv(\006)518 5053 y FB(in)h(\(13\).)37 b(T)-7 b(aking)27 b(in)n(to)g(accoun)n(t)g(these)g(facts,)h(one)f(can)g (also)g(de\014ne)h(the)g(analogous)d(functions)j FA(A)3664 5065 y Fx(l)3690 5053 y FB(.)195 5153 y(The)g(solutions)f(of)g(the)h (Hamilton-Jacobi)f(equation)g(\(66\))g(w)n(ere)f(studied)j(in)e([Bal06) o(])h(in)f(the)h(complex)g(domains)1318 5327 y Fw(D)1384 5287 y Fy(+)p Fx(;u)1382 5352 y(\024;\022)1521 5327 y FB(=)23 b Fw(f)p FA(z)j Fw(2)d Ft(C)p FB(;)14 b Fw(j)p FB(Im)g FA(z)t Fw(j)23 b FA(>)f(\022)16 b FB(Re)e FA(z)22 b FB(+)c FA(\024)p Fw(g)1326 5492 y(D)1392 5452 y Fy(+)p Fx(;s)1390 5517 y(\024;\022)1521 5492 y FB(=)1609 5399 y Fz(n)1664 5492 y FA(z)27 b Fw(2)c Ft(C)p FB(;)14 b Fw(\000)p FA(z)26 b Fw(2)d(D)2179 5452 y Fy(+)p Fx(;u)2177 5517 y(\024;\022)2294 5399 y Fz(o)3703 5419 y FB(\(69\))p Black 1919 5753 a(35)p Black eop end %%Page: 36 36 TeXDict begin 36 35 bop Black Black Black Black Black 112 1606 a /PSfrag where{pop(t)[[0(Bl)1 0]](k1)[[1(Bl)1 0]](k2)[[2(Bl)1 0]](D1)[[3(Bl)1 0]](D2)[[4(Bl)1 0]]5 0 -1/Begin PSfrag}{userdict /PSfrag{pop}put}ifelse 112 1606 a @beginspecial 39 @llx 615 @lly 545 @urx 809 @ury 1700 @rhi @setspecial %%BeginDocument: DomEqInner.eps %!PS-Adobe-3.0 EPSF-3.0 %%Creator: 0.46 %%Pages: 1 %%Orientation: Portrait %%BoundingBox: 39 615 545 809 %%HiResBoundingBox: 39.6 615.6 544.4 808.4 %%EndComments %%Page: 1 1 0 842 translate 0.8 -0.8 scale 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap gsave [1 0 0 1 0 0] concat gsave [1 0 0 1 360 -10] concat 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 200 52.362183 moveto 200 292.36218 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 40 172.36218 moveto 320 172.36218 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 40 102.36218 moveto 250 172.36218 lineto 40 242.36218 lineto stroke grestore gsave [-1 0 0 1 370 -9.9999974] concat 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 200 52.362183 moveto 200 292.36218 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 40 172.36218 moveto 320 172.36218 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 40 102.36218 moveto 250 172.36218 lineto 40 242.36218 lineto stroke grestore gsave [1 0 0 -1 151.54097 142.54602] concat gsave /newlatin1font {findfont dup length dict copy dup /Encoding ISOLatin1Encoding put definefont} def /Sans-ISOLatin1 /Sans newlatin1font 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (k1) show grestore grestore gsave [1 0 0 -1 146.28607 192.95392] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (k2) show grestore grestore gsave [1 0 0 -1 560.23407 191.89687] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (k2) show grestore grestore gsave [1 0 0 -1 560.50641 142.45146] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (k1) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 203.57143 134.50504 moveto 208.64487 140.86017 210.64006 150.29355 210 162.36218 curveto stroke gsave [1 0 0 -1 210.42857 155.93361] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (t) show grestore grestore gsave [1 0 0 -1 60 102.36218] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D1) show grestore grestore gsave [1 0 0 -1 630 102.36218] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D2) show grestore grestore grestore showpage %%EOF %%EndDocument @endspecial 112 1606 a /End PSfrag 112 1606 a 112 976 a /Hide PSfrag 112 976 a -628 1034 a FB(PSfrag)26 b(replacemen)n(ts)p -628 1063 741 4 v 112 1066 a /Unhide PSfrag 112 1066 a -174 1166 a { -174 1166 a Black FB(arctan)13 b FA(\022)p Black -174 1166 a } 0/Place PSfrag -174 1166 a 36 1266 a { 36 1266 a Black FA(i\024)p Black 36 1266 a } 1/Place PSfrag 36 1266 a -29 1358 a { -29 1358 a Black Fw(\000)p FA(i\024)p Black -29 1358 a } 2/Place PSfrag -29 1358 a -68 1450 a { -68 1450 a Black Fw(D)-2 1410 y Fy(+)p Fx(;u)-4 1475 y(\024;\022)p Black -68 1450 a } 3/Place PSfrag -68 1450 a -60 1570 a { -60 1570 a Black Fw(D)6 1530 y Fy(+)p Fx(;s)4 1595 y(\024;\022)p Black -60 1570 a } 4/Place PSfrag -60 1570 a Black 968 1872 a FB(Figure)27 b(9:)36 b(The)28 b(domains)f Fw(D)1896 1832 y Fy(+)p Fx(;u)1894 1897 y(\024;\022)2038 1872 y FB(and)h Fw(D)2266 1832 y Fy(+)p Fx(;s)2264 1897 y(\024;\022)2400 1872 y FB(de\014ned)g(in)g (\(69\))o(.)p Black Black 71 2149 a(for)i FA(\024)e(>)f FB(0)j(and)h FA(\022)f(>)e FB(0.)45 b(Let)31 b(us)f(observ)n(e)f(that,) j(for)e(an)n(y)f FA(c)f(>)g FB(0,)j Fw(D)2253 2119 y Fy(in)p Fx(;)p Fy(+)p Fx(;)p Fv(\003)2251 2169 y Fx(\024;c)2465 2149 y Fw(\032)d(D)2624 2109 y Fy(+)p Fx(;)p Fv(\003)2622 2174 y Fx(\024;)p Fy(tan)10 b Fx(\014)2825 2182 y Fu(2)2892 2149 y FB(for)30 b Fw(\003)e FB(=)f FA(u;)14 b(s)p FB(.)45 b(Nev)n(ertheless,)71 2269 y(since)32 b(through)f(the)h(pro)r(of)f(w)n (e)h(will)g(ha)n(v)n(e)f(to)g(c)n(hange)g(the)h(slop)r(e)g(of)g(the)g (domains)f Fw(D)2839 2229 y Fy(+)p Fx(;)p Fv(\003)2837 2294 y Fx(\024;\022)2948 2269 y FB(,)i(w)n(e)f(start)f(with)i(a)e (certain)71 2369 y(\014xed)d(slop)r(e)f FA(\022)521 2381 y Fy(0)581 2369 y FA(<)c FB(tan)13 b FA(\014)849 2381 y Fy(2)914 2369 y FB(whic)n(h)28 b(will)g(b)r(e)g(determined)g Fs(a)i(p)l(osteriori)p FB(.)195 2468 y(The)k(di\013erence)f(b)r(et)n(w) n(een)h(the)f(stable)g(and)h(unstable)f(manifolds)g(of)g(the)h(inner)f (equation)g(w)n(as)g(studied)g(in)h(the)71 2568 y(in)n(tersection)27 b(domain)1219 2668 y Fw(R)1289 2632 y Fy(+)1289 2693 y Fx(\024;\022)1409 2668 y FB(=)c Fw(D)1563 2628 y Fy(+)p Fx(;u)1561 2693 y(\024;\022)1695 2668 y Fw(\\)c(D)1835 2628 y Fy(+)p Fx(;s)1833 2693 y(\024;\022)1960 2668 y Fw(\\)g(f)o FA(z)27 b Fw(2)c Ft(C)p FB(;)14 b(Im)g FA(z)26 b(<)d FB(0)p Fw(g)13 b FA(:)1001 b FB(\(70\))102 2811 y(The)32 b(next)f(theorem)g(giv)n(es)g(the)g(main)h(results)f(obtained) g(in)g([Bal06)o(])h(ab)r(out)f(the)h(solutions)e(of)i(equation)f (\(66\))g(and)p Black Black Black 885 4344 a /PSfrag where{pop(k)[[0(Bl)1 0]](D1)[[1(Bl)1 0]](D2)[[2(Bl)1 0]](R)[[3(Bl)1 0]]4 0 -1/Begin PSfrag}{userdict /PSfrag{pop}put}ifelse 885 4344 a @beginspecial 127 @llx 599 @lly 385 @urx 769 @ury 1700 @rhi @setspecial %%BeginDocument: DomEqInnerInter.eps %!PS-Adobe-3.0 EPSF-3.0 %%Creator: 0.46 %%Pages: 1 %%Orientation: Portrait %%BoundingBox: 127 599 385 769 %%HiResBoundingBox: 127.2 599.6 384.8 768.4 %%EndComments %%Page: 1 1 0 842 translate 0.8 -0.8 scale 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap gsave [1 0 0 1 0 0] concat 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 160 172.36218 moveto 480 172.36218 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 320 92.362183 moveto 320 302.36218 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 160 282.36218 moveto 320 212.36218 lineto 480 282.36218 lineto stroke gsave [1 0 0 -1 340 277.36218] concat gsave /newlatin1font {findfont dup length dict copy dup /Encoding ISOLatin1Encoding put definefont} def /Sans-ISOLatin1 /Sans newlatin1font 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (R) show grestore grestore gsave [1 0 0 -1 320 208.36218] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (k) show grestore grestore grestore showpage %%EOF %%EndDocument @endspecial 885 4344 a /End PSfrag 885 4344 a 885 3797 a /Hide PSfrag 885 3797 a 145 3854 a FB(PSfrag)26 b(replacemen)n(ts)p 145 3884 741 4 v 885 3887 a /Unhide PSfrag 885 3887 a 744 3980 a { 744 3980 a Black Fw(\000)p FA(i\024)p Black 744 3980 a } 0/Place PSfrag 744 3980 a 705 4071 a { 705 4071 a Black Fw(D)771 4031 y Fy(+)p Fx(;u)769 4096 y(\024;\022)p Black 705 4071 a } 1/Place PSfrag 705 4071 a 713 4191 a { 713 4191 a Black Fw(D)779 4151 y Fy(+)p Fx(;s)777 4216 y(\024;\022)p Black 713 4191 a } 2/Place PSfrag 713 4191 a 719 4308 a { 719 4308 a Black Fw(R)789 4272 y Fy(+)789 4333 y Fx(\024;\022)p Black 719 4308 a } 3/Place PSfrag 719 4308 a Black 1151 4610 a FB(Figure)h(10:)36 b(The)28 b(domain)f Fw(R)2092 4574 y Fy(+)2092 4635 y Fx(\024;\022)2216 4610 y FB(de\014ned)h(in)g(\(70\).)p Black Black 71 4820 a(their)f(di\013erence.)p Black 71 4977 a Fp(Theorem)38 b(4.12.)p Black 43 w Fs(L)l(et)d(us)f(c)l(onsider) i(any)g(\014xe)l(d)f FA(\022)1720 4989 y Fy(0)1789 4977 y FA(>)e FB(0)p Fs(.)54 b(Then,)37 b(for)43 b FB(^)-49 b FA(\026)33 b Fw(2)g FA(B)t FB(\()7 b(^)-49 b FA(\026)2714 4989 y Fy(0)2752 4977 y FB(\))35 b Fs(the)g(fol)t(lowing)j(statements)c (ar)l(e)71 5077 y(satis\014e)l(d:)p Black 169 5234 a(1.)p Black 42 w(Ther)l(e)g(exists)e FA(\024)798 5246 y Fy(4)862 5234 y FA(>)c FB(0)j Fs(such)i(that,)g(e)l(quation)39 b FB(\(66\))32 b Fs(has)h(solutions)g FA( )2497 5204 y Fv(\003)2494 5255 y Fy(0)2563 5234 y FB(:)27 b Fw(D)2679 5194 y Fy(+)p Fx(;)p Fv(\003)2677 5259 y Fx(\024)2716 5267 y Fu(4)2749 5259 y Fx(;\022)2801 5267 y Fu(0)2857 5234 y Fw(\002)20 b Ft(T)2997 5246 y Fx(\033)3070 5234 y Fw(!)27 b Ft(C)p Fs(,)34 b Fw(\003)27 b FB(=)g FA(u;)14 b(s)p Fs(,)33 b(of)g(the)278 5334 y(form)944 5468 y FA( )1001 5428 y Fx(u;s)998 5490 y Fy(0)1095 5468 y FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))24 b(=)f Fw(\000)1692 5412 y FB(1)p 1471 5449 486 4 v 1471 5525 a(\(2)p FA(r)d Fw(\000)e FB(1\))p FA(z)1802 5501 y Fy(2)p Fx(r)r Fv(\000)p Fy(1)1984 5468 y FB(+)25 b(^)-49 b FA(\026)p 2117 5400 58 4 v( )2175 5414 y Fx(u;s)2175 5488 y Fy(0)2269 5468 y FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))19 b(+)f FA(K)2637 5434 y Fx(u;s)2731 5468 y FA(;)99 b(K)2930 5434 y Fx(u;s)3047 5468 y Fw(2)23 b Ft(C)518 b FB(\(71\))p Black 1919 5753 a(36)p Black eop end %%Page: 37 37 TeXDict begin 37 36 bop Black Black 278 272 a Fs(wher)l(e)p 504 204 58 4 v 22 w FA( )561 218 y Fx(u;s)561 293 y Fy(0)677 272 y Fs(ar)l(e)22 b(analytic)g(functions)g(in)f(al)t(l)h(their)g (variables.)38 b(Mor)l(e)l(over,)25 b(the)c(derivatives)j(of)p 3246 204 V 22 w FA( )3303 218 y Fx(u;s)3303 293 y Fy(0)3419 272 y Fs(ar)l(e)d(uniquely)278 372 y(determine)l(d)31 b(by)f(the)g(c)l(ondition)1608 575 y FB(sup)1410 660 y Fy(\()p Fx(z)r(;\034)7 b Fy(\))p Fv(2D)1653 632 y Fu(+)p Fm(;)p Fl(\003)1651 681 y Fm(\024)1685 693 y Fu(4)1717 681 y Fm(;\022)1764 693 y Fu(0)1800 660 y Fv(\002)p Fn(T)1891 668 y Fm(\033)1944 479 y Fz(\014)1944 529 y(\014)1944 579 y(\014)1972 575 y FA(z)2015 540 y Fx(`)p Fy(+1)2130 575 y FA(@)2174 587 y Fx(z)p 2212 507 V 2212 575 a FA( )2270 521 y Fv(\003)2270 595 y Fy(0)2308 575 y FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))2497 479 y Fz(\014)2497 529 y(\014)2497 579 y(\014)2548 575 y FA(<)22 b Fw(1)278 871 y Fs(for)31 b Fw(\003)23 b FB(=)f FA(u;)14 b(s)p Fs(.)38 b(In)30 b(fact,)g(one)g(c)l(an)g(cho)l(ose)p 1620 804 V 31 w FA( )1677 818 y Fx(u;s)1677 892 y Fy(0)1802 871 y Fs(such)f(that)1691 1079 y FB(sup)1494 1164 y Fy(\()p Fx(z)r(;\034)7 b Fy(\))p Fv(2D)1737 1136 y Fu(+)p Fm(;)p Fl(\003)1735 1185 y Fm(\024)1769 1197 y Fu(4)1800 1185 y Fm(;\022)1847 1197 y Fu(0)1883 1164 y Fv(\002)p Fn(T)1974 1172 y Fm(\033)2027 983 y Fz(\014)2027 1033 y(\014)2027 1083 y(\014)2055 1079 y FA(z)2098 1044 y Fx(`)p 2129 1011 V 2129 1079 a FA( )2186 1025 y Fv(\003)2186 1099 y Fy(0)2224 1079 y FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))2413 983 y Fz(\014)2413 1033 y(\014)2413 1083 y(\014)2465 1079 y FA(<)22 b Fw(1)278 1354 y Fs(for)31 b Fw(\003)23 b FB(=)f FA(u;)14 b(s)p Fs(.)p Black 169 1520 a(2.)p Black 42 w(Ther)l(e)29 b(exists)f FA(\024)789 1532 y Fy(5)849 1520 y FA(>)23 b(\024)985 1532 y Fy(4)1022 1520 y Fs(,)29 b(analytic)g(functions)1747 1453 y Fz(\010)1796 1520 y FA(\037)1848 1490 y Fy([)p Fx(k)q Fy(])1926 1520 y FB(\()7 b(^)-49 b FA(\026)p FB(\))2040 1453 y Fz(\011)2089 1553 y Fx(k)q Fv(2)p Fn(Z)2209 1537 y Fl(\000)2290 1520 y Fs(de\014ne)l(d)29 b(on)f FA(B)t FB(\()7 b(^)-49 b FA(\026)2835 1532 y Fy(0)2873 1520 y FB(\))28 b Fs(and)h FA(g)c FB(:)e Fw(R)3274 1485 y Fy(+)3274 1545 y Fx(\024)3313 1553 y Fu(5)3346 1545 y Fx(;)p Fy(2)p Fx(\022)3431 1553 y Fu(0)3482 1520 y Fw(\002)14 b Ft(T)3616 1532 y Fx(\033)3684 1520 y Fw(!)23 b Ft(C)278 1631 y Fs(such)30 b(that)g(two)g(solutions)g FA( )1197 1591 y Fx(u;s)1194 1653 y Fy(0)1321 1631 y Fs(of)h(e)l(quation)36 b FB(\(66\))29 b Fs(of)i(the)f(form)g(given)h (in)36 b FB(\(71\))29 b Fs(with)i FA(K)3129 1601 y Fx(u)3194 1631 y FB(=)23 b FA(K)3359 1601 y Fx(s)3394 1631 y Fs(,)30 b(satisfy)1221 1832 y FB(\()p FA( )1310 1797 y Fx(u)1307 1852 y Fy(0)1372 1832 y Fw(\000)18 b FA( )1512 1797 y Fx(s)1509 1852 y Fy(0)1547 1832 y FB(\))d(\()p FA(z)t(;)f(\034)9 b FB(\))23 b(=)29 b(^)-48 b FA(\026)1958 1753 y Fz(X)1958 1932 y Fx(k)q(<)p Fy(0)2093 1832 y FA(\037)2145 1797 y Fy([)p Fx(k)q Fy(])2223 1832 y FB(\()7 b(^)-49 b FA(\026)q FB(\))p FA(e)2377 1797 y Fx(ik)q Fy(\()q Fx(z)r Fv(\000)p Fx(\034)7 b Fy(+)e(^)-38 b Fx(\026)o(g)r Fy(\()p Fx(z)r(;\034)7 b Fy(\)\))2885 1832 y FA(:)795 b FB(\(72\))278 2093 y Fs(Mor)l(e)l(over,)32 b(the)e(function)g FA(g)i Fs(satis\014es)e(that) 1241 2276 y FB(sup)1026 2361 y Fy(\()p Fx(z)r(;\034)7 b Fy(\))p Fv(2R)1272 2334 y Fu(+)1272 2381 y Fm(\024)1306 2393 y Fu(5)1338 2381 y Fm(;)p Fu(2)p Fm(\022)1413 2393 y Fu(0)1450 2361 y Fv(\002)p Fn(T)1541 2369 y Fm(\033)1594 2205 y Fz(\014)1594 2255 y(\014)1621 2276 y FA(z)1664 2242 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)1813 2276 y FA(g)s FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))2045 2205 y Fz(\014)2045 2255 y(\014)2096 2276 y FA(<)23 b Fw(1)657 b Fs(if)31 b FA(`)22 b(>)h FB(2)p FA(r)1093 2530 y FB(sup)879 2615 y Fy(\()p Fx(z)r(;\034)7 b Fy(\))p Fv(2R)1125 2588 y Fu(+)1125 2635 y Fm(\024)1159 2647 y Fu(5)1191 2635 y Fm(;)p Fu(2)p Fm(\022)1266 2647 y Fu(0)1303 2615 y Fv(\002)p Fn(T)1394 2623 y Fm(\033)1447 2435 y Fz(\014)1447 2484 y(\014)1447 2534 y(\014)1474 2530 y FB(\()q(ln)14 b Fw(j)p FA(z)t Fw(j)p FB(\))1710 2488 y Fv(\000)p Fy(1)1813 2530 y FA(g)s FB(\()p FA(z)t(;)g(\034)9 b FB(\))2045 2435 y Fz(\014)2045 2484 y(\014)2045 2534 y(\014)2096 2530 y FA(<)23 b Fw(1)657 b Fs(if)31 b FA(`)22 b FB(=)h(2)p FA(r)n(:)195 2842 y FB(The)28 b(pro)r(of)f(of)h(Theorem)e(4.12)h(is)g (giv)n(en)g(in)h([Bal06)n(].)p Black 71 3008 a Fp(Remark)21 b(4.13.)p Black 32 w Fs(F)-6 b(ol)t(lowing)23 b(the)e(pr)l(o)l(ofs)i (of)f([Bal06)r(],)i(it)d(c)l(an)g(b)l(e)h(e)l(asily)g(se)l(en)f(that)h (the)f(analytic)h(functions)3431 2941 y Fz(\010)3479 3008 y FA(\037)3531 2978 y Fy([)p Fx(k)q Fy(])3609 3008 y FB(\()7 b(^)-49 b FA(\026)q FB(\))3724 2941 y Fz(\011)3773 3042 y Fx(k)q Fv(2)p Fn(Z)3893 3025 y Fl(\000)71 3108 y Fs(ar)l(e)30 b(entir)l(e.)195 3274 y FB(F)-7 b(or)27 b(the)i(case)d FA(`)19 b Fw(\000)f FB(2)p FA(r)25 b FB(=)e(0)k(w)n(e)h (will)g(need)f(b)r(etter)i(kno)n(wledge)d(of)i(the)g(function)g FA(g)i FB(giv)n(en)d(b)n(y)h(Theorem)f(4.12.)35 b(The)71 3374 y(next)27 b(prop)r(osition)e(giv)n(es)h(its)h(\014rst)f (asymptotic)g(terms.)37 b(First,)27 b(w)n(e)f(de\014ne)h(certain)f (functions)h(whic)n(h)g(will)f(b)r(e)i(used)e(in)71 3473 y(the)i(statemen)n(t)f(of)h(the)g(next)g(prop)r(osition.)36 b(Let)27 b(us)h(consider)e(the)i(functions)g FA(A)2634 3485 y Fx(j)2697 3473 y FB(de\014ned)g(in)g(\(67\),)f(then)i(w)n(e)e (de\014ne)1482 3740 y FA(Q)1548 3752 y Fx(j)1583 3740 y FB(\()p FA(\034)9 b FB(\))24 b(=)1834 3636 y Fx(N)1804 3661 y Fz(X)1805 3840 y Fx(k)q Fy(=)p Fx(j)1938 3623 y Fz(\022)2040 3690 y FA(k)2044 3789 y(j)2128 3623 y Fz(\023)2203 3740 y FA(A)2265 3752 y Fx(k)2306 3740 y FB(\()p FA(\034)9 b FB(\))p FA(;)1265 b FB(\(73\))71 4009 y(and)27 b(functions)h FA(F)643 4021 y Fx(j)706 4009 y FB(suc)n(h)g(that)1482 4108 y FA(@)1526 4120 y Fx(\034)1567 4108 y FA(F)1620 4120 y Fx(j)1679 4108 y FB(=)22 b FA(Q)1832 4120 y Fx(j)1922 4108 y FB(and)55 b Fw(h)p FA(F)2196 4120 y Fx(j)2232 4108 y Fw(i)23 b FB(=)g(0)p FA(;)1263 b FB(\(74\))71 4258 y(whic)n(h)27 b(are)g(p)r(erio)r(dic)g(since)h Fw(h)p FA(Q)1067 4270 y Fx(j)1102 4258 y Fw(i)23 b FB(=)g(0.)p Black 71 4437 a Fp(Remark)42 b(4.14.)p Black 46 w Fs(The)d(functions)f FA(Q)1301 4449 y Fx(j)1336 4437 y FB(\()p FA(\034)9 b FB(\))39 b Fs(c)l(an)g(b)l(e)f(also)h(de\014ne)l(d)g(intrinsic)l(al)t (ly)h(either)2956 4416 y Fz(b)2936 4437 y FA(H)3012 4407 y Fy(1)3005 4458 y(1)3087 4437 y Fs(is)f(a)g(p)l(olynomial)h(or)f(a)71 4537 y(trigonometric)31 b(p)l(olynomial,)h(as)970 4754 y FA(Q)1036 4766 y Fx(j)1070 4754 y FB(\()p FA(\034)9 b FB(\))24 b(=)1311 4698 y(1)p 1301 4735 63 4 v 1301 4811 a FA(j)5 b FB(!)1373 4754 y FA(C)1438 4714 y Fx(j)s Fv(\000)p Fy(2)1432 4774 y(+)1597 4754 y FB(lim)1572 4806 y Fx(u)p Fv(!)p Fx(ia)1737 4754 y FB(\()p FA(u)18 b Fw(\000)g FA(ia)p FB(\))2023 4720 y Fx(`)p Fv(\000)p Fx(r)r(j)2170 4754 y FA(@)2219 4720 y Fx(j)2214 4774 y(p)2273 4733 y Fz(b)2254 4754 y FA(H)2330 4720 y Fy(1)2323 4774 y(1)2367 4754 y FB(\()p FA(q)2436 4766 y Fy(0)2474 4754 y FB(\()p FA(u)p FB(\))p FA(;)c(p)2665 4766 y Fy(0)2702 4754 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))p FA(;)71 4999 y Fs(wher)l(e)325 4978 y Fz(b)305 4999 y FA(H)381 4969 y Fy(1)374 5020 y(1)448 4999 y Fs(is)30 b(the)g(Hamiltonian)h(de\014ne)l(d)f(in)36 b FB(\(35\))29 b Fs(and)i FA(C)1941 5011 y Fy(+)2026 4999 y Fs(is)f(given)g(in)36 b FB(\(12\))30 b Fs(and)39 b FB(\(13\))o Fs(.)p Black 71 5165 a Fp(Prop)s(osition)30 b(4.15.)p Black 40 w Fs(L)l(et)f(us)g(c) l(onsider)i(the)f(c)l(onstant)1524 5348 y FA(b)23 b FB(=)g(2)p FA(r)16 b Fw(h)p FA(Q)1864 5360 y Fy(0)1901 5348 y FA(F)1954 5360 y Fy(1)2010 5348 y FB(+)i(2)p FA(F)2188 5360 y Fy(0)2225 5348 y FA(Q)2291 5360 y Fy(2)2328 5348 y Fw(i)c FA(;)1306 b FB(\(75\))p Black 1919 5753 a(37)p Black eop end %%Page: 38 38 TeXDict begin 38 37 bop Black Black 71 272 a Fs(wher)l(e)36 b FA(Q)377 284 y Fx(j)447 272 y Fs(and)f FA(F)666 284 y Fx(j)737 272 y Fs(ar)l(e)g(the)h(functions)f(de\014ne)l(d)g(in)42 b FB(\(73\))35 b Fs(and)44 b FB(\(74\))35 b Fs(r)l(esp)l(e)l(ctively.) 56 b(Then,)37 b(when)f FA(`)22 b Fw(\000)g FB(2)p FA(r)35 b FB(=)e(0)p Fs(,)j(the)71 372 y(function)30 b FA(g)i Fs(obtaine)l(d)f(in)f(The)l(or)l(em)g(4.12,)i(is)e(of)h(the)f(form)1327 552 y FA(g)s FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))23 b(=)f Fw(\000)p FA(F)1787 564 y Fy(1)1825 552 y FB(\()p FA(\034)9 b FB(\))19 b Fw(\000)25 b FB(^)-49 b FA(\026b)14 b FB(ln)g FA(z)22 b FB(+)p 2363 506 43 4 v 18 w FA(g)r FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))71 732 y Fs(and)p 232 687 V 30 w FA(g)32 b Fs(satis\014es)1623 832 y FB(sup)1408 916 y Fy(\()p Fx(z)r(;\034)7 b Fy(\))p Fv(2R)1654 890 y Fu(+)1654 937 y Fm(\024)1688 949 y Fu(5)1720 937 y Fm(;)p Fu(2)p Fm(\022)1795 949 y Fu(0)1832 916 y Fv(\002)p Fn(T)1923 924 y Fm(\033)1976 832 y Fw(j)p FA(z)p 2042 786 V 4 w(g)r FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))p Fw(j)23 b FA(<)g Fw(1)p FA(:)195 1088 y FB(T)-7 b(o)27 b(ha)n(v)n(e)g(a)g(b)r(etter)g(kno)n(wledge)g(of)g(the)h (parameterizations)d(of)i(the)h(in)n(v)-5 b(arian)n(t)26 b(manifolds)h(in)h(the)g(inner)f(domains)71 1187 y Fw(D)137 1157 y Fy(in)p Fx(;)p Fy(+)p Fx(;)p Fv(\003)135 1208 y Fx(\024;c)321 1187 y FB(,)e Fw(\003)e FB(=)g FA(u;)14 b(s)23 b FB(in)i(\(64\),)g(w)n(e)f(need)g(to)g(compare)f(the)i (parameterizations)d FA( )2563 1157 y Fx(u;s)2658 1187 y FB(,)j(whic)n(h)f(are)g(solutions)f(of)31 b(\(62\))24 b(with)71 1287 y FA( )128 1247 y Fx(u;s)125 1309 y Fy(0)250 1287 y FB(whic)n(h)k(are)e(solutions)h(of)34 b(\(65\))27 b(and)h(ha)n(v)n(e)e(b)r(een)i(giv)n(en)f(in)h(Theorem)f(4.12.)195 1387 y(Since)19 b(w)n(e)f(ha)n(v)n(e)g(to)g(use)g(the)h(functions)g (and)g(results)f(obtained)g(in)h(Theorem)e(4.12,)i(w)n(e)g(need)f(that) h Fw(D)3303 1357 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)3301 1407 y(\024;c)3516 1387 y Fw(\032)j(D)3669 1347 y Fy(+)p Fx(;u)3667 1412 y(\024;)p Fy(2)p Fx(\022)3791 1420 y Fu(0)3827 1387 y FB(.)71 1486 y(T)-7 b(o)27 b(this)h(end,)g(w)n(e)f(imp)r(ose)1737 1625 y FA(\022)1776 1637 y Fy(0)1836 1625 y FB(=)1934 1568 y(tan)13 b FA(\014)2114 1580 y Fy(2)p 1934 1605 218 4 v 2022 1681 a FB(2)2161 1625 y FA(:)195 1796 y FB(W)-7 b(e)22 b(state)f(the)g(next)g(theorem)g(for)f(the)i(unstable)f (in)n(v)-5 b(arian)n(t)20 b(manifold.)35 b(The)21 b(stable)f(manifold)i (satis\014es)e(analogous)71 1895 y(prop)r(erties.)p Black 71 2060 a Fp(Theorem)34 b(4.16.)p Black 41 w Fs(L)l(et)e FA(\015)g Fw(2)27 b FB(\(0)p FA(;)14 b FB(1\))p Fs(,)33 b(the)f(c)l(onstants)f FA(\024)1819 2072 y Fy(3)1888 2060 y Fs(and)i FA(\024)2100 2072 y Fy(5)2169 2060 y Fs(de\014ne)l(d)f(in)h(The)l(or)l(ems)g(4.8)g(and)f(4.12,)j FA(c)3489 2072 y Fy(1)3554 2060 y FA(>)26 b FB(0)32 b Fs(and)71 2159 y FA(")110 2171 y Fy(0)176 2159 y FA(>)c FB(0)k Fs(smal)t(l)i(enough)g(and)f FA(\024)1065 2171 y Fy(6)1131 2159 y FA(>)28 b FB(max)p Fw(f)p FA(\024)1469 2171 y Fy(3)1505 2159 y FA(;)14 b(\024)1590 2171 y Fy(5)1627 2159 y Fw(g)33 b Fs(big)g(enough,)i(which)f(might)g(dep)l(end)g(on)f (the)g(pr)l(evious)h(c)l(onstants.)71 2259 y(Then,)d(for)f FA(")23 b Fw(2)h FB(\(0)p FA(;)14 b(")736 2271 y Fy(0)772 2259 y FB(\))30 b Fs(and)37 b FB(^)-49 b FA(\026)24 b Fw(2)f FA(B)t FB(\()7 b(^)-49 b FA(\026)1296 2271 y Fy(0)1334 2259 y FB(\))p Fs(,)30 b(ther)l(e)g(exists)f(a)h(c)l(onstant)f FA(b)2294 2271 y Fy(10)2387 2259 y FA(>)23 b FB(0)29 b Fs(such)h(that)g(for)g FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))24 b Fw(2)f(D)3394 2229 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)3392 2279 y(\024)3431 2287 y Fu(6)3464 2279 y Fx(;c)3514 2287 y Fu(1)3602 2259 y Fw(\002)18 b Ft(T)3740 2271 y Fx(\033)1305 2522 y Fw(j)p FA(@)1372 2534 y Fx(z)1411 2522 y FA( )1468 2488 y Fx(u)1511 2522 y FB(\()p FA(z)t(;)c(\034)9 b FB(\))19 b Fw(\000)f FA(@)1846 2534 y Fx(z)1884 2522 y FA( )1941 2488 y Fx(u)1938 2542 y Fy(0)1985 2522 y FB(\()p FA(z)t(;)c(\034)9 b FB(\))p Fw(j)23 b(\024)2349 2466 y FA(b)2385 2478 y Fy(10)2455 2466 y FA(")2507 2406 y Fu(1)p 2503 2415 35 3 v 2503 2449 a Fm(\014)p 2318 2503 266 4 v 2318 2603 a Fw(j)p FA(z)t Fw(j)2406 2562 y Fy(2)p Fx(r)r Fv(\000)2537 2539 y Fu(1)p 2534 2548 35 3 v 2534 2582 a Fm(\014)2593 2522 y FA(;)71 2782 y Fs(wher)l(e)30 b FA(\015)k Fs(enters)29 b(in)g(the)h(de\014nition)g(of) g Fw(D)1404 2752 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)1402 2803 y(\024)1441 2811 y Fu(6)1473 2803 y Fx(;c)1523 2811 y Fu(1)1594 2782 y Fs(,)g FA(r)c FB(=)c FA(\013=\014)34 b Fs(has)c(b)l(e)l(en)f(de\014ne)l(d)h(in)f(Hyp)l(othesis)i Fo(HP2)p Fs(,)g FA( )3401 2752 y Fx(u)3398 2803 y Fy(0)3473 2782 y Fs(is)f(given)g(in)71 2882 y(The)l(or)l(em)h(4.12)g(and)f FA( )816 2852 y Fx(u)890 2882 y Fs(is)g(the)g(sc)l(aling)g(of)h(the)f (gener)l(ating)g(function)f FA(T)2413 2852 y Fx(u)2486 2882 y Fs(given)h(in)36 b FB(\(61\))p Fs(.)195 3046 y FB(The)28 b(pro)r(of)f(of)h(this)g(theorem)f(is)g(giv)n(en)g(in)h (Section)f(8.)71 3278 y Fq(4.7)112 b(Study)38 b(of)g(the)f (di\013erence)h(b)s(et)m(w)m(een)g(the)g(in)m(v)-6 b(arian)m(t)38 b(manifolds)71 3431 y FB(Once)d(w)n(e)h(ha)n(v)n(e)f(obtained)h (parameterizations)d(of)j(the)h(in)n(v)-5 b(arian)n(t)35 b(manifolds)h(of)g(the)g(form)g(\(42\))f(in)i(the)f(domains)71 3531 y FA(D)142 3501 y Fx(s)140 3555 y(\024)179 3563 y Fu(3)211 3555 y Fx(;d)266 3563 y Fu(2)328 3531 y FB(and)27 b FA(D)560 3501 y Fx(u)558 3555 y(\024)597 3563 y Fu(3)629 3555 y Fx(;d)684 3563 y Fu(2)746 3531 y FB(and)g(studied)g(their)g (\014rst)f(order)f(appro)n(ximations)g(close)h(to)g(the)h (singularities,)f(the)h(next)g(step)g(is)71 3631 y(to)g(study)h(their)g (di\013erence.)195 3730 y(W)-7 b(e)31 b(dev)n(ote)f(Section)h(4.7.1)f (to)g(study)h(the)g(\(easier\))f(case)g FA(`)20 b Fw(\000)g FB(2)p FA(r)30 b(<)e FB(0)i(and)h(then)g(in)g(Section)g(4.7.2)e(w)n(e)i (consider)71 3830 y(the)d(case)f FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b Fw(\025)e FB(0.)71 4045 y Fp(4.7.1)94 b(Study)32 b(of)g(the)g(di\013erence)f(b)s(et)m(w)m(een)g(the)h(in)m(v) -5 b(arian)m(t)33 b(manifolds)e(for)h(the)f(case)h FA(`)19 b Fw(\000)f FB(2)p FA(r)25 b(<)e FB(0)71 4199 y(W)-7 b(e)22 b(are)e(going)g(to)h(pro)r(ceed)g(to)g(study)h(the)f (di\013erence)h FA(@)1815 4211 y Fx(u)1858 4199 y FA(T)1919 4168 y Fx(u)1962 4199 y FB(\()p FA(u;)14 b(\034)9 b FB(\))d Fw(\000)g FA(@)2277 4211 y Fx(u)2320 4199 y FA(T)2381 4168 y Fx(s)2416 4199 y FB(\()p FA(u;)14 b(\034)9 b FB(\).)35 b(Recall)21 b(that)h(in)g(the)f(case)g FA(`)6 b Fw(\000)g FB(2)p FA(r)24 b(<)f FB(0,)71 4298 y(Hyp)r(othesis)k Fp(HP5)h FB(b)r(ecomes)f FA(\021)f Fw(\025)d FB(0.)36 b(Therefore)27 b(our)g(study)g(includes)h(the)g(non)g(p)r(erturbativ)n (e)e(case)h FA(\021)f FB(=)d(0.)195 4398 y(T)-7 b(o)28 b(study)f(the)h(di\013erence)g(b)r(et)n(w)n(een)f(the)h(manifolds,)g(w) n(e)f(de\014ne)1428 4578 y(\001\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)f FA(T)1864 4544 y Fx(u)1906 4578 y FB(\()p FA(u;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(T)2264 4544 y Fx(s)2298 4578 y FB(\()p FA(u;)c(\034)9 b FB(\))1211 b(\(76\))71 4758 y(in)28 b(the)g(domain)f FA(R)669 4770 y Fx(\024;d)789 4758 y FB(=)c FA(D)948 4728 y Fx(s)946 4782 y(\024;d)1061 4758 y Fw(\\)c FA(D)1206 4728 y Fx(u)1204 4782 y(\024;d)1329 4758 y FB(whic)n(h)27 b(is)h(de\014ned)g(in)g (\(27\))o(.)195 4858 y(W)-7 b(e)37 b(recall)d(that)i FA(p)806 4870 y Fy(0)843 4858 y FB(\()p FA(u)p FB(\))h Fw(6)p FB(=)g(0)e(if)h FA(u)g Fw(2)h FA(R)1494 4870 y Fx(\024;d)1628 4858 y FB(and)e(hence)h(w)n(e)f(can)h(use)f(the)i (Hamilton-Jacobi)d(equation)h(in)h(this)71 4958 y(domain.)195 5057 y(Subtracting)e(equation)g(\(41\))g(for)f(b)r(oth)i FA(T)1579 5027 y Fx(u)1656 5057 y FB(and)f FA(T)1885 5027 y Fx(s)1919 5057 y FB(,)i(one)e(can)g(see)g(that)g(\001)g (satis\014es)g(the)g(partial)g(di\013eren)n(tial)71 5157 y(equation)1824 5235 y Fz(e)1807 5256 y Fw(L)1864 5268 y Fx(")1899 5256 y FA(\030)28 b FB(=)22 b(0)p FA(;)1588 b FB(\(77\))71 5404 y(where)1441 5483 y Fz(e)1424 5504 y Fw(L)1481 5516 y Fx(")1540 5504 y FB(=)22 b FA(")1666 5470 y Fv(\000)p Fy(1)1755 5504 y FA(@)1799 5516 y Fx(\034)1859 5504 y FB(+)c(\(1)h(+)f FA(G)p FB(\()p FA(u;)c(\034)9 b FB(\)\))p FA(@)2453 5516 y Fx(u)3703 5504 y FB(\(78\))p Black 1919 5753 a(38)p Black eop end %%Page: 39 39 TeXDict begin 39 38 bop Black Black 71 272 a FB(with)443 460 y FA(G)p FB(\()p FA(u;)14 b(\034)9 b FB(\))25 b(=)897 403 y(1)p 801 440 233 4 v 801 516 a(2)p FA(p)885 528 y Fy(0)922 516 y FB(\()p FA(u)p FB(\))1057 460 y(\()q FA(@)1134 472 y Fx(u)1177 460 y FA(T)1238 425 y Fx(u)1226 480 y Fy(1)1281 460 y FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b(+)f FA(@)1621 472 y Fx(u)1664 460 y FA(T)1725 425 y Fx(s)1713 480 y Fy(1)1760 460 y FB(\()p FA(u;)c(\034)9 b FB(\)\))810 700 y(+)933 644 y FA(\026")1022 614 y Fx(\021)p 903 681 192 4 v 903 757 a FA(p)945 769 y Fy(0)982 757 y FB(\()p FA(u)p FB(\))1117 587 y Fz(Z)1200 608 y Fy(1)1164 776 y(0)1252 700 y FA(@)1296 712 y Fx(p)1353 679 y Fz(b)1334 700 y FA(H)1403 712 y Fy(1)1454 583 y Fz(\022)1515 700 y FA(q)1552 712 y Fy(0)1590 700 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1781 712 y Fy(0)1818 700 y FB(\()p FA(u)p FB(\))k(+)2041 644 y FA(s@)2124 656 y Fx(u)2167 644 y FA(T)2228 614 y Fx(u)2216 665 y Fy(1)2271 644 y FB(\()p FA(u;)c(\034)9 b FB(\))19 b(+)f(\(1)g Fw(\000)g FA(s)p FB(\))p FA(@)2857 656 y Fx(u)2901 644 y FA(T)2962 614 y Fx(s)2950 665 y Fy(1)2997 644 y FB(\()p FA(u;)c(\034)9 b FB(\))p 2041 681 1151 4 v 2521 757 a FA(p)2563 769 y Fy(0)2600 757 y FB(\()p FA(u)p FB(\))3201 700 y FA(;)14 b(\034)3283 583 y Fz(\023)3373 700 y FA(ds;)3703 586 y FB(\(79\))71 938 y(where)333 917 y Fz(b)314 938 y FA(H)383 950 y Fy(1)451 938 y FB(is)30 b(the)h(function)h(de\014ned)f(in)g (\(34\))f(and)g FA(T)1804 898 y Fx(u;s)1792 960 y Fy(1)1898 938 y FB(\()p FA(u;)14 b(\034)9 b FB(\))29 b(=)e FA(T)2274 908 y Fx(u;s)2368 938 y FB(\()p FA(u;)14 b(\034)9 b FB(\))21 b Fw(\000)f FA(T)2717 950 y Fy(0)2754 938 y FB(\()p FA(u)p FB(\))31 b(with)g FA(@)3133 950 y Fx(u)3176 938 y FA(T)3225 950 y Fy(0)3262 938 y FB(\()p FA(u)p FB(\))d(=)g FA(p)3537 908 y Fy(2)3537 959 y(0)3574 938 y FB(\()p FA(u)p FB(\))j(and)71 1038 y FA(T)132 1007 y Fx(u;s)253 1038 y FB(are)c(giv)n(en)f(in)i (Theorems)f(4.4)g(and)g(4.8.)195 1137 y(F)-7 b(ollo)n(wing)31 b([Bal06)n(],)i(to)e(obtain)h(the)g(asymptotic)f(expression)f(of)h(the) h(di\013erence)g(\001,)h(w)n(e)e(tak)n(e)g(adv)-5 b(an)n(tage)30 b(from)71 1237 y(the)f(fact)f(that)h(it)f(is)h(a)f(solution)f(of)i(the) f(homogeneous)f(linear)g(partial)h(di\013eren)n(tial)g(equation)g (\(77\))o(.)39 b(In)29 b([Bal06)n(])g(it)g(is)71 1336 y(seen)i(that)g(if)38 b(\(77\))31 b(has)g(a)f(solution)h FA(\030)1281 1348 y Fy(0)1350 1336 y FB(suc)n(h)f(that)i(\()p FA(\030)1792 1348 y Fy(0)1830 1336 y FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))33 b(is)e(injectiv)n(e)g (in)g FA(R)2754 1348 y Fx(\024;d)2872 1336 y Fw(\002)21 b Ft(T)3013 1348 y Fx(\033)3058 1336 y FB(,)32 b(then)g(an)n(y)e (solution)h(of)71 1436 y(equation)c(\(77\))g(de\014ned)h(in)g FA(R)1031 1448 y Fx(\024;d)1147 1436 y Fw(\002)18 b Ft(T)1285 1448 y Fx(\033)1358 1436 y FB(can)27 b(b)r(e)h(written)g(as)e FA(\030)i FB(=)22 b(\007)c Fw(\016)g FA(\030)2345 1448 y Fy(0)2411 1436 y FB(for)27 b(some)g(function)h(\007.)195 1536 y(F)-7 b(ollo)n(wing)27 b(this)h(approac)n(h,)d(w)n(e)j(b)r(egin)f (b)n(y)h(lo)r(oking)e(for)h(a)h(solution)f(of)g(the)h(form)1438 1705 y FA(\030)1474 1717 y Fy(0)1511 1705 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)e FA(")1855 1670 y Fv(\000)p Fy(1)1944 1705 y FA(u)c Fw(\000)g FA(\034)28 b FB(+)18 b Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))1220 b(\(80\))71 1874 y(b)r(eing)28 b Fw(C)k FB(a)27 b(function)h(2)p FA(\031)s FB(-p)r(erio)r(dic)f(in)h FA(\034)9 b FB(,)28 b(suc)n(h)g(that)g(\()p FA(\030)1832 1886 y Fy(0)1869 1874 y FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))30 b(is)d(injectiv)n(e)h(in)g FA(R)2780 1886 y Fx(\024;d)2896 1874 y Fw(\002)18 b Ft(T)3034 1886 y Fx(\033)3079 1874 y FB(.)195 1973 y(F)-7 b(rom)29 b(no)n(w)f(on)h(the)g (parameter)f FA(\024)h FB(will)g(b)r(e)g(pla)n(y)g(an)f(imp)r(ortan)n (t)h(role)f(in)h(our)f(computations.)41 b(The)29 b(next)g(results)71 2073 y(will)37 b(deal)g(with)g(big)g(v)-5 b(alues)37 b(of)g FA(\024)i FB(=)f FA(\024)p FB(\()p FA(")p FB(\))f(suc)n(h)g (that)g FA(\024")i(<)f(a)p FB(.)65 b(In)37 b(particular,)h(in)g (Theorem)e(4.19)g(w)n(e)g(will)i(use)71 2173 y FA(\024)23 b FB(=)f Fw(O)r FB(\(log)q(\(1)p FA(=")p FB(\)\).)p Black 71 2328 a Fp(Theorem)37 b(4.17.)p Black 43 w Fs(L)l(et)d FA(d)914 2340 y Fy(2)982 2328 y FA(>)d FB(0)j Fs(and)h FA(\024)1368 2340 y Fy(3)1436 2328 y FA(>)c FB(0)j Fs(the)g(c)l (onstants)g(de\014ne)l(d)h(in)f(The)l(or)l(em)h(4.8,)i FA(d)3074 2340 y Fy(3)3143 2328 y FA(<)31 b(d)3282 2340 y Fy(2)3320 2328 y Fs(,)k FA(")3419 2340 y Fy(0)3488 2328 y FA(>)c FB(0)i Fs(smal)t(l)71 2427 y(enough)h(and)h FA(\024)571 2439 y Fy(7)639 2427 y FA(>)c(\024)783 2439 y Fy(3)854 2427 y Fs(big)k(enough,)h(which)g(might)f(dep)l(end)g(on)g (the)f(pr)l(evious)h(c)l(onstants.)51 b(Then,)37 b(for)e FA(")c Fw(2)g FB(\(0)p FA(;)14 b(")3756 2439 y Fy(0)3793 2427 y FB(\))p Fs(,)71 2527 y FA(\026)30 b Fw(2)h FA(B)t FB(\()p FA(\026)386 2539 y Fy(0)423 2527 y FB(\))j Fs(and)g(any)g FA(\024)c Fw(\025)g FA(\024)1038 2539 y Fy(7)1109 2527 y Fs(such)k(that)f FA("\024)d(<)g(a)p Fs(,)35 b(ther)l(e)e(exists)h(a)g (r)l(e)l(al-analytic)h(function)e Fw(C)i FB(:)30 b FA(R)3310 2539 y Fx(\024;d)3404 2547 y Fu(3)3461 2527 y Fw(\002)21 b Ft(T)3602 2539 y Fx(\033)3677 2527 y Fw(!)30 b Ft(C)71 2627 y Fs(such)g(that)f FA(\030)465 2639 y Fy(0)503 2627 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)e FA(")847 2597 y Fv(\000)p Fy(1)936 2627 y FA(u)c Fw(\000)g FA(\034)28 b FB(+)18 b Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))30 b Fs(is)g(a)h(solution)f(of)48 b FB(\(77\))29 b Fs(and)1285 2796 y FB(\()p FA(\030)1353 2808 y Fy(0)1390 2796 y FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))25 b(=)1811 2729 y Fz(\000)1849 2796 y FA(")1888 2762 y Fv(\000)p Fy(1)1977 2796 y FA(u)18 b Fw(\000)g FA(\034)28 b FB(+)18 b Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)2598 2729 y Fz(\001)71 2965 y Fs(is)30 b(inje)l(ctive.)195 3065 y(Mor)l(e)l(over,)f(ther)l(e)e(exists)f(a)g(c) l(onstant)g FA(b)1443 3077 y Fy(11)1536 3065 y FA(>)d FB(0)i Fs(indep)l(endent)i(of)h FA(\026)p Fs(,)f FA(")f Fs(and)h FA(\024)p Fs(,)g(such)g(that)f(for)h FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)f FA(R)3508 3077 y Fx(\024;d)3602 3085 y Fu(3)3649 3065 y Fw(\002)11 b Ft(T)3780 3077 y Fx(\033)3825 3065 y Fs(,)1541 3236 y Fw(jC)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p Fw(j)24 b(\024)e FA(b)1977 3248 y Fy(11)2047 3236 y Fw(j)p FA(\026)p Fw(j)p FA(")2182 3201 y Fx(\021)1454 3371 y Fw(j)p FA(@)1521 3383 y Fx(u)1564 3371 y Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p Fw(j)24 b(\024)e FA(b)1977 3383 y Fy(11)2047 3371 y FA(\024)2095 3336 y Fv(\000)p Fy(1)2184 3371 y Fw(j)p FA(\026)p Fw(j)p FA(")2319 3336 y Fx(\021)r Fv(\000)p Fy(1)2444 3371 y FA(:)195 3536 y FB(T)-7 b(o)24 b(study)h(the)f (\014rst)g(order)f(of)h(the)h(di\013erence)f(b)r(et)n(w)n(een)g(the)h (in)n(v)-5 b(arian)n(t)23 b(manifolds,)h(w)n(e)g(need)g(a)g(b)r(etter)h (kno)n(wledge)71 3635 y(of)31 b(the)h(b)r(eha)n(vior)f(of)g(the)h (function)h Fw(C)j FB(in)c(the)g(inner)f(domains)g(de\014ned)h(in)g (\(30\).)49 b(The)31 b(next)h(prop)r(osition)f(giv)n(es)f(the)71 3735 y(\014rst)25 b(order)f(asymptotic)i(terms)f(of)g Fw(C)31 b FB(close)25 b(to)g FA(u)e FB(=)f FA(ia)p FB(,)k(that)g(is)f (in)h FA(D)2278 3705 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)2276 3756 y(\024;c)2482 3735 y Fw(\\)15 b FA(D)2623 3705 y Fy(in)o Fx(;)p Fy(+)p Fx(;s)2621 3756 y(\024;c)2804 3735 y FB(.)36 b(The)26 b(study)g(close)f(to)g FA(u)e FB(=)f Fw(\000)p FA(ia)71 3835 y FB(can)27 b(b)r(e)h(done)f(analogously)-7 b(.)p Black 71 3990 a Fp(Prop)s(osition)37 b(4.18.)p Black 44 w Fs(L)l(et)e FA(\024)1033 4002 y Fy(7)1105 3990 y Fs(b)l(e)h(given)g(by)g(The)l(or)l(em)h(4.17)g(and)f FA(c)2291 4002 y Fy(1)2362 3990 y FA(>)d FB(0)p Fs(.)55 b(Then,)38 b(for)f(any)e FA(")3173 4002 y Fy(0)3244 3990 y FA(>)e FB(0)i Fs(and)h FA(\024)d(>)g(\024)3813 4002 y Fy(7)71 4090 y Fs(such)f(that)h FA(\024")27 b(<)g(a)p Fs(,)34 b(ther)l(e)e(exist)g(a)h(c)l(onstant)e FA(C)6 b FB(\()p FA(\026;)14 b(")p FB(\))33 b Fs(de\014ne)l(d)g(for)g FB(\()p FA(\026;)14 b(")p FB(\))28 b Fw(2)g FA(B)t FB(\()p FA(\026)2713 4102 y Fy(0)2751 4090 y FB(\))21 b Fw(\002)e FB(\(0)p FA(;)14 b(")3038 4102 y Fy(0)3075 4090 y FB(\))33 b Fs(and)g(dep)l(ending)g(r)l(e)l(al-)71 4189 y(anal)t(lytic)l(al)t(ly) 24 b(in)d FA(\026)h Fs(and)f(a)h(c)l(onstant)f FA(b)1267 4201 y Fy(12)1360 4189 y FA(>)h FB(0)f Fs(such)g(that)g Fw(j)p FA(C)6 b FB(\()p FA(\026;)14 b(")p FB(\))p Fw(j)24 b(\024)f FA(b)2300 4201 y Fy(12)2370 4189 y Fw(j)p FA(\026)p Fw(j)p FA(")2505 4159 y Fx(\021)2566 4189 y Fs(and,)h(if)e FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)3115 4122 y Fz(\000)3153 4189 y FA(D)3224 4159 y Fy(in)o Fx(;)p Fy(+)p Fx(;u)3222 4210 y(\024;c)3311 4218 y Fu(1)3431 4189 y Fw(\\)19 b FA(D)3576 4159 y Fy(in)p Fx(;)p Fy(+)p Fx(;s)3574 4210 y(\024;c)3663 4218 y Fu(1)3758 4122 y Fz(\001)3796 4189 y Fw(\002)71 4289 y Ft(T)126 4301 y Fx(\033)171 4289 y Fs(,)1420 4432 y Fw(jC)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b Fw(\000)f FA(C)6 b FB(\()p FA(\026;)14 b(")p FB(\))p Fw(j)23 b(\024)2187 4376 y FA(b)2223 4388 y Fy(12)2293 4376 y Fw(j)p FA(\026)p Fw(j)p FA(")2428 4345 y Fx(\021)p 2187 4413 282 4 v 2304 4489 a FA(\024)2478 4432 y(:)195 4610 y FB(The)28 b(pro)r(ofs)f(of)g (Theorem)g(4.17)f(and)i(Prop)r(osition)e(4.18)g(are)h(done)g(in)h (Section)f(9.3.)195 4710 y(As)j(w)n(e)f(ha)n(v)n(e)g(explained,)g (since)h(\001)c(=)g FA(T)1487 4680 y Fx(u)1550 4710 y Fw(\000)19 b FA(T)1695 4680 y Fx(s)1759 4710 y FB(is)30 b(a)f(solution)g(of)g(the)h(same)f(homogeneous)f(partial)h(di\013eren)n (tial)71 4809 y(equation)e(as)g FA(\030)548 4821 y Fy(0)613 4809 y FB(giv)n(en)g(in)h(Theorem)e(4.17,)h(there)g(exists)g(a)h (function)g(\007)f(suc)n(h)g(that)h(\001)23 b(=)g(\007)18 b Fw(\016)g FA(\030)3130 4821 y Fy(0)3168 4809 y FB(,)27 b(whic)n(h)h(giv)n(es)1344 4978 y(\001\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)f(\007)1798 4911 y Fz(\000)1835 4978 y FA(")1874 4944 y Fv(\000)p Fy(1)1963 4978 y FA(u)18 b Fw(\000)g FA(\034)28 b FB(+)18 b Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))2502 4911 y Fz(\001)2554 4978 y FA(:)1126 b FB(\(81\))71 5148 y(Since)38 b(\001)g(is)f(2)p FA(\031)s FB(-p)r(erio)r(dic)g(in)h FA(\034)9 b FB(,)41 b(w)n(e)d(notice)f(that)h (the)g(function)h(\007)e(is)h(2)p FA(\031)s FB(-p)r(erio)r(dic)f(in)h (its)f(v)-5 b(ariable.)67 b(Therefore,)71 5247 y(considering)26 b(the)i(F)-7 b(ourier)27 b(series)f(of)i(\007)f(w)n(e)g(obtain)1302 5433 y(\001\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)1677 5354 y Fz(X)1676 5533 y Fx(k)q Fv(2)p Fn(Z)1810 5433 y FB(\007)1875 5399 y Fy([)p Fx(k)q Fy(])1953 5433 y FA(e)1992 5397 y Fx(ik)2051 5403 y FB(\()2085 5397 y Fx(")2116 5372 y Fl(\000)p Fu(1)2193 5397 y Fx(u)p Fv(\000)p Fx(\034)7 b Fy(+)p Fv(C)s Fy(\()p Fx(u;\034)g Fy(\))2561 5403 y FB(\))2596 5433 y FA(:)1084 b FB(\(82\))p Black 1919 5753 a(39)p Black eop end %%Page: 40 40 TeXDict begin 40 39 bop Black Black 195 272 a FB(No)n(w)36 b(w)n(e)g(are)g(going)f(to)i(\014nd)f(the)h(\014rst)f(asymptotic)g (term)h(of)f(\001.)64 b(Let)37 b(us)f(\014rst)g(observ)n(e)f(that)i (the)g(Melnik)n(o)n(v)71 372 y(P)n(oten)n(tial)26 b(de\014ned)i(in)g (\(16\))f(can)g(b)r(e)h(de\014ned)g(through)f(the)h(functions)g Fw(T)2418 332 y Fx(u;s)2396 394 y Fy(0)2512 372 y FB(,)g(giv)n(en)e(in) i(\(57\),)g(as)1320 554 y Fw(T)1386 520 y Fx(u)1365 575 y Fy(0)1430 554 y FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b Fw(\000)f(T)1792 520 y Fx(s)1771 575 y Fy(0)1828 554 y FB(\()p FA(u;)c(\034)9 b FB(\))24 b(=)e Fw(\000)p FA(\026")2287 520 y Fx(\021)2327 554 y FA(L)p FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(:)1102 b FB(\(83\))71 737 y(Moreo)n(v)n(er)25 b(b)n(y)j(\(17\),)1415 858 y FA(L)p FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b(=)1777 779 y Fz(X)1777 958 y Fx(k)q Fv(2)p Fn(Z)1911 858 y FA(M)2001 824 y Fy([)p Fx(k)q Fy(])2079 858 y FA(e)2118 822 y Fx(ik)2177 828 y FB(\()2210 822 y Fx(")2241 797 y Fl(\000)p Fu(1)2319 822 y Fx(u)p Fv(\000)p Fx(\034)2448 828 y FB(\))2483 858 y FA(:)1197 b FB(\(84\))71 1091 y(In)31 b([DS97])g(\(for)g(the)h(h)n(yp)r(erb)r (olic)f(case\))g(and)g([BF04)o(])g(\(for)g(the)h(parab)r(olic)e (case\),)i(it)f(w)n(as)g(seen)g(that)g(for)g FA(\021)h(>)d(`)p FB(,)j(the)71 1191 y(function)c FA(L)f FB(giv)n(es)f(the)h(leading)g (term)g(of)g(the)h(di\013erence)f(b)r(et)n(w)n(een)g(manifolds.)37 b(Nev)n(ertheless,)26 b(for)h(the)g(general)f(case)71 1290 y FA(\021)g Fw(\025)d FB(0,)k(one)g(has)g(to)h(mo)r(dify)g(sligh)n (tly)f(this)h(function)g(to)f(obtain)h(the)g(correct)e(\014rst)h (order.)36 b(Let)27 b(us)h(de\014ne)1283 1490 y(\001)1352 1502 y Fy(0)1390 1490 y FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b(=)1695 1411 y Fz(X)1695 1590 y Fx(k)q Fv(2)p Fn(Z)1829 1490 y FB(\007)1894 1447 y Fy([)p Fx(k)q Fy(])1894 1512 y(0)1972 1490 y FA(e)2011 1454 y Fx(ik)2070 1460 y FB(\()2103 1454 y Fx(")2134 1429 y Fl(\000)p Fu(1)2212 1454 y Fx(u)p Fv(\000)p Fx(\034)7 b Fy(+)p Fv(C)s Fy(\()p Fx(u;\034)g Fy(\))2580 1460 y FB(\))2615 1490 y FA(;)1065 b FB(\(85\))71 1751 y(where)1269 1929 y(\007)1334 1886 y Fy([)p Fx(k)q Fy(])1334 1951 y(0)1435 1929 y FB(=)23 b Fw(\000)p FA(\026")1677 1895 y Fx(\021)1717 1929 y FA(M)1807 1895 y Fy([)p Fx(k)q Fy(])1885 1929 y FA(e)1924 1895 y Fv(\000)p Fx(ik)q(C)t Fy(\()p Fx(\026;")p Fy(\))2309 1929 y FB(if)74 b FA(k)26 b(<)c FB(0)1273 2079 y(\007)1338 2036 y Fy([0])1338 2101 y(0)1435 2079 y FB(=)h(0)1269 2230 y(\007)1334 2187 y Fy([)p Fx(k)q Fy(])1334 2252 y(0)1435 2230 y FB(=)g Fw(\000)p FA(\026")1677 2196 y Fx(\021)1717 2230 y FA(M)1807 2196 y Fy([)p Fx(k)q Fy(])1885 2230 y FA(e)1924 2196 y Fv(\000)p Fx(ik)p 2035 2148 52 3 v 1 w(C)t Fy(\()p Fx(\026;")p Fy(\))2309 2230 y FB(if)74 b FA(k)26 b(>)c FB(0)p FA(;)3703 2073 y FB(\(86\))71 2425 y(where)29 b FA(C)6 b FB(\()p FA(\026;)14 b(")p FB(\))29 b(is)h(the)f(constan)n(t)g(obtained)g(in)g (Prop)r(osition)f(4.18)g(and)p 2394 2358 66 4 v 29 w FA(C)6 b FB(\()p FA(\026;)14 b(")p FB(\))30 b(is)f(its)g(complex)g (conjugate.)41 b(Let)30 b(us)71 2524 y(p)r(oin)n(t)e(out)f(that,)h(b)n (y)f(Prop)r(osition)f(4.18,)h(these)g(co)r(e\016cien)n(ts)h(satisfy) 1386 2719 y(\007)1451 2675 y Fy([)p Fx(k)q Fy(])1451 2741 y(0)1552 2719 y FB(=)22 b Fw(\000)p FA(\026")1793 2684 y Fx(\021)1833 2719 y FA(M)1923 2684 y Fy([)p Fx(k)q Fy(])2015 2719 y FB(\(1)c(+)g Fw(O)f FB(\()p FA(\026")2394 2684 y Fx(\021)2434 2719 y FB(\)\))d FA(:)71 2901 y FB(Next)27 b(theorem)f(sho)n(ws)f(that)i(this)g(function)g(\001)1565 2913 y Fy(0)1629 2901 y FB(giv)n(es)f(the)h(\014rst)f(asymptotic)g (order)f(of)33 b(\(76\).)k(F)-7 b(rom)26 b(no)n(w)g(on,)g(in)h(this)71 3001 y(subsection,)36 b(w)n(e)f(consider)e(real)h(v)-5 b(alues)35 b(of)f FA(\034)45 b Fw(2)35 b Ft(T)h FB(=)e Ft(T)1906 3013 y Fx(\033)1975 3001 y Fw(\\)23 b Ft(R)p FB(.)59 b(In)35 b(this)g(setting)g(it)g(can)f(b)r(e)i(easily)d(seen)i (that)g(the)71 3100 y(function)28 b(\001)465 3112 y Fy(0)530 3100 y FB(is)g(real-analytic)d(in)j FA(u)p FB(.)p Black 71 3267 a Fp(Theorem)41 b(4.19.)p Black 45 w Fs(L)l(et)c(us)g(c)l (onsider)h(the)g(me)l(an)f(value)h(of)g FB(\007)p Fs(,)i FB(\007)2227 3236 y Fy([0])2301 3267 y Fs(,)g(de\014ne)l(d)e(in)44 b FB(\(82\))o Fs(,)c FA(s)d(<)f(\027)3205 3236 y Fv(\003)3281 3267 y Fs(wher)l(e)j FA(\027)3570 3236 y Fv(\003)3645 3267 y Fs(is)f(the)71 3366 y(c)l(onstant)26 b(de\014ne)l(d)h(in)g(Pr)l (op)l(osition)i(4.10)g(and)e FA(")1591 3378 y Fy(0)1651 3366 y FA(>)c FB(0)j Fs(smal)t(l)i(enough.)38 b(Then,)29 b(ther)l(e)e(exists)g(a)g(c)l(onstant)f FA(b)3442 3378 y Fy(13)3535 3366 y FA(>)d FB(0)j Fs(such)71 3466 y(that)32 b(for)h FA(")27 b Fw(2)h FB(\(0)p FA(;)14 b(")677 3478 y Fy(0)714 3466 y FB(\))32 b Fs(and)h FA(\026)28 b Fw(2)g FA(B)t FB(\()p FA(\026)1252 3478 y Fy(0)1289 3466 y FB(\))21 b Fw(\\)f Ft(R)33 b Fs(and)f FB(\()p FA(u;)14 b(\034)9 b FB(\))29 b Fw(2)1978 3399 y Fz(\000)2016 3466 y FA(R)2079 3481 y Fx(s)12 b Fy(ln\(1)p Fx(=")p Fy(\))p Fx(;d)2383 3489 y Fu(3)2437 3466 y Fw(\\)19 b Ft(R)2571 3399 y Fz(\001)2629 3466 y Fw(\002)h Ft(T)p Fs(,)34 b(the)e(fol)t(lowing)j(statements)c(ar) l(e)71 3565 y(satis\014e)l(d:)1104 3704 y Fz(\014)1104 3754 y(\014)1104 3804 y(\014)1132 3800 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))19 b Fw(\000)f FB(\007)1562 3765 y Fy([0])1655 3800 y Fw(\000)g FB(\001)1807 3812 y Fy(0)1844 3800 y FB(\()p FA(u;)c(\034)9 b FB(\))2038 3704 y Fz(\014)2038 3754 y(\014)2038 3804 y(\014)2090 3800 y Fw(\024)2188 3743 y FA(b)2224 3755 y Fy(13)2293 3743 y Fw(j)p FA(\026)p Fw(j)p FA(")2428 3713 y Fx(\021)r Fy(+1)p Fv(\000)p Fx(`)p 2188 3780 446 4 v 2319 3856 a Fw(j)14 b FB(ln)g FA(")p Fw(j)2642 3800 y FA(e)2681 3762 y Fv(\000)2743 3729 y Fx(a)p 2743 3743 37 4 v 2745 3791 a(")1179 4037 y Fw(j)p FA(@)1246 4049 y Fx(u)1290 4037 y FB(\001\()p FA(u;)g(\034)9 b FB(\))19 b Fw(\000)f FA(@)1699 4049 y Fx(u)1743 4037 y FB(\001)1812 4049 y Fy(0)1849 4037 y FB(\()p FA(u;)c(\034)9 b FB(\))p Fw(j)24 b(\024)2188 3981 y FA(b)2224 3993 y Fy(13)2293 3981 y Fw(j)p FA(\026)p Fw(j)p FA(")2428 3951 y Fx(\021)r Fv(\000)p Fx(`)p 2188 4018 361 4 v 2277 4094 a Fw(j)14 b FB(ln)g FA(")p Fw(j)2558 4037 y FA(e)2597 4000 y Fv(\000)2659 3967 y Fx(a)p 2659 3981 37 4 v 2661 4028 a(")1170 4204 y Fz(\014)1170 4254 y(\014)1198 4275 y FA(@)1247 4240 y Fy(2)1242 4295 y Fx(u)1285 4275 y FB(\001\()p FA(u;)g(\034)9 b FB(\))20 b Fw(\000)e FA(@)1700 4240 y Fy(2)1695 4295 y Fx(u)1738 4275 y FB(\001)1807 4287 y Fy(0)1844 4275 y FB(\()p FA(u;)c(\034)9 b FB(\))2038 4204 y Fz(\014)2038 4254 y(\014)2090 4275 y Fw(\024)2188 4218 y FA(b)2224 4230 y Fy(13)2293 4218 y Fw(j)p FA(\026)p Fw(j)p FA(")2428 4188 y Fx(\021)r Fv(\000)p Fy(1)p Fv(\000)p Fx(`)p 2188 4256 446 4 v 2320 4332 a Fw(j)14 b FB(ln)g FA(")p Fw(j)2643 4275 y FA(e)2682 4237 y Fv(\000)2744 4204 y Fx(a)p 2744 4218 37 4 v 2746 4266 a(")2794 4275 y FA(:)195 4501 y FB(W)-7 b(e)24 b(observ)n(e)d(that,)j(assuming)e(Hyp)r(othesis)h Fp(HP6)p FB(,)h(Lemma)e(2.4)g(giv)n(es)g(the)h(\014rst)g(order)f(of)h (the)g(function)g FA(L)p FB(.)35 b(Then,)71 4612 y(using)c(this)h (lemma,)g(the)g(de\014nition)g(of)f(the)h(co)r(e\016cien)n(ts)f(\007) 2007 4569 y Fy([)p Fx(k)q Fy(])2007 4635 y(0)2117 4612 y FB(in)h(\(86\))f(and)g(Prop)r(osition)f(4.18,)h(one)g(can)g(deduce)h (a)71 4712 y(simpler)27 b(leading)g(term)h(of)f(\001)h(in)g(\(76\).)36 b(F)-7 b(or)27 b(this)h(purp)r(ose)f(let)h(us)g(de\014ne)g(the)g (function)976 4936 y(\001)1045 4948 y Fy(00)1116 4936 y FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b(=)1431 4879 y(2)p FA(\026")1562 4849 y Fx(\021)p 1431 4916 171 4 v 1439 4992 a FA(")1478 4968 y Fx(`)p Fv(\000)p Fy(1)1612 4936 y FA(e)1651 4898 y Fw(\000)1726 4865 y Fx(a)p 1725 4879 37 4 v 1727 4927 a(")1775 4936 y FB(Re)1901 4843 y Fz(\020)1950 4936 y FA(f)1991 4948 y Fy(0)2028 4936 y FA(e)2067 4901 y Fx(iC)t Fy(\()p Fx(\026;")p Fy(\))2289 4936 y FA(e)2328 4898 y Fv(\000)p Fx(i)2403 4904 y FB(\()2446 4865 y Fx(u)p 2446 4879 40 4 v 2450 4927 a(")2495 4898 y Fv(\000)p Fx(\034)7 b Fy(+)p Fv(C)s Fy(\()p Fx(u;\034)g Fy(\))2824 4904 y FB(\))2859 4843 y Fz(\021)2922 4936 y FA(;)758 b FB(\(87\))71 5146 y(where)27 b FA(C)6 b FB(\()p FA(\026;)14 b(")p FB(\))28 b(is)g(the)g(constan)n(t)e(giv)n(en) h(in)h(Prop)r(osition)e(4.18)g(and)i Fw(C)k FB(is)27 b(the)h(function)h(giv)n(en)d(b)n(y)i(Theorem)e(4.17.)p Black 1919 5753 a(40)p Black eop end %%Page: 41 41 TeXDict begin 41 40 bop Black Black Black 71 272 a Fp(Corollary)36 b(4.20.)p Black 41 w Fs(Ther)l(e)d(exists)f(a)h(c)l(onstant)e FA(b)1654 284 y Fy(14)1752 272 y FA(>)c FB(0)k Fs(such)i(that)f(for)h FA(")27 b Fw(2)h FB(\(0)p FA(;)14 b(")2715 284 y Fy(0)2752 272 y FB(\))p Fs(,)33 b FA(\026)28 b Fw(2)f FA(B)t FB(\()p FA(\026)3151 284 y Fy(0)3189 272 y FB(\))21 b Fw(\\)f Ft(R)32 b Fs(and)h FB(\()p FA(u;)14 b(\034)9 b FB(\))28 b Fw(2)71 304 y Fz(\000)109 372 y FA(R)172 387 y Fx(s)12 b Fy(ln)o(\(1)p Fx(=")p Fy(\))p Fx(;d)475 395 y Fu(3)530 372 y Fw(\\)19 b Ft(R)664 304 y Fz(\001)720 372 y Fw(\002)f Ft(T)p Fs(,)31 b(the)e(fol)t(lowing)k(statements)28 b(ar)l(e)i (satis\014e)l(d.)1074 524 y Fz(\014)1074 574 y(\014)1074 623 y(\014)1101 619 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))20 b Fw(\000)e FB(\007)1532 585 y Fy([0])1625 619 y Fw(\000)g FB(\001)1777 631 y Fy(00)1847 619 y FB(\()p FA(u;)c(\034)9 b FB(\))2041 524 y Fz(\014)2041 574 y(\014)2041 623 y(\014)2093 619 y Fw(\024)2190 563 y FA(b)2226 575 y Fy(14)2310 563 y Fw(j)p FA(\026)p Fw(j)14 b FA(")2459 533 y Fx(\021)r Fy(+1)p Fv(\000)p Fx(`)p 2190 600 473 4 v 2336 676 a Fw(j)g FB(ln)g FA(")p Fw(j)2673 619 y FA(e)2712 582 y Fv(\000)2773 549 y Fx(a)p 2773 563 37 4 v 2775 610 a(")1149 857 y Fw(j)p FA(@)1216 869 y Fx(u)1259 857 y FB(\001\()p FA(u;)g(\034)9 b FB(\))20 b Fw(\000)e FA(@)1669 869 y Fx(u)1712 857 y FB(\001)1781 869 y Fy(00)1852 857 y FB(\()p FA(u;)c(\034)9 b FB(\))p Fw(j)24 b(\024)2190 801 y FA(b)2226 813 y Fy(14)2310 801 y Fw(j)p FA(\026)p Fw(j)14 b FA(")2459 770 y Fx(\021)r Fv(\000)p Fx(`)p 2190 838 389 4 v 2294 914 a Fw(j)g FB(ln)g FA(")p Fw(j)2589 857 y FA(e)2628 819 y Fv(\000)2689 787 y Fx(a)p 2689 801 37 4 v 2691 848 a(")1140 1024 y Fz(\014)1140 1074 y(\014)1167 1094 y FA(@)1216 1060 y Fy(2)1211 1115 y Fx(u)1255 1094 y FB(\001\()p FA(u;)g(\034)9 b FB(\))19 b Fw(\000)f FA(@)1669 1060 y Fy(2)1664 1115 y Fx(u)1708 1094 y FB(\001)1777 1106 y Fy(00)1847 1094 y FB(\()p FA(u;)c(\034)9 b FB(\))2041 1024 y Fz(\014)2041 1074 y(\014)2093 1094 y Fw(\024)2190 1038 y FA(b)2226 1050 y Fy(14)2310 1038 y Fw(j)p FA(\026)p Fw(j)14 b FA(")2459 1008 y Fx(\021)r Fv(\000)p Fy(1)p Fv(\000)p Fx(`)p 2190 1075 474 4 v 2336 1151 a Fw(j)g FB(ln)g FA(")p Fw(j)2674 1094 y FA(e)2713 1057 y Fv(\000)2774 1024 y Fx(a)p 2774 1038 37 4 v 2776 1085 a(")2824 1094 y FA(:)195 1321 y FB(W)-7 b(e)22 b(dev)n(ote)f(the)h(rest)g(of)f(this)h(section)f(to)h (pro)n(v)n(e)e(Theorem)h(4.19,)g(from)h(whic)n(h,)h(using)e(also)f (Lemma)i(2.4,)g(Corollary)71 1421 y(4.20)k(is)i(a)f(direct)g (consequence.)p Black 71 1587 a Fs(Pr)l(o)l(of)k(of)f(The)l(or)l(em)h (4.19.)p Black 44 w FB(F)-7 b(or)24 b(the)h(\014rst)f(part)g(of)h(the)g (pro)r(of)f(w)n(e)g(consider)g(complex)g(v)-5 b(alues)24 b(of)h FA(\026)e Fw(2)g FA(B)t FB(\()p FA(\026)3431 1599 y Fy(0)3469 1587 y FB(\))i(and)f(later)71 1686 y(w)n(e)j(will)h (restrict)f(to)g FA(\026)c Fw(2)h FA(B)t FB(\()p FA(\026)1039 1698 y Fy(0)1076 1686 y FB(\))19 b Fw(\\)g Ft(R)p FB(.)37 b(W)-7 b(e)28 b(de\014ne)1610 1865 y Fz(e)1601 1886 y FB(\007\()p FA(\020)6 b FB(\))24 b(=)1883 1807 y Fz(X)1883 1985 y Fx(k)q Fv(2)p Fn(Z)2026 1865 y Fz(e)2017 1886 y FB(\007)2082 1851 y Fy([)p Fx(k)q Fy(])2160 1886 y FA(e)2199 1851 y Fx(ik)q(\020)2297 1886 y FA(;)71 2176 y FB(where)320 2155 y Fz(e)311 2176 y FB(\007)376 2146 y Fy([)p Fx(k)q Fy(])477 2176 y FB(=)f(\007)630 2146 y Fy([)p Fx(k)q Fy(])726 2176 y Fw(\000)18 b FB(\007)874 2133 y Fy([)p Fx(k)q Fy(])874 2198 y(0)952 2176 y FB(.)37 b(By)29 b(\(82\))e(and)g(\(85\),)h(the)g(function)2158 2155 y Fz(e)2147 2176 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)e(\001\()p FA(u;)14 b(\034)9 b FB(\))20 b Fw(\000)e FB(\001)2956 2188 y Fy(0)2993 2176 y FB(\()p FA(u;)c(\034)9 b FB(\))28 b(can)g(b)r(e)g(written)f(as)847 2361 y Fz(e)836 2382 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)1219 2361 y Fz(e)1210 2382 y FB(\007)1289 2314 y Fz(\000)1327 2382 y FA(")1366 2347 y Fv(\000)p Fy(1)1455 2382 y FA(u)17 b Fw(\000)i FA(\034)28 b FB(+)18 b Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))1994 2314 y Fz(\001)2055 2382 y FB(=)2143 2303 y Fz(X)2143 2481 y Fx(k)q Fv(2)p Fn(Z)2286 2361 y Fz(e)2277 2382 y FB(\007)2342 2347 y Fy([)p Fx(k)q Fy(])2420 2382 y FA(e)2459 2346 y Fx(ik)2518 2352 y FB(\()2551 2346 y Fx(")2582 2321 y Fl(\000)p Fu(1)2660 2346 y Fx(u)p Fv(\000)p Fx(\034)e Fy(+)p Fv(C)s Fy(\()p Fx(u;\034)g Fy(\))3028 2352 y FB(\))3063 2382 y FA(:)617 b FB(\(88\))71 2681 y(Therefore,)26 b(to)i(obtain)f (the)h(b)r(ounds)g(of)f(Theorem)g(4.19,)f(it)i(is)g(crucial)f(to)g(b)r (ound)2697 2585 y Fz(\014)2697 2635 y(\014)2697 2685 y(\014)2734 2660 y(e)2725 2681 y FB(\007)2790 2651 y Fy([)p Fx(k)q Fy(])2868 2585 y Fz(\014)2868 2635 y(\014)2868 2685 y(\014)2895 2681 y FB(.)195 2824 y(The)33 b(\014rst)g(step)g(is)f (to)h(obtain)g(a)f(b)r(ound)h(of)1634 2803 y Fz(e)1623 2824 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))34 b(for)e(\()p FA(u;)14 b(\034)9 b FB(\))32 b Fw(2)g FA(R)2428 2845 y Fx(s)12 b Fy(ln)2548 2822 y Fu(1)p 2548 2831 29 3 v 2548 2865 a Fm(")2586 2845 y Fx(;d)2641 2853 y Fu(3)2699 2824 y Fw(\002)21 b Ft(T)p FB(.)53 b(First)33 b(w)n(e)f(b)r(ound)h (this)h(term)71 2968 y(for)25 b(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)492 2876 y Fz(\020)542 2968 y FA(R)605 2989 y Fx(s)11 b Fy(ln)724 2967 y Fu(1)p 724 2976 V 724 3009 a Fm(")762 2989 y Fx(;d)817 2997 y Fu(3)872 2968 y Fw(\\)19 b FA(D)1017 2929 y Fy(in)o Fx(;)p Fy(+)p Fx(;s)1015 3002 y(s)11 b Fy(ln)1134 2980 y Fu(1)p 1134 2989 V 1134 3022 a Fm(")1172 3002 y Fx(;c)1222 3010 y Fu(1)1276 2968 y Fw(\\)19 b FA(D)1421 2929 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)1419 3002 y(s)12 b Fy(ln)1538 2980 y Fu(1)p 1538 2989 V 1538 3022 a Fm(")1577 3002 y Fx(;c)1627 3010 y Fu(1)1662 2876 y Fz(\021)1727 2968 y Fw(\002)i Ft(T)p FB(.)37 b(Recalling)25 b(the)h(de\014nitions)g(in)g(\(76\))o(,)g(\(57\),)g(\(83\),)g(\(84\),)g (\(85\))71 3112 y(and)h(\(86\),)h(w)n(e)f(split)749 3091 y Fz(e)738 3112 y FB(\001)h(as)1027 3284 y Fz(e)1016 3305 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)1402 3284 y Fz(e)1390 3305 y FB(\001)1459 3270 y Fx(u)1459 3325 y Fy(1)1503 3305 y FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b Fw(\000)1811 3284 y Fz(e)1799 3305 y FB(\001)1868 3270 y Fx(s)1868 3325 y Fy(1)1906 3305 y FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b(+)2213 3284 y Fz(e)2202 3305 y FB(\001)2271 3317 y Fy(2)2308 3305 y FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b(+)2616 3284 y Fz(e)2604 3305 y FB(\001)2673 3317 y Fy(3)2711 3305 y FB(\()p FA(u;)14 b(\034)9 b FB(\))71 3487 y(with)772 3649 y Fz(e)761 3670 y FB(\001)830 3630 y Fx(u;s)830 3692 y Fy(1)925 3670 y FB(\()p FA(u;)14 b(\034)9 b FB(\))83 b(=)g FA(T)1411 3636 y Fx(u;s)1504 3670 y FB(\()p FA(u;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(T)1850 3682 y Fy(0)1886 3670 y FB(\()p FA(u)p FB(\))h Fw(\000)f(T)2166 3630 y Fx(u;s)2145 3692 y Fy(0)2261 3670 y FB(\()p FA(u;)c(\034)9 b FB(\))p Black 1248 w(\(89\))p Black 830 3800 a Fz(e)818 3821 y FB(\001)887 3833 y Fy(2)925 3821 y FB(\()p FA(u;)14 b(\034)9 b FB(\))83 b(=)g Fw(\000)p FA(\026")1504 3786 y Fx(\021)1558 3742 y Fz(X)1557 3921 y Fx(k)q(<)p Fy(0)1693 3821 y FA(M)1783 3786 y Fy([)p Fx(k)q Fy(])1861 3821 y FA(e)1900 3785 y Fx(ik)1959 3791 y FB(\()1992 3785 y Fx(")2023 3760 y Fl(\000)p Fu(1)2101 3785 y Fx(u)p Fv(\000)p Fx(\034)2230 3791 y FB(\))2280 3729 y Fz(\020)2329 3821 y FB(1)18 b Fw(\000)g FA(e)2511 3786 y Fx(ik)q Fy(\()q Fv(C)s Fy(\()p Fx(u;\034)7 b Fy(\))p Fv(\000)p Fx(C)t Fy(\()p Fx(\026;")p Fy(\)\))3061 3729 y Fz(\021)p Black 3703 3821 a FB(\(90\))p Black 830 4041 a Fz(e)818 4062 y FB(\001)887 4074 y Fy(3)925 4062 y FB(\()p FA(u;)14 b(\034)9 b FB(\))83 b(=)g Fw(\000)p FA(\026")1504 4028 y Fx(\021)1558 3983 y Fz(X)1557 4162 y Fx(k)q(>)p Fy(0)1693 4062 y FA(M)1783 4028 y Fy([)p Fx(k)q Fy(])1861 4062 y FA(e)1900 4026 y Fx(ik)1959 4032 y FB(\()1992 4026 y Fx(")2023 4001 y Fl(\000)p Fu(1)2101 4026 y Fx(u)p Fv(\000)p Fx(\034)2230 4032 y FB(\))2280 3970 y Fz(\020)2329 4062 y FB(1)18 b Fw(\000)g FA(e)2511 4026 y Fx(ik)2570 4032 y FB(\()2603 4026 y Fv(C)s Fy(\()p Fx(u;\034)7 b Fy(\))p Fv(\000)p 2843 3978 52 3 v Fx(C)s Fy(\()p Fx(\026;")p Fy(\))3037 4032 y FB(\))3074 3970 y Fz(\021)3137 4062 y FA(:)p Black 543 w FB(\(91\))p Black 71 4361 a(Applying)28 b(Prop)r(osition)e(4.10,)g(one)h(can)g(see)h(that)f(for)g(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)2118 4269 y Fz(\020)2167 4361 y FA(R)2230 4382 y Fx(s)12 b Fy(ln)2350 4360 y Fu(1)p 2350 4369 29 3 v 2350 4402 a Fm(")2388 4382 y Fx(;d)2443 4390 y Fu(3)2497 4361 y Fw(\\)19 b FA(D)2642 4321 y Fy(in)o Fx(;)p Fy(+)p Fx(;s)2640 4395 y(s)11 b Fy(ln)2759 4373 y Fu(1)p 2759 4382 V 2759 4415 a Fm(")2797 4395 y Fx(;c)2847 4403 y Fu(1)2902 4361 y Fw(\\)19 b FA(D)3047 4321 y Fy(in)o Fx(;)p Fy(+)p Fx(;u)3045 4395 y(s)11 b Fy(ln)3164 4373 y Fu(1)p 3164 4382 V 3164 4415 a Fm(")3202 4395 y Fx(;c)3252 4403 y Fu(1)3288 4269 y Fz(\021)3356 4361 y Fw(\002)18 b Ft(T)p FB(,)1416 4507 y Fz(\014)1416 4556 y(\014)1416 4606 y(\014)1444 4602 y FA(@)1488 4614 y Fx(u)1543 4581 y Fz(e)1531 4602 y FB(\001)1600 4562 y Fx(u;s)1600 4624 y Fy(1)1695 4602 y FB(\()p FA(u;)c(\034)9 b FB(\))1889 4507 y Fz(\014)1889 4556 y(\014)1889 4606 y(\014)1940 4602 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2240 4568 y Fx(\021)r Fv(\000)p Fx(`)p Fy(+)p Fx(\027)2444 4543 y Fl(\003)2482 4602 y FA(;)71 4805 y FB(where)27 b FA(\027)357 4775 y Fv(\003)418 4805 y FA(>)c FB(0)k(is)h(a)f(constan) n(t)g(de\014ned)h(in)g(that)g(prop)r(osition.)195 4905 y(T)-7 b(o)31 b(b)r(ound)592 4884 y Fz(e)581 4905 y FB(\001)650 4917 y Fy(2)687 4905 y FB(,)h(it)g(is)f(enough)g(to)g(apply)g(Lemma)f (2.4,)i(Theorem)e(4.17)g(and)h(Prop)r(osition)e(4.18)h(to)h(obtain)g (that)71 5025 y(for)c(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)494 4932 y Fz(\020)544 5025 y FA(R)607 5045 y Fx(s)11 b Fy(ln)726 5023 y Fu(1)p 726 5032 V 726 5065 a Fm(")764 5045 y Fx(;d)819 5053 y Fu(3)874 5025 y Fw(\\)18 b FA(D)1018 4985 y Fy(in)p Fx(;)p Fy(+)p Fx(;s)1016 5058 y(s)12 b Fy(ln)1136 5036 y Fu(1)p 1136 5045 V 1136 5078 a Fm(")1174 5058 y Fx(;c)1224 5066 y Fu(1)1278 5025 y Fw(\\)19 b FA(D)1423 4985 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)1421 5058 y(s)11 b Fy(ln)1540 5036 y Fu(1)p 1540 5045 V 1540 5078 a Fm(")1578 5058 y Fx(;c)1628 5066 y Fu(1)1664 4932 y Fz(\021)1732 5025 y Fw(\002)18 b Ft(T)p FB(,)1420 5201 y Fz(\014)1420 5251 y(\014)1420 5301 y(\014)1447 5297 y FA(@)1491 5309 y Fx(u)1546 5276 y Fz(e)1535 5297 y FB(\001)1604 5309 y Fy(2)1641 5297 y FB(\()p FA(u;)c(\034)9 b FB(\))1835 5201 y Fz(\014)1835 5251 y(\014)1835 5301 y(\014)1887 5297 y Fw(\024)1984 5240 y FA(K)d Fw(j)p FA(\026)p Fw(j)2157 5210 y Fy(2)2194 5240 y FA(")2233 5210 y Fy(2)p Fx(\021)r Fv(\000)p Fx(`)p Fy(+)p Fx(s)p 1984 5277 485 4 v 2142 5354 a Fw(j)q FB(ln)14 b FA(")p Fw(j)2478 5297 y FA(:)p Black 1919 5753 a FB(41)p Black eop end %%Page: 42 42 TeXDict begin 42 41 bop Black Black 71 272 a FB(Finally)-7 b(,)22 b(to)f(b)r(ound)g FA(@)752 284 y Fx(u)807 251 y Fz(e)795 272 y FB(\001)864 284 y Fy(3)902 272 y FB(,)h(it)f(is)g (enough)f(to)h(tak)n(e)f(in)n(to)h(accoun)n(t)f(again)f(Lemma)i(2.4,)g (Theorem)f(4.17)f(and)i(Prop)r(osition)71 392 y(4.18.)35 b(Then,)28 b(one)f(can)h(see)f(that)h(for)f(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)1560 300 y Fz(\020)1610 392 y FA(R)1673 412 y Fx(s)11 b Fy(ln)1792 390 y Fu(1)p 1792 399 29 3 v 1792 432 a Fm(")1830 412 y Fx(;d)1885 420 y Fu(3)1940 392 y Fw(\\)18 b FA(D)2084 352 y Fy(in)p Fx(;)p Fy(+)p Fx(;s)2082 426 y(s)12 b Fy(ln)2202 403 y Fu(1)p 2202 412 V 2202 446 a Fm(")2240 426 y Fx(;c)2290 434 y Fu(1)2344 392 y Fw(\\)19 b FA(D)2489 352 y Fy(in)o Fx(;)p Fy(+)p Fx(;u)2487 426 y(s)11 b Fy(ln)2606 403 y Fu(1)p 2606 412 V 2606 446 a Fm(")2644 426 y Fx(;c)2694 434 y Fu(1)2730 300 y Fz(\021)2798 392 y Fw(\002)18 b Ft(T)p FB(,)1337 548 y Fz(\014)1337 598 y(\014)1337 648 y(\014)1365 644 y FA(@)1409 656 y Fx(u)1464 623 y Fz(e)1453 644 y FB(\001)1522 656 y Fy(3)1559 644 y FB(\()p FA(u;)c(\034)9 b FB(\))1753 548 y Fz(\014)1753 598 y(\014)1753 648 y(\014)1804 644 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)2065 609 y Fy(2)2102 644 y FA(")2141 609 y Fy(2)p Fx(\021)r Fv(\000)p Fx(`)p Fv(\000)p Fx(s)2377 644 y FA(e)2416 606 y Fv(\000)2478 573 y Fy(2)p Fx(a)p 2478 587 70 4 v 2496 635 a(")2561 644 y FA(:)71 869 y FB(Therefore,)30 b(from)g(the)i(b)r(ounds)e(of)1220 848 y Fz(e)1209 869 y FB(\001)1278 829 y Fx(u;s)1278 891 y Fy(1)1373 869 y FB(,)1439 848 y Fz(e)1427 869 y FB(\001)1496 881 y Fy(2)1564 869 y FB(and)1740 848 y Fz(e)1729 869 y FB(\001)1798 881 y Fy(3)1866 869 y FB(and)h(recalling)e(that)i(b)n(y)f(h)n(yp)r (othesis)g FA(s)f(<)e(\027)3285 839 y Fv(\003)3324 869 y FB(,)k(w)n(e)g(ha)n(v)n(e)e(that)71 995 y(for)e(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)494 902 y Fz(\020)544 995 y FA(R)607 1015 y Fx(s)11 b Fy(ln)726 993 y Fu(1)p 726 1002 29 3 v 726 1035 a Fm(")764 1015 y Fx(;d)819 1023 y Fu(3)874 995 y Fw(\\)18 b FA(D)1018 955 y Fy(in)p Fx(;)p Fy(+)p Fx(;s)1016 1028 y(s)12 b Fy(ln)1136 1006 y Fu(1)p 1136 1015 V 1136 1048 a Fm(")1174 1028 y Fx(;c)1224 1036 y Fu(1)1278 995 y Fw(\\)19 b FA(D)1423 955 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)1421 1028 y(s)11 b Fy(ln)1540 1006 y Fu(1)p 1540 1015 V 1540 1048 a Fm(")1578 1028 y Fx(;c)1628 1036 y Fu(1)1664 902 y Fz(\021)1732 995 y Fw(\002)18 b Ft(T)p FB(,)1474 1170 y Fz(\014)1474 1220 y(\014)1474 1269 y(\014)1501 1265 y FA(@)1545 1277 y Fx(u)1600 1244 y Fz(e)1589 1265 y FB(\001\()p FA(u;)c(\034)9 b FB(\))1852 1170 y Fz(\014)1852 1220 y(\014)1852 1269 y(\014)1903 1265 y Fw(\024)2001 1209 y FA(K)d Fw(j)p FA(\026)p Fw(j)p FA(")2213 1179 y Fx(\021)r Fv(\000)p Fx(`)p Fy(+)p Fx(s)p 2001 1246 414 4 v 2124 1322 a Fw(j)p FB(ln)14 b FA(")p Fw(j)2425 1265 y FA(:)1255 b FB(\(92\))71 1491 y(Reasoning)26 b(analogously)-7 b(,)25 b(one)j(can)f(see)g(that)h(for)1113 1692 y(\()p FA(u;)14 b(\034)9 b FB(\))23 b Fw(2)1409 1600 y Fz(\020)1458 1692 y FA(R)1521 1713 y Fx(s)12 b Fy(ln)1641 1690 y Fu(1)p 1641 1699 29 3 v 1641 1733 a Fm(")1679 1713 y Fx(;d)1734 1721 y Fu(3)1788 1692 y Fw(\\)19 b FA(D)1933 1652 y Fy(in)p Fx(;)p Fv(\000)p Fx(;s)1931 1726 y(s)12 b Fy(ln)2050 1704 y Fu(1)p 2050 1713 V 2050 1746 a Fm(")2089 1726 y Fx(;c)2139 1734 y Fu(1)2193 1692 y Fw(\\)19 b FA(D)2338 1652 y Fy(in)o Fx(;)p Fv(\000)p Fx(;u)2336 1726 y(s)11 b Fy(ln)2455 1704 y Fu(1)p 2455 1713 V 2455 1746 a Fm(")2493 1726 y Fx(;c)2543 1734 y Fu(1)2579 1600 y Fz(\021)2647 1692 y Fw(\002)18 b Ft(T)p FA(;)71 1917 y FB(the)28 b(function)g FA(@)583 1929 y Fx(u)638 1896 y Fz(e)626 1917 y FB(\001)g(satis\014es)1474 1969 y Fz(\014)1474 2019 y(\014)1474 2069 y(\014)1501 2065 y FA(@)1545 2077 y Fx(u)1600 2044 y Fz(e)1589 2065 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))1852 1969 y Fz(\014)1852 2019 y(\014)1852 2069 y(\014)1903 2065 y Fw(\024)2001 2009 y FA(K)d Fw(j)p FA(\026)p Fw(j)p FA(")2213 1978 y Fx(\021)r Fv(\000)p Fx(`)p Fy(+)p Fx(s)p 2001 2046 414 4 v 2124 2122 a Fw(j)p FB(ln)14 b FA(")p Fw(j)2425 2065 y FA(:)1255 b FB(\(93\))195 2295 y(Finally)-7 b(,)22 b(for)d(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)904 2203 y Fz(\020)953 2295 y FA(R)1016 2316 y Fx(s)12 b Fy(ln)1136 2293 y Fu(1)p 1136 2302 29 3 v 1136 2336 a Fm(")1174 2316 y Fx(;d)1229 2324 y Fu(3)1283 2295 y Fw(\\)19 b FA(D)1428 2255 y Fy(out)p Fx(;s)1426 2314 y(c)1456 2322 y Fu(1)1488 2314 y Fx(")1519 2298 y Fm(\015)1557 2314 y Fx(;\032)1611 2322 y Fu(4)1666 2295 y Fw(\\)g FA(D)1811 2255 y Fy(out)p Fx(;u)1809 2314 y(c)1839 2322 y Fu(1)1871 2314 y Fx(")1902 2298 y Fm(\015)1940 2314 y Fx(;\032)1994 2322 y Fu(4)2031 2203 y Fz(\021)2084 2295 y Fw(\002)s Ft(T)p FB(,)j(w)n(e)d(decomp)r(ose)2788 2274 y Fz(e)2776 2295 y FB(\001)q(\()p FA(u;)14 b(\034)9 b FB(\))23 b(=)g(\()p FA(T)3244 2265 y Fx(u)3287 2295 y FB(\()p FA(u;)14 b(\034)9 b FB(\))s Fw(\000)s FA(T)3601 2307 y Fy(0)3638 2295 y FB(\()p FA(u)p FB(\)\))s Fw(\000)71 2420 y FB(\()p FA(T)164 2390 y Fx(s)199 2420 y FB(\()p FA(u;)14 b(\034)9 b FB(\))16 b Fw(\000)g FA(T)539 2432 y Fy(0)576 2420 y FB(\()p FA(u)p FB(\)\))g Fw(\000)g FB(\001)886 2432 y Fy(0)923 2420 y FB(\()p FA(u;)e(\034)9 b FB(\).)37 b(Using)27 b(Theorems)e(4.4,)h(4.8,)g(and)g(4.17)f(and)h (also)f(Lemma)h(2.4,)g(one)g(can)g(easily)g(see)71 2519 y(that)1412 2619 y Fw(j)p FA(@)1479 2631 y Fx(u)1522 2619 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))p Fw(j)24 b(\024)f FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2132 2585 y Fx(\021)r Fy(+1)p Fv(\000)p Fx(\015)t Fy(\()p Fx(`)p Fy(+1\))71 2768 y FB(pro)n(vided)24 b Fw(j)p FA(u)13 b Fw(\000)g FA(ia)p Fw(j)23 b(\025)f(O)r FB(\()p FA(")917 2738 y Fx(\015)960 2768 y FB(\).)37 b(This)25 b(b)r(ound)g(is)g (smaller)f(than)i(\(92\))e(and)h(\(93\))g(due)g(to)g(the)h(fact)f(that) g(\()p FA(`)13 b FB(+)g(1\)\(1)g Fw(\000)g FA(\015)5 b FB(\))24 b FA(>)71 2867 y(\027)117 2837 y Fv(\003)178 2867 y FA(>)f(s)28 b FB(\(see)f(Prop)r(osition)f(4.10)g(for)h(the)h (de\014nition)g(of)g FA(\026)1905 2837 y Fv(\003)1943 2867 y FB(\).)195 2967 y(T)-7 b(aking)27 b(in)n(to)g(accoun)n(t)g (\(92\))g(and)h(\(93\),)f(one)g(can)h(conclude)f(that)h(for)f FA(\026)c Fw(2)g FA(B)t FB(\()p FA(\026)2735 2979 y Fy(0)2773 2967 y FB(\))c Fw(\\)g Ft(R)p FB(,)1474 3109 y Fz(\014)1474 3159 y(\014)1474 3209 y(\014)1501 3204 y FA(@)1545 3216 y Fx(u)1600 3183 y Fz(e)1589 3204 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))1852 3109 y Fz(\014)1852 3159 y(\014)1852 3209 y(\014)1903 3204 y Fw(\024)2001 3148 y FA(K)d Fw(j)p FA(\026)p Fw(j)p FA(")2213 3118 y Fx(\021)r Fv(\000)p Fx(`)p Fy(+)p Fx(s)p 2001 3185 414 4 v 2124 3261 a Fw(j)p FB(ln)14 b FA(")p Fw(j)2425 3204 y FA(:)1255 b FB(\(94\))71 3434 y(The)24 b(second)f(step)i(of)f(the)g(pro)r(of)f(is)h(to)g (consider)f(the)i(c)n(hange)e(of)h(v)-5 b(ariables)22 b(\()p FA(w)r(;)14 b(\034)9 b FB(\))25 b(=)e(\()p FA(u)11 b FB(+)g FA(")p Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\).)37 b(By)23 b(Theorem)71 3544 y(4.17,)i(one)i(can)f(easily)g(see)g(that)h(it)g(is)f(a)g (di\013eomorphism)h(from)f FA(R)2185 3559 y Fx(s)12 b Fy(ln)o(\(1)p Fx(=")p Fy(\))p Fx(;d)2488 3567 y Fu(3)2541 3544 y Fw(\002)k Ft(T)27 b FB(on)n(to)f(its)g(image)3259 3523 y Fz(e)3243 3544 y FA(R)17 b Fw(\002)f Ft(T)p FB(.)37 b(Denoting)71 3663 y(b)n(y)195 3642 y Fz(e)186 3663 y FB(\007)251 3633 y Fv(0)302 3663 y FB(the)28 b(deriv)-5 b(ativ)n(e)26 b(of)i(the)g(function)1397 3642 y Fz(e)1388 3663 y FB(\007)f(\(see)h(\(88\))o(\),)h(w)n(e)e(de\014ne)h(the)g (function)1493 3855 y(\002\()p FA(w)r(;)14 b(\034)9 b FB(\))24 b(=)1886 3834 y Fz(e)1877 3855 y FB(\007)1942 3821 y Fv(0)1979 3788 y Fz(\000)2017 3855 y FA(")2056 3821 y Fv(\000)p Fy(1)2145 3855 y FA(w)d Fw(\000)d FA(\034)2353 3788 y Fz(\001)2405 3855 y FA(;)71 4055 y FB(on)202 4034 y Fz(e)186 4055 y FA(R)h Fw(\002)f Ft(T)28 b FB(whic)n(h,)g(b)n(y)f (construction,)g(satis\014es)1037 4298 y(\002\()p FA(u)18 b FB(+)g FA(")p Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))24 b(=)1791 4181 y Fz(\022)1862 4242 y FB(1)p 1862 4279 42 4 v 1863 4355 a FA(")1932 4298 y FB(+)18 b FA(@)2059 4310 y Fx(u)2102 4298 y Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))2345 4181 y Fz(\023)2407 4199 y Fv(\000)p Fy(1)2510 4298 y FA(@)2554 4310 y Fx(u)2609 4277 y Fz(e)2597 4298 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))p FA(:)820 b FB(\(95\))71 4530 y(Moreo)n(v)n(er,)25 b(as)i(\002\()p FA(w)r(;)14 b(\034)9 b FB(\))29 b(is)e(p)r(erio)r(dic)h (in)g FA(\034)9 b FB(,)28 b(it)g(can)f(b)r(e)h(also)f(written)h(as)1485 4728 y(\002\()p FA(w)r(;)14 b(\034)9 b FB(\))25 b(=)1869 4649 y Fz(X)1869 4827 y Fx(k)q Fv(2)p Fn(Z)2003 4728 y FB(\002)2068 4693 y Fy([)p Fx(k)q Fy(])2146 4728 y FB(\()p FA(w)r FB(\))p FA(e)2310 4693 y Fx(ik)q(\034)2413 4728 y FA(:)71 5011 y FB(Then,)j(for)f(an)n(y)g FA(w)e Fw(2)773 4990 y Fz(e)758 5011 y FA(R)p FB(,)j(the)g(F)-7 b(ourier)27 b(co)r(e\016cien)n(ts)g(satisfy)1512 5213 y FA(ik)1596 5192 y Fz(e)1587 5213 y FB(\007)1652 5178 y Fy([)p Fx(k)q Fy(])1753 5213 y FB(=)22 b(\002)1905 5178 y Fy([)p Fv(\000)p Fx(k)q Fy(])2035 5213 y FB(\()p FA(w)r FB(\))p FA(e)2199 5175 y Fv(\000)p Fx(ik)2322 5142 y(w)p 2322 5156 50 4 v 2331 5204 a(")2386 5213 y FA(:)71 5404 y FB(No)n(w,)33 b(taking)f(adv)-5 b(an)n(tage)31 b(of)h(the)h(fact)f(that)h(the)g(co)r(e\016cien)n(ts)2129 5383 y Fz(e)2120 5404 y FB(\007)2185 5374 y Fy([)p Fx(k)q Fy(])2295 5404 y FB(do)g(not)f(dep)r(end)h(on)f FA(w)r FB(,)j(w)n(e)d(will)g(obtain)g(sharp)71 5504 y(b)r(ounds)40 b(for)g(the)g(co)r(e\016cien)n(ts)1109 5483 y Fz(e)1100 5504 y FB(\007)1165 5474 y Fy([)p Fx(k)q Fy(])1283 5504 y FB(with)g FA(k)47 b(<)c FB(0.)74 b(Since)40 b(w)n(e)g(are)f(dealing)g (with)i(real)e(analytic)h(functions,)j(the)p Black 1919 5753 a(42)p Black eop end %%Page: 43 43 TeXDict begin 43 42 bop Black Black 71 272 a FB(co)r(e\016cien)n(ts)501 251 y Fz(e)492 272 y FB(\007)557 242 y Fy([)p Fx(k)q Fy(])664 272 y FB(with)29 b FA(k)f(>)d FB(0)j(will)h(satisfy)g(the)g (same)f(b)r(ounds.)41 b(Let)29 b(us)g(consider)e FA(w)h FB(=)d FA(w)3002 242 y Fv(\003)3066 272 y FB(=)f FA(u)3203 242 y Fv(\003)3260 272 y FB(+)19 b FA(")p Fw(C)5 b FB(\()p FA(u)3512 242 y Fv(\003)3549 272 y FA(;)14 b FB(0\))29 b(with)71 372 y FA(u)119 342 y Fv(\003)180 372 y FB(=)22 b FA(i)p FB(\()p FA(a)c Fw(\000)g FA(s")c FB(ln\(1)p FA(=")p FB(\)\).)37 b(Then,)842 477 y Fz(\014)842 527 y(\014)842 576 y(\014)878 551 y(e)869 572 y FB(\007)934 538 y Fy([)p Fx(k)q Fy(])1012 477 y Fz(\014)1012 527 y(\014)1012 576 y(\014)1063 572 y Fw(\024)23 b(j)p FA(k)s Fw(j)1243 538 y Fv(\000)p Fy(1)1355 572 y FB(sup)1346 659 y Fx(w)r Fv(2)1452 644 y Fk(e)1441 659 y Fx(R)1504 477 y Fz(\014)1504 527 y(\014)1504 576 y(\014)1532 572 y FB(\002)1597 538 y Fy([)p Fv(\000)p Fx(k)q Fy(])1727 572 y FB(\()p FA(w)r FB(\))1852 477 y Fz(\014)1852 527 y(\014)1852 576 y(\014)1894 572 y FA(e)1933 535 y Fv(\000)1995 495 y(j)p Fx(k)q Fv(j)p 1995 516 77 4 v 2017 563 a Fx(")2081 541 y FB(\()2113 535 y Fx(a)p Fv(\000)p Fx(s")12 b Fy(ln)2352 512 y Fu(1)p 2352 521 29 3 v 2352 555 a Fm(")2390 541 y FB(\))2422 535 y Fv(\000j)p Fx(k)q Fv(j)p Fy(Im)f(\()p Fv(C)s Fy(\()p Fx(u)2770 510 y Fl(\003)2805 535 y Fx(;)p Fy(0\)\))1063 826 y Fw(\024)23 b(j)p FA(k)s Fw(j)1243 792 y Fv(\000)p Fy(1)1455 826 y FB(sup)1346 913 y Fy(\()p Fx(w)r(;\034)7 b Fy(\))p Fv(2)1561 898 y Fk(e)1551 913 y Fx(R)n Fv(\002)p Fn(T)1704 826 y Fw(j)p FB(\002\()p FA(w)r(;)14 b(\034)9 b FB(\))p Fw(j)15 b FA(e)2076 789 y Fv(\000)2137 748 y(j)p Fx(k)q Fv(j)p 2137 770 77 4 v 2160 817 a Fx(")2224 795 y FB(\()2256 789 y Fx(a)p Fv(\000)p Fx(s")d Fy(ln)2495 766 y Fu(1)p 2495 775 29 3 v 2495 809 a Fm(")2533 795 y FB(\))2565 789 y Fv(\000j)p Fx(k)q Fv(j)p Fy(Im)f(\()p Fv(C)s Fy(\()p Fx(u)2913 764 y Fl(\003)2948 789 y Fx(;)p Fy(0\)\))3057 826 y FA(:)71 1056 y FB(Then,)28 b(taking)f(in)n(to)g(accoun)n(t)g(\(95\))g(and)g (Theorem)g(4.17,)g(w)n(e)g(ha)n(v)n(e)f(that)i(for)f FA(k)f(<)d FB(0,)684 1162 y Fz(\014)684 1211 y(\014)684 1261 y(\014)721 1236 y(e)712 1257 y FB(\007)777 1223 y Fy([)p Fx(k)q Fy(])855 1162 y Fz(\014)855 1211 y(\014)855 1261 y(\014)906 1257 y Fw(\024)f FA(K)6 b(")270 b FB(sup)1123 1331 y Fy(\()p Fx(u;\034)7 b Fy(\))p Fv(2)p Fx(R)1367 1342 y Fm(s)h Fu(ln\(1)p Fm(=")p Fu(\))p Fm(;d)1632 1354 y Fu(3)1669 1331 y Fv(\002)p Fn(T)1774 1162 y Fz(\014)1774 1211 y(\014)1774 1261 y(\014)1801 1257 y FA(@)1845 1269 y Fx(u)1900 1236 y Fz(e)1889 1257 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))2152 1162 y Fz(\014)2152 1211 y(\014)2152 1261 y(\014)2194 1257 y FA(e)2233 1220 y Fv(\000)2295 1179 y(j)p Fx(k)q Fv(j)p 2295 1200 77 4 v 2317 1248 a Fx(")2381 1226 y FB(\()2413 1220 y Fx(a)p Fv(\000)p Fx(s")j Fy(ln)2652 1197 y Fu(1)p 2652 1206 29 3 v 2652 1240 a Fm(")2690 1226 y FB(\))2722 1220 y Fv(\000j)p Fx(k)q Fv(j)p Fy(Im)f(\()p Fv(C)s Fy(\()p Fx(u)3070 1194 y Fl(\003)3105 1220 y Fx(;)p Fy(0\)\))3214 1257 y FA(:)71 1502 y FB(Therefore,)22 b(to)g(obtain)f(the)i(b)r(ounds)f(for)1366 1481 y Fz(e)1356 1502 y FB(\007)1421 1472 y Fy([)p Fx(k)q Fy(])1522 1502 y FB(with)g FA(k)k(<)d FB(0,)f(it)h(only)e(remains)h(to)f(use)h(b)r (ounds)g(\(94\))g(and)g(the)g(prop)r(erties)71 1602 y(of)27 b Fw(C)33 b FB(giv)n(en)26 b(in)i(Theorem)f(4.17)f(and)i(Prop)r (osition)e(4.18.)35 b(Then,)28 b(w)n(e)f(obtain)h(that)f(for)h FA(k)d(<)e FB(0)1094 1707 y Fz(\014)1094 1757 y(\014)1094 1807 y(\014)1131 1782 y(e)1121 1803 y FB(\007)1186 1768 y Fy([)p Fx(k)q Fy(])1264 1707 y Fz(\014)1264 1757 y(\014)1264 1807 y(\014)1315 1803 y Fw(\024)1413 1746 y FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1625 1716 y Fx(\021)1665 1746 y FA(e)1704 1716 y Fv(\000)1765 1694 y Fm(a)p 1765 1703 33 3 v 1767 1736 a(")p 1413 1784 399 4 v 1443 1860 a Fw(j)14 b FB(ln)g FA(")p Fw(j)p FA(")1664 1836 y Fx(`)p Fv(\000)p Fy(1)1821 1803 y FA(e)1860 1765 y Fv(\000)1922 1736 y Fl(j)p Fm(k)q Fl(j\000)p Fu(1)p 1922 1752 144 3 v 1979 1785 a Fm(")2075 1697 y Fz(\000)2113 1765 y Fx(a)p Fy(+)p Fx("s)e Fy(log)g Fx(")p Fy(+)p Fx(b)2482 1773 y Fu(11)2544 1765 y Fv(j)p Fx(\026)p Fv(j)p Fx(")2655 1740 y Fm(\021)q Fu(+1)2762 1697 y Fz(\001)2804 1803 y FA(:)71 2020 y FB(Finally)-7 b(,)30 b(the)h(b)r(ounds)f(of)915 1999 y Fz(e)905 2020 y FB(\007)970 1990 y Fy([)p Fx(k)q Fy(])1078 2020 y FB(lead)g(easily)f(to)h(the)g(desired)f(b)r(ounds)h (of)2420 1999 y Fz(e)2408 2020 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))31 b(for)e(\()p FA(u;)14 b(\034)9 b FB(\))28 b Fw(2)3135 1952 y Fz(\000)3173 2020 y FA(R)3236 2035 y Fx(s)12 b Fy(ln)o(\(1)p Fx(=")p Fy(\))p Fx(;d)3539 2043 y Fu(3)3594 2020 y Fw(\\)19 b Ft(R)3728 1952 y Fz(\001)3786 2020 y Fw(\002)71 2119 y Ft(T)p FB(.)p 3790 2119 4 57 v 3794 2067 50 4 v 3794 2119 V 3843 2119 4 57 v 71 2329 a Fp(4.7.2)94 b(Study)32 b(of)g(the)g(di\013erence)f(b)s(et)m(w)m(een)g (the)h(in)m(v)-5 b(arian)m(t)33 b(manifolds)e(for)h(the)f(case)h FA(`)19 b Fw(\000)f FB(2)p FA(r)25 b Fw(\025)e FB(0)71 2483 y(Recall)j(that)i(when)f FA(`)17 b Fw(\000)g FB(2)p FA(r)25 b Fw(\025)e FB(0,)j(Hyp)r(othesis)h Fp(HP5)g FB(b)r(ecomes)g FA(\021)f Fw(\025)d FA(`)17 b Fw(\000)g FB(2)p FA(r)r FB(.)37 b(F)-7 b(or)26 b(this)h(reason,)f(as)g(w)n(e)h (did)g(in)g(Section)71 2582 y(4.6.2,)f(w)n(e)h(will)h(denote)34 b(^)-48 b FA(\026)23 b FB(=)g FA(\026")1089 2552 y Fx(\021)r Fv(\000)p Fx(`)p Fy(+2)p Fx(r)1325 2582 y FB(.)195 2682 y(As)g(w)n(e)g(ha)n(v)n(e)e(done)i(for)f(the)h(case)f FA(`)9 b Fw(\000)g FB(2)p FA(r)24 b(<)e FB(0)h(in)g(Section)f(4.7.1,)h (w)n(e)f(consider)g(the)h(function)g(\001\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)f FA(T)3540 2652 y Fx(u)3582 2682 y FB(\()p FA(u;)14 b(\034)9 b FB(\))g Fw(\000)71 2782 y FA(T)132 2751 y Fx(s)166 2782 y FB(\()p FA(u;)14 b(\034)9 b FB(\))29 b(de\014ned)f(in)g(\(76\))f(in)h(the)g(domain)f FA(R)1545 2794 y Fx(\024;d)1665 2782 y FB(=)c FA(D)1824 2751 y Fx(s)1822 2805 y(\024;d)1937 2782 y Fw(\\)c FA(D)2082 2751 y Fx(u)2080 2805 y(\024;d)2205 2782 y FB(de\014ned)28 b(in)g(\(27\))f(\(see)h(also)e(Figure)h(3\).)195 2881 y(No)n(w)h(\001)f(satis\014es)g(the)h(partial)f(di\013eren)n(tial)g (equation)1824 3010 y Fz(e)1807 3031 y Fw(L)1864 3043 y Fx(")1899 3031 y FA(\030)h FB(=)22 b(0)p FA(;)1588 b FB(\(96\))71 3191 y(where)328 3170 y Fz(e)311 3191 y Fw(L)368 3203 y Fx(")431 3191 y FB(is)28 b(the)g(op)r(erator)e (de\014ned)i(in)g(\(78\))f(and)g FA(G)h FB(no)n(w)f(is)420 3380 y FA(G)p FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)873 3324 y(1)p 777 3361 233 4 v 777 3437 a(2)p FA(p)861 3449 y Fy(0)898 3437 y FB(\()p FA(u)p FB(\))1034 3380 y(\()p FA(@)1110 3392 y Fx(u)1154 3380 y FA(T)1215 3346 y Fx(u)1203 3401 y Fy(1)1257 3380 y FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b(+)f FA(@)1597 3392 y Fx(u)1641 3380 y FA(T)1702 3346 y Fx(s)1690 3401 y Fy(1)1736 3380 y FB(\()p FA(u;)c(\034)9 b FB(\)\))786 3621 y(+)886 3565 y(^)-49 b FA(\026")968 3535 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)p 879 3602 239 4 v 903 3678 a FA(p)945 3690 y Fy(0)982 3678 y FB(\()p FA(u)p FB(\))1141 3508 y Fz(Z)1224 3528 y Fy(1)1187 3697 y(0)1275 3621 y FA(@)1319 3633 y Fx(p)1377 3600 y Fz(b)1358 3621 y FA(H)1427 3633 y Fy(1)1478 3504 y Fz(\022)1539 3621 y FA(q)1576 3633 y Fy(0)1613 3621 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1804 3633 y Fy(0)1841 3621 y FB(\()p FA(u)p FB(\))19 b(+)2065 3565 y FA(s@)2148 3577 y Fx(u)2191 3565 y FA(T)2252 3535 y Fx(u)2240 3585 y Fy(1)2294 3565 y FB(\()p FA(u;)14 b(\034)9 b FB(\))20 b(+)e(\(1)g Fw(\000)g FA(s)p FB(\))p FA(@)2881 3577 y Fx(u)2925 3565 y FA(T)2986 3535 y Fx(s)2974 3585 y Fy(1)3020 3565 y FB(\()p FA(u;)c(\034)9 b FB(\))p 2065 3602 1151 4 v 2544 3678 a FA(p)2586 3690 y Fy(0)2623 3678 y FB(\()p FA(u)p FB(\))3225 3621 y FA(;)14 b(\034)3307 3504 y Fz(\023)3396 3621 y FA(ds;)3703 3506 y FB(\(97\))71 3839 y(where)333 3818 y Fz(b)314 3839 y FA(H)383 3851 y Fy(1)451 3839 y FB(is)30 b(the)h(function)h (de\014ned)f(in)g(\(34\))f(and)g FA(T)1804 3809 y Fx(u;s)1898 3839 y FB(\()p FA(u;)14 b(\034)9 b FB(\))29 b(=)e FA(T)2262 3851 y Fy(0)2299 3839 y FB(\()p FA(u)p FB(\))21 b(+)f FA(T)2578 3799 y Fx(u;s)2566 3861 y Fy(1)2671 3839 y FB(\()p FA(u;)14 b(\034)9 b FB(\))32 b(with)f FA(@)3133 3851 y Fx(u)3176 3839 y FA(T)3225 3851 y Fy(0)3262 3839 y FB(\()p FA(u)p FB(\))d(=)g FA(p)3537 3809 y Fy(2)3537 3860 y(0)3574 3839 y FB(\()p FA(u)p FB(\))j(and)71 3939 y FA(T)132 3899 y Fx(u;s)120 3961 y Fy(1)250 3939 y FB(are)24 b(giv)n(en)g(in)h(Theorems)f(4.4)g(and)h(4.8.)35 b(Let)25 b(us)g(p)r(oin)n(t)g(out)g(that)g(the)g(only)g(di\013erence)g(b)r(et)n (w)n(een)f(the)i(function)f FA(G)71 4038 y FB(de\014ned)i(in)h(\(97\))f (from)f(the)i(one)f(de\014ned)g(in)h(\(79\))f(is)g(the)g(dep)r(endence) h(on)f(the)g(parameters.)35 b(The)27 b(\014rst)g(one)g(dep)r(ends)71 4138 y(on)g FA(\026)h FB(and)f FA(")h FB(whereas)e(the)i(second)f(one)g (dep)r(ends)h(on)34 b(^)-48 b FA(\026)p FB(,)27 b(whic)n(h)h(has)f(b)r (een)h(de\014ned)g(in)g(terms)f(of)h FA(\026)f FB(and)h FA(")f FB(in)h(\(58\).)195 4238 y(As)d(w)n(e)g(ha)n(v)n(e)e(done)i(in)g (Section)g(4.7.1,)f(to)h(obtain)f(the)h(asymptotic)g(expression)e(of)i (the)g(di\013erence)g(\001,)g(w)n(e)g(lo)r(ok)f(for)71 4337 y(a)j(solution)g FA(\030)490 4349 y Fy(0)555 4337 y FB(of)34 b(\(77\))28 b(of)f(the)h(form)1438 4437 y FA(\030)1474 4449 y Fy(0)1511 4437 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)e FA(")1855 4403 y Fv(\000)p Fy(1)1944 4437 y FA(u)c Fw(\000)g FA(\034)28 b FB(+)18 b Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))71 4567 y(with)30 b Fw(C)k FB(a)c(function)g(2)p FA(\031)s FB(-p)r(erio)r(dic)f (in)h FA(\034)9 b FB(,)31 b(suc)n(h)e(that)h(\()p FA(\030)1816 4579 y Fy(0)1854 4567 y FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))31 b(is)f(injectiv)n(e)f(in)h FA(R)2772 4579 y Fx(\024;d)2890 4567 y Fw(\002)19 b Ft(T)3029 4579 y Fx(\033)3074 4567 y FB(.)43 b(Then,)31 b(w)n(e)e(will)h(write)71 4666 y(\001)e(as)f FA(\030)g FB(=)22 b(\007)d Fw(\016)e FA(\030)599 4678 y Fy(0)665 4666 y FB(for)27 b(some)g(function)h(\007.) p Black 71 4806 a Fp(Theorem)36 b(4.21.)p Black 42 w Fs(L)l(et)d(us)g(c)l(onsider)h(the)g(c)l(onstants)e FA(d)1864 4818 y Fy(2)1931 4806 y FA(>)d FB(0)k Fs(de\014ne)l(d)h(in)f(The)l(or)l (em)i(4.8)f(and)g FA(\024)3194 4818 y Fy(6)3261 4806 y FA(>)29 b FB(0)k Fs(in)g(The)l(or)l(em)71 4906 y(4.16,)39 b FA(d)329 4918 y Fy(3)399 4906 y FA(<)33 b(d)540 4918 y Fy(2)612 4906 y Fs(and)j FA(")818 4918 y Fy(0)888 4906 y FA(>)c FB(0)j Fs(smal)t(l)h(enough)g(and)g FA(\024)1791 4918 y Fy(8)1861 4906 y FA(>)c(\024)2006 4918 y Fy(6)2078 4906 y Fs(big)k(enough,)i(which)f(might)e(dep)l(end)i(on)e(the)g(pr)l (evious)71 5005 y(c)l(onstants.)j(Then,)30 b(for)g FA(")23 b Fw(2)g FB(\(0)p FA(;)14 b(")1133 5017 y Fy(0)1170 5005 y FB(\))p Fs(,)30 b FA(\026)23 b Fw(2)h FA(B)t FB(\()p FA(\026)1558 5017 y Fy(0)1595 5005 y FB(\))30 b Fs(and)g(any)f FA(\024)23 b Fw(\025)g FA(\024)2183 5017 y Fy(8)2249 5005 y Fs(such)30 b(that)f FA("\024)23 b(<)f(a)p Fs(,)30 b(ther)l(e)f(exists)g(a)h(r)l(e)l(al-analytic)71 5105 y(function)g Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b(:)g FA(R)773 5117 y Fx(\024;d)867 5125 y Fu(3)921 5105 y Fw(\002)18 b Ft(T)1059 5117 y Fx(\033)1127 5105 y Fw(!)23 b Ft(C)30 b Fs(such)g(that)g FA(\030)1718 5117 y Fy(0)1755 5105 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)f FA(")2100 5075 y Fv(\000)p Fy(1)2188 5105 y FA(u)18 b Fw(\000)g FA(\034)29 b FB(+)18 b Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))30 b Fs(is)g(solution)g(of)48 b FB(\(96\))29 b Fs(and)1285 5255 y FB(\()p FA(\030)1353 5267 y Fy(0)1390 5255 y FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))25 b(=)1811 5187 y Fz(\000)1849 5255 y FA(")1888 5220 y Fv(\000)p Fy(1)1977 5255 y FA(u)18 b Fw(\000)g FA(\034)28 b FB(+)18 b Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)2598 5187 y Fz(\001)71 5404 y Fs(is)30 b(inje)l(ctive.)195 5504 y(Mor)l(e)l(over,)f(ther)l(e)e (exists)f(a)g(c)l(onstant)g FA(b)1443 5516 y Fy(15)1536 5504 y FA(>)d FB(0)i Fs(indep)l(endent)i(of)h FA(\026)p Fs(,)f FA(")f Fs(and)h FA(\024)p Fs(,)g(such)g(that)f(for)h FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)f FA(R)3508 5516 y Fx(\024;d)3602 5524 y Fu(3)3649 5504 y Fw(\002)11 b Ft(T)3780 5516 y Fx(\033)3825 5504 y Fs(,)p Black 1919 5753 a FB(43)p Black eop end %%Page: 44 44 TeXDict begin 44 43 bop Black Black Black 195 272 a Fw(\017)p Black 41 w Fs(If)31 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b(>)e FB(0)p Fs(,)1624 506 y Fw(jC)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p Fw(j)24 b(\024)2057 450 y FA(b)2093 462 y Fy(15)2177 450 y Fw(j)6 b FB(^)-48 b FA(\026)p Fw(j)14 b FA(")2326 420 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)p 2034 487 464 4 v 2034 579 a Fw(j)p FA(u)2105 555 y Fy(2)2161 579 y FB(+)k FA(a)2288 555 y Fy(2)2325 579 y Fw(j)2348 537 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)1537 760 y Fw(j)p FA(@)1604 772 y Fx(u)1647 760 y Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p Fw(j)24 b(\024)2045 704 y FA(b)2081 716 y Fy(15)2165 704 y Fw(j)7 b FB(^)-49 b FA(\026)p Fw(j)14 b FA(")2314 674 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fv(\000)p Fy(1)p 2034 741 525 4 v 2034 833 a FA(\024)g Fw(j)p FA(u)2167 809 y Fy(2)2222 833 y FB(+)k FA(a)2349 809 y Fy(2)2386 833 y Fw(j)2409 791 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)2569 760 y FA(:)p Black 195 1036 a Fw(\017)p Black 41 w Fs(If)31 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b FB(=)e(0)p Fs(,)1590 1215 y Fw(jC)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p Fw(j)24 b(\024)e FA(b)2026 1227 y Fy(15)2110 1215 y Fw(j)7 b FB(^)-49 b FA(\026)p Fw(j)14 b FB(ln)2303 1144 y Fz(\014)2303 1194 y(\014)2331 1215 y FA(u)2379 1181 y Fy(2)2434 1215 y FB(+)k FA(a)2561 1181 y Fy(2)2598 1144 y Fz(\014)2598 1194 y(\014)1503 1396 y Fw(j)p FA(@)1570 1408 y Fx(u)1613 1396 y Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p Fw(j)24 b(\024)2049 1340 y FA(b)2085 1352 y Fy(15)2169 1340 y Fw(j)7 b FB(^)-49 b FA(\026)p Fw(j)p 2000 1377 314 4 v 2000 1453 a(j)p FA(u)2071 1429 y Fy(2)2127 1453 y FB(+)18 b FA(a)2254 1429 y Fy(2)2291 1453 y Fw(j)2324 1396 y FA(:)195 1656 y FB(T)-7 b(o)30 b(study)g(the)h(\014rst)f(order)e(of)i(the)h (di\013erence)f(b)r(et)n(w)n(een)g(the)h(in)n(v)-5 b(arian)n(t)28 b(manifolds)i(when)h FA(`)19 b Fw(\000)h FB(2)p FA(r)29 b FB(=)e(0,)k(w)n(e)e(need)71 1756 y(a)j(b)r(etter)h(kno)n(wledge)e(of) h(the)h(b)r(eha)n(vior)e(of)i(the)f(function)i Fw(C)j FB(in)32 b(the)h(inner)f(domains)g(\(30\).)51 b(The)33 b(next)g(prop)r(osition)71 1856 y(giv)n(es)f(the)j(\014rst)e(order)g (asymptotic)g(terms)h(of)f Fw(C)39 b FB(close)33 b(to)h FA(u)e FB(=)h FA(ia)p FB(.)56 b(The)33 b(study)h(close)f(to)h FA(u)f FB(=)g Fw(\000)p FA(ia)g FB(can)h(b)r(e)g(done)71 1955 y(analogously)-7 b(.)p Black 71 2121 a Fp(Prop)s(osition)30 b(4.22.)p Black 40 w Fs(Assume)f FA(`)22 b FB(=)h(2)p FA(r)r Fs(.)39 b(L)l(et)29 b FA(c)1604 2133 y Fy(1)1671 2121 y Fs(b)l(e)h(a)g(c)l(onstant)f(as)h(in)g(The)l(or)l(em)h(4.16.)40 b(We)30 b(c)l(onsider)h FA(c)3460 2133 y Fy(2)3520 2121 y FA(>)23 b(c)3644 2133 y Fy(1)3711 2121 y Fs(and)1768 2286 y FA(\014)p 1697 2323 195 4 v 1697 2399 a(\014)f FB(+)d(1)1924 2343 y FA(<)k(\015)k(<)c FB(1)p FA(;)1468 b FB(\(98\))71 2569 y Fs(wher)l(e)30 b FA(r)c FB(=)d FA(\013=\014)34 b Fs(has)c(b)l(e)l(en)g(de\014ne)l(d)g(in)g(Hyp)l (othesis)h Fo(HP2)p Fs(.)195 2669 y(Then,)d(for)f(any)f FA(")757 2681 y Fy(0)817 2669 y FA(>)c FB(0)p Fs(,)27 b(ther)l(e)f(exist)f(a)h(c)l(onstant)f FA(C)6 b FB(\()h(^)-49 b FA(\026)q(;)14 b(")p FB(\))25 b Fs(de\014ne)l(d)h(for)h FB(\()7 b(^)-49 b FA(\026;)14 b(")p FB(\))23 b Fw(2)h FA(B)t FB(\()7 b(^)-49 b FA(\026)2912 2681 y Fy(0)2949 2669 y FB(\))10 b Fw(\002)g FB(\(0)p FA(;)k(")3216 2681 y Fy(0)3252 2669 y FB(\))26 b Fs(dep)l(ending)h(r)l(e)l(al-)71 2768 y(analytic)l(al)t(ly)j(in)35 b FB(^)-49 b FA(\026)29 b Fs(and)f(a)h(c)l(onstant)e FA(b)1282 2780 y Fy(16)1375 2768 y FA(>)22 b FB(0)28 b Fs(such)g(that)g Fw(j)p FA(C)6 b FB(\()h(^)-49 b FA(\026)q(;)14 b(")p FB(\))p Fw(j)23 b(\024)f FA(b)2335 2780 y Fy(16)2405 2768 y Fw(j)7 b FB(^)-49 b FA(\026)q Fw(j)28 b Fs(and,)h(if)g FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)3090 2701 y Fz(\000)3128 2768 y FA(D)3199 2738 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)3197 2789 y(\024)3236 2797 y Fu(8)3268 2789 y Fx(;c)3318 2797 y Fu(2)3407 2768 y Fw(\\)19 b FA(D)3552 2738 y Fy(in)o Fx(;)p Fy(+)p Fx(;s)3550 2789 y(\024)3589 2797 y Fu(8)3621 2789 y Fx(;c)3671 2797 y Fu(2)3733 2701 y Fz(\001)3786 2768 y Fw(\002)71 2868 y Ft(T)126 2880 y Fx(\033)171 2868 y Fs(,)949 2940 y Fz(\014)949 2990 y(\014)977 3011 y Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b Fw(\000)f FA(C)6 b FB(\()h(^)-49 b FA(\026;)14 b(")p FB(\))19 b(+)f FA(\026F)1782 3023 y Fy(1)1819 3011 y FB(\()p FA(\034)9 b FB(\))20 b(+)25 b(^)-49 b FA(\026)2081 2977 y Fy(2)2118 3011 y FA(b)14 b FB(ln\()p FA(u)k Fw(\000)g FA(ia)p FB(\))2523 2940 y Fz(\014)2523 2990 y(\014)2574 3011 y Fw(\024)2685 2955 y FA(b)2721 2967 y Fy(16)2791 2955 y Fw(j)7 b FB(^)-49 b FA(\026)p Fw(j)p FA(")p 2671 2992 268 4 v 2671 3068 a Fw(j)p FA(u)18 b Fw(\000)g FA(ia)p Fw(j)2949 3011 y FA(:)71 3216 y Fs(We)28 b(r)l(e)l(c)l(al)t(l)h(that)e FA(\015)33 b Fs(enters)27 b(in)h(the)g(de\014nitions)h(of)g FA(D)1733 3186 y Fy(in)o Fx(;)p Fy(+)p Fx(;u)1731 3237 y(\024)1770 3245 y Fu(8)1802 3237 y Fx(;c)1852 3245 y Fu(2)1950 3216 y Fs(and)f FA(D)2180 3186 y Fy(in)p Fx(;)p Fy(+)p Fx(;s)2178 3237 y(\024)2217 3245 y Fu(8)2249 3237 y Fx(;c)2299 3245 y Fu(2)2362 3216 y Fs(,)g Fw(C)33 b Fs(is)28 b(the)g(function)g(given)h(in)f(The)l(or)l(em)h(4.21)71 3316 y(and)h(the)g(function)g FA(F)750 3328 y Fy(1)817 3316 y Fs(and)g(the)g(c)l(onstant)f FA(b)h Fs(have)h(b)l(e)l(en)e (de\014ne)l(d)h(in)36 b FB(\(74\))30 b Fs(and)38 b FB(\(75\))30 b Fs(r)l(esp)l(e)l(ctively.)195 3415 y(Ther)l(efor)l(e,)38 b(if)e(we)f(c)l(onsider)g(the)g(function)g FA(g)i Fs(given)e(in)f(The)l (or)l(em)i(4.12,)i(by)d(Pr)l(op)l(osition)h(4.15,)i(ther)l(e)d(exists)f (a)71 3515 y(c)l(onstant)29 b FA(b)438 3527 y Fy(17)531 3515 y FA(>)23 b FB(0)29 b Fs(such)h(that,)g(if)g FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)1451 3448 y Fz(\000)1489 3515 y FA(D)1560 3485 y Fy(in)o Fx(;)p Fy(+)p Fx(;u)1558 3535 y(\024)1597 3543 y Fu(8)1629 3535 y Fx(;c)1679 3543 y Fu(2)1767 3515 y Fw(\\)19 b FA(D)1912 3485 y Fy(in)p Fx(;)p Fy(+)p Fx(;s)1910 3535 y(\024)1949 3543 y Fu(8)1981 3535 y Fx(;c)2031 3543 y Fu(2)2094 3448 y Fz(\001)2150 3515 y Fw(\002)f Ft(T)2288 3527 y Fx(\033)2333 3515 y Fs(,)852 3686 y Fz(\014)852 3736 y(\014)880 3756 y Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))18 b Fw(\000)g FA(C)6 b FB(\()h(^)-49 b FA(\026)q(;)14 b(")p FB(\))k(+)25 b(^)-49 b FA(\026)1631 3722 y Fy(2)1669 3756 y FA(b)14 b FB(ln)f FA(")18 b Fw(\000)25 b FB(^)-49 b FA(\026g)2048 3689 y Fz(\000)2086 3756 y FA(")2125 3722 y Fv(\000)p Fy(1)2214 3756 y FB(\()p FA(u)18 b Fw(\000)g FA(ia)p FB(\))p FA(;)c(\034)2582 3689 y Fz(\001)2620 3686 y(\014)2620 3736 y(\014)2671 3756 y Fw(\024)2782 3700 y FA(b)2818 3712 y Fy(17)2888 3700 y Fw(j)7 b FB(^)-49 b FA(\026)p Fw(j)p FA(")p 2769 3737 V 2769 3813 a Fw(j)p FA(u)18 b Fw(\000)g FA(ia)p Fw(j)3046 3756 y FA(:)195 3983 y FB(The)28 b(pro)r(ofs)f(of)g(Theorem)g (4.21)f(and)i(Prop)r(osition)e(4.22)g(are)h(done)g(in)h(Section)f(9.3.) 