%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: ARNDDFU1.dvi %%CreationDate: Tue Jul 22 10:21:16 2003 %%Pages: 53 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX10 CMMIB10 CMSSBX10 CMR7 CMCSC10 CMR8 CMTI8 CMR10 %%+ CMMI10 CMTI10 CMSY8 CMSY10 MSBM10 CMMI7 CMR5 CMEX10 EUFB10 CMSY7 %%+ EUFB7 CMSY5 CMMI5 MSBM7 MSAM10 CMSL10 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: D:\TEX\MIKTEX\TEXMF\MIKTEX\BIN\DVIPS.EXE -P pdf -G0 %+ ARNDDFU1 %DVIPSParameters: dpi=8000, compressed %DVIPSSource: TeX output 2003.07.22:1018 %%BeginProcSet: tex.pro %! /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: alt-rule.pro %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). TeXDict begin /QV { gsave newpath /ruleY X /ruleX X Rx Ry gt { ruleX ruleY Ry 2 div sub moveto Rx 0 rlineto Ry } { ruleX Rx 2 div add ruleY moveto 0 Ry neg rlineto Rx } ifelse setlinewidth 0 setlinecap stroke grestore } bind def end %%EndProcSet %%BeginProcSet: texc.pro %! /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/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/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]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/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: texps.pro %! 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]/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[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 %%BeginFont: CMSL10 %!PS-AdobeFont-1.1: CMSL10 1.0 %%CreationDate: 1991 Aug 20 16:40:20 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSL10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 102 /f put dup 105 /i put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 117 /u put readonly def /FontBBox{-62 -250 1123 750}readonly def /UniqueXX 5000798 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd800bcb325468c79ffacb9cd3036f2ed7bb65d c6f12e33dd6d45b61272314f98990d08c0f8274e999f4b46af4b5c5da923683c 960face50087915d2044d7211a18e0c22967ef89552662ba990e2dfb9267e09e 354f41e74d44bd80a2d905d2edd60d8829e820c6ab0fdf322964c1ecb6569818 a465d5266d3366d748b412446e602f0cc52f3cad2213ff6ad92c04f6ce8bba94 890fb44eadc41a0c42eda4fe9722e6d4b17a3f6d35cf755d0a1c90d0c2033b16 ba76c456137acee0112e35edfb7206a9cb18dd9274401c70c3039059a583940f 44cc19ffed32940fe8203292e103d2abcda173df85995aee98bf4a1f65ead55f b80bc0c737589fa10cf51999c634e05d80a7dffb61433bbdba17ea9a7b92f763 586105c8277d6a961d092617c64745450b7df4efa70884f6017d9b0331814cf7 02f1140425eb8bf4c269ec042fde895a7f625c4832c1a143a14abeb798e23b59 f88ebd1414a66520435a4c8f5dd7a548a7d6a2fa96f0995bc0fbcf91b1a1eb71 1026a99d2404b2103bb3626978385830a00a02b6aa820cf33785f7aa45e267e6 ebe7d737ded351c1d4e944733feb30a3e6f81c1a7b74ddc9148b46b02a525539 c54a7eb25dec18807557b6ea94c36db0c91121160f61bd37cff40bb2b11aceb7 5f146be38109daeb77803d8f580f9086bc460e5c96f764c1254e36feb77c48a4 419b9a9cb5ec916b73efcc4c5a698909d5c98ca014b7c8e2ebcde195f5ad44f5 75bd2dcebce29979618bee325c726f5d4967cf0e3f7d35058e04dbb7fbcf60f1 436fade30418ba3c7dde0b6aab803140e6aa7c9808ab4d3d31432f5d4cbc007d 32d7541ccb867f6ccba8bd18c3d18524a692ed2ad29db54d7ca84a618052d3af 3999354cd44c306a7b8a7e1c6fb384bc2bc9995bc11001daa5833aa4add9def7 6e61a035fab5ddbc6e75a49ef0958a8cec616d08d04a69dce12d2fcc33f3c1ab 13f949e390d3d4893ab9edd3803ed193d9214120380915b3bb60e6f04d2c5955 37e40f0cca171a2efd0d5322e4ed824fbb2dbb186edb589ca98bb7cdd8138910 c60600146ff457dd3286bb292a7ca72d7438f6651325e557b1c49787e5497a9e d7906648ec13e5ffc42071576fefcedc505529ce084b92eb0159ba5ecc76411e 58194edcb5da0d08ad8ca1055ee8746a3b27089b84fa57508dc30d6332943627 98530f5bad51339c8ef812fe0edaac024a7d55a9351367f4433a1f3dbb77f5dd 67d1c54e87a590f21354a4a77914dda8add279bd50d1b2e86b03f2f6d82d5332 4afa642dd41227e7ee238d68fae89f2b1dd430baeb0d8756ca87d17bf2ea31db 25a30167a3cc357deec0e411444caadbe3bb72b3f2e0392055430dfb2a2bd2a3 7f254d287bf3c2ec3a98ca616f7cf26ede7ec8543c66b35eb5fd974c58540b05 6e2616f57b183f59ebba7cd7fb3222566cd080af0b3a9efe743ce1145075a964 b77a3e60cc38195ed2f0579f6e32ef82f9ed9e1abb12f5d63de69d677bf13c5c b839f7989f33f5ef04445752283bafbbacccfdd872f20f1eff30b9750933ecf0 408658417f60cfee3a221de658b51e9751ecdd1a0cd44df38617ba9c6f499d69 d9855f5a8c89d0397bdc4f7ff7dd5db32cdb7b64ab464bc40ac0fb23c84c4541 3b03462691cc8cc2bcc780c43759ae0c560df6122d993184b7219d0ddafb1012 c3257f29e935f92eac609dcccbd44c924370a45a19f22020f2491024aecb23da f42da4d75450362cc7534b13cec301482bed8753574180102b428fd4cc34158f 29cd004a8c7c621a1a5b6c282cd81a4d890b10429cb73b630e208e0f3470bd3f 95c7dde9b5ff88a59144c5d00e8d3d304ac1144ced192c7cf750be0adf752778 dcc7a28e77c06149e3d8b290fd2cd3aa100c976aeb79f1305787e2064f508d81 d5a4c91748e081523f980988eb1853e353968d08afe855ed3f558d672db50c9a 4ec795d8ce1611c6d22a423a039567f87991772c34c78b75175679b87a9955ff b9ba5267e2b14175a37886285e1387213bfac9d4db5c646adcc7c75aed4f223d 28fcbbb341aa41adfb021011835e033ad3d497b7488056f5483ac15d73890ad5 b2162cd4473761efab7567ca447fbff5fe26b79deb5cdcc63bef888d2cd5d37c 2efdcad81530ea00a113cb06eea17e8f94207499b811cf8cf9be7d8bacfb4f6c 17705c3d397b15bc2434c50b8d2ab54bf094f348e45d5aeeb5e98bff182a2a6a 3b9f51e5513e59a2e5907a9175833c10479b7637d254431613af7c3f8093dfea d57ea143509259373028b86cac0b0c225e960eb8763a18ce7dc80fcbcda62409 3df7aa2c06270b5f51a9012a7364461e5bd574fbb352286fec3691244688a9aa 645609c86daf8bd648e87afa0c9915304b01ceda584fde8d41344a692b522f25 8a1c7fecaa0f190795acb775671c9a203ed8de01acdae2919a218fe715558bd9 78ad41ae7f9710e4a74f600503b6ba9c016ba03704710ffe9906a2b0c94398ff 361003686f1eebc0278bda28c9f28b0d7950175bd12fbc9e63e737fb3f95c23a a9121d54ce1182a6625887095532c658ee9e0a41eee42bfd63f89a5e7385a732 8b80cdc36ea0f2a36077dc2fd29332a1e13e7b86dc129dccc259b174db0a7635 a2d6c73cc5ca66e494c007b2eecc3bd9016007b938d25bb1b5d456360b94c119 28d5b27620d54d3f9a92d72bedc5b4f5f3b3bd52e9051e48d1b820fb967d7498 42acb691edee8056ccd0355add675dd9e532b8ab005bcf69d600a0f37190db82 ff483d2fd0156110fafbed5016c48cf0932cbbb69c288a 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueXX 5031981 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e3200f0935ea98f1a0cfaff49cfa2 d877ff805620cba84126dfd175a82e1b4c6ec58461fce97cdd749a012e2c4243 86b199a97c306b282350cbe5af90cf1d8e7371dcaedbd5fa3346dd3e558072c5 fd51a2edfee6890f62efd1b552699123f4f52c2817dadd5ece30b3bc178a3ef4 f45fd8ad0ee5bf30db5f9eee9ddefe249f3c349a886cf9130cba68ae559ad719 c439930ece1d52d38366116c390cbeabb8fdb36811cf3b8e2416825e8c880b24 6521ec6880ebe7f127c8741c756b29e1f89e3fd5d638a7bbade8c3a0fdbbf7ba ec37cd372afb2d9dd25d509726d20a05914e418f6b7e759c440beb61519df02a 06734ec1226890d17c51f25f775e6879fdc231bc4d7fcb4b33d04fa273bdb8e8 f124171714614f8a94e98a6c2d6bdc1254913999f9c181e42c0886556cdf48af 92a265c1eb06bb30fb2745840c17bd31c7b8bdfd37dfb9643a418a2eaae14094 6752e0a315207aef85c3c2630538907971ec1b05687208571a3504bb3369b5c8 9a43e5abe343d0599fd8d41a1bf01de49f5856bdd67d63a2b3ecea75d721a6ca 0c8751ab41e892e9cf674218dd44acfd40019bf59f4d69ea40d5cf9068a6ff96 9f8cc772b920baf17334833d732431a2969b9a30fe38bd44772cef653935a957 af61dd568f041847ccce3c587656cba7bbf46d8cefec936c6d34dbe51bc04929 0440354f3b29ed8e6b4f21ad53b5809779d67339f284563d54ed4d2a504b014d 974a31c0ceb57b364af519bad4e85c41d521a3ffc26dbac555b42dc191367152 1d60c30b95b5bbafb3c7d87a1c043ea5f0e6dab83a4de99a742e75f02837b923 c50eb478ac4131ee604e3d7b784bd8c93eb9430291d452c21be0495718546333 66a602a9f142a563fdae3c8d40de6a8aa581ff3172367ca325e6d23102c5c6a2 02cc99580287dbd76a05bb4cca591b12a9dd3d602d7957376ec30b16051a63c7 e26ddf009e21fdca5a31cc973f0b1f6967a66b30b973ddd9d9a3095b673ea8f8 5db8dbf3acb7bb2d85c04a0ea8853e4592e4f8e581adaadb78bc3ae58a2d4c80 3b09839eafc7bc3e7cdf784e852d6590ab1c4b7558e28505887c6858072f09dc 2caba91b6871920e5256b602a0de02b87e5eb456878d3634e8eb0e069fa2bb5c c10d5cd98401de1cf07ec9f7de62aaa20b583e2f7aad2fb7e727ebc43f712c74 e8eed1fa76133428ac71589b933cc37ad200c516045fc1a06d0086c541103b5d 4ec54dc77b 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 66 /B put dup 84 /T put dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueXX 5032014 def currentdict end currentfile eexec b5e2457af4a6caaa4536453c498a618ecdce7b5f6f2d5944265309f833bd3d50 3df8bd14c071f404d2177500681a801856cb8c9c9cf0112c9bd17ccba9ab8944 1759a320b2078df0fafdd3e357994bd81740d1a7522ddfafb75d58f5def046f7 084d10fa2f702e570b09a7073d70efe37b48c355a7089772238b7b4538c4912f cf6703a14fa416057ea2e1b94455913dd271b42846ee10d78270a8118e7e4bcb 8deb048bb280c240d59409d47046c890e5f79e5dd4b0be65a28a77aeb53dd33e ac506a6da87bb55e329f433627e06c3c94b078838b908538d93f888e504e25e0 3a8ccf7d2c02122ab3d485525564295390d1154e15fee35edff4eb016fadd32c 26d1b4c09de87943780954cebc4965aa890fe1f80b03eb07103e13d468245c7b c26455f229e8859585c6eeafd9474acdb27290fd62950dc52f893c99a2a614d4 2a7e3618266f3fb0c68a01a30373e048385ffe7dd5844e208a55d1a9263ab2db cb9b19038724f95af00349840157be45cb337871b55a304b87ade64b10ff0eba 21aa0eaeef74aa01f049303a8b25de0acc26bb3a33fcf267f479a2a73f7dece3 2aa23083687322b417fe86e8f5afc2136d045ad6b392364811a17630a94ff097 038eedeedb8da4ebf8fa5a96ce27a2ebf771d14f4a11fb1f9f48cdd207209934 885ef9118463552984b0fd3cf15d55de967fbb051549dd65f60dfe07f299690e 58252f032bca78935b7d2df852ed688bd13dd171b91cf7bd46d5433aef9f50f1 41a2517c4912b837793f84ddcf1116f70f7e9b1e6ec752dbc9183d4791d92602 397bafe29ac6f879ceb72c673c8a1aa0135d6f92d0db7791f9a2f1a5b55347e0 3e63723efd258754f5f4a9ad35873a98d54b6c54b17a39892dc1d38660597448 fdba9258493a0db0d560c01d49f2d889a99eaa8a5b5919cbd33f6b3d97bf27b3 e0b94b4dcda24573dd2df8f3307fb09870dca0c3e60150e64234204155d9e5f7 aa79dabe87315ce570ee206034708f56f23db2901c8c0d907b9fd41b16405d2e 0419506d9e406f27263513f1b3a69164813a6baa7d489eab955cfc82b4a73b24 54efda69e2536137fbdd784b6128e36279cab174a4f6386801a8d464977b6f07 a8e3439c9eee680ae9088e7695df125bc29b942eea32c54ea3d36c9a168faad9 d7902b265318d2447ad96a04c713a451e2010df358a8ad9e432f18ed4334fbb2 de3cd55f57765e2c55891bc441556f2d9312dd57c75fff381395d1740220fb8f 8e47b45e8f876ad15a70111ebc76cf03087d51b62f9400a5f1415acfb5f137ab e199b76d339ed7403ee044376a76892a7c5ad8ce97b8f329f7c7fb46846c979f 60efad69db49bc1d9537911d857d33ae455e4edffd5c87324d863d3a5ec3f792 214b4a4256b4e9cce002d85f076369e230e2317ac2d9d79e7c46b4d8f1cbaec5 28337fddb736b84e8075f0d92863b8b94005e5fe064ecac7b95b55b24f388779 8aff5400a6dab446c96be5190b8a16ae28aa4950bbcbe5e35a92ea90195401cb 83f184d54dacc40f8c3916e0ccaca31bdb720bc82ce5d79c273c25a082501508 70ef259908f4336d23f1fb9424a116d616469f62523fc6108e1b3cc003d19d79 5c69c34f7379a762cfff8e00704b1473dcc7e5f2b9a37689ab1520a222ef8bd8 6343700a530178773290aa77ab54a6e1d7df9bc1cc0af4ed15323c2b06b6d0dd b68f29643c9816217c98647760d62286d6a447ec74a2034ed370fb98b6480775 b88dc65b68a45b8bc5d0687fb39075291a76f1bc8a2589d3adf9e247d6892b72 eed99b23bd0b5b2d653facc2faffe99fd16b9e243b9a0e78b1264c34f2857c85 c8ccf11acbafcc63b17920f1e9d7da5db88845c170abdb45ab5cba313f42704d de128c05f50609b869566a8c6b43dda7aafd79dacbf0e97df59dc908df1c17bf 4f21d22ad2381df2f74d22ce4cf8c11d37b55cb2e31f566b361cdf35fbfbf6c4 91ae23aedee8401c646741745a12a09951385fbe99196bf8e9d55a3895cb9922 f9b1eccbcfa3471d8d827941bc23e137f2749a1677d61bca802fdac4b8a882ec 0d6024a706f4939c88c2f9c272f07074eb0e5f976d11c91c9c7661bbac6e4e04 dee6fafc40eaf1c7375e67afe580cbede551ac1b1cc44b07b4ffa7805621bd47 898d0ace4314f974b9563c0d1efcfe9efa8a2e53ea6f0766617bcf57ae7ece19 5db85621f0f5b2496a489cd306b97356e76d0ac29d9543821e608473826e83de 4ba72c39f5914ae32b30e8a43e405b92243f9f60b152264fb4570fa1ffacf3e2 a22ba8d99251f4f6aeba04717bb353e3a06f184a8ef3377ac003eff38835afb6 0b9790080b469d4fb726959266b529033e5a287f696e846897714749d7705469 796cf1bfd619196fffed7bbe47dd3626003706b3df4a78fb57d5083090a67434 4108f2026b7c03d545f5404f67beca038225d0d79e2dd598 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 15 /epsilon1 put dup 17 /eta put dup 22 /mu put dup 27 /sigma put dup 59 /comma put dup 72 /H put dup 75 /K put dup 78 /N put dup 96 /lscript put dup 99 /c put dup 100 /d put dup 105 /i put dup 106 /j put dup 107 /k put dup 114 /r put dup 115 /s put dup 117 /u put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueXX 5087380 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311e252234d51741e18db3a68e8ad10242 29e5817a10e796a78d2c7f7c1f50961b9a57aa604c9f821dbf5a9295197bc666 31678d7d2c7e1f8f2151ce0c29efce46270570f4301c5dad1b38884732e53dad 05dfc36b020e726ca575f71f04abf2b49e1f1d6d2a08a477658ca9a48f1c8cce 14382f0a42201de56e2821ee2b0e91a818b1b753d810ec2428dae5153b66af66 9442192024becf743a8f6264a58a0b3aa823510a4957d1bab3e1b83135362888 3e392f49926471a8021fdb48637baf0abcf31204b1b214f48c99f44af7782323 69d772ae4f49be53a1bf584747bf29b94f530d59a5be5370b8dd0691e32a21ba 9d0d592cf3bb50a99a3ecd18d6c4114cb4cfecc49705da90d7edb10e3a28f9eb a49347a09d31eec896f2a28e9607d99d42914aea16545c16e136bdd0b75fe486 61e24c7a9e9b233a76754c760822d0a88c575e08ae3e3f081d7bfb80ec4c03ac a08b65fb6474f71fc95c7c2462f2eb3754e647c29498288f2ced0bd3a2bca5c3 2f80a61fa667802d376cacd34de1f9572d1b3ab09d6f352750db4a768f869638 a87457889aa502850427b40e1294c5ba33e6b17532b2d13b5f50e71582f93daa 231e86597872a2a968127afb0f36257a76ce1eb24e6c4f29fce2aaf6532fb5cf df84da03f03a45b915209f67d398c7c4f7a3b0623750259087a478dca684b29e a8b7f52060bae9394369ed1cecc645d113cca3772a8dc36c6d916b5c5e65454c 0a188a33395fd3da67ad0a31ddab6a8434dd80f0bd0ea1a2a62eecc331d88b3e 369552daa1c575a98fe84383548626b65be7c24ae7eb61159f149872cfd87cf7 6c1a7c155c7186c7600e58cb61b8cb0c5dea98268a39e928bf6fc53985eaa763 2b28c1eb3d82e4d3fe225223f4266d2b6147e344aad2d4884a9308aadc7801af a606838925e3e50ee7aaf73f090baefcfd4e34b795cd92d83c944d91a4940f2f c955d727ea5bd76f0c8057b12621eceb0947b0a5dd10dbcbc52957dd978d7db0 c794660afdd4e946b3d7fc1ba60907fb3f756aab0406c0e7bce5a2238fb3e12e 07e2c0a7f465e90057642d704434ae0b57bd46c8ea91ab2a386e24c6deaf9cb0 8cdaa8fc4ec3fadc06bf28659cb878453db16f4b03925b2127b83be8fc240f33 23ea950b51d70c8fe861e30c346efa0a0125137598caca5478991e35f5bbbc19 4c787488ba339830bca15f8333431902e82b66a514a41c0bff92719a6a0d9587 4cff62c622fc61a0308a69f9fb7ef51bb274ce9e059c04b64e817abdb810556f 1ad6e9d9dd1de1cd971107f405255d600457c4e4b3d2b6b285a186100471f995 d1fbd0674df45cceaff5d26a386bcb542f1839c18f9343c98991e17ebe95e8ef 2120ed932ea73c03bf0c26b0bea5a3125f3e7dfb144916ce092f6453c6c4c863 4e4ab40645b5b9ad4d8dbbe037a401c69cb396c8405122b9a70b774612357671 ad78ec77ee30ce3095c40acf6960833833de3b9d0609c6077ded13fb6b85a7a3 b5fa76c70f9df3ee6c2b04f033607ded09a2e013185e2775d73b484450a3587c 9a423b4884c89a5e39c3cb5f2be913f05395766bb9d6a0cecb8683cdc3fbc8fd 865d7f5a450dbaa0c82a0aeb8fce61fbd582f0dbece0c1fa9c6c15dbfd174e26 6766912f814ddba1b06075988771d7afe5471b16f149b570061e2ae61e6e622d cfaed0ba3a685dd17194f32a5dbcd27353a43da084f59a697320f077ec1d261c 24555a1236e10c68736990d232595df30eb55432209f2b2c44484ac24c03397c de8796f662a974de8fa2de38a791c5fe74325639147200bb03aa32d513ba1bd4 d3c7a22e51bd15a4ac0040d73c27d336c9e8f6574bfc6c5b350d1cfb46e45b35 b1aa819a2621398a86382d5518c22177132fb5d05ce30c6f209441e9c7ab7bb5 e34a648339f6ca594244b8631bdcf2707660ecdd37c0ce75674295ba3098695e c77a70fa51cfc193c1935949b0fc04756fd09fdef94f0a86fe5532e73bcf6453 8a056415d1f2212062ad5679ba19996239a0a97d419928dcb236f467353f01a9 af5fa48e1ff15ef3b6098810fb6987ed4ac12aa46c1f46182495ba6a8454516f 7c81364fa3e1c5354ebcde776befd4dc71f53b7598907837f05bae91249af5e5 9597fc1b6b9177c0c51e98384427a68d71aebc94b5bbca7f0e1ec4a722783089 ac4e5d6f3af03a08a3b0de90d8e51038a18498b589c3a87cc01a69e98ff5c6af 302246bea73afd9d69ef40c4e18f709471cd35abdf4291a2b6d1cb71d0505b10 1dbe8037d9dfcc9627eac33c43879575e87e78f831187410f2967babcf10fa20 b1f2e4f6b0efe319bd05367e9fe7d80d984f0e32e66e001db795284df09ec8c7 9304cc28d22cea6f250e0ba6d51783696571235869eef744689c0b39eef33fe8 36a9a67b972c1af3bd5632e2218c570e934bed74693fa5f68c77f68624dedcf6 1b5df9db27bdf1b53bbea7ac39351bd30691abb68c81f6862ab3cb103ae39b91 bae4aea360e36e1fa9242ba9f163c845f251c7bc92d251abebd303f74cfdacfd eaa2dfe99c98b1aa124a5bf54008766277ec983dd25252d0ef1d24bb81f8f8a9 d2cb4aa04e59b1e93839ab29326c3d567eb206b5939c077b1b782ac38b6cd991 74314693cb948ec61e4c12b07772f5a0691d463ce1cbf80972db689817aa292e c950e8cbc5fe219ccb9c2325f409bbdb2f385e28de9bb8f938eb55562f49df77 30b7c04a12b4d3e09947d953242f61ae806d0b4c40e3840d2dcff87862435d81 a61ba1c751992f060f838847d3bb809a43905cc0c3461be58dc115a2a37bd4a0 7632260c74cc2b008df3d0d3d140ac346fdefaa5fd5f0fc57db78fadf6c7b8ec aa3e26ea6d285731326f965daaa189f96ca7e44c6e7d0efde1b5a50bf7c4559b ba67fb66321ca43b65eba91de2e85afde05b263f2297fd246600f951c278112f 48152db69f1f6a3343860ddd0641aa09d5d5724610831295ecdc6b88fd46f812 5ca4b57b035f7fc1426a7729e5014cdf7c6e177100b9b551244047cebf0a7275 4a674ab622153c1ba0f0c53e672bf39298cb278fc1c6754668c3fae0fd5dc80a 8cad38347e36c12243cf8c4445ad13bf1feef15801aa4df5f43e9d8d7560e0e3 ecc9fedb4ca3526bec2d7c0a205b8d75e665766eedac5fcc64022e0c595b3711 7348e1c9d0dae244477245398ac9b88c9eb92518b50b9c8ff963952cd7509d9a 2d6e3dce62056887355cd27d7bb50e1dd4740ff2233f3dfdcc0d96ed939ae190 33f0bbc0f7ea2df29cabeaef872847d554f695829588b180714e13704ebf1922 372a335f7e4d89290fbdaa06a7a74d478c3982ee59feaf946d0713817ca2855e 2871a2aae23bb30fbbbdce238f22e25cd6c2946de8bebb03a8ed2e262245ca63 539621a9d19f781a890afa183f5416a0a8d331fabccd6935a4a71c442109fcb6 ee333460abcb21b4c654895ae37e7942de5ed69dd18f98be0a2327332c4333af e13925384affb0cc82adb29c5a5aef2419bae28d8bb9907c102801f80a1cc3bb 056008c937e91da6be0874fe1c0532115787d608fd6bcea6900146308d520420 12c3d4835388dc6a86c44c7891b350f5ac36dd0d207cd4bcffa1881be51d8bb0 f12af86c2f5aa2bc137062082a8044ec22a8d8088796b4bb94a6bf82233ba729 6eae49e404fd57a2f94e0de3461cf64c5a2cbc096b9a9a1de3b3b86a1579d8b1 66b5eafcc7516d2fec23e354cc2dee53d414e4b15870be487838792faf7bcc97 8dd1ae1125665d7dced8474a3ad2104d72abf1f58d6ba6e33351b302a848351a 958e42618e5013bcf97f8d7c10ac8bc5f4b375f2db467ae00859daa09b17c19d a796b0a3195f98d7439575b50a936026f80cd86bcbe934d26d9a94e58cc4d895 d35cd4b544e3cf7e7ad647bb8ecc0e0b5649834f7f444988a90b72a812af04e2 95d892985767b0ce11a412950c4c14ac66b9d3a5f4ba1fb8ef954e1b6b2f5f98 f2f1ae7457121baa5fce037a08717ca2c668bd75a87cc4a120fa9e95a07407a1 f19ea6cc7dc7d63254e5eb98218add1a3491e2624165cb9e82c2e449e611e330 fd0a6f70c60a2ca51b0f15197cf082f86d8f11d3ddc0ad649e16a38610916694 631dbe1d9f28eeac4f839e0a2b5d26856a707c5f5940cf360a847686747df5ae 32986185ff62632b3465eb62bf2b539b35e4a5a24ddb96f1090ffa6505b9d884 e2a49aed14cfc1d03b52a0d8e7aedb080ea135662676ff151f33c4047b0ae35c 12c35c8a65cf6d48e415f4601d4dd98efae2008a638ad884ea5beeebd33deb3b 4a96403ff8326772ed2670ff590de21a4f4faa00ee2d3e1eb5fe438a7df59e25 9ce47d0602b3f683432a96978feadc1c62f3867a89648f009a4d0ac7f7dca291 5ac6b9e05b9e6226159bff5364bfb2f46a57f50deb6cec5281dc295307e57818 0304c5d4cf3e7c95cd729cf38d2596d9d8df55b0a8efc13bf3fe96466cb90b88 ff352c1c07e17c9f5d9303a8d46fe17bf68c12fa062519b4311df06867c90cb4 7c18f869e5cad0cb7cc7a6c3b9b971727fa03ddd7b8cf4851aba2000d4a0b2c6 3f1011b8c5c9a7db2073600432cba1bfe84bd8f46e9f03322b008a8186fb69f1 61f2793d3788e0be26a899da143ca6e448adb735a730328cf6b33a5f3db57d29 e698ef4a5b128cf22dba0d69eb02dbcb959b90ed6c6626b0b1da357a30f337fe f72e88ffd6e68d5192a7ab1e2963f583a50db77aab76149902101f2deeed113f 6eaaf2f825d448c7c43be9cff3eff78478209ee9878541fcfb06139365f96dbd a79101c4be9b1371adc776cbca536081160c54b82dc24c73bd5cfc0c8201dd9f b5e0fbf95892eefaac7c25e93c64c7fcccda60ab1de94cf3c0326f1a699ed909 3b0053184855e43e478bcc915e330683d7b504d786a0418ab27f0e06a47c2331 18a84e7efd85c8d1e47b12d137b5a662c7728cf5e6eac869398c28ee9442c20f 9b5bcbc171516debd1547a8d9831eec89a4ed177debdaecfd5cb1c60a5181799 fb780dfe33e1e8d0d42cc77f35b8f0a24e96939a08f6718862700daaf4ad69dd 0089450c01fb2f730e4fba9748e9ace4e23489c7134e742b8e0a592846b2da1c c33e1ec22c053eff87fdb98c5b89505a6e895162fcc136d4bb736a18b5a6d44c 680c90277b831678b1921fdd275cc7c93736ab499e6b4103ed8a3382b00105fd 8624ecf4e8121a631b9a42d48d68fa4d3a31037f13e6384e617e68c77e42a40c 50182bdb74bf2b3d542ee81078518795881706b7272ae2c3d80a562eb866e388 ed0100835f227dd08e52ee11da0208c82ecdbd03f314da82ba2e6eaa02ef8eee 9b0619c823b53b618f440670ad6fdd893ff0ce23e5f8e32d72cbd3e400abdf70 353f212f370354191b16c0007b78c051ac4264f6c66d08f3f02e848d2643d42b a7fc3a0d6ce91a14a71208e2d1089f4106fcf4da210485a14f7114346f90071c 35dbc9973fc363efb37380d7cdc604f165ea9abbea1e9970496b728434640aed db01a49e27b7f2226bb0c5fc9ee1ef8c2df708011ecfcac9713e287d68d0fca7 6fb90016f34edbfca60e15b60c9d10d447c9831d23bd79833014371816c071b7 ab77d87b9966af88496ed79ec27260563e3feb7e92ced0ecbdf6be15cea684ff 31722ba4149d4c594848fd4af4034fb09e332cd4f50437b74fc196e68cd686d0 2fe845ec3cf0b2c9d338b77e18d5e0eca9692494621a6f40b9862f6af68b4ddd 8cad8c6bbfe1fe15ece514944b530076be39819b60e2e6c8922525124f920eca 99f6d3571dc9274f152e635c565c2871f083f5363eb8f05881a5b928a6d9a3ae 148e0c6d8ef9cd82ee4dd190ab7a799dfd65408cd831550617733ea0c596808e 8342cdadd013d1a17b20fc4ff25869d64316eef18a1fc64142838e519391630a 7fbd3a7a2569759a60c4b0af4b423db61d2f6496f46d25a2bcbe04e57a3ab3cb 58c06d31876931d543e31d92929a1e8a3e148d00bac02fb43d4f18acbf196a43 72e248f8bdfa739b9f0c6569fd6c73fb9589b5b1fb5f63998dd21865b88a4703 d49b4abf2c612d310e8df3c36ffcdfcc8b43e7851bbc27fd1d2b20bcba5e1bb1 86e2754268330f6c18353f24a59dba55c4949b93e795286f39b7e19dcb5368e5 496277aa86027bcccc20f19defd128793f0d1ef7851578a9b049b7940b27b462 dd135927ed41f00d5030815a4ad35664e53de146db62aa3d9f74ea325dd450e0 c6cb6f770f7be70563e4118010f2c8604f9b2f6632f992d6f4daef30da388120 bb1a2456dc9c982655697701b4acdd83226c6415cfb6733f9ad9504a1bca33de 5903021de80ea0752d91ab413704fbe1897383ca39a508e941e9921d56f2c26c b48a45c05a11a0aac2c276058a8d00556503f054eefbc213a8f88bd7d694f530 880c29958e8d572ade2a8f5685ed730a02cf36516bf3051ad7e5554d42c771ac b2c3fea3577e25e8554425d3a79c781bfa7f0377d698ea1cf5a2ef9d9a60518b 6c25d3dc2cb0db8d65367391cbb87e51d4f5fd570ca847d78b78ff4c0e7d3ece 3849bfb7375ad3e8421ba382fe757365c053f9aa55396e0d778044cbeada7b5f 57552356e5c4843fc2fa788ecf1e98a1d201d687831ba76536d3b641223f890e cac8a942c66bdfcffb0a97afcd06b876531a9fbff9e7af32ffe780c590b058da df5ddd10af3519d148084dda1c21a7497d64af64f0b5c5f3b907151588171bb3 c46fc248d7981fb60de0082ed298b1cddab8cccb82cbd5da755ad4c545af3733 5f3d2b219fd0844b0d6df5 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 6 /plusminus put dup 48 /prime put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueXX 5000815 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6784decd2fb36dec8f 7478da853762e00bc9c52ba86ef50d222ff142abc18e4fb95b86b3e57d9390ca 86e8de6347bf7f48f93322a372c5d7830521b14e8fa0771961b027c81365426d d95d16b950e216a90f28ce806244ff28dd969cb512a23e05045e7b2bcaf7d6f0 ca8b51494f2ec0681bdfde60420628d9bce98a0ca1ecca9140f452a88db19a30 ed46ae881b098c843a51d06f9a153489357fd24de6a0c36cb5a9eb9b3b64dd75 18f83018ae7cefa1c12fd205df6880a705e28457af7b869a6dd28f6b37d82cc3 e51cba348dcdb34efb119b8ce5a996f15492b0f5d34538850a2c1b6a509fe5fa 015eceaf28b0223cb57ac422cc75ea5af6535ee85ace82cd3674b7407f29093e bdedc359e654b1b8f6fd7ec00f592fe46a806ef91ca0d902c83031b8043976f4 8f7fdc818c2b9dc1d11fb997865314bc2bc382270efd30a9ab4fb8f113142fbf ea9a3a876ec1259982fa24aa57639d8815ee6d26282a68f0eeccac15b3fce189 a9637ea9599d2029c93c733252fcf45824c3886204fc496e3c41dd5395c09ace bd57e9f7ebe4660226821de01148dd9973f5e2f91269084b3b25f84e755a3982 e5336f8a6446140251e13a70a7321ff963368cffd5bc3e58e8da6778be91043f 1ab3fc605b25e2a94c37a0d99c862efbde612500889c6f072aed395e5596a088 8a9e04f40b23cdb96b601bd3574b4470612e4bb8ae7e1ab1f0bd3c3d58756deb a6061fcea0cb21de366a225d4910ae18166ad53c3cbf79c0640cfa168bd71aec a1a3e33823fe7315d7ba1e107310d4b6ad01201f425f9409a548a46b54357005 fe00e552548473c7345303cb072906ac653c10b39cd247d6806380b234e161cc dc3021d830dd434d93f09146444ced6af8d4c0ae05399e2dbb3bcdedfe4a4549 c9e716d4f49a16e2b196d46bd25db5c0afa1895b598c708e9b91204d47b2dc7c 34669a01b6e38c1dde595adc62f5e0ff230bb8445779183867d31963a81f7400 3946ec44d0d0f6c8fac561f5b07e22b31845fb627c7092affd665475673e0a6b 03892ab7b3803f6214c1deb5d998f36504a936bb2ba67a26156760e73d0d8f06 26d5da1a1cb43970c02cc4ab5942758e4b3d9879704edf18478f9334e0429daa c8b40380cbb3c4897004c266c663032c7d8e11d3530e0e94c8e3db98680316b7 d78629b7d3a4fe807c2951d0e50599715ca0d8e439f23eda656e9096851695e6 f726f73530a03c34564b821a8cda2c1c1730affb35c0f9e160d49f26464f983e 5562f84fbc66571e66 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: EUFB7 %!PS-AdobeFont-1.1: EUFB7 2.1 %%CreationDate: 1992 Nov 20 17:36:18 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFB7) readonly def /FamilyName (Euler) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFB7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 77 /M put readonly def /FontBBox{0 -250 1327 750}readonly def /UniqueXX 5031993 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e320fe93acc64a69e2abbb107d230 6f952b5533efa450ec38390709158e36e135615078216b667bd7d4611b9d0873 1ee78f27b66b665486e2597e2fb17b6fa4e2bcfcfbb4b6131b94582db80d3067 88be19598f9569571b5a80343c534ef5ccf87095e7badee9d76d4911ec99bcd3 b7e0494e74139216885be8cfc78c136f11e964b12bd8beaa565aed5d60e89111 431d111cc16be4b231c470c7f478e2d0a427a1a824df7900ec852bb72851a3ed 07d8efeaef2f108b33a164559fd4eba2c20a4b43ffccb171084b0a8a805888e3 25340efe7789b5f23bf38bf91c30c19136b0cb31ad7caf35c1dd819d7fb09970 fb55553ceb5f3a855720a31cc3b71c5aef792924744486e6545178caecef724c 6ea27e9177cb962e600bf0faec72c452ea4a8136265b23017edbb6efe1d73d54 28df489ba50a1bd735fa8481614b30b7d974fc38c4ec1d1e44b4630c860e0db5 35b233b95dc2a09514cd3d4b49162c2dbad6d077273c9510299e993652c55493 5ab0f5ebaa505c30664baeb5909339bbda0353fc1be5aeebb19cbec735802d54 22810ec1bb4e6e8d57ccbf82894599f1b96a862e547dbab0d01e38f032d6b548 8a322104ca1a379f4668bac864fa42d1c75a8fb0695984cb5c613213f0c39253 1ee0595b4fc7d7341178c2e4f3e429340338ff0d2df67924f88235277902a3c7 6b6e33c188700fa5d6d7ac0b510e5d60f2a238e95fbdca7b8efe9a2927a36ff5 baf1acf2f9d1e076d68575813207a01d82073fc2bfce45f0142c0d2ba97aab24 77bce53f291a5f45464037d0bd90ad03c99c4fec726e5d5a69aa254a1fe3a1d1 d19683f0d26e3836ba90d553d86cf2df073240736b4f2ac41ca80810d2c00c35 abd963bc03b3bc92ca368b5a6c6be34f382d4a59e1e0562046a62d81c6dacca0 53495fe8f4daab74279ac6e622b66a3f279a4429e1a59a56034a7779d61c97c6 2b83d8f4265bc0ee1907e5f72587216d835f82794536f0f9c5c1e064fd7da5a7 ed2f59921ba27a41e4a26c7b75efd7dd4131ae6542ea99ca8574a3808177317f c379c8b6c0b6ed9f86ac1211a2ac747fc534ddf669f301d0333fae3015652627 238dbfbabb4a35621097e29ae533e29b8b2ba6ae54c4e39c4fc278d67fa2fce9 d4c2d0bc678fc2af22846e65e7f15ba8d11825217a726bd145b77807618d591c e1abcba1d9181fbed5f23c68b7f5095370d7b1149d32bc50e3e153285b0d2d80 337acef40fbe476eba44fd93aba7ec7aa3a43d06584ebfc3e85a6ff9bf93317b 73d6b5e959fe8600e9f1b2fc981ca1bfb5537383c092b1ce4898368dbc366d25 ceb54b94966124d6c485cf586469b026ef586cfa761a33b78bca5edab8c86638 53ed48def48f06fc83baa8ab2ebcb009f65db238fd6b810164d483be54a17cee 014df942fcf03d8e29ac4be1eedcd5901fad3feb26c2d7b8e4482970188af5cb 51d97eeeca584766fb618d23a1d94ce1968a6f53d89e90dd52d93c4e56c5567d 56466bb41cce7d1252397d440bea401cbad18e4e9fa54b195ae641f884b23868 97882a79a55a3bbea03b069bb789a60bf732b6e64e0f6a2b92404f9a327f4576 4d2fdb5e7845cfef6672889a33b4695b612553532b835a7a82f176ef71bccf54 514822441de15395671c2453c56a26568cd4b8363b59f9cb3e8d99b6c2fd28da d2bc1349a11b257423e18e61e163a996e5736a1e43ad7b1fa0a6d4ce6c24efc0 2a8261b722325c8bc771c5a171076c030ec99e990b1616c5c7abfaebf07bd2f0 e05f175ddab17094721313257db46ab888fbf4eb69f74623a2f57a31d937e623 425a2b6dc9ceda368d71f67fe1c71879e2bab889104db7dfb72d7e9306849553 1315eac4983bfa1186ace72a45e709b846d2d5861af4745d360b8e39f14e3757 5041c4f72697af00ec75f46c4021e4b03a6e1fe26326667b08f6fb584f6b7e01 33d5fcdc9cb6c5b67bb5c7c52d115d189b796c 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] 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 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 63 /perpendicular put dup 66 /B put dup 77 /M put dup 95 /logicalor put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueXX 5000817 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f73e9b697ddc0351763500e23c4d170ea8a2d2 75ad0b6b23097f02fa803c1f46f9ac10fe0e527c431b11b1b9a50e874896cca6 17fe3e255960846bc319ca68b9b930a45d5d0fc36b352c3443ab6f7793b38d2c 7fb9c409ab7980ead55e9f8f6b83c70088d1fa6d7f2214da8928e40b96334ad2 72d1b5db9d1355a579b26a8dac21f021e393d7d837a95deb8de92ed818a8b80b 98ee9cdd8a3c57ad7d6eb7a82f43eaa6ec520eefa7302e750edc4b495f70bba2 24a2182eec37f1a1c5dc8e20973007150237daa6bf03d498826777eeebe09014 ebfe05a8731e1a421cfc513d47d673e3d94d805f5d0668ffa265e0962cb5a75a 4a93d9b872d01d1cf15c588f109254105491ed6878db2cad0a40ac6883b3b0b4 de62566d8a369aebf50dc268d21a68e409fed3e377c7497f2cf247e0c09ea052 4b117464f6f0e13b753456b6a80c1ac1766a9d58f92d08a64692c674167e1f6e de2da29bf0746b116e8d61c962945c49d43dc9f161f173e2df83a6a6f6dc477c eee2a7f6cc52e5f39b6035a00abdf1195f79d742284b7ff23b4ce0d3ec2fe616 5ffc8a837e8d19346f91f3ed749c76cf12ed5d7c89615a2297bab7ad17bfca45 fdbc0ae472635e225448e0d54baee12693ab28bfa21968c4e03cd7eedf41bfd0 dfefcfa54bd1148f44c74ff2da824b97569a0ba471fb4e8212d9b692cd5c2cfa 8d0f38b2cf1f40c6af9154842023ea0b68eaacd5763dd6918dc8b5ad316ff56b 030175b7cb595bf30c049680025ebe640ef7822598807b2e90886791a5942a40 a9e436a42e952240c132d52640e8599a49bf2ff451d4e12bcb8509c9c7dc59ba 050c469f26af5c884cc5264daae36e0d24fea688de802f91df50d70cefff5b65 9a6d762a6ab50bc75e661fec93f1cc5e020d0611f67dc65ec5d4f7ced11496b9 375ce23ad38ab9c30b63f4075365bef6bcbe8eeb76b3c9555e3c5cc174db0f92 42747aa896dbc362290819ec83a4ce53d8b5d565ac16d82d85f7f1b91b2e9897 d095e13b4d9f553e521853379148fc7d344004d06887e68800a595c976db606f 13d0494efd25c7eb0a54a9cbb0e1a55164dbd54f782ec75fb68fcc96aa42a50e af4d6532391b06de83fe39c02df60cb9c34a6188e6b345f4cab25497e42ec26f 45cc36f6520c66e39333e536b19bd340c5ff157b0e3bf9e4a48311dc88d8cb1f 164ba6644d7c0ac3c0c7e95064bd935796af3985a4c2f91c4d8056d009a98479 94b54b556d66883492e0884f766d1b7ad5c39d4e8a7d304597157b046cff384b cf7f0d8ab754878a8dd4d22c537d4ab75ba1c3ccfaaf644524c73a0eed09225c 23c03f4d8f6b648bcba26f103f89c3bc483502bd6ae3487f32a5023c69074cb2 b0c237e01a9656b07c33cb3723c8f104464deb5331fffe8c6d6adb0fdb3c2096 3a35f232d21333a7f9f430dab8be4bbda24896511cc6d49fcc29bf912d2230ae 5879832aeaf7b46e9aa726d5d44f93cbc2f40c33f497a06b0a2c8a6c52e312ea fdb51dd4aba481db1d58350834461f4eff244f29382964c34b8522b58e9a8be6 9a440834e3a0315f336c5441a6b37ac8ce7d088aaf760ada98cb0a129dfb8d74 291b6ccb03e3245616e3b9df2f2aac85d7996399a1e6f32e23a636a48835a240 da83880999cc731d7a4d18289e67d28cb44a0ac3371d4383be5ddff8ac1950e8 4ed391001c7172e88c01c248278afb3290084e33d10a9ce6b29317b27257127a 04d4a33db2ce5a76743303726f32732dbc28da963084b69712700dbfea07fc18 cf0d72fa07358970e2bf2fe2683d99177d0c930d01c28ea7bde4df6aac49a9ec 90f83ca7ec1484ca90905aacd8100eae5fe5f6474812a68d45cd18bfe2f04a09 2547fd99c422e4144a46e0bbe0cd722057381b232d5519e9c9fc48ae86197380 df4ca14c9f15a41e3b287f753b59ad500544ea712ae8b0935732a83c97fcc0aa 5ec8225afb7871403d4aba72471464da5119cd669ce3e0df968c6604f944d0d4 a06aa3cdabcd4c409a3e7cb90b218d654af1f000b6cd5042be80f924dd8fa96c 52be16ebb373b87239c8da79817be1476acad6d165e67ca110367530bca31e3f fcc750dca5394d7b5aea65693ecc6143c2c5a544e91706a4e7992b22af63b0a6 1975b1b4ee48b48254872ee817ec7fb5cda58e1728d92664c68dccc8f022390b 582a63a4849cb03566dcda5a9a7f4100d2610300f7d8eb7fb6202a2ace6c2c36 6d5b3a8da6fae2a9a405cfdb241a3903851e062687e010612a4a85461df29b03 877b27930da74cb8790df4dceb96e77f3a362bc9a022662ca03b8b8d433f8dd8 61c8a00c62de5a55a8416da215c5357578670e44b8634c98aa4d31a24e200ad1 ba80c12f6c63d526bd3c456afa03f3f5cac8d77ee6cdc2cb616c892c6d24158f 3b44fb6272c95d70a0815f4b4f8dcf6f52535dcba0e27d24e3c4f4208945be9f cf95724dbffdd40835d328cd925c1caefbbd93ef6de225eb3973d8378f52c815 f2b358b643af55b54bb36812b664ea1b82e2164b5628d5c7f37649a190eac62d c2971ee002eb87e44a6faf8ebe43dd73d63fad587625ce89af40e257622adca0 9efc86a18a4cbb2472614be2f298bd0701a10099f4e442c5bbde85fd792036d6 3035689034760be9481f29263462b5b398a58671e4be0cd1bc8193158288baa6 946eca0438ced9eb0964ec1a52af4d1a460ea31b6b2906db82a02f0361c4d070 48df647130b39074608eb634a76be7ac46e28360d5afdb5ba827b63c9320b98c ed136d9beeced6e657ec5e99a3ebcb0feef58f2752eb2c30656a21b272f8d538 cd5f9eef0c9a8079e0a39adf68b9943370dd319ea0dd711dca57a530ed303e21 0ff3294f6112578cbfd01fb7ab2687c7c123e22a6e82737b23efb63cc001ee4a 22ea6c52409aca115388aabb9bf6c71d8723c8716054eef476be600bdca9b573 b2e6bf1af88a03a22932c8a2031f08c87862443713e32fb0a6cc2aa06fca3cde 6e18cfb0cbd81bca301193bb4a4e0fc9baee1894bfd312bab83a2e8676737cde d3fb066764e0d336c919c71899ac6f7c210d093e929239602db57450193b5434 df5c4b512c951bd78f6036f4b424e851141d9bd491cb33a153d8b389062e206e e3ba7ca2209794048d81f83b1ad591514623088e41a98e854e58c4e8f7d92438 7926945e13c597c67275737b9670343dfb5da7c099da73767992a044d9cd15d9 8caa33bc8f9be141d7981a9152679a3d8a9a14697eda34607f6cac106c522b7e 0140f3cc223e45592fffb1d892e0b6451f0455bfaa71d22260d202ac0c9bbf3d ee81bf739b451f3e632f4dc7a128cf7c3f67752ab7ff2489a078f104286ed015 a6bab45e4b45c09d0a06668f8108438cf0ea772900ef01dfdee1e4ad93701322 5cd215ec7c1a62dbf0e9402794ec0a7bf63ba972cf7f117e7a4e4723ca052760 41188938b62d92a86ff84d4acbb54c532d93e4aed79b6575ebbd05cc01455a0b d805ffd1fa04da9d0b76f3ea0500eecf736b124d5e8ab93d1549731e4093365a 55a90e587386edea33d7f25069b62db32afdedbf0a9fbaffff19ee168933b41b 44fada7919a1bb030a8c91a500f7331f40c4b86318ebd1d75f0d16a7cb17bf94 e271dd23648a8bf49fa11f5b50ccf316aa98822e6ee3214e0e857daae6d4bcb3 5f4ab37e7dc388b8173d16b67101a126b5c8e21c6a5815842dc1cd2d2820b603 23f2167706abe91e173c1c08c59daf59fd86a4c45f052e8bf49e6a866faedc40 469fac4714765fa873b834c0238cfc365d62960b239d5435703eb9090f4735be 30ac2989289f2482068d6850a3863b5bb8f2ccaf63a215246465b45c7f532efa 97e86573530cd0f42ccb1af0f992e532bed01a1cfab6f7a7219af3a147f2e207 2c01eabca1cd71a8119117f502ac3bf5a6d3e8abfd741bb8fca260bfe5b0643b 17b80c8f8fde653b7f76ebd07bdded99f9d576d07b007deaaf2c9ab22fee15cd ea8f9badc9d82e2c38e1e5ce6d0603a94ccc151c619957e40a8be9265db854a9 8bf90885b6509732b4041597a8d7be04f3eb0fad7bdf91b6a4f55f8b02debe0f fd58f911b88329399d9552fe27933b603910e114d6ecea9820834a81cf5d1388 73f6a41e7e85e5847b0eded5e2b887b54481ac9bc4cc59bc00dd99199677de4a 73563fa37a169c5cbad1b4670a3a3ed2972bc258500e5aeeae2616fb72dc5d37 37f4727468a1c78dd08ef9f5f33260139706387b699198887efd3639782edd4b 066cef406e1bf9ea6ee86ff253ba10d3e5e8774cd3a303e433f6b84c7032057a d85a41df3212f1f9486bd4f52662ec98b2195487a0e7d95905b1a3f029666fd6 52c5dedfbaf68a6ce978714556cf0893685e4d6207dd451fe47a7a6b882fb60b 78a56f6ab398893bd71d3e4f02f13ead7aa59969b03ab21a3051c0ea15dd751c ba509910f1e686692270015e8c637a8404f2440fe48e33b5aba291e1995ba940 375a9fe2883840edc7b176493e9f1547da425f21fb386b30c48bf95180a7ff75 8af1d1688b9caa0bfc57b1cd0d340e39e3cd2dc0997cf01faa82b749c5693603 a620afa68e51021bb9aa989be4b59fc723e8f4a8c0e0e5c1ad9574e4a5f88c69 2c8fbb90e5259469933b9957d41d01b7b843b6b095bad4c3746a514425bf4a6b 007512d3f89d45afaf3cad3d296c615f44671c5cfcf46d24c66c25e7d4123d47 47b1d9785eb25939d308b485c90810184f520c9dc4114c28df954e45813ac8e7 e856b38643ae2fff901746ac2704da6aef77723cb6dec2081236bd1642d18c1e b33817031e699c3cb66d24b516e487a378857d95f28dcf 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: EUFB10 %!PS-AdobeFont-1.1: EUFB10 2.1 %%CreationDate: 1992 Nov 20 17:36:14 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFB10) readonly def /FamilyName (Euler) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFB10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 70 /F put dup 72 /H put dup 77 /M put dup 78 /N put dup 83 /S put dup 90 /Z put dup 104 /h put readonly def /FontBBox{-24 -223 1240 741}readonly def /UniqueXX 5031987 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e320fe93acc64a69e2abbb107d230 6f952b5533efa450ec38390709158e36e135615078216b667bd7d4611b9d0873 1ee78f27b66b665486e2597e2fb17b6fa4e2bcfcfbb4b6131b94582db80d3067 88be19598f94387a8995633650dfbf84b2e71891ef24d420d868d05d7ed911fc fb6c8c68b1993b44cf4638dce8f3b85b28ddca390cdaa8f9a1c76eb49145edb2 c55676f6c2436077edbf11efa3df3c50b51bd437e6ad8a7642ea751fa5d457a2 83abca54283369d7d3665dba84ce6c88ecf2a58126aa64910d2e810ffc30094f 64b320e096d91d5687b6deba20c94213e426cfba95227bb4009c1fc84d0b97a9 d4efba051b1f56c309290dc0f63c5acced4ae3a4cf4f565de35421a7bda1d8c0 f1eb8a088bfccc7e14753faf87ecd32b01289d4b22d3f46e2a0074b18540c618 525a453c40bafe0fde387a68f2579bc1376221b9dae3970b4e7c6ce36e201704 0d85e3583fffb25a77e9777aeeb12b2abb06e5b5e2fd0512e2ef1819625662e9 afb187a8c6ee777af34e6e7426e12393a01f0ee4e2c2d64673a21df41401cacc b1480fe8f6a1e6321d07bdd37a3cee74566061fe6c10aa93508a824b2436b9f8 56a5e3fc462fe614a14ed536aa31c988dc8aba17a25d3467c537185e57d35544 b42d6dffdbdcae0016ddff9aaa53254b1e08792b7f53b68cc0d3841ba2f482bb 07897ff2b40693798cc7c3cff99598bb2bc6da8d2b70b7988f65905a6f03b58a e2a8844c332e7812d02e704f3cce300758a4487c6df9b21aea1d83c0f18db47a 643d3032c031928211ef9aee8a7b2c87b7f506b8cd728dd3fb36a6bf99210309 3bf1f95b9702c7cfcbef6bd68aa64d6d81142096a771a326807b08efc46347e0 cb4bf44782fa923ad2eeae161388c3be3aaa0855976e3aaa5f417f71652053e6 011f38e3187959c052aa01899322d153826b77869231ee9f41985861d1cf57c8 b6b14f9502b97383ffe787b46d679daa268134857c83315ea65fc35409452748 5e094c2c8bbad8d65512d2c2710ea0c9757c75b7c1920dccf04b1bf09d8ba2c7 3a664017701729bc689b8c315e1f1a7ad5b75e3e38b09da9a17b65ed47c3eeaa 5a711bad43d8a89e671f976ea6b6e36b501afbba7aa61992d0f63d4e046f36c7 5a7b83d1ae9fef2daa2a8590f490e42317316fc97cd0dbd3e829621286dea718 16a4a9634feba13be16edcd024c5178e6533bf95e1cc483c3ebf5c79914abcd2 72f1b6171b3fce66be0e083610b635b8ccc5ae178d789ba48c50acd9532ded25 ee11b8f969c6a60f9d2e7ac68f5a38a2ee42c19ce0d1070e5a9baa22192e6036 e6d4e16713ab1d489d33733ef703d5527593ef7504484960e44d25cf621ea77d ee95fcd9191f20e6e160f02571204fffa337fd579c06d992673a0106e9ddaf9e ae09d8d09fbabaa0d49e3e5a90862e280e6039c861ee6a8e53d851aa39a69c10 dc71f9757f7bcf76b63b4393afbb2bce15d0790892ba435abae23cd0b7f54aee bef10c4607e8375d92fe3ebf719d403d140756ecc4674c8c9d1e7a35e50b6732 fb41b569b05356d31cb0b1707cbe1fc4a8f464b956b9045d169fda5cbd183e07 80d3a9401619148537e7cd8c5bdd33ac8b1d6a112dea8f08b10e58aab2fdf35f 2f910a48f64c75f15e275a19d4d17677b06b260f689ab705d7d17ffdb837a369 f81d053033985bc20c498ed2ac49c5f3e0676a5711441df5de07dc313016779f 4d82292520972ed0cf6431dcac205c2e76371114d2b010f317a247e8e30c8918 dff19955d566f5d5e9c5973fc2d4f0f0300092fd6a95c3e16616ff38bea1b826 5937638b57d54fdca44af2e7f183625d25b2e5773d496c6f7026e61d1262aba7 ddf01ccd9f6f84b1b1727231f212e0bc7da50e1c3810d399f1e0f9f5aa6160b0 0c2ee0d5931b602cd2d30e27ecd3fa4d919c54c50bb944879f50fdc906ab4f3a aae628c0612127db598508e04b6bce0e2f950093a4f2578efe90288c514e176f 3964ff969ac90e5ce4c0f51f450dcd5098cbd080c96c9e23a20c1a5d7a835bf9 68477db1bfdb8da972e2ea5fd2c3c253155d2e60557e090be9ba5eadad5c1fc6 a72f01ff91c13c7917dfc5a33eb105ae2b34b2ffd864362b0c4f8a1bed92c143 0cdbdcb61f9d9278e86f4e11be920996f96442292b0f57b836cb81c989451ac7 fd698a15a9fc08736f9e4360b4c307917a509ad0e10129d73ba7e12173799738 58987a06a883883933c15b121d2870c5f07bf11044b5d19df28d1108118b5d7f 256fa0325561429dceca05880a8c2643b8f786452a177dd8f288d1f1f634fd44 ea64c08bda68af2dac5603b53b3244df7bcbe769c2e5902f914b0cbdef8388f9 43c1684e70f635548b8a33406c5767749a59802dec1d975fb854ec52818bf003 5101fb4a97a28fda981de3eb024420f41154dcd8c848ab58ddc6c180b5a61780 b302e3e74c2a22b0d621ab4601b1472e956ccbe0d5e987cfac949ca64b734b51 329e3cde70f43b6b9849e567e7c6ba1b56d5fc39eff01597e4dd2b188b0b7d9e e8174a7adb6dd50502f4f77d9e5c346dc2e1259504992cfe20a8dd951fbc74e5 01c3a6e9d16c8f1f9356479192cb3dbdc33079c81d07c131b5e08ae6a6f347ab 59d5271beda3d20402eb33ae1347e84f8d4b1b634c6c6f6375ccfa3b20f9ea7c 9f1fe3d166a9ed131c82663281a3bcbd56100b4c4ca8266d801fc3de6dda657c 42e6034e28c3c77cac19bfe629b84f6af9a2cbc7445cc32159d60fb0dd33853b 7ee377f720033852cd9464afdfdfc744e135e3a506ddab245f61a21e8fecadd2 7954538ade6b5119b3e081adaf13f55095971599893ebbe00260c823ee8b4812 a9f7b2fffbc8cdac445f9f5f38132b263fe53a389769ea1661a8d1c17011905c 2f08b1e8b473b090f62aa363dfe89ad8b3f3a7dab1f55e748e8350eb1e7e6e7d bb0c7e24f49ed9806edd79d24468838f75e3c9cb516bc1ea0790666a98ae2e3b 362b1c66072018b51b2b3b3d06b4b16d2ffb14d71f07ec98d51f905ffc07dc7b a8830cc6c864faaa46dd8e0f040fb64cd32c70e76ac2fcdb028689164f8fe12f 47c0d0f01fff95c765923c1a96c5849e0f747685de63b4d4b0143735c3a9b3ef bccd7fb90df81f2e7bce1591937214f84e76cb6405d84236eb88e2a09d461922 1c28de924c8f92e0cc0667eee0e7c78d7af700343f7d68a5eb4980b9ae76a601 afdf84b5fe6bb3c36e6e87277fdc95fc8ed60ad53e55b144c541f906e515b8e9 73d8aec2c76386aa66762bba42889be93dcb14f0be3199620618cdce039a4a4c 5873f65a8860029a2279cc191cab876710224a3eb1e0f2c533a561363ff16a62 c9a0cdee44af119fe7e402a9e68094db835f07916e45427a387bafcb3dc798e7 4dfcd03649ba5beceaaaebb2dffc1ba479e6a456a4551cbef337cc2a46c64a60 d87a821f02582d55162f34e5aa143643f73e0f3e29a179722b2c61a3785a4def 6606d60c5d99a29f8c0ee64c00b92cdfe652b192b4384cb19791f9e0a6e5c7af 6488b231659a4ad81dc2d1f685c7b5cd27a6b1f21690f2a1e44e1cbb3124e8ad 19673b8a01f7f03c55817e5361b4e06f7438b4cbf82a8cb30fc92a7dbcf84a4d 510073be69214b50345eeda521df059da6af85639015a2cf039377f6de3075ad 7af30cf6e82211d49b0cc947154ef0c85684fa986f41db8c4bb66cb27eb6d13d 8241193693bd85db380dcbfb58d0e7a712dc0381d70adffbdc03614cd8c0b295 38a8b5e87bc6dfb1e3b6d2eea718285437f9d1be08fe04ab36659d5fd5bfa8de 83c840c1022845411333aa829381d1902661e7c39179176027e9b9f01961c348 ee6cc341accfa47f45f312d318fc2a04ee140d3ee82f095bb0cc5d29507191bb 7318f8d3599344ab03e455315b26f20166cac6ce9ae8c405305a2457016b0176 e144c4295a95c85eae73ba5a114af1179fbeddc84fd1ec0c7eccad2ccafae971 9c6e743fb9738295aa8b2644747990b744fc6847a61d9781cb4c4005aeffbb0e 6499396af7d32ef1c02710d88ef009fd1ea56bec14f9e494339e223275d2d53c 73ffebaa5fe916aef3e93b0721a9153ff0db96e7b8d54d13d7a0f0498b6e4b85 790c95fe91064d6fab17ed49b858da3645a2082290f093c567ec9082084cde77 7c3cc7b86a61610528d1910d9eaa3bc318c22e76efb9589d3d081f746b7246df d8a3bfcfd9710c7def73929b0ed7b036c2773a3cdef5f7fb53bc539a639a72f1 7f227ce2cd30b0da1062007aa94d3d5fa11aac262ceca84cd7347b9009b75ac5 1fa7cb3c71eb8933d1f502795917225551eb1b717031096d78016cc626171a55 faec70e0ef361143d5ce2a33f7176c386aa109da3373ddc3a01ca07999d35167 ad9892bcdf825e5cb5441c9574c076c9fe4e2850c52fe6fafc3dfe8e489cb1fd 2d6ae4986ca80e4305c2f54d2ba62a8a4e8bbde78dba92c352987d3d61846c47 6aadcc0e5054c2cf9481eadbbd5d6043378b23d7b633250273e1663d9c30c09e e2c84f91d69e91ae8cc7712c06614191b80ace474611d681bbaa4af5c29135ab 5938ab6eff116a9a43ad0ab53fbc86bf58d30d6255422348f4696edb22108f9c 59148761722a5d9fd48b482dffc5c3c3f16f0e5d63bfd1452c778f274388f3d5 53b8e9a6784b9a61d82eaa4e4296b5e99617e87791f9c03dde5df7ce4cdba951 52d26cfffb4a3d3fbf978cb0ccf6bf8bdcc2f0e1b8b06b31c9368685224e2801 fc62b99e06d3418504bd68871cd42cfcbe75477cefad067d60f4a954331c15e5 9dad9728b5d1357c6b3598b5bd6fa34be674b92fe52925162dd9241bb2008e7a b9f7ebbdb53679e04103fc3c2d1d2d0250957d650b55f3b3a128dd9e3f5c1f82 78feb6a1a71caae54ed9287886510084101f9786112e877e8b8488af 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg 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 73 /contintegraldisplay put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 92 /intersectiondisplay put dup 110 /braceleftBig put dup 111 /bracerightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 6d73c998789ff258a07a3c8c2057325784514c845500bfd1a971310cfc11d41c 1a167dbd5ff012c60add4e87325f6e5299032a839de65fb1473a166aae1876a4 414a434f22c1d241591fb36f857df6fa930608750ffc0c54f44994662b1f00f1 400bf752ea8d83ffc4cb77a290bc2d99981ae59a191748ba5c7ba1a9d2583fd2 1398452b6ff5d83a059f7eadcd2ef744e9dd22bdf9c79d049bf06835e878c32b 7765c69bdd8ef4deb4ea7cfff4cf9354a4ddffa689de961d16772491c7afbd7f ffde42400764c68e954ee5c455a5687959829bc3b319b2147deaab3628662c80 30c5e02fea09609abe4eaa12e217bc3af673f1bc36a7039eb13fcacb4218fe0f c5a3f9452d4edf46cc91db67b624d4f2d37502fb9f11af4da18ca40b61097f95 d44329375467ed13c5cb585ec53f62b83ef9502cc755af44bf32b87b8ae9f3f2 f8dbf72dab90acafbacd280db6aaffaefdff6d5eff26669bac56280a950560e3 d01714ba38659b0e4e28bf8cdc1f15ff7e3ea9d95a6afbbe3d8e6905aa6d531d 6b6dcab89e2036cc492c65f76ddcc3cfcff33fde3d9ffd10f7a90530fd5c7683 4329baa902e701ca42717476298c9ff511b1941244bb1dfd0e5cab92a43970cd 1bdb292eadd055c97972844fdcdb0aa06401e7b2632b9d134f96bce0064860f3 9859da7a9a82be0a840672a1331212752ae1b521b693b384a72dba8d37a86eaf a0341e7fc5f6ef7a8c566aacb91e8bf5005c18837894c4eacff41dba54e071ca 83fe311791c87b2ff9202147208dc8ad4a1f61199acdb6cd55a68bc25dc4f25f 51c70b695223d143a8b36f63440a9f959310b1684f06b11bc4814024b671a3a6 590acc6dca586d9d1fe21152f63e93761212654a82333df4654788e8286cfa1c 5c1dfd56a0cd510b3f0fa2768e698256b88509589da0ca0e5e2f38f09f1f00a3 6c8f522e33549c790f4b77860405e920b1da934bb0ff51dc06dbe25d92987b39 0e985a8c81d4e0b6ff9bd0524e58ba0d597e07e0d4cf5b3c02a30de9d805e256 ae0e09a52baffc967c3b9b6368d4e4cee3c055b90805861026ed789bc44a0b75 01967d19956253f2535a6accf154e276a09bf9d4add577170955abd2c6c1b6c1 44bb4b04d18d0fe84b9c65ff8c62f3d187554e5d8cc99619540d37c776f490be 7ac46c2f55d46361254daa0c3ba6a69ec5aaf98f05c83a75590cb5af682b59f0 85759085e2cf57b93b59b8d5b441f3789c3020034ca353deb15e06d563e7c73a e59a2bedba5b0a96ba4f0968b8808601af0e9dadaf33c70d17c0121045893874 7e82a7fcc9f95eefdc93ef7c1cfbbecc39c75db31a5a69051ef2137ccc31d0ca 8f9467f9ee2495e0f1698c9033113e99c88b7d982100a32af6aa17a77e02df5c 13061c7853eeb71314ea0df2a546dfec93d51369dfedc093bb2cf3dde052c262 2710c94517f3abdc411b8bb81c3143911b6406b730612a8100c77f85bf2ab5bd 6b89c7cd47e0a0a61b7d2040cd63fe917ed8d9ed311cc1874432d07ea7ff69b7 e5a799921e7194b36d180fd2a5a3db45987f47135755e4eeace455ffe528cb94 e9aad0e3a7ef5756f9bc6a4126ab1912826e2969c5d6211fb248b1b1e582f84e 54779c977e0255d3e2f967eee419f72b0d5a5d9b0efe83930059b6055738afca 4d7b83a63ee99eff1f530312a1e0151b9150d9fb7c491fecf9f016b88e2d816b cd967622b48b4bbd330b0b59fb308562294452f7d8be2606ca137f42f7d2b8fd 7530a02ed58a0ff58bedac729368cb13febe7258dbf9845f5a1c3d44cd98449c ab31c64f867c04b61b5748fa8e88acbee744fb93338af41111cf44f125e9967f 9fd63ef84f098fbb9d92e15b19ad310aae56f3083a3be415f8a7bbe1e5742873 7b87263d20297b5279d1eb82123270c84547d34f10f526a0fe7059ff874b67f6 cb7d38cc32975cdcc6a5ba0e0cb166da66de0bfad1a1e2f679cd827997c19601 289f03f2f2ce8618c282684052f3ed27a75138ab87300fe6b54e17a7acf73242 11a7a8f4f72974a6a0b5982aa8cb318081b19c61a0a06941a384e76466785b2c dd8a876a28e6813017218492a84010fb83678ea8ade75cd53b54a099b62504c0 4d040402fc26c7b6f820257321583b16eee13aadedff4481d0d44355aeef3804 4e8aae87c9180501783e2d8e296544e81809afc4921350a788f9810ea43a4fb2 1813f8514e30b663a3954ee892f8abe33d02f26a3844c7fb2f62dd027b45cafb b336ea3b19fb5c2ce95c187726646535dc930d7464cb1aaa939548337fe78769 76863d0ae75aaf3d026445c6a82ed36e54afd200bd0ff27664b7463a0bf3d39b 0a8243f54e4ce403897b04c27f94e01cc85ff26424794536dbcb2b1b17c3d71b 0be38a832423992174f4a34605c0656611edcc65b6514a83d124b753719706d7 3e2490f9af0aa3632be1ccd3c20278ba61cff7b9066924a1ced310f7c51aeb91 c4fb134d809cef1bcb894c378f6008f2746d892e5e1eb424255f0e83e76d14d3 3e9b8195fdc530fb51cdd4d228bc9b4dce24dfbee3b43464e5e033b03e17058e cd2584bc9d0d8594c8a788b2485903f81a58de48654d70b6c48fb0f171a2e547 11bc77e3a49e42107eb0f24e7ea8b1e9de25002f2dac49206d782f545b8c9ce1 16dfc4905fb4136e4092015a78aae739b564adab7a98679e09c1b4c98daa2a3e 4c662b7a83477809a0884e90fb651aa0cd9cba1db7da3d8bbc831713bcb1edfa 9217c0de72506606ff50f3f47b8bc8f5f181df7707b385544e8ec8d89f4759fb 124166bf66f2ea4921408effc501a8fd43c1ad459927f480c641185b869e2f3b cf6c5e70429eb0122e5b2d999c694ddf1e78773daa488b36d0986ebd34a5f361 25e964a97e3e4d193284836dd431859fd8336bb95ae898e47a034ccf6514f52a f21373f36edb5a2e03ec543ce18721c051d94418da26e198857886f3c72ddd4a 2f77f51b900d745e9a99ce1f77721940cfd5f8d4c198194cb20866143b35bf8f 03571f296b76623e4d6125cc4cda4e1a5a2a002c246fe4ceb1e0f632d87b4709 191d262da38cc3c47a188402dd0ea48443a46d226fba33649eaf03776d90a2b3 54125488c629392fca2adcb2553c2dd59130707635a54a6737a7717f1e15e3d5 2622e3632375b202cabcf84a37bac9e4183b4e1b8dc59936020a63eb26d5c1df da8e7994042eb49afb018da53c5fad9bfe8ef8bd7603c7962c07eefcd7e28656 1a98024e06a7d140340cde5b5e3b29fe64295af33b1bde2b900c37fd7f7809af e200b8e40551297bf4fd2c299ceed8912209bc9e919155f6f17ccd9731a9ff64 ebbd17c3db0d7a9b936933ef38cc5b88a54051bcc3745dd7428e984417b437a3 bc43cd103b684de53aa1555d813b059a0e390f614c95cc04f4df576909f3b66e 59b7826df8a0c7c042e84c5a5c611c59af697a7fa691d9c5eae637a5f0365f2d 9462dac9074ce4691122eeb51f2f6b12717d1b127354c9f738a6fa630c8dc244 5b0133a300324371c911ef01c1c39608e217240e35bd7d0028dd5ef902525112 f74658fe7aac6bfc65cf7cb2b17ab7377e262d09b69fa27d02c435cce00c7542 4ad38f2e5bb6daa536d4262fa1dbd278465b2848f24a0ec00871d4f7b0f32524 5f8c6836dd2ce74f65e32b52fd639c44b114a628bbc2b6d19db0af0e2d1eb0e9 da1a758de2099112a6fe9e6aec158b16836158257c04a160fa4bb255df3a589a c135e069ed8b7a117b387c2f58594e48f1e5e08774efe2d659b79e8475c42ccb c83f310019366823d05f1c53b780dfa2a622a2cdbfb0418a0e2be9f079da795e 942d52fdeec02d68363db83093f1aacc4cc0fab009c866a28c43e32eefc108f4 10da9001894e8e2039de764e29b38d90a7911d3fec25913d9d01c23b89aae3ab e4a78bdfbda40431a7f68a59cf397df286fa578cf206cfc43d08d171cd549497 4adcebc62063d873fc96960a319300f328fb9936112b63a1c2441e5277a14c92 3609bb49a65b5a9cf72887334538c9e79cf87e33cf6db7518f21a6f5ccbd29e7 fb1f7849e0a109a270262d25724c6f0b6cc67e2ec6bbc6716d4991d9dfa6d39d fd5e3de88c5852479caebb6e1ac9f0e8ac5ddcf62ba2e75604788a0530e46a41 16d34a21a060770e80945dd109abfcb401d87f804462661b02f2d91e58b7e1a2 2521444965e8ec9a89ccb84256ada7a5f463aae6eb0641a5dcbed1bb490f2b9e 5d3a4367c0033f69b52c4953ce8fb489b42ee651e7ccb4998626c09b26964060 e48863b89d5da10c52683ea6f453e66d8d4a7a353278f8a2133dc37401cea709 b16392cbb687e6ed1b879d32ed747b1e70390a3c4181c1afb1a3187bd0f344ec af25c0572bc7276450bd82a933cddbf72bc0f5bbfa962180c820272cbee7e552 29c3c5a75490285d27b6495f721df86d804aaf80e4d3935b0d0905485c356e58 0cea3f108173e78dafa212956921b912be3924ec75406d599a84862743927d80 73de2b7f7e966bffcc427a63d56ae530689c99af8beefd1bc78706bf8c8052d6 6f4a3ed127a5a9001b25f49c12ec2615b7dac6dbc8d236c5783d43e4b0f3bbee ef4f15dfbb6a7813a6082ed6968a3dcd6addbd287417cb7d0a442fb2a1cedbcb 36370bfd5c035cfea795cc3681ab7efa2350e100e739cedb1ead917779a8153a e19dfdd34dcb4cdf136494e93a8c20f1f374a3ad04f1a55b3f655251f97c38b0 4c335837f7777071e91c4333722de877a60a9ac35b9cc353e0a469bcdb701d42 177fb5658bcb31e978debc924d1b31ca7313a8b00cad09734f2e30c5d6e9080f bedb5da3b6d3ef80faa8cd5b0851d28548fd820d506d2f73e48004aaba321b60 7e1f6d4c6c1861c9e73904ae633dd69a9f49e98bffcd004101e3b37a9efbefb5 954cedbff3cca3687f5c0957e3dd554d2ceaf1c8364844a5842d8058d820009c ad36d54b4ae4cac5a7e123fecc3fe097f16311e7831d0370f24929e498bcadfe c1299ebbe6d87a1153273332bc87fec94cea0f845d26a11068312544340686f2 11e6f24f3a057909d46a327d97aa5495c0e33d614a9d6026e015f3b2404a5212 82badc543afb6517907ea8ddf6ff892758a3bbb3a8b84bb414c99080bb0a9e3b bbe533191cd3397fe2a54c6329311c890dcd8994e73e8705f4e1f8b58135f305 176c045a8ac6561e63d54c7ee20acbbd57f9efd3ebfcd2b1a9a7a064c84858ac 229b3dfdc0081af9f97113d1c5219afb3a54fc77009f386fe1bf18cad9a7620f a0362f37974dbec72bf061b0556b30b2b74e192f340b96642f6df6179ec29965 a09d5212f2dda3cff79c5b89df2734264865c6d91f7ae138c5a290e75a22f218 9b1cd694ad226d9eeb3d7a20df75c042fe9b08133214c5dfab7b90309efd2046 7f9a7e3ad0cacd2316b44f78464663b65f2b2bbe7c5b490b5083c0284dad52cc f401577da7f44f4d9e38e3ee5dc076fe9dcd94fcf07a2187297abe75885cf5b9 a20f66eab918a2ff544ee86439f334fee93e5b986e4db4f251ed40ddb4d6db81 d4f808e7a62c642d7831f83bf3b354f49c51b414e3aa5503c58be432b318168b 50e81b37f684e3ffd3479436393f2545472d4d55ca6f51b5f204cfdc53c032c7 c87f5263c6c0d4f03a682b58ed153a25eb852955d3e0bc5613369ff01740000b 186123e8261bbe9d1dc245b062c5e43d9b26542a48fef8385b90f7e210d93eea 53fcfd49c66e57ee77e2fa07ee57931d948349368d84ad3385fbf9a481decce8 fadc911c301579bb8ac4d248248acf291a55dd93d95f63a7a1389fc7cbf276a3 8f9e30e185ce92c511b9116bd46ab081ad120ef4ea952bc398607e3c717e18b7 2d4c92c9ff866a5de1c003867199f4aa8a68586f412ffa974d02a3f09dc5507c 9320322d3a025166ff24dcb89fd1ef3c251d26321f64ac81a222171bfa8b47a4 5f47e94d1bf995945bc30cf650ab0e1fbef68fb09bb1c581f243c9ca5d1638aa bd4faaf2a4f06683c029efdd803ce178e1e7d203f521d7eb455cd923691620af 59177a0b5deacd40c4012d98f2452f94087e76dd646d07828ba137405ccffac1 691885b1fcd07f6ee544e5ecd6ad9b6bcdc20760a9e9fa9c444d29db3c08d4b6 a930f3eb6564427076ccecaeb1021da2b15aa0f065f9f5486b7152faec89bb6f cc9ee18e71cbc3dbbc861c6448097b0889ec369ab7ac71b55d93d9789b2d413c d818ebfb3422928fb283b615add2990cacc11ec65d14a601014e72fe1dd2b5bb 52d1db24c89198c96021f2c20012cd2f97c588a0ecf77eb5638c9cde4093fd71 ffc4ca286866e74a5c4b7b3e7c1fb3d71836ba041fe4eb3a525ff0af0900f30d f8eec3f143737962aad68f711ff0b3457029d6663e5fcefd878924b4ce626dbe 6653be349eda736e38669cebce9651b7d3f39cbcbd546a8bdfe1c8ce1fd7de57 a044a30bff2fc1 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] 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 readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueXX 5000788 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 3a7c0e66de8ed85c14f1f8492167357f51a7e84cc5d92e0fee4a71813d2ffb26 445026f89b4787516ecd1afc78f8bd19e91e9ccc9402e8c36d2449c1ff850a8a f61165aac3fe931332dd28e261b91b05edd18f44ea7d58a8f35fe88493b64aad 6bfac3a0136215fc2f4ca8e91d70c5010e6f4013e6d63b44f6fb00afdbd7cdb4 5ec5b1d9736f45cefc8a0124b815987cebd81bbe0d44dedb2d5ef37923b3d551 abb6a1cefca7868fc7bf3814ef7d7b6b1ae6e869cb77aa29e3d90b12b0dc3ff6 ec945922b5899bbf2f12e92731486d2ef1230c528bf8d7e0ae09ad7632a38966 5963de49d1ef3d65bc483e4a577b927c940f5e121169ba52f6576c85793e5fd7 7f5863c488e55bdefd5d8b2514795533aabbc12e7f51816c7e1484f1c441aa9a 66a5dc77158e79cd6692d299b95e8058b35e771a6972d2b5eb1cd6d2bb8e835d 361a6eac0c90c906aea1cd75c4412a1339dc1439f93e59917311bf20af4f2bc8 bba5a4626f75f28a47d4595c37f4019df480d9385975159142618e22e964f019 d88125210e2ef21d65baf500adf802297b24eebbf4e93c617a613f856bd0d45f e17a9480980ef83950bdb8ebf208c895ebc4a7132f0765b7c19a332ebbf8a3c9 ab65063a42e19ca9d43dec3de7aba46b9a5d24fb93d615c0a1726ab7e31c378e ab77150858535b91362ab0e2b968dee1c539964126d937f9d4980ccefa92bc81 b054996a65a13995c9c9d819ff05511cfc89cbfe67447c940405001d61be633b b771eff383fc6977ec06af30f981504260de7b9ec97359974734f495ac073935 fb30e3700679092ef5d971b6b2da0e15960f2391fa6b63c33e7f56b4a9b86951 1d3686a5e8e613a69a0add8b8c73b127ddcd397a48d785079e786d24b6868c73 c8868545d5bdf432f8f31894c97a60150969cb73e1e76a399c601763359f5a5a 7c6a5aa7a71c76fc13cb9cfde253d76c906b30c308c9e7a72d64757de4a91ea5 4e44f4ce198306ec50cefd94ad932e3e7646ee275251f9383fe2536f444d8d0d 980a6a25721548831ef18a4dfe478c6589918a46ebb64fe7727fe6be9a5c5f0b e5e47cb3a55adb8cb10b56d8f56af3d40d672c347be7e61688869455d572569f 6bf6b96d66d63f0be5076d479eb90e776c6eae00891c4c499c6e9a6506d8ca62 90ce7d0c9f440f25d2d5a6a2a874ec429a6574416dfe36a7a320a6f9e2124351 9d39bb4bb418afee48d12e0874ccdae2fcf36925f8bb6a97dd0a59c862c51b85 3a712b0f740e1cb50e877ac84f954660c4c3ce573ddef3724041b787db50b040 0f6fee3d1c4a46f2a2fe6a601cf2f6632e8bd03eb1e5bece9f1a382e149d8c3f cc9afd9e2ad7e8f167d2aca0190863c3236233cd3f23e1323783b3396c5bcc14 63366ee6a64bb1b0421353fdc433887454f887e1fe15844bb711e4624c7255a0 77242f1936defe19d03b84b58c65eee1eff08f186dd4677848c50aa411a65035 e5aa4623a80ef2b855ea578873763cdccee3e604c71a26316af45cefd560a29d 1de749f62dd048fe136a6a0dd56ae8626fb27110670ecd5c3dc98a4c1c7db63c 82266961646e079fed3df55f0296ae458efd876cb16a20a3d1794d57e208f693 50df30eb8bd8dd5d0e3f9e575bfe3c814ec9b276518da9c36bc98a8643bbc094 80e4b3562e21215b646daab2ad71aedc403d23fde2bf421d4829a631d62d4c11 0d3da1e0fd205833b55e870ab7e9e21ac322488029541d4fe01a07be20d12341 9e70d799f3e791c7571623ff5694a1e2a5f39709fbc182353ae713b1a5f26a16 0ae50d0a69a4447a193c8eb6bf13e70e5175d59b6ed1dae92818326cd40477f9 1a9ca170c697691efc2ddfa39ce2d8780c93bdb4d524f7ab0281b3f3b01e419a bbeac96b3d5547d201fd31c91ed3526712ef5c792859da271923912ca32dd23d 674a4306ab90da86b1100aa8904903c85f1c578856068834c221e2af39b4c3c7 4a92a70e005446ac17992e58124192cd2d36e475b1d4ff7191999e1bab998d50 f68cbcfc44cf7c004e2a71cd1b29bf920214dc0d0634061d6ee0eac39471d163 d21e732729c4f11009f1e8c9e8009d58ca2c5c2672d25826780eda9444f60a2d af654515dae339a14a3e6ee07156b67129384faf9864bcd822f79ddcbeaa31d4 2f39cc072bce449d404bfff05d57ff02a0e033582334e1c3290f210b 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta 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 33 /omega put dup 34 /epsilon 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 70 /F 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 81 /Q put dup 82 /R put dup 84 /T put dup 85 /U put dup 87 /W put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d 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 120 /x put dup 122 /z put dup 126 /vector put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueXX 5087382 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930f01bdaae649990604a33aa9eaffbe5b5489 e5c7c9ff9d9be01b08220832c41caac64816b53bbc087ae4b621d2b44b21ae5e 3f7fe4ddb05c675abfe30510eee4a7c874bb57b2ffe521a6875eddfdfd18c781 25bfca5a097aad361dd94df46f14026c25ea988194899b95001209cb7009cead 4b4ea03473ea120e7a7e341d4e3926796347fdb3a124c29660ac09f4648b313c b58e824884382f24ce08d2edc686923acace1201c16a07a42803288cd0b946f7 546510e95b1fa1cc96f0bf72aa823d8e83d7c68c752c82a54c752eed2b1e699d e9db1830272ffbf2f4996ccc2d6fe2ae272798989525ef3b67b0d09bffcef749 a805e5f76578222b9c4a8a09b13189a16ab746ddef7875f1ecf83e568f493d82 ff729baa1e0834dbec30a35d3c49c9b10c5e1d90c6e3c8fb737df5ceb3299d96 0fb79632f91f257753b4d2e34e3f54a26c7b950981fb7fe4dee3315db63e75b0 24b4d318baec8aac9cde186f6d65767df1dba35124287c2e805bc660a3a22772 0ae4fa097f1d75107441c0a0fbb9eec17d5516fa03b98756b7df7ac7eff9c7a2 364b9f691ed5cb692dfff9dd80c761e9c4d5d6061782b327302c053ce1cd09ba 0605ebf761cf9b730524e1994e7dfaa956799d522ec53d5acba27a35c4d28279 617771938ece58c5ab664aa00c1bc4efba2d15504247a68e3af0f9bd59a12ac1 d8764421dfef5dfa13fbee58f5ecd4deeb4cda9fe9483c229ab29460b05559b0 35c73d6b88f6a384de0e2dfae4b1886f91e522d800b0ad40eae57f680490fea4 a2eda9f8f57aea1a4315674be45989c2293362dfbd0b5d90a523a7b79de04d6f 7b9ffa9f9ff2f2f7a1645e6b6e199c88c8430208a7df1f8f8839769a44b3922b 18294062582dfb8eb65d087635f1b602efbd5aa1d63676616822ac40947e58d4 49f3226143efd388f810db12077f5cbbd8d6cfee0ad3ef2426bdd29bc2cff5ef 5e788cc13d25fba235413d11d4ad98599be4bf92bb1f63e9435d2900f90e0d5d 4e9225094d9a39e37e74226517669b48e18370a5b6f42735b62ee15590f623d4 cb270c59a4e6798e5478abe0ab9aea587a976114323a4dd478dfc7d1b40d2dbc ce4c93746c635f14a1c8846f0824ba355c0c7d931c78afe56fc2bddd264b3127 4bcaebd41ea7b3acd4c99ab7e17f27a94622384a7fdf805ad8a395a413480fc1 3dd75690422d76e3965235cb89a2289c47dc8893cab0afe4b73a128ac626f2c9 bc2153de7e527602d17815e3fc92d7916860bf0af5f783e59e96f8962a3ca34e a006bb5d244e65f7a50a2df93efdbe91187b7b0e2c38a0a7b17c0958a2ea5c4b 1b3fe858f88ba81eb5767201823d1d4b5c634a2494a10c16d4ea3f0ea7818651 8f6f2dfc6d200160ac12eb2ce41f33714f21428f5a1d750febf0eb32b512bf6d b5f03cfc4a404d8d9eb916a9a71e122d02bc991d21643ddedcbf2b5361e3d7c1 15e01bd25eccb6f427dec3be2d60a6c8ca9ac1324380fbfcf37cd8293b322d63 9a54a7c027368b60afe27c735857916f9d5d6b9bc24691bea6b8a425286940f9 407a00247200e9786e393c3bdfd7b3018fe0e6651b4ea216c5fe91910bd65538 9714b29aba0bc388ddebc9e0b0456f7e6217fb3e6b89a5795ccd64112d0a2bc5 d04b2c2077ce41525a6084a932624f297edd79a207bd94cc9e161a034561082e 1e6a582d6d4bb49752e21860e0e782a10c573d7194d82add85b58d26cc40124f 89caecb2fb5e059c14161cb00d25001bbff5b1fb89ab5204492e11d268750ae1 d70ee4609c81116f5889bc15bceefd0e8073f438e5b5925ae076d3279a5d75a9 c81c3bc8ff1e689a3920dfa1f5b76e1263a015d9539e0396e97f52c73d287e3e 9245c7a27b4d8ee8da68363d42cbc1e6bff1ff351302981a1f11d6ab049208b9 522e088125261265f2690087a485d0a199a62e2983e7f47ea86f51744a5481a0 0a5335e85c846e4f2eb628211ee0fcbc93df6f780a61b17260857a6b21534675 685c207e2b17bdc4a5ae247d995c846b803cd58adb5a92db860517d83d8a285d ef4d325e5906c57f452909833233c4ace2b0573b92cd1bef3220434b26ddddd6 185f8de371f09bc0caad05bbc99a799c2d515442db9493447a578c0e62df1b15 605974f349be80b358f7562acff272b365979f1d1f6dd3fd86fd2f146d43252b 02643e54b4d6ce9137b20403e92570c0a08169690a9bc05bc0e5b2150200adc7 c89a5fd87bac32c94ce959e2eda40ed6fbc276582586ffe1d69d7def99c85ca4 778974a56814b0bb551397c4961280d7b98b4fde2afae02d0dc0466b8846b81c 9f37a86bfd91eec7e2056dc2b1e00ef5049d3d934c3eda23e341987a445ab28a 203c4bc48f8cbf3bbf4940f51b0cd1e125bcc904df48151ff5f35e2babd84591 592634b61270a5d8eaa45c0807f346b7c1520a2b6a590e2177da53b99aec09f5 05e9f6d339d5574cd28f1896bb40270841e12752684e3bc5d3e8904311ac3c04 bd8ef14c90073610cd812b8ffaa957440953881d775e6685a61180282ba0ef3a 7262850ff23b3a469f7a776fbe3c36e68e83d4e60c327bb7d46129f152a66b06 0f6a64a5d026fac568f0669adeae0be3f355704298ad26007c021f649eb5c543 5e557b08ccaef5dca94ca11969a3dfcdfd6b29d0f9deb61c609814b971c99777 5a09a937a5c08ed1f158c568ff6414536ec622f7af98e60caec86b81b4d76998 f1ca9181ce67e7519739dc592e93a0f0f7704793f4ad9289b75e59779dd3c2d2 d72ab02cacf24af49a66ada7ea276e60aaf5cb3ee0b8a447f3d14e6af999b294 9b68363d45b3a615582fcfd7434441f53fd368a947831a0f0eaf8fd944c4422c 47db504a686f7526621e530ae9fda10ace1c56e62bfb99b499849e4e3c363e0c 6544298c837c1b17813c80d80e6660fba402efda31ffdf1789000ee130ee27bf 7491e58efb0b21af8fc97cc3f95b6236f11ff4b45e29e2255ec15c8ccc15a522 bc46182fbfa17cde6ce0e244fce2addee2a75a8000964f929e67868a183e50c5 20d09e987f4d3e01be0b97b0a9001fde15e686a283efdfa0dfeb98cad502a659 be7c9696a9e51a4e8de0a826d386e17ed5058e5a6497b784baee09f4d25b4f37 11c36fcf729f076f09ca4ca83665013167d0ba870867f825bc792027e8d06cfb 5e7ebfd65f0f1de21e39aaf341469dbbc1782f3063a52927df0c82a81ad4ad36 e0f2b0355d7536211d0ed35af411ca89d6f9edc0780e1f03af88871cb40187ea 762af7dce6e5144d0143a648556e04f2f4eb0008de6d43a99a314ce827b804c3 6ce1340c90f218bbf08bd7fc527e242410f7b1e1671ec928e74b87f05e458007 a9ba70cc1bf492843bfb95167b935e778eed5ea965f7d28d7dc56f9a996fea78 c5ac7af7eb8b9be4134dc1fdd81b2c9f926b5ab9880a6aa89809f2dda12ca2a2 b84d03927efe073beaa043fb34b7ef3e79bf0366d1ca3142785b4f5f03311cbe cc391fe1fcaf341afd329a049005dab899a5df082b5afe06a8e3b56f33e98086 c0f25b9f734e68ee3f4463f5beffc36f32663664b07276253037894c7bb080fd 749f732f6d70d0ce0d427813162ba8328eabfd7396570ebc4b75912d6b2739cf ffbd083e4a6172be70b92b899566d8250bf15cb543cbff07187ac57e03392c03 11a55e884de63c428806a078e9b6b972c25f55281d1f4115ed6c3ce16f498134 c8ae62b64dc842567efa3e1ebff45c479572fbd9f7003572fd1e01e73077e64f 564438817dce0d1e25a8bc85d81ceb15e247bcbfdaa3662a35ab031244bf292c 324d4001d33d11d0767766e1c9ffd8773c2a8db8ff49dcd7749cff1e4563913d eeac101112192429daa7efdc1df99112d9b3ea307fdf3ea1c617adf0797bc255 564f549e73561ccffb7d340c863d4e8c6d0333e8e339b13e798fe7a9baa176c8 6dd673c5debccbda8583a34730551e0f0a81e06ac3100a2d9f7c84abcada6434 2257b60bd57fbc1e6413b41e6fa0a237e5f2e2e0d8f627a0251792680b42ab85 c3d93a302430c630af6bf6c11713fe0fe9333490e418c69e22cf4bad832b9f94 a545f9f3167f149dc8208a260a1fdb208b7c38b1a2701d13887056b0a62a664f 81c29eb4c204a992a90947c951b42c0aabef8c109eb5b56ee5f55b8ef4c3281a 37673acc4ca68fd1c4e4d5b060fc9be8620172e26729c7322afa23e05f02b413 607937adba1c7d14551f503727ee107e07d6b2cf9a698384102cb4383e973210 f2f12d4382f4098250c1a9b650a932f5c92e5e50f82ecdf288ef751583bb8fe0 899a32fc4f11ba97c982a98e65b6754999309d9c6e301e903f5e3204edb0cb66 a64a2d932920498701314a3a1056d2ada7f58a71dff8e2f9d380abdc23a7b762 dd666aa4610a9814f0ec548055293794af5bab83344aabfd127f9d65133d87d6 9549bafe4fc1e42d9f784d503704405f6468e02c79423b7eebb93c248e9ba732 bd1998ee723e207d79fc360585964b1a480e1f999c6c65fbafa00ddc8861f56c 42f903f9a681d90241272831033eee127feb6af6177a0c8cf03fdecb630920e2 30a3cdd2564e71358133e723ed47b90064dd59e9da0ac88db419169c99209411 fa06fd4183cfca7c43d69284b46f42d11fe8898dce6abf4bde095d0e65f9d6dc e17ac5be2a3e9a8c3ad3adeb9c2055600dcb75b87392031de73830243f249e48 3c3e3329e73ccd8015ddc683d8376572458c62907b7710ece1dd4d228ff0b349 125a824785d646da91ea2ce82549990539dc1b534e084523981afefbd8f8c3f4 275477cb47e73b88db88bf1523981d8531c3067ecbb09eb435298a3ba7681824 5c113c8097708fd36715459c438214a4aaa78eba124d34a382ec929869ed8ef6 848ced8141fcb7cc642876be4b2bc458128a212b2c2ebef7a926b2af4599b1bb cc82ae1126e0ef299e6c749c1f6c23556ef07d544e38182eaa2d3c23bbdda8a9 ceaa26fd629136cd1340f6bd6700bc999b4120baa34f0cddeae5abf471d75867 88d34aca087d7709f34f9ee37ee155bdde155f866e819406795f6cb5e11f503c ad5ea9969414d96b55baa3070d71d32e0c212d5aa94914190fc6842b20710e1d 7725fc712bd6df83de6d7a2f74045ea81d3aadc20346ec26d5d1ab10919b5186 81888561135c0a796fd19f0b0b1ab1e5a0a7c457cba36d0cf0b2df88e3ee9396 446c97cd88ac7449d82339cac03e8dbf30d78cf2dd2a1d9df0ed1b2ae9e2e472 ff71b8e4d1555ae2438ff787a2a8d98b5e32ee601119d8b16219616a55ee63bf e45f02d2027654f1729941c3831497b29525a0945ab566f57079062661c8441f 1cdea761735360f9cbd40ecad3fe9a8fe70e592a9e2e97e5b8086e38fd2fc94f 2a5aadaca9ea1f42e90304f49b12d4861a6bf9bb1482b7bf336e52ff98ecde0e b3b62f65463791529db1a61529aa4ed5bc00796f078d619cc05d22a872861e7a 5f884566beb693d0c3aab1dadfc6aa60711c1c7855fa1620391dfe6e7d734e47 dd1c7333a466c7a7f1243ff24a4895703674159fbf6f4b6071332c9c9fbe65dd e7ab72ae09a19cbcff96bc75fd7bf0d4abfab87872669200be7afd9fb2c162f6 1191a62076bdbb7735d830f5bab2a50a04171c1c2600cbcbe68fd7b4e9b6c6d3 81436b4d270017f8e2162f4bad4ee4e770439a0be644cdd13d58146a3998f87f 444644816c8ef5c32d1b8c0fcd00cdb27983ffe1ac927b7ca111c27a1dbd34b7 5f95f253aabedca820e146f571e136a4cc1b32031b133fa1c29db0e9df11cbcd 03f8fc8edbd7ef898c826e0a3bd3cd72a9babe853d2c6516960c10eeb6772a65 ea323a20f7c717122bc6e3882f9dbc6d138fafcd70301328865f1bf45fef1798 c332606500b32a1825359cb4d89cce55b7b2953706203ef8bef0208de21de1ef 7275f26371cbd4882c24bc34992159d2d960ae565058ac9f001ba320b52c5059 4333af6fe1fdae34d7ca09f7e6e29e5595ba3aa55ccd215ab39c8e1b94549bf8 36669646a9fce23d62969bdeb37efcd78b4d6cb622cde02df38b75644ed00302 8d7fb0dccf2e912dddbac924a744ade5ba25e0838293a0c67857a622609412c4 2256c19b2c9ecaf3dd9bd56cbfb1b7eee2d1d97d1a7e4edb2481439021d7e710 f1d815b762ecb9181005fa69cc886db3d513c78d37455cc4c4ba3147528c14b9 be4015154de34e2641dd3551cbe8c45789cbc1fcaed27815c7fa2c55cb55756f cf1b42254a8d91ce36d82fe314883175d2814ae39e68223ce606e6385824d5e1 9d8eaa8434c70ed51f5851b3eed59d052f4ed1ec04803db14ce5b640489e1a2b e27cc56a40bb3e6b8b59dc9f7a7b990c4cceca6aec270a227dcc5b212f63d2e3 0d56ca1f2852ab656f77637833e6fd2e22a601d6e9234ebc68c259aae88cea98 a9e17f78501e23e70d19464dbfabdd1506f7c7e6e4d449b1419abe5b05349348 5d78f9b62512495da6f94a8652f0ecd69e4cfef987bc0e1ee74faa8e0f4623ea 162ec5a71eb827a357f5a2d1546aba8f22627ba422057eefd9eeca2c84fd9f2a 9f499bd130459f5f6b07f5e258209df44ecb66e100e134f0ccc8e2c2693ff239 2f37b3fc5a3e6ecaf2033780717dd0a92d85af2f0d29763467b1a20b5f989b7e 464caebd895e2adcd98e12f3e8ea09b19168ea11a4d7eddf417f3763008fa12b c78f9fa2101d2af60a96484a92b063f94d79365e05fdf1f2bed0524cbdcf3536 2f910f1f1b3bb9ad1a91fd3d8f045457a3a826e04b38ac6e3af48348e90233a3 2ff933239737b18d2676da08bf9bbc66ca8a530833f1ba19e66f6e1b556e97d5 8248d7672d1c6ee36ec310440777cd01f23da035af3c38ef175b9722a6f19c62 32d08349beffc42bd9285d9f8c97711664b5b78269eb2706c2bf8f2828e6942c c7f076512778ea7589f0e1d01e058060aca8cbd1863b696b4ec6c72006d67a5f 8cf81a330b52db1afa5abfb7d154e2d9d30d17860ba4a2c900249736e373775e 66e2ccd17c769595ff0de6c7794c5068c91e089fdf2d7a8d02efc7b394e4579a bd2a416348b46a382661a0afb18f9e8eb379308c20872f71b1b77b7e15f2d5d3 7581630c3d75ffa8965b1776799e94506161ba58e1a3014a29d890d97550c4f0 9559b05d5776b08511c3da4fc152e222948fedcc5bb019157d8ae09fb99139db 5d97ef608b77043175954c3ddebc8aab9ac91dd9a3e524344d8e4fae3314312b 82f52946235d0bbea167274e75100e63f389a91ded67a03629edb28557ae5d46 dd9e6b9fa04574544f26a39c3584149c01e408653e630cdb3d60c1e272c9b055 7d67b6f07685aa9215c75da9251b6a75be1ffb8d296e0da525ee2ef4afe379cd 11482f57a677a45c05b8b50193be62ac107706d44ae169dc31b78cbb18cabe16 4861dfefd12f1beab0fc489750804c2b9c0c124b478354508db9e6127afe701b 0499a9831e5c4c0de68e72b6dfb5a64d8bac2541db45a2e8bbe7272494114ee5 a99be9de9f810476a93d81112148156582bcc7b439002f7dd4bd7d7ed0a0a251 5adaa1d673882c4d1213f5c94ad013ae95686e6c9c8049dcf40f108522839496 f9ba3233c524600bfa02b8bec1725c05056171a7f1a8742dfe6b39aee0761df3 a9f44a6802ec5ccdc37d2f06ab491ee7838cd8fc492e508110eb6b1d65c2d329 ad460b4b870134df65b12a4a0b4f7b2d83cd3ba2f59f13c24804bd0d23a7d0a0 500e8e7dc51406bcd45f14b1c188270b6374af5310fbf88d9003278fcf0fcc1f 8fe3ff52a91628bd3233cde319f79d893ca390087b684f018b862a5fc440c20d 4f1742d2b8d90083439f55549a6c512137faa949c646396218c3621f0656a3bc 90fed7c5aab809f258977f7f123daeeb90700fb4ad37c451ac4dd8c2db7f3bbc e153a3464b062001db0433011de270e8fb742e4a40a905540405d934568c929f 045cdb0ce13c879ec516cec04c0f4a0c3875b7575d67a385a860c2495e090fcd f2d640e0b07d8a5c083a0c04e4d47035268b0e114b8af9d554930169658fa138 b19bb1d63600949cb4f30d79ed354047622adba279d2318edecd5394c451cb8e 2414252434925278fba635d5c5cb17e8705628d518dc22be18de79ce74a74996 9590d4a4da4041dba967f5d919a69bef612ccb227ba85e32324634096f891820 fd185a0f0dc2e8ae76fdbae2bfacad286b0d208935574df908932be86e6e4f24 edfdc8446f14ead375e74fb71f2005788dd4106e5f4896715d5a50decddc209b 4410cb2a41e2cde54aa6375cab7f1f249d18743fc7f49393963f37db19ea5ebe 5c2a9cf7fbe28f69ddb76b27337c4d80e0a7ba6b2647926b4afa40d034c2ddd5 2e6c126c18c09b8a87e583ccfeed82a5995b15691ccfa69286b4c51ee964db89 81b3a31abc93a877bd7a04011dca6522dbf02d37af67484fd01246ce56cf846c 7a32f6cc773ca4240c95a8cd54b416b2777bb6ca198170076ab167d69ff17d97 0e1f510c0d34474a0c7cc38d8b765afeb8e2710dcc35af8e3f5837bc4700d62e 950635a29320ca7a11c36428efa95a148e2b46f7334b2446f60dc98345233ba6 2d020ac563cb5210c0bdbbfaa1fe53213c560a1ee10c2a4ac83f165245aef546 03a13424f7e6b020db03a49fe1975f18dae3c6e58e8ba12261b4fad049bc784a fde5567600539ffc999890e8014e0401fbf6d1411bc2e06b75f2c38856177388 26a1390d33825ff60a2d736e4b94a7b9e40ceea38ab62f4e630653c536727294 238f91381353623f42f1d8afd5b45e8c0026dcdaf31d0f312ed798b1b14c2031 9a0acbab4a52494050a3660a6dced5fb993dc7958fdbeaeb0b501b51c215fcf8 889884056ce4a13c43c7c7dbc62f54db2cb368df73af088551166b4e5a642934 5f54af5da76d8babe28385de5a77363cd59e97db95bd8e902c27046601169eb1 d92fb7be8a23a0661f5a77b13637b6f282ba68fc23ddff1770694bd3c1542ff6 f57b13c7d4d912ff0de1baa2a5730b0078a56fc7b30f437fd02e4d8940faeb73 1a11368142aceb2141a0f98b835f553994d68402091dcfae0c60b88e333635d7 180bdcb3cd3e905671af55021b4567815e3288de30882ee175160f408dcb1282 809be153bcd56c71d5dd03e84a051bf063c1c44feb1089e0870525436b22df29 5c0f0c5b9237a440039d80d6837656b63a1c450e2a92357d8655f2703c206374 a321ec976a9ce2f07ed024b0528fb8055d413019a8120c88ade591f46947f7a1 1e5a252882e176d792bc93b0409e3d270e0b3cebbbf17f0df720940f89899c0c 079807f4039267c0ac3b8e6edcac391eb167fa0b927f0493ee3ffe5882f6171e 63e5833f0beb27435027a91ffd23f07a23d6a312ac9e7c6946f2d243fd1d6678 37043cf050eef0c9b89d7f236c7f6a1f20e283583580449a95e31c03cf9bcd41 20332ac6671c6a3f3c02b02060c1b9607b5356ae7014e5934e69dc5450dcca1e 43d0acbec08040d05550712392259688963f99f6d36b63f0304151aed5d944ff 29854db4dbea899bcf23ffe6ed10e45d794d8221430fe2fbb8070d753f09a6a3 e99e7f9125284fbc6ffe02a1c04d3ca62309e55a4e516241c3d26589f5fc3a8d 5fac85b63bfc010f03a2ce156a8109270bfad827e9db395d985dacd677e6a0a6 d689fdf1fe8a97a90494069b6845b020d0c25cbc2c44bbc25fed545dbf0b68f3 7c50f5fc4a23345bd3ef01fa345fd7c34035941f28ec6be4a7b4b10381bebd2b 3aa673db02bdd29795b03ad92f1efba24fc3cd0589bdeb9a19eb02dbcf1c3667 194327f4f174ca96f701283a86dafe2a6a5d7256d0c25b50125b5b0d257e439c fc6fb05e088be6862c306ab15013309add20c40761985b111285ab2efcd81cb2 67cdbd356cd9ca12d6989fdf06e15a4f29473a82cbf9e710c07d0477107a6521 ec9c870defba334903b66e5e0e35b5697246a7af093442aee2290742037cfaf6 ad9ba12133548b4905e3524151c93f52774f086e482a42bc56d27cbd0b5ef9fc 2f38599b4bee1aa82115633ae420f823863946496b5603fdd06814c5bade6138 b2d73dd2b6f9e7662691320aeaf8379d282ce3e691174e6724ceb0b8c0a0864d 935a1f8bb2ea0894e6fae5b52e0f6143a718849e6b89e1069d3ed53eada7824e 74f8a927e5e4ba586c02c0383c270df3fdeab82b9a87e6257ed7eaf46a27f520 f8927ba7c05614603d4b1f991c64af3d9fbf60fff9f6628dfe1d9789ec8d0348 705f75bac84c4e8df18b7514afb846ed0bffad6e22a6b003a51ff96527c33bdf 4464af4c97b2da6c764f1ca7012f255bc3c2181dba631d2be3bf2b67a6050b7a 2518ebdef21d2e043c5bfae9ba4111098081322b08ce92256a1c07f77207f581 db4f4048340397bf8290487ba1452103e80009dad95e616c9611768a59caea61 ca9a82630149bc8110e7ea98b76d160d4c9b86b923cf03b67756c636987f0e73 580560f34192aab488e7d3d9122754aad44714de9969a375c18192bbb2b91349 f1fa16b96c45bfda392d24b5ed73f8dffa8fe39c93113e4f924ff3fc58254022 05d20600f0e567cf259fcdded8450a9fef15297cb4781d0b86ac03309f875f9b 3d33d908a821f206e3eb0239f42e82d292cac4fbc67561c040f7510e0cfeac3d c8c1f521389650c3102232bba2ed1fbf572dbdb0ce4f74bab41512b6dc567975 875452832953312d3e803565c51323b6a335dcb32ecd9541df45ad7e7ad6be99 ea8c7ae36528f2b76b55853a3408d968e4da2d8d53a4c08389071e0b467d72b3 6d31f10e6f45865dfd98acf8625413e77783695f795d7ed0683df328fad19e21 1bf0386718b7df6531833ee81bbc619b9ed224f98877371c644d29cf59d822af b804477994e52df0ce8180a7f20629c002edc6f2ac9dbd067a785004ced7f275 e20e5b0f6546dce6035acd9829c808e822c507777b59e7a1525148ed9dcca8f2 6ff4e1c672483038443f9c8cbdf59f1cb1896405b9c4f425a6bf8b4e5461f995 81a7044f85373167b958742fe5fbbf2945d0e3b7cef1a03a08b6663c4c7ed566 60b9c6ead2e5d17fbe83c33892d9c090875cd13afcdd81b8affba16855ac0f8f 1423512058b09edf46f519df0e55878cae45a9c517bc8a17fe4f4db6cb8e33f4 8b679f4d32674beb64bde7ac8d4cedb5abcfe6c7851997073d0bc5d52de57b38 6a1e53eb1024256eb4e4b784d834997314fc6ab6de8b54974cd0a9fd98bef1a6 47aa5e649d6849adee2179e0799cc9a1818800b9a0339c254d33516e4e85ecab 214e86ddddba90e8862e69f6132aec6efee5f8741fe64c168f321fdab7b618d0 c594ba4ff70fef6a92b577ddfa7ce813f847ed8881e0216c94673b27dc275153 1c387fe06462c563003e4c3b95dd5543473e60fa14ab07d23a4e7a68c01b77db afd3594727cc25b72b7cf6b78d91b7de9121322b39027427a07fb375d1e6d5e8 ea96cb62789de81487e2c807d9edf99918ea47f28c1b249cac21d577d219feb0 9a0a749cdcace73ad9641467526f76305c800661357c80517b891d1b9a44a33b 2eafc23d1fc696cbb484a09b8d9e2a61e534c7c4b84f68d9ae03c2648832df31 71a6455d8a18134c4c1101c8e630b7049e664dc94bfa40e1aab56b4b3d823f2e 4d2771c909cefedfb6204043ea560fdc92dc6e60f214256efb677e944674dde4 483dc1211e56807074bd294d753dcda2d52bc58485a47ad027cd0ebd0cc3a0f1 e8fc7efdf019e14db4399896599b974d96522c00666475c468ff461311044714 23cdb4e3ccb51d9e6526c4c84386ee8ff098807c13707561f271799dadd1cef9 cdb3e2fb66af9c1f6199647da55907c13357ded77dc5fc89b3a5bd0fbfb9f716 32e4b090dded794fdf2a463a88b27b72eb74eb8e6b00249604360d73339e32e8 b5226ce84b0aecc14e2bd318edfdf227cae824604c81775ef978fe1b666d33c2 84f4a7a362924ee75529d9f7de958543ce853acbdca6da588d1a0f58d13561cc 9e0828ea430740346ecb654b58ce228fe6e25f74cd14b90484c4c2faf954e07b b4f79e3e24d84cefc5624399cb0746510ad57ac777efc2793fc3f2bc99e0b134 215885877a19ad4996ef5727bbbdc75246c9c12891530170dece7f7a638565be 0fde839f6d26427fc76e32e4268d72a3fd19d85dcb20f4ef323c8e01a32988d4 150402f7b3b9c419c78f75fed7609943b6dae16b8e0ee71bbbdd88a84c4d849c 2616c5c2d4fba2d829d9e3316ebbc6ee9f31b9f8e960a3b8649a54f7452c8ef2 41db5b3691f9905e84d8b5d4419e0c0dfc26c807678d35432df5459330188b14 cc2db547a3c9677c0f912b24f81ee2938483da3f920d9eb3ec550ffdab3e9262 3fa994b519d20ed683c751a03d0ae4e7d446ab3219c9ff0bbecf8db861f7de7a 82cf0e3bc2a278466033ca2cb8964150dc9ac3f77bad9f9c08ffd9ec52bccd7f 4dc6120e7077d8205656a1cec7d9582cf7444ec7e060930ee493dc377d76a24c b342c53e8dae03315d444d7f1a3f79c09d4bdd5c3f63322382c1dba9c59ff363 d45e59a761449c3e2f37bfa7a49daa55be5470920d2d363820f1245c05597dd7 d4495d389df20fbe0e7d6ead65ed1884f9f706af979ce5e01cbe141e2664b039 cec0ab3719d4bc21c389116eb4e606f7dc1c8d91ffa0886e8227a3f1d1157d73 9a2466f09ca30dc83b4cf4c21ce3d32cef03804919dfd7e409caf6e3e8fdfdeb 374b0c9ae4660e5b14029d35644e55cc87f53781d2a1103171d617d740214c74 66500950ec6898bf0f36df3e8baa618e6e3e14a2e8862b3d8acf6f3791add33f dd7939cc4389720f35e3e49bcbc5edc4c7a81f48c76840f3e92ebd31ee7594b3 3e95f9823689ba0df3114266891048b9839f290efa830dc69b1bc707b962b7e2 e99c86e7c608959ac2e418f43d35664c77d10f46bf312d77236be67dcaec4902 d15ec9e96cc1b869c1306ca68bdf833b4b6a86b84847f076ec53da7420ff6881 5897b32e918d829d6182000c26b016cd623e00306cc0a94af89fb4262d7ab918 fedd53ae038c074c1f2c5c01fecf87d10cd25b463a015562cf51ba7c42f466d1 3687d6b7b1808f614fce6a1f2315412ed3b9a40a93cbb19579ca7055df8fced3 9242699d92b310802cde9cdf098a80d1c24188143b2d837948024385ff1a18d9 72d388a4d32fc6f7579547f8352a3b2c639be6274292492592b00251ddfa463d 9c70904d3d0bcab6099b94676a996d9319c090057432e0f477862329fa6102f0 4f0a9519790309e3ec1180ba4aa92e2e3a309d93c1ba109dc2fc35f1d688062b 6460a7bcf2c27541ac65a116f788506cfa822adf8892e703922281cc5454a56d d27bbc4993dab66f1750eff2da3f6ab0556ba590aec17d2e7015cce4a6ef59f6 c677de0c3f3e51894ed2e20cb7fa1c6cc8145b9387660a75eb7ffcbc0d0b927a 8df24474d1d981ade98555fdbefdcbab315cb0bd2386e8404002ad775059976e b863999573bb54afbfa8cf403a886423d7d3b7b83d72b9e933f2af10e1c65008 364e1196464be5b38988e6fd41945f87ee8ce364a47bb6332b0dd3ffbc344ac8 be31837259e7421f62eeda9d116665520eb969982141527c63741ef236c8a934 81669fffc65e7f819eb39a095ebb4b1210aa35412c5aab93aa50c11d842f4a0d 9abd8e62e61f1b707ea668ace7a0fcd6bfcd94c6f096ffdfc6d2e6c6de525813 e76a2e8f82a56e6e0304e825d86e7d827cc76be909484ebe53e4f940f6cc212a c160a00c7e2464d34274fb5b4478738ce6b11fbe4c3646b618e35fedeaf8b452 3c68adfc54da13bcf09ff706e533b679d9b166b190a58fb353869a8665581f88 d6c80881af99310ffc836731aa44978d67b9960ee08281e286b93b9017bf9ebf b6df7d08f8c41913f2112d0ec93b4acedb781416b53b982cc013d3093cc3d609 0819ad29976f22c8959256c8bab1e3ac8aa34ead58e09fed7e7a4c3cf6713ba8 981cc1b48aeb04b830b10806158cc6e52cafecda598a0293750e80cdea8ef58c 4b77cab021bda18ea8db5d6d7f8f0b4fb590a748ded819fd98a6326e058487f9 c6e91e3222e5795448ab4f0ff2b8a9581f46a1951a075f5ab820ecce0347af3d abf9b6de8255a485c0afa07a2249f4e91bcfbdf300ff81da3e27425c1e97decc 5067fe547ed9cbf94809af52608a15c1c37404ed64f448bd0214f1417d7d9c86 238e40693f29dee4ea1edd7acfe20718a0ae2b6251c4e82654c8250e7fdbacc6 34d9e52a087b3eb6b8301b3887709768c1e9d427d150563ae023e0b00ceec008 ca93513c6699aa0e0a9b24702a5bd14d518cae21253a732c30f34ad6959ec202 de6a21d02545052550f225cf78af9fbf535dff0835cd7a4c1cd36a0487ac0743 a0512f08a2cbca65e71d9280c86d6edca25eced08d59948eaeee1f7f3df266bf 5aabbd82e3c642db4741fe2a880a17df6e6b264174c80c77006d691ca47e38de a45e379136d27357783cfac6ca6f3b68990ad886ea4b15e1eb1321bea1f30c3a 95c2bb126fe630f4debe0efdc824150ec37f2af5e89805ab5fa2f2f02d363e1f c5386e64a91f04aa32b81e7b6d691166acafb9d6dd5c283437a163e0da196d1d 1a1252ab0539707b340e6e79a3b057e2aa2779d02098254cb184147774d24286 165a6c56ae4ed4bca670222be2b978684e22aab663c2804c3335d177d61b9b13 ca8f91fa7b2c004591e5e347a81851d4cb47d16346ae78a15791ad6766058fbe 17197f40ebfaa99006b92ac389c16d023d7e79b05a86525af6d50634f0c9aa20 bcfc7564d0efd787c94c068946f6094b945c1a50b1a3c61b3c83e153981dcde0 bf7be7e4e022015e14c341548482062f041d2ce6e518be4b9e78ea4312d8a837 882c85352829626803095880a79e6491f9d263fe5e68cdf94b7dee92e7314a38 fad61d66dddc6e8e2316854d64463d3bd9f83a8a20728f3fe42fdb5a962908e3 8be807c331e4f3bfd1b92ca7995361934c0b2a0a112eda92c8b0fab67e82f587 bfe0f36a73824805e9d8b854724cc0f584af983bf32f50a3618773b9aad5617c 1e587b7ae949aecaaece7eb6d5c343a738286d75e4adb98c9aab151756eb834e ed9cfd8a6cb58beeb85d3de1850c30437458380d33ad91b2b0cc3f1c549cd64a 871ab963eeb96e0d09b618f1b9510e5173575f5a6980247dc057e604d463d2bf 6af01b362cc704198a5a189d5957db475a9448f039fb463786ffe8df665cd684 278f0aa895c414f18c2ea8f98e49d303cd37a9c3f68fcda2897bf9476001327b 606e6decf809d3da40559f78f7c1aa9254c84e1a3351017179cb38d7ac67b6fc b310620a02b92a8c10d4a25d45f5987d9f4a911aa3a1c3df29051d7c07a50277 e5041487f2c0dfbb203bc2d25e8f667746e17dcb14db47165f2b6e1b7329d3f4 ee911a33a6b036c723527862d3438d5ee39be55e04f14bb003303619be797289 b14822ea4055f9fdbba368e7c3f3e0c43fddbebf17d78c3743a3af3ca42cf706 61ae917034ff26ac80828dd504e3f4059553a955e77ab648ae11acc895deb65d b3873e7d134f1546e1d0b21c4981f376bfd3030c57c57d0aecafb7b025a76d8a 3ccc2da20712dbddc697e6e3e2afbebaf685e6b74112afa8eccbf96daae110d9 00b492b75c5e9bdcde9cf05f56ef6afc73f14cdaa2dcfbd88661500ad6bab6ad f877e78bb2b81488df0e0609c1289508a258a8e4877b78445749dd7c6ca4dbd8 8be0cbd7b0007f368879ce2811bfb2f0a25f1664b221b40c71cf36ba48be66b7 19b219dbf76e2dc4e7fab39f2454a4cfe2d5105cc03939998fb884a83d3d1ba9 0968bff7a52b020dc7cbe52df89bfefa8b6fc3c38b9443dc797ed60896d7673c 6cd5e319dfb2742fb8d00f57427238ec4946623e9df64ffce943f07e02d5ef52 70e6d9f5e6a28a5a8348fe3187e3acf4664a2e4eb3698d321c21c3b44b20c5c0 f3786965fceda8b3fc51752dcbe040ec2c724f44ce54b820dca0176ca1a606ae 44eca4ef3f2f48a074296ab10921bf9e3be6273709006516293386b11842fc7b 054e7d05e22d35c0ece932584e1ac053d55edb15d01abc92932bd1da8b9534fd 6359e3ff763844b5e98c8e7edad22c762466d72e876eaeb4c90bb5007ee8aad3 7add245297de5938d545bc314fc9c67f7e7ef950a89f3c31c6c06ce632499875 66257fbe586095512e335321b082af123214754a4b5e2da8a60978872af105c5 f5f32e1f3cb5108547e791de7ca8856a4702d5c86060a8184e6b0a4305af9bc5 04056d05a0dafe9679d05f9e091db401a98c1dd91061fd647937f21d9bf410a9 ffb7e0b5e3c9e475803965e15d4a3a93c7526646d663dd770f766f131d16372b e5551bc0291111a239a4d0bbf274ea3e58070f6cb210841cc7695eacdffe098b 74808b772473916d486da026ab2a5dcf0591a2d9154fbd3aa5d474cba6ed1c9f 09874a08459e80f733108fca070db984867c72b998c7bfab8e102c8b2f88c650 3d11cee83f52d56fbe8d050c4d01d582f3b916490c2c2c7a768e2f1ee48fcac8 b92a715eac393fb972dac2357a16e805a962039083423c0eee45767f075cdaad b0014a69dd4acb7b888ba47fb5b01a84291e4ca99b186387ec598c67b8e15b7f f5874241c7188aca3b1d10803c9a1704c999df183c369e48b4c68422633781af 90149fa6b30d1302a0a34738f49d880fbd0d0e4159fd06b61e451e86c2cadfdb 6830a167123b4ea97fa80616e2311ca424bb13df26a10186f9c608779af5d01c f458b5aafd381f3ce05e891ceacd7d8a2d4e0efc3143fb2d1053fc76f5b29f15 ebdf74ec5753b437e65b513c09f6a7c4ca60f7f53c835021092c7899bb78265c 7122cab77f11d8426130e033b3d817fbe1407010719272b78c06926a509ba9be 3608e071ac08a0c2586f6130e161d283ecd9ceb6b55d13621adebfcda203e7de 90341d5599af94cb9334ef54581b819b55e0f0e819865b719332ce2f759d6896 29d0d5b0e41f7080f61259fec7d58a0bbc8266d2659d0b27698a1a7866e17619 38c99bcffe980cb5038f0e821d71f5b1fb4bf3100e9145ae61116a12fb74c469 bf0dc178116487e6aced7ad83cfa386834778e9024f2ffa26064c6a9f8eec77b 0efcafa46e267b6127d11b52f93a02bf677a8802993d63f29142a0323cab4620 844b02ccb13177566388814410f9ff97a641e207e071624375e81a92bed9cc00 60a178eb8bfc374d90005d8bfa7fc5e53b74f9768e2c69d5b4986bb81457a249 eb046e87f91f0f607d8a3c2893913aee81faefcb7a24cd4050e57c99dab4dfdc 3d5d2fa14899390af859a7496b3182c966e91b47db45203b2bfebb3990599aa2 b72f2e92a907a9784145d5bbdaa06e63bcc515c45a0d00ac2a154fcf4218ec92 f11204a7b21bc233f8234fc4e4611a64f0f429ab6716ca752ec7240eb0e4f7a8 f0046b1e0065c75a08142f8ac0a4eaf780701e06e5fb07ba77cc1f26b90d8390 2b3a04c8ba87e2857b512ade2880bab7734ec0ef0445c720f598ff728df0d110 8b2c47abb56265e2deeeafe9182c0cd77ac4299b95e95a344595649660c4dfb2 4a09566a56c116242cc14b3348f66f69d2739dc0b86e7b15f821726d251f0d27 7532b3c13373c943de829b015e6dd63d9b19411dafb0e20dda0fccefd80ae200 190c00e7b19e3fbc225dfba8eaa4ae0c29b62de4103cfd147c678dbd813a2d44 2325eef2be784948a1eedd5bd0acb1b8fcd3d8e911c2a4177156349d2949bb17 d1c9dab7d59a476beffb0c644eea3e28e32dd9494e73eca1014400e2f5d65629 b4c8cc6c58e40a5d0cc769ae374a69611905387ecf74e2ee90466c291362cf45 499be1105d7cdbc2b62d97549e7b1ae5788bb857cc3a160e22c89e9c05a4e0ce 81191fb53146cd7f66b467c925e8a256f4a58a689a5d601e07a079ceac8ecb4f 8a137b1b214e5530a20d354e98a950ce21c3076f89250e0f3a2e799af9de14eb 034c2ab21ee93c04d5e2da0dacbf9912c9e8458da8ca576ef50bfcb3a8bbdf3c 675b2546d9a291b643cb6dc800954e1c1a3dfe7af65d7f30d0bff09e8687d736 1ed999143f1902ac338af2bc1e3384f5fd8a95f573e8882a8fc36f62519de460 90d59130f2a33c4493d591ca3e97ef48379b4e92bbf6f6d989234ec494abdc56 4a203dd72f9b6509e066952c5e545fc2b4e27fc385dfb54097bc18580a8c3e59 1ee77278a270eb7f6c07a1631ace8939f193215cc0efaa1430a23cd18d9feff5 8df6f418c1ca76d2cd0c081decdf25ce35a70eae86cfd9efdfebc0e6519bd98b cf3a3fafa7754bf44cecd6bb863413b53dfae87902abaf18cbaa6bf62cbb5df2 363e579ba37a0545b6e5f3b1daf51d507fec30490d1ef9ffa71c1f9cbcbe623f a2708ba2ad79ad634967ddf76faae71616c624dc3c596e8eea9bf9d2b702dfaf 8c87e0273fde463bee6b97aabe20b9069ee54c6aa01fcb4b2735e1b148e4efbb c9b1fa7176dc105a0c85e9d4ac9b312131afe557bf845ea8a0f8f71b44872c78 26567cd22dfdcc3d2bb1a5d88196a646052e5b9fec13c082baa9d7f7e7980351 e13ef76726f9c44c091ef4eb3a85746fe30259577516fa7f862c34a0d3f45b68 3f4cfe0cf2e477929f1a954d08c29be30d84dd7ac8ed2b048de88c23afafd1b8 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /subsetnoteql put dup 42 /notsubseteql put dup 63 /emptyset put dup 66 /B put dup 78 /N put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueXX 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 858d8b430efef0ed914a80ffc9818aba0fb30dae3694e5b31df3855892d59b89 7f82ff79fce8444c7926ffc4ea008e63bc518936c9098afc2d1c14030a2c15bc c9285c1b57e80570d2f6301009e63d9ddeba44e3251c75ca9616a2c5de9ade66 1a15db37b40b67c1bd71f77343912c62f47dd24be0e03dd833ea60d2eb020407 eaf042fed30f0c2f01956c83d46d982b50dd979c2026ed5f71979dd98c9c1eee 25054000a5eed5ce9498b2cb59a7ee55e0538ab47d86e69b2ac79229c8c6bfc9 3a24e822a2f1c9964fe630344579a25f66646113f73d1315c447f070c575de76 266c6773b01137e4a039df5e4c2f199e8e3396ef9fd001eebe090e04384c4089 ec9f13a07ad8733c6af775f432f154ef50ec4477e08760c03c006e4d1eb1c7cd acb8955ca4b0d62c7de046ff58d431bb9883d28586552c51809e3ced4dfc2198 b97ad9e52ea33c91b9e335a80d9b77d3430e0487930f992459664abca5dd9177 150b0fb502fbb04ccaa94ffe2697ed07906f95fd98e7b325bfbf10ea6b39fd76 c1e5295022cb39e9cd471b7b0b3f9d72da99deb0615d35cc22f42dbf5cfb70dc 6279fd520d99f32201abf25e621cfe1d37961d9c0ad36ef3d9264c547638152a be30812b175f2e06c1d76ccd18c61cac90de08b1b44dbb075abb9f9321d670ed a8f7eed2a8de0c93c1c1cf3fbff392ef1775454654e38b2fb2014981ca3ae85e 259053c58c8de2a1cea6b16b9df113f6f86c9ab948114d0f4241574033c78f33 6c560ff935f7a9cc52fe5bc59e864a74877ab5256f6f0b40631420e6d7db88e9 816d9fdc2d2f7f264a4de99be97a3efe7fd54f8b81d92f6947f52f6d7d106371 477310e2a9d3a095171b5f8249a211100a508b0cead6668c04c51f1913ff92a7 d5c81a1444b8071d16a2d911635e102114e2d6190c7b45357df695a743ea5e07 add8854a0aec7497a871dd31e87030ab433666c6a20bae93450a9fc4bffb5b5e 974487eb05297392a133e0db9aa703859680f515bce4d39d5230471a0421f215 12a597e5132347fd9837ea85f10d4754bec03041d75558a71c0fc262644d2b64 a295e9eb6aedfceb249d1d07545133ff9a7e4b28859908e4f7cf8baa97c80992 6fcd1373c7a9a949fd91e0aacd16680387790071018d9f39a4629ba025a1b755 77eb1e4866c036c5904309902ef5e7cff12087c2c35257912ffd6df92c8f864c b40f5aea3a2d7065805809ded84635e3a5cf127c15fea29d607497e9166f532b d2cdb29c73f7bf25c7773052396735e8cf8eea5f57f70cdc43f5372f09ba14f8 3277c72f15628400a577d9c6e66b5d17bb5625ab2b245390b20f9aee8f4698ef f523acf2d3dfae2ee0ba789a08ab5bed77b68cce3d7d59ab69bbbde7bc3d62fb 438ef93ff5291211f56d0ef5c61bdc2dbe692ace2d8c128ad5b705f14949e2f0 8e17c961981c662999bf25f4342a0fc95062f039231a09f8e21ed5a907fd753e b3bee1f7a56b374d3411081c91d94308f8526a46a5c369ba5e4fd556144c5127 851e16e903cd020cb0676a7737ae446de3514d63dc775e56ee4681508e3c8fc5 25a859cea233d1da355aeb04ab9345d69014b36d3c2d5c45dd3b24189fc30af9 329c1d9827557dd20fb3c3823ace1bfb60db4c02a9edce2cdcc6960effdac04a 69193f4319f1b45934708ed350df3e323cde2b046611f5b6927d9a3c95f0760a d5986e95f6da8ecdeba0555b8d4f973aafc5e1cdb73b1e82b2b6c13bed552f1d 8205c50535d352cacdbaeeb79181e87cd2d3d29ac1d977d9ee78f65d791d4467 acf09002466d46807401aaf986170308d14de1e58c2dba425d8b13dd762761b9 f60956d6c03ef5f3ab535459d6da7f2bb6d645723f9fde879570a9141249d9f6 94eb47d88bb9489b45afce53afb00957a013ce84d5105ec238d448b7fade43af 11a6897af9e1afb6671db6e809e81ff79d6c77689cc5073c28abb6dbc4df64b3 db1237f2c9869000a484970ae927849e6ce73d98db1369f1b0da94ec7e182001 869b1886d1767e96dac7679d37501a198b54198d85e9784eee127fc6ca1fc849 1dd105ceaa7c99d4af08d6e39149d1ce80eadc00e7a7f9e94c523cbbec69ce9e 486cd9dc64bf2ad8ae2f8935f4111fdee0d9c1812cad0976e5f585d5c9832f50 bd37ff7dea8e5123eef8cd4ecbee3133bd2bce988f295ca015f0a4820e6c903f e968c24e8ec7fa2c42b03696657cbc939719d3c01b1f8ca55e2a37bf834696af 9a5c5e7fdb21af3572f511a431540b2b389d7646b5c1093e2de6c00d0c185db5 2c5fcf1a105c8a0234083f58117f2360a03a5700b45737c1b0b22c534fa9d4f1 ceeca2150f02b5814cba8d70ecd54e391b3dd23c8527d42a074e7c504cc35648 0dde249940bda297a2b4addfe0a4cf9d5d6959d5184e82f263c819a19432cc8d 8d00035f0aa1b75c0266bf3a41b63f6c84be29ba052fcc71d5c413c997109192 864b7c4bad3086508fd902336c9a1e4c611a60b33eee2dcf0b9a52e889baeb79 e737edc1b04a2652bf0492fa0847bde13758b4550e3a26a141370cc605edc75b b5336ed7ea0566bf7454a480860d9560d9aeeb7b4e7d3cab18a6a902d6231899 7c5b688c741470f657150dabdc846ae8264ce544669e86b222948014382134ad feb5d245d82dfc5a2382bd55e86dbe770f932c11548f390bdfeb6aa38fb5e1c2 eb0f73e0e87d50e57f596f100841bbc038354d649559e0a9dc7fed51d67ce6b6 46c8082036fb055c24771e374530f939bafcb39ea2e078d730052ab2f080106e 33766eb3758d023cd92a487dc83104bb92ac0dd42a6777c8d9b373fe48750f12 ae26c2b9ad0fadc9259fec912b53599470c60f7e7209240d0b8a273f3690ca55 446a753f6aee2e1b9ef497c7cd19241f545e60807a1073ec9e71ce405dd9b805 739267497da0ed885b4b1a7a18783b60a21845401114410954911a3a05ecde49 c06d31dbea32843b2756446dfdfb2fda85ff9037f35a7a8385ec398b06f6fad3 8eb63db4f4a1e8f38ee87517a1b4eb661ea03bbf0fe2a28618f7ce0543f50215 001d99143ef5582b92a2ee6e82bfbc9490d84cf4c42033568347b628b949eacf fe6c64756672019b9cb9b80671826ccd33eb442199fdf4b1f243dcd08702e5b1 165778e0128e5faf5525b663954272e1cd8d71d45cb73e4a5d216c458e8c8f50 65d0804378cb088b19d61c9ae0957935fc9f707ece8bfb1fbc43abe731688eb7 4282939599814c65ebe9f80977c1aa03f344ef75f65b36a84aeba0021a768e21 82bd78b905223dcb24ef5b1986e0e4775d1d7ebb82bcb760fbb2a1b64e2d79ff d4161013f23c440dea6c239bfaf61f4774e6a2a8649bd2b65d5dff4ee62c6a52 b8a447223e5b4581c9be262aa63c5031319db81207f17ce6b28a283af80b01af ad57e3d09ed989a76f96b0d94a0bb6a5cc4f012408eded51897b31b70748ce6b 30eeb574a0d42c64f522ee6401f12a2d762c85fb19cbbf3f4ba9bddadc16010c 0ee5af82dea37289357351117d7fd13a9f5a6bbb52270ed12dd949070493575a f5b65067407a5c4a0ee19fab58244169478f44018fb34293778b0b2e2c70f7d9 5a165e464e0e7c168a0f79abe022fa3e3c6489fd51fccc6f350e613ff0277cf3 791561c922df736af534d86393e36457db6c736bc02073e90e76d64608adf718 bfc8993c70f0061e9a76f7e33ee78088ad5af9beab570c5eab31fa9df0d64fa7 806a02cd0de83af0228ef80630a3be660dd211b7bcc78c87507fa1115a5d863e 2b215d8dc8c6e928b7d088232e790b86cb9e70965cc2e14e7a0e59fddb8e3650 1ba414b0a19a542ea53afe3e42df4c61047161c09aa7671db28518bd224dd95b 1252215e4476887ce046d52bccb5a16f7db0f26021e8002bc677db853b9e38ed 1ad993adf8b8eb816da39d07ec78bf5514936830927aa4939d7f9ee2d8bb2112 2672323a7a6462e6d0548e64ca8d7e78f9b2891e50bce147fbe4a11bae070d2e 5bf1cc4f411775756154e2932d61b7631fbdfa95546ffb1eb68b06a264d98ee6 cbfc157db8b89788c6dbfacc0301123037150af4258f4c2047c7a07bd027e90f 9e2ecdd457a044c5da1fecc7f41278dad82bf6f6ae4fd7112dc300d1876334fa a3040e2f223cdc45e706767e96b4c0224eed792ee9e925accf721616299dbde8 c16a9eabcdffd53130e3e5b7cede6126da57756c6d0f369acb0572ab140bd32b c1909fbaf68c86cd37b86d452299adc519aba5ce7b0f29136b15dccb3a7d012f 2d0266f1e92c6ade0bb4a8ba14cbdae6d2f9aad7a4b06854db380a65477c69df 6820085580e6ba9595e4afc0f9cda2cd85c87cc46399b80e488d642e0245e8a8 2852fc26847c9c6eacc71504bbade43228d4beec9e0e0753aa9637b7bddb61cc 556791d953abd19ef5b438c514b62586b6a250021de7a7f294ea4370cc3dd8a2 7cf1d45312633b99a4e7bca294441af4a6957b4440f654e8753c51e0d0336d68 9e61461336d806ceeee44075f184d61e675537178e404213e11cf5e021f43b31 e2f6ffe6d4da39895e6b07080633150e0de9f1826b645407d69d987478a61a8b e32ef29b8d921f348d1beab5f0ca2d3ce4066db436829ce409417a60b71762f0 29c197c197c941115504c4715ca32fc3ac75c13d3ddc159b51f5366230b981d7 86a1ecdde301eb91135516a79b6354abf44bcc79818b83b212f1da5b8433778e 59dc2c9507cfa8ba3effd03306ad966ef920de34e366d21ee4807131fe59ab18 0123bebb39e1f23ddb489d72cb3e1b4bba73d938570f07213da25c3b01b89681 a5c0ef7f9d6c2cf389ff16078fc85a3f332a3dd40fcaa31d87c5a5c9763278de 3606b1bae69e8232d3c98d8269a6345eee9257a3b2a8cd552765daf0fe2d9c1a d3740d54c5ea9264e231b17744a22429eb39226eecf3cd7cf3d4ed021a8d4cf0 d88b3a0313edeae7f26e05f1b86ff227 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] 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 6 /plusminus put dup 14 /openbullet put dup 17 /equivalence put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 27 /propersuperset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 56 /universal put dup 57 /existential put dup 65 /A put dup 66 /B put dup 71 /G put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 83 /S put dup 91 /union put dup 92 /intersection put dup 94 /logicaland put dup 95 /logicalor 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 110 /backslash put dup 112 /radical put dup 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4 9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa 4d739b772f8d674e567534c6c5bbf1cf615372be20b18472f7aa58be8c216dbd df81cc0a86b6d8318ca68fe22c8af13b54d7576fe4ca5a7af9005ea5cc4edb79 c0ab668e4fec4b7f5a9eb5f0e4c088cd818ecc4feb4b40ec8bd2981bf2336074 b64c43078b090eb68d3192ee14d751103975a05f6af6e4efb4281ad6a860bc38 0f806bd924f7d7f3ccabd85072eeff7d07aeac9ed13c27089a069d0333923e86 894924068fc431b37462c62ddf9118accbf3b5e872f0c0f52a910e71ce7699bd 69c2877cfe0e9e71f3cab04d4b9da9bbe03d5277f10056b468af26f6d6a06e33 29c9ba396e3d5d98e442f6e0de47989e22b264ccdff4c0a2727f375a54e5d1b8 768fc43215353b9cb9acae4b4ba5b728ac308db487a46f3860e876a312027b3c 9bf232e928ae9d54fe71547dbb3d3a00acd5f8314b3e9ebce0c8f1cb6463b71b 29b94dfc190d6c9e193422800191844d30179b9aa7c9008946b42ac6b6912dc4 e6d83b20ab613c887578e8be5eed545e99f526908000a370cfcd27608564a4e3 c24eccae99a23f74b1233a0127be382f19fbafb647209248d1f387ca1816e5aa 8656f51a41276d3a26263b9f1ae1477f8d5427595db7229d78be4fdc35645105 d4f73229819cf730bd2a4a5d4fc23d7799947def8b180a982ade7cf196a137ba e7a4b5b9e8d42ca1ce4f301eecef5be92d0ec17cd40c5dfb9b67459174f23c0c 82948e5148b4f5fcf3e717057204bcb794173029e592a8e8190ed939a0f1ee20 a1179b6b9dc433150449528a3b4f8fd58acfb963656bbef3d175585aaf9d2d30 8b8faf502235d14122cba5838e8bb57b0d6b190b32cd31a13b527762d6eb85b7 3b3899eedf893a83bfed0743b0a8255c30ff4449d99dd7b2d8f03a5d2b80da54 0303792ae783f18b473cce8151eba0a467982c9283cf46eaa4c240ff8f4a1705 2b23a447f441d5a741ee507c9e22b34ca773a1f2487a9a10d09c430c933bde56 594ccc1e9b6b671c5d08667f9aecacf56fd55c91a91827ed2f7bf952adffae50 b71f2d6ec91553e5bbdab94320255914873b2d80515c72f051be163c69154f47 a3c09a9a38265f2530dc05e7c05e96cf2164286a754882a2d21a1cc5e5661cf9 10dbfc439a83ce3fb0a222f324a41b70af9521f9ca920e3c00e055c1e0225071 4fad7ff082e80e339adb78fac1b44dc5a67c398027ddaa6848d95d0b4da1c042 2dd3f68ca3323ff975aff0c52b54b8306191b281f9d8077e92205bc1ef37eded dd2af8e47766e4da97e1901784e6baa8b8ebbc4bed1e823699ab620118a3a9f4 d2d14f99ca1331eb7db5a37a99bc2f244227341bea7466b6d5bd9fe7ba61fc55 3309bb9edaf8e6bec1bb3db3cb89fdf92da0b1f70add76b3d7c60f666c5b967d 59380f4caffa98c5711031a165d4908e05d1dae21d7e0cc828566170db098e78 a07c699df5a45b8ab75c15d7f7a7e64718b7d66eac4eedcdc6a5348ad3cec606 64d833c9e4846de21be3dfd55ec0d8a08cece51f4f88ccdce4e735af18fd4117 4eaa258a7f8cdd27dfa8de1ff72ea0cf87716fa36bb5c63c749ae6f345dc5a86 1fa956d78f1b278271a1a6ab2660317c30b2dea2d725e6487bc317b4e5ff715a 3378958c592feebd81d6c7208c790e84096524a92a8d947f16a3c24225753bd1 287df1757fff1575610b37163d399a368d645766201581ca5c422e364a4d2f22 fd8e57370b90022249091c89167949cf5c3c4adbd41827a8d322a58e8f8d2bda 6465a29fb4d4d9b066397d045f96f8a4d44d8315784a404f4a8e2a3c860745cb 0b686cf5f662252c29d69c16076e48dee4fe74ecf3870133590a7b6cd3eedb83 c940b11a3738c291256cd5afc09c80e8f9a22fa5b88c80fbe7187d4df467f8cf 8021eb16a2c308580b383d14a1f814547871c353463a7eed807023f66c943149 e5e4cc24cae39e84b6a32528a063b8480224af2bb2b3c319f68db4dab435b3fb 0c118f5ef33fe8a4817754adf7adc50295a4ddd57bc69fb9a7dd4261f47e857a 042c91c573aad67edcf875989886ab6e2d7c4629848f93fd8ade849af85e21f1 1403d8aa2611d2cb36954629f31a3615598e4f4d52ff2f8d87059816fef066f0 8c02e1a367e945e537b1b27830fa6d18332868e9ca278e0fda9063c4281d7cc4 6308eae2250b46f27f5e42a36fe3154f3ec1df56d23c62052af7b3901f368601 f8b449c925e8b4f1c730f09df35ade10d04edfc8e574d812bf166460e017114e 6d73b6a4622b6ac2c4a23dd4afe654ae566f959ef12c8255bead837b89e5a178 f658e04292573c8486b68bd8a8d85b13511920262dc0e44d856f943ac6c00b87 869821ecf56c16d62bcb8d80a59288a7d12205d4af33d07ca8ed42166c1b549e dc216ffe74f54877177b3bc82b88ad4ca77379d2f0099f2c046d51d4e4750d55 47bf5c45c6be7aa98593c3c0e1cdf09fc4d9f1e7b6ec92baabc37e37f6d34f40 f9a64c7f9f865a87bd29636717c7e1303bc3d2b465d34683cc0a0f56d6b5e186 3bc42bf971c576ed3a447cef0999a609eab027a600be25bef9234a1076d363f4 c01c7b4f3a3178a9337472b721fc971974fc49323b6a464faf3d92dc386b731f 485c819560552cb277b38d1c3c404fdd8839d8b230a8eec990cb1fada8e38ad0 8d3495de23d475c8ee53a86f3c1b0cd55ef75d930b86badc8143dec5b9dcb758 c39b4aa29d924079d4d25b56f40d20835cf08870fffc9f5308d280fc373c00e6 9fa1725154ed9f098464afc05d1bfa5f59b9bf372b12b03c6bf8c816c4f92bbf f03221be59036849e8a7c6c9d6e89680382ea44a62b755d9d5430efb6dbb4f3f 442c12a2018cda3b760febd50c1be32bf4ea92ca6e8bc362d6630928cc7e08c3 e90ad96d0fc86d91d56ddd9555a989bd0ac622306c6a947b8706d0c76f3974f2 4ec859aba7e6c9b1284494ee3ff68eebaf29540b86dc6f02fed96684eac3750b 3f02d0ae2c0726a08dd28c0b739f76c3b347f329648f7f6e96acc1e73ebbe4e0 49ba83d4a2f9bb3badce9332916ba6423c5f04d74ed0d485441978adcb6b81bf cb8cff52044c020873c2d484e19af2f941d35729b45e829d4ae7c019aa906844 68d28459d60b7cf477836152f8c991b9bb8cc5c6e249b9ec1cff5d1c9b9d04fd af237e271be7e254229026824bff8788fb703b55be6180c4d0c89f8a19baab34 b20f3e3fb22b1b473ccd4c48e3a4dac1aeac9e4b52f5b10c3110fe805983b3e1 1d3bd8ea5b53351bf9b9baad95adaf948eea2567f781c5ae29dfb10cd41b3ab1 52ced4dc160dac7d315dd91fe87122060c591a07240418f828e0979f68a1d1aa 1172a03cc5f6b16f4e402568f1a7cb4fa37fb1d87248914f50365279580cb020 f8e3eb841978f18f7b2b7bdff7abe8663d905c83320ba015672e4d288b55718b 2781d4122dae20563ec26c1ea1bf10aa499f57d5ab3cfcec4d77ca32b83cfd04 858bfce90e8816068924e39a26cd96636ab6dd072a478997fbeb5b2e92cca339 25fc27ce16faa39fed246c8bf92a4b742cd2b345649f5f892535d07320006cbc 57c1dfb2e9f5a681eba86171d97075262a97fecda629dcf53315bacf0b5f0db0 52b1fbe1129703772f06ae4dc0e4faff8aa09db0c3720473b4c7db5eec32fab3 4126f78332e8f187b5f5aec95e7484cb298ef79721834fa46c529ef91f8dc642 e7b8111a105d3fa62c1885db7a89c05acf99c25c17f55dc394a36c7806ab14a8 b8ceab174c1c29ca148ce335db9a6213e238a9c03a2b1ac03ed7810ac788b2bf 3b981422adb3e06619440734a141eb58f6e2ef0e385ceae8fd537fba7afc1f14 9a92f6a1acb6fa6a2e527baa846bc8a1ff4bddd49f55f36de72b33eea75e25cd f9caab84993ac55b84af283b078da8f6656e8b32d6d73937854cf8dee9193280 ae6f57f47e76e8428a03c65a1fd9b7f4e7202497f9541a1da5590e042e60ca00 0cbdf6a18bd6085abb568d935f0ddd97dddb9f61c2540c2c0ec8af399700ee29 248a500245489794e242e8d477728e83b353e48f23406b8b1a60e832d1d4363d c233c92302b816e1729b1433607243fdc05c8054777ab1dd9563ecede88b2271 d60f029157342a97108976abb54ba6f8dab6f0a961d7f5bdbce7f745f678dad4 8df89e5fad11e2eb1013c7499d9cddde47179c705a44a8187b4267c9e680aa91 4bb581137c11b305c7b54f9e5880dfc730e97373c4ba503c787edb259f2b49e7 ee8db971e918ba91e8c177df8cc289e56c0069e5c623e058224f908489f228d8 f317fc651b11d7156846645c356944c26dec1660bef596f702640318671f110a 21f5019c83fef2818b0852f6679c8dd9bed4b9f6c111560ec2600379c2ae51d7 0f0cb1c97cb65d09d4cb3e50d7afbc552308784652c0a3164469348a8f17f2ad 8d7f196633f53a83eb3e28dbbc4ec905323395d35aaba69f5d012819618ba724 506809e9849bc3ead5d7e22fa548e69e258a24d6e6d1f36b7695fb3b1a3f79d4 b3c879e00af0ba0ee16970486fffa8f04d2e5a3be9d194d2e9c4cc7cb68592cd 9d56cec5bf1b1777c9c44634585a46eaf4fc00ae8ac63e7ae62428da80fd4e2a 6708b416e5e6c09e9a802d2ac892f825ba743f28cfdf85993b86818c13fb7eb6 ace1c46cd82273304964404b4d89a88b23ad036c1e3e3d57689c3f1a5ff976fc 1571e784a0ea8584b1f5134b17624b587caecb34587c4648037a58be020171c8 0d18809b1538a96fc8499f9d0456995a5f133aa92790060d758c0f9d60eb3c4f 409b0291fbaa6d5bb99162fa513a21c3c94803af553b894e5666397cdfa2a662 b483af8095f1bd4a185873546dcd08ee5c2e4120268be2b1f0f61429924ac25d aaab172dab277249859409879537f08364c76f008a734e125b05a13b81301147 f40c7e7f69f93f3212a0438d304dda551b56c14bc68ac7a932b514e2b6227942 5fc76eb52b1c0c1cc6781a64e81b2ea2d9d77f85bff84e413295cbba66af44bc edc133f2f40aa860c8a6244175c422e4ebea78d9c66292ff7eec6192c52745de e7d285120ab26167f2bdb047f30e15f7429a8b347e2120700409584c4870f91c ec578971fd40608e74457612035602fa087f3808be7b1c44b38cdde581f2444d da7bb1483988f3d4d3f3db27aeef522998931173d7a6642cc0498179d934dc45 fbdcd7499344f03285f5b46ac3e61692a9bd8670c6396e324de159339f2d6c75 c03217d5e1a8adace97c1c97bbef5a1938f0aee0f888befa287990c51d3157c9 ecc6af473b052fc6adc324ae2651c75fbaca607980a51c8686d10c25ab886f08 d031cb9d13717749aefc5c9fdfa2dfa92d6d134f58253514b9f34ad09ef6bb34 a42c699420d463110b6b911333f98b236a5c8dc8bd0230d2a3a49449e3e8087c f0dbaae7611bdda87e3b4ef4c837d35521e6c6bde2c649d3b3de51cc0546d04c 12c7bef55c0c20ae456e2393e02b868664234cb7deee7dd82cafc58d36c32689 a6d7e100afde76e8ba0d78b11e5bbf9969038ba826d56cd1535a6cefb8a94376 d09ff67efdc9b030b0a6e687294cf8c25483b8a0b50abe441c8405034e195a19 0cfde1e3dc8cd97a3f74665df94a8587cdb2c62bf73fd552317f971bac825cf3 34161734c5c55be42611c2abf6f52ee903bbb25a16f47f24a8cfe64e286d976a 3b90c5487a729418c3d1a9bca63b338613bf0924544ea647062103a695914e26 8ea4dd0ba23945a145152be211081d3f761b671076ba67fd1bc3db08717f3264 1cacfc0551559563c4f7b19db3c38cd353d6e45046934ac22b384f78fcf4fa32 4432ae5cc50fae7cb933e3c9ccac30ab0dfde913cdd6abd1a14c3f2e4009b532 9d91ee3e0155bdd40c5eb54004d94f92b1a4ca9fd923cb8504b3ab772885e754 c07bcb9e73a54c5b8d56ad3a04077017d8961119f60ca8adff0f29f022b33d74 d3c4071698b9fac873af5c64487011b8d3c2208f968bf2f5760ec44d9e15cb97 a969b3c58b84cb4b14bb9005f7c18c0b5d8293ce5e7ef62cb4b83f8a409fd077 00c426b254f477732e17cb2c5fc8b51d98f0a89405e71c756e0b6fea81af9061 b94f71b9a2e7c33c646651a5b6c736408ab76cf0def8fc7a346332f6c5dd1e35 2dcb674c66e06f894683fc20f28bd86d9a1765800c49d17db68402c01dbac61f 7b4229c4dbf60db4044ec3d92a4272514dcb65135a6367c1ed38098c7a1c73b5 6a54df605e4d393c26b5f80a49b17d7549e1aea43dd79d7f9d0aca192e29c2f1 88d417fc7462bb3f86cc9eb63cb4839ca0727a7321733fe5cd988cd445aaa131 a76ca509f3fe2967ce29056ffe5039575e12dce6217e8bcdbc7faa06b8568409 0a9f7c428f8263305c48d1adea43857f59c63f67d78345efdd87e11104773886 c785092476ec236895d3d26d7454e1f770c4c1dbb40c21ee4df292fe831f2ea8 1eaa956f497f9f1a8f94106eaadaba6c9ae674756e5f135cd507b464fc8d396d 5cdc0ff9515347b37a06642893436dacad7187d3cccb18f38c4d9b7fd564db97 c8b964e78d28d7cf037b62595c92408fdcec66527542a561f68406d2a2d73568 b0e4ce6e0b09518501bd08c14904dc5294b274c2ed63a888268c8a874ebae618 05d14eeabd0387016fb7380d404a06dd4a1ab1e1473e43e1486fc691ac951bba 7579ece5a019be993958af0d9f1df5da0ed8955dbd930def8a9c30d4b8362d4a 9d382611bc6eb0d45cbc56c2facae12249019a0dcfc5e39a9305e3a5cf7def26 6fba57e2318f81b9eb3630327deb0d54b5630bbfa2f15d0af43d93047a9b3ad4 bad1a76f8f525a9738f03e4fa2899501e504536fddf41ddb3b5e4cad683667a7 b1402b09939cde0e9d6b9c1295cedafbe3c3c248f2e589f35baea16151f2a83d 06424a89e22e67653e740e1943aed928dbe8033ae3b5dd36530600b6653fc00d 46a0e8b1e4969ca9647c30d701a94fbad1cd4d08f3412ad59678aa55ff5b0c00 fff27a6470aa7ee6b835e2f66c618ecee058a407dda4e69fb6caf1a83bf8097e d3d830d6662da168f386679f17b0b61f7f97741f24230193054beadb9c0556cc ee950fdf4d43cb0685ada7134fac8d0f7a58f46c2fe6f0a8faea3dc19be44e13 f3f4bd885414d968d1edda1aaef96c145d63f5c06083b7eedb11f2aaab82c5db 72cf62244d21083b1e4075a80c36d197d9d50bd259db934e1761b91e1f992acd 335a3f112ef0baf62f9cfa41a4fc3d438928c851adde01dfecae4e4e33521c3a 9017c8cbc2dd40390c81be933e9e4a4663071560171963736b63d5ee9e193486 236dc78ef9e8b27b9354f5d7c643cf8089865abcbe6a8810121db99523723eb2 bb69993e42d7ae5ebb6861d1ed32f83afcfa9d3ffc6553eaa603bc6a2dc28209 2194fdbb9eb1617408c46e9efc2055aae3ef0dda419875a8421b5d41c186beaa bd3c6ebb29ff395ac2700c088923aad6a5a5089ad63a281eb48ed40b51a49b56 816e549359aa8b1538d1e1045bc3c0a1fb45b04e7161f3c40f9be1d29010a5ac ff7cae0e706e73badb6ebf6024761a85dd9cc1bdb307b34d588f5956e978fe5e 741f37327ce6b01c6ec97b4236d5e2e9aa19fb651f11fc1e1edebb49adb8edba 7fa56415a732f468ef5637da9805e70dfa284242e85b2422e7f9bdb94f170e4a fcf81652145da15048664f7f8bfc41bf236ab09d9c052d28c85ef31457366c3b d4f661d969d73724f7b6af402a39b9258133eec5fe22df7cdbe4c23215161c28 56b10582bbc9a3a5188845b24b7bcf7421bae512c4d482cbe1f42e317056b009 ab09237b98f940f417cfc730fe6f60ced5818b520d3e9f01782e24332a90546e 0ffed68778e6b88886b85f9d604757cf5ccf20c0c9bd96f31803a7ab8998cb84 b5ac7829214d894c529bf412494db1362d87613573ec303145ffa50cf435f80f 90bb2f9f0cbf21a9da27bb5d7c432566f4fc5d1d5757bf494e01047dc4c934db 5080d4a1a52390f73f94686581bfa616bb87890f3ecaca7bf661ccd17a1ef6dd fc6ee62bf99b4a4459555f8d2f8e3456ab8f018218cef2db20e8c12e34c38808 f44530483e21c6f0dc89a4ad9f89f2a863df4556fd87f3e79414d24ec9d5ced0 1fbc809f3ace35bac8de6a72fadc36188fcb0b2ea31426d20c4d0dbb32ae4a9d 9f17dfbaf859de05f8616d9f6f2f7797d68ad3996ad37cf53aa37f7c4d39c588 ea71dc9ed7d9c6a64b078aca7005b7d4c9ebb0947418c04e6df1241965b98b89 05c371c0d582e96de58d6682c67bc735192f233d4b91120077f72fb27bc110a4 3311cbc05de602ef8e2bce35f5a4ee06cf16a97630171f9510829b85209db2c3 60b30979fb198110fb027e9d533ad2166a9e11069cfd25db7861bba20fe468d4 936d3b46770ee49db4bc696e354eb6b44acc72a2310d6e09062db6d8a3ff2136 898bb331d12d316195ec23664a326311c202d15edf23fa497f8ee57f84d45b61 93ff72825c55d94109a604b96655f3fe8cfbb4e69810d008b8ea2319937b2b47 c5ab2122d71cac56498cbe30ac4cd6f6fb38504967fd7266553f7ecb8a930ab8 04bd017fb908760c11af0f2018ac4736506a51adc8f1105209b759785dda1207 9fc36b848b7efbe8c9641657f63c96a084ed17f2ad593dd7d0a1b277f69f28b5 579fa70e7169d6 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 65 /A put dup 77 /M put dup 83 /S put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueXX 5000818 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b66566ca5ba42b1911a 5d7f1bf343015eece988b7a93bce0c7aa61344d48aed9c92c8698d4b7c9951c8 7d103f2414b39e1437f9d2e50c4ee5f218f2e6716926a79ea978f13b1f855345 191dd7d31d8f82c2e3343c7a5894d95bdc492c28226834efcb5c12fea36ac5cc 430e0aa604961e34888adf6c1f3954cbc2498e225d953cf5685852162346f474 5a2a7087d5d7ad486de16d2ca8e15cee26e012671ba3bdc7d95cc8c98bb774f5 08625e968aee27ff7d1a06e63bcfb5aa4876c3f8f13b30ccccee73c3caf4e70d 98e6ed2f422dbb4950bf789680e064150995941a9f4dd68a575949847a7d012b b910bf03a18bfd18d2ff279844b29842a5e4038099fec03d478a0d01a204f1df cc67700a27d122e922da3fe8baa0dfe65dd43abccc0d11e84d7a80c1e21721ad 7b588260d0ccded9ae54990d79700d22e1bac3828c49489d58acbb2c6fa0288b 54b61e13fce2202e1b2d5c57ad1ff076eb1759ea165000de4403497fde46f046 469887475a0e04f63dc9ecb312fd0b5ae89e9617b7cec3d68165735cd8703fba 3091ae5a2b2498bfca4c6c27379170171cadcf02ae7e4ad461d4869e025b4623 c1bf6e98cf53a51b6752a19b5830fb0e2df339f86b5dff30c1a6831a4a3e331d 548c32a5dd6de87b069c2cb630af9f0671d8f6871dc1f17ce486dadb906d1eff d44a38d822fe4f92bfa4495b59201f4836f9bc0ee7f32a78e2b94c418c40ef3a a611ed72a04ac597c1abdc5275de74c4d1b630c0e1f7e429e9d63063ef0f13df b40c6060da2a1b47438358eb0853bf32b63cd85ccd845581ab2e0dab31a0f9a9 ed60747e2a539eecb4d6849d5941d0cddd70b3ddb17cca7d2e8e8e51c616cca1 81f1966f66561efbe8b8f0b0f0cf42eb92d78fda41c250b96aebd915151871a6 3d6e86561fac97340205a99743f387ac0060df93cd919a73b525dd6e02c607ad 44049a8099e4d7731fea8267ac4953174f90442ff36613940a013bd78f3d5bbd 8f462620f26474578f25cfad4cc0e763e5949803854a79e8be528fbc7e8891bd e85b77e501e7e144bd97132561f486e91eac3d35c71ca1709908c7a57916fa71 9c109b32eeec21888c82f7799ab37810765feb0147c376e87fe214647cf36361 90bc7576f02d0e78c39fcf7fc4d6b32bfdacc91a7d262735fe64e257e01f1060 f525160341def5e1d004f91278d126fe19ca0f92929f6ee1bea40e0e0635ef37 6491b2dc03eb1f59237829bf7f3bd84563da9e0111539307e2a8d137d89d9f10 9c8d3bfc98066939182e42290fecf43760f53309fb18812a9fa57419efddc67c 3b1d7a9080ab2e2de99c960130dc3f2ad973da60c14041b14f2eeb050c5edc68 47a734d0a168669a187c48996ec5e9f6a34adcf3af34408f85000d6b8ced7dc6 fd35abb0151378f5b19bf8038575f54434ae83a072f1664b74458060e9c0d014 9f968bbc7f4346759599cf9cc0e594dd82d6b52737b7d141f94cbdcbaf7de75e 2cf2dc30c70bce4d8f4a8ea3e1b4ad4a67344f1c57bb100f672be9aa4b8619f3 e59472fb 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 51 /three put dup 58 /colon put dup 59 /semicolon 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 76 /L put dup 77 /M put dup 79 /O 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 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 127 /dieresis put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueXX 5000828 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3ffd20326947b1065649b533eb5e9e3 a88a87fe987918a687950b26f647d6e08bf14d983ed1d4a8ca1e2da8d985f944 2eb916f5b6ed0697ac7c33e1e36a3bf460d34ce45f1631871097cb04f18e3889 4cf4ac1538eb19481311d24fe3be7beaa4a3730e8b4831fe59d6d9ce2e46116b 629c7ba2f9ce3ecba2f43bc162a5a077ca1b2882a42afdcec3f4b75b5d63e0bc 8e5dc95257766d8ea467ad9cbaf47be60f797580cced6884b3a68f70c91f4fdf 80fd00ed9139e7f480dc3a76af72ad9b434187730bdfaefe4cbfe5c7edcaaf24 9204fd703011932e5a3c27be468b7dab69daa18dbbb6335ab8ddfdc607961c7c 02582763fa069d43563ba17704029945cf42fcb19cf78c51df0ec4d851086d43 c1ab38e865ef36865fff3a08b01cbcb070ba4893f7482dc7819ad03d337e520e 7d8cd83bb8ac7aff4df36751f4f12aaaea2b6d7260c09a26e987c904003e0723 7dbcc4a8e4e85cd3259f40d3caf55c742cbb40ad0db1044c20a4f122a63f7b81 1d945bfa69662189ce4d089464d2ead9f6ed60b581b04114d5d45e97be27a576 35154e9b8464f2a0df18da855eb51b8f64d3054b0b2e84621f025a0776d16fed b988a43d2d64c32768dd106b9f541ae8e68ac0bb7399e16f4d4ba074b2152d0a 4e9b5bfc21bfa0d531e28165e518ffc054dd4ed0d74eca079d20e79670419f92 e03bcbfeecf3554d661596f6f29e12170804d15a250d917ffe7a8b6e8fa53047 1ed97ba56481589a65ca13619129f1e26affe6c22f0c38540f8ce37efb6e9e9b 60b51a5a9cb4454fd7c7f92f2f9437159687ab590f35b946a54491a7739c9269 e0f1fb48b129811c82edb3171cbd5105fe65906974db8a7da92f4f43dc7f6b36 6981146d86d23613a0b38d9d6191db9fea0172e982cb1cc50e382f69da1b377c 1be9407f88a2a761b56f398467a057f35d18c01cc377e006182c891ec80272e2 2e3a5c7123cd27eccdc87392a9fdf63bb40f02b7649cfc53ef933c6518744cb8 9cbbd437830ef5b1d58b00b67bbcf4426dc1e6ef583b0b4ad3aebd741e3ff0f7 29ae271f032019b1e7268043fd215be590f9cf24da14a90f1683afcba8006cef 5d5632c2481fbc1bd8224e8952debb7c4596a237e5505ac143180956ca1f603e ec70cdb4a33e1c4ca174436a38b06fd9d2b9196f02114b77a9a5f66595727faa ba323668573ae328f1f84e5a3e0e7634d559479c134207aad7a949da589970c8 da77d71a20a8ffe72beb0643b81044daf53e79809e17a14cce790765c281b4c5 cd8c35a9f87a34fe5981c199ffae7fe41503e38d5f437d0134d3a0a7bbda0350 4bbe59c8585052ac7c18ce6b421449d73b16961298c14548b796d7a2198080f3 45be1b247c5a2e454e47886dbaa56f166a4b875a9cc7b735bd0686a3570da5a5 ae911fcd2159fef835f73b4721f47d4734828e4c7e30997976d6c457941ec11a e59c8de8fdcd954c78ea7937031d7fc78b2773ff600cfe51766b2d67175e6d43 69d4b874cffb5495dd49dc34fd71466cffa2750eb4d989f14db03f19dc377716 6a0cc548125f1cefb68d46f380cf498d95d68ab7d027af957421bce6fa8addc2 aa8d516379f76c43a9aac57282ec2a99edc3426f299ab1f7d064042a3ae7d736 0527ce04029e4a27fa47b3e52e3fa9383c1fae5416599a7ae3951cf9a1cad2d2 ddf9c25212e0e9e7e4f7697064d0c8fbbfdeb4f7af58226c83cfeb52da79fecb 3e113a3b6d3f2f38e02292a45274cda51988894c34037a017cff131a14fd0724 d941761086e38e2ad668e388ad5def84660b4665bb5232418a5f6165fe6abca4 83daf33d9a00e77b064e79a2fcc1e7b9ed30e7e84badac927926a956a3c13328 666ec9eddcdadad5dc1517952feb2d32666e7253fd588bcae884413f9821a4fb 63523e3b5abaebc82b600f8401e63f2ab8044399ed5c1b270ca7bb4a758564d9 98f6278f3b59f77ad7e8ace1083ea4ac7a10b94f42a0c045b443e2009505935d 9b9689dbadd0069387c64cd9b12e64bf60e6041e51c1c10fcb901a9486c7f740 c9d99ea8cb3e3aeec31b3e1e332ab36ae8ecd180a99a5d8bf68837d9e91a7571 dc0e3a9e06f160f981edda2b74f7ca0a871f7bfeb7148d176bacc40c997d354e 1f7152f6e9d65af4a03c3898f3a48aba82980ec71dbd4438497e6a46010a7b12 2999e8c0d0d63e73cfe0af7a2e14ad4be89f94fd741ca5455abcf527a061c567 843fafc161bd56c7963b7013d7cc943a1b9dfe289fbc2d252a3ac9c57786cde6 521636c321a9d5d5169268317877e66a6cde9544d3443dc8f16e8ed46bb407e7 448ed6f180c1a25c5f9e3bd2dec13378e10fafc94af33d6ef6a9cc239ee81acc d2f4bc09dd86f5df617b5f4303010649b55bd89d7c631ad1f50add40aee481a6 4053bc88885fc76fc30d540f3e6e1650a6c9b1184b0f79f8bbeada251aa6122d dcb808696f120f8070939c2afdd021f296e15ce41f958738007bfff013d5056c 57d6c06e96aa44ea842d957ed94e3d25efc052ef4c8b4af8f729413ec59ecc01 c46a983ff1ef98b6752cb69d54fec82ef87ca703934e3ed8bf1f1c061af93345 038b0e8e8594c80f41cdbcbefc475b89efbbde5ea9b0917e9722bcb21c1535bc c4bede5a9c76b7d1e9dfc2745bdc5ff48aa41b5e21263acfdf7ab67e48872bd5 fce05259d55f4a89497312577c8e787a913eb191e4d85d71abedb98cf4f727fb 393b3259d773ab78c4f9613e78d1d4f1d2fe2bba31fce8f93e23189cfdfe69bf 5bcfc208fcd5f182355baa116eacb24cb9ca493c88d0075cdfa22a21204138d0 b696a09e0ee4fb338ac4df0d7f4ee457abece777f7e149bfabf6d7ad88d77608 f330bd9d5757ef7a7b85631c3af27a65a27b930e2099315081cd2d74fb6e7353 e49579604fd3d6418bc2a24edf5e76b8b1ba54f3d621268e0d60f4aa05d56d1a 9332d1905eb6e6ba76a91f2b55205e2a5602afddf5dd6d1dc52c67ebcf15ca48 5a3879d830788de613960cfa99ef9aa8b5306b2f1de16bf0704b13c5a33996c8 99b23561533fd3a88eb967146f26a8474f5daabfcab4f867f131a642f3a59b00 f1ce8b8024f8c64e8641d6ba140e4798c1b513c77b11628909aee95a5df848f7 94eb8d380f01c4e233452e13eb732bc73cd0b2b5302a22278dcf38930989a7c0 9e23c54a247851880e87b2f093db0c8ca432b1e80e7aa1bf560787df3d76c0d8 de8a0d0d68a25ecc8af367c32962a6efd74c6eb2a018bf1aad53ba80197abe1d 439758f96bc97107524fcd5f1e7511114c1d6656fca033cd0e7b676086abb7b8 da230b3bc8d911a902c88fc8771c4911604c101561eb243806787231f665a0f0 cd6e789c3d1169a6f884c83bc2fcd9a66c7b8c76e415be546249779693ad94d1 3a6fa7f2a245ff8c49e8bedac0dd68a4da5f241baefc00491537b4c4b2d79521 27ae229f0b120243faabba67946c6cb854e8543f5cf9960148ba889f4ae60582 642e6b40513f5f26ac2d36bcce5aab30dc384a598721d90608024d366395668d 5822d8f632c37a19ae4bd41c283f8e057f8f274bc33d9a6f25a5d1f80f7da73e 9ebeb32e30941a8ea9b65fc6845b7c7ca9afb04347f3b478f84725bb004f4f4f 8fec10ee6ef252454e45d8bf6b4469e7ac67b28e6f4eb9191c19a378b7e69f44 be24a9966fa86dc33059790232d31ff5b2f6a76b28443083ca26dd33d027edca c16579dbff6bbd0cbf7b209acc131d88a26950b5d48a39389b001b96759e78c7 068041afcdcfe7ce3370390f87830bf1cfb2e10fe7d7985c2dbfc27a13214686 62c3b65ee50041c27f1d030b2dbcb07884efbda7b543e4cbc8c1e1659d59848a 903bb63dc1948d9c7b657da23b17c1c23157ac754da221d9befda170dab0ec62 dbe75727e146ae10368242c88ef6d71e4fff231fd7886a5a80504d6bdfc868bf a4f9a50ca4bd28bef8fe0bbb5330180ac7258ba77c29ce4c3c2ec8ef71301f81 1e9ca33b6801657a19a999c2736a70461fb0d23240baff9faabc06f9cd82088b 8d9a9df46f41907bc9e2132998299fa08d34039712e8fe36fde051099e03cd7b 2037240a03b5d2b69e90fd6aa708d6c1153be31f0d8014b31e832c55148ae1b6 5b6507ae6fd6c2cd3428ab1b082d17de42add9e42a45a03d9a85198b8d73a494 0ab9ec2039b0afd62558256eb2178547f229e9a408bb697f28b923c86bfd85ad ca97c357abb0c524c092c8e7eb370af06669df022b9b0be95fba10f1a5d56956 dfbc132d12191e26981ada32b1d5090e38878277a420abc1a2e94793bb0c7bbc 8c1dedb22e1c19e999209a6805c9a6a1c4b392c074edbff060d78738572ab561 57eba50c2f89b2b78abf71c0c9e7afdc7a5f0b7dca0483a3e5967f1a9a85e21a c72f6b4d9122f1c383c7084159e9cf4be6c9775233f1a285d4d114b8c71f63e4 d9fba8bcf0f5a6839a52a8b19697754e944459a061cb708ee86aa7d0743d0461 72df3d73b0758b147705111c954bdc5f9f3fbc4d12ad224dc1f76e371372e3e1 4ed5867d5aa3f242ce5d0ad91f564b22fc397d03789e368ea570a16479785acf 67c6d53d9df7f3e9db5fd75b8c05e7c493fad4e9fbf531b411019044182f03c2 a6cdf0bd55597752dcb6593efe29c694bbf2dad60fc99470994528f7081bc3d6 e20d6e4527ea5570043aef34c3e2e376bdad85625ec723b7e448a91e9ce8d8bf 8a641b1837aa58d900c7419410bfc99d2206473b3ccf900ea1547496df711137 1d44020284e25968ade72f2b8b6a6912f688cc4ee1edbb8940dd0ad894d912a9 34b1169562fe030cdb53e60463bb37556e497a82eb306f1daf2064048f341ed5 d2c8b2d32de25aa3891068ddb89347899ed2a31efea55957732a1b57f53eda50 53bd37b8972499b0a200b9685380530b12c63a53c7645a2b8cae35736567e7a3 edbf0cea45e5850cd25c35c5330e0b37752640f8563fa0c1ad36d3cb2bca8fd6 9cdf17c98a37ba6cb572de84c5a1eeeae3ecc282184bf6467000ad7050e6a68c e721361945d6ce1d3ea1d8de7478dc3c8d0cafdb3ca28fb7387a07793f2f467d 5f8a2e7b3bd9c8f81e6cdbb617997f834d10b0bb2370fac7a80db222e462b9dc a9f8c5c441470e3d0ddcec12e2a55810782debaeaf6771f24560d82abe598dba b4bfa2c370701e8f487aa44324070a51ebd03aa5499e646ec1f3a3ca4aecaecc e7a4738ed092455989f649fddc1528e7e90491560b785bc3c0f73a8deb6d665e 78219785eb60d7f25bae444ce6f328cded20a1d2e8489ace17cce3182c17f1e4 215efdded69077ee26ff6cc0068ed191858ddb4070856f0d749334d7da748cd2 fbc94f86103505bc8b9d102c87686d3eace549e5088b754081e517a70457a19a 6c227865d07d1d27b03cf53363115a68ea08f9d6aa02e6b4273c5b19ae5d0206 21a276612f35f9edbfb8e901af1439c5a5c55e7b2610fd04abaf11198ff7bf66 e0018b303cae5f52f774e14eaaae75336d2cd9cbedbfdf451ba09a84fee27175 3a1b037dad9da9b8979153e49c61b09daee53667ca55239292b361e9a949923b c352142c67fc9368de3a8f60a49e7e4a159b35206684df532fee2110c11d326e c7df15ab9675a831a2bcf1cce1aa2a6da6d226a2b18bbda77d6a50fd9af0bea4 3fce9185472c7453fc24f50370e35cfc899797507c2d4a1141e0ffa31a35fde0 ddb527fa1b7574bd34ccfab8f6451d0ffbe8d3adf0e417bd7342c5f7d9dc831c 6bbcf9b866aeff442d5859d877cbbe68e0f8037035c3bb71c65e3650748f14cc 52dab7ec390530e559ae7b2c37a53d40a7762fec0357390fe715e548463b4436 2cfb0546eeb34caf37ac3968e24fb58e1db8a2ea1a0b0723751a85922b6397c3 37ae445e7f621137354b74c6fa99e6ab2ce20f4f573be169917075c9a4edb76f 3a220c10ac6517df2622fdfff892f3ff0cc1154dff3bd23ddeb2822a30e5512e 39aee7b897b0032df3539ee819a3cca239595fb754fd7a1e80c7c94a59d92939 5b84c1d06ce3ca4ec5966ee450517cf4a753741e36cabd8d58ab605195e92d76 590943809f890c4cb4a3bfe662051d7209650e78e31e3f558a3be33d66dc7250 939fd25d1fbb13cf5ccb71dcb7b78e717425486259652ab7e12a23d28f252c7b f4592d36b5d74339ad67a1ae13d96250da20ac421858bc269bc98512efe585ee 217b1afd4e63eb1b39e48f81b5baab6320f79b1fdb786c5ab9a0bf74e9a271fe 8232573efd89b5ec91d9332f39d9b10490ecbbb9179938a09c6e11bebb70c658 109f6ef977000fd0196f961ebf9dd8ea9f1e2b81e6a272461347cb8af71316b9 4d7d42274533ffef8760dc0d9d16d89940b54f68f4ff2b95a4b254804455a1cf 73ec866505452fde05379f5899ef029ea679de8add26bc3d3a0af01039454e84 5655e39ca32ccf8ab617954a2acee8ded1dea577eb7fc98c43304c76ee06a775 96d94341c2d336271fd5cb04b076fdf713654054afdb0a67077abc8bc5653747 71fcabbf12ba75c27846afcae645c27605468066336af2f5ae04d81d2e476787 2a3108bb56358efac71daa52a1f24c56b0f251bddbc74f002f951db47607f102 e9fdea491402b766d773a2373b8c17e94ab2c759d49aec33cb09e677c53a3bbe 1c66acca11aedc1f0c64330693a0d1962fb8232d1e9062c534c386669c6b3ae3 c46413b8f5f382fe77d7a4d549ca5b3968e8b9bd8e5a15c9326c8ec3a77cf93a 248a5b6098e07ed106ba730fdce9ada65e5031edadbe3fdfbfc60ff6c8504f5d aea354c173d670a7748c64ec519b3c524dda6a8f6054fdcd2563a12b5e77d111 0ae75553a628366722ed04f6687837a10a749f58888b44e8e0b3a2c043d880f4 e75ab1f18496f4de7f91dda3292c629b9a7bb7cb117166ee9255dc0087b1af34 f9dbe028f69eeb398f16e0b6e20c7ece2011238161edf12499550af7bdefde4f 4d1732f9be1d1a9704e7b746dae1dcdeef23289ed8c5d824b0054dcb52aa5f46 6c506e6684631a2b6e089b3c07492d4516c776feab3f776be2c25c925d17bcc1 a8cdc435b5e6915fd7f995c987cf3d339549c63bf48eef7a9e69478f81f2684e 72badf5df48ab57bb7eb573c155172356a6e0192b038a21be662486c0fce5a39 cb1ada06a8ad158a6b558e44b2076eae748a3cfc9bb6ff4edf347500d53edac5 670ce40369aece2bdab05100e369a8565206b37495ec37e7cad5d92d117f8856 58879bfb90f5b9ffb482cb142995faaa94a8a06dd29636ffe3772b7ff4d1b05d c8ae2c6ecf2a964edba37647e763cf294638374739296ce63fd1259b2ccc4b5f 4f78d58b23983cf25d416f82e0f60d3999503942c4c4b3018f39fbc50f4141ef ac419d36760c8aa07fc5b6084a9416a1870860e1048d9412a8defd6ea0a04fd8 d3abd3611f173d13bafc4342b94b2f670a33e677a357a3e13e0ec4ff90cfd479 4b50310526588d7d89fee3852e2daa94ff5351c369a952f5cd3b5d2de1074a0f 0eabe1050b19aa79cd0e013e52c523ace66528ca9c0ed174bc4ffecfd09f77fa 6bd80d8593555db83cf32f790dadf07ab5713cddfabe5410ba3cfd4f78773646 9cf9efe040789d019a9a7bcff4604a8c2213a9f570005553d304b82288028ce3 88aa20a03c238a17709ee9b04e30857292cb9c2cfa6d2c25d23d66ae4ec04a24 965e1c935a038f83cd1ceffda67452f4029bdff2f32bdec888ee4be2beded396 4c37c12133e07e4501540ae53e24a485379f13a11bb839978dfa0316064a2ea1 3c07056890c8c21cc7daaa4dd58a1e7556b051ed73e0280189aa28438c3fea4d 569af5924b46151a50f44a1dc4da5980feefe2c12a47625ad530329e6e2fbd9c f9b7e6d304007920c9231790f5ea8baa1b4fe7fbccc7d39fb53555732c0f5c81 6c16aff75edee4c5c6e97dd13b97a67c3bf412ff9e406eace2d0c5d2c2a2beaf c220bd9b90b26c006522304ed4b9b77d05d10bc569ad1940787821959e5b5546 317905c3795bff78ab2bfc10dfd2c2d17804992329a554528928dbac8d4833ac 4412dddcb837146f8ab9ad386444331d37894c6c1abbbe6e93022caa9b520503 463eb0f14a2ae80c10b0864733ada0d6b20ad11428badb2ee242e6288cd122ac b2c7a08843560220fcd00c7e62440931a0cc5f404e381678912f07da1238bbb0 286e0b55cdffc77620b3d7ebcf76f0c14c0bbb49c7c01c98bfc33f9d5ee22b22 03240ee9b988161f8b4ca0923a4a3cd0d87ba6de47513957d14f4aae81fb7f95 2cf97c0465ac6295c85b5c3501850543f4a4e42e478d334dbb3b76a2c1993297 db12bf1d06d9b4d10f4c2a9dfe8bcaa38eeb6e4d10996cbd0f984f2113c7d2ee 283bad74a076056019696c0fcb8d341f056d4ade73e87125cdce78791956278e b392012ee3b382d6640b0210205da1b3a20f5252f90038f22ba982d110731ef0 5d54c590412c1044fbf0d27b2eeeec4323df1329d292198268e271a3a21c1f7c 92517810f4e5b2bb6b0032cba0509687c50c19761daefe1f2af496c600e22762 221c75cd3822f32558046a082c5c273ea0f69928546d8ae2c763626f97df6998 46b9270d80f891f9c2c3908ed79392c6f30f0c0a0e8a920416866b12a8800cf8 443e10c606b33a7a99fdb336a8ab517d3acd20aa231be15f5d3d4b02a175e638 be071625a28269a541551897cf270c7e854c1be4f3294230074ce96a0ac88686 071fad33361b0c47b5e545640fe2f245be096ffcc4e0d0bfcadb5afeadb81c99 0b1b3d4af1e94f3170cdf1e932ecb2e1ecf7ac2a0711a9d804ed7c6b7a7c1a70 a62c7b8026a499883425f7be318e600d19cfa5c1a8d9b87dada882ee13d072cc cffcb0a5a6344050be0978b6418cc3569e9aa5be9c8b144023a5e0e28fc63e14 d96cced89a2469b6572f8fd8064d175a80eeec944b7dc418beb9a36166a016fe 9fd905c32a8f92c2f47977bb5a8e66ebb19a0424e38e84345afe06e8c1a61aa0 f59c03f6957d52701e8b1a80db15bdd144157f4f5f4a48b9da4e06a2edeb55ad f1afc14421d176a60efdf3b2030e746cc597794a8b5181f3701dc29625ce4204 732b3c10cd6a790c1718396ef7c49d751f3fb0666c506f6baad78d331d386cc1 1985f66c306409ff99a80b14dbd289e401a26dfbeb3710088e0484923b0979e3 7d1fd0d0a71c981df2d3ff1015ad72619a71b61f9b1283d12a7714f8fa2ab9c0 1a12fae2cfcc64e6f78270ef3a479a77e8008731f691e6b1abad1fd08be0cb40 3ce98362e45491f4405a97549fcfaea9bd5cd6cd0fb58a6653d5904bfad3b1ba b42c1f4d2d39fd0fecdc69505e328c09df641937330a17c28dfee888e919ba08 f04d23d72602e5e9217f6ba4e3cb85c6d75ef4ea7e57bd2ace1e5bf2400b2187 525d16cecfba40db6208980d5e7864987bb90d0b1570dd15f4e2197d61167b23 bebf63132394253f89ea79bdbd17bdfdc3edca515dd25c28207aaa1d02ebadd1 9ac975fc738e37fc2e0a460390d5eab31d6539d19fd7b8b3278467e3f422bbac 04fe99105b665a7668d290ad69d06d20d79cfb9044da47b2d0fd0c954c3a68e0 7c642e390116ab3a7bc33a9bf5f4c5f2f3ccda722b842d7ed79ff9bd0d93d903 ba538b0b4ac5e85d37b43752e3d998927645406d45071edbad0097ee6dad11e7 93e4f0d9c639471f7b9f0fd4743490c8c6df5a84225abb54e79d43830ef0e193 be30ce8d0a69d030556839e5cd99ee8baf273eff240b7ddf6cea172a6c11be70 a79bcd37e87a5a0fb3f55fa714f5807644d50598319e28867257cc60a2063d76 1646068a670cc9e8d6a897fbc813e8425524377e1cf39aba1d08e46f85a9192f 6ed92540a93db6e5a91614b0dbad89f6753dab5fecacebe8ab291e45c1bca5ce 45d56f53267df7b27d0dfe479fe728921468b45c8502b03d8845305cf9b6429c 51a9ba94ad1d7cda524891e548e967e7c6e5b44e9094f4594d4e4d38a52093db 3b5c4b9b23a93ac9d96ff45b190084bb8219b20ff7852d2327ec76399cceb850 f8355f9f967bff65025b14a00ad026797f878afcb0cb2dac05b94441ddfe6deb 888bb7e353110e15425db7708680548fe76ff76e20ca4675f6d2a3bb67d068b7 b6885f26368872a490484bdc3e3d856ba9e3830fdcce919dccd415dc23591027 d7d122f4c634c3827513ce7ec2365abf253e84c7fdc43e857deefcaf478d4b77 bd7d0abd0c1838dad6a170e97b8fd3efe2a6ef7df55d1e8265283645ea2cc221 0039a2102d04fce75ddf8c683f4c8ee4d029b7016d13608f6b33b9ed9f04e520 be7b7f4891f8d280836004ad3bb5990cb33f095c06462d62e053d6083f24f2ae bd215cf225c1d5c92be403bfeeac600c4ded5b0ee510940446598bd05d9e2891 efa2d1da18ea0689de24a96f28c4ff87062b2ab56c6903785a4b2e27357b3a7e 9128212de0b147735b0dd1f278bb34be31c8fdadb04709b7d300a38ddfa18ad1 0b0500886697919cbaf227adc6d5d83d0db6c93b3a85788b8006e3b0bf5c54c2 2538c36d04b0e41d0999a478407d54984de2ffbbcdb2312cb7ddb80e0181d767 0722a90f91464f527a30de1aff06a8d80f6d696af5d88319d30fd2d12cf90bbd 6093b92328dfb5fea66cae5497134c545673efb0e8e1813d92cd55a7562d5833 63439045feef5b76bd0f4ff964aa3b5b8d8fc42eb09b284a7aaf426d3b32d2c2 09f20a4796e5a71bde60e64eadd891f8785f18759d640ee8fe853b28c2f36192 7718997c66dc7189c2097dba5553fdae88bf78063b4025e5fd455fae22422a85 4b22b168a3ed643a58d15f9f2a5a005c946f9a63710c1748ae6730d340968cf6 eeed332b0ff630133ece2ab40455514621cdf936c139ba5309a3a6b10e59affc 4f70aa23edb07837396cd01b88995bcb9de98753c8b719d9e5dc1ec05163b559 b9eb35f7df0ed330794e0823f530220a87236abb650377c660403275a2051765 b5b2225b30425456589db04ffcf59413511b5b2a54a24ea476f29e38238cd50f e6fc809114bee926455feb6ef550bccafa832b70ff82973b59b8b55d57695ef3 8664e91213400c6548e729b8d2185890d716341d73f14c44aeb1e27b9f6d3035 7031a20d26351e8173e02ed410f28b2d25d8b0a34d09f0549d388c9606b9f5e7 50b8f7c4d55478b5842deda87ce5ba3e14c84306e48bcbaa5aa04ac487ddf6e0 4ec99181c6501eb52c9bb42328eaad3b5d1ba734be589c9843f05c45d2fc8107 a5cc1eae12d395b7b941e370828ad5a3cce4d5d6a1e3e88333ab1d332d5d4d25 3cea33a82fe6b043ced9b047f13ef642a18272509651492486955ccfb3908321 825c499032e41afef5d6d727cedfc75ccabcf4862f116668435bd90cafedcedf fe37b85806c6951e7e2b09e72e4598a79ff78472c6e199ef832ea07d40f867df f7828cc4763024172329246ecfd5d56d94c9f890bd0e924a0efd7670905ed33e 9b3c6861ece72d79c69ae68840f5ff48b9a91afb6d146b092601cbf1d569597d fdc0c9b4c1618ab6894977e8e386b5fd4deb839646ea5562121956beed9250da d730496fef52b259e9d175e0eb8fddaa3b0774925c75174e9bb60d02262cbbfa 1e23b407e743ce64de516923bcb0dd6bf5d07248c48e33f0c57b2e1db6d4e987 ef91034a52a3f1fd710020730604eaf957b3295e07871dfc84e604a393ba0bd2 1809e1def9f2cd41723f1f7a6f4b195681e9e9791adae7b86c9959a50dde5577 45c7c8112622518e3c35c3e1f41206d75627fcd05d15e7ea082801b01b52b4a6 c7c51e9f224182a5ae5a305fc2feeb3b587c912efd2c1ccb9fbc1e0b59a16ac7 cfc67c45428da579d7d7b27bc8163618746a59435e71ec6277c3a44b09d8afa6 3b8de4c338d017baab0305109c1f996c469b0a821c267f6475070adc756036b7 b07fc3de4861d19bf7b40267460839a07982e1f2dd99ae044cfdd5c2e939b962 4ef618f2375ecd22d84f951f46ae8ff2beaf6f996715fb0a9acddc83dc415498 5f167166602868f07d6f6610206fc02425be3cb983399588568cfb19363ccb2c 42a9938bfdd86aed9d3ae5bb7eddff56f51c3792fe53b7f942fe3f62eb8d9304 db97ef9330a56a22d15b8d5bf23d73b7989cc4885751d8034da1fb28a2a114fe aede824205664d8b9c90993dbae92c53d3851a744d8c4611d598203011009d25 ed15366e8e47fc1e89bd12bb1b2e264d78c481ee56d3a515ac4124590bfdf44b fccaa50e037714087e48be8bce46fba72c7a416b7323183d684f38f5c6d071fc 1c5036b3152d6661069f2cb38c8e7648c33a2f3f6c1ef579214c2e59932a150a 34bfe31d77a7d001137cd20b25d3b6a7e461d00b1e2a33fc2d3b21eb4b714c35 07b762e43a7a63802f66b102a7dbbfd816d69333f8464bd9b060162afb05c42e 6c03fa2d76e6133e9671ee333b7044b210d51cb5b2613e724675dd2fd268c0c3 e6fdc2867b06e104df737c8d60c4425ba6f6a58af2221a498417382febc2d5ad 6f82632cf4802a1bdb88de0e84714d54dc7901a373c1a64190481266ccf8731f 01025d3f00644d6d04b55df7b18eba672ff49182045b8baf8fc8c2e0c3189cf9 a2264749389bd83ddd5038d40238e6160297d3e7cd6c8a6f0559c8e42d358152 b5ec787d4c13a92f93cf30966c019933984f9095f7a10a3d854f8a6f95ea4aa7 04bcd217afda0b8eb4b65409516dafffd30296e8597879ddf600c42904465cb4 750a760daf04bc8fbb4971d513dd768aa7e11755fa9e00464c203a2b6cf8536d 231740134158e523e99f8410eeb996838feef53d30e0635a8884bc7474be7411 0f171908722f3cf8005fba0b44e4f70a850ae8c7d9267c6f055d4cabbf856bb2 900ead7494777e0fbb541843a1ae721128948a13857292e00071d5d70fbd33a3 a0646296001fb10f877849950b14326669988b42fab65dd28c10795029254318 3b41a8a9a6f725c3b1acf007f0c6aab16d2ef0780c832021f56deae51c8dd554 786b96c2edaf6e19d2dd35112ab0108281aa7ed2469903b107b14f54a2f1a62a 5ba324dc102b2488f6b71b541da5d6ac4147effe547dae732c71027f580b5852 46a59abbfa068268f1ae87066d4c2ed38cf366adf22347f0dee43ed29f14f38c 1a03874d8eca26211b6c036b8baec56de892ee803490d307596e34d0283baf68 5a16fcf6d68afd2c5ccfea20e88016bc7fb811abdaa40a18aca40b640e6d6245 823b04ec00041ac3d16090bcfc90fb43615ecc5ff03f8cd7f51b31f879db0b64 65ff887afd1c9b1d9e6a9987ac55a28e99fdbd28d7c1c83578dd1ffa5acfc378 fbd5f275622944d3dfc1fe80a211f08b5aa38e8c36d2f4c5a97f0be198a7352e 414a3e5a2f6a4b6ebd85d2d3f0a77d223b0b58df4ed27b862f1cd22073f8d6e9 9d72b9df9ed35767104a31a6e70b99d2f0f11c3a948d294520a6c2076029e893 208ed808c0d06502b1d1906f6f437ec90977f8094e7005ada508495d66426da3 92fc05e36bd54305252a3720b36b5c479c2801c93853a1461d38454b333cc98e c89171c8730e173734c079cc828f1c27fe3e0a630e4efc361da2cf271a1a9b55 3f96d25747f00ac6bb9a7b54d5bde2d09e0b6df716ead47af37bf94a273693dd dbc2ba381f2b173259f0cf53c2b33655defb4bb3ec210ce8f312d16f41167ec9 3e0ceb185c133aa9ccd7dae72766d76bbebb9e80dd8518b7e58f5f36c7c3eee7 736eca8b9802e57b3ec0c5ace67d04f6314ffedf305780ff96f2662f4710982c 485eba0cce15dcd1e383b0f0cf4b6bbb579f6116729d80e0a462597b8717005b deac4bf720292660f7cb800ee757af005536bdeff4662d62e0a8c2aeea22e02c 982e9882c0628d206f3415c27783925b5506ca5fda8fd22817be8640ad581642 f28a59addb4ee1ddaea34d520156a449d7c6cbe35bf104b9634ea7e9fdfdcbd2 e6ad6accc210ce2cf680ac5a4440f0bbd8553ce091a4a69c57c497cfdc237af7 805cfe90d1d3a0766f906081595245e23493b4ec5b9882bdf7f62e23e16f525d 14b166c779a96fb812ee6fd61547cae553d0cb57d08a688c909b0ca13d3db70f 5ee739ca7718f239374f8a557960c0cda80eac36097c6fefdcaa4d49801c517a d7f35a0d3d476d88dff484fb1a2c30f89b5b2f4696fd73d9d272b64e5e5bcb1b 4a4ace8b47e1891fecaf6d6ca4c0d1e2f181f63580eb1828dcb6967f10d0d6a8 a146258882f3b9fc1df30d7bf53230540b65b4f1c00d30612d81ea981a170ff3 d8f92a315433be575144933f87dc6df79f0b17d302a23dc7173c8a055e242c01 b7042bae89e00f1d88d30326ce4eed168a821ce523d3ccb687f6060409bf5277 29455998407167cf214424a859f632c634904058f146ace112c54bf7b4700892 dd2ca75ddda55670324aed748f0b0a4a86076da3a0de7ded7739f10e532349d6 7bab0922e618b73fd8b3ac844cb21927157567b2a32b6796f69688e44eb05316 51793edd396c199af12002df982dd8e4f75a51eea632d98ac036a33c4c202d8a eda947d32c9548bf03d643419496459b0bb1fe0802669229afa6db67c6ae5013 940153ddc858b9adf52896df8491022e0f53f2b32b0ed3ce9ea57c3acdfdd43f 4b95f78a4f5e84d8f93ee7536ab487a2e98d13e5d24d0cd2f22a4393762f16fd 89ae91d6d8112bfa93fc7bed8bb93d0644d6d861436bb4a5c2c8ba3a93e26a6c 7c64ed5e055c77f15a8a60dd77a666ffa49024a87bd7eb48c855df241648685e 15fcbe17755ee5525fe4c6e3f875913f8537039b353016a440d78240a2bf804f 82897e12ec710103c91c19e1dafdae0375d64990e0dc5e14b0b99394a955fe07 8de507f7704f8a7c06d3adc41b9c4fb896854809c86e558100b45ff5f8a2a7a2 789c1941abd5b08236e3c7b226ca4e99e15161fa57a5e7369250e81ca86d9e48 29214b310306982463a8220a24cd8b71bd070e858df18a3af7c58ffb1cca0ae3 8e52c3560172b5e7c709f32ce014348c49952399693b7021eb877738d6c37e34 28e82f883ae11045aa951d4c982b23e052d7669f2626cd370e4f15e257500f81 6214410937d4a6390f98e9a19374b9881a7cc6bb926bc5f75d91c369731f1e4b a3bdd572ba26c5d0ac6badba90766dc5791c5fa0d8a4ba296a99d2250e3b13be 97ed878e5c33df27bf48488ce443a4db3f6cb9f5a0e75b6ff72aeb8b708b3884 207753fb24b1a1cf5b258a2f5bea9353d6d4a193f6b2765dc0c9b78c5ddcab5f e124c7ac45d4cec84ccb7070f27acb28efd056e6f6373c14ef8648d3598965e7 b7709f57e694ec06279a3b9f7e525610d1503428fa26e3807b5e3b70e8761af7 a833a62437ed375c5845c8d4005a30b16ac3b8c04aee9f3f0fc35dce9ea8a134 9bd744adcb0d5b1a42accd4daba28e1d178bfc45bfeff112f99ec0c80b615bb0 e68ae4289e89c3852860885a077f5b197b70be6984771e96f4a018424eb82232 31f02d8027427ae2963b42a1d508bb163226ebf08382d43b2ff018342745703b a1409f20f71e7ed746ac58235d55de47d95f6b320cac21db46042c1c3679104c b1580b7311e2521bd5c29980aa8d8625813ff3327f29d960b4029e19ab874f1f 67ad3011ff0b634c1a852d06e81e02f6d969d4886eb34cf26e42a074fed50fd5 f12b5d779d0f2830a8f989a0bb3e43f90c73c239be137759590613234afe793e d0fcb174edbac815ae5503d4053e9c3c4258fbc2b04db0cf62e0efc24a692e40 17da90086214d0743f7e48ea730fffdaec900b48bcd30dce53747c0be2c007ff 5ab445c3b70e0a25ef79a9213a1bd57ccff93dbe17affd7ab84fbb14f13c2e20 b80ac59dc0f7c30bec787436d991091fc3c6eb699955d164930531a1630ca98b 8b10a4a2669ee2fbe218078f8c817418e86f487a210e2b51a88ed9c81518a2f3 c93b7181b7fa8c08a79055c1038207eb09a3d5514cafcca686be41e89eb2f18a 1590a5b654b09e53cd2c7627488956c652d52ca1d1da0c0968032fcbb4fb8f82 102cdf3852d3ec082a9c50d7c283bc566a3568c26dac8e8459ed91777e44aaec 9b48e8ad262189d9896cb12e6d0dedd28477bfdcebcaeb12e28d1cadf0f757e1 3a22305aa274145bd958ca2cc5fc5820af4bb95d71934944015805f941191350 4f76b81a26e833e8cf0370a37ae8e22b32c61303d2daa32d51fdedf1fae495e1 1bfae6290090f4933dbddf5d580653f068b3fd7aedf07d21543093ee38116a26 2c682e8425272a6677d20cb1a03bd12493bb143cae3a4407f57cf68a258d0197 a06381ad7853ef83d12eb775d26c9bef55b7169619cf39377b3aeb1b26055efe a2b03c914bb67a28c25bc059736a8c40c5fe2dc2cdcbdcbf46f5795e3b45ebb8 073f85fe543c96592f6fa9619fd993d36db15f7b4062e8873f1cc8318eb1a56c 51a0b0d84a560d025166b0d6e6d9037d0343e6eb7e549af097ceda304d6c262c 91c1691b62a8d484edc2be40e298567463d7721f8734790cd6ff712567eb4849 e9ed1a4fdfc16bbd782accb4bec0485210ec9b3b6ce6fb9f730d61bbf46ae6ce b247cfe8c5b77f1e020a2762585b28fa250ba3b7470ed033730ce4038606735e 9ee34daf636c2afb2d6c3fab84526e0a1f8b70dd837c14057e131514532ff263 5a97881eb6aa02bd782fd6cebd230e29aa68d112c781aa64a2f733ab57033b22 2f94d5e22cd383fd7be7a228ad4cb89b26656c3cf69b3f2fa73fb1ee57d5d293 436b6a30aa1e60a61d0f11ec69e464e813562f374620d3f07fe4da1864af11c7 95b0c7c7fa95189fb7a10346286c3c4b23ffeef59eb30885f85cc1b68ed57afb 1f9bf230d3aa3a7b94e92f66c62d07790164b9dec1f0de5e1aa9706277649bb6 2874fea73de19b408dd75fa1d4d1c1b6d8ebf839b9a8657aee189066fa6887d4 352d78ca2b187c8e5b45c0f3ac73b949af497797b3e10028e7c2fe88a2bef339 28124ba655fe2698940cffc91716fc9ae6f3b8446cc0de60f570c507364da311 48d4c724809f911509929fe1d81f9faaab6951d4ec13ce507579b7e13aa03fd7 2e1f49f7e23b9df882e47b4eaf1ceed85750eaad5a411eb06ce2b28f0f6589dd 46cb754d4c1cb503fc44b7bc707b3ddd955372f0b4ae637e54cd8c9ce35c0d73 5bed22f63943f37f81c8cbf424fec67217d3d2c4271a5b3b8b98c892c9fb5b06 09d35fb6abb587635ce9837a4f33c76a07557921cec626553d90946e350d6450 912b0c13991f2050da34046e58443c11e43507e00b0fefa7c79e075d5fb27060 855056a4a5675e7180e6dba5bf120e2419b3f0ddebb03aedbe765f6c36e99339 ad03da0df57f36ade625eef3294521c3939f709d76c671373466274b7c4f4eff bd4dfc2f2c0ee1a06192c6949892394fabe04452b641b205a57b0effb9fa957f 2b863dd272f696f563b64ae1537469e8a02bf1e3569ac2539a6a6032d29cfa16 90867714ae8aeaee071e20e2f13196b773cfc4eb8b8a0336ea13f03a9cc8a90d cdb7138e7769fc1a8b8b0b825f75519d33cc40edb7b36878b70584bbbe867ced 38b4d11bca6a397a58544eda42af892457bf5a30aae3ca1fbbe9c080e2922530 2ddb6328e972496e1c99149fb6860792030bf3ebbd0a160e7c215572976267f3 ad1b292fead53164a209ee15f5ad296604e3ce1ef4980f89b25b75d0e64011da f47a2f1c889143dd35b323f6fbfd9aa183cf9e05af73c6bcb1bb9e2ec6638db6 de410bd98b2e72d225b5b555e9736b03c369e2923a76aca5e6fc44c102c970f8 f4f4ff92e41cff5b2f700d7413c54a5ac24ac63c15f8b8b2b238494d3d58970b 6bf84f29622ba2b9a217b3a121e1e3136a0b0f4f3163c0054d01c4af57580a90 291ac2a6b8d88c526b36ac44b7d7632c86dceb01a8c31ec481ffbf6e3873cb09 33309e08b910c690911b62630f3adf0f7b0985ba43bb36eac66d4e9a8911bec4 147b1e95321f0f0cf42c2fa263824b41540688cdc71a781ec74f330cb7560d3a 802ef69d164d374679805c5ca11679c893d069daa3df5870db0e5576ee682cf9 293d8adf64d3cddb14c164393b854dab976d1cc4941dc54c8218865581842a19 0729c094efb09c5702eb4d2aefd2ab744b179efbe0ad6e90a2ce4075eb83ce7b c6530a5e3c312a8b3ceffd9501d74c3fd410e03a7fa66b2e11f1a5db3a6b8330 c42f9fa2aba9e9435d63d672d2dbe119954c8118ba3923342692048526970a8a 1d144525f84bf5a0caa2591f754b4ff66b48c7d7084cac0ef59dd8d00a05ea26 86b8cfe20a095709f93afac9c6bdef5acf25c420e138f3ecde38613ae4e7b366 91f46eafd2906bfae8425a7f9610e455141cd79414fe6baf9523881cec61d27a 2c314b1d139e32a3c70d4af9f36bc38564084a82baa1cc2b8fa39da7f61e90bd 7061836988377e6b17715f0f606bff5ed3c484ddc501844e2c1991f46e748968 6c071f7102ee 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] 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 15 /epsilon1 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 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 37 /rho1 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 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 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 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d 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 120 /x put dup 121 /y put dup 122 /z put dup 126 /vector put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueXX 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930da4f8d58abefd45db119b10eb409dd89792 3c6e705479464a4b310b58348c4b42393988fef4925cf984423aaf65fea9f0e6 4629953bcf50b919d968d99bd185f83112b2759cc411764e9bde677f57c5ee5a c555448d5f81a16259ded1e11bf4119d53e8ab07a802df900d3a2d5ccc1c6876 d29c5e9effb7af3ef83400b0910c07873a8c56fa7b1eb1ba426043b00cc95dbe dc6e136cbbbcb3f67509964f9f281ebf81fe5b018122eaf66c4a838487e82e18 6e006093042848a903efb3469ab6b4049767aadb95c30408dfd58d8a10f4cb22 168decd9f3ee100f07b49aa44c92139b669cc312ba20192454eb2375be6284b0 26659d964b96ae82d4942e758027fcf23c25ed01115af27ce7f20efe2a822bb6 84004f20243a49c9e93301fc21b80815c033c3e2ba58ef53da2157d524b395f2 b37abca13bc6a2f42e824ab7e47106176b0d6db267fbb795ac7425582df2e3dc 55863468a9200742bd7b552c48f8cf58bc21343bd3b95abfa140f33f37c6f3f7 8b0d8a5154eb7c1f62ec598267f13e841a3e64172663935ac8b665d86540d316 ddece329c008049c5e74b27d59022c5515059bc3b89370b1bc6a169c888bb325 e0b74282d6f053a50da4024ed1e433271a32ab8c17d41c632b41cf9f3ecd5fe1 24daf7aea7ce8a63047b245822930d517df7baaafa69d2d17f7d93cebd47deb8 5405d720de0bb4dcae731a8e27bdbb6e6be38d168b7daff815acfc32ce367374 bf9fd98c0ae5ba1bccdb84d94b5c173abc92f7d18e19a68f64a2241a0a163910 8dc46b8a04ead17662b7bbff3d3d9a95d5937698573bfac64973dc8d8b8d9f68 c52d0c133119eca4fae72f7553d276c817003234a98337d7c037c0f1bce8a804 57113bcd97515c002d20f836fa086a3277727873ec2b64a92bedb01193491134 b6692a08de4a201a3f11dbe070d797f7629ad487d2b8dbf89a0631956c93864a 428e5d6a63f13f70187e35fb779d678e687df1fd75aede7a4e9181956212bd51 e144f81f7fe1d2b3c1b008ab938c6603880c454d048be2b92478e08255f2fd06 176cc4b24b7f9404c81d9aabf74dd76f992bccb6ca3ccfae8c55ec6ea0164941 6fd9b00fa5335e87fa7cbcd323d10a610fc2513efd23e06916019098eadf3adc 39ab7934cfea60bfa17b53a8e4c348a06aa0523694ba8f2c74200537fd323db5 bb34a9278bbb06c40dc65e96d882c559fff9ec1d06cdeb1bfb55ba1afc8b65f5 a25e2d72e6d98e5fea27965e798c9294823b4a7fd4afe522466c1c225c116732 fe4adacdb3e6eef9d0b0f690192559d6f70a730be21a4dc795ad22892daa01fa 253f678a7958edd19ef2104d45f9ac4d6832368b123b4f48214f6b842efaffdb a217f0489445d864dbd01026dd20fed9424796ef2476074aeb646fc31d9e467d b02d31d31a63b15e309ef6f453bc1bfdf4d81422133e1a127bc4464f964930f0 1c049e7c37218770c8bd35fa800eb66fdea4a1fe2864b7683f7153441ba4caeb f58ffa49d71c01e5c38b2cf3207044c3afd58dda1446361d82a0682936efc195 5217d8a4116dde468b31c9e68b7618cd966936fdc1d0dd6b1793b56dd4c79b4b 3bd21dd6e1336fb4fcea1fe9d64ee1787e2ca3c0672b6b821a1a877afcb18857 56f246944bebbe64c46e4b34af5abc81e20232f02b6e1c1f3a2a2b8576723c4e c78c49663d6985f7badad2cf2e335867c491fc7ae5c621b04957ad798d7b5407 85fc79b31f971d199c234064390c3b628e2792896dc60e058d1909d5913bc413 935daeadc9128fd641b41fb12055621a5d25bfbf5c9e1e036acf0fc72f7caed6 c06c2aa63064c8ff0fee67e8b6d971db6e72115bf9f6e1cf51f1eab96ae6b490 2ce008cbe65c613694ca5d379f97574c4519778b073c7ac584794481bbdc5308 7b553319cb0e63dfd7732d82013c6547ab7530238fc6a0ecf73c378ac3fb0023 d0d85c3479e69780526bab9c329dcaf2cdcbd3c9e0e0fc70acff73e8ccacfa03 c3e8b23ff9da61547bc8e7a3a7b92fe6023e02d105f4318c2b854db4abfd686b a523cfd8101d25a8e76891399882b401e355a3a03188e68a187e3f28f0319784 cc29ea7548558662955c3d3a0349b19a9dcdabcda4531da949d67724d141b5a5 2481546c1ff2cb8e94ae11ad855f84fd513cfb92f995e5aecbd0ca8723fa9953 93221a8bf7ab0b68adf50cffbf70178c64d7da1b0f1ec4c06356127cde321bf5 1b4eb036ff7d2a420ab69418efd3af8747c6cd43cd6871074192227f1da12704 b49bfa4bd98c3732fd88dcd84530621b76b2b5ae1a5d582c8cd37ef163092e32 e954913553e1733935ae26a6f0fe13e22fb5031dc9c78bdd58496aeb5ad847b5 78e2276dca140515fdad9ce5a3a8e7c901c238e390301b4496e0c51d8deea29d bb8b70d77a02c42ff2cc322b56c22aa12e5a542ca36f8c829175c9a83ff0cfa6 a892df3e56d1abf4c8dbe1540489f6fb9cc280db4cfa899ab17436b5ab6f881a 2b10841a0ff5d4fc8fe77ff32aef20e9e7485e9d99c493a4012ae1375479781a 41cb0e9bc4c6a0479c801ee0b602d4b32e7818356bb1dfef5759c3b1a46d011e 64c59cfcafd3bb0bd878996a32ae919241516a06b50661df6e712b370d97c612 5b50ab2c2f572d0b9d0761608e441a01a56a488ff2f622fc87f51a42d09445a4 7b38558202703cd48725ce4ddbe7795a361b126ecffa3d142e73558600fda6d6 29a2020186e6a64ea1245503aa7a7e5df2ef761150074477ec990d7e56415a4c d862015d44c3bfa8b5b4c86280ef7f3b20cf891c18540b046a51be68c1710e26 6fec12270783a203082a5f02a92500a2d61c13b52bc973cf0ae0966c10c8192f 018aaf9714f8789d0dff86d241e3ce5e9e5ad802068b5a80a0fb7315094b31c5 d7651414f48b4db65710b31fc775f7a49f6e5cc6e7b9a0522eeef09f1ef87823 1f948a2933ab17a7fb6f6ee573cef4b3ad82a8e42427be86f86181ad0762d323 fb557cc8800f8a79e520853c71e2ca918a96ea26d2938d872903feafa2d706f5 79a54c53b3633488d125ed5a1e4427ae9f12d25fcda029738aec7687e3c30d6e 43ff9664f05b974f06ff0295a69bb5420a25dbbd37e1dda0bfce98534263d9fa 8b1815bfc2310f25ce4faba638b1f4959d0b764fd7811dcf7fd763ee40698f5c 1c0967d04abb911d14bccd9427268918afaff5775f56b2ad5b55e4a2033a32a7 9dba1a33ebb82accfed27103cf3e1556cad0420ca1d72ddbd49cf40f6e542abf f5f1c4bcd8efa9a9368a2b5f8db768be30d8c524dc5a38d6a71a6ae961904265 904e19e6543e43bd8a2872ba9ea48b5ea3e64ffcbf2e41ff3228cdd50689bf51 6ce4f49409a5b505f258553db0d3cee22bb57c709a3836e1327f8d3103104c21 768f16c80a8cf39858a3df47401ce5edf3ce92bd75b20113d0621988f193f73f 68b964e392d244ef903fb7705ceaac1a5994b998ccb2b44006b269b383e10ddb a4e1dba262c1332f667ebca8849fe87b7d318546f99c695f744a772b26666eeb 0ca0b0e2fd1eb2851588a6211fc5a921151799e03bf375cbffb29a138691a7e8 2ee0fc7cef2102bc3feda90d09c204cb8534f38049ba8bc6872591264aedbcb1 c9314040133579b770d5a84de536592b5002a759eacc410a78d6dbde91204058 e83af9cc2d7c23abbc78b3244b9029547ca3d4075fdb4a6a313517a1109ec6f6 74ee656814d98d7c784f57aebc187983335eda17c3f6775dd42fa952413240cf 5900b3f1b278b09a6d85ad6d0cf38ffedd50c8cd02e1ee1c5256ba220b34b676 8164f2bde36dcb6eb7f9c0e5ae5cdbb6d1ef0d197851197eabda2547b928528a cf472a36068f9816ee25422e2249d3d5df3b7399160eed1e65fe7ebc75f0762e 9bf6ceb705a412529b8d0d795a1de2019c9e3c17a2a3fa2a0bd425f24f2490c9 8ff783d249ccb022af80fed4fe54d2d7998f97476a8fed73d795b5faf8d1a191 0c3f957a7002e8bc2b6396714fbc4377a4856bea47befb37e3ee02bb1efd68f3 51de843285faae3e42aca50dfc28eb2b11abe90679fe99e2c1ddee85243bb541 3af19074ed3e6f3908c32ba797fdaa1d7bcf8a6bb66b6d45a59447dfc88944e0 a82c0fb45ddf26ac98ce4305298bc317e5771652fb25d14ce115dadf42ed0214 4cc0bfed2302233fd3c62b68e93e683b4fcf2cb9e5f2b7feec79bdfcad979b33 a73aeffd1a4ef06d582ad46003762e063842707a7a38da43070205fb6e7eb5cd 36794230b36623c1a59596658a4fabea44185c7be81d5f62112c3b9c9752745c 031f779d5e788abba0e34514d93403abf174ffc5d3e5ff8f2937beb7c62b87d4 673f6aa904dc43b944ce0af897f3573d4d4143dbd6b7947c0efa843528870e20 86caef2a6fcdd42252d18f7206aeb972cb6dc88256912c415b8319e42e3c0474 5629939d78de797926877b8ae31fbc288f825482c4db2b3d79172cc213c09e25 091863954cfdbf26770dbebcf5c9a1c5b723f73a40a8d394a375a9b9ca858c69 0e2e29b4dfb83dd64d57803ba381127a7ef77eb6c08cb3345377c5a9fc24dc10 b120b1d0548f41485fe250c3d6dd3dfa2eac0f9630fc59e1657843fdef6779d8 f1da4f216e19452b0b396566f14a2333ff8fb9159c7e22311052d1430a6a8b96 b2c9e99a8ae9ce6088326217accec28a3650b677145e87750616524e5e6233dc a8ecc94457a52280b2cec80847e815c1eb9d4e3ef4fae877f082be3b9f3c07e3 e47ae089cd5e68e1fb29b45be36ab6ba369ec32e2c230f7e12c5bd5b8e33f0cc 05ce82a531b85759ac6eaf771108611f2da2f0f7bddca3c2178bc5061230492d ca15ea770474bfc1ff68875c0e244e0ea63144d8f5ec408c96b753cdd9401803 af5b38a26cb59eb63e760bcb830c3da6747c5a93513a85fc69482f9b0de73d58 89966247c71ff065cd808251ff9b7596d5c4708a8a1ab859267aca0d5e4a6158 ec69339fa389be93eda20132ea8e5344a303b87b885deeea3a5b8a482ed64270 e051ebfaadac09b653a5c29fd53a06a93fa79eb4a772698dbc26a09b900300e4 a1407eeddeea10c699f4fb897111b1b697e533914f5f193901bcbbf820768612 d5780264f1c5c0939cec1940d2ac7c32444e1e2dceee356dcb5fde6640bbd847 94169f2b679944259f4c2ebdaebd4cc9e69818c8fc045d874113adc88a59e329 9b05123f0d2d2bd3376d1b0cf2712ad520e70f292c47f2b2a50c775d5c87ec5d fc9e472c0f14b7124b8b155f919e877493e2910da0d1c675235e5121b39acc5f 4570b461ba83d2b85f590d3a18a38e85b9fe04621811a675b0a33f1d4f8d1061 a89f64370432c851f123a873bacb23e9953d08de4dfdc2c2ead690ef93b93ce3 4172851049b43bcb51bca9ab73cb68c0869ca7d829481e7b79dde70d4f93ee6b 9cb4976644fcc2eb02e5f7a3c332d2d34367c8c0b233461cd756f233b05070e0 a3bcba928725ed308823c463815fab8fdd5a7e954598a1af5e9525c5ca38c0a8 0537f5662e701f8202cd39fa6985edb0c0023a3f32e48d38eb2b6aea20ee37bb e8b9c0f1a759bf6bced33a0dfcd67826ea760a40f282cf360b58348014ed2603 8a6e4095b65dcb42bcfe774e91d38702d0fbbb6545297313d9aaf7ceaeb2fb1c 2d3bd08a5b2ba7c9134d8308f8dc9a6da398c3dc06a2b107da708cbbac13cb1b aa0cd00d9b14cb35f3bc3ef7035f47e2ff0db5e282e0181cfb8d91ef3c118743 f7608dd780418298ac925bb5c779c0ee59c604321ba48998652e3ec46891127e daade8d1fb8e86af6dfefb1b165f530159d697470daab1d0462fb1ab3b8258d1 a5c48967dfff32b9241031e6fac6b73385788a5ba5a117999cd49972ca1226f4 f8a31cdf15ee53653bdfa51aa0cbb3afa574c8f5599ed93278e4161c1f1573b9 158b2ecc02f9346d914fcd366a7f0dde7c0afab7cd93c15a1dcca3682bc92ebb 6e43e82879589f800def966dc5b1d9973b344c351afdc0613ff8ff92f8117aa8 73e084bdd856246cdb12a906b9da6e7f72fcfa11e692d38cb205124e4581b7e4 fdfd53fa0b2c54670db44ca89dda5fd360b3a99f3567e952e59be5acfb1924d1 c03bf2d17d053bdd2d5731b52082036d1049a36670102c224b5d30ecc6d7d33b d271a3ce1463d0a342b2cffc6656b6184242621cfa46b74958425db52a3c8fa1 5c57dcc98aae57a491a76753dbc8f8cba42c046aa87304b3171c0e7fc709ca01 ac1a2f2424d0a47d3b7393e43987e7c3a669e3112b14b9a9323d88e4a1997b71 10247be9996c0c85d5cc5712a871897c6eaec87f5442df885a24951d8ff01cae e308f8caed1a349e2bd5530d5e27dbba95ec4fc7bcf1335ecb03d470b709c072 66a563c6e22e92e68f6edec3a63142bc3ffafa6d000cb7e8b7d0a671bc9e8f69 56951ab91298d11e37bd91cbc534680e9006dbe7fc227f9a2b83d74c898fa488 060ea29307cf8deea69dba198fba6658a87d20a1a666ee2c952b1a0069767b3c d069f06e962841ae2923b681750808394a68685b9ce4c8d4490dc1fa766f2a88 6bfd321c0cc7c0239979ca9160b07bd071eabc119f2b430c41b809a92987cf6b 560c5c784b141becc06b204ba598c7910d108ca8d252c9909fa43950fa6b2964 b1038e279129bdf93addeb45297f97561d5d405231a37d5afebc060645a8694f 34061728ff796b537c4c3f413d52fe006cdfc4c5c1a376047544cbe8239cf0cf 7a2d5382af3cf39d7d3ef11cd6c9327ea9340a0938a8404df2e9fb225aa15a05 f1cd4d1ddd79ba145e46cc5743966bbf9490ef7e4d65cd36fdff386a7bf82274 b8f97901f3ced5d2e8112ecfa1c93946f59601235b9a392e355157ee972c0392 d69cb9f6a5219bb30e8a28fe6ef023f47c6b626da10ac93cbccb3134d071b2c2 753af032a7885f3e5c80d383eddcfea385f9cbcc3f20c075e9452f60ea9777fa 602798cd999ea483bedf80d1ca298d8f8c739ef4bbcd54274b80ae5e1b55c747 7a9f1c0ed4a98dd32e903d1449c54446c050d997e3ef5b4e069266a5395b298e 066de6f77301d570620ec3b30934c13ed7d9cb81f9943e6b2f1c80dc2782b2b8 c946313a73cf9e1b362d1a846be9312358b95fc107756f401c1421b1cf184df7 eb27ad6571d5895b8f5a8c4247ffaabfddd7da32ead94904d7f7c33964a6956e 0597309d5360d9761f840efcc952594918eb024d21a40597664da7074d499ab1 55475c1636d37c40135ccbbfeb1d5f5d1e2a59a08245078b36bc04ecb692bf80 0aa90a8363ba67259408324a7c8d540352936ff3d66a8aa5923940d7e7299386 b75dab58400205c97e32870049d0743ae22b183b809557ff5fbed25b5c06f1a8 80f463c5b8d9f39884504a52521144b49c935eb603fc44b7a2eca5d42ac3d167 a0ff31a40c995bf75a63c80d542688d50b86251aac12dcc9b2cf822f58d9227c 7f6ef31a1be79900951cd9eaf6f63dc022799cb01702134eac2ac6d7bee85973 9041e793393098bd25778ad0e9b72f5a83b0e25a396313bb9a41f22eca07c1ef 551bc66068628eeb3e44b7e324feeacae060205c56e3a8ceb6dda064d5206e12 dc044858ea37b4f228cc7867c18459454d9b30e6088260121bb9690aa7a4ed74 0b80085c3ba29726f252c01f5e5cff7d9e2976d2d9a78154b1778c4fdaad1341 43a353d76a9d201922c2470f3dc3b32336711db2ff1bc136309f7a5ee602aef3 3d489e3e4c7551015b714e9391468d1d273e11c7fa7cb9a185ba8af1f99252da b23332fef9fef9f40f8e900d14cefa6138ff30ad30bd5aa0d7f106f4158fb071 1c5bef3246cca792eba51cdaabf9c9e7dba84c5bf8304728de78aeace2616278 780b4684a00bfa4e1a9478b4723645d80c2ee0e6c5d73a5aa3db213db9f6eba1 bb17fb9494f7d106bef9ce3db224c3281c31301dc4d43b1035277c8a9fed95a5 e5a318c95251fdf9ffbea02c9a53d22106159f6c2f710d3b3152115c6e928d35 988166de7a92b0c3ccc9f9626cf8f6f40b64ea0d7a5c47196a0f49778d941aa5 64764dc74477fa8d8d61624c5a99cb942e2c4b3a67a9ec01a6f48d7ffd42b8b9 bc1efdec145bb7244ae16daa426a652207500fd43c2c6716650c9e98c548c307 5b1e7d283d9349fd0190bad6cbf688b2d38795952388559e05bbb63efe4ee4e9 0ddcd8d5ca2ddad0b4754959a4d8502c46081f2ae31507280e5a1868cf60b686 deb40c6e23a525fc9c5ed396246609d1cc66897ae12db93832f0134d2ba1a645 3f99e685edc489a2a1f12ab51d1d18101cd90caecc7f4fbb2951b40f6e402ff4 db109fb1389b8c71f8bc3d9b00b3b74ec82da1f15ee976c1bf1589e6ee490bd6 4ab3f6fee319d26122c56b831f3f6e3268122bee182db4d53219d4236fa18399 b176fb6cb2134e5d4b0aa6d927ecb56c4a94612de34340548a7d8165826ea45f 573903e149a9e9ef6da2acd23543e25749b8924f92abdec279dc1c01f181757d 16ca69321ce0a1decf81dfb2643d2e2fc64bafcadab0d10be8c74cad4ac4b59a 0ae25dfed3dce37529980c27aed7ac0a4bc4432a80e117350f64a132befcc0ac 98e7af6b5bd475f2e9ee2c15214ef2370417a8afaa734c2a9d1e0320a51e5e06 1b780e9fa07d9885359d91536aa1189f906f64432b65d4b89c3aa531fa8b0699 7f4d40c5e93320e9fa739a425b639229584033f2651d15af70f6e742dfb18234 c045df4e130fd2671877692407e22b2c16e46bd45d9f50d5267559e116db5296 6c32fe56e1f6f27e3616195951ef1d4e080188627c7fa8511b88987ca06b330c 2cc0112200aa39be1a086c53e392dbdbe8f49908259e7934ec387d23dd135f83 77357fbf377caa5d31f378443a2af89293ff61f4d8896ee6a2f598d521a9aae8 ae427e828cb490c8a727927891fba7af2cd86f78cbb7c12263298a3a18b7b76a 61e150e8a01d40b4a14cfacc2915e97fa3bfe0acc34226c01bfcac2ff6a507e3 fca32c0d8d623b879ee04adc47abd737c2465044afbabc63f5adb3562b5ab094 000be46b822af3d4cb82f659be6953862b4d80b87294ba9af98080e3fedd5b26 80eeb12ee99c21f8567851ac0147e7536d641c37bfa43a4e5f355622ce55b007 745a525036a9ab53b983986cd267044ea3837afe5575fd770d3c95e36739b1ca b356d16fc8bca434d7547be997685be9a74e2f684f298488ebf289454a0baefb 3b3c0f1001e1fc326bd036a54fdc1515aa4ff7eb97e64c3ccd3b158eecdc180a 551f9b65839128a3a73ceb96f212fd63ca09ebe831dc89ce3ee2ab89a6fbe480 1a110947b1b9d69fbe70ef9c413d05afeb9112f3daed1493ebfa24f9a1dadb54 c35bd0661a23d571fbedad4e779ffd7384053e1b1d916498977cfa9800511ffb cacf0d91c7269c2b364eb11dee4cfe484539820bc8dcd49c25e75a561ef60896 f062695ad8199481c648b5bdb0d6ac13ce5238f83a070987e1bdc201d1945228 036ff03c2f35e6e4d06048c90409a66f5eaffdc39fd1d9247d48bb4bed006d41 973e6e7b0513c45a3effb1c1c0e915af9d545e86538ca8eafbde0189d8db1bc9 0cffba3b3206cb3b56bd808d486889978c8afa05b4b8efd39b9cb12ba938d40d 578b7dac3577872a9deb45e0cff02f35314732d3947bfc67428adb588550889c d6d3315ba6db61d9d56e00be3e2c65b9ed0807e1c21c4e65af33601859e09363 cb9534e433821013542ed06b5fb0b9bb006a3977ae3f501670e80069064724d0 9ad5b40918f3da9cdf63881aa43b7c5c671e18a752fe129b4a89997e3d80dbcc 922aec08960d291d9d0a7204ae971924591d4aa251d1b027820f73e46772d23c 3e2da7b9fd3f7104b36661d1fc336fc7900fb82e910706ac61c6ba3f25eb6903 9b9a456a9661b54ba3dc1830e35c90516e237719a92cee792f24573f83b42420 a01f4e8e092eb220a71d2331ebaaf268a34f3ca8c6887eb212f82fee0822ed7d 9f68f36fa817a05337c0ab886b3340d9a03ae64dd64a0db88f872b9f5c59e21c 9512750e74717209ada1346024cea0af7fac3a1e60bdbbe837f60bc17c735368 041127f242ffa208a0acb03179da9f138cb1253cf9f7b4399603e0452fa123c6 bc214961c22327b3682864bba74e644eff047f69c3958038614572b19ad5c90a 657151c3d0da4feaf29a23f8659cc85fec6616eb15f0434915799ab58c8349ae e63e6fdeb34dd6fbf9fc10a9711e8bd2094897673317573372234e9b48ae9822 dcc6e40c5c54d79ada243e16c849cd74f16e08c3a79ae0387f539ee817344f2c 737ef84d103303699c05ef1f23ac4b8b01ac200d9324ff6f1a884041d89ae290 e4b0b6a78415f7d029c7b0fe8df97b5b0d3a5cc5c414261d277c5b50c909fe33 248acc0692959fb57c163de87b2bf1642d1498c2c6f2f0d159b1b867a1fc49c9 0ffda258e558604d6d683f0aa70d1cce9bdc0170963e6af1ea7d1295b21a12b8 58b0d0c808ebfc723ded04aa97019a0df61eb419e19263e27130486bda77c348 54271ee5f224e44e4555fb0a2cd520545f67edd9cdbd1765009bb732a0db6faa 3bdb060c02d422584f7f19d44266f11a671db179f2e317bd1988ae5575363031 92fe6fd7046c4d31ce018bec894aeeeea9124364354eddaf45e3a3ba5354c130 e356c33ba98e351deda0aabeec216e66d6bb348ab78778f6a57c33370190b410 e69cd80fe4db98a960cc72a7e03bcda4c82cd701d4098957e3f552fbfe6bcf64 b9de4ce0f2cacb569d91fbd466e20ee1ef5118934f5c2e1382c7ed21c71845f4 b408f74eb159c9b0ca66c764798c791976eb5f1b494bd3187dcb51cf968b6c34 399a68cc0e100c4d76713849d1ca451aa8ddf40fe3f8447b44d838cdc0aa8b71 2aed0b27e00f3e8db27c8c99b34b44ae7b23f9299a4674c95cec366209d5dcff f1cbeb1851f5087042373c516a73b09ead904557d5ece56ec7e0f0d38fb73707 2df062e0dca2cb65ccc8c1dbecaa8e4cff6c6dce8ec231925f4be06203bc6a62 e0654e8cdf80badda75e98dab745a94c2447020218a0799ed006b0b47e5ac128 7220d4d361f301696828b944d292593d5dd67b62037a0eebcd934f2398c27e6c dd564fc1d85d3c87d457b1e367f59f02f92e99627c951a172b9fea3c60453870 7fbe4ad484afb10c5aac396ea0786bd118ee61172f86535079d1d978c4d7bc8f d2477e1d0be27c9e529ab33f4b0349c7df3ea22cf8f3bc5427b9afacf599dd7c cb4dfc829a2667278d08f5a5bdd74072493d726f46767183e37d97995ec1bc76 52fb0b219efec033ceea19f06cdec5f2b87db244cd3a70e3ba8b13461ee95401 301dbeff46c41189a673ab2b912172386ea64fb1e71f7df5b3cbb1d8e74833b1 64c7558c06303d9680cc29679bf2289f48a97cd9879559c79561a949b9923b00 11b5e8143b725c0c5cf16581a064d42c0428393f3f25a512bfd6784288b4d89c 58c7b247184c54fead0e80250fd2a621df6cbee49a27394123405395217707f7 fbf8e8acad450bf8c9f46bec1751d73a332e7ca1a7be3909b5b064a25452cabd 66b43990bba5e706dce48bd419779a32fd67691b29638bb14ac03e73a8dac3d3 bdcdc087ad810f22b24ce6521e729a4e0942965c9d7bb7cd975027e819137702 2c229a613667ee0943626d8159d80382c8ea62c85274c8c47165f3229d3940ff f256f9ee7adc29ae326824c22022284370240df0bfd83e1cac54e1220acfe970 3848fa7b0ad87496b76fdecd691da63e66d835359356cdce5728f70668b54554 bc366c738fc687c3f8c039e05f2bb558054ac83ab9de369140f802f064aa2fe1 7b26a45196e1a50911b45ed1781453d57a892115da9a870fa362fd043b00a5a1 efe64ea7891f5212847b22015322ad1f9916b7c06da5f2efe3f93df2f1871ae7 a84178c33ea5c6dc48c2565214c7ddaa60db50edf0e958f13dc2452c16cc9199 8559d1636d638f372a69e1faf7e87c1585112d4316a2a000e0531d5a644e5baf 3c9b8159ac65d6a82677d7948feb44822580568cfbffb857d7cb8f572652f230 eaa0906cca437ad347eb4b8e2969d4b9c7f3029490a7af398e415ca5b4c17570 d34e043bf0c513a5480ea527038eaec1611aad576f7896152b8c616513b164fb 94daedc33fc452c12a9e6c11647ccbc3414157b1a3ed3803cef28c10a45645d8 8388565c2ec6adf25ac73b797a5b44705d89ae4c31f546f8619d02864790c3da a382466ef4116645116a09bb102da47fb71e30c247f2ecd95f1f3bd7637fde84 d4d0ade768ff952e8b9b2d8f0ffba87f7c20187025a52b79a6db346f24fc9412 9a70e22d91e74b4d4ccc98febfac55f590ce9f0b144ace9e296cbe1ab3538b13 30ddd13eac8adc4c95d60d08e74b43d9f2db5bb526b9f8b221d2193c9c2c26a3 d153689d4845eaf0e381d4f6d02d05afdcfab622ca8302c3e5475cb8807229ff 360418b63d1a8f6d530ffbe9bba4d611116ac441002262aabba068d3dc764ae1 64b9f3c706ba0d6e6566cf2953ae9f5f0c427d8a72a05198eb50e75829dacc62 219048a84073504746ea90718ea0d564aa2f24e438af641b7408faeebf06d469 69fbbe0582e2025473eaa614997f479675ec21f195455ff4ef3f2f0216cea0b8 18256d020ddb14965f145303866cb6237699ae8eba2ea8015bf8db90911b5e0e 50724d732b5b8f9671d5521d302400006f0d0e2fef56dec9769b41107d5dd4b3 6d7a82439a1286d156eaebc31453a1cfb079efaefae5c153f18e17d761355109 ba906cf77eb5c093e9268d7bb042831c176f05ac07e04fc84708b9172741a198 40ba0aab270b2d96534711fa6cf2cfa3cd7cf9bb7dc9acd2b7ce1c415a1c7be2 9fb56ef46afbeb2ff5d471418aaa3e5cf0f8745672eca741ee222ef36e29950b 8fbaae280197c57161a63494724653797fee75a5efff85c2609a833a26ca99b8 7d323762522c783a663e532f405df9d7da9d815aa8b4dc227e84fd1cd8fddcde 14bad79500d5542db3fbe2c2058e1bd64f62358e8f6f6b20bbb2020e0afd28eb 87444d00a95b58ba207a1ebadeb5ced25f795fce32d54a5b9657b99c11b963bc da48d7bb76fb9e1ba8f906bad59da697a8d24e7b7bc39e9c64e1f047cf43c112 5d3e863d3d01f2b6c87177bf632ae6855c17a0db5901c3e41e694d9e94b2d030 e3755f4aa9827f7e9ada990ffc5cbdbecf8adfabfa3996ebf117c78105971193 5919e6bc9abb9bf21fa9294e93fceb6da3df2f55d723d41ea5945f9231646bb8 4347be8bc56173565f667bc57c0f72662366dff5a81f73d92efd19acec50f338 16d4f4dc3d99cf6696b8c88fcd90b9b7c088e432aa6a1cc4b3a73f3e0daa0f31 5a784306b9338d265802a60f6c8cef82d6aa88a829eaba9b528a67f10c82e9d2 b29fc694b84fdf453ed28e163a99e66397f4db450bac7699930e4550927023b2 408b85a7772c590a832a3aacb0d2d156ee674c49c215ec882b46d5eb55477f11 a84f2523f44bca9b06a7fe403113969b04222905e1876ab06d881bb839184d4a 79eb0818ffc629b93ccb41dc254b9d27a4b7606cbda5c506594efe778af2ad62 7bd60216d32e96337a952b807019c9deb90ef013d4a1f285b21c6261c443ac7d 4b8418b19563c27f30321642ad3cd335a87e46b2ab7c7d419b110ae961eff461 bb1492f6e49436b80d454feefb4721ed362cc08f2475bc104270ef43865a7c35 c12b0f201c7d9824635339dc6d6f197ca38df3b2310943f2c592a14dc148bebc ea037133e264cd967a8a7417169fb813d1451db3e5b2193d7fd77805ff4c3ba9 9f06ac132b2e0b4256737a91c33882c84794b9d6f82fe493b229554e25dbf007 c6b6ad1ffc736397321630cfa6046d6ad2946b692de0701ad18ec0ecd17f6839 76f8e2e9d06b64aed12414a424f8f8d0ac9a7c8fde9e530391bb26906927af87 99f3292c8cc98a1a47885eda29d48a4cd7313b0e9d32ebe6daaac982db5bc5f1 f3062a7673e4058d6cfeef20246984f30987c8d465e7defd1eb665b1086b42c8 f8adc680bae2b0e90b72685f6a177f4cdbc3005ce952e9e82d005a2f16c229d7 5181c9bc30df94c0837c495da2916f1e1d65171962a8744b8f4dd45af325ec18 f41f1c2a53480dcf6a53223cdcaec02bbfc79b9cb994822d56d21a0620096202 4666b817d07078ede2a9dd0565ef08b6270ffc2a4a8bfa2aafeda49f0227f944 77b0b1194960c0878ce8f63b555ded471a3bb37899e89fefa16769db6dc212ca 39378e8ee2379d36a848f58c37127a675161b41a49a7385d8cf4f6ad29f6e631 de48a0b9a3154626448fa5bb10b72631660070f3364f58702d3d7c00c3e384f8 fe75409f6cd563de33067048e95eb9cde3dbd66e0d016141a79ec820a0140c7d 685887d75680d8d834577e354115db56842ba0e6428b227adbbcd8e3b29d9987 3407d8eca5e05edc3f0de8688e50daf999e0d2812cb7c879ded0aea399badb47 08b404d2c831895abb4b0c13182d1763a37bd8834a2b5d8d060af5be02cc2199 3cc211a3042eeb8ef74cd4632f984b9beb85832ad8d24ab553ae2510577347dc 5a81cb397d26fcfb7d0dfd2461f387835a9fc4eead4c4b994c59bf9c4ebc90bf f4c0ab8f81e2aa5a217e53f08ed0e99b90e401e96104812e1d770416de42c54d 347682545257fc5ade4186e598866be93d2b4676dd446b998ab3ded9d2b92a87 1922ccbec73e9c245f1f8743dec6ea7a7948eaf39b4785c14824b7f1b1420c5f 30cb4c3c3051b4dcd92c7917ec0dbea3d0600af30a1a844ac1ca3adc85b001ce a019596b598725056fa4a582379adda4dd93a4e15d350349240b8384d1857cff 6e934f244daa8df93247810bd3a2c28ffacab9fa44b098f1eeb5f8e5e83c206c abab6b44b250b8a0ac2cb80ba4ae76c51d69ad4c1cf5bfec1824d2546e3535eb fa6d8c102466515cdadca17a397dedf13da17e50ededb7f2bb790b8c47df7293 f01cd9ad70b5e6b8679c0220f003b52c4d424624e83c5a2022a4285be7f0879a 9967c44c75c25599fc88129e1fe2638c120bccbc6532cd8e76f4b9e4bf068add 172391298b92e107c6892eb1ca550ede4991fd3c2cae84e47de6695b68c3883c 9e923d0e9434234471a93dfa6a19645c2459283c6938812242ec502218acce54 291a0034cf7744e2ff7db6aff72817a49c25923a946dcda5b39721a0468de437 5508c73d8582f7d57b95032048a416412024aa2d8f2bf605fb738306a722d68a d510468db5c0587dc5eec854d55e4f84a783adf9b7a6364ae0247eef02e44098 ce2c2265dd299dbbd04fa108a5f861bee8ca902af5d4bc97d38a4b599a52ed98 efb5ee71985b3a7a7794e6907c14748b5a73d7f2e4d68aa1646167990680f7b9 bb73a01f20f0f159f4bb35f140b6a57300583d69ff2c426d3c22276691b351b3 48f5bc4aff63cc75ce27f333b615154cf79a6791e703dbdc0039b9011bdbb33a 9177e18e59593ca4b569c250e437addf4b5c7c62f1e15ef9d0b2f397dbc572d7 67d7123862f5130c9ac536878aa420d3623ae28f563f6d6062ffd99890c7458e ed03bd758a4941642102437287428678e5a8270d2074003acebf4e144d495fb6 28c3b9897118d120239c570a9ee75c02689715127fd95b599659b32b05d7c3d4 864fbafe2138a7dfc36e69a1839fe84ba565bd3d2dfce0cd8232633b033fe991 c6a9f2252f0900848067866143b784fa59c6155875fccc12351792c713ff5b7d c364619059caddb1e9a180b251c60b17eecf5a9017bd93511319bad326893e38 eefd2ba517ccbb0ee7a212c60e65b090afa27c62398af5d7e61747935856baf0 c28367d37e2660398a8da17fe7d01cb0d0cb449e94a554682b3d2637d90db514 6b23185e7a95a8c6b7df2aba92030c479501315f6040923512c8e36a816af3af 4d144104142169c6ba26961bcd3ae2bf39abe704ba94ea62271976cb480e106f 436c8a9c47effeffec54e5c4e3140823082a71be5598a81312280f82ce63db3e c1f358c69731908837f0adce7504888237d16dc76a63cde48d42fb106c0bf4e0 8a9ee329fbe3667353e8e8f29ab23d156c5eff1c67212ab73d6f7cbf55f4432b 1c7b8b36495d1e7ed6c075cf3eb7e2012acdce4c90fd47be6a5f2d349e7c9b47 cb97a37b03fe61296301c5ae973ff55e3aeb108c1fbcbf33fecc300f77ee443d 2b9c920bfd331bd163e6c1398d6b2041ffb8fb2b0b4da15d25e98f5142d9810c 3c4928535ab946e4c09e1efbcc94f8040477fe16596ed693615aec7422166064 2c80ff62092aa1b4960204768b89c5ef32347b88768981f296981f4aaa2b864a 949b17a7f4fbe91af3a788ec847d848c53216fee2a71ff8b0f62c9b2c02dfb10 277db0a0904341ddec5f2b763a3d06f6ab2ea9dd1535a53d98fbb423bbea4efb 97ba2af0c9c124c14800b91be7019f8e87a06638567279d63a04252e7b64da99 89a48da82aa6b130e802f6e6df4d294e2a134ff3126feaf093a575a26c18a0f5 84a371b94325bfe873ada28eb9e62bfb8e1490b48c808e428c550ed055c00a78 43c617deb33c2882c06bbebdee0564899e5e047ae1935bbcd09a60fcf4778a96 bbea09faf45ad187f5573e99fe0f579df8709181e64ccfc7f8ef0c861559f638 c8093bd2355c617e7f42358d2b91a3a4dcb1c2a7dca5dbd5eb532cab4e81c93e 39354b8bfb2c9b1b3969947d78eb14e95e57721d6c173928f9cafe0d9b3046ef e9307ea60f64cfbdadbc784381c0e5245242a6e418a8274e56dc4136cb9a8596 da071e49a05826fed6377a312a54cb74740c68d9fda3463852c48102364dedf8 dc578ab2984a5c1ca80be9259e0a67103329bda2010fd2baae271426a4d760f5 bd4f1677ff60b31d6afa35c8d66a651cbcf0df2bc031314a9c84700536acaf51 a38cc62dbe520eca003a5f178fa756dcab2b633a74f89b5b21dfd3dca4a0b60f 79d3b9b4fd76edf1a53b806912ad19d502917691d3d6df0748eadc2681139574 b6b035af831868e8eca634a9e3d3d7fa76c694a9f4d21df6c391b0cf4621f656 b8225dff0198777014208f86d02eeb6d3367debb810c97e700f5e2f62829ca28 bc10342005a7a14d1f42bad749e84bf31890b3ea3945802a4712f37ed374bb00 7cab0b49911a0a4d5c979e6110acde599ebc868979f811939f834dd416b50d13 79f56da3d8dc4416d70f49ba233ad3c439d3cb810c9baae06bf52194ed46dc5e f687c1deb7ccc7ef03b0d2bc8ee726e13df23342cc0a99c7f33b99ea1a438ea6 bc7f7c304276becead270bc542c25e3fadcbbf2f5cb8af332ddac1a4b83ad94b f1bf0ab564227285f9529b8baf8c0b2acf4b7dcea3a6b4c84897ae11c9f517a2 50be4b3eeb2d82c431ac64d1f9b9954d68656b07c568b9930d91d60933a41210 99ed3396989cbd0c44ed8ebcaef9721f1aae796708c85515c985108e04badaa3 15c3980c75f23c69e4215adca84d50f600a1e80ee2300ab67c5e86973ef01894 3f4b6c3d549d6544cf725d01ff122420f39de1071eec813bd15c272bac389cd3 05d2f98308cef577346942809da398605a0a67a28d583820876ec506b98e9bc7 2a346ce872fc3423452f69c6d5df5d73250a4e9c1a8545dc8f30d6d51502ad20 3886c9f0a331bb8423a1e78e32690b9f57b221fa446e427de75a2ec2215d5d52 5e0a78ba24724e5a6c690333d56c2d410a0141a6498f0f5a1279636e7ae618df c741b4fc21262edd17d4c67684e514f5e42b354e77d9591475a460cd272e84fe bd0b7b4bf2f30eb82f59c079eee10df711d04b7273bc4dc742fd7c628bb4b9ce e47cdd9bffd00591e70b84ed385b3a5206fdacdfece493f2c96397cf7de5c723 afc1b582e7fa49d078e36236801ef4054ce26e97177fec53d728a88da027ccae 12b9aa2863e6d985a5457bde4f8a63e85ba6ca437aee0bca6fdc4164394f25e5 19198b9171d8b1bc4caf04b1689f930656ff21e05326c7f6fa20fb6c5c4b4491 146d04c22b8bc9d82e9e915e6662a27424b19b6f148682f4d191c4f0bea69b68 cc006d40767289b0b2e0248e4b10a35f2b8fb8ab62d3592d4e69b33380523b58 dc7b2b89c239e25ebbb1d8828cdf78f8e9c889e4cfa7c85c0b2db6bc4558935e 0aec55fd8bdcb87d1db089795fd55021b618d93328274ed705563425e050067f 533e51604af2e7a257e5084887be1d74432252648fb07ef999ee246e02f83841 fbeea34cbe447c9e903b63d8d21ca03011c0c227a99d96642ac72cdbee36be2c 615f93d87ca6f0e10ad8a53ec342b2d0b33294feb6e4344d3c3f558a04e36acc 1db5bd45a89e534c551b25fc37ce501b636d9d7213ce7bf4ecfee28fdde190ee 50089f55768e220eda00c06cd053379552cbe47dac4073bc0a962cdac5d0f734 77c1ce5232a939c4211388a05514131039a507253f1f81a3eabfc9802aa582a0 c31433047c5bf66ae7e04f2333d413b06f99e8d161875ef2b78181c8fde59205 b9254e1dfab58644faf880d90838135d063fee714db4f07f3165bc7d7665f068 6e6b22cb52c76c16d90e9c427523c02563aa42fd6bc68af934b2c1e60af66fd3 bb808cb080ad8117d96014f77c9f97c2c5b16f7de8b4abd8ceec6cdacb525ab1 0dfb98f07d49e048c6a4accf772ba45d51f14ff2204e9407ebfb32b6bc6003c3 02161010709de4029b0c60962593ae08bfed6348e087525585c92118c8bf7899 9ef316fa101f01e72d8b1329912b87ed37df4c0b51c4e085b483c1f9ea455b00 3947d1de3fa5fe89c238c5b27d2fed1a84a499a120e7b16c719dbe1c85e8a8a9 3d915915fcec821ab115e6180df78e00891ed1491d1b72845e3d3826b74a8871 4eddb1408063a370a44cbe01d1e745fca25ccff46884c72efbc6e306c7746ef8 f21c1f7cc288a05d3143caa1f5798f5382e4981e8a48c796aba3f524009c23f4 558fc080c1f076bc76e56f06c2583dd7fc614442e172d2963e67f83fea563a8e 85003a4883854ebf7db0fa7b6db03037b2fd0cecee5a4be43067583336922c24 b221cde98ff5cbe47b5e79f6f6697273ba803d5a93e150e9350140a76d512b92 d27bd905caf8bb1f49cb3797c6e0a0c5c8811e6287599c121c6d251ed3df4f07 8c5a116a558b95853d6ad547966e037c550ecf330a9ce97fd02dfd744fd65a4e e9e026ca2cfb8ded2534c8ff6e701fcb20a5f688981f81e5a927f550101aeaf4 929192eacfae848733cbe92677dd2c3429e5d889fc0a2d0c7bd8f56434797d7a 5713d7ce92a4ab8fa1cc5e64971986de6cdab440a6047bd48173fdf9c3f70b6e 8c93f25e18137102700826823eb9733804c6250476d703e6b7d48e5242c5f370 753c0d7586b743e149fd4b00251d85aa3044036ce8a3fba3305ae1dbf2042f93 def2152a7eaeab685639ed1b934772191c77875e6ee2ac4215ba8e21ad17f28d b29a8533c056c4033802d6da64049b9344805ed2ab5429d0503313da08be3615 fa34c00e8d9bc0377f9a9cfc67e51c01c31b71b321f5b4aaba8397f8dc908650 512dcfcf43a8816c744b1f73662e92dfc8996a439a70ea51831a77177d81d138 086d48bad1bde324c85d96a392f8d08cc5eccb8a9e019db95a504cae7739a3d3 d4d66a7541159b187d69f1b50ec9991ac5aaf810b61d9101d228848e9947e866 e10f26aade9afb1b96813cb2929418e89e99abf7ac20cebb85422b0a29e33484 ded313c438343910676cc7693c6644087495e53d6d4ae4c405527270317d3ea9 3698e1b3b3b546fa4b784ec9590d791ab9835034b51075793faab45225eccd1e 0278d733920c829466e28914e2686ab2e1f9a9fad88b1d09466d52f27b8ddda3 e1bf0858ed04d77ad375e94ce05fb90a02445cb4e183abbb4696b7b59f47dbde 46bbf2a4bebe42a437ccd0b7d4d1ee29a1c711ac9b06b6f091736d3a9ca89223 a1193022ee51023b219997cbd2f72af66629cf0d9a538dc0f5a329746c8f737e eb014c0d7a59dc0263882aaa1775d9c943b2f91ce6779a210b4e0dd3e7c23237 300ac99b989ff8571874eae7948be146d15a3131b664835d9c838d8c86c18407 379f70a56cf6048b0a2005a2a21efc31c6eb9e57474202523539a22d553883a1 b67b779f2b6bfd8598f3fb9eda9e71e04a7eb96bbaeb31c85d1364914d88dc56 7fe0a0e0567aac19bf52b6b854cd555735900acebc7e05d9132a5fa379820642 407f6b7fc836d4ff328f6416c6d480dddf9be4dcd1c4877381959d74192e1213 dbabd7b2712c5a69dcd6196777552a31d9d711a5a1e14674d688e493d12036ec 90e18d117bd28ea96c7af738cb365cdf800995f57816981c02e710613875f9a9 89f7d82aa090237cf6a7ea67e885474e40beebb00bdda98f2604bec863d90ebf 6ff553e5890f72572a31386d614d9ff79f7ab5d05af413deac7000df1d0f1f76 0cb66f44fa195c37a719307eb6d566fc9045bf2d88e44e5a1cd1770a4b4acf76 7d1bda4b7f490b6c56f301ac43dd938978499e5048ea37a06d241a5c9f28516b f99bcbe0e54496751c7514e157bb7ffbe4eb8ed365fdd48ac1f1409326fca4bc 41a7f1dc912495ab68cf24842a3972ba621e7da310f32282bd34645044b31352 0f797d95e095963554a497106def6697c1ccb2c929bed22a2dfd1623e792dcc9 2465b53cffa0f6fe800ec72e0996e4190f3cdc574b87f7d2c4fdff1a7f9428bc 0b65e29935abae5e4f00553e93bba5a194b51a3dd2d5425c6699ff4e9fb2ed99 5d893584dde23e4049972b5894dee05ffad55fc2474a57e7972639b808cb6462 9f5207511694d755c607b4de86bfec7bae50af7c291d14c6da9193b3174e5bc4 f147e770843bc54220d56ad2e283f4641a38747464e7e8378ea525ceeba17a8e fcb28ea4d8c9d896a83e2a63e54f84b1b4551c56da40ffe44fb77d23e8cf7fe2 b0dcd30efd13fa2a0b61f24c8088263a4b237743e617064c50a7cdd2ae2d671d a925714104b72b4cb96464a557efb7b1ba9eb37d1b41429079d8380d5a8165d6 fb2396d15a89e43b7b1c64ecf8b9708f5147119b877536336ab304151e190580 903ad9756360b93836f5a89eed89bd1139810b2bd8b8d08a7679fb2aa9f3b4c2 0adc98e076104c6bb1a0a7629d0a11b77e8efcb18e5f1454d88b11073b2f0dca 93d1515086f4a386b5068d7bddb4ac247646d995abe74834bcfd94be5e51a059 89dd623335cc8a37e7bc790aca2a4cb0ba776aa51591c7ee918d971a95a6b4a3 c391f4ef9671b9366b0ff99b7787998d0783911f45100f87e2694ceabcf68b58 a72f585ef7394ec698ac9cf9d981c958b04075cb97b9bf6033918a9b7ab0e378 3f2e409b1831776bda8027656cc1a003d0a3f24bfd802e3d7dcbb15990a1ded4 1c07eaee10b991bded7d53b25b5a0626b0aecedacfcfd8287fc2c07111b70ba2 c2e583b4ef5a9673adbb5bd5fb2947deae3c58800b4b7379f3d210e2076df4ca fe4187b78b61294626fa9cd674c0f2c29a3577b022059ab244d671c1f93f1250 57e6b1bcd63b4bd8ec500c15c87f222873ccf2e9562049be5a92ec4047e92d9e ce1da5116ec64cf6662514ab2458d574bc3c6d424d2f433db083a29e9008d2b3 adf5dd9db81513325d03a57eb07ebb7436b89bd44799523f995b9d629636995e 241f550352c0c3226fac19189a9fd0ab0cbb092cb2e76cf59ad66a7bebca2932 e4c4079ca0399ee295c328228587e7c7547ec7c2f32c8d4ba31f6022eec22824 4a8f99f2c7b6690e4936b3ab13dcd2d1206f5a4b42c02013c9db97fb038026c2 6fb460890f2bafafdddd0334fbc01d6b2386a83e6c276ff4f885b8b3ecf43ffa 904bd2b031f7fda230442fd94f832e7159dbf2dbc8e196b78cc4310ff5de0d83 cfa471d499bb6981defd5bebfee5dcd38845c1470251ac2d9f05df066524be23 effe0425a7bbacd8c2b32ffa49e38cf08838c6dab92fba66c2963bbe0ca1a175 d6e2f29098a2dbd697c4a872679e0b26303232a0337d91592bce874a2600440a 8071a3d5a8856a53f22e945be59b8a8e7829a56b4c2f8159f4ea3e03f5290bd1 7c1e161ac665ef1b2040f199f2e53f491946a25379c3f9254839c85cb8778dd3 4dc465b346e89c70ea5ecd3a91286fbbe66c01ad7f0411851e2591b4d40b8f49 4b6ecbb59526340d1fd1801785a4a0fe9bc5eadf5bfed8de902108e8c1f6da43 b71063dc05c4bcb8c3dbdd4e95fb67f7f0789179f7af784b2cff7841a3f74ea9 80e5894a850b1ce04fd779a3bac5fc99b3e38998bccf901627dc29770ec67044 888bbd7509001cfe677df09850e32ef54513e500cd415eeec19e9b06ba8f2dc6 480945e8b737ee8253fa3e67c704f6b13a1303bc6ee12cf667c378445ae15775 32aad97a027e4e83d59931e952487f12645552c9a16ec110ce90f432a17b2b26 f1cc2ff56e119f81b35383ad4e2ef4e80682e40a58829162c2b426b0c1721f40 4cc3e2f2238edd3b82db177bc0366c2232ebda369b310f16e085327931c57757 94b3de362cea9b66c0b5b876144302e33d535962a705a04ac0d457b788123458 a609282b64559677c7f4167a265d0d43058a259f1c3f1111fc617a73b9aa661a 5113361641c7c9da260f1a1ad048abba43171db1ea3e420f1cbfaf2ec7c74bbf b6acede582b5806ef4d4644686bf5bf5a3c5c6a9c66276c238e9c9f21d32ca33 7e9fcca8bb7dcf8ce426667af38e62873c9ae95139158f0e0d44f95505255ee5 3124d9efa5b34189e5a3380cff1701bf5729bafe9ba8e764bf9bfa57de6d4bad 20d7219b357c4fdc520bb62582b80150bc9f58bc0d7223b171a9f1e0ecc42629 ce18deee4f037e7cc4fead8ca52d9fa4ccedfaf0f1dbc65636652923147e537d fa46d941348ebb25521d8eed73729d1460ceb010ce72932e152ea7070759b6e8 4dc62a10ae95cbcfa0498c469c9a7655d671cf185da9a91b3d6aaa7e33ad69d8 2cae2732307a9342223639f1fe1abb68c06ac64a8383258f31af4627942aeb07 223a2e78c1aabd658b9bfe8c1ddfe879c4dd08408761b25c55cbbfbc94a224c8 2853041c3da77b741129ea9c3a34ee5d750fd9027fbd7526a2 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 3 /Lambda put dup 5 /Pi put dup 6 /Sigma put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 21 /breve put dup 22 /macron put dup 33 /exclam put dup 34 /quotedblright put dup 38 /ampersand 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 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 63 /question 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 88 /X put dup 90 /Z 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 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 3a7c0e66de8ed85c14f1f8492167357f51a7e84cc5d92e0fee4d81cf7fbc8de5 2d2e7bb57142033993f9c08c315abade8dbc4a732e84e142d3bee51557910e12 cd8aa37c459a5e6b7f5269f59078aba3be4641a11ac48d0b625c8325b38ec08e 4c9e5e7fed976a5650d99d82114f449b9ca14c4ec957702295a39a93ef93f618 99b8ea06b092c3c1e503e6e436e0a9fa22576c8930ab3dc8c20f5d82b69cddf8 ff4dacfa9c54bed5a3aa3ea5b129fe96be632843b9b6bc91b615581a985db56b 1e01ca60ee69ca92cf5c0882ece62edad3e106d835348822400f0b66af658f2a e56ed08f8b0010571807009b73ab12a8cf14ca6c71f03c2a48c500f9d62266af 154a6375ff600d9bac3f05ce34142d6867a79581c533176bb2f3117336671e2e 44638a97167e2ea9644e31ea16c2ad2990ea33c54001e0c8156e6de8ab6a4d40 a7137ba275f39589fea2e2db8256adc103d6f9cc038037a47e8fd469c5f98a5e 3c15bd4ace40d340018b1cff7d1ed8abb0ac57b5b5a2c20a51957b96c453edb7 dae5affd91a46d938fe0a13363001d844ded4323f1ee6d30012aea19b024a552 315505535c85dc26bad31e09c50e6512802976d298c4e90d0044c362e6bf3ab3 62a454ee93de25ce54411090c29e9d75c80ce26a84404bd9de3aee0e3f921ac5 87f907572b8354a5c3165eea7e8b2ba4e333802132f760b41a900c698eb3d273 8bea0aaf7aae96a558f08b61aec7c7d65b2889bb7b99800cffee5ba19ac90fce 11437be1ddfcf0bca5670b543b9644a71c8726d0c1005abdb96f6e0cf7caaa31 95017ca81519151c2c0b04f81e717c0691b63a76e9b25329ce2625b711cb7bf9 9d23ec2bb5ec8aefb8c1a3b422ad617b3da4d49685fd44cc21a1fe5e1541163a b3e1594601d63c7e98801d7746474377ce93f81404f3570afbd5dbfe77578605 917ab96963987e6efc2d3f99fa56ecae6422579852567f90cc90e4ef6d6f71d1 858270205477e221018a375ca83c572b3f972c5181d18643cecb04909d84cf76 7174b3c0c8ec69bf1e9fdfbc0fc7ed97ec67d878ea4ddac8caa522fdc6932df7 21b81af837a200e1d96e90a7ad7589fa917658d7a173a511db53af1848892270 44c4603be1a5ac1893c15a21fe5979a09aed3473a98e06c98356dc01ed3f5e14 1c3c057c3f97e2f9e8718b12bea7715a3432abe07df50291d1a395e7ba2f06ba 7cf20988904310104fa43c7a1f6d32f5fd290d4d572cdb5ab4b433bb849dd1a3 58b292cbd5c762262751c26d8756c7a3c4e6bef14ad9a628fa2ef5c0010ee4a8 d530f3786bd4e0af07d5587e90e19687d1a31921f975bff26ec2cf07f3efd4fc 54fd4f92f08a60ce2b3beb2dc8d81a303c8103283e6e5341eaf61191f0122624 2a4a88caca606438a767277972e2ad7ddc9a36f43d5b2d010b30cac07a383c1f 9c18c89749e610d026f72e5ee2477286eb57b77567af9c4f20f7454a5719b0f1 9581ebf64277e8f2ddf1ce08ee33f2a532228509a9ec0d95bec2b7beb23319d4 236fb1446ff7df90952101435fdb0a0fe7622a6a3972afa4b2cd99b5822b1bbd c28f623efa3532adbe2bb8f50ee214454f95d282af029d174edec357747b4d50 b02e2f47b1e9d736f85d789b598b19994b4bd1216a964efa28c5125a6ff6d16a 9e89fd1b4fdada3be2a18b5a305b22290c90980dba0562ab6d2ac1435683e40d 0d978602ba184a88e40ec0c33d17bdc368ffc918a3bfe4e2c8615950b45a6153 79e242cce3f5586ff100efe5e4e91c83678ec855bb8ef048f4d3e4af0cb56503 11178f8928bbac72e594effd9b0b6f1fa8bb0753a6e677f0cb709f9df41d3961 598f9e7e873eac311efb9b55706d89e0d82ef720a47ba58eae5bdfaa880a1b05 162ca0489a3730572a59b7731b8959d167e775fb07fd908b63f9529efa6607a0 d7dc30ccd04f643758b7c0539ee3dad62979168a0177c3a3eae5d1bc600eb740 03425bc2b16f3464562837b09edb51f5cda22660b503c147cbed3f5f14e6d4bc 08ed7d088744cadddb0daa6ad59d1bf41acd72d83e7560afcfeafa21c3043170 aabcabaf2e351988f8f87b8ef50e4474552dfdd8cde761440aaa7af123f4b364 4231b2f347bfcc39eadc63ed4aac86f421026140028e8354a9dd676ae1418376 8451c4ea7185936fc36921d48b579610388be4cd119595f8c8604d9ecee70f41 80357538110d001831e8fe2c0b3f53486a3ae64904b1a69a37126369f754c145 d0df01a4cecbcb6a2a7bd6daaf2aba5f41002bcce37694dedbfcb5a3b9076094 d0dd4280e79cc4b1576d4dd0894c8fbb4c23811e4f49aef049253fe45e699a22 085c3566e9edaeb91bdd6b563069ca3560069b8f8147c5805dc5cd7a8db50136 40da1df5e91a087199b1b5b9834ea0c512663ecc934d69d990001cddc316151c 6d7d87efc69ffdac29eb7f9f34c69a32903b05cbcbd562dcd5c3f7e9f3ff7e8e dbcb262684e7a45d55ab53d7823384d4e2550fc82e2a40c3547eda037d199418 11e7ca5209fcbec91720012e8700e2191301f8248b25dcfe048cfebfcefa92a3 ad818f63ad269517890476a718980bfadbe478d25e7c591337e33fe0acff35f1 6134570fe49f1e65d8e3c3caa9061abdbd9e002d379845b96222834e651d3894 2b43370842dad222765a7d2edd41e5d64b32ca3d8daf0860255522bac1484606 7f36294343d9feeb061c42716c27e9c2a73a987189c00f6b509027d1ddbd6867 27e3ce6b540c78eeb01a88fcf17847977e3154631402fedc12c390e68f3944be 092a99667c06da6873893706b516b9b908d52d3b0f458c3ee477332d7d7430f2 6efebe8d7ff4556ce29bf9d5f16c45af4a7c71866ec011da156d06c6bbfb9291 5801351ca0e1a5e06cfbb9a6dbd761d0fb2e1d3d21ca312c0e93e81c946fef0b 73de93d58717b388722429b578771fd3e705159b3155a8b0272681adde5dbe13 177b8875d03ecc6c3e1c2ddc9d24ef737516de5f2ec7ec3a2384afa96f907aa3 829c2b2fc992f49f88348dacd117b7a649430b2e53e2c5a8ab6a016ed0748532 4fda7afe7acddf4bded975e49907c7521534986a369f4a6652f7f9baecc7f912 8a7290297570d89d1195d2cd4d9a69ae8996c3736f58e5e4e27604f031fac122 4e04ab5b02982f97ea2e0f975c156738fac7663f4104b70c86eb2988528f07fb f6b883cc1e17f36567be7128dc88b135b83d876cc8250bf2a30d9a48fdd721b3 9d9d21c5d24cd615a016cc7ddb968edb7d76dd8a0408dc822e8930701b7e88e3 17b4df8f8a1af5e4eab655b9cbf14181c18988ac50f0421ce9b6957afa48d735 37d8672327ba43b93681e65f97a721504d5cd1fa9a6c1ebc9e66d182b4699714 fc23eefc07a262e2916d33341ea7f9d424105bd2442016f95909dd2ad864a3cb c087b6c27be4d7919f775688cb496a0fb884799daae31726c3460cafcf6b94a0 a5eef50d8a9e5004c2f0838885361fa883bc883456cb91a63fbb1d72e75e8bdb 9a47c3a8ec5d2623426c4cc7ea4edca5c6fc822302900b1bed4f0176e00dabf9 6506c433a31425235c2185a0fd30d7782fb75fb351db16ee57a6aa0529aad550 bde1ae0c3c9cb52cd8d4dfeaabacd345107461f3d67b8f3c2d736ba29f570945 3a0be4e35b828d8a96eb63f037a916f908caae56c7c0c9849fc6cd3d56a43d3f 8453e6641d8a7811608310e2166357c0963c68fa9f2cdd13c450dc9e92e54b0e 2f365c904631c1681e872cf20168b4b47836b431d4ef5f9e36abcb69b4acea7b 1c09ea6a34b9778b0669833a45700dd5970f4f332dc5aed455d52236a82e9271 afab1e36452376a86b1d928d6086199568692ded0fc37a3a215543d11abf51f2 df75a82869e04cce58664ab45edb97cfbffde1a13bdefbe80adb4a472a345ee0 fa19cdea6bb18300dbae7b72c89cd2b0cc96241c8a01a9db37d167d9fd855f65 2004707b015a3ea0903b6877248633f0666744ab558d8e429fd6d5a8a4c106d3 3ddca8726c646701ca4ba76e56dd32c9b026d78e704fe5b212a32cdcc979581c 8f9dde303c269e79d0c4583d1268c19e8f2b18ad06e25313f11b2facd01c732b 2ee85d4e1b16564cafa33a1313dfc777f6699273155f48038bf6b469c9ff3da5 f29ff104f78bd24d83967757e53e27b314e01d016f48e750c3057cddce6fddb6 17ebda03403f169816c2f88055d05acb1c7bca581b1e1c5bd5b459ab3c8b9d7c 3bcb10e613c57499164a3fa88cc344f147b99f4f9b4ded93bf506124a629b079 eb4fc32a4251fa3b51ae9b241146194da47a4e9a371a60892c1523e6f57ed401 3a0b0e0b1e875dea4654e6eda203e69a8bb58952d0d92422ee5a7071ec80fe50 ea3392340701b28139cbda826472637e21015f9d49edcc153ce699c44f31c978 be39ed3555657ac0725842b7eb31503a7f2440b2ecd15ae1116b022da62f34fe 00c11c15206e7b8323dc11bcddb4cd133c50cb6b62ab35ff6172333098f164cc 8f065c4cb98bd365bceae695eff6431d34107d33b807bd776d89d1bee7053086 1116d20b27006af26779b9266d2fdffbbf0d1f34a6915a1b9682b5eac6b6bfc4 210d90149cf25f43ea8434a1d90d24a281b31e74fa0ff5596961e8275aa7ec1e 702ba84bffdb1b95a956a138cfe481f88f4cf9742fed719372e211494b469113 d3f8da8ed980f277eae61a82be238254190ad4c67b66b196472652f6427dda14 83c8071d0b141425a33b157bd0a3085f027b36cbbdeae3de87e72d57acf798eb 0fc76e373fa4f099a29c74cf6758625a24d80f29d4e95edb7b61f896145f5125 b981cd3b0e9f10cfe5d0926f8bb9182ad518b9af1ec5b1301f4f0d09fb77ab22 41bb98d991e9fe4c1204788d28a07921efa39d3fdf8852d2cfd6557f4c4ecf4d 7f92a2f8f71cb1ab6ac11191e018a79692c950708002b0b08fff9c00cb65d329 632d425a034d40f3083fdbbfbb39d93fa2e2f9da9cec64035535b0f635b739da 2309d8ee8142f052235d3f2c4f69d66ded926eb929e8e759f1302fdb8eb8c455 651223b02b2b9f36fef4db66689f4f521c546d45e3adb20f777dd2abb8a1c3fd 34cd615e43aeb26df567ff3060022ac65cf92f3dc6169f5ed229da51f89b38c5 b0547c2a3a7ba56cc66e40304a32e37bea58ca4f03f0e23402b0e73f2ba798ed cc037b133fcc63d917f77394242ef22f3ca406c8bccd8fdf43dc15ccc35c7ba7 1a2910c8e7f64236cc91fed15640cafa08f9c5615ea6de9a06c6f7b3d621015d 1889a468e2c3979cee9a05b619d83d2a51c49a74c9b15fea66625b5a5812a15a 3dde0388bc4e4174630898281e63cf2cc8556e37af8bc4285189160a5339b9a6 ae52eeb77407ce056d3c8d2ff37b09892290919aff7198914dd37ff8d7d6ddd8 2f434cad1fc24b2e1d29e94f8ac5d37026286e553f71a5ee19e85e38b14d2fa2 2c6a1682c3777dea9f71e0c0048caadf3c2041aec67c2d0e857af38b11ca61d3 81c8d290ad01caff7b9125e23325ec5b35a4efae2716874756e0d38f71955fd7 b9ceabdd28c5421ca39dce9beaaaae0f04196a81be41335303cfadf2303ec4b6 8b31b4bd73d0329d6a077478cc10653a92b0b6b14cddbe5dfe34ea7332053c95 f9b8283aafe4f848d4439212793ddcc8401e76fdf24d143687e6d3bb1a06bb24 974510d0017ff748a0b4fa7834778f0bcea73a198f25f1b65de960f25338b60f 5a4f972415f7f1dbc44c7cbf094dd0571fd1cd948283c46e514f298265f90598 958a8ee77108cd86a9c6ea67c095f0e0732a92f19951c96dc3c2f82d9ef8a8e8 79572107786a2967f64b6452176666a540fefa8dbeaa79f6f0fe60d42a1f178d b1121e3c3b51a5cb69435539644aac5961c187ba6d72d6cee8c821a803ff833b 253746c88e3d6a1baee33592fd65780d210abb2fefd387106040408df8beaab6 b4a0b96ecacdea63210832e99bb07dfbb6fc6e06c5d253aecd1b329e96578e3c ce9b29bfda477a69fc7b3f1321ee315ecb49bf4ee60d30850c94c4fe1d454655 aa8b8e09d35dcee5676f5f890134a6a9d5d0c86b9e52f4b3f32418bcbdb4b679 a4052c4760c8530ea5d2174968c2d3d1a9fe92404f212542523774afe741e704 5dd36485dbd96f0ec78e6522ab1b35c3fc37e50cd0b03da3f5cfb02a79b41ff3 c7b0db46fe8ca5d23c8b574c723adf17999ba3ddd8d70be97c10eea2f1be8a58 ac88fb20873c55ea2c74a8ac1c8b389a4ab4b809e597b87441693dd5903d0233 18c3c68c89816c4ca96006029900726e61008a5a6553c7048e8975536a6e8a28 325eb06c512125cbb5bd25b4acb661ad167f95345d5b8d402eba47a0edad2c76 e0488dfb305d47e3d79167bff32517e27820a2fa07ed9dad84712aa918549992 25034ae9d61250dc9dcc7acf8b96b34aeeb3a44b86c7ef1e20132f8300d6b306 f4ec2f1a36f94c2e32bf25807669ea6f1dc226829f4b6b4a29f23f72bd80a80b 0c4b5aa1f8c8c5e6418eb77c1d279512e599080113ad0c760b8431fc7470faa7 87a9dac577c7c2570d14e82eb954372168fdbfce50c8db9ab79c06f2cf2b5ec3 142c8c41293102f4d1b203af835eae9950fd895bf29d5157f70a97e418adb4fe 27fac33bb0cf683d291d28918ff02a98183d947010121097b206677b3a8a8d50 00aa237afb46857e8120feed93ea064cecbe7d115e75bb508462a33e5640dd28 15205be10db428efa41583388678c83fde533a43c2c0a05cacf846e3a36034a6 77e0accfbb49c5e67374e74f8dea11814f854356ee22cc3751ba04f15749c34b 3631477063cd20bc2876ca31b684ee562dee323fd1e4b3ec731cbec4c89f298a 65ef2c1d9ab089a8b490d5cfecfb615643aceb6cea5b0e13182f6423d4593ba2 c8d9b3e7d00a16f39e49a3e9a917c38db5c2d65a1c119fff444e2803ba95fdfc 49951df49aff374a92db2f1fe35f26694e87813d75135d09b38799c4be231387 2eac3ea6931283d2afb69ccaad99f4a1d52605764cc33d645f6beb8197378390 543f6a4ab476e24621d2db972b83bcb359705cecb80c4611a1361ec48164fc76 bb44b7843574d74eb3384cf2288f12aa8a988ff73f4ae28bd932c32ba2e69ef4 286278996b1ce0ea779c88edaf7d115777dd80aea6041d6c244cacae1a32250f d168a7b9e0dcbf0abadba8471756637d7a89fe59ad48f111155678612f0314fe 47679b97c8853cbba95527a9a0113aac74d5a56c480a9c7fc84e509dc56d2ad9 536d2932a747eabfea0b81dc6e6505fc5d03eb9aca81e841ab331c2a9e1eb001 bfa11a09cc3b172c502ed28bbec580e3c04ae3bb3e9571b232b944d59fc63400 6a5d28a6984b64cc425b2384537a4ca2be91f8e6bdf39773b66ad09d5295a77d aa4eb49f1d86e3154313f129b31369eb0d6ffe606b7b6d84d8f7a3369e7406d8 a6a400a7cb87ca51c5fff0e6fb9c88714287d03de2eecd8e6dcf1c512ccafce1 156b2d1171572c352afc0710eaea98cc76cfb6c482d73b8791a762c044f4dfc1 c9fc54fcfeef85d65c12aa5072da5a668176bbec123af2cd0d049129759ef41a 1093a7b4fb301735e3999ef2943239a618afab9de800e922e23138f1a04170e3 d4ad80016755d17a6f4e2ea25e7e1d563bba21fb4947c3c5f7144c9fabf9472c fa33eab494d14833828fe5d3a4b34222994a2cb826ee189e6792b35b27a56a00 696968da00616f493118f26c1221f06321f1c8aafa1b7868a7c6a510088452d6 11161f1e9695fdbf44b0b6e0bda510686d60983a3435c6fc474338de14876773 c994f875cc2fd9de615a876c0c4f521ab5f2e515a8af025fa0f339b674181349 01cf52358e2268eb32f90dec855f0e553e956132488bbe557abcf2846906f007 d5877893ca9a8eb7aa042f1521130ea75db0041a9dda2e1b30a2dacdca84329b baa171b89db4a5c52dfbc972c9e8c93baddb93fcb07b48855bf30a9b932b4b0a cd1d4665a7a8fa4caec053ea97cc0de8eb482b7158a555b8b141af9774feedc8 3c7458f0b96fcc6998032347388c3d2a0aff120a924fedb5a760506dcd68f4d8 e8473cc0f70c7d6d48a804d6fd25f08040bec444ede503f3717419bb08142cb7 7fd393d168ffb6450421bcce3f66b45c62e288dcf3b8b6996a17b9221823ab7b 265b3691f057159ceb3ad76ac54c63ddf5d83b4bf48274249e9b99e8c74e04c5 508815005d12e2e9129c7910ed447c3209b6f665762064971cbe41bdb5bf9cc7 ed6a9d7387fd88bdcb65c3eee7499078c5a8ce5c790ae5e65b15c5524d1959de 54fc3c1944ed2e0f37bdb67e1af79213236a44f1bba8cbe95d4bd5f4fafc6517 ec65a332ab4c08822b0ccbe3ea0f99965573c65ea7b71e6186384d2313209328 f13ded297df3a73d53dc1e71d3df96106bd4e647bb733ee781af484598a4c469 636082ebec5547ec25273598134894a0a03f11f090ff28813c8a95f8000066d8 f4725917e6a8bbe05e5bd53cf17a170692b4676eaf80f4841bb076c0d121e8a0 7a108e934eda997409165acf03a92b15978b84a4807906f75f8f8c957d78f5b9 71b5ceb5e5a601bd1a5d74e2477bf6561eab3e9b6bcd00d21ac4a2734ac4be69 55bb9c20bb26118af610ee0759f82dfee9898191ebc479703d16c081a00f079f 7ca98096d25f97e52f2d7ca9a4147cfbbfeb4712ac926e90638d364460d70c48 ecd6bd2ac1153c1648d2ac958479b8c3315074693f8656833926f753e40742ef 99bb5f5a5d26ddf1e0b113b1a5940a246e0f70e9caaa3d37aaa0df6bb7a9c5e6 70570b61c39520a6182c3124a54864b140f6c2f9531f0631483afc2844f0d0c6 9f7ebfeec150ed8254ae975b039750b73425f57daaa9bec579fb22e9a90ac889 e366e47968931a0c2bfbfdb5144acea46e6c9dc0364b63492f58032ebe61ee1a ba78d289ed6c523eee4a6d5d66bc3d87914e54360b567a49807fddb67ba1bffe b1b95fb25d1564e5d4a7a490a55921974242a365d1e14cb71b83d0eb0ff9dd8b 9b1a295b3c709d1cd3f422e767eb8e8b422e25ba6e5310e6a239874318a03588 99604ad99cff215d11f8c7e651153716f2c7e8436bc51e91c8247a8a00636222 c8fc96b400bf4f488c7c377b316e9166e103249e1441011e92a4aff5cb648f4d 6f59c35d555d5d57c848d7e3c9ef0afac1263f657ce611730f4903d4fbce0e01 f717f9f67ba83864f1f168e5d9dad215bd12a2df0e96b5ad608be8543119981a e2501c1166c5db0f4aa1a34febc38cfb5bc25f7ea414ccc8be07988bbb1cd3b0 216947cc79ad4ba079e158b0f11f471beec31958ae4d8fe8207480a95772df99 2d21ee8a374d87e65248f7eab3e7ece103e8480adbabb2841941c30dd8b0a8cc c7ff61c2418ec1566c15ed9fff493a18b782d598371342adf5f9d0f12a321a96 149d5e866558460a6fd658d712c8eeaf64544d9b580944dab5363e2a17c2a6a7 f23378f01242daffc9cc48e4226c7cbddfe5852c7293ec551fb04b4b4fbe5248 565c0f24a6f890cc4837779558f2982211f73c5eafb9a236ac26d8072cad4945 e29c9a7d344f0cb3cb77044b8eff84997cce1f2923acba4079ab297128b8ef5e 86c6527c03f6ae24a0139a9e4369099d76354f9d911c0c97698c0311eb69d06d d9d42bf248c514f905a360c602887d13ee6520de5e854ddc00ee96d9d1bacc40 1625ff222c6c89a460a60b0a66c51bb7cc930b80f2661f4cecd63c74d334789c c815164844d57a6c9ba98bc1547d311c04a50c1acae3a8859ddcd8ffd25c4506 502c2218c3bdd9a7a9e296abc4462d0a1925a7a4214826c6e4c1ce9dde1d1e72 78e3bc8a8ccc95a7186f65bd17196facd5baa43cfdf5b32beccbb15ba90fe125 9a4ba2444cdc21606179ff1afa92315789a4979ec0289e1780ee053e4932e137 e6683339e177566662dd0de4fb12b219b263cd04edcceafb816f943fb7343168 e505be7782854df26b564c168a598a78f9b262b0f5af68456cad25d025ea211d e559e1a61fd0b2c324f0bc5d8f1c2cebce51714731fa15e5d54ae7939c8afe5f 57642158392f3ea4dea7e268e0efbe997ef881503deb1e48347cce11d85a974b 6db1618579dae025aa671b02aa26309988c3ba109cb173842e105c8585e62f4e 13686906c04d9a18f17e801a99ac55aa05a856831287d28c7c7a7131452cf1d2 47969222a1ee9e3bd894e01272f1ff42e614eceff568f8107c31c8758383f8a0 b7ce9fe11b1311eb5355c91b71334fa3fc4597d1156826c405ac1f893982b9a6 37f52b6484cbf3df80566fc91d49643344ffc992566cadaea64fee5f5ebb4c64 f48df623865c36d77582ada328500bf8688d2a998eebfd37c73134cee5899e84 3ecf559c9f9de20d062441a37e310e085ecb793e8909535014469daac440a705 613dff7bf92eeb59ec3f7a42438f8879d3398bb15c2999fa61092b6a4feac317 e275733860f3567dd83a7f95cbfb4ce97d4655b1502ec7ecd02467271606d5b2 37333cae41b2ddebb4e433a2c1a3c92d89dbdedcd6f068a55c5e04d61d903ab6 8f2af3d367ac475391ed79c0e0a5aca5285c6a521862b0977ac1bbddba526b86 b34bdcf72fc4a52beb9db2918e8bc53bc765e7fc59ab4163186e4e259152d770 bb58d596edee0b41bd3ad87ad670aaad9f8a7fe2810c26d987e1439cee0eb86f d2c2837b1ef3e0a6f00debed8c8374db1b6f66eb87b8f4e47c5c96d19a1bee5d a28647203e63ca5d912db9c4295c0d9cb3a3afaf2d7062ca71954d220de1de14 e4071a2817dc0e35dd467df047aee3638cf543881b9d4d7a416d0baae64f1018 c26667ccbcd203b03a6b7f5b02435a4a647e91772413da8c0ad0f73ebf264763 69f679a1f1fe339b94d5ab8baa7c45d51d4322017ac5ae88157ef555beece413 133f45556960d55004a7cf7b5d2b66b631e08938f57bb66745e1d01247148531 9246ebc59bd964ab631e657f001119cd1406d78c23744f00797b2aac94172f2a 0369b7569b16cfebc9d45882c4d509580a27b88e287f3d71b0bcfc26892d5182 d22c57ad50c4ffda241cb5d8910262647d5a7dd104b4760c9a34fdfac163e33f f4066e80c496eb00037a7478b0e9164e1249da34574d1b6cd7b1eb6d21614b87 ca0f3c9e32ff0d1034b699133b6d6db439058b29b34829cfec59ddf220c6dc55 11ca4c256a2e371f0d3744dfd90558e96677c21560834e5cec4088e7f0b5a9bf 52323135d23b794f94ab598eee28f151005f25daeadd50ad25b4dea9d4409a8a fe87ed8b8601bb9dbd7d0af4cb0cf5f6eaf9a1333ec272eade3dc254be1c9c83 19ddb88a5c7fe017078d7ce2f0c548e65595ec5bc8cc8f47a767801752fc4bfd 58ab9a68610277d870498a79b1fb4e809323ebaa5e3115347fbcb5ecc8add124 9f10d7bdf1e65bd3e50588e8e82e1a750920cbfd74b2b2d5e6db3bd78924b95a 370220bdfbbcadb94cb263e333c98cd7d0c4e7ded82f928d7cca67a8407a33d9 ed5c8fbc59e9af2fff8a1132ed3756371c9f5239de705641b9eb0cb78c7a9fab fc3b20127a118c6fc92e4da19d5c5f1c6b6172a83d4287a9a394d9e4a317ff07 cfd2c6ca446e2ecbcd5f4eb9221349b8956bfb889d9ee85b9aeddc69f5e10f4e a7dc29cd4d87ac58575a3dd1c4548df4fdb29442d7f17f9d11102996eca04f30 50292d2673f1d7c5945f98ee0fcf753476f1f78441876fb43ce39721b57ed365 2a1986fc57f008f4f436fbbcd45d8740867faed323febf9ad9fa4d22ea6feca8 0e89a70a1ee46beec3db866c1b412a74d8739fd934d16d4d7c7675539c333682 92a339a30ab5938506d0378b08ba0c6f4b58f5b619ac4f2a5c132ced9ba292be 081d321e1bd4571ab0520a8bdcd391b0174067fe25c788a13bdc94823e17d709 a8a946c4640cf5af72f1f090f33a83ca11ef67acf7b7b30eae6322f236af7f32 9ea9b58f79c8d613784105f597bca1e6f590ac61df5b1793c6e16f364984756b 628eaf4ed67049934f130d1260d637e3a8f3d8a8d466d0be26b19b812ab3c0ae a8d64cc1c78d97f1104fa090ed9855f7817b1ccd2c87a258b1b0e6064ffe4263 83b862e3ebb74228e82881da7e86c10ba8289b98e1723e16fe66e3db48bb14c4 298a15b785485e3263ac431b34a35f60f70514d405f92138b28e1e54de59644e d48b75dff09e10dc0815dba92b00665fad7ee12d141eafcbe802b723a2dbe3bf 78b668d35c5f17ac49fe86bdcd50d12ea200622d4927e789a4bd6fa0ea1f3b18 e8e9b5fdb2bb78cc784504e4f09f1d6a039def9fbb7fb8291c70d72622e6e7bd 32c07946dc3b2787a4c05c2ea2965cf81a1238f1780edc1e5a516172c7a3f2ce 77d3e435b812edddbd0cc9f645816f5967e6985cba8465b78e369ff9f735cffc 03302575f4a69da95b52d1ddaf4d7f71e5ea0f6a23baed201d5d1cef3d448887 dfada7d87d8ec3509ea20873a20245205e31845f047e8aa09c354c8de4ec4055 12b9ef161e34cf74660129aa34bde1d99993670b9b46fe71120415b32ea204ca ba177560a955e9f84bdf5c918b9cecd3f254dfaab7cedb10d649839523449a2b 9fb7606b2741e637f9187f62a31c12c27d277de39f6ec4ab8d2b5e92ee6870a2 353c797ce977fdeb73766315e7fbe5d796af65ee519b9c0167b606de3f26469f 11df9328a82a06d15c77cc55239d7f7b92bb95a188319746385b8143409b207a c025ee8af4049c771a6cabe17287f1daebee7f5948c2ec1e7f432e8a994773d5 f8fbbd26f5488d7376dd6b9f972e79f793358325f94784b0a3f740f5b7c3719f 803735fc0f0cf5d26a44988fcec44fa3a994ac0c146e0ebd2dba304078a7fe76 e8ace7436146eb1377f7335d0efe83697fff0c70417d0b98d941a81a3d472afc 6f98f71fc421232663707003cc669862401266b295833a51e3b1ffa8ccdb66f7 d30a8f5050b3170fea91479196fea09c459419efdd4f3147bf8a63b6a01b4e5b a4c78db4c99ad26e1b1f941c3219684d3ccd208414e7908ac5de41b0bf97aa68 3d2bfe2a9f8babc2262e25cfe848c9d09cb475a47c3e32fcd9e8d8c48244f2fd fdd1d11a06e0cd5f08acaa51ba7ec007c9b8613dabfbb815c595bae28d97b5a4 7615c34b7f67be5014c567f057f1bcbc6a998333408951b8e0a5dc4af9edeeb1 040f4ad1645d4a07464a086481e5d6d88314cba1ac115dbecd67ee8c0a2a83ff 3e0debd94cd27de6dd6b6e96dfeccd5f9a3d16557778a1dab4d4e04c0233b4ab 109bc5fb463f0fa5a3c6cf96615fd6cb711b3b2af225e5902aa676f76c4ceb87 fb0ac858d5a588275ce16057cc814384d84760b785279496f853fdfa3d9af77e f9fa6a87ab3b85cd4a1a62d812928427a1a430fce8ea0420bd8b235adad283bf 2293407d1946581ece2322e1082bfaedaf06b17ed79a60a05d9b7dc0602492f6 8bd4fae78091e7bd98e22ab434f25d24c520ffec9b51fb7aa6d3f69d8c77ce42 032e0ae95dffe04f9635284183809aca58a2a1fcf6902f78d541ada21ee1d7f0 1b9b263ee4c2f2f7e43aaa4f12e8b44f6f23691759ae7e027d7f625a82df0dc8 3458bcf5831426c7de134d1ced25190d3a849bfb62dadc3f2878a20e6040bcd4 da134085cf50d8602554e5581ab8167054bab475678d2816056a49dc7efdda7d 343c6b6d158c74790caa6d5893c8b7ab1dcb5ac6520a4ea812293d4682ea6894 6e78a95b6ac3b8f7b9b9174935d7fbb226d44390705151185da8096454be4c5b b7c0c2d8bfc3c322fef5888fc80780a453546bcf11d5f09b8b6454e6463bb5ab f1bd89932900c4b6384f2dd84b8e24bb5fa659f8b9d059c7f7d6f7acb4d61dc7 24e98bc00effba34493b02e53d6ffdf30c44f4ba2db022318e28b33013fe9109 0068272464178cf1e32ebad452232fb01f5eaf6dc4538ab1f39b0afcfe04b129 d5e57ce3e9c0a9e75a611c254b48ff3b6dbe39d70182bf90181c76cd2cf0ac4a 2d09c953cbb588d696a1e9a0597c3f16fdd836d44afa945c3e9524bc03abc25e 41edf1f7a2b04080a980c7315ef1e3a259177e1b0e7f8651538737f782991163 a343243ffdcb24aac18047d024b612e8a2c29b3486f3eed68f0fc8a63a71503f 26f543536694173c32ea13058526618391ad01af5b30bcf328a6471bf3e3829f 86f03d4aa084a6cb7fc864a09545f2ec1dfad1c88f79a45d0d38c4eb5efd1934 094db88ccd0cd35395ba6efdc992f1c3e58c1fb669367ce2d7cb76e17f1dfb9d 6b3626cda920a6c638f949adf09a6588905ab9e7c9ae54c8d1f85a0239fa3128 b57f032415472425a9db5230272a51602fb54c74083feeaad8b4523332045b2f db4e07759af4dac6ce4b2a9a86b7df5c296086deabf7cdd28826a413cca5f231 7b39c023cc739a8f39e02c9de79187763ffd8122d76d0039019e61be03569841 f5f7f0933ef22f9ca0c9d2abefd6d9a1d15e06bf760a9df75cc0110e07acb7ea 894d4afca091d99faa62408676f42ef1a62dcda8a75eb8ddaf2165825387a311 5c4e4e5ec801ba9de0e9e5ddff644214f72c37e0808034bbbeac407697385575 810c33781ac34bc4bff836ca3ea92abd1478baf6361d7801dba2b301c9dd76a3 00b89826f53de7124c3c470f9efd4de230e4b9145bade2fc9b26f4e60dbdb891 22463b3320adb9a08c5f8076b98928d35a1c6fcdc80273e02bb3e863803870b0 98edef4519674295d9a0effbdf25c4c085a5c4ccd38b64ba865b2670aef29462 e39e7b56b8226ca14cbdab4b9d6d4734a4420c83e729aaf6607fba3bd52eec9b 7034081bc401995177c798861c0de8c877e3bfd6d6003c4ca5148afae6ee19ce 7f4b69c6cf11b5715ab7a70df399dc473c1d8cffa3d45710cfd7cbb07c6bd3cb fdcb8837ab38260dc2f2b6369991cc8003e0874251c0a0d2fd5ec15e6cc45376 896d303699e09425721a7adf7e44296f3762f10bf609e460fd3382702cf6f048 4ebe2ab24a3195c3299061ae9dec630a0f3a0890f64831c436a3ffca7e1707ef bba8ca26130cbf2379ea83efc381bc567cc501c1c90f59829cb4dbf12f664de5 848cc77159d35537bc0da32dcb0e1207e9cd817d742a974a536d112b25b10b52 c05aabfe5a743ded6805b264d658a934798e755ac570d04d62fa2c02b4582816 0e7636dc7673a6c338625fa4891a32070dad41042b5e8498fc2b3f0d8bc199a5 ebf1a3b81becd9ef32f235fd14b2a3a1edadf2ac8107b88ee0bdb7a5d1448d72 dca1fa30c7730ecdb34c8283eda37654ea7f3bb39fb27d599a06db1f6f1cfc86 b742575bf4eb41992c23240db83a84b51d6cfaa88961b2fc30abf5d652748b50 33816fbe24b01776eef0d493adc43247135cd0e292679651ea25c8bfa0a820d8 376ce3f0096f8ff7647ed4ebd3eb59e48d7a850cf749916dc6eca6cdeb83f35b b910aa334594cde937bc1a65addea651091934d6b3c26714937ade20e03dec38 50777449af58ecf218d443e0f960cdc5f07ba681f92033400e0bca87f095f281 7249645e3701617645e5a0b151a7354762f5e3075120a3ef316574835a6ef060 9f592d45007244bbab85114c9eb68fc7568d6e7e0c5a43e6be0e1da4e3699f2a 635ce9cc444cb575a506e13a0449be94031b7cf0bb0424d8f272f8ab82d1159b 9fa7276d37c21ef7ddeb6b9efe056c5984f617255db47eefc47e5ef95c9a007e 388ebf71f219a9f05c76d6131c7e9853e68339ebc203e102e4eaf80aacc12325 ea270dfb91ceb2e4841e1b0d43dd731664dff749fc759050ee37e139c5a71507 d44f21102bf673d84ea7c6e80b4ee64b5119de878e4bca724f4a5bdabf46e3fb 59e843bc6191bce76554f9ba649bafb09e80438c10d8baff5774953521939639 2301e8ca39089c20b38112f317e06f2cbceddce55be7548a15b24d38446d97c3 02e9f77e8906a423e1db98261e0764eb4e81b41de1a571bf9713e1ec24425a76 7f5ee5747c99acad05a59d7ed0a219531e41d4a0b39394d773dd679c3c425513 19ddf1665faa639f18eae279195db93465f7a65c7bec59833a67b6421b4ea636 1dbd9b0c73dc97fdd126449ba68811b3207c3498d02d41a0d0e333ca43c12a51 856f380b56543857cd882f8b6ea521941370c31f40d7658e8478a7f0df8d1899 ef64132b07026f0caf49f58fb56818d31836b6be39bd009d50ee0dc9fbbd63e5 7c3b8b79679366d7f1b7ade8aabc88753bd8ba66f3c35fa661359de2181e495c 348092ff04aea5a1951b8a52baecd011c7a6fb968f371de0ee1f8322d81f99cd 940e2f697c52cb99afd67d7b65d25c72d6ca952102b0a3cc18a09d92395e6620 a55a1e0ab5659dd62514398c0483259a64292ef166e1ae776065f16b44386e37 96160bf569207e18babfeb7bf1a411caf1fbda789f6dc587a14f1f062ce3724a 1c2a6c25cb74e545285dbf7a15dc06ca3356ea7b99c765c14987da9e9e6a3c53 e433c4b3b50ef6c59fb6235f12b0f527bb5d5e524b12e39c4f528c2f4f29be20 3b5d7a41afc2e62ba00af9569480870af89120e8cd3bd84b957584743a96872c 8cc0345bb6dad12d20ae4f4353f761302a6ad90f0aaebf6d80f6069a964879f5 e53898a04e94e084bbd3e921d7a3f680067d4b7746d0495cd04c447f4b7404a7 cc019e15cb4e7bb77b01fd0f96ebfc6d9cb97b4297ada4f26848189f7ba078be aee6699aad6436eb378e5f10f5ee47bc311438421703d8ae155934e8d6d8404d f0f755d8d61688c0980163da2d72cb48c6a75481b3b82a956fd768f0332c36a7 fcb78a847d4174d27eaca5d8a2d7a2fe15953e45be0f9627cc426ba772ae3bde 5f2927eed8b92f4fe60205518b48b6de2c16a896f1b9e67f0568f85f6d9f8d27 3ddcb5b6f1fb8ff9d34a2e272effb64517ed187c43c17c617f38aabdc6c9af06 53a5a8c89fd8802f434544c668ad3daafa6fbade734cbfb3f4d5d5016b943f6b 534b37df109273a9b539e1f4acc1f70c7d6d43e2d095f5f738c4d071e62c2eb5 834619683574ef4a000d5198435b038ca8e5b5c978e4ff64cab1bcd9107391a5 9fa5852aca8cf4931882c3b6f788b137bf4a0ced9eba652bb2cba5ff11df77e4 03ae583f6108947867439e4f1f04db329178dd5dbd6e2b1626b9c2c139fdab8e 572b90f6d68ed609577e67c89bea3dded73f2c768730db6030ac5ef1c0406bc8 5790ad6d994e9ffd4257af59215bbec99e2ba38caed7a3faf501353574aa5c91 697da917f1d143fb55e6a7a1ae0cd15e6918c2548d864dde05108c2a7e516ce8 87e304cd6c1aaf1766e7a08e4d60af9fdea8adff197f905e75a06f51effcbe27 978ab3df5c4c0d57ed3e7466c4e2c6a2376b1f8dae8f10f17205be931d99bd2c f616d7f381427690191fe5f2171b5076f96022b4c2a30ca2029d87cfed6e5669 ca406ab71dc12051733ea23c18802be86c72a3868566e43f9d36041328fbfd27 0040db706b68396509bac3452e11e03edc130377c6b2a92789eed6b7ffc43c10 e1f9169bae0149da2a8a45aa7a9da748a6d8f362d51795d7671bbdbf02b35906 6bf393a53eb22c58c4b8c4dc50b57311b01b7707ef6b6c5c3ed8b3ebea5468de 0d6ce288f72b970847859c24bc60501633a06cf7a14ba65484c828b8f44de109 b94aa520848a2b45bea9261f052052ce5770b2afce7ceb8047375bc0b3d07c1f 48bfca04efedc00da70d086bef1650d3356f9461e7e67a210fde69e70f7de84d 316923d8c98c75bf77ef15f92238033f13ea752658a3bd9793378bde9ed1dcd6 5213c74300a34bcc9ca04b96bc70136ad8ddd1bbd9a43d17befbcba5ef1b61e6 be85e9632aa986ce0f70760b40129beff54e125e4e6855b5e235d1ccf39570a8 ca148c749ed6ec6bd197f2d97ad24e7c2ddf66bc476fd9d20cc46536e8c44509 