195 4083 y(As)k(w)n(e)f(ha)n(v)n(e)f(explained)h(in)h(Section)f(4.7.1,) g(since)g(\001)g(is)h(a)f(solution)f(of)i(the)f(same)g(homogeneous)f (linear)g(partial)71 4182 y(di\013eren)n(tial)h(equation)g(as)g FA(\030)972 4194 y Fy(0)1040 4182 y FB(giv)n(en)f(b)n(y)h(Theorem)g (4.21,)g(there)g(exists)g(a)g(2)p FA(\031)s FB(-p)r(erio)r(dic)g (function)h(\007)f(suc)n(h)g(that)h(\001)d(=)71 4282 y(\007)18 b Fw(\016)g FA(\030)250 4294 y Fy(0)287 4282 y FB(,)28 b(whic)n(h)g(giv)n(es)1344 4382 y(\001\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)f(\007)1798 4314 y Fz(\000)1835 4382 y FA(")1874 4347 y Fv(\000)p Fy(1)1963 4382 y FA(u)18 b Fw(\000)g FA(\034)28 b FB(+)18 b Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))2502 4314 y Fz(\001)2554 4382 y FA(:)1126 b FB(\(99\))71 4531 y(and)27 b(considering)g(its)g(F)-7 b(ourier)27 b(series)f(w)n(e)i(ha)n(v)n(e)1302 4731 y(\001\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)1677 4652 y Fz(X)1676 4830 y Fx(k)q Fv(2)p Fn(Z)1810 4731 y FB(\007)1875 4696 y Fy([)p Fx(k)q Fy(])1953 4731 y FA(e)1992 4695 y Fx(ik)2051 4701 y FB(\()2085 4695 y Fx(")2116 4670 y Fl(\000)p Fu(1)2193 4695 y Fx(u)p Fv(\000)p Fx(\034)7 b Fy(+)p Fv(C)s Fy(\()p Fx(u;\034)g Fy(\))2561 4701 y FB(\))2596 4731 y FA(:)1042 b FB(\(100\))195 4997 y(No)n(w)31 b(w)n(e)h(are)e(going)h(to)g(\014nd)h (the)g(\014rst)f(asymptotic)g(term)h(of)g(\001)f(whic)n(h)h(will)g(b)r (e)g(strongly)e(related)h(with)h(\()p FA( )3721 4967 y Fx(u)3718 5017 y Fy(0)3786 4997 y Fw(\000)71 5096 y FA( )128 5066 y Fx(s)125 5117 y Fy(0)163 5096 y FB(\)\()p FA(")266 5066 y Fv(\000)p Fy(1)356 5096 y FB(\()p FA(u)18 b Fw(\000)g FA(ia)p FB(\))p FA(;)c(\034)9 b FB(\),)29 b(b)r(eing)f FA( )1089 5056 y Fx(u;s)1086 5118 y Fy(0)1212 5096 y FB(the)g(solutions)f(of)h(the)g(inner)g(equation)f(giv)n(en)g (in)h(Theorem)f(4.12.)37 b(W)-7 b(e)28 b(in)n(tro)r(duce)71 5196 y(the)g(auxiliary)e(function)1285 5312 y(\001)1354 5277 y Fy(+)1354 5334 y(0)1409 5312 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)1716 5233 y Fz(X)1715 5412 y Fx(k)q(<)p Fy(0)1850 5312 y FB(\007)1915 5269 y Fy([)p Fx(k)q Fy(])1915 5334 y(0)1993 5312 y FA(e)2032 5276 y Fx(ik)2091 5282 y FB(\()2124 5276 y Fx(")2155 5251 y Fl(\000)p Fu(1)2233 5276 y Fx(u)p Fv(\000)p Fx(\034)7 b Fy(+)p Fv(C)s Fy(\()p Fx(u;\034)g Fy(\))2601 5282 y FB(\))3661 5312 y(\(101\))p Black 1919 5753 a(44)p Black eop end %%Page: 45 45 TeXDict begin 45 44 bop Black Black 71 272 a FB(with)615 454 y(\007)680 411 y Fy([)p Fx(k)q Fy(])680 477 y(0)781 454 y FB(=)893 395 y FA(C)958 364 y Fy(2)952 415 y(+)1014 395 y FB(^)-49 b FA(\026)p 878 435 194 4 v 878 511 a(")917 487 y Fy(2)p Fx(r)r Fv(\000)p Fy(1)1082 454 y FA(\037)1134 420 y Fy([)p Fx(k)q Fy(])1212 454 y FB(\()7 b(^)-49 b FA(\026)q FB(\))p FA(e)1366 417 y Fv(\000)1427 377 y(j)p Fx(k)q Fv(j)p Fx(a)p 1427 398 113 4 v 1468 446 a(")2861 454 y FB(if)28 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b(>)e FB(0)354 b(\(102\))615 672 y(\007)680 629 y Fy([)p Fx(k)q Fy(])680 694 y(0)781 672 y FB(=)893 612 y FA(C)958 582 y Fy(2)952 633 y(+)1014 612 y FB(^)-49 b FA(\026)p 878 653 194 4 v 878 729 a(")917 705 y Fy(2)p Fx(r)r Fv(\000)p Fy(1)1082 672 y FA(\037)1134 638 y Fy([)p Fx(k)q Fy(])1212 672 y FB(\()7 b(^)-49 b FA(\026)q FB(\))p FA(e)1366 634 y Fv(\000)1427 594 y(j)p Fx(k)q Fv(j)p Fx(a)p 1427 615 113 4 v 1468 663 a(")1549 634 y Fv(\000)p Fx(i)p Fv(j)p Fx(k)q Fv(j)p Fy(\()p Fv(\000)p Fx(C)t Fy(\()5 b(^)-38 b Fx(\026)q(;")p Fy(\)+)5 b(^)-38 b Fx(\026)2065 609 y Fu(2)2098 634 y Fx(b)11 b Fy(ln)g Fx(")p Fy(\))2838 672 y FB(if)28 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b FB(=)e(0)p FA(;)354 b FB(\(103\))71 854 y(where)305 787 y Fz(\010)354 854 y FA(\037)406 824 y Fx(k)446 854 y FB(\()7 b(^)-49 b FA(\026)q FB(\))561 787 y Fz(\011)609 888 y Fx(k)q(<)p Fy(0)757 854 y FB(are)21 b(the)h(co)r(e\016cien)n(ts)g (giv)n(en)f(in)h(Theorem)f(4.12)f(and)i FA(C)6 b FB(\()h(^)-49 b FA(\026)q(;)14 b(")p FB(\))22 b(and)f FA(b)h FB(are)f(the)h(constan)n (ts)f(obtained)71 968 y(in)28 b(Prop)r(ositions)d(4.22)i(and)g(4.15)f (resp)r(ectiv)n(ely)-7 b(.)36 b(The)28 b(scaling)f FA(C)2146 938 y Fy(2)2140 988 y(+)2195 968 y FA(=")2276 938 y Fy(2)p Fx(r)r Fv(\000)p Fy(1)2458 968 y FB(comes)g(from)g(the)h(inner)f(c)n (hange)g(in)h(\(61\))o(.)195 1068 y(W)-7 b(e)28 b(also)f(in)n(tro)r (duce)1285 1167 y(\001)1354 1132 y Fv(\000)1354 1189 y Fy(0)1410 1167 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)1716 1088 y Fz(X)1715 1267 y Fx(k)q(>)p Fy(0)1851 1167 y FB(\007)1916 1124 y Fy([)p Fx(k)q Fy(])1916 1189 y(0)1994 1167 y FA(e)2033 1131 y Fx(ik)2092 1137 y FB(\()2125 1131 y Fx(")2156 1106 y Fl(\000)p Fu(1)2234 1131 y Fx(u)p Fv(\000)p Fx(\034)7 b Fy(+)p Fv(C)s Fy(\()p Fx(u;\034)g Fy(\))2602 1137 y FB(\))71 1374 y(with)598 1579 y(\007)663 1536 y Fy([)p Fx(k)q Fy(])663 1601 y(0)764 1579 y FB(=)p 873 1452 66 4 v 873 1519 a FA(C)938 1466 y Fy(2)938 1539 y(+)1000 1519 y FB(^)-49 b FA(\026)p 861 1560 194 4 v 861 1636 a(")900 1612 y Fy(2)p Fx(r)r Fv(\000)p Fy(1)p 1065 1533 52 4 v 1065 1579 a FA(\037)1117 1544 y Fy([)p Fv(\000)p Fx(k)q Fy(])1247 1579 y FB(\()7 b(^)-49 b FA(\026)p FB(\))p FA(e)1400 1541 y Fv(\000)1462 1501 y(j)p Fx(k)q Fv(j)p Fx(a)p 1462 1522 113 4 v 1503 1570 a(")2878 1579 y FB(if)28 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b(>)e FB(0)337 b(\(104\))598 1819 y(\007)663 1776 y Fy([)p Fx(k)q Fy(])663 1841 y(0)764 1819 y FB(=)p 873 1692 66 4 v 873 1759 a FA(C)938 1706 y Fy(2)938 1780 y(+)1000 1759 y FB(^)-49 b FA(\026)p 861 1800 194 4 v 861 1876 a(")900 1852 y Fy(2)p Fx(r)r Fv(\000)p Fy(1)p 1065 1773 52 4 v 1065 1819 a FA(\037)1117 1785 y Fy([)p Fv(\000)p Fx(k)q Fy(])1247 1819 y FB(\()7 b(^)-49 b FA(\026)p FB(\))p FA(e)1400 1781 y Fv(\000)1462 1741 y(j)p Fx(k)q Fv(j)p Fx(a)p 1462 1762 113 4 v 1503 1810 a(")1584 1781 y Fy(+)p Fx(i)p Fv(j)p Fx(k)q Fv(j)p Fy(\()p Fv(\000)p 1812 1733 52 3 v Fx(C)t Fy(\()5 b(^)-38 b Fx(\026)q(;")p Fy(\)+)5 b(^)-38 b Fx(\026)2099 1756 y Fu(2)p 2132 1733 30 3 v 2132 1781 a Fx(b)11 b Fy(ln)g Fx(")p Fy(\))2855 1819 y FB(if)28 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b FB(=)e(0)p FA(:)337 b FB(\(105\))71 2002 y(The)21 b(function)h(\001)623 1967 y Fv(\000)623 2025 y Fy(0)679 2002 y FB(\()p FA(u;)14 b(\034)9 b FB(\))22 b(corresp)r(onds)d(to)i(the)g(di\013erence)g(of)g (the)h(solutions)e(of)h(the)h(inner)e(equation)h(close)f(to)h FA(u)i FB(=)f Fw(\000)p FA(ia)71 2102 y FB(if)40 b(^)-49 b FA(\026;)14 b(\034)41 b Fw(2)32 b Ft(R)p FB(.)52 b(W)-7 b(e)34 b(note)e(that,)j(taking)d FA(\034)5 b(;)20 b FB(^)-48 b FA(\026)31 b Fw(2)h Ft(R)p FB(,)i(\001)1779 2067 y Fv(\000)1779 2124 y Fy(0)1868 2102 y FB(is)f(nothing)f(but)i(the)f (complex)f(conjugate)g(of)h(\001)3452 2067 y Fy(+)3452 2124 y(0)3507 2102 y FB(.)52 b(In)33 b(fact,)71 2202 y(as)28 b(w)n(e)g(kno)n(w)g(that)h(\001)g(is)g(a)g(real)e(analytic)i (function)g(in)g(the)g FA(u)f FB(v)-5 b(ariable)28 b(for)g(real)g(v)-5 b(alues)29 b(of)35 b(^)-49 b FA(\026)q(;)14 b(\034)9 b FB(,)29 b(w)n(e)g(can)f(de\014ne)h(\001)3794 2166 y Fv(\000)3794 2224 y Fy(0)71 2301 y FB(as)i(the)h(function)h(that)f (satis\014es)f(that)h(\001)1401 2313 y Fy(0)1468 2301 y FB(=)e(\001)1632 2266 y Fy(+)1632 2323 y(0)1708 2301 y FB(+)21 b(\001)1863 2266 y Fv(\000)1863 2323 y Fy(0)1951 2301 y FB(is)32 b(also)f(a)g(real)g(analytic)g(function)h(in)g(the)h (same)e(sense)g(as)71 2401 y(explained)c(b)r(efore)g(for)h(\001.)195 2501 y(W)-7 b(e)28 b(will)g(see)f(that)h(the)g(\014rst)g(order)e(of)h (\001)h(is)g(giv)n(en)f(b)n(y)1373 2647 y(\001)1442 2659 y Fy(0)1479 2647 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)f(\001)1854 2611 y Fy(+)1854 2669 y(0)1909 2647 y FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b(+)f(\001)2274 2611 y Fv(\000)2274 2669 y Fy(0)2331 2647 y FB(\()p FA(u;)c(\034)9 b FB(\))p FA(:)1113 b FB(\(106\))71 2793 y(Let)28 b(us)f(p)r(oin)n(t)h (out)g(that)f(it)h(can)g(b)r(e)g(written)f(as)1216 2956 y(\001)1285 2968 y Fy(0)1322 2956 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)1695 2877 y Fz(X)1627 3059 y Fx(k)q Fv(2)p Fn(Z)p Fv(nf)p Fy(0)p Fv(g)1897 2956 y FB(\007)1962 2913 y Fy([)p Fx(k)q Fy(])1962 2978 y(0)2040 2956 y FA(e)2079 2920 y Fx(ik)2138 2926 y FB(\()2171 2920 y Fx(")2202 2895 y Fl(\000)p Fu(1)2280 2920 y Fx(u)p Fv(\000)p Fx(\034)7 b Fy(+)p Fv(C)s Fy(\()p Fx(u;\034)g Fy(\))2648 2926 y FB(\))2683 2956 y FA(;)955 b FB(\(107\))71 3224 y(where)26 b(\007)375 3181 y Fy([)p Fx(k)q Fy(])375 3246 y(0)480 3224 y FB(are)g(de\014ned)h(either)g(b)n(y)h(\(102\))e (and)h(\(104\))f(in)h(the)g(case)f FA(`)17 b Fw(\000)g FB(2)p FA(r)25 b(>)e FB(0)j(or)g(b)n(y)i(\(103\))e(and)h(\(105\))f(in)h (the)g(case)71 3341 y FA(`)21 b Fw(\000)f FB(2)p FA(r)33 b FB(=)c(0.)49 b(F)-7 b(or)31 b(con)n(v)n(enience)g(w)n(e)g(in)n(tro)r (duce)h(\007)1709 3298 y Fy([0])1709 3363 y(0)1813 3341 y FB(=)e(0.)49 b(F)-7 b(rom)31 b(no)n(w)g(on,)i(in)f(this)g (subsection,)g(w)n(e)g(consider)e(real)71 3440 y(v)-5 b(alues)27 b(of)h FA(\034)k Fw(2)24 b Ft(T)615 3452 y Fx(\033)678 3440 y Fw(\\)19 b Ft(R)p FB(.)p Black 71 3577 a Fp(Theorem)30 b(4.23.)p Black 38 w Fs(L)l(et)e(us)g(c)l(onsider) i(the)f(me)l(an)f(value)h(of)h FB(\007)p Fs(,)f FB(\007)2137 3547 y Fy([0])2211 3577 y Fs(,)g(de\014ne)l(d)g(in)35 b FB(\(100\))o Fs(,)29 b FA(s)23 b(<)g FB(1)p FA(=\014)t Fs(,)29 b(wher)l(e)g FA(r)d FB(=)c FA(\013=\014)33 b Fs(is)71 3677 y(de\014ne)l(d)27 b(in)g(Hyp)l(othesis)h Fo(HP2)p Fs(,)g(and)g FA(")1297 3689 y Fy(0)1357 3677 y FA(>)22 b FB(0)k Fs(smal)t(l)i(enough.)38 b(Then,)29 b(ther)l(e)e(exists)f(a)h(c)l(onstant)f FA(b)3146 3689 y Fy(18)3239 3677 y FA(>)d FB(0)j Fs(such)h(that)f(for)71 3777 y FA(")d Fw(2)g FB(\(0)p FA(;)14 b(")361 3789 y Fy(0)398 3777 y FB(\))30 b Fs(and)37 b FB(^)-49 b FA(\026)23 b Fw(2)g FA(B)t FB(\()7 b(^)-49 b FA(\026)921 3789 y Fy(0)959 3777 y FB(\))19 b Fw(\\)g Ft(R)29 b Fs(and)i FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b Fw(2)1631 3709 y Fz(\000)1669 3777 y FA(R)1732 3792 y Fx(s)11 b Fy(ln\(1)p Fx(=")p Fy(\))p Fx(;d)2035 3800 y Fu(3)2090 3777 y Fw(\\)18 b Ft(R)2223 3709 y Fz(\001)2280 3777 y Fw(\002)g Ft(T)p Fs(,)30 b(the)g(fol)t(lowing)j(statements)28 b(ar)l(e)i(satis\014e)l (d.)p Black 195 3914 a Fw(\017)p Black 41 w Fs(If)h FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b(>)e FB(0)p Fs(,)1169 4008 y Fz(\014)1169 4058 y(\014)1169 4107 y(\014)1197 4103 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))19 b Fw(\000)f FB(\007)1627 4069 y Fy([0])1720 4103 y Fw(\000)g FB(\001)1872 4115 y Fy(0)1909 4103 y FB(\()p FA(u;)c(\034)9 b FB(\))2103 4008 y Fz(\014)2103 4058 y(\014)2103 4107 y(\014)2155 4103 y Fw(\024)2414 4047 y FA(b)2450 4059 y Fy(18)2520 4047 y Fw(j)e FB(^)-49 b FA(\026)p Fw(j)p 2252 4084 525 4 v 2252 4160 a FA(")2291 4136 y Fy(2)p Fx(r)r Fv(\000)p Fy(1)2446 4160 y Fw(j)14 b FB(ln)g FA(")p Fw(j)2628 4136 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)2787 4103 y FA(e)2826 4066 y Fv(\000)2888 4033 y Fx(a)p 2888 4047 37 4 v 2890 4094 a(")1244 4333 y Fw(j)p FA(@)1311 4345 y Fx(u)1355 4333 y FB(\001\()p FA(u;)g(\034)9 b FB(\))19 b Fw(\000)f FA(@)1764 4345 y Fx(u)1807 4333 y FB(\001)1876 4345 y Fy(0)1914 4333 y FB(\()p FA(u;)c(\034)9 b FB(\))p Fw(j)24 b(\024)2371 4276 y FA(b)2407 4288 y Fy(18)2477 4276 y Fw(j)7 b FB(^)-49 b FA(\026)p Fw(j)p 2252 4313 440 4 v 2252 4389 a FA(")2291 4365 y Fy(2)p Fx(r)2361 4389 y Fw(j)14 b FB(ln)g FA(")p Fw(j)2543 4365 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)2702 4333 y FA(e)2741 4295 y Fv(\000)2803 4262 y Fx(a)p 2803 4276 37 4 v 2805 4324 a(")1235 4491 y Fz(\014)1235 4541 y(\014)1263 4562 y FA(@)1312 4528 y Fy(2)1307 4582 y Fx(u)1350 4562 y FB(\001\()p FA(u;)g(\034)9 b FB(\))19 b Fw(\000)f FA(@)1764 4528 y Fy(2)1759 4582 y Fx(u)1803 4562 y FB(\001)1872 4574 y Fy(0)1909 4562 y FB(\()p FA(u;)c(\034)9 b FB(\))2103 4491 y Fz(\014)2103 4541 y(\014)2155 4562 y Fw(\024)2413 4506 y FA(b)2449 4518 y Fy(18)2519 4506 y Fw(j)e FB(^)-49 b FA(\026)p Fw(j)p 2252 4543 524 4 v 2252 4619 a FA(")2291 4595 y Fy(2)p Fx(r)r Fy(+1)2445 4619 y Fw(j)14 b FB(ln)g FA(")p Fw(j)2627 4595 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)2786 4562 y FA(e)2825 4524 y Fv(\000)2887 4492 y Fx(a)p 2887 4506 37 4 v 2889 4553 a(")2937 4562 y FA(:)p Black 195 4778 a Fw(\017)p Black 41 w Fs(If)31 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b FB(=)e(0)p Fs(,)1068 4872 y Fz(\014)1068 4922 y(\014)1068 4972 y(\014)1095 4968 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))20 b Fw(\000)e FB(\007)1526 4933 y Fy([0])1618 4968 y Fw(\000)g FB(\001)1770 4980 y Fy(0)1808 4968 y FB(\()p FA(u;)c(\034)9 b FB(\))2002 4872 y Fz(\014)2002 4922 y(\014)2002 4972 y(\014)2053 4968 y Fw(\024)2238 4911 y FA(b)2274 4923 y Fy(18)2343 4911 y Fw(j)e FB(^)-49 b FA(\026)q Fw(j)p 2151 4949 376 4 v 2151 5025 a FA(")2190 5001 y Fy(2)p Fx(r)r Fv(\000)p Fy(1)2344 5025 y Fw(j)14 b FB(ln)g FA(")p Fw(j)2536 4968 y FA(e)2575 4930 y Fv(\000)2637 4897 y Fx(a)p 2637 4911 37 4 v 2639 4959 a(")2683 4930 y Fy(+)5 b(^)-38 b Fx(\026)2774 4905 y Fu(2)2806 4930 y Fy(Im)11 b Fx(b)h Fy(ln)f Fx(")1143 5197 y Fw(j)p FA(@)1210 5209 y Fx(u)1253 5197 y FB(\001\()p FA(u;)j(\034)9 b FB(\))20 b Fw(\000)e FA(@)1663 5209 y Fx(u)1706 5197 y FB(\001)1775 5209 y Fy(0)1813 5197 y FB(\()p FA(u;)c(\034)9 b FB(\))p Fw(j)23 b(\024)2195 5141 y FA(b)2231 5153 y Fy(18)2301 5141 y Fw(j)7 b FB(^)-49 b FA(\026)p Fw(j)p 2151 5178 291 4 v 2151 5254 a FA(")2190 5230 y Fy(2)p Fx(r)2260 5254 y Fw(j)14 b FB(ln)f FA(")p Fw(j)2451 5197 y FA(e)2490 5159 y Fv(\000)2552 5127 y Fx(a)p 2552 5141 37 4 v 2554 5188 a(")2598 5159 y Fy(+)5 b(^)-38 b Fx(\026)2689 5134 y Fu(2)2721 5159 y Fy(Im)11 b Fx(b)h Fy(ln)f Fx(")1134 5356 y Fz(\014)1134 5406 y(\014)1161 5426 y FA(@)1210 5392 y Fy(2)1205 5447 y Fx(u)1249 5426 y FB(\001\()p FA(u;)j(\034)9 b FB(\))19 b Fw(\000)f FA(@)1663 5392 y Fy(2)1658 5447 y Fx(u)1701 5426 y FB(\001)1770 5438 y Fy(0)1808 5426 y FB(\()p FA(u;)c(\034)9 b FB(\))2002 5356 y Fz(\014)2002 5406 y(\014)2053 5426 y Fw(\024)2237 5370 y FA(b)2273 5382 y Fy(18)2343 5370 y Fw(j)e FB(^)-49 b FA(\026)p Fw(j)p 2151 5407 375 4 v 2151 5483 a FA(")2190 5459 y Fy(2)p Fx(r)r Fy(+1)2344 5483 y Fw(j)14 b FB(ln)f FA(")p Fw(j)2535 5426 y FA(e)2574 5389 y Fv(\000)2636 5356 y Fx(a)p 2636 5370 37 4 v 2638 5417 a(")2682 5389 y Fy(+)5 b(^)-38 b Fx(\026)2773 5364 y Fu(2)2805 5389 y Fy(Im)11 b Fx(b)h Fy(ln)f Fx(")3038 5426 y FA(:)p Black 1919 5753 a FB(45)p Black eop end %%Page: 46 46 TeXDict begin 46 45 bop Black Black 195 276 a FB(W)-7 b(e)31 b(observ)n(e)d(that)i FA(@)865 288 y Fx(u)908 276 y FB(\001)977 288 y Fy(0)1044 276 y FB(giv)n(es)f(the)h(correct)e (asymptotic)h(prediction)h(of)g FA(@)2639 288 y Fx(u)2682 276 y FB(\001)g(if)g(\007)2924 233 y Fy([)p Fv(\000)p Fy(1])2924 298 y(0)3077 276 y Fw(6)p FB(=)c(0.)43 b(In)30 b(fact,)h(w)n(e)e(only)71 375 y(need)k(this)g(co)r(e\016cien)n(t)f(to)h (giv)n(e)e(a)i(simpler)f(leading)g(term)h(of)f(the)h(asymptotic)f(form) n(ula.)51 b(F)-7 b(or)33 b(this)f(purp)r(ose)h(let)g(us)71 475 y(de\014ne)28 b(the)g(function)1587 575 y FA(f)22 b FB(\()7 b(^)-48 b FA(\026)p FB(\))23 b(=)g FA(C)1941 540 y Fy(2)1935 595 y(+)1990 575 y FA(\037)2042 540 y Fy([)p Fv(\000)p Fy(1])2183 575 y FB(\()7 b(^)-49 b FA(\026)p FB(\))14 b FA(;)1327 b FB(\(108\))71 734 y(where)25 b FA(C)368 746 y Fy(+)449 734 y FB(is)h(the)g(constan)n(t)f(de\014ned)h (in)g(\(12\))f(or)g(\(13\))g(and)g FA(\037)2041 704 y Fy([)p Fv(\000)p Fy(1])2168 734 y FB(\()7 b(^)-49 b FA(\026)p FB(\))26 b(is)g(the)g(constan)n(t)f(giv)n(en)g(in)h(Theorem)e(4.12.)35 b(Let)71 834 y(us)27 b(p)r(oin)n(t)h(out)g(that)g(the)g(zeros)e(of)h FA(f)9 b FB(\()e(^)-49 b FA(\026)p FB(\))28 b(corresp)r(ond)e(to)i(the) g(zeros)e(of)i FA(\037)2382 803 y Fy([)p Fv(\000)p Fy(1])2508 834 y FB(\()7 b(^)-49 b FA(\026)p FB(\).)38 b(W)-7 b(e)28 b(de\014ne)295 1059 y(\001)364 1071 y Fy(00)435 1059 y FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b(=)801 1003 y(2)7 b(^)-49 b FA(\026)p 750 1040 194 4 v 750 1116 a(")789 1092 y Fy(2)p Fx(r)r Fv(\000)p Fy(1)954 1059 y FA(e)993 1022 y Fw(\000)1068 989 y Fx(a)p 1067 1003 37 4 v 1069 1051 a(")1117 1059 y FB(Re)1243 967 y Fz(\020)1292 1059 y FA(f)9 b FB(\()e(^)-49 b FA(\026)p FB(\))p FA(e)1495 1022 y Fv(\000)p Fx(i)1570 1028 y FB(\()1613 989 y Fx(u)p 1613 1003 40 4 v 1617 1051 a(")1662 1022 y Fv(\000)p Fx(\034)7 b Fy(+)p Fv(C)s Fy(\()p Fx(u;\034)g Fy(\))1991 1028 y FB(\))2026 967 y Fz(\021)2991 1059 y FB(if)28 b FA(`)18 b Fw(\000)h FB(2)p FA(r)25 b(>)d FB(0)224 b(\(109\))295 1263 y(\001)364 1275 y Fy(00)435 1263 y FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b(=)801 1207 y(2)7 b(^)-49 b FA(\026)p 750 1244 194 4 v 750 1320 a(")789 1296 y Fy(2)p Fx(r)r Fv(\000)p Fy(1)954 1263 y FA(e)993 1226 y Fw(\000)1068 1193 y Fx(a)p 1067 1207 37 4 v 1069 1255 a(")1117 1263 y FB(Re)1243 1171 y Fz(\020)1292 1263 y FA(f)9 b FB(\()e(^)-49 b FA(\026)p FB(\))p FA(e)1495 1228 y Fv(\000)p Fx(i)1570 1234 y FB(\()1608 1228 y Fy(^)-38 b Fx(\026)1643 1202 y Fu(2)1675 1228 y Fx(b)12 b Fy(ln)f Fx(")p Fv(\000)p Fx(C)t Fy(\()5 b(^)-38 b Fx(\026;")p Fy(\))2061 1234 y FB(\))2097 1263 y FA(e)2136 1226 y Fv(\000)p Fx(i)2211 1232 y FB(\()2254 1193 y Fx(u)p 2254 1207 40 4 v 2258 1255 a(")2303 1226 y Fv(\000)p Fx(\034)7 b Fy(+)p Fv(C)s Fy(\()p Fx(u;\034)g Fy(\))2632 1232 y FB(\))2667 1171 y Fz(\021)2968 1263 y FB(if)28 b FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b FB(=)c(0)p FA(;)224 b FB(\(110\))71 1474 y(where)25 b FA(b)g FB(is)g(the)g(constan)n(t)g(de\014ned)h(in)f (\(75\),)h FA(C)6 b FB(\()p FA(\026;)14 b(")p FB(\))25 b(the)h(constan)n(t)f(giv)n(en)f(in)i(Prop)r(osition)d(4.22)h(and)h Fw(C)30 b FB(the)c(function)71 1574 y(giv)n(en)h(b)n(y)g(Theorem)g (4.21.)p Black 71 1740 a Fp(Corollary)36 b(4.24.)p Black 41 w Fs(Ther)l(e)d(exists)f(a)h(c)l(onstant)e FA(b)1654 1752 y Fy(19)1752 1740 y FA(>)c FB(0)k Fs(such)i(that)f(for)h FA(")27 b Fw(2)h FB(\(0)p FA(;)14 b(")2715 1752 y Fy(0)2752 1740 y FB(\))p Fs(,)40 b FB(^)-49 b FA(\026)28 b Fw(2)f FA(B)t FB(\()7 b(^)-49 b FA(\026)3151 1752 y Fy(0)3189 1740 y FB(\))21 b Fw(\\)f Ft(R)32 b Fs(and)h FB(\()p FA(u;)14 b(\034)9 b FB(\))28 b Fw(2)71 1772 y Fz(\000)109 1839 y FA(R)172 1854 y Fx(s)12 b Fy(ln)o(\(1)p Fx(=")p Fy(\))p Fx(;d)475 1862 y Fu(3)530 1839 y Fw(\\)19 b Ft(R)664 1772 y Fz(\001)720 1839 y Fw(\002)f Ft(T)p Fs(,)31 b(the)e(fol)t (lowing)k(statements)28 b(ar)l(e)i(satis\014e)l(d.)p Black 195 2005 a Fw(\017)p Black 41 w Fs(If)h FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b(>)e FB(0)p Fs(,)1152 2136 y Fz(\014)1152 2186 y(\014)1152 2235 y(\014)1180 2231 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))19 b Fw(\000)f FB(\007)1610 2197 y Fy([0])1703 2231 y Fw(\000)g FB(\001)1855 2243 y Fy(00)1926 2231 y FB(\()p FA(u;)c(\034)9 b FB(\))2120 2136 y Fz(\014)2120 2186 y(\014)2120 2235 y(\014)2171 2231 y Fw(\024)2430 2175 y FA(b)2466 2187 y Fy(19)2536 2175 y Fw(j)e FB(^)-49 b FA(\026)p Fw(j)p 2269 2212 525 4 v 2269 2288 a FA(")2308 2264 y Fy(2)p Fx(r)r Fv(\000)p Fy(1)2462 2288 y Fw(j)14 b FB(ln)g FA(")p Fw(j)2644 2264 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)2804 2231 y FA(e)2843 2194 y Fv(\000)2904 2161 y Fx(a)p 2904 2175 37 4 v 2906 2222 a(")1228 2461 y Fw(j)p FA(@)1295 2473 y Fx(u)1338 2461 y FB(\001\()p FA(u;)g(\034)9 b FB(\))19 b Fw(\000)f FA(@)1747 2473 y Fx(u)1791 2461 y FB(\001)1860 2473 y Fy(00)1930 2461 y FB(\()p FA(u;)c(\034)9 b FB(\))p Fw(j)24 b(\024)2388 2404 y FA(b)2424 2416 y Fy(19)2494 2404 y Fw(j)7 b FB(^)-49 b FA(\026)p Fw(j)p 2269 2441 440 4 v 2269 2518 a FA(")2308 2494 y Fy(2)p Fx(r)2377 2518 y Fw(j)14 b FB(ln)g FA(")p Fw(j)2559 2494 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)2719 2461 y FA(e)2758 2423 y Fv(\000)2819 2390 y Fx(a)p 2819 2404 37 4 v 2821 2452 a(")1218 2619 y Fz(\014)1218 2669 y(\014)1246 2690 y FA(@)1295 2656 y Fy(2)1290 2710 y Fx(u)1333 2690 y FB(\001\()p FA(u;)g(\034)9 b FB(\))20 b Fw(\000)e FA(@)1748 2656 y Fy(2)1743 2710 y Fx(u)1786 2690 y FB(\001)1855 2702 y Fy(00)1926 2690 y FB(\()p FA(u;)c(\034)9 b FB(\))2120 2619 y Fz(\014)2120 2669 y(\014)2171 2690 y Fw(\024)2430 2634 y FA(b)2466 2646 y Fy(19)2536 2634 y Fw(j)e FB(^)-49 b FA(\026)p Fw(j)p 2269 2671 524 4 v 2269 2747 a FA(")2308 2723 y Fy(2)p Fx(r)r Fy(+1)2461 2747 y Fw(j)14 b FB(ln)g FA(")p Fw(j)2643 2723 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)2803 2690 y FA(e)2842 2652 y Fv(\000)2903 2620 y Fx(a)p 2903 2634 37 4 v 2905 2681 a(")2953 2690 y FA(:)p Black 195 2950 a Fw(\017)p Black 41 w Fs(If)31 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b FB(=)e(0)p Fs(,)1051 3080 y Fz(\014)1051 3130 y(\014)1051 3180 y(\014)1079 3176 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))19 b Fw(\000)f FB(\007)1509 3141 y Fy([0])1602 3176 y Fw(\000)g FB(\001)1754 3188 y Fy(00)1824 3176 y FB(\()p FA(u;)c(\034)9 b FB(\))2018 3080 y Fz(\014)2018 3130 y(\014)2018 3180 y(\014)2070 3176 y Fw(\024)2247 3120 y FA(b)2283 3132 y Fy(19)2367 3120 y Fw(j)e FB(^)-49 b FA(\026)p Fw(j)p 2167 3157 376 4 v 2167 3233 a FA(")2206 3209 y Fy(2)p Fx(r)r Fv(\000)p Fy(1)2361 3233 y Fw(j)14 b FB(ln)g FA(")p Fw(j)2553 3176 y FA(e)2592 3138 y Fv(\000)2653 3105 y Fx(a)p 2653 3119 37 4 v 2655 3167 a(")2699 3138 y Fy(+)5 b(^)-38 b Fx(\026)2790 3113 y Fu(2)2823 3138 y Fy(Im)11 b Fx(b)g Fy(ln)g Fx(")1126 3405 y Fw(j)p FA(@)1193 3417 y Fx(u)1237 3405 y FB(\001\()p FA(u;)j(\034)9 b FB(\))19 b Fw(\000)f FA(@)1646 3417 y Fx(u)1690 3405 y FB(\001)1759 3417 y Fy(00)1829 3405 y FB(\()p FA(u;)c(\034)9 b FB(\))p Fw(j)24 b(\024)2205 3349 y FA(b)2241 3361 y Fy(19)2324 3349 y Fw(j)7 b FB(^)-48 b FA(\026)p Fw(j)p 2167 3386 291 4 v 2167 3462 a FA(")2206 3438 y Fy(2)p Fx(r)2276 3462 y Fw(j)14 b FB(ln)g FA(")p Fw(j)2468 3405 y FA(e)2507 3367 y Fv(\000)2568 3335 y Fx(a)p 2568 3349 37 4 v 2570 3396 a(")2614 3367 y Fy(+)5 b(^)-38 b Fx(\026)2705 3342 y Fu(2)2738 3367 y Fy(Im)11 b Fx(b)g Fy(ln)g Fx(")1117 3564 y Fz(\014)1117 3614 y(\014)1145 3634 y FA(@)1194 3600 y Fy(2)1189 3655 y Fx(u)1232 3634 y FB(\001\()p FA(u;)j(\034)9 b FB(\))19 b Fw(\000)f FA(@)1646 3600 y Fy(2)1641 3655 y Fx(u)1685 3634 y FB(\001)1754 3646 y Fy(00)1824 3634 y FB(\()p FA(u;)c(\034)9 b FB(\))2018 3564 y Fz(\014)2018 3614 y(\014)2070 3634 y Fw(\024)2247 3578 y FA(b)2283 3590 y Fy(19)2366 3578 y Fw(j)e FB(^)-48 b FA(\026)p Fw(j)p 2167 3615 375 4 v 2167 3691 a FA(")2206 3667 y Fy(2)p Fx(r)r Fy(+1)2360 3691 y Fw(j)14 b FB(ln)g FA(")p Fw(j)2552 3634 y FA(e)2591 3597 y Fv(\000)2652 3564 y Fx(a)p 2652 3578 37 4 v 2654 3625 a(")2698 3597 y Fy(+)5 b(^)-38 b Fx(\026)2789 3572 y Fu(2)2822 3597 y Fy(Im)11 b Fx(b)g Fy(ln)g Fx(")3055 3634 y FA(:)195 3894 y FB(W)-7 b(e)40 b(dev)n(ote)e(the)i(rest)f(of)g(this)g(section)g (to)g(pro)n(v)n(e)e(Theorem)i(4.23,)h(from)f(whic)n(h)g(Corollary)e (4.24)h(is)h(a)g(direct)71 3994 y(consequence.)p Black 71 4160 a Fs(Pr)l(o)l(of)31 b(of)f(The)l(or)l(em)h(4.23.)p Black 44 w FB(F)-7 b(or)24 b(the)h(\014rst)f(part)g(of)h(the)g(pro)r (of)f(w)n(e)g(consider)g(complex)g(v)-5 b(alues)24 b(of)31 b(^)-48 b FA(\026)23 b Fw(2)g FA(B)t FB(\()7 b(^)-49 b FA(\026)3431 4172 y Fy(0)3469 4160 y FB(\))25 b(and)f(later)71 4270 y(w)n(e)j(will)h(restrict)f(to)35 b(^)-49 b FA(\026)24 b Fw(2)f FA(B)t FB(\()7 b(^)-49 b FA(\026)1040 4282 y Fy(0)1078 4270 y FB(\))19 b Fw(\\)g Ft(R)p FB(.)38 b(By)28 b(\(100\))f(and)h(\(107\))o(,)g(the)g(function)2554 4249 y Fz(e)2543 4270 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)f(\001\()p FA(u;)14 b(\034)9 b FB(\))19 b Fw(\000)g FB(\001)3353 4282 y Fy(0)3390 4270 y FB(\()p FA(u;)14 b(\034)9 b FB(\))29 b(can)e(b)r(e)71 4370 y(written)h(as)847 4448 y Fz(e)836 4469 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)1219 4448 y Fz(e)1210 4469 y FB(\007)1289 4402 y Fz(\000)1327 4469 y FA(")1366 4435 y Fv(\000)p Fy(1)1455 4469 y FA(u)17 b Fw(\000)i FA(\034)28 b FB(+)18 b Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))1994 4402 y Fz(\001)2055 4469 y FB(=)2143 4390 y Fz(X)2143 4569 y Fx(k)q Fv(2)p Fn(Z)2286 4448 y Fz(e)2277 4469 y FB(\007)2342 4435 y Fy([)p Fx(k)q Fy(])2420 4469 y FA(e)2459 4433 y Fx(ik)2518 4439 y FB(\()2551 4433 y Fx(")2582 4408 y Fl(\000)p Fu(1)2660 4433 y Fx(u)p Fv(\000)p Fx(\034)e Fy(+)p Fv(C)s Fy(\()p Fx(u;\034)g Fy(\))3028 4439 y FB(\))3063 4469 y FA(;)575 b FB(\(111\))71 4735 y(where)320 4714 y Fz(e)310 4735 y FB(\007)375 4705 y Fy([)p Fx(k)q Fy(])476 4735 y FB(=)23 b(\007)629 4705 y Fy([)p Fx(k)q Fy(])725 4735 y Fw(\000)17 b FB(\007)872 4692 y Fy([)p Fx(k)q Fy(])872 4757 y(0)950 4735 y FB(.)36 b(Therefore,)26 b(to)h(obtain)g(the)g(b)r(ounds)h(of)f(Theorem)f(4.23,)g(it)h(is)g (crucial)g(to)g(b)r(ound)3629 4640 y Fz(\014)3629 4690 y(\014)3629 4740 y(\014)3666 4714 y(e)3657 4735 y FB(\007)3722 4705 y Fy([)p Fx(k)q Fy(])3800 4640 y Fz(\014)3800 4690 y(\014)3800 4740 y(\014)3827 4735 y FB(.)195 4879 y(The)h(\014rst)e (step)i(is)f(to)g(obtain)g(a)f(b)r(ound)i(of)1584 4858 y Fz(e)1572 4879 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))28 b(for)f(\()p FA(u;)14 b(\034)9 b FB(\))23 b Fw(2)h FA(R)2349 4899 y Fx(s)12 b Fy(ln)2468 4877 y Fu(1)p 2468 4886 29 3 v 2468 4919 a Fm(")2507 4899 y Fx(;d)2562 4907 y Fu(3)2615 4879 y Fw(\002)17 b Ft(T)p FB(.)37 b(First)27 b(w)n(e)g(b)r(ound)g (this)h(term)f(for)71 5023 y(\()p FA(u;)14 b(\034)9 b FB(\))29 b Fw(2)378 4931 y Fz(\020)428 5023 y FA(R)491 5043 y Fx(s)11 b Fy(ln)610 5021 y Fu(1)p 610 5030 V 610 5063 a Fm(")648 5043 y Fx(;d)703 5051 y Fu(3)758 5023 y Fw(\\)18 b FA(D)902 4983 y Fy(in)p Fx(;)p Fy(+)p Fx(;s)900 5057 y(s)12 b Fy(ln)1019 5034 y Fu(1)p 1019 5043 V 1019 5077 a Fm(")1058 5057 y Fx(;c)1108 5065 y Fu(2)1162 5023 y Fw(\\)19 b FA(D)1307 4983 y Fy(in)o Fx(;)p Fy(+)p Fx(;u)1305 5057 y(s)11 b Fy(ln)1424 5034 y Fu(1)p 1424 5043 V 1424 5077 a Fm(")1462 5057 y Fx(;c)1512 5065 y Fu(2)1548 4931 y Fz(\021)1618 5023 y Fw(\002)21 b Ft(T)p FB(.)47 b(Recalling)30 b(the)h(de\014nitions)g(of)38 b(\(76\))o(,)32 b(\(106\))o(,)g(\(101\))e (and)h(\(72\))o(,)71 5166 y(w)n(e)c(split)390 5145 y Fz(e)378 5166 y FB(\001)h(as)1027 5245 y Fz(e)1016 5266 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)1402 5245 y Fz(e)1390 5266 y FB(\001)1459 5231 y Fx(u)1459 5286 y Fy(1)1503 5266 y FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b Fw(\000)1811 5245 y Fz(e)1799 5266 y FB(\001)1868 5231 y Fx(s)1868 5286 y Fy(1)1906 5266 y FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b(+)2213 5245 y Fz(e)2202 5266 y FB(\001)2271 5278 y Fy(2)2308 5266 y FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b(+)2616 5245 y Fz(e)2604 5266 y FB(\001)2673 5278 y Fy(3)2711 5266 y FB(\()p FA(u;)14 b(\034)9 b FB(\))p Black 1919 5753 a(46)p Black eop end %%Page: 47 47 TeXDict begin 47 46 bop Black Black 71 272 a FB(with)745 452 y Fz(e)734 473 y FB(\001)803 433 y Fx(u;s)803 495 y Fy(1)897 473 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)f FA(T)1264 439 y Fx(u;s)1357 473 y FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b Fw(\000)1703 413 y FA(C)1768 383 y Fy(2)1762 434 y(+)p 1663 454 194 4 v 1663 530 a FA(")1702 506 y Fy(2)p Fx(r)r Fv(\000)p Fy(1)1867 473 y FA( )1924 433 y Fx(u;s)1921 495 y Fy(0)2032 356 y Fz(\022)2103 417 y FA(u)f Fw(\000)g FA(ia)p 2103 454 222 4 v 2195 530 a(")2335 473 y(;)c(\034)2417 356 y Fz(\023)1115 713 y FB(=)1252 653 y FA(C)1317 623 y Fy(2)1311 673 y(+)p 1213 693 194 4 v 1213 770 a FA(")1252 746 y Fy(2)p Fx(r)r Fv(\000)p Fy(1)1430 596 y Fz(\022)1491 713 y FA( )1548 678 y Fx(u;s)1657 596 y Fz(\022)1728 656 y FA(u)k Fw(\000)g FA(ia)p 1728 693 222 4 v 1819 770 a(")1959 713 y(;)c(\034)2041 596 y Fz(\023)2121 713 y Fw(\000)k FA( )2261 673 y Fx(u;s)2258 735 y Fy(0)2370 596 y Fz(\022)2441 656 y FA(u)g Fw(\000)g FA(ia)p 2441 693 V 2532 770 a(")2672 713 y(;)c(\034)2754 596 y Fz(\023)q(\023)3661 713 y FB(\(112\))803 931 y Fz(e)791 952 y FB(\001)860 964 y Fy(2)897 952 y FB(\()p FA(u;)g(\034)9 b FB(\))24 b(=)1252 892 y FA(C)1317 862 y Fy(2)1311 913 y(+)p 1213 933 194 4 v 1213 1009 a FA(")1252 985 y Fy(2)p Fx(r)r Fv(\000)p Fy(1)1430 835 y Fz(\022)1491 952 y FA( )1548 918 y Fx(u)1545 973 y Fy(0)1606 835 y Fz(\022)1677 896 y FA(u)18 b Fw(\000)g FA(ia)p 1677 933 222 4 v 1768 1009 a(")1908 952 y(;)c(\034)1990 835 y Fz(\023)2070 952 y Fw(\000)k FA( )2210 918 y Fx(s)2207 973 y Fy(0)2260 835 y Fz(\022)2331 896 y FA(u)g Fw(\000)g FA(ia)p 2331 933 V 2422 1009 a(")2562 952 y(;)c(\034)2644 835 y Fz(\023)q(\023)2785 952 y Fw(\000)k FB(\001)2937 917 y Fy(+)2937 974 y(0)2993 952 y FB(\()p FA(u;)c(\034)9 b FB(\))474 b(\(113\))803 1124 y Fz(e)791 1145 y FB(\001)860 1157 y Fy(3)897 1145 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)f Fw(\000)p FB(\001)1337 1110 y Fv(\000)1337 1167 y Fy(0)1393 1145 y FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(:)2051 b FB(\(114\))71 1344 y(Applying)28 b(Theorem)e(4.16,)h(one)g (can)g(see)g(that)h(for)f(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)2021 1252 y Fz(\020)2071 1344 y FA(R)2134 1364 y Fx(s)12 b Fy(ln)2254 1342 y Fu(1)p 2254 1351 29 3 v 2254 1385 a Fm(")2292 1364 y Fx(;d)2347 1372 y Fu(3)2401 1344 y Fw(\\)19 b FA(D)2546 1304 y Fy(in)o Fx(;)p Fy(+)p Fx(;s)2544 1378 y(s)11 b Fy(ln)2663 1355 y Fu(1)p 2663 1364 V 2663 1398 a Fm(")2701 1378 y Fx(;c)2751 1386 y Fu(2)2806 1344 y Fw(\\)18 b FA(D)2950 1304 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)2948 1378 y(s)12 b Fy(ln)3068 1355 y Fu(1)p 3068 1364 V 3068 1398 a Fm(")3106 1378 y Fx(;c)3156 1386 y Fu(2)3192 1252 y Fz(\021)3260 1344 y Fw(\002)18 b Ft(T)p FB(,)1461 1518 y Fz(\014)1461 1568 y(\014)1461 1618 y(\014)1489 1614 y FA(@)1533 1626 y Fx(u)1588 1593 y Fz(e)1576 1614 y FB(\001)1645 1574 y Fx(u;s)1645 1636 y Fy(1)1740 1614 y FB(\()p FA(u;)c(\034)9 b FB(\))1934 1518 y Fz(\014)1934 1568 y(\014)1934 1618 y(\014)1985 1614 y Fw(\024)2109 1558 y FA(K)d(")2238 1498 y Fu(1)p 2234 1507 35 3 v 2234 1541 a Fm(\014)2279 1521 y Fv(\000)p Fy(2)p Fx(r)p 2083 1595 345 4 v 2083 1695 a Fw(j)p FB(ln)14 b FA(")p Fw(j)2251 1654 y Fy(2)p Fx(r)r Fv(\000)2382 1631 y Fu(1)p 2378 1640 35 3 v 2378 1674 a Fm(\014)2437 1614 y FA(:)71 1871 y FB(T)-7 b(o)27 b(b)r(ound)461 1850 y Fz(e)449 1871 y FB(\001)518 1883 y Fy(2)556 1871 y FB(,)h(one)f(has)g(to)h(pro)r(ceed)f(in)h(di\013eren)n(t)g(w)n(a)n(ys,) e(dep)r(ending)i(on)g(whether)f FA(`)19 b Fw(\000)f FB(2)p FA(r)25 b(>)e FB(0)28 b(or)e FA(`)19 b Fw(\000)f FB(2)p FA(r)25 b FB(=)e(0.)37 b(F)-7 b(or)71 1970 y(the)28 b(\014rst)f(case,)g (let)h(us)f(p)r(oin)n(t)h(out)g(that,)698 2135 y Fz(e)687 2156 y FB(\001)756 2168 y Fy(2)793 2156 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)1099 2077 y Fz(X)1098 2255 y Fx(k)q(<)p Fy(0)1234 2156 y FB(\007)1299 2113 y Fy([)p Fx(k)q Fy(])1299 2178 y(0)1391 2063 y Fz(\020)1440 2156 y FA(e)1479 2120 y Fx(ik)1538 2126 y FB(\()1571 2120 y Fx(")1602 2095 y Fl(\000)p Fu(1)1680 2120 y Fx(u)p Fv(\000)p Fx(\034)7 b Fy(+)e(^)-38 b Fx(\026g)1934 2126 y FB(\()1967 2120 y Fx(")1998 2095 y Fl(\000)p Fu(1)2076 2120 y Fy(\()p Fx(u)p Fv(\000)p Fx(ia)p Fy(\))p Fx(;\034)2336 2126 y FB(\))o(\))2422 2156 y Fw(\000)18 b FA(e)2544 2120 y Fx(ik)2603 2126 y FB(\()2636 2120 y Fx(")2667 2095 y Fl(\000)p Fu(1)2745 2120 y Fx(u)p Fv(\000)p Fx(\034)7 b Fy(+)p Fv(C)s Fy(\()p Fx(u;\034)g Fy(\))3113 2126 y FB(\))3148 2063 y Fz(\021)3212 2156 y FA(:)71 2404 y FB(Then,)44 b(applying)c(Theorems)f(4.12)g(and)i(4.21)e(and)h(the)h (mean)f(v)-5 b(alue)41 b(theorem)f(one)g(obtains)g(that)h(for)f(\()p FA(u;)14 b(\034)9 b FB(\))45 b Fw(2)71 2436 y Fz(\020)120 2529 y FA(R)183 2549 y Fx(s)12 b Fy(ln)303 2527 y Fu(1)p 303 2536 29 3 v 303 2569 a Fm(")341 2549 y Fx(;d)396 2557 y Fu(3)450 2529 y Fw(\\)19 b FA(D)595 2489 y Fy(in)p Fx(;)p Fy(+)p Fx(;s)593 2562 y(s)12 b Fy(ln)712 2540 y Fu(1)p 712 2549 V 712 2582 a Fm(")751 2562 y Fx(;c)801 2570 y Fu(2)855 2529 y Fw(\\)19 b FA(D)1000 2489 y Fy(in)o Fx(;)p Fy(+)p Fx(;u)998 2562 y(s)11 b Fy(ln)1117 2540 y Fu(1)p 1117 2549 V 1117 2582 a Fm(")1155 2562 y Fx(;c)1205 2570 y Fu(2)1241 2436 y Fz(\021)1309 2529 y Fw(\002)18 b Ft(T)p FB(,)1461 2684 y Fz(\014)1461 2734 y(\014)1461 2784 y(\014)1489 2780 y FA(@)1533 2792 y Fx(u)1588 2759 y Fz(e)1576 2780 y FB(\001)1645 2792 y Fy(2)1683 2780 y FB(\()p FA(u;)c(\034)9 b FB(\))1877 2684 y Fz(\014)1877 2734 y(\014)1877 2784 y(\014)1928 2780 y Fw(\024)2026 2724 y FA(K)d Fw(j)h FB(^)-49 b FA(\026)o Fw(j)2198 2694 y Fy(2)2235 2724 y FA(")2274 2694 y Fx(s)p Fv(\000)p Fy(2)p Fx(r)p 2026 2761 402 4 v 2068 2853 a Fw(j)p FB(ln)14 b FA(")p Fw(j)2236 2811 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)2437 2780 y FA(:)71 3034 y FB(F)-7 b(or)27 b(the)h(case)f FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b FB(=)e(0,)k(taking)g(in)n(to)g (accoun)n(t)g(the)h(de\014nition)g(of)f(\007)2361 2991 y Fy([)p Fx(k)q Fy(])2361 3056 y(0)2467 3034 y FB(in)h(\(103\))o(,)116 3236 y Fz(e)104 3257 y FB(\001)173 3269 y Fy(2)210 3257 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)540 3197 y FA(C)605 3167 y Fy(2)599 3218 y(+)661 3197 y FB(^)-48 b FA(\026)p 526 3238 194 4 v 526 3314 a(")565 3290 y Fy(2)p Fx(r)r Fv(\000)p Fy(1)744 3178 y Fz(X)743 3357 y Fx(k)q(<)p Fy(0)879 3257 y FA(\037)931 3223 y Fy([)p Fx(k)q Fy(])1009 3257 y FB(\()7 b(^)-49 b FA(\026)p FB(\))1137 3165 y Fz(\020)1187 3257 y FA(e)1226 3221 y Fx(ik)1285 3227 y FB(\()1318 3221 y Fx(")1349 3196 y Fl(\000)p Fu(1)1427 3221 y Fy(\()p Fx(u)p Fv(\000)p Fx(ia)p Fy(\))p Fv(\000)p Fx(\034)7 b Fy(+)e(^)-38 b Fx(\026g)1844 3227 y FB(\()1877 3221 y Fx(")1908 3196 y Fl(\000)p Fu(1)1985 3221 y Fy(\()p Fx(u)p Fv(\000)p Fx(ia)p Fy(\))p Fx(;\034)2245 3227 y FB(\)\))2332 3257 y Fw(\000)18 b FA(e)2454 3221 y Fx(ik)2513 3227 y FB(\()2546 3221 y Fx(")2577 3196 y Fl(\000)p Fu(1)2655 3221 y Fy(\()p Fx(u)p Fv(\000)p Fx(ia)p Fy(\))p Fv(\000)p Fx(\034)7 b Fy(+)p Fv(C)s Fy(\()p Fx(u;\034)g Fy(\))p Fv(\000)p Fx(C)t Fy(\()e(^)-38 b Fx(\026)o(;")p Fy(\)+)5 b(^)-38 b Fx(\026)3523 3196 y Fu(2)3556 3221 y Fx(b)11 b Fy(ln)g Fx(")3694 3227 y FB(\))3731 3165 y Fz(\021)3794 3257 y FA(:)71 3539 y FB(By)29 b(Theorems)f(4.12)g(and)h(4.21)f(and)h (Prop)r(osition)f(4.22)g(for)h(\()p FA(u;)14 b(\034)9 b FB(\))26 b Fw(2)2324 3447 y Fz(\020)2374 3539 y FA(R)2437 3559 y Fx(s)12 b Fy(ln)2557 3537 y Fu(1)p 2557 3546 29 3 v 2557 3579 a Fm(")2595 3559 y Fx(;d)2650 3567 y Fu(3)2704 3539 y Fw(\\)19 b FA(D)2849 3499 y Fy(in)o Fx(;)p Fy(+)p Fx(;s)2847 3572 y(s)11 b Fy(ln)2966 3550 y Fu(1)p 2966 3559 V 2966 3592 a Fm(")3004 3572 y Fx(;c)3054 3580 y Fu(2)3109 3539 y Fw(\\)18 b FA(D)3253 3499 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)3251 3572 y(s)12 b Fy(ln)3371 3550 y Fu(1)p 3371 3559 V 3371 3592 a Fm(")3409 3572 y Fx(;c)3459 3580 y Fu(2)3495 3447 y Fz(\021)3564 3539 y Fw(\002)19 b Ft(T)p FA(;)30 b FB(w)n(e)71 3659 y(ha)n(v)n(e)c(that)1412 3695 y Fz(\014)1412 3745 y(\014)1412 3795 y(\014)1440 3791 y FA(@)1484 3803 y Fx(u)1539 3770 y Fz(e)1527 3791 y FB(\001)1596 3803 y Fy(2)1634 3791 y FB(\()p FA(u;)14 b(\034)9 b FB(\))1828 3695 y Fz(\014)1828 3745 y(\014)1828 3795 y(\014)1879 3791 y Fw(\024)2026 3734 y FA(K)d Fw(j)h FB(^)-49 b FA(\026)o Fw(j)2198 3704 y Fy(2)2235 3734 y FA(")2274 3704 y Fx(s)p Fv(\000)p Fy(2)p Fx(r)p 1977 3772 500 4 v 1977 3867 a Fw(j)p FB(ln)14 b FA(")p Fw(j)2144 3825 y Fy(1+Im)d(\()6 b(^)-39 b Fx(\026)2384 3808 y Fu(2)2417 3825 y Fx(b)p Fy(\))2486 3791 y FA(:)71 4016 y FB(Finally)-7 b(,)39 b(to)d(b)r(ound)i FA(@)801 4028 y Fx(u)856 3995 y Fz(e)844 4016 y FB(\001)913 4028 y Fy(3)950 4016 y FB(,)i(it)d(is)f(enough)g(to)h(tak)n(e)f(in)n(to)g(accoun)n(t)g (\(72\).)64 b(Then,)39 b(one)d(can)h(see)f(that)h(for)f(\()p FA(u;)14 b(\034)9 b FB(\))39 b Fw(2)71 4048 y Fz(\020)120 4141 y FA(R)183 4161 y Fx(s)12 b Fy(ln)303 4139 y Fu(1)p 303 4148 29 3 v 303 4181 a Fm(")341 4161 y Fx(;d)396 4169 y Fu(3)450 4141 y Fw(\\)19 b FA(D)595 4101 y Fy(in)p Fx(;)p Fy(+)p Fx(;s)593 4174 y(s)12 b Fy(ln)712 4152 y Fu(1)p 712 4161 V 712 4194 a Fm(")751 4174 y Fx(;c)801 4182 y Fu(2)855 4141 y Fw(\\)19 b FA(D)1000 4101 y Fy(in)o Fx(;)p Fy(+)p Fx(;u)998 4174 y(s)11 b Fy(ln)1117 4152 y Fu(1)p 1117 4161 V 1117 4194 a Fm(")1155 4174 y Fx(;c)1205 4182 y Fu(2)1241 4048 y Fz(\021)1309 4141 y Fw(\002)18 b Ft(T)p FB(,)383 4280 y Fz(\014)383 4330 y(\014)383 4380 y(\014)410 4376 y FA(@)454 4388 y Fx(u)509 4355 y Fz(e)498 4376 y FB(\001)567 4388 y Fy(3)604 4376 y FB(\()p FA(u;)c(\034)9 b FB(\))798 4280 y Fz(\014)798 4330 y(\014)798 4380 y(\014)850 4376 y Fw(\024)22 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FA(")1149 4342 y Fv(\000)p Fx(s)p Fv(\000)p Fy(2)p Fx(r)1354 4376 y FA(e)1393 4338 y Fv(\000)1454 4306 y Fy(2)p Fx(a)p 1454 4320 70 4 v 1473 4367 a(")2827 4376 y FB(pro)n(vided)27 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b(>)e FB(0)383 4475 y Fz(\014)383 4525 y(\014)383 4575 y(\014)410 4571 y FA(@)454 4583 y Fx(u)509 4550 y Fz(e)498 4571 y FB(\001)567 4583 y Fy(3)604 4571 y FB(\()p FA(u;)14 b(\034)9 b FB(\))798 4475 y Fz(\014)798 4525 y(\014)798 4575 y(\014)850 4571 y Fw(\024)22 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FA(")1149 4536 y Fv(\000)p Fx(s)p Fv(\000)p Fy(2)p Fx(r)1354 4571 y FA(e)1393 4533 y Fv(\000)1454 4500 y Fy(2)p Fx(a)p 1454 4514 V 1473 4562 a(")1533 4533 y Fy(+2Im)1707 4539 y FB(\()1745 4533 y Fy(^)-39 b Fx(\026)1779 4508 y Fu(2)1812 4533 y Fx(b)1841 4539 y FB(\))1885 4533 y Fy(ln)11 b Fx(")p Fy(+Im)2124 4539 y FB(\()2162 4533 y Fy(^)-39 b Fx(\026)2196 4508 y Fu(2)2229 4533 y Fx(b)2258 4539 y FB(\))2302 4533 y Fy(ln)11 b(ln)2445 4500 y(1)p 2445 4514 34 4 v 2446 4562 a Fx(")2804 4571 y FB(pro)n(vided)27 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b FB(=)e(0)p FA(:)71 4779 y FB(Therefore,)k(from)h(the)h(b)r(ounds)f(of)1208 4758 y Fz(e)1197 4779 y FB(\001)1266 4739 y Fx(u;s)1266 4801 y Fy(1)1360 4779 y FB(,)1423 4758 y Fz(e)1412 4779 y FB(\001)1481 4791 y Fy(2)1547 4779 y FB(and)1720 4758 y Fz(e)1709 4779 y FB(\001)1778 4791 y Fy(3)1844 4779 y FB(and)g(recalling)f(that)h(b)n(y)g(h)n(yp)r(othesis)g FA(s)c(<)g FB(1)p FA(=\014)t FB(,)k(w)n(e)g(ha)n(v)n(e)f(that)71 4905 y(for)g(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)494 4813 y Fz(\020)544 4905 y FA(R)607 4926 y Fx(s)11 b Fy(ln)726 4903 y Fu(1)p 726 4912 29 3 v 726 4946 a Fm(")764 4926 y Fx(;d)819 4934 y Fu(3)874 4905 y Fw(\\)18 b FA(D)1018 4865 y Fy(in)p Fx(;)p Fy(+)p Fx(;s)1016 4939 y(s)12 b Fy(ln)1136 4917 y Fu(1)p 1136 4926 V 1136 4959 a Fm(")1174 4939 y Fx(;c)1224 4947 y Fu(2)1278 4905 y Fw(\\)19 b FA(D)1423 4865 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)1421 4939 y(s)11 b Fy(ln)1540 4917 y Fu(1)p 1540 4926 V 1540 4959 a Fm(")1578 4939 y Fx(;c)1628 4947 y Fu(2)1664 4813 y Fz(\021)1732 4905 y Fw(\002)18 b Ft(T)p FB(,)740 5061 y Fz(\014)740 5111 y(\014)740 5161 y(\014)768 5156 y FA(@)812 5168 y Fx(u)867 5135 y Fz(e)855 5156 y FB(\001)q(\()p FA(u;)c(\034)9 b FB(\))1119 5061 y Fz(\014)1119 5111 y(\014)1119 5161 y(\014)1170 5156 y Fw(\024)1292 5100 y FA(K)d(")1408 5070 y Fx(s)p Fv(\000)p Fy(2)p Fx(r)p 1268 5137 318 4 v 1268 5229 a Fw(j)p FB(ln)14 b FA(")p Fw(j)1436 5188 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)2470 5156 y FB(pro)n(vided)26 b FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b(>)c FB(0)740 5312 y Fz(\014)740 5361 y(\014)740 5411 y(\014)768 5407 y FA(@)812 5419 y Fx(u)867 5386 y Fz(e)855 5407 y FB(\001)q(\()p FA(u;)14 b(\034)9 b FB(\))1119 5312 y Fz(\014)1119 5361 y(\014)1119 5411 y(\014)1170 5407 y Fw(\024)1383 5351 y FA(K)d(")1499 5321 y Fx(s)p Fv(\000)p Fy(2)p Fx(r)p 1268 5388 500 4 v 1268 5483 a Fw(j)p FB(ln)14 b FA(")p Fw(j)1436 5442 y Fy(1+Im)c(\()c(^)-39 b Fx(\026)1675 5425 y Fu(2)1708 5442 y Fx(b)p Fy(\))2447 5407 y FB(pro)n(vided)26 b FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b FB(=)c(0)p FA(:)p Black 1919 5753 a FB(47)p Black eop end %%Page: 48 48 TeXDict begin 48 47 bop Black Black 71 285 a FB(Moreo)n(v)n(er,)17 b(taking)h(in)n(to)g(accoun)n(t)g(that)g FA(@)1369 297 y Fx(u)1424 264 y Fz(e)1413 285 y FB(\001\()p FA(u;)c(\034)9 b FB(\))19 b(dep)r(ends)g(analytically)e(on)25 b(^)-49 b FA(\026)19 b FB(and)f(moreo)n(v)n(er)e(satis\014es)27 b FA(@)3471 297 y Fx(u)3526 264 y Fz(e)3515 285 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))3778 189 y Fz(\014)3778 239 y(\014)3778 289 y(\014)3812 343 y Fy(^)-39 b Fx(\026)q Fy(=0)3958 285 y FB(=)71 420 y(0,)27 b(one)g(can)h(apply)f(Sc)n(h)n(w)n (artz)f(Lemma)h(to)h(obtain)740 556 y Fz(\014)740 605 y(\014)740 655 y(\014)768 651 y FA(@)812 663 y Fx(u)867 630 y Fz(e)855 651 y FB(\001)q(\()p FA(u;)14 b(\034)9 b FB(\))1119 556 y Fz(\014)1119 605 y(\014)1119 655 y(\014)1170 651 y Fw(\024)1268 595 y FA(K)d Fw(j)h FB(^)-49 b FA(\026)o Fw(j)p FA(")1479 565 y Fx(s)p Fv(\000)p Fy(2)p Fx(r)p 1268 632 365 4 v 1291 724 a Fw(j)p FB(ln)14 b FA(")p Fw(j)1459 682 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)2470 651 y FB(pro)n(vided)26 b FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b(>)c FB(0)480 b(\(115\))740 806 y Fz(\014)740 856 y(\014)740 906 y(\014)768 902 y FA(@)812 914 y Fx(u)867 881 y Fz(e)855 902 y FB(\001)q(\()p FA(u;)14 b(\034)9 b FB(\))1119 806 y Fz(\014)1119 856 y(\014)1119 906 y(\014)1170 902 y Fw(\024)1335 846 y FA(K)d Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FA(")1547 816 y Fx(s)p Fv(\000)p Fy(2)p Fx(r)p 1268 883 500 4 v 1268 978 a Fw(j)p FB(ln)14 b FA(")p Fw(j)1436 936 y Fy(1+Im)c(\()c(^)-39 b Fx(\026)1675 920 y Fu(2)1708 936 y Fx(b)p Fy(\))2447 902 y FB(pro)n(vided)26 b FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b FB(=)c(0)p FA(:)480 b FB(\(116\))71 1186 y(Reasoning)27 b(analogously)-7 b(,)25 b(one)j(can)f(see)h(that)g(for)g(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)1981 1093 y Fz(\020)2031 1186 y FA(R)2094 1206 y Fx(s)11 b Fy(ln)2213 1184 y Fu(1)p 2213 1193 29 3 v 2213 1226 a Fm(")2251 1206 y Fx(;d)2306 1214 y Fu(3)2361 1186 y Fw(\\)19 b FA(D)2506 1146 y Fy(in)o Fx(;)p Fv(\000)p Fx(;s)2504 1219 y(s)11 b Fy(ln)2623 1197 y Fu(1)p 2623 1206 V 2623 1239 a Fm(")2661 1219 y Fx(;c)2711 1227 y Fu(2)2765 1186 y Fw(\\)19 b FA(D)2910 1146 y Fy(in)p Fx(;)p Fv(\000)p Fx(;u)2908 1219 y(s)12 b Fy(ln)3027 1197 y Fu(1)p 3027 1206 V 3027 1239 a Fm(")3065 1219 y Fx(;c)3115 1227 y Fu(2)3151 1093 y Fz(\021)3220 1186 y Fw(\002)18 b Ft(T)p FB(,)29 b(the)f(function)71 1329 y FA(@)115 1341 y Fx(u)170 1308 y Fz(e)158 1329 y FB(\001)g(satis\014es)736 1461 y Fz(\014)736 1511 y(\014)736 1560 y(\014)763 1556 y FA(@)807 1568 y Fx(u)862 1535 y Fz(e)851 1556 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))1114 1461 y Fz(\014)1114 1511 y(\014)1114 1560 y(\014)1166 1556 y Fw(\024)1263 1500 y FA(K)d Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FA(")1475 1470 y Fx(s)p Fv(\000)p Fy(2)p Fx(r)p 1263 1537 365 4 v 1287 1629 a Fw(j)p FB(ln)14 b FA(")p Fw(j)1454 1588 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)2474 1556 y FB(pro)n(vided)27 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b(>)e FB(0)475 b(\(117\))736 1711 y Fz(\014)736 1761 y(\014)736 1811 y(\014)763 1807 y FA(@)807 1819 y Fx(u)862 1786 y Fz(e)851 1807 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))1114 1711 y Fz(\014)1114 1761 y(\014)1114 1811 y(\014)1166 1807 y Fw(\024)1338 1751 y FA(K)d Fw(j)h FB(^)-49 b FA(\026)o Fw(j)p FA(")1549 1721 y Fx(s)p Fv(\000)p Fy(2)p Fx(r)p 1263 1788 514 4 v 1263 1896 a Fw(j)p FB(ln)14 b FA(")p Fw(j)1431 1854 y Fy(1)p Fv(\000)p Fy(Im)1606 1860 y FB(\()1644 1854 y Fy(^)-39 b Fx(\026)1678 1837 y Fu(2)p 1711 1805 30 3 v 1711 1854 a Fx(b)1740 1860 y FB(\))2451 1807 y(pro)n(vided)27 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b FB(=)e(0)p FA(:)475 b FB(\(118\))71 2089 y(Finally)-7 b(,)28 b(b)n(y)g(Theorems)f(4.4,)h(4.8,)g(4.12)f(and) h(4.21,)f(one)h(can)g(easily)f(see)h(that)g(the)h(b)r(ound)g(of)f FA(@)3113 2101 y Fx(u)3168 2068 y Fz(e)3156 2089 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))29 b(for)f(\()p FA(u;)14 b(\034)9 b FB(\))25 b Fw(2)71 2121 y Fz(\020)120 2213 y FA(R)183 2234 y Fx(s)12 b Fy(ln)303 2211 y Fu(1)p 303 2220 29 3 v 303 2254 a Fm(")341 2234 y Fx(;d)396 2242 y Fu(3)450 2213 y Fw(\\)19 b FA(D)595 2173 y Fy(out)p Fx(;s)593 2232 y(c)623 2240 y Fu(2)655 2232 y Fx(")686 2216 y Fm(\015)724 2232 y Fx(;\032)778 2240 y Fu(4)834 2213 y Fw(\\)f FA(D)978 2173 y Fy(out)p Fx(;u)976 2232 y(c)1006 2240 y Fu(2)1038 2232 y Fx(")1069 2216 y Fm(\015)1108 2232 y Fx(;\032)1162 2240 y Fu(4)1198 2121 y Fz(\021)1271 2213 y Fw(\002)k Ft(T)34 b FB(is)g(smaller)f(than)i(\(115\))e(and)h(\(117\))f(\(case)g FA(`)23 b Fw(\000)f FB(2)p FA(r)36 b(>)e FB(0\))g(and)f(\(116\))h(and) 71 2338 y(\(118\))27 b(\(case)g FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b FB(=)e(0\),)k(pro)n(vided)g Fw(j)p FA(u)18 b Fw(\000)g FA(ia)p Fw(j)k(\025)h(O)r FB(\()p FA(")1807 2308 y Fx(\015)1850 2338 y FB(\).)195 2437 y(T)-7 b(aking)26 b(in)n(to)g(accoun)n(t)g(\(115\))g(and)g(\(117\))g(\(case)g FA(`)16 b Fw(\000)g FB(2)p FA(r)25 b(>)e FB(0\))k(and)f(\(116\))g(and)g (\(118\))g(\(case)g FA(`)16 b Fw(\000)g FB(2)p FA(r)25 b FB(=)e(0\),)k(one)f(can)71 2537 y(conclude)h(that)h(for)34 b(^)-49 b FA(\026)23 b Fw(2)h FA(B)t FB(\()7 b(^)-49 b FA(\026)1020 2549 y Fy(0)1057 2537 y FB(\))19 b Fw(\\)g Ft(R)p FB(,)758 2677 y Fz(\014)758 2727 y(\014)758 2777 y(\014)785 2773 y FA(@)829 2785 y Fx(u)884 2752 y Fz(e)873 2773 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))1136 2677 y Fz(\014)1136 2727 y(\014)1136 2777 y(\014)1187 2773 y Fw(\024)1285 2717 y FA(K)d Fw(j)h FB(^)-49 b FA(\026)o Fw(j)p FA(")1496 2686 y Fx(s)p Fv(\000)p Fy(2)p Fx(r)p 1285 2754 365 4 v 1308 2846 a Fw(j)q FB(ln)14 b FA(")p Fw(j)1476 2804 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)2453 2773 y FB(pro)n(vided)26 b FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b(>)c FB(0)497 b(\(119\))758 2928 y Fz(\014)758 2978 y(\014)758 3028 y(\014)785 3023 y FA(@)829 3035 y Fx(u)884 3002 y Fz(e)873 3023 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))1136 2928 y Fz(\014)1136 2978 y(\014)1136 3028 y(\014)1187 3023 y Fw(\024)1327 2967 y FA(K)d Fw(j)h FB(^)-49 b FA(\026)o Fw(j)p FA(")1538 2937 y Fx(s)p Fv(\000)p Fy(2)p Fx(r)p 1285 3004 448 4 v 1285 3099 a Fw(j)p FB(ln)14 b FA(")p Fw(j)1453 3058 y Fy(1+)5 b(^)-38 b Fx(\026)1577 3041 y Fu(2)1610 3058 y Fy(Im)10 b Fx(b)2430 3023 y FB(pro)n(vided)26 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b FB(=)e(0)p FA(:)497 b FB(\(120\))71 3274 y(Analogously)23 b(to)i(the)g(pro)r(of)g(of)g (Theorem)f(4.19,)g(the)h(second)g(step)g(is)f(to)h(consider)f(the)i(c)n (hange)d(of)i(v)-5 b(ariables)24 b(\()p FA(w)r(;)14 b(\034)9 b FB(\))25 b(=)71 3373 y(\()p FA(u)18 b FB(+)g FA(")p Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))29 b(and)e(the)h(auxiliary)e(function)1493 3566 y(\002\()p FA(w)r(;)14 b(\034)9 b FB(\))24 b(=)1886 3545 y Fz(e)1877 3566 y FB(\007)1942 3532 y Fv(0)1979 3499 y Fz(\000)2017 3566 y FA(")2056 3532 y Fv(\000)p Fy(1)2145 3566 y FA(w)d Fw(\000)d FA(\034)2353 3499 y Fz(\001)2405 3566 y FA(;)71 3768 y FB(to)27 b(obtain)h(a)f(b)r(ound)h (for)f(the)h(F)-7 b(ourier)26 b(co)r(e\016cien)n(ts)i(of)1837 3747 y Fz(e)1828 3768 y FB(\007:)684 3885 y Fz(\014)684 3935 y(\014)684 3985 y(\014)721 3960 y(e)712 3981 y FB(\007)777 3946 y Fy([)p Fx(k)q Fy(])855 3885 y Fz(\014)855 3935 y(\014)855 3985 y(\014)906 3981 y Fw(\024)22 b FA(K)6 b(")270 b FB(sup)1123 4054 y Fy(\()p Fx(u;\034)7 b Fy(\))p Fv(2)p Fx(R)1367 4065 y Fm(s)h Fu(ln\(1)p Fm(=")p Fu(\))p Fm(;d)1632 4077 y Fu(3)1669 4054 y Fv(\002)p Fn(T)1774 3885 y Fz(\014)1774 3935 y(\014)1774 3985 y(\014)1801 3981 y FA(@)1845 3993 y Fx(u)1900 3960 y Fz(e)1889 3981 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))2152 3885 y Fz(\014)2152 3935 y(\014)2152 3985 y(\014)2194 3981 y FA(e)2233 3943 y Fv(\000)2295 3903 y(j)p Fx(k)q Fv(j)p 2295 3924 77 4 v 2317 3972 a Fx(")2381 3949 y FB(\()2413 3943 y Fx(a)p Fv(\000)p Fx(s")j Fy(ln)2652 3921 y Fu(1)p 2652 3930 29 3 v 2652 3963 a Fm(")2690 3949 y FB(\))2722 3943 y Fv(\000j)p Fx(k)q Fv(j)p Fy(Im)f(\()p Fv(C)s Fy(\()p Fx(u)3070 3918 y Fl(\003)3105 3943 y Fx(;)p Fy(0\)\))3214 3981 y FA(:)195 4258 y FB(Therefore,)30 b(to)g(obtain)g(the)g(b)r (ounds)h(for)1538 4237 y Fz(e)1529 4258 y FB(\007)1594 4228 y Fy([)p Fx(k)q Fy(])1702 4258 y FB(with)g FA(k)f(<)d FB(0,)j(it)h(only)f(remains)f(to)h(use)g(b)r(ounds)g(\(119\))f(and)h (\(120\))71 4358 y(and)d(the)h(prop)r(erties)f(of)g Fw(C)33 b FB(giv)n(en)27 b(in)g(Theorem)g(4.21)f(and)i(Prop)r(osition)e(4.22.) 35 b(Then,)28 b(w)n(e)f(obtain)h(that)g(for)f FA(k)f(<)c FB(0)286 4488 y Fz(\014)286 4538 y(\014)286 4588 y(\014)323 4563 y(e)314 4584 y FB(\007)379 4549 y Fy([)p Fx(k)q Fy(])457 4488 y Fz(\014)457 4538 y(\014)457 4588 y(\014)507 4584 y Fw(\024)781 4528 y FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p 605 4565 525 4 v 605 4657 a FA(")644 4633 y Fy(2)p Fx(r)r Fv(\000)p Fy(1)812 4657 y Fw(j)p FB(ln)14 b FA(")p Fw(j)980 4615 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)1140 4584 y FA(e)1179 4546 y Fv(\000j)p Fx(k)q Fv(j)1316 4513 y Fx(a)p 1316 4527 37 4 v 1318 4575 a(")1362 4546 y Fy(+\()p Fv(j)p Fx(k)q Fv(j\000)p Fy(1\))p Fx(s)e Fy(ln)1746 4524 y Fu(1)p 1746 4533 29 3 v 1746 4566 a Fm(")2924 4584 y FB(pro)n(vided)27 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b(>)e FB(0)286 4734 y Fz(\014)286 4783 y(\014)286 4833 y(\014)323 4808 y(e)314 4829 y FB(\007)379 4795 y Fy([)p Fx(k)q Fy(])457 4734 y Fz(\014)457 4783 y(\014)457 4833 y(\014)507 4829 y Fw(\024)706 4773 y FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p 605 4810 376 4 v 605 4886 a FA(")644 4862 y Fy(2)p Fx(r)r Fv(\000)p Fy(1)812 4886 y Fw(j)p FB(ln)14 b FA(")p Fw(j)990 4829 y FA(e)1029 4792 y Fv(\000j)p Fx(k)q Fv(j)1157 4798 y FB(\()1199 4759 y Fx(a)p 1199 4773 37 4 v 1201 4820 a(")1245 4792 y Fv(\000)p Fy(Im)1387 4798 y FB(\()1425 4792 y Fy(^)-39 b Fx(\026)1459 4766 y Fu(2)1492 4792 y Fx(b)1521 4798 y FB(\))1565 4792 y Fy(ln)11 b Fx(")1663 4798 y FB(\))1695 4792 y Fy(+\()p Fv(j)p Fx(k)q Fv(j\000)p Fy(1\))1959 4798 y FB(\()1991 4792 y Fx(s)h Fy(ln)2111 4769 y Fu(1)p 2111 4778 29 3 v 2111 4812 a Fm(")2149 4792 y Fy(+Im)2290 4798 y FB(\()2328 4792 y Fy(^)-39 b Fx(\026)2362 4766 y Fu(2)2395 4792 y Fx(b)2424 4798 y FB(\))2468 4792 y Fy(ln)11 b(ln)2611 4769 y Fu(1)p 2611 4778 V 2611 4812 a Fm(")2650 4798 y FB(\))2901 4829 y(pro)n(vided)27 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b FB(=)e(0)p FA(:)71 5079 y FB(Since)34 b FA(@)338 5091 y Fx(u)393 5058 y Fz(e)382 5079 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))35 b(and)f Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))34 b(are)g(real-analytic)e(for)h(\()p FA(\026;)14 b(\034)9 b FB(\))36 b Fw(2)e Ft(R)p FB(,)i(the)e(co)r(e\016cien)n(ts)2913 5058 y Fz(e)2904 5079 y FB(\007)2969 5049 y Fy([)p Fx(k)q Fy(])3081 5079 y FB(for)g FA(k)j(>)d FB(0)f(satisfy)h(the)71 5189 y(same)39 b(b)r(ounds.)75 b(Finally)-7 b(,)43 b(the)e(b)r(ounds)f (of)1537 5168 y Fz(e)1528 5189 y FB(\007)1593 5159 y Fy([)p Fx(k)q Fy(])1711 5189 y FB(lead)g(easily)f(to)h(the)h(desired)e (b)r(ounds)i(of)3124 5168 y Fz(e)3113 5189 y FB(\001\()p FA(u;)14 b(\034)9 b FB(\))41 b(for)e(\()p FA(u;)14 b(\034)9 b FB(\))45 b Fw(2)71 5221 y Fz(\000)109 5289 y FA(R)172 5304 y Fx(s)12 b Fy(ln)o(\(1)p Fx(=")p Fy(\))p Fx(;d)475 5312 y Fu(3)530 5289 y Fw(\\)19 b Ft(R)664 5221 y Fz(\001)720 5289 y Fw(\002)f Ft(T)p FB(.)p 3790 5289 4 57 v 3794 5236 50 4 v 3794 5289 V 3843 5289 4 57 v Black 1919 5753 a(48)p Black eop end %%Page: 49 49 TeXDict begin 49 48 bop Black Black 71 272 a Fq(4.8)112 b(Computation)39 b(of)e(the)h(area)g(of)f(the)h(lob)s(es:)51 b(pro)s(of)37 b(of)h(Theorems)g(2.5)g(and)g(2.6)71 425 y FB(Once)30 b(w)n(e)f(ha)n(v)n(e)g(obtained)h(the)h(\014rst)f (asymptotic)f(term)h(of)g(the)h(di\013erence)f(b)r(et)n(w)n(een)g(the)h (manifolds)f(in)g(Corollaries)71 525 y(4.20)38 b(and)h(4.24,)i(w)n(e)e (use)h(it)g(to)f(pro)n(v)n(e)f(the)i(existence)f(of)g(transv)n(ersal)e (homo)r(clinic)i(p)r(oin)n(ts)h(and)f(to)g(compute)h(an)71 625 y(asymptotic)27 b(form)n(ula)g(of)g(the)h(area)e(of)i(the)g(lob)r (es)f(b)r(et)n(w)n(een)h(t)n(w)n(o)e(consecutiv)n(e)h(ones.)195 724 y(Let)d(us)g(\014x)g FA(\026)f Fw(2)h Ft(R)g FB(suc)n(h)f(that)h FA(f)9 b FB(\()p FA(\026)p FB(\))24 b Fw(6)p FB(=)e(0.)35 b(Let)24 b(us)g(\014x)g(also)f(a)g(transv)n(ersal)f(section)h(corresp)r (onding)f(to)i FA(\034)33 b FB(=)22 b FA(\034)3628 736 y Fy(0)3689 724 y Fw(2)h Ft(R)p FB(.)71 824 y(It)j(can)g(b)r(e)g (easily)g(seen)f(that)i(the)f(consecutiv)n(e)f(zeros)g(of)h FA(@)1928 836 y Fx(u)1971 824 y FB(\001)2040 836 y Fy(00)2111 824 y FB(\()p FA(u;)14 b(\034)2264 836 y Fy(0)2301 824 y FB(\))27 b(\(see)f(\(87\))o(,)h(\(109\))e(and)h(\(110\))o(\))g(are)f Fw(O)r FB(\()p FA(")p FB(\))i(close)71 923 y(and)20 b(therefore)g (taking)h FA(")f FB(small)h(enough,)g(in)1480 856 y Fz(\000)1518 923 y FA(R)1581 938 y Fx(s)12 b Fy(ln)o(\(1)p Fx(=")p Fy(\))p Fx(;d)1884 946 y Fu(3)1939 923 y Fw(\\)19 b Ft(R)2073 856 y Fz(\001)2132 923 y FB(there)h(exist)h(at)g(least)f(t)n(w)n(o)g (consecutiv)n(e)g(zeros)f FA(u)3640 935 y Fv(\000)3717 923 y FB(and)71 1032 y FA(u)119 1044 y Fy(+)204 1032 y FB(in)303 965 y Fz(\000)341 1032 y FA(R)404 1047 y Fx(s)12 b Fy(ln\(1)p Fx(=")p Fy(\))p Fx(;d)708 1055 y Fu(3)762 1032 y Fw(\\)19 b Ft(R)896 965 y Fz(\001)934 1032 y FB(,)31 b(whic)n(h)f(dep)r(end)h(on)f FA(\034)1673 1044 y Fy(0)1711 1032 y FB(.)45 b(It)31 b(can)f(b)r(e)g(easily)g(c)n (hec)n(k)n(ed)f(that)h(the)h(function)g(\001)3406 1044 y Fy(00)3507 1032 y FB(ev)-5 b(aluated)71 1131 y(at)27 b(these)h(p)r(oin)n(ts)f(satis\014es)1455 1231 y(\001)1524 1243 y Fy(00)1595 1231 y FB(\()p FA(u)1675 1243 y Fy(+)1730 1231 y FA(;)14 b(\034)1803 1243 y Fy(0)1840 1231 y FB(\))24 b(=)e Fw(\000)p FB(\001)2117 1243 y Fy(00)2187 1231 y FB(\()p FA(u)2267 1243 y Fv(\000)2323 1231 y FA(;)14 b(\034)2396 1243 y Fy(0)2434 1231 y FB(\))1195 b(\(121\))71 1381 y(and)521 1563 y Fw(j)p FB(\001)613 1575 y Fy(00)683 1563 y FB(\()p FA(u)763 1575 y Fv(\006)819 1563 y FA(;)14 b(\034)892 1575 y Fy(0)930 1563 y FB(\))p Fw(j)23 b FB(=)g(2)p FA(\026")1227 1529 y Fx(\021)r Fy(+1)p Fv(\000)p Fx(`)1444 1493 y Fz(\014)1444 1542 y(\014)1472 1563 y FA(f)9 b FB(\()p FA(\026")1643 1529 y Fx(\021)r Fv(\000)p Fx(`)p Fy(+2)p Fx(r)1879 1563 y FB(\))1911 1493 y Fz(\014)1911 1542 y(\014)1953 1563 y FA(e)1992 1526 y Fw(\000)2067 1493 y Fx(a)p 2066 1507 37 4 v 2068 1554 a(")2955 1563 y FB(if)28 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b(>)e FB(0)260 b(\(122\))521 1722 y Fw(j)p FB(\001)613 1734 y Fy(00)683 1722 y FB(\()p FA(u)763 1734 y Fv(\006)819 1722 y FA(;)14 b(\034)892 1734 y Fy(0)930 1722 y FB(\))p Fw(j)23 b FB(=)g(2)p FA(\026")1227 1687 y Fx(\021)r Fy(+1)p Fv(\000)p Fx(`)1444 1626 y Fz(\014)1444 1676 y(\014)1444 1726 y(\014)1472 1722 y FA(f)9 b FB(\()p FA(\026)p FB(\))p FA(e)1675 1687 y Fx(iC)t Fy(\()p Fx(\026;")p Fy(\))1897 1626 y Fz(\014)1897 1676 y(\014)1897 1726 y(\014)1939 1722 y FA(e)1978 1684 y Fw(\000)2053 1652 y Fx(a)p 2052 1666 V 2054 1713 a(")2098 1684 y Fy(+)p Fx(\026)2189 1659 y Fu(2)2222 1684 y Fy(Im)h Fx(b)i Fy(ln)f Fx(")2955 1722 y FB(if)28 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b FB(=)e(0)260 b(\(123\))521 1904 y Fw(j)p FB(\001)613 1916 y Fy(00)683 1904 y FB(\()p FA(u)763 1916 y Fv(\006)819 1904 y FA(;)14 b(\034)892 1916 y Fy(0)930 1904 y FB(\))p Fw(j)23 b FB(=)g(2)p FA(\026")1227 1870 y Fx(\021)r Fy(+1)p Fv(\000)p Fx(`)1444 1809 y Fz(\014)1444 1859 y(\014)1444 1909 y(\014)1472 1904 y FA(f)1513 1916 y Fy(0)1549 1904 y FA(e)1588 1870 y Fx(iC)t Fy(\()p Fx(\026;")p Fy(\))1811 1809 y Fz(\014)1811 1859 y(\014)1811 1909 y(\014)1852 1904 y FA(e)1891 1867 y Fw(\000)1966 1834 y Fx(a)p 1966 1848 V 1968 1895 a(")2932 1904 y FB(if)28 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b(<)e FB(0)p FA(:)260 b FB(\(124\))71 2112 y(In)28 b(the)g(case)e FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b(>)c FB(0)28 b(w)n(e)f(ha)n(v)n(e)f(undone)i(the)g(c)n(hange)e (of)i(parameter)e(\(58\))o(.)195 2212 y(By)31 b(Corollary)d(4.24,)i (since)h(b)n(y)f(h)n(yp)r(othesis)g FA(f)9 b FB(\()p FA(\026)p FB(\))29 b Fw(6)p FB(=)e(0,)k(w)n(e)g(can)f(apply)g(the)h (implicit)h(function)f(theorem)f(to)h(see)71 2311 y(that)f(close)e(to)i FA(u)605 2323 y Fv(\000)690 2311 y FB(and)f FA(u)901 2323 y Fy(+)956 2311 y FB(,)h(there)f(exist)h(zeros)e FA(u)1680 2281 y Fv(\003)1680 2332 y(\000)1765 2311 y FB(and)h FA(u)1976 2281 y Fv(\003)1976 2332 y Fy(+)2061 2311 y FB(of)g(the)h(function)g FA(@)2673 2323 y Fx(u)2717 2311 y FB(\001\()p FA(u;)14 b(\034)2939 2323 y Fy(0)2976 2311 y FB(\))30 b(de\014ned)g(in)f(\(76\).)43 b(These)71 2411 y(zeros)26 b(corresp)r(ond)g(to)i(transv)n(ersal)d(homo)r(clinic)i (p)r(oin)n(ts)h(and)f(they)h(satisfy)1496 2639 y FA(u)1544 2604 y Fv(\003)1544 2659 y(\006)1623 2639 y FB(=)23 b FA(u)1759 2651 y Fv(\006)1833 2639 y FB(+)18 b Fw(O)1998 2522 y Fz(\022)2174 2583 y FA(")p 2069 2620 249 4 v 2069 2696 a Fw(j)c FB(ln)g FA(")p Fw(j)2251 2672 y Fx(\027)2284 2681 y Fm(`)2327 2522 y Fz(\023)2402 2639 y FA(;)1236 b FB(\(125\))71 2867 y(where)27 b FA(\027)352 2879 y Fx(`)407 2867 y FB(=)c FA(`)18 b Fw(\000)g FB(2)p FA(r)30 b FB(for)d FA(`)22 b(>)h FB(2)p FA(r)30 b FB(and)d FA(\027)1323 2879 y Fx(`)1378 2867 y FB(=)c(1)k(for)g FA(`)c Fw(\024)g FB(2)p FA(r)r FB(.)195 2966 y(T)-7 b(o)35 b(compute)g(the)g(area)f Fw(A)h FB(of)g(the)g(lob)r(es)g(b)r(et)n(w)n(een)f FA(u)1953 2978 y Fv(\000)2044 2966 y FB(and)h FA(u)2261 2978 y Fy(+)2350 2966 y FB(it)h(is)e(enough)h(to)f(p)r(oin)n(t)h(out)g(that,)i (since)e(the)71 3066 y(c)n(hange)30 b(\(39\))h(is)g(symplectic,)h(it)f (preserv)n(es)e(area.)46 b(Then)31 b(recalling)f(the)i(de\014nition)f (of)g(\001)h(in)f(\(76\),)h(the)f(area)f(is)h(just)71 3165 y(giv)n(en)c(b)n(y)1010 3335 y Fw(A)c FB(=)1187 3190 y Fz(\014)1187 3240 y(\014)1187 3290 y(\014)1187 3339 y(\014)1187 3389 y(\014)1214 3222 y(Z)1297 3243 y Fx(u)1336 3218 y Fl(\003)1336 3259 y Fu(+)1260 3411 y Fx(u)1299 3391 y Fl(\003)1299 3430 y(\000)1401 3335 y FA(@)1445 3347 y Fx(u)1489 3335 y FB(\001\()p FA(u;)14 b(\034)1711 3347 y Fy(0)1748 3335 y FB(\))g FA(du)1885 3190 y Fz(\014)1885 3240 y(\014)1885 3290 y(\014)1885 3339 y(\014)1885 3389 y(\014)1936 3335 y FB(=)2023 3265 y Fz(\014)2023 3314 y(\014)2051 3335 y FB(\001\()p FA(u)2200 3301 y Fv(\003)2200 3356 y Fy(+)2255 3335 y FA(;)g(\034)2328 3347 y Fy(0)2366 3335 y FB(\))k Fw(\000)g FB(\001\()p FA(u)2648 3301 y Fv(\003)2648 3356 y(\000)2704 3335 y FA(;)c(\034)2777 3347 y Fy(0)2815 3335 y FB(\))2847 3265 y Fz(\014)2847 3314 y(\014)2889 3335 y FA(:)71 3559 y FB(Finally)-7 b(,)32 b(if)g FA(`)e(<)f FB(2)p FA(r)k FB(w)n(e)f(use)f(Corollary)e(4.20)h(and)h(Theorem)g(4.19)f(and)i(also)e (form)n(ulas)g(\(121\))o(,)j(\(124\))e(and)g(\(125\))g(to)71 3659 y(obtain)e(the)g(asymptotic)g(form)n(ula)f(for)h Fw(A)g FB(stated)h(in)f(Theorem)f(2.5.)41 b(In)30 b(the)f(same)g(w)n(a) n(y)-7 b(,)28 b(if)i FA(`)25 b Fw(\025)h FB(2)p FA(r)31 b FB(using)e(Corollary)71 3759 y(4.24,)c(Theorem)g(4.21)g(and)h(form)n (ulas)f(\(121\))o(,)i(\(122\))e(\(if)i FA(`)15 b Fw(\000)g FB(2)p FA(r)25 b(>)e FB(0\))j(or)f(\(123\))g(\(if)i FA(`)15 b Fw(\000)g FB(2)p FA(r)25 b FB(=)e(0\))j(and)g(\(125\))o(,)h(w)n(e)e (deriv)n(e)71 3858 y(the)j(results)f(in)h(Theorem)f(2.6.)71 4133 y FC(5)135 b(Existence)50 b(of)f(the)g(p)t(erio)t(dic)f(orbit)i (in)e(the)h(h)l(yp)t(erb)t(olic)g(case:)68 b(pro)t(of)273 4282 y(of)45 b(Theorem)g(4.1)71 4464 y FB(In)29 b(this)g(section)f(w)n (e)g(pro)n(v)n(e)f(Theorem)h(4.1.)39 b(W)-7 b(e)29 b(lo)r(ok)f(for)h(a) f(p)r(erio)r(dic)g(orbit)h(\()p FA(x;)14 b(y)s FB(\))25 b(=)f(\()p FA(x)2919 4476 y Fx(p)2959 4464 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(y)3146 4476 y Fx(p)3185 4464 y FB(\()p FA(\034)9 b FB(\)\))30 b(whic)n(h)f(is)f(close)71 4564 y(to)f(the)h(h)n(yp)r(erb)r(olic)f(critical)g(p)r(oin)n(t)h(of)g (the)g(unp)r(erturb)r(ed)g(system)f(\(0)p FA(;)14 b FB(0\).)195 4663 y(By)29 b Fp(HP1.1)p FB(,)e(the)h(di\013eren)n(tial)f(of)g(the)h (unp)r(erturb)r(ed)h(h)n(yp)r(erb)r(olic)e(critical)g(p)r(oin)n(t)g(is) 1584 4891 y FA("A)1685 4903 y Fy(0)1745 4891 y FB(=)c FA(")1886 4774 y Fz(\022)2010 4841 y FB(0)105 b(1)1988 4940 y FA(\025)2036 4910 y Fy(2)2157 4940 y FB(0)2240 4774 y Fz(\023)2314 4891 y FA(:)1324 b FB(\(126\))71 5124 y(Then,)28 b(de\014ning)f FA(z)g FB(=)22 b(\()p FA(x;)14 b(y)s FB(\))29 b(and)e(considering)f(the)i(di\013eren)n(tial)g (op)r(erator)1636 5350 y Fw(D)1700 5362 y Fy(0)1737 5350 y FA(z)t FB(\()p FA(\034)9 b FB(\))24 b(=)2033 5293 y FA(d)p 2011 5330 89 4 v 2011 5406 a(d\034)2110 5350 y(z)t FB(\()p FA(\034)9 b FB(\))p FA(;)1376 b FB(\(127\))p Black 1919 5753 a(49)p Black eop end %%Page: 50 50 TeXDict begin 50 49 bop Black Black 71 272 a FB(w)n(e)27 b(lo)r(ok)g(for)g(the)h(p)r(erio)r(dic)g(orbit)f(as)g(a)g(2)p FA(\031)s FB(-p)r(erio)r(dic)g(solution)g(of)g(the)h(follo)n(wing)f (equation,)1516 455 y(\()q Fw(D)1613 467 y Fy(0)1668 455 y Fw(\000)18 b FA("A)1852 467 y Fy(0)1890 455 y FB(\))c FA(z)26 b FB(=)d FA("F)12 b FB(\()p FA(z)t(;)i(\034)9 b FB(\))p FA(;)1256 b FB(\(128\))71 637 y(where)1019 766 y FA(F)12 b FB(\()p FA(z)t(;)i(\034)9 b FB(\))24 b(=)1384 649 y Fz(\022)1487 715 y FA(\026")1576 685 y Fx(\021)1616 715 y FA(@)1660 727 y Fx(y)1700 715 y FA(H)1769 727 y Fy(1)1806 715 y FB(\()p FA(x;)14 b(y)s(;)g(\034)9 b FB(\))1487 816 y Fw(\000)p FA(\026")1641 786 y Fx(\021)1681 816 y FA(@)1725 828 y Fx(x)1767 816 y FA(H)1836 828 y Fy(1)1873 816 y FB(\()p FA(x;)14 b(y)s(;)g(\034)9 b FB(\))20 b Fw(\000)2250 749 y Fz(\000)2288 816 y FA(V)2354 786 y Fv(0)2378 816 y FB(\()p FA(x)p FB(\))f(+)f FA(\025)2639 786 y Fy(2)2677 816 y FA(x)2724 749 y Fz(\001)2804 649 y(\023)2879 766 y FA(:)71 962 y FB(W)-7 b(e)28 b(split)g FA(F)39 b FB(in)28 b(constan)n(t,)f(linear)g(and)g(higher)g(order)g (terms)g(with)h(resp)r(ect)f(to)h FA(z)1315 1144 y(F)12 b FB(\()p FA(z)t(;)i(\034)9 b FB(\))23 b(=)g FA(F)1733 1156 y Fy(0)1770 1144 y FB(\()p FA(\034)9 b FB(\))20 b(+)e FA(F)2035 1156 y Fy(1)2073 1144 y FB(\()p FA(\034)9 b FB(\))p FA(z)23 b FB(+)18 b FA(F)2380 1156 y Fy(2)2417 1144 y FB(\()p FA(z)t(;)c(\034)9 b FB(\))1055 b(\(129\))71 1327 y(with)1011 1539 y FA(F)1064 1551 y Fy(0)1101 1539 y FB(\()p FA(\034)9 b FB(\))25 b(=)1322 1421 y Fz(\022)1425 1488 y FA(\026")1514 1458 y Fx(\021)1554 1488 y FA(@)1598 1500 y Fx(y)1638 1488 y FA(H)1707 1500 y Fy(1)1744 1488 y FB(\(0)p FA(;)14 b FB(0)p FA(;)g(\034)9 b FB(\))1425 1588 y Fw(\000)p FA(\026")1579 1557 y Fx(\021)1619 1588 y FA(@)1663 1600 y Fx(x)1704 1588 y FA(H)1773 1600 y Fy(1)1811 1588 y FB(\(0)p FA(;)14 b FB(0)p FA(;)g(\034)9 b FB(\))2119 1421 y Fz(\023)3661 1539 y FB(\(130\))1011 1771 y FA(F)1064 1783 y Fy(1)1101 1771 y FB(\()p FA(\034)g FB(\))25 b(=)1322 1654 y Fz(\022)1425 1720 y FA(\026")1514 1690 y Fx(\021)1554 1720 y FA(@)1598 1732 y Fx(y)r(x)1676 1720 y FA(H)1745 1732 y Fy(1)1782 1720 y FB(\(0)p FA(;)14 b FB(0)p FA(;)g(\034)9 b FB(\))149 b FA(\026")2287 1690 y Fx(\021)2328 1720 y FA(@)2372 1732 y Fx(y)r(y)2447 1720 y FA(H)2516 1732 y Fy(1)2554 1720 y FB(\(0)p FA(;)14 b FB(0)p FA(;)g(\034)9 b FB(\))1425 1820 y Fw(\000)p FA(\026")1579 1790 y Fx(\021)1619 1820 y FA(@)1663 1832 y Fx(xx)1742 1820 y FA(H)1811 1832 y Fy(1)1848 1820 y FB(\(0)p FA(;)14 b FB(0)p FA(;)g(\034)9 b FB(\))83 b Fw(\000)p FA(\026")2352 1790 y Fx(\021)2392 1820 y FA(@)2436 1832 y Fx(xy)2514 1820 y FA(H)2583 1832 y Fy(1)2620 1820 y FB(\(0)p FA(;)14 b FB(0)p FA(;)g(\034)9 b FB(\))2929 1654 y Fz(\023)3661 1771 y FB(\(131\))931 1953 y FA(F)984 1965 y Fy(2)1022 1953 y FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))24 b(=)e FA(F)12 b FB(\()p FA(z)t(;)i(\034)9 b FB(\))19 b Fw(\000)f FA(F)1731 1965 y Fy(0)1769 1953 y FB(\()p FA(\034)9 b FB(\))19 b Fw(\000)f FA(F)2033 1965 y Fy(1)2071 1953 y FB(\()p FA(\034)9 b FB(\))p FA(z)t(:)1415 b FB(\(132\))195 2135 y(W)-7 b(e)38 b(dev)n(ote)e(the)h(rest)f(of)h (the)h(section)e(to)h(obtain)f(a)h(solution)f(of)h(equation)f(\(128\))o (.)65 b(First)37 b(in)g(Section)g(5.1)f(w)n(e)71 2235 y(de\014ne)27 b(a)f(Banac)n(h)f(space)h(w)n(e)h(will)g(use)f(and)h(w)n (e)f(state)g(some)g(tec)n(hnical)h(prop)r(erties.)35 b(Then,)27 b(in)g(Section)g(5.2)f(w)n(e)g(pro)n(v)n(e)71 2335 y(Theorem)h(4.1.)71 2567 y Fq(5.1)112 b(Banac)m(h)38 b(spaces)h(and)f(tec)m(hnical)g(lemmas)71 2720 y FB(F)-7 b(or)27 b(analytic)g(functions)h FA(z)e FB(:)d Ft(T)1060 2732 y Fx(\033)1128 2720 y Fw(!)g Ft(C)p FB(,)28 b FA(z)t FB(\()p FA(\034)9 b FB(\))24 b(=)1608 2658 y Fz(P)1696 2745 y Fx(k)q Fv(2)p Fn(Z)1834 2720 y FA(z)1877 2690 y Fy([)p Fx(k)q Fy(])1955 2720 y FA(e)1994 2690 y Fx(ik)q(\034)2095 2720 y FB(,)k(w)n(e)f(de\014ne)h(the)g(F)-7 b(ourier)26 b(norm)1567 2936 y Fw(k)p FA(z)t Fw(k)1694 2948 y Fx(\033)1760 2936 y FB(=)1848 2858 y Fz(X)1848 3036 y Fx(k)q Fv(2)p Fn(Z)1982 2841 y Fz(\014)1982 2891 y(\014)1982 2941 y(\014)2009 2936 y FA(z)2052 2902 y Fy([)p Fx(k)q Fy(])2130 2841 y Fz(\014)2130 2891 y(\014)2130 2941 y(\014)2172 2936 y FA(e)2211 2902 y Fv(j)p Fx(k)q Fv(j)p Fx(\033)2331 2936 y FA(:)71 3198 y FB(Then,)i(w)n(e)f(de\014ne)h(the)g(function)g (space)f(endo)n(w)n(ed)f(with)j(the)f(previous)e(norm)1145 3381 y Fw(S)1195 3393 y Fx(\033)1263 3381 y FB(=)c Fw(f)p FA(z)k FB(:)d Ft(T)1558 3393 y Fx(\033)1626 3381 y Fw(!)g Ft(C)p FB(;)42 b(real-analytic)26 b FA(;)14 b Fw(k)p FA(z)t Fw(k)2499 3393 y Fx(\033)2565 3381 y FA(<)22 b Fw(1g)884 b FB(\(133\))71 3563 y(whic)n(h)27 b(is)h(a)f(Banac)n(h)f (algebra.)36 b(W)-7 b(e)27 b(also)g(consider)g(the)h(pro)r(duct)f (space)g Fw(S)2440 3575 y Fx(\033)2503 3563 y Fw(\002)18 b(S)2636 3575 y Fx(\033)2709 3563 y FB(with)28 b(the)g(induced)g(norm) 1416 3746 y Fw(k)o FB(\()p FA(z)1528 3758 y Fy(1)1565 3746 y FA(;)14 b(z)1641 3758 y Fy(2)1678 3746 y FB(\))p Fw(k)1752 3771 y Fy(1)p Fx(;\033)1872 3746 y FB(=)23 b Fw(k)p FA(z)2041 3758 y Fy(1)2077 3746 y Fw(k)2119 3771 y Fx(\033)2182 3746 y FB(+)18 b Fw(k)p FA(z)2346 3758 y Fy(2)2382 3746 y Fw(k)2424 3771 y Fx(\033)2483 3746 y FA(:)p Black 71 3929 a Fp(Remark)32 b(5.1.)p Black 40 w Fs(L)l(et)d(us)g(c)l(onsider)i(the)f(classic)l(al)h(supr)l(emmum)e (norm)1572 4111 y Fw(k)p FA(z)t Fw(k)1699 4123 y Fv(1)p Fx(;\033)1851 4111 y FB(=)41 b(sup)1939 4195 y Fx(\034)7 b Fv(2)p 2022 4147 39 3 v Fn(T)2060 4203 y Fm(\033)2113 4111 y Fw(j)p FA(z)t FB(\()p FA(\034)i FB(\))p Fw(j)15 b FA(:)71 4366 y Fs(Then,)35 b(it)e(is)g(a)g(wel)t(l)h(known)f(fact)h (\(se)l(e)e(for)i(instanc)l(e)f([Sau01)q(]\))g(that)g(for)h(any)f FA(\033)2662 4378 y Fy(1)2729 4366 y FA(<)28 b(\033)2869 4378 y Fy(2)2907 4366 y Fs(,)34 b(the)f(supr)l(emmum)f(and)h(the)71 4465 y(F)-6 b(ourier)30 b(norm)g(satisfy)h(the)f(fol)t(lowing)i(r)l (elation)1332 4693 y Fw(k)p FA(z)t Fw(k)1459 4705 y Fx(\033)1497 4713 y Fu(1)1555 4693 y FA(<)23 b(K)1733 4576 y Fz(\022)1794 4693 y FB(1)18 b(+)2062 4637 y(1)p 1947 4674 271 4 v 1947 4750 a FA(\033)1994 4762 y Fy(2)2050 4750 y Fw(\000)g FA(\033)2180 4762 y Fy(1)2228 4576 y Fz(\023)2303 4693 y Fw(k)p FA(z)t Fw(k)2430 4705 y Fv(1)p Fx(;\033)2554 4713 y Fu(2)71 4926 y Fs(Ther)l(efor)l(e,)33 b(sinc)l(e)e(we)h(ar)l(e)f (assuming)g(that)g(ther)l(e)g(exists)f FA(\033)1967 4938 y Fy(0)2030 4926 y FA(>)25 b FB(0)30 b Fs(such)h(that)g(the)g (functions)g FA(a)3098 4938 y Fx(k)q(l)3191 4926 y Fs(de\014ne)l(d)g (in)38 b FB(\(8\))31 b Fs(and)71 5025 y FB(\(9\))d Fs(ar)l(e)h Fw(C)394 4995 y Fy(0)460 5025 y Fs(in)p 560 4958 56 4 v 28 w Ft(T)616 5037 y Fx(\033)654 5045 y Fu(0)719 5025 y Fs(and)g(analytic)h(in)f Ft(T)1348 5037 y Fx(\033)1386 5045 y Fu(0)1423 5025 y Fs(,)g(we)g(c)l(an)g(de)l(duc)l(e)g(that)f(for) i(any)e FA(\033)f(<)c(\033)2681 5037 y Fy(0)2747 5025 y Fs(such)28 b(that)h FA(\033)3150 5037 y Fy(0)3203 5025 y Fw(\000)16 b FA(\033)32 b Fs(has)d(a)g(p)l(ositive)71 5125 y(lower)i(b)l(ound)e(indep)l(endent)i(of)f FA(")p Fs(,)g(they)g(satisfy)1741 5307 y Fw(k)p FA(a)1827 5319 y Fx(k)q(l)1889 5307 y Fw(k)1931 5319 y Fx(\033)1998 5307 y FA(<)22 b(K)q(:)71 5490 y Fs(We)30 b(wil)t(l)h(use)e(this)h (fact)h(without)e(mentioning)h(it,)h(in)e(the)h(r)l(est)g(of)g(the)g (se)l(ction)g(and)g(also)h(in)f(Se)l(ctions)f(6.1)i(to)f(9.)p Black 1919 5753 a FB(50)p Black eop end %%Page: 51 51 TeXDict begin 51 50 bop Black Black 195 272 a FB(Since)28 b(w)n(e)e(deal)h(with)h(v)n(ector)e(functions,)h(w)n(e)g(also)f (consider)g(the)i(norm)e(for)h(2)17 b Fw(\002)g FB(2)27 b(matrices)f(induced)i(b)n(y)f Fw(k)17 b(\001)g(k)3730 284 y Fy(1)p Fx(;\033)3827 272 y FB(.)71 372 y(Let)27 b(us)g(consider)f FA(B)h FB(=)826 304 y Fz(\000)864 372 y FA(b)900 342 y Fx(ij)958 304 y Fz(\001)1023 372 y FB(a)g(2)17 b Fw(\002)g FB(2)26 b(matrix)h(suc)n(h)g(that)g FA(b)1973 342 y Fx(ij)2054 372 y Fw(2)d(S)2183 384 y Fx(\033)2227 372 y FB(.)37 b(Then,)28 b(the)f(induced)h(matrix)e(norm)h(is)g(giv)n (en)f(b)n(y)1332 565 y Fw(k)p FA(B)t Fw(k)1483 577 y Fy(1)p Fx(;\033)1603 565 y FB(=)j(max)1691 617 y Fx(j)s Fy(=1)p Fx(;)p Fy(2)1872 498 y Fz(\010)1921 494 y(\015)1921 544 y(\015)1967 565 y FA(b)2003 531 y Fy(1)p Fx(j)2071 494 y Fz(\015)2071 544 y(\015)2117 598 y Fx(\033)2180 565 y FB(+)2263 494 y Fz(\015)2263 544 y(\015)2309 565 y FA(b)2345 531 y Fy(2)p Fx(j)2413 494 y Fz(\015)2413 544 y(\015)2459 598 y Fx(\033)2504 498 y Fz(\011)2566 565 y FA(:)71 786 y FB(The)f(next)f(lemma)h(giv)n(es)e(some)h(prop)r (erties)g(of)g(this)h(norm.)p Black 71 952 a Fp(Lemma)j(5.2.)p Black 40 w Fs(The)g(fol)t(lowing)h(statements)d(ar)l(e)h(satis\014e)l (d.)p Black 169 1118 a(1.)p Black 42 w(If)h FA(h)22 b Fw(2)i(S)565 1130 y Fx(\033)628 1118 y Fw(\002)18 b(S)761 1130 y Fx(\033)836 1118 y Fs(and)30 b FA(B)d FB(=)1175 1051 y Fz(\000)1213 1118 y FA(b)1249 1088 y Fx(ij)1307 1051 y Fz(\001)1375 1118 y Fs(is)j(a)g FB(2)18 b Fw(\002)g FB(2)29 b Fs(matrix)h(with)g FA(b)2233 1088 y Fx(ij)2315 1118 y Fw(2)23 b(S)2443 1130 y Fx(\033)2488 1118 y Fs(,)30 b(then)g FA(B)t(h)23 b Fw(2)g(S)2994 1130 y Fx(\033)3058 1118 y Fw(\002)18 b(S)3191 1130 y Fx(\033)3265 1118 y Fs(and)1598 1300 y Fw(k)o FA(B)t(h)p Fw(k)1796 1325 y Fy(1)p Fx(;\033)1916 1300 y Fw(\024)23 b(k)p FA(B)t Fw(k)2154 1325 y Fy(1)p Fx(;\033)2266 1300 y Fw(k)o FA(h)p Fw(k)2396 1325 y Fy(1)p Fx(;\033)2508 1300 y FA(:)p Black 169 1557 a Fs(2.)p Black 42 w(If)31 b FA(B)429 1569 y Fy(1)489 1557 y FB(=)577 1465 y Fz(\020)626 1557 y FA(b)662 1517 y Fx(ij)662 1579 y Fy(1)720 1465 y Fz(\021)799 1557 y Fs(and)g FA(B)1024 1569 y Fy(2)1084 1557 y FB(=)1172 1465 y Fz(\020)1221 1557 y FA(b)1257 1517 y Fx(ij)1257 1579 y Fy(2)1315 1465 y Fz(\021)1395 1557 y Fs(ar)l(e)f FB(2)18 b Fw(\002)g FB(2)29 b Fs(matric)l(es)h(which)h(satisfy)g FA(b)2612 1517 y Fx(ij)2612 1579 y Fy(1)2670 1557 y FA(;)14 b(b)2743 1517 y Fx(ij)2743 1579 y Fy(2)2824 1557 y Fw(2)24 b(S)2953 1569 y Fx(\033)2998 1557 y Fs(,)30 b(then)1526 1764 y Fw(k)p FA(B)1631 1776 y Fy(1)1668 1764 y FA(B)1731 1776 y Fy(2)1768 1764 y Fw(k)1810 1776 y Fy(1)p Fx(;\033)1930 1764 y Fw(\024)23 b(k)p FA(B)2123 1776 y Fy(1)2160 1764 y Fw(k)2202 1776 y Fy(1)p Fx(;\033)2299 1764 y Fw(k)p FA(B)2404 1776 y Fy(2)2440 1764 y Fw(k)2482 1776 y Fy(1)p Fx(;\033)2579 1764 y FA(:)195 1980 y FB(Throughout)28 b(this)h(section,)f(w)n(e)g(will)h(need)g(to)f(solv)n(e)g(equations)f (of)i(the)g(form)f(\()p Fw(D)2810 1992 y Fy(0)2867 1980 y Fw(\000)18 b FA("A)3051 1992 y Fy(0)3089 1980 y FB(\))p FA(z)28 b FB(=)c FA(w)r FB(.)41 b(F)-7 b(or)28 b(that,)h(w)n(e)71 2080 y(will)21 b(in)n(v)n(ert)e(the)i(op)r(erator)e Fw(D)979 2092 y Fy(0)1021 2080 y Fw(\000)t FA("A)1191 2092 y Fy(0)1249 2080 y FB(acting)h(on)g Fw(S)1649 2092 y Fx(\033)1699 2080 y Fw(\002)t(S)1818 2092 y Fx(\033)1863 2080 y FB(.)34 b(Considering)20 b(the)h(F)-7 b(ourier)19 b(series)h(of)g FA(z)t FB(\()p FA(\034)9 b FB(\))24 b(=)f(\()p FA(z)3425 2092 y Fy(1)3462 2080 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(z)3647 2092 y Fy(2)3685 2080 y FB(\()p FA(\034)9 b FB(\)\),)71 2179 y(one)27 b(has)g(that)1499 2279 y Fw(D)1563 2291 y Fy(0)1600 2279 y FB(\()p FA(z)t FB(\)\()p FA(\034)9 b FB(\))24 b(=)1929 2200 y Fz(X)1928 2378 y Fx(\024)p Fv(2)p Fn(Z)2064 2279 y FA(ik)s(z)2182 2245 y Fy([)p Fx(k)q Fy(])2259 2279 y FA(e)2298 2245 y Fx(ik)q(\034)2399 2279 y FA(:)71 2507 y FB(Then,)k(one)f(can)g(in)n(v)n(ert)g Fw(D)917 2519 y Fy(0)973 2507 y Fw(\000)18 b FA("A)1157 2519 y Fy(0)1222 2507 y FB(as)961 2760 y Fw(G)1010 2772 y Fy(0)1048 2760 y FB(\()p FA(w)r FB(\)\()p FA(\034)9 b FB(\))26 b(=)c Fw(\000)1474 2681 y Fz(X)1474 2860 y Fx(k)q Fv(2)p Fn(Z)1770 2704 y FB(1)p 1618 2741 347 4 v 1618 2817 a FA(k)1664 2793 y Fy(2)1719 2817 y FB(+)c FA(\025)1850 2793 y Fy(2)1888 2817 y FA(")1927 2793 y Fy(2)1988 2618 y Fz( )2095 2709 y FA(ik)s(w)2231 2666 y Fy([)p Fx(k)q Fy(])2229 2731 y(1)2328 2709 y FB(+)g FA("w)2511 2666 y Fy([)p Fx(k)q Fy(])2509 2731 y(2)2095 2826 y FA("\025)2182 2796 y Fy(2)2219 2826 y FA(w)2280 2783 y Fy([)p Fx(k)q Fy(])2278 2848 y(1)2378 2826 y FB(+)g FA(ik)s(w)2597 2783 y Fy([)p Fx(k)q Fy(])2595 2848 y(2)2717 2618 y Fz(!)2796 2760 y FA(e)2835 2725 y Fx(ik)q(\034)2937 2760 y FA(:)701 b FB(\(134\))p Black 71 3026 a Fp(Lemma)31 b(5.3.)p Black 40 w Fs(The)g(op)l(er)l(ator)g Fw(G)1145 3038 y Fy(0)1206 3026 y FB(:)23 b Fw(S)1302 3038 y Fx(\033)1365 3026 y Fw(\002)18 b(S)1498 3038 y Fx(\033)1566 3026 y Fw(!)23 b(S)1722 3038 y Fx(\033)1786 3026 y Fw(\002)18 b(S)1919 3038 y Fx(\033)1993 3026 y Fs(in)37 b FB(\(134\))28 b Fs(is)j(wel)t(l)f(de\014ne)l(d,)h(and)f(for)h FA(w)25 b Fw(2)f(S)3392 3038 y Fx(\033)3455 3026 y Fw(\002)18 b(S)3588 3038 y Fx(\033)3633 3026 y Fs(,)1528 3251 y Fw(k)o(G)1618 3263 y Fy(0)1656 3251 y FB(\()p FA(w)r FB(\))p Fw(k)1824 3276 y Fy(1)p Fx(;\033)1944 3251 y Fw(\024)2042 3195 y FA(K)p 2042 3232 77 4 v 2061 3308 a(")2128 3251 y Fw(k)p FA(w)r Fw(k)2273 3263 y Fy(1)p Fx(;\033)2370 3251 y FA(:)71 3461 y Fs(Mor)l(e)l(over,)31 b(if)g Fw(h)p FA(w)r Fw(i)24 b FB(=)f(0)p Fs(,)1538 3561 y Fw(k)o(G)1628 3573 y Fy(0)1666 3561 y FB(\()p FA(w)r FB(\))p Fw(k)1834 3586 y Fy(1)p Fx(;\033)1954 3561 y Fw(\024)g FA(K)6 b Fw(k)p FA(w)r Fw(k)2264 3573 y Fy(1)p Fx(;\033)2360 3561 y FA(:)195 3727 y FB(W)-7 b(e)28 b(\014nally)g (state)f(a)g(tec)n(hnical)h(lemma)f(whic)n(h)h(will)f(b)r(e)h(used)g (in)g(Section)f(5.2.)36 b(Its)28 b(pro)r(of)f(is)h(straigh)n(tforw)n (ard.)p Black 71 3893 a Fp(Lemma)41 b(5.4.)p Black 46 w Fs(The)e(functions)e FA(F)1214 3905 y Fy(0)1252 3893 y Fs(,)j FA(F)1370 3905 y Fy(1)1445 3893 y Fs(and)f FA(F)1668 3905 y Fy(2)1743 3893 y Fs(de\014ne)l(d)f(in)44 b FB(\(130\))p Fs(,)c FB(\(131\))d Fs(and)47 b FB(\(132\))36 b Fs(r)l(esp)l(e)l (ctively)j(satisfy)g(the)71 3993 y(fol)t(lowing)32 b(pr)l(op)l(erties.) p Black 169 4159 a(1.)p Black 42 w FA(F)331 4171 y Fy(0)392 4159 y Fw(2)24 b(S)521 4171 y Fx(\033)584 4159 y Fw(\002)18 b(S)717 4171 y Fx(\033)762 4159 y Fs(,)30 b Fw(h)p FA(F)902 4171 y Fy(0)940 4159 y Fw(i)23 b FB(=)g(0)29 b Fs(and)1736 4258 y Fw(k)p FA(F)1831 4270 y Fy(0)1868 4258 y Fw(k)1910 4283 y Fy(1)p Fx(;\033)2030 4258 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2330 4224 y Fx(\021)2370 4258 y FA(:)p Black 169 4481 a Fs(2.)p Black 42 w FA(F)331 4493 y Fy(1)392 4481 y FB(=)480 4389 y Fz(\020)529 4481 y FA(F)594 4442 y Fx(ij)582 4504 y Fy(1)653 4389 y Fz(\021)732 4481 y Fs(satis\014es)30 b FA(F)1109 4442 y Fx(ij)1097 4504 y Fy(1)1190 4481 y Fw(2)24 b(S)1319 4493 y Fx(\033)1364 4481 y Fs(,)30 b Fw(h)p FA(F)1516 4442 y Fx(ij)1504 4504 y Fy(1)1575 4481 y Fw(i)23 b FB(=)g(0)29 b Fs(and)1736 4689 y Fw(k)p FA(F)1831 4701 y Fy(1)1868 4689 y Fw(k)1910 4714 y Fy(1)p Fx(;\033)2030 4689 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2330 4655 y Fx(\021)2370 4689 y FA(:)p Black 169 4905 a Fs(3.)p Black 42 w(If)31 b FA(z)t(;)14 b(z)489 4875 y Fv(0)533 4905 y Fw(2)24 b FA(B)t FB(\()p FA(\027)5 b FB(\))24 b Fw(\032)e(S)950 4917 y Fx(\033)1025 4905 y Fs(with)31 b FA(\027)d Fw(\034)23 b FB(1)p Fs(,)30 b(then)1361 5087 y Fw(k)p FA(F)1456 5099 y Fy(2)1493 5087 y FB(\()p FA(z)1568 5053 y Fv(0)1591 5087 y FA(;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(F)1861 5099 y Fy(2)1898 5087 y FB(\()p FA(z)t(;)c(\034)9 b FB(\))p Fw(k)2129 5112 y Fx(\033)2197 5087 y Fw(\024)23 b FA(K)6 b(\027)f Fw(k)p FA(z)2493 5053 y Fv(0)2533 5087 y Fw(\000)18 b FA(z)t Fw(k)2701 5099 y Fx(\033)2744 5087 y FA(:)p Black 1919 5753 a FB(51)p Black eop end %%Page: 52 52 TeXDict begin 52 51 bop Black Black 71 272 a Fq(5.2)112 b(Pro)s(of)37 b(of)h(Theorem)g(4.1)71 425 y FB(W)-7 b(e)28 b(rewrite)f(Theorem)f(4.1)h(in)h(terms)f(of)h(the)g(Banac)n(h)e(space)h (\(133\))o(.)p Black 71 589 a Fp(Prop)s(osition)46 b(5.5.)p Black 48 w Fs(L)l(et)c FA(")996 601 y Fy(0)1079 589 y FA(>)j FB(0)d Fs(smal)t(l)h(enough.)77 b(Then,)47 b(for)c FA(")j Fw(2)g FB(\(0)p FA(;)14 b(")2599 601 y Fy(0)2636 589 y FB(\))p Fs(,)46 b(e)l(quation)j FB(\(128\))41 b Fs(has)i(a)g(solution)71 689 y FB(\()p FA(x)150 701 y Fx(p)189 689 y FA(;)14 b(y)267 701 y Fx(p)305 689 y FB(\))23 b Fw(2)h(S)489 701 y Fx(\033)534 689 y Fs(.)38 b(Mor)l(e)l(over,)32 b(ther)l(e)e(exists)f(a)h(c)l(onstant)f FA(b)1861 701 y Fy(0)1921 689 y FA(>)23 b FB(0)29 b Fs(such)h(that)1504 869 y Fw(k)p FB(\()p FA(x)1625 881 y Fx(p)1664 869 y FA(;)14 b(y)1742 881 y Fx(p)1780 869 y FB(\))p Fw(k)1854 897 y Fy(1)p Fx(;\033)1974 869 y Fw(\024)23 b FA(b)2098 881 y Fy(0)2135 869 y Fw(j)p FA(\026)p Fw(j)p FA(")2270 835 y Fx(\021)r Fy(+1)2394 869 y FA(:)p Black 71 1059 a Fp(Corollary)34 b(5.6.)p Black 41 w Fs(The)e(change)g(of)f(variables) 40 b FB(\(32\))30 b Fs(tr)l(ansforms)h(the)g(Hamiltonian)h(system)f (with)g(Hamiltonian)38 b FB(\(6\))71 1159 y Fs(to)30 b(a)g(new)f(Hamiltonian)i(system)f(with)g(Hamiltonian)37 b FB(\(33\))p Fs(.)195 1259 y(Mor)l(e)l(over,)32 b(the)e(functions)g FA(c)1121 1271 y Fx(ij)1209 1259 y Fs(in)g(the)f(de\014nition)i(of)48 b FB(\(33\))29 b Fs(\(se)l(e)h(also)36 b FB(\(37\))o Fs(\))30 b(satisfy)1657 1439 y Fw(k)p FA(c)1735 1451 y Fx(ij)1793 1439 y Fw(k)1835 1451 y Fx(\033)1902 1439 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2202 1404 y Fx(\021)2241 1439 y FA(:)195 1619 y FB(W)-7 b(e)29 b(dev)n(ote)e(the)i(rest)e(of)i(the)f(section)g(to)g(pro)n(v)n(e)e (Prop)r(osition)h(5.5.)37 b(W)-7 b(e)29 b(obtain)f(the)g(solution)g(of) g(equation)f(\(128\))71 1718 y(through)h(a)h(\014xed)g(p)r(oin)n(t)h (argumen)n(t.)40 b(T)-7 b(o)29 b(obtain)g(a)f(con)n(tractiv)n(e)g(op)r (erator,)g(\014rst)g(w)n(e)h(ha)n(v)n(e)f(to)h(p)r(erform)g(a)g(c)n (hange)f(of)71 1818 y(v)-5 b(ariables,)26 b(whic)n(h)i(actually)f(it)h (is)f(only)g(needed)h(in)g(the)g(case)f FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b FB(=)e(0.)195 1918 y(Let)32 b(us)f(consider)f(a)h (function)p 1186 1851 65 4 v 31 w FA(F)1251 1930 y Fy(1)1320 1918 y FB(whic)n(h)g(satis\014es)f Fw(h)p 1902 1851 V FA(F)1967 1930 y Fy(1)2005 1918 y Fw(i)f FB(=)g(0)h(and)h FA(@)2441 1930 y Fx(\034)p 2483 1851 V 2483 1918 a FA(F)2548 1930 y Fy(1)2614 1918 y FB(=)e FA(F)2761 1930 y Fy(1)2798 1918 y FB(,)j(where)f FA(F)3150 1930 y Fy(1)3219 1918 y FB(is)g(the)g(function)h(in)71 2017 y(\(131\))o(.)37 b(The)28 b(function)p 816 1951 V 28 w FA(F)881 2029 y Fy(1)946 2017 y FB(can)f(b)r(e)h(de\014ned)g(as)p 1475 2170 V 1475 2236 a FA(F)1540 2248 y Fy(1)1578 2236 y FB(\()p FA(\034)9 b FB(\))24 b(=)1866 2158 y Fz(X)1799 2339 y Fx(k)q Fv(2)p Fn(Z)p Fv(nf)p Fy(0)p Fv(g)2094 2180 y FB(1)p 2078 2217 75 4 v 2078 2293 a FA(ik)2162 2236 y(F)2227 2193 y Fy([)p Fx(k)q Fy(])2215 2259 y(1)2305 2236 y FA(e)2344 2202 y Fx(ik)q(\034)71 2509 y FB(and)j(satis\014es) 1605 2538 y Fz(\015)1605 2588 y(\015)p 1651 2542 65 4 v 20 x FA(F)1716 2620 y Fy(1)1754 2538 y Fz(\015)1754 2588 y(\015)1800 2642 y Fy(1)p Fx(;\033)1920 2608 y Fw(\024)c(k)o FA(F)2102 2620 y Fy(1)2140 2608 y Fw(k)2182 2633 y Fy(1)p Fx(;\033)2293 2608 y FA(:)1345 b FB(\(135\))71 2767 y(W)-7 b(e)28 b(p)r(erform)f(the)h(c)n(hange)e(of)i(v)-5 b(ariables)1604 2867 y FA(z)26 b FB(=)1757 2799 y Fz(\000)1795 2867 y FB(Id)18 b(+)g FA(")p 2011 2800 V(F)2076 2879 y Fy(1)2113 2867 y FB(\()p FA(\034)9 b FB(\))2222 2799 y Fz(\001)p 2275 2821 43 4 v 2275 2867 a FA(z)1347 b FB(\(136\))71 3015 y(and)27 b(then)h(equation)f(\(128\))g(b)r(ecomes)1536 3114 y(\()p Fw(D)1632 3126 y Fy(0)1688 3114 y Fw(\000)18 b FA("A)1872 3126 y Fy(0)1909 3114 y FB(\))p 1955 3069 V 14 w FA(z)26 b FB(=)p 2108 3048 65 4 v 23 w FA(F)12 b FB(\()p 2205 3069 43 4 v FA(z)t(;)i(\034)9 b FB(\))p FA(;)1276 b FB(\(137\))71 3262 y(where)p 704 3375 65 4 v 704 3441 a FA(F)12 b FB(\()p 801 3396 43 4 v FA(z)s(;)i(\034)9 b FB(\))24 b(=)p FA(")1099 3374 y Fz(\000)1137 3441 y FB(Id)18 b(+)g FA(")p 1353 3375 65 4 v(F)1418 3453 y Fy(1)1455 3441 y FB(\()p FA(\034)9 b FB(\))1564 3374 y Fz(\001)1603 3388 y Fv(\000)p Fy(1)1706 3441 y FA(F)1759 3453 y Fy(0)1797 3441 y FB(\()p FA(\034)g FB(\))1064 3595 y(+)18 b FA(")1186 3560 y Fy(2)1237 3527 y Fz(\000)1275 3595 y FB(Id)h(+)f FA(")p 1492 3528 V(F)1556 3607 y Fy(1)1594 3595 y FB(\()p FA(\034)9 b FB(\))1703 3527 y Fz(\001)1742 3542 y Fv(\000)p Fy(1)1845 3527 y Fz(\000)1883 3595 y FA(A)1945 3607 y Fy(0)p 1983 3528 V 1983 3595 a FA(F)2047 3607 y Fy(1)2085 3595 y FB(\()p FA(\034)g FB(\))19 b Fw(\000)p 2296 3528 V 18 w FA(F)2361 3607 y Fy(1)2399 3595 y FB(\()p FA(\034)9 b FB(\))p FA(A)2570 3607 y Fy(0)2627 3595 y FB(+)p 2710 3528 V 18 w FA(F)2775 3607 y Fy(1)2812 3595 y FB(\()p FA(\034)g FB(\))p FA(F)2974 3607 y Fy(1)3013 3595 y FB(\()p FA(\034)g FB(\))3122 3527 y Fz(\001)p 3175 3549 43 4 v 3175 3595 a FA(z)1064 3748 y FB(+)18 b FA(")1200 3681 y Fz(\000)1238 3748 y FB(Id)h(+)f FA(")p 1455 3681 65 4 v(F)1519 3760 y Fy(1)1556 3748 y FB(\()p FA(\034)9 b FB(\))1665 3681 y Fz(\001)1705 3695 y Fv(\000)p Fy(1)1808 3748 y FA(F)1861 3760 y Fy(2)1912 3681 y Fz(\000\000)1988 3748 y FB(Id)19 b(+)f FA(")p 2205 3681 V(F)2270 3760 y Fy(1)2307 3748 y FB(\()p FA(\034)9 b FB(\))2416 3681 y Fz(\001)p 2469 3702 43 4 v 2469 3748 a FA(z)s(;)14 b(\034)2593 3681 y Fz(\001)2646 3748 y FA(:)3661 3586 y FB(\(138\))71 3929 y(Since)32 b(the)h(op)r(erator)d Fw(G)828 3941 y Fy(0)898 3929 y FB(de\014ned)i(in)h(\(134\))e(is)h(a)g (left)h(in)n(v)n(erse)d(of)i Fw(D)2265 3941 y Fy(0)2324 3929 y Fw(\000)21 b FA("A)2511 3941 y Fy(0)2548 3929 y FB(,)34 b(w)n(e)e(lo)r(ok)f(for)h(a)f(solution)h(of)g(equation)71 4028 y(\(137\))27 b(as)g(a)g(\014xed)g(p)r(oin)n(t)h(of)g(the)g(op)r (erator)1730 4128 y Fw(F)1790 4140 y Fy(0)1850 4128 y FB(=)23 b Fw(G)1987 4140 y Fy(0)2043 4128 y Fw(\016)p 2103 4061 65 4 v 18 w FA(F)12 b(:)1470 b FB(\(139\))195 4276 y(Then)28 b(Prop)r(osition)e(5.5)h(follo)n(ws)f(from)i(the)g (follo)n(wing)e(lemma.)p Black 71 4440 a Fp(Lemma)35 b(5.7.)p Black 42 w Fs(L)l(et)d FA(")790 4452 y Fy(0)855 4440 y FA(>)27 b FB(0)32 b Fs(smal)t(l)i(enough.)47 b(Then,)34 b(ther)l(e)f(exists)f(a)h(c)l(onstant)f FA(b)2699 4452 y Fy(0)2764 4440 y FA(>)27 b FB(0)32 b Fs(such)h(that,)g(for)h FA(")28 b Fw(2)g FB(\(0)p FA(;)14 b(")3756 4452 y Fy(0)3793 4440 y FB(\))p Fs(,)71 4540 y(the)30 b(op)l(er)l(ator)h Fw(F)596 4552 y Fy(0)662 4540 y Fs(in)36 b FB(\(139\))29 b Fs(is)h(c)l(ontr)l(active)g(fr)l(om)p 1696 4473 68 4 v 31 w FA(B)1777 4472 y Fz(\000)1815 4540 y FA(b)1851 4552 y Fy(0)1888 4540 y Fw(j)p FA(\026)p Fw(j)p FA(")2023 4510 y Fx(\021)r Fy(+1)2147 4472 y Fz(\001)2208 4540 y Fw(\032)23 b(S)2346 4552 y Fx(\033)2409 4540 y Fw(\002)18 b(S)2542 4552 y Fx(\033)2617 4540 y Fs(to)30 b(itself.)195 4650 y(Then,)h Fw(F)497 4662 y Fy(0)564 4650 y Fs(has)f(a)h(unique)e (\014xe)l(d)g(p)l(oint)p 1453 4605 43 4 v 30 w FA(z)1495 4619 y Fv(\003)1557 4650 y Fw(2)p 1635 4584 68 4 v 23 w FA(B)1716 4583 y Fz(\000)1754 4650 y FA(b)1790 4662 y Fy(0)1827 4650 y Fw(j)p FA(\026)p Fw(j)p FA(")1962 4620 y Fx(\021)r Fy(+1)2086 4583 y Fz(\001)2147 4650 y Fw(\032)23 b(S)2285 4662 y Fx(\033)2349 4650 y Fw(\002)18 b(S)2482 4662 y Fx(\033)2527 4650 y Fs(.)p Black 71 4814 a(Pr)l(o)l(of.)p Black 43 w FB(It)25 b(is)g(easily)g(c)n(hec)n(k)n(ed)e (that)j Fw(F)1265 4826 y Fy(0)1327 4814 y FB(sends)f Fw(S)1597 4826 y Fx(\033)1655 4814 y Fw(\002)13 b(S)1783 4826 y Fx(\033)1853 4814 y FB(in)n(to)25 b(itself.)36 b(T)-7 b(o)25 b(see)g(that)g(it)h(is)f(con)n(tractiv)n(e)e(w)n(e)i (\014rst)g(consider)71 4914 y Fw(F)131 4926 y Fy(0)168 4914 y FB(\(0\),)j(whic)n(h)f(can)g(b)r(e)h(split)g(as)1104 5119 y Fw(F)1164 5131 y Fy(0)1201 5119 y FB(\(0\))c(=)e FA(")p Fw(G)1506 5131 y Fy(0)1557 5119 y FB(\()q FA(F)1643 5131 y Fy(0)1680 5119 y FB(\))d Fw(\000)f FA(")1853 5085 y Fy(2)1890 5119 y Fw(G)1939 5131 y Fy(0)1990 5027 y Fz(\020)2040 5052 y(\000)2078 5119 y FB(Id)h(+)f FA(")p 2295 5052 65 4 v(F)2359 5131 y Fy(1)2397 5052 y Fz(\001)2435 5066 y Fv(\000)p Fy(1)p 2538 5052 V 2538 5119 a FA(F)2602 5131 y Fy(1)2640 5119 y FA(F)2693 5131 y Fy(0)2730 5027 y Fz(\021)2794 5119 y FA(:)71 5324 y FB(By)27 b(Lemma)h(5.4,)e Fw(h)p FA(F)739 5336 y Fy(0)777 5324 y Fw(i)e FB(=)e(0)27 b(and)h Fw(k)p FA(F)1246 5336 y Fy(0)1283 5324 y Fw(k)1325 5336 y Fy(1)p Fx(;\033)1445 5324 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1745 5294 y Fx(\021)1784 5324 y FB(.)37 b(Then,)28 b(applying)f(Lemma)h(5.3,)e(one)i(has)f(that) 1550 5504 y Fw(k)o(G)1640 5516 y Fy(0)1692 5504 y FB(\()p FA(F)1777 5516 y Fy(0)1815 5504 y FB(\))p Fw(k)1888 5529 y Fy(1)p Fx(;\033)2009 5504 y Fw(\024)c FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2309 5470 y Fx(\021)2348 5504 y FA(:)p Black 1919 5753 a FB(52)p Black eop end %%Page: 53 53 TeXDict begin 53 52 bop Black Black 71 272 a FB(F)-7 b(or)30 b(the)g(second)g(term,)h(considering)e(also)h(\(135\))f(and)i (Lemmas)f(5.2,)g(5.3)f(and)i(5.4,)f(one)g(can)g(pro)r(ceed)g (analogously)71 372 y(to)d(obtain)1169 376 y Fz(\015)1169 426 y(\015)1169 475 y(\015)1215 471 y Fw(G)1264 483 y Fy(0)1316 379 y Fz(\020)1365 404 y(\000)1403 471 y FB(Id)19 b(+)f FA(")p 1620 405 65 4 v(F)1684 483 y Fy(1)1722 404 y Fz(\001)1760 419 y Fv(\000)p Fy(1)p 1863 405 V 1863 471 a FA(F)1928 483 y Fy(1)1965 471 y FA(F)2018 483 y Fy(0)2055 379 y Fz(\021)2105 376 y(\015)2105 426 y(\015)2105 475 y(\015)2151 529 y Fy(1)p Fx(;\033)2272 471 y Fw(\024)k FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2571 437 y Fy(2)p Fx(\021)r Fv(\000)p Fy(1)2729 471 y FA(:)71 657 y FB(Therefore,)26 b(there)i(exists)f(a)g(constan)n(t)g FA(b)1352 669 y Fy(0)1412 657 y FA(>)c FB(0)k(suc)n(h)g(that)1526 878 y Fw(k)o(F)1627 890 y Fy(0)1664 878 y FB(\(0\))p Fw(k)1812 903 y Fy(1)p Fx(;\033)1933 878 y Fw(\024)2030 822 y FA(b)2066 834 y Fy(0)p 2030 859 73 4 v 2046 935 a FB(2)2113 878 y Fw(j)p FA(\026)p Fw(j)p FA(")2248 844 y Fx(\021)r Fy(+1)2372 878 y FA(:)71 1099 y FB(Let)h(us)g(consider)g(no)n(w)f FA(z)868 1069 y Fy(1)905 1099 y FA(;)14 b(z)985 1069 y Fy(2)1045 1099 y Fw(2)p 1125 1033 68 4 v 25 w FA(B)1206 1032 y Fz(\000)1244 1099 y FA(b)1280 1111 y Fy(0)1317 1099 y Fw(j)p FA(\026)p Fw(j)p FA(")1452 1069 y Fx(\021)r Fy(+1)1576 1032 y Fz(\001)1638 1099 y Fw(\032)24 b(S)1777 1111 y Fx(\033)1841 1099 y Fw(\002)18 b(S)1974 1111 y Fx(\033)2019 1099 y FB(.)39 b(Then,)29 b(b)n(y)f(Lemmas)f(5.3,)h(5.2)f (and)h(5.4,)g(and)g(reducing)71 1199 y FA(")f FB(if)h(necessary)-7 b(,)26 b(one)i(can)f(see)g(that,)1093 1308 y Fz(\015)1093 1357 y(\015)1139 1378 y Fw(F)1199 1390 y Fy(0)1250 1311 y Fz(\000)1288 1378 y FA(z)1331 1344 y Fy(2)1368 1311 y Fz(\001)1424 1378 y Fw(\000)18 b(F)1567 1390 y Fy(0)1618 1311 y Fz(\000)1656 1378 y FA(z)1699 1344 y Fy(1)1736 1311 y Fz(\001)1774 1308 y(\015)1774 1357 y(\015)1820 1411 y Fy(1)p Fx(;\033)1940 1378 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2240 1344 y Fx(\021)r Fy(+1)2378 1308 y Fz(\015)2378 1357 y(\015)2424 1378 y FA(z)2467 1344 y Fy(2)2522 1378 y Fw(\000)18 b FA(z)2648 1344 y Fy(1)2684 1308 y Fz(\015)2684 1357 y(\015)2730 1411 y Fy(1)p Fx(;\033)1940 1566 y Fw(\024)2038 1509 y FB(1)p 2038 1547 42 4 v 2038 1623 a(2)2103 1495 y Fz(\015)2103 1545 y(\015)2149 1566 y FA(z)2192 1531 y Fy(2)2247 1566 y Fw(\000)g FA(z)2373 1531 y Fy(1)2410 1495 y Fz(\015)2410 1545 y(\015)2456 1599 y Fy(1)p Fx(;\033)2567 1566 y FA(:)71 1787 y FB(Then,)27 b Fw(F)370 1799 y Fy(0)430 1787 y FB(:)p 476 1721 68 4 v 23 w FA(B)557 1720 y Fz(\000)595 1787 y FA(b)631 1799 y Fy(0)668 1787 y Fw(j)p FA(\026)p Fw(j)p FA(")803 1757 y Fx(\021)r Fy(+1)927 1720 y Fz(\001)989 1787 y Fw(!)p 1095 1721 V 23 w FA(B)1176 1720 y Fz(\000)1214 1787 y FA(b)1250 1799 y Fy(0)1287 1787 y Fw(j)p FA(\026)p Fw(j)p FA(")1422 1757 y Fx(\021)r Fy(+1)1546 1720 y Fz(\001)1607 1787 y Fw(\032)c(S)1745 1799 y Fx(\033)1806 1787 y Fw(\002)17 b(S)1938 1799 y Fx(\033)2010 1787 y FB(and)26 b(is)h(con)n(tractiv)n (e.)35 b(Therefore,)26 b(it)h(has)f(a)g(unique)h(\014xed)71 1887 y(p)r(oin)n(t)p 288 1841 43 4 v 28 w FA(z)330 1855 y Fv(\003)368 1887 y FB(.)p 3790 1887 4 57 v 3794 1834 50 4 v 3794 1887 V 3843 1887 4 57 v Black 71 2053 a Fs(Pr)l(o)l(of)k (of)f(Pr)l(op)l(osition)i(5.5.)p Black 43 w FB(It)c(is)f(enough)g(to)h (tak)n(e)1444 2236 y FA(z)1487 2201 y Fv(\003)1524 2236 y FB(\()p FA(\034)9 b FB(\))24 b(=)1745 2168 y Fz(\000)1783 2236 y FB(Id)19 b(+)f FA(")p 2000 2169 65 4 v(F)2064 2248 y Fy(1)2102 2236 y FB(\()p FA(\034)9 b FB(\))2211 2168 y Fz(\001)p 2264 2190 43 4 v 2264 2236 a FA(z)2306 2201 y Fv(\003)2344 2236 y FB(\()p FA(\034)g FB(\))p FA(;)71 2418 y FB(whic)n(h)27 b(satis\014es)g(equation)g(\(128\))g(and) g(satis\014es)g(the)h(desired)f(b)r(ound)h(\(increasing)f FA(b)2775 2430 y Fy(0)2839 2418 y FB(sligh)n(tly)g(if)h(necessary\).)p 3790 2418 4 57 v 3794 2365 50 4 v 3794 2418 V 3843 2418 4 57 v 71 2693 a FC(6)135 b(Lo)t(cal)45 b(in)l(v)-7 b(arian)l(t)46 b(manifolds:)61 b(pro)t(of)45 b(of)g(Theorem)g(4.3)71 2875 y FB(Since)29 b(the)f(pro)r(of)g(for)g(b)r(oth)h(in)n(v)-5 b(arian)n(t)28 b(manifolds)g(is)g(analogous,)f(w)n(e)h(only)g(deal)g (with)h(the)g(unstable)g(case.)38 b(W)-7 b(e)29 b(lo)r(ok)71 2974 y(for)d(a)g(solution)g(of)h(equation)f(\(41\))h(satisfying)f(the)h (asymptotic)f(condition)g(\(49\).)36 b(W)-7 b(e)28 b(lo)r(ok)d(for)i (it)g(as)f(a)g(p)r(erturbation)71 3074 y(of)h(the)h(unp)r(erturb)r(ed)h (separatrix)1572 3216 y FA(T)1621 3228 y Fy(0)1657 3216 y FB(\()p FA(u)p FB(\))24 b(=)1880 3103 y Fz(Z)1963 3123 y Fx(u)1926 3291 y Fv(\0001)2062 3216 y FA(p)2104 3182 y Fy(2)2104 3236 y(0)2141 3216 y FB(\()p FA(v)s FB(\))14 b FA(dv)1316 b FB(\(140\))71 3419 y(and)27 b(therefore)g(w)n(e)g(w)n (ork)f(with)i FA(T)1144 3431 y Fy(1)1181 3419 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)f FA(T)12 b FB(\()p FA(u;)i(\034)9 b FB(\))18 b Fw(\000)g FA(T)1892 3431 y Fy(0)1929 3419 y FB(\()p FA(u)p FB(\).)195 3518 y(Replacing)24 b FA(T)36 b FB(in)24 b(equation)g(\(41\))g(and)g(taking)g(in)n(to)g (accoun)n(t)f(that)i FA(V)19 b FB(\()p FA(q)2454 3530 y Fy(0)2492 3518 y FB(\()p FA(u)p FB(\)\))k(=)g Fw(\000)p FA(p)2854 3488 y Fy(2)2854 3539 y(0)2890 3518 y FB(\()p FA(u)p FB(\))p FA(=)p FB(2,)h(it)h(is)f(straigh)n(tforw)n(ard)71 3618 y(to)j(see)h(that)f(the)h(equation)f(for)g FA(T)1144 3630 y Fy(1)1209 3618 y FB(reads)1554 3801 y Fw(L)1611 3813 y Fx(")1647 3801 y FA(T)1696 3813 y Fy(1)1756 3801 y FB(=)c Fw(F)e FB(\()q FA(@)2002 3813 y Fx(u)2045 3801 y FA(T)2094 3813 y Fy(1)2131 3801 y FA(;)14 b(u;)g(\034)9 b FB(\))14 b FA(;)1294 b FB(\(141\))71 3983 y(where)27 b Fw(L)368 3995 y Fx(")431 3983 y FB(is)h(the)g(op)r(erator)e (de\014ned)i(in)g(\(45\))f(and)373 4219 y Fw(F)8 b FB(\()p FA(w)r(;)14 b(u;)g(\034)9 b FB(\))24 b(=)83 b Fw(\000)1046 4163 y FA(w)1107 4133 y Fy(2)p 979 4200 233 4 v 979 4276 a FB(2)p FA(p)1063 4247 y Fy(2)1063 4298 y(0)1100 4276 y FB(\()p FA(u)p FB(\))1240 4219 y Fw(\000)1323 4102 y Fz(\022)1385 4219 y FA(V)18 b FB(\()p FA(q)1520 4231 y Fy(0)1558 4219 y FB(\()p FA(u)p FB(\))h(+)f FA(x)1819 4231 y Fx(p)1857 4219 y FB(\()p FA(\034)9 b FB(\)\))20 b Fw(\000)e FA(V)h FB(\()p FA(x)2247 4231 y Fx(p)2286 4219 y FB(\()p FA(\034)9 b FB(\)\))20 b Fw(\000)f FA(V)f FB(\()p FA(q)2666 4231 y Fy(0)2704 4219 y FB(\()p FA(u)p FB(\)\))h Fw(\000)f FA(V)3017 4185 y Fv(0)3040 4219 y FB(\()p FA(x)3119 4231 y Fx(p)3158 4219 y FB(\()p FA(\034)9 b FB(\)\))p FA(q)3336 4231 y Fy(0)3375 4219 y FB(\()p FA(u)p FB(\))3487 4102 y Fz(\023)905 4419 y Fw(\000)p FA(\026")1059 4384 y Fx(\021)1118 4398 y Fz(b)1099 4419 y FA(H)1168 4431 y Fy(1)1219 4302 y Fz(\022)1280 4419 y FA(q)1317 4431 y Fy(0)1354 4419 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1545 4431 y Fy(0)1582 4419 y FB(\()p FA(u)p FB(\))19 b(+)1870 4362 y FA(w)p 1806 4399 192 4 v 1806 4476 a(p)1848 4488 y Fy(0)1885 4476 y FB(\()p FA(u)p FB(\))2007 4419 y FA(;)14 b(\034)2089 4302 y Fz(\023)2164 4419 y FA(;)71 4670 y FB(where)330 4649 y Fz(b)311 4670 y FA(H)380 4682 y Fy(1)445 4670 y FB(is)27 b(the)h(function)h (de\014ned)e(in)h(\(34\).)195 4769 y(W)-7 b(e)28 b(split)g Fw(F)36 b FB(in)n(to)27 b(constan)n(t,)g(linear)g(and)g(higher)g(order) g(terms)g(in)h FA(w)i FB(as)888 4952 y Fw(F)8 b FB(\()p FA(w)r(;)14 b(u;)g(\034)9 b FB(\))25 b(=)d FA(A)p FB(\()p FA(u;)14 b(\034)9 b FB(\))20 b(+)e(\()p FA(B)1814 4964 y Fy(1)1851 4952 y FB(\()p FA(u;)c(\034)9 b FB(\))19 b(+)f FA(B)2210 4964 y Fy(2)2247 4952 y FB(\()p FA(u;)c(\034)9 b FB(\)\))15 b FA(w)21 b FB(+)d FA(C)6 b FB(\()p FA(w)r(;)14 b(u;)g(\034)9 b FB(\))p FA(;)630 b FB(\(142\))p Black 1919 5753 a(53)p Black eop end %%Page: 54 54 TeXDict begin 54 53 bop Black Black 71 272 a FB(with)705 449 y FA(A)p FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)18 b Fw(\000)g FB(\()p FA(V)h FB(\()p FA(q)1319 461 y Fy(0)1357 449 y FB(\()p FA(u)p FB(\))f(+)g FA(x)1617 461 y Fx(p)1656 449 y FB(\()p FA(\034)9 b FB(\)\))20 b Fw(\000)e FA(V)h FB(\()p FA(x)2046 461 y Fx(p)2085 449 y FB(\()p FA(\034)9 b FB(\)\))20 b Fw(\000)e FA(V)h FB(\()p FA(q)2465 461 y Fy(0)2503 449 y FB(\()p FA(u)p FB(\)\))f Fw(\000)g FA(V)2815 415 y Fv(0)2839 449 y FB(\()p FA(x)2918 461 y Fx(p)2957 449 y FB(\()p FA(\034)9 b FB(\)\))p FA(q)3135 461 y Fy(0)3174 449 y FB(\()p FA(u)p FB(\)\))1068 594 y Fw(\000)18 b FA(\026")1240 559 y Fx(\021)1300 573 y Fz(b)1280 594 y FA(H)1349 606 y Fy(1)1400 594 y FB(\()p FA(q)1469 606 y Fy(0)1507 594 y FB(\()p FA(u)p FB(\))p FA(;)c(p)1698 606 y Fy(0)1735 594 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))15 b FA(;)1662 b FB(\(143\))667 738 y FA(B)730 750 y Fy(1)767 738 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)18 b Fw(\000)g FA(\026")1240 703 y Fx(\021)1280 738 y FA(p)1322 702 y Fv(\000)p Fy(1)1322 760 y(0)1411 738 y FB(\()p FA(u)p FB(\))p FA(@)1567 750 y Fx(p)1625 717 y Fz(b)1606 738 y FA(H)1682 703 y Fy(1)1675 758 y(1)1719 738 y FB(\()p FA(q)1788 750 y Fy(0)1825 738 y FB(\()p FA(u)p FB(\))p FA(;)c(p)2016 750 y Fy(0)2053 738 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))p FA(;)1359 b FB(\(144\))667 882 y FA(B)730 894 y Fy(2)767 882 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)18 b Fw(\000)g FA(\026")1240 847 y Fx(\021)r Fy(+1)1364 882 y FA(p)1406 846 y Fv(\000)p Fy(1)1406 904 y(0)1495 882 y FB(\()p FA(u)p FB(\))p FA(@)1651 894 y Fx(p)1709 861 y Fz(b)1690 882 y FA(H)1766 847 y Fy(2)1759 902 y(1)1803 882 y FB(\()p FA(q)1872 894 y Fy(0)1909 882 y FB(\()p FA(u)p FB(\))p FA(;)c(p)2100 894 y Fy(0)2137 882 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))p FA(;)1275 b FB(\(145\))603 1068 y FA(C)6 b FB(\()p FA(w)r(;)14 b(u;)g(\034)9 b FB(\))25 b(=)18 b Fw(\000)1228 1012 y FA(w)1289 982 y Fy(2)p 1161 1049 233 4 v 1161 1125 a FB(2)p FA(p)1245 1097 y Fy(2)1245 1148 y(0)1281 1125 y FB(\()p FA(u)p FB(\))1422 1068 y Fw(\000)g FA(\026")1594 1034 y Fx(\021)1654 1047 y Fz(b)1634 1068 y FA(H)1703 1080 y Fy(1)1754 951 y Fz(\022)1815 1068 y FA(q)1852 1080 y Fy(0)1890 1068 y FB(\()p FA(u)p FB(\))p FA(;)c(p)2081 1080 y Fy(0)2118 1068 y FB(\()p FA(u)p FB(\))k(+)2406 1012 y FA(w)p 2341 1049 192 4 v 2341 1125 a(p)2383 1137 y Fy(0)2420 1125 y FB(\()p FA(u)p FB(\))2542 1068 y FA(;)c(\034)2624 951 y Fz(\023)1068 1273 y FB(+)k FA(\026")1240 1238 y Fx(\021)1355 1217 y FA(w)p 1290 1254 V 1290 1330 a(p)1332 1342 y Fy(0)1369 1330 y FB(\()p FA(u)p FB(\))1491 1273 y FA(@)1535 1285 y Fx(p)1593 1252 y Fz(b)1574 1273 y FA(H)1643 1285 y Fy(1)1680 1273 y FB(\()p FA(q)1749 1285 y Fy(0)1786 1273 y FB(\()p FA(u)p FB(\))p FA(;)c(p)1977 1285 y Fy(0)2014 1273 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))20 b(+)e FA(\026")2432 1238 y Fx(\021)2492 1252 y Fz(b)2472 1273 y FA(H)2541 1285 y Fy(1)2592 1273 y FB(\()p FA(q)2661 1285 y Fy(0)2699 1273 y FB(\()p FA(u)p FB(\))p FA(;)c(p)2890 1285 y Fy(0)2927 1273 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))15 b FA(;)470 b FB(\(146\))71 1517 y(where)330 1496 y Fz(b)311 1517 y FA(H)387 1487 y Fy(1)380 1538 y(1)452 1517 y FB(and)632 1496 y Fz(b)613 1517 y FA(H)689 1487 y Fy(2)682 1538 y(1)754 1517 y FB(are)26 b(the)i(functions)g (de\014ned)g(in)g(\(35\))f(and)h(\(37\))f(resp)r(ectiv)n(ely)-7 b(.)71 1749 y Fq(6.1)112 b(Lo)s(cal)38 b(in)m(v)-6 b(arian)m(t)39 b(manifolds)g(in)f(the)f(h)m(yp)s(erb)s(olic)i(case)71 1902 y FB(In)d(this)g(section)f(w)n(e)g(pro)n(v)n(e)g(the)h(existence)f (of)h(suitable)f(represen)n(tations)f(of)i(the)g(unstable)g(and)f (stable)h(in)n(v)-5 b(arian)n(t)71 2002 y(manifolds)18 b(in)h(the)g(domains)f FA(D)1050 1972 y Fx(u)1048 2022 y Fv(1)p Fx(;\032)1173 2002 y Fw(\002)q Ft(T)1294 2014 y Fx(\033)1358 2002 y FB(and)g FA(D)1581 1972 y Fx(s)1579 2022 y Fv(1)p Fx(;\032)1704 2002 y Fw(\002)q Ft(T)1825 2014 y Fx(\033)1888 2002 y FB(resp)r(ectiv)n(ely)g(under)g(the)i(h)n (yp)r(othesis)e(that)h(the)g(unp)r(erturb)r(ed)71 2101 y(Hamiltonian)27 b(system)h(has)f(a)g(h)n(yp)r(erb)r(olic)g(critical)g (p)r(oin)n(t)h(at)f(the)h(origin.)71 2316 y Fp(6.1.1)94 b(Banac)m(h)33 b(spaces)f(and)g(tec)m(hnical)h(lemmas)71 2470 y FB(This)25 b(subsection)g(is)g(dev)n(oted)g(to)g(de\014ne)h(the) f(Banac)n(h)f(spaces)h(whic)n(h)g(will)g(b)r(e)h(used)f(in)h(Section)f (6.1.2.)35 b(W)-7 b(e)25 b(also)g(state)71 2569 y(some)i(of)g(their)h (useful)g(prop)r(erties.)195 2669 y(W)-7 b(e)31 b(de\014ne)f(some)f (norms)g(for)g(functions)i(de\014ned)f(in)g(a)f(domain)h FA(D)2362 2639 y Fx(u)2360 2689 y Fv(1)p Fx(;\032)2514 2669 y FB(with)g FA(\032)d Fw(\025)f FB(0.)44 b(Giv)n(en)29 b FA(\013)f Fw(\025)e FB(0,)k FA(\032)d Fw(\025)f FB(0)k(and)71 2768 y(an)d(analytic)g(function)h FA(h)23 b FB(:)g FA(D)1015 2738 y Fx(u)1013 2789 y Fv(1)p Fx(;\032)1161 2768 y Fw(!)g Ft(C)p FB(,)k(w)n(e)h(consider)1444 2957 y Fw(k)p FA(h)p Fw(k)1576 2969 y Fx(\013;\032)1699 2957 y FB(=)84 b(sup)1787 3027 y Fx(u)p Fv(2)p Fx(D)1927 3010 y Fm(u)1925 3043 y Fl(1)p Fm(;\032)2048 2886 y Fz(\014)2048 2936 y(\014)2076 2957 y FA(e)2115 2923 y Fv(\000)p Fx(\013u)2253 2957 y FA(h)p FB(\()p FA(u)p FB(\))2413 2886 y Fz(\014)2413 2936 y(\014)2455 2957 y FA(:)71 3203 y FB(Moreo)n(v)n(er)29 b(for)h(analytic)h(functions)h FA(h)d FB(:)h FA(D)1453 3173 y Fx(u)1451 3224 y Fv(1)p Fx(;\032)1596 3203 y Fw(\002)20 b Ft(T)1736 3215 y Fx(\033)1811 3203 y Fw(!)29 b Ft(C)j FB(whic)n(h)f(are)g(2)p FA(\031)s FB(-p)r(erio)r(dic)f(in)i FA(\034)9 b FB(,)33 b(w)n(e)e(consider)g(the)g(corre-)71 3303 y(sp)r(onding)c(F)-7 b(ourier)27 b(norm)1434 3423 y Fw(k)p FA(h)p Fw(k)1566 3435 y Fx(\013;\032;\033)1749 3423 y FB(=)1837 3344 y Fz(X)1837 3523 y Fx(k)q Fv(2)p Fn(Z)1971 3327 y Fz(\015)1971 3377 y(\015)1971 3427 y(\015)2017 3423 y FA(h)2065 3389 y Fy([)p Fx(k)q Fy(])2143 3327 y Fz(\015)2143 3377 y(\015)2143 3427 y(\015)2190 3481 y Fx(\013;\032)2305 3423 y FA(e)2344 3389 y Fv(j)p Fx(k)q Fv(j)p Fx(\033)2464 3423 y FA(:)71 3648 y FB(W)-7 b(e)28 b(consider,)e(th)n(us,)i(the)g(follo)n(wing)f(function)h(space)867 3847 y Fw(H)937 3859 y Fx(\013;\032;\033)1122 3847 y FB(=)23 b Fw(f)p FA(h)f FB(:)h FA(D)1439 3813 y Fx(u)1437 3868 y Fv(1)p Fx(;\032)1580 3847 y Fw(\002)18 b Ft(T)1718 3859 y Fx(\033)1786 3847 y Fw(!)23 b Ft(C)p FB(;)42 b(real-analytic)m FA(;)14 b Fw(k)p FA(h)p Fw(k)2635 3859 y Fx(\013;\032;\033)2819 3847 y FA(<)23 b Fw(1g)p FA(;)606 b FB(\(147\))71 3994 y(whic)n(h)27 b(can)h(b)r(e)g(c)n(hec)n(k)n(ed)e(that)i(is)f(a)h(Banac) n(h)e(space)h(for)g(an)n(y)g(\014xed)g FA(\013)d(>)e FB(0)28 b(and)f FA(\033)g(>)22 b FB(0.)195 4093 y(In)28 b(the)g(next)g(lemma,)g(w)n(e)f(state)g(some)g(prop)r(erties)g(of)g (these)h(Banac)n(h)e(spaces.)p Black 71 4255 a Fp(Lemma)31 b(6.1.)p Black 40 w Fs(The)g(fol)t(lowing)h(statements)d(hold:)p Black 169 4417 a(1.)p Black 42 w(If)i FA(\013)419 4429 y Fy(1)479 4417 y Fw(\025)23 b FA(\013)620 4429 y Fy(2)680 4417 y Fw(\025)g FB(0)p Fs(,)29 b(then)h Fw(H)1119 4429 y Fx(\013)1162 4437 y Fu(1)1195 4429 y Fx(;\032;\033)1336 4417 y Fw(\032)23 b(H)1494 4429 y Fx(\013)1537 4437 y Fu(2)1570 4429 y Fx(;\032;\033)1718 4417 y Fs(and)1673 4594 y Fw(k)p FA(h)p Fw(k)1805 4606 y Fx(\013)1848 4614 y Fu(2)1879 4606 y Fx(;\032;\033)2021 4594 y Fw(\024)f(k)p FA(h)p Fw(k)2240 4606 y Fx(\013)2283 4614 y Fu(1)2315 4606 y Fx(;\032;\033)2433 4594 y FA(:)p Black 169 4804 a Fs(2.)p Black 42 w(If)31 b FA(\013)419 4816 y Fy(1)456 4804 y FA(;)14 b(\013)546 4816 y Fy(2)606 4804 y Fw(\025)23 b FB(0)p Fs(,)30 b(then,)g(for)g FA(h)23 b Fw(2)h(H)1353 4816 y Fx(\013)1396 4824 y Fu(1)1428 4816 y Fx(;\032;\033)1576 4804 y Fs(and)31 b FA(g)25 b Fw(2)f(H)1952 4816 y Fx(\013)1995 4824 y Fu(2)2027 4816 y Fx(;\032;\033)2146 4804 y Fs(,)30 b(we)g(have)h(that)f FA(hg)25 b Fw(2)f(H)2947 4816 y Fx(\013)2990 4824 y Fu(1)3023 4816 y Fy(+)p Fx(\013)3117 4824 y Fu(2)3149 4816 y Fx(;\032;\033)3297 4804 y Fs(and)1428 4981 y Fw(k)p FA(hg)s Fw(k)1603 4993 y Fx(\013)1646 5001 y Fu(1)1677 4993 y Fy(+)p Fx(\013)1771 5001 y Fu(2)1804 4993 y Fx(;\032;\033)1945 4981 y Fw(\024)f(k)p FA(h)p Fw(k)2165 4993 y Fx(\013)2208 5001 y Fu(1)2239 4993 y Fx(;\032;\033)2358 4981 y Fw(k)p FA(g)s Fw(k)2485 4993 y Fx(\013)2528 5001 y Fu(2)2559 4993 y Fx(;\032;\033)2678 4981 y FA(:)p Black 169 5191 a Fs(3.)p Black 42 w(L)l(et)29 b(us)g(c)l(onsider)i FA(\013)23 b Fw(\025)g FB(0)28 b Fs(and)i FA(\032)1295 5161 y Fv(0)1342 5191 y FA(>)22 b(\032)h(>)g FB(0)29 b Fs(such)g(that)g FA(\032)2054 5161 y Fv(0)2095 5191 y Fw(\000)17 b FA(\032)30 b Fs(has)g(a)g(p)l (ositive)g(lower)h(b)l(ound)e(indep)l(endent)h(of)g FA(")p Fs(.)278 5290 y(Then)h(for)f FA(h)23 b Fw(2)h(H)847 5302 y Fx(\013;\032;\033)1038 5290 y Fs(we)30 b(have)h(that)f FA(@)1566 5302 y Fx(u)1610 5290 y FA(h)22 b Fw(2)i(H)1829 5302 y Fx(\013;\032)1926 5286 y Fl(0)1949 5302 y Fx(;\033)2043 5290 y Fs(and)1612 5468 y Fw(k)o FA(@)1697 5480 y Fx(u)1741 5468 y FA(h)p Fw(k)1830 5493 y Fx(\013;\032)1927 5476 y Fl(0)1950 5493 y Fx(;\033)2037 5468 y Fw(\024)f FA(K)6 b Fw(k)p FA(h)p Fw(k)2334 5480 y Fx(\013;\032;\033)2494 5468 y FA(:)p Black 1919 5753 a FB(54)p Black eop end %%Page: 55 55 TeXDict begin 55 54 bop Black Black 195 272 a FB(Throughout)40 b(this)h(section)f(w)n(e)h(are)f(going)f(to)i(solv)n(e)e(equations)h (of)h(the)g(form)g Fw(L)2904 284 y Fx(")2939 272 y FA(h)k FB(=)g FA(g)s FB(,)f(where)c Fw(L)3562 284 y Fx(")3638 272 y FB(is)h(the)71 372 y(di\013eren)n(tial)25 b(op)r(erator)e (de\014ned)i(in)g(\(45\).)36 b(Note)25 b(that)g(if)g FA(\013)f(>)f FB(0,)i(Ker)o Fw(L)2294 384 y Fx(")2353 372 y FB(=)d Fw(f)p FB(0)p Fw(g)i FB(and)g(hence)h Fw(L)3033 384 y Fx(")3094 372 y FB(is)g(in)n(v)n(ertible.)35 b(It)26 b(turns)71 471 y(out)h(that)h(its)g(in)n(v)n(erse)e(is)i Fw(G)922 483 y Fx(")985 471 y FB(de\014ned)g(b)n(y)1247 702 y Fw(G)1296 714 y Fx(")1331 702 y FB(\()p FA(h)p FB(\)\()p FA(u;)14 b(\034)9 b FB(\))25 b(=)1749 589 y Fz(Z)1832 610 y Fy(0)1795 778 y Fv(\0001)1931 702 y FA(h)p FB(\()p FA(u)18 b FB(+)h FA(t;)14 b(\034)28 b FB(+)18 b FA(")2414 668 y Fv(\000)p Fy(1)2502 702 y FA(t)p FB(\))c FA(dt:)987 b FB(\(148\))71 927 y(W)-7 b(e)28 b(also)e(in)n(tro)r(duce)p 1419 960 55 4 v 1419 1027 a Fw(G)1473 1039 y Fx(")1508 1027 y FB(\()p FA(h)p FB(\)\()p FA(u;)14 b(\034)9 b FB(\))25 b(=)d FA(@)1970 1039 y Fx(u)2028 1027 y FB([)p Fw(G)2100 1039 y Fx(")2136 1027 y FB(\()p FA(h)p FB(\)\()p FA(u;)14 b(\034)9 b FB(\)])15 b FA(:)1158 b FB(\(149\))71 1172 y(W)-7 b(e)28 b(will)g(consider)e Fw(G)743 1184 y Fx(")807 1172 y FB(de\014ned)i(in)g Fw(H)1260 1184 y Fx(\013;\032;\033)1449 1172 y FB(with)g FA(\013)c(>)e FB(0)28 b(in)f(order)g(the)h(in)n (tegral)e(in)i(\(148\))f(to)g(b)r(e)h(con)n(v)n(ergen)n(t.)p Black 71 1333 a Fp(Lemma)42 b(6.2.)p Black 46 w Fs(L)l(et)c FA(\013)g(>)h FB(0)p Fs(.)63 b(Then,)42 b(the)c(op)l(er)l(ators)i Fw(G)1911 1345 y Fx(")1985 1333 y Fs(and)p 2155 1266 V 39 w Fw(G)2209 1345 y Fx(")2283 1333 y Fs(in)k FB(\(148\))38 b Fs(and)47 b FB(\(149\))37 b Fs(r)l(esp)l(e)l(ctively)j(satisfy)f(the) 71 1432 y(fol)t(lowing)32 b(pr)l(op)l(erties.)p Black 169 1593 a(1.)p Black 42 w Fw(G)327 1605 y Fx(")393 1593 y Fs(is)e(line)l(ar)h(fr)l(om)f Fw(H)983 1605 y Fx(\013;\032;\033)1174 1593 y Fs(to)g(itself,)h(c)l(ommutes)e(with)i FA(@)2112 1605 y Fx(u)2185 1593 y Fs(and)f Fw(L)2403 1605 y Fx(")2457 1593 y Fw(\016)18 b(G)2566 1605 y Fx(")2625 1593 y FB(=)23 b(Id)p Fs(.)p Black 169 1757 a(2.)p Black 42 w(If)31 b FA(h)22 b Fw(2)i(H)585 1769 y Fx(\013;\032;\033)747 1757 y Fs(,)30 b(then)1592 1856 y Fw(k)o(G)1682 1868 y Fx(")1718 1856 y FB(\()p FA(h)p FB(\))p Fw(k)1872 1881 y Fx(\013;\032;\033)2057 1856 y Fw(\024)23 b FA(K)6 b Fw(k)p FA(h)p Fw(k)2354 1868 y Fx(\013;\032;\033)2514 1856 y FA(:)278 2002 y Fs(F)-6 b(urthermor)l(e,)30 b(if)h Fw(h)p FA(h)p Fw(i)24 b FB(=)e(0)p Fs(,)30 b(then)1573 2101 y Fw(k)o(G)1663 2113 y Fx(")1699 2101 y FB(\()p FA(h)p FB(\))p Fw(k)1853 2126 y Fx(\013;\032;\033)2038 2101 y Fw(\024)22 b FA(K)6 b(")p Fw(k)p FA(h)p Fw(k)2373 2113 y Fx(\013;\032;\033)2533 2101 y FA(:)p Black 169 2297 a Fs(3.)p Black 42 w(If)31 b FA(h)22 b Fw(2)i(H)585 2309 y Fx(\013;\032;\033)747 2297 y Fs(,)30 b(then)p 987 2230 V 30 w Fw(G)1041 2309 y Fx(")1076 2297 y FB(\()p FA(h)p FB(\))24 b Fw(2)f(H)1360 2309 y Fx(\013;\032;\033)1552 2297 y Fs(and)1585 2402 y Fz(\015)1585 2452 y(\015)p 1631 2406 V 21 x Fw(G)1685 2485 y Fx(")1721 2473 y FB(\()p FA(h)p FB(\))1833 2402 y Fz(\015)1833 2452 y(\015)1879 2506 y Fx(\013;\032;\033)2064 2473 y Fw(\024)g FA(K)6 b Fw(k)p FA(h)p Fw(k)2361 2485 y Fx(\013;\032;\033)2521 2473 y FA(:)p Black 71 2696 a Fs(Pr)l(o)l(of.)p Black 43 w FB(It)28 b(follo)n(ws)f(the)h(same)f(lines)g(as)g(the)h(pro)r(of)f (of)h(Lemma)f(5.5)g(in)h([GOS10)n(].)p 3790 2696 4 57 v 3794 2644 50 4 v 3794 2696 V 3843 2696 4 57 v 195 2861 a(Finally)-7 b(,)38 b(w)n(e)e(state)f(a)h(tec)n(hnical)f(lemma)h(ab)r (out)g(estimates)f(of)h(the)g(functions)h FA(A)p FB(,)h FA(B)2982 2873 y Fy(1)3019 2861 y FB(,)g FA(B)3143 2873 y Fy(2)3216 2861 y FB(and)e FA(C)42 b FB(de\014ned)36 b(in)71 2961 y(\(143\))o(,)24 b(\(144\))o(,)g(\(145\))f(and)f(\(146\))h (resp)r(ectiv)n(ely)-7 b(.)34 b(First)23 b(w)n(e)g(\014x)g FA(\032)2073 2973 y Fy(0)2133 2961 y FB(big)g(enough)f(suc)n(h)h(that)g FA(p)2948 2973 y Fy(0)2986 2961 y FB(\()p FA(u)p FB(\))g Fw(6)p FB(=)f(0)h(in)g FA(D)3436 2930 y Fx(u)3434 2981 y Fv(1)p Fx(;\032)3554 2989 y Fu(0)3591 2961 y FB(,)h(where)71 3060 y FA(D)142 3030 y Fx(u)140 3081 y Fv(1)p Fx(;\032)260 3089 y Fu(0)324 3060 y FB(is)k(the)g(domain)f(de\014ned)h(in)g(\(28\))o (.)p Black 71 3221 a Fp(Lemma)21 