05f6c6d767f56c0930884dd760b315fae71a8ee5f61c17546e56eeb14933cf28 806a51771c381b39c6fa0d8fad55536c8baab244fce0f5986c209efad1a903d5 6505481626c64dad3361f916d6231579172570395c1f87274a7120fa90693cfc 78eabfce1ea89002f0d4bca0d5f3f8abe3642ef628134d7cc4fb4469c2273dc9 7e93cfa82028cc91c0dd4b2058dd736e3b21a475b8f300191bd84ecd22d3db41 1b025afa23467f841f555b67242e933b2e5339a44ccdbc3fd472003bbeb92ad5 42226405da1070a9d1a103939c0512713a01df2b15ad471238848bf570ab6043 5aaaf64531de8a5a780799a56d82d8996572bc148be2c09e266f1d3e5263c246 3d04708fbb49e0b9c2849ce852fcf11bfcaadc98d55f645b2056ee71369c36e0 b4cb920da514ad79a1e8ad9cd840b4b5d41141e8ef6a9be362f619c6f6fce68f d250b2a30e22a9e7efa048b9fc3fc95928c77099ca6459d25bcdaa7b113a6e87 a710a119f30be9d0e4b2dda679dc0e05c9c40fdfd2b9b4037b14dce93ddc1346 6612d29a3f766420ca553d140aa399b32174cae14f69b0c92ccdf6651b775914 b09491e66db3242438f88714fa04e4c2a660b9fd882d6711889bf5ff4a74831b cc0e65a9c65c0e324d4693485d711b46cb4bca5d9f82690c8c53a2ac90220954 d7918ef1de7d82aae12ff48a05c3134dfa878e77926eed6c100399ddefadbb86 92e35839b20724f07448a526b36d6c7154bbdf8c1f557e2e7dc1d17d24cb5dff f5197112f55da24762ee349ce62e01bed09557c3410923b4a8425c8a076dd14c dbe5109333129c451f60eb5990c3b1e1389ed76e0879b1ee0cadb09a35651a5f 99ad90838dfb76f6f788842d991097147117e7772663b161abfb4f6ba5ccd3c6 dbb57a57e716a985c426f7e15f856caf122a2e59bf7bd284723aab5f699924d2 96c6bb0461e3f1ab03b532d9f83b9cda7aea318f5bc1a322c6418817388bb342 2e7f5ab26ab8e565bb596a48432ac31284ac379fcb30eaed4252f7c1f1f12674 073291d5f84757a1c9a91bcaeadc9de26c5c8b58d3a00621f23637ddb2237dc6 308eee06ff8844ab7810f1cce120cdcec6324380325525960d10af406fa61e4a d69c78b3842ec4ef5580fd64a6f3edc8e322ea9b9ac5e132f7839987b63233ce ffcb627ff75882f043ee632ac96bc8537c53d3aa84e12a3205e0e3ab20cf5441 9f06138e83f0fed90dce53747c0bf8bd3ebbcd2e99512d2b4fe02647e291bfa3 99e7eabddb1c12c9b2fd8c7f23cb07a275ccebc18cf1f2436e594a9b2ec5407b 2c9638b461368236d90432fc2ad5a744687d4c367e4748c2988704a328466d37 8a90a7d97144a46312fd04a0ba636cce4be41ce7865b0aab9ddc2bc0496bb8e1 2011b60b9852dd59bc03cf0055af1420597e6e20427794aded79739cd474d8aa 0d1acfded2924154513feb415c837b6348a7499222d6ec64b32bab665071a081 d97068c0ba6f499d59c5cef261c91e8c07ee95fa251b8db9d00168ab57dc0968 13d402300f9acf0d7717d2e8e602441015ef87e291430d10501216fddb7670db 421f83f16af8bc9b633a4519c765452e97f9ca0039a55dc6ecc0031cc96a9a1d 96d998753f02d1530fdf7d78c8c0047d3bf512b1243550007ce10f8f4fc55739 eed0c38d4b2f6be7fd9bb5420b6f829b4cd8e4847f815daf1f3086d46d967e58 932c9c39436a106f986509ffc31379c3e10e0d1130061b3738a7c220badfb0db 53e96583fe60958fd676042b4f3141f5c7e8912b60c43ef4b0d9935f4a293a59 b0e90e8bd34ec915e26e1beffd6085ee53dfabd097fb8d1f8519156a93a6fee3 b53a8c7c855c492fe206464b6c229383884c440699eb1cf19e2d99a20d82654f bcd6f5d1292e76138581052b6d8b6376f26679ac1c0f285b35610fff4e92576f c2650cc63a196eef75bd2f5b576d0780913b789fc75ee940684f22e352e5533c bf46d925529c97965ffb9ed00822fbfe96348d63babc723ba2ef815fb9343684 86c6299d4a0170415e0721c763e10d50eef1a51e59a98fc044098b9befe202e2 a5857e63ee2405ef170195017ba83455a164b642f41f6ccfb0f2115236eff806 b60400fb066f501c24e11801e2b39aeb40fb4e8661e543ed3d1dbed63e83c773 82bdea011729d5828e058a2c1aff079da3095274966f40440b7815ae278ec43d d7cd0ab3190d92f5bf55a75ef429de2d0f9118cb3763da366ff4ed0482837a00 26478fb54d6282af4c8ffbac9f1d965633fb69d678741b20356f640cfbfc7cf5 0f8f2c1806782941d628cce2a481d20dd8f4402ceddd4ecfb907242d240f3385 d33d708049ced55889ac8155a067be1ce6dd7f601f19ffd1f9fbdceb339c86e8 674b3e84071c2eacde319b10dfcc34e91ed51bf406c406d66c271a3482fff677 734ca0d361fe7f4e8055f138fcc7138300dd1ab15aba6fd0fde1ce01a0855dc9 5ca7ef08e76e258f100d5b8800019530c336ceb8a9bd53df3edf115145f29245 d83670c295f5784172010df58d43b86463a9dd8f273af78e9e5b4a91195c0851 5a45811d5219d7cb9fd783f637d585a80edb311920c5a6c1fa8033958e72f90a e477cf3e0cbd5228faed687c8049f939c6dff32cc52bc764e2b67e7e1d537a8d 99e1eb265d65e99ff4c5372d1de069c5651a80d9767a871f8559e29deab8f93d 5a1eb9380b793a25f19fc41c43f452b08c08a2fda1ce257a56c42251df1c5edc 3f7a5b0e776d31b7e4c7a538ce962b23e25343bac7c8ccdbe67b3c3b8e0e4eb2 1297431c0c51ae90e0fd1b76af9272ed8cfacc0a6ea605faaf14de9bc76875f0 5e41ffdc698814c4406db06d7ca72fc886ee3f35cd3f5f4948305cc21c55c0a1 23a43930f02443fdb4304c3e1df8d1664db68eb0ee17bfa2ba200771801caee8 554a9066918503fc3b8f82e512699d0ed62433f493d3c996a5872157ea26ae69 c66810f3c1c468b244e5d336aa762c49fd5cfdb0f916ffa643fbe4d452a9b8b1 bdaa35db25af583084f9dc6ce76e193ef2f71f357e46e6c10e57c29014351049 8f563ce3030f1bfdc08654a3a9ddd68ade63036c8ca18932b046b32bc0ce15df 01835ad7f52aa113d77a071e79f169b5813ea6efc9dd0fb3a34fc8d721dc5c20 09d62126ec7280eb18b6a514a0451bb2737772cd523e2f3e2f08b6fea8dd9de6 7237d03a33c1d34b7f15b0f3594f8944971978e783e2779a5406e6c4e7d9f78e ab0b75d3424d3061f6b3599e88faa8612a7ccad0181fffcd95c607bfafab959f 31ca05311d4fa1e05140446cf0b872d754aa8730d439c05f9176812a4da1f767 aa9ec45b91198490d1feca56b03bc96be1ff1ad62f5f7187c16801f7134aea79 a2483d6c03f23e41ebf652f9693031fa9df794328c628429cb0cdf17e6b837a0 8bc56d709df5da788c8a21399498c058b66e57ff01685776ac758101faafb6fb 230f180f43c1155586d3a4594908bd4fa9ccefc2273906d1610762663fe326bb 5f3a0edf400e38edc3b8f6c59c3e5ff4c883e6aa741216f0ef512d53b3418997 b47f5be2bae41c8bdd861e5a7d638bca391cab0490ca6bacb7df6286478b7c68 9e95c4be2fb7d372fc5c806a8b200fcf3b0c8299b8639630fa7904595b9c0b1e 26553a801f63b9f1ea8bd82b28c679f0fad20ecf4684bc9c6c9c93828563fde3 ad1b1701a24995235fe4daed2180cc96b5654196b6b393753661672ca290f623 95d9b349c53dc33cdc1d0e771fa2426c4afdb5b3a78a032326fa20a8e0cb39ae f1a694d3afc6e88822de283fbe4dd500774cd36db6103e23842b2cd172128ba2 0173c30b0f86b23f97d48afae8439fbde61fb747f13ba9b7b1ab9d317cdcbd10 fdbc0a67d6688a3c0fc33ff32f2ec37c04440372c43585efe9f995b43d440313 e3b2c50f8ff6e48d8dae99c525780a6e7d5533796b560a1b808cda05ab834e2c 7b5e7813ba421cb7501a4d0a6bd3d9baa223f464dffaf8f8fabb86643cb99900 639c0297abba00849e29ea7d8001a43a8e7391d850987f293f61aa6273b277ae 9f96d8c13171a3ee8da37c20d1f2000829c7f8cc3fd7d621a87ec93a6b546e0e c7bcf8a4506978a48dcf2dec34c8a346745afd5e0aad7b9f2a9669745feeaf8a 90c56376587d08bc4bfa80d723149c7a8582b35954bd03123310fae52a473943 67b53cea707ff4c5b4a52b6fa4df0bf11442232d662442747aa8b3117e8b0632 cabe3439fbe58bad46af39ed360b14204d06dbf2464494c4a39c8057c49b1773 7705a793907e4412f838d54243d61d9e4923e279cbe4b84e774a95682304bcf4 c9b83710c4788569f52c42966cc0bbf1c960bc6b51ffc15781d0305dac6166f9 fa4471ad7b434eb5e920c81ad5e4bb62b40066d595c782440dbf9c3ed085ae36 8445dacd37c40de2d5e2026dfc8a0dbe938c9af0a85f5d4d2559d8c24e47219c c2f045ba28262ed857ab0de68b238a9aa4cfdbbc425fe28a218a6aa57ca90dd5 d8a6b6b1285bc360642457f9b913be8373a4d32831c99ef65e80eb86ee27a13e 8d5db69709e213f03a9839be6c059f97d70a9ab48cfd6fe79c115dd531e70924 23d1e8f48e272b3ea4d076ae6e264653b7a261bbe3f95b9658e4500b906cdd71 3883045f9c7142813c9819a5470c4ec09aff1afa973c5c53d914201a584689c0 739b67782dd767a21ea8878c9f67ed7cb9285cb643bc5f077938716a76eff55d 0ed2f310910853dc20ce63f92805b7d7bdebd7e3c8bbe1f2c5d90766ccef1b3c ab2876e53e2c6e33cdd32e5222ff5dd19ced88ca5abcbaea25b489aecf37708c f165f3869e312f648b40d291ae7c946cfc627c26d03c4845d450285e140dac3b 66fd79a020fa13ef4f49471df2ae24f674a92c483965fde5d6832e5b85dfdca1 bf1a69f94599fab16f7dd3226af2deaa0444d4c42aec12feedb0c871ba4394d6 4e4fa884845fa1c2a6dca422165379add8c95d360e5684a7240ec18175cbc802 7b5fa2e8ccedf41294c8cd480b611be8a073b41a2898ede4d4d90601d1fd9bd2 8f204e7be940e3d2ec00c5c959e4c223ec7cceebabe13ab4b4083187db921016 e660a033c2bdae725d04926a7ec860300abaae940a7eeb607b6a831937307b39 39dbcdf43c692dfdb371719fc37dfdc5e915ee5ecf2e071879c9c3e636e0b0ad a3ed0651ff6a4894104d15b09fccb19c3217cf3d01ca9311629548396370c798 dd6dfe5a8ec779deb0c29f7563b207e679fe7e65f4bc847c852f4616211e0447 82326784f33e2a79c57f152259a79a25159bcdd46b3ee99c8b2d637440754b3c 9b4b0837314ec4e63253bfb7cdc6ed963399eb55cf488728d127f4b0531fc21a 72643ef7ab284ca85d5df23f0bc2cb87e8d33470df727994ebe9238618a62846 62145b3ec26f8d120dd6dc32778bb970c3e2c363305a42aa9444abdc07bffb6c 30bd028d36c909c2f258158973ca0766daa2f69d55e2fde979b799d44357dd14 2bf1c2ee8589b25869ba0b882d8369ec7ddea7e90b0ee5b7136a7e69147c46b4 839db2de8c1dd5480120e7e01fb02dd46c39132ded9c623cfbced05f785715f1 ad2a163e322daf248846ca6bcbf8e6e16060fbf18a40ee6046c7cf1c31c9569d 1515047b0da3007d191ed6114461dbfc1fae5630abf25f60444c2bc35c9f006b b9e6fafe50ac4e77a19fe5ee937d22659f32983a7500603e40eea772171309ca 1d70a940d6350ce7d054f6c77986316ea7973fe5607bb9565005c583926c89b0 be8b14cdab096543d9 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi 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 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M 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 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 /FontBBox{-301 -250 1164 946}readonly def /UniqueXX 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ece66fcc9c831843ecae 1fefc1a232e2724f7baee428ae03aadb95c3035345c15e9922fe49e1f2cfc980 237316572dbc57064edac9b0db8913a5e2d45e97e19a91435ccf8adfc835b585 48e74b291446d689c7f2f8c4325e8356e974ae30c3e2977477baeaa33d141fb7 80f59351e84bfc88c87b3a4a1d25e168b9d134554f6a581378c7d2d6eca8ac09 045cc3a5e0ed86f147133f094029e3483bebc81cfde69942cf645ea20d0a2b64 5140ccc2f9cb4a3d969f4b0a79cd3329b440dfcff5285494406f3dd08d804c86 df191e57232f6f1f83df6430f214e03fa5412b1e13d83c60c52a698bc9a38159 19a64162af7b83a9468fae9507c6cb55ea80df8c696dfbc3296c663902d77e89 a181502627281181ac8624f65be9de5fb32f056cc7dd7cbc7366a91717b5892a 34ef6b441871c6993da3a81afc9bf70ff3842a21a29c0e49da1ccba37dd6b920 2702db510b8b1109acf4c27ee8d676bdbd16abe189f34a9b61532eb777457698 cee0561838eaab2d3939c5093afbb215bac433d1918c739b62566906bb2187f3 378521fb52e70c18886e1aa15f53a1f8a169528cf2ddd54c203613ff88e619f2 6c9c50c2c3b8b8c868a88afab1dc8472e7b6f3f992a161723a3c0904865a7c42 3281601eee91f35222c4a71f343739f102eb43145a47e4fac3bd666e4c114a90 5950618ecb1bdc29f90f6ebed1d8325fe6b52e14a8870e9b4bb088e585c71b91 493c12845a101937e3473ac5e9ae24f7c358f06d590d7618e1a1d5834d87621f e3f5432b655eebd4ca61a7b00dde0aa740796ce88a43d1c6952f8d2ca63d30b6 dcbadbb0dadf13150a442714675dbe51ff838f2e47ebe8cd44f1eac6ae5b6ceb 0b07af03f5d3896ff935a198aa635e337e367e1e128d15e31afe3ffca8245e41 a3e17011340e49d52fa1e322b435b4f7f2f49ae01f6a319f68d3d786c24be810 1e1c975a8feb5dd7e4922e00e3a08cacae5a945e1e8f8a20f1fe9534153307aa 5ac7424759743e08af90b9bcca7447375222a66934e1e47f97f8985b0c694a87 1ee7eea9e0d7928e555a83f7456ba5f235a1b5ef8bad622dea313825a43446bf 0b300d86f38133965120d1dc43eb1118d1fbb2112412711a576c4cbbe90ca05a 8865da40d9bca6bf94a9dee1596b6ce5af218492f0f13797305d5ce027a15f62 e97b1a96f52bfdb35d7fb3291fc270beede6eefa93e4979c0133bf56ae6ba3f1 5780cd62e482645875d638a9a3452665b2f8c9428faf9fa1530a7f486ddeb242 d6b88dee354fe924ab2f20bf5c8b3c55d4b01ee47fc61d6de927cf6b293311c8 25941c35ba0064d3467c5dfb92a474a142800f1e823aa6803f498d0a84509eca 0a647b5bc98d591c2e10895888f13aaceee95af56243c8df4131bdb1d7676296 7822c2b6f25b1f0471f995dfb9177fe3731671a4796ab1ef1ca8251aa4defb2c 1b6e6549e399009c8a83711ae45c4822eaf0f96d3d7b0799fe7ab9544f9992fc c34802a21f0adc9b53ea6cc4eff7caaecbcd465a1d2e8d121e4ed99eaa0ac035 3c1df112cad17ec8e1ab31f303a461112b806da6c08dc2f450aa787b0dbb5cf9 99b512a80563dd34aa240c096e8feed2046550079df84c12c44ca7c3e0de96fe caff1ec43798da864e7688865d3dea1d690e0261fa46e64a81b8cf0636396826 10b445d5a2f9f3831bc311e8d3dc51a9f6913cb12c553cfb252f812a95acc3d6 e2ac2c23f61650c42ddec2341376a8fe3ab8c19af465dd8c55f513a4d6efb72d ffe4d82e847b6ac6496a5ae7a4837b4031e56fc6f10fda33814c90d912649155 e7613cf7d2076f3ef6afaa6406ed2ddc54a686487c2944d33b4d9717219ece80 b7a2d037dfd39c50a87a216d2c24b08be11212b92cc8670eb6567bf7df3b1115 629711b41ed372956dc93858f859e66f13e40ced75810d2971f4c202a7b01fd5 d5c8c58329e5512f04c6576b96f20f522b84484ea73140ed68506ba5467f3603 c665252f60cc83868aa08eb1e54c80d2ce7ae883e6c903f54e228ce7caf97bb9 2292e339bfa699e20747c612f6959fa8d1f9d6a0676cb393a75860e50d01df05 4f538eefba15b299df0af505849062425134faf4184abddcd881541575923a05 4bb979c1d9a55c5b80e71276ab2db72d3ae4687e4ce0cda29d0a1cd650f387cf e3b058d77ab2144e83e67806efae04ec52a66e1f82fdd877fe368bec40e80edb bda430e6d56192124fead0ef576d6065bb27bbfd43a8a6b758333483826f4152 b659c367a031b6a9fa0241e9af69fef02bd2f0ff4656a2e8bac52a17edf1af81 8f63ad2d02e5eb3b4d53d56e19c0d835009ec532c70491a6cc60c75fb20f0f02 6576fd8a8492c44e01be6cb7e33807fa9a580b742f31d65609ef6edd495fe344 74acd113bfc953b478ac4131ee7727bb88c223247c048c993e34e6728b91c3fc 3b16ec8fc9f55831263680d5d1e07de9a2379fda19581344b963ac8dde85913c 3bb72fbff91fa29d236bab99a52711c2f894ed85de9ebd13e5afa809716d9e8f 708605945ef9512d506dac7b459f6970c6d94516f4f5c8b36b515518aa59aec0 52cdbda4c075e305f0efd565fce972b39e6fae7aeed043d2e3b1fd9d50d0d09f b56424a2210db7875090679ee232e23389f80785e433146740dbb36eb326ee95 ae6f54a9909788092760810be77cce1b56405f53ea90099aefac8d499dbaae60 7d4b9900b5cb90e3d15cb4bd95923fd6fd798b7e26a9997a63652487356cc073 9ed371ee1a2184b419fb30bb0e4d96be4187aff22ad02770fac4e61faa9b2b9b 0979563caaf8caed01fc6919dd1c527d586c7b0a7a12b4158b85fa3e2d08796c 2c842b69600ba7f85529793f88548c4ebd44de2487bdf6aa2f967a40cb5bbcc3 014a07c872b7c034958b9cb41828cb16287f271c9122277ee7c2aae0d75c2c8d f9184c2338c22b883a948cfed5502b449c6c4db377e8be2c4ea333b9f486c224 76309ce30c5533606e43ebdea54dd1093161e6316cc24ea3fbd9fea9c44686ec 332a056ff8ee236087d0ad1d6840c665e004716b16f3cbfdbcbafbcf8726b3ea d565618516a7556cb432df91c081bbd875c966b19020f055f4ec9b68c3b41e67 12fb4445a778357af66a15384c9c0e9cfdf30a64850c881440dbf512b47c6d0f 7f26e7aabc8a2ab9bc1056fb551f803d654ad8d4004f877ee89e3dce34d244c9 a41a27d973288daefcf9b37ab9ade0bb4da5886fe907ff646eb1704591ae0dc8 6dcf6b5db701d5bd0b925abab407aa48302d91dda8bc2f293544ab88e5c8dfe2 b28a6621e61a05a55487db403211321654f12c2b3632d0915abb14c66eab2180 acb09014f88af38a236ddd0e4ffe463a3312a4c5d772b0f9f1e7a8cadb6138b1 bb62e18c378b71542a881d05ed5a3428a95ed7b1f28ba8e1d89d69d2e3ffbb71 1d58f8424ec18c8034c9361c997e0435b9788c910471654e40f733c6809facbd ef2afbcb84320dad5cb921c26af13bbb2eeb140c089c137525dffdde6edfbee1 46ed012ac6893789f821cb57028de222a56753dbc74eb1686b96c1b31d10a444 c8e72fb98a10f45bbe9200b2873ea4294952338251e3e3fff0c1a4ebe59e2b5b 6ead194ba0b029d93f859c639ca7cb34d3ee78075d8ceca8c2ad22dcb0095f9b f66c08fa04bb5cb3f320f5def4abbc78ef23ca02b61c77709b4df4fdda4b7f09 493c56b07e5e511cbd9afcabb0fcb726f5a1093a4718d25b772362ab8b62dc50 5a22e7b972c1902c1d4c2d71a2698db277358529d22c1ccc17caf94ff4fd69a9 27bccf49d89f181f0b21397f1dcb5ff951bd9a486822c01c43eb04591a1e70f9 f912e48834f2fa4ef375ee545b40b9560fb7619d28682ebd58bb98533680d4fc 8962534bdcd4457c1f0f1f4fa894ac944e739c918a478487914d43cea2609ec4 78053fbb063a3c6754a459b4e9b4ce53dc96a79d63a6082e21d589267880dfc4 cee197726f6c4726a536439568fa4588bdbaa239d8d3b5139475aecfb4861daf c1da22b47d972a4c2478e0f8d287b2c6374929ce1ed0937bf2e2e0f8862b9bbf 473e31a3b0f3d4a504a31826000023aa3c147df0b5645c8c0be5eef7ed598ac3 4fa783f4e90a8c6c50f250e81c884545e99f0bdaae394792432580708a8f681d f60c9b40a2076a0c3c87332e5b14481bdc6cb8d3d8098ecfd095e00b419462d9 f3f3a8d02d6f9029cce33a064957f627cfc1439e3b4719577f6681e9f3358e8d 27b986abef48178319488ec9d38746bef0044d89cc5600a8fac022ac3e923bff 1a12119b31fdf40c036bdecef428b5761acde231c3bdbf9dd0d464cedcbbbb1e 387c6cf80dbe6d4c1303c2bce492ff8ecf36b7934f1de450c5dde1eb1a13c547 31fdc3703cbc72d1ad8e755a18aa85831f57f82a2434b868e0a4340863bf55f3 a6b37cccf3eedc328e0285ae63f7662fea31e92b3a1bf310754603bad74245e6 7f64aac1c5bc5e84da30708d195a4534c66821134d6c9bf0d9c115d70d0a3845 f0360cbad82b4eb8ab527ee473a98cc2c4cbb690aaa61aa49a6df306b53820e2 ea455da2b6d47abbca89e689f823589ef60449e8247a7f74232d6ec1c5d1047b badcf7d1cdb42eb82a6d4d0a092486b8ec5848332b7b881114acc5b0f46f5fd1 83a3489de376cfdfb6ab3d596744a2bbcc12b3b279b4f248faf382739f3fb3f7 c05fd242b6bc1a0faeda5ad1ce1187a56235aef479a567badcfd498ee8a584c9 b928d4582db19b0a51bf3f0ddd789f0c004255ecd8297f6bc0a39b045e997198 a8e4768489565562d9d03d1e27934594245975635c6fa564e83b97e9001511c7 95b0bbcf5d0fcc78c83f4bc636a3804324154e98de739bf0b015e005686b2e42 7ae6973a1e36068e7163cd99911640fb16178dde0671dd17e5107db2956f9a7f c4e7bb57ef240ad4a4d31d2ae68fce4ec0c9b95b61ce578d5396249295a85d3a 8159ca6ee59960265cb96545d31f514fb22291d5327a099a8348146143466b9b 96c10b87a1e031ee012edcd8ce32e3b674a032798e256f29ab3122d86623c405 5928afd510e39d27cbc282b0bd23fddcc1d9b576144b05930718ec271fd98a4e 9a9015ea7fc78199d3e28913bba6e8f06c27b71597f682c70ac60cf0ca3701d2 ca8256f30f405490a61ba8ab5b26c339c853aeed3b871f61f47d430a8b7983ac f0f316fa6f4c7801c65dc62d81bfb7bf60a14f962cde3966d854943e70a917dd bdfaa5b4eecc5bce181761f2bc9eab89ea30603a3e4c4c6ee9c2f4814e8b8800 7864cea441087354a978968321a74a6884ac5952238f6080f03d36f5a0d51fc6 51409255e933931b08c2dbde629a488971acec7cbe637c592aba0ebf55e197b3 4130bd75efeddf1dda3552ba7504592ce71f4416502f48dfcdd241fa0c06e937 59aeee2212b9869d5c416b9edbc827795efe5f0bec2b5def3b8505f634b31e15 3d7d0851033d1e6af477aa925627eb5793d3c7f3ecc1f481fb8d670471e7c748 afbb415f9bb46ef333d3ff62b1e01f3585b5e07994200bcc590b641d25921807 d1baaa438725dd23927b88a4427f4b5111468fe1f2bec03ae8a7d6e89bcdf865 91a553be451b438b6efe78624f99c22a5287539ef183ed9dbda76b807ac5ec1a beed9e40267f15dc59866d5faab2dc60427b542e3ac3b982fdbcc0d1351d48b3 f33718c01dde5d5321f122c2d95706351cc86d07ad76981dbff03a2f8e4936ca d7964e515d39bbedf6bb970a86c7da5c5f01cb65e1fb923ce15f883194b1dbc9 7f4ffee774a8b8584196b0362ac9870d47de29b56e50679b65eb9bf4c6fefe90 0a65d856d95b828a93947c3260c2c4ee78817dc4044662bc55414b094de78a2f a19d551c181d2c330d2691b482120e2569a61ad92affbd5fb8ab932e842dd0ae a1bea8cbce2565123b9711ceda98b31971839f6b9fe42f5426ab7389d8cf04c2 9e8776ff6a56a7ab02d00bc8cb61db71a3f01096316ce884c8471783e5f20f20 4bbbcb78097f2e94bf61dfc676f32a48664a92c66519d264ca11fac00c14d259 c3d7f637c28a06a3fe3962c211d71b7cf902b4ebfb36e639bd7979f2d95c1952 5d9b21a06a1225bec0347d43158f112368ff1f3cd711938b81afc864ee9bb298 38deeeaf8679920ebbe00b9f80d934832a9a6d79dcffd4e421129c4a4f6c8986 fa9a2b8640ccbb36bc65989d7c0112269c6419d31cfa42a738ab2e7c78c497ea 26ca1fe8c0119e783650621db73c2d7837991b373df097600dc0d328b2a51b0d 3142cb86e2f1b00e5d7b8d7aa481b338dce8b28029e1381527255266da0085c4 62d9188822675a858a3178959f3ab20c8b9c1d9da790fa391c5abfd233924d5e 254466a7f0746fc02015fcf82cdf52e34f666ddfada5d41723d7080a4c0c16e9 2859917bbdd51be3c505b9683f6bb9945b1287459d577e40eb69bf74d580b887 944c3810f0e5d18e78c5dd3b2a5b13a79b8867d9da6c18d5aa4272dac51b799c 65e4282039c1cdc41a629371b1bc62481891a6bfeb5daa0526fdbe08131f9de8 0880e1512bd7d3a2cc9e53a6c802da62cd3f882dd7c4556a6c2455ca11652cd8 6896a5a6269977406887482beea099955de05e783a7abb2ea5016ebdcec98424 563a089e78e74f2239ea0ba3461ab5e622800865ff5a6090015942bdcea91953 64dd6bc93c1925c8a2f55e185863ac775681db53b4cc603a468e4ff01d6c8d8f 00aaa25307ebbd5081a22d7447ce2dcf99a65492ed98afc4e9a8d421e5da83a8 1b1bcc3b6f362a3d2ffd337edb594e4e80da700dd003a161bd3896fff47fdcb3 b72740acb29e74c3e759862379cc335037a59ec4357004344212072302c2d13c ed9dc07943a5d4db1443eab060b45391ac05c006f146a0bedeaaa731bd838065 4877c8a706f369c7fa9bb7bc9312087b922980c93281aa232c08c702f59452d9 ab309b4ba15084f2e25ed5f522f5b74ca5d97f462abe7bacabab36229421b196 c54033e28e957613a012228ac4ad1ea1508bdd280aacc1a377ba5b173f910060 c535b64a49ba6d4b2336a59600ea0d5c16efcbb3f78c4e7b5a32cb7e368ea132 675a415f36bf2c03e95f3cb5bd283b8a781d783aa3f56c7d00811543562d34f8 bfb594c5f8b330e319e775c5a6fa821dc232e3ae45657d585f1a05962339248f 34be58ecf8c96c311d84bcae8dc9bfc5637eab4da5c592acacc266fe1092a4a9 9c40e99e82b1291e1f2be90dc919b31115c0f968aa38434b57b70b6b0c523810 ca9612cbd9f9216589093c704ff9dce3f399c0516a41335eeedf585eb62215d1 6dc9f045c36450123d24995dea69e77f97c33e06517b28bca22e2eaab3d4b90f 9b50f723c96b6dda450bae42b6d05ff150a564f64d72099a34b34f21f7bfe176 56e088263216b8f88e7a0e67462a4f61da409cfa94bdcaf6424069b1b40bcbe2 a7103000e9e0a56fa7eec3ca8a5478dc5c6ff6df44d3de76f422687db9a0fb36 192437298148ca58eadd502079e2ff019bb979fb72abd2fc20157811afc57b78 9717e12b74107c9e23c8b8170f8c4e3f6eb42227b7e335d81eab256f319f67bd 33d6d640ca62f9671c8c899d4b3136d8888043a768dad23cddf3bf2ff7316709 e1729c3bc032d8e1b3e43deb31803eab9725874b2e930ab24241e946ab3bdac8 dc658f2e0bc83d1ba3a61b307ca23af15e0e47c522aecf4da0b8863a5da6d2ce bbfbb71a89ca02a2bff417a3284aab12506ec9e259162fe1c9dc4767e88d7e23 e13c6ac2442aa32a94df6cfeb83a24eb3bb4dae57a4194348b954971699d943d a975fdf0fd5d3c3e420c8b9af23fe94ff69194f1c7c7f90af8dd1edd1404ad40 564a1fa75a70a0cfa5b97118f8a329ea0366b29445be6ef6363c0b51da6f0312 2c1b618e86b5bf3c341ed925d898304f1b29785a02f63523e9d80b096bdd17ce 88e408ac201e87de5d2baa0049f9a0d024e0f2de6bbfd3b3fa82da3a745bfd1d bfbc2121c299ddef77f78ec20e1032004561232c5ac9aba4600f39c9fbb6f8a1 dc61d07062f14376be56ce9b38f48586957ffb0008f58a664d885b17ad47f4a3 1cca6fa7905140971ef2e728cfb455df81f009b3e0696ae191ff69b11778f77c b31d06e601fdb7f42040c92f8c45ebfb6c7d3743d7524c7a9f18db6a3e32b368 f71d5a96e47860711a1ad619769c4703f863285b7226abb9a0b9a7754211f537 8b29da7f25655474bea5bd042ee1f955e1c8c3a609d131481acad68764502e83 49d1b52bd5764b28cd24671b98f7c40de68569c7d10aeef7cabb0afd4f497838 cc0ee23aac20497ff62dc97fc52846aa3c6154707447d4a0ce7564b788bc5d5f e838a92059b18fdfcf2289f96f85da9d23f1e1f53d86c21c2e769b716fc5ff0c 6c9f5a03cdbd811a0c9edc66cba09655eba09ff3e9aa9ed6b8e3694f40d18f3a b335c648a557cb23e0465e97d7bc8af423e20fd0e5bb55ff68df8f0822d701f2 80b6a941bf4eb365fac71fd55a00bb7284b9a70a184111f1d43095b2600c4639 14c8e2153d64eef23328d42421c65ce378614dc250ce8a96d640e8fd80e9eed8 4d5665324a04c6f92cc2b19a319dfb21c8a65209e3565b4a8deef7de7f16d6bd 75a001a8e984d2194cb7aae81aeb4addc3fe0309f562e47563044ce5fa92ad36 d8494147881721f92ca055022100ed4f672c31390bc90695417764439baaf361 51bff48f6cd6d76c2fbd98faeb01b9465d86609eb485d3ddd23fd680cfb51f6e 3274d2cce3848a3d80b52238972a1c59de0ab45cb8cbf4d412767ae466892d2b 845094014df780e8f510e7ed61f0450d63c69f9b4fec591163942e22f66368d9 8f057a541084da57fee74614d192a58e83b63c8f35baf1a4325d0e3680143a76 2520cd0cae84da26acc9ed0acb8b757d04e89bc2901ce149368d18df04c3988b 55b2fc5fce125227d07d6516c011998af1b8727f221a21c55d31d3b78cf2a631 5f344a9f8cf8483ee06746deff00f718cea88986712f35a531182068ea8ed34c f03aed671b0f0992b4fe0031ffc5205461c0325a5cf54c662ef6c4f5dff8c01d a340494fa37f30f5d4334a0a2abf86ef42fd2ac760058c6f04f6fb274ef0df73 d7a74acf23e6c55e8ffd313c066b9d50f526a83e26f70720d0e896436fcb2a9a 6819423beb4011171c342bd8f546ae4add2a030e5eee24e1169de09339730eb9 7bc43d4df651e7525cbe4de2f496cec38ee110fbb7224d5fd18b9aed4e4d439e e192db55e75329d2e86b7efb0188e53a747292e7ee2bb730b4d1a3534bbe1763 2958fee9b350db0ee6be05c8ae66d0a44b682b0d41fd8f012af3635bf4788164 47347d0e70da9d12a1b41e7a27d0a373e87c2b91aaf1b90238188542e0c968a0 299ca5feaee7fd50e4b6e8266a40641b38379e9a5177e0c26ae68743d673900d b4a09fabb1db2d748b3539fbe58bad468d8f87a3274ebaebad079972ad8d4334 93d43f68603cd9e394622b2b662b6736e12a6af88308068e3bf34918ccc2d19f d887a47dde8dd19d132c4425dd0ae3994d2dc78a9046847edc4fe25debdae007 74693740ece1977e187972c51c48e3d8b003a01dfe55a957ec2581dcb7e69426 7d0f79af175aa0e4632611fc7a1e334e9e61967af1dd76e9ed7d1ed72732fe90 a8c981bd2fc30ccdc285bccc5b541d152f1b91c8a2370b85708fafe899316343 56e9153fc03bf3b41ae9d9742337534223c8d6ac413062c7d19f106d85e72433 7a1aeb82b934274cdeed9d4eb102b9af0ce0b1bb4f95be4c394723e4e7689ed1 ae8395131d90afd47af9f2908a432596ee0730f51316cf3f85c6cf6a16cda877 4989851aab8fd175d3f30a664514d53c24a5c329979787285acb886170af7ae2 9a6286a34bfc5e9b1d0fe9364830460b9c6a9f13de0538662fef46b13d04225d 361db0fc5a263427e02e0effc7ae6b40ee8a7aa53d8d05370f763af7ca0c9387 dcaf8d98e6beba61b6b1b03da06177d61b51eb1ffef1a8fe6906e91b6a9a7a3f f6c2ce0bfe8f8ee788a6bc99b6f7615857653f85e373181c7482f114003c42f1 ba76616b2b667bcbe6573584eb4269dd07026af1faf44f87e46e5412e44c9932 ea6102464d568dfd3719a15caf9f62e9eb8b00a008225689498f7a6b0d26d7ce d81da86392908665bd32eb6ea822cca3e471fa7c287503815a695ee3774510f8 5d412ea9fa4acabd313aedfde2138c28c4fda084de697c5622848622417e61cd c73633904f2a12eea1630afe69586c8ad37401493851f0b6e36d92445031cc48 4f5abf03c61e0b3e263081b577875c9ba41512d7465a5680183aa0817dafb879 f3dcd70786091f10071e759e2296d3e4546cd04016e2fd5bac61e2fde1a50016 baf1fcb574caa1acde9abc7f807bc413ccc77dd2c42775a1b24082aa2c667265 f2588137d70baed9a71215c1573103b5447d10d98c0654fcabfd41df1468a08a 0865ea294234579b5d404528728508388306e33503fe24c344985825c67cec76 ffb1019763901e84202e5e31cb27e09da0ae1d590c3a491b0ce09c079df796c2 22578093812f42e6b4976ec69895099ae40518493ff4b421c101a73b4abca453 01a283d6bd741210463fee37e293b82e481808b40e45a429deedb94bea7d9255 8627fa82ac4bf28aaf6e32e94dc6f659500ea861a1b4c39b4a6fa1f49ab33e2b 9ef3fc46790dd9629af92ec9385f741f47ad83c5a86be85cfae2f8dcecf0f7b6 40becfe6779fc48f3890df20133ba255c5532e7ea09499d6939f24a15ac0182f bfe49b1b89ca86fca6129ddf9adb7c0c5d64b121a5517cca7f54f6c55acdbde5 5d71d5cab725de5c0525d33235f7272bfeb1156307e504b8f349271a0c6a906f 27963621a4a491fcbb6de38dbfa47c8b8ba38e7487df0a6915b33020256e71f4 df8499f2a9734c144ee06389a425faa3411317d16ae8a28499af4d4527c690dc d5da66ea802d15a71b7c406fc07c7c3b0ae6af875cba506e1c32746d833f27b3 3d7e31bf064f9be3a5ec69f507ca402fa68df7dccfa16038d433cd7e69a8e50b 3ad8111b6675257cc01dba330e1c89d7099a88681b6151d35edd0979e2a0dcbe 36e7c34a19b39af92adb7e56242ca5b15faa078029b0d8ce72f692f0f2487c76 92b4ff80ac1fc8ed46dd2957007e1f04c5b21dc64393959a809c71c92824efae c25f09265232cbf4fac4599913b0c7c68bb2f8621af56b864ed3f5f0aca8f84d 20ae7353e8ce7a44a002ec17144845c22e27594e323b965eb5a9a0f5c20cba32 e5098d35cf569c7c4a1d3457a2f2a382fe6a71ceadd17da2341d4cf7acd6688e 5d0313b8f7b36bef3497fb38f270423eb945ff13079cc4bc9b443ce1922b16ad fb707a6d3bdd43a261c45b4144e64ab5ffad1959b9da24a1f61da847af0c0c04 b4a586f2d455398c0692398476e3211cdd39f1f85435d5e251bef55b38cabf39 37328722b472d23ff17151239af414fad6877d84f3ccfd16b002d39c0e370b06 e7e5545b2c00f7cc74c7f9b0478441fbf1ce3f05fb5d6a371462a1d3e1dc264c 9b78c956f9050354db4a2b0eb4de8ecb94673c21f6c3b5216bc2615f4e6f714d ba47aa23ea926b8f2a0124ae7370605b49e540b68cecb4b931462f29b374a103 564e87e87ea8609e44db372a328a5a6bac22b168ccfb274559d5e6067d42b64a 3f7c820728c39bb629d58f7a5b0a775bf763453957785022839daa109d106ece 172edd4f0eecb86d47148abd540028a593c51aefccf4eb97973d630a6830a25e cee68fd250a588f94f4ced19e1d19663cfa8c4acc662b4b280b1a7bed5b554e3 38b6865099005a08e77d03f685d7363298642234fb63509ae9a2287eb02b75fd fc61293a142f5337ff4c3cc66a7680fa9954323d493e8c8e951100d06e7f764d f2aecd9db1971db2640f023aadf45b9532ec006dac0cc51a9356840fba640729 ab641242742194e8b86c7ae4ae063fa1702bb938ffa76e249957c125dfce0a5a 46d6e6fd7016d75f0fe2a887db519cb458998d33a6247cd1fadaa3f9efebb894 add87dacb2a982d17e8764ec6518c7861cb8bcdc6034d34113e97d380e090f09 5ace6d1f9db202c106d0649799143c9b29225237d13520944eacd2df5c89ee14 a4f2e58c0f975023f78740f1cc601bfdb4584acac563244179938238d1485390 a6334d2cd2fac7afec71bee14f4ab9038c9d84c10fb807fde2773d240d80f57d 7b2b4f95e619fd75426a56da9b0ece0490332bdd60127efadfe4bbf2c8ae7a35 358a955cf5855a2e9ca760384f4beed2a0d88004054b8f1b89a7aba39d0f274c d59a31da76c7752c533aa8fb97003cca743d2b0e2de1485b1dc406172a3b6785 ebfba75995486a9704c27d88044842dd8e4dff16343aeafacb044c60d4cf84ae 7a356fdbeab7926aa70ddb77c1cf9bc6e3b796e98d1779882eb24ae85bc1edb9 1b37ff8f50fcd9277528ccf182649918c2f5df9ae3f107648b10ea743207c2ba 70adabb92d5ec76e7acfa5101cd6fe02cc7721f50a26a2e4fe5a2ee64eb4c321 208d1d2c2cbc702ba66cf4585a1464f1d85e1e1b105b3bb96c3d857b651fe403 1856f0eb4ad90cb8b60fd02ca5789441d420671036e771440fd318b4656dc70f cf66d592915140a8677a6b665031362735b8a3cfe14009daf9c47513c6f4737a cf69a249c7dc0ebc69023e25cf288230e01e15120c077c249f5b2f6f8c63dadb ca6395aedcab66201de19dc74e35d876eb41d9c0d013a0d5c9d285a6102a8973 342c6ab5b12b2312e2748fe82cf1e9007c6cc8bfbadbc284574cb748c5c2c2a8 f44208f3d1ac2b84df121bca6d109b2a013f03207f3c13a9263c5e3a2718f38b 9322718a2fece28188cf5e897786f1a61b176ece388e6f7a05d53bcfe7a7657c 8adec9b4c6801cce46dae165794bb41a0ba78bdedc9f4af84d4c4105def56b74 12cea35dfccb20b1f07067dbe3f5a259f1980a631587954084adb7617c4bb499 f92e5b86cf3bb0b1b18ac06a0964ee98c552bf5a9983ab7d1516109930842eac 3532fda6bb18ed8062bf1ce89a600e564575db7bc90ef2548975ebe3b987a98d 5c4535cd20148a2fdfba59aaa39224116ddf539d63cd4a30cdb98d613d93d153 be0e7652fb762dab7dff03e32f610af06becacdb2f8d9a158c0bcc27a49eaca1 e9096f2202eee113b2b8d41449fbc295c556338c46adcd735a2ea8efbf77576b c0d5ec030f760528ff6c159ea76f618a8594c3c9a627ef7b43625f057735f039 01d505a6ae782968346df3b1ab9fe68eefccf5ec68545e1d1ca43542d096d9e0 b413272c2480a1dbf637de9b2b0982da9c0feb58fbe7f30b564a7c99ba6c0728 86b8d8bd8a5a19b47852f5f3f5922f85fc65b7726a70b0562936db6c20616fa0 7eae768e1f8815eea5b2aeef5c4b6cba069e155a48c9e91bc0800467b002ed60 6070cd0c176588ef5c21e59c78644f57fc9917d1ccdd1c8e2b9e844678f4df63 41ff65d7c0544c1d876565818081877d60adc7e1c8e972471ae80798efd26ffa b11a9c9746327fa87a3eae8b714a53b2ea5e2cdbde8225a63079f466acc45da1 07806c041095da3b494364a694156d021f5ec8d64c881222206136d7c538f09e 48fd82ee967fb550ddeebae67f310dc054d9882beefede5524b761ba96a93eff 86f9346bb303f96a5bd897d4cbad0fc45e8fd87da72e5a7926d5bc86132dd08e 1841bd4de907f866bfa938c52ffa178f66bd7dbe7be88c158891de378c567123 8e9fcf1751bc5a8d60338917c85fca4669eb4b1de1e0d5956ca1785101d8ce56 28ee36c680da9ef5c53b925f5082186d52d7eb5464fe00aa4cb20d1a7f1457bc d1463e3f348bf77a2a732943d26ce4bc04c5c507ac820891764422a85ac2e715 c3a0d6e6b776a83ea19d70564bc2334a6df2d33904eaedf7e47a4887edced8ec acdd39c9d062732dd3bdbe0a2e44013d52f12e2f10eba4fe0b8b53b9507cba97 46f2cd72890598cbbb4b54e82a07b799fbd924430cfaeee620f16f6383a5f876 7e1f1d499ed7193dbfd90b539511037da73985fbb6f6963aad46b744b61a2d8d 69e6d5d45e2c7c57dc3108d93b31b3c85410e1f7831a8458754e2a935a5cb197 b013a379d2a050103a4e8331b35067963ff25be5f263eeed397d1fd5473a1b1d 20e3c7f87cea6e0848fe364cff0ce8aafc5544ccccfadc3ccdacf562e1b25857 a58faf9a33e391e2bf8723e4ce1d838c4411ad0ad993951c75266d91236d4079 e496f6e4513af5cc2189672af0641d941a0984b41dc3a4d6533457130641dbae a95274773e192f00af3fab5ec516c189f917740d87db6634c98a7aa5b4e53904 48ec77d1480238c7a54d6d08e860212f96023c5429fe092bb7fca9fdd4ff6681 aad67ec70c80831ee5af2318996b82a1011576841dff0e323b0ead71c527d64a c638bb731ab15dc4a1db35443aa20f2c31f12d341c6d50f183497355f4fab65d 0bc940d1c26f5abb8e80b79b99dc2173fb05a37558af41d3b188727dbb800395 9cedb5fa3f60d0c3890890f3c1ac2ae996b561c2af42526a05de9d0ba33258ff afcc195fde03a9aa5904de901e0fdd0786b4a3eb83001f3640e482e5f08cfe1c 1267d29ac7c4cee060a4b2bbc581fae51e4eda683c81266fc1c800b4a1fb7add cd918cded6a85e36278f3aeeeb61189252c63c46be95c6ad9806a7e28e080373 28a4876e70d76e112ac877074a4fbded37045f6537e5026a9b0ffa3ddceceddc dbdb11acdecd6ebb50d643bec487897de85d52c74433d138467858ceb4d60848 fd3618f05861b2685239aabfdeda7a5e4d359e0e671bc6100547a68b18fb915e 2211dfe0376ef548c3dbbe9ff412a8c55db8c211f9c7f4399efc6b7a1e043670 ccce40f68bca89e123038cdfd2da7cc1e57c3c67706b145ddbd2b1f1fee7be68 684441d28a067cafd734d12037e4086a199c7f239eef1ce0ab60cff1975000e6 693133aeed74d889b7ef3b186f879f33e1dcc90b4c8311776f2f18e0e22cce2b 6ba7e9b216b56ee4afcf90aecb9cf714ec42291958b9db87e1bf14922b288daa d2556819a6780c173f2c153adab195d397bec3805c0e4f208b892d2f1ecdb70f 10273a38949e3b372b2e4d5e9a635b8c0100b1550b4b5a96eff5811745c8efba 28efc723154d40697b672088a8e64ce7e6cf9bef 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI8 %!PS-AdobeFont-1.1: CMTI8 1.0 %%CreationDate: 1991 Aug 18 21:07:42 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 97 /a put dup 105 /i put dup 111 /o put dup 112 /p put dup 114 /r put readonly def /FontBBox{-35 -250 1190 750}readonly def /UniqueXX 5000826 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd803271dc8bb1656323089e9e2bb50f2b95088 87d31dc36f6372968dbd0df8505aef5b22e57f2b9834fef2c2798f9d715aa323 3e09b183f8d92797bbd3a990980982018887612cdc5507a47581f07a1cf99a5c 587021d6b67b0524e780517e7a088a6388097ced01e5a59aa8779c6ad82af704 f6283c0a8b62d57cc5d24220e4d52a7c736389ff9d77df48c0de3d3db6482774 72a5e726bf4d83d29751b70eca9ee9d04a18fb7d011542a2b2737eff052a4eff 2800d8dbc1aeeb7273d9172927c2912b597c90faf4eb15f30201e026c5d06c94 122d69e4c81882d9f10b74f81d8a85a329a5946464c9765706a4e979e3b76913 83ea8ff66e4608e5c6fa0171e0c14bba3efb8fbcd0285e6ca408671bf068e1b9 f1c040e0417a92557f6b43a2b49c772c5d6e1e6fe0315378d71f1b762c987f29 6ada445624744ef29f727436345875bab3b78eff3a86027026983eae46ee5cb6 baefe7a95b00007094774550ad81f1b2c7e3c042a77548631e4baeacfb244b62 80922062c83c0f96d439448951a8f4e8bdf68d60245c60c8a8cf3e340644ff00 efbbcbd9abfe6018af21b3fe10b1381325d2599e22b2c5ad47bd6d5ecf1f55fc 7b382f2bf4889af9fb6beb58dc03087e9de3624e2e171c669acc08ff9a88f879 48749ad40d3c781c14e04ad93b147a89b1f29c82f756d9de7c26ded896aa4918 46e3376cb7cad2c19cd70be96eb4cc3509e812fe2aac63a59c3f5016fe6380ea fa2042e6b070a6a0e9ea6d4fdedbd9147fcc6eedee7b48bcfb29d09c214fd2d1 6297d9220ae3467eeecbaa9efd677e0964f8b737d0db9a5fafc7ecc4ff00bb8c c06359af9c0db731419186f985c3ab3f726a465d094c89610c370b44b90a79d1 8f892c1109d94d52726d85a70d8dd42d8fdbcedca8acf535ebb1a483ed4ae688 57910ff580a634135432bcc38f653a3fd54bcaadb73e15e3e24b60ad5c421b8c b722b5a2d1da6465d3d6b290ec0fa9f0591c602e7005b7fcd95fc1da1c3d5b26 bd36336bd57cd17e87d252325a1a9d768711eaf5545b327741f6f338c117d008 1fda5be781c9accd874a49b6149932816134d314cd5a112970b7dd4a22493d1f 92c3776bba1161d8b4e1f9dcd1698317980272192a7043d56e624167b36d7d24 b8ef111cc6d728280840705144f4fab833a2d0813ada736ad0cf530af9a69493 c4b22f95d66c71b9fdc35cbd5ae475b9bff93d94863c2ae9130bb07e16a648f4 626fab2d85a9766808650b0dc26ff9bef9e155b10cacb4c965095f57e2b03a20 3b864ba5a8dc515f91fb246d939e071f2ca954d7a395f32c6ea043b939c78599 6a46aa8e0cec17e10745322379e9ea0dc447f1d915822af28a270571296c1667 a81ded035fb2380bd15d5dfa0c69b7d381d7ff989d382330599fffbb4e94c9c5 12fe2d16808e015710201d2008409bc4f864ac828d0b9611494548aa46c954ab b5655228f3d48dbf701f90148136a9f3b55b21a43bbd4b8f815e14a23b7f4fcc 40acdbb29a0529537f226472317f33cca7d8bde0e5fe78e6d8afa298e593d36d 0d2fc86b4d799d9cf1ab125a3e2bb396006697c7c31635d10d8b1374b13348f8 d7cb063cf404ac40592999240b7790c833d7993aea8ea0ea164b1d449ff856e0 f5d50ad6a063fd5568734fac3fc38d3bc370d10113af84178757e3d6a2e2b608 2b12e7ef98f35ba670f5e9a9060b27ecf37173f7319e8363d979138b6c153f0c 75c81b5cbf13238a167e0de4aee8a2afe587e1dd6d49521672724d410fa212fc 77adefbb816efe7621f6731dce965c3e70d49b3c00406850ad7277d81b82d032 f6351421443ddf479b1f619ebbf70b70c29968fe5e0259d7861f2f67f3853e40 a356d1a2f8db29d678fb55b8aa4b2bf3db6e6b957829232cae10d8ec56187e91 c7b305f90b75d034b6b031ef1cf2ed05b32d732a7e50964a44e7b71022b348ed 418eca870c3ceb2dc7366f55d382304f4a4a4599e39268038c26ff892b9e1813 913457e9f4678224b9f476ee444fe2f1fe00efb83814e660a23216b1ec990f6a 966ef295ac67e0afc68790d58b362dcee6a1a95d8187fcf837111a6a56082051 e4f58217f74504989e11620977b68df2609467706eae1cc3c9264df37e85bbfa 17bd423c2e9cc09174c106da8cc27033dd6fcb73eca0e2f4e387f4b23b1dda74 da47678c0c251d42bdf37285491d86df922db470072f78bbe7de5b18d9f8ca6d dd8fdaedf903b86fac5f1025254095b6d0ad0b95f4ae6285563295543dd01002 a0036fb2f0e130843187c00df413e4e18ae55adf9d853a7384facefd865af948 7e4c2c5bbf1573e98d8abf309ea323062abdf9db8d75 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff 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 65 /A put dup 69 /E put dup 72 /H put dup 84 /T put dup 85 /U put dup 88 /X 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 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 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b094541f345ec367ae85ca9bd 24568a01d3b9f8095f7420e6c423c414b3dcce6da48dd1c89a56d078e0d0e2f2 62a13640a06d17e44ee3866c3471fb58fedf5a3b77294517651c16bdd7267d39 a54e7171752dbde63ac19bb4b3021ce95eb5fe67390b09ae4d9ed4d704a67443 f55dce17acd996c1f5e023c9e5a18cbeecc3097f23763acb86cdd7cd13381ae7 4e48495ec7fa520539d87f8a8dcb3c826275469b6800876a457e7d1e5be867c7 b1ccad69742a8c9b0ad943482bf2a4ad0aed40baeb69a0233bad36b4ca2d2da7 322956c70375d152653500b2f22d2ab6990cadde2da14b4917f7515e64bc3d96 bf775258fc7dae4e42a4c9b6da8eddec4a800c8aadc8d75e48cae52137e05c03 677f5d6a82fa46d9f2fc7f56d62e5c605a1b7898b8d1401c2cac1a0122a2c8a7 aae09607f2c5f29293a09b9959399283be89051452898238b777db9830ff4318 a298b221c4a820c700ca964fc99e6b1d9eb0bafc39be9aa9dffa2fa326b2a466 1e67261bb3dbc85844e76f9313ab533ee16bcb478166de5502b68e6ec1ea6110 f53206cf1cfea419571b0243244b32ea38b828a01e62b03bcf476a32c2341cec b8a0fa2a3f3f339315d5f3aa7704408f6617d026ffee4925aa48537a4b2e556d 225be09392546ad561cd46cf766f0ba14c6854c5af1a528c3842b0c5aeefefcf 19f19cbce5d967153c4f88e51c776cfbe10d39328466f7a3b83d85601ef4d1ce 0cc136610fa6a74d9676e4e99291d565cad27b9eb2337b0eba4148efd0eb2866 ebf0a4209ba135ad1376cf9cc2d4110f36121055caa3f2e006ca9e87e164a07d 68bd46aeab400608d46c8fadd00ce3d10c4f3425ec1473dede1393f390db489b c163ada2dad50a8b86e2c5797925c43bc8282b40122b48fd110ba1bce307af1f 18a66e9cec73ace914e144bf8fad1f3a96c50a01d09c59e991f268caac57d0cf fbac9004381a7084f8d9ab2c95e999ac7d616ca649d0000c6b1c6cb6f601dd79 6fcbb4619b71fd69ebf160b42a874fa9ba25ac70be25dc81387781e8152a8a5f 98ee050acaedfb36cf8c8bbb72fa67b3829cf9518519e1ad767cffb74c9f6f99 75b7beb66e012b79dacc8c2aa42e498c6f4d0153f50ffd8b607ba329409ddbef 5b8b63b4a5a593d58a41a7be05de4fa3a88813439200a278d8eb63f1e1409db3 b34dc82f22fd8bd500c234a5b2a5a6d900b1baf916e7efb80bd6ed9bfbfd51b8 f51594d9b7141de14e042e733e62737c438815e56339ee552cb7009a5cdbe916 5dde4d3e982dcdfc1f23df8f9a883acf2ed7d1f57c2b012d7affbd91f73eb75a 4c692327448ae09aefc12c8f495ee8715cda6a93ce773bd8e0a59e28871fcafc 5feba0067bad4588959724cede8c6892d40226ef0467a147363bba861b969414 3727684d843976c2afc0219dacc2e74f005d551dc331ad85b71157a51b77fee8 872fa78d922aa69583a19ba947a0d5eebae2bd649771f0aa4abb6fd338f31d7c 3a08d6ed344d13e78ebb503056b1c4ecd65cf764cb3dd16340ea1153a874ea81 41a9551871cc3d58ccbea94adc84d72043b154889b8c06b47d824c7cd1a98cb8 c73f9de9f47cdb2b4fe01bc511331156b90d110813e25626b4e9295e2a810005 aae47f553af25723c2a3933850f2f53200bba2951504220a45485b576d4e52e9 977f6a68c48fec540231c38f222c8f47dac4f5dc3361544bfb6641d4e134d99b a785ab4ffc6d578e795a100610f17e8166a92ca1f46b67b61eb70f3186a7e84e 1e3218419632865675de676bf74dec923c2109d8c450d3dd24e23384f0f3daaa aaf33f6a7b2c75c218708210efefd8d9e9a6c232f2291da101c51eb41891392b 9cacbccf371af3f6e6931ec310b7c2c61ad1f6fb0eea7d7a22d3e345c231c5db 3b0c4c144b72675b83f3edffe94ee15583bec0a1179971cea7a2de935821ef7e e1a2927782cb4720ad3b3c79ab0fd2f1de4e4bae1b1ac2c88a9550750e6f6bec 45700971bf50ecfb4bdd1126e2e81e96c17d726c52089be541aa3c5f7f072947 e36969fcc64a73483411c48df3e9a590fea5f66f10331b1b2234826a9358bfb1 477f869a6393c2a07b3ccd0ff27764d2800fa19ba409c827c80604710ca7dfe0 0850f2a66d9c024da865b0496e37c3aa7849111b49bc987fd52151ff8f977d86 d532551a6820e7980d2b4d0a77356af577dbc8173a1a7d19219933fcb00c4814 9b9590dfa364b8626264e4e2ed87895c8936ede98424981847f559fa5b9fd923 02d471a07d7cd581666a1214d27407798d0231bad80b49ff363de615cc9f6b95 40182be0b45389ccbbe81a9d28d36953e27cd85dcab22febf409c4c65eb760f9 1e84cc36d2beea63e101383b56c44847e4f260c9b856954c18d651480dc90437 9d1e59a1d6187acbd8e0620f264ab91e1b3aa8595f0a53b7938132db5b71ed66 a5b6d5b8a688db16a7d48cfe09822beb6d6108e801defa4c85a1129d4365ecb4 0ca0a51d88415b61f89647fcb8067336649ab0ea6f81b8998e49723592ec0828 12052d28412f2ddd863353e38118f411e95c18580a160b5ccb877e9fe629f88b 35439fdcd84631d27d5d0935c502b9b5cb5995b2c6f8a71bef7087633429df7e 9029e63ea4fb95dad9b4cfd1d37bb67fada6f5dd055fa58af7849950da4a97f9 644229602abc69328cff0dc09ac18eeb050ea00cbbd217fb5e6304d8c56ac2ec 38243727bc714f15dc37e0ecc43ba5cb7244537d41773491da18a564af0d4767 553288c8ff8981ed091d6343981f59a74c2a18487dc78caa0e8f06efd2c068b0 24eb533118d30d2c844ffcf4c5d500acc4d58acbf50587950536c26939d703e4 588afc34176e138d2789dcb1222cc82106920ffe68f96cf55eeccd2b95af8673 a40db4511f955ad832e247dfb0ebe114d0dc749ea0ae7e56c806b8e1a7794bb4 30496349c705f85e03c2d0833ebffbee24a8fe123446701d0bf656d24be3e9b0 99932d3ec40d1396c061e06a65dacfa630fa0caeaf6bf0be4c11705787170fba c991664ec7a3fc140b4c7a7f0060479a08216296c8fbed0e8fe84dd23cbc337c eba7da72de1caf3dc8926ee1d3ef939dac25a412b3b2729939d0bc0ca1bde96f c2dcd1607e600b2797d0249fc161664d488d1f8af68a20c3a71e1363915a221d 3c11831f48f3d8c4d2ea9e0456cc50d79547c9a1569b59d1116b97b2061f5f24 c6fa25f303c4289cef8132bc1995c36951424fcc4230c720d5bc58acb7cab4a7 d506a79710443113240f1db050739646f43b89452cce00f461f38d89176c53d9 95d1ed13f338c7e7a6a624f01b6253c1caebc6e6bb65f903d7415cf10b79bc2a 335a533c12ad1a8f325e6b94225691b3b6820d5397406ef52db26b5bc6aab9a3 d2250a54bb5c63b34ad7d723981f5dd9bf63c281f5b43966fc8086b9116ade72 4c88770deacac82bb9067d8e1ead2b05958e405c9328c303a517ecf348c68b53 2c2fdc1f95c3d6ae4f34cead679898f28ecfb3072755a7feb8d4846c1e693b4d 5ebd1893a2977a0776302861d3c991c2698ea56b8625daab2ddaa67cc61d97dc c001df243814dcc5b008a49b532cdd51c8c527ec1c012560a766f52ca51c9b17 adb2428882f632d70bbe91427964d1090c943edb97bb58fbdbae12deb592a6b5 90410f52927a712b874be60340bf13b4588a4a0e7f298904da6015824a15d102 c9939d863b8d6216f19f52558677abd4f793be596fbee5be0fa73bc14d077ffb b6374eb063a491e070e48005215737bccdbd03a8fe5ba4fd2203575a56507646 66ae1136f32c690332e725758e499f04d8ad955b4779b1960509b98d726bf63c 1912fa5e7cb11a589cff6b077f7ade5569ba22e48990f43ae963736d63a199d2 9504bd529ba4184e1d924ab96f7e4feb995cbff1973a163683bd28d8e7a88faa fe0dc66929d72cdc444331433b4647fb4e9e9e4ce943f67f392f2eb5d1f3b875 982cca8cb89769da9e116e7782e53df09f711ad1f9a9d3df75a2a2becba1b72d 44887cb489d5d750f8ddfbb5f33b80279ac413b33e2f99133a73b9bb84a68644 eb4e00e55daf964cc9cc07b3d5ef2471e49c605d3395348df5ee18dcf82929f9 ec1082b8a50078c8afe5ec34e81a56c66b829c0afa4cfd9817897a193e0bd542 9753cf87c50c6bfd577257f05f5ab872b08160b62178b5fa07731b87e84504d8 7bd00a14dda97cebd78827c01075ef4be9391e3575edd2fb2881a17818811a89 c62c6083e2dec9b8c1466db02043db211c1d4231d471de297297db64b1b32f9e 35ad3ef99878d5276936a192f4563f23b9c60e7053be87966e50ecb4051544b9 c081632e69e55a893d575bf6ed2cf00e3a17733753636318f6a8833aa89e3623 e3d2f242557c55f39c7055b408631eeb6ce2af02fd21a9d515804590ba9b8a88 2a70a2e3891f1e298883093f98c4d2e5de21e6103617e042c15e2cf4f6f601af 50b71e3c1b1f293c4d378e89e9f2b0e4176faaf66ec5f2909221cf6735d955d6 317fd93759b32a7422d8c2af4b70d97c4a39a6919781f202bf6e676066deb0c8 85d090ae6e3cb8f5e31b09c1fd9a8c6b65013df7b7ed6f1810ad9ea92c724934 727f545bdb696f874b5fdf4cb94a6881052c6a92624f249b20e549c19ab28cf2 ffeca9ec1bf3410c4d173a4696888e25f9953ce2d8e0f72f5ce81bf06c9f21df d49b6a5b6ea869952dd78df721814faa85b15c562051d8f8a1ebc7a2116f098c a8c23df08997c0e297e7c9a094a3ca8bcef37b872c94d0e21d253d5cbc5e3c45 ecb9352ed4a15eb45368531b4882b5ea277abafa6e985481c6dfc09402415cd3 6aa5b17b5e1d91e8aefb8bbf26f87cb2da2ccee6babfa49ccd0eab8b7f5f5369 c3d68c60133655799afc8a68acd10d040682ec5d1f7833ded6526aac983e1b4e a02fd5f882f8869bedcde9c734eaf08b376381d7b74b9b406eac946b049a46c0 911ed14e7672c9ade90f8508a36e6f67de0732c3dbebe678a4fd8ab2becbacbd 45aeff136059462ae0d382e7856c197f09c967294b3da0483e896118484b6986 0b58c303f6e5b53aea3132d70bf9729d92e8dc930d000b9ca34322650f3b06e9 1f98ee4d7d27add3c839ce282240f1731ebc8ec356a398a9857d657c42e5e5e1 655fbc3f7edd2bfd67f18cf50a8aa3e9f0066d63314fea6844bfe9ce1d179ac0 2f6dd73834123d53720430ea7ab73834132821c956a954e106239de0a475b3db 85d8964af8ef04b1cd315d3d414f582b6883c84cd41278fefb23e49217c38ab2 c2354770f52bb2502bc7876ee47c3c5d9b76a33bbd473c71b4dd4d808fe426f2 75570ae40bab25103560e6d08abcc84acae3d18d27237c12532068a38f8be0d7 11abc3199d58675f59b2965106253e145a9f18c498a45c07154a42a17154d109 8ab23700091b833420880200f56dc4604d4c786b55ed5b65683ab8aec2d5e2af 4f689dfe676aba006319eb1f33cea9feed0935492fd6843f0f676786edf002e4 062e230f5287a267fa7390c681b46b04b42297df26623e40f8e12917e61c6542 b7307407eafe70cc89a67ab91f01c851989781ea8790e2ae43602693f86c7584 84e04e172af56520c1c56425f775711ba82ca3ce2ccc32272184ad3b2f2af5f3 a0cf31e4b09dea60768afea565aa036109f2383e8bad1051ce4b56dfe0906850 234a9bfd8e67a06525fc136d073f13831eca950773a19adcf9d2ad2aeae4459a 7fcc687942a0edfefe626f8c092f7c02317fe43f1cd235408690d1bcfdf93fc4 da28434c129afd5cc1c466458d48d025cc0faf44c71ba69d8a4e77989b0b4f41 b69a0d9d1188a59da1e0f87033d5ec87fc704c80839045d5a0abfd058ff1a25e ab6550e9a76b5cdbae19129d74c540a8d23eb9bd04abf36cba55417f3fe7a226 cf43187e04d50ad45f66a86cb6f32416136c69114e9fe5a8abdbd655800973e6 099a3c3ab8ba2e08587fe332285cee5352fbe65c6076fbc25cda8a82dc782d7f 166b4b67a6e44a4d696b0228b33c44440e6711ea833da507b5037b6128d48043 3f6f9f1c6233751e40d8200f842256c360788c96825f13167f0840f3cfef50e6 ef8c37661c3cb914de458682f763de4d4c6b48a524b04f7546e29c18c8095e42 76931613151c6c17d1fedaa30461ecf0e439d01a96c901efb0e3c3ef9b95af45 aa7aa8c5ebddf47ebc2c5c6e6c843971193f814530d27bfa8b81e53207956183 192947b478353dd8af6aab4ddf0774abc01c9107e019b4a4b2783f2b651a37a4 103d76207bc789ce6c0cc65115f4786e8539368c37f7f2eb7ec2d68fc3fda7c9 2f296e4f881474c0ca2a7fcb3fa665ceb96bf8056d522ba9a3ebb688c8469b2c 489107236b44d83abf6350db5f86a648b287050bacd691cf480a0475bc9e2dab abed21d0389c4b7b2550607b971ff39ad5a3c31e1d1cccae3ea57148896875e8 82fe3dcf931ce554c15e342a06acb91040ad86083f112c54d448f397e6b3aee1 3acedbdef988b34fd81a73bbfa8bd6acdefcb499717ec6cd58c95fe328e875d3 5945f7917d3993f602b4e4ff7954fca1560c7bbda01cdfdea84b1559b9153ae5 5da6f3518cc1559e6d60f07491405ee01eea8a8e8f2367f8a824ff7c867b78fb dc3f4c6bddbb732312362d7472f3f34963ffab5b403bf0b13e52c491e9b8f0ab 3c9c0226a5a7a71037123f53f9ed697a1cfccb49a1e39d7ef4ee3e26b61ae38d b2140927b0b9d44854774da68a88c6a519f433a90d413024faeda470768602f5 741f99b1c119129f66ebe3bdf1b21210fd141a7895508955f23bb31cb0e91cb8 03f63870e773528e330e2d0f01732697b690a5059a85becf125957850f22180d be692cfdad32af3e9456c8dd1717adaf591a375d0b3aecfd87bdf8c0a6775b20 0da80a6706b73644c0c91c8317278f9b93acec5824fe30ec54069e557040a023 67dc8ae55753f4002dffb1d604d5bbf5c305be3db89eb4bd95bc8fc4c85ed87c 098453df2c4f4125c3b1bd78be6601c68e95d442174971ab088dad563a238317 3ae1721ec83ab497d37c7cf015228e6aafe7dfd9248465c452b957dfaef6fa0b 57d133c1d5dac6602e3fd1afa254a20374c296bc8f241a8d10fa515b4b5613a1 e0f00806d05fa614db213bbf3bb1c5f81388a42ad80f2704828925689d4f9286 2354f4118254e62a402aae81862d0259c15203e8eb001e064a3c52873ad67e87 846ef39590dfa364b21607425680372edde2ff978809c5fd3018404ea768adb4 2af2c52829a31c09bae22ce22e9e730ecf316080c7dffdb4cebaa37e319e8563 886994e38154b2a4195306ebd3472d3960509c0ce2ef356d77d560a5e098388c ba304780ee209ca0bd9222129728e83f5ce274b8f8f70656f1ab8772ab9be000 155493d6a6c9db7040a49b2285c9409c53cbf54705cf4d8f2c006ffa548f9e68 ba20daddba068e83fb63311123ec9397316615a17424d15d2090495bd4f678b2 cae909ad3dd34ad09ddeeb1180e2bedc21c9d6637514ef4c079fdedc13b39d57 34639c60dd46f196f456344838af8531b2603801ca222694af4c0b3f226477ec 7842e9756ded047cf8f328eae5c19611ec6369ba2fc1eaf2644a8afe1e232596 39c1ddab1f26f2222ad30000553b2a9e78fc6a0ca80613214b1578032df2bffc 5ace8df1577d14072bf8d3365c6f0eae47a60b0e9c978d8271f0c353c55f215b 0b7a62041590e4eb0c61fcc03fba97a290a643fc6e70caea7a4d6b3c3d73478c e9f1247ca7c95f3ee22d9243f082447ad5c80d1a86d9a9740f3c8fecf29dec55 1297e04fb040ed9ee88b421b3293c7d4ebf775be071e6e62c86bb1fae036c2f3 f9e24545445b182bbf2b66a2b38227867d8c47bc64428d2570691dc01531fdf2 3a18219adb7e163a3f458da04227d2b3476a730111b694ae3e7a75cd3991709d ce1680e6970e541d4b7e8ae1f9a3cdf456a6c31730af51808ad1e05871ae275f 91aabb41a7da0cdbf2645057280ac1551c9047130dca0a9915b529d89fdf0fa1 6d8d02e927815a50ed5f273243b77f947a834bf921234f9e0810a9e6154b9227 c2714b9c512bfc997cef554b297f7b3c8eea6f894225e72110ea6a74662a1bc9 a3d2dcff5243e960e698f25d181c1576c1ab30d1bb6920b39e58d0fcde9f6c0b 9d345b9dec69ca48914e6941199f0e29c46acd7324054db03074a9922de298c8 54f125fe64c0a9242b5962b9a8b6b30424bda7f55faead8cad4aaa8efb13843b 1cd960121c2e7b16dda9a2ef0bc202347482b83bfab5df1122c8ed16602a2684 ccf87b96477497aa55042181629227c7581db137391dec317551364482def45d 8517423d9f45213db654de2c713010a076ef9899a68257de3f530fbe05f40d7c b4dd58fff984adb7bcc6f38a9589a475c251ce4d5eb94c433cf06f794ebbca86 4d5bc826c1ff02919398043d629f5495c860d8bd79c6d426078dc7797b9743f0 c1ceab78d7e35065257faccf4dd2cc620b59ab296e4072887717c196fbf36888 3f5032569516e5e40c82317c64f78fff9ff7daf38285da3e066ccaa07868f2d8 59464423455e613a863600773ad6c91e07417096d9f5f795b87d4ae00332c996 87bf02820a872a3f136d92cce957c87581b530353a220fd72c630404db9435b2 4bc6966ddb6c0cec1f070a269eb74970fb64a39ac6306091ce12f3fe95490476 965b6a649cf08ea44927c7bc078102a58b6400f788fde68b314da2db84c78d2b 9202acb688dbb70a0e35857d42c9acd3724f84b59fb3b2d4f39eefb1752fe2ba 1f0f6c605d1a5e5ef75fc016565f052f635d14445818b000cc3758ea4a1a6895 1f3dd46db0cc79355fd82db4983a83804337ac974b9da53856759d0d3f2ce0f6 74ff004153ce0c309b5529cbc0a931a5679fa80096c1fc950d6b5224bbced742 40b672505a4756f517113527c63ae4de9bd6ab7669bb7bf91b38281289c8a7fc d0f9c2bb83906bdc00eed222d1b341f29353cfc726d1d3ae34f5726624cfced7 ed6816dd9d1c0a3257651fa34a1b0f16f795b614e7c2a0a2581402df0de71b35 a4837b4be3ec9415e595ef454069d9f3818caa53f1e018e57a46f3206773b1d3 580b2c8824c1692dca92c46043c5feb4704d48d91ec5585030800e6c15e1e6be fce1ae56bb64b31b6c1729a90a96dbba770d3149c15380a4a8619583739a4f96 2bf599115b297f3de99410b83203cbd153f85bdf5123efc3904add4c9386f87e a45b1c407bae6597cbdf746f4f20060c51a4cfc3eb3ce7b212eb72feaa8fcb6f b9d9ce2134ca963144d55ed9cec4488fb718c36cea14c6372e5e468ffff988fa 75b78f8c8daab60b95429686e4f86e799ae86f8fcc8045097be846ac593fa0b6 1a4585cc826cc94f121c87900c7540bee7f841ac1f50078b8c553dd257d1253c 79c760d707c13c43909442aaaf8b39c9bed6c132da5e46508995c1d2da42f221 c71a60882a7060f2fd4680b9bfefa3a70fa0a9479f45b2d744d180f7c85a3c4d a658218db75979192aab56a9ab4886105834c4c814d15d3427fa23444fd4cd67 336a355f3353fc118e373395ac00ccf78a77ea7c7871ad01c4b60bc3411b8f08 76b1970801a49e313104d9e368abb88306dd4bb6ca01877b76430afb399c16fc 6b187f68d2be979a04afa72c3d736161a2f6b2eba69edcada9cf8b43a8d32c3d 89c473392f5272648ad26edbb43af4eacc699763ded547cdaa37be986281e02f 25cd49a14e92626a8d18878c72d0682b911b67c3547754234089a613a49ed8f3 65618a78928ba3aa05580adfc50a8dd493488e2866b691c08c8e13e176f003f6 da62f3959012f10b1419409b3769ec361397f573462a64cc5f3cae4b2c9d912e da481fe5537d032549875a50d8f9fcd2bab50b332f3fc8d76f0b1a45edfd9d92 0d7fea0d2475940d4f890af5c64269d4e95df88a6d2d5154186925d3679c1f3a 1983ef191a03d33052b8dcd51bcabfccb3a8338ef284947a1473f8ed08dfa810 4bf6a8df52ef1da5cb0b59cfcda33ae5fad47cfaa725ab0b132f0cea81a64537 df418a81a65e5bbdde98e3755a8ca4b2f60ae0339006219adaf2b37a34447c8a 8869d7b0e947c3470f9f35ae7d9e9a4b6940ae6aac5facc7f7ce2fac31bf0769 03044c51f33f045e2b75b729587b9794 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMCSC10 %!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 46 /period put 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 /FontBBox{14 -250 1077 750}readonly def /UniqueXX 5000772 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd800bcb325468c79ffacb9cd3037a112066957 7889d67fff57549064b985d67ede2fb03272083723541051ebe936eff01a4ab3 cd845d565dfdf25d9ddd2792db17f52f894bea484480f06c832140fdd6c3af0e fef0584c6b78b5e2bbb9a752f10581fd1c48cb498a822afb18cddebba34695b9 b418010368fdfb7bf758d50f3af03360b81c776325c33c9c81130b1e1be840cf 68c71ffd091dccc0df7b507fe8953dc3836af0da08ff9ff3e9cf7288d9eb8baf ffac546a0fd987ae8e0bcf7b759f4408e13524f478de5896d816cb0ad2904271 cbec82a5ca50f0598502b434ecd94531b4a65181b5b3afcb650904b15157737a a4386f1a7f1252caf2adbd5c8162bf3b065c01ca471d520c884ebbe1b29f9549 134e285dea2fe158c15a08a744cdda00bb6c62158262e24702803d9e1dc1d3a3 1383085631154b077fc20488ee3aa03f47ed2766644330b94133f2f9389b99b8 96d3707d1c5bb227fd3bbdf3a6ff572bf07164107c7a1c0710a5c7cfb67282f9 c238100fa138b75c553d7cbef17e2292a320d3d86a6210e6e3a1e590b10dac6e 177094d5dd163db623d7b344478752f6f3d7d5b3fa92e6bc72dc6fc343d2f946 9153e3d14c5a6d6b1d9ca52947b789cb4cf6196a02973aff0b097333f8d615c0 a1726ab7e31c378eab77150faa28cce535afe3ae5798402ca51cbd18dedc2c43 5281d0781351df30f4d11761579676449ae4a47a10f677cdd31ca9120fc2606a 9025d3cc09d58856a9bbfb17669f706ff8c7efe37a3656c9b20609cccb58b309 0595ca5666833f2bbf884aac1cd632fce69b4e0ea2e0a25b9c4d471f0e322e09 4cefda321359fdbe0a82656d3ecaeb68b7e0c7e733d85066ed7474c22b67169a 1bf1bdf7dfd0ce0aaeef78ee393ce6ea65edc1fdcdf3f744f714005d8221a156 4e7afb117b9edb8b470bbad346f856507852a367877a199f0a199d704449538c ed349240ffa8898c77e5de3c7a36282ad76007e7c464dabdaede85a15b0354eb 06fdd3328991036e1081af3f436ad305ed24665d0725339e9dcc3c44b516a0e2 c91f8f4cc1ed7824d4efa363b503be97a3631fb4207727dfcf0ea73e8f19dc44 85386cf32533d52f57d10758dff2a1c6c8dc54a7d97cbf2e7a5070bc9541f569 f78b864fff16d4c8915767ab4d6053ee0c6b8fb4899c5e5fd460d1108c03eb9c f63f08d9123f6e6eb7a96c913b403ba3f5811bfadce7dd9a7b2cac73f903bb22 065c548bace04881b09ceebc928a285e726adcc0228d51b0573170bd07980246 75a11799e33afe4cb0025553ea2f8334600fd311366583fd9df70934b81e3b61 b7a7680b1d74ff3457f03090b0adc1df4e2261c23a57482c3d8873f4fa3dc381 3aac05724781413a2f215491c1f090f481f637c2a8e0b0ab4fd08ce012ebcd71 e7f4364bef1a7ab31e0ba8567aceee0980c6af355965b44cd6ccfbfcbf760a34 f8b105720cf50d32e1870e55f4708229e184f75e5f2f8f22a9d8cb911928b46c b5e1b22c00682399814dc61d7256c1e3f44ed42573d0affb65fb8e00789ab713 88811691dbea044284c251791774ee45f802d26d6e0cc16c2bee9cdaacaa9461 916338b625f2bbb7b2893abdfd194a5955ab3a6deb07de830d332e87234dd0ed a7f8a650547b4a5d58a08fe1329cdfd204e28b055fc6b30046786eb07b65f1c0 5cae6398d7ad3a56f892d6535be94d84b59a4bf7e120b1470319796fcc5eb190 3a4e26ea4f338b5a7be46db260302ea562d2e17d23a3265c329667d4a590997d 6e1ac98116a07aa735fdc8a8b8c8c316832abada3e5a399ec127ce5ca3ba6de9 d52ef98eda31c5a7ed0f0abe024ed95de9323b2ac8cb935d5075537bc9ef96d4 0516d10cefc86b5eaafdb99d07f59ac8e69f9156a5835ea37e3967bb1c9dac38 7ae08a4211a73612acf2e8db4550793c24b9235ffc92902a3d010629ed73ab04 4f0189ca190ad0a7dc55d71935d48be0f1310bac41945fc73e512815fc7ce822 8152ae94e4a0b807ea9af0f978b4cce77cbbc01b760c28796ff6cc778217a5fa 7c759438c99b0c6d1bd327818ba915dd8ffb6a5d5e5333b166e5f2d2b254a8df 5ce15b74a13988747fe688f0e2c33ec62dcc1e46012dc3c377b4e6b360b20199 01d900a26575712bd4350bc68fb576130c25333b5bc0fbb248949109a211ee8e a08238df32527c2d9213265aba880792e2b8b1dd40a8099a8b09bf49f3781b5b c8ac33b256183759f73dd7aee4986344963a09bcd40026d7126534e4823c82c7 d982e71d7e4c6a1f2186ec62653e3360ac5ae38480bec879abfe7e5b88d5ddf2 45144e423fb5630bf16a2a7c2ce0fedb067b645e2b93e6eb58255fa2227dad42 79efbfce1befd3658e982e059bcd8967948d054cdcdae474347db7834dde43d3 ccbcae558b7dc0e74302e8231942850a1c23bb7026cc683b36b00bb62448ff7a 1c07bbae8d062bc07e54a4c885a6e7c9f19b97248ad9cee1943ff698db1dc1db 4322fa3807c80f8526a0309f58c357f5aa5f7c97186b9db1d6025aa932abf298 8b81cc299278839e3c448f91d402a999d2b808a33177bdfa576bb0011030d5c8 9cc5af8f05420ca05a5214d63c782e8335c0a66949413994febebbce5f1d4c0a 0dfec20824a2ff5ed03bb05937bd67d4da66ccf1a3312a60f16c1a8e09c1aba8 7091a9641fd14ab95721fa030f498842784c7afb0e661f064b491ade93832e1e b03da8bbbc297a8c4ce6162ccdd14dc2b961a628be673b82fb49907c9d7a9767 8247dd523ae369394358be71a5537078fe6ce1fa5c91181e55a726d0b0c396d1 ee2f2c81b5267d398df8761a9cc933f824cf102ee564bc74d26eef96ddb65152 8f1dadf2c6f1401d01f9cae738db58c6d3e6097892dc032b074178fa498753 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 3 /Lambda put dup 6 /Sigma put dup 8 /Phi put dup 21 /breve put 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 52 /four put dup 55 /seven put dup 56 /eight put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 97 /a put dup 100 /d put dup 109 /m put dup 111 /o put dup 120 /x put dup 126 /tilde put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueXX 5000790 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685940729e6e22f4be0e08c6505cba868f7950 93f556b64b808dd710eb936d3ac83e5783b5f7e56d363a59a026a5619a357d21 c4e77bea121eb24b09027d1aad93158edf912940e91cd2f17922b35885894644 7773730bb19af9e70453e1b78d5edc123b273a979891d6d5e624f4a388f3897e 13b045e34b4ee2f5e4fc5e996d3549640010c671de14d18b0b3cd300e57dac03 22e59bfbf7b29422230870f5897fcfaad4b50c7c1c58edcd119218163d6f6e56 25ccb882db7b6d09a1586508e8cb642a11c29a46915e4a96e282079cb21922c1 c2e360b487a45302fd22ec8c5fab63e54b5e844d4b17ca2fff37c69c366dd72a d02922c14c0932f65ed03e4219c117962edbad2dcdeaa9c10ce8af38a4ae52e2 b377245b0be19a77d6c936e7530cc4d0b78d0cc4a92698fa2870fa54f2d8503e 2d17b3d52fb2febb09f2b2af0c2a1892039ebe19a690098799a858e3d39631bd 6925a154d161df3918074ada6bd52baddd0adc3f07e2d9f15e27cbf7fe8b98c4 07205c811121fa91e059f2f99322fed63f359ac9da97aec383f067f23e5de331 51e80f0a88ab50fe8fdae4a5de93c1ae2fdca06150b37246140c0e87cb2325a6 0d2349162ae3ac93144eee1e665a1289105318fdfe86b6e76251cb25adc967d3 d0b97fe5e279e1161736ab22b4ca510b964342383a840defd38f96a7280e6ac1 34e48d740607ff2e7804164a16d47735864db847c97335e6d4215cb99911a1ec 015a3edaac1f28fedd56d2467130d07bae9416c15f0827d27c6c79f59054282a 418c12c157c91223a829947f47592f7cafd93ca182b25a73a9419127e3b12a9e 5167ac3963f2b019b338ac46d63880f94dda4b538835884d2a5538c85528d6bd 977f844d32b43b0e48caed5a4bdabcefe71695d69ca784dc738a8d7bbbf2c609 e4a229ecd7e6b8708911ba60696abc682e1738e07d5f90055b07fb6f4b054a64 4823eadc0e45f26dd59aea81e488c939b706588be53ffa610db5879ebf6b9926 63727251a854a9a464212db89a539f3cf98939eea455b70af790176d309847d6 fe73af7773b4df7e686f588360d3791d0b580bd0953293b7a2895c7d09e45686 ef85aef4632a3899be3a987477799dadd1ac5b52a5172d4dc3d25e42c53463cc d19ca26b78f98fcc558dd1bb9a6877e8fe95a397727eebeb24fa4165686e34ac 712a62fe0e128e172e91dc9868dae4e60cdb18daa96f8d325a7a0b98d468cd28 d1c8714bbe88cc53a4382db8db142d2a7d9e78c5f6c5c8e781842d0fa994ce1c 086f9e26070d8b6d5c219be96548b874305de05a90ec62f6766fc41b511c7360 8f28f55c3b1530c36ad966eeac4ff46797393e9f85b529c4a7044570e3af418e 9ef7460b9d5d9d4ef4c30cc11f62cd5cb0baff71c39cf00a972b54b91d47008f 52889b44565d4c1f50bfa8453c8bb18dd45bbba5c0c6ac7700b12b645895bf77 c45a8fe6c8ff77c7bd06da28cbbb8b83c0a37b1a40b2fef989ec076677d29fc2 78034ca9ea785aeda6947294739a57f82b0db6ec03605f1413fd84485852506d 9d1607a5371ddf95deb1292b06a3ac9468e87a6c1e6257d20709bd259bd0b4d8 ec2f9441bf9d2fa6fa71e66e9980a835d93f98afca75ceb7e9f0c938d001627d 71f9933e2e2f21874d425d3e93b94e6863f86793137c7075765e60bb4522f336 17e9074731c10c8cc1d62153c5cb1e022f66ef9ab4f9c2c6fa8ea7d580404dd7 1de0f27e20042d15fd3c2892ffa4190cd7f4945fe3fd399a648750ebdae31a4f 0c876dc354a8faf723a2c14135799348baa116c4e5abf8b0a2c807df059f3ddf 277def97cf31a30bb3165bc6311773965164c985d25f2521d4ae038bbb1c824a 976160555003804bae824349d28374dcd6df895a3ae2bb7ebe6d78d30e6b9197 0bf09d674b2941338766c1640c08afa9331ca9b95195f94f94a51a8b664152ae 6353e31bd4a63f9705f41b8d1166d73d1a7029a90e2f3077e41c63e33577ffbe 2e38b27eedeae80ccee7320e6062144bcf6a5c2bc1f63e266c308621360f7a48 3ff6047dcc08da963aa3a1fcd5f6d10d0fdc91756f220b75b11fa0c0eeeb609b da1f0cdb8fda449b0d4ec90c35f6ed65e0ed9b5be7150382b03c0f78470acb57 d2e63a8a8e98ea62dcb36689235308b63d21031f859128fb03da7644b6dd8d71 0c0dc816dccf23f15a20c369958ba3f13e6b4a8e9558d311d6bd1a46e1c63df3 8bdbededd0f48ba3a12f4071c1a6096eead6c8a2355b0b557de3f7f451aa8a4b abe4ebe2cd94d16478b1e094940dba5063f5d6cc1ef8597875b2c7ac98826b79 cc3c7ecfa2510adce2fec33c516bec9d6f918b30363869f600040c75cfa895b7 006a049ff31e6b2b400a93f3048d90df46b29ac3b3fb5a5238f4c390272d8eb3 5dbcbcc057d20c276d1659f41b517b18e9b00a0893c1b370f0f1c3b46ddc46bb e0b01b49b02ae09e22f66eaa12d47a4f7f436fc1f58993d07cba58587f885e73 581f506fb87b44618d89150b19cd25ef9d58c0e12c8498f6d0541813a30ca0b1 fecc449c7918bcb325af8f0d7f6ab345954b81f2a457d395e624af42411a6857 a3dcd0afcfa1b11498154fbb4082335f7e363d05171dc2044a1f2f7daf6dbbe0 08d2f8ef3ae5b3af8ef49d8f2da75b22f837bdc709a103068fa230d43a9a1bab 5daac53a2162af9c05a787d342ef2cb2b837999653c6338f8dbff588e48833a1 bac767b82751efbc5c66a808ee07550d8188f5ed180af5d6c89e430f2905c8e2 1987b0cbe12ad191225f4a9447422adab783c4fa2d9cfe3bda8b24bf7ed379bb e2a58ae2f0d58ead42fee5ab60860e83f8ac579733d234a10cd60854bd609f82 a4f07b0109d839a8d1b4a2d44fcbe58d59556778705134de6a022ca5b7c260d8 f34ab66f10171fa2a0ba8737da8699c45b797640f123b8abd399584913f003f8 81049734235f1d452d9ed92f89781beac67f4fb325a94a2f79c9811e03d08984 98094b3c3e8e12296f01750985e33bae1c5b24862791bd49f3577a02c1d8ab72 ed365099fad2bc96397b1ad3a7634e001a1b206bce4ae0527c7bfc34b94f3794 7e5e7d892aa3344cbeba1c7d1f74bd8e12628d471e5f70f6224a1c82c0106cbf 689d277f3fb78e87021bb7bdb3095f50a79cff8528346a347909d2377a6d78af 67fc52236095bdfc8dac7662024e984531df5ad5fb80e48a34ddd03222efb753 2d26c93bf439d53b7c411df7fc9f31ab85b4499dccddaa7f248f04276c2423b7 88139bac24d32138e1d4681063c95c1e1a5172336118f2e972ed4b80f8f5d4fc ba51d1b5c1c929c8d0e08f7324689bee8515ab12c29f7d9e4da11f6955fe1097 fd2f1c5a45f8715efea2aaf921b1f9dceea989bb766cf92cb66752b35419b19c 71803d4089ac14654ad861e6832f1dff9a080dde6958b2263154e8dea5d2ecb4 6a5003bf15deb7dc6101e35c584ede7ee21e15c0d55e1960440b6b69f8c890d6 fdf8d4d3c3b6e60f033c8b86f91e115710debbdc5d93426654010a9e50a2ba92 5baec664352df00b8139ffcf82b8fbf1959a2fc78fd2588b75bdbf11771351d4 f6a1f2319bb5292a899fe6a6891cba5c6dead9d09d283c02970a2362c527b429 7656c50026684f4be69663a5f780e99818c0b9ac38d52b135ec135b936131277 ea6989d8947cc86d2ac643672c1892f8397c12e949a1045a97bbdb44ab84ee56 b9c97a50ba66bdf478737a3c6eaed1fe93adf6d410cefeccb10adba185ce6880 f9930f0c1e75a975fa472c279da73bb78198bb4823bc626a233549eafebe7b17 f40225440454d514aaf4062d94c361be7035949e0b5ae8dd25e7d864f5bece1e 46151ce2d985c5ce3454999e8f948aaaec796f3276fcb7e85f96942776555962 d90f57afba355ffe11d934cfffacebe273ebf759533fd6b55e421d1df7e61360 770b467e1229eb436123d6da1cd0f3f56b291451f2e10c8a82464813cb6d0535 d1ae846a6fc64cea7c91f0f41c86f3c7022766c9bc7aaad3e2b66e0e1c17365c bb7bd3c4c3d4546046c683059f88d8bdf1705f2be106250fcfe0f99a2df3ac7b 5815397d66a86970cea7fd1b7cf68f58713ec79d508ef98528ce0b9feb29ddd3 82b6580c293752c3c70e47d55c4f3f03893252328af64ca8910d1a9a039bd3c9 778ebde06bfe6ebfdd3d0fabe6c4d727e170c39257c08d153745ecf399610ac6 4bc2a4822b3f93ac857e8a121818f3a18fe092b59e5e70828ccc8ddc93c77bc7 e903ef0c3f35fccd0a9fa734dd40b01ae94775ccb2a1558dc6645d96339292f2 4a34f5403ed04a0a084b34efd05300bf8965d06aa822a091dead156a146c318c a807be5853efa1b49e9c4b7bdfd4cf0963cc9b668855530e8e62861e298b5bd2 571ba8490f8889464a069ff808590257e99c52c530fbb610a372cbefe4cc565f b76b056c1c9a4bee81e6b687adf4d478c4c521d5386a8a626306fd0a410a2bd7 9bb690d13c36ec5e37a3287fbbd4a4a61b5f5564db701fdec3a4fc3e9e8426ec d3a7e7aa06b907dc592f929a46cfa44ef241c9c31fb062d371b32ca0f858938e 8362fbd1c2beb4e2df3219c5f6990e8161574d8077f085fd5ad23ea014c29cfb 19712aea5f3d1271f941ac13f0c3b943eea25c58d0959b64be5c3b562c2b04a8 736d6cf2e859c23279c32bc2f718f760e78f944ecbb02d662d92eab8aa9d42bd 3009940c4ebb5f481c0cc5ac0c7a87b705f40cca3f90ea5bc24b2da405f2133d 85ad3f8b12d3a9446749d3ce6c8a175d34ee18bf7a3a8ced9586170e4ac5057e 9b7d1fb779ef882e5f688087a5aa2fa6429221a6368f032888044c2dd41e8806 beca5cd1f11bf84beaba147b50e14d98c0ab7a04c8e6fccab81999af42831c9c ceca939fd8725b3205b322941266c540dcc6e10f64b3c527248bb6a065f621ec 8596a3f4553ba0e1bf7b4e4450c4f59f6500e0edf5c3ba0ef2b95c7ae5e9e360 6b031b352c5f176143e8965d35f6f1c1d3e06efc0f42223e48bac31a049c8f9f 64f39bb54c923edc341394955159d2f344f008e500dd63d7f588d4277e5ef669 6381292365ad5e70ddc2525511b2dcae24953286a57907bf61d10e39dd4cdad8 10e9c0ad3db4e96715e3b70e3a840d4a64683d8be22d73bdcd8f5c36fb2d454d be0bd371af29f77a6ec65ad961f7a7f332e56a9064bd75c30a6a63b208b36e25 bfb1bad0b10c7a58c6416d14ee3baaed2c54d174f53e1ee69e90ec418607f64c 9df1cbf918ba92005532cf1816bf511087fa9997101bb40171c874ae26ad0dbc 9cd92796b70569cb34d5a5e69d1a2d4fa94be71a285fd0f3d3864e06227351c5 a3b651be72082ddef7bc823b14be780c6280fedb1d828327fa485a2c1d4bdab7 d2c3b3a1ed1d2a960a82e29085f9b67a8d5568d09ac1ddbbd07fb03aa92e3cc2 9c8b0fec2dab6a76b5a600c8966b84fc95ffcd9802f1471599e38eef7cdf371e e450d25d981c2f89d10d53f89ec7d445b78006ab1505d6c990ce3914c809b64d 0a44a3aa0b4313e7c959ec7df2ca57b1d531a0cadcf5d06768748ddde4b5948a 08c1c723ba5c6c0917b24812064c9f90bbb7f081f4a2213506bbb0d01ee5c68a 37482019de64b83d972035c3b257d772c36b84b7dc7034d446f5d889bd78e916 bed0c679027479165a41b2c259b8b579821114e9754c3848ca918534856d05b2 df7d289ea7089c4d969cc6be5f6541e657fe1d985ca3b9e4bd3d6b84f4caf10e fc11dac7cee033ccbff50c72e60e7029bae521aba6e608fb6007fbefda08915a b93b1e7369fdb3eec83dfd4120c040873f57f31ccc27065fae974ceeddfc1853 d7143a40745460eb52c3c7f8391e16c3f84412a23082e46f0eb8255c27ebfc36 6621bf6b6d7c9598f8c738231c0d36cc038793bedc58605fc87610dc8886916a 216d04f8deeda3f664f4d45e632e2897df78c7a8f927c06d891084 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSSBX10 %!PS-AdobeFont-1.1: CMSSBX10 1.0 %%CreationDate: 1991 Aug 20 16:43:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSSBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMSSBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 38 /ampersand put dup 45 /hyphen put dup 46 /period put dup 64 /at put dup 65 /A put dup 67 /C put dup 69 /E put dup 71 /G put dup 72 /H put dup 74 /J put dup 78 /N put dup 81 /Q put dup 89 /Y put dup 99 /c put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 110 /n put dup 111 /o put dup 113 /q put dup 117 /u put readonly def /FontBBox{-71 -250 1099 780}readonly def /UniqueXX 5000806 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd800bcb325468d2c43f9469c605847b026bcd2 1cbf124128e8ee31aa6d4286fa6d465315407a865001dbf5f9a84d3c7974880b 04bb3fdcc784bea5fdd113f8fbd8a454347aa30bf2f5369abf0829dcd5378296 a42091f24af75e9e8310e61e74704d7d88f586944710ad7adb098ad4ef8fb0f6 34873391837c2456f1126b8b6f2a19c5f43e5c47b746fbfbe14db596bd4a5000 adaba65a30556845ae06c6e20efc20647fd0cc730d587ddcea79c40388105b0c 20782fee3f0f2257b942de0853a1ee99805fe77e1796c3d2420f5658fdf30766 0c5525634b2b530a17b104e9a21ecc2f6a30e22520eefdb6a62600acc6b6e690 358036a18b2f414ae204fd429694fd212be6ede4bf941e3257f88a03a75aaba1 59aac9d0ba10cbbb4d4458c8bb0b3894d367d6256d5577e71a7c409a6f955909 e0f05b89175e89e50d833178e230f23553979497371147f764d7d5f8c7060278 d4ff7b299631604cf2b13e03038c1d2f8a726cff45122f37597dbb6eb8dab326 f200eccbb85c7e45d16c0aef3213e6b9c6190f6cbd1d0c9b24892fe97e57f2f3 a35a9ef6f0564ae852620bbd8d333c1fe8f6423157ad465809c536e9bf60700a 6d86aa05bc4257a31cff4282186a1540d0c9f91526c436def5a58caf23890bd1 e4f862076a77ff2339272d758ade8ce76d712d09a66a486038c7526ba4ec8188 dfcb3554fc97d7896893656c3b0dd357ab78098b97b42d507072e35112e7bade 4f03b0a7802093c546be61c5c9a61562f13364ce4b089a976f9f5cb84f83a077 44631b9532e1532309e42c60407656ee331dc020c2c04dfcaeef8de0145ed640 cc6e4f5111c718d64ffd3b1c32278711bd32e1ae3ef02a322ae3e68759fef5a1 14de58a3b524cccf6a063f46fa1c4d0e1aeb7aa966ea1e187ade0c92948eefe1 d0e76066738d7fcc7529ad6c2c2cec8bba03383f0ebe3379b57b6d618400215f d6b6c2ee3c04cd1b4e15862fe3e863ed2c5b03b5a9bd463ea767fe9bb7c7b6a8 7b6f62067b4b47c115f978c4ba9e46929a8737650d1ab30fba4efc7f91a94f5a 657925e56064c14694eed9eb8d58a90dbbf8a2912c3949644710eb2fa73c2179 f56666a9848d37bd428af010f6ab4b804d483e7519b13de4dfee0e5ddf78520f cd9096f1862391a796fb1e948a9ed70e11b1b99bdd892ca23f1c4ceccc7ccb8d 4fdcd480564e82a4298d4067776e6048f4bdf4b4ee60c0f6cee745332f18b41f 4c43952fd6288b0600809e8fd0f77ad587ad3556abee8f8f857831d9aba79b69 233c5b89472366d07607b643c74893ba5d38374c89daf3801663cc850ed54fb5 9ff8b9426069cb5d6e0098c6dcbcaecd50c73b3ebc0b38eb815bb3eb75ce13db d61905319804a846bed63216e00eaf000679a125ec3141f5cb8d11277f3ca548 a2d392d502e55fc23b8fb0f88e6d9d87ea9a8ee7c688456cd17c0b18ea2f36e8 1be0494adc055dd3fada0435f57fa3675d354dc69ab037fefc6b4aa2dda6c6fe 23e46d5a3bb7b824419db8ee13be4112a0c7056b531254afb2966f53a3d32bcc 2eca57b85df4cae165c66d0d5ee6fda5e321f76a936c4e3cd628d8c45badfa22 4315733e83eefee0e6424b9d217482fc784ed0f1c99760d863a7e7bd000c711a ffb5ce5c211e34d5e6dce0565d220a541b0c37d162eded01adac8cbb22544940 3f868126b799d5521b58c87c7ee2b4257e4b662c4d38d5c74de650943917faa7 90e1ee58f1608c5e9acc220c65549642379cc41906f760bad88fd2eeb39c3d0c ea0da07c43e6023ae0bceec0ed9d04763830f814fd0bdf95863662468183d820 470ab2e1bdf568316051109202731bc9ec56398ab079d325c6cb22ea424f8ecb 882188b36dadfb7c352bdd102f10c4d1a1774a0b0a9090f60ff92631a314bd67 5af9703f0f9668d77eb4eee928bc55b016cadef07d9ffdd60252f965daf6d4fb b50f14a4ca0b18b1467119a867ac4688a2d1aa8c46305102d0dbe0ab97f11a33 798880f9ccd2dd1e29b752fd5fcfc14016900604e7c0e24877d1e7dd0df54b8e 9737a2119164801205b6faa0bf0c00ac57de84ded82cf6f3159cdcb652b5b601 ac2f472e29f99fe4b646c5c84b5e982f335f3c5e912ccff03d3c10df130000b7 7cd4f5884528ca10772361c03f4b5b854c5baca43e01d4e907739d6d56168d20 53572e8a8b226cd4ec3489c94f2a9a332efab774209dea59a8fe00a70badcc3d ace8c332c9de99b2248acfbb534f3e681e015fb837be75b49885975bb1a00c6f 79c059f5606db71e63e8641a5e7785761d51ad4f00d669edd5e8bc9bc1393773 98b94e962c1015647eaf6ac674522810386a3a0983da4ca3f44dccfa4a949a3f 3b49d02f25af10d8cf2016f9b17af578545a2262e91edbadbf59374a6e59f2a2 cc1e7fd6f9d95f74f646e914ea30e634688a1dc859326954062a1714636f295a 7251931ec1acc413f861085b79036618444efd790f1a1db2dc6bd8e4a868a011 06bb688178425907e1448b9bc333bed693e853acb39e7b049ffab6bfe89e8540 136427123fd72e873ef4f90b4a9386d054a81e40c7edfcfd1d9b48cf2267bc31 0445f0dabcb0d6353020bcc2a654efebc46a3416948c73d918ae721a723d2932 88b102148c7faaa9178df0c9871c51c83de3979d56c3bb95041ea0eff995fad5 62cf63e6266920931d2653a16c43f223d801564a7a67cf2e5aec6dd0d49b0837 df431db53b653521c89e74b102d17dbe0e996e70a86277b107b7fbf5b90fd428 2b155e9d46cee86baa783b4805480361e808ce72c80e6245fab86b02da7dbd0e 9186758388c6c0a3c7e973e2a22244b622796b58b12b878988b4b1ca28be6cfe 8affd27850bd08f3303c2f24661b0a23f0b325c67905f723a6175efc861bb4b3 8fcea0d8da1c7adb0cf98e92b2f8ecb2da09e89e62f370a195c81d0432a91bd4 df91f5e43a04692e5a068676e4907c55eb36359d3352963228a46dcaf0deb20c 70c798ececb7a1b74f21dba829eebd537713a197f1ac56ab54380313a7d17ff4 9f5856d2fed2ab97d6633b80cc9dc7582632475edb53bf76fd1208f549d170bb b49c56f4002e0b1df9995b78088a5fa61a89e9a1868c02eede1297f40fb21be4 21572d94a398932512f05fc07ab653d3c246d2be2efa9d555657144cbee5ca8f 76b73444b3b19d10a6992e5283203e391065aa39ffbab054059602accf8d9d41 1c4431a3b0a22b2e1c3cefdbdb5c4f77dc19b360cec7a08339c49ae568cc872d f73f5ef6c5832a816e307ad60befaea8e85b6bfa84701b5f8db543ac76d3d64e 40dd83414dc6bb421db3e2cf41781e24368fd715116c6440c84cbae611164df7 ba654c3688815e2f64922ba28b6e78a2930e4483d44807df788ddedd563179a7 00073c65f69150e90bf5dabccc081f5e2a39f0b27ae51e8b877cea8adb9336c8 51f5f265711fedb618a6a77d5d54613de5809fd13ff13a4243fd175e7f6a15a2 e9cea95b34b425d6c85f53e7f4563b90e61cd3031729f3fc7511ceb201c1d82a a9afe8f1e2d6455129d6245594be7682dde4078233b9122114f17b4a9401ac8d 446af4f61c56400b55808619c4d8d4a25d47f44f41e0a39f65fa8ccd85916d44 407522cd97363cee09613df6fbc6d75454a33560d069cd8d2663ea3bd512a870 f51268c02c042d0e48208d212958b3f26b74020e553a5e2ed1329ca81a9a0473 33b02a9a71df6d22a07c46fafacc30e403d306002f1819fa74ce3bdaa81f9c1f a066d82942a611a9056246e842c97d89bf70986d5921cda49a881b2d2660e5c6 ba2d839d8c2dfe1208e8e310002090cc4032ad1e5c61037b545d265014e0982d c832d6b403706fd481f38df8463edbb189f508674fe0d1c362e16df270475851 6dc454ef7a6872e6bc1ef0a5ef5e04191917b4a256732027af73efd89c99f529 a4231e87622dff4f4a85d86c0e4bfda6f51f5950e3cd006fff425cd0d3cb1f30 17dd77305298c87e4282bb72e689215e4515088c1ca644cb1ea8801f945e7cd6 416260bef840169565023857a3b758808c341da7b30ecc35153902b4631a544c 5a659eedec582f15884f78cf37863c5d7daaa4f1eab8a6c848c8483c877c9ffa 4fb3042209e541ef435b4b461d2d414dfb7e699613b72fe05a94aa84ff8db414 7b7f4059f30fff0ab05e170e803c8f676aa41b59678345f8c4d4005f8ae0ff19 0be53cc78b2bb1aa4e9a666cf121057d9618fa77cff41ed69e1a6d55a1513fa4 8d59a763274b05cd58cacee3d80b333b95478a707e78d05b89b29c5e070cc709 55920e494840897c7b038c3e6915d7f141ebdd6a12e13c209989048f885fbada 3702329dd92b9fff6aac345a11b80221f90448d1e7bd57e866cb48f0b9645f86 5e83bfd25f1e60745d748f74a5780b04824e55ed66d700cefa4eb06e6140def5 2b90cb9beff9d488e61a3fcb121d54f6dfb2f1ebc19362b7c4a8395be8e331e9 60062811da5afd2629aa763d366892c56e3ded90c2d965794576b07396ce96fc 09d66286e294f9be3f3177037c9530e83601821789db4af2a45098087023d342 2806a95df9c1a513ecddeed76edfaa8f527731e8e3d37c6bec052abd1fa8325d 9a22ae08b25522b185c788f21ebeaf5af0b24be548888661f3204a54e3a505b2 b5839a485feebbb11b6c6c799b2ba77add936f5fd8fa94775c26db2ba163e6fa d53cc95401f216debea1ac4458d95c43b652273ad9981d1a25868eeb3467aae5 c114ad78af2d0918176a5a639871e6b218f836d176af6681a40c8f4f889d5930 4097e1cd3a1809aaa63f1fc988726809ba7e637e95ecba622acdfb69f6c0e0c3 7bbb5c906c5c1fe8b8f30e777e21689fcbe9bf5321055bbb5931140f9d003e40 23b191208941074e1c8eec1aa0d2f9cc04136fc07409e9fa2ab8ca129f96608b 8cc1c60b67bd2d8b01a2dc496be881649f72aa870738521ca84f1112c064692f a47a4c17d563ae273fd80c4f9f0202f340c2d3ca68fa9391cff5e7338cea0c99 2dd20d67aa0b2a0d322b5f41501f2898e80737cf578a66df0ce23f8ef9d26840 fa0fe398cd4c66795852f7d96ae7dec1433ece1d3698fef0a38a4e8d847d79f2 4ef9ae3a5fa0a453eec08dda9b49bbe7b3f07834303732010ef3f697d192eba5 1cdab588943ea736d35161d52eaaffcae6a283f1aba9ae0e1360cdef28a49776 beea463b353ac4232af18acd68d21a700e7f25433d9de684c6b575ec56bdb35e f4bbacc24d361a834a9803a57747bab59a699f181aef39b11cdb1da7e73a5618 bec23d849128099e319b1c3801a76301dbe02995b4e5b8c33b788999698194fe eafcb4b61ba09c4917db877cfd99ab11ff7a97427333fbfee5b9d2ee276b59c6 4a5d1129039ebb475545738d8ee1203fc7e1e6ebad959cb4159b8f374eca28ad 820da6a4f76019c23979e17f60ea5d686fab9e7f39f20bdee085465ed8573097 98d8231217936796e5fd7ea012232907d2ee1fedfaf82a27d9cc577bd6d6d1d8 3bf78940a2eeb65b3263389544ef84e439af0b3ef5e7e539998083939d89cea0 5d1cc1a91207f16a3aa2a37b68d38a20071930574c8e0fc4c92f12b95d470b9c 6a5d4ca59587f6deb33bd6a512afe11116559aa9a0dd341b06e1bb6b4f536abc 1eef303198d4b08efe9991e8d8f459f42de2799ac8a4490df106fce9ae62c472 d85cbefcc708d699b1da5f7390c1c7329b79f140e566ecb008b5d177e0e8db97 4f37190cbf68af0fcc506cb2a6ebd38fc35aad9ec220b2442463eed9a6fa473f 70d2a59cc1e8a0fd09efa16970e3de45a927785fd654aaa81fc8d85c1c6caf95 2388ff18587e81bd5fa6409b5ae320d45735cd72bc1646d2ee11cbd344ce0fab 4abb40c414a5f95c6fa61c9c0e970efa3968132f3d2965bfa641d62ac5e0825c 3ad4c1bb0d1dd109c1d00855f46d245d10808480e956f98cab8f353e1f639dbe ee57344f926854b4ac7717f79f882342f96a1261ca854704876f6b9b7194d371 44a4af152b368a6ba4329401b43c81e5e3 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMIB10 %!PS-AdobeFont-1.1: CMMIB10 1.100 %%CreationDate: 1996 Jul 23 07:54:00 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMIB10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMIB10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 97 /a put dup 105 /i put dup 111 /o put dup 112 /p put dup 114 /r put readonly def /FontBBox{-15 -250 1216 750}readonly def /UniqueXX 5087392 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d78041490e364babe4b85935b8c9c7758c2dbe253653c 892b1fc2c6a01103e0b8553f847bb38d8e4fcd563ebe51cb7d5f0877af29249d b61636618fb31eeb6cbb4ea4db687fd77ed1574da8c9f9d33b592ec0e38d798c 498a205cab252f7e2f1ddb82a38f331f917b8452af11b2b93e34c1ef55ee005c e8418dbcd0fb9d4d3fa8cb388ec91f16c8864913ad3fc8960d3c424650bdd12c 4db4049367faae97c660f00021f712e9eb6366b7ce74255ff62b72396ca43287 7c77fecdc53883427c90c1c3ff7bd46e472cea9bd72eef0e1be36abb3a50a7bf 6f5d460fa2fc81466014df9ee1e15365642c2de78706dfa740e035b4f28f290d 79b8ba16e5bca13fd9c3eb7eb54006ac2c1cf0bf363b3b69e923797a95b76515 073892c75cf733da7c07da8ed1a8d1ded902f784764c6df6fcd0fd28083d88c8 4c41aeaa6c2c2c1bb10219f4feaab75ee904f4314fc0b9de5e96054f2472ad86 42140315385400686cb07a0585d922ae4b1b25ae722edc6b4afe053560fc9ac9 20a1d2fb8d0921c52c9468d34dba9bed42e3a8308742d36186ed0a3b43f690a4 dca6bb2af8bd698fb303fd68df53edc5e01912cf007dd31ca7481ce844be1258 7002d4aafb204a8075fc6df86f22249d61dd840e7ef87362d72e35f4e8ece044 95ae41ee5c02d6aafaad37141935867a8770fbd9e9e894ea6cc5cf3770f35d98 09be73c56fb053ebb622cd4a3826bc95c192bc2248e1a30e641799994debd48f c5a0ca09be0b2831672271e6a087154d1541899171bc12c8ae00dac586f5c0dc a9c8f4bc7910fe68531c2adf25f29002b74151ec41cfe904b25dc975f97ca163 59d6ea5787a3b0529b3a7a7f8b5b8606cf11dfa1029106881e56be3b79089194 3f00ab0020388dc3ce3ce7fe3efd18cdf3ad919e01751813f98c179ff379582c 9fa2cb8231072175b8223c010d0c2be918887247152c6ae5df400a3106e0cc3e ec2757c5c7a3cdf50ac98846a2b1f5532608065a253ef019d57b9476a5aa7ac3 ee33798bfc0a39fdf0cb68acac05b15b4935bd1f5ea61b107a1539281ecc7296 9d9f3eb2eb1c896f7d7fdfb41b1051fa79fc4eb8b889989f3beebcdd2347665e e53ed443298b2c0ac652411177da6232342bb886376a0ea8ba21dd4c053039c2 187049de130c74d190e3f77b3158844bc1fe63b2fa0c5422339be666e83b573f f8036ad34ad5154d4b4f7cd5f923b1e9dbc2c355fdbdc969b264a012128e2942 8cc30947f642f139a4a77b9c1221e5a1752ac4fb57b583bfc0860920e1863c3a 6d57983ddb185845af4f2e12704ffc4cecef4133dcae33a88b94cd222f4f8373 f91393e606932077eed4e390526ed3262ae5db03c7c58e376302dc5adf8ce628 f109e4edefb893820e1da70a72006934a0dad058e0858e431899b5a3b0ca62b0 ee756f22a76a2f1f2819a14c4ef1d14382f04142fbb5e52ecf917b511b1f01a3 77fdea0971442b3964fda7d73f8465af1201f61cae7ef5b49fddf36accd46f80 328e64852ab63df8f9632725d49b5712a9e2bf1bb50ea8f99f0b83f011c69bde cb89790ae213be883e711d3cf536571b9ed4da16ce3828c19d2f34102185f83d ab23ef454780bc2d3769afe29d974a7008246bd300a3278db0867de7ebd7c3f9 0cca850019886eccc67bd18e87d6cd9e7f20be0e15776400e6485cface780455 df553c84f3d846191d72c9621b95ee1832cce6a292fb570a93b8c09e526b502e 29553e4e44fe50eb2bd851c00518c88b565f34fda21b270062ff1b302688e638 4db1c3cb7936ef7a4aee29fa7cea0bc594b0815d9c8b50969bf8332d9d5b1803 9e25aedaeb6eda9c216aae3892118e8f807f65c8073e0a53fc09457203397afe b383050a990fe81d841861c590886d664391ac4208fad1b0e1b1f6495e492cd3 906fc92d9fbdad3a9fbdd45daabfe7b8d5972b4947edd4d2c32d36ee941e338b c1e760fe1236f4d5ceaac3866faa8b85d2d1d081c2614ee943b96edf1da80594 31d65ddc2da0ff0451a6b871e9e4eca11f54b7737af985182e671cc67783f22c a82cda71b28513c1caa216de3c160165f2f9aa5cd444c5e3fae16423bacfb5d0 d852312131be020a0e859306b72a37ab13d0b5ac8f85d664589aa675c522735d de43be8e2b7f8456d4d35a97d67cdf2917a672e09c0c4fd0501608f0aab911cd 7cfb9509ef776de4ffffcc56e6fd66ac8ed7bf036ff474d428de3105bfe4bec9 bcc662a65ecb15dddf349dfb0dd5b51eb66b20a67b94cb9db0f28092dfa63a0c bec688d3a52875d850208e75bca1ea9114cc7ef469e3a6c4571bf9fe4b76c89c 6c78a7133889f9ff056b5b6c7435e69cb22b12f4a9e3ce8e72e194a0b97810ef 7dc51a88066930fa0590b4adb369ebb5037664ddc979856809ff20c20bc14ed5 841ad59134 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 39158280 55380996 1200 8000 8000 (ARNDDFU1.dvi) @start /Fa 138[738 1[524 520 1[738 664 5[369 2[406 102[{}7 1328.35 /CMSL10 rf /Fb 252[1033 3[{}1 1328.35 /MSAM10 rf /Fc 165[620 5[620 17[620 66[{}3 929.846 /MSBM7 rf /Fd 138[562 1[445 446 6[504 391 354 4[491 431 2[397 17[714 2[756 2[738 12[304 31[541 4[564 4[494 1[408 15[{}17 664.176 /CMMI5 rf /Fe 207[293 41[720 2[489 2[720{}4 664.176 /CMSY5 rf /Ff 178[1128 77[{}1 797.011 /EUFB7 rf /Fg 145[544 2[544 315 2[544 544 6[716 17[1248 10[701 2[830 12[716 1059 306 14[1059 8[830 2[830 830 13[830 2[544 830 315 830{}21 929.846 /CMSY7 rf /Fh 151[818 13[944 6[1300 4[1305 1646 4[1130 1[959 70[{}7 1328.35 /EUFB10 rf /Fi 144[886 886 17[1476 1476 738 1[1919 5[627 1[1402 6[738 7[1162 1162 1[1181 1[1181 1[1181 1[1181 6[1162 1162 14[1052 1052 12[978 978 793 793 3[443 12[{}24 1328.35 /CMEX10 rf /Fj 205[452 452 452 4[683 1[360 360 40[{}6 664.176 /CMR5 rf /Fk 129[544 3[508 1[602 2[628 401 501 493 487 548 524 657 943 336 565 440 376 622 518 518 1[553 475 467 576 445 8[999 1[724 627 1[800 837 2[841 1013 729 893 1[471 870 1[674 1[869 762 803 799 567 1[830 544 830 315 24[502 669 4[490 616 551 621 483 533 646 4[547 483 443 483 558 1[691 11[{}60 929.846 /CMMI7 rf /Fl 165[886 5[886 738 959 1033 2[959 11[886 2[1033 20[1033 1[1033 40[{}10 1328.35 /MSBM10 rf /Fm 141[1107 1[1107 1[664 2[664 369 517 517 664 664 6[886 886 1[886 886 7[804 3[1058 1090 1595 916 4[790 4[872 1061 7[738 738 1[0 3[886 1328 15[1328 5[1033 1033 1[1033 2[1033 1033 1033 1033 1033 2[664 7[1033 3[1033 369 1033{}40 1328.35 /CMSY10 rf /Fn 172[684 5[1344 11[903 65[{}3 1062.68 /CMSY8 rf /Fo 128[679 4[543 645 616 883 611 713 441 543 560 611 679 679 747 1086 339 611 407 407 679 611 407 611 679 611 611 679 9[1327 987 987 951 747 969 2[1018 1[1191 833 2[512 987 1028 867 901 1003 951 935 987 5[407 407 6[679 1[679 2[407 475 407 2[543 543 25[1171 781 747 815 11[{}58 1328.35 /CMTI10 rf /Fp 129[664 3[618 651 759 3[480 623 599 593 668 644 797 1166 396 692 547 458 765 634 650 1[691 575 570 702 553 5[907 771 1101 1255 775 907 776 815 1009 1050 853 1013 1067 1289 904 1128 737 584 1104 1044 854 1[1100 949 1008 996 705 1[1033 664 1033 369 369 20[687 2[619 827 865 1[791 1[581 759 687 757 581 656 800 775 765 1[624 660 581 539 590 688 751 850 11[{}77 1328.35 /CMMI10 rf /Fq 128[664 664 3[590 701 701 959 701 738 517 524 520 701 738 664 738 1107 369 701 406 369 738 664 406 590 738 590 738 664 1[369 664 369 664 369 812 1[996 1365 996 996 959 738 978 1[904 1033 996 1218 830 1033 683 480 996 1042 867 904 1015 959 941 996 1[627 1[1033 1[369 369 664 664 664 664 664 664 664 664 664 664 1[369 443 369 1033 1[517 517 369 1033 3[664 369 10[664 664 1[664 4[1107 738 738 775 959 1033 959 1[959 996 1[922 1[1107 830{}96 1328.35 /CMR10 rf /Fr 134[806 806 1103 806 849 594 603 629 1[849 764 849 1273 424 806 1[424 849 764 467 700 849 679 849 743 10[1155 1[1063 849 1146 1[1044 2[1450 919 2[579 1196 1201 3[1103 1087 1155 7[764 764 764 764 764 764 764 764 764 764 1[424 509 424 2[594 594 27[849 12[{}52 1328.35 /CMBX10 rf /Fs 141[481 1[583 583 5[350 7[583 97[{}5 1062.68 /CMTI8 rf /Ft 134[596 596 815 1[627 439 445 440 1[627 565 627 941 314 2[314 627 565 345 502 627 502 627 565 8[846 2[846 815 11[846 2[768 3[846 7[565 565 565 565 565 565 565 565 565 565 1[314 376 33[659 11[{}40 1062.68 /CMR8 rf /Fu 139[653 509 663 14[653 639 673 31[912 18[351 46[{}8 1062.68 /CMCSC10 rf /Fv 129[529 5[558 8[529 1[873 8[587 2[529 1[301 529 301 1[301 29[815 4[529 529 2[529 1[529 529 529 4[815 1[415 415 17[529 529 12[758 1[758 2[727 2[657{}26 929.846 /CMR7 rf /Fw 138[745 3[745 1[731 745 3[380 339 745 731 1[679 745 649 9[974 7[1055 2[1055 3[690 1[1055 974 1[852 1[934 1[974 974 17[406 487 6[1103 38[{}24 1328.35 /CMSSBX10 rf /Fx 141[1011 1[1149 1118 5[774 7[1211 97[{}5 1912.83 /CMMIB10 rf /Fy 134[1161 1161 2[1222 855 868 906 2[1100 1222 1833 611 2[611 2[672 1008 1[978 1222 1069 11[1692 1[1222 3[1652 6[1722 2[1445 1687 56[1283 11[{}23 1912.83 /CMBX10 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 8000dpi TeXDict begin %%BeginPaperSize: a4 a4 %%EndPaperSize %%EndSetup %%Page: 1 1 1 0 bop 11588 5427 a Fy(Existence)734 b(of)f(Di\013usion)h(Orbits)5093 7808 y(in)f Fx(a)g(priori)i Fy(Unstable)g(Hamiltonian)f(Systems)15045 11607 y Fw(Chong-Qing)488 b(CHENG)27646 11125 y Fv(1)28242 11607 y Fw(&)g(Jun)f(Y)-122 b(AN)35381 11125 y Fv(2)4290 15167 y Fu(Abstra)-24 b(ct.)617 b Ft(Under)500 b(op)31 b(en)500 b(and)g(dense)h(conditions)e(w)-31 b(e)500 b(sho)-31 b(w)500 b(that)g(Arnold)g(di\013usion)f(orbits)4290 16495 y(exist)409 b(in)f Fs(a)438 b(priori)408 b Ft(unstable)h(and)g(time-p) 31 b(erio)g(dic)408 b(Hamiltonian)f(systems)i(with)g(t)-31 b(w)g(o)408 b(degrees)h(of)4290 17824 y(freedom.)1102 23584 y Fr(1,)510 b(In)-42 b(tro)42 b(duction)508 b(and)g(Results)2696 26033 y Fq(By)577 b(the)g(KAM)g(\(Kolmogoro)-37 b(v,)612 b(Arnold)577 b(and)h(Moser\))f(theory)g(w)-37 b(e)578 b(kno)-37 b(w)578 b(that)e(there)h(are)1102 27627 y(man)-37 b(y)545 b(in)-37 b(v)-74 b(arian)-37 b(t)544 b(tori)g(in)g(nearly)h(in) -37 b(tegrable)544 b(Hamiltonian)h(systems)e(with)h(arbitrary)h Fp(n)f Fq(de-)1102 29221 y(grees)506 b(of)e(freedom.)776 b(These)505 b(tori)f(are)h(of)g Fp(n)f Fq(dimension)h(and)f(o)37 b(ccup)-37 b(y)505 b(a)g(nearly)g(full)f(Leb)37 b(esgue)1102 30815 y(measure)495 b(set)f(in)g(the)f(phase)h(space.)744 b(As)494 b(an)g(imp)37 b(ortan)-37 b(t)494 b(consequence,)508 b(all)494 b(orbits)g(are)g(stable)1102 32409 y(in)579 b(autonomous)g(system)g(with)f(t)-37 b(w)g(o)579 b(degrees)i(of)d (freedom,)614 b(or)579 b(time-p)37 b(erio)g(dic)580 b(system)e(with) 1102 34003 y(one)500 b(degree)h(of)e(freedom,)514 b(in)499 b(the)g(sense)g(that)g(the)g(actions)g(do)g(not)g(c)-37 b(hange)500 b(m)-37 b(uc)g(h)500 b(along)f(the)1102 35597 y(orbits.)571 b(Ho)-37 b(w)g(ev)g(er,)399 b(the)384 b(KAM)h(theory)g (dose)g(not)g(guaran)-37 b(tee)385 b(suc)-37 b(h)385 b(stabilit)-37 b(y)384 b(when)i(the)e(system)1102 37191 y(has)430 b(three)g(or)g(more)g(degrees)h(of)f(freedom)h(for)e(the)h (autonomous)g(case)g(or)g(when)g(it)f(has)h(t)-37 b(w)g(o)430 b(or)1102 38785 y(more)487 b(degrees)h(of)f(freedom)g(for)g(the)f (time-p)37 b(erio)g(dic)487 b(case,)498 b(simply)486 b(b)37 b(ecause)488 b(the)e(KAM)g(torus)1102 40379 y(can)433 b(not)g(separate)g(the)f(phase)h(space)h(\(or)e(in)-37 b(tegral)433 b(manifold\))g(in)-37 b(to)432 b(t)-37 b(w)g(o)433 b(disconnected)g(parts.)2696 42828 y(In)478 b(his)g(celebrated)i(pap)37 b(er)478 b([Ar],)488 b(Arnold)478 b(constructed)h(an)f(example)h(of)f (nearly)h(in)-37 b(tegrable)1102 44422 y(Hamiltonian)563 b(system,)593 b(where)563 b(some)h(orbits)e(are)h(unstable.)950 b(His)563 b(example)g(is)g(a)f(time)h(p)37 b(eri-)1102 46016 y(o)g(dic)590 b(system)f(with)g(t)-37 b(w)g(o)589 b(degrees)i(of)e(freedom.)1030 b(In)589 b(Arnold's)h(example)g(the)f(p) 37 b(erturbations)1102 47610 y(are)436 b(c)-37 b(hosen)436 b(so)f(sp)37 b(eci\014cally)436 b(that)e(all)i(h)-37 b(yp)37 b(erb)g(olic)436 b(in)-37 b(v)-74 b(arian)-37 b(t)435 b(tori)g(preserv)-37 b(e)437 b(in)e(the)g(p)37 b(erturb)g(ed)1102 49204 y(system.)831 b(Hence)525 b(one)e(can)h(use)f (so)g(called)h(P)-37 b(oincar)g(\023)-627 b(e-Melnik)-37 b(o)g(v)525 b(metho)37 b(d)522 b(to)h(construct)g(tran-)1102 50798 y(sition)557 b(c)-37 b(hain)557 b(along)h(whic)-37 b(h)557 b(the)g(action)g(has)g(substan)-37 b(tial)556 b(v)-74 b(ariation.)933 b(Ho)-37 b(w)g(ev)g(er,)588 b(in)557 b(generic)1102 52392 y(case)404 b(the)f(p)37 b(erturb)g(ed)403 b(systems)g(do)g(not)g(p)37 b(ossess)403 b(suc)-37 b(h)403 b(a)g(go)37 b(o)g(d)403 b(prop)37 b(ert)-37 b(y)-111 b(,)412 b(some)403 b(resonan)-37 b(t)404 b(gaps)1102 53986 y(b)37 b(et)-37 b(w)g(een)482 b(in)-37 b(v)-74 b(arian)-37 b(t)482 b(tori)f(break)h(up)e(the)h(transition)g(c)-37 b(hain,)491 b(th)-37 b(us)480 b(it)h(seems)g(unclear)h(whether)1102 55580 y(one)597 b(can)h(apply)e(Arnold's)i(metho)37 b(d)596 b(to)g(\014nd)h(di\013usion)f(orbits.)1052 b(Despite)597 b(of)f(this)g(tec)-37 b(hnical)1102 57174 y(di\016cult)g(y)-111 b(,)586 b(Arnold)557 b(ask)-37 b(ed)558 b(whether)g(there)f(is)g(suc) -37 b(h)557 b(a)g(phenomenon)h(for)f(a)g(\\t)-37 b(ypical")557 b(small)1102 58768 y(p)37 b(erturbation.)844 b(After)528 b(near)g(four)f(decades)i(of)e(study)g(some)g(remark)-74 b(able)529 b(generalizations)g(of)1102 60362 y(Arnold's)464 b(result)f(ha)-37 b(v)g(e)463 b(b)37 b(een)463 b(announced)h (\([X1],[DLS],[Ma5]\).)650 b(A)463 b(few)g(y)-37 b(ears)464 b(ago,)k(Xia)463 b([X1])1102 61956 y(announced)378 b(that)e(Arnol'd)i (di\013usion)e(exists)h(in)g(generic)h Fo(a)415 b(priori)377 b Fq(unstable)g(systems,)390 b(recen)-37 b(tly)1102 63550 y(Mather)413 b(announced)g(\([Ma5]\))e(that,)418 b(under)413 b(so-called)g(cusp)f(residual)h(condition,)419 b(Arnold)413 b(dif-)1102 65144 y(fusion)448 b(exists)f(in)h Fo(a)479 b(priori)448 b Fq(stable)g(systems)f(with)g(t)-37 b(w)g(o)448 b(degrees)h(of)f(freedom)g(in)g(time-p)37 b(erio)g(dic)1102 66738 y(case,)444 b(or)f(with)g(three)g(degrees)i(of)e(freedom)g(in)g (autonomous)g(case.)2696 69186 y(In)579 b(this)f(pap)37 b(er,)613 b(w)-37 b(e)580 b(study)e(generic)i(p)37 b(erturbations)579 b(of)g Fo(a)600 b(priori)579 b Fq(unstable)g(Hamiltonian)1102 70780 y(systems)319 b(whic)-37 b(h)319 b(ha)-37 b(v)g(e)319 b(t)-37 b(w)g(o)319 b(degrees)h(of)e(freedom)i(and)e(are)h(time-p)37 b(erio)g(dic,)344 b(and)319 b(giv)-37 b(e)319 b(a)f(complete)1102 72374 y(pro)37 b(of)417 b(of)g(the)g(existence)g(of)g(di\013usion)g (orbits.)581 b(The)418 b(approac)-37 b(h)417 b(of)g(our)g(pro)37 b(of)417 b(is)g(di\013eren)-37 b(t)417 b(from)38973 74965 y Ft(T)-31 b(yp)31 b(eset)377 b(b)-31 b(y)376 b Fn(A)45559 75194 y(M)46761 74965 y(S)p Ft(-T)48448 75194 y(E)49076 74965 y(X)25230 76559 y(1)p eop %%Page: 2 2 2 1 bop 1102 2239 a Ft(2)1102 5427 y Fq(the)484 b(approac)-37 b(hes)486 b(prop)37 b(osed)485 b(b)-37 b(y)484 b(Mather)h(and)f(b)-37 b(y)484 b(Xia)h(\(cf.)714 b([Ma5])484 b(and)g([X2]\).)715 b(The)485 b(starting)1102 7021 y(p)37 b(oin)-37 b(t)549 b(of)h(our)g(pro)37 b(of)550 b(is)f(based)h(on)f(the)h(previous)g(w)-37 b(ork)550 b(of)g(Mather)f(\([Ma3],)576 b([Ma4]\).)909 b(With)1102 8615 y(his)393 b(profound)g(insigh)-37 b(t,)402 b(Mather)393 b(op)37 b(ened)393 b(a)g(w)-37 b(a)g(y)393 b(to)g(study)f(Hamiltonian)h(dynamics)g(in)f(higher)1102 10209 y(dimensions.)922 b(In)552 b([Ma3])h(Mather)g(established)h(the)f (v)-74 b(ariational)553 b(set-up)g(of)g(time-dep)37 b(enden)-37 b(t)1102 11803 y(p)37 b(ositiv)-37 b(e)431 b(de\014nite)g(Lagrangian)h (systems)f(and)g(sho)-37 b(w)g(ed)432 b(the)f(existence)g(of)g(minimal) g(measures.)1102 13397 y(By)459 b(exploiting)f(the)g(prop)37 b(erties)459 b(of)g(barrier)g(functions)f(in)h([Ma4],)j(he)c(in)-37 b(tro)37 b(duced)459 b(the)f(idea)h(of)1102 14991 y Fp(C)95 b Fq(-equiv)-74 b(alence)530 b(and)d(p)37 b(oin)-37 b(ted)527 b(out)g(a)h(p)37 b(ossible)527 b(w)-37 b(a)g(y)528 b(to)f(construct)h (connecting)g(orbits.)844 b(The)1102 16585 y(di\016cult)-37 b(y)532 b(to)f(apply)g(this)g(metho)37 b(d)531 b(to)g(in)-37 b(teresting)531 b(problem)h(in)f(higher)h(dimensions)g(is)f(that)1102 18179 y(w)-37 b(e)472 b(do)f(not)f(kno)-37 b(w)472 b(the)e(structures)h (of)g(related)h Fp(c)p Fq(-minimal)e(orbit)h(sets.)674 b(In)470 b(this)g(pap)37 b(er)472 b(w)-37 b(e)471 b(are)1102 19773 y(succeeded)375 b(in)d(getting)h(su\016cien)-37 b(t)372 b(information)h(of)g(the)f(top)37 b(ological)373 b(structure)g(of)g(the)f(relev)-74 b(an)-37 b(t)1102 21367 y(Ma)37 b(~)-701 b(n)-37 b(\023)-627 b(e)429 b(sets)g(and)g(in)f (pro)-37 b(viding)430 b(the)f(pro)37 b(of)429 b(of)f(a)h(theorem)h(of)f (connecting)g Fp(C)95 b Fq(-equiv)-74 b(alen)-37 b(t)430 b(Ma)37 b(~)-701 b(n)-37 b(\023)-627 b(e)1102 22961 y(sets)611 b(form)-37 b(ulated)611 b(b)-37 b(y)611 b(Mather)g(in)f([Ma4].)1095 b(Consequen)-37 b(tly)-111 b(,)654 b(w)-37 b(e)611 b(are)h(able)f(to)f (construct)h(the)1102 24555 y(di\013usion)640 b(orbits)g(crossing)h (the)e(gaps.)1182 b(Ho)-37 b(w)g(ev)g(er,)691 b(it)640 b(app)37 b(ears)640 b(unclear)h(whether)g(suc)-37 b(h)640 b Fp(C)95 b Fq(-)1102 26149 y(equiv)-74 b(alence)583 b(can)f(b)37 b(e)581 b(established)g(at)f(the)h(place)h(where)g(uncoun) -37 b(tably)581 b(man)-37 b(y)581 b(whisk)-37 b(er)582 b(tori)1102 27743 y(cluster)508 b(together.)783 b(F)-111 b(ortunately)g(,)524 b(this)506 b(is)h(the)g(place)g(where)i(there)e (is)g(no)g(big)g(gap.)783 b(Arnold's)1102 29337 y(mec)-37 b(hanism)439 b(can)f(b)37 b(e)438 b(used)f(here)i(b)37 b(ecause)439 b(a)e(transition)h(c)-37 b(hain)438 b(of)f(whisk)-37 b(er)439 b(tori)e(clearly)i(exists)1102 30931 y(in)485 b(this)f(case.)716 b(Crucially)486 b(relying)f(on)g(suc)-37 b(h)485 b(geometric)h(structure,)495 b(w)-37 b(e)486 b(are)f(able)g(to)f(establish)1102 32525 y(lo)37 b(cal)443 b(v)-74 b(ariational)443 b(principle)g(\(cf.)590 b([Bs],)443 b([BCV]\),)f(the)g(lo)37 b(cal)443 b(minim)-37 b(um)442 b(corresp)37 b(onds)444 b(to)e(some)1102 34119 y(di\013usion)500 b(orbits)f(crossing)i(these)f(whisk)-37 b(er)500 b(tori.)761 b(It)499 b(is)h(the)f(v)-74 b(ariational)500 b(v)-37 b(ersion)501 b(of)f(Arnold's)1102 35713 y(mec)-37 b(hanism.)928 b(Our)555 b(another)g(w)-37 b(ork)556 b(is)f(to)g(sho)-37 b(w)555 b(that)f(w)-37 b(e)556 b(can)f(join)g(the)g(orbits)g (constructed)1102 37307 y(b)-37 b(y)524 b Fp(C)95 b Fq(-equiv)-74 b(alence)525 b(smo)37 b(othly)523 b(with)g(the)g(orbits)g(whic)-37 b(h)524 b(realize)h(the)e(minim)-37 b(um)523 b(of)h(the)f(lo)37 b(cal)1102 38901 y(v)-74 b(ariational)400 b(principle.)576 b(In)399 b(this)f(w)-37 b(a)g(y)399 b(w)-37 b(e)400 b(do)f(\014nd)f (some)h(di\013usion)g(orbits)g(in)f(generic)i(systems.)2696 41445 y(Giv)-37 b(en)444 b(a)f(Hamiltonian)g(function)g Fp(H)108 b Fq(\()p Fp(p;)221 b(q)48 b(;)221 b(t)p Fq(\))442 b(the)h(Hamiltonian)g(equation)g(has)g(the)g(form:)18698 45478 y(_)-615 b Fp(q)416 b Fq(=)21023 44579 y Fp(@)74 b(H)p 21023 45172 1992 54 v 21295 46389 a(@)g(p)23173 45478 y(;)3139 b Fq(_)-630 b Fp(p)370 b Fq(=)f Fm(\000)30052 44579 y Fp(@)74 b(H)p 30052 45172 V 30338 46389 a(@)g(q)32203 45478 y(:)14619 b Fq(\(1.1\))1102 49450 y(The)444 b(Hamiltonian)f (function)g(studied)f(here)i(has)f(the)g(follo)-37 b(wing)443 b(form:)13951 52790 y Fp(H)108 b Fq(\()p Fp(p;)221 b(q)48 b(;)221 b(t)p Fq(\))369 b(=)g Fp(f)143 b Fq(\()p Fp(p)22915 52989 y Fv(1)23511 52790 y Fq(\))295 b(+)g Fp(g)48 b Fq(\()p Fp(p)27518 52989 y Fv(2)28113 52790 y Fp(;)221 b(q)29296 52989 y Fv(2)29893 52790 y Fq(\))294 b(+)i Fp(P)184 b Fq(\()p Fp(p;)221 b(q)48 b(;)221 b(t)p Fq(\))10118 b(\(1.2\))1102 56131 y(where)498 b Fp(p)458 b Fq(=)g(\()p Fp(p)8799 56330 y Fv(1)9394 56131 y Fp(;)221 b(p)10652 56330 y Fv(2)11249 56131 y Fq(\))457 b Fm(2)g Fl(R)14525 55649 y Fv(2)15122 56131 y Fq(,)509 b Fp(q)c Fq(=)458 b(\()p Fp(q)19699 56330 y Fv(1)20295 56131 y Fp(;)221 b(q)21478 56330 y Fv(2)22074 56131 y Fq(\))457 b Fm(2)g Fl(T)25277 55649 y Fv(2)25873 56131 y Fq(,)509 b Fp(H)566 b Fm(2)457 b Fp(C)30808 55649 y Fk(r)31893 56131 y Fq(\()p Fp(r)494 b Fm(\025)458 b Fq(3\),)509 b Fp(P)681 b Fq(is)495 b(a)i(time-1-p)37 b(erio)g(dic)1102 57725 y(small)443 b(p)37 b(erturbation.)591 b(W)-111 b(e)444 b(assume)f(it)f(satis\014es) h(follo)-37 b(wing)443 b(conditions:)2696 60269 y(1,)546 b Fp(f)493 b Fq(+)350 b Fp(g)573 b Fq(is)525 b(a)g(con)-37 b(v)g(ex)526 b(function)f(in)g Fp(p)g Fq(i.e.)837 b(the)525 b(Hessian)h(matrix)f Fp(@)38104 60468 y Fk(pp)39265 60269 y Fq(\()p Fp(f)493 b Fq(+)350 b Fp(g)48 b Fq(\))524 b(is)h(p)37 b(ositiv)-37 b(e)1102 61863 y(de\014nite,)634 b(\014nite)596 b(ev)-37 b(erywhere)598 b(and)d(has)h(sup)37 b(erlinear)597 b(gro)-37 b(wth)596 b(in)f Fp(p)p Fq(,)634 b(\()p Fp(f)540 b Fq(+)397 b Fp(g)48 b Fq(\))p Fp(=)p Fm(k)p Fp(p)p Fm(k)623 b(!)h(1)596 b Fq(as)1102 63457 y Fm(k)p Fp(p)p Fm(k)370 b(!)f(1)p Fq(;)2696 66001 y(2,)617 b(it)582 b(is)g Fo(a)603 b(priori)583 b Fq(unstable)f(in)g(the)g(sense)h(that)e Fp(g)629 b Fq(has)582 b(non-degenerate)i(saddle)e(critical)1102 67595 y(p)37 b(oin)-37 b(t,)655 b(i.e.)1098 b Fp(@)8652 67794 y Fk(p)9200 67927 y Fj(2)9719 67794 y Fk(q)10206 67927 y Fj(2)10790 67595 y Fp(g)11472 67113 y Fv(2)12476 67595 y Fm(\000)408 b Fp(@)14622 67794 y Fk(p)15170 67927 y Fj(2)15688 67794 y Fk(p)16236 67927 y Fj(2)16821 67595 y Fp(g)48 b(@)18208 67794 y Fk(q)18695 67927 y Fj(2)19212 67794 y Fk(q)19699 67927 y Fj(2)20284 67595 y Fp(g)699 b(>)651 b Fq(0)612 b(at)g(\()p Fp(p)27555 67794 y Fv(2)28151 67595 y Fp(;)221 b(q)29334 67794 y Fv(2)29930 67595 y Fq(\).)1098 b(Without)612 b(loss)g(of)h(generalit)-37 b(y)-111 b(,)655 b(w)-37 b(e)1102 69189 y(assume)509 b(\()p Fp(p)6943 69388 y Fv(2)7539 69189 y Fp(;)221 b(q)8722 69388 y Fv(2)9318 69189 y Fq(\))477 b(=)h(0.)786 b(As)508 b(the)g(Hamiltonian)g(\015o)-37 b(w)508 b Fp(\036)29349 68707 y Fk(t)29349 69517 y(g)30476 69189 y Fq(induced)g(b)-37 b(y)508 b Fp(g)556 b Fq(is)507 b(in)-37 b(tegrable,)526 b(it)507 b(has)1102 70827 y(t)-37 b(w)g(o)443 b(homo)37 b(clinic)444 b(lo)37 b(ops)443 b(on)g(the)g(cylinder)h(\()p Fp(p)24106 71026 y Fv(2)24701 70827 y Fp(;)221 b(q)25884 71026 y Fv(2)26481 70827 y Fq(\))368 b Fm(2)h Fl(R)295 b Fm(\002)h Fl(T)p Fq(.)2696 73371 y(As)404 b Fp(f)360 b Fq(+)217 b Fp(g)450 b Fq(is)404 b(an)g(in)-37 b(tegrable)404 b(system)f(and)h Fp(H)512 b Fq(is)403 b(its)g(small)h(p)37 b(erturbation,)411 b(ev)-37 b(ery)405 b(solution)e(of)1102 74965 y Fp(H)551 b Fq(is)443 b(w)-37 b(ell)444 b(de\014ned)f(for)g Fp(t)369 b Fm(2)f Fl(R)p Fq(.)p eop %%Page: 3 3 3 2 bop 49357 2239 a Ft(3)2696 5427 y Fq(Let)327 b Fm(B)5832 5626 y Fk(\017;K)7941 5427 y Fq(denote)f(a)h(ball)f(in)g(the)g (function)g(space)i Fp(C)28729 4945 y Fk(r)29317 5427 y Fq(\()p Fm(f)p Fq(\()p Fp(p;)221 b(q)48 b Fq(\))368 b Fm(2)h Fl(T)35940 4945 y Fv(2)36597 5427 y Fm(\002)62 b Fl(R)38651 4945 y Fv(2)39617 5427 y Fq(:)369 b Fm(k)p Fp(p)p Fm(k)g(\024)h Fp(K)95 b Fm(g)369 b(!)g Fl(R)p Fq(\),)1102 7021 y(cen)-37 b(tered)550 b(at)d(the)h(origin)h(with)f (radius)g(of)g Fp(\017)p Fq(.)906 b(No)-37 b(w)549 b(w)-37 b(e)548 b(can)h(state)e(the)h(theorem)h(whic)-37 b(h)549 b(w)-37 b(as)1102 8615 y(form)g(ulated)444 b(b)-37 b(y)443 b(Arnold)g(in)g([Ar]:)1102 10876 y Fr(Theorem)509 b(1.1.)764 b Fo(L)-68 b(et)477 b Fp(A)369 b(<)h(B)542 b Fo(b)-68 b(e)475 b(two)h(arbitr)-68 b(arily)477 b(given)f(numb)-68 b(ers)476 b(and)f(assume)h Fp(H)583 b Fo(satis\014es)1102 12470 y(the)507 b(ab)-68 b(ove)506 b(two)g(c)-68 b(onditions.)706 b(Ther)-68 b(e)506 b(exist)h(a)f(smal)68 b(l)508 b(numb)-68 b(er)506 b Fp(\017)426 b(>)f Fq(0)p Fo(,)515 b(a)506 b(lar)-68 b(ge)507 b(numb)-68 b(er)506 b Fp(K)521 b(>)425 b Fq(0)1102 14064 y Fo(and)456 b(an)f(op)-68 b(en)455 b(and)g(dense)h(set)g(in)g Fp(B)19523 14263 y Fk(\017;K)21760 14064 y Fo(such)g(that)g(for)g(e)-68 b(ach)455 b Fp(P)553 b Fm(2)369 b Fp(B)36196 14263 y Fk(\017;K)38433 14064 y Fo(ther)-68 b(e)456 b(exists)h(an)e(orbit)1102 15658 y(of)583 b(the)f(Hamiltonian)h(\015ow)f(which)g(c)-68 b(onne)g(cts)582 b(the)g(r)-68 b(e)g(gion)583 b(with)f Fp(p)35102 15857 y Fv(1)36260 15658 y Fp(<)563 b(A)582 b Fo(to)g(the)h(r)-68 b(e)g(gion)582 b(with)1102 17253 y Fp(p)1770 17452 y Fv(1)2736 17253 y Fp(>)369 b(B)67 b Fo(.)2696 19514 y Fq(W)-111 b(e)348 b(shall)f(use)f(v)-74 b(ariational)348 b(argumen)-37 b(t)347 b(to)f(complete)i(the)e(pro)37 b(of.)559 b(In)346 b(the)g(section)h(2,)366 b(b)-37 b(y)347 b(using)1102 21108 y(Legender)311 b(transformation)d(w)-37 b(e)310 b(follo)-37 b(w)309 b(Mather's)g(w)-37 b(ork)309 b([Ma4])g(and)f(put)h(this)f(problem)h(in)-37 b(to)308 b(the)1102 22702 y(Lagrangian)496 b(formalism.)745 b(The)495 b(di\013usion)g(orbits)f(are)h(found)f(b)-37 b(y)495 b(searc)-37 b(hing)495 b(for)g(the)f(minimal)1102 24296 y(action)569 b(of)f(the)g(Lagrangian.)968 b(Some)568 b(prop)37 b(erties)569 b(suc)-37 b(h)569 b(as)f(upp)37 b(er)568 b(semi-con)-37 b(tin)g(uit)g(y)569 b(of)f(some)1102 25890 y(set-v)-74 b(alued)481 b(functions)f(are)h(also)g(pro)-37 b(v)g(ed)481 b(in)f(this)g(section.)703 b(In)479 b(the)h(section)h(3,) 490 b(w)-37 b(e)481 b(in)-37 b(v)g(estigate)1102 27484 y(the)558 b(top)37 b(ological)558 b(structure)f(of)h(some)f(relev)-74 b(an)-37 b(t)559 b(Ma)37 b(~)-701 b(n)-37 b(\023)-627 b(e)557 b(sets.)934 b(The)558 b(section)g(4)f(is)h(dev)-37 b(oted)557 b(to)1102 29078 y(the)570 b(study)g(of)g(the)g(barrier)i (function)e(when)g(the)g(Aubry)h(set)f(con)-37 b(tains)571 b(a)f(co)37 b(dimension)571 b(one)1102 30672 y(torus.)1090 b(In)609 b(the)g(section)g(5,)651 b(b)-37 b(y)609 b(making)h(use)f(of)g (the)h(semi-con)-37 b(tin)g(uit)g(y)609 b(prop)37 b(ert)-37 b(y)610 b(sho)-37 b(wn)609 b(in)1102 32266 y(the)j(section)f(2)h(w)-37 b(e)612 b(obtain)f(the)g(pro)37 b(of)612 b(of)f(a)g(theorem)h(of)g (connecting)g Fp(C)95 b Fq(-equiv)-74 b(alen)-37 b(t)612 b(Ma)37 b(~)-701 b(n)-37 b(\023)-627 b(e)1102 33860 y(sets,)439 b(form)-37 b(ulated)438 b(b)-37 b(y)438 b(Mather)g(in)f([Ma4].)589 b(Based)438 b(on)g(the)f(understanding)h(of)g(the)g(top)37 b(ological)1102 35454 y(structure)659 b(of)e(the)h(relev)-74 b(an)-37 b(t)659 b(Ma)37 b(~)-701 b(n)-37 b(\023)-627 b(e)658 b(sets)g(sho)-37 b(wn)658 b(in)g(the)g(section)g(3,)712 b(w)-37 b(e)658 b(establish)g(the)g Fp(C)95 b Fq(-)1102 37048 y(equiv)-74 b(alence)426 b(among)e(those)g(relev)-74 b(an)-37 b(t)425 b(Ma)37 b(~)-701 b(n)-37 b(\023)-627 b(e)424 b(sets)g(and)g(use)g(this)f Fp(C)95 b Fq(-equiv)-74 b(alence)426 b(to)e(construct)1102 38642 y(the)467 b(di\013usion)h (orbits)f(crossing)h(resonan)-37 b(t)467 b(gaps.)664 b(In)467 b(virtue)h(of)f(the)g(tec)-37 b(hniques)468 b(dev)-37 b(elop)37 b(ed)468 b(in)1102 40236 y([BCV])493 b(and)g(the)g(analytic)g(expression)g(of)g(the)g(barrier)h(function)e (obtained)h(in)f(the)h(section)g(4)1102 41830 y(w)-37 b(e)361 b(join)f(the)g(orbits)g(constructed)h(b)-37 b(y)360 b Fp(C)95 b Fq(-equiv)-74 b(alence)363 b(smo)37 b(othly)360 b(with)g(the)g(orbits)g(constructed)1102 43424 y(via)492 b(transition)f(c)-37 b(hain.)736 b(Th)-37 b(us)492 b(w)-37 b(e)492 b(obtain)f(the)g(di\013usion)h(orbits.)735 b(In)491 b(the)g(section)h(6)g(w)-37 b(e)492 b(sho)-37 b(w)1102 45018 y(the)443 b(op)37 b(en)443 b(and)g(dense)h(prop)37 b(ert)-37 b(y)-111 b(.)1102 48187 y Fr(2,)510 b(V)-127 b(ariational)508 b(set-up)2696 50448 y Fq(Roughly)496 b(sp)37 b(eaking,)509 b(the)495 b(di\013usion)g(orbits)g(are)h (constructed)g(b)-37 b(y)496 b(connecting)g(di\013eren)-37 b(t)496 b Fp(c)p Fq(-)1102 52042 y(minimal)472 b(orbit)g(sets,)478 b(along)472 b(whic)-37 b(h)473 b(the)e(Lagrange)j(action)d(tak)-37 b(es)472 b(its)f(minim)-37 b(um.)677 b(Therefore,)1102 53636 y(w)-37 b(e)357 b(shall)f(study)f(the)h(Lagrangian)h(equation)f (equiv)-74 b(alen)-37 b(t)357 b(to)e(the)h(Hamiltonian)g(equation)g (\(1.1\):)19862 56139 y Fp(d)p 19622 56733 1172 54 v 19622 57949 a(dt)21174 55165 y Fi(\022)22311 56139 y Fp(@)74 b(L)p 22311 56733 1684 54 v 22443 57949 a(@)320 b Fq(_)-615 b Fp(q)24154 55165 y Fi(\023)25427 57038 y Fm(\000)26914 56139 y Fp(@)74 b(L)p 26914 56733 V 27046 57949 a(@)g(q)29126 57038 y Fq(=)369 b(0)p Fp(;)15630 b Fq(\(2.1\))1102 60440 y(where)503 b(the)e(Lagrangian)i(function)e Fp(L)p Fq(\()246 b(_)-615 b Fp(q)47 b(;)221 b(q)48 b(;)221 b(t)p Fq(\))501 b(is)g(obtained)g(from)h(the)f(Hamiltonian)g(function) 1102 62034 y(\(1.1\))443 b(b)-37 b(y)443 b(using)f(Legendre)j (transformation)e Fm(L)p Fq(:)591 b(\()p Fp(p;)221 b(q)48 b(;)221 b(t)p Fq(\))368 b Fm(!)h Fq(\()247 b(_)-616 b Fp(q)48 b(;)221 b(q)48 b(;)221 b(t)p Fq(\))442 b(suc)-37 b(h)443 b(that)15217 64638 y Fp(L)p Fq(\()246 b(_)-615 b Fp(q)47 b(;)221 b(q)48 b(;)221 b(t)p Fq(\))369 b(=)g(max)22829 65435 y Fk(p)24339 64638 y Fm(fh)p Fp(p;)468 b Fq(_)-616 b Fp(q)48 b Fm(i)295 b(\000)g Fp(H)108 b Fq(\()p Fp(p;)221 b(q)48 b(;)221 b(t)p Fq(\))p Fm(g)p Fp(:)11384 b Fq(\(2.2\))1102 68061 y(where)812 b(_)-615 b Fp(q)619 b Fq(=)818 b(_)-616 b Fp(q)48 b Fq(\()p Fp(p;)221 b(q)48 b(;)221 b(t)p Fq(\))564 b(is)g(implicitly)g(determined)h(b)-37 b(y)811 b(_)-615 b Fp(q)619 b Fq(=)32613 67538 y Fk(@)57 b(H)p 32613 67756 1566 54 v 32809 68519 a(@)g(p)34338 68061 y Fq(.)954 b(Since)565 b(w)-37 b(e)565 b(study)f(a)g(nearly)1102 69755 y(in)-37 b(tegrable)444 b(system,)f(the)g(Lagrangian)h(has)f(the) f(form)h(of)18149 72360 y Fp(L)369 b Fq(=)g Fp(L)21728 72559 y Fv(0)22324 72360 y Fq(\()p Fp(q)23434 72559 y Fv(2)24029 72360 y Fp(;)468 b Fq(_)-616 b Fp(q)48 b Fq(\))295 b(+)g Fp(L)28304 72559 y Fv(1)28900 72360 y Fq(\()p Fp(q)48 b(;)467 b Fq(_)-615 b Fp(q)47 b(;)221 b(t)p Fq(\))1102 74965 y(where)445 b Fp(L)5848 75164 y Fv(0)6886 74965 y Fq(corresp)37 b(onds)444 b(to)f Fp(f)438 b Fq(+)296 b Fp(g)490 b Fq(through)443 b(the)g(Legender)h(transformation.)p eop %%Page: 4 4 4 3 bop 1102 2239 a Ft(4)2696 5427 y Fq(Throughout)479 b(this)f(pap)37 b(er,)488 b(w)-37 b(e)479 b(use)g Fp(\036)21998 4945 y Fk(t)22944 5427 y Fq(to)f(denote)h(the)f(Euler-Lagrange)i(\015o) -37 b(w)479 b(determined)1102 7021 y(b)-37 b(y)396 b Fp(L)p Fq(,)405 b(use)396 b(\010)7785 6539 y Fk(t)8649 7021 y Fq(to)f(denote)h(the)g(Hamiltonian)g(\015o)-37 b(w)396 b(determined)g(b)-37 b(y)396 b Fp(H)108 b Fq(.)575 b(T)-111 b(o)396 b(sp)37 b(ecify)396 b(the)g(Euler-)1102 8615 y(Lagrange)561 b(\(Hamiltonian\))e(\015o)-37 b(w)560 b(determined)g(b)-37 b(y)559 b(other)h(functions)f(w)-37 b(e)560 b(add)f(the)g(subscript,)1102 10209 y(e.g.)591 b Fp(\036)4476 9727 y Fk(t)4476 10575 y(L)5205 10708 y Fj(0)5790 10209 y Fq(,)443 b(\010)7561 9727 y Fk(t)7561 10585 y(f)104 b Fv(+)p Fk(g)9617 10209 y Fq(,)443 b(etc.)2696 12674 y(Clearly)-111 b(,)445 b(the)d(equation)i(\(2.1\))e(corresp)37 b(onds)444 b(to)f(the)f(critical)i(p)37 b(oin)-37 b(t)442 b(of)h(the)g(functional)19126 16460 y Fp(A)p Fq(\()p Fp(\015)74 b Fq(\))367 b(=)23688 14652 y Fi(Z)25237 16460 y Fp(L)p Fq(\()p Fp(\015)74 b(;)417 b Fq(_)-565 b Fp(\015)73 b(;)221 b(t)p Fq(\))p Fp(dt:)1102 20331 y Fq(W)-111 b(e)615 b(can)g(think)e(that)h Fp(L)f Fq(is)h(a)h(function)e(de\014ned)i(on)f Fp(T)184 b(M)555 b Fm(\002)409 b Fl(T)614 b Fq(where)h Fp(M)799 b Fq(=)654 b Fl(T)43752 19849 y Fv(2)44348 20331 y Fq(.)1103 b(By)614 b(the)1102 21925 y(assumption)296 b(on)f Fp(H)404 b Fq(w)-37 b(e)296 b(see)g(that)f Fp(L)h Fq(satis\014es)f(the)g(follo)-37 b(wing)297 b(conditions)e(in)-37 b(tro)37 b(duced)297 b(b)-37 b(y)295 b(Mather)1102 23519 y([Ma3]:)1102 25892 y Fr(P)-42 b(ositiv)g(e)453 b(de\014niteness.)573 b Fq(F)-111 b(or)394 b(ev)-37 b(ery)395 b(\()p Fp(q)48 b(;)221 b(t)p Fq(\))368 b Fm(2)g Fp(M)341 b Fm(\002)197 b Fl(T)p Fq(,)403 b(the)394 b(Lagrangian)g(function)g(is)f(strictly) 1102 27486 y(con)-37 b(v)g(ex)444 b(in)f(v)-37 b(elo)37 b(cit)-37 b(y:)591 b(the)443 b(Hessian)h Fp(L)20667 27685 y Fv(_)-497 b Fk(q)228 b Fv(_)-496 b Fk(q)22020 27486 y Fq(is)443 b(p)37 b(ositiv)-37 b(e)443 b(de\014nite.)1102 29859 y Fr(Sup)42 b(erlinear)658 b(gro)-42 b(wth.)979 b Fq(W)-111 b(e)573 b(supp)37 b(ose)572 b(that)g Fp(L)g Fq(has)g(\014b)37 b(er-wise)574 b(sup)37 b(erlinear)573 b(gro)-37 b(wth:)850 b(for)1102 31453 y(ev)-37 b(ery)444 b(\()p Fp(q)48 b(;)221 b(t)p Fq(\))368 b Fm(2)h Fp(M)440 b Fm(\002)295 b Fl(T)p Fq(,)442 b(w)-37 b(e)444 b(ha)-37 b(v)g(e)443 b Fp(L=)p Fm(k)247 b Fq(_)-616 b Fp(q)48 b Fm(k)370 b(!)f(1)443 b Fq(as)g Fm(k)247 b Fq(_)-616 b Fp(q)48 b Fm(k)369 b(!)g(1)p Fq(.)1102 33826 y Fr(Completeness.)549 b Fq(All)319 b(solutions)f(of)h(the)g(Lagrange)h(equations)f(are)g(w) -37 b(ell)320 b(de\014ned)f(for)g(all)g Fp(t)368 b Fm(2)h Fl(R)p Fq(.)2696 36200 y(Under)354 b(these)g(conditions)f(Mather)h (established)f(the)g(theory)h(of)f Fp(c)p Fq(-minimal)g(measure)h(and)g Fp(c)p Fq(-)1102 37794 y(minimal)363 b(orbits)g([Ma3,)379 b(Ma4].)564 b(T)-111 b(o)363 b(in)-37 b(tro)37 b(duce)364 b(some)f(basic)h(results)f(of)f(Mather,)379 b(let)363 b(us)g(observ)-37 b(e)1102 39388 y(the)552 b(fact)f(that)f(the)i (functional)17954 38318 y Fi(R)19061 39388 y Fp(Ldt)f Fq(has)h(the)f(same)h(critical)g(p)37 b(oin)-37 b(t)551 b(as)39854 38318 y Fi(R)40739 39388 y Fq(\()p Fp(L)368 b Fm(\000)f Fp(\021)44588 39587 y Fk(c)45129 39388 y Fq(\))p Fp(dt)551 b Fq(do)37 b(es)1102 40982 y(if)507 b Fp(\021)3044 41181 y Fk(c)4092 40982 y Fq(is)g(a)g(closed)g(1-form)h (on)f Fp(M)482 b Fm(\002)338 b Fl(T)p Fq(,)523 b(whose)507 b(\014rst)g(de)g(Rham)g(co-homology)h(class)f(is)g Fp(c)p Fq(,)523 b(i.e.)1102 42576 y([)p Fp(\021)2131 42775 y Fk(c)2672 42576 y Fq(])369 b(=)g Fp(c)p Fq(,)443 b(in)g(other)g(w)-37 b(ords,)443 b(their)g(Lagrange)i(equations)e(are)g(the)g(same.)2696 44949 y(Let)549 b Fp(I)650 b Fq(=)546 b([)p Fp(a;)221 b(b)p Fq(])549 b(b)37 b(e)549 b(a)g(compact)h(in)-37 b(terv)-74 b(al)549 b(of)g(time.)908 b(A)549 b(curv)-37 b(e)550 b Fp(\015)619 b Fm(2)545 b Fp(C)39055 44467 y Fv(1)39652 44949 y Fq(\()p Fp(I)104 b(;)221 b(M)145 b Fq(\))548 b(is)g(called)i(a)1102 46543 y Fp(c)p Fq(-minimizer)581 b(or)g(a)g Fp(c)p Fq(-minimal)f(curv)-37 b(e)581 b(if)f(it)g(minimizes) h(the)g(action)f(among)h(all)g(curv)-37 b(es)581 b Fp(\030)660 b Fm(2)1102 48137 y Fp(C)2146 47655 y Fv(1)2743 48137 y Fq(\()p Fp(I)104 b(;)221 b(M)145 b Fq(\))442 b(whic)-37 b(h)443 b(satisfy)g(the)f(same)i(b)37 b(oundary)443 b(conditions:)13914 52240 y Fp(A)14910 52439 y Fk(c)15451 52240 y Fq(\()p Fp(\015)74 b Fq(\))368 b(=)1641 b(min)19017 53159 y Fk(\030)32 b Fv(\()p Fk(a)p Fv(\)=)p Fk(\015)57 b Fv(\()p Fk(a)p Fv(\))19126 54224 y Fk(\030)32 b Fv(\()p Fk(b)p Fv(\)=)p Fk(\015)57 b Fv(\()p Fk(b)p Fv(\))23997 50432 y Fi(Z)25325 50760 y Fk(b)24735 53450 y(a)25859 52240 y Fq(\()p Fp(L)295 b Fm(\000)g Fp(\021)29563 52439 y Fk(c)30104 52240 y Fq(\)\()p Fp(d\030)61 b Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))p Fp(dt:)10079 b Fq(\(2.3\))1102 57317 y(As)635 b(w)-37 b(e)636 b(ha)-37 b(v)g(e)635 b(the)g(condition)g(of)f (completeness)i(the)f(minimizer)g(m)-37 b(ust)634 b(b)37 b(e)635 b(a)g Fp(C)43742 56835 y Fv(1)44338 57317 y Fq(-curv)-37 b(e)636 b(b)-37 b(y)1102 58911 y(T)-111 b(onelli's)334 b(theorem.)554 b(Without)332 b(the)g(completeness)h(the)g(minimizer)g (can)g(only)f(b)37 b(e)333 b(an)f(absolutely)1102 60505 y(con)-37 b(tin)g(uous)490 b(curv)-37 b(e)490 b(\([BM]\).)e(If)h Fp(J)617 b Fq(is)489 b(a)g(non)g(compact)h(in)-37 b(terv)-74 b(al,)502 b(the)489 b(curv)-37 b(e)490 b Fp(\015)520 b Fm(2)446 b Fp(C)44169 60023 y Fv(1)44765 60505 y Fq(\()p Fp(J)-20 b(;)221 b(M)145 b Fq(\))488 b(is)1102 62099 y(said)424 b(a)h Fp(c)p Fq(-minimizer)f(if)g Fp(\015)74 b Fm(j)14441 62298 y Fk(I)15474 62099 y Fq(is)424 b Fp(c)p Fq(-minimal)g(for)g(an)-37 b(y)424 b(compact)h(in)-37 b(terv)-74 b(al)425 b Fp(I)473 b Fm(\032)369 b Fp(J)128 b Fq(.)583 b(An)425 b(orbit)f Fp(X)104 b Fq(\()p Fp(t)p Fq(\))1102 63693 y(of)443 b Fp(\036)3406 63211 y Fk(t)4316 63693 y Fq(is)f(called)h Fp(c)p Fq(-minimizing)f(if)g(the)g(curv)-37 b(e)444 b Fp(\031)341 b Fm(\016)294 b Fp(X)546 b Fq(is)442 b Fp(c)p Fq(-minimizing,)h(where)g(the)f(op)37 b(erator)443 b Fp(\031)1102 65287 y Fq(is)468 b(the)h(standard)f(pro)74 b(jection)469 b(from)f(tangen)-37 b(t)468 b(bundle)h(to)e(the)i (underling)f(manifold)h(along)f(the)1102 66881 y(\014b)37 b(ers,)523 b(a)507 b(p)37 b(oin)-37 b(t)506 b(\()p Fp(z)58 b(;)221 b(s)p Fq(\))475 b Fm(2)f Fp(T)184 b(M)483 b Fm(\002)338 b Fl(R)507 b Fq(is)f Fp(c)p Fq(-minimizing)h(if)f(its)g(orbit)h Fp(\036)36818 66399 y Fk(t)37286 66881 y Fq(\()p Fp(z)58 b(;)221 b(s)p Fq(\))506 b(is)g Fp(c)p Fq(-minimizing.)1102 68569 y(W)-111 b(e)392 b(use)5831 68233 y(~)5581 68569 y Fm(G)6371 68768 y Fk(L)7166 68569 y Fq(\()p Fp(c)p Fq(\))368 b Fm(\032)h Fp(T)184 b(M)336 b Fm(\002)191 b Fl(R)391 b Fq(to)g(denote)g(the)f(set)h(of)f(minimal)h(orbits)g(of)f Fp(L)191 b Fm(\000)g Fp(\021)40497 68768 y Fk(c)41428 68569 y Fq(\(the)391 b Fp(c)p Fq(-minimal)1102 70163 y(orbits)582 b(of)f Fp(L)p Fq(\).)1006 b(W)-111 b(e)582 b(shall)g(drop)f(the)h(subscript)f Fp(L)h Fq(when)g(it)f(is)g(clear)h (whic)-37 b(h)583 b(Lagrangian)f(is)1102 71757 y(under)504 b(consideration.)773 b(It)502 b(is)i(not)f(necessary)h(to)f(assume)h (the)f(p)37 b(erio)g(dicit)-37 b(y)504 b(of)f Fp(L)h Fq(in)f Fp(t)g Fq(for)g(the)1102 73371 y(de\014nition)539 b(of)8962 73035 y(~)8712 73371 y Fm(G)79 b Fq(.)879 b(When)540 b(it)e(is)h(p)37 b(erio)g(dic)540 b(in)f Fp(t)p Fq(,)26156 73035 y(~)25906 73371 y Fm(G)79 b Fq(\()p Fp(c)p Fq(\))529 b Fm(\032)g Fp(T)184 b(M)505 b Fm(\002)360 b Fl(R)539 b Fq(is)g(a)g(nonempt)-37 b(y)540 b(compact)1102 74965 y(subset)443 b(of)g Fp(T)184 b(M)440 b Fm(\002)296 b Fl(T)p Fq(,)442 b(in)-37 b(v)-74 b(arian)-37 b(t)443 b(for)h(the)e(Euler-Lagrange)j(\015o)-37 b(w)443 b Fp(\036)35201 74483 y Fk(t)35670 74965 y Fq(.)p eop %%Page: 5 5 5 4 bop 49357 2239 a Ft(5)2696 5427 y Fq(W)-111 b(e)462 b(can)f(extend)g(the)f(de\014nition)h(of)f(action)h(along)g(a)g Fp(C)31279 4945 y Fv(1)31875 5427 y Fq(-curv)-37 b(e)461 b(to)g(the)f(action)h(on)f(a)h(prob-)1102 7021 y(abilit)-37 b(y)507 b(measure.)783 b(Let)507 b Fh(M)g Fq(b)37 b(e)507 b(the)f(set)h(of)g(Borel)g(probabilit)-37 b(y)507 b(measures)h(on)f Fp(T)184 b(M)483 b Fm(\002)338 b Fl(T)p Fq(.)781 b(F)-111 b(or)1102 8615 y(eac)-37 b(h)444 b Fp(\027)454 b Fm(2)368 b Fh(M)q Fq(,)442 b(the)h(action)g Fp(A)16182 8814 y Fk(c)16724 8615 y Fq(\()p Fp(\027)85 b Fq(\))441 b(is)i(de\014ned)g(as) g(the)g(follo)-37 b(wing:)19062 12039 y Fp(A)20058 12238 y Fk(c)20599 12039 y Fq(\()p Fp(\027)85 b Fq(\))368 b(=)24144 10231 y Fi(Z)25473 12039 y Fq(\()p Fp(L)295 b Fm(\000)g Fp(\021)29177 12238 y Fk(c)29718 12039 y Fq(\))p Fp(d\027)11 b(:)15229 b Fq(\(2.4\))1102 15464 y(Mather)481 b(has)g(pro)-37 b(v)g(ed)482 b([Ma3])f(that)f(for)h(eac)-37 b(h)482 b(\014rst)f(de)g (Rham)g(cohomology)h(class)f Fp(c)g Fq(there)g(is)g(a)1102 17058 y(probabilit)-37 b(y)443 b(measure)h Fp(\026)g Fq(whic)-37 b(h)443 b(minimizes)g(the)g(actions)g(o)-37 b(v)g(er)444 b Fh(M)17702 20408 y Fp(A)18698 20607 y Fk(c)19240 20408 y Fq(\()p Fp(\026)p Fq(\))368 b(=)781 b(inf)22844 21205 y Fk(\027)62 b Fg(2)p Ff(M)25504 18600 y Fi(Z)26833 20408 y Fq(\()p Fp(L)295 b Fm(\000)g Fp(\021)30537 20607 y Fk(c)31078 20408 y Fq(\))p Fp(d\027)11 b(:)1102 24059 y Fq(This)456 b Fp(\026)f Fq(is)g(in)-37 b(v)-74 b(arian)-37 b(t)456 b(to)f(the)g(Euler-Lagrange)i(\015o)-37 b(w.)628 b(W)-111 b(e)456 b(use)34209 23723 y(~)33559 24059 y Fm(M)p Fq(\()p Fp(c)p Fq(\))e(to)h(denote)g(the)g(supp)37 b(ort)1102 25653 y(of)415 b(the)f(measure,)421 b(use)415 b Fp(\013)5 b Fq(\()p Fp(c)p Fq(\))368 b(=)h Fp(A)17866 25852 y Fk(c)18407 25653 y Fq(\()p Fp(\026)p Fq(\))414 b(to)g(denote)h(the)g(minim)-37 b(um)414 b Fp(c)p Fq(-action.)581 b(It)414 b(de\014nes)h(a)f(func-)1102 27247 y(tion)536 b Fp(\013)5 b Fq(:)775 b Fp(H)7137 26765 y Fv(1)7734 27247 y Fq(\()p Fp(M)71 b(;)221 b Fl(R)p Fq(\))523 b Fm(!)h Fl(R)p Fq(,)559 b(usually)536 b(called)g Fp(\013)5 b Fq(-function.)869 b(Its)535 b(Legendre)i(transformation)f Fp(\014)70 b Fq(:)1102 28841 y Fp(H)2206 29040 y Fv(1)2802 28841 y Fq(\()p Fp(M)h(;)221 b Fl(R)p Fq(\))391 b Fm(!)g Fl(R)457 b Fq(is)f(usually)g(called)g Fp(\014)70 b Fq(-function.)631 b(Both)455 b(functions)h(are)h(con)-37 b(v)g(ex,)460 b(\014nite)c(ev)-37 b(ery-)1102 30529 y(where)444 b(and)e(ha)-37 b(v)g(e)443 b(sup)37 b(er-linear)443 b(gro)-37 b(wth)442 b([Ma3].)590 b(As)29193 30193 y(~)28543 30529 y Fm(M)p Fq(\()p Fp(c)p Fq(\))441 b(is)h(de\014ned)h(as)f(the)g(limit)f(measure) 1102 32123 y(of)575 b Fp(c)p Fq(-minimal)f(orbits,)608 b(the)574 b(follo)-37 b(wing)575 b(lemma)g(is)g(a)g(straigh)-37 b(tforw)g(ard)575 b(result)g(of)g(top)37 b(ological)1102 33717 y(dynamics:)1102 36010 y Fr(Lemma)379 b(2.1.)764 b Fo(F)-102 b(or)371 b(e)-68 b(ach)372 b(c)-68 b(o-homolo)g(gic)g(al) 372 b(class)g Fp(c)f Fo(and)g(e)-68 b(ach)371 b(p)-68 b(ositive)373 b(numb)-68 b(er)371 b Fp(\017)p Fo(,)392 b(ther)-68 b(e)372 b(exists)1102 37604 y(a)665 b(p)-68 b(ositive)666 b(numb)-68 b(er)665 b Fp(T)13123 37803 y Fv(0)14431 37604 y Fq(=)713 b Fp(T)16953 37803 y Fv(0)17549 37604 y Fq(\()p Fp(c;)221 b(\017)p Fq(\))p Fo(,)712 b(if)665 b Fp(T)897 b Fm(\025)712 b Fp(T)27079 37803 y Fv(0)28340 37604 y Fo(and)665 b Fp(\015)74 b Fo(:)989 b Fq([0)p Fp(;)221 b(T)184 b Fq(])714 b Fm(!)e Fp(M)577 b Fm(\002)433 b Fl(T)664 b Fo(is)i(a)e(curve)1102 39198 y(minimizing)477 b(the)f(action)g(of)f Fp(L)295 b Fm(\000)h Fp(\021)19023 39397 y Fk(c)19563 39198 y Fo(,)476 b Fq([)p Fp(\021)21475 39397 y Fk(c)22016 39198 y Fq(])369 b(=)g Fp(c)p Fo(,)475 b(then)h(ther)-68 b(e)475 b(is)h Fp(t)369 b Fm(2)f Fq([0)p Fp(;)221 b(T)184 b Fq(])477 b Fo(such)e(that)19930 42037 y Fp(d)p Fq(\()p Fp(d\015)74 b Fq(\()p Fp(t)p Fq(\))p Fp(;)25344 41701 y Fq(~)24695 42037 y Fm(M)p Fq(\()p Fp(c)p Fq(\)\))367 b Fm(\024)i Fp(\017:)2696 45481 y Fq(Before)501 b(to)d(start)h(the)g(existence)h(pro)37 b(of)500 b(of)f(di\013usion)g(orbits)g(w)-37 b(e)500 b(need)g(to)f(in)-37 b(tro)37 b(duce)500 b(some)1102 47075 y(more)469 b(concepts)g(and)g(in)-37 b(v)g(estigate)469 b(some)f(relev)-74 b(an)-37 b(t)470 b(prop)37 b(erties,)475 b(whic)-37 b(h)469 b(shall)f(b)37 b(e)469 b(made)g(use)f(of)1102 48669 y(b)37 b(elo)-37 b(w)444 b(for)f(our)g(purp)37 b(ose.)2696 50982 y(W)-111 b(e)465 b(ha)-37 b(v)g(e)464 b(de\014ned)g(the)g(set)17737 50646 y(~)17087 50982 y Fm(M)p Fq(\()p Fp(c)p Fq(\))f(and)23607 50646 y(~)23357 50982 y Fm(G)79 b Fq(\()p Fp(c)p Fq(\).)652 b(It)463 b(is)g(easy)h(to)f(see)i(that)39976 50646 y(~)39326 50982 y Fm(M)q Fq(\()p Fp(c)p Fq(\))d(is)i(con)-37 b(tained)1102 52669 y(in)428 b(the)g(set)7218 52334 y(~)6968 52669 y Fm(G)79 b Fq(\()p Fp(c)p Fq(\).)585 b(Bet)-37 b(w)g(een)429 b(the)e(set)20261 52334 y(~)20011 52669 y Fm(G)507 b Fq(and)428 b(set)26584 52334 y(~)25934 52669 y Fm(M)g Fq(w)-37 b(e)429 b(can)f(also)g(de\014ne)g(so-called)h(Aubry)f(set)1559 54022 y(~)1102 54357 y Fm(A)p Fq(\()p Fp(c)p Fq(\))442 b(and)h(Ma)37 b(~)-701 b(n)-37 b(\023)-627 b(e)443 b(set)12945 54022 y(~)12523 54357 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\))442 b(as)h(w)-37 b(ell)443 b(as)g(the)g(limit)f(p)37 b(oin)-37 b(t)443 b(set)33129 54022 y(~)32819 54357 y Fm(L)p Fq(\()p Fp(c)p Fq(\).)2696 56650 y(As)369 b(all)f(orbits)g(are)h(w)-37 b(ell)369 b(de\014ned)f(on)g(the)g(whole)h Fl(R)p Fq(,)384 b(they)368 b(ha)-37 b(v)g(e)368 b Fp(!)48 b Fq(-limit)367 b(set)h(and)g Fp(\013)5 b Fq(-limit)367 b(set.)1102 58244 y(Let)521 b(~)-769 b Fp(!)47 b Fq(\()p Fp(c)p Fq(\))414 b(b)37 b(e)415 b(the)g(union)g(of)g Fp(!)48 b Fq(-limit)414 b(p)37 b(oin)-37 b(ts)415 b(of)g Fp(c)p Fq(-minimal)g(orbits)g Fp(X)104 b Fq(\()p Fp(t)p Fq(\))367 b(:)415 b([0)p Fp(;)221 b Fm(1)p Fq(\))370 b Fm(!)f Fp(T)184 b(M)385 b Fm(\002)240 b Fl(T)p Fq(,)1102 59838 y(let)514 b(~)-796 b Fp(\013)5 b Fq(\()p Fp(c)p Fq(\))380 b(b)37 b(e)382 b(the)g(union)g(of)g Fp(\013)5 b Fq(-limit)380 b(p)37 b(oin)-37 b(ts)382 b(of)g Fp(c)p Fq(-minimal)f(orbits)h Fp(X)104 b Fq(\()p Fp(t)p Fq(\))367 b(:)382 b(\()p Fm(\0001)p Fp(;)221 b Fq(0])370 b Fm(!)f Fp(T)184 b(M)318 b Fm(\002)173 b Fl(T)p Fq(.)1102 61526 y(W)-111 b(e)444 b(call)5982 61190 y(~)5825 61526 y Fp(L)p Fq(\()p Fp(c)p Fq(\))368 b(=)475 b(~)-769 b Fp(!)47 b Fq(\()p Fp(c)p Fq(\))294 b Fm([)427 b Fq(~)-796 b Fp(\013)5 b Fq(\()p Fp(c)p Fq(\))441 b(the)i(limit)g(set.)2696 63818 y(T)-111 b(o)444 b(de\014ne)g(the)e(Aubry)i(set)e(and)h(the)g(Ma) 37 b(~)-701 b(n)-37 b(\023)-627 b(e)443 b(set)g(let)f(us)h(de\014ne) 4687 67610 y Fp(h)5452 67809 y Fk(c)5993 67610 y Fq(\()p Fp(x;)221 b(x)8618 67062 y Fg(0)8991 67610 y Fp(;)g(t;)g(t)11131 67062 y Fg(0)11504 67610 y Fq(\))369 b(=)3044 b(min)14067 68661 y Fk(\015)57 b Fg(2)p Fk(C)16225 68260 y Fj(1)16743 68661 y Fv(\([)p Fk(t;t)18576 68260 y Fe(0)18936 68661 y Fv(])p Fk(;M)100 b Fv(\))13792 69799 y Fk(\015)57 b Fv(\()p Fk(t)p Fv(\)=)p Fk(x;\015)g Fv(\()p Fk(t)18801 69398 y Fe(0)19163 69799 y Fv(\)=)p Fk(x)20995 69398 y Fe(0)21576 65802 y Fi(Z)22905 66131 y Fk(t)23306 65730 y Fe(0)22314 68821 y Fk(t)23732 67610 y Fq(\()p Fp(L)295 b Fm(\000)g Fp(\021)27436 67809 y Fk(c)27977 67610 y Fq(\)\()p Fp(d\015)74 b Fq(\()p Fp(s)p Fq(\))p Fp(;)221 b(s)p Fq(\))p Fp(ds)292 b Fm(\000)k Fq(\()p Fp(t)37783 67062 y Fg(0)38450 67610 y Fm(\000)f Fp(t)p Fq(\))p Fp(\013)5 b Fq(\()p Fp(c)p Fq(\))p Fp(;)3583 b Fq(\(2.5\))14137 72513 y Fp(F)14991 72712 y Fk(c)15532 72513 y Fq(\()p Fp(x;)221 b(x)18157 71965 y Fg(0)18530 72513 y Fp(;)g(s;)g(s)20956 71965 y Fg(0)21329 72513 y Fq(\))368 b(=)2235 b(inf)23976 73380 y Fk(s)p Fv(=)p Fk(t)182 b Fv(mo)29 b(d)181 b(1)23616 74285 y Fk(s)24117 73885 y Fe(0)24477 74285 y Fv(=)p Fk(t)25693 73885 y Fe(0)26234 74285 y Fv(mo)29 b(d)181 b(1)24621 75191 y Fk(t)25022 74790 y Fe(0)25382 75191 y Fg(\025)p Fk(t)p Fv(+1)29184 72513 y Fp(h)29949 72712 y Fk(c)30491 72513 y Fq(\()p Fp(x;)221 b(x)33116 71965 y Fg(0)33489 72513 y Fp(;)g(t;)g(t)35629 71965 y Fg(0)36002 72513 y Fq(\))p Fp(:)p eop %%Page: 6 6 6 5 bop 1102 2239 a Ft(6)18366 5427 y Fp(h)19131 5626 y Fk(c)19673 5427 y Fq(\()p Fp(x;)221 b(x)22298 4879 y Fg(0)22671 5427 y Fq(\))368 b(=)h Fp(h)25723 5626 y Fk(c)26265 5427 y Fq(\()p Fp(x;)221 b(x)28890 4879 y Fg(0)29263 5427 y Fp(;)g Fq(0)p Fp(;)g Fq(1\))p Fp(:)14534 b Fq(\(2.6\))1102 7904 y(Let)10090 11918 y Fp(h)10855 11370 y Fk(n)10855 12246 y(c)11579 11918 y Fq(\()p Fp(x;)221 b(x)14204 11370 y Fg(0)14577 11918 y Fq(\))368 b(=)h(min)19300 10444 y Fi(n)20407 10258 y Fk(n)p Fg(\000)p Fv(1)20455 10656 y Fi(X)20554 13485 y Fk(i)p Fv(=0)22644 11918 y Fp(h)23409 12117 y Fk(c)23951 11918 y Fq(\()p Fp(m)25634 12117 y Fk(i)26076 11918 y Fp(;)221 b(m)27832 12117 y Fk(i)p Fg(\000)p Fv(1)29635 11918 y Fq(\))368 b(:)h Fp(m)32424 12117 y Fv(0)33389 11918 y Fq(=)g Fp(x;)221 b(m)37306 12117 y Fk(n)38399 11918 y Fq(=)370 b Fp(x)40561 11370 y Fg(0)20185 15678 y Fq(and)443 b Fp(m)23934 15877 y Fk(i)24746 15678 y Fm(2)368 b Fp(M)587 b Fq(for)444 b(0)369 b Fm(\024)g Fp(i)g Fm(\024)g Fp(n)35371 14203 y Fi(o)36257 15678 y Fp(;)1102 19085 y Fq(and)443 b(let)15706 22168 y Fp(h)16471 21619 y Fg(1)16471 22496 y Fk(c)17597 22168 y Fq(\()p Fp(x;)221 b(x)20222 21619 y Fg(0)20595 22168 y Fq(\))369 b(=)g(lim)221 b(inf)23337 22965 y Fk(n)p Fg(!1)26787 22168 y Fp(h)27552 21619 y Fk(n)27552 22496 y(c)28275 22168 y Fq(\()p Fp(x;)g(x)30900 21619 y Fg(0)31273 22168 y Fq(\))15401 b(\(2.7\))16365 24695 y Fp(d)17056 24894 y Fk(c)17597 24695 y Fq(\()p Fp(x;)221 b(x)20222 24146 y Fg(0)20595 24695 y Fq(\))369 b(=)g Fp(h)23648 24146 y Fg(1)23648 25023 y Fk(c)24774 24695 y Fq(\()p Fp(x;)221 b(x)27399 24146 y Fg(0)27772 24695 y Fq(\))294 b(+)i Fp(h)30677 24146 y Fg(1)30677 25023 y Fk(c)31803 24695 y Fq(\()p Fp(x)33079 24146 y Fg(0)33451 24695 y Fp(;)221 b(x)p Fq(\))11874 b(\(2.8\))1102 27844 y(Mather)310 b(sho)-37 b(w)g(ed)312 b(in)e([Ma4])g(that)f Fp(d)18557 28043 y Fk(c)19409 27844 y Fq(is)h(a)g(pseudo-metric)h(on)f(the)g(set)g Fm(f)p Fp(x)369 b Fm(2)g Fp(M)513 b Fq(:)369 b Fp(h)42185 27362 y Fg(1)42185 28172 y Fk(c)43311 27844 y Fq(\()p Fp(x;)221 b(x)p Fq(\))369 b(=)g(0)p Fm(g)p Fq(.)1102 29438 y(A)443 b(curv)-37 b(e)444 b Fp(\015)f Fm(2)368 b Fp(C)9516 28956 y Fv(1)10113 29438 y Fq(\()p Fl(R)p Fp(;)221 b(M)145 b Fq(\))442 b(is)h(called)g Fp(c)p Fq(-semi-static)g (if)8935 32520 y Fp(A)9931 32719 y Fk(c)10472 32520 y Fq(\()p Fp(\015)74 b Fm(j)12120 32760 y Fv([)p Fk(a;b)p Fv(])14146 32520 y Fq(\))295 b Fm(\000)g Fp(\013)5 b Fq(\()p Fp(c)p Fq(\)\()p Fp(b)294 b Fm(\000)h Fp(a)p Fq(\))369 b(=)g Fp(F)25303 32719 y Fk(c)25844 32520 y Fq(\()p Fp(\015)74 b Fq(\()p Fp(a)p Fq(\))p Fp(;)221 b(\015)74 b Fq(\()p Fp(b)p Fq(\))p Fp(;)221 b(a)g Fq(mo)37 b(d)220 b(1)p Fp(;)h(b)g Fq(mo)37 b(d)223 b(1\))1102 35696 y(for)444 b(eac)-37 b(h)443 b([)p Fp(a;)221 b(b)p Fq(])370 b Fm(\032)f Fl(R)p Fq(.)591 b(A)443 b(curv)-37 b(e)444 b Fp(\015)e Fm(2)369 b Fp(C)20829 35213 y Fv(1)21425 35696 y Fq(\()p Fl(R)p Fp(;)221 b(M)145 b Fq(\))442 b(is)h(called)h Fp(c)p Fq(-static)e(if)8419 38778 y Fp(A)9415 38977 y Fk(c)9956 38778 y Fq(\()p Fp(\015)74 b Fm(j)11604 39018 y Fv([)p Fk(a;b)p Fv(])13630 38778 y Fq(\))294 b Fm(\000)i Fp(\013)5 b Fq(\()p Fp(c)p Fq(\)\()p Fp(b)293 b Fm(\000)j Fp(a)p Fq(\))368 b(=)i Fm(\000)p Fp(F)25820 38977 y Fk(c)26361 38778 y Fq(\()p Fp(\015)74 b Fq(\()p Fp(b)p Fq(\))p Fp(;)221 b(\015)74 b Fq(\()p Fp(a)p Fq(\))p Fp(;)221 b(b)g Fq(mo)37 b(d)220 b(1)p Fp(;)h(a)g Fq(mo)37 b(d)222 b(1)q(\))1102 41868 y(for)498 b(eac)-37 b(h)498 b([)p Fp(a;)221 b(b)p Fq(])461 b Fm(\032)e Fl(R)p Fq(.)755 b(An)497 b(orbit)g Fp(X)104 b Fq(\()p Fp(t)p Fq(\))459 b(=)g(\()p Fp(d\015)74 b Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)g Fq(mo)37 b(d)442 b(1\))497 b(is)g(called)h(static)f(\(semi-static\))f(if)1102 43556 y Fp(\015)h Fq(is)424 b(static)f(\(semi-static\).)583 b(W)-111 b(e)424 b(call)g(the)f(Ma)37 b(~)-701 b(n)-37 b(\023)-627 b(e)424 b(set)28470 43220 y(~)28048 43556 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\))423 b(the)g(union)g(of)h(global)g Fp(c)p Fq(-semi-static)1102 45244 y(orbits,)448 b(the)e(set)10076 44908 y(~)9620 45244 y Fm(A)o Fq(\()p Fp(c)p Fq(\))g(is)g(de\014ned)h (as)g(the)f(union)h(of)f(global)h Fp(c)p Fq(-static)f(orbits,)h(w)-37 b(e)448 b(call)f(it)f(Aubry)1102 46838 y(set.)2696 49206 y(W)-111 b(e)579 b(use)e Fm(M)p Fq(\()p Fp(c)p Fq(\),)611 b Fm(L)p Fq(\()p Fp(c)p Fq(\),)f Fm(A)p Fq(\()p Fp(c)p Fq(\),)g Fm(N)196 b Fq(\()p Fp(c)p Fq(\))576 b(and)h Fm(G)79 b Fq(\()p Fp(c)p Fq(\))576 b(to)i(denote)f(the)h(standard)f (pro)74 b(jection)578 b(of)1752 50558 y(~)1102 50894 y Fm(M)q Fq(\()p Fp(c)p Fq(\),)5420 50558 y(~)5110 50894 y Fm(L)p Fq(\()p Fp(c)p Fq(\),)8894 50558 y(~)8438 50894 y Fm(A)p Fq(\()p Fp(c)p Fq(\),)12332 50558 y(~)11911 50894 y Fm(N)195 b Fq(\()p Fp(c)p Fq(\))432 b(and)18061 50558 y(~)17811 50894 y Fm(G)78 b Fq(\()p Fp(c)p Fq(\))433 b(from)g Fp(T)184 b(M)421 b Fm(\002)276 b Fl(T)433 b Fq(to)g Fp(M)420 b Fm(\002)276 b Fl(T)432 b Fq(resp)37 b(ectiv)-37 b(ely)-111 b(.)589 b(W)-111 b(e)434 b(ha)-37 b(v)g(e)434 b(the)1102 52488 y(follo)-37 b(wing)444 b(inclusions)f (\([Be]\))15552 55328 y(~)14902 55664 y Fm(M)p Fq(\()p Fp(c)p Fq(\))368 b Fm(\022)20187 55328 y Fq(~)19876 55664 y Fm(L)p Fq(\()p Fp(c)p Fq(\))g Fm(\022)24628 55328 y Fq(~)24171 55664 y Fm(A)p Fq(\()p Fp(c)p Fq(\))g Fm(\022)29033 55328 y Fq(~)28611 55664 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\))368 b Fm(\022)33526 55328 y Fq(~)33276 55664 y Fm(G)79 b Fq(\()p Fp(c)p Fq(\))p Fp(:)11068 b Fq(\(2.9\))1102 58841 y(The)564 b(set)6397 58505 y(~)6147 58841 y Fm(G)79 b Fq(\()p Fp(c)p Fq(\))563 b(and)12312 58505 y(~)11891 58841 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\))562 b(ha)-37 b(v)g(e)564 b(the)f(go)37 b(o)g(d)564 b(prop)37 b(ert)-37 b(y)563 b(of)h(upp)37 b(er)563 b(semi-con)-37 b(tin)g(uit)g(y)564 b(in)f Fp(c)p Fq(.)951 b(T)-111 b(o)1102 60435 y(describ)37 b(e)478 b(these)e(minimal)g(orbit)g(sets,)484 b(Mather)477 b(in)-37 b(tro)37 b(duced)476 b(t)-37 b(w)g(o)477 b(kinds)f(of)g (barrier)h(functions)1102 62029 y Fp(B)2110 62228 y Fk(c)3094 62029 y Fq(and)443 b Fp(B)6752 61547 y Fg(\003)6685 62357 y Fk(c)7362 62029 y Fq(,)f(it)h(is)g(de\014ned)g(as)g(follo)-37 b(ws)20617 65111 y Fp(B)21625 65310 y Fk(c)22166 65111 y Fq(\()p Fp(q)48 b Fq(\))368 b(=)h Fp(h)26376 64563 y Fg(1)26376 65440 y Fk(c)27502 65111 y Fq(\()p Fp(q)48 b(;)221 b(q)48 b Fq(\))3872 68194 y Fp(B)4947 67646 y Fg(\003)4880 68522 y Fk(c)5557 68194 y Fq(\()p Fp(q)g Fq(\))367 b(=)j(min)p Fm(f)p Fp(h)12645 67646 y Fg(1)12645 68522 y Fk(c)13770 68194 y Fq(\()p Fp(\030)61 b(;)221 b(q)48 b Fq(\))295 b(+)g Fp(h)19065 67646 y Fg(1)19065 68522 y Fk(c)20191 68194 y Fq(\()p Fp(q)48 b(;)221 b(\021)48 b Fq(\))294 b Fm(\000)h Fp(h)25551 67646 y Fg(1)25551 68522 y Fk(c)26677 68194 y Fq(\()p Fp(\030)61 b(;)221 b(\021)48 b Fq(\))368 b(:)h Fm(8)443 b Fp(\030)61 b(;)221 b(\021)417 b Fm(2)368 b Fp(\031)48 b Fq(\()p Fm(M)p Fq(\()p Fp(c)p Fq(\))p Fm(j)40396 68393 y Fk(t)p Fv(=0)42207 68194 y Fq(\))p Fm(g)p Fp(:)2770 b Fq(\(2.10\))1102 70671 y(Clearly)-111 b(,)613 b(w)-37 b(e)579 b(ha)-37 b(v)g(e)578 b(0)594 b Fm(\024)g Fp(B)15393 70189 y Fg(\003)15326 70999 y Fk(c)16596 70671 y Fm(\024)g Fp(B)19231 70870 y Fk(c)19772 70671 y Fq(.)995 b(When)578 b Fp(d)25836 70870 y Fk(c)26377 70671 y Fq(\()p Fp(\030)61 b(;)221 b(\021)48 b Fq(\))593 b(=)h(0)578 b(for)g(all)f Fp(\030)61 b(;)221 b(\021)642 b Fm(2)593 b(M)p Fq(\()p Fp(c)p Fq(\))p Fm(j)44546 70870 y Fk(t)p Fv(=0)46358 70671 y Fq(,)611 b(then)1102 72265 y Fp(B)2110 72464 y Fk(c)3091 72265 y Fq(=)441 b Fp(B)5640 71783 y Fg(\003)5573 72593 y Fk(c)6735 72265 y Fq(\([Ma4]\).)718 b(It)484 b(is)i(not)f(hard)h(to)f(see)h(that) f Fm(A)p Fq(\()p Fp(c)p Fq(\))p Fm(j)30366 72464 y Fk(t)p Fv(=0)32618 72265 y Fq(=)440 b Fm(f)p Fp(x)h Fm(2)e Fp(M)585 b Fq(:)440 b Fp(B)40971 72464 y Fk(c)41512 72265 y Fq(\()p Fp(x)p Fq(\))f(=)i(0)p Fm(g)p Fq(.)719 b(The)1102 73859 y(follo)-37 b(wing)444 b(lemma)f(is)g(a)g(mo)37 b(di\014ed)443 b(v)-37 b(ersion)444 b(of)f(the)f(prop)37 b(osition)443 b(2.1)g(in)g([Be].)p eop %%Page: 7 7 7 6 bop 49357 2239 a Ft(7)1102 5427 y Fr(Lemma)666 b(2.2.)764 b Fo(L)-68 b(et)602 b Fp(M)746 b Fo(b)-68 b(e)601 b(a)g(c)-68 b(omp)g(act,)634 b(c)-68 b(onne)g(cte)g(d)601 b(R)-34 b(iemanian)602 b(manifold.)990 b(Assume)602 b Fp(L)597 b Fm(2)1102 7021 y Fp(C)2146 6539 y Fk(r)2735 7021 y Fq(\()p Fp(T)184 b(M)497 b Fm(\002)352 b Fl(R)p Fp(;)221 b Fl(R)p Fq(\))555 b(\()p Fp(r)549 b Fm(\025)511 b Fq(2\))554 b Fo(satis\014es)h(the)f(p)-68 b(ositive)555 b(de\014nite,)575 b(sup)-68 b(erline)g(ar-gr)g(owth)555 b(and)f(c)-68 b(om-)1102 8615 y(pleteness)672 b(c)-68 b(onditions.)1197 b(Consider)-68 b(e)g(d)670 b(as)g(the)h(function)f(of)h Fp(t)p Fo(,)719 b Fp(L)670 b Fo(is)h(assume)-68 b(d)671 b(p)-68 b(erio)g(dic)671 b(for)1102 10229 y Fp(t)737 b Fm(2)g Fq(\()p Fm(\0001)p Fp(;)221 b Fq(0])680 b Fo(and)e(for)h Fp(t)737 b Fm(2)g Fq([1)p Fp(;)221 b Fm(1)p Fq(\))p Fo(.)1222 b(Then)678 b(the)h(map)g Fp(L)737 b Fm(!)35324 9893 y Fq(~)35074 10229 y Fm(G)35864 10428 y Fk(L)37397 10229 y Fm(\032)h Fp(T)184 b(M)587 b Fm(\002)443 b Fl(R)679 b Fo(is)g(upp)-68 b(er)1102 11917 y(semi-c)g(ontinuous.)610 b(As)466 b(an)f(imme)-68 b(diate)467 b(c)-68 b(onse)g(quenc)g(e,)29762 11581 y Fq(~)29512 11917 y Fm(G)79 b Fq(\()p Fp(c)p Fq(\))464 b Fo(is)i(a)f(non-empty)h(c)-68 b(omp)g(act)466 b(set)g(in)1102 13605 y Fp(T)184 b(M)441 b Fm(\002)295 b Fl(T)475 b Fo(and)g(the)g(map) h Fp(c)369 b Fm(!)17078 13269 y Fq(~)16828 13605 y Fm(G)79 b Fq(\()p Fp(c)p Fq(\))474 b Fo(is)i(upp)-68 b(er)475 b(semi-c)-68 b(ontinuous)477 b(if)f Fp(L)f Fo(is)h(p)-68 b(erio)g(dic)476 b(in)f Fp(t)p Fo(.)2696 15934 y Fq(W)-111 b(e)354 b(can)f(consider)g Fp(t)f Fq(is)h(de\014ned)g(on)f(\()p Fl(T)114 b Fm(_)g Fq([0)p Fp(;)221 b Fq(1])114 b Fm(_)g Fl(T)p Fq(\))p Fp(=)369 b Fm(\030)p Fq(,)i(where)354 b Fm(\030)e Fq(is)h(de\014ned)g(b)-37 b(y)352 b(iden)-37 b(tifying)1102 17528 y Fm(f)p Fq(0)p Fm(g)370 b(2)e Fq([0)p Fp(;)221 b Fq(1])349 b(with)e(some)h(p)37 b(oin)-37 b(t)348 b(on)f(one)h(circle,)368 b(and)348 b(iden)-37 b(tifying)348 b Fm(f)p Fq(1)p Fm(g)369 b(2)g Fq([0)p Fp(;)221 b Fq(1])348 b(with)f(some)i(p)37 b(oin)-37 b(t)1102 19122 y(on)514 b(another)g(circle.)803 b(Let)514 b Fp(U)15521 19321 y Fk(k)16661 19122 y Fq(=)486 b Fm(f)p Fq(\()p Fp(\020)98 b(;)221 b(q)48 b(;)221 b(t)p Fq(\))487 b(:)f(\()p Fp(q)48 b(;)221 b(t)p Fq(\))486 b Fm(2)g Fp(M)g Fm(\002)343 b Fq(\()p Fl(T)e Fm(_)h Fq([0)p Fp(;)221 b Fq(1])343 b Fm(_)f Fl(T)p Fq(\))p Fp(=)486 b Fm(\030)p Fp(;)443 b Fm(k)p Fp(\020)98 b Fm(k)487 b(\024)f Fp(k)42 b(;)221 b Fm(g)p Fq(,)1102 20716 y Fm([)1988 20234 y Fg(1)1988 21092 y Fk(k)22 b Fv(=1)3986 20716 y Fp(U)4893 20915 y Fk(k)5926 20716 y Fq(=)379 b Fp(T)184 b(M)445 b Fm(\002)299 b Fl(R)p Fq(.)609 b(Let)449 b Fp(L)16591 20915 y Fk(i)17413 20716 y Fm(2)378 b Fp(C)19721 20234 y Fk(r)20310 20716 y Fq(\()p Fp(T)184 b(M)444 b Fm(\002)299 b Fl(T)p Fp(;)221 b Fl(R)p Fq(\).)609 b(W)-111 b(e)450 b(sa)-37 b(y)449 b Fp(L)34281 20915 y Fk(i)35172 20716 y Fq(con)-37 b(v)g(erges)451 b(to)d Fp(L)h Fq(if)g(for)g(eac)-37 b(h)1102 22310 y Fp(\017)370 b(>)f Fq(0)443 b(and)g(eac)-37 b(h)443 b Fp(U)10998 22509 y Fk(k)12095 22310 y Fq(there)g(exists)g Fp(i)19619 22509 y Fv(0)20657 22310 y Fq(suc)-37 b(h)443 b(that)f Fm(k)p Fp(L)296 b Fm(\000)f Fp(L)30627 22509 y Fk(i)31069 22310 y Fm(k)31733 22509 y Fk(U)32457 22650 y Fd(k)33473 22310 y Fm(\024)369 b Fp(\017)443 b Fq(if)g Fp(i)369 b Fm(\025)g Fp(i)39762 22509 y Fv(0)40357 22310 y Fq(.)1102 24640 y Fr(Pro)42 b(of)p Fq(:)719 b(Since)507 b Fp(M)651 b Fq(is)507 b(connected)g(and)g(compact,)523 b(an)-37 b(y)507 b(t)-37 b(w)g(o)507 b(p)37 b(oin)-37 b(t)507 b Fp(x)36881 24839 y Fv(1)37983 24640 y Fp(x)38742 24839 y Fv(2)39814 24640 y Fm(2)475 b Fp(M)651 b Fq(can)507 b(b)37 b(e)507 b(con-)1102 26234 y(nected)429 b(b)-37 b(y)429 b(a)g(geo)37 b(desic.)587 b(Let)428 b Fp(`)p Fq(\()p Fp(x)18134 26433 y Fv(1)18730 26234 y Fp(;)221 b(x)20079 26433 y Fv(2)20676 26234 y Fq(\))428 b(b)37 b(e)428 b(the)h(length)f(of)h(the)f(shortest)h(geo)37 b(desic)429 b(connecting)1102 27828 y(these)613 b(t)-37 b(w)g(o)612 b(p)37 b(oin)-37 b(ts,)655 b(there)612 b(is)h(uniformly)f (upp)37 b(er)613 b(b)37 b(ound)612 b Fp(K)32590 28027 y Fv(1)33838 27828 y Fp(>)651 b Fq(0)612 b(of)h Fp(`)p Fq(\()p Fp(x)40310 28027 y Fv(1)40906 27828 y Fp(;)221 b(x)42255 28027 y Fv(2)42851 27828 y Fq(\))612 b(for)h(all)f Fp(x)48956 28027 y Fv(1)49552 27828 y Fq(,)1102 29422 y Fp(x)1861 29621 y Fv(2)2826 29422 y Fm(2)369 b Fp(M)145 b Fq(.)590 b(Let)15459 31016 y Fp(K)464 b Fq(=)4935 b(max)18453 32066 y Fv(\()p Fk(q)33 b(;t)p Fv(\))p Fg(2)p Fc(T)21855 31665 y Fj(2)22375 32066 y Fg(\002)p Fv(\()p Fc(T)p Fg(_)p Fv([0)p Fk(;)p Fv(1])p Fg(_)p Fc(T)p Fv(\))p Fk(=)p Fg(\030)22318 33131 y(k)p Fk(\020)60 b Fg(k\024)p Fk(K)25672 33264 y Fj(1)30278 31016 y Fp(L)p Fq(\()p Fp(q)48 b(;)221 b(\020)98 b(;)221 b(t)p Fq(\))p Fp(:)1102 35432 y Fq(Giv)-37 b(en)342 b(time)e(in)-37 b(terv)-74 b(al)342 b([)p Fp(a;)221 b(b)p Fq(])341 b(with)f Fp(b)91 b Fm(\000)g Fp(a)369 b Fm(\025)g Fq(1,)361 b(if)341 b(w)-37 b(e)341 b(reparametrize)i(the)e(shortest)f (geo)37 b(desic)342 b Fp(\015)74 b Fq(\()p Fp(s)p Fq(\))1102 37026 y(b)-37 b(y)563 b(\026)-713 b Fp(\015)74 b Fq(\()p Fp(t)p Fq(\))486 b(=)i Fp(\015)74 b Fq(\()p Fp(`)p Fq(\()p Fp(x)10409 37225 y Fv(1)11004 37026 y Fp(;)221 b(x)12353 37225 y Fv(2)12950 37026 y Fq(\)\()p Fp(t)341 b Fm(\000)i Fp(a)p Fq(\))p Fp(=)p Fq(\()p Fp(b)f Fm(\000)h Fp(a)p Fq(\)\),)531 b(then)563 b(\026)-713 b Fp(\015)73 b Fq(\()p Fp(t)p Fq(\))513 b(is)h(a)g Fp(C)33019 36544 y Fv(1)33616 37026 y Fq(-curv)-37 b(e)514 b(suc)-37 b(h)515 b(that)562 b(\026)-713 b Fp(\015)74 b Fq(\()p Fp(a)p Fq(\))486 b(=)i Fp(x)48956 37225 y Fv(1)49552 37026 y Fq(,)1151 38620 y(\026)-713 b Fp(\015)74 b Fq(\()p Fp(b)p Fq(\))567 b(=)h Fp(x)6395 38819 y Fv(2)6991 38620 y Fq(.)948 b(Clearly)-111 b(,)594 b(the)562 b(action)g(of)h Fp(L)f Fq(along)g(this)g(curv)-37 b(e)563 b(is)f(not)g(bigger)h(than)f Fp(K)95 b Fq(\()p Fp(b)375 b Fm(\000)g Fp(a)p Fq(\).)1102 40214 y(Ob)-37 b(viously)-111 b(,)514 b(there)499 b(is)g(a)g(upp)37 b(er)500 b(b)37 b(ound)498 b(uniformly)i(for)f(all)g(minimizing)g (action)g(of)g Fp(L)45230 39732 y Fg(0)46102 40214 y Fq(if)f(they)1102 41808 y(close)444 b(to)f Fp(L)f Fq(on)h Fm(fk)p Fp(\020)98 b Fm(k)370 b(\024)f Fp(K)14669 42007 y Fv(1)15265 41808 y Fm(g)p Fq(,)443 b(still)f(denoted)i(b)-37 b(y)443 b Fp(K)95 b Fq(\()p Fp(b)295 b Fm(\000)g Fp(a)p Fq(\).)2696 44138 y(Since)519 b(the)f(sup)37 b(er-linear)519 b(gro)-37 b(wth)519 b(is)f(assumed,)537 b(there)519 b(are)g(t)-37 b(w)g(o)518 b(constan)-37 b(t)518 b Fp(C)614 b Fq(and)518 b Fp(D)555 b Fq(suc)-37 b(h)1102 45732 y(that)456 b Fp(L)4898 45250 y Fg(0)5271 45732 y Fq(\()p Fp(q)48 b(;)467 b Fq(_)-615 b Fp(q)47 b(;)221 b(t)p Fq(\))392 b Fm(\025)g Fp(C)95 b Fm(k)247 b Fq(_)-616 b Fp(q)48 b Fm(k)304 b(\000)h Fp(D)493 b Fq(for)457 b(all)f(\()p Fp(q)48 b(;)468 b Fq(_)-616 b Fp(q)48 b(;)221 b(t)p Fq(\))391 b Fm(2)g Fp(T)184 b(M)450 b Fm(\002)305 b Fq([)p Fp(a;)221 b(b)p Fq(])457 b(and)f(for)h(all)g Fp(L)41360 45250 y Fg(0)42189 45732 y Fq(close)g(to)f Fp(L)p Fq(.)632 b(It)1102 47326 y(follo)-37 b(ws)444 b(that)12606 48685 y(dist\()p Fp(\015)74 b Fq(\()p Fp(a)p Fq(\))p Fp(;)221 b(\015)74 b Fq(\()p Fp(b)p Fq(\)\))p 12606 49278 8633 54 v 15475 50495 a Fp(b)295 b Fm(\000)g Fp(a)21767 49584 y Fm(\024)24445 48685 y Fq(1)p 23329 49278 2896 54 v 23329 50495 a Fp(b)g Fm(\000)h Fp(a)26606 47776 y Fi(Z)27934 48104 y Fk(b)27344 50794 y(a)28689 49584 y Fm(k)p Fp(d\015)74 b Fm(k)369 b(\024)33401 48685 y Fq(\()p Fp(K)390 b Fq(+)295 b Fp(D)37 b Fq(\))p 33401 49278 5017 54 v 35387 50495 a Fp(C)46527 49584 y Fq(\(2.11\))1102 52730 y(if)548 b Fp(\015)622 b Fq(is)548 b(a)h(minimizer.)907 b(As)549 b(\(2.11\))f(holds)g(for)h (an)-37 b(y)548 b Fp(b)366 b Fm(\000)f Fp(a)545 b Fm(\025)g Fq(1,)575 b(it)548 b(implies)g(that)g(there)h(m)-37 b(ust)1102 54324 y(b)37 b(e)554 b(some)g Fp(\034)7041 54523 y Fk(i)8036 54324 y Fm(2)f Fq([)p Fp(a)369 b Fq(+)g Fp(i;)221 b(a)369 b Fq(+)g Fp(i)f Fq(+)h(1])554 b(\()p Fp(i)e Fm(2)h Fl(Z)p Fq(\))g(suc)-37 b(h)553 b(that)g Fm(k)196 b Fq(_)-565 b Fp(\015)74 b Fq(\()p Fp(\034)33194 54523 y Fk(i)33635 54324 y Fq(\))p Fm(k)553 b(\024)h Fp(C)38000 53842 y Fg(\000)p Fv(1)39426 54324 y Fq(\()p Fp(K)464 b Fq(+)369 b Fp(D)37 b Fq(\).)921 b(By)553 b(the)1102 55918 y(compactness)526 b(of)f Fp(M)494 b Fm(\002)350 b Fq(\()p Fl(T)f Fm(_)g Fq([0)p Fp(;)221 b Fq(1])351 b Fm(_)e Fl(T)p Fq(\))p Fp(=)505 b Fm(\030)525 b Fq(w)-37 b(e)525 b(see)h(that)e(there)h (exists)g Fp(K)40579 56117 y Fv(2)41680 55918 y Fp(>)506 b Fq(0)525 b(suc)-37 b(h)525 b(that)1102 57512 y Fm([)1988 57752 y Fk(s)p Fg(2)p Fv([0)p Fk(;)p Fv(1])5247 57512 y Fp(\036)6038 57030 y Fk(s)6607 57512 y Fq(\()p Fm(f)p Fp(q)48 b(;)221 b(\030)61 b(;)221 b(t)449 b Fq(:)h(\()p Fp(q)48 b(;)221 b(t)p Fq(\))448 b Fm(2)h Fp(M)472 b Fm(\002)328 b Fq(\()p Fl(T)e Fm(_)h Fq([0)p Fp(;)221 b Fq(1])328 b Fm(_)f Fl(T)p Fq(\))p Fp(=)449 b Fm(\030)p Fp(;)221 b Fm(k)p Fp(\030)61 b Fm(k)451 b(\024)f Fp(C)35875 57030 y Fg(\000)p Fv(1)37301 57512 y Fq(\()p Fp(K)422 b Fq(+)328 b Fp(D)37 b Fq(\))p Fm(g)p Fq(\))449 b Fm(\032)g(f)p Fp(q)48 b(;)221 b(\030)61 b(;)221 b(t)450 b Fq(:)1102 59113 y(\()p Fp(q)48 b(;)221 b(t)p Fq(\))368 b Fm(2)h Fp(M)439 b Fm(\002)296 b Fq(\()p Fl(T)e Fm(_)h Fq([0)p Fp(;)221 b Fq(1])296 b Fm(_)f Fl(T)p Fq(\))p Fp(=)368 b Fm(\030)p Fp(;)221 b Fm(k)p Fp(\030)61 b Fm(k)370 b(\024)f Fp(K)24466 59312 y Fv(2)25062 59113 y Fm(g)p Fq(.)2696 61443 y(Let)422 b Fp(L)5959 61642 y Fk(i)6770 61443 y Fm(2)368 b Fp(C)9068 60961 y Fk(r)9657 61443 y Fq(\()p Fp(T)184 b(M)397 b Fm(\002)251 b Fl(R)p Fp(;)221 b Fl(R)p Fq(\))422 b(b)37 b(e)421 b(a)h(sequence)g(con)-37 b(v)g(erging)423 b(to)e Fp(L)p Fq(,)k(let)c Fp(\015)38378 61642 y Fk(i)38820 61443 y Fq(:)580 b([)p Fp(a;)221 b(b)p Fq(])369 b Fm(!)g Fp(M)566 b Fq(b)37 b(e)421 b(the)1102 63037 y(minimizer)519 b(of)f Fp(L)9872 63236 y Fk(i)10832 63037 y Fq(with)g Fp(b)345 b Fm(\000)g Fp(a)494 b Fm(\025)h Fq(1.)815 b(By)518 b(the)g(argumen)-37 b(t)518 b(ab)37 b(o)-37 b(v)g(e,)537 b(w)-37 b(e)519 b(see)g(there)f(exists)g(some)1102 64631 y Fp(U)2009 64830 y Fk(k)3032 64631 y Fm(\033)369 b(f)p Fp(\030)61 b(;)221 b(q)48 b(;)221 b(t)370 b Fq(:)f(\()p Fp(q)48 b(;)221 b(t)p Fq(\))367 b Fm(2)i Fp(M)278 b Fm(\002)133 b Fl(R)p Fp(;)221 b Fm(k)p Fp(\030)61 b Fm(k)370 b(\024)g Fp(K)22669 64830 y Fv(2)23265 64631 y Fm(g)p Fq(.)563 b(So)362 b Fm(k)p Fq(\()p Fp(L)p Fq(\()p Fp(z)58 b(;)221 b(t)p Fq(\))133 b Fm(\000)g Fp(L)33693 64830 y Fk(i)34135 64631 y Fq(\()p Fp(z)58 b(;)221 b(t)p Fq(\))p Fm(k)37579 64830 y Fk(U)38303 64971 y Fd(k)39319 64631 y Fm(\024)369 b Fp(\017)41260 64830 y Fk(i)41702 64631 y Fq(,)378 b(where)363 b Fp(\017)46748 64830 y Fk(i)47560 64631 y Fm(!)369 b Fq(0)1102 66225 y(as)443 b Fp(i)369 b Fm(!)g(1)p Fq(.)591 b(Th)-37 b(us)15249 66675 y Fi(Z)16577 67003 y Fk(b)15987 69693 y(a)17332 68483 y Fp(L)p Fq(\()p Fp(d\015)20132 68682 y Fk(i)20574 68483 y Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))p Fp(dt)368 b Fm(\024)h Fq(\()p Fp(K)390 b Fq(+)295 b Fp(\017)30518 68682 y Fk(i)30960 68483 y Fq(\)\()p Fp(b)g Fm(\000)g Fp(a)p Fq(\))p Fp(:)10752 b Fq(\(2.12\))1102 71714 y(As)412 b(all)f Fp(\015)5535 71913 y Fk(i)6388 71714 y Fq(is)g(a)g Fp(C)9811 71232 y Fv(1)10408 71714 y Fq(-curv)-37 b(e)412 b(and)f(the)g(actions)g(of)g Fp(L)g Fq(on)h(eac)-37 b(h)412 b Fp(\015)31904 71913 y Fk(i)32757 71714 y Fq(are)g(b)37 b(ounded)411 b(b)-37 b(y)411 b([2.12],)418 b(the)411 b(set)1102 73308 y Fm(f)p Fp(\015)2454 73507 y Fk(i)2897 73308 y Fm(g)3561 73507 y Fk(i)p Fg(2)p Fc(Z)5273 73640 y Fj(+)6556 73308 y Fq(is)469 b(compact)g(in)g(the)f Fp(C)18190 72826 y Fv(0)18786 73308 y Fq(-top)37 b(ology)469 b(\(cf)f([Ma3]\).)668 b(Moreo)-37 b(v)g(er)470 b(this)e(set)h(is)f(compact)h(in)1102 74965 y(the)453 b Fp(C)4444 74483 y Fv(1)5040 74965 y Fq(\([)p Fp(a;)221 b(b)p Fq(])p Fp(;)g(M)145 b Fq(\)-top)37 b(ology)452 b(as)h(w)-37 b(e)453 b(ha)-37 b(v)g(e)453 b Fm(k)196 b Fq(_)-565 b Fp(\015)24641 75164 y Fk(i)25083 74965 y Fm(k)385 b(\024)g Fp(K)28678 75164 y Fv(2)29727 74965 y Fq(and)452 b(as)h Fp(@)34739 74483 y Fv(2)35335 74965 y Fp(L=@)321 b Fq(_)-616 b Fp(q)38323 74483 y Fv(2)39371 74965 y Fq(is)452 b(p)37 b(ositiv)-37 b(e)453 b(de\014nite)p eop %%Page: 8 8 8 7 bop 1102 2239 a Ft(8)1102 5427 y Fq(so)404 b(w)-37 b(e)405 b(can)f(write)h(the)f(Lagrange)h(equations)f(in)g(the)g(form)g (of)503 b(\177)-763 b Fp(q)417 b Fq(=)369 b Fp(f)143 b Fq(\()p Fp(q)48 b(;)467 b Fq(_)-615 b Fp(q)47 b(;)221 b(t)p Fq(\),)412 b(whic)-37 b(h)404 b(implies)g Fp(\015)49479 5626 y Fk(i)1102 7021 y Fq(is)443 b(b)37 b(ounded)443 b(in)g Fp(C)10456 6539 y Fv(2)11052 7021 y Fq(-top)37 b(ology)-111 b(.)2696 9269 y(Let)337 b Fp(\015)74 b Fq(:)536 b([)p Fp(a;)221 b(b)p Fq(])370 b Fm(!)f Fp(M)481 b Fq(b)37 b(e)336 b(one)h(of)f(the)g(accum)-37 b(ulate)337 b(p)37 b(oin)-37 b(t)336 b(of)g(this)g(set.)554 b(Clearly)-111 b(,)359 b Fp(\015)74 b Fq(:)537 b([)p Fp(a;)221 b(b)p Fq(])369 b Fm(!)h Fp(M)1102 10863 y Fq(is)443 b(the)g(minimizer)h(of)e Fp(L)h Fq(and)g(w)-37 b(e)444 b(ha)-37 b(v)g(e)16399 14073 y Fp(A)17395 14272 y Fk(c)17936 14073 y Fq(\()p Fp(\015)74 b Fq(\))368 b(=)694 b(lim)21502 14910 y Fk(i)p Fg(!1)24217 12265 y Fi(Z)25546 12594 y Fk(b)24955 15284 y(a)26301 14073 y Fp(L)27205 14272 y Fk(i)27647 14073 y Fq(\()p Fp(d\015)29543 14272 y Fk(i)29985 14073 y Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))p Fp(dt:)1102 17365 y Fq(W)-111 b(e)533 b(let)g Fp(I)6072 17564 y Fk(i)7032 17365 y Fq(=)518 b([)p Fm(\000)p Fp(T)10761 17564 y Fk(i)11203 17365 y Fp(;)221 b(T)12569 17564 y Fk(i)13012 17365 y Fq(])532 b(and)h(let)f Fp(T)19370 17564 y Fk(i)20330 17365 y Fm(!)518 b(1)p Fq(,)555 b(there)533 b(is)f(a)g(sequence)i(of)e (minimizers)h(of)g Fp(L)47057 17564 y Fk(i)47499 17365 y Fq(,)554 b Fp(\015)49110 17564 y Fk(i)49552 17365 y Fq(:)1102 18959 y Fp(I)1686 19158 y Fk(i)2649 18959 y Fm(!)521 b Fp(M)145 b Fq(.)862 b(By)534 b(diagonal)g(extraction)g (argumen)-37 b(t)534 b(w)-37 b(e)535 b(can)f(\014nd)f(a)h(subsequence)h (of)f Fp(\015)45588 19158 y Fk(i)46563 18959 y Fq(whic)-37 b(h)1102 20553 y(con)g(v)g(erges)475 b Fp(C)8128 20071 y Fv(1)9197 20553 y Fq(uniformly)d(on)h(eac)-37 b(h)474 b(compact)f(set)f(to)g(a)h Fp(C)31455 20071 y Fv(1)32051 20553 y Fq(-curv)-37 b(e)474 b Fp(\015)74 b Fq(:)649 b Fl(R)419 b Fm(!)g Fp(M)617 b Fq(whic)-37 b(h)473 b(is)f(the)1102 22147 y(minimizer)425 b(of)e Fp(L)h Fq(on)f(an)-37 b(y)424 b(compact)g(in)-37 b(terv)-74 b(al)425 b(of)e Fl(R)p Fq(.)584 b(This)424 b(pro)-37 b(v)g(es)425 b(the)e(upp)37 b(er)424 b(semi-con)-37 b(tin)g(uit)g(y)1102 23761 y(of)443 b Fp(L)369 b Fm(!)5836 23425 y Fq(~)5586 23761 y Fm(G)6376 23960 y Fk(L)7171 23761 y Fq(.)2696 26009 y(Giv)-37 b(en)443 b Fp(L)f Fq(p)37 b(erio)g(dic)443 b(in)e Fp(t)p Fq(,)h(w)-37 b(e)443 b(let)e Fp(L)20598 26208 y Fk(c)21508 26009 y Fq(=)370 b Fp(L)293 b Fm(\000)g Fp(\021)26094 26208 y Fk(c)27077 26009 y Fq(where)443 b Fp(\021)31577 26208 y Fk(c)32559 26009 y Fq(is)f(a)g(closed)g(one)h(form)f(suc)-37 b(h)442 b(that)1102 27603 y([)p Fp(\021)2131 27802 y Fk(c)2672 27603 y Fq(])458 b(=)g Fp(c)p Fq(.)751 b Fp(\021)7345 27802 y Fk(c)8382 27603 y Fq(is)496 b(a)g(linear)h(function)g(in)742 b(_)-615 b Fp(q)47 b Fq(.)751 b(If)496 b Fp(c)25255 27802 y Fk(i)26155 27603 y Fm(!)458 b Fp(c)p Fq(,)510 b(w)-37 b(e)497 b(can)g(c)-37 b(ho)37 b(ose)497 b(a)f(sequence)i(of)e(closed) 1102 29197 y(1-form)354 b Fp(\021)5920 29396 y Fk(c)6395 29529 y Fd(i)7235 29197 y Fq(suc)-37 b(h)354 b(that)f([)p Fp(\021)13960 29396 y Fk(c)14435 29529 y Fd(i)14922 29197 y Fq(])369 b(=)g Fp(c)17637 29396 y Fk(i)18432 29197 y Fq(and)354 b Fm(j)p Fp(\021)21955 29396 y Fk(c)22430 29529 y Fd(i)23033 29197 y Fm(\000)117 b Fp(\021)24843 29396 y Fk(c)25384 29197 y Fm(j)25753 29437 y Fg(k)196 b Fv(_)-497 b Fk(q)33 b Fg(k\024)p Fk(K)29084 29570 y Fj(1)30038 29197 y Fm(!)369 b Fq(0.)561 b(In)354 b(this)f(case)h Fp(L)41028 29396 y Fk(c)41503 29529 y Fd(i)42359 29197 y Fm(!)369 b Fp(L)44960 29396 y Fk(c)45855 29197 y Fq(implies)1102 30791 y Fp(c)1677 30990 y Fk(i)2515 30791 y Fm(!)396 b Fp(c)p Fq(.)638 b(Since)460 b(the)f Fp(c)p Fq(-minimal)f(orbits)h (are)h(indep)37 b(enden)-37 b(t)459 b(of)g(the)g(c)-37 b(hoice)460 b(of)f Fp(\021)41363 30990 y Fk(i)41804 30791 y Fq(,)k(applying)c(the)1102 32405 y(argumen)-37 b(t)444 b(ab)37 b(o)-37 b(v)g(e)443 b(w)-37 b(e)444 b(obtain)f(the)f(upp)37 b(er)444 b(semi-con)-37 b(tin)g(uit)g(y)443 b Fp(c)368 b Fm(!)35188 32069 y Fq(~)34938 32405 y Fm(G)79 b Fq(\()p Fp(c)p Fq(\).)2164 b Fb(\003)2696 34747 y Fq(In)493 b(the)g (application,)506 b(the)493 b(set)18752 34412 y(~)18502 34747 y Fm(G)79 b Fq(\()p Fp(c)p Fq(\))492 b(seems)i(to)37 b(o)493 b(bigger)h(to)f(b)37 b(e)493 b(used)g(for)h(the)f(construction) 1102 36341 y(of)545 b(connecting)g(orbits)g(in)f(in)-37 b(teresting)545 b(problems.)896 b(Ma)37 b(~)-701 b(n)-37 b(\023)-627 b(e)545 b(sets)f(seem)i(go)37 b(o)g(d)544 b(candidates.)897 b(In)1102 37955 y(the)473 b(time-p)37 b(erio)g(dic)473 b(case,)481 b(Ma)37 b(~)-701 b(n)-37 b(\023)-627 b(e)473 b(set)g(can)g(b)37 b(e)473 b(a)g(prop)37 b(er)473 b(subset)g(of)36193 37620 y(~)35943 37955 y Fm(G)79 b Fq(\()p Fp(c)p Fq(\),)39690 37620 y(~)39269 37955 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\))417 b Fl(\()44283 37620 y Fq(~)44033 37955 y Fm(G)79 b Fq(\()p Fp(c)p Fq(\).)679 b(It)472 b(is)1102 39549 y(closely)422 b(related)f(to)f(the)h(problem)g (whether)g(the)g(Lax-Oleinik)g(semi-group)g(con)-37 b(v)g(erges)423 b(or)e(not,)1102 41144 y(some)438 b(example)g(can)f(b)37 b(e)437 b(found)g(in)g([FM].)g(T)-111 b(o)438 b(establish)f(the)g (connection)h(b)37 b(et)-37 b(w)g(een)438 b(t)-37 b(w)g(o)437 b(Ma)37 b(~)-701 b(n)-37 b(\023)-627 b(e)1102 42738 y(sets)443 b(w)-37 b(e)444 b(consider)g(a)f(mo)37 b(di\014ed)443 b(Lagrangian)20541 45283 y Fp(L)21445 45482 y Fk(\021)33 b(;\026)23422 45283 y Fq(=)369 b Fp(L)295 b Fm(\000)h Fp(\021)342 b Fm(\000)296 b Fp(\026)1102 47829 y Fq(where)483 b Fp(\021)528 b Fq(is)482 b(a)f(closed)h(1-form)g(on)f Fp(M)626 b Fq(suc)-37 b(h)481 b(that)g([)p Fp(\021)48 b Fq(])432 b(=)i Fp(c)p Fq(,)490 b Fp(\026)482 b Fq(is)f(a)h(1-form)f (dep)37 b(ending)482 b(on)f Fp(t)g Fq(in)1102 49423 y(the)390 b(w)-37 b(a)g(y)391 b(that)e(the)h(restriction)h(of)f Fp(\026)g Fq(on)g Fm(f)p Fp(t)369 b Fm(\024)g Fq(0)p Fm(g)391 b Fq(is)f(0,)400 b(the)390 b(restriction)h(on)f Fm(f)p Fp(t)369 b Fm(\025)g Fq(1)p Fm(g)390 b Fq(is)g(a)g(closed)1102 51017 y(1-form)548 b(\026)-769 b Fp(\026)444 b Fq(on)f Fp(M)587 b Fq(with)443 b([)105 b(\026)-769 b Fp(\026)p Fq(])369 b(=)g Fp(c)17224 50535 y Fg(0)17892 51017 y Fm(\000)295 b Fp(c)p Fq(.)590 b(Let)443 b Fp(m)24300 51216 y Fv(0)24896 51017 y Fp(;)221 b(m)26652 51216 y Fv(1)27618 51017 y Fm(2)369 b Fp(M)145 b Fq(,)442 b(w)-37 b(e)444 b(de\014ne)9591 54549 y Fp(h)10356 54001 y Fk(T)10983 54134 y Fj(0)11502 54001 y Fk(;T)12444 54134 y Fj(1)10356 54878 y Fk(\021)33 b(;\026)13030 54549 y Fq(\()p Fp(m)14713 54748 y Fv(0)15308 54549 y Fp(;)221 b(m)17064 54748 y Fv(1)17661 54549 y Fq(\))369 b(=)2262 b(inf)19801 55468 y Fk(\015)57 b Fv(\()p Fg(\000)p Fk(T)22288 55601 y Fj(0)22808 55468 y Fv(\)=)p Fk(m)24981 55601 y Fj(0)20216 56533 y Fk(\015)g Fv(\()p Fk(T)21873 56666 y Fj(1)22392 56533 y Fv(\)=)p Fk(m)24565 56666 y Fj(1)25721 52741 y Fi(Z)27049 53069 y Fk(T)27676 53202 y Fj(1)26459 55759 y Fg(\000)p Fk(T)27916 55892 y Fj(0)28501 54549 y Fq(\()p Fp(L)295 b Fm(\000)g Fp(\021)343 b Fm(\000)295 b Fp(\026)p Fq(\)\()p Fp(d\015)74 b Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))p Fp(dt)19875 58496 y Fm(\000)295 b Fp(T)21979 58695 y Fv(0)22575 58496 y Fp(\013)5 b Fq(\()p Fp(c)p Fq(\))294 b Fm(\000)i Fp(T)27438 58695 y Fv(1)28034 58496 y Fp(\013)5 b Fq(\()p Fp(c)29981 57947 y Fg(0)30353 58496 y Fq(\))p Fp(:)15288 b Fq(\(2.13\))1102 61041 y(Clearly)528 b(there)e(is)g Fp(m)11868 60559 y Fg(\003)12987 61041 y Fm(2)508 b Fp(M)670 b Fq(and)526 b(some)h(constan)-37 b(t)526 b Fp(C)28808 61240 y Fk(\026)29521 61041 y Fq(,)547 b Fp(C)31386 61240 y Fk(\021)33 b(;\026)32995 61041 y Fq(,)546 b(indep)37 b(enden)-37 b(t)527 b(of)f Fp(T)43893 61240 y Fv(0)44489 61041 y Fp(;)221 b(T)45855 61240 y Fv(1)46452 61041 y Fq(,)547 b(suc)-37 b(h)1102 62635 y(that)10290 65181 y Fp(h)11055 64633 y Fk(T)11682 64766 y Fj(0)12201 64633 y Fk(;T)13143 64766 y Fj(1)11055 65510 y Fk(\021)33 b(;\026)13729 65181 y Fq(\()p Fp(m)15412 65380 y Fv(0)16008 65181 y Fp(;)221 b(m)17764 65380 y Fv(1)18360 65181 y Fq(\))369 b Fm(\024)g Fp(h)21413 64633 y Fk(T)22040 64766 y Fj(0)21413 65510 y Fk(\021)22626 65181 y Fq(\()p Fp(m)24309 65380 y Fv(0)24904 65181 y Fp(;)221 b(m)26660 64633 y Fg(\003)27272 65181 y Fq(\))295 b(+)g Fp(h)30177 64591 y Fk(T)30804 64724 y Fj(2)30177 65512 y Fk(\021)33 b Fv(+)87 b(\026)-616 b Fk(\026)32286 65181 y Fq(\()p Fp(m)33969 64633 y Fg(\003)34579 65181 y Fp(;)221 b(m)36335 65380 y Fv(2)36932 65181 y Fq(\))295 b(+)g Fp(C)40021 65380 y Fk(\026)19246 67174 y Fm(\024)369 b Fp(C)21597 67373 y Fk(\021)33 b(;\026)23206 67174 y Fp(:)1102 69720 y Fq(Th)-37 b(us)444 b(its)e(limit)g(in\014m) -37 b(um)443 b(is)g(b)37 b(ounded)14172 72266 y Fp(h)14937 71718 y Fg(1)14937 72595 y Fk(\021)c(;\026)16546 72266 y Fq(\()p Fp(m)18229 72465 y Fv(0)18825 72266 y Fp(;)221 b(m)20581 72465 y Fv(1)21178 72266 y Fq(\))368 b(=)743 b(lim)221 b(inf)23318 73123 y Fk(T)23945 73256 y Fj(0)24464 73123 y Fk(;T)25406 73256 y Fj(1)25925 73123 y Fg(!1)28264 72266 y Fp(h)29029 71718 y Fk(T)29656 71851 y Fj(0)30175 71718 y Fk(;T)31117 71851 y Fj(1)29029 72595 y Fk(\021)33 b(;\026)31703 72266 y Fq(\()p Fp(m)33386 72465 y Fv(0)33982 72266 y Fp(;)221 b(m)35738 72465 y Fv(1)36335 72266 y Fq(\))22063 74965 y Fm(\024)p Fp(C)24045 75164 y Fk(\021)33 b(;\026)25654 74965 y Fp(:)20504 b Fq(\(2.14\))p eop %%Page: 9 9 9 8 bop 49357 2239 a Ft(9)1102 5427 y Fq(Let)498 b Fm(f)p Fp(T)5161 4945 y Fk(i)4977 5757 y Fv(0)5604 5427 y Fm(g)6268 5626 y Fk(i)p Fg(2)p Fc(Z)7980 5759 y Fj(+)9292 5427 y Fq(and)g Fm(f)p Fp(T)13554 4945 y Fk(i)13370 5757 y Fv(1)13997 5427 y Fm(g)14661 5626 y Fk(i)p Fg(2)p Fc(Z)16373 5759 y Fj(+)17685 5427 y Fq(b)37 b(e)498 b(the)f(sequence)i(of)e(p)37 b(ositiv)-37 b(e)497 b(in)-37 b(tegers)498 b(suc)-37 b(h)498 b(that)e Fp(T)45902 4945 y Fk(i)45718 5773 y(j)46805 5427 y Fm(!)460 b(1)1102 7083 y Fq(\()p Fp(j)445 b Fq(=)369 b(0)p Fp(;)221 b Fq(1\))443 b(as)g Fp(i)369 b Fm(!)g(1)443 b Fq(and)g(the)g(follo)-37 b(wing)443 b(limit)g(exists)15309 10243 y(lim)14984 11080 y Fk(i)p Fg(!1)17700 10243 y Fp(h)18465 9520 y Fk(T)19235 9119 y Fd(i)19092 9785 y Fj(0)19656 9520 y Fk(;T)20741 9119 y Fd(i)20598 9785 y Fj(1)18465 10398 y Fk(\021)33 b(;\026)21229 10243 y Fq(\()p Fp(m)22912 10442 y Fv(0)23508 10243 y Fp(;)221 b(m)25264 10442 y Fv(1)25861 10243 y Fq(\))368 b(=)h Fp(h)28913 9695 y Fg(1)28913 10571 y Fk(\021)33 b(;\026)30522 10243 y Fq(\()p Fp(m)32205 10442 y Fv(0)32801 10243 y Fp(;)221 b(m)34557 10442 y Fv(1)35154 10243 y Fq(\))p Fp(:)1102 13571 y Fq(Let)443 b Fp(\015)4170 13770 y Fk(i)4612 13571 y Fq(\()p Fp(t;)221 b(m)7365 13770 y Fv(0)7961 13571 y Fp(;)g(m)9717 13770 y Fv(1)10314 13571 y Fq(\):)590 b([)p Fm(\000)p Fp(T)14152 13089 y Fk(i)13968 13901 y Fv(0)14595 13571 y Fp(;)221 b(T)16145 13089 y Fk(i)15961 13901 y Fv(1)16588 13571 y Fq(])369 b Fm(!)h Fp(M)587 b Fq(b)37 b(e)443 b(the)g(minimizer)g(connecting)h Fp(m)39006 13770 y Fv(0)40045 13571 y Fq(and)f Fp(m)43794 13770 y Fv(1)6364 17488 y Fp(h)7129 16765 y Fk(T)7899 16364 y Fd(i)7756 17031 y Fj(0)8321 16765 y Fk(;T)9406 16364 y Fd(i)9263 17031 y Fj(1)7129 17643 y Fk(\021)33 b(;\026)9894 17488 y Fq(\()p Fp(m)11577 17687 y Fv(0)12173 17488 y Fp(;)221 b(m)13929 17687 y Fv(1)14525 17488 y Fq(\))369 b(=)16813 15680 y Fi(Z)18141 16008 y Fk(T)18911 15608 y Fd(i)18768 16274 y Fj(1)17551 18698 y Fg(\000)p Fk(T)19151 18378 y Fd(i)19008 18987 y Fj(0)19639 17488 y Fq(\()p Fp(L)295 b Fm(\000)g Fp(\021)342 b Fm(\000)296 b Fp(\026)p Fq(\)\()p Fp(d\015)28227 17687 y Fk(i)28669 17488 y Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))p Fp(dt)293 b Fm(\000)j Fp(T)35523 16940 y Fk(i)35339 17816 y Fv(0)35966 17488 y Fp(\013)5 b Fq(\()p Fp(c)p Fq(\))294 b Fm(\000)h Fp(T)41012 16940 y Fk(i)40828 17816 y Fv(1)41455 17488 y Fp(\013)5 b Fq(\()p Fp(c)43402 16940 y Fg(0)43774 17488 y Fq(\))p Fp(:)1102 21245 y Fq(F)-111 b(rom)399 b(the)e(pro)37 b(of)398 b(of)g(the)f(lemma)h(2.2)g(w)-37 b(e)398 b(can)g(see)h(that)d (for)i(an)-37 b(y)398 b(compact)g(in)-37 b(terv)-74 b(al)399 b([)p Fp(a;)221 b(b)p Fq(])398 b(there)1102 22839 y(is)443 b(some)g Fp(I)474 b Fm(2)368 b Fl(Z)8964 23038 y Fv(+)10288 22839 y Fq(suc)-37 b(h)443 b(that)g(the)f(set)h Fm(f)p Fp(\015)21876 23038 y Fk(i)22318 22839 y Fm(g)22982 23038 y Fk(i)p Fg(\025)p Fk(I)25241 22839 y Fq(is)f(pre-compact)i(in)f Fp(C)36775 22357 y Fv(1)37371 22839 y Fq(\([)p Fp(a;)221 b(b)p Fq(])p Fp(;)g(M)145 b Fq(\).)1102 25126 y Fr(Lemma)509 b(2.3.)764 b Fo(L)-68 b(et)476 b Fp(\015)74 b Fo(:)611 b Fl(R)369 b Fm(!)h Fp(M)619 b Fo(b)-68 b(e)476 b(an)f(ac)-68 b(cumulation)476 b(p)-68 b(oint)476 b(of)f Fm(f)p Fp(\015)36584 25325 y Fk(i)37026 25126 y Fm(g)p Fo(.)612 b(If)476 b Fp(s)368 b Fm(\025)i Fq(1)475 b Fo(then)9358 28600 y Fp(A)10354 28799 y Fk(L)11083 28932 y Fd(\021)24 b(;\026)12602 28600 y Fq(\()p Fp(\015)74 b Fm(j)p Fq([)p Fp(s;)221 b(\034)150 b Fq(]\))368 b(=)2627 b(inf)19072 29462 y Fk(\034)19562 29595 y Fj(1)20080 29462 y Fg(\000)p Fk(\034)107 b Fg(2)p Fc(Z)p Fk(;\034)23648 29595 y Fj(1)24167 29462 y Fk(>s)19653 30488 y(\015)20268 30087 y Fe(\003)20824 30488 y Fv(\()p Fk(s)p Fv(\)=)p Fk(\015)57 b Fv(\()p Fk(s)p Fv(\))19352 31566 y Fk(\015)19967 31165 y Fe(\003)20522 31566 y Fv(\()p Fk(\034)21427 31699 y Fj(1)21945 31566 y Fv(\)=)p Fk(\015)g Fv(\()p Fk(\034)107 b Fv(\))25720 26792 y Fi(Z)27048 27120 y Fk(\034)27538 27253 y Fj(1)26458 29810 y Fk(s)28123 28600 y Fq(\()p Fp(L)295 b Fm(\000)g Fp(\021)343 b Fm(\000)295 b Fp(\026)p Fq(\)\()p Fp(d\015)36785 28051 y Fg(\003)37395 28600 y Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))p Fp(dt)19145 33528 y Fm(\000)296 b Fq(\()p Fp(\034)21572 33727 y Fv(1)22462 33528 y Fm(\000)g Fp(\034)150 b Fq(\))p Fp(\013)5 b Fq(\()p Fp(c)26986 32980 y Fg(0)27357 33528 y Fq(\);)17619 b(\(2.15a\))1102 36251 y Fo(if)476 b Fp(\034)519 b Fm(\024)370 b Fq(0)475 b Fo(then)9552 39651 y Fp(A)10548 39850 y Fk(L)11277 39983 y Fd(\021)24 b(;\026)12796 39651 y Fq(\()p Fp(\015)74 b Fm(j)p Fq([)p Fp(s;)221 b(\034)150 b Fq(]\))368 b(=)2639 b(inf)19266 40513 y Fk(s)19767 40646 y Fj(1)20285 40513 y Fg(\000)p Fk(s)p Fg(2)p Fc(Z)p Fk(;s)23768 40646 y Fj(1)24288 40513 y Fk(<\034)19599 41539 y(\015)20214 41138 y Fe(\003)20770 41539 y Fv(\()p Fk(s)21686 41672 y Fj(1)22205 41539 y Fv(\)=)p Fk(\015)57 b Fv(\()p Fk(s)p Fv(\))19763 42617 y Fk(\015)20378 42216 y Fe(\003)20933 42617 y Fv(\()p Fk(\034)107 b Fv(\)=)p Fk(\015)57 b Fv(\()p Fk(\034)107 b Fv(\))25937 37843 y Fi(Z)27265 38171 y Fk(\034)26675 40861 y(s)27176 40994 y Fj(1)27929 39651 y Fq(\()p Fp(L)295 b Fm(\000)g Fp(\021)343 b Fm(\000)295 b Fp(\026)p Fq(\)\()p Fp(d\015)36591 39102 y Fg(\003)37201 39651 y Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))p Fp(dt)19339 44510 y Fq(+)296 b(\()p Fp(s)21808 44709 y Fv(1)22698 44510 y Fm(\000)g Fp(s)p Fq(\))p Fp(\013)5 b Fq(\()p Fp(c)p Fq(\);)17789 b(\(2.15b\))1102 47233 y Fo(if)476 b Fp(s)369 b Fm(\024)g Fq(0)476 b Fo(and)f Fp(\034)519 b Fm(\025)369 b Fq(1)475 b Fo(then)8684 50633 y Fp(A)9680 50832 y Fk(L)10409 50965 y Fd(\021)24 b(;\026)11929 50633 y Fq(\()p Fp(\015)74 b Fm(j)p Fq([)p Fp(s;)221 b(\034)150 b Fq(]\))368 b(=)3300 b(inf)18398 51494 y Fk(s)18899 51627 y Fj(1)19418 51494 y Fg(\000)p Fk(s)p Fg(2)p Fc(Z)p Fk(;\034)22890 51627 y Fj(1)23409 51494 y Fg(\000)p Fk(\034)107 b Fg(2)p Fc(Z)19754 52448 y Fk(s)20255 52581 y Fj(1)20773 52448 y Fg(\024)p Fv(0)p Fk(;\034)22937 52581 y Fj(1)23457 52448 y Fg(\025)p Fv(1)19394 53474 y Fk(\015)20009 53073 y Fe(\003)20564 53474 y Fv(\()p Fk(s)21480 53607 y Fj(1)21999 53474 y Fv(\)=)p Fk(\015)57 b Fv(\()p Fk(s)p Fv(\))19352 54552 y Fk(\015)19967 54151 y Fe(\003)20522 54552 y Fv(\()p Fk(\034)21427 54685 y Fj(1)21945 54552 y Fv(\)=)p Fk(\015)g Fv(\()p Fk(\034)107 b Fv(\))26394 48824 y Fi(Z)27722 49153 y Fk(\034)28212 49286 y Fj(1)27132 51843 y Fk(s)27633 51976 y Fj(1)28797 50633 y Fq(\()p Fp(L)295 b Fm(\000)g Fp(\021)342 b Fm(\000)296 b Fp(\026)p Fq(\)\()p Fp(d\015)37459 50084 y Fg(\003)38069 50633 y Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))p Fp(dt)18472 56514 y Fq(+)295 b(\()p Fp(s)20940 56713 y Fv(1)21831 56514 y Fm(\000)g Fp(s)p Fq(\))p Fp(\013)5 b Fq(\()p Fp(c)p Fq(\))293 b Fm(\000)i Fq(\()p Fp(\034)29482 56713 y Fv(1)30373 56514 y Fm(\000)g Fp(\034)150 b Fq(\))p Fp(\013)5 b Fq(\()p Fp(c)34896 55966 y Fg(0)35267 56514 y Fq(\))p Fp(:)9783 b Fq(\(2.15c\))1102 59997 y Fr(Pro)42 b(of)p Fq(:)558 b(T)-111 b(o)379 b(sho)-37 b(w)379 b(that)e(let)h(us)g (supp)37 b(ose)378 b(the)g(con)-37 b(trary)-111 b(,)393 b(for)378 b(instance,)391 b(\(2.15b\))378 b(do)37 b(es)378 b(not)g(hold.)1102 61591 y(Th)-37 b(us)502 b(there)h(w)-37 b(ould)502 b(exist)g(\001)467 b Fp(>)g Fq(0,)517 b Fp(s)467 b(<)g(\034)617 b Fm(\024)468 b Fq(0,)516 b Fp(s)27205 61790 y Fv(1)28268 61591 y Fp(<)468 b(\034)617 b Fm(\024)467 b Fq(0,)517 b Fp(s)34640 61790 y Fv(1)35570 61591 y Fm(\000)334 b Fp(s)467 b Fm(2)g Fl(Z)501 b Fq(and)h(a)g(curv)-37 b(e)503 b Fp(\015)48942 61109 y Fg(\003)49552 61591 y Fq(:)1102 63185 y([)p Fp(s)2094 63384 y Fv(1)2690 63185 y Fp(;)221 b(\034)150 b Fq(])369 b Fm(!)h Fp(M)587 b Fq(with)443 b Fp(\015)12111 62703 y Fg(\003)12721 63185 y Fq(\()p Fp(s)13861 63384 y Fv(1)14456 63185 y Fq(\))369 b(=)g Fp(\015)74 b Fq(\()p Fp(s)18646 63384 y Fv(1)19240 63185 y Fq(\),)443 b Fp(\015)21331 62703 y Fg(\003)21941 63185 y Fq(\()p Fp(\034)150 b Fq(\))368 b(=)h Fp(\015)74 b Fq(\()p Fp(\034)150 b Fq(\))442 b(suc)-37 b(h)443 b(that)6972 66659 y Fp(A)7968 66858 y Fk(L)8697 66991 y Fd(\021)24 b(;\026)10216 66659 y Fq(\()p Fp(\015)74 b Fm(j)p Fq([)p Fp(s;)221 b(\034)150 b Fq(]\))368 b Fm(\025)16833 64851 y Fi(Z)18162 65179 y Fk(\034)17571 67869 y(s)18072 68002 y Fj(1)18825 66659 y Fq(\()p Fp(L)295 b Fm(\000)h Fp(\021)342 b Fm(\000)295 b Fp(\026)p Fq(\)\()p Fp(d\015)27487 66110 y Fg(\003)28097 66659 y Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))p Fp(dt)294 b Fq(+)i(\()p Fp(s)35132 66858 y Fv(1)36022 66659 y Fm(\000)f Fp(s)p Fq(\))p Fp(\013)5 b Fq(\()p Fp(c)p Fq(\))293 b(+)j(\001)p Fp(:)1102 70477 y Fq(Let)342 b Fp(\017)369 b Fq(=)5851 69954 y Fv(1)p 5851 70172 530 54 v 5851 70935 a(4)6540 70477 y Fq(\001.)556 b(By)342 b(the)f(de\014nition)h(of)f(limit)g(in\014m)-37 b(um)342 b(there)g(exist)g(subsequences)h Fp(T)43358 69879 y Fk(i)43734 70020 y Fd(k)43174 70847 y Fk(j)44723 70477 y Fq(\()p Fp(j)444 b Fq(=)370 b(0)p Fp(;)221 b Fq(1,)1102 72157 y Fp(k)411 b Fq(=)369 b(0)p Fp(;)221 b Fq(1)p Fp(;)g Fq(2)p Fp(;)g Fm(\001)g(\001)g(\001)j Fq(\))442 b(suc)-37 b(h)443 b(that)16147 74965 y Fp(T)17107 74366 y Fk(i)17483 74507 y Fd(k)16923 75319 y Fv(0)18426 74965 y Fm(\000)295 b Fp(T)20714 74374 y Fk(i)21090 74507 y Fj(0)20530 75319 y Fv(0)22044 74965 y Fm(\025)370 b Fp(s)294 b Fm(\000)i Fp(s)26316 75164 y Fv(1)26911 74965 y Fp(;)2878 b Fm(8)443 b Fp(k)411 b Fm(\025)369 b Fq(1)p Fp(;)11650 b Fq(\(2.16\))p eop %%Page: 10 10 10 9 bop 1102 2239 a Ft(10)5592 5427 y Fp(h)6357 4879 y Fk(T)6984 5012 y Fj(0)7503 4879 y Fk(;T)8445 5012 y Fj(1)6357 5755 y Fk(\021)33 b(;\026)9031 5427 y Fq(\()p Fp(m)10714 5626 y Fv(0)11310 5427 y Fp(;)221 b(m)13066 5626 y Fv(1)13662 5427 y Fq(\))369 b Fm(\025)g Fp(h)16715 4879 y Fg(1)16715 5755 y Fk(\021)33 b(;\026)18324 5427 y Fq(\()p Fp(m)20007 5626 y Fv(0)20603 5427 y Fp(;)221 b(m)22359 5626 y Fv(1)22955 5427 y Fq(\))295 b Fm(\000)g Fp(\017;)2879 b Fm(8)885 b Fp(T)31281 5626 y Fv(0)32246 5427 y Fm(\025)370 b Fp(T)34609 4837 y Fk(i)34985 4970 y Fj(0)34425 5781 y Fv(0)35570 5427 y Fp(;)664 b(T)37379 5626 y Fv(1)38344 5427 y Fm(\025)369 b Fp(T)40706 4837 y Fk(i)41082 4970 y Fj(0)40522 5781 y Fv(1)41668 5427 y Fp(;)4490 b Fq(\(2.17\))1102 7789 y(and)11730 9663 y Fm(j)p Fp(h)12864 8917 y Fk(T)13634 8364 y Fd(i)13988 8597 y(k)13491 9205 y Fj(0)14637 8917 y Fk(;T)15722 8364 y Fd(i)16076 8597 y(k)15579 9205 y Fj(1)12864 9818 y Fk(\021)33 b(;\026)16790 9663 y Fq(\()p Fp(m)18473 9862 y Fv(0)19069 9663 y Fp(;)221 b(m)20825 9862 y Fv(1)21422 9663 y Fq(\))295 b Fm(\000)g Fp(h)24327 9114 y Fg(1)24327 9991 y Fk(\021)33 b(;\026)25936 9663 y Fq(\()p Fp(m)27619 9862 y Fv(0)28215 9663 y Fp(;)221 b(m)29971 9862 y Fv(1)30567 9663 y Fq(\))p Fm(j)369 b Fp(<)g(\017;)2878 b Fm(8)443 b Fp(k)42 b(:)7233 b Fq(\(2.18\))1102 12025 y(By)475 b(taking)f(subsequence)i(further)f(w)-37 b(e)475 b(can)g(assume)g Fp(\015)29151 12224 y Fk(i)29527 12365 y Fd(k)30596 12025 y Fm(!)422 b Fp(\015)74 b Fq(.)685 b(In)474 b(this)g(case,)483 b(w)-37 b(e)475 b(can)g(c)-37 b(ho)37 b(ose)1102 13619 y(su\016cien)-37 b(tly)474 b(large)h Fp(k)515 b Fq(suc)-37 b(h)474 b(that)e Fp(\015)18894 13818 y Fk(i)19270 13959 y Fd(k)19917 13619 y Fq(\()p Fp(s)p Fq(\))h(and)g Fp(\015)25348 13818 y Fk(i)25724 13959 y Fd(k)26371 13619 y Fq(\()p Fp(\034)150 b Fq(\))473 b(are)h(so)g(close)g(to)f Fp(\015)74 b Fq(\()p Fp(s)p Fq(\))472 b(and)i Fp(\015)74 b Fq(\()p Fp(\034)150 b Fq(\))472 b(resp)37 b(ec-)1102 15213 y(tiv)-37 b(ely)450 b(that)e(w)-37 b(e)450 b(can)g(construct)f(a)h(curv)-37 b(e)450 b Fp(\015)23325 14731 y Fg(\003)23251 15558 y Fk(i)23627 15699 y Fd(k)24274 15213 y Fq(:)603 b([)p Fp(s)26238 15412 y Fv(1)26834 15213 y Fp(;)221 b(\034)150 b Fq(])380 b Fm(!)g Fp(M)593 b Fq(whic)-37 b(h)450 b(has)f(the)h(same)f (endp)37 b(oin)-37 b(ts)1102 16829 y(as)443 b Fp(\015)3421 17028 y Fk(i)3797 17169 y Fd(k)4444 16829 y Fq(:)590 b Fp(\015)6165 16346 y Fg(\003)6091 17174 y Fk(i)6776 16829 y Fq(\()p Fp(s)7916 17028 y Fv(1)8511 16829 y Fq(\))368 b(=)h Fp(\015)11486 17028 y Fk(i)11928 16829 y Fq(\()p Fp(s)p Fq(\),)442 b Fp(\015)15158 16346 y Fg(\003)15084 17174 y Fk(i)15768 16829 y Fq(\()p Fp(\034)150 b Fq(\))368 b(=)i Fp(\015)19992 17028 y Fk(i)20434 16829 y Fq(\()p Fp(\034)150 b Fq(\))442 b(and)g(satis\014es)h(the)g(follo)-37 b(wing)4139 20416 y Fp(A)5135 20615 y Fk(L)5864 20748 y Fd(\021)24 b(;\026)7384 20416 y Fq(\()p Fp(\015)8589 20615 y Fk(i)8965 20756 y Fd(k)9611 20416 y Fm(j)p Fq([)p Fp(s;)221 b(\034)150 b Fq(]\))369 b Fm(\025)14950 18608 y Fi(Z)16278 18937 y Fk(\034)15688 21627 y(s)16189 21760 y Fj(1)16942 20416 y Fq(\()p Fp(L)295 b Fm(\000)g Fp(\021)342 b Fm(\000)296 b Fp(\026)p Fq(\)\()p Fp(d\015)25604 19868 y Fg(\003)25530 20745 y Fk(i)25906 20886 y Fd(k)26552 20416 y Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))p Fp(dt)294 b Fq(+)i(\()p Fp(s)33587 20615 y Fv(1)34477 20416 y Fm(\000)f Fp(s)p Fq(\))p Fp(\013)5 b Fq(\()p Fp(c)p Fq(\))293 b(+)41190 19518 y(3)p 41190 20111 665 54 v 41190 21328 a(4)42014 20416 y(\001)p Fp(:)3037 b Fq(\(2.19\))1102 24335 y(Let)443 b Fp(T)4442 23853 y Fg(0)4258 24665 y Fv(0)5223 24335 y Fq(=)370 b Fp(T)7586 23737 y Fk(i)7962 23878 y Fd(k)7402 24690 y Fv(0)8905 24335 y Fq(+)295 b(\()p Fp(s)f Fm(\000)i Fp(s)13619 24534 y Fv(1)14214 24335 y Fq(\),)443 b(if)f(w)-37 b(e)444 b(extend)f Fp(\015)23795 23853 y Fg(\003)23721 24681 y Fk(i)24097 24822 y Fd(k)25186 24335 y Fq(to)g Fl(R)369 b Fm(!)h Fp(M)587 b Fq(suc)-37 b(h)443 b(that)13791 29306 y Fp(\015)14553 28757 y Fg(\003)14479 29634 y Fk(i)14855 29775 y Fd(k)15871 29306 y Fq(=)17273 26184 y Fi(8)17273 27380 y(>)17273 27778 y(<)17273 30169 y(>)17273 30568 y(:)18897 27302 y Fp(\015)19585 27501 y Fk(i)19961 27642 y Fd(k)20607 27302 y Fq(\()p Fp(t)295 b Fm(\000)g Fp(s)23850 27501 y Fv(1)24741 27302 y Fq(+)g Fp(s)p Fq(\))p Fp(;)3056 b(t)368 b Fm(\024)h Fp(s)33507 27501 y Fv(1)34103 27302 y Fp(;)18897 29295 y(\015)19659 28813 y Fg(\003)19585 29641 y Fk(i)19961 29782 y Fd(k)20607 29295 y Fq(\()p Fp(t)p Fq(\))p Fp(;)8095 b(s)31208 29494 y Fv(1)32172 29295 y Fm(\024)369 b Fp(t)g Fm(\024)g Fp(\034)77 b(;)18897 31310 y(\015)19585 31509 y Fk(i)19961 31650 y Fd(k)20607 31310 y Fq(\()p Fp(t)p Fq(\))p Fp(;)8095 b(t)368 b Fm(\025)h Fp(\034)77 b(;)1102 34122 y Fq(then)443 b(w)-37 b(e)444 b(obtain)e(from)h(\(2.18\))g(and)g(\(2.19\))f(that)5748 37557 y Fp(h)6513 36811 y Fk(T)7283 36411 y Fe(0)7140 37077 y Fj(0)7659 36811 y Fk(;T)8744 36259 y Fd(i)9098 36491 y(k)8601 37100 y Fj(1)6513 37713 y Fk(\021)33 b(;\026)9813 37557 y Fq(\()p Fp(m)11496 37756 y Fv(0)12092 37557 y Fp(;)221 b(m)13848 37756 y Fv(1)14445 37557 y Fq(\))368 b Fm(\024)p Fp(A)17359 37756 y Fk(L)18088 37889 y Fd(\021)24 b(;\026)19608 37557 y Fq(\()p Fp(\015)20887 37009 y Fg(\003)20813 37886 y Fk(i)21189 38027 y Fd(k)21835 37557 y Fm(j)p Fq([)p Fm(\000)p Fp(T)24566 37009 y Fg(0)24382 37886 y Fv(0)24978 37557 y Fp(;)221 b(T)26528 36959 y Fk(i)26904 37100 y Fd(k)26344 37912 y Fv(1)27552 37557 y Fq(]\))295 b Fm(\000)h Fp(T)31022 36959 y Fk(i)31398 37100 y Fd(k)30838 37912 y Fv(1)32045 37557 y Fp(\013)5 b Fq(\()p Fp(c)33992 37009 y Fg(0)34364 37557 y Fq(\))295 b Fm(\000)g Fp(T)37464 37009 y Fg(0)37280 37886 y Fv(0)37876 37557 y Fp(\013)5 b Fq(\()p Fp(c)p Fq(\))15330 40375 y Fm(\024)p Fp(A)17359 40574 y Fk(L)18088 40707 y Fd(\021)24 b(;\026)19608 40375 y Fq(\()p Fp(\015)20813 40574 y Fk(i)21189 40715 y Fd(k)21835 40375 y Fm(j)p Fq([)p Fm(\000)p Fp(T)24566 39777 y Fk(i)24942 39918 y Fd(k)24382 40730 y Fv(0)25590 40375 y Fp(;)221 b(T)27140 39777 y Fk(i)27516 39918 y Fd(k)26956 40730 y Fv(1)28164 40375 y Fq(]\))295 b Fm(\000)g Fp(T)31633 39777 y Fk(i)32009 39918 y Fd(k)31449 40730 y Fv(1)32657 40375 y Fp(\013)5 b Fq(\()p Fp(c)34604 39827 y Fg(0)34976 40375 y Fq(\))294 b Fm(\000)i Fp(T)38076 39777 y Fk(i)38452 39918 y Fd(k)37892 40730 y Fv(0)39099 40375 y Fp(\013)5 b Fq(\()p Fp(c)p Fq(\))294 b Fm(\000)43345 39477 y Fq(3)p 43345 40070 V 43345 41286 a(4)44169 40375 y(\001)15330 42815 y Fm(\024)p Fp(h)17128 42266 y Fg(1)17128 43143 y Fk(\021)33 b(;\026)18737 42815 y Fq(\()p Fp(m)20420 43014 y Fv(0)21016 42815 y Fp(;)221 b(m)22772 43014 y Fv(1)23368 42815 y Fq(\))295 b Fm(\000)h Fq(2)p Fp(\017:)1102 45922 y Fq(but)611 b(this)g(con)-37 b(tradicts)611 b(\(2.17\))g(since)h Fp(T)21859 45440 y Fg(0)21675 46252 y Fv(0)22920 45922 y Fm(\025)650 b Fp(T)25563 45332 y Fk(i)25939 45465 y Fj(0)25379 46276 y Fv(0)27136 45922 y Fq(and)611 b Fp(T)30847 45323 y Fk(i)31223 45464 y Fd(k)30663 46276 y Fv(1)32520 45922 y Fm(\025)650 b Fp(T)35163 45332 y Fk(i)35539 45465 y Fj(0)34979 46276 y Fv(1)36124 45922 y Fq(,)j(guaran)-37 b(teed)612 b(b)-37 b(y)612 b(\(2.16\).)1102 47516 y(\(2.15a\))443 b(and)g(\(2.15c\))g(can)g(b)37 b(e)443 b(pro)-37 b(v)g(ed)444 b(in)f(the)f(same)i(w)-37 b(a)g(y)-111 b(.)2166 b Fb(\003)2696 49826 y Fq(W)-111 b(e)444 b(de\014ne)g(a)f(set)f(for)h Fp(L)14939 50025 y Fk(\021)33 b(;\026)16991 49826 y Fq(somewhat)443 b(similar)g(to)g(the)f(de\014nition)h(of)g(Ma)37 b(~)-701 b(n)-37 b(\023)-627 b(e)443 b(set)9570 52455 y(~)9149 52791 y Fm(N)10239 52990 y Fk(\021)33 b(;\026)12216 52791 y Fq(=)369 b Fm(f)p Fp(d\015)443 b Fm(2)17609 52455 y Fq(~)17359 52791 y Fm(G)18149 52990 y Fk(L)18878 53123 y Fd(\021)24 b(;\026)20766 52791 y Fq(:)812 b(\(2)p Fp(:)p Fq(15)p Fp(a)p Fq(\))443 b(\(2)p Fp(:)p Fq(15)p Fp(b)p Fq(\))f(and)h(\(2)p Fp(:)p Fq(15)p Fp(c)p Fq(\))g(hold)f Fm(g)p Fp(:)1102 55837 y Fr(Lemma)673 b(2.4.)764 b Fo(The)605 b(map)i Fq(\()p Fp(\021)48 b(;)221 b(\026)p Fq(\))605 b Fm(!)21813 55502 y Fq(~)21392 55837 y Fm(N)22482 56036 y Fk(\021)33 b(;\026)24696 55837 y Fo(is)607 b(upp)-68 b(er)606 b(semi-c)-68 b(ontinuous.)41201 55502 y Fq(~)40779 55837 y Fm(N)41869 56036 y Fk(\021)33 b(;)p Fv(0)43967 55837 y Fq(=)46028 55502 y(~)45607 55837 y Fm(N)195 b Fq(\()p Fp(c)p Fq(\))606 b Fo(if)1102 57573 y Fq([)p Fp(\021)48 b Fq(])369 b(=)g Fp(c)p Fo(,)475 b(c)-68 b(onse)g(quently,) 476 b(the)g(map)f Fp(c)369 b Fm(!)22010 57237 y Fq(~)21589 57573 y Fm(N)195 b Fq(\()p Fp(c)p Fq(\))474 b Fo(is)i(upp)-68 b(er)475 b(semi-c)-68 b(ontinuous.)1102 59977 y Fr(Pro)42 b(of)p Fq(:)656 b(Let)475 b Fp(\021)8909 60176 y Fk(i)9774 59977 y Fm(!)423 b Fp(\021)523 b Fq(and)475 b Fp(\026)16123 60176 y Fk(i)16989 59977 y Fm(!)423 b Fp(\026)p Fq(,)484 b(let)475 b Fp(\015)23032 60176 y Fk(i)23897 59977 y Fm(2)25627 59641 y Fq(~)25206 59977 y Fm(N)26296 60176 y Fk(\021)26843 60309 y Fd(i)27264 60176 y Fk(;\026)28225 60309 y Fd(i)29187 59977 y Fq(and)h(let)f Fp(\015)549 b Fq(b)37 b(e)475 b(an)g(accum)-37 b(ulation)477 b(p)37 b(oin)-37 b(t)1102 61713 y(of)560 b(the)g(set)f Fm(f)p Fp(\015)8679 61912 y Fk(i)9685 61713 y Fm(2)11556 61377 y Fq(~)11135 61713 y Fm(N)12225 61912 y Fk(\021)12772 62045 y Fd(i)13193 61912 y Fk(;\026)14154 62045 y Fd(i)14641 61713 y Fm(g)15305 61964 y Fk(i)p Fg(2)p Fc(Z)17017 61698 y Fj(+)17832 61713 y Fq(.)941 b(Clearly)-111 b(,)591 b Fp(\015)637 b Fm(2)27359 61377 y Fq(~)26937 61713 y Fm(N)28027 61912 y Fk(\021)33 b(;\026)29636 61713 y Fq(.)941 b(If)559 b Fp(\015)785 b(=)-812 b Fm(2)35587 61377 y Fq(~)35166 61713 y Fm(N)36256 61912 y Fk(\021)33 b(;\026)38424 61713 y Fq(there)560 b(w)-37 b(ould)560 b(b)37 b(e)560 b(t)-37 b(w)g(o)1102 63307 y(p)37 b(oin)-37 b(t)467 b Fp(\015)74 b Fq(\()p Fp(s)p Fq(\),)p Fp(\015)g Fq(\()p Fp(\034)150 b Fq(\))406 b Fm(2)i Fp(M)611 b Fq(suc)-37 b(h)467 b(that)f(one)i(of)e(the)h(follo)-37 b(wing)468 b(three)f(p)37 b(ossible)467 b(cases)g(tak)-37 b(es)467 b(place.)1102 64901 y(Either)525 b Fp(\015)74 b Fq(\()p Fp(s)p Fq(\))522 b(and)j Fp(\015)74 b Fq(\()p Fp(\034)150 b Fq(\))503 b Fm(2)h Fp(M)669 b Fq(can)524 b(b)37 b(e)525 b(connected)g(b)-37 b(y)524 b(another)h(curv)-37 b(e)525 b Fp(\015)39206 64419 y Fg(\003)39817 64901 y Fq(:)753 b([)p Fp(s)349 b Fq(+)g Fp(n;)221 b(\034)150 b Fq(])506 b Fm(!)f Fp(M)1102 66495 y Fq(with)443 b(smaller)h(action)13218 69318 y Fp(A)14214 69517 y Fk(\021)33 b(;\026)15823 69318 y Fq(\()p Fp(\015)74 b Fm(j)p Fq([)p Fp(s;)221 b(\034)150 b Fq(]\))368 b Fp(<)h(A)23436 69517 y Fk(\021)33 b(;\026)25045 69318 y Fq(\()p Fp(\015)26324 68770 y Fg(\003)26933 69318 y Fm(j)p Fq([)p Fp(s)295 b Fq(+)h Fp(n;)221 b(\034)150 b Fq(]\))295 b Fm(\000)g Fp(n\013)5 b Fq(\()p Fp(c)p Fq(\))1102 72141 y(in)443 b(the)g(case)h Fp(\034)519 b(<)369 b Fq(0;)443 b(or)g(there)g(w)-37 b(ould)444 b(a)f(curv)-37 b(e)444 b Fp(\015)26045 71659 y Fg(\003)26655 72141 y Fq(:)590 b([)p Fp(s;)221 b(\034)445 b Fq(+)296 b Fp(n)p Fq(])369 b Fm(!)g Fp(M)588 b Fq(suc)-37 b(h)443 b(that)13032 74965 y Fp(A)14028 75164 y Fk(\021)33 b(;\026)15637 74965 y Fq(\()p Fp(\015)74 b Fm(j)p Fq([)p Fp(s;)221 b(\034)150 b Fq(]\))367 b Fp(<)j(A)23250 75164 y Fk(\021)33 b(;\026)24858 74965 y Fq(\()p Fp(\015)26137 74416 y Fg(\003)26747 74965 y Fm(j)p Fq([)p Fp(s;)221 b(\034)445 b Fq(+)296 b Fp(n)p Fq(]\))f Fm(\000)g Fp(n\013)5 b Fq(\()p Fp(c)37103 74416 y Fg(0)37475 74965 y Fq(\))p eop %%Page: 11 11 11 10 bop 48792 2239 a Ft(11)1102 5427 y Fq(in)310 b(the)g(case)g Fp(s)369 b Fm(\025)g Fq(1,)337 b(or)310 b(when)g Fp(s)369 b Fm(\024)g Fq(0)310 b(and)g Fp(\034)519 b Fm(\025)369 b Fq(1)310 b(there)g(w)-37 b(ould)311 b(b)37 b(e)310 b(a)g(curv)-37 b(e)311 b Fp(\015)39070 4945 y Fg(\003)39680 5427 y Fq(:)524 b([)p Fp(s)29 b Fq(+)g Fp(n)43453 5626 y Fv(1)44049 5427 y Fp(;)221 b(\034)179 b Fq(+)29 b Fp(n)47258 5626 y Fv(2)47855 5427 y Fq(])369 b Fm(!)1102 7021 y Fp(M)588 b Fq(suc)-37 b(h)443 b(that)8188 9016 y Fp(A)9184 9215 y Fk(\021)33 b(;\026)10793 9016 y Fq(\()p Fp(\015)74 b Fm(j)p Fq([)p Fp(s;)221 b(\034)150 b Fq(]\))367 b Fp(<)i(A)18405 9215 y Fk(\021)33 b(;\026)20014 9016 y Fq(\()p Fp(\015)21293 8468 y Fg(\003)21903 9016 y Fm(j)p Fq([)p Fp(s)295 b Fq(+)g Fp(n)25684 9215 y Fv(1)26280 9016 y Fp(;)221 b(\034)446 b Fq(+)295 b Fp(n)30022 9215 y Fv(2)30619 9016 y Fq(]\))f Fm(\000)i Fp(n)33925 9215 y Fv(1)34521 9016 y Fp(\013)5 b Fq(\()p Fp(c)p Fq(\))294 b Fm(\000)h Fp(n)39404 9215 y Fv(2)40000 9016 y Fp(\013)5 b Fq(\()p Fp(c)41947 8468 y Fg(0)42319 9016 y Fq(\))1102 11011 y(where)462 b Fp(s)307 b Fq(+)g Fp(n)8028 11210 y Fv(1)9023 11011 y Fm(\024)399 b Fq(0,)465 b Fp(\034)457 b Fq(+)308 b Fp(n)15129 11210 y Fv(2)16123 11011 y Fm(\025)399 b Fq(1.)644 b(Since)462 b Fp(\015)534 b Fq(is)460 b(an)h(accum)-37 b(ulation)461 b(p)37 b(oin)-37 b(t)461 b(of)f Fp(\015)41034 11210 y Fk(i)41476 11011 y Fq(,)465 b(for)c(an)-37 b(y)461 b(small)1102 12605 y Fp(\017)370 b(>)f Fq(0,)440 b(there)h(w)-37 b(ould)440 b(b)37 b(e)440 b(su\016cien)-37 b(tly)441 b(large)g Fp(i)e Fq(suc)-37 b(h)441 b(that)e Fm(k)p Fp(\015)363 b Fm(\000)290 b Fp(\015)34345 12804 y Fk(i)34786 12605 y Fm(k)35450 12845 y Fk(C)36277 12579 y Fj(1)36796 12845 y Fv([)p Fk(s;t)p Fv(])39051 12605 y Fp(<)369 b(\017)p Fq(,)441 b(it)e(follo)-37 b(ws)441 b(that)1102 14433 y Fp(\015)1790 14632 y Fk(i)2749 14433 y Fp(=)-812 b Fm(2)4277 14097 y Fq(~)3856 14433 y Fm(N)4946 14632 y Fk(\021)5493 14765 y Fd(i)5914 14632 y Fk(;\026)6875 14765 y Fd(i)7805 14433 y Fq(but)443 b(that)f(is)g(absurd.)2696 16559 y(Let)485 b(us)g(consider)h(the)f(case)h(that)e Fp(\026)439 b Fq(=)g(0.)717 b(In)485 b(this)f(case,)496 b Fp(L)323 b Fm(\000)h Fp(\021)532 b Fq(is)485 b(p)37 b(erio)g(dic)485 b(in)g Fp(t)p Fq(.)716 b(If)485 b(some)1102 18173 y(orbit)451 b Fp(\015)457 b Fm(2)7195 17837 y Fq(~)6774 18173 y Fm(N)7864 18372 y Fk(\021)33 b(;)p Fv(0)9356 18173 y Fq(:)607 b Fl(R)383 b Fm(!)g Fp(M)595 b Fq(is)451 b(not)g(semi-static,)i(then)e(there)g (exist)g Fp(s)382 b(<)h(\034)533 b Fm(2)382 b Fl(R)p Fq(,)454 b Fp(n)383 b Fm(2)f Fl(Z)p Fq(,)453 b(\001)382 b Fp(>)h Fq(0)1102 19767 y(and)443 b(a)g(curv)-37 b(e)444 b Fp(\015)9100 19285 y Fg(\003)9710 19767 y Fq(:)591 b([)p Fp(s;)221 b(\034)445 b Fq(+)295 b Fp(n)p Fq(])370 b Fm(!)f Fp(M)587 b Fq(suc)-37 b(h)443 b(that)g Fp(\015)26352 19285 y Fg(\003)26962 19767 y Fq(\()p Fp(s)p Fq(\))368 b(=)h Fp(\015)74 b Fq(\()p Fp(s)p Fq(\),)441 b Fp(\015)34380 19285 y Fg(\003)34990 19767 y Fq(\()p Fp(\034)k Fq(+)295 b Fp(n)p Fq(\))369 b(=)g Fp(\015)74 b Fq(\()p Fp(\034)150 b Fq(\))442 b(and)11785 21762 y Fp(A)12781 21961 y Fk(\021)33 b(;)p Fv(0)14273 21762 y Fq(\()p Fp(\015)74 b Fm(j)p Fq([)p Fp(s;)221 b(\034)150 b Fq(]\))368 b Fm(\025)h Fp(A)21886 21961 y Fk(\021)33 b(;)p Fv(0)23378 21762 y Fq(\()p Fp(\015)24657 21214 y Fg(\003)25267 21762 y Fm(j)p Fq([)p Fp(s;)221 b(\034)445 b Fq(+)296 b Fp(n)p Fq(]\))f Fm(\000)g Fp(n\013)5 b Fq(\()p Fp(c)p Fq(\))294 b(+)i(\001)p Fp(:)1102 23757 y Fq(W)-111 b(e)411 b(can)g(extend)f Fp(\015)10806 23275 y Fg(\003)11826 23757 y Fq(to)f([)p Fp(s)14408 23956 y Fv(1)15004 23757 y Fp(;)221 b(\034)16175 23956 y Fv(1)17000 23757 y Fq(+)230 b Fp(n)p Fq(])369 b Fm(!)g Fp(M)555 b Fq(suc)-37 b(h)410 b(that)f Fp(s)29770 23956 y Fv(1)30735 23757 y Fm(\024)369 b Fq(min)p Fm(f)p Fp(s;)221 b Fq(0)p Fm(g)p Fq(,)417 b(min)p Fm(f)p Fp(\034)41801 23956 y Fv(1)42397 23757 y Fp(;)221 b(\034)43568 23956 y Fv(1)44393 23757 y Fq(+)229 b Fp(n)p Fm(g)370 b(\025)f Fq(1,)1102 25351 y Fp(\034)1683 25550 y Fv(1)2648 25351 y Fm(\025)g Fp(\034)593 b Fq(and)13938 28805 y Fp(\015)14700 28256 y Fg(\003)15679 28805 y Fq(=)17081 25683 y Fi(8)17081 26879 y(>)17081 27277 y(<)17081 29668 y(>)17081 30067 y(:)18705 26812 y Fp(\015)74 b Fq(\()p Fp(t)p Fq(\))p Fp(;)4314 b(s)26287 27011 y Fv(1)27251 26812 y Fm(\024)369 b Fp(t)g Fm(\024)g Fp(s;)18705 28805 y(\015)19467 28323 y Fg(\003)20077 28805 y Fq(\()p Fp(t)p Fq(\))p Fp(;)3685 b(s)368 b Fm(\024)h Fp(t)g Fm(\024)g Fp(\034)445 b Fq(+)296 b Fp(n;)18705 30798 y(\015)74 b Fq(\()p Fp(t)293 b Fm(\000)j Fp(n)p Fq(\))p Fp(;)1796 b(\034)445 b Fq(+)295 b Fp(n)370 b Fm(\024)f Fp(t)g Fm(\024)g Fp(\034)33320 30997 y Fv(1)34211 30798 y Fq(+)295 b Fp(n:)1102 32717 y Fq(Since)444 b Fp(L)295 b Fm(\000)g Fp(\021)490 b Fq(is)443 b(p)37 b(erio)g(dic)444 b(in)f Fp(t)p Fq(,)f(w)-37 b(e)444 b(w)-37 b(ould)443 b(ha)-37 b(v)g(e)10378 34712 y Fp(A)11374 34911 y Fk(\021)33 b(;)p Fv(0)12866 34712 y Fq(\()p Fp(\015)74 b Fm(j)p Fq([)p Fp(s)15506 34911 y Fv(1)16101 34712 y Fp(;)221 b(\034)17272 34911 y Fv(1)17868 34712 y Fq(]\))368 b Fm(\025)i Fp(A)21521 34911 y Fk(\021)33 b(;)p Fv(0)23013 34712 y Fq(\()p Fp(\034)24261 34164 y Fg(\003)24871 34712 y Fp(\015)74 b Fm(j)p Fq([)p Fp(s)26994 34911 y Fv(1)27589 34712 y Fp(;)221 b(\034)28760 34911 y Fv(1)29651 34712 y Fq(+)296 b Fp(n)p Fq(]\))f Fm(\000)g Fp(n\013)5 b Fq(\()p Fp(c)p Fq(\))294 b(+)i(\001)p Fp(:)1102 36707 y Fq(but)443 b(this)f(con)-37 b(tradicts)443 b(to)g(\(2.15c\).)1575 b Fb(\003)2696 38927 y Fq(The)577 b(upp)37 b(er)576 b(semi-con)-37 b(tin)g(uit)g(y)576 b(of)g Fp(c)590 b Fm(!)24087 38591 y Fq(~)23666 38927 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\))574 b(will)i(b)37 b(e)576 b(fully)g(exploited)g(to)g(build)f(the)h Fp(C)95 b Fq(-)1102 40615 y(equiv)-74 b(alence)669 b(among)f(some)16815 40279 y(~)16394 40615 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\),)722 b(the)666 b(construction)i(of)f(di\013usion)f(orbits)h(in)g(this)g(pap) 37 b(er)1102 42209 y(dep)g(ends)619 b(crucially)g(on)g(this)e(prop)37 b(ert)-37 b(y)-111 b(.)1117 b(T)-111 b(o)-37 b(w)g(ards)620 b(that,)661 b(w)-37 b(e)619 b(shall)g(also)f(mak)-37 b(e)619 b(use)f(of)g(the)1102 43803 y(Lipsc)-37 b(hitz)504 b(prop)37 b(ert)-37 b(y)503 b(of)g(the)g(Aubry)g(sets.)770 b(Let)503 b Fp(\031)48 b Fq(:)710 b Fp(T)184 b(M)481 b Fm(\002)335 b Fl(T)468 b Fm(!)i Fp(M)479 b Fm(\002)336 b Fl(T)502 b Fq(b)37 b(e)503 b(the)g(pro)74 b(jection)1102 45397 y(along)444 b(the)e(\014b)37 b(ers,)444 b(Mather)e(disco)-37 b(v)g(ered)445 b(the)e(follo)-37 b(wing)443 b(\(cf.)591 b([Ma3,4]\):)1102 47617 y Fr(Lemma)520 b(2.5.)764 b Fp(\031)48 b Fo(:)11967 47281 y Fq(~)11510 47617 y Fm(A)p Fq(\()p Fp(c)p Fq(\))384 b Fm(!)h Fp(M)446 b Fm(\002)301 b Fl(T)484 b Fo(is)g(inje)-68 b(ctive.)639 b(Its)484 b(inverse)h Fq(\()p Fo(c)-68 b(onsider)g(e)g(d)484 b(as)g(a)f(map)h(fr)-68 b(om)1102 49305 y Fm(A)p Fq(\()p Fp(c)p Fq(\))368 b(=)h Fp(\031)6804 48969 y Fq(~)6347 49305 y Fm(A)p Fq(\()p Fp(c)p Fq(\))i Fo(to)11337 48969 y Fq(~)10881 49305 y Fm(A)o Fq(\()p Fp(c)p Fq(\)\))g Fo(is)i(Lipschitz,)395 b(i.e.)578 b Fm(9)372 b Fo(a)h(c)-68 b(onstant)372 b Fp(C)32419 49504 y Fk(L)33587 49305 y Fo(such)h(that)g(for)f(any)h Fp(x;)221 b(y)417 b Fm(2)369 b(A)p Fq(\()p Fp(c)p Fq(\))1102 50899 y Fo(we)475 b(have)15076 52493 y Fq(dist\()p Fp(\031)18546 51944 y Fg(\000)p Fv(1)19971 52493 y Fq(\()p Fp(x)p Fq(\))p Fp(;)221 b(\031)23159 51944 y Fg(\000)p Fv(1)24585 52493 y Fq(\()p Fp(y)48 b Fq(\)\))368 b Fm(\024)h Fp(C)29554 52692 y Fk(L)30349 52493 y Fq(dist\()p Fp(x;)221 b(y)48 b Fq(\))p Fp(:)2696 55018 y Fq(The)410 b(concept)g(of)g(of)f(regular)i (Lagrangian)f(is)f(useful)h(for)f(us)g(in)h(this)e(pap)37 b(er.)580 b Fp(L)409 b Fq(is)h(said)f(to)g(b)37 b(e)1102 56612 y Fp(c)p Fq(-regular)444 b(if)f(the)f(follo)-37 b(wing)444 b(limit)e(exists)h(for)g(all)g(\()p Fp(x;)221 b(x)29087 56130 y Fg(0)29460 56612 y Fp(;)g(s;)g(s)31886 56130 y Fg(0)32259 56612 y Fq(\))13875 58607 y Fp(h)14640 58806 y Fk(c)15182 58607 y Fq(\()p Fp(x;)g(x)17807 58059 y Fg(0)18180 58607 y Fp(;)g(s;)g(s)20606 58059 y Fg(0)20978 58607 y Fq(\))369 b(=)2381 b(lim)23625 59474 y Fk(s)p Fv(=)p Fk(t)444 b Fv(mo)29 b(d)443 b(1)23266 60380 y Fk(s)23767 59979 y Fe(0)24127 60380 y Fv(=)p Fk(t)25343 59979 y Fe(0)26146 60380 y Fv(mo)29 b(d)442 b(1)24146 61285 y Fk(t)24547 60884 y Fe(0)24906 61285 y Fg(\000)p Fk(t)p Fg(!1)29357 58607 y Fp(F)30211 58806 y Fk(c)30752 58607 y Fq(\()p Fp(x;)221 b(x)33377 58059 y Fg(0)33750 58607 y Fp(;)g(t;)g(t)35890 58059 y Fg(0)36263 58607 y Fq(\))p Fp(:)9378 b Fq(\(2.20\))1102 63285 y Fr(Lemma)340 b(2.6.)764 b Fq(\(Bernard)443 b(2002\))340 b Fo(If)20583 62949 y Fq(~)19933 63285 y Fm(M)p Fq(\()p Fp(c)p Fq(\))f Fo(is)h(minimal)h(in)f(the)g(sense)g(of)f(top)-68 b(olo)g(gic)g(al)342 b(dynamics)1102 64879 y(and)531 b(if)h(ther)-68 b(e)531 b(exists)h(a)f(se)-68 b(quenc)g(e)531 b Fp(\015)19493 65078 y Fk(n)20747 64879 y Fo(of)g Fp(n)p Fo(-p)-68 b(erio)g(dic)533 b(curves)e(such)g Fp(A)36919 65078 y Fk(c)37460 64879 y Fq(\()p Fp(\015)38665 65078 y Fk(n)39387 64879 y Fq(\))470 b Fm(!)g Fq(0)531 b Fo(as)g Fp(n)470 b Fm(!)h(1)p Fo(,)1102 66567 y(then)476 b Fp(L)4960 66766 y Fk(c)5976 66567 y Fo(is)g(r)-68 b(e)g(gular,)476 b(henc)-68 b(e)16345 66231 y Fq(~)15889 66567 y Fm(A)o Fq(\()p Fp(c)p Fq(\))368 b(=)20750 66231 y(~)20328 66567 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\))368 b(=)25243 66231 y(~)24993 66567 y Fm(G)79 b Fq(\()p Fp(c)p Fq(\))p Fo(.)2696 68693 y Fq(F)-111 b(or)490 b(the)f(completeness)h(sak)-37 b(e,)502 b(w)-37 b(e)489 b(shall)h(presen)-37 b(t)489 b(his)g(pro)37 b(of)489 b(in)g(the)g(app)37 b(endix.)730 b(Applying)1102 70287 y(this)443 b(lemma)g(to)f(the)h(area-preserving)i(t)-37 b(wist)443 b(map)f(w)-37 b(e)444 b(ha)-37 b(v)g(e)444 b(the)e(follo)-37 b(wing:)1102 72413 y Fr(Corollary)684 b(2.7.)764 b Fo(L)-68 b(et)616 b Fp(!)669 b Fm(2)621 b Fl(R)p Fm(n)p Fl(Q)615 b Fo(b)-68 b(e)615 b(the)g(r)-68 b(otation)615 b(numb)-68 b(er)615 b(and)g Fp(c)621 b Fq(=)h Fp(\014)40481 71931 y Fg(0)40854 72413 y Fq(\()p Fp(!)48 b Fq(\))p Fo(,)648 b(then)615 b Fp(L)47815 72612 y Fk(c)48971 72413 y Fo(is)1102 74101 y(r)-68 b(e)g(gular)476 b(and)8345 73765 y Fq(~)8095 74101 y Fm(G)79 b Fq(\()p Fp(c)p Fq(\))368 b(=)12800 73765 y(~)12343 74101 y Fm(A)p Fq(\()p Fp(c)p Fq(\))p Fo(.)p eop %%Page: 12 12 12 11 bop 1102 2239 a Ft(12)1102 5427 y Fr(3,)510 b(Structure)e(of)h (some)g Fp(c)p Fr(-minimal)g(orbit)g(sets)2696 7794 y Fq(Our)303 b(construction)g(of)g(connecting)g(orbits)g(b)37 b(et)-37 b(w)g(een)303 b(di\013eren)-37 b(t)303 b Fp(c)p Fq(-minimal)f(orbit)h(sets)f(exploit)1102 9408 y(fully)321 b(the)f(upp)37 b(er-semi)321 b(con)-37 b(tin)g(uit)g(y)320 b(of)g(the)h(set-v)-74 b(alued)320 b(function)h Fp(c)368 b Fm(!)36855 9072 y Fq(~)36434 9408 y Fm(N)195 b Fq(\()p Fp(c)p Fq(\),)344 b(and)321 b(the)f(structure)1102 11002 y(of)443 b(the)g(relev)-74 b(an)-37 b(t)444 b(Ma)37 b(~)-701 b(n)-37 b(\023)-627 b(e)443 b(sets.)2696 13369 y(Let)328 b(us)g(consider)g(the)g(Hamiltonian)g(\015o)-37 b(w)328 b(\010)24919 12887 y Fk(t)25714 13369 y Fq(whic)-37 b(h)329 b(is)e(a)h(small)g(p)37 b(erturbation)327 b(of)h(\010)45007 12887 y Fk(t)45007 13746 y(f)104 b Fv(+)p Fk(g)47063 13369 y Fq(.)552 b(Let)1102 15056 y(\010)489 b(and)g(\010)6138 15255 y Fk(f)104 b Fv(+)p Fk(g)8683 15056 y Fq(b)37 b(e)489 b(their)g(time-1-map.)728 b(As)489 b(the)f(cylinder)i Fl(T)326 b Fm(\002)f Fl(R)i Fm(\002)f(f)p Fq(\()p Fp(q)37935 15255 y Fv(2)38530 15056 y Fp(;)221 b(p)39788 15255 y Fv(2)40385 15056 y Fq(\))445 b(=)h(\(0)p Fp(;)221 b Fq(0\))p Fm(g)445 b Fq(=)h(\006)49325 15255 y Fv(0)1102 16650 y Fq(is)604 b(the)g(normally)g(h)-37 b(yp)37 b(erb)g(olic)605 b(in)-37 b(v)-74 b(arian)-37 b(t)605 b(manifold)f(for)g(\010)31992 16849 y Fk(f)104 b Fv(+)p Fk(g)34652 16650 y Fq(and)604 b(the)f Fo(a)624 b(priori)604 b Fq(unstable)1102 18244 y(condition)337 b(is)f(assumed,)358 b(it)336 b(follo)-37 b(ws)337 b(from)g(the)f(fundamen)-37 b(tal)337 b(theorem)g(of)f (normally)h(h)-37 b(yp)37 b(erb)g(olic)1102 19838 y(in)-37 b(v)-74 b(arian)-37 b(t)475 b(manifold)e(\(cf.)683 b([HPS]\))473 b(that)g(there)h(is)f Fp(\017)421 b Fq(=)f Fp(\017)p Fq(\()p Fp(A;)221 b(B)67 b Fq(\))420 b Fp(>)g Fq(0)474 b(suc)-37 b(h)474 b(that)f(if)g Fm(k)p Fp(P)184 b Fm(k)46079 20037 y Fk(C)46906 19771 y Fd(r)47929 19838 y Fm(\024)420 b Fp(\017)1102 21432 y Fq(on)512 b(the)f(region)h Fm(fj)p Fp(p)p Fm(j)484 b(\024)g Fq(max\()p Fm(j)p Fp(A)p Fm(j)p Fp(;)221 b Fm(j)p Fp(B)67 b Fm(j)p Fq(\))340 b(+)h(1)p Fm(g)511 b Fq(the)h(map)f(\010)31032 20950 y Fk(t)31500 21432 y Fm(j)31869 21631 y Fk(t)p Fv(=)p Fk(s)p Fv(+)p Fk(k)35568 21432 y Fq(\()p Fp(k)524 b Fm(2)483 b Fl(Z)p Fq(\))510 b(also)i(has)f(a)h Fp(C)47973 20950 y Fk(r)29 b Fg(\000)p Fv(1)1102 23026 y Fq(in)-37 b(v)-74 b(arian)-37 b(t)594 b(manifold)g(\006\()p Fp(s)p Fq(\))619 b Fm(\032)i Fl(R)18325 22544 y Fv(2)19317 23026 y Fm(\002)396 b Fl(T)21632 22544 y Fv(2)22227 23026 y Fq(.)1043 b(This)593 b(manifold)h(is)f(a)h (small)g(deformation)g(of)f(the)1102 24620 y(manifold)334 b(\006)7450 24819 y Fv(0)8047 24620 y Fm(j)8416 24860 y Fg(fj)p Fk(p)9823 24993 y Fj(1)10341 24860 y Fg(j\024)p Fv(max)q(\()p Fg(j)p Fk(A)p Fg(j)p Fk(;)p Fg(j)p Fk(B)45 b Fg(j)p Fv(\)+1)p Fg(g)19457 24620 y Fq(,)356 b(and)334 b(is)f(also)i(normally)f(h)-37 b(yp)37 b(erb)g(olic)335 b(and)f(time-1-p)37 b(erio)g(dic.)1102 26306 y(Let)637 b(\006)692 b(=)h(\006\(0\),)684 b(it)636 b(can)i(b)37 b(e)636 b(considered)i(as)f(the)g(image)g(of)g(a)f(map)h Fp( )48 b Fq(:)978 b(\006)41325 26505 y Fv(0)42613 26306 y Fm(!)693 b Fl(R)45593 25824 y Fv(2)46613 26306 y Fm(\002)425 b Fl(T)48957 25824 y Fv(2)49552 26306 y Fq(,)1102 27900 y(\006)370 b(=)f Fm(f)p Fp(p)5165 28099 y Fv(1)5761 27900 y Fp(;)221 b(q)6944 28099 y Fv(1)7541 27900 y Fp(;)g(p)8799 28099 y Fv(2)9395 27900 y Fq(\()p Fp(p)10580 28099 y Fv(1)11176 27900 y Fp(;)g(q)12359 28099 y Fv(1)12955 27900 y Fq(\))p Fp(;)g(q)14655 28099 y Fv(2)15251 27900 y Fq(\()p Fp(p)16436 28099 y Fv(1)17032 27900 y Fp(;)g(q)18215 28099 y Fv(1)18811 27900 y Fq(\))p Fm(g)p Fq(.)590 b(This)443 b(map)g(induces)h(a)f(2-form)g Fp( )37934 27418 y Fg(\003)38544 27900 y Fp(!)491 b Fq(on)443 b(\006)42666 28099 y Fv(0)15767 31778 y Fp( )16680 31229 y Fg(\003)17291 31778 y Fp(!)416 b Fq(=)19936 29905 y Fi(\022)20914 31778 y Fq(1)296 b(+)23361 30879 y Fp(@)74 b Fq(\()p Fp(p)25325 31078 y Fv(2)25921 30879 y Fp(;)221 b(q)27104 31078 y Fv(2)27700 30879 y Fq(\))p 23361 31472 4856 54 v 23361 32689 a Fp(@)74 b Fq(\()p Fp(p)25325 32888 y Fv(1)25921 32689 y Fp(;)221 b(q)27104 32888 y Fv(1)27700 32689 y Fq(\))28376 29905 y Fi(\023)29576 31778 y Fp(dp)30935 31977 y Fv(1)31826 31778 y Fm(^)295 b Fp(dq)34291 31977 y Fv(1)34887 31778 y Fp(:)1102 35582 y Fq(Since)647 b(the)g(second)g(de)g(Rham)f (co-homology)h(group)g(of)g(\006)32427 35781 y Fv(0)33669 35582 y Fq(is)g(trivial,)697 b(b)-37 b(y)646 b(using)h(Moser's)1102 37176 y(argumen)-37 b(t)427 b(on)f(the)g(isotop)-37 b(y)426 b(of)h(symplectic)f(forms)h([Mo],)i(w)-37 b(e)427 b(\014nd)f(that)f (there)i(exists)f(a)g(di\013eo-)1102 38770 y(morphism)443 b Fp( )8177 38969 y Fv(1)9216 38770 y Fq(on)g(\006)12020 38969 y Fv(0)12617 38770 y Fm(j)12986 39010 y Fg(fj)p Fk(p)14393 39143 y Fj(1)14911 39010 y Fg(j\024)p Fv(max)q(\()p Fg(j)p Fk(A)p Fg(j)p Fk(;)p Fg(j)p Fk(B)45 b Fg(j)p Fv(\)+1)p Fg(g)24470 38770 y Fq(suc)-37 b(h)443 b(that)18712 41858 y(\()p Fp( )343 b Fm(\016)295 b Fp( )22261 42057 y Fv(1)22858 41858 y Fq(\))23375 41310 y Fg(\003)23985 41858 y Fp(!)416 b Fq(=)369 b Fp(dp)27989 42057 y Fv(1)28881 41858 y Fm(^)295 b Fp(dq)31346 42057 y Fv(1)31942 41858 y Fp(:)14880 b Fq(\(3.1\))1102 44939 y(Since)444 b(\006)f(is)g(in)-37 b(v)-74 b(arian)-37 b(t)443 b(for)g(\010)g(and)g(\010)19861 44457 y Fg(\003)20472 44939 y Fp(!)417 b Fq(=)369 b Fp(!)48 b Fq(,)442 b(w)-37 b(e)444 b(ha)-37 b(v)g(e)11107 46974 y Fi(\020)11900 48449 y Fq(\()p Fp( )343 b Fm(\016)295 b Fp( )15449 48648 y Fv(1)16045 48449 y Fq(\))16562 47900 y Fg(\000)p Fv(1)18283 48449 y Fm(\016)g Fq(\010)h Fm(\016)f Fq(\()p Fp( )343 b Fm(\016)295 b Fp( )25005 48648 y Fv(1)25602 48449 y Fq(\))26119 46974 y Fi(\021)26912 47249 y Fg(\003)27522 48449 y Fp(dp)28881 48648 y Fv(1)29773 48449 y Fm(^)g Fp(dq)32238 48648 y Fv(1)33203 48449 y Fq(=)369 b Fp(dp)35964 48648 y Fv(1)36856 48449 y Fm(^)295 b Fp(dq)39321 48648 y Fv(1)1102 52013 y Fq(i.e.)859 b(\()p Fp( )402 b Fm(\016)355 b Fp( )7326 52212 y Fv(1)7923 52013 y Fq(\))8440 51531 y Fg(\000)p Fv(1)10220 52013 y Fm(\016)g Fq(\010)g Fm(\016)g Fq(\()p Fp( )402 b Fm(\016)355 b Fp( )17240 52212 y Fv(1)17836 52013 y Fq(\))532 b(preserv)-37 b(es)534 b(the)e(standard)g(area.)860 b(Clearly)-111 b(,)556 b(it)531 b(is)h(exact)h(and)1102 53607 y(t)-37 b(wist)426 b(since)h(it)f(is)g(a)g(small)g(p)37 b(erturbation)427 b(of)f(\010)25027 53806 y Fk(f)25716 53607 y Fq(.)585 b(In)426 b(this)f(sense,)431 b(w)-37 b(e)427 b(sa)-37 b(y)426 b(that)g(the)g(restriction)1102 55201 y(of)473 b(\010)f(on)h(\006)f(is)g(ob)-37 b(viously)473 b(area-preserving)h(and)f(t)-37 b(wist.)678 b(If)472 b Fp(r)455 b(>)418 b Fq(4)472 b(there)h(are)g(man)-37 b(y)473 b(in)-37 b(v)-74 b(arian)-37 b(t)1102 56795 y(homotopically)542 b(non-trivial)f(curv)-37 b(es,)567 b(including)542 b(man)-37 b(y)541 b(KAM)g(curv)-37 b(es.)887 b(As)541 b(it)g(still)g(remains)1102 58389 y(op)37 b(en)594 b(whether)g(the)e(in)-37 b(v)-74 b(arian)-37 b(t)594 b(curv)-37 b(es)594 b(m)-37 b(ust)593 b(b)37 b(e)593 b(di\013eren)-37 b(tiable,)631 b(w)-37 b(e)594 b(can)g(only)f(assume)g(all)1102 59983 y(these)339 b(curv)-37 b(es)339 b(are)h(of)e(Lipsc)-37 b(hitz.)556 b(Giv)-37 b(en)339 b Fp(\032)369 b Fm(2)g Fl(R)338 b Fq(there)h(is)g(an)f(Aubry-Mather)h(set)f(with)g(rotation)1102 61577 y(n)-37 b(um)g(b)37 b(er)424 b Fp(\032)p Fq(,)i(whic)-37 b(h)424 b(is)f(either)g(an)g(in)-37 b(v)-74 b(arian)-37 b(t)424 b(circle,)k(or)423 b(a)g(Denjo)-37 b(y)424 b(set)e(if)h Fp(\032)369 b Fm(2)f Fl(R)p Fm(n)p Fl(Q)p Fq(,)428 b(or)423 b(p)37 b(erio)g(dic)1102 63171 y(orbits)375 b(if)g Fp(\032)368 b Fm(2)h Fl(Q)p Fq(.)568 b(Under)375 b(the)g(generic)h(condition)f(w) -37 b(e)376 b(can)f(assume)g(there)g(is)g(no)g(homotopically)1102 64765 y(non-trivial)451 b(in)-37 b(v)-74 b(arian)-37 b(t)451 b(curv)-37 b(es)451 b(with)f(rational)h(rotation)f(n)-37 b(um)g(b)37 b(er)451 b(for)g(\010)p Fm(j)38611 64964 y Fv(\006)39436 64765 y Fq(,)h(there)f(is)f(only)h(one)1102 66359 y(minimal)443 b(p)37 b(erio)g(dic)444 b(orbit)f(on)g(\006)g(for)g (eac)-37 b(h)444 b(rational)f(rotation)g(n)-37 b(um)g(b)37 b(er.)2696 68726 y(Let)665 b(us)g(consider)h(the)f(Legendre)h (transformation)f Fm(L)p Fq(.)1257 b(By)665 b(abuse)g(of)g(terminology) g(w)-37 b(e)1102 70320 y(con)g(tin)g(ue)591 b(to)f(denote)g(\006\()p Fp(s)p Fq(\))f(and)i(its)e(image)i(under)f(the)g(Legendre)i (transformation)e(b)-37 b(y)590 b(the)1102 71914 y(same)444 b(sym)-37 b(b)37 b(ol.)590 b(Let)20760 73365 y(~)20613 73700 y(\006)369 b(=)23524 72438 y Fi([)23343 75293 y Fk(s)p Fg(2)p Fc(T)25180 73700 y Fq(\(\006\()p Fp(s)p Fq(\))p Fp(;)221 b(s)p Fq(\))p Fp(;)p eop %%Page: 13 13 13 12 bop 48792 2239 a Ft(13)1102 5427 y Fq(whic)-37 b(h)334 b(has)f(the)f(normal)h(h)-37 b(yp)37 b(erb)g(olicit)-37 b(y)334 b(as)e(w)-37 b(ell.)555 b(Under)333 b(the)g(Legendre)h (transformation)f(those)1102 7021 y(Aubry-Mather)511 b(sets)g(\(in)-37 b(v)-74 b(arian)-37 b(t)510 b(curv)-37 b(es,)529 b(Denjo)-37 b(y)511 b(sets)f(or)h(minimal)g(p)37 b(erio)g(dic)511 b(orbits\))f(on)h(\006)1102 8615 y(corresp)37 b(ond)450 b(to)e(the)g(supp)37 b(ort)448 b(of)h(some)f Fp(c)p Fq(-minimal)g(measures.)608 b(Recall)449 b Fp(H)39217 8133 y Fv(1)39813 8615 y Fq(\()p Fp(M)71 b(;)221 b Fl(R)p Fq(\))378 b(=)g Fl(R)46504 8133 y Fv(2)47101 8615 y Fq(.)606 b(W)-111 b(e)1102 10209 y(claim)461 b(that)f(eac)-37 b(h)462 b(of)e(these)h(sets)f(corresp)37 b(onds)462 b(to)e(an)h(in)-37 b(terv)-74 b(al)461 b(or)g(a)f(rectangle)i(in)f Fp(H)45014 9727 y Fv(1)45610 10209 y Fq(\()p Fp(M)71 b(;)221 b Fl(R)p Fq(\),)1102 11803 y(in)411 b(other)g(w)-37 b(ords,)417 b(for)411 b(all)g Fp(c)f Fq(in)g(this)g(in)-37 b(terv)-74 b(al)411 b(\(rectangle\),)418 b(the)410 b(time-1-section)h(of)f(the)g (supp)37 b(ort)1102 13397 y(of)443 b(the)g Fp(c)p Fq(-minimal)f (measure)i(is)f(exactly)g(this)g(Aubry-Mather)g(set.)2696 15537 y(T)-111 b(o)-37 b(w)g(ards)519 b(that)e(goal,)536 b(w)-37 b(e)518 b(in)-37 b(tro)37 b(duce)518 b(the)f(co)37 b(ordinate)518 b(transformation)f(\()p Fp(p)41574 15736 y Fv(1)42170 15537 y Fp(;)221 b(q)43353 15736 y Fv(2)43949 15537 y Fp(;)g(p)45207 15736 y Fv(2)45804 15537 y Fp(;)g(q)46987 15736 y Fv(2)47583 15537 y Fq(\))493 b Fm(!)1102 17131 y Fq(\()p Fp(p)2287 17330 y Fv(1)2883 17131 y Fp(;)221 b(q)4066 17330 y Fv(2)4662 17131 y Fp(;)g(p)5920 17330 y Fv(2)6812 17131 y Fq(+)296 b Fp(\020)98 b Fq(\()p Fp(q)9930 17330 y Fv(2)10525 17131 y Fq(\))p Fp(;)221 b(q)12225 17330 y Fv(2)12821 17131 y Fq(\))443 b(where)h Fp(\020)541 b Fq(is)442 b(de\014ned)i(in)f(the)f(w)-37 b(a)g(y)444 b(suc)-37 b(h)443 b(that)20523 18982 y Fp(@)74 b(g)p 20231 19575 2044 54 v 20231 20792 a(@)g(p)21678 20991 y Fv(2)22434 19880 y Fq(\()p Fp(\020)98 b Fq(\()p Fp(q)24740 20079 y Fv(2)25335 19880 y Fq(\))p Fp(;)221 b(q)27035 20079 y Fv(2)27631 19880 y Fq(\))369 b(=)g(0)p Fp(;)16239 b Fq(\(3.2\))1102 22645 y(and)443 b(let)g Fp(g)6286 22163 y Fg(0)6658 22645 y Fq(\()p Fp(p)7843 22844 y Fv(2)8439 22645 y Fp(;)221 b(q)9622 22844 y Fv(2)10218 22645 y Fq(\))369 b(=)g Fp(g)48 b Fq(\()p Fp(p)14373 22844 y Fv(2)15263 22645 y Fq(+)295 b Fp(\020)98 b Fq(\()p Fp(q)18380 22844 y Fv(2)18976 22645 y Fq(\))p Fp(;)221 b(q)20676 22844 y Fv(2)21272 22645 y Fq(\).)590 b(By)443 b(the)f(assumption)h(on) g Fp(g)490 b Fq(w)-37 b(e)443 b(no)-37 b(w)444 b(ha)-37 b(v)g(e)14675 24654 y Fp(@)15454 24172 y Fv(2)16050 24654 y Fp(g)16732 24172 y Fg(0)p 14675 25248 2429 54 v 14881 26464 a Fp(@)74 b(q)16301 26007 y Fv(2)16253 26818 y(2)17263 25553 y Fq(\(0)p Fp(;)221 b Fq(0\))369 b Fp(<)g Fq(0)p Fp(;)26848 24654 y(@)27627 24172 y Fv(2)28223 24654 y Fp(g)28905 24172 y Fg(0)p 26057 25248 4012 54 v 26057 26464 a Fp(@)74 b(p)27504 26663 y Fv(2)28100 26464 y Fp(@)g(q)29472 26663 y Fv(2)30227 25553 y Fq(\(0)p Fp(;)221 b(q)32591 25752 y Fv(2)33187 25553 y Fq(\))369 b(=)g(0)p Fp(:)10683 b Fq(\(3.3\))1102 28413 y(T)-111 b(o)604 b(simplify)g(the)f (notation)g(w)-37 b(e)604 b(still)e(use)i Fp(g)650 b Fq(to)603 b(denote)h(the)f(function)h Fp(g)39652 27931 y Fg(0)40023 28413 y Fq(.)1072 b(Let)604 b Fp(L)44909 28612 y Fv(0)46108 28413 y Fq(b)37 b(e)603 b(the)1102 30007 y(Lagrangian)444 b(obtained)g(from)f Fp(f)438 b Fq(+)295 b Fp(g)490 b Fq(b)-37 b(y)443 b(Legendre)i(transformation,)e (it)f(has)h(the)g(form)17077 32065 y Fp(L)17981 32264 y Fv(0)18577 32065 y Fq(\()p Fp(q)19687 32264 y Fv(2)20282 32065 y Fp(;)468 b Fq(_)-616 b Fp(q)48 b Fq(\))369 b(=)g Fp(`)24354 32264 y Fv(1)24950 32065 y Fq(\()247 b(_)-616 b Fp(q)26060 32264 y Fv(1)26656 32065 y Fq(\))295 b(+)g Fp(`)29349 32264 y Fv(2)29945 32065 y Fq(\()p Fp(q)31055 32264 y Fv(2)31651 32065 y Fp(;)468 b Fq(_)-616 b Fp(q)32834 32264 y Fv(2)33430 32065 y Fq(\))1102 34122 y(where)554 b Fp(`)5606 34321 y Fv(1)6755 34122 y Fq(and)e Fp(`)10000 34321 y Fv(2)11149 34122 y Fq(are)h(the)f(Legendre)i(transformation)e (of)h Fp(f)695 b Fq(and)552 b Fp(g)600 b Fq(resp)37 b(ectiv)-37 b(ely)-111 b(.)920 b(As)553 b Fp(g)599 b Fq(is)1102 35716 y(a)541 b(con)-37 b(v)g(ex)541 b(function)g(in)f Fp(p)14374 35915 y Fv(2)14970 35716 y Fq(,)811 b(_)-615 b Fp(q)16497 35915 y Fv(2)17624 35716 y Fq(=)779 b(_)-616 b Fp(q)19782 35915 y Fv(2)20378 35716 y Fq(\()p Fp(p)21563 35915 y Fv(2)22159 35716 y Fp(;)221 b(q)23342 35915 y Fv(2)23938 35716 y Fq(\))531 b(=)h Fp(@)27256 35915 y Fk(p)27804 36048 y Fj(2)28388 35716 y Fp(g)48 b Fq(\()p Fp(p)30255 35915 y Fv(2)30851 35716 y Fp(;)221 b(q)32034 35915 y Fv(2)32630 35716 y Fq(\),)564 b(w)-37 b(e)541 b(\014nd)f(from)h (\(3.2\))f(and)g(the)1102 37310 y(con)-37 b(v)g(exit)g(y)401 b(of)f Fp(g)447 b Fq(that)645 b(_)-615 b Fp(q)12942 37509 y Fv(2)13538 37310 y Fq(\(0)p Fp(;)221 b(q)15902 37509 y Fv(2)16498 37310 y Fq(\))368 b(=)h(0)400 b(and)g Fp(@)321 b Fq(_)-616 b Fp(q)23761 37509 y Fv(2)24357 37310 y Fp(=@)74 b(p)26468 37509 y Fv(2)27434 37310 y Fp(>)369 b Fq(0,)408 b(th)-37 b(us)399 b Fp(`)33709 37509 y Fv(2)34706 37310 y Fq(can)h(b)37 b(e)400 b(written)g(in)f(the)h(form)17066 39368 y Fp(`)17619 39567 y Fv(2)18216 39368 y Fq(\()p Fp(q)19326 39567 y Fv(2)19921 39368 y Fp(;)468 b Fq(_)-616 b Fp(q)21104 39567 y Fv(2)21701 39368 y Fq(\))368 b(=)h Fp(V)295 b Fq(\()p Fp(q)26168 39567 y Fv(2)26764 39368 y Fq(\))g(+)g Fp(U)145 b Fq(\()p Fp(q)31066 39567 y Fv(2)31661 39368 y Fp(;)468 b Fq(_)-616 b Fp(q)32844 39567 y Fv(2)33441 39368 y Fq(\))13233 b(\(3.4\))1102 41425 y(where)450 b Fp(V)295 b Fq(\()p Fp(q)7129 41624 y Fv(2)7724 41425 y Fq(\))378 b(=)g Fm(\000)p Fp(g)48 b Fq(\(0)p Fp(;)221 b(q)14109 41624 y Fv(2)14705 41425 y Fq(\),)449 b Fp(U)523 b Fm(\025)378 b Fq(0)448 b(is)g(a)g(con)-37 b(v)g(ex)450 b(function)e(in)695 b(_)-616 b Fp(q)34162 41624 y Fv(2)35206 41425 y Fq(with)448 b(sup)37 b(er-linear)450 b(gro)-37 b(wth,)1102 43019 y(attains)646 b(its)f(minim)-37 b(um)646 b(at)892 b(_)-616 b Fp(q)16359 43218 y Fv(2)17663 43019 y Fq(=)707 b(0)646 b(\()p Fm(8)p Fp(q)22561 43218 y Fv(2)23864 43019 y Fm(2)707 b Fl(T)p Fq(\).)1199 b(By)646 b(the)g(assumption,)696 b Fp(V)941 b Fq(has)646 b(a)g(global)1102 44613 y(minim)-37 b(um)443 b(at)g Fp(q)9260 44812 y Fv(2)10225 44613 y Fq(=)369 b(0)443 b(whic)-37 b(h)443 b(is)g(non-degenerate.)2696 46753 y(No)-37 b(w)627 b(let)e(us)g(consider)i(the)e Fp(\014)696 b Fq(function)625 b(of)h Fp(L)27155 46952 y Fv(0)27751 46753 y Fq(.)1138 b(Under)626 b(the)g(\015o)-37 b(w)626 b Fp(\036)39678 46271 y Fk(t)39678 47119 y(L)40407 47252 y Fj(0)41617 46753 y Fq(eac)-37 b(h)627 b(in)-37 b(v)-74 b(arian)-37 b(t)1102 48347 y(circle)604 b(on)e(\006)h(with)f (rotation)g(n)-37 b(um)g(b)37 b(er)603 b Fp(\032)f Fq(is)h(the)f(supp) 37 b(ort)602 b(of)g(the)g(minimal)h(measure)g Fp(\026)47628 48587 y Fv(\()p Fk(\032;)p Fv(0\))1102 49948 y Fq(whose)524 b(rotation)e(v)-37 b(ector)524 b(is)e(\()p Fp(\032;)221 b Fq(0\).)829 b(If)523 b(w)-37 b(e)523 b(consider)h(the)e Fp(\013)527 b Fq(function)c(of)f Fp(L)40157 50147 y Fv(0)40753 49948 y Fq(,)543 b(it)522 b(corresp)37 