b(6.3.)p Black 32 w Fs(L)l(et)h Fw(f)p FA(\025;)14 b Fw(\000)p FA(\025)p Fw(g)21 b Fs(b)l(e)g(the)h(eigenvalues)h(of)g(the)e(hyp)l(erb)l(olic)j (critic)l(al)f(p)l(oint)f(of)h(the)f(unp)l(erturb)l(e)l(d)e (Hamiltonian)71 3321 y(system)26 b(and)p 497 3254 55 4 v 27 w Fw(G)551 3333 y Fx(")613 3321 y Fs(the)h(op)l(er)l(ator)g (de\014ne)l(d)g(in)f(L)l(emma)h(6.2.)39 b(L)l(et)26 b(us)f(\014x)h FA(\032)2309 3333 y Fy(0)2372 3321 y Fs(big)i(enough)e(such)h(that)f FA(p)3173 3333 y Fy(0)3210 3321 y FB(\()p FA(u)p FB(\))d Fw(6)p FB(=)g(0)j Fs(in)g FA(D)3670 3290 y Fx(u)3668 3341 y Fv(1)p Fx(;\032)3788 3349 y Fu(0)3825 3321 y Fs(.)71 3420 y(Then,)35 b(for)g(any)f FA(\032)29 b(>)h(\032)827 3432 y Fy(0)864 3420 y Fs(,)35 b(the)e(functions)h FA(A)p Fs(,)h FA(B)1615 3432 y Fy(1)1652 3420 y Fs(,)g FA(B)1775 3432 y Fy(2)1845 3420 y Fs(and)f FA(C)40 b Fs(de\014ne)l(d)34 b(in)40 b FB(\(143\))o Fs(,)35 b FB(\(144\))o Fs(,)g FB(\(145\))d Fs(and)43 b FB(\(146\))32 b Fs(satisfy)71 3520 y(the)e(fol)t(lowing)i(pr)l(op)l(erties,)p Black 169 3681 a(1.)p Black 42 w FA(A;)14 b(@)421 3693 y Fx(u)465 3681 y FA(A)23 b Fw(2)h(H)699 3693 y Fy(2)p Fx(\025;\032;\033)920 3681 y Fs(and)30 b(satisfy)1133 3792 y Fz(\015)1133 3842 y(\015)p 1180 3796 V 1180 3862 a Fw(G)1234 3875 y Fx(")1269 3863 y FB(\()p FA(A)p FB(\))1395 3792 y Fz(\015)1395 3842 y(\015)1442 3896 y Fy(2)p Fx(\025;\032;\033)1656 3863 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1956 3832 y Fx(\021)r Fy(+1)2080 3863 y FA(;)83 b Fw(k)p FA(@)2272 3875 y Fx(u)2315 3863 y FA(A)p Fw(k)2419 3887 y Fy(2)p Fx(\025;\032;\033)2633 3863 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2933 3832 y Fx(\021)2972 3863 y FA(:)3661 3869 y FB(\(150\))p Black 169 4077 a Fs(2.)p Black 42 w FA(B)341 4089 y Fy(1)379 4077 y FA(;)14 b(@)460 4089 y Fx(u)503 4077 y FA(B)566 4089 y Fy(1)603 4077 y FA(;)g(B)703 4089 y Fy(2)763 4077 y Fw(2)24 b(H)912 4089 y Fy(0)p Fx(;\032;\033)1093 4077 y Fs(and)31 b(satisfy)815 4252 y Fw(k)p FA(B)920 4264 y Fy(1)957 4252 y Fw(k)999 4264 y Fy(0)p Fx(;\032;\033)1173 4252 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1473 4222 y Fx(\021)1512 4252 y FA(;)83 b Fw(k)p FA(@)1704 4264 y Fx(u)1747 4252 y FA(B)1810 4264 y Fy(1)1848 4252 y Fw(k)1890 4264 y Fy(0)p Fx(;\032;\033)2064 4252 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2364 4222 y Fx(\021)2403 4252 y FA(;)83 b Fw(k)p FA(B)2614 4264 y Fy(2)2651 4252 y Fw(k)2693 4264 y Fy(0)p Fx(;\032;\033)2867 4252 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")3167 4222 y Fx(\021)r Fy(+1)3291 4252 y FA(:)3661 4253 y FB(\(151\))p Black 169 4460 a Fs(3.)p Black 42 w(L)l(et)30 b(us)f(c)l(onsider)i FA(h)906 4472 y Fy(1)943 4460 y FA(;)14 b(h)1028 4472 y Fy(2)1088 4460 y Fw(2)23 b FA(B)t FB(\()p FA(\027)5 b FB(\))24 b Fw(\032)f(H)1525 4472 y Fy(2)p Fx(\025;\032;\033)1716 4460 y Fs(.)39 b(Then,)1094 4637 y Fw(k)o FA(C)6 b FB(\()p FA(h)1280 4649 y Fy(2)1318 4637 y FA(;)14 b(u;)g(\034)9 b FB(\))18 b Fw(\000)h FA(C)6 b FB(\()p FA(h)1764 4649 y Fy(1)1801 4637 y FA(;)14 b(u;)g(\034)9 b FB(\))p Fw(k)2042 4661 y Fy(2)p Fx(\025;\032;\033)2256 4637 y Fw(\024)23 b FA(K)6 b(\027)f Fw(k)p FA(h)2557 4649 y Fy(2)2611 4637 y Fw(\000)19 b FA(h)2743 4649 y Fy(1)2780 4637 y Fw(k)2822 4649 y Fy(2)p Fx(\025;\032;\033)3012 4637 y FA(:)p Black 71 4844 a Fs(Pr)l(o)l(of.)p Black 43 w FB(F)-7 b(or)27 b(the)h(\014rst)f(b)r(ounds,)h(w)n(e)f(split)h FA(A)c FB(=)e FA(A)1648 4856 y Fy(1)1704 4844 y FB(+)c FA(A)1849 4856 y Fy(2)1905 4844 y FB(+)g FA(A)2050 4856 y Fy(3)2115 4844 y FB(as)635 5020 y FA(A)697 5032 y Fy(1)735 5020 y FB(\()p FA(u;)c(\034)9 b FB(\))24 b(=)18 b Fw(\000)g FB(\()p FA(V)h FB(\()p FA(q)1287 5032 y Fy(0)1325 5020 y FB(\()p FA(u)p FB(\))f(+)g FA(x)1585 5032 y Fx(p)1624 5020 y FB(\()p FA(\034)9 b FB(\)\))20 b Fw(\000)e FA(V)h FB(\()p FA(x)2014 5032 y Fx(p)2053 5020 y FB(\()p FA(\034)9 b FB(\)\))20 b Fw(\000)e FA(V)h FB(\()p FA(q)2433 5032 y Fy(0)2470 5020 y FB(\()p FA(u)p FB(\)\))g Fw(\000)f FA(V)2783 4986 y Fv(0)2806 5020 y FB(\()p FA(x)2885 5032 y Fx(p)2925 5020 y FB(\()p FA(\034)9 b FB(\)\))p FA(q)3103 5032 y Fy(0)3141 5020 y FB(\()p FA(u)p FB(\)\))376 b(\(152\))635 5164 y FA(A)697 5176 y Fy(2)735 5164 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)18 b Fw(\000)g FA(\026")1208 5130 y Fx(\021)1267 5143 y Fz(b)1248 5164 y FA(H)1324 5130 y Fy(1)1317 5185 y(1)1361 5164 y FB(\()p FA(q)1430 5176 y Fy(0)1468 5164 y FB(\()p FA(u)p FB(\))p FA(;)c(p)1659 5176 y Fy(0)1696 5164 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))1739 b(\(153\))635 5309 y FA(A)697 5321 y Fy(3)735 5309 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)18 b Fw(\000)g FA(\026")1208 5274 y Fx(\021)r Fy(+1)1351 5288 y Fz(b)1332 5309 y FA(H)1408 5274 y Fy(2)1401 5329 y(1)1445 5309 y FB(\()p FA(q)1514 5321 y Fy(0)1552 5309 y FB(\()p FA(u)p FB(\))p FA(;)c(p)1743 5321 y Fy(0)1780 5309 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))p FA(;)1632 b FB(\(154\))71 5504 y(where)330 5483 y Fz(b)311 5504 y FA(H)387 5474 y Fy(1)380 5525 y(1)452 5504 y FB(and)632 5483 y Fz(b)613 5504 y FA(H)689 5474 y Fy(2)682 5525 y(1)754 5504 y FB(are)26 b(the)i(functions)g(de\014ned)g(in)g(\(35\))f (and)h(\(37\))o(.)p Black 1919 5753 a(55)p Black eop end %%Page: 56 56 TeXDict begin 56 55 bop Black Black 195 272 a FB(F)-7 b(or)27 b FA(A)406 284 y Fy(1)444 272 y FB(,)h(using)f(the)h(mean)f(v) -5 b(alue)28 b(theorem)f(and)g(Hyp)r(othesis)h Fp(HP1.1)p FB(,)f(one)g(can)g(see)g(that)633 509 y FA(A)695 521 y Fy(1)733 509 y FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b(=)18 b Fw(\000)g FA(q)1156 475 y Fy(2)1153 530 y(0)1194 509 y FB(\()p FA(u)p FB(\))1320 396 y Fz(Z)1403 417 y Fy(1)1366 585 y(0)1454 417 y Fz(\020)1503 509 y FA(V)1570 450 y Fl(00)1629 509 y FB(\()p FA(x)1708 521 y Fx(p)1747 509 y FB(\()p FA(\034)9 b FB(\))20 b(+)e FA(s)1998 521 y Fy(1)2035 509 y FA(q)2072 521 y Fy(0)2109 509 y FB(\()p FA(u)p FB(\)\))h Fw(\000)f FA(V)2422 450 y Fl(00)2481 509 y FB(\()p FA(s)2552 521 y Fy(1)2589 509 y FA(q)2626 521 y Fy(0)2663 509 y FB(\()p FA(u)p FB(\)\))2808 417 y Fz(\021)2871 509 y FB(\(1)h Fw(\000)f FA(s)3086 521 y Fy(1)3123 509 y FB(\))c FA(ds)3251 521 y Fy(1)950 748 y FB(=)k Fw(\000)g FA(q)1156 714 y Fy(2)1153 769 y(0)1194 748 y FB(\()p FA(u)p FB(\))p FA(x)1353 760 y Fx(p)1392 748 y FB(\()p FA(\034)9 b FB(\))1515 635 y Fz(Z)1599 656 y Fy(1)1562 824 y(0)1650 635 y Fz(Z)1733 656 y Fy(1)1696 824 y(0)1784 748 y FA(V)1851 689 y Fl(000)1928 748 y FB(\()p FA(s)1999 760 y Fy(2)2036 748 y FA(x)2083 760 y Fx(p)2122 748 y FB(\()p FA(\034)g FB(\))20 b(+)e FA(s)2373 760 y Fy(1)2410 748 y FA(q)2447 760 y Fy(0)2484 748 y FB(\()p FA(u)p FB(\)\))d(\(1)j Fw(\000)g FA(s)2857 760 y Fy(1)2894 748 y FB(\))c FA(ds)3022 760 y Fy(1)3059 748 y FA(ds)3141 760 y Fy(2)3179 748 y FA(:)3661 623 y FB(\(155\))71 989 y(Therefore,)29 b FA(A)535 1001 y Fy(1)599 989 y Fw(2)d(H)750 1001 y Fy(2)p Fx(\025;\032;\033)971 989 y FB(and)k Fw(k)p FA(A)1239 1001 y Fy(1)1276 989 y Fw(k)1318 1001 y Fy(2)p Fx(\025;\032;\033)1535 989 y Fw(\024)c FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1838 958 y Fx(\021)r Fy(+1)1962 989 y FB(.)42 b(Applying)30 b(Lemma)g(6.2,)f(w)n(e)g(obtain)g Fw(k)p 3273 922 55 4 v(G)3327 1001 y Fx(")3363 989 y FB(\()p FA(A)3457 1001 y Fy(1)3495 989 y FB(\))p Fw(k)3569 1001 y Fy(2)p Fx(\025;\032;\033) 3786 989 y Fw(\024)71 1088 y FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")283 1058 y Fx(\021)r Fy(+1)407 1088 y FB(.)195 1198 y(F)-7 b(or)32 b(the)h(other)f(terms,)h(let)g(us)f(p)r(oin)n(t)g (out)h(that,)h(b)n(y)e(construction,)2442 1177 y Fz(b)2423 1198 y FA(H)2499 1168 y Fy(1)2492 1219 y(1)2568 1198 y FB(and)2754 1177 y Fz(b)2734 1198 y FA(H)2810 1168 y Fy(2)2803 1219 y(1)2880 1198 y FB(are)f(quadratic)g(in)i(\()p FA(q)s(;)14 b(p)p FB(\))33 b(and)71 1298 y(therefore)28 b FA(A)483 1310 y Fy(2)521 1298 y FA(;)14 b(A)620 1310 y Fy(3)684 1298 y Fw(2)26 b(H)835 1310 y Fy(2)p Fx(\025;\032;\033)1027 1298 y FB(.)42 b(T)-7 b(o)29 b(b)r(ound)p 1474 1231 V 30 w Fw(G)1529 1310 y Fx(")1564 1298 y FB(\()p FA(A)1658 1310 y Fy(2)1696 1298 y FB(\),)h(let)g(us)g(p)r(oin)n(t)g(out)f(that)h Fw(h)p FA(A)2656 1310 y Fy(2)2694 1298 y Fw(i)c FB(=)g(0)j(and)h (therefore,)f(taking)g(in)n(to)71 1397 y(accoun)n(t)e(that)g FA(A)619 1409 y Fy(2)685 1397 y FB(is)g(analytic)g(in)h FA(D)1252 1367 y Fx(u)1250 1418 y Fv(1)p Fx(;\032)1370 1426 y Fu(0)1425 1397 y Fw(\002)18 b Ft(T)1563 1409 y Fx(\033)1636 1397 y FB(and)27 b FA(\032)c(>)g(\032)1994 1409 y Fy(0)2031 1397 y FB(,)28 b(b)n(y)f(Lemmas)g(\(6.1\))g(and)h (6.2,)1162 1594 y Fw(k)p 1204 1527 V(G)1258 1606 y Fx(")1293 1594 y FB(\()p FA(A)1387 1606 y Fy(2)1425 1594 y FB(\))p Fw(k)1499 1606 y Fy(2)p Fx(\025;\032;\033)1713 1594 y Fw(\024)23 b FA(K)6 b(")p Fw(k)p FA(A)2021 1606 y Fy(2)2057 1594 y Fw(k)2099 1606 y Fy(2)p Fx(\025;\032;\033)2313 1594 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2612 1560 y Fx(\021)r Fy(+1)2736 1594 y FA(:)71 1777 y FB(On)36 b(the)h(other)f(hand,)j(since)d(b)n(y)g(Corollary)e (5.6,)k Fw(k)p FA(A)1822 1789 y Fy(3)1859 1777 y Fw(k)1901 1789 y Fy(2)p Fx(\025;\032;\033)2129 1777 y Fw(\024)g FA(K)6 b Fw(j)p FA(\026)p Fw(j)2405 1746 y Fy(2)2442 1777 y FA(")2481 1746 y Fy(2)p Fx(\021)r Fy(+1)2638 1777 y FB(,)39 b(w)n(e)d(ha)n(v)n(e)f(that)i Fw(k)p 3262 1710 V(G)3316 1789 y Fx(")3351 1777 y FB(\()p FA(A)3445 1789 y Fy(3)3483 1777 y FB(\))p Fw(k)3557 1789 y Fy(2)p Fx(\025;\032;\033) 3786 1777 y Fw(\024)71 1876 y FA(K)6 b Fw(j)p FA(\026)p Fw(j)244 1846 y Fy(2)281 1876 y FA(")320 1846 y Fy(2)p Fx(\021)r Fy(+1)477 1876 y FB(.)37 b(Therefore)1476 1905 y Fz(\015)1476 1955 y(\015)p 1522 1909 V 21 x Fw(G)1576 1988 y Fx(")1612 1976 y FB(\()p FA(A)p FB(\))1738 1905 y Fz(\015)1738 1955 y(\015)1785 2009 y Fy(2)p Fx(\025;\032;\033)1999 1976 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2299 1942 y Fx(\021)r Fy(+1)2422 1976 y FA(:)71 2136 y FB(The)28 b(b)r(ound)g(for)f FA(@)669 2148 y Fx(u)712 2136 y FA(A)h FB(can)f(b)r(e)h(obtained)g(just)g(di\013eren)n(tiating)f FA(A)2170 2148 y Fx(i)2198 2136 y FB(,)h FA(i)22 b FB(=)h(1)p FA(;)14 b FB(2)p FA(;)g FB(3.)195 2236 y(The)28 b(other)f(b)r(ounds)h (are)e(straigh)n(tforw)n(ard.)p 3790 2236 4 57 v 3794 2183 50 4 v 3794 2236 V 3843 2236 4 57 v 71 2452 a Fp(6.1.2)94 b(Pro)s(of)31 b(of)h(Theorem)f(4.3)g(in)h(the)f(h)m(yp)s(erb)s(olic)h (case)71 2605 y FB(W)-7 b(e)31 b(dev)n(ote)f(this)h(section)g(to)f(pro) n(v)n(e)f(Theorem)h(4.3)g(for)h(the)g(case)f(in)h(whic)n(h)g(the)g(unp) r(erturb)r(ed)g(Hamiltonian)g(has)f(a)71 2704 y(h)n(yp)r(erb)r(olic)d (critical)g(p)r(oin)n(t.)37 b(First)28 b(w)n(e)f(rewrite)g(it)h(in)f (terms)h(of)f(the)h(Banac)n(h)f(spaces)f(de\014ned)i(in)g(\(147\))o(.)p Black 71 2870 a Fp(Prop)s(osition)40 b(6.4.)p Black 44 w Fs(L)l(et)d Fw(f)p FA(\025;)14 b Fw(\000)p FA(\025)p Fw(g)37 b Fs(b)l(e)g(the)h(eigenvalues)g(of)h(the)e(unp)l(erturb)l(e)l (d)f(hyp)l(erb)l(olic)k(critic)l(al)f(p)l(oint,)h FA(\032)3634 2882 y Fy(1)3708 2870 y FA(>)c FB(0)71 2970 y Fs(big)f(enough)h(and)f FA(")700 2982 y Fy(0)769 2970 y FA(>)d FB(0)i Fs(smal)t(l)i(enough.)55 b(Then,)37 b(for)e FA(")d Fw(2)h FB(\(0)p FA(;)14 b(")2194 2982 y Fy(0)2231 2970 y FB(\))p Fs(,)36 b(ther)l(e)f(exists)g(a)g (function)g FA(T)3227 2982 y Fy(1)3264 2970 y FB(\()p FA(u;)14 b(\034)9 b FB(\))35 b Fs(de\014ne)l(d)g(in)71 3070 y FA(D)142 3040 y Fx(u)140 3090 y Fv(1)p Fx(;\032)260 3098 y Fu(1)317 3070 y Fw(\002)21 b Ft(T)458 3082 y Fx(\033)536 3070 y Fs(which)35 b(satis\014es)e(e)l(quation)40 b FB(\(141\))32 b Fs(and)i(the)f(asymptotic)h(c)l(ondition)41 b FB(\(49\))o Fs(.)49 b(Mor)l(e)l(over,)36 b(ther)l(e)d(exists)g(a)71 3169 y(c)l(onstant)c FA(b)438 3181 y Fy(1)498 3169 y FA(>)22 b FB(0)30 b Fs(such)f(that)1488 3269 y Fw(k)p FA(@)1574 3281 y Fx(u)1617 3269 y FA(T)1666 3281 y Fy(1)1703 3269 y Fw(k)1745 3281 y Fy(2)p Fx(\025;\032)1871 3289 y Fu(1)1903 3281 y Fx(;\033)1991 3269 y Fw(\024)22 b FA(b)2114 3281 y Fy(1)2151 3269 y Fw(j)p FA(\026)p Fw(j)p FA(")2286 3235 y Fx(\021)r Fy(+1)2410 3269 y FA(:)195 3435 y FB(Theorem)27 b(4.3)g(is)g(a)h(straigh)n(tforw)n(ard)c (consequence)j(of)g(this)h(prop)r(osition.)195 3535 y(Let)h(us)g (observ)n(e)e(that)j(the)f(op)r(erator)e Fw(F)37 b FB(de\014ned)29 b(in)g(\(142\))f(has)h(linear)f(terms)g(in)i FA(w)h FB(whic)n(h)e(are)f (not)h(small)f(when)71 3634 y FA(\021)e FB(=)d(0.)36 b(Therefore,)26 b(if)h(one)g(w)n(an)n(ts)f(to)h(pro)n(v)n(e)e(the)i (existence)g(of)g FA(T)38 b FB(through)26 b(a)h(\014xed)g(p)r(oin)n(t)g (argumen)n(t,)f(\014rst)h(w)n(e)f(m)n(ust)71 3734 y(lo)r(ok)f(for)g(a)h (c)n(hange)e(of)i(v)-5 b(ariables.)35 b(Let)26 b(us)g(p)r(oin)n(t)g (out)g(that)g(this)g(c)n(hange)f(of)h(v)-5 b(ariables)24 b(is)i(not)g(necessary)e(for)h(the)h(case)71 3834 y FA(\021)g(>)d FB(0.)p Black 71 4000 a Fp(Lemma)f(6.5.)p Black 33 w Fs(L)l(et)g(us)g(c)l(onsider)i FA(\032)1184 4012 y Fy(1)1244 4000 y FA(>)f(\032)1375 3969 y Fv(0)1375 4020 y Fy(0)1435 4000 y FA(>)g(\032)1566 4012 y Fy(0)1626 4000 y FA(>)f FB(0)p Fs(,)i(wher)l(e)f FA(\032)2074 4012 y Fy(0)2134 4000 y Fs(is)g(big)g(enough)g(such)g(that)f FA(p)3001 4012 y Fy(0)3038 4000 y FB(\()p FA(u)p FB(\))h Fw(6)p FB(=)g(0)f Fs(for)h FA(u)g Fw(2)g FA(D)3670 3969 y Fx(u)3668 4020 y Fv(1)p Fx(;\032)3788 4028 y Fu(0)3825 4000 y Fs(.)71 4099 y(Then,)31 b(for)f FA(")23 b(>)g FB(0)29 b Fs(smal)t(l)i(enough,)f (ther)l(e)g(exists)f(a)i(function)e FA(g)d Fw(2)d(H)2241 4113 y Fy(0)p Fx(;\032)2328 4094 y Fl(0)2328 4132 y Fu(0)2361 4113 y Fx(;\033)2455 4099 y Fs(such)30 b(that)g Fw(h)p FA(g)s Fw(i)23 b FB(=)g(0)29 b Fs(and)h(is)g(solution)g(of)1648 4282 y Fw(L)1705 4294 y Fx(")1741 4282 y FA(g)c FB(=)d Fw(\000)p FA(B)2023 4294 y Fy(1)2059 4282 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)1390 b FB(\(156\))71 4464 y Fs(wher)l(e)33 b Fw(L)365 4476 y Fx(")434 4464 y Fs(is)g(the)g(op)l(er)l(ator)h (de\014ne)l(d)f(in)39 b FB(\(45\))33 b Fs(and)g FA(B)1799 4476 y Fy(1)1869 4464 y Fs(is)g(the)g(function)g(de\014ne)l(d)g(in)39 b FB(\(144\))o Fs(.)48 b(Mor)l(e)l(over,)35 b(it)e(satis\014es)71 4564 y(that)1110 4662 y Fw(k)p FA(g)s Fw(k)1237 4676 y Fy(0)p Fx(;\032)1324 4656 y Fl(0)1324 4694 y Fu(0)1355 4676 y Fx(;\033)1442 4662 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1742 4631 y Fx(\021)r Fy(+1)1866 4662 y FA(;)83 b Fw(k)p FA(@)2058 4674 y Fx(v)2097 4662 y FA(g)s Fw(k)2182 4676 y Fy(0)p Fx(;\032)2269 4656 y Fl(0)2269 4694 y Fu(0)2301 4676 y Fx(;\033)2388 4662 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2688 4631 y Fx(\021)r Fy(+1)71 4813 y Fs(and)30 b FA(v)22 b FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))24 b Fw(2)f FA(D)782 4783 y Fx(u)780 4834 y Fv(1)p Fx(;\032)900 4842 y Fu(0)967 4813 y Fs(for)30 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b Fw(2)e FA(D)1462 4783 y Fx(u)1460 4839 y Fv(1)p Fx(;\032)1580 4819 y Fl(0)1580 4857 y Fu(0)1635 4813 y Fw(\002)18 b Ft(T)1773 4825 y Fx(\033)1818 4813 y Fs(.)195 4928 y(F)-6 b(urthermor)l(e,)26 b FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b(=)g(\()p FA(v)8 b FB(+)d FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))27 b Fs(is)d(invertible)h(and)f(its)g(inverse)g(is)g (of)h(the)f(form)g FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)d(\()p FA(u)5 b FB(+)g FA(h)p FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))p Fs(,)71 5028 y(wher)l(e)30 b FA(h)g Fs(is)g(a)g(function)g(de\014ne)l(d)g(for)g FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)g FA(D)1651 4998 y Fx(u)1649 5048 y Fv(1)p Fx(;\032)1769 5056 y Fu(1)1824 5028 y Fw(\002)18 b Ft(T)1962 5040 y Fx(\033)2037 5028 y Fs(and)30 b(satis\014es)g(that)f FA(h)23 b Fw(2)h(H)2899 5040 y Fy(0)p Fx(;\032)2986 5048 y Fu(1)3018 5040 y Fx(;\033)3083 5028 y Fs(,)1580 5223 y Fw(k)p FA(h)p Fw(k)1712 5235 y Fy(0)p Fx(;\032)1799 5243 y Fu(1)1830 5235 y Fx(;\033)1918 5223 y Fw(\024)f FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2218 5188 y Fx(\021)r Fy(+1)71 5405 y Fs(and)30 b(that)g FA(u)18 b FB(+)g FA(h)p FB(\()p FA(u;)c(\034)9 b FB(\))23 b Fw(2)h FA(D)966 5375 y Fx(u)964 5431 y Fv(1)p Fx(;\032)1084 5411 y Fl(0)1084 5449 y Fu(0)1150 5405 y Fs(for)31 b FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)f FA(D)1650 5375 y Fx(u)1648 5426 y Fv(1)p Fx(;\032)1768 5434 y Fu(1)1823 5405 y Fw(\002)18 b Ft(T)1961 5417 y Fx(\033)2006 5405 y Fs(.)p Black 1919 5753 a FB(56)p Black eop end %%Page: 57 57 TeXDict begin 57 56 bop Black Black Black 71 272 a Fs(Pr)l(o)l(of.)p Black 43 w FB(F)-7 b(rom)24 b(the)h(de\014nition)g(of)g FA(B)1207 284 y Fy(1)1269 272 y FB(in)g(\(144\))e(w)n(e)i(ha)n(v)n(e)e (that)i Fw(h)p FA(B)2156 284 y Fy(1)2194 272 y Fw(i)e FB(=)g(0.)35 b(On)24 b(the)h(other)g(hand,)g(using)f(the)h (de\014nition)71 382 y(of)185 361 y Fz(b)165 382 y FA(H)241 352 y Fy(1)234 403 y(1)306 382 y FB(and)j FA(\025)g FB(in)f(\(35\))h (and)f(\(46\))g(resp)r(ectiv)n(ely)-7 b(,)27 b FA(B)1684 394 y Fy(1)1749 382 y FB(can)g(b)r(e)h(split)g(as)1414 539 y FA(B)1477 551 y Fy(1)1514 539 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)d FA(B)1878 551 y Fy(10)1949 539 y FB(\()p FA(\034)9 b FB(\))19 b(+)f FA(B)2223 551 y Fy(11)2294 539 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)71 696 y FB(where,)27 b(using)h(\(47\),)951 845 y FA(B)1014 857 y Fy(10)1084 845 y FB(\()p FA(\034)9 b FB(\))24 b(=)120 b(lim)1305 899 y Fy(Re)11 b Fx(v)r Fv(!\0001)1627 845 y FA(B)1690 857 y Fy(1)1728 845 y FB(\()p FA(v)s(;)j(\034)9 b FB(\))24 b(=)f Fw(\000)p FA(\026")2183 811 y Fx(\021)2236 728 y Fz(\022)2307 789 y FA(a)2351 801 y Fy(11)2422 789 y FB(\()p FA(\034)9 b FB(\))p 2307 826 225 4 v 2395 902 a FA(\025)2560 845 y FB(+)18 b(2)p FA(a)2729 857 y Fy(02)2799 845 y FB(\()p FA(\034)9 b FB(\))2908 728 y Fz(\023)71 1029 y FB(and)31 b FA(B)299 1041 y Fy(11)369 1029 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))31 b(=)e FA(B)746 1041 y Fy(1)784 1029 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))22 b Fw(\000)f FA(B)1144 1041 y Fy(10)1214 1029 y FB(\()p FA(\034)9 b FB(\).)50 b(Both)31 b(terms)g(ha)n(v)n(e)f(zero)h(mean.)48 b(Moreo)n(v)n(er,)30 b FA(B)2937 1041 y Fy(10)3036 1029 y Fw(2)g(H)3191 1043 y Fy(0)p Fx(;\032)3278 1024 y Fl(0)3278 1062 y Fu(0)3311 1043 y Fx(;\033)3407 1029 y FB(and)h(satis\014es)71 1129 y Fw(k)p FA(B)176 1141 y Fy(10)246 1129 y Fw(k)288 1143 y Fy(0)p Fx(;\032)375 1123 y Fl(0)375 1161 y Fu(0)407 1143 y Fx(;\033)494 1129 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")794 1099 y Fx(\021)861 1129 y FB(and)28 b FA(B)1086 1141 y Fy(11)1179 1129 y Fw(2)c(H)1328 1143 y Fx(\025;\032)1421 1123 y Fl(0)1421 1161 y Fu(0)1454 1143 y Fx(;\033)1546 1129 y FB(and)j(satis\014es)g Fw(k)p FA(B)2118 1141 y Fy(11)2188 1129 y Fw(k)2230 1143 y Fx(\025;\032)2323 1123 y Fl(0)2323 1161 y Fu(0)2355 1143 y Fx(;\033)2443 1129 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2742 1099 y Fx(\021)2782 1129 y FB(.)195 1256 y(Since)31 b FA(B)478 1268 y Fy(10)549 1256 y FB(\()p FA(\034)9 b FB(\))30 b(=)781 1194 y Fz(P)868 1281 y Fx(k)q Fv(2)p Fn(Z)p Fv(nf)p Fy(0)p Fv(g)1142 1256 y FA(B)1209 1213 y Fy([)p Fx(k)q Fy(])1205 1279 y(10)1287 1256 y FA(e)1326 1226 y Fx(ik)q(\034)1459 1256 y FB(has)g(zero)g(a)n(v)n(erage,)f(w)n(e) h(can)h(de\014ne)g(a)f(2)p FA(\031)s FB(-p)r(erio)r(dic)g(primitiv)n(e) h(with)h(zero)71 1356 y(a)n(v)n(erage)25 b(as)p 1494 1457 68 4 v 1494 1523 a FA(B)1561 1535 y Fy(10)1631 1523 y FB(\()p FA(\034)9 b FB(\))25 b(=)1920 1445 y Fz(X)1852 1627 y Fx(k)q Fv(2)p Fn(Z)p Fv(nf)p Fy(0)p Fv(g)2131 1467 y FA(B)2198 1424 y Fy([)p Fx(k)q Fy(])2194 1489 y(10)p 2131 1504 146 4 v 2167 1580 a FA(ik)2287 1523 y(e)2326 1489 y Fx(ik)q(\034)71 1765 y FB(whic)n(h)i(satis\014es)g Fw(k)p 656 1699 68 4 v FA(B)723 1777 y Fy(10)793 1765 y Fw(k)835 1779 y Fy(0)p Fx(;\032)922 1760 y Fl(0)922 1798 y Fu(0)954 1779 y Fx(;\033)1042 1765 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1341 1735 y Fx(\021)1381 1765 y FB(.)195 1865 y(By)28 b(the)g(linearit)n(y)e(of)i(equation)f (\(156\))o(,)h(w)n(e)f(can)g(tak)n(e)g FA(g)j FB(as)1301 2022 y FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)e Fw(\000)p FA(")p 1748 1955 V(B)1815 2034 y Fy(10)1885 2022 y FB(\()p FA(\034)9 b FB(\))20 b Fw(\000)e(G)2146 2034 y Fx(")2195 2022 y FB(\()q FA(B)2291 2034 y Fy(11)2361 2022 y FB(\))c(\()p FA(v)s(;)g(\034)9 b FB(\))p FA(;)71 2179 y FB(where)25 b Fw(G)358 2191 y Fx(")420 2179 y FB(is)h(the)g(op)r(erator)f(de\014ned)h(in)g(\(148\))o(.)37 b(Moreo)n(v)n(er,)23 b(using)j(the)g(\014rst)g(statemen)n(t)g(of)g (Lemma)g(6.1)f(and)h(Lemma)71 2279 y(6.2,)405 2378 y Fw(k)p FA(g)s Fw(k)532 2392 y Fy(0)p Fx(;\032)619 2373 y Fl(0)619 2411 y Fu(0)650 2392 y Fx(;\033)738 2378 y Fw(\024)c FA(")878 2308 y Fz(\015)878 2357 y(\015)p 924 2312 V 21 x FA(B)991 2390 y Fy(10)1061 2308 y Fz(\015)1061 2357 y(\015)1108 2411 y Fy(0)p Fx(;\032)1195 2391 y Fl(0)1195 2429 y Fu(0)1227 2411 y Fx(;\033)1310 2378 y FB(+)c Fw(kG)1484 2390 y Fx(")1533 2378 y FB(\()q FA(B)1629 2390 y Fy(11)1699 2378 y FB(\))p Fw(k)1773 2403 y Fx(\025;\032)1866 2383 y Fl(0)1866 2421 y Fu(0)1899 2403 y Fx(;\033)1986 2378 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2286 2344 y Fx(\021)r Fy(+1)2428 2378 y FB(+)18 b FA(K)6 b(")14 b Fw(k)o FA(B)2745 2390 y Fy(11)2815 2378 y Fw(k)2856 2403 y Fx(\025;\032)2949 2383 y Fl(0)2949 2421 y Fu(0)2982 2403 y Fx(;\033)3070 2378 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")3369 2344 y Fx(\021)r Fy(+1)3493 2378 y FA(:)71 2530 y FB(Moreo)n(v)n(er,)25 b(b)n(y)i(Lemma)g(6.2,)1342 2629 y FA(@)1386 2641 y Fx(v)1426 2629 y FA(g)e FB(=)e Fw(\000)p FA(@)1688 2641 y Fx(v)1727 2629 y Fw(G)1776 2641 y Fx(")1826 2629 y FB(\()p FA(B)1921 2641 y Fy(11)1991 2629 y FB(\))h(=)e Fw(\000G)2248 2641 y Fx(")2298 2629 y FB(\()p FA(@)2374 2641 y Fx(v)2413 2629 y FA(B)2476 2641 y Fy(11)2547 2629 y FB(\))71 2764 y(and)27 b(then,)520 2920 y Fw(k)p FA(@)606 2932 y Fx(v)645 2920 y FA(g)s Fw(k)730 2934 y Fy(0)p Fx(;\032)817 2915 y Fl(0)817 2953 y Fu(0)849 2934 y Fx(;\033)936 2920 y Fw(\024)c(k)p FA(@)1110 2932 y Fx(v)1149 2920 y FA(g)s Fw(k)1234 2934 y Fx(\025;\032)1327 2915 y Fl(0)1327 2953 y Fu(0)1359 2934 y Fx(;\033)1447 2920 y FB(=)f Fw(kG)1625 2932 y Fx(")1674 2920 y FB(\()q FA(@)1751 2932 y Fx(v)1790 2920 y FA(B)1853 2932 y Fy(11)1924 2920 y FB(\))14 b Fw(k)2012 2934 y Fx(\025;\032)2105 2915 y Fl(0)2105 2953 y Fu(0)2137 2934 y Fx(;\033)2225 2920 y Fw(\024)22 b FA(K)6 b(")14 b Fw(k)o FA(@)2527 2932 y Fx(v)2566 2920 y FA(B)2629 2932 y Fy(11)2700 2920 y Fw(k)2741 2945 y Fx(\025;\032)2834 2925 y Fl(0)2834 2963 y Fu(0)2867 2945 y Fx(;\033)2955 2920 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")3254 2886 y Fx(\021)r Fy(+1)3378 2920 y FA(:)71 3097 y FB(Since)26 b Fw(k)p FA(g)s Fw(k)413 3111 y Fy(0)p Fx(;\032)500 3091 y Fl(0)500 3129 y Fu(0)532 3111 y Fx(;\033)619 3097 y Fw(\024)d FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")919 3067 y Fx(\021)r Fy(+1)1043 3097 y FB(,)26 b(w)n(e)g(ha)n(v)n(e)f(that)i FA(v)19 b FB(+)c FA(g)s FB(\()p FA(v)s(;)f(\034)9 b FB(\))24 b Fw(2)g FA(D)2127 3067 y Fx(u)2125 3117 y Fv(1)p Fx(;\032)2245 3125 y Fu(0)2308 3097 y FB(for)h(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b Fw(2)e FA(D)2796 3067 y Fx(u)2794 3123 y Fv(1)p Fx(;\032)2914 3103 y Fl(0)2914 3141 y Fu(0)2967 3097 y Fw(\002)15 b Ft(T)3102 3109 y Fx(\033)3173 3097 y FB(pro)n(vided)26 b FA(")g FB(is)g(small)71 3212 y(enough)h(and)g FA(\032)561 3182 y Fv(0)561 3232 y Fy(0)621 3212 y FA(>)c(\032)752 3224 y Fy(0)789 3212 y FB(.)195 3311 y(T)-7 b(o)28 b(obtain)f(the)h(in) n(v)n(erse)e(c)n(hange)g(and)i(its)g(prop)r(erties)e(it)i(is)g(straigh) n(tforw)n(ard.)p 3790 3311 4 57 v 3794 3259 50 4 v 3794 3311 V 3843 3311 4 57 v 195 3472 a(If)g(w)n(e)g(apply)f(the)h(c)n (hange)e(of)i(v)-5 b(ariables)26 b FA(u)d FB(=)f FA(v)g FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))29 b(to)e(equation)g (\(141\))o(,)h(one)f(can)h(see)f(that)1469 3619 y Fz(b)1455 3640 y FA(T)1504 3652 y Fy(1)1541 3640 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f FA(T)1891 3652 y Fy(1)1941 3640 y FB(\()q FA(v)e FB(+)d FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))71 3797 y(is)27 b(solution)g(of)1622 3896 y Fw(L)1679 3908 y Fx(")1729 3875 y Fz(b)1715 3896 y FA(T)1764 3908 y Fy(1)1824 3896 y FB(=)1932 3875 y Fz(b)1912 3896 y Fw(F)1993 3804 y Fz(\020)2043 3896 y FA(@)2087 3908 y Fx(v)2140 3875 y Fz(b)2127 3896 y FA(T)2176 3908 y Fy(1)2212 3804 y Fz(\021)2276 3896 y FA(;)1362 b FB(\(157\))71 4050 y(where)987 4129 y Fz(b)967 4150 y Fw(F)8 b FB(\()p FA(h)p FB(\)\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)1468 4129 y Fz(b)1449 4150 y FA(A)p FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b(+)1820 4129 y Fz(b)1803 4150 y FA(B)t FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(h)p FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))21 b(+)2417 4129 y Fz(b)2400 4150 y FA(C)6 b FB(\()p FA(h)p FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)14 b(v)s(;)g(\034)9 b FB(\))p FA(;)710 b FB(\(158\))71 4284 y(with)954 4420 y Fz(b)934 4441 y FA(A)p FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)d FA(A)14 b FB(\()q FA(v)22 b FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))1764 b(\(159\))947 4602 y Fz(b)929 4623 y FA(B)t FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)1307 4566 y FA(B)1370 4578 y Fy(1)1422 4566 y FB(\()p FA(v)c FB(+)d FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))21 b Fw(\000)d FA(B)2111 4578 y Fy(1)2148 4566 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))20 b(+)e FA(B)2503 4578 y Fy(2)2554 4566 y FB(\()p FA(v)k FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))p 1307 4604 1772 4 v 1963 4680 a(1)18 b(+)g FA(@)2150 4692 y Fx(v)2190 4680 y FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))3661 4623 y(\(160\))849 4833 y Fz(b)833 4854 y FA(C)d FB(\()p FA(w)r(;)14 b(v)s(;)g(\034)9 b FB(\))25 b(=)d FA(C)1377 4737 y Fz(\022)1657 4798 y FB(1)p 1448 4835 460 4 v 1448 4911 a(1)c(+)g FA(@)1635 4923 y Fx(v)1674 4911 y FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))1917 4854 y FA(w)r(;)14 b(v)22 b FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)2474 4737 y Fz(\023)2551 4854 y FA(:)1087 b FB(\(161\))71 5061 y(where)27 b(the)h(functions)g FA(A)p FB(\()p FA(u;)14 b(\034)9 b FB(\),)29 b FA(B)1183 5073 y Fy(1)1220 5061 y FB(\()p FA(u;)14 b(\034)9 b FB(\))28 b(and)g FA(B)1667 5073 y Fy(2)1704 5061 y FB(\()p FA(u;)14 b(\034)9 b FB(\))28 b(are)f(de\014ned)h(in)g(\(143\))o(,)f(\(144\))g (and)h(\(145\))o(.)195 5171 y(W)-7 b(e)27 b(lo)r(ok)f(for)656 5150 y Fz(b)642 5171 y FA(T)691 5183 y Fy(1)754 5171 y FB(b)n(y)h(using)f(a)g(\014xed)g(p)r(oin)n(t)h(argumen)n(t)e(for)h FA(@)2108 5183 y Fx(v)2162 5150 y Fz(b)2148 5171 y FA(T)2197 5183 y Fy(1)2260 5171 y FB(instead)h(of)2653 5150 y Fz(b)2639 5171 y FA(T)2688 5183 y Fy(1)2751 5171 y FB(itself.)37 b(Therefore,)26 b(w)n(e)g(lo)r(ok)g(for)g(a)71 5270 y(\014xed)i(p)r (oin)n(t)f(of)h(the)g(op)r(erator)p 1742 5303 68 4 v 1742 5370 a Fw(F)j FB(=)p 1920 5303 55 4 v 22 w Fw(G)1975 5382 y Fx(")2029 5370 y Fw(\016)2109 5349 y Fz(b)2089 5370 y Fw(F)7 b FA(;)1482 b FB(\(162\))71 5504 y(where)p 311 5437 V 27 w Fw(G)365 5516 y Fx(")428 5504 y FB(is)28 b(the)g(op)r(erator)e(in)i(\(149\))o(,)f(in)h(the)g(Banac)n(h)f(space)g Fw(H)2155 5518 y Fy(2)p Fx(\025;\032)2281 5498 y Fl(0)2281 5537 y Fu(0)2314 5518 y Fx(;\033)2406 5504 y FB(de\014ned)h(in)g (\(147\))o(.)p Black 1919 5753 a(57)p Black eop end %%Page: 58 58 TeXDict begin 58 57 bop Black Black Black 71 272 a Fp(Lemma)28 b(6.6.)p Black 37 w Fs(L)l(et)f FA(\032)777 242 y Fv(0)777 293 y Fy(0)841 272 y Fs(b)l(e)h(de\014ne)l(d)f(in)g(L)l(emma)g(6.5)i (and)e FA(")1943 284 y Fy(0)2003 272 y FA(>)c FB(0)j Fs(smal)t(l)i(enough.)39 b(Then,)28 b(for)g FA(")23 b Fw(2)h FB(\(0)p FA(;)14 b(")3353 284 y Fy(0)3389 272 y FB(\))28 b Fs(ther)l(e)f(exists)71 382 y(a)g(function)477 361 y Fz(b)463 382 y FA(T)512 394 y Fy(1)549 382 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))28 b Fs(de\014ne)l(d)f(in)g FA(D)1213 352 y Fx(u)1211 408 y Fv(1)p Fx(;\032)1331 388 y Fl(0)1331 426 y Fu(0)1379 382 y Fw(\002)12 b Ft(T)1511 394 y Fx(\033)1581 382 y Fs(such)27 b(that)g FA(@)1978 394 y Fx(v)2031 361 y Fz(b)2017 382 y FA(T)2066 394 y Fy(1)2126 382 y Fw(2)c(H)2274 396 y Fy(2)p Fx(\025;\032)2400 377 y Fl(0)2400 415 y Fu(0)2433 396 y Fx(;\033)2524 382 y Fs(is)k(a)g(\014xe)l(d)f(p)l(oint)h(of)h(the)e(op)l(er)l(ator)37 b FB(\(162\))o Fs(.)71 492 y(F)-6 b(urthermor)l(e,)30 b(ther)l(e)g(exists)f(a)h(c)l(onstant)f FA(b)1449 504 y Fy(1)1509 492 y FA(>)23 b FB(0)29 b Fs(such)h(that,)1485 570 y Fz(\015)1485 620 y(\015)1485 670 y(\015)1531 665 y FA(@)1575 677 y Fx(v)1629 644 y Fz(b)1615 665 y FA(T)1664 677 y Fy(1)1701 570 y Fz(\015)1701 620 y(\015)1701 670 y(\015)1747 724 y Fy(2)p Fx(\025;\032)1873 704 y Fl(0)1873 742 y Fu(0)1906 724 y Fx(;\033)1993 665 y Fw(\024)23 b FA(b)2117 677 y Fy(1)2154 665 y Fw(j)p FA(\026)p Fw(j)p FA(")2289 631 y Fx(\021)r Fy(+1)2413 665 y FA(:)p Black 71 876 a Fs(Pr)l(o)l(of.)p Black 43 w FB(It)e(is)g(straigh)n(tforw)n (ard)d(to)j(see)g(that)p 1465 810 68 4 v 21 w Fw(F)29 b FB(is)21 b(w)n(ell)g(de\014ned)g(from)g Fw(H)2332 890 y Fy(2)p Fx(\025;\032)2458 871 y Fl(0)2458 909 y Fu(0)2491 890 y Fx(;\033)2577 876 y FB(to)f(itself.)36 b(W)-7 b(e)21 b(are)f(going)g(to)h(pro)n(v)n(e)e(that)71 991 y(there)31 b(exists)f(a)g(constan)n(t)h FA(b)966 1003 y Fy(1)1031 991 y FA(>)d FB(0)i(suc)n(h)h(that)p 1570 924 V 31 w Fw(F)39 b FB(sends)p 1894 924 V 30 w FA(B)5 b FB(\()p FA(b)2030 1003 y Fy(1)2067 991 y Fw(j)p FA(\026)p Fw(j)p FA(")2202 960 y Fx(\021)r Fy(+1)2326 991 y FB(\))29 b Fw(\032)f(H)2550 1005 y Fy(2)p Fx(\025;\032)2676 985 y Fl(0)2676 1023 y Fu(0)2709 1005 y Fx(;\033)2804 991 y FB(to)j(itself)h(and)e(it)h(is)g(con)n(tractiv)n(e)71 1090 y(there.)195 1190 y(Let)26 b(us)f(\014rst)f(consider)p 935 1123 V 24 w Fw(F)8 b FB(\(0\).)36 b(F)-7 b(rom)25 b(the)g(de\014nition)h(of)p 1983 1123 V 25 w Fw(F)33 b FB(in)25 b(\(162\))f(and)h(the)h(de\014nition)f(of)3162 1169 y Fz(b)3142 1190 y Fw(F)33 b FB(in)25 b(\(158\))o(,)h(w)n(e)f(ha)n (v)n(e)71 1289 y(that)p 851 1322 V 851 1389 a Fw(F)8 b FB(\(0\)\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)p 1326 1322 55 4 v 23 w Fw(G)1380 1401 y Fx(")1430 1297 y Fz(\020)1499 1368 y(b)1479 1389 y FA(A)1541 1297 y Fz(\021)1605 1389 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)p 1906 1322 V 23 w Fw(G)1960 1401 y Fx(")1996 1389 y FB(\()p FA(A)p FB(\)\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b(+)p 2414 1322 V 18 w Fw(G)2469 1401 y Fx(")2518 1297 y Fz(\020)2587 1368 y(b)2568 1389 y FA(A)e Fw(\000)g FA(A)2793 1297 y Fz(\021)2857 1389 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(:)71 1541 y FB(The)30 b(\014rst)f(term)h(w)n(as)f(already) f(b)r(ounded)i(in)h(Lemma)e(6.3.)43 b(F)-7 b(or)29 b(the)h(second)f (one,)h(it)h(is)e(enough)h(to)f(use)h(mean)g(v)-5 b(alue)71 1641 y(theorem)27 b(and)g(Lemmas)g(6.3)g(and)h(6.5)e(to)i(b)r(ound)g FA(@)1715 1653 y Fx(u)1758 1641 y FA(A)g FB(and)g FA(g)i FB(resp)r(ectiv)n(ely)-7 b(,)27 b(to)g(obtain)1067 1793 y Fw(k)o FA(A)p FB(\()p FA(v)c FB(+)18 b FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(A)p FB(\()p FA(v)s(;)c(\034)9 b FB(\))p Fw(k)2091 1818 y Fy(2)p Fx(\025;\032)2217 1798 y Fl(0)2217 1836 y Fu(0)2250 1818 y Fx(;\033)2338 1793 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)2598 1759 y Fy(2)2635 1793 y FA(")2674 1759 y Fy(2)p Fx(\021)r Fy(+1)2831 1793 y FA(:)71 1956 y FB(Th)n(us,)27 b(applying)g(Lemma)h(6.2,)e(there)i (exists)f(constan)n(t)g FA(b)1909 1968 y Fy(1)1969 1956 y FA(>)22 b FB(0)28 b(suc)n(h)f(that)1473 2076 y Fz(\015)1473 2126 y(\015)p 1519 2080 68 4 v 21 x Fw(F)8 b FB(\(0\))1693 2076 y Fz(\015)1693 2126 y(\015)1739 2180 y Fy(2)p Fx(\025;\032)1865 2160 y Fl(0)1865 2198 y Fu(0)1898 2180 y Fx(;\033)1986 2147 y Fw(\024)2083 2091 y FA(b)2119 2103 y Fy(1)p 2083 2128 73 4 v 2099 2204 a FB(2)2166 2147 y Fw(j)p FA(\026)p Fw(j)p FA(")2301 2113 y Fx(\021)r Fy(+1)2425 2147 y FA(:)195 2339 y FB(Let)32 b(us)g(consider)e(no)n(w,)i FA(h)1035 2351 y Fy(1)1072 2339 y FA(;)14 b(h)1157 2351 y Fy(2)1224 2339 y Fw(2)p 1309 2272 68 4 v 30 w FA(B)t FB(\()p FA(b)1444 2351 y Fy(1)1481 2339 y Fw(j)p FA(\026)p Fw(j)p FA(")1616 2309 y Fx(\021)r Fy(+1)1741 2339 y FB(\))30 b Fw(2)g(H)1958 2353 y Fy(2)p Fx(\025;\032)2084 2333 y Fl(0)2084 2371 y Fu(0)2117 2353 y Fx(;\033)2181 2339 y FB(.)49 b(Then,)33 b(using)e(the)h(prop)r(erties)f(of)p 3359 2272 55 4 v 31 w Fw(G)3414 2351 y Fx(")3481 2339 y FB(in)h(Lemma)71 2461 y(6.2)27 b(and)g(the)h(de\014nition)g(of)993 2440 y Fz(b)973 2461 y Fw(F)35 b FB(in)28 b(\(158\))543 2568 y Fz(\015)543 2618 y(\015)p 589 2572 68 4 v 20 x Fw(F)8 b FB(\()p FA(h)737 2650 y Fy(2)774 2638 y FB(\))19 b Fw(\000)p 908 2572 V 18 w(F)8 b FB(\()p FA(h)1056 2650 y Fy(1)1093 2638 y FB(\))1125 2568 y Fz(\015)1125 2618 y(\015)1171 2672 y Fy(2)p Fx(\025;\032)1297 2652 y Fl(0)1297 2690 y Fu(0)1331 2672 y Fx(;\033)1418 2638 y Fw(\024)23 b FA(K)1596 2543 y Fz(\015)1596 2593 y(\015)1596 2643 y(\015)1662 2617 y(b)1642 2638 y Fw(F)8 b FB(\()p FA(h)1790 2650 y Fy(2)1827 2638 y FB(\))19 b Fw(\000)1981 2617 y Fz(b)1961 2638 y Fw(F)8 b FB(\()p FA(h)2109 2650 y Fy(1)2146 2638 y FB(\))2178 2543 y Fz(\015)2178 2593 y(\015)2178 2643 y(\015)2225 2696 y Fy(2)p Fx(\025;\032)2351 2676 y Fl(0)2351 2714 y Fu(0)2384 2696 y Fx(;\033)1418 2843 y Fw(\024)23 b FA(K)1596 2748 y Fz(\015)1596 2798 y(\015)1596 2847 y(\015)1659 2822 y(b)1642 2843 y FA(B)g Fw(\001)18 b FB(\()p FA(h)1849 2855 y Fy(2)1905 2843 y Fw(\000)g FA(h)2036 2855 y Fy(1)2073 2843 y FB(\))h(+)2223 2822 y Fz(b)2207 2843 y FA(C)6 b FB(\()p FA(h)2352 2855 y Fy(2)2389 2843 y FA(;)14 b(u;)g(\034)9 b FB(\))19 b Fw(\000)2707 2822 y Fz(b)2690 2843 y FA(C)6 b FB(\()p FA(h)2835 2855 y Fy(1)2873 2843 y FA(;)14 b(u;)g(\034)9 b FB(\))3072 2748 y Fz(\015)3072 2798 y(\015)3072 2847 y(\015)3118 2901 y Fy(2)p Fx(\025;\032)3244 2881 y Fl(0)3244 2919 y Fu(0)3277 2901 y Fx(;\033)3356 2843 y FA(:)71 3062 y FB(T)-7 b(aking)23 b(in)n(to)g(accoun)n(t)g(the)i(de\014nitions) f(of)1457 3041 y Fz(b)1439 3062 y FA(B)k FB(and)1705 3041 y Fz(b)1688 3062 y FA(C)i FB(in)24 b(\(160\))f(and)h(\(161\))f (resp)r(ectiv)n(ely)g(and)h(applying)f(Lemmas)g(6.1,)71 3161 y(6.3)k(and)g(6.5,)g(w)n(e)g(obtain)1011 3244 y Fz(\015)1011 3294 y(\015)p 1057 3248 V 20 x Fw(F)8 b FB(\()p FA(h)1205 3326 y Fy(2)1242 3314 y FB(\))19 b Fw(\000)p 1376 3248 V 18 w(F)8 b FB(\()p FA(h)1524 3326 y Fy(1)1561 3314 y FB(\))1593 3244 y Fz(\015)1593 3294 y(\015)1639 3347 y Fy(2)p Fx(\025;\032)1765 3327 y Fl(0)1765 3366 y Fu(0)1799 3347 y Fx(;\033)1886 3314 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2186 3280 y Fx(\021)r Fy(+1)2309 3314 y Fw(k)p FA(h)2399 3326 y Fy(2)2454 3314 y Fw(\000)19 b FA(h)2586 3326 y Fy(1)2623 3314 y Fw(k)2665 3328 y Fy(2)p Fx(\025;\032)2791 3309 y Fl(0)2791 3347 y Fu(0)2823 3328 y Fx(;\033)2888 3314 y FA(:)71 3500 y FB(Therefore,)h(reducing)g FA(")f FB(if)i(necessary)-7 b(,)20 b(Lip)p 1420 3434 V Fw(F)31 b(\024)22 b FB(1)p FA(=)p FB(2)d(and)h(therefore)p 2238 3434 V 19 w Fw(F)28 b FB(is)20 b(con)n(tractiv)n(e)e(from)i(the)g(ball)p 3299 3434 V 20 w FA(B)t FB(\()p FA(b)3434 3512 y Fy(1)3471 3500 y Fw(j)p FA(\026)p Fw(j)p FA(")3606 3470 y Fx(\021)r Fy(+1)3730 3500 y FB(\))k Fw(\032)71 3600 y(H)141 3614 y Fy(2)p Fx(\025;\032)267 3594 y Fl(0)267 3632 y Fu(0)300 3614 y Fx(;\033)392 3600 y FB(in)n(to)j(itself,)i(and)e(it)h(has)f(a)g (unique)h(\014xed)g(p)r(oin)n(t)g FA(h)1982 3570 y Fv(\003)2020 3600 y FB(.)37 b(Since)27 b(it)h(satis\014es)1458 3769 y Fw(j)p FA(h)1529 3734 y Fv(\003)1567 3769 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p Fw(j)25 b(\024)d FA(b)1927 3781 y Fy(1)1964 3769 y Fw(j)p FA(\026)p Fw(j)p FA(")2099 3734 y Fx(\021)r Fy(+1)2224 3769 y FA(e)2263 3734 y Fy(2)p Fx(\025)p Fy(Re)11 b Fx(v)71 3932 y FB(for)27 b(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b Fw(2)g FA(D)561 3902 y Fx(u)559 3958 y Fv(1)p Fx(;\032)679 3938 y Fl(0)679 3976 y Fu(0)734 3932 y Fw(\002)18 b Ft(T)872 3944 y Fx(\033)917 3932 y FB(,)28 b(w)n(e)f(can)g(tak)n(e)1436 3911 y Fz(b)1422 3932 y FA(T)1471 3944 y Fy(1)1535 3932 y FB(as)1472 4134 y Fz(b)1458 4155 y FA(T)1507 4167 y Fy(1)1544 4155 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)1845 4042 y Fz(Z)1928 4063 y Fx(v)1891 4231 y Fv(\0001)2027 4155 y FA(h)2075 4121 y Fv(\003)2113 4155 y FB(\()p FA(w)r(;)14 b(\034)9 b FB(\))14 b FA(dw)r(:)p 3790 4355 4 57 v 3794 4303 50 4 v 3794 4355 V 3843 4355 4 57 v 195 4515 a FB(Finally)-7 b(,)26 b(to)g(pro)n(v)n(e)d(Prop)r(osition)h(6.4)h(from)g(Lemma)g(6.6,) g(it)h(is)g(enough)f(to)g(consider)f(the)i(c)n(hange)e FA(v)j FB(=)22 b FA(u)14 b FB(+)g FA(h)p FB(\()p FA(u;)g(\034)9 b FB(\))71 4625 y(obtained)27 b(in)h(Lemma)f(6.5,)g(tak)n(e)g FA(T)1191 4637 y Fy(1)1228 4625 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)1547 4604 y Fz(b)1533 4625 y FA(T)1582 4637 y Fy(1)1619 4625 y FB(\()p FA(u)18 b FB(+)g FA(h)p FB(\()p FA(u;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))29 b(and)f(increase)e(sligh)n(tly)h FA(b)2988 4637 y Fy(1)3053 4625 y FB(if)h(necessary)-7 b(.)71 4853 y Fq(6.2)112 b(Lo)s(cal)38 b(in)m(v)-6 b(arian)m(t)39 b(manifolds)g(in)f(the)f (parab)s(olic)i(case)71 5006 y FB(W)-7 b(e)40 b(dev)n(ote)g(this)g (section)g(to)g(pro)n(v)n(e)f(the)h(existence)g(of)h(suitable)f (represen)n(tations)e(of)i(the)h(unstable)f(and)g(stable)71 5106 y(in)n(v)-5 b(arian)n(t)30 b(manifolds)h(in)h(the)g(domains)f FA(D)1454 5075 y Fx(u)1452 5126 y Fv(1)p Fx(;\032)1597 5106 y Fw(\002)21 b Ft(T)1738 5118 y Fx(\033)1815 5106 y FB(and)31 b FA(D)2051 5075 y Fx(s)2049 5126 y Fv(1)p Fx(;\032)2194 5106 y Fw(\002)21 b Ft(T)2335 5118 y Fx(\033)2411 5106 y FB(resp)r(ectiv)n(ely)-7 b(,)32 b(under)f(the)h(h)n(yp)r (otheses)f(that)71 5205 y(the)c(unp)r(erturb)r(ed)f(Hamiltonian)h (system)f(has)f(a)h(parab)r(olic)f(critical)h(p)r(oin)n(t)g(at)h(the)f (origin.)36 b(W)-7 b(e)26 b(pro)r(ceed)g(as)g(w)n(e)g(ha)n(v)n(e)71 5305 y(done)i(in)h(Section)f(6.1)g(for)g(the)h(h)n(yp)r(erb)r(olic)f (case,)g(that)h(is,)f(solving)g(equation)g(\(141\))o(.)39 b(Let)29 b(us)f(p)r(oin)n(t)h(out)g(that)f(in)h(the)71 5404 y(parab)r(olic)i(case,)j(b)n(y)e(Hyp)r(othesis)h Fp(HP4.2)p FB(,)g(the)g(p)r(erturbation)f(is)h(tak)n(en)f(in)h(suc)n(h) f(a)h(w)n(a)n(y)e(that)i(the)g(p)r(erio)r(dic)g(orbit)71 5504 y(remains)27 b(at)g(the)h(origin.)p Black 1919 5753 a(58)p Black eop end %%Page: 59 59 TeXDict begin 59 58 bop Black Black 71 272 a Fp(6.2.1)94 b(Banac)m(h)33 b(spaces)f(and)g(tec)m(hnical)h(lemmas)71 425 y FB(Giv)n(en)27 b FA(\013)d Fw(\025)e FB(0,)28 b FA(\032)23 b Fw(\025)f FB(0)27 b(and)h(an)f(analytic)g(function)h FA(h)23 b FB(:)g FA(D)1896 395 y Fx(u)1894 446 y Fv(1)p Fx(;\032)2042 425 y Fw(!)g Ft(C)p FB(,)k(w)n(e)h(de\014ne)1489 597 y Fw(k)p FA(h)p Fw(k)1621 609 y Fx(\013;\032)1745 597 y FB(=)84 b(sup)1832 667 y Fx(u)p Fv(2)p Fx(D)1972 650 y Fm(u)1970 684 y Fl(1)p Fm(;\032)2094 597 y Fw(j)p FA(u)2165 563 y Fx(\013)2212 597 y FA(h)p FB(\()p FA(u)p FB(\))p Fw(j)14 b FA(:)71 838 y FB(Moreo)n(v)n(er)j(for)i(analytic)g (functions)h FA(h)j FB(:)g FA(D)1393 808 y Fx(u)1391 859 y Fv(1)p Fx(;\032)1518 838 y Fw(\002)s Ft(T)1641 850 y Fx(\033)1708 838 y Fw(!)g Ft(C)d FB(whic)n(h)f(are)g(2)p FA(\031)s FB(-p)r(erio)r(dic)g(in)h FA(\034)9 b FB(,)22 b(w)n(e)d(de\014ne)h(the)g(corresp)r(onding)71 938 y(F)-7 b(ourier)26 b(norm)1434 1038 y Fw(k)p FA(h)p Fw(k)1566 1050 y Fx(\013;\032;\033)1749 1038 y FB(=)1837 959 y Fz(X)1837 1137 y Fx(k)q Fv(2)p Fn(Z)1971 942 y Fz(\015)1971 992 y(\015)1971 1042 y(\015)2017 1038 y FA(h)2065 1003 y Fy([)p Fx(k)q Fy(])2143 942 y Fz(\015)2143 992 y(\015)2143 1042 y(\015)2190 1096 y Fx(\013;\032)2305 1038 y FA(e)2344 1003 y Fv(j)p Fx(k)q Fv(j)p Fx(\033)2464 1038 y FA(:)71 1260 y FB(W)-7 b(e)28 b(in)n(tro)r(duce,)f(th)n(us,)h(the)g(follo)n (wing)e(function)j(space)873 1459 y Fw(P)931 1471 y Fx(\013;\032;\033) 1116 1459 y FB(=)22 b Fw(f)p FA(h)h FB(:)g FA(D)1433 1425 y Fx(u)1431 1479 y Fv(1)p Fx(;\032)1574 1459 y Fw(\002)18 b Ft(T)1712 1471 y Fx(\033)1780 1459 y Fw(!)23 b Ft(C)p FB(;)41 b(real-analytic)n FA(;)14 b Fw(k)p FA(h)p Fw(k)2629 1471 y Fx(\013;\032;\033)2813 1459 y FA(<)22 b Fw(1g)p FA(;)613 b FB(\(163\))71 1602 y(whic)n(h)27 b(can)h(b)r(e)g(c)n(hec)n (k)n(ed)e(that)i(is)f(a)h(Banac)n(h)e(space)h(for)g(an)n(y)g(\014xed)g FA(\013)d Fw(\025)e FB(0.)195 1702 y(In)28 b(the)g(next)g(lemma,)g(w)n (e)f(state)g(some)g(prop)r(erties)g(of)g(these)h(Banac)n(h)e(spaces.)p Black 71 1859 a Fp(Lemma)31 b(6.7.)p Black 40 w Fs(The)g(fol)t(lowing)h (statements)d(hold:)p Black 169 2017 a(1.)p Black 42 w(If)i FA(\013)419 2029 y Fy(1)479 2017 y Fw(\025)23 b FA(\013)620 2029 y Fy(2)680 2017 y Fw(\025)g FB(0)p Fs(,)29 b(then)h Fw(P)1107 2029 y Fx(\013)1150 2037 y Fu(1)1182 2029 y Fx(;\032;\033)1324 2017 y Fw(\032)23 b(P)1470 2029 y Fx(\013)1513 2037 y Fu(2)1545 2029 y Fx(;\032;\033)1693 2017 y Fs(and)1673 2189 y Fw(k)p FA(h)p Fw(k)1805 2201 y Fx(\013)1848 2209 y Fu(2)1879 2201 y Fx(;\032;\033)2021 2189 y Fw(\024)f(k)p FA(h)p Fw(k)2240 2201 y Fx(\013)2283 2209 y Fu(1)2315 2201 y Fx(;\032;\033)2433 2189 y FA(:)p Black 169 2392 a Fs(2.)p Black 42 w(If)31 b FA(\013)419 2404 y Fy(1)456 2392 y FA(;)14 b(\013)546 2404 y Fy(2)606 2392 y Fw(\025)23 b FB(0)p Fs(,)30 b(then,)g(for)g FA(h)23 b Fw(2)h(P)1341 2404 y Fx(\013)1384 2412 y Fu(1)1416 2404 y Fx(;\032;\033)1564 2392 y Fs(and)30 b FA(g)c Fw(2)d(P)1927 2404 y Fx(\013)1970 2412 y Fu(2)2003 2404 y Fx(;\032;\033)2121 2392 y Fs(,)30 b(we)h(have)f(that)g FA(hg)c Fw(2)d(P)2910 2404 y Fx(\013)2953 2412 y Fu(1)2985 2404 y Fy(+)p Fx(\013)3079 2412 y Fu(2)3112 2404 y Fx(;\032;\033)3260 2392 y Fs(and)1428 2564 y Fw(k)p FA(hg)s Fw(k)1603 2576 y Fx(\013)1646 2584 y Fu(1)1677 2576 y Fy(+)p Fx(\013)1771 2584 y Fu(2)1804 2576 y Fx(;\032;\033)1945 2564 y Fw(\024)g(k)p FA(h)p Fw(k)2165 2576 y Fx(\013)2208 2584 y Fu(1)2239 2576 y Fx(;\032;\033)2358 2564 y Fw(k)p FA(g)s Fw(k)2485 2576 y Fx(\013)2528 2584 y Fu(2)2559 2576 y Fx(;\032;\033)2678 2564 y FA(:)195 2767 y FB(As)42 b(in)g(Section)f(6.1,)j(w)n(e)d(need)h (to)f(use)g(the)h(op)r(erators)d Fw(G)2124 2779 y Fx(")2202 2767 y FB(and)2393 2746 y(\026)2377 2767 y Fw(G)2426 2779 y Fx(")2503 2767 y FB(formally)i(de\014ned)h(in)f(\(148\))g(and)g (\(149\))71 2867 y(resp)r(ectiv)n(ely)-7 b(.)p Black 71 3024 a Fp(Lemma)43 b(6.8.)p Black 47 w Fs(The)d(op)l(er)l(ators)g Fw(G)1216 3036 y Fx(")1291 3024 y Fs(and)p 1462 2957 55 4 v 40 w Fw(G)1516 3036 y Fx(")1591 3024 y Fs(acting)g(on)f(the)g (sp)l(ac)l(es)h Fw(P)2442 3036 y Fx(\013;\032;\033)2643 3024 y Fs(with)g FA(\013)h(>)e FB(1)g Fs(satisfy)h(the)g(fol)t(lowing) 71 3124 y(pr)l(op)l(erties.)p Black 169 3281 a(1.)p Black 42 w(F)-6 b(or)31 b(any)h FA(\013)25 b(>)g FB(1)p Fs(,)31 b Fw(G)910 3293 y Fx(")971 3281 y FB(:)25 b Fw(P)1077 3293 y Fx(\013;\032;\033)1263 3281 y Fw(!)g(P)1429 3293 y Fx(\013)p Fv(\000)p Fy(1)p Fx(;\032;\033)1707 3281 y Fs(is)31 b(wel)t(l)h(de\014ne)l(d)f(and)g(line)l(ar)h(c)l(ontinuous.) 41 b(Mor)l(e)l(over,)33 b(c)l(ommutes)278 3381 y(with)e FA(@)503 3393 y Fx(u)576 3381 y Fs(and)f Fw(L)794 3393 y Fx(")848 3381 y Fw(\016)18 b(G)957 3393 y Fx(")1016 3381 y FB(=)23 b(Id)p Fs(.)p Black 169 3543 a(2.)p Black 42 w(If)31 b FA(h)22 b Fw(2)i(P)573 3555 y Fx(\013;\032;\033)764 3543 y Fs(for)31 b(some)f FA(\013)23 b(>)g FB(1)p Fs(,)30 b(then)1549 3715 y Fw(kG)1640 3727 y Fx(")1676 3715 y FB(\()p FA(h)p FB(\))p Fw(k)1830 3740 y Fx(\013)p Fv(\000)p Fy(1)p Fx(;\032;\033)2100 3715 y Fw(\024)22 b FA(K)6 b Fw(k)p FA(h)p Fw(k)2396 3727 y Fx(\013;\032;\033)2556 3715 y FA(:)278 3887 y Fs(F)-6 b(urthermor)l(e,)30 b(if)h FA(h)23 b Fw(2)g(P)1071 3899 y Fx(\013;\032;\033)1263 3887 y Fs(for)30 b(some)g FA(\013)24 b(>)e FB(0)30 b Fs(and)g Fw(h)p FA(h)p Fw(i)23 b FB(=)g(0)p Fs(,)30 b(then)1573 4059 y Fw(k)o(G)1663 4071 y Fx(")1699 4059 y FB(\()p FA(h)p FB(\))p Fw(k)1853 4084 y Fx(\013;\032;\033)2038 4059 y Fw(\024)22 b FA(K)6 b(")p Fw(k)p FA(h)p Fw(k)2373 4071 y Fx(\013;\032;\033)2533 4059 y FA(:)p Black 169 4280 a Fs(3.)p Black 42 w(If)31 b FA(h)22 b Fw(2)i(P)573 4292 y Fx(\013;\032;\033)764 4280 y Fs(for)31 b(some)f FA(\013)23 b Fw(\025)g FB(1)p Fs(,)30 b(then)p 1554 4213 V 29 w Fw(G)1609 4292 y Fx(")1644 4280 y FB(\()p FA(h)p FB(\))24 b Fw(2)f(P)1916 4292 y Fx(\013;\032;\033)2107 4280 y Fs(and)1585 4381 y Fz(\015)1585 4431 y(\015)p 1631 4385 V 21 x Fw(G)1685 4464 y Fx(")1721 4452 y FB(\()p FA(h)p FB(\))1833 4381 y Fz(\015)1833 4431 y(\015)1879 4485 y Fx(\013;\032;\033)2064 4452 y Fw(\024)g FA(K)6 b Fw(k)p FA(h)p Fw(k)2361 4464 y Fx(\013;\032;\033)2521 4452 y FA(:)195 4671 y FB(W)-7 b(e)24 b(also)f(state)g(a)g(tec)n (hnical)g(lemma)g(ab)r(out)h(prop)r(erties)e(of)h(the)h(functions)g FA(A)p FB(,)h FA(B)2744 4683 y Fy(1)2804 4671 y FB(and)f FA(C)29 b FB(de\014ned)24 b(in)g(\(143\))o(,)g(\(144\))71 4770 y(and)h(\(146\))g(resp)r(ectiv)n(ely)-7 b(.)36 b(Let)26 b(us)g(p)r(oin)n(t)g(out)f(that)h(no)n(w)g(the)g(function)g FA(B)2412 4782 y Fy(2)2475 4770 y FB(de\014ned)g(in)g(\(145\))f (satis\014es)g FA(B)3436 4782 y Fy(2)3497 4770 y FB(=)d(0)k(since,)71 4870 y(b)n(y)h(h)n(yp)r(othesis,)g(the)h(p)r(erturbation)f(\014xes)h (the)g(p)r(erio)r(dic)f(orbit)g(at)h(the)g(origin.)195 4970 y(W)-7 b(e)28 b(\014rst)g(\014x)f FA(\032)670 4982 y Fy(0)731 4970 y FA(>)22 b FB(0)27 b(suc)n(h)h(that)g FA(p)1297 4982 y Fy(0)1334 4970 y FB(\()p FA(u)p FB(\))f(do)r(es)h(not) f(v)-5 b(anish)28 b(in)f FA(D)2232 4939 y Fx(u)2230 4990 y Fv(1)p Fx(;\032)2350 4998 y Fu(0)2415 4970 y FB(and)g(w)n(e)g (de\014ne)h(the)g(constan)n(t)1655 5192 y FA(\013)1708 5204 y Fy(0)1768 5192 y FB(=)1928 5136 y(2)p FA(n)p 1866 5173 216 4 v 1866 5249 a(m)18 b Fw(\000)g FB(2)2114 5192 y FA(>)23 b FB(1)p FA(;)1394 b FB(\(164\))71 5399 y(where)30 b FA(m)h FB(is)f(the)h(order)f(of)g(the)h(p)r(oten)n(tial)g(\(11\))f (and)h FA(n)f FB(is)h(the)g(order)e(of)i(the)g(p)r(erturbation)f (\(8\).)46 b(W)-7 b(e)31 b(observ)n(e)e(that)71 5498 y FA(q)108 5510 y Fy(0)145 5498 y FB(\()p FA(u)p FB(\))23 b Fw(2)h(P)474 5496 y Fu(2)p 426 5505 125 3 v 426 5539 a Fm(m)p Fl(\000)p Fu(2)561 5519 y Fx(;\032;\033)707 5498 y FB(and)j FA(p)910 5510 y Fy(0)947 5498 y FB(\()p FA(u)p FB(\))c Fw(2)h(P)1265 5488 y Fm(m)p 1229 5497 V 1229 5531 a(m)p Fl(\000)p Fu(2)1363 5510 y Fx(;\032;\033)1509 5498 y FB(for)j(an)n(y)g FA(\032)h FB(big)f(enough)g(and)g(an)n(y)g FA(\033)g(>)22 b FB(0.)p Black 1919 5753 a(59)p Black eop end %%Page: 60 60 TeXDict begin 60 59 bop Black Black Black 71 272 a Fp(Lemma)39 b(6.9.)p Black 44 w Fs(L)l(et)d(us)g(c)l(onsider)h FA(\032)d(>)g(\032) 1429 284 y Fy(0)1466 272 y Fs(.)58 b(Then,)39 b(the)d(functions)g FA(A)p Fs(,)i FA(B)2498 284 y Fy(1)2572 272 y Fs(and)e FA(C)43 b Fs(de\014ne)l(d)36 b(in)43 b FB(\(143\))o Fs(,)38 b FB(\(144\))d Fs(and)71 372 y FB(\(146\))29 b Fs(satisfy)i(the)e(fol)t (lowing)k(pr)l(op)l(erties,)p Black 169 538 a(1.)p Black 42 w FA(A)24 b Fw(2)f(P)500 550 y Fx(\013)543 558 y Fu(0)575 550 y Fx(;\032;\033)724 538 y Fs(and)30 b FA(@)929 550 y Fx(u)972 538 y FA(A)24 b Fw(2)f(P)1194 550 y Fx(\013)1237 558 y Fu(0)1270 550 y Fy(+1)p Fx(;\032;\033)1472 538 y Fs(.)39 b(Mor)l(e)l(over,)31 b Fw(h)p FA(A)p Fw(i)24 b FB(=)f Fw(h)p FA(@)2240 550 y Fx(u)2284 538 y FA(A)p Fw(i)g FB(=)g(0)29 b Fs(and)1046 727 y Fw(k)p FA(@)1132 739 y Fx(u)1175 727 y FA(A)p Fw(k)1279 751 y Fx(\013)1322 759 y Fu(0)1355 751 y Fy(+1)p Fx(;\032;\033)1580 727 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1880 696 y Fx(\021)1920 727 y FA(;)2026 656 y Fz(\015)2026 705 y(\015)p 2072 659 55 4 v 21 x Fw(G)2126 739 y Fx(")2162 727 y FB(\()p FA(A)p FB(\))2288 656 y Fz(\015)2288 705 y(\015)2335 759 y Fx(\013)2378 767 y Fu(0)2410 759 y Fy(+1)p Fx(;\032;\033)2636 727 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2936 696 y Fx(\021)r Fy(+1)3059 727 y FA(:)3661 733 y FB(\(165\))p Black 169 948 a Fs(2.)p Black 42 w FA(B)341 960 y Fy(1)402 948 y Fw(2)23 b(P)548 947 y Fu(2)p Fm(n)p Fl(\000)p Fm(m)p Fl(\000)p Fu(2)p 548 960 234 3 v 603 993 a Fm(m)p Fl(\000)p Fu(2)792 973 y Fx(;\032;\033)940 948 y Fs(and)30 b FA(@)1145 960 y Fx(u)1189 948 y FA(B)1252 960 y Fy(1)1312 948 y Fw(2)23 b(P)1458 947 y Fu(2)p Fm(n)p Fl(\000)p Fm(m)p Fl(\000)p Fu(2)p 1458 960 V 1513 993 a Fm(m)p Fl(\000)p Fu(2)1702 973 y Fy(+1)p Fx(;\032;\033)1905 948 y Fs(.)38 b(Mor)l(e)l(over,)32 b(they)e(satisfy)996 1161 y Fw(k)p FA(B)1101 1173 y Fy(1)1138 1161 y Fw(k)1190 1160 y Fu(2)p Fm(n)p Fl(\000)p Fm(m)p Fl(\000)p Fu(2)p 1189 1173 V 1244 1206 a Fm(m)p Fl(\000)p Fu(2)1433 1186 y Fx(;\032;\033)1575 1161 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1874 1131 y Fx(\021)1914 1161 y FA(;)83 b Fw(k)p FA(@)2106 1173 y Fx(u)2149 1161 y FA(B)2212 1173 y Fy(1)2249 1161 y Fw(k)2301 1160 y Fu(2)p Fm(n)p Fl(\000)p Fm(m)p Fl(\000)p Fu(2)p 2301 1173 V 2356 1206 a Fm(m)p Fl(\000)p Fu(2)2545 1186 y Fy(+1)p Fx(;\032;\033)2770 1161 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")3070 1131 y Fx(\021)3110 1161 y FA(:)3661 1173 y FB(\(166\))p Black 169 1400 a Fs(3.)p Black 42 w(L)l(et)30 b(us)f(c)l(onsider)i FA(h)906 1412 y Fy(1)943 1400 y FA(;)14 b(h)1028 1412 y Fy(2)1088 1400 y Fw(2)23 b FA(B)t FB(\()p FA(\027)5 b FB(\))24 b Fw(\032)f(P)1513 1412 y Fx(\013)1556 1420 y Fu(0)1588 1412 y Fy(+1)p Fx(;\032;\033)1821 1400 y Fs(with)30 b FA(\027)e Fw(\034)23 b FB(1)p Fs(.)39 b(Then,)1006 1582 y Fw(k)p FA(C)6 b FB(\()p FA(h)1193 1594 y Fy(2)1231 1582 y FA(;)14 b(u;)g(\034)9 b FB(\))18 b Fw(\000)g FA(C)6 b FB(\()p FA(h)1676 1594 y Fy(1)1714 1582 y FA(;)14 b(u;)g(\034)9 b FB(\))p Fw(k)1955 1607 y Fx(\013)1998 1615 y Fu(0)2030 1607 y Fy(+1)p Fx(;\032;\033)2256 1582 y Fw(\024)23 b FA(K)6 b(\027)f Fw(k)p FA(h)2557 1594 y Fy(2)2611 1582 y Fw(\000)19 b FA(h)2743 1594 y Fy(1)2780 1582 y Fw(k)2822 1594 y Fx(\013)2865 1602 y Fu(0)2897 1594 y Fy(+1)p Fx(;\032;\033)3099 1582 y FA(:)p Black 71 1798 a Fs(Pr)l(o)l(of.)p Black 43 w FB(W)-7 b(e)25 b(pro)n(v)n(e)f(the)h(lemma)g(in)g(the)h(p)r (olynomial)e(case.)35 b(The)25 b(trigonometric)f(one)h(can)f(b)r(e)i (done)e(analogously)-7 b(.)34 b(F)-7 b(or)71 1898 y(the)29 b(\014rst)f(statemen)n(t,)h(let)g(us)g(p)r(oin)n(t)f(out)h(that)g(in)g (the)g(parab)r(olic)e(case)h(the)h(p)r(erio)r(dic)f(orbit)g(is)h(lo)r (cated)f(at)h(the)g(origin)71 1997 y(b)n(y)e(Hyp)r(othesis)h Fp(HP4.2)p FB(.)36 b(Then)1334 2097 y FA(A)p FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)f Fw(\000)p FA(\026")1856 2063 y Fx(\021)1896 2097 y FA(H)1965 2109 y Fy(1)2002 2097 y FB(\()p FA(q)2071 2109 y Fy(0)2109 2097 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2300 2109 y Fy(0)2337 2097 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))p FA(;)71 2246 y FB(where)36 b FA(H)389 2258 y Fy(1)464 2246 y FB(is)h(the)g(function)h(de\014ned)f(in)g(\(8\))g(and)g(has)g(zero)f (mean.)65 b(On)36 b(the)i(other)e(hand,)j(it)f(is)f(clear)f(that)h(the) 71 2346 y(monomial)27 b(with)h(lo)n(w)n(est)e(order)h(as)f(Re)14 b FA(u)23 b Fw(!)g FB(+)p Fw(1)k FB(corresp)r(onds)f(to)h FA(a)2276 2358 y Fx(n)p Fy(0)2355 2346 y FA(q)2395 2316 y Fx(n)2392 2367 y Fy(0)2440 2346 y FB(\()p FA(u)p FB(\))h(whic)n(h)f (b)r(eha)n(v)n(es)g(as)1605 2568 y FA(a)1649 2580 y Fx(n)p Fy(0)1727 2568 y FB(\()p FA(\034)9 b FB(\))p FA(q)1876 2534 y Fx(n)1873 2588 y Fy(0)1923 2568 y FB(\()p FA(u)p FB(\))23 b Fw(\030)2199 2512 y FB(1)p 2156 2549 128 4 v 2156 2625 a FA(u)2204 2601 y Fx(\013)2247 2609 y Fu(0)2293 2568 y FA(:)71 2774 y FB(Then)28 b FA(A)23 b Fw(2)g(P)509 2786 y Fx(\013)552 2794 y Fu(0)585 2786 y Fx(;\032;\033)703 2774 y FB(,)28 b(that)g(implies)g FA(@)1260 2786 y Fx(u)1303 2774 y FA(A)c Fw(2)f(P)1525 2786 y Fx(\013)1568 2794 y Fu(0)1600 2786 y Fy(+1)p Fx(;\032;\033)1831 2774 y FB(and)1512 2956 y Fw(k)p FA(@)1598 2968 y Fx(u)1641 2956 y FA(A)p Fw(k)1745 2981 y Fx(\013)1788 2989 y Fu(0)1821 2981 y Fy(+1)p Fx(;\032;\033)2046 2956 y Fw(\024)g FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2346 2922 y Fx(\021)2386 2956 y FA(:)71 3139 y FB(Moreo)n(v)n(er,)25 b(b)n(y)i(Lemma)g(6.8,)1047 3251 y Fz(\015)1047 3301 y(\015)p 1093 3255 55 4 v 21 x Fw(G)1147 3334 y Fx(")1183 3322 y FB(\()p FA(A)p FB(\))1309 3251 y Fz(\015)1309 3301 y(\015)1356 3355 y Fx(\013)1399 3363 y Fu(0)1431 3355 y Fy(+1)p Fx(;\032;\033)1657 3322 y FB(=)c Fw(k)o(G)1835 3334 y Fx(")1871 3322 y FB(\()p FA(@)1947 3334 y Fx(u)1991 3322 y FA(A)p FB(\))p Fw(k)2127 3347 y Fx(\013)2170 3355 y Fu(0)2202 3347 y Fy(+1)p Fx(;\032;\033)2428 3322 y Fw(\024)g FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2728 3287 y Fx(\021)r Fy(+1)2851 3322 y FA(:)71 3515 y FB(F)-7 b(or)27 b(the)h(second)f(statemen)n(t,)g(let)h(us)g(recall)f(that)1201 3774 y FA(B)1264 3786 y Fy(1)1301 3774 y FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b(=)g Fw(\000)p FA(\026")1760 3740 y Fx(\021)1883 3671 y(N)1852 3696 y Fz(X)1814 3872 y Fx(i)p Fy(+)p Fx(j)s Fy(=)p Fx(n)1854 3932 y(j)s Fv(\025)p Fy(1)2025 3774 y FA(a)2069 3786 y Fx(ij)2127 3774 y FB(\()p FA(\034)9 b FB(\))p FA(q)2276 3740 y Fx(i)2273 3795 y Fy(0)2312 3774 y FB(\()p FA(u)p FB(\))p FA(p)2466 3735 y Fx(j)s Fv(\000)p Fy(2)2466 3797 y(0)2586 3774 y FB(\()p FA(u)p FB(\))p FA(:)71 4101 y FB(As)28 b(Re)13 b FA(u)23 b Fw(!)g(\0001)p FB(,)k(the)h(monomials)f(of)h FA(B)1397 4113 y Fy(1)1461 4101 y FB(b)r(eha)n(v)n(e)f(as)1134 4300 y FA(a)1178 4312 y Fx(ij)1236 4300 y FB(\()p FA(\034)9 b FB(\))p FA(q)1385 4266 y Fx(i)1382 4321 y Fy(0)1421 4300 y FB(\()p FA(u)p FB(\))p FA(p)1575 4260 y Fx(j)s Fv(\000)p Fy(2)1575 4322 y(0)1694 4300 y FB(\()p FA(u)p FB(\))24 b Fw(\030)e FA(u)1965 4264 y Fv(\000)2017 4270 y FB(\()2107 4242 y Fu(2)p 2059 4251 125 3 v 2059 4284 a Fm(m)p Fl(\000)p Fu(2)2193 4264 y Fx(i)p Fy(+)2267 4270 y FB(\()2358 4242 y Fu(2)p 2310 4251 V 2310 4284 a Fm(m)p Fl(\000)p Fu(2)2444 4264 y Fy(+1)2528 4270 y FB(\))2560 4264 y Fy(\()p Fx(j)s Fv(\000)p Fy(2\))2727 4270 y FB(\))2764 4300 y FA(:)71 4483 y FB(T)-7 b(aking)27 b(in)n(to)g(accoun)n(t)g(that)h(2)p FA(n)17 b Fw(\000)h FB(2)23 b Fw(\025)g FA(m)k FB(b)n(y)g(Hyp)r(othesis)h Fp(HP5)f FB(and)h(that)g FA(i)18 b FB(+)g FA(j)28 b Fw(\025)22 b FA(n)28 b FB(and)f FA(j)h Fw(\025)23 b FB(1,)965 4654 y(2)p 878 4691 216 4 v 878 4767 a FA(m)18 b Fw(\000)g FB(2)1104 4710 y FA(i)g FB(+)1234 4593 y Fz(\022)1392 4654 y FB(2)p 1305 4691 V 1305 4767 a FA(m)g Fw(\000)g FB(2)1549 4710 y(+)g(1)1674 4593 y Fz(\023)1749 4710 y FB(\()p FA(j)24 b Fw(\000)18 b FB(2\))k(=)2203 4654 y(2)p 2116 4691 V 2116 4767 a FA(m)c Fw(\000)g FB(2)2342 4710 y(\()p FA(i)g FB(+)g FA(j)5 b FB(\))19 b(+)f FA(j)23 b Fw(\000)2878 4654 y FB(2)p FA(m)p 2827 4691 V 2827 4767 a(m)c Fw(\000)f FB(2)2018 4932 y Fw(\025)2178 4876 y FB(2)p FA(n)p 2116 4913 V 2116 4989 a(m)g Fw(\000)g FB(2)2360 4932 y(+)g(1)g Fw(\000)2647 4876 y FB(2)p FA(m)p 2596 4913 V 2596 4989 a(m)h Fw(\000)f FB(2)2822 4932 y FA(:)71 5150 y FB(Therefore)29 b FA(B)513 5162 y Fy(1)578 5150 y Fw(2)g(P)730 5149 y Fu(2)p Fm(n)p Fl(\000)p Fm(m)p Fl(\000)p Fu(2)p 729 5162 234 3 v 784 5194 a Fm(m)p Fl(\000)p Fu(2)973 5174 y Fx(;\032;\033)1123 5150 y FB(and)h(satis\014es)g Fw(k)p FA(B)1701 5162 y Fy(1)1738 5150 y Fw(k)1790 5149 y Fu(2)p Fm(n)p Fl(\000)p Fm(m)p Fl(\000)p Fu(2)p 1789 5162 V 1844 5194 a Fm(m)p Fl(\000)p Fu(2)2033 5174 y Fx(;\032;\033)2180 5150 y Fw(\024)d FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2484 5120 y Fx(\021)2524 5150 y FB(.)46 b(F)-7 b(or)30 b FA(@)2789 5162 y Fx(u)2832 5150 y FA(B)2895 5162 y Fy(1)2932 5150 y FB(,)i(it)f(is)f(enough)g(to)h(di\013eren-)71 5267 y(tiate.)50 b(F)-7 b(or)31 b(the)h(case)f(2)p FA(n)20 b Fw(\000)h FB(2)30 b FA(>)g(m)h FB(w)n(e)h(ha)n(v)n(e)e(that)i FA(@)1809 5279 y Fx(u)1853 5267 y FA(B)1916 5279 y Fy(1)1983 5267 y Fw(2)f(P)2137 5266 y Fu(2)p Fm(n)p Fl(\000)p Fm(m)p Fl(\000)p Fu(2)p 2136 5279 V 2191 5312 a Fm(m)p Fl(\000)p Fu(2)2380 5292 y Fy(+1)p Fx(;\032;\033)2583 5267 y FB(.)50 b(In)32 b(the)g(case)f(2)p FA(n)20 b Fw(\000)h FB(2)30 b(=)f FA(m)j FB(w)n(e)g(ha)n(v)n(e)71 5385 y(that)1290 5485 y FA(@)1334 5497 y Fx(u)1377 5485 y FA(B)1440 5497 y Fy(1)1501 5485 y Fw(2)23 b(P)1695 5483 y Fu(1)p 1647 5492 125 3 v 1647 5525 a Fm(m)p Fl(\000)p Fu(2)1781 5505 y Fy(+1)p Fx(;\032;\033)2007 5485 y Fw(\032)f(P)2162 5484 y Fu(2)p Fm(n)p Fl(\000)p Fm(m)p Fl(\000)p Fu(2)p 2162 5497 234 3 v 2217 5529 a Fm(m)p Fl(\000)p Fu(2)2406 5509 y Fy(+1)p Fx(;\032;\033)2608 5485 y FA(:)p Black 1919 5753 a FB(60)p Black eop end %%Page: 61 61 TeXDict begin 61 60 bop Black Black 71 272 a FB(In)28 b(b)r(oth)g(cases,)e(w)n(e)h(ha)n(v)n(e)g(that)h Fw(k)p FA(@)1182 284 y Fx(u)1225 272 y FA(B)1288 284 y Fy(1)1325 272 y Fw(k)1377 271 y Fu(2)p Fm(n)p Fl(\000)p Fm(m)p Fl(\000)p Fu(2)p 1376 284 234 3 v 1431 317 a Fm(m)p Fl(\000)p Fu(2)1620 297 y Fy(+1)p Fx(;\032;\033)1846 272 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2146 242 y Fx(\021)2185 272 y FB(.)195 390 y(W)-7 b(e)28 b(b)r(ound)g(the)f(third)h(term)f(in)h (the)f(p)r(olynomial)g(case.)36 b(The)27 b(trigonometric)f(case)g (could)h(b)r(e)h(done)f(analogously)-7 b(.)71 489 y(W)g(e)28 b(split)g FA(C)h FB(=)23 b FA(C)634 501 y Fy(1)690 489 y FB(+)18 b FA(C)832 501 y Fy(2)897 489 y FB(as)612 721 y FA(C)671 733 y Fy(1)709 721 y FB(\()p FA(w)r(;)c(u;)g(\034)9 b FB(\))24 b(=)f Fw(\000)1254 664 y FA(w)1315 634 y Fy(2)p 1187 701 233 4 v 1187 778 a FB(2)p FA(p)1271 749 y Fy(2)1271 800 y(0)1308 778 y FB(\()p FA(u)p FB(\))612 985 y FA(C)671 997 y Fy(2)709 985 y FB(\()p FA(w)r(;)14 b(u;)g(\034)9 b FB(\))24 b(=)f Fw(\000)p FA(\026")1267 950 y Fx(\021)1389 881 y(N)1359 906 y Fz(X)1320 1083 y Fx(i)p Fy(+)p Fx(j)s Fy(=)p Fx(n)1361 1142 y(j)s Fv(\025)p Fy(1)1531 985 y FA(a)1575 997 y Fx(ij)1634 985 y FB(\()p FA(\034)9 b FB(\))p FA(q)1783 950 y Fx(i)1780 1005 y Fy(0)1818 985 y FB(\()p FA(u)p FB(\))p FA(p)1972 945 y Fx(j)1972 1007 y Fy(0)2009 985 y FB(\()p FA(u)p FB(\))2135 843 y Fz( )2201 868 y(\022)2262 985 y FB(1)18 b(+)2480 928 y FA(w)p 2415 966 192 4 v 2415 1042 a(p)2457 1013 y Fy(2)2457 1064 y(0)2494 1042 y FB(\()p FA(u)p FB(\))2616 868 y Fz(\023)2677 885 y Fx(j)2731 985 y Fw(\000)g FB(1)g Fw(\000)g FA(j)3070 928 y(w)p 3006 966 V 3006 1042 a(p)3048 1013 y Fy(2)3048 1064 y(0)3085 1042 y FB(\()p FA(u)p FB(\))3207 843 y Fz(!)3286 985 y FA(:)71 1315 y FB(Let)28 b FA(h)268 1327 y Fy(1)305 1315 y FA(;)14 b(h)390 1327 y Fy(2)450 1315 y Fw(2)23 b FA(B)t FB(\()p FA(\027)5 b FB(\))24 b Fw(\032)f(P)875 1327 y Fx(\013)918 1335 y Fu(0)950 1327 y Fy(+1)p Fx(;\032;\033)1153 1315 y FB(.)37 b(Then,)28 b(for)f(the)h(\014rst)f(term,)462 1502 y Fw(k)p FA(C)563 1514 y Fy(1)600 1502 y FB(\()p FA(h)680 1514 y Fy(2)717 1502 y FA(;)14 b(u;)g(\034)9 b FB(\))19 b Fw(\000)f FA(C)1077 1514 y Fy(1)1115 1502 y FB(\()p FA(h)1195 1514 y Fy(1)1232 1502 y FA(;)c(u;)g(\034)9 b FB(\))p Fw(k)1473 1527 y Fx(\013)1516 1535 y Fu(0)1549 1527 y Fy(+1)p Fx(;\032;\033)1774 1502 y Fw(\024)23 b FA(K)1952 1432 y Fz(\015)1952 1481 y(\015)1998 1502 y FA(p)2040 1514 y Fy(0)2077 1502 y FB(\()p FA(u)p FB(\))2189 1468 y Fv(\000)p Fy(2)2279 1502 y FB(\()p FA(h)2359 1514 y Fy(2)2414 1502 y FB(+)18 b FA(h)2545 1514 y Fy(1)2582 1502 y FB(\))2614 1432 y Fz(\015)2614 1481 y(\015)2661 1535 y Fy(0)p Fx(;\032;\033)2826 1502 y Fw(k)p FA(h)2916 1514 y Fy(2)2971 1502 y Fw(\000)g FA(h)3102 1514 y Fy(1)3140 1502 y Fw(k)3181 1527 y Fx(\013)3224 1535 y Fu(0)3257 1527 y Fy(+1)p Fx(;\032;\033)1774 1650 y Fw(\024)23 b FA(K)c Fw(k)p FA(h)2042 1662 y Fy(2)2097 1650 y FB(+)f FA(h)2228 1662 y Fy(1)2265 1650 y Fw(k)2307 1675 y Fy(2)p Fx(m=)p Fy(\()p Fx(m)p Fv(\000)p Fy(2\))p Fx(;\032;\033)2761 1650 y Fw(k)p FA(h)2851 1662 y Fy(2)2906 1650 y Fw(\000)g FA(h)3037 1662 y Fy(1)3074 1650 y Fw(k)3116 1675 y Fx(\013)3159 1683 y Fu(0)3191 1675 y Fy(+1)p Fx(;\032;\033)3408 1650 y FA(:)71 1843 y FB(By)27 b(Hyp)r(otheses)h Fp(HP5)p FB(,)f(w)n(e)g(ha)n(v)n(e)g(2)p FA(n)17 b Fw(\000)h FB(2)23 b Fw(\025)g FA(m)k FB(whic)n(h)h(implies)g(2)p FA(m=)p FB(\()p FA(m)17 b Fw(\000)h FB(2\))23 b Fw(\024)g FA(\013)2760 1855 y Fy(0)2815 1843 y FB(+)18 b(1)28 b(and)f(therefore)1065 2025 y Fw(k)p FA(h)1155 2037 y Fy(2)1210 2025 y FB(+)18 b FA(h)1341 2037 y Fy(1)1379 2025 y Fw(k)1420 2050 y Fy(2)p Fx(m=)p Fy(\()p Fx(m)p Fv(\000)p Fy(2\))p Fx(;\032;\033)1883 2025 y Fw(\024)23 b(k)p FA(h)2061 2037 y Fy(2)2116 2025 y FB(+)18 b FA(h)2247 2037 y Fy(1)2284 2025 y Fw(k)2326 2050 y Fx(\013)2369 2058 y Fu(0)2401 2050 y Fy(+1)p Fx(;\032;\033)2627 2025 y Fw(\024)23 b FA(K)6 b(\027)q(:)71 2208 y FB(Reasoning)26 b(analogously)-7 b(,)25 b(one)j(can)f(see)g(that)770 2391 y Fw(k)o FA(C)870 2403 y Fy(2)908 2391 y FB(\()p FA(h)988 2403 y Fy(2)1025 2391 y FA(;)14 b(u;)g(\034)9 b FB(\))19 b Fw(\000)f FA(C)1385 2403 y Fy(2)1423 2391 y FB(\()p FA(h)1503 2403 y Fy(1)1540 2391 y FA(;)c(u;)g(\034)9 b FB(\))p Fw(k)1781 2416 y Fx(\013)1824 2424 y Fu(0)1856 2416 y Fy(+1)p Fx(;\032;\033)2082 2391 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2382 2356 y Fx(\021)2422 2391 y FA(\027)19 b Fw(k)o FA(h)2571 2403 y Fy(2)2627 2391 y Fw(\000)f FA(h)2758 2403 y Fy(1)2795 2391 y Fw(k)2836 2416 y Fx(\013)2879 2424 y Fu(0)2912 2416 y Fy(+1)p Fx(;\032;\033)3128 2391 y FA(:)p 3790 2573 4 57 v 3794 2521 50 4 v 3794 2573 V 3843 2573 4 57 v 71 2789 a Fp(6.2.2)94 b(Pro)s(of)31 b(of)h(Theorem)f(4.3)g(in)h(the)f(parab)s(olic)h(case)71 2942 y FB(W)-7 b(e)31 b(dev)n(ote)f(this)h(section)g(to)f(pro)n(v)n(e)f (Theorem)h(4.3)g(for)h(the)g(case)f(in)h(whic)n(h)g(the)g(unp)r(erturb) r(ed)g(Hamiltonian)g(has)f(a)71 3042 y(parab)r(olic)c(critical)h(p)r (oin)n(t.)37 b(First)28 b(w)n(e)f(rewrite)g(it)h(in)g(terms)f(of)g(the) h(Banac)n(h)f(spaces)f(de\014ned)i(in)g(\(163\))o(.)p Black 71 3208 a Fp(Prop)s(osition)39 b(6.10.)p Black 44 w Fs(L)l(et)e(us)f(c)l(onsider)i(the)f(c)l(onstant)f FA(\013)1975 3220 y Fy(0)2049 3208 y Fs(de\014ne)l(d)i(in)43 b FB(\(164\))o Fs(,)c FA(\032)2748 3220 y Fy(1)2821 3208 y FA(>)d FB(0)g Fs(big)i(enough)f(and)g FA(")3635 3220 y Fy(0)3708 3208 y FA(>)f FB(0)71 3308 y Fs(smal)t(l)28 b(enough.)38 b(Then,)29 b(for)f FA(")23 b Fw(2)g FB(\(0)p FA(;)14 b(")1264 3320 y Fy(0)1301 3308 y FB(\))p Fs(,)28 b(ther)l(e)g(exists)f(a)g(function)g FA(T)2258 3320 y Fy(1)2295 3308 y FB(\()p FA(u;)14 b(\034)9 b FB(\))28 b Fs(de\014ne)l(d)g(in)f FA(D)2965 3278 y Fx(u)2963 3328 y Fv(1)p Fx(;\032)3083 3336 y Fu(1)3133 3308 y Fw(\002)13 b Ft(T)3266 3320 y Fx(\033)3337 3308 y Fs(which)29 b(satis\014es)71 3407 y(e)l(quation)49 b FB(\(141\))41 b Fs(and)i(the)f(asymptotic)i(c)l (ondition)50 b FB(\(49\))o Fs(.)77 b(Mor)l(e)l(over,)47 b FA(@)2488 3419 y Fx(u)2531 3407 y FA(T)2580 3419 y Fy(1)2663 3407 y Fw(2)f(P)2822 3419 y Fx(\013)2865 3427 y Fu(0)2897 3419 y Fy(+1)p Fx(;\032)3035 3427 y Fu(1)3068 3419 y Fx(;\033)3175 3407 y Fs(and)d(ther)l(e)f(exists)g(a)71 3507 y(c)l(onstant)29 b FA(b)438 3519 y Fy(1)498 3507 y FA(>)22 b FB(0)30 b Fs(such)f(that)1444 3607 y Fw(k)p FA(@)1530 3619 y Fx(u)1573 3607 y FA(T)1622 3619 y Fy(1)1659 3607 y Fw(k)1701 3619 y Fx(\013)1744 3627 y Fu(0)1776 3619 y Fy(+1)p Fx(;\032)1914 3627 y Fu(1)1947 3619 y Fx(;\033)2034 3607 y Fw(\024)23 b FA(b)2158 3619 y Fy(1)2195 3607 y Fw(j)p FA(\026)p Fw(j)p FA(")2330 3572 y Fx(\021)r Fy(+1)2454 3607 y FA(:)195 3773 y FB(Theorem)k(4.3)g(is)g(a)h(straigh)n (tforw)n(ard)c(consequence)j(of)g(this)h(prop)r(osition.)195 3872 y(The)39 b(pro)r(of)f(of)g(this)h(prop)r(osition)e(follo)n(ws)g (the)i(same)f(steps)g(as)g(the)h(pro)r(of)e(of)i(Prop)r(osition)e(6.4.) 68 b(W)-7 b(e)39 b(c)n(ho)r(ose)71 3972 y(constan)n(ts)26 b FA(\032)481 3984 y Fy(0)542 3972 y FA(<)c(\032)672 3942 y Fv(0)672 3992 y Fy(0)733 3972 y FA(<)g(\032)863 3984 y Fy(1)900 3972 y FB(.)195 4071 y(The)33 b(\014rst)f(step)g(is)h (to)f(p)r(erform)g(a)g(c)n(hange)f(of)h(v)-5 b(ariables)32 b(whic)n(h)g(reduces)g(the)g(size)g(of)h(the)g(linear)e(term)h(of)h Fw(F)40 b FB(in)71 4171 y(\(142\))o(.)d(This)27 b(c)n(hange)g(is)g(not) h(necessary)e(for)h(the)h(case)f FA(\021)f(>)d FB(0.)p Black 71 4337 a Fp(Lemma)k(6.11.)p Black 36 w Fs(L)l(et)e(us)h(c)l (onsider)h FA(\032)1250 4307 y Fv(0)1250 4358 y Fy(0)1310 4337 y FA(>)c(\032)1441 4349 y Fy(0)1478 4337 y Fs(.)37 b(Then,)28 b(for)f FA(")c(>)f FB(0)k Fs(smal)t(l)h(enough,)g(ther)l(e)g (exists)e(a)i(function)f FA(g)f Fw(2)f(P)3667 4351 y Fy(0)p Fx(;\032)3754 4332 y Fl(0)3754 4370 y Fu(0)3786 4351 y Fx(;\033)71 4437 y Fs(such)30 b(that)f Fw(h)p FA(g)s Fw(i)24 b FB(=)e(0)29 b Fs(and)i(is)f(a)g(solution)g(of)48 b FB(\(156\))o Fs(.)39 b(Mor)l(e)l(over,)32 b(it)d(satis\014es)h(that) 1098 4617 y Fw(k)p FA(g)s Fw(k)1225 4631 y Fy(0)p Fx(;\032)1312 4612 y Fl(0)1312 4650 y Fu(0)1343 4631 y Fx(;\033)1431 4617 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1730 4587 y Fx(\021)r Fy(+1)1854 4617 y FA(;)83 b Fw(k)p FA(@)2046 4629 y Fx(v)2085 4617 y FA(g)s Fw(k)2170 4631 y Fy(0)p Fx(;\032)2257 4612 y Fl(0)2257 4650 y Fu(0)2289 4631 y Fx(;\033)2377 4617 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2676 4587 y Fx(\021)r Fy(+1)2800 4617 y FA(;)71 4802 y Fs(and)30 b FA(v)22 b FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))24 b Fw(2)f FA(D)782 4772 y Fx(u)780 4823 y Fv(1)p Fx(;\032)900 4831 y Fu(0)967 4802 y Fs(for)30 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b Fw(2)e FA(D)1462 4772 y Fx(u)1460 4828 y Fv(1)p Fx(;\032)1580 4808 y Fl(0)1580 4846 y Fu(0)1635 4802 y Fw(\002)18 b Ft(T)1773 4814 y Fx(\033)1818 4802 y Fs(.)195 4917 y(F)-6 b(urthermor)l(e,)26 b FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b(=)g(\()p FA(v)8 b FB(+)d FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))27 b Fs(is)d(invertible)h (and)f(its)g(inverse)g(is)g(of)h(the)f(form)g FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)d(\()p FA(u)5 b FB(+)g FA(h)p FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))p Fs(,)71 5016 y(wher)l(e)30 b FA(h)g Fs(is)g(a)g(function)g (de\014ne)l(d)g(for)g FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)g FA(D)1651 4986 y Fx(u)1649 5037 y Fv(1)p Fx(;\032)1769 5045 y Fu(1)1824 5016 y Fw(\002)18 b Ft(T)1962 5028 y Fx(\033)2037 5016 y Fs(and)30 b(satis\014es)g(that)f FA(h)23 b Fw(2)h(P)2887 5028 y Fy(0)p Fx(;\032)2974 5036 y Fu(1)3006 5028 y Fx(;\033)3071 5016 y Fs(,)1580 5211 y Fw(k)p FA(h)p Fw(k)1712 5223 y Fy(0)p Fx(;\032)1799 5231 y Fu(1)1830 5223 y Fx(;\033)1918 5211 y Fw(\024)f FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2218 5177 y Fx(\021)r Fy(+1)71 5394 y Fs(and)30 b(that)g FA(u)18 b FB(+)g FA(h)p FB(\()p FA(u;)c(\034)9 b FB(\))23 b Fw(2)h FA(D)966 5364 y Fx(u)964 5420 y Fv(1)p Fx(;\032)1084 5400 y Fl(0)1084 5438 y Fu(0)1150 5394 y Fs(for)31 b FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)f FA(D)1650 5364 y Fx(u)1648 5415 y Fv(1)p Fx(;\032)1768 5423 y Fu(1)1823 5394 y Fw(\002)18 b Ft(T)1961 5406 y Fx(\033)2006 5394 y Fs(.)p Black 1919 5753 a FB(61)p Black eop end %%Page: 62 62 TeXDict begin 62 61 bop Black Black Black 71 272 a Fs(Pr)l(o)l(of.)p Black 43 w FB(Since)28 b FA(B)611 284 y Fy(1)671 272 y Fw(2)c(P)818 271 y Fu(2)p Fm(n)p Fl(\000)p Fm(m)p Fl(\000)p Fu(2)p 818 284 234 3 v 872 317 a Fm(m)p Fl(\000)p Fu(2)1062 297 y Fx(;\032;\033)1208 272 y FB(and)j(it)i(migh)n(t)e(happ)r(en)h (that)2173 239 y Fy(2)p Fx(n)p Fv(\000)p Fx(m)p Fv(\000)p Fy(2)p 2173 253 270 4 v 2236 301 a Fx(m)p Fv(\000)p Fy(2)2476 272 y FA(<)23 b FB(1,)k(w)n(e)h(cannot)f(apply)g(directly)h(Lemma)71 405 y(6.8)23 b(to)h(in)n(v)n(ert)g Fw(L)591 417 y Fx(")626 405 y FB(.)36 b(Let)25 b(us)f(observ)n(e)e(that,)k(b)n(y)d(Lemma)h (6.9,)g Fw(h)p FA(B)2080 417 y Fy(1)2118 405 y Fw(i)f FB(=)g(0)h(and)g(then)g(w)n(e)g(can)g(de\014ne)h(a)e(function)p 3562 339 68 4 v 25 w FA(B)3629 417 y Fy(1)3691 405 y FB(suc)n(h)71 505 y(that)1452 605 y FA(@)1496 617 y Fx(\034)p 1538 538 V 1538 605 a FA(B)1605 617 y Fy(1)1665 605 y FB(=)g FA(B)1816 617 y Fy(1)1922 605 y FB(and)69 b Fw(h)p 2157 538 V FA(B)2224 617 y Fy(1)2262 605 y Fw(i)23 b FB(=)g(0)p FA(;)71 744 y FB(whic)n(h)k(satis\014es)g Fw(k)p 656 678 V FA(B)723 756 y Fy(1)760 744 y Fw(k)812 743 y Fu(2)p Fm(n)p Fl(\000)p Fm(m)p Fl(\000)p Fu(2)p 812 756 234 3 v 866 789 a Fm(m)p Fl(\000)p Fu(2)1056 769 y Fx(;\032;\033)1197 744 y Fw(\024)c FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1497 714 y Fx(\021)1537 744 y FB(.)195 862 y(W)-7 b(e)28 b(can)g(de\014ne)f FA(g)k FB(as)1225 962 y FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)e Fw(\000)p FA(")p 1672 895 68 4 v(B)1739 974 y Fy(1)1776 962 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b(+)e FA(")p Fw(G)2156 974 y Fx(")2205 894 y Fz(\000)2243 962 y FA(@)2287 974 y Fx(v)p 2327 895 V 2327 962 a FA(B)2394 974 y Fy(1)2431 894 y Fz(\001)2483 962 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(:)71 1101 y FB(Then,)28 b(applying)f(Lemmas)g(6.8)g(and)g (6.9)g(one)g(obtains)g(the)h(b)r(ounds)g(for)f FA(g)j FB(and)d FA(@)2685 1113 y Fx(v)2725 1101 y FA(g)s FB(.)195 1201 y(The)h(pro)r(of)f(of)h(the)g(other)f(statemen)n(ts)g(is)g (analogous)f(to)h(the)h(pro)r(of)f(of)h(Lemma)f(6.5.)p 3790 1201 4 57 v 3794 1148 50 4 v 3794 1201 V 3843 1201 4 57 v 195 1363 a(As)h(in)g(Section)g(6.1.2,)e(w)n(e)h(de\014ne)1464 1442 y Fz(b)1450 1463 y FA(T)1499 1475 y Fy(1)1536 1463 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)d FA(T)1886 1475 y Fy(1)1923 1463 y FB(\()p FA(v)g FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))p FA(;)71 1613 y FB(whic)n(h)26 b(is)g(a)g(solution)g(of)32 b(\(157\).)k(Then,)27 b(w)n(e)f(lo)r(ok)f(for)h FA(@)1826 1625 y Fx(v)1879 1592 y Fz(b)1866 1613 y FA(T)1915 1625 y Fy(1)1978 1613 y FB(as)f(a)h(\014xed)g(p)r(oin)n(t)h(of)f(the)h(op)r (erator)d(\(162\))i(in)g(the)h(Banac)n(h)71 1713 y(space)g Fw(P)351 1727 y Fx(\013)394 1735 y Fu(0)426 1727 y Fy(+1)p Fx(;\032)564 1707 y Fl(0)564 1745 y Fu(0)597 1727 y Fx(;\033)661 1713 y FB(.)p Black 71 1866 a Fp(Lemma)39 b(6.12.)p Black 43 w Fs(L)l(et)c(us)g(c)l(onsider)i(the)f(c)l(onstant)f FA(\013)1789 1878 y Fy(0)1862 1866 y Fs(de\014ne)l(d)g(in)43 b FB(\(164\))34 b Fs(and)i FA(")2692 1878 y Fy(0)2763 1866 y FA(>)d FB(0)i Fs(smal)t(l)i(enough.)56 b(Then,)38 b(for)71 1976 y FA(")28 b Fw(2)g FB(\(0)p FA(;)14 b(")371 1988 y Fy(0)408 1976 y FB(\))32 b Fs(ther)l(e)h(exists)f(a)h(function) 1331 1955 y Fz(b)1317 1976 y FA(T)1366 1988 y Fy(1)1402 1976 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))34 b Fs(de\014ne)l(d)f(in)g FA(D)2084 1946 y Fx(u)2082 2002 y Fv(1)p Fx(;\032)2202 1982 y Fl(0)2202 2020 y Fu(0)2259 1976 y Fw(\002)20 b Ft(T)2399 1988 y Fx(\033)2476 1976 y Fs(such)33 b(that)f FA(@)2884 1988 y Fx(v)2938 1955 y Fz(b)2924 1976 y FA(T)2973 1988 y Fy(1)3038 1976 y Fw(2)c(P)3179 1990 y Fx(\013)3222 1998 y Fu(0)3255 1990 y Fy(+1)p Fx(;\032)3393 1970 y Fl(0)3393 2008 y Fu(0)3425 1990 y Fx(;\033)3522 1976 y Fs(is)33 b(a)g(\014xe)l(d)71 2090 y(p)l(oint)d(of)h(the)e(op)l(er)l (ator)40 b FB(\(162\))o Fs(.)f(F)-6 b(urthermor)l(e,)30 b(ther)l(e)g(exists)f(a)h(c)l(onstant)f FA(b)2483 2102 y Fy(1)2543 2090 y FA(>)23 b FB(0)29 b Fs(such)h(that)1417 2186 y Fz(\015)1417 2236 y(\015)1417 2286 y(\015)1463 2282 y FA(@)1507 2294 y Fx(v)1560 2261 y Fz(b)1547 2282 y FA(T)1596 2294 y Fy(1)1632 2186 y Fz(\015)1632 2236 y(\015)1632 2286 y(\015)1678 2340 y Fx(\013)1721 2348 y Fu(0)1754 2340 y Fy(+1)p Fx(;\032)1892 2320 y Fl(0)1892 2358 y Fu(0)1925 2340 y Fx(;\033)o(;)p Fy(0)2061 2282 y Fw(\024)23 b FA(b)2185 2294 y Fy(1)2222 2282 y Fw(j)p FA(\026)p Fw(j)p FA(")2357 2247 y Fx(\021)r Fy(+1)2481 2282 y FA(:)p Black 71 2506 a Fs(Pr)l(o)l(of.)p Black 43 w FB(It)k(is)g(straigh)n(tforw)n(ard)c(to)k(see)f(that)p 1499 2439 68 4 v 27 w Fw(F)35 b FB(is)27 b(w)n(ell)f(de\014ned)h(from)f Fw(P)2382 2520 y Fx(\013)2425 2528 y Fu(0)2458 2520 y Fy(+1)p Fx(;\032)2596 2500 y Fl(0)2596 2538 y Fu(0)2628 2520 y Fx(;\033)2720 2506 y FB(to)g(itself.)37 b(W)-7 b(e)27 b(are)f(going)g(to)g(pro)n(v)n(e)71 2620 y(that)i(there)f (exists)g(a)h(constan)n(t)e FA(b)1132 2632 y Fy(1)1192 2620 y FA(>)d FB(0)k(suc)n(h)g(that)p 1716 2553 V 28 w Fw(F)36 b FB(is)27 b(con)n(tractiv)n(e)f(in)p 2419 2553 V 28 w FA(B)t FB(\()p FA(b)2554 2632 y Fy(1)2591 2620 y Fw(j)p FA(\026)p Fw(j)p FA(")2726 2590 y Fx(\021)r Fy(+1)2851 2620 y FB(\))d Fw(\032)g(P)3052 2634 y Fx(\013)3095 2642 y Fu(0)3127 2634 y Fy(+1)p Fx(;\032)3265 2615 y Fl(0)3265 2653 y Fu(0)3298 2634 y Fx(;\033)3362 2620 y FB(.)195 2742 y(Let)j(us)f(consider)f(\014rst)p 935 2675 V 24 w Fw(F)8 b FB(\(0\).)36 b(F)-7 b(rom)25 b(the)g(de\014nition) h(of)p 1983 2675 V 25 w Fw(F)33 b FB(in)25 b(\(162\))f(and)h(the)h (de\014nition)f(of)3162 2721 y Fz(b)3142 2742 y Fw(F)33 b FB(in)25 b(\(158\))o(,)h(w)n(e)f(ha)n(v)n(e)71 2842 y(that)p 606 2874 V 606 2941 a Fw(F)8 b FB(\(0\)\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)p 1081 2874 55 4 v 23 w Fw(G)1135 2953 y Fx(")1184 2849 y Fz(\020)1254 2920 y(b)1234 2941 y FA(A)p FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))1485 2849 y Fz(\021)1559 2941 y FB(=)p 1647 2874 V 23 w Fw(G)1701 2953 y Fx(")1751 2941 y FB(\()p FA(A)p FB(\()p FA(v)s(;)14 b(\034)9 b FB(\)\))20 b(+)p 2169 2874 V 18 w Fw(G)2224 2953 y Fx(")2273 2941 y FB(\()p FA(A)p FB(\()p FA(v)i FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))21 b Fw(\000)d FA(A)p FB(\()p FA(v)s(;)c(\034)9 b FB(\)\))16 b FA(:)71 3101 y FB(The)26 b(\014rst)g(term)h(has)f(b)r(een)g(b)r(ounded)h(in)g(Lemma) f(6.9.)35 b(F)-7 b(or)26 b(the)h(second)f(one,)g(w)n(e)g(apply)g (Lemmas)g(6.9)f(and)i(6.11)e(and)71 3201 y(the)j(mean)f(v)-5 b(alue)28 b(theorem)f(to)g(obtain)541 3367 y Fw(k)p FA(A)p FB(\()p FA(v)22 b FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(A)p FB(\()p FA(v)s(;)c(\034)9 b FB(\))p Fw(k)1566 3392 y Fx(\013)1609 3400 y Fu(0)1641 3392 y Fy(+1)p Fx(;\032)1779 3372 y Fl(0)1779 3410 y Fu(0)1812 3392 y Fx(;\033)1899 3367 y Fw(\024)23 b(k)p FA(@)2073 3379 y Fx(u)2116 3367 y FA(A)p Fw(k)2220 3379 y Fx(\013)2263 3387 y Fu(0)2295 3379 y Fy(+1)p Fx(;\032)2433 3387 y Fu(0)2466 3379 y Fx(;\033)2530 3367 y Fw(k)p FA(g)s Fw(k)2657 3381 y Fy(0)p Fx(;\032)2744 3361 y Fl(0)2744 3399 y Fu(0)2776 3381 y Fx(;\033)2863 3367 y Fw(\024)g FA(K)6 b Fw(j)p FA(\026)p Fw(j)3124 3333 y Fy(2)3161 3367 y FA(")3200 3333 y Fy(2)p Fx(\021)r Fy(+1)3357 3367 y FA(:)71 3542 y FB(Th)n(us,)27 b(applying)g(Lemma)h(6.8,)e(there)i(exists)f(a)g(constan)n(t)g FA(b)1978 3554 y Fy(1)2038 3542 y FA(>)c FB(0)k(suc)n(h)g(that)1473 3677 y Fz(\015)1473 3726 y(\015)p 1519 3681 68 4 v 21 x Fw(F)8 b FB(\(0\))1693 3677 y Fz(\015)1693 3726 y(\015)1739 3780 y Fx(\013)1782 3788 y Fu(0)1814 3780 y Fy(+1)p Fx(;\033)1986 3747 y Fw(\024)2083 3691 y FA(b)2119 3703 y Fy(1)p 2083 3728 73 4 v 2099 3804 a FB(2)2166 3747 y Fw(j)p FA(\026)p Fw(j)p FA(")2301 3713 y Fx(\021)r Fy(+1)2426 3747 y FA(:)71 3952 y FB(Let)27 b(us)g(consider)e(no)n(w,)i FA(h)891 3964 y Fy(1)928 3952 y FA(;)14 b(h)1013 3964 y Fy(2)1073 3952 y Fw(2)p 1151 3886 68 4 v 23 w FA(B)5 b FB(\()p FA(b)1287 3964 y Fy(1)1324 3952 y Fw(j)p FA(\026)p Fw(j)p FA(")1459 3922 y Fx(\021)r Fy(+1)1583 3952 y FB(\))23 b Fw(\032)g(P)1784 3966 y Fx(\013)1827 3974 y Fu(0)1859 3966 y Fy(+1)p Fx(;\032)1997 3947 y Fl(0)1997 3985 y Fu(0)2030 3966 y Fx(;\033)2094 3952 y FB(.)37 b(Then,)27 b(using)g(the)g(prop)r(erties)f(of)p 3236 3886 55 4 v 27 w Fw(G)3290 3964 y Fx(")3353 3952 y FB(in)h(Lemma)f(6.8)71 4074 y(and)h(the)h(de\014nition)g(of)859 4053 y Fz(b)839 4074 y Fw(F)35 b FB(in)28 b(\(158\))o(,)460 4195 y Fz(\015)460 4245 y(\015)p 506 4199 68 4 v 20 x Fw(F)8 b FB(\()p FA(h)654 4277 y Fy(2)691 4265 y FB(\))19 b Fw(\000)p 825 4199 V 18 w(F)8 b FB(\()p FA(h)973 4277 y Fy(1)1010 4265 y FB(\))1042 4195 y Fz(\015)1042 4245 y(\015)1089 4299 y Fx(\013)1132 4307 y Fu(0)1164 4299 y Fy(+1)p Fx(;\032)1302 4279 y Fl(0)1302 4317 y Fu(0)1335 4299 y Fx(;\033)1422 4265 y Fw(\024)23 b FA(K)1600 4170 y Fz(\015)1600 4220 y(\015)1600 4269 y(\015)1666 4244 y(b)1646 4265 y Fw(F)8 b FB(\()p FA(h)1794 4277 y Fy(2)1832 4265 y FB(\))18 b Fw(\000)1986 4244 y Fz(b)1965 4265 y Fw(F)8 b FB(\()p FA(h)2113 4277 y Fy(1)2151 4265 y FB(\))2183 4170 y Fz(\015)2183 4220 y(\015)2183 4269 y(\015)2229 4323 y Fx(\013)2272 4331 y Fu(0)2305 4323 y Fy(+1)p Fx(;\032)2443 4303 y Fl(0)2443 4341 y Fu(0)2475 4323 y Fx(;\033)1422 4470 y Fw(\024)23 b FA(K)1600 4375 y Fz(\015)1600 4424 y(\015)1600 4474 y(\015)1664 4449 y(b)1646 4470 y FA(B)g Fw(\001)18 b FB(\()p FA(h)1853 4482 y Fy(2)1909 4470 y Fw(\000)g FA(h)2040 4482 y Fy(1)2077 4470 y FB(\))h(+)2228 4449 y Fz(b)2211 4470 y FA(C)6 b FB(\()p FA(h)2356 4482 y Fy(2)2394 4470 y FA(;)14 b(v)s(;)g(\034)9 b FB(\))19 b Fw(\000)2707 4449 y Fz(b)2690 4470 y FA(C)6 b FB(\()p FA(h)2835 4482 y Fy(1)2873 4470 y FA(;)14 b(v)s(;)g(\034)9 b FB(\))3067 4375 y Fz(\015)3067 4424 y(\015)3067 4474 y(\015)3114 4528 y Fx(\013)3157 4536 y Fu(0)3190 4528 y Fy(+1)p Fx(;\032)3328 4508 y Fl(0)3328 4546 y Fu(0)3360 4528 y Fx(;\033)3438 4470 y FA(:)71 4702 y FB(T)-7 b(aking)29 b(in)n(to)i(accoun)n(t)e(the)i(de\014nitions)g(of)1497 4681 y Fz(b)1479 4702 y FA(B)k FB(and)1758 4681 y Fz(b)1741 4702 y FA(C)i FB(in)31 b(\(160\))e(and)i(\(161\))o(,)g(recalling)e (that)i FA(B)3145 4714 y Fy(2)3210 4702 y FB(=)d(0)i(and)g(applying)71 4802 y(Lemmas)d(6.7,)g(6.9)g(and)g(6.11,w)n(e)f(obtain)923 4897 y Fz(\015)923 4947 y(\015)p 970 4901 V 970 4968 a Fw(F)8 b FB(\()p FA(h)1118 4980 y Fy(2)1155 4968 y FB(\))19 b Fw(\000)p 1289 4901 V 18 w(F)8 b FB(\()p FA(h)1437 4980 y Fy(1)1474 4968 y FB(\))1506 4897 y Fz(\015)1506 4947 y(\015)1552 5001 y Fx(\013)1595 5009 y Fu(0)1628 5001 y Fy(+1)p Fx(;\032)1766 4981 y Fl(0)1766 5019 y Fu(0)1799 5001 y Fx(;\033)1886 4968 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2186 4934 y Fx(\021)r Fy(+1)2309 4968 y Fw(k)p FA(h)2399 4980 y Fy(2)2454 4968 y Fw(\000)19 b FA(h)2586 4980 y Fy(1)2623 4968 y Fw(k)2665 4982 y Fx(\013)2708 4990 y Fu(0)2740 4982 y Fy(+1)p Fx(;\032)2878 4962 y Fl(0)2878 5001 y Fu(0)2910 4982 y Fx(;\033)2975 4968 y FA(:)71 5167 y FB(Then,)33 b(reducing)e FA(")h FB(if)h(necessary)-7 b(,)31 b(Lip)p 1319 5101 V Fw(F)39 b FA(<)30 b FB(1)p FA(=)p FB(2)g(and)i(then)p 2028 5101 V 32 w Fw(F)40 b FB(is)32 b(con)n(tractiv)n(e)e(from)p 2848 5101 V 32 w FA(B)2929 5100 y Fz(\000)2967 5167 y FA(b)3003 5179 y Fy(1)3040 5167 y Fw(j)p FA(\026)p Fw(j)p FA(")3175 5137 y Fx(\021)r Fy(+1)3299 5100 y Fz(\001)3368 5167 y Fw(\032)g(P)3521 5179 y Fx(\013)3564 5187 y Fu(0)3596 5179 y Fy(+1)p Fx(;\033)3777 5167 y FB(to)71 5267 y(itself)e(and)f(has) g(a)h(unique)f(\014xed)h(p)r(oin)n(t)g FA(h)1386 5237 y Fv(\003)1424 5267 y FB(.)37 b(Moreo)n(v)n(er,)25 b(since)i(it)h (satis\014es)1441 5468 y Fw(j)p FA(h)1512 5434 y Fv(\003)1550 5468 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p Fw(j)25 b(\024)e FA(b)1911 5480 y Fy(1)1947 5468 y Fw(j)p FA(\026)p Fw(j)p FA(")2082 5434 y Fx(\021)r Fy(+1)2323 5412 y FB(1)p 2217 5449 254 4 v 2217 5525 a Fw(j)p FA(v)s Fw(j)2306 5501 y Fx(\013)2349 5509 y Fu(0)2382 5501 y Fy(+1)p Black 1919 5753 a FB(62)p Black eop end %%Page: 63 63 TeXDict begin 63 62 bop Black Black 71 272 a FB(for)27 b(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b Fw(2)g FA(D)561 242 y Fx(u)559 298 y Fv(1)p Fx(;\032)679 278 y Fl(0)679 316 y Fu(0)734 272 y Fw(\002)18 b Ft(T)872 284 y Fx(\033)917 272 y FB(,)28 b(w)n(e)f(can)g(de\014ne)1496 251 y Fz(b)1482 272 y FA(T)1531 284 y Fy(1)1595 272 y FB(as)1472 504 y Fz(b)1458 525 y FA(T)1507 537 y Fy(1)1544 525 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)1845 412 y Fz(Z)1928 433 y Fx(v)1891 601 y Fv(\0001)2027 525 y FA(h)2075 491 y Fv(\003)2113 525 y FB(\()p FA(w)r(;)14 b(\034)9 b FB(\))14 b FA(dw)r(:)p 3790 755 4 57 v 3794 702 50 4 v 3794 755 V 3843 755 4 57 v 195 921 a FB(T)-7 b(o)31 b(pro)n(v)n(e)e(Prop)r (osition)g(6.10)g(from)i(Lemma)f(6.12,)h(as)f(w)n(e)g(ha)n(v)n(e)g(pro) r(ceeded)g(in)h(Section)f(6.1.2,)h(it)g(is)f(enough)h(to)71 1021 y(consider)21 b(the)i(c)n(hange)e(of)h(v)-5 b(ariables)21 b FA(v)26 b FB(=)d FA(u)8 b FB(+)g FA(h)p FB(\()p FA(u;)14 b(\034)9 b FB(\))21 b(obtained)h(in)g(Lemma)g(6.11,)g(tak)n(e)f FA(T)2905 1033 y Fy(1)2942 1021 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)3261 1000 y Fz(b)3248 1021 y FA(T)3297 1033 y Fy(1)3333 1021 y FB(\()p FA(u)8 b FB(+)g FA(h)p FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))71 1120 y(and)27 b(increase)g(sligh)n(tly)g FA(b)874 1132 y Fy(1)938 1120 y FB(if)h(necessary)-7 b(.)71 1395 y FC(7)135 b(In)l(v)-7 b(arian)l(t)43 b(manifolds)g(in)g(the)g(outer)g (domains:)59 b(pro)t(of)42 b(of)h(Theorems)273 1544 y(4.4)i(and)g(4.8) 71 1743 y Fq(7.1)112 b(In)m(v)-6 b(arian)m(t)38 b(manifolds)h(in)f(the) g(outer)f(domains)i(when)326 1859 y Fh(p)375 1874 y Fj(0)415 1859 y FF(\()p Fh(u)p FF(\))27 b Ff(6)p FF(=)g(0)p Fq(:)50 b(pro)s(of)38 b(of)f(Theorem)h(4.4)71 2012 y FB(In)28 b(this)g(section)g(w)n(e)f(pro)n(v)n(e)g(the)h(existence)g(of)g(the)g (in)n(v)-5 b(arian)n(t)27 b(manifolds)h(in)g(the)g(domains)f FA(D)3062 1982 y Fy(out)p Fx(;)p Fv(\003)3060 2033 y Fx(\032;\024)3235 2012 y Fw(\002)18 b Ft(T)3373 2024 y Fx(\033)3446 2012 y FB(for)28 b Fw(\003)23 b FB(=)g FA(u;)14 b(s)71 2112 y FB(de\014ned)36 b(in)f(\(29\))g(pro)n(vided)g FA(p)1044 2124 y Fy(0)1081 2112 y FB(\()p FA(u)p FB(\))h Fw(6)p FB(=)g(0)f(in)g(these)h(domains.)59 b(Since)36 b(the)g(pro)r(of)e(for)h(b)r(oth)h(in)n(v)-5 b(arian)n(t)34 b(manifolds)i(is)71 2212 y(analogous,)25 b(w)n(e)i(deal)h(only)f(with)h (the)g(unstable)g(case.)195 2311 y(First)j(in)g(Section)g(7.1.1)f(w)n (e)h(de\014ne)g(some)f(Banac)n(h)g(spaces)g(and)g(w)n(e)h(state)g(some) f(tec)n(hnical)h(lemmas.)46 b(Then,)32 b(in)71 2411 y(Section)e(7.1.2)e (w)n(e)h(pro)n(v)n(e)f(Theorem)h(4.4.)43 b(Finally)-7 b(,)30 b(w)n(e)f(dev)n(ote)h(Sections)f(8.1)g(and)g(7.3)g(to)h(pro)n(v) n(e)e(Prop)r(ositions)g(4.10)71 2510 y(and)f(7.23.)71 2726 y Fp(7.1.1)94 b(Banac)m(h)33 b(spaces)f(and)g(tec)m(hnical)h (lemmas)71 2879 y FB(W)-7 b(e)27 b(start)f(b)n(y)g(de\014ning)g(some)g (norms.)36 b(Giv)n(en)26 b FA(\027)j Fw(2)23 b Ft(R)k FB(and)f(an)g(analytic)g(function)h FA(h)c FB(:)g FA(D)2901 2849 y Fy(out)p Fx(;u)2899 2900 y(\032;\024)3083 2879 y Fw(!)g Ft(C)p FB(,)k(where)f FA(D)3609 2849 y Fy(out)p Fx(;u)3607 2900 y(\032;\024)3795 2879 y FB(is)71 2979 y(the)i(domain)f(de\014ned)h(in)g(\(29\))o(,)g(w)n(e)f(consider)1292 3187 y Fw(k)p FA(h)p Fw(k)1424 3199 y Fx(\027;\032;\024)1597 3187 y FB(=)100 b(sup)1684 3270 y Fx(u)p Fv(2)p Fx(D)1824 3242 y Fu(out)q Fm(;u)1822 3279 y(\032;\024)1978 3091 y Fz(\014)1978 3141 y(\014)1978 3191 y(\014)2006 3119 y(\000)2044 3187 y FA(u)2092 3152 y Fy(2)2147 3187 y FB(+)18 b FA(a)2274 3152 y Fy(2)2311 3119 y Fz(\001)2349 3135 y Fx(\027)2405 3187 y FA(h)p FB(\()p FA(u)p FB(\))2565 3091 y Fz(\014)2565 3141 y(\014)2565 3191 y(\014)2606 3187 y FA(:)71 3453 y FB(Moreo)n(v)n(er)39 b(for)h(analytic)h (functions)h FA(h)k FB(:)g FA(D)1526 3423 y Fy(out)p Fx(;u)1524 3474 y(\032;\024)1713 3453 y Fw(\002)27 b Ft(T)1860 3465 y Fx(\033)1951 3453 y Fw(!)46 b Ft(C)c FB(whic)n(h)f(are)f(2)p FA(\031)s FB(-p)r(erio)r(dic)h(in)h FA(\034)9 b FB(,)46 b(w)n(e)41 b(consider)f(the)71 3553 y(corresp)r(onding)26 b(F)-7 b(ourier)26 b(norm)1385 3673 y Fw(k)p FA(h)p Fw(k)1517 3685 y Fx(\027;\032;\024;\033)1749 3673 y FB(=)1837 3594 y Fz(X)1837 3772 y Fx(k)q Fv(2)p Fn(Z)1971 3577 y Fz(\015)1971 3627 y(\015)1971 3677 y(\015)2017 3673 y FA(h)2065 3638 y Fy([)p Fx(k)q Fy(])2143 3577 y Fz(\015)2143 3627 y(\015)2143 3677 y(\015)2190 3731 y Fx(\027;\032;\024)2354 3673 y FA(e)2393 3638 y Fv(j)p Fx(k)q Fv(j)p Fx(\033)2513 3673 y FA(:)71 3901 y FB(W)-7 b(e)28 b(consider,)e(th)n(us,)i(the)g(follo)n(wing)f(function)h(space) 813 4100 y Fw(E)857 4112 y Fx(\027;\032;\024;\033)1090 4100 y FB(=)23 b Fw(f)p FA(h)g FB(:)g FA(D)1408 4066 y Fy(out)o Fx(;u)1406 4121 y(\032;\024)1585 4100 y Fw(\002)18 b Ft(T)1723 4112 y Fx(\033)1791 4100 y Fw(!)23 b Ft(C)p FB(;)42 b(real-analytic)n FA(;)14 b Fw(k)p FA(h)p Fw(k)2641 4112 y Fx(\027;\032;\024;\033)2873 4100 y FA(<)23 b Fw(1g)p FA(;)552 b FB(\(167\))71 4250 y(whic)n(h)27 b(can)h(b)r(e)g(c)n(hec)n (k)n(ed)e(that)i(is)f(a)h(Banac)n(h)e(space)h(for)g(an)n(y)g FA(\027)h Fw(2)23 b Ft(R)p FB(.)195 4349 y(If)28 b(there)g(is)f(no)h (danger)e(of)i(confusion)f(ab)r(out)g(the)h(domain)f FA(D)2170 4319 y Fy(out)p Fx(;u)2168 4370 y(\032;\024)2329 4349 y FB(,)h(w)n(e)f(will)h(denote)1165 4542 y Fw(k)18 b(\001)h(k)1309 4554 y Fx(\027;\033)1429 4542 y FB(=)k Fw(k)18 b(\001)g(k)1660 4554 y Fx(\027;\032;\024;\033)1981 4542 y FB(and)111 b Fw(E)2270 4554 y Fx(\027;\033)2391 4542 y FB(=)22 b Fw(E)2522 4554 y Fx(\027;\032;\024;\033)2733 4542 y FA(:)195 4726 y FB(In)30 b(the)g(next)g(lemma,)f(w)n(e)h(state)f (some)g(prop)r(erties)f(of)i(these)f(Banac)n(h)f(spaces.)42 b(In)29 b(the)h(estimates)g(w)n(e)f(will)g(mak)n(e)71 4825 y(explicit)f(the)g(dep)r(endence)g(of)f(the)h(constan)n(ts)f(with) h(resp)r(ect)f(to)h FA(\024)p FB(.)p Black 71 4991 a Fp(Lemma)j(7.1.)p Black 40 w Fs(The)g(fol)t(lowing)h(statements)d (hold:)p Black 169 5157 a(1.)p Black 42 w(If)i FA(\027)407 5169 y Fy(1)467 5157 y Fw(\025)23 b FA(\027)596 5169 y Fy(2)633 5157 y Fs(,)30 b(then)g Fw(E)917 5169 y Fx(\027)950 5177 y Fu(1)982 5169 y Fx(;\033)1070 5157 y Fw(\032)22 b(E)1201 5169 y Fx(\027)1234 5177 y Fu(2)1267 5169 y Fx(;\033)1361 5157 y Fs(and)30 b(mor)l(e)l(over)h(if)f FA(h)23 b Fw(2)h(E)2155 5169 y Fx(\027)2188 5177 y Fu(1)2220 5169 y Fx(;\033)2284 5157 y Fs(,)1529 5340 y Fw(k)p FA(h)p Fw(k)1661 5352 y Fx(\027)1694 5360 y Fu(2)1726 5352 y Fx(;\033)1813 5340 y Fw(\024)f FA(K)6 b FB(\()p FA(\024")p FB(\))2129 5306 y Fx(\027)2162 5314 y Fu(2)2194 5306 y Fv(\000)p Fx(\027)2279 5314 y Fu(1)2316 5340 y Fw(k)p FA(h)p Fw(k)2448 5352 y Fx(\027)2481 5360 y Fu(1)2512 5352 y Fx(;\033)2577 5340 y FA(:)p Black 1919 5753 a FB(63)p Black eop end %%Page: 64 64 TeXDict begin 64 63 bop Black Black Black 169 272 a Fs(2.)p Black 42 w(If)31 b FA(\027)407 284 y Fy(1)467 272 y Fw(\024)23 b FA(\027)596 284 y Fy(2)633 272 y Fs(,)30 b(then)g Fw(E)917 284 y Fx(\027)950 292 y Fu(1)982 284 y Fx(;\033)1070 272 y Fw(\032)22 b(E)1201 284 y Fx(\027)1234 292 y Fu(2)1267 284 y Fx(;\033)1361 272 y Fs(and)30 b(mor)l(e)l(over)h(if)f FA(h)23 b Fw(2)h(E)2155 284 y Fx(\027)2188 292 y Fu(1)2220 284 y Fx(;\033)2284 272 y Fs(,)1698 455 y Fw(k)p FA(h)p Fw(k)1830 467 y Fx(\027)1863 475 y Fu(2)1895 467 y Fx(;\033)1982 455 y Fw(\024)f FA(K)6 b Fw(k)p FA(h)p Fw(k)2279 467 y Fx(\027)2312 475 y Fu(1)2343 467 y Fx(;\033)2407 455 y FA(:)p Black 169 671 a Fs(3.)p Black 42 w(If)31 b FA(h)22 b Fw(2)i(E)559 683 y Fx(\027)592 691 y Fu(1)624 683 y Fx(;\033)719 671 y Fs(and)30 b FA(g)25 b Fw(2)f(E)1068 683 y Fx(\027)1101 691 y Fu(2)1133 683 y Fx(;\033)1198 671 y Fs(,)30 b(then)g FA(hg)25 b Fw(2)e(E)1673 683 y Fx(\027)1706 691 y Fu(1)1739 683 y Fy(+)p Fx(\027)1823 691 y Fu(2)1856 683 y Fx(;\033)1950 671 y Fs(and)1529 853 y Fw(k)p FA(hg)s Fw(k)1704 865 y Fx(\027)1737 873 y Fu(1)1768 865 y Fy(+)p Fx(\027)1852 873 y Fu(2)1885 865 y Fx(;\033)1972 853 y Fw(\024)g(k)p FA(h)p Fw(k)2192 865 y Fx(\027)2225 873 y Fu(1)2257 865 y Fx(;\033)2321 853 y Fw(k)p FA(g)s Fw(k)2448 865 y Fx(\027)2481 873 y Fu(2)2512 865 y Fx(;\033)2577 853 y FA(:)p Black 169 1069 a Fs(4.)p Black 42 w(L)l(et)37 b FA(\032)472 1039 y Fv(0)532 1069 y FA(<)g(\032)g Fs(b)l(e)g(such)h(that)f FA(\032)24 b Fw(\000)f FA(\032)1395 1039 y Fv(0)1456 1069 y Fs(has)38 b(a)g(p)l(ositive)g(lower)h(b)l(ound)e(indep)l(endent) h(of)g FA(")p Fs(,)i FA(\024)3200 1039 y Fv(0)3260 1069 y Fs(and)e FA(\024)f Fs(such)g(that)278 1169 y FA(\024)23 b(<)g(\024)485 1139 y Fv(0)531 1169 y FA(<)g FB(0)29 b Fs(and)h FA(h)23 b Fw(2)h(E)1045 1181 y Fx(\027;\032;\024;\033)1255 1169 y Fs(.)39 b(Then)30 b FA(@)1579 1181 y Fx(u)1623 1169 y FA(h)23 b Fw(2)g(E)1816 1181 y Fx(\027;\032)1903 1165 y Fl(0)1926 1181 y Fx(;\024)1985 1165 y Fl(0)2007 1181 y Fx(;\033)2101 1169 y Fs(and)30 b(satis\014es)1420 1397 y Fw(k)p FA(@)1506 1409 y Fx(u)1549 1397 y FA(h)p Fw(k)1639 1409 y Fx(\027;\032)1726 1393 y Fl(0)1749 1409 y Fx(;\024)1808 1393 y Fl(0)1830 1409 y Fx(;\033)1917 1397 y Fw(\024)2136 1341 y FA(K)p 2015 1378 320 4 v 2015 1454 a(")14 b Fw(j)o FA(\024)2138 1430 y Fv(0)2180 1454 y Fw(\000)k FA(\024)p Fw(j)2344 1397 y(k)p FA(h)p Fw(k)2476 1409 y Fx(\027;\032;\024;\033)2686 1397 y FA(:)195 1657 y FB(Throughout)40 b(this)h(section)f(w)n(e)h(are)f(going)f(to)i(solv)n (e)e(equations)h(of)h(the)g(form)g Fw(L)2904 1669 y Fx(")2939 1657 y FA(h)k FB(=)g FA(g)s FB(,)f(where)c Fw(L)3562 1669 y Fx(")3638 1657 y FB(is)h(the)71 1756 y(di\013eren)n(tial)25 b(op)r(erator)e(de\014ned)i(in)h(\(45\))o(.)37 b(Note)25 b(that)g Fw(L)1833 1768 y Fx(")1894 1756 y FB(acting)g(on)f Fw(E)2297 1768 y Fx(\027;\032)2414 1756 y FB(is)h(not)g(in)n(v)n (ertible.)36 b(Indeed)25 b(for)f(an)n(y)h(smo)r(oth)71 1856 y(function)j FA(f)9 b FB(,)27 b FA(f)9 b FB(\()p FA(u=")17 b Fw(\000)h FA(\034)9 b FB(\))24 b Fw(2)g FB(Ker)n Fw(L)1177 1868 y Fx(")1213 1856 y FB(.)37 b(W)-7 b(e)28 b(consider)f(a)g(left-in)n(v)n(erse)f(of)i(the)f(op)r(erator)f Fw(L)2858 1868 y Fx(")2894 1856 y FB(,)i(whic)n(h)f(w)n(e)g(call)h Fw(G)3506 1868 y Fx(")3541 1856 y FB(,)g(de\014ned)71 1955 y(acting)22 b(on)g(the)h(F)-7 b(ourier)22 b(co)r(e\016cien)n(ts.) 35 b(Let)22 b(us)h(consider)e FA(u)1908 1967 y Fy(1)1945 1955 y FA(;)e FB(\026)-47 b FA(u)2030 1967 y Fy(1)2090 1955 y Fw(2)23 b Ft(C)g FB(the)g(v)n(ertices)e(of)i(the)g(domain)f FA(D)3273 1925 y Fy(out)p Fx(;u)3271 1976 y(\032;\024)3455 1955 y FB(\(see)g(Figure)71 2055 y(4\).)37 b(Then,)27 b(w)n(e)h(de\014ne)g Fw(G)856 2067 y Fx(")919 2055 y FB(as)1367 2171 y Fw(G)1416 2183 y Fx(")1452 2171 y FB(\()p FA(h)p FB(\)\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)1869 2092 y Fz(X)1869 2271 y Fx(k)q Fv(2)p Fn(Z)2003 2171 y Fw(G)2052 2183 y Fx(")2088 2171 y FB(\()p FA(h)p FB(\))2200 2137 y Fy([)p Fx(k)q Fy(])2279 2171 y FB(\()p FA(u)p FB(\))p FA(e)2430 2137 y Fx(ik)q(\034)2532 2171 y FA(;)1106 b FB(\(168\))71 2400 y(where)27 b(its)h(F)-7 b(ourier)26 b(co)r(e\016cien)n(ts)h(are)g(giv)n(en)g(b)n(y)763 2625 y Fw(G)812 2637 y Fx(")848 2625 y FB(\()p FA(h)p FB(\))960 2590 y Fy([)p Fx(k)q Fy(])1039 2625 y FB(\()p FA(u)p FB(\))c(=)1262 2512 y Fz(Z)1345 2532 y Fx(u)1313 2700 y Fy(\026)-38 b Fx(u)1347 2708 y Fu(1)1402 2625 y FA(e)1441 2590 y Fx(ik)q(")1531 2565 y Fl(\000)p Fu(1)1610 2590 y Fy(\()p Fx(t)p Fv(\000)p Fx(u)p Fy(\))1782 2625 y FA(h)1830 2590 y Fy([)p Fx(k)q Fy(])1908 2625 y FB(\()p FA(t)p FB(\))14 b FA(dt)744 b FB(for)27 b FA(k)f(<)c FB(0)767 2859 y Fw(G)816 2871 y Fx(")852 2859 y FB(\()p FA(h)p FB(\))964 2825 y Fy([0])1039 2859 y FB(\()p FA(u)p FB(\))h(=)1262 2746 y Fz(Z)1345 2767 y Fx(u)1308 2935 y Fv(\000)p Fx(\032)1412 2859 y FA(h)1460 2825 y Fy([0])1535 2859 y FB(\()p FA(t)p FB(\))14 b FA(dt)763 3097 y Fw(G)812 3109 y Fx(")848 3097 y FB(\()p FA(h)p FB(\))960 3063 y Fy([)p Fx(k)q Fy(])1039 3097 y FB(\()p FA(u)p FB(\))23 b(=)1262 2984 y Fz(Z)1345 3004 y Fx(u)1308 3173 y(u)1347 3181 y Fu(1)1402 3097 y FA(e)1441 3063 y Fx(ik)q(")1531 3038 y Fl(\000)p Fu(1)1610 3063 y Fy(\()p Fx(t)p Fv(\000)p Fx(u)p Fy(\))1782 3097 y FA(h)1830 3063 y Fy([)p Fx(k)q Fy(])1908 3097 y FB(\()p FA(t)p FB(\))14 b FA(dt)721 b FB(for)27 b FA(k)f(>)c FB(0)p FA(:)p Black 71 3330 a Fp(Remark)34 b(7.2.)p Black 42 w Fs(L)l(et)d(us)h(observe)h(that)f(the)g(de\014nition)g(of)h(the)f (op)l(er)l(ator)h Fw(G)2483 3342 y Fx(")2551 3330 y Fs(dep)l(ends)g(on) f(the)g(domain,)i(sinc)l(e)e(in)g(its)71 3429 y(de\014nition)e(we)g (use)g(its)f(vertic)l(es)i FA(u)1177 3441 y Fy(1)1213 3429 y Fs(,)36 b FB(\026)-48 b FA(u)1316 3441 y Fy(1)1383 3429 y Fs(and)30 b(also)h FA(\032)p Fs(.)p Black 71 3596 a Fp(Lemma)g(7.3.)p Black 40 w Fs(The)g(op)l(er)l(ator)g Fw(G)1145 3608 y Fx(")1211 3596 y Fs(in)36 b FB(\(168\))29 b Fs(satis\014es)g(the)h(fol)t(lowing)j(pr)l(op)l(erties.)p Black 169 3762 a(1.)p Black 42 w(If)e FA(h)22 b Fw(2)i(E)559 3774 y Fx(\027;\033)686 3762 y Fs(for)31 b(some)f FA(\027)f Fw(\025)22 b FB(0)p Fs(,)30 b(then)f Fw(G)1518 3774 y Fx(")1554 3762 y FB(\()p FA(h)p FB(\))24 b Fw(2)f(E)1812 3774 y Fx(\027;\033)1940 3762 y Fs(and)1656 3944 y Fw(kG)1747 3956 y Fx(")1782 3944 y FB(\()p FA(h)p FB(\))p Fw(k)1936 3956 y Fx(\027;\033)2057 3944 y Fw(\024)g FA(K)6 b Fw(k)p FA(h)p Fw(k)2354 3956 y Fx(\027;\033)2450 3944 y FA(:)278 4127 y Fs(F)-6 b(urthermor)l(e,)30 b(if)h Fw(h)p FA(h)p Fw(i)24 b FB(=)e(0)p Fs(,)1623 4226 y Fw(k)o(G)1713 4238 y Fx(")1749 4226 y FB(\()p FA(h)p FB(\))p Fw(k)1903 4251 y Fx(\027;\033)2024 4226 y Fw(\024)h FA(K)6 b(")14 b Fw(k)n FA(h)p Fw(k)2371 4251 y Fx(\027;\033)2483 4226 y FA(:)p Black 169 4409 a Fs(2.)p Black 42 w(If)31 b FA(h)22 b Fw(2)i(E)559 4421 y Fx(\027;\033)686 4409 y Fs(for)31 b(some)f FA(\027)f(>)22 b FB(1)p Fs(,)30 b(then)f Fw(G)1518 4421 y Fx(")1554 4409 y FB(\()p FA(h)p FB(\))24 b Fw(2)f(E)1812 4421 y Fx(\027)t Fv(\000)p Fy(1)p Fx(;\033)2028 4409 y Fs(and)1611 4592 y Fw(kG)1702 4604 y Fx(")1738 4592 y FB(\()p FA(h)p FB(\))p Fw(k)1892 4617 y Fx(\027)t Fv(\000)p Fy(1)p Fx(;\033)2101 4592 y Fw(\024)g FA(K)6 b Fw(k)p FA(h)p Fw(k)2398 4604 y Fx(\027;\033)2494 4592 y FA(:)p Black 169 4808 a Fs(3.)p Black 42 w(If)31 b FA(h)22 b Fw(2)i(E)559 4820 y Fx(\027;\033)686 4808 y Fs(for)31 b(some)f FA(\027)f Fw(2)23 b FB(\(0)p FA(;)14 b FB(1\))p Fs(,)30 b(then)f Fw(G)1652 4820 y Fx(")1688 4808 y FB(\()p FA(h)p FB(\))24 b Fw(2)f(E)1946 4820 y Fy(0)p Fx(;\033)2073 4808 y Fs(and)1656 4990 y Fw(k)o(G)1746 5002 y Fx(")1782 4990 y FB(\()p FA(h)p FB(\))p Fw(k)1936 5015 y Fy(0)p Fx(;\033)2057 4990 y Fw(\024)f FA(K)6 b Fw(k)p FA(h)p Fw(k)2353 5002 y Fx(\027;\033)2450 4990 y FA(:)p Black 169 5206 a Fs(4.)p Black 42 w(If)31 b FA(h)22 b Fw(2)i(E)559 5218 y Fx(\027;\033)686 5206 y Fs(for)31 b(some)f FA(\027)f Fw(\025)22 b FB(0)p Fs(,)30 b(then)f Fw(G)1518 5218 y Fx(")1554 5206 y FB(\()p FA(@)1630 5218 y Fx(u)1674 5206 y FA(h)p FB(\))23 b Fw(2)h(E)1900 5218 y Fx(\027;\033)2027 5206 y Fs(and)1612 5389 y Fw(k)o(G)1702 5401 y Fx(")1738 5389 y FB(\()p FA(@)1814 5401 y Fx(u)1858 5389 y FA(h)p FB(\))p Fw(k)1980 5414 y Fx(\027;\033)2101 5389 y Fw(\024)e FA(K)6 b Fw(k)p FA(h)p Fw(k)2397 5401 y Fx(\027;\033)2494 5389 y FA(:)p Black 1919 5753 a FB(64)p Black eop end %%Page: 65 65 TeXDict begin 65 64 bop Black Black Black 169 272 a Fs(5.)p Black 42 w(If)31 b FA(h)22 b Fw(2)i(X)574 284 y Fx(\027;\033)702 272 y Fs(for)30 b(some)h FA(\027)d Fw(\025)23 b FB(0)p Fs(,)29 b Fw(L)1357 284 y Fx(")1411 272 y Fw(\016)18 b(G)1520 284 y Fx(")1556 272 y FB(\()p FA(h)p FB(\))24 b(=)e FA(h)30 b Fs(and)427 462 y Fw(G)476 474 y Fx(")530 462 y Fw(\016)18 b(L)647 474 y Fx(")683 462 y FB(\()p FA(h)p FB(\)\()p FA(v)s(;)c(\034)9 b FB(\))24 b(=)f FA(h)p FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b Fw(\000)1437 383 y Fz(X)1436 562 y Fx(k)q(<)p Fy(0)1571 462 y FA(e)1610 428 y Fx(ik)q(")1700 403 y Fl(\000)p Fu(1)1779 428 y Fy(\()p Fv(\000)p Fx(u)1896 436 y Fu(1)1928 428 y Fv(\000)p Fx(u)p Fy(\))2050 462 y FA(h)2098 428 y Fy([)p Fx(k)q Fy(])2176 462 y FB(\()p Fw(\000)p FA(u)2321 474 y Fy(1)2358 462 y FB(\))e Fw(\000)g FA(h)2539 428 y Fy([0])2614 462 y FB(\()p FA(u)2694 474 y Fy(0)2731 462 y FB(\))h Fw(\000)2866 383 y Fz(X)2865 562 y Fx(k)q(>)p Fy(0)3000 462 y FA(e)3039 428 y Fx(ik)q(")3129 403 y Fl(\000)p Fu(1)3208 428 y Fy(\()p Fx(u)3273 436 y Fu(1)3305 428 y Fv(\000)p Fx(u)p Fy(\))3427 462 y FA(h)3475 428 y Fy([)p Fx(k)q Fy(])3553 462 y FB(\()p FA(u)3633 474 y Fy(1)3670 462 y FB(\))p Black 169 747 a Fs(6.)p Black 42 w(If)31 b FA(h)22 b Fw(2)i(X)574 759 y Fx(\027;\033)702 747 y Fs(for)30 b(some)h FA(\027)d Fw(\025)23 b FB(0)p Fs(,)29 b Fw(L)1357 759 y Fx(")1411 747 y Fw(\016)18 b(G)1520 759 y Fx(")1556 747 y FB(\()p FA(h)p FB(\))24 b(=)e FA(h)30 b Fs(and)427 937 y Fw(G)476 949 y Fx(")530 937 y Fw(\016)18 b(L)647 949 y Fx(")683 937 y FB(\()p FA(h)p FB(\)\()p FA(v)s(;)c(\034)9 b FB(\))24 b(=)f FA(h)p FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b Fw(\000)1437 858 y Fz(X)1436 1037 y Fx(k)q(<)p Fy(0)1571 937 y FA(e)1610 903 y Fx(ik)q(")1700 878 y Fl(\000)p Fu(1)1779 903 y Fy(\()p Fv(\000)p Fx(u)1896 911 y Fu(1)1928 903 y Fv(\000)p Fx(u)p Fy(\))2050 937 y FA(h)2098 903 y Fy([)p Fx(k)q Fy(])2176 937 y FB(\()p Fw(\000)p FA(u)2321 949 y Fy(1)2358 937 y FB(\))e Fw(\000)g FA(h)2539 903 y Fy([0])2614 937 y FB(\()p FA(u)2694 949 y Fy(0)2731 937 y FB(\))h Fw(\000)2866 858 y Fz(X)2865 1037 y Fx(k)q(>)p Fy(0)3000 937 y FA(e)3039 903 y Fx(ik)q(")3129 878 y Fl(\000)p Fu(1)3208 903 y Fy(\()p Fx(u)3273 911 y Fu(1)3305 903 y Fv(\000)p Fx(u)p Fy(\))3427 937 y FA(h)3475 903 y Fy([)p Fx(k)q Fy(])3553 