b(onds)1102 51557 y(to)638 b(the)f(supp)37 b(ort)637 b(of)g(those)h Fp(c)p Fq(-minimal)f(measures)h(that)f Fp(c)693 b Fm(2)g(f)p Fp(c)35326 51756 y Fv(1)35922 51557 y Fm(g)425 b(\002)g Fq([)p Fp(c)39413 50989 y Fg(\000)39413 51911 y Fv(2)40310 51557 y Fp(;)221 b(c)41475 50989 y Fv(+)41475 51911 y(2)42357 51557 y Fq(])693 b Fm(\032)h Fl(R)46105 51075 y Fv(2)47338 51557 y Fq(with)1102 53188 y Fm(\0001)448 b Fp(<)f(c)5966 52620 y Fg(\000)5966 53542 y Fv(2)7310 53188 y Fp(<)g Fq(0)g Fp(<)h(c)11957 52620 y Fv(+)11957 53542 y(2)13285 53188 y Fp(<)g Fm(1)p Fq(.)731 b(W)-111 b(e)490 b(ha)-37 b(v)g(e)491 b Fp(c)23213 52620 y Fg(\000)23213 53542 y Fv(2)24556 53188 y Fp(<)448 b(c)26612 52620 y Fv(+)26612 53542 y(2)27983 53188 y Fq(since)490 b(the)g Fp(\014)560 b Fq(function)490 b(of)f(the)h(t)-37 b(wist)489 b(map)1102 54782 y(has)357 b(corner)h(at)e(rational)h(n)-37 b(um)g(b)37 b(ers.)562 b Fp(\014)427 b Fq(is)356 b(di\013eren)-37 b(tiable)358 b(at)e(some)h(rational)g(n)-37 b(um)g(b)37 b(er)357 b Fp(p=q)405 b Fq(if)356 b(and)1102 56376 y(only)415 b(there)g(exists)g(a)f(homotopically)i(non-trivial)e(in)-37 b(v)-74 b(arian)-37 b(t)415 b(curv)-37 b(e)416 b(of)e(rotation)h(n)-37 b(um)g(b)37 b(er)415 b Fp(p=q)48 b Fq(,)1102 57985 y(consists)365 b(en)-37 b(tirely)366 b(of)f(p)37 b(erio)g(dic)366 b(orbits)f(of)g(p)37 b(erio)g(d)366 b Fp(q)412 b Fq(\([Ba],[Ma2]\).)564 b(W)-111 b(e)366 b(obtain)f(that)f Fm(\0001)370 b Fp(<)f(c)49025 57417 y Fg(\000)49025 58339 y Fv(2)1102 59616 y Fq(and)506 b Fp(c)4323 59048 y Fv(+)4323 59970 y(2)5679 59616 y Fp(<)474 b Fm(1)p Fq(,)522 b(from)507 b(the)e(prop)37 b(ert)-37 b(y)507 b(that)e(b)37 b(oth)506 b Fp(\013)k Fq(and)c Fp(\014)576 b Fq(functions)506 b(are)h(\014nite)f(ev)-37 b(erywhere)1102 61210 y(and)443 b(has)g(sup)37 b(er-linear)444 b(gro)-37 b(wth.)2696 63350 y(Next,)464 b(let)c(us)f(consider)h(the)g Fp(\013)k Fq(function)459 b(of)g Fp(L)p Fq(.)641 b(W)-111 b(e)460 b(use)g Fp(c)396 b Fq(=)h(\()p Fp(c)35573 63549 y Fv(1)36168 63350 y Fp(;)221 b(c)37333 63549 y Fv(2)37929 63350 y Fq(\))396 b Fm(2)h Fl(R)41084 62868 y Fv(2)42139 63350 y Fq(to)459 b(denote)h(the)1102 64959 y(\014rst)338 b(de)f(Rham)h(cohomology)g(class)g(of)g Fp(M)145 b Fq(.)555 b(F)-111 b(or)338 b(eac)-37 b(h)338 b Fp(c)369 b Fm(2)g Fl(R)84 b Fm(\002)g Fq(\()p Fp(c)33779 64390 y Fg(\000)33779 65313 y Fv(2)34676 64959 y Fp(;)221 b(c)35841 64390 y Fv(+)35841 65313 y(2)36723 64959 y Fq(\))337 b(the)h(action)f(v)-74 b(ariable)339 b(on)1102 66553 y(eac)-37 b(h)459 b Fp(c)p Fq(-minimal)e(orbit)g(of)h Fp(L)16001 66752 y Fv(0)17054 66553 y Fq(tak)-37 b(es)458 b(v)-74 b(alue)458 b(\()p Fp(p)25102 66752 y Fv(1)25698 66553 y Fp(;)221 b Fq(0\))457 b(whic)-37 b(h)459 b(is)e(indep)37 b(enden)-37 b(t)458 b(of)f Fp(t)p Fq(.)634 b(Let)458 b Fp(A)47035 66070 y Fg(\003)47646 66553 y Fp(;)221 b(B)49311 66070 y Fg(\003)1102 68161 y Fq(b)37 b(e)466 b(suc)-37 b(h)466 b(n)-37 b(um)g(b)37 b(ers)466 b(that)f(for)h(eac)-37 b(h)466 b Fp(c)407 b Fm(2)g Fq([)p Fp(A)22944 67679 y Fg(\003)23555 68161 y Fp(;)221 b(B)25220 67679 y Fg(\003)25830 68161 y Fq(])310 b Fm(\002)h Fq(\()p Fp(c)28945 67593 y Fg(\000)28945 68516 y Fv(2)29841 68161 y Fp(;)221 b(c)31006 67593 y Fv(+)31006 68516 y(2)31888 68161 y Fq(\))465 b(the)g(corresp)37 b(onding)467 b Fp(p)44409 68360 y Fv(1)45471 68161 y Fq(satis\014es)1102 69755 y(the)443 b(condition)19601 71349 y Fp(A)296 b Fm(\000)f Fq(1)369 b Fm(\024)g Fp(p)25324 71548 y Fv(1)26289 71349 y Fm(\024)h Fp(B)361 b Fq(+)295 b(1)p Fp(:)1102 73371 y Fq(In)474 b(the)g(follo)-37 b(wing,)482 b(w)-37 b(e)475 b(shall)f(use)19075 73035 y(\026)18580 73371 y Fp(M)565 b Fq(=)421 b Fm(f)p Fq(\()99 b(\026)-763 b Fp(q)23662 73570 y Fv(1)24258 73371 y Fp(;)320 b Fq(\026)-763 b Fp(q)25441 73570 y Fv(2)26037 73371 y Fq(\))421 b Fm(2)f Fl(T)315 b Fm(\002)i Fl(R)p Fm(g)474 b Fq(to)g(denote)g(a)g(co)-37 b(v)g(ering)475 b(space)g(of)1102 74965 y Fp(M)145 b Fq(.)590 b(F)-111 b(or)444 b(a)f(curv)-37 b(e)444 b Fp(\015)74 b Fq(:)590 b Fp(I)473 b Fm(!)369 b Fp(M)145 b Fq(,)442 b(w)-37 b(e)444 b(use)492 b(\026)-713 b Fp(\015)517 b Fq(to)442 b(denote)h(its)g(lift)f(to)g(this)h(co)-37 b(v)g(ering)444 b(space.)p eop %%Page: 14 14 14 13 bop 1102 2239 a Ft(14)1102 5427 y Fr(Lemma)493 b(3.1.)764 b Fo(Ther)-68 b(e)462 b(exists)h Fp(\017)17612 5626 y Fv(0)18577 5427 y Fp(>)369 b Fq(0)p Fo(,)465 b(if)e Fm(k)p Fp(P)184 b Fm(k)25157 5626 y Fk(r)26115 5427 y Fm(\024)369 b Fp(\017)28056 5626 y Fv(0)29115 5427 y Fo(on)462 b(the)g(r)-68 b(e)g(gion)463 b Fm(fj)p Fp(p)p Fm(j)369 b(\024)g Fq(max\()p Fm(j)p Fp(A)p Fm(j)p Fp(;)221 b Fm(j)p Fp(B)67 b Fm(j)p Fq(\))266 b(+)1102 7021 y(1)p Fm(g)p Fo(,)472 b(ther)-68 b(e)469 b(is)h(a)f(strip)h Fp(C)465 b Fq(=)369 b([)p Fp(A)16461 6539 y Fg(\003)17072 7021 y Fp(;)221 b(B)18737 6539 y Fg(\003)19347 7021 y Fq(])282 b Fm(\002)g Fq([)p Fm(\000)p Fp(c)23290 6539 y Fg(\003)23290 7351 y Fv(2)23901 7021 y Fp(;)221 b(c)25066 6539 y Fg(\003)25066 7351 y Fv(2)25677 7021 y Fq(])369 b Fm(\032)g Fp(H)29029 6539 y Fv(1)29625 7021 y Fq(\()p Fp(M)71 b(;)221 b Fl(R)p Fq(\))469 b(\()p Fp(c)35129 6539 y Fg(\003)35129 7351 y Fv(2)36108 7021 y Fp(>)369 b Fq(0\))p Fo(,)471 b(such)e(that)h(for)f(e)-68 b(ach)1102 8709 y Fp(c)369 b Fm(2)g Fp(C)95 b Fo(,)476 b(the)f Fp(c)p Fo(-minimal)i(orbit)f(set)19207 8373 y Fq(~)18957 8709 y Fm(G)79 b Fq(\()p Fp(c)p Fq(\))368 b Fm(\032)23353 8373 y Fq(~)23205 8709 y(\006)p Fo(.)1102 10981 y Fr(Pro)42 b(of)p Fq(:)853 b(Note)574 b(the)g(Lagrange)i(\015o)-37 b(w)574 b(of)g Fp(L)23150 11180 y Fv(0)24320 10981 y Fq(is)g(in)-37 b(tegrable)575 b(and)f(is)g(decoupled)h(b)37 b(et)-37 b(w)g(een)575 b(t)-37 b(w)g(o)1102 12575 y(phase)501 b(sub-space)f(\()p Fp(q)12016 12774 y Fv(1)12611 12575 y Fp(;)468 b Fq(_)-616 b Fp(q)13794 12774 y Fv(1)14391 12575 y Fq(\))499 b(and)h(\()p Fp(q)19157 12774 y Fv(2)19753 12575 y Fp(;)467 b Fq(_)-615 b Fp(q)20936 12774 y Fv(2)21532 12575 y Fq(\).)761 b(Let)500 b(us)g(consider)g Fp(\036)33402 12093 y Fk(t)33402 12951 y(`)33847 13084 y Fj(2)34432 12575 y Fq(,)514 b(the)500 b(second)h(comp)37 b(onen)-37 b(t)500 b(of)1102 14275 y(the)604 b(\015o)-37 b(w)604 b Fp(\036)7270 13793 y Fk(t)7270 14640 y(L)7999 14773 y Fj(0)8584 14275 y Fq(.)1073 b(It)603 b(has)h(t)-37 b(w)g(o)604 b(homo)37 b(clinic)604 b(lo)37 b(ops)604 b(\000)28095 13793 y Fv(+)29581 14275 y Fq(and)g(\000)33155 13793 y Fg(\000)34052 14275 y Fq(,)643 b(whic)-37 b(h)605 b(can)f(b)37 b(e)604 b(though)-37 b(t)603 b(as)1102 16050 y(the)548 b(graph)g(of)f(the)h(functions)f Fp(G)18252 15568 y Fg(\006)19149 16050 y Fq(\()p Fp(q)20259 16249 y Fv(2)20855 16050 y Fq(\),)573 b(i.e.)905 b(\000)25746 15568 y Fg(\006)27186 16050 y Fq(=)544 b Fm(f)p Fp(q)30020 16249 y Fv(2)30616 16050 y Fp(;)221 b(G)32250 15568 y Fg(\006)33148 16050 y Fq(\()p Fp(q)34258 16249 y Fv(2)34853 16050 y Fq(\))p Fm(g)p Fq(.)904 b(The)549 b(orbit)e Fp(dq)44830 15482 y Fv(+)44782 16404 y(2)46259 16050 y Fq(on)h(\000)49039 15568 y Fv(+)1102 17681 y Fq(encircles)530 b(the)d(cylinder)h Fl(T)352 b Fm(\002)f Fl(R)528 b Fq(in)g(coun)-37 b(ter)528 b(clo)37 b(c)-37 b(kwise)529 b(direction)f(\()247 b(_)-616 b Fp(q)37237 17880 y Fv(2)38343 17681 y Fp(>)510 b Fq(0\),)549 b(the)527 b(orbit)h Fp(dq)49025 17113 y Fg(\000)48977 18035 y Fv(2)1102 19275 y Fq(on)507 b(\000)3841 18793 y Fg(\000)5245 19275 y Fq(encircles)h(the)f(cylinder)h(in)e(clo)37 b(c)-37 b(kwise)509 b(direction)e(\()246 b(_)-615 b Fp(q)32298 19474 y Fv(2)33369 19275 y Fp(<)476 b Fq(0\).)782 b(Clearly)508 b(w)-37 b(e)507 b(ha)-37 b(v)g(e)508 b(some)1102 20884 y(p)37 b(ositiv)-37 b(e)443 b(n)-37 b(um)g(b)37 b(ers)444 b Fp(C)12423 20315 y Fg(\006)12328 21274 y Fk(A)13689 20884 y Fp(>)369 b Fq(0,)443 b Fp(C)17611 20315 y Fg(\006)17516 21274 y Fk(K)18909 20884 y Fp(>)369 b Fq(0)443 b(suc)-37 b(h)444 b(that)7065 22538 y Fi(Z)8393 22866 y Fg(1)7803 25556 y(\0001)9980 24346 y Fp(`)10533 24545 y Fv(2)11129 24346 y Fq(\()p Fp(q)12287 23778 y Fg(\006)12239 24700 y Fv(2)13183 24346 y Fq(\()p Fp(t)p Fq(\))p Fp(;)467 b Fq(_)-615 b Fp(q)15928 23778 y Fg(\006)15880 24700 y Fv(2)16824 24346 y Fq(\()p Fp(t)p Fq(\)\))p Fp(dt)367 b Fq(=)i Fp(C)22839 23778 y Fg(\006)22744 24736 y Fk(A)23736 24346 y Fp(;)26983 22538 y Fi(Z)28312 22866 y Fg(1)27721 25556 y(\0001)29898 24346 y Fp(U)145 b Fq(\()p Fp(q)32108 23778 y Fg(\006)32060 24700 y Fv(2)33004 24346 y Fq(\()p Fp(t)p Fq(\))p Fp(;)467 b Fq(_)-615 b Fp(q)35749 23778 y Fg(\006)35701 24700 y Fv(2)36645 24346 y Fq(\()p Fp(t)p Fq(\)\))p Fp(dt)367 b Fq(=)i Fp(C)42660 23778 y Fg(\006)42565 24736 y Fk(K)43590 24346 y Fp(;)1102 27781 y Fq(it)443 b(is)f(easy)i(to)e(see)i(that)16230 29927 y Fp(c)16805 29359 y Fv(+)16805 30281 y(2)18056 29927 y Fq(=)20020 29028 y(1)p 19618 29621 1470 54 v 19618 30838 a(2)p Fp(\031)21246 29927 y(C)22290 29359 y Fv(+)22195 30317 y Fk(A)23172 29927 y Fp(;)2879 b(c)26995 29359 y Fg(\000)26995 30281 y Fv(2)28260 29927 y Fq(=)369 b Fm(\000)31257 29028 y Fq(1)p 30854 29621 V 30854 30838 a(2)p Fp(\031)32483 29927 y(C)33527 29359 y Fg(\000)33432 30317 y Fk(A)34424 29927 y Fp(:)1102 32807 y Fq(Let)469 b(us)g(single)g(out)f(one)h(lift)f (of)g(these)h(homo)37 b(clinic)470 b(lo)37 b(ops)468 b Fp(d)99 b Fq(\026)-763 b Fp(q)32406 32239 y Fv(+)32358 33161 y(2)33757 32807 y Fq(and)468 b Fp(d)99 b Fq(\026)-763 b Fp(q)37697 32239 y Fg(\000)37649 33161 y Fv(2)39063 32807 y Fq(with)567 b(\026)-763 b Fp(q)42755 32239 y Fv(+)42707 33161 y(2)43636 32807 y Fq(\()p Fm(\0001)p Fq(\))412 b(=)g(0,)1201 34438 y(\026)-763 b Fp(q)1743 33870 y Fv(+)1695 34792 y(2)2625 34438 y Fq(\()p Fm(1)p Fq(\))680 b(=)g(2)p Fp(\031)48 b Fq(,)775 b(\026)-763 b Fp(q)10535 33870 y Fg(\000)10487 34792 y Fv(2)11431 34438 y Fq(\()p Fm(\0001)p Fq(\))680 b(=)g(2)p Fp(\031)48 b Fq(,)775 b(\026)-762 b Fp(q)20375 33870 y Fg(\000)20327 34792 y Fv(2)21271 34438 y Fq(\()p Fm(1)p Fq(\))679 b(=)i(0.)1150 b(F)-111 b(or)631 b Fp(a)31482 34637 y Fv(1)32078 34438 y Fp(;)221 b(a)33370 34637 y Fv(2)34646 34438 y Fp(>)681 b Fq(0,)676 b(w)-37 b(e)630 b(assume)729 b(\026)-763 b Fp(q)45629 33870 y Fv(+)45581 34792 y(2)46511 34438 y Fq(\(0\))679 b(=)1102 36069 y Fp(a)1804 36268 y Fv(1)2400 36069 y Fq(,)826 b(\026)-763 b Fp(q)4137 35501 y Fg(\000)4089 36423 y Fv(2)5034 36069 y Fq(\(0\))747 b(=)h(2)p Fp(\031)495 b Fm(\000)447 b Fp(a)13358 36268 y Fv(2)13954 36069 y Fq(.)1272 b(Let)670 b Fp(T)18978 36268 y Fv(+)19861 36069 y Fq(\()p Fp(a)21080 36268 y Fv(1)21675 36069 y Fp(;)221 b(a)22967 36268 y Fv(2)23564 36069 y Fq(\))669 b(and)i Fp(T)28337 36268 y Fg(\000)29233 36069 y Fq(\()p Fp(a)30452 36268 y Fv(1)31048 36069 y Fp(;)221 b(a)32340 36268 y Fv(2)32937 36069 y Fq(\))669 b(b)37 b(e)671 b(the)f(n)-37 b(um)g(b)37 b(ers)670 b(suc)-37 b(h)671 b(that)1201 37700 y(\026)-763 b Fp(q)1743 37131 y Fv(+)1695 38054 y(2)2625 37700 y Fq(\()p Fp(T)3918 37899 y Fv(+)4800 37700 y Fq(\()p Fp(a)6019 37899 y Fv(1)6614 37700 y Fp(;)221 b(a)7906 37899 y Fv(2)8503 37700 y Fq(\)\))629 b(=)i(2)p Fp(\031)447 b Fm(\000)400 b Fp(a)15833 37899 y Fv(2)17029 37700 y Fq(and)699 b(\026)-763 b Fp(q)20410 37131 y Fg(\000)20362 38054 y Fv(2)21306 37700 y Fq(\()p Fp(T)22599 37899 y Fg(\000)23495 37700 y Fq(\()p Fp(a)24714 37899 y Fv(1)25310 37700 y Fp(;)221 b(a)26602 37899 y Fv(2)27199 37700 y Fq(\)\))629 b(=)h Fp(a)31227 37899 y Fv(1)31824 37700 y Fq(.)1060 b(Clearly)-111 b(,)641 b Fp(T)39100 37899 y Fg(\006)39997 37700 y Fq(\()p Fp(a)41216 37899 y Fv(1)41811 37700 y Fp(;)221 b(a)43103 37899 y Fv(2)43700 37700 y Fq(\))630 b Fm(!)g(1)600 b Fq(as)1102 39294 y(one)444 b(of)f Fp(a)5753 39493 y Fk(i)6564 39294 y Fm(!)369 b Fq(0.)591 b(There)444 b(exists)f Fp(a)18096 39493 y Fv(0)19061 39294 y Fp(>)369 b Fq(0)443 b(for)g Fp(a)24305 39493 y Fk(i)25116 39294 y Fm(\024)369 b Fp(a)27220 39493 y Fv(0)28259 39294 y Fq(\()p Fp(i)g Fq(=)g(1)p Fp(;)221 b Fq(2\))443 b(w)-37 b(e)443 b(ha)-37 b(v)g(e)14443 41187 y Fi(Z)15772 41515 y Fk(T)16399 41648 y Fe(\006)17185 41515 y Fv(\()p Fk(a)18176 41648 y Fj(1)18695 41515 y Fk(;a)19586 41648 y Fj(2)20105 41515 y Fv(\))15181 44205 y(0)20808 42995 y Fp(`)21361 43194 y Fv(2)21957 42995 y Fq(\()p Fp(q)23115 42426 y Fg(\006)23067 43349 y Fv(2)24011 42995 y Fq(\()p Fp(t)p Fq(\))p Fp(;)467 b Fq(_)-615 b Fp(q)26756 42426 y Fg(\006)26708 43349 y Fv(2)27652 42995 y Fq(\()p Fp(t)p Fq(\)\))p Fp(dt)367 b Fm(\025)33115 42096 y Fq(9)p 32783 42689 1329 54 v 32783 43906 a(10)34270 42995 y Fp(C)35314 42426 y Fg(\006)35219 43385 y Fk(A)36211 42995 y Fp(:)10611 b Fq(\(3.5\))1102 46454 y(Let)507 b(\012)475 b(=)h Fm(f)p Fq(\()p Fp(q)8263 46653 y Fv(2)8858 46454 y Fp(;)468 b Fq(_)-616 b Fp(q)10041 46653 y Fv(2)10637 46454 y Fq(\))475 b(:)g Fp(G)13517 45972 y Fg(\000)14414 46454 y Fq(\()p Fp(q)15524 46653 y Fv(2)16120 46454 y Fq(\))f Fm(\024)722 b Fq(_)-616 b Fp(q)19212 46653 y Fv(2)20283 46454 y Fm(\024)475 b Fp(G)22835 45972 y Fv(+)23718 46454 y Fq(\()p Fp(q)24828 46653 y Fv(2)25423 46454 y Fq(\))p Fm(g)p Fq(,)522 b(it)506 b(is)h(a)f(\014sh-shap)37 b(ed)507 b(area)g(and)g(\014lled)f(with)1102 48048 y(p)37 b(erio)g(dic)379 b(orbits)e(of)h Fp(`)11843 48247 y Fv(2)12439 48048 y Fq(.)569 b(The)378 b(circles)h(made)f(up)f(b)-37 b(y)378 b(these)g(p)37 b(erio)g(dic)378 b(orbits)f(are)i(con)-37 b(tractible)378 b(on)1102 49643 y(the)453 b(cylinder.)622 b(F)-111 b(or)454 b(small)f Fp(a)15588 49842 y Fv(2)16570 49643 y Fp(>)386 b Fq(0)454 b(there)f(exists)g Fp(a)26895 49842 y Fv(1)27877 49643 y Fq(=)386 b Fp(a)29998 49842 y Fv(1)30594 49643 y Fq(\()p Fp(a)31813 49842 y Fv(2)32409 49643 y Fq(\))g Fp(>)g Fq(0)453 b(suc)-37 b(h)453 b(that)g(if)f(the)h (p)37 b(erio)g(dic)1102 51237 y(orbit)422 b Fp(dq)5616 51436 y Fv(2)6212 51237 y Fq(\()p Fp(t)p Fq(\))f(in)-37 b(tersects)422 b(the)g(line)g Fm(f)p Fp(q)20023 51436 y Fv(2)20988 51237 y Fq(=)369 b Fp(a)23092 51436 y Fv(1)23688 51237 y Fm(g)422 b Fq(with)668 b(_)-615 b Fp(q)28372 51436 y Fv(2)28968 51237 y Fq(\(0\))368 b Fm(\025)h Fq(0)422 b(then)f(it)h(can)g(in)-37 b(tersect)422 b(the)g(line)1102 52831 y Fp(q)1695 53030 y Fv(2)2660 52831 y Fq(=)369 b(2)p Fp(\031)156 b Fm(\000)6941 52286 y Fk(a)7517 52419 y Fj(2)p 6941 52525 1095 54 v 7223 53289 a Fv(2)8544 52831 y Fq(also.)560 b(Let)368 b(\026)-683 b Fp(a)14682 53030 y Fv(2)15647 52831 y Fq(=)389 b(\026)-683 b Fp(a)17752 53030 y Fv(2)18348 52831 y Fq(\()p Fp(a)19567 53030 y Fv(0)20162 52831 y Fq(\))369 b Fm(\024)g Fp(a)23152 53030 y Fv(0)24098 52831 y Fq(b)37 b(e)349 b(the)h(biggest)f(n)-37 b(um)g(b)37 b(er)350 b(suc)-37 b(h)350 b(that)f Fp(a)43556 53030 y Fv(1)44152 52831 y Fq(\()19 b(\026)-683 b Fp(a)45371 53030 y Fv(2)45966 52831 y Fq(\))369 b Fm(\024)g Fp(a)48956 53030 y Fv(0)49552 52831 y Fq(.)1102 54425 y(Let)483 b Fp(T)4298 54624 y Fv(+)5180 54425 y Fq(\()p Fp(a)6399 54624 y Fv(0)6995 54425 y Fq(\))435 b Fp(>)h Fq(0)483 b(b)37 b(e)482 b(the)h(smallest)g(n)-37 b(um)g(b)37 b(er)483 b(suc)-37 b(h)483 b(that)f Fp(q)31309 54624 y Fv(2)31905 54425 y Fq(\(0\))435 b(=)g Fp(a)36208 54624 y Fv(1)36804 54425 y Fq(\()19 b(\026)-683 b Fp(a)38023 54624 y Fv(2)38619 54425 y Fq(\))482 b(and)h Fp(q)42834 54624 y Fv(2)43430 54425 y Fq(\()p Fp(T)44723 54624 y Fv(+)45605 54425 y Fq(\()p Fp(a)46824 54624 y Fv(0)47419 54425 y Fq(\)\))435 b(=)1102 56019 y(2)p Fp(\031)352 b Fm(\000)324 b Fq(\026)-683 b Fp(a)4915 56218 y Fv(2)5511 56019 y Fq(.)630 b(Since)456 b Fp(`)10544 56218 y Fv(2)11597 56019 y Fq(is)f(con)-37 b(v)g(ex)458 b(in)702 b(_)-615 b Fp(q)19469 56218 y Fv(2)20065 56019 y Fq(,)459 b(it)c(is)h(easy)h(to)f(see)g(that)g Fp(T)33984 56218 y Fv(+)34866 56019 y Fq(\()p Fp(a)36085 56218 y Fv(0)36680 56019 y Fq(\))391 b Fp(>)g(T)39788 56218 y Fv(+)40670 56019 y Fq(\()p Fp(a)41889 56218 y Fv(1)42485 56019 y Fq(\()19 b(\026)-683 b Fp(a)43704 56218 y Fv(2)44299 56019 y Fq(\))p Fp(;)240 b Fq(\026)-683 b Fp(a)46108 56218 y Fv(2)46704 56019 y Fq(\()p Fp(a)47923 56218 y Fv(0)48519 56019 y Fq(\)\).)1102 57613 y(Similarly)-111 b(,)538 b(w)-37 b(e)519 b(can)g(de\014ne)538 b(\026)-683 b Fp(a)16261 57812 y Fv(1)17352 57613 y Fq(=)514 b(\026)-683 b Fp(a)19582 57812 y Fv(1)20178 57613 y Fq(\()p Fp(a)21397 57812 y Fv(0)21993 57613 y Fq(\))494 b Fm(\024)h Fp(a)25234 57812 y Fv(0)26349 57613 y Fq(and)518 b Fp(T)29783 57812 y Fg(\000)30680 57613 y Fq(\()p Fp(a)31899 57812 y Fv(0)32495 57613 y Fq(\))g(so)g(that)g Fp(q)38783 57812 y Fv(2)39378 57613 y Fq(\(0\))495 b(=)g(2)p Fp(\031)393 b Fm(\000)346 b Fp(a)46994 57812 y Fv(2)47590 57613 y Fq(\()19 b(\026)-683 b Fp(a)48809 57812 y Fv(1)49405 57613 y Fq(\))1102 59207 y(and)512 b Fp(q)4347 59406 y Fv(2)4943 59207 y Fq(\()p Fp(T)6236 59406 y Fg(\000)7132 59207 y Fq(\()19 b(\026)-683 b Fp(a)8351 59406 y Fv(0)8947 59207 y Fq(\)\))482 b(=)502 b(\026)-683 b Fp(a)12681 59406 y Fv(1)13277 59207 y Fq(.)796 b(Ob)-37 b(viously)-111 b(,)529 b(there)512 b(exists)530 b(\026)-683 b Fp(a)28934 59406 y Fv(0)30014 59207 y Fp(>)483 b Fq(0)511 b(and)h Fp(\034)633 b(>)483 b Fq(0)511 b(suc)-37 b(h)512 b(that)f(for)g(eac)-37 b(h)1102 60801 y Fp(a)1804 61000 y Fv(0)2769 60801 y Fm(\024)389 b Fq(\026)-683 b Fp(a)4874 61000 y Fv(0)5912 60801 y Fq(and)443 b(for)g(all)g Fp(T)13149 61000 y Fv(+)14032 60801 y Fq(\()p Fp(a)15251 61000 y Fv(1)15846 60801 y Fp(;)221 b(a)17138 61000 y Fv(2)17735 60801 y Fq(\))294 b Fm(\000)i Fp(\034)519 b Fm(\024)369 b Fp(T)554 b Fm(\024)369 b Fp(T)25885 61000 y Fv(+)26767 60801 y Fq(\()p Fp(a)27986 61000 y Fv(0)28582 60801 y Fq(\))442 b(w)-37 b(e)444 b(ha)-37 b(v)g(e)16711 64440 y(inf)15705 65195 y Fv(\026)15612 65440 y Fk(\020)61 b Fv(\(0\)=)p Fk(a)18906 65573 y Fj(1)14578 66342 y Fv(\026)14485 66588 y Fk(\020)f Fv(\()p Fk(T)143 b Fv(\)=2)p Fk(\031)33 b Fg(\000)p Fk(a)20032 66721 y Fj(2)20775 62632 y Fi(Z)22103 62960 y Fk(T)21513 65650 y Fv(0)23161 64440 y Fp(`)23714 64639 y Fv(2)24311 64440 y Fq(\()p Fp(\020)98 b Fq(\()p Fp(t)p Fq(\))p Fp(;)27876 64089 y Fq(_)27611 64440 y Fp(\020)f Fq(\()p Fp(t)p Fq(\)\))p Fp(dt)367 b Fm(\025)33420 63541 y Fq(4)p 33420 64134 665 54 v 33420 65351 a(5)34244 64440 y Fp(C)35288 63871 y Fv(+)35193 64830 y Fk(A)36170 64440 y Fp(;)10652 b Fq(\(3.6\))1102 69141 y(where)489 b Fp(a)5690 69340 y Fv(2)6728 69141 y Fq(=)462 b(\026)-683 b Fp(a)8906 69340 y Fv(2)9502 69141 y Fq(\()p Fp(a)10721 69340 y Fv(0)11317 69141 y Fq(\))486 b(and)h Fp(a)15649 69340 y Fv(1)16688 69141 y Fq(=)443 b Fp(a)18866 69340 y Fv(1)19462 69141 y Fq(\()19 b(\026)-683 b Fp(a)20681 69340 y Fv(2)21276 69141 y Fq(\).)723 b(Similarly)-111 b(,)498 b(for)488 b(all)f Fp(T)33590 69340 y Fg(\000)34487 69141 y Fq(\()p Fp(a)35706 69340 y Fv(1)36301 69141 y Fp(;)221 b(a)37593 69340 y Fv(2)38190 69141 y Fq(\))324 b Fm(\000)h Fp(\034)592 b Fm(\024)443 b Fp(T)627 b Fm(\024)443 b Fp(T)46693 69340 y Fg(\000)47590 69141 y Fq(\()p Fp(a)48809 69340 y Fv(0)49405 69141 y Fq(\))1102 70735 y(w)-37 b(e)444 b(ha)-37 b(v)g(e)16704 72983 y(inf)14691 73738 y Fv(\026)14598 73984 y Fk(\020)61 b Fv(\(0\)=2)p Fk(\031)33 b Fg(\000)p Fk(a)19905 74117 y Fj(2)15577 74885 y Fv(\026)15485 75131 y Fk(\020)60 b Fv(\()p Fk(T)143 b Fv(\)=)p Fk(a)19019 75264 y Fj(1)20647 71175 y Fi(Z)21975 71503 y Fk(T)21385 74193 y Fv(0)23034 72983 y Fp(`)23587 73182 y Fv(2)24183 72983 y Fq(\()p Fp(\020)98 b Fq(\()p Fp(t)p Fq(\))p Fp(;)27748 72632 y Fq(_)27483 72983 y Fp(\020)f Fq(\()p Fp(t)p Fq(\)\))p Fp(dt)368 b Fm(\025)33292 72084 y Fq(4)p 33292 72677 V 33292 73894 a(5)34116 72983 y Fp(C)35160 72415 y Fg(\000)35065 73373 y Fk(A)36057 72983 y Fp(;)10396 b Fq(\(3.6'\))p eop %%Page: 15 15 15 14 bop 48792 2239 a Ft(15)1102 5427 y Fq(where)561 b Fp(a)5762 5626 y Fv(1)6921 5427 y Fq(=)582 b(\026)-683 b Fp(a)9219 5626 y Fv(1)9815 5427 y Fq(\()p Fp(a)11034 5626 y Fv(0)11630 5427 y Fq(\))558 b(and)i Fp(a)16107 5626 y Fv(2)17266 5427 y Fq(=)j Fp(a)19564 5626 y Fv(2)20160 5427 y Fq(\()19 b(\026)-683 b Fp(a)21379 5626 y Fv(1)21974 5427 y Fq(\).)939 b(On)560 b(the)f(other)g(hand,)589 b(for)559 b(eac)-37 b(h)560 b Fp(a)41914 5626 y Fv(1)42510 5427 y Fp(;)221 b(a)43802 5626 y Fv(2)44962 5427 y Fp(>)563 b Fq(0)c(and)1102 7036 y(eac)-37 b(h)560 b Fp(T)747 b(>)563 b Fq(0)c(there)h(exists)e Fp(\030)16489 6468 y Fg(\006)16428 7426 y Fk(T)17386 7036 y Fq(:)823 b([0)p Fp(;)221 b(T)184 b Fq(])564 b Fm(!)f Fl(R)c Fq(suc)-37 b(h)559 b(that)f Fp(\030)32251 6468 y Fv(+)32190 7426 y Fk(T)33133 7036 y Fq(\(0\))k(=)h Fp(a)37691 7235 y Fv(1)38846 7036 y Fq(and)c Fp(\030)42187 6468 y Fv(+)42126 7426 y Fk(T)43069 7036 y Fq(\()p Fp(T)184 b Fq(\))562 b(=)h Fm(\000)p Fp(a)48956 7235 y Fv(2)49552 7036 y Fq(,)1102 8703 y Fp(\030)1744 8135 y Fg(\000)1683 9093 y Fk(T)2641 8703 y Fq(\(0\))369 b(=)g Fm(\000)p Fp(a)7845 8902 y Fv(2)8884 8703 y Fq(and)443 b Fp(\030)12109 8135 y Fg(\000)12048 9093 y Fk(T)13006 8703 y Fq(\()p Fp(T)184 b Fq(\))368 b(=)i Fp(a)17473 8902 y Fv(1)18069 8703 y Fq(,)442 b(whic)-37 b(h)444 b(minimizes)f(the)g(action)13281 12485 y Fp(A)14277 12684 y Fv(2)14874 12485 y Fq(\()p Fp(\030)16033 11917 y Fv(+)15972 12876 y Fk(T)16914 12485 y Fq(\))369 b(=)19055 10677 y Fi(Z)20383 11006 y Fk(T)19792 13696 y Fv(0)21441 12485 y Fp(`)21994 12684 y Fv(2)22590 12485 y Fq(\()p Fp(\030)23749 11917 y Fv(+)23688 12876 y Fk(T)24631 12485 y Fq(\()p Fp(t)p Fq(\))p Fp(;)27019 12135 y Fq(_)26735 12485 y Fp(\030)27377 11917 y Fv(+)27316 12876 y Fk(T)28259 12485 y Fq(\()p Fp(t)p Fq(\)\))p Fp(dt)17800 16343 y Fq(=)2230 b(inf)19590 17408 y Fk(\020)20133 17007 y Fj(+)20883 17408 y Fv(\(0\)=)p Fk(a)23633 17541 y Fj(1)19055 18619 y Fk(\020)19598 18218 y Fj(+)20347 18619 y Fv(\()p Fk(T)143 b Fv(\)=)p Fg(\000)p Fk(a)24168 18752 y Fj(2)24910 14535 y Fi(Z)26238 14863 y Fk(T)25648 17553 y Fv(0)27296 16343 y Fp(`)27849 16542 y Fv(2)28446 16343 y Fq(\()p Fp(\020)29642 15774 y Fv(+)29544 16733 y Fk(T)30523 16343 y Fq(\()p Fp(t)p Fq(\))p Fp(;)32892 15992 y Fq(_)32627 16343 y Fp(\020)33306 15774 y Fv(+)33208 16733 y Fk(T)34187 16343 y Fq(\()p Fp(t)p Fq(\)\))p Fp(dt:)13267 21632 y(A)14263 21831 y Fv(2)14859 21632 y Fq(\()p Fp(\030)16018 21063 y Fg(\000)15957 22022 y Fk(T)16914 21632 y Fq(\))369 b(=)19055 19823 y Fi(Z)20383 20152 y Fk(T)19792 22842 y Fv(0)21441 21632 y Fp(`)21994 21831 y Fv(2)22590 21632 y Fq(\()p Fp(\030)23749 21063 y Fg(\000)23688 22022 y Fk(T)24646 21632 y Fq(\()p Fp(t)p Fq(\))p Fp(;)27033 21281 y Fq(_)26750 21632 y Fp(\030)27392 21063 y Fg(\000)27331 22022 y Fk(T)28288 21632 y Fq(\()p Fp(t)p Fq(\)\))p Fp(dt)17800 25489 y Fq(=)2128 b(inf)19055 26554 y Fk(\020)19598 26153 y Fe(\000)20384 26554 y Fv(\(0\)=)p Fg(\000)p Fk(a)23964 26687 y Fj(2)19349 27765 y Fk(\020)19892 27365 y Fe(\000)20679 27765 y Fv(\()p Fk(T)143 b Fv(\)=)p Fk(a)23670 27898 y Fj(1)24706 23681 y Fi(Z)26034 24009 y Fk(T)25444 26699 y Fv(0)27092 25489 y Fp(`)27645 25688 y Fv(2)28242 25489 y Fq(\()p Fp(\020)29438 24921 y Fg(\000)29340 25879 y Fk(T)30334 25489 y Fq(\()p Fp(t)p Fq(\))p Fp(;)32703 25138 y Fq(_)32438 25489 y Fp(\020)33117 24921 y Fg(\000)33019 25879 y Fk(T)34013 25489 y Fq(\()p Fp(t)p Fq(\)\))p Fp(dt:)1102 30552 y Fq(It)472 b(is)h(easy)g(to)f(see)h(that)f Fp(dist)p Fq(\()p Fp(d\030)17729 29983 y Fg(\006)17668 30942 y Fk(T)18625 30552 y Fp(;)221 b Fq(\000)20045 30070 y Fg(\006)20942 30552 y Fq(\))418 b Fm(!)h Fq(0)473 b(as)f Fp(T)603 b Fm(!)419 b(1)p Fq(,)480 b(so)473 b(there)g(exists)f Fp(T)41286 30751 y Fk(a)42348 30552 y Fp(>)418 b Fq(0,)480 b(for)473 b(eac)-37 b(h)1102 32146 y Fp(T)554 b Fm(\025)369 b Fp(T)4610 32345 y Fk(a)5696 32146 y Fq(w)-37 b(e)444 b(ha)-37 b(v)g(e)11589 35947 y Fp(A)12585 36146 y Fv(2)13181 35947 y Fq(\()p Fp(\030)14340 35378 y Fg(\006)14279 36337 y Fk(T)15236 35947 y Fq(\))369 b Fm(\024)p Fq(2)18040 33675 y Fi( )19092 34139 y(Z)20421 34467 y Fv(0)19830 37157 y Fg(\0001)22007 35947 y Fq(+)23261 34139 y Fi(Z)24590 34467 y Fg(1)24000 37157 y Fk(T)24627 37290 y Fe(\006)25413 37157 y Fv(\()p Fk(a)26404 37290 y Fj(1)26923 37157 y Fk(;a)27814 37290 y Fj(2)28333 37157 y Fv(\))28814 33675 y Fi(!)30087 35947 y Fp(`)30640 36146 y Fv(2)31237 35947 y Fq(\()p Fp(q)32347 36146 y Fv(2)32942 35947 y Fq(\()p Fp(t)p Fq(\))p Fp(;)467 b Fq(_)-615 b Fp(q)35639 36146 y Fv(2)36234 35947 y Fq(\()p Fp(t)p Fq(\)\))p Fp(dt)15827 39135 y Fm(!)p Fq(0)4725 b(\(as)442 b Fp(a)25393 39334 y Fv(1)26358 39135 y Fm(!)369 b Fq(0)p Fp(;)665 b(a)30455 39334 y Fv(2)31420 39135 y Fm(!)369 b Fq(0\))p Fp(:)12524 b Fq(\(3.7\))1102 41940 y(It)620 b(follo)-37 b(ws)621 b(from)g(\(3.5\))f(and)g(\(3.7\))g(that)g (there)h(exists)f(0)665 b Fp(<)g(a)34244 41458 y Fg(\003)34244 42270 y Fv(0)35520 41940 y Fm(\024)684 b Fq(\026)-683 b Fp(a)37920 42139 y Fv(0)38516 41940 y Fq(,)665 b(if)620 b Fp(a)41647 42139 y Fk(i)42755 41940 y Fm(\024)665 b Fp(a)45155 41458 y Fg(\003)45155 42270 y Fv(0)46386 41940 y Fq(and)620 b(if)1102 43534 y Fp(T)2062 43052 y Fg(0)1878 43862 y Fv(+)3129 43534 y Fm(\025)370 b Fp(T)5308 43733 y Fv(+)6190 43534 y Fq(\()p Fp(a)7409 43733 y Fv(1)8004 43534 y Fp(;)221 b(a)9296 43733 y Fv(2)9893 43534 y Fq(\))295 b Fm(\000)g Fp(\034)150 b Fq(,)443 b Fp(T)14536 43052 y Fg(0)14352 43862 y(\000)15618 43534 y Fm(\025)369 b Fp(T)17796 43733 y Fg(\000)18693 43534 y Fq(\()p Fp(a)19912 43733 y Fv(1)20507 43534 y Fp(;)221 b(a)21799 43733 y Fv(2)22396 43534 y Fq(\))295 b Fm(\000)g Fp(\034)593 b Fq(then)14348 47005 y Fp(A)15344 47204 y Fv(2)15940 47005 y Fq(\()p Fp(\030)17099 46437 y Fv(+)17038 47449 y Fk(T)17808 47129 y Fe(0)17665 47752 y Fj(+)18481 47005 y Fq(\))369 b Fm(\024)20928 46106 y Fq(1)p 20928 46699 665 54 v 20928 47916 a(5)21752 47005 y Fp(C)22796 46437 y Fv(+)22701 47395 y Fk(A)23678 47005 y Fp(;)2878 b(A)27921 47204 y Fv(2)28517 47005 y Fq(\()p Fp(\030)29676 46437 y Fg(\000)29615 47449 y Fk(T)30385 47129 y Fe(0)30242 47752 y(\000)31095 47005 y Fq(\))368 b Fm(\024)33542 46106 y Fq(1)p 33542 46699 V 33542 47916 a(5)34365 47005 y Fp(C)35409 46437 y Fg(\000)35314 47395 y Fk(A)36306 47005 y Fp(:)10516 b Fq(\(3.8\))2696 50115 y(Let)380 b(us)f(no)-37 b(w)379 b(consider)i(those)e(orbits)g(of)g Fp(L)g Fq(of)g(whic)-37 b(h)380 b(the)f(second)h(comp)37 b(onen)-37 b(t)379 b Fp(d\015)44064 50314 y Fv(2)45040 50115 y Fq(encircles)1102 51709 y(around)652 b(the)g(phase)g(cylinder)g (\()p Fp(q)18596 51908 y Fv(2)19191 51709 y Fp(;)468 b Fq(_)-616 b Fp(q)20374 51908 y Fv(2)20971 51709 y Fq(\))716 b Fm(2)g Fl(T)434 b Fm(\002)h Fl(R)651 b Fq(at)g(least)h(once.)1217 b(W)-111 b(e)652 b(assume)g Fp(d\015)46194 51908 y Fv(2)47442 51709 y Fq(go)37 b(es)1102 53303 y(around)588 b(the)e(cylinder)i(in)f (coun)-37 b(ter)587 b(clo)37 b(c)-37 b(k-wise)589 b(direction.)1023 b(In)586 b(this)g(case,)624 b(there)587 b(is)g Fp(t)46263 53502 y Fv(0)47467 53303 y Fm(2)609 b Fl(R)1102 54898 y Fq(suc)-37 b(h)600 b(that)647 b(\026)-713 b Fp(\015)7977 55097 y Fv(2)8573 54898 y Fq(\()p Fp(t)9570 55097 y Fv(0)10165 54898 y Fq(\))629 b(=)g Fp(a)13675 55097 y Fv(1)14271 54898 y Fq(\()19 b(\026)-683 b Fp(a)15490 55097 y Fv(2)16086 54898 y Fq(\()p Fp(a)17305 54415 y Fg(\003)17305 55227 y Fv(0)17915 54898 y Fq(\)\))598 b(with)648 b(\026)-713 b Fp(\015)23417 55097 y Fv(2)24013 54898 y Fq(\()p Fp(t)25010 55097 y Fv(0)26004 54898 y Fq(+)400 b Fp(t)p Fq(\))628 b Fp(>)i(a)31427 55097 y Fv(1)32023 54898 y Fq(\()19 b(\026)-683 b Fp(a)33242 55097 y Fv(2)33838 54898 y Fq(\()p Fp(a)35057 54415 y Fg(\003)35057 55227 y Fv(0)35667 54898 y Fq(\)\))598 b(for)h(small)h Fp(t)629 b(>)g Fq(0.)1059 b(Let)1102 56492 y Fp(t)1582 56691 y Fv(1)2600 56492 y Fq(=)421 b(min)p Fm(f)p Fp(t)h Fm(2)f Fl(R)h Fq(:)643 b Fp(t)421 b(>)h(t)14370 56691 y Fv(0)14965 56492 y Fp(;)270 b Fq(\026)-713 b Fp(\015)16243 56691 y Fv(2)16839 56492 y Fq(\()p Fp(t)17836 56691 y Fv(1)18432 56492 y Fq(\))421 b(=)h(2)p Fp(\031)364 b Fm(\000)335 b Fq(\026)-683 b Fp(a)24661 56691 y Fv(2)25257 56492 y Fq(\()p Fp(a)26476 56009 y Fg(\003)26476 56821 y Fv(0)27087 56492 y Fq(\))p Fm(g)p Fq(.)685 b(Based)474 b(on)h(the)f(understanding)h(on)f(the)1102 58086 y(phase)354 b(p)37 b(ortraits)352 b(of)h Fp(`)12108 58285 y Fv(2)13057 58086 y Fq(and)g(the)g(smo)37 b(oth)353 b(dep)37 b(endence)354 b(of)f(solutions)g(of)f(ODE's)h(on)g(parameter,) 1102 59680 y(w)-37 b(e)444 b(see)g(that)e(for)h(giv)-37 b(en)443 b Fp(\016)420 b(>)369 b Fq(0)443 b(there)g(exists)g Fp(\017)369 b(>)g Fq(0)443 b(suc)-37 b(h)443 b(that)g(if)f Fm(k)p Fp(P)184 b Fm(k)370 b(\024)f Fp(\017)443 b Fq(then)19412 62418 y Fp(t)19892 62617 y Fv(1)20783 62418 y Fm(\000)295 b Fp(t)22591 62617 y Fv(0)23556 62418 y Fm(\024)369 b Fp(T)25734 62617 y Fv(+)26616 62418 y Fq(\()p Fp(a)27835 61869 y Fg(\003)27835 62746 y Fv(0)28445 62418 y Fq(\))295 b(+)g Fp(\034)77 b(;)15579 b Fq(\(3.9\))1102 65156 y(and)443 b(if)g(the)g(initial)f(v)-74 b(alue)444 b(is)e(c)-37 b(hosen)444 b(so)f(that)24751 64974 y(_)24604 65156 y(\026)-713 b Fp(\015)25317 65481 y Fv(2)25913 65156 y Fq(\()p Fp(t)26910 65355 y Fv(0)27505 65156 y Fq(\))368 b Fm(\024)i Fp(G)30837 64674 y Fv(+)31719 65156 y Fq(\()49 b(\026)-713 b Fp(\015)32924 65355 y Fv(2)33519 65156 y Fq(\()p Fp(t)34516 65355 y Fv(0)35111 65156 y Fq(\)\))295 b(+)g Fp(\016)50 b Fq(,)443 b(w)-37 b(e)444 b(ha)-37 b(v)g(e)444 b(further)14963 67894 y Fp(t)15443 68093 y Fv(1)16334 67894 y Fm(\000)296 b Fp(t)18143 68093 y Fv(0)19107 67894 y Fm(\025)369 b Fp(T)21285 68093 y Fv(+)22167 67894 y Fq(\()p Fp(a)23386 68093 y Fv(1)23982 67894 y Fq(\()19 b(\026)-683 b Fp(a)25201 68093 y Fv(2)25797 67894 y Fq(\()p Fp(a)27016 67346 y Fg(\003)27016 68223 y Fv(0)27626 67894 y Fq(\)\))p Fp(;)240 b Fq(\026)-683 b Fp(a)29952 68093 y Fv(2)30548 67894 y Fq(\()p Fp(a)31767 67346 y Fg(\003)31767 68223 y Fv(0)32377 67894 y Fq(\)\))295 b Fm(\000)g Fp(\034)77 b(:)10466 b Fq(\(3.10\))1102 70633 y(Similarly)-111 b(,)511 b(in)496 b(the)h(case)g(of)f Fp(d\015)16832 70832 y Fv(2)17925 70633 y Fq(going)h(around)g(the)f(cylinder)i(in)e(clo)37 b(c)-37 b(kwise)498 b(direction,)511 b(there)1102 72227 y(is)430 b Fp(t)2905 71744 y Fg(0)2905 72556 y Fv(0)3501 72227 y Fq(,)i Fp(t)4782 71744 y Fg(0)4782 72556 y Fv(1)5747 72227 y Fm(2)368 b Fl(R)431 b Fq(suc)-37 b(h)430 b(that)478 b(\026)-713 b Fp(\015)14927 72426 y Fv(2)15523 72227 y Fq(\()p Fp(t)16520 71744 y Fg(0)16520 72556 y Fv(0)17115 72227 y Fq(\))368 b(=)i(2)p Fp(\031)317 b Fm(\000)270 b Fp(a)23146 72426 y Fv(2)23742 72227 y Fq(\()19 b(\026)-683 b Fp(a)24961 72426 y Fv(1)25556 72227 y Fq(\()p Fp(a)26775 71744 y Fg(\003)26775 72556 y Fv(0)27386 72227 y Fq(\)\))429 b(and)478 b(\026)-712 b Fp(\015)32107 72426 y Fv(2)32702 72227 y Fq(\()p Fp(t)33699 71744 y Fg(0)33699 72556 y Fv(1)34295 72227 y Fq(\))368 b(=)388 b(\026)-683 b Fp(a)37284 72426 y Fv(1)37880 72227 y Fq(\()p Fp(a)39099 71744 y Fg(\003)39099 72556 y Fv(0)39710 72227 y Fq(\),)432 b(if)d Fm(k)p Fp(P)184 b Fm(k)370 b(\024)f Fp(\017)430 b Fq(then)19405 74965 y Fp(t)19885 75164 y Fv(1)20775 74965 y Fm(\000)296 b Fp(t)22584 75164 y Fv(0)23548 74965 y Fm(\024)369 b Fp(T)25726 75164 y Fg(\000)26623 74965 y Fq(\()p Fp(a)27842 74416 y Fg(\003)27842 75293 y Fv(0)28453 74965 y Fq(\))294 b(+)i Fp(\034)77 b(;)15202 b Fq(\(3.9'\))p eop %%Page: 16 16 16 15 bop 1102 2239 a Ft(16)1102 5427 y Fq(and)443 b(if)g(the)g (initial)f(v)-74 b(alue)444 b(is)e(c)-37 b(hosen)444 b(so)f(that)24751 5245 y(_)24604 5427 y(\026)-713 b Fp(\015)25317 5752 y Fv(2)25913 5427 y Fq(\()p Fp(t)26910 5626 y Fv(0)27505 5427 y Fq(\))368 b Fm(\025)i Fp(G)30837 4945 y Fg(\000)31734 5427 y Fq(\()49 b(\026)-713 b Fp(\015)32939 5626 y Fv(2)33534 5427 y Fq(\()p Fp(t)34531 5626 y Fv(0)35126 5427 y Fq(\)\))294 b Fm(\000)i Fp(\016)50 b Fq(,)443 b(w)-37 b(e)444 b(ha)-37 b(v)g(e)443 b(further)14956 8790 y Fp(t)15436 8989 y Fv(1)16327 8790 y Fm(\000)295 b Fp(t)18135 8989 y Fv(0)19100 8790 y Fm(\025)369 b Fp(T)21278 8989 y Fg(\000)22175 8790 y Fq(\()p Fp(a)23394 8989 y Fv(1)23989 8790 y Fq(\()19 b(\026)-683 b Fp(a)25208 8989 y Fv(2)25804 8790 y Fq(\()p Fp(a)27023 8242 y Fg(\003)27023 9119 y Fv(0)27634 8790 y Fq(\)\))p Fp(;)239 b Fq(\026)-682 b Fp(a)29960 8989 y Fv(2)30555 8790 y Fq(\()p Fp(a)31774 8242 y Fg(\003)31774 9119 y Fv(0)32385 8790 y Fq(\)\))294 b Fm(\000)h Fp(\034)77 b(:)10090 b Fq(\(3.10'\))2696 12212 y(Let)482 b(us)e(consider)j(the)d (minimizer)i(of)f Fp(L)p Fq(\()246 b(_)-615 b Fp(q)48 b(;)221 b(q)48 b(;)221 b(t)p Fq(\))320 b Fm(\000)h Fp(c)29319 12411 y Fv(1)30161 12212 y Fq(_)-616 b Fp(q)30507 12411 y Fv(1)31584 12212 y Fq(in)481 b(the)g(co)-37 b(v)g(ering)483 b(space)f Fl(T)320 b Fm(\002)h Fl(R)481 b Fq(for)1102 13806 y(\()p Fp(q)2212 14005 y Fv(1)2808 13806 y Fp(;)221 b(q)3991 14005 y Fv(2)4587 13806 y Fq(\).)590 b(Let)443 b Fp(\030)61 b Fq(:)591 b([0)p Fp(;)221 b(T)184 b Fq(])370 b Fm(!)f Fl(T)295 b Fm(\002)g Fl(R)444 b Fq(b)37 b(e)443 b(a)g(curv)-37 b(e)444 b(whic)-37 b(h)443 b(minimize)g(the)g(action:) 11703 18155 y Fp(A)p Fq(\()p Fp(\030)61 b Fm(j)p Fq([0)p Fp(;)221 b(T)184 b Fq(]\))370 b(=)19320 16347 y Fi(Z)20648 16675 y Fk(T)20058 19365 y Fv(0)21485 18155 y Fq(\()p Fp(L)p Fq(\()p Fp(d\030)61 b Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))294 b Fm(\000)h Fp(c)30054 18354 y Fv(1)30934 17804 y Fq(_)30650 18155 y Fp(\030)31231 18354 y Fv(1)31827 18155 y Fq(\()p Fp(t)p Fq(\)\))p Fp(dt)18066 22012 y Fq(=)1132 b(min)19718 22931 y Fk(\020)60 b Fv(\(0\)=)81 b(^)-610 b Fk(q)19320 24009 y(\020)60 b Fv(\()p Fk(T)143 b Fv(\)=)p Fk(q)22798 23608 y Fe(\003)23577 20204 y Fi(Z)24905 20533 y Fk(T)24315 23223 y Fv(0)25742 22012 y Fq(\()p Fp(L)p Fq(\()p Fp(d\020)98 b Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))293 b Fm(\000)j Fp(c)34348 22211 y Fv(1)35209 21662 y Fq(_)34943 22012 y Fp(\020)35524 22211 y Fv(1)36121 22012 y Fq(\()p Fp(t)p Fq(\)\))p Fp(dt)1102 27199 y Fq(where)556 b(^)-763 b Fp(q)5549 27398 y Fv(2)6534 27199 y Fq(=)389 b Fm(\000)p Fp(a)9691 27398 y Fv(2)10742 27199 y Fq(and)455 b Fp(q)13978 26717 y Fg(\003)13930 27528 y Fv(2)14978 27199 y Fq(=)389 b Fp(a)17102 27398 y Fv(1)17698 27199 y Fq(.)627 b(Since)455 b(there)h(is)f(a)g(normally)g(h)-37 b(yp)37 b(erb)g(olic)456 b(structure)g(around)1250 28548 y(~)1102 28884 y(\006)q Fm(j)2431 29124 y Fg(fj)p Fk(p)3838 29257 y Fj(1)4357 29124 y Fg(j\024)p Fv(max\()p Fg(j)p Fk(A)p Fg(j)p Fk(;)p Fg(j)p Fk(B)45 b Fg(j)p Fv(\)+1)p Fg(g)13472 28884 y Fq(,)443 b(there)h(is)e Fp(T)19978 28402 y Fg(\003)19794 29213 y Fv(+)21045 28884 y Fp(>)370 b Fq(0)443 b(suc)-37 b(h)443 b(that)10638 32146 y(_)10354 32496 y Fp(\030)10935 32695 y Fv(2)11531 32496 y Fq(\()p Fp(T)184 b Fq(\))369 b Fp(<)g(G)16340 31948 y Fv(+)17222 32496 y Fq(\()p Fp(a)18441 32695 y Fv(1)19037 32496 y Fq(\))295 b(+)g Fp(\016)-24 b(;)2879 b Fm(8)443 b Fp(a)26874 32695 y Fv(1)27470 32496 y Fp(;)221 b(a)28762 32695 y Fv(2)29727 32496 y Fm(\024)369 b Fp(a)31831 31948 y Fg(\003)31831 32825 y Fv(0)32885 32496 y Fq(and)886 b Fp(T)553 b Fm(\025)370 b Fp(T)39603 31948 y Fg(\003)39419 32825 y Fv(+)40301 32496 y Fp(:)5857 b Fq(\(3.11\))2696 35918 y(No)-37 b(w)339 b(w)-37 b(e)338 b(are)h(on)f(the)f(stage)h(to)f(claim)h(that)f(the)h (second)g(comp)37 b(onen)-37 b(t)339 b Fp(d\015)38956 36117 y Fv(2)39889 35918 y Fq(of)f(eac)-37 b(h)339 b Fp(c)p Fq(-minimal)1102 37512 y(orbit)375 b(of)f Fp(L)g Fq(can)h(not)f(go)h(around)g(the)f(cylinder)h(t)-37 b(wice)375 b(for)g(small)f Fm(j)p Fp(c)34695 37711 y Fv(2)35291 37512 y Fm(j)p Fq(.)567 b(If)374 b(our)h(claim)g(is)f(not)g(true,)1102 39106 y(there)501 b(exist)e Fp(t)8238 39305 y Fv(0)8834 39106 y Fp(;)221 b(T)649 b Fm(2)463 b Fl(R)500 b Fq(suc)-37 b(h)500 b(that)548 b(\026)-713 b Fp(\015)20333 39305 y Fv(2)20929 39106 y Fq(\()p Fp(t)21926 39305 y Fv(0)22521 39106 y Fq(\))463 b(=)h(0,)563 b(\026)-713 b Fp(\015)27233 39305 y Fv(2)27829 39106 y Fq(\()p Fp(t)28826 39305 y Fv(0)29754 39106 y Fq(+)333 b(2)p Fp(T)184 b Fq(\))464 b(=)g(4)p Fp(\031)48 b Fq(.)761 b(If)500 b(w)-37 b(e)500 b(c)-37 b(ho)37 b(ose)501 b(another)1102 40700 y(curv)-37 b(e)394 b Fp(\015)5360 40218 y Fg(\003)5970 40700 y Fq(\()p Fp(t)p Fq(\))368 b(=)h(\()p Fp(\015)10459 40899 y Fv(1)11054 40700 y Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(q)13799 40218 y Fk(T)13751 41030 y Fv(2)14635 40700 y Fq(\()p Fp(t=)p Fq(2\)\))391 b(where)j(\()p Fp(q)23334 40218 y Fk(T)23286 41030 y Fv(2)24170 40700 y Fq(\()p Fp(t)p Fq(\))p Fp(;)467 b Fq(_)-615 b Fp(q)26915 40218 y Fk(T)26867 41030 y Fv(2)27751 40700 y Fq(\()p Fp(t)p Fq(\)\))391 b(is)h(an)h(orbit)f(of)h Fp(\036)38707 40218 y Fk(t)38707 41076 y(`)39152 41209 y Fj(2)40129 40700 y Fq(going)g(around)g(the)1102 42338 y(cylinder)478 b(with)f(p)37 b(erio)g(d)477 b Fp(T)184 b Fq(,)486 b(then)477 b Fp(\015)74 b Fq(\()p Fp(t)20059 42537 y Fv(0)20654 42338 y Fq(\))425 b(=)h Fp(\015)23817 41856 y Fg(\003)24427 42338 y Fq(\()p Fp(t)25424 42537 y Fv(0)26019 42338 y Fq(\))477 b(and)g Fp(\015)74 b Fq(\()p Fp(t)31389 42537 y Fv(0)32301 42338 y Fq(+)318 b(2)p Fp(T)184 b Fq(\))427 b(=)e Fp(\015)38440 41856 y Fg(\003)39051 42338 y Fq(\()p Fp(t)40048 42537 y Fv(0)40961 42338 y Fq(+)318 b(2)p Fp(T)184 b Fq(\).)693 b(Since)477 b Fp(L)1102 43932 y Fq(is)443 b(a)g(small)g(p)37 b(erturbation)443 b(of)g Fp(L)17231 44131 y Fv(0)17827 43932 y Fq(,)g(i.e.)8472 48485 y Fp(L)p Fq(\()p Fp(q)48 b(;)467 b Fq(_)-615 b Fp(q)48 b(;)221 b(t)p Fq(\))294 b Fm(\000)i(h)p Fp(c;)467 b Fq(_)-615 b Fp(q)47 b Fm(i)369 b Fq(=)g Fp(`)20138 48684 y Fv(1)20734 48485 y Fq(\()246 b(_)-615 b Fp(q)21844 48684 y Fv(1)22440 48485 y Fq(\))295 b(+)g Fp(`)25133 48684 y Fv(2)25729 48485 y Fq(\()p Fp(q)26839 48684 y Fv(2)27435 48485 y Fp(;)468 b Fq(_)-616 b Fp(q)28618 48684 y Fv(2)29214 48485 y Fq(\))295 b Fm(\000)32049 46824 y Fv(2)31354 47223 y Fi(X)31453 50052 y Fk(i)p Fv(=1)33494 48485 y Fp(c)34069 48684 y Fk(i)34758 48485 y Fq(_)-616 b Fp(q)35104 48684 y Fk(i)35842 48485 y Fq(+)295 b Fp(P)184 b Fq(\()p Fp(q)48 b(;)468 b Fq(_)-616 b Fp(q)48 b(;)221 b(t)p Fq(\))p Fp(;)1102 53009 y Fq(w)-37 b(e)544 b(can)g(compare)h(the)e(action)g(of)h Fp(L)f Fq(along)h(these)f(t)-37 b(w)g(o)544 b(curv)-37 b(es.)892 b(T)-111 b(o)-37 b(w)g(ard)546 b(that)c(w)-37 b(e)544 b(consider)1102 54603 y Fp(q)1743 54121 y Fk(T)1695 54933 y Fv(2)2580 54603 y Fq(\()p Fp(t)p Fq(\),)416 b(a)411 b Fp(T)184 b Fq(-p)37 b(erio)g(dic)412 b(orbit)e(of)h Fp(`)17673 54802 y Fv(2)18680 54603 y Fq(with)657 b(_)-615 b Fp(q)22315 54121 y Fk(T)22267 54933 y Fv(2)23520 54603 y Fp(>)369 b Fq(0,)417 b(the)411 b(action)g(of)g Fp(`)34615 54802 y Fv(2)35622 54603 y Fq(on)f Fp(q)38075 54121 y Fk(T)38027 54933 y Fv(2)38912 54603 y Fq(\()p Fp(t)p Fq(\))p Fm(j)p Fq([0)p Fp(;)221 b(T)184 b Fq(])411 b(minimizes)1102 56198 y(the)436 b(action)f(of)g Fp(`)9418 56397 y Fv(2)10450 56198 y Fq(along)h(all)f(curv)-37 b(es)437 b Fp(\030)61 b Fq(:)587 b([0)p Fp(;)221 b(T)184 b Fq(])370 b Fm(!)f Fl(R)436 b Fq(suc)-37 b(h)436 b(that)33893 55847 y(\026)33756 56198 y Fp(\030)61 b Fq(\(0\))369 b(=)g(0)435 b(and)41678 55847 y(\026)41542 56198 y Fp(\030)61 b Fq(\()p Fp(T)184 b Fq(\))369 b(=)g(2)p Fp(\031)48 b Fq(,)437 b(i.e.)9112 60547 y Fp(A)10108 60746 y Fv(2)10704 60547 y Fq(\()p Fp(q)11862 59998 y Fk(T)11814 60875 y Fv(2)12698 60547 y Fq(\()p Fp(t)p Fq(\))p Fm(j)p Fq([0)p Fp(;)221 b(T)184 b Fq(]\))369 b(=)19674 58739 y Fi(Z)21002 59067 y Fk(T)20412 61757 y Fv(0)21839 60547 y Fm(f)p Fp(V)295 b Fq(\()p Fp(q)24731 59998 y Fk(T)24683 60875 y Fv(2)25567 60547 y Fq(\()p Fp(t)p Fq(\)\))f(+)h Fp(U)145 b Fq(\()p Fp(q)31430 59998 y Fk(T)31382 60875 y Fv(2)32266 60547 y Fq(\()p Fp(t)p Fq(\))p Fp(;)467 b Fq(_)-615 b Fp(q)35011 59998 y Fk(T)34963 60875 y Fv(2)35847 60547 y Fq(\()p Fp(t)p Fq(\)\))p Fm(g)p Fp(dt)18419 64404 y Fq(=)1471 b(inf)20229 65159 y Fv(\026)20121 65405 y Fk(\030)33 b Fv(\(0\)=0)19781 66307 y(\026)19674 66552 y Fk(\030)f Fv(\()p Fk(T)143 b Fv(\)=2)p Fk(\031)24010 62596 y Fi(Z)25339 62924 y Fk(T)24748 65614 y Fv(0)26175 64404 y Fm(f)p Fp(V)296 b Fq(\()p Fp(\030)61 b Fq(\()p Fp(t)p Fq(\)\))293 b(+)j Fp(U)145 b Fq(\()p Fp(\030)61 b Fq(\()p Fp(t)p Fq(\))p Fp(;)37320 64053 y Fq(_)37037 64404 y Fp(\030)f Fq(\()p Fp(t)p Fq(\)\))p Fm(g)p Fp(dt:)1102 69937 y Fq(Let)542 b(^)-763 b Fp(q)4123 69455 y Fk(T)4075 70266 y Fv(2)4960 69937 y Fq(\()p Fp(t)p Fq(\))368 b(=)h Fp(q)8885 69455 y Fk(T)8837 70266 y Fv(2)9721 69937 y Fq(\()p Fp(t=)p Fq(2\),)442 b(then)h(w)-37 b(e)444 b(ha)-37 b(v)g(e)7054 74286 y Fp(A)8050 74485 y Fv(2)8646 74286 y Fq(\()p Fp(q)9804 73737 y Fk(T)9756 74614 y Fv(2)10640 74286 y Fq(\()p Fp(t)p Fq(\))p Fm(j)p Fq([0)p Fp(;)221 b Fq(2)p Fp(T)184 b Fq(]\))295 b Fm(\000)h Fp(A)19276 74485 y Fv(2)19872 74286 y Fq(\()99 b(^)-763 b Fp(q)21030 73737 y Fk(T)20982 74614 y Fv(2)21866 74286 y Fq(\()p Fp(t)p Fq(\))p Fm(j)p Fq([0)p Fp(;)221 b Fq(2)p Fp(T)184 b Fq(]\))369 b Fm(\025)29812 73387 y Fq(1)p 29812 73980 665 54 v 29812 75197 a(2)30857 72478 y Fi(Z)32186 72806 y Fv(2)p Fk(T)31595 75496 y Fv(0)33773 74286 y Fp(U)145 b Fq(\()p Fp(q)35983 73737 y Fk(T)35935 74614 y Fv(2)36819 74286 y Fq(\()p Fp(t)p Fq(\))p Fp(;)467 b Fq(_)-615 b Fp(q)39564 73737 y Fk(T)39516 74614 y Fv(2)40400 74286 y Fq(\()p Fp(t)p Fq(\)\))p Fp(dt;)p eop %%Page: 17 17 17 16 bop 48792 2239 a Ft(17)1102 5427 y Fq(where)550 b(w)-37 b(e)550 b(ha)-37 b(v)g(e)549 b(used)g(the)g(facts)g(suc)-37 b(h)549 b(as)f Fp(U)145 b Fq(\(0)p Fp(;)221 b(q)27316 5626 y Fv(2)27912 5427 y Fq(\))545 b Fm(\021)h Fq(0,)575 b Fp(@)33062 5626 y Fv(_)-497 b Fk(q)33353 5759 y Fj(2)33938 5427 y Fp(U)145 b Fq(\(0)p Fp(;)221 b(q)37354 5626 y Fv(2)37950 5427 y Fq(\))545 b Fm(\021)g Fq(0)k(as)g(w)-37 b(ell)550 b(as)e(the)1102 7021 y(p)37 b(ositiv)-37 b(e)390 b(de\014niteness)f(of)h Fp(@)15248 6539 y Fv(2)15369 7377 y(_)-496 b Fk(q)15661 7510 y Fj(2)16246 7021 y Fp(U)145 b Fq(.)572 b(Rep)37 b(eating)389 b(this)g(pro)37 b(cess)390 b(b)-37 b(y)390 b(replacing)g Fp(T)574 b Fq(with)389 b(2)p Fp(T)184 b Fq(,)401 b(4)p Fp(T)574 b Fq(and)1102 8808 y(so)443 b(on,)g(note)g Fp(A)8895 9007 y Fv(2)9491 8808 y Fq(\()99 b(^)-763 b Fp(q)10649 8326 y Fk(T)10601 9138 y Fv(2)11485 8808 y Fm(j)p Fq([0)p Fp(;)221 b Fq(2)p Fp(T)184 b Fq(]\))370 b Fm(\025)g Fp(A)18756 9007 y Fv(2)19352 8808 y Fq(\()p Fp(q)20510 8326 y Fv(2)p Fk(T)20462 9138 y Fv(2)21875 8808 y Fm(j)p Fq([0)p Fp(;)221 b Fq(2)p Fp(T)184 b Fq(]\))444 b(w)-37 b(e)444 b(obtain)12550 12745 y Fp(A)13546 12944 y Fv(2)14142 12745 y Fq(\()p Fp(q)15300 12197 y Fk(T)15252 13074 y Fv(2)16136 12745 y Fq(\()p Fp(t)p Fq(\))p Fm(j)p Fq([0)p Fp(;)221 b Fq(2)p Fp(T)184 b Fq(]\))295 b Fm(\000)g Fp(A)24771 12944 y Fv(2)25367 12745 y Fq(\()99 b(^)-763 b Fp(q)26525 12197 y Fk(T)26477 13074 y Fv(2)27361 12745 y Fq(\()p Fp(t)p Fq(\))p Fm(j)p Fq([0)p Fp(;)221 b Fq(2)p Fp(T)184 b Fq(]\))369 b Fm(\025)35308 11847 y Fq(1)p 35308 12440 665 54 v 35308 13657 a(2)36131 12745 y Fp(C)37175 12177 y Fv(+)37080 13136 y Fk(K)38105 12745 y Fp(:)1102 16429 y Fq(It)442 b(follo)-37 b(ws)444 b(that)3744 19741 y Fp(A)4740 19940 y Fk(c)5281 19741 y Fq(\()p Fp(\015)74 b Fm(j)p Fq([)p Fp(t)7778 19940 y Fv(0)8373 19741 y Fp(;)221 b(t)9443 19940 y Fv(0)10334 19741 y Fq(+)295 b(2)p Fp(T)184 b Fq(]\))p Fm(\000)q Fp(A)16202 19940 y Fk(c)16743 19741 y Fq(\()p Fp(\015)18022 19193 y Fg(\003)18632 19741 y Fm(j)p Fq([)p Fp(t)19850 19940 y Fv(0)20446 19741 y Fp(;)221 b(t)21516 19940 y Fv(0)22407 19741 y Fq(+)295 b(2)p Fp(T)184 b Fq(]\))14173 21988 y Fm(\025)p Fp(A)16202 22187 y Fv(2)16798 21988 y Fq(\()p Fp(\015)18003 22187 y Fv(2)18598 21988 y Fm(j)p Fq([)p Fp(t)19816 22187 y Fv(0)20412 21988 y Fp(;)221 b(t)21482 22187 y Fv(0)22373 21988 y Fq(+)295 b(2)p Fp(T)184 b Fq(]\))296 b Fm(\000)f Fp(A)28831 22187 y Fv(2)29428 21988 y Fq(\()99 b(^)-763 b Fp(q)30586 21440 y Fk(T)30538 22317 y Fv(2)31422 21988 y Fm(j)p Fq([)p Fp(t)32640 22187 y Fv(0)33235 21988 y Fp(;)221 b(t)34305 22187 y Fv(0)35196 21988 y Fq(+)296 b(2)p Fp(T)184 b Fq(]\))295 b Fm(\000)h Fq(4)p Fp(T)184 b(\017)296 b Fm(\000)f Fq(2)p Fp(\031)48 b(c)46490 22187 y Fv(2)14173 24235 y Fm(\025)p Fp(A)16202 24434 y Fv(2)16798 24235 y Fq(\()p Fp(q)17956 23687 y Fk(T)17908 24564 y Fv(2)18792 24235 y Fm(j)p Fq([)p Fp(t)20010 24434 y Fv(0)20606 24235 y Fp(;)221 b(t)21676 24434 y Fv(0)22567 24235 y Fq(+)295 b(2)p Fp(T)184 b Fq(]\))296 b Fm(\000)f Fp(A)29025 24434 y Fv(2)29621 24235 y Fq(\()99 b(^)-763 b Fp(q)30779 23687 y Fk(T)30731 24564 y Fv(2)31615 24235 y Fm(j)p Fq([)p Fp(t)32833 24434 y Fv(0)33429 24235 y Fp(;)221 b(t)34499 24434 y Fv(0)35390 24235 y Fq(+)296 b(2)p Fp(T)184 b Fq(]\))295 b Fm(\000)h Fq(4)p Fp(T)184 b(\017)296 b Fm(\000)f Fq(2)p Fp(\031)48 b(c)46684 24434 y Fv(2)14173 27053 y Fm(\025)15365 26155 y Fq(1)p 15365 26748 V 15365 27965 a(2)16189 27053 y Fp(C)17233 26485 y Fv(+)17138 27444 y Fk(K)18458 27053 y Fm(\000)295 b Fq(4)p Fp(T)184 b(\017)296 b Fm(\000)g Fq(2)p Fp(\031)48 b(c)25618 27252 y Fv(2)14173 29426 y Fp(>)p Fq(0)1102 32952 y(if)492 b Fp(T)637 b Fm(\024)451 b Fp(T)6226 32470 y Fg(\003)6042 33280 y Fv(+)7252 32952 y Fq(+)329 b Fp(T)9390 33151 y Fv(+)10272 32952 y Fq(\()p Fp(a)11491 32470 y Fg(\003)11491 33282 y Fv(0)12101 32952 y Fq(\))f(+)g Fp(\034)150 b Fq(,)505 b Fm(j)p Fp(c)16856 33151 y Fv(2)17452 32952 y Fm(j)451 b(\024)20243 32429 y Fv(1)p 19916 32646 1185 54 v 19916 33410 a(8)p Fk(\031)21259 32952 y Fp(C)22303 32384 y Fv(+)22208 33342 y Fk(K)23725 32952 y Fq(and)492 b Fp(\017)452 b Fm(\024)29257 32429 y Fv(1)p 28992 32646 1059 54 v 28992 33410 a(17)30210 32952 y Fp(C)31254 32384 y Fv(+)31159 33342 y Fk(K)32184 32952 y Fq(\()p Fp(T)33661 32470 y Fg(\003)33477 33280 y Fv(+)34687 32952 y Fq(+)328 b Fp(T)36824 33151 y Fv(+)37706 32952 y Fq(\()p Fp(a)38925 32470 y Fg(\003)38925 33282 y Fv(0)39536 32952 y Fq(\))f(+)i Fp(\034)150 b Fq(\))42990 32470 y Fg(\000)p Fv(1)44415 32952 y Fq(,)505 b(but)492 b(this)1102 34546 y(con)-37 b(tradicts)444 b(to)e(the)h(minimalit)-37 b(y)443 b(of)f Fp(\015)74 b Fq(.)2696 37061 y(Let)632 b(us)f(consider)i(the)f(case)g (when)g Fp(T)868 b Fm(\025)684 b Fp(T)25979 36579 y Fg(\003)25795 37389 y Fv(+)27098 37061 y Fq(+)421 b Fp(T)29328 37260 y Fv(+)30210 37061 y Fq(\()p Fp(a)31429 36579 y Fg(\003)31429 37390 y Fv(0)32040 37061 y Fq(\).)1156 b(Since)632 b Fp(d\015)39118 37260 y Fv(2)40346 37061 y Fq(encircles)h(around)1102 38655 y(the)559 b(cylinder,)589 b(there)560 b(exist)f Fp(t)16334 38173 y Fg(0)17265 38655 y Fq(and)g Fp(T)20924 38173 y Fg(0)21860 38655 y Fm(\025)k Fp(T)744 b Fq(suc)-37 b(h)559 b(that)607 b(\026)-712 b Fp(\015)31771 38854 y Fv(2)32366 38655 y Fq(\()p Fp(t)33363 38173 y Fg(0)33735 38655 y Fq(\))562 b(=)h Fm(\000)19 b Fq(\026)-683 b Fp(a)38145 38854 y Fv(2)38741 38655 y Fq(\()p Fp(a)39960 38173 y Fg(\003)39960 38984 y Fv(0)40571 38655 y Fq(\),)636 b(\026)-713 b Fp(\015)42732 38854 y Fv(2)43328 38655 y Fq(\()p Fp(t)44325 38173 y Fg(0)45070 38655 y Fq(+)372 b Fp(T)47435 38173 y Fg(0)47809 38655 y Fq(\))562 b(=)1102 40249 y(2)p Fp(\031)344 b Fm(\000)314 b Fq(\026)-683 b Fp(a)4897 40448 y Fv(2)5493 40249 y Fq(\()p Fp(a)6712 39767 y Fg(\003)6712 40578 y Fv(0)7322 40249 y Fq(\))443 b(and)491 b(\026)-712 b Fp(\015)11553 40448 y Fv(2)12148 40249 y Fq(\()p Fp(t)p Fq(\))368 b Fp(<)h Fq(2)p Fp(\031)344 b Fm(\000)314 b Fq(\026)-683 b Fp(a)19227 40448 y Fv(2)19823 40249 y Fq(\()p Fp(a)21042 39767 y Fg(\003)21042 40578 y Fv(0)21652 40249 y Fq(\))443 b(for)g Fp(t)369 b(<)g(t)27376 39767 y Fg(0)28043 40249 y Fq(+)296 b Fp(T)30332 39767 y Fg(0)30705 40249 y Fq(.)590 b(Let)8910 43354 y(^)8892 43599 y Fp(t)368 b Fq(=)i(max)p Fm(f)p Fp(t)f Fm(2)f Fq(\()p Fp(t)17379 43050 y Fg(0)17751 43599 y Fp(;)221 b(t)18821 43050 y Fg(0)19489 43599 y Fq(+)295 b Fp(T)21777 43050 y Fg(0)22151 43599 y Fq(\))368 b(:)861 b(\026)-713 b Fp(\015)24905 43798 y Fv(2)25501 43599 y Fq(\()p Fp(t)p Fq(\))368 b(=)h Fp(a)29487 43798 y Fv(1)30083 43599 y Fq(\()19 b(\026)-683 b Fp(a)31302 43798 y Fv(2)31897 43599 y Fq(\()p Fp(a)33116 43050 y Fg(\003)33116 43927 y Fv(0)33727 43599 y Fq(\)\))p Fp(;)35989 43417 y Fq(_)35842 43599 y(\026)-713 b Fp(\015)36555 43924 y Fv(2)37151 43599 y Fq(\()p Fp(t)p Fq(\))367 b Fm(\025)j Fq(0)p Fm(g)p Fp(:)1102 46949 y Fq(Clearly)-111 b(,)445 b(there)e(is)g(suc)-37 b(h)443 b(a)14832 46704 y(^)14814 46949 y Fp(t)f Fq(whic)-37 b(h)444 b(satis\014es)f(the)f (follo)-37 b(wing:)9830 50261 y Fp(T)10606 50460 y Fv(+)11488 50261 y Fq(\()p Fp(a)12707 50460 y Fv(1)13302 50261 y Fq(\()19 b(\026)-683 b Fp(a)14521 50460 y Fv(2)15117 50261 y Fq(\()p Fp(a)16336 49712 y Fg(\003)16336 50589 y Fv(0)16947 50261 y Fq(\)\))p Fp(;)239 b Fq(\026)-682 b Fp(a)19273 50460 y Fv(2)19868 50261 y Fq(\()p Fp(a)21087 49712 y Fg(\003)21087 50589 y Fv(0)21698 50261 y Fq(\)\))294 b Fm(\000)h Fp(\034)519 b Fm(\024)369 b Fp(t)27336 49712 y Fg(0)28004 50261 y Fq(+)295 b Fp(T)30292 49712 y Fg(0)30961 50261 y Fm(\000)32308 50015 y Fq(^)32289 50261 y Fp(t)369 b Fm(\024)g Fp(T)35316 50460 y Fv(+)36198 50261 y Fq(\()p Fp(a)37417 49712 y Fg(\003)37417 50589 y Fv(0)38028 50261 y Fq(\))294 b(+)i Fp(\034)77 b(:)5332 b Fq(\(3.12\))1102 53611 y(The)407 b(second)f(inequalit)-37 b(y)406 b(is)g(from)g(\(3.9\)) f(and)g(the)h(c)-37 b(hoice)407 b(of)f(this)34463 53366 y(^)34445 53611 y Fp(t)p Fq(.)577 b(T)-111 b(o)407 b(v)-37 b(erify)406 b(the)g(\014rst,)413 b(let)405 b(us)1102 55243 y(supp)37 b(ose)388 b(the)e(con)-37 b(trary)-111 b(.)573 b(If)387 b Fp(t)15735 54761 y Fg(0)16291 55243 y Fq(+)183 b Fp(T)18467 54761 y Fg(0)19024 55243 y Fm(\000)20259 54998 y Fq(^)20240 55243 y Fp(t)369 b(<)g(T)23267 55442 y Fv(+)24149 55243 y Fq(\()p Fp(a)25368 55442 y Fv(1)25964 55243 y Fq(\()19 b(\026)-683 b Fp(a)27183 55442 y Fv(2)27779 55243 y Fq(\()p Fp(a)28998 54761 y Fg(\003)28998 55573 y Fv(0)29608 55243 y Fq(\)\))p Fp(;)240 b Fq(\026)-683 b Fp(a)31934 55442 y Fv(2)32530 55243 y Fq(\()p Fp(a)33749 54761 y Fg(\003)33749 55573 y Fv(0)34359 55243 y Fq(\)\))183 b Fm(\000)g Fp(\034)150 b Fq(,)398 b(then)41279 54998 y(^)41260 55243 y Fp(t)183 b Fm(\000)g Fp(t)43619 54761 y Fg(0)44361 55243 y Fp(>)369 b(T)46723 54761 y Fg(\003)46539 55572 y Fv(+)47605 55243 y Fq(+)183 b Fp(\034)150 b Fq(.)1102 56983 y(In)475 b(this)f(case)i(w)-37 b(e)476 b(obtain)f(from)g (\(3.11\))g(that)24562 56801 y(_)24415 56983 y(\026)-713 b Fp(\015)25128 57307 y Fv(2)25724 56983 y Fq(\()26259 56738 y(^)26241 56983 y Fp(t)o Fq(\))422 b Fp(<)h(G)30159 56501 y Fv(+)31041 56983 y Fq(\()49 b(\026)-713 b Fp(\015)32246 57182 y Fv(2)32841 56983 y Fq(\()33376 56738 y(^)33358 56983 y Fp(t)p Fq(\)\))315 b(+)i Fp(\016)50 b Fq(.)688 b(But)474 b(this)h(con)-37 b(tradicts)1102 58615 y(to)439 b(the)g(de\014nition)g(of)12434 58370 y(^)12415 58615 y Fp(t)g Fq(and)g Fp(\016)50 b Fq(,)440 b(since)g Fp(d)49 b Fq(\026)-713 b Fp(\015)21992 58814 y Fv(2)23027 58615 y Fq(can)440 b(not)e(reac)-37 b(h)441 b(the)e(line)g Fp(q)36704 58814 y Fv(2)37669 58615 y Fq(=)369 b(2)p Fp(\031)336 b Fm(\000)306 b Fq(\026)-683 b Fp(a)42850 58814 y Fv(2)43446 58615 y Fq(\()p Fp(a)44665 58133 y Fg(\003)44665 58945 y Fv(0)45276 58615 y Fq(\))438 b(within)1102 60209 y(time)443 b Fp(T)4904 60408 y Fv(+)5786 60209 y Fq(\()p Fp(a)7005 60408 y Fv(1)7601 60209 y Fq(\()19 b(\026)-683 b Fp(a)8820 60408 y Fv(2)9416 60209 y Fq(\()p Fp(a)10635 59727 y Fg(\003)10635 60539 y Fv(0)11245 60209 y Fq(\)\))p Fp(;)239 b Fq(\026)-682 b Fp(a)13571 60408 y Fv(2)14167 60209 y Fq(\()p Fp(a)15386 59727 y Fg(\003)15386 60539 y Fv(0)15996 60209 y Fq(\)\))294 b Fm(\000)i Fp(\034)150 b Fq(,)442 b(whic)-37 b(h)444 b(v)-37 b(eri\014es)444 b(the)f(\014rst)f(inequalit)-37 b(y)443 b(of)g(\(3.12\).)2696 62724 y(Let)g(us)g(construct)g(a)g(curv)-37 b(e)444 b Fp(\020)98 b Fq(\()p Fp(t)p Fq(\))368 b(=)h(\()p Fp(\020)22336 62923 y Fv(1)22932 62724 y Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(\020)25617 62923 y Fv(2)26212 62724 y Fq(\()p Fp(t)p Fq(\)\))441 b(in)i(the)g(w)-37 b(a)g(y)443 b(that)g(its)f(lift)g(has)h (the)g(form)12960 67678 y(\026)12842 68028 y Fp(\020)13423 68227 y Fv(2)14019 68028 y Fq(\()p Fp(t)p Fq(\))368 b(=)17303 64906 y Fi(8)17303 66102 y(>)17303 66500 y(<)17303 68892 y(>)17303 69290 y(:)18975 66034 y Fq(\026)-713 b Fp(\015)19614 66233 y Fv(2)20210 66034 y Fq(\()p Fp(t)p Fq(\))7873 b Fp(t)368 b Fm(\024)31866 65790 y Fq(^)31848 66035 y Fp(t)18926 68067 y(\030)19507 68266 y Fv(2)20103 68067 y Fq(\()p Fp(t)p Fq(\))29509 67821 y(^)29490 68067 y Fp(t)h Fm(\024)g Fp(t)g Fm(\024)g Fp(t)34472 67585 y Fg(0)35139 68067 y Fq(+)296 b Fp(T)37428 67585 y Fg(0)18975 70060 y Fq(\026)-713 b Fp(\015)19614 70259 y Fv(2)20210 70060 y Fq(\()p Fp(t)p Fq(\))294 b Fm(\000)h Fq(2)p Fp(\031)4773 b(t)368 b Fm(\025)i Fp(t)32271 69578 y Fg(0)32938 70060 y Fq(+)295 b Fp(T)35226 69578 y Fg(0)46527 68028 y Fq(\(3.13\))1102 73332 y(and)466 b Fp(\020)4289 73531 y Fv(1)4885 73332 y Fq(\()p Fp(t)p Fq(\))406 b(=)h Fp(\015)8933 73531 y Fv(1)9528 73332 y Fq(\()p Fp(t)p Fq(\),)470 b(where)d Fp(\030)16326 73531 y Fv(2)17388 73332 y Fq(is)e(the)h(extremal)g(of)g (the)f(Lagrangian)i Fp(`)37962 73531 y Fv(2)39024 73332 y Fq(with)e(the)h(b)37 b(oundary)1102 74965 y(condition)445 b Fp(\030)7515 75164 y Fv(2)8111 74965 y Fq(\()8646 74720 y(^)8628 74965 y Fp(t)o Fq(\))371 b(=)g Fp(a)12101 75164 y Fv(1)12697 74965 y Fq(\()19 b(\026)-683 b Fp(a)13916 75164 y Fv(2)14512 74965 y Fq(\()p Fp(a)15731 74483 y Fg(\003)15731 75294 y Fv(0)16341 74965 y Fq(\)\))443 b(and)h Fp(\030)20983 75164 y Fv(2)21579 74965 y Fq(\()p Fp(t)22576 74483 y Fg(0)23244 74965 y Fq(+)296 b Fp(T)25533 74483 y Fg(0)25907 74965 y Fq(\))371 b(=)g Fm(\000)19 b Fq(\026)-683 b Fp(a)29934 75164 y Fv(2)30530 74965 y Fq(\()p Fp(a)31749 74483 y Fg(\003)31749 75294 y Fv(0)32359 74965 y Fq(\).)594 b(Th)-37 b(us,)445 b Fp(\015)f Fq(=)372 b Fp(\020)542 b Fq(at)444 b Fp(t)370 b Fq(=)45133 74720 y(^)45115 74965 y Fp(t)443 b Fq(as)h(w)-37 b(ell)p eop %%Page: 18 18 18 17 bop 1102 2239 a Ft(18)1102 5427 y Fq(as)443 b(at)g Fp(t)369 b Fq(=)g Fp(t)7088 4945 y Fg(0)7755 5427 y Fq(+)296 b Fp(T)10044 4945 y Fg(0)10417 5427 y Fq(.)590 b(The)444 b(di\013erence)g(b)37 b(et)-37 b(w)g(een)444 b(the)f(action)g(of)g Fp(L)295 b Fm(\000)g Fp(\021)36153 5626 y Fk(c)37137 5427 y Fq(along)443 b Fp(\015)516 b Fq(and)443 b Fp(\020)541 b Fq(is)7521 8643 y Fp(A)8517 8842 y Fk(c)9059 8643 y Fq(\()p Fp(\015)74 b Fq(\))294 b Fm(\000)h Fp(A)13473 8842 y Fk(c)14014 8643 y Fq(\()p Fp(\020)98 b Fq(\))369 b(=)17350 6835 y Fi(Z)18679 7164 y Fk(t)19080 6763 y Fe(0)19439 7164 y Fv(+)p Fk(T)21024 6763 y Fe(0)18110 9682 y Fv(^)18088 9854 y Fk(t)21672 7169 y Fi(n)22558 8643 y Fp(`)23111 8842 y Fv(2)23707 8643 y Fq(\()p Fp(d\015)25603 8842 y Fv(2)26199 8643 y Fq(\()p Fp(t)p Fq(\)\))293 b Fm(\000)j Fp(`)30405 8842 y Fv(2)31001 8643 y Fq(\()p Fp(d\020)32790 8842 y Fv(2)33386 8643 y Fq(\()p Fp(t)p Fq(\)\))35417 7169 y Fi(o)36301 8643 y Fp(dt)17424 12635 y Fq(+)18752 10827 y Fi(Z)20081 11155 y Fk(t)20482 10755 y Fe(0)20841 11155 y Fv(+)p Fk(T)22426 10755 y Fe(0)19512 13673 y Fv(^)19490 13845 y Fk(t)22853 12635 y Fm(f)p Fp(P)184 b Fq(\()p Fp(d\015)74 b Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))294 b Fm(\000)h Fp(P)184 b Fq(\()p Fp(d\020)98 b Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))p Fm(g)p Fp(dt)294 b Fm(\000)i Fq(2)p Fp(\031)48 b(c)42774 12834 y Fv(2)16096 16132 y Fm(\025)17288 15233 y Fq(3)p 17288 15826 665 54 v 17288 17043 a(5)18112 16132 y Fp(C)19156 15564 y Fv(+)19061 16522 y Fk(A)20333 16132 y Fm(\000)296 b Fq(2\()p Fp(T)23619 16331 y Fv(+)24501 16132 y Fq(\()p Fp(a)25720 15584 y Fg(\003)25720 16460 y Fv(0)26330 16132 y Fq(\))f(+)g Fp(\034)150 b Fq(\))p Fp(\017)295 b Fm(\000)g Fq(2)p Fp(\031)48 b(c)33924 16331 y Fv(2)28152 18571 y Fq(\(b)-37 b(y)443 b(\(3)p Fp(:)p Fq(6\))p Fp(;)f Fq(\(3)p Fp(:)p Fq(8\))221 b(and)443 b(\(3)p Fp(:)p Fq(12\)\))16096 20564 y Fp(>)p Fq(0)1102 22948 y(if)485 b Fm(j)p Fp(c)3306 23147 y Fv(2)3902 22948 y Fm(j)439 b(\024)6933 22425 y Fv(1)p 6341 22643 1714 54 v 6341 23406 a(10)p Fk(\031)8214 22948 y Fp(C)9258 22380 y Fv(+)9163 23339 y Fk(A)10625 22948 y Fq(and)485 b Fp(\017)439 b Fm(\024)16124 22425 y Fv(1)p 15860 22643 1059 54 v 15860 23406 a(10)17078 22948 y Fp(C)18122 22380 y Fv(+)18027 23339 y Fk(A)19004 22948 y Fq(\()p Fp(T)20297 23147 y Fv(+)21179 22948 y Fq(\()p Fp(a)22398 22466 y Fg(\003)22398 23278 y Fv(0)23009 22948 y Fq(\)\))322 b(+)h Fp(\034)150 b Fq(\))26969 22466 y Fg(\000)p Fv(1)28395 22948 y Fq(,)495 b(but)485 b(this)f(con)-37 b(tradicts)485 b(the)g(fact)g(that)f Fp(\015)1102 24542 y Fq(is)443 b(a)g Fp(c)p Fq(-minimal)g(orbit.)2696 26707 y(Let)533 b Fp(a)5868 26225 y Fg(\003)6997 26707 y Fq(=)519 b(min)p Fm(f)p Fp(a)12129 26906 y Fv(1)12725 26707 y Fq(\()19 b(\026)-683 b Fp(a)13944 26906 y Fv(2)14539 26707 y Fq(\()p Fp(a)15758 26225 y Fg(\003)15758 27037 y Fv(0)16369 26707 y Fq(\)\))p Fp(;)239 b Fq(\026)-682 b Fp(a)18695 26906 y Fv(2)19290 26707 y Fq(\()p Fp(a)20509 26225 y Fg(\003)20509 27037 y Fv(0)21120 26707 y Fq(\))p Fp(;)240 b Fq(\026)-683 b Fp(a)22929 26906 y Fv(1)23525 26707 y Fq(\()p Fp(a)24744 26225 y Fg(\003)24744 27037 y Fv(0)25354 26707 y Fq(\))p Fp(;)221 b(a)27163 26906 y Fv(2)27759 26707 y Fq(\()19 b(\026)-683 b Fp(a)28978 26906 y Fv(1)29574 26707 y Fq(\()p Fp(a)30793 26225 y Fg(\003)30793 27037 y Fv(0)31403 26707 y Fq(\)\))p Fm(g)p Fq(,)555 b(let)532 b Fp(\017)518 b(>)h Fq(0)532 b(b)37 b(e)533 b(suitably)f(small)1102 28301 y(suc)-37 b(h)444 b(that)15451 29895 y(min)p Fm(f)p Fp(V)295 b Fq(\()p Fp(q)20509 30094 y Fv(2)21105 29895 y Fq(\))368 b(:)h Fm(j)p Fp(q)23690 30094 y Fv(2)24581 29895 y Fm(\000)296 b Fq(2)p Fp(k)42 b(\031)48 b Fm(j)368 b(\025)h Fp(a)30954 29347 y Fg(\003)31565 29895 y Fm(g)g(\025)g Fq(3)p Fp(\017:)1102 31966 y Fq(F)-111 b(or)489 b(eac)-37 b(h)489 b(lift)e(of)h Fp(\015)561 b Fq(w)-37 b(e)489 b(claim)g(there)f(is)g Fp(k)d Fm(2)444 b Fl(Z)487 b Fq(suc)-37 b(h)488 b(that)g Fm(j)49 b Fq(\026)-713 b Fp(\015)32835 32165 y Fv(2)33430 31966 y Fq(\()p Fp(t)p Fq(\))324 b Fm(\000)i Fq(2)p Fp(k)42 b(\031)48 b Fm(j)443 b(\025)h Fp(a)41821 31484 y Fg(\003)41821 32296 y Fv(0)42920 31966 y Fq(holds)488 b(for)g(all)1102 33560 y Fp(t)369 b Fm(2)g Fl(R)p Fq(.)590 b(Let)443 b(us)g(supp)37 b(ose)443 b(the)g(con)-37 b(trary)-111 b(,)444 b(th)-37 b(us)442 b(w)-37 b(e)444 b(ha)-37 b(v)g(e)444 b(t)-37 b(w)g(o)443 b(alternativ)-37 b(es:)2696 35725 y(1\))443 b(F)-111 b(or)444 b(all)f Fp(k)410 b Fm(2)368 b Fl(Z)443 b Fq(and)g(all)g Fp(t)368 b Fm(2)h Fl(R)443 b Fq(w)-37 b(e)444 b(ha)-37 b(v)g(e)443 b Fm(j)49 b Fq(\026)-713 b Fp(\015)26242 35924 y Fv(2)26838 35725 y Fq(\()p Fp(t)p Fq(\))294 b Fm(\000)h Fq(2)p Fp(k)42 b(\031)48 b Fm(j)369 b Fp(>)g(a)35019 35243 y Fg(\003)35630 35725 y Fq(,)442 b(or)2696 37319 y(2\))h(There)h(exists)f Fp(t)12309 37518 y Fv(0)13273 37319 y Fm(2)369 b Fl(R)443 b Fq(suc)-37 b(h)443 b(that)f Fm(j)49 b Fq(\026)-713 b Fp(\015)22861 37518 y Fv(2)23457 37319 y Fq(\()p Fp(t)p Fq(\))294 b Fm(\000)h Fq(2)p Fp(k)42 b(\031)48 b Fm(j)369 b Fq(=)g Fp(a)31638 36837 y Fg(\003)32692 37319 y Fq(for)443 b(some)g Fp(k)410 b Fm(2)369 b Fl(Z)p Fq(.)2696 39484 y(In)472 b(the)h(\014rst)f(case,)481 b(w)-37 b(e)473 b(consider)h(those)e(lift)g (of)g Fp(\015)27753 39683 y Fv(2)28821 39484 y Fq(whic)-37 b(h)474 b(tak)-37 b(es)472 b(v)-74 b(alue)474 b(b)37 b(et)-37 b(w)g(een)473 b(0)g(and)f(2)p Fp(\031)1102 41078 y Fq(and)443 b(construct)g(a)g(curv)-37 b(e)444 b Fp(\020)98 b Fq(\()p Fp(t)p Fq(\))368 b(=)h(\()p Fp(\015)19347 41277 y Fv(1)19943 41078 y Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(\020)22628 41277 y Fv(1)23223 41078 y Fq(\()p Fp(t)p Fq(\)\))441 b(in)i(the)g(w)-37 b(a)g(y)443 b(suc)-37 b(h)443 b(that)g(its)f(lift)g(has)h(the)g(form)10785 46750 y(\026)10667 47101 y Fp(\020)11248 47300 y Fv(2)11844 47101 y Fq(\()p Fp(t)p Fq(\))368 b(=)15128 42385 y Fi(8)15128 43581 y(>)15128 43979 y(>)15128 44378 y(>)15128 44776 y(>)15128 45175 y(>)15128 45573 y(<)15128 47964 y(>)15128 48363 y(>)15128 48761 y(>)15128 49160 y(>)15128 49558 y(>)15128 49957 y(:)16801 43115 y Fq(\026)-713 b Fp(\015)17440 43314 y Fv(2)18035 43115 y Fq(\()p Fp(t)p Fq(\))9921 b Fp(t)368 b Fm(\024)h Fp(t)32200 43314 y Fv(1)16752 45108 y Fq(\(1)295 b(+)g Fp(t)20036 45307 y Fv(1)20927 45108 y Fm(\000)g Fp(t)p Fq(\))49 b(\026)-713 b Fp(\015)23940 45307 y Fv(2)24535 45108 y Fq(\()p Fp(t)p Fq(\))3306 b Fp(t)29835 45307 y Fv(1)30800 45108 y Fm(\024)369 b Fp(t)f Fm(\024)i Fp(t)34933 45307 y Fv(1)35823 45108 y Fq(+)296 b(1)16752 47101 y(0)11811 b Fp(t)29707 47300 y Fv(1)30598 47101 y Fq(+)295 b(1)369 b Fm(\024)g Fp(t)g Fm(\024)g Fp(t)37092 47300 y Fv(2)37983 47101 y Fm(\000)295 b Fq(1)16752 49094 y(\()p Fp(t)17749 49293 y Fv(2)18639 49094 y Fm(\000)h Fp(t)p Fq(\))49 b(\026)-713 b Fp(\015)21653 49293 y Fv(2)22247 49094 y Fq(\()p Fp(t)p Fq(\))5669 b Fp(t)29910 49293 y Fv(2)30800 49094 y Fm(\000)296 b Fq(1)369 b Fm(\024)g Fp(t)g Fm(\024)g Fp(t)16801 51087 y Fq(\026)-713 b Fp(\015)17440 51286 y Fv(2)18035 51087 y Fq(\()p Fp(t)p Fq(\))9921 b Fp(t)368 b Fm(\025)h Fp(t)32200 51286 y Fv(2)32796 51087 y Fp(:)1102 53050 y Fq(Comparing)444 b(the)f(action)g(of)g Fp(L)295 b Fm(\000)g Fp(\021)18942 53249 y Fk(c)19926 53050 y Fq(along)443 b Fp(\015)516 b Fq(with)443 b(that)f(along)i Fp(\020)98 b Fq(,)443 b(w)-37 b(e)443 b(\014nd)8009 56096 y Fp(A)9005 56295 y Fk(c)9547 56096 y Fq(\()p Fp(\015)74 b Fq(\))294 b Fm(\000)h Fp(A)13961 56295 y Fk(c)14502 56096 y Fq(\()p Fp(\020)98 b Fq(\))369 b(=)17838 54223 y Fi(\022)18816 54288 y(Z)20144 54616 y Fk(t)20545 54749 y Fj(1)21064 54616 y Fv(+1)19554 57306 y Fk(t)19955 57439 y Fj(1)22697 56096 y Fq(+)23951 54288 y Fi(Z)25280 54616 y Fk(t)25681 54749 y Fj(2)24689 57306 y Fk(t)25090 57439 y Fj(2)25609 57306 y Fg(\000)p Fv(1)27035 54223 y Fi(\023)28234 54622 y(n)29120 56096 y Fp(`)29673 56295 y Fv(2)30269 56096 y Fq(\()p Fp(d\015)74 b Fq(\()p Fp(t)p Fq(\)\))293 b Fm(\000)j Fp(`)36445 56295 y Fv(2)37041 56096 y Fq(\()p Fp(d\020)98 b Fq(\()p Fp(t)p Fq(\)\))40959 54622 y Fi(o)41843 56096 y Fp(dt)17912 60023 y Fq(+)19240 58215 y Fi(Z)20569 58543 y Fk(t)20970 58676 y Fj(2)19978 61233 y Fk(t)20379 61366 y Fj(1)21776 58548 y Fi(n)22662 60023 y Fp(P)184 b Fq(\()p Fp(d\015)74 b Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))293 b Fm(\000)j Fp(P)184 b Fq(\()p Fp(d\020)98 b Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))36417 58548 y Fi(o)37301 60023 y Fp(dt)17912 63977 y Fq(+)19240 62169 y Fi(Z)20569 62497 y Fk(t)20970 62630 y Fj(2)21488 62497 y Fg(\000)p Fv(1)19978 65187 y Fk(t)20379 65320 y Fj(1)20898 65187 y Fv(+1)23136 63977 y Fp(`)23689 64176 y Fv(2)24285 63977 y Fq(\()p Fp(d\015)74 b Fq(\()p Fp(t)p Fq(\)\))p Fp(dt)16584 67931 y Fm(\025)17838 66058 y Fi(\022)18816 66123 y(Z)20144 66451 y Fk(t)20545 66584 y Fj(1)21064 66451 y Fv(+1)19554 69141 y Fk(t)19955 69274 y Fj(1)22697 67931 y Fq(+)23951 66123 y Fi(Z)25280 66451 y Fk(t)25681 66584 y Fj(2)24689 69141 y Fk(t)25090 69274 y Fj(2)25609 69141 y Fg(\000)p Fv(1)27035 66058 y Fi(\023)28234 66456 y(n)29120 67931 y Fp(`)29673 68130 y Fv(2)30269 67931 y Fq(\()p Fp(d\015)g Fq(\()p Fp(t)p Fq(\)\))293 b Fm(\000)j Fp(`)36445 68130 y Fv(2)37041 67931 y Fq(\()p Fp(d\020)98 b Fq(\()p Fp(t)p Fq(\)\))40959 66456 y Fi(o)41843 67931 y Fp(dt)17912 70802 y Fq(+)295 b Fp(\017)p Fq(\()p Fp(t)20776 71001 y Fv(2)21667 70802 y Fm(\000)g Fp(t)23475 71001 y Fv(1)24366 70802 y Fm(\000)g Fq(6\))16584 72795 y Fp(>)p Fq(0)p Fp(;)1102 74965 y Fq(if)443 b Fp(t)2800 75164 y Fv(2)3691 74965 y Fm(\000)295 b Fp(t)5499 75164 y Fv(1)6538 74965 y Fq(is)442 b(su\016cien)-37 b(tly)443 b(large.)592 b(Th)-37 b(us)443 b(the)g(\014rst)f(alternativ)-37 b(e)444 b(do)37 b(es)443 b(not)f(o)37 b(ccur.)p eop %%Page: 19 19 19 18 bop 48792 2239 a Ft(19)2696 5427 y Fq(Let)428 b(us)f(consider)i (the)e(second)h(p)37 b(ossibilit)-37 b(y)-111 b(.)585 b(By)428 b(taking)f(suitable)h(lift)e(of)i Fp(\015)74 b Fq(,)430 b(w)-37 b(e)428 b(can)g(assume)1102 7021 y Fm(j)49 b Fq(\026)-713 b Fp(\015)2159 7220 y Fv(2)2755 7021 y Fq(\()p Fp(t)3752 7220 y Fv(0)4347 7021 y Fq(\))p Fm(j)369 b Fq(=)g Fp(a)7706 6539 y Fg(\003)7706 7351 y Fv(0)8317 7021 y Fq(.)555 b(If)335 b(there)i(is)f(some)h(p)37 b(oin)-37 b(t)336 b Fp(t)368 b Fm(2)h Fl(R)336 b Fq(suc)-37 b(h)337 b(that)532 b(_)-566 b Fp(\015)31314 7220 y Fv(2)31910 7021 y Fq(\()p Fp(t)p Fq(\))82 b Fm(\000)g Fp(G)35665 6539 y Fv(+)36546 7021 y Fq(\()p Fp(\015)37751 7220 y Fv(2)38347 7021 y Fq(\()p Fp(t)p Fq(\)\))367 b Fp(>)i(\016)50 b Fq(,)358 b(then)336 b Fp(d\015)47812 7220 y Fv(2)48408 7021 y Fq(\()p Fp(t)p Fq(\))1102 8615 y(shall)536 b(encircle)i(around)f (the)e(cylinder)i(for)g(at)e(least)h(t)-37 b(wice,)560 b(it)536 b(has)g(b)37 b(een)537 b(sho)-37 b(wn)536 b(imp)37 b(ossible.)1102 10209 y(Th)-37 b(us)561 b(w)-37 b(e)561 b(ha)-37 b(v)g(e)757 b(_)-565 b Fp(\015)10526 10408 y Fv(2)11122 10209 y Fq(\()p Fp(t)12119 10408 y Fv(0)12714 10209 y Fq(\))373 b Fm(\000)g Fp(G)16054 9727 y Fv(+)16937 10209 y Fq(\()p Fp(\015)18142 10408 y Fv(2)18737 10209 y Fq(\()p Fp(t)19734 10408 y Fv(0)20329 10209 y Fq(\)\))564 b Fp(<)h(\016)50 b Fq(.)943 b(In)560 b(this)f(case,)591 b Fp(d\015)34669 10408 y Fv(2)35265 10209 y Fq(\()p Fp(t)p Fq(\))559 b(k)-37 b(eeps)561 b(close)g(to)f(\000)46874 9727 y Fg(\006)48331 10209 y Fq(for)1102 11803 y Fp(t)1582 12002 y Fv(0)2625 11803 y Fm(\024)446 b Fp(t)h Fm(\024)g Fp(t)6991 12002 y Fv(0)7912 11803 y Fq(+)327 b Fp(T)10048 12002 y Fv(+)10930 11803 y Fq(\()p Fp(a)12149 12002 y Fv(1)12745 11803 y Fq(\()19 b(\026)-683 b Fp(a)13964 12002 y Fv(2)14559 11803 y Fq(\()p Fp(a)15778 11321 y Fg(\003)15778 12133 y Fv(0)16389 11803 y Fq(\)\))p Fp(;)239 b Fq(\026)-682 b Fp(a)18715 12002 y Fv(2)19310 11803 y Fq(\()p Fp(a)20529 11321 y Fg(\003)20529 12133 y Fv(0)21140 11803 y Fq(\)\),)500 b(or)490 b(for)f Fp(t)27276 12002 y Fv(0)28198 11803 y Fm(\000)327 b Fp(T)30334 12002 y Fv(+)31216 11803 y Fq(\()p Fp(a)32435 12002 y Fv(1)33030 11803 y Fq(\()19 b(\026)-683 b Fp(a)34249 12002 y Fv(2)34845 11803 y Fq(\()p Fp(a)36064 11321 y Fg(\003)36064 12133 y Fv(0)36674 11803 y Fq(\)\))p Fp(;)240 b Fq(\026)-683 b Fp(a)39000 12002 y Fv(2)39596 11803 y Fq(\()p Fp(a)40815 11321 y Fg(\003)40815 12133 y Fv(0)41425 11803 y Fq(\)\))446 b Fm(\024)h Fp(t)f Fm(\024)h Fp(t)47271 12002 y Fv(0)47867 11803 y Fq(,)501 b(or)1102 13397 y(for)469 b Fp(t)3641 13596 y Fv(0)4649 13397 y Fm(\024)411 b Fp(t)h Fm(\024)g Fp(t)8910 13596 y Fv(0)9818 13397 y Fq(+)312 b Fp(T)11939 13596 y Fg(\000)12836 13397 y Fq(\()19 b(\026)-683 b Fp(a)14055 13596 y Fv(1)14651 13397 y Fq(\()p Fp(a)15870 12915 y Fg(\003)15870 13727 y Fv(0)16480 13397 y Fq(\))p Fp(;)221 b(a)18289 13596 y Fv(2)18885 13397 y Fq(\()19 b(\026)-683 b Fp(a)20104 13596 y Fv(1)20700 13397 y Fq(\()p Fp(a)21919 12915 y Fg(\003)21919 13727 y Fv(0)22529 13397 y Fq(\)\)\),)474 b(or)469 b(for)f Fp(t)29114 13596 y Fv(0)30022 13397 y Fm(\000)313 b Fp(T)32144 13596 y Fg(\000)33041 13397 y Fq(\()19 b(\026)-683 b Fp(a)34260 13596 y Fv(1)34855 13397 y Fq(\()p Fp(a)36074 12915 y Fg(\003)36074 13727 y Fv(0)36685 13397 y Fq(\))p Fp(;)221 b(a)38494 13596 y Fv(2)39090 13397 y Fq(\()19 b(\026)-683 b Fp(a)40309 13596 y Fv(1)40904 13397 y Fq(\()p Fp(a)42123 12915 y Fg(\003)42123 13727 y Fv(0)42734 13397 y Fq(\)\)\))410 b Fm(\024)i Fp(t)g Fm(\024)g Fp(t)48957 13596 y Fv(0)49552 13397 y Fq(.)1102 14991 y(In)364 b(eac)-37 b(h)364 b(case,)381 b(w)-37 b(e)364 b(can)g(construct)g(a)g(curv)-37 b(e)365 b(as)f(w)-37 b(e)364 b(did)g(in)g(\(3.13\))f(and)h(\014nd)f Fp(\015)438 b Fq(is)363 b(not)h Fp(c)p Fq(-minimal,)1102 16585 y(but)488 b(this)f(is)h(absurd.)726 b(Therefore,)501 b(all)488 b Fp(c)p Fq(-minimal)f(orbits)h(should)g(b)37 b(e)488 b(in)g Fp(a)39410 16103 y Fg(\003)40021 16585 y Fq(-neigh)-37 b(b)37 b(orho)g(o)g(d)488 b(of)1250 17863 y(~)1102 18199 y(\006)q(.)590 b(Since)6740 17863 y(~)6490 18199 y Fm(G)79 b Fq(\()p Fp(c)p Fq(\))441 b(is)i(in)-37 b(v)-74 b(arian)-37 b(t,)443 b(b)-37 b(y)443 b(the)g(normal)g(h)-37 b(yp)37 b(erb)g(olicit)-37 b(y)-111 b(,)444 b(it)f(is)f(con)-37 b(tained)444 b(in)44009 17863 y(~)43862 18199 y(\006.)2165 b Fb(\003)2696 20570 y Fq(Although)502 b(the)f(structure)g(of)g (minimal)g(measures)h(is)f(unclear)i(in)e(general)h(case,)517 b(w)-37 b(e)502 b(kno)-37 b(w)1102 22184 y(v)g(ery)643 b(w)-37 b(ell)642 b(the)f(structures)g(of)h(those)22053 21848 y(~)21403 22184 y Fm(M)p Fq(\()p Fp(c)p Fq(\))699 b Fm(\032)27186 21848 y Fq(~)27039 22184 y(\006)641 b(since)h(the)f (time-1-map)g(\010)h(is)f(an)g(area-)1102 23778 y(preserving)426 b(t)-37 b(wist)423 b(map)h(when)h(it)f(is)f(restricted)i(on)f(\006.)585 b(Under)425 b(the)e(pro)74 b(jection)425 b(from)f Fp(T)184 b(M)403 b Fm(\002)258 b Fl(T)1102 25372 y Fq(to)511 b Fp(T)184 b(M)486 b Fm(\002)341 b(f)p Fp(t)483 b Fq(=)f(0)p Fm(g)p Fq(,)529 b(the)511 b(supp)37 b(ort)511 b(of)g(those)g Fp(c)p Fq(-minimal)f(measures)j(are)e(the)g(image)h(of)f(those)1102 26966 y(An)-37 b(bry-Mather)550 b(sets)f(suc)-37 b(h)549 b(as)h(homotopically)f(non-trivial)g(in)-37 b(v)-74 b(arian)-37 b(t)550 b(curv)-37 b(es,)577 b(Denjo)-37 b(y)549 b(sets)1102 28560 y(or)564 b(minimal)f(p)37 b(erio)g(dic)564 b(orbits)f(on)h(\006)f (under)h(the)f(Legendre)i(transformation)e Fm(L)p Fq(.)952 b(W)-111 b(e)564 b(use)f(\000)1102 30154 y(to)548 b(denote)f(those)h (Aubry-Mather)g(sets)f(on)g(\006)h(in)g(the)f(Hamitonian)g(formalism,) 574 b(let)548 b(\000\()p Fp(t)p Fq(\))542 b(=)1102 31842 y(\010)2061 31360 y Fk(t)2061 32208 y(H)3069 31842 y Fq(\(\000\))369 b Fm(\032)g Fq(\006\()p Fp(t)p Fq(\),)10071 31506 y(~)9988 31842 y(\000)g(=)g Fm([)13475 32041 y Fk(t)p Fg(2)p Fc(T)15278 31842 y Fq(\()p Fm(L)p Fq(\(\000\()p Fp(t)p Fq(\)\))p Fp(;)221 b(t)p Fq(\).)2696 34213 y(Before)337 b(going)g(on)f(the)g(study)g(of)g(some)g Fp(c)p Fq(-minimal)g (measures,)358 b(let)336 b(us)g(note)g(a)g(fact)g(as)h(follo)-37 b(ws:)1102 36584 y Fr(Prop)42 b(osition)610 b(3.2.)764 b Fo(L)-68 b(et)557 b Fp(c)15551 36101 y Fg(0)15923 36584 y Fp(;)221 b(c)17088 36101 y Fg(\003)18214 36584 y Fm(2)515 b Fp(H)20827 36101 y Fv(1)21423 36584 y Fq(\()p Fp(M)71 b(;)221 b Fl(R)p Fq(\))p Fo(,)576 b Fp(\026)27149 36101 y Fg(0)28078 36584 y Fo(and)556 b Fp(\026)31539 36101 y Fg(\003)32706 36584 y Fo(b)-68 b(e)556 b(the)g(c)-68 b(orr)g(esp)g(onding)557 b(minimal)1102 38178 y(me)-68 b(asur)g(es)476 b(r)-68 b(esp)g(e)g(ctively.)614 b(If)475 b Fm(h)p Fp(c)16760 37695 y Fg(0)17428 38178 y Fm(\000)295 b Fp(c)19331 37695 y Fg(\003)19942 38178 y Fp(;)221 b(\032)p Fq(\()p Fp(\026)22536 37695 y Fg(0)22908 38178 y Fq(\))p Fm(i)368 b Fq(=)i Fm(h)p Fp(c)26805 37695 y Fg(0)27472 38178 y Fm(\000)295 b Fp(c)29375 37695 y Fg(\003)29986 38178 y Fp(;)221 b(\032)p Fq(\()p Fp(\026)32580 37695 y Fg(\003)33191 38178 y Fq(\))p Fm(i)368 b Fq(=)h(0)p Fo(,)476 b(then)f Fp(\013)5 b Fq(\()p Fp(c)42442 37695 y Fg(0)42814 38178 y Fq(\))368 b(=)i Fp(\013)5 b Fq(\()p Fp(c)47049 37695 y Fg(\003)47658 38178 y Fq(\))p Fo(.)1102 40548 y Fq(Pro)37 b(of.)591 b(By)443 b(the)g(de\014nition)f(of)h(the)g Fp(\013)448 b Fq(function)442 b(w)-37 b(e)444 b(\014nd)f(that)14079 44250 y Fp(\013)5 b Fq(\()p Fp(c)16026 43702 y Fg(0)16398 44250 y Fq(\))369 b(=)632 b(inf)18538 45047 y Fk(\027)62 b Fg(2)p Ff(M)21198 42442 y Fi(Z)22527 44250 y Fq(\()p Fp(L)294 b Fm(\000)i Fp(\021)26231 44449 y Fk(c)26706 44184 y Fe(0)27131 44250 y Fq(\))p Fp(d\027)17284 47734 y Fq(=)18538 45926 y Fi(Z)19866 47734 y Fq(\()p Fp(L)f Fm(\000)h Fp(\021)23571 47933 y Fk(c)24046 47668 y Fe(0)24471 47734 y Fq(\))p Fp(d\026)26479 47186 y Fg(0)17284 51218 y Fq(=)18538 49410 y Fi(Z)19866 51218 y Fq(\()p Fp(L)f Fm(\000)h Fp(\021)23571 51417 y Fk(c)24046 51152 y Fe(\003)24667 51218 y Fq(\))p Fp(d\026)26675 50670 y Fg(0)27343 51218 y Fq(+)f Fm(h)p Fp(c)29763 50670 y Fg(\003)30669 51218 y Fm(\000)g Fp(c)32572 50670 y Fg(0)32944 51218 y Fp(;)221 b(\032)p Fq(\()p Fp(\026)35538 50670 y Fg(0)35911 51218 y Fq(\))p Fm(i)17284 53890 y(\025)p Fp(\013)5 b Fq(\()p Fp(c)20264 53342 y Fg(\003)20874 53890 y Fq(\))p Fp(:)1102 57056 y Fq(In)443 b(the)g(same)g(w)-37 b(a)g(y)443 b(w)-37 b(e)444 b(\014nd)f(that)f Fp(\013)5 b Fq(\()p Fp(c)20510 56574 y Fg(\003)21120 57056 y Fq(\))368 b Fm(\025)h Fp(\013)5 b Fq(\()p Fp(c)25354 56574 y Fg(0)25726 57056 y Fq(\).)2165 b Fb(\003)1102 59520 y Fr(Lemma)601 b(3.3.)764 b Fo(Assume)14867 59184 y Fq(~)14784 59520 y(\000)502 b Fm(2)18154 59184 y Fq(~)17504 59520 y Fm(M)p Fq(\()29 b(\026)-693 b Fp(c)o Fq(\))549 b Fo(for)g(some)578 b Fq(\026)-693 b Fp(c)502 b Fq(=)g(\()29 b(\026)-693 b Fp(c)30623 59719 y Fv(1)31218 59520 y Fp(;)251 b Fq(\026)-694 b Fp(c)32383 59719 y Fv(2)32980 59520 y Fq(\))501 b Fm(2)h Fq([)p Fp(A)36751 59038 y Fg(\003)37362 59520 y Fp(;)221 b(B)39027 59038 y Fg(\003)39637 59520 y Fq(])349 b Fm(\002)f Fq([)p Fm(\000)p Fp(c)43713 59038 y Fg(\003)43713 59850 y Fv(2)44324 59520 y Fp(;)221 b(c)45489 59038 y Fg(\003)45489 59850 y Fv(2)46100 59520 y Fq(])p Fo(,)567 b(then)1102 61114 y(ther)-68 b(e)521 b(is)g(an)f(interval)h Fp(I)554 b Fq(=)451 b Fp(I)104 b Fq(\()29 b(\026)-693 b Fp(c)17192 61313 y Fv(1)17787 61114 y Fq(\))450 b(=)g Fm(f)p Fq(\()p Fp(c)21993 61313 y Fv(1)22589 61114 y Fp(;)221 b(c)23754 61313 y Fv(2)24350 61114 y Fq(\))450 b Fm(2)g Fp(H)27865 60632 y Fv(1)28461 61114 y Fq(\()p Fp(M)71 b(;)221 b Fl(R)p Fq(\))450 b(:)g Fp(c)34248 61313 y Fv(1)35294 61114 y Fq(=)480 b(\026)-693 b Fp(c)37353 61313 y Fv(1)37948 61114 y Fp(;)221 b(a)p Fq(\()p Fp(c)40332 61313 y Fv(1)40928 61114 y Fq(\))450 b Fm(\024)h Fp(c)43954 61313 y Fv(2)45000 61114 y Fm(\024)f Fp(b)p Fq(\()p Fp(c)48145 61313 y Fv(1)48740 61114 y Fq(\))p Fm(g)1102 62802 y Fo(with)634 b Fm(\0001)655 b Fp(<)f(a)p Fq(\()p Fp(c)10643 63001 y Fv(1)11239 62802 y Fq(\))g Fp(<)h Fq(0)g Fp(<)g(b)p Fq(\()p Fp(c)18767 63001 y Fv(1)19362 62802 y Fq(\))f Fp(<)h Fm(1)p Fo(,)673 b(such)633 b(that)31332 62466 y Fq(~)30682 62802 y Fm(M)p Fq(\()p Fp(c)p Fq(\))653 b(=)36310 62466 y(~)36227 62802 y(\000)633 b Fo(for)h(al)68 b(l)633 b Fp(c)655 b Fm(2)f Fq(In)-37 b(t)o Fp(I)738 b Fo(and)1752 64154 y Fq(~)1102 64490 y Fm(M)q Fq(\()p Fp(c)p Fq(\))527 b Fm(\023)6478 64154 y Fq(~)6395 64490 y(\000)563 b Fo(for)h Fp(c)528 b Fm(2)f Fp(@)74 b(I)104 b Fo(;)608 b(if)564 b(ther)-68 b(e)563 b(is)h(an)f(invariant)h(curve)f(c)-68 b(ontaining)563 b Fq(\000)h Fo(we)e(have)h(further)1752 65842 y Fq(~)1102 66178 y Fm(M)q Fq(\()p Fp(c)p Fq(\))368 b(=)6160 65842 y(~)6077 66178 y(\000)475 b Fo(for)h(al)68 b(l)476 b Fp(c)369 b Fm(2)f Fp(I)104 b Fo(.)1102 68549 y Fr(Pro)42 b(of)p Fq(:)620 b(Let)563 b(\026)-769 b Fp(\026)457 b Fq(b)37 b(e)458 b(a)487 b(\026)-693 b Fp(c)p Fq(-minimal)457 b(measure.)635 b(W)-111 b(e)458 b(ha)-37 b(v)g(e)459 b(sho)-37 b(wn)458 b(in)f(the)g(lemma)h(3.1)g(that)f(there)1102 70163 y(can)403 b(not)e(b)37 b(e)402 b(an)-37 b(y)402 b(supp)37 b(ort)402 b(of)507 b(\026)-769 b Fp(\026)402 b Fq(outside)g(of)23765 69827 y(~)23618 70163 y(\006.)577 b(F)-111 b(rom)403 b(the)e(w)-37 b(ell)403 b(kno)-37 b(wn)402 b(theory)h(for)f(t)-37 b(wist)401 b(map)1102 71777 y(it)439 b(follo)-37 b(ws)440 b(that)544 b(\026)-769 b Fp(\026)439 b Fq(is)h(uniquely)f(ergo)37 b(dic,)441 b(i.e.)590 b(supp\()105 b(\026)-769 b Fp(\026)p Fq(\))368 b(=)31310 71441 y(~)31227 71777 y(\000,)440 b(since)g(there)f(is)h(a)f (dense)h(orbit)f(in)1102 73371 y(the)400 b(Aubry-Mather)f(set)g(b)37 b(ecause)401 b(w)-37 b(e)400 b(ha)-37 b(v)g(e)400 b(assumed)f(that)g (there)h(is)f(only)g(one)h(minimal)f(orbit)1102 74965 y(when)375 b(the)f(rotation)f(n)-37 b(um)g(b)37 b(er)375 b(is)f(rational.)567 b(Let)374 b Fp(\036)26326 74483 y Fk(t)26795 74965 y Fq(\()p Fp(z)58 b(;)221 b(\022)37 b Fq(\))368 b Fm(2)g Fp(T)184 b(M)303 b Fm(\002)157 b Fl(T)374 b Fq(b)37 b(e)374 b(the)f(Lagrangian)j(\015o)-37 b(w,)p eop %%Page: 20 20 20 19 bop 1102 2239 a Ft(20)1102 5427 y Fp(z)1720 5626 y Fk(t)2612 5427 y Fq(b)37 b(e)424 b(the)g Fp(T)184 b(M)569 b Fq(comp)37 b(onen)-37 b(t,)524 b(^)-759 b Fp(\021)416 b Fq(=)369 b Fp(dq)20394 5626 y Fv(2)20990 5427 y Fq(.)585 b(F)-111 b(or)424 b(an)-37 b(y)425 b(in)-37 b(v)-74 b(arian)-37 b(t)424 b(measure)i Fp(\026)p Fq(,)i(if)c(supp\()p Fp(\026)p Fq(\))368 b Fm(\032)46800 5091 y Fq(~)46652 5427 y(\006)q(,)427 b(w)-37 b(e)1102 7021 y(ha)g(v)g(e)15759 8896 y Fi(Z)17404 10704 y Fq(^)-759 b Fp(\021)47 b(d\026)370 b Fq(=)21218 9805 y(1)p 21070 10398 961 54 v 21070 11615 a Fp(T)22411 8896 y Fi(Z)23739 9224 y Fk(T)23149 11914 y Fv(0)24798 10704 y Fp(ds)26333 8896 y Fi(Z)27661 10704 y Fq(\()95 b(^)-759 b Fp(\021)342 b Fm(\016)296 b Fp(\036)30931 10156 y Fk(s)31499 10704 y Fq(\))p Fp(d\026)19877 14561 y Fq(=)21218 13663 y(1)p 21070 14256 V 21070 15472 a Fp(T)22411 12753 y Fi(Z)23739 13082 y Fk(T)23149 15772 y Fv(0)24798 14561 y Fp(ds)26333 12753 y Fi(Z)27661 14561 y Fm(h)95 b Fq(^)-759 b Fp(\021)47 b(;)221 b(z)30093 14760 y Fk(s)30661 14561 y Fm(i)p Fp(d\026)p Fq(\()p Fp(z)58 b Fq(\))19877 18419 y Fm(\024)21218 17520 y Fq(1)p 21070 18113 V 21070 19330 a Fp(T)22411 16610 y Fi(Z)23961 16891 y(\014)23961 17688 y(\014)23961 18485 y(\014)24625 16610 y(Z)25953 16939 y Fk(T)25363 19629 y Fv(0)26790 18419 y Fm(h)95 b Fq(^)-759 b Fp(\021)47 b(;)221 b(z)29222 18618 y Fk(s)29790 18419 y Fm(i)p Fp(ds)31621 16891 y Fi(\014)31621 17688 y(\014)31621 18485 y(\014)32063 18419 y Fp(d\026)p Fq(\()p Fp(z)58 b Fq(\))19877 21915 y Fm(\024)21070 21017 y Fq(2)p Fp(\031)p 21070 21610 1470 54 v 21324 22827 a(T)23067 21915 y Fm(!)369 b Fq(0)1102 25465 y(as)488 b Fp(T)627 b Fm(!)444 b(1)p Fq(.)724 b(Since)11887 24395 y Fi(R)13089 25465 y Fq(^)-759 b Fp(\021)47 b(d\026)488 b Fq(is)f(indep)37 b(enden)-37 b(t)488 b(of)f Fp(T)184 b Fq(,)28017 24395 y Fi(R)29219 25465 y Fq(^)-759 b Fp(\021)47 b(d\026)444 b Fq(=)f(0.)724 b(Therefore,)500 b(it)487 b(follo)-37 b(ws)488 b(from)1102 27226 y(the)571 b(prop)37 b(osition)570 b(3.2)h(that)f Fp(\013)5 b Fq(\()29 b(\026)-693 b Fp(c)o Fq(\))581 b(=)h Fp(\013)5 b Fq(\()29 b(^)-693 b Fp(c)o Fq(\))570 b(if)h(b)37 b(oth)599 b(\026)-693 b Fp(c)p Fq(-)570 b(and)600 b(^)-694 b Fp(c)p Fq(-minimal)570 b(measures)i(are)f(on)49109 26891 y(~)48962 27226 y(\006)1102 28914 y(with)542 b(\026)-693 b Fp(c)342 b Fm(\000)372 b Fq(^)-694 b Fp(c)486 b Fq(=)g(\(0)p Fp(;)221 b(c)11416 29113 y Fv(2)12012 28914 y Fq(\).)801 b(As)513 b(the)g Fp(\014)584 b Fq(function)513 b(for)g(a)g(t)-37 b(wist)513 b(map)g(is)g(strictly)g(con)-37 b(v)g(ex,)45849 28578 y(~)45199 28914 y Fm(M)p Fq(\()29 b(\026)-693 b Fp(c)o Fq(\))486 b(=)1752 30266 y(~)1102 30602 y Fm(M)q Fq(\()29 b(^)-693 b Fp(c)o Fq(\).)652 b(As)464 b(the)f Fp(\013)468 b Fq(function)c(is)f(con)-37 b(v)g(ex)465 b(and)f(has)f(sup)37 b(er-linear)465 b(gro)-37 b(wth,)469 b Fp(I)568 b Fq(is)463 b(connected)i(and)1102 32196 y Fm(\0001)580 b Fp(<)f(a)h(<)f Fq(0)g Fp(<)h(b)f(<)g Fm(1)p Fq(.)969 b(What)569 b(remains)h(to)e(sho) -37 b(w)570 b(is)f Fp(I)673 b Fq(is)569 b(closed.)969 b(If)569 b(not,)600 b(there)569 b(is)g(a)1102 33810 y(sequence)519 b Fp(c)7257 34009 y Fk(k)8403 33810 y Fm(!)494 b Fp(c)517 b Fq(suc)-37 b(h)517 b(that)17423 33474 y(~)17340 33810 y(\000)493 b Fm(\032)20839 33474 y Fq(~)20189 33810 y Fm(M)p Fq(\()p Fp(c)22876 34009 y Fk(k)23529 33810 y Fq(\))517 b(and)27303 33474 y(~)27220 33810 y(\000)494 b Fl(*)30720 33474 y Fq(~)30070 33810 y Fm(M)p Fq(\()p Fp(c)p Fq(\),)535 b(so)517 b(there)h(exists)f Fp(\026)g Fq(suc)-37 b(h)518 b(that)1102 35568 y Fp(A)2098 35767 y Fk(c)2640 35568 y Fq(\()p Fp(\026)4021 35736 y Fv(~)3957 35971 y(\000)4680 35568 y Fq(\))380 b Fp(>)h(A)7987 35767 y Fk(c)8529 35568 y Fq(\()p Fp(\026)p Fq(\),)451 b(where)g Fp(\026)15895 35736 y Fv(~)15831 35971 y(\000)17005 35568 y Fq(is)e(the)h(in)-37 b(v)-74 b(arian)-37 b(t)451 b(measure)g(on)33344 35232 y(~)33261 35568 y(\000.)612 b(Let)450 b Fp(k)491 b Fq(b)37 b(e)450 b(su\016cien)-37 b(tly)450 b(large)1102 37162 y(so)443 b(that)f Fp(c)6186 37361 y Fk(k)7283 37162 y Fq(is)g(su\016cien)-37 b(tly)443 b(close)h(to)f Fp(c)p Fq(,)6894 40796 y Fp(A)7890 40995 y Fk(c)8365 41136 y Fd(k)9013 40796 y Fq(\()p Fp(\026)p Fq(\))368 b(=)12617 38988 y Fi(Z)14167 40796 y Fp(Ld\026)296 b Fm(\000)f(h)p Fp(\032)p Fq(\()p Fp(\026)p Fq(\))p Fp(;)221 b(c)22389 40995 y Fk(k)23042 40796 y Fm(i)369 b Fq(=)g Fp(A)26326 40995 y Fk(c)26867 40796 y Fq(\()p Fp(\026)p Fq(\))295 b Fm(\000)g(h)p Fp(\032)p Fq(\()p Fp(\026)p Fq(\))p Fp(;)221 b(c)34527 40995 y Fk(k)35475 40796 y Fm(\000)295 b Fp(c)p Fm(i)369 b Fp(<)g(A)40662 40995 y Fk(c)41203 40796 y Fq(\()p Fp(\026)42584 40964 y Fv(~)42520 41199 y(\000)43244 40796 y Fq(\))p Fp(:)1102 44504 y Fq(On)340 b(the)f(other)h(hand,)360 b(it)338 b(follo)-37 b(ws)340 b(from)g Fm(h)p Fp(c)88 b Fm(\000)g Fp(c)23768 44703 y Fk(k)24421 44504 y Fp(;)221 b(\032)p Fq(\()p Fp(\026)27079 44672 y Fv(~)27015 44907 y(\000)27738 44504 y Fq(\))p Fm(i)368 b Fq(=)i(0)339 b(that)f Fp(A)35316 44703 y Fk(c)35858 44504 y Fq(\()p Fp(\026)37239 44672 y Fv(~)37175 44907 y(\000)37898 44504 y Fq(\))369 b(=)g Fp(A)41182 44703 y Fk(c)41657 44844 y Fd(k)42304 44504 y Fq(\()p Fp(\026)43685 44672 y Fv(~)43621 44907 y(\000)44344 44504 y Fq(\),)360 b(th)-37 b(us)338 b(w)-37 b(e)1102 46098 y(ha)g(v)g(e)384 b Fp(A)5101 46297 y Fk(c)5576 46438 y Fd(k)6223 46098 y Fq(\()p Fp(\026)7604 46266 y Fv(~)7540 46501 y(\000)8263 46098 y Fq(\))369 b Fp(>)g(A)11547 46297 y Fk(c)12022 46438 y Fd(k)12669 46098 y Fq(\()p Fp(\026)p Fq(\),)395 b(but)382 b(it)h(con)-37 b(tradicts)383 b(to)g(the)g(fact)g(that)f Fp(\026)35722 46266 y Fv(~)35658 46501 y(\000)36765 46098 y Fq(is)h Fp(c)38616 46297 y Fk(k)39269 46098 y Fq(-minimal)g(measure.)2696 48454 y(If)593 b(there)i(is)e(another)h(measure)h Fp(\026)e Fq(whic)-37 b(h)595 b(can)f(also)g(minimize)g(the)f Fp(c)p Fq(-action)g(of)h Fp(L)g Fq(when)1102 50048 y Fp(a)p Fq(\()p Fp(c)2896 50247 y Fv(1)3492 50048 y Fq(\))368 b Fp(<)i(c)6355 50247 y Fv(2)7319 50048 y Fp(<)f(b)p Fq(\()p Fp(c)10383 50247 y Fv(1)10979 50048 y Fq(\),)442 b(then)h Fm(h)p Fp(dq)17134 50247 y Fv(2)17730 50048 y Fp(;)221 b(\026)p Fm(i)369 b Fq(=)g(0.)591 b(Indeed,)443 b(for)g(all)g Fp(a)p Fq(\()p Fp(c)33390 50247 y Fv(1)33986 50048 y Fq(\))368 b Fp(<)h(c)36848 49566 y Fg(0)36848 50378 y Fv(2)37813 50048 y Fp(<)g(b)p Fq(\()p Fp(c)40877 50247 y Fv(1)41473 50048 y Fq(\))442 b(w)-37 b(e)444 b(ha)-37 b(v)g(e)5192 51948 y Fi(Z)6742 52282 y(\020)7535 53756 y Fp(L)296 b Fm(\000)f Fp(c)10638 53955 y Fv(1)11480 53756 y Fq(_)-615 b Fp(q)11827 53955 y Fv(1)12718 53756 y Fm(\000)295 b Fp(c)14621 53208 y Fg(0)14621 54085 y Fv(2)15463 53756 y Fq(_)-615 b Fp(q)15810 53955 y Fv(2)16406 52282 y Fi(\021)17199 53756 y Fp(d\026)18690 53955 y Fv(\000)19783 53756 y Fq(=)21038 51948 y Fi(Z)22587 52282 y(\020)23381 53756 y Fp(L)295 b Fm(\000)g Fp(c)26483 53955 y Fv(1)27326 53756 y Fq(_)-616 b Fp(q)27672 53955 y Fv(1)28563 53756 y Fm(\000)296 b Fp(c)30467 53955 y Fv(2)31309 53756 y Fq(_)-616 b Fp(q)31655 53955 y Fv(2)32251 52282 y Fi(\021)33045 53756 y Fp(d\026)34536 53955 y Fv(\000)19783 57240 y Fq(=)21038 55432 y Fi(Z)22587 55766 y(\020)23381 57240 y Fp(L)295 b Fm(\000)g Fp(c)26483 57439 y Fv(1)27326 57240 y Fq(_)-616 b Fp(q)27672 57439 y Fv(1)28563 57240 y Fm(\000)296 b Fp(c)30467 57439 y Fv(2)31309 57240 y Fq(_)-616 b Fp(q)31655 57439 y Fv(2)32251 55766 y Fi(\021)33045 57240 y Fp(d\026)1575 b Fq(\(b)-37 b(y)443 b(assumption)o(\))19783 60724 y(=)21038 58916 y Fi(Z)22587 59249 y(\020)23381 60724 y Fp(L)295 b Fm(\000)g Fp(c)26483 60923 y Fv(1)27326 60724 y Fq(_)-616 b Fp(q)27672 60923 y Fv(1)28563 60724 y Fm(\000)296 b Fp(c)30467 60175 y Fg(0)30467 61052 y Fv(2)31309 60724 y Fq(_)-616 b Fp(q)31655 60923 y Fv(2)32251 59249 y Fi(\021)33045 60724 y Fp(d\026)295 b Fq(+)h(\()p Fp(c)37252 60175 y Fg(0)37252 61052 y Fv(2)38142 60724 y Fm(\000)g Fp(c)40046 60923 y Fv(2)40641 60724 y Fq(\))p Fm(h)p Fp(dq)42959 60923 y Fv(2)43555 60724 y Fp(;)221 b(\026)p Fm(i)p Fp(;)1102 64434 y Fq(th)-37 b(us)460 b(w)-37 b(e)460 b(can)g(c)-37 b(ho)37 b(ose)461 b Fp(c)13272 63952 y Fg(0)13272 64764 y Fv(2)14327 64434 y Fq(in)f(the)f(w)-37 b(a)g(y)460 b(that)f(\()p Fp(c)24895 63952 y Fg(0)24895 64764 y Fv(2)25797 64434 y Fm(\000)307 b Fp(c)27712 64633 y Fv(2)28307 64434 y Fq(\))p Fm(h)p Fp(dq)30625 64633 y Fv(2)31221 64434 y Fp(;)221 b(\026)p Fm(i)397 b Fp(>)g Fq(0)460 b(if)f Fm(h)p Fp(dq)39114 64633 y Fv(2)39710 64434 y Fp(;)221 b(\026)p Fm(i)398 b(6)p Fq(=)f(0.)641 b(But)459 b(this)1102 66028 y(con)-37 b(tradicts)444 b(to)e(the)h(minimalit)-37 b(y)443 b(of)f Fp(\026)20879 66227 y Fv(\000)21603 66028 y Fq(.)590 b(Consequen)-37 b(tly)-111 b(,)444 b(w)-37 b(e)444 b(alw)-37 b(a)g(ys)444 b(ha)-37 b(v)g(e)15340 67855 y Fi(Z)16668 69663 y Fq(\()p Fp(L)295 b Fm(\000)g Fp(c)20287 69862 y Fv(1)21129 69663 y Fq(_)-615 b Fp(q)21476 69862 y Fv(1)22072 69663 y Fq(\))p Fp(d\026)369 b Fq(=)25851 67855 y Fi(Z)27180 69663 y Fq(\()p Fp(L)294 b Fm(\000)i Fp(c)30799 69862 y Fv(1)31641 69663 y Fq(_)-615 b Fp(q)31988 69862 y Fv(1)32583 69663 y Fq(\))p Fp(d\026)34655 69831 y Fv(~)34591 70066 y(\000)35315 69663 y Fp(;)1102 73371 y Fq(whic)-37 b(h)513 b(is)f(indep)37 b(enden)-37 b(t)513 b(of)f(the)g(v)-74 b(alue)513 b Fp(c)21989 73570 y Fv(2)23097 73371 y Fq(tak)-37 b(es)513 b(b)37 b(et)-37 b(w)g(een)513 b Fp(a)p Fq(\()p Fp(c)33561 73570 y Fv(1)34156 73371 y Fq(\))f(and)g Fp(b)p Fq(\()p Fp(c)39499 73570 y Fv(1)40095 73371 y Fq(\),)529 b(it)512 b(implies)g(that)1102 74965 y Fp(\026)370 b Fq(=)f Fp(\026)4538 75133 y Fv(~)4474 75368 y(\000)5640 74965 y Fq(as)443 b Fp(\026)8135 75133 y Fv(~)8071 75368 y(\000)9238 74965 y Fq(is)f(the)h(only)g Fp(c)p Fq(-minimal)g(measure)g (when)h Fm(j)p Fp(c)31549 75164 y Fv(2)32145 74965 y Fm(j)369 b(\024)g Fp(c)34860 74483 y Fg(\003)34860 75294 y Fv(2)35470 74965 y Fq(.)p eop %%Page: 21 21 21 20 bop 48792 2239 a Ft(21)2696 5427 y Fq(Let)390 b(us)f(consider)i (the)e(case)h(when)g(\000)g(is)f(con)-37 b(tained)391 b(in)e(an)g(in)-37 b(v)-74 b(arian)-37 b(t)390 b(curv)-37 b(e)391 b(and)f Fp(c)44076 5626 y Fv(2)45040 5427 y Fm(2)369 b Fp(@)74 b(I)104 b Fq(.)573 b(In)1102 7021 y(this)405 b(case,)413 b(the)405 b(Aubry)g(set)15646 6685 y(~)15189 7021 y Fm(A)p Fq(\()p Fp(c)p Fq(\))f(con)-37 b(tains)405 b(a)g(co-dimensional)h(one)f(torus.)578 b(As)405 b(an)-37 b(y)406 b Fp(c)p Fq(-minimal)1102 8709 y(curv)-37 b(e)538 b Fp(\015)599 b Fm(\032)525 b(A)p Fq(\()p Fp(c)p Fq(\))536 b(can)h(not)f(cross)h(the)g(2-torus)f Fp(\031)27012 8373 y Fq(~)26929 8709 y(\000)526 b Fm(\032)f Fp(T)30803 8227 y Fv(2)31758 8709 y Fm(\002)358 b Fl(T)p Fq(,)559 b(where)538 b Fp(\031)585 b Fq(is)536 b(the)h(pro)74 b(jection)1102 10397 y Fp(T)184 b(M)360 b Fm(\002)214 b Fl(T)368 b Fm(!)i Fp(M)358 b Fm(\002)214 b Fl(T)p Fq(,)410 b(b)37 b(ecause)403 b(the)f(in)-37 b(v)g(erse)404 b(map)e(to)g(the)g(Aubry)h(sets)f(is)g (of)g(Lipsc)-37 b(hitz)403 b(and)f Fp(\031)49174 10061 y Fq(~)49091 10397 y(\000)1102 11991 y(is)443 b(a)g(co)37 b(dimension)444 b(1)f(torus.)590 b(Th)-37 b(us)443 b(there)h(exist)e Fp(\016)420 b(>)369 b Fq(0)443 b(suc)-37 b(h)443 b(that)f(for)h(an)-37 b(y)444 b Fp(T)553 b(>)369 b Fq(0)17113 16316 y Fm(\000)p Fp(\016)420 b Fm(\024)20558 13992 y Fi(\014)20558 14789 y(\014)20558 15586 y(\014)20558 16383 y(\014)20558 17180 y(\014)21001 14508 y(Z)22329 14837 y Fk(T)21739 17527 y Fg(\000)p Fk(T)23823 16134 y Fq(_)23676 16316 y(\026)-713 b Fp(\015)24389 16641 y Fv(2)24984 16316 y Fq(\()p Fp(t)p Fq(\))p Fp(dt)27669 13992 y Fi(\014)27669 14789 y(\014)27669 15586 y(\014)27669 16383 y(\014)27669 17180 y(\014)28480 16316 y Fm(\024)369 b Fq(2)p Fp(\031)344 b Fq(+)295 b Fp(\016)-24 b(:)1102 20718 y Fq(So,)443 b(if)g Fp(\026)g Fq(is)g(also)g(a)g Fp(c)p Fq(-minimal)f(measure)i(and)f Fp(c)25402 20236 y Fg(0)26144 20718 y Fq(=)369 b(\()p Fp(c)28638 20917 y Fv(1)29233 20718 y Fp(;)221 b Fq(0\),)443 b(then)11536 24601 y Fp(A)12532 24800 y Fk(c)13007 24535 y Fe(0)13433 24601 y Fq(\()p Fp(\026)14814 24769 y Fv(~)14750 25004 y(\000)15473 24601 y Fq(\))368 b(=)17613 22793 y Fi(Z)18941 24601 y Fq(\()p Fp(L)295 b Fm(\000)g Fp(c)22560 24800 y Fv(1)23403 24601 y Fq(_)-616 b Fp(q)23749 24800 y Fv(1)24345 24601 y Fq(\))p Fp(d\026)26417 24769 y Fv(~)26353 25004 y(\000)16358 28085 y Fq(=)17613 26277 y Fi(Z)18941 28085 y Fq(\()p Fp(L)295 b Fm(\000)g Fp(c)22560 28284 y Fv(1)23403 28085 y Fq(_)-616 b Fp(q)23749 28284 y Fv(1)24640 28085 y Fm(\000)296 b Fp(c)26544 28284 y Fv(2)27386 28085 y Fq(_)-616 b Fp(q)27732 28284 y Fv(2)28328 28085 y Fq(\))p Fp(d\026)886 b Fq(\(b)-37 b(y)443 b(condition\))16358 31569 y(=)17613 29761 y Fi(Z)18941 31569 y Fq(\()p Fp(L)295 b Fm(\000)g Fp(c)22560 31768 y Fv(1)23403 31569 y Fq(_)-616 b Fp(q)23749 31768 y Fv(1)24345 31569 y Fq(\))p Fp(d\026)886 b Fq(\(b)-37 b(y)885 b(\(3)p Fp(:)p Fq(13\)\))16358 34241 y(=)q Fp(A)18388 34440 y Fk(c)18863 34175 y Fe(0)19288 34241 y Fq(\()p Fp(\026)p Fq(\))1102 37445 y(it)443 b(implies)g(that)f (the)h(only)g(minimal)f(measure)i(is)f Fp(\026)27563 37613 y Fv(~)27499 37848 y(\000)28223 37445 y Fq(.)2165 b Fb(\003)2696 39875 y Fq(It)306 b(follo)-37 b(ws)307 b(from)g(the)f(lemma)h(3.3)g(w)-37 b(e)307 b(see)g(that)f(there)h(is)f (a)h(strip)f Fm(S)469 b Fq(=)369 b Fm(f)p Fq(\()p Fp(c)38936 40074 y Fv(1)39531 39875 y Fp(;)221 b(c)40696 40074 y Fv(2)41292 39875 y Fq(\))369 b Fm(2)g Fp(H)44645 39393 y Fv(1)45241 39875 y Fq(\()p Fp(M)71 b(;)221 b Fl(R)p Fq(\))368 b(:)1102 41469 y Fp(c)1677 41668 y Fv(1)2653 41469 y Fm(2)379 b Fl(R)p Fp(;)443 b(a)p Fq(\()p Fp(c)7483 41668 y Fv(1)8079 41469 y Fq(\))379 b Fm(\024)h Fp(c)10963 41668 y Fv(2)11938 41469 y Fm(\024)g Fp(b)p Fq(\()p Fp(c)15013 41668 y Fv(1)15608 41469 y Fq(\))p Fp(;)443 b(A)17933 40987 y Fg(\003)18923 41469 y Fp(<)380 b(c)20911 41668 y Fv(1)21886 41469 y Fp(<)g(B)24374 40987 y Fg(\003)24984 41469 y Fp(;)443 b Fm(\0001)380 b Fp(<)g(a)p Fq(\()p Fp(c)31744 41668 y Fv(1)32339 41469 y Fq(\))f Fp(<)h Fq(0)g Fp(<)g(b)p Fq(\()p Fp(c)38767 41668 y Fv(1)39362 41469 y Fq(\))f Fp(<)h Fm(1g)p Fq(,)451 b(suc)-37 b(h)450 b(that)1102 43157 y(if)515 b Fp(c)488 b Fm(2)f Fq(in)-37 b(t)p Fm(S)100 b Fq(,)532 b(the)514 b Fp(c)p Fq(-minimal)g(measure)i (is)e(on)25554 42821 y(~)25406 43157 y(\006)h(and)f(is)g(uniquely)h (ergo)37 b(dic.)806 b(If)514 b Fp(c)488 b Fm(2)g Fp(@)74 b Fm(S)614 b Fq(and)1102 44771 y(\000)418 b Fm(\032)4451 44435 y Fq(~)3801 44771 y Fm(M)q Fq(\()p Fp(c)p Fq(\))471 b(is)h(an)g(in)-37 b(v)-74 b(arian)-37 b(t)473 b(curv)-37 b(e,)480 b(the)472 b Fp(c)p Fq(-minimal)g(measure)h(is)f(also)h (uniquely)f(ergo)37 b(dic.)680 b(In)1102 46459 y(these)443 b(cases,)h(w)-37 b(e)444 b(ha)-37 b(v)g(e)13684 46123 y(~)13227 46459 y Fm(A)p Fq(\()p Fp(c)p Fq(\))368 b(=)18089 46123 y(~)17667 46459 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\).)2696 48866 y(In)485 b(the)g(follo)-37 b(wing,)497 b(w)-37 b(e)486 b(use)f Fp(I)104 b Fq(\()p Fp(c)18875 49065 y Fv(1)19471 48866 y Fq(\))439 b(=)g Fm(f)p Fp(c)h Fq(=)f(\()p Fp(c)26142 49065 y Fv(1)26738 48866 y Fp(;)221 b(c)27903 49065 y Fv(2)28499 48866 y Fq(\))439 b(:)g Fp(a)p Fq(\()p Fp(c)32057 49065 y Fv(1)32653 48866 y Fq(\))g Fm(\024)h Fp(c)35657 49065 y Fv(2)36692 48866 y Fm(\024)g Fp(b)p Fq(\()p Fp(c)39827 49065 y Fv(1)40422 48866 y Fq(\))p Fm(g)485 b Fq(to)g(denote)g(the)1102 50460 y(maximal)368 b(in)-37 b(terv)-74 b(al)369 b(in)e(the)h(follo)-37 b(wing)368 b(sense:)554 b(for)368 b(eac)-37 b(h)368 b Fp(c)29732 49978 y Fg(0)30474 50460 y Fq(=)h(\()p Fp(c)32968 50659 y Fv(1)33563 50460 y Fp(;)221 b(c)34728 49978 y Fg(0)34728 50789 y Fv(2)35324 50460 y Fq(\))368 b(with)f Fp(a)p Fq(\()p Fp(c)40953 50659 y Fv(1)41549 50460 y Fq(\))h Fp(<)h(c)44411 49978 y Fg(0)44411 50789 y Fv(2)45376 50460 y Fp(<)g(b)p Fq(\()p Fp(c)48440 50659 y Fv(1)49036 50460 y Fq(\),)1102 52147 y(the)537 b Fp(c)4059 51665 y Fg(0)4431 52147 y Fq(-minimal)f(measure)i(has)e(some)21371 51812 y(~)21288 52147 y(\000)525 b Fm(\032)24349 51812 y Fq(~)24201 52147 y(\006)537 b(as)f(its)g(supp)37 b(ort,)560 b(this)37538 51812 y(~)37455 52147 y(\000)537 b(is)f(not)g(con)-37 b(tained)537 b(in)1102 53742 y(the)450 b(supp)37 b(ort)449 b(of)g(an)-37 b(y)450 b Fp(c)12932 53259 y Fg(\003)13542 53742 y Fq(-minimal)g(measure)g(where)h Fp(c)28764 53259 y Fg(\003)29754 53742 y Fq(=)380 b(\()p Fp(c)32259 53941 y Fv(1)32855 53742 y Fp(;)221 b(c)34020 53259 y Fg(\003)34020 54071 y Fv(2)34631 53742 y Fq(\))449 b(with)g(either)h Fp(c)42978 53259 y Fg(\003)42978 54070 y Fk(c)43968 53742 y Fp(<)380 b(a)p Fq(\()p Fp(c)47175 53941 y Fv(1)47771 53742 y Fq(\))449 b(or)1102 55336 y Fp(c)1677 54853 y Fg(\003)1677 55665 y Fv(2)2657 55336 y Fp(>)369 b(b)p Fq(\()p Fp(c)5721 55535 y Fv(1)6316 55336 y Fq(\).)1102 57836 y Fr(Lemma)539 b(3.4.)765 b Fo(L)-68 b(et)12090 57501 y Fq(~)12007 57836 y(\000)413 b Fm(\032)14844 57501 y Fq(~)14696 57836 y(\006)500 b Fo(b)-68 b(e)500 b(the)g(supp)-68 b(ort)500 b(of)g(some)g(minimal)h(me)-68 b(asur)g(e)500 b(for)529 b Fq(\026)-693 b Fp(c)413 b Fm(2)f Fp(I)104 b Fq(\()p Fp(c)46402 58035 y Fv(1)46998 57836 y Fq(\))p Fo(,)506 b(we)1102 59524 y(assume)546 b(that)g(it)h(has)e(dense)h (orbit.)823 b(Then)24069 59188 y Fq(~)23647 59524 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\))495 b Fm(\032)28714 59188 y Fq(~)28567 59524 y(\006)546 b Fo(for)g(e)-68 b(ach)545 b Fp(c)496 b Fm(2)g Fp(intI)104 b Fq(\()p Fp(c)41289 59723 y Fv(1)41884 59524 y Fq(\))496 b(=)h Fm(f)p Fq(\()p Fp(c)46183 59723 y Fv(1)46778 59524 y Fp(;)221 b(c)47943 59723 y Fv(2)48539 59524 y Fq(\))496 b(:)1102 61118 y Fp(a)p Fq(\()p Fp(c)2896 61317 y Fv(1)3492 61118 y Fq(\))603 b Fp(<)g(c)6823 61317 y Fv(2)8023 61118 y Fp(<)g(b)p Fq(\()p Fp(c)11321 61317 y Fv(1)11917 61118 y Fq(\))p Fm(g)p Fo(.)999 b(If)605 b Fq(\000)g Fo(is)g(an)f(invariant)i(curve)f(or)f(a)h(Denjoy)f(set)i(c) -68 b(ontaine)g(d)605 b(in)f(an)1102 62712 y(invariant)484 b(curve,)g(and)f(if)g Fp(c)382 b Fm(2)g Fp(@)74 b(I)486 b Fq(=)382 b Fm(f)p Fq(\()p Fp(c)21991 62911 y Fv(1)22587 62712 y Fp(;)221 b(c)23752 62911 y Fv(2)24348 62712 y Fq(\))382 b(:)g Fp(c)26573 62911 y Fv(2)27551 62712 y Fq(=)g Fp(a)p Fq(\()p Fp(c)30760 62911 y Fv(1)31356 62712 y Fq(\))482 b Fa(or)h Fp(c)34597 62911 y Fv(2)35575 62712 y Fq(=)382 b Fp(b)p Fq(\()p Fp(c)38652 62911 y Fv(1)39248 62712 y Fq(\))p Fm(g)482 b Fo(we)g(have)h(further)1102 64400 y(that)4240 64064 y Fq(~)3818 64400 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\))474 b Fo(c)-68 b(onsists)476 b(of)13754 64064 y Fq(~)13671 64400 y(\000)f Fo(and)g(the)h Fp(c)p Fo(-minimal)h(orbits)f(homo)-68 b(clinic)476 b(to)38090 64064 y Fq(~)38007 64400 y(\000)p Fo(.)1102 66807 y Fq(Pro)37 b(of.)1040 b(Let)592 b(us)g(consider)i(a)e Fp(c)p Fq(-minimal)g(orbit)g Fp(d\015)666 b Fq(with)593 b Fp(c)617 b Fm(2)h Fp(intI)104 b Fq(\()p Fp(c)37837 67006 y Fv(1)38432 66807 y Fq(\))592 b(\()p Fp(c)618 b Fm(2)f Fp(I)104 b Fq(\()p Fp(c)44534 67006 y Fv(1)45130 66807 y Fq(\))592 b(if)g(\000)g(is)1102 68495 y(an)570 b(in)-37 b(v)-74 b(arian)-37 b(t)570 b(curv)-37 b(e\).)972 b(If)569 b(this)g(orbit)g(is)h(not)f(con)-37 b(tained)570 b(in)33704 68159 y(~)33054 68495 y Fm(M)p Fq(\()p Fp(c)p Fq(\))579 b(=)38533 68159 y(~)38450 68495 y(\000)q(,)601 b(then)569 b Fp(d\015)644 b Fq(is)569 b(semi-)1102 70183 y(asymptotic)445 b(to)9646 69847 y(~)9563 70183 y(\000)g(as)g Fp(t)371 b Fm(!)i(\0061)p Fq(.)596 b(W)-111 b(e)446 b(sa)-37 b(y)445 b(an)f(orbit)h(is)f(semi-asymptotic)h (to)f(an)h(in)-37 b(v)-74 b(arian)-37 b(t)445 b(set)1102 71777 y(\000)534 b(as)f Fp(t)519 b Fm(!)h(1)533 b Fq(if)g(ev)-37 b(ery)534 b(in)-37 b(v)-74 b(arian)-37 b(t)534 b(subset)f(of)g(its)g Fp(!)48 b Fq(-limit)531 b(set)i(that)g(is)g(minimal)g(in)g(Birkho\013) 1102 73371 y(sense)420 b(is)g(con)-37 b(tained)420 b(in)f(\000.)583 b(W)-111 b(e)420 b(use)g(the)f(argumen)-37 b(t)420 b(in)f([Bo])g(to)g (sho)-37 b(w)420 b(it.)582 b(Let)420 b Fp(N)564 b Fq(is)420 b(a)f(minimal)1102 74965 y(\(in)488 b(Birkho\013)h(sense\))f(in)-37 b(v)-74 b(arian)-37 b(t)489 b(subset)f(of)h(the)f Fp(!)48 b Fq(-limit)487 b(set)h(of)h Fp(d\015)74 b Fq(,)499 b(there)489 b(exists)f(a)h(sequence)p eop %%Page: 22 22 22 21 bop 1102 2239 a Ft(22)1102 5427 y Fp(t)1582 5626 y Fk(k)2663 5427 y Fm(!)427 b(1)478 b Fq(suc)-37 b(h)478 b(that)f(dist\()p Fp(d\015)74 b Fq(\()p Fp(t)17283 5626 y Fk(k)17935 5427 y Fq(\))p Fp(;)221 b(N)145 b Fq(\))427 b Fm(!)g Fq(0.)696 b(W)-111 b(e)478 b(claim)h(that)d(there)j(is)e(a)h (sequence)h Fp(T)45756 5626 y Fk(k)46837 5427 y Fm(!)428 b(1)1102 7021 y Fq(suc)-37 b(h)444 b(that)11955 8615 y(lim)221 b(sup)12636 9740 y Fk(k)22 b Fg(!1)16021 8615 y Fm(f)p Fq(dist\()p Fp(d\015)74 b Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(N)145 b Fq(\))368 b(:)590 b Fp(t)26443 8814 y Fk(k)27465 8615 y Fm(\024)369 b Fp(t)g Fm(\024)g Fp(t)31598 8814 y Fk(k)32547 8615 y Fq(+)295 b Fp(T)34651 8814 y Fk(k)35305 8615 y Fm(g)369 b(!)g Fq(0)p Fp(:)7459 b Fq(\(3.14\))1102 11383 y(If)508 b(not,)524 b(there)509 b(exist)f Fp(d)479 b(>)f Fq(0,)525 b Fp(T)662 b(>)479 b Fq(0)508 b(and)g(a)h(subsequence)g Fp(t)32213 11582 y Fk(j)33281 11383 y Fq(of)f(the)g(sequence)i Fp(t)43263 11582 y Fk(k)44424 11383 y Fq(suc)-37 b(h)509 b(that)1102 12977 y(dist\()p Fp(d\015)74 b Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(N)145 b Fq(\))592 b Fm(\025)h Fp(d)578 b Fq(for)f(ev)-37 b(ery)579 b Fp(j)653 b Fq(and)578 b(some)f Fp(s)26354 13176 y Fk(j)27507 12977 y Fm(2)593 b Fq([)p Fp(t)29835 13176 y Fk(j)30394 12977 y Fp(;)221 b(t)31464 13176 y Fk(j)32409 12977 y Fq(+)385 b Fp(T)184 b Fq(].)995 b(As)577 b Fp(\015)74 b Fq(\()p Fp(t)p Fq(\))576 b(is)h(a)h Fp(c)p Fq(-minimal)1102 14571 y(curv)-37 b(e,)523 b Fp(d\015)580 b Fq(lies)506 b(in)g(a)g(b)37 b(ounded)506 b(region)h(of)f Fp(T)184 b(M)483 b Fm(\002)337 b Fl(T)p Fq(,)521 b(the)506 b(closure)h(of)f(the)g(orbit)g(is)g(compact.)1102 16165 y(Th)-37 b(us,)426 b(for)421 b(some)g(subsequence)i Fp(t)18100 16364 y Fk(i)18962 16165 y Fq(of)e(the)g(sequence)h Fp(t)28682 16364 y Fk(j)29242 16165 y Fq(,)j(the)c(sequence)h Fp(d\015)74 b Fq(\()p Fp(t)40235 16364 y Fk(i)40676 16165 y Fq(\))421 b(and)g Fp(d\015)74 b Fq(\()p Fp(s)46768 16364 y Fk(i)47209 16165 y Fq(\))420 b(are)1102 17760 y(con)-37 b(v)g(ergen)g(t)373 b(to)e(some)h(p)37 b(oin)-37 b(ts)371 b Fp(x)e Fm(2)f Fp(N)517 b Fq(and)371 b Fp(y)417 b Fm(2)369 b Fp(T)184 b(M)297 b Fm(\002)152 b Fl(T)371 b Fq(resp)37 b(ectiv)-37 b(ely)-111 b(,)387 b(where)373 b(dist)o(\()p Fp(y)48 b(;)221 b(N)145 b Fq(\))369 b Fm(\025)g Fp(d)p Fq(.)1102 19354 