937 y FB(\()p FA(u)3633 949 y Fy(1)3670 937 y FB(\))p Black 71 1222 a Fs(Pr)l(o)l(of.)p Black 43 w FB(It)28 b(is)f(a)h(consequence)e(of)i(Lemma)f(5.5)g(in)h ([GOS10)o(].)p 3790 1222 4 57 v 3794 1169 50 4 v 3794 1222 V 3843 1222 4 57 v 71 1436 a Fp(7.1.2)94 b(Pro)s(of)31 b(of)h(Theorem)f(4.4)71 1589 y FB(W)-7 b(e)33 b(pro)n(v)n(e)f(Theorem)g (4.4,)i(lo)r(oking)e(for)g(the)i(analytic)e(con)n(tin)n(uation)h(of)g (the)g(function)h FA(T)2981 1601 y Fy(1)3050 1589 y FB(=)e FA(T)h Fw(\000)21 b FA(T)3364 1601 y Fy(0)3434 1589 y FB(obtained)33 b(in)71 1689 y(Prop)r(ositions)g(6.4)h(and)g(6.10)g(as)g (a)g(solution)g(of)h(equation)f(\(141\))o(.)58 b(First)35 b(w)n(e)f(rewrite)g(the)i(result)e(in)h(terms)f(of)h(the)71 1788 y(Banac)n(h)26 b(spaces)h(de\014ned)h(in)g(\(167\))o(.)p Black 71 1945 a Fp(Prop)s(osition)i(7.4.)p Black 39 w Fs(L)l(et)f FA(\032)962 1957 y Fy(1)1029 1945 y Fs(b)l(e)g(the)h(c)l (onstant)f(c)l(onsider)l(e)l(d)h(in)g(The)l(or)l(em)g(4.3)h(and)f(let)f (us)g(c)l(onsider)h FA(\032)3339 1957 y Fy(2)3400 1945 y FA(>)22 b(\032)3530 1957 y Fy(1)3567 1945 y Fs(,)30 b FA(")3661 1957 y Fy(0)3721 1945 y FA(>)23 b FB(0)71 2044 y Fs(smal)t(l)38 b(enough)f(and)h FA(\024)805 2056 y Fy(1)878 2044 y FA(>)e FB(0)h Fs(big)h(enough.)61 b(Then,)40 b(for)e FA(")e Fw(2)h FB(\(0)p FA(;)14 b(")2243 2056 y Fy(0)2280 2044 y FB(\))p Fs(,)39 b(ther)l(e)f(exists)e(a)i(function)f FA(T)3288 2056 y Fy(1)3361 2044 y Fw(2)g(E)3497 2056 y Fx(`)p Fy(+1)p Fx(;\032)3663 2064 y Fu(2)3695 2056 y Fx(;\024)3754 2064 y Fu(1)3786 2056 y Fx(;\033)71 2144 y Fs(which)h(satis\014es)f(e)l(quation)43 b FB(\(141\))35 b Fs(and)i(is)g(the)g(analytic)h(c)l(ontinuation)e(of)h(the)g(analytic) h(function)e FA(T)3369 2156 y Fy(1)3443 2144 y Fs(obtaine)l(d)h(in)71 2243 y(Pr)l(op)l(ositions)31 b(6.4)g(and)f(6.10.)41 b(Mor)l(e)l(over,) 31 b(ther)l(e)f(exists)g(a)g(c)l(onstant)f FA(b)2328 2255 y Fy(2)2388 2243 y FA(>)22 b FB(0)30 b Fs(such)f(that)1423 2414 y Fw(k)o FA(@)1508 2426 y Fx(u)1552 2414 y FA(T)1601 2426 y Fy(1)1637 2414 y Fw(k)1679 2439 y Fx(`)p Fy(+1)p Fx(;\032)1845 2447 y Fu(2)1877 2439 y Fx(;\024)1936 2447 y Fu(1)1968 2439 y Fx(;\033)2056 2414 y Fw(\024)22 b FA(b)2179 2426 y Fy(2)2216 2414 y Fw(j)p FA(\026)p Fw(j)p FA(")2351 2380 y Fx(\021)r Fy(+1)2476 2414 y FA(:)195 2595 y FB(This)28 b(prop)r(osition)f(giv)n(es)f(the)i(existence)f(of)h (the)g(in)n(v)-5 b(arian)n(t)26 b(manifolds)i(in)f FA(D)2651 2565 y Fy(out)p Fx(;)p Fv(\003)2649 2616 y Fx(\032)2683 2624 y Fu(2)2716 2616 y Fx(;\024)2775 2624 y Fu(1)2830 2595 y Fw(\002)18 b Ft(T)2968 2607 y Fx(\033)3013 2595 y FB(,)27 b Fw(\003)c FB(=)g FA(u;)14 b(s)p FB(.)195 2695 y(W)-7 b(e)28 b(dev)n(ote)f(the)h(rest)f(of)h(the)g(section)f(to)h (pro)n(v)n(e)e(Prop)r(osition)f(7.4.)195 2794 y(First,)h(w)n(e)e(state) h(a)g(tec)n(hnical)f(lemma)h(ab)r(out)g(prop)r(erties)f(of)h(the)g (functions)h FA(A)p FB(,)g FA(B)2801 2806 y Fy(1)2838 2794 y FB(,)f FA(B)2949 2806 y Fy(2)3012 2794 y FB(and)f FA(C)32 b FB(de\014ned)25 b(in)g(\(143\))o(,)71 2894 y(\(144\))o(,)j(\(145\))e(and)i(\(146\))f(resp)r(ectiv)n(ely)-7 b(.)p Black 71 3050 a Fp(Lemma)36 b(7.5.)p Black 43 w Fs(L)l(et)d(us)g(c)l(onsider)i(any)f FA(\032)d(>)e FB(0)34 b Fs(and)g FA(\024)c(>)g FB(0)p Fs(.)50 b(Then,)36 b(the)e(functions)f FA(A)p Fs(,)j FA(B)2999 3062 y Fy(1)3036 3050 y Fs(,)f FA(B)3159 3062 y Fy(2)3230 3050 y Fs(and)f FA(C)40 b Fs(de\014ne)l(d)34 b(in)71 3150 y FB(\(143\))o Fs(,)c FB(\(144\))o Fs(,)g FB(\(145\))f Fs(and)39 b FB(\(146\))29 b Fs(satisfy)i(the)f(fol)t(lowing)i(pr)l(op)l(erties.)p Black 169 3306 a(1.)p Black 42 w FA(A)24 b Fw(2)f(E)486 3318 y Fx(`;\032;\024;\033)721 3306 y Fs(and)30 b FA(@)926 3318 y Fx(u)969 3306 y FA(A)24 b Fw(2)f(E)1177 3318 y Fx(`)p Fy(+1)p Fx(;\032;\024;\033)1466 3306 y Fs(.)39 b(Mor)l(e)l(over)31 b FA(@)1939 3318 y Fx(u)1982 3306 y FA(A)f Fs(satis\014es)1643 3479 y Fw(k)p FA(@)1729 3491 y Fx(u)1772 3479 y FA(A)p Fw(k)1876 3504 y Fx(`)p Fy(+1)p Fx(;\032;\024;\033)2188 3479 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2488 3445 y Fx(\021)1494 3620 y Fw(k)o(G)1584 3632 y Fx(")1620 3620 y FB(\()p FA(@)1696 3632 y Fx(u)1740 3620 y FA(A)p FB(\))p Fw(k)1876 3645 y Fx(`)p Fy(+1)p Fx(;\032;\024;\033)2188 3620 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2488 3586 y Fx(\021)r Fy(+1)2612 3620 y FA(:)3661 3554 y FB(\(169\))p Black 169 3829 a Fs(2.)p Black 42 w(If)31 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b(<)e FB(0)p Fs(,)29 b FA(B)853 3841 y Fy(1)891 3829 y FA(;)14 b(@)972 3841 y Fx(u)1015 3829 y FA(B)1078 3841 y Fy(1)1115 3829 y FA(;)g(B)1215 3841 y Fy(2)1275 3829 y Fw(2)24 b(E)1398 3841 y Fy(0)p Fx(;\032;\024;\033)1637 3829 y Fs(and)31 b(satisfy)g Fw(h)p FA(B)2153 3841 y Fy(1)2190 3829 y Fw(i)23 b FB(=)g(0)29 b Fs(and)1861 4001 y Fw(k)p FA(B)1966 4013 y Fy(1)2003 4001 y Fw(k)2045 4013 y Fy(0)p Fx(;\032;\024;\033)2278 4001 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2577 3967 y Fx(\021)1334 4126 y Fw(k)p FA(@)1420 4138 y Fx(u)1463 4126 y FA(B)1526 4138 y Fy(1)1563 4126 y Fw(k)1605 4141 y Fy(max)o Fv(f)p Fy(0)p Fx(;`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fv(g)p Fx(;\032;\024;\033)2278 4126 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2577 4092 y Fx(\021)1861 4261 y Fw(k)p FA(B)1966 4273 y Fy(2)2003 4261 y Fw(k)2045 4273 y Fy(0)p Fx(;\032;\024;\033)2278 4261 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)2538 4227 y Fy(2)2575 4261 y FA(")2614 4227 y Fy(2)p Fx(\021)r Fy(+1)2772 4261 y FA(:)3661 4132 y FB(\(170\))p Black 169 4463 a Fs(3.)p Black 42 w(If)31 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b Fw(\025)e FB(0)p Fs(,)29 b FA(B)853 4475 y Fy(1)891 4463 y FA(;)14 b(B)991 4475 y Fy(2)1051 4463 y Fw(2)23 b(E)1173 4475 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\032;\024;\033)1492 4463 y Fs(,)30 b FA(@)1591 4475 y Fx(u)1635 4463 y FA(B)1698 4475 y Fy(1)1758 4463 y Fw(2)23 b(E)1880 4475 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;\032;\024;\033)2316 4463 y Fs(and)31 b(satisfy)f Fw(h)p FA(B)2831 4475 y Fy(1)2869 4463 y Fw(i)23 b FB(=)g(0)29 b Fs(and)1631 4636 y Fw(k)p FA(B)1736 4648 y Fy(1)1773 4636 y Fw(k)1815 4648 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\032;\024;\033) 2156 4636 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2456 4601 y Fx(\021)1456 4760 y Fw(k)p FA(@)1542 4772 y Fx(u)1585 4760 y FA(B)1648 4772 y Fy(1)1685 4760 y Fw(k)1727 4772 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;\032;\024;\033)2156 4760 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2456 4726 y Fx(\021)1631 4895 y Fw(k)p FA(B)1736 4907 y Fy(2)1773 4895 y Fw(k)1815 4907 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\032;\024;\033)2156 4895 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)2417 4861 y Fy(2)2454 4895 y FA(")2493 4861 y Fy(2)p Fx(\021)r Fy(+1)2650 4895 y FA(:)3661 4767 y FB(\(171\))p Black 169 5097 a Fs(4.)p Black 42 w(L)l(et)30 b(us)f(c)l(onsider)i FA(h)906 5109 y Fy(1)943 5097 y FA(;)14 b(h)1028 5109 y Fy(2)1088 5097 y Fw(2)23 b FA(B)t FB(\()p FA(\027)5 b FB(\))24 b Fw(\032)f(E)1499 5109 y Fx(`)p Fy(+1)p Fx(;\032;\024;\033) 1818 5097 y Fs(with)30 b FA(\027)e Fw(\034)23 b FB(1)p Fs(.)38 b(Then,)p Black 378 5258 a Fw(\017)p Black 41 w Fs(If)30 b FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b(<)c FB(0)p Fs(,)842 5446 y Fw(k)p FA(C)6 b FB(\()p FA(h)1029 5458 y Fy(2)1066 5446 y FA(;)14 b(u;)g(\034)9 b FB(\))19 b Fw(\000)f FA(C)6 b FB(\()p FA(h)1512 5458 y Fy(1)1550 5446 y FA(;)14 b(u;)g(\034)9 b FB(\))p Fw(k)1791 5471 y Fx(`)p Fy(+1)p Fx(;\032;\024;\033)2103 5446 y Fw(\024)22 b FA(K)2511 5389 y(\027)p 2277 5426 516 4 v 2277 5504 a(")2316 5480 y Fy(max)o Fv(f)p Fy(0)p Fx(;`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fv(g)2802 5446 y Fw(k)p FA(h)2892 5458 y Fy(2)2947 5446 y Fw(\000)c FA(h)3078 5458 y Fy(1)3116 5446 y Fw(k)3158 5458 y Fx(`)p Fy(+1)p Fx(;\032;\024;\033)3446 5446 y FA(:)p Black 1919 5753 a FB(65)p Black eop end %%Page: 66 66 TeXDict begin 66 65 bop Black Black Black 378 272 a Fw(\017)p Black 41 w Fs(If)30 b FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b Fw(\025)c FB(0)p Fs(,)1012 454 y Fw(k)o FA(C)6 b FB(\()p FA(h)1198 466 y Fy(2)1236 454 y FA(;)14 b(u;)g(\034)9 b FB(\))18 b Fw(\000)h FA(C)6 b FB(\()p FA(h)1682 466 y Fy(1)1719 454 y FA(;)14 b(u;)g(\034)9 b FB(\))p Fw(k)1960 479 y Fy(2)p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+2)p Fx(;\032;\024;\033) 2423 454 y Fw(\024)22 b FA(K)6 b(\027)f Fw(k)p FA(h)2723 466 y Fy(2)2778 454 y Fw(\000)18 b FA(h)2909 466 y Fy(1)2946 454 y Fw(k)2988 466 y Fx(`)p Fy(+1)p Fx(;\032;\024;\033)3277 454 y FA(:)p Black 71 652 a Fs(Pr)l(o)l(of.)p Black 43 w FB(F)-7 b(or)29 b(the)h(\014rst)f(b)r(ounds,)h(w)n(e)f(split)h FA(A)d FB(=)f FA(A)1667 664 y Fy(1)1724 652 y FB(+)19 b FA(A)1870 664 y Fy(2)1928 652 y FB(+)g FA(A)2074 664 y Fy(3)2111 652 y FB(,)31 b(where)e FA(A)2469 664 y Fx(i)2496 652 y FB(,)i FA(i)25 b FB(=)h(1)p FA(;)14 b FB(2)p FA(;)g FB(3,)29 b(are)f(the)i(functions)g(de\014ned)71 751 y(in)e(\(152\))o(,) g(\(153\))e(and)i(\(154\))f(resp)r(ectiv)n(ely)-7 b(.)195 851 y(Using)29 b(\(155\))e(and)g(\(15\),)g(one)h(can)f(see)g(that)h FA(A)1687 863 y Fy(1)1748 851 y Fw(2)23 b(E)1870 863 y Fx(r)r Fy(+1)p Fx(;\032;\016)o(;\033)2177 851 y Fw(\032)f(E)2308 863 y Fx(`)p Fy(+1)p Fx(;\032;\016)o(;\033)2615 851 y FB(and)1207 1033 y Fw(k)p FA(A)1311 1045 y Fy(1)1348 1033 y Fw(k)1390 1045 y Fx(`)p Fy(+1)p Fx(;\032;\016)o(;\033)1691 1033 y Fw(\024)h(k)p FA(A)1883 1045 y Fy(1)1920 1033 y Fw(k)1962 1045 y Fx(r)r Fy(+1)p Fx(;\032;\016)o(;\033)2268 1033 y Fw(\024)g FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2568 999 y Fx(\021)r Fy(+1)2692 1033 y FA(:)946 b FB(\(172\))71 1214 y(Applying)28 b(Lemma)f(6.2,)g(w)n(e)g(obtain)g Fw(kG)1355 1226 y Fx(")1391 1214 y FB(\()p FA(@)1467 1226 y Fx(u)1510 1214 y FA(A)1572 1226 y Fy(1)1610 1214 y FB(\))p Fw(k)1684 1226 y Fx(`)p Fy(+1)p Fx(;\032;\016)o(;\033)1986 1214 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2285 1184 y Fx(\021)r Fy(+1)2409 1214 y FB(.)195 1314 y(F)-7 b(or)42 b(the)g(other)g(terms,)j(let)e(us)f(p)r(oin)n(t)g (out)g(that,)k(b)n(y)c(the)h(de\014nition)f(of)g FA(`)p FB(,)k FA(A)2836 1326 y Fy(2)2873 1314 y FA(;)14 b(A)2972 1326 y Fy(3)3057 1314 y Fw(2)48 b(E)3204 1326 y Fx(`;\032;\016)o(;\033) 3398 1314 y FB(.)81 b(Therefore)71 1414 y FA(@)115 1426 y Fx(u)158 1414 y FA(A)220 1426 y Fy(2)258 1414 y FA(;)14 b(@)339 1426 y Fx(u)382 1414 y FA(A)444 1426 y Fy(3)505 1414 y Fw(2)23 b(E)627 1426 y Fx(`)p Fy(+1)p Fx(;\032;\016)o(;\033)934 1414 y FB(and)k(satisfy)g Fw(k)p FA(@)1440 1426 y Fx(u)1483 1414 y FA(A)1545 1426 y Fy(2)1583 1414 y Fw(k)1625 1426 y Fx(`)p Fy(+1)p Fx(;\032;\016)o(;\033)1926 1414 y Fw(\024)c FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2226 1384 y Fx(\021)2293 1414 y FB(and)28 b Fw(k)p FA(@)2541 1426 y Fx(u)2584 1414 y FA(A)2646 1426 y Fy(3)2683 1414 y Fw(k)2725 1426 y Fx(`)p Fy(+1)p Fx(;\032;\016)o(;\033)3027 1414 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)3287 1384 y Fy(2)3324 1414 y FA(")3363 1384 y Fy(2)p Fx(\021)r Fy(+1)3520 1414 y FB(.)195 1513 y(T)-7 b(o)28 b(b)r(ound)g Fw(G)623 1525 y Fx(")659 1513 y FB(\()p FA(A)753 1525 y Fy(2)790 1513 y FB(\),)g(let)g(us)g(p)r(oin)n(t)g(out)f(that)h Fw(h)p FA(A)1738 1525 y Fy(2)1776 1513 y Fw(i)23 b FB(=)g(0)k(and)g(then,)i(b) n(y)e(Lemma)g(6.2,)982 1695 y Fw(kG)1073 1707 y Fx(")1123 1695 y FB(\()p FA(@)1199 1707 y Fx(u)1242 1695 y FA(A)1304 1707 y Fy(2)1342 1695 y FB(\))p Fw(k)1416 1720 y Fx(`)p Fy(+1)p Fx(;\032;\016)o(;\033)1718 1695 y Fw(\024)22 b FA(K)6 b(")p Fw(k)p FA(@)2007 1707 y Fx(u)2049 1695 y FA(A)2111 1707 y Fy(2)2149 1695 y Fw(k)2191 1707 y Fx(`)p Fy(+1)p Fx(;\032;\016)o(;\033)2492 1695 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2792 1661 y Fx(\021)r Fy(+1)2916 1695 y FA(:)71 1889 y FB(Applying)28 b(again)e(Lemma)h(6.2)g (w)n(e)g(ha)n(v)n(e)g Fw(kG)1487 1901 y Fx(")1522 1889 y FB(\()p FA(@)1598 1901 y Fx(u)1642 1889 y FA(A)1704 1901 y Fy(3)1741 1889 y FB(\))p Fw(k)1815 1901 y Fx(`)p Fy(+1)p Fx(;\032;\016)o(;\033)2117 1889 y Fw(\024)c FA(K)6 b Fw(j)p FA(\026)p Fw(j)2378 1859 y Fy(2)2415 1889 y FA(")2454 1859 y Fy(2)p Fx(\021)r Fy(+1)2611 1889 y FB(.)37 b(Therefore)1360 2071 y Fw(kG)1451 2083 y Fx(")1487 2071 y FB(\()p FA(@)1563 2083 y Fx(u)1606 2071 y FA(A)p FB(\))p Fw(k)1742 2096 y Fx(`)p Fy(+1)p Fx(;\032;\016)o(;\033)2044 2071 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)2305 2037 y Fy(2)2342 2071 y FA(")2381 2037 y Fy(2)p Fx(\021)r Fy(+1)2538 2071 y FA(:)71 2253 y FB(The)28 b(other)f(b)r(ounds)g(are)g (straigh)n(tforw)n(ard.)p 3790 2253 4 57 v 3794 2200 50 4 v 3794 2253 V 3843 2253 4 57 v 195 2418 a(T)-7 b(o)29 b(pro)n(v)n(e)e(Prop)r(osition)g(7.4,)i(w)n(e)g(pro)r(ceed)f(as)g(in)h (the)h(pro)r(ofs)e(of)h(Prop)r(ositions)e(6.4)h(and)g(6.10.)40 b(That)29 b(is,)g(w)n(e)g(\014rst)71 2518 y(p)r(erform)e(a)g(c)n(hange) f(of)i(v)-5 b(ariables)26 b(whic)n(h)i(reduces)e(the)i(size)f(of)h(the) g(linear)f(terms)g(of)g Fw(F)36 b FB(in)28 b(\(142\))o(.)37 b(Let)27 b(us)h(p)r(oin)n(t)f(out,)71 2618 y(that)33 b(for)g(the)g(pro)r(of)g(of)g(Prop)r(osition)f(7.4)g(w)n(e)h(could)g (lo)r(ok)f(for)h(this)g(c)n(hange)f(as)h(the)g(analytic)g(con)n(tin)n (uation)f(of)h(the)71 2717 y(c)n(hanges)25 b(obtained)h(in)h(Lemmas)f (6.5)f(and)i(6.11.)35 b(Nev)n(ertheless,)25 b(since)i(w)n(e)f(w)n(an)n (t)f(the)i(pro)r(of)f(of)h(Theorem)e(4.4)h(b)r(e)h(also)71 2817 y(v)-5 b(alid)29 b(for)g(Theorem)g(4.8,)g(w)n(e)h(lo)r(ok)e(for)h (a)h(c)n(hange)e FA(g)k FB(whic)n(h)e(is)f(not)h(necessarily)e(con)n (tin)n(uation)g(of)i(the)g(one)f(obtained)71 2917 y(in)f(Lemmas)f(6.5)g (and)g(6.11.)p Black 71 3082 a Fp(Lemma)j(7.6.)p Black 39 w Fs(L)l(et)e(us)g(c)l(onsider)i FA(\024)1221 3094 y Fy(1)1281 3082 y FA(>)23 b(\024)1417 3052 y Fv(0)1417 3102 y Fy(0)1477 3082 y FA(>)g(\024)1613 3094 y Fy(0)1673 3082 y FA(>)f FB(0)29 b Fs(and)g FA(\032)2034 3052 y Fv(00)2034 3102 y Fy(1)2099 3082 y FA(>)23 b(\032)2230 3052 y Fv(0)2230 3102 y Fy(1)2290 3082 y FA(>)g(\032)2421 3094 y Fy(2)2481 3082 y FA(>)g(\032)2612 3052 y Fv(0)2612 3102 y Fy(0)2649 3082 y Fs(,)29 b(wher)l(e)g FA(\032)2979 3052 y Fv(0)2979 3102 y Fy(0)3045 3082 y Fs(is)g(the)g(c)l(onstant)f (de\014ne)l(d)71 3181 y(in)36 b(L)l(emmas)f(6.5)i(and)g(6.11.)58 b(Then,)38 b(for)f FA(")c(>)h FB(0)h Fs(smal)t(l)h(enough)h(and)f FA(\024)2426 3151 y Fv(0)2426 3202 y Fy(0)2498 3181 y Fs(big)h(enough,)h(ther)l(e)e(exists)f(a)h(function)g FA(g)71 3281 y Fs(which)31 b(is)f(solution)g(of)49 b FB(\(156\))28 b Fs(and)j(satis\014es:)p Black 195 3446 a Fw(\017)p Black 41 w Fs(If)g FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b(<)e FB(0)p Fs(,)29 b FA(g)d Fw(2)d(E)978 3460 y Fy(0)p Fx(;\032)1065 3441 y Fl(0)1065 3479 y Fu(1)1098 3460 y Fx(;\024)1157 3441 y Fl(0)1157 3479 y Fu(0)1189 3460 y Fx(;\033)1283 3446 y Fs(and)1661 3641 y Fw(k)p FA(g)s Fw(k)1788 3655 y Fy(0)p Fx(;\032)1875 3635 y Fl(0)1875 3673 y Fu(1)1907 3655 y Fx(;\024)1966 3635 y Fl(0)1966 3673 y Fu(0)1998 3655 y Fx(;\033)2085 3641 y Fw(\024)g FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2385 3607 y Fx(\021)r Fy(+1)1620 3778 y Fw(k)p FA(@)1706 3790 y Fx(v)1745 3778 y FA(g)s Fw(k)1830 3792 y Fy(0)p Fx(;\032)1917 3773 y Fl(0)1917 3811 y Fu(1)1949 3792 y Fx(;\016)o(;\033)2085 3778 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2385 3744 y Fx(\021)r Fy(+1)p Black 195 3992 a Fw(\017)p Black 41 w Fs(If)31 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b Fw(\025)e FB(0)p Fs(,)29 b FA(g)d Fw(2)d(E)978 4006 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\032)1174 3986 y Fl(0)1174 4024 y Fu(1)1207 4006 y Fx(;\024)1266 3986 y Fl(0)1266 4024 y Fu(0)1298 4006 y Fx(;\033)1392 3992 y Fs(and)1693 4187 y Fw(k)p FA(g)s Fw(k)1820 4201 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\032)2016 4181 y Fl(0)2016 4219 y Fu(1)2047 4201 y Fx(;\016)o(;\033)2183 4187 y Fw(\024)g FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2483 4152 y Fx(\021)r Fy(+1)1522 4324 y Fw(k)p FA(@)1608 4336 y Fx(v)1647 4324 y FA(g)s Fw(k)1732 4338 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;\032)2016 4318 y Fl(0)2016 4356 y Fu(1)2047 4338 y Fx(;\016)o(;\033)2183 4324 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2483 4290 y Fx(\021)r Fy(+1)71 4555 y Fs(Mor)l(e)l(over,)31 b FA(v)22 b FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))24 b Fw(2)g FA(D)1012 4516 y Fy(out)o Fx(;u)1010 4583 y(\032)1044 4563 y Fl(00)1044 4601 y Fu(1)1085 4583 y Fx(;\024)1144 4591 y Fu(0)1209 4555 y Fs(for)31 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b Fw(2)g FA(D)1705 4516 y Fy(out)p Fx(;u)1703 4583 y(\032)1737 4563 y Fl(0)1737 4601 y Fu(1)1769 4583 y Fx(;\024)1828 4563 y Fl(0)1828 4601 y Fu(0)1883 4555 y Fw(\002)18 b Ft(T)2021 4567 y Fx(\033)2066 4555 y Fs(.)195 4672 y(F)-6 b(urthermor)l(e,)27 b(the)f(change)g(of)g (variables)i FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)e(\()p FA(v)12 b FB(+)d FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))28 b Fs(is)e(invertible)g(and)g(its) g(inverse)g(is)g(of)g(the)g(form)71 4771 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f(\()p FA(u)18 b FB(+)g FA(h)p FB(\()p FA(u;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))p Fs(.)40 b(The)31 b(function)f FA(h)f Fs(is)h(de\014ne)l(d)g(in)g(the)g (domain)h FA(D)2525 4741 y Fy(out)p Fx(;u)2523 4792 y(\032)2557 4800 y Fu(2)2590 4792 y Fx(;\024)2649 4800 y Fu(1)2703 4771 y Fw(\002)18 b Ft(T)2841 4783 y Fx(\033)2916 4771 y Fs(and)30 b(it)g(satis\014es)p Black 195 4937 a Fw(\017)p Black 41 w Fs(If)h FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b(<)e FB(0)1627 5036 y Fw(k)p FA(h)p Fw(k)1759 5048 y Fy(0)p Fx(;\032)1846 5056 y Fu(2)1877 5048 y Fx(;\024)1936 5056 y Fu(1)1968 5048 y Fx(;\033)2056 5036 y Fw(\024)f FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2355 5002 y Fx(\021)r Fy(+1)2479 5036 y FA(:)p Black 195 5218 a Fw(\017)p Black 41 w Fs(If)31 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b Fw(\025)e FB(0)1572 5318 y Fw(k)p FA(h)p Fw(k)1704 5330 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\032)1900 5338 y Fu(2)1932 5330 y Fx(;\024)1991 5338 y Fu(1)2023 5330 y Fx(;\033)2110 5318 y Fw(\024)g FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2410 5284 y Fx(\021)r Fy(+1)2533 5318 y FA(:)71 5500 y Fs(Mor)l(e)l(over,)31 b FA(u)18 b FB(+)g FA(h)p FB(\()p FA(u;)c(\034)9 b FB(\))24 b Fw(2)f FA(D)1025 5460 y Fy(out)p Fx(;u)1023 5528 y(\032)1057 5508 y Fl(0)1057 5546 y Fu(1)1090 5528 y Fx(;\024)1149 5508 y Fl(0)1149 5546 y Fu(0)1215 5500 y Fs(for)30 b FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)g FA(D)1715 5470 y Fy(out)o Fx(;u)1713 5520 y(\032)1747 5528 y Fu(2)1779 5520 y Fx(;\024)1838 5528 y Fu(1)1893 5500 y Fw(\002)18 b Ft(T)2031 5512 y Fx(\033)2076 5500 y Fs(.)p Black 1919 5753 a FB(66)p Black eop end %%Page: 67 67 TeXDict begin 67 66 bop Black Black 195 272 a FB(In)28 b(the)g(case)e FA(`)18 b Fw(\000)f FB(2)p FA(r)26 b(<)c FB(0)27 b(w)n(e)g(need)h(more)e(precise)h(b)r(ounds)h(of)f(b)r(oth)h (functions)f FA(g)j FB(and)e FA(h)f FB(restricted)g(to)g(the)h(inner)71 372 y(domain)f FA(D)437 342 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)435 392 y(\024)474 400 y Fu(1)506 392 y Fx(;c)654 372 y FB(de\014ned)h(in)g (\(30\))o(.)37 b(These)28 b(b)r(ounds)f(are)g(giv)n(en)g(in)h(next)f (corollary)-7 b(.)p Black 71 538 a Fp(Corollary)34 b(7.7.)p Black 41 w Fs(L)l(et)d(us)f(assume)h FA(`)19 b Fw(\000)g FB(2)p FA(r)28 b(<)c FB(0)31 b Fs(and)g(let)g(us)g(c)l(onsider)h(a)f(c) l(onstant)f FA(c)2802 550 y Fy(1)2865 538 y FA(>)25 b FB(0)p Fs(.)41 b(Then,)33 b(the)e(functions)g FA(g)71 637 y Fs(ad)f FA(h)g Fs(obtaine)l(d)h(in)f(L)l(emma)g(7.6)h(r)l (estricte)l(d)e(to)h(the)g(inner)g(domain)h FA(D)2318 607 y Fy(in)o Fx(;)p Fy(+)p Fx(;u)2316 658 y(\024)2355 666 y Fu(1)2387 658 y Fx(;c)2437 666 y Fu(1)2537 637 y Fs(satisfy)g(the)f(fol)t(lowing)i(b)l(ounds)396 839 y FB(sup)14 b Fw(j)p FA(g)s FB(\()p FA(u;)g(\034)9 b FB(\))p Fw(j)818 867 y Fy(\()p Fx(u;\034)e Fy(\))p Fv(2)p Fx(D)1068 839 y Fu(in)p Fm(;)p Fu(+)p Fm(;u)1066 876 y(\024)1100 888 y Fu(1)1133 876 y Fm(;c)1179 888 y Fu(1)1236 867 y Fv(\002)p Fn(T)1327 875 y Fm(\033)1393 839 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1692 805 y Fx(\021)r Fy(+1+)p Fx(\027)1900 780 y Fl(\003)1896 822 y Fu(1)1996 839 y Fs(and)45 b FB(sup)13 b Fw(j)p FA(h)p Fw(j)2404 867 y Fy(\()p Fx(u;\034)7 b Fy(\))p Fv(2)p Fx(D)2654 839 y Fu(in)o Fm(;)p Fu(+)p Fm(;u)2652 876 y(\024)2686 888 y Fu(1)2718 876 y Fm(;c)2764 888 y Fu(1)2821 867 y Fv(\002)p Fn(T)2912 875 y Fm(\033)2979 839 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")3278 805 y Fx(\021)r Fy(+1+)p Fx(\027)3486 780 y Fl(\003)3482 822 y Fu(1)71 1042 y Fs(with)30 b FA(\027)297 1012 y Fv(\003)292 1063 y Fy(1)359 1042 y FB(=)22 b(min)q Fw(f)p FB(2)p FA(r)e Fw(\000)e FA(`)p FB(\))p FA(\015)5 b(;)14 b FB(1)p Fw(g)p Fs(.)p Black 71 1208 a(Pr)l(o)l(of)31 b(of)f(L)l(emma)g(7.6)h(and)g(c)l (or)l(ol)t(lary)g(7.7.)p Black 44 w FB(T)-7 b(o)20 b(de\014ne)i FA(g)s FB(,)g(let)f(us)g(recall)f(\014rst)h(that,)i(b)n(y)e(Lemma)g (7.5,)g Fw(h)p FA(B)3381 1220 y Fy(1)3419 1208 y Fw(i)i FB(=)g(0.)34 b(Then)71 1308 y(w)n(e)d(can)f(de\014ne)h(a)g(function)p 997 1241 68 4 v 32 w FA(B)1064 1320 y Fy(1)1132 1308 y FB(suc)n(h)g(that)g FA(@)1550 1320 y Fx(\034)p 1592 1241 V 1592 1308 a FA(B)1659 1320 y Fy(1)1725 1308 y FB(=)d FA(B)1881 1320 y Fy(1)1949 1308 y FB(and)j Fw(h)p 2146 1241 V FA(B)2214 1320 y Fy(1)2251 1308 y Fw(i)e FB(=)f(0.)47 b(Then,)32 b(one)f(can)f(see)h(that)g(a)g(solution)g(of)71 1407 y(equation)c(\(156\))o(,)h(can)f(b)r(e)h(giv)n(en)f(b)n(y)1244 1590 y FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f Fw(\000)p FA(")p 1692 1523 V(B)1758 1602 y Fy(1)1796 1590 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))19 b(+)f FA(")p Fw(G)2175 1602 y Fx(")2211 1590 y FB(\()p FA(@)2287 1602 y Fx(v)p 2327 1523 V 2327 1590 a FA(B)2394 1602 y Fy(1)2431 1590 y FB(\)\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)986 b FB(\(173\))71 1773 y(where)27 b Fw(G)360 1785 y Fx(")424 1773 y FB(is)g(the)h(in)n(tegral)e(op)r(erator)g(de\014ned)i (in)g(\(168\))o(.)195 1872 y(By)g(Lemma)f(7.5,)g(if)h FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b Fw(\025)e FB(0)k(one)g(has:)1563 1983 y Fz(\015)1563 2032 y(\015)p 1609 1986 V 21 x FA(B)1677 2065 y Fy(1)1714 1983 y Fz(\015)1714 2032 y(\015)1760 2086 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\032)1956 2094 y Fu(2)1988 2086 y Fx(;\024)2047 2066 y Fl(0)2047 2104 y Fu(0)2079 2086 y Fx(;\033)2167 2053 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2466 2019 y Fx(\021)1392 2140 y Fz(\015)1392 2190 y(\015)1438 2211 y FA(@)1482 2223 y Fx(v)p 1522 2144 V 1522 2211 a FA(B)1589 2223 y Fy(1)1626 2140 y Fz(\015)1626 2190 y(\015)1672 2244 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;\032)1956 2252 y Fu(2)1988 2244 y Fx(;\024)2047 2224 y Fl(0)2047 2262 y Fu(0)2079 2244 y Fx(;\033)2167 2211 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2466 2177 y Fx(\021)2506 2211 y FA(:)3661 2142 y FB(\(174\))71 2411 y(If)28 b Fw(\000)p FB(1)22 b Fw(\024)h FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b(<)e FB(0)1672 2516 y Fz(\015)1672 2566 y(\015)p 1718 2520 V 21 x FA(B)1785 2599 y Fy(1)1822 2516 y Fz(\015)1822 2566 y(\015)1869 2620 y Fy(0)p Fx(;\032)1956 2628 y Fu(2)1988 2620 y Fx(;\024)2047 2600 y Fl(0)2047 2638 y Fu(0)2079 2620 y Fx(;\033)2167 2587 y Fw(\024)f FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2466 2553 y Fx(\021)1392 2674 y Fz(\015)1392 2724 y(\015)1438 2745 y FA(@)1482 2757 y Fx(v)p 1522 2678 V 1522 2745 a FA(B)1589 2757 y Fy(1)1626 2674 y Fz(\015)1626 2724 y(\015)1672 2778 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;\032)1956 2786 y Fu(2)1988 2778 y Fx(;\024)2047 2758 y Fl(0)2047 2796 y Fu(0)2079 2778 y Fx(;\033)2167 2745 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2466 2711 y Fx(\021)2506 2745 y FA(:)3661 2676 y FB(\(175\))71 2945 y(Finally)-7 b(,)28 b(if)g FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b(<)e Fw(\000)p FB(1)1574 3055 y Fz(\015)1574 3105 y(\015)p 1620 3059 V 21 x FA(B)1687 3138 y Fy(1)1724 3055 y Fz(\015)1724 3105 y(\015)1770 3159 y Fy(0)p Fx(;\032)1857 3167 y Fu(2)1890 3159 y Fx(;\024)1949 3139 y Fl(0)1949 3177 y Fu(0)1981 3159 y Fx(;\033)2069 3126 y Fw(\024)f FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2368 3092 y Fx(\021)1490 3213 y Fz(\015)1490 3263 y(\015)1536 3284 y FA(@)1580 3296 y Fx(v)p 1620 3217 V 1620 3284 a FA(B)1687 3296 y Fy(1)1724 3213 y Fz(\015)1724 3263 y(\015)1770 3317 y Fy(0)p Fx(;\032)1857 3325 y Fu(2)1890 3317 y Fx(;\024)1949 3297 y Fl(0)1949 3335 y Fu(0)1981 3317 y Fx(;\033)2069 3284 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2368 3249 y Fx(\021)2408 3284 y FA(:)3661 3215 y FB(\(176\))195 3484 y(F)-7 b(rom)28 b(these)f(inequalities,)h(using)f(Lemma)g(7.3)g(w)n(e)g(conclude)g (that:)1011 3596 y Fz(\015)1011 3646 y(\015)1058 3666 y FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))19 b(+)f FA(")p 1431 3600 V(B)1498 3678 y Fy(1)1535 3666 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))1724 3596 y Fz(\015)1724 3646 y(\015)1772 3700 y Fy(max)o Fv(f)p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;)p Fy(0)p Fv(g)p Fx(;\032)2299 3708 y Fu(2)2331 3700 y Fx(;\024)2390 3680 y Fl(0)2390 3718 y Fu(0)2422 3700 y Fx(;\033)2509 3666 y Fw(\024)23 b FA(K)6 b(\026")2763 3632 y Fx(\021)r Fy(+2)2887 3666 y FA(;)71 3871 y FB(whic)n(h,)27 b(together)f(with)i(\(174\))e(when)h FA(`)17 b Fw(\000)g FB(2)p FA(r)26 b Fw(\025)c FB(0)27 b(and)g(with)g(\(175\))g(and)f(\(176\))h(when)g FA(`)17 b Fw(\000)g FB(2)p FA(r)25 b(<)e FB(0,)k(giv)n(es)f(the)h(desired)71 3971 y(b)r(ounds)33 b(for)f FA(g)s FB(.)51 b(F)-7 b(or)32 b(the)h(pro)r(of)f(of)h(the)g(b)r(ound)g(of)f FA(@)1790 3983 y Fx(v)1830 3971 y FA(g)j FB(it)e(is)f(enough)g(to)h(apply)f (again)g(Lemmas)g(7.3)f(and)i(7.5)f(and)71 4071 y(\(174\))o(.)195 4170 y(The)c(rest)f(of)h(the)g(statemen)n(ts)f(are)g(straigh)n(tforw)n (ard.)195 4270 y(T)-7 b(o)34 b(pro)r(of)g(corollary)d(7.7)j(w)n(e)f (just)i(need)f(to)g(use)g(the)h(de\014nition)f(of)g FA(B)2485 4282 y Fy(1)2557 4270 y FB(in)g(\(144\))o(,)i(and)e(observ)n(e)e(that)j (it)f(has)g(a)71 4369 y(singularit)n(y)26 b(or)h(order)f FA(`)18 b Fw(\000)g FB(2)p FA(r)30 b FB(if)e FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b Fw(\025)c FB(0)27 b(and)h(a)f(zero)f(of)i (order)e(2)p FA(r)21 b Fw(\000)d FA(`)27 b FB(if)h FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b Fw(\024)c FB(0.)p 3790 4369 4 57 v 3794 4317 50 4 v 3794 4369 V 3843 4369 4 57 v 195 4535 a(Once)28 b(w)n(e)f(ha)n(v)n(e)f(the)i(c)n(hange)f FA(g)s FB(,)g(w)n(e)g(pro)r(ceed)g(as)g(in)h(Section)f(6.1.2,)g (de\014ning)1476 4697 y Fz(b)1462 4718 y FA(T)1511 4730 y Fy(1)1548 4718 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f FA(T)1898 4730 y Fy(1)1935 4718 y FB(\()p FA(v)e FB(+)e FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))1203 b(\(177\))71 4901 y(whic)n(h)27 b(is)h(solution)f(of)34 b(\(157\))o(,)28 b(that)g(is:)1622 5025 y Fw(L)1679 5037 y Fx(")1729 5004 y Fz(b)1715 5025 y FA(T)1764 5037 y Fy(1)1824 5025 y FB(=)1932 5004 y Fz(b)1912 5025 y Fw(F)1993 4933 y Fz(\020)2043 5025 y FA(@)2087 5037 y Fx(v)2140 5004 y Fz(b)2127 5025 y FA(T)2176 5037 y Fy(1)2212 4933 y Fz(\021)2276 5025 y FA(:)71 5218 y FB(W)-7 b(e)41 b(lo)r(ok)f(for)g(it)h(using)g(a)f(\014xed)h(p)r(oin)n (t)g(argumen)n(t)e(on)i FA(@)1970 5230 y Fx(v)2023 5197 y Fz(b)2009 5218 y FA(T)2058 5230 y Fy(1)2095 5218 y FB(.)76 b(Nev)n(ertheless,)43 b(since)e(w)n(e)f(w)n(an)n(t)g FA(@)3324 5230 y Fx(u)3368 5218 y FA(T)3417 5230 y Fy(1)3494 5218 y FB(to)h(b)r(e)g(the)71 5318 y(analytic)36 b(con)n(tin)n(uation)g (of)h(the)g(function)h FA(@)1520 5330 y Fx(u)1563 5318 y FA(T)1612 5330 y Fy(1)1686 5318 y FB(obtained)f(in)g(Prop)r(ositions) e(6.4)h(and)h(6.10,)h(w)n(e)e(ha)n(v)n(e)g(to)h(imp)r(ose)71 5418 y Fs(initial)31 b(c)l(onditions)p FB(.)40 b(Nev)n(ertheless,)27 b(since)h(w)n(e)f(in)n(v)n(ert)h Fw(L)1868 5430 y Fx(")1932 5418 y FB(b)n(y)f(using)h(the)h(op)r(erator)d Fw(G)2793 5430 y Fx(")2857 5418 y FB(de\014ned)i(in)h(\(168\))e(adapted)h(to)p Black 1919 5753 a(67)p Black eop end %%Page: 68 68 TeXDict begin 68 67 bop Black Black 71 272 a FB(the)34 b(domain)f FA(D)592 232 y Fy(out)o Fx(;u)590 300 y(\032)624 280 y Fl(0)624 318 y Fu(1)656 300 y Fx(;\016)773 272 y Fw(\002)22 b Ft(T)915 284 y Fx(\033)960 272 y FB(,)35 b(w)n(e)e(consider)f(a)h(di\013eren)n(t)h(initial)f(condition)h(dep)r (ending)f(on)g(the)h(F)-7 b(ourier)33 b(co)r(e\016cien)n(t.)71 407 y(Recall)27 b(that)h(w)n(e)f(are)g(lo)r(oking)f(for)h FA(@)1223 419 y Fx(v)1277 386 y Fz(b)1263 407 y FA(T)1312 419 y Fy(1)1376 407 y FB(de\014ned)h(in)g FA(D)1830 367 y Fy(out)p Fx(;u)1828 435 y(\032)1862 415 y Fl(0)1862 453 y Fu(1)1895 435 y Fx(;\016)2008 407 y Fw(\002)18 b Ft(T)2146 419 y Fx(\033)2191 407 y FB(.)37 b(Th)n(us,)27 b(w)n(e)g(de\014ne)1153 631 y FA(A)1215 643 y Fy(0)1253 631 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)1545 553 y Fz(X)1545 731 y Fx(k)q(<)p Fy(0)1680 631 y FA(@)1724 643 y Fx(v)1777 610 y Fz(b)1764 631 y FA(T)1825 588 y Fy([)p Fx(k)q Fy(])1813 654 y(1)1916 631 y FB(\()p 1948 586 44 4 v FA(v)1991 643 y Fy(1)2029 631 y FB(\))14 b FA(e)2114 597 y Fv(\000)p Fx(ik)q(")2256 572 y Fl(\000)p Fu(1)2334 597 y Fy(\()p Fx(v)r Fv(\000)p 2447 564 36 3 v Fx(v)2483 605 y Fu(1)2515 597 y Fy(\))2545 631 y FA(e)2584 597 y Fx(ik)q(\034)1549 864 y FB(+)1633 785 y Fz(X)1632 964 y Fx(k)q(>)p Fy(0)1768 864 y FA(@)1812 876 y Fx(v)1865 843 y Fz(b)1851 864 y FA(T)1912 821 y Fy([)p Fx(k)q Fy(])1900 886 y(1)2004 864 y FB(\()p FA(v)2076 876 y Fy(1)2113 864 y FB(\))g FA(e)2198 830 y Fv(\000)p Fx(ik)q(")2340 805 y Fl(\000)p Fu(1)2419 830 y Fy(\()p Fx(v)r Fv(\000)p Fx(v)2565 838 y Fu(1)2598 830 y Fy(\))2628 864 y FA(e)2667 830 y Fx(ik)q(\034)1549 1097 y FB(+)k FA(@)1676 1109 y Fx(v)1730 1076 y Fz(b)1716 1097 y FA(T)1777 1054 y Fy([0])1765 1119 y(1)1851 1097 y FB(\()p Fw(\000)p FA(\032)1991 1063 y Fv(0)1991 1117 y Fy(1)2028 1097 y FB(\))p FA(;)3661 857 y FB(\(178\))71 1299 y(where)23 b FA(v)347 1311 y Fy(1)385 1299 y FA(;)p 422 1253 44 4 v 14 w(v)465 1311 y Fy(1)526 1299 y FB(are)g(the)i(v)n(ertices)d(of)i (the)h(outer)e(domain)h FA(D)1904 1259 y Fy(out)p Fx(;u)1902 1327 y(\032)1936 1307 y Fl(0)1936 1345 y Fu(1)1968 1327 y Fx(;\016)2087 1299 y FB(\(see)g(Figure)f(4\))h(and)g FA(@)2808 1311 y Fx(v)2861 1278 y Fz(b)2847 1299 y FA(T)2896 1311 y Fy(1)2957 1299 y FB(can)g(b)r(e)g(obtained)g(di\013eren-)71 1416 y(tiating)k(\(177\))o(,)g(since)f FA(T)831 1428 y Fy(1)895 1416 y FB(is)g(already)f(kno)n(wn)h(in)g(a)g(neigh)n(b)r (orho)r(o)r(d)f(of)h(these)h(p)r(oin)n(ts.)36 b(Note)28 b(that)f FA(v)3228 1428 y Fy(1)3266 1416 y FA(;)p 3303 1370 V 14 w(v)3346 1428 y Fy(1)3383 1416 y FA(;)14 b(\032)3463 1385 y Fv(0)3463 1436 y Fy(1)3523 1416 y Fw(2)24 b FA(D)3673 1385 y Fx(u)3671 1436 y Fv(1)p Fx(;\032)3791 1444 y Fu(1)3827 1416 y FB(.)71 1515 y(Applying)k(the)g(b)r(ounds)f(obtained)h(in)g (Prop)r(ositions)d(6.4)i(and)g(6.10)g(and)g(Lemma)g(7.6,)g(one)g(can)h (see)f(that)1497 1698 y Fw(k)p FA(A)1601 1710 y Fy(0)1638 1698 y Fw(k)1680 1712 y Fy(0)p Fx(;\032)1767 1692 y Fl(0)1767 1730 y Fu(1)1799 1712 y Fx(;\024)1858 1692 y Fl(0)1858 1730 y Fu(0)1890 1712 y Fx(;\033)1978 1698 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2277 1663 y Fx(\021)r Fy(+1)2401 1698 y FA(:)1237 b FB(\(179\))71 1880 y(Let)28 b(us)f(de\014ne)h FA(S)5 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))29 b(as)e(the)h(solution)f(of)1238 2088 y FA(S)5 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f FA(A)1657 2100 y Fy(0)1694 2088 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b(+)e Fw(G)2035 2100 y Fx(")2085 1996 y Fz(\020)2135 2088 y FA(@)2179 2100 y Fx(v)2238 2067 y Fz(b)2218 2088 y Fw(F)8 b FB(\()p FA(S)d FB(\))2406 1996 y Fz(\021)2470 2088 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)71 2314 y FB(where)34 b Fw(G)367 2326 y Fx(")439 2314 y FB(and)628 2293 y Fz(b)607 2314 y Fw(F)44 b FB(are)34 b(the)h(op)r(erators)e(de\014ned)j(in)f(\(168\))f(and)h(\(158\))f(resp) r(ectiv)n(ely)-7 b(.)59 b(Let)35 b(us)g(p)r(oin)n(t)h(out)f(that)g(the) 71 2424 y(de\014nition)f(of)566 2403 y Fz(b)546 2424 y Fw(F)42 b FB(in)n(v)n(olv)n(es)31 b(the)j(functions)1498 2403 y Fz(b)1478 2424 y FA(A)p FB(,)1616 2403 y Fz(b)1599 2424 y FA(B)j FB(and)1883 2403 y Fz(b)1867 2424 y FA(C)i FB(de\014ned)34 b(in)g(\(159\))o(,)h(\(160\))e(and)g(\(161\))o(.)55 b(Ev)n(en)33 b(if)h(w)n(e)f(k)n(eep)71 2524 y(the)27 b(same)g(notation,)f(no)n(w)g(the)i(de\014nitions)f(in)n(v)n(olv)n(e)e (the)i(function)h FA(g)i FB(obtained)c(in)h(Lemma)g(7.6)f(instead)h(of) g(the)g(ones)71 2623 y(giv)n(en)g(in)h(Lemmas)f(6.5)f(and)i(Lemma)f (6.11.)195 2723 y(W)-7 b(e)31 b(will)f(see)g(that)h FA(S)k FB(is)30 b(the)h(analytic)e(con)n(tin)n(uation)g(of)i(the)f(function)h FA(@)2553 2735 y Fx(u)2597 2723 y FA(T)2646 2735 y Fy(1)2682 2723 y FB(\()p FA(v)24 b FB(+)c FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\)\(1)22 b(+)d FA(@)3433 2735 y Fx(v)3473 2723 y FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\)\))3737 2693 y Fv(\000)p Fy(1)3827 2723 y FB(,)71 2823 y(where)27 b FA(T)360 2835 y Fy(1)424 2823 y FB(is)h(obtained)f(from)g(Prop)r(ositions)f(6.4)h(and)g(6.10.) 195 2922 y(Th)n(us,)h(w)n(e)f(lo)r(ok)g(for)g(a)g(\014xed)h(p)r(oin)n (t)g FA(S)f Fw(2)d(E)1546 2936 y Fx(`)p Fy(+1)p Fx(;\032)1712 2917 y Fl(0)1712 2955 y Fu(1)1744 2936 y Fx(;\024)1803 2917 y Fl(0)1803 2955 y Fu(0)1835 2936 y Fx(;\033)1927 2922 y FB(of)k(the)g(op)r(erator)1170 3142 y Fw(J)16 b FB(\()p FA(S)5 b FB(\)\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f FA(A)1725 3154 y Fy(0)1762 3142 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b(+)e Fw(G)2103 3154 y Fx(")2153 3050 y Fz(\020)2203 3142 y FA(@)2247 3154 y Fx(v)2306 3121 y Fz(b)2286 3142 y Fw(F)8 b FB(\()p FA(S)d FB(\))2474 3050 y Fz(\021)2538 3142 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(:)911 b FB(\(180\))p Black 71 3349 a Fp(Lemma)33 b(7.8.)p Black 40 w Fs(L)l(et)d(us)g(c)l(onsider)i FA(")1222 3361 y Fy(0)1284 3349 y FA(>)24 b FB(0)30 b Fs(smal)t(l)i(enough)f(and) g FA(\024)2160 3319 y Fv(0)2160 3370 y Fy(0)2222 3349 y FA(>)24 b(\024)2359 3361 y Fy(0)2427 3349 y Fs(big)31 b(enough.)42 b(Then,)32 b(for)g FA(")24 b Fw(2)h FB(\(0)p FA(;)14 b(")3548 3361 y Fy(0)3585 3349 y FB(\))p Fs(,)31 b(ther)l(e)71 3449 y(exists)c(a)i(function)f FA(S)f Fw(2)d(E)893 3463 y Fx(`)p Fy(+1)p Fx(;\032)1059 3443 y Fl(0)1059 3481 y Fu(1)1091 3463 y Fx(;\024)1150 3443 y Fl(0)1150 3481 y Fu(0)1182 3463 y Fx(;\033)1274 3449 y Fs(de\014ne)l(d)k(in)g FA(D)1723 3409 y Fy(out)p Fx(;u)1721 3477 y(\032)1755 3457 y Fl(0)1755 3495 y Fu(1)1788 3477 y Fx(;\024)1847 3457 y Fl(0)1847 3495 y Fu(0)1898 3449 y Fw(\002)14 b Ft(T)2032 3461 y Fx(\033)2105 3449 y Fs(such)28 b(that)f(it)h(is)h(a)f (\014xe)l(d)f(p)l(oint)i(of)f(the)g(op)l(er)l(ator)38 b FB(\(180\))71 3571 y Fs(and)28 b(is)g(the)g(analytic)g(c)l (ontinuation)g(of)g(the)g(function)f FA(@)1843 3583 y Fx(u)1887 3571 y FA(T)1936 3583 y Fy(1)1973 3571 y FB(\()p FA(v)17 b FB(+)d FA(g)s FB(\()p FA(v)s(;)g(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\)\(1)14 b(+)g FA(@)2698 3583 y Fx(v)2737 3571 y FA(g)s FB(\()p FA(v)s(;)g(\034)9 b FB(\)\))3001 3541 y Fv(\000)p Fy(1)3119 3571 y Fs(wher)l(e)28 b FA(T)3400 3583 y Fy(1)3464 3571 y Fs(is)g(obtaine)l(d)71 3670 y(fr)l(om)i(Pr)l (op)l(ositions)h(6.4)g(and)g(6.10)g(and)f FA(g)j Fs(is)d(given)g(in)g (L)l(emma)g(7.6.)195 3770 y(Mor)l(e)l(over,)i(ther)l(e)e(exists)f(a)h (c)l(onstant)f FA(b)1459 3782 y Fy(2)1519 3770 y FA(>)23 b FB(0)29 b Fs(such)h(that)1481 3953 y Fw(k)p FA(S)5 b Fw(k)1621 3967 y Fx(`)p Fy(+1)p Fx(;\032)1787 3947 y Fl(0)1787 3985 y Fu(1)1818 3967 y Fx(;\024)1877 3947 y Fl(0)1877 3985 y Fu(0)1909 3967 y Fx(;\033)1997 3953 y Fw(\024)23 b FA(b)2121 3965 y Fy(2)2157 3953 y Fw(j)p FA(\026)p Fw(j)p FA(")2292 3918 y Fx(\021)r Fy(+1)2417 3953 y FA(:)p Black 71 4135 a Fs(Pr)l(o)l(of.)p Black 43 w FB(W)-7 b(e)28 b(recall)f(that,)h(during)f(the)h(pro)r(of,)f FA(g)j FB(is)d(the)h(function)h(giv)n(en)d(in)i(Lemma)g(7.6.)195 4235 y(It)e(is)e(straigh)n(tforw)n(ard)e(to)j(see)f(that)i Fw(J)40 b FB(is)25 b(w)n(ell)g(de\014ned)g(from)f Fw(E)2216 4249 y Fx(`)p Fy(+1)p Fx(;\032)2382 4229 y Fl(0)2382 4267 y Fu(1)2414 4249 y Fx(;\016)o(;\033)2552 4235 y FB(to)h(itself.)36 b(W)-7 b(e)26 b(are)e(going)f(to)i(pro)n(v)n(e)e (that)71 4349 y(there)k(exists)g(a)h(constan)n(t)f FA(b)953 4361 y Fy(2)1012 4349 y FA(>)c FB(0)k(suc)n(h)h(that)f Fw(J)44 b FB(is)27 b(con)n(tractiv)n(e)f(in)p 2243 4282 68 4 v 28 w FA(B)t FB(\()p FA(b)2378 4361 y Fy(2)2415 4349 y Fw(j)p FA(\026)p Fw(j)p FA(")2550 4319 y Fx(\021)r Fy(+1)2675 4349 y FB(\))d Fw(\032)g(E)2862 4363 y Fx(`)p Fy(+1)p Fx(;\032)3028 4343 y Fl(0)3028 4381 y Fu(1)3060 4363 y Fx(;\024)3119 4343 y Fl(0)3119 4381 y Fu(0)3151 4363 y Fx(;\033)3215 4349 y FB(.)195 4471 y(First,)34 b(let)f(us)g(consider)f Fw(J)15 b FB(\(0\).)52 b(F)-7 b(rom)33 b(the)g(de\014nition)g(of)g Fw(J)48 b FB(in)33 b(\(180\))f(and)g(the)h(de\014nition)g(of)3327 4450 y Fz(b)3307 4471 y Fw(F)41 b FB(in)33 b(\(158\))o(,)h(w)n(e)71 4570 y(ha)n(v)n(e)1240 4670 y Fw(J)16 b FB(\(0\)\()p FA(v)s(;)e(\034)9 b FB(\))24 b(=)f FA(A)1781 4682 y Fy(0)1818 4670 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b(+)e Fw(G)2159 4682 y Fx(")2209 4578 y Fz(\020)2259 4670 y FA(@)2303 4682 y Fx(v)2362 4649 y Fz(b)2342 4670 y FA(A)p FB(\()p FA(v)s(;)c(\034)9 b FB(\))2593 4578 y Fz(\021)2658 4670 y FA(;)71 4863 y FB(where)331 4842 y Fz(b)311 4863 y FA(A)28 b FB(is)f(the)h(function)h(in)e(\(159\))o(.)195 4973 y(T)-7 b(aking)27 b(in)n(to)g(accoun)n(t)g(the)h(de\014nition)g (of)1573 4952 y Fz(b)1554 4973 y FA(A)p FB(,)g(w)n(e)f(split)h Fw(J)16 b FB(\(0\))27 b(as)583 5156 y Fw(J)15 b FB(\(0\)\()p FA(v)s(;)f(\034)9 b FB(\))25 b(=)d FA(A)1123 5168 y Fy(0)1161 5156 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b(+)e Fw(G)1502 5168 y Fx(")1552 5156 y FB(\()p FA(@)1628 5168 y Fx(v)1667 5156 y FA(A)p FB(\()p FA(v)s(;)c(\034)9 b FB(\)\))21 b(+)d Fw(G)2103 5168 y Fx(")2152 5156 y FB(\()q FA(@)2229 5168 y Fx(v)2282 5156 y FB([)p FA(A)p FB(\()p FA(v)k FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))21 b Fw(\000)d FA(A)p FB(\()p FA(v)s(;)c(\034)9 b FB(\)])q(\))15 b FA(;)71 5338 y FB(where)22 b FA(a)g FB(is)h(giv)n(en)e(in)i(\(143\))o(.)35 b(The)23 b(\014rst)f(term)h(has)f(already)f(b)r(een)i(b)r(ounded)f(in)h(\(179\)) f(and)g(the)h(second)f(one)g(in)h(Lemma)71 5438 y(7.5.)47 b(F)-7 b(or)31 b(the)g(third)h(one,)g(using)f FA(\032)1203 5408 y Fv(00)1203 5459 y Fy(1)1276 5438 y FB(de\014ned)h(in)f(Lemma)h (7.6,)f(and)g(applying)g(Lemmas)g(7.3,)g(7.5)f(and)i(7.6)e(and)h(the)p Black 1919 5753 a(68)p Black eop end %%Page: 69 69 TeXDict begin 69 68 bop Black Black 71 272 a FB(mean)27 b(v)-5 b(alue)28 b(theorem,)374 437 y Fw(k)o(G)464 449 y Fx(")514 437 y FB(\()p FA(@)590 449 y Fx(v)644 437 y FB([)p FA(A)p FB(\()p FA(v)22 b FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(A)p FB(\()p FA(v)s(;)c(\034)9 b FB(\)])r(\))p Fw(k)1705 462 y Fx(`)p Fy(+1)p Fx(;\032)1871 442 y Fl(0)1871 480 y Fu(1)1903 462 y Fx(;\024)1962 442 y Fl(0)1962 480 y Fu(0)1994 462 y Fx(;\033)2082 437 y Fw(\024)22 b(k)p FA(A)p FB(\()p FA(v)g FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(A)p FB(\()p FA(v)s(;)c(\034)9 b FB(\))p Fw(k)3194 462 y Fx(`)p Fy(+1)p Fx(;\032)3360 442 y Fl(0)3360 480 y Fu(1)3392 462 y Fx(;\024)3451 442 y Fl(0)3451 480 y Fu(0)3483 462 y Fx(;\033)2082 583 y Fw(\024)22 b(k)p FA(@)2255 595 y Fx(u)2298 583 y FA(A)p Fw(k)2402 597 y Fx(`)p Fy(+1)p Fx(;\032)2568 578 y Fl(00)2568 616 y Fu(1)2609 597 y Fx(;\024)2668 605 y Fu(0)2700 597 y Fx(";\033)2796 583 y Fw(k)p FA(g)s Fw(k)2923 597 y Fy(0)p Fx(;\032)3010 578 y Fl(0)3010 616 y Fu(1)3041 597 y Fx(;\024)3100 578 y Fl(0)3100 616 y Fu(0)3132 597 y Fx(;\033)2082 721 y Fw(\024)g FA(K)6 b Fw(j)p FA(\026)p Fw(j)2342 686 y Fy(2)2379 721 y FA(")2418 686 y Fy(2)p Fx(\021)r Fy(+1)71 887 y FB(Th)n(us,)27 b(there)h(exists)f(a)g(constan)n(t)g FA(b)1186 899 y Fy(2)1246 887 y FA(>)22 b FB(0)28 b(suc)n(h)f(that)1410 1097 y Fw(kJ)16 b FB(\(0\))p Fw(k)1671 1122 y Fx(`)p Fy(+1)p Fx(;\032)1837 1102 y Fl(0)1837 1140 y Fu(1)1870 1122 y Fx(;\024)1929 1102 y Fl(0)1929 1140 y Fu(0)1961 1122 y Fx(;\033)2048 1097 y Fw(\024)2146 1041 y FA(b)2182 1053 y Fy(2)p 2146 1078 73 4 v 2161 1154 a FB(2)2228 1097 y Fw(j)p FA(\026)p Fw(j)p FA(")2363 1062 y Fx(\021)r Fy(+1)2488 1097 y FA(:)71 1314 y FB(Let)34 b(us)g(consider)f(no)n(w,)i FA(h)921 1326 y Fy(1)958 1314 y FA(;)14 b(h)1043 1326 y Fy(2)1114 1314 y Fw(2)p 1203 1247 68 4 v 34 w FA(B)t FB(\()p FA(b)1338 1326 y Fy(2)1375 1314 y Fw(j)p FA(\026)p Fw(j)p FA(")1510 1284 y Fx(\021)r Fy(+1)1635 1314 y FB(\))34 b Fw(\032)f(E)1843 1328 y Fx(`)p Fy(+1)p Fx(;\032)2009 1308 y Fl(0)2009 1346 y Fu(1)2041 1328 y Fx(;\024)2100 1308 y Fl(0)2100 1346 y Fu(0)2132 1328 y Fx(;\033)2197 1314 y FB(.)56 b(Then,)36 b(using)d(the)i(de\014nitions)f(of)g Fw(J)49 b FB(and)3699 1293 y Fz(b)3679 1314 y Fw(F)42 b FB(in)71 1414 y(\(180\))27 b(and)g(\(158\))g(resp)r(ectiv)n(ely)-7 b(,)27 b(and)g(applying)g(Lemma)g(7.3,)417 1609 y Fw(kJ)16 b FB(\()p FA(h)611 1621 y Fy(2)648 1609 y FB(\))j Fw(\000)f(J)d FB(\()p FA(h)933 1621 y Fy(1)971 1609 y FB(\))p Fw(k)1044 1634 y Fx(`)p Fy(+1)p Fx(;\032)1210 1614 y Fl(0)1210 1652 y Fu(1)1243 1634 y Fx(;\024)1302 1614 y Fl(0)1302 1652 y Fu(0)1334 1634 y Fx(;\033)1421 1609 y Fw(\024)23 b FA(K)1599 1514 y Fz(\015)1599 1563 y(\015)1599 1613 y(\015)1666 1588 y(b)1645 1609 y Fw(F)8 b FB(\()p FA(h)1793 1621 y Fy(2)1831 1609 y FB(\))18 b Fw(\000)1985 1588 y Fz(b)1964 1609 y Fw(F)8 b FB(\()p FA(h)2112 1621 y Fy(1)2150 1609 y FB(\))2182 1514 y Fz(\015)2182 1563 y(\015)2182 1613 y(\015)2228 1667 y Fx(`)p Fy(+1)p Fx(;\032)2394 1647 y Fl(0)2394 1685 y Fu(1)2427 1667 y Fx(;\024)2486 1647 y Fl(0)2486 1685 y Fu(0)2518 1667 y Fx(;\033)1421 1814 y Fw(\024)23 b FA(K)1599 1718 y Fz(\015)1599 1768 y(\015)1599 1818 y(\015)1663 1793 y(b)1645 1814 y FA(B)g Fw(\001)c FB(\()p FA(h)1853 1826 y Fy(2)1908 1814 y Fw(\000)f FA(h)2039 1826 y Fy(1)2076 1814 y FB(\))h(+)2227 1793 y Fz(b)2210 1814 y FA(C)6 b FB(\()p FA(h)2355 1826 y Fy(2)2393 1814 y FA(;)14 b(v)s(;)g(\034)9 b FB(\))19 b Fw(\000)2706 1793 y Fz(b)2689 1814 y FA(C)7 b FB(\()p FA(h)2835 1826 y Fy(1)2872 1814 y FA(;)14 b(v)s(;)g(\034)9 b FB(\))3066 1718 y Fz(\015)3066 1768 y(\015)3066 1818 y(\015)3113 1872 y Fx(`)p Fy(+1)p Fx(;\032)3279 1852 y Fl(0)3279 1890 y Fu(1)3311 1872 y Fx(;\024)3370 1852 y Fl(0)3370 1890 y Fu(0)3402 1872 y Fx(;\033)3481 1814 y FA(:)71 2050 y FB(T)-7 b(o)28 b(b)r(ound)h(the)g(Lipsc)n(hitz)g (constan)n(t)f(of)g Fw(J)16 b FB(,)29 b(one)f(has)g(to)h(tak)n(e)f(in)n (to)g(accoun)n(t)g(the)h(de\014nitions)f(of)3229 2029 y Fz(b)3211 2050 y FA(B)33 b FB(and)3486 2029 y Fz(b)3469 2050 y FA(C)i FB(in)29 b(\(160\))71 2150 y(and)37 b(\(161\))f(resp)r (ectiv)n(ely)-7 b(,)39 b(and)d(to)h(apply)g(Lemmas)g(7.5)f(and)h(7.6.) 64 b(W)-7 b(e)37 b(b)r(ound)h(it)f(in)g(di\013eren)n(t)h(w)n(a)n(ys)d (dep)r(ending)71 2250 y(whether)27 b FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b(<)c FB(0)28 b(or)e FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b Fw(\025)c FB(0.)37 b(In)28 b(the)g(\014rst)f(case)g(w) n(e)g(obtain)605 2420 y Fw(kJ)15 b FB(\()p FA(h)798 2432 y Fy(2)835 2420 y FB(\))k Fw(\000)f(J)e FB(\()p FA(h)1121 2432 y Fy(1)1158 2420 y FB(\))p Fw(k)1232 2445 y Fx(`)p Fy(+1)p Fx(;\032)1398 2425 y Fl(0)1398 2463 y Fu(1)1430 2445 y Fx(;\024)1489 2425 y Fl(0)1489 2463 y Fu(0)1521 2445 y Fx(;\033)1609 2420 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1909 2386 y Fx(\021)r Fy(+1)p Fv(\000)p Fy(max)o Fv(f)p Fy(0)p Fx(;`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fv(g)2571 2420 y Fw(k)o FA(h)2660 2432 y Fy(2)2716 2420 y Fw(\000)18 b FA(h)2847 2432 y Fy(1)2884 2420 y Fw(k)2926 2445 y Fx(`)p Fy(+1)p Fx(;\032)3092 2425 y Fl(0)3092 2463 y Fu(1)3124 2445 y Fx(;\024)3183 2425 y Fl(0)3183 2463 y Fu(0)3215 2445 y Fx(;\033)3293 2420 y FA(;)71 2600 y FB(and)27 b(in)h(the)g(second,)747 2825 y Fw(kJ)15 b FB(\()p FA(h)940 2837 y Fy(2)978 2825 y FB(\))k Fw(\000)f(J)d FB(\()p FA(h)1263 2837 y Fy(1)1301 2825 y FB(\))p Fw(k)1374 2850 y Fx(`)p Fy(+1)p Fx(;\032)1540 2830 y Fl(0)1540 2868 y Fu(1)1573 2850 y Fx(;\024)1632 2830 y Fl(0)1632 2868 y Fu(0)1664 2850 y Fx(;\033)1751 2825 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)2049 2769 y FA(")2088 2739 y Fx(\021)r Fv(\000)p Fy(\()p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(\))p 2021 2806 384 4 v 2021 2898 a FB(\()q FA(\024)2102 2870 y Fv(0)2102 2921 y Fy(0)2139 2898 y FB(\))2171 2857 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)2429 2825 y Fw(k)o FA(h)2518 2837 y Fy(2)2574 2825 y Fw(\000)18 b FA(h)2705 2837 y Fy(1)2742 2825 y Fw(k)2783 2850 y Fx(`)p Fy(+1)p Fx(;\032)2949 2830 y Fl(0)2949 2868 y Fu(1)2982 2850 y Fx(;\024)3041 2830 y Fl(0)3041 2868 y Fu(0)3073 2850 y Fx(;\033)3151 2825 y FA(:)71 3062 y FB(Therefore,)39 b(since)e FA(\021)42 b Fw(\025)d FB(max)p Fw(f)p FB(0)p FA(;)14 b(`)23 b Fw(\000)i FB(2)p FA(r)r Fw(g)p FB(,)40 b(taking)c FA(")k(<)f(")1980 3074 y Fy(0)2054 3062 y FB(and)e FA(\024)2273 3032 y Fv(0)2273 3083 y Fy(0)2348 3062 y FB(big)g(enough,)i(Lip)p Fw(J)55 b FA(<)39 b FB(1)p FA(=)p FB(2)d(and)i(then)f Fw(J)54 b FB(is)71 3162 y(con)n(tractiv)n(e)26 b(in)p 595 3095 68 4 v 28 w FA(B)t FB(\()p FA(b)730 3174 y Fy(2)767 3162 y Fw(j)p FA(\026)p Fw(j)p FA(")902 3132 y Fx(\021)r Fy(+1)1026 3162 y FB(\))e Fw(\032)e(E)1213 3176 y Fx(`)p Fy(+1)p Fx(;\032)1379 3156 y Fl(0)1379 3194 y Fu(1)1411 3176 y Fx(;\024)1470 3156 y Fl(0)1470 3194 y Fu(0)1502 3176 y Fx(;\033)1595 3162 y FB(and)27 b(it)h(has)f(a)g(unique)h(\014xed)g(p) r(oin)n(t)f FA(S)5 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\).)195 3284 y(No)n(w,)33 b(w)n(e)e(ha)n(v)n(e)g(to)h(pro)n(v)n(e)e(that)i FA(S)5 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))33 b(is)e(the)i(analytic) e(con)n(tin)n(uation)g(of)g(the)i(function)3157 3263 y Fz(e)3145 3284 y FA(S)5 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))31 b(=)f FA(@)3560 3296 y Fx(u)3603 3284 y FA(T)3652 3296 y Fy(1)3689 3284 y FB(\()p FA(v)25 b FB(+)71 3383 y FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\)\(1)23 b(+)d FA(@)643 3395 y Fx(v)683 3383 y FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\)\))947 3353 y Fv(\000)p Fy(1)1069 3383 y FB(obtained)31 b(from)g(Prop)r (ositions)e(6.4)i(and)g(6.10.)47 b(First)32 b(let)f(us)h(observ)n(e)d (that)j(op)r(erator)71 3508 y(\(180\))21 b(is)h(w)n(ell)g(de\014ned)g (for)g(functions)g(in)1369 3416 y Fz(\020)1418 3508 y FA(D)1489 3478 y Fx(u)1487 3528 y Fv(1)p Fx(;\032)1607 3536 y Fu(1)1662 3508 y Fw(\\)d FA(D)1807 3468 y Fy(out)p Fx(;u)1805 3536 y(\032)1839 3516 y Fl(0)1839 3554 y Fu(1)1872 3536 y Fx(;\024)1931 3516 y Fl(0)1931 3554 y Fu(0)1967 3416 y Fz(\021)2024 3508 y Fw(\002)7 b Ft(T)2151 3520 y Fx(\033)2196 3508 y FB(.)35 b(Moreo)n(v)n(er,)20 b(b)r(oth)j (functions)f FA(S)5 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))23 b(and)3616 3487 y Fz(e)3604 3508 y FA(S)5 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))71 3666 y(are)22 b(de\014ned)i(in)580 3573 y Fz(\020)629 3666 y FA(D)700 3635 y Fx(u)698 3686 y Fv(1)p Fx(;\032)818 3694 y Fu(1)874 3666 y Fw(\\)18 b FA(D)1018 3626 y Fy(out)p Fx(;u)1016 3693 y(\032)1050 3673 y Fl(0)1050 3711 y Fu(1)1083 3693 y Fx(;\024)1142 3673 y Fl(0)1142 3711 y Fu(0)1178 3573 y Fz(\021)1238 3666 y Fw(\002)10 b Ft(T)1368 3678 y Fx(\033)1436 3666 y FB(and)23 b(for)g(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(in)f(this)g(domain)f(b)r(oth)h(are)e(\014xed)i(p)r(oin)n(ts) f(of)h(op)r(erator)d(\(180\))71 3785 y(and)1589 3790 y Fz(\015)1589 3839 y(\015)1589 3889 y(\015)1646 3864 y(e)1635 3885 y FA(S)1690 3790 y Fz(\015)1690 3839 y(\015)1690 3889 y(\015)1737 3943 y Fx(`)p Fy(+1)p Fx(;\033)1936 3885 y Fw(\024)h FA(b)2059 3897 y Fy(1)2096 3885 y FA(\026")2185 3851 y Fx(\021)r Fy(+1)2310 3885 y FA(:)195 4101 y FB(Then,)32 b(using)e(the)h(norms)e(de\014ned)i(in)g(Section)f(7.1.1)f(but)i(for)f (functions)h(de\014ned)g(in)2974 4009 y Fz(\020)3024 4101 y FA(D)3095 4071 y Fx(u)3093 4121 y Fv(1)p Fx(;\032)3213 4129 y Fu(1)3268 4101 y Fw(\\)19 b FA(D)3413 4061 y Fy(out)o Fx(;u)3411 4129 y(\032)3445 4109 y Fl(0)3445 4147 y Fu(1)3477 4129 y Fx(;\024)3536 4109 y Fl(0)3536 4147 y Fu(0)3572 4009 y Fz(\021)3642 4101 y Fw(\002)h Ft(T)3782 4113 y Fx(\033)3827 4101 y FB(,)71 4221 y(one)27 b(can)g(see)g(that)876 4300 y Fz(\015)876 4350 y(\015)876 4400 y(\015)922 4396 y FA(S)5 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))19 b Fw(\000)1281 4375 y Fz(e)1269 4396 y FA(S)5 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))1514 4300 y Fz(\015)1514 4350 y(\015)1514 4400 y(\015)1562 4454 y Fx(`)p Fy(+1)p Fx(;\033)1761 4396 y Fw(\024)1849 4300 y Fz(\015)1849 4350 y(\015)1849 4400 y(\015)1895 4396 y Fw(J)29 b FB(\()q FA(S)5 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\)\))19 b Fw(\000)f(J)2478 4303 y Fz(\020)2539 4375 y(e)2527 4396 y FA(S)5 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))2772 4303 y Fz(\021)2823 4300 y(\015)2823 4350 y(\015)2823 4400 y(\015)2869 4454 y Fx(`)p Fy(+1)p Fx(;\033)1761 4608 y Fw(\024)1859 4552 y FB(1)p 1859 4589 42 4 v 1859 4665 a(2)1924 4512 y Fz(\015)1924 4562 y(\015)1924 4612 y(\015)1970 4608 y FA(S)c FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b Fw(\000)2329 4587 y Fz(e)2318 4608 y FA(S)t FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))2562 4512 y Fz(\015)2562 4562 y(\015)2562 4612 y(\015)2610 4666 y Fx(`)p Fy(+1)p Fx(;\033)2800 4608 y FA(:)71 4852 y FB(Then)26 b FA(S)5 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)654 4831 y Fz(e)642 4852 y FA(S)5 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))27 b(for)e(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b Fw(2)1331 4760 y Fz(\020)1381 4852 y FA(D)1452 4822 y Fx(u)1450 4873 y Fv(1)p Fx(;\032)1570 4881 y Fu(1)1625 4852 y Fw(\\)18 b FA(D)1769 4812 y Fy(out)p Fx(;u)1767 4880 y(\032)1801 4860 y Fl(0)1801 4898 y Fu(1)1834 4880 y Fx(;\024)1893 4860 y Fl(0)1893 4898 y Fu(0)1929 4760 y Fz(\021)1993 4852 y Fw(\002)c Ft(T)2127 4864 y Fx(\033)2198 4852 y FB(and)25 b FA(S)5 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))27 b(is)e(the)h(analytic)f(con)n(tin)n(uation)g(of)g(the)71 4990 y(function)j FA(@)440 5002 y Fx(u)484 4990 y FA(T)533 5002 y Fy(1)569 4990 y FB(\()p FA(v)22 b FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\)\(1)20 b(+)e FA(@)1313 5002 y Fx(v)1353 4990 y FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\)\))1617 4960 y Fv(\000)p Fy(1)1735 4990 y FB(to)28 b FA(D)1908 4950 y Fy(out)o Fx(;u)1906 5018 y(\032)1940 4998 y Fl(0)1940 5036 y Fu(1)1972 5018 y Fx(;\024)2031 4998 y Fl(0)2031 5036 y Fu(0)2086 4990 y Fw(\002)18 b Ft(T)2224 5002 y Fx(\033)2269 4990 y FB(.)37 b(Finally)-7 b(,)28 b(it)g(is)f(enough)g (to)h(p)r(oin)n(t)f(out)h(that)g(one)71 5125 y(can)f(easily)g(reco)n(v) n(er)750 5104 y Fz(b)736 5125 y FA(T)785 5137 y Fy(1)849 5125 y FB(from)h FA(S)5 b FB(.)p 3790 5125 4 57 v 3794 5072 50 4 v 3794 5125 V 3843 5125 4 57 v Black 71 5289 a Fs(Pr)l(o)l(of)31 b(of)f(Pr)l(op)l(osition)i(7.4.)p Black 43 w FB(T)-7 b(o)27 b(pro)n(v)n(e)f(Prop)r(osition)f(7.4)i(from)g (Lemma)g(7.8,)g(it)g(is)h(enough)e(to)i(consider)e(the)i(c)n(hange)71 5388 y(of)k(v)-5 b(ariables)30 b FA(v)j FB(=)d FA(u)20 b FB(+)h FA(h)p FB(\()p FA(u;)14 b(\034)9 b FB(\))32 b(obtained)g(in)g(Lemma)f(7.6)g(and)h(to)f(tak)n(e)g FA(T)2503 5400 y Fy(1)2540 5388 y FB(\()p FA(u;)14 b(\034)9 b FB(\))31 b(=)2873 5367 y Fz(b)2859 5388 y FA(T)2908 5400 y Fy(1)2945 5388 y FB(\()p FA(u)21 b FB(+)g FA(h)p FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))33 b(whic)n(h)f(b)n(y)71 5488 y(construction)27 b(is)g(the)h(analytic)f(con)n(tin)n(uation)g(of)g(the)h(function)g FA(T)2180 5500 y Fy(1)2245 5488 y FB(obtained)f(in)h(Prop)r(ositions)e (6.4)g(and)i(6.10.)p 3790 5488 V 3794 5435 50 4 v 3794 5488 V 3843 5488 4 57 v Black 1919 5753 a(69)p Black eop end %%Page: 70 70 TeXDict begin 70 69 bop Black Black 71 272 a Fq(7.2)112 b(In)m(v)-6 b(arian)m(t)46 b(manifolds)h(in)f(the)f(outer)g(domains)j (in)d(the)h(general)g(case:)66 b(pro)s(of)46 b(of)326 388 y(Theorems)38 b(4.5,)g(4.6,)g(4.7)f(and)i(4.8)71 542 y FB(W)-7 b(e)27 b(dev)n(ote)f(this)h(section)g(to)g(pro)n(v)n(e)e (the)i(existence)f(of)h(the)g(in)n(v)-5 b(arian)n(t)26 b(manifolds)h(in)g(the)g(outer)f(domains,)h(in)g(general)71 641 y(case,)42 b(that)e(is)g(assuming)f(that)h FA(p)1180 653 y Fy(0)1217 641 y FB(\()p FA(u)p FB(\))g(can)f(v)-5 b(anish.)74 b(W)-7 b(e)40 b(do)f(it)i(in)f(four)f(di\013eren)n(t)h (sections,)i(dev)n(oted)d(to)h(pro)n(v)n(e)71 741 y(Theorems)26 b(4.5,)h(4.6,)g(4.7)g(and)g(4.8)71 957 y Fp(7.2.1)94 b(The)32 b(v)-5 b(ariational)32 b(equation)f(along)h(the)f(separatrix) 71 1110 y FB(In)19 b(order)e(to)h(pro)n(v)n(e)f(the)i(existence)f(of)h (the)g(p)r(erturb)r(ed)g(stable)f(and)g(unstable)h(in)n(v)-5 b(arian)n(t)17 b(manifolds)h(in)h(certain)f(domains,)71 1209 y(w)n(e)31 b(will)h(need)f(to)h(consider)e(a)h(real-analytic)f (fundamen)n(tal)i(matrix)f(solution)g(of)g(the)h(v)-5 b(ariational)30 b(equations)h(along)71 1309 y(the)d(unp)r(erturb)r(ed)g (separatrix)1784 1387 y(_)1766 1409 y FA(\030)g FB(=)22 b FA(A)p FB(\()p FA(u)p FB(\))p FA(\030)t(;)1507 b FB(\(181\))71 1558 y(where)1434 1687 y FA(A)p FB(\()p FA(u)p FB(\))23 b(=)1719 1570 y Fz(\022)2020 1636 y FB(0)281 b(1)1822 1736 y Fw(\000)p FA(V)1953 1702 y Fv(00)2009 1736 y FB(\()q FA(q)2079 1748 y Fy(0)2116 1736 y FB(\()p FA(u)p FB(\)\))83 b(0)2426 1570 y Fz(\023)3661 1687 y FB(\(182\))71 1886 y(and)27 b(\()p FA(q)301 1898 y Fy(0)339 1886 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)530 1898 y Fy(0)567 1886 y FB(\()p FA(u)p FB(\)\))28 b(is)f(the)h(parametrization)e(of)i(the)g (unp)r(erturb)r(ed)g(separatrix)d(giv)n(en)i(in)h(Hyp)r(othesis)g Fp(HP2)p FB(.)195 1986 y(It)21 b(is)f(a)f(w)n(ell)h(kno)n(wn)g(fact)g (that)g(the)h(deriv)-5 b(ativ)n(e)19 b(of)h(the)g(parametrization)e(of) i(the)h(separatrix,)e(that)i(is)f(\()p FA(p)3441 1998 y Fy(0)3478 1986 y FB(\()p FA(u)p FB(\))p FA(;)30 b FB(_)-39 b FA(p)3669 1998 y Fy(0)3706 1986 y FB(\()p FA(u)p FB(\)\))71 2085 y(\(recall)39 b(that)55 b(_)-38 b FA(q)566 2097 y Fy(0)603 2085 y FB(\()p FA(u)p FB(\))43 b(=)f FA(p)907 2097 y Fy(0)944 2085 y FB(\()p FA(u)p FB(\)\),)h(is)d(a)f(solution)g (of)46 b(\(181\))o(.)73 b(A)40 b(second)f(indep)r(enden)n(t)h(solution) f(can)g(b)r(e)h(giv)n(en)f(b)n(y)71 2185 y(\()p FA(\020)6 b FB(\()p FA(u)p FB(\))p FA(;)311 2163 y FB(_)294 2185 y FA(\020)h FB(\()p FA(u)p FB(\)\),)28 b(where)1490 2332 y FA(\020)6 b FB(\()p FA(u)p FB(\))24 b(=)f FA(p)1798 2344 y Fy(0)1835 2332 y FB(\()p FA(u)p FB(\))1961 2219 y Fz(Z)2044 2239 y Fx(u)2007 2407 y(u)2046 2415 y Fu(0)2183 2275 y FB(1)p 2111 2312 187 4 v 2111 2389 a FA(p)2153 2360 y Fy(2)2153 2411 y(0)2190 2389 y FB(\()p FA(v)s FB(\))2321 2332 y FA(dv)s(;)1231 b FB(\(183\))71 2536 y(where)27 b FA(u)359 2548 y Fy(0)419 2536 y Fw(2)c Ft(R)28 b FB(is)f(suc)n(h)h(that)f FA(p)1077 2548 y Fy(0)1114 2536 y FB(\()p FA(u)1194 2548 y Fy(0)1232 2536 y FB(\))c Fw(6)p FB(=)f(0.)37 b(W)-7 b(e)28 b(consider)f(then)h(the)g(follo)n (wing)e(fundamen)n(tal)i(matrix)1484 2772 y(\010\()p FA(u)p FB(\))23 b(=)1767 2655 y Fz(\022)1869 2718 y FA(p)1911 2730 y Fy(0)1948 2718 y FB(\()p FA(u)p FB(\))83 b FA(\020)6 b FB(\()p FA(u)p FB(\))1885 2825 y(_)-39 b FA(p)1911 2837 y Fy(0)1948 2825 y FB(\()p FA(u)p FB(\))2160 2803 y(_)2143 2825 y FA(\020)7 b FB(\()p FA(u)p FB(\))2339 2655 y Fz(\023)2414 2772 y FA(:)1224 b FB(\(184\))p Black 71 3008 a Fp(Remark)41 b(7.9.)p Black 45 w Fs(L)l(et)36 b(us)h(p)l(oint)g(out)g(that)g(the)g(function)h FA(\020)43 b Fs(de\014ne)l(d)37 b(in)44 b FB(\(183\))36 b Fs(is)i(wel)t(l)g (de\014ne)l(d)f(and)h(analytic)g(even)71 3108 y(if)d FA(p)198 3120 y Fy(0)235 3108 y FB(\()p FA(u)p FB(\))e Fs(c)l(an)h(vanish)h(for)g(some)f FA(u)c Fw(2)h Ft(C)j Fs(and)42 b FB(a)32 b(priori)h Fs(it)g(c)l(ould)i(se)l(em)f(that)f(the) h(inte)l(gr)l(al)g(dep)l(ends)h(on)f(the)g(p)l(ath)h(of)71 3208 y(inte)l(gr)l(ation.)195 3307 y(However,)41 b(sinc)l(e)k FB(\177)-49 b FA(p)827 3319 y Fy(0)864 3307 y FB(\()p FA(u)p FB(\))38 b(=)e Fw(\000)p FA(V)1247 3277 y Fv(00)1289 3307 y FB(\()p FA(q)1358 3319 y Fy(0)1396 3307 y FB(\()p FA(u)p FB(\)\))p FA(p)1582 3319 y Fy(0)1619 3307 y FB(\()p FA(u)p FB(\))p Fs(,)k(one)e(c)l(an)g(se)l(e)g(that)f(the)h(T)-6 b(aylor)39 b(exp)l(ansion)f(ar)l(ound)g(any)g(zer)l(o)71 3407 y FA(u)119 3377 y Fv(\003)180 3407 y Fw(2)23 b Ft(C)30 b Fs(of)g FA(p)487 3419 y Fy(0)524 3407 y FB(\()p FA(u)p FB(\))g Fs(is)g(of)h(the)f(form)1272 3598 y FA(p)1314 3610 y Fy(0)1351 3598 y FB(\()p FA(u)p FB(\))23 b(=)39 b(_)-40 b FA(p)1615 3610 y Fy(0)1666 3598 y FB(\()p FA(u)1746 3564 y Fv(\003)1784 3598 y FB(\))14 b(\()q FA(u)k Fw(\000)g FA(u)2060 3564 y Fv(\003)2097 3598 y FB(\))h(+)f Fw(O)e FB(\()q FA(u)i Fw(\000)g FA(u)2543 3564 y Fv(\003)2580 3598 y FB(\))2613 3556 y Fy(3)71 3781 y Fs(\(observe)28 b(that)44 b FB(_)-39 b FA(p)606 3793 y Fy(0)643 3781 y FB(\()p FA(u)723 3751 y Fv(\003)761 3781 y FB(\))23 b Fw(6)p FB(=)g(0\))p Fs(\))k(and)h(then,)g(the)f(r)l(esidue)h(of)g (the)g(inte)l(gr)l(and)f(app)l(e)l(aring)j(in)d(the)h(de\014nition)f (of)i FA(\020)k Fs(in)h FB(\(183\))71 3880 y Fs(is)g(zer)l(o.)50 b(Final)t(ly,)36 b(let)e(us)f(observe)h(that)g(even)f(if)i(the)e(inte)l (gr)l(al)h(might)g(b)l(e)g(diver)l(gent)g(if)g(one)g(takes)f FA(u)3329 3850 y Fv(\003)3401 3880 y Fs(as)g(the)h(upp)l(er)71 3980 y(limit)c(of)h(inte)l(gr)l(ation,)f FB(lim)923 3992 y Fx(u)p Fv(!)p Fx(u)1067 3976 y Fl(\003)1121 3980 y FA(\020)6 b FB(\()p FA(u)p FB(\))23 b(=)g Fw(\000)p FB(1)p FA(=p)1577 3992 y Fy(0)1613 3980 y FB(\()p FA(u)1693 3950 y Fv(\003)1731 3980 y FB(\))p Fs(.)71 4196 y Fp(7.2.2)94 b(Pro)s(of)31 b(of)h(Theorem)f(4.5)71 4349 y FB(In)25 b(this)g(section)g(w)n(e)f(pro)n(v)n(e)g(the)h(existence)g(of)g(a)f(c)n (hange)g(of)h(v)-5 b(ariables)24 b(whic)n(h)h(allo)n(w)e(us)i(to)g (obtain)g(a)f(parametrization)71 4449 y(of)32 b(the)g(in)n(v)-5 b(arian)n(t)30 b(manifolds)i(whic)n(h)g(satis\014es)f(equation)g (\(44\))g(from)h(the)g(parametrization)e(obtained)h(in)h(Theorem)71 4548 y(4.3.)195 4648 y(Applying)26 b Fw(L)610 4660 y Fx(")669 4648 y FB(=)d FA(")796 4618 y Fv(\000)p Fy(1)885 4648 y FA(@)929 4660 y Fx(\034)985 4648 y FB(+)14 b FA(@)1108 4660 y Fx(v)1173 4648 y FB(to)25 b(b)r(oth)h(comp)r(onen)n(ts)g(of)f (equation)g(\(44\),)h(it)g(is)f(straigh)n(tforw)n(ard)e(to)j(see)f (that)h(\(53\))71 4748 y(satis\014es)h(these)g(equations)g(pro)n(vided) g Fw(U)1363 4718 y Fx(u)1434 4748 y FB(satis\014es)1500 4930 y Fw(L)1557 4942 y Fx(")1593 4930 y FA(h)22 b FB(=)h FA(M)g FB(\()p FA(v)e FB(+)d FA(h)p FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))16 b FA(;)1240 b FB(\(185\))71 5113 y(where)548 5242 y FA(M)9 b FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b(=)1028 5185 y(1)p 953 5223 192 4 v 953 5299 a FA(p)995 5270 y Fy(2)995 5321 y(0)1032 5299 y FB(\()p FA(u)p FB(\))1154 5242 y FA(@)1198 5254 y Fx(u)1241 5242 y FA(T)1290 5254 y Fy(1)1327 5242 y FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b(+)1664 5185 y FA(\026")1753 5155 y Fx(\021)p 1633 5223 V 1633 5299 a FA(p)1675 5311 y Fy(0)1712 5299 y FB(\()p FA(u)p FB(\))1834 5242 y FA(@)1878 5254 y Fx(p)1936 5221 y Fz(b)1917 5242 y FA(H)1986 5254 y Fy(1)2037 5125 y Fz(\022)2098 5242 y FA(q)2135 5254 y Fy(0)2172 5242 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2363 5254 y Fy(0)2400 5242 y FB(\()p FA(u)p FB(\))19 b(+)2699 5185 y(1)p 2624 5223 V 2624 5299 a FA(p)2666 5311 y Fy(0)2703 5299 y FB(\()p FA(u)p FB(\))2825 5242 y FA(@)2869 5254 y Fx(u)2912 5242 y FA(T)2961 5254 y Fy(1)2998 5242 y FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)3274 5125 y Fz(\023)3350 5242 y FA(;)288 b FB(\(186\))90 5439 y Fz(b)71 5460 y FA(H)140 5472 y Fy(1)205 5460 y FB(is)27 b(the)h(Hamiltonian)g(de\014ned)g(in)f(\(34\))h(and)f FA(T)1677 5472 y Fy(1)1742 5460 y FB(is)g(the)h(function)g(obtained)g (in)f(Prop)r(osition)f(6.4.)p Black 1919 5753 a(70)p Black eop end %%Page: 71 71 TeXDict begin 71 70 bop Black Black 195 272 a FB(Splitting)33 b(the)f(righ)n(t)f(hand)h(side)g(of)f(equation)h(\(185\))f(in)n(to)g (constan)n(t,)h(linear)f(and)h(higher)f(order)g(terms)g(in)h FA(h)p FB(,)h(it)71 372 y(can)27 b(b)r(e)h(rewritten)f(as)1722 471 y Fw(L)1779 483 y Fx(")1815 471 y FA(h)c FB(=)g FA(M)9 b FB(\()p FA(h)p FB(\))p FA(;)1462 b FB(\(187\))71 621 y(where)985 720 y FA(M)9 b FB(\()p FA(h)p FB(\))23 b(=)g FA(M)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))19 b(+)f(\()q FA(N)1779 732 y Fy(1)1816 720 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))19 b(+)f FA(N)2174 732 y Fy(2)2211 720 y FB(\()p FA(v)s(;)c(\034)9 b FB(\)\))16 b FA(h)i FB(+)g FA(R)q FB(\()p FA(h;)c(v)s(;)g(\034)9 b FB(\))726 b(\(188\))71 870 y(and)1107 1082 y FA(N)1174 1094 y Fy(1)1211 1082 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f FA(\026")1601 1047 y Fx(\021)1641 1082 y FA(@)1685 1094 y Fx(v)1739 964 y Fz(\024)1865 1025 y FB(1)p 1792 1062 187 4 v 1792 1139 a FA(p)1834 1151 y Fy(0)1871 1139 y FB(\()p FA(v)s FB(\))1989 1082 y FA(@)2033 1094 y Fx(p)2091 1061 y Fz(b)2072 1082 y FA(H)2148 1047 y Fy(1)2141 1102 y(1)2198 1082 y FB(\()q FA(q)2268 1094 y Fy(0)2305 1082 y FB(\()p FA(v)s FB(\))p FA(;)14 b(p)2491 1094 y Fy(0)2529 1082 y FB(\()p FA(v)s FB(\))p FA(;)g(\034)9 b FB(\))2751 964 y Fz(\025)3661 1082 y FB(\(189\))1107 1263 y FA(N)1174 1275 y Fy(2)1211 1263 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f FA(@)1556 1275 y Fx(v)1596 1263 y FA(M)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))19 b Fw(\000)f FA(N)2044 1275 y Fy(1)2081 1263 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))1391 b(\(190\))1063 1388 y FA(R)q FB(\()p FA(h;)14 b(v)s(;)g(\034)9 b FB(\))23 b(=)g FA(M)9 b FB(\()p FA(v)22 b FB(+)c FA(h;)c(\034)9 b FB(\))19 b Fw(\000)f FA(@)2087 1400 y Fx(v)2126 1388 y FA(M)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(h)20 b Fw(\000)e FA(M)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)803 b FB(\(191\))71 1590 y(where)330 1569 y Fz(b)311 1590 y FA(H)387 1560 y Fy(1)380 1611 y(1)452 1590 y FB(and)27 b FA(M)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))29 b(are)d(de\014ned)i (in)g(\(35\))f(and)h(\(186\))f(resp)r(ectiv)n(ely)-7 b(.)195 1690 y(W)g(e)27 b(no)n(w)e(de\014ne)h(appropriate)e(Banac)n(h)h (spaces.)36 b(F)-7 b(or)25 b(analytic)g(functions)h FA(h)d FB(:)g FA(I)2751 1660 y Fx(u)2744 1710 y(\032)2778 1718 y Fu(3)2812 1710 y Fx(;\032)2866 1718 y Fu(4)2918 1690 y Fw(\002)14 b Ft(T)3052 1702 y Fx(\033)3121 1690 y Fw(!)23 b Ft(C)p FB(,)j(where)f FA(I)3617 1660 y Fx(u)3610 1710 y(\032)3644 1718 y Fu(3)3678 1710 y Fx(;\032)3732 1718 y Fu(4)3795 1690 y FB(is)71 1789 y(the)j(domain)f(de\014ned)h(in)g (\(52\))o(,)g(w)n(e)f(de\014ne)h(the)g(F)-7 b(ourier)27 b(norm)1508 1997 y Fw(k)p FA(h)p Fw(k)1640 2009 y Fx(\033)1706 1997 y FB(=)1794 1918 y Fz(X)1794 2097 y Fx(k)q Fv(2)p Fn(Z)1928 1901 y Fz(\015)1928 1951 y(\015)1928 2001 y(\015)1974 1997 y FA(h)2022 1963 y Fy([)p Fx(k)q Fy(])2101 1901 y Fz(\015)2101 1951 y(\015)2101 2001 y(\015)2147 2055 y Fv(1)2231 1997 y FA(e)2270 1963 y Fv(j)p Fx(k)q Fv(j)p Fx(\033)2390 1997 y FA(;)71 2263 y FB(where)g Fw(k)18 b(\001)g(k)454 2275 y Fv(1)552 2263 y FB(is)27 b(the)h(classical)f (suprem)n(um)g(norm)g(in)h FA(I)1859 2233 y Fx(u)1852 2284 y(\032)1886 2292 y Fu(3)1919 2284 y Fx(;\032)1973 2292 y Fu(4)2010 2263 y FB(.)37 b(W)-7 b(e)28 b(consider)e(the)i(follo) n(wing)f(function)h(space)990 2457 y Fw(A)1056 2469 y Fx(\033)1124 2457 y FB(=)1212 2389 y Fz(\010)1260 2457 y FA(h)23 b FB(:)g FA(I)1420 2422 y Fx(u)1413 2477 y(\032)1447 2485 y Fu(3)1481 2477 y Fx(;\032)1535 2485 y Fu(4)1590 2457 y Fw(\002)18 b Ft(T)1728 2469 y Fx(\033)1796 2457 y Fw(!)23 b Ft(C)p FB(;)42 b(real-analytic)m FA(;)14 b Fw(k)p FA(h)p Fw(k)2645 2469 y Fx(\033)2712 2457 y FA(<)23 b Fw(1)2883 2389 y Fz(\011)3661 2457 y FB(\(192\))71 2639 y(whic)n(h)k(is)h(straigh)n(tforw)n(ard)c(to)k(see)f(that)h(is)g (a)f(Banac)n(h)f(algebra.)195 2739 y(Throughout)18 b(this)h(section)g (w)n(e)f(will)h(need)g(to)g(solv)n(e)e(equations)h(of)h(the)g(form)f Fw(L)2630 2751 y Fx(")2666 2739 y FA(h)23 b FB(=)g FA(g)s FB(,)d(where)e Fw(L)3199 2751 y Fx(")3254 2739 y FB(is)h(the)g (di\013eren)n(tial)71 2839 y(op)r(erator)k(de\014ned)i(in)g(\(45\).)36 b(W)-7 b(e)25 b(tak)n(e)f(the)h(op)r(erator)f Fw(G)1826 2851 y Fx(")1886 2839 y FB(de\014ned)i(in)f(\(168\))f(as)g(righ)n(t)g (in)n(v)n(erse)f(of)i Fw(L)3196 2851 y Fx(")3232 2839 y FB(.)36 b(In)25 b(Section)g(7.1.1)71 2938 y(it)30 b(w)n(as)f(applied) h(to)g(functions)g(b)r(elonging)f(to)h Fw(E)1601 2950 y Fx(\027;\032;\016)o(;\033)1831 2938 y FB(\(see)g(\(167\))o(\))g(but)h (it)f(is)g(clear)f(that)h(it)g(can)g(also)e(b)r(e)j(applied)f(to)71 3038 y(functions)e(in)g Fw(A)592 3050 y Fx(\033)664 3038 y FB(if)g(w)n(e)g(tak)n(e)f FA(v)1083 3050 y Fy(1)1148 3038 y FB(and)p 1309 2992 44 4 v 27 w FA(v)1352 3050 y Fy(1)1417 3038 y FB(the)h(v)n(ertices)f(of)g(the)h(domain)f FA(I)2436 3008 y Fx(u)2429 3058 y(\032)2463 3066 y Fu(3)2497 3058 y Fx(;\032)2551 3066 y Fu(4)2615 3038 y FB(and)h(also)e Fw(\000)p FA(\032)3051 3050 y Fy(4)3116 3038 y FB(\(see)h(Figure)g (6\).)p Black 71 3204 a Fp(Lemma)k(7.10.)p Black 40 w Fs(The)g(op)l(er)l(ator)g Fw(G)1193 3216 y Fx(")1258 3204 y Fs(in)36 b FB(\(168\))29 b Fs(satis\014es)h(the)g(fol)t(lowing)i (pr)l(op)l(erties.)p Black 195 3370 a Fw(\017)p Black 41 w(G)327 3382 y Fx(")393 3370 y Fs(is)e(line)l(ar)h(fr)l(om)f Fw(A)979 3382 y Fx(\033)1054 3370 y Fs(to)f(itself)i(and)f(satis\014es) g Fw(L)1885 3382 y Fx(")1939 3370 y Fw(\016)18 b(G)2048 3382 y Fx(")2107 3370 y FB(=)23 b(Id)p Fs(.)p Black 195 3536 a Fw(\017)p Black 41 w Fs(If)31 b FA(h)22 b Fw(2)i(A)581 3548 y Fx(\033)626 3536 y Fs(,)30 b(then)1709 3636 y Fw(kG)1800 3648 y Fx(")1835 3636 y FB(\()p FA(h)p FB(\))p Fw(k)1989 3648 y Fx(\033)2057 3636 y Fw(\024)23 b FA(K)6 b Fw(k)p FA(h)p Fw(k)2354 3648 y Fx(\033)2397 3636 y FA(:)278 3785 y Fs(F)-6 b(urthermor)l(e,)30 b(if)h Fw(h)p FA(h)p Fw(i)24 b FB(=)e(0)p Fs(,)30 b(then)1690 3885 y Fw(kG)1781 3897 y Fx(")1816 3885 y FB(\()p FA(h)p FB(\))p Fw(k)1970 3897 y Fx(\033)2038 3885 y Fw(\024)22 b FA(K)6 b(")p Fw(k)p FA(h)p Fw(k)2373 3897 y Fx(\033)2416 3885 y FA(:)195 4067 y FB(Finally)-7 b(,)32 b(w)n(e)f(state)f(a)h(tec)n (hnical)g(lemma)f(whic)n(h)h(giv)n(es)f(some)g(prop)r(erties)g(of)h (the)g(functions)h FA(M)9 b FB(,)31 b FA(N)3395 4079 y Fy(1)3432 4067 y FB(,)h FA(N)3554 4079 y Fy(2)3622 4067 y FB(and)f FA(R)71 4167 y FB(de\014ned)d(in)g(\(186\))o(,)g (\(189\))o(,)f(\(190\))g(and)h(\(191\))e(resp)r(ectiv)n(ely)-7 b(.)p Black 71 4333 a Fp(Lemma)40 b(7.11.)p Black 45 w Fs(The)e(functions)f FA(M)9 b Fs(,)39 b FA(N)1427 4345 y Fy(1)1464 4333 y Fs(,)g FA(N)1595 4345 y Fy(2)1669 4333 y Fs(and)e FA(R)h Fs(de\014ne)l(d)f(in)43 b FB(\(186\))p Fs(,)c FB(\(189\))o Fs(,)g FB(\(190\))d Fs(and)46 b FB(\(191\))36 b Fs(satisfy)i(the)71 4433 y(fol)t(lowing)32 b(pr)l(op)l(erties:)p Black 169 4599 a(1.)p Black 42 w FA(M)g Fw(2)23 b(A)535 4611 y Fx(\033)610 4599 y Fs(and)30 b(satis\014es)1303 4698 y Fw(k)p FA(M)9 b Fw(k)1477 4710 y Fx(\033)1544 4698 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1843 4664 y Fx(\021)1883 4698 y FA(;)83 b Fw(kG)2080 4710 y Fx(")2116 4698 y FB(\()p FA(M)9 b FB(\))p Fw(k)2312 4710 y Fx(\033)2379 4698 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2679 4664 y Fx(\021)r Fy(+1)2802 4698 y FA(:)836 b FB(\(193\))p Black 169 4881 a Fs(2.)p Black 42 w FA(N)345 4893 y Fy(1)382 4881 y FA(;)14 b(N)486 4893 y Fy(2)546 4881 y Fw(2)24 b(A)691 4893 y Fx(\033)736 4881 y Fs(.)38 b(Mor)l(e)l(over)31 b(they)g(satisfy)g Fw(h)p FA(N)1701 4893 y Fy(1)1738 4881 y Fw(i)23 b FB(=)g(0)29 b Fs(and)1364 5064 y Fw(k)p FA(N)1473 5076 y Fy(1)1509 5064 y Fw(k)1551 5076 y Fx(\033)1618 5064 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1918 5030 y Fx(\021)1958 5064 y FA(;)83 b Fw(k)p FA(N)2173 5076 y Fy(2)2209 5064 y Fw(k)2251 5076 y Fx(\033)2319 5064 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2618 5030 y Fx(\021)r Fy(+1)2742 5064 y FA(:)896 b FB(\(194\))p Black 169 5279 a Fs(3.)p Black 42 w(L)l(et)30 b(us)f(c)l(onsider)i FA(h)906 5291 y Fy(1)943 5279 y FA(;)14 b(h)1028 5291 y Fy(2)1088 5279 y Fw(2)23 b FA(B)t FB(\()p FA(\027)5 b FB(\))24 b Fw(\032)f(A)1521 5291 y Fx(\033)1596 5279 y Fs(with)30 b FA(\027)e Fw(\034)23 b FB(1)p Fs(.)39 b(Then,)1246 5462 y Fw(k)o FA(R)q FB(\()p FA(h)1431 5474 y Fy(2)1468 5462 y FA(;)14 b(v)s(;)g(\034)9 b FB(\))20 b Fw(\000)e FA(R)q FB(\()p FA(h)1909 5474 y Fy(1)1946 5462 y FA(;)c(v)s(;)g(\034)9 b FB(\))p Fw(k)2182 5487 y Fx(\033)2250 5462 y Fw(\024)23 b FA(K)6 b(\027)f Fw(k)p FA(h)2551 5474 y Fy(2)2606 5462 y Fw(\000)18 b FA(h)2737 5474 y Fy(1)2774 5462 y Fw(k)2816 5474 y Fx(\033)2860 5462 y FA(:)p Black 1919 5753 a FB(71)p Black eop end %%Page: 72 72 TeXDict begin 72 71 bop Black Black Black 71 272 a Fs(Pr)l(o)l(of.)p Black 43 w FB(The)38 b(\014rst)f(b)r(ound)h(is)g(straigh)n(tforw)n(ard) d(taking)i(in)n(to)h(accoun)n(t)e(the)j(b)r(ounds)e(for)h FA(c)3035 284 y Fx(lk)3135 272 y FB(and)f FA(T)3355 284 y Fy(1)3430 272 y FB(obtained)g(in)71 372 y(Corollary)26 b(5.6)j(and)f(Prop)r(ositions)f(6.4)i(and)f(6.10.)40 b(F)-7 b(or)29 b(the)g(second)f(one,)h(one)g(has)f(to)h(split)h FA(M)37 b FB(as)29 b FA(M)k FB(=)25 b FA(M)3569 384 y Fy(1)3626 372 y FB(+)19 b FA(M)3791 384 y Fy(2)3827 372 y FB(,)71 471 y(where)1191 589 y FA(M)1272 601 y Fy(1)1309 589 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f FA(\026")1699 555 y Fx(\021)1824 533 y FB(1)p 1749 570 192 4 v 1749 646 a FA(p)1791 658 y Fy(0)1828 646 y FB(\()p FA(u)p FB(\))1950 589 y FA(@)1994 601 y Fx(p)2052 568 y Fz(b)2033 589 y FA(H)2109 555 y Fy(1)2102 610 y(1)2146 589 y FB(\()p FA(q)2215 601 y Fy(0)2252 589 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2443 601 y Fy(0)2480 589 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))p FA(;)71 801 y FB(where)328 780 y Fz(b)309 801 y FA(H)385 771 y Fy(1)378 822 y(1)447 801 y FB(is)25 b(the)h(Hamiltonian)f(in)h(\(35\))o(,)g(and)g FA(M)1676 813 y Fy(2)1735 801 y FB(=)d FA(M)g Fw(\000)13 b FA(M)2086 813 y Fy(1)2123 801 y FB(.)36 b(Since)26 b Fw(h)p FA(M)2510 813 y Fy(1)2547 801 y Fw(i)d FB(=)g(0)i(and)g (satis\014es)f Fw(k)p FA(M)3342 813 y Fy(1)3379 801 y Fw(k)3421 813 y Fx(\033)3488 801 y Fw(\024)f FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")3788 771 y Fx(\021)3827 801 y FB(,)71 901 y(b)n(y)39 b(Lemma)h(7.10)f(w)n(e)g(ha)n(v)n(e)g (that)h Fw(kG)1316 913 y Fx(")1351 901 y FB(\()p FA(M)1464 913 y Fy(1)1501 901 y FB(\))p Fw(k)1575 913 y Fx(\033)1663 901 y Fw(\024)k FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1984 870 y Fx(\021)r Fy(+1)2107 901 y FB(.)74 b(On)40 b(the)g(other)f(hand,) k(b)n(y)d(the)g(b)r(ound)g(of)g FA(c)3679 913 y Fx(lk)3781 901 y FB(in)71 1000 y(Corollary)f(5.6)h(and)i(the)f(b)r(ound)h(of)g FA(T)1359 1012 y Fy(1)1437 1000 y FB(giv)n(en)f(b)n(y)g(Prop)r(osition) e(6.4,)45 b FA(M)2513 1012 y Fy(2)2591 1000 y FB(satis\014es)40 b Fw(k)p FA(M)3033 1012 y Fy(2)3070 1000 y Fw(k)3112 1012 y Fx(\033)3202 1000 y Fw(\024)46 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")3525 970 y Fx(\021)r Fy(+1)3649 1000 y FB(,)45 b(and)71 1100 y(therefore)27 b Fw(kG)511 1112 y Fx(")546 1100 y FB(\()p FA(M)659 1112 y Fy(2)696 1100 y FB(\))p Fw(k)770 1112 y Fx(\033)838 1100 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1137 1070 y Fx(\021)r Fy(+1)1261 1100 y FB(.)195 1199 y(The)27 b(b)r(ounds)g(of)f FA(N)813 1211 y Fy(1)850 1199 y FB(,)h FA(N)967 1211 y Fy(2)1031 1199 y FB(and)f FA(R)h FB(can)f(b)r(e)h(obtained)g (analogously)d(taking)i(in)n(to)g(accoun)n(t)g(the)h(de\014nition)f(of) h FA(M)35 b FB(in)71 1299 y(\(186\))27 b(and)g(that)h FA(R)g FB(is)g(quadratic)e(in)i FA(h)p FB(.)p 3790 1299 4 57 v 3794 1246 50 4 v 3794 1299 V 3843 1299 4 57 v 195 1463 a(W)-7 b(e)30 b(split)f(Theorem)f(4.5)g(in)h(the)g(follo)n (wing)f(prop)r(osition)f(and)i(corollary)-7 b(,)27 b(whic)n(h)i(are)e (rewritten)i(in)g(terms)f(of)h(the)71 1563 y(Banac)n(h)d(space)h (de\014ned)h(in)g(\(192\))o(.)37 b(Theorem)27 b(4.5)f(follo)n(ws)h (directly)g(from)h(those)f(results.)p Black 71 1722 a Fp(Prop)s(osition)34 b(7.12.)p Black 42 w Fs(L)l(et)e FA(\032)1020 1734 y Fy(1)1090 1722 y Fs(b)l(e)h(the)g(c)l(onstant)f(c)l (onsider)l(e)l(d)j(in)e(Pr)l(op)l(osition)h(6.4)g(and)g(let)f(us)f(c)l (onsider)i FA(\032)3536 1734 y Fy(3)3606 1722 y Fs(and)f FA(\032)3813 1734 y Fy(4)71 1822 y Fs(such)j(that)h FA(\032)486 1834 y Fy(4)558 1822 y FA(>)e(\032)701 1834 y Fy(3)774 1822 y FA(>)f(\032)916 1834 y Fy(1)990 1822 y Fs(and)j FA(")1197 1834 y Fy(0)1269 1822 y FA(>)e FB(0)h Fs(smal)t(l)h(enough)g (\(which)h(might)f(dep)l(end)h(on)e FA(\032)2926 1834 y Fx(i)2954 1822 y Fs(,)i FA(i)d FB(=)g(1)p FA(;)14 b FB(3)p FA(;)g FB(4)p Fs(\).)57 b(Then,)39 b(for)71 1922 y FA(")23 b Fw(2)g FB(\(0)p FA(;)14 b(")361 1934 y Fy(0)398 1922 y FB(\))28 b Fs(ther)l(e)g(exists)g(a)g(function)g Fw(U)1344 1891 y Fx(u)1411 1922 y Fw(2)23 b(A)1555 1934 y Fx(\033)1629 1922 y Fs(de\014ne)l(d)28 b(in)g FA(I)2050 1891 y Fx(u)2043 1942 y(\032)2077 1950 y Fu(3)2110 1942 y Fx(;\032)2164 1950 y Fu(4)2216 1922 y Fw(\002)14 b Ft(T)2350 1934 y Fx(\033)2423 1922 y Fs(that)28 b(satis\014es)h(e)l (quation)34 b FB(\(187\))o Fs(.)39 b(Mor)l(e)l(over,)71 2021 y(for)30 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b Fw(2)e FA(I)538 1991 y Fx(u)531 2042 y(\032)565 2050 y Fu(3)599 2042 y Fx(;\032)653 2050 y Fu(4)708 2021 y Fw(\002)18 b Ft(T)846 2033 y Fx(\033)891 2021 y Fs(,)30 b FA(v)22 b FB(+)c Fw(U)1151 1991 y Fx(u)1194 2021 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))25 b Fw(2)e FA(D)1557 1991 y Fx(u)1555 2042 y Fv(1)p Fx(;\032)1675 2050 y Fu(1)1742 2021 y Fs(and)30 b(ther)l(e)g(exists)f(a)h(c)l(onstant)f FA(b)2776 2033 y Fy(3)2836 2021 y FA(>)23 b FB(0)29 b Fs(such)h(that)1612 2208 y Fw(kU)1714 2173 y Fx(u)1757 2208 y Fw(k)1799 2220 y Fx(\033)1866 2208 y Fw(\024)23 b FA(b)1990 2220 y Fy(3)2027 2208 y Fw(j)p FA(\026)p Fw(j)p FA(")2162 2173 y Fx(\021)r Fy(+1)2286 2208 y FA(:)p Black 71 2382 a Fp(Corollary)39 b(7.13.)p Black 43 w Fs(L)l(et)34 b(us)g(c)l(onsider)i(the)e(c)l(onstants)g FA(\032)1893 2394 y Fy(3)1965 2382 y Fs(and)h FA(\032)2174 2394 y Fy(4)2246 2382 y Fs(given)g(by)g(Pr)l(op)l(osition)h(7.12)g(and) f FA(")3418 2394 y Fy(0)3487 2382 y FA(>)c FB(0)j Fs(smal)t(l)71 2481 y(enough.)39 b(Then,)31 b(for)f FA(")23 b Fw(2)h FB(\(0)p FA(;)14 b(")1052 2493 y Fy(0)1088 2481 y FB(\))30 b Fs(ther)l(e)g(exist)g(p)l(ar)l(ameterizations)h(of)f(the)g(invariant) h(manifolds)920 2656 y FB(\()p FA(Q)1018 2621 y Fx(u)1061 2656 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)14 b(P)1352 2621 y Fx(u)1397 2656 y FB(\()p FA(v)s(;)g(\034)9 b FB(\)\))24 b(=)f(\()p FA(q)1799 2668 y Fy(0)1837 2656 y FB(\()p FA(v)s FB(\))c(+)f FA(Q)2112 2621 y Fx(u)2112 2676 y Fy(1)2155 2656 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(p)2423 2668 y Fy(0)2461 2656 y FB(\()p FA(v)s FB(\))19 b(+)g FA(P)2736 2621 y Fx(u)2724 2676 y Fy(1)2779 2656 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\)\))71 2830 y Fs(which)32 b(ar)l(e)f(solution)g(of)h(e)l(quation)37 b FB(\(44\))o Fs(.)42 b(Mor)l(e)l(over)32 b FB(\()p FA(Q)1877 2800 y Fx(u)1877 2850 y Fy(1)1920 2830 y FA(;)14 b(P)2022 2800 y Fx(u)2010 2850 y Fy(1)2065 2830 y FB(\))25 b Fw(2)g(A)2268 2842 y Fx(\033)2333 2830 y Fw(\002)18 b(A)2482 2842 y Fx(\033)2558 2830 y Fs(ar)l(e)31 b(de\014ne)l(d)g(in)g FA(I)3127 2800 y Fx(u)3120 2850 y(\032)3154 2858 y Fu(3)3187 2850 y Fx(;\032)3241 2858 y Fu(4)3297 2830 y Fw(\002)18 b Ft(T)3435 2842 y Fx(\033)3511 2830 y Fs(and)31 b(ther)l(e)71 2929 y(exists)e(a)h(c)l(onstant)f FA(b)737 2941 y Fy(4)797 2929 y FA(>)23 b FB(0)29 b Fs(such)h(that)1610 3020 y Fw(k)o FA(Q)1717 2990 y Fx(u)1717 3040 y Fy(1)1760 3020 y Fw(k)1802 3045 y Fx(\033)1869 3020 y Fw(\024)23 b FA(b)1993 3032 y Fy(4)2030 3020 y Fw(j)p FA(\026)p Fw(j)p FA(")2165 2990 y Fx(\021)r Fy(+1)1610 3119 y Fw(k)o FA(P)1716 3089 y Fx(u)1704 3140 y Fy(1)1759 3119 y Fw(k)1801 3144 y Fx(\033)1869 3119 y Fw(\024)f FA(b)1992 3131 y Fy(4)2029 3119 y Fw(j)p FA(\026)p Fw(j)p FA(")2164 3089 y Fx(\021)r Fy(+1)2289 3119 y FA(:)195 3275 y FB(The)28 b(pro)r(of)f(of)h(this)g (corollary)d(is)i(a)g(straigh)n(tforw)n(ard)e(consequence)h(of)i(Prop)r (osition)e(7.12.)195 3374 y(W)-7 b(e)33 b(pro)n(v)n(e)e(Prop)r(osition) g(7.12)g(through)h(a)g(\014xed)g(p)r(oin)n(t)h(argumen)n(t.)50 b(Nev)n(ertheless,)33 b(the)g(op)r(erator)d FA(M)42 b FB(in)32 b(\(188\))71 3474 y(has)d(linear)h(terms)g(in)g FA(h)g FB(whic)n(h)g(are)f(not)i(small)e(when)i FA(\021)f FB(=)d(0.)44 b(Therefore,)30 b(w)n(e)g(ha)n(v)n(e)f(\014rst)h(to)g (consider)f(a)h(c)n(hange)f(of)71 3574 y(v)-5 b(ariables)30 b(to)h(obtain)g(a)g(con)n(tractiv)n(e)f(op)r(erator.)47 b(F)-7 b(or)31 b(this)g(purp)r(ose,)h(let)g(us)f(consider)p 2889 3507 76 4 v 31 w FA(N)2965 3586 y Fy(1)3031 3574 y FB(=)e Fw(G)3174 3586 y Fx(")3210 3574 y FB(\()p FA(N)3309 3586 y Fy(1)3346 3574 y FB(\),)k(where)e Fw(G)3727 3586 y Fx(")3795 3574 y FB(is)71 3673 y(the)j(op)r(erator)e(in)i(\(168\))e (and)i FA(N)1121 3685 y Fy(1)1191 3673 y FB(the)g(function)h(in)e (\(189\).)55 b(T)-7 b(aking)32 b(in)n(to)i(accoun)n(t)e(that)i Fw(h)p FA(N)3096 3685 y Fy(1)3133 3673 y Fw(i)g FB(=)e(0)i(and)f (applying)71 3773 y(Lemmas)27 b(7.11)f(and)i(7.10,)e(w)n(e)h(ha)n(v)n (e)g(that)1355 3877 y Fz(\015)1355 3926 y(\015)p 1401 3880 V 21 x FA(N)1477 3959 y Fy(1)1514 3877 y Fz(\015)1514 3926 y(\015)1560 3980 y Fx(\033)1628 3947 y FB(=)c Fw(k)o(G)1806 3959 y Fx(")1842 3947 y FB(\()p FA(N)1941 3959 y Fy(1)1978 3947 y FB(\))p Fw(k)2052 3972 y Fx(\033)2120 3947 y Fw(\024)f FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2419 3913 y Fx(\021)r Fy(+1)2543 3947 y FA(:)1095 b FB(\(195\))71 4121 y(Then,)28 b(w)n(e)f(consider)f(the)i(c)n(hange)1684 4221 y FA(h)23 b FB(=)1843 4154 y Fz(\000)1881 4221 y FB(1)18 b(+)p 2024 4154 V 18 w FA(N)2100 4233 y Fy(1)2137 4154 y Fz(\001)p 2189 4153 48 4 v 2189 4221 a FA(h)1424 b FB(\(196\))71 4377 y(whic)n(h,)27 b(b)n(y)i(\(195\))o(,)f(is)f(in)n(v)n(ertible)g (for)g(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b Fw(2)e FA(I)1600 4347 y Fx(u)1593 4398 y(\032)1627 4406 y Fu(3)1660 4398 y Fx(;\032)1714 4406 y Fu(4)1770 4377 y Fw(\002)18 b Ft(T)1908 4389 y Fx(\033)1953 4377 y FB(.)37 b(By)28 b(\(187\))f(and)g(\(196\))o(,)p 2762 4310 V 28 w FA(h)h FB(is)f(solution)g(of)1722 4575 y Fw(L)1779 4587 y Fx(")p 1815 4508 V 1815 4575 a FA(h)c FB(=)1984 4554 y Fz(c)1974 4575 y FA(M)8 b FB(\()p 2095 4508 V FA(h)p FB(\))p FA(;)71 4749 y FB(where)929 4828 y Fz(c)919 4849 y FA(M)1022 4782 y Fz(\000)p 1060 4781 V 67 x FA(h)1108 4782 y Fz(\001)1160 4849 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)1471 4828 y Fz(c)1461 4849 y FA(M)8 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b(+)1864 4828 y Fz(b)1842 4849 y FA(N)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p 2107 4781 V FA(h)q FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b(+)2464 4828 y Fz(b)2448 4849 y FA(R)2526 4782 y Fz(\000)p 2564 4781 V 67 x FA(h)p FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)14 b(v)s(;)g(\034)2963 4782 y Fz(\001)3661 4849 y FB(\(197\))71 4993 y(with)995 5146 y Fz(c)985 5167 y FA(M)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)1376 5100 y Fz(\000)1414 5167 y FB(1)18 b(+)p 1557 5101 76 4 v 18 w FA(N)1633 5179 y Fy(1)1670 5167 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))1859 5100 y Fz(\001)1898 5115 y Fv(\000)p Fy(1)2001 5167 y FA(M)g FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))1381 b(\(198\))1021 5300 y Fz(b)999 5321 y FA(N)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)1376 5254 y Fz(\000)1414 5321 y FB(1)18 b(+)p 1557 5254 V 18 w FA(N)1633 5333 y Fy(1)1670 5321 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))1859 5254 y Fz(\001)1898 5268 y Fv(\000)p Fy(1)2001 5321 y FA(N)2068 5333 y Fy(1)2105 5321 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p 2294 5254 V FA(N)2371 5333 y Fy(1)2409 5321 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b(+)e FA(N)2768 5333 y Fy(2)2804 5321 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))668 b(\(199\))942 5453 y Fz(b)926 5474 y FA(R)q FB(\()p 1022 5407 48 4 v FA(h;)14 b(v)s(;)g(\034)9 b FB(\))24 b(=)1376 5407 y Fz(\000)1414 5474 y FB(1)18 b(+)p 1557 5407 76 4 v 18 w FA(N)1633 5486 y Fy(1)1670 5474 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))1859 5407 y Fz(\001)1898 5421 y Fv(\000)p Fy(1)2001 5474 y FA(R)2079 5407 y Fz(\000\000)2155 5474 y FB(1)18 b(+)p 2298 5407 V 18 w FA(N)2374 5486 y Fy(1)2411 5474 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))2600 5407 y Fz(\001)p 2653 5407 48 4 v 2653 5474 a FA(h;)14 b(v)s(;)g(\034)2863 5407 y Fz(\001)2916 5474 y FA(:)722 b FB(\(200\))p Black 1919 5753 a(72)p Black eop end %%Page: 73 73 TeXDict begin 73 72 bop Black Black 71 272 a FB(T)-7 b(o)27 b(\014nd)h(a)f(solution)g(of)h(this)g(equation,)f(w)n(e)g(lo)r (ok)g(for)g(a)g(\014xed)h(p)r(oin)n(t)p 2277 204 48 4 v 28 w FA(h)23 b Fw(2)g(A)2492 284 y Fx(\033)2565 272 y FB(of)k(the)h(op)r(erator)p 1723 379 90 4 v 1723 446 a FA(M)j FB(=)23 b Fw(G)1972 458 y Fx(")2026 446 y Fw(\016)2096 425 y Fz(c)2086 446 y FA(M)9 b(;)1462 b FB(\(201\))71 632 y(where)36 b Fw(G)369 644 y Fx(")442 632 y FB(and)623 611 y Fz(c)613 632 y FA(M)45 b FB(are)36 b(the)h(op)r(erators)e (\(168\))h(and)h(\(197\))o(.)65 b(Then,)39 b(Prop)r(osition)d(7.12)f (is)i(a)f(consequence)g(of)h(the)71 732 y(follo)n(wing)26 b(lemma.)p Black 71 891 a Fp(Lemma)h(7.14.)p Black 36 w Fs(L)l(et)e(us)h(c)l(onsider)h FA(")1246 903 y Fy(0)1306 891 y FA(>)c FB(0)i Fs(smal)t(l)i(enough.)38 b(Then,)28 b(for)f FA(")22 b Fw(2)i FB(\(0)p FA(;)14 b(")2651 903 y Fy(0)2687 891 y FB(\))p Fs(,)28 b(ther)l(e)e(exists)g(a)g(function)p 3590 823 48 4 v 26 w FA(h)d Fw(2)g(A)3805 903 y Fx(\033)71 990 y Fs(de\014ne)l(d)30 b(in)g FA(I)496 960 y Fx(u)489 1011 y(\032)523 1019 y Fu(3)556 1011 y Fx(;\032)610 1019 y Fu(4)665 990 y Fw(\002)18 b Ft(T)803 1002 y Fx(\033)848 990 y Fs(,)31 b(such)e(that)h(it)g(is)g(a)g(\014xe)l(d)f(p)l(oint)h(of) h(the)f(op)l(er)l(ator)40 b FB(\(201\))o Fs(.)e(Mor)l(e)l(over,)32 b(it)e(satis\014es)1645 1108 y Fz(\015)1645 1158 y(\015)p 1691 1111 V 21 x FA(h)1739 1108 y Fz(\015)1739 1158 y(\015)1785 1212 y Fx(\033)1853 1179 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2152 1145 y Fx(\021)r Fy(+1)71 1369 y Fs(and)30 b(then)g FA(u)22 b FB(=)h FA(v)f FB(+)720 1301 y Fz(\000)758 1369 y FB(1)c(+)p 901 1302 76 4 v 18 w FA(N)976 1381 y Fy(1)1014 1369 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))1203 1301 y Fz(\001)p 1256 1301 48 4 v 1256 1369 a FA(h)p FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b Fw(2)g FA(D)1667 1339 y Fx(u)1665 1389 y Fv(1)p Fx(;\032)1785 1397 y Fu(1)1851 1369 y Fs(for)31 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b Fw(2)f FA(I)2318 1339 y Fx(u)2311 1389 y(\032)2345 1397 y Fu(3)2379 1389 y Fx(;\032)2433 1397 y Fu(4)2488 1369 y Fw(\002)18 b Ft(T)2626 1381 y Fx(\033)2671 1369 y Fs(.)p Black 71 1541 a(Pr)l(o)l(of.)p Black 43 w FB(It)32 b(is)f(straigh)n(tforw)n(ard)d(to)j(see)g(that)h (the)g(op)r(erator)p 2012 1475 90 4 v 29 w FA(M)40 b FB(sends)31 b Fw(A)2425 1553 y Fx(\033)2502 1541 y FB(to)g(itself.)48 b(W)-7 b(e)32 b(are)e(going)g(to)i(pro)n(v)n(e)d(that)71 1641 y(there)e(exists)g(a)h(constan)n(t)f FA(b)953 1653 y Fy(3)1012 1641 y FA(>)c FB(0)k(suc)n(h)h(that)p 1536 1574 V 27 w FA(M)37 b FB(is)27 b(con)n(tractiv)n(e)f(in)p 2261 1574 68 4 v 28 w FA(B)t FB(\()p FA(b)2396 1653 y Fy(3)2433 1641 y Fw(j)p FA(\026)p Fw(j)p FA(")2568 1611 y Fx(\021)r Fy(+1)2693 1641 y FB(\))d Fw(\032)g(A)2902 1653 y Fx(\033)2947 1641 y FB(.)195 1753 y(Let)40 b(us)g(consider)f (\014rst)p 994 1687 90 4 v 39 w FA(M)9 b FB(\(0\))43 b(=)g Fw(G)1390 1765 y Fx(")1452 1753 y Fw(\016)1530 1732 y Fz(c)1520 1753 y FA(M)9 b FB(\(0\).)73 b(F)-7 b(rom)39 b(the)h(de\014nitions)g(of)2728 1732 y Fz(c)2718 1753 y FA(M)48 b FB(and)3031 1732 y Fz(c)3021 1753 y FA(M)g FB(in)40 b(\(197\))f(and)g(\(198\))71 1853 y(resp)r(ectiv)n(ely) -7 b(,)27 b(w)n(e)g(ha)n(v)n(e)f(that)p 603 1980 V 603 2047 a FA(M)8 b FB(\(0\))23 b(=)g Fw(G)958 2059 y Fx(")994 2047 y FB(\()1036 2026 y Fz(c)1026 2047 y FA(M)9 b FB(\))23 b(=)g Fw(G)1308 2059 y Fx(")1357 1955 y Fz(\020)1407 1980 y(\000)1445 2047 y FB(1)18 b(+)p 1588 1980 76 4 v 18 w FA(N)1664 2059 y Fy(1)1701 1980 y Fz(\001)1739 1994 y Fv(\000)p Fy(1)1842 2047 y FA(M)1932 1955 y Fz(\021)2004 2047 y FB(=)23 b Fw(G)2141 2059 y Fx(")2191 2047 y FB(\()p FA(M)9 b FB(\))18 b Fw(\000)g(G)2495 2059 y Fx(")2545 1955 y Fz(\020)2595 1980 y(\000)2633 2047 y FB(1)g(+)p 2776 1980 V 18 w FA(N)2851 2059 y Fy(1)2889 1980 y Fz(\001)2927 1994 y Fv(\000)p Fy(1)p 3030 1980 V 3030 2047 a FA(N)3105 2059 y Fy(1)3143 2047 y FA(M)3233 1955 y Fz(\021)3296 2047 y FA(:)71 2251 y FB(The)31 b(\014rst)g(term)g(has)f(already)g(b)r (een)h(b)r(ounded)g(in)h(Lemma)e(7.11,)h(and)g(satis\014es)f Fw(kG)2779 2263 y Fx(")2814 2251 y FB(\()p FA(M)9 b FB(\))p Fw(k)3010 2263 y Fx(\033)3083 2251 y Fw(\024)29 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")3389 2221 y Fx(\021)r Fy(+1)3513 2251 y FB(.)47 b(F)-7 b(or)30 b(the)71 2351 y(second)d(one)g(has)g(to)h(tak)n(e)e(in)n(to)i(accoun)n(t)f(Lemma)g (7.10,)f(and)i(then)g(\(195\))f(and)g(Lemma)g(7.11,)g(to)g(obtain)863 2454 y Fz(\015)863 2504 y(\015)863 2553 y(\015)909 2549 y Fw(G)958 2561 y Fx(")1008 2457 y Fz(\020)1058 2482 y(\000)1096 2549 y FB(1)18 b(+)p 1239 2483 V 18 w FA(N)1315 2561 y Fy(1)1352 2482 y Fz(\001)1390 2496 y Fv(\000)p Fy(1)p 1493 2483 V 1493 2549 a FA(N)1569 2561 y Fy(1)1606 2549 y FA(M)1696 2457 y Fz(\021)1745 2454 y(\015)1745 2504 y(\015)1745 2553 y(\015)1791 2607 y Fx(\033)1859 2549 y Fw(\024)23 b FA(K)2037 2479 y Fz(\015)2037 2528 y(\015)p 2083 2483 V 21 x FA(N)2159 2561 y Fy(1)2196 2479 y Fz(\015)2196 2528 y(\015)2242 2582 y Fx(\033)2301 2549 y Fw(k)o FA(M)9 b Fw(k)2473 2574 y Fx(\033)2541 2549 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)2802 2515 y Fy(2)2839 2549 y FA(")2878 2515 y Fy(2)p Fx(\021)r Fy(+1)3035 2549 y FA(:)71 2747 y FB(Therefore,)26 b(there)i(exists)f(a) g(constan)n(t)g FA(b)1352 2759 y Fy(3)1412 2747 y FA(>)c FB(0)k(suc)n(h)g(that)1551 2889 y Fz(\015)1551 2939 y(\015)p 1597 2893 90 4 v 21 x FA(M)9 b FB(\(0\))1793 2889 y Fz(\015)1793 2939 y(\015)1839 2993 y Fx(\033)1907 2960 y Fw(\024)2005 2904 y FA(b)2041 2916 y Fy(3)p 2005 2941 73 4 v 2020 3017 a FB(2)2087 2960 y Fw(j)p FA(\026)p Fw(j)p FA(")2222 2925 y Fx(\021)r Fy(+1)2347 2960 y FA(:)71 3173 y FB(Let)29 b(us)h(consider)e(no)n(w)p 830 3106 48 4 v 29 w FA(h)878 3185 y Fy(1)915 3173 y FA(;)p 952 3106 V 14 w(h)1000 3185 y Fy(2)1063 3173 y Fw(2)p 1145 3107 68 4 v 27 w FA(B)t FB(\()p FA(b)1280 3185 y Fy(3)1317 3173 y Fw(j)p FA(\026)p Fw(j)p FA(")1452 3143 y Fx(\021)r Fy(+1)1576 3173 y FB(\))f Fw(\032)e(A)1791 3185 y Fx(\033)1836 3173 y FB(.)43 b(Then)29 b(using)g(the)h(prop)r(erties)f(of)g Fw(G)3022 3185 y Fx(")3087 3173 y FB(giv)n(en)g(in)h(Lemma)f(7.10)71 3286 y(and)e(the)h(de\014nition)g(of)849 3265 y Fz(c)839 3286 y FA(M)36 b FB(in)28 b(\(197\))o(,)606 3414 y Fz(\015)606 3463 y(\015)p 652 3418 90 4 v 21 x FA(M)9 b FB(\()p 774 3417 48 4 v FA(h)822 3496 y Fy(2)859 3484 y FB(\))19 b Fw(\000)p 993 3418 90 4 v 18 w FA(M)9 b FB(\()p 1115 3417 48 4 v FA(h)1163 3496 y Fy(1)1200 3484 y FB(\))1232 3414 y Fz(\015)1232 3463 y(\015)1278 3517 y Fx(\033)1406 3484 y Fw(\024)23 b FA(K)1584 3389 y Fz(\015)1584 3439 y(\015)1584 3488 y(\015)1640 3463 y(c)1630 3484 y FA(M)9 b FB(\()p 1752 3417 V FA(h)1800 3496 y Fy(2)1837 3484 y FB(\))19 b Fw(\000)1981 3463 y Fz(c)1971 3484 y FA(M)8 b FB(\()p 2092 3417 V FA(h)2141 3496 y Fy(1)2178 3484 y FB(\))2210 3389 y Fz(\015)2210 3439 y(\015)2210 3488 y(\015)2256 3542 y Fx(\033)1406 3638 y Fw(\024)23 b FA(K)1584 3542 y Fz(\015)1584 3592 y(\015)1584 3642 y(\015)1652 3617 y(b)1630 3638 y FA(N)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\)\()p 1927 3570 V FA(h)1976 3650 y Fy(2)2032 3638 y Fw(\000)p 2115 3570 V 18 w FA(h)2163 3650 y Fy(1)2200 3638 y FB(\))19 b(+)2350 3617 y Fz(b)2334 3638 y FA(R)q FB(\()p 2430 3570 V FA(h)2478 3650 y Fy(2)2515 3638 y FA(;)14 b(v)s(;)g(\034)9 b FB(\))19 b Fw(\000)2827 3617 y Fz(b)2811 3638 y FA(R)q FB(\()p 2907 3570 V FA(h)2955 3650 y Fy(1)2992 3638 y FA(;)14 b(v)s(;)g(\034)9 b FB(\))3186 3542 y Fz(\015)3186 3592 y(\015)3186 3642 y(\015)3234 3696 y Fx(\033)3292 3638 y FA(:)71 3859 y FB(T)-7 b(aking)25 b(in)n(to)i(accoun)n(t)e(the)i(de\014nitions)f(of)1477 3838 y Fz(b)1455 3859 y FA(N)35 b FB(and)1733 3838 y Fz(b)1717 3859 y FA(R)27 b FB(in)g(\(199\))f(and)g(\(200\))g(and)g (applying)g(Lemma)g(7.11)f(and)h(b)r(ound)71 3959 y(\(195\))o(,)i(one)f (obtains)1168 3988 y Fz(\015)1168 4038 y(\015)p 1214 3992 90 4 v 21 x FA(M)8 b FB(\()p 1335 3991 48 4 v FA(h)1384 4071 y Fy(2)1421 4059 y FB(\))19 b Fw(\000)p 1555 3992 90 4 v 18 w FA(M)8 b FB(\()p 1676 3991 48 4 v FA(h)1724 4071 y Fy(1)1762 4059 y FB(\))1794 3988 y Fz(\015)1794 4038 y(\015)1840 4092 y Fx(\033)1908 4059 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2207 4024 y Fx(\021)r Fy(+1)2331 4059 y Fw(k)p 2373 3991 V FA(h)2420 4071 y Fy(2)2476 4059 y Fw(\000)p 2559 3991 V 18 w FA(h)2607 4071 y Fy(1)2644 4059 y Fw(k)2686 4071 y Fx(\033)2730 4059 y FA(:)71 4218 y FB(Therefore,)19 b(reducing)f FA(")g FB(if)h(necessary)-7 b(,)18 b(Lip)p 1412 4151 90 4 v FA(M)32 b Fw(\024)23 b FB(1)p FA(=)p FB(2)17 b(and)h(therefore)p 2247 4151 V 17 w FA(M)27 b FB(is)19 b(con)n(tractiv)n(e)d(from)i(the)h (ball)f FA(B)t FB(\()p FA(b)3455 4230 y Fy(3)3493 4218 y Fw(j)p FA(\026)p Fw(j)p FA(")3628 4188 y Fx(\021)r Fy(+1)3752 4218 y FB(\))23 b Fw(\032)71 4317 y(A)137 4329 y Fx(\033)210 4317 y FB(in)n(to)k(itself)h(and)f(it)h(has)f(a)h (unique)g(\014xed)f(p)r(oin)n(t)p 1728 4250 48 4 v 28 w FA(h)p FB(.)p 3790 4317 4 57 v 3794 4265 50 4 v 3794 4317 V 3843 4317 4 57 v Black 71 4482 a Fs(Pr)l(o)l(of)k(of)f(Pr)l(op)l (osition)i(7.12.)p Black 43 w FB(T)-7 b(o)30 b(pro)r(of)g(Prop)r (osition)f(7.12)g(from)h(Lemma)g(6.6,)g(it)h(is)f(enough)g(to)g(undo)g (the)h(c)n(hange)71 4581 y(of)36 b(v)-5 b(ariables)36 b(\(196\))g(to)g(obtain)g Fw(U)1191 4551 y Fx(u)1273 4581 y FB(=)1376 4514 y Fz(\000)1414 4581 y FB(1)18 b(+)p 1557 4515 76 4 v 18 w FA(N)1633 4593 y Fy(1)1670 4514 y Fz(\001)p 1722 4514 48 4 v 1722 4581 a FA(h)p FB(.)64 b(Then,)39 b(using)d(b)r(ound)h(\(195\))f(and)g(increasing)f(sligh)n (tly)h FA(b)3728 4593 y Fy(3)3802 4581 y FB(if)71 4681 y(necessary)-7 b(,)26 b(w)n(e)h(obtain)g(the)h(b)r(ound)g(for)f Fw(U)1421 4651 y Fx(u)1465 4681 y FB(.)p 3790 4681 4 57 v 3794 4628 50 4 v 3794 4681 V 3843 4681 4 57 v 71 4895 a Fp(7.2.3)94 b(Pro)s(of)31 b(of)h(Theorem)f(4.6)71 5048 y FB(W)-7 b(e)36 b(pro)n(v)n(e)e(Theorem)h(4.6)g(lo)r(oking)g(for) g(a)h(solution)f(of)42 b(\(44\))36 b(through)f(a)g(\014xed)h(p)r(oin)n (t)g(argumen)n(t,)h(taking)e(the)h(pa-)71 5148 y(rameterizations)26 b(of)i(the)h(in)n(v)-5 b(arian)n(t)27 b(manifolds)h(as)f(p)r (erturbations)h(of)g(the)g(parameterizations)e(of)i(the)h(unp)r(erturb) r(ed)71 5248 y(separatrix.)35 b(Since)28 b(w)n(e)f(only)g(deal)g(with)h (the)g(unstable)g(manifold,)g(w)n(e)f(omit)h(the)g(sup)r(erscript)f FA(u)p FB(.)36 b(W)-7 b(e)28 b(consider)1284 5350 y Fz(\022)1386 5416 y FA(Q)p FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))1387 5516 y FA(P)j FB(\()p FA(v)s(;)i(\034)9 b FB(\))1684 5350 y Fz(\023)1768 5467 y FB(=)1856 5350 y Fz(\022)1958 5416 y FA(q)1995 5428 y Fy(0)2033 5416 y FB(\()p FA(v)s FB(\))19 b(+)f FA(Q)2308 5428 y Fy(1)2345 5416 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))1962 5516 y FA(p)2004 5528 y Fy(0)2041 5516 y FB(\()p FA(v)s FB(\))19 b(+)f FA(P)2303 5528 y Fy(1)2341 5516 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))2577 5350 y Fz(\023)p Black 1919 5753 a FB(73)p Black eop end %%Page: 74 74 TeXDict begin 74 73 bop Black Black 71 272 a FB(and)27 b(th)n(us)h(w)n(e)f(lo)r(ok)g(for)g(\()p FA(Q)942 284 y Fy(1)979 272 y FA(;)14 b(P)1069 284 y Fy(1)1107 272 y FB(\))28 b(as)f(solutions)f(of)1316 495 y(\()p Fw(L)1405 507 y Fx(")1460 495 y Fw(\000)18 b FA(A)p FB(\()p FA(u)p FB(\)\))1763 378 y Fz(\022)1866 445 y FA(Q)1932 457 y Fy(1)1866 544 y FA(P)1919 556 y Fy(1)2010 378 y Fz(\023)2094 495 y FB(=)23 b Fw(K)2260 378 y Fz(\022)2363 445 y FA(Q)2429 457 y Fy(1)2363 544 y FA(P)2416 556 y Fy(1)2507 378 y Fz(\023)2582 495 y FA(;)1056 b FB(\(202\))71 718 y(where)27 b Fw(L)368 730 y Fx(")431 718 y FB(is)h(the)g(op)r(erator)e(de\014ned)i (in)g(\(45\))o(,)g FA(A)g FB(is)f(the)h(matrix)f(de\014ned)h(in)g (\(182\))o(,)745 966 y Fw(K)q FB(\()p FA(\030)t FB(\)\()p FA(u;)14 b(\034)9 b FB(\))25 b(=)1220 824 y Fz( )1327 916 y FA(\026")1416 881 y Fx(\021)1456 916 y FA(@)1500 928 y Fx(p)1558 895 y Fz(b)1539 916 y FA(H)1608 928 y Fy(1)1659 916 y FB(\()p FA(q)1728 928 y Fy(0)1766 916 y FB(\()p FA(u)p FB(\))18 b(+)g FA(\030)2015 928 y Fy(1)2053 916 y FA(;)c(p)2132 928 y Fy(0)2169 916 y FB(\()p FA(u)p FB(\))k(+)g FA(\030)2418 928 y Fy(2)2456 916 y FA(;)c(\034)9 b FB(\))1327 1027 y FA(G)p FB(\()p FA(\030)1460 1039 y Fy(1)1498 1027 y FB(\)\()p FA(u;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(\026")1916 992 y Fx(\021)1956 1027 y FA(@)2000 1039 y Fx(q)2056 1006 y Fz(b)2037 1027 y FA(H)2106 1039 y Fy(1)2157 1027 y FB(\()p FA(q)2226 1039 y Fy(0)2263 1027 y FB(\()p FA(u)p FB(\))h(+)f FA(\030)2513 1039 y Fy(1)2551 1027 y FA(;)c(p)2630 1039 y Fy(0)2667 1027 y FB(\()p FA(u)p FB(\))k(+)g FA(\030)2916 1039 y Fy(2)2954 1027 y FA(;)c(\034)9 b FB(\))3110 824 y Fz(!)71 1210 y FB(and)494 1309 y FA(G)p FB(\()p FA(\030)627 1321 y Fy(1)665 1309 y FB(\)\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)f Fw(\000)14 b FB(\()p FA(V)1181 1275 y Fv(0)1204 1309 y FB(\()p FA(x)1283 1321 y Fx(p)1322 1309 y FB(\()p FA(\034)9 b FB(\))20 b(+)e FA(q)1571 1321 y Fy(0)1608 1309 y FB(\()p FA(u)p FB(\))h(+)f FA(\030)1858 1321 y Fy(1)1895 1309 y FB(\))h Fw(\000)f FA(V)2096 1275 y Fv(0)2119 1309 y FB(\()p FA(x)2198 1321 y Fx(p)2237 1309 y FB(\()p FA(\034)9 b FB(\)\))20 b Fw(\000)e FA(V)2548 1275 y Fv(0)2571 1309 y FB(\()p FA(q)2640 1321 y Fy(0)2678 1309 y FB(\()p FA(u)p FB(\)\))h Fw(\000)f FA(V)2991 1275 y Fv(00)3033 1309 y FB(\()p FA(q)3102 1321 y Fy(0)3140 1309 y FB(\()p FA(u)p FB(\)\))p FA(\030)3320 1321 y Fy(1)3358 1309 y FB(\))c FA(;)234 b FB(\(203\))71 1453 y(where)27 b(for)g(shortness)f(w) n(e)i(ha)n(v)n(e)e(put)i FA(\030)1301 1465 y Fy(1)1366 1453 y FB(and)g FA(\030)1564 1465 y Fy(2)1629 1453 y FB(for)f FA(\030)1792 1465 y Fy(1)1830 1453 y FB(\()p FA(u;)14 b(\034)9 b FB(\))28 b(and)g FA(\030)2250 1465 y Fy(2)2287 1453 y FB(\()p FA(u;)14 b(\034)9 b FB(\).)195 1553 y(W)-7 b(e)28 b(split)g Fw(K)h FB(considering)e(constan)n(t)g(and) g(higher)g(order)f(terms)h(in)h FA(\030)k FB(as)789 1726 y Fw(K)q FB(\()p FA(\030)t FB(\)\()p FA(u;)14 b(\034)9 b FB(\))25 b(=)d FA(L)p FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b(+)f(\()p FA(M)1729 1738 y Fy(1)1766 1726 y FB(\()p FA(u;)c(\034)9 b FB(\))19 b(+)f FA(M)2143 1738 y Fy(2)2180 1726 y FB(\()p FA(u;)c(\034)9 b FB(\)\))15 b FA(\030)t FB(\()p FA(u;)f(\034)9 b FB(\))19 b(+)f FA(N)9 b FB(\()p FA(\030)t FB(\)\()p FA(u;)14 b(\034)9 b FB(\))530 b(\(204\))71 1899 y(with)719 2127 y FA(L)p FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b(=)g FA(\026")1170 2092 y Fx(\021)1224 1985 y Fz( )1363 2076 y FA(@)1407 2088 y Fx(p)1465 2055 y Fz(b)1445 2076 y FA(H)1514 2088 y Fy(1)1551 2076 y FB(\()p FA(q)1620 2088 y Fy(0)1658 2076 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1849 2088 y Fy(0)1886 2076 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))1331 2187 y Fw(\000)p FA(@)1440 2199 y Fx(q)1496 2166 y Fz(b)1477 2187 y FA(H)1546 2199 y Fy(1)1583 2187 y FB(\()p FA(q)1652 2199 y Fy(0)1690 2187 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1881 2199 y Fy(0)1918 2187 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))2186 1985 y Fz(!)2270 2127 y FB(+)2353 2010 y Fz(\022)2618 2076 y FB(0)2456 2176 y FA(G)p FB(\(0\)\()p FA(u;)14 b(\034)9 b FB(\))2863 2010 y Fz(\023)3661 2127 y FB(\(205\))658 2409 y FA(M)739 2421 y Fy(1)776 2409 y FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b(=)g FA(\026")1170 2375 y Fx(\021)1224 2267 y Fz( )1363 2358 y FA(@)1407 2370 y Fx(q)r(p)1497 2337 y Fz(b)1478 2358 y FA(H)1554 2324 y Fy(1)1547 2379 y(1)1605 2358 y FB(\()p FA(q)1674 2370 y Fy(0)1711 2358 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1902 2370 y Fy(0)1939 2358 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))147 b FA(@)2356 2370 y Fx(pp)2448 2337 y Fz(b)2429 2358 y FA(H)2505 2328 y Fy(1)2498 2379 y(1)2542 2358 y FB(\()p FA(q)2611 2370 y Fy(0)2648 2358 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2839 2370 y Fy(0)2876 2358 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))1331 2469 y Fw(\000)p FA(@)1440 2481 y Fx(q)r(q)1529 2448 y Fz(b)1509 2469 y FA(H)1585 2435 y Fy(1)1578 2490 y(1)1636 2469 y FB(\()p FA(q)1705 2481 y Fy(0)1743 2469 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1934 2481 y Fy(0)1971 2469 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))84 b Fw(\000)p FA(@)2390 2481 y Fx(q)r(p)2480 2448 y Fz(b)2460 2469 y FA(H)2536 2439 y Fy(1)2529 2490 y(1)2573 2469 y FB(\()p FA(q)2642 2481 y Fy(0)2680 2469 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2871 2481 y Fy(0)2908 2469 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))3176 2267 y Fz(!)3661 2409 y FB(\(206\))658 2691 y FA(M)739 2703 y Fy(2)776 2691 y FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b(=)g FA(\026")1170 2657 y Fx(\021)r Fy(+1)1308 2549 y Fz( )1447 2641 y FA(@)1491 2653 y Fx(q)r(p)1581 2620 y Fz(b)1562 2641 y FA(H)1638 2606 y Fy(2)1631 2661 y(1)1689 2641 y FB(\()p FA(q)1758 2653 y Fy(0)1795 2641 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1986 2653 y Fy(0)2023 2641 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))147 b FA(@)2440 2653 y Fx(pp)2532 2620 y Fz(b)2513 2641 y FA(H)2589 2610 y Fy(2)2582 2661 y(1)2626 2641 y FB(\()p FA(q)2695 2653 y Fy(0)2732 2641 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2923 2653 y Fy(0)2960 2641 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))1415 2751 y Fw(\000)p FA(@)1524 2763 y Fx(q)r(q)1613 2730 y Fz(b)1593 2751 y FA(H)1669 2717 y Fy(2)1662 2772 y(1)1720 2751 y FB(\()p FA(q)1789 2763 y Fy(0)1827 2751 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2018 2763 y Fy(0)2055 2751 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))84 b Fw(\000)p FA(@)2474 2763 y Fx(q)r(p)2564 2730 y Fz(b)2544 2751 y FA(H)2620 2721 y Fy(2)2613 2772 y(1)2657 2751 y FB(\()p FA(q)2726 2763 y Fy(0)2764 2751 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2955 2763 y Fy(0)2992 2751 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))3260 2549 y Fz(!)3661 2691 y FB(\(207\))595 2898 y FA(N)g FB(\()p FA(\030)t FB(\)\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)f FA(L)p FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b(+)f(\()p FA(M)1547 2910 y Fy(1)1584 2898 y FB(\()p FA(u;)c(\034)9 b FB(\))19 b(+)f FA(M)1961 2910 y Fy(2)1998 2898 y FB(\()p FA(u;)c(\034)9 b FB(\)\))15 b FA(\030)t FB(\()p FA(u;)f(\034)9 b FB(\))19 b Fw(\000)f(K)q FB(\()p FA(\030)t FB(\)\()p FA(u;)c(\034)9 b FB(\))p FA(:)701 b FB(\(208\))195 3071 y(First)28 b(step)g(is)f(to)h(de\014ne)f (the)h(follo)n(wing)f(function)h(space)975 3265 y Fw(Y)1030 3277 y Fx(\033)1098 3265 y FB(=)1186 3173 y Fz(n)1241 3265 y FA(h)23 b FB(:)1375 3244 y Fz(e)1358 3265 y FA(D)1429 3225 y Fy(out)p Fx(;u)1427 3290 y(\032;d;\024)1606 3265 y Fw(\002)18 b Ft(T)24 b Fw(!)f Ft(C)p FB(;)41 b(real-analytic)n FA(;)14 b Fw(k)p FA(h)p Fw(k)2617 3277 y Fx(\033)2684 3265 y FA(<)22 b Fw(1)2854 3173 y Fz(o)2924 3265 y FA(;)71 3482 y FB(where)328 3461 y Fz(e)311 3482 y FA(D)382 3442 y Fy(out)p Fx(;u)380 3507 y(\032;d;\024)568 3482 y FB(is)28 b(the)g(domain)f(de\014ned)h(in)g(\(54\))f(and)1508 3695 y Fw(k)p FA(h)p Fw(k)1640 3707 y Fx(\033)1706 3695 y FB(=)1794 3617 y Fz(X)1794 3795 y Fx(k)q Fv(2)p Fn(Z)1928 3600 y Fz(\015)1928 3650 y(\015)1928 3700 y(\015)1974 3695 y FA(h)2022 3661 y Fy([)p Fx(k)q Fy(])2101 3600 y Fz(\015)2101 3650 y(\015)2101 3700 y(\015)2147 3754 y Fv(1)2231 3695 y FA(e)2270 3661 y Fv(j)p Fx(k)q Fv(j)p Fx(\033)2390 3695 y FA(;)1248 b FB(\(209\))71 3952 y(where)22 b Fw(k)8 b(\001)g(k)429 3964 y Fv(1)520 3952 y FB(is)23 b(the)f(classical)f(supremm)n(um)h(norm.)35 b(It)23 b(is)f(a)g(w)n(ell) g(kno)n(wn)f(fact)i(that)f(this)h(function)g(space)f(is)g(a)g(Banac)n (h)71 4052 y(algebra)k(\(see)h(for)g(instance)h([Sau01)n(]\).)38 b(W)-7 b(e)28 b(also)e(de\014ne)i(the)g(pro)r(duct)g(space)249 4236 y Fw(Y)304 4248 y Fx(\033)368 4236 y Fw(\002)18 b(Y)506 4248 y Fx(\033)574 4236 y FB(=)23 b Fw(f)p FA(h)f FB(=)h(\()p FA(h)942 4248 y Fy(1)979 4236 y FA(;)14 b(h)1064 4248 y Fy(2)1101 4236 y FB(\))23 b(:)1219 4215 y Fz(e)1202 4236 y FA(D)1273 4196 y Fy(out)p Fx(;u)1271 4261 y(\032;d;\024)1451 4236 y Fw(\002)18 b Ft(T)1589 4248 y Fx(\033)1657 4236 y Fw(!)23 b Ft(C)1823 4201 y Fy(2)1860 4236 y FB(;)42 b(real-analytic)n FA(;)14 b Fw(k)p FA(h)p Fw(k)2544 4248 y Fx(\033)2610 4236 y FB(=)23 b Fw(k)p FA(h)2788 4248 y Fy(1)2824 4236 y Fw(k)2866 4248 y Fx(\033)2929 4236 y FB(+)18 b Fw(k)p FA(h)3102 4248 y Fy(2)3139 4236 y Fw(k)3181 4248 y Fx(\033)3248 4236 y FA(<)23 b Fw(1g)p FA(:)177 b FB(\(210\))71 4409 y(Since)31 b(w)n(e)f(deal)g(with)h(the)g (Banac)n(h)e(space)h Fw(Y)1511 4421 y Fx(\033)1577 4409 y Fw(\002)20 b(Y)1717 4421 y Fx(\033)1762 4409 y FB(,)31 b(it)g(is)g(also)e(useful)i(to)g(consider)e(the)i(norm)f(for)g(2)20 b Fw(\002)g FB(2)30 b(matrices)71 4508 y(induced)e(b)n(y)f Fw(k)18 b(\001)g(k)638 4520 y Fx(\033)682 4508 y FB(.)37 b(Let)28 b(us)f(consider)g FA(B)g FB(=)1499 4441 y Fz(\000)1537 4508 y FA(b)1573 4478 y Fx(ij)1631 4441 y Fz(\001)1696 4508 y FB(a)g(2)18 b Fw(\002)g FB(2)27 b(matrix)g(suc)n(h)g(that)h FA(b)2650 4478 y Fx(ij)2731 4508 y Fw(2)23 b(Y)2864 4520 y Fx(\033)2910 4508 y FB(.)37 b(Then,)27 b(the)h(induced)g(norm)71 4608 y(with)g(resp)r(ect)f(to)h(the)g(norm)f(of)g Fw(Y)1155 4620 y Fx(\033)1219 4608 y Fw(\002)18 b(Y)1357 4620 y Fx(\033)1402 4608 y FB(,)28 b(whic)n(h)f(w)n(e)h(also)e(denote)i Fw(k)18 b(\001)g(k)2390 4620 y Fx(\033)2462 4608 y FB(abusing)27 b(notation,)g(is)h(giv)n(en)e(b)n(y)1359 4781 y Fw(k)p FA(B)t Fw(k)1510 4793 y Fx(\033)1577 4781 y FB(=)j(max)1664 4834 y Fx(j)s Fy(=1)p Fx(;)p Fy(2)1846 4714 y Fz(\010)1894 4711 y(\015)1894 4761 y(\015)1940 4781 y FA(b)1976 4747 y Fy(1)p Fx(j)2044 4711 y Fz(\015)2044 4761 y(\015)2090 4815 y Fx(\033)2153 4781 y FB(+)2236 4711 y Fz(\015)2236 4761 y(\015)2283 4781 y FA(b)2319 4747 y Fy(2)p Fx(j)2386 4711 y Fz(\015)2386 4761 y(\015)2432 4815 y Fx(\033)2477 4714 y Fz(\011)2539 4781 y FA(:)1099 b FB(\(211\))71 4993 y(The)28 b(next)f(lemma)h(giv)n(es)e(some)h(prop)r(erties)g(of)g (this)h(induced)g(norm.)p Black 71 5151 a Fp(Lemma)j(7.15.)p Black 40 w Fs(The)g(fol)t(lowing)h(statements)d(ar)l(e)h(satis\014e)l (d)p Black 169 5310 a(1.)p Black 42 w(If)h FA(h)22 b Fw(2)i(Y)570 5322 y Fx(\033)634 5310 y Fw(\002)18 b(Y)772 5322 y Fx(\033)846 5310 y Fs(and)31 b FA(B)c FB(=)c(\()p FA(b)1254 5280 y Fx(ij)1312 5310 y FB(\))30 b Fs(is)g FB(2)18 b Fw(\002)g FB(2)29 b Fs(matrix)h(with)g FA(b)2160 5280 y Fx(ij)2241 5310 y Fw(2)24 b(Y)2375 5322 y Fx(\033)2420 5310 y Fs(,)30 b(then)g FA(B)t(h)23 b Fw(2)g(Y)2931 5322 y Fx(\033)2995 5310 y Fw(\002)18 b(Y)3133 5322 y Fx(\033)3208 5310 y Fs(and)1691 5483 y Fw(k)p FA(B)t(h)p Fw(k)1890 5495 y Fx(\033)1957 5483 y Fw(\024)k(k)p FA(B)t Fw(k)2195 5495 y Fx(\033)2239 5483 y Fw(k)p FA(h)p Fw(k)2371 5495 y Fx(\033)2415 5483 y FA(:)p Black 1919 5753 a FB(74)p Black eop end %%Page: 75 75 TeXDict begin 75 74 bop Black Black Black 169 272 a Fs(2.)p Black 42 w(If)34 b FA(B)432 284 y Fy(1)499 272 y FB(=)29 b(\()p FA(b)661 232 y Fx(ij)661 294 y Fy(1)720 272 y FB(\))k Fs(and)h FA(B)1013 284 y Fy(2)1080 272 y FB(=)29 b(\()p FA(b)1242 232 y Fx(ij)1242 294 y Fy(2)1301 272 y FB(\))k Fs(ar)l(e)h FB(2)21 b Fw(\002)f FB(2)33 b Fs(matric)l(es)h (which)h(satisfy)f FA(b)2607 232 y Fx(ij)2607 294 y Fy(1)2695 272 y Fw(2)c(Y)2835 284 y Fx(\033)2914 272 y Fs(and)j FA(b)3114 232 y Fx(ij)3114 294 y Fy(2)3202 272 y Fw(2)d(Y)3342 284 y Fx(\033)3421 272 y Fs(r)l(esp)l(e)l(ctively,)278 381 y(then)g FA(B)526 393 y Fy(3)586 381 y FB(=)23 b(\()p FA(b)742 341 y Fx(ij)742 403 y Fy(3)800 381 y FB(\))h(=)e FA(B)1006 393 y Fy(1)1044 381 y FA(B)1107 393 y Fy(2)1173 381 y Fs(satis\014es)30 b FA(b)1521 341 y Fx(ij)1521 403 y Fy(3)1602 381 y Fw(2)24 b(E)1725 393 y Fx(\033)1799 381 y Fs(and)1656 562 y Fw(k)p FA(B)1761 574 y Fy(3)1797 562 y Fw(k)1839 574 y Fx(\033)1907 562 y Fw(\024)e(k)p FA(B)2099 574 y Fy(1)2136 562 y Fw(k)2178 574 y Fx(\033)2222 562 y Fw(k)p FA(B)2327 574 y Fy(2)2364 562 y Fw(k)2406 574 y Fx(\033)2450 562 y FA(:)195 777 y FB(Second)i(step)f(is)h(to)f (lo)r(ok)g(for)g(a)g(righ)n(t)f(in)n(v)n(erse)g(of)i Fw(L)1800 789 y Fx(")1846 777 y Fw(\000)10 b FA(A)p FB(\()p FA(u)p FB(\),)24 b(where)f FA(A)h FB(is)f(de\014ned)h(in)g(\(182\))o(.) 