y(Consequen)-37 b(tly)-111 b(,)453 b Fp(\036)10396 18871 y Fk(t)10797 19004 y Fj(0)11383 19354 y Fq(\()p Fp(x)p Fq(\))379 b(=)h Fp(y)498 b Fq(for)450 b(some)g(0)380 b Fp(<)h(t)24430 19553 y Fv(0)25405 19354 y Fm(\024)g Fp(T)184 b Fq(.)611 b(This)450 b(con)-37 b(tradicts)450 b(to)f(the)h(in)-37 b(v)-74 b(ariance)451 b(of)1102 20948 y Fp(N)588 b Fq(to)443 b(the)g(Euler-Lagrange)h(\015o)-37 b(w.)2696 23101 y(Let)511 b Fp(\026)5944 23300 y Fk(n)7178 23101 y Fq(b)37 b(e)511 b(the)g(probabilit)-37 b(y)510 b(measure)i(ev)-37 b(enly)512 b(distributed)e(along)h Fp(d\015)74 b Fq([)p Fp(t)40459 23300 y Fk(k)41112 23101 y Fp(;)221 b(t)42182 23300 y Fk(k)43176 23101 y Fq(+)341 b Fp(T)45326 23300 y Fk(k)45979 23101 y Fq(],)528 b Fp(\026)511 b Fq(b)37 b(e)1102 24695 y(an)490 b(accum)-37 b(ulation)491 b(p)37 b(oin)-37 b(t)489 b(of)h Fm(f)p Fp(\026)17735 24894 y Fk(n)18458 24695 y Fm(g)p Fq(.)731 b(As)490 b Fp(d\015)564 b Fq(is)489 b(a)h Fp(c)p Fq(-minimal)f(orbit)h(of)g(the)f (Lagrange)i(system)1102 26383 y Fp(\026)456 b Fq(is)f(a)g Fp(c)p Fq(-minimal)g(measure,)460 b(i.e.)628 b Fp(\026)390 b Fq(=)f Fp(\026)22384 26551 y Fv(~)22320 26786 y(\000)23044 26383 y Fq(.)628 b(F)-111 b(rom)456 b(\(3.14\))f(w)-37 b(e)456 b(see)g(dist)o(\()p Fp(N)71 b(;)39997 26047 y Fq(~)39914 26383 y(\000)q(\))389 b(=)h(0.)627 b(As)46793 26047 y(~)46710 26383 y(\000)455 b(has)1102 28142 y(dense)460 b(orbit,)i Fp(N)540 b Fq(=)11499 27806 y(~)11416 28142 y(\000)q(,)462 b(i.e.)638 b(the)458 b Fp(!)48 b Fq(-limit)457 b(set)i(of)f Fp(d\015)533 b Fq(has)458 b(only)h(one)g(minimal)f(in)-37 b(v)-74 b(arian)-37 b(t)459 b(subset)1185 29420 y(~)1102 29756 y(\000)506 b(\(in)f(Birkho\013)g(sense\).)779 b(In)505 b(the)g(same)h(w)-37 b(a)g(y)506 b(w)-37 b(e)506 b(can)g(sho)-37 b(w)506 b(that)f(the)g Fp(\013)5 b Fq(-limit)504 b(set)h(of)h Fp(d\015)579 b Fq(has)1102 31444 y(only)443 b(one)h(minimal)f(in)-37 b(v)-74 b(arian)-37 b(t)443 b(subset)21351 31108 y(~)21268 31444 y(\000)g(also.)2696 33617 y(Let)424 b Fp(c)369 b Fm(2)g Fq(in)-37 b(t)o Fp(I)528 b Fq(and)424 b Fp(d\015)443 b Fm(2)16015 33281 y Fq(~)15594 33617 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\).)583 b(F)-111 b(rom)424 b(the)g(lemma)g(3.3)g(w)-37 b(e)424 b(see)g(that)38908 33281 y(~)38486 33617 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\))368 b(=)43607 33281 y(~)43151 33617 y Fm(A)p Fq(\()p Fp(c)p Fq(\))422 b(in)i(this)1102 35211 y(case.)587 b(W)-111 b(e)429 b(claim)g(that)e(for)i(eac)-37 b(h)429 b Fp(\030)i Fm(2)368 b Fp(\031)48 b Fq(\(\000\))428 b(there)g(exist)h(t)-37 b(w)g(o)428 b(sequences)i Fp(k)39160 35410 y Fk(ij)40464 35211 y Fm(!)369 b(1)429 b Fq(as)g Fp(j)445 b Fm(!)369 b(1)p Fq(,)1102 36805 y(\()p Fp(i)g Fq(=)g(1)p Fp(;)221 b Fq(2\),)443 b(suc)-37 b(h)443 b(that)5322 39921 y Fp(d\015)74 b Fq(\()p Fm(\000)p Fp(k)9017 40120 y Fv(1)p Fk(j)10105 39921 y Fq(\))p Fp(;)221 b(d\015)74 b Fq(\()p Fp(k)13874 40120 y Fv(2)p Fk(j)14962 39921 y Fq(\))369 b Fm(!)g Fp(\031)18350 39372 y Fg(\000)p Fv(1)19776 39921 y Fq(\()p Fp(\030)61 b Fq(\))368 b Fm(2)23158 39585 y Fq(~)23075 39921 y(\000)p Fm(j)24274 40120 y Fk(t)p Fv(=0)26087 39921 y Fp(;)3261 b Fq(lim)29334 40757 y Fk(j)54 b Fg(!1)32167 38113 y Fi(Z)33495 38441 y Fk(k)34060 38574 y Fj(2)p Fd(j)32905 41131 y Fg(\000)p Fk(k)34300 41264 y Fj(1)p Fd(j)35732 39921 y Fq(_)-565 b Fp(\015)36224 40120 y Fv(2)36819 39921 y Fq(\()p Fp(t)p Fq(\))p Fp(dt)368 b Fq(=)h(0)p Fp(;)4220 b Fq(\(3.15\))1102 43274 y(where)538 b Fp(\031)5842 42792 y Fg(\000)p Fv(1)7792 43274 y Fq(:)524 b Fm(A)p Fq(\()p Fp(c)p Fq(\))f Fm(!)14187 42938 y Fq(~)13730 43274 y Fm(A)p Fq(\()p Fp(c)p Fq(\).)868 b(In)536 b(fact,)559 b(for)536 b(an)-37 b(y)536 b Fp(\030)586 b Fm(2)523 b Fp(\031)48 b Fq(\(\000\))535 b(there)i(exist)f(t)-37 b(w)g(o)536 b(subsequences)1102 44868 y Fp(k)1794 45067 y Fk(ij)3098 44868 y Fm(!)370 b(1)415 b Fq(as)g Fp(j)445 b Fm(!)370 b(1)415 b Fq(\()p Fp(i)368 b Fq(=)h(1)p Fp(;)221 b Fq(2\))416 b(suc)-37 b(h)415 b(that)f Fp(d\015)74 b Fq(\()p Fm(\000)p Fp(k)27684 45067 y Fk(ij)28619 44868 y Fq(\))369 b Fm(!)g Fp(\031)32007 44386 y Fg(\000)p Fv(1)33433 44868 y Fq(\()p Fp(\030)61 b Fq(\))414 b(and)i Fp(d\015)74 b Fq(\()p Fp(k)40741 45067 y Fv(2)p Fk(j)41829 44868 y Fq(\))368 b Fm(!)i Fp(\031)45217 44386 y Fg(\000)p Fv(1)46643 44868 y Fq(\()p Fp(\030)61 b Fq(\))414 b(as)1102 46462 y Fp(j)445 b Fm(!)370 b(1)p Fq(.)590 b(It)443 b(follo)-37 b(ws)443 b(from)g(the)g(fact)g(that)f Fp(\015)516 b Fq(is)443 b Fp(c)p Fq(-static)f(that)12631 48581 y Fp(h)13396 48033 y Fk(k)13961 48166 y Fj(1)p Fd(j)13396 48910 y Fk(c)14976 48581 y Fq(\()p Fp(\015)74 b Fq(\()p Fm(\000)p Fp(k)18497 48780 y Fv(1)p Fk(j)19584 48581 y Fq(\))p Fp(;)221 b(\015)74 b Fq(\(0\)\))294 b(+)h Fp(h)26055 48033 y Fk(k)26620 48166 y Fj(2)p Fd(j)26055 48910 y Fk(c)27635 48581 y Fq(\()p Fp(\015)74 b Fq(\(0\))p Fp(;)221 b(\015)74 b Fq(\()p Fp(k)33173 48780 y Fv(2)p Fk(j)34260 48581 y Fq(\)\))368 b Fm(!)h Fq(0)p Fp(:)1102 50701 y Fq(If)448 b(the)f(equalit)-37 b(y)448 b(in)g(\(3.15\))g(do)37 b(es)448 b(not)f(hold,)i(b)-37 b(y)448 b(c)-37 b(ho)37 b(osing)449 b(a)f(subsequence)h(again)f(\(w)-37 b(e)448 b(use)g(the)1102 52295 y(same)c(sym)-37 b(b)37 b(ol\))442 b(w)-37 b(e)444 b(w)-37 b(ould)443 b(ha)-37 b(v)g(e)16930 53285 y Fi(\014)16930 54082 y(\014)16930 54879 y(\014)16930 55676 y(\014)16930 56473 y(\014)17756 55610 y Fq(lim)17373 56446 y Fk(j)54 b Fg(!1)20206 53802 y Fi(Z)21534 54130 y Fk(k)22099 54263 y Fj(2)22617 54130 y Fk(j)20944 56820 y Fg(\000)p Fk(k)22339 56953 y Fj(1)22857 56820 y Fk(j)23835 55610 y Fq(_)-566 b Fp(\015)24326 55809 y Fv(2)24922 55610 y Fq(\()p Fp(t)p Fq(\))p Fp(dt)27607 53285 y Fi(\014)27607 54082 y(\014)27607 54879 y(\014)27607 55676 y(\014)27607 56473 y(\014)28418 55610 y Fm(\025)369 b Fq(2)p Fp(\031)417 b(>)369 b Fq(0)p Fp(:)1102 58927 y Fq(In)663 b(this)f(case,)719 b(let)663 b(us)g(consider)h(the)f(barrier)h(function)f Fp(B)32301 58445 y Fg(\003)32234 59292 y Fk(c)32709 59026 y Fe(0)33797 58927 y Fq(where)h Fp(c)38433 58445 y Fg(0)39542 58927 y Fq(=)736 b(\()p Fp(c)42403 59126 y Fv(1)42998 58927 y Fp(;)221 b(c)44163 58445 y Fg(0)44163 59256 y Fv(2)44759 58927 y Fq(\).)1251 b(Since)1102 60521 y Fp(c)295 b Fm(\000)h Fp(c)3876 60039 y Fg(0)4617 60521 y Fq(=)369 b(\(0)p Fp(;)221 b(c)8365 60720 y Fv(2)9256 60521 y Fm(\000)296 b Fp(c)11160 60039 y Fg(0)11160 60850 y Fv(2)11756 60521 y Fq(\),)442 b(w)-37 b(e)444 b(obtain)e(from)h(the)g(prop)37 b(osition)443 b(3.2)g(that)f Fp(\013)5 b Fq(\()p Fp(c)38550 60039 y Fg(0)38922 60521 y Fq(\))369 b(=)g Fp(\013)5 b Fq(\()p Fp(c)p Fq(\),)441 b(so)6344 63636 y Fp(B)7352 63835 y Fk(c)7827 63570 y Fe(0)8252 63636 y Fq(\()p Fp(\015)74 b Fq(\(0\)\))367 b Fm(\024)222 b Fq(lim)f(inf)13903 64473 y Fk(j)54 b Fg(!1)17271 61828 y Fi(Z)18600 62157 y Fk(k)19165 62290 y Fj(2)p Fd(j)18009 64847 y Fg(\000)p Fk(k)19404 64980 y Fj(1)p Fd(j)20640 63636 y Fq(\()p Fp(L)p Fq(\()p Fp(d\015)74 b Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))293 b Fm(\000)j Fp(c)29329 63835 y Fv(1)30121 63636 y Fq(_)-566 b Fp(\015)30612 63835 y Fv(1)31208 63636 y Fq(\()p Fp(t)p Fq(\))294 b Fm(\000)h Fp(c)34919 63088 y Fg(0)34919 63965 y Fv(2)35711 63636 y Fq(_)-565 b Fp(\015)36203 63835 y Fv(2)36799 63636 y Fq(\()p Fp(t)p Fq(\))294 b Fm(\000)h Fp(\013)5 b Fq(\()p Fp(c)41882 63088 y Fg(0)42254 63636 y Fq(\)\))221 b Fp(dt)12113 67766 y Fm(\024)h Fq(lim)f(inf)13903 68603 y Fk(j)54 b Fg(!1)17271 65958 y Fi(Z)18600 66286 y Fk(k)19165 66419 y Fj(2)p Fd(j)18009 68976 y Fg(\000)p Fk(k)19404 69109 y Fj(1)p Fd(j)20640 67766 y Fq(\()p Fp(L)p Fq(\()p Fp(d\015)74 b Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))293 b Fm(\000)j Fp(c)29329 67965 y Fv(1)30121 67766 y Fq(_)-566 b Fp(\015)30612 67965 y Fv(1)31208 67766 y Fq(\()p Fp(t)p Fq(\))294 b Fm(\000)h Fp(c)34919 67965 y Fv(2)35711 67766 y Fq(_)-565 b Fp(\015)36203 67965 y Fv(2)36799 67766 y Fq(\()p Fp(t)p Fq(\))294 b Fm(\000)h Fp(\013)5 b Fq(\()p Fp(c)p Fq(\)\))220 b Fp(dt)13441 71896 y Fq(+)296 b(\()p Fp(c)15862 72095 y Fv(2)16752 71896 y Fm(\000)g Fp(c)18656 71347 y Fg(0)18656 72224 y Fv(2)19251 71896 y Fq(\))605 b(lim)19989 72732 y Fk(j)54 b Fg(!1)22822 70087 y Fi(Z)24150 70416 y Fk(k)24715 70549 y Fj(2)p Fd(j)23560 73106 y Fg(\000)p Fk(k)24955 73239 y Fj(1)p Fd(j)26387 71896 y Fq(_)-565 b Fp(\015)26879 72095 y Fv(2)27475 71896 y Fq(\()p Fp(t)p Fq(\))p Fp(dt)12113 74965 y Fm(\024)295 b(\000)h Fq(2)p Fm(j)p Fp(c)16378 75164 y Fv(2)17269 74965 y Fm(\000)f Fp(c)19172 74416 y Fg(0)19172 75293 y Fv(2)19768 74965 y Fm(j)p Fp(\031)417 b(<)369 b Fq(0)p eop %%Page: 23 23 23 22 bop 48792 2239 a Ft(23)1102 5427 y Fq(as)549 b(w)-37 b(e)549 b(can)f(c)-37 b(ho)37 b(ose)549 b Fp(c)12334 4945 y Fg(0)12334 5757 y Fv(2)13475 5427 y Fp(>)544 b(c)15627 5626 y Fv(2)16771 5427 y Fq(or)549 b Fp(c)19079 4945 y Fg(0)19079 5757 y Fv(2)20219 5427 y Fp(<)c(c)22372 5626 y Fv(2)23515 5427 y Fq(accordingly)-111 b(.)909 b(But)547 b(this)h(is)g(absurd)g(since)h(barrier)1102 7021 y(function)443 b(is)g(non-negativ)-37 b(e.)2696 9390 y(No)g(w)539 b(let)f(us)g(deriv)-37 b(e)539 b(from)f(\(3.15\))g (that)f(there)i(is)e(no)h(suc)-37 b(h)539 b(a)f Fp(c)p Fq(-semi-static)f(orbit)h(that)f(is)1102 11078 y(not)501 b(con)-37 b(tained)503 b(in)11351 10742 y(~)11204 11078 y(\006.)766 b(It)501 b(is)g(easy)h(to)f(see)h(that)f(there)h(is)f(no)h (other)f(suc)-37 b(h)502 b(orbits)f(whic)-37 b(h)502 b(sta)-37 b(ys)1102 12692 y(in)560 b(the)g(small)g(neigh)-37 b(b)37 b(orho)g(o)g(d)560 b(of)19004 12356 y(~)18856 12692 y(\006)g(for)g(all)g Fp(t)k Fm(2)f Fl(R)d Fq(except)h(for)f(the)g (orbit)f(in)42458 12356 y(~)42375 12692 y(\000)h(as)g(there)g(is)1102 14286 y(normally)519 b(h)-37 b(yp)37 b(erb)g(olic)519 b(structure)f(nearb)-37 b(y)-111 b(.)817 b(If)518 b(there)g(is)g(a)g Fp(c)p Fq(-minimal)f(curv)-37 b(e)520 b Fp(\015)591 b Fq(whic)-37 b(h)519 b(is)e(not)1102 15880 y(con)-37 b(tained)365 b(in)f(the)g(supp)37 b(ort)364 b(of)g(the)g(minimal)g(measure,)381 b(guaran)-37 b(teed)365 b(b)-37 b(y)364 b(the)g(semi-asymptotic)1102 17474 y(prop)37 b(ert)-37 b(y)533 b(of)f Fp(d\015)74 b Fq(,)554 b(there)533 b(are)f(still)g(t)-37 b(w)g(o)532 b(sequences)h Fp(t)28271 17673 y Fv(1)p Fk(j)29878 17474 y Fm(!)518 b(\0001)p Fq(,)554 b Fp(t)35488 17673 y Fv(2)p Fk(j)37095 17474 y Fm(!)518 b(1)532 b Fq(as)g Fp(j)594 b Fm(!)518 b(1)532 b Fq(suc)-37 b(h)1102 19210 y(that)649 b Fp(d\015)74 b Fq(\()p Fp(t)6637 19409 y Fk(ij)7571 19210 y Fq(\))649 b(\()p Fp(i)712 b Fq(=)g(1)p Fp(;)221 b Fq(2\))650 b(approac)-37 b(hes)650 b(to)24210 18874 y(~)24127 19210 y(\000)f(as)g Fp(j)789 b Fm(!)713 b(1)p Fq(.)1209 b(Let)649 b Fp(R)722 b Fq(=)713 b Fm(f)p Fq(\()p Fp(q)41563 19409 y Fv(2)42159 19210 y Fp(;)468 b Fq(_)-616 b Fp(q)43342 19409 y Fv(2)43938 19210 y Fq(\))712 b(:)h Fm(j)p Fp(q)47211 19409 y Fv(2)47807 19210 y Fm(j)f(\024)1102 20804 y Fq(min)p Fm(f)p Fp(a)4682 21003 y Fv(1)5279 20804 y Fq(\()19 b(\026)-683 b Fp(a)6498 21003 y Fv(2)7093 20804 y Fq(\()p Fp(a)8312 20322 y Fg(\003)8312 21133 y Fv(0)8923 20804 y Fq(\)\))p Fp(;)221 b(a)11249 21003 y Fv(2)11844 20804 y Fq(\()19 b(\026)-683 b Fp(a)13063 21003 y Fv(1)13659 20804 y Fq(\()p Fp(a)14878 20322 y Fg(\003)14878 21133 y Fv(0)15488 20804 y Fq(\)\))p Fm(g)p Fp(;)221 b Fm(j)247 b Fq(_)-616 b Fp(q)18738 21003 y Fv(2)19334 20804 y Fm(j)369 b(\024)g Fp(\016)50 b Fm(g)414 b Fq(\(for)f(the)f(de\014nition)h(of)g Fp(a)36028 20322 y Fg(\003)36028 21133 y Fv(0)37052 20804 y Fq(and)g Fp(\016)50 b Fq(,)419 b(see)414 b(the)f(pro)37 b(of)413 b(of)1102 22398 y(the)510 b(lemma)f(3.1\).)790 b(F)-111 b(orced)511 b(b)-37 b(y)510 b(\(3.15\))e(there)i(is)g(an)f(in\014nite)g (subsequence)i Fp(t)40759 22597 y Fk(k)41921 22398 y Fq(suc)-37 b(h)510 b(that)f(one)1102 23992 y(lift)488 b(of)g Fp(d\015)562 b Fq(has)489 b(the)f(prop)37 b(ert)-37 b(y)489 b Fp(d)49 b Fq(\026)-713 b Fp(\015)18354 24191 y Fv(2)18950 23992 y Fq(\()p Fp(t)19947 24191 y Fk(k)20600 23992 y Fq(\))444 b Fm(2)g Fp(R)10 b Fq(.)727 b(On)488 b(the)g(other)h(hand,)500 b(since)489 b Fp(d\015)562 b Fq(do)37 b(es)488 b(not)g(sta)-37 b(y)1102 25680 y(in)2746 25344 y(~)2663 25680 y(\000)q(,)456 b(there)f(exists)11426 25435 y(^)11408 25680 y Fp(t)387 b Fm(2)g Fl(R)454 b Fq(suc)-37 b(h)455 b(that)e Fp(d)49 b Fq(\026)-713 b Fp(\015)22237 25879 y Fv(2)22833 25680 y Fq(\()23368 25435 y(^)23350 25680 y Fp(t)o Fq(\))535 b Fp(=)-812 b Fm(2)388 b Fp(R)10 b Fq(.)623 b(Therefore,)459 b(there)c(m)-37 b(ust)453 b(b)37 b(e)454 b Fp(t)43434 25879 y Fv(2)44418 25680 y Fp(>)387 b(t)46318 25879 y Fv(1)47368 25680 y Fq(suc)-37 b(h)1102 27274 y(that)475 b Fp(d)49 b Fq(\026)-713 b Fp(\015)5392 27473 y Fv(2)6463 27274 y Fq(lea)-37 b(v)g(es)476 b Fp(R)485 b Fq(at)475 b Fp(t)422 b Fq(=)h Fp(t)16291 27473 y Fv(1)17362 27274 y Fq(and)475 b Fp(d)49 b Fq(\026)-713 b Fp(\015)21356 27473 y Fv(2)22427 27274 y Fq(gets)475 b(in)-37 b(to)475 b Fp(R)485 b Fq(at)475 b Fp(t)422 b Fq(=)h Fp(t)33911 27473 y Fv(2)34982 27274 y Fq(from)475 b(the)g(same)h(side)f(of)g(the)1102 28868 y(line)443 b Fp(q)4204 29067 y Fv(2)5169 28868 y Fq(=)370 b(0)442 b(in)h(the)g(follo)-37 b(wing)444 b(sense:)2696 31237 y(either)623 b(\026)-713 b Fp(\015)7282 31436 y Fv(2)7878 31237 y Fq(\()p Fp(t)8875 31436 y Fv(1)9470 31237 y Fq(\))586 b(=)h Fp(a)12895 31436 y Fv(1)13491 31237 y Fq(\()19 b(\026)-683 b Fp(a)14710 31436 y Fv(2)15306 31237 y Fq(\()p Fp(a)16525 30755 y Fg(\003)16525 31566 y Fv(0)17135 31237 y Fq(\)\))573 b(and)g(0)587 b Fm(\024)24522 31055 y Fq(_)24374 31237 y(\026)-712 b Fp(\015)25087 31561 y Fv(2)25683 31237 y Fq(\()p Fp(t)26680 31436 y Fv(1)27275 31237 y Fq(\))586 b Fp(<)h(G)31042 30755 y Fv(+)31924 31237 y Fq(\()49 b(\026)-713 b Fp(\015)33129 31436 y Fv(2)33725 31237 y Fq(\()p Fp(t)34722 31436 y Fv(1)35317 31237 y Fq(\)\))381 b(+)i Fp(\016)624 b Fq(\(lea)-37 b(ving)573 b(to)h(righ)-37 b(t\),)1102 32831 y(together)529 b(with)576 b(\026)-712 b Fp(\015)10230 33030 y Fv(2)10825 32831 y Fq(\()p Fp(t)11822 33030 y Fv(2)12418 32831 y Fq(\))510 b(=)530 b(\026)-683 b Fp(a)15691 33030 y Fv(1)16287 32831 y Fq(\()p Fp(a)17506 32349 y Fg(\003)17506 33160 y Fv(0)18117 32831 y Fq(\))527 b(and)h Fp(G)22873 32349 y Fg(\000)23770 32831 y Fq(\()49 b(\026)-713 b Fp(\015)24975 33030 y Fv(2)25570 32831 y Fq(\()p Fp(t)26567 33030 y Fv(2)27163 32831 y Fq(\)\))351 b Fm(\000)h Fp(\016)561 b(<)32825 32649 y Fq(_)32677 32831 y(\026)-713 b Fp(\015)33390 33155 y Fv(2)33986 32831 y Fq(\()p Fp(t)34983 33030 y Fv(2)35578 32831 y Fq(\))511 b Fm(\024)g Fq(0)528 b(\(getting)f(in)-37 b(to)528 b(from)1102 34425 y(righ)-37 b(t\).)2696 36794 y(or)512 b(\026)-712 b Fp(\015)5032 36993 y Fv(2)5627 36794 y Fq(\()p Fp(t)6624 36993 y Fv(1)7220 36794 y Fq(\))402 b(=)g Fm(\000)19 b Fq(\026)-683 b Fp(a)11309 36993 y Fv(2)11905 36794 y Fq(\()p Fp(a)13124 36312 y Fg(\003)13124 37123 y Fv(0)13735 36794 y Fq(\)\))462 b(with)h Fp(G)19321 36312 y Fg(\000)20218 36794 y Fq(\()49 b(\026)-713 b Fp(\015)21423 36993 y Fv(2)22018 36794 y Fq(\()p Fp(t)23015 36993 y Fv(1)23610 36794 y Fq(\)\))308 b Fm(\000)h Fp(\016)453 b Fm(\024)28968 36612 y Fq(_)28821 36794 y(\026)-713 b Fp(\015)29534 37118 y Fv(2)30130 36794 y Fq(\()p Fp(t)31127 36993 y Fv(1)31722 36794 y Fq(\))402 b Fp(<)g Fq(0)464 b(\(lea)-37 b(ving)463 b(to)f(left\),)468 b(together)1102 38388 y(with)k(\026)-712 b Fp(\015)4797 38587 y Fv(2)5392 38388 y Fq(\()p Fp(t)6389 38587 y Fv(2)6985 38388 y Fq(\))368 b(=)h Fm(\000)p Fp(a)11007 38587 y Fv(2)11603 38388 y Fq(\()19 b(\026)-683 b Fp(a)12822 38587 y Fv(1)13418 38388 y Fq(\()p Fp(a)14637 37906 y Fg(\003)14637 38717 y Fv(0)15248 38388 y Fq(\)\))422 b(with)h Fp(G)20754 37906 y Fg(\000)21652 38388 y Fq(\()49 b(\026)-713 b Fp(\015)22857 38587 y Fv(2)23452 38388 y Fq(\()p Fp(t)24449 38587 y Fv(2)25044 38388 y Fq(\)\))256 b Fm(\000)g Fp(\016)420 b(>)30231 38206 y Fq(_)30084 38388 y(\026)-713 b Fp(\015)30797 38712 y Fv(2)31392 38388 y Fq(\()p Fp(t)32389 38587 y Fv(2)32985 38388 y Fq(\))368 b Fm(\025)h Fq(0)424 b(\(getting)f(in)-37 b(to)423 b(from)h(left\).)2696 40757 y(Without)551 b(losing)g(of)f (generalit)-37 b(y)552 b(w)-37 b(e)552 b(study)e(the)h(\014rst)f (alternativ)-37 b(e,)578 b(in)551 b(the)g(same)g(w)-37 b(a)g(y)551 b(w)-37 b(e)1102 42351 y(can)448 b(study)e(the)h(second)g (alternativ)-37 b(e.)603 b(F)-111 b(rom)448 b(\(3.9\))e(and)h(\(3.10\)) f(w)-37 b(e)448 b(\014nd)f(that)f Fp(d\015)42945 42550 y Fv(2)43988 42351 y Fq(shall)g(cross)1102 43945 y(the)d(line)542 b(\026)-763 b Fp(q)6492 44144 y Fv(2)7457 43945 y Fq(=)369 b(2)p Fp(\031)343 b Fm(\000)315 b Fq(\026)-683 b Fp(a)12654 44144 y Fv(2)13250 43945 y Fq(\()p Fp(a)14469 43463 y Fg(\003)14469 44274 y Fv(0)15079 43945 y Fq(\))443 b(at)f(the)h(time)g Fp(t)368 b Fq(=)h Fp(t)25706 44144 y Fv(1)26597 43945 y Fq(+)296 b Fp(T)28702 44144 y Fv(1)29740 43945 y Fq(with)13240 47033 y Fp(T)14016 47232 y Fv(+)14898 47033 y Fq(\()p Fp(a)16117 47232 y Fv(1)16713 47033 y Fq(\()19 b(\026)-683 b Fp(a)17932 47232 y Fv(2)18528 47033 y Fq(\()p Fp(a)19747 46485 y Fg(\003)19747 47362 y Fv(0)20357 47033 y Fq(\)\))p Fp(;)240 b Fq(\026)-683 b Fp(a)22683 47232 y Fv(2)23279 47033 y Fq(\()p Fp(a)24498 46485 y Fg(\003)24498 47362 y Fv(0)25108 47033 y Fq(\)\))294 b Fm(\000)i Fp(\034)519 b Fm(\024)369 b Fp(T)31043 47232 y Fv(1)32008 47033 y Fm(\024)g Fp(T)34186 47232 y Fv(+)35068 47033 y Fq(\()p Fp(a)36287 46485 y Fg(\003)36287 47362 y Fv(0)36898 47033 y Fq(\))p Fp(:)1102 50122 y Fq(F)-111 b(rom)459 b(\(3.9'\))f(and)g (\(3.10'\))f(w)-37 b(e)459 b(\014nd)f Fp(d\015)21007 50321 y Fv(2)22061 50122 y Fq(has)g(crossed)h(the)f(line)557 b(\026)-763 b Fp(q)34474 50321 y Fv(2)35464 50122 y Fq(=)395 b(2)p Fp(\031)353 b Fm(\000)305 b Fp(a)40706 50321 y Fv(2)41302 50122 y Fq(\()19 b(\026)-683 b Fp(a)42521 50321 y Fv(1)43117 50122 y Fq(\()p Fp(a)44336 49640 y Fg(\003)44336 50451 y Fv(0)44947 50122 y Fq(\)\))457 b(at)g(the)1102 51716 y(time)443 b Fp(t)369 b Fq(=)g Fp(t)6859 51915 y Fv(2)7750 51716 y Fm(\000)295 b Fp(T)9854 51915 y Fv(2)10893 51716 y Fq(with)12967 54804 y Fp(T)13743 55003 y Fg(\000)14640 54804 y Fq(\()19 b(\026)-683 b Fp(a)15859 55003 y Fv(1)16455 54804 y Fq(\()p Fp(a)17674 54256 y Fg(\003)17674 55133 y Fv(0)18284 54804 y Fq(\)\))p Fp(;)221 b(a)20610 55003 y Fv(2)21206 54804 y Fq(\()19 b(\026)-683 b Fp(a)22425 55003 y Fv(1)23020 54804 y Fq(\()p Fp(a)24239 54256 y Fg(\003)24239 55133 y Fv(0)24850 54804 y Fq(\)\)\))294 b Fm(\000)h Fp(\034)519 b Fm(\024)369 b Fp(T)31301 55003 y Fv(2)32266 54804 y Fm(\024)h Fp(T)34445 55003 y Fg(\000)35341 54804 y Fq(\()p Fp(a)36560 54256 y Fg(\003)36560 55133 y Fv(0)37171 54804 y Fq(\))p Fp(:)1102 57893 y Fq(Let)443 b(us)g(construct)g(a)g(curv)-37 b(e)444 b Fp(\020)98 b Fq(\()p Fp(t)p Fq(\))368 b(=)h(\()p Fp(\020)20742 58092 y Fv(1)21338 57893 y Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(\020)24023 58092 y Fv(2)24618 57893 y Fq(\()p Fp(t)p Fq(\)\))441 b(in)i(the)g(w)-37 b(a)g(y)443 b(that)g(its)f(lift)g(has)h (the)g(form)11008 64484 y(\026)10890 64835 y Fp(\020)11471 65034 y Fv(2)12067 64835 y Fq(\()p Fp(t)p Fq(\))368 b(=)15351 60119 y Fi(8)15351 61315 y(>)15351 61713 y(>)15351 62112 y(>)15351 62510 y(>)15351 62909 y(>)15351 63307 y(<)15351 65698 y(>)15351 66097 y(>)15351 66495 y(>)15351 66894 y(>)15351 67292 y(>)15351 67691 y(:)17023 60849 y Fq(\026)-712 b Fp(\015)17663 61048 y Fv(2)18258 60849 y Fq(\()p Fp(t)p Fq(\))7873 b Fp(t)369 b Fm(\024)g Fp(t)30376 61048 y Fv(1)16975 62842 y Fp(\030)17556 63041 y Fv(2)18152 62842 y Fq(\()p Fp(t)p Fq(\))7873 b Fp(t)28019 63041 y Fv(1)28983 62842 y Fm(\024)369 b Fp(t)g Fm(\024)g Fp(t)33116 63041 y Fv(1)34007 62842 y Fq(+)295 b Fp(T)36111 63041 y Fv(1)17023 64835 y Fq(\026)-712 b Fp(\015)17663 65034 y Fv(2)18258 64835 y Fq(\()p Fp(t)p Fq(\))294 b Fm(\000)i Fq(2)p Fp(\031)4772 b(t)28068 65034 y Fv(1)28959 64835 y Fq(+)295 b Fp(T)31063 65034 y Fv(1)32028 64835 y Fm(\024)370 b Fp(t)e Fm(\024)h Fp(t)36161 65034 y Fv(2)37052 64835 y Fm(\000)296 b Fp(T)39157 65034 y Fv(2)16975 66828 y Fp(\021)17635 67027 y Fv(2)18230 66828 y Fq(\()p Fp(t)p Fq(\))7873 b Fp(t)28097 67027 y Fv(2)28988 66828 y Fm(\000)295 b Fp(T)31092 67027 y Fv(2)32057 66828 y Fm(\024)369 b Fp(t)g Fm(\024)g Fp(t)36190 67027 y Fv(2)17023 68821 y Fq(\026)-712 b Fp(\015)17663 69020 y Fv(2)18258 68821 y Fq(\()p Fp(t)p Fq(\))7873 b Fp(t)369 b Fm(\025)g Fp(t)30376 69020 y Fv(2)1102 71777 y Fq(and)466 b Fp(\020)4289 71976 y Fv(1)4885 71777 y Fq(\()p Fp(t)p Fq(\))406 b(=)h Fp(\015)8933 71976 y Fv(1)9528 71777 y Fq(\()p Fp(t)p Fq(\),)470 b(where)d Fp(\030)16326 71976 y Fv(2)17388 71777 y Fq(is)e(the)h(extremal)g(of)g(the)f (Lagrangian)i Fp(`)37962 71976 y Fv(2)39024 71777 y Fq(with)e(the)h(b) 37 b(oundary)1102 73371 y(condition)512 b Fp(\030)7582 73570 y Fv(2)8178 73371 y Fq(\()p Fp(t)9175 73570 y Fv(1)9771 73371 y Fq(\))483 b(=)h Fp(a)12990 73570 y Fv(1)13587 73371 y Fq(\()19 b(\026)-683 b Fp(a)14806 73570 y Fv(2)15401 73371 y Fq(\()p Fp(a)16620 72889 y Fg(\003)16620 73700 y Fv(0)17231 73371 y Fq(\)\))511 b(and)h Fp(\030)22009 73570 y Fv(2)22605 73371 y Fq(\()p Fp(t)23602 73570 y Fv(1)24538 73371 y Fq(+)341 b Fp(T)26688 73570 y Fv(1)27285 73371 y Fq(\))483 b(=)h Fm(\000)19 b Fq(\026)-683 b Fp(a)31537 73570 y Fv(2)32134 73371 y Fq(\()p Fp(a)33353 72889 y Fg(\003)33353 73700 y Fv(0)33963 73371 y Fq(\);)546 b Fp(\021)36055 73570 y Fv(2)37162 73371 y Fq(is)512 b(the)g(extremal)g (of)g(the)1102 74965 y(Lagrangian)623 b Fp(`)8793 75164 y Fv(2)10011 74965 y Fq(with)e(the)g(b)37 b(oundary)622 b(condition)g Fp(\021)28510 75164 y Fv(2)29105 74965 y Fq(\()p Fp(t)30102 75164 y Fv(2)31112 74965 y Fm(\000)414 b Fp(T)33335 75164 y Fv(2)33931 74965 y Fq(\))666 b(=)h Fm(\000)p Fp(a)38549 75164 y Fv(2)39145 74965 y Fq(\()19 b(\026)-683 b Fp(a)40364 75164 y Fv(1)40960 74965 y Fq(\()p Fp(a)42179 74483 y Fg(\003)42179 75294 y Fv(0)42789 74965 y Fq(\)\),)665 b Fp(\021)45517 75164 y Fv(2)46113 74965 y Fq(\()p Fp(t)47110 75164 y Fv(2)47705 74965 y Fq(\))h(=)p eop %%Page: 24 24 24 23 bop 1102 2239 a Ft(24)1121 5427 y Fq(\026)-683 b Fp(a)1804 5626 y Fv(1)2400 5427 y Fq(\()p Fp(a)3619 4945 y Fg(\003)3619 5757 y Fv(0)4230 5427 y Fq(\).)588 b(Th)-37 b(us,)488 b(\026)-713 b Fp(\015)443 b Fq(=)12085 5076 y(\026)11967 5427 y Fp(\020)535 b Fq(at)438 b Fp(t)369 b Fq(=)g Fp(t)17433 5626 y Fv(1)18466 5427 y Fq(as)438 b(w)-37 b(ell)438 b(as)g(at)f Fp(t)369 b Fq(=)g Fp(t)28755 5626 y Fv(2)29351 5427 y Fq(.)588 b(The)438 b(di\013erence)i(b)37 b(et)-37 b(w)g(een)438 b(the)g(action)1102 7021 y(of)443 b Fp(L)295 b Fm(\000)h Fp(\021)5803 7220 y Fk(c)6786 7021 y Fq(along)444 b Fp(\015)516 b Fq(and)443 b Fp(\020)541 b Fq(is)5922 11156 y Fp(A)6918 11355 y Fk(c)7459 11156 y Fq(\()p Fp(\015)74 b Fq(\))294 b Fm(\000)h Fp(A)11873 11355 y Fk(c)12415 11156 y Fq(\()p Fp(\020)98 b Fq(\))368 b Fm(\025)15750 9348 y Fi(Z)17079 9677 y Fk(t)17480 9810 y Fj(1)17998 9677 y Fv(+)p Fk(T)19440 9810 y Fj(1)16488 12367 y Fk(t)16889 12500 y Fj(1)20248 9682 y Fi(n)21133 11156 y Fp(`)21686 11355 y Fv(2)22283 11156 y Fq(\()p Fp(d\015)24179 11355 y Fv(2)24774 11156 y Fq(\()p Fp(t)p Fq(\)\))294 b Fm(\000)h Fp(`)28980 11355 y Fv(2)29577 11156 y Fq(\()p Fp(d\020)31366 11355 y Fv(2)31962 11156 y Fq(\()p Fp(t)p Fq(\)\))33993 9682 y Fi(o)34877 11156 y Fp(dt)15824 15083 y Fq(+)17153 13275 y Fi(Z)18481 13603 y Fk(t)18882 13736 y Fj(2)17891 16293 y Fk(t)18292 16426 y Fj(2)18810 16293 y Fg(\000)p Fk(T)20267 16426 y Fj(2)21074 13608 y Fi(n)21960 15083 y Fp(`)22513 15282 y Fv(2)23109 15083 y Fq(\()p Fp(d\015)25005 15282 y Fv(2)25601 15083 y Fq(\()p Fp(t)p Fq(\)\))e Fm(\000)j Fp(`)29807 15282 y Fv(2)30403 15083 y Fq(\()p Fp(d\020)32192 15282 y Fv(2)32788 15083 y Fq(\()p Fp(t)p Fq(\)\))34819 13608 y Fi(o)35703 15083 y Fp(dt)f Fm(\000)h Fq(2\()p Fp(T)40455 15282 y Fv(1)41346 15083 y Fq(+)f Fp(T)43450 15282 y Fv(2)44046 15083 y Fq(\))p Fp(\017)14496 18713 y Fm(\025)15688 17814 y Fq(6)p 15688 18407 665 54 v 15688 19624 a(5)16512 18713 y Fp(C)17556 18144 y Fv(+)17461 19103 y Fk(A)18733 18713 y Fm(\000)h Fq(2\()p Fp(T)22019 18912 y Fv(+)22901 18713 y Fq(\()p Fp(a)24120 18164 y Fg(\003)24120 19041 y Fv(0)24730 18713 y Fq(\))f(+)g Fp(T)27646 18912 y Fg(\000)28543 18713 y Fq(\()p Fp(a)29762 18164 y Fg(\003)29762 19041 y Fv(0)30373 18713 y Fq(\))f(+)i(2)p Fp(\034)150 b Fq(\))p Fp(\017)26553 21152 y Fq(\(b)-37 b(y)442 b(\(3)p Fp(:)p Fq(6\))p Fp(;)g Fq(\(3)p Fp(:)p Fq(8\))221 b(and)443 b(\(3)p Fp(:)p Fq(12\)\))14496 23145 y Fp(>)p Fq(0)1102 26582 y(if)g Fp(\017)369 b Fm(\024)4790 26059 y Fv(1)p 4790 26277 530 54 v 4790 27040 a(2)5479 26582 y Fp(C)6523 26014 y Fv(+)6428 26973 y Fk(A)7405 26582 y Fq(\()p Fp(T)8698 26781 y Fv(+)9580 26582 y Fq(\()p Fp(a)10799 26100 y Fg(\003)10799 26912 y Fv(0)11409 26582 y Fq(\))295 b(+)g Fp(T)14325 26781 y Fg(\000)15222 26582 y Fq(\()p Fp(a)16441 26100 y Fg(\003)16441 26912 y Fv(0)17051 26582 y Fq(\))g(+)h(2)p Fp(\034)150 b Fq(\))21104 26100 y Fg(\000)p Fv(1)22530 26582 y Fq(,)442 b(but)h(this)f(con)-37 b(tradicts)443 b(the)g(minimalit)-37 b(y)443 b(of)f Fp(\015)74 b Fq(.)2696 29008 y(Finally)-111 b(,)685 b(let)636 b(us)g(consider)h(the)f(case)h (when)f Fp(c)691 b Fm(2)g Fp(@)74 b(I)740 b Fq(and)636 b(there)g(is)g(an)g(in)-37 b(v)-74 b(arian)-37 b(t)637 b(circle)1102 30602 y(con)-37 b(taining)384 b(\000.)571 b(In)383 b(this)f(case,)396 b(w)-37 b(e)384 b(obtain)f(from)h(the)f (lemma)h(3.3)f(that)g Fp(\026)37796 30770 y Fv(~)37732 31005 y(\000)38839 30602 y Fq(is)g(the)g(only)g(minimal)1102 32196 y(measure)644 b(still.)1189 b(According)644 b(to)e(the)h(upp)37 b(er)643 b(semi-con)-37 b(tin)g(uit)g(y)642 b(of)h(the)f(set-v)-74 b(alued)643 b(function)1102 33810 y Fp(c)474 b Fm(!)4375 33475 y Fq(~)3953 33810 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\))504 b(that)10715 33475 y(~)10293 33810 y Fm(N)196 b Fq(\()p Fp(c)12671 33328 y Fg(0)13043 33810 y Fq(\))505 b(should)h(b)37 b(e)506 b(in)g(a)g(small)f(neigh)-37 b(b)37 b(orho)g(o)g(d)507 b(of)36937 33475 y(~)36516 33810 y Fm(N)195 b Fq(\()p Fp(c)p Fq(\))505 b(if)h Fp(c)41771 33328 y Fg(0)42649 33810 y Fq(is)f(close)i(to)e Fp(c)p Fq(.)1102 35498 y(It)577 b(implies)h(that)10754 35163 y(~)10332 35498 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\))577 b(should)g(con)-37 b(tain)578 b(some)g(orbits)f(outside)h(of)36858 35163 y(~)36710 35498 y(\006.)995 b(If)577 b(this)g(is)g(not)g(true,)1524 36850 y(~)1102 37186 y Fm(N)196 b Fq(\()p Fp(c)3480 36704 y Fg(0)3852 37186 y Fq(\))477 b(w)-37 b(ould)478 b(b)37 b(e)477 b(in)g(a)g(small)h(neigh)-37 b(b)37 b(orho)g(o)g(d)478 b(of)26904 36850 y(~)26757 37186 y(\006)f(for)h(some)f Fp(c)34198 36704 y Fg(0)34997 37186 y Fq(=)427 b(\()p Fp(c)37549 37385 y Fv(1)38144 37186 y Fp(;)221 b(c)39309 36704 y Fg(0)39309 37516 y Fv(2)39905 37186 y Fq(\))477 b(with)g Fp(c)44534 37385 y Fv(2)45556 37186 y Fp(<)426 b(a)p Fq(\()p Fp(c)48809 37385 y Fv(1)49405 37186 y Fq(\))1102 38874 y(or)486 b Fp(c)3347 39073 y Fv(2)4384 38874 y Fp(>)440 b(b)p Fq(\()p Fp(c)7519 39073 y Fv(1)8115 38874 y Fq(\).)718 b(As)486 b(w)-37 b(e)487 b(ha)-37 b(v)g(e)486 b(normally)g(h)-37 b(yp)37 b(erb)g(olic)487 b(structure)f(in)g(the)f (neigh)-37 b(b)37 b(orho)g(o)g(d)487 b(of)48740 38538 y(~)48593 38874 y(\006,)1102 40562 y(an)-37 b(y)557 b(in)-37 b(v)-74 b(arian)-37 b(t)557 b(set)f(should)g(b)37 b(e)557 b(on)19955 40226 y(~)19808 40562 y(\006,)585 b(consequen)-37 b(tly)-111 b(,)586 b(w)-37 b(e)557 b(w)-37 b(ould)556 b(ha)-37 b(v)g(e)39871 40226 y(~)39220 40562 y Fm(M)q Fq(\()p Fp(c)41908 40080 y Fg(0)42280 40562 y Fq(\))557 b(=)45029 40226 y(~)44946 40562 y(\000)f(as)g(the)1102 42156 y(map)600 b(induced)f(b)-37 b(y)600 b(the)f(Euler-Lagrange)i (\015o)-37 b(w)600 b(on)f(this)g(manifold)g(corresp)37 b(onds)600 b(to)f(a)h(t)-37 b(wist)1102 43750 y(area-preserving)445 b(map)e(on)g(\006.)591 b(But)442 b(this)h(con)-37 b(tradicts)443 b(to)f(the)h(de\014nition)g(of)g Fp(I)104 b Fq(\()p Fp(c)42272 43949 y Fv(1)42867 43750 y Fq(\).)2696 46176 y(A)-37 b(t)487 b(the)f(b)37 b(eginning)486 b(of)h(the)f(pro)37 b(of)486 b(w)-37 b(e)487 b(ha)-37 b(v)g(e)487 b(sho)-37 b(wn)487 b(that)f(an)-37 b(y)486 b Fp(c)p Fq(-minimal)g(orbits)g(m)-37 b(ust)486 b(b)37 b(e)1102 47770 y(semi-asymptotic)305 b(to)f(the)g(supp)37 b(ort)304 b(of)h(the)f(minimal)h(measure)g(if)g (it)f(is)g(uniquely)h(ergo)37 b(dic.)545 b(What)1102 49364 y(remain)402 b(to)f(sho)-37 b(w)402 b(is)f(suc)-37 b(h)401 b(orbit)g(is)g(homo)37 b(clinic)402 b(to)f(the)g(in)-37 b(v)-74 b(arian)-37 b(t)402 b(circle)h(in)e(this)f(case.)578 b(As)401 b(\000)g(is)1102 50958 y(con)-37 b(tained)359 b(in)f(an)h(in)-37 b(v)-74 b(arian)-37 b(t)358 b(circle,)377 b(denoted)358 b(b)-37 b(y)359 b(\000)27046 50476 y Fg(\003)27657 50958 y Fq(,)375 b(the)358 b(Aubry)h(set)f(con)-37 b(tains)358 b(a)g(co)37 b(dimension)1102 52572 y(1)548 b(torus)5908 52236 y(~)5825 52572 y(\000)6655 52090 y Fg(\003)7810 52572 y Fq(=)c Fm([)10273 52812 y Fk(t)p Fg(2)p Fv([0)p Fk(;)p Fv(1])13432 52572 y Fq(\()p Fp(\036)14740 52090 y Fk(t)15208 52572 y Fq(\()p Fm(L)p Fq(\(\000)17988 52090 y Fg(\003)18598 52572 y Fq(\)\))p Fp(;)221 b(t)p Fq(\),)573 b(b)37 b(ecause)548 b Fp(P)28033 52771 y Fk(!)28802 52572 y Fq(\()p Fp(q)48 b Fq(\))543 b(=)h Fp(B)33605 52771 y Fk(c)34146 52572 y Fq(\()p Fp(q)48 b Fq(\))546 b(for)i(all)g Fp(q)591 b Fm(2)544 b Fp(\031)48 b Fq(\(\000)45221 52090 y Fg(\003)45831 52572 y Fq(\))548 b(when)1102 54174 y Fp(!)518 b Fq(=)470 b Fp(@)4655 54373 y Fv(1)5251 54174 y Fp(\013)5 b Fq(\()p Fp(c)p Fq(\))502 b(is)h(irrational,)520 b(and)503 b(b)37 b(ecause)505 b(the)e(necessary)i(and)e(su\016cien)-37 b(t)504 b(condition)g(for)f(the)1102 55768 y(existence)379 b(of)f(in)-37 b(v)-74 b(arian)-37 b(t)378 b(circle)h(is)f(the)f(P)-37 b(eierl's)379 b(barrier)g(function)f(is)f(iden)-37 b(tically)378 b(equal)h(to)e(zero.)1102 57362 y(Due)498 b(to)f(the)h(Lipsc)-37 b(hitz)498 b(prop)37 b(ert)-37 b(y)498 b(of)g(the)g(Aubry)g(set,)511 b(an)-37 b(y)498 b Fp(c)p Fq(-minimal)f(curv)-37 b(e)499 b(can)f(not)f(cross)1102 58976 y Fp(\031)48 b Fq(\()2507 58640 y(~)2424 58976 y(\000)3254 58494 y Fg(\003)3865 58976 y Fq(\),)442 b(so)13167 59793 y Fi(Z)14496 60121 y Fk(k)13905 62811 y Fg(\000)p Fk(k)15807 61601 y Fq(_)-565 b Fp(\015)16299 61800 y Fv(2)16894 61601 y Fq(\()p Fp(t)p Fq(\))p Fp(dt)368 b Fm(\024)h Fq(2)p Fp(\031)343 b Fq(+)296 b Fm(O)37 b Fq(\()p Fm(k)p Fp(P)184 b Fm(k)p Fq(\))p Fp(;)2877 b Fm(8)443 b Fp(k)410 b Fm(2)369 b Fl(Z)36606 61053 y Fv(+)37487 61601 y Fp(:)1102 65243 y Fq(As)380 b Fp(d\015)452 b Fq(is)379 b(semi-asymptotic)g(to)17551 64907 y(~)17468 65243 y(\000,)392 b Fp(d\015)453 b Fq(en)-37 b(ters)379 b(the)g(small)g(neigh)-37 b(b)37 b(orho)g(o)g(d)380 b(of)40224 64907 y(~)40076 65243 y(\006)q(.)569 b(If)378 b Fp(d\015)453 b Fq(do)37 b(es)379 b(not)1102 66837 y(fall)322 b(either)h(on)f(the)g(stable)g(manifold)g(or)g(on)g(the)g(unstable)g (manifold,)346 b(then)322 b(it)g(will)g(go)g(outside)g(of)1102 68431 y(the)405 b(neigh)-37 b(b)37 b(orho)g(o)g(d)406 b(again.)579 b(It)404 b(implies)h(that)f Fp(d\015)479 b Fq(is)405 b(a)h(m)-37 b(ulti-bump)404 b(solution)h(of)g(the)g (Lagrange)1102 70025 y(equation.)804 b(As)513 b(w)-37 b(e)515 b(did)e(in)h(the)f(pro)37 b(of)514 b(of)g(the)g(lemma)g(3.1,) 531 b(w)-37 b(e)515 b(can)f(construct)g(a)g(curv)-37 b(e)514 b Fp(\020)612 b Fq(b)-37 b(y)1102 71619 y(cutting)405 b(o\013)g(all)g(other)g(bumps)f(and)h(lea)-37 b(v)g(e)406 b(only)f(one)g(bump.)578 b(In)404 b(this)h(case)g(the)g Fp(c)p Fq(-action)g(of)f Fp(\020)503 b Fq(is)1102 73233 y(smaller)468 b(than)g(that)e(of)i Fp(\015)74 b Fq(,)473 b(but)467 b(this)f(is)i(absurd.)664 b(Th)-37 b(us,)474 b Fp(d\015)74 b Fq(\()p Fp(t)p Fq(\))409 b Fm(2)g Fp(W)36147 72751 y Fk(s)35963 73609 y(l)5 b(oc)37370 73233 y Fq(\()37970 72897 y(~)37887 73233 y(\000)38717 72751 y Fg(\003)39327 73233 y Fq(\))311 b Fm([)g Fp(W)42791 72751 y Fk(u)42607 73609 y(l)5 b(oc)44014 73233 y Fq(\()44614 72897 y(~)44531 73233 y(\000)45361 72751 y Fg(\003)45971 73233 y Fq(\))p Fm(nf)47899 72897 y Fq(~)47816 73233 y(\000)48646 72751 y Fg(\003)49257 73233 y Fm(g)1102 74965 y Fq(when)444 b Fp(d\015)74 b Fq(\()p Fp(t)p Fq(\))441 b(is)i(in)g(a)g(small)g(neigh) -37 b(b)37 b(orho)g(o)g(d)444 b(of)25451 74629 y(~)25303 74965 y(\006)q(.)1574 b Fb(\003)p eop %%Page: 25 25 25 24 bop 48792 2239 a Ft(25)2696 5427 y Fq(W)-111 b(e)318 b(can)f(see)h(from)f(this)f(lemma)h(that)f(only)h(the)g(prime)g(in)-37 b(tersection)317 b(p)37 b(oin)-37 b(t)317 b(of)g(the)f(stable)h(and) 1102 7041 y(unstable)378 b(manifolds)f(of)h Fm(L)p Fq(\(\000\))f(can)h (b)37 b(e)377 b(the)h(candidates)g(for)32153 6705 y(~)31731 7041 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\))377 b(when)h(there)g(is)f(an)h (in)-37 b(v)-74 b(arian)-37 b(t)1102 8635 y(circle)364 b(con)-37 b(taining)363 b(\000.)564 b(In)362 b(general,)379 b(the)363 b(time-1-section)f Fm(N)196 b Fq(\()p Fp(c)p Fq(\))p Fm(j)33056 8834 y Fk(t)p Fv(=0)34867 8635 y Fm(n)p Fp(\031)48 b Fq(\()p Fm(L)p Fq(\(\000\)\))361 b(is)i(homotopically)1102 10229 y(trivial.)715 b(By)484 b(de\014nition)h(w)-37 b(e)485 b(mean)f(that)g(there)h(exists)f(an)g(op)37 b(en)485 b(neigh)-37 b(b)37 b(orho)g(o)g(d)485 b Fp(U)583 b Fq(=)438 b Fm([)46785 9747 y Fk(m)46785 10575 y(i)p Fv(=1)48572 10229 y Fp(U)49479 10428 y Fk(i)1102 11823 y Fq(of)i Fm(N)196 b Fq(\()p Fp(c)p Fq(\))p Fm(j)5876 12022 y Fk(t)p Fv(=0)8126 11823 y Fq(suc)-37 b(h)440 b(that)f Fp(U)14901 12022 y Fk(i)15631 11823 y Fm(\\)288 b Fp(U)17712 12022 y Fk(j)18641 11823 y Fq(=)369 b Fl(?)440 b Fq(if)f Fp(i)368 b Fm(6)p Fq(=)i Fp(j)76 b Fq(,)440 b Fp(U)27298 12022 y Fv(0)28333 11823 y Fq(is)f(an)h(op)37 b(en)439 b(neigh)-37 b(b)37 b(orho)g(o)g(d)441 b(of)e Fm(L)p Fq(\(\000\))g(and)1102 13417 y(eac)-37 b(h)444 b Fp(U)4998 13616 y Fk(i)5883 13417 y Fq(\()p Fp(i)368 b Fm(6)p Fq(=)i(0\))442 b(is)h(con)-37 b(tractible)444 b(to)e(one)h(p)37 b(oin)-37 b(t.)590 b(In)443 b(this)f(case)i(w)-37 b(e)443 b(ha)-37 b(v)g(e)18112 16355 y Fp(i)18570 16554 y Fg(\003)19181 16355 y Fp(H)20285 16554 y Fv(1)20881 16355 y Fq(\()p Fp(U)m(;)221 b Fl(R)p Fq(\))369 b Fm(\032)g Fq(span\([)p Fp(@)74 b(q)31061 16554 y Fv(1)31657 16355 y Fq(]\))p Fp(;)1102 19387 y Fq(where)446 b Fp(i)e Fq(is)g(the)g(standard)g(inclusion)h(map.)595 b(By)444 b(the)g(Lipsc)-37 b(hitz)445 b(prop)37 b(ert)-37 b(y)445 b(of)41488 19051 y(~)41032 19387 y Fm(A)p Fq(\()p Fp(c)p Fq(\))370 b(=)45898 19051 y(~)45476 19387 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\))443 b(in)1102 21075 y(this)474 b(case,)482 b(w)-37 b(e)475 b(ma)-37 b(y)474 b(c)-37 b(ho)37 b(ose)475 b(b)37 b(ounded,)482 b(m)-37 b(utually)473 b(disjoin)-37 b(t)474 b(op)37 b(en)474 b(sets)38418 20739 y(~)38187 21075 y Fp(U)39094 21274 y Fk(i)40010 21075 y Fq(in)g Fp(T)184 b(M)619 b Fq(suc)-37 b(h)474 b(that)1102 22689 y Fp(\031)2138 22353 y Fq(~)1907 22689 y Fp(U)2814 22888 y Fk(i)3625 22689 y Fq(=)370 b Fp(U)5935 22888 y Fk(i)6820 22689 y Fq(and)442 b Fm([)10519 22353 y Fq(~)10288 22689 y Fp(U)11195 22888 y Fk(i)12006 22689 y Fm(\033)13830 22353 y Fq(~)13408 22689 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\).)589 b(Under)444 b(this)e(assumption)h(w)-37 b(e)444 b(ha)-37 b(v)g(e)1102 25118 y Fr(Lemma)639 b(3.5.)764 b Fo(Assume)580 b Fp(c)557 b Fq(=)h(\()p Fp(c)18667 25317 y Fv(1)19262 25118 y Fp(;)221 b(b)p Fq(\()p Fp(c)21514 25317 y Fv(1)22110 25118 y Fq(\)\))p Fo(,)24806 24782 y Fq(~)24156 25118 y Fm(M)p Fq(\()p Fp(c)p Fq(\))556 b(=)29590 24782 y(~)29506 25118 y(\000)580 b Fo(and)f Fm(N)196 b Fq(\()p Fp(c)p Fq(\))p Fm(j)36864 25317 y Fk(t)p Fv(=0)38675 25118 y Fm(n)p Fp(\031)48 b Fq(\()40744 24782 y(~)40661 25118 y(\000\))579 b Fo(is)g(homotopi-)1102 26712 y(c)-68 b(al)68 b(ly)723 b(trivial.)1354 b(L)-68 b(et)723 b Fp(c)12758 26230 y Fg(0)13947 26712 y Fq(=)816 b(\()p Fp(c)16888 26911 y Fv(1)17484 26712 y Fp(;)221 b(c)18649 26230 y Fg(0)18649 27042 y Fv(2)19245 26712 y Fq(\))722 b Fo(with)g Fp(c)24191 26230 y Fg(0)24191 27042 y Fv(2)25261 26712 y Fm(\000)474 b Fp(b)p Fq(\()p Fp(c)28430 26911 y Fv(1)29026 26712 y Fq(\))816 b Fp(>)g Fq(0)722 b Fo(b)-68 b(eing)723 b(su\016ciently)g(smal)68 b(l.)1354 b(If)1752 28064 y Fq(~)1102 28400 y Fm(M)q Fq(\()p Fp(c)3790 27918 y Fg(0)4162 28400 y Fq(\))536 b Fo(is)i(uniquely)g(er)-68 b(go)g(dic,)553 b(then)537 b(ther)-68 b(e)538 b(exists)g(a)f(neighb)-68 b(orho)g(o)g(d)538 b Fp(N)37333 28599 y Fk(c)37808 28333 y Fe(0)38771 28400 y Fo(of)f Fm(N)41484 28599 y Fv(0)42080 28400 y Fq(\()p Fp(c)43172 27918 y Fg(0)43544 28400 y Fq(\))g Fo(such)g(that)1102 29994 y Fp(i)1560 30193 y Fg(\003)2171 29994 y Fp(H)108 b Fq(\()p Fp(N)4967 30193 y Fk(c)5442 29927 y Fe(0)5867 29994 y Fp(;)221 b Fl(R)p Fq(\))369 b(=)h(0)p Fo(.)1102 32423 y Fr(Pro)42 b(of)p Fq(.)868 b(By)535 b(assumption,)559 b(w)-37 b(e)536 b(can)f(c)-37 b(ho)37 b(ose)24849 32087 y(~)24619 32423 y Fp(U)667 b Fq(=)524 b Fm([)28636 31941 y Fk(m)28636 32769 y(i)p Fv(=0)30653 32087 y Fq(~)30422 32423 y Fp(U)31329 32622 y Fk(i)31772 32423 y Fq(,)558 b(a)535 b(neigh)-37 b(b)37 b(orho)g(o)g(d)537 b(of)44360 32087 y(~)43939 32423 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\))534 b(suc)-37 b(h)1102 34125 y(that)513 b Fp(\031)48 b Fq(\()5604 33789 y(~)5373 34125 y Fp(U)6280 34324 y Fk(i)6722 34125 y Fq(\))342 b Fm(\\)g Fp(\031)48 b Fq(\()10362 33789 y(~)10131 34125 y Fp(U)11038 34324 y Fk(j)11597 34125 y Fq(\))487 b(=)h Fl(?)514 b Fq(if)g Fp(i)487 b Fm(6)p Fq(=)g Fp(j)76 b Fq(,)21177 33789 y(~)20947 34125 y Fp(U)21854 34324 y Fv(0)22963 34125 y Fq(is)514 b(an)g(op)37 b(en)514 b(neigh)-37 b(b)37 b(orho)g(o)g(d)515 b(of)f Fm(L)p Fq(\(\000\))f(and)h (eac)-37 b(h)515 b Fp(U)49479 34324 y Fk(i)1102 35860 y Fq(\()p Fp(i)614 b Fm(6)p Fq(=)h(0\))590 b(is)g(con)-37 b(tractible)591 b(to)f(one)h(p)37 b(oin)-37 b(t.)1032 b(By)590 b(the)g(upp)37 b(er-semi)591 b(con)-37 b(tin)g(uit)g(y)590 b(of)g Fp(c)615 b Fm(!)47080 35525 y Fq(~)46659 35860 y Fm(N)195 b Fq(\()p Fp(c)p Fq(\),)1524 37212 y(~)1102 37548 y Fm(N)h Fq(\()p Fp(c)3480 37066 y Fg(0)3852 37548 y Fq(\))412 b Fm(\032)6458 37212 y Fq(~)6227 37548 y Fp(U)614 b Fq(if)469 b Fp(c)9567 37066 y Fg(0)9567 37878 y Fv(2)10475 37548 y Fm(\000)313 b Fp(b)p Fq(\()p Fp(c)13483 37747 y Fv(1)14079 37548 y Fq(\))468 b(su\016cien)-37 b(tly)469 b(small.)670 b(W)-111 b(e)469 b(claim)h(that)e(for)i(eac)-37 b(h)470 b Fp(z)g Fm(2)42289 37212 y Fq(~)42059 37548 y Fp(U)42966 37747 y Fv(0)43874 37548 y Fm(\\)45494 37212 y Fq(~)45072 37548 y Fm(N)196 b Fq(\()p Fp(c)47450 37066 y Fg(0)47822 37548 y Fq(\),)475 b Fm(9)1102 39236 y Fq(an)522 b(in)-37 b(teger)522 b Fp(k)42 b Fq(\()p Fp(z)58 b Fq(\))498 b Fm(2)h Fl(Z)12712 39435 y Fv(+)14115 39236 y Fq(suc)-37 b(h)521 b(that)g Fp(\036)20937 38754 y Fk(k)22 b Fv(\()p Fk(z)38 b Fv(\))22967 39236 y Fq(\()p Fp(z)58 b Fq(\))646 b Fp(=)-811 b Fm(2)26791 38900 y Fq(~)26561 39236 y Fp(U)27468 39435 y Fv(0)28585 39236 y Fq(and)521 b(there)h(is)f(a)g(uniform)h(upp) 37 b(er)521 b(b)37 b(ound)1102 40924 y Fp(K)514 b Fm(2)418 b Fl(Z)4934 41123 y Fv(+)6288 40924 y Fq(for)473 b(all)g(these)g Fp(k)42 b Fq(\()p Fp(z)58 b Fq(\).)678 b(If)472 b(this)g(is)h(not)f (true,)480 b(for)473 b(an)-37 b(y)473 b Fp(k)460 b(>)418 b Fq(0,)481 b Fp(k)459 b Fm(2)419 b Fl(Z)472 b Fq(there)h(is)f Fp(z)46042 41123 y Fk(k)47114 40924 y Fm(2)48649 40588 y Fq(~)48418 40924 y Fp(U)49325 41123 y Fv(0)1102 42612 y Fq(suc)-37 b(h)581 b(that)f Fp(\036)8043 42130 y Fk(l)8452 42612 y Fq(\()p Fp(z)9587 42811 y Fk(k)10240 42612 y Fq(\))598 b Fm(2)13070 42276 y Fq(~)12839 42612 y Fp(U)13746 42811 y Fv(0)14342 42612 y Fq(,)615 b Fm(8)p Fq(0)599 b Fm(\024)g Fp(l)625 b Fm(\024)599 b Fp(k)42 b Fq(.)1004 b(Let)581 b Fp(\027)26893 42811 y Fk(k)28127 42612 y Fq(b)37 b(e)581 b(a)g(probabilit)-37 b(y)581 b(measure)h(distributed) 1102 44206 y(ev)-37 b(enly)477 b(on)e Fp(\036)7899 43724 y Fk(t)8367 44206 y Fq(\()p Fp(z)58 b Fq(\))475 b(\(0)423 b Fm(\024)h Fp(t)f Fm(\024)g Fp(k)42 b Fq(\))475 b(and)g(let)h Fp(k)464 b Fm(!)424 b(1)p Fq(,)484 b(w)-37 b(e)476 b(\014nd)f(there)h (is)g(an)f(accum)-37 b(ulation)477 b(p)37 b(oin)-37 b(t)1102 45894 y Fp(\027)85 b Fq(,)406 b(supp\()p Fp(\027)85 b Fq(\))368 b Fm(\032)9132 45558 y Fq(~)8901 45894 y Fp(U)9808 46093 y Fv(0)10404 45894 y Fq(.)575 b(Ob)-37 b(viously)-111 b(,)407 b Fp(\027)454 b Fm(2)20828 45558 y Fq(~)20178 45894 y Fm(M)p Fq(\()p Fp(c)p Fq(\).)574 b(As)398 b(there)f(is)g (normally)h(h)-37 b(yp)37 b(erb)g(olic)399 b(structure)e(on)1250 47246 y(~)1102 47581 y(\006)q(,)413 b(the)405 b(in)-37 b(v)-74 b(arian)-37 b(t)407 b(set)e(in)14413 47246 y(~)14182 47581 y Fp(U)15089 47780 y Fv(0)16090 47581 y Fq(m)-37 b(ust)406 b(b)37 b(e)406 b(on)23071 47246 y(~)22923 47581 y(\006)q(,)413 b(it)405 b(follo)-37 b(ws)406 b(that)33771 47246 y(~)33121 47581 y Fm(M)q Fq(\()p Fp(c)p Fq(\))368 b Fm(\032)38243 47246 y Fq(~)38096 47581 y(\006,)413 b(but)405 b(it)h(con)-37 b(tradicts)1102 49175 y(the)443 b(de\014nition)g(of)g Fp(I)104 b Fq(\()p Fp(c)12587 49374 y Fv(1)13182 49175 y Fq(\).)2696 51605 y(By)401 b(the)f(upp)37 b(er)401 b(semi-con)-37 b(tin)g(uit)g(y)400 b(of)h Fp(c)369 b Fm(!)24542 51269 y Fq(~)24121 51605 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\))399 b(and)h(the)h(assumption)f(on)g(the)h(in)-37 b(tersection)1102 53293 y(of)485 b(the)g(stable)g(and)f(unstable)h (manifolds)g(w)-37 b(e)485 b(see)h(that)30453 52957 y(~)30032 53293 y Fm(N)195 b Fq(\()p Fp(c)32409 52811 y Fg(0)32781 53293 y Fq(\))p Fm(n)34193 52957 y Fq(~)33962 53293 y Fp(U)34869 53492 y Fv(0)35950 53293 y Fq(can)485 b(b)37 b(e)485 b(co)-37 b(v)g(ered)487 b(b)-37 b(y)484 b(\014nite)1102 54887 y(m)-37 b(utually)559 b(disjoin)-37 b(t)558 b(op)37 b(en)559 b(sets,)588 b(eac)-37 b(h)560 b(of)e(them)h(is)g(homotopic)g (to)f(a)h(p)37 b(oin)-37 b(t.)938 b(As)559 b(eac)-37 b(h)560 b(p)37 b(oin)-37 b(t)1102 56501 y(in)2906 56165 y(~)2675 56501 y Fp(U)3582 56700 y Fv(0)4644 56501 y Fq(shall)466 b(go)h(outside)f(under)g(the)g(time-1-map)g Fp(\036)28175 56019 y Fv(1)28771 56501 y Fq(,)472 b(the)466 b(whole)36132 56165 y(~)35710 56501 y Fm(N)196 b Fq(\()p Fp(c)38088 56019 y Fg(0)38460 56501 y Fq(\))465 b(can)i(b)37 b(e)466 b(co)-37 b(v)g(ered)468 b(b)-37 b(y)1102 58189 y(\014nite)486 b(m)-37 b(utually)485 b(disjoin)-37 b(t,)495 b(homotopically)486 b(trivial)g(op)37 b(en)486 b(sets.)718 b(Because)40246 57853 y(~)39596 58189 y Fm(M)p Fq(\()p Fp(c)42283 57706 y Fg(0)42656 58189 y Fq(\))485 b(is)g(assumed)1102 59876 y(uniquely)505 b(ergo)37 b(dic)506 b(w)-37 b(e)505 b(obtain)f(from)g(the)g(lemma)h(2.5)g(that)32895 59541 y(~)32473 59876 y Fm(N)196 b Fq(\()p Fp(c)34851 59394 y Fg(0)35223 59876 y Fq(\))471 b(=)38172 59541 y(~)37716 59876 y Fm(A)o Fq(\()p Fp(c)39868 59394 y Fg(0)40240 59876 y Fq(\).)775 b(The)505 b(Lipsc)-37 b(hitz)1102 61470 y(prop)37 b(ert)-37 b(y)444 b(of)f Fm(A)p Fq(\()p Fp(c)10200 60988 y Fg(0)10571 61470 y Fq(\))g(guaran)-37 b(tees)444 b(that)e Fm(N)196 b Fq(\()p Fp(c)23403 60988 y Fg(0)23775 61470 y Fq(\))368 b(=)h Fm(A)p Fq(\()p Fp(c)28215 60988 y Fg(0)28587 61470 y Fq(\))442 b(is)h(also)g(homotopically)h (trivial.)2165 b Fb(\003)1102 64639 y Fr(4,)510 b(Some)f(Barrier)g (functions)2696 66995 y Fq(In)499 b(this)f(section)h(w)-37 b(e)500 b(consider)g(that)e(co-homology)i(class)f Fp(c)462 b Fq(=)g(\()p Fp(c)36291 67194 y Fv(1)36887 66995 y Fp(;)221 b(b)p Fq(\()p Fp(c)39139 67194 y Fv(1)39734 66995 y Fq(\)\))498 b(suc)-37 b(h)499 b(that)47709 66659 y(~)47253 66995 y Fm(A)o Fq(\()p Fp(c)p Fq(\))1102 68589 y(con)-37 b(tains)531 b(a)g(2-torus)f(in)h Fl(T)14719 68107 y Fv(2)15668 68589 y Fm(\002)353 b Fl(T)p Fq(,)552 b(i.e.)854 b(its)530 b(time-1-sections)h(has)f(an)h(in)-37 b(v)-74 b(arian)-37 b(t)531 b(circle)h(on)e(the)1102 70183 y(cylinder,)396 b(and)382 b(study)g(the)g(relev)-74 b(an)-37 b(t)384 b(barrier)f(functions)g(in)-37 b(tro)37 b(duced)383 b(in)f([Ma4].)570 b(The)383 b(study)f(for)1102 71777 y Fp(c)369 b Fq(=)g(\()p Fp(c)4540 71976 y Fv(1)5136 71777 y Fp(;)221 b(a)p Fq(\()p Fp(c)7520 71976 y Fv(1)8116 71777 y Fq(\)\))403 b(is)h(the)g(same.)578 b(As)405 b(w)-37 b(e)405 b(ha)-37 b(v)g(e)405 b(assumed,)412 b(the)405 b(rotation)f(n)-37 b(um)g(b)37 b(er)405 b(of)f(this)g(circle) h(is)1102 73371 y(irrational.)568 b(T)-111 b(o)372 b(go)h(further)f (for)g(our)g(pro)37 b(of,)387 b(let)371 b(us)h(turn)g(bac)-37 b(k)372 b(to)g(the)g(Hamiltonian)g(formalism)1102 74965 y(temp)37 b(orarily)444 b(to)e(lo)37 b(ok)444 b(at)e(something.)p eop %%Page: 26 26 26 25 bop 1102 2239 a Ft(26)2696 5427 y Fq(Let)553 b(\010)6145 5626 y Fk(H)7704 5427 y Fq(=)e(\010)10247 4945 y Fv(1)10247 5793 y Fk(H)11807 5427 y Fq(b)37 b(e)552 b(the)g(time-1-map)g(of)g(the) g(Hamiltonian)g(\015o)-37 b(w)553 b(\010)38917 4945 y Fk(t)38917 5793 y(H)39925 5427 y Fq(,)579 b(it)551 b(has)i(in)-37 b(v)-74 b(arian)-37 b(t)1102 7021 y(cylinder)354 b(\006.)561 b(Restricted)353 b(to)f(the)g(cylinder)i(\006)f(this)f(map)h(is)f (clearly)i(t)-37 b(wist)352 b(and)h(area-preserving,)1102 8615 y(th)-37 b(us)403 b(the)g(in)-37 b(v)-74 b(arian)-37 b(t)404 b(circle)h(\000)e(is)g(at)g(least)h(of)f(Lipsc)-37 b(hitz.)578 b(When)403 b Fp(P)554 b Fq(=)369 b(0,)411 b(w)-37 b(e)404 b(ha)-37 b(v)g(e)404 b(the)f(cylinder)1102 10209 y Fl(T)372 b Fm(\002)h Fl(R)f Fm(\002)h(f)p Fp(q)7760 10408 y Fv(2)8918 10209 y Fq(=)562 b Fp(p)11181 10408 y Fv(2)12339 10209 y Fq(=)g(0)p Fm(g)c Fq(as)h(the)f(normally)h(h)-37 b(yp)37 b(erb)g(olic)560 b(manifold)f(for)f(\010)40996 10408 y Fk(f)104 b Fv(+)p Fk(g)43052 10209 y Fq(,)588 b(eac)-37 b(h)559 b(orbit)1102 11803 y(on)422 b(this)g(manifold)g(mak) -37 b(es)423 b(up)e(an)h(in)-37 b(v)-74 b(arian)-37 b(t)423 b(circle)g(and)f(has)g(zero)i(Ly)-37 b(apuno)g(v)422 b(exp)37 b(onen)-37 b(t)423 b(only)-111 b(.)1102 13397 y(Both)559 b(stable)g(and)g(unstable)g(manifold)g(ha)-37 b(v)g(e)560 b(t)-37 b(w)g(o)559 b(branc)-37 b(hes,)589 b(eac)-37 b(h)560 b(of)f(them)g(has)g(in)-37 b(v)-74 b(arian)-37 b(t)1102 15011 y(\014bration)399 b Fm(f)p Fp(q)7706 15210 y Fv(1)8671 15011 y Fq(=)369 b Fp(p)10741 15210 y Fv(1)11706 15011 y Fq(=)g(0)p Fp(;)221 b(p)15030 15210 y Fv(2)15996 15011 y Fq(=)17699 14675 y(~)17398 15011 y Fp(G)18442 14529 y Fg(\006)19339 15011 y Fq(\()p Fp(q)20449 15210 y Fv(2)21045 15011 y Fq(\))p Fm(g)398 b Fq(if)g(w)-37 b(e)399 b(use)f Fm(f)p Fp(q)29215 15210 y Fv(2)29812 15011 y Fp(;)30703 14675 y Fq(~)30402 15011 y Fp(G)31446 14529 y Fg(\006)32343 15011 y Fq(\()p Fp(q)33453 15210 y Fv(2)34048 15011 y Fq(\))p Fm(g)g Fq(to)g(denote)h(the)f(homo) 37 b(clinic)1102 16605 y(lo)g(ops)372 b(of)g(\010)6871 16804 y Fk(g)7861 16605 y Fq(in)g(the)g(space)g(of)g(\()p Fp(q)17587 16804 y Fv(2)18183 16605 y Fp(;)221 b(p)19441 16804 y Fv(2)20037 16605 y Fq(\).)566 b(Under)373 b(the)f(small)g(p)37 b(erturbation,)386 b(the)371 b(in)-37 b(v)-74 b(arian)-37 b(t)373 b(circle)1102 18199 y(on)547 b(\006)g(is)f(the)h(graph)g(of)g (small)f(functions,)573 b(i.e.)902 b(\000)542 b(=)g Fm(f)p Fp(q)30484 18398 y Fv(1)31622 18199 y Fm(2)f Fl(T)p Fp(;)221 b(p)542 b Fq(=)g Fp(p)37978 18398 y Fv(\000)38702 18199 y Fq(\()p Fp(q)39812 18398 y Fv(1)40407 18199 y Fq(\))p Fp(;)221 b(q)42107 18398 y Fv(2)43245 18199 y Fq(=)542 b Fp(q)45413 18398 y Fv(2\000)46666 18199 y Fq(\()p Fp(q)47776 18398 y Fv(1)48371 18199 y Fq(\))p Fm(g)p Fq(.)1102 19793 y(F)-111 b(rom)557 b(the)e(theory)h(of)g(normally)g(h)-37 b(yp)37 b(erb)g(olic)556 b(manifolds)g(w)-37 b(e)556 b(kno)-37 b(w)556 b(that)f(the)h(\014bration)f(has)1102 21387 y Fp(C)2146 20905 y Fk(r)29 b Fg(\000)p Fv(2)4095 21387 y Fq(-smo)37 b(othness)383 b(on)g(the)h(base)f(p)37 b(oin)-37 b(ts.)571 b(As)383 b(\000)h(is)f(an)h(in)-37 b(v)-74 b(arian)-37 b(t)384 b(circle,)396 b(all)384 b(stable)f (\(unstable\))1102 22981 y(\014b)37 b(ers)443 b(with)f(base)h(p)37 b(oin)-37 b(ts)441 b(on)i(\000)f(constitute)g(the)g(lo)37 b(cal)443 b(stable)f(\(unstable\))f(manifold)h Fp(W)46546 22343 y Fk(s;u)46362 23372 y(H)48058 22981 y Fq(\(\000\))1102 24575 y(of)410 b(\000.)580 b(Both)409 b(the)h(stable)g(and)f(the)h (unstable)g(manifold)g(ha)-37 b(v)g(e)410 b(t)-37 b(w)g(o)410 b(branc)-37 b(hes)411 b(corresp)37 b(onding)411 b(to)1102 26169 y(\()p Fp(c)2194 26368 y Fv(1)2790 26169 y Fp(;)221 b(b)p Fq(\()p Fp(c)5042 26368 y Fv(1)5638 26169 y Fq(\)\))506 b(and)h(\()p Fp(c)10917 26368 y Fv(1)11512 26169 y Fp(;)221 b(a)p Fq(\()p Fp(c)13896 26368 y Fv(1)14492 26169 y Fq(\)\))506 b(resp)37 b(ectiv)-37 b(ely)-111 b(.)784 b(Let)508 b(us)e(consider)i (the)f(branc)-37 b(h)508 b(corresp)37 b(onding)508 b(to)1102 27763 y(\()p Fp(c)2194 27962 y Fv(1)2790 27763 y Fp(;)221 b(b)p Fq(\()p Fp(c)5042 27962 y Fv(1)5638 27763 y Fq(\)\).)758 b(In)499 b(the)g(co)-37 b(v)g(ering)501 b(space)f Fp(T)184 b Fq(\()p Fl(T)333 b Fm(\002)g Fl(R)p Fq(\),)513 b(one)500 b(lift)f(of)g(unstable)g(manifold)h(originates)1102 29357 y(from)600 b Fm(f)p Fp(p)631 b Fq(=)g Fp(p)8694 29556 y Fv(\000)9418 29357 y Fq(\()p Fp(q)10528 29556 y Fv(1)11123 29357 y Fq(\))p Fp(;)221 b(q)12823 29556 y Fv(2)14049 29357 y Fq(=)631 b Fp(q)16306 29556 y Fv(2\000)17559 29357 y Fq(\()p Fp(q)18669 29556 y Fv(1)19264 29357 y Fq(\))p Fm(g)600 b Fq(and)g(extends)g(to)f(righ)-37 b(t,)639 b(one)600 b(lift)g(of)f(stable)h(manifold)1102 30951 y(originates)513 b(from)f Fm(f)p Fp(p)485 b Fq(=)g Fp(p)14446 31150 y Fv(\000)15169 30951 y Fq(\()p Fp(q)16279 31150 y Fv(1)16875 30951 y Fq(\))p Fp(;)221 b(q)18575 31150 y Fv(2)19655 30951 y Fq(=)484 b Fp(q)21765 31150 y Fv(2\000)23018 30951 y Fq(\()p Fp(q)24128 31150 y Fv(1)24724 30951 y Fq(\))341 b(+)g(2)p Fp(\031)48 b Fm(g)512 b Fq(and)g(extends)h(to)e (left.)798 b(When)513 b Fp(P)669 b Fq(=)484 b(0,)1102 32545 y(these)493 b(t)-37 b(w)g(o)493 b(manifolds)g(totally)f(coincide) i(with)e(eac)-37 b(h)494 b(other)f(and)g(k)-37 b(eep)493 b(horizon)-37 b(tal)494 b(when)f(0)452 b Fm(\024)1102 34139 y Fp(q)1695 34338 y Fv(2)2697 34139 y Fm(\024)406 b Fq(2)p Fp(\031)48 b Fq(.)657 b(Th)-37 b(us,)470 b(for)c(suitably)e (small)h Fp(a)406 b(>)g Fq(0,)471 b(there)465 b(exists)g Fp(\017)406 b(>)g Fq(0)465 b(suc)-37 b(h)465 b(that)f(if)h Fm(k)p Fp(P)184 b Fm(k)406 b(\024)g Fp(\017)465 b Fq(the)1102 35733 y(unstable)537 b(manifold)f(k)-37 b(eeps)538 b(horizon)-37 b(tal)537 b(in)f(the)h(region)g Fm(f)p Fp(q)31449 35932 y Fv(2\000)32702 35733 y Fq(\()p Fp(q)33812 35932 y Fv(1)34407 35733 y Fq(\))525 b Fm(\024)g Fp(q)37600 35932 y Fv(2)38721 35733 y Fm(\024)h Fq(2)p Fp(\031)405 b Fm(\000)358 b Fp(a)p Fm(g)537 b Fq(and)f(the)1102 37327 y(stable)447 b(manifold)g(k)-37 b(eeps)447 b(horizon)-37 b(tal)447 b(in)g(the)f(region)i Fm(f)p Fp(a)375 b Fm(\024)h Fp(q)31830 37526 y Fv(2)32801 37327 y Fm(\024)f Fp(q)34802 37526 y Fv(2\000)36055 37327 y Fq(\()p Fp(q)37165 37526 y Fv(1)37760 37327 y Fq(\))297 b(+)h(2)p Fp(\031)48 b Fm(g)p Fq(,)448 b(i.e.)602 b(they)446 b(are)1102 38921 y(the)d(graphs)g(of)g(some)h (functions)e(in)h(the)g(relev)-74 b(an)-37 b(t)444 b(regions,)10725 42194 y Fp(W)12164 41646 y Fk(u)12858 42194 y Fq(\(\000\))368 b(=)q Fm(f)p Fp(q)48 b(;)221 b(p)18687 41646 y Fk(u)19382 42194 y Fq(\()p Fp(q)48 b Fq(\))367 b(:)i Fp(q)22755 42393 y Fv(1)23720 42194 y Fm(2)g Fl(T)p Fp(;)442 b(q)27265 42393 y Fv(2\000)28518 42194 y Fq(\()p Fp(q)29628 42393 y Fv(1)30223 42194 y Fq(\))369 b Fm(\024)g Fp(q)33104 42393 y Fv(2)34069 42194 y Fm(\024)g Fq(2)p Fp(\031)343 b Fm(\000)296 b Fp(a)p Fm(g)p Fp(;)10851 44254 y(W)12290 43705 y Fk(s)12858 44254 y Fq(\(\000\))368 b(=)q Fm(f)p Fp(q)48 b(;)221 b(p)18687 43705 y Fk(s)19255 44254 y Fq(\()p Fp(q)48 b Fq(\))368 b(:)h Fp(q)22629 44453 y Fv(1)23594 44254 y Fm(2)f Fl(T)p Fp(;)443 b(a)369 b Fm(\024)g Fp(q)29612 44453 y Fv(2)30577 44254 y Fm(\024)g Fp(q)32572 44453 y Fv(2\000)33825 44254 y Fq(\()p Fp(q)34935 44453 y Fv(1)35530 44254 y Fq(\))295 b(+)g(2)p Fp(\031)48 b Fm(g)p Fp(:)7019 b Fq(\(4.1\))1102 47678 y(Although)420 b(eac)-37 b(h)420 b(stable)f(\(unstable\))f(\014b)37 b(er)419 b(has)h Fp(C)26494 47196 y Fk(r)29 b Fg(\000)p Fv(2)28442 47678 y Fq(-smo)37 b(othness,)424 b(the)419 b(base)g(p)37 b(oin)-37 b(ts)419 b(of)g(these)1102 49272 y(\014b)37 b(ers)476 b(fall)g(on)g(a)f(circle)i(for)f(whic)-37 b(h)477 b(w)-37 b(e)476 b(can)g(only)g(assume)g(the)f(Lipsc)-37 b(hitz)476 b(smo)37 b(othness,)484 b(these)1102 50866 y(manifolds)624 b(are)g(at)e(least)i(of)f(Lipsc)-37 b(hitz,)669 b(i.e.)1131 b Fp(p)26249 50384 y Fk(s;u)27761 50866 y Fq(\()p Fp(q)48 b Fq(\))622 b(in)h(\(4.1\))g(are)g(at)g(least)h(of)f (Lipsc)-37 b(hitz.)1102 52460 y(W)-111 b(e)564 b(c)-37 b(ho)37 b(ose)565 b(suitably)d(small)i Fp(a)569 b(>)h Fq(0)563 b(suc)-37 b(h)564 b(that)e(the)h(time)h(for)f(an)-37 b(y)563 b Fp(d\015)38556 52659 y Fv(2)39716 52460 y Fq(to)f(cross)i (the)f(strip)1102 54054 y Fm(f)p Fp(a)514 b Fm(\024)f Fp(q)5121 54253 y Fv(2)6231 54054 y Fm(\024)g Fq(2)p Fp(\031)401 b Fm(\000)353 b Fp(a)p Fm(g)530 b Fq(is)f(longer)h(than)f (1.)850 b(Suc)-37 b(h)530 b(assumption)f(is)h(feasible)g(as)f(\010) 42735 53572 y Fk(t)42735 54420 y(H)44272 54054 y Fq(is)h(a)f(small)1102 55648 y(p)37 b(erturbation)443 b(of)g(\010)11367 55166 y Fk(t)11367 56025 y(f)104 b Fv(+)p Fk(g)13866 55648 y Fq(for)443 b(whic)-37 b(h)444 b(this)e(assumption)h(is)f(clearly)i (true.)2696 58143 y(If)538 b(there)g(is)f(another)i(in)-37 b(v)-74 b(arian)-37 b(t)538 b(circle)h(\000)24080 58342 y Fv(1)25214 58143 y Fq(v)-37 b(ery)538 b(close)h(to)e(\000,)562 b(b)-37 b(y)537 b(the)h(smo)37 b(othness)538 b(of)f(the)1102 59737 y(in)-37 b(v)-74 b(arian)-37 b(t)507 b(\014bration)f(w)-37 b(e)507 b(see)f Fp(W)17863 59098 y Fk(s;u)17679 60127 y(H)19375 59737 y Fq(\(\000)20722 59936 y Fv(1)21318 59737 y Fq(\))f(can)i(also)f(k)-37 b(eep)507 b(horizon)-37 b(tal)506 b(in)g(the)g(relev)-74 b(an)-37 b(t)507 b(region.)1102 61331 y(Let)367 b(\000\()p Fp(A)p Fq(\))f(b)37 b(e)367 b(the)g(highest)f(circle)i(on)f(\006)g(where)h Fp(p)26006 61530 y Fv(1)26971 61331 y Fm(\024)h Fp(A)p Fq(,)383 b(let)366 b(\000\()p Fp(B)67 b Fq(\))365 b(b)37 b(e)367 b(the)g(lo)-37 b(w)g(est)367 b(circle)h(where)1102 62925 y Fp(p)1770 63124 y Fv(1)2746 62925 y Fm(\025)380 b Fp(B)67 b Fq(.)608 b(As)449 b(all)h(in)-37 b(v)-74 b(arian)-37 b(t)449 b(circles)i(on)e(\006)g(mak)-37 b(e)450 b(up)f(a)g(closed)h (set,)h(it)e(is)g(reasonable)h(to)f(assert)1102 64519 y(that)350 b(w)-37 b(e)350 b(ha)-37 b(v)g(e)351 b(some)f Fp(\017)369 b(>)g Fq(0)350 b(suc)-37 b(h)351 b(that)e(if)g Fm(k)p Fp(P)184 b Fm(k)370 b(\024)f Fp(\017)p Fq(,)g(the)350 b(stable)f(and)h(unstable)g(manifolds)g(of)g(all)1102 66113 y(\000)424 b(b)37 b(et)-37 b(w)g(een)424 b(\000\()p Fp(A)p Fq(\))e(and)h(\000\()p Fp(B)67 b Fq(\))422 b(can)h(k)-37 b(eep)424 b(horizon)-37 b(tal)424 b(in)f(the)g(region)h Fm(f)p Fp(a)369 b Fm(\024)g Fp(q)39809 66312 y Fv(2)40774 66113 y Fm(\024)g Fp(q)42769 66312 y Fv(2\000)44022 66113 y Fq(\()p Fp(q)45132 66312 y Fv(1)45727 66113 y Fq(\))255 b(+)h(2)p Fp(\031)48 b Fm(g)1102 67707 y Fq(and)443 b Fm(f)p Fp(q)4942 67906 y Fv(2\000)6195 67707 y Fq(\()p Fp(q)7305 67906 y Fv(1)7901 67707 y Fq(\))368 b Fm(\024)h Fp(q)10781 67906 y Fv(2)11746 67707 y Fm(\024)g Fq(2)p Fp(\031)344 b Fm(\000)295 b Fp(a)p Fm(g)443 b Fq(resp)37 b(ectiv)-37 b(ely)-111 b(.)2696 70183 y(As)486 b(the)f(Hamiltonian)h (system)g(under)g(study)f(has)g(standard)h(symplectic)g(structure,)496 b(eac)-37 b(h)1102 71777 y(horizon)g(tal)464 b(Lagrangian)g (sub-manifold)f(is)f(a)h(graph)h(of)e(some)i(closed)f(1-form)g (de\014ned)h(on)f Fp(M)145 b Fq(.)1102 73371 y(W)-111 b(e)517 b(kno)-37 b(w)517 b(that)e(the)h(stable)f(\(unstable\))g (manifold)h(of)g(some)g(smo)37 b(oth)516 b(isotropic)g(manifold)g(is) 1102 74965 y(a)555 b(Lagrangian)i(manifold,)583 b(therefore,)h(if)554 b(w)-37 b(e)556 b(use)f(\()p Fp(q)48 b(;)221 b(p)p Fq(\()p Fp(q)48 b Fq(\)\))553 b(\()p Fp(p)p Fq(\()p Fp(q)48 b Fq(\))555 b Fm(2)g Fp(C)38353 74483 y Fv(1)38950 74965 y Fq(\))f(to)h(denote)g(suc)-37 b(h)555 b(a)p eop %%Page: 27 27 27 26 bop 48792 2239 a Ft(27)1102 5427 y Fq(smo)37 b(oth)443 b(manifold,)g(then)22239 6780 y Fp(@)74 b(p)23686 6979 y Fv(1)p 22239 7373 2044 54 v 22277 8590 a Fp(@)g(q)23649 8789 y Fv(2)24811 7679 y Fq(=)26372 6780 y Fp(@)g(p)27819 6979 y Fv(2)p 26372 7373 V 26410 8590 a Fp(@)g(q)27782 8789 y Fv(1)28575 7679 y Fp(;)18247 b Fq(\(4.2\))1102 11032 y(it)443 b(follo)-37 b(ws)443 b(that)f(there)i(exists)e(a)h Fp(C)18887 10550 y Fv(2)19484 11032 y Fq(-function)f Fp(S)77 b Fq(\()p Fp(q)48 b Fq(\))441 b(and)i(constan)-37 b(t)442 b(v)-37 b(ector)444 b Fp(c)369 b Fm(2)g Fl(R)43224 10550 y Fv(2)44263 11032 y Fq(suc)-37 b(h)443 b(that)15896 14053 y Fp(@)74 b(S)p 15747 14646 1968 54 v 15747 15863 a(@)g(q)17119 16062 y Fv(1)18169 14952 y Fq(+)296 b Fp(c)20073 15151 y Fv(1)21037 14952 y Fq(=)370 b Fp(p)23108 15151 y Fv(1)23704 14952 y Fp(;)27259 14053 y(@)74 b(S)p 27110 14646 V 27110 15863 a(@)g(q)28482 16062 y Fv(2)29533 14952 y Fq(+)295 b Fp(c)31436 15151 y Fv(2)32401 14952 y Fq(=)369 b Fp(p)34471 15151 y Fv(2)35067 14952 y Fp(:)11755 b Fq(\(4.3\))1102 18969 y(If)577 b(w)-37 b(e)578 b(consider)g(the)f (manifold)g(as)g(the)g(graph)h(of)f(some)g(closed)h(1-form,)611 b Fp(c)592 b Fm(2)h Fp(H)43913 18487 y Fv(1)44509 18969 y Fq(\()p Fp(M)71 b(;)221 b Fl(R)p Fq(\))576 b(is)1102 20563 y(the)455 b(cohomology)h(class)f(of)f(this)g(closed)i(1-form.)626 b(Since)455 b(Lipsc)-37 b(hitz)455 b(function)g(is)f(di\013eren)-37 b(tiable)1102 22157 y(almost)455 b(ev)-37 b(erywhere,)460 b(w)-37 b(e)455 b(claim)h(that)e(there)h(exists)f(a)h Fp(C)30357 21675 y Fv(1)p Fk(;)p Fv(1)31798 22157 y Fq(-function)g Fp(S)530 b Fq(so)455 b(that)f(\(4.3\))g(holds,)1102 23751 y(here)444 b(w)-37 b(e)444 b(use)f Fp(C)9279 23269 y Fk(k)22 b(;\013)11382 23751 y Fq(to)442 b(denote)i(those)e(functions)h (whose)h Fp(k)42 b Fq(-th)441 b(deriv)-74 b(ativ)-37 b(e)444 b(is)f(of)g Fp(\013)5 b Fq(-H\177)-664 b(older.)1102 26182 y Fr(Lemma)736 b(4.1.)764 b Fo(L)-68 b(et)657 b Fq(\000)g Fo(b)-68 b(e)657 b(an)f(invariant)i(cir)-68 b(cle)657 b(on)g(the)g(cylinder)g Fq(\006)p Fo(,)703 b(let)657 b Fp(W)42687 25700 y Fk(s;u)44199 26182 y Fq(\(\000\))f Fo(b)-68 b(e)656 b(its)1102 27776 y(stable)613 b(\(unstable\))f (manifold,)647 b(which)612 b(is)g(horizontal)g(over)g(a)g(c)-68 b(onne)g(cte)g(d)611 b(op)-68 b(en)612 b(set)g Fp(U)761 b Fm(\032)616 b Fp(M)1102 29370 y Fo(with)506 b Fp(\031)48 b Fq(\(\000\))424 b Fm(2)g Fp(U)145 b Fo(,)513 b(then)506 b(ther)-68 b(e)507 b(exists)g Fp(C)21430 28888 y Fv(1)p Fk(;)p Fv(1)23377 29370 y Fo(functions)f Fp(S)30070 28888 y Fk(s;u)31581 29370 y Fo(:)672 b Fp(U)569 b Fm(!)425 b Fl(R)506 b Fo(and)g(a)f(c)-68 b(onstant)507 b(ve)-68 b(ctor)1102 30964 y Fp(c)369 b Fm(2)g Fl(R)4260 30482 y Fv(2)5331 30964 y Fo(such)476 b(that)f Fm(f)p Fp(W)13171 30326 y Fk(s;u)12987 31354 y(H)15052 30964 y Fq(:)369 b Fp(q)416 b Fm(2)369 b Fp(U)145 b Fm(g)369 b Fq(=)g Fm(f)p Fq(\()p Fp(q)48 b(;)221 b(dS)25536 30482 y Fk(s;u)27047 30964 y Fq(\()p Fp(q)48 b Fq(\)\))293 b(+)j Fp(c)368 b Fq(:)591 b Fp(q)416 b Fm(2)369 b Fp(U)145 b Fm(g)p Fo(.)1102 33395 y Fr(Pro)42 b(of)p Fq(:)861 b(Let)578 b(us)g(consider)h(the)e(case)i(for)f(stable)g(manifold.)995 b(By)578 b(the)g(condition)g(there)g(is)g(a)1102 34989 y(Lipsc)-37 b(hitz)536 b(function)f Fp(p)524 b Fq(=)f(\()p Fp(p)16094 35188 y Fv(1)16690 34989 y Fp(;)221 b(p)17948 35188 y Fv(2)18544 34989 y Fq(\):)775 b Fp(U)668 b Fm(!)523 b Fp(R)24650 34507 y Fv(2)25781 34989 y Fq(suc)-37 b(h)536 b(that)e Fp(W)33279 34507 y Fk(s)33847 34989 y Fq(\(\000\))522 b(=)h Fm(f)p Fq(\()p Fp(q)48 b(;)221 b(p)40869 34507 y Fk(s)41437 34989 y Fq(\()p Fp(q)48 b Fq(\)\))522 b(:)744 b Fp(q)571 b Fm(2)522 b Fp(U)145 b Fm(g)p Fq(.)1102 36583 y(Let)544 b Fp(\015)617 b Fq(b)37 b(e)544 b(a)f(closed)i(curv)-37 b(e)544 b(whic)-37 b(h)545 b(is)e(the)g(b)37 b(oundary)544 b(of)g(some)g(top)37 b(ological)544 b(disk)g Fp(\033)591 b Fq(on)543 b Fp(W)48984 36101 y Fk(s)49552 36583 y Fq(.)1102 38177 y(Since)467 b Fp(\015)539 b Fq(is)465 b(on)h(the)g(stable)f (manifold,)472 b(\010)22080 37695 y Fk(k)22080 38543 y(H)23088 38177 y Fq(\()p Fp(\015)74 b Fq(\))464 b(approac)-37 b(hes)467 b(uniformly)f(to)f(\000,)472 b(it)465 b(implies)h(that)1102 39803 y(\010)2061 39321 y Fk(k)2061 40169 y(H)3069 39803 y Fq(\()p Fp(\015)74 b Fq(\))611 b(is)h(auc)-37 b(h)613 b(a)f(closed)h(curv)-37 b(e)613 b(going)f(from)h(a)f(p)37 b(oin)-37 b(t)611 b(to)h(another)h(p)37 b(oin)-37 b(t)612 b(and)g(returning)1102 41397 y(bac)-37 b(k)435 b(along)e(almost)h(the)f (same)h(path)f(when)h Fp(k)475 b Fq(is)434 b(su\016cien)-37 b(tly)433 b(large.)589 b(As)433 b(\010)39960 41596 y Fk(H)41402 41397 y Fq(is)g(a)h(symplectic)1102 42991 y(di\013eomorphism)444 b(w)-37 b(e)443 b(ha)-37 b(v)g(e)14112 45016 y Fi(Z)148 b(Z)15736 48034 y Fk(\033)16673 46824 y Fp(dp)296 b Fm(^)e Fp(dq)417 b Fq(=)22611 45016 y Fi(I)23349 48034 y Fk(\015)24252 46824 y Fp(pdq)h Fq(=)28024 45016 y Fi(I)28762 48034 y Fv(\010)29520 47714 y Fd(k)29520 48349 y(H)30373 48034 y Fv(\()p Fk(\015)57 b Fv(\))32107 46824 y Fp(pdq)417 b Fq(=)369 b(0)p Fp(:)10280 b Fq(\(4.4\))1102 51111 y(Note)399 b Fp(p)f Fq(is)g(of)g(Lipsc)-37 b(hitz,)407 b(b)-37 b(y)398 b(the)g(theorem)h(of)f(Rademac)-37 b(her)399 b(\([R]\),)e Fp(p)i Fq(is)e(di\013eren)-37 b(tiable)399 b(almost)1102 52705 y(ev)-37 b(erywhere)432 b(in)c Fp(U)145 b Fq(.)586 b(As)429 b Fp(\015)502 b Fq(is)429 b(arbitrarily)g(c)-37 b(hosen,)433 b(\(4.2\))428 b(holds)h(for)g(almost)f(all)h Fp(q)417 b Fm(2)368 b Fp(U)145 b Fq(.)586 b(Conse-)1102 54299 y(quen)-37 b(tly)-111 b(,)507 b(there)493 b(exists)g(a)h Fp(C)15552 53817 y Fv(1)p Fk(;)p Fv(1)16993 54299 y Fq(-function)f Fp(S)23581 53817 y Fk(s)24641 54299 y Fq(and)g Fp(c)453 b Fm(2)g Fl(R)30600 53817 y Fv(2)31689 54299 y Fq(suc)-37 b(h)494 b(that)e Fp(p)38332 53817 y Fk(s)39353 54299 y Fq(=)453 b Fp(dS)42422 53817 y Fk(s)43318 54299 y Fq(+)329 b Fp(c)p Fq(.)741 b(In)493 b(the)1102 55893 y(same)520 b(w)-37 b(a)g(y)-111 b(,)539 b(w)-37 b(e)520 b(obtain)f(a)g Fp(C)16022 55411 y Fv(1)p Fk(;)p Fv(1)17463 55893 y Fq(-function)g Fp(S)24077 55411 y Fk(u)25290 55893 y Fq(and)g Fp(c)28524 55411 y Fg(0)29392 55893 y Fm(2)496 b Fl(R)31733 55411 y Fv(2)32848 55893 y Fq(suc)-37 b(h)520 b(that)e Fp(p)39543 55411 y Fk(u)40734 55893 y Fq(=)496 b Fp(dS)43846 55411 y Fk(u)44886 55893 y Fq(+)346 b Fp(c)46840 55411 y Fg(0)47213 55893 y Fq(.)819 b(As)1102 57487 y Fp(W)2541 57005 y Fk(s)2357 57853 y(H)3807 57487 y Fq(in)-37 b(tersects)444 b Fp(W)11132 57005 y Fk(u)10948 57853 y(H)12397 57487 y Fq(at)f(the)g(whole)g(\000,)g Fp(c)22289 57005 y Fg(0)23031 57487 y Fq(=)369 b Fp(c)p Fq(.)2165 b Fb(\003)2696 59951 y Fq(In)449 b(fact,)i(for)f(almost)f(all)h(initial)f(p)37 b(oin)-37 b(t)449 b(\()p Fp(q)48 b(;)221 b(p)25281 59469 y Fk(s)25848 59951 y Fq(\()p Fp(q)48 b Fq(\)\),)450 b Fp(p)f Fq(is)g(di\013eren)-37 b(tiable)451 b(at)e(all)g(\010)43739 59469 y Fk(k)43739 60317 y(H)44747 59951 y Fq(\()p Fp(q)48 b(;)221 b(p)47163 59469 y Fk(s)47731 59951 y Fq(\()p Fp(q)48 b Fq(\)\))1102 61545 y(\()p Fm(8)p Fp(k)481 b Fm(2)440 b Fl(Z)5742 61063 y Fv(+)6624 61545 y Fq(\).)718 b(T)-111 b(o)486 b(see)g(that,)496 b(let)485 b Fp(O)523 b Fq(b)37 b(e)486 b(an)f(op)37 b(en)486 b(set)g(in)f Fp(U)145 b Fq(,)496 b(for)486 b(eac)-37 b(h)486 b Fp(k)c Fm(2)440 b Fl(Z)40324 61063 y Fv(+)41205 61545 y Fq(,)496 b(there)486 b(is)g(a)f(full)1102 63139 y(Leb)37 b(esgue)522 b(measure)f(set)f Fp(O)15343 63338 y Fk(k)16495 63139 y Fm(\032)499 b Fp(\031)48 b Fq(\(\010)20308 62657 y Fk(k)20308 63505 y(H)21315 63139 y Fm(f)p Fp(O)37 b(;)221 b(p)p Fq(\()p Fp(O)37 b Fq(\))p Fm(g)p Fq(\))520 b(where)i Fp(p)e Fq(is)h(di\013eren)-37 b(tiable,)540 b(since)521 b(\010)g(is)f(a)1102 64733 y(di\013eomorphism,)444 b(the)f(set)16541 68951 y Fp(O)17591 68403 y Fg(\003)18571 68951 y Fq(=)20409 67291 y Fg(1)20201 67689 y Fi(\\)19973 70549 y Fk(k)22 b Fv(=0)22126 68951 y Fp(\031)22931 67477 y Fi(\020)23725 68951 y Fq(\010)24684 68383 y Fg(\000)p Fk(k)24684 69342 y(H)26168 68951 y Fm(f)p Fp(O)27845 69150 y Fk(k)28499 68951 y Fp(;)221 b(p)29757 68403 y Fk(s)30325 68951 y Fq(\()p Fp(O)31855 69150 y Fk(k)32509 68951 y Fq(\))p Fm(g)33690 67477 y Fi(\021)1102 73371 y Fq(is)484 b(a)f(full)g(Leb)37 b(esgue)485 b(measure)f(subset)f(of)h Fp(O)37 b Fq(.)712 b(F)-111 b(or)484 b(an)-37 b(y)484 b(p)37 b(oin)-37 b(t)483 b Fp(q)h Fm(2)436 b Fp(O)36725 72889 y Fg(\003)37336 73371 y Fq(,)493 b Fp(p)484 b Fq(is)f(di\013eren)-37 b(tiable)484 b(at)1102 74965 y(the)443 b(p)37 b(oin)-37 b(ts)443 b Fp(\031)48 b Fq(\(\010)9664 74483 y Fk(k)9664 75331 y(H)10671 74965 y Fq(\()p Fp(q)g(;)221 b(p)13087 74483 y Fk(s)13655 74965 y Fq(\()p Fp(q)48 b Fq(\)\)\))440 b(for)j(all)g Fp(k)411 b Fm(2)368 b Fl(Z)23925 74483 y Fv(+)24807 74965 y Fq(.)p eop %%Page: 28 28 28 27 bop 1102 2239 a Ft(28)2696 5427 y Fq(Let)515 b(us)g(consider)h (the)f(Hamiltonian)f(\015o)-37 b(w.)807 b(If)515 b(the)f(lo)37 b(cally)516 b(horizon)-37 b(tal)515 b(stable)g(\(unstable\))1102 7021 y(manifold)443 b(has)g(the)g(form)13840 9689 y Fp(W)15279 9051 y Fk(s;u)15095 10080 y(H)17160 9689 y Fq(=)369 b Fm(f)p Fq(\()p Fp(q)48 b(;)221 b(p)21642 9141 y Fk(s;u)23153 9689 y Fq(\()p Fp(q)48 b(;)221 b(t)p Fq(\))p Fp(;)g(t)p Fq(\))368 b(:)h(\()p Fp(q)48 b(;)221 b(t)p Fq(\))368 b Fm(2)g Fp(U)440 b Fm(\002)295 b Fl(T)p Fm(g)1102 12357 y Fq(and)638 b(if)e(w)-37 b(e)638 b(call)g(the)f(2-form)g(\012)693 b(=)20372 11361 y Fi(P)21996 12357 y Fp(dp)23355 12556 y Fk(i)24222 12357 y Fm(^)425 b Fp(dq)26817 12556 y Fk(i)27684 12357 y Fm(\000)g Fp(dH)533 b Fm(^)424 b Fp(dt)p Fq(,)686 b(then)637 b(\()p Fp(p)39411 11875 y Fk(s;u)40922 12357 y Fp(;)221 b(t)p Fq(\))42509 11875 y Fg(\003)43119 12357 y Fq(\012)693 b(=)g(0.)1174 b(In)1102 13952 y(the)622 b(co)-37 b(v)g(ering)623 b(space)f Fl(R)13641 13469 y Fv(2)14651 13952 y Fm(\002)415 b Fl(R)621 b Fq(w)-37 b(e)623 b(\014nd)e(that)f(there)i(exists)33353 13616 y(\026)33129 13952 y Fp(S)34021 13469 y Fk(s;u)35531 13952 y Fq(\()p Fp(q)48 b(;)221 b(t)p Fq(\))620 b(suc)-37 b(h)622 b(that)f Fp(d)46043 13616 y Fq(\026)45819 13952 y Fp(S)46711 13469 y Fk(s;u)48888 13952 y Fq(=)1102 15546 y Fp(p)1770 15063 y Fk(s;u)3282 15546 y Fq(\()p Fp(q)48 b(;)221 b(t)p Fq(\))p Fp(dq)100 b Fm(\000)53 b Fp(H)108 b Fq(\()p Fp(p)10894 15063 y Fk(s;u)12405 15546 y Fq(\()p Fp(q)48 b(;)221 b(t)p Fq(\))p Fp(;)g(q)48 b(;)221 b(t)p Fq(\))p Fp(dt)p Fq(.)548 b(By)321 b(applying)h(the)g(standard)f (argumen)-37 b(t)322 b(\(see)g(for)g(instance)1102 17140 y(the)443 b(app)37 b(endix)443 b(2)g(in)g([Ma3]\),)f(w)-37 b(e)444 b(\014nd)f(that)16576 19808 y Fp(L)17480 19259 y Fk(s;u)19360 19808 y Fq(=)370 b Fp(L)295 b Fm(\000)g(h)p Fp(@)24512 20007 y Fk(q)25323 19472 y Fq(\026)25099 19808 y Fp(S)25991 19259 y Fk(s;u)27501 19808 y Fp(;)468 b Fq(_)-616 b Fp(q)48 b Fm(i)295 b(\000)g Fp(@)31577 20007 y Fk(t)32269 19472 y Fq(\026)32045 19808 y Fp(S)32937 19259 y Fk(s;u)47191 19808 y Fq(\(4.5\))1102 22485 y(attains)606 b(its)g(minim)-37 b(um)607 b(at)f Fp(@)16314 22684 y Fk(q)17125 22149 y Fq(\026)16901 22485 y Fp(S)17793 22003 y Fk(s;u)19910 22485 y Fq(as)g(the)g(function)h(of)853 b(_)-616 b Fp(q)48 b Fq(.)1081 b(Note)607 b Fp(L)37567 21847 y Fk(s;u)37763 22865 y Fv(_)-497 b Fk(q)39720 22485 y Fq(=)642 b Fp(L)42495 22684 y Fv(_)-497 b Fk(q)43290 22485 y Fm(\000)405 b Fp(@)45433 22684 y Fk(q)46244 22149 y Fq(\026)46019 22485 y Fp(S)46911 22003 y Fk(s;u)49028 22485 y Fq(is)1102 24218 y(of)529 b(Lipsc)-37 b(hitz,)551 b Fp(dL)10444 23579 y Fk(s;u)10640 24598 y Fv(_)-497 b Fk(q)11956 24218 y Fp(=dt)529 b Fq(and)g Fp(L)17893 23736 y Fk(s;u)17893 24546 y(q)19933 24218 y Fq(exist)f(almost)h(ev)-37 b(erywhere.)851 b(Since)529 b Fp(W)40211 23736 y Fk(s;u)42251 24218 y Fq(is)g(a)g(manifold)1102 25834 y(consisting)519 b(of)g(the)f(tra)74 b(jectories)520 b(of)e(the)h(Euler-Lagrange)h (\015o)-37 b(w,)538 b(it)518 b(follo)-37 b(ws)520 b(from)e(the)h (Euler-)1102 27428 y(Lagrange)556 b(equation)f Fp(dL)14319 27627 y Fv(_)-497 b Fk(q)14710 27428 y Fp(=dt)554 b Fq(=)h Fp(L)19591 27627 y Fk(q)20732 27428 y Fq(and)f(\(4.2\))g(that)f Fp(L)30604 26946 y Fk(s;u)30604 27757 y(q)32670 27428 y Fq(=)i(0)f(almost)g(ev)-37 b(erywhere.)928 b(The)1102 29022 y(absolute)587 b(con)-37 b(tin)g(uit)g(y)587 b(of)f Fp(L)h Fq(implies)f(that)g Fp(L)24635 28540 y Fk(s;u)26733 29022 y Fq(is)h(a)f(function)h(of)f Fp(t)g Fq(alone.)1023 b(Therefore,)624 b(b)-37 b(y)1102 30616 y(adding)594 b(some)g(function)g(of)g Fp(t)f Fq(to)19176 30280 y(\026)18952 30616 y Fp(S)19844 30134 y Fk(s;u)21354 30616 y Fq(,)631 b(w)-37 b(e)595 b(can)f(mak)-37 b(e)595 b Fp(L)31571 30134 y Fk(s;u)33702 30616 y Fq(=)621 b(0.)1043 b(Note)594 b(the)g(lo)37 b(cal)594 b(stable)1102 32210 y(\(unstable\))360 b(manifold)h(can)g(b)37 b(e)360 b(though)-37 b(t)360 b(as)h(the)f(graph)h(of)g(some)g(function)f(de\014ned)h(on)g Fm(f)p Fq(\()p Fp(q)48 b(;)221 b(t)p Fq(\))368 b Fm(2)1102 33804 y Fl(T)1988 33322 y Fv(2)2720 33804 y Fm(\002)136 b Fl(T)369 b Fq(:)g Fp(a)g Fm(\024)g Fp(q)8948 34003 y Fv(2)9913 33804 y Fm(\024)h Fp(q)11909 34003 y Fv(2\000)13161 33804 y Fq(\()p Fp(q)14271 34003 y Fv(1)14867 33804 y Fp(;)221 b(t)p Fq(\))136 b(+)g(2)p Fp(\031)48 b Fm(g)364 b Fq(\()p Fm(ff)p Fq(\()p Fp(q)48 b(;)221 b(t)p Fq(\))368 b Fm(2)g Fl(T)27354 33322 y Fv(2)28086 33804 y Fm(\002)136 b Fl(T)369 b Fq(:)g Fp(q)31841 34003 y Fv(2\000)33094 33804 y Fq(\()p Fp(q)34204 34003 y Fv(1)34799 33804 y Fp(;)221 b(t)p Fq(\))369 b Fm(\024)g Fp(q)38750 34003 y Fv(2)39715 33804 y Fm(\024)g Fq(2)p Fp(\031)184 b Fm(\000)136 b Fp(a)p Fm(g)p Fq(\),)380 b(where)1102 35492 y Fp(q)1695 35691 y Fv(2\000)2948 35492 y Fq(\()p Fp(q)4058 35691 y Fv(1)4654 35492 y Fp(;)221 b(t)p Fq(\))392 b(is)h(suc)-37 b(h)393 b(a)g(function)g(that)f Fp(\031)48 b Fq(\()21308 35156 y(~)21225 35492 y(\000)o(\))369 b(=)g Fm(f)p Fq(\()p Fp(q)48 b(;)221 b(t)p Fq(\))368 b(:)h Fp(q)29450 35691 y Fv(2)30415 35492 y Fq(=)g Fp(q)32410 35691 y Fv(2\000)33663 35492 y Fq(\()p Fp(q)34773 35691 y Fv(1)35368 35492 y Fp(;)221 b(t)p Fq(\))p Fm(g)p Fq(,)403 b Fp(q)38984 35691 y Fv(2\000)40237 35492 y Fq(\()p Fp(q)41347 35691 y Fv(1)41942 35492 y Fq(\))369 b(=)g Fp(q)44823 35691 y Fv(2\000)46076 35492 y Fq(\()p Fp(q)47186 35691 y Fv(1)47781 35492 y Fp(;)221 b Fq(0\).)1102 37086 y(The)482 b(\014rst)f(co-homology)g (group)h(is)f Fl(R)321 b Fm(\002)f(f)p Fq(0)p Fm(g)h(\002)g Fl(R)p Fq(.)705 b(Th)-37 b(us,)491 b(there)481 b(exists)g(a)g(function) g Fp(S)46115 36604 y Fk(u)46808 37086 y Fq(\()p Fp(q)48 b(;)221 b(t)p Fq(\):)1102 38680 y Fm(ff)p Fq(\()p Fp(q)48 b(;)221 b(t)p Fq(\))496 b Fm(2)g Fl(T)7939 38198 y Fv(2)8881 38680 y Fm(\002)346 b Fl(T)496 b Fq(:)g Fp(q)13100 38879 y Fv(2\000)14353 38680 y Fq(\()p Fp(q)15463 38879 y Fv(1)16058 38680 y Fp(;)221 b(t)p Fq(\))496 b Fm(\024)h Fp(q)20264 38879 y Fv(2)21356 38680 y Fm(\024)g Fq(2)p Fp(\031)394 b Fm(\000)346 b Fp(a)p Fm(g)497 b(!)f Fl(R)p Fq(,)539 b Fp(S)32526 38198 y Fk(s)33093 38680 y Fq(\()p Fp(q)48 b(;)221 b(t)p Fq(\):)742 b Fm(f)p Fq(\()p Fp(q)48 b(;)221 b(t)p Fq(\))495 b Fm(2)h Fl(T)43121 38198 y Fv(2)44063 38680 y Fm(\002)346 b Fl(T)496 b Fq(:)h Fp(a)f Fm(\024)1102 40274 y Fp(q)1695 40473 y Fv(2)2660 40274 y Fm(\024)369 b Fp(q)4655 40473 y Fv(2\000)5908 40274 y Fq(\()p Fp(q)7018 40473 y Fv(1)7614 40274 y Fp(;)221 b(t)p Fq(\))185 b(+)g(2)p Fp(\031)48 b Fm(g)370 b(!)f Fl(R)388 b Fq(and)g(\()p Fp(c)19771 39792 y Fg(\003)19771 40604 y Fv(1)20381 40274 y Fp(;)221 b Fq(0)p Fp(;)g(\013)23080 39792 y Fg(\003)23692 40274 y Fq(\))387 b(suc)-37 b(h)389 b(that)30585 39938 y(\026)30361 40274 y Fp(S)31253 39792 y Fk(s;u)32763 40274 y Fq(\()p Fp(q)48 b(;)221 b(t)p Fq(\))368 b(=)h Fp(S)38170 39792 y Fk(s;u)39681 40274 y Fq(\()p Fp(q)48 b(;)221 b(t)p Fq(\))185 b(+)g Fp(c)44404 39792 y Fg(\003)44404 40604 y Fv(1)45014 40274 y Fp(q)45607 40473 y Fv(1)46389 40274 y Fq(+)g Fp(\013)48462 39792 y Fg(\003)49073 40274 y Fp(t)p Fq(,)1102 41868 y(where)374 b(w)-37 b(e)373 b(ha)-37 b(v)g(e)372 b(used)h(the)f(fact)g(that)f(b)37 b(oth)372 b(the)g(stable)g(and)g(the)g(unstable)g(manifolds)g(coincide) 1102 43462 y(with)443 b(eac)-37 b(h)444 b(other)f(at)12296 43126 y(~)12213 43462 y(\000.)590 b(In)443 b(this)f(case)i(w)-37 b(e)443 b(obtain)g(from)g(\(4.5\))f(that)12896 46130 y Fp(L)13800 45582 y Fk(s;u)15680 46130 y Fq(=)369 b Fp(L)295 b Fm(\000)h(h)p Fq(\()p Fp(c)21219 45582 y Fg(\003)21219 46459 y Fv(1)21828 46130 y Fp(;)221 b Fq(0\))p Fp(;)468 b Fq(_)-616 b Fp(q)49 b Fm(i)294 b(\000)i(h)p Fp(@)28193 46329 y Fk(q)28779 46130 y Fp(S)29671 45582 y Fk(s;u)31181 46130 y Fp(;)468 b Fq(_)-616 b Fp(q)49 b Fm(i)294 b(\000)i Fp(@)35258 46329 y Fk(t)35726 46130 y Fp(S)36618 45582 y Fk(s;u)1102 48799 y Fq(attains)447 b(its)h(minim)-37 b(um)447 b(at)h Fp(W)16413 48317 y Fk(s;u)18371 48799 y Fq(as)g(the)g(function)f(of)695 b(_)-616 b Fp(q)495 b Fq(with)448 b Fp(L)34048 48317 y Fk(s;u)35559 48799 y Fm(j)35928 48998 y Fk(W)37070 48732 y Fd(s;u)38892 48799 y Fq(=)377 b Fp(\013)41157 48317 y Fg(\003)41767 48799 y Fq(.)605 b(Th)-37 b(us,)449 b(for)f(all)1102 50487 y Fp(d\015)517 b Fq(on)4926 50151 y(~)4843 50487 y(\000)443 b(w)-37 b(e)444 b(ha)-37 b(v)g(e)1767 52024 y Fi(Z)3095 52352 y Fg(1)2505 55042 y(\0001)4682 52357 y Fi(\020)5475 53832 y Fp(L)p Fq(\()p Fp(d\015)74 b Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))55 b Fm(\000)g(h)p Fq(\()p Fp(c)14202 53283 y Fg(\003)14202 54160 y Fv(1)14811 53832 y Fp(;)221 b Fq(0\))p Fp(;)417 b Fq(_)-565 b Fp(\015)74 b Fm(i)55 b(\000)g(h)p Fp(@)20816 54031 y Fk(q)21403 53832 y Fp(S)22295 53283 y Fk(s;u)23805 53832 y Fq(\()p Fp(\015)74 b Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))p Fp(;)416 b Fq(_)-564 b Fp(\015)72 b Fm(i)55 b(\000)g Fp(@)31900 54031 y Fk(t)32368 53832 y Fp(S)33260 53283 y Fk(s;u)34771 53832 y Fq(\()p Fp(\015)74 b Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))55 b Fm(\000)g Fp(\013)41149 53283 y Fg(\003)41758 52357 y Fi(\021)42551 53832 y Fp(dt)369 b Fq(=)g(0)p Fp(:)665 b Fq(\(4.6\))2696 57282 y(W)-111 b(e)368 b(ha)-37 b(v)g(e)368 b(men)-37 b(tioned)367 b(b)37 b(efore)367 b(that)f(the)h(Euler-Lagrange)i(equation)e(for)g Fp(L)143 b Fm(\000)g Fp(\021)42657 57481 y Fk(c)43564 57282 y Fq(is)367 b(the)g(same)1102 58876 y(as)407 b(that)f(for)h Fp(L)g Fq(if)f Fp(\021)10688 59075 y Fk(c)11635 58876 y Fq(is)h(a)g(closed)g(1-form.)579 b(In)406 b(lo)37 b(cal)407 b(co)37 b(ordinates)408 b(w)-37 b(e)407 b(can)g(write)h Fp(\021)42728 59075 y Fk(c)43637 58876 y Fq(=)369 b Fm(h)p Fp(c)p Fq(\()p Fp(q)48 b Fq(\))p Fp(;)467 b Fq(_)-615 b Fp(q)47 b Fm(i)p Fq(.)1102 60470 y(If)443 b(w)-37 b(e)443 b(use)g Fp(H)7785 60669 y Fk(\021)8332 60802 y Fd(c)8897 60470 y Fq(\()p Fp(p;)221 b(q)48 b(;)221 b(t)p Fq(\))442 b(to)h(denote)g(the)g(Legendre)i(transformation)12693 63585 y Fp(H)13797 63784 y Fk(\021)14344 63917 y Fd(c)14909 63585 y Fq(\()p Fp(p;)221 b(q)48 b(;)221 b(t)p Fq(\))368 b(=)i(max)21645 64382 y Fk(p)23376 62111 y Fi(n)24262 63585 y Fm(h)p Fp(p;)468 b Fq(_)-616 b Fp(q)48 b Fm(i)294 b(\000)28818 62111 y Fi(\020)29611 63585 y Fp(L)h Fm(\000)h(h)p Fp(c)p Fq(\()p Fp(q)48 b Fq(\))p Fp(;)466 b Fq(_)-614 b Fp(q)46 b Fm(i)36652 62111 y Fi(\021o)1102 66825 y Fq(then)443 b(w)-37 b(e)444 b(obtain)21171 68779 y Fp(p)295 b Fq(+)g Fp(c)p Fq(\()p Fp(q)48 b Fq(\))368 b(=)27642 67881 y Fp(@)74 b(L)p 27642 68474 1684 54 v 27774 69691 a(@)320 b Fq(_)-615 b Fp(q)29484 68779 y(:)1102 71777 y Fq(It)472 b(implies)g(that)g Fp(H)11121 71976 y Fk(\021)11668 72109 y Fd(c)12233 71777 y Fq(\()p Fp(p;)221 b(q)48 b(;)221 b(t)p Fq(\))417 b(=)i Fp(H)108 b Fq(\()p Fp(p)315 b Fq(+)g Fp(c)p Fq(\()p Fp(q)48 b Fq(\))p Fp(;)221 b(q)48 b(;)221 b(t)p Fq(\).)677 b(As)472 b Fp(\021)30931 71976 y Fk(c)31944 71777 y Fq(is)h(closed,)480 b(the)472 b(co)37 b(ordinate)473 b(trans-)1102 73371 y(lation)519 b(\()p Fp(p;)221 b(q)48 b Fq(\))494 b Fm(!)h Fq(\()p Fp(p)346 b Fq(+)f Fp(c)p Fq(\()p Fp(q)48 b Fq(\))p Fp(;)221 b(q)48 b Fq(\))517 b(is)h(symplectic.)818 b(Under)519 b(suc)-37 b(h)518 b(a)h(co)37 b(ordinate)519 b(translation)f(the)1102 74965 y(horizon)-37 b(tal)444 b(stable)f(\(unstable\))f(manifold)h(is)f(the)h (graph)h(of)e(the)h(function)g Fp(p)40101 74483 y Fk(s;u)41612 74965 y Fq(\()p Fp(q)48 b Fq(\))294 b Fm(\000)i Fp(c)p Fq(\()p Fp(q)48 b Fq(\).)p eop %%Page: 29 29 29 28 bop 48792 2239 a Ft(29)2696 5427 y Fq(W)-111 b(e)322 b(kno)-37 b(w)321 b(that)11047 5091 y(~)10964 5427 y(\000)f(is)h(con) -37 b(tained)321 b(in)f(some)h(Aubry)g(set)g Fm(A)p Fq(\()p Fp(c)p Fq(\))367 b(=)j Fm(f)p Fp(B)35931 5626 y Fk(c)36840 5427 y Fq(=)g(0)p Fm(g)320 b Fq(where)i Fp(c)369 b Fq(=)g(\()p Fp(c)47048 5626 y Fv(1)47644 5427 y Fp(;)221 b(c)48809 4945 y Fg(0)48809 5757 y Fv(2)49405 5427 y Fq(\))1102 7021 y(with)475 b Fp(a)p Fq(\()p Fp(c)5954 7220 y Fv(1)6549 7021 y Fq(\))422 b Fm(\024)g Fp(c)9518 6539 y Fg(0)9518 7351 y Fv(2)10536 7021 y Fm(\024)g Fp(b)p Fq(\()p Fp(c)13653 7220 y Fv(1)14249 7021 y Fq(\).)685 b(F)-111 b(rom)476 b(ab)37 b(o)-37 b(v)g(e)475 b(argumen)-37 b(ts)476 b(and)e(the)h(prop) 37 b(osition)475 b(3.2)g(w)-37 b(e)475 b(can)h(see)1102 8615 y(that)442 b Fp(c)4555 8814 y Fv(1)5520 8615 y Fq(=)369 b Fp(c)7497 8133 y Fg(\003)7497 8945 y Fv(1)8551 8615 y Fq(and)443 b Fp(\013)11989 8133 y Fg(\003)12968 8615 y Fq(=)369 b Fp(\013)5 b Fq(\()p Fp(c)p Fq(\).)2696 11032 y(T)-111 b(o)447 b(study)e(the)g(barrier)i(function)e Fp(B)21256 10550 y Fg(\003)21189 11360 y Fk(c)21866 11032 y Fq(,)h(w)-37 b(e)447 b(consider)f(the)g(co)-37 b(v)g(ering)447 b(of)e Fl(T)39720 10550 y Fv(2)40761 11032 y Fq(giv)-37 b(en)447 b(b)-37 b(y)445 b Fl(T)297 b Fm(\002)g Fl(R)p Fq(,)1102 12716 y(let)3016 12380 y(~)2933 12716 y(\000)3763 12915 y Fk(k)4772 12716 y Fq(b)37 b(e)356 b(the)f(lift)f(of)12216 12380 y(~)12133 12716 y(\000)h(whic)-37 b(h)356 b(is)f(close)h(to)f Fl(T)120 b Fm(\002)g(f)p Fq(2)p Fp(k)42 b(\031)48 b Fm(g)120 b(\002)g(f)p Fp(p)31203 12915 y Fv(1)32167 12716 y Fq(=)369 b(const.)p Fp(;)221 b(p)38229 12915 y Fv(2)39195 12716 y Fq(=)369 b(0)p Fm(g)120 b(\002)g Fl(T)p Fq(.)561 b(Without)1102 14310 y(lose)338 b(of)g(generalit)-37 b(y)339 b(w)-37 b(e)338 b(single)g(out)g(one)g(lift)f(of)g(the)h(unstable)f(manifold)h Fp(W)38683 13828 y Fk(u)39715 14310 y Fq(that)f(extends)h(from)1185 15588 y(~)1102 15924 y(\000)1932 16123 y Fv(0)3037 15924 y Fq(and)509 b(k)-37 b(eep)510 b(horizon)-37 b(tal)510 b(o)-37 b(v)g(er)510 b Fm(f)p Fq(\()p Fp(q)48 b(;)221 b(t)p Fq(\))478 b Fm(2)h Fl(T)24133 15442 y Fv(2)25067 15924 y Fm(\002)340 b Fl(T)478 b Fq(:)h Fp(q)29245 16123 y Fv(2\000)30498 15924 y Fq(\()p Fp(q)31608 16123 y Fv(1)32204 15924 y Fp(;)221 b(t)p Fq(\))478 b Fm(\024)i Fp(q)36375 16123 y Fv(2)37450 15924 y Fm(\024)f Fq(2)p Fp(\031)387 b Fm(\000)340 b Fp(a)p Fm(g)509 b Fq(and)g(single)1102 17612 y(out)399 b(one)h(lift)f(of)g(the)g(stable)h(manifold)f Fp(W)22280 17130 y Fk(s)23247 17612 y Fq(that)g(extends)g(from)34059 17276 y(~)33976 17612 y(\000)34806 17811 y Fv(1)35801 17612 y Fq(and)h(k)-37 b(eep)400 b(horizon)-37 b(tal)400 b(o)-37 b(v)g(er)1102 19206 y Fm(f)p Fq(\()p Fp(q)48 b(;)221 b(t)p Fq(\))368 b Fm(2)h Fl(T)7020 18724 y Fv(2)7911 19206 y Fm(\002)295 b Fl(T)369 b Fq(:)f Fp(a)i Fm(\024)f Fp(q)14298 19405 y Fv(2)15263 19206 y Fm(\024)g Fp(q)17258 19405 y Fv(2\000)18511 19206 y Fq(\()p Fp(q)19621 19405 y Fv(1)20216 19206 y Fp(;)221 b(t)p Fq(\))295 b(+)g(2)p Fp(\031)48 b Fm(g)p Fq(.)2696 21623 y(Recall)622 b Fp(c)665 b Fq(=)h(\()p Fp(c)10909 21822 y Fv(1)11504 21623 y Fp(;)221 b(b)p Fq(\()p Fp(c)13756 21822 y Fv(1)14352 21623 y Fq(\)\).)1123 b(Since)621 b Fp(L)21428 21141 y Fk(s;u)23560 21623 y Fq(attains)f(its)g(minim)-37 b(um)621 b(on)g(the)f(lo)37 b(cal)621 b(horizon)-37 b(tal)1102 23217 y(stable)524 b(\(unstable\))e(manifold,)543 b(for)524 b Fp(q)551 b Fm(2)502 b Fl(T)349 b Fm(\002)g Fq(\()p Fp(a;)221 b Fq(2)p Fp(\031)397 b Fm(\000)349 b Fp(a)p Fq(\))522 b(there)i(exists)f(only)h (one)g Fp(c)p Fq(-minimal)1102 24811 y(orbit)427 b Fp(d\015)5790 24329 y Fk(s)5716 25139 y(c)6358 24811 y Fq(:)582 b Fl(R)8268 24329 y Fv(+)9519 24811 y Fm(!)370 b Fp(T)184 b(M)571 b Fq(as)427 b(w)-37 b(ell)428 b(as)e(only)h(one)g Fp(c)p Fq(-minimal)f(orbit)h Fp(d\015)36117 24329 y Fk(u)36043 25139 y(c)36811 24811 y Fq(:)583 b Fl(R)38722 24329 y Fg(\000)39988 24811 y Fm(!)369 b Fp(T)184 b(M)572 b Fq(suc)-37 b(h)427 b(that)1102 26405 y Fp(\015)1864 25923 y Fk(s;u)3375 26405 y Fq(\(0\))381 b(=)h Fp(q)48 b Fq(.)613 b(In)450 b(fact,)j(suc)-37 b(h)451 b(an)f(orbit)h Fp(d\015)22727 25923 y Fk(s;u)22653 26733 y(c)24688 26405 y Fq(lies)g(on)g(the)f(lo)37 b(cal)451 b(stable)f(\(unstable\))g(manifold.)1102 27999 y(Note)444 b Fp(L)5217 27517 y Fk(s;u)7097 27999 y Fq(=)369 b Fp(\013)5 b Fq(\()p Fp(c)p Fq(\))441 b(on)i Fp(W)14688 27517 y Fk(s;u)16200 27999 y Fq(,)g(for)g(arbitrary)g Fp(T)554 b(>)369 b Fq(0)443 b(w)-37 b(e)444 b(ha)-37 b(v)g(e)4178 30355 y Fi(Z)5507 30683 y Fv(0)4916 33373 y Fg(\000)p Fk(T)6805 30688 y Fi(\020)7598 32163 y Fp(L)p Fq(\()p Fp(d\015)10472 31614 y Fk(u)10398 32491 y(c)11166 32163 y Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))294 b Fm(\000)h(h)p Fp(c;)417 b Fq(_)-565 b Fp(\015)18333 31614 y Fk(u)18259 32491 y(c)19027 32163 y Fq(\()p Fp(t)p Fq(\))p Fm(i)293 b(\000)j Fp(\013)5 b Fq(\()p Fp(c)p Fq(\))25144 30688 y Fi(\021)25936 32163 y Fp(dt)369 b Fq(=)p Fp(S)29401 31614 y Fk(u)30095 32163 y Fq(\()p Fp(\015)31374 31614 y Fk(u)31300 32491 y(c)32067 32163 y Fq(\(0\))p Fp(;)221 b Fq(0\))295 b Fm(\000)g Fp(S)38051 31614 y Fk(u)38745 32163 y Fq(\()p Fp(\015)40024 31614 y Fk(u)39950 32491 y(c)40717 32163 y Fq(\()p Fm(\000)p Fp(T)184 b Fq(\))p Fp(;)221 b Fm(\000)p Fp(T)184 b Fq(\))28804 35012 y Fm(\000)296 b Fp(b)p Fq(\()p Fp(c)31795 35211 y Fv(1)32390 35012 y Fq(\)\()49 b(\026)-713 b Fp(\015)34186 34463 y Fk(u)34112 35340 y(c)p Fv(2)35182 35012 y Fq(\(0\))294 b Fm(\000)344 b Fq(\026)-713 b Fp(\015)39264 34463 y Fk(u)39190 35340 y(c)p Fv(2)40261 35012 y Fq(\()p Fm(\000)p Fp(T)184 b Fq(\)\))p Fp(;)4672 36382 y Fi(Z)6000 36711 y Fk(T)5410 39400 y Fv(0)7058 36716 y Fi(\020)7851 38190 y Fp(L)p Fq(\()p Fp(d\015)10725 37642 y Fk(s)10651 38519 y(c)11293 38190 y Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))293 b Fm(\000)j(h)p Fp(c;)417 b Fq(_)-565 b Fp(\015)18460 37642 y Fk(s)18386 38519 y(c)19027 38190 y Fq(\()p Fp(t)p Fq(\))p Fm(i)293 b(\000)j Fp(\013)5 b Fq(\()p Fp(c)p Fq(\))25144 36716 y Fi(\021)25936 38190 y Fp(dt)369 b Fq(=)p Fp(S)29401 37642 y Fk(s)29968 38190 y Fq(\()p Fp(\015)31247 37642 y Fk(s)31173 38519 y(c)31814 38190 y Fq(\()p Fp(T)184 b Fq(\))p Fp(;)221 b(T)184 b Fq(\))296 b Fm(\000)f Fp(S)38391 37642 y Fk(s)38958 38190 y Fq(\()p Fp(\015)40237 37642 y Fk(s)40163 38519 y(c)40804 38190 y Fq(\(0\))p Fp(;)221 b Fq(0\))28804 40929 y Fm(\000)296 b Fp(b)p Fq(\()p Fp(c)31795 41128 y Fv(1)32390 40929 y Fq(\)\()49 b(\026)-713 b Fp(\015)34186 40380 y Fk(s)34112 41257 y(c)p Fv(2)35182 40929 y Fq(\()p Fp(T)184 b Fq(\))295 b Fm(\000)344 b Fq(\026)-713 b Fp(\015)39561 40380 y Fk(s)39487 41257 y(c)p Fv(2)40557 40929 y Fq(\(0\)\))p Fp(:)4050 b Fq(\(4.7\))1102 44143 y(Since)510 b(\010)e(is)h(an)f (area-preserving)j(t)-37 b(wist)508 b(map)g(when)i(it)e(is)g (restricted)h(on)g(the)f(cylinder,)526 b(from)1102 45737 y(the)552 b(lemma)g(2.6)f(and)h(the)f(corollary)i(2.7)f(w)-37 b(e)552 b(see)g(that)f Fp(L)31298 45936 y Fk(c)32391 45737 y Fq(is)g(regular.)917 b(Therefore,)581 b(for)552 b(an)-37 b(y)1102 47331 y Fp(")370 b(>)f Fq(0,)443 b(0)369 b Fm(\024)g Fp(s)g(<)g Fq(1,)443 b(0)369 b Fm(\024)g Fp(t)g(<)g Fq(1)443 b(and)g Fp(q)20291 46849 y Fg(0)20663 47331 y Fp(;)221 b(q)21894 46849 y Fg(\003)22874 47331 y Fm(2)368 b Fp(M)145 b Fq(,)443 b(there)g(exists)g Fp(K)34568 47530 y Fv(0)35533 47331 y Fm(2)368 b Fl(Z)37673 46849 y Fv(+)38998 47331 y Fq(suc)-37 b(h)443 b(that)9334 50545 y Fm(j)p Fp(h)10468 49996 y Fg(1)10468 50873 y Fk(c)11593 50545 y Fq(\()p Fp(q)12751 49996 y Fg(0)13123 50545 y Fp(;)221 b(q)14354 49996 y Fg(\003)14965 50545 y Fp(;)g(s;)g(t)p Fq(\))295 b Fm(\000)g Fp(h)20153 49996 y Fk(K)20153 50873 y(c)21178 50545 y Fq(\()p Fp(q)22336 49996 y Fg(0)22708 50545 y Fp(;)221 b(q)23939 49996 y Fg(\003)24549 50545 y Fp(;)g(s;)g(t)p Fq(\))p Fm(j)369 b(\024)g Fp(";)2879 b Fm(8)p Fp(K)35222 50744 y Fv(0)36187 50545 y Fm(\024)369 b Fp(K)464 b Fm(2)368 b Fl(Z)p Fp(:)1102 53758 y Fq(Since)473 b Fm(M)p Fq(\()p Fp(c)p Fq(\))e(is)i(uniquely)f(ergo)37 b(dic)473 b(in)f(this)g(case,)480 b(for)473 b(an)-37 b(y)472 b Fp(\016)d(>)418 b Fq(0,)479 b(0)418 b Fm(\024)g Fp(t)g(<)g Fq(1,)479 b Fp(\015)42897 53276 y Fk(s)43465 53758 y Fq(:)648 b Fl(R)45441 53276 y Fv(+)46742 53758 y Fm(!)418 b Fp(M)1102 55352 y Fq(with)453 b Fp(\015)4900 54870 y Fk(s)5467 55352 y Fq(\(0\))385 b(=)g Fp(q)433 b Fm(2)384 b Fl(T)302 b Fm(\002)f Fq(\()p Fp(a;)221 b Fq(2)p Fp(\031)350 b Fm(\000)302 b Fp(a)p Fq(\))452 b(and)h Fp(q)23606 54870 y Fg(\003)24601 55352 y Fm(2)385 b(M)27467 55551 y Fk(t)27935 55352 y Fq(\()p Fp(c)p Fq(\))452 b(there)h(exists)f (a)h(sequence)h(of)e Fm(f)p Fp(K)47027 55551 y Fk(i)47470 55352 y Fm(g)48134 54870 y Fg(1)48134 55698 y Fk(i)p Fv(=1)1102 56946 y Fq(\()p Fp(K)2747 57145 y Fk(i)3558 56946 y Fm(2)369 b Fl(Z)5699 56464 y Fv(+)6580 56946 y Fq(\))443 b(suc)-37 b(h)443 b(that)19357 58566 y Fp(d)p Fq(\()p Fp(\015)21327 58018 y Fk(s)21894 58566 y Fq(\()p Fp(t)295 b Fq(+)g Fp(K)25642 58765 y Fk(i)26085 58566 y Fq(\))p Fp(;)221 b(q)27833 58018 y Fg(\003)28443 58566 y Fq(\))368 b Fm(\024)i Fp(\016)-24 b(:)1102 61116 y Fq(It)480 b(is)g(easy)h(to)f(construct)h(an)f(absolutely)h(con)-37 b(tin)g(uous)480 b(curv)-37 b(e)482 b Fp(\020)98 b Fq(:)665 b([)p Fp(s;)221 b(K)37382 61315 y Fk(i)38145 61116 y Fq(+)320 b Fp(t)p Fq(])432 b Fm(!)f Fp(M)625 b Fq(suc)-37 b(h)481 b(that)1102 62710 y Fp(\020)98 b Fq(\()p Fp(t)p Fq(\))368 b(=)i Fp(\015)5828 62228 y Fk(s)6395 62710 y Fq(\()p Fp(t)p Fq(\))405 b(as)i Fp(s)369 b Fm(\024)g Fp(t)f Fm(\024)h Fp(K)15681 62909 y Fk(i)16347 62710 y Fq(+)222 b Fp(t)h Fm(\000)g Fq(2,)414 b Fp(d)p Fq(\()p Fp(d\020)98 b Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(d\015)27143 62228 y Fk(s)27710 62710 y Fq(\()p Fp(t)p Fq(\)\))368 b Fm(\024)h Fp(\016)457 b Fq(as)407 b Fp(K)35281 62909 y Fk(i)35946 62710 y Fq(+)223 b Fp(t)f Fm(\000)h Fq(2)369 b Fm(\024)g Fp(t)g Fm(\024)g Fp(K)44974 62909 y Fk(i)45639 62710 y Fq(+)223 b Fp(t)406 b Fq(and)1102 64304 y Fp(\020)98 b Fq(\()p Fp(K)3426 64503 y Fk(i)4118 64304 y Fq(+)250 b Fp(t)p Fq(\))368 b(=)h Fp(q)8809 63822 y Fg(\003)9420 64304 y Fq(.)582 b(As)12468 63968 y(\026)12312 64304 y Fp(L)13216 63822 y Fk(s)14203 64304 y Fq(attains)420 b(its)f(minim)-37 b(um)420 b(at)g Fp(W)29403 63822 y Fk(s)30391 64304 y Fq(for)g(eac)-37 b(h)421 b(\()p Fp(q)48 b(;)221 b(t)p Fq(\))368 b Fm(2)g Fp(U)145 b Fq(,)425 b(it)419 b(follo)-37 b(ws)421 b(from)1102 65898 y(the)443 b(con)-37 b(v)g(exit)g(y)444 b(of)f Fp(L)g Fq(in)689 b(_)-616 b Fp(q)491 b Fq(and)443 b(\(4.7\))f(that)10652 70098 y(0)369 b Fm(\024)12940 68290 y Fi(Z)14268 68618 y Fk(K)15161 68751 y Fd(i)15581 68618 y Fv(+)p Fk(t)13678 71308 y(s)17086 68623 y Fi(\020)17879 70098 y Fp(L)18783 70297 y Fk(c)19324 70098 y Fq(\()p Fp(d\020)98 b Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\)\))294 b Fm(\000)h Fp(\013)5 b Fq(\()p Fp(c)p Fq(\))28915 68623 y Fi(\021)29707 70098 y Fp(dt)13013 72905 y Fm(\000)296 b Fp(S)15234 72357 y Fk(s)15801 72905 y Fq(\()p Fp(q)16959 72357 y Fg(\003)17864 72905 y Fq(+)f(\(0)p Fp(;)221 b Fq(2)p Fp(\031)48 b Fq(\))p Fp(;)221 b(t)p Fq(\))295 b(+)g Fp(S)27051 72357 y Fk(s)27618 72905 y Fq(\()p Fp(q)28776 72357 y Fg(0)29148 72905 y Fp(;)221 b(s)p Fq(\))295 b Fm(\000)g Fp(b)p Fq(\()p Fp(c)34163 73104 y Fv(1)34758 72905 y Fq(\)\()p Fp(q)36433 72357 y Fg(\003)36385 73234 y Fv(2)37338 72905 y Fm(\000)g Fp(q)39259 73104 y Fv(2)39855 72905 y Fq(\))11685 74965 y Fm(\024)p Fp(o)p Fq(\()p Fp(\016)50 b Fq(\))p Fp(;)p eop %%Page: 30 30 30 29 bop 1102 2239 a Ft(30)1102 5427 y Fq(where)532 b Fp(L)5935 5626 y Fk(c)6990 5427 y Fq(=)515 b Fp(L)353 b Fm(\000)h(h)p Fp(c;)467 b Fq(_)-615 b Fp(q)47 b Fm(i)p Fq(.)852 b(If)529 b Fp(\015)17419 4945 y Fk(s)17345 5793 y(K)18238 5926 y Fd(i)18725 5427 y Fq(:)765 b([)p Fp(s;)221 b(K)22569 5626 y Fk(i)23365 5427 y Fq(+)353 b Fp(t)p Fq(])514 b Fm(!)h Fp(M)675 b Fq(is)530 b(the)f(minimizer)i(of)f Fp(h)42373 4945 y Fk(K)43266 5078 y Fd(i)42373 5755 y Fk(c)43754 5427 y Fq(\()p Fp(q)48 b(;)221 b(q)46143 4945 y Fg(\003)46753 5427 y Fp(;)g(s;)g(t)p Fq(\),)1102 7021 y(then)8562 10644 y(0)369 b Fm(\024)10850 8836 y Fi(Z)12178 9164 y Fk(K)13071 9297 y Fd(i)13492 9164 y Fv(+)p Fk(t)11588 11854 y(s)14996 9169 y Fi(\020)15790 10644 y Fp(L)16694 10843 y Fk(c)17235 10644 y Fq(\()p Fp(d\015)19205 10096 y Fk(s)19131 10972 y(K)20024 11105 y Fd(i)20510 10644 y Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\)\))294 b Fm(\000)h Fp(\013)5 b Fq(\()p Fp(c)p Fq(\))28214 9169 y Fi(\021)29006 10644 y Fp(dt)10924 14501 y Fm(\000)295 b Fp(S)13144 13953 y Fk(s)13711 14501 y Fq(\()p Fp(q)14869 13953 y Fg(\003)15774 14501 y Fq(+)g(\(0)p Fp(;)221 b Fq(2)p Fp(\031)48 b Fq(\))p Fp(;)221 b(t)p Fq(\))295 b(+)g Fp(S)24961 13953 y Fk(s)25528 14501 y Fq(\()p Fp(q)48 b(;)221 b(s)p Fq(\))294 b(+)i Fp(b)p Fq(\()p Fp(c)31701 14700 y Fv(1)32296 14501 y Fq(\))33034 12693 y Fi(Z)34362 13022 y Fk(K)35255 13155 y Fd(i)35676 13022 y Fv(+)p Fk(t)33772 15712 y(s)37377 14501 y Fq(_)-566 b Fp(\015)37942 13953 y Fk(s)37868 14830 y(K)38761 14963 y Fd(i)39181 14830 y Fv(2)39777 14501 y Fq(\()p Fp(t)p Fq(\))p Fp(dt)9595 18359 y Fm(\024)10850 16550 y Fi(Z)12178 16879 y Fk(K)13071 17012 y Fd(i)13492 16879 y Fv(+)p Fk(t)11588 19569 y(s)14996 16884 y Fi(\020)15790 18359 y Fp(L)16694 18558 y Fk(c)17235 18359 y Fq(\()p Fp(d\020)98 b Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\)\))293 b Fm(\000)j Fp(\013)5 b Fq(\()p Fp(c)p Fq(\))26826 16884 y Fi(\021)27618 18359 y Fp(dt)10924 21097 y Fm(\000)295 b Fp(S)13144 20549 y Fk(s)13711 21097 y Fq(\()p Fp(q)14869 20549 y Fg(\003)15774 21097 y Fq(+)g(\(0)p Fp(;)221 b Fq(2)p Fp(\031)48 b Fq(\))p Fp(;)221 b(t)p Fq(\))295 b(+)g Fp(S)24961 20549 y Fk(s)25528 21097 y Fq(\()p Fp(q)48 b(;)221 b(s)p Fq(\))294 b(+)i Fp(b)p Fq(\()p Fp(c)31701 21296 y Fv(1)32296 21097 y Fq(\)\()p Fp(q)33971 20549 y Fg(\003)33923 21425 y Fv(2)34876 21097 y Fm(\000)f Fp(q)36797 21296 y Fv(2)37393 21097 y Fq(\))9595 23157 y Fm(\024)q Fp(o)p Fq(\()p Fp(\016)50 b Fq(\))p Fp(:)1102 26192 y Fq(It)456 b(is)h(easy)g(to)f(see)i(that)d Fp(d\015)14984 25710 y Fk(s)14910 26558 y(K)15803 26691 y Fd(i)16290 26192 y Fq(\()p Fp(t)p Fq(\))h(k)-37 b(eeps)457 b(close)h(to)e(the)h(branc)-37 b(h)457 b(of)g(the)f(stable)h(manifold)f (whic)-37 b(h)1102 27820 y(corresp)37 b(onds)481 b(to)f(the)f (cohomology)i(class)f Fp(c)430 b Fq(=)h(\()p Fp(c)26433 28019 y Fv(1)27028 27820 y Fp(;)221 b(b)p Fq(\()p Fp(c)29280 28019 y Fv(1)29876 27820 y Fq(\)\))479 b(if)g Fp(K)33771 28019 y Fk(i)34693 27820 y Fq(is)h(su\016cien)-37 b(tly)480 b(large.)702 b(Th)-37 b(us,)1102 29414 y(w)g(e)444 b(ha)-37 b(v)g(e)16150 29854 y Fi(Z)17478 30182 y Fk(K)18371 30315 y Fd(i)18792 30182 y Fv(+)p Fk(t)16888 32872 y(s)20493 31662 y Fq(_)-566 b Fp(\015)21058 31113 y Fk(s)20984 31990 y(K)21877 32123 y Fd(i)22297 31990 y Fv(2)22893 31662 y Fq(\()p Fp(t)p Fq(\))p Fp(dt)368 b Fq(=)h Fp(q)27989 31113 y Fg(\003)27941 31990 y Fv(2)28895 31662 y Fq(+)295 b(2)p Fp(\031)343 b Fm(\000)296 b Fp(q)33909 31861 y Fv(2)34505 31662 y Fp(:)1102 35060 y Fq(Therefore,)445 b(w)-37 b(e)444 b(assert)f(that)f(for)h(all)g Fp(q)417 b Fm(2)368 b Fl(T)295 b Fm(\002)g Fq(\()p Fp(a;)221 b Fq(2)p Fp(\031)344 b Fm(\000)295 b Fp(a)p Fq(\),)443 b Fp(q)32334 34578 y Fg(\003)33313 35060 y Fm(2)368 b Fp(\031)48 b Fq(\()35972 34725 y(~)35889 35060 y(\000)36719 35259 y Fk(t)37187 35060 y Fq(\()p Fp(c)p Fq(\)\))441 b(and)i Fp(s;)221 b(t)369 b Fm(2)f Fl(T)6222 38029 y Fp(h)6987 37481 y Fg(1)6987 38358 y Fk(c)8113 38029 y Fq(\()p Fp(q)48 b(;)221 b(q)10502 37481 y Fg(\003)11112 38029 y Fp(;)g(s;)g(t)p Fq(\))368 b(=)i Fp(S)16575 37481 y Fk(s)17141 38029 y Fq(\()p Fp(q)18299 37481 y Fg(\003)19205 38029 y Fq(+)295 b(\(0)p Fp(;)221 b Fq(2)p Fp(\031)48 b Fq(\))p Fp(;)221 b(t)p Fq(\))294 b Fm(\000)i Fp(S)28392 37481 y Fk(s)28959 38029 y Fq(\()p Fp(q)48 b(;)221 b(s)p Fq(\))294 b Fm(\000)h Fp(b)p Fq(\()p Fp(c)35131 38228 y Fv(1)35726 38029 y Fq(\)\()p Fp(q)37401 37481 y Fg(\003)37353 38358 y Fv(2)38306 38029 y Fq(+)h(2)p Fp(\031)343 b Fm(\000)295 b Fp(q)43320 38228 y Fv(2)43916 38029 y Fq(\))p Fp(;)6222 40089 y(h)6987 39541 y Fg(1)6987 40417 y Fk(c)8113 40089 y Fq(\()p Fp(q)9271 39541 y Fg(\003)9881 40089 y Fp(;)221 b(q)48 b(;)221 b(s;)g(t)p Fq(\))368 b(=)i Fp(S)16575 39541 y Fk(u)17268 40089 y Fq(\()p Fp(q)48 b(;)221 b(s)p Fq(\))294 b Fm(\000)i Fp(S)22671 39541 y Fk(u)23364 40089 y Fq(\()p Fp(q)24522 39541 y Fg(\003)25132 40089 y Fp(;)221 b(t)p Fq(\))295 b Fm(\000)g Fp(b)p Fq(\()p Fp(c)30004 40288 y Fv(1)30600 40089 y Fq(\)\()p Fp(q)32227 40288 y Fv(2)33117 40089 y Fm(\000)g Fp(q)35086 39541 y Fg(\003)35038 40417 y Fv(2)35697 40089 y Fq(\))p Fp(:)10608 b Fq(\(4.8\))1102 43351 y(In)348 b(fact,)367 b(w)-37 b(e)349 b(ha)-37 b(v)g(e)349 b(seen)g(that)f(\(4.8\))g(holds)g(for)g Fp(q)25024 42869 y Fg(\003)26004 43351 y Fm(2)368 b(M)28853 43550 y Fk(t)29321 43351 y Fq(\()p Fp(c)p Fq(\),)e Fp(q)417 b Fm(2)368 b Fl(T)106 b Fm(\002)g Fq(\()p Fp(a;)221 b Fq(2)p Fp(\031)154 b Fm(\000)106 b Fp(a)p Fq(\))348 b(or)h Fp(q)417 b Fm(2)368 b Fp(\031)48 b Fq(\()47352 43015 y(~)47269 43351 y(\000)p Fm(j)48468 43550 y Fk(s)49036 43351 y Fq(\).)1102 44945 y(As)431 b(there)g(exists)f(an)h(in)-37 b(v)-74 b(arian)-37 b(t)431 b(circle)g(on)g(whic)-37 b(h)431 b(the)f(rotation)h(n)-37 b(um)g(b)37 b(er)431 b(is)f(irrational,)j(w)-37 b(e)432 b(see)1102 46539 y(that)417 b Fp(B)4963 46738 y Fk(c)5504 46539 y Fq(\()p Fp(q)48 b Fq(\))368 b(=)h Fp(P)9802 46738 y Fk(!)10571 46539 y Fq(\()p Fp(q)48 b Fq(\))367 b Fm(\021)i Fq(0)418 b(for)g(all)f Fp(q)g Fm(2)368 b Fp(\031)48 b Fq(\(\000\),)422 b(th)-37 b(us)417 b Fp(d)28236 46738 y Fk(c)28778 46539 y Fq(\()99 b(^)-763 b Fp(q)47 b(;)221 b(q)31166 46057 y Fg(\003)31777 46539 y Fq(\))368 b(=)h(0)418 b(for)g(all)f Fp(q)39614 46057 y Fg(\003)40593 46539 y Fm(2)369 b(M)p Fq(\()p Fp(c)p Fq(\))417 b(and)516 b(^)-763 b Fp(q)417 b Fm(2)1102 48133 y Fp(\031)48 b Fq(\(\000\),)500 b(where)491 b Fp(!)j Fq(=)446 b Fp(@)12033 48332 y Fv(1)12629 48133 y Fp(\013)5 b Fq(\()p Fp(c)p Fq(\),)500 b Fp(P)16815 48332 y Fk(!)18073 48133 y Fq(is)489 b(the)g(P)-37 b(eierl's)490 b(barrier)h(function.)729 b(Consequen)-37 b(tly)-111 b(,)502 b(w)-37 b(e)490 b(ha)-37 b(v)g(e)1102 49727 y Fp(h)1867 49245 y Fg(1)1867 50056 y Fk(c)2993 49727 y Fq(\()p Fp(q)48 b(;)320 b Fq(^)-763 b Fp(q)47 b Fq(\))369 b(=)g Fp(h)8434 49245 y Fg(1)8434 50056 y Fk(c)9560 49727 y Fq(\()p Fp(q)48 b(;)221 b(q)11949 49245 y Fg(\003)12559 49727 y Fq(\))92 b(+)g Fp(h)15058 49245 y Fg(1)15058 50056 y Fk(c)16183 49727 y Fq(\()p Fp(q)17341 49245 y Fg(\003)17951 49727 y Fp(;)320 b Fq(^)-763 b Fp(q)48 b Fq(\).)556 b(Therefore)344 b(w)-37 b(e)342 b(obtain)f(\(4.8\))f(for)i (an)-37 b(y)341 b Fp(q)417 b Fm(2)369 b Fl(T)92 b Fm(\002)g Fq(\()p Fp(a;)221 b Fq(2)p Fp(\031)140 b Fm(\000)92 b Fp(a)p Fq(\))1102 51415 y(and)515 b(an)-37 b(y)515 b Fp(q)6979 50933 y Fg(\003)8078 51415 y Fm(2)488 b Fp(\031)48 b Fq(\()10857 51079 y(~)10774 51415 y(\000)11604 51614 y Fk(t)12071 51415 y Fq(\).)805 b(As)515 b Fp(dS)17380 50933 y Fk(s)17947 51415 y Fm(j)18316 51655 y Fk(\031)33 b Fv(\(\000\))21012 51415 y Fq(=)489 b Fp(dS)24117 50933 y Fk(u)24811 51415 y Fm(j)25180 51655 y Fk(\031)33 b Fv(\(\000\))27388 51415 y Fq(,)532 b(b)-37 b(y)515 b(adding)g(a)f (constan)-37 b(t)515 b(w)-37 b(e)515 b(can)g(assume)1102 53243 y(that)586 b Fp(S)5016 52761 y Fk(s)5583 53243 y Fq(\()p Fp(q)439 b Fq(+)391 b(\(0)p Fp(;)221 b Fq(2)p Fp(\031)48 b Fq(\))p Fp(;)221 b(t)p Fq(\))609 b(=)g Fp(S)17043 52761 y Fk(u)17736 53243 y Fq(\()p Fp(q)48 b(;)221 b(t)p Fq(\))586 b(if)h(\()p Fp(q)48 b(;)221 b(t)p Fq(\))607 b Fm(2)i Fp(\031)48 b Fq(\()28681 52907 y(~)28598 53243 y(\000\).)1022 b(Since)587 b(the)g Fp(c)p Fq(-minimal)f(measure)i(is) 1102 54837 y(uniquely)444 b(ergo)37 b(dic,)444 b(b)-37 b(y)443 b(the)f(lemma)i(3.6,)f(w)-37 b(e)443 b(ha)-37 b(v)g(e)444 b(the)f(follo)-37 b(wing)1102 57180 y Fr(Lemma)509 b(4.2.)764 b Fo(L)-68 b(et)476 b Fp(q)417 b Fm(2)368 b Fl(T)295 b Fm(\002)g Fq(\()p Fp(a;)221 b Fq(2)p Fp(\031)344 b Fm(\000)295 b Fp(a)p Fq(\))p Fo(,)475 b(then)14567 60149 y Fp(B)15642 59600 y Fg(\003)15575 60477 y Fk(c)16252 60149 y Fq(\()p Fp(q)48 b Fq(\))367 b(=)j Fp(S)20589 59600 y Fk(u)21282 60149 y Fq(\()p Fp(q)48 b(;)221 b Fq(0\))295 b Fm(\000)g Fp(S)26726 59600 y Fk(s)27293 60149 y Fq(\()p Fp(q)48 b(;)221 b Fq(0\))294 b Fm(\000)i Fq(2)p Fp(\031)48 b(b)p Fq(\()p Fp(c)34976 60348 y Fv(1)35571 60149 y Fq(\))p Fp(:)10734 b Fq(\(4.9\))1102 63960 y Fr(Pro)42 b(of)p Fq(:)860 b(Since)10294 63624 y(~)9644 63960 y Fm(M)p Fq(\()p Fp(c)p Fq(\))576 b(is)h(uniquely)h(ergo)37 b(dic,)612 b(b)-37 b(y)578 b(de\014nition)f(of)g Fp(B)36309 63478 y Fg(\003)36242 64288 y Fk(c)36919 63960 y Fq(,)611 b(the)577 b(prop)37 b(ert)-37 b(y)578 b Fp(S)46779 63478 y Fk(s)47346 63960 y Fq(\()p Fp(q)432 b Fq(+)1102 65648 y(\(0)p Fp(;)221 b Fq(2)p Fp(\031)48 b Fq(\))p Fp(;)221 b(t)p Fq(\))369 b(=)g Fp(S)9109 65166 y Fk(u)9802 65648 y Fq(\()p Fp(q)48 b(;)221 b(t)p Fq(\))442 b(if)h(\()p Fp(q)48 b(;)221 b(t)p Fq(\))368 b Fm(2)g Fp(\031)48 b Fq(\()19979 65312 y(~)19896 65648 y(\000\))442 b(and)h(\(4.8\))f(w)-37 b(e)444 b(ha)-37 b(v)g(e)8096 69016 y Fp(B)9171 68467 y Fg(\003)9104 69344 y Fk(c)9780 69016 y Fq(\()p Fp(q)48 b Fq(\))368 b(=)222 b(min)13479 69883 y Fk(\030)32 b(;\021)15513 67541 y Fi(n)16399 69016 y Fp(h)17164 68467 y Fg(1)17164 69344 y Fk(c)18289 69016 y Fq(\()p Fp(\030)61 b(;)221 b(q)48 b Fq(\))295 b(+)g Fp(h)23584 68467 y Fg(1)23584 69344 y Fk(c)24710 69016 y Fq(\()p Fp(q)48 b(;)221 b(\021)48 b Fq(\))294 b Fm(\000)h Fp(h)30070 68467 y Fg(1)30070 69344 y Fk(c)31196 69016 y Fq(\()p Fp(\030)61 b(;)221 b(\021)48 b Fq(\))368 b(:)h Fp(\030)61 b(;)221 b(\021)417 b Fm(2)368 b(M)p Fq(\()p Fp(c)p Fq(\))42043 67541 y Fi(o)11823 72256 y Fq(=)222 b(min)13927 73123 y Fk(\030)15513 70781 y Fi(n)16399 72256 y Fp(h)17164 71707 y Fg(1)17164 72584 y Fk(c)18289 72256 y Fq(\()p Fp(\030)61 b(;)221 b(q)48 b Fq(\))295 b(+)g Fp(h)23584 71707 y Fg(1)23584 72584 y Fk(c)24710 72256 y Fq(\()p Fp(q)48 b(;)221 b(\030)61 b Fq(\))369 b(:)g Fp(\030)430 b Fm(2)368 b(M)p Fq(\()p Fp(c)p Fq(\))34193 70781 y Fi(o)11823 74965 y Fq(=)p Fp(S)13748 74416 y Fk(u)14442 74965 y Fq(\()p Fp(q)48 b(;)221 b Fq(0\))295 b Fm(\000)g Fp(S)19886 74416 y Fk(s)20453 74965 y Fq(\()p Fp(q)48 b(;)221 b Fq(0\))294 b Fm(\000)i Fq(2)p Fp(\031)48 b(b)p Fq(\()p Fp(c)28136 75164 y Fv(1)28731 74965 y Fq(\))p Fp(:)10078 b Fb(\003)p eop %%Page: 31 31 31 30 bop 48792 2239 a Ft(31)2696 5427 y Fq(Next,)441 b(w)-37 b(e)441 b(consider)f(the)g(stable)g(\(unstable\))f(manifold)g (of)h(all)g(in)-37 b(v)-74 b(arian)-37 b(t)440 b(circles.)591 b(Di\013eren)-37 b(t)1102 7021 y(in)g(v)-74 b(arian)-37 b(t)343 b(circle)h(determines)f(di\013eren)-37 b(t)343 b(stable)f(and)g(unstable)h(manifold,)362 b(so)343 b(w)-37 b(e)343 b(ha)-37 b(v)g(e)343 b(a)f(family)1102 8615 y(of)606 b(these)g(manifolds.)1080 b(W)-111 b(e)607 b(claim)f(that)f(this)g (family)h(of)g(stable)g(\(unstable\))f(manifolds)h(can)1102 10209 y(b)37 b(e)589 b(parameterized)g(b)-37 b(y)589 b(some)f(parameter)h Fp(\033)636 b Fq(so)588 b(that)f(b)37 b(oth)588 b Fp(p)33949 9571 y Fk(s;u)33949 10563 y Fv(1)36049 10209 y Fq(and)g Fp(p)39445 9571 y Fk(s;u)39445 10563 y Fv(2)41544 10209 y Fq(ha)-37 b(v)g(e)44911 9686 y Fv(1)p 44911 9904 530 54 v 44911 10667 a(2)45600 10209 y Fq(-H\177)-664 b(older)1102 11803 y(con)-37 b(tin)g(uit)g(y)413 b(in)f Fp(\033)48 b Fq(.)580 b(Indeed)413 b(w)-37 b(e)413 b(arbitrarily)g(c) -37 b(ho)37 b(ose)413 b(one)g(circle)h(\000)34056 12002 y Fv(0)35064 11803 y Fq(and)e(parameterize)j(another)1102 13397 y(circle)445 b(\000)5405 13596 y Fk(\033)6563 13397 y Fq(b)-37 b(y)443 b(the)g(algebraic)h(area)g(b)37 b(et)-37 b(w)g(een)444 b(\000)25149 13596 y Fk(\033)26307 13397 y Fq(and)f(\000)29720 13596 y Fv(0)30316 13397 y Fq(,)17100 17461 y Fp(\033)417 b Fq(=)19678 15653 y Fi(Z)21006 15981 y Fv(1)20416 18671 y(0)21602 17461 y Fq(\(\000)22949 17660 y Fk(\033)23664 17461 y Fq(\()p Fp(q)24774 17660 y Fv(1)25370 17461 y Fq(\))294 b Fm(\000)i Fq(\000)28340 17660 y Fv(0)28936 17461 y Fq(\()p Fp(q)30046 17660 y Fv(1)30641 17461 y Fq(\)\))p Fp(dq)32959 17660 y Fv(1)33555 17461 y Fp(:)12603 b Fq(\(4.10\))1102 21321 y(This)617 b(in)-37 b(tegration)618 b(is)e(in)h(the)g(sense)g(that)g(w)-37 b(e)617 b(pull)g(it)f(bac)-37 b(k)618 b(to)f(the)f(standard)h(cylinder) h(b)-37 b(y)1102 22915 y Fp( )207 b Fm(\016)159 b Fp( )3862 23114 y Fv(1)4829 22915 y Fm(2)368 b Fq(di\013\(\006)9441 23114 y Fv(0)10037 22915 y Fp(;)221 b Fq(\006\))375 b(\(cf.)568 b(\(3.1\)\).)e(In)375 b(this)f(w)-37 b(a)g(y)376 b(w)-37 b(e)376 b(obtain)f(an)g(one-parameter)h(family)f(curv)-37 b(es)1102 24509 y(\000:)542 b Fl(T)98 b Fm(\002)g Fl(S)369 b Fm(!)g Fq(\006)345 b(in)f(whic)-37 b(h)345 b Fl(S)369 b Fm(\032)g Fq([)p Fp(A)18093 24027 y Fg(0)18466 24509 y Fp(;)221 b(B)20131 24027 y Fg(0)20504 24509 y Fq(])344 b(is)g(a)h(closed)g(set.)557 b(Usually)-111 b(,)365 b Fl(S)344 b Fq(is)g(a)h(Can)-37 b(tor)345 b(with)f(p)37 b(ositiv)-37 b(e)1102 26103 y(Leb)37 b(esgue)426 b(measure,)k Fp(A)13357 25621 y Fg(0)14154 26103 y Fq(and)425 b Fp(B)17794 25621 y Fg(0)18590 26103 y Fq(corresp)37 b(ond)426 b(to)e(the)h(curv) -37 b(es)426 b(where)g(the)e(action)h Fp(p)43993 26302 y Fv(1)44958 26103 y Fm(\024)369 b Fp(A)425 b Fq(and)1102 27697 y Fp(p)1770 27896 y Fv(1)2792 27697 y Fm(\025)h Fp(B)543 b Fq(resp)37 b(ectiv)-37 b(ely)-111 b(.)693 b(Clearly)-111 b(,)487 b(for)477 b(eac)-37 b(h)478 b Fp(\033)473 b Fm(2)425 b Fl(S)p Fq(,)485 b(there)477 b(is)g(only)g(one)g Fp(c)38489 27896 y Fv(1)39510 27697 y Fq(=)426 b Fp(c)41544 27896 y Fv(1)42139 27697 y Fq(\()p Fp(\033)48 b Fq(\))476 b(suc)-37 b(h)477 b(that)1102 29385 y(\000)1932 29584 y Fk(\033)3017 29385 y Fq(=)5069 29049 y(~)4419 29385 y Fm(M)p Fq(\()p Fp(c)p Fq(\))p Fm(j)7992 29584 y Fk(t)p Fv(=0)10167 29385 y Fq(for)363 b(all)h Fp(c)k Fm(2)h Fp(I)104 b Fq(\()p Fp(c)17864 29584 y Fv(1)18459 29385 y Fq(\()p Fp(\033)48 b Fq(\)\))362 b(as)h(the)g(rotation)g(n)-37 b(um)g(b)37 b(er)364 b(is)f(irrational.) 