35 b(T)-7 b(o)24 b(obtain)f(it)h(w)n(e)f(use)71 877 y(the)i(op)r (erator)e Fw(G)592 889 y Fx(")653 877 y FB(de\014ned)j(in)f(\(168\))o (,)g(whic)n(h)g(is)g(w)n(ell)g(de\014ned)g(for)f(functions)i(b)r (elonging)e(to)h Fw(Y)3040 889 y Fx(\033)3085 877 y FB(,)g(if)h(w)n(e)e (tak)n(e)h FA(u)3552 889 y Fy(1)3588 877 y FA(;)p 3625 831 48 4 v 14 w(u)3673 889 y Fy(1)3735 877 y FB(the)71 987 y(v)n(ertices)30 b(of)g(the)i(domain)934 966 y Fz(e)917 987 y FA(D)988 947 y Fy(out)p Fx(;u)986 1012 y(\032;d;\024)1178 987 y FB(de\014ned)f(in)g(\(54\))g(\(see)g(Figure)f(7\).)47 b(Recalling)30 b(that)h(\010)g(de\014ned)g(in)g(\(184\))f(satis\014es) 71 1086 y Fw(L)128 1098 y Fx(")164 1086 y FB(\010)23 b(=)f FA(A)p FB(\010,)28 b(w)n(e)f(can)h(de\014ne)g(a)f(righ)n(t)g(in)n (v)n(erse)f(of)h Fw(L)1718 1098 y Fx(")1773 1086 y Fw(\000)18 b FA(A)p FB(\()p FA(v)s FB(\))28 b(as)1278 1297 y Fz(b)1265 1318 y Fw(G)1314 1330 y Fx(")1350 1318 y FB(\()p FA(h)p FB(\))23 b(=)g(\010)p Fw(G)1682 1330 y Fx(")1718 1318 y FB(\(\010)1810 1283 y Fv(\000)p Fy(1)1899 1318 y FA(h)p FB(\))p FA(;)42 b FB(for)27 b FA(h)c FB(=)2329 1201 y Fz(\022)2432 1267 y FA(h)2480 1279 y Fy(1)2432 1367 y FA(h)2480 1379 y Fy(2)2559 1201 y Fz(\023)2633 1318 y FA(:)1005 b FB(\(212\))p Black 71 1568 a Fp(Lemma)31 b(7.16.)p Black 40 w Fs(The)g(op)l(er)l(ator)1157 1547 y Fz(b)1144 1568 y Fw(G)1193 1580 y Fx(")1258 1568 y Fs(in)36 b FB(\(212\))29 b Fs(satis\014es)h(the)g(fol)t(lowing)i(pr)l (op)l(erties.)p Black 169 1743 a(1.)p Black 42 w(If)f FA(h)22 b Fw(2)i(Y)570 1755 y Fx(\033)634 1743 y Fw(\002)18 b(Y)772 1755 y Fx(\033)817 1743 y Fs(,)30 b(then)1070 1722 y Fz(b)1057 1743 y Fw(G)1106 1755 y Fx(")1141 1743 y FB(\()p FA(h)p FB(\))24 b Fw(2)f(Y)1410 1755 y Fx(\033)1474 1743 y Fw(\002)18 b(Y)1612 1755 y Fx(\033)1687 1743 y Fs(and)1704 1854 y Fz(\015)1704 1904 y(\015)1704 1954 y(\015)1764 1929 y(b)1751 1950 y Fw(G)1800 1962 y Fx(")1835 1950 y FB(\()p FA(h)p FB(\))1947 1854 y Fz(\015)1947 1904 y(\015)1947 1954 y(\015)1994 2008 y Fx(\033)2062 1950 y Fw(\024)k FA(K)6 b Fw(k)p FA(h)p Fw(k)2358 1962 y Fx(\033)2401 1950 y FA(:)p Black 169 2194 a Fs(2.)p Black 42 w(F)-6 b(urthermor)l(e,)30 b(if)h Fw(h)p FA(h)p Fw(i)24 b FB(=)e(0)p Fs(,)30 b(then)1685 2223 y Fz(\015)1685 2272 y(\015)1685 2322 y(\015)1744 2297 y(b)1731 2318 y Fw(G)1780 2330 y Fx(")1816 2318 y FB(\()p FA(h)p FB(\))1928 2223 y Fz(\015)1928 2272 y(\015)1928 2322 y(\015)1975 2376 y Fx(\033)2042 2318 y Fw(\024)23 b FA(K)6 b(")p Fw(k)p FA(h)p Fw(k)2378 2330 y Fx(\033)2421 2318 y FA(:)195 2529 y FB(W)-7 b(e)28 b(rewrite)f(Theorem)g(4.6)g(in)h(terms)f(of)g (equation)g(\(202\))g(and)h(the)g(Banac)n(h)e(spaces)h(de\014ned)h(in)f (\(210\).)p Black 71 2694 a Fp(Prop)s(osition)41 b(7.17.)p Black 45 w Fs(L)l(et)d FA(\032)1036 2706 y Fy(4)1111 2694 y Fs(and)h FA(\024)1329 2706 y Fy(1)1404 2694 y Fs(b)l(e)f(the)g(c)l(onstant)g(c)l(onsider)l(e)l(d)h(in)f(Pr)l(op)l (osition)i(7.12)g(and)f(7.4)g(and)g(let)f(us)71 2794 y(c)l(onsider)f(also)f FA(d)623 2806 y Fy(0)695 2794 y FA(>)d FB(0)i Fs(and)h FA(")1076 2806 y Fy(0)1147 2794 y FA(>)e FB(0)h Fs(smal)t(l)h(enough.)57 b(Then,)39 b(for)d FA(")e Fw(2)g FB(\(0)p FA(;)14 b(")2583 2806 y Fy(0)2620 2794 y FB(\))36 b Fs(ther)l(e)f(exist)h(functions)f FB(\()p FA(Q)3564 2806 y Fy(1)3601 2794 y FA(;)14 b(P)3691 2806 y Fy(1)3729 2794 y FB(\))34 b Fw(2)71 2893 y(Y)126 2905 y Fx(\033)183 2893 y Fw(\002)12 b(Y)315 2905 y Fx(\033)386 2893 y Fs(which)28 b(satisfy)g(e)l(quation)33 b FB(\(202\))26 b Fs(and)h(ar)l(e)g(the)g(analytic)h(c)l(ontinuation)f(of)g(the)g (functions)g FB(\()p FA(Q)3328 2905 y Fy(1)3365 2893 y FA(;)14 b(P)3455 2905 y Fy(1)3492 2893 y FB(\))27 b Fs(obtaine)l(d)71 2993 y(in)j(Cor)l(ol)t(lary)i(7.13.)40 b(Mor)l(e)l(over,)32 b(ther)l(e)e(exists)f(a)h(c)l(onstant)f FA(b)2022 3005 y Fy(5)2082 2993 y FA(>)23 b FB(0)29 b Fs(such)h(that)1516 3175 y Fw(k)p FB(\()p FA(Q)1656 3187 y Fy(1)1693 3175 y FA(;)14 b(P)1783 3187 y Fy(1)1820 3175 y FB(\))p Fw(k)1894 3187 y Fx(\033)1962 3175 y Fw(\024)23 b FA(b)2086 3187 y Fy(5)2123 3175 y Fw(j)p FA(\026)p Fw(j)p FA(")2258 3140 y Fx(\021)r Fy(+1)2382 3175 y FA(:)195 3356 y FB(Before)31 b(pro)n(ving)e(the)j(prop)r(osition,)f(w)n(e)g (state)g(and)g(pro)n(v)n(e)e(the)j(follo)n(wing)e(tec)n(hnical)h(lemma) g(whic)n(h)g(giv)n(es)f(some)71 3456 y(prop)r(erties)d(of)g FA(L)p FB(,)g FA(M)744 3468 y Fy(1)781 3456 y FB(,)h FA(M)913 3468 y Fy(2)977 3456 y FB(and)g FA(N)36 b FB(de\014ned)28 b(in)g(\(205\))o(,)g(\(206\))o(,)g(\(207\))e(and)i(\(208\))f(resp)r (ectiv)n(ely)-7 b(.)p Black 71 3621 a Fp(Lemma)36 b(7.18.)p Black 42 w Fs(The)e(functions)f FA(L)p Fs(,)i FA(M)1389 3633 y Fy(1)1425 3621 y Fs(,)g FA(M)1566 3633 y Fy(2)1636 3621 y Fs(and)f FA(N)42 b Fs(de\014ne)l(d)33 b(in)40 b FB(\(205\))o Fs(,)35 b FB(\(206\))o Fs(,)g FB(\(207\))d Fs(and)42 b FB(\(208\))33 b Fs(r)l(esp)l(e)l(ctively,)71 3721 y(have)e(the)f(fol)t(lowing)i(pr)l(op)l(erties,)p Black 169 3886 a(1.)p Black 42 w FA(L)23 b Fw(2)g(Y)491 3898 y Fx(\033)555 3886 y Fw(\002)18 b(Y)693 3898 y Fx(\033)768 3886 y Fs(and)30 b(satis\014es)1332 4088 y Fw(k)o FA(L)p Fw(k)1471 4113 y Fx(\033)1539 4088 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1839 4058 y Fx(\021)1878 4088 y FA(;)1985 3992 y Fz(\015)1985 4042 y(\015)1985 4092 y(\015)2044 4067 y(b)2031 4088 y Fw(G)2080 4100 y Fx(")2116 4088 y FB(\()p FA(L)p FB(\))2237 3992 y Fz(\015)2237 4042 y(\015)2237 4092 y(\015)2283 4146 y Fx(\033)2351 4088 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2650 4058 y Fx(\021)r Fy(+1)2774 4088 y FA(:)p Black 169 4365 a Fs(2.)p Black 42 w FA(M)359 4377 y Fy(1)419 4365 y FB(=)507 4272 y Fz(\020)556 4365 y FA(m)629 4325 y Fx(ij)629 4387 y Fy(1)688 4272 y Fz(\021)767 4365 y Fs(and)30 b FA(M)1009 4377 y Fy(2)1069 4365 y FB(=)1157 4272 y Fz(\020)1206 4365 y FA(m)1279 4325 y Fx(ij)1279 4387 y Fy(2)1338 4272 y Fz(\021)1417 4365 y Fs(satisfy)h FA(m)1749 4325 y Fx(ij)1749 4387 y Fy(1)1807 4365 y FA(;)14 b(m)1917 4325 y Fx(ij)1917 4387 y Fy(2)1999 4365 y Fw(2)23 b(Y)2132 4377 y Fx(\033)2196 4365 y Fw(\002)18 b(Y)2334 4377 y Fx(\033)2379 4365 y Fs(,)30 b Fw(h)p FA(M)2547 4377 y Fy(1)2584 4365 y Fw(i)23 b FB(=)g(0)29 b Fs(and)1315 4578 y Fw(k)o FA(M)1437 4590 y Fy(1)1474 4578 y Fw(k)1515 4603 y Fx(\033)1583 4578 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1883 4548 y Fx(\021)1923 4578 y FA(;)83 b Fw(k)o FA(M)2151 4590 y Fy(2)2188 4578 y Fw(k)2230 4603 y Fx(\033)2297 4578 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)2558 4548 y Fy(2)2595 4578 y FA(")2634 4548 y Fy(2)p Fx(\021)r Fy(+1)2791 4578 y FA(:)p Black 169 4794 a Fs(3.)p Black 42 w(If)31 b FA(\030)t(;)14 b(\030)483 4764 y Fv(0)529 4794 y Fw(2)24 b FA(B)t FB(\()p FA(\027)5 b FB(\))24 b Fw(\032)e(Y)951 4806 y Fx(\033)1015 4794 y Fw(\002)c(Y)1153 4806 y Fx(\033)1198 4794 y Fs(,)30 b(then)1449 4975 y Fw(k)p FA(N)9 b FB(\()p FA(\030)1639 4941 y Fv(0)1662 4975 y FB(\))19 b Fw(\000)f FA(N)9 b FB(\()p FA(\030)t FB(\))p Fw(k)2018 5000 y Fx(\033)2086 4975 y Fw(\024)22 b FA(K)6 b(\027)19 b Fw(k)o FA(\030)2391 4941 y Fv(0)2433 4975 y Fw(\000)f FA(\030)t Fw(k)2598 5000 y Fx(\033)2657 4975 y FA(:)p Black 71 5190 a Fs(Pr)l(o)l(of.)p Black 43 w FB(F)-7 b(or)27 b(the)h(\014rst)f(statemen)n(t)h(let)g(us)f (split)h FA(L)f FB(as)g FA(L)c FB(=)g FA(L)2002 5202 y Fy(1)2057 5190 y FB(+)18 b FA(L)2197 5202 y Fy(2)2252 5190 y FB(+)g FA(L)2392 5202 y Fy(3)2456 5190 y FB(with)939 5442 y FA(L)996 5454 y Fx(i)1023 5442 y FB(\()p FA(u;)c(\034)9 b FB(\))24 b(=)1328 5300 y Fz( )1467 5391 y FA(\026")1556 5361 y Fx(\021)r Fy(+)p Fx(i)p Fv(\000)p Fy(1)1756 5391 y FA(@)1800 5403 y Fx(p)1858 5370 y Fz(b)1838 5391 y FA(H)1914 5361 y Fx(i)1907 5412 y Fy(1)1944 5391 y FB(\()p FA(q)2013 5403 y Fy(0)2051 5391 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2242 5403 y Fy(0)2279 5391 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))1436 5502 y Fw(\000)p FA(\026")1590 5472 y Fx(\021)r Fy(+)p Fx(i)p Fv(\000)p Fy(1)1789 5502 y FA(@)1833 5514 y Fx(q)1889 5481 y Fz(b)1870 5502 y FA(H)1946 5472 y Fx(i)1939 5523 y Fy(1)1976 5502 y FB(\()p FA(q)2045 5514 y Fy(0)2082 5502 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2273 5514 y Fy(0)2310 5502 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))2579 5300 y Fz(!)2659 5442 y FA(;)41 b(i)23 b FB(=)f(1)p FA(;)14 b FB(2)p Black 1919 5753 a(75)p Black eop end %%Page: 76 76 TeXDict begin 76 75 bop Black Black 71 272 a FB(and)1457 401 y FA(L)1514 413 y Fy(3)1551 401 y FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b(=)1856 284 y Fz(\022)2121 350 y FB(0)1959 450 y FA(G)p FB(\(0\)\()p FA(u;)14 b(\034)9 b FB(\))2366 284 y Fz(\023)2441 401 y FA(;)71 613 y FB(where)324 592 y Fz(b)305 613 y FA(H)381 583 y Fy(1)374 633 y(1)418 613 y FB(,)483 592 y Fz(b)463 613 y FA(H)539 583 y Fy(2)532 633 y(1)598 613 y FB(and)21 b FA(G)g FB(are)f(the)i(functions)g (de\014ned)f(in)h(\(35\))o(,)h(\(37\))e(and)g(\(203\))f(resp)r(ectiv)n (ely)-7 b(.)34 b(One)21 b(can)g(easily)f(see)h(that)71 712 y FA(L)128 724 y Fy(1)165 712 y FA(;)14 b(L)259 724 y Fy(2)318 712 y Fw(2)24 b(Y)452 724 y Fx(\033)509 712 y Fw(\002)12 b(Y)641 724 y Fx(\033)686 712 y FB(,)25 b Fw(h)p FA(L)823 724 y Fy(1)860 712 y Fw(i)e FB(=)g(0)h(and)g Fw(k)p FA(L)1326 724 y Fy(1)1362 712 y Fw(k)1404 724 y Fx(\033)1471 712 y Fw(\024)f FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1771 682 y Fx(\021)1835 712 y FB(and,)25 b(using)f(Corollary)e(5.6,)i(also)f(that)i Fw(k)p FA(L)3188 724 y Fy(2)3224 712 y Fw(k)3266 724 y Fx(\033)3334 712 y Fw(\024)d FA(K)6 b Fw(j)p FA(\026)p Fw(j)3594 682 y Fy(2)3631 712 y FA(")3670 682 y Fy(2)p Fx(\021)r Fy(+1)3827 712 y FB(.)71 822 y(Th)n(us,)27 b(applying)g(Lemma)h(7.16)e(one)h (obtains)g Fw(k)1613 801 y Fz(b)1600 822 y Fw(G)1649 834 y Fx(")1685 822 y FB(\()p FA(L)1774 834 y Fx(i)1801 822 y FB(\))p Fw(k)1875 834 y Fx(\033)1943 822 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2242 792 y Fx(\021)r Fy(+1)2394 822 y FB(for)27 b FA(i)22 b FB(=)h(1)p FA(;)14 b FB(2.)195 922 y(T)-7 b(o)28 b(obtain)f(analogous)e(prop)r(erties)i (for)g FA(L)1534 934 y Fy(3)1571 922 y FB(,)g(it)h(is)g(enough)f(to)g (apply)h(Mean)f(V)-7 b(alue)28 b(Theorem)e(to)i(obtain)782 1193 y FA(L)839 1205 y Fy(3)876 1193 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)1181 1026 y Fz(0)1181 1175 y(@)2121 1089 y FB(0)1296 1249 y Fw(\000)1375 1136 y Fz(Z)1457 1156 y Fy(1)1420 1325 y(0)1508 1249 y FA(V)1575 1190 y Fl(000)1652 1249 y FB(\()p FA(s)1723 1261 y Fy(1)1760 1249 y FA(x)1807 1261 y Fx(p)1846 1249 y FB(\()p FA(\034)9 b FB(\))20 b(+)e FA(s)2097 1261 y Fy(2)2134 1249 y FA(q)2171 1261 y Fy(0)2208 1249 y FB(\()p FA(u)p FB(\)\))d FA(ds)2449 1261 y Fy(1)2486 1249 y FA(ds)2568 1261 y Fy(2)2605 1249 y FA(q)2642 1261 y Fy(0)2680 1249 y FB(\()p FA(u)p FB(\))p FA(x)2839 1261 y Fx(p)2878 1249 y FB(\()p FA(\034)9 b FB(\))3029 1026 y Fz(1)3029 1175 y(A)3116 1193 y FA(:)71 1487 y FB(Then,)32 b Fw(k)p FA(L)414 1499 y Fy(3)450 1487 y Fw(k)492 1499 y Fx(\033)565 1487 y Fw(\024)c FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")870 1457 y Fx(\021)r Fy(+1)994 1487 y FB(.)46 b(Therefore,)31 b(applying)f(Lemma)h(7.16)e(w)n(e)i(ha)n (v)n(e)e(that)i Fw(k)2845 1466 y Fz(b)2832 1487 y Fw(G)2881 1499 y Fx(")2917 1487 y FB(\()p FA(L)3006 1499 y Fy(3)3043 1487 y FB(\))p Fw(k)3117 1499 y Fx(\033)3190 1487 y Fw(\024)d FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")3495 1457 y Fx(\021)r Fy(+1)3619 1487 y FB(.)47 b(This)71 1586 y(\014nishes)27 b(the)h(pro)r(of)f(of)h(the)g(\014rst)f(statemen)n(t.)195 1686 y(The)h(pro)r(of)f(of)h(the)g(other)f(statemen)n(ts)g(is)g (straigh)n(tforw)n(ard.)p 3790 1686 4 57 v 3794 1633 50 4 v 3794 1686 V 3843 1686 4 57 v 195 1850 a(T)-7 b(o)28 b(pro)n(v)n(e)e(Prop)r(osition)g(7.17,)h(one)g(has)h(to)f(p)r(erform)h (\014rst)f(a)h(c)n(hange)f(of)g(v)-5 b(ariables)27 b(to)h(equation)f (\(202\))g(to)h(obtain)71 1950 y(a)f(con)n(tractiv)n(e)f(op)r(erator.) 35 b(In)28 b(fact,)g(this)g(c)n(hange)e(is)h(only)h(necessary)d(in)j (the)g(case)f FA(\021)f FB(=)d(0.)36 b(Let)28 b(us)g(consider)p 1307 2076 90 4 v 1307 2143 a FA(M)1397 2155 y Fy(1)1457 2143 y FB(=)1545 2050 y Fz(\020)p 1594 2097 73 4 v 1594 2143 a FA(m)1667 2103 y Fx(ij)1667 2165 y Fy(1)1726 2050 y Fz(\021)1817 2143 y FB(with)p 2006 2097 V 28 w FA(m)2079 2103 y Fx(ij)2079 2165 y Fy(1)2160 2143 y FB(=)23 b Fw(G)2297 2155 y Fx(")2347 2050 y Fz(\020)2396 2143 y FA(m)2469 2103 y Fx(ij)2469 2165 y Fy(1)2528 2050 y Fz(\021)2591 2143 y FA(;)1047 b FB(\(213\))71 2373 y(where)30 b Fw(G)363 2385 y Fx(")429 2373 y FB(is)g(the)g(op)r(erator)f(de\014ned)h(in)h (\(168\))e(and)h FA(M)1850 2385 y Fy(1)1914 2373 y FB(=)2006 2281 y Fz(\020)2056 2373 y FA(m)2129 2333 y Fx(ij)2129 2395 y Fy(1)2187 2281 y Fz(\021)2267 2373 y FB(is)g(the)h(matrix)e (de\014ned)i(in)f(\(206\))o(.)45 b(By)30 b(Lemmas)71 2493 y(7.18)c(and)i(7.3,)e(one)i(can)f(see)g(that)1599 2593 y Fw(k)p 1641 2526 90 4 v FA(M)1730 2605 y Fy(1)1767 2593 y Fw(k)1809 2605 y Fx(\033)1876 2593 y Fw(\024)c FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2176 2559 y Fx(\021)r Fy(+1)2300 2593 y FA(:)1338 b FB(\(214\))71 2736 y(W)-7 b(e)28 b(consider)e(the)i(c)n(hange)f(of)g(v)-5 b(ariables)1668 2836 y FA(\030)27 b FB(=)1819 2769 y Fz(\000)1857 2836 y FB(Id)18 b(+)p 2034 2769 V 18 w FA(M)2124 2848 y Fy(1)2161 2769 y Fz(\001)p 2213 2768 41 4 v 2213 2836 a FA(\030)1412 b FB(\(215\))71 2991 y(whic)n(h)27 b(is)h(in)n(v)n(ertible.)36 b(Using)29 b(\(202\))d(and)i(\(215\))o(,)p 1644 2924 V 28 w FA(\030)j FB(is)d(solution)f(of)h(equation)1566 3174 y(\()p Fw(L)1655 3186 y Fx(")1709 3174 y Fw(\000)18 b FA(A)p FB(\()p FA(u)p FB(\)\))p 2013 3107 V 15 w FA(\030)27 b FB(=)2177 3153 y Fz(b)2163 3174 y Fw(K)r FB(\()p 2260 3107 V FA(\030)t FB(\))p FA(;)1306 b FB(\(216\))71 3347 y(where)1537 3426 y Fz(b)1523 3447 y Fw(K)r FB(\()p 1620 3379 V FA(\030)t FB(\))23 b(=)1811 3426 y Fz(b)1803 3447 y FA(L)18 b FB(+)1971 3426 y Fz(c)1961 3447 y FA(M)p 2051 3379 V 9 w(\030)k FB(+)2214 3426 y Fz(b)2192 3447 y FA(N)2282 3379 y Fz(\000)p 2320 3379 V 68 x FA(\030)2360 3379 y Fz(\001)3661 3447 y FB(\(217\))71 3590 y(with)947 3742 y Fz(b)940 3763 y FA(L)g FB(=)1107 3695 y Fz(\000)1145 3763 y FB(Id)c(+)p 1322 3696 90 4 v 18 w FA(M)1412 3775 y Fy(1)1449 3695 y Fz(\001)1487 3710 y Fv(\000)p Fy(1)1590 3763 y FA(L)2014 b FB(\(218\))917 3895 y Fz(c)906 3916 y FA(M)32 b FB(=)1107 3849 y Fz(\000)1145 3916 y FB(Id)18 b(+)p 1322 3849 V 18 w FA(M)1412 3928 y Fy(1)1449 3849 y Fz(\001)1487 3863 y Fv(\000)p Fy(1)1590 3849 y Fz(\000)1628 3916 y FA(M)1709 3928 y Fy(1)p 1746 3849 V 1746 3916 a FA(M)1836 3928 y Fy(1)1891 3916 y FB(+)g FA(A)p 2036 3849 V(M)2126 3928 y Fy(1)2182 3916 y Fw(\000)p 2265 3849 V 18 w FA(M)2355 3928 y Fy(1)2392 3916 y FA(A)g FB(+)h FA(M)2637 3928 y Fy(2)2687 3849 y Fz(\000)2725 3916 y FB(Id)g(+)p 2903 3849 V 18 w FA(M)2992 3928 y Fy(1)3030 3849 y Fz(\001\001)3661 3916 y FB(\(219\))837 4048 y Fz(b)816 4069 y FA(N)8 b FB(\()p 923 4002 41 4 v FA(\030)d FB(\))23 b(=)1107 4002 y Fz(\000)1145 4069 y FB(Id)18 b(+)p 1322 4003 90 4 v 18 w FA(M)1412 4081 y Fy(1)1449 4002 y Fz(\001)1487 4016 y Fv(\000)p Fy(1)1590 4069 y FA(N)1680 4002 y Fz(\000\000)1756 4069 y FB(Id)g(+)p 1933 4003 V 18 w FA(M)2023 4081 y Fy(1)2060 4002 y Fz(\001)p 2112 4002 41 4 v 2112 4069 a FA(\030)2152 4002 y Fz(\001)3661 4069 y FB(\(220\))195 4242 y(Since)26 b(w)n(e)g(w)n(an)n(t)f(to)h (obtain)f(the)h(analytic)g(con)n(tin)n(uation)f(of)g(the)i (parameterizations)c(of)j(the)g(manifolds)g(obtained)71 4342 y(in)h(Corollary)d(7.13,)i(w)n(e)g(need)h(to)f(imp)r(ose)h Fs(initial)j(c)l(onditions)p FB(.)38 b(Nev)n(ertheless,)25 b(since)i(w)n(e)f(in)n(v)n(ert)g Fw(L)3212 4354 y Fx(")3264 4342 y Fw(\000)16 b FA(A)p FB(\()p FA(u)p FB(\))27 b(b)n(y)g(using)71 4452 y(the)c(op)r(erator)553 4431 y Fz(b)540 4452 y Fw(G)589 4464 y Fx(")648 4452 y FB(in)g(\(212\))f(whic)n(h)i(is)f(de\014ned)g (acting)g(on)f(the)i(F)-7 b(ourier)22 b(co)r(e\016cien)n(ts,)i(w)n(e)e (need)i(to)f(consider)f(a)g(di\013eren)n(t)71 4551 y(initial)35 b(condition)f(dep)r(ending)h(on)g(the)g(F)-7 b(ourier)34 b(co)r(e\016cien)n(t,)i(that)f(is)g(in)g FA(u)2512 4563 y Fy(1)2583 4551 y FB(or)f(in)40 b(\026)-47 b FA(u)2844 4563 y Fy(1)2916 4551 y FB(\(see)34 b(Figure)g(7\).)59 b(Th)n(us,)36 b(w)n(e)71 4651 y(de\014ne)28 b(the)g(follo)n(wing)e (function)978 4846 y FA(L)1035 4858 y Fy(0)1072 4846 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)1365 4767 y Fz(X)1364 4945 y Fx(k)q(<)p Fy(0)1499 4846 y FB(\010\()p FA(v)s FB(\)\010)1726 4811 y Fv(\000)p Fy(1)1816 4846 y FB(\()p 1848 4800 44 4 v FA(v)1892 4858 y Fy(1)1929 4846 y FB(\))p 1961 4778 41 4 v FA(\030)2001 4792 y Fy([)p Fx(k)q Fy(])2094 4846 y FB(\()p 2126 4800 44 4 v FA(v)2169 4858 y Fy(1)2206 4846 y FB(\))15 b FA(e)2292 4811 y Fv(\000)p Fx(ik)q(")2434 4786 y Fl(\000)p Fu(1)2512 4811 y Fy(\()p Fx(v)r Fv(\000)s Fy(\026)-36 b Fx(v)2658 4819 y Fu(1)2691 4811 y Fy(\))2721 4846 y FA(e)2760 4811 y Fx(ik)q(\034)1369 5089 y FB(+)1452 5010 y Fz(X)1452 5188 y Fx(k)q Fv(\025)p Fy(0)1587 5089 y FB(\010\()p FA(v)s FB(\)\010)1814 5054 y Fv(\000)p Fy(1)1904 5089 y FB(\()p FA(v)1976 5101 y Fy(1)2014 5089 y FB(\))p 2046 5021 41 4 v FA(\030)2086 5035 y Fy([)p Fx(k)q Fy(])2178 5089 y FB(\()q FA(v)2251 5101 y Fy(1)2288 5089 y FB(\))14 b FA(e)2373 5054 y Fv(\000)p Fx(ik)q(")2515 5029 y Fl(\000)p Fu(1)2594 5054 y Fy(\()p Fx(v)r Fv(\000)p Fx(v)2740 5062 y Fu(1)2773 5054 y Fy(\))2803 5089 y FA(e)2842 5054 y Fx(ik)q(\034)1369 5319 y FB(+)k(\010\()p FA(v)s FB(\)\010)1679 5284 y Fv(\000)p Fy(1)1782 5319 y FB(\()q Fw(\000)p FA(\032)1923 5331 y Fy(4)1959 5319 y FB(\))c FA(\030)2045 5284 y Fy([0])2134 5319 y FB(\()p Fw(\000)p FA(\032)2274 5331 y Fy(4)2311 5319 y FB(\))g FA(:)3661 5070 y FB(\(221\))71 5504 y(Recall)27 b(that)p 501 5436 V 28 w FA(\030)t FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))29 b(is)f(already)e(kno)n(wn)h(for)g FA(v)f FB(=)d FA(v)1720 5516 y Fy(1)1757 5504 y FA(;)p 1794 5458 44 4 v 14 w(v)1837 5516 y Fy(1)1875 5504 y FA(;)14 b Fw(\000)p FA(\032)2020 5516 y Fy(4)2084 5504 y FB(using)28 b(\(215\))o(,)g (\(213\))f(and)g(Corollary)e(7.13.)p Black 1919 5753 a(76)p Black eop end %%Page: 77 77 TeXDict begin 77 76 bop Black Black Black 71 272 a Fp(Lemma)31 b(7.19.)p Black 40 w Fs(The)g(function)f FA(L)1201 284 y Fy(0)1237 272 y FB(\()p FA(u;)14 b(\034)9 b FB(\))31 b Fs(in)36 b FB(\(221\))29 b Fs(satis\014es)h(de)g(fol)t(lowing)i(pr)l (op)l(erties:)p Black 195 410 a Fw(\017)p Black 41 w FB(\()q Fw(L)368 422 y Fx(")422 410 y Fw(\000)18 b FA(A)p FB(\()p FA(v)s FB(\)\))d FA(L)778 422 y Fy(0)838 410 y FB(=)23 b(0)p Fs(,)29 b(wher)l(e)i Fw(L)1314 422 y Fx(")1379 410 y Fs(is)f(the)g(op)l(er)l(ator)h(in)36 b FB(\(45\))p Fs(.)p Black 195 562 a Fw(\017)p Black 41 w FA(L)335 574 y Fy(0)395 562 y Fw(2)24 b(Y)529 574 y Fx(\033)592 562 y Fw(\002)18 b(Y)730 574 y Fx(\033)805 562 y Fs(and)1719 662 y Fw(k)p FA(L)1818 674 y Fy(0)1854 662 y Fw(k)1896 674 y Fx(\033)1963 662 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2263 628 y Fx(\021)r Fy(+1)2387 662 y FA(:)195 817 y FB(The)23 b(function)p 681 749 41 4 v 23 w FA(\030)j FB(satis\014es)c(equation)f(\(216\))h (and)g(the)h(initial)f(conditions)g(on)g(the)h(F)-7 b(ourier)21 b(co)r(e\016cien)n(ts)h FA(L)3511 829 y Fy(0)3570 817 y FB(in)g(\(221\))71 916 y(if)28 b(and)f(only)h(if)g(it)g(is)f (solution)g(of)h(the)g(in)n(tegral)e(equation)1360 992 y Fz(\022)1463 1059 y FA(Q)1529 1071 y Fy(1)1463 1158 y FA(P)1516 1170 y Fy(1)1607 992 y Fz(\023)1691 1109 y FB(=)d FA(L)1836 1121 y Fy(0)1891 1109 y FB(+)1988 1088 y Fz(b)1974 1109 y Fw(G)2023 1121 y Fx(")2078 1109 y Fw(\016)18 b(K)2216 992 y Fz(\022)2319 1059 y FA(Q)2385 1071 y Fy(1)2319 1158 y FA(P)2372 1170 y Fy(1)2463 992 y Fz(\023)2538 1109 y FA(;)71 1326 y FB(where)321 1305 y Fz(b)308 1326 y Fw(G)357 1338 y Fx(")418 1326 y FB(and)24 b Fw(K)i FB(are)e(the)h(op)r(erators)e(de\014ned)i(in)g(\(212\))f(and)g (\(204\))g(resp)r(ectiv)n(ely)-7 b(.)35 b(Th)n(us,)25 b(w)n(e)f(lo)r(ok)g(for)g(a)h(\014xed)f(p)r(oin)n(t)71 1425 y FA(\030)j FB(=)c(\()p FA(Q)320 1437 y Fy(1)357 1425 y FA(;)14 b(P)447 1437 y Fy(1)484 1425 y FB(\))24 b Fw(2)f(Y)673 1437 y Fx(\033)737 1425 y Fw(\002)18 b(Y)875 1437 y Fx(\033)948 1425 y FB(of)27 b(the)h(op)r(erator)p 1650 1517 65 4 v 1650 1584 a Fw(K)d FB(=)d FA(L)1882 1596 y Fy(0)1937 1584 y FB(+)2034 1563 y Fz(b)2020 1584 y Fw(G)2069 1596 y Fx(")2124 1584 y Fw(\016)2197 1563 y Fz(b)2184 1584 y Fw(K)q FA(:)1390 b FB(\(222\))71 1731 y(Therefore,)26 b(Prop)r(osition)g(7.17)g(is)i(a)f(straigh)n(tforw)n (ard)e(consequence)h(of)i(the)g(follo)n(wing)e(lemma.)p Black 71 1870 a Fp(Lemma)44 b(7.20.)p Black 46 w Fs(L)l(et)c(us)f(c)l (onsider)i FA(")1315 1882 y Fy(0)1393 1870 y FA(>)g FB(0)e Fs(smal)t(l)i(enough.)69 b(Then,)43 b(for)e FA(")g Fw(2)g FB(\(0)p FA(;)14 b(")2880 1882 y Fy(0)2917 1870 y FB(\))p Fs(,)43 b(ther)l(e)d(exists)f(a)h(function)p 71 1912 41 4 v 71 1980 a FA(\030)27 b Fw(2)c(Y)267 1992 y Fx(\033)331 1980 y Fw(\002)18 b(Y)469 1992 y Fx(\033)544 1980 y Fs(de\014ne)l(d)30 b(in)943 1959 y Fz(e)926 1980 y FA(D)997 1940 y Fy(out)p Fx(;u)995 2005 y(\032)1029 2013 y Fu(4)1062 2005 y Fx(;d)1117 2013 y Fu(0)1148 2005 y Fx(;\024)1207 2013 y Fu(1)1262 1980 y Fw(\002)18 b Ft(T)1400 1992 y Fx(\033)1475 1980 y Fs(such)29 b(that)h(is)g(a)g(\014xe)l(d)g(p)l(oint)g(of)g(the)g(op)l (er)l(ator)40 b FB(\(222\))29 b Fs(and)h(satis\014es)1639 2076 y Fz(\015)1639 2126 y(\015)p 1685 2079 V 21 x FA(\030)1725 2076 y Fz(\015)1725 2126 y(\015)1771 2180 y Fx(\033)1839 2147 y Fw(\024)23 b FA(b)1963 2159 y Fy(5)2000 2147 y Fw(j)p FA(\026)p Fw(j)p FA(")2135 2112 y Fx(\021)r Fy(+1)2259 2147 y FA(:)71 2311 y Fs(for)39 b(a)f(c)l(ertain)g(c)l(onstant)f FA(b)954 2323 y Fy(5)1029 2311 y FA(>)h FB(0)f Fs(indep)l(endent)i(of)f FA(")g Fs(and)g FA(\026)p Fs(.)64 b(Mor)l(e)l(over,)42 b FA(\030)g FB(=)37 b(\(Id)25 b(+)p 2973 2244 90 4 v 24 w FA(M)3062 2323 y Fy(1)3100 2311 y FB(\))p 3132 2243 41 4 v FA(\030)t Fs(,)41 b(wher)l(e)p 3480 2244 90 4 v 38 w FA(M)3570 2323 y Fy(1)3645 2311 y Fs(is)d(the)71 2410 y(function)29 b(de\014ne)l(d)g(in)36 b FB(\(213\))o Fs(,)29 b(is)h(the)f(analytic)h(c)l(ontinuation)f(of)h(the)f(function)g FA(\030)e FB(=)c(\()p FA(Q)2856 2422 y Fy(1)2893 2410 y FA(;)14 b(P)2983 2422 y Fy(1)3020 2410 y FB(\))30 b Fs(obtaine)l(d)g(in)f(Cor)l(ol)t(lary)71 2510 y(7.13.)p Black 71 2648 a(Pr)l(o)l(of.)p Black 43 w FB(T)-7 b(o)28 b(pro)n(v)n(e)g(the)h(lemma,)g(\014rst)g(w)n(e)f(see)h(that)g(there)f (exists)h(a)f(constan)n(t)h FA(b)2616 2660 y Fy(5)2678 2648 y FA(>)24 b FB(0)29 b(suc)n(h)f(that)h(the)h(op)r(erator)3704 2627 y(\026)3688 2648 y Fw(K)g FB(in)71 2748 y(\(222\))e(is)i(con)n (tractiv)n(e)d(from)p 1001 2681 68 4 v 29 w FA(B)5 b FB(\()p FA(b)1137 2760 y Fy(5)1174 2748 y Fw(j)p FA(\026)p Fw(j)p FA(")1309 2718 y Fx(\021)r Fy(+1)1433 2748 y FB(\))26 b Fw(\032)g(Y)1637 2760 y Fx(\033)1702 2748 y Fw(\002)19 b(Y)1841 2760 y Fx(\033)1916 2748 y FB(to)29 b(itself)h(and)f(th)n(us)h (that)f(it)h(has)f(a)g(\014xed)g(p)r(oin)n(t.)43 b(Then,)30 b(w)n(e)71 2847 y(will)g(see)f(that)g FA(\030)i FB(=)26 b(\(Id)20 b(+)p 917 2781 90 4 v 19 w FA(M)1007 2859 y Fy(1)1044 2847 y FB(\))p 1076 2780 41 4 v FA(\030)t FB(,)30 b(where)p 1411 2781 90 4 v 29 w FA(M)1501 2859 y Fy(1)1568 2847 y FB(is)f(the)h(function)g(de\014ned)g(in)g(\(213\))o(,)g(is)f (the)h(analytic)f(con)n(tin)n(uation)g(of)71 2947 y(the)f (parameterizations)d(of)j(the)g(manifolds)f(whic)n(h)h(ha)n(v)n(e)e(b)r (een)i(obtained)f(in)h(Corollary)d(7.13.)195 3047 y(Let)f(us)g(\014rst) g(consider)p 930 2980 65 4 v 22 w Fw(K)r FB(\(0\).)35 b(Using)24 b(the)g(de\014nitions)g(of)p 2017 2980 V 24 w Fw(K)q FB(,)2143 3026 y Fz(b)2129 3047 y Fw(K)h FB(and)2382 3026 y Fz(b)2375 3047 y FA(L)e FB(in)h(\(222\))o(,)h(\(204\))e(and)g (\(218\),)h(w)n(e)g(ha)n(v)n(e)e(that)p 1104 3153 V 1104 3219 a Fw(K)q FB(\(0\))i(=)e FA(L)1442 3231 y Fy(0)1497 3219 y FB(+)1594 3198 y Fz(b)1580 3219 y Fw(G)1629 3231 y Fx(")1679 3127 y Fz(\020)1736 3198 y(b)1729 3219 y FA(L)1785 3127 y Fz(\021)1298 3402 y FB(=)p 1385 3335 57 4 v 22 w FA(L)1442 3414 y Fy(0)1497 3402 y FB(+)1594 3381 y Fz(b)1580 3402 y Fw(G)1629 3414 y Fx(")1679 3402 y FB(\()q FA(L)p FB(\))c(+)1915 3381 y Fz(b)1902 3402 y Fw(G)1951 3414 y Fx(")2001 3310 y Fz(\020)p 2050 3335 90 4 v 2050 3402 a FA(M)2140 3414 y Fy(1)2191 3335 y Fz(\000)2229 3402 y FB(Id)g(+)p 2406 3335 V 18 w FA(M)2496 3414 y Fy(1)2533 3335 y Fz(\001)2571 3349 y Fv(\000)p Fy(1)2674 3402 y FA(L)2731 3310 y Fz(\021)2794 3402 y FA(:)195 3586 y FB(F)-7 b(rom)22 b(Lemmas)f(7.19,)h(7.16)e(and)h(7.18,) h(and)g(applying)f(also)f(the)j(b)r(ound)f(of)p 2568 3519 V 21 w FA(M)2658 3598 y Fy(1)2717 3586 y FB(in)g(\(214\))o(,)h(it) f(is)g(straigh)n(tforw)n(ard)d(to)71 3685 y(see)27 b(that)h Fw(k)p 427 3619 65 4 v(K)q FB(\(0\))p Fw(k)639 3697 y Fx(\033)706 3685 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1006 3655 y Fx(\021)r Fy(+1)1130 3685 y FB(,)27 b(and)h(th)n(us)g(there)f(exists)g(a)g(constan)n(t)g FA(b)2406 3697 y Fy(5)2466 3685 y FA(>)c FB(0)k(suc)n(h)g(that)h Fw(k)p 3032 3619 V(K)q FB(\(0\))p Fw(k)3244 3697 y Fx(\033)3311 3685 y Fw(\024)23 b FA(b)3435 3697 y Fy(5)3472 3685 y Fw(j)p FA(\026)p Fw(j)p FA(")3607 3655 y Fx(\021)r Fy(+1)3731 3685 y FA(=)p FB(2.)195 3806 y(Let)j(us)g(consider)f(no)n(w)p 940 3738 41 4 v 25 w FA(\030)980 3752 y Fy(1)1017 3806 y FA(;)p 1054 3738 V 14 w(\030)1094 3752 y Fy(2)1155 3806 y Fw(2)p 1233 3739 68 4 v 23 w FA(B)t FB(\()p FA(b)1368 3818 y Fy(5)1405 3806 y Fw(j)p FA(\026)p Fw(j)p FA(")1540 3776 y Fx(\021)r Fy(+1)1665 3806 y FB(\))e Fw(\032)g(Y)1863 3818 y Fx(\033)1923 3806 y Fw(\002)14 b(Y)2057 3818 y Fx(\033)2102 3806 y FB(.)37 b(Then)26 b(using)f(the)h(de\014nitions)g (of)p 3226 3739 65 4 v 26 w Fw(K)h FB(and)3490 3785 y Fz(b)3476 3806 y Fw(K)g FB(in)f(\(222\))71 3905 y(and)h(\(217\))o(,)h (and)g(applying)f(Lemma)g(7.16,)814 3983 y Fz(\015)814 4032 y(\015)814 4082 y(\015)p 860 4011 V -4 x Fw(K)938 3986 y Fz(\020)p 988 4010 41 4 v 92 x FA(\030)1028 4024 y Fy(1)1065 3986 y Fz(\021)1133 4078 y Fw(\000)p 1216 4011 65 4 v 18 w(K)1294 3986 y Fz(\020)p 1344 4010 41 4 v 92 x FA(\030)1384 4024 y Fy(2)1421 3986 y Fz(\021)1471 3983 y(\015)1471 4032 y(\015)1471 4082 y(\015)1517 4136 y Fx(\033)1585 4078 y Fw(\024)o FA(K)1740 3983 y Fz(\015)1740 4032 y(\015)1740 4082 y(\015)1800 4057 y(b)1786 4078 y Fw(K)1864 3986 y Fz(\020)p 1914 4010 V 92 x FA(\030)1954 4024 y Fy(1)1991 3986 y Fz(\021)2059 4078 y Fw(\000)2156 4057 y Fz(b)2142 4078 y Fw(K)2220 3986 y Fz(\020)p 2270 4010 V 92 x FA(\030)2310 4024 y Fy(2)2347 3986 y Fz(\021)2397 3983 y(\015)2397 4032 y(\015)2397 4082 y(\015)2443 4136 y Fx(\033)1585 4265 y Fw(\024)o FA(K)1740 4169 y Fz(\015)1740 4219 y(\015)1740 4269 y(\015)1796 4244 y(c)1786 4265 y FA(M)1889 4173 y Fz(\020)p 1939 4197 V 92 x FA(\030)1979 4211 y Fy(2)2035 4265 y Fw(\000)p 2118 4197 V 18 w FA(\030)2158 4211 y Fy(1)2195 4173 y Fz(\021)2263 4265 y FB(+)2368 4244 y Fz(b)2346 4265 y FA(N)2436 4173 y Fz(\020)p 2485 4197 V 2485 4265 a FA(\030)2525 4211 y Fy(1)2563 4173 y Fz(\021)2631 4265 y Fw(\000)2735 4244 y Fz(b)2714 4265 y FA(N)2803 4173 y Fz(\020)p 2853 4197 V 92 x FA(\030)2893 4211 y Fy(2)2930 4173 y Fz(\021)2980 4169 y(\015)2980 4219 y(\015)2980 4269 y(\015)3026 4323 y Fx(\033)3085 4265 y FA(:)71 4463 y FB(Then,)h(using)g(the)g(de\014nitions)g(of)1180 4442 y Fz(c)1169 4463 y FA(M)37 b FB(and)1471 4442 y Fz(b)1449 4463 y FA(N)g FB(in)28 b(\(219\))f(and)h(\(220\))f(and)h (applying)g(Lemma)f(7.18)g(and)h(b)r(ound)g(\(214\))o(,)71 4562 y(one)f(can)g(see)g(that)1141 4567 y Fz(\015)1141 4616 y(\015)1141 4666 y(\015)p 1187 4595 65 4 v -4 x Fw(K)1266 4570 y Fz(\020)p 1315 4594 41 4 v 1315 4662 a FA(\030)1355 4608 y Fy(1)1393 4570 y Fz(\021)1461 4662 y Fw(\000)p 1544 4595 65 4 v 18 w(K)1622 4570 y Fz(\020)p 1672 4594 41 4 v 92 x FA(\030)1712 4608 y Fy(2)1749 4570 y Fz(\021)1799 4567 y(\015)1799 4616 y(\015)1799 4666 y(\015)1845 4720 y Fx(\033)1913 4662 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2212 4628 y Fx(\021)r Fy(+1)2350 4567 y Fz(\015)2350 4616 y(\015)2350 4666 y(\015)p 2396 4594 V -4 x FA(\030)2436 4608 y Fy(1)2492 4662 y Fw(\000)p 2575 4594 V 18 w FA(\030)2615 4608 y Fy(2)2652 4567 y Fz(\015)2652 4616 y(\015)2652 4666 y(\015)2698 4720 y Fx(\033)2757 4662 y FA(:)71 4831 y FB(Therefore,)29 b(reducing)g FA(")g FB(if)i(necessary)-7 b(,)28 b(Lip)p 1466 4764 65 4 v Fw(K)g FA(<)e FB(1)p FA(=)p FB(2)j(and)g(then)p 2157 4764 V 30 w Fw(K)i FB(is)f(con)n(tractiv)n(e)e(from)p 2964 4764 68 4 v 29 w FA(B)t FB(\()p FA(b)3099 4843 y Fy(5)3137 4831 y Fw(j)p FA(\026)p Fw(j)p FA(")3272 4801 y Fx(\021)r Fy(+1)3396 4831 y FB(\))f Fw(\032)f(Y)3601 4843 y Fx(\033)3666 4831 y Fw(\002)19 b(Y)3805 4843 y Fx(\033)71 4930 y FB(to)27 b(itself)h(and)g(it)g(has)f(a)g(unique)h (\014xed)g(p)r(oin)n(t)p 1522 4863 41 4 v 27 w FA(\030)5 b FB(.)195 5030 y(T)-7 b(o)31 b(pro)n(v)n(e)e(that)j FA(\030)h FB(=)28 b(\(Id)21 b(+)p 1109 4963 90 4 v 21 w FA(M)1198 5042 y Fy(1)1235 5030 y FB(\))p 1267 4962 41 4 v FA(\030)36 b FB(is)31 b(the)g(analytic)g(con)n(tin)n(uation)f (of)h(the)g(function)h FA(\030)h FB(=)28 b(\()p FA(Q)3208 5042 y Fy(1)3246 5030 y FA(;)14 b(P)3336 5042 y Fy(1)3373 5030 y FB(\))31 b(obtained)g(in)71 5130 y(Corollary)25 b(7.13,)h(one)i(can)f(pro)r(ceed)g(as)g(in)g(the)h(pro)r(of)f(of)h (Lemma)f(7.8.)p 3790 5130 4 57 v 3794 5077 50 4 v 3794 5130 V 3843 5130 4 57 v Black 71 5289 a Fs(Pr)l(o)l(of)k(of)f(Pr)l(op)l (osition)i(7.17.)p Black 43 w FB(It)26 b(is)f(enough)f(to)h(undo)g(c)n (hange)f(\(215\))o(.)36 b(F)-7 b(or)24 b(the)i(b)r(ound)f(of)g FA(\030)i FB(=)c(\()p FA(Q)3209 5301 y Fy(1)3246 5289 y FA(;)14 b(P)3336 5301 y Fy(1)3373 5289 y FB(\))26 b(it)f(is)g(enough) 71 5388 y(to)37 b(consider)e(the)j(b)r(ound)f(of)p 1036 5322 90 4 v 37 w FA(M)1125 5400 y Fy(1)1200 5388 y FB(in)g(\(214\))f (and)h(the)g(b)r(ound)g(of)p 2224 5321 41 4 v 37 w FA(\030)k FB(in)c(Lemma)g(7.20)e(and)i(increase)f(sligh)n(tly)g FA(b)3728 5400 y Fy(5)3802 5388 y FB(if)71 5488 y(necessary)-7 b(.)p 3790 5488 4 57 v 3794 5435 50 4 v 3794 5488 V 3843 5488 4 57 v Black 1919 5753 a(77)p Black eop end %%Page: 78 78 TeXDict begin 78 77 bop Black Black 71 272 a Fp(7.2.4)94 b(Pro)s(of)31 b(of)h(Theorem)f(4.7)71 425 y FB(This)i(section)h(is)f (dev)n(oted)g(to)h(obtain)f(a)h(parametrization)d(of)j(the)g(in)n(v)-5 b(arian)n(t)32 b(manifolds)i(of)f(the)i(form)e(\(42\))g(in)h(the)71 525 y(domains)27 b(\(26\))o(.)37 b(T)-7 b(o)28 b(this)f(end,)h(w)n(e)g (lo)r(ok)e(for)i(c)n(hanges)e(of)h(v)-5 b(ariables)27 b FA(v)f FB(=)d FA(u)17 b FB(+)h Fw(V)2606 495 y Fx(u;s)2701 525 y FB(\()p FA(u;)c(\034)9 b FB(\))28 b(whic)n(h)g(satisfy)g(\(56\).) 195 625 y(Since)39 b(the)g(pro)r(of)f(of)g(Theorem)g(4.7)g(is)g (analogous)e(for)i(b)r(oth)h(in)n(v)-5 b(arian)n(t)38 b(manifolds,)j(w)n(e)d(only)g(deal)g(with)h(the)71 724 y(unstable)27 b(case)g(and)h(w)n(e)f(omit)h(the)g(sup)r(erscript)f FA(u)g FB(to)g(simplify)h(notation.)195 824 y(Splitting)h FA(Q)p FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)e FA(q)940 836 y Fy(0)978 824 y FB(\()p FA(v)s FB(\))d(+)f FA(Q)1253 836 y Fy(1)1290 824 y FB(\()p FA(v)s(;)c(\034)9 b FB(\),)29 b(equation)e(\(56\))g(reads)1101 1004 y FA(q)1138 1016 y Fy(0)1189 1004 y FB(\()p FA(u)18 b FB(+)g Fw(V)7 b FB(\()p FA(u;)14 b(\034)9 b FB(\)\))19 b Fw(\000)f FA(q)1793 1016 y Fy(0)1831 1004 y FB(\()p FA(u)p FB(\))23 b(=)g Fw(\000)p FA(Q)2185 1016 y Fy(1)2235 1004 y FB(\()p FA(u)18 b FB(+)g Fw(V)7 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))15 b FA(:)71 1185 y FB(T)-7 b(aking)31 b(in)n(to)h(accoun)n(t)g(that)48 b(_)-39 b FA(q)1058 1197 y Fy(0)1096 1185 y FB(\()p FA(u)p FB(\))30 b(=)h FA(p)1376 1197 y Fy(0)1413 1185 y FB(\()p FA(u)p FB(\),)j(to)e(obtain)g(a)f(solution)h(of)g(this)h(equation)f(is)g (equiv)-5 b(alen)n(t)32 b(to)g(obtain)g(a)71 1285 y(\014xed)c(p)r(oin)n (t)f(of)h(the)g(op)r(erator)454 1500 y Fw(N)12 b FB(\()p FA(h)p FB(\)\()p FA(u;)i(\034)9 b FB(\))25 b(=)d Fw(\000)1102 1444 y FB(1)p 1027 1481 192 4 v 1027 1557 a FA(p)1069 1569 y Fy(0)1106 1557 y FB(\()p FA(u)p FB(\))1242 1500 y(\()p FA(Q)1340 1512 y Fy(1)1391 1500 y FB(\()p FA(u)c FB(+)g FA(h)p FB(\()p FA(u;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))20 b(+)e FA(q)2068 1512 y Fy(0)2119 1500 y FB(\()p FA(u)g FB(+)g FA(h)p FB(\()p FA(u;)c(\034)9 b FB(\)\))19 b Fw(\000)g FA(q)2714 1512 y Fy(0)2751 1500 y FB(\()p FA(u)p FB(\))f Fw(\000)g FA(p)3006 1512 y Fy(0)3043 1500 y FB(\()p FA(u)p FB(\))p FA(h)p FB(\()p FA(u;)c(\034)9 b FB(\)\))15 b FA(:)194 b FB(\(223\))195 1724 y(Let)28 b(us)g(consider)e(the)i(function)g(space)847 1925 y Fw(Q)915 1937 y Fx(\024;d;\033)1096 1925 y FB(=)1183 1833 y Fz(n)1239 1925 y FA(h)23 b FB(:)g FA(I)1399 1885 y Fy(out)o Fx(;u)1392 1950 y(\024;d)1576 1925 y Fw(\002)18 b Ft(T)1714 1937 y Fx(\033)1782 1925 y Fw(!)23 b Ft(C)p FB(;)42 b(real-analytic)n FA(;)14 b Fw(k)p FA(h)p Fw(k)2632 1937 y Fx(\024;d;\033)2811 1925 y FA(<)23 b Fw(1)2982 1833 y Fz(o)3051 1925 y FA(;)587 b FB(\(224\))71 2144 y(where)27 b Fw(k)18 b(\001)g(k)454 2156 y Fx(\024;d;\033)639 2144 y FB(is)28 b(the)g(F)-7 b(ourier)26 b(norm)h(de\014ned)h(in)g(\(209\))f(but)h(applied)g(to)f (functions)h(de\014ned)g(in)g FA(I)3298 2104 y Fy(out)p Fx(;u)3291 2169 y(\024;d)3475 2144 y Fw(\002)18 b Ft(T)3613 2156 y Fx(\033)3658 2144 y FB(.)195 2244 y(W)-7 b(e)34 b(split)f(Theorem)g(4.7)f(in)h(the)h(follo)n(wing)e(prop)r(osition)g (and)h(corollary)-7 b(,)32 b(whic)n(h)h(are)f(written)h(in)g(terms)g (of)g(the)71 2343 y(Banac)n(h)26 b(space)h(de\014ned)h(in)g(\(224\))o (.)p Black 71 2508 a Fp(Prop)s(osition)g(7.21.)p Black 38 w Fs(L)l(et)f(us)g(c)l(onsider)i(the)f(c)l(onstant)f FA(\024)1907 2520 y Fy(1)1972 2508 y Fs(given)i(in)f(Pr)l(op)l(osition) h(7.17,)i FA(d)2976 2520 y Fy(0)3036 2508 y FA(>)23 b(d)3167 2520 y Fy(1)3227 2508 y FA(>)g FB(0)p Fs(,)28 b FA(\024)3458 2520 y Fy(2)3518 2508 y FA(>)23 b(\024)3654 2520 y Fy(1)3719 2508 y Fs(and)71 2607 y FA(")110 2619 y Fy(0)170 2607 y FA(>)g FB(0)j Fs(smal)t(l)j(enough,)f(which)i(might)d(dep)l(end)i(on) e(the)h(pr)l(evious)g(c)l(onstants.)37 b(Then,)29 b(ther)l(e)f(exists)f (a)h(c)l(onstant)e FA(b)3661 2619 y Fy(6)3721 2607 y FA(>)d FB(0)71 2707 y Fs(such)h(that)h(for)g FA(")e Fw(2)g FB(\(0)p FA(;)14 b(")836 2719 y Fy(0)873 2707 y FB(\))24 b Fs(and)h FA(\024)1133 2719 y Fy(1)1194 2707 y Fs(and)g FA(\024)1398 2719 y Fy(2)1460 2707 y Fs(big)g(enough,)h(the)f(op)l(er)l (ator)g Fw(N)37 b Fs(is)24 b(c)l(ontr)l(active)h(fr)l(om)p 3138 2640 68 4 v 25 w FA(B)3219 2640 y Fz(\000)3257 2707 y FA(b)3293 2719 y Fy(6)3330 2707 y Fw(j)p FA(\026)p Fw(j)p FA(")3465 2677 y Fx(\021)r Fy(+1)3589 2640 y Fz(\001)3650 2707 y Fw(\032)e(Q)3806 2719 y Fx(\033)71 2807 y Fs(to)30 b(itself.)195 2906 y(Then,)35 b Fw(N)45 b Fs(has)33 b(a)g(unique)f (\014xe)l(d)g(p)l(oint)h Fw(V)i(2)p 1626 2840 V 28 w FA(B)1707 2839 y Fz(\000)1746 2906 y FA(b)1782 2918 y Fy(6)1818 2906 y Fw(j)p FA(\026)p Fw(j)p FA(")1953 2876 y Fx(\021)r Fy(+1)2078 2839 y Fz(\001)2144 2906 y Fw(\032)28 b(Q)2305 2918 y Fx(\033)2349 2906 y Fs(,)34 b(which)g(satis\014es)f (that)f FA(u)20 b FB(+)g Fw(V)7 b FB(\()p FA(u;)14 b(\034)9 b FB(\))29 b Fw(2)g FA(I)3692 2866 y Fy(out)o Fx(;u)3685 2931 y(\024)3724 2939 y Fu(1)3756 2931 y Fx(;d)3811 2939 y Fu(0)71 3027 y Fs(for)h FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)g FA(I)543 2987 y Fy(out)o Fx(;u)536 3052 y(\024)575 3060 y Fu(2)607 3052 y Fx(;d)662 3060 y Fu(1)720 3027 y Fw(\002)18 b Ft(T)858 3039 y Fx(\033)903 3027 y Fs(.)p Black 71 3212 a Fp(Corollary)33 b(7.22.)p Black 40 w Fs(Ther)l(e)d(exists)g(a)g(function)g FA(T)k FB(:)23 b FA(I)1772 3172 y Fy(out)p Fx(;u)1765 3237 y(\024)1804 3245 y Fu(2)1837 3237 y Fx(;d)1892 3245 y Fu(1)1949 3212 y Fw(\002)18 b Ft(T)2087 3224 y Fx(\033)2155 3212 y Fw(!)23 b Ft(C)30 b Fs(such)g(that)1320 3392 y FA(@)1364 3404 y Fx(u)1408 3392 y FA(T)12 b FB(\()p FA(u;)i(\034)9 b FB(\))23 b(=)f FA(p)1815 3404 y Fy(0)1852 3392 y FB(\()p FA(u)p FB(\))p FA(P)12 b FB(\()p FA(u)19 b FB(+)f Fw(V)7 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))p FA(;)71 3573 y Fs(wher)l(e)28 b FA(P)39 b Fs(and)27 b Fw(V)34 b Fs(ar)l(e)28 b(the)f(functions)g(obtaine)l(d)h(in)f(The)l (or)l(em)i(4.6)f(and)g(Pr)l(op)l(osition)g(7.21)h(r)l(esp)l(e)l (ctively,)h(and)d(satis\014es)71 3672 y(e)l(quation)36 b FB(\(41\))p Fs(.)i(Mor)l(e)l(over,)32 b(it)e(b)l(elongs)g(to)g Fw(Q)1545 3684 y Fx(\033)1619 3672 y Fs(and)g(satis\014es)1364 3853 y Fw(k)o FA(@)1449 3865 y Fx(u)1493 3853 y FA(T)f Fw(\000)18 b FA(@)1698 3865 y Fx(u)1742 3853 y FA(T)1791 3865 y Fy(0)1828 3853 y Fw(k)1869 3878 y Fx(\024)1908 3886 y Fu(2)1940 3878 y Fx(;d)1995 3886 y Fu(1)2027 3878 y Fx(;\033)2115 3853 y Fw(\024)k FA(b)2238 3865 y Fy(7)2275 3853 y Fw(j)p FA(\026)p Fw(j)p FA(")2410 3819 y Fx(\021)r Fy(+1)2534 3853 y FA(:)71 4034 y Fs(for)30 b(c)l(ertain)g(c)l(onstant)f FA(b)849 4046 y Fy(7)909 4034 y FA(>)23 b FB(0)p Fs(.)195 4198 y FB(W)-7 b(e)28 b(dev)n(ote)f(the)h(rest)f(of)h(this)g(section)f (to)h(pro)n(v)n(e)d(Prop)r(osition)h(7.21)h(and)g(Corollary)e(7.22.)p Black 71 4364 a Fs(Pr)l(o)l(of)31 b(of)f(Pr)l(op)l(osition)i(7.21.)p Black 43 w FB(The)d(op)r(erator)e Fw(N)40 b FB(sends)29 b Fw(Q)1960 4376 y Fx(\024)1999 4384 y Fu(2)2031 4376 y Fx(;d)2086 4384 y Fu(1)2117 4376 y Fx(;\033)2210 4364 y FB(to)f(itself.)40 b(T)-7 b(o)28 b(see)g(that)h(exists)f(a)g(constan) n(t)g FA(b)3659 4376 y Fy(6)3720 4364 y FA(>)c FB(0)71 4463 y(suc)n(h)e(that)h Fw(N)34 b FB(is)23 b(con)n(tractiv)n(e)d(in)p 1122 4396 V 23 w FA(B)1203 4396 y Fz(\000)1241 4463 y FA(b)1277 4475 y Fy(6)1314 4463 y Fw(j)p FA(\026)p Fw(j)p FA(")1449 4433 y Fx(\021)r Fy(+1)1573 4396 y Fz(\001)1634 4463 y Fw(\032)j(Q)1790 4475 y Fx(\024)1829 4483 y Fu(2)1861 4475 y Fx(;d)1916 4483 y Fu(1)1948 4475 y Fx(;\033)2012 4463 y FB(,)h(w)n(e)e(\014rst)g(consider)f Fw(N)12 b FB(\(0\).)36 b(By)22 b(Prop)r(osition)f(7.17,)h(there)71 4563 y(exists)27 b(a)g(constan)n(t)g FA(b)740 4575 y Fy(6)800 4563 y FA(>)c FB(0)k(suc)n(h)g(that)993 4778 y Fw(kN)12 b FB(\(0\))p Fw(k)1263 4803 y Fx(\024)1302 4811 y Fu(2)1334 4803 y Fx(;d)1389 4811 y Fu(1)1421 4803 y Fx(;\033)1508 4778 y FB(=)1596 4708 y Fz(\015)1596 4758 y(\015)1642 4778 y FA(p)1684 4743 y Fv(\000)p Fy(1)1684 4800 y(0)1773 4778 y FB(\()p FA(v)s FB(\))p FA(Q)1946 4790 y Fy(1)1983 4778 y FB(\()p FA(v)s(;)i(\034)9 b FB(\))2172 4708 y Fz(\015)2172 4758 y(\015)2220 4812 y Fx(\024)2259 4820 y Fu(2)2291 4812 y Fx(;d)2346 4820 y Fu(1)2378 4812 y Fx(;\033)2465 4778 y Fw(\024)2563 4722 y FA(b)2599 4734 y Fy(6)p 2563 4759 73 4 v 2579 4835 a FB(2)2646 4778 y Fw(j)p FA(\026)p Fw(j)p FA(")2781 4744 y Fx(\021)r Fy(+1)2905 4778 y FA(:)71 4998 y FB(T)-7 b(o)28 b(see)g(that)h Fw(N)41 b FB(is)29 b(con)n(tractiv)n(e,)e(let)i(us)g(consider)e FA(h)1756 5010 y Fy(1)1793 4998 y FA(;)14 b(h)1878 5010 y Fy(2)1940 4998 y Fw(2)p 2020 4931 68 4 v 25 w FA(B)2101 4931 y Fz(\000)2139 4998 y FA(b)2175 5010 y Fy(6)2212 4998 y Fw(j)p FA(\026)p Fw(j)p FA(")2347 4968 y Fx(\021)r Fy(+1)2471 4931 y Fz(\001)2534 4998 y Fw(\032)25 b(Q)2692 5010 y Fx(\024)2731 5018 y Fu(2)2763 5010 y Fx(;d)2818 5018 y Fu(1)2849 5010 y Fx(;\033)2914 4998 y FB(.)40 b(By)28 b(Prop)r(osition)f(7.17,)h(w)n(e)71 5111 y(kno)n(w)k(that)h FA(Q)544 5123 y Fy(1)581 5111 y FB(\()p FA(u;)14 b(\034)9 b FB(\))33 b(is)g(de\014ned)g(in)g FA(I)1333 5071 y Fy(out)o Fx(;u)1326 5136 y(\024)1365 5144 y Fu(1)1398 5136 y Fx(;d)1453 5144 y Fu(0)1524 5111 y FB(and)g(satis\014es)f Fw(k)p FA(Q)2110 5123 y Fy(1)2146 5111 y Fw(k)2188 5123 y Fx(\024)2227 5131 y Fu(1)2259 5123 y Fx(;d)2314 5131 y Fu(0)2346 5123 y Fx(;\033)2442 5111 y Fw(\024)f FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2750 5081 y Fx(\021)r Fy(+1)2906 5111 y FB(in)33 b(this)g(domain.)52 b(Applying)71 5231 y(Cauc)n(h)n(y)26 b(estimates)i(in)g(the)g(nested)f(domains)g FA(I)1603 5191 y Fy(out)p Fx(;u)1596 5259 y Fy(2)p Fx(\024)1668 5267 y Fu(1)1701 5259 y Fx(;d)1756 5267 y Fu(0)1788 5259 y Fx(=)p Fy(2)1882 5231 y Fw(\032)c FA(I)2013 5191 y Fy(out)o Fx(;u)2006 5256 y(\024)2045 5264 y Fu(1)2077 5256 y Fx(;d)2132 5264 y Fu(0)2172 5231 y FB(,)k(one)g(has)h(that)1481 5476 y Fw(k)p FA(@)1567 5488 y Fx(v)1606 5476 y FA(Q)1672 5488 y Fy(1)1709 5476 y Fw(k)1751 5491 y Fy(2)p Fx(\024)1823 5499 y Fu(1)1854 5491 y Fx(;d)1909 5499 y Fu(0)1941 5491 y Fx(=)p Fy(2)p Fx(;\033)2096 5476 y Fw(\024)2198 5420 y FA(K)p 2193 5457 86 4 v 2193 5533 a(\024)2241 5545 y Fy(1)2288 5476 y FA(\026")2377 5442 y Fx(\021)2418 5476 y FA(:)p Black 1919 5753 a FB(78)p Black eop end %%Page: 79 79 TeXDict begin 79 78 bop Black Black 71 272 a FB(Then,)22 b(de\014ning)e FA(h)659 242 y Fx(s)694 272 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)d FA(sh)1082 284 y Fy(2)1119 272 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))t(+)t(\(1)t Fw(\000)t FA(s)p FB(\))p FA(h)1647 284 y Fy(1)1683 272 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))22 b(for)e FA(s)j Fw(2)g FB(\(0)p FA(;)14 b FB(1\),)21 b(using)f(the)h(mean)f(v)-5 b(alue)20 b(theorem,)h(increasing)71 372 y FA(\024)119 384 y Fy(1)184 372 y FB(if)28 b(necessary)e(and)h(taking)g FA(\024)1093 384 y Fy(2)1153 372 y FA(>)c FB(2)p FA(\024)1331 384 y Fy(1)1367 372 y FB(,)375 606 y Fw(kN)12 b FB(\()p FA(h)577 618 y Fy(2)614 606 y FB(\))19 b Fw(\000)f(N)12 b FB(\()p FA(h)908 618 y Fy(1)946 606 y FB(\))p Fw(k)1020 631 y Fx(\024)1059 639 y Fu(2)1091 631 y Fx(;d)1146 639 y Fu(1)1178 631 y Fx(;\033)1265 606 y Fw(\024)1344 486 y Fz(\015)1344 536 y(\015)1344 586 y(\015)1344 635 y(\015)1390 606 y FA(p)1432 571 y Fv(\000)p Fy(1)1432 628 y(0)1521 606 y FB(\()p FA(v)s FB(\))1642 493 y Fz(Z)1725 514 y Fy(1)1688 682 y(0)1776 606 y FB(\()q FA(@)1853 618 y Fx(u)1896 606 y FA(Q)1962 618 y Fy(1)2013 606 y FB(\()p FA(v)22 b FB(+)c FA(h)2238 572 y Fx(s)2273 606 y FA(;)c(\034)9 b FB(\))19 b(+)f FA(p)2531 618 y Fy(0)2568 606 y FB(\()p FA(v)k FB(+)c FA(h)2793 572 y Fx(s)2829 606 y FB(\))g Fw(\000)h FA(p)3005 618 y Fy(0)3042 606 y FB(\()p FA(v)s FB(\)\))14 b FA(ds)3277 486 y Fz(\015)3277 536 y(\015)3277 586 y(\015)3277 635 y(\015)3324 689 y Fx(\024)3363 697 y Fu(2)3395 689 y Fx(;d)3450 697 y Fu(1)3482 689 y Fx(;\033)1348 804 y Fw(\001)19 b(k)o FA(h)1479 816 y Fy(2)1535 804 y Fw(\000)f FA(h)1666 816 y Fy(1)1703 804 y Fw(k)1744 828 y Fx(\024)1783 836 y Fu(2)1816 828 y Fx(;d)1871 836 y Fu(1)1902 828 y Fx(;\033)1265 991 y Fw(\024)1340 935 y FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1552 905 y Fx(\021)p 1340 972 252 4 v 1423 1048 a FA(\024)1471 1060 y Fy(1)1615 991 y Fw(k)p FA(h)1705 1003 y Fy(2)1760 991 y Fw(\000)18 b FA(h)1891 1003 y Fy(1)1928 991 y Fw(k)1970 1016 y Fx(\024)2009 1024 y Fu(2)2041 1016 y Fx(;d)2096 1024 y Fu(1)2128 1016 y Fx(;\033)1265 1204 y Fw(\024)1340 1148 y FB(1)p 1340 1185 42 4 v 1340 1261 a(2)1405 1204 y Fw(k)o FA(h)1494 1216 y Fy(2)1550 1204 y Fw(\000)g FA(h)1681 1216 y Fy(1)1718 1204 y Fw(k)1760 1229 y Fx(\033)1818 1204 y FA(:)71 1422 y FB(Then,)28 b Fw(N)36 b FB(:)p 462 1356 68 4 v 24 w FA(B)543 1355 y Fz(\000)581 1422 y FA(b)617 1434 y Fy(6)654 1422 y Fw(j)p FA(\026)p Fw(j)p FA(")789 1392 y Fx(\021)r Fy(+1)913 1355 y Fz(\001)975 1422 y Fw(!)p 1081 1356 V 23 w FA(B)1162 1355 y Fz(\000)1200 1422 y FA(b)1236 1434 y Fy(6)1273 1422 y Fw(j)p FA(\026)p Fw(j)p FA(")1408 1392 y Fx(\021)r Fy(+1)1533 1355 y Fz(\001)1594 1422 y Fw(\032)23 b(Q)1750 1434 y Fx(\024)1789 1442 y Fu(2)1821 1434 y Fx(;d)1876 1442 y Fu(1)1908 1434 y Fx(;\033)2001 1422 y FB(and)k(is)h(con)n(tractiv)n(e.)36 b(Therefore,)27 b(it)h(has)g(a)f(unique)h(\014xed)71 1522 y(p)r(oin)n(t)g(whic)n(h)f (satis\014es)g(the)h(prop)r(erties)f(stated)g(in)h(Prop)r(osition)e (7.21.)p 3790 1522 4 57 v 3794 1469 50 4 v 3794 1522 V 3843 1522 4 57 v Black 71 1688 a Fs(Pr)l(o)l(of)31 b(of)f(Cor)l(ol)t(lary)i(7.22.)p Black 44 w FB(Prop)r(osition)26 b(7.21,)g(giv)n(es)h(a)g(parametrization)f(of)h(form)632 1867 y(\()p FA(q)s(;)14 b(p)p FB(\))23 b(=)g(\()p FA(Q)p FB(\()p FA(u)18 b FB(+)g Fw(V)7 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))p FA(;)14 b(P)e FB(\()p FA(u)19 b FB(+)f Fw(V)7 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\)\))25 b(=)d(\()p FA(q)2434 1879 y Fy(0)2472 1867 y FB(\()p FA(u)p FB(\))p FA(;)14 b(P)e FB(\()p FA(u)18 b FB(+)g Fw(V)7 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\)\))p FA(:)71 2047 y FB(W)-7 b(e)22 b(w)n(an)n(t)g(to)g(ha)n(v)n(e)f(a)g (parametrization)f(of)i(the)h(form)e(\(42\),)i(where)f FA(T)33 b FB(is)22 b(a)g(function)g(whic)n(h)g(satis\014es)g(\(41\))o (.)35 b(T)-7 b(o)22 b(reco)n(v)n(er)71 2147 y(this)27 b(function)g(it)g(is)f(enough)g(to)h(p)r(oin)n(t)g(out)f(that,)h(since) g(w)n(e)f(w)n(an)n(t)g(it)h(to)f(b)r(e)h(solution)f(of)33 b(\(41\),)27 b(w)n(e)f(kno)n(w)g(its)h(gradien)n(t)445 2327 y(\()p FA(@)521 2339 y Fx(u)564 2327 y FA(T)12 b FB(\()p FA(u;)i(\034)9 b FB(\))p FA(;)14 b(@)900 2339 y Fx(\034)942 2327 y FA(T)e FB(\()p FA(u;)i(\034)9 b FB(\)\))23 b(=)1339 2259 y Fz(\000)1377 2327 y FA(p)1419 2339 y Fy(0)1456 2327 y FB(\()p FA(u)p FB(\))p FA(P)12 b FB(\()p FA(u)19 b FB(+)f Fw(V)7 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))p FA(;)14 b Fw(\000)p FA(")p 2322 2260 76 4 v(H)7 b FB(\()p FA(u;)14 b(p)2557 2339 y Fy(0)2594 2327 y FB(\()p FA(u)p FB(\))p FA(P)e FB(\()p FA(u)18 b FB(+)g Fw(V)7 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)3400 2259 y Fz(\001)3454 2327 y FA(:)71 2507 y FB(Then,)28 b(it)g(is)f(enough)g(to)h(c)n(hec)n(k)e(the)i(compatibilit) n(y)g(condition)712 2686 y FA(@)756 2698 y Fx(\034)811 2686 y FB([)q FA(p)877 2698 y Fy(0)914 2686 y FB(\()p FA(u)p FB(\))p FA(P)12 b FB(\()p FA(u)18 b FB(+)g Fw(V)7 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\)])24 b(=)f Fw(\000)p FA(@)1882 2698 y Fx(u)1939 2619 y Fz(\002)1973 2686 y FA(")p 2012 2620 V(H)e FB(\()p FA(u;)14 b(p)2261 2698 y Fy(0)2297 2686 y FB(\()p FA(u)p FB(\))p FA(P)e FB(\()p FA(u)19 b FB(+)f Fw(V)7 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))3138 2619 y Fz(\003)3186 2686 y FA(:)452 b FB(\(225\))71 2866 y(Di\013eren)n(tiating)27 b(equation)g(\(56\),)h(one)f(has)g(that)h Fw(V)34 b FB(satis\014es)1469 3045 y FA(@)1513 3057 y Fx(v)1553 3045 y FA(Q)p FB(\()p FA(u)18 b FB(+)g Fw(V)7 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))14 b(\()q(1)k(+)g FA(@)2400 3057 y Fx(u)2443 3045 y Fw(V)7 b FB(\()p FA(u;)14 b(\034)9 b FB(\)\))24 b(=)f FA(p)2881 3057 y Fy(0)2918 3045 y FB(\()p FA(u)p FB(\))891 3170 y FA(@)935 3182 y Fx(v)975 3170 y FA(Q)p FB(\()p FA(u)18 b FB(+)g Fw(V)7 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))p FA(@)1632 3182 y Fx(\034)1675 3170 y Fw(V)e FB(\()p FA(u;)14 b(\034)9 b FB(\))18 b(+)g FA(@)2072 3182 y Fx(\034)2114 3170 y FA(Q)p FB(\()p FA(u)g FB(+)g Fw(V)7 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))24 b(=)f(0)71 3350 y(Then,)28 b(using)f(this)h(equalities)f(and)g (equation)g(\(44\),)h(one)f(can)g(pro)n(v)n(e)f(\(225\))o(.)195 3450 y(Finally)-7 b(,)29 b(recalling)e(that)h FA(@)1052 3462 y Fx(u)1096 3450 y FA(T)1145 3462 y Fy(0)1181 3450 y FB(\()p FA(u)p FB(\))d(=)e FA(p)1448 3420 y Fy(2)1448 3471 y(0)1485 3450 y FB(\()p FA(u)p FB(\))29 b(and)f FA(P)12 b FB(\()p FA(v)s(;)i(\034)9 b FB(\))25 b(=)e FA(p)2197 3462 y Fy(0)2234 3450 y FB(\()p FA(v)s FB(\))d(+)e FA(P)2497 3462 y Fy(1)2535 3450 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))30 b(and)e(applying)f(Prop)r(osition)g(7.17)71 3550 y(and)g(the)h(mean)g(v)-5 b(alue)27 b(theorem,)554 3724 y Fw(k)o FA(@)639 3736 y Fx(u)683 3724 y FA(T)i Fw(\000)18 b FA(@)888 3736 y Fx(u)932 3724 y FA(T)981 3736 y Fy(0)1018 3724 y Fw(k)1059 3749 y Fx(\024)1098 3757 y Fu(2)1130 3749 y Fx(;d)1185 3757 y Fu(1)1217 3749 y Fx(;\033)1304 3724 y Fw(\024)23 b(k)p FA(p)1476 3736 y Fy(0)1513 3724 y FB(\()p FA(u)p FB(\))14 b(\()p FA(P)1724 3736 y Fy(1)1761 3724 y FB(\()p FA(u)19 b FB(+)f Fw(V)7 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))19 b(+)f FA(p)2453 3736 y Fy(0)2490 3724 y FB(\()p FA(u)h FB(+)f Fw(V)7 b FB(\()p FA(u;)14 b(\034)9 b FB(\)\))19 b Fw(\000)f FA(p)3100 3736 y Fy(0)3137 3724 y FB(\()p FA(u)p FB(\)\))p Fw(k)3323 3749 y Fx(\033)1304 3865 y Fw(\024)23 b FA(b)1428 3877 y Fy(7)1465 3865 y Fw(j)p FA(\026)p Fw(j)p FA(")1600 3831 y Fx(\021)r Fy(+1)1724 3865 y FA(:)p 3790 4040 4 57 v 3794 3987 50 4 v 3794 4040 V 3843 4040 4 57 v 71 4255 a Fp(7.2.5)94 b(Pro)s(of)31 b(of)h(Theorem)f(4.8)71 4408 y FB(The)g(pro)r(of)g(of)h(Theorem)f(4.8)f (follo)n(ws)h(the)h(same)f(steps)g(as)g(the)h(pro)r(of)f(of)g(Theorem)g (4.4.)48 b(F)-7 b(or)31 b(this)g(reason,)g(in)h(this)71 4508 y(section)27 b(w)n(e)g(only)g(explain)h(whic)n(h)f(are)g(the)h (main)f(di\013erences.)195 4607 y(First,)j(let)f(us)h(p)r(oin)n(t)f (out)g(that)h(the)f(op)r(erator)e Fw(G)1731 4619 y Fx(")1797 4607 y FB(de\014ned)i(in)h(\(168\))e(can)h(b)r(e)g(also)f(applied)i(to) f(functions)g(de\014ned)71 4707 y(in)c FA(D)236 4677 y Fx(u)234 4731 y(\024)273 4739 y Fu(3)305 4731 y Fx(;d)360 4739 y Fu(2)410 4707 y Fw(\002)14 b Ft(T)544 4719 y Fx(\033)614 4707 y FB(if)25 b(one)g(tak)n(es)g(as)f FA(u)1195 4719 y Fy(1)1232 4707 y FA(;)19 b FB(\026)-47 b FA(u)1317 4719 y Fy(1)1379 4707 y FB(the)25 b(v)n(ertices)g(of)g FA(D)1981 4677 y Fx(u)1979 4731 y(\024)2018 4739 y Fu(3)2050 4731 y Fx(;d)2105 4739 y Fu(2)2166 4707 y FB(\(see)g(Figure)g(2\))g (and)g(as)g FA(\032)g FB(the)g(left)h(endp)r(oin)n(t)g(of)f(the)71 4807 y(in)n(terv)-5 b(al)26 b FA(D)441 4777 y Fx(u)439 4830 y(\024)478 4838 y Fu(3)510 4830 y Fx(;d)565 4838 y Fu(2)618 4807 y Fw(\\)18 b Ft(R)p FB(.)36 b(No)n(w)27 b(the)g(paths)g(of)g(in)n(tegration)f(cannot)g(b)r(e)h(straigh)n(t)f (lines.)37 b(Nev)n(ertheless,)26 b(it)h(is)g(easy)f(to)h(see)71 4906 y(that)f Fw(G)298 4918 y Fx(")361 4906 y FB(satis\014es)f(the)i (same)e(prop)r(erties)h(as)f(the)i(ones)f(stated)g(in)g(Lemma)g(7.3)g (but)h(applied)f(to)g(functions)h(de\014ned)f(in)71 5006 y(the)i(new)g(domain.)195 5106 y(Then,)38 b(if)e(one)f(considers)g (Banac)n(h)f(spaces)g(analogous)g(to)h Fw(E)2165 5118 y Fx(\027;\033)2263 5106 y FB(,)i(with)f FA(\027)42 b(>)36 b FB(0,)h(giv)n(en)e(in)g(\(167\),)i(for)e(functions)71 5205 y(de\014ned)26 b(in)f FA(D)520 5175 y Fx(u)518 5229 y(\024)557 5237 y Fu(3)589 5229 y Fx(;d)644 5237 y Fu(2)694 5205 y Fw(\002)14 b Ft(T)828 5217 y Fx(\033)873 5205 y FB(,)26 b(one)f(can)g(pro)n(v)n(e)e(the)j(Prop)r(osition)e(7.4,)h (but)h(lo)r(oking)e(for)h(the)h(function)f FA(T)3259 5217 y Fy(1)3322 5205 y FB(as)f(the)i(analytic)71 5305 y(con)n(tin)n(uation)j(of)h(the)g(function)h(obtained)f(Corollary)d (7.22)i(instead)g(of)h(the)h(function)f FA(T)2921 5317 y Fy(1)2988 5305 y FB(obtained)g(in)g(Prop)r(osition)71 5404 y(6.4)d(and)g(Prop)r(osition)f(6.10.)195 5504 y(The)i(rest)f(of)h (the)g(pro)n(v)n(e)e(follo)n(ws)g(the)i(same)f(lines)h(as)f(the)h(pro)r (of)f(of)g(Prop)r(osition)f(7.4.)p Black 1919 5753 a(79)p Black eop end %%Page: 80 80 TeXDict begin 80 79 bop Black Black 71 272 a Fq(7.3)112 b(The)26 b(\014rst)g(asymptotic)g(term)g(of)g(the)g(in)m(v)-6 b(arian)m(t)27 b(manifolds)h(near)e(the)g(singularities)326 388 y(for)38 b(the)f(case)h Fh(`)28 b FF(=)f(2)p Fh(r)71 542 y FB(In)c(the)g(case)f FA(\021)k FB(=)c(0)h(and)f FA(`)9 b Fw(\000)g FB(2)p FA(r)24 b FB(=)e(0,)i(w)n(e)e(need)h(a)f(b)r (etter)h(kno)n(wledge)e(of)i(the)g(\014rst)f(asymptotic)h(terms)f(of)h (the)g(in)n(v)-5 b(arian)n(t)71 641 y(manifolds)32 b(close)g(the)h (singularities)e(of)h(the)h(unp)r(erturb)r(ed)g(separatrix)d FA(u)h FB(=)g Fw(\006)p FA(ia)p FB(.)50 b(In)33 b(the)g(next)f(result,) i(w)n(e)e(obtain)71 741 y(them)c(for)f(the)h(unstable)g(in)n(v)-5 b(arian)n(t)26 b(manifold)i(close)f(to)g FA(u)c FB(=)f FA(ia)p FB(.)37 b(The)27 b(other)g(cases)g(can)g(b)r(e)h(done)f (analogously)-7 b(.)195 840 y(This)28 b(prop)r(osition)f(will)g(b)r(e)h (used)g(later)f(in)h(Section)f(9.)195 940 y(F)-7 b(or)27 b(real)g(analytic)g(functions)h FA(h)23 b FB(:)g FA(D)1368 910 y Fx(u)1366 964 y(\024)1405 972 y Fu(3)1437 964 y Fx(;d)1492 972 y Fu(2)1546 940 y Fw(\002)18 b Ft(T)1684 952 y Fx(\033)1752 940 y Fw(!)23 b Ft(C)p FB(,)28 b(2)p FA(\031)s FB(-)f(p)r(erio)r(dic)h(in)g FA(\034)37 b FB(w)n(e)27 b(de\014ne)h(the)g(F)-7 b(ourier)27 b(norm)1043 1147 y Fw(k)p FA(h)p Fw(k)1175 1159 y Fx(\027;\033)1295 1147 y FB(=)1383 1068 y Fz(X)1383 1247 y Fx(k)q Fv(2)p Fn(Z)1719 1147 y FB(sup)1517 1221 y Fy(\()p Fx(u;\034)7 b Fy(\))p Fv(2)p Fx(D)1767 1201 y Fm(u)1765 1241 y(\024)1799 1253 y Fu(3)1831 1241 y Fm(;d)1881 1253 y Fu(2)1917 1221 y Fv(\002)p Fn(T)2008 1229 y Fm(\033)2061 1147 y Fw(j)p FB(\()p FA(u)2164 1113 y Fy(2)2219 1147 y FB(+)18 b FA(a)2346 1113 y Fy(2)2384 1147 y FB(\))2416 1113 y Fx(\027)2457 1147 y FA(h)2505 1113 y Fy([)p Fx(k)q Fy(])2583 1147 y FB(\()p FA(u)p FB(\))p Fw(j)p FA(e)2757 1113 y Fv(j)p Fx(k)q Fv(j)p Fx(\033)71 1421 y FB(b)r(eing,)28 b(as)f(usual,)g FA(h)708 1390 y Fy([)p Fx(k)q Fy(])814 1421 y FB(the)h FA(k)s FB(-F)-7 b(ourier)26 b(co)r(e\016cien)n(t)h(of)h FA(h)p FB(.)p Black 71 1582 a Fp(Prop)s(osition)35 b(7.23.)p Black 42 w Fs(L)l(et)f(us)f(assume)g FA(`)21 b Fw(\000)g FB(2)p FA(r)33 b FB(=)d(0)p Fs(,)k(and)h(let)e(us)g(c)l(onsider)i(the)f (functions)g FA(Q)3139 1594 y Fx(j)3207 1582 y Fs(and)g FA(F)3425 1594 y Fx(j)3494 1582 y Fs(de\014ne)l(d)g(in)71 1681 y FB(\(73\))29 b Fs(and)39 b FB(\(74\))29 b Fs(r)l(esp)l(e)l (ctively)i(\(se)l(e)f(also)h(R)l(emark)e(4.14\))j(and)e(the)g(c)l (onstant)f FA(C)2582 1693 y Fy(+)2667 1681 y Fs(given)h(in)37 b FB(\(12\))29 b Fs(and)39 b FB(\(13\))o Fs(.)195 1781 y(Then,)31 b(ther)l(e)f(exists)f(a)i(r)l(e)l(al)f(analytic)h(function)e FA(\030)f FB(:)23 b FA(D)1923 1751 y Fx(u)1921 1804 y(\024)1960 1812 y Fu(3)1992 1804 y Fx(;d)2047 1812 y Fu(2)2101 1781 y Fw(\002)18 b Ft(T)2239 1793 y Fx(\033)2307 1781 y Fw(!)23 b Ft(C)p Fs(,)30 b FB(2)p FA(\031)s Fs(-)g(p)l(erio)l(dic)h(in)f FA(\034)40 b Fs(satisfying)31 b(that:)1481 1973 y Fw(k)p FA(\030)t Fw(k)1605 1988 y Fy(2)p Fx(r)r Fy(+1)p Fv(\000)p Fy(1)p Fx(=q)r(;\033)1993 1973 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2293 1938 y Fx(\021)r Fy(+1)2417 1973 y FA(;)71 2149 y Fs(wher)l(e)38 b FA(r)i FB(=)d FA(\013=\014)42 b Fs(has)c(b)l(e)l(en)f(de\014ne)l(d)h(in)g(Hyp) l(othesis)g Fo(HP2)h Fs(and,)h(for)f FB(\()p FA(u;)14 b(\034)9 b FB(\))38 b Fw(2)f FA(D)2800 2119 y Fx(u)2798 2170 y(\024)2837 2178 y Fu(3)2869 2170 y Fx(;c)2919 2178 y Fu(2)2979 2149 y Fw(\002)24 b Ft(T)3123 2161 y Fx(\033)3168 2149 y Fs(,)40 b(the)d(functions)h FA(T)3808 2119 y Fx(u)71 2249 y Fs(obtaine)l(d)c(r)l(esp)l(e)l(ctively)h(in)e(Pr)l(op)l(osition) i(7.4)g(\(c)l(ase)f FA(\013)1804 2261 y Fy(0)1841 2249 y FB(\()p FA(u)p FB(\))c Fw(6)p FB(=)f(0)p Fs(\))k(and)g(Pr)l(op)l (osition)i(7.17)h(\(gener)l(al)d(c)l(ase\),)j(ar)l(e)d(such)71 2348 y(that)595 2364 y Fz(\015)595 2414 y(\015)595 2463 y(\015)595 2513 y(\015)642 2484 y FA(@)686 2496 y Fx(u)729 2484 y FA(T)778 2496 y Fy(1)815 2484 y FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b Fw(\000)1136 2424 y FB(2)p FA(r)r(\026")1306 2394 y Fx(\021)r Fy(+1)1431 2424 y FA(C)1496 2394 y Fy(2)1490 2445 y(+)p 1121 2465 440 4 v 1121 2541 a FB(\()p FA(u)f Fw(\000)g FA(ia)p FB(\))1407 2517 y Fy(2)p Fx(r)r Fy(+1)1585 2484 y FB(\()p FA(F)1670 2496 y Fy(0)1708 2484 y FB(\()p FA(\034)9 b FB(\))19 b(+)f FA(\026)p Fw(h)p FA(Q)2067 2496 y Fy(0)2104 2484 y FA(F)2157 2496 y Fy(1)2195 2484 y Fw(i)p FB(\))h(+)f FA(\030)t FB(\()p FA(u;)c(\034)9 b FB(\))2595 2364 y Fz(\015)2595 2414 y(\015)2595 2463 y(\015)2595 2513 y(\015)2642 2567 y Fy(2)p Fx(r)r Fy(+2)p Fx(;\033)2879 2484 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")3179 2450 y Fx(\021)r Fy(+2)3303 2484 y FA(:)335 b FB(\(226\))p Black 71 2706 a Fs(Pr)l(o)l(of.)p Black 43 w FB(W)-7 b(e)25 b(pro)n(v)n(e)f(Prop)r(osition)f(7.23)h(in)h(the)h (p)r(olynomial)e(case.)35 b(T)-7 b(aking)24 b(in)n(to)h(accoun)n(t)f (Remark)g(4.14,)h(the)g(pro)r(of)g(of)71 2805 y(the)j(trigonometric)e (case)h(is)g(completely)h(analogous.)195 2905 y(W)-7 b(e)32 b(only)f(deal)f(with)i(the)g(case)e FA(p)1267 2917 y Fy(0)1304 2905 y FB(\()p FA(u)p FB(\))f Fw(6)p FB(=)g(0)h(b)r(eing)i(the)f(other)g(case)f(analogous.)45 b(F)-7 b(or)31 b(that)g(reason)f(w)n(e)h(will)g(only)71 3005 y(tak)n(e)c(in)n(to)h(accoun)n(t)g(the)h(previous)e(results)g(in)i (this)g(case.)38 b(In)28 b(fact)h(w)n(e)e(will)i(see)f(that)g(Prop)r (osition)f(7.23)g(is)h(also)f(v)-5 b(alid)71 3104 y(for)27 b(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)f FA(D)565 3064 y Fx(out;u)563 3132 y(\032)597 3112 y Fl(0)597 3150 y Fu(1)630 3132 y Fx(;\024)689 3112 y Fl(0)689 3150 y Fu(0)753 3104 y FB(where)k FA(\032)1036 3074 y Fv(0)1036 3125 y Fy(1)1101 3104 y FB(and)g FA(\024)1310 3074 y Fv(0)1310 3125 y Fy(0)1375 3104 y FB(are)g(the)h(constan)n(ts)e(for)h (whic)n(h)h(Prop)r(osition)e(7.4)h(holds.)195 3239 y(W)-7 b(e)38 b(\014rst)g(obtain)f(the)h(asymptotic)f(expansion)g(for)g(the)h (function)g FA(@)2453 3251 y Fx(v)2506 3218 y Fz(b)2492 3239 y FA(T)2541 3251 y Fy(1)2578 3239 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))39 b(obtained)e(in)h(Prop)r(osition)e(7.4,)71 3339 y(whic)n(h)27 b(is)g(de\014ned)g(for)g(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b Fw(2)g FA(D)1166 3299 y Fy(out)o Fx(;u)1164 3367 y(\032)1198 3347 y Fl(0)1198 3385 y Fu(1)1230 3367 y Fx(;\024)1289 3347 y Fl(0)1289 3385 y Fu(0)1343 3339 y Fw(\002)16 b Ft(T)1479 3351 y Fx(\033)1552 3339 y FB(and)26 b(then)i(w)n(e)f(use)g(the)g(c)n(hange)f(v)-5 b(ariables)26 b FA(v)g FB(=)c FA(u)17 b FB(+)g FA(h)p FB(\()p FA(u;)d(\034)9 b FB(\))28 b(de\014ned)f(in)71 3450 y(Lemma)g(7.6.)195 3549 y(T)-7 b(o)30 b(obtain)g(the)h(asymptotic) f(expansion,)g(w)n(e)g(split)h FA(@)1926 3561 y Fx(v)1979 3528 y Fz(b)1966 3549 y FA(T)2015 3561 y Fy(1)2082 3549 y FB(in)f(sev)n(eral)f(parts)h(taking)g(in)n(to)g(accoun)n(t)f(that)i FA(@)3639 3561 y Fx(v)3692 3528 y Fz(b)3678 3549 y FA(T)3727 3561 y Fy(1)3795 3549 y FB(is)71 3649 y(\014xed)k(p)r(oin)n(t)g(a)f(of) h(the)g(op)r(erator)e Fw(J)51 b FB(in)35 b(\(180\))f(and)h(that)g(w)n (e)g(kno)n(w)f(explicitly)h Fw(J)15 b FB(\(0\).)59 b(W)-7 b(e)35 b(use)g(the)g(functions)g FA(A)3822 3661 y Fx(i)71 3749 y FB(de\014ned)i(in)h(\(152\))o(,)i(\(153\))o(,)f(\(154\))e(resp)r (ectiv)n(ely)-7 b(,)38 b(the)g(c)n(hange)e(of)h(v)-5 b(ariables)36 b FA(g)k FB(obtained)d(in)g(Lemma)g(7.6)f(and)h(the)71 3848 y(op)r(erator)26 b Fw(J)43 b FB(in)28 b(\(180\))o(.)37 b(W)-7 b(e)28 b(tak)n(e)1610 4027 y FA(@)1654 4039 y Fx(v)1708 4006 y Fz(b)1694 4027 y FA(T)1743 4039 y Fy(1)1803 4027 y FB(=)1934 3923 y Fy(7)1890 3948 y Fz(X)1897 4125 y Fx(i)p Fy(=1)2024 4027 y FA(D)2093 4039 y Fx(i)2120 4027 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))71 4245 y(with)742 4422 y FA(D)811 4434 y Fy(1)848 4422 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f FA(A)1211 4434 y Fy(0)1249 4422 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))2223 b(\(227\))742 4546 y FA(D)811 4558 y Fy(2)848 4546 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f Fw(G)1198 4558 y Fx(")1248 4546 y FB(\()p FA(@)1324 4558 y Fx(v)1364 4546 y FA(A)1426 4558 y Fy(1)1463 4546 y FB(\()p FA(v)f FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\)\))1643 b(\(228\))742 4671 y FA(D)811 4683 y Fy(3)848 4671 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f Fw(G)1198 4683 y Fx(")1248 4671 y FB(\()p FA(@)1324 4683 y Fx(v)1364 4671 y FA(A)1426 4683 y Fy(2)1463 4671 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\)\))1977 b(\(229\))742 4795 y FA(D)811 4807 y Fy(4)848 4795 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f Fw(G)1198 4807 y Fx(")1248 4795 y FB(\()p FA(@)1324 4807 y Fx(v)1377 4795 y FB([)p FA(@)1444 4807 y Fx(v)1484 4795 y FA(A)1546 4807 y Fy(2)1583 4795 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\)])s(\))1599 b(\(230\))742 4920 y FA(D)811 4932 y Fy(5)848 4920 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f Fw(G)1198 4932 y Fx(")1248 4920 y FB(\()p FA(@)1324 4932 y Fx(v)1377 4920 y FB([)p FA(A)1462 4932 y Fy(2)1500 4920 y FB(\()p FA(v)f FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(@)2170 4932 y Fx(v)2210 4920 y FA(A)2272 4932 y Fy(2)2309 4920 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))21 b Fw(\000)d FA(A)2896 4932 y Fy(2)2934 4920 y FB(\()p FA(v)s(;)c(\034)9 b FB(\)])q(\))482 b(\(231\))742 5044 y FA(D)811 5056 y Fy(6)848 5044 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f Fw(G)1198 5056 y Fx(")1248 5044 y FB(\()p FA(@)1324 5056 y Fx(v)1377 5044 y FB([)p FA(A)1462 5056 y Fy(3)1500 5044 y FB(\()p FA(v)f FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\)])r(\))1581 b(\(232\))742 5203 y FA(D)811 5215 y Fy(7)848 5203 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f Fw(J)1234 5111 y Fz(\020)1284 5203 y FA(@)1328 5215 y Fx(v)1381 5182 y Fz(b)1368 5203 y FA(T)1417 5215 y Fy(1)1453 5111 y Fz(\021)1517 5203 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b Fw(\000)e(J)29 b FB(\(0\))14 b(\()p FA(v)s(;)g(\034)9 b FB(\))p FA(:)1435 b FB(\(233\))71 5404 y(Let)25 b(us)g(p)r(oin)n(t)g(out)g(that)g(the)g(sum)g(of)g(the)h (\014rst)e(six)h(terms)g(is)f Fw(J)16 b FB(\(0\).)36 b(W)-7 b(e)25 b(b)r(ound)h(eac)n(h)e(term.)36 b(F)-7 b(or)24 b(the)h(second)g(to)f(\014fth)71 5504 y(terms,)j(w)n(e)g(follo) n(w)g(the)h(pro)r(of)f(of)h(Lemma)f(7.5,)g(where)g(the)h(functions)g FA(A)2398 5516 y Fy(1)2435 5504 y FB(,)g FA(A)2548 5516 y Fy(2)2613 5504 y FB(and)g FA(A)2837 5516 y Fy(3)2902 5504 y FB(ha)n(v)n(e)e(b)r(een)i(b)r(ounded.)p Black 1919 5753 a(80)p Black eop end %%Page: 81 81 TeXDict begin 81 80 bop Black Black 195 272 a FB(T)-7 b(o)28 b(b)r(ound)g(\(227\))o(,)f(it)h(is)g(enough)f(to)g(recall)g (that,)h(b)n(y)g(\(179\),)f FA(D)2217 284 y Fy(1)2277 272 y Fw(2)d(E)2400 284 y Fy(0)p Fx(;\032)2487 292 y Fu(2)2519 284 y Fx(;\024)2578 292 y Fu(1)2610 284 y Fx(";\033)2729 272 y Fw(\032)f(E)2861 287 y Fy(2)p Fx(r)r Fy(+1)p Fv(\000)p Fy(1)p Fx(=\014)s(;\032)3225 295 y Fu(2)3256 287 y Fx(;\024)3315 295 y Fu(1)3347 287 y Fx(;\033)3412 272 y FB(,)28 b(to)f(obtain)1272 453 y Fw(k)p FA(D)1383 465 y Fy(1)1420 453 y Fw(k)1462 474 y Fy(2)p Fx(r)r Fy(+1)p Fv(\000)1676 452 y Fu(1)p 1673 461 35 3 v 1673 494 a Fm(\014)1718 474 y Fx(;\033)1805 453 y Fw(\024)c(k)p FA(D)2004 465 y Fy(1)2040 453 y Fw(k)2082 465 y Fy(0)p Fx(;\033)2202 453 y Fw(\024)g FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2502 419 y Fx(\021)r Fy(+1)2626 453 y FA(:)71 644 y FB(T)-7 b(o)30 b(b)r(ound)h(\(228\))o(,)g(w)n(e)f (apply)g(the)h(b)r(ound)g(of)f FA(A)1618 656 y Fy(1)1686 644 y FB(obtained)g(in)h(\(172\))f(and)g(use)g FA(r)h Fw(\025)c FB(1)j(to)g(see)h(that)f FA(D)3385 656 y Fy(2)3450 644 y Fw(2)e(E)3577 656 y Fx(r)r Fy(+1)p Fx(;\033)3786 644 y Fw(\032)71 744 y(E)115 759 y Fy(2)p Fx(r)r Fy(+1)p Fv(\000)p Fy(1)p Fx(=\014)s(;\033)516 744 y FB(and)1230 854 y Fw(k)p FA(D)1341 866 y Fy(2)1378 854 y Fw(k)1420 874 y Fy(2)p Fx(r)r Fy(+1)p Fv(\000)1634 852 y Fu(1)p 1631 861 V 1631 894 a Fm(\014)1676 874 y Fx(;\033)1763 854 y Fw(\024)23 b(k)p FA(D)1962 866 y Fy(2)1999 854 y Fw(k)2041 866 y Fx(r)r Fy(+1)p Fx(;\033)2244 854 y Fw(\024)g FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2544 819 y Fx(\021)r Fy(+1)2668 854 y FA(:)71 1027 y FB(Since)25 b Fw(h)p FA(A)379 1039 y Fy(2)416 1027 y Fw(i)f FB(=)e(0,)j(w)n(e)f (can)h(de\014ne)g(a)f(function)p 1543 960 63 4 v 25 w FA(A)1605 1039 y Fy(2)1667 1027 y FB(suc)n(h)g(that)h FA(@)2072 1039 y Fx(\034)p 2114 960 V 2114 1027 a FA(A)2176 1039 y Fy(2)2236 1027 y FB(=)e FA(A)2386 1039 y Fy(2)2448 1027 y FB(and)i Fw(h)p 2639 960 V FA(A)2701 1039 y Fy(2)2738 1027 y Fw(i)f FB(=)e(0.)36 b(Moreo)n(v)n(er,)22 b(one)i(can)h(write) 1294 1201 y FA(D)1363 1213 y Fy(3)1423 1201 y FB(=)e Fw(G)1560 1213 y Fx(")1596 1201 y FB(\()p FA(@)1672 1213 y Fx(v)1711 1201 y FA(A)1773 1213 y Fy(2)1811 1201 y FB(\))g(=)g Fw(G)2003 1213 y Fx(")2053 1134 y Fz(\000)2091 1201 y FA(@)2140 1167 y Fy(2)2135 1221 y Fx(\034)7 b(v)p 2211 1134 V 2211 1201 a FA(A)2274 1213 y Fy(2)2311 1134 y Fz(\001)1423 1337 y FB(=)23 b FA(")p Fw(G)1599 1349 y Fx(")1648 1270 y Fz(\000)1686 1337 y Fw(L)1743 1349 y Fx(")1793 1270 y Fz(\000)1831 1337 y FA(@)1875 1349 y Fx(v)p 1914 1271 V 1914 1337 a FA(A)1977 1349 y Fy(2)2014 1270 y Fz(\001\001)2108 1337 y Fw(\000)18 b FA(")p Fw(G)2279 1349 y Fx(")2329 1270 y Fz(\000)2367 1337 y FA(@)2416 1303 y Fy(2)2411 1358 y Fx(v)p 2453 1271 V 2453 1337 a FA(A)2515 1349 y Fy(2)2552 1270 y Fz(\001)2604 1337 y FA(:)71 1510 y FB(Then,)24 b(using)e(the)h(de\014nition)g(of)g Fw(G)1160 1522 y Fx(")1219 1510 y FB(in)g(\(168\))f(and)g(applying)g (Lemma)h(7.3,)g(one)f(can)g(see)h(that)g(there)f(exists)h(a)f(function) 77 1599 y Fz(e)71 1621 y FA(\030)107 1633 y Fy(3)167 1621 y Fw(2)i(E)290 1633 y Fy(0)p Fx(;\033)410 1621 y Fw(\032)f(E)542 1636 y Fy(2)p Fx(r)r Fy(+1)p Fv(\000)p Fy(1)p Fx(=\014)s(;\033)916 1621 y FB(,)k(whic)n(h)h(satis\014es,)1266 1812 y Fw(k)1314 1790 y Fz(e)1308 1812 y FA(\030)1344 1824 y Fy(3)1382 1812 y Fw(k)1424 1832 y Fy(2)p Fx(r)r Fy(+1)p Fv(\000)1638 1810 y Fu(1)p 1635 1819 35 3 v 1635 1852 a Fm(\014)1680 1832 y Fx(;\033)1767 1812 y Fw(\024)23 b FA(K)6 b Fw(k)1980 1790 y Fz(e)1974 1812 y FA(\030)2010 1824 y Fy(3)2046 1812 y Fw(k)2088 1824 y Fy(0)p Fx(;\033)2208 1812 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2508 1778 y Fx(\021)r Fy(+1)2632 1812 y FA(;)71 1998 y FB(suc)n(h)27 b(that)1271 2002 y Fz(\015)1271 2052 y(\015)1271 2102 y(\015)1317 2098 y FA(D)1386 2110 y Fy(3)1441 2098 y Fw(\000)18 b FA("@)1607 2110 y Fx(v)p 1646 2031 63 4 v 1646 2098 a FA(A)1709 2110 y Fy(2)1764 2098 y Fw(\000)1854 2076 y Fz(e)1847 2098 y FA(\030)1883 2110 y Fy(3)1921 2002 y Fz(\015)1921 2052 y(\015)1921 2102 y(\015)1967 2156 y Fy(2)p Fx(r)r Fy(+2)p Fx(;\033)2204 2098 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2504 2064 y Fx(\021)r Fy(+2)2628 2098 y FA(:)71 2281 y FB(Moreo)n(v)n(er,)25 b(recalling)h(the)i(de\014nition)g(of)g FA(A)1460 2293 y Fy(2)1525 2281 y FB(in)g(\(153\))e(and)i(de\014ning)g (functions)p 2671 2235 44 4 v 27 w FA(a)2715 2293 y Fx(k)q(l)2805 2281 y FB(suc)n(h)f(that)1505 2452 y FA(@)1549 2464 y Fx(\034)p 1591 2407 V 1591 2452 a FA(a)1635 2464 y Fx(k)q(l)1720 2452 y FB(=)22 b(0)55 b(and)g Fw(h)p 2125 2407 V FA(a)2170 2464 y Fx(k)q(l)2232 2452 y Fw(i)23 b FB(=)g(0)1244 b(\(234\))71 2624 y(w)n(e)27 b(ha)n(v)n(e)g(that)1064 2723 y FA(@)1108 2735 y Fx(v)p 1148 2657 63 4 v 1148 2723 a FA(A)1210 2735 y Fy(2)1247 2723 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)d Fw(\000)p FA(\026)1769 2645 y Fz(X)1677 2823 y Fy(2)p Fv(\024)p Fx(k)q Fy(+)p Fx(l)p Fv(\024)p Fx(N)p 1995 2678 44 4 v 1995 2723 a FA(a)2039 2735 y Fx(k)q(l)2101 2723 y FB(\()p FA(\034)9 b FB(\))p FA(@)2254 2735 y Fx(v)2309 2656 y Fz(\000)2347 2723 y FA(q)2384 2735 y Fy(0)2421 2723 y FB(\()p FA(v)s FB(\))2528 2689 y Fx(k)2570 2723 y FA(p)2612 2735 y Fy(0)2649 2723 y FB(\()p FA(v)s FB(\))2756 2689 y Fx(l)2782 2656 y Fz(\001)2834 2723 y FA(:)71 2956 y FB(Then,)30 b(recalling)e(the)i(de\014nition)g (of)f(the)h(functions)g FA(Q)1830 2968 y Fx(j)1894 2956 y FB(and)g FA(F)2111 2968 y Fx(j)2176 2956 y FB(in)f(\(73\))h(and)f (\(74\))g(and)g(the)h(constan)n(t)f FA(C)3496 2968 y Fy(+)3581 2956 y FB(in)h(\(12\))o(,)71 3056 y FA(@)115 3068 y Fx(v)p 154 2989 63 4 v 154 3056 a FA(A)217 3068 y Fy(2)282 3056 y FB(satis\014es)927 3222 y FA("@)1010 3234 y Fx(v)p 1049 3155 V 1049 3222 a FA(A)1112 3234 y Fy(2)1149 3222 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)1460 3162 y(2)p FA(r)r(\026")1630 3132 y Fx(\021)r Fy(+1)1754 3162 y FA(C)1819 3132 y Fy(2)1813 3182 y(+)1869 3162 y FA(F)1922 3174 y Fy(0)1960 3162 y FB(\()p FA(\034)9 b FB(\))p 1460 3203 610 4 v 1547 3279 a(\()p FA(v)22 b Fw(\000)c FA(ia)p FB(\))1829 3255 y Fy(2)p Fx(r)r Fy(+1)2098 3222 y FB(+)g Fw(O)2263 3080 y Fz( )2504 3166 y FA(\026")2593 3135 y Fx(\021)r Fy(+1)p 2339 3203 543 4 v 2339 3299 a FB(\()p FA(v)k Fw(\000)c FA(ia)p FB(\))2621 3262 y Fy(2)p Fx(r)r Fy(+1)p Fv(\000)2836 3239 y Fu(1)p 2832 3248 35 3 v 2832 3282 a Fm(\014)2892 3080 y Fz(!)2971 3222 y FA(:)71 3434 y FB(Therefore,)26 b(there)i(exists)f FA(\030)948 3446 y Fy(3)1009 3434 y Fw(2)c(E)1131 3449 y Fy(2)p Fx(r)r Fy(+1)p Fv(\000)p Fy(1)p Fx(=\014)s(;\032)1495 3457 y Fu(2)1527 3449 y Fx(;\024)1586 3457 y Fu(1)1618 3449 y Fx(";\033)1741 3434 y FB(satisfying)1487 3616 y Fw(k)p FA(\030)1565 3628 y Fy(3)1602 3616 y Fw(k)1644 3636 y Fy(2)p Fx(r)r Fy(+1)p Fv(\000)1859 3614 y Fu(1)p 1855 3623 V 1855 3656 a Fm(\014)1900 3636 y Fx(;\033)1988 3616 y Fw(\024)f FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2287 3582 y Fx(\021)r Fy(+1)2411 3616 y FA(;)71 3802 y FB(suc)n(h)27 b(that)876 3817 y Fz(\015)876 3867 y(\015)876 3917 y(\015)876 3967 y(\015)922 3938 y FA(D)991 3950 y Fy(3)1028 3938 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b Fw(\000)1330 3878 y FB(2)p FA(r)r(\026")1500 3848 y Fx(\021)r Fy(+1)1625 3878 y FA(C)1690 3848 y Fy(2)1684 3899 y(+)1739 3878 y FA(F)1792 3890 y Fy(0)1830 3878 y FB(\()p FA(\034)9 b FB(\))p 1330 3919 610 4 v 1417 3995 a(\()p FA(v)22 b Fw(\000)c FA(ia)p FB(\))1699 3971 y Fy(2)p Fx(r)r Fy(+1)1968 3938 y Fw(\000)g FA(\030)2087 3950 y Fy(3)2125 3938 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))2314 3817 y Fz(\015)2314 3867 y(\015)2314 3917 y(\015)2314 3967 y(\015)2361 4021 y Fy(2)p Fx(r)r Fy(+2)p Fx(;\033)2599 3938 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2898 3904 y Fx(\021)r Fy(+2)3022 3938 y FA(:)71 4146 y FB(T)-7 b(o)23 b(b)r(ound)g(\(230\),)h (w)n(e)f(\014rst)g(subtract)g(its)g(a)n(v)n(eraged)d(term.)36 b(Then,)24 b(using)f(Lemma)g(7.6)g(to)g(b)r(ound)g FA(g)j FB(and)d FA(@)3452 4158 y Fx(v)3492 4146 y FA(g)s FB(,)h(Lemma)71 4245 y(7.18)i(to)i(b)r(ound)g(the)g(\014rst)f(and)g(second)g(deriv)-5 b(ativ)n(es)27 b(of)g FA(A)1917 4257 y Fy(2)1983 4245 y FB(and)g(Lemma)g(7.3,)g(w)n(e)g(obtain)1156 4417 y Fw(k)p FA(D)1267 4429 y Fy(4)1322 4417 y Fw(\000)18 b(G)1454 4429 y Fx(")1504 4417 y FB(\()p FA(@)1580 4429 y Fx(v)1620 4417 y Fw(h)p FA(@)1696 4429 y Fx(v)1736 4417 y FA(A)1798 4429 y Fy(2)1854 4417 y Fw(\001)g FA(g)s Fw(i)p FB(\))p Fw(k)2044 4442 y Fy(2)p Fx(r)r Fy(+2)p Fx(;\033)2281 4417 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)2542 4383 y Fy(2)2579 4417 y FA(")2618 4383 y Fx(\021)r Fy(+2)2742 4417 y FA(:)71 4588 y FB(On)27 b(the)h(other)f(hand,)h(using)f(the)h (de\014nition)g(of)g Fw(G)1673 4600 y Fx(")1736 4588 y FB(in)g(\(168\))975 4760 y Fw(G)1024 4772 y Fx(")1074 4760 y FB(\()p FA(@)1150 4772 y Fx(v)1190 4760 y Fw(h)p FA(@)1266 4772 y Fx(v)1305 4760 y FA(A)1367 4772 y Fy(2)1423 4760 y Fw(\001)19 b FA(g)s Fw(i)p FB(\))14 b(\()p FA(v)s FB(\))24 b(=)e Fw(h)p FA(@)1880 4772 y Fx(v)1920 4760 y FA(A)1982 4772 y Fy(2)2038 4760 y Fw(\001)d FA(g)s Fw(i)p FB(\()p FA(v)s FB(\))g Fw(\000)f(h)p FA(@)2440 4772 y Fx(v)2480 4760 y FA(A)2542 4772 y Fy(2)2597 4760 y Fw(\001)h FA(g)s Fw(i)p FB(\()p Fw(\000)p FA(\032)2854 4725 y Fv(0)2854 4780 y Fy(1)2891 4760 y FB(\))p FA(:)71 4931 y FB(T)-7 b(o)32 b(obtain)f(its)i(leading)e(term,)i(\014rst)f(w)n (e)g(lo)r(ok)f(for)h(the)g(\014rst)g(order)e(of)i(the)h(function)g FA(g)h FB(giv)n(en)d(in)i(\(173\))o(.)50 b(Using)32 b(the)71 5031 y(de\014nition)27 b(of)f FA(B)595 5043 y Fy(1)659 5031 y FB(in)h(\(144\))o(,)f(the)h(functions)g(\(234\))o(,)g(the)g(b)r (ounds)f(of)h FA(@)2298 5043 y Fx(v)2337 5031 y FA(B)2400 5043 y Fy(1)2464 5031 y FB(in)g(\(151\))e(and)i(Lemma)f(7.3,)g(w)n(e)g (ha)n(v)n(e)f(that)809 5132 y Fz(\015)809 5181 y(\015)809 5231 y(\015)809 5281 y(\015)809 5331 y(\015)809 5381 y(\015)809 5430 y(\015)809 5480 y(\015)856 5352 y FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))19 b Fw(\000)f FA(\026")1279 5317 y Fx(\021)r Fy(+1)1509 5273 y Fz(X)1417 5451 y Fy(2)p Fv(\024)p Fx(k)q Fy(+)p Fx(l)p Fv(\024)p Fx(N)1516 5511 y(l)p Fv(\025)p Fy(1)1736 5352 y FA(l)p 1763 5306 44 4 v 2 w(a)1806 5364 y Fx(k)q(l)1868 5352 y FB(\()p FA(\034)9 b FB(\))p FA(q)2014 5364 y Fy(0)2053 5352 y FB(\()p FA(v)s FB(\))2160 5317 y Fx(k)2201 5352 y FA(p)2243 5364 y Fy(0)2280 5352 y FB(\()p FA(v)s FB(\))2387 5317 y Fx(l)p Fv(\000)p Fy(2)2499 5132 y Fz(\015)2499 5181 y(\015)2499 5231 y(\015)2499 5281 y(\015)2499 5331 y(\015)2499 5381 y(\015)2499 5430 y(\015)2499 5480 y(\015)2545 5534 y Fy(1)p Fx(;\033)2665 5352 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2965 5317 y Fx(\021)r Fy(+2)3089 5352 y FA(:)549 b FB(\(235\))p Black 1919 5753 a(81)p Black eop end %%Page: 82 82 TeXDict begin 82 81 bop Black Black 71 272 a FB(Then,)31 b(using)f(the)g(functions)h FA(Q)1106 284 y Fx(j)1170 272 y FB(and)f FA(F)1387 284 y Fx(j)1453 272 y FB(de\014ned)g(in)h (\(73\))f(and)g(\(74\))f(resp)r(ectiv)n(ely)-7 b(,)30 b(and)g(taking)g(in)n(to)g(accoun)n(t)f(the)71 372 y(de\014nition)f(of) f FA(A)596 384 y Fy(2)662 372 y FB(in)g(\(153\),)g(there)h(exists)f(a)g (function)h FA(\030)1870 384 y Fy(4)1931 372 y Fw(2)23 b(E)2053 387 y Fy(2)p Fx(r)r Fy(+1)p Fv(\000)p Fy(1)p Fx(=\014)s(;\032)2417 395 y Fu(2)2449 387 y Fx(;\024)2508 395 y Fu(1)2540 387 y Fx(";\033)2664 372 y FB(satisfying)1487 552 y Fw(k)p FA(\030)1565 564 y Fy(4)1602 552 y Fw(k)1644 572 y Fy(2)p Fx(r)r Fy(+1)p Fv(\000)1859 550 y Fu(1)p 1855 559 35 3 v 1855 592 a Fm(\014)1900 572 y Fx(;\033)1988 552 y Fw(\024)f FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2287 518 y Fx(\021)r Fy(+1)2411 552 y FA(;)71 737 y FB(suc)n(h)27 b(that)1032 873 y Fw(G)1081 885 y Fx(")1130 873 y FB(\()q FA(@)1207 885 y Fx(v)1246 873 y Fw(h)p FA(@)1322 885 y Fx(v)1362 873 y FA(A)1424 885 y Fy(2)1480 873 y Fw(\001)19 b FA(g)s Fw(i)p FB(\))k(=)1749 813 y(2)p FA(r)r(\026)1880 783 y Fy(2)1918 813 y FA(")1957 783 y Fy(2)p Fx(\021)r Fy(+1)2114 813 y FA(C)2179 783 y Fy(2)2173 833 y(+)2229 813 y Fw(h)p FA(Q)2327 825 y Fy(0)2364 813 y FA(F)2417 825 y Fy(1)2454 813 y Fw(i)p 1749 853 738 4 v 1900 930 a FB(\()p FA(v)f Fw(\000)c FA(ia)p FB(\))2182 906 y Fy(2)p Fx(r)r Fy(+1)2515 873 y FB(+)g FA(\030)2634 885 y Fy(4)2672 873 y FB(\()p FA(u;)c(\034)9 b FB(\))p FA(:)71 1059 y FB(Therefore,)26 b(one)i(can)f(see)g(that)775 1161 y Fz(\015)775 1210 y(\015)775 1260 y(\015)775 1310 y(\015)821 1281 y FA(D)890 1293 y Fy(4)927 1281 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b Fw(\000)1229 1221 y FB(2)p FA(r)r(\026)1360 1191 y Fy(2)1397 1221 y FA(")1436 1191 y Fy(2)p Fx(\021)r Fy(+1)1594 1221 y FA(C)1659 1191 y Fy(2)1653 1242 y(+)1708 1221 y Fw(h)p FA(Q)1806 1233 y Fy(0)1843 1221 y FA(F)1896 1233 y Fy(1)1934 1221 y Fw(i)p 1229 1262 V 1380 1338 a FB(\()p FA(v)i Fw(\000)c FA(ia)p FB(\))1662 1314 y Fy(2)p Fx(r)r Fy(+1)1995 1281 y Fw(\000)g FA(\030)2114 1293 y Fy(4)2151 1281 y FB(\()p FA(u;)c(\034)9 b FB(\))2345 1161 y Fz(\015)2345 1210 y(\015)2345 1260 y(\015)2345 1310 y(\015)2392 1364 y Fy(2)p Fx(r)r Fy(+2)p Fx(;\033)2629 1281 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)2890 1247 y Fy(2)2927 1281 y FA(")2966 1247 y Fy(2)p Fx(\021)r Fy(+2)3123 1281 y FA(:)71 1516 y FB(F)-7 b(or)29 b(\(231\))o(,)h(it)g (is)f(enough)g(to)g(apply)g(Lemmas)g(7.3)f(and)i(7.6,)f(the)g (de\014nition)h(of)g FA(A)2718 1528 y Fy(2)2784 1516 y FB(and)g(the)f(mean)h(v)-5 b(alue)29 b(theorem,)71 1616 y(to)e(obtain)1489 1716 y Fw(k)p FA(D)1600 1728 y Fy(5)1637 1716 y Fw(k)1679 1728 y Fy(2)p Fx(r)r Fy(+2)p Fx(;\033)1915 1716 y Fw(\024)c FA(K)6 b Fw(j)p FA(\026)p Fw(j)2176 1681 y Fy(3)2213 1716 y FA(")2252 1681 y Fy(3)p Fx(\021)r Fy(+2)2409 1716 y FA(:)71 1871 y FB(T)-7 b(o)25 b(b)r(ound)i(\(232\))o(,)f(let)h(us)f(recall)f(the)h(de\014nitions)g (of)g FA(A)1824 1883 y Fy(3)1888 1871 y FB(and)2067 1850 y Fz(b)2047 1871 y FA(H)2123 1841 y Fy(2)2116 1892 y(1)2186 1871 y FB(in)h(\(154\))e(and)h(\(37\))o(.)37 b(Then,)26 b(it)h(is)f(enough)f(to)h(apply)71 1971 y(Lemma)h(7.3,)g(to)h(obtain) 1258 2070 y Fw(k)p FA(D)1369 2082 y Fy(6)1405 2070 y Fw(k)1447 2091 y Fy(2)p Fx(r)r Fy(+1)p Fv(\000)1662 2068 y Fu(1)p 1658 2077 35 3 v 1658 2111 a Fm(\014)1703 2091 y Fx(;\033)1791 2070 y Fw(\024)22 b(k)p FA(D)1989 2082 y Fy(6)2026 2070 y Fw(k)2068 2082 y Fy(2)p Fx(r)n(;\033)2217 2070 y Fw(\024)h FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2517 2036 y Fx(\021)r Fy(+1)2640 2070 y FA(:)71 2250 y FB(Finally)-7 b(,)33 b(for)e(\(233\))o(,)i(it)f(is)f(enough)g(to)h(tak)n(e)f(in)n(to) g(accoun)n(t)g(the)h(de\014nitions)g(of)g Fw(J)47 b FB(and)2929 2229 y Fz(b)2909 2250 y Fw(F)39 b FB(in)32 b(\(180\))f(and)h(\(158\))f (and)71 2350 y(apply)c(Lemmas)g(7.3,)g(7.5)g(and)g(7.8,)g(whic)n(h)h (giv)n(e,)597 2445 y Fz(\015)597 2495 y(\015)597 2545 y(\015)643 2540 y Fw(J)728 2448 y Fz(\020)778 2540 y FA(@)822 2552 y Fx(v)875 2519 y Fz(b)861 2540 y FA(T)910 2552 y Fy(1)947 2448 y Fz(\021)1015 2540 y Fw(\000)18 b(J)30 b FB(\(0\))1290 2445 y Fz(\015)1290 2495 y(\015)1290 2545 y(\015)1336 2599 y Fy(2)p Fx(r)r Fy(+2)p Fx(;\033)1573 2540 y Fw(\024)1661 2445 y Fz(\015)1661 2495 y(\015)1661 2545 y(\015)1727 2519 y(b)1707 2540 y Fw(F)1789 2448 y Fz(\020)1838 2540 y FA(@)1882 2552 y Fx(v)1936 2519 y Fz(b)1922 2540 y FA(T)1971 2552 y Fy(1)2007 2448 y Fz(\021)2075 2540 y Fw(\000)2179 2519 y Fz(b)2158 2540 y Fw(F)22 b FB(\()q(0\))2346 2445 y Fz(\015)2346 2495 y(\015)2346 2545 y(\015)2392 2599 y Fy(2)p Fx(r)r Fy(+2)p Fx(;\033)1573 2739 y Fw(\024)1661 2643 y Fz(\015)1661 2693 y(\015)1661 2743 y(\015)1724 2718 y(b)1707 2739 y FA(B)g Fw(\001)d FA(@)1878 2751 y Fx(v)1931 2718 y Fz(b)1917 2739 y FA(T)1966 2751 y Fy(1)2022 2739 y FB(+)2121 2718 y Fz(b)2105 2739 y FA(C)2184 2646 y Fz(\020)2233 2739 y FA(@)2277 2751 y Fx(v)2331 2718 y Fz(b)2317 2739 y FA(T)2366 2751 y Fy(1)2403 2739 y FA(;)14 b(v)s(;)g(\034)2565 2646 y Fz(\021)2633 2739 y Fw(\000)2733 2718 y Fz(b)2716 2739 y FA(C)21 b FB(\(0)p FA(;)14 b(v)s(;)g(\034)9 b FB(\))3064 2643 y Fz(\015)3064 2693 y(\015)3064 2743 y(\015)3110 2797 y Fy(2)p Fx(r)r Fy(+2)p Fx(;\033)1573 2937 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1873 2902 y Fx(\021)r Fy(+1)2010 2841 y Fz(\015)2010 2891 y(\015)2010 2941 y(\015)2056 2937 y FA(@)2100 2949 y Fx(v)2154 2916 y Fz(b)2140 2937 y FA(T)2189 2949 y Fy(1)2226 2841 y Fz(\015)2226 2891 y(\015)2226 2941 y(\015)2272 2995 y Fy(2)p Fx(r)r Fy(+1)p Fx(;\033)2509 2937 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)2770 2902 y Fy(2)2807 2937 y FA(")2846 2902 y Fy(2)p Fx(\021)r Fy(+2)3003 2937 y FA(:)71 3143 y FB(Considering)26 b(all)h(the)h(b)r(ounds)g(of)g FA(D)1239 3155 y Fx(i)1266 3143 y FB(,)g(w)n(e)f(de\014ne)866 3313 y FA(\030)t FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)e FA(D)1280 3325 y Fy(1)1317 3313 y FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b(+)f FA(D)1682 3325 y Fy(2)1719 3313 y FB(\()p FA(u;)c(\034)9 b FB(\))19 b(+)f FA(\030)2051 3325 y Fy(3)2089 3313 y FB(\()p FA(u;)c(\034)9 b FB(\))19 b(+)f FA(\030)2421 3325 y Fy(4)2459 3313 y FB(\()p FA(u;)c(\034)9 b FB(\))19 b(+)f FA(D)2824 3325 y Fy(6)2861 3313 y FB(\()p FA(u;)c(\034)9 b FB(\))71 3483 y(Then,)28 b FA(\030)f Fw(2)c(E)496 3498 y Fy(2)p Fx(r)r Fy(+1)p Fv(\000)p Fy(1)p Fx(=\014)s(;\033)898 3483 y FB(satisfying)1504 3592 y Fw(k)p FA(\030)t Fw(k)1628 3613 y Fy(2)p Fx(r)r Fy(+1)p Fv(\000)1842 3591 y Fu(1)p 1839 3600 V 1839 3633 a Fm(\014)1883 3613 y Fx(;\033)1971 3592 y Fw(\024)g FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2271 3558 y Fx(\021)r Fy(+1)2394 3592 y FA(;)71 3749 y FB(and)27 b(then)h(w)n(e)g(ha)n(v)n(e)602 3835 y Fz(\015)602 3885 y(\015)602 3935 y(\015)602 3985 y(\015)648 3955 y FA(@)692 3967 y Fx(v)745 3934 y Fz(b)731 3955 y FA(T)780 3967 y Fy(1)817 3955 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b Fw(\000)1132 3896 y FB(2)p FA(r)r(\026")1302 3866 y Fx(\021)r Fy(+1)1427 3896 y FA(C)1492 3866 y Fy(2)1486 3916 y(+)p 1119 3936 436 4 v 1119 4012 a FB(\()p FA(v)i Fw(\000)c FA(ia)p FB(\))1401 3988 y Fy(2)p Fx(r)r Fy(+1)1578 3955 y FB(\()q FA(F)1664 3967 y Fy(0)1701 3955 y FB(\()p FA(\034)9 b FB(\))20 b(+)e FA(\026)p Fw(h)p FA(Q)2061 3967 y Fy(0)2098 3955 y FA(F)2151 3967 y Fy(1)2189 3955 y Fw(i)p FB(\))h Fw(\000)f FA(\030)t FB(\()p FA(u;)c(\034)9 b FB(\))2589 3835 y Fz(\015)2589 3885 y(\015)2589 3935 y(\015)2589 3985 y(\015)2636 4039 y Fy(2)p Fx(r)r Fy(+2)p Fx(;\033)2873 3955 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")3173 3921 y Fx(\021)r Fy(+2)3297 3955 y FA(:)341 b FB(\(236\))71 4191 y(T)-7 b(o)23 b(\014nish)h(the)g(pro)r(of)f(of)h (Prop)r(osition)e(7.23,)h(one)g(has)h(to)f(consider)g(the)h(c)n(hange)e (of)i(v)-5 b(ariables)23 b FA(v)j FB(=)c FA(u)11 b FB(+)g FA(h)p FB(\()p FA(u;)j(\034)9 b FB(\))23 b(de\014ned)71 4291 y(in)28 b(Lemma)f(7.6)g(to)g(obtain)1042 4461 y FA(@)1086 4473 y Fx(u)1130 4461 y FA(T)1179 4473 y Fy(1)1216 4461 y FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b(=)g(\(1)18 b(+)g FA(@)1740 4473 y Fx(u)1784 4461 y FA(h)p FB(\()p FA(u;)c(\034)9 b FB(\)\))2058 4426 y Fv(\000)p Fy(1)2148 4461 y FA(@)2192 4473 y Fx(v)2245 4440 y Fz(b)2231 4461 y FA(T)2280 4473 y Fy(1)2317 4461 y FB(\()p FA(u)18 b FB(+)g FA(h)p FB(\()p FA(u;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))p FA(:)71 4631 y FB(Then,)28 b(the)g(b)r(ounds)f(of)h FA(h)f FB(and)h FA(@)1118 4643 y Fx(u)1161 4631 y FA(h)g FB(in)g(Lemma)f(7.6)g(and)g(\(236\))o(,)h(\014nish)g(the)g(pro)r(of)f (of)h(the)g(prop)r(osition.)p 3790 4631 4 57 v 3794 4578 50 4 v 3794 4631 V 3843 4631 4 57 v 71 4903 a FC(8)135 b(Appro)l(ximation)66 b(of)f(the)h(in)l(v)-7 b(arian)l(t)67 b(manifolds)f(in)f(the)h(inner)g(do-)273 5053 y(mains.)71 5251 y Fq(8.1)112 b(Case)38 b Fh(`)28 b(<)f FF(2)p Fh(r)40 b Fq(:)50 b(pro)s(of)38 b(of)g(Prop)s(osition)f(4.10)71 5404 y FB(W)-7 b(e)22 b(pro)n(v)n(e)e(the)i(results)f(stated)h(in)f (Prop)r(osition)f(4.10)h(concerning)f(the)i(unstable)g(manifold.)35 b(The)21 b(pro)r(of)g(of)h(the)g(results)71 5504 y(concerning)31 b(the)i(stable)f(one)g(follo)n(ws)f(the)i(same)f(lines.)51 b(T)-7 b(o)33 b(obtain)f(the)g(b)r(ound)h(of)g FA(@)2858 5516 y Fx(u)2901 5504 y FA(T)2962 5474 y Fx(u)2950 5525 y Fy(1)3005 5504 y FB(\()p FA(u;)14 b(\034)9 b FB(\))22 b Fw(\000)f FA(@)3351 5516 y Fx(u)3395 5504 y Fw(T)3461 5474 y Fx(u)3440 5525 y Fy(0)3504 5504 y FB(\()p FA(u;)14 b(\034)9 b FB(\),)35 b(w)n(e)p Black 1919 5753 a(82)p Black eop end %%Page: 83 83 TeXDict begin 83 82 bop Black Black 71 272 a FB(\014rst)26 b(b)r(ound)g FA(@)539 284 y Fx(v)593 251 y Fz(b)579 272 y FA(T)640 242 y Fx(u)628 293 y Fy(1)682 272 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))17 b Fw(\000)e FA(@)1012 284 y Fx(v)1052 272 y Fw(T)1118 242 y Fx(u)1097 293 y Fy(0)1161 272 y FB(\()p FA(v)s(;)f(\034)9 b FB(\))28 b(where)1630 251 y Fz(b)1617 272 y FA(T)1678 242 y Fx(u)1666 293 y Fy(1)1746 272 y FB(is)e(the)h(function)f(obtained)g(in)h (Theorems)e(4.4)g(and)h(4.8,)g(whic)n(h)g(is)71 372 y(de\014ned)i(for)f (\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b Fw(2)f FA(D)848 342 y Fx(u)846 395 y(\024)885 403 y Fu(3)917 395 y Fx(;d)972 403 y Fu(2)1026 372 y Fw(\002)18 b Ft(T)1164 384 y Fx(\033)1209 372 y FB(,)28 b(and)g Fw(T)1488 342 y Fx(u)1467 392 y Fy(0)1559 372 y FB(is)g(the)g(function)g(de\014ned)h(in)f(\(57\))o(.)38 b(Then,)28 b(w)n(e)f(will)h(use)g(the)g(c)n(hange)f(of)71 471 y(v)-5 b(ariables)26 b FA(v)g FB(=)d FA(u)18 b FB(+)g FA(h)p FB(\()p FA(u;)c(\034)9 b FB(\))28 b(de\014ned)g(in)g(Lemma)f (7.6)g(to)h(obtain)f(the)h(b)r(ound)g(stated)f(in)h(Prop)r(osition)e (4.10.)195 571 y(Let)37 b(us)e(de\014ne)i(\014rst)e FA(v)935 583 y Fy(3)1009 571 y FB(and)h FA(v)1219 583 y Fy(4)1292 571 y FB(the)h(leftmost)f(and)g(righ)n(tmost)f(v)n(ertices)g(of)h(the)g (inner)g(domain)f FA(D)3486 541 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)3484 591 y(\024)3523 599 y Fu(3)3555 591 y Fx(;c)3605 599 y Fu(1)3712 571 y FB(\(see)71 671 y(Figure)27 b(5\).)37 b(Then,)27 b(w)n(e)h(can)f(de\014ne)h(the)g(op)r(erator)1384 846 y Fz(e)1371 867 y Fw(G)1420 879 y Fx(")1456 867 y FB(\()p FA(h)p FB(\)\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)1869 788 y Fz(X)1869 966 y Fx(k)q Fv(2)p Fn(Z)2017 846 y Fz(e)2003 867 y Fw(G)2052 879 y Fx(")2088 867 y FB(\()p FA(h)p FB(\))2200 832 y Fy([)p Fx(k)q Fy(])2279 867 y FB(\()p FA(v)s FB(\))p FA(e)2425 832 y Fx(ik)q(\034)2527 867 y FA(;)1111 b FB(\(237\))71 1125 y(where)27 b(its)h(F)-7 b(ourier)26 b(co)r(e\016cien)n(ts)h(are)g(giv)n(en)g(b)n(y)781 1326 y Fz(e)767 1347 y Fw(G)816 1359 y Fx(")852 1347 y FB(\()p FA(h)p FB(\))964 1312 y Fy([)p Fx(k)q Fy(])1043 1347 y FB(\()p FA(v)s FB(\))d(=)1262 1234 y Fz(Z)1345 1254 y Fx(v)1308 1422 y(v)1341 1430 y Fu(3)1398 1347 y FA(e)1437 1312 y Fx(ik)q(")1527 1287 y Fl(\000)p Fu(1)1605 1312 y Fy(\()p Fx(t)p Fv(\000)p Fx(v)r Fy(\))1774 1347 y FA(h)1822 1312 y Fy([)p Fx(k)q Fy(])1900 1347 y FB(\()p FA(t)p FB(\))14 b FA(dt)748 b FB(for)27 b FA(k)f(>)c FB(0)784 1560 y Fz(e)771 1581 y Fw(G)820 1593 y Fx(")856 1581 y FB(\()p FA(h)p FB(\))968 1547 y Fy([0])1043 1581 y FB(\()p FA(v)s FB(\))i(=)1262 1468 y Fz(Z)1345 1489 y Fx(v)1308 1657 y(v)1341 1665 y Fu(4)1398 1581 y FA(h)1446 1547 y Fy([0])1521 1581 y FB(\()p FA(t)p FB(\))14 b FA(dt)781 1795 y Fz(e)767 1816 y Fw(G)816 1828 y Fx(")852 1816 y FB(\()p FA(h)p FB(\))964 1782 y Fy([)p Fx(k)q Fy(])1043 1816 y FB(\()p FA(v)s FB(\))24 b(=)1262 1703 y Fz(Z)1345 1723 y Fx(v)1308 1892 y(v)1341 1900 y Fu(4)1398 1816 y FA(e)1437 1782 y Fx(ik)q(")1527 1757 y Fl(\000)p Fu(1)1605 1782 y Fy(\()p Fx(t)p Fv(\000)p Fx(v)r Fy(\))1774 1816 y FA(h)1822 1782 y Fy([)p Fx(k)q Fy(])1900 1816 y FB(\()p FA(t)p FB(\))14 b FA(dt)725 b FB(for)27 b FA(k)f(<)c FB(0)p FA(:)71 2046 y FB(It)j(can)f(b)r(e)g(easily)g(seen)g(that)h (this)f(op)r(erator)f(satis\014es)g(analogous)f(prop)r(erties)i(to)g (the)h(ones)e(satis\014ed)h(b)n(y)g(the)h(op)r(erator)71 2145 y Fw(G)120 2157 y Fx(")183 2145 y FB(de\014ned)j(in)g(\(168\))o(,) g(whic)n(h)g(are)e(giv)n(en)h(in)h(Lemma)f(7.3.)36 b(Let)28 b(us)g(consider)e(also)h(the)h(F)-7 b(ourier)26 b(expansions)200 2341 y FA(h)248 2353 y Fy(1)285 2341 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)d FA(H)655 2353 y Fy(1)693 2341 y FB(\()p FA(q)762 2353 y Fy(0)799 2341 y FB(\()p FA(v)s FB(\))p FA(;)14 b(p)985 2353 y Fy(0)1023 2341 y FB(\()p FA(v)s FB(\))p FA(;)g(\034)9 b FB(\))25 b(=)1356 2262 y Fz(X)1356 2441 y Fx(k)q Fv(2)p Fn(Z)1490 2341 y FA(H)1566 2298 y Fy([)p Fx(k)q Fy(])1559 2363 y(1)1645 2341 y FB(\()p FA(v)s FB(\))p FA(e)1791 2307 y Fx(ik)q(\034)1948 2341 y FB(and)2157 2320 y Fz(b)2137 2341 y FA(A)p FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)d FA(A)p FB(\()p FA(v)h FB(+)18 b FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))25 b(=)3198 2262 y Fz(X)3198 2441 y Fx(k)q Fv(2)p Fn(Z)3352 2320 y Fz(b)3332 2341 y FA(A)3394 2307 y Fy([)p Fx(k)q Fy(])3473 2341 y FB(\()p FA(v)s FB(\))p FA(e)3619 2307 y Fx(ik)q(\034)71 2604 y FB(where)j FA(H)381 2616 y Fy(1)446 2604 y FB(is)g(the)h(function)g (de\014ned)f(in)h(\(8\))f(and)g(\(9\),)h FA(A)f FB(is)g(the)h(function) g(de\014ned)f(in)h(\(143\))e(and)h FA(g)j FB(has)d(b)r(een)g(giv)n(en) 71 2704 y(in)g(Lemma)f(7.6.)195 2803 y(First,)h(w)n(e)f(observ)n(e)f (that,)i(since)f FA(@)1289 2815 y Fx(v)1343 2782 y Fz(b)1329 2803 y FA(T)1378 2815 y Fy(1)1438 2803 y FB(=)22 b Fw(J)16 b FB(\()p FA(@)1673 2815 y Fx(v)1727 2782 y Fz(b)1713 2803 y FA(T)1762 2815 y Fy(1)1799 2803 y FB(\))28 b(where)f(the)h(op)r (erator)d Fw(J)44 