564 b(W)-111 b(e)364 b(can)g(think)1102 30979 y(\000)1932 31178 y Fk(\033)3088 30979 y Fq(as)439 b(a)h(map)g(to)g(function)f (space)i Fp(C)20179 30497 y Fv(0)21215 30979 y Fq(equipp)37 b(ed)440 b(with)g(suprem)-37 b(um)440 b(norm)g(\000:)589 b Fl(S)369 b Fm(!)g Fp(C)45487 30497 y Fv(0)46084 30979 y Fq(\()p Fl(T)p Fp(;)221 b Fl(R)p Fq(\),)13410 34167 y Fm(k)p Fq(\000)14904 34366 y Fk(\033)15520 34499 y Fj(1)16400 34167 y Fm(\000)295 b Fq(\000)18558 34366 y Fk(\033)19174 34499 y Fj(2)19759 34167 y Fm(k)369 b Fq(=)g(max)22260 35029 y Fk(q)22747 35162 y Fj(1)23265 35029 y Fg(2)p Fc(T)24888 34167 y Fm(j)p Fq(\000\()p Fp(q)27197 34366 y Fv(1)27793 34167 y Fp(;)221 b(\033)29142 34366 y Fv(1)29738 34167 y Fq(\))295 b Fm(\000)g Fq(\000\()p Fp(q)33818 34366 y Fv(1)34414 34167 y Fp(;)221 b(\033)35763 34366 y Fv(2)36359 34167 y Fq(\))p Fm(j)p Fp(:)1102 37992 y Fq(Direct)443 b(calculation)h(sho)-37 b(ws)11573 41915 y Fm(j)p Fp(\033)12701 42114 y Fv(1)13592 41915 y Fm(\000)296 b Fp(\033)15680 42114 y Fv(2)16276 41915 y Fm(j)369 b(\025)19394 41017 y Fq(1)p 18575 41610 2302 54 v 18575 42827 a(2)p Fp(C)20188 43026 y Fk(h)21258 40042 y Fi(\022)22236 41915 y Fq(max)22301 42777 y Fk(q)22788 42910 y Fj(1)23307 42777 y Fg(2)p Fc(T)24929 41915 y Fm(j)p Fq(\000\()p Fp(q)27238 42114 y Fv(1)27834 41915 y Fp(;)221 b(\033)29183 42114 y Fv(1)29779 41915 y Fq(\))295 b Fm(\000)h Fq(\000\()p Fp(q)33860 42114 y Fv(1)34455 41915 y Fp(;)221 b(\033)35804 42114 y Fv(2)36401 41915 y Fq(\))p Fm(j)37287 40042 y Fi(\023)38264 40318 y Fv(2)39081 41915 y Fp(;)1102 45827 y Fq(where)445 b Fp(C)5893 46026 y Fk(h)7024 45827 y Fq(is)e(the)f(Lipsc)-37 b(hitz)444 b(constan)-37 b(t)443 b(for)g(the)f(t)-37 b(wist)443 b(map,)g(it)f(follo)-37 b(ws)443 b(that)17407 49145 y Fm(k)p Fq(\000)18901 49344 y Fk(\033)19517 49477 y Fj(1)20397 49145 y Fm(\000)295 b Fq(\000)22555 49344 y Fk(\033)23171 49477 y Fj(2)23756 49145 y Fm(k)369 b(\024)g Fp(C)27140 49344 y Fk(s)27708 49145 y Fm(j)p Fp(\033)28836 49344 y Fv(1)29727 49145 y Fm(\000)296 b Fp(\033)31815 49344 y Fv(2)32411 49145 y Fm(j)32939 48240 y Fj(1)p 32939 48387 453 46 v 32939 48917 a(2)46527 49145 y Fq(\(4.11\))1102 52336 y(where)405 b Fp(C)5853 52535 y Fk(s)6790 52336 y Fq(=)8192 51311 y Fm(p)p 9299 51311 2302 54 v 1025 x Fq(2)p Fp(C)10912 52535 y Fk(h)11601 52336 y Fq(.)577 b(Since)404 b(the)f(stable)g (\(unstable\))f(\014b)37 b(ers)404 b(ha)-37 b(v)g(e)404 b Fp(C)35961 51854 y Fk(r)29 b Fg(\000)p Fv(2)37909 52336 y Fq(-smo)37 b(othness)403 b(on)g(their)1102 53930 y(base)362 b(p)37 b(oin)-37 b(ts)362 b(on)f(\006,)379 b Fp(p)12030 53448 y Fk(s;u)12030 54258 y(\033)13903 53930 y Fq(is)361 b(also)17899 53407 y Fv(1)p 17899 53624 530 54 v 17899 54388 a(2)18588 53930 y Fq(-H\177)-664 b(older)363 b(con)-37 b(tin)g(uous)362 b(in)f Fp(\033)48 b Fq(.)563 b(Th)-37 b(us,)378 b(there)362 b(exist)g(t)-37 b(w)g(o)362 b(families)1102 55602 y(of)569 b Fp(C)3785 55120 y Fv(1)p Fk(;)p Fv(1)5794 55602 y Fq(functions)g Fp(S)12539 55120 y Fk(u)12462 55930 y(\033)13232 55602 y Fq(\()p Fp(q)48 b(;)221 b(t)p Fq(\):)840 b Fm(f)p Fq(\()p Fp(q)48 b(;)221 b(t)p Fq(\))578 b(:)g Fp(q)22713 55801 y Fv(2\000)23899 55934 y Fd(\033)24597 55602 y Fq(\()p Fp(q)25707 55801 y Fv(1)26303 55602 y Fp(;)221 b(t)p Fq(\))578 b Fm(\024)g Fp(q)30672 55801 y Fv(2)31846 55602 y Fm(\024)g Fq(2)p Fp(\031)427 b Fm(\000)379 b Fp(a)p Fm(g)579 b(!)f Fp(M)713 b Fq(and)569 b Fp(S)46171 55120 y Fk(s)46094 55930 y(\033)46808 55602 y Fq(\()p Fp(q)48 b(;)221 b(t)p Fq(\):)1102 57196 y Fm(f)p Fq(\()p Fp(q)48 b(;)221 b(t)p Fq(\))472 b(:)h Fp(a)g Fm(\024)g Fp(q)9099 57395 y Fv(2)10167 57196 y Fm(\024)g Fp(q)12266 57395 y Fv(2\000)13452 57528 y Fd(\033)14150 57196 y Fq(\()p Fp(q)15260 57395 y Fv(1)15856 57196 y Fp(;)221 b(t)p Fq(\))336 b(+)h(2)p Fp(\031)48 b Fm(g)473 b(!)g Fp(M)145 b Fq(,)520 b(whic)-37 b(h)506 b(are)g(also)34907 56673 y Fv(1)p 34907 56890 V 34907 57654 a(2)35596 57196 y Fq(-H\177)-664 b(older)506 b(con)-37 b(tin)g(uous)506 b(in)f Fp(\033)48 b Fq(.)1102 58790 y(Remem)-37 b(b)37 b(er)501 b(for)e(eac)-37 b(h)501 b Fp(\033)510 b Fm(2)463 b Fl(S)p Fq(,)513 b Fp(B)18271 58308 y Fg(\003)18204 59218 y Fk(c)p Fv(\()p Fk(\033)33 b Fv(\))20224 58790 y Fq(\()p Fp(q)48 b Fq(\))498 b(can)i(alw)-37 b(a)g(ys)500 b(tak)-37 b(e)500 b(the)f(v)-74 b(alue)500 b(zero)g(as)g(its)e(minim) -37 b(um)1102 60705 y(in)424 b(the)f(region)h Fm(f)p Fp(a)369 b Fm(\024)g Fp(q)12600 60904 y Fv(2)13565 60705 y Fm(\024)h Fq(2)p Fp(\031)304 b Fm(\000)256 b Fp(a)p Fm(g)p Fq(,)428 b(it)422 b(follo)-37 b(ws)424 b(from)g(the)31343 60182 y Fv(1)p 31343 60400 V 31343 61163 a(2)32032 60705 y Fq(-H\177)-664 b(older)424 b(con)-37 b(tin)g(uit)g(y)424 b(of)f Fp(S)45415 60067 y Fk(s;u)45338 61158 y(c)p Fv(\()p Fk(\033)33 b Fv(\))47781 60705 y Fq(and)1102 62645 y(the)497 b(expression)h(of)e Fp(B)12541 62163 y Fg(\003)12474 63073 y Fk(c)p Fv(\()p Fk(\033)33 b Fv(\))14991 62645 y Fq(giv)-37 b(en)497 b(b)-37 b(y)497 b(\(4.9\))f(that)g Fp(b)p Fq(\()p Fp(c)28233 62844 y Fv(1)28828 62645 y Fq(\()p Fp(\033)48 b Fq(\)\))495 b(also)i(has)36981 62122 y Fv(1)p 36981 62340 V 36981 63103 a(2)37670 62645 y Fq(-H\177)-664 b(older)498 b(con)-37 b(tin)g(uit)g(y)496 b(in)1102 64661 y Fp(\033)48 b Fq(.)835 b(F)-111 b(or)526 b Fp(z)563 b Fm(2)504 b Fl(T)p Fq(,)545 b(there)525 b(is)f(unique)h Fp(z)19864 64860 y Fk(\033)20579 64661 y Fq(\()p Fp(t)p Fq(\))504 b Fm(2)g Fp(\031)48 b Fq(\()25392 64326 y(~)25309 64661 y(\000)26139 64860 y Fk(\033)33 b(t)27256 64661 y Fq(\))524 b(suc)-37 b(h)524 b(that)g Fp(z)34952 64860 y Fk(\033)35667 64661 y Fq(\()p Fp(t)p Fq(\))504 b(=)h(\()p Fp(z)58 b(;)221 b(q)41599 64860 y Fv(2\000)42785 64993 y Fd(\033)43484 64661 y Fq(\()p Fp(z)58 b(;)221 b(t)p Fq(\)\).)834 b(Let)1102 66255 y Fp(c)p Fq(\()p Fp(\033)48 b Fq(\))368 b(=)h(\()p Fp(c)6380 66454 y Fv(1)6976 66255 y Fq(\()p Fp(\033)48 b Fq(\))p Fp(;)221 b(b)p Fq(\()p Fp(c)11069 66454 y Fv(1)11663 66255 y Fq(\()p Fp(\033)48 b Fq(\)\)\),)441 b(w)-37 b(e)444 b(ha)-37 b(v)g(e:)1102 68646 y Fr(Lemma)548 b(4.3.)764 b Fo(F)-102 b(or)507 b(al)68 b(l)507 b Fp(q)473 b Fm(2)426 b Fl(T)317 b Fm(\002)h Fq(\()p Fp(a;)221 b Fq(2)p Fp(\031)366 b Fm(\000)318 b Fp(a)p Fq(\))p Fo(,)514 b Fp(z)484 b Fm(2)425 b Fl(T)506 b Fo(and)g Fp(s;)221 b(t)426 b Fm(2)f Fl(T)506 b Fo(the)g(functions)i Fp(S)46330 68164 y Fk(s;u)46253 68975 y(\033)47840 68646 y Fq(\()p Fp(q)48 b Fq(\))p Fo(,)1102 70240 y Fp(h)1867 69758 y Fg(1)1867 70668 y Fk(c)p Fv(\()p Fk(\033)33 b Fv(\))3888 70240 y Fq(\()p Fp(q)48 b(;)221 b(z)6254 70439 y Fk(\033)6969 70240 y Fq(\()p Fp(t)p Fq(\))p Fp(;)g(s;)g(t)p Fq(\))p Fo(,)427 b Fp(h)12882 69758 y Fg(1)12882 70668 y Fk(c)p Fv(\()p Fk(\033)33 b Fv(\))14902 70240 y Fq(\()p Fp(z)16037 70439 y Fk(\033)16752 70240 y Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(q)48 b(;)221 b(s;)g(t)p Fq(\))415 b Fo(and)h Fp(B)26308 69758 y Fg(\003)26241 70668 y Fk(c)p Fv(\()p Fk(\033)33 b Fv(\))28261 70240 y Fq(\()p Fp(q)48 b Fq(\))416 b Fo(ar)-68 b(e)32710 69717 y Fv(1)p 32710 69935 V 32710 70698 a(2)33399 70240 y Fo(-H\177)-679 b(older)417 b(c)-68 b(ontinuous)417 b(in)g Fp(\033)f Fm(2)369 b Fl(S)p Fo(.)2696 72827 y Fq(Di\013eren)-37 b(t)460 b(from)f Fp(B)12464 72345 y Fg(\003)12397 73155 y Fk(c)13074 72827 y Fq(,)k Fp(h)14671 72345 y Fg(1)14671 73155 y Fk(c)16256 72827 y Fq(dep)37 b(ends)460 b(on)f(the)g(c)-37 b(hoice)461 b(of)e(the)g(closed)h(1-form)g Fp(\021)42231 73026 y Fk(c)43230 72827 y Fq(\(cf.)640 b([Ma4]\).)1102 74421 y(T)-111 b(o)444 b(guaran)-37 b(tee)444 b(the)f(H\177)-664 b(older)444 b(con)-37 b(tin)g(uit)g(y)442 b(w)-37 b(e)444 b(c)-37 b(ho)37 b(ose)444 b Fp(\021)28861 74620 y Fk(c)29770 74421 y Fq(=)370 b Fm(h)p Fp(c)p Fq(\()p Fp(\033)48 b Fq(\))p Fp(;)466 b Fq(_)-614 b Fp(q)46 b Fm(i)442 b Fq(in)h(ab)37 b(o)-37 b(v)g(e)444 b(lemma.)p eop %%Page: 32 32 32 31 bop 1102 2239 a Ft(32)1102 5427 y Fr(5,)510 b(Construction)e(of)h (connection)f(orbits)2696 8013 y Fq(Throughout)408 b(this)f(section)h (w)-37 b(e)408 b(shall)f(mak)-37 b(e)408 b(the)f(follo)-37 b(wing)408 b(h)-37 b(yp)37 b(otheses,)415 b(their)407 b(v)-37 b(eri\014cation)1102 9607 y(shall)443 b(b)37 b(e)443 b(p)37 b(ostp)g(oned)443 b(to)g(the)f(section)i(6.)2696 12192 y Fr(\(H1\))p Fq(:)582 b(F)-111 b(or)428 b(eac)-37 b(h)428 b Fp(\033)417 b Fm(2)368 b Fl(S)h Fm(\032)g Fq([)p Fp(A)18440 11710 y Fg(0)18813 12192 y Fp(;)221 b(B)20478 11710 y Fg(0)20850 12192 y Fq(],)430 b(the)d(set)g Fm(f)p Fp(B)28087 11710 y Fg(\003)28020 12620 y Fk(c)p Fv(\()p Fk(\033)33 b Fv(\))30409 12192 y Fq(=)369 b(0)p Fm(g)264 b(\\)f(f)p Fp(a)369 b Fm(\024)g Fp(q)38282 12391 y Fv(2)39247 12192 y Fm(\024)g Fq(2)p Fp(\031)312 b Fm(\000)263 b Fp(a)p Fm(g)427 b Fq(consists)1102 13908 y(of)443 b(discrete)h(p)37 b(oin)-37 b(ts.)1102 16494 y Fr(Remark)p Fq(:)613 b(By)453 b(the)h(c)-37 b(hoice)455 b(of)f Fp(a)p Fq(,)j(the)c(set)h Fm(f)p Fp(B)24908 16011 y Fg(\003)24841 16921 y Fk(c)p Fv(\()p Fk(\033)33 b Fv(\))27248 16494 y Fq(=)388 b(0)p Fm(g)303 b(\\)f(f)p Fp(a)387 b Fm(\024)h Fp(q)35255 16693 y Fv(2)36238 16494 y Fm(\024)g Fq(2)p Fp(\031)350 b Fm(\000)303 b Fp(a)p Fm(g)454 b Fq(is)g(not)f(empt)-37 b(y)1102 18283 y(since)524 b Fp(d\015)5816 18482 y Fv(2)6935 18283 y Fq(can)f(not)f(cross)i(the)f(strip)f Fm(f)p Fp(a)503 b Fm(\024)f Fp(q)24792 18482 y Fv(2)25890 18283 y Fm(\024)h Fq(2)p Fp(\031)396 b Fm(\000)349 b Fp(a)p Fm(g)523 b Fq(under)g(one)h(step)e(of)h(the)g(map)g Fp(\036)p Fq(,)1102 19877 y(there)449 b(m)-37 b(ust)447 b(b)37 b(e)447 b(some)h(p)37 b(oin)-37 b(ts)448 b(on)f(time-1-section)h(of)g(the)f(minimal)h(orbits) f(whose)i(pro)74 b(jection)1102 21491 y(fall)538 b(in)-37 b(to)537 b(the)g(strip.)873 b(By)538 b(the)f(de\014nition)g(of)g Fl(S)p Fq(,)561 b(for)538 b(eac)-37 b(h)538 b Fp(\033)574 b Fm(2)526 b Fl(S)p Fq(,)36444 21155 y(~)35987 21491 y Fm(A)p Fq(\()p Fp(c)p Fq(\()p Fp(\033)48 b Fq(\)\))p Fm(j)40867 21690 y Fk(t)p Fv(=0)43215 21491 y Fq(con)-37 b(tains)537 b(an)1102 23085 y(in)-37 b(v)-74 b(arian)-37 b(t)348 b(circle)h(on)e(the)g(cylinder.)559 b(In)347 b(this)g(case)h(w)-37 b(e)348 b(ha)-37 b(v)g(e)347 b(an)h(explicit)f (expression)h(of)f Fp(B)46183 22603 y Fg(\003)46116 23413 y Fk(c)46793 23085 y Fq(\()p Fp(q)48 b Fq(\))346 b(in)1102 24679 y(the)427 b(strip.)584 b(The)428 b(h)-37 b(yp)37 b(othesis)426 b(\(H1\))g(implies)h(the)f(minimal)g(critical)i(p)37 b(oin)-37 b(t)426 b(set)g(of)h Fp(S)43586 24197 y Fk(s)43509 25107 y(c)p Fv(\()p Fk(\033)33 b Fv(\))45791 24679 y Fm(\000)262 b Fp(S)47978 24197 y Fk(u)47901 25107 y(c)p Fv(\()p Fk(\033)33 b Fv(\))1102 26395 y Fq(consists)443 b(of)g(discrete)h(p)37 b(oin)-37 b(ts,)443 b(and)g(there)g(m)-37 b(ust)443 b(b)37 b(e)443 b(some)g(minimal)g(p)37 b(oin)-37 b(ts)443 b(in)g(the)g(in)-37 b(terior)443 b(of)1102 27989 y(this)g(strip.)2696 30574 y Fr(\(H2\))p Fq(:)1006 b(If)650 b(the)g(rotation)g(n)-37 b(um)g(b)37 b(er)651 b(of)g(\000)g(is)f (rational,)703 b(then)650 b(the)g(asso)37 b(ciated)652 b Fp(c)p Fq(-minimal)1102 32168 y(measure)528 b(consists)e(of)h(single) g(p)37 b(erio)g(dic)527 b(orbit.)841 b(There)528 b(is)e(no)h(homo)37 b(clinic)527 b(lo)37 b(op)526 b(in)h(\006)g(to)f(this)1102 33762 y(orbit.)2696 36348 y(Before)572 b(making)g(the)f(third)f(h)-37 b(yp)37 b(othesis)571 b(let)g(us)g(note)g(that)f(all)h(in)-37 b(v)-74 b(arian)-37 b(t)572 b(circles)g(on)f(the)1102 37942 y(cylinder)465 b(form)g(a)f(closed)h(set.)654 b(These)465 b(circles)g(do)f(not)g(in)-37 b(tersect)465 b(eac)-37 b(h)465 b(other,)k(so)c(the)f(comple-)1102 39536 y(men)-37 b(t)443 b(set)g(consists)g(of)f(coun)-37 b(tably)444 b(man)-37 b(y)443 b(in)-37 b(v)-74 b(arian)-37 b(t)443 b(ann)-37 b(ulus.)2696 42122 y Fr(\(H3\))p Fq(:)654 b(Let)475 b(\000)g(b)37 b(e)475 b(an)g(in)-37 b(v)-74 b(arian)-37 b(t)476 b(circle)g(on)f(\006,)483 b(asso)37 b(ciated)476 b(with)e(co-homology)i(class)f Fp(c)p Fq(.)687 b(If)1102 43716 y(this)469 b(circle)h(is)f(on)f(the)h(b)37 b(oundary)469 b(of)g(a)g(gap,)476 b(then)468 b(for)i(su\016cien)-37 b(tly)469 b(small)g Fp(\016)463 b(>)412 b Fq(0)469 b(there)g(exists) 1102 45310 y Fp(c)1677 44828 y Fg(0)2419 45310 y Fq(=)369 b(\()p Fp(c)4913 45509 y Fv(1)5508 45310 y Fp(;)221 b(c)6673 44828 y Fg(0)6673 45639 y Fv(2)7270 45310 y Fq(\))419 b(with)g(either)h(0)369 b Fp(<)g(c)17962 44828 y Fg(0)17962 45639 y Fv(2)18807 45310 y Fm(\000)249 b Fp(b)p Fq(\()p Fp(c)21751 45509 y Fv(1)22346 45310 y Fq(\))369 b Fp(<)g(\016)470 b Fq(or)420 b Fm(\000)p Fp(\016)g(<)369 b(c)31318 44828 y Fg(0)31318 45639 y Fv(2)32162 45310 y Fm(\000)249 b Fp(a)p Fq(\()p Fp(c)35238 45509 y Fv(1)35834 45310 y Fq(\))368 b Fp(<)h Fq(0)420 b(suc)-37 b(h)420 b(that)f Fm(M)p Fq(\()p Fp(c)47720 44828 y Fg(0)48092 45310 y Fq(\))g(is)1102 46904 y(uniquely)444 b(ergo)37 b(dic.)2696 49509 y(Let)5671 49174 y(~)5250 49509 y Fm(N)6340 49708 y Fk(s)6908 49509 y Fq(\()p Fp(c)p Fq(\))657 b(=)11287 49174 y(~)10865 49509 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\))p Fm(j)14129 49708 y Fk(t)p Fv(=)p Fk(s)15912 49509 y Fq(,)660 b Fm(N)18031 49708 y Fk(s)18599 49509 y Fq(\()p Fp(c)p Fq(\))d(=)i Fm(N)196 b Fq(\()p Fp(c)p Fq(\))p Fm(j)25821 49708 y Fk(t)p Fv(=)p Fk(s)27604 49509 y Fq(.)1111 b(According)618 b(to)e(the)g(study)g(in)g(the)h(last)1102 51197 y(section)350 b(w)-37 b(e)351 b(kno)-37 b(w)350 b(that)13889 50861 y(~)13467 51197 y Fm(N)14557 51396 y Fv(0)15153 51197 y Fq(\()p Fp(c)16245 50715 y Fg(0)16617 51197 y Fq(\))f(is)h (homotopically)g(trivial,)368 b(but)350 b(this)f(do)37 b(es)350 b(not)f(guaran)-37 b(tee)351 b(that)1102 52885 y Fm(N)2192 53084 y Fv(0)2788 52885 y Fq(\()p Fp(c)3880 52403 y Fg(0)4252 52885 y Fq(\))475 b(is)h(also)f(homotopically)h (trivial)g(on)f Fp(M)145 b Fq(,)483 b(since)477 b(the)e(pro)74 b(jection)476 b(from)41635 52549 y(~)41214 52885 y Fm(N)195 b Fq(\()p Fp(c)43591 52403 y Fg(0)43963 52885 y Fq(\))423 b Fm(!)h(N)196 b Fq(\()p Fp(c)49033 52403 y Fg(0)49405 52885 y Fq(\))1102 54573 y(is)513 b(not)f(necessarily)j(injectiv)-37 b(e.)800 b(If)19684 54237 y(~)19034 54573 y Fm(M)p Fq(\()p Fp(c)21721 54091 y Fg(0)22093 54573 y Fq(\))512 b(is)h(uniquely)g(ergo) 37 b(dic,)531 b(then)38574 54237 y(~)38153 54573 y Fm(N)195 b Fq(\()p Fp(c)40530 54091 y Fg(0)40903 54573 y Fq(\))485 b(=)43880 54237 y(~)43423 54573 y Fm(A)p Fq(\()p Fp(c)45576 54091 y Fg(0)45948 54573 y Fq(\).)800 b(The)1102 56167 y(Lipsc)-37 b(hitze)433 b(prop)37 b(ert)-37 b(y)433 b(of)f Fm(A)p Fq(\()p Fp(c)16429 55685 y Fg(0)16801 56167 y Fq(\))f(implies)h(that)f Fm(N)26204 56366 y Fv(0)26800 56167 y Fq(\()p Fp(c)27892 55685 y Fg(0)28264 56167 y Fq(\))h(is)g(homotopically)g(trivial)g(in)g(this)f(case.)1102 57761 y(Giv)-37 b(en)607 b(arbitrary)g(small)f Fp(d)641 b(>)g Fq(0,)646 b(there)607 b(are)g(only)f(\014nite)f(in)-37 b(v)-74 b(arian)-37 b(t)607 b(circles)g(whic)-37 b(h)607 b(are)f(the)1102 59355 y(b)37 b(oundary)488 b(of)f(some)h(ann)-37 b(ulus)487 b(with)g(width)g(not)g(smaller)h(than)e Fp(d)p Fq(.)724 b(Actually)-111 b(,)499 b(w)-37 b(e)488 b(require)g(the)1102 60949 y(third)443 b(h)-37 b(yp)37 b(othesis)443 b(only)g(for)g(these)g (tori.)2696 63535 y(The)503 b(\014rst)e(task)g(in)g(this)g(section)h (is)f(to)g(build)h(a)f Fp(C)95 b Fq(-equiv)-74 b(alen)-37 b(t)503 b(sequence)g Fm(f)p Fp(c)42298 63053 y Fv(\()p Fk(i)p Fv(\))43570 63535 y Fm(g)44234 63053 y Fk(m)44234 63880 y(i)p Fv(=1)46523 63535 y Fq(where)1102 65401 y Fp(c)1677 64711 y Fv(\(1\))1677 65755 y(1)3596 65401 y Fq(=)492 b Fp(c)5696 65600 y Fv(1)6292 65401 y Fq(\()p Fp(\033)7616 64919 y Fg(0)7988 65401 y Fq(\),)535 b Fp(a)p Fq(\()p Fp(c)11203 64711 y Fv(\(1\))11203 65755 y(1)12629 65401 y Fq(\))491 b Fm(\024)i Fp(c)15738 64711 y Fv(\(1\))15738 65755 y(2)17656 65401 y Fm(\024)g Fp(b)p Fq(\()p Fp(c)20844 64711 y Fv(\(1\))20844 65755 y(1)22269 65401 y Fq(\),)535 b Fp(c)24265 64711 y Fv(\()p Fk(m)p Fv(\))24265 65755 y(1)26597 65401 y Fq(=)492 b Fp(c)28697 65600 y Fv(1)29293 65401 y Fq(\()p Fp(\033)30617 64919 y Fg(\003)31227 65401 y Fq(\),)535 b Fp(a)p Fq(\()p Fp(c)34442 64711 y Fv(\()p Fk(m)p Fv(\))34442 65755 y(1)36281 65401 y Fq(\))492 b Fm(\024)g Fp(c)39390 64711 y Fv(\()p Fk(m)p Fv(\))39390 65755 y(2)41721 65401 y Fm(\024)h Fp(b)p Fq(\()p Fp(c)44909 64711 y Fv(\()p Fk(m)p Fv(\))44909 65755 y(1)46748 65401 y Fq(\))516 b(and)1102 66995 y Fp(\033)1909 66513 y Fg(0)2665 66995 y Fp(<)383 b(\033)4888 66513 y Fg(\003)5950 66995 y Fq(corresp)37 b(ond)453 b(to)e(t)-37 b(w)g(o)451 b(in)-37 b(v)-74 b(arian)-37 b(t)452 b(circles)h(whic)-37 b(h)452 b(mak)-37 b(e)452 b(up)f(the)h(whole)g(b)37 b(oundary)452 b(of)f(a)1102 68589 y(gap.)645 b(Th)-37 b(us)462 b(a)f(theorem)g(of)g (connecting)h Fp(C)95 b Fq(-equiv)-74 b(alen)-37 b(t)462 b(Ma)37 b(~)-701 b(n)-37 b(\023)-627 b(e)461 b(sets)g(is)f(used)i(to)e (construct)h(the)1102 70183 y(di\013usion)422 b(orbits)g(crossing)g (this)f(gap.)584 b(This)422 b(kind)g(of)g(theorem)g(w)-37 b(as)422 b(disco)-37 b(v)g(ered)424 b(b)-37 b(y)422 b(Mather)f(in)1102 71777 y([Ma4])515 b(where)i(the)e(pro)37 b(of)515 b(w)-37 b(as)516 b(sk)-37 b(etc)g(hed.)807 b(T)-111 b(o)516 b(mak)-37 b(e)516 b(use)f(of)g(this)g(theorem,)533 b(w)-37 b(e)516 b(shall)f(giv)-37 b(e)516 b(a)1102 73371 y(complete)453 b(pro)37 b(of)452 b(\014rst.)616 b(A)452 b(theorem)g(of)g(connecting)g (di\013eren)-37 b(t)452 b Fm(G)79 b Fq(\()p Fp(c)p Fq(\))451 b(w)-37 b(as)452 b(pro)-37 b(v)g(ed)453 b(b)-37 b(y)451 b(Bernard)1102 74965 y(recen)-37 b(tly)444 b(\([Be]\).)p eop %%Page: 33 33 33 32 bop 48792 2239 a Ft(33)2696 5427 y Fq(T)-111 b(o)444 b(an)-37 b(y)443 b(subset)g Fp(A)g Fq(of)g Fp(M)587 b Fq(w)-37 b(e)444 b(asso)37 b(ciate)443 b(a)g(subspace)h(of)e Fp(H)32878 5626 y Fv(1)33475 5427 y Fq(\()p Fp(M)71 b(;)221 b Fl(R)p Fq(\))7881 8806 y Fp(V)295 b Fq(\()p Fp(A)p Fq(\))368 b(=)12751 7544 y Fi(\\)14449 7332 y(n)15334 8806 y Fp(i)15792 9005 y Fk(U)100 b Fg(\003)17226 8806 y Fp(H)18330 9005 y Fv(1)18926 8806 y Fq(\()p Fp(U)m(;)221 b Fl(R)p Fq(\))369 b(:)g Fp(U)588 b Fq(is)442 b(an)h(op)37 b(en)443 b(neigh)-37 b(b)37 b(orho)g(o)g(d)444 b(of)f Fp(A)42257 7332 y Fi(o)47191 8806 y Fq(\(5.1\))1102 12186 y(where)366 b Fp(i)5323 12385 y Fk(U)100 b Fg(\003)6757 12186 y Fq(:)551 b Fp(H)8781 12385 y Fv(1)9377 12186 y Fq(\()p Fp(U)m(;)221 b Fl(R)p Fq(\))369 b Fm(!)h Fp(H)16035 12385 y Fv(1)16631 12186 y Fq(\()p Fp(M)71 b(;)221 b Fl(R)p Fq(\))364 b(is)h(the)f(map)h(induced)g(b)-37 b(y)365 b(the)f(inclusion.)565 b(Clearly)-111 b(,)382 b(there)1102 13780 y(exists)399 b(an)f(op)37 b(en)399 b(neigh)-37 b(b)37 b(orho)g(o)g(d)400 b Fp(U)543 b Fq(of)399 b Fp(A)g Fq(suc)-37 b(h)398 b(that)g Fp(V)295 b Fq(\()p Fp(A)p Fq(\))369 b(=)g Fp(i)33419 13979 y Fk(U)100 b Fg(\003)34853 13780 y Fp(H)35957 13979 y Fv(1)36553 13780 y Fq(\()p Fp(U)145 b Fq(\).)574 b(Let)399 b Fp(V)42988 13298 y Fg(?)43885 13780 y Fq(\()p Fp(A)p Fq(\))f(b)37 b(e)398 b(the)1102 15374 y(annihilator)535 b(of)g Fp(V)295 b Fq(\()p Fp(A)p Fq(\).)866 b(In)534 b(other)h(w)-37 b(ords,)559 b(if)535 b Fp(c)522 b Fm(2)g Fp(H)28566 14892 y Fv(1)29162 15374 y Fq(\()p Fp(M)71 b(;)221 b Fl(R)p Fq(\),)557 b(then)535 b Fp(c)522 b Fm(2)f Fp(V)40724 14892 y Fg(?)42155 15374 y Fq(if)535 b(and)f(only)h(if)1102 16968 y Fm(h)p Fp(c;)221 b(h)p Fm(i)369 b Fq(=)g(0)443 b(for)g(all)g Fp(h)370 b Fm(2)e Fp(V)295 b Fq(\()p Fp(A)p Fq(\).)590 b(Giv)-37 b(en)444 b Fp(c)369 b Fm(2)f Fp(H)24527 16486 y Fv(1)25123 16968 y Fq(\()p Fp(M)71 b(;)221 b Fl(R)p Fq(\))442 b(w)-37 b(e)444 b(de\014ne)18570 20214 y Fp(R)10 b Fq(\()p Fp(c)p Fq(\))367 b(=)22968 18952 y Fi(X)23058 21807 y Fk(t)p Fg(2)p Fc(T)24886 20214 y Fq(\()p Fp(V)295 b Fq(\()p Fm(N)28080 20413 y Fk(t)28547 20214 y Fq(\()p Fp(c)p Fq(\)\)\))31190 19666 y Fg(?)32085 20214 y Fp(:)14737 b Fq(\(5.2\))1102 24526 y(In)443 b([Be])g Fp(R)10 b Fq(\()p Fp(c)p Fq(\))442 b(is)g(de\014ned)i(b)-37 b(y)443 b(using)g Fm(G)79 b Fq(\()p Fp(c)p Fq(\))441 b(instead)i(of)g(using)g Fm(N)196 b Fq(\()p Fp(c)p Fq(\).)2696 26871 y(W)-111 b(e)433 b(sa)-37 b(y)432 b(a)g(con)-37 b(tin)g(uous)431 b(curv)-37 b(e)433 b(\000:)585 b Fl(R)370 b Fm(!)f Fp(H)24547 26389 y Fv(1)25143 26871 y Fq(\()p Fp(M)71 b(;)221 b Fl(R)p Fq(\))431 b(is)h(admissible)g(if)f(for)h(eac)-37 b(h)432 b Fp(t)369 b Fm(2)g Fl(R)432 b Fq(there)1102 28465 y(exists)497 b Fp(\016)509 b(>)458 b Fq(0)497 b(suc)-37 b(h)497 b(that)e(\000\()p Fp(t)p Fq(\))330 b Fm(\000)h Fq(\000\()p Fp(t)20421 28664 y Fv(0)21017 28465 y Fq(\))458 b Fm(2)g Fp(R)10 b Fq(\(\000\()p Fp(t)26699 28664 y Fv(0)27293 28465 y Fq(\)\))496 b(for)h(all)f Fp(t)458 b Fm(2)g Fq([)p Fp(t)35939 28664 y Fv(0)36866 28465 y Fm(\000)331 b Fp(\016)-24 b(;)221 b(t)39866 28664 y Fv(0)40793 28465 y Fq(+)332 b Fp(\016)50 b Fq(].)751 b(W)-111 b(e)498 b(sa)-37 b(y)496 b Fp(c)p Fq(,)1102 30059 y Fp(c)1677 29577 y Fg(0)2537 30059 y Fm(2)486 b Fp(H)5121 29577 y Fv(1)5717 30059 y Fq(\()p Fp(M)71 b(;)221 b Fl(R)p Fq(\))513 b(are)i Fp(C)95 b Fq(-equiv)-74 b(alen)-37 b(t)514 b(if)g(there)g(is)f(an)h (admissible)f(curv)-37 b(e)515 b(\000:)732 b([0)p Fp(;)221 b Fq(1])488 b Fm(!)f Fp(M)658 b Fq(suc)-37 b(h)1102 31653 y(that)442 b(\000\(0\))369 b(=)g Fp(c)443 b Fq(and)g(\000\(1\))368 b(=)h Fp(c)16753 31171 y Fg(0)17126 31653 y Fq(.)2696 33998 y(Let)451 b Fp(U)595 b Fq(b)37 b(e)451 b(an)g(op)37 b(en)451 b(subset)f(of)g Fp(M)445 b Fm(\002)301 b Fl(T)p Fq(,)452 b(w)-37 b(e)451 b(can)g(think)f(it)g(as)h(the)f(op)37 b(en)451 b(subset)f(in)h Fp(M)444 b Fm(\002)301 b Fl(R)1102 35592 y Fq(of)401 b(p)37 b(oin)-37 b(ts)400 b(\()p Fp(q)48 b(;)221 b(t)p Fq(\))400 b(suc)-37 b(h)401 b(that)e(\()p Fp(q)48 b(;)221 b(t)400 b Fq(mo)37 b(d)401 b(1\))369 b Fm(2)f Fp(U)145 b Fq(.)576 b(The)402 b(1-form)e Fp(\026)h Fq(on)g Fp(M)355 b Fm(\002)211 b Fl(R)401 b Fq(is)g(called)g(a)g Fp(U)145 b Fq(-step)1102 37186 y(form)504 b(if)f(there)h(is)g(a)g (closed)g(form)609 b(\026)-769 b Fp(\026)503 b Fq(on)h Fp(M)480 b Fm(\002)336 b Fl(T)p Fq(,)518 b(also)504 b(considered)h(as)e (a)h(p)37 b(erio)g(dic)504 b(1-form)g(on)1102 38780 y Fp(M)441 b Fm(\002)296 b Fl(R)p Fq(,)444 b(suc)-37 b(h)444 b(that)f(the)h(restriction)h(of)e Fp(\026)i Fq(to)e Fp(t)371 b Fm(\024)f Fq(0)444 b(is)g(0,)g(the)g(restriction)g(of)g Fp(\026)g Fq(to)g Fp(t)370 b Fm(\025)h Fq(1)444 b(is)549 b(\026)-769 b Fp(\026)p Fq(,)1102 40374 y(and)473 b(suc)-37 b(h)474 b(that)e(the)h(restriction)g(of)g Fp(\026)g Fq(to)g(the)g(set)f Fp(U)461 b Fm([)314 b(f)p Fp(t)419 b Fm(\024)h Fq(0)p Fm(g)315 b([)g(f)p Fp(t)419 b Fm(\025)g Fq(1)p Fm(g)473 b Fq(is)g(closed.)682 b(In)472 b(the)1102 41968 y(application)444 b(in)e(this)h(pap)37 b(er,)548 b(\026)-769 b Fp(\026)443 b Fq(is)g(c)-37 b(hosen)444 b(as)f(a)g(closed)g(form)g(on)g Fp(M)145 b Fq(.)2696 44313 y(If)411 b(the)f(\014rst)h(de)g(Rham)f (cohomology)i Fp(d)369 b Fm(2)g Fp(R)10 b Fq(\()p Fp(c)p Fq(\),)416 b(then)411 b(there)g(exist)f(an)h(op)37 b(en)411 b(neigh)-37 b(b)37 b(orho)g(o)g(d)1102 45907 y Fp(U)646 b Fq(of)500 b Fm(N)196 b Fq(\()p Fp(c)p Fq(\))500 b(and)g(a)h Fp(U)145 b Fq(-step)500 b(form)h Fp(\026)g Fq(suc)-37 b(h)501 b(that)f([)105 b(\026)-769 b Fp(\026)p Fq(])465 b(=)h Fp(d)p Fq(.)764 b(Suc)-37 b(h)501 b(a)f(neigh)-37 b(b)37 b(orho)g(o)g(d)502 b Fp(U)645 b Fq(will)501 b(b)37 b(e)1102 47501 y(called)583 b(an)f(adapted)g(neigh)-37 b(b)37 b(orho)g(o)g(d.)1009 b(Indeed,)617 b(similar)582 b(to)g(the)f(argumen)-37 b(ts)583 b(in)f([Be],)616 b(let)582 b(us)1102 49095 y(\014x)564 b(a)f(time)h Fp(t)569 b Fm(2)h Fq([0)p Fp(;)221 b Fq(1])564 b(and)f(a)h(cohomology)g(class)g Fp(d)570 b Fm(2)g Fp(V)295 b Fq(\()p Fm(N)33153 49294 y Fk(t)33620 49095 y Fq(\()p Fp(c)p Fq(\)\))35746 48613 y Fg(?)36641 49095 y Fq(.)952 b(There)565 b(exist)e(an)h(op)37 b(en)1102 50689 y(neigh)-37 b(b)37 b(orho)g(o)g(d)568 b(\012)f(of)f Fm(N)13821 50888 y Fk(t)14289 50689 y Fq(\()p Fp(c)p Fq(\))g(and)g(a)h Fp(\016)626 b(>)575 b Fq(0)567 b(suc)-37 b(h)567 b(that)f Fp(V)295 b Fq(\(\012\))575 b(=)g Fp(V)295 b Fq(\()p Fm(N)38501 50888 y Fk(t)38968 50689 y Fq(\()p Fp(c)p Fq(\)\))565 b(and)i(suc)-37 b(h)567 b(that)1102 52283 y Fm(N)2192 52482 y Fk(s)2760 52283 y Fq(\()p Fp(c)p Fq(\))419 b Fm(\032)h Fq(\012)473 b(for)h(all)f Fp(s)420 b Fm(2)f Fq([)p Fp(t)315 b Fm(\000)h Fp(\016)-24 b(;)221 b(t)316 b Fq(+)g Fp(\016)50 b Fq(].)682 b(As)474 b Fp(d)420 b Fm(2)f Fp(R)10 b Fq(\()p Fp(c)p Fq(\),)480 b(w)-37 b(e)474 b(can)g(tak)-37 b(e)474 b(a)f(closed)h(form)579 b(\026)-769 b Fp(\026)473 b Fq(on)h Fp(M)1102 53877 y Fq(whose)458 b(supp)37 b(ort)456 b(is)g(disjoin)-37 b(t)456 b(from)h(\012)g(and)f(suc)-37 b(h)457 b(that)f([)105 b(\026)-769 b Fp(\026)p Fq(])392 b(=)g Fp(d)p Fq(.)631 b(Let)457 b Fp(\032)p Fq(:)617 b Fl(R)392 b Fm(!)g Fl(R)457 b Fq(b)37 b(e)457 b(a)g(smo)37 b(oth)1102 55471 y(function)500 b(suc)-37 b(h)499 b(that)g Fp(\032)462 b Fq(=)i(0)499 b(on)g(\()p Fm(\0001)p Fp(;)221 b(t)333 b Fm(\000)g Fp(\016)50 b Fq(],)514 b Fp(\032)463 b Fq(=)g(1)499 b(on)h([)p Fp(t)332 b Fq(+)h Fp(\016)-24 b(;)221 b Fm(1)p Fq(\))501 b(and)e(0)463 b Fm(\024)g Fp(\032)g Fm(\024)g Fq(1)500 b(for)f(all)1102 57065 y Fp(t)369 b Fm(2)g Fl(R)421 b Fq(and)g(let)g Fp(U)513 b Fq(=)370 b Fp(M)395 b Fm(\002)252 b Fq(\(\(0)p Fp(;)221 b(t)251 b Fm(\000)g Fp(\016)50 b Fq(\))252 b Fm([)f Fq(\()p Fp(t)f Fq(+)i Fp(\016)-24 b(;)221 b Fq(1\)\))252 b Fm([)f Fq(\012)h Fm(\002)f Fq([)p Fp(t)g Fm(\000)h Fp(\016)-24 b(;)221 b(t)252 b Fq(+)f Fp(\016)50 b Fq(].)584 b(Ob)-37 b(viously)-111 b(,)426 b(the)421 b(form)22726 60046 y Fp(\026)369 b Fq(=)h Fp(\032)p Fq(\()p Fp(t)p Fq(\))105 b(\026)-769 b Fp(\026)1102 63027 y Fq(is)443 b(an)g Fp(U)145 b Fq(-step)442 b(form)h(satis\014es)g(the)g(required)h(conditions.)2696 65372 y(Let)359 b(\000:)549 b([0)p Fp(;)221 b Fq(1])369 b Fm(!)h Fp(H)12675 64890 y Fv(1)13271 65372 y Fq(\()p Fp(M)71 b(;)221 b Fl(R)p Fq(\))358 b(b)37 b(e)359 b(an)g(admissible)f (curv)-37 b(e)360 b(suc)-37 b(h)359 b(that)f(\000\(0\))369 b(=)g Fp(c)358 b Fq(and)h(\000\(1\))369 b(=)g Fp(c)49180 64890 y Fg(0)49552 65372 y Fq(.)1102 66966 y(F)-111 b(or)440 b(eac)-37 b(h)439 b Fp(t)369 b Fm(2)f Fq([0)p Fp(;)221 b Fq(1])440 b(and)e(an)h(adapted)f(neigh)-37 b(b)37 b(orho)g(o)g(d)440 b Fp(U)145 b Fq(\()p Fp(t)p Fq(\),)438 b(let)g Fp(\021)48 b Fq(\()p Fp(t)p Fq(\))437 b(b)37 b(e)438 b(a)h(closed)g(1-form)g(on) 1102 68560 y Fp(M)595 b Fq(suc)-37 b(h)451 b(that)e([)p Fp(\021)48 b Fq(\()p Fp(t)p Fq(\)])379 b(=)j(\000\()p Fp(t)p Fq(\),)451 b(there)f(exists)g Fp(\016)50 b Fq(\()p Fp(t)p Fq(\))381 b Fp(>)g Fq(0)451 b(suc)-37 b(h)450 b(that)g(\000\()p Fp(s)p Fq(\))299 b Fm(\000)h Fq(\000\()p Fp(t)p Fq(\))380 b Fm(2)h Fp(R)10 b Fq(\(\000\()p Fp(t)p Fq(\)\))448 b(and)1102 70154 y(a)498 b Fp(U)145 b Fq(-step)497 b(form)g Fp(\026)p Fq(\()p Fp(s)p Fq(\))g(with)g([)105 b(\026)-769 b Fp(\026)q Fq(\()p Fp(s)p Fq(\)])459 b(=)h(\000\()p Fp(s)p Fq(\))330 b Fm(\000)i Fq(\000\()p Fp(t)p Fq(\))497 b(if)g Fp(s)459 b Fm(2)h Fq(\()p Fp(t)331 b Fm(\000)h Fp(\016)-24 b(;)221 b(t)332 b Fq(+)g Fp(\016)50 b Fq(\).)754 b(According)499 b(to)e(the)1102 71842 y(upp)37 b(er)444 b(semi-con)-37 b(tin)g(uit)g(y)443 b(\()p Fp(\021)48 b(;)221 b(\026)p Fq(\))368 b Fm(!)19831 71506 y Fq(~)19410 71842 y Fm(N)20500 72041 y Fk(\021)33 b(;\026)22551 71842 y Fq(pro)-37 b(v)g(ed)444 b(in)f(the)f(lemma)i(2.4,)f(w)-37 b(e)443 b(can)h(assume)f(that)17956 74965 y Fp(\031)48 b Fq(\()19699 74629 y(~)19278 74965 y Fm(N)20368 75205 y Fk(\021)33 b Fv(\()p Fk(t)p Fv(\))p Fk(;\026)p Fv(\()p Fk(s)p Fv(\))24539 74965 y Fq(\))295 b(+)g Fp(\017)p Fq(\()p Fp(t)p Fq(\))368 b Fm(\032)i Fp(U)145 b Fq(\()p Fp(t)p Fq(\))14122 b(\(5.3\))p eop %%Page: 34 34 34 33 bop 1102 2239 a Ft(34)1102 5427 y Fq(if)421 b(w)-37 b(e)422 b(tak)-37 b(e)422 b(suitably)e(small)i Fp(\016)50 b Fq(\()p Fp(t)p Fq(\).)582 b(In)421 b(this)g(pap)37 b(er)421 b(w)-37 b(e)422 b(use)g Fp(U)396 b Fq(+)252 b Fp(a)421 b Fq(to)g(denote)h(the)f(set)g Fm(f)p Fp(x)369 b Fm(2)g Fp(M)513 b Fq(:)1102 7021 y(dist\()p Fp(x;)221 b(U)145 b Fq(\))368 b Fm(\024)i Fp(a)p Fm(g)p Fq(.)590 b(Clearly)-111 b(,)445 b(there)e(is)g(a)g(\014nite)f(increasing)i (sequence)h Fm(f)p Fp(t)37751 7220 y Fk(i)38193 7021 y Fm(g)38857 7220 y Fv(0)p Fg(\024)p Fk(i)p Fg(\024)p Fk(N)42873 7021 y Fq(suc)-37 b(h)443 b(that)18473 9976 y Fk(N)18205 10374 y Fi([)18083 13203 y Fk(i)p Fv(=0)19804 11636 y Fq(\()p Fp(t)20801 11835 y Fk(i)21537 11636 y Fm(\000)296 b Fp(\016)50 b Fq(\()p Fp(t)24503 11835 y Fk(i)24945 11636 y Fq(\))p Fp(;)221 b(t)26532 11835 y Fk(i)27269 11636 y Fq(+)295 b Fp(\016)50 b Fq(\()p Fp(t)30234 11835 y Fk(i)30676 11636 y Fq(\)\))369 b Fm(\033)g Fq([0)p Fp(;)221 b Fq(1])p Fp(;)14517 14864 y(t)14997 15063 y Fk(i)p Fg(\000)p Fv(1)17168 14864 y Fp(>)369 b(t)19050 15063 y Fk(i)19787 14864 y Fm(\000)295 b Fp(\016)50 b Fq(\()p Fp(t)22752 15063 y Fk(i)23195 14864 y Fq(\))p Fp(;)2877 b(t)27438 15063 y Fk(i)p Fv(+1)29594 14864 y Fp(<)369 b(t)31476 15063 y Fk(i)32213 14864 y Fq(+)296 b Fp(\016)50 b Fq(\()p Fp(t)35179 15063 y Fk(i)35621 14864 y Fq(\))p Fp(:)10684 b Fq(\(5.4\))1102 18207 y(and)497 b(\(5.3\))e(holds)h(for)h(eac)-37 b(h)497 b Fp(t)16103 18406 y Fk(i)16545 18207 y Fq(,)509 b(and)496 b(eac)-37 b(h)497 b Fp(s)457 b Fm(2)h Fq(\()p Fp(t)26522 18406 y Fk(i)27294 18207 y Fm(\000)331 b Fp(\016)50 b Fq(\()p Fp(t)30295 18406 y Fk(i)30737 18207 y Fq(\))p Fp(;)221 b(t)32324 18406 y Fk(i)33097 18207 y Fq(+)330 b Fp(\016)50 b Fq(\()p Fp(t)36097 18406 y Fk(i)36540 18207 y Fq(\)\).)749 b(In)496 b(the)g(follo)-37 b(wing)496 b(w)-37 b(e)1102 19801 y(shall)404 b(use)h Fp(\017)6966 20000 y Fk(i)7408 19801 y Fq(,)412 b Fp(\016)8779 20000 y Fk(i)9221 19801 y Fq(,)g Fp(U)10909 20000 y Fk(i)11351 19801 y Fq(,)g Fp(\021)12792 20000 y Fk(i)13637 19801 y Fq(and)405 b Fp(\026)16982 20000 y Fk(i)17828 19801 y Fq(to)f(denote)g Fp(\017)p Fq(\()p Fp(t)25190 20000 y Fk(i)25632 19801 y Fq(\),)411 b Fp(\016)50 b Fq(\()p Fp(t)28566 20000 y Fk(i)29008 19801 y Fq(\),)411 b Fp(U)145 b Fq(\()p Fp(t)32354 20000 y Fk(i)32796 19801 y Fq(\),)411 b Fp(\021)48 b Fq(\()p Fp(t)35798 20000 y Fk(i)36239 19801 y Fq(\))403 b(and)h Fp(\026)p Fq(\()p Fp(t)41500 20000 y Fk(i)41942 19801 y Fq(\))g(resp)37 b(ectiv)-37 b(ely)-111 b(.)1102 21395 y(Th)-37 b(us)444 b(w)-37 b(e)443 b(ha)-37 b(v)g(e)20701 24024 y Fp(\021)21361 24223 y Fk(i)22172 24024 y Fq(=)369 b Fp(\021)24234 24223 y Fv(0)25125 24024 y Fq(+)26545 22364 y Fk(i)p Fg(\000)p Fv(1)26453 22762 y Fi(X)26493 25592 y Fk(j)54 b Fv(=0)28698 24024 y Fq(\026)-769 b Fp(\026)29393 24223 y Fk(j)29953 24024 y Fp(:)16869 b Fq(\(5.5\))2696 28203 y(Let)635 b(us)g(\014x)g(some)g(0)689 b Fm(\024)g Fp(i)f Fm(\024)h Fp(N)780 b Fq(and)635 b(consider)h(the)e (function)h Fp(h)37332 27720 y Fk(T)37959 27853 y Fj(0)38478 27720 y Fk(;T)39420 27853 y Fj(1)37332 28531 y Fk(\021)37879 28664 y Fd(i)38300 28531 y Fk(;\026)39261 28664 y Fd(i)40006 28203 y Fq(\()p Fp(m)41689 28402 y Fv(0)42285 28203 y Fp(;)221 b(m)44041 28402 y Fv(1)44637 28203 y Fq(\))635 b(de\014ned)1102 30103 y(in)575 b(\(2.13\).)985 b(F)-111 b(or)575 b(eac)-37 b(h)576 b(small)e Fp(\017)17315 29621 y Fg(\003)17315 30449 y Fk(i)18515 30103 y Fp(>)588 b Fq(0)575 b(and)g(\()p Fp(m)25773 30302 y Fv(0)26368 30103 y Fp(;)221 b(m)28124 30302 y Fv(1)28721 30103 y Fq(\))588 b Fm(2)g Fp(M)527 b Fm(\002)384 b Fp(M)719 b Fq(there)575 b(exists)f(\()44646 29767 y(\025)44387 30103 y Fp(T)45347 29621 y Fk(i)45163 30432 y Fv(0)45790 30103 y Fp(;)46639 29767 y Fq(\025)46380 30103 y Fp(T)47340 29621 y Fk(i)47156 30432 y Fv(1)47783 30103 y Fq(\))588 b(=)1102 31840 y(\()1878 31504 y(\025)1619 31840 y Fp(T)2579 31358 y Fk(i)2395 32169 y Fv(0)3022 31840 y Fp(;)3871 31504 y Fq(\025)3612 31840 y Fp(T)4572 31358 y Fk(i)4388 32169 y Fv(1)5015 31840 y Fq(\)\()p Fp(\017)6588 31358 y Fg(\003)6588 32185 y Fk(i)7198 31840 y Fp(;)221 b(m)8954 32039 y Fv(0)9551 31840 y Fp(;)g(m)11307 32039 y Fv(1)11904 31840 y Fq(\))368 b Fm(2)h Fl(Z)14930 31358 y Fv(+)16254 31840 y Fq(suc)-37 b(h)443 b(that)7676 35258 y Fp(h)8441 34710 y Fk(T)9068 34843 y Fj(0)9587 34710 y Fk(;T)10529 34843 y Fj(1)8441 35587 y Fk(\021)8988 35720 y Fd(i)9409 35587 y Fk(;\026)10370 35720 y Fd(i)11115 35258 y Fq(\()p Fp(m)12798 35457 y Fv(0)13394 35258 y Fp(;)221 b(m)15150 35457 y Fv(1)15746 35258 y Fq(\))369 b Fm(\025)g Fp(h)18799 34710 y Fg(1)18799 35587 y Fk(\021)19346 35720 y Fd(i)19767 35587 y Fk(;\026)20728 35720 y Fd(i)21216 35258 y Fq(\()p Fp(m)22899 35457 y Fv(0)23495 35258 y Fp(;)221 b(m)25251 35457 y Fv(1)25847 35258 y Fq(\))295 b Fm(\000)h Fp(\017)28527 34710 y Fg(\003)28527 35587 y Fk(i)29137 35258 y Fp(;)2879 b Fm(8)p Fp(T)33899 35457 y Fk(j)34827 35258 y Fm(\025)36489 34923 y Fq(\025)36230 35258 y Fp(T)37190 34710 y Fk(i)37006 35587 y(j)37633 35258 y Fp(;)664 b(j)445 b Fq(=)369 b(0)p Fp(;)221 b Fq(1)p Fp(:)3844 b Fq(\(5.6\))1102 38841 y(Ob)-37 b(viously)-111 b(,)444 b(there)f(are)h(in\014nite)f Fp(T)18498 39040 y Fk(j)19427 38841 y Fm(\025)21088 38505 y Fq(\025)20829 38841 y Fp(T)21789 38359 y Fk(i)21605 39186 y(j)22674 38841 y Fq(\()p Fp(j)i Fq(=)369 b(0)p Fp(;)221 b Fq(1\))443 b(suc)-37 b(h)443 b(that)14228 42366 y Fm(j)p Fp(h)15362 41817 y Fk(T)15989 41950 y Fj(0)16508 41817 y Fk(;T)17450 41950 y Fj(1)15362 42694 y Fk(\021)15909 42827 y Fd(i)16330 42694 y Fk(;\026)17291 42827 y Fd(i)18036 42366 y Fq(\()p Fp(m)19719 42565 y Fv(0)20314 42366 y Fp(;)221 b(m)22070 42565 y Fv(1)22667 42366 y Fq(\))295 b Fm(\000)g Fp(h)25572 41817 y Fg(1)25572 42694 y Fk(\021)26119 42827 y Fd(i)26540 42694 y Fk(;\026)27501 42827 y Fd(i)27989 42366 y Fq(\()p Fp(m)29672 42565 y Fv(0)30268 42366 y Fp(;)221 b(m)32024 42565 y Fv(1)32620 42366 y Fq(\))p Fm(j)369 b(\024)g Fp(\017)35816 41817 y Fg(\003)35816 42694 y Fk(i)36427 42366 y Fp(:)10395 b Fq(\(5.7\))1102 45807 y(Let)498 b Fp(\015)4225 46006 y Fk(i)4667 45807 y Fq(\()p Fp(t;)221 b(m)7420 46006 y Fv(0)8016 45807 y Fp(;)g(m)9772 46006 y Fv(1)10368 45807 y Fp(;)g(T)11734 46006 y Fv(0)12331 45807 y Fp(;)g(T)13697 46006 y Fv(1)14293 45807 y Fq(\):)699 b([)p Fm(\000)p Fp(T)18056 46006 y Fv(0)18652 45807 y Fp(;)221 b(T)20018 46006 y Fv(1)20615 45807 y Fq(])459 b Fm(!)h Fp(M)642 b Fq(b)37 b(e)497 b(the)g(minimizer)h(of)f Fp(h)37955 45325 y Fk(T)38582 45458 y Fj(0)39101 45325 y Fk(;T)40043 45458 y Fj(1)37955 46136 y Fk(\021)38502 46269 y Fd(i)38924 46136 y Fk(;\026)39885 46269 y Fd(i)40629 45807 y Fq(\()p Fp(m)42312 46006 y Fv(0)42908 45807 y Fp(;)221 b(m)44664 46006 y Fv(1)45260 45807 y Fq(\),)511 b(it)496 b(fol-)1102 47708 y(lo)-37 b(ws)301 b(from)g(the)g(lemma)g (2.3)f(that)g(if)h Fp(\017)19512 47225 y Fg(\003)19512 48053 y Fk(i)20491 47708 y Fp(>)370 b Fq(0)300 b(is)h(su\016cien)-37 b(tly)301 b(small,)34549 47372 y(\025)34290 47708 y Fp(T)35250 47225 y Fk(i)35066 48053 y(j)35993 47708 y Fq(\()p Fp(j)445 b Fq(=)369 b(0)p Fp(;)221 b Fq(1\))301 b(are)g(su\016cien)-37 b(tly)1102 49363 y(large,)444 b(and)f Fp(T)8081 49562 y Fv(0)8677 49363 y Fq(,)g Fp(T)10265 49562 y Fv(1)11304 49363 y Fq(are)h(c)-37 b(hosen)443 b(so)g(that)f(\(5.7\))h(holds,)f (then)10053 52733 y Fp(d\015)11432 52932 y Fk(i)11875 52733 y Fq(\()p Fp(t;)221 b(m)14628 52932 y Fv(0)15224 52733 y Fp(;)g(m)16980 52932 y Fv(1)17576 52733 y Fp(;)g(T)18942 52932 y Fv(0)19539 52733 y Fp(;)g(T)20905 52932 y Fv(1)21501 52733 y Fq(\))369 b Fm(2)24063 52397 y Fq(~)23641 52733 y Fm(N)24731 52932 y Fk(\021)25278 53065 y Fd(i)25699 52932 y Fk(;\026)26660 53065 y Fd(i)27148 52733 y Fq(\()p Fp(t)p Fq(\))294 b(+)h Fp(\017)30823 52932 y Fk(i)31266 52733 y Fp(;)2878 b Fm(8)p Fq(0)369 b Fm(\024)g Fp(t)g Fm(\024)g Fq(1)p Fp(:)6221 b Fq(\(5.8\))1102 56045 y(F)-111 b(rom)447 b(the)f(Lipsc)-37 b(hitze)447 b(prop)37 b(ert)-37 b(y)446 b(of)g Fp(h)20867 55563 y Fk(T)21494 55696 y Fj(0)22013 55563 y Fk(;T)22955 55696 y Fj(1)20867 56374 y Fk(\021)21414 56507 y Fd(i)21836 56374 y Fk(;\026)22797 56507 y Fd(i)23541 56045 y Fq(\()p Fp(m)25224 56244 y Fv(0)25820 56045 y Fp(;)221 b(m)27576 56244 y Fv(1)28173 56045 y Fq(\))445 b(in)h(\()p Fp(m)32371 56244 y Fv(0)32967 56045 y Fp(;)221 b(m)34723 56244 y Fv(1)35319 56045 y Fq(\))446 b(and)g(the)f(compactness)i(of)1102 57946 y Fp(M)145 b Fq(,)553 b(w)-37 b(e)532 b(see)g(that)f(there)h(are)16757 57610 y(\025)16498 57946 y Fp(T)17458 57463 y Fk(i)17274 58291 y(j)18417 57946 y Fq(=)20226 57610 y(\025)19967 57946 y Fp(T)20927 57463 y Fk(i)20743 58291 y(j)21370 57946 y Fq(\()p Fp(\017)22426 58145 y Fk(i)22868 57946 y Fq(\))f(\()p Fp(j)592 b Fq(=)516 b(0)p Fp(;)221 b Fq(1\),)554 b(indep)37 b(enden)-37 b(t)532 b(of)f(\()p Fp(m)41379 58145 y Fv(0)41975 57946 y Fp(;)221 b(m)43731 58145 y Fv(1)44328 57946 y Fq(\),)553 b(so)531 b(that)1102 59863 y(\(5.6\))400 b(holds)g(for)g(all)g Fp(T)12234 60062 y Fk(j)13163 59863 y Fm(\025)14824 59527 y Fq(\025)14565 59863 y Fp(T)15525 59381 y Fk(i)15341 60209 y(j)15968 59863 y Fq(.)577 b(W)-111 b(e)401 b(can)f(see)h(also)f(that)f(there)i (exist)35828 59527 y(^)35569 59863 y Fp(T)36529 59381 y Fk(i)36345 60209 y(j)36972 59863 y Fq(\()p Fp(\017)38028 60062 y Fk(i)38470 59863 y Fq(\))368 b Fp(>)41017 59527 y Fq(\025)40758 59863 y Fp(T)41718 59381 y Fk(i)41534 60209 y(j)42161 59863 y Fq(\()p Fp(\017)43217 60062 y Fk(i)43659 59863 y Fq(\))399 b(\()p Fp(j)445 b Fq(=)369 b(0)p Fp(;)221 b Fq(1\))1102 61781 y(so)474 b(that)f(for)i(an)-37 b(y)474 b(\()p Fp(m)11961 61980 y Fv(0)12557 61781 y Fp(;)221 b(m)14313 61980 y Fv(1)14909 61781 y Fq(\))420 b Fm(2)h Fp(M)460 b Fm(\002)316 b Fp(M)145 b Fq(,)481 b(there)475 b(are)f Fp(T)28989 61980 y Fk(j)30023 61781 y Fq(with)33339 61445 y(\025)33080 61781 y Fp(T)34040 61299 y Fk(i)33856 62127 y(j)34903 61781 y Fm(\024)421 b Fp(T)37133 61980 y Fk(j)38114 61781 y Fm(\024)39827 61445 y Fq(^)39568 61781 y Fp(T)40528 61299 y Fk(i)40344 62127 y(j)41445 61781 y Fq(\()p Fp(j)496 b Fq(=)421 b(0)p Fp(;)221 b Fq(1\))474 b(suc)-37 b(h)1102 63436 y(that)543 b(\(5.7\))g(and)h(consequen)-37 b(tly)545 b(\(5.8\))e(holds.)893 b(Note)544 b(that)f(for)h(di\013eren)-37 b(t)544 b(\()p Fp(m)41027 63635 y Fv(0)41623 63436 y Fp(;)221 b(m)43379 63635 y Fv(1)43975 63436 y Fq(\),)568 b(w)-37 b(e)545 b(ma)-37 b(y)1102 65160 y(need)444 b(di\013eren)-37 b(t)443 b Fp(T)10221 65359 y Fk(j)11150 65160 y Fm(\025)12811 64824 y Fq(\025)12552 65160 y Fp(T)13512 64678 y Fk(i)13328 65505 y(j)13955 65160 y Fq(.)2696 67700 y(W)-111 b(e)357 b(are)f(no)-37 b(w)356 b(on)g(the)f(stage)h(to)f(construct)h(a)f (connecting)i(orbit)e(joining)h Fm(N)196 b Fq(\()p Fp(c)41689 67899 y Fv(0)42284 67700 y Fq(\))355 b(and)g Fm(N)196 b Fq(\()p Fp(c)48029 67899 y Fk(N)49036 67700 y Fq(\).)1102 69294 y(W)-111 b(e)479 b(consider)g Fp(\034)9218 69493 y Fk(i)10137 69294 y Fq(as)f(the)g(time)f(translation)h(\()p Fp(q)48 b(;)221 b(t)p Fq(\))425 b Fm(!)j Fq(\()p Fp(q)48 b(;)221 b(t)318 b Fq(+)g Fp(\034)33352 69493 y Fk(i)33794 69294 y Fq(\))477 b(on)h Fp(M)463 b Fm(\002)319 b Fl(R)p Fq(,)486 b(and)478 b(de\014ne)g(the)1102 70888 y(mo)37 b(di\014ed)443 b(Lagrangian)17822 73460 y(~)17665 73796 y Fp(L)369 b Fq(=)g Fp(L)295 b Fm(\000)g Fp(\021)23527 73995 y Fv(0)24418 73796 y Fm(\000)25746 72136 y Fk(N)100 b Fg(\000)p Fv(1)25938 72534 y Fi(X)26036 75363 y Fk(i)p Fv(=0)28047 73796 y Fq(\()p Fm(\000)p Fp(\034)30178 73995 y Fk(i)30620 73796 y Fq(\))31137 73248 y Fg(\003)31747 73796 y Fp(\026)32547 73995 y Fk(i)32990 73796 y Fp(:)13832 b Fq(\(5.9\))p eop %%Page: 35 35 35 34 bop 48792 2239 a Ft(35)1102 5427 y Fq(F)-111 b(or)444 b(eac)-37 b(h)412 b Fp(~)-632 b(\034)519 b Fq(=)369 b(\()p Fp(\034)10075 5626 y Fv(0)10670 5427 y Fp(;)221 b(\034)11841 5626 y Fv(1)12437 5427 y Fp(;)g Fm(\001)g(\001)g(\001)444 b Fp(;)221 b(\034)16191 5626 y Fk(N)100 b Fg(\000)p Fv(1)18558 5427 y Fq(\))443 b(the)f(follo)-37 b(wing)444 b(v)-74 b(ariational)444 b(problem)2239 9688 y Fp(h)3004 9049 y Fk(T)3631 9182 y Fj(0)4150 9049 y Fk(;T)5092 9183 y Fd(N)3132 10068 y Fv(~)3004 10303 y Fk(L)6012 9688 y Fq(\()p Fp(m;)221 b(m)9451 9139 y Fg(0)9825 9688 y Fp(;)190 b(~)-633 b(\034)150 b Fq(\))369 b(=)3600 b(inf)14883 10607 y Fk(\015)57 b Fv(\()p Fg(\000)p Fk(T)17370 10740 y Fj(0)17890 10607 y Fv(\)=)p Fk(m)13286 11744 y(\015)g Fv(\()p Fk(T)14943 11878 y Fd(N)15797 11744 y Fv(+)p Fk(\034)17102 11878 y Fd(N)72 b Fe(\000)p Fj(1)19127 11744 y Fv(\)=)p Fk(m)21300 11343 y Fe(0)21881 7880 y Fi(Z)23210 8208 y Fk(T)23837 8342 y Fd(N)24690 8208 y Fv(+)p Fk(\034)25995 8342 y Fd(N)g Fe(\000)p Fj(1)22619 10898 y Fg(\000)p Fk(T)24076 11031 y Fj(0)28308 8213 y Fi(\020)29102 9688 y Fp(L)295 b Fm(\000)g Fp(\021)32289 9887 y Fv(0)33180 9688 y Fm(\000)34508 8027 y Fk(N)100 b Fg(\000)p Fv(1)34699 8426 y Fi(X)34798 11255 y Fk(i)p Fv(=0)36809 9688 y Fq(\()p Fm(\000)p Fp(\034)38940 9887 y Fk(i)39382 9688 y Fq(\))39899 9139 y Fg(\003)40509 9688 y Fp(\026)41309 9887 y Fk(i)41752 8213 y Fi(\021)42545 9688 y Fq(\()p Fp(d\015)74 b Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))p Fp(dt)13360 15082 y Fm(\000)14688 13422 y Fk(N)100 b Fg(\000)p Fv(1)14879 13820 y Fi(X)14978 16649 y Fk(i)p Fv(=1)16989 15082 y Fq(\()p Fp(\034)18087 15281 y Fk(i)18824 15082 y Fm(\000)295 b Fp(\034)20733 15281 y Fk(i)p Fg(\000)p Fv(1)22535 15082 y Fq(\))p Fp(\013)5 b Fq(\()p Fp(c)24999 15281 y Fk(i)25440 15082 y Fq(\))295 b Fm(\000)g Fp(T)28356 15281 y Fv(0)28952 15082 y Fp(\013)5 b Fq(\()p Fp(c)30899 15281 y Fv(0)31494 15082 y Fq(\))295 b Fm(\000)g Fp(T)34410 15281 y Fk(N)35418 15082 y Fp(\013)5 b Fq(\()p Fp(c)37365 15281 y Fk(N)38372 15082 y Fq(\))1102 19363 y(has)510 b(a)g Fp(C)5756 18881 y Fv(1)6352 19363 y Fq(-minimizer)g Fp(\015)74 b Fq(\()p Fp(t;)221 b(m;)g(m)18335 18881 y Fg(0)18708 19363 y Fp(;)190 b(~)-633 b(\034)151 b(;)221 b(T)21396 19562 y Fv(0)21992 19363 y Fp(;)g(T)23358 19562 y Fk(N)24366 19363 y Fq(\))509 b(whic)-37 b(h)511 b(is)e(clearly)i(the)e(solution)h(of)f(the)h(Euler-)1102 21051 y(Lagrangian)424 b(equation)f(determined)g(b)-37 b(y)22346 20715 y(~)22189 21051 y Fp(L)p Fq(.)584 b(W)-111 b(e)423 b(need)g(to)f(sho)-37 b(w)423 b(it)f(can)h(b)37 b(e)422 b(the)h(extremal)g(of)f Fp(L)1102 22645 y Fq(if)443 b(w)-37 b(e)444 b(suitably)e(c)-37 b(ho)37 b(ose)412 b Fp(~)-632 b(\034)150 b Fq(,)443 b Fp(T)15871 22844 y Fv(0)16910 22645 y Fq(and)g Fp(T)20269 22844 y Fk(N)21276 22645 y Fq(.)591 b(W)-111 b(e)444 b(de\014ne)5804 25931 y(\003)370 b(=)8498 24457 y Fi(n)9352 25931 y Fp(~)-633 b(\034)519 b Fm(2)369 b Fl(Z)12624 25383 y Fk(N)14000 25931 y Fq(:)221 b(max)q Fm(f)17986 25595 y Fq(\025)17727 25931 y Fp(T)18687 25383 y Fk(i)18503 26260 y Fv(0)19130 25931 y Fp(;)19979 25595 y Fq(\025)19720 25931 y Fp(T)20680 25363 y Fk(i)p Fg(\000)p Fv(1)20496 26285 y(1)22778 25931 y Fq(+)295 b(1)p Fm(g)370 b(\024)f Fp(\034)27787 26130 y Fk(i)28524 25931 y Fm(\000)295 b Fp(\034)30433 26130 y Fk(i)p Fg(\000)p Fv(1)32604 25931 y Fm(\024)369 b Fq(max)p Fm(f)37401 25595 y Fq(^)37142 25931 y Fp(T)38102 25383 y Fk(i)37918 26260 y Fv(0)38545 25931 y Fp(;)39395 25595 y Fq(^)39135 25931 y Fp(T)40095 25363 y Fk(i)p Fg(\000)p Fv(1)39911 26285 y(1)42194 25931 y Fq(+)295 b(1)p Fm(g)p Fp(;)14369 28854 y Fm(8)p Fq(1)369 b Fm(\024)g Fp(i)g Fm(\024)g Fp(N)440 b Fm(\000)296 b Fq(1)p Fp(;)221 b(\034)24442 29053 y Fv(0)25407 28854 y Fq(=)369 b(0)27473 27380 y Fi(o)28359 28854 y Fp(;)1102 32491 y Fq(and)443 b(tak)-37 b(e)443 b(the)g(minim)-37 b(um)443 b(of)g Fp(h)17070 31853 y Fk(T)17697 31986 y Fj(0)18216 31853 y Fk(;T)19158 31987 y Fd(N)17198 32872 y Fv(~)17070 33107 y Fk(L)20078 32491 y Fq(\()p Fp(m;)221 b(m)23517 32009 y Fg(0)23891 32491 y Fp(;)190 b(~)-633 b(\034)150 b Fq(\))443 b(o)-37 b(v)g(er)443 b(\003)13355 35881 y Fp(F)14337 36049 y Fv(~)14209 36283 y Fk(L)15004 35881 y Fq(\()p Fp(m;)221 b(m)18443 35332 y Fg(0)18816 35881 y Fp(;)g(T)20182 36080 y Fv(0)20779 35881 y Fp(;)g(T)22145 36080 y Fk(N)23153 35881 y Fq(\))368 b(=)i(min)25511 36770 y Fk(~)-527 b(\034)107 b Fg(2)p Fv(\003)27876 35881 y Fp(h)28641 35242 y Fk(T)29268 35375 y Fj(0)29787 35242 y Fk(;T)30729 35376 y Fd(N)28770 36261 y Fv(~)28641 36496 y Fk(L)31650 35881 y Fq(\()p Fp(m;)221 b(m)35089 35332 y Fg(0)35462 35881 y Fp(;)190 b(~)-633 b(\034)150 b Fq(\))p Fp(:)8858 b Fq(\(5.10\))1102 39472 y(w)-37 b(e)460 b(call)428 b Fp(~)-633 b(\034)6256 38990 y Fg(\003)6867 39472 y Fq(\()p Fp(T)8160 39671 y Fv(0)8756 39472 y Fp(;)221 b(T)10122 39671 y Fk(N)11130 39472 y Fq(\))458 b(the)h(v)-74 b(alue)460 b(where)g(the)f(minim)-37 b(um)458 b(is)h(reac)-37 b(hed.)640 b(If)458 b Fp(\015)74 b Fq(\()p Fp(t;)221 b(m;)g(m)43343 38990 y Fg(0)43716 39472 y Fp(;)g(T)45082 39671 y Fv(0)45679 39472 y Fp(;)g(T)47045 39671 y Fk(N)48053 39472 y Fq(\))458 b(is)1102 41066 y(the)349 b(minimizer)h(of)e Fp(F)11805 41234 y Fv(~)11677 41469 y Fk(L)12473 41066 y Fq(\()p Fp(m;)221 b(m)15912 40584 y Fg(0)16285 41066 y Fp(;)g(T)17651 41265 y Fv(0)18248 41066 y Fp(;)g(T)19614 41265 y Fk(N)20622 41066 y Fq(\),)367 b(w)-37 b(e)349 b(claim)g(that)f(for)h Fp(t)369 b Fm(2)g Fq([)p Fp(\034)34961 41265 y Fk(i)35403 41066 y Fp(;)221 b(\034)36574 41265 y Fk(i)37123 41066 y Fq(+)107 b(1])349 b(and)g(0)369 b Fp(<)g(i)g(<)g(N)252 b Fm(\000)107 b Fq(1)12218 44497 y Fp(d\015)74 b Fq(\()p Fp(t;)221 b(m;)g(m)18180 43948 y Fg(0)18553 44497 y Fp(;)g(T)19919 44696 y Fv(0)20516 44497 y Fp(;)g(T)21882 44696 y Fk(N)22890 44497 y Fq(\))369 b Fm(2)f Fq(\()p Fm(\000)p Fp(\034)27161 44696 y Fk(i)27603 44497 y Fq(\))28120 43948 y Fg(\003)28730 43022 y Fi(\020)29945 44161 y Fq(~)29523 44497 y Fm(N)30613 44696 y Fk(\021)31160 44829 y Fd(i)p Fe(\000)p Fj(1)32753 44696 y Fk(;\026)33714 44829 y Fd(i)34202 44497 y Fm(j)34571 44696 y Fk(t)35038 43022 y Fi(\021)36127 44497 y Fq(+)295 b Fp(\017)37994 44696 y Fk(i)38437 44497 y Fp(:)7721 b Fq(\(5.11\))1102 47859 y(In)509 b(fact,)525 b(let)509 b(us)g(to)g(c)-37 b(ho)37 b(ose)510 b Fp(m)16792 48058 y Fk(i)17714 47859 y Fq(=)479 b Fp(\015)74 b Fq(\()p Fp(\034)21086 48058 y Fk(i)p Fg(\000)p Fv(1)23226 47859 y Fq(+)340 b(1\),)525 b Fp(m)27840 47377 y Fg(0)27840 48204 y Fk(i)28762 47859 y Fq(=)479 b Fp(\015)74 b Fq(\()p Fp(\034)32134 48058 y Fk(i)p Fv(+1)33920 47859 y Fq(\))508 b(for)i(0)479 b Fp(<)g(i)g(<)h(N)k Fm(\000)340 b Fq(1.)789 b(Since)1102 49453 y Fp(\015)74 b Fq(\()p Fp(t;)221 b(m;)g(m)6373 48971 y Fg(0)6746 49453 y Fp(;)g(T)8112 49652 y Fv(0)8709 49453 y Fp(;)g(T)10075 49652 y Fk(N)11083 49453 y Fq(\))442 b(is)h(the)g(minimizer)g(of)g Fp(F)24363 49621 y Fv(~)24235 49856 y Fk(L)25031 49453 y Fq(\()p Fp(m;)221 b(m)28470 48971 y Fg(0)28843 49453 y Fp(;)g(T)30209 49652 y Fv(0)30805 49453 y Fp(;)g(T)32171 49652 y Fk(N)33180 49453 y Fq(\),)442 b(th)-37 b(us)5961 53471 y Fp(A)6957 53670 y Fk(\021)7504 53803 y Fd(i)p Fe(\000)p Fj(1)9097 53670 y Fk(;\026)10058 53803 y Fd(i)10545 53471 y Fq(\(\()p Fm(\000)p Fp(\034)13193 53670 y Fk(i)13634 53471 y Fq(\))14151 52922 y Fg(\003)14762 53471 y Fp(\015)74 b Fm(j)15893 52770 y Fk(\034)16383 52903 y Fd(i)p Fj(+1)15893 53825 y Fk(\034)16383 53958 y Fd(i)p Fe(\000)p Fj(1)17974 53825 y Fv(+1)19386 53471 y Fq(\))368 b(=)2492 b(inf)21526 54402 y Fk(\015)22141 54001 y Fe(\003)22696 54402 y Fv(\()p Fg(\000)p Fk(T)24568 54535 y Fj(0)25088 54402 y Fv(\)=)p Fk(m)27261 54535 y Fd(i)21941 55540 y Fk(\015)22556 55139 y Fe(\003)23112 55540 y Fv(\()p Fk(T)24154 55673 y Fj(1)24673 55540 y Fv(\)=)p Fk(m)26846 55139 y Fe(0)26846 55805 y Fd(i)22216 56587 y Fv(\025)22009 56822 y Fk(T)22779 56421 y Fd(i)22636 57087 y Fj(0)23201 56822 y Fg(\024)p Fk(T)24658 56955 y Fj(0)25177 56822 y Fg(\024)26213 56587 y Fv(^)26007 56822 y Fk(T)26777 56421 y Fd(i)26634 57087 y Fj(0)22216 57869 y Fv(\025)22009 58104 y Fk(T)22779 57703 y Fd(i)22636 58369 y Fj(1)23201 58104 y Fg(\024)p Fk(T)24658 58237 y Fj(1)25177 58104 y Fg(\024)26213 57869 y Fv(^)26007 58104 y Fk(T)26777 57703 y Fd(i)26634 58369 y Fj(1)27903 51663 y Fi(Z)29231 51991 y Fk(T)29858 52124 y Fj(1)28641 54681 y Fg(\000)p Fk(T)30098 54814 y Fj(0)30683 53471 y Fq(\()p Fp(L)295 b Fm(\000)h Fp(\021)34388 53670 y Fk(i)35125 53471 y Fm(\000)f Fp(\026)37253 53670 y Fk(i)37696 53471 y Fq(\)\()p Fp(d\015)40183 52922 y Fg(\003)40792 53471 y Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))p Fp(dt)21600 60030 y Fm(\000)295 b Fp(T)23704 60229 y Fv(0)24300 60030 y Fp(\013)5 b Fq(\()p Fp(c)26247 60229 y Fk(i)26689 60030 y Fq(\))294 b Fm(\000)i Fp(T)29605 60229 y Fv(1)30201 60030 y Fp(\013)5 b Fq(\()p Fp(c)32148 60229 y Fk(i)p Fv(+1)33934 60030 y Fq(\))p Fp(:)11707 b Fq(\(5.12\))1102 63320 y(So,)380 b(w)-37 b(e)365 b(obtain)f(\(5.11\)) f(from)h(\(5.6)p Fm(\030)p Fq(8\),)380 b(\(5.12\))363 b(and)h(the)g(c)-37 b(hoice)365 b(of)35213 62984 y(\025)34954 63320 y Fp(T)35914 62838 y Fk(i)35730 63666 y(j)36721 63320 y Fq(as)f(w)-37 b(ell)364 b(as)42698 62984 y(^)42439 63320 y Fp(T)43399 62838 y Fk(i)43215 63666 y(j)44206 63320 y Fq(\()p Fp(j)445 b Fq(=)369 b(0)p Fp(;)221 b Fq(1\).)1102 64978 y(W)-111 b(e)444 b(de\014ne)g(the)e(in\014m)-37 b(um)443 b(limit)f(of)h Fp(F)20386 65146 y Fv(~)20258 65381 y Fk(L)21054 64978 y Fq(\()p Fp(m;)221 b(m)24493 64496 y Fg(0)24866 64978 y Fp(;)g(T)26232 65177 y Fv(0)26828 64978 y Fp(;)g(T)28194 65177 y Fk(N)29202 64978 y Fq(\))13701 67946 y Fp(h)14466 67397 y Fg(1)14594 68235 y Fv(~)14466 68470 y Fk(L)15592 67946 y Fq(\()p Fp(m;)g(m)19031 67397 y Fg(0)19404 67946 y Fq(\))369 b(=)890 b(lim)221 b(inf)21692 68803 y Fk(T)22319 68936 y Fj(0)22838 68803 y Fk(;T)23780 68936 y Fj(1)24299 68803 y Fg(!1)26638 67946 y Fp(F)27620 68114 y Fv(~)27492 68349 y Fk(L)28288 67946 y Fq(\()p Fp(m;)g(m)31727 67397 y Fg(0)32100 67946 y Fp(;)g(T)33466 68145 y Fv(0)34063 67946 y Fp(;)g(T)35429 68145 y Fk(N)36437 67946 y Fq(\))p Fp(:)9204 b Fq(\(5.13\))1102 71744 y(Let)443 b Fp(T)4442 71262 y Fk(k)4258 72090 y(j)5539 71744 y Fq(\()p Fp(j)i Fq(=)369 b(0)p Fp(;)221 b(N)145 b Fq(\))443 b(b)37 b(e)443 b(the)g(subsequences)h(suc)-37 b(h)443 b(that)f Fp(T)30835 71262 y Fk(k)30651 72090 y(j)31859 71744 y Fm(!)369 b(1)443 b Fq(as)g Fp(k)410 b Fm(!)370 b(1)8877 74965 y(j)p Fp(F)10228 75133 y Fv(~)10100 75368 y Fk(L)10896 74965 y Fq(\()p Fp(m;)221 b(m)14335 74416 y Fg(0)14708 74965 y Fp(;)g(T)16258 74416 y Fk(k)16074 75293 y Fv(0)16913 74965 y Fp(;)g(T)18463 74416 y Fk(k)18279 75293 y(N)19287 74965 y Fq(\))295 b Fm(\000)g Fp(h)22192 74416 y Fg(1)22320 75254 y Fv(~)22192 75489 y Fk(L)23318 74965 y Fq(\()p Fp(m;)221 b(m)26757 74416 y Fg(0)27130 74965 y Fq(\))p Fm(j)369 b(\024)g Fq(min)p Fm(f)p Fp(\017)33204 74416 y Fg(\003)33204 75293 y Fv(0)33815 74965 y Fp(;)221 b(\017)34944 74416 y Fg(\003)34944 75293 y Fk(N)35952 74965 y Fm(g)p Fp(;)2878 b Fm(8)443 b Fp(k)42 b(;)p eop %%Page: 36 36 36 35 bop 1102 2239 a Ft(36)1102 5427 y Fq(as)443 b(w)-37 b(ell)444 b(as)15020 7021 y(lim)14590 7888 y Fk(k)22 b Fg(!1)17517 7021 y Fp(F)18499 7189 y Fv(~)18371 7424 y Fk(L)19166 7021 y Fq(\()p Fp(m;)221 b(m)22605 6473 y Fg(0)22978 7021 y Fp(;)g(T)24528 6473 y Fk(k)24344 7349 y Fv(0)25183 7021 y Fp(;)g(T)26733 6473 y Fk(k)26549 7349 y(N)27557 7021 y Fq(\))369 b(=)g Fp(h)30610 6473 y Fg(1)30738 7310 y Fv(~)30610 7546 y Fk(L)31736 7021 y Fq(\()p Fp(m;)221 b(m)35175 6473 y Fg(0)35548 7021 y Fq(\))p Fp(;)1102 10017 y Fq(and)626 b(let)f Fp(\015)6657 10216 y Fk(k)7310 10017 y Fq(\()p Fp(t;)221 b(m;)g(m)11819 9534 y Fg(0)12193 10017 y Fq(\))673 b(=)g Fp(\015)74 b Fq(\()p Fp(t;)221 b(m;)g(m)20360 9534 y Fg(0)20733 10017 y Fp(;)g(T)22283 9534 y Fk(k)22099 10346 y Fv(0)22938 10017 y Fp(;)g(T)24488 9534 y Fk(k)24304 10382 y(N)25312 10017 y Fq(\))625 b(b)37 b(e)625 b(the)h(minimizer)g(of)f Fp(F)39977 10185 y Fv(~)39849 10419 y Fk(L)40645 10017 y Fq(\()p Fp(m;)221 b(m)44084 9534 y Fg(0)44457 10017 y Fp(;)g(T)46007 9534 y Fk(k)45823 10346 y Fv(0)46662 10017 y Fp(;)g(T)48212 9534 y Fk(k)48028 10382 y(N)49036 10017 y Fq(\).)1102 11611 y(It)613 b(is)g(easy)h(to)f(see)h(that)f (\(5.11\))g(holds)g(also)g(for)h Fp(i)653 b Fq(=)g(0)p Fp(;)221 b(N)145 b Fq(.)1102 b(F)-111 b(rom)615 b(\(5.3\),)655 b(\(5.12\))613 b(and)g(the)1102 13205 y(de\014nition)309 b(of)g Fp(U)9158 13404 y Fk(i)9909 13205 y Fq(w)-37 b(e)309 b(obtain)g(that)f Fp(d\015)19852 13404 y Fk(k)20506 13205 y Fq(\()p Fp(t)p Fq(\))f(is)i(the)g(extremal)g(of)g Fp(L)g Fq(with)f(the)h(b)37 b(oundary)309 b(condition)1102 14799 y Fp(\015)1790 14998 y Fk(k)2444 14799 y Fq(\()p Fm(\000)p Fp(T)4954 14317 y Fk(k)4770 15128 y Fv(0)5608 14799 y Fq(\))416 b(=)g Fp(m)p Fq(,)479 b Fp(\015)10692 14998 y Fk(k)11346 14799 y Fq(\()p Fp(T)12823 14317 y Fk(k)12639 15164 y(N)13960 14799 y Fq(+)314 b Fp(\034)16038 14317 y Fg(\003)15888 15164 y Fk(N)100 b Fg(\000)p Fv(1)18255 14799 y Fq(\))416 b(=)h Fp(m)21804 14317 y Fg(0)22177 14799 y Fq(.)676 b(Clearly)-111 b(,)480 b(for)472 b(an)-37 b(y)471 b(compact)h(in)-37 b(terv)-74 b(al)472 b([)p Fp(a;)221 b(b)p Fq(])472 b(the)f(set)1102 16513 y Fm(f)p Fp(\015)2454 16712 y Fk(k)3108 16513 y Fm(g)3772 16834 y Fk(k)22 b Fg(\025)5218 16589 y Fv(\026)5189 16834 y Fk(k)6391 16513 y Fq(is)549 b(pre-compact)g(in)g(the)f Fp(C)20636 16031 y Fv(1)21233 16513 y Fq(\([)p Fp(a;)221 b(b)p Fq(])p Fp(;)g(M)145 b Fq(\))548 b(top)37 b(ology)549 b(if)34364 16162 y(\026)34329 16513 y Fp(k)590 b Fq(is)548 b(suitably)h(large.)908 b(Let)549 b Fp(\015)74 b Fq(:)1102 18136 y Fl(R)585 b Fm(!)f Fp(M)716 b Fq(b)37 b(e)572 b(the)g(accum)-37 b(ulation)573 b(p)37 b(oin)-37 b(t)571 b(of)h Fm(f)p Fp(\015)25792 18335 y Fk(k)26446 18136 y Fm(g)p Fq(,)604 b(then)571 b Fp(d\015)646 b Fq(is)572 b(the)g(solution)f(of)h(the)g(Euler-)1102 19730 y(Lagrange)445 b(equation)e(determined)g(b)-37 b(y)443 b Fp(L)g Fq(and)15112 22500 y Fp(\013)5 b Fq(\()p Fp(d\015)74 b Fq(\))367 b Fm(\022)20680 22164 y Fq(~)20223 22500 y Fm(A)p Fq(\()p Fp(c)22376 22699 y Fv(0)22971 22500 y Fq(\))p Fp(;)2878 b(!)48 b Fq(\()p Fp(d\015)74 b Fq(\))368 b Fm(\022)32323 22164 y Fq(~)31867 22500 y Fm(A)o Fq(\()p Fp(c)34019 22699 y Fk(N)35026 22500 y Fq(\))p Fp(:)2696 25249 y Fq(Consider)596 b(a)f(bi-in\014nite)f(sequence)i(\()p Fm(\001)221 b(\001)g(\001)443 b Fp(;)221 b(c)25187 25448 y Fk(i)25629 25249 y Fp(;)g Fm(\001)g(\001)g(\001)i Fq(\))594 b(of)g Fp(C)95 b Fq(-equiv)-74 b(alen)-37 b(t)596 b(cohomology)f (classes)1102 26843 y(and)489 b(a)f(sequence)h(\()p Fm(\001)221 b(\001)g(\001)443 b Fp(;)221 b(")14151 27042 y Fk(i)14594 26843 y Fp(;)g Fm(\001)g(\001)g(\001)i Fq(\))487 b(of)h(small)h(p)37 b(ositiv)-37 b(e)488 b(n)-37 b(um)g(b)37 b(ers.)726 b(Let)488 b Fm(f)p Fp(\034)37683 27042 y Fk(i)38126 26843 y Fm(g)38790 26361 y Fg(1)38790 27172 y(\0001)41233 26843 y Fq(b)37 b(e)489 b(a)f(monotone)1102 28437 y(sequence)445 b(of)e(in)-37 b(tegers)443 b(suc)-37 b(h)443 b Fp(\034)16616 28636 y Fv(0)17581 28437 y Fq(=)369 b(0,)443 b Fp(\034)21040 28636 y Fk(i)21851 28437 y Fm(!)369 b(\0061)444 b Fq(as)e Fp(i)369 b Fm(!)g(\0061)p Fq(.)591 b(Let)19657 32413 y Fp(\027)20313 32612 y Fk(N)21690 32413 y Fq(=)24103 30752 y Fk(N)23614 31151 y Fi(X)23092 34000 y Fk(i)p Fv(=)p Fg(\000)p Fk(N)26055 32413 y Fq(\()p Fm(\000)p Fp(\034)28186 32612 y Fk(i)28627 32413 y Fq(\))29144 31864 y Fg(\003)29755 32413 y Fp(\026)30555 32612 y Fk(i)30997 32413 y Fp(:)1102 36528 y Fq(F)-111 b(or)444 b(eac)-37 b(h)412 b Fp(~)-632 b(\034)7056 36727 y Fk(N)8432 36528 y Fq(=)369 b(\()p Fp(\034)10932 36727 y Fg(\000)p Fk(N)12769 36528 y Fp(;)221 b Fm(\001)g(\001)g(\001)444 b Fp(;)221 b(\034)16523 36727 y Fk(N)100 b Fg(\000)p Fv(1)18890 36528 y Fq(\))443 b(w)-37 b(e)443 b(consider)h(the)f(follo)-37 b(wing)444 b(v)-74 b(ariational)443 b(problem)3435 40264 y Fp(h)4200 39562 y Fk(T)4827 39696 y Fe(\000)p Fd(N)6401 39562 y Fk(;T)7343 39696 y Fd(N)4329 40644 y Fv(~)4200 40879 y Fk(L)8263 40264 y Fq(\()p Fp(m;)221 b(m)11702 39715 y Fg(0)12075 40264 y Fp(;)190 b(~)-633 b(\034)13246 40463 y Fk(N)14254 40264 y Fq(\))369 b(=)3977 b(inf)16394 41182 y Fk(\015)57 b Fv(\()p Fg(\000)p Fk(T)18881 41316 y Fe(\000)p Fd(N)20455 41182 y Fg(\000)p Fk(\034)21775 41316 y Fe(\000)p Fd(N)23348 41182 y Fv(\)=)p Fk(m)16771 42332 y(\015)g Fv(\()p Fk(T)18428 42466 y Fd(N)19282 42332 y Fv(+)p Fk(\034)20587 42466 y Fd(N)72 b Fe(\000)p Fj(1)22612 42332 y Fv(\)=)p Fk(m)24785 41931 y Fe(0)25742 38455 y Fi(Z)27071 38784 y Fk(T)27698 38918 y Fd(N)28551 38784 y Fv(+)p Fk(\034)29856 38918 y Fd(N)g Fe(\000)p Fj(1)26480 41474 y Fg(\000)p Fk(T)27937 41608 y Fe(\000)p Fd(N)29511 41474 y Fg(\000)p Fk(\034)30831 41608 y Fe(\000)p Fd(N)32692 38789 y Fi(\020)33485 40264 y Fp(L)295 b Fm(\000)g Fp(\021)36672 40463 y Fv(0)37563 40264 y Fm(\000)g Fp(\027)39547 40463 y Fk(N)40555 38789 y Fi(\021)41349 40264 y Fq(\()p Fp(d\015)74 b Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))p Fp(dt)16468 45670 y Fm(\000)18800 44009 y Fk(N)100 b Fg(\000)p Fv(1)18991 44408 y Fi(X)17796 47257 y Fk(i)p Fv(=)p Fg(\000)p Fk(N)g Fv(+1)22104 45670 y Fq(\()p Fp(\034)23202 45869 y Fk(i)23939 45670 y Fm(\000)295 b Fp(\034)25848 45869 y Fk(i)p Fg(\000)p Fv(1)27650 45670 y Fq(\))p Fp(\013)5 b Fq(\()p Fp(c)30114 45869 y Fk(i)30555 45670 y Fq(\))294 b Fm(\000)i Fp(T)33471 45869 y Fg(\000)p Fk(N)35309 45670 y Fp(\013)5 b Fq(\()p Fp(c)37256 45869 y Fg(\000)p Fk(N)39093 45670 y Fq(\))294 b Fm(\000)i Fp(T)42009 45869 y Fk(N)43016 45670 y Fp(\013)5 b Fq(\()p Fp(c)44963 45869 y Fk(N)45970 45670 y Fq(\))1102 49711 y(Let)455 b(\003)4416 49910 y Fk(N)5879 49711 y Fq(b)37 b(e)454 b(the)h(set)f(of)h(2)p Fp(N)600 b Fq(dimensional)455 b(in)-37 b(teger)455 b(v)-37 b(ectors)456 b(de\014ned)f(in)f(the)h(same)g(w)-37 b(a)g(y)455 b(as)g(for)1102 51305 y(\003)484 b(with)e(the)h(subscripts)g(ranging)g (o)-37 b(v)g(er)484 b(\()p Fm(\000)p Fp(N)71 b(;)221 b Fm(\001)g(\001)g(\001)444 b Fp(;)221 b(N)468 b Fm(\000)322 b Fq(1\))482 b(instead)h(of)g(\(0)p Fp(;)221 b Fm(\001)g(\001)g(\001) 444 b Fp(;)221 b(N)467 b Fm(\000)322 b Fq(1\).)710 b(Let)1102 52899 y Fp(\015)1790 53098 y Fk(N)2798 52899 y Fq(\()p Fp(t;)221 b(m;)g(m)7307 52417 y Fg(0)7680 52899 y Fp(;)g(T)9046 53098 y Fg(\000)p Fk(N)10884 52899 y Fp(;)g(T)12250 53098 y Fk(N)13258 52899 y Fq(\))443 b(b)37 b(e)443 b(the)g(minimizer)g(of) 11438 55857 y Fp(F)12420 56025 y Fv(~)12292 56260 y Fk(L)13088 55857 y Fq(\()p Fp(m;)221 b(m)16527 55309 y Fg(0)16900 55857 y Fp(;)g(T)18266 56056 y Fg(\000)p Fk(N)20104 55857 y Fp(;)g(T)21470 56056 y Fk(N)22478 55857 y Fq(\))369 b(=)709 b(min)24750 56747 y Fk(~)-528 b(\034)107 b Fg(2)p Fv(\003)26806 56881 y Fd(N)27881 55857 y Fp(h)28646 55155 y Fk(T)29273 55289 y Fe(\000)p Fd(N)30846 55155 y Fk(;T)31788 55289 y Fd(N)28774 56237 y Fv(~)28646 56473 y Fk(L)32709 55857 y Fq(\()p Fp(m;)221 b(m)36148 55309 y Fg(0)36521 55857 y Fp(;)190 b(~)-633 b(\034)37692 56056 y Fk(N)38700 55857 y Fq(\))p Fp(:)1102 59300 y Fq(With)354 b(the)g(same)h(argumen) -37 b(ts)354 b(ab)37 b(o)-37 b(v)g(e,)373 b(w)-37 b(e)354 b(can)h(mak)-37 b(e)355 b Fp(\015)28606 59499 y Fk(N)29613 59300 y Fq(\()p Fp(t;)221 b(m;)g(m)34122 58818 y Fg(0)34495 59300 y Fp(;)g(T)35861 59499 y Fg(\000)p Fk(N)37700 59300 y Fp(;)g(T)39066 59499 y Fk(N)40074 59300 y Fq(\))353 b(b)37 b(e)355 b(the)e(extremal)1102 60894 y(of)417 b Fp(L)f Fq(b)-37 b(y)416 b(c)-37 b(ho)37 b(osing)417 b(suitably)f(large) h Fp(T)20131 61093 y Fg(\000)p Fk(N)21969 60894 y Fq(,)k(and)c Fp(T)26092 61093 y Fk(N)27099 60894 y Fq(.)582 b(F)-111 b(rom)417 b(\(5.3\))e(and)i(\(5.11\))e(w)-37 b(e)417 b(can)g(see)g(that)1102 62617 y Fp(d\015)2481 62816 y Fk(N)3869 62617 y Fq(passes)380 b(within)g(a)h(distance)f(of)g Fp(")20107 62816 y Fk(i)20930 62617 y Fq(of)g(eac)-37 b(h)25616 62282 y(~)25306 62617 y Fm(L)p Fq(\()p Fp(c)27314 62816 y Fk(i)27756 62617 y Fq(\))379 b(for)i Fm(\000)p Fp(N)514 b Fm(\024)369 b Fp(i)g Fm(\024)g Fp(N)525 b Fq(if)380 b(w)-37 b(e)381 b(set)43777 62282 y(^)43519 62617 y Fp(T)44479 62135 y Fk(i)44295 62963 y(j)45302 62617 y Fq(suitably)1102 64211 y(large)495 b(for)f(eac)-37 b(h)494 b Fp(j)529 b Fq(=)454 b(0)p Fp(;)221 b Fq(1)494 b(and)g(eac)-37 b(h)494 b Fm(\000)p Fp(N)599 b Fm(\024)453 b Fp(i)g Fm(\024)h Fp(N)145 b Fq(.)742 b(Let)494 b Fp(\015)74 b Fq(:)691 b Fl(R)454 b Fm(!)f Fp(M)638 b Fq(b)37 b(e)494 b(an)f(accum)-37 b(ulation)1102 65805 y(p)37 b(oin)-37 b(t)393 b(of)h(the)f(set)h Fm(f)p Fq(\000)11742 66004 y Fk(N)12750 65805 y Fm(g)13414 65323 y Fg(1)13414 66173 y Fk(N)100 b Fg(\025)p Fk(N)16026 66306 y Fj(0)16611 65805 y Fq(,)404 b Fp(d\015)467 b Fq(clearly)395 b(determines)f(a)f (tra)74 b(jectory)394 b(of)g(the)f(Euler-Lagrange)1102 67701 y(\015o)-37 b(w)468 b(of)f Fp(L)g Fq(whic)-37 b(h)468 b(passes)f(within)g(a)g(distance)g(of)g Fp(")27299 67900 y Fk(i)28209 67701 y Fq(of)g(eac)-37 b(h)33215 67366 y(~)32759 67701 y Fm(A)p Fq(\()p Fp(c)34912 67900 y Fk(i)35353 67701 y Fq(\))467 b(for)g(all)g Fp(i)409 b Fm(2)g Fl(Z)p Fq(.)662 b(Therefore)1102 69295 y(w)-37 b(e)444 b(ha)-37 b(v)g(e)444 b(pro)-37 b(v)g(ed)443 b(the)g(theorem)1102 71589 y Fr(Theorem)599 b(5.1.)764 b Fq(\(Mather)442 b(1993\))548 b Fo(Supp)-68 b(ose)547 b Fp(c)25451 71788 y Fv(0)26593 71589 y Fo(and)g Fp(c)29820 71788 y Fk(N)31375 71589 y Fo(ar)-68 b(e)547 b Fp(C)95 b Fo(-e)-68 b(quivalent)549 b(classes.)828 b(Ther)-68 b(e)1102 73277 y(ther)g(e)507 b(is)f(a)g(tr)-68 b(aje)g(ctory)507 b(of)g(the)f(Euler-L)-68 b(agr)g(ange)507 b(\015ow)e(of)h Fp(L)g Fo(whose)g Fp(\013)5 b Fo(-limit)507 b(set)g(lies)g(in)47113 72941 y Fq(~)46657 73277 y Fm(A)o Fq(\()p Fp(c)48809 73476 y Fv(0)49405 73277 y Fq(\))1102 74965 y Fo(and)475 b(whose)g Fp(!)48 b Fo(-limit)477 b(set)f(lies)g(in)18592 74629 y Fq(~)18135 74965 y Fm(A)p Fq(\()p Fp(c)20288 75164 y Fk(N)21295 74965 y Fq(\))p Fo(.)p eop %%Page: 37 37 37 36 bop 48792 2239 a Ft(37)2696 5427 y Fo(Consider)640 b(a)g(bi-in\014nite)i(se)-68 b(quenc)g(e)640 b Fq(\()p Fm(\001)221 b(\001)g(\001)443 b Fp(;)221 b(c)25310 5626 y Fk(i)25752 5427 y Fp(;)g Fm(\001)g(\001)g(\001)i Fq(\))640 b Fo(of)g Fp(C)95 b Fo(-e)-68 b(quivalent)642 b(c)-68 b(ohomolo)g(gy)640 b(classes)1102 7021 y(and)479 b(a)g(se)-68 b(quenc)g(e)480 b Fq(\()p Fm(\001)221 b(\001)g(\001)443 b Fp(;)221 b(")13964 7220 y Fk(i)14407 7021 y Fp(;)g Fm(\001)g(\001)g(\001)i Fq(\))478 b Fo(of)i(smal)68 b(l)480 b(p)-68 b(ositive)481 b(numb)-68 b(ers.)623 b(Then)479 b(ther)-68 b(e)480 b(is)f(a)g(tr)-68 b(aje)g(ctory)481 b(of)1102 8709 y(the)515 b(Euler-L)-68 b(agr)g(ange)516 b(\015ow)e(of)i Fp(L)e Fo(which)h(p)-68 b(asses)515 b(within)g(a)g (distanc)-68 b(es)516 b(of)f Fp(")39556 8908 y Fk(i)40513 8709 y Fo(of)h(e)-68 b(ach)45598 8373 y Fq(~)45141 8709 y Fm(A)p Fq(\()p Fp(c)47294 8908 y Fk(i)47736 8709 y Fq(\))514 b Fo(in)1102 10303 y(turn.)2696 12940 y Fq(The)363 b(next)g(step)f(is)g(to)g(establish)g Fp(C)95 b Fq(-equiv)-74 b(alence)365 b(among)d(some)h(Ma)37 b(~)-701 b(n)-37 b(\023)-627 b(e)363 b(sets)f(of)g(the)g(sp)37 b(ecial)363 b Fp(L)1102 14535 y Fq(giv)-37 b(en)322 b(b)-37 b(y)321 b(\(2.2\).)549 b(Let)321 b(us)g(consider)h(the)e(\014rst)h(de)g(Rham)g (cohomology)h(class)f Fp(c)369 b Fm(2)f Fp(H)42508 14052 y Fv(1)43104 14535 y Fq(\()p Fp(M)71 b(;)221 b Fl(R)p Fq(\))321 b(suc)-37 b(h)1102 16222 y(that)524 b(the)g(supp)37 b(ort)525 b(of)f Fp(c)p Fq(-minimal)g(measure)h(uniquely)g(sits)f(on) 34486 15887 y(~)34403 16222 y(\000)505 b Fm(\032)37423 15887 y Fq(~)37276 16222 y(\006.)835 b(First,)545 b(w)-37 b(e)525 b(consider)1102 17816 y(the)542 b(case)h(that)f(\000)g(is)g(a)g (Denjo)-37 b(y)543 b(set)f(and)g(there)h(is)f(no)g(in)-37 b(v)-74 b(arian)-37 b(t)543 b(circle)g(con)-37 b(taining)543 b(\000.)889 b(The)1102 19410 y(rotation)569 b(n)-37 b(um)g(b)37 b(er)570 b(of)f(\000)g(is)f(irrational.)970 b(By)568 b(the)h(w)-37 b(ell-kno)g(wn)570 b(kno)-37 b(wledge)571 b(w)-37 b(e)570 b(see)f(that)f(the)1102 21004 y Fp(\014)70 b Fq(-function)568 b(for)g(the)g(t)-37 b(wist)568 b(map)f(is)h (di\013eren)-37 b(tiable)569 b(at)f(the)f(p)37 b(oin)-37 b(t)568 b(of)g(irrational)g(n)-37 b(um)g(b)37 b(er,)600 b(it)1102 22618 y(implies)551 b(that)f(there)h(is)f(only)h(one)g Fp(c)19795 22817 y Fv(1)20941 22618 y Fq(suc)-37 b(h)550 b(that)27113 22283 y(~)27030 22618 y(\000)h(is)f(the)g(supp)37 b(ort)551 b(of)f Fp(c)p Fq(-minimal)g(measure)1102 24233 y(if)509 b Fp(c)479 b Fm(2)f Fp(intI)104 b Fq(\()p Fp(c)8319 24432 y Fv(1)8914 24233 y Fq(\).)788 b(W)-111 b(e)510 b(see)f(from)g(the)g(lemma)g(3.4)g(that)30632 23897 y(~)30211 24233 y Fm(N)195 b Fq(\()p Fp(c)p Fq(\))478 b(=)35745 23897 y(~)35095 24233 y Fm(M)p Fq(\()p Fp(c)p Fq(\))508 b(when)i Fp(a)p Fq(\()p Fp(c)44136 24432 y Fv(1)44731 24233 y Fq(\))478 b Fp(<)h(c)47813 24432 y Fv(2)48888 24233 y Fp(<)1102 25920 y(b)p Fq(\()p Fp(c)2764 26119 y Fv(1)3360 25920 y Fq(\).)712 b(By)484 b(the)f(upp)37 b(er)484 b(semi-con)-37 b(tin)g(uit)g(y)484 b(of)f Fp(c)437 b Fm(!)27356 25585 y Fq(~)26934 25920 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\))483 b(w)-37 b(e)484 b(\014nd)f(that)g(there)h(exists)g Fp(\016)j(>)437 b Fq(0,)494 b(if)1102 27514 y Fp(c)1677 27032 y Fg(0)2419 27514 y Fm(2)368 b Fp(J)497 b Fq(=)369 b(\(\()p Fp(c)7918 27713 y Fv(1)8715 27514 y Fm(\000)202 b Fp(\016)-24 b(;)221 b(c)11681 27713 y Fv(1)12481 27514 y Fq(+)202 b Fp(\016)50 b Fq(\))202 b Fm(\002)g Fq(\()p Fp(a)p Fq(\()p Fp(c)18621 27713 y Fv(1)19218 27514 y Fq(\))g(+)g Fp(\016)-24 b(;)221 b(b)p Fq(\()p Fp(c)23990 27713 y Fv(1)24588 27514 y Fq(\))202 b Fm(\000)g Fp(\016)50 b Fq(\))397 b(then)g Fm(N)196 b Fq(\()p Fp(c)33454 27032 y Fg(0)33825 27514 y Fq(\))396 b(is)h(in)f(a)h(small)f(neigh)-37 b(b)37 b(orho)g(o)g(d)1102 29202 y(of)516 b Fp(\031)48 b Fq(\()4093 28866 y(~)4010 29202 y(\000\),)534 b(th)-37 b(us)516 b(eac)-37 b(h)517 b(of)f(suc)-37 b(h)516 b Fm(N)18063 29401 y Fv(0)18659 29202 y Fq(\()p Fp(c)p Fq(\))f(is)h(homotopically)g (trivial.)810 b(Therefore,)537 b(all)516 b Fp(c)44526 28720 y Fg(0)45390 29202 y Fm(2)490 b Fp(J)643 b Fq(are)1102 30796 y Fp(C)95 b Fq(-equiv)-74 b(alen)-37 b(t.)2696 33434 y(Next,)496 b(let)485 b(us)h(consider)g(the)f(case)h(when)g(\000) f(consists)g(of)g(single)h(p)37 b(erio)g(dic)486 b(orbit.)717 b(Since)486 b(the)1102 35028 y Fp(\014)70 b Fq(-function)457 b(of)f(the)g(t)-37 b(wist)456 b(map)h(has)f(a)h(corner)g(at)f(the)h (rational)f(rotation)h(n)-37 b(um)g(b)37 b(er,)460 b(there)d(is)f(a) 1102 36622 y(\015at)379 b(piece)h(of)e(the)h Fp(\013)5 b Fq(-function)378 b(of)h(the)g(t)-37 b(wist)378 b(map,)391 b(o)-37 b(v)g(er)380 b(the)f(in)-37 b(terv)-74 b(al)379 b([)p Fp(c)37619 36054 y Fg(\000)37619 36976 y Fv(1)38516 36622 y Fp(;)221 b(c)39681 36054 y Fv(+)39681 36976 y(1)40563 36622 y Fq(].)569 b(Consequen)-37 b(tly)-111 b(,)1102 38253 y(there)644 b(is)f(a)g(rectangle)h(\()p Fp(c)14522 37685 y Fg(\000)14522 38607 y Fv(1)15418 38253 y Fp(;)221 b(c)16583 37685 y Fv(+)16583 38607 y(1)17466 38253 y Fq(\))428 b Fm(\002)h Fq(\()p Fp(a)p Fq(\()p Fp(c)22184 38452 y Fv(1)22779 38253 y Fq(\))p Fp(;)221 b(b)p Fq(\()p Fp(c)25548 38452 y Fv(1)26143 38253 y Fq(\)\))702 b Fm(2)g Fp(H)30679 37771 y Fv(1)31275 38253 y Fq(\()p Fp(M)71 b(;)221 b Fl(R)p Fq(\))643 b(suc)-37 b(h)643 b(that)f(all)i Fp(c)p Fq(-minimal)1102 39963 y(measures)585 b(ha)-37 b(v)g(e)584 b(their)f(supp)37 b(ort)583 b(on)20592 39627 y(~)20509 39963 y(\000)g(if)g Fp(c)g Fq(is)h(in)f(this)g(rectangle.) 1013 b(When)584 b Fp(c)41550 39395 y Fg(\000)41550 40317 y Fv(1)43049 39963 y Fp(<)603 b(c)45260 40162 y Fv(1)46459 39963 y Fp(<)h(c)48671 39395 y Fv(+)48671 40317 y(1)49552 39963 y Fq(,)1102 41673 y Fp(a)p Fq(\()p Fp(c)2896 41872 y Fv(1)3492 41673 y Fq(\))690 b Fp(<)h(c)6998 41872 y Fv(2)8284 41673 y Fp(<)g(b)p Fq(\()p Fp(c)11670 41872 y Fv(1)12266 41673 y Fq(\),)14257 41337 y(~)13835 41673 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\))690 b(=)19794 41337 y(~)19144 41673 y Fm(M)p Fq(\()p Fp(c)p Fq(\).)1168 b(When)637 b Fp(a)p Fq(\()p Fp(c)29747 41872 y Fv(1)30342 41673 y Fq(\))690 b Fp(<)h(c)33848 41872 y Fv(2)35135 41673 y Fp(<)f(b)p Fq(\()p Fp(c)38520 41872 y Fv(1)39116 41673 y Fq(\))635 b(and)h Fp(c)43619 41872 y Fv(1)44906 41673 y Fq(=)691 b Fp(c)47205 41105 y Fg(\000)47205 42027 y Fv(1)48737 41673 y Fq(or)1102 43383 y Fp(c)1677 43582 y Fv(1)2723 43383 y Fq(=)450 b Fp(c)4781 42815 y Fv(+)4781 43737 y(1)5663 43383 y Fq(,)6957 43047 y(~)6535 43383 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\))449 b(=)12012 43047 y(~)11362 43383 y Fm(M)p Fq(\()p Fp(c)p Fq(\))327 b Fm([)g(f)p Fq(minimal)443 b(homo)37 b(clinic)444 b(orbit)f(in)33611 43047 y(~)33464 43383 y(\006)p Fm(g)p Fq(.)736 b(Due)492 b(to)f(the)g(upp)37 b(er)492 b(semi-)1102 45093 y(con)-37 b(tin)g(uit)g(y)399 b(of)g Fp(c)369 b Fm(!)11863 44757 y Fq(~)11441 45093 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\))398 b(and)h(the)f(h)-37 b(yp)37 b(othesis)399 b(of)g(\(H2\),)407 b(w)-37 b(e)400 b(\014nd)f(that)f(there)h(exists)g Fp(\016)419 b(>)370 b Fq(0,)407 b(if)1102 46702 y Fp(c)1677 46220 y Fg(0)2419 46702 y Fm(2)368 b Fp(J)497 b Fq(=)369 b(\(\()p Fp(c)7918 46133 y Fg(\000)7918 47056 y Fv(1)9084 46702 y Fm(\000)271 b Fp(\016)-24 b(;)221 b(c)12119 46133 y Fv(+)12119 47056 y(1)13272 46702 y Fq(+)271 b Fp(\016)50 b Fq(\))271 b Fm(\002)f Fq(\()p Fp(a)p Fq(\()p Fp(c)19618 46901 y Fv(1)20213 46702 y Fq(\))g(+)h Fp(\016)-24 b(;)221 b(b)p Fq(\()p Fp(c)25122 46901 y Fv(1)25719 46702 y Fq(\))270 b Fm(\000)g Fp(\016)50 b Fq(\))431 b(then)f Fm(N)33500 46901 y Fv(0)34096 46702 y Fq(\()p Fp(c)p Fq(\))g(is)g(homotopically)h (trivial,)1102 48296 y(th)-37 b(us)443 b(all)g Fp(c)6445 47814 y Fg(0)7186 48296 y Fm(2)369 b Fp(J)570 b Fq(are)443 b Fp(C)95 b Fq(-equiv)-74 b(alen)-37 b(t.)2696 50933 y(Finally)604 b(w)-37 b(e)604 b(consider)h(the)e(case)i(when)f(\000)f (is)h(con)-37 b(tained)604 b(in)f(an)h(in)-37 b(v)-74 b(arian)-37 b(t)604 b(circle)h(on)e(the)1102 52547 y(b)37 b(oundary)560 b(of)g(a)f(gap.)940 b(In)559 b(this)g(case)20923 52211 y(~)20840 52547 y(\000)h(is)f(the)g(supp)37 b(ort)559 b(of)g(that)g Fp(c)p Fq(-minimal)g(measure)h(with)1102 54141 y Fp(c)369 b Fm(2)g Fp(I)104 b Fq(\()29 b(\026)-693 b Fp(c)5081 54340 y Fv(1)5676 54141 y Fq(\))369 b(=)g Fm(f)p Fq(\()29 b(\026)-693 b Fp(c)9720 54340 y Fv(1)10316 54141 y Fp(;)221 b(c)11481 54340 y Fv(2)12077 54141 y Fq(\))368 b(:)h Fp(a)p Fq(\()29 b(\026)-693 b Fp(c)15494 54340 y Fv(1)16090 54141 y Fq(\))368 b Fm(\024)h Fp(c)18952 54340 y Fv(2)19917 54141 y Fm(\024)g Fp(b)p Fq(\()29 b(\026)-693 b Fp(c)22981 54340 y Fv(1)23577 54141 y Fq(\))p Fm(g)p Fq(.)578 b(Because)407 b(of)f(the)f(h)-37 b(yp)37 b(otheses)407 b(\(H1\),)413 b(\(H3\))405 b(and)1102 55735 y(in)496 b(virtue)g(of)g(the)g(lemma)g(3.4)p Fm(\030)p Fq(6,)509 b(w)-37 b(e)497 b(ha)-37 b(v)g(e)497 b Fm(N)25363 55934 y Fv(0)25958 55735 y Fq(\()p Fp(c)p Fq(\))457 b Fm(\032)g Fp(U)602 b Fq(=)457 b Fm([)33399 55253 y Fk(m)33399 56081 y(i)p Fv(=0)35186 55735 y Fp(U)36093 55934 y Fk(i)36535 55735 y Fq(,)509 b(where)497 b Fp(U)42214 55934 y Fk(i)42987 55735 y Fm(\\)330 b Fp(U)45110 55934 y Fk(j)46127 55735 y Fq(=)457 b Fl(?)496 b Fq(if)1102 57329 y Fp(i)369 b Fm(6)p Fq(=)g Fp(j)76 b Fq(,)416 b Fp(U)5646 57528 y Fv(0)6651 57329 y Fq(is)410 b(an)f(op)37 b(en)410 b(neigh)-37 b(b)37 b(orho)g(o)g(d)410 b(of)g(\000,)416 b(all)409 b(other)h Fp(U)30503 57528 y Fk(i)31355 57329 y Fq(\()p Fp(i)368 b Fm(6)p Fq(=)h(0\))409 b(are)h(op)37 b(en)410 b(set)f(con)-37 b(tractible)1102 58923 y(to)577 b(one)g(p)37 b(oin)-37 b(t.)991 b(Let)576 b Fp(J)719 b Fq(=)592 b(\()29 b(\026)-693 b Fp(c)16501 59122 y Fv(1)17481 58923 y Fm(\000)384 b Fp(\016)-24 b(;)252 b Fq(\026)-695 b Fp(c)20629 59122 y Fv(1)21610 58923 y Fq(+)385 b Fp(\016)50 b Fq(\))384 b Fm(\002)h Fq(\()p Fp(a)p Fq(\()29 b(\026)-693 b Fp(c)28298 59122 y Fv(1)28893 58923 y Fq(\))384 b Fm(\000)g Fp(\016)-24 b(;)221 b(b)p Fq(\()29 b(\026)-693 b Fp(c)34029 59122 y Fv(1)34626 58923 y Fq(\))384 b(+)g Fp(\016)50 b Fq(\).)992 b(Due)576 b(to)g(the)h(upp)37 b(er)1102 60611 y(semi-con)-37 b(tin)g(uit)g(y)600 b(of)f Fp(c)629 b Fm(!)15818 60275 y Fq(~)15396 60611 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\),)637 b(w)-37 b(e)600 b(can)g(see)g(that)e Fm(N)30429 60810 y Fv(0)31025 60611 y Fq(\()p Fp(c)p Fq(\))629 b Fm(\032)g Fp(U)744 b Fq(for)600 b(all)f Fp(c)629 b Fm(2)h Fp(J)726 b Fq(if)599 b Fp(\016)680 b(>)630 b Fq(0)1102 62205 y(is)641 b(su\016cien)-37 b(tly)642 b(small.)1185 b(T)-111 b(o)641 b(establish)g(the)g Fp(C)95 b Fq(-equiv)-74 b(alen)-37 b(t)642 b(relationship)g(b)37 b(et)-37 b(w)g(een)642 b(an)-37 b(y)641 b(t)-37 b(w)g(o)1102 63799 y Fp(c;)221 b(c)2842 63317 y Fg(0)3584 63799 y Fm(2)369 b Fp(J)128 b Fq(,)434 b(let)e(us)g(consider)i(\014rst)e(the)h(sp)37 b(ecial)433 b(case)h(when)f Fp(c;)221 b(c)32599 63317 y Fg(0)33341 63799 y Fm(2)368 b Fp(J)560 b Fq(and)433 b Fp(c)274 b Fm(\000)h Fp(c)41197 63317 y Fg(0)41938 63799 y Fq(=)370 b(\(0)p Fp(;)221 b(c)45687 63998 y Fv(2)46557 63799 y Fm(\000)275 b Fp(c)48440 63317 y Fg(0)48440 64129 y Fv(2)49036 63799 y Fq(\).)1102 65393 y(Let)495 b(\000\()p Fp(s)p Fq(\))454 b(=)h Fm(f)p Fq(\()p Fp(c)9719 65592 y Fv(1)10315 65393 y Fp(;)221 b(sc)12103 65592 y Fv(2)13028 65393 y Fq(+)330 b(\(1)g Fm(\000)f Fp(s)p Fq(\)\))p Fp(c)19496 64911 y Fg(0)19496 65723 y Fv(2)20546 65393 y Fq(:)677 b(0)455 b Fm(\024)g Fp(s)g Fm(\024)g Fq(1)p Fm(g)p Fq(,)508 b(ob)-37 b(viously)-111 b(,)508 b([)p Fp(dq)36820 65592 y Fv(2)37416 65393 y Fq(])495 b(is)f(the)g(annihilator)h(of)1102 66987 y Fp(V)1877 67227 y Fv(\000\()p Fk(s)p Fv(\))3932 66987 y Fq(\()p Fp(t)p Fq(\))460 b Fm(8)p Fp(s)400 b Fm(2)g Fq([0)p Fp(;)221 b Fq(1],)466 b Fp(t)400 b Fm(2)g Fl(T)p Fq(.)646 b(Th)-37 b(us)462 b(\000\()p Fp(s)p Fq(\))e(is)i(an)f (admissible)h(curv)-37 b(e.)647 b(Second,)467 b(let)462 b(us)f(consider)1102 68589 y(the)496 b(case)h(when)f Fp(c)457 b Fq(=)h(\()p Fp(c)13444 68788 y Fv(1)14039 68589 y Fp(;)221 b(c)15204 68788 y Fv(2)15800 68589 y Fq(\))457 b Fm(2)g Fp(J)623 b Fq(but)496 b Fp(c)22541 68788 y Fv(2)23594 68589 y Fp(>)457 b(b)p Fq(\()29 b(\026)-693 b Fp(c)26746 68788 y Fv(1)27342 68589 y Fq(\))495 b(or)h Fp(c)30609 68788 y Fv(2)31662 68589 y Fp(<)458 b(a)p Fq(\()29 b(\026)-693 b Fp(c)34947 68788 y Fv(1)35542 68589 y Fq(\).)749 b(Under)497 b(the)f(h)-37 b(yp)37 b(otheses)1102 70183 y(\(H1\))514 b(and)g(\(H3\),)532 b(for)514 b(an)-37 b(y)514 b Fp(\016)539 b(>)488 b Fq(0,)532 b(there)514 b(exists)g Fp(c)488 b Fq(=)g(\()29 b(\026)-693 b Fp(c)30342 70382 y Fv(1)30937 70183 y Fp(;)221 b(c)32102 70382 y Fv(2)32698 70183 y Fq(\))514 b(with)g Fp(b)p Fq(\()29 b(\026)-693 b Fp(c)38488 70382 y Fv(1)39083 70183 y Fq(\))488 b Fp(<)g(c)42184 70382 y Fv(2)43267 70183 y Fp(<)g(b)p Fq(\()29 b(\026)-693 b Fp(c)46450 70382 y Fv(1)47045 70183 y Fq(\))343 b(+)g Fp(\016)1102 71777 y Fq(or)568 b Fp(a)p Fq(\()29 b(\026)-693 b Fp(c)4648 71976 y Fv(1)5244 71777 y Fq(\))377 b Fm(\000)i Fp(\016)627 b(<)576 b(c)10951 71976 y Fv(2)12123 71777 y Fp(<)h(a)p Fq(\()29 b(\026)-693 b Fp(c)15527 71976 y Fv(1)16122 71777 y Fq(\))567 b(suc)-37 b(h)568 b(that)e Fm(N)24419 71976 y Fk(t)24887 71777 y Fq(\()p Fp(c)p Fq(\))g(is)h(homop)37 b(otically)568 b(trivial)g(for)f(an)-37 b(y)568 b Fp(t)576 b Fm(2)g Fl(T)p Fq(.)1102 73371 y(Therefore,)655 b Fm(9)610 b Fp(\016)9691 72889 y Fg(0)10712 73371 y Fp(>)648 b Fq(0)611 b(suc)-37 b(h)610 b(that)g(for)h(all)f Fp(c)24665 72889 y Fg(0)25686 73371 y Fm(2)647 b Fp(B)28227 73570 y Fk(\016)28744 73304 y Fe(0)29169 73371 y Fq(\()p Fp(c)p Fq(\),)k Fm(G)32588 73570 y Fv(0)33184 73371 y Fq(\()p Fp(c)34276 72889 y Fg(0)34648 73371 y Fq(\))610 b(is)g(homotopically)h (trivial.)1102 74965 y(Replacing)417 b Fp(\016)466 b Fq(with)416 b Fp(\016)11914 74483 y Fg(0)12703 74965 y Fq(in)g(the)f(de\014nition)h(of)g Fp(J)128 b Fq(,)420 b(w)-37 b(e)417 b(\014nd)e(that)g(all)h Fp(c)369 b Fm(2)g Fp(J)543 b Fq(are)416 b Fp(C)95 b Fq(-equiv)-74 b(alen)-37 b(t.)583 b(In)p eop %%Page: 38 38 38 37 bop 1102 2239 a Ft(38)1102 5427 y Fq(fact,)491 b(giv)-37 b(en)481 b(an)-37 b(y)481 b(t)-37 b(w)g(o)481 b Fp(c)p Fq(,)490 b Fp(c)14748 4945 y Fg(0)15553 5427 y Fm(2)432 b Fp(J)128 b Fq(,)490 b(w)-37 b(e)481 b(can)g(construct)g (the)g(admissible)g(curv)-37 b(e)482 b(as)f(follo)-37 b(ws.)705 b(Let)1102 7021 y(\000:)591 b([0)p Fp(;)221 b Fq(3])370 b Fm(!)f Fp(H)8827 6539 y Fv(1)9423 7021 y Fq(\()p Fp(M)71 b(;)221 b Fl(R)p Fq(\),)12902 12134 y(\000\()p Fp(s)p Fq(\))368 b(=)17159 9012 y Fi(8)17159 10207 y(>)17159 10606 y(<)17159 12997 y(>)17159 13395 y(:)18783 10140 y Fp(sc)295 b Fq(+)g(\(1)g Fm(\000)g Fp(s)p Fq(\))29 b(~)-693 b Fp(c)o(;)5756 b Fq(0)370 b Fm(\024)f Fp(s)f Fm(\024)i Fq(1)18783 12134 y(\()p Fp(s)294 b Fm(\000)i Fq(1\))29 b(~)-693 b Fp(c)294 b Fq(+)i(\(2)f Fm(\000)g Fp(s)p Fq(\))29 b(~)-693 b Fp(c)29444 11651 y Fg(0)29816 12134 y Fp(;)1992 b Fq(1)369 b Fm(\024)h Fp(s)e Fm(\024)h Fq(2)18783 14127 y(\()p Fp(s)294 b Fm(\000)i Fq(2\))29 b(~)-693 b Fp(c)23302 13645 y Fg(0)23969 14127 y Fq(+)295 b(\(3)g Fm(\000)h Fp(s)p Fq(\))p Fp(c)29817 13645 y Fg(0)30189 14127 y Fp(;)1549 b Fq(2)369 b Fm(\024)h Fp(s)e Fm(\024)h Fq(3)1102 17248 y(in)540 b(whic)-37 b(h)570 b(~)-693 b Fp(c)539 b Fq(and)569 b(~)-693 b Fp(c)11016 16766 y Fg(0)11919 17248 y Fm(2)530 b Fp(J)667 b Fq(are)541 b(de\014ned)f(in)g(the)g(w)-37 b(a)g(y)569 b(~)-693 b Fp(c)29123 17447 y Fv(2)30249 17248 y Fq(=)560 b(~)-693 b Fp(c)32388 16766 y Fg(0)32388 17578 y Fv(2)33514 17248 y Fp(>)531 b(b)p Fq(\()29 b(\026)-693 b Fp(c)36740 17447 y Fv(1)37336 17248 y Fq(\))539 b(or)569 b(~)-693 b Fp(c)40691 17447 y Fv(2)41817 17248 y Fq(=)560 b(~)-693 b Fp(c)43956 16766 y Fg(0)43956 17578 y Fv(2)45082 17248 y Fp(<)531 b(a)p Fq(\()29 b(\026)-693 b Fp(c)48440 17447 y Fv(1)49036 17248 y Fq(\),)1132 18842 y(~)f Fp(c)1677 19041 y Fv(1)2642 18842 y Fq(=)369 b Fp(c)4619 19041 y Fv(1)5658 18842 y Fq(and)472 b(~)-693 b Fp(c)8816 18360 y Fg(0)8816 19172 y Fv(1)9780 18842 y Fq(=)370 b Fp(c)11758 18360 y Fg(0)11758 19172 y Fv(1)12353 18842 y Fq(,)443 b(b)37 b(oth)443 b Fm(N)17392 19041 y Fk(t)17859 18842 y Fq(\()29 b(~)-693 b Fp(c)p Fq(\))442 b(and)h Fm(N)23583 19041 y Fk(t)24050 18842 y Fq(\()29 b(~)-693 b Fp(c)25142 18360 y Fg(0)25515 18842 y Fq(\))442 b(are)i(homotopically)f(trivial.)1102 21218 y Fr(Lemma)782 b(5.2.)765 b Fo(We)693 b(assume)i(the)f(hyp)-68 b(otheses)694 b(\(H1)p Fm(\030)p Fo(3\).)1269 b(L)-68 b(et)723 b Fq(^)-693 b Fp(c)765 b Fq(=)g(\()p Fp(c)39476 21417 y Fv(1)40072 21218 y Fq(\()p Fp(\033)41396 20736 y Fg(0)41768 21218 y Fq(\))p Fp(;)221 b Fq(0\))693 b Fo(and)723 b Fq(\026)-693 b Fp(c)765 b Fq(=)1102 22906 y(\()p Fp(c)2194 23105 y Fv(1)2790 22906 y Fq(\()p Fp(\033)4114 22424 y Fg(\003)4724 22906 y Fq(\))p Fp(;)221 b Fq(0\))475 b Fo(b)-68 b(e)475 b(two)g(c)-68 b(o-homolo)g(gy)477 b(classes)f(such)f(that)29600 22570 y Fq(~)29179 22906 y Fm(N)196 b Fq(\()29 b(^)-693 b Fp(c)o Fq(\))p Fm(j)32442 23105 y Fk(t)p Fv(=0)34730 22906 y Fo(and)37731 22570 y Fq(~)37310 22906 y Fm(N)195 b Fq(\()29 b(\026)-693 b Fp(c)p Fq(\))p Fm(j)40573 23105 y Fk(t)p Fv(=0)42860 22906 y Fo(make)476 b(up)f(the)1102 24500 y(whole)h(b)-68 b(oundary)475 b(of)g(some)h(given)f(gap)g(with)h Fp(\033)25044 24018 y Fg(0)25785 24500 y Fp(<)369 b(\033)27994 24018 y Fg(\003)28605 24500 y Fo(.)611 b(Then)504 b Fq(^)-693 b Fp(c)475 b Fo(and)504 b Fq(\026)-693 b Fp(c)475 b Fo(ar)-68 b(e)475 b Fp(C)95 b Fo(-e)-68 b(quivalent.)1102 26876 y Fr(Pro)42 b(of)p Fq(:)1045 b(By)671 b(assumption,)726 b(there)671 b(is)f(no)g(other)g(in)-37 b(v)-74 b(arian)-37 b(t)671 b(circle)h(b)37 b(et)-37 b(w)g(een)671 b Fm(N)196 b Fq(\()29 b(^)-693 b Fp(c)o Fq(\))p Fm(j)45299 27075 y Fk(t)p Fv(=0)47781 26876 y Fq(and)1102 28470 y Fm(N)196 b Fq(\()29 b(\026)-693 b Fp(c)o Fq(\))p Fm(j)4365 28669 y Fk(t)p Fv(=0)6178 28470 y Fq(.)939 b(In)559 b(this)f(case,)589 b(w)-37 b(e)560 b(ha)-37 b(v)g(e)560 b(sho)-37 b(wn)560 b(that)e(for)i(an)-37 b(y)559 b Fp(c)33036 28669 y Fv(1)33632 28470 y Fq(\()p Fp(\033)34956 27988 y Fg(0)35328 28470 y Fq(\))k Fm(\024)g Fp(c)38579 28669 y Fv(1)39737 28470 y Fm(\024)g Fp(c)41908 28669 y Fv(1)42504 28470 y Fq(\()p Fp(\033)43828 27988 y Fg(\003)44438 28470 y Fq(\))c(there)g(is)1102 30064 y(an)532 b(op)37 b(en)532 b(rectangle)h Fp(J)128 b Fq(\()p Fp(c)14067 30263 y Fv(1)14662 30064 y Fq(\))517 b Fm(\032)g Fp(H)18458 29582 y Fv(1)19054 30064 y Fq(\()p Fp(M)71 b(;)221 b Fl(R)p Fq(\))532 b(con)-37 b(taining)532 b(\()p Fp(c)31167 30263 y Fv(1)31762 30064 y Fp(;)221 b Fq(0\))532 b(suc)-37 b(h)532 b(that)g(all)f Fp(c)518 b Fm(2)e Fp(J)128 b Fq(\()p Fp(c)46503 30263 y Fv(1)47098 30064 y Fq(\))531 b(are)1102 31658 y Fp(C)95 b Fq(-equiv)-74 b(alen)-37 b(t.)629 b(By)456 b(the)f(compactness)h(of)g(the)f(in)-37 b(terv)-74 b(al)456 b([)p Fp(c)31136 31857 y Fv(1)31732 31658 y Fq(\()p Fp(\033)33056 31176 y Fg(0)33428 31658 y Fq(\))p Fp(;)221 b(c)35110 31857 y Fv(1)35705 31658 y Fq(\()p Fp(\033)37029 31176 y Fg(\003)37639 31658 y Fq(\)])455 b(there)h(is)f(a)h(sequence)1102 33511 y Fm(f)p Fp(c)2341 32821 y Fv(\()p Fk(i)p Fv(\))2341 33865 y(1)3614 33511 y Fm(g)4278 33028 y Fk(m)4278 33856 y(i)p Fv(=0)6464 33511 y Fq(suc)-37 b(h)399 b(that)f Fm([)13136 33028 y Fk(m)13136 33856 y(i)p Fv(=0)14923 33511 y Fp(J)128 b Fq(\()p Fp(c)16880 32821 y Fv(\()p Fk(i)p Fv(\))16880 33865 y(1)18151 33511 y Fq(\))369 b Fm(\033)g Fq([)p Fp(c)21383 33710 y Fv(1)21979 33511 y Fq(\()p Fp(\033)23303 33028 y Fg(0)23675 33511 y Fq(\))p Fp(;)221 b(c)25357 33710 y Fv(1)25952 33511 y Fq(\()p Fp(\033)27276 33028 y Fg(\003)27886 33511 y Fq(\)])207 b Fm(\002)g(f)p Fq(0)p Fm(g)p Fq(.)576 b(Ob)-37 b(viously)-111 b(,)408 b(the)399 b Fp(C)95 b Fq(-equiv)-74 b(alence)1102 35105 y(has)443 b(transitivit)-37 b(y)-111 b(.)2165 b Fb(\003)2696 37480 y Fq(This)504 b Fp(C)95 b Fq(-equiv)-74 b(alence)505 b(establishes)e(the)h(existence)g(of)f(the)g(di\013usion)g(orbits)g (crossing)h(gaps)1102 39074 y(as)443 b(w)-37 b(e)444 b(ha)-37 b(v)g(e)444 b(the)e(theorem)i(5.1.)2696 41450 y(T)-111 b(o)417 b(go)f(further,)421 b(w)-37 b(e)416 b(need)g(to)g(kno)-37 b(w)416 b(more)g(details)g(of)f Fp(U)145 b Fq(-step)415 b(forms.)582 b(Let)415 b Fp(\021)41658 41649 y Fk(j)42633 41450 y Fq(b)37 b(e)416 b(an)-37 b(y)416 b(giv)-37 b(en)1102 43143 y(closed)468 b(1-form)g(suc)-37 b(h)468 b(that)e([)p Fp(\021)16269 43342 y Fk(j)16828 43143 y Fq(])410 b(=)g Fp(c)19625 42661 y Fv(\()p Fk(j)54 b Fv(\))21482 43143 y Fq(for)468 b Fp(j)485 b Fq(=)410 b(1)p Fp(;)221 b(k)42 b Fq(.)664 b(A)468 b(natural)f(question)g(is)g (whether)i(there)1102 44737 y(exists)441 b(suc)-37 b(h)441 b(kind)f(of)h Fp(\026)p Fq(\()p Fp(t)p Fq(\))f(so)h(that)e Fp(\026)p Fq(\()p Fp(t)p Fq(\))369 b(=)g Fp(\021)24262 44936 y Fv(1)25298 44737 y Fq(for)441 b Fp(t)368 b Fm(\024)i Fq(0)440 b(and)h Fp(\026)p Fq(\()p Fp(t)p Fq(\))368 b(=)h Fp(\021)38009 44936 y Fk(k)39103 44737 y Fq(for)441 b Fp(t)369 b Fm(\025)g Fp(\034)43966 44936 y Fk(k)44910 44737 y Fq(+)291 b(1)440 b(ev)-37 b(en)1102 46381 y(though)521 b Fp(c)6257 45899 y Fv(\(1\))8205 46381 y Fq(is)g(equiv)-74 b(alen)-37 b(t)522 b(to)f Fp(c)18284 45899 y Fv(\()p Fk(j)54 b Fv(\))19674 46381 y Fq(?)825 b(In)521 b(general,)542 b(w)-37 b(e)522 b(do)g(not)f(kno)-37 b(w)521 b(whether)i(it)d(is)h (true)h(or)1102 47975 y(not,)443 b(but)f(in)h(our)g(case,)h(the)f(answ) -37 b(er)444 b(is)e(y)-37 b(es.)1102 50535 y Fr(Lemma)733 b(5.3.)764 b Fo(L)-68 b(et)655 b Fp(c)12930 50053 y Fv(\(1\))15049 50535 y Fq(=)693 b(\()p Fp(c)17867 49845 y Fv(\(1\))17867 50890 y(1)19293 50535 y Fp(;)221 b(c)20458 49845 y Fv(\(1\))20458 50890 y(2)21884 50535 y Fq(\))p Fo(,)699 b Fp(c)24082 50053 y Fv(\()p Fk(k)22 b Fv(\))26259 50535 y Fq(=)693 b(\()p Fp(c)29077 49845 y Fv(\()p Fk(k)22 b Fv(\))29077 50890 y(1)30560 50535 y Fp(;)221 b(c)31725 49845 y Fv(\()p Fk(k)22 b Fv(\))31725 50890 y(2)33209 50535 y Fq(\))654 b Fo(b)-68 b(e)654 b(two)g(c)-68 b(ohomolo)g(gy)655 b(classes)1102 52410 y(c)-68 b(onne)g(cte)g(d)521 b(by)g(an)f(admissible)i(curve)f Fq(\000)p Fo(,)532 b(wher)-68 b(e)521 b Fp(a)p Fq(\()p Fp(c)28293 51720 y Fv(\(1\))28293 52764 y(1)29718 52410 y Fq(\))451 b Fm(\024)g Fp(c)32745 51720 y Fv(\(1\))32745 52764 y(2)34622 52410 y Fm(\024)g Fp(b)p Fq(\()p Fp(c)37768 51720 y Fv(\(1\))37768 52764 y(1)39194 52410 y Fq(\))p Fo(,)531 b Fp(a)p Fq(\()p Fp(c)42443 51720 y Fv(\()p Fk(k)22 b Fv(\))42443 52764 y(1)43927 52410 y Fq(\))450 b Fm(\024)h Fp(c)46953 51720 y Fv(\()p Fk(k)22 b Fv(\))46953 52764 y(2)48888 52410 y Fm(\024)1102 54284 y Fp(b)p Fq(\()p Fp(c)2764 53594 y Fv(\()p Fk(k)g Fv(\))2764 54639 y(1)4248 54284 y Fq(\))p Fo(,)609 b(and)9119 53949 y Fq(~)8469 54284 y Fm(M)p Fq(\()p Fp(c)11156 53802 y Fv(\(1\))12581 54284 y Fq(\))p Fo(,)14765 53949 y Fq(~)14115 54284 y Fm(M)p Fq(\()p Fp(c)16802 53802 y Fv(\(2\))18228 54284 y Fq(\))563 b Fm(\032)21053 53949 y Fq(~)20905 54284 y(\006)p Fo(.)934 b(L)-68 b(et)584 b Fp(\021)26266 54483 y Fv(1)26861 54284 y Fo(,)610 b Fp(\021)28538 54483 y Fk(k)29774 54284 y Fo(b)-68 b(e)583 b(two)f(close)-68 b(d)584 b(one)e(forms)i(such)f(that)1102 55951 y Fq([)p Fp(\021)2131 56150 y Fv(1)2727 55951 y Fq(])464 b(=)g Fp(c)5632 55469 y Fv(\(1\))7058 55951 y Fo(,)542 b Fq([)p Fp(\021)9036 56150 y Fk(k)9689 55951 y Fq(])464 b(=)h Fp(c)12595 55469 y Fv(\()p Fk(k)22 b Fv(\))14078 55951 y Fo(.)769 b(Then)527 b(ther)-68 b(e)529 b(exists)g(a)f(c)-68 b(omp)g(osition)529 b(of)f(\014nite)g Fp(U)145 b Fo(-step)528 b(forms)h Fp(\026)p Fq(\()p Fp(t)p Fq(\))1102 57545 y Fo(such)476 b(that)f Fp(\026)p Fq(\()p Fp(t)p Fq(\))369 b(=)g Fp(\021)11584 57744 y Fv(1)12654 57545 y Fo(for)476 b Fp(t)369 b Fm(\024)g Fq(0)475 b Fo(and)g Fp(\026)p Fq(\()p Fp(t)p Fq(\))368 b(=)i Fp(\021)25491 57744 y Fk(k)26619 57545 y Fo(for)476 b Fp(t)368 b Fm(\025)h Fp(\034)31572 57744 y Fk(k)32521 57545 y Fq(+)295 b(1)p Fo(.)1102 59921 y Fr(Pro)42 b(of)p Fq(.)576 b(Since)399 b(\010)g(is)f(an)g(area-preserving)j(t)-37 b(wist)397 b(map)i(when)g(it)e(is)i(restricted)g(on)f(the)g(cylinder,)1102 61699 y(b)-37 b(y)512 b(the)f(h)-37 b(yp)37 b(othesis)512 b(\(H2\),)528 b(there)512 b(is)f(some)h Fp(c)f Fq(with)h Fp(c)28602 61009 y Fv(\(1\))28602 62054 y(1)30511 61699 y Fp(<)483 b(c)32602 61898 y Fv(1)33681 61699 y Fp(<)h(c)35773 61009 y Fv(\()p Fk(k)22 b Fv(\))35773 62054 y(1)37257 61699 y Fq(,)528 b Fp(c)38729 61898 y Fv(2)39808 61699 y Fq(=)484 b(0)511 b(suc)-37 b(h)512 b(that)f(its)1102 63293 y(semi-static)488 b(minimal)g(orbit)g(set)f(consists)h(of)g (single)g Fp(m)p Fq(-p)37 b(erio)g(dic)489 b(orbit)f(with)g Fp(m)444 b(>)g Fq(1.)726 b(Th)-37 b(us,)1102 64887 y(for)457 b(eac)-37 b(h)458 b Fp(s)392 b Fm(2)g Fl(T)p Fq(,)459 b Fm(N)11249 65086 y Fk(s)12274 64887 y Fq(consists)d(of)h(sev)-37 b(eral)458 b(p)37 b(oin)-37 b(ts,)460 b Fm(N)28555 65086 y Fk(s)29123 64887 y Fq(\()p Fp(c)p Fq(\))391 b(=)h Fm([f)p Fp(q)34691 65086 y Fk(i)35133 64887 y Fq(\()p Fp(s)p Fq(\))p Fm(g)p Fq(.)631 b(Consequen)-37 b(tly)-111 b(,)462 b(there)1102 66481 y(exist)443 b Fp(\016)420 b(>)369 b Fq(0,)443 b(and)g(0)369 b Fp(<)g(s)13775 66680 y Fv(1)14740 66481 y Fp(<)g(s)16765 66680 y Fv(2)17729 66481 y Fp(<)g Fq(1)443 b(suc)-37 b(h)444 b(that)12765 68452 y Fi(\020)13854 69926 y Fm([)295 b(f)p Fp(q)16292 70125 y Fk(i)16734 69926 y Fq(\()p Fp(s)17874 70125 y Fv(1)18469 69926 y Fq(\))p Fm(g)g Fq(+)h(3)p Fp(\016)22578 68452 y Fi(\021)23667 69926 y Fm(\\)24848 68452 y Fi(\020)25936 69926 y Fm([)f(f)p Fp(q)28374 70125 y Fk(i)28816 69926 y Fq(\()p Fp(s)29956 70125 y Fv(2)30551 69926 y Fq(\))p Fm(g)g Fq(+)h(3)p Fp(\016)34660 68452 y Fi(\021)35823 69926 y Fq(=)369 b Fl(?)1102 73371 y Fq(and)554 b(there)g(exists)g Fp(\017)g(>)g Fq(0)g(suc)-37 b(h)554 b(that)f(0)h Fp(<)g(s)24506 73570 y Fv(1)25471 73371 y Fm(\000)369 b Fp(\017)554 b(<)g(s)30176 73570 y Fv(1)31141 73371 y Fq(+)369 b Fp(\017)554 b(<)g(s)35846 73570 y Fv(2)36811 73371 y Fm(\000)369 b Fp(\017)554 b(<)g(s)41516 73570 y Fv(2)42481 73371 y Fq(+)369 b Fp(\017)554 b(<)g Fq(1)g(and)1102 74965 y Fm([f)p Fp(q)3245 75164 y Fk(i)3687 74965 y Fq(\()p Fp(s)p Fq(\))p Fm(g)368 b(\032)h([f)p Fp(q)9921 75164 y Fk(i)10363 74965 y Fq(\()p Fp(s)11503 75164 y Fk(j)12062 74965 y Fq(\))p Fm(g)295 b Fq(+)15026 74442 y Fv(1)p 15026 74659 530 54 v 15026 75423 a(2)15715 74965 y Fp(\016)493 b Fq(when)444 b Fm(j)p Fp(s)295 b Fm(\000)g Fp(s)23505 75164 y Fk(j)24065 74965 y Fm(j)368 b Fp(<)i(\017)443 b Fq(for)g Fp(j)i Fq(=)369 b(1)p Fp(;)221 b Fq(2.)p eop %%Page: 39 39 39 38 bop 48792 2239 a Ft(39)2696 5427 y Fq(Let)547 b Fp(\021)593 b Fq(b)37 b(e)546 b(an)h(an)-37 b(y)546 b(exact)h(1-form,) 572 b(w)-37 b(e)547 b(claim)g(there)f(exists)g(a)h Fp(U)145 b Fq(-step)545 b(form)i Fp(\027)630 b Fq(suc)-37 b(h)547 b(that)1102 7021 y Fp(\027)85 b Fq(\()p Fp(t)p Fq(\))536 b(=)h(0)544 b(for)g Fp(t)537 b Fm(\024)g Fq(0)544 b(and)g Fp(\027)85 b Fq(\()p Fp(t)p Fq(\))535 b(=)i Fp(\021)591 b Fq(for)544 b Fp(t)537 b Fm(\025)g Fq(1,)569 b(where)545 b Fp(U)689 b Fq(is)543 b(a)h(neigh)-37 b(b)37 b(orho)g(o)g(d)545 b(of)e Fm(N)196 b Fq(\()p Fp(c)p Fq(\))536 b(=)1102 8615 y Fm([)1988 8814 y Fk(s)p Fg(2)p Fc(T)3994 8615 y Fm([)4880 8814 y Fk(i)5424 8615 y Fm(f)p Fp(q)6681 8814 y Fk(i)7123 8615 y Fq(\()p Fp(s)p Fq(\))p Fm(g)p Fq(.)558 b(Let)346 b Fp(F)184 b Fq(:)543 b Fp(M)513 b Fm(!)370 b Fl(R)346 b Fq(b)37 b(e)347 b(the)f(function)h(suc)-37 b(h)346 b(that)g Fp(\021)416 b Fq(=)369 b Fp(dF)184 b Fq(.)560 b(Let)346 b Fp(\025)42294 8814 y Fk(\016)42877 8615 y Fq(\()p Fp(q)48 b Fq(\):)541 b Fp(M)514 b Fm(!)369 b Fl(R)1102 10209 y Fq(b)37 b(e)540 b(a)g(smo)37 b(oth)540 b(function)f Fp(\025)15076 10408 y Fk(\016)16190 10209 y Fq(=)530 b(1)540 b(when)g Fm(k)p Fp(q)48 b Fm(k)531 b(\024)g Fp(\016)50 b Fq(,)564 b(0)531 b Fp(<)f(\025)31692 10408 y Fk(\016)32806 10209 y Fp(<)g Fq(1)540 b(when)h Fp(\016)581 b(<)530 b Fm(k)p Fp(q)48 b Fm(k)531 b Fp(<)g Fq(2)p Fp(\016)590 b Fq(and)1102 11803 y Fp(\025)1877 12002 y Fk(\016)2829 11803 y Fq(=)369 b(0)443 b(when)h Fm(k)p Fp(q)48 b Fm(k)369 b(\025)g Fq(2)p Fp(\016)50 b Fq(.)591 b(Let)6774 15885 y Fp(F)7812 15336 y Fg(\003)8792 15885 y Fq(=)10195 14410 y Fi(\020)10988 15885 y Fq(1)295 b Fm(\000)13941 14224 y Fk(k)13276 14623 y Fi(X)13375 17452 y Fk(i)p Fv(=1)15416 15885 y Fp(\025)16191 16084 y Fk(\016)16774 15885 y Fq(\()p Fp(q)342 b Fm(\000)295 b Fp(q)20147 16084 y Fk(i)20590 15885 y Fq(\()p Fp(s)21730 16084 y Fv(1)22325 15885 y Fq(\)\))23359 14410 y Fi(\021)24151 15885 y Fp(F)37 b(;)28587 15549 y Fq(~)28289 15885 y Fp(F)554 b Fq(=)31099 14410 y Fi(\020)32780 14224 y Fk(k)32114 14623 y Fi(X)32213 17452 y Fk(i)p Fv(=1)34254 15885 y Fp(\025)35029 16084 y Fk(\016)35612 15885 y Fq(\()p Fp(q)342 b Fm(\000)296 b Fp(q)38986 16084 y Fk(i)39428 15885 y Fq(\()p Fp(s)40568 16084 y Fv(1)41163 15885 y Fq(\)\))42197 14410 y Fi(\021)42989 15885 y Fp(F)37 b(;)1102 20118 y Fq(ob)-37 b(viously)-111 b(,)510 b(supp\()p Fp(dF)12285 19636 y Fg(\003)12897 20118 y Fq(\))330 b Fm(\\)g Fq(\()p Fm([f)p Fp(q)17620 20317 y Fk(i)18062 20118 y Fq(\()p Fp(s)19202 20317 y Fv(1)19797 20118 y Fq(\))p Fm(g)g Fq(+)h(2)p Fp(\016)50 b Fq(\))458 b(=)g Fl(?)p Fq(,)510 b(supp\()p Fp(d)32597 19783 y Fq(~)32300 20118 y Fp(F)184 b Fq(\))330 b Fm(\\)h Fq(\()p Fm([f)p Fp(q)38062 20317 y Fk(i)38503 20118 y Fq(\()p Fp(s)39643 20317 y Fv(2)40238 20118 y Fq(\))p Fm(g)g Fq(+)g(2)p Fp(\016)50 b Fq(\))458 b(=)f Fl(?)p Fq(.)751 b(If)1102 21712 y(w)-37 b(e)444 b(c)-37 b(ho)37 b(ose)14020 23306 y Fp(\027)453 b Fq(=)370 b Fp(\032)p Fq(\()p Fp(t)294 b Fm(\000)h Fp(s)20461 23505 y Fv(1)21352 23306 y Fq(+)g Fp(\017)p Fq(\))p Fp(dF)25465 22758 y Fg(\003)26372 23306 y Fq(+)g Fp(\032)p Fq(\()p Fp(t)g Fm(\000)g Fp(s)31630 23505 y Fv(2)32521 23306 y Fq(+)g Fp(\017)p Fq(\))p Fp(d)35894 22971 y Fq(~)35596 23306 y Fp(F)185 b(;)1102 25645 y Fq(where)387 b Fp(\032)369 b Fq(=)g(0)386 b(for)g Fp(t)369 b Fm(\024)g Fq(0,)397 b(0)370 b Fp(<)f(\032)g(<)g Fq(1)386 b(for)g(0)369 b Fp(<)g(t)g(<)g Fq(2)p Fp(\017)386 b Fq(and)g Fp(\032)368 b Fq(=)i(1)385 b(for)i(all)e Fp(t)369 b Fm(\025)g Fq(2)p Fp(\017)p Fq(,)398 b(then)385 b Fp(\027)85 b Fq(\()p Fp(t)p Fq(\))368 b(=)h(0)1102 27239 y(for)429 b Fp(t)369 b Fm(\024)g Fq(0)429 b(and)f Fp(\027)85 b Fq(\()p Fp(t)p Fq(\))368 b(=)h Fp(dF)614 b Fq(for)429 b Fp(t)368 b Fm(\025)h Fq(1.)586 b(Let)429 b Fp(U)514 b Fq(=)369 b Fm([)27180 27438 y Fk(j)54 b Fv(=1)p Fk(;)p Fv(2)29929 27239 y Fq(\()p Fm([f)p Fp(q)32589 27438 y Fk(i)33031 27239 y Fq(\()p Fp(s)34171 27438 y Fk(j)34730 27239 y Fq(\))p Fm(g)266 b Fq(+)h(2)p Fp(\016)50 b Fq(\))267 b Fm(\002)g Fq([)p Fp(s)41857 27438 y Fk(j)42683 27239 y Fm(\000)f Fp(\017;)221 b(s)45734 27438 y Fk(j)46561 27239 y Fq(+)267 b Fp(\017)p Fq(])g Fm([)1102 28833 y Fp(M)440 b Fm(\002)295 b Fq(\([0)p Fp(;)221 b(s)6922 29032 y Fv(1)7813 28833 y Fm(\000)296 b Fp(\017)p Fq(])f Fm([)g Fq([)p Fp(s)12518 29032 y Fv(1)13409 28833 y Fq(+)g Fp(\017;)221 b(s)16489 29032 y Fv(2)17380 28833 y Fm(\000)296 b Fp(\017)p Fq(])f Fm([)g Fq([)p Fp(s)22085 29032 y Fv(2)22975 28833 y Fq(+)h Fp(\017;)221 b Fq(1]\),)443 b(then)g Fp(d\027)85 b Fm(j)32622 29032 y Fk(U)33880 28833 y Fq(=)370 b(0.)2696 31132 y(Since)387 b(b)37 b(oth)385 b([)p Fp(\021)10216 31331 y Fv(1)10812 31132 y Fq(])h(and)g([)p Fp(\021)15122 31331 y Fv(2)15717 31132 y Fq(])g(are)h Fp(C)95 b Fq(-equiv)-74 b(alen)-37 b(t)387 b(to)e Fp(c)p Fq(,)397 b(there)387 b(are)g(t)-37 b(w)g(o)386 b(comp)37 b(osition)386 b(of)g Fp(U)145 b Fq(-step)1102 32726 y(forms)443 b Fp(\027)5422 32925 y Fv(1)6019 32726 y Fq(,)f Fp(\027)7486 32925 y Fv(2)8525 32726 y Fq(suc)-37 b(h)443 b(that)13602 35897 y Fp(\021)14262 36096 y Fv(1)15152 35897 y Fq(+)296 b Fp(\027)17137 36096 y Fv(1)17733 35897 y Fq(\()p Fp(t)p Fq(\))368 b(=)h Fm(h)p Fp(c;)221 b(dq)48 b Fm(i)294 b Fq(+)i Fp(dF)27716 36096 y Fv(1)28312 35897 y Fp(;)2657 b(t)369 b Fm(\025)g Fp(\034)34170 36096 y Fv(1)34765 35897 y Fq(;)13602 37956 y Fp(\027)14258 38155 y Fv(2)14854 37956 y Fq(\()p Fp(t)p Fq(\))f(=)h(0)p Fp(;)12167 b(t)369 b Fm(\024)g Fp(\034)34170 38155 y Fv(1)35060 37956 y Fq(+)296 b(1)p Fp(;)13602 40016 y Fm(h)p Fp(c;)221 b(dq)48 b Fm(i)294 b Fq(+)i Fp(\027)19412 40215 y Fv(2)20008 40016 y Fq(\()p Fp(t)p