b FB(is)27 b(de\014ned)h(in)g(\(180\))o(,)1223 3062 y FA(@)1267 3074 y Fx(v)1320 3041 y Fz(b)1306 3062 y FA(T)1355 3074 y Fy(1)1392 3062 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)1706 3041 y Fz(e)1693 3062 y Fw(G)1742 3074 y Fx(")1778 3062 y FB(\()p FA(@)1854 3074 y Fx(v)1894 3062 y FA(A)p FB(\)\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b(+)2324 2958 y Fy(4)2280 2983 y Fz(X)2286 3160 y Fx(i)p Fy(=1)2414 3062 y FA(N)2481 3074 y Fx(i)2508 3062 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))71 3314 y(with:)1183 3493 y FA(N)1250 3505 y Fy(1)1286 3493 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)o FA(A)1626 3505 y Fy(0)1664 3493 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))1808 b(\(238\))1183 3652 y FA(N)1250 3664 y Fy(2)1286 3652 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)o Fw(J)1650 3559 y Fz(\020)1700 3652 y FA(@)1744 3664 y Fx(v)1797 3631 y Fz(b)1783 3652 y FA(T)1832 3664 y Fy(1)1869 3559 y Fz(\021)1932 3652 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b Fw(\000)e(J)30 b FB(\(0\))13 b(\()p FA(v)s(;)h(\034)9 b FB(\))p FA(:)1020 b FB(\(239\))1183 3820 y FA(N)1250 3832 y Fy(3)1286 3820 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)18 b Fw(\000)1679 3799 y Fz(e)1666 3820 y Fw(G)1715 3832 y Fx(")1751 3820 y FB(\()p FA(@)1827 3832 y Fx(v)1886 3799 y Fz(b)1867 3820 y FA(A)p FB(\)\()p FA(v)s(;)c(\034)9 b FB(\))20 b(+)e Fw(G)2302 3832 y Fx(")2338 3820 y FB(\()p FA(@)2414 3832 y Fx(v)2473 3799 y Fz(b)2454 3820 y FA(A)p FB(\)\()p FA(v)s(;)c(\034)9 b FB(\))924 b(\(240\))1183 3964 y FA(N)1250 3976 y Fy(4)1286 3964 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)1578 3943 y Fz(e)1564 3964 y Fw(G)1613 3976 y Fx(")1649 3964 y FB(\()p FA(@)1725 3976 y Fx(v)1785 3943 y Fz(b)1765 3964 y FA(A)p FB(\)\()p FA(v)s(;)14 b(\034)9 b FB(\))21 b Fw(\000)2165 3943 y Fz(e)2152 3964 y Fw(G)2201 3976 y Fx(")2237 3964 y FB(\()p FA(@)2313 3976 y Fx(v)2352 3964 y FA(A)p FB(\)\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(:)1003 b FB(\(241\))195 4143 y(Second)28 b(w)n(e)f(split)h FA(@)828 4155 y Fx(v)867 4143 y Fw(T)934 4113 y Fx(u)912 4164 y Fy(0)1005 4143 y FB(as:)1355 4253 y FA(@)1399 4265 y Fx(v)1438 4253 y Fw(T)1505 4219 y Fx(u)1483 4274 y Fy(0)1571 4253 y FB(=)23 b Fw(\000)p FA(\026")1813 4219 y Fx(\021)1866 4232 y Fz(e)1853 4253 y Fw(G)1902 4265 y Fx(")1937 4253 y FB(\()p FA(@)2013 4265 y Fx(v)2053 4253 y FA(h)2101 4265 y Fy(1)2138 4253 y FB(\)\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(N)2529 4265 y Fy(5)71 4401 y FB(where)1066 4606 y FA(N)1133 4618 y Fy(5)1170 4606 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))84 b(=)f FA(\026")1680 4572 y Fx(\021)1734 4527 y Fz(X)1734 4706 y Fx(k)q(>)p Fy(0)1869 4493 y Fz(Z)1952 4514 y Fx(v)1985 4522 y Fu(3)1915 4682 y Fv(\0001)2051 4606 y FA(e)2090 4572 y Fx(ik)q(")2180 4547 y Fl(\000)p Fu(1)2259 4572 y Fy(\()p Fx(t)p Fv(\000)p Fx(v)r Fy(\))2427 4606 y FA(@)2471 4618 y Fx(v)2511 4606 y FA(H)2587 4563 y Fy([)p Fx(k)q Fy(])2580 4628 y(1)2665 4606 y FB(\()p FA(t)p FB(\))p FA(dt)1591 4869 y(\026")1680 4835 y Fx(\021)1734 4790 y Fz(X)1734 4969 y Fx(k)q Fv(\024)p Fy(0)1869 4756 y Fz(Z)1952 4777 y Fx(v)1985 4785 y Fu(4)1915 4945 y Fv(\0001)2051 4869 y FA(e)2090 4835 y Fx(ik)q(")2180 4810 y Fl(\000)p Fu(1)2259 4835 y Fy(\()p Fx(t)p Fv(\000)p Fx(v)r Fy(\))2427 4869 y FA(@)2471 4881 y Fx(v)2511 4869 y FA(H)2587 4826 y Fy([)p Fx(k)q Fy(])2580 4891 y(1)2665 4869 y FB(\()p FA(t)p FB(\))p FA(dt:)p Black 806 w FB(\(242\))p Black 195 5157 a(Finally)-7 b(,)33 b(w)n(e)e(use)g(the)h(de\014nition)g (of)f FA(A)h FB(in)g(\(143\))e(and)1990 5136 y Fz(b)1971 5157 y FA(H)2040 5169 y Fy(1)2109 5157 y FB(in)h(\(34\),)h(and)g(the)g (fact)f(that,)i(as)e(the)h(p)r(erio)r(dic)f(orbit)71 5257 y(do)r(es)c(not)h(dep)r(end)g(on)f FA(v)s FB(,)1251 5357 y FA(@)1295 5369 y Fx(v)1349 5357 y FB(\()p FA(V)19 b FB(\()p FA(x)1527 5369 y Fx(p)1566 5357 y FB(\()p FA(\034)9 b FB(\)\))20 b(+)e FA(H)1879 5369 y Fy(1)1916 5357 y FB(\()p FA(x)1995 5369 y Fx(p)2034 5357 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(y)2221 5369 y Fx(p)2260 5357 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\)\))26 b(=)d(0)71 5504 y(to)k(obtain)p Black 1919 5753 a(83)p Black eop end %%Page: 84 84 TeXDict begin 84 83 bop Black Black 839 421 a Fz(e)826 442 y Fw(G)875 454 y Fx(")911 442 y FB(\()p FA(@)987 454 y Fx(v)1027 442 y FA(A)p FB(\)\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b(+)e FA(\026")1502 407 y Fx(\021)1556 421 y Fz(e)1542 442 y Fw(G)1591 454 y Fx(")1627 442 y FB(\()p FA(@)1703 454 y Fx(v)1743 442 y FA(h)1791 454 y Fy(1)1828 442 y FB(\)\()p FA(v)s(;)c(\034)9 b FB(\))25 b(=)e Fw(\000)p FA(y)2268 454 y Fx(p)2305 442 y FB(\()p FA(\034)9 b FB(\))p FA(p)2456 454 y Fy(0)2494 442 y FB(\()p FA(u)p FB(\))19 b(+)f FA(x)2755 454 y Fx(p)2794 442 y FB(\()p FA(\034)9 b FB(\))17 b(_)-40 b FA(p)2945 454 y Fy(0)2983 442 y FB(\()p FA(u)p FB(\))566 b(\(243\))2069 566 y(+)18 b FA(N)2219 578 y Fy(6)2275 566 y FB(+)g FA(N)2425 578 y Fy(7)2480 566 y FB(+)g FA(N)2630 578 y Fy(8)3661 566 y FB(\(244\))71 736 y(with)559 905 y FA(N)626 917 y Fy(6)745 905 y FB(=)83 b Fw(\000)p FA(\026")1047 871 y Fx(\021)1100 884 y Fz(e)1087 905 y Fw(G)1136 917 y Fx(")1172 905 y FA(@)1216 917 y Fx(v)1255 813 y Fz(\020)1305 905 y FA(H)1374 917 y Fy(1)1411 905 y FB(\()p FA(q)1480 917 y Fy(0)1518 905 y FB(\()p FA(v)s FB(\))19 b(+)f FA(x)1774 917 y Fx(p)1813 905 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(p)2001 917 y Fy(0)2039 905 y FB(\()p FA(v)s FB(\))19 b(+)f FA(y)2289 917 y Fx(p)2327 905 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))21 b Fw(\000)d FA(H)2723 917 y Fy(1)2760 905 y FB(\()p FA(q)2829 917 y Fy(0)2867 905 y FB(\()p FA(v)s FB(\))p FA(;)c(p)3053 917 y Fy(0)3090 905 y FB(\()p FA(v)s FB(\))p FA(;)g(\034)9 b FB(\))3311 813 y Fz(\021)p Black 3661 905 a FB(\(245\))p Black 559 1088 a FA(N)626 1100 y Fy(7)745 1088 y FB(=)83 b Fw(\000)971 1067 y Fz(e)958 1088 y Fw(G)1007 1100 y Fx(")1043 1088 y FA(@)1087 1100 y Fx(v)1126 996 y Fz(\020)1176 1088 y FA(V)19 b FB(\()p FA(q)1312 1100 y Fy(0)1349 1088 y FB(\()p FA(u)p FB(\))g(+)f FA(x)1610 1100 y Fx(p)1649 1088 y FB(\()p FA(\034)9 b FB(\)\))20 b Fw(\000)e FA(V)g FB(\()p FA(q)2028 1100 y Fy(0)2066 1088 y FB(\()p FA(u)p FB(\)\))h Fw(\000)f FA(V)2379 1054 y Fv(0)2402 1088 y FB(\()p FA(q)2471 1100 y Fy(0)2509 1088 y FB(\()p FA(u)p FB(\)\))p FA(x)2700 1100 y Fx(p)2739 1088 y FB(\()p FA(\034)9 b FB(\))2848 996 y Fz(\021)2899 1088 y FA(:)p Black 739 w FB(\(246\))p Black 559 1271 a FA(N)626 1283 y Fy(8)745 1271 y FB(=)906 1250 y Fz(e)893 1271 y Fw(G)942 1283 y Fx(")978 1271 y FA(@)1022 1283 y Fx(v)1062 1179 y Fz(\020)1130 1271 y Fw(\000)18 b FA(V)1279 1236 y Fv(0)1303 1271 y FB(\()p FA(q)1372 1283 y Fy(0)1409 1271 y FB(\()p FA(u)p FB(\)\))p FA(x)1600 1283 y Fx(p)1640 1271 y FB(\()p FA(\034)9 b FB(\))19 b(+)f FA(V)1918 1236 y Fv(0)1941 1271 y FB(\()p FA(x)2020 1283 y Fx(p)2060 1271 y FB(\()p FA(\034)9 b FB(\)\))p FA(q)2238 1283 y Fy(0)2276 1271 y FB(\()p FA(u)p FB(\))745 1453 y(+)83 b FA(\026")982 1419 y Fx(\021)1036 1453 y FB(\()p FA(q)1105 1465 y Fy(0)1143 1453 y FB(\()p FA(u)p FB(\))p FA(@)1299 1465 y Fx(x)1341 1453 y FA(H)1410 1465 y Fy(1)1447 1453 y FB(\()p FA(x)1526 1465 y Fx(p)1565 1453 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(y)1752 1465 y Fx(p)1791 1453 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))21 b(+)d FA(p)2160 1465 y Fy(0)2197 1453 y FB(\()p FA(u)p FB(\))p FA(@)2353 1465 y Fx(y)2393 1453 y FA(H)2462 1465 y Fy(1)2499 1453 y FB(\()p FA(x)2578 1465 y Fx(p)2617 1453 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(y)2804 1465 y Fx(p)2844 1453 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\)\))3115 1361 y Fz(\021)745 1603 y FB(+)83 b FA(y)934 1615 y Fx(p)972 1603 y FB(\()p FA(\034)9 b FB(\))p FA(p)1123 1615 y Fy(0)1161 1603 y FB(\()p FA(u)p FB(\))19 b Fw(\000)f FA(x)1422 1615 y Fx(p)1461 1603 y FB(\()p FA(\034)9 b FB(\))17 b(_)-40 b FA(p)1612 1615 y Fy(0)1650 1603 y FB(\()p FA(u)p FB(\))p FA(:)p Black 1876 w FB(\(247\))p Black 71 1772 a(Finally)27 b(w)n(e)h(obtain:)840 2016 y FA(@)884 2028 y Fx(v)937 1995 y Fz(b)923 2016 y FA(T)972 2028 y Fy(1)1009 2016 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(@)1345 2028 y Fx(v)1384 2016 y Fw(T)1451 1982 y Fx(u)1429 2037 y Fy(0)1517 2016 y FB(=)23 b Fw(\000)p FA(y)1711 2028 y Fx(p)1748 2016 y FB(\()p FA(\034)9 b FB(\))p FA(p)1899 2028 y Fy(0)1938 2016 y FB(\()p FA(u)p FB(\))18 b(+)g FA(x)2198 2028 y Fx(p)2237 2016 y FB(\()p FA(\034)9 b FB(\))18 b(_)-41 b FA(p)2388 2028 y Fy(0)2426 2016 y FB(\()p FA(u)p FB(\))19 b(+)2683 1912 y Fy(8)2640 1937 y Fz(X)2646 2114 y Fx(i)p Fy(=1)2774 2016 y FA(N)2841 2028 y Fx(i)2868 2016 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(:)195 2258 y FB(No)n(w,)28 b(w)n(e)f(pro)r(ceed)g(to)g(b)r (ound)h FA(N)1261 2270 y Fy(1)1298 2258 y FA(;)14 b(:)g(:)g(:)g(;)g(N) 1550 2270 y Fy(8)1587 2258 y FB(.)195 2358 y(T)-7 b(o)28 b(b)r(ound)g FA(N)641 2370 y Fy(1)705 2358 y FB(in)g(\(238\))o(,)g(it)g (is)f(enough)g(to)h(recall)f(that,)g(b)n(y)i(\(179\))o(,)f FA(N)2444 2370 y Fy(1)2504 2358 y Fw(2)23 b(E)2626 2372 y Fy(0)p Fx(;\032)2713 2353 y Fl(0)2713 2391 y Fu(1)2746 2372 y Fx(;\024)2805 2353 y Fl(0)2805 2391 y Fu(0)2837 2372 y Fx(;\033)2929 2358 y FB(and)1584 2541 y Fw(k)p FA(N)1693 2553 y Fy(1)1729 2541 y Fw(k)1771 2553 y Fy(0)p Fx(;\033)1891 2541 y Fw(\024)g FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2191 2506 y Fx(\021)r Fy(+1)2315 2541 y FA(:)195 2724 y FB(F)-7 b(or)24 b FA(N)408 2736 y Fy(2)470 2724 y FB(in)h(\(239\))o(,)g(it)g(is)g(enough)f(to)g(consider)g(the)h(b)r (ound)g(of)g FA(@)2193 2736 y Fx(v)2246 2703 y Fz(b)2232 2724 y FA(T)2281 2736 y Fy(1)2343 2724 y FB(giv)n(en)e(in)i(Prop)r (osition)e(7.4)h(and)h(the)g(Lipsc)n(hitz)71 2823 y(constan)n(t)37 b(of)h(the)g(op)r(erator)e Fw(J)53 b FB(in)38 b(\(180\))f(restricted)h (to)f(the)h(ball)p 2279 2756 68 4 v 38 w FA(B)t FB(\()p Fw(j)p FA(\026)p Fw(j)p FA(")2513 2793 y Fx(\021)r Fy(+1)2638 2823 y FB(\))i Fw(\032)g(E)2859 2837 y Fx(`)p Fy(+1)p Fx(;\032)3025 2818 y Fl(0)3025 2856 y Fu(1)3057 2837 y Fx(;\024)3116 2818 y Fl(0)3116 2856 y Fu(0)3148 2837 y Fx(;\033)3213 2823 y FB(,)g(whic)n(h)e(has)f(b)r(een)71 2923 y(obtained)27 b(in)h(the)g(pro)r(of)f(of)h(Lemma)f(7.8.)36 b(Then,)985 3147 y Fw(k)p FA(N)1094 3159 y Fy(2)1131 3147 y Fw(k)1173 3159 y Fy(0)p Fx(;\033)1293 3147 y Fw(\024)o FA(K)1447 3091 y(")1486 3061 y Fv(\000)p Fy(\()p Fx(`)p Fy(+1\))p 1444 3128 266 4 v 1444 3204 a FB(\()p FA(\024)1524 3175 y Fv(0)1524 3226 y Fy(0)1561 3204 y FB(\))1593 3180 y Fx(`)p Fy(+1)1719 3147 y Fw(k)p FA(N)1828 3159 y Fy(2)1865 3147 y Fw(k)1907 3159 y Fx(`)p Fy(+1)p Fx(;\033)1293 3355 y Fw(\024)o FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1569 3321 y Fv(\000)p Fy(\()p Fx(`)p Fy(+1\)+)p Fx(\021)r Fy(+1)p Fv(\000)p Fy(max)o Fv(f)p Fy(0)p Fx(;`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fv(g)2498 3259 y Fz(\015)2498 3309 y(\015)2498 3359 y(\015)2544 3355 y FA(@)2588 3367 y Fx(v)2642 3334 y Fz(b)2628 3355 y FA(T)2677 3367 y Fy(1)2714 3259 y Fz(\015)2714 3309 y(\015)2714 3359 y(\015)2760 3413 y Fx(`)p Fy(+1)p Fx(;\033)1293 3535 y Fw(\024)o FA(K)g Fw(j)p FA(\026)p Fw(j)1530 3501 y Fy(2)1567 3535 y FA(")1606 3501 y Fy(2)p Fx(\021)r Fv(\000)p Fx(`)p Fy(+1)p Fv(\000)p Fy(max)o Fv(f)p Fy(0)p Fx(;`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fv(g)2368 3535 y FA(:)195 3706 y FB(T)-7 b(o)28 b(b)r(ound)g FA(N)641 3718 y Fy(3)705 3706 y FB(in)g(\(240\))f(w)n(e)g(observ)n(e)f(that)i Fw(h)p FA(N)1716 3718 y Fy(3)1753 3706 y Fw(i)23 b FB(=)g(0)k(and)568 3922 y FA(N)644 3879 y Fy([)p Fx(k)q Fy(])635 3944 y(3)722 3922 y FB(\()p FA(v)s FB(\))d(=)e FA(e)979 3888 y Fx(ik)q(")1069 3863 y Fl(\000)p Fu(1)1148 3888 y Fy(\()p Fx(v)1207 3896 y Fu(3)1240 3888 y Fv(\000)p Fx(v)r Fy(\))1371 3809 y Fz(Z)1454 3830 y Fx(v)1487 3838 y Fu(3)1417 3998 y Fx(u)1456 4006 y Fu(1)1537 3922 y FA(e)1576 3888 y Fx(ik)q(")1666 3863 y Fl(\000)p Fu(1)1745 3888 y Fy(\()p Fx(t)p Fv(\000)p Fx(v)1881 3896 y Fu(3)1913 3888 y Fy(\))1957 3830 y Fz(\020)2007 3922 y FA(@)2051 3934 y Fx(v)2110 3901 y Fz(b)2090 3922 y FA(A)2152 3888 y Fy([)p Fx(k)q Fy(])2231 3830 y Fz(\021)2295 3922 y FB(\()p FA(t)p FB(\))p FA(dt)497 b FB(for)96 b FA(k)26 b(>)d FB(0)571 4126 y FA(N)647 4083 y Fy([0])638 4148 y(3)722 4126 y FB(\()p FA(v)s FB(\))h(=)960 4105 y Fz(b)940 4126 y FA(A)1002 4092 y Fy([0])1078 4126 y FB(\()p FA(v)s FB(\))19 b Fw(\000)1306 4105 y Fz(b)1287 4126 y FA(A)1349 4092 y Fy([0])1424 4126 y FB(\()p FA(v)1496 4138 y Fy(4)1534 4126 y FB(\))568 4307 y FA(N)644 4264 y Fy([)p Fx(k)q Fy(])635 4329 y(3)722 4307 y FB(\()p FA(v)s FB(\))24 b(=)e FA(e)979 4272 y Fx(ik)q(")1069 4247 y Fl(\000)p Fu(1)1148 4272 y Fy(\()p Fx(v)1207 4280 y Fu(4)1240 4272 y Fv(\000)p Fx(v)r Fy(\))1371 4194 y Fz(Z)1454 4214 y Fx(v)1487 4222 y Fu(4)1422 4382 y Fy(\026)-38 b Fx(u)1456 4390 y Fu(1)1537 4307 y FA(e)1576 4272 y Fx(ik)q(")1666 4247 y Fl(\000)p Fu(1)1745 4272 y Fy(\()p Fx(t)p Fv(\000)p Fx(v)1881 4280 y Fu(4)1913 4272 y Fy(\))1957 4215 y Fz(\020)2007 4307 y FA(@)2051 4319 y Fx(v)2110 4286 y Fz(b)2090 4307 y FA(A)2152 4272 y Fy([)p Fx(k)q Fy(])2231 4215 y Fz(\021)2295 4307 y FB(\()p FA(t)p FB(\))p FA(dt)497 b FB(for)96 b FA(k)26 b(<)d FB(0)71 4550 y(T)-7 b(aking)28 b(in)n(to)g(accoun)n(t)g(that)h(the)g(op)r(erator)1500 4529 y Fz(e)1486 4550 y Fw(G)1535 4562 y Fx(")1600 4550 y FB(sati\014es)f(also)f(the)i(prop)r(erties)f(of)h(the)g(op)r(erator)e Fw(G)3202 4562 y Fx(")3266 4550 y FB(giv)n(en)h(in)h(Lemma)71 4649 y(7.3,)23 b(and)h(using)f(the)h(b)r(ounds)g(of)f FA(g)k FB(and)c FA(@)1377 4661 y Fx(v)1417 4649 y FA(A)h FB(giv)n(en)e(in)i(Lemmas)f(7.6)g(and)h(7.5)e(resp)r(ectiv)n(ely)-7 b(,)24 b(w)n(e)f(obtain)h(the)g(follo)n(wing)71 4749 y(b)r(ounds.)37 b(F)-7 b(or)27 b FA(k)f Fw(6)p FB(=)c(0,)1153 4843 y Fz(\015)1153 4893 y(\015)1153 4943 y(\015)1199 4939 y FA(N)1275 4896 y Fy([)p Fx(k)q Fy(])1266 4961 y(3)1353 4843 y Fz(\015)1353 4893 y(\015)1353 4943 y(\015)1399 4997 y Fy(0)p Fx(;\033)1520 4939 y Fw(\024)1607 4843 y Fz(\015)1607 4893 y(\015)1607 4943 y(\015)1667 4918 y(e)1653 4939 y Fw(G)1702 4951 y Fx(")1752 4847 y Fz(\020)1802 4939 y FA(@)1846 4951 y Fx(v)1905 4918 y Fz(b)1885 4939 y FA(A)1947 4905 y Fy([)p Fx(k)q Fy(])2026 4939 y FB(\()p FA(v)s FB(\))p FA(e)2172 4905 y Fx(ik)q(\034)2274 4847 y Fz(\021)2324 4843 y(\015)2324 4893 y(\015)2324 4943 y(\015)2370 4997 y Fy(0)p Fx(;\033)1520 5137 y Fw(\024)g FA(K)6 b(")1737 5042 y Fz(\015)1737 5091 y(\015)1737 5141 y(\015)1782 5137 y FA(@)1826 5149 y Fx(v)1886 5116 y Fz(b)1866 5137 y FA(A)1928 5103 y Fy([)p Fx(k)q Fy(])2007 5137 y FB(\()p FA(v)s FB(\))p FA(e)2153 5103 y Fx(ik)q(\034)2255 5042 y Fz(\015)2255 5091 y(\015)2255 5141 y(\015)2301 5195 y Fy(0)p Fx(;\033)1520 5335 y Fw(\024)22 b FA(K)6 b(")1723 5301 y Fy(1)p Fv(\000)p Fy(\()p Fx(`)p Fy(+1\))p Fx(\015)2028 5240 y Fz(\015)2028 5290 y(\015)2028 5339 y(\015)2074 5335 y FA(@)2118 5347 y Fx(v)2177 5314 y Fz(b)2157 5335 y FA(A)2219 5301 y Fy([)p Fx(k)q Fy(])2298 5335 y FB(\()p FA(v)s FB(\))p FA(e)2444 5301 y Fx(ik)q(\034)2546 5240 y Fz(\015)2546 5290 y(\015)2546 5339 y(\015)2592 5393 y Fx(`)p Fy(+1)p Fx(;\033)1520 5516 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1819 5481 y Fx(\021)r Fy(+1)p Fv(\000)p Fy(\()p Fx(`)p Fy(+1\))p Fx(\015)2197 5516 y FA(:)p Black 1919 5753 a FB(84)p Black eop end %%Page: 85 85 TeXDict begin 85 84 bop Black Black 71 272 a FB(F)-7 b(or)27 b FA(k)f FB(=)c(0,)28 b(w)n(e)f(ha)n(v)n(e)f(that)1213 464 y Fw(k)p FA(N)1331 421 y Fy([0])1322 486 y(3)1405 464 y Fw(k)1447 476 y Fy(0)p Fx(;\033)1567 464 y Fw(\024)d FA(K)1745 368 y Fz(\015)1745 418 y(\015)1745 468 y(\015)1811 443 y(b)1791 464 y FA(A)1853 430 y Fy([0])1928 368 y Fz(\015)1928 418 y(\015)1928 468 y(\015)1974 522 y Fy(0)p Fx(;\033)1567 662 y Fw(\024)g FA(K)6 b(")1771 628 y Fv(\000)p Fx(`\015)1906 566 y Fz(\015)1906 616 y(\015)1906 666 y(\015)1972 641 y(b)1952 662 y FA(A)2014 628 y Fy([0])2089 566 y Fz(\015)2089 616 y(\015)2089 666 y(\015)2135 720 y Fx(`;\033)2251 662 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2550 628 y Fx(\021)r Fv(\000)p Fx(`\015)71 869 y FB(Finally)-7 b(,)32 b(note)f(that)h(in)f(the)h(case)e FA(`)f FB(=)f(0,)k(w)n(e)f(ha)n(v)n(e)f(that)h(the)h(c)n(hange)e FA(g)k FB(obtained)c(in)i(Lemma)f(7.6)f(satis\014es)h FA(g)g FB(=)e(0.)71 980 y(Then)309 959 y Fz(b)290 980 y FA(A)e FB(=)f FA(A)p FB(,)k(that)g(implies)g Fw(h)1103 959 y Fz(b)1083 980 y FA(A)q Fw(i)d FB(=)f(0.)43 b(Therefore)28 b(when)i FA(`)c FB(=)h(0)i(w)n(e)g(ha)n(v)n(e)g(that)h FA(N)2801 937 y Fy([0])2792 1002 y(3)2902 980 y FB(=)c(0.)43 b(T)-7 b(aking)29 b(this)h(fact)g(in)n(to)71 1080 y(accoun)n(t,)d(w)n (e)g(can)g(b)r(ound)h FA(N)998 1092 y Fy(3)1063 1080 y FB(b)n(y)1536 1180 y Fw(k)p FA(N)1645 1192 y Fy(3)1681 1180 y Fw(k)1723 1192 y Fy(0)p Fx(;\033)1843 1180 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2143 1145 y Fx(\021)r Fv(\000)p Fx(`)p Fy(+)p Fx(\027)2347 1120 y Fl(\003)2343 1162 y Fu(2)71 1322 y FB(where)1449 1451 y FA(\027)1495 1417 y Fv(\003)1490 1471 y Fy(2)1557 1451 y FB(=)1644 1334 y Fz(\032)1748 1400 y FA(`)p FB(\(1)18 b Fw(\000)g FA(\015)5 b FB(\))83 b(if)28 b FA(`)23 b(>)f FB(0)1748 1500 y(1)c Fw(\000)g FA(\015)187 b FB(if)28 b FA(`)23 b FB(=)f(0)2449 1451 y FA(:)71 1643 y FB(F)-7 b(or)25 b FA(N)285 1655 y Fy(4)348 1643 y FB(in)g(\(241\))o(,)i(one)e(has)g(to) g(consider)g(the)h(b)r(ound)g(of)g FA(@)1930 1655 y Fx(v)1969 1643 y FA(A)g FB(giv)n(en)f(in)h(Lemma)f(7.5)g(and)g(the)h(b)r(ound)g (of)g FA(g)i FB(restricted)71 1743 y(to)e(the)g(inner)g(domain)f(giv)n (en)g(in)i(Corollary)c(7.7.)36 b(Then,)26 b(using)g(again)f(the)h(b)r (ounds)g(analogous)e(to)h(the)i(ones)e(giv)n(en)g(in)71 1843 y(Lemma)i(7.3,)g(but)h(to)g(the)g(op)r(erator)1269 1822 y Fz(e)1256 1843 y Fw(G)1305 1855 y Fx(")836 2014 y Fw(k)p FA(N)945 2026 y Fy(4)981 2014 y Fw(k)1023 2026 y Fy(0)p Fx(;\033)1143 2014 y Fw(\024)23 b FA(K)6 b Fw(k)1368 1993 y Fz(b)1350 2014 y FA(A)17 b Fw(\000)h FA(A)p Fw(k)1616 2026 y Fy(0)p Fx(;\033)1737 2014 y Fw(\024)k FA(K)6 b Fw(k)p FA(@)1987 2026 y Fx(v)2026 2014 y FA(A)p Fw(k)2130 2026 y Fy(0)p Fx(;\033)2227 2014 y Fw(k)p FA(g)s Fw(k)2354 2026 y Fy(0)p Fx(;\033)2473 2014 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)2734 1980 y Fy(2)2771 2014 y FA(")2810 1980 y Fy(2)p Fx(\021)r Fv(\000)p Fx(`)p Fy(+)p Fx(\027)3047 1955 y Fl(\003)3043 1996 y Fu(1)71 2186 y FB(with)28 b FA(\027)306 2155 y Fv(\003)301 2206 y Fy(1)372 2186 y FB(is)g(de\014ned)g(in)f(Corollary)f(7.7.)195 2285 y(F)-7 b(or)25 b FA(N)409 2297 y Fy(5)472 2285 y FB(in)h(\(242\))o(,)g(it)h(is)e(enough)g(to)h(tak)n(e)f(in)n(to)g (accoun)n(t)g(that)h Fw(h)p FA(h)2259 2297 y Fy(1)2296 2285 y Fw(i)e FB(=)e(0,)k(that)g FA(h)2756 2297 y Fy(1)2819 2285 y FB(has)f(a)g(rami\014ed)g(p)r(oin)n(t)h(of)g(order)71 2385 y FA(`)i FB(at)h FA(u)24 b FB(=)h FA(ia)j FB(and)h(that)g(b)r(oth) g FA(v)1081 2397 y Fy(3)1147 2385 y FB(and)g FA(v)1350 2397 y Fy(4)1416 2385 y FB(satisfy)f Fw(j)p FA(v)1739 2397 y Fx(i)1786 2385 y Fw(\000)19 b FA(ia)p Fw(j)25 b FB(=)f Fw(O)16 b FB(\()q FA(")2234 2355 y Fx(\015)2276 2385 y FB(\),)29 b FA(i)c FB(=)g(3)p FA(;)14 b FB(4.)39 b(Then,)29 b(b)r(ounding)g(the)g(in)n(tegrals)e(as)71 2484 y(in)h(Lemma)f(6.2)g(and)g(6.8,)g(one)g(has)g(that)1059 2656 y Fw(k)p FA(N)1168 2668 y Fy(5)1204 2656 y Fw(k)1246 2668 y Fy(0)p Fx(;\033)1366 2656 y Fw(\024)c FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1666 2622 y Fx(\021)r Fy(+1)1790 2656 y Fw(k)p FA(@)1876 2668 y Fx(v)1915 2656 y FA(h)1963 2668 y Fy(1)2000 2656 y Fw(k)2042 2668 y Fy(0)p Fx(;\033)2162 2656 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2462 2622 y Fx(\021)r Fy(+1)p Fv(\000)p Fx(\015)t Fy(\()p Fx(`)p Fy(+1\))2839 2656 y FA(:)195 2827 y FB(T)-7 b(o)28 b(b)r(ound)g FA(N)641 2839 y Fy(6)705 2827 y FB(in)g(\(245\))f (w)n(e)g(\014st)h(use)f(the)h(mean)g(v)-5 b(alue)27 b(theorem)g(to)h (obtain)630 2999 y Fw(k)p FA(H)741 3011 y Fy(1)778 2999 y FB(\()p FA(q)847 3011 y Fy(0)884 2999 y FB(\()p FA(v)s FB(\))20 b(+)e FA(x)1141 3011 y Fx(p)1179 2999 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(p)1367 3011 y Fy(0)1406 2999 y FB(\()p FA(v)s FB(\))19 b(+)f FA(y)1656 3011 y Fx(p)1694 2999 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))21 b Fw(\000)d FA(H)2090 3011 y Fy(1)2127 2999 y FB(\()p FA(q)2196 3011 y Fy(0)2233 2999 y FB(\()p FA(v)s FB(\))p FA(;)c(p)2419 3011 y Fy(0)2457 2999 y FB(\()p FA(v)s FB(\))p FA(;)g(\034)9 b FB(\))p Fw(k)2721 3027 y Fy(0)p Fx(;\033)2842 2999 y Fw(\024)23 b(j)p FA(\026)p Fw(j)p FA(")3065 2964 y Fx(\021)r Fv(\000)p Fx(`)p Fy(+)p Fx(r)3268 2999 y FA(:)71 3199 y FB(Then,)k(using)f(that)717 3178 y Fz(e)704 3199 y Fw(G)753 3211 y Fx(")815 3199 y FB(has)g(similar)g(prop)r(erties)f(to)h(the)h(ones)e(giv)n(en)h(in)g (Lemma)g(7.3)g(for)f(the)i(op)r(erator)e Fw(G)3437 3211 y Fx(")3499 3199 y FB(w)n(e)h(obtain)1509 3370 y Fw(k)p FA(N)1618 3382 y Fy(6)1654 3370 y Fw(k)1696 3382 y Fy(0)p Fx(;\033)1816 3370 y Fw(\024)d FA(K)6 b Fw(j)p FA(\026)p Fw(j)2077 3336 y Fy(2)2114 3370 y FA(")2153 3336 y Fy(2)p Fx(\021)r Fv(\000)p Fx(`)p Fy(+)p Fx(r)2389 3370 y FA(:)195 3542 y FB(The)28 b(b)r(ound)g(for)f FA(N)816 3554 y Fy(7)881 3542 y FB(in)h(\(246\))e(comes)h(from)h(applying)f(the)h(mean)f(b)r (ound)h(theorem)f(to)g(the)h(function)1124 3713 y FA(V)19 b FB(\()p FA(q)1260 3725 y Fy(0)1297 3713 y FB(\()p FA(u)p FB(\))g(+)f FA(x)1558 3725 y Fx(p)1597 3713 y FB(\()p FA(\034)9 b FB(\)\))20 b Fw(\000)e FA(V)h FB(\()p FA(q)1977 3725 y Fy(0)2014 3713 y FB(\()p FA(u)p FB(\)\))g Fw(\000)f FA(V)2327 3679 y Fv(0)2350 3713 y FB(\()p FA(q)2419 3725 y Fy(0)2457 3713 y FB(\()p FA(u)p FB(\)\))p FA(x)2648 3725 y Fx(p)2687 3713 y FB(\()p FA(\034)9 b FB(\))71 3885 y(and)28 b(using)g(that)g FA(V)698 3855 y Fv(00)740 3885 y FB(\()p FA(q)809 3897 y Fy(0)847 3885 y FB(\()p FA(u)p FB(\)\))h(has)e(a)h(p)r(ole)g(of)g(second)g(order,)f(the)h(b)r (ound)h(of)f(the)g(p)r(erio)r(dic)g(orbit)g(and)g(the)g(prop)r(erties) 71 3995 y(of)179 3974 y Fz(e)165 3995 y Fw(G)214 4007 y Fx(")250 3995 y FB(.)37 b(Then,)28 b(w)n(e)f(obtain)195 4166 y Fw(k)p FA(N)304 4178 y Fy(7)341 4166 y Fw(k)383 4178 y Fy(0)p Fx(;\033)503 4166 y Fw(\024)22 b FA(K)6 b Fw(k)p FA(V)18 b FB(\()p FA(q)844 4178 y Fy(0)882 4166 y FB(\()p FA(u)p FB(\))g(+)g FA(x)1142 4178 y Fx(p)1181 4166 y FB(\()p FA(\034)9 b FB(\)\))20 b Fw(\000)e FA(V)h FB(\()p FA(q)1561 4178 y Fy(0)1599 4166 y FB(\()p FA(u)p FB(\)\))g Fw(\000)f FA(V)1912 4132 y Fv(0)1935 4166 y FB(\()p FA(q)2004 4178 y Fy(0)2041 4166 y FB(\()p FA(u)p FB(\)\))p FA(x)2232 4178 y Fx(p)2272 4166 y FB(\()p FA(\034)9 b FB(\))p Fw(k)2423 4178 y Fy(0)p Fx(;\033)2544 4166 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)2805 4132 y Fy(2)2841 4166 y FA(")2880 4132 y Fy(2)p Fx(\021)2977 4166 y FB(=)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)3237 4132 y Fy(2)3274 4166 y FA(")3313 4132 y Fy(\()p Fx(\021)r Fv(\000)p Fx(`)p Fy(\)+\()p Fx(\021)r Fy(+)p Fx(`)p Fy(\))3703 4166 y FA(:)195 4338 y FB(T)-7 b(o)28 b(b)r(ound)g FA(N)641 4350 y Fy(8)705 4338 y FB(in)g(\(247\))o(,)g(w)n(e)f(write)h(it)g(as) 1182 4519 y FA(N)1249 4531 y Fy(8)1309 4519 y FB(=)1410 4498 y Fz(e)1397 4519 y Fw(G)1446 4531 y Fx(")1482 4519 y FA(@)1526 4531 y Fx(v)1579 4452 y Fz(\000)1617 4519 y FA(N)1693 4485 y Fy(0)1684 4540 y(8)1730 4452 y Fz(\001)1787 4519 y FB(+)18 b FA(y)1911 4531 y Fx(p)1949 4519 y FB(\()p FA(\034)9 b FB(\))p FA(p)2100 4531 y Fy(0)2138 4519 y FB(\()p FA(u)p FB(\))19 b Fw(\000)f FA(x)2399 4531 y Fx(p)2438 4519 y FB(\()p FA(\034)9 b FB(\))17 b(_)-40 b FA(p)2589 4531 y Fy(0)2627 4519 y FB(\()p FA(u)p FB(\))71 4691 y(with)592 4843 y FA(N)668 4808 y Fy(0)659 4863 y(8)705 4843 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)18 b Fw(\000)g FA(V)1151 4808 y Fv(0)1175 4843 y FB(\()p FA(q)1244 4855 y Fy(0)1281 4843 y FB(\()p FA(u)p FB(\)\))p FA(x)1472 4855 y Fx(p)1511 4843 y FB(\()p FA(\034)9 b FB(\))20 b(+)e FA(V)1790 4808 y Fv(0)1813 4843 y FB(\()p FA(x)1892 4855 y Fx(p)1931 4843 y FB(\()p FA(\034)9 b FB(\)\))p FA(q)2109 4855 y Fy(0)2148 4843 y FB(\()p FA(u)p FB(\))1001 4967 y(+)18 b FA(\026")1173 4933 y Fx(\021)1227 4967 y FB(\()q FA(q)1297 4979 y Fy(0)1334 4967 y FB(\()p FA(u)p FB(\))p FA(@)1490 4979 y Fx(x)1532 4967 y FA(H)1601 4979 y Fy(1)1638 4967 y FB(\()p FA(x)1717 4979 y Fx(p)1756 4967 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(y)1943 4979 y Fx(p)1983 4967 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))20 b(+)e FA(p)2351 4979 y Fy(0)2388 4967 y FB(\()p FA(u)p FB(\))p FA(@)2544 4979 y Fx(y)2584 4967 y FA(H)2653 4979 y Fy(1)2691 4967 y FB(\()p FA(x)2770 4979 y Fx(p)2809 4967 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(y)2996 4979 y Fx(p)3035 4967 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\)\))16 b FA(:)71 5140 y FB(Using)32 b(that)g Fw(\000)p FA(V)625 5110 y Fv(0)648 5140 y FB(\()p FA(q)717 5152 y Fy(0)754 5140 y FB(\()p FA(u)p FB(\)\))f(=)47 b(_)-40 b FA(p)1066 5152 y Fy(0)1103 5140 y FB(\()p FA(u)p FB(\),)49 b(_)-38 b FA(q)1309 5152 y Fy(0)1346 5140 y FB(\()p FA(u)p FB(\))31 b(=)f FA(p)1626 5152 y Fy(0)1663 5140 y FB(\()p FA(u)p FB(\))i(and)g(that)g(the)h(p)r (erio)r(dic)f(orbit)f(satis\014es)h(equations)f(\(31\),)i(one)71 5239 y(has)634 5391 y FA(N)710 5357 y Fy(0)701 5412 y(8)747 5391 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)15 b(_)-39 b FA(p)1067 5403 y Fy(0)1104 5391 y FB(\()p FA(u)p FB(\))p FA(x)1263 5403 y Fx(p)1302 5391 y FB(\()p FA(\034)9 b FB(\))20 b Fw(\000)e FA(")1553 5357 y Fv(\000)p Fy(1)1642 5391 y FA(@)1686 5403 y Fx(\034)1727 5391 y FA(y)1768 5403 y Fx(p)1806 5391 y FB(\()p FA(\034)9 b FB(\))p FA(q)1952 5403 y Fy(0)1991 5391 y FB(\()p FA(u)p FB(\))18 b Fw(\000)h FA(p)2247 5403 y Fy(0)2284 5391 y FB(\()p FA(u)p FB(\))p FA(y)2437 5403 y Fx(p)2475 5391 y FB(\()p FA(\034)9 b FB(\))19 b(+)f FA(")2725 5357 y Fv(\000)p Fy(1)2814 5391 y FA(@)2858 5403 y Fx(\034)2900 5391 y FA(x)2947 5403 y Fx(p)2986 5391 y FB(\()p FA(\034)9 b FB(\))p FA(p)3137 5403 y Fy(0)3175 5391 y FB(\()p FA(u)p FB(\))961 5516 y(=)18 b Fw(\000)g(L)1184 5528 y Fx(")1219 5516 y FB(\()p FA(y)1292 5528 y Fx(p)1331 5516 y FB(\()p FA(\034)9 b FB(\))p FA(q)1477 5528 y Fy(0)1515 5516 y FB(\()p FA(u)p FB(\)\))19 b(+)f Fw(L)1818 5528 y Fx(")1854 5516 y FB(\()p FA(x)1933 5528 y Fx(p)1972 5516 y FB(\()p FA(\034)9 b FB(\))p FA(p)2123 5528 y Fy(0)2161 5516 y FB(\()p FA(u)p FB(\)\))p Black 1919 5753 a(85)p Black eop end %%Page: 86 86 TeXDict begin 86 85 bop Black Black 71 272 a FB(Therefore)26 b FA(N)514 284 y Fy(8)579 272 y FB(can)h(b)r(e)h(written)g(as)1231 449 y FA(N)1298 461 y Fy(8)1358 449 y FB(=)1435 428 y Fz(e)1422 449 y Fw(G)1471 461 y Fx(")1507 449 y FA(@)1551 461 y Fx(v)1591 449 y Fw(L)1648 461 y Fx(")1697 449 y FB(\()p Fw(\000)p FA(y)1835 461 y Fx(p)1873 449 y FB(\()p FA(\034)9 b FB(\))p FA(q)2019 461 y Fy(0)2058 449 y FB(\()p FA(u)p FB(\))18 b(+)g FA(x)2318 461 y Fx(p)2357 449 y FB(\()p FA(\034)9 b FB(\))p FA(p)2508 461 y Fy(0)2546 449 y FB(\()p FA(u)p FB(\)\))1441 574 y(+)18 b FA(y)1565 586 y Fx(p)1603 574 y FB(\()p FA(\034)9 b FB(\))p FA(p)1754 586 y Fy(0)1792 574 y FB(\()p FA(u)p FB(\))18 b Fw(\000)h FA(x)2053 586 y Fx(p)2091 574 y FB(\()p FA(\034)9 b FB(\))18 b(_)-41 b FA(p)2242 586 y Fy(0)2281 574 y FB(\()p FA(u)p FB(\))1358 718 y(=)1435 697 y Fz(e)1422 718 y Fw(G)1471 730 y Fx(")1507 718 y Fw(L)1564 730 y Fx(")1614 718 y FB(\()p Fw(\000)p FA(y)1752 730 y Fx(p)1790 718 y FB(\()p FA(\034)9 b FB(\))p FA(p)1941 730 y Fy(0)1979 718 y FB(\()p FA(u)p FB(\))18 b(+)g FA(x)2239 730 y Fx(p)2278 718 y FB(\()p FA(\034)9 b FB(\))18 b(_)-41 b FA(p)2429 730 y Fy(0)2468 718 y FB(\()p FA(u)p FB(\)\))1441 842 y Fw(\000)18 b FB(\()p Fw(\000)p FA(y)1662 854 y Fx(p)1700 842 y FB(\()p FA(\034)9 b FB(\))p FA(p)1851 854 y Fy(0)1889 842 y FB(\()p FA(u)p FB(\))18 b(+)g FA(x)2149 854 y Fx(p)2188 842 y FB(\()p FA(\034)9 b FB(\))18 b(_)-41 b FA(p)2339 854 y Fy(0)2377 842 y FB(\()p FA(u)p FB(\)\))p FA(:)71 1037 y FB(Then,)25 b(using)e(that)711 1016 y Fz(e)697 1037 y Fw(G)746 1049 y Fx(")806 1037 y FB(satis\014es)g(an)h(analogous)e (prop)r(ert)n(y)g(to)i(the)g(one)g(giv)n(en)f(for)g Fw(G)2706 1049 y Fx(")2766 1037 y FB(in)h(the)h(last)e(item)i(of)f(Lemma)f(7.3:) 1466 1212 y Fw(k)p FA(N)1575 1224 y Fy(8)1611 1212 y Fw(k)1653 1224 y Fy(0)p Fx(;\033)1773 1212 y Fw(\024)g FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2073 1178 y Fx(\021)r Fy(+1)p Fv(\000)p Fy(\()p Fx(r)r Fy(+1\))p Fx(\015)71 1387 y FB(No)n(w,)27 b(c)n(ho)r(osing)f FA(\015)32 b FB(suc)n(h)c(that)1637 1487 y(1)18 b Fw(\000)g FB(\()p FA(r)j FB(+)d(1\))p FA(\015)28 b(>)22 b Fw(\000)p FA(`)71 1632 y FB(that)28 b(is,)1780 1770 y FA(\015)g(<)1951 1714 y(`)18 b FB(+)g(1)p 1948 1751 183 4 v 1948 1827 a FA(r)k FB(+)c(1)71 1945 y(and)27 b(considering)g(all)g(the)h(b)r (ounds)g(of)f FA(N)1375 1957 y Fx(i)1430 1945 y FB(and)h(taking)819 2120 y FA(\027)865 2086 y Fv(\003)926 2120 y FB(=)23 b(min)14 b Fw(f)p FA(\027)1254 2086 y Fv(\003)1249 2141 y Fy(2)1292 2120 y FA(;)g(\027)1375 2086 y Fv(\003)1370 2141 y Fy(1)1414 2120 y FA(;)g FB(1)j Fw(\000)h FB(max)p Fw(f)p FB(0)p FA(;)c(`)j Fw(\000)h FB(2)p FA(r)j FB(+)d(1)p Fw(g)p FA(;)c(r)n(;)g(`;)g(`)i FB(+)i(1)g Fw(\000)g FB(\()p FA(r)k FB(+)c(1\))p FA(\015)5 b Fw(g)12 b FA(;)71 2295 y FB(w)n(e)27 b(obtain)1163 2299 y Fz(\015)1163 2349 y(\015)1163 2399 y(\015)1209 2395 y FA(@)1253 2407 y Fx(v)1307 2374 y Fz(b)1293 2395 y FA(T)1342 2407 y Fy(1)1378 2395 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(@)1714 2407 y Fx(v)1754 2395 y Fw(T)1799 2407 y Fy(0)1836 2395 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))2025 2299 y Fz(\015)2025 2349 y(\015)2025 2399 y(\015)2073 2453 y Fy(0)p Fx(;\033)2193 2395 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2493 2360 y Fx(\021)r Fv(\000)p Fx(`)p Fy(+)p Fx(\027)2697 2335 y Fl(\003)2735 2395 y FA(:)71 2580 y FB(T)-7 b(o)30 b(\014nish)g(the)h(pro)r(of)f(of)g(Prop)r (osition)f(4.10,)g(it)i(is)f(enough)g(to)g(consider)f(the)i(c)n(hange)e (of)h(v)-5 b(ariables)29 b FA(v)i FB(=)c FA(u)20 b FB(+)g FA(h)p FB(\()p FA(u;)14 b(\034)9 b FB(\))71 2680 y(de\014ned)28 b(in)g(Lemma)f(7.6)g(and)g(its)h(b)r(ounds)g(restricted)f(to)g(the)h (inner)f(domains)g(giv)n(en)g(in)h(Corollary)d(7.7.)71 2911 y Fq(8.2)112 b(Case)38 b Fh(`)28 b Ff(\025)g FF(2)p Fh(r)s Fq(:)49 b(pro)s(of)38 b(of)g(Theorem)g(4.16)71 3064 y FB(This)31 b(section)f(is)g(dev)n(oted)g(to)h(obtain)f(go)r(o)r (d)g(appro)n(ximations)f(of)i(the)g(in)n(v)-5 b(arian)n(t)29 b(manifolds)i(in)g(the)g Fs(inner)h(domains)71 3164 y FB(de\014ned)c(in)g(\(30\))f(for)g(the)h(case)f FA(`)22 b Fw(\025)h FB(2)p FA(r)r FB(.)195 3263 y(First)29 b(in)h(Section)f (8.2.1)e(w)n(e)i(de\014ne)g(the)h(Banac)n(h)d(spaces)h(that)i(will)f(b) r(e)g(used)g(in)h(the)f(forthcoming)f(sections)h(and)71 3363 y(w)n(e)e(state)g(some)h(tec)n(hnical)f(lemmas.)36 b(In)28 b(Section)g(8.2.2)e(w)n(e)h(pro)n(v)n(e)f(Theorem)h(4.16.)71 3577 y Fp(8.2.1)94 b(Banac)m(h)33 b(spaces)f(and)g(tec)m(hnical)h (lemmas)71 3731 y FB(W)-7 b(e)28 b(start)g(b)n(y)g(de\014ning)g(some)g (norms.)38 b(Giv)n(en)28 b FA(\027)h Fw(2)c Ft(R)j FB(and)g(an)g (analytic)g(function)h FA(h)24 b FB(:)g Fw(D)2922 3700 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)2920 3751 y(\024;c)3135 3731 y Fw(!)h Ft(C)p FB(,)j(where)g Fw(D)3661 3700 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)3659 3751 y(\024;c)71 3830 y FB(is)f(the)h(domain)g(de\014ned)f(in)h(\(30\),)g(w)n(e)f(consider) 1449 4005 y Fw(k)p FA(h)p Fw(k)1581 4017 y Fx(\027;\024;c)1749 4005 y FB(=)110 b(sup)1837 4092 y Fx(z)r Fv(2D)1970 4063 y Fu(in)o Fm(;)p Fu(+)p Fm(;u)1968 4101 y(\024;c)2151 4005 y Fw(j)p FA(z)2217 3971 y Fx(\027)2257 4005 y FA(h)p FB(\()p FA(z)t FB(\))p Fw(j)14 b FA(:)71 4270 y FB(Then,)24 b(for)f(analytic)g(functions)g FA(h)g FB(:)g Fw(D)1278 4240 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)1276 4291 y(\024;c)1478 4270 y Fw(\002)10 b Ft(T)1608 4282 y Fx(\033)1676 4270 y Fw(!)23 b Ft(C)g FB(whic)n(h)g(are)g(2)p FA(\031)s FB(-p)r(erio)r(dic)f(in)i FA(\034)9 b FB(,)25 b(w)n(e)e(de\014ne)h(the) f(corresp)r(onding)71 4370 y(F)-7 b(ourier)26 b(norm)1413 4469 y Fw(k)p FA(h)p Fw(k)1545 4481 y Fx(\027;\024;c;\033)1773 4469 y FB(=)1860 4390 y Fz(X)1860 4569 y Fx(k)q Fv(2)p Fn(Z)1994 4469 y Fw(k)p FA(h)2084 4435 y Fy([)p Fx(k)q Fy(])2162 4469 y Fw(k)2204 4481 y Fx(\027;\024;c)2349 4469 y FA(e)2388 4435 y Fv(j)p Fx(k)q Fv(j)p Fx(\033)71 4693 y FB(and)h(the)h(function)g(space)882 4868 y Fw(Z)942 4880 y Fx(\027;\024;c;\033)1171 4868 y FB(=)1259 4801 y Fz(\010)1307 4868 y FA(h)23 b FB(:)g Fw(D)1490 4834 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)1488 4889 y(\024;c)1698 4868 y Fw(\002)18 b Ft(T)1836 4880 y Fx(\033)1904 4868 y Fw(!)24 b Ft(C)p FB(;)41 b(analytic)o FA(;)14 b Fw(k)p FA(h)p Fw(k)2592 4880 y Fx(\027;\024;c;\033)2820 4868 y FA(<)23 b Fw(1)2991 4801 y Fz(\011)3661 4868 y FB(\(248\))71 5043 y(whic)n(h)k(can)h(b)r(e)g(c)n(hec)n(k)n(ed)e(that)i(is)f(a)h (Banac)n(h)e(space)h(for)g(an)n(y)g FA(\027)h Fw(2)23 b Ft(R)p FB(.)195 5143 y(If)28 b(there)g(is)f(no)h(danger)e(of)i (confusion)f(ab)r(out)g(the)h(de\014nition)g(domain)f Fw(D)2534 5113 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)2532 5163 y(\024;c)2752 5143 y FB(w)n(e)g(will)h(denote)1154 5328 y Fw(k)18 b(\001)g(k)1297 5340 y Fx(\027;\033)1418 5328 y FB(=)k Fw(k)c(\001)h(k)1649 5340 y Fx(\027;\024;c;\033)1965 5328 y FB(and)110 b Fw(Z)2269 5340 y Fx(\027;\033)2391 5328 y FB(=)22 b Fw(Z)2538 5340 y Fx(\027;\024;c;\033)2745 5328 y FA(:)195 5504 y FB(The)28 b(next)g(lemma)f(giv)n(es)g(some)g (prop)r(erties)f(of)i(these)g(Banac)n(h)e(spaces.)p Black 1919 5753 a(86)p Black eop end %%Page: 87 87 TeXDict begin 87 86 bop Black Black Black 71 272 a Fp(Lemma)31 b(8.1.)p Black 40 w Fs(L)l(et)f(us)f(c)l(onsider)i FA(c;)14 b(\024)22 b(>)h FB(0)p Fs(.)p Black 169 438 a(1.)p Black 42 w(If)31 b FA(\027)407 450 y Fy(1)467 438 y Fw(\024)23 b FA(\027)596 450 y Fy(2)633 438 y Fs(,)30 b Fw(Z)748 450 y Fx(\027)781 458 y Fu(2)814 450 y Fx(;\033)901 438 y Fw(\032)23 b(Z)1049 450 y Fx(\027)1082 458 y Fu(1)1115 450 y Fx(;\033)1179 438 y Fs(.)39 b(Mor)l(e)l(over,)1609 666 y Fw(k)p FA(h)p Fw(k)1741 678 y Fx(\027)1774 686 y Fu(2)1806 678 y Fx(;\033)1893 666 y Fw(\024)2070 610 y FA(K)p 1991 647 236 4 v 1991 723 a(\024)2039 699 y Fx(\027)2072 707 y Fu(2)2104 699 y Fv(\000)p Fx(\027)2189 707 y Fu(1)2236 666 y Fw(k)p FA(h)p Fw(k)2368 678 y Fx(\027)2401 686 y Fu(1)2432 678 y Fx(;\033)2497 666 y FA(:)p Black 169 905 a Fs(2.)p Black 42 w(If)31 b FA(h)22 b Fw(2)i(Z)575 917 y Fx(\027)608 925 y Fu(1)641 917 y Fx(;\033)735 905 y Fs(and)30 b FA(g)c Fw(2)d(Z)1100 917 y Fx(\027)1133 925 y Fu(2)1166 917 y Fx(;\033)1230 905 y Fs(,)31 b(the)f(pr)l(o)l (duct)f FA(hg)d Fw(2)d(Z)1969 917 y Fx(\027)2002 925 y Fu(1)2035 917 y Fy(+)p Fx(\027)2119 925 y Fu(2)2152 917 y Fx(;\033)2246 905 y Fs(and)1529 1088 y Fw(k)p FA(hg)s Fw(k)1704 1100 y Fx(\027)1737 1108 y Fu(1)1768 1100 y Fy(+)p Fx(\027)1852 1108 y Fu(2)1885 1100 y Fx(;\033)1972 1088 y Fw(\024)g(k)p FA(h)p Fw(k)2192 1100 y Fx(\027)2225 1108 y Fu(1)2257 1100 y Fx(;\033)2321 1088 y Fw(k)p FA(g)s Fw(k)2448 1100 y Fx(\027)2481 1108 y Fu(2)2512 1100 y Fx(;\033)2577 1088 y FA(:)p Black 169 1304 a Fs(3.)p Black 42 w(L)l(et)30 b(us)f(c)l(onsider)i FA(h)23 b Fw(2)g(Z)1067 1316 y Fx(\027;\024;c;\033)1303 1304 y Fs(and)30 b Fz(b)-46 b FA(c)23 b(<)g(c)p Fs(,)30 b(then,)g FA(@)1956 1316 y Fx(x)1998 1304 y FA(h)23 b Fw(2)g(X)2206 1317 y Fx(\027;)p Fy(2)p Fx(\024;)p Fk(b)-38 b Fx(c;\033)2475 1304 y Fs(and)1553 1533 y Fw(k)p FA(@)1639 1545 y Fx(x)1680 1533 y FA(h)p Fw(k)1770 1546 y Fx(\027;)p Fy(2)p Fx(\024;)p Fk(b)g Fx(c)o(;\033)2032 1533 y Fw(\024)2129 1476 y FA(K)p 2129 1514 77 4 v 2144 1590 a(\024)2216 1533 y Fw(k)p FA(h)p Fw(k)2348 1545 y Fx(\027;\024;c;\033)2553 1533 y FA(:)195 1772 y FB(Throughout)27 b(this)h(section)f(w)n(e)g(are)g(going)f(to)i (solv)n(e)e(equations)h(of)g(the)h(form)g Fw(L)p FA(h)23 b FB(=)f FA(g)31 b FB(and)c Fw(L)p FA(h)c FB(=)g FA(@)3396 1784 y Fx(z)3434 1772 y FA(g)s FB(,)28 b(where)1730 1954 y Fw(L)c FB(=)e FA(@)1942 1966 y Fx(z)1999 1954 y FB(+)c FA(@)2126 1966 y Fx(\034)2168 1954 y FA(:)1470 b FB(\(249\))71 2137 y(T)-7 b(o)27 b(solv)n(e)f(these)h(equations)g(w)n(e)g(consider)f (op)r(erators)f Fw(G)33 b FB(and)p 2035 2070 55 4 v 27 w Fw(G)5 b FB(,)28 b(whic)n(h)f(are)f(de\014ned)i(\\acting)e(on)h(the)h (F)-7 b(ourier)26 b(co)r(e\016-)71 2237 y(cien)n(ts".)p Black Black Black 1234 3754 a /PSfrag where{pop(D)[[0(Bl)1 0]](I)[[1(Bl)1 0]](z1)[[2(Bl)1 0]](z2)[[3(Bl)1 0]](b0)[[4(Bl)1 0]](b1)[[5(Bl)1 0]](b2)[[6(Bl)1 0]](a)[[7(Bl)1 0]](a1)[[8(Bl)1 0]](a2)[[9(Bl)1 0]]10 0 -1/Begin PSfrag}{userdict /PSfrag{pop}put}ifelse 1234 3754 a @beginspecial 62 @llx 599 @lly 305 @urx 836 @ury 1700 @rhi @setspecial %%BeginDocument: InnerVarInner.eps %!PS-Adobe-3.0 EPSF-3.0 %%Creator: 0.46 %%Pages: 1 %%Orientation: Portrait %%BoundingBox: 62 599 305 836 %%HiResBoundingBox: 62.910712 599.6 304.4 835.45714 %%EndComments %%Page: 1 1 0 842 translate 0.8 -0.8 scale 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap gsave [1 0 0 1 0 0] concat 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 230 22.362183 moveto 230 302.36218 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 2 setlinewidth 0 setlinejoin 0 setlinecap newpath 90 42.362183 moveto 230 62.362183 lineto 280 232.36218 lineto 90 42.362183 lineto closepath stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 100 22.362183 moveto 380 302.36218 lineto stroke gsave [1 0 0 -1 77.85714 36.790752] concat gsave /newlatin1font {findfont dup length dict copy dup /Encoding ISOLatin1Encoding put definefont} def /Sans-ISOLatin1 /Sans newlatin1font 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (z2) show grestore grestore gsave [1 0 0 -1 256.42856 243.79076] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (z1) show grestore grestore 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 300 302.36218 moveto 301.07143 302.36218 lineto 301.07143 302.36218 moveto 260 302.36218 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 229.8097 61.907611 moveto 300.52038 302.32392 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 293.19678 277.82772 moveto 287.54952 282.54629 279.94283 286.25718 277.79195 301.81884 curveto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 361.88715 284.39371 moveto 357.08237 288.22866 354.24248 293.78289 354.81608 302.32392 curveto stroke gsave [1 0 0 -1 260 295.36218] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (b2) show grestore grestore gsave [1 0 0 -1 332 295.36218] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (b1) show grestore grestore 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 86.893424 41.70456 moveto 375 82.362183 lineto 325 82.362183 lineto stroke 0 0 0 setrgbcolor [3 1] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 380 302.36218 moveto 340 302.36218 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 329.28571 75.93361 moveto 327.16764 78.703127 327.90863 80.519622 328.57143 82.362181 curveto stroke gsave [1 0 0 -1 303 86.362183] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (b0) show grestore grestore gsave [1 0 0 -1 172 90.219322] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (D) show grestore grestore 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 120 99.505038 moveto 160 94.147895 lineto 146.42857 92.362181 lineto stroke 0 0 0 setrgbcolor [] 0 setdash 1 setlinewidth 0 setlinejoin 0 setlinecap newpath 159.28571 94.147895 moveto 149.64286 98.076467 lineto stroke gsave [1 0 0 -1 100.85714 118.57647] concat gsave /Sans-ISOLatin1 findfont 40 scalefont setfont 0 0 0 setrgbcolor newpath 0 0 moveto (I) show grestore grestore grestore showpage %%EOF %%EndDocument @endspecial 1234 3754 a /End PSfrag 1234 3754 a 1234 2638 a /Hide PSfrag 1234 2638 a 494 2695 a FB(PSfrag)g(replacemen)n(ts)p 494 2725 741 4 v 1234 2728 a /Unhide PSfrag 1234 2728 a 978 2806 a { 978 2806 a Black Fw(D)1044 2775 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)1042 2826 y(\024;c)p Black 978 2806 a } 0/Place PSfrag 978 2806 a 1069 2922 a { 1069 2922 a Black Fw(I)1120 2882 y Fy(+)p Fx(;u)1114 2945 y(c;)p 1164 2912 30 3 v(c)p Black 1069 2922 a } 1/Place PSfrag 1069 2922 a 1158 3044 a { 1158 3044 a Black FA(z)1197 3056 y Fy(1)p Black 1158 3044 a } 2/Place PSfrag 1158 3044 a 1158 3144 a { 1158 3144 a Black FA(z)1197 3156 y Fy(2)p Black 1158 3144 a } 3/Place PSfrag 1158 3144 a 1150 3239 a { 1150 3239 a Black FA(\014)1197 3251 y Fy(0)p Black 1150 3239 a } 4/Place PSfrag 1150 3239 a 1150 3339 a { 1150 3339 a Black FA(\014)1197 3351 y Fy(1)p Black 1150 3339 a } 5/Place PSfrag 1150 3339 a 1150 3439 a { 1150 3439 a Black FA(\014)1197 3451 y Fy(2)p Black 1150 3439 a } 6/Place PSfrag 1150 3439 a 1162 3554 a { 1162 3554 a Black FA(ia)p Black 1162 3554 a } 7/Place PSfrag 1162 3554 a 1097 3647 a { 1097 3647 a Black Fw(\000)p FA(ia)p Black 1097 3647 a } 8/Place PSfrag 1097 3647 a 909 3733 a { 909 3733 a Black FA(i)p FB(\()p FA(a)18 b Fw(\000)g FA(\024")p FB(\))p Black 909 3733 a } 9/Place PSfrag 909 3733 a Black 107 4019 a FB(Figure)27 b(11:)36 b(The)28 b Fs(inner)i(domain)35 b Fw(D)1272 3989 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)1270 4040 y(\024;c)1489 4019 y FB(de\014ned)28 b(in)g(\(64\))f(and)h(the)g (transition)f(domain)g Fw(I)3077 3979 y Fy(+)p Fx(;u)3071 4043 y(c;)p 3121 4010 30 3 v(c)3219 4019 y FB(de\014ned)h(in)g(\(253\)) o(.)p Black Black 195 4254 a(Let)h(us)g(consider)f FA(z)817 4266 y Fy(1)883 4254 y FB(and)g FA(z)1084 4266 y Fy(2)1150 4254 y FB(the)h(v)n(ertices)f(of)g(the)i(inner)e(domain)h Fw(D)2412 4224 y Fy(in)o Fx(;)p Fy(+)p Fx(;u)2410 4274 y(\024;c)2630 4254 y FB(\(see)g(Figure)f(11\).)40 b(As)29 b(w)n(e)f(ha)n(v)n(e)g(done)71 4353 y(in)f(Section)f(7.2.2)g(to)g(in)n (v)n(ert)g(the)h(op)r(erator)e Fw(L)1524 4365 y Fx(")1583 4353 y FB(=)d FA(")1709 4323 y Fv(\000)p Fy(1)1798 4353 y FA(@)1842 4365 y Fx(\034)1900 4353 y FB(+)16 b FA(@)2025 4365 y Fx(v)2065 4353 y FB(,)27 b(w)n(e)f(in)n(v)n(ert)g Fw(L)h FB(in)n(tegrating)e(from)h FA(z)3212 4365 y Fy(1)3276 4353 y FB(or)g FA(z)3416 4365 y Fy(2)3479 4353 y FB(dep)r(ending)71 4453 y(on)h(the)h(harmonic.)195 4553 y(W)-7 b(e)28 b(de\014ne)g(the)g (op)r(erators)1403 4664 y Fw(G)5 b FB(\()p FA(h)p FB(\)\()p FA(z)t(;)14 b(\034)9 b FB(\))24 b(=)1869 4585 y Fz(X)1869 4764 y Fx(k)q Fv(2)p Fn(Z)2003 4664 y Fw(G)5 b FB(\()p FA(h)p FB(\))2169 4630 y Fy([)p Fx(k)q Fy(])2249 4664 y FB(\()p FA(z)t FB(\))p FA(e)2395 4630 y Fx(ik)q(\034)2496 4664 y FA(;)1142 b FB(\(250\))71 4893 y(where)27 b(the)h(F)-7 b(ourier)27 b(co)r(e\016cien)n(ts)g(are)f(giv)n(en)h(b)n(y)797 5117 y Fw(G)5 b FB(\()p FA(h)p FB(\))963 5083 y Fy([)p Fx(k)q Fy(])1042 5117 y FB(\()p FA(z)t FB(\))23 b(=)1260 5004 y Fz(Z)1343 5025 y Fx(z)1306 5193 y(z)1338 5201 y Fu(1)1395 5117 y FA(e)1434 5083 y Fv(\000)p Fx(ik)q Fy(\()p Fx(z)r Fv(\000)p Fx(s)p Fy(\))1719 5117 y FA(h)1767 5083 y Fy([)p Fx(k)q Fy(])1845 5117 y FB(\()p FA(s)p FB(\))14 b FA(ds)755 b FB(for)27 b FA(k)f(<)d FB(0)797 5352 y Fw(G)5 b FB(\()p FA(h)p FB(\))963 5318 y Fy([)p Fx(k)q Fy(])1042 5352 y FB(\()p FA(z)t FB(\))23 b(=)1260 5239 y Fz(Z)1343 5260 y Fx(z)1306 5428 y(z)1338 5436 y Fu(2)1395 5352 y FA(e)1434 5318 y Fv(\000)p Fx(ik)q Fy(\()p Fx(z)r Fv(\000)p Fx(s)p Fy(\))1719 5352 y FA(h)1767 5318 y Fy([)p Fx(k)q Fy(])1845 5352 y FB(\()p FA(s)p FB(\))14 b FA(ds)755 b FB(for)27 b FA(k)f Fw(\025)d FB(0)p Black 1919 5753 a(87)p Black eop end %%Page: 88 88 TeXDict begin 88 87 bop Black Black 71 272 a FB(and)p 1403 305 55 4 v 1403 372 a Fw(G)5 b FB(\()p FA(h)p FB(\)\()p FA(z)t(;)14 b(\034)9 b FB(\))24 b(=)1869 293 y Fz(X)1869 472 y Fx(k)q Fv(2)p Fn(Z)p 2003 305 V 2003 372 a Fw(G)6 b FB(\()p FA(h)p FB(\))2170 337 y Fy([)p Fx(k)q Fy(])2249 372 y FB(\()p FA(z)t FB(\))p FA(e)2395 337 y Fx(ik)q(\034)2496 372 y FA(;)1142 b FB(\(251\))71 600 y(where)27 b(its)h(F)-7 b(ourier)26 b(co)r(e\016cien)n(ts)h(are)g(giv)n(en)g(b)n(y)p 407 758 V 407 825 a Fw(G)5 b FB(\()p FA(h)p FB(\))573 791 y Fy([)p Fx(k)q Fy(])652 825 y FB(\()p FA(z)t FB(\))23 b(=)g FA(h)918 791 y Fy([)p Fx(k)q Fy(])996 825 y FB(\()p FA(z)t FB(\))18 b Fw(\000)g FA(e)1243 791 y Fv(\000)p Fx(ik)q Fy(\()p Fx(z)r Fv(\000)p Fx(z)1498 799 y Fu(1)1531 791 y Fy(\))1561 825 y FA(h)1609 791 y Fy([)p Fx(k)q Fy(])1687 825 y FB(\()p FA(z)1758 837 y Fy(1)1795 825 y FB(\))h Fw(\000)f FA(ik)2018 712 y Fz(Z)2101 733 y Fx(z)2064 901 y(z)2096 909 y Fu(1)2153 825 y FA(e)2192 791 y Fv(\000)p Fx(ik)q Fy(\()p Fx(z)r Fv(\000)p Fx(s)p Fy(\))2477 825 y FA(h)2525 791 y Fy([)p Fx(k)q Fy(])2603 825 y FB(\()p FA(s)p FB(\))c FA(ds)387 b FB(for)27 b FA(k)f(<)d FB(0)p 411 953 V 411 1020 a Fw(G)5 b FB(\()p FA(h)p FB(\))577 986 y Fy([0])652 1020 y FB(\()p FA(z)t FB(\))23 b(=)g FA(h)918 986 y Fy([0])992 1020 y FB(\()p FA(z)t FB(\))c Fw(\000)f FA(h)1249 986 y Fy([0])1323 1020 y FB(\()p FA(z)1394 1032 y Fy(2)1431 1020 y FB(\))p 407 1134 V 407 1201 a Fw(G)5 b FB(\()p FA(h)p FB(\))573 1166 y Fy([)p Fx(k)q Fy(])652 1201 y FB(\()p FA(z)t FB(\))23 b(=)g FA(h)918 1166 y Fy([)p Fx(k)q Fy(])996 1201 y FB(\()p FA(z)t FB(\))18 b Fw(\000)g FA(e)1243 1166 y Fv(\000)p Fx(ik)q Fy(\()p Fx(z)r Fv(\000)p Fx(z)1498 1174 y Fu(2)1531 1166 y Fy(\))1561 1201 y FA(h)1609 1166 y Fy([)p Fx(k)q Fy(])1687 1201 y FB(\()p FA(z)1758 1213 y Fy(2)1795 1201 y FB(\))h Fw(\000)f FA(ik)2018 1088 y Fz(Z)2101 1108 y Fx(z)2064 1276 y(z)2096 1284 y Fu(2)2153 1201 y FA(e)2192 1166 y Fv(\000)p Fx(ik)q Fy(\()p Fx(z)r Fv(\000)p Fx(s)p Fy(\))2477 1201 y FA(h)2525 1166 y Fy([)p Fx(k)q Fy(])2603 1201 y FB(\()p FA(s)p FB(\))c FA(ds)364 b FB(for)27 b FA(k)f(>)d FB(0)p FA(:)71 1434 y FB(The)32 b(next)g(lemma)f(giv)n(es)g (some)g(prop)r(erties)g(of)h(these)f(op)r(erators.)48 b(Its)32 b(pro)r(of)f(is)g(analogous)f(to)i(the)g(one)f(of)h(Lemma)71 1533 y(5.5)27 b(in)g([GOS10)o(].)p Black 71 1699 a Fp(Lemma)k(8.2.)p Black 40 w Fs(L)l(et)f(us)f(c)l(onsider)i FA(\024;)14 b(c;)g(\027)28 b(>)22 b FB(0)30 b Fs(and)g FA(\015)d Fw(2)d FB(\(0)p FA(;)14 b FB(1\))p Fs(.)38 b(Then,)p Black 169 1865 a(1.)p Black 42 w(The)31 b(op)l(er)l(ator)g Fw(G)d FB(:)23 b Fw(Z)958 1877 y Fx(\027)t Fy(+1)p Fx(;\033)1167 1865 y Fw(!)g(Z)1333 1877 y Fx(\027;\033)1461 1865 y Fs(is)30 b(wel)t(l)h(de\014ne)l(d.)39 b(Mor)l(e)l(over,)32 b(if)e FA(h)23 b Fw(2)h(Z)2715 1877 y Fx(\027)t Fy(+1)p Fx(;\033)2900 1865 y Fs(,)1627 2048 y Fw(kG)5 b FB(\()p FA(h)p FB(\))p Fw(k)1877 2073 y Fx(\027;\033)1998 2048 y Fw(\024)23 b FA(K)6 b Fw(k)p FA(h)p Fw(k)2295 2060 y Fx(\027)t Fy(+1)p Fx(;\033)2479 2048 y FA(:)p Black 169 2264 a Fs(2.)p Black 42 w(The)31 b(op)l(er)l(ator)g Fw(G)d FB(:)23 b Fw(Z)958 2276 y Fx(\027;\033)1080 2264 y Fw(!)g(Z)1246 2276 y Fx(\027;\033)1374 2264 y Fs(is)30 b(wel)t(l)g(de\014ne)l(d.)39 b(Mor)l(e)l(over,)32 b(if)f FA(h)23 b Fw(2)g(Z)2627 2276 y Fx(\027;\033)2725 2264 y Fs(,)1588 2446 y Fw(k)o(G)5 b FB(\()p FA(h)p FB(\))p Fw(k)1838 2471 y Fx(\027;\033)1959 2446 y Fw(\024)22 b FA(K)6 b(")2162 2412 y Fx(\015)t Fv(\000)p Fy(1)2289 2446 y Fw(k)p FA(h)p Fw(k)2421 2458 y Fx(\027;\033)2518 2446 y FA(:)p Black 169 2680 a Fs(3.)p Black 42 w(The)31 b(op)l(er)l(ator)p 775 2614 V 31 w Fw(G)d FB(:)23 b Fw(Z)958 2692 y Fx(\027;\033)1080 2680 y Fw(!)g(Z)1246 2692 y Fx(\027;\033)1374 2680 y Fs(is)30 b(wel)t(l)g(de\014ne)l(d.)39 b(Mor)l(e)l(over,)32 b(if)f FA(h)23 b Fw(2)g(Z)2627 2692 y Fx(\027;\033)2725 2680 y Fs(,)1666 2793 y Fz(\015)1666 2842 y(\015)p 1713 2796 V 1713 2863 a Fw(G)5 b FB(\()p FA(h)p FB(\))1879 2793 y Fz(\015)1879 2842 y(\015)1925 2896 y Fx(\027;\033)2046 2863 y Fw(\024)23 b FA(K)6 b Fw(k)p FA(h)p Fw(k)2343 2875 y Fx(\027;\033)2439 2863 y FA(:)71 3090 y Fp(8.2.2)94 b(Pro)s(of)31 b(of)h(Theorem)f(4.16)71 3243 y FB(W)-7 b(e)28 b(rewrite)f(Theorem)f(4.16)h(in)h(terms)f(of)g (the)h(Banac)n(h)f(space)g(\(248\))o(.)p Black 71 3409 a Fp(Prop)s(osition)j(8.3.)p Black 40 w Fs(L)l(et)f(us)h(c)l(onsider)g FA(\015)e Fw(2)23 b FB(\(0)p FA(;)14 b(\015)1659 3421 y Fy(2)1696 3409 y FB(\))p Fs(,)31 b(wher)l(e)1534 3640 y FA(\015)1577 3652 y Fy(2)1637 3640 y FB(=)1806 3583 y FA(\014)t FB(\()p FA(`)19 b Fw(\000)f FB(2)p FA(r)j FB(+)d(1\))p 1735 3621 620 4 v 1735 3697 a FA(\014)t FB(\()p FA(`)h Fw(\000)f FB(2)p FA(r)i FB(+)e(1\))h(+)f(1)2364 3640 y FA(;)1274 b FB(\(252\))71 3871 y FA(c)107 3883 y Fy(1)176 3871 y FA(>)31 b FB(0)p Fs(,)36 b FA(")414 3883 y Fy(0)483 3871 y FA(>)31 b FB(0)j Fs(smal)t(l)i(enough)f(and)g FA(\024)1382 3883 y Fy(6)1451 3871 y FA(>)c FB(max)o Fw(f)p FA(\024)1791 3883 y Fy(3)1828 3871 y FA(;)14 b(\024)1913 3883 y Fy(5)1950 3871 y Fw(g)34 b Fs(big)h(enough,)i(wher)l(e)e FA(\024)2763 3883 y Fy(5)2835 3871 y Fs(ar)l(e)g(the)g(c)l(onstants)e (de\014ne)l(d)i(in)71 3971 y(The)l(or)l(ems)c(4.8)g(and)f(4.12)h(r)l (esp)l(e)l(ctively.)40 b(L)l(et,)1715 4070 y FA(')24 b FB(=)e FA( )1937 4036 y Fx(u)1999 4070 y Fw(\000)c FA( )2139 4036 y Fx(u)2136 4091 y Fy(0)2183 4070 y FA(;)71 4220 y Fs(wher)l(e)31 b FA( )363 4190 y Fx(u)437 4220 y Fs(is)g(the)g(function)f(in)37 b FB(\(61\))p Fs(and)31 b FA( )1466 4232 y Fy(0)1534 4220 y Fs(is)g(the)f(function)h(obtaine)l (d)h(in)e(The)l(or)l(em)i(4.12.)43 b(Then,)31 b(for)h FA(")24 b Fw(2)h FB(\(0)p FA(;)14 b(")3756 4232 y Fy(0)3793 4220 y FB(\))p Fs(,)71 4319 y(we)30 b(have)h FA(')23 b Fw(2)h(Z)601 4340 y Fy(2)p Fx(r)r Fv(\000)732 4317 y Fu(1)p 728 4326 35 3 v 728 4360 a Fm(\014)773 4340 y Fx(;\024)832 4348 y Fu(6)864 4340 y Fx(;c)914 4348 y Fu(1)946 4340 y Fx(;\033)1040 4319 y Fs(and)30 b(ther)l(e)g(exists)g (a)g(c)l(onstant)f FA(b)2075 4331 y Fy(10)2168 4319 y FA(>)22 b FB(0)30 b Fs(such)f(that)1477 4545 y Fw(k)p FA(@)1563 4557 y Fx(z)1601 4545 y FA(')p Fw(k)1697 4570 y Fy(2)p Fx(r)r Fv(\000)1828 4548 y Fu(1)p 1824 4557 V 1824 4590 a Fm(\014)1869 4570 y Fx(;\024)1928 4578 y Fu(6)1960 4570 y Fx(;c)2010 4578 y Fu(1)2042 4570 y Fx(;\033)2130 4545 y Fw(\024)22 b FA(b)2253 4557 y Fy(10)2323 4545 y FA(")2375 4486 y Fu(1)p 2372 4495 V 2372 4528 a Fm(\014)2421 4545 y FA(;)71 4752 y Fs(wher)l(e)30 b FA(r)c FB(=)d FA(\013=\014)34 b Fs(has)c(b)l(e)l(en)g(de\014ne)l(d)g (in)36 b FB(\(12\))p Fs(.)p Black 71 4918 a Fp(Remark)i(8.4.)p Black 43 w Fs(We)c(emphasize)j(that)d(Pr)l(op)l(osition)i(8.3)g (implies)g(str)l(aightforwar)l(d)t(ly)h(The)l(or)l(em)f(4.16.)55 b(Inde)l(e)l(d,)36 b(we)71 5017 y(observe)27 b(that)f(the)g(only)g(r)l (estriction)g(is)g(ab)l(out)g(the)g(r)l(ange)g(of)h(values)f(of)h FA(\015)h Fw(2)23 b FB(\(0)p FA(;)14 b(\015)2627 5029 y Fy(2)2664 5017 y FB(\))p Fs(.)38 b(L)l(et)25 b(us)g(denote)i(by)f FA(D)3439 4987 y Fy(in)3437 5038 y Fx(\015)3524 5017 y Fs(the)g(inner)71 5127 y(domain)j(de\014ne)l(d)f(by)g FA(\015)5 b Fs(.)38 b(It)27 b(is)h(cle)l(ar)h(that,)f(if)h FA(\015)f Fw(\025)22 b FA(\015)1707 5139 y Fy(2)1768 5127 y FA(>)g(\015)1898 5139 y Fy(1)1935 5127 y Fs(,)29 b(then)f FA(D)2243 5097 y Fy(in)2241 5147 y Fx(\015)2325 5127 y Fw(\032)23 b FA(D)2484 5097 y Fy(in)2482 5147 y Fx(\015)2517 5155 y Fu(1)2581 5127 y Fs(and)28 b(henc)l(eforth)h(the) f(r)l(esult)f(holds)i(also)71 5226 y(for)h(values)h(of)f FA(\015)e Fw(\025)23 b FA(\015)751 5238 y Fy(2)788 5226 y Fs(.)195 5326 y(We)30 b(ne)l(e)l(d)g(to)g(imp)l(ose)g(this)h(c)l (ondition)f(ab)l(out)g FA(\015)k Fs(just)c(for)g(te)l(chnic)l(al)h(r)l (e)l(asons.)p Black 1919 5753 a FB(88)p Black eop end %%Page: 89 89 TeXDict begin 89 88 bop Black Black 195 272 a FB(In)31 b(the)f(pro)r(of)g(of)g(this)g(prop)r(osition)g(w)n(e)f(will)i(refer)e (sev)n(eral)g(times)h(to)g(the)h(b)r(ounds)f(giv)n(en)f(in)i(Theorem)e (4.12.)43 b(In)71 372 y(fact,)26 b(w)n(e)g(need)g(these)g(b)r(ounds)g (expressed)f(in)i(terms)e(of)h(the)h(F)-7 b(ourier)25 b(norm,)g(whic)n(h)h(are)f(giv)n(en)h(in)g(Prop)r(osition)e(4.8)i(of)71 471 y([Bal06)n(],)i(instead)g(of)f(the)h(ones)f(giv)n(en)g(in)h(this)g (theorem,)f(whic)n(h)g(use)h(the)g(classical)e(supremm)n(um)h(norm.)195 571 y(Let)34 b(us)f(p)r(oin)n(t)g(out)h(that)f(using)g(the)h(b)r(ounds) f(of)g(Prop)r(osition)f(4.8)g(of)h([Bal06)o(])g(and)g(Corollary)e(7.22) h(leads)h(to)g(a)71 671 y(b)r(ound)k(of)f FA(@)483 683 y Fx(z)522 671 y FA(')h FB(of)f(order)g(1)g(with)h(resp)r(ect)f(to)h FA(")p FB(.)64 b(Nev)n(ertheless,)37 b(this)g(b)r(ound)g(is)g(to)r(o)f (rough)g(to)g(pro)n(v)n(e)f(later)h(the)71 770 y(asymptotic)c(form)n (ula)g(for)h(the)g(splitting)h(of)f(separatrices)e(and)h(therefore)h(w) n(e)f(will)h(need)h(the)f(impro)n(v)n(ed)f(estimates)71 870 y(giv)n(en)27 b(in)h(Prop)r(osition)d(8.3.)195 969 y(The)31 b(pro)r(of)g(of)g(Prop)r(osition)e(8.3)i(go)r(es)f(as)g(follo) n(ws.)46 b(First)31 b(in)g(Section)g(8.2.2)f(w)n(e)h(obtain)g(a)f (\(non-homogeneous\))71 1069 y(linear)24 b(partial)f(di\013eren)n(tial) h(equation)g(satis\014ed)g(b)n(y)g FA(')g FB(=)e FA( )16 b Fw(\000)c FA( )2105 1081 y Fy(0)2142 1069 y FB(.)36 b(Then,)25 b(in)g(Section)f(8.2.2,)g(w)n(e)g(obtain)g(quan)n(titativ)n (e)71 1169 y(estimates)j(of)h FA(@)575 1181 y Fx(z)613 1169 y FA(')g FB(in)g(the)g Fs(tr)l(ansition)i(domain)35 b Fw(I)1666 1129 y Fy(+)p Fx(;u)1660 1193 y(c;)p 1710 1159 30 3 v(c)1808 1169 y FB(de\014ned)28 b(as)1144 1383 y Fw(I)1195 1344 y Fv(\006)p Fx(;u)1189 1407 y(c;)p 1239 1374 V(c)1333 1383 y FB(=)1421 1291 y Fz(n)1476 1383 y FA(z)f Fw(2)c Ft(C)p FB(;)14 b FA(ia)k FB(+)g FA("z)26 b Fw(2)d FA(D)2144 1344 y Fy(out)p Fx(;u)2142 1407 y(\032)2176 1415 y Fu(2)2209 1407 y Fx(;)p 2229 1374 V(c)o(")2289 1391 y Fm(\015)2350 1383 y Fw(\\)c FA(D)2495 1349 y Fy(in)o Fx(;)p Fv(\006)p Fx(;u)2493 1404 y(\024;c)2685 1291 y Fz(o)2754 1383 y FA(;)884 b FB(\(253\))71 1584 y(where)25 b Fw(\003)d FB(=)h FA(u;)14 b(s)25 b FB(\(see)h(Figure)f(11\),)g(whic)n (h)h(allo)n(w)e(us)h(to)h(obtain)f(an)g(in)n(tegral)g(equation)g (satis\014ed)g(b)n(y)g FA(@)3332 1596 y Fx(z)3371 1584 y FA(')p FB(.)36 b(Finally)-7 b(,)26 b(in)71 1683 y(Sections)j(8.2.2)g (and)h(8.2.2)e(w)n(e)i(obtain)g(the)g(impro)n(v)n(ed)e(b)r(ound)j(for)e FA(@)2288 1695 y Fx(z)2326 1683 y FA(')i FB(for)e(the)h(cases)f FA(`)20 b Fw(\000)f FB(2)p FA(r)29 b(>)e FB(0)i(and)h FA(`)20 b Fw(\000)f FB(2)p FA(r)29 b FB(=)e(0)71 1783 y(resp)r(ectiv)n(ely)-7 b(,)27 b(pro)n(ving)f(Prop)r(osition)g(8.3.)71 1998 y Fp(The)32 b(Hamilton-Jacobi)f(equation)83 b FB(First)27 b(w)n(e)g(lo)r(ok)g(for)g(the)h(equation)f(satis\014ed)h(b)n(y)1742 2173 y FA(')23 b FB(=)g FA( )e Fw(\000)d FA( )2119 2185 y Fy(0)2157 2173 y FA(:)1481 b FB(\(254\))71 2348 y(Subtracting)27 b(the)h(Hamilton-Jacobi)e(equations)h(\(62\))g(and)h(\(65\))o(,)g(one)f (obtains)1093 2524 y FA(@)1137 2536 y Fx(\034)1179 2524 y FA(')19 b FB(+)f Fw(H)q FB(\()p FA(@)1482 2536 y Fx(z)1520 2524 y FA( )1574 2536 y Fy(0)1630 2524 y FB(+)g FA(@)1757 2536 y Fx(z)1795 2524 y FA(';)c(z)t(;)g(\034)9 b FB(\))19 b Fw(\000)f(H)2215 2536 y Fy(0)2253 2524 y FB(\()p FA(@)2329 2536 y Fx(z)2367 2524 y FA( )2421 2536 y Fy(0)2459 2524 y FA(;)c(z)t(;)g(\034)9 b FB(\))23 b(=)f(0)p FA(:)71 2699 y FB(T)-7 b(aking)27 b(in)n(to)g(accoun)n(t)g(that)h(w)n(e)f (already)f(kno)n(w)h(the)h(existence)f(of)h FA(')p FB(,)g(w)n(e)f(kno)n (w)g(that)g(it)h(is)g(also)e(solution)i(of)1612 2875 y Fw(L)p FA(')c FB(=)e Fw(W)7 b FB(\()p FA(@)1999 2887 y Fx(z)2038 2875 y FA(';)14 b(z)t(;)g(\034)9 b FB(\))p FA(;)1352 b FB(\(255\))71 3050 y(where)27 b Fw(L)h FB(is)f(the)h(op)r (erator)e(de\014ned)i(in)g(\(249\))f(and)998 3275 y Fw(W)7 b FB(\()p FA(w)r(;)14 b(z)t(;)g(\034)9 b FB(\))24 b(=)e Fw(\000)p FA(L)p FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))18 b Fw(\000)1897 3158 y Fz(\022)1958 3275 y FA(Q)2024 3287 y Fy(1)2061 3275 y FB(\()p FA(\034)9 b FB(\))2259 3219 y(^)-49 b FA(\026)p 2180 3256 192 4 v 2180 3332 a(z)2223 3308 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)2402 3275 y FB(+)18 b FA(M)9 b FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))2764 3158 y Fz(\023)2839 3275 y FA(w)r(;)738 b FB(\(256\))71 3501 y(where)27 b FA(Q)377 3513 y Fy(1)441 3501 y FB(is)h(the)g (function)g(de\014ned)g(in)g(\(73\))f(and)718 3676 y FA(L)p FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))23 b(=)g Fw(H)q FB(\()p FA(@)1222 3688 y Fx(z)1261 3676 y FA( )1315 3688 y Fy(0)1352 3676 y FA(;)14 b(z)t(;)g(\034)9 b FB(\))18 b Fw(\000)g(H)1717 3688 y Fy(0)1755 3676 y FB(\()p FA(@)1831 3688 y Fx(z)1869 3676 y FA( )1923 3688 y Fy(0)1961 3676 y FA(;)c(z)t(;)g(\034)9 b FB(\))1506 b(\(257\))685 3869 y FA(M)9 b FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))23 b(=)1075 3756 y Fz(Z)1158 3776 y Fy(1)1121 3945 y(0)1209 3869 y FA(@)1253 3881 y Fx(w)1307 3869 y Fw(H)15 b FB(\()p FA(@)1468 3881 y Fx(z)1506 3869 y FA( )1560 3881 y Fy(0)1598 3869 y FB(\()p FA(z)t(;)f(\034)9 b FB(\))19 b(+)f FA(s@)1972 3881 y Fx(z)2010 3869 y FA(')p FB(\()p FA(z)t(;)c(\034)9 b FB(\))p FA(;)14 b(z)t(;)g(\034)9 b FB(\))14 b FA(ds)19 b Fw(\000)f FB(1)g Fw(\000)g FA(Q)2854 3881 y Fy(1)2891 3869 y FB(\()p FA(\034)9 b FB(\))3089 3813 y(^)-49 b FA(\026)p 3010 3850 V 3010 3926 a(z)3053 3902 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)3213 3869 y FA(;)425 b FB(\(258\))71 4091 y(where)32 b Fw(H)h FB(and)f Fw(H)655 4103 y Fy(0)725 4091 y FB(are)g(the)h(Hamiltonian)f(de\014ned)h(in)f(\(63\))g(and)h (\(68\))f(resp)r(ectiv)n(ely)-7 b(.)50 b(Ev)n(en)32 b(if)h FA(M)41 b FB(dep)r(ends)33 b(on)f FA(')p FB(,)71 4191 y(since)g(its)g(existence)f(is)h(already)f(kno)n(wn,)h FA(M)41 b FB(can)31 b(b)r(e)i(seen)e(as)h(a)f(function)i(dep)r(ending)f (on)g(the)g(v)-5 b(ariables)31 b FA(z)k FB(and)d FA(\034)9 b FB(,)71 4290 y(and)26 b(then)g(equation)g(\(255\))f(can)g(b)r(e)i (seen)f(as)f(a)h(linear)f(equation.)36 b(This)26 b(fact)g(simpli\014es) g(considerably)e(the)j(obten)n(tion)71 4390 y(of)g(the)h(estimates)g (for)f FA(')p FB(.)195 4489 y(Let)37 b(us)g(p)r(oin)n(t)g(out)g(that)g (the)g(term)44 b(^)-49 b FA(\026Q)1517 4501 y Fy(1)1554 4489 y FB(\()p FA(\034)9 b FB(\))p FA(z)1706 4459 y Fv(\000)p Fy(\()p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(\))1997 4489 y FB(in)37 b(\(256\))f(b)r(eha)n(v)n(es)g(completely)g(di\013eren)n(t)h (in)g(the)h(cases)71 4589 y FA(`)20 b Fw(\000)g FB(2)p FA(r)31 b(>)c FB(0)k(and)f FA(`)20 b Fw(\000)g FB(2)p FA(r)31 b FB(=)d(0,)j(since)f(in)h(the)g(\014rst)g(case)e(is)i(small)f (for)g FA(z)i Fw(2)c(D)2551 4559 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)2549 4610 y(\024;c)2772 4589 y FB(and)i(in)h(the)g(second)f(is)h (not.)46 b(F)-7 b(or)71 4689 y(this)28 b(reason,)e(w)n(e)h(split)h(the) g(pro)r(of)f(of)h(Prop)r(osition)d(8.3)i(in)h(these)g(t)n(w)n(o)f (cases.)195 4788 y(Finally)34 b(in)g(this)g(section,)h(w)n(e)f(state)f (the)i(follo)n(wing)d(lemma,)k(whic)n(h)d(giv)n(es)g(some)g(prop)r (erties)g(of)h(the)g(functions)71 4888 y(in)n(v)n(olv)n(ed)26 b(in)i(equation)f(\(255\))o(.)p Black 71 5048 a Fp(Lemma)33 b(8.5.)p Black 41 w Fs(L)l(et)d(us)g(c)l(onsider)i(any)g FA(\024)25 b Fw(\025)f FA(\024)1555 5060 y Fy(5)1623 5048 y Fs(and)32 b(any)f FA(c)25 b(>)g FB(0)p Fs(.)41 b(The)32 b(functions)f FA(L)f Fs(and)i FA(M)39 b Fs(de\014ne)l(d)32 b(in)37 b FB(\(257\))30 b Fs(and)71 5148 y FB(\(258\))f Fs(r)l(esp)l(e)l(ctively,)i(satisfy)g(the)f(fol)t(lowing)i(pr)l(op)l (erties.)p Black 169 5308 a(1.)p Black 42 w FA(L)23 b Fw(2)g(Z)496 5329 y Fy(2)p Fx(r)r Fv(\000)627 5306 y Fu(1)p 624 5315 35 3 v 624 5349 a Fm(\014)669 5329 y Fx(;\024;c;\033)871 5308 y Fs(and)30 b(satis\014es)1668 5451 y Fw(k)p FA(L)p Fw(k)1808 5476 y Fy(2)p Fx(r)r Fv(\000)1939 5454 y Fu(1)p 1935 5463 V 1935 5496 a Fm(\014)1980 5476 y Fx(;\024;c;\033)2176 5451 y Fw(\024)22 b FA(K)6 b(")2392 5392 y Fu(1)p 2388 5401 V 2388 5434 a Fm(\014)2438 5451 y FA(:)p Black 1919 5753 a FB(89)p Black eop end %%Page: 90 90 TeXDict begin 90 89 bop Black Black Black 169 272 a Fs(2.)p Black 42 w FA(M)32 b Fw(2)23 b(Z)529 284 y Fy(0)p Fx(;\024;c;\033)765 272 y Fs(and)30 b(satis\014es)1658 417 y Fw(k)o FA(M)9 b Fw(k)1830 442 y Fy(0)p Fx(;\024;c;\033)2059 417 y Fw(\024)2259 361 y FA(K)p 2157 398 282 4 v 2157 474 a(\024)2205 450 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)2448 417 y FA(:)p Black 71 616 a Fs(Pr)l(o)l(of.)p Black 43 w FB(W)-7 b(e)30 b(pro)n(v)n(e)f(the)h(lemma)g(in)g(the)g(p)r(olynomial)g(case.)43 b(In)30 b(the)g(trigonometric)e(case)h(can)h(b)r(e)g(done)g (analogously)71 715 y(taking)d(in)n(to)g(accoun)n(t)g(Remark)g(and)g (4.14.)195 815 y(First)g(w)n(e)g(b)r(ound)g FA(L)p FB(.)37 b(Using)27 b(the)g(de\014nitions)g(of)g Fw(H)q FB(,)p 1882 748 76 4 v 27 w FA(H)7 b FB(,)2028 794 y Fz(b)2009 815 y FA(H)33 b FB(and)27 b Fw(H)2342 827 y Fy(0)2407 815 y FB(in)g(\(63\),)g(\(40\),)g(\(33\))g(and)f(\(68\))h(resp)r(ectiv) n(ely)-7 b(,)71 915 y(w)n(e)27 b(split)h(it)g(as)f FA(L)c FB(=)f FA(L)787 927 y Fy(1)842 915 y FB(+)c FA(L)982 927 y Fy(2)1038 915 y FB(+)g FA(L)1178 927 y Fy(3)1233 915 y FB(+)g FA(L)1373 927 y Fy(4)1437 915 y FB(with)592 1141 y FA(L)649 1153 y Fy(1)686 1141 y FB(\()p FA(z)t(;)c(\034)9 b FB(\))23 b(=)973 1084 y(1)p 973 1122 42 4 v 973 1198 a(2)1038 1024 y Fz(\022)1306 1081 y FA(C)1371 1051 y Fy(2)1365 1101 y(+)p 1109 1122 508 4 v 1109 1198 a FA(")1148 1174 y Fy(2)p Fx(r)1218 1198 y FA(p)1260 1169 y Fy(2)1260 1220 y(0)1297 1198 y FB(\()p FA(ia)18 b FB(+)g FA("z)t FB(\))1645 1141 y Fw(\000)g FA(z)1771 1106 y Fy(2)p Fx(r)1840 1024 y Fz(\023)1915 1141 y FB(\()p FA(@)1991 1153 y Fx(z)2030 1141 y FA( )2084 1153 y Fy(0)2121 1141 y FB(\))2153 1099 y Fy(2)592 1377 y FA(L)649 1389 y Fy(2)686 1377 y FB(\()p FA(z)t(;)c(\034)9 b FB(\))23 b(=)976 1320 y FA(")1015 1290 y Fy(2)p Fx(r)p 973 1358 115 4 v 973 1434 a FA(C)1038 1405 y Fy(2)1032 1454 y(+)1111 1377 y FB(\()p FA(V)c FB(\()p FA(q)1279 1389 y Fy(0)1331 1377 y FB(\()p FA(ia)f FB(+)g FA("z)t FB(\))g(+)g FA(x)1799 1389 y Fx(p)1838 1377 y FB(\()p FA(\034)9 b FB(\)\))20 b Fw(\000)e FA(V)32 b FB(\()q FA(x)2242 1389 y Fx(p)2280 1377 y FB(\()p FA(\034)9 b FB(\)\))20 b Fw(\000)e FA(V)2591 1342 y Fv(0)2628 1377 y FB(\()q FA(x)2708 1389 y Fx(p)2747 1377 y FB(\()p FA(\034)9 b FB(\)\))15 b FA(q)2940 1389 y Fy(0)2977 1377 y FB(\()p FA(ia)k FB(+)f FA("z)t FB(\)\))981 1604 y Fw(\000)1130 1548 y FB(1)p 1074 1585 154 4 v 1074 1661 a(2)p FA(z)1159 1637 y Fy(2)p Fx(r)592 1821 y FA(L)649 1833 y Fy(3)686 1821 y FB(\()p FA(z)t(;)c(\034)9 b FB(\))23 b(=)980 1765 y(^)-49 b FA(\026")1062 1735 y Fx(`)p 973 1802 121 4 v 976 1878 a FA(C)1041 1849 y Fy(2)1035 1898 y(+)1123 1800 y Fz(b)1104 1821 y FA(H)1180 1787 y Fy(1)1173 1841 y(1)1230 1754 y Fz(\000)1268 1821 y FA(q)1305 1833 y Fy(0)1343 1821 y FB(\()p FA(ia)18 b FB(+)g FA("z)t FB(\))p FA(;)c(C)1765 1787 y Fy(2)1759 1841 y(+)1814 1821 y FA(")1853 1787 y Fv(\000)p Fy(2)p Fx(r)1974 1821 y FA(@)2018 1833 y Fx(z)2056 1821 y FA( )2110 1833 y Fy(0)2148 1821 y FB(\()p FA(z)t(;)g(\034)9 b FB(\))p FA(;)14 b(\034)2419 1754 y Fz(\001)981 2074 y Fw(\000)1093 2018 y FB(^)-48 b FA(\026)p 1074 2055 75 4 v 1074 2131 a(z)1117 2107 y Fx(`)1307 1996 y Fz(X)1172 2178 y Fy(\()p Fx(r)r Fv(\000)p Fy(1\))p Fx(k)q Fy(+)p Fx(r)r(l)p Fy(=)p Fx(`)1576 2074 y FA(a)1620 2086 y Fx(k)q(l)1682 2074 y FB(\()p FA(\034)9 b FB(\))1815 2014 y FA(C)1880 1979 y Fx(k)q Fy(+)p Fx(l)p Fv(\000)p Fy(2)1874 2035 y(+)p 1802 2055 289 4 v 1802 2131 a FB(\(1)19 b Fw(\000)f FA(r)r FB(\))2049 2107 y Fx(k)2114 2007 y Fz(\000)2152 2074 y FA(z)2195 2040 y Fy(2)p Fx(r)2265 2074 y FA(@)2309 2086 y Fx(z)2347 2074 y FA( )2401 2086 y Fy(0)2438 2074 y FB(\()p FA(z)t(;)c(\034)9 b FB(\))2627 2007 y Fz(\001)2666 2023 y Fx(l)592 2360 y FA(L)649 2372 y Fy(4)686 2360 y FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))23 b(=)980 2304 y(^)-49 b FA(\026")1062 2274 y Fx(`)p Fy(+1)p 973 2341 205 4 v 1018 2417 a FA(C)1083 2389 y Fy(2)1077 2438 y(+)1207 2339 y Fz(b)1188 2360 y FA(H)1264 2326 y Fy(2)1257 2381 y(1)1314 2293 y Fz(\000)1353 2360 y FA(q)1390 2372 y Fy(0)1427 2360 y FB(\()p FA(ia)18 b FB(+)g FA("z)t FB(\))p FA(;)c(C)1849 2326 y Fy(2)1843 2381 y(+)1898 2360 y FA(")1937 2326 y Fv(\000)p Fy(2)p Fx(r)2058 2360 y FA(@)2102 2372 y Fx(z)2141 2360 y FA( )2195 2372 y Fy(0)2232 2360 y FB(\()p FA(z)t(;)g(\034)9 b FB(\))2421 2293 y Fz(\001)2473 2360 y FA(:)71 2589 y FB(T)-7 b(aking)27 b(in)n(to)g(accoun)n(t)g(the)h(prop)r(erties)e(of)i FA(p)1493 2601 y Fy(0)1530 2589 y FB(\()p FA(u)p FB(\))g(in)g(\(12\))f(and)g (Theorem)g(4.12,)f(one)i(can)f(see)g(that)1546 2779 y Fw(k)o FA(L)1644 2791 y Fy(1)1681 2779 y Fw(k)1723 2803 y Fy(2)p Fx(r)r Fv(\000)1854 2781 y Fu(1)p 1850 2790 35 3 v 1850 2824 a Fm(\014)1895 2803 y Fx(;\024;c;\033)2091 2779 y Fw(\024)22 b FA(K)6 b(")2307 2719 y Fu(1)p 2303 2728 V 2303 2762 a Fm(\014)2352 2779 y FA(:)71 2982 y FB(F)-7 b(or)36 b FA(L)286 2994 y Fy(2)360 2982 y FB(one)g(has)g(to)h (tak)n(e)f(in)n(to)g(accoun)n(t)g(that)h FA(V)19 b FB(\()p FA(q)1796 2994 y Fy(0)1833 2982 y FB(\()p FA(u)p FB(\)\))39 b(=)f Fw(\000)p FA(p)2226 2951 y Fy(2)2226 3002 y(0)2263 2982 y FB(\()p FA(u)p FB(\))p FA(=)p FB(2,)g(use)e(\(15\))h(and)f(the)i (b)r(ound)f(of)f FA(x)3595 2994 y Fx(p)3634 2982 y FB(\()p FA(\034)9 b FB(\))38 b(in)71 3081 y(Prop)r(osition)26 b(5.5.)36 b(Then,)28 b(one)f(can)g(obtain)1564 3266 y Fw(k)p FA(L)p Fw(k)1704 3291 y Fy(2)p Fx(r)r Fv(\000)1835 3268 y Fu(1)p 1832 3277 V 1832 3311 a Fm(\014)1876 3291 y Fx(;\024;c;\033)2072 3266 y Fw(\024)c FA(K)6 b(")2289 3206 y Fu(1)p 2285 3215 V 2285 3249 a Fm(\014)2334 3266 y FA(:)71 3482 y FB(T)-7 b(o)27 b(b)r(ound)h(the)g(third)g(term,)f (using)h(the)g(de\014nition)g(of)1865 3461 y Fz(b)1845 3482 y FA(H)1921 3452 y Fy(1)1914 3503 y(1)1986 3482 y FB(in)g(\(35\))f(and)g(also)g(\(12\),)g(one)h(can)f(rewrite)g(it)h (as)548 3750 y FA(L)605 3762 y Fy(3)642 3750 y FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))23 b(=)6 b(^)-48 b FA(\026")1008 3716 y Fx(`)p Fv(\000)p Fy(\()p Fx(r)r Fv(\000)p Fy(1\))p Fx(k)q Fv(\000)p Fx(r)r(l)1417 3671 y Fz(X)1551 3750 y FA(a)1595 3762 y Fx(k)q(l)1657 3750 y FB(\()p FA(\034)9 b FB(\))1789 3690 y FA(C)1854 3655 y Fx(k)q Fy(+)p Fx(l)p Fv(\000)p Fy(2)1848 3711 y(+)p 1777 3731 289 4 v 1777 3807 a FB(\(1)18 b Fw(\000)g FA(r)r FB(\))2023 3783 y Fx(k)2089 3608 y Fz( )2226 3694 y FB(1)p 2165 3731 165 4 v 2165 3807 a FA(z)2208 3783 y Fx(r)r Fv(\000)p Fy(1)2357 3750 y FB(+)g Fw(O)2522 3608 y Fz( )2685 3694 y FA(")2737 3635 y Fu(1)p 2733 3644 35 3 v 2733 3677 a Fm(\014)p 2598 3731 271 4 v 2598 3827 a FA(z)2641 3790 y Fx(r)r Fv(\000)p Fy(1)p Fv(\000)2823 3768 y Fu(1)p 2820 3777 35 3 v 2820 3810 a Fm(\014)2879 3608 y Fz(!!)3010 3626 y Fx(k)3065 3750 y FB(\()p FA(z)3140 3716 y Fx(r)3176 3750 y FA(@)3220 3762 y Fx(z)3259 3750 y FA( )s FB(\))3348 3709 y Fx(l)937 4023 y Fw(\000)1049 3967 y FB(^)-49 b FA(\026)p 1030 4004 75 4 v 1030 4080 a(z)1073 4056 y Fx(`)1263 3944 y Fz(X)1128 4126 y Fy(\()p Fx(r)r Fv(\000)p Fy(1\))p Fx(k)q Fy(+)p Fx(r)r(l)p Fy(=)p Fx(`)1532 4023 y FA(a)1576 4035 y Fx(k)q(l)1638 4023 y FB(\()p FA(\034)9 b FB(\))1770 3963 y FA(C)1835 3928 y Fx(k)q Fy(+)p Fx(l)p Fv(\000)p Fy(2)1829 3984 y(+)p 1758 4004 289 4 v 1758 4080 a FB(\(1)18 b Fw(\000)g FA(r)r FB(\))2004 4056 y Fx(k)2070 3956 y Fz(\000)2108 4023 y FA(z)2151 3989 y Fy(2)p Fx(r)2220 4023 y FA(@)2264 4035 y Fx(z)2302 4023 y FA( )2356 4035 y Fy(0)2394 4023 y FB(\()p FA(z)t(;)c(\034)9 b FB(\))2583 3956 y Fz(\001)2621 3972 y Fx(l)2661 4023 y FA(:)71 4289 y FB(Then,)28 b(it)g(is)f(easy)g(to)g(see)h(that)f FA(L)1132 4301 y Fy(3)1192 4289 y Fw(2)d(Z)1331 4310 y Fx(`)p Fv(\000)1424 4287 y Fu(1)p 1420 4296 35 3 v 1420 4330 a Fm(\014)1465 4310 y Fx(;\024;c;\033)1661 4289 y Fw(\032)e(Z)1808 4310 y Fy(2)p Fx(r)r Fv(\000)1940 4287 y Fu(1)p 1936 4296 V 1936 4330 a Fm(\014)1981 4310 y Fx(;\024;c;\033)2181 4289 y FB(and)1203 4506 y Fw(k)p FA(L)1302 4518 y Fy(3)1339 4506 y Fw(k)1380 4531 y Fy(2)p Fx(r)r Fv(\000)1511 4509 y Fu(1)p 1508 4518 V 1508 4551 a Fm(\014)1553 4531 y Fx(;\024;c;\033)1748 4506 y Fw(\024)h FA(K)c Fw(k)p FA(L)2025 4518 y Fy(3)2061 4506 y Fw(k)2103 4531 y Fx(`)p Fv(\000)2196 4509 y Fu(1)p 2193 4518 V 2193 4551 a Fm(\014)2237 4531 y Fx(;\024;c;\033)2433 4506 y Fw(\024)k FA(K)6 b(")2650 4447 y Fu(1)p 2646 4456 V 2646 4489 a Fm(\014)2695 4506 y FA(:)71 4699 y FB(The)28 b(b)r(ound)g(of)f FA(L)649 4711 y Fy(4)713 4699 y FB(is)h(straigh)n (tforw)n(ard.)195 4799 y(F)-7 b(or)27 b(the)h(b)r(ound)g(of)g FA(M)9 b FB(,)27 b(w)n(e)g(split)h(it)g(as)f FA(M)32 b FB(=)23 b FA(M)1752 4811 y Fy(1)1807 4799 y FB(+)18 b FA(M)1971 4811 y Fy(2)2026 4799 y FB(+)g FA(M)2190 4811 y Fy(3)2255 4799 y FB(with)676 5011 y FA(M)757 5023 y Fy(1)794 5011 y FB(\()p FA(z)t(;)c(\034)9 b FB(\))23 b(=)g FA(@)1138 5023 y Fx(w)1192 5011 y Fw(H)1262 5023 y Fy(0)1313 5011 y FB(\()p FA(@)1389 5023 y Fx(z)1427 5011 y FA( )1481 5023 y Fy(0)1519 5011 y FA(;)14 b(z)t(;)g(\034)9 b FB(\))18 b Fw(\000)g FA(Q)1880 5023 y Fy(1)1917 5011 y FB(\()p FA(\034)9 b FB(\))2115 4955 y(^)-49 b FA(\026)p 2036 4992 192 4 v 2036 5068 a(z)2079 5044 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)2258 5011 y Fw(\000)18 b FB(1)676 5231 y FA(M)757 5243 y Fy(2)794 5231 y FB(\()p FA(z)t(;)c(\034)9 b FB(\))23 b(=)1094 5118 y Fz(Z)1177 5139 y Fy(1)1140 5307 y(0)1228 5231 y FB(\()p FA(@)1304 5243 y Fx(w)1358 5231 y Fw(H)1428 5243 y Fy(0)1479 5231 y FB(\()p FA(@)1555 5243 y Fx(z)1594 5231 y FA( )1648 5243 y Fy(0)1704 5231 y FB(+)18 b FA(s@)1870 5243 y Fx(z)1908 5231 y FA(';)c(z)t(;)g(\034)9 b FB(\))19 b Fw(\000)f FA(@)2302 5243 y Fx(w)2355 5231 y Fw(H)2425 5243 y Fy(0)2477 5231 y FB(\()p FA(@)2553 5243 y Fx(z)2591 5231 y FA( )2645 5243 y Fy(0)2683 5231 y FA(;)c(z)t(;)g(\034)9 b FB(\)\))14 b FA(ds)676 5470 y(M)757 5482 y Fy(3)794 5470 y FB(\()p FA(z)t(;)g(\034)9 b FB(\))23 b(=)1094 5357 y Fz(Z)1177 5377 y Fy(1)1140 5546 y(0)1228 5470 y FB(\()p FA(@)1304 5482 y Fx(w)1358 5470 y Fw(H)15 b FB(\()p FA(@)1519 5482 y Fx(z)1557 5470 y FA( )1611 5482 y Fy(0)1667 5470 y FB(+)j FA(s@)1833 5482 y Fx(z)1871 5470 y FA(';)c(z)t(;)g(\034)9 b FB(\))19 b Fw(\000)f FA(@)2265 5482 y Fx(w)2319 5470 y Fw(H)2389 5482 y Fy(0)2440 5470 y FB(\()p FA(@)2516 5482 y Fx(z)2555 5470 y FA( )2609 5482 y Fy(0)2665 5470 y FB(+)g FA(s@)2831 5482 y Fx(z)2869 5470 y FA(';)c(z)t(;)g(\034)9 b FB(\)\))14 b FA(ds)p Black 1919 5753 a FB(90)p Black eop end %%Page: 91 91 TeXDict begin 91 90 bop Black Black 71 272 a FB(and)27 b(w)n(e)h(b)r(ound)g(eac)n(h)e(term.)195 372 y(T)-7 b(aking)27 b(in)n(to)g(accoun)n(t)g(the)h(de\014nitions)g(of)f Fw(H)1656 384 y Fy(0)1721 372 y FB(and)h FA(Q)1949 384 y Fx(j)2011 372 y FB(in)g(\(68\))f(and)g(\(73\))h(resp)r(ectiv)n(ely)-7 b(,)27 b(and)g(the)h(prop)r(erties)f(of)71 471 y FA( )125 483 y Fy(0)187 471 y FB(giv)n(en)c(b)n(y)h(Theorem)f(4.12,)h(one)g(can) g(see)g(that)g FA(M)1741 483 y Fy(1)1801 471 y Fw(2)g(Z)1940 483 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;\024;c;\033)2366 471 y FB(and)g Fw(k)p FA(M)2647 483 y Fy(1)2683 471 y Fw(k)2725 483 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;\024;c;\033)3150 471 y Fw(\024)e FA(K)6 b FB(,)25 b(whic)n(h)f(implies)1540 699 y Fw(k)p FA(M)1663 711 y Fy(1)1699 699 y Fw(k)1741 711 y Fy(0)p Fx(;\024;c;\033)1969 699 y Fw(\024)2169 643 y FA(K)p 2067 680 282 4 v 2067 756 a(\024)2115 732 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)2358 699 y FA(:)71 910 y FB(F)-7 b(or)21 b(the)h(second)f(term,)i (let)f(us)g(recall)f(that,)i(using)e(the)h(de\014nition)g(of)g FA(T)2308 922 y Fy(0)2367 910 y FB(in)g(\(51\))o(,)h(b)n(y)f(Theorems)e (4.4)h(\(see)h(also)e(Section)71 1009 y(7.2.5\))26 b(and)i(4.12,)e(w)n (e)h(ha)n(v)n(e)g(an)g Fs(a)j(priori)38 b FB(estimate)28 b(for)f FA(@)1908 1021 y Fx(z)1946 1009 y FA(')p FB(,)1606 1192 y Fw(k)p FA(@)1692 1204 y Fx(z)1730 1192 y FA(')p Fw(k)1826 1204 y Fx(`)p Fy(+1)p Fx(;\024;c;\033)2133 1192 y Fw(\024)22 b FA(K)q(:)71 1375 y FB(Then,)36 b(it)f(is)f(enough)g (to)g(apply)g(again)g(the)g(mean)h(v)-5 b(alue)34 b(theorem)g(and)g (the)h(b)r(ounds)f(of)h FA( )3062 1387 y Fy(0)3133 1375 y FB(in)g(Theorem)f(4.12)f(to)71 1474 y(obtain)1540 1596 y Fw(k)p FA(M)1663 1608 y Fy(2)1699 1596 y Fw(k)1741 1608 y Fy(0)p Fx(;\024;c;\033)1969 1596 y Fw(\024)2169 1539 y FA(K)p 2067 1577 V 2067 1653 a(\024)2115 1629 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)2358 1596 y FA(:)71 1768 y FB(F)-7 b(or)27 b FA(M)301 1780 y Fy(3)338 1768 y FB(,)g(it)h(is)g(enough)f(to)g(pro)r(ceed)g(as)g(in)h(the)g(b)r(ound) g(for)f FA(L)g FB(to)h(obtain)1604 1951 y Fw(k)p FA(M)1727 1963 y Fy(3)1763 1951 y Fw(k)1805 1963 y Fy(0)p Fx(;\024;c;\033)2033 1951 y Fw(\024)23 b FA(K)6 b(")2247 1887 y Fm(\015)p 2246 1901 35 3 v 2246 1934 a(\014)2295 1951 y FA(:)p 3790 2134 4 57 v 3794 2081 50 4 v 3794 2134 V 3843 2134 4 57 v 71 2349 a Fp(The)43 b(initial)f(condition)g(in)h(the)g (transition)f(domains)82 b FB(T)-7 b(o)37 b(obtain)g(b)r(etter)h (estimates)f(of)g FA(@)3347 2361 y Fx(z)3386 2349 y FA(')h FB(w)n(e)f(use)g(an)71 2449 y(in)n(tegral)g(equation.)70 b(T)-7 b(o)38 b(obtain)h(it)g(from)f(\(255\))g(w)n(e)g(need)h(initial)g (conditions.)70 b(Therefore,)40 b(w)n(e)e(tak)n(e)g(constan)n(ts)71 2549 y FA(c)107 2561 y Fy(1)167 2549 y FA(<)23 b(c)291 2519 y Fv(0)291 2569 y Fy(0)351 2549 y FA(<)g(c)475 2561 y Fy(0)539 2549 y FB(an)k(w)n(e)g(lo)r(ok)g(for)g(them)h(in)f(the)h (transition)f(domains)f Fw(I)2291 2509 y Fy(+)p Fx(;u)2285 2576 y(c)2315 2584 y Fu(0)2348 2576 y Fx(;c)2398 2556 y Fl(0)2398 2594 y Fu(0)2452 2549 y Fw(\002)17 b Ft(T)2589 2561 y Fx(\033)2634 2549 y FB(,)28 b(de\014ned)f(in)h(\(253\))f(\(see)g (also)f(Figure)71 2665 y(11\).)40 b(In)29 b(this)g(domain,)g(the)h (next)f(lemma)g(giv)n(es)e(sharp)i(estimates)f(of)h(the)g(function)h FA(@)2850 2677 y Fx(z)2888 2665 y FA(')p FB(.)41 b(W)-7 b(e)30 b(abuse)e(notation)h(and)71 2765 y(w)n(e)e(use)h(the)g(norms)e (de\014ned)i(in)g(Section)g(7.2.4,)e(ev)n(en)h(if)h(here)f(the)h (suprema)f(are)g(tak)n(en)g(in)h Fw(I)3058 2725 y Fy(+)p Fx(;u)3052 2792 y(c)3082 2800 y Fu(0)3114 2792 y Fx(;c)3164 2772 y Fl(0)3164 2810 y Fu(0)3200 2765 y FB(.)p Black 71 2947 a Fp(Lemma)k(8.6.)p Black 40 w Fs(L)l(et)e(us)f(c)l(onsider)j FA(\015)c Fw(2)c FB(\(0)p FA(;)14 b(\015)1485 2959 y Fy(2)1522 2947 y FB(\))p Fs(,)31 b(wher)l(e)g FA(\015)1888 2959 y Fy(2)1955 2947 y Fs(is)g(de\014ne)l(d)f(in)37 b FB(\(252\))o Fs(,)31 b(and)f FA(")2879 2959 y Fy(0)2940 2947 y FA(>)23 b FB(0)30 b Fs(smal)t(l)h(enough.)40 b(Then,)71 3047 y(for)30 b FA(")23 b Fw(2)h FB(\(0)p FA(;)14 b(")494 3059 y Fy(0)530 3047 y FB(\))p Fs(,)31 b(the)f(function)f FA(@)1126 3059 y Fx(z)1165 3047 y FA(')h Fs(r)l(estricte)l(d)g(to)f Fw(I)1762 3007 y Fy(+)p Fx(;u)1756 3075 y(c)1786 3083 y Fu(0)1819 3075 y Fx(;c)1869 3055 y Fl(0)1869 3093 y Fu(0)1934 3047 y Fs(satis\014es)1502 3266 y Fw(k)p FA(@)1588 3278 y Fx(z)1626 3266 y FA(')p Fw(k)1722 3291 y Fy(0)p Fx(;\033)1842 3266 y Fw(\024)23 b FA(K)6 b(")2046 3229 y Fy(2)p Fx(r)r Fy(\(1)p Fv(\000)p Fx(\015)t Fy(\)+)2348 3202 y Fm(\015)p 2347 3216 35 3 v 2347 3249 a(\014)2396 3266 y FA(:)p Black 71 3449 a Fs(Pr)l(o)l(of.)p Black 43 w FB(Considering)26 b(the)i(functions)g FA(T)34 b FB(=)23 b FA(T)1509 3461 y Fy(0)1564 3449 y FB(+)18 b FA(T)1696 3461 y Fy(1)1733 3449 y FB(,)28 b(obtained)f(in)h(Prop)r (osition)e(7.4)h(\(see)g(also)g(Section)g(7.2.5\),)g(and)1164 3670 y FA( )1218 3682 y Fy(0)1255 3670 y FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))24 b(=)e Fw(\000)1852 3614 y FB(1)p 1630 3651 486 4 v 1630 3727 a(\(2)p FA(r)f Fw(\000)d FB(1\))p FA(z)1962 3703 y Fy(2)p Fx(r)r Fv(\000)p Fy(1)2144 3670 y FB(+)25 b(^)-49 b FA(\026)p 2277 3603 58 4 v( )2334 3691 y Fy(0)2372 3670 y FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))18 b(+)g FA(K)q(;)71 3907 y FB(obtained)27 b(in)h(Theorem)f (4.12,)f(and)i(recalling)e(that)i FA(@)1776 3919 y Fx(u)1819 3907 y FA(T)1868 3919 y Fy(0)1905 3907 y FB(\()p FA(u)p FB(\))23 b(=)g FA(p)2170 3877 y Fy(2)2170 3928 y(0)2207 3907 y FB(\()p FA(u)p FB(\),)28 b(w)n(e)f(split)h FA(@)2721 3919 y Fx(z)2759 3907 y FA(')g FB(as)944 4089 y FA(@)988 4101 y Fx(z)1026 4089 y FA(')p FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))24 b(=)p FA(@)1402 4101 y Fx(z)1440 4089 y FA( )s FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))19 b Fw(\000)f FA(@)1832 4101 y Fx(z)1870 4089 y FA( )1924 4101 y Fy(0)1962 4089 y FB(\()p FA(z)t(;)c(\034)9 b FB(\))1293 4272 y(=)p FA(")1397 4238 y Fy(2)p Fx(r)1466 4272 y FA(C)1531 4238 y Fy(2)1525 4293 y(+)1581 4155 y Fz(\022)1642 4272 y FA(@)1686 4284 y Fx(u)1729 4272 y FA(T)j FB(\()p FA("z)21 b FB(+)d FA(ia;)c(\034)9 b FB(\))19 b Fw(\000)f FA(@)2337 4284 y Fx(u)2381 4272 y FA(T)2430 4284 y Fy(0)2466 4272 y FB(\()p FA("z)k FB(+)c FA(ia)p FB(\))2786 4155 y Fz(\023)1376 4505 y FB(+)1459 4388 y Fz(\022)1520 4505 y FA(")1559 4470 y Fy(2)p Fx(r)1629 4505 y FA(C)1694 4470 y Fy(2)1688 4525 y(+)1743 4505 y FA(p)1785 4470 y Fy(2)1785 4525 y(0)1822 4505 y FB(\()p FA("z)k FB(+)c FA(ia)p FB(\))g Fw(\000)2289 4448 y FB(1)p 2253 4486 113 4 v 2253 4562 a FA(z)2296 4538 y Fy(2)p Fx(r)2375 4388 y Fz(\023)2455 4505 y Fw(\000)25 b FB(^)-49 b FA(\026@)2632 4517 y Fx(z)p 2670 4437 58 4 v 2670 4505 a FA( )2727 4525 y Fy(0)2765 4505 y FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))p FA(:)71 4733 y FB(W)-7 b(e)30 b(b)r(ound)h(eac)n(h)e (term.)44 b(F)-7 b(or)30 b(the)g(\014rst)g(term)g(it)h(is)e(enough)h (to)g(apply)g(the)g(result)g(obtained)g(in)g(Prop)r(osition)e(7.4)i(to) 71 4832 y(obtain)837 4836 y Fz(\015)837 4886 y(\015)837 4936 y(\015)883 4932 y FA(")922 4897 y Fy(2)p Fx(r)991 4932 y FA(C)1056 4897 y Fy(2)1050 4952 y(+)1106 4840 y Fz(\020)1155 4932 y FA(@)1199 4944 y Fx(u)1243 4932 y FA(T)12 b FB(\()p FA("z)21 b FB(+)d FA(ia;)c(\034)9 b FB(\))18 b Fw(\000)g FA(@)1850 4944 y Fx(u)1894 4932 y FA(T)1943 4944 y Fy(0)1980 4932 y FB(\()p FA("z)j FB(+)d FA(ia)p FB(\))2299 4840 y Fz(\021)2349 4836 y(\015)2349 4886 y(\015)2349 4936 y(\015)2395 4990 y Fy(0)p Fx(;\033)2516 4932 y Fw(\024)k FA(K)6 b(")2719 4897 y Fy(\(1)p Fv(\000)p Fx(\015)t Fy(\)\()p Fx(`)p Fy(+1\))3062 4932 y FA(:)71 5122 y FB(Then,)26 b(since)e FA(\015)k Fw(2)c FB(\(0)p FA(;)14 b(\015)813 5134 y Fy(2)849 5122 y FB(\),)26 b(\()p FA(`)13 b FB(+)g(1\)\(1)g Fw(\000)g FA(\015)5 b FB(\))23 b Fw(\025)g FB(2)p FA(r)r FB(\(1)13 b Fw(\000)g FA(\015)5 b FB(\))13 b(+)1946 5085 y Fx(\015)p 1945 5102 41 4 v 1945 5150 a(\014)1996 5122 y FB(,)26 b(w)n(e)e(obtain)h(the)g(desired)g (b)r(ound.)36 b(F)-7 b(or)25 b(the)g(second)g(term)71 5232 y(w)n(e)30 b(use)h(\(12\))o(.)47 b(Finally)-7 b(,)31 b(the)g(b)r(ound)g(of)g(the)g(third)g(term)g(is)f(a)g(direct)h (consequence)f(of)g(Prop)r(osition)f(4.8)h(of)h([Bal06)n(].)71 5332 y(This)22 b(prop)r(osition)f(states)g(the)i(same)e(results)h(of)g (Theorem)f(4.12)f(but)j(b)r(ounds)p 2548 5264 58 4 v 22 w FA( )2605 5352 y Fy(0)2642 5332 y FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))23 b(using)f(F)-7 b(ourier)20 b(norms)i(instead)71 5431 y(of)27 b(using)h(classical)e(supremm)n(um)h(norms.)p 3790 5431 4 57 v 3794 5379 50 4 v 3794 5431 V 3843 5431 4 57 v Black 1919 5753 a(91)p Black eop end %%Page: 92 92 TeXDict begin 92 91 bop Black Black 71 272 a Fp(The)25 b(\014xed)g(p)s(oin)m(t)f(equation)g(for)h FA(`)6 b Fw(\000)g FB(2)p FA(r)25 b(>)d FB(0)83 b(In)21 b(this)h(section)f(w)n(e)g(pro)n (v)n(e)f(Prop)r(osition)f(8.3)i(under)g(the)h(h)n(yp)r(othesis)71 372 y FA(`)c Fw(\000)g FB(2)p FA(r)25 b(>)e FB(0.)36 b(Let)28 b(us)g(de\014ne)g FA(\036)23 b FB(=)g FA(@)1200 384 y Fx(z)1238 372 y FA(')p FB(,)28 b(whic)n(h,)g(using)g(\(255\))o(,) g(is)f(solution)g(of)1351 554 y(\()q Fw(L)p FA(\036)p FB(\))15 b(\()p FA(z)t(;)f(\034)9 b FB(\))23 b(=)g FA(@)1881 566 y Fx(z)1933 554 y FB([)p Fw(W)7 b FB(\()p FA(\036)p FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))p FA(;)14 b(z)t(;)g(\034)9 b FB(\)])15 b FA(;)1091 b FB(\(259\))71 737 y(where)25 b Fw(L)f FB(=)e FA(@)521 749 y Fx(\034)578 737 y FB(+)15 b FA(@)702 749 y Fx(z)767 737 y FB(and)26 b Fw(W)33 b FB(is)26 b(the)g(op)r(erator)f(de\014ned)h(in)g(\(256\).)36 b(W)-7 b(e)26 b(use)g(this)h(equation)e(to)h(obtain)g(b)r(ounds)g(for)g FA(\036)p FB(.)195 837 y(T)-7 b(o)29 b(in)n(v)n(ert)f(the)h(op)r (erator)e Fw(L)e FB(=)f FA(@)1253 849 y Fx(\034)1314 837 y FB(+)19 b FA(@)1442 849 y Fx(z)1480 837 y FB(,)29 b(w)n(e)g(consider)e(the)j(op)r(erator)p 2461 770 55 4 v 27 w Fw(G)k FB(de\014ned)29 b(in)g(\(251\))o(.)40 b(Since)29 b(the)g(op)r(erator)p 71 870 V 71 936 a Fw(G)h FB(is)c(de\014ned)f(acting)g(on)g(the)g(F)-7 b(ourier)24 b(harmonics,)h(w)n(e)g(imp)r(ose)g(a)g(di\013eren)n(t)g(initial)g (condition)g(for)g(eac)n(h.)35 b(Recall)25 b(that)71 1036 y(for)30 b(the)g(negativ)n(e)g(harmonics)f(w)n(e)h(in)n(tegrate)f (from)h FA(z)1791 1048 y Fy(1)1855 1036 y Fw(2)e(D)2004 996 y Fx(u;)p Fy(+)2002 1064 y Fx(\024)2041 1044 y Fl(0)2041 1082 y Fu(5)2073 1064 y Fx(;c)2123 1072 y Fu(0)2190 1036 y FB(and)i(for)g(the)g(p)r(ositiv)n(e)g(and)h(zero)e(harmonics)g(from) 71 1166 y FA(z)110 1178 y Fy(2)170 1166 y Fw(2)23 b(D)314 1126 y Fx(u;)p Fy(+)312 1194 y Fx(\024)351 1174 y Fl(0)351 1212 y Fu(5)384 1194 y Fx(;c)434 1202 y Fu(0)497 1166 y FB(\(see)28 b(Figure)f(11\))g(for)g(a)g(\014xed)g FA(\024)1513 1136 y Fv(0)1513 1186 y Fy(5)1574 1166 y FA(>)22 b(\024)1709 1178 y Fy(5)1746 1166 y FB(.)37 b(Then,)28 b(w)n(e)f(de\014ne)h(the)g (function)702 1390 y FA(W)780 1402 y Fy(0)818 1390 y FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))23 b(=)1119 1311 y Fz(X)1118 1490 y Fx(k)q(<)p Fy(0)1254 1390 y FA(@)1298 1402 y Fx(z)1336 1390 y FA(')1390 1356 y Fy([)p Fx(k)q Fy(])1469 1390 y FB(\()p FA(z)1540 1402 y Fy(1)1577 1390 y FB(\))p FA(e)1648 1356 y Fv(\000)p Fx(ik)q Fy(\()p Fx(z)r Fv(\000)p Fx(z)1903 1364 y Fu(1)1936 1356 y Fy(\))1966 1390 y FA(e)2005 1356 y Fx(ik)q(\034)2125 1390 y FB(+)2209 1311 y Fz(X)2208 1490 y Fx(k)q Fv(\025)p Fy(0)2343 1390 y FA(@)2387 1402 y Fx(z)2426 1390 y FA(')2480 1356 y Fy([)p Fx(k)q Fy(])2558 1390 y FB(\()p FA(z)2629 1402 y Fy(2)2666 1390 y FB(\))p FA(e)2737 1356 y Fv(\000)p Fx(ik)q Fy(\()p Fx(z)r Fv(\000)p Fx(z)2992 1364 y Fu(2)3026 1356 y Fy(\))3056 1390 y FA(e)3095 1356 y Fx(ik)q(\034)3196 1390 y FA(;)442 b FB(\(260\))71 1658 y(where)23 b FA(@)351 1670 y Fx(z)390 1658 y FA(')h FB(is)g(the)h(function)g(b)r(ounded)f(in) h(Lemma)e(8.6.)35 b(The)25 b(next)f(lemma,)h(whose)e(pro)r(of)h(is)g (straigh)n(tforw)n(ard,)e(giv)n(es)71 1758 y(some)27 b(prop)r(erties)g(of)g(this)h(function.)p Black 71 1924 a Fp(Lemma)j(8.7.)p Black 40 w Fs(The)g(function)f FA(W)1174 1936 y Fy(0)1241 1924 y Fs(de\014ne)l(d)g(in)37 b FB(\(260\))28 b Fs(satis\014es:)p Black 169 2090 a(1.)p Black 42 w Fw(L)p FA(W)413 2102 y Fy(0)474 2090 y FB(=)23 b(0)p Fs(,)30 b(wher)l(e)g Fw(L)23 b FB(=)g FA(@)1105 2102 y Fx(\034)1165 2090 y FB(+)18 b FA(@)1292 2102 y Fx(x)1334 2090 y Fs(.)p Black 169 2256 a(2.)p Black 42 w FA(W)356 2268 y Fy(0)417 2256 y Fw(2)24 b(Z)556 2276 y Fy(2)p Fx(r)r Fv(\000)687 2254 y Fu(1)p 683 2263 35 3 v 683 2296 a Fm(\014)728 2276 y Fx(;\033)822 2256 y Fs(and)1693 2399 y Fw(k)o FA(W)1812 2411 y Fy(0)1850 2399 y Fw(k)1891 2424 y Fy(2)p Fx(r)r Fv(\000)2022 2401 y Fu(1)p 2019 2410 V 2019 2444 a Fm(\014)2064 2424 y Fx(;\033)2151 2399 y Fw(\024)f FA(K)6 b(")2368 2339 y Fu(1)p 2364 2348 V 2364 2382 a Fm(\014)2413 2399 y FA(:)195 2601 y FB(Then,)28 b(the)g(function)g FA(\036)h FB(is)e(a)g(solution)g(of)h(the)g(in)n (tegral)e(equation)1593 2783 y FA(\036)d FB(=)g FA(W)1831 2795 y Fy(0)1887 2783 y FB(+)p 1970 2717 55 4 v 18 w Fw(G)h(\016)18 b(W)7 b FB(\()p FA(\036)p FB(\))p FA(:)71 2966 y FB(W)-7 b(e)28 b(use)f(a)g(\014xed)h(p)r(oin)n(t)g(argumen)n(t)e (to)i(obtain)f(go)r(o)r(d)g(estimates)g(of)h FA(\036)p FB(.)37 b(W)-7 b(e)28 b(study)g FA(\036)c Fw(2)f(Z)2930 2986 y Fy(2)p Fx(r)r Fv(\000)3061 2964 y Fu(1)p 3058 2973 35 3 v 3058 3006 a Fm(\014)3103 2986 y Fx(;\033)3195 2966 y FB(as)k(a)g(\014xed)h(p)r(oin)n(t)g(of)71 3080 y(the)g(op)r(erator)p 1630 3113 89 4 v 1630 3180 a Fw(W)i FB(=)23 b FA(W)1908 3192 y Fy(0)1964 3180 y FB(+)p 2047 3113 55 4 v 18 w Fw(G)g(\016)18 b(W)7 b FA(:)1370 b FB(\(261\))p Black 71 3346 a Fp(Lemma)28 b(8.8.)p Black 37 w Fs(L)l(et)e(us)g(c)l (onsider)i FA(\015)g Fw(2)23 b FB(\(0)p FA(;)14 b(\015)1466 3358 y Fy(2)1503 3346 y FB(\))p Fs(,)28 b FA(")1627 3358 y Fy(0)1691 3346 y Fs(smal)t(l)f(enough)g(and)h FA(\024)2394 3316 y Fv(0)2394 3367 y Fy(5)2454 3346 y FA(>)22 b(\024)2589 3358 y Fy(5)2653 3346 y Fs(big)28 b(enough.)38 b(Then,)28 b(for)g FA(")23 b Fw(2)g FB(\(0)p FA(;)14 b(")3756 3358 y Fy(0)3793 3346 y FB(\))p Fs(,)71 3446 y(the)30 b(op)l(er)l(ator)p 536 3379 89 4 v 31 w Fw(W)36 b Fs(is)30 b(c)l(ontr)l(active)g(fr)l(om)h Fw(Z)1421 3466 y Fy(2)p Fx(r)r Fv(\000)1552 3444 y Fu(1)p 1548 3453 35 3 v 1548 3486 a Fm(\014)1593 3466 y Fx(;\033)1687 3446 y Fs(to)f(itself.)195 3576 y(Then,)h(ther)l(e)f(exists)f(a)i(c)l (onstant)e FA(b)1311 3588 y Fy(10)1404 3576 y FA(>)22 b FB(0)29 b Fs(such)h(that)g FA(\036)p Fs(,)h(the)f(unique)f(\014xe)l (d)g(p)l(oint)h(of)p 2930 3509 89 4 v 31 w Fw(W)7 b Fs(,)30 b(satis\014es)1607 3770 y Fw(k)p FA(\036)p Fw(k)1740 3790 y Fy(2)p Fx(r)r Fv(\000)1871 3768 y Fu(1)p 1867 3777 35 3 v 1867 3810 a Fm(\014)1912 3790 y Fx(;\033)2000 3770 y Fw(\024)22 b FA(b)2123 3782 y Fy(10)2193 3770 y FA(")2245 3710 y Fu(1)p 2242 3719 V 2242 3753 a Fm(\014)2291 3770 y FA(:)p Black 71 3983 a Fs(Pr)l(o)l(of.)p Black 331 3916 89 4 v 43 w Fw(W)39 b FB(sends)32 b Fw(Z)739 4003 y Fy(2)p Fx(r)r Fv(\000)870 3981 y Fu(1)p 867 3990 35 3 v 867 4023 a Fm(\014)912 4003 y Fx(;\033)1008 3983 y FB(to)g(itself.)52 b(T)-7 b(o)32 b(see)f(that)p 1812 3916 89 4 v 33 w Fw(W)39 b FB(is)32 b(con)n(tractiv)n(e)f(from)g Fw(Z)2713 4003 y Fy(2)p Fx(r)r Fv(\000)2845 3981 y Fu(1)p 2841 3990 35 3 v 2841 4023 a Fm(\014)2886 4003 y Fx(;\033)2983 3983 y FB(to)h(itself,)i(let)e(us)h(consider)71 4102 y FA(\036)120 4114 y Fy(1)158 4102 y FA(;)14 b(\036)244 4114 y Fy(2)307 4102 y Fw(2)26 b(Z)448 4122 y Fy(2)p Fx(r)r Fv(\000)579 4100 y Fu(1)p 575 4109 V 575 4142 a Fm(\014)620 4122 y Fx(;\033)685 4102 y FB(.)41 b(Then,)30 b(applying)e(Lemmas)h(8.2)f(and)h(8.5)f(and)h(the)h(de\014nition)f(of)g Fw(W)36 b FB(in)29 b(\(256\))o(,)h(and)f(increasing)71 4221 y FA(\024)119 4191 y Fv(0)119 4242 y Fy(5)179 4221 y FA(>)23 b FB(0)k(if)h(necessary)-7 b(,)577 4336 y Fz(\015)577 4386 y(\015)p 623 4340 89 4 v 20 x Fw(W)7 b FB(\()p FA(\036)793 4418 y Fy(2)831 4406 y FB(\))18 b Fw(\000)p 964 4340 V 18 w(W)7 b FB(\()p FA(\036)1134 4418 y Fy(1)1172 4406 y FB(\))1204 4336 y Fz(\015)1204 4386 y(\015)1251 4440 y Fy(2)p Fx(r)r Fv(\000)1381 4417 y Fu(1)p 1378 4426 35 3 v 1378 4460 a Fm(\014)1423 4440 y Fx(;\033)1511 4406 y Fw(\024)22 b FA(K)d Fw(kW)7 b FB(\()p FA(\036)1900 4418 y Fy(2)1938 4406 y FB(\))19 b Fw(\000)f(W)7 b FB(\()p FA(\036)2242 4418 y Fy(1)2279 4406 y FB(\))p Fw(k)2353 4431 y Fy(2)p Fx(r)r Fv(\000)2484 4409 y Fu(1)p 2481 4418 V 2481 4451 a Fm(\014)2526 4431 y Fx(;\033)1511 4621 y Fw(\024)22 b FA(K)1688 4501 y Fz(\015)1688 4551 y(\015)1688 4600 y(\015)1688 4650 y(\015)1735 4504 y(\022)1796 4621 y FA(Q)1862 4633 y Fy(1)1899 4621 y FB(\()p FA(\034)9 b FB(\))2096 4565 y(^)-48 b FA(\026)p 2018 4602 192 4 v 2018 4678 a(z)2061 4654 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)2239 4621 y FB(+)18 b FA(M)9 b FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))2601 4504 y Fz(\023)2681 4621 y Fw(\001)18 b FB(\()p FA(\036)2803 4633 y Fy(2)2860 4621 y Fw(\000)g FA(\036)2992 4633 y Fy(1)3029 4621 y FB(\))3061 4501 y Fz(\015)3061 4551 y(\015)3061 4600 y(\015)3061 4650 y(\015)3108 4704 y Fy(2)p Fx(r)r Fv(\000)3239 4682 y Fu(1)p 3235 4691 35 3 v 3235 4724 a Fm(\014)3280 4704 y Fx(;\033)1511 4878 y Fw(\024)1720 4822 y FA(K)p 1608 4859 300 4 v 1608 4935 a FB(\()p FA(\024)1688 4907 y Fv(0)1688 4958 y Fy(5)1726 4935 y FB(\))1758 4911 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)1931 4878 y Fw(k)p FA(\036)2022 4890 y Fy(2)2078 4878 y Fw(\000)g FA(\036)2210 4890 y Fy(1)2248 4878 y Fw(k)2289 4903 y Fy(2)p Fx(r)r Fv(\000)2420 4881 y Fu(1)p 2417 4890 35 3 v 2417 4923 a Fm(\014)2462 4903 y Fx(;\033)1511 5100 y Fw(\024)1608 5044 y FB(1)p 1608 5081 42 4 v 1608 5157 a(2)1673 5100 y Fw(k)p FA(\036)1764 5112 y Fy(2)1820 5100 y Fw(\000)g FA(\036)1952 5112 y Fy(1)1990 5100 y Fw(k)2031 5125 y Fy(2)p Fx(r)r Fv(\000)2162 5103 y Fu(1)p 2159 5112 35 3 v 2159 5145 a Fm(\014)2204 5125 y Fx(;\033)2282 5100 y FA(:)71 5322 y FB(Then)p 288 5255 89 4 v 28 w Fw(W)34 b FB(is)28 b(con)n(tractiv)n(e)e(from)h Fw(Z)1171 5342 y Fy(2)p Fx(r)r Fv(\000)1302 5320 y Fu(1)p 1298 5329 35 3 v 1298 5362 a Fm(\014)1343 5342 y Fx(;\033)1436 5322 y FB(to)g(itself,)h(and)g(then)g(it)g(has)f(a)g(unique)h(\014xed)f (p)r(oin)n(t)h FA(\036)p FB(.)p Black 1919 5753 a(92)p Black eop end %%Page: 93 93 TeXDict begin 93 92 bop Black Black 195 272 a FB(T)-7 b(o)27 b(obtain)g(a)f(b)r(ound)i(for)e FA(\036)p FB(,)i(it)f(is)g (enough)g(to)g(tak)n(e)f(in)n(to)h(accoun)n(t)f(that)h Fw(k)p FA(\036)p Fw(k)2642 293 y Fy(2)p Fx(r)r Fv(\000)2773 270 y Fu(1)p 2769 279 35 3 v 2769 313 a Fm(\014)2814 293 y Fx(;\033)2902 272 y Fw(\024)22 b FB(2)p Fw(k)p 3073 205 89 4 v(W)6 b FB(\(0\))p Fw(k)3309 293 y Fy(2)p Fx(r)r Fv(\000)3440 270 y Fu(1)p 3436 279 35 3 v 3436 313 a Fm(\014)3481 293 y Fx(;\033)3546 272 y FB(.)36 b(By)27 b(the)71 402 y(de\014nition)33 b(of)p 545 336 89 4 v 33 w Fw(W)40 b FB(in)34 b(\(261\))o(,)g(w)n(e)f(ha)n(v)n(e)f (that)p 1526 336 V 33 w Fw(W)7 b FB(\(0\))32 b(=)g FA(W)1928 414 y Fy(0)1988 402 y FB(+)p 2074 336 55 4 v 21 w Fw(G)6 b FB(\()p FA(L)p FB(\).)53 b(Then,)35 b(applying)d(Lemmas)h(8.2,)g(8.5) f(and)h(8.6,)71 502 y(there)27 b(exists)g(a)h(constan)n(t)f FA(b)953 514 y Fy(10)1046 502 y FA(>)22 b FB(0)27 b(suc)n(h)h(that,)937 629 y Fz(\015)937 679 y(\015)p 983 633 89 4 v 21 x Fw(W)7 b FB(\(0\))1178 629 y Fz(\015)1178 679 y(\015)1224 733 y Fy(2)p Fx(r)r Fv(\000)1355 710 y Fu(1)p 1352 719 35 3 v 1352 753 a Fm(\014)1396 733 y Fx(;\033)1484 700 y Fw(\024)23 b(k)o FA(W)1691 712 y Fy(0)1729 700 y Fw(k)1770 724 y Fy(2)p Fx(r)r Fv(\000)1901 702 y Fu(1)p 1898 711 V 1898 744 a Fm(\014)1943 724 y Fx(;\033)2026 700 y FB(+)2109 629 y Fz(\015)2109 679 y(\015)p 2155 633 55 4 v 21 x Fw(G)c FB(\()p FA(L)p FB(\))2344 629 y Fz(\015)2344 679 y(\015)2390 733 y Fy(2)p Fx(r)r Fv(\000)2521 710 y Fu(1)p 2518 719 35 3 v 2518 753 a Fm(\014)2563 733 y Fx(;\033)2650 700 y Fw(\024)2748 643 y FA(b)2784 655 y Fy(10)p 2748 680 106 4 v 2780 757 a FB(2)2864 700 y FA(")2916 640 y Fu(1)p 2912 649 35 3 v 2912 683 a Fm(\014)2961 700 y FA(:)71 902 y FB(Let)35 b(us)g(p)r(oin)n(t)g(out)g(that)g(since)g (the)h(\014xed)f(p)r(oin)n(t)g(of)p 1803 835 89 4 v 35 w Fw(W)42 b FB(is)35 b(unique)g(in)g Fw(Z)2459 922 y Fy(2)p Fx(r)r Fv(\000)2590 900 y Fu(1)p 2587 909 35 3 v 2587 942 a Fm(\014)2632 922 y Fx(;\033)2696 902 y FB(,)i(the)f (obtained)e(function)i FA(\036)f FB(m)n(ust)71 1021 y(coincide)26 b(with)h FA(\036)d FB(=)e FA( )793 991 y Fx(u)853 1021 y Fw(\000)16 b FA( )991 991 y Fx(u)988 1042 y Fy(0)1034 1021 y FB(,)27 b(where)f FA( )1380 991 y Fx(u)1450 1021 y FB(is)g(the)h(function)g(de\014ned)g(in)g(\(61\))f(and)g FA( )2770 991 y Fx(u)2767 1042 y Fy(0)2841 1021 y FB(is)g(the)h(one)f (giv)n(en)g(in)g(Theorem)71 1121 y(4.12.)p 3790 1121 4 57 v 3794 1068 50 4 v 3794 1121 V 3843 1121 4 57 v 71 1332 a Fp(The)33 b(\014xed)f(p)s(oin)m(t)g(equation)h(for)f FA(`)19 b Fw(\000)f FB(2)p FA(r)26 b FB(=)e(0)83 b(W)-7 b(e)28 b(dev)n(ote)g(this)g(section)g(to)g(pro)n(v)n(e)e(Prop)r (osition)h(8.3)g(under)h(the)71 1432 y(h)n(yp)r(othesis)j FA(`)21 b Fw(\000)g FB(2)p FA(r)32 b FB(=)e(0.)49 b(No)n(w,)32 b(the)g(term)38 b(^)-48 b FA(\026Q)1629 1444 y Fy(1)1666 1432 y FB(\()p FA(\034)9 b FB(\))p FA(z)1818 1402 y Fv(\000)p Fy(\()p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(\))2102 1432 y FB(=)36 b(^)-49 b FA(\026Q)2312 1444 y Fy(1)2349 1432 y FB(\()p FA(\034)9 b FB(\))33 b(in)f Fw(W)39 b FB(\(see)32 b(\(256\))o(\))g(is)f(not)h(small.)49 b(Then,)71 1532 y(follo)n(wing)26 b([Bal06)o(],)i(the)g(\014rst)f(step)h(is)f(to)h(p)r (erform)f(the)h(c)n(hange)e(of)i(v)-5 b(ariables)1673 1691 y FA(z)26 b FB(=)d FA(x)c FB(+)24 b(^)-48 b FA(\026F)2078 1703 y Fy(1)2115 1691 y FB(\()p FA(\034)9 b FB(\))p FA(;)1414 b FB(\(262\))71 1850 y(where)27 b FA(F)364 1862 y Fy(1)429 1850 y FB(is)h(the)g(function)g(de\014ned)g(in)g(\(74\))o(.)37 b(Then,)28 b(w)n(e)f(de\014ne)1466 2009 y Fz(b)-57 b FA(')p FB(\()p FA(x;)14 b(\034)9 b FB(\))25 b(=)d FA(')14 b FB(\()q FA(x)19 b FB(+)24 b(^)-48 b FA(\026F)2167 2021 y Fy(1)2204 2009 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))16 b FA(;)71 2168 y FB(whic)n(h)27 b(satis\014es)g (equation)1608 2267 y Fw(L)11 b Fz(b)-57 b FA(')23 b FB(=)1840 2246 y Fz(c)1830 2267 y Fw(W)7 b FB(\()p FA(@)1995 2279 y Fx(x)2048 2267 y Fz(b)-57 b FA(';)14 b(x;)g(\034)9 b FB(\))p FA(;)1349 b FB(\(263\))71 2403 y(with)1011 2481 y Fz(c)1001 2502 y Fw(W)7 b FB(\()p FA(w)r(;)14 b(x;)g(\034)9 b FB(\))25 b(=)e FA(L)p FB(\()p FA(x)c FB(+)24 b(^)-48 b FA(\026F)1835 2514 y Fy(1)1872 2502 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))21 b(+)d FA(M)9 b FB(\()p FA(x)19 b FB(+)24 b(^)-48 b FA(\026F)2573 2514 y Fy(1)2610 2502 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))p FA(w)r(:)744 b FB(\(264\))71 2638 y(W)-7 b(e)35 b(study)h(this)f(equation)f(through)h(a)f(\014xed)h(p)r (oin)n(t)h(argumen)n(t,)g(as)e(w)n(e)h(ha)n(v)n(e)f(done)g(in)i (Section)f(8.2.2.)58 b(Then,)37 b(w)n(e)71 2737 y(de\014ne)319 2715 y Fz(b)311 2737 y FA(\036)23 b FB(=)g FA(@)515 2749 y Fx(x)568 2737 y Fz(b)-57 b FA(')p FB(,)28 b(whic)n(h)f(is)h(a)f (solution)g(of)1517 2917 y Fw(L)1583 2895 y Fz(b)1574 2917 y FA(\036)c FB(=)g FA(@)1778 2929 y Fx(x)1834 2824 y Fz(h)1883 2896 y(c)1873 2917 y Fw(W)7 b FB(\()p FA(@)2038 2929 y Fx(x)2088 2895 y Fz(b)2080 2917 y FA(\036;)14 b(x;)g(\034)9 b FB(\))2327 2824 y Fz(i)2382 2917 y FA(:)71 3120 y FB(Let)28 b(us)f(tak)n(e)g FA(c)542 3090 y Fv(00)542 3141 y Fy(0)607 3120 y Fw(2)d FB(\()p FA(c)754 3090 y Fv(0)754 3141 y Fy(0)791 3120 y FA(;)14 b(c)864 3132 y Fy(0)901 3120 y FB(\))28 b(and)g FA(\024)1171 3090 y Fv(00)1171 3141 y Fy(5)1236 3120 y FA(>)23 b(\024)1372 3132 y Fy(5)1409 3120 y FB(.)37 b(Then,)28 b(w)n(e)f(lo)r(ok)g(for)2146 3098 y Fz(b)2138 3120 y FA(\036)h FB(de\014ned)g(for)f(\()p FA(x;)14 b(\034)9 b FB(\))25 b Fw(2)e(D)2990 3080 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)2988 3148 y(\024)3027 3128 y Fl(00)3027 3166 y Fu(5)3068 3148 y Fx(;c)3118 3128 y Fl(00)3118 3166 y Fu(0)3198 3120 y Fw(\002)18 b Ft(T)3336 3132 y Fx(\033)3381 3120 y FB(.)195 3248 y(T)-7 b(o)25 b(in)n(v)n(ert)f(the)h (op)r(erator)f Fw(L)f FB(=)f FA(@)1234 3260 y Fx(\034)1289 3248 y FB(+)13 b FA(@)1411 3260 y Fx(x)1453 3248 y FB(,)25 b(w)n(e)g(consider)f(the)h(op)r(erator)p 2414 3181 55 4 v 23 w Fw(G)30 b FB(de\014ned)c(in)f(\(251\))f(and)h(initial)g (conditions)71 3347 y(as)i(w)n(e)g(ha)n(v)n(e)f(done)i(in)g(Section)f (8.2.2.)36 b(Th)n(us,)27 b(w)n(e)g(de\014ne)1023 3497 y Fz(c)1020 3518 y FA(W)1098 3530 y Fy(0)1136 3518 y FB(\()p FA(x;)14 b(\034)9 b FB(\))24 b(=)1433 3439 y Fz(X)1432 3618 y Fx(k)q(<)p Fy(0)1567 3518 y FA(@)1611 3530 y Fx(z)1650 3518 y FA(')1704 3484 y Fy([)p Fx(k)q Fy(])1782 3518 y FB(\()p FA(x)1861 3530 y Fy(1)1918 3518 y FB(+)g(^)-48 b FA(\026F)2104 3530 y Fy(1)2141 3518 y FB(\()p FA(\034)9 b FB(\)\))p FA(e)2321 3484 y Fv(\000)p Fx(ik)q Fy(\()p Fx(x)p Fv(\000)p Fx(x)2586 3492 y Fu(1)2620 3484 y Fy(\))2650 3518 y FA(e)2689 3484 y Fx(ik)q(\034)1436 3751 y FB(+)1520 3672 y Fz(X)1519 3851 y Fx(k)q Fv(\025)p Fy(0)1655 3751 y FA(@)1699 3763 y Fx(z)1737 3751 y FA(')1791 3717 y Fy([)p Fx(k)q Fy(])1870 3751 y FB(\()p FA(x)1949 3763 y Fy(2)2005 3751 y FB(+)25 b(^)-49 b FA(\026F)2191 3763 y Fy(1)2229 3751 y FB(\()p FA(\034)9 b FB(\)\))p FA(e)2409 3717 y Fv(\000)p Fx(ik)q Fy(\()p Fx(x)p Fv(\000)p Fx(x)2674 3725 y Fu(2)2708 3717 y Fy(\))2738 3751 y FA(e)2777 3717 y Fx(ik)q(\034)2878 3751 y FA(;)3661 3672 y FB(\(265\))71 4017 y(where)37 b FA(x)368 4029 y Fy(1)443 4017 y FB(and)g FA(x)661 4029 y Fy(2)737 4017 y FB(are)f(the)i(v)n(ertices)e(of)i Fw(D)1519 3977 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)1517 4045 y(\024)1556 4025 y Fl(00)1556 4063 y Fu(5)1597 4045 y Fx(;c)1647 4025 y Fl(00)1647 4063 y Fu(0)1709 4017 y FB(.)66 b(Since)38 b FA(c)2061 3987 y Fv(00)2061 4037 y Fy(0)2143 4017 y Fw(2)i FB(\()p FA(c)2306 3987 y Fv(0)2306 4037 y Fy(0)2343 4017 y FA(;)14 b(c)2416 4029 y Fy(0)2453 4017 y FB(\),)41 b FA(x)2596 4029 y Fy(1)2633 4017 y FA(;)14 b(x)2717 4029 y Fy(2)2795 4017 y Fw(2)39 b(I)2940 3977 y Fy(+)p Fx(;u)2934 4045 y(c)2964 4053 y Fu(0)2997 4045 y Fx(;c)3047 4025 y Fl(0)3047 4063 y Fu(0)3120 4017 y FB(and)f(then)g FA(@)3535 4029 y Fx(z)3573 4017 y FA(')g FB(is)f(al-)71 4133 y(ready)30 b(de\014ned)i(in)f FA(x)740 4145 y Fx(i)790 4133 y FB(+)20 b FA(\026F)978 4145 y Fy(1)1016 4133 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(i)30 b FB(=)f(1)p FA(;)14 b FB(2)30 b(and)h(moreo)n(v)n(er,)f(w)n(e)h (can)g(use)g(the)h(b)r(ounds)f(in)h(Lemma)f(8.6.)47 b(Then,)33 b(it)f(is)71 4246 y(straigh)n(tforw)n(ard)24 b(to)k(see)f(that)1075 4225 y Fz(c)1071 4246 y FA(W)1149 4258 y Fy(0)1214 4246 y FB(satis\014es)g(the)h(same)f(prop)r(erties)g(as)g(the)h(function)g FA(W)2910 4258 y Fy(0)2975 4246 y FB(giv)n(en)f(in)h(Lemma)f(8.7.)195 4345 y(The)h(function)700 4323 y Fz(b)691 4345 y FA(\036)g FB(is)g(a)f(solution)g(of)h(the)g(in)n(tegral)e(equation)1601 4482 y Fz(b)1593 4504 y FA(\036)d FB(=)1756 4483 y Fz(c)1753 4504 y FA(W)1831 4516 y Fy(0)1887 4504 y FB(+)p 1970 4438 V 18 w Fw(G)h(\016)2112 4483 y Fz(c)2103 4504 y Fw(W)6 b FB(\()2232 4482 y Fz(b)2223 4504 y FA(\036)q FB(\))p FA(:)71 4675 y FB(W)-7 b(e)28 b(study)451 4653 y Fz(b)443 4675 y FA(\036)23 b Fw(2)h(Z)654 4695 y Fy(2)p Fx(r)r Fv(\000)785 4673 y Fu(1)p 781 4682 35 3 v 781 4715 a Fm(\014)826 4695 y Fx(;\033)918 4675 y FB(as)j(a)g(\014xed)h(p)r (oin)n(t)g(of)f(the)h(op)r(erator)1640 4853 y Fz(f)1630 4874 y Fw(W)i FB(=)1833 4853 y Fz(c)1830 4874 y FA(W)1908 4886 y Fy(0)1964 4874 y FB(+)p 2047 4807 55 4 v 18 w Fw(G)23 b(\016)2189 4853 y Fz(c)2179 4874 y Fw(W)7 b FA(:)1370 b FB(\(266\))p Black 71 5033 a Fp(Lemma)41 b(8.9.)p Black 45 w Fs(L)l(et)d(us)f(c)l(onsider)h FA(\015)k Fw(2)c FB(\(0)p FA(;)14 b(\015)1549 5045 y Fy(2)1586 5033 y FB(\))p Fs(,)40 b FA(")1722 5045 y Fy(0)1796 5033 y FA(>)d FB(0)g Fs(smal)t(l)i(enough)f(and)g FA(\024)2713 5003 y Fv(00)2713 5054 y Fy(5)2792 5033 y FA(>)f(\024)2942 5045 y Fy(5)3017 5033 y Fs(big)h(enough.)63 b(Then,)40 b(for)71 5145 y FA(")23 b Fw(2)g FB(\(0)p FA(;)14 b(")361 5157 y Fy(0)398 5145 y FB(\))p Fs(,)30 b(the)g(op)l(er)l(ator)960 5124 y Fz(f)950 5145 y Fw(W)36 b Fs(is)30 b(c)l(ontr)l(active)h(fr)l (om)f Fw(Z)1835 5166 y Fy(2)p Fx(r)r Fv(\000)1966 5144 y Fu(1)p 1963 5153 35 3 v 1963 5186 a Fm(\014)2007 5166 y Fx(;\024)2066 5146 y Fl(00)2066 5184 y Fu(5)2107 5166 y Fx(;c)2157 5146 y Fl(00)2157 5184 y Fu(0)2197 5166 y Fx(;\033)2291 5145 y Fs(to)g(itself.)195 5286 y(Then,)h(ther)l(e)f (exists)f(a)i(c)l(onstant)e FA(b)1311 5298 y Fy(10)1404 5286 y FA(>)22 b FB(0)29 b Fs(such)h(that)1930 5264 y Fz(b)1921 5286 y FA(\036)p Fs(,)h(the)f(unique)f(\014xe)l(d)g(p)l(oint) h(of)2940 5265 y Fz(f)2930 5286 y Fw(W)7 b Fs(,)30 b(satis\014es)1512 5455 y Fw(k)1563 5434 y Fz(b)1554 5455 y FA(\036)p Fw(k)1645 5476 y Fy(2)p Fx(r)r Fv(\000)1776 5454 y Fu(1)p 1772 5463 V 1772 5496 a Fm(\014)1817 5476 y Fx(;\024)1876 5456 y Fl(00)1876 5494 y Fu(5)1917 5476 y Fx(;c)1967 5456 y Fl(00)1967 5494 y Fu(0)2007 5476 y Fx(;\033)2094 5455 y Fw(\024)23 b FA(b)2218 5467 y Fy(10)2288 5455 y FA(")2340 5396 y Fu(1)p 2337 5405 V 2337 5438 a Fm(\014)2386 5455 y FA(:)p Black 1919 5753 a FB(93)p Black eop end %%Page: 94 94 TeXDict begin 94 93 bop Black Black Black 71 272 a Fs(Pr)l(o)l(of.)p Black 43 w FB(The)30 b(pro)r(of)e(of)i(this)g(lemma)f(is)h(completely)f (analogous)e(to)j(the)f(pro)r(of)g(of)h(Lemma)f(8.8.)42 b(The)29 b(only)h(fact)f(that)71 372 y(one)24 b(has)f(to)h(tak)n(e)f (in)n(to)h(accoun)n(t)f(is)h(that)h(the)f(functions)h FA(L)p FB(\()p FA(x)11 b FB(+)18 b(^)-49 b FA(\026F)2183 384 y Fy(1)2221 372 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))26 b(and)e FA(M)9 b FB(\()p FA(x)i FB(+)18 b(^)-49 b FA(\026)q(F)2988 384 y Fy(1)3026 372 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))26 b(satisfy)e(the)g(same) 71 471 y(prop)r(erties)j(as)f FA(L)p FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))28 b(and)f FA(M)9 b FB(\()p FA(z)t(;)14 b(\034)9 b FB(\),)28 b(whic)n(h)g(are)e(giv)n(en)h(in)h(Lemma)f(8.5.)p 3790 471 4 57 v 3794 419 50 4 v 3794 471 V 3843 471 4 57 v 195 636 a(T)-7 b(o)32 b(pro)n(v)n(e)e(Prop)r(osition)h(8.3)g(for)h FA(`)21 b Fw(\000)g FB(2)p FA(r)32 b FB(=)e(0,)j(it)g(is)e(enough)h(to) g(undo)g(the)g(c)n(hange)f(of)h(v)-5 b(ariables)31 b(\(262\))o(.)50 b(Then,)71 747 y(taking)22 b FA(\036)p FB(\()p FA(z)t(;)14 b(\034)9 b FB(\))24 b(=)681 725 y Fz(b)672 747 y FA(\036)p FB(\()p FA(x)9 b Fw(\000)17 b FB(^)-50 b FA(\026)r(F)988 759 y Fy(1)1025 747 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\),)27 b(w)n(e)22 b(reco)n(v)n(er)f FA(@)1739 759 y Fx(z)1777 747 y FA(')j FB(whic)n(h)f(is)g(de\014ned)g(for)g(\()p FA(z)t(;)14 b(\034)9 b FB(\))23 b Fw(2)h FA(D)2933 716 y Fy(in)o Fx(;)p Fy(+)p Fx(;u)2931 767 y(\024)2970 775 y Fu(6)3002 767 y Fx(;c)3052 775 y Fu(1)3131 747 y Fw(\002)9 b Ft(T)3260 759 y Fx(\033)3305 747 y FB(,)25 b(where)d FA(c)3624 759 y Fy(1)3684 747 y FA(<)h(c)3808 716 y Fv(00)3808 767 y Fy(0)71 846 y FB(and)k FA(\024)280 858 y Fy(6)340 846 y FA(>)c(\024)476 816 y Fv(00)476 867 y Fy(5)518 846 y FB(.)71 1119 y FC(9)135 b(An)39 b(injectiv)l(e)i(solution)g(of)f (the)g(partial)i(di\013eren)l(tial)g(equation)3569 1089 y Fe(e)3544 1119 y Fd(L)3626 1137 y FA(")3669 1119 y Fc(\030)d Fb(=)273 1269 y(0)71 1451 y FB(In)f(this)h(section)f(w)n(e)f (pro)n(v)n(e)g(the)i(existence)f(and)g(pro)n(vide)f(useful)h(prop)r (erties)g(of)g(a)g(solution)f FA(\030)3204 1463 y Fy(0)3280 1451 y FB(of)h(the)h(equation)88 1529 y Fz(e)71 1550 y Fw(L)128 1562 y Fx(")164 1550 y FA(\030)27 b FB(=)22 b(0)28 b(\(see)f(\(77\)\))h(of)f(the)h(form)1426 1650 y FA(\030)1462 1662 y Fy(0)1500 1650 y FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b(=)g FA(")1844 1616 y Fv(\000)p Fy(1)1933 1650 y FA(u)18 b Fw(\000)g FA(\034)28 b FB(+)18 b Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(:)71 1794 y FB(The)28 b(function)g Fw(C)k FB(m)n(ust)c(satisfy)1524 1894 y Fw(L)1581 1906 y Fx(")1617 1894 y Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b(=)g Fw(F)f FB(\()p Fw(C)5 b FB(\))13 b(\()p FA(u;)h(\034)9 b FB(\))p FA(;)1265 b FB(\(267\))71 2038 y(where)27 b Fw(L)368 2050 y Fx(")431 2038 y FB(is)h(the)g(op)r(erator)e(in)i(\(45\))o(,)1145 2212 y Fw(F)8 b FB(\()p Fw(C)d FB(\)\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)f Fw(\000)p FA(")1736 2178 y Fv(\000)p Fy(1)1824 2212 y FA(G)p FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b Fw(\000)f FA(G)p FB(\()p FA(u;)c(\034)9 b FB(\))p FA(@)2488 2224 y Fx(u)2533 2212 y Fw(C)c FB(\()p FA(u;)14 b(\034)9 b FB(\))885 b(\(268\))71 2386 y(and)27 b FA(G)h FB(is)f(the)h(function) g(de\014ned)f(in)h(\(79\))f(\(case)g FA(`)17 b Fw(\000)h FB(2)p FA(r)25 b(<)e FB(0\))k(and)g(\(97\))g(\(case)g FA(`)17 b Fw(\000)h FB(2)p FA(r)25 b Fw(\025)e FB(0\).)37 b(W)-7 b(e)27 b(dev)n(ote)g(the)h(rest)f(of)71 2485 y(the)h(section)f (to)h(obtain)f(a)g(solution)g(of)h(this)g(equation)f(in)g(b)r(oth)h (cases.)71 2716 y Fq(9.1)112 b(Banac)m(h)38 b(spaces)h(and)f(tec)m (hnical)g(lemmas)71 2870 y FB(This)32 b(section)g(is)h(dev)n(oted)f(to) g(de\014ne)h(the)f(Banac)n(h)g(spaces)f(and)h(to)h(state)f(some)g(tec)n (hnical)g(lemmas)g(whic)n(h)g(will)h(b)r(e)71 2969 y(used)27 b(in)h(Sections)g(9.2)e(and)i(9.3.)195 3069 y(W)-7 b(e)29 b(start)e(b)n(y)h(de\014ning)g(some)f(norms.)37 b(Giv)n(en)28 b FA(\027)g Fw(\025)c FB(0)j(and)h(an)f(analytic)h(function)g FA(h)c FB(:)f FA(R)3027 3081 y Fx(\024;d)3148 3069 y Fw(!)h Ft(C)p FB(,)k(where)f FA(R)3669 3081 y Fx(\024;d)3795 3069 y FB(is)71 3168 y(the)h(domain)f(de\014ned)h(in)g(\(27\))o(,)g(w)n (e)f(consider)1254 3367 y Fw(k)p FA(h)p Fw(k)1386 3379 y Fx(\027;\024;d)1558 3367 y FB(=)72 b(sup)1646 3437 y Fx(u)p Fv(2)p Fx(R)1780 3446 y Fm(\024;d)1882 3272 y Fz(\014)1882 3322 y(\014)1882 3372 y(\014)1910 3300 y(\000)1948 3367 y FA(u)1996 3333 y Fy(2)2051 3367 y FB(+)18 b FA(a)2178 3333 y Fy(2)2215 3300 y Fz(\001)2253 3316 y Fx(\027)2308 3367 y FA(h)p FB(\()p FA(u)p FB(\))2468 3272 y Fz(\014)2468 3322 y(\014)2468 3372 y(\014)1232 3570 y Fw(k)p FA(h)p Fw(k)1364 3582 y Fy(ln)n Fx(;\024;d)1558 3570 y FB(=)72 b(sup)1646 3640 y Fx(u)p Fv(2)p Fx(R)1780 3649 y Fm(\024;d)1882 3499 y Fz(\014)1882 3549 y(\014)1910 3570 y FB(ln)1979 3533 y Fv(\000)p Fy(1)2082 3499 y Fz(\014)2082 3549 y(\014)2110 3570 y FA(u)2158 3536 y Fy(2)2213 3570 y FB(+)18 b FA(a)2340 3536 y Fy(2)2377 3499 y Fz(\014)2377 3549 y(\014)2423 3570 y Fw(\001)h FA(h)p FB(\()p FA(u)p FB(\))2625 3499 y Fz(\014)2625 3549 y(\014)2666 3570 y FA(:)71 3805 y FB(Moreo)n(v)n(er)30 b(for)i(analytic)h(functions)g FA(h)f FB(:)g FA(R)1456 3817 y Fx(\024;d)1575 3805 y Fw(\002)22 b Ft(T)1717 3817 y Fx(\033)1794 3805 y Fw(!)32 b Ft(C)h FB(whic)n(h)g(are)f(2)p FA(\031)s FB(-p)r(erio)r(dic)g(in)h FA(\034)9 b FB(,)36 b(w)n(e)c(consider)g(the)h(corre-)71 3905 y(sp)r(onding)27 b(F)-7 b(ourier)27 b(norms)1385 4099 y Fw(k)p FA(h)p Fw(k)1517 4111 y Fx(\027;\024;d;\033)1749 4099 y FB(=)1837 4020 y Fz(X)1837 4199 y Fx(k)q Fv(2)p Fn(Z)1971 4004 y Fz(\015)1971 4053 y(\015)1971 4103 y(\015)2017 4099 y FA(h)2065 4065 y Fy([)p Fx(k)q Fy(])2143 4004 y Fz(\015)2143 4053 y(\015)2143 4103 y(\015)2190 4157 y Fx(\027;\024;d)2354 4099 y FA(e)2393 4065 y Fv(j)p Fx(k)q Fv(j)p Fx(\033)1363 4340 y Fw(k)p FA(h)p Fw(k)1495 4352 y Fy(ln)n Fx(;\024;d;\033)1749 4340 y FB(=)1837 4261 y Fz(X)1837 4440 y Fx(k)q Fv(2)p Fn(Z)1971 4245 y Fz(\015)1971 4295 y(\015)1971 4345 y(\015)2017 4340 y FA(h)2065 4306 y Fy([)p Fx(k)q Fy(])2143 4245 y Fz(\015)2143 4295 y(\015)2143 4345 y(\015)2190 4398 y Fy(ln)o Fx(;\024;d)2376 4340 y FA(e)2415 4306 y Fv(j)p Fx(k)q Fv(j)p Fx(\033)2535 4340 y FA(:)71 4593 y FB(W)-7 b(e)28 b(consider,)e(th)n(us,)i(the)g (follo)n(wing)f(function)h(spaces)840 4762 y Fw(X)899 4774 y Fx(\027;\024;d;\033)1133 4762 y FB(=)23 b Fw(f)p FA(h)f FB(:)h FA(R)1442 4774 y Fx(\024;d)1558 4762 y Fw(\002)18 b Ft(T)1696 4774 y Fx(\033)1764 4762 y Fw(!)23 b Ft(C)p FB(;)42 b(real-analytic)n FA(;)14 b Fw(k)p FA(h)p Fw(k)2614 4774 y Fx(\027;\024;d;\033)2846 4762 y FA(<)23 b Fw(1g)818 4886 y(X)877 4898 y Fy(ln)p Fx(;\024;d;\033)1133 4886 y FB(=)g Fw(f)p FA(h)f FB(:)h FA(R)1442 4898 y Fx(\024;d)1558 4886 y Fw(\002)18 b Ft(T)1696 4898 y Fx(\033)1764 4886 y Fw(!)23 b Ft(C)p FB(;)42 b(real-analytic)n FA(;)14 b Fw(k)p FA(h)p Fw(k)2614 4898 y Fy(ln)n Fx(;\024;d;\033)2868 4886 y FA(<)23 b Fw(1g)p FA(;)3661 4825 y FB(\(269\))71 5056 y(whic)n(h)k(can)h(b)r(e)g(c)n(hec)n(k)n(ed)e(that)i(are)f(a)g (Banac)n(h)f(spaces.)195 5156 y(If)i(there)g(is)f(no)h(danger)e(of)i (confusion)f(ab)r(out)g(the)h(de\014nition)g(domain)f FA(R)2531 5168 y Fx(\024;d)2656 5156 y FB(w)n(e)h(will)f(denote)1150 5329 y Fw(k)18 b(\001)g(k)1293 5341 y Fx(\027;\033)1414 5329 y FB(=)k Fw(k)c(\001)h(k)1645 5341 y Fx(\027;\024;d;\033)1966 5329 y FB(and)110 b Fw(X)2269 5341 y Fx(\027;\033)2391 5329 y FB(=)22 b Fw(X)2537 5341 y Fx(\027;\024;d;\033)2748 5329 y FA(:)195 5504 y FB(In)28 b(the)g(next)g(lemma,)g(w)n(e)f(state)g (some)g(prop)r(erties)g(of)g(these)h(Banac)n(h)e(spaces.)p Black 1919 5753 a(94)p Black eop end %%Page: 95 95 TeXDict begin 95 94 bop Black Black Black 71 272 a Fp(Lemma)31 b(9.1.)p Black 40 w Fs(The)g(fol)t(lowing)h(statements)d(hold:)p Black 169 435 a(1.)p Black 42 w(If)i FA(\027)407 447 y Fy(1)467 435 y Fw(\025)23 b FA(\027)596 447 y Fy(2)656 435 y Fw(\025)f FB(0)p Fs(,)30 b Fw(X)899 447 y Fx(\027)932 455 y Fu(1)965 447 y Fx(;\033)1053 435 y Fw(\032)22 b(X)1199 447 y Fx(\027)1232 455 y Fu(2)1265 447 y Fx(;\033)1359 435 y Fs(and)30 b(mor)l(e)l(over)h(if)g FA(h)23 b Fw(2)g(X)2168 447 y Fx(\027)2201 455 y Fu(1)2234 447 y Fx(;\033)2298 435 y Fs(,)1529 614 y Fw(k)p FA(h)p Fw(k)1661 626 y Fx(\027)1694 634 y Fu(2)1726 626 y Fx(;\033)1813 614 y Fw(\024)g FA(K)6 b FB(\()p FA(\024")p FB(\))2129 580 y Fx(\027)2162 588 y Fu(2)2194 580 y Fv(\000)p Fx(\027)2279 588 y Fu(1)2316 614 y Fw(k)p FA(h)p Fw(k)2448 626 y Fx(\027)2481 634 y Fu(1)2512 626 y Fx(;\033)2577 614 y FA(:)p Black 169 826 a Fs(2.)p Black 42 w(If)31 b FB(0)22 b Fw(\024)h FA(\027)559 838 y Fy(1)619 826 y Fw(\024)g FA(\027)748 838 y Fy(2)785 826 y Fs(,)30 b Fw(X)899 838 y Fx(\027)932 846 y Fu(1)965 838 y Fx(;\033)1053 826 y Fw(\032)22 b(X)1199 838 y Fx(\027)1232 846 y Fu(2)1265 838 y Fx(;\033)1359 826 y Fs(and)30 b(mor)l(e)l(over)h(if)g FA(h)23 b Fw(2)g(X)2168 838 y Fx(\027)2201 846 y Fu(1)2234 838 y Fx(;\033)2298 826 y Fs(,)1698 1005 y Fw(k)p FA(h)p Fw(k)1830 1017 y Fx(\027)1863 1025 y Fu(2)1895 1017 y Fx(;\033)1982 1005 y Fw(\024)g FA(K)6 b Fw(k)p FA(h)p Fw(k)2279 1017 y Fx(\027)2312 1025 y Fu(1)2343 1017 y Fx(;\033)2407 1005 y FA(:)p Black 169 1217 a Fs(3.)p Black 42 w(If)31 b FA(h)22 b Fw(2)i(X)574 1229 y Fx(\027)607 1237 y Fu(1)640 1229 y Fx(;\033)734 1217 y Fs(and)30 b FA(g)c Fw(2)d(X)1098 1229 y Fx(\027)1131 1237 y Fu(2)1164 1229 y Fx(;\033)1229 1217 y Fs(,)30 b(then)f FA(hg)d Fw(2)d(X)1719 1229 y Fx(\027)1752 1237 y Fu(1)1785 1229 y Fy(+)p Fx(\027)1869 1237 y Fu(2)1902 1229 y Fx(;\033)1996 1217 y Fs(and)1529 1396 y Fw(k)p FA(hg)s Fw(k)1704 1408 y Fx(\027)1737 1416 y Fu(1)1768 1408 y Fy(+)p Fx(\027)1852 1416 y Fu(2)1885 1408 y Fx(;\033)1972 1396 y Fw(\024)g(k)p FA(h)p Fw(k)2192 1408 y Fx(\027)2225 1416 y Fu(1)2257 1408 y Fx(;\033)2321 1396 y Fw(k)p FA(g)s Fw(k)2448 1408 y Fx(\027)2481 1416 y Fu(2)2512 1408 y Fx(;\033)2577 1396 y FA(:)p Black 169 1608 a Fs(4.)p Black 42 w(L)l(et)38 b(us)g(c)l(onsider)h FA(d)f(>)g(d)1110 1577 y Fv(0)1171 1608 y FA(>)g FB(0)p Fs(,)i(such)e(that)g FA(d)25 b Fw(\000)f FA(d)1956 1577 y Fv(0)2017 1608 y Fs(has)39 b(a)g(p)l(ositive)g(lower)g(b)l(ound)f(indep)l(endent)h(of)g FA(")p Fs(,)h(and)278 1707 y FA(h)23 b Fw(2)h(X)487 1719 y Fx(\027;\024;d;\033)698 1707 y Fs(.)38 b(Then,)31 b FA(@)1047 1719 y Fx(u)1091 1707 y FA(h)23 b Fw(2)g(X)1299 1719 y Fx(\027;)p Fy(2)p Fx(\024;d)1479 1703 y Fl(0)1501 1719 y Fx(;\033)1595 1707 y Fs(and)31 b(satis\014es)1531 1932 y Fw(k)p FA(@)1617 1944 y Fx(u)1660 1932 y FA(h)p Fw(k)1750 1944 y Fx(\027;)p Fy(2)p Fx(\024;d)1930 1928 y Fl(0)1951 1944 y Fx(;\033)2039 1932 y Fw(\024)2142 1876 y FA(K)p 2136 1913 87 4 v 2136 1989 a(\024")2233 1932 y Fw(k)p FA(h)p Fw(k)2365 1944 y Fx(\027;\024;d;\033)2575 1932 y FA(:)195 2167 y FB(Throughout)22 b(this)i(section)e(w)n(e)h(are) f(going)g(to)h(solv)n(e)e(equations)i(of)g(the)g(form)g Fw(L)2691 2179 y Fx(")2726 2167 y FA(h)g FB(=)g FA(g)s FB(,)h(where)e Fw(L)3267 2179 y Fx(")3326 2167 y FB(is)h(the)g(op)r (erator)71 2266 y(de\014ned)k(in)h(\(45\).)36 b(T)-7 b(o)27 b(\014nd)h(a)f(righ)n(t-in)n(v)n(erse)d(of)k(this)f(op)r(erator) f(in)h FA(R)2242 2278 y Fx(\024;d)2367 2266 y FB(let)g(us)g(consider)g FA(u)2964 2278 y Fy(1)3023 2266 y FB(=)c FA(i)p FB(\()p FA(a)18 b Fw(\000)f FA(\024")p FB(\))27 b(and)g FA(u)3671 2278 y Fy(0)3735 2266 y FB(the)71 2366 y(left)h(endp)r(oin)n(t)g(of)g FA(R)720 2378 y Fx(\024;d)835 2366 y Fw(\\)19 b Ft(R)p FB(.)37 b(Then,)28 b(w)n(e)f(de\014ne)h(the)g(op)r(erator)e Fw(G)2158 2378 y Fx(")2221 2366 y FB(as)1367 2565 y Fw(G)1416 2577 y Fx(")1452 2565 y FB(\()p FA(h)p FB(\)\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)1869 2486 y Fz(X)1869 2664 y Fx(k)q Fv(2)p Fn(Z)2003 2565 y Fw(G)2052 2577 y Fx(")2088 2565 y FB(\()p FA(h)p FB(\))2200 2530 y Fy([)p Fx(k)q Fy(])2279 2565 y FB(\()p FA(u)p FB(\))p FA(e)2430 2530 y Fx(ik)q(\034)2532 2565 y FA(;)1106 b FB(\(270\))71 2823 y(where)27 b(its)h(F)-7 b(ourier)26 b(co)r(e\016cien)n(ts)h(are)g (giv)n(en)g(b)n(y)731 3044 y Fw(G)780 3056 y Fx(")816 3044 y FB(\()p FA(h)p FB(\))928 3010 y Fy([)p Fx(k)q Fy(])1007 3044 y FB(\()p FA(u)p FB(\))c(=)1230 2931 y Fz(Z)1313 2952 y Fx(u)1276 3120 y Fv(\000)p Fx(u)1367 3128 y Fu(1)1417 3044 y FA(e)1456 3010 y Fx(ik)q(")1546 2985 y Fl(\000)p Fu(1)1625 3010 y Fy(\()p Fx(v)r Fv(\000)p Fx(u)p Fy(\))1807 3044 y FA(h)1855 3010 y Fy([)p Fx(k)q Fy(])1934 3044 y FB(\()p FA(v)s FB(\))14 b FA(dv)778 b FB(if)28 b FA(k)e(<)d FB(0)735 3279 y Fw(G)784 3291 y Fx(")820 3279 y FB(\()p FA(h)p FB(\))932 3244 y Fy([0])1007 3279 y FB(\()p FA(u)p FB(\))g(=)1230 3166 y Fz(Z)1313 3186 y Fx(u)1276 3354 y(u)1315 3362 y Fu(0)1370 3279 y FA(h)1418 3244 y Fy([0])1493 3279 y FB(\()p FA(v)s FB(\))14 b FA(dv)731 3513 y Fw(G)780 3525 y Fx(")816 3513 y FB(\()p FA(h)p FB(\))928 3479 y Fy([)p Fx(k)q Fy(])1007 3513 y FB(\()p FA(u)p FB(\))23 b(=)g Fw(\000)1309 3400 y Fz(Z)1391 3421 y Fx(u)1430 3429 y Fu(1)1354 3589 y Fx(u)1481 3513 y FA(e)1520 3479 y Fx(ik)q(")1610 3454 y Fl(\000)p Fu(1)1688 3479 y Fy(\()p Fx(v)r Fv(\000)p Fx(u)p Fy(\))1871 3513 y FA(h)1919 3479 y Fy([)o Fx(k)q Fy(])1997 3513 y FB(\()p FA(v)s FB(\))14 b FA(dv)692 b FB(if)28 b FA(k)e(>)d FB(0)p FA(:)71 3734 y FB(where)k(w)n(e)g(mak)n (e)g(the)h(in)n(tegrals)e(along)g(an)n(y)h(path)h(con)n(tained)f(in)h FA(R)2236 3746 y Fx(\024;d)2333 3734 y FB(.)195 3834 y(Let)i(us)f(p)r(oin)n(t)h(that)f(w)n(e)g(will)h(apply)f(this)h(op)r (erator)d(to)j(functions)f(de\014ned)h(in)g FA(R)2780 3846 y Fx(\024;d)2897 3834 y Fw(\002)19 b Ft(T)3036 3846 y Fx(\033)3110 3834 y FB(with)30 b(di\013eren)n(t)g(v)-5 b(alues)71 3934 y(of)27 b FA(\024)h FB(and)f FA(d)h FB(and)g(then)g (the)g(de\014nition)g(of)f Fw(G)1479 3946 y Fx(")1543 3934 y FB(dep)r(ends)h(on)f(the)h(domain.)p Black 71 4097 a Fp(Lemma)j(9.2.)p Black 40 w Fs(The)g(op)l(er)l(ator)g Fw(G)1145 4109 y Fx(")1211 4097 y Fs(in)36 b FB(\(270\))29 b Fs(satis\014es)g(the)h(fol)t(lowing)j(pr)l(op)l(erties.)p Black 169 4260 a(1.)p Black 42 w(If)e FA(h)22 b Fw(2)i(X)574 4272 y Fx(\027;\033)702 4260 y Fs(for)30 b(some)h FA(\027)d Fw(\025)23 b FB(0)p Fs(,)29 b(then)h Fw(G)1534 4272 y Fx(")1570 4260 y FB(\()p FA(h)p FB(\))23 b Fw(2)h(X)1843 4272 y Fx(\027;\033)1971 4260 y Fs(and)1656 4439 y Fw(kG)1747 4451 y Fx(")1782 4439 y FB(\()p FA(h)p FB(\))p Fw(k)1936 4451 y Fx(\027;\033)2057 4439 y Fw(\024)f FA(K)6 b Fw(k)p FA(h)p Fw(k)2354 4451 y Fx(\027;\033)2450 4439 y FA(:)278 4618 y Fs(F)-6 b(urthermor)l(e,)30 b(if)h Fw(h)p FA(h)p Fw(i)24 b FB(=)e(0)p Fs(,)1623 4718 y Fw(k)o(G)1713 4730 y Fx(")1749 4718 y FB(\()p FA(h)p FB(\))p Fw(k)1903 4743 y Fx(\027;\033)2024 4718 y Fw(\024)h FA(K)6 b(")14 b Fw(k)n FA(h)p Fw(k)2371 4743 y Fx(\027;\033)2483 4718 y FA(:)p Black 169 4898 a Fs(2.)p Black 42 w(If)31 b FA(h)22 b Fw(2)i(X)574 4910 y Fx(\027;\033)702 4898 y Fs(for)30 b(some)h FA(\027)d(>)23 b FB(1)p Fs(,)29 b(then)h Fw(G)1534 4910 y Fx(")1570 4898 y FB(\()p FA(h)p FB(\))23 b Fw(2)h(X)1843 4910 y Fx(\027)t Fv(\000)p Fy(1)p Fx(;\033)2059 4898 y Fs(and)1611 5077 y Fw(kG)1702 5089 y Fx(")1738 5077 y FB(\()p FA(h)p FB(\))p Fw(k)1892 5102 y Fx(\027)t Fv(\000)p Fy(1)p Fx(;\033)2101 5077 y Fw(\024)f FA(K)6 b Fw(k)p FA(h)p Fw(k)2398 5089 y Fx(\027;\033)2494 5077 y FA(:)p Black 169 5289 a Fs(3.)p Black 42 w(If)31 b FA(h)22 b Fw(2)i(X)574 5301 y Fx(\027;\033)702 5289 y Fs(for)30 b(some)h FA(\027)d Fw(2)23 b FB(\(0)p FA(;)14 b FB(1\))p Fs(,)30 b(then)g Fw(G)1668 5301 y Fx(")1704 5289 y FB(\()p FA(h)p FB(\))23 b Fw(2)g(X)1976 5301 y Fy(0)p Fx(;\033)2104 5289 y Fs(and)1656 5468 y Fw(k)o(G)1746 5480 y Fx(")1782 5468 y FB(\()p FA(h)p FB(\))p Fw(k)1936 5493 y Fy(0)p Fx(;\033)2057 5468 y Fw(\024)f FA(K)6 b Fw(k)p FA(h)p Fw(k)2353 5480 y Fx(\027;\033)2450 5468 y FA(:)p Black 1919 5753 a FB(95)p Black eop end %%Page: 96 96 TeXDict begin 96 95 bop Black Black Black 169 272 a Fs(4.)p Black 42 w(If)31 b FA(h)22 b Fw(2)i(X)574 284 y Fy(1)p Fx(;\033)672 272 y Fs(,)30 b(then)f Fw(G)960 284 y Fx(")996 272 y FB(\()p FA(h)p FB(\))24 b Fw(2)f(X)1269 284 y Fy(ln)p Fx(;\033)1419 272 y Fs(and)1645 440 y Fw(kG)1736 452 y Fx(")1771 440 y FB(\()p FA(h)p FB(\))p Fw(k)1925 465 y Fy(ln)p Fx(;\033)2068 440 y Fw(\024)g FA(K)6 b Fw(k)p FA(h)p Fw(k)2365 452 y Fy(1)p Fx(;\033)2461 440 y FA(:)p Black 169 639 a Fs(5.)p Black 42 w(If)31 b FA(h)22 b Fw(2)i(X)574 651 y Fx(\027;\033)702 639 y Fs(for)30 b(some)h FA(\027)d Fw(\025)23 b FB(0)p Fs(,)29 b(then)h Fw(G)1534 651 y Fx(")1570 639 y FB(\()p FA(@)1646 651 y Fx(u)1690 639 y FA(h)p FB(\))23 b Fw(2)g(X)1930 651 y Fx(\027;\033)2058 639 y Fs(and)1612 807 y Fw(k)o(G)1702 819 y Fx(")1738 807 y FB(\()p FA(@)1814 819 y Fx(u)1858 807 y FA(h)p FB(\))p Fw(k)1980 832 y Fx(\027;\033)2101 807 y Fw(\024)f FA(K)6 b Fw(k)p FA(h)p Fw(k)2397 819 y Fx(\027;\033)2494 807 y FA(:)p Black 169 1005 a Fs(6.)p Black 42 w(If)31 b FA(h)22 b Fw(2)i(X)574 1017 y Fx(\027;\033)702 1005 y Fs(for)30 b(some)h FA(\027)d Fw(\025)23 b FB(0)p Fs(,)29 b(then)h FA(@)1529 1017 y Fx(u)1572 1005 y Fw(G)1621 1017 y Fx(")1657 1005 y FB(\()p FA(h)p FB(\))24 b Fw(2)f(X)1930 1017 y Fx(\027;\033)2058 1005 y Fs(and)1612 1173 y Fw(k)o FA(@)1697 1185 y Fx(u)1741 1173 y Fw(G)1790 1185 y Fx(")1826 1173 y FB(\()p FA(h)p FB(\))p Fw(k)1980 1198 y Fx(\027;\033)2101 1173 y Fw(\024)f FA(K)6 b Fw(k)p FA(h)p Fw(k)2397 1185 y Fx(\027;\033)2494 1173 y FA(:)p Black 169 1372 a Fs(7.)p Black 42 w(If)31 b FA(h)22 b Fw(2)i(X)574 1384 y Fx(\027;\033)702 1372 y Fs(for)30 b(some)h FA(\027)d Fw(\025)23 b FB(0)p Fs(,)29 b Fw(L)1357 1384 y Fx(")1411 1372 y Fw(\016)18 b(G)1520 1384 y Fx(")1556 1372 y FB(\()p FA(h)p FB(\))24 b(=)e FA(h)30 b Fs(and)427 1559 y Fw(G)476 1571 y Fx(")530 1559 y Fw(\016)18 b(L)647 1571 y Fx(")683 1559 y FB(\()p FA(h)p FB(\)\()p FA(v)s(;)c(\034)9 b FB(\))24 b(=)f FA(h)p FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b Fw(\000)1437 1481 y Fz(X)1436 1659 y Fx(k)q(<)p Fy(0)1571 1559 y FA(e)1610 1525 y Fx(ik)q(")1700 1500 y Fl(\000)p Fu(1)1779 1525 y Fy(\()p Fv(\000)p Fx(u)1896 1533 y Fu(1)1928 1525 y Fv(\000)p Fx(u)p Fy(\))2050 1559 y FA(h)2098 1525 y Fy([)p Fx(k)q Fy(])2176 1559 y FB(\()p Fw(\000)p FA(u)2321 1571 y Fy(1)2358 1559 y FB(\))e Fw(\000)g FA(h)2539 1525 y Fy([0])2614 1559 y FB(\()p FA(u)2694 1571 y Fy(0)2731 1559 y FB(\))h Fw(\000)2866 1481 y Fz(X)2865 1659 y Fx(k)q(>)p Fy(0)3000 1559 y FA(e)3039 1525 y Fx(ik)q(")3129 1500 y Fl(\000)p Fu(1)3208 1525 y Fy(\()p Fx(u)3273 1533 y Fu(1)3305 1525 y Fv(\000)p Fx(u)p Fy(\))3427 1559 y FA(h)3475 1525 y Fy([)p Fx(k)q Fy(])3553 1559 y FB(\()p FA(u)3633 1571 y Fy(1)3670 1559 y FB(\))p Black 71 1837 a Fs(Pr)l(o)l(of.)p Black 43 w FB(The)26 b(\014rst)f(four)h(statemen)n(ts)g(are)e(straigh)n (tforw)n(ard.)34 b(F)-7 b(or)25 b(the)h(\014fth)h(one,)f(one)f(has)h (to)f(in)n(tegrate)g(b)n(y)h(parts)f(and)71 1936 y(for)i(the)h(sixth)g (one)f(has)g(to)g(apply)h(Leibnitz)g(rule.)p 3790 1936 4 57 v 3794 1884 50 4 v 3794 1936 V 3843 1936 4 57 v 71 2166 a Fq(9.2)112 b(Case)38 b Fh(`)28 b(<)f FF(2)p Fh(r)s Fq(:)50 b(pro)s(of)38 b(of)f(Theorem)h(4.17)g(and)g(Prop)s (osition)g(4.18)71 2320 y Fp(9.2.1)94 b(Pro)s(of)31 b(of)h(Theorem)f (4.17)71 2473 y FB(Theorem)c(4.17)f(is)h(a)h(straigh)n(tforw)n(ard)c (consequence)j(of)g(the)h(follo)n(wing)f(prop)r(osition.)p Black 71 2627 a Fp(Prop)s(osition)36 b(9.3.)p Black 43 w Fs(L)l(et)d(us)h(c)l(onsider)h(the)g(c)l(onstants)e FA(d)1933 2639 y Fy(2)2002 2627 y FA(>)d FB(0)k Fs(and)h FA(\024)2387 2639 y Fy(3)2455 2627 y FA(>)30 b FB(0)k Fs(de\014ne)l(d)h(in)f(The)l(or)l(em)h(4.8,)i FA(d)3580 2639 y Fy(3)3649 2627 y FA(<)30 b(d)3787 2639 y Fy(2)3825 2627 y Fs(,)71 2727 y FA(")110 2739 y Fy(0)173 2727 y FA(>)c FB(0)k Fs(smal)t(l)j(enough)f(and)g FA(\024)1054 2739 y Fy(7)1117 2727 y FA(>)25 b(\024)1255 2739 y Fy(3)1324 2727 y Fs(big)32 b(enough,)h(which)g(might)f(dep)l(end)g(on)g(the)f(pr) l(evious)i(c)l(onstants.)43 b(Then,)32 b(for)71 2826 y FA(")e Fw(2)g FB(\(0)p FA(;)14 b(")375 2838 y Fy(0)412 2826 y FB(\))33 b Fs(and)i(any)f FA(\024)29 b Fw(\025)h FA(\024)1026 2838 y Fy(7)1097 2826 y Fs(such)j(that)h FA("\024)c(<)f(a)p Fs(,)35 b(ther)l(e)f(exists)f(a)h(function)f Fw(C)i FB(:)30 b FA(R)2821 2838 y Fx(\024;d)2915 2846 y Fu(3)2972 2826 y Fw(\002)21 b Ft(T)3113 2838 y Fx(\033)3188 2826 y Fw(!)30 b Ft(C)k Fs(that)f(satis\014es)71 2926 y(e)l(quation)j FB(\(267\))o Fs(.)195 3026 y(Mor)l(e)l(over,)1285 3125 y FB(\()p FA(\030)1353 3137 y Fy(0)1390 3125 y FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))25 b(=)1811 3058 y Fz(\000)1849 3125 y FA(")1888 3091 y Fv(\000)p Fy(1)1977 3125 y FA(u)18 b Fw(\000)g FA(\034)28 b FB(+)18 b Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)2598 3058 y Fz(\001)71 3266 y Fs(is)30 b(inje)l(ctive)h(and)f(ther)l(e)g(exists)f(a)h(c)l(onstant)f FA(b)1523 3278 y Fy(11)1616 3266 y FA(>)23 b FB(0)29 b Fs(indep)l(endent)i(of)f FA(")p Fs(,)g FA(\026)g Fs(and)g FA(\024)g Fs(such)f(that)1571 3429 y Fw(kC)5 b(k)1703 3454 y Fy(0)p Fx(;\033)1823 3429 y Fw(\024)23 b FA(b)1947 3441 y Fy(11)2017 3429 y Fw(j)p FA(\026)p Fw(j)p FA(")2152 3394 y Fx(\021)1484 3570 y Fw(k)o FA(@)1569 3582 y Fx(u)1613 3570 y Fw(C)5 b(k)1703 3595 y Fy(0)p Fx(;\033)1823 3570 y Fw(\024)23 b FA(b)1947 3582 y Fy(11)2017 3570 y FA(\024)2065 3536 y Fv(\000)p Fy(1)2154 3570 y Fw(j)p FA(\026)p Fw(j)p FA(")2289 3536 y Fx(\021)r Fv(\000)p Fy(1)2414 3570 y FA(:)195 3741 y FB(T)-7 b(o)41 b(pro)n(v)n(e)f(this)i(prop)r(osition,)i (\014rst)d(w)n(e)g(split)g FA(G)h FB(in)g(sev)n(eral)d(parts.)78 b(Recall)41 b(that,)k(since)c FA(`)28 b Fw(\000)f FB(2)p FA(r)48 b(<)e FB(0,)e(the)71 3840 y(p)r(erturbation)578 3819 y Fz(b)559 3840 y FA(H)628 3852 y Fy(1)694 3840 y FB(in)29 b(\(34\))f(is)h(a)f(p)r(olynomial)g(of)h(degree)f(one)g(in)h FA(p)p FB(.)40 b(Then,)29 b FA(G)h FB(can)e(b)r(e)h(split)g(as)f FA(G)d FB(=)g FA(G)3402 3852 y Fy(1)3459 3840 y FB(+)18 b FA(G)3607 3852 y Fy(2)3664 3840 y FB(+)h FA(G)3813 3852 y Fy(3)71 3940 y FB(with)1124 4108 y FA(G)1189 4120 y Fy(1)1227 4108 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)e FA(\026")1621 4074 y Fx(\021)1662 4108 y FA(p)1704 4120 y Fy(0)1741 4108 y FB(\()p FA(u)p FB(\))1853 4074 y Fv(\000)p Fy(1)1942 4108 y FA(@)1986 4120 y Fx(p)2044 4087 y Fz(b)2024 4108 y FA(H)2100 4074 y Fy(1)2093 4128 y(1)2151 4108 y FB(\()p FA(q)2220 4120 y Fy(0)2258 4108 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2449 4120 y Fy(0)2486 4108 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))949 b(\(271\))1124 4252 y FA(G)1189 4264 y Fy(2)1227 4252 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)e FA(\026")1621 4218 y Fx(\021)r Fy(+1)1746 4252 y FA(p)1788 4264 y Fy(0)1825 4252 y FB(\()p FA(u)p FB(\))1937 4218 y Fv(\000)p Fy(1)2026 4252 y FA(@)2070 4264 y Fx(p)2128 4231 y Fz(b)2108 4252 y FA(H)2184 4218 y Fy(2)2177 4273 y(1)2235 4252 y FB(\()p FA(q)2304 4264 y Fy(0)2342 4252 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2533 4264 y Fy(0)2570 4252 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))865 b(\(272\))1124 4434 y FA(G)1189 4446 y Fy(3)1227 4434 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)1542 4377 y FA(@)1586 4389 y Fx(u)1630 4377 y FA(T)1691 4347 y Fx(s)1679 4398 y Fy(1)1725 4377 y FB(\()p FA(u;)14 b(\034)9 b FB(\))20 b(+)e FA(@)2066 4389 y Fx(u)2109 4377 y FA(T)2170 4347 y Fx(u)2158 4398 y Fy(1)2212 4377 y FB(\()p FA(u;)c(\034)9 b FB(\))p 1542 4414 865 4 v 1858 4490 a(2)p FA(p)1942 4462 y Fy(2)1942 4513 y(0)1979 4490 y FB(\()p FA(u)p FB(\))2417 4434 y FA(:)1221 b FB(\(273\))71 4647 y(Next)28 b(lemma)f(giv)n(es)g(sev)n (eral)f(prop)r(erties)g(of)i(these)f(functions.)p Black 71 4801 a Fp(Lemma)d(9.4.)p Black 34 w Fs(L)l(et)f(us)g(c)l(onsider)i (any)f FA(\024)f(>)g(\024)1507 4813 y Fy(3)1567 4801 y Fs(and)h FA(d)g(<)e(d)1919 4813 y Fy(2)1957 4801 y Fs(,)j(wher)l(e)g FA(\024)2284 4813 y Fy(3)2344 4801 y Fs(and)f FA(d)2542 4813 y Fy(2)2604 4801 y Fs(ar)l(e)g(the)f(c)l (onstants)g(given)i(in)e(The)l(or)l(em)71 4901 y(4.8.)38 b(Then,)28 b(the)d(functions)g FA(G)1036 4913 y Fy(1)1074 4901 y Fs(,)i FA(G)1191 4913 y Fy(2)1253 4901 y Fs(and)f FA(G)1475 4913 y Fy(3)1538 4901 y Fs(de\014ne)l(d)g(in)31 b FB(\(271\))o Fs(,)c FB(\(272\))d Fs(and)35 b FB(\(273\))24 b Fs(r)l(esp)l(e)l(ctively,)k(have)e(the)g(fol)t(lowing)71 5001 y(pr)l(op)l(erties.)p Black 169 5155 a(1.)p Black 42 w FA(G)343 5167 y Fy(1)404 5155 y Fw(2)d(X)541 5167 y Fy(0)p Fx(;\033)669 5155 y Fs(and)30 b(it)g(satis\014es)g Fw(h)p FA(G)1322 5167 y Fy(1)1360 5155 y Fw(i)23 b FB(=)g(0)29 b Fs(and)1992 5325 y Fw(k)o FA(G)2098 5337 y Fy(1)2136 5325 y Fw(k)2177 5350 y Fy(0)p Fx(;\033)2298 5325 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2597 5291 y Fx(\021)1469 5465 y Fw(k)o FA(@)1554 5477 y Fx(v)1594 5465 y FA(G)1659 5477 y Fy(1)1696 5465 y Fw(k)1738 5490 y Fy(max)o Fv(f)p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;)p Fy(0)p Fv(g)p Fx(;\033)2298 5465 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2597 5430 y Fx(\021)2637 5465 y FA(:)p Black 1919 5753 a FB(96)p Black eop end %%Page: 97 97 TeXDict begin 97 96 bop Black Black Black 169 272 a Fs(2.)p Black 42 w FA(G)343 284 y Fy(2)404 272 y Fw(2)23 b(X)541 284 y Fy(0)p Fx(;\033)669 272 y Fs(and)30 b(it)g(satis\014es)1688 372 y Fw(k)p FA(G)1795 384 y Fy(2)1832 372 y Fw(k)1874 397 y Fy(0)p Fx(;\033)1994 372 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2294 337 y Fx(\021)r Fy(+1)2418 372 y FA(:)p Black 169 554 a Fs(3.)p Black 42 w FA(G)343 566 y Fy(3)404 554 y Fw(2)23 b(X)541 569 y Fy(max)p Fv(f)p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;)p Fy(0)p Fv(g)p Fx(;\033)1109 554 y Fs(and)30 b(it)g(satis\014es)1468 747 y Fw(k)p FA(G)1575 759 y Fy(3)1612 747 y Fw(k)1654 772 y Fy(max)o Fv(f)p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;)p Fy(0)p Fv(g)p Fx(;\033)2214 747 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2514 713 y Fx(\021)r Fy(+1)2638 747 y FA(:)p Black 71 963 a Fs(Pr)l(o)l(of.)p Black 43 w FB(The)26 b(pro)r(of)e(of)i(the)g(statemen)n(ts)f(ab)r(out)g FA(G)1664 975 y Fy(1)1727 963 y FB(and)h FA(G)1952 975 y Fy(2)2015 963 y FB(are)e(straigh)n(tforw)n(ard,)f(taking)i(in)n(to)g (accoun)n(t)g(for)g FA(G)3672 975 y Fy(2)3735 963 y FB(the)71 1062 y(b)r(ounds)31 b(obtained)f(in)h(Corollary)d(5.6.)45 b(F)-7 b(or)30 b FA(G)1569 1074 y Fy(3)1606 1062 y FB(,)i(one)e(has)g (to)g(tak)n(e)g(in)n(to)g(accoun)n(t)g(the)h(b)r(ounds)g(for)f FA(T)3364 1032 y Fx(u)3352 1083 y Fy(1)3437 1062 y FB(obtained)g(in)71 1162 y(Prop)r(osition)c(7.4)h(and)g(the)h(analogous)d(b)r(ounds)j(that) g FA(T)1869 1132 y Fx(s)1857 1183 y Fy(1)1931 1162 y FB(satis\014es.)p 3790 1162 4 57 v 3794 1109 50 4 v 3794 1162 V 3843 1162 4 57 v 195 1328 a(T)-7 b(o)34 b(pro)n(v)n(e)f(Prop)r (osition)g(9.3,)i(w)n(e)f(\014rst)g(p)r(erform)g(a)g(c)n(hange)f(of)h (v)-5 b(ariables)33 b(whic)n(h)i(reduces)e(the)i(linear)e(terms)i(of)71 1428 y(equation)27 b(\(267\))o(.)p Black 71 1594 a Fp(Lemma)38 b(9.5.)p Black 43 w Fs(L)l(et)d(us)g(c)l(onsider)h FA(\024)1253 1606 y Fy(7)1322 1594 y FA(>)d(\024)1468 1564 y Fv(0)1468 1614 y Fy(3)1537 1594 y FA(>)f(\024)1682 1606 y Fy(3)1755 1594 y Fs(and)j FA(d)1964 1606 y Fy(3)2034 1594 y FA(<)d(d)2174 1564 y Fv(0)2174 1614 y Fy(2)2244 1594 y FA(<)h(d)2385 1606 y Fy(2)2422 1594 y Fs(.)55 b(Then,)37 b(for)f FA(")c(>)h FB(0)h Fs(smal)t(l)i(enough,)h(ther)l(e)71 1693 y(exists)29 b(a)h(function)g FA(g)i Fs(which)g(is)e(solution)g(of)g(the)g(e)l (quation)1584 1876 y Fw(L)1641 1888 y Fx(")1677 1876 y FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f FA(G)2086 1888 y Fy(1)2123 1876 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)71 2059 y Fs(wher)l(e)30 b FA(G)370 2071 y Fy(1)438 2059 y Fs(is)g(the)g(function)f(de\014ne)l(d)h(in)37 b FB(\(271\))o Fs(.)h(Mor)l(e)l(over,)32 b(it)e(satis\014es)g(that)799 2239 y Fw(k)p FA(g)s Fw(k)926 2253 y Fy(0)p Fx(;\024)1018 2233 y Fl(0)1018 2271 y Fu(3)1048 2253 y Fx(;d)1103 2233 y Fl(0)1103 2271 y Fu(2)1135 2253 y Fx(;\033)1223 2239 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1522 2209 y Fx(\021)r Fy(+1)1646 2239 y FA(;)83 b Fw(k)p FA(@)1838 2251 y Fx(v)1877 2239 y FA(g)s Fw(k)1962 2254 y Fy(max)o Fv(f)p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;)p Fy(0)p Fv(g)p Fx(;\024)2494 2234 y Fl(0)2494 2272 y Fu(3)2525 2254 y Fx(;d)2580 2234 y Fl(0)2580 2272 y Fu(2)2612 2254 y Fx(;\033)2699 2239 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2999 2209 y Fx(\021)r Fy(+1)71 2424 y Fs(and)30 b(that)g FA(u)22 b FB(=)h FA(v)f FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))24 b Fw(2)f FA(R)1102 2436 y Fx(\024)1141 2444 y Fu(3)1173 2436 y Fx(;d)1228 2444 y Fu(2)1294 2424 y Fs(for)31 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b Fw(2)f FA(R)1781 2438 y Fx(\024)1820 2418 y Fl(0)1820 2456 y Fu(3)1853 2438 y Fx(;d)1908 2418 y Fl(0)1908 2456 y Fu(2)1962 2424 y Fw(\002)18 b Ft(T)2100 2436 y Fx(\033)2145 2424 y Fs(.)195 2524 y(F)-6 b(urthermor)l(e,)34 b(the)f(change)g FB(\()p FA(u;)14 b(\034)9 b FB(\))29 b(=)f(\()p FA(v)c FB(+)c FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))34 b Fs(is)f(invertible)h(and)f(its) g(inverse)g(is)g(of)g(the)g(form)g FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))30 b(=)71 2623 y(\()p FA(u)18 b FB(+)g FA(h)p FB(\()p FA(u;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))p Fs(.)40 b(The)31 b(function)f FA(h)f Fs(is)h(de\014ne)l(d)g (in)g(the)g(domain)h FA(R)2216 2635 y Fx(\024)2255 2643 y Fu(7)2287 2635 y Fx(;d)2342 2643 y Fu(3)2397 2623 y Fw(\002)18 b Ft(T)2535 2635 y Fx(\033)2610 2623 y Fs(and)30 b(it)g(satis\014es)1534 2806 y Fw(k)p FA(h)p Fw(k)1666 2818 y Fy(0)p Fx(;\024)1758 2826 y Fu(7)1789 2818 y Fx(;d)1844 2826 y Fu(3)1876 2818 y Fx(;\033)1963 2806 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2263 2772 y Fx(\021)r Fy(+1)71 2989 y Fs(and)30 b(that)g FA(u)18 b FB(+)g FA(h)p FB(\()p FA(u;)c(\034)9 b FB(\))23 b Fw(2)h FA(R)958 3003 y Fx(\024)997 2983 y Fl(0)997 3021 y Fu(3)1029 3003 y Fx(;d)1084 2983 y Fl(0)1084 3021 y Fu(2)1150 2989 y Fs(for)30 b FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)f FA(R)1641 3001 y Fx(\024)1680 3009 y Fu(7)1713 3001 y Fx(;d)1768 3009 y Fu(3)1822 2989 y Fw(\002)18 b Ft(T)1960 3001 y Fx(\033)2005 2989 y Fs(.)195 3165 y FB(F)-7 b(urthermore,)24 b(w)n(e)f(need)g(precise)g(b)r(ounds)h(of)f(b)r(oth)h(functions)g FA(g)i FB(and)d FA(h)h FB(restricted)f(to)g(the)h(inner)f(domain)g FA(D)3661 3135 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)3659 3186 y(\024)3698 3194 y Fu(7)3730 3186 y Fx(;c)71 3265 y FB(de\014ned)36 b(in)f(\(30\).)61 b(These)35 b(b)r(ounds)h(are)e(giv)n(en)h(in)h(next)f (corollary)-7 b(,)35 b(whose)g(pro)r(of)g(is)g(straigh)n(tforw)n(ard.) 58 b(W)-7 b(e)36 b(abuse)71 3364 y(notation)27 b(and)g(w)n(e)h(use)f (the)h(norms)f(de\014ned)h(in)g(Section)f(9.1)g(for)g(functions)h (restricted)f(to)g(the)h(inner)g(domain.)p Black 71 3530 a Fp(Corollary)40 b(9.6.)p Black 44 w Fs(L)l(et)35 b FA(c)883 3542 y Fy(1)954 3530 y FA(>)f FB(0)h Fs(b)l(e)h(the)f(c)l (onstant)g(de\014ne)l(d)h(in)g(Cor)l(ol)t(lary)i(7.7)f(and)f(let)g(us)f (c)l(onsider)i(also)f FA(c)3582 3542 y Fy(2)3653 3530 y FA(>)e(c)3788 3542 y Fy(1)3825 3530 y Fs(.)71 3630 y(Then,)g(the)f(functions)f FA(g)j Fs(ad)e FA(h)f Fs(obtaine)l(d)i(in)e (L)l(emma)h(9.5)g(r)l(estricte)l(d)g(to)f(the)h(inner)f(domains)i FA(D)3205 3590 y Fy(in)o Fx(;)p Fy(+)p Fx(;u)3203 3658 y(\024)3242 3638 y Fl(0)3242 3676 y Fu(3)3274 3658 y Fx(;c)3324 3666 y Fu(1)3426 3630 y Fs(and)f FA(D)3661 3600 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)3659 3650 y(\024)3698 3658 y Fu(7)3730 3650 y Fx(;c)3780 3658 y Fu(2)71 3742 y Fs(r)l(esp)l(e)l(ctively)e(satisfy)g(the)e(fol)t(lowing)k(b)l(ounds) 703 3924 y Fw(k)p FA(g)s Fw(k)830 3938 y Fy(0)p Fx(;\024)922 3919 y Fl(0)922 3957 y Fu(3)953 3938 y Fx(;d)1008 3919 y Fl(0)1008 3957 y Fu(2)1040 3938 y Fx(;\033)1127 3924 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1427 3890 y Fx(\021)r Fy(+1+\(2)p Fx(r)r Fv(\000)p Fx(`)p Fy(\))p Fx(\015)1895 3924 y Fs(and)30 b Fw(k)p FA(h)p Fw(k)2188 3936 y Fy(0)p Fx(;\024)2280 3944 y Fu(7)2311 3936 y Fx(;d)2366 3944 y Fu(3)2397 3936 y Fx(;\033)2485 3924 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2785 3890 y Fx(\021)r Fy(+1+\(2)p Fx(r)r Fv(\000)p Fx(`)p Fy(\))p Fx(\015)3195 3924 y FA(:)p Black 71 4121 a Fs(Pr)l(o)l(of)31 b(of)f(L)l(emma)g(9.5.)p Black 44 w FB(F)-7 b(rom)27 b(Lemma)g(9.4,)g Fw(h)p FA(G)1631 4133 y Fy(1)1669 4121 y Fw(i)c FB(=)g(0)k(and)g(then)h(w)n(e)g(can)f (de\014ne)h(a)f(function)p 3140 4055 66 4 v 28 w FA(G)3205 4133 y Fy(1)3270 4121 y FB(suc)n(h)g(that)1467 4304 y FA(@)1511 4316 y Fx(\034)p 1552 4237 V 1552 4304 a FA(G)1617 4316 y Fy(1)1678 4304 y FB(=)22 b FA(G)1830 4316 y Fy(1)1923 4304 y FB(and)55 b Fw(h)p 2144 4237 V FA(G)2210 4316 y Fy(1)2247 4304 y Fw(i)23 b FB(=)g(0)p FA(;)1206 b FB(\(274\))71 4487 y(whic)n(h)27 b(satis\014es)1795 4581 y Fz(\015)1795 4631 y(\015)p 1841 4585 V 20 x FA(G)1906 4663 y Fy(1)1943 4581 y Fz(\015)1943 4631 y(\015)1989 4685 y Fy(0)p Fx(;\024)2081 4665 y Fl(0)2081 4703 y Fu(3)2113 4685 y Fx(;d)2168 4665 y Fl(0)2168 4703 y Fu(2)2200 4685 y Fx(;\033)2288 4651 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2587 4617 y Fx(\021)1271 4739 y Fz(\015)1271 4788 y(\015)1317 4809 y FA(@)1361 4821 y Fx(v)p 1401 4743 V 1401 4809 a FA(G)1466 4821 y Fy(1)1503 4739 y Fz(\015)1503 4788 y(\015)1550 4842 y Fy(max)o Fv(f)p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;)p Fy(0)p Fv(g)p Fx(;\024)2082 4822 y Fl(0)2082 4860 y Fu(3)2113 4842 y Fx(;d)2168 4822 y Fl(0)2168 4860 y Fu(2)2200 4842 y Fx(;\033)2288 4809 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2587 4775 y Fx(\021)2627 4809 y FA(:)3661 4740 y FB(\(275\))71 5009 y(Then,)28 b(w)n(e)f(can)g(de\014ne)h FA(g)i FB(as)1259 5109 y FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)e FA(")p 1641 5042 V(G)1706 5121 y Fy(1)1744 5109 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(")p Fw(G)2124 5121 y Fx(")2173 5042 y Fz(\000)2211 5109 y FA(@)2255 5121 y Fx(v)p 2295 5042 V 2295 5109 a FA(G)2360 5121 y Fy(1)2397 5042 y Fz(\001)2449 5109 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)1000 b FB(\(276\))71 5258 y(where)27 b Fw(G)360 5270 y Fx(")424 5258 y FB(is)g(the)h(op)r (erator)e(de\014ned)i(in)g(\(270\))e(adapted)i(to)f(the)h(domain)f FA(R)2505 5272 y Fx(\024)2544 5253 y Fl(0)2544 5291 y Fu(3)2577 5272 y Fx(;d)2632 5253 y Fl(0)2632 5291 y Fu(2)2686 5258 y Fw(\002)18 b Ft(T)2824 5270 y Fx(\033)2869 5258 y FB(.)195 5358 y(Finally)-7 b(,)33 b(applying)e(Lemma)g(9.4)f(and)i (9.2,)f(one)g(obtains)g(the)h(b)r(ounds)g(for)e FA(g)35 b FB(and)c FA(@)2911 5370 y Fx(v)2950 5358 y FA(g)s FB(.)48 b(The)32 b(other)f(statemen)n(ts)71 5458 y(are)26 b(straigh)n(tforw)n (ard.)p 3790 5458 4 57 v 3794 5405 50 4 v 3794 5458 V 3843 5458 4 57 v Black 1919 5753 a(97)p Black eop end %%Page: 98 98 TeXDict begin 98 97 bop Black Black 195 272 a FB(W)-7 b(e)35 b(p)r(erform)e(the)h(c)n(hange)f(of)g(v)-5 b(ariables)33 b FA(u)g FB(=)f FA(v)26 b FB(+)c FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))35 b(giv)n(en)e(in)h(Lemma)g(9.5)f(to)g(equation)g (\(268\))g(and)h(w)n(e)71 372 y(obtain)1701 471 y Fw(L)1758 483 y Fx(")1807 450 y Fz(b)1794 471 y Fw(C)28 b FB(=)1974 450 y Fz(b)1953 471 y Fw(F)2035 379 y Fz(\020)2098 450 y(b)2085 471 y Fw(C)2133 379 y Fz(\021)2197 471 y FA(;)1441 b FB(\(277\))71 659 y(where)324 638 y Fz(b)311 659 y Fw(C)32 b FB(is)c(the)g(unkno)n(wn)1512 737 y Fz(b)1499 758 y Fw(C)5 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f Fw(C)5 b FB(\()p FA(v)21 b FB(+)d FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))1241 b(\(278\))71 902 y(and)1446 980 y Fz(b)1426 1001 y Fw(F)8 b FB(\()p FA(h)p FB(\))23 b(=)g FA(M)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))19 b(+)f FA(N)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(@)2407 1013 y Fx(v)2448 1001 y FA(h)1165 b FB(\(279\))71 1145 y(with)1212 1318 y FA(M)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f Fw(\000)p FA(")1707 1284 y Fv(\000)p Fy(1)1795 1318 y FA(G)14 b FB(\()q FA(v)22 b FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))1263 b(\(280\))1226 1500 y FA(N)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f Fw(\000)1678 1443 y FA(G)14 b FB(\()p FA(v)22 b FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(G)2448 1455 y Fy(1)2485 1443 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))p 1678 1480 999 4 v 1947 1557 a(1)18 b(+)g FA(@)2134 1569 y Fx(v)2173 1557 y FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))2686 1500 y FA(:)952 b FB(\(281\))71 1717 y(Next)28 b(lemma)f(giv)n(es)g (some)g(prop)r(erties)f(of)i(these)g(functions)p Black 71 1875 a Fp(Lemma)j(9.7.)p Black 40 w Fs(The)g(functions)f FA(M)38 b Fs(and)30 b FA(N)39 b Fs(de\014ne)l(d)30 b(in)36 b FB(\(280\))29 b Fs(and)39 b FB(\(281\))29 b Fs(satisfy)h(the)g(fol)t (lowing)j(pr)l(op)l(erties.)p Black 195 2033 a Fw(\017)p Black 41 w(G)327 2045 y Fx(")363 2033 y FB(\()p FA(M)9 b FB(\))24 b Fw(2)f(X)678 2047 y Fy(0)p Fx(;\024)770 2028 y Fl(0)770 2066 y Fu(3)802 2047 y Fx(;d)857 2028 y Fl(0)857 2066 y Fu(2)889 2047 y Fx(;\033)983 2033 y Fs(and)30 b(it)g(satis\014es)1573 2206 y Fw(kG)1664 2218 y Fx(")1699 2206 y FB(\()p FA(M)9 b FB(\))p Fw(k)1895 2231 y Fy(0)p Fx(;\024)1987 2211 y Fl(0)1987 2249 y Fu(3)2019 2231 y Fx(;d)2074 2211 y Fl(0)2074 2249 y Fu(2)2106 2231 y Fx(;\033)2193 2206 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2493 2172 y Fx(\021)2533 2206 y FA(:)p Black 195 2424 a Fw(\017)p Black 41 w(h)p FA(M)j Fw(i)24 b(2)f(X)593 2439 y Fy(max)p Fv(f)p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;)p Fy(0)p Fv(g)p Fx(;\024)1126 2419 y Fl(0)1126 2457 y Fu(3)1157 2439 y Fx(;d)1212 2419 y Fl(0)1212 2457 y Fu(2)1244 2439 y Fx(;\033)1338 2424 y Fs(and)30 b(it)g(satis\014es) 1396 2597 y Fw(k)o(h)p FA(M)9 b Fw(ik)1633 2622 y Fy(max)o Fv(f)p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;)p Fy(0)p Fv(g)p Fx(;\024)2165 2602 y Fl(0)2165 2640 y Fu(3)2197 2622 y Fx(;d)2252 2602 y Fl(0)2252 2640 y Fu(2)2283 2622 y Fx(;\033)2371 2597 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2670 2563 y Fx(\021)2710 2597 y FA(:)p Black 195 2811 a Fw(\017)p Black 41 w FA(@)322 2823 y Fx(v)362 2811 y FA(M)32 b Fw(2)23 b(X)612 2826 y Fy(max)p Fv(f)p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;)p Fy(0)p Fv(g)p Fx(;\024)1145 2806 y Fl(0)1145 2844 y Fu(3)1176 2826 y Fx(;d)1231 2806 y Fl(0)1231 2844 y Fu(2)1263 2826 y Fx(;\033)1357 2811 y Fs(and)30 b(it)g(satis\014es)1344 2997 y Fw(k)o FA(@)1429 3009 y Fx(v)1469 2997 y FA(M)9 b Fw(k)1600 3022 y Fy(max)o Fv(f)p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;)p Fy(0)p Fv(g)p Fx(;\024)2132 3002 y Fl(0)2132 3040 y Fu(3)2164 3022 y Fx(;d)2219 3002 y Fl(0)2219 3040 y Fu(2)2250 3022 y Fx(;\033)2338 2997 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2637 2963 y Fx(\021)r Fv(\000)p Fy(1)2762 2997 y FA(:)p Black 195 3223 a Fw(\017)p Black 41 w Fs(R)l(estricte)l(d)30 b(to)766 3156 y Fz(\000)805 3223 y FA(D)876 3193 y Fy(in)o Fx(;)p Fy(+)p Fx(;u)874 3244 y(\024;c)963 3252 y Fu(1)1083 3223 y Fw(\\)19 b FA(D)1228 3193 y Fy(in)p Fx(;)p Fy(+)p Fx(;s)1226 3244 y(\024;c)1315 3252 y Fu(1)1409 3156 y Fz(\001)1466 3223 y Fw(\002)f Ft(T)1604 3235 y Fx(\033)1649 3223 y Fs(,)30 b(the)g(function)g FA(M)38 b Fs(satis\014es)1825 3409 y Fw(k)o FA(M)9 b Fw(k)1997 3434 y Fy(0)p Fx(;\024)2089 3414 y Fl(0)2089 3452 y Fu(3)2122 3434 y Fx(;d)2177 3414 y Fl(0)2177 3452 y Fu(2)2208 3434 y Fx(;\033)2296 3409 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2595 3374 y Fx(\021)r Fy(+)p Fx(\027)t Fv(\000)p Fy(1)1321 3555 y Fw(k)o(h)p FA(M)j Fw(ik)1558 3579 y Fy(max)o Fv(f)p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;)p Fy(0)p Fv(g)p Fx(;\024)2090 3559 y Fl(0)2090 3597 y Fu(3)2122 3579 y Fx(;d)2177 3559 y Fl(0)2177 3597 y Fu(2)2208 3579 y Fx(;\033)2296 3555 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2595 3520 y Fx(\021)278 3737 y Fs(wher)l(e)1271 3836 y FA(\027)28 b FB(=)23 b(min)p Fw(f)p FB(1)18 b Fw(\000)g FB(max)o Fw(f)p FA(`)g Fw(\000)g FB(2)p FA(r)j FB(+)d(1)p FA(;)c FB(0)p Fw(g)p FA(;)g FB(\(2)p FA(r)19 b Fw(\000)f FA(`)p FB(\))p FA(\015)5 b Fw(g)p FA(:)802 b FB(\(282\))p Black 195 4011 a Fw(\017)p Black 41 w FA(N)32 b Fw(2)24 b(X)515 4026 y Fy(max)o Fv(f)p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;)p Fy(0)p Fv(g)p Fx(;\024)1047 4006 y Fl(0)1047 4044 y Fu(3)1079 4026 y Fx(;d)1134 4006 y Fl(0)1134 4044 y Fu(2)1165 4026 y Fx(;\033)1259 4011 y Fs(and)31 b(it)e(satis\014es)1446 4198 y Fw(k)p FA(N)9 b Fw(k)1606 4213 y Fy(max)n Fv(f)p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;)p Fy(0)p Fv(g)p Fx(;\024)2137 4193 y Fl(0)2137 4231 y Fu(3)2169 4213 y Fx(;d)2224 4193 y Fl(0)2224 4231 y Fu(2)2255 4213 y Fx(;\033)2343 4198 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2642 4163 y Fx(\021)r Fy(+1)1363 4382 y Fw(k)o FA(@)1448 4394 y Fx(v)1488 4382 y FA(N)j Fw(k)1605 4407 y Fy(max)o Fv(f)p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;)p Fy(0)p Fv(g)p Fx(;\024)2137 4387 y Fl(0)2137 4425 y Fu(3)2169 4407 y Fx(;d)2224 4387 y Fl(0)2224 4425 y Fu(2)2255 4407 y Fx(;\033)2343 4382 y Fw(\024)22 b FA(K)2517 4326 y Fw(j)p FA(\026)p Fw(j)p FA(")2652 4296 y Fx(\021)p 2517 4363 176 4 v 2562 4439 a FA(\024)2610 4411 y Fv(0)2610 4461 y Fy(3)2702 4382 y FA(:)p Black 71 4632 a Fs(Pr)l(o)l(of.)p Black 43 w FB(W)-7 b(e)28 b(split)g FA(M)36 b FB(as)27 b FA(M)32 b FB(=)22 b FA(M)1159 4644 y Fy(1)1215 4632 y FB(+)c FA(M)1379 4644 y Fy(2)1443 4632 y FB(with)337 4802 y FA(M)418 4814 y Fy(1)455 4802 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))24 b(=)f Fw(\000)p FA(")860 4767 y Fv(\000)p Fy(1)948 4802 y FA(G)1013 4814 y Fy(1)1051 4802 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))337 4936 y FA(M)418 4948 y Fy(2)455 4936 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f Fw(\000)p FA(")860 4902 y Fv(\000)p Fy(1)962 4936 y FB(\()p FA(G)1059 4948 y Fy(1)1111 4936 y FB(\()p FA(v)f FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(G)1802 4948 y Fy(1)1839 4936 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))20 b(+)e FA(G)2196 4948 y Fy(2)2248 4936 y FB(\()p FA(v)k FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))20 b(+)e FA(G)2939 4948 y Fy(3)2990 4936 y FB(\()q FA(v)j FB(+)d FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))r(\))14 b FA(:)71 5106 y FB(Then,)28 b(for)g(the)g(\014rst)g(statemen)n(t)f(it)i(is)f(enough)f(to)h(use)f (the)i(prop)r(erties)e(of)h(the)g(functions)g FA(G)3034 5118 y Fy(1)3072 5106 y FB(,)g FA(G)3188 5118 y Fy(2)3253 5106 y FB(and)g FA(G)3480 5118 y Fy(3)3546 5106 y FB(giv)n(en)f(b)n(y) 71 5205 y(Lemma)e(9.4)f(and)h(apply)f(also)g(Lemmas)h(9.2,)f(9.1)h(and) f(9.5.)36 b(F)-7 b(or)24 b(the)h(second)g(and)f(the)i(third)f(one)g (has)f(to)h(apply)g(again)71 5305 y(Lemmas)i(9.4,)g(9.1)g(and)h(9.5,)f (taking)g(also)g(in)n(to)h(accoun)n(t)f(for)g(the)h(second)f(that)i Fw(h)p FA(M)2741 5317 y Fy(1)2778 5305 y Fw(i)23 b FB(=)g(0.)38 b(Besides,)27 b(these)h(lemmas,)71 5404 y(for)f(the)g(fourth)h (statemen)n(t,)f(one)g(has)g(to)g(consider)g(also)f(the)i(b)r(ound)g (of)f(the)h(c)n(hange)e FA(g)k FB(in)e(the)f(inner)g(domain,)h(whic)n (h)71 5504 y(is)23 b(giv)n(en)g(in)h(Corollary)d(9.6.)35 b(F)-7 b(or)23 b(the)h(last)g(statemen)n(t,)g(it)g(is)g(enough)f(to)h (apply)f(again)f(Lemmas)i(9.4,)f(9.1)g(and)h(9.5.)p 3790 5504 4 57 v 3794 5451 50 4 v 3794 5504 V 3843 5504 4 57 v Black 1919 5753 a(98)p Black eop end %%Page: 99 99 TeXDict begin 99 98 bop Black Black 195 272 a FB(With)26 b(the)f(b)r(ounds)g(obtained)f(in)h(Lemma)f(9.7,)g(w)n(e)h(can)f(lo)r (ok)g(for)g(a)g(solution)g(of)h(equation)f(\(277\))f(through)h(a)g (\014xed)71 372 y(p)r(oin)n(t)k(argumen)n(t.)35 b(F)-7 b(or)27 b(that)h(purp)r(ose,)f(w)n(e)h(de\014ne)f(the)h(op)r(erator) 1776 533 y Fz(e)1756 554 y Fw(F)j FB(=)22 b Fw(G)1983 566 y Fx(")2038 554 y Fw(\016)2118 533 y Fz(b)2098 554 y Fw(F)1503 b FB(\(283\))71 747 y(where)25 b Fw(G)358 759 y Fx(")419 747 y FB(and)599 726 y Fz(b)579 747 y Fw(F)33 b FB(are)25 b(the)h(op)r(erators)d(de\014ned)j(in)g(\(270\))f (and)g(\(279\))g(resp)r(ectiv)n(ely)-7 b(.)35 b(F)-7 b(or)25 b(con)n(v)n(enience,)g(w)n(e)g(rewrite)3803 726 y Fz(b)3782 747 y Fw(F)71 847 y FB(as)856 926 y Fz(b)836 947 y Fw(F)7 b FB(\()p FA(h)p FB(\)\()p FA(u;)14 b(\034)9 b FB(\))25 b(=)d FA(M)9 b FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b(+)f FA(@)1751 959 y Fx(v)1804 947 y FB(\()q FA(N)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(h)p FB(\()p FA(v)s(;)14 b(\034)9 b FB(\)\))21 b(+)d FA(@)2519 959 y Fx(v)2558 947 y FA(N)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(h)p FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(:)578 b FB(\(284\))p Black 71 1123 a Fp(Lemma)26 b(9.8.)p Black 36 w Fs(L)l(et)f(us)h(c)l(onsider)g FA(")1196 1135 y Fy(0)1256 1123 y FA(>)d FB(0)i Fs(smal)t(l)i(enough)f(and)g FA(\024)2111 1093 y Fv(0)2111 1144 y Fy(3)2171 1123 y FA(>)d(\024)2307 1135 y Fy(3)2369 1123 y Fs(big)k(enough.)37 b(Then,)28 b(the)e(op)l(er)l(ator)3527 1102 y Fz(e)3507 1123 y Fw(F)33 b Fs(de\014ne)l(d)71 1223 y(in)j FB(\(283\))29 b Fs(is)h(c)l(ontr)l(active)g(fr)l(om)g Fw(X)1163 1237 y Fy(0)p Fx(;\024)1255 1217 y Fl(0)1255 1255 y Fu(3)1288 1237 y Fx(;d)1343 1217 y Fl(0)1343 1255 y Fu(2)1374 1237 y Fx(;\033)1469 1223 y Fs(to)f(itself.)195 1322 y(Thus,)i(it)f(has)g(a) g(unique)g(\014xe)l(d)f(p)l(oint,)h(which)i(mor)l(e)l(over)e (satis\014es)1549 1430 y Fz(\015)1549 1480 y(\015)1549 1529 y(\015)1608 1504 y(b)1595 1525 y Fw(C)1644 1430 y Fz(\015)1644 1480 y(\015)1644 1529 y(\015)1690 1583 y Fy(0)p Fx(;\024)1782 1563 y Fl(0)1782 1601 y Fu(3)1814 1583 y Fx(;d)1869 1563 y Fl(0)1869 1601 y Fu(2)1901 1583 y Fx(;\033)1988 1525 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2288 1491 y Fx(\021)1466 1663 y Fz(\015)1466 1713 y(\015)1466 1763 y(\015)1512 1758 y FA(@)1556 1770 y Fx(v)1608 1737 y Fz(b)1595 1758 y Fw(C)1644 1663 y Fz(\015)1644 1713 y(\015)1644 1763 y(\015)1690 1816 y Fy(0)p Fx(;\024)1782 1796 y Fl(0)1782 1834 y Fu(3)1814 1816 y Fx(;d)1869 1796 y Fl(0)1869 1834 y Fu(2)1901 1816 y Fx(;\033)1988 1758 y Fw(\024)23 b FA(K)2162 1702 y Fw(j)p FA(\026)p Fw(j)p FA(")2297 1672 y Fx(\021)r Fv(\000)p Fy(1)p 2162 1739 261 4 v 2250 1815 a FA(\024)2298 1787 y Fv(0)2298 1837 y Fy(3)2433 1758 y FA(:)p Black 71 2010 a Fs(Pr)l(o)l(of.)p Black 43 w FB(T)-7 b(o)30 b(see)g(that)795 1989 y Fz(e)775 2010 y Fw(F)38 b FB(is)30 b(con)n(tractiv)n(e,)f(let)i (us)f(consider)f FA(h)2018 2022 y Fy(1)2055 2010 y FA(;)14 b(h)2140 2022 y Fy(2)2204 2010 y Fw(2)28 b(X)2346 2024 y Fy(0)p Fx(;\024)2438 2004 y Fl(0)2438 2042 y Fu(3)2470 2024 y Fx(;d)2525 2004 y Fl(0)2525 2042 y Fu(2)2557 2024 y Fx(;\033)2621 2010 y FB(.)45 b(Then,)31 b(recalling)d(the)j (de\014nition)g(of)91 2110 y Fz(e)71 2131 y Fw(F)k FB(and)348 2110 y Fz(b)328 2131 y Fw(F)g FB(in)28 b(\(283\))f(and)h(\(284\))e (resp)r(ectiv)n(ely)h(and)g(applying)g(Lemmas)h(9.2,)e(9.1)h(and)h (9.7,)71 2244 y Fz(\015)71 2293 y(\015)71 2343 y(\015)137 2318 y(e)117 2339 y Fw(F)8 b FB(\()p FA(h)265 2351 y Fy(2)302 2339 y FB(\))19 b Fw(\000)456 2318 y Fz(e)436 2339 y Fw(F)8 b FB(\()p FA(h)584 2351 y Fy(1)621 2339 y FB(\))653 2244 y Fz(\015)653 2293 y(\015)653 2343 y(\015)700 2397 y Fy(0)p Fx(;\024)792 2377 y Fl(0)792 2415 y Fu(3)824 2397 y Fx(;d)879 2377 y Fl(0)879 2415 y Fu(2)910 2397 y Fx(;\033)998 2339 y Fw(\024)23 b(k)o(G)1176 2351 y Fx(")1212 2339 y FA(@)1256 2351 y Fx(v)1309 2339 y FB(\()q FA(N)k Fw(\001)18 b FB(\()p FA(h)1557 2351 y Fy(2)1613 2339 y Fw(\000)g FA(h)1744 2351 y Fy(1)1781 2339 y FB(\)\))q Fw(k)1887 2364 y Fy(0)p Fx(;\024)1979 2344 y Fl(0)1979 2382 y Fu(3)2011 2364 y Fx(;d)2066 2344 y Fl(0)2066 2382 y Fu(2)2098 2364 y Fx(;\033)2181 2339 y FB(+)g Fw(kG)2355 2351 y Fx(")2404 2339 y FB(\()q FA(@)2481 2351 y Fx(v)2520 2339 y FA(N)27 b Fw(\001)19 b FB(\()p FA(h)2736 2351 y Fy(2)2792 2339 y Fw(\000)f FA(h)2923 2351 y Fy(1)2960 2339 y FB(\)\))p Fw(k)3066 2364 y Fy(0)p Fx(;\024)3158 2344 y Fl(0)3158 2382 y Fu(3)3190 2364 y Fx(;d)3245 2344 y Fl(0)3245 2382 y Fu(2)3277 2364 y Fx(;\033)998 2518 y Fw(\024)23 b(k)o FA(N)9 b Fw(k)1244 2543 y Fy(0)p Fx(;\024)1336 2523 y Fl(0)1336 2561 y Fu(3)1368 2543 y Fx(;d)1423 2523 y Fl(0)1423 2561 y Fu(2)1455 2543 y Fx(;\033)1533 2518 y Fw(k)p FA(h)1623 2530 y Fy(2)1678 2518 y Fw(\000)18 b FA(h)1809 2530 y Fy(1)1847 2518 y Fw(k)1888 2543 y Fy(0)p Fx(;\024)1980 2523 y Fl(0)1980 2561 y Fu(3)2012 2543 y Fx(;d)2067 2523 y Fl(0)2067 2561 y Fu(2)2099 2543 y Fx(;\033)2182 2518 y FB(+)g Fw(k)o FA(@)2350 2530 y Fx(v)2390 2518 y FA(N)9 b Fw(k)2507 2543 y Fy(max)p Fv(f)p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;)p Fy(0)p Fv(g)p Fx(;\024)3040 2523 y Fl(0)3040 2561 y Fu(3)3071 2543 y Fx(;d)3126 2523 y Fl(0)3126 2561 y Fu(2)3157 2543 y Fx(;\033)3236 2518 y Fw(k)o FA(h)3325 2530 y Fy(2)3381 2518 y Fw(\000)18 b FA(h)3512 2530 y Fy(1)3549 2518 y Fw(k)3590 2543 y Fy(0)p Fx(;\024)3682 2523 y Fl(0)3682 2561 y Fu(6)3714 2543 y Fx(;d)3769 2523 y Fl(0)3769 2561 y Fu(2)3801 2543 y Fx(;\033)998 2713 y Fw(\024)1096 2657 y FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")1308 2626 y Fx(\021)p 1096 2694 252 4 v 1179 2770 a FA(\024)1227 2741 y Fv(0)1227 2792 y Fy(3)1371 2713 y Fw(k)p FA(h)1461 2725 y Fy(2)1516 2713 y Fw(\000)18 b FA(h)1647 2725 y Fy(1)1684 2713 y Fw(k)1726 2738 y Fy(0)p Fx(;\024)1818 2718 y Fl(0)1818 2756 y Fu(3)1850 2738 y Fx(;d)1905 2718 y Fl(0)1905 2756 y Fu(2)1937 2738 y Fx(;\033)2015 2713 y FA(:)71 2964 y FB(Then,)28 b(increasing)e FA(\024)747 2934 y Fv(0)747 2985 y Fy(3)812 2964 y FB(if)i(necessary)-7 b(,)1292 2943 y Fz(e)1272 2964 y Fw(F)35 b FB(is)28 b(con)n(tractiv)n(e)e(from)h Fw(X)2133 2978 y Fy(0)p Fx(;\024)2225 2959 y Fl(0)2225 2997 y Fu(3)2257 2978 y Fx(;d)2312 2959 y Fl(0)2312 2997 y Fu(2)2344 2978 y Fx(;\033)2436 2964 y FB(and)g(then)h(it)h(has)e(a)g (unique)h(\014xed)f(p)r(oin)n(t.)195 3086 y(T)-7 b(o)28 b(obtain)f(a)g(b)r(ound)h(for)f(the)h(\014xed)g(p)r(oin)n(t)1601 3065 y Fz(b)1589 3086 y Fw(C)t FB(,)g(it)g(is)f(enough)g(to)h(recall)e (that)1380 3193 y Fz(\015)1380 3243 y(\015)1380 3293 y(\015)1439 3268 y(b)1427 3289 y Fw(C)1475 3193 y Fz(\015)1475 3243 y(\015)1475 3293 y(\015)1521 3347 y Fy(0)p Fx(;\024)1613 3327 y Fl(0)1613 3365 y Fu(3)1645 3347 y Fx(;d)1700 3327 y Fl(0)1700 3365 y Fu(2)1732 3347 y Fx(;\033)1819 3289 y Fw(\024)d FB(2)1963 3193 y Fz(\015)1963 3243 y(\015)1963 3293 y(\015)2029 3268 y(e)2009 3289 y Fw(F)8 b FB(\(0\))2183 3193 y Fz(\015)2183 3243 y(\015)2183 3293 y(\015)2229 3347 y Fy(0)p Fx(;\024)2321 3327 y Fl(0)2321 3365 y Fu(3)2353 3347 y Fx(;d)2408 3327 y Fl(0)2408 3365 y Fu(2)2439 3347 y Fx(;\033)2518 3289 y FA(:)71 3537 y FB(By)28 b(the)h(de\014nition)f (of)831 3516 y Fz(e)811 3537 y Fw(F)36 b FB(in)28 b(\(283\),)1265 3516 y Fz(e)1245 3537 y Fw(F)8 b FB(\(0\))24 b(=)g Fw(G)1581 3549 y Fx(")1617 3537 y FB(\()p FA(M)9 b FB(\).)39 b(Then,)29 b(applying)f(Lemma)g(9.7,)g(w)n(e)g(obtain)g(the)g(b)r(ound)h(for)3792 3516 y Fz(b)3779 3537 y Fw(C)t FB(.)71 3647 y(F)-7 b(or)29 b(the)h(b)r(ound)h(of)e FA(@)766 3659 y Fx(v)819 3626 y Fz(b)806 3647 y Fw(C)34 b FB(it)d(is)f(enough)f(to)g(reduce)h(sligh)n (tly)f(the)h(domain)g(and)g(apply)f(fourth)h(statemen)n(t)g(of)g(Lemma) 71 3747 y(9.1.)p 3790 3747 4 57 v 3794 3694 50 4 v 3794 3747 V 3843 3747 4 57 v Black 71 3913 a Fs(Pr)l(o)l(of)h(of)f(Pr)l(op)l (osition)i(9.3.)p Black 43 w FB(T)-7 b(o)31 b(reco)n(v)n(er)e Fw(C)36 b FB(from)1717 3892 y Fz(b)1704 3913 y Fw(C)g FB(it)31 b(is)h(enough)e(to)i(consider)e(the)i(c)n(hange)e(of)h(v)-5 b(ariables)30 b FA(v)j FB(=)28 b FA(u)21 b FB(+)71 4012 y FA(h)p FB(\()p FA(u;)14 b(\034)9 b FB(\))35 b(obtained)g(in)g(Lemma)g (9.5,)h(whic)n(h)f(is)f(de\014ned)i(for)e(\()p FA(u;)14 b(\034)9 b FB(\))36 b Fw(2)g FA(R)2418 4024 y Fx(\024)2457 4032 y Fu(7)2489 4024 y Fx(;d)2544 4032 y Fu(3)2603 4012 y Fw(\002)23 b Ft(T)2746 4024 y Fx(\033)2826 4012 y FB(with)35 b FA(\024)3070 4024 y Fy(7)3143 4012 y FA(>)g(\024)3291 3982 y Fv(0)3291 4033 y Fy(3)3363 4012 y FB(and)f FA(d)3574 4024 y Fy(3)3647 4012 y FA(<)h(d)3790 3982 y Fv(0)3790 4033 y Fy(2)3827 4012 y FB(.)71 4112 y(Applying)30 b(this)h(c)n(hange,) f(one)g(obtains)g Fw(C)35 b FB(whic)n(h)30 b(satis\014es)g(the)g(b)r (ounds)h(of)f Fw(C)35 b FB(and)30 b FA(@)2797 4124 y Fx(u)2841 4112 y Fw(C)35 b FB(stated)30 b(in)h(Prop)r(osition)d(9.3.)71 4212 y(T)-7 b(o)27 b(c)n(hec)n(k)g(that)h(\()p FA(\030)665 4224 y Fy(0)703 4212 y FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))29 b(is)e(injectiv)n(e,)h(it)g(is)g(enough)f (to)g(see)g(that)h(for)f(\()p FA(u)2551 4224 y Fy(1)2588 4212 y FA(;)14 b(\034)9 b FB(\))p FA(;)14 b FB(\()p FA(u)2819 4224 y Fy(2)2857 4212 y FA(;)g(\034)9 b FB(\))24 b Fw(2)f FA(R)3136 4224 y Fx(\024)3175 4232 y Fu(7)3208 4224 y Fx(;d)3263 4232 y Fu(3)3317 4212 y Fw(\002)18 b Ft(T)3455 4224 y Fx(\033)3500 4212 y FB(,)1164 4394 y FA(")1203 4360 y Fv(\000)p Fy(1)1291 4394 y FA(u)1339 4406 y Fy(2)1395 4394 y Fw(\000)g FA(\034)28 b FB(+)18 b Fw(C)5 b FB(\()p FA(u)1754 4406 y Fy(2)1790 4394 y FA(;)14 b(\034)9 b FB(\))24 b(=)f FA(")2055 4360 y Fv(\000)p Fy(1)2144 4394 y FA(u)2192 4406 y Fy(1)2247 4394 y Fw(\000)18 b FA(\034)28 b FB(+)18 b Fw(C)5 b FB(\()p FA(u)2606 4406 y Fy(1)2643 4394 y FA(;)14 b(\034)9 b FB(\))71 4577 y(implies)32 b FA(u)405 4589 y Fy(2)471 4577 y FB(=)e FA(u)614 4589 y Fy(1)650 4577 y FB(.)50 b(T)-7 b(o)31 b(pro)n(v)n(e)f(this)i(fact,)h (it)f(is)f(enough)g(to)h(tak)n(e)f(in)n(to)g(accoun)n(t)g(the)h(just)g (obtained)g(b)r(ound)g(of)f FA(@)3735 4589 y Fx(u)3779 4577 y Fw(C)5 b FB(,)71 4677 y(whic)n(h)27 b(giv)n(es)1366 4854 y Fw(j)p FA(u)1437 4866 y Fy(2)1492 4854 y Fw(\000)18 b FA(u)1623 4866 y Fy(1)1660 4854 y Fw(j)23 b FB(=)g FA(")14 b Fw(jC)5 b FB(\()p FA(u)1999 4866 y Fy(2)2035 4854 y FA(;)14 b(\034)9 b FB(\))20 b Fw(\000)e(C)5 b FB(\()p FA(u)2381 4866 y Fy(1)2417 4854 y FA(;)14 b(\034)9 b FB(\))p Fw(j)1706 5035 y(\024)1804 4979 y FA(K)d Fw(j)p FA(\026)p Fw(j)p FA(")2016 4949 y Fx(\021)p 1804 5016 252 4 v 1887 5092 a FA(\024)1935 5104 y Fy(7)2066 5035 y Fw(j)p FA(u)2137 5047 y Fy(2)2192 5035 y Fw(\000)18 b FA(u)2323 5047 y Fy(1)2360 5035 y Fw(j)p FA(:)71 5254 y FB(Then,)28 b(increasing)e FA(\024)747 5266 y Fy(7)812 5254 y FB(if)i(necessary)-7 b(,)26 b(one)h(can)g(see)g(that)h FA(u)1938 5266 y Fy(2)1998 5254 y FB(=)23 b FA(u)2134 5266 y Fy(1)2170 5254 y FB(.)p 3790 5254 4 57 v 3794 5201 50 4 v 3794 5254 V 3843 5254 4 57 v Black 1919 5753 a(99)p Black eop end %%Page: 100 100 TeXDict begin 100 99 bop Black Black 71 272 a Fp(9.2.2)94 b(Pro)s(of)31 b(of)h(Prop)s(osition)e(4.18)71 425 y FB(T)-7 b(o)27 b(pro)n(v)n(e)g(Prop)r(osition)f(4.18)h(it)h(is)g(enough)f(to)h (study)g(the)g(\014rst)g(asymptotic)g(terms)f(of)h(the)g(function)3376 404 y Fz(b)3363 425 y Fw(C)33 b FB(obtained)27 b(in)71 525 y(Lemma)g(9.5.)36 b(F)-7 b(or)27 b(that)h(purp)r(ose,)f(w)n(e)g (de\014ne)1454 664 y Fz(f)1444 685 y FA(M)8 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)d FA(M)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b Fw(\000)e(h)p FA(M)9 b Fw(i)p FB(\()p FA(v)s FB(\))1184 b(\(285\))71 856 y(and)27 b(w)n(e)h(split)552 835 y Fz(b)539 856 y Fw(C)33 b FB(as)730 835 y Fz(b)718 856 y Fw(C)27 b FB(=)c FA(E)938 868 y Fy(1)994 856 y FB(+)18 b FA(E)1138 868 y Fy(2)1194 856 y FB(+)g FA(E)1338 868 y Fy(3)1403 856 y FB(with)1584 1017 y FA(E)1645 1029 y Fy(1)1683 1017 y FB(\()p FA(v)s FB(\))23 b(=)g Fw(G)1950 1029 y Fx(")2000 1017 y FB(\()p Fw(h)p FA(M)9 b Fw(i)p FB(\))14 b(\()p FA(v)s FB(\))1322 b(\(286\))1501 1175 y FA(E)1562 1187 y Fy(2)1600 1175 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f Fw(G)1950 1187 y Fx(")2000 1083 y Fz(\020)2060 1154 y(f)2049 1175 y FA(M)2139 1083 y Fz(\021)2202 1175 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))1270 b(\(287\))1501 1358 y FA(E)1562 1370 y Fy(3)1600 1358 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)1921 1337 y Fz(e)1901 1358 y Fw(F)1983 1266 y Fz(\020)2045 1337 y(b)2032 1358 y Fw(C)2081 1266 y Fz(\021)2149 1358 y Fw(\000)2252 1337 y Fz(e)2232 1358 y Fw(F)e FB(\(0\))1241 b(\(288\))71 1562 y(Let)24 b(us)g(p)r(oin)n(t)g (out)g(that)g(the)g(sum)g(of)g(the)h(\014rst)e(t)n(w)n(o)g(terms)h (corresp)r(onds)e(to)2512 1541 y Fz(e)2492 1562 y Fw(F)8 b FB(\(0\).)35 b(W)-7 b(e)25 b(study)f(eac)n(h)f(term)h(separately)-7 b(.)71 1662 y(W)g(e)29 b(abuse)g(notation)g(and)g(w)n(e)f(use)h(the)h (same)e(norms)h(as)f(in)i(the)f(previous)f(section)h(but)h(no)n(w)e (for)h(functions)g(de\014ned)71 1761 y(in)168 1694 y Fz(\000)206 1761 y FA(D)277 1731 y Fy(in)o Fx(;)p Fy(+)p Fx(;u)275 1782 y(\024;c)364 1790 y Fu(1)485 1761 y Fw(\\)18 b FA(D)629 1731 y Fy(in)p Fx(;)p Fy(+)p Fx(;s)627 1782 y(\024;c)716 1790 y Fu(1)811 1694 y Fz(\001)867 1761 y Fw(\002)g Ft(T)1005 1773 y Fx(\033)1050 1761 y FB(.)195 1861 y(F)-7 b(or)27 b FA(E)405 1873 y Fy(1)443 1861 y FB(,)h(using)f(the)h(de\014nition)g(of)f Fw(G)1366 1873 y Fx(")1430 1861 y FB(in)h(\(270\))o(,)g(one)f(has)g(that)1507 2068 y FA(E)1568 2080 y Fy(1)1605 2068 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)1907 1955 y Fz(Z)1990 1976 y Fx(v)1953 2144 y(v)1986 2152 y Fu(0)2029 2068 y Fw(h)p FA(M)9 b Fw(i)p FB(\()p FA(w)r FB(\))p FA(dw)71 2284 y FB(and)31 b(then,)i(if)f(w)n(e)g(consider)e FA(v)1027 2296 y Fy(1)1094 2284 y FB(=)g FA(i)p FB(\()p FA(a)21 b Fw(\000)f FA(\024)1448 2254 y Fv(0)1448 2304 y Fy(3)1485 2284 y FA(")p FB(\))32 b(the)g(upp)r(er)g(v)n(ertex)e(of)i(the)g (domain)f FA(R)2840 2298 y Fx(\024)2879 2278 y Fl(0)2879 2316 y Fu(3)2911 2298 y Fx(;d)2966 2306 y Fu(3)3034 2284 y FB(\(see)g(Figure)g(3\),)i(w)n(e)e(can)71 2383 y(de\014ne)1497 2509 y FA(C)6 b FB(\()p FA(\026;)14 b(")p FB(\))24 b(=)1863 2396 y Fz(Z)1946 2417 y Fx(v)1979 2425 y Fu(1)1909 2585 y Fx(v)1942 2593 y Fu(0)2016 2509 y Fw(h)p FA(M)9 b Fw(i)p FB(\()p FA(w)r FB(\))p FA(dw)r(;)1239 b FB(\(289\))71 2695 y(whic)n(h)27 b(b)n(y)h(Lemmas)f(9.2)g(and)g(9.7)g(satis\014es) 1550 2856 y Fw(k)o FA(C)6 b FB(\()p FA(\026;)14 b(")p FB(\))p Fw(k)1888 2881 y Fy(0)p Fx(;\033)2009 2856 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2309 2822 y Fx(\021)2348 2856 y FA(:)71 3016 y FB(Then)1307 3116 y Fw(k)o FA(E)1409 3128 y Fy(1)1465 3116 y Fw(\000)18 b FA(C)6 b FB(\()p FA(\026;)14 b(")p FB(\))p Fw(k)1845 3141 y Fy(0)p Fx(;\033)1966 3116 y Fw(\024)22 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2265 3082 y Fx(\021)r Fy(+\(2)p Fx(r)r Fv(\000)p Fx(`)p Fy(\))p Fx(\015)2592 3116 y FA(:)71 3280 y FB(T)-7 b(o)27 b(b)r(ound)h FA(E)510 3292 y Fy(2)575 3280 y FB(de\014ned)g(in)g(\(287\))o(,)g(w)n(e)f (\014rst)h(recall)e(that)i Fw(h)1935 3259 y Fz(f)1925 3280 y FA(M)9 b Fw(i)23 b FB(=)g(0.)36 b(Then)28 b(w)n(e)f(can)h (de\014ne)f(a)h(function)p 3385 3214 90 4 v 28 w FA(M)36 b FB(suc)n(h)27 b(that)1486 3453 y FA(@)1530 3465 y Fx(\034)p 1572 3387 V 1572 3453 a FA(M)k FB(=)1782 3432 y Fz(f)1772 3453 y FA(M)64 b FB(and)55 b Fw(h)p 2138 3387 V FA(M)9 b Fw(i)23 b FB(=)g(0)p FA(;)71 3614 y FB(whic)n(h)k(satis\014es)g(that) h(for)f(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b Fw(2)1213 3547 y Fz(\000)1251 3614 y FA(D)1322 3584 y Fy(in)p Fx(;)p Fy(+)p Fx(;u)1320 3634 y(\024;c)1409 3642 y Fu(1)1530 3614 y Fw(\\)19 b FA(D)1675 3584 y Fy(in)o Fx(;)p Fy(+)p Fx(;s)1673 3634 y(\024;c)1762 3642 y Fu(1)1856 3547 y Fz(\001)1913 3614 y Fw(\002)f Ft(T)2051 3626 y Fx(\033)2096 3614 y FB(,)1553 3717 y Fz(\015)1553 3767 y(\015)p 1599 3721 V 21 x FA(M)1689 3717 y Fz(\015)1689 3767 y(\015)1735 3821 y Fy(0)p Fx(;\033)1856 3788 y Fw(\024)k FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2155 3754 y Fx(\021)r Fy(+)p Fx(\027)t Fv(\000)p Fy(1)71 3964 y FB(where)27 b FA(\027)33 b FB(is)27 b(the)h(constan)n(t)f(de\014ned)h(in)g(\(282\))o(.)37 b(Then,)28 b(w)n(e)f(can)g(write)h FA(E)2366 3976 y Fy(2)2431 3964 y FB(as)1369 4124 y FA(E)1430 4136 y Fy(2)1491 4124 y FB(=)22 b FA(")p Fw(G)1666 4136 y Fx(")1720 4124 y Fw(\016)c(L)1837 4136 y Fx(")1873 4124 y FB(\()p 1905 4058 V FA(M)9 b FB(\))19 b Fw(\000)f FA(")p Fw(G)2217 4136 y Fx(")2266 4057 y Fz(\000)2304 4124 y FA(@)2348 4136 y Fx(v)p 2388 4058 V 2388 4124 a FA(M)2477 4057 y Fz(\001)2529 4124 y FA(:)71 4285 y FB(and)27 b(therefore,)g(b)n(y)g (Lemma)h(9.2,)1596 4384 y Fw(k)o FA(E)1698 4396 y Fy(2)1736 4384 y Fw(k)1777 4409 y Fy(0)p Fx(;\033)1898 4384 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2198 4350 y Fx(\021)r Fy(+)p Fx(\027)71 4546 y FB(F)-7 b(or)27 b FA(E)281 4558 y Fy(3)347 4546 y FB(in)h(\(288\),)g(it)g(is)g(enough)g(to)g(consider)f (the)h(b)r(ound)h(of)f(the)h(Lipsc)n(hitz)f(constan)n(t)f(of)h(the)g (op)r(erator)3488 4525 y Fz(e)3468 4546 y Fw(F)36 b FB(giv)n(en)27 b(in)71 4646 y(the)h(pro)r(of)f(of)g(Lemma)h(9.8,)f(whic)n(h)g(giv)n (es)1602 4794 y Fw(k)p FA(E)1705 4806 y Fy(3)1742 4794 y Fw(k)1784 4806 y Fy(0)p Fx(;\033)1904 4794 y Fw(\024)c FA(K)2078 4738 y Fw(j)p FA(\026)p Fw(j)p FA(")2213 4708 y Fy(2)p Fx(\021)p 2078 4775 209 4 v 2140 4851 a FA(\024)2188 4823 y Fv(0)2188 4873 y Fy(3)2296 4794 y FA(:)71 4976 y FB(Th)n(us,)k(w)n(e)g(ha)n(v)n(e)g(that)1460 5023 y Fz(\015)1460 5073 y(\015)1460 5123 y(\015)1519 5098 y(b)1506 5119 y Fw(C)c(\000)18 b FA(C)6 b FB(\()p FA(\026;)14 b(")p FB(\))1911 5023 y Fz(\015)1911 5073 y(\015)1911 5123 y(\015)1958 5177 y Fy(0)p Fx(;\033)2079 5119 y Fw(\024)22 b FA(K)2253 5062 y Fw(j)p FA(\026)p Fw(j)p FA(")2388 5032 y Fx(\021)p 2253 5100 176 4 v 2298 5176 a FA(\024)2346 5147 y Fv(0)2346 5198 y Fy(3)2438 5119 y FA(:)71 5305 y FB(T)-7 b(o)30 b(\014nish)g(the)h(pro)r(of)f(of)g(Prop)r(osition)f (4.18,)g(it)i(is)f(enough)g(to)g(consider)f(the)i(c)n(hange)e(of)h(v)-5 b(ariables)29 b FA(v)i FB(=)c FA(u)20 b FB(+)g FA(h)p FB(\()p FA(u;)14 b(\034)9 b FB(\))71 5404 y(obtained)29 b(in)i(Lemma)e(9.5.)43 b(Since)30 b FA(h)g FB(restricted)f(to)h(the)g (inner)g(domains)f(satis\014es)g(the)i(b)r(ounds)f(giv)n(en)f(in)h (Corollary)71 5504 y(9.6,)d(this)h(c)n(hange)e(of)i(v)-5 b(ariables)26 b(do)r(es)h(not)h(c)n(hange)e(the)i(asymptotic)f(\014rst) h(order)e(of)h Fw(C)5 b FB(.)p Black 1898 5753 a(100)p Black eop end %%Page: 101 101 TeXDict begin 101 100 bop Black Black 71 272 a Fq(9.3)112 b(Case)38 b Fh(`)28 b Ff(\025)g FF(2)p Fh(r)s Fq(:)49 b(Pro)s(of)38 b(of)f(Theorem)h(4.21)g(and)g(Prop)s(osition)g(4.22)71 425 y Fp(9.3.1)94 b(Pro)s(of)31 b(of)h(Theorem)f(4.21)71 579 y FB(This)j(section)f(is)h(dev)n(oted)g(to)f(pro)n(v)n(e)g(Theorem) g(4.21.)54 b(This)34 b(theorem)g(is)g(a)f(straigh)n(tforw)n(ard)e (consequence)i(of)h(the)71 678 y(follo)n(wing)26 b(prop)r(osition.)p Black 71 842 a Fp(Prop)s(osition)i(9.9.)p Black 38 w Fs(L)l(et)g(us)f(c)l(onsider)i(the)f(c)l(onstants)g FA(d)1890 854 y Fy(2)1950 842 y FA(>)23 b FB(0)k Fs(and)i FA(\024)2315 854 y Fy(6)2375 842 y FA(>)22 b FB(0)28 b Fs(de\014ne)l(d)g(in)g(The)l (or)l(em)h(4.8)h(and)e(Pr)l(op)l(osi-)71 942 y(tion)g(8.3,)h FA(d)447 954 y Fy(3)508 942 y FA(<)23 b(d)639 954 y Fy(2)676 942 y Fs(,)29 b FA(")769 954 y Fy(0)829 942 y FA(>)22 b FB(0)27 b Fs(smal)t(l)i(enough)f(and)g FA(\024)1691 954 y Fy(8)1751 942 y FA(>)23 b(\024)1887 954 y Fy(6)1952 942 y Fs(big)28 b(enough,)h(which)h(might)e(dep)l(end)g(on)g(the)g(pr)l (evious)h(c)l(on-)71 1042 y(stants.)37 b(Then,)29 b(for)f FA(")23 b Fw(2)g FB(\(0)p FA(;)14 b(")1006 1054 y Fy(0)1043 1042 y FB(\))27 b Fs(and)h(any)g FA(\024)23 b Fw(\025)f FA(\024)1624 1054 y Fy(8)1689 1042 y Fs(such)27 b(that)g FA("\024)c(<)g(a)p Fs(,)28 b(ther)l(e)f(exists)g(a)h(function)f Fw(C)h FB(:)23 b FA(R)3341 1054 y Fx(\024;d)3435 1062 y Fu(3)3484 1042 y Fw(\002)13 b Ft(T)3617 1054 y Fx(\033)3684 1042 y Fw(!)23 b Ft(C)71 1141 y Fs(that)30 b(satis\014es)f(e)l(quation) 37 b FB(\(267\))o Fs(.)195 1241 y(Mor)l(e)l(over,)1326 1341 y FB(\()p FA(\030)1394 1353 y Fy(0)1432 1341 y FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))24 b(=)1852 1273 y Fz(\000)1890 1341 y FA(")1929 1306 y Fv(\000)p Fy(1)2018 1341 y FA(u)18 b Fw(\000)g FA(\034)28 b FB(+)18 b Fw(C)5 b FB(\()p FA(u;)14 b(\034)9 b FB(\))2557 1273 y Fz(\001)71 1489 y Fs(is)30 b(inje)l(ctive)h(and)f(ther)l(e)g (exists)f(a)h(c)l(onstant)f FA(b)1523 1501 y Fy(15)1616 1489 y FA(>)23 b FB(0)29 b Fs(such)h(that)p Black 195 1653 a Fw(\017)p Black 41 w Fs(If)h FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b(>)e FB(0)p Fs(,)1566 1833 y Fw(kC)5 b(k)1698 1857 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)1927 1833 y Fw(\024)23 b FA(b)2051 1845 y Fy(15)2121 1833 y Fw(j)7 b FB(^)-49 b FA(\026)p Fw(j)p FA(")2256 1798 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)1479 1977 y Fw(k)o FA(@)1564 1989 y Fx(u)1608 1977 y Fw(C)5 b(k)1698 2002 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)1927 1977 y Fw(\024)23 b FA(b)2051 1989 y Fy(15)2121 1977 y FA(\024)2169 1942 y Fv(\000)p Fy(1)2258 1977 y Fw(j)7 b FB(^)-49 b FA(\026)p Fw(j)p FA(")2393 1942 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fv(\000)p Fy(1)2627 1977 y FA(:)p Black 195 2192 a Fw(\017)p Black 41 w Fs(If)31 b FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b FB(=)e(0)p Fs(,)1803 2367 y Fw(kC)5 b(k)1935 2392 y Fy(ln)o Fx(;\033)2078 2367 y Fw(\024)22 b FA(b)2201 2379 y Fy(15)2271 2367 y Fw(j)7 b FB(^)-49 b FA(\026)q Fw(j)1738 2506 y(k)p FA(@)1824 2518 y Fx(u)1867 2506 y Fw(C)5 b(k)1957 2531 y Fy(1)p Fx(;\033)2078 2506 y Fw(\024)22 b FA(b)2201 2518 y Fy(15)2271 2506 y Fw(j)7 b FB(^)-49 b FA(\026)q Fw(j)p FA(:)195 2722 y FB(W)-7 b(e)29 b(split)f(the)g(pro)r(of)f(in)n (to)g(t)n(w)n(o)g(cases.)37 b(First)27 b(in)h(Section)g(9.3.1)e(w)n(e)i (pro)n(v)n(e)e(it)i(under)g(the)g(h)n(yp)r(othesis)f FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b(>)d FB(0.)71 2822 y(Then,)34 b(in)f(Section)g(9.3.1)e(w)n(e)i(giv)n(e)e(a)i(tec)n(hnical) f(lemma,)i(whic)n(h)f(is)f(used)h(in)g(Section)g(9.3.1)e(to)i(deal)f (with)h(the)g(case)71 2921 y FA(`)18 b Fw(\000)g FB(2)p FA(r)25 b FB(=)e(0.)195 3021 y(Nev)n(ertheless)k(w)n(e)g(need)h(to)f (state)h(some)f(useful)h(prop)r(erties)e(of)i(function)g FA(G)g FB(de\014ned)g(in)g(\(97\).)71 3236 y Fp(Prop)s(erties)j(of)h (the)f(function)h FA(G)84 b FB(Let)27 b(us)h(split)g(the)g(function)g FA(G)g FB(in)g(\(97\))f(as)g FA(G)c FB(=)g FA(G)2952 3248 y Fy(1)3008 3236 y FB(+)18 b FA(G)3156 3248 y Fy(2)3212 3236 y FB(+)g FA(G)3360 3248 y Fy(3)3416 3236 y FB(+)g FA(G)3564 3248 y Fy(4)3629 3236 y FB(with)642 3427 y FA(G)707 3439 y Fy(1)745 3427 y FB(\()p FA(u;)c(\034)9 b FB(\))24 b(=)29 b(^)-49 b FA(\026")1139 3393 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)1289 3427 y FA(p)1331 3439 y Fy(0)1368 3427 y FB(\()p FA(u)p FB(\))1480 3393 y Fv(\000)p Fy(1)1569 3427 y FA(@)1613 3439 y Fx(p)1671 3406 y Fz(b)1651 3427 y FA(H)1727 3393 y Fy(1)1720 3447 y(1)1778 3427 y FB(\()p FA(q)1847 3439 y Fy(0)1885 3427 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2076 3439 y Fy(0)2113 3427 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))1322 b(\(290\))642 3571 y FA(G)707 3583 y Fy(2)745 3571 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)29 b(^)-49 b FA(\026")1139 3537 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)1373 3571 y FA(p)1415 3583 y Fy(0)1452 3571 y FB(\()p FA(u)p FB(\))1564 3537 y Fv(\000)p Fy(1)1653 3571 y FA(@)1697 3583 y Fx(p)1755 3550 y Fz(b)1735 3571 y FA(H)1811 3537 y Fy(2)1804 3592 y(1)1862 3571 y FB(\()p FA(q)1931 3583 y Fy(0)1969 3571 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2160 3583 y Fy(0)2197 3571 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))1238 b(\(291\))642 3753 y FA(G)707 3765 y Fy(3)745 3753 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)1060 3696 y(1)p 1060 3733 42 4 v 1060 3809 a(2)1126 3660 y Fz(\020)1175 3753 y FB(1)18 b(+)25 b(^)-49 b FA(\026")1407 3718 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)1556 3753 y FA(@)1605 3718 y Fy(2)1600 3773 y Fx(p)1662 3732 y Fz(b)1642 3753 y FA(H)1718 3718 y Fy(1)1711 3773 y(1)1769 3753 y FB(\()p FA(q)1838 3765 y Fy(0)1876 3753 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)2067 3765 y Fy(0)2104 3753 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))2331 3660 y Fz(\021)2404 3696 y FA(@)2448 3708 y Fx(u)2492 3696 y FA(T)2553 3666 y Fx(s)2541 3717 y Fy(1)2587 3696 y FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b(+)f FA(@)2927 3708 y Fx(u)2971 3696 y FA(T)3032 3666 y Fx(u)3020 3717 y Fy(1)3074 3696 y FB(\()p FA(u;)c(\034)9 b FB(\))p 2404 3733 865 4 v 2741 3809 a FA(p)2783 3781 y Fy(2)2783 3832 y(0)2820 3809 y FB(\()p FA(u)p FB(\))3661 3753 y(\(292\))642 3935 y FA(G)707 3947 y Fy(4)745 3935 y FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)e FA(G)p FB(\()p FA(u;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(G)1477 3947 y Fy(1)1514 3935 y FB(\()p FA(u;)c(\034)9 b FB(\))19 b Fw(\000)f FA(G)1875 3947 y Fy(2)1913 3935 y FB(\()p FA(u;)c(\034)9 b FB(\))19 b Fw(\000)f FA(G)2274 3947 y Fy(3)2312 3935 y FB(\()p FA(u;)c(\034)9 b FB(\))p FA(;)1132 b FB(\(293\))71 4134 y(where)332 4113 y Fz(b)313 4134 y FA(H)389 4104 y Fy(1)382 4155 y(1)455 4134 y FB(and)637 4113 y Fz(b)618 4134 y FA(H)694 4104 y Fy(2)687 4155 y(1)760 4134 y FB(are)28 b(the)i(functions)f(de\014ned)h(in)f(\(35\))g (and)g(\(37\).)41 b(The)30 b(next)f(lemma)g(giv)n(es)f(prop)r(erties)g (of)h(these)71 4234 y(functions.)p Black 71 4398 a Fp(Lemma)40 b(9.10.)p Black 44 w Fs(L)l(et)c(us)g(c)l(onsider)i(any)f FA(\024)e(>)g(\024)1657 4410 y Fy(6)1731 4398 y Fs(and)i FA(d)e(<)g(d)2120 4410 y Fy(2)2158 4398 y Fs(,)k(wher)l(e)e FA(\024)2511 4410 y Fy(6)2584 4398 y Fs(an)g FA(d)2753 4410 y Fy(0)2827 4398 y Fs(ar)l(e)g(the)g(c)l(onstants)e(de\014ne)l(d)i (in)71 4498 y(The)l(or)l(ems)g(8.3)g(and)g(4.7.)59 b(Then,)39 b(the)d(functions)g FA(G)1789 4510 y Fx(i)1817 4498 y Fs(,)i FA(i)c FB(=)g(1)p FA(;)14 b FB(2)p FA(;)g FB(3)p FA(;)g FB(4)34 b Fs(de\014ne)l(d)i(in)43 b FB(\(290\))o Fs(,)38 b FB(\(291\))o Fs(,)g FB(\(292\))d Fs(and)45 b FB(\(293\))71 4597 y Fs(r)l(esp)l(e)l(ctively,)31 b(have)g(the)f(fol) t(lowing)i(pr)l(op)l(erties.)p Black 169 4762 a(1.)p Black 42 w FA(G)343 4774 y Fy(1)404 4762 y Fw(2)23 b(X)541 4774 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)778 4762 y Fs(and)30 b(it)g(satis\014es)f Fw(h)p FA(G)1430 4774 y Fy(1)1468 4762 y Fw(i)24 b FB(=)e(0)29 b Fs(and)1621 4942 y Fw(k)p FA(G)1728 4954 y Fy(1)1765 4942 y Fw(k)1807 4967 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)2036 4942 y Fw(\024)23 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)p FA(")2335 4908 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)2485 4942 y FA(:)278 5122 y Fs(Mor)l(e)l(over)p Black 378 5287 a Fw(\017)p Black 41 w Fs(If)30 b FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b(>)c FB(0)p Fs(,)30 b FA(@)1017 5299 y Fx(u)1060 5287 y FA(G)1125 5299 y Fy(1)1186 5287 y Fw(2)23 b(X)1323 5299 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;\033)1647 5287 y Fs(and)30 b(it)g(satis\014es)1625 5468 y Fw(k)o FA(@)1710 5480 y Fx(u)1754 5468 y FA(G)1819 5480 y Fy(1)1856 5468 y Fw(k)1898 5493 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;\033)2215 5468 y Fw(\024)23 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)p FA(")2514 5434 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)2663 5468 y FA(:)p Black 1898 5753 a FB(101)p Black eop end %%Page: 102 102 TeXDict begin 102 101 bop Black Black Black 378 272 a Fw(\017)p Black 41 w Fs(If)30 b FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b FB(=)c(0)p Fs(,)30 b FA(@)1017 284 y Fx(u)1060 272 y FA(G)1125 284 y Fy(1)1186 272 y Fw(2)23 b(X)1323 293 y Fy(1)p Fv(\000)1422 270 y Fu(1)p 1419 279 35 3 v 1419 313 a Fm(\014)1464 293 y Fx(;\033)1558 272 y Fs(and)30 b(it)g(satis\014es)1764 462 y Fw(k)o FA(@)1849 474 y Fx(u)1893 462 y FA(G)1958 474 y Fy(1)1995 462 y Fw(k)2037 487 y Fy(1)p Fv(\000)2135 465 y Fu(1)p 2132 474 V 2132 507 a Fm(\014)2177 487 y Fx(;\033)2264 462 y Fw(\024)23 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)p FA(:)p Black 169 666 a Fs(2.)p Black 42 w FA(G)343 678 y Fy(2)404 666 y Fw(2)23 b(X)541 678 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)778 666 y Fs(and)30 b(it)g(satis\014es)1518 776 y Fw(k)p FA(G)1625 788 y Fy(2)1662 776 y Fw(k)1704 801 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)1933 776 y Fw(\024)23 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)2193 742 y Fy(2)2230 776 y FA(")2269 742 y Fy(2\()p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(\)+1)2588 776 y FA(:)278 919 y Fs(Mor)l(e)l(over)p Black 378 1080 a Fw(\017)p Black 41 w Fs(If)30 b FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b(>)c FB(0)p Fs(,)30 b FA(@)1017 1092 y Fx(u)1060 1080 y FA(G)1125 1092 y Fy(2)1186 1080 y Fw(2)23 b(X)1323 1092 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;\033)1647 1080 y Fs(and)30 b(it)g(satis\014es)1522 1261 y Fw(k)o FA(@)1607 1273 y Fx(u)1651 1261 y FA(G)1716 1273 y Fy(2)1753 1261 y Fw(k)1795 1286 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;\033)2112 1261 y Fw(\024)22 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)2372 1226 y Fy(2)2409 1261 y FA(")2448 1226 y Fy(2\()p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(\)+1)2767 1261 y FA(:)p Black 378 1446 a Fw(\017)p Black 41 w Fs(If)30 b FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b FB(=)c(0)p Fs(,)30 b FA(@)1017 1458 y Fx(u)1060 1446 y FA(G)1125 1458 y Fy(2)1186 1446 y Fw(2)23 b(X)1323 1466 y Fy(1)p Fv(\000)1422 1444 y Fu(1)p 1419 1453 V 1419 1486 a Fm(\014)1464 1466 y Fx(;\033)1558 1446 y Fs(and)30 b(it)g(satis\014es)1726 1645 y Fw(k)o FA(@)1811 1657 y Fx(u)1855 1645 y FA(G)1920 1657 y Fy(2)1957 1645 y Fw(k)1999 1670 y Fy(1)p Fv(\000)2097 1648 y Fu(1)p 2094 1657 V 2094 1690 a Fm(\014)2139 1670 y Fx(;\033)2226 1645 y Fw(\024)23 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)2486 1611 y Fy(2)2524 1645 y FA(":)p Black 169 1849 a Fs(3.)p Black 42 w FA(G)343 1861 y Fy(3)404 1849 y Fw(2)23 b(X)541 1861 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;\033)865 1849 y Fs(and)30 b(satis\014es)1535 1949 y Fw(k)p FA(G)1642 1961 y Fy(3)1679 1949 y Fw(k)1721 1974 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;\033)2038 1949 y Fw(\024)22 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FA(")2337 1915 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)2570 1949 y FA(:)278 2091 y Fs(Mor)l(e)l(over,)p Black 378 2252 a Fw(\017)p Black 41 w Fs(If)30 b FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b(>)c FB(0)p Fs(,)30 b FA(@)1017 2264 y Fx(u)1060 2252 y FA(G)1125 2264 y Fy(3)1186 2252 y Fw(2)23 b(X)1323 2264 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;\033)1647 2252 y Fs(and)30 b(it)g(satis\014es)1556 2423 y Fw(k)p FA(@)1642 2435 y Fx(u)1685 2423 y FA(G)1750 2435 y Fy(3)1788 2423 y Fw(k)1830 2448 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;\033)2146 2423 y Fw(\024)23 b FA(K)6 b(\024)2359 2389 y Fv(\000)p Fy(1)2448 2423 y Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FA(")2583 2389 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)2732 2423 y FA(:)p Black 378 2608 a Fw(\017)p Black 41 w Fs(If)30 b FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b FB(=)c(0)p Fs(,)30 b FA(@)1017 2620 y Fx(u)1060 2608 y FA(G)1125 2620 y Fy(3)1186 2608 y Fw(2)23 b(X)1323 2620 y Fy(2)p Fx(;\033)1451 2608 y Fs(and)30 b(it)g(satis\014es)1798 2779 y Fw(k)o FA(@)1883 2791 y Fx(u)1927 2779 y FA(G)1992 2791 y Fy(3)2029 2779 y Fw(k)2071 2804 y Fy(2)p Fx(;\033)2192 2779 y Fw(\024)22 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FA(":)p Black 169 2964 a Fs(4.)p Black 42 w FA(G)343 2976 y Fy(4)381 2964 y FA(;)14 b(@)462 2976 y Fx(u)505 2964 y FA(G)570 2976 y Fy(4)631 2964 y Fw(2)23 b(X)768 2979 y Fy(3\()p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(\)+2)p Fx(;\033)1177 2964 y Fs(and)30 b(satisfy)1407 3150 y Fw(k)p FA(G)1514 3162 y Fy(4)1551 3150 y Fw(k)1593 3175 y Fy(3\()p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(\)+2)p Fx(;\033)1994 3150 y Fw(\024)23 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)2255 3116 y Fy(3)2292 3150 y FA(")2331 3116 y Fy(3\()p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(\)+2)1320 3301 y Fw(k)o FA(@)1405 3313 y Fx(u)1449 3301 y FA(G)1514 3313 y Fy(4)1551 3301 y Fw(k)1593 3326 y Fy(3\()p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(\)+2)p Fx(;\033)1994 3301 y Fw(\024)23 b FA(K)6 b(\024)2207 3266 y Fv(\000)p Fy(1)2296 3301 y Fw(j)h FB(^)-49 b FA(\026)p Fw(j)2392 3266 y Fy(3)2429 3301 y FA(")2468 3266 y Fy(3\()p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(\)+1)2786 3301 y FA(:)p Black 71 3508 a Fs(Pr)l(o)l(of.)p Black 43 w FB(The)32 b(pro)r(of)f(of)h(the)g(statemen)n(ts)f(ab)r(out)h FA(G)1702 3520 y Fy(1)1771 3508 y FB(and)f FA(G)2001 3520 y Fy(2)2071 3508 y FB(are)f(straigh)n(tforw)n(ard,)g(taking)h(in)n(to)h(accoun)n (t,)g(for)f FA(G)3790 3520 y Fy(2)3827 3508 y FB(,)71 3608 y(the)e(b)r(ounds)f(obtained)g(in)h(Corollary)c(5.6.)38 b(F)-7 b(or)28 b FA(G)1694 3620 y Fy(3)1732 3608 y FB(,)g(one)g(has)g (to)g(tak)n(e)g(in)n(to)g(accoun)n(t)f(the)i(b)r(ounds)f(for)g FA(T)3466 3577 y Fx(u)3454 3628 y Fy(1)3537 3608 y FB(obtained)71 3707 y(in)h(Prop)r(osition)f(7.4)g(and)h(Corollary)e(7.22)h(and)h(the)g (analogous)e(b)r(ounds)j(that)f FA(T)2687 3677 y Fx(s)2675 3728 y Fy(1)2751 3707 y FB(satis\014es.)41 b(T)-7 b(o)29 b(obtain)g(the)g(b)r(ound)71 3807 y(for)j(its)h(deriv)-5 b(ativ)n(e,)34 b(one)e(can)h(apply)f(the)i(fourth)f(statemen)n(t)f(of)h (Lemma)g(9.1.)52 b(Analogously)-7 b(,)32 b(one)h(can)f(obtain)h(the)71 3906 y(b)r(ounds)28 b(for)f FA(G)552 3918 y Fy(4)617 3906 y FB(and)g FA(@)822 3918 y Fx(u)866 3906 y FA(G)931 3918 y Fy(4)968 3906 y FB(.)p 3790 3906 4 57 v 3794 3854 50 4 v 3794 3906 V 3843 3906 4 57 v 71 4120 a Fp(Case)k FA(`)17 b Fw(\000)g FB(2)p FA(r)25 b(>)e FB(0)82 b(T)-7 b(o)27 b(pro)n(v)n(e)e(Prop)r(osition)g(9.9)i(for)f FA(`)17 b Fw(\000)g FB(2)p FA(r)25 b(>)e FB(0,)k(w)n(e)f(lo)r(ok)g(for)h Fw(C)32 b FB(as)26 b(a)h(\014xed)g(p)r(oin)n(t)g(of)g(the)g(op)r (erator)p 1744 4224 68 4 v 1744 4291 a Fw(F)k FB(=)23 b Fw(G)1972 4303 y Fx(")2026 4291 y Fw(\016)18 b(F)8 b FA(;)1484 b FB(\(294\))71 4461 y(where)25 b Fw(G)358 4473 y Fx(")419 4461 y FB(and)h Fw(F)33 b FB(are)25 b(the)h(op)r (erators)d(de\014ned)j(in)g(\(270\))f(and)g(\(268\))g(resp)r(ectiv)n (ely)-7 b(.)35 b(F)-7 b(or)25 b(con)n(v)n(enience,)g(w)n(e)g(rewrite)f Fw(F)71 4561 y FB(as)748 4661 y Fw(F)8 b FB(\()p Fw(C)d FB(\)\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)f Fw(\000)p FA(")1339 4626 y Fv(\000)p Fy(1)1427 4661 y FA(G)p FB(\()p FA(u;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(@)1833 4673 y Fx(u)1890 4661 y FB(\()p FA(G)p FB(\()p FA(u;)c(\034)9 b FB(\))p Fw(C)c FB(\()p FA(u;)14 b(\034)9 b FB(\)\))20 b(+)e FA(@)2603 4673 y Fx(u)2647 4661 y FA(G)p FB(\()p FA(u;)c(\034)9 b FB(\))p Fw(C)c FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(:)489 b FB(\(295\))71 4803 y(Then)28 b(Prop)r(osition)e (9.9)g(is)i(a)f(consequence)g(of)g(the)h(follo)n(wing)f(lemma.)p Black 71 4959 a Fp(Lemma)34 b(9.11.)p Black 40 w Fs(L)l(et)e(us)e(c)l (onsider)j FA(")1274 4971 y Fy(0)1337 4959 y FA(>)26 b FB(0)31 b Fs(smal)t(l)h(enough)g(and)g FA(\024)2218 4971 y Fy(8)2282 4959 y FA(>)25 b(\024)2420 4971 y Fy(6)2489 4959 y Fs(big)32 b(enough.)45 b(Then,)33 b(for)f FA(")26 b Fw(2)g FB(\(0)p FA(;)14 b(")3618 4971 y Fy(0)3655 4959 y FB(\))32 b Fs(and)71 5059 y(any)e FA(\024)23 b Fw(\025)g FA(\024)437 5071 y Fy(8)503 5059 y Fs(such)30 b(that)g FA("\024)22 b(<)h(a)p Fs(,)30 b(the)g(op)l(er)l(ator)p 1623 4992 V 31 w Fw(F)37 b Fs(de\014ne)l(d)30 b(in)37 b FB(\(294\))28 b Fs(is)j(c)l(ontr)l(active)f(fr)l(om)g Fw(X)3093 5071 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)3329 5059 y Fs(to)g(itself.)195 5159 y(Then,)h(it)f(has)h(a)f(unique)f (\014xe)l(d)g(p)l(oint)h Fw(C)e(2)23 b(X)1618 5171 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)1825 5159 y Fs(,)30 b(which)h(mor)l(e)l(over)g(satis\014es)1477 5336 y Fw(kC)5 b(k)1610 5348 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)1838 5336 y Fw(\024)23 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)p FA(")2137 5302 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)1390 5474 y Fw(k)p FA(@)1476 5486 y Fx(u)1519 5474 y Fw(C)5 b(k)1610 5486 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)1838 5474 y Fw(\024)23 b FA(K)6 b(\024)2051 5440 y Fv(\000)p Fy(1)2139 5474 y Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FA(")2274 5440 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fv(\000)p Fy(1)2508 5474 y FA(:)p Black 1898 5753 a FB(102)p Black eop end %%Page: 103 103 TeXDict begin 103 102 bop Black Black 195 272 a FB(Before)40 b(pro)n(ving)f(Lemma)h(9.11,)j(w)n(e)d(state)g(the)h(follo)n(wing)f (tec)n(hnical)g(lemma)h(ab)r(out)f(the)h(prop)r(erties)f(of)g(the)71 372 y(function)28 b FA(G)g FB(de\014ned)g(in)g(\(97\))o(.)p Black 71 538 a Fp(Lemma)46 b(9.12.)p Black 47 w Fs(L)l(et)40 b(us)h(assume)g FA(`)26 b Fw(\000)g FB(2)p FA(r)46 b(>)d FB(0)p Fs(.)73 b(Then,)45 b(the)c(function)g FA(G)g Fs(de\014ne)l(d)h (in)47 b FB(\(97\))41 b Fs(has)h(the)f(fol)t(lowing)71 637 y(pr)l(op)l(erties:)p Black 169 803 a(1.)p Black 42 w FA(G)24 b Fw(2)f(X)504 815 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)740 803 y Fs(and)31 b(satis\014es)1640 903 y Fw(k)p FA(G)p Fw(k)1789 915 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)2017 903 y Fw(\024)23 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FA(")2317 869 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)2466 903 y FA(:)p Black 169 1086 a Fs(2.)p Black 42 w FA(@)322 1098 y Fx(u)366 1086 y FA(G)23 b Fw(2)h(X)592 1098 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;\033)915 1086 y Fs(and)31 b(satis\014es)1552 1185 y Fw(k)p FA(@)1638 1197 y Fx(u)1681 1185 y FA(G)p Fw(k)1788 1197 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)p Fx(;\033)2105 1185 y Fw(\024)23 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)p FA(")2404 1151 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)2553 1185 y FA(:)p Black 169 1368 a Fs(3.)p Black 42 w Fw(G)327 1380 y Fx(")363 1368 y FB(\()p FA(G)p FB(\))24 b Fw(2)g(X)654 1380 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)890 1368 y Fs(and)30 b(satis\014es)1523 1468 y Fw(kG)1614 1480 y Fx(")1649 1468 y FB(\()p FA(G)p FB(\))p Fw(k)1820 1480 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)2050 1468 y Fw(\024)23 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)p FA(")2349 1433 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)2583 1468 y FA(:)p Black 71 1650 a Fs(Pr)l(o)l(of.)p Black 43 w FB(The)33 b(b)r(ounds)h(for)f FA(G)g FB(and)h FA(@)1245 1662 y Fx(u)1288 1650 y FA(G)g FB(are)e(a)h(direct)g(consequence)f(of)i (Lemma)f(9.10.)53 b(T)-7 b(o)33 b(obtain)g(the)g(b)r(ound)h(for)71 1750 y Fw(G)120 1762 y Fx(")156 1750 y FB(\()p FA(G)p FB(\),)28 b(it)g(is)g(enough)f(to)g(apply)h(Lemma)f(9.2)g(and)g(to)h (tak)n(e)e(in)n(to)i(accoun)n(t)f(that)g Fw(h)p FA(G)2743 1762 y Fy(1)2781 1750 y Fw(i)d FB(=)e(0.)p 3790 1750 4 57 v 3794 1697 50 4 v 3794 1750 V 3843 1750 4 57 v 195 1916 a(Using)28 b(the)g(b)r(ounds)f(giv)n(en)g(in)h(this)g(lemma,)g (w)n(e)f(can)g(pro)n(v)n(e)f(Lemma)h(9.11.)p Black 71 2082 a Fs(Pr)l(o)l(of)k(of)f(L)l(emma)g(9.11.)p Black 44 w FB(T)-7 b(o)28 b(see)g(that)p 1344 2015 68 4 v 29 w Fw(F)37 b FB(is)28 b(con)n(tractiv)n(e,)f(let)i(us)g(consider)e Fw(C)2573 2094 y Fy(1)2610 2082 y FA(;)14 b Fw(C)2691 2094 y Fy(2)2753 2082 y Fw(2)25 b(X)2892 2094 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)3098 2082 y FB(.)40 b(Then,)29 b(recalling)e(the)71 2182 y(de\014nition)h(of)f Fw(F)36 b FB(in)28 b(\(295\))f(and)g(applying)g(Lemmas)g(9.2,)g(9.1)g(and)g (9.12,)409 2296 y Fz(\015)409 2346 y(\015)p 455 2300 V 21 x Fw(F)8 b FB(\()p Fw(C)599 2379 y Fy(2)636 2367 y FB(\))19 b Fw(\000)p 770 2300 V 18 w(F)8 b FB(\()p Fw(C)914 2379 y Fy(1)951 2367 y FB(\))983 2296 y Fz(\015)983 2346 y(\015)1029 2400 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)1259 2367 y Fw(\024)22 b(kG)1437 2379 y Fx(")1486 2367 y FB(\()q FA(@)1563 2379 y Fx(u)1620 2367 y FB(\()p FA(G)d Fw(\001)g FB(\()p Fw(C)1854 2379 y Fy(2)1909 2367 y Fw(\000)f(C)2036 2379 y Fy(1)2073 2367 y FB(\))q(\)\))p Fw(k)2212 2392 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)2436 2367 y FB(+)g Fw(kG)2610 2379 y Fx(")2660 2367 y FB(\()p FA(@)2736 2379 y Fx(u)2779 2367 y FA(G)h Fw(\001)g FB(\()p Fw(C)2981 2379 y Fy(2)3036 2367 y Fw(\000)f(C)3163 2379 y Fy(1)3200 2367 y FB(\))q(\))p Fw(k)3306 2392 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)1259 2514 y Fw(\024)k FA(K)d Fw(k)p FA(G)p Fw(k)1585 2539 y Fy(0)p Fx(;\033)1696 2514 y Fw(kC)1782 2526 y Fy(2)1837 2514 y Fw(\000)f(C)1964 2526 y Fy(1)2001 2514 y Fw(k)2043 2539 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)2267 2514 y FB(+)g FA(K)i Fw(k)o FA(@)2526 2526 y Fx(u)2570 2514 y FA(G)p Fw(k)2676 2539 y Fy(1)p Fx(;\033)2788 2514 y Fw(k)o(C)2873 2526 y Fy(2)2929 2514 y Fw(\000)e(C)3056 2526 y Fy(1)3093 2514 y Fw(k)3134 2539 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)1259 2702 y Fw(\024)1369 2646 y FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)p 1356 2683 198 4 v 1356 2769 a FA(\024)1404 2734 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)1404 2791 y Fy(7)1577 2702 y Fw(kC)1663 2714 y Fy(2)1718 2702 y Fw(\000)18 b(C)1845 2714 y Fy(1)1882 2702 y Fw(k)1924 2727 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)2144 2702 y FA(:)71 2956 y FB(Then,)32 b(increasing)e FA(\024)755 2968 y Fy(8)823 2956 y FB(if)h(necessary)-7 b(,)p 1290 2889 68 4 v 30 w Fw(F)40 b FB(is)30 b(con)n(tractiv)n(e)g(from)g Fw(X)2165 2968 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)2403 2956 y FB(to)h(itself,)h(and)f(then)h(it)f(has)g(a)f(unique)h(\014xed) 71 3055 y(p)r(oin)n(t)d Fw(C)f(2)d(X)497 3067 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)703 3055 y FB(.)195 3155 y(T)-7 b(o)28 b(obtain)f(a)g(b)r(ound)h(for)f(the)h(\014xed)g(p)r(oin)n (t)g Fw(C)k FB(it)c(is)f(enough)g(to)h(recall)e(that)1454 3338 y Fw(kC)5 b(k)1586 3363 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)1815 3338 y Fw(\024)23 b FB(2)1959 3267 y Fz(\015)1959 3317 y(\015)p 2004 3271 V 2004 3338 a Fw(F)8 b FB(\(0\))2178 3267 y Fz(\015)2178 3317 y(\015)2224 3371 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;\033)2444 3338 y FA(:)71 3547 y FB(By)29 b(the)g(de\014nition)h(of)p 814 3480 V 29 w Fw(F)37 b FB(in)30 b(\(294\))o(,)p 1251 3480 V 29 w Fw(F)8 b FB(\(0\))26 b(=)f Fw(\000)p FA(")1645 3517 y Fv(\000)p Fy(1)1734 3547 y Fw(G)1783 3559 y Fx(")1819 3547 y FB(\()p FA(G)p FB(\).)42 b(Then,)30 b(applying)f(Lemma)g(9.12,)f (w)n(e)h(obtain)g(the)g(b)r(ound)71 3647 y(for)34 b Fw(C)5 b FB(.)57 b(Finally)-7 b(,)36 b(to)e(obtain)g(the)h(b)r(ound)g(for)f FA(@)1605 3659 y Fx(u)1648 3647 y Fw(C)39 b FB(it)c(is)f(enough)g(to)g (reduce)g(sligh)n(tly)g(the)h(domain)f(and)g(apply)g(the)71 3746 y(fourth)27 b(statemen)n(t)h(of)g(Lemma)f(9.1.)p 3790 3746 4 57 v 3794 3693 50 4 v 3794 3746 V 3843 3746 4 57 v Black 71 3912 a Fs(Pr)l(o)l(of)k(of)f(Pr)l(op)l(osition)i(9.9)f (for)f FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b(>)c FB(0)p Fs(.)p Black 42 w FB(T)-7 b(o)31 b(pro)n(v)n(e)f(Prop)r(osition)g(9.9)h (from)g(Lemma)h(9.11,)f(it)h(only)g(remains)f(to)71 4012 y(c)n(hec)n(k)e(that)h(\()p FA(\030)547 4024 y Fy(0)585 4012 y FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))31 b(is)f(injectiv)n(e)g(in)g FA(R)1504 4024 y Fx(\024;d)1598 4032 y Fu(3)1654 4012 y Fw(\002)20 b Ft(T)1794 4024 y Fx(\033)1839 4012 y FB(.)44 b(W)-7 b(e)30 b(pro)n(v)n(e)e(this)i(fact)g(as)f(in)i(the)f(pro)r(of)f(of)h(Prop)r (osition)e(9.3,)71 4111 y(that)g(is,)f(w)n(e)h(c)n(hec)n(k)e(that)i(if) 1164 4211 y FA(")1203 4177 y Fv(\000)p Fy(1)1291 4211 y FA(u)1339 4223 y Fy(2)1395 4211 y Fw(\000)18 b FA(\034)28 b FB(+)18 b Fw(C)5 b FB(\()p FA(u)1754 4223 y Fy(2)1790 4211 y FA(;)14 b(\034)9 b FB(\))24 b(=)f FA(")2055 4177 y Fv(\000)p Fy(1)2144 4211 y FA(u)2192 4223 y Fy(1)2247 4211 y Fw(\000)18 b FA(\034)28 b FB(+)18 b Fw(C)5 b FB(\()p FA(u)2606 4223 y Fy(1)2643 4211 y FA(;)14 b(\034)9 b FB(\))71 4361 y(for)27 b(\()p FA(u)278 4373 y Fy(1)315 4361 y FA(;)14 b(\034)9 b FB(\))p FA(;)14 b FB(\()p FA(u)546 4373 y Fy(2)584 4361 y FA(;)g(\034)9 b FB(\))25 b Fw(2)f FA(R)865 4373 y Fx(\024;d)959 4381 y Fu(3)1014 4361 y Fw(\002)18 b Ft(T)1152 4373 y Fx(\033)1197 4361 y FB(,)29 b(then)f(w)n(e)g(ha)n(v)n(e)f(that)h FA(u)1981 4373 y Fy(2)2042 4361 y FB(=)23 b FA(u)2178 4373 y Fy(1)2215 4361 y FB(.)38 b(T)-7 b(o)28 b(pro)n(v)n(e)f(this)h(fact,)g(it)h(is)f (enough)f(to)h(tak)n(e)g(in)n(to)71 4460 y(accoun)n(t)f(the)h(b)r(ound) g(of)f FA(@)915 4472 y Fx(u)959 4460 y Fw(C)32 b FB(giv)n(en)27 b(in)g(Lemma)h(9.11,)e(whic)n(h)i(giv)n(es)1366 4637 y Fw(j)p FA(u)1437 4649 y Fy(2)1492 4637 y Fw(\000)18 b FA(u)1623 4649 y Fy(1)1660 4637 y Fw(j)23 b FB(=)g FA(")14 b Fw(jC)5 b FB(\()p FA(u)1999 4649 y Fy(2)2035 4637 y FA(;)14 b(\034)9 b FB(\))20 b Fw(\000)e(C)5 b FB(\()p FA(u)2381 4649 y Fy(1)2417 4637 y FA(;)14 b(\034)9 b FB(\))p Fw(j)1706 4819 y(\024)1858 4763 y FA(K)d Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p 1804 4800 282 4 v 1804 4886 a FA(\024)1852 4850 y Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(+1)1852 4908 y(8)2095 4819 y Fw(j)p FA(u)2166 4831 y Fy(2)2222 4819 y Fw(\000)18 b FA(u)2353 4831 y Fy(1)2389 4819 y Fw(j)p FA(:)71 5057 y FB(Then,)28 b(increasing)e FA(\024)747 5069 y Fy(8)812 5057 y FB(if)i(necessary)-7 b(,)26 b(one)h(can)g(see)g(that)h FA(u)1938 5069 y Fy(2)1998 5057 y FB(=)23 b FA(u)2134 5069 y Fy(1)2170 5057 y FB(.)p 3790 5057 4 57 v 3794 5004 50 4 v 3794 5057 V 3843 5057 4 57 v Black 1898 5753 a(103)p Black eop end %%Page: 104 104 TeXDict begin 104 103 bop Black Black 71 272 a Fp(Case)28 b FA(`)13 b Fw(\000)g FB(2)p FA(r)25 b FB(=)e(0)82 b(W)-7 b(e)26 b(dev)n(ote)e(this)h(section)g(to)g(pro)n(v)n(e)e(Prop)r (osition)g(9.9)h(under)h(the)g(h)n(yp)r(othesis)g FA(`)13 b Fw(\000)g FB(2)p FA(r)25 b FB(=)d(0.)36 b(No)n(w,)71 372 y(as)c(happ)r(ened)g(in)h(Section)f(9.2,)h(the)g(linear)f(term)g FA(G)1770 384 y Fy(1)1840 372 y FB(in)h(\(290\))e(of)i Fw(F)40 b FB(in)33 b(\(268\))e(is)i(not)f(small.)51 b(Then,)34 b(w)n(e)e(p)r(erform)71 471 y(again)26 b(a)h(c)n(hange)g(of)g(v)-5 b(ariables.)p Black 71 624 a Fp(Lemma)36 b(9.13.)p Black 43 w Fs(L)l(et)d(us)g(c)l(onsider)h FA(\024)1293 636 y Fy(8)1361 624 y FA(>)29 b(\024)1503 594 y Fv(0)1503 645 y Fy(6)1570 624 y FA(>)h(\024)1713 636 y Fy(6)1784 624 y Fs(and)k FA(d)1992 636 y Fy(3)2059 624 y FA(<)c(d)2197 594 y Fv(0)2197 645 y Fy(2)2264 624 y FA(<)g(d)2402 636 y Fy(2)2439 624 y Fs(.)51 b(Then,)35 b(for)g FA(")30 b(>)f FB(0)k Fs(smal)t(l)i(enough,)g(ther)l(e)71 724 y(exists)29 b(a)h(function)g FA(g)i Fs(which)g(is)e(solution)g(of)g (the)g(e)l(quation)1584 890 y Fw(L)1641 902 y Fx(")1677 890 y FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f FA(G)2086 902 y Fy(1)2123 890 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)71 1056 y Fs(wher)l(e)30 b FA(G)370 1068 y Fy(1)438 1056 y Fs(is)g(the)g(function)f(de\014ne)l(d)h(in)37 b FB(\(290\))o Fs(.)h(Mor)l(e)l(over,)32 b(it)e(satis\014es)g(that)1090 1222 y Fw(k)p FA(g)s Fw(k)1217 1236 y Fy(0)p Fx(;\024)1309 1216 y Fl(0)1309 1254 y Fu(6)1339 1236 y Fx(;d)1394 1216 y Fl(0)1394 1254 y Fu(2)1426 1236 y Fx(;\033)1513 1222 y Fw(\024)23 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FA(";)83 b Fw(k)p FA(@)2005 1234 y Fx(v)2044 1222 y FA(g)s Fw(k)2129 1242 y Fy(1)p Fv(\000)2226 1220 y Fu(1)p 2223 1229 35 3 v 2223 1262 a Fm(\014)2268 1242 y Fx(;\024)2327 1222 y Fl(0)2327 1260 y Fu(6)2358 1242 y Fx(;d)2413 1222 y Fl(0)2413 1260 y Fu(2)2445 1242 y Fx(;\033)2533 1222 y Fw(\024)22 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FA(")71 1407 y Fs(and)30 b(that)g FA(u)22 b FB(=)h FA(v)f FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))24 b Fw(2)f FA(R)1102 1419 y Fx(\024)1141 1427 y Fu(6)1173 1419 y Fx(;d)1228 1427 y Fu(2)1294 1407 y Fs(for)31 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b Fw(2)f FA(R)1781 1421 y Fx(\024)1820 1402 y Fl(0)1820 1440 y Fu(6)1853 1421 y Fx(;d)1908 1402 y Fl(0)1908 1440 y Fu(2)1962 1407 y Fw(\002)18 b Ft(T)2100 1419 y Fx(\033)2145 1407 y Fs(.)195 1507 y(F)-6 b(urthermor)l(e,)34 b(the)f(change)g FB(\()p FA(u;)14 b(\034)9 b FB(\))29 b(=)f(\()p FA(v)c FB(+)c FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))34 b Fs(is)f(invertible)h(and)f(its)g(inverse)g(is)g (of)g(the)g(form)g FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))30 b(=)71 1607 y(\()p FA(u)18 b FB(+)g FA(h)p FB(\()p FA(u;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))p Fs(.)40 b(The)31 b(function)f FA(h)f Fs(is)h(de\014ne)l(d)g(in)g(the)g(domain)h FA(R)2216 1619 y Fx(\024)2255 1627 y Fu(8)2287 1619 y Fx(;d)2342 1627 y Fu(3)2397 1607 y Fw(\002)18 b Ft(T)2535 1619 y Fx(\033)2610 1607 y Fs(and)30 b(it)g(satis\014es)1597 1773 y Fw(k)p FA(h)p Fw(k)1729 1785 y Fy(0)p Fx(;\024)1821 1793 y Fu(8)1852 1785 y Fx(;d)1907 1793 y Fu(3)1938 1785 y Fx(;\033)2026 1773 y Fw(\024)22 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FA(")71 1939 y Fs(and)30 b(that)g FA(u)18 b FB(+)g FA(h)p FB(\()p FA(u;)c(\034)9 b FB(\))23 b Fw(2)h FA(R)958 1953 y Fx(\024)997 1934 y Fl(0)997 1972 y Fu(6)1029 1953 y Fx(;d)1084 1934 y Fl(0)1084 1972 y Fu(2)1150 1939 y Fs(for)30 b FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b Fw(2)f FA(R)1641 1951 y Fx(\024)1680 1959 y Fu(8)1713 1951 y Fx(;d)1768 1959 y Fu(3)1822 1939 y Fw(\002)18 b Ft(T)1960 1951 y Fx(\033)2005 1939 y Fs(.)p Black 71 2107 a(Pr)l(o)l(of.)p Black 43 w FB(F)-7 b(rom)27 b(Lemma)h(9.10,)e Fw(h)p FA(G)1141 2119 y Fy(1)1179 2107 y Fw(i)d FB(=)g(0)k(and)g(then)h(w)n(e)g(can)f(de\014ne)h(a)f(function) p 2650 2040 66 4 v 28 w FA(G)2715 2119 y Fy(1)2780 2107 y FB(suc)n(h)g(that)1467 2273 y FA(@)1511 2285 y Fx(\034)p 1552 2206 V 1552 2273 a FA(G)1617 2285 y Fy(1)1678 2273 y FB(=)22 b FA(G)1830 2285 y Fy(1)1923 2273 y FB(and)55 b Fw(h)p 2144 2206 V FA(G)2210 2285 y Fy(1)2247 2273 y Fw(i)23 b FB(=)g(0)p FA(;)1206 b FB(\(296\))71 2439 y(whic)n(h)27 b(satis\014es)1668 2517 y Fz(\015)1668 2566 y(\015)p 1714 2521 V 21 x FA(G)1779 2599 y Fy(1)1816 2517 y Fz(\015)1816 2566 y(\015)1862 2620 y Fy(0)p Fx(;\024)1954 2600 y Fl(0)1954 2638 y Fu(6)1986 2620 y Fx(;d)2041 2600 y Fl(0)2041 2638 y Fu(2)2073 2620 y Fx(;\033)2161 2587 y Fw(\024)22 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)1477 2674 y Fz(\015)1477 2724 y(\015)1523 2745 y FA(@)1567 2757 y Fx(v)p 1607 2678 V 1607 2745 a FA(G)1672 2757 y Fy(1)1709 2674 y Fz(\015)1709 2724 y(\015)1756 2778 y Fy(1)p Fv(\000)1854 2756 y Fu(1)p 1851 2765 35 3 v 1851 2798 a Fm(\014)1895 2778 y Fx(;\024)1954 2758 y Fl(0)1954 2796 y Fu(6)1986 2778 y Fx(;d)2041 2758 y Fl(0)2041 2796 y Fu(2)2073 2778 y Fx(;\033)2161 2745 y Fw(\024)22 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FA(:)3661 2681 y FB(\(297\))71 2939 y(Then,)28 b(w)n(e)f(can)g(de\014ne)h FA(g)i FB(as)1259 3038 y FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)e FA(")p 1641 2972 66 4 v(G)1706 3050 y Fy(1)1744 3038 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(")p Fw(G)2124 3050 y Fx(")2173 2971 y Fz(\000)2211 3038 y FA(@)2255 3050 y Fx(v)p 2295 2972 V 2295 3038 a FA(G)2360 3050 y Fy(1)2397 2971 y Fz(\001)2449 3038 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)1000 b FB(\(298\))71 3178 y(where)27 b Fw(G)360 3190 y Fx(")424 3178 y FB(is)g(the)h(op)r(erator)e(de\014ned)i(in)g(\(270\))e(adapted)i (to)f(the)h(domain)f FA(R)2505 3192 y Fx(\024)2544 3173 y Fl(0)2544 3211 y Fu(6)2577 3192 y Fx(;d)2632 3173 y Fl(0)2632 3211 y Fu(2)2686 3178 y Fw(\002)18 b Ft(T)2824 3190 y Fx(\033)2869 3178 y FB(.)195 3278 y(Finally)-7 b(,)30 b(applying)f(Lemma)g(9.10)e(and)i(9.2,)g(one)g(obtains)g(the)g (b)r(ounds)h(for)e FA(g)k FB(and)d FA(@)2922 3290 y Fx(v)2962 3278 y FA(g)s FB(.)41 b(The)29 b(other)g(statemen)n(ts)71 3377 y(are)d(straigh)n(tforw)n(ard.)p 3790 3377 4 57 v 3794 3325 50 4 v 3794 3377 V 3843 3377 4 57 v 195 3540 a(W)-7 b(e)32 b(p)r(erform)g(the)g(c)n(hange)e(of)i(v)-5 b(ariables)30 b FA(u)f FB(=)h FA(v)24 b FB(+)d FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))33 b(giv)n(en)d(in)i(Lemma)g(9.13) e(to)h(equation)h(\(268\))e(and)i(w)n(e)71 3640 y(obtain)1433 3739 y Fw(L)1490 3751 y Fx(")1539 3718 y Fz(b)1526 3739 y Fw(C)c FB(=)22 b FA(M)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b(+)e FA(N)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(@)2376 3751 y Fx(v)2429 3718 y Fz(b)2416 3739 y Fw(C)c FA(;)1173 b FB(\(299\))71 3889 y(where)324 3868 y Fz(b)311 3889 y Fw(C)32 b FB(is)c(the)g(unkno)n(wn)1512 3968 y Fz(b)1499 3989 y Fw(C)5 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f Fw(C)5 b FB(\()p FA(v)21 b FB(+)d FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))1241 b(\(300\))71 4129 y(and)1212 4295 y FA(M)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f Fw(\000)p FA(")1707 4260 y Fv(\000)p Fy(1)1795 4295 y FA(G)14 b FB(\()q FA(v)22 b FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))1263 b(\(301\))1226 4476 y FA(N)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f Fw(\000)1678 4420 y FA(G)14 b FB(\()p FA(v)22 b FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(G)2448 4432 y Fy(1)2485 4420 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))p 1678 4457 999 4 v 1947 4533 a(1)18 b(+)g FA(@)2134 4545 y Fx(v)2173 4533 y FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))2686 4476 y FA(:)952 b FB(\(302\))71 4710 y(Moreo)n(v)n(er,)29 b(w)n(e)h(w)n(an)n(t)g(to)g(ha)n(v)n(e)g(the)h(\014rst)f(order)f(terms) h(in)1980 4689 y Fz(b)1967 4710 y Fw(C)5 b FB(,)31 b(coming)f(from)g FA(G)2623 4722 y Fy(1)2661 4710 y FB(,)h FA(G)2780 4722 y Fy(2)2849 4710 y FB(and)f FA(G)3078 4722 y Fy(3)3116 4710 y FB(,)h(explicitly)-7 b(.)46 b(F)-7 b(or)30 b(this)71 4809 y(purp)r(ose,)d(w)n(e)g(de\014ne)1193 4960 y Fz(b)1180 4981 y Fw(C)1224 4993 y Fy(0)1261 4981 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)18 b Fw(\000)p 1641 4915 66 4 v 18 w FA(G)1706 4993 y Fy(1)1743 4981 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))20 b Fw(\000)e FA(")2074 4947 y Fv(\000)p Fy(1)2163 4981 y Fw(G)2212 4993 y Fx(")2262 4981 y FB(\()p Fw(h)p FA(@)2370 4993 y Fx(v)2410 4981 y FA(G)2475 4993 y Fy(1)2512 4981 y FA(g)s Fw(i)p FB(\))c(\()p FA(v)s FB(\))1558 5116 y Fw(\000)k FA(")1680 5082 y Fv(\000)p Fy(1)1769 5116 y Fw(G)1818 5128 y Fx(")1867 5116 y FB(\()q Fw(h)p FA(G)1997 5128 y Fy(2)2053 5116 y FB(+)g FA(G)2201 5128 y Fy(3)2238 5116 y Fw(i)p FB(\))d(\()p FA(v)s FB(\))p FA(;)3661 5045 y FB(\(303\))71 5294 y(where)p 317 5228 V 33 w FA(G)382 5306 y Fy(1)453 5294 y FB(is)33 b(the)g(function)h(de\014ned)g(in)f (\(296\),)h FA(g)i FB(is)d(the)h(function)g(giv)n(en)e(b)n(y)h(Lemma)h (9.13)d(and)j FA(G)3366 5306 y Fy(2)3436 5294 y FB(and)g FA(G)3669 5306 y Fy(3)3739 5294 y FB(are)71 5394 y(the)e(functions)h (de\014ned)f(in)g(\(291\))g(and)g(\(292\))f(resp)r(ectiv)n(ely)-7 b(.)49 b(The)32 b(next)h(lemma,)g(whose)e(pro)r(of)h(is)f(straigh)n (tforw)n(ard)71 5504 y(applying)c(Lemmas)g(9.2,)g(9.10)f(and)h(9.13,)g (giv)n(es)f(some)h(prop)r(erties)g(of)2340 5483 y Fz(b)2327 5504 y Fw(C)2371 5516 y Fy(0)2408 5504 y FB(.)p Black 1898 5753 a(104)p Black eop end %%Page: 105 105 TeXDict begin 105 104 bop Black Black Black 71 272 a Fp(Lemma)31 b(9.14.)p Black 40 w Fs(The)g(function)1156 251 y Fz(b)1144 272 y Fw(C)1188 284 y Fy(0)1254 272 y Fs(de\014ne)l(d)f(in)37 b FB(\(303\))28 b Fs(satis\014es)i(that)1111 357 y Fz(\015)1111 407 y(\015)1111 457 y(\015)1170 432 y(b)1158 453 y Fw(C)1202 465 y Fy(0)1239 357 y Fz(\015)1239 407 y(\015)1239 457 y(\015)1285 511 y Fy(ln)o Fx(;\024)1399 491 y Fl(0)1399 529 y Fu(6)1431 511 y Fx(;d)1486 491 y Fl(0)1486 529 y Fu(2)1518 511 y Fx(;\033)1605 453 y Fw(\024)23 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)p FA(;)1972 357 y Fz(\015)1972 407 y(\015)1972 457 y(\015)2018 453 y FA(@)2062 465 y Fx(v)2114 432 y Fz(b)2101 453 y Fw(C)2145 465 y Fy(0)2182 357 y Fz(\015)2182 407 y(\015)2182 457 y(\015)2228 511 y Fy(1)p Fx(;\024)2320 491 y Fl(0)2320 529 y Fu(6)2352 511 y Fx(;d)2407 491 y Fl(0)2407 529 y Fu(2)2439 511 y Fx(;\033)2527 453 y Fw(\024)22 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FA(:)195 651 y FB(Then,)28 b(w)n(e)f(de\014ne)1751 729 y Fz(b)1738 750 y Fw(C)1782 762 y Fy(1)1842 750 y FB(=)1942 729 y Fz(b)1929 750 y Fw(C)c(\000)2092 729 y Fz(b)2079 750 y Fw(C)2123 762 y Fy(0)2160 750 y FA(:)71 884 y FB(T)-7 b(aking)27 b(in)n(to)g(accoun)n (t)g(equation)g(\(299\))o(,)1414 863 y Fz(b)1402 884 y Fw(C)1446 896 y Fy(1)1510 884 y FB(is)h(a)f(solution)g(of)1669 1064 y Fw(L)1726 1076 y Fx(")1775 1043 y Fz(b)1762 1064 y Fw(C)1806 1076 y Fy(1)1866 1064 y FB(=)1974 1043 y Fz(b)1953 1064 y Fw(F)2035 972 y Fz(\020)2098 1043 y(b)2085 1064 y Fw(C)2129 1076 y Fy(1)2166 972 y Fz(\021)2229 1064 y FA(;)1409 b FB(\(304\))71 1240 y(where)1446 1319 y Fz(b)1426 1340 y Fw(F)8 b FB(\()p FA(h)p FB(\))23 b(=)1727 1319 y Fz(c)1717 1340 y FA(M)8 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b(+)e FA(N)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(@)2407 1352 y Fx(v)2448 1340 y FA(h)1165 b FB(\(305\))71 1473 y(and)1220 1551 y Fz(c)1210 1572 y FA(M)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f FA(M)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))19 b Fw(\000)f(L)2039 1584 y Fx(")2088 1551 y Fz(b)2075 1572 y Fw(C)2119 1584 y Fy(0)2174 1572 y FB(+)g FA(N)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(@)2566 1584 y Fx(v)2620 1551 y Fz(b)2607 1572 y Fw(C)2651 1584 y Fy(0)2688 1572 y FA(:)950 b FB(\(306\))71 1716 y(W)-7 b(e)28 b(obtain)485 1695 y Fz(b)472 1716 y Fw(C)516 1728 y Fy(1)581 1716 y FB(through)f(a)g(\014xed)g(p)r(oin)n(t)h(argumen)n (t.)36 b(F)-7 b(or)27 b(this)h(purp)r(ose)f(w)n(e)g(de\014ne)h(the)g (op)r(erator)1764 1851 y Fz(e)1744 1872 y Fw(F)j FB(=)23 b Fw(G)1972 1884 y Fx(")2026 1872 y Fw(\016)2106 1851 y Fz(b)2086 1872 y Fw(F)8 b FA(;)1484 b FB(\(307\))71 2038 y(where)331 2017 y Fz(b)311 2038 y Fw(F)36 b FB(and)27 b Fw(G)617 2050 y Fx(")681 2038 y FB(are)f(the)i(op)r(erators)e (de\014ned)i(\(305\))e(and)i(\(270\))o(.)37 b(F)-7 b(or)27 b(con)n(v)n(enience,)f(w)n(e)i(rewrite)e(it)i(as)860 2185 y Fz(b)840 2206 y Fw(F)8 b FB(\()p FA(h)p FB(\)\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)1331 2185 y Fz(c)1321 2206 y FA(M)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b(+)e FA(@)1747 2218 y Fx(v)1800 2206 y FB(\()p FA(N)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(h)p FB(\()p FA(v)s(;)14 b(\034)9 b FB(\)\))21 b Fw(\000)d FA(@)2514 2218 y Fx(v)2554 2206 y FA(N)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(h)p FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(:)582 b FB(\(308\))p Black 71 2372 a Fp(Lemma)35 b(9.15.)p Black 42 w Fs(L)l(et)e(us)f(c)l(onsider)i FA(")1281 2384 y Fy(0)1347 2372 y FA(>)28 b FB(0)k Fs(smal)t(l)i(enough)g(and)f FA(\024)2236 2342 y Fv(0)2236 2393 y Fy(6)2302 2372 y FA(>)28 b(\024)2443 2384 y Fy(6)2513 2372 y Fs(big)34 b(enough.)48 b(Then,)35 b(the)e(op)l(er)l(ator)3710 2351 y Fz(e)3690 2372 y Fw(F)41 b Fs(is)71 2472 y(c)l(ontr)l(active)30 b(fr)l(om)g Fw(X)747 2486 y Fy(1)p Fx(;\024)839 2466 y Fl(0)839 2504 y Fu(6)872 2486 y Fx(;d)927 2466 y Fl(0)927 2504 y Fu(2)958 2486 y Fx(;\033)1052 2472 y Fs(to)g(itself.)195 2571 y(Thus,)h(it)f(has)g(a)g(unique)g(\014xe)l(d)f(p)l(oint,)h(which)i (mor)l(e)l(over)e(satis\014es)g(that)1615 2652 y Fz(\015)1615 2702 y(\015)1615 2751 y(\015)1674 2726 y(b)1661 2747 y Fw(C)1705 2759 y Fy(1)1742 2652 y Fz(\015)1742 2702 y(\015)1742 2751 y(\015)1788 2805 y Fy(1)p Fx(;\024)1880 2785 y Fl(0)1880 2823 y Fu(6)1912 2805 y Fx(;d)1967 2785 y Fl(0)1967 2823 y Fu(2)1999 2805 y Fx(;\033)2087 2747 y Fw(\024)22 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FA(")1532 2879 y Fz(\015)1532 2929 y(\015)1532 2979 y(\015)1578 2975 y FA(@)1622 2987 y Fx(v)1674 2954 y Fz(b)1661 2975 y Fw(C)1705 2987 y Fy(1)1742 2879 y Fz(\015)1742 2929 y(\015)1742 2979 y(\015)1788 3033 y Fy(1)p Fx(;\024)1880 3013 y Fl(0)1880 3051 y Fu(6)1912 3033 y Fx(;d)1967 3013 y Fl(0)1967 3051 y Fu(2)1999 3033 y Fx(;\033)2087 2975 y Fw(\024)22 b FA(K)2261 2919 y Fw(j)7 b FB(^)-49 b FA(\026)p Fw(j)p 2261 2956 97 4 v 2267 3032 a FA(\024)2315 3003 y Fv(0)2315 3054 y Fy(6)2367 2975 y FA(:)195 3176 y FB(Before)24 b(pro)n(ving)g(this)h(lemma,)g(w)n(e)g(state)g(the)g(follo)n(wing)f (lemma,)h(whose)g(pro)r(of)f(is)h(p)r(ostp)r(oned)g(to)g(the)g(end)g (of)g(this)71 3276 y(section.)p Black 71 3420 a Fp(Lemma)41 b(9.16.)p Black 44 w Fs(The)d(functions)1217 3399 y Fz(c)1207 3420 y FA(M)45 b Fs(and)38 b FA(N)46 b Fs(de\014ne)l(d)37 b(in)44 b FB(\(306\))36 b Fs(and)46 b FB(\(302\))37 b Fs(r)l(esp)l(e)l(ctively,)j(satisfy)f(the)e(fol)t(lowing)71 3520 y(pr)l(op)l(erties.)p Black 195 3664 a Fw(\017)p Black 41 w(G)327 3676 y Fx(")363 3664 y FB(\()405 3643 y Fz(c)395 3664 y FA(M)9 b FB(\))24 b Fw(2)f(X)678 3678 y Fy(1)p Fx(;\024)770 3659 y Fl(0)770 3697 y Fu(6)802 3678 y Fx(;d)857 3659 y Fl(0)857 3697 y Fu(2)889 3678 y Fx(;\033)983 3664 y Fs(and)30 b(satis\014es)1589 3761 y Fz(\015)1589 3811 y(\015)1589 3861 y(\015)1635 3856 y Fw(G)1684 3868 y Fx(")1720 3856 y FB(\()1762 3835 y Fz(c)1752 3856 y FA(M)9 b FB(\))1874 3761 y Fz(\015)1874 3811 y(\015)1874 3861 y(\015)1920 3914 y Fy(1)p Fx(;\024)2012 3894 y Fl(0)2012 3933 y Fu(6)2044 3914 y Fx(;d)2099 3894 y Fl(0)2099 3933 y Fu(2)2131 3914 y Fx(;\033)2218 3856 y Fw(\024)23 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)p FA(":)p Black 195 4082 a Fw(\017)p Black 41 w FA(N)t(;)14 b(@)430 4094 y Fx(v)470 4082 y FA(N)32 b Fw(2)23 b(X)706 4096 y Fy(1)p Fx(;\024)798 4077 y Fl(0)798 4115 y Fu(6)831 4096 y Fx(;d)886 4077 y Fl(0)886 4115 y Fu(2)917 4096 y Fx(;\033)1011 4082 y Fs(and)31 b(satisfy)1726 4249 y Fw(k)p FA(N)9 b Fw(k)1885 4274 y Fy(1)p Fx(;\024)1977 4254 y Fl(0)1977 4292 y Fu(6)2009 4274 y Fx(;d)2064 4254 y Fl(0)2064 4292 y Fu(2)2096 4274 y Fx(;\033)2183 4249 y Fw(\024)23 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)p FA(")1643 4443 y Fw(k)o FA(@)1728 4455 y Fx(v)1768 4443 y FA(N)9 b Fw(k)1885 4468 y Fy(1)p Fx(;\024)1977 4448 y Fl(0)1977 4486 y Fu(6)2009 4468 y Fx(;d)2064 4448 y Fl(0)2064 4486 y Fu(2)2096 4468 y Fx(;\033)2183 4443 y Fw(\024)23 b FA(K)2357 4387 y Fw(j)7 b FB(^)-49 b FA(\026)p Fw(j)p 2357 4424 V 2363 4500 a FA(\024)2411 4472 y Fv(0)2411 4522 y Fy(6)2463 4443 y FA(:)p Black 71 4695 a Fs(Pr)l(o)l(of)31 b(of)f(L)l(emma)g(9.15.)p Black 44 w FB(The)38 b(op)r(erator)1452 4675 y Fz(e)1432 4695 y Fw(F)46 b FB(sends)38 b Fw(X)1830 4709 y Fy(1)p Fx(;\024)1922 4690 y Fl(0)1922 4728 y Fu(6)1954 4709 y Fx(;d)2009 4690 y Fl(0)2009 4728 y Fu(2)2041 4709 y Fx(;\033)2144 4695 y FB(to)g(itself.)69 b(T)-7 b(o)38 b(see)g(that)h(it)f(is)h(con)n(tractiv)n(e,)g(let)g(us)71 4817 y(consider)28 b FA(h)444 4829 y Fy(1)481 4817 y FA(;)14 b(h)566 4829 y Fy(2)628 4817 y Fw(2)25 b(X)767 4831 y Fy(1)p Fx(;\024)859 4812 y Fl(0)859 4850 y Fu(6)891 4831 y Fx(;d)946 4812 y Fl(0)946 4850 y Fu(2)978 4831 y Fx(;\033)1042 4817 y FB(.)41 b(Then,)29 b(recalling)f(the)h (de\014nitions)g(of)2344 4796 y Fz(e)2324 4817 y Fw(F)36 b FB(and)2603 4796 y Fz(b)2583 4817 y Fw(F)h FB(in)29 b(\(307\))f(and)g(\(308\))g(and)h(applying)71 4917 y(Lemmas)e(9.2)g (and)g(9.16,)f(one)i(can)f(see)g(that)283 4997 y Fz(\015)283 5047 y(\015)283 5097 y(\015)349 5072 y(e)329 5093 y Fw(F)8 b FB(\()p FA(h)477 5105 y Fy(2)515 5093 y FB(\))18 b Fw(\000)668 5072 y Fz(e)648 5093 y Fw(F)8 b FB(\()p FA(h)796 5105 y Fy(1)834 5093 y FB(\))866 4997 y Fz(\015)866 5047 y(\015)866 5097 y(\015)912 5151 y Fy(1)p Fx(;\024)1004 5131 y Fl(0)1004 5169 y Fu(6)1036 5151 y Fx(;d)1091 5131 y Fl(0)1091 5169 y Fu(2)1123 5151 y Fx(;\033)1210 5093 y Fw(\024)23 b(k)o(G)1388 5105 y Fx(")1424 5093 y FA(@)1468 5105 y Fx(v)1522 5093 y FB(\()p FA(N)k Fw(\001)19 b FB(\()p FA(h)1770 5105 y Fy(2)1825 5093 y Fw(\000)f FA(h)1956 5105 y Fy(1)1994 5093 y FB(\)\))p Fw(k)2100 5118 y Fy(1)p Fx(;\024)2192 5098 y Fl(0)2192 5136 y Fu(6)2224 5118 y Fx(;d)2279 5098 y Fl(0)2279 5136 y Fu(2)2310 5118 y Fx(;\033)2393 5093 y FB(+)g Fw(kG)2567 5105 y Fx(")2617 5093 y FB(\()p FA(@)2693 5105 y Fx(v)2732 5093 y FA(N)28 b Fw(\001)18 b FB(\()p FA(h)2948 5105 y Fy(2)3004 5093 y Fw(\000)g FA(h)3135 5105 y Fy(1)3172 5093 y FB(\)\))q Fw(k)3278 5118 y Fy(1)p Fx(;\024)3370 5098 y Fl(0)3370 5136 y Fu(6)3402 5118 y Fx(;d)3457 5098 y Fl(0)3457 5136 y Fu(2)3489 5118 y Fx(;\033)1210 5272 y Fw(\024)23 b(k)o FA(N)9 b Fw(k)1457 5297 y Fy(0)p Fx(;\024)1549 5277 y Fl(0)1549 5315 y Fu(6)1581 5297 y Fx(;d)1636 5277 y Fl(0)1636 5315 y Fu(2)1667 5297 y Fx(;\033)1746 5272 y Fw(k)o FA(h)1835 5284 y Fy(2)1891 5272 y Fw(\000)18 b FA(h)2022 5284 y Fy(1)2059 5272 y Fw(k)2100 5297 y Fy(1)p Fx(;\024)2192 5277 y Fl(0)2192 5315 y Fu(6)2224 5297 y Fx(;d)2279 5277 y Fl(0)2279 5315 y Fu(2)2311 5297 y Fx(;\033)2394 5272 y FB(+)g Fw(k)o FA(@)2562 5284 y Fx(v)2602 5272 y FA(N)9 b Fw(k)2719 5297 y Fy(1)p Fx(;\024)2811 5277 y Fl(0)2811 5315 y Fu(6)2843 5297 y Fx(;d)2898 5277 y Fl(0)2898 5315 y Fu(2)2930 5297 y Fx(;\033)3008 5272 y Fw(k)p FA(h)3098 5284 y Fy(2)3153 5272 y Fw(\000)18 b FA(h)3284 5284 y Fy(1)3321 5272 y Fw(k)3363 5297 y Fy(1)p Fx(;\024)3455 5277 y Fl(0)3455 5315 y Fu(6)3487 5297 y Fx(;d)3542 5277 y Fl(0)3542 5315 y Fu(2)3574 5297 y Fx(;\033)1210 5466 y Fw(\024)1308 5410 y FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)p 1308 5447 173 4 v 1352 5523 a FA(\024)1400 5495 y Fv(0)1400 5546 y Fy(6)1504 5466 y Fw(k)p FA(h)1594 5478 y Fy(2)1649 5466 y Fw(\000)18 b FA(h)1780 5478 y Fy(1)1817 5466 y Fw(k)1859 5491 y Fy(1)p Fx(;\024)1951 5471 y Fl(0)1951 5509 y Fu(6)1983 5491 y Fx(;d)2038 5471 y Fl(0)2038 5509 y Fu(2)2070 5491 y Fx(;\033)2148 5466 y FA(:)p Black 1898 5753 a FB(105)p Black eop end %%Page: 106 106 TeXDict begin 106 105 bop Black Black 71 272 a FB(and)30 b(therefore,)h(increasing)e FA(\024)1050 242 y Fv(0)1050 293 y Fy(6)1117 272 y FB(if)i(necessary)-7 b(,)1604 251 y Fz(e)1584 272 y Fw(F)38 b FB(is)31 b(con)n(tractiv)n(e)d(in)j Fw(X)2357 286 y Fy(1)p Fx(;\024)2449 267 y Fl(0)2449 305 y Fu(6)2481 286 y Fx(;d)2536 267 y Fl(0)2536 305 y Fu(2)2568 286 y Fx(;\033)2663 272 y FB(and)f(has)g(a)g(unique)h (\014xed)g(p)r(oin)n(t)3759 251 y Fz(b)3746 272 y Fw(C)3790 284 y Fy(1)3827 272 y FB(.)71 394 y(T)-7 b(o)27 b(obtain)g(b)r(ounds)h (for)880 373 y Fz(b)867 394 y Fw(C)911 406 y Fy(1)976 394 y FB(it)g(is)f(enough)g(to)h(recall)e(that)1364 463 y Fz(\015)1364 513 y(\015)1364 563 y(\015)1423 538 y(b)1410 559 y Fw(C)1454 571 y Fy(1)1491 463 y Fz(\015)1491 513 y(\015)1491 563 y(\015)1537 617 y Fy(1)p Fx(;\024)1629 597 y Fl(0)1629 635 y Fu(6)1662 617 y Fx(;d)1717 597 y Fl(0)1717 635 y Fu(2)1748 617 y Fx(;\033)1836 559 y Fw(\024)c FB(2)1979 463 y Fz(\015)1979 513 y(\015)1979 563 y(\015)2045 538 y(e)2025 559 y Fw(F)8 b FB(\(0\))2199 463 y Fz(\015)2199 513 y(\015)2199 563 y(\015)2245 617 y Fy(1)p Fx(;\024)2337 597 y Fl(0)2337 635 y Fu(6)2369 617 y Fx(;d)2424 597 y Fl(0)2424 635 y Fu(2)2456 617 y Fx(;\033)2534 559 y FA(:)71 771 y FB(By)27 b(the)h(de\014nition)g(of) 828 750 y Fz(e)808 771 y Fw(F)35 b FB(in)28 b(\(307\))o(,)1260 750 y Fz(e)1240 771 y Fw(F)8 b FB(\(0\))23 b(=)g Fw(G)1574 783 y Fx(")1610 771 y FB(\()1652 750 y Fz(c)1642 771 y FA(M)8 b FB(\).)38 b(Then,)28 b(it)g(is)f(enough)g(to)g(apply)h (Lemma)f(9.16)f(to)i(obtain)1564 845 y Fz(\015)1564 895 y(\015)1564 945 y(\015)1623 920 y(b)1610 941 y Fw(C)1654 953 y Fy(1)1691 845 y Fz(\015)1691 895 y(\015)1691 945 y(\015)1737 999 y Fy(1)p Fx(;\024)1829 979 y Fl(0)1829 1017 y Fu(6)1861 999 y Fx(;d)1916 979 y Fl(0)1916 1017 y Fu(2)1948 999 y Fx(;\033)2035 941 y Fw(\024)23 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FA(":)71 1151 y FB(F)-7 b(or)27 b(the)h(b)r(ound)g(of)f FA(@)757 1163 y Fx(v)810 1130 y Fz(b)797 1151 y Fw(C)841 1163 y Fy(1)906 1151 y FB(it)h(is)f(enough)g(to)g(apply)h(the)g(fourth)f(statemen)n(t)h (of)g(Lemma)f(9.1)g(and)g(rename)g FA(\024)3493 1121 y Fv(0)3493 1172 y Fy(6)3530 1151 y FB(.)p 3790 1151 4 57 v 3794 1098 50 4 v 3794 1151 V 3843 1151 4 57 v Black 71 1309 a Fs(Pr)l(o)l(of)k(of)f(Pr)l(op)l(osition)i(9.9)f(for)f FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b FB(=)c(0)p Fs(.)p Black 42 w FB(By)27 b(Lemmas)h(9.14)e(and)i(9.15,)e(w)n(e)i(ha)n(v)n(e) f(that)h(there)f(exists)h(a)f(constan)n(t)71 1409 y FA(b)107 1421 y Fy(15)200 1409 y FA(>)c FB(0)k(suc)n(h)g(that)1693 1475 y Fz(\015)1693 1524 y(\015)1693 1574 y(\015)1752 1549 y(b)1739 1570 y Fw(C)1788 1475 y Fz(\015)1788 1524 y(\015)1788 1574 y(\015)1834 1628 y Fy(ln)o Fx(;\033)1977 1570 y Fw(\024)22 b FA(b)2100 1582 y Fy(15)2170 1570 y Fw(j)7 b FB(^)-49 b FA(\026)p Fw(j)1632 1673 y Fz(\015)1632 1723 y(\015)1632 1772 y(\015)1678 1768 y FA(@)1722 1780 y Fx(v)1774 1747 y Fz(b)1761 1768 y Fw(C)1810 1673 y Fz(\015)1810 1723 y(\015)1810 1772 y(\015)1856 1826 y Fy(1)p Fx(;\033)1977 1768 y Fw(\024)22 b FA(b)2100 1780 y Fy(15)2170 1768 y Fw(j)7 b FB(^)-49 b FA(\026)p Fw(j)p FA(:)71 1953 y FB(T)-7 b(o)31 b(reco)n(v)n(er)e Fw(C)37 b FB(it)32 b(is)f(enough)h(to)f(consider)g(the)h(c)n(hange)e(of)i(v)-5 b(ariables)31 b FA(v)i FB(=)c FA(u)21 b FB(+)g FA(h)p FB(\()p FA(u;)14 b(\034)9 b FB(\))32 b(obtained)f(in)h(Lemma)g(9.13,)71 2053 y(whic)n(h)e(is)g(de\014ned)h(for)e(\()p FA(u;)14 b(\034)9 b FB(\))29 b Fw(2)e FA(R)1183 2065 y Fx(\024)1222 2073 y Fu(8)1255 2065 y Fx(;d)1310 2073 y Fu(3)1366 2053 y Fw(\002)19 b Ft(T)1505 2065 y Fx(\033)1581 2053 y FB(with)30 b FA(\024)1820 2065 y Fy(8)1885 2053 y FA(>)d(\024)2025 2023 y Fv(0)2025 2073 y Fy(6)2092 2053 y FB(and)j FA(d)2299 2065 y Fy(3)2364 2053 y FA(<)d(d)2499 2023 y Fv(0)2499 2073 y Fy(2)2537 2053 y FB(.)45 b(Applying)30 b(this)g(c)n(hange,)g (one)g(obtains)71 2152 y Fw(C)37 b FB(whic)n(h)32 b(satis\014es)g(the)g (b)r(ounds)h(stated)f(in)h(Prop)r(osition)e(9.9.)50 b(T)-7 b(o)32 b(c)n(hec)n(k)f(that)i(\()p FA(\030)2742 2164 y Fy(0)2780 2152 y FB(\()p FA(u;)14 b(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))34 b(is)e(injectiv)n(e,)i(one)e(can)71 2252 y(pro)r(ceed)e(as)f(in)i(the)f(pro)r(of)g(of)g(Prop)r(osition)f (9.9)g(for)h FA(`)20 b Fw(\000)g FB(2)p FA(r)30 b(>)d FB(0.)44 b(Finally)30 b(let)h(us)f(p)r(oin)n(t)h(out)f(that)g(it)h(is)f (easy)g(to)g(see)71 2352 y(that)e(this)g(prop)r(osition)e(is)i(also)e (satis\014ed)h(taking)g(an)n(y)g FA(\024)c Fw(\025)g FA(\024)2038 2364 y Fy(8)2102 2352 y FB(suc)n(h)28 b(that)g FA("\024)22 b(<)h(a)p FB(.)p 3790 2352 V 3794 2299 50 4 v 3794 2352 V 3843 2352 4 57 v 195 2510 a(It)28 b(only)g(remains)e (to)i(pro)n(v)n(e)e(Lemma)h(9.16.)p Black 71 2668 a Fs(Pr)l(o)l(of)k (of)f(L)l(emma)g(9.16.)p Black 44 w FB(W)-7 b(e)35 b(start)f(pro)n (ving)g(the)h(second)f(statemen)n(t.)58 b(Let)35 b(us)g(split)g(the)g (function)g FA(N)44 b FB(de\014ned)35 b(in)71 2768 y(\(302\))27 b(as)g FA(N)k FB(=)23 b FA(N)643 2780 y Fy(1)698 2768 y FB(+)18 b FA(N)848 2780 y Fy(2)913 2768 y FB(with)218 2921 y FA(N)285 2933 y Fy(1)322 2921 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))24 b(=)f Fw(\000)14 b FB(\()o(1)k(+)g FA(@)920 2933 y Fx(v)960 2921 y FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\)\))1225 2880 y Fv(\000)p Fy(1)1328 2921 y FB(\()p FA(G)1425 2933 y Fy(1)1463 2921 y FB(\()p FA(v)22 b FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(G)2154 2933 y Fy(1)2192 2921 y FB(\()p FA(v)s(;)c(\034)9 b FB(\)\))1248 b(\(309\))218 3064 y FA(N)285 3076 y Fy(2)322 3064 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f Fw(\000)14 b FB(\()o(1)k(+)g FA(@)920 3076 y Fx(v)960 3064 y FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\)\))1225 3022 y Fv(\000)p Fy(1)1328 3064 y FB(\()p FA(G)1425 3076 y Fy(2)1463 3064 y FB(\()p FA(v)22 b FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))20 b(+)e FA(G)2154 3076 y Fy(3)2192 3064 y FB(\()p FA(v)k FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))20 b(+)e FA(G)2883 3076 y Fy(4)2921 3064 y FB(\()p FA(v)j FB(+)d FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\)\))16 b FA(:)147 b FB(\(310\))71 3208 y(T)-7 b(o)27 b(b)r(ound)h FA(N)516 3220 y Fy(1)553 3208 y FB(,)g(w)n(e)f(apply)g(Lemmas)g(9.13)g(and)g (9.10)f(and)i(mean)f(v)-5 b(alue)28 b(theorem,)f(obtaining)1485 3353 y Fw(k)p FA(N)1594 3365 y Fy(1)1630 3353 y Fw(k)1672 3373 y Fy(1)p Fv(\000)1770 3351 y Fu(1)p 1767 3360 35 3 v 1767 3393 a Fm(\014)1812 3373 y Fx(;\024)1871 3353 y Fl(0)1871 3391 y Fu(6)1903 3373 y Fx(;d)1958 3353 y Fl(0)1958 3391 y Fu(2)1990 3373 y Fx(;\033)2077 3353 y Fw(\024)c FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)2337 3318 y Fy(2)2374 3353 y FA(":)71 3512 y FB(Applying)28 b(the)g(same)f(lemmas,)g(one)g(can)g(see)h(that)1557 3656 y Fw(k)p FA(N)1666 3668 y Fy(2)1702 3656 y Fw(k)1744 3670 y Fy(1)p Fx(;\024)1836 3651 y Fl(0)1836 3689 y Fu(6)1868 3670 y Fx(;d)1923 3651 y Fl(0)1923 3689 y Fu(2)1955 3670 y Fx(;\033)2042 3656 y Fw(\024)23 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)p FA(":)71 3801 y FB(whic)n(h)30 b(giv)n(es)e(the)j(b) r(ound)f(for)f FA(N)9 b FB(.)44 b(T)-7 b(o)30 b(obtain)f(the)i(b)r (ound)f(for)f FA(@)2154 3813 y Fx(v)2194 3801 y FA(N)38 b FB(it)31 b(is)f(enough)f(to)h(apply)f(the)i(fourth)e(statemen)n(t)71 3901 y(of)e(Lemma)h(9.1)f(and)g(rename)g FA(\024)1096 3871 y Fv(0)1096 3921 y Fy(6)1133 3901 y FB(.)195 4013 y(F)-7 b(or)20 b(the)i(\014rst)e(statemen)n(t,)i(taking)e(in)n(to)h (accoun)n(t)e(the)i(de\014nitions)g(of)2380 3992 y Fz(c)2370 4013 y FA(M)29 b FB(and)21 b FA(M)29 b FB(in)21 b(\(301\))f(and)g (\(306\))g(resp)r(ectiv)n(ely)-7 b(,)71 4112 y(and)32 b(using)g(the)h(functions)f FA(G)1034 4124 y Fx(i)1062 4112 y FB(,)i FA(i)c FB(=)h(1)p FA(;)14 b FB(2)p FA(;)g FB(3)p FA(;)g FB(4)30 b(and)p 1749 4046 66 4 v 32 w FA(G)1814 4124 y Fy(1)1884 4112 y FB(de\014ned)j(in)f(\(290\))o(,)i(\(291\))o(,)g (\(292\))o(,)g(\(293\))d(and)h(\(296\))o(,)i(let)f(us)71 4225 y(split)266 4204 y Fz(c)256 4225 y FA(M)j FB(as)1559 4378 y Fz(c)1549 4399 y FA(M)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)1983 4295 y Fy(6)1940 4320 y Fz(X)1946 4497 y Fx(i)p Fy(=1)2084 4378 y Fz(c)2074 4399 y FA(M)2155 4411 y Fx(i)2182 4399 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))71 4598 y(with)554 4721 y Fz(c)543 4742 y FA(M)624 4754 y Fy(1)661 4742 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)d FA(@)1006 4754 y Fx(v)p 1046 4676 V 1046 4742 a FA(G)1111 4754 y Fy(1)1148 4742 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(")1479 4708 y Fv(\000)p Fy(1)1582 4742 y FB(\()p FA(@)1658 4754 y Fx(v)1698 4742 y FA(G)1763 4754 y Fy(1)1800 4742 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))21 b Fw(\000)d(h)p FA(@)2401 4754 y Fx(v)2441 4742 y FA(G)2506 4754 y Fy(1)2543 4742 y FA(g)s Fw(i)p FB(\()p FA(v)s FB(\)\))904 b(\(311\))554 4868 y Fz(c)543 4889 y FA(M)624 4901 y Fy(2)661 4889 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)d Fw(\000)p FA(")1066 4854 y Fv(\000)p Fy(1)1168 4889 y FB(\()q FA(G)1266 4901 y Fy(1)1303 4889 y FB(\()p FA(v)g FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))21 b Fw(\000)d FA(G)1995 4901 y Fy(1)2032 4889 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))20 b Fw(\000)e FA(@)2368 4901 y Fx(v)2407 4889 y FA(G)2472 4901 y Fy(1)2510 4889 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\)\))698 b(\(312\))554 5014 y Fz(c)543 5035 y FA(M)624 5047 y Fy(3)661 5035 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)d Fw(\000)p FA(")1066 5001 y Fv(\000)p Fy(1)1168 5035 y FB(\()q FA(G)1266 5047 y Fy(2)1303 5035 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b(+)e FA(G)1660 5047 y Fy(3)1698 5035 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))19 b Fw(\000)f(h)p FA(G)2086 5047 y Fy(2)2143 5035 y FB(+)g FA(G)2291 5047 y Fy(3)2328 5035 y Fw(i)p FB(\()p FA(v)s FB(\)\))1162 b(\(313\))554 5160 y Fz(c)543 5181 y FA(M)624 5193 y Fy(4)661 5181 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)d Fw(\000)p FA(")1066 5147 y Fv(\000)p Fy(1)1168 5181 y FB(\()q FA(G)1266 5193 y Fy(2)1303 5181 y FB(\()p FA(v)g FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))21 b(+)d FA(G)1995 5193 y Fy(3)2032 5181 y FB(\()p FA(v)k FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(G)2723 5193 y Fy(2)2761 5181 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))20 b Fw(\000)e FA(G)3118 5193 y Fy(3)3155 5181 y FB(\()p FA(v)s(;)c(\034)9 b FB(\)\))285 b(\(314\))554 5307 y Fz(c)543 5328 y FA(M)624 5340 y Fy(5)661 5328 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)d Fw(\000)p FA(")1066 5294 y Fv(\000)p Fy(1)1155 5328 y FA(G)1220 5340 y Fy(4)1257 5328 y FB(\()p FA(v)g FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))1881 b(\(315\))554 5453 y Fz(c)543 5474 y FA(M)624 5486 y Fy(6)661 5474 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)d FA(N)9 b FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(@)1271 5486 y Fx(v)1325 5453 y Fz(b)1312 5474 y Fw(C)1356 5486 y Fy(0)1393 5474 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(:)2056 b FB(\(316\))p Black 1898 5753 a(106)p Black eop end %%Page: 107 107 TeXDict begin 107 106 bop Black Black 71 272 a FB(W)-7 b(e)32 b(b)r(ound)h(eac)n(h)e(term.)50 b(F)-7 b(or)31 b(the)i(\014rst)e(one,)i(b)n(y)f(Lemmas)f(9.13)g(and)h(9.10,)g(w)n(e)f (ha)n(v)n(e)g(that)3091 251 y Fz(c)3080 272 y FA(M)3161 284 y Fy(1)3229 272 y Fw(2)f(X)3373 293 y Fy(1)p Fv(\000)3472 270 y Fu(1)p 3468 279 35 3 v 3468 313 a Fm(\014)3513 293 y Fx(;\024)3572 273 y Fl(0)3572 311 y Fu(6)3604 293 y Fx(;d)3659 273 y Fl(0)3659 311 y Fu(2)3691 293 y Fx(;\033)3786 272 y Fw(\032)71 391 y(X)130 405 y Fy(1)p Fx(;\024)222 386 y Fl(0)222 424 y Fu(6)254 405 y Fx(;d)309 386 y Fl(0)309 424 y Fu(2)341 405 y Fx(;\033)405 391 y FB(.)37 b(Moreo)n(v)n(er,)25 b(taking)i(also)g(in)n(to)g(accoun)n(t)g(\(297\))o(,)1565 513 y Fz(\015)1565 563 y(\015)1565 612 y(\015)1621 587 y(c)1611 608 y FA(M)1692 620 y Fy(1)1729 513 y Fz(\015)1729 563 y(\015)1729 612 y(\015)1775 666 y Fy(1)p Fx(;\024)1867 646 y Fl(0)1867 684 y Fu(6)1899 666 y Fx(;d)1954 646 y Fl(0)1954 684 y Fu(2)1986 666 y Fx(;\033)2073 608 y Fw(\024)c FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)p FA(:)71 857 y FB(and)27 b(therefore,)g(since)g Fw(h)849 836 y Fz(c)839 857 y FA(M)920 869 y Fy(1)958 857 y Fw(i)c FB(=)g(0,)k(b)n(y)g(Lemma)g(9.2,)1447 966 y Fz(\015)1447 1016 y(\015)1447 1066 y(\015)1493 1062 y Fw(G)1542 1074 y Fx(")1591 970 y Fz(\020)1651 1041 y(c)1641 1062 y FA(M)1722 1074 y Fy(1)1759 970 y Fz(\021)1808 966 y(\015)1808 1016 y(\015)1808 1066 y(\015)1854 1120 y Fy(1)p Fx(;\024)1946 1100 y Fl(0)1946 1138 y Fu(6)1979 1120 y Fx(;d)2034 1100 y Fl(0)2034 1138 y Fu(2)2065 1120 y Fx(;\033)2153 1062 y Fw(\024)22 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FA(":)71 1310 y FB(F)-7 b(or)19 b(the)h(term)f(\(312\))o(,)i(it)f(is)g (enough)f(to)g(apply)g(Lemmas)g(9.13)f(and)i(T)-7 b(a)n(ylor's)17 b(form)n(ula)i(to)g(obtain)3113 1289 y Fz(c)3102 1310 y FA(M)3183 1322 y Fy(2)3243 1310 y Fw(2)24 b(X)3381 1330 y Fy(2)p Fv(\000)3479 1308 y Fu(1)p 3476 1317 V 3476 1350 a Fm(\014)3521 1330 y Fx(;\024)3580 1310 y Fl(0)3580 1348 y Fu(6)3612 1330 y Fx(;d)3667 1310 y Fl(0)3667 1348 y Fu(2)3698 1330 y Fx(;\033)3786 1310 y Fw(\032)71 1424 y(X)130 1438 y Fy(2)p Fx(;\024)222 1419 y Fl(0)222 1457 y Fu(6)254 1438 y Fx(;d)309 1419 y Fl(0)309 1457 y Fu(2)341 1438 y Fx(;\033)433 1424 y FB(and)1527 1465 y Fz(\015)1527 1515 y(\015)1527 1565 y(\015)1583 1540 y(c)1573 1561 y FA(M)1654 1573 y Fy(2)1691 1465 y Fz(\015)1691 1515 y(\015)1691 1565 y(\015)1737 1619 y Fy(2)p Fx(;\024)1829 1599 y Fl(0)1829 1637 y Fu(6)1861 1619 y Fx(;d)1916 1599 y Fl(0)1916 1637 y Fu(2)1948 1619 y Fx(;\033)2035 1561 y Fw(\024)f FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)2295 1526 y Fy(3)2333 1561 y FA(":)71 1751 y FB(Then,)28 b(applying)f(again) f(Lemma)h(9.2,)g(w)n(e)g(ha)n(v)n(e)g(that,)1428 1856 y Fz(\015)1428 1906 y(\015)1428 1956 y(\015)1474 1952 y Fw(G)1523 1964 y Fx(")1573 1860 y Fz(\020)1633 1931 y(c)1622 1952 y FA(M)1703 1964 y Fy(2)1740 1860 y Fz(\021)1790 1856 y(\015)1790 1906 y(\015)1790 1956 y(\015)1836 2010 y Fy(1)p Fx(;\024)1928 1990 y Fl(0)1928 2028 y Fu(6)1960 2010 y Fx(;d)2015 1990 y Fl(0)2015 2028 y Fu(2)2047 2010 y Fx(;\033)2134 1952 y Fw(\024)c FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)2394 1917 y Fy(3)2432 1952 y FA(":)71 2179 y FB(T)-7 b(o)27 b(b)r(ound)h(\(313\))o(,)g(it)g(is)f(enough)g(to) h(Lemma)f(9.10)f(to)i(see)f(that)h FA(M)2211 2191 y Fy(3)2271 2179 y Fw(2)23 b(X)2408 2193 y Fy(1)p Fx(;\024)2500 2173 y Fl(0)2500 2211 y Fu(6)2533 2193 y Fx(;d)2588 2173 y Fl(0)2588 2211 y Fu(0)2619 2193 y Fx(;\033)2711 2179 y FB(and)1576 2300 y Fz(\015)1576 2350 y(\015)1576 2400 y(\015)1633 2375 y(c)1622 2396 y FA(M)1703 2408 y Fy(3)1740 2300 y Fz(\015)1740 2350 y(\015)1740 2400 y(\015)1786 2454 y Fy(1)p Fx(;\024)1878 2434 y Fl(0)1878 2472 y Fu(6)1910 2454 y Fx(;d)1965 2434 y Fl(0)1965 2472 y Fu(2)1997 2454 y Fx(;\033)2085 2396 y Fw(\024)f FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)71 2644 y FB(whic)n(h,)27 b(using)h(that)g Fw(h)771 2623 y Fz(c)761 2644 y FA(M)842 2656 y Fy(3)879 2644 y Fw(i)23 b FB(=)g(0,)k(implies)1447 2754 y Fz(\015)1447 2804 y(\015)1447 2853 y(\015)1493 2849 y Fw(G)1542 2861 y Fx(")1591 2757 y Fz(\020)1651 2828 y(c)1641 2849 y FA(M)1722 2861 y Fy(3)1759 2757 y Fz(\021)1808 2754 y(\015)1808 2804 y(\015)1808 2853 y(\015)1854 2907 y Fy(1)p Fx(;\024)1946 2887 y Fl(0)1946 2925 y Fu(6)1979 2907 y Fx(;d)2034 2887 y Fl(0)2034 2925 y Fu(2)2065 2907 y Fx(;\033)2153 2849 y Fw(\024)22 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FA(":)71 3076 y FB(Applying)22 b(mean)g(v)-5 b(alue)23 b(theorem,)g(using)f(the)g(de\014nition)h(of)f FA(G)2060 3088 y Fy(3)2120 3076 y FB(in)h(\(292\))e(and)h(Prop)r(osition)f(7.23,) h(and)g(the)h(de\014nition)71 3189 y(of)k FA(G)230 3201 y Fy(2)296 3189 y FB(in)g(\(291\),)g(Lemmas)g(9.13)g(and)g(9.10,)f(one) i(can)f(see)g(that)2125 3168 y Fz(c)2115 3189 y FA(M)2196 3201 y Fy(4)2260 3189 y FB(in)h(\(314\))f(satis\014es)1538 3298 y Fz(\015)1538 3348 y(\015)1538 3398 y(\015)1595 3373 y(c)1585 3394 y FA(M)1666 3406 y Fy(4)1702 3298 y Fz(\015)1702 3348 y(\015)1702 3398 y(\015)1748 3452 y Fy(2)p Fx(;\024)1840 3432 y Fl(0)1840 3470 y Fu(6)1873 3452 y Fx(;d)1928 3432 y Fl(0)1928 3470 y Fu(2)1959 3452 y Fx(;\033)2047 3394 y Fw(\024)22 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)2307 3360 y Fy(2)2344 3394 y FA(")71 3617 y FB(and)27 b(then,)1428 3641 y Fz(\015)1428 3691 y(\015)1428 3741 y(\015)1474 3737 y Fw(G)1523 3749 y Fx(")1573 3644 y Fz(\020)1633 3716 y(c)1622 3737 y FA(M)1703 3749 y Fy(4)1740 3644 y Fz(\021)1790 3641 y(\015)1790 3691 y(\015)1790 3741 y(\015)1836 3795 y Fy(1)p Fx(;\024)1928 3775 y Fl(0)1928 3813 y Fu(6)1960 3795 y Fx(;d)2015 3775 y Fl(0)2015 3813 y Fu(2)2047 3795 y Fx(;\033)2134 3737 y Fw(\024)c FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)2394 3702 y Fy(2)2432 3737 y FA(":)71 3953 y FB(F)-7 b(or)230 3932 y Fz(c)220 3953 y FA(M)301 3965 y Fy(5)365 3953 y FB(in)28 b(\(315\))o(,)g(it)g(is)g(enough)e(to)i(recall)f(that,)h(b)n (y)f(Lemma)g(9.10)f(and)i(9.2,)1538 4062 y Fz(\015)1538 4112 y(\015)1538 4162 y(\015)1595 4137 y(c)1585 4158 y FA(M)1666 4170 y Fy(5)1702 4062 y Fz(\015)1702 4112 y(\015)1702 4162 y(\015)1748 4216 y Fy(2)p Fx(;\024)1840 4196 y Fl(0)1840 4234 y Fu(6)1873 4216 y Fx(;d)1928 4196 y Fl(0)1928 4234 y Fu(2)1959 4216 y Fx(;\033)2047 4158 y Fw(\024)22 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)2307 4123 y Fy(3)2344 4158 y FA(")71 4380 y FB(and)1428 4384 y Fz(\015)1428 4434 y(\015)1428 4484 y(\015)1474 4480 y Fw(G)1523 4492 y Fx(")1573 4388 y Fz(\020)1633 4459 y(c)1622 4480 y FA(M)1703 4492 y Fy(5)1740 4388 y Fz(\021)1790 4384 y(\015)1790 4434 y(\015)1790 4484 y(\015)1836 4538 y Fy(1)p Fx(;\024)1928 4518 y Fl(0)1928 4556 y Fu(0)1960 4538 y Fx(;d)2015 4518 y Fl(0)2015 4556 y Fu(2)2047 4538 y Fx(;\033)2134 4480 y Fw(\024)23 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)2394 4446 y Fy(3)2432 4480 y FA(":)71 4675 y FB(Finally)-7 b(,)29 b(for)f(the)h(last)g(term)g(\(316\))o(,)g (one)g(has)f(to)g(apply)h(the)g(b)r(ound)g(of)g FA(N)38 b FB(already)27 b(obtained)i(and)f(Lemma)h(9.14,)f(to)71 4775 y(see)f(that)982 4779 y Fz(\015)982 4829 y(\015)982 4878 y(\015)1038 4853 y(c)1028 4874 y FA(M)1109 4886 y Fy(6)1146 4779 y Fz(\015)1146 4829 y(\015)1146 4878 y(\015)1192 4932 y Fy(2)p Fx(;\024)1284 4912 y Fl(0)1284 4950 y Fu(6)1316 4932 y Fx(;d)1371 4912 y Fl(0)1371 4950 y Fu(2)1402 4932 y Fx(;\033)1490 4874 y Fw(\024)c(k)o FA(N)9 b Fw(k)1736 4899 y Fy(1)p Fx(;\024)1828 4879 y Fl(0)1828 4917 y Fu(6)1860 4899 y Fx(;d)1915 4879 y Fl(0)1915 4917 y Fu(0)1947 4899 y Fx(;\033)2025 4779 y Fz(\015)2025 4829 y(\015)2025 4878 y(\015)2072 4874 y FA(@)2116 4886 y Fx(v)2168 4853 y Fz(b)2155 4874 y Fw(C)2199 4886 y Fy(0)2236 4779 y Fz(\015)2236 4829 y(\015)2236 4878 y(\015)2282 4932 y Fy(1)p Fx(;\024)2374 4912 y Fl(0)2374 4950 y Fu(6)2406 4932 y Fx(;d)2461 4912 y Fl(0)2461 4950 y Fu(2)2493 4932 y Fx(;\033)2580 4874 y Fw(\024)23 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)2841 4840 y Fy(2)2878 4874 y FA(":)71 5065 y FB(Then,)28 b(b)n(y)f(Lemma)g(9.2,)g(w)n(e)g(ha)n(v)n(e)g(that,) 1428 5170 y Fz(\015)1428 5220 y(\015)1428 5269 y(\015)1474 5265 y Fw(G)1523 5277 y Fx(")1573 5173 y Fz(\020)1633 5244 y(c)1622 5265 y FA(M)1703 5277 y Fy(6)1740 5173 y Fz(\021)1790 5170 y(\015)1790 5220 y(\015)1790 5269 y(\015)1836 5323 y Fy(1)p Fx(;\024)1928 5303 y Fl(0)1928 5341 y Fu(6)1960 5323 y Fx(;d)2015 5303 y Fl(0)2015 5341 y Fu(2)2047 5323 y Fx(;\033)2134 5265 y Fw(\024)c FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)2394 5231 y Fy(2)2432 5265 y FA(":)71 5488 y FB(Joining)27 b(all)g(these)h(b)r(ounds,)f(w)n (e)h(pro)n(v)n(e)d(the)j(\014rst)g(statemen)n(t)f(of)h(Lemma)f(9.16)p 3790 5488 4 57 v 3794 5435 50 4 v 3794 5488 V 3843 5488 4 57 v Black 1898 5753 a(107)p Black eop end %%Page: 108 108 TeXDict begin 108 107 bop Black Black 71 272 a Fp(9.3.2)94 b(Pro)s(of)31 b(of)h(Prop)s(osition)e(4.22)71 425 y FB(T)-7 b(o)28 b(pro)n(v)n(e)f(Prop)r(osition)g(4.22,)h(it)h(is)f(enough)g(to)h (obtain)f(the)h(\014rst)g(asymptotic)f(terms)g(of)h(the)g(function)3440 404 y Fz(b)3427 425 y Fw(C)3471 437 y Fy(0)3537 425 y FB(obtained)71 525 y(in)d(Lemma)g(9.14.)35 b(F)-7 b(rom)26 b(them,)h(w)n(e)f(can)g(deduce)g(the)h(\014rst)f(order)f(terms)h(of) 2529 504 y Fz(b)2517 525 y Fw(C)h FB(=)2689 504 y Fz(b)2676 525 y Fw(C)2720 537 y Fy(0)2773 525 y FB(+)2865 504 y Fz(b)2853 525 y Fw(C)2897 537 y Fy(1)2934 525 y FB(,)f(where)3235 504 y Fz(b)3222 525 y Fw(C)3266 537 y Fy(1)3329 525 y FB(is)g(the)h(function)71 625 y(b)r(ounded)h(in)g(Lemma)f(9.15,)f(and)i (from)f(them,)h(using)h(\(300\))o(,)f(the)g(ones)e(of)i Fw(C)5 b FB(.)195 735 y(Recall)28 b(that)638 714 y Fz(b)625 735 y Fw(C)669 747 y Fy(0)734 735 y FB(has)f(b)r(een)h(de\014ned)g(in)g (\(303\))f(as)1792 714 y Fz(b)1780 735 y Fw(C)1824 747 y Fy(0)1884 735 y FB(=)22 b FA(E)2032 747 y Fy(1)2088 735 y FB(+)c FA(E)2232 747 y Fy(2)2288 735 y FB(+)g FA(E)2432 747 y Fy(3)2488 735 y FB(+)g FA(E)2632 747 y Fy(4)2698 735 y FB(with)1369 917 y FA(E)1430 929 y Fy(1)1467 917 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))25 b(=)18 b Fw(\000)p 1847 851 66 4 v 18 w FA(G)1912 929 y Fy(1)1949 917 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))1523 b(\(317\))1451 1052 y FA(E)1512 1064 y Fy(2)1550 1052 y FB(\()p FA(v)s FB(\))24 b(=)18 b Fw(\000)g FA(")1886 1018 y Fv(\000)p Fy(1)1974 1052 y Fw(G)2023 1064 y Fx(")2073 1052 y FB(\()q Fw(h)p FA(@)2182 1064 y Fx(v)2221 1052 y FA(G)2286 1064 y Fy(1)2324 1052 y FA(g)s Fw(i)p FB(\))c(\()p FA(v)s FB(\))1109 b(\(318\))1451 1187 y FA(E)1512 1199 y Fy(3)1550 1187 y FB(\()p FA(v)s FB(\))24 b(=)18 b Fw(\000)g FA(")1886 1153 y Fv(\000)p Fy(1)1974 1187 y Fw(G)2023 1199 y Fx(")2073 1187 y FB(\()q Fw(h)p FA(G)2203 1199 y Fy(2)2240 1187 y Fw(i)p FB(\))d(\()p FA(v)s FB(\))1235 b(\(319\))1451 1322 y FA(E)1512 1334 y Fy(4)1550 1322 y FB(\()p FA(v)s FB(\))24 b(=)18 b Fw(\000)g FA(")1886 1287 y Fv(\000)p Fy(1)1974 1322 y Fw(G)2023 1334 y Fx(")2073 1322 y FB(\()q Fw(h)p FA(G)2203 1334 y Fy(3)2240 1322 y Fw(i)p FB(\))d(\()p FA(v)s FB(\))p FA(;)1212 b FB(\(320\))71 1516 y(where)26 b FA(G)375 1528 y Fy(1)412 1516 y FB(,)h FA(G)527 1528 y Fy(2)564 1516 y FB(,)g FA(G)679 1528 y Fy(3)742 1516 y FB(and)p 902 1450 V 26 w FA(G)968 1528 y Fy(1)1031 1516 y FB(are)e(the)i(functions)f(de\014ned)h(in)f(\(290\),)g(\(291\))o (,)h(\(292\))e(and)h(\(296\))f(resp)r(ectiv)n(ely)h(and)g FA(g)j FB(is)71 1616 y(the)f(function)g(giv)n(en)f(b)n(y)g(Lemma)g (9.13.)195 1736 y(W)-7 b(e)32 b(analyze)f(eac)n(h)f(of)i(the)g(four)f (terms)g(that)h(form)1889 1715 y Fz(b)1876 1736 y Fw(C)1920 1748 y Fy(0)1989 1736 y FB(for)f(\()p FA(v)s(;)14 b(\034)9 b FB(\))30 b Fw(2)2424 1644 y Fz(\020)2474 1736 y FA(D)2545 1696 y Fy(in)o Fx(;)p Fy(+)p Fx(;u)2543 1764 y(\024)2582 1744 y Fl(0)2582 1782 y Fu(6)2614 1764 y Fx(;c)2664 1772 y Fu(1)2753 1736 y Fw(\\)19 b FA(D)2898 1696 y Fy(in)o Fx(;)p Fy(+)p Fx(;s)2896 1764 y(\024)2935 1744 y Fl(0)2935 1782 y Fu(6)2967 1764 y Fx(;c)3017 1772 y Fu(1)3079 1644 y Fz(\021)3150 1736 y Fw(\002)h Ft(T)3290 1748 y Fx(\033)3335 1736 y FB(.)49 b(F)-7 b(or)31 b(the)h(\014rst)71 1860 y(one)27 b(\(317\))o(,)h(it)g(is)f(enough)g(to)h(recall)e(that,)i(b)n (y)g(de\014nition,)g(the)g(function)g FA(F)2470 1872 y Fy(1)2535 1860 y FB(de\014ned)g(in)g(\(74\))f(satis\014es)g(that)1625 2043 y(^)-48 b FA(\026F)1722 2055 y Fy(1)1759 2043 y FB(\()p FA(\034)9 b FB(\))25 b(=)p 1980 1976 V 22 w FA(G)2046 2055 y Fy(1)2083 2043 y FB(\()p FA(ia;)14 b(\034)9 b FB(\))71 2226 y(and)27 b(therefore,)1102 2336 y FA(E)1163 2348 y Fy(1)1201 2336 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f Fw(\000)p 1567 2269 V FA(G)1632 2348 y Fy(1)1669 2336 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f Fw(\000)7 b FB(^)-49 b FA(\026F)2138 2348 y Fy(1)2175 2336 y FB(\()p FA(\034)9 b FB(\))20 b(+)e Fw(O)r FB(\()p FA(v)k Fw(\000)c FA(ia)p FB(\))2750 2277 y Fu(1)p 2747 2286 35 3 v 2747 2319 a Fm(\014)2796 2336 y FA(:)71 2486 y FB(Then,)28 b(using)g(\(98\))f(and)h(that)g Fw(j)p FA(v)21 b Fw(\000)d FA(ia)p Fw(j)23 b(\024)g FA(K)6 b(")1536 2455 y Fx(\015)1577 2486 y FB(,)1527 2668 y Fw(k)p FA(E)1630 2680 y Fy(1)1686 2668 y FB(+)18 b FA(\026F)1872 2680 y Fy(1)1910 2668 y Fw(k)1951 2693 y Fy(1)p Fx(;\033)2072 2668 y Fw(\024)k FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FA(":)71 2851 y FB(F)-7 b(or)30 b(the)h(second)e(term,)j(let)e (us)h(recall)e(that)i(b)n(y)g(\(298\))f(and)g(applying)g(Lemma)g(9.2,)h (w)n(e)f(ha)n(v)n(e)f(that)i(the)g(function)g FA(g)s FB(,)71 2951 y(obtained)c(in)h(Lemma)f(9.13,)g(satis\014es)1383 3063 y Fz(\015)1383 3112 y(\015)1429 3133 y FA(g)22 b Fw(\000)c FA(")p 1613 3066 66 4 v(G)1678 3145 y Fy(1)1715 3133 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))1904 3063 y Fz(\015)1904 3112 y(\015)1951 3166 y Fy(1)p Fv(\000)2050 3144 y Fu(1)p 2046 3153 35 3 v 2046 3186 a Fm(\014)2091 3166 y Fx(;\033)2179 3133 y Fw(\024)22 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FA(")2478 3099 y Fy(2)2515 3133 y FA(:)71 3343 y FB(Then,)28 b(b)n(y)f(Lemma)g(9.10,)g(one)g(can)g (see)g(that)1290 3456 y Fz(\015)1290 3505 y(\015)1336 3526 y FA(@)1380 3538 y Fx(v)1433 3459 y Fz(\000)1471 3526 y FA(g)21 b Fw(\000)d FA(")p 1654 3459 66 4 v(G)1719 3538 y Fy(1)1756 3526 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))1945 3459 y Fz(\001)1985 3456 y(\015)1985 3505 y(\015)2031 3559 y Fy(2)p Fv(\000)2129 3537 y Fu(2)p 2126 3546 35 3 v 2126 3579 a Fm(\014)2171 3559 y Fx(;\033)2258 3526 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)2519 3492 y Fy(2)2556 3526 y FA(")2595 3492 y Fy(2)71 3736 y FB(and)27 b(therefore,)g(using)g(Lemma)h(9.2,)1199 3848 y Fz(\015)1199 3898 y(\015)1245 3919 y FA(")1284 3885 y Fv(\000)p Fy(1)1373 3919 y Fw(G)1422 3931 y Fx(")1471 3852 y Fz(\000)1509 3919 y FA(@)1553 3931 y Fx(v)1607 3852 y Fz(\000)1645 3919 y FA(g)21 b Fw(\000)d FA(")p 1828 3852 66 4 v(G)1893 3931 y Fy(1)1930 3919 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))2119 3852 y Fz(\001)r(\001)2197 3848 y(\015)2197 3898 y(\015)2243 3952 y Fy(1)p Fx(;\033)2363 3919 y Fw(\024)23 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)2623 3885 y Fy(2)2661 3919 y FA(":)71 4112 y FB(No)n(w)27 b(remains)g(to)g(b)r(ound,)h(the)g (\014rst)g(order)e(of)h FA(E)1637 4124 y Fy(3)1675 4112 y FB(,)h(whic)n(h)f(is)h(giv)n(en)e(b)n(y)1525 4337 y Fw(\000)p FA(\026)1654 4224 y Fz(Z)1736 4245 y Fx(v)1699 4413 y(v)1732 4421 y Fu(0)1776 4337 y Fw(h)p FA(@)1852 4349 y Fx(v)1891 4337 y FA(G)1956 4349 y Fy(1)p 1994 4271 V 1994 4337 a FA(G)2059 4349 y Fy(1)2096 4337 y Fw(i)p FB(\()p FA(w)r FB(\))14 b FA(dw)r(;)71 4570 y FB(where)27 b(w)n(e)g(recall)g(that)h FA(v)875 4582 y Fy(0)935 4570 y Fw(2)c FA(R)1077 4584 y Fx(\024)1116 4565 y Fl(0)1116 4603 y Fu(6)1148 4584 y Fx(;d)1203 4592 y Fu(3)1257 4570 y Fw(\002)18 b Ft(T)1395 4582 y Fx(\033)1440 4570 y FB(.)195 4697 y(Since)28 b Fw(h)p FA(@)488 4709 y Fx(v)528 4697 y FA(G)593 4709 y Fy(1)p 631 4631 V 631 4697 a FA(G)696 4709 y Fy(1)733 4697 y Fw(i)23 b FB(=)g Fw(O)r FB(\()p FA(v)f Fw(\000)c FA(ia)p FB(\))1226 4660 y Fy(1)p Fv(\000)1324 4638 y Fu(1)p 1321 4647 35 3 v 1321 4680 a Fm(\014)1370 4697 y FB(,)28 b(w)n(e)f(can)g(de\014ne)h(the) g(constan)n(t)1363 4940 y FA(C)1422 4952 y Fy(2)1460 4940 y FB(\()p FA(\026)p FB(\))23 b(=)g Fw(\000)7 b FB(^)-49 b FA(\026)1813 4827 y Fz(Z)1896 4848 y Fx(ia)1860 5016 y(v)1893 5024 y Fu(0)1960 4940 y Fw(h)p FA(@)2036 4952 y Fx(v)2076 4940 y FA(G)2141 4952 y Fy(1)p 2178 4874 66 4 v 2178 4940 a FA(G)2244 4952 y Fy(1)2281 4940 y Fw(i)p FB(\()p FA(w)r FB(\))14 b FA(dw)71 5178 y FB(and)27 b(then,)h(using)h(\(98\))o(,)f(one)f(has)g(that)1474 5360 y Fw(k)o FA(E)1576 5372 y Fy(2)1632 5360 y Fw(\000)18 b FA(C)1774 5372 y Fy(2)1812 5360 y FB(\()7 b(^)-49 b FA(\026)p FB(\))p Fw(k)1968 5385 y Fy(1)p Fx(;\033)2088 5360 y Fw(\024)23 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)2349 5326 y Fy(2)2386 5360 y FA(":)p Black 1898 5753 a FB(108)p Black eop end %%Page: 109 109 TeXDict begin 109 108 bop Black Black 71 272 a FB(F)-7 b(or)27 b(the)h(third)g(term,)f(b)n(y)h(the)g(de\014nitions)f(of)h FA(G)1612 284 y Fy(2)1677 272 y FB(in)g(\(291\))f(and)g Fw(G)2201 284 y Fx(")2265 272 y FB(in)h(\(270\))o(,)f(w)n(e)h(ha)n(v)n (e)e(that)1193 491 y FA(E)1254 503 y Fy(3)1292 491 y FB(\()p FA(v)s FB(\))e(=)e Fw(\000)7 b FB(^)-49 b FA(\026)1639 378 y Fz(Z)1722 399 y Fx(v)1685 567 y(v)1718 575 y Fu(0)1761 491 y Fw(h)1813 470 y Fz(b)1793 491 y FA(H)1869 457 y Fy(2)1862 512 y(1)1907 491 y Fw(i)p FB(\()p FA(w)r FB(\))14 b FA(dw)1423 739 y FB(=)22 b Fw(\000)7 b FB(^)-49 b FA(\026)1639 626 y Fz(Z)1722 647 y Fx(ia)1685 815 y(v)1718 823 y Fu(0)1785 739 y Fw(h)1837 718 y Fz(b)1817 739 y FA(H)1893 705 y Fy(2)1886 760 y(1)1931 739 y Fw(i)p FB(\()p FA(w)r FB(\))p FA(dw)23 b FB(+)18 b Fw(O)r FB(\()p FA(v)k Fw(\000)c FA(ia)p FB(\))2659 680 y Fu(1)p 2656 689 35 3 v 2656 722 a Fm(\014)2705 739 y FA(:)71 962 y FB(Then,)28 b(pro)r(ceeding)e (as)h(for)g FA(E)1018 974 y Fy(2)1056 962 y FB(,)h(w)n(e)f(de\014ne) 1413 1190 y FA(C)1472 1202 y Fy(3)1510 1190 y FB(\()p FA(\026;)14 b(")p FB(\))23 b(=)g Fw(\000)7 b FB(^)-49 b FA(\026)1939 1077 y Fz(Z)2022 1097 y Fx(ia)1985 1266 y(v)2018 1274 y Fu(0)2086 1190 y Fw(h)2137 1169 y Fz(b)2118 1190 y FA(H)2194 1156 y Fy(2)2187 1210 y(1)2231 1190 y Fw(i)p FB(\()p FA(w)r FB(\))14 b FA(dw)71 1417 y FB(and)27 b(using)i(\(98\))o(,)f(w)n(e)f(ha)n(v)n(e)g(that)1454 1517 y Fw(k)p FA(E)1557 1529 y Fy(3)1613 1517 y Fw(\000)18 b FA(C)1755 1529 y Fy(3)1793 1517 y FB(\()p FA(\026;)c(")p FB(\))p Fw(k)2024 1542 y Fy(1)p Fx(;\033)2145 1517 y Fw(\024)22 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FA(":)71 1672 y FB(T)-7 b(o)27 b(b)r(ound)h FA(E)510 1684 y Fy(4)548 1672 y FB(,)f(using)h(Prop)r(osition)e(7.23,)g(w)n(e)h (split)h FA(G)1833 1684 y Fy(3)1898 1672 y FB(in)g(t)n(w)n(o)f(parts)g (as)g FA(G)2532 1684 y Fy(3)2592 1672 y FB(=)c FA(G)2745 1642 y Fy(1)2745 1693 y(3)2801 1672 y FB(+)18 b FA(G)2949 1642 y Fy(2)2949 1693 y(3)270 1901 y FA(G)335 1867 y Fy(1)335 1922 y(3)373 1901 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))24 b(=)674 1809 y Fz(\020)723 1901 y FB(1)18 b(+)25 b(^)-49 b FA(\026@)965 1867 y Fy(2)960 1922 y Fx(p)1022 1880 y Fz(b)1002 1901 y FA(H)1078 1867 y Fy(1)1071 1922 y(1)1129 1901 y FB(\()p FA(q)1198 1913 y Fy(0)1236 1901 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1427 1913 y Fy(0)1464 1901 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))1691 1809 y Fz(\021)1754 1901 y FA(p)1796 1913 y Fy(0)1833 1901 y FB(\()p FA(u)p FB(\))1945 1867 y Fv(\000)p Fy(2)2048 1784 y Fz(\022)2195 1842 y FB(2)p FA(r)g FB(^)-49 b FA(\026"C)2430 1811 y Fy(2)2424 1862 y(+)p 2119 1882 436 4 v 2119 1958 a FB(\()p FA(v)22 b Fw(\000)c FA(ia)p FB(\))2401 1934 y Fy(2)p Fx(r)r Fy(+1)2579 1901 y FB(\()p FA(F)2664 1913 y Fy(0)2702 1901 y FB(\()p FA(\034)9 b FB(\))19 b(+)25 b(^)-49 b FA(\026)p Fw(h)p FA(Q)3061 1913 y Fy(0)3099 1901 y FA(F)3152 1913 y Fy(1)3189 1901 y Fw(i)p FB(\))19 b(+)f FA(\030)t FB(\()p FA(u;)c(\034)9 b FB(\))3589 1784 y Fz(\023)71 2129 y FB(and)27 b FA(G)297 2099 y Fy(2)297 2149 y(3)358 2129 y FB(=)c FA(G)511 2141 y Fy(3)566 2129 y Fw(\000)c FA(G)715 2099 y Fy(1)715 2149 y(3)752 2129 y FB(.)37 b(By)27 b(Prop)r(osition)f(7.23,)h Fw(k)p FA(G)1695 2099 y Fy(2)1695 2149 y(3)1732 2129 y Fw(k)1774 2141 y Fy(2)p Fx(;\033)1894 2129 y Fw(\024)22 b FA(K)13 b FB(^)-49 b FA(\026")2147 2099 y Fy(2)2211 2129 y FB(and)28 b(therefore)1458 2230 y Fz(\015)1458 2280 y(\015)1504 2301 y FA(")1543 2267 y Fv(\000)p Fy(1)1632 2301 y Fw(G)1681 2313 y Fx(")1731 2234 y Fz(\000)1769 2301 y Fw(h)p FA(G)1866 2267 y Fy(2)1866 2322 y(3)1904 2301 y Fw(i)1936 2234 y Fz(\001)1974 2230 y(\015)1974 2280 y(\015)2020 2334 y Fy(2)p Fx(;\033)2141 2301 y Fw(\024)23 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)p FA(":)71 2507 y FB(F)-7 b(or)29 b(the)h(other)f(term,)h(using)f(the)h (de\014nitions)g(of)1693 2486 y Fz(b)1674 2507 y FA(H)1750 2477 y Fy(1)1743 2528 y(1)1787 2507 y FB(,)g FA(b)p FB(,)g FA(Q)1995 2519 y Fx(j)2059 2507 y FB(and)g FA(F)2276 2519 y Fx(j)2341 2507 y FB(in)f(\(35\),)h(\(75\),)g(\(73\))f(and)h (\(74\))o(,)g(and)g(recalling)71 2621 y(that)e(b)n(y)f(Prop)r(osition)f (7.23,)g FA(\030)i Fw(2)23 b(X)1212 2642 y Fy(1)p Fv(\000)1310 2620 y Fu(1)p 1307 2629 35 3 v 1307 2662 a Fm(\014)1352 2642 y Fx(;\033)1416 2621 y FB(,)28 b(there)g(exist)f(a)g(function)2277 2599 y Fz(b)2270 2621 y FA(\030)h Fw(2)23 b(X)2471 2642 y Fy(1)p Fv(\000)2570 2620 y Fu(1)p 2566 2629 V 2566 2662 a Fm(\014)2611 2642 y Fx(;\033)2676 2621 y FB(,)k(suc)n(h)h(that) 1458 2807 y Fz(\012)1497 2875 y FA(G)1562 2840 y Fy(1)1562 2895 y(3)1600 2807 y Fz(\013)1653 2875 y FB(\()p FA(v)s FB(\))c(=)1909 2818 y FA(b)7 b FB(^)-49 b FA(\026)1995 2788 y Fy(2)2032 2818 y FA(")p 1881 2855 218 4 v 1881 2932 a(v)22 b Fw(\000)c FA(ia)2127 2875 y FB(+)2216 2853 y Fz(b)2210 2875 y FA(\030)t FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(:)71 3082 y FB(Then,)28 b(one)f(can)g(see)g(that)h(there)g (exist)f(a)g(constan)n(t)g FA(C)1801 3094 y Fy(4)1839 3082 y FB(\()7 b(^)-49 b FA(\026;)14 b(")p FB(\))28 b(satisfying)f Fw(j)p FA(C)2509 3094 y Fy(4)2546 3082 y FB(\()7 b(^)-49 b FA(\026;)14 b(")p FB(\))p Fw(j)24 b(\024)e FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FB(,)27 b(suc)n(h)h(that,)1101 3183 y Fz(\015)1101 3233 y(\015)1147 3254 y FA(E)1208 3266 y Fy(4)1246 3254 y FB(\()p FA(v)s FB(\))19 b(+)f FA(b)7 b FB(^)-49 b FA(\026)1541 3220 y Fy(2)1592 3254 y FB(ln\()p FA(v)22 b Fw(\000)c FA(ia)p FB(\))g Fw(\000)g FA(C)2103 3266 y Fy(4)2141 3254 y FB(\()7 b(^)-49 b FA(\026;)14 b(")p FB(\))2331 3183 y Fz(\015)2331 3233 y(\015)2377 3287 y Fy(1)p Fx(;\033)2498 3254 y Fw(\024)23 b FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)o Fw(j)p FA(":)71 3437 y FB(T)-7 b(aking)27 b FA(C)i FB(=)23 b FA(C)583 3449 y Fy(2)639 3437 y FB(+)18 b FA(C)781 3449 y Fy(3)837 3437 y FB(+)g FA(C)979 3449 y Fy(4)1044 3437 y FB(one)27 b(obtains)g(that) 924 3534 y Fz(\015)924 3584 y(\015)924 3634 y(\015)983 3609 y(b)971 3630 y Fw(C)t FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b(+)25 b(^)-49 b FA(\026F)1414 3642 y Fy(1)1452 3630 y FB(\()p FA(\034)9 b FB(\))19 b Fw(\000)f FA(C)6 b FB(\()h(^)-49 b FA(\026)q(;)14 b(")p FB(\))k(+)g FA(b)7 b FB(^)-49 b FA(\026)2106 3595 y Fy(2)2157 3630 y FB(ln\()p FA(v)22 b Fw(\000)c FA(ia)p FB(\))2508 3534 y Fz(\015)2508 3584 y(\015)2508 3634 y(\015)2554 3688 y Fy(1)p Fx(;\033)2675 3630 y Fw(\024)k FA(K)6 b Fw(j)h FB(^)-49 b FA(\026)p Fw(j)p FA(":)71 3842 y FB(T)-7 b(o)30 b(\014nish)g(the)h(pro)r(of)f(of) g(Prop)r(osition)f(4.22,)g(it)i(is)f(enough)g(to)g(consider)f(the)i(c)n (hange)e(of)h(v)-5 b(ariables)29 b FA(v)i FB(=)c FA(u)20 b FB(+)g FA(h)p FB(\()p FA(u;)14 b(\034)9 b FB(\))71 3942 y(obtained)27 b(in)h(Lemma)f(9.13,)g(whic)n(h)g(do)r(es)h(not)f(c) n(hange)g(the)h(asymptotic)f(\014rst)g(order)f(of)i Fw(C)5 b FB(.)71 4172 y Fq(9.4)112 b(An)37 b(asymptotic)h(form)m(ula)h(for)e Fh(C)7 b FF(\()p Fh(\026;)17 b(")p FF(\))71 4326 y FB(When)j FA(\021)26 b FB(=)d(0,)e(the)f(constan)n(ts)f FA(C)6 b FB(\()p FA(\026;)14 b(")p FB(\))20 b(considered)f(in)h(Theorems)f (2.5)g(and)h(2.6)f(satisfy)g(that)h(lim)3124 4338 y Fx(")p Fv(!)p Fy(0)3273 4326 y FA(C)6 b FB(\()p FA(\026;)14 b(")p FB(\))24 b(=)e FA(C)3698 4338 y Fy(0)3736 4326 y FB(\()p FA(\026)p FB(\))71 4425 y(for)j(certain)g(function)h FA(C)853 4437 y Fy(0)890 4425 y FB(\()p FA(\026)p FB(\))g(analytic)f (in)h FA(\026)p FB(.)36 b(W)-7 b(e)26 b(dev)n(ote)f(this)g(section)h (to)f(pro)n(v)n(e)f(this)h(fact)h(for)f(the)h(case)e FA(`)14 b Fw(\000)g FB(2)p FA(r)25 b(<)e FB(0.)71 4525 y(The)28 b(pro)r(of)f(for)g(the)h(case)e FA(`)18 b Fw(\000)g FB(2)p FA(r)26 b FB(=)c(0)28 b(is)f(completely)g(analagous.)195 4625 y(The)e(pro)r(of)g(for)f(the)h(case)f FA(`)13 b Fw(\000)g FB(2)p FA(r)24 b(<)f FB(0)h(follo)n(ws)g(the)h(same)g(lines)g (as)f(the)h(one)f(of)h(Prop)r(osition)e(4.18)h(in)h(Section)g(9.2.2)71 4724 y(and,)i(therefore,)g(w)n(e)g(only)h(sk)n(etc)n(h)e(it.)38 b(Recall)27 b(that)h(throughout)f(this)g(section)h(w)n(e)f(assume)g FA(\021)f FB(=)d(0.)195 4824 y(W)-7 b(e)34 b(split)g(the)f(constan)n(t) g FA(C)6 b FB(\()p FA(\026;)14 b(")p FB(\))34 b(as)e FA(C)6 b FB(\()p FA(\026;)14 b(")p FB(\))33 b(=)f FA(C)1870 4794 y Fy(1)1908 4824 y FB(\()p FA(\026;)14 b(")p FB(\))22 b(+)g FA(C)2272 4794 y Fy(2)2310 4824 y FB(\()p FA(\026;)14 b(")p FB(\))22 b(+)g FA(C)2674 4794 y Fy(3)2711 4824 y FB(\()p FA(\026;)14 b(")p FB(\))34 b(and)f(w)n(e)g(obtain)g(the)g (corre-)71 4923 y(sp)r(onding)28 b(\014rst)g(orders)f(in)i FA(")p FB(,)g(whic)n(h)f(w)n(e)g(call)g FA(C)1616 4893 y Fx(i)1610 4944 y Fy(0)1648 4923 y FB(\()p FA(\026)p FB(\))h(for)f FA(i)c FB(=)h(1)p FA(;)14 b FB(2)p FA(;)g FB(3.)38 b(Then,)29 b(the)g(function)g FA(C)3093 4935 y Fy(0)3130 4923 y FB(\()p FA(\026)p FB(\))g(will)g(b)r(e)g(giv)n(en)f (b)n(y)71 5023 y FA(C)130 5035 y Fy(0)167 5023 y FB(\()p FA(\026)p FB(\))c(=)f FA(C)458 4993 y Fy(1)452 5044 y(0)495 5023 y FB(\()p FA(\026)p FB(\))c(+)f FA(C)776 4993 y Fy(2)770 5044 y(0)814 5023 y FB(\()p FA(\026)p FB(\))h(+)f FA(C)1095 4993 y Fy(3)1089 5044 y(0)1132 5023 y FB(\()p FA(\026)p FB(\).)195 5123 y(Recall)j(that)g FA(C)6 b FB(\()p FA(\026;)14 b(")p FB(\))21 b(has)f(b)r(een)h(de\014ned)g(as)f (\(289\))g(where)h FA(v)2076 5135 y Fy(0)2134 5123 y FB(is)f(the)h(left)h(endp)r(oin)n(t)f(of)g FA(R)2975 5137 y Fx(\024)3014 5117 y Fl(0)3014 5155 y Fu(3)3046 5137 y Fx(;d)3101 5145 y Fu(3)3141 5123 y Fw(\\)5 b Ft(R)21 b FB(,)h FA(v)3367 5135 y Fy(1)3428 5123 y FB(=)h FA(i)p FB(\()p FA(a)5 b Fw(\000)g FA(\024)3744 5093 y Fv(0)3744 5143 y Fy(3)3779 5123 y FA(")p FB(\))71 5222 y(is)24 b(the)g(upp)r(er)h(v)n(ertex)e(of)h(the)g(domain)g FA(R)1358 5236 y Fx(\024)1397 5217 y Fl(0)1397 5255 y Fu(3)1430 5236 y Fx(;d)1485 5244 y Fu(3)1545 5222 y FB(\(see)g(Figure)f(3\))h (and)g FA(M)33 b FB(is)24 b(the)g(function)h(de\014ned)g(in)f(\(280\))o (.)36 b(T)-7 b(o)24 b(obtain)71 5332 y(the)k(constan)n(ts)e FA(C)646 5302 y Fx(i)702 5332 y FB(w)n(e)h(split)h FA(M)36 b FB(as)27 b FA(M)32 b FB(=)23 b FA(M)1519 5302 y Fy(1)1574 5332 y FB(+)18 b FA(M)1747 5302 y Fy(2)1802 5332 y FB(+)g FA(M)1975 5302 y Fy(3)2040 5332 y FB(with)1054 5504 y FA(M)1144 5470 y Fx(i)1171 5504 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))25 b(=)d Fw(\000)p FA(")1576 5470 y Fv(\000)p Fy(1)1665 5504 y FA(G)1730 5516 y Fx(i)1772 5504 y FB(\()p FA(v)g FB(+)c FA(g)s FB(\()p FA(v)s(;)c(\034)9 b FB(\))p FA(;)14 b(\034)9 b FB(\))84 b(for)27 b FA(i)c FB(=)g(1)p FA(;)14 b FB(2)p FA(;)g FB(3)p FA(;)792 b FB(\(321\))p Black 1898 5753 a(109)p Black eop end %%Page: 110 110 TeXDict begin 110 109 bop Black Black 71 272 a FB(where)29 b FA(G)378 284 y Fx(i)406 272 y FB(,)i FA(i)c FB(=)g(1)p FA(;)14 b FB(2)p FA(;)g FB(3,)29 b(are)g(the)i(functions)f(de\014ned)h (in)f(\(271\))o(,)h(\(272\))f(and)f(\(273\))h(and)g FA(g)j FB(is)d(the)g(function)h(obtained)71 372 y(in)d(Lemma)f(9.5.)36 b(Then,)1467 514 y FA(C)1532 479 y Fx(i)1560 514 y FB(\()p FA(\026;)14 b(")p FB(\))23 b(=)1861 401 y Fz(Z)1944 421 y Fx(v)1977 429 y Fu(1)1907 589 y Fx(v)1940 597 y Fu(0)2027 446 y Fz(\012)2067 514 y FA(M)2157 479 y Fx(i)2184 446 y Fz(\013)2237 514 y FB(\()p FA(v)s FB(\))p FA(dv)s(:)195 722 y FB(T)-7 b(o)27 b(de\014ne)g FA(C)621 692 y Fy(1)615 743 y(0)659 722 y FB(,)g(w)n(e)g(expand)g FA(M)1209 692 y Fy(1)1273 722 y FB(with)g(resp)r(ect)g(to)g FA(")p FB(.)37 b(Using)26 b(the)i(form)n(ulas)e(\(276\))g(for)h FA(g)i FB(and)e(\(271\))f(for)h FA(G)3638 734 y Fy(1)3676 722 y FB(,)g(one)71 822 y(can)g(easily)g(see)g(that)h(for)f(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b Fw(2)g FA(R)1248 836 y Fx(\024)1287 816 y Fl(0)1287 854 y Fu(3)1319 836 y Fx(;d)1374 844 y Fu(3)1428 822 y Fw(\002)18 b Ft(T)1566 834 y Fx(\033)1612 822 y FB(,)616 1065 y FA(M)706 1030 y Fy(1)743 1065 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))24 b(=)f Fw(\000)p FA(")1148 1030 y Fv(\000)p Fy(1)1236 1065 y FA(G)1301 1077 y Fy(1)1339 1065 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))19 b Fw(\000)f FA(@)1674 1077 y Fx(v)1714 1065 y FA(G)1779 1077 y Fy(1)1816 1065 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))p 2005 998 66 4 v FA(G)2072 1077 y Fy(1)2109 1065 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b(+)e Fw(O)2483 947 y Fz(\022)2850 1008 y FA(\026")p 2555 1045 680 4 v 2555 1123 a FB(\()p FA(v)j Fw(\000)e FA(ia)p FB(\))2837 1099 y Fy(max)o Fv(f)p Fy(0)p Fx(;)p Fy(2)p Fv(\000)p Fx(\027)3164 1107 y Fu(1)3196 1099 y Fv(g)3244 947 y Fz(\023)71 1296 y FB(for)29 b(certain)f FA(\027)519 1308 y Fy(1)583 1296 y FA(>)d FB(0.)42 b(Recall)29 b(that)g(b)n(y)g (Lemma)g(9.4,)g(w)n(e)g(ha)n(v)n(e)f(that)i Fw(h)p FA(G)2383 1308 y Fy(1)2421 1296 y Fw(i)c FB(=)f(0)k(and)g(therefore)g(this)g (\014rst)h(term)f(do)r(es)71 1395 y(not)c(con)n(tribute)g(to)g FA(C)771 1407 y Fy(1)808 1395 y FB(\()p FA(\026;)14 b(")p FB(\).)36 b(The)26 b(second)e(term,)i(that)f(is)g Fw(\000)p FA(@)2078 1407 y Fx(v)2117 1395 y FA(G)2182 1407 y Fy(1)2219 1395 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p 2408 1329 66 4 v FA(G)2475 1407 y Fy(1)2512 1395 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\),)27 b(is)e(indep)r(enden)n(t)h(of)f FA(")p FB(.)36 b(Moreo)n(v)n(er,)71 1495 y(using)24 b(the)h(prop)r (erties)f(of)g FA(G)969 1507 y Fy(1)1032 1495 y FB(stated)g(in)h(Lemma) g(9.4,)f(one)g(can)g(see)h(that)g(it)g(can)f(b)r(e)h(analytically)e (extended)i(to)g(reac)n(h)71 1595 y FA(v)h FB(=)d FA(ia)k FB(and)g(that)h(it)g(satis\014es)1086 1805 y Fw(\000)p FA(@)1195 1817 y Fx(v)1234 1805 y FA(G)1299 1817 y Fy(1)1337 1805 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p 1526 1738 V FA(G)1592 1817 y Fy(1)1630 1805 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f Fw(O)2013 1688 y Fz(\022)2399 1748 y FA(\026)p 2084 1786 680 4 v 2084 1866 a FB(\()p FA(v)f Fw(\000)c FA(ia)p FB(\))2366 1839 y Fy(max)o Fv(f)p Fy(0)p Fx(;)p Fy(1)p Fv(\000)p Fx(\027)2697 1819 y Fl(0)2693 1857 y Fu(1)2726 1839 y Fv(g)2774 1688 y Fz(\023)71 2039 y FB(for)27 b(certain)g FA(\027)521 2009 y Fv(0)516 2059 y Fy(1)576 2039 y FA(>)c FB(0.)36 b(Therefore,)27 b(one)g(can)g (de\014ne)1242 2280 y FA(C)1307 2246 y Fy(1)1301 2301 y(0)1344 2280 y FB(\()p FA(\026)p FB(\))d(=)f Fw(\000)1649 2167 y Fz(Z)1731 2188 y Fx(ia)1694 2356 y(v)1727 2364 y Fu(0)1808 2213 y Fz(\012)1848 2280 y FA(@)1892 2292 y Fx(v)1931 2280 y FA(G)1996 2292 y Fy(1)2034 2280 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p 2223 2213 66 4 v FA(G)2289 2292 y Fy(1)2327 2280 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))2516 2213 y Fz(\013)2570 2280 y FA(dv)s(;)982 b FB(\(322\))71 2511 y(whic)n(h)27 b(is)h(a)f(constan)n(t)g(indep)r (enden)n(t)h(of)g FA(")p FB(.)37 b(Finally)27 b(it)h(can)f(b)r(e)h (easily)f(seen)g(that)1425 2622 y Fz(\014)1425 2672 y(\014)1453 2692 y FA(C)1518 2658 y Fy(1)1555 2692 y FB(\()p FA(\026;)14 b(")p FB(\))19 b Fw(\000)f FA(C)1912 2658 y Fy(1)1906 2713 y(0)1950 2692 y FB(\()p FA(\026)p FB(\))2064 2622 y Fz(\014)2064 2672 y(\014)2115 2692 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2415 2658 y Fx(\027)2452 2633 y Fl(00)2448 2675 y Fu(1)3661 2692 y FB(\(323\))71 2873 y(for)27 b(certain)g FA(\027)521 2843 y Fv(00)516 2894 y Fy(1)587 2873 y FA(>)22 b FB(0.)195 2973 y(T)-7 b(o)27 b(obtain)g FA(C)640 2943 y Fy(2)634 2994 y(0)678 2973 y FB(\()p FA(\026)p FB(\),)h(let)g(us)f(\014rst)g(p)r(oin)n(t)h (out)f(that,)h(follo)n(wing)e(the)i(pro)r(of)e(of)i(Theorem)e(4.1,)h (one)g(can)g(see)f(that)i(the)71 3073 y(parameterization)d(of)j(the)g (p)r(erio)r(dic)f(orbit)h(satis\014es)1162 3254 y(\()p FA(x)1241 3266 y Fx(p)1280 3254 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(y)1467 3266 y Fx(p)1506 3254 y FB(\()p FA(\034)9 b FB(\)\))25 b(=)1759 3186 y Fz(\000)1797 3254 y FA("x)1883 3219 y Fy(0)1883 3274 y Fx(p)1922 3254 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b("y)2151 3219 y Fy(0)2148 3274 y Fx(p)2189 3254 y FB(\()p FA(\034)9 b FB(\))2298 3186 y Fz(\001)2355 3254 y FB(+)18 b Fw(O)2521 3186 y Fz(\000)2559 3254 y FA(\026")2648 3219 y Fy(2)2685 3186 y Fz(\001)2737 3254 y FA(;)901 b FB(\(324\))71 3446 y(where)315 3378 y Fz(\000)353 3446 y FA(x)400 3416 y Fy(0)400 3466 y Fx(p)439 3446 y FB(\()p FA(\034)9 b FB(\))p FA(;)14 b(y)629 3416 y Fy(0)626 3466 y Fx(p)667 3446 y FB(\()p FA(\034)9 b FB(\))776 3378 y Fz(\001)847 3446 y FB(is)32 b(indep)r(enden)n(t)g(of)g FA(")p FB(.)49 b(Using)32 b(that)g(fact,)h(one)e(can)g(easily)g(deduce)h(that)g(the)g (functions)g FA(c)3788 3458 y Fx(k)q(l)71 3567 y FB(in)n(v)n(olv)n(ed) 26 b(in)i(the)g(de\014nition)g(of)1119 3546 y Fz(b)1099 3567 y FA(H)1175 3537 y Fy(2)1168 3587 y(1)1240 3567 y FB(in)g(\(37\))f(satisfy)1524 3748 y FA(c)1560 3760 y Fx(k)q(l)1622 3748 y FB(\()p FA(\034)9 b FB(\))24 b(=)f FA(c)1879 3713 y Fy(0)1879 3768 y Fx(k)q(l)1941 3748 y FB(\()p FA(\034)9 b FB(\))20 b(+)e Fw(O)r FB(\()p FA(\026")p FB(\))p FA(;)71 3939 y FB(for)27 b(certain)g(functions)h FA(c)869 3909 y Fy(0)869 3963 y Fx(k)q(l)931 3939 y FB(\()p FA(\034)9 b FB(\))29 b(indep)r(enden)n(t)f(of)g FA(")p FB(.)37 b(Therefore,)2151 3918 y Fz(b)2132 3939 y FA(H)2208 3909 y Fy(2)2201 3960 y(1)2273 3939 y FB(satis\014es)1204 4112 y Fz(b)1184 4133 y FA(H)1260 4099 y Fy(2)1253 4154 y(1)1297 4133 y FB(\()p FA(q)s(;)14 b(p;)g(\034)9 b FB(\))24 b(=)f FA(")1732 4112 y Fz(b)1713 4133 y FA(H)1789 4099 y Fy(20)1782 4154 y(1)1859 4133 y FB(\()p FA(q)s(;)14 b(p;)g(\034)9 b FB(\))19 b(+)f FA(")2265 4099 y Fy(2)2321 4112 y Fz(b)2302 4133 y FA(H)2378 4099 y Fy(22)2371 4154 y(1)2448 4133 y FB(\()p FA(q)s(;)c(p;)g(\034)9 b FB(\))p FA(;)925 b FB(\(325\))71 4325 y(where)334 4304 y Fz(b)315 4325 y FA(H)391 4295 y Fy(20)384 4345 y(1)461 4325 y FB(\()p FA(q)s(;)14 b(p;)g(\034)9 b FB(\))32 b(is)g(indep)r(enden)n(t)g (of)f FA(")p FB(.)48 b(T)-7 b(aking)31 b(in)n(to)g(accoun)n(t)g(the)h (de\014nition)f(of)h FA(M)2991 4337 y Fy(2)3059 4325 y FB(in)g(\(321\))e(and)i(recalling)71 4424 y(that)c(for)f(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b Fw(2)g FA(R)733 4438 y Fx(\024)772 4419 y Fl(0)772 4457 y Fu(3)804 4438 y Fx(;d)859 4446 y Fu(3)913 4424 y Fw(\002)18 b Ft(T)1051 4436 y Fx(\033)1096 4424 y FB(,)1105 4627 y FA(p)1147 4639 y Fy(0)1184 4627 y FB(\()p FA(v)s FB(\))1291 4593 y Fv(\000)p Fy(1)1400 4606 y Fz(b)1380 4627 y FA(H)1456 4593 y Fy(20)1449 4648 y(1)1527 4627 y FB(\()p FA(q)1596 4639 y Fy(0)1633 4627 y FB(\()p FA(v)s FB(\))p FA(;)c(p)1819 4639 y Fy(0)1857 4627 y FB(\()p FA(v)s FB(\))p FA(;)g(\034)9 b FB(\))25 b(=)d Fw(O)2272 4560 y Fz(\000)2311 4627 y FB(\()p FA(v)f Fw(\000)d FA(ia)p FB(\))2592 4593 y Fy(2)p Fx(r)r Fv(\000)p Fx(`)2742 4560 y Fz(\001)2794 4627 y FA(;)71 4808 y FB(w)n(e)27 b(can)g(de\014ne)1052 4948 y FA(C)1117 4913 y Fy(2)1111 4968 y(0)1155 4948 y FB(\()p FA(\026)p FB(\))d(=)e Fw(\000)p FA(\026)1509 4835 y Fz(Z)1592 4855 y Fx(ia)1555 5023 y(v)1588 5031 y Fu(0)1669 4856 y Fz(D)1720 4948 y FA(p)1762 4960 y Fy(0)1799 4948 y FB(\()p FA(v)s FB(\))1906 4913 y Fv(\000)p Fy(1)2015 4927 y Fz(b)1996 4948 y FA(H)2072 4913 y Fy(20)2065 4968 y(1)2142 4948 y FB(\()p FA(q)2211 4960 y Fy(0)2248 4948 y FB(\()p FA(v)s FB(\))p FA(;)14 b(p)2434 4960 y Fy(0)2472 4948 y FB(\()p FA(v)s FB(\))p FA(;)g(\034)9 b FB(\))2693 4856 y Fz(E)2759 4948 y FA(dv)s(:)793 b FB(\(326\))71 5156 y(Then,)27 b(the)f(constan)n(t)g FA(C)850 5126 y Fy(2)844 5177 y(0)887 5156 y FB(\()p FA(\026)p FB(\))h(is)f(indep)r (enden)n(t)h(of)g FA(")p FB(.)36 b(Moreo)n(v)n(er,)24 b(using)i(Lemmas)f(9.2)h(and)g(9.4)f(and)h(9.5,)g(one)g(can)g(see)71 5256 y(that)1431 5285 y Fz(\014)1431 5335 y(\014)1459 5356 y FA(C)1524 5321 y Fy(2)1562 5356 y FB(\()p FA(\026;)14 b(")p FB(\))k Fw(\000)g FA(C)1918 5321 y Fy(2)1912 5376 y(0)1956 5356 y FB(\()p FA(\026)p FB(\))2070 5285 y Fz(\014)2070 5335 y(\014)2121 5356 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2421 5321 y Fx(\027)2454 5329 y Fu(2)3661 5356 y FB(\(327\))71 5504 y(for)27 b(certain)g(constan)n(t)g FA(\027)851 5516 y Fy(2)911 5504 y FA(>)c FB(0.)p Black 1898 5753 a(110)p Black eop end %%Page: 111 111 TeXDict begin 111 110 bop Black Black 195 272 a FB(T)-7 b(o)26 b(obtain)f FA(C)637 242 y Fy(0)631 293 y(3)675 272 y FB(\()p FA(\026)p FB(\))i(w)n(e)e(need)h(a)g(careful)f(study)h (of)g(the)g(function)h FA(G)2314 284 y Fy(3)2377 272 y FB(in)f(\(273\))o(.)37 b(T)-7 b(o)25 b(this)h(end,)h(w)n(e)e(ha)n(v)n (e)g(to)h(expand)71 382 y(asymptotically)d(the)i(functions)g FA(@)1165 394 y Fx(v)1218 361 y Fz(b)1205 382 y FA(T)1266 342 y Fx(u;s)1254 404 y Fy(1)1359 382 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))26 b(obtained)e(in)h(Theorems)e(4.4)h(and)g(4.8.)35 b(T)-7 b(o)24 b(obtain)h(this)f(expansion)g(w)n(e)71 482 y(consider)i(equation)h(\(157\))g(for)g(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b Fw(2)e FA(R)1432 496 y Fx(\024)1471 476 y Fl(0)1471 514 y Fu(3)1503 496 y Fx(;d)1558 504 y Fu(3)1613 482 y Fw(\002)18 b Ft(T)1751 494 y Fx(\033)1796 482 y FB(.)195 581 y(As)28 b(a)f(\014rst)h(step)f(w)n(e)h(expand)f(the) h(function)g FA(A)p FB(\()p FA(u;)14 b(\034)9 b FB(\))29 b(de\014ned)f(in)f(\(143\).)37 b(It)27 b(can)h(b)r(e)g(seen)f(that)h (it)g(satis\014es)1093 799 y FA(A)p FB(\()p FA(u;)14 b(\034)9 b FB(\))24 b(=)e FA(A)1522 765 y Fy(0)1560 799 y FB(\()p FA(u;)14 b(\034)9 b FB(\))19 b(+)f FA("A)1957 765 y Fy(1)1994 799 y FB(\()p FA(u;)c(\034)9 b FB(\))19 b(+)f Fw(O)2373 682 y Fz(\022)2538 743 y FA(\026")2627 713 y Fy(2)p 2444 780 314 4 v 2444 856 a FB(\()p FA(v)k Fw(\000)c FA(ia)p FB(\))2726 832 y Fx(`)2767 682 y Fz(\023)71 1009 y FB(where)829 1173 y FA(A)891 1185 y Fy(0)929 1173 y FB(\()p FA(u;)c(\034)9 b FB(\))23 b(=)g Fw(\000)p FA(\026)1368 1152 y Fz(b)1349 1173 y FA(H)1425 1139 y Fy(1)1418 1193 y(1)1475 1173 y FB(\()q FA(q)1545 1185 y Fy(0)1582 1173 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1773 1185 y Fy(0)1810 1173 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))1625 b(\(328\))829 1317 y FA(A)891 1329 y Fy(1)929 1317 y FB(\()p FA(u;)14 b(\034)9 b FB(\))23 b(=)g Fw(\000)p FA(\026)1368 1296 y Fz(b)1349 1317 y FA(H)1425 1283 y Fy(20)1418 1337 y(1)1509 1317 y FB(\()p FA(q)1578 1329 y Fy(0)1615 1317 y FB(\()p FA(u)p FB(\))p FA(;)14 b(p)1806 1329 y Fy(0)1843 1317 y FB(\()p FA(u)p FB(\))p FA(;)g(\034)9 b FB(\))20 b Fw(\000)e FA(V)2239 1283 y Fv(0)2262 1317 y FB(\()p FA(q)2331 1329 y Fy(0)2368 1317 y FB(\()p FA(u)p FB(\)\))p FA(x)2559 1283 y Fy(0)2559 1337 y Fx(p)2599 1317 y FB(\()p FA(\034)9 b FB(\))19 b(+)f FA(\025)2858 1283 y Fy(2)2896 1317 y FA(x)2943 1283 y Fy(0)2943 1337 y Fx(p)2982 1317 y FB(\()p FA(\034)9 b FB(\))570 b(\(329\))71 1503 y(where)323 1482 y Fz(b)304 1503 y FA(H)380 1473 y Fy(1)373 1523 y(1)417 1503 y FB(,)482 1482 y Fz(b)462 1503 y FA(H)538 1473 y Fy(20)531 1523 y(1)629 1503 y FB(and)20 b FA(x)830 1473 y Fy(0)830 1523 y Fx(p)890 1503 y FB(are)f(the)i(functions)g(de\014ned)g(in)g(\(35\),)h(\(325\))e (and)g(\(324\))g(resp)r(ectiv)n(ely)-7 b(,)21 b(and)g FA(\025)g FB(is)f(the)h(constan)n(t)71 1602 y(de\014ned)27 b(in)g(Hyp)r(othesis)g Fp(HP1.1)p FB(.)36 b(Recall)27 b(that)g(in)g(the)g(parab)r(olic)f(case,)g(w)n(e)h(ha)n(v)n(e)f(that)h FA(x)2967 1572 y Fy(0)2967 1623 y Fx(p)3006 1602 y FB(\()p FA(\034)9 b FB(\))24 b(=)f(0.)36 b(It)27 b(is)g(clear)f(that)71 1702 y(b)r(oth)i FA(A)329 1672 y Fy(0)394 1702 y FB(and)g FA(A)618 1672 y Fy(1)683 1702 y FB(are)e(indep)r(enden)n(t)j(of)e FA(")p FB(.)195 1802 y(F)-7 b(rom)31 b(this)h(expansion,)f(one)g(can)g (deduce)h(the)g(expansion)e(of)h(the)h(function)2719 1781 y Fz(b)2699 1802 y FA(A)g FB(de\014ned)g(in)f(\(159\))o(.)49 b(Let)31 b(us)h(\014rst)71 1901 y(recall)26 b(that)i(the)g(c)n(hange)f (of)g(v)-5 b(ariables)27 b FA(g)j FB(obtained)d(in)h(Lemma)f(7.6)g(can) g(b)r(e)h(written)g(as)1031 2114 y FA(g)s FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)e Fw(\000)p FA(")p 1478 2048 68 4 v(B)1545 2126 y Fy(1)1582 2114 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b(+)e Fw(O)1956 1997 y Fz(\022)2342 2058 y FA(\026")2431 2028 y Fy(2)p 2027 2095 756 4 v 2027 2173 a FB(\()p FA(v)k Fw(\000)c FA(ia)p FB(\))2309 2149 y Fy(max)p Fv(f)p Fy(1+)p Fx(`)p Fv(\000)p Fy(2)p Fx(r)n(;)p Fy(0)p Fv(g)2792 1997 y Fz(\023)2867 2114 y FA(;)71 2340 y FB(where)p 311 2273 68 4 v 27 w FA(B)378 2352 y Fy(1)443 2340 y FB(is)27 b(the)h(function)h(de\014ned)f(on)f (the)h(pro)r(of)f(of)g(Lemma)h(7.6,)f(whic)n(h)g(is)h(indep)r(enden)n (t)g(of)g FA(")p FB(.)195 2440 y(Therefore,)918 2567 y Fz(b)898 2588 y FA(A)p FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))25 b(=)1281 2567 y Fz(b)1261 2588 y FA(A)1323 2553 y Fy(0)1361 2588 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b(+)e FA(")1711 2567 y Fz(b)1692 2588 y FA(A)1754 2553 y Fy(1)1791 2588 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))20 b(+)e Fw(O)2165 2471 y Fz(\022)2513 2532 y FA(\026")2602 2501 y Fy(2)p 2236 2569 680 4 v 2236 2646 a FB(\()p FA(v)k Fw(\000)c FA(ia)p FB(\))2518 2622 y Fx(`)p Fy(+2+2\()p Fx(`)p Fv(\000)p Fy(2)p Fx(r)r Fy(\))2925 2471 y Fz(\023)3000 2588 y FA(;)71 2772 y FB(with)1239 2905 y Fz(b)1219 2926 y FA(A)1281 2938 y Fy(0)1319 2926 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))24 b(=)f FA(A)1682 2938 y Fy(0)1719 2926 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))1239 3049 y Fz(b)1219 3070 y FA(A)1281 3082 y Fy(1)1319 3070 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f FA(A)1682 3082 y Fy(1)1719 3070 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b Fw(\000)e FA(@)2055 3082 y Fx(v)2095 3070 y FA(A)2157 3082 y Fy(0)2194 3070 y FB(\()p FA(v)s(;)c(\034)9 b FB(\))p 2383 3003 68 4 v FA(B)2452 3082 y Fy(1)2489 3070 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))p FA(:)71 3254 y FB(Using)34 b(this)g(fact)g(and)g (the)h(prop)r(erties)e(of)h(the)h(functions)1997 3233 y Fz(b)1979 3254 y FA(B)j FB(and)2265 3233 y Fz(b)2248 3254 y FA(C)j FB(in)34 b(\(160\))g(and)f(\(161\),)i(one)f(can)g(see)g (that)g(the)71 3364 y(functions)443 3343 y Fz(b)429 3364 y FA(T)490 3324 y Fx(u;s)478 3386 y Fy(1)583 3364 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))29 b(obtained)f(in)g(Theorems)e(4.4)h(and) g(4.8)g(satisfy)g(that)944 3583 y FA(@)988 3595 y Fx(v)1041 3562 y Fz(b)1027 3583 y FA(T)1088 3543 y Fx(u;s)1076 3605 y Fy(1)1182 3583 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))24 b(=)f FA("@)1566 3595 y Fx(v)1619 3562 y Fz(b)1605 3583 y FA(T)1666 3548 y Fy(0)1654 3603 y(1)1702 3583 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b(+)e Fw(O)2077 3466 y Fz(\022)2464 3526 y FA(\026")2553 3496 y Fy(2)p 2148 3564 759 4 v 2148 3641 a FB(\()p FA(v)k Fw(\000)c FA(ia)p FB(\))2430 3617 y Fy(max)o Fv(f)p Fy(0)p Fx(;)p Fy(2+)p Fx(`)p Fv(\000)p Fx(\027)2836 3625 y Fu(3)2868 3617 y Fv(g)2916 3466 y Fz(\023)71 3816 y FB(for)28 b(certain)g FA(\027)518 3828 y Fy(3)580 3816 y FA(>)c FB(0.)40 b(The)29 b(\014rst)f(order)f FA(@)1380 3828 y Fx(v)1434 3795 y Fz(b)1420 3816 y FA(T)1481 3786 y Fy(0)1469 3836 y(1)1517 3816 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))30 b(is)f(de\014ned)g(b)n (y)f FA(@)2268 3828 y Fx(v)2322 3795 y Fz(b)2308 3816 y FA(T)2369 3786 y Fy(0)2357 3836 y(1)2405 3816 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))26 b(=)f FA(@)2754 3828 y Fx(v)p 2793 3749 63 4 v 2793 3816 a FA(A)2855 3828 y Fy(0)2893 3816 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))20 b(+)f Fw(h)3238 3795 y Fz(b)3218 3816 y FA(A)3280 3828 y Fy(1)3318 3816 y Fw(i)p FB(\()p FA(v)s FB(\),)30 b(where)p 3751 3749 V 28 w FA(A)3813 3828 y Fy(0)71 3926 y FB(is)f(a)g(function)g (satisfying)g(that)g FA(@)1150 3938 y Fx(\034)p 1192 3859 V 1192 3926 a FA(A)1254 3938 y Fy(0)1317 3926 y FB(=)c FA(A)1469 3938 y Fy(0)1536 3926 y FB(and)k Fw(h)p FA(A)1793 3938 y Fy(0)1831 3926 y Fw(i)d FB(=)f(0.)41 b(Then,)30 b FA(@)2371 3938 y Fx(v)2424 3905 y Fz(b)2410 3926 y FA(T)2471 3896 y Fy(0)2459 3946 y(1)2508 3926 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))30 b(is)f(indep)r(enden)n(t)h (of)f FA(")g FB(and)g(can)g(b)r(e)71 4025 y(analytically)d(extended)i (to)g(reac)n(h)e FA(v)g FB(=)d FA(ia)p FB(.)195 4125 y(T)-7 b(aking)23 b(in)n(to)g(accoun)n(t)g(the)h(prop)r(erties)f(of)g (the)h(c)n(hange)f FA(g)j FB(stated)e(in)f(Lemma)h(7.6,)f(one)h(can)f (see)g(that)h(the)g(function)71 4225 y FA(@)115 4237 y Fx(u)158 4225 y FA(T)207 4237 y Fy(1)244 4225 y FB(\()p FA(u;)14 b(\034)9 b FB(\))28 b(has)g(the)g(same)f(expansion)f(as)h(the) h(function)g FA(@)1965 4237 y Fx(v)2019 4204 y Fz(b)2005 4225 y FA(T)2054 4237 y Fy(1)2090 4225 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\).)195 4324 y(W)-7 b(e)28 b(can)g(de\014ne)1253 4464 y FA(C)1318 4429 y Fy(3)1312 4484 y(0)1355 4464 y FB(\()p FA(\026)p FB(\))c(=)f Fw(\000)1660 4351 y Fz(Z)1742 4371 y Fx(ia)1705 4539 y(v)1738 4547 y Fu(0)1820 4396 y Fz(\012)1859 4464 y FA(p)1901 4476 y Fy(0)1938 4464 y FB(\()p FA(v)s FB(\))2045 4429 y Fv(\000)p Fy(2)2135 4464 y FA(@)2179 4476 y Fx(v)2218 4464 y FA(T)2279 4429 y Fy(0)2267 4484 y(1)2316 4464 y FB(\()p FA(v)s(;)14 b(\034)9 b FB(\))2505 4396 y Fz(\013)2559 4464 y FA(dv)s(;)993 b FB(\(330\))71 4652 y(whic)n(h)27 b(is)h(a)f(constan)n(t)g(indep)r (enden)n(t)h(of)g FA(")p FB(.)37 b(Doing)27 b(little)h(e\013ort,)g(it)g (can)f(b)r(e)h(seen)f(also)g(that)1431 4746 y Fz(\014)1431 4796 y(\014)1459 4817 y FA(C)1524 4782 y Fy(3)1562 4817 y FB(\()p FA(\026;)14 b(")p FB(\))k Fw(\000)g FA(C)1918 4782 y Fy(3)1912 4837 y(0)1956 4817 y FB(\()p FA(\026)p FB(\))2070 4746 y Fz(\014)2070 4796 y(\014)2121 4817 y Fw(\024)23 b FA(K)6 b Fw(j)p FA(\026)p Fw(j)p FA(")2421 4782 y Fx(\027)2458 4757 y Fl(0)2454 4799 y Fu(3)3661 4817 y FB(\(331\))71 4981 y(for)27 b(certain)g FA(\027)521 4951 y Fv(0)516 5002 y Fy(3)576 4981 y FA(>)c FB(0.)195 5081 y(Finally)-7 b(,)27 b(it)g(is)f(enough)f(to)h(de\014ne)h FA(C)1339 5093 y Fy(0)1376 5081 y FB(\()p FA(\026)p FB(\))d(=)f FA(C)1667 5051 y Fy(1)1661 5101 y(0)1704 5081 y FB(\()p FA(\026)p FB(\))16 b(+)g FA(C)1980 5051 y Fy(2)1974 5101 y(0)2017 5081 y FB(\()p FA(\026)p FB(\))g(+)g FA(C)2293 5051 y Fy(3)2287 5101 y(0)2330 5081 y FB(\()p FA(\026)p FB(\))27 b(where)f FA(C)2775 5051 y Fx(i)2769 5101 y Fy(0)2806 5081 y FB(\()p FA(\026)p FB(\))h(are)e(the)i(constan)n(ts)e (de\014ned)71 5180 y(in)34 b(\(322\))o(,)i(\(326\))d(and)h(\(330\))o(.) 56 b(It)35 b(is)f(straigh)n(tforw)n(ard)d(to)j(see)f(that)i FA(C)2353 5192 y Fy(0)2390 5180 y FB(\()p FA(\026)p FB(\))g(is)f(an)g (en)n(tire)f(function.)57 b(Moreo)n(v)n(er,)33 b(b)n(y)71 5280 y(\(323\))o(,)28 b(\(327\))e(and)i(\(331\))o(,)g(it)g(is)f(clear)g (that)1596 5444 y(lim)1588 5496 y Fx(")p Fv(!)p Fy(0)1733 5444 y FA(C)6 b FB(\()p FA(\026;)14 b(")p FB(\))23 b(=)g FA(C)2158 5456 y Fy(0)2195 5444 y FB(\()p FA(\026)p FB(\))p FA(:)p Black 1898 5753 a FB(111)p Black eop end %%Page: 112 112 TeXDict begin 112 111 bop Black Black 71 272 a FC(References)p Black 71 454 a FB([AKN88])p Black 69 w(V.I.)22 b(Arnold,)h(V.V.)g (Kozlo)n(v,)e(and)h(A.I.)h(Neish)n(tadt.)28 b Fs(Dynamic)l(al)d (Systems)f(III)p FB(,)f(v)n(olume)e(3)h(of)g Fs(Encyclop)l(ae-)459 554 y(dia)31 b(Math.)g(Sci.)38 b FB(Springer,)26 b(Berlin,)h(1988.)p Black 71 720 a([Ang93])p Black 108 w(S.)39 b(Angenen)n(t.)71 b(A)40 b(v)-5 b(ariational)37 b(in)n(terpretation)i(of)g(Mel)2316 689 y Fv(0)2339 720 y FB(nik)n(o)n(v's)f(function)h(and)g(exp)r(onen)n (tially)g(small)459 819 y(separatrix)28 b(splitting.)46 b(In)31 b Fs(Symple)l(ctic)i(ge)l(ometry)p FB(,)f(v)n(olume)d(192)g(of) i Fs(L)l(ondon)i(Math.)g(So)l(c.)g(L)l(e)l(ctur)l(e)e(Note)459 919 y(Ser.)p FB(,)d(pages)e(5{35.)g(Cam)n(bridge)g(Univ.)i(Press,)f (Cam)n(bridge,)f(1993.)p Black 71 1085 a([Bal06])p Black 134 w(I.)g(Baldom\023)-42 b(a.)34 b(The)26 b(inner)g(equation)g(for)g (one)g(and)g(a)g(half)g(degrees)f(of)i(freedom)f(rapidly)f(forced)h (Hamilto-)459 1185 y(nian)h(systems.)37 b Fs(Nonline)l(arity)p FB(,)28 b(19\(6\):1415{1445,)23 b(2006.)p Black 71 1351 a([BF04])p Black 145 w(I.)30 b(Baldom\023)-42 b(a)30 b(and)g(E.)g(F)-7 b(on)n(tic)n(h.)45 b(Exp)r(onen)n(tially)29 b(small)h(splitting)h(of)f(in)n(v)-5 b(arian)n(t)29 b(manifolds)h(of)h (parab)r(olic)459 1450 y(p)r(oin)n(ts.)37 b Fs(Mem.)30 b(A)n(mer.)g(Math.)h(So)l(c.)p FB(,)d(167\(792\):x{83,)c(2004.)p Black 71 1616 a([BF05])p Black 145 w(I.)e(Baldom\023)-42 b(a)20 b(and)i(E.)f(F)-7 b(on)n(tic)n(h.)27 b(Exp)r(onen)n(tially)21 b(small)h(splitting)g(of)f(separatrices)f(in)i(a)f(w)n(eakly)g(h)n(yp)r (erb)r(olic)459 1716 y(case.)36 b Fs(J.)30 b(Di\013er)l(ential)g (Equations)p FB(,)e(210\(1\):106{134,)23 b(2005.)p Black 71 1882 a([BG10])p Black 134 w(N.)36 b(Br\177)-42 b(annstrom)34 b(and)h(V.)h(Gelfreic)n(h.)59 b(Asymptotic)36 b(series)e(for)h(the)h (splitting)g(of)f(separatrices)e(near)i(a)459 1982 y(hamiltonian)27 b(bifurcation.)37 b(Preprin)n(t,)26 b(2010.)p Black 71 2148 a([BO93])p Black 134 w(A.)21 b(Bensen)n(y)f(and)h(C.)g(Oliv)n (\023)-39 b(e.)25 b(High)c(precision)f(angles)g(b)r(et)n(w)n(een)h(in)n (v)-5 b(arian)n(t)19 b(manifolds)i(for)g(radpidly)f(forced)459 2247 y(hamiltonian)27 b(systems.)36 b Fs(Pr)l(o)l(c)l(e)l(e)l(dings)31 b(Equadi\01391)p FB(,)f(pages)c(315{319,)f(1993.)p Black 71 2413 a([BS06])p Black 153 w(I.)h(Baldom\023)-42 b(a)25 b(and)h(T.)g(M.)h(Seara.)33 b(Breakdo)n(wn)24 b(of)i(hetero)r(clinic)g (orbits)f(for)h(some)f(analytic)h(unfoldings)g(of)459 2513 y(the)i(Hopf-zero)e(singularit)n(y)-7 b(.)36 b Fs(J.)30 b(Nonline)l(ar)g(Sci.)p FB(,)e(16\(6\):543{582,)c(2006.)p Black 71 2679 a([BS08])p Black 153 w(I.)19 b(Baldom\023)-42 b(a)18 b(and)h(T.)h(M.)f(Seara.)j(The)d(inner)g(equation)g(for)g (generic)f(analytic)h(unfoldings)g(of)g(the)h(Hopf-zero)459 2779 y(singularit)n(y)-7 b(.)35 b Fs(Discr)l(ete)30 b(Contin.)g(Dyn.)g (Syst.)g(Ser.)g(B)p FB(,)e(10\(2-3\):323{347,)23 b(2008.)p Black 71 2945 a([BSSV98])p Black 45 w(Carles)39 b(Bonet,)k(Da)n(vid)d (Sauzin,)j(T)-7 b(ere)39 b(Seara,)j(and)e(Marta)f(V)-7 b(al)n(\022)-39 b(encia.)73 b(Adiabatic)40 b(in)n(v)-5 b(arian)n(t)39 b(of)h(the)459 3044 y(harmonic)35 b(oscillator,)i (complex)g(matc)n(hing)f(and)g(resurgence.)61 b Fs(SIAM)38 b(J.)g(Math.)h(A)n(nal.)p FB(,)h(29\(6\):1335{)459 3144 y(1360)26 b(\(electronic\),)h(1998.)p Black 71 3310 a([CG94])p Black 133 w(L.)33 b(Chierc)n(hia)f(and)g(G.)i(Galla)n(v)n(otti.)51 b(Drift)34 b(and)e(di\013usion)h(in)h(phase)e(space.)52 b Fs(A)n(nn.)34 b(Inst.)g(H.)h(Poinc)l(ar)n(\023)-40 b(e)459 3410 y(Phys.)31 b(Th)n(\023)-40 b(eor.)p FB(,)30 b(60\(1\):144,)25 b(1994.)p Black 71 3576 a([DG00])p Black 130 w(A.)39 b(Delshams)f(and)h(P)-7 b(.)38 b(Guti)n(\023)-39 b(errez.)68 b(Splitting)40 b(p)r(oten)n(tial)e(and)h(the)g(Poincar)n (\023)-39 b(e-Melnik)n(o)n(v)34 b(metho)r(d)39 b(for)459 3675 y(whisk)n(ered)26 b(tori)h(in)h(Hamiltonian)g(systems.)36 b Fs(J.)30 b(Nonline)l(ar)g(Sci.)p FB(,)f(10\(4\):433{476,)23 b(2000.)p Black 71 3841 a([DGJS97])p Black 41 w(A.)30 b(Delshams,)h(V.)g(Gelfreic)n(h,)1478 3820 y(\022)1467 3841 y(A.)g(Jorba,)e(and)h(T.M.)h(Seara.)43 b(Exp)r(onen)n(tially)29 b(small)h(splitting)g(of)g(sepa-)459 3941 y(ratrices)c(under)h(fast)h (quasip)r(erio)r(dic)f(forcing.)36 b Fs(Comm.)30 b(Math.)h(Phys.)p FB(,)f(189\(1\):35{71,)24 b(1997.)p Black 71 4107 a([DGS04])p Black 84 w(A.)31 b(Delshams,)g(P)-7 b(.)30 b(Guti)n(\023)-39 b(errez,)30 b(and)g(T.M.)h(Seara.)44 b(Exp)r(onen)n(tially)30 b(small)g(splitting)h(for)f(whisk)n(ered)f(tori)459 4207 y(in)d(Hamiltonian)h(sysems:)35 b(\015o)n(w-b)r(o)n(x)25 b(co)r(ordinates)g(and)h(upp)r(er)g(b)r(ounds.)35 b Fs(Discr)l(ete)29 b(Contin.)g(Dyn.)f(Syst.)p FB(,)459 4306 y(11\(4\):785{826,)23 b(2004.)p Black 71 4472 a([DR97])p Black 134 w(A.)37 b(Delshams)f(and)g(R.)h(Ram)-9 b(\023)-32 b(\020rez-Ros.)60 b(Melnik)n(o)n(v)36 b(p)r(oten)n(tial)g(for)g(exact)g(symplectic)g (maps.)63 b Fs(Comm.)459 4572 y(Math.)31 b(Phys.)p FB(,)e (190\(1\):213{245,)23 b(1997.)p Black 71 4738 a([DR98])p Black 134 w(A.)28 b(Delshams)f(and)g(R.)g(Ram)-9 b(\023)-32 b(\020rez-Ros.)34 b(Exp)r(onen)n(tially)26 b(small)h(splitting)g(of)g (separatrices)e(for)i(p)r(erturb)r(ed)459 4837 y(in)n(tegrable)f (standard-lik)n(e)g(maps.)37 b Fs(J.)29 b(Nonline)l(ar)i(Sci.)p FB(,)d(8\(3\):317{352,)c(1998.)p Black 71 5004 a([DS92])p Black 149 w(A.)32 b(Delshams)g(and)g(T.)g(M.)h(Seara.)48 b(An)33 b(asymptotic)f(expression)e(for)i(the)g(splitting)h(of)f (separatrices)e(of)459 5103 y(the)e(rapidly)f(forced)g(p)r(endulum.)38 b Fs(Comm.)30 b(Math.)i(Phys.)p FB(,)d(150\(3\):433{463,)23 b(1992.)p Black 71 5269 a([DS97])p Black 149 w(A.)30 b(Delshams)f(and)g(T.M.)h(Seara.)41 b(Splitting)31 b(of)e(separatrices) e(in)j(Hamiltonian)f(systems)g(with)i(one)e(and)459 5369 y(a)e(half)h(degrees)e(of)i(freedom.)36 b Fs(Math.)31 b(Phys.)g(Ele)l(ctr)l(on.)g(J.)p FB(,)c(3:P)n(ap)r(er)f(4,)h(40)g(pp.)h (\(electronic\),)f(1997.)p Black 1898 5753 a(112)p Black eop end %%Page: 113 113 TeXDict begin 113 112 bop Black Black Black 71 272 a FB([)101 251 y(\023)94 272 y(Eca81a])p Black 80 w(Jean)675 251 y(\023)667 272 y(Ecalle.)76 b Fs(L)l(es)42 b(fonctions)h(r)n(\023) -40 b(esur)l(gentes.)42 b(Tome)h(I)p FB(,)e(v)n(olume)f(5)h(of)g Fs(Public)l(ations)i(Math)n(\023)-40 b(ematiques)459 372 y(d'Orsay)29 b(81)f([Mathematic)l(al)j(Public)l(ations)e(of)f (Orsay)g(81])p FB(.)35 b(Univ)n(ersit)n(\023)-39 b(e)24 b(de)h(P)n(aris-Sud)f(D)n(\023)-39 b(epartemen)n(t)24 b(de)459 471 y(Math)n(\023)-39 b(ematique,)26 b(Orsa)n(y)-7 b(,)26 b(1981.)34 b(Les)27 b(alg)n(\022)-39 b(ebres)25 b(de)i(fonctions)g(r)n(\023)-39 b(esurgen)n(tes.)25 b([The)i(algebras)f (of)h(resurgen)n(t)459 571 y(functions],)h(With)g(an)g(English)f(forew) n(ord.)p Black 71 737 a([)101 716 y(\023)94 737 y(Eca81b])p Black 76 w(Jean)672 716 y(\023)665 737 y(Ecalle.)68 b Fs(L)l(es)40 b(fonctions)g(r)n(\023)-40 b(esur)l(gentes.)40 b(Tome)g(II)p FB(,)f(v)n(olume)f(6)g(of)g Fs(Public)l(ations)j(Math)n (\023)-40 b(ematiques)459 837 y(d'Orsay)38 b(81)f([Mathematic)l(al)i (Public)l(ations)f(of)g(Orsay)f(81])p FB(.)60 b(Univ)n(ersit)n(\023)-39 b(e)34 b(de)h(P)n(aris-Sud)f(D)n(\023)-39 b(epartemen)n(t)459 936 y(de)29 b(Math)n(\023)-39 b(ematique,)29 b(Orsa)n(y)-7 b(,)27 b(1981.)40 b(Les)29 b(fonctions)g(r)n(\023)-39 b(esurgen)n(tes)26 b(appliqu)n(\023)-39 b(ees)28 b(\022)-42 b(a)29 b(l'it)n(\023)-39 b(eration.)27 b([Resurgen)n(t)459 1036 y(functions)h(applied)f(to)h(iteration].)p Black 71 1202 a([EKS93])p Black 90 w(James)i(A.)i(Ellison,)g(Martin)f (Kummer,)h(and)g(A.)g(W.)g(S\023)-42 b(aenz.)48 b(T)-7 b(ranscenden)n(tally)30 b(small)h(transv)n(ersalit)n(y)459 1302 y(in)d(the)g(rapidly)f(forced)g(p)r(endulum.)38 b Fs(J.)29 b(Dynam.)h(Di\013er)l(ential)h(Equations)p FB(,)d(5\(2\):241{277,)c(1993.)p Black 71 1468 a([Eli94])p Black 155 w(L.)g(H.)g(Eliasson.)k(Biasymptotic)23 b(solutions)g(of)h(p) r(erturb)r(ed)g(in)n(tegrable)e(Hamiltonian)h(systems.)30 b Fs(Bol.)e(So)l(c.)459 1567 y(Br)l(asil.)j(Mat.)g(\(N.S.\))p FB(,)d(25\(1\):57{76,)c(1994.)p Black 71 1733 a([F)-7 b(on93])p Black 123 w(E.)18 b(F)-7 b(on)n(tic)n(h.)21 b(Exp)r(onen)n(tially)d(small)g(upp)r(er)g(b)r(ounds)h(for)f(the)h (splitting)f(of)h(separatrices)d(for)i(high)g(frequency)459 1833 y(p)r(erio)r(dic)27 b(p)r(erturbations.)36 b Fs(Nonline)l(ar)31 b(A)n(nal.)p FB(,)d(20\(6\):733{744,)23 b(1993.)p Black 71 1999 a([F)-7 b(on95])p Black 123 w(E.)33 b(F)-7 b(on)n(tic)n(h.)55 b(Rapidly)34 b(forced)f(planar)g(v)n(ector)f(\014elds)i(and)g (splitting)g(of)f(separatrices.)53 b Fs(J.)36 b(Di\013er)l(ential)459 2099 y(Equations)p FB(,)28 b(119\(2\):310{335,)23 b(1995.)p Black 71 2265 a([FS90])p Black 158 w(E.)31 b(F)-7 b(on)n(tic)n(h)32 b(and)g(C.)g(Sim\023)-42 b(o.)49 b(The)32 b(splitting)g(of)g (separatrices)e(for)h(analytic)g(di\013eomorphisms.)49 b Fs(Er)l(go)l(dic)459 2364 y(The)l(ory)31 b(Dynam.)f(Systems)p FB(,)e(10\(2\):295{318,)23 b(1990.)p Black 71 2530 a([FS96])p Black 158 w(B.)31 b(Fiedler)g(and)g(J.)g(Sc)n(heurle.)47 b(Discretization)31 b(of)g(homo)r(clinic)g(orbits,)h(rapid)e(forcing)h (and)g(\\in)n(visible")459 2630 y(c)n(haos.)k Fs(Mem.)c(A)n(mer.)f (Math.)h(So)l(c.)p FB(,)d(119\(570\):viii+79,)c(1996.)p Black 71 2796 a([Gal94])p Black 128 w(G.)g(Galla)n(v)n(otti.)30 b(Twistless)23 b(KAM)h(tori,)h(quasi)e(\015at)h(homo)r(clinic)g(in)n (tersections,)f(and)h(other)g(cancellations)459 2896 y(in)d(the)h(p)r(erturbation)e(series)g(of)h(certain)g(completely)g(in) n(tegrable)f(Hamiltonian)h(systems.)f(A)i(review.)j Fs(R)l(ev.)459 2995 y(Math.)31 b(Phys.)p FB(,)e(6\(3\):343{411,)24 b(1994.)p Black 71 3161 a([Gel94])p Black 133 w(V.)32 b(G.)h(Gelfreic)n(h.)49 b(Separatrices)30 b(splitting)j(for)e(the)i(rapidly)e(forced)g(p)r (endulum.)52 b(In)32 b Fs(Seminar)i(on)g(Dy-)459 3261 y(namic)l(al)29 b(Systems)f(\(St.)f(Petersbur)l(g,)i(1991\))p FB(,)f(v)n(olume)e(12)f(of)g Fs(Pr)l(o)l(gr.)k(Nonline)l(ar)g(Di\013er) l(ential)g(Equations)459 3361 y(Appl.)p FB(,)g(pages)d(47{67.)g (Birkh\177)-42 b(auser,)25 b(Basel,)i(1994.)p Black 71 3527 a([Gel97a])p Black 91 w(V.)h(G.)h(Gelfreic)n(h.)37 b(Melnik)n(o)n(v)27 b(metho)r(d)i(and)f(exp)r(onen)n(tially)f(small)h (splitting)g(of)g(separatrices.)36 b Fs(Phys.)c(D)p FB(,)459 3626 y(101\(3-4\):227{248,)22 b(1997.)p Black 71 3792 a([Gel97b])p Black 87 w(V.)28 b(G.)g(Gelfreic)n(h.)36 b(Reference)28 b(systems)f(for)g(splittings)h(of)f(separatrices.)35 b Fs(Nonline)l(arity)p FB(,)29 b(10\(1\):175{193,)459 3892 y(1997.)p Black 71 4058 a([Gel99])p Black 133 w(V.)i(G.)h (Gelfreic)n(h.)46 b(A)31 b(pro)r(of)g(of)g(the)g(exp)r(onen)n(tially)g (small)f(transv)n(ersalit)n(y)f(of)h(the)i(separatrices)d(for)h(the)459 4158 y(standard)c(map.)37 b Fs(Comm.)31 b(Math.)g(Phys.)p FB(,)e(201\(1\):155{216,)23 b(1999.)p Black 71 4324 a([Gel00])p Black 133 w(V.)h(G.)g(Gelfreic)n(h.)30 b(Separatrix)22 b(splitting)i(for)f(a)g(high-frequency)g(p)r(erturbation)g(of)h(the)g (p)r(endulum.)31 b Fs(R)n(uss.)459 4423 y(J.)f(Math.)h(Phys.)p FB(,)e(7\(1\):48{71,)c(2000.)p Black 71 4589 a([GG10])p Black 128 w(J.)31 b(P)-7 b(.)32 b(Gaiv)-5 b(ao)31 b(and)g(V.)h (Gelfreic)n(h.)49 b(Splitting)33 b(of)f(separatrices)d(for)i(the)i (hamiltonian-hopf)e(bifurcation)459 4689 y(with)d(the)g(swift-hohen)n (b)r(erg)f(equation)g(as)g(an)g(example.)36 b(2010.)f(Preprin)n(t.)p Black 71 4855 a([GGM99])p Black 52 w(G.)f(Galla)n(v)n(otti,)g(G.)g(Gen) n(tile,)i(and)e(V.)g(Mastropietro.)53 b(Separatrix)33 b(splitting)h(for)f(systems)g(with)i(three)459 4955 y(time)28 b(scales.)36 b Fs(Comm.)31 b(Math.)g(Phys.)p FB(,)e(202\(1\):197{236,) 23 b(1999.)p Black 71 5121 a([GL)-7 b(T91])p Black 88 w(V.)26 b(G.)g(Gelfreic)n(h,)g(V.)h(F.)f(Lazutkin,)g(and)g(M.)g(B.)g(T) -7 b(abano)n(v.)32 b(Exp)r(onen)n(tially)25 b(small)h(splittings)f(in)i (Hamil-)459 5220 y(tonian)g(systems.)36 b Fs(Chaos)p FB(,)30 b(1\(2\):137{142,)24 b(1991.)p Black 71 5386 a([GOS10])p Black 82 w(M.)d(Guardia,)g(C.)g(Oliv)n(\023)-39 b(e,)20 b(and)h(T.)g(Seara.)j(Exp)r(onen)n(tially)19 b(small)i(splitting)g(for)f(the)h(p)r(endulum:)35 b(a)20 b(classical)459 5486 y(problem)27 b(revisited.)36 b Fs(J.)30 b(Nonline)l(ar)g(Sci.)p FB(,)f(20\(5\):595{685,)23 b(2010.)p Black 1898 5753 a(113)p Black eop end %%Page: 114 114 TeXDict begin 114 113 bop Black Black Black 71 272 a FB([GS01])p Black 147 w(V.)25 b(Gelfreic)n(h)f(and)g(D.)h(Sauzin.)32 b(Borel)23 b(summation)h(and)h(splitting)g(of)f(separatrices)e(for)i (the)h(H)n(\023)-39 b(enon)24 b(map.)459 372 y Fs(A)n(nn.)29 b(Inst.)g(F)-6 b(ourier)31 b(\(Gr)l(enoble\))p FB(,)d(51\(2\):513{567,) c(2001.)p Black 71 535 a([HMS88])p Black 74 w(P)-7 b(.)33 b(Holmes,)h(J.)g(Marsden,)f(and)h(J.)f(Sc)n(heurle.)53 b(Exp)r(onen)n(tially)32 b(small)h(splittings)h(of)f(separatrices)e (with)459 635 y(applications)26 b(to)i(KAM)f(theory)g(and)g(degenerate) f(bifurcations.)36 b(In)28 b Fs(Hamiltonian)i(dynamic)l(al)i(systems)p FB(,)459 734 y(v)n(olume)27 b(81)g(of)g Fs(Contemp.)k(Math.)e FB(1988.)p Black 71 898 a([Laz84])p Black 127 w(V.)35 b(F.)g(Lazutkin.)59 b(Splitting)35 b(of)g(separatrices)e(for)h(the)i (Chirik)n(o)n(v)d(standard)h(map.)58 b(VINITI)36 b(6372/82,)459 997 y(1984.)f(Preprin)n(t)26 b(\(Russian\).)p Black 71 1161 a([Laz03])p Black 127 w(V.)k(F.)g(Lazutkin.)42 b(Splitting)30 b(of)g(separatrices)d(for)i(the)h(Chirik)n(o)n(v)e(standard)g(map.)43 b Fs(Zap.)32 b(Nauchn.)g(Sem.)459 1260 y(S.-Peterbur)l(g.)42 b(Otdel.)h(Mat.)g(Inst.)e(Steklov.)i(\(POMI\))p FB(,)f(300\(T)-7 b(eor.)38 b(Predst.)j(Din.)g(Sist.)h(Sp)r(ets.)f(Vyp.)459 1360 y(8\):25{55,)25 b(285,)h(2003.)p Black 71 1523 a([LM88])p Black 130 w(P)-7 b(.)31 b(Lo)r(c)n(hak)g(and)h(C.)f(Meunier.)49 b Fs(Multiphase)36 b(A)n(ver)l(aging)e(for)g(Classic)l(al)i(Systems)p FB(,)c(v)n(olume)g(72)e(of)i Fs(Appl.)459 1623 y(Math.)f(Sci.)38 b FB(Springer,)26 b(New)i(Y)-7 b(ork,)27 b(1988.)p Black 71 1786 a([LMS03])p Black 84 w(P)-7 b(.)34 b(Lo)r(c)n(hak,)i(J.-P)-7 b(.)33 b(Marco,)j(and)e(D.)h(Sauzin.)58 b(On)35 b(the)g(splitting)g(of) f(in)n(v)-5 b(arian)n(t)34 b(manifolds)g(in)h(m)n(ultidi-)459 1886 y(mensional)29 b(near-in)n(tegrable)e(Hamiltonian)j(systems.)42 b Fs(Mem.)33 b(A)n(mer.)f(Math.)h(So)l(c.)p FB(,)e (163\(775\):viii+145,)459 1986 y(2003.)p Black 71 2149 a([Lom00])p Black 95 w(E.)24 b(Lom)n(bardi.)31 b Fs(Oscil)t(latory)d (inte)l(gr)l(als)f(and)h(phenomena)g(b)l(eyond)g(al)t(l)g(algebr)l(aic) h(or)l(ders)p FB(,)d(v)n(olume)e(1741)f(of)459 2249 y Fs(L)l(e)l(ctur)l(e)j(Notes)g(in)h(Mathematics)p FB(.)34 b(Springer-V)-7 b(erlag,)22 b(Berlin,)j(2000.)30 b(With)25 b(applications)e(to)i(homo)r(clinic)459 2348 y(orbits)i(in)h(rev)n (ersible)e(systems.)p Black 71 2512 a([LS80])p Black 160 w(Jaume)f(Llibre)g(and)g(Carlos)f(Sim\023)-42 b(o.)33 b(Oscillatory)23 b(solutions)i(in)h(the)f(planar)g(restricted)f (three-b)r(o)r(dy)h(prob-)459 2611 y(lem.)37 b Fs(Math.)31 b(A)n(nn.)p FB(,)c(248\(2\):153{184,)c(1980.)p Black 71 2775 a([Mel63])p Black 122 w(V.)i(K.)f(Melnik)n(o)n(v.)31 b(On)24 b(the)h(stabilit)n(y)f(of)h(the)g(cen)n(ter)e(for)h(time)h(p)r (erio)r(dic)g(p)r(erturbations.)31 b Fs(T)-6 b(r)l(ans.)27 b(Mosc)l(ow)459 2874 y(Math.)k(So)l(c.)p FB(,)d(12:1{57,)d(1963.)p Black 71 3038 a([MP94])p Black 125 w(Regina)35 b(Mart)-9 b(\023)-32 b(\020nez)34 b(and)h(Conxita)g(Pin)n(y)n(ol.)59 b(P)n(arab)r(olic)33 b(orbits)i(in)h(the)g(elliptic)g(restricted)f (three)g(b)r(o)r(dy)459 3137 y(problem.)h Fs(J.)30 b(Di\013er)l(ential) g(Equations)p FB(,)f(111\(2\):299{339,)23 b(1994.)p Black 71 3300 a([MSS10a])p Black 48 w(P)-7 b(.)33 b(Mart)-9 b(\023)-32 b(\020n,)33 b(D.)g(Sauzin,)i(and)e(T.)g(M.)g(Seara.)52 b(Exp)r(onen)n(tially)32 b(small)h(splitting)g(of)g(separatrices)e(in)i (the)459 3400 y(p)r(erturb)r(ed)28 b(mcmillan)g(map.)36 b(Preprin)n(t,)27 b(2010.)p Black 71 3563 a([MSS10b])p Black 44 w(P)-7 b(.)26 b(Mart)-9 b(\023)-32 b(\020n,)25 b(D.)i(Sauzin,)g(and)f(T.)g(M.)h(Seara.)33 b(Resurgence)25 b(of)i(inner)f(solutions)f(for)h(p)r(erturbations)g(of)g(the)459 3663 y(mcmillan)i(map.)36 b(Preprin)n(t,)27 b(2010.)p Black 71 3826 a([Ne)-9 b(\025)-32 b(\02084])p Black 135 w(A.)24 b(I.)g(Ne)-9 b(\025)-32 b(\020sh)n(tadt.)30 b(The)24 b(separation)f(of)g(motions)h(in)g(systems)g(with)g(rapidly)f(rotating) g(phase.)30 b Fs(Prikl.)e(Mat.)459 3926 y(Mekh.)p FB(,)h (48\(2\):197{204,)24 b(1984.)p Black 71 4089 a([Oli06])p Black 147 w(C.)d(Oliv)n(\023)-39 b(e.)25 b Fs(C\022)-42 b(alcul)24 b(de)h(l'escissi\023)-42 b(o)26 b(de)e(sep)l(ar)l(atrius)g (usant)f(t)n(\022)-40 b(ecniques)24 b(de)g(matching)h(c)l(omplex)f(i)g (r)l(essur)l(g)n(\022)-40 b(encia)459 4189 y(aplic)l(ades)32 b(a)e(l'e)l(quaci\023)-42 b(o)32 b(de)e(Hamilton-Jac)l(obi)p FB(.)38 b Fa(http://www.tdx.ca)o(t/)o(TD)o(X-0)o(91)o(710)o(7-)o(12)o (595)o(0)p FB(,)22 b(2006.)p Black 71 4352 a([OSS03])p Black 101 w(C.)k(Oliv)n(\023)-39 b(e,)25 b(D.)h(Sauzin,)g(and)g(T.)g (M.)g(Seara.)33 b(Resurgence)25 b(in)h(a)f(Hamilton-Jacobi)g(equation.) 33 b(In)26 b Fs(Pr)l(o)l(c)l(e)l(e)l(d-)459 4452 y(ings)31 b(of)i(the)e(International)h(Confer)l(enc)l(e)g(in)f(Honor)h(of)g(F)-6 b(r)n(\023)-40 b(ed)n(\023)g(eric)32 b(Pham)h(\(Nic)l(e,)f(2002\))p FB(,)g(v)n(olume)c(53\(4\),)459 4552 y(pages)e(1185{1235,)e(2003.)p Black 71 4715 a([P)n(oi99])p Black 138 w(H.)h(P)n(oincar)n(\023)-39 b(e.)28 b Fs(L)l(es)f(m)n(\023)-40 b(etho)l(des)28 b(nouvel)t(les)g(de) f(la)h(m)n(\023)-40 b(ec)l(anique)27 b(c)n(\023)-40 b(eleste)p FB(,)26 b(v)n(olume)e(1,)h(2,)f(3.)32 b(Gauthier-Villars,)459 4815 y(P)n(aris,)26 b(1892{1899.)p Black 71 4978 a([Sau95])p Black 124 w(D.)31 b(Sauzin.)46 b(R)n(\023)-39 b(esurgence)29 b(param)n(\023)-39 b(etrique)28 b(et)j(exp)r(onen)n(tielle)g(p)r (etitesse)g(de)g(l')n(\023)-39 b(ecart)29 b(des)i(s)n(\023)-39 b(eparatrices)27 b(du)459 5078 y(p)r(endule)h(rapidemen)n(t)f(forc)n (\023)-39 b(e.)35 b Fs(A)n(nn.Ins.F)-6 b(ourier)p FB(,)28 b(45\(2\):453{511,)c(1995.)p Black 71 5241 a([Sau01])p Black 124 w(D.)k(Sauzin.)37 b(A)28 b(new)g(metho)r(d)g(for)g(measuring) e(the)i(splitting)g(of)g(in)n(v)-5 b(arian)n(t)27 b(manifolds.)36 b Fs(A)n(nn.)30 b(Sci.)3670 5220 y(\023)3658 5241 y(Ec)l(ole)459 5341 y(Norm.)g(Sup.)g(\(4\))p FB(,)e(34,)f(2001.)p Black 71 5504 a([Sc)n(h89])p Black 131 w(J.)g(Sc)n(heurle.)36 b(Chaos)27 b(in)h(a)f(rapidly)g(forced)g(p)r(endulum)i(equation.)36 b Fs(Contemp.)31 b(Math.)g(AMS)p FB(,)c(97,)g(1989.)p Black 1898 5753 a(114)p Black eop end %%Page: 115 115 TeXDict begin 115 114 bop Black Black Black 71 272 a FB([Sim94])p Black 120 w(C.)21 b(Sim\023)-42 b(o.)25 b(Av)n(eraging)19 b(under)i(fast)f(quasip)r(erio)r(dic)g(forcing.)25 b(In)c Fs(Hamiltonian)j(me)l(chanics)h(\(T)-6 b(oru)r(\023)-45 b(n,)26 b(1993\))p FB(,)459 372 y(v)n(olume)h(331)f(of)i Fs(NA)-6 b(TO)28 b(A)l(dv.)j(Sci.)f(Inst.)g(Ser.)g(B)g(Phys.)p FB(,)f(pages)d(13{34.)g(Plen)n(um,)h(New)h(Y)-7 b(ork,)27 b(1994.)p Black 71 538 a([SMH91])p Black 74 w(J)r(\177)-44 b(urgen)31 b(Sc)n(heurle,)i(Jerrold)e(E.)h(Marsden,)h(and)g(Philip)f (Holmes.)51 b(Exp)r(onen)n(tially)32 b(small)g(estimates)g(for)459 637 y(separatrix)e(splittings.)51 b(In)32 b Fs(Asymptotics)i(b)l(eyond) h(al)t(l)g(or)l(ders)g(\(L)l(a)f(Jol)t(la,)j(CA,)d(1991\))p FB(,)h(v)n(olume)d(284)f(of)459 737 y Fs(NA)-6 b(TO)29 b(A)l(dv.)h(Sci.)g(Inst.)g(Ser.)g(B)g(Phys.)p FB(,)f(pages)e(187{195.)d (Plen)n(um,)j(New)h(Y)-7 b(ork,)27 b(1991.)p Black 71 903 a([SV09])p Black 150 w(C.)32 b(Sim\023)-42 b(o)32 b(and)h(A.)f(Vieiro.)50 b(Resonan)n(t)32 b(zones,)g(inner)g(and)g (outer)g(splittings)g(in)h(generic)e(and)h(lo)n(w)g(order)459 1003 y(resonances)26 b(of)h(area)f(preserving)g(maps.)37 b Fs(Nonline)l(arity)p FB(,)28 b(22\(5\):1191{1245,)23 b(2009.)p Black 71 1169 a([T)-7 b(re94])p Black 135 w(D.)36 b(T)-7 b(resc)n(hev.)59 b(Hyp)r(erb)r(olic)36 b(tori)f(and)h (asymptotic)f(surfaces)f(in)i(Hamiltonian)g(systems.)60 b Fs(R)n(ussian)36 b(J.)459 1268 y(Math.)31 b(Phys.)p FB(,)e(2\(1\):93{110,)c(1994.)p Black 71 1434 a([T)-7 b(re97])p Black 135 w(D.)35 b(T)-7 b(resc)n(hev.)56 b(Separatrix)34 b(splitting)g(for)h(a)f(p)r(endulum)h(with)h(rapidly)e(oscillating)f (susp)r(ension)h(p)r(oin)n(t.)459 1534 y Fs(R)n(uss.)29 b(J.)h(Math.)h(Phys.)p FB(,)e(5\(1\):63{98,)c(1997.)p Black 1898 5753 a(115)p Black eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------1102081820642--