Fq(\))368 b(=)h Fp(\021)23952 40215 y Fv(2)24842 40016 y Fq(+)296 b Fp(dF)27716 40215 y Fv(2)28312 40016 y Fp(;)2657 b(t)369 b Fm(\025)g Fp(\034)34170 40215 y Fv(2)34765 40016 y Fp(:)1102 42854 y Fq(By)435 b(the)f(demonstration)h(ab)37 b(o)-37 b(v)g(e,)437 b(there)e(is)g(a)f Fp(U)145 b Fq(-step)434 b(form)h Fp(\027)519 b Fq(suc)-37 b(h)435 b(that)f Fp(\027)85 b Fq(\()p Fp(t)p Fq(\))368 b(=)h Fm(\000)p Fp(d)p Fq(\()p Fp(F)45768 43053 y Fv(1)46643 42854 y Fq(+)279 b Fp(F)48809 43053 y Fv(2)49405 42854 y Fq(\))1102 44448 y(when)425 b Fp(t)368 b Fm(\025)h Fq(1.)585 b(Clearly)-111 b(,)429 b(the)423 b(1-form)h Fp(\026)369 b Fq(=)g(\()p Fm(\000)p Fp(\034)24477 44647 y Fv(1)25073 44448 y Fq(\))25590 43966 y Fg(\003)26200 44448 y Fp(\027)341 b Fq(+)257 b Fp(\027)29143 44647 y Fv(1)29996 44448 y Fq(+)g Fp(\027)31942 44647 y Fv(2)32962 44448 y Fq(is)423 b(what)h(w)-37 b(e)424 b(are)h(lo)37 b(oking)424 b(for.)584 b Fb(\003)2696 46747 y Fq(The)463 b(follo)-37 b(wing)462 b(w)-37 b(ork)463 b(in)e(this)h(section)g(is)f (to)h(join)f(the)h(orbit)f(crossing)i(the)f(gaps)f(smo)37 b(othly)1102 48341 y(with)387 b(the)g(orbit)g(constructed)h(via)f (Arnold's)h(mec)-37 b(hanism.)572 b(W)-111 b(e)388 b(shall)f(mak)-37 b(e)388 b(use)f(of)g(some)h(ideas)1102 49935 y(dev)-37 b(elop)37 b(ed)372 b(in)f([Bs])f(and)h(in)g([BCV],)g(it)f(is)g(sho)-37 b(w)g(ed)372 b(that)e(the)g(di\013usion)h(orbits)f(in)h(sev)-37 b(eral)372 b(exam-)1102 51529 y(ples,)471 b(constructed)465 b(b)-37 b(y)465 b(transition)g(c)-37 b(hain,)471 b(are)465 b(actually)g(the)g(orbits)g(whic)-37 b(h)465 b(lo)37 b(cally)466 b(minimize)1102 53123 y(the)443 b(Lagrange)h(action.)2696 55421 y(Let)434 b(us)g(consider)h(the)f(barrier)h(function)f(of)g (those)g(cohomology)h(classes)g(corresp)37 b(onding)435 b(to)1102 57015 y(an)466 b(in)-37 b(v)-74 b(arian)-37 b(t)466 b(circle)g(\000)12892 57214 y Fk(c)13899 57015 y Fq(on)g(the)f(cylinder.)659 b(In)465 b(this)g(case,)471 b Fm(M)p Fq(\()p Fp(c)p Fq(\))p Fm(j)34797 57214 y Fk(t)p Fv(=0)37016 57015 y Fm(\022)406 b Fq(\000)39285 57214 y Fk(c)40292 57015 y Fq(and)466 b Fp(d)43589 57214 y Fk(c)44130 57015 y Fq(\()p Fp(\030)61 b(;)221 b(\030)46521 56533 y Fg(0)46894 57015 y Fq(\))406 b(=)h(0)1102 58609 y(for)444 b(all)e Fp(\030)61 b(;)221 b(\030)6854 58127 y Fg(0)7597 58609 y Fm(2)369 b Fp(\031)48 b Fq(\(\000)11004 58808 y Fk(c)11544 58609 y Fq(\).)590 b(Th)-37 b(us)11131 61381 y Fp(B)12206 60832 y Fg(\003)12139 61709 y Fk(c)12816 61381 y Fq(\()p Fp(q)48 b Fq(\))368 b(=)1455 b(min)16113 62300 y Fk(\030)32 b(;\021)h Fg(2M)p Fv(\()p Fk(c)p Fv(\))20794 61381 y Fm(f)p Fp(h)22223 60832 y Fg(1)22223 61709 y Fk(c)23349 61381 y Fq(\()p Fp(\030)61 b(;)221 b(q)48 b Fq(\))294 b(+)i Fp(h)28644 60832 y Fg(1)28644 61709 y Fk(c)29770 61381 y Fq(\()p Fp(q)48 b(;)221 b(\021)48 b Fq(\))293 b Fm(\000)j Fp(h)35130 60832 y Fg(1)35130 61709 y Fk(c)36255 61381 y Fq(\()p Fp(\030)61 b(;)221 b(\021)48 b Fq(\))p Fm(g)14859 64193 y Fq(=)p Fp(h)16657 63645 y Fg(1)16657 64522 y Fk(c)17783 64193 y Fq(\()p Fp(\030)61 b(;)221 b(q)48 b Fq(\))294 b(+)i Fp(h)23078 63645 y Fg(1)23078 64522 y Fk(c)24203 64193 y Fq(\()p Fp(q)48 b(;)221 b(\030)61 b Fq(\))p Fp(;)2878 b Fm(8)443 b Fp(\030)430 b Fm(2)369 b Fp(\031)48 b Fq(\(\000)35956 64392 y Fk(c)36496 64193 y Fq(\))p Fp(:)9145 b Fq(\(5.14\))2696 67031 y(Under)649 b(the)f(h)-37 b(yp)37 b(othesis)648 b(\(H1\),)699 b(the)648 b(set)f Fm(f)p Fp(B)26443 66549 y Fg(\003)26376 67459 y Fk(c)p Fv(\()p Fk(\033)33 b Fv(\))29107 67031 y Fq(=)711 b(0)p Fm(g)432 b(\\)g Fl(T)f Fm(\002)i Fq(\()p Fp(a;)221 b Fq(2)p Fp(\031)480 b Fm(\000)432 b Fp(a)p Fq(\))647 b(consists)h(of)1102 68747 y(discrete)382 b(p)37 b(oin)-37 b(ts)381 b(for)g(all)g Fp(\033)417 b Fm(2)368 b Fl(S)p Fq(.)570 b(Th)-37 b(us,)393 b(for)382 b(an)-37 b(y)381 b(giv)-37 b(en)381 b Fp(\033)417 b Fm(2)369 b Fl(S)380 b Fq(and)h(an)-37 b(y)382 b Fp(\017)369 b(>)g Fq(0,)393 b(there)382 b(are)f(\014nite)1102 70341 y(and)515 b(m)-37 b(utual)514 b(disjoin)-37 b(t)515 b(balls)f Fm(B)17220 70540 y Fk(\017)17730 70341 y Fq(\()p Fp(q)18840 70540 y Fk(i)19282 70341 y Fq(\))g(and)g Fp(\016)540 b Fq(=)488 b Fp(\016)50 b Fq(\()p Fp(\033)-26 b(;)221 b(\017)p Fq(\))489 b Fp(>)g Fq(0)515 b(suc)-37 b(h)515 b(that)f Fm([B)40120 70540 y Fk(\017)40629 70341 y Fq(\()p Fp(q)41739 70540 y Fk(i)42181 70341 y Fq(\))488 b Fm(\033)g(f)p Fp(B)46446 69859 y Fg(\003)46379 70769 y Fk(c)p Fv(\()p Fk(\033)33 b Fv(\))48888 70341 y Fq(=)1102 72130 y(0)p Fm(g)296 b(\\)f Fl(T)f Fm(\002)i Fq(\()p Fp(a;)221 b Fq(2)p Fp(\031)343 b Fm(\000)295 b Fp(a)p Fq(\))443 b(and)9622 74902 y(min)p Fm(f)p Fp(B)13575 74354 y Fg(\003)13508 75264 y Fk(c)p Fv(\()p Fk(\033)33 b Fv(\))15528 74902 y Fq(\()p Fp(q)48 b Fq(\))368 b(:)h Fp(q)416 b Fm(2)369 b Fp(@)74 b Fm(B)22224 75101 y Fk(\017)22733 74902 y Fq(\()p Fp(q)23843 75101 y Fk(i)24285 74902 y Fq(\))p Fp(;)221 b Fm(8)p Fp(i)p Fm(g)369 b(\025)g Fq(2)p Fp(\016)-24 b(;)2879 b(B)34576 74354 y Fg(\003)34509 75264 y Fk(c)p Fv(\()p Fk(\033)33 b Fv(\))36529 74902 y Fq(\()p Fp(q)37639 75101 y Fk(i)38081 74902 y Fq(\))368 b(=)i(0)p Fp(:)p eop %%Page: 40 40 40 39 bop 1102 2239 a Ft(40)1102 5427 y Fq(In)463 b(other)f(w)-37 b(ords,)469 b(as)462 b(a)h(function)g(of)f Fp(q)48 b Fq(.)649 b Fp(B)22746 4945 y Fg(\003)22679 5855 y Fk(c)p Fv(\()p Fk(\033)33 b Fv(\))25162 5427 y Fq(reac)-37 b(hes)464 b(its)e(minim)-37 b(um)462 b(in)h Fm(f)p Fp(a)402 b Fm(\024)g Fp(q)43003 5626 y Fv(2)44001 5427 y Fm(\024)g Fq(2)p Fp(\031)356 b Fm(\000)309 b Fp(a)p Fm(g)1102 7143 y Fq(at)443 b(these)g(discrete)h(p)37 b(oin)-37 b(ts)14910 9830 y(min)13631 10749 y Fk(q)33 b Fg(2)p Fk(@)57 b Fg(B)16192 10882 y Fd(\017)16666 10749 y Fv(\()p Fk(q)17568 10882 y Fd(i)17989 10749 y Fv(\))18625 9830 y Fp(B)19700 9282 y Fg(\003)19633 10192 y Fk(c)p Fv(\()p Fk(\033)33 b Fv(\))21653 9830 y Fq(\()p Fp(q)48 b Fq(\))294 b Fm(\000)1262 b Fq(min)24950 10749 y Fk(q)33 b Fg(2B)26887 10882 y Fd(\017)27361 10749 y Fv(\()p Fk(q)28263 10882 y Fd(i)28684 10749 y Fv(\))29320 9830 y Fp(B)30395 9282 y Fg(\003)30328 10192 y Fk(c)p Fv(\()p Fk(\033)g Fv(\))32348 9830 y Fq(\()p Fp(q)48 b Fq(\))368 b Fm(\025)h Fq(2)p Fp(\016)-24 b(:)9135 b Fq(\(5.15\))1102 13337 y(Recall)465 b(for)f(eac)-37 b(h)465 b Fp(z)d Fm(2)404 b Fl(T)p Fq(,)469 b(there)464 b(is)g(unique)g Fp(z)24017 13536 y Fk(\033)25136 13337 y Fm(2)404 b Fp(\031)48 b Fq(\(\000)28578 13536 y Fk(\033)29293 13337 y Fq(\))463 b(suc)-37 b(h)465 b(that)e Fp(z)36808 13536 y Fk(\033)37927 13337 y Fq(=)404 b(\()p Fp(z)58 b(;)221 b(q)41740 13536 y Fv(2\000)42926 13669 y Fd(\033)43625 13337 y Fq(\()p Fp(z)58 b Fq(\)\).)652 b(F)-111 b(rom)1102 14931 y(\(5.14\),)506 b(\(5.15\))494 b(and)f(the)h(H\177)-664 b(older)495 b(con)-37 b(tin)g(uit)g(y)493 b(guaran)-37 b(teed)495 b(b)-37 b(y)494 b(Lemma)g(4.3)g(w)-37 b(e)494 b(\014nd)g(that)f(for)1102 16525 y(eac)-37 b(h)444 b Fp(z)427 b Fm(2)369 b Fl(T)15006 19213 y Fq(min)13726 20132 y Fk(q)33 b Fg(2)p Fk(@)57 b Fg(B)16287 20265 y Fd(\017)16761 20132 y Fv(\()p Fk(q)17663 20265 y Fd(i)18084 20132 y Fv(\))18499 19213 y Fp(h)19264 18664 y Fg(1)19264 19574 y Fk(c)p Fv(\()p Fk(\033)33 b Fv(\))21285 19213 y Fq(\()p Fp(z)22420 19412 y Fk(\033)23135 19213 y Fp(;)221 b(q)48 b Fq(\))294 b(+)i Fp(h)27271 18664 y Fg(1)27271 19574 y Fk(c)p Fv(\()p Fk(\033)28810 19309 y Fe(0)29169 19574 y Fv(\))29651 19213 y Fq(\()p Fp(q)48 b(;)221 b(z)32017 19412 y Fk(\033)32666 19147 y Fe(0)33091 19213 y Fq(\))p Fm(\000)15318 22784 y Fq(min)14351 23702 y Fk(q)33 b Fg(2B)16288 23835 y Fd(\017)16761 23702 y Fv(\()p Fk(q)17663 23835 y Fd(i)18084 23702 y Fv(\))18499 22784 y Fp(h)19264 22235 y Fg(1)19264 23145 y Fk(c)p Fv(\()p Fk(\033)g Fv(\))21285 22784 y Fq(\()p Fp(z)22420 22983 y Fk(\033)23135 22784 y Fp(;)221 b(q)48 b Fq(\))294 b(+)i Fp(h)27271 22235 y Fg(1)27271 23145 y Fk(c)p Fv(\()p Fk(\033)28810 22879 y Fe(0)29169 23145 y Fv(\))29651 22784 y Fq(\()p Fp(q)48 b(;)221 b(z)32017 22983 y Fk(\033)32666 22717 y Fe(0)33091 22784 y Fq(\))368 b Fm(\025)35538 21885 y Fq(3)p 35538 22478 665 54 v 35538 23695 a(2)36361 22784 y Fp(\016)-24 b(:)9231 b Fq(\(5.16\))1102 26291 y(As)600 b(these)g(functions)g(dep)37 b(end)600 b(on)g(the)g(c)-37 b(hoice)601 b(of)f(closed)g(1-form)g Fp(\021)36795 26490 y Fk(c)37336 26291 y Fq(,)639 b(to)599 b(obtain)h(\(5.16\))f(w)-37 b(e)1102 27885 y(c)g(ho)37 b(ose)428 b Fp(\021)5960 28084 y Fk(c)6870 27885 y Fq(=)369 b Fm(h)p Fp(c)p Fq(\()p Fp(\033)48 b Fq(\))p Fp(;)466 b Fq(_)-614 b Fp(q)46 b Fm(i)p Fq(.)585 b(In)426 b(general,)431 b Fp(h)21249 27402 y Fg(1)21249 28312 y Fk(c)p Fv(\()p Fk(\033)33 b Fv(\))23270 27885 y Fq(\()p Fp(z)24405 28084 y Fk(\033)25119 27885 y Fp(;)221 b(q)48 b Fq(\))263 b(+)g Fp(h)29191 27402 y Fg(1)29191 28312 y Fk(c)p Fv(\()p Fk(\033)30730 28047 y Fe(0)31089 28312 y Fv(\))31571 27885 y Fq(\()p Fp(q)48 b(;)221 b(z)33937 28084 y Fk(\033)34586 27818 y Fe(0)35011 27885 y Fq(\))426 b(is)g(also)h(the)g(function)f(of)h Fp(z)58 b Fq(,)1102 29676 y(but)342 b(its)f(v)-74 b(ariation)342 b(o)-37 b(v)g(er)343 b Fp(z)427 b Fm(2)369 b Fl(T)341 b Fq(is)h(v)-37 b(ery)343 b(small)e(if)h Fp(\033)26284 29194 y Fg(0)26998 29676 y Fq(is)g(su\016cien)-37 b(tly)342 b(close)h(to)e Fp(\033)48 b Fq(,)362 b(b)37 b(ecasue)343 b Fp(q)46328 29875 y Fv(2\000)47514 30008 y Fd(\033)48212 29676 y Fq(\()p Fp(z)58 b Fq(\))1102 31270 y(has)3700 30747 y Fv(1)p 3700 30965 530 54 v 3700 31728 a(2)4389 31270 y Fq(-H\177)-664 b(older)513 b(con)-37 b(tin)g(uit)g(y)512 b(in)g Fp(\033)48 b Fq(.)798 b(Since)512 b Fl(S)g Fq(is)g(compact,)530 b(there)513 b(exists)f Fp(\016)535 b Fq(=)484 b Fp(\016)50 b Fq(\()p Fp(\017)p Fq(\),)530 b Fp(\017)44616 31469 y Fv(1)45696 31270 y Fq(=)485 b Fp(\017)47753 31469 y Fv(1)48349 31270 y Fq(\()p Fp(\017)p Fq(\))1102 32864 y(indep)37 b(enden)-37 b(t)444 b(of)f Fp(\033)48 b Fq(,)442 b(suc)-37 b(h)443 b(that)f(\(5.15\))h(and)g(\(5.16\))f(hold)h(if)g Fm(j)p Fp(\033)343 b Fm(\000)295 b Fp(\033)35669 32382 y Fg(0)36041 32864 y Fm(j)369 b(\024)g Fp(\017)38720 33063 y Fv(1)39317 32864 y Fq(.)2696 35144 y(W)-111 b(e)352 b(sa)-37 b(y)351 b Fp(\033)7854 35343 y Fk(j)8764 35144 y Fq(is)g(link)-37 b(ed)351 b(with)f Fp(\033)17519 35343 y Fk(j)54 b Fv(+1)19775 35144 y Fq(b)-37 b(y)351 b(transition)f(torus)g (with)h(some)g(p)37 b(ersistency)352 b(if)e Fp(\033)45655 35343 y Fk(j)54 b Fv(+1)47929 35144 y Fm(2)368 b Fl(S)1102 36738 y Fq(is)443 b(so)g(close)h(to)e Fp(\033)9632 36937 y Fk(j)10635 36738 y Fq(suc)-37 b(h)443 b(that)18291 40100 y Fm(j)p Fp(c)19235 40299 y Fv(1)19831 40100 y Fq(\()p Fp(\033)21107 40299 y Fk(j)21666 40100 y Fq(\))295 b Fm(\000)g Fp(c)24381 40299 y Fv(1)24977 40100 y Fq(\()p Fp(\033)26253 40299 y Fk(j)54 b Fv(+1)28157 40100 y Fq(\))p Fm(j)369 b(\024)30973 39201 y Fq(1)p 30973 39794 665 54 v 30973 41011 a(4)31797 40100 y Fp(\016)-24 b(;)13795 b Fq(\(5.17\))1102 43236 y(and)413 b(\(5.16\))f(hold)h(where)h(w)-37 b(e)414 b(replace)g Fp(\033)460 b Fq(and)413 b Fp(\033)25175 42754 y Fg(0)25960 43236 y Fq(b)-37 b(y)413 b Fp(\033)28534 43435 y Fk(j)29506 43236 y Fq(and)g Fp(\033)32818 43435 y Fk(j)54 b Fv(+1)35135 43236 y Fq(resp)37 b(ectiv)-37 b(ely)-111 b(.)582 b(W)-111 b(e)414 b(sa)-37 b(y)413 b Fp(\033)48056 43435 y Fk(j)49028 43236 y Fq(is)1102 44830 y(link)-37 b(ed)434 b(with)e Fp(\033)8778 45029 y Fk(k)9864 44830 y Fq(b)-37 b(y)433 b(transition)g(c)-37 b(hain)433 b(with)f(some)i(p)37 b(ersistency)433 b(if)g(there)g(there)g (is)g(a)g(sequence)1102 46424 y Fp(\033)1861 46623 y Fk(j)2421 46424 y Fp(;)221 b(\033)3770 46623 y Fk(j)54 b Fv(+1)5675 46424 y Fp(;)221 b Fm(\001)g(\001)g(\001)444 b Fp(;)221 b(\033)9607 46623 y Fk(k)22 b Fg(\000)p Fv(1)11621 46424 y Fp(;)221 b(\033)12970 46623 y Fk(k)14174 46424 y Fq(in)551 b Fl(S)g Fq(suc)-37 b(h)551 b(that)f(for)h(eac)-37 b(h)551 b Fp(j)625 b Fm(\024)549 b Fp(i)f(<)h(k)592 b(\033)35833 46623 y Fk(i)36826 46424 y Fq(is)551 b(link)-37 b(ed)551 b(with)f Fp(\033)46181 46623 y Fk(i)p Fv(+1)48519 46424 y Fq(b)-37 b(y)1102 48018 y(transition)387 b(torus)h(with)f(some)h(p)37 b(ersistency)-111 b(.)573 b(T)-111 b(o)388 b(b)37 b(e)387 b(brief,)399 b(w)-37 b(e)388 b(shall)g(sa)-37 b(y)387 b(in)h(the)f(follo)-37 b(wing)388 b(that)1102 49612 y(they)376 b(are)g(link)-37 b(ed)376 b(b)-37 b(y)375 b(transition)g(torus)h(\(c) -37 b(hain\).)567 b(Note)376 b(that)f Fl(S)369 b Fm(\032)g Fq([)p Fp(A)35989 49130 y Fg(0)36362 49612 y Fp(;)221 b(B)38027 49130 y Fg(0)38399 49612 y Fq(])376 b(is)f(compact,)389 b(w)-37 b(e)377 b(can)1102 51206 y(\014nd)509 b(\014nite)f Fp(\033)8044 51405 y Fk(k)9177 51206 y Fm(2)478 b Fl(S)509 b Fq(\(0)478 b Fm(\024)h Fp(k)520 b Fm(\024)479 b Fp(K)95 b Fq(\))508 b(suc)-37 b(h)509 b(that)f(w)-37 b(e)509 b(ha)-37 b(v)g(e)510 b(one)f(of)g(the)f(follo)-37 b(wing)510 b(alternativ)-37 b(es)1102 52800 y(for)457 b(eac)-37 b(h)457 b Fp(k)432 b(<)391 b(K)95 b Fq(:)617 b(either)457 b Fp(\033)15448 52999 y Fk(k)16558 52800 y Fq(is)e(link)-37 b(ed)457 b(with)f Fp(\033)25629 52999 y Fk(k)22 b Fv(+1)28083 52800 y Fq(b)-37 b(y)457 b(transition)e(c)-37 b(hain,)460 b(or)d(\000)42379 52999 y Fk(\033)42995 53140 y Fd(k)44098 52800 y Fq(and)f(\000)47524 52999 y Fk(\033)48140 53140 y Fd(k)11 b Fj(+1)1102 54394 y Fq(mak)-37 b(e)554 b(up)f(the)g(b)37 b(oundary)554 b(of)f(an)g(ann)-37 b(ulus)553 b(of)g(Birkho\013)g (instabilit)-37 b(y)-111 b(,)581 b(i.e.)921 b(there)554 b(is)f(no)g(other)1102 55988 y(in)-37 b(v)-74 b(arian)-37 b(t)522 b(circle)g(b)37 b(et)-37 b(w)g(een)522 b(\000)16343 56187 y Fk(\033)16959 56328 y Fd(k)18126 55988 y Fq(and)f(\000)21617 56187 y Fk(\033)22233 56328 y Fd(k)11 b Fj(+1)24014 55988 y Fq(.)824 b(In)521 b(the)f(follo)-37 b(wing)522 b(w)-37 b(e)521 b(shall)g(use)g(\000)43419 56187 y Fk(i)44382 55988 y Fq(to)g(denote)1102 57582 y(\000)1932 57781 y Fk(\033)2548 57914 y Fd(i)3478 57582 y Fq(and)443 b(use)g Fp(z)8974 57781 y Fk(i)9859 57582 y Fq(to)f(denote)i Fp(z)16381 57781 y Fk(\033)16997 57914 y Fd(i)17483 57582 y Fq(.)2696 59862 y(Let)576 b(us)g(consider)g(a)g(sequence)h(of)f(in) -37 b(v)-74 b(arian)-37 b(t)576 b(circles)h(\000)31547 60061 y Fk(i)32565 59862 y Fq(\()p Fp(i)589 b Fq(=)h(0)p Fp(;)221 b Fq(1)p Fp(;)g Fm(\001)g(\001)g(\001)445 b Fp(;)221 b(`;)g(`)385 b Fq(+)f(1\))576 b(on)f(the)1102 61456 y(cylinder)445 b(\006)f(suc)-37 b(h)444 b(that)f(\000)14271 61655 y Fv(1)15311 61456 y Fq(is)g(link)-37 b(ed)445 b(with)e(\000)24416 61655 y Fk(`)25371 61456 y Fq(through)h(the)f (transition)h(c)-37 b(hain)444 b(\000)43082 61655 y Fv(2)43678 61456 y Fp(;)221 b Fm(\001)g(\001)g(\001)444 b Fp(;)221 b Fq(\000)47681 61655 y Fk(`)p Fg(\000)p Fv(1)49552 61456 y Fq(,)1102 63050 y(and)415 b(there)h(are)g(t)-37 b(w)g(o)415 b(ann)-37 b(ulus)415 b(of)g(Birkho\013)h(instabilit)-37 b(y)-111 b(,)420 b(one)415 b(has)g(\000)35289 63249 y Fv(0)36301 63050 y Fq(and)g(\000)39686 63249 y Fv(1)40697 63050 y Fq(as)g(its)f(b)37 b(oundary)-111 b(,)1102 64644 y(another)448 b(one)f(has)g(\000)11623 64843 y Fk(`)12582 64644 y Fq(and)g(\000)15999 64843 y Fk(`)p Fv(+1)18302 64644 y Fq(as)g(its)f(b)37 b(oundary)-111 b(.)604 b(By)447 b(the)g(construction)g(of)g(this)g(transition)1102 66238 y(c)-37 b(hain)482 b(w)-37 b(e)482 b(kno)-37 b(w)482 b(that)f(for)g(eac)-37 b(h)482 b(1)433 b Fm(\024)h Fp(i)e(<)h(`)482 b Fq(there)g(is)f Fp(x)29687 66437 y Fk(i)30562 66238 y Fm(2)433 b(f)p Fp(B)33620 65756 y Fg(\003)33553 66666 y Fk(c)p Fv(\()p Fk(\033)35059 66799 y Fd(i)35479 66666 y Fv(\))36393 66238 y Fq(=)h(0)p Fm(g)321 b(\\)f Fq(\()p Fp(a;)221 b Fq(2)p Fp(\031)369 b Fm(\000)321 b Fp(a)p Fq(\))481 b(suc)-37 b(h)1102 67954 y(that)442 b(for)i(an)-37 b(y)443 b Fp(z)427 b Fm(2)368 b Fl(T)14011 70641 y Fq(min)12674 71560 y Fk(q)33 b Fg(2)p Fk(@)57 b Fg(B)15235 71693 y Fd(\017)15709 71560 y Fv(\()p Fk(x)16726 71693 y Fd(i)17147 71560 y Fv(\))17562 70641 y Fp(h)18327 70093 y Fg(1)18327 71003 y Fk(c)p Fv(\()p Fk(\033)19833 71136 y Fd(i)20254 71003 y Fv(\))20735 70641 y Fq(\()p Fp(z)21870 70840 y Fk(i)22312 70641 y Fp(;)221 b(q)48 b Fq(\))295 b(+)g Fp(h)26448 70093 y Fg(1)26448 71003 y Fk(c)p Fv(\()p Fk(\033)27954 71136 y Fd(i)p Fj(+1)29509 71003 y Fv(\))29991 70641 y Fq(\()p Fp(q)48 b(;)221 b(z)32357 70840 y Fk(i)p Fv(+1)34143 70641 y Fq(\))p Fm(\000)14323 74212 y Fq(min)13298 75131 y Fk(q)33 b Fg(2B)15235 75264 y Fd(\017)15709 75131 y Fv(\()p Fk(x)16726 75264 y Fd(i)17147 75131 y Fv(\))17562 74212 y Fp(h)18327 73664 y Fg(1)18327 74573 y Fk(c)p Fv(\()p Fk(\033)19833 74706 y Fd(i)20254 74573 y Fv(\))20735 74212 y Fq(\()p Fp(z)21870 74411 y Fk(i)22312 74212 y Fp(;)221 b(q)48 b Fq(\))295 b(+)g Fp(h)26448 73664 y Fg(1)26448 74573 y Fk(c)p Fv(\()p Fk(\033)27954 74706 y Fd(i)p Fj(+1)29509 74573 y Fv(\))29991 74212 y Fq(\()p Fp(q)48 b(;)221 b(z)32357 74411 y Fk(i)p Fv(+1)34143 74212 y Fq(\))369 b Fm(\025)36590 73313 y Fq(3)p 36590 73907 V 36590 75123 a(2)37414 74212 y Fp(\016)-24 b(:)7809 b Fq(\(5.16i\))p eop %%Page: 41 41 41 40 bop 48792 2239 a Ft(41)1102 5427 y Fq(As)601 b(in)f([BCV],)h(let) f(us)g(consider)h(the)f(co)-37 b(v)g(ering)602 b(of)e Fl(T)29171 4945 y Fv(2)30367 5427 y Fq(giv)-37 b(en)601 b(b)-37 b(y)36490 5091 y(\026)35995 5427 y Fp(M)776 b Fq(=)631 b Fl(T)400 b Fm(\002)g Fl(R)p Fq(.)1063 b(F)-111 b(or)601 b(eac)-37 b(h)1102 7021 y Fp(x)1861 7220 y Fk(i)2915 7021 y Fq(w)g(e)612 b(iden)-37 b(tify)612 b(it)f(with)g(its)g(lift)f (in)i(the)f(region)h Fl(T)407 b Fm(\002)h Fq(\(0)p Fp(;)221 b Fq(2)p Fp(\031)48 b Fq(\))611 b(and)h(single)f(out)h(a)f(p)37 b(oin)-37 b(t)611 b(on)1102 8615 y(its)586 b(lift,)707 b(\026)-749 b Fp(x)6509 8814 y Fk(i)7560 8615 y Fq(=)609 b Fp(x)9961 8814 y Fk(i)10794 8615 y Fq(+)391 b(\(0)p Fp(;)221 b Fq(2)p Fp(i\031)48 b Fq(\),)622 b(w)-37 b(e)588 b(also)e(iden)-37 b(tify)587 b(eac)-37 b(h)587 b Fp(z)31059 8814 y Fk(i)32088 8615 y Fq(with)f(its)g(lift)g Fp(z)40118 8814 y Fk(i)40951 8615 y Fq(+)391 b(\(0)p Fp(;)221 b Fq(2)p Fp(i\031)48 b Fq(\).)1021 b(F)-111 b(or)1102 10209 y Fp(i)432 b Fm(2)h Fq(\(1)p Fp(;)221 b Fq(2)p Fp(;)g(:)g(:)g(:)h(;)f (`)322 b Fm(\000)f Fq(1\))480 b(w)-37 b(e)482 b(in)-37 b(tro)37 b(duce)482 b(a)f(smo)37 b(oth)480 b(function)h(\011)32677 10408 y Fk(i)33119 10209 y Fq(:)667 b Fl(T)320 b Fm(\002)h Fl(R)433 b Fm(!)g Fl(R)481 b Fq(whic)-37 b(h)482 b(v)-74 b(anishes)1102 11803 y(outside)443 b Fm(f)p Fp(q)417 b Fq(:)590 b Fm(j)p Fp(q)343 b Fm(\000)380 b Fq(\026)-748 b Fp(x)11711 12002 y Fk(i)12153 11803 y Fm(j)369 b(\024)g Fq(2)p Fp(\017)p Fm(g)443 b Fq(and)g(suc)-37 b(h)443 b(that)11971 14788 y Fm(r)p Fq(\011)14111 14987 y Fk(i)14553 14788 y Fq(\()p Fp(q)48 b Fq(\))368 b(=)h Fp(c)p Fq(\()p Fp(\033)19849 14987 y Fk(i)p Fv(+1)21636 14788 y Fq(\))294 b Fm(\000)i Fp(c)p Fq(\()p Fp(\033)25627 14987 y Fk(i)26068 14788 y Fq(\))2657 b Fm(8)p Fp(q)416 b Fq(:)812 b Fm(j)p Fp(q)343 b Fm(\000)380 b Fq(\026)-749 b Fp(x)35562 14987 y Fk(i)36005 14788 y Fm(j)369 b(\024)g Fp(\017:)7474 b Fq(\(5.18\))1102 17773 y(If)443 b(w)-37 b(e)443 b(set)18781 19367 y(\026)-693 b Fp(c)19327 19566 y Fk(i)19769 19367 y Fq(\()p Fp(q)48 b Fq(\))368 b(=)h Fp(c)p Fq(\()p Fp(\033)25065 19566 y Fk(i)25507 19367 y Fq(\))294 b(+)i Fm(r)p Fq(\011)29787 19566 y Fk(i)30229 19367 y Fq(\()p Fp(q)48 b Fq(\))p Fp(;)14254 b Fq(\(5.19\))1102 21800 y(then)5376 24785 y Fp(h)6141 24237 y Fg(1)6171 25114 y Fv(\026)-559 b Fk(c)6616 25247 y Fd(i)7266 24785 y Fq(\()p Fp(z)58 b(;)221 b(q)48 b Fq(\))295 b(+)g Fp(h)12595 24237 y Fg(1)12625 25114 y Fv(\026)-559 b Fk(c)13070 25247 y Fd(i)p Fj(+1)14692 24785 y Fq(\()p Fp(q)48 b(;)221 b(z)58 b Fq(\))368 b(=)i Fp(h)20169 24237 y Fg(1)20169 25147 y Fk(c)p Fv(\()p Fk(\033)21675 25280 y Fd(i)22095 25147 y Fv(\))22577 24785 y Fq(\()p Fp(z)58 b(;)221 b(q)48 b Fq(\))294 b(+)i Fp(h)27906 24237 y Fg(1)27906 25147 y Fk(c)p Fv(\()p Fk(\033)29412 25280 y Fd(i)p Fj(+1)30967 25147 y Fv(\))31448 24785 y Fq(\()p Fp(q)48 b(;)221 b(z)58 b Fq(\))295 b(+)g(\011)37045 24984 y Fk(i)p Fv(+1)38833 24785 y Fq(\()p Fp(q)48 b Fq(\))294 b Fm(\000)h Fq(\011)43163 24984 y Fk(i)43605 24785 y Fq(\()p Fp(q)48 b Fq(\))p Fp(:)1102 27910 y Fq(Note)443 b(\011)5345 28109 y Fk(i)p Fv(+1)7132 27910 y Fq(\()p Fp(q)48 b Fq(\))368 b(=)h(0)442 b(as)g Fp(q)417 b Fm(2)368 b(B)16449 28109 y Fk(\017)16959 27910 y Fq(\()p Fp(q)18069 28109 y Fk(i)18511 27910 y Fq(\).)589 b(If)442 b(w)-37 b(e)442 b(require)h(further)g(that)e Fp(\033)35847 28109 y Fk(i)p Fv(+1)38076 27910 y Fq(is)h(so)g(close)g(to)g Fp(\033)46602 28109 y Fk(i)47486 27910 y Fq(that)1102 29504 y(\(5.17\))h(holds,)g(w)-37 b(e)443 b(obtain)g(from)g(\(5.16i\))g (and)f(\(5.18\))h(that)15867 32489 y(min)14529 33408 y Fk(q)33 b Fg(2)p Fk(@)57 b Fg(B)17090 33541 y Fd(\017)17564 33408 y Fv(\()p Fk(x)18581 33541 y Fd(i)19003 33408 y Fv(\))19418 32489 y Fp(h)20183 31941 y Fg(1)20213 32818 y Fv(\026)-559 b Fk(c)20658 32951 y Fd(i)21309 32489 y Fq(\()p Fp(z)22444 32688 y Fk(i)22885 32489 y Fp(;)221 b(q)48 b Fq(\))295 b(+)g Fp(h)27021 31941 y Fg(1)27051 32818 y Fv(\026)-559 b Fk(c)27496 32951 y Fd(i)p Fj(+1)29118 32489 y Fq(\()p Fp(q)48 b(;)221 b(z)31484 32688 y Fk(i)p Fv(+1)33271 32489 y Fq(\))p Fm(\000)16179 35302 y Fq(min)15154 36220 y Fk(q)33 b Fg(2B)17091 36353 y Fd(\017)17564 36220 y Fv(\()p Fk(x)18581 36353 y Fd(i)19003 36220 y Fv(\))19418 35302 y Fp(h)20183 34753 y Fg(1)20213 35630 y Fv(\026)-559 b Fk(c)20658 35763 y Fd(i)21309 35302 y Fq(\()p Fp(z)22444 35501 y Fk(i)22885 35302 y Fp(;)221 b(q)48 b Fq(\))295 b(+)g Fp(h)27021 34753 y Fg(1)27051 35630 y Fv(\026)-559 b Fk(c)27496 35763 y Fd(i)p Fj(+1)29118 35302 y Fq(\()p Fp(q)48 b(;)221 b(z)31484 35501 y Fk(i)p Fv(+1)33271 35302 y Fq(\))368 b Fm(\025)h Fp(\016)-24 b(:)10034 b Fq(\(5.20\))1102 39106 y(Let)442 b Fl(B)369 b Fq(=)g Fm(B)7010 39305 y Fk(\017)7519 39106 y Fq(\()p Fp(x)8795 39305 y Fv(1)9391 39106 y Fq(\))292 b Fm(\002)h(B)12398 39305 y Fk(\017)12907 39106 y Fq(\()p Fp(x)14183 39305 y Fv(2)14779 39106 y Fq(\))f Fm(\002)h(\001)221 b(\001)g(\001)293 b(\002)f(B)20953 39305 y Fk(\017)21463 39106 y Fq(\()p Fp(x)22739 39305 y Fk(`)p Fg(\000)p Fv(1)24610 39106 y Fq(\),)441 b Fp(Q)369 b Fq(=)g(\()p Fp(q)29868 39305 y Fv(1)30464 39106 y Fp(;)221 b(:)g(:)g(:)h(;)f(q)34008 39305 y Fk(`)p Fg(\000)p Fv(1)35879 39106 y Fq(\),)406 b Fp(~)-628 b(n)369 b Fq(=)g(\()p Fp(n)41089 39305 y Fv(0)41685 39106 y Fp(;)221 b(n)43072 39305 y Fv(1)43668 39106 y Fp(;)g Fm(\001)g(\001)g(\001)444 b Fp(;)221 b(n)47638 39305 y Fk(`)48150 39106 y Fq(\))369 b Fm(2)1102 40700 y Fl(Z)1988 40218 y Fk(`)p Fv(+1)4287 40700 y Fq(and)443 b(de\014ne)9933 44662 y Fh(h)p Fq(\()p Fp(Q;)221 b(z)13526 44861 y Fv(1)14121 44662 y Fp(;)g(z)15329 44861 y Fk(`)15841 44662 y Fp(;)186 b(~)-629 b(n)p Fq(\))369 b(=)19426 43002 y Fk(`)p Fg(\000)p Fv(2)19368 43400 y Fi(X)19467 46230 y Fk(i)p Fv(=1)21509 44662 y Fp(h)22274 43961 y Fk(n)22931 44094 y Fd(i)p Fj(+1)24486 43961 y Fg(\000)p Fk(n)25973 44094 y Fd(i)22304 44951 y Fv(\026)-559 b Fk(c)22749 45084 y Fd(i)p Fj(+1)26460 44662 y Fq(\()p Fp(q)27570 44861 y Fk(i)28012 44662 y Fp(;)221 b(q)29195 44861 y Fk(i)p Fv(+1)30983 44662 y Fq(\))19442 48179 y(+)296 b Fp(h)21536 47589 y Fk(n)22193 47722 y Fj(1)22711 47589 y Fg(\000)p Fk(n)24198 47722 y Fj(0)21566 48468 y Fv(\026)-559 b Fk(c)22011 48601 y Fj(1)24783 48179 y Fq(\()p Fp(z)25918 48378 y Fv(1)26513 48179 y Fp(;)221 b(q)27696 48378 y Fv(1)28292 48179 y Fq(\))295 b(+)g Fp(h)31197 47470 y Fk(n)31854 47611 y Fd(`)32317 47470 y Fg(\000)p Fk(n)33804 47611 y Fd(`)p Fe(\000)p Fj(1)31227 48468 y Fv(\026)-559 b Fk(c)31672 48609 y Fd(`)35505 48179 y Fq(\()p Fp(q)36615 48378 y Fk(`)p Fg(\000)p Fv(1)38486 48179 y Fp(;)221 b(z)39694 48378 y Fk(`)40205 48179 y Fq(\))p Fp(;)5436 b Fq(\(5.21\))1102 51164 y(W)-111 b(e)493 b(see)f(that)f Fh(h)p Fq(,)503 b(as)492 b(the)f(function)h(of)f Fp(Q)p Fq(,)504 b(tak)-37 b(es)492 b(its)f(lo)37 b(cal)492 b(minim)-37 b(um)492 b(in)f(the)g(in)-37 b(terior)493 b(of)e Fl(B)g Fq(if)1102 52758 y Fp(n)1899 52957 y Fk(i)p Fv(+1)3982 52758 y Fm(\000)296 b Fp(n)6108 52957 y Fk(i)6993 52758 y Fq(is)443 b(su\016cien)-37 b(tly)444 b(large)g(for)f(all)g(1)370 b Fm(\024)g Fp(i)f Fm(\024)g Fp(`)296 b Fm(\000)g Fq(1.)591 b(In)443 b(fact,)g(let)g Fp(x)37904 52276 y Fg(\003)37904 53104 y Fk(i)38958 52758 y Fq(b)37 b(e)443 b(the)g(p)37 b(oin)-37 b(t)443 b(where)1102 54352 y(the)528 b(function)f(of)g Fp(q)576 b(h)12293 53870 y Fg(1)12323 54680 y Fv(\026)-559 b Fk(c)12768 54813 y Fd(i)13418 54352 y Fq(\()p Fp(z)14553 54551 y Fk(i)14995 54352 y Fp(;)221 b(q)48 b Fq(\))351 b(+)h Fp(h)19244 53870 y Fg(1)19274 54680 y Fv(\026)-559 b Fk(c)19719 54813 y Fd(i)p Fj(+1)21341 54352 y Fq(\()p Fp(q)48 b(;)221 b(z)23707 54551 y Fk(i)p Fv(+1)25493 54352 y Fq(\))527 b(attains)g(its)g(lo)37 b(cal)528 b(minim)-37 b(um)527 b(in)h Fm(B)44747 54551 y Fk(\017)45256 54352 y Fq(\()p Fp(x)46532 54551 y Fk(i)46974 54352 y Fq(\),)548 b(w)-37 b(e)1102 55979 y(\014nd)443 b(that)f(the)h(function)g(of)f Fp(Q)16920 58540 y Fk(`)p Fg(\000)p Fv(1)16862 58938 y Fi(X)16961 61768 y Fk(i)p Fv(=1)19003 60200 y Fp(h)19768 59652 y Fg(1)19798 60529 y Fv(\026)-559 b Fk(c)20243 60662 y Fd(i)20893 60200 y Fq(\()p Fp(z)22028 60399 y Fk(i)22470 60200 y Fp(;)221 b(q)23653 60399 y Fk(i)24095 60200 y Fq(\))295 b(+)h Fp(h)27001 59652 y Fg(1)27031 60529 y Fv(\026)-559 b Fk(c)27476 60662 y Fd(i)p Fj(+1)29097 60200 y Fq(\()p Fp(q)30207 60399 y Fk(i)30649 60200 y Fp(;)221 b(z)31857 60399 y Fk(i)p Fv(+1)33645 60200 y Fq(\))1102 64420 y(tak)-37 b(es)636 b(its)g(lo)37 b(cal)636 b(minim)-37 b(um)635 b(at)h(the)f(p)37 b(oin)-37 b(t)636 b(\()p Fp(x)25440 63938 y Fg(\003)25440 64750 y Fv(1)26050 64420 y Fp(;)221 b(x)27399 63938 y Fg(\003)27399 64750 y Fv(2)28010 64420 y Fp(;)g(:)g(:)g(:)i(;)e(x)31721 63938 y Fg(\003)31721 64797 y Fk(`)p Fg(\000)p Fv(1)33592 64420 y Fq(\))635 b(whic)-37 b(h)637 b(is)f(ob)-37 b(viously)636 b(in)f(the)1102 66014 y(in)-37 b(terior)523 b(of)e Fl(B)p Fq(.)827 b(Th)-37 b(us,)541 b(the)522 b(lo)37 b(cal)522 b(minim)-37 b(um)521 b(of)h Fh(h)f Fq(is)g(in)h(the)f(in)-37 b(terior)523 b(of)e Fl(B)g Fq(if)h(the)f(follo)-37 b(wing)1102 67608 y(holds)7306 70593 y(lim)5110 71390 y Fk(n)5767 71523 y Fd(i)p Fj(+1)7322 71390 y Fg(\000)p Fk(n)8809 71523 y Fd(i)9229 71390 y Fg(!1)11569 70593 y Fp(h)12334 69892 y Fk(n)12991 70025 y Fd(i)p Fj(+1)14546 69892 y Fg(\000)p Fk(n)16033 70025 y Fd(i)12364 70882 y Fv(\026)-559 b Fk(c)12809 71015 y Fd(i)p Fj(+1)16520 70593 y Fq(\()p Fp(q)17630 70792 y Fk(i)18072 70593 y Fp(;)221 b(q)19255 70792 y Fk(i)p Fv(+1)21043 70593 y Fq(\))368 b(=)h Fp(h)24095 70045 y Fg(1)24125 70922 y Fv(\026)-559 b Fk(c)24570 71055 y Fd(i)p Fj(+1)26192 70593 y Fq(\()p Fp(q)27302 70792 y Fk(i)27744 70593 y Fp(;)221 b(z)28952 70792 y Fk(i)p Fv(+1)30740 70593 y Fq(\))294 b(+)i Fp(h)33645 70045 y Fg(1)33675 70922 y Fv(\026)-559 b Fk(c)34120 71055 y Fd(i)p Fj(+1)35742 70593 y Fq(\()p Fp(z)36877 70792 y Fk(i)p Fv(+1)38663 70593 y Fp(;)221 b(q)39846 70792 y Fk(i)p Fv(+1)41634 70593 y Fq(\))p Fp(:)4007 b Fq(\(5.22\))1102 74213 y(T)-111 b(o)444 b(sho)-37 b(w)443 b(this)g(let)f(us)h(state)g(a)g(lemma:)p eop %%Page: 42 42 42 41 bop 1102 2239 a Ft(42)1102 5427 y Fr(Lemma)402 b(5.4.)764 b Fo(Assume)15076 5091 y Fq(~)14426 5427 y Fm(M)p Fq(\()p Fp(c)p Fq(\))389 b Fo(has)h(a)g(dense)g(orbit.)583 b(F)-102 b(or)391 b(any)e Fp(m)34669 5626 y Fv(0)35265 5427 y Fp(;)221 b(m)37021 5626 y Fv(1)37987 5427 y Fm(2)368 b Fp(M)145 b Fo(,)407 b(let)391 b Fp(\015)74 b Fo(:)568 b Fq([0)p Fp(;)221 b(K)95 b Fq(])370 b Fm(!)1102 7021 y Fp(M)736 b Fo(b)-68 b(e)592 b Fp(c)p Fo(-minimal)h(curve)f(c)-68 b(onne)g(cting)592 b Fp(m)22799 7220 y Fv(0)23987 7021 y Fo(and)f Fp(m)27849 7220 y Fv(1)28445 7021 y Fo(,)621 b Fp(\015)74 b Fq(\(0\))579 b(=)h Fp(m)35291 7220 y Fv(0)35887 7021 y Fo(,)621 b Fp(\015)74 b Fq(\()p Fp(K)95 b Fq(\))578 b(=)i Fp(m)43291 7220 y Fv(1)43887 7021 y Fo(.)961 b(F)-102 b(or)591 b(any)1102 8615 y Fp(\016)420 b(>)369 b Fq(0)p Fo(,)470 b(any)d Fp(K)8721 8814 y Fv(1)9686 8615 y Fm(2)368 b Fl(Z)11826 8133 y Fv(+)13175 8615 y Fo(and)468 b(any)f Fp(z)427 b Fm(2)368 b(M)p Fq(\()p Fp(c)p Fq(\))p Fm(j)24158 8814 y Fk(t)p Fv(=0)25970 8615 y Fo(,)470 b Fm(9)d Fp(K)29180 8814 y Fv(0)30145 8615 y Fm(2)369 b Fl(Z)32286 8133 y Fv(+)33167 8615 y Fo(,)470 b(if)e Fp(K)c Fm(\025)369 b Fp(K)39448 8814 y Fv(0)40512 8615 y Fo(then)467 b(ther)-68 b(e)468 b(exists)1102 10209 y Fp(T)554 b Fm(2)369 b Fl(Z)4573 9727 y Fv(+)5929 10209 y Fo(such)476 b(that)f Fp(\015)74 b Fq(\()p Fp(T)184 b Fq(\))369 b Fm(2)f Fp(B)17053 10408 y Fk(\016)17636 10209 y Fq(\()p Fp(z)58 b Fq(\))p Fo(,)475 b Fp(T)553 b Fm(\025)370 b Fp(K)24088 10408 y Fv(1)25159 10209 y Fo(and)475 b Fp(K)28867 10408 y Fv(0)29758 10209 y Fm(\000)296 b Fp(T)553 b Fm(\025)369 b Fp(K)34946 10408 y Fv(1)35542 10209 y Fo(.)1102 12617 y Fr(Pro)42 b(of)p Fq(:)546 b(F)-111 b(or)355 b(an)-37 b(y)354 b Fp(\016)11082 12135 y Fg(\003)12062 12617 y Fp(>)369 b Fq(0)354 b(there)h(is)e Fp(K)20166 12816 y Fv(0)21131 12617 y Fm(2)369 b Fl(Z)23272 12816 y Fv(+)24507 12617 y Fq(and)354 b Fp(k)410 b Fm(2)369 b Fl(Z)30244 12135 y Fv(+)31479 12617 y Fq(suc)-37 b(h)354 b(that)f Fp(d\015)74 b Fq(\()p Fp(k)42 b Fq(\))367 b Fm(2)42668 12282 y Fq(~)42018 12617 y Fm(M)p Fq(\()p Fp(c)p Fq(\))p Fm(j)45591 12816 y Fk(t)p Fv(=0)47520 12617 y Fq(+)117 b Fp(\016)49310 12135 y Fg(\003)1102 14211 y Fq(if)635 b Fp(K)784 b Fm(\025)689 b Fp(K)7274 14410 y Fv(0)7870 14211 y Fq(,)682 b(otherwise)636 b(there)f(w)-37 b(ould)635 b(b)37 b(e)635 b(another)g Fp(c)p Fq(-minimal)g(measure.) 1166 b(F)-111 b(or)636 b(an)-37 b(y)635 b Fp(z)747 b Fm(2)1102 15806 y(M)p Fq(\()p Fp(c)p Fq(\))p Fm(j)4675 16005 y Fk(t)p Fv(=0)6487 15806 y Fq(,)575 b(b)-37 b(y)549 b(c)-37 b(ho)37 b(osing)549 b(su\016cien)-37 b(tly)549 b(small)f Fp(\016)25858 15323 y Fg(\003)27018 15806 y Fq(and)h(su\016cien)-37 b(tly)548 b(large)i Fp(K)40946 16005 y Fv(0)42090 15806 y Fq(there)f(is)g(some)1102 17400 y Fp(T)777 b Fm(2)591 b Fl(Z)5018 16917 y Fv(+)6476 17400 y Fq(so)576 b(that)g Fp(\015)74 b Fq(\()p Fp(T)184 b Fq(\))576 b(is)g(in)g Fp(\016)50 b Fq(-neigh)-37 b(b)37 b(orho)g(o)g(d)578 b(of)e Fp(z)58 b Fq(.)991 b(Clearly)-111 b(,)612 b(for)576 b(an)-37 b(y)577 b Fp(K)41957 17599 y Fv(1)43145 17400 y Fm(2)591 b Fl(Z)45508 16917 y Fv(+)46966 17400 y Fq(there)1102 18994 y(exists)449 b(suc)-37 b(h)450 b Fp(T)634 b Fq(so)449 b(that)f Fp(T)565 b Fm(\025)379 b Fp(K)17591 19193 y Fv(1)18637 18994 y Fq(and)449 b Fp(K)22354 19193 y Fv(0)23249 18994 y Fm(\000)300 b Fp(T)564 b Fm(\025)380 b Fp(K)28463 19193 y Fv(1)29059 18994 y Fq(,)450 b(otherewise)h(there)f(w)-37 b(ould)449 b(b)37 b(e)450 b(another)1102 20588 y Fp(c)p Fq(-minimal)443 b(measure)h(also.)2165 b Fb(\003)2696 22902 y Fq(Applying)423 b(the)f(lemma)h(to)f(this)f(problem)i(w)-37 b(e)423 b(\014nd)f(that)g (for)g(eac)-37 b(h)423 b Fp(m)370 b Fm(2)e(M)p Fq(\()p Fp(c)42166 23101 y Fk(i)42608 22902 y Fq(\),)426 b(eac)-37 b(h)423 b(small)1102 24496 y Fp(\016)d(>)369 b Fq(0)403 b(and)f(eac)-37 b(h)403 b(large)g Fp(K)465 b(>)369 b Fq(0)402 b(there)h(exist)f Fp(n)24600 24014 y Fg(\003)25580 24496 y Fm(2)369 b Fl(Z)27721 24014 y Fv(+)29004 24496 y Fq(suc)-37 b(h)403 b(that)f(if)g Fp(n)369 b Fm(\025)g Fp(n)39340 24014 y Fg(\003)40353 24496 y Fq(then)402 b(there)h(exists)1102 26090 y Fp(z)1720 26289 y Fk(n)2812 26090 y Fm(2)369 b(B)4939 26289 y Fk(\016)5522 26090 y Fq(\()p Fp(m)p Fq(\))442 b(suc)-37 b(h)444 b(that)15514 28934 y Fp(h)16279 28386 y Fk(n)16279 29263 y(c)16754 29396 y Fd(i)17241 28934 y Fq(\()p Fp(q)48 b(;)221 b(q)19630 28386 y Fg(0)20002 28934 y Fq(\))369 b(=)g Fp(h)23055 28386 y Fk(n)23712 28519 y Fj(1)23055 29263 y Fk(c)23530 29396 y Fd(i)24297 28934 y Fq(\()p Fp(q)48 b(;)221 b(z)26663 29133 y Fk(n)27385 28934 y Fq(\))295 b(+)g Fp(h)30290 28386 y Fk(n)30947 28519 y Fj(2)30290 29263 y Fk(c)30765 29396 y Fd(i)31532 28934 y Fq(\()p Fp(z)32667 29133 y Fk(n)33389 28934 y Fp(;)221 b(q)34620 28386 y Fg(0)34993 28934 y Fq(\))1102 31778 y(where)492 b Fp(n)449 b Fq(=)g Fp(n)8516 31977 y Fv(1)9440 31778 y Fq(+)327 b Fp(n)11597 31977 y Fv(2)12684 31778 y Fq(with)490 b Fp(n)16554 31977 y Fv(1)17150 31778 y Fp(;)221 b(n)18537 31977 y Fv(2)19583 31778 y Fm(\025)449 b Fp(K)95 b Fq(.)734 b(Using)491 b(the)f(Lipsc)-37 b(hitz)491 b(prop)37 b(ert)-37 b(y)492 b(of)f Fp(h)43033 31296 y Fk(n)43033 32107 y(c)43508 32240 y Fd(i)43995 31778 y Fq(\()p Fp(m;)221 b(m)47434 31296 y Fg(0)47807 31778 y Fq(\))490 b(in)1102 33372 y(\()p Fp(m;)221 b(m)4541 32890 y Fg(0)4915 33372 y Fq(\))442 b(w)-37 b(e)444 b(\014nd)e(that)g(for)i(eac)-37 b(h)443 b(small)g Fp(\017)369 b(>)h Fq(0)442 b(the)h(follo)-37 b(wing)444 b(holds)14164 35087 y Fi(\014)14164 35884 y(\014)14607 36216 y Fp(h)15372 35668 y Fk(n)15372 36545 y(c)15847 36678 y Fd(i)16334 36216 y Fq(\()p Fp(q)48 b(;)221 b(q)18723 35668 y Fg(0)19095 36216 y Fq(\))295 b Fm(\000)g Fp(h)22000 35668 y Fk(n)22657 35801 y Fj(1)22000 36545 y Fk(c)22475 36678 y Fd(i)23242 36216 y Fq(\()p Fp(q)48 b(;)221 b(m)p Fq(\))295 b(+)g Fp(h)29061 35668 y Fk(n)29718 35801 y Fj(2)29061 36545 y Fk(c)29536 36678 y Fd(i)30303 36216 y Fq(\()p Fp(m;)221 b(q)33217 35668 y Fg(0)33590 36216 y Fq(\))34107 35087 y Fi(\014)34107 35884 y(\014)34918 36216 y Fp(<)369 b(\017)1102 39060 y Fq(if)515 b Fp(\016)565 b Fq(is)515 b(su\016cien)-37 b(tly)515 b(small)f(and)h Fp(n)18675 38578 y Fg(\003)19800 39060 y Fq(is)g(su\016cien)-37 b(tly)515 b(large.)807 b(Since)515 b(w)-37 b(e)515 b(consider)h(those)f Fm(M)p Fq(\()p Fp(c)48963 39259 y Fk(i)49405 39060 y Fq(\))1102 40654 y(whic)-37 b(h)384 b(is)g(on)f(the)g(cylinder)h(with)g (irrational)f(n)-37 b(um)g(b)37 b(er,)396 b(thanks)383 b(to)g(the)g(corollary)i(2.7,)395 b(w)-37 b(e)385 b(kno)-37 b(w)1102 42248 y(that)453 b Fp(L)302 b Fm(\000)h(h)29 b Fq(\026)-693 b Fp(c)7625 42447 y Fk(i)8066 42248 y Fq(\()p Fp(q)48 b Fq(\))p Fp(;)467 b Fq(_)-615 b Fp(q)47 b Fm(i)453 b Fq(is)g(regular)i(for)e(eac)-37 b(h)455 b(0)386 b Fm(\024)h Fp(i)f Fm(\024)h Fp(`)302 b Fq(+)h(1,)456 b(\(5.22\))d(follo)-37 b(ws)454 b(from)f(the)h(prop)37 b(ert)-37 b(y)1102 43842 y(that)512 b Fp(d)4741 44041 y Fk(c)5216 44174 y Fd(i)5703 43842 y Fq(\()p Fp(m;)221 b(m)9142 43360 y Fg(0)9515 43842 y Fq(\))485 b(=)f(0)513 b(for)f(all)g Fp(m;)221 b(m)20149 43360 y Fg(0)21008 43842 y Fm(2)484 b Fp(\031)48 b Fq(\(\000)24530 44041 y Fk(i)24972 43842 y Fq(\).)798 b(Denote)512 b(the)g(corresp)37 b(onding)514 b(minimizer)f(b)-37 b(y)1102 45436 y Fp(\015)74 b Fq(:)581 b([)p Fp(n)3980 45635 y Fv(0)4576 45436 y Fp(;)221 b(n)5963 45635 y Fk(`)6475 45436 y Fq(])424 b Fm(!)369 b Fp(M)145 b Fq(,)427 b(w)-37 b(e)425 b(use)f Fp(\015)16096 45635 y Fk(i)16538 45436 y Fq(\()p Fp(t)p Fq(\))f(to)g(denote)i(its)e(restriction)i(on)f(the)f(time)h(in)-37 b(terv)-74 b(al)425 b([)p Fp(n)45566 45635 y Fk(i)46008 45436 y Fp(;)221 b(n)47395 45635 y Fk(i)p Fv(+1)49183 45436 y Fq(].)1102 47031 y(Once)486 b Fp(\015)74 b Fq(\()p Fp(t)p Fq(\))482 b(reac)-37 b(hes)486 b(its)e(lo)37 b(cal)485 b(minim)-37 b(um)484 b(in)g(the)h(in)-37 b(terior)485 b(of)f Fl(B)p Fq(,)494 b(standard)485 b(argumen)-37 b(t)485 b(sho)-37 b(ws)1102 48625 y(that)15145 49742 y Fp(@)74 b(L)16858 49941 y Fv(\026)-559 b Fk(c)17303 50074 y Fd(i)p 15145 50335 2645 54 v 15757 51552 a Fp(@)321 b Fq(_)-616 b Fp(q)17949 50641 y Fq(\()p Fp(d\015)19845 50840 y Fk(i)20287 50641 y Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))367 b(=)25317 49680 y Fp(@)74 b(L)27030 49879 y Fv(\026)-559 b Fk(c)27475 50012 y Fd(i)p Fj(+1)p 25317 50335 3780 54 v 26497 51552 a Fp(@)321 b Fq(_)-616 b Fp(q)29256 50641 y Fq(\()p Fp(d\015)31152 50840 y Fk(i)p Fv(+1)32939 50641 y Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))1102 53716 y(holds)604 b(at)e Fp(t)636 b Fq(=)h Fp(n)10105 53915 y Fk(i)p Fv(+1)11892 53716 y Fq(.)1072 b(Note)603 b(that)g Fp(L)20676 53915 y Fv(\026)-559 b Fk(c)21121 54048 y Fd(i)22244 53716 y Fq(=)636 b Fp(L)24847 53915 y Fv(\026)-559 b Fk(c)25292 54048 y Fd(i)p Fj(+1)27517 53716 y Fq(in)603 b(the)g(neigh)-37 b(b)37 b(orho)g(o)g(d)604 b(of)g Fm(B)42724 53915 y Fk(\017)43233 53716 y Fq(\()84 b(\026)-748 b Fp(x)44509 53915 y Fk(i)44951 53716 y Fq(\))603 b(b)-37 b(y)603 b(the)1102 55310 y(de\014nition)443 b(of)g(\011)9552 55509 y Fk(i)9994 55310 y Fq(\()p Fp(q)48 b Fq(\),)442 b(w)-37 b(e)444 b(get)19218 56904 y(_)-565 b Fp(\015)19710 57103 y Fk(i)20152 56904 y Fq(\()p Fp(n)21466 57103 y Fk(i)p Fv(+1)23253 56904 y Fq(\))368 b(=)566 b(_)-566 b Fp(\015)26228 57103 y Fk(i)p Fv(+1)28015 56904 y Fq(\()p Fp(n)29329 57103 y Fk(i)p Fv(+1)31116 56904 y Fq(\))p Fp(;)1102 59275 y Fq(th)-37 b(us)426 b Fp(\015)74 b Fq(\()p Fp(t)p Fq(\))425 b(is)h(a)h(solution)f(of)g(the)g(Euler-Lagrange)j (equation)d(o)-37 b(v)g(er)428 b(the)e(time)g(in)-37 b(terv)-74 b(al)427 b([)p Fp(n)46688 59474 y Fv(0)47285 59275 y Fp(;)221 b(n)48672 59474 y Fk(`)49183 59275 y Fq(].)2696 61590 y(In)547 b(fact,)572 b(w)-37 b(e)547 b(can)g(remo)-37 b(v)g(e)548 b(the)f(restriction)g(on)f Fp(z)28246 61789 y Fv(1)29389 61590 y Fq(and)g Fp(z)32693 61789 y Fk(`)33751 61590 y Fq(that)f(there)i(is)g Fp(z)600 b Fm(2)541 b Fl(T)546 b Fq(so)h(that)1102 63184 y Fp(z)1720 63383 y Fk(j)2840 63184 y Fq(=)560 b(\()p Fp(z)58 b(;)221 b(q)6809 63383 y Fv(2\000)7995 63516 y Fd(j)8558 63184 y Fq(\()p Fp(z)58 b Fq(\)\))556 b(for)i Fp(j)636 b Fq(=)560 b(1)p Fp(;)221 b(`)p Fq(,)587 b(W)-111 b(e)558 b(can)g(replace)h Fp(z)29218 63383 y Fk(j)30335 63184 y Fq(b)-37 b(y)557 b(an)-37 b(y)558 b(p)37 b(oin)-37 b(t)557 b Fp(z)39177 62702 y Fg(\003)39119 63530 y Fk(j)40348 63184 y Fm(2)i(M)p Fq(\()p Fp(c)44480 63383 y Fk(j)45039 63184 y Fq(\))e(simply)1102 64842 y(b)37 b(ecause)444 b Fp(d)6708 65041 y Fk(c)7183 65174 y Fd(j)7746 64842 y Fq(\()p Fp(m;)221 b(m)11185 64360 y Fg(0)11558 64842 y Fq(\))369 b(=)g(0)443 b(for)g(all)g Fp(m;)221 b(m)21753 64360 y Fg(0)22496 64842 y Fm(2)368 b Fp(\031)48 b Fq(\(\000)25902 65041 y Fk(j)26462 64842 y Fq(\))442 b(th)-37 b(us)442 b(the)h(function)g(of)g Fp(Q)369 b Fq(=)g(\()p Fp(q)43278 65041 y Fv(1)43873 64842 y Fp(;)221 b(:)g(:)g(:)i(;)e(q)47418 65041 y Fk(`)p Fg(\000)p Fv(1)49289 64842 y Fq(\))9951 67465 y Fk(`)p Fg(\000)p Fv(2)9894 67863 y Fi(X)9993 70692 y Fk(i)p Fv(=2)12034 69125 y Fp(h)12799 68576 y Fg(1)12829 69453 y Fv(\026)-559 b Fk(c)13274 69586 y Fd(i)13925 69125 y Fq(\()p Fp(z)15060 69324 y Fk(i)15501 69125 y Fp(;)221 b(q)16684 69324 y Fk(i)17127 69125 y Fq(\))295 b(+)g Fp(h)20032 68576 y Fg(1)20062 69453 y Fv(\026)-559 b Fk(c)20507 69586 y Fd(i)p Fj(+1)22129 69125 y Fq(\()p Fp(q)23239 69324 y Fk(i)23681 69125 y Fp(;)221 b(z)24889 69324 y Fk(i)p Fv(+1)26676 69125 y Fq(\))295 b(+)g Fp(h)29581 68576 y Fg(1)29611 69453 y Fv(\026)-559 b Fk(c)30056 69586 y Fj(1)30707 69125 y Fq(\()p Fp(z)31900 68576 y Fg(\003)31842 69453 y Fv(1)32510 69125 y Fp(;)221 b(q)33693 69324 y Fv(1)34290 69125 y Fq(\))294 b(+)i Fp(h)37195 68576 y Fg(1)37225 69453 y Fv(\026)-559 b Fk(c)37670 69594 y Fd(`)38320 69125 y Fq(\()p Fp(q)39430 69324 y Fk(`)39941 69125 y Fp(;)221 b(z)41207 68576 y Fg(\003)41149 69453 y Fk(`)41818 69125 y Fq(\))8639 73796 y(=)9951 72136 y Fk(`)p Fg(\000)p Fv(1)9894 72534 y Fi(X)9993 75363 y Fk(i)p Fv(=1)12034 73796 y Fp(h)12799 73248 y Fg(1)12829 74124 y Fv(\026)-559 b Fk(c)13274 74257 y Fd(i)13925 73796 y Fq(\()p Fp(z)15060 73995 y Fk(i)15501 73796 y Fp(;)221 b(q)16684 73995 y Fk(i)17127 73796 y Fq(\))295 b(+)g Fp(h)20032 73248 y Fg(1)20062 74124 y Fv(\026)-559 b Fk(c)20507 74257 y Fd(i)p Fj(+1)22129 73796 y Fq(\()p Fp(q)23239 73995 y Fk(i)23681 73796 y Fp(;)221 b(z)24889 73995 y Fk(i)p Fv(+1)26676 73796 y Fq(\))295 b(+)g Fp(h)29581 73248 y Fg(1)29611 74124 y Fv(\026)-559 b Fk(c)30056 74257 y Fj(1)30707 73796 y Fq(\()p Fp(z)31900 73248 y Fg(\003)31842 74124 y Fv(1)32510 73796 y Fp(;)221 b(z)33718 73995 y Fv(1)34314 73796 y Fq(\))295 b(+)g Fp(h)37219 73248 y Fg(1)37249 74124 y Fv(\026)-559 b Fk(c)37694 74265 y Fd(`)38345 73796 y Fq(\()p Fp(z)39480 73995 y Fk(`)39991 73796 y Fp(;)221 b(z)41257 73248 y Fg(\003)41199 74124 y Fk(`)41868 73796 y Fq(\))p eop %%Page: 43 43 43 42 bop 48792 2239 a Ft(43)1102 5427 y Fq(also)507 b(reac)-37 b(hes)509 b(its)d(lo)37 b(cal)507 b(minim)-37 b(um)506 b(at)h(the)f(p)37 b(oin)-37 b(t)507 b(\()p Fp(x)28486 4945 y Fg(\003)28486 5757 y Fv(1)29096 5427 y Fp(;)221 b(x)30445 4945 y Fg(\003)30445 5757 y Fv(2)31056 5427 y Fp(;)g Fm(\001)g(\001)g(\001)444 b Fp(;)221 b(x)34988 4945 y Fg(\003)34988 5803 y Fk(`)p Fg(\000)p Fv(1)36860 5427 y Fq(\).)781 b(Th)-37 b(us,)523 b(the)507 b(Lipsc)-37 b(hitz)1102 7043 y(prop)37 b(ert)-37 b(y)547 b(of)e Fp(h)9017 6561 y Fk(n)9017 7371 y(c)10286 7043 y Fq(enable)i(us)e(to)h(assert)g (that)f(there)h(exist)f(large)i(\(\001)p Fp(n)37444 7242 y Fv(1)38040 7043 y Fp(;)221 b Fq(\001)p Fp(n)40534 7242 y Fv(2)41130 7043 y Fp(;)g Fm(\001)g(\001)g(\001)444 b Fp(;)221 b Fq(\001)p Fp(n)46207 7242 y Fk(`)46719 7043 y Fq(\))545 b(and)1102 8637 y(small)413 b Fp(\016)5188 8155 y Fg(\003)6168 8637 y Fp(>)369 b Fq(0,)419 b(if)413 b Fp(n)11007 8836 y Fk(i)11684 8637 y Fm(\000)235 b Fp(n)13749 8836 y Fk(i)p Fg(\000)p Fv(1)15920 8637 y Fm(\025)369 b Fq(\001)p Fp(n)19226 8836 y Fk(i)19669 8637 y Fq(,)418 b Fp(z)21074 8836 y Fk(j)22003 8637 y Fm(2)368 b(B)24129 8836 y Fk(\016)24646 8571 y Fe(\003)25268 8637 y Fq(\()p Fp(z)26461 8155 y Fg(\003)26403 8983 y Fk(j)27072 8637 y Fq(\))412 b(for)h Fp(j)445 b Fq(=)369 b(1)p Fp(;)221 b(`)p Fq(,)419 b(then)413 b(as)g(the)f(function)g(of)h Fp(Q)p Fq(,)1102 10293 y Fh(h)p Fq(\()p Fp(Q;)221 b(z)4695 10492 y Fv(1)5291 10293 y Fp(;)g(z)6499 10492 y Fk(`)7010 10293 y Fp(;)186 b(~)-629 b(n)q Fq(\))442 b(reac)-37 b(hes)444 b(its)f(lo)37 b(cal)443 b(minim)-37 b(um)443 b(in)g(the)f(in)-37 b(terior)444 b(of)f Fl(B)p Fq(.)2696 12628 y(No)-37 b(w)422 b(w)-37 b(e)422 b(are)g(ready)g(to)f(construct)g (some)g(di\013usion)g(orbit)g Fp(\015)74 b Fq(:)579 b Fl(R)370 b Fm(!)f Fp(M)566 b Fq(suc)-37 b(h)421 b(that)f Fp(\013)5 b Fq(\()p Fp(d\015)74 b Fq(\))368 b Fm(\033)1102 14222 y Fq(\000)1932 14421 y Fv(0)2971 14222 y Fq(and)443 b Fp(!)48 b Fq(\()p Fp(d\015)74 b Fq(\))368 b Fm(\033)h Fq(\000)11516 14421 y Fk(`)p Fv(+1)13372 14222 y Fq(.)2696 16557 y(By)448 b(the)f(condition,)i(\000)14113 16756 y Fk(i)15002 16557 y Fq(and)f(\000)18420 16756 y Fk(i)p Fv(+1)20655 16557 y Fq(mak)-37 b(e)448 b(up)f(the)g(b)37 b(oundary)448 b(of)f(the)g(resonan)-37 b(t)448 b(zone)h Fp(Z)47442 16756 y Fk(i)48331 16557 y Fq(for)1102 18179 y Fp(i)369 b Fq(=)g(0)p Fp(;)221 b(`)p Fq(.)550 b(F)-111 b(or)317 b Fp(i)369 b Fq(=)g(0)p Fp(;)221 b(`)43 b Fq(+)g(1)318 b(w)-37 b(e)318 b(tak)-37 b(e)317 b Fp(c)19608 17697 y Fv(\()p Fk(i)p Fv(\))21197 18179 y Fq(b)37 b(e)317 b(suc)-37 b(h)317 b(a)g(co-homology)h(class)f(that)f Fm(f)p Fp(B)41795 17697 y Fg(\003)41728 18674 y Fk(c)42203 18408 y Fj(\()p Fd(i)p Fj(\))43778 18179 y Fq(=)369 b(0)p Fm(g)h Fq(=)f(\000)49110 18378 y Fk(i)49552 18179 y Fq(.)1102 20194 y(F)-111 b(or)452 b Fp(i)383 b Fq(=)g(1)p Fp(;)221 b(`)452 b Fq(w)-37 b(e)453 b(tak)-37 b(e)451 b Fp(c)13436 19712 y Fv(\()p Fk(i)p Fv(\))15160 20194 y Fq(b)37 b(e)451 b(suc)-37 b(h)452 b(a)f(co-homology)h(class)g(that)e Fp(c)35400 19712 y Fv(\()p Fk(i)p Fv(\))37056 20194 y Fq(=)383 b(\()p Fp(c)39564 19504 y Fv(\()p Fk(i)p Fv(\))39564 20548 y(1)40836 20194 y Fp(;)221 b(b)p Fq(\()p Fp(c)43088 19504 y Fv(\()p Fk(i)p Fv(\))43088 20548 y(1)44360 20194 y Fq(\)\),)453 b(in)e(this)1102 21861 y(case,)360 b Fm(f)p Fp(B)5938 21379 y Fg(\003)5871 22355 y Fk(c)6346 22089 y Fj(\()p Fd(i)p Fj(\))7921 21861 y Fq(=)369 b(0)p Fm(g)g Fq(=)h(\000)13253 22060 y Fk(i)13779 21861 y Fm([)84 b(f)p Fq(its)443 b Fp(c)17841 21379 y Fv(\()p Fk(i)p Fv(\))19113 21861 y Fq(-minimal)g(homo)37 b(clinic)443 b(orbits)p Fm(g)p Fq(.)556 b(Since)338 b(the)f Fp(C)95 b Fq(-equiv)-74 b(alence)1102 23668 y(b)37 b(et)-37 b(w)g(een)563 b Fp(c)6925 23186 y Fv(\()p Fk(i)p Fv(\))8759 23668 y Fq(and)e Fp(c)12035 23186 y Fv(\()p Fk(i)p Fv(+1\))14652 23668 y Fq(has)h(b)37 b(een)562 b(established)f(for)h Fp(i)k Fq(=)h(0)p Fp(;)221 b(`)p Fq(,)592 b(analogous)562 b(to)f(the)g(pro)37 b(of)562 b(of)1102 25262 y(Theorem)445 b(5.1)e(w)-37 b(e)443 b(can)h(\014nd)e(the)h(comp)37 b(osition)443 b(of)g(\014nite)g Fp(U)145 b Fq(-step)442 b(forms)h Fp(\027)39148 25461 y Fk(j)16098 29690 y Fp(\027)16754 29889 y Fk(j)17683 29690 y Fq(=)19376 27886 y Fk(N)20217 28019 y Fd(j)19085 28428 y Fi(X)19184 31257 y Fk(i)p Fv(=0)21004 29690 y Fq(\()p Fm(\000)p Fp(\034)23285 29052 y Fk(j)23135 30060 y(i)23845 29690 y Fq(\))24362 29142 y Fg(\003)24972 29690 y Fp(\026)25772 29052 y Fk(j)25772 30060 y(i)26332 29690 y Fp(;)2878 b Fq(\()p Fp(j)445 b Fq(=)369 b(1)p Fp(;)221 b Fq(2\))1102 34045 y(suc)-37 b(h)512 b(that)e(their)h(cohomology)h(classes)f(are)h([)p Fp(\027)25302 34244 y Fv(1)25898 34045 y Fq(\()p Fp(t)p Fq(\))p Fm(j)27781 34244 y Fk(t)p Fg(\024)p Fv(0)29607 34045 y Fq(])482 b(=)h(0,)528 b([)p Fp(\027)34560 34244 y Fv(1)35156 34045 y Fq(\()p Fp(t)p Fq(\))p Fm(j)37039 34336 y Fk(t)p Fg(\025)p Fk(\034)38867 34016 y Fj(1)38760 34650 y Fd(N)39474 34849 y Fj(1)40058 34336 y Fv(+1)41469 34045 y Fq(])483 b(=)f Fp(c)44411 33563 y Fv(\(1\))46178 34045 y Fm(\000)340 b Fp(c)48126 33563 y Fv(\(0\))49552 34045 y Fq(,)1102 36236 y([)p Fp(\027)2127 36435 y Fv(2)2723 36236 y Fq(\()p Fp(t)p Fq(\))p Fm(j)4606 36435 y Fk(t)p Fg(\024)p Fv(0)6433 36236 y Fq(])532 b(=)517 b(0)533 b(and)f([)p Fp(\027)13778 36435 y Fv(2)14374 36236 y Fq(\()p Fp(t)p Fq(\))p Fm(j)16257 36527 y Fk(t)p Fg(\025)p Fk(\034)18085 36206 y Fj(2)17978 36841 y Fd(N)18692 37040 y Fj(2)19276 36527 y Fv(+1)20688 36236 y Fq(])517 b(=)h Fp(c)23700 35754 y Fv(\()p Fk(`)p Fv(+1\))26741 36236 y Fm(\000)354 b Fp(c)28703 35754 y Fv(\()p Fk(`)p Fv(\))30045 36236 y Fq(,)554 b(where)533 b Fp(\034)35629 35597 y Fk(j)35479 36606 y(i)36721 36236 y Fq(is)f(the)g(time)f(translation) 1102 38427 y(\()p Fp(q)48 b(;)221 b(t)p Fq(\))463 b Fm(!)h Fq(\()p Fp(q)48 b(;)221 b(t)333 b Fq(+)g Fp(\034)10760 37788 y Fk(j)10610 38797 y(i)11320 38427 y Fq(\).)761 b(Moreo)-37 b(v)g(er,)515 b(b)-37 b(y)500 b(the)g(lemma)g(5.3,)515 b(w)-37 b(e)500 b(can)g(c)-37 b(ho)37 b(ose)501 b(those)f Fp(\027)43373 38626 y Fk(j)44433 38427 y Fq(suc)-37 b(h)500 b(that)1102 40021 y Fp(\027)1758 40220 y Fv(1)2354 40021 y Fq(\()p Fp(t)p Fq(\))p Fm(j)4237 40220 y Fk(t)p Fg(\024)p Fv(0)6509 40021 y Fq(=)446 b(0,)500 b Fp(\027)10177 40220 y Fv(1)10773 40021 y Fq(\()p Fp(t)p Fq(\))p Fm(j)12656 40311 y Fk(t)p Fg(\025)p Fk(\034)14484 39991 y Fj(1)14377 40626 y Fd(N)15091 40825 y Fj(1)15676 40311 y Fv(+1)17532 40021 y Fq(=)446 b Fm(h)29 b Fq(\026)-693 b Fp(c)20103 40220 y Fv(1)20698 40021 y Fq(\()p Fp(q)48 b Fq(\))325 b Fm(\000)355 b Fq(\026)-693 b Fp(c)24632 40220 y Fv(0)25228 40021 y Fq(\()p Fp(q)48 b Fq(\))p Fp(;)221 b(dq)48 b Fm(i)p Fq(,)499 b Fp(\027)30866 40220 y Fv(2)31462 40021 y Fq(\()p Fp(t)p Fq(\))p Fm(j)33345 40220 y Fk(t)p Fg(\024)p Fv(0)35616 40021 y Fq(=)446 b(0)489 b(and)g Fp(\027)41533 40220 y Fv(2)42129 40021 y Fq(\()p Fp(t)p Fq(\))p Fm(j)44012 40311 y Fk(t)p Fg(\025)p Fk(\034)45840 39991 y Fj(2)45733 40626 y Fd(N)46447 40825 y Fj(2)47031 40311 y Fv(+1)48888 40021 y Fq(=)1102 42039 y Fm(h)29 b Fq(\026)-693 b Fp(c)2194 42238 y Fk(`)p Fv(+1)4050 42039 y Fq(\()p Fp(q)48 b Fq(\))400 b Fm(\000)431 b Fq(\026)-694 b Fp(c)8134 42238 y Fk(`)8645 42039 y Fq(\()p Fp(q)48 b Fq(\))p Fp(;)221 b(dq)48 b Fm(i)p Fq(,)641 b(\(see)602 b(\(5.18\))g(for)g(the)g(de\014nition)g(of) 632 b(\026)-694 b Fp(c)33538 42238 y Fk(i)33980 42039 y Fq(\()p Fp(q)48 b Fq(\)\).)1067 b(Let)602 b Fp(\021)40855 41557 y Fv(1)40807 42369 y(0)42085 42039 y Fq(=)634 b Fm(h)29 b Fq(\026)-693 b Fp(c)44844 42238 y Fv(0)45439 42039 y Fq(\()p Fp(q)48 b Fq(\))p Fp(;)221 b(dq)48 b Fm(i)p Fq(,)1102 43787 y Fp(\021)1810 43305 y Fv(2)1762 44117 y(0)2774 43787 y Fq(=)370 b Fm(h)29 b Fq(\026)-693 b Fp(c)5269 43986 y Fk(`)5779 43787 y Fq(\()p Fp(q)48 b Fq(\))p Fp(;)221 b(dq)48 b Fm(i)p Fq(,)424 b Fp(\021)11394 43149 y Fk(j)11346 44157 y(i)12322 43787 y Fq(=)369 b Fp(\021)14432 43149 y Fk(j)14384 44141 y Fv(0)15242 43787 y Fq(+)16526 42791 y Fi(P)17928 43119 y Fk(i)p Fg(\000)p Fv(1)17928 44185 y Fk(k)22 b Fv(=0)20253 43787 y Fq(\026)-769 b Fp(\026)20948 43149 y Fk(j)20948 44188 y(k)22023 43787 y Fq(and)421 b Fp(c)25159 43149 y Fk(j)25159 44157 y(i)26087 43787 y Fq(=)370 b([)p Fp(\021)28567 43149 y Fk(j)28519 44157 y(i)29125 43787 y Fq(],)426 b(then)420 b Fp(\021)34000 43305 y Fv(1)33952 44153 y Fk(N)34793 44286 y Fj(1)35312 44153 y Fv(+1)37092 43787 y Fq(=)369 b Fm(h)29 b Fq(\026)-693 b Fp(c)39586 43986 y Fv(1)40181 43787 y Fq(\()p Fp(q)48 b Fq(\))p Fp(;)221 b(dq)48 b Fm(i)p Fq(,)424 b Fp(\021)45796 43305 y Fv(2)45748 44153 y Fk(N)46589 44286 y Fj(2)47108 44153 y Fv(+1)48888 43787 y Fq(=)1102 45672 y Fm(h)29 b Fq(\026)-693 b Fp(c)2194 45871 y Fk(`)p Fv(+1)4050 45672 y Fq(\()p Fp(q)48 b Fq(\))p Fp(;)221 b(dq)48 b Fm(i)p Fq(.)572 b(Based)392 b(on)f(the)h(pro)37 b(of)392 b(of)f(Theorem)i(5.1)f(w)-37 b(e)393 b(can)f(c)-37 b(ho)37 b(ose)392 b(eac)-37 b(h)393 b Fp(\026)41705 45034 y Fk(j)41705 46042 y(i)42265 45672 y Fq(,)402 b(the)391 b(adapted)1102 47429 y(neigh)-37 b(b)37 b(orho)g(o)g(d)444 b Fp(U)10497 46791 y Fk(j)10352 47799 y(i)11499 47429 y Fq(and)f Fp(\017)14621 46791 y Fk(j)14621 47799 y(i)15550 47429 y Fp(>)369 b Fq(0)443 b(so)g(that)11478 50528 y Fp(\031)48 b Fq(\()13221 50193 y(~)12800 50528 y Fm(N)13890 50963 y Fk(\021)14470 50513 y Fd(j)14437 51266 y(i)14966 50963 y Fk(;\026)15927 50513 y Fd(j)15927 51266 y(i)16490 50528 y Fq(\))295 b(+)g Fp(\017)19169 49890 y Fk(j)19169 50899 y(i)20098 50528 y Fm(\032)369 b Fp(U)22552 49890 y Fk(j)22407 50899 y(i)23112 50528 y Fp(;)2878 b Fm(8)443 b Fp(j)i Fq(=)369 b(1)p Fp(;)221 b Fq(2)p Fp(;)665 b Fq(0)369 b Fm(\024)g Fp(i)g Fm(\024)g Fp(N)38617 50727 y Fk(j)39177 50528 y Fp(:)6870 b Fq(\(5.6ij\))1102 54000 y(F)-111 b(or)444 b(eac)-37 b(h)444 b Fp(\017)7014 53361 y Fk(j)54 b Fg(\003)7014 54370 y Fk(i)8487 54000 y Fp(>)369 b Fq(0,)443 b(there)g(exist)18166 53664 y(^)17907 54000 y Fp(T)18867 53361 y Fk(j)18683 54400 y(k)22 b(i)19713 54000 y Fp(;)20562 53664 y Fq(\025)20303 54000 y Fp(T)21263 53361 y Fk(j)21079 54400 y(k)g(i)22478 54000 y Fm(2)368 b Fl(Z)24618 54199 y Fv(+)25943 54000 y Fq(with)29227 53664 y(^)28968 54000 y Fp(T)29928 53361 y Fk(j)29744 54400 y(k)22 b(i)31143 54000 y Fp(>)32804 53664 y Fq(\025)32545 54000 y Fp(T)33505 53361 y Fk(j)33321 54400 y(k)g(i)34351 54000 y Fq(,)443 b(\()p Fp(k)409 b Fq(=)370 b(0)p Fp(;)221 b Fq(1\))443 b(suc)-37 b(h)443 b(that)6524 57150 y Fp(h)7289 56511 y Fk(T)7916 56644 y Fj(0)8435 56511 y Fk(;T)9377 56644 y Fj(1)7289 57797 y Fk(\021)7869 57347 y Fd(j)7836 58100 y(i)8366 57797 y Fk(;\026)9327 57347 y Fd(j)9327 58100 y(i)9962 57150 y Fq(\()p Fp(m)11645 57349 y Fv(0)12241 57150 y Fp(;)221 b(m)13997 57349 y Fv(1)14594 57150 y Fq(\))368 b Fm(\025)i Fp(h)17647 56601 y Fg(1)17647 57706 y Fk(\021)18227 57257 y Fd(j)18194 58009 y(i)18724 57706 y Fk(;\026)19685 57257 y Fd(j)19685 58009 y(i)20248 57150 y Fq(\()p Fp(m)21931 57349 y Fv(0)22527 57150 y Fp(;)221 b(m)24283 57349 y Fv(1)24879 57150 y Fq(\))295 b Fm(\000)g Fp(\017)27558 56511 y Fk(j)54 b Fg(\003)27558 57520 y Fk(i)28663 57150 y Fp(;)2656 b Fm(8)443 b Fp(T)33645 57349 y Fk(k)34668 57150 y Fm(\025)36329 56814 y Fq(\025)36070 57150 y Fp(T)37030 56511 y Fk(j)36846 57551 y(k)22 b(i)37876 57150 y Fp(;)221 b Fq(\()p Fp(k)410 b Fq(=)369 b(0)p Fp(;)221 b Fq(1\))p Fp(;)31688 59628 y Fm(8)443 b Fq(\()p Fp(m)34552 59827 y Fv(0)35148 59628 y Fp(;)221 b(m)36904 59827 y Fv(1)37500 59628 y Fq(\))369 b Fm(2)g Fp(M)439 b Fm(\002)296 b Fp(M)145 b Fq(;)1102 62893 y(for)453 b(an)-37 b(y)452 b(giv)-37 b(en)452 b(\()p Fp(m)10823 63092 y Fv(0)11419 62893 y Fp(;)221 b(m)13175 63092 y Fv(1)13772 62893 y Fq(\))384 b Fm(2)f Fp(M)446 b Fm(\002)302 b Fp(M)596 b Fq(there)452 b(exists)g Fp(T)28757 63092 y Fk(k)29795 62893 y Fq(=)385 b Fp(T)31989 63092 y Fk(k)32642 62893 y Fq(\()p Fp(m)34325 63092 y Fv(0)34921 62893 y Fp(;)221 b(m)36677 63092 y Fv(1)37274 62893 y Fq(\))451 b(with)41536 62557 y(\025)41277 62893 y Fp(T)42237 62255 y Fk(j)42053 63294 y(k)22 b(i)43467 62893 y Fm(\024)384 b Fp(T)45660 63092 y Fk(k)46698 62893 y Fm(\024)48375 62557 y Fq(^)48116 62893 y Fp(T)49076 62255 y Fk(j)48892 63294 y(k)22 b(i)1102 64487 y Fq(suc)-37 b(h)444 b(that)13815 64620 y Fi(\014)13815 65417 y(\014)13815 66214 y(\014)14258 66148 y Fp(h)15023 65509 y Fk(T)15650 65642 y Fj(0)16169 65509 y Fk(;T)17111 65642 y Fj(1)15023 66795 y Fk(\021)15603 66345 y Fd(j)15570 67098 y(i)16100 66795 y Fk(;\026)17061 66345 y Fd(j)17061 67098 y(i)17697 66148 y Fq(\()p Fp(m)19380 66347 y Fv(0)19976 66148 y Fp(;)221 b(m)21732 66347 y Fv(1)22328 66148 y Fq(\))295 b Fm(\000)g Fp(h)25233 65599 y Fg(1)25233 66704 y Fk(\021)25813 66254 y Fd(j)25780 67007 y(i)26311 66704 y Fk(;\026)27272 66254 y Fd(j)27272 67007 y(i)27835 66148 y Fq(\()p Fp(m)29518 66347 y Fv(0)30113 66148 y Fp(;)221 b(m)31869 66347 y Fv(1)32466 66148 y Fq(\))32983 64620 y Fi(\014)32983 65417 y(\014)32983 66214 y(\014)33794 66148 y Fm(\024)369 b Fp(\017)35735 65509 y Fk(j)54 b Fg(\003)35735 66518 y Fk(i)36840 66148 y Fp(:)9207 b Fq(\(5.7ij\))1102 69323 y(Let)370 b Fp(\015)4171 68684 y Fk(j)4097 69693 y(i)4730 69323 y Fq(\()p Fp(t;)221 b(m)7483 69522 y Fv(0)8079 69323 y Fp(;)g(m)9835 69522 y Fv(1)10431 69323 y Fp(;)g(T)11797 69522 y Fv(0)12394 69323 y Fp(;)g(T)13760 69522 y Fv(1)14357 69323 y Fq(\):)553 b([)p Fm(\000)p Fp(T)17974 69522 y Fv(0)18570 69323 y Fp(;)221 b(T)19936 69522 y Fv(1)20533 69323 y Fq(])369 b Fm(!)g Fp(M)514 b Fq(b)37 b(e)369 b(the)g(minimizer)h(of)f Fp(h)37052 68684 y Fk(T)37679 68817 y Fj(0)38199 68684 y Fk(;T)39141 68817 y Fj(1)37052 69970 y Fk(\021)37632 69520 y Fd(j)37599 70273 y(i)38130 69970 y Fk(;\026)39091 69520 y Fd(j)39091 70273 y(i)39726 69323 y Fq(\()p Fp(m)41409 69522 y Fv(0)42005 69323 y Fp(;)221 b(m)43761 69522 y Fv(1)44358 69323 y Fq(\).)565 b(Let)48375 68987 y(\025)48116 69323 y Fp(T)49076 68684 y Fk(j)48892 69723 y(k)22 b(i)1102 71659 y Fq(b)37 b(e)443 b(set)g(so)g(large)h(and)f Fp(\017)12988 71021 y Fk(j)54 b Fg(\003)12988 72029 y Fk(i)14461 71659 y Fp(>)369 b Fq(0)443 b(b)37 b(e)443 b(set)g(so)g(small)f(suc)-37 b(h)444 b(that)e(if)g(\(5.7ij\))h(holds,)f(then)10342 74759 y Fp(d\015)11795 74120 y Fk(j)11721 75129 y(i)12355 74759 y Fq(\()p Fp(t;)221 b(m)15108 74958 y Fv(0)15703 74759 y Fp(;)g(m)17459 74958 y Fv(1)18056 74759 y Fp(;)g(T)19422 74958 y Fv(0)20019 74759 y Fp(;)g(T)21385 74958 y Fv(1)21981 74759 y Fq(\))369 b Fm(2)24543 74423 y Fq(~)24121 74759 y Fm(N)25211 75193 y Fk(\021)25791 74744 y Fd(j)25758 75496 y(i)26288 75193 y Fk(;\026)27249 74744 y Fd(j)27249 75496 y(i)28107 74759 y Fq(+)296 b Fp(\017)29975 74120 y Fk(j)29975 75129 y(i)30535 74759 y Fp(;)2878 b Fm(8)442 b Fq(0)370 b Fm(\024)f Fp(t)f Fm(\024)h Fq(1)p Fp(:)5735 b Fq(\(5.8ij\))p eop %%Page: 44 44 44 43 bop 1102 2239 a Ft(44)1102 5427 y Fq(W)-111 b(e)444 b(de\014ne)g(the)e(index)h(set)g(for)411 b Fp(~)-632 b(\034)17932 4945 y Fk(j)18861 5427 y Fq(=)369 b(\()p Fp(\034)21511 4789 y Fk(j)21361 5781 y Fv(0)22071 5427 y Fp(;)221 b(\034)23392 4789 y Fk(j)23242 5781 y Fv(1)23952 5427 y Fp(;)g Fm(\001)g(\001)g(\001)444 b Fp(;)221 b(\034)27856 4789 y Fk(j)27706 5817 y(N)28547 5950 y Fd(j)29110 5427 y Fq(\))5582 9381 y(\003)6504 8833 y Fk(j)7433 9381 y Fq(=)8835 7907 y Fi(n)9689 9381 y Fp(~)-632 b(\034)10452 8833 y Fk(j)11381 9381 y Fm(2)368 b Fl(Z)13521 8833 y Fk(N)14362 8966 y Fd(j)15294 9381 y Fq(:)221 b(max)p Fm(f)19280 9045 y Fq(\025)19020 9381 y Fp(T)19980 8743 y Fk(j)19796 9752 y Fv(0)p Fk(i)20769 9381 y Fp(;)21618 9045 y Fq(\025)21359 9381 y Fp(T)22319 8743 y Fk(j)22135 9752 y Fv(1)p Fk(i)23402 9381 y Fq(+)295 b(1)p Fm(g)370 b(\024)f Fp(\034)28561 8743 y Fk(j)28411 9752 y(i)29416 9381 y Fm(\000)295 b Fp(\034)31475 8743 y Fk(j)31325 9752 y(i)p Fg(\000)p Fv(1)33496 9381 y Fm(\024)369 b Fq(max)p Fm(f)38293 9045 y Fq(^)38034 9381 y Fp(T)38994 8743 y Fk(j)38810 9752 y Fv(0)p Fk(i)39782 9381 y Fp(;)40632 9045 y Fq(^)40372 9381 y Fp(T)41332 8743 y Fk(j)41148 9752 y Fv(1)p Fk(i)42416 9381 y Fq(+)295 b(1)p Fm(g)p Fp(;)15663 12304 y Fm(8)p Fq(1)369 b Fm(\024)g Fp(i)g Fm(\024)g Fp(N)22132 12503 y Fk(j)22692 12304 y Fp(;)221 b(\034)24013 11666 y Fk(j)23863 12658 y Fv(0)24942 12304 y Fq(=)369 b(0)27008 10830 y Fi(o)27894 12304 y Fp(;)1102 16039 y Fq(and)423 b(in)-37 b(tro)37 b(duce)424 b(a)f(mo)37 b(di\014ed)423 b(Lagrangian)i(dep)37 b(ending)423 b(on)g(the)g (parameters)393 b Fp(~)-633 b(\034)41029 15557 y Fk(j)42012 16039 y Fq(\()p Fp(j)445 b Fq(=)369 b(1)p Fp(;)221 b Fq(2\))423 b(and)1067 17633 y Fp(~)-629 b(n)3163 22193 y Fq(~)3006 22529 y Fp(L)369 b Fq(=)5682 19408 y Fi(8)5682 20603 y(>)5682 21002 y(<)5682 23393 y(>)5682 23791 y(:)7305 20479 y Fp(L)295 b Fm(\000)h(h)29 b Fq(\026)-693 b Fp(c)10925 20678 y Fv(0)11520 20479 y Fq(\()p Fp(q)48 b Fq(\))p Fp(;)467 b Fq(_)-615 b Fp(q)47 b Fm(i)295 b(\000)g Fq(\()p Fp(\034)17813 19997 y Fv(1)17663 20845 y Fk(N)18504 20978 y Fj(1)19384 20479 y Fq(+)g(1\))21893 19997 y Fg(\003)22503 20479 y Fp(\027)23159 20678 y Fv(1)23756 20479 y Fp(;)6520 b(t)369 b Fm(\024)g Fp(n)33693 20678 y Fv(1)34289 20479 y Fp(;)7305 22506 y(L)295 b Fm(\000)h(h)p Fp(c)p Fq(\()p Fp(\033)12201 22705 y Fk(j)12759 22506 y Fq(\))f(+)h Fp(\045)15587 22705 y Fk(j)16146 22506 y Fq(\()p Fp(t)p Fq(\))p Fm(r)p Fq(\011)19800 22705 y Fk(j)20359 22506 y Fq(\()p Fp(q)48 b Fq(\))p Fp(;)466 b Fq(_)-614 b Fp(q)47 b Fm(i)p Fp(;)6333 b(n)31280 22705 y Fk(j)32209 22506 y Fm(\024)369 b Fp(t)f Fm(\024)i Fp(n)36659 22705 y Fk(j)54 b Fv(+1)38564 22506 y Fp(;)664 b Fq(1)369 b Fm(\024)g Fp(j)445 b Fm(\024)369 b Fp(`)296 b Fm(\000)f Fq(2)p Fp(;)7305 24546 y(L)g Fm(\000)h(h)29 b Fq(\026)-693 b Fp(c)10925 24064 y Fk(`)p Fg(\000)p Fv(1)12795 24546 y Fq(\()p Fp(q)48 b Fq(\))p Fp(;)467 b Fq(_)-615 b Fp(q)47 b Fm(i)294 b Fq(+)i(\()p Fp(n)19154 24745 y Fk(`)19960 24546 y Fq(+)f Fp(\034)22019 24064 y Fv(2)21869 24912 y Fk(N)22710 25045 y Fj(2)23590 24546 y Fq(+)h(1\))26100 24064 y Fg(\003)26710 24546 y Fp(\027)27366 24745 y Fv(2)27962 24546 y Fp(;)2111 b(t)369 b Fm(\025)g Fp(n)33490 24745 y Fk(`)p Fg(\000)p Fv(1)35361 24546 y Fp(;)1102 27883 y Fq(where)470 b Fp(\045)5656 28082 y Fk(j)6684 27883 y Fq(is)e(suc)-37 b(h)469 b(a)f(smo)37 b(oth)468 b(function)g(that)g Fp(\045)25737 28082 y Fk(j)26297 27883 y Fq(\()p Fp(t)p Fq(\))410 b(=)i(0)468 b(for)h Fp(t)411 b Fm(\024)35352 27360 y Fv(1)p 35352 27578 530 54 v 35352 28341 a(2)36041 27883 y Fq(\()p Fp(n)37355 28082 y Fk(j)54 b Fv(+1)39572 27883 y Fm(\000)312 b Fp(n)41714 28082 y Fk(j)42274 27883 y Fq(\),)474 b(0)412 b Fp(<)f(\045)46841 28082 y Fk(j)47812 27883 y Fp(<)h Fq(1)1102 29596 y(when)4763 29073 y Fv(1)p 4763 29291 V 4763 30055 a(2)5452 29596 y Fq(\()p Fp(n)6766 29795 y Fk(j)54 b Fv(+1)8988 29596 y Fm(\000)317 b Fp(n)11135 29795 y Fk(j)11695 29596 y Fq(\))423 b Fp(<)h(t)g(<)16612 29073 y Fv(1)p 16612 29291 V 16612 30055 a(2)17301 29596 y Fq(\()p Fp(n)18615 29795 y Fk(j)54 b Fv(+1)20837 29596 y Fm(\000)317 b Fp(n)22984 29795 y Fk(j)23544 29596 y Fq(\))g(+)g(1)476 b(and)g Fp(\045)30171 29795 y Fk(j)31154 29596 y Fq(=)424 b(1)476 b(when)g Fp(t)424 b Fm(\025)39772 29073 y Fv(1)p 39772 29291 V 39772 30055 a(2)40461 29596 y Fq(\()p Fp(n)41775 29795 y Fk(j)54 b Fv(+1)43997 29596 y Fm(\000)317 b Fp(n)46144 29795 y Fk(j)46704 29596 y Fq(\))g(+)g(1,)1102 31410 y(this)443 b(function)f(is)h(w)-37 b(ell)444 b(de\014ned)f(if)g Fp(n)19515 31609 y Fk(j)20370 31410 y Fm(\000)295 b Fp(n)22495 31609 y Fk(j)54 b Fg(\000)p Fv(1)24784 31410 y Fm(\025)369 b Fq(4.)591 b(Clearly)-111 b(,)32841 31074 y(~)32684 31410 y Fp(L)443 b Fq(is)g(smo)37 b(oth)442 b(in)7175 36004 y(\()246 b(_)-615 b Fp(q)47 b(;)221 b(q)48 b(;)221 b(t)p Fq(\))369 b Fm(2)g Fp(T)184 b(M)440 b Fm(\002)16791 34529 y Fi(n)17677 36004 y Fl(R)p Fm(n)19521 34343 y Fk(`)p Fg(\000)p Fv(1)19686 34742 y Fi([)19564 37571 y Fk(i)p Fv(=1)21326 36004 y Fm(f)p Fp(n)22787 36203 y Fk(i)23230 36004 y Fm(g)23894 34529 y Fi(o)25075 36004 y Fm([)26256 34343 y Fk(`)p Fg(\000)p Fv(1)26420 34742 y Fi([)26297 37571 y Fk(i)p Fv(=1)28281 36004 y Fp(T)184 b(B)30249 36203 y Fk(\017)30759 36004 y Fq(\()p Fp(x)32035 36203 y Fk(i)32477 36004 y Fq(\))295 b Fm(\002)34617 34529 y Fi(\020)35410 36004 y Fp(n)36207 36203 y Fk(i)36945 36004 y Fm(\000)g Fq(1)p Fp(;)221 b(n)40324 36203 y Fk(i)41063 36004 y Fq(+)295 b(1)43055 34529 y Fi(\021)1102 40477 y Fq(F)-111 b(or)444 b(eac)-37 b(h)444 b(\()p Fp(m;)221 b(m)9914 39995 y Fg(0)10287 40477 y Fq(\))369 b Fm(2)f Fp(M)440 b Fm(\002)295 b Fp(M)145 b Fq(,)443 b Fp(Q)369 b Fq(=)g(\()p Fp(q)21661 40676 y Fv(1)22256 40477 y Fp(;)221 b Fm(\001)g(\001)g(\001)444 b Fp(;)221 b(q)26022 40676 y Fk(`)p Fg(\000)p Fv(1)27893 40477 y Fq(\))369 b Fm(2)f Fl(B)443 b Fq(let)4026 44773 y Fp(h)4791 44063 y Fk(T)5418 44196 y Fj(0)5937 44063 y Fk(;T)6879 44204 y Fd(`)p Fj(+1)4920 45153 y Fv(~)4791 45388 y Fk(L)8544 44773 y Fq(\()p Fp(m;)221 b(m)11983 44224 y Fg(0)12357 44773 y Fp(;)g(Q;)190 b(~)-633 b(\034)15318 44224 y Fv(1)15914 44773 y Fp(;)190 b(~)-633 b(\034)17235 44224 y Fv(2)17832 44773 y Fp(;)186 b(~)-629 b(n)q Fq(\))368 b(=)2308 b(inf)21574 45704 y Fk(\015)57 b Fv(\()p Fg(\000)p Fk(T)24204 45303 y Fe(\003)24061 45970 y Fj(0)24760 45704 y Fv(\)=)p Fk(m)21360 46875 y(\015)g Fv(\()p Fk(T)23160 46474 y Fe(\003)23017 47141 y Fd(`)p Fj(+1)24615 46875 y Fv(\)=)p Fk(m)26788 46474 y Fe(0)22055 48084 y Fk(\015)g Fv(\()p Fk(n)23742 48217 y Fd(j)24238 48084 y Fv(\)=)p Fk(q)25955 48217 y Fd(j)21553 49127 y Fk(j)d Fv(=1)p Fk(;)p Fg(\001\001\001)183 b Fk(;`)p Fg(\000)p Fv(1)27369 42965 y Fi(Z)28697 43293 y Fk(T)29467 42892 y Fe(\003)29324 43559 y Fd(`)p Fj(+1)28107 45983 y Fg(\000)p Fk(T)29707 45663 y Fe(\003)29564 46271 y Fj(0)31367 44437 y Fq(~)31210 44773 y Fp(L)p Fq(\()p Fp(d\015)74 b Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))p Fp(dt)21433 51368 y Fm(\000)23754 50106 y Fi(X)22762 52957 y Fv(1)p Fg(\024)p Fk(i)p Fg(\024)p Fk(N)26168 53090 y Fd(j)23372 53970 y Fk(j)54 b Fv(=1)p Fk(;)p Fv(2)26665 51368 y Fq(\()p Fp(\034)27913 50729 y Fk(j)27763 51738 y(i)28768 51368 y Fm(\000)295 b Fp(\034)30827 50729 y Fk(j)30677 51738 y(i)p Fg(\000)p Fv(1)32479 51368 y Fq(\))p Fp(\013)5 b Fq(\()p Fp(c)34943 50729 y Fk(j)34943 51738 y(i)35501 51368 y Fq(\))295 b Fm(\000)g Fp(n)38438 51567 y Fv(0)39034 51368 y Fp(\013)5 b Fq(\()p Fp(c)40981 51567 y Fv(1)41576 51368 y Fq(\))21433 55812 y Fm(\000)296 b Fq(\()p Fp(n)24076 56011 y Fk(`)24882 55812 y Fm(\000)f Fp(n)27007 56011 y Fk(`)p Fg(\000)p Fv(1)28878 55812 y Fq(\))p Fp(\013)5 b Fq(\()p Fp(c)31342 56011 y Fk(`)31852 55812 y Fq(\))295 b Fm(\000)g Fp(T)34768 56011 y Fv(0)35364 55812 y Fp(\013)5 b Fq(\()p Fp(c)37311 56011 y Fv(0)37906 55812 y Fq(\))295 b Fm(\000)h Fp(T)40823 56011 y Fk(`)p Fv(+1)42679 55812 y Fp(\013)5 b Fq(\()p Fp(c)44626 56011 y Fk(`)p Fv(+1)46481 55812 y Fq(\))1102 59199 y(where)506 b Fp(T)5965 58716 y Fg(\003)5781 59528 y Fv(0)7048 59199 y Fq(=)472 b Fp(T)9329 59398 y Fv(0)10262 59199 y Fq(+)336 b Fp(\034)12362 58716 y Fv(1)12212 59564 y Fk(N)13053 59697 y Fj(1)13974 59199 y Fq(+)h(1,)520 b Fp(T)17857 58716 y Fg(\003)17673 59575 y Fk(`)p Fv(+1)20001 59199 y Fq(=)472 b Fp(T)22282 59398 y Fk(`)p Fv(+1)24474 59199 y Fq(+)337 b Fp(n)26641 59398 y Fk(`)27489 59199 y Fq(+)f Fp(\034)29589 58716 y Fv(2)29439 59564 y Fk(N)30280 59697 y Fj(2)31201 59199 y Fq(+)h(1)504 b(and)h Fp(T)37160 59398 y Fv(0)37756 59199 y Fp(;)221 b(T)39122 59398 y Fk(`)p Fv(+1)41450 59199 y Fp(>)472 b Fq(0.)776 b(In)504 b(virtue)1102 60829 y(of)480 b(the)g(lemma)h(5.4,)490 b(w)-37 b(e)481 b(can)f(tak)-37 b(e)481 b(su\016cien)-37 b(tly)480 b(large)h Fp(n)30003 60347 y Fg(0)30003 61158 y Fv(1)31079 60829 y Fq(so)f(that)g(an)-37 b(y)480 b Fp(c)p Fq(\()p Fp(\033)40060 61028 y Fv(1)40656 60829 y Fq(\)-minimal)f(curv)-37 b(e)1102 62423 y Fp(\015)1790 62622 y Fv(1)2386 62423 y Fq(:)749 b([0)p Fp(;)221 b(n)5924 62622 y Fv(1)6521 62423 y Fq(])501 b Fm(!)h Fp(M)667 b Fq(with)522 b Fp(n)15079 62622 y Fv(1)16176 62423 y Fm(\025)502 b Fp(n)18508 61941 y Fg(0)18508 62752 y Fv(1)19626 62423 y Fq(has)523 b(a)f(p)37 b(oin)-37 b(t)522 b Fp(\015)27497 62622 y Fv(1)28093 62423 y Fq(\()p Fp(n)29407 62622 y Fv(0)30002 62423 y Fq(\))501 b Fm(2)g Fp(B)33415 62622 y Fk(\016)33932 62356 y Fe(\003)34553 62423 y Fq(\()p Fp(z)35746 61941 y Fg(\003)35688 62752 y Fv(1)36356 62423 y Fq(\))522 b(with)g Fp(n)41297 62622 y Fv(1)42242 62423 y Fm(\000)348 b Fp(n)44420 62622 y Fv(0)45518 62423 y Fm(\025)501 b Fq(\001)p Fp(n)48956 62622 y Fv(1)49552 62423 y Fq(.)1102 64017 y(Similarly)-111 b(,)675 b(w)-37 b(e)629 b(can)f(tak)-37 b(e)628 b(su\016cien)-37 b(tly)628 b(large)h(\001)p Fp(n)27239 63535 y Fg(0)27239 64393 y Fk(`)28378 64017 y Fq(so)f(that)f(an)-37 b(y)628 b Fp(c)p Fq(\()p Fp(\033)37802 64216 y Fk(`)38312 64017 y Fq(\)-minimal)g(curv)-37 b(e)628 b Fp(\015)49041 64216 y Fk(`)49552 64017 y Fq(:)1102 65611 y([)p Fp(n)2268 65810 y Fk(`)p Fg(\000)p Fv(1)4139 65611 y Fp(;)221 b(n)5526 65129 y Fg(0)5526 65987 y Fk(`)6038 65611 y Fq(])369 b Fm(!)h Fp(M)467 b Fq(with)323 b Fp(n)13933 65129 y Fg(0)13933 65987 y Fk(`)14500 65611 y Fm(\000)56 b Fp(n)16386 65810 y Fk(`)p Fg(\000)p Fv(1)18626 65611 y Fm(\025)369 b Fq(\001)p Fp(n)21932 65129 y Fg(0)21932 65987 y Fk(`)22767 65611 y Fq(has)323 b(a)g(p)37 b(oin)-37 b(t)323 b Fp(\015)30040 65810 y Fk(`)30551 65611 y Fq(\()p Fp(n)31865 65810 y Fk(`)32376 65611 y Fq(\))368 b Fm(2)h Fp(B)35524 65810 y Fk(\016)36041 65544 y Fe(\003)36662 65611 y Fq(\()p Fp(z)37797 65810 y Fk(`)p Fv(+1)39652 65611 y Fq(\))323 b(with)g Fp(n)44195 65129 y Fg(0)44195 65987 y Fk(`)44762 65611 y Fm(\000)56 b Fp(n)46648 65810 y Fk(`)p Fg(\000)p Fv(1)48888 65611 y Fp(>)1102 67205 y(n)1899 67404 y Fk(`)2775 67205 y Fm(\000)365 b Fp(n)4970 67404 y Fk(`)p Fg(\000)p Fv(1)7383 67205 y Fm(\025)543 b Fq(\001)p Fp(n)10863 67404 y Fk(`)11374 67205 y Fq(.)903 b(W)-111 b(e)547 b(can)h(also)f(tak)-37 b(e)547 b(suitable)g(large)g Fp(n)32534 67404 y Fk(i)33524 67205 y Fq(\()p Fp(i)541 b Fq(=)i(2)p Fp(;)221 b Fq(3)p Fp(;)g Fm(\001)g(\001)g(\001)444 b Fp(;)221 b(`)366 b Fm(\000)e Fq(1\))547 b(so)g(that)1102 68799 y Fp(n)1899 68998 y Fk(i)p Fv(+1)3858 68799 y Fm(\000)171 b Fp(n)5859 68998 y Fk(i)6671 68799 y Fm(\025)369 b Fq(\001)p Fp(n)9977 68998 y Fk(i)10800 68799 y Fq(for)381 b(eac)-37 b(h)382 b(1)369 b Fm(\024)g Fp(i)f Fm(\024)i Fp(`)171 b Fm(\000)g Fq(1.)570 b(Under)381 b(these)g(conditions)g(w)-37 b(e)382 b(tak)-37 b(e)381 b(the)f(minim)-37 b(um)1102 70586 y(of)443 b Fp(h)3380 69877 y Fk(T)4007 70010 y Fj(0)4526 69877 y Fk(;T)5468 70018 y Fd(`)p Fj(+1)3509 70967 y Fv(~)3380 71202 y Fk(L)7133 70586 y Fq(\()p Fp(m;)221 b(m)10572 70104 y Fg(0)10946 70586 y Fp(;)g(Q;)190 b(~)-633 b(\034)13907 70104 y Fv(1)14504 70586 y Fp(;)190 b(~)-633 b(\034)15825 70104 y Fv(2)16421 70586 y Fp(;)186 b(~)-629 b(n)q Fq(\))442 b(o)-37 b(v)g(er)444 b Fl(B)8334 74321 y Fp(h)9099 73611 y Fk(T)9726 73744 y Fj(0)10246 73611 y Fk(;T)11188 73752 y Fd(`)p Fj(+1)9228 74701 y Fv(~)9099 74936 y Fk(L)12853 74321 y Fq(\()p Fp(m;)221 b(m)16292 73772 y Fg(0)16665 74321 y Fp(;)190 b(~)-633 b(\034)17986 73772 y Fv(1)18582 74321 y Fp(;)190 b(~)-633 b(\034)19903 73772 y Fv(2)20500 74321 y Fp(;)186 b(~)-629 b(n)p Fq(\))369 b(=)g(min)24196 75182 y Fk(Q)p Fg(2)p Fc(B)26610 74321 y Fp(h)27375 73611 y Fk(T)28002 73744 y Fj(0)28521 73611 y Fk(;T)29463 73752 y Fd(`)p Fj(+1)27504 74701 y Fv(~)27375 74936 y Fk(L)31129 74321 y Fq(\()p Fp(m;)221 b(m)34568 73772 y Fg(0)34941 74321 y Fp(;)g(Q;)190 b(~)-633 b(\034)37902 73772 y Fv(1)38499 74321 y Fp(;)190 b(~)-633 b(\034)39820 73772 y Fv(2)40416 74321 y Fp(;)186 b(~)-629 b(n)q Fq(\))p Fp(:)p eop %%Page: 45 45 45 44 bop 48792 2239 a Ft(45)1102 5443 y Fq(Let)457 b Fp(\015)74 b Fq(\()p Fp(t)p Fq(\))390 b(=)i Fp(\015)74 b Fq(\()p Fp(t;)221 b(m;)g(m)12858 4961 y Fg(0)13230 5443 y Fp(;)190 b(~)-633 b(\034)14551 4961 y Fv(1)15148 5443 y Fp(;)190 b(~)-633 b(\034)16469 4961 y Fv(2)17065 5443 y Fp(;)186 b(~)-629 b(n)q Fq(\))456 b(b)37 b(e)456 b(the)g(minimizer)i(of)e Fp(h)32056 4734 y Fk(T)32683 4867 y Fj(0)33202 4734 y Fk(;T)34144 4875 y Fd(`)p Fj(+1)32185 5824 y Fv(~)32056 6059 y Fk(L)35809 5443 y Fq(\()p Fp(m;)221 b(m)39248 4961 y Fg(0)39622 5443 y Fp(;)190 b(~)-633 b(\034)40943 4961 y Fv(1)41539 5443 y Fp(;)190 b(~)-633 b(\034)42860 4961 y Fv(2)43456 5443 y Fp(;)186 b(~)-629 b(n)q Fq(\).)631 b(Recall)1102 7114 y(the)412 b(supp)37 b(ort)412 b(of)g(\011)10762 7313 y Fk(i)11616 7114 y Fq(is)g(a)h(small)f(ball.)580 b(F)-111 b(or)413 b(eac)-37 b(h)413 b(cohomology)g(class)f(under)h(our)f(consideration)1102 8708 y(here,)422 b(the)416 b(supp)37 b(ort)415 b(of)h(the)f(minimal)h (measure)g(is)g(on)f(the)h(cylinder,)422 b(the)415 b(h)-37 b(yp)37 b(erb)g(olicit)-37 b(y)417 b(of)e(the)1102 10302 y(cylinder)357 b(let)e(us)g(see)g(that)g Fp(\015)74 b Fq(\()p Fp(t)p Fq(\))353 b(is)i(outside)g(of)h(the)f(supp)37 b(ort)355 b(of)g Fm(r)p Fq(\011)34766 10501 y Fk(i)35563 10302 y Fq(if)g(b)37 b(oth)355 b Fp(t)120 b Fm(\000)g Fp(n)42292 10501 y Fk(i)43089 10302 y Fq(and)355 b Fp(n)46381 10501 y Fk(i)p Fv(+1)48289 10302 y Fm(\000)120 b Fp(t)1102 11896 y Fq(are)470 b(suitably)f(large,)476 b(in)469 b(other)g(w)-37 b(ords,)476 b(for)469 b Fp(t)412 b Fm(2)g Fq([)26151 11373 y Fv(1)p 26152 11591 530 54 v 26152 12354 a(2)26841 11896 y Fq(\()p Fp(n)28155 12095 y Fk(i)p Fv(+1)30254 11896 y Fm(\000)313 b Fp(n)32397 12095 y Fk(i)32840 11896 y Fq(\))p Fp(;)34106 11373 y Fv(1)p 34106 11591 V 34106 12354 a(2)34795 11896 y Fq(\()p Fp(n)36109 12095 y Fk(i)p Fv(+1)38208 11896 y Fm(\000)g Fp(n)40351 12095 y Fk(i)40794 11896 y Fq(\))f(+)h(1],)475 b Fp(\015)74 b Fq(\()p Fp(t)p Fq(\))467 b(falls)1102 13490 y(in)-37 b(to)508 b(the)g(area)h(where)h Fm(h)p Fp(\045)14272 13689 y Fk(i)14713 13490 y Fq(\()p Fp(t)p Fq(\))p Fm(r)p Fq(\011)18367 13689 y Fk(i)18808 13490 y Fq(\()p Fp(q)48 b Fq(\))p Fp(;)221 b(dq)48 b Fm(i)507 b Fq(is)h(exact.)787 b(Th)-37 b(us,)525 b Fp(d\015)582 b Fq(solv)-37 b(es)508 b(the)g(Euler-Lagrange)1102 15084 y(equation)345 b(of)f Fp(L)g Fq(for)g Fp(t)369 b Fm(2)g Fq([)p Fp(n)14294 15283 y Fv(0)14890 15084 y Fp(;)221 b(n)16277 15283 y Fk(`)16789 15084 y Fq(])344 b(if)g(w)-37 b(e)345 b(rep)37 b(eat)344 b(the)g(argumen)-37 b(t)345 b(for)f(the)g(function)g Fh(h)p Fq(\()p Fp(Q;)221 b(z)45333 15283 y Fv(1)45929 15084 y Fp(;)g(z)47137 15283 y Fk(`)47648 15084 y Fp(;)186 b(~)-629 b(n)q Fq(\).)2696 17545 y(Next,)641 b(b)-37 b(y)602 b(c)-37 b(ho)37 b(osing)602 b(su\016cien)-37 b(tly)601 b(larger)h(v)-74 b(alue)602 b(for)30844 17209 y(\025)30585 17545 y Fp(T)31545 17063 y Fv(1)31361 17911 y(1)p Fk(N)32731 18044 y Fj(1)33317 17545 y Fq(,)34585 17209 y(^)34326 17545 y Fp(T)35286 17063 y Fv(1)35102 17911 y(1)p Fk(N)36472 18044 y Fj(1)37058 17545 y Fq(,)38327 17209 y(\025)38068 17545 y Fp(T)39028 17063 y Fv(2)38844 17875 y(00)40571 17545 y Fq(and)43571 17209 y(^)43312 17545 y Fp(T)44272 17063 y Fv(2)44088 17875 y(00)45815 17545 y Fq(w)-37 b(e)602 b(can)1102 19435 y(assume)5948 19099 y(\025)5689 19435 y Fp(T)6649 18953 y Fv(1)6465 19801 y(1)p Fk(N)7835 19934 y Fj(1)8790 19435 y Fm(\025)369 b Fp(n)10989 18953 y Fg(0)10989 19764 y Fv(1)12025 19435 y Fq(and)14863 19099 y(\025)14604 19435 y Fp(T)15564 18953 y Fv(2)15380 19764 y(00)16875 19435 y Fm(\025)g Fp(n)19074 19634 y Fk(`)p Fg(\000)p Fv(1)21233 19435 y Fq(+)288 b(\001)p Fp(n)24458 18953 y Fg(0)24458 19811 y Fk(`)24970 19435 y Fq(.)589 b(In)439 b(this)g(case,)h(let)g(us)f (consider)h(the)f(minim)-37 b(um)1102 21411 y(of)389 b Fp(h)3326 20702 y Fk(T)3953 20835 y Fj(0)4472 20702 y Fk(;T)5414 20843 y Fd(`)p Fj(+1)3455 21791 y Fv(~)3326 22026 y Fk(L)7080 21411 y Fq(\()p Fp(m;)221 b(m)10519 20929 y Fg(0)10892 21411 y Fp(;)190 b(~)-633 b(\034)12213 20929 y Fv(1)12809 21411 y Fp(;)190 b(~)-633 b(\034)14130 20929 y Fv(2)14726 21411 y Fp(;)186 b(~)-629 b(n)q Fq(\))389 b(o)-37 b(v)g(er)389 b(\003)20732 20929 y Fv(1)21516 21411 y Fm(\002)187 b Fq(\003)23658 20929 y Fv(2)24442 21411 y Fm(\002)g(f)26586 21075 y Fq(\025)26326 21411 y Fp(T)27286 20929 y Fv(1)27102 21777 y(1)p Fk(N)28472 21910 y Fj(1)29428 21411 y Fm(\024)369 b Fp(n)31627 21610 y Fv(1)32592 21411 y Fm(\024)34253 21075 y Fq(^)33994 21411 y Fp(T)34954 20929 y Fv(1)34770 21777 y(1)p Fk(N)36140 21910 y Fj(1)36726 21411 y Fm(g)187 b(\002)g(f)39721 21075 y Fq(\025)39461 21411 y Fp(T)40421 20929 y Fv(2)40237 21741 y(00)41551 21411 y Fq(+)g Fp(n)43568 21610 y Fk(`)p Fg(\000)p Fv(1)45809 21411 y Fm(\024)369 b Fp(n)48008 21610 y Fk(`)48888 21411 y Fm(\024)1361 23082 y Fq(^)1102 23417 y Fp(T)2062 22935 y Fv(2)1878 23747 y(00)3299 23417 y Fq(+)296 b Fp(n)5425 23616 y Fk(`)p Fg(\000)p Fv(1)7296 23417 y Fm(g)6778 26554 y Fp(h)7543 25845 y Fk(T)8170 25978 y Fj(0)8689 25845 y Fk(;T)9631 25986 y Fd(`)p Fj(+1)7672 26935 y Fv(~)7543 27170 y Fk(L)11296 26554 y Fq(\()p Fp(m;)221 b(m)14735 26006 y Fg(0)15109 26554 y Fq(\))368 b(=)5357 b(min)20240 27604 y Fk(~)-528 b(\034)20853 27204 y Fj(1)21372 27604 y Fg(2)p Fv(\003)22815 27204 y Fj(1)23333 27604 y Fk(;)-16 b(~)-528 b(\034)24245 27204 y Fj(2)24765 27604 y Fg(2)p Fv(\003)26208 27204 y Fj(2)19901 28566 y Fv(\025)19694 28802 y Fk(T)20464 28401 y Fj(1)20321 29067 y(1)p Fd(N)21487 29266 y Fj(1)22073 28802 y Fg(\024)p Fk(n)23560 28935 y Fj(1)24079 28802 y Fg(\024)25115 28566 y Fv(^)24909 28802 y Fk(T)25679 28401 y Fj(1)25536 29067 y(1)p Fd(N)26702 29266 y Fj(1)17603 30048 y Fv(\025)17396 30283 y Fk(T)18166 29882 y Fj(2)18023 30548 y(00)18994 30283 y Fv(+)p Fk(n)20466 30424 y Fd(`)p Fe(\000)p Fj(1)22101 30283 y Fg(\024)p Fk(n)23588 30424 y Fd(`)24051 30283 y Fg(\024)25087 30048 y Fv(^)24881 30283 y Fk(T)25651 29882 y Fj(2)25508 30548 y(00)26479 30283 y Fv(+)p Fk(n)27951 30424 y Fd(`)p Fe(\000)p Fj(1)29807 26554 y Fp(h)30572 25845 y Fk(T)31199 25978 y Fj(0)31718 25845 y Fk(;T)32660 25986 y Fd(`)p Fj(+1)30701 26935 y Fv(~)30572 27170 y Fk(L)34325 26554 y Fq(\()p Fp(m;)221 b(m)37764 26006 y Fg(0)38138 26554 y Fp(;)190 b(~)-633 b(\034)39459 26006 y Fv(1)40055 26554 y Fp(;)190 b(~)-633 b(\034)41376 26006 y Fv(2)41972 26554 y Fp(;)186 b(~)-629 b(n)q Fq(\))p Fp(:)1102 33239 y Fq(Denote)443 b(b)-37 b(y)412 b Fp(~)-633 b(\034)8235 32757 y Fk(j)54 b Fg(\003)9339 33239 y Fq(,)443 b Fp(n)10948 32757 y Fg(\003)10948 33568 y Fv(1)12002 33239 y Fq(and)g Fp(n)15382 32757 y Fg(\003)15382 33615 y Fk(`)16436 33239 y Fq(where)h(the)f(the)f(minim)-37 b(um)443 b(is)g(reac)-37 b(hed.)592 b(Let)443 b Fp(\015)74 b Fq(\()p Fp(t)p Fq(\))367 b(=)i Fp(\015)74 b Fq(\()p Fp(t;)221 b(m;)g(m)49179 32757 y Fg(0)49552 33239 y Fp(;)1102 35092 y(T)1878 35291 y Fv(0)2475 35092 y Fp(;)g(T)3841 35291 y Fk(`)p Fv(+1)5697 35092 y Fq(\))525 b(b)37 b(e)526 b(the)f(minimizer)h(of)g Fp(h)19646 34383 y Fk(T)20273 34516 y Fj(0)20792 34383 y Fk(;T)21734 34524 y Fd(`)p Fj(+1)19774 35473 y Fv(~)19646 35708 y Fk(L)23399 35092 y Fq(\()p Fp(m;)221 b(m)26838 34610 y Fg(0)27211 35092 y Fq(\).)838 b(Let)526 b Fp(\034)31979 35291 y Fv(1)p Fk(j)33574 35092 y Fq(=)507 b Fp(\034)35845 34610 y Fv(1)35695 35438 y Fk(j)36791 35092 y Fm(\000)351 b Fp(\034)38906 34610 y Fv(1)38756 35458 y Fk(N)39597 35591 y Fj(1)40532 35092 y Fm(\000)f Fq(1,)546 b Fp(\034)44075 35291 y Fv(2)p Fk(j)45671 35092 y Fq(=)507 b Fp(\034)47942 34610 y Fv(2)47792 35438 y Fk(j)48888 35092 y Fq(+)1102 36922 y Fp(\034)1833 36440 y Fv(2)1683 37288 y Fk(N)2524 37421 y Fj(2)3248 36922 y Fq(+)139 b(1)g(+)g Fp(n)7192 37121 y Fk(`)7703 36922 y Fq(.)564 b(F)-111 b(rom)366 b(the)e(pro)37 b(of)365 b(of)g(Theorem)h(5.1)f(w)-37 b(e)365 b(can)g(see)h(that)e(\(5.6ij\),) 379 b(\(5.7ij\))364 b(and)h(\(5.8ij\))1102 38550 y(hold)443 b(for)g(\()p Fm(\000)p Fp(\034)8218 38749 y Fk(ij)9153 38550 y Fq(\))9670 38068 y Fg(\003)10281 38550 y Fp(\015)516 b Fq(at)442 b Fp(j)k Fq(=)369 b(1)p Fp(;)221 b Fq(2,)443 b(0)369 b Fm(\024)g Fp(i)g Fm(\024)g Fp(N)23964 38749 y Fk(j)24967 38550 y Fq(except)444 b(for)f(\()p Fp(i;)221 b(j)76 b Fq(\))368 b(=)h(\(0)p Fp(;)221 b Fq(1\))p Fp(;)g Fq(\()p Fp(N)40771 38749 y Fv(2)41367 38550 y Fp(;)g Fq(2\).)2696 40881 y(As)443 b(the)g(third)g(step)f(w)-37 b(e)444 b(consider)g(the)f(limit)f(in\014m)-37 b(um)14626 43944 y Fp(h)15391 43396 y Fg(1)15519 44234 y Fv(~)15391 44469 y Fk(L)16517 43944 y Fq(\()p Fp(m;)221 b(m)19956 43396 y Fg(0)20329 43944 y Fq(\))369 b(=)699 b(lim)221 b(inf)23157 44801 y Fk(T)23784 44934 y Fj(0)24302 44801 y Fg(!1)22617 45705 y Fk(T)23244 45846 y Fd(`)p Fj(+1)24842 45705 y Fg(!1)27181 43944 y Fp(h)27946 43235 y Fk(T)28573 43368 y Fj(0)29093 43235 y Fk(;T)30035 43376 y Fd(`)p Fj(+1)28075 44325 y Fv(~)27946 44560 y Fk(L)31700 43944 y Fq(\()p Fp(m;)g(m)35139 43396 y Fg(0)35512 43944 y Fq(\))p Fp(:)1102 48677 y Fq(Let)443 b Fp(T)4442 48195 y Fk(k)4258 49007 y Fv(0)5097 48677 y Fq(,)f Fp(T)6868 48195 y Fk(k)6684 49053 y(`)p Fv(+1)8983 48677 y Fq(b)37 b(e)444 b(the)e(subsequence)i(so)f(that)f Fp(T)26053 48195 y Fk(k)25869 49007 y Fv(0)27077 48677 y Fm(!)369 b(1)p Fq(,)443 b Fp(T)31874 48195 y Fk(k)31690 49053 y(`)p Fv(+1)33915 48677 y Fm(!)370 b(1)443 b Fq(as)g Fp(k)410 b Fm(!)370 b(1)p Fq(,)9156 52320 y Fm(j)p Fp(h)10290 51486 y Fk(T)11060 51085 y Fd(k)10917 51752 y Fj(0)11641 51486 y Fk(;T)12726 51085 y Fd(k)12583 51752 y(`)p Fj(+1)10418 52700 y Fv(~)10290 52935 y Fk(L)14248 52320 y Fq(\()p Fp(m;)221 b(m)17687 51772 y Fg(0)18060 52320 y Fq(\))295 b Fm(\000)h Fp(h)20966 51772 y Fg(1)21094 52609 y Fv(~)20966 52845 y Fk(L)22091 52320 y Fq(\()p Fp(m;)221 b(m)25530 51772 y Fg(0)25904 52320 y Fq(\))p Fm(j)368 b(\024)h Fq(min)p Fm(f)p Fp(\017)31977 51772 y Fv(1)p Fg(\003)31977 52648 y Fv(0)33118 52320 y Fp(;)221 b(\017)34247 51772 y Fv(2)p Fg(\003)34247 52648 y Fk(N)35088 52781 y Fj(2)35674 52320 y Fm(g)p Fp(;)2878 b Fm(8)443 b Fp(k)42 b(;)1102 55318 y Fq(and)15773 57147 y(lim)15343 58014 y Fk(k)22 b Fg(!1)18269 57147 y Fp(h)19034 56313 y Fk(T)19804 55912 y Fd(k)19661 56579 y Fj(0)20385 56313 y Fk(;T)21470 55912 y Fd(k)21327 56579 y(`)p Fj(+1)19163 57527 y Fv(~)19034 57762 y Fk(L)22993 57147 y Fq(\()p Fp(m;)221 b(m)26432 56598 y Fg(0)26805 57147 y Fq(\))368 b(=)i Fp(h)29858 56598 y Fg(1)29986 57436 y Fv(~)29858 57671 y Fk(L)30983 57147 y Fq(\()p Fp(m;)221 b(m)34422 56598 y Fg(0)34796 57147 y Fq(\))p Fp(;)1102 60787 y Fq(and)557 b(let)f Fp(\015)6519 60986 y Fk(k)7172 60787 y Fq(:)818 b([)p Fm(\000)p Fp(T)10721 60305 y Fk(k)22 b Fg(\003)10537 61117 y Fv(0)11919 60787 y Fp(;)221 b(T)13469 60305 y Fk(k)22 b Fg(\003)13285 61163 y Fk(`)p Fv(+1)15142 60787 y Fq(])558 b Fm(!)h Fp(M)700 b Fq(b)37 b(e)557 b(the)f(minimizer)h(of)g Fp(h)32976 59953 y Fk(T)33746 59552 y Fd(k)33603 60219 y Fj(0)34327 59953 y Fk(;T)35412 59552 y Fd(k)35269 60219 y(`)p Fj(+1)33104 61167 y Fv(~)32976 61402 y Fk(L)36934 60787 y Fq(\()p Fp(m;)221 b(m)40373 60305 y Fg(0)40746 60787 y Fq(\),)585 b(where)557 b Fp(T)47131 60305 y Fk(k)22 b Fg(\003)46947 61117 y Fv(0)48888 60787 y Fq(=)1102 62663 y Fp(T)2062 62181 y Fk(k)1878 62993 y Fv(0)2891 62663 y Fq(+)174 b Fp(\034)4829 62181 y Fv(1)p Fg(\003)4679 63029 y Fk(N)100 b Fv(1)6391 62663 y Fq(+)174 b(1,)395 b Fp(T)9986 62181 y Fk(k)22 b Fg(\003)9802 63039 y Fk(`)p Fv(+1)12027 62663 y Fq(=)369 b Fp(T)14389 62181 y Fk(k)14205 63039 y(`)p Fv(+1)16236 62663 y Fq(+)174 b Fp(n)18240 62181 y Fg(\003)18240 63039 y Fk(`)19026 62663 y Fq(+)g Fp(\034)20964 62181 y Fv(2)p Fg(\003)20814 63029 y Fk(N)21655 63162 y Fj(2)22415 62663 y Fq(+)g(1.)571 b(By)382 b(the)g(similar)h (argumen)-37 b(t)383 b(to)f(pro)-37 b(v)g(e)384 b(Theorem)1102 64291 y(5.1)494 b(w)-37 b(e)494 b(can)g(see)h(\(5.6ij\),)505 b(\(5.7ij\))493 b(and)g(\(5.8ij\))g(hold)h(for)g(\()p Fm(\000)p Fp(\034)32213 64490 y Fk(ij)33148 64291 y Fq(\))33665 63809 y Fg(\003)34275 64291 y Fp(\015)34963 64490 y Fk(k)36110 64291 y Fq(at)f(\()p Fp(i;)221 b(j)76 b Fq(\))452 b(=)i(\(0)p Fp(;)221 b Fq(1\))p Fp(;)g Fq(\()p Fp(N)47554 64490 y Fv(2)48150 64291 y Fp(;)g Fq(2\))1102 65885 y(also.)893 b(In)543 b(this)f(case,)570 b Fp(d\015)13722 66084 y Fk(k)14919 65885 y Fq(is)543 b(a)g(solution)g(of)h(the)f (Euler-Lagrange)i(equation)f(induced)g(b)-37 b(y)543 b Fp(L)p Fq(.)1102 67499 y(F)-111 b(or)466 b(eac)-37 b(h)466 b(small)f Fp(\016)50 b Fq(,)471 b Fp(d\015)12876 67698 y Fk(k)13994 67499 y Fq(connects)19494 67163 y(~)19411 67499 y(\000)20241 67698 y Fv(0)21147 67499 y Fq(+)310 b Fp(\016)516 b Fq(with)26726 67163 y(~)26643 67499 y(\000)27473 67698 y Fk(`)p Fv(+1)29640 67499 y Fq(+)310 b Fp(\016)515 b Fq(if)465 b Fp(k)506 b Fq(is)465 b(su\016cien)-37 b(tly)465 b(large.)658 b(Let)465 b Fp(\015)74 b Fq(:)1102 69093 y Fl(R)375 b Fm(!)f Fp(M)590 b Fq(b)37 b(e)446 b(the)f(accum)-37 b(ulation)447 b(p)37 b(oin)-37 b(t)445 b(of)h Fm(f)p Fp(\015)24615 69292 y Fk(k)25268 69093 y Fm(g)25932 69344 y Fk(k)22 b Fg(2)p Fc(Z)27855 69078 y Fj(+)28671 69093 y Fq(,)446 b(then)g Fp(\013)5 b Fq(\()p Fp(d\015)74 b Fq(\))372 b(=)i(\000)38466 69292 y Fv(0)39508 69093 y Fq(and)446 b Fp(!)48 b Fq(\()p Fp(d\015)74 b Fq(\))372 b(=)i(\000)48065 69292 y Fk(`)p Fv(+1)1102 70781 y Fq(since)4813 70445 y(~)4357 70781 y Fm(A)o Fq(\()p Fp(c)6509 70980 y Fk(i)6951 70781 y Fq(\))369 b(=)g(\000)10069 70980 y Fk(i)10954 70781 y Fq(for)443 b Fp(i)369 b Fq(=)g(0)p Fp(;)221 b(`)296 b Fq(+)g(1.)2696 73112 y(In)420 b(the)g(case)h(that)f (there)h(are)g(\014nite)f(resonan)-37 b(t)420 b(gaps,)425 b(the)420 b(construction)h(of)f(di\013usion)g(orbits)1102 74706 y(can)444 b(b)37 b(e)443 b(done)g(in)g(the)f(same)i(w)-37 b(a)g(y)-111 b(.)p eop %%Page: 46 46 46 45 bop 1102 2239 a Ft(46)1102 5427 y Fr(6,)510 b(Generic)f(prop)42 b(ert)-42 b(y)2696 7732 y Fq(Up)493 b(to)f(no)-37 b(w)493 b(w)-37 b(e)493 b(ha)-37 b(v)g(e)493 b(sho)-37 b(wn)493 b(the)f(existence)i(of)e(di\013usion)g(orbits)h(under)g(the)f(h)-37 b(yp)37 b(otheses)1102 9326 y(\(H1\),)362 b(\(H2\))342 b(and)g(\(H3\).)556 b(The)342 b(task)g(of)g(this)f(section)i(is)f(to)f (sho)-37 b(w)343 b(these)f(h)-37 b(yp)37 b(otheses)342 b(are)h(actually)1102 10920 y(dense)519 b(prop)37 b(erties)519 b(in)e Fp(C)13795 10438 y Fk(r)14384 10920 y Fq(-top)37 b(ology)518 b(for)g Fp(r)531 b Fm(\025)495 b Fq(3.)815 b(Since)519 b(w)-37 b(e)519 b(are)f(in)-37 b(terested)519 b(in)f(the)f(di\013usion)1102 12514 y(from)546 b Fp(p)5013 12713 y Fv(1)6150 12514 y Fp(<)541 b(A)546 b Fq(to)f Fp(p)11660 12713 y Fv(1)12797 12514 y Fp(>)c(B)67 b Fq(,)571 b(a)545 b(compact)i(domain)f(for)g Fm(fk)p Fp(p)p Fm(k)541 b(\024)g Fp(K)95 b Fm(g)364 b(\002)g Fl(T)39283 12032 y Fv(2)40424 12514 y Fq(satis\014es)546 b(suc)-37 b(h)546 b(an)1102 14108 y(requiremen)-37 b(t)473 b(if)f Fp(K)512 b(>)417 b Fq(0)472 b(is)g(su\016cien)-37 b(tly)472 b(large.)678 b(The)472 b Fp(C)29691 13626 y Fk(r)30280 14108 y Fq(-top)37 b(ology)471 b(is)h(endo)-37 b(w)g(ed)473 b(in)f(the)f(usual)1102 15702 y(sense)444 b(for)f(functions)g Fm(fk)p Fp(p)p Fm(k)369 b(\024)h Fp(K)95 b Fm(g)295 b(\002)h Fl(T)21101 15220 y Fv(2)22065 15702 y Fm(!)369 b Fl(R)p Fq(.)2696 18006 y(The)540 b(h)-37 b(yp)37 b(othesis)540 b(\(H1\))f(is)g(made)h (only)f(for)h(those)f(co-homology)h(classes)g Fp(c)530 b Fq(=)g(\()p Fp(c)45076 18205 y Fv(1)45671 18006 y Fp(;)221 b(b)p Fq(\()p Fp(c)47923 18205 y Fv(1)48519 18006 y Fq(\)\),)1102 19600 y(the)469 b(time-1-section)g(of)g(the)f(minimal)h(measure)h(is)f (con)-37 b(tained)469 b(in)g(an)g(in)-37 b(v)-74 b(arian)-37 b(t)469 b(circle)h(on)f(the)1102 21194 y(cylinder,)624 b(and)587 b(its)f(semi-static)g(orbit)h(set)24266 20859 y(~)23845 21194 y Fm(N)196 b Fq(\()p Fp(c)p Fq(\))585 b(consists)i(of)f(the)h(in)-37 b(v)-74 b(arian)-37 b(t)587 b(circle)h(and)f(its)1102 22788 y(minimal)426 b(homo)37 b(clinic)426 b(orbits.)585 b(It)425 b(is)g(precisely)i(the)f(set)f Fm(f)p Fp(B)31422 22306 y Fg(\003)31355 23117 y Fk(c)32401 22788 y Fq(=)369 b(0)p Fm(g)p Fq(.)585 b(Let)426 b(us)g(lo)37 b(ok)426 b(at)f(this)g(issue)1102 24382 y(from)443 b(the)g(Hamiltonian) g(dynamics)h(p)37 b(oin)-37 b(t)442 b(of)h(view.)2696 26687 y(Since)510 b(the)f(system)g(is)g(p)37 b(ositiv)-37 b(e)509 b(de\014nite)g(in)g(the)f(action)i(v)-74 b(ariable)510 b Fp(p)p Fq(,)526 b(it)508 b(has)h(a)g(generating)1102 28281 y(function)443 b Fp(G)p Fq(\()p Fp(q)48 b(;)221 b(q)9738 27799 y Fg(0)10110 28281 y Fq(\))12505 32030 y Fp(G)p Fq(\()p Fp(q)48 b(;)221 b(q)15938 31481 y Fg(0)16310 32030 y Fq(\))369 b(=)3209 b(inf)18790 33080 y Fk(\015)57 b Fg(2)p Fk(C)20948 32679 y Fj(1)21467 33080 y Fv(\([0)p Fk(;)p Fv(1])p Fk(;)24551 32845 y Fv(\026)24172 33080 y Fk(M)101 b Fv(\))18598 34218 y Fk(\015)57 b Fv(\(0\)=)p Fk(q)33 b(;\015)57 b Fv(\(1\)=)p Fk(q)25531 33817 y Fe(0)26115 30222 y Fi(Z)27443 30550 y Fv(1)26853 33240 y(0)28261 32030 y Fp(L)p Fq(\()p Fp(\015)74 b Fq(\()p Fp(s)p Fq(\))p Fp(;)415 b Fq(_)-563 b Fp(\015)72 b Fq(\()p Fp(s)p Fq(\))p Fp(;)221 b(s)p Fq(\))p Fp(ds;)8670 b Fq(\(6.1\))1102 37011 y(where)410 b(\()p Fp(q)48 b(;)221 b(q)7298 36529 y Fg(0)7670 37011 y Fq(\))409 b(is)f(in)h(the)g(co)-37 b(v)g(ering)410 b(space)22850 36675 y(\026)22355 37011 y Fp(M)513 b Fq(=)369 b Fl(R)26518 36529 y Fv(2)27341 37011 y Fm(\002)228 b Fl(R)29561 36529 y Fv(2)30157 37011 y Fq(.)579 b(Clearly)-111 b(,)417 b Fp(G)p Fq(\()p Fp(q)275 b Fq(+)227 b(2)p Fp(m\031)48 b(;)221 b(q)43507 36529 y Fg(0)44107 37011 y Fq(+)227 b(2)p Fp(m\031)48 b Fq(\))369 b(=)1102 38605 y Fp(G)p Fq(\()p Fp(q)48 b(;)221 b(q)4535 38123 y Fg(0)4908 38605 y Fq(\))442 b(for)h(all)g Fp(m)369 b Fm(2)g Fl(Z)13421 38123 y Fv(2)14016 38605 y Fq(.)591 b(The)443 b(map)g(\010)21617 38804 y Fk(H)22625 38605 y Fq(:)591 b(\()p Fp(p;)221 b(q)48 b Fq(\))368 b Fm(!)h Fq(\()p Fp(p)29768 38123 y Fg(0)30141 38605 y Fp(;)221 b(q)31372 38123 y Fg(0)31745 38605 y Fq(\))442 b(is)h(giv)-37 b(en)443 b(b)-37 b(y)14768 41404 y Fp(p)15436 40855 y Fg(0)16178 41404 y Fq(=)369 b Fp(@)18285 41603 y Fk(q)18805 41337 y Fe(0)19231 41404 y Fp(G)p Fq(\()p Fp(q)48 b(;)221 b(q)22664 40855 y Fg(0)23037 41404 y Fq(\))p Fp(;)2877 b(p)370 b Fq(=)f Fm(\000)p Fp(@)30978 41603 y Fk(q)31565 41404 y Fp(G)p Fq(\()p Fp(q)48 b(;)221 b(q)34998 40855 y Fg(0)35370 41404 y Fq(\))p Fp(:)10935 b Fq(\(6.2\))1102 44203 y(Let)443 b Fp(\031)4239 44402 y Fv(1)5278 44203 y Fq(b)37 b(e)443 b(the)g(standard)g(pro)74 b(jection)443 b(from)g Fl(R)25332 43721 y Fv(2)26371 44203 y Fq(to)g Fl(T)28881 43721 y Fv(2)29476 44203 y Fq(,)g(let)g Fp(c)369 b Fm(2)f Fl(R)35364 43721 y Fv(2)36403 44203 y Fq(and)16484 47002 y Fp(G)17528 47201 y Fk(c)18070 47002 y Fq(\()p Fp(q)48 b(;)221 b(q)20459 46453 y Fg(0)20831 47002 y Fq(\))368 b(=)h Fp(G)p Fq(\()p Fp(q)48 b(;)221 b(q)26551 46453 y Fg(0)26924 47002 y Fq(\))294 b Fm(\000)i(h)p Fp(c;)221 b(q)31387 46453 y Fg(0)32054 47002 y Fm(\000)295 b Fp(q)48 b Fm(i)1102 49801 y Fq(then)10849 51395 y Fp(h)11614 51594 y Fk(c)12155 51395 y Fq(\()p Fp(x;)221 b(x)14780 50846 y Fg(0)15153 51395 y Fq(\))369 b(=)1575 b(min)17800 52314 y Fk(\031)18421 52447 y Fj(1)18940 52314 y Fv(\()p Fk(q)33 b Fv(\)=)p Fk(x)17441 53451 y(\031)18062 53584 y Fj(1)18580 53451 y Fv(\()p Fk(q)19515 53051 y Fe(0)19875 53451 y Fv(\)=)p Fk(x)21707 53051 y Fe(0)22289 51395 y Fp(G)23333 51594 y Fk(c)23874 51395 y Fq(\()p Fp(q)48 b(;)221 b(q)26263 50846 y Fg(0)26635 51395 y Fq(\))295 b Fm(\000)g Fp(\013)5 b Fq(\()p Fp(c)p Fq(\))p Fp(:)2656 b Fq(\(see)443 b(\(2)p Fp(:)p Fq(6\)\))7015 b(\(6.3\))1102 55788 y(As)497 b(the)f(system)g(is)g(nearly)h(in)-37 b(tegrable,)511 b(the)496 b(matrix)h Fp(@)29484 55306 y Fv(2)29410 56153 y Fk(q)29930 55887 y Fe(0)30289 56153 y Fk(q)30876 55788 y Fp(G)g Fq(is)f(non-degenerate)i(ev)-37 b(erywhere.)1102 57463 y(Th)g(us)532 b(w)-37 b(e)532 b(can)g(solv)-37 b(e)532 b(the)f(second)h(equation)f(in)g(\(6.2\))g (and)h(obtain)f(someho)-37 b(w)532 b(more)g(explicit)1102 59057 y(form)443 b(of)g(the)g(map)g(\(6.2\))14756 62548 y Fp(p)15424 61999 y Fg(0)16166 62548 y Fq(=)17728 61649 y Fp(@)74 b(G)p 17728 62242 1824 54 v 17744 63459 a(@)g(q)19164 63075 y Fg(0)19711 62548 y Fq(\()p Fp(q)48 b(;)221 b(q)22100 61999 y Fg(0)22472 62548 y Fq(\()p Fp(p;)g(q)48 b Fq(\)\))p Fp(;)2877 b(q)29809 61999 y Fg(0)30550 62548 y Fq(=)369 b Fp(q)32593 61999 y Fg(0)32966 62548 y Fq(\()p Fp(p;)221 b(q)48 b Fq(\))p Fp(:)10923 b Fq(\(6.4\))1102 66054 y(Let)413 b(us)g(consider)h(a)g(small)f(p)37 b(erturbation)413 b(of)g(the)g(generating)h(function)e Fp(G)p Fq(\()p Fp(q)48 b(;)221 b(q)41375 65572 y Fg(0)41748 66054 y Fq(\))368 b Fm(!)i Fp(G)p Fq(\()p Fp(q)48 b(;)221 b(q)47764 65572 y Fg(0)48136 66054 y Fq(\))235 b(+)1102 67648 y Fp(\024)p Fq(\()p Fp(q)112 b Fm(\000)64 b Fp(q)4827 67166 y Fg(0)5199 67648 y Fq(\))p Fp(G)6760 67847 y Fv(1)7356 67648 y Fq(\()p Fp(q)8514 67166 y Fg(0)8886 67648 y Fq(\))327 b(in)g(whic)-37 b(h)328 b(0)369 b Fm(\024)g Fp(\024)p Fq(\()p Fp(q)112 b Fm(\000)64 b Fp(q)21009 67166 y Fg(0)21381 67648 y Fq(\))369 b Fm(\024)g Fq(1,)350 b Fp(\024)p Fq(\()p Fp(q)112 b Fm(\000)64 b Fp(q)28777 67166 y Fg(0)29149 67648 y Fq(\))369 b(=)g(1)328 b(if)f Fm(j)p Fp(q)112 b Fm(\000)64 b Fp(q)36343 67166 y Fg(0)36715 67648 y Fm(j)369 b(\024)g Fp(K)422 b Fq(and)327 b Fp(\024)p Fq(\()p Fp(q)112 b Fm(\000)64 b Fp(q)46597 67166 y Fg(0)46969 67648 y Fq(\))369 b(=)g(0)1102 69242 y(if)430 b Fm(j)p Fp(q)318 b Fm(\000)270 b Fp(q)5531 68760 y Fg(0)5903 69242 y Fm(j)369 b(\025)g Fp(K)c Fq(+)270 b(1.)586 b(W)-111 b(e)431 b(c)-37 b(ho)37 b(ose)432 b(su\016cien)-37 b(tly)430 b(large)h Fp(K)525 b Fq(so)431 b(that)e Fm(fk)p Fp(p)p Fm(k)370 b(\024)f Fq(max\()p Fm(j)p Fp(A)p Fm(j)p Fp(;)221 b Fm(j)p Fp(B)67 b Fm(j)p Fq(\))269 b(+)h(1)p Fm(g)1102 70836 y Fq(is)443 b(con)-37 b(tained)444 b(in)e(the)h(set)g(where)h Fm(j)p Fp(q)343 b Fm(\000)295 b Fp(q)21479 70354 y Fg(0)21851 70836 y Fm(j)369 b(\024)h Fp(K)95 b Fq(.)590 b(In)443 b(this)f(set)h(the)f(map)h(will)g(ha)-37 b(v)g(e)444 b(the)f(form)10086 74327 y Fp(p)10754 73778 y Fg(0)11496 74327 y Fq(=)13058 73428 y Fp(@)74 b(G)p 13058 74021 V 13074 75238 a(@)g(q)14494 74854 y Fg(0)15040 74327 y Fq(\()p Fp(q)48 b(;)221 b(q)17429 73778 y Fg(0)17801 74327 y Fq(\()p Fp(p;)g(q)48 b Fq(\)\))295 b(+)23033 73428 y Fp(@)74 b(G)24856 73627 y Fv(1)p 23033 74021 2420 54 v 23347 75238 a Fp(@)g(q)24767 74854 y Fg(0)25612 74327 y Fq(\()p Fp(q)26770 73778 y Fg(0)27142 74327 y Fq(\()p Fp(p;)221 b(q)48 b Fq(\)\))p Fp(;)2877 b(q)34479 73778 y Fg(0)35221 74327 y Fq(=)369 b Fp(q)37264 73778 y Fg(0)37636 74327 y Fq(\()p Fp(p;)221 b(q)48 b Fq(\))p Fp(:)6253 b Fq(\(6.5\))p eop %%Page: 47 47 47 46 bop 48792 2239 a Ft(47)1102 5427 y Fq(Note)595 b(that)f(b)37 b(oth)594 b(stable)h(and)f(unstable)h(manifolds)f(of)h (\000)g(k)-37 b(eep)595 b(horizon)-37 b(tal)595 b(o)-37 b(v)g(er)596 b(the)e(strip)1102 7021 y Fp(U)514 b Fq(=)369 b Fm(f)p Fp(a)h Fm(\024)f Fp(q)7656 7220 y Fv(2)8621 7021 y Fm(\024)g Fq(2)p Fp(\031)341 b Fm(\000)294 b Fp(a)p Fm(g)p Fq(,)442 b(restricting)h(\010)f(to)g Fp(W)26113 6539 y Fk(s)27122 7021 y Fq(and)g(to)g Fp(W)32766 6539 y Fk(u)33902 7021 y Fq(where)i(they)e(k)-37 b(eep)442 b(horizon)-37 b(tal,)1102 8615 y(and)474 b(pro)74 b(jecting)473 b(it)g(to)g(the)g(underline)h(manifold)f Fp(M)618 b Fq(along)473 b(the)g(\014b)37 b(ers)474 b(w)-37 b(e)474 b(obtain)f(t)-37 b(w)g(o)473 b(maps)1102 10209 y Fp(f)1895 9727 y Fk(s)3062 10209 y Fq(and)598 b Fp(f)6593 9727 y Fk(u)7886 10209 y Fq(on)g Fp(M)743 b Fq(suc)-37 b(h)598 b(that)f Fp(\031)447 b Fm(\016)399 b Fq(\010)628 b(=)g Fp(f)24410 9727 y Fk(s;u)26320 10209 y Fm(\016)399 b Fp(\031)48 b Fq(.)1056 b(W)-111 b(e)599 b(c)-37 b(ho)37 b(ose)600 b Fp(G)37470 10408 y Fv(1)38694 10209 y Fm(2)627 b Fp(C)41251 9727 y Fk(r)42438 10209 y Fq(satisfying)598 b(its)1102 11803 y(supp)37 b(ort)520 b Fa(supp)o Fq(\()p Fp(G)10396 12002 y Fv(1)10992 11803 y Fq(\))497 b(=)f Fm(B)14407 12002 y Fk(b)14941 11803 y Fq(\()p Fp(q)16099 11321 y Fg(\003)16709 11803 y Fq(\))519 b(mo)37 b(d)520 b(2)p Fp(\031)48 b(m)497 b Fm(\032)g Fp(U)664 b Fq(mo)37 b(d)519 b(2)p Fp(\031)48 b(m)520 b Fq(where)h Fp(m)497 b Fm(2)f Fl(Z)41113 11321 y Fv(2)41708 11803 y Fq(.)820 b(W)-111 b(e)521 b(see)f(that)1102 13397 y(\()p Fp(f)2412 12915 y Fk(u)3107 13397 y Fq(\))3624 12915 y Fg(\000)p Fv(1)5049 13397 y Fq(\()p Fm(B)6438 13596 y Fk(b)6972 13397 y Fq(\()p Fp(q)8130 12915 y Fg(\003)8740 13397 y Fq(\)\))265 b Fm(\\)h(B)12063 13596 y Fk(b)12597 13397 y Fq(\()p Fp(q)13755 12915 y Fg(\003)14365 13397 y Fq(\))368 b(=)h Fl(?)429 b Fq(and)f Fp(f)21475 12915 y Fk(s)22043 13397 y Fq(\()p Fm(B)23432 13596 y Fk(b)23966 13397 y Fq(\()p Fp(q)25124 12915 y Fg(\003)25734 13397 y Fq(\)\))265 b Fm(\\)h(B)29057 13596 y Fk(b)29591 13397 y Fq(\()p Fp(q)30749 12915 y Fg(\003)31358 13397 y Fq(\))369 b(=)g Fl(?)429 b Fq(if)f Fp(b)369 b(>)g Fq(0)428 b(is)g(c)-37 b(hosen)430 b(suitably)1102 14991 y(small.)1038 b(Let)593 b(us)f(consider)h(the)f(problem)g(in)g(the)g(co)-37 b(v)g(ering)594 b(space)f Fl(T)394 b Fm(\002)h Fl(R)592 b Fq(and)g(assume)h(one)1102 16585 y(lift)480 b(of)h(the)f(unstable)g(manifold)h(starting)f(from)h Fp(q)26777 16784 y Fv(2)27804 16585 y Fq(=)432 b(0)480 b(to)h(the)f(righ)-37 b(t,)490 b(one)481 b(lift)e(of)i(the)f(stable) 1102 18179 y(manifold)575 b(starting)e(from)i Fp(q)15683 18378 y Fv(2)16867 18179 y Fq(=)588 b(2)p Fp(\031)622 b Fq(to)574 b(the)g(left.)984 b(F)-111 b(rom)575 b(\(6.5\))f(w)-37 b(e)575 b(can)f(see)h(that)f(the)g(lo)37 b(cal)1102 19773 y(stable)489 b(manifold)f(is)g(not)g(deformed)h Fp(W)21605 19291 y Fk(s)22173 19773 y Fm(j)p Fq([)p Fp(q)23552 19291 y Fg(\003)23504 20103 y Fv(2)24488 19773 y Fm(\000)326 b Fp(b)444 b Fm(\024)h Fp(q)28932 19972 y Fv(2)29973 19773 y Fm(\024)g Fq(2)p Fp(\031)373 b Fq(+)326 b Fp(q)35197 19972 y Fv(2\000)36450 19773 y Fq(\()p Fp(q)37560 19972 y Fv(1)38155 19773 y Fq(\)])444 b(=)h Fm(f)p Fp(q)48 b(;)221 b(dS)44441 19291 y Fk(s)45334 19773 y Fq(+)326 b Fp(c)p Fq(\()p Fp(\033)48 b Fq(\))443 b(:)1102 21367 y Fp(q)1743 20885 y Fg(\003)1695 21697 y Fv(2)2552 21367 y Fm(\000)199 b Fp(b)369 b Fm(\024)g Fp(q)6718 21566 y Fv(2)7683 21367 y Fm(\024)g Fq(2)p Fp(\031)247 b Fq(+)199 b Fp(q)12578 21566 y Fv(2\000)13830 21367 y Fq(\()p Fp(q)14940 21566 y Fv(1)15536 21367 y Fq(\))p Fm(g)p Fq(,)403 b(but)394 b(the)h(unstable)f(manifold)h(undergo)37 b(es)395 b(sligh)-37 b(t)395 b(deformation,)1102 22961 y Fp(W)2541 22479 y Fk(u)3236 22961 y Fm(j)p Fq([)p Fp(q)4567 23160 y Fv(2\000)5820 22961 y Fq(\()p Fp(q)6930 23160 y Fv(1)7525 22961 y Fq(\))475 b Fm(\024)g Fp(q)10618 23160 y Fv(2)11689 22961 y Fm(\024)h Fp(q)13839 22479 y Fg(\003)13791 23291 y Fv(2)14787 22961 y Fq(+)338 b Fp(b)p Fq(])475 b(=)g Fm(f)p Fp(q)48 b(;)221 b(dS)22558 22479 y Fk(u)23590 22961 y Fq(+)338 b Fp(dG)26696 23160 y Fv(1)27630 22961 y Fq(+)g Fp(c)p Fq(\()p Fp(\033)48 b Fq(\))474 b(:)697 b Fp(q)33550 23160 y Fv(2\000)34803 22961 y Fq(\()p Fp(q)35913 23160 y Fv(1)36508 22961 y Fq(\))475 b Fm(\024)g Fp(q)39601 23160 y Fv(2)40672 22961 y Fm(\024)h Fp(q)42822 22479 y Fg(\003)42774 23291 y Fv(2)43770 22961 y Fq(+)338 b Fp(b)p Fm(g)p Fq(.)781 b(It)506 b(is)1102 24555 y(easy)444 b(to)e(see)i(that)e(the)h(barrier)h (function)e(has)h(the)g(form:)3990 28007 y Fp(B)5065 27459 y Fg(\003)4998 28369 y Fk(c)p Fv(\()p Fk(\033)33 b Fv(\))7018 28007 y Fq(\()p Fp(q)48 b Fq(\))368 b(=)h Fp(S)11355 27459 y Fk(u)11278 28369 y(c)p Fv(\()p Fk(\033)33 b Fv(\))13298 28007 y Fq(\()p Fp(q)48 b Fq(\))294 b Fm(\000)h Fp(S)17487 27459 y Fk(s)17410 28369 y(c)p Fv(\()p Fk(\033)33 b Fv(\))19430 28007 y Fq(\()p Fp(q)48 b Fq(\))294 b Fm(\000)i Fp(G)23772 28206 y Fv(1)24368 28007 y Fq(\()p Fp(q)48 b Fq(\))294 b(+)h(2)p Fp(\031)48 b(b)p Fq(\()p Fp(c)30796 28206 y Fv(1)31392 28007 y Fq(\()p Fp(\033)g Fq(\)\))p Fp(;)2876 b Fa(if)443 b Fp(q)417 b Fm(2)368 b Fp(B)41485 28206 y Fk(b)42018 28007 y Fq(\()p Fp(q)43176 27459 y Fg(\003)43786 28007 y Fq(\))2888 b(\(6.6\))1102 31515 y(W)-111 b(e)418 b(should)g(note)f(the)g(total)f(action)i(of)f(the)g (minimal)g(orbit)g(ma)-37 b(y)417 b(b)37 b(e)418 b(c)-37 b(hanged)418 b(b)37 b(ecause)418 b(of)f(the)1102 33109 y(p)37 b(erturbation,)411 b(in)403 b(other)h(w)-37 b(ords,)411 b(the)403 b(asso)37 b(ciated)404 b(cohomology)g(class)f(ma)-37 b(y)403 b(b)37 b(e)403 b(sub)74 b(jected)403 b(to)g(a)1102 34703 y(small)443 b(p)37 b(erturbation)443 b(\()p Fp(c)13463 34902 y Fv(1)14059 34703 y Fp(;)221 b(b)p Fq(\()p Fp(c)16311 34902 y Fv(1)16906 34703 y Fq(\)\))369 b Fm(!)g Fq(\()p Fp(c)21098 34902 y Fv(1)21693 34703 y Fp(;)221 b(b)p Fq(\()p Fp(c)23945 34902 y Fv(1)24541 34703 y Fq(\))295 b Fm(\006)g Fp(")p Fq(\).)2696 37358 y(Let)333 b Fp(R)5975 37557 y Fk(d)6963 37358 y Fq(=)369 b Fm(f)p Fp(q)417 b Fm(2)368 b Fp(M)514 b Fq(:)369 b Fm(j)p Fp(q)14796 37557 y Fv(1)15466 37358 y Fm(\000)74 b Fp(q)17214 36876 y Fg(\003)17166 37688 y Fv(1)17824 37358 y Fm(j)369 b(\024)h Fp(d;)443 b Fm(j)p Fp(q)22430 37557 y Fv(2)23100 37358 y Fm(\000)74 b Fp(q)24848 36876 y Fg(\003)24800 37688 y Fv(2)25458 37358 y Fm(j)369 b(\024)h Fp(d)p Fm(g)f(\032)g(B)31597 37557 y Fk(b)32131 37358 y Fq(\()p Fp(q)33289 36876 y Fg(\003)33899 37358 y Fq(\),)354 b(let)332 b Fp(S)37762 37557 y Fk(\033)38846 37358 y Fq(=)369 b Fp(S)41140 36876 y Fk(u)41063 37786 y(c)p Fv(\()p Fk(\033)33 b Fv(\))43157 37358 y Fm(\000)74 b Fp(S)45156 36876 y Fk(s)45079 37786 y(c)p Fv(\()p Fk(\033)33 b Fv(\))47174 37358 y Fm(\000)74 b Fp(G)49325 37557 y Fv(1)1102 39073 y Fq(w)-37 b(e)444 b(de\014ne)15086 40932 y Fp(Z)95 b Fq(\()p Fp(\033)48 b Fq(\))368 b(=)h Fm(f)p Fp(q)417 b Fm(2)369 b Fp(R)23637 41131 y Fk(d)24625 40932 y Fq(:)590 b Fp(S)26399 41131 y Fk(\033)27114 40932 y Fq(\()p Fp(q)48 b Fq(\))368 b(=)559 b(min)30559 41788 y Fk(q)33 b Fg(2)p Fk(R)32595 41929 y Fd(d)33374 40932 y Fp(S)34189 41131 y Fk(\033)34904 40932 y Fm(g)p Fp(:)1102 44425 y Fq(W)-111 b(e)405 b(sa)-37 b(y)404 b(a)g(connected)h(set)f Fp(V)699 b Fq(is)404 b(non-trivial)g(for)g Fp(R)27140 44624 y Fk(d)28163 44425 y Fq(if)f(either)i(\005)34066 44624 y Fv(1)34662 44425 y Fq(\()p Fp(V)512 b Fm(\\)217 b Fp(R)38578 44624 y Fk(d)39197 44425 y Fq(\))368 b(=)h Fm(f)p Fp(q)42789 43943 y Fg(\003)42741 44755 y Fv(1)43617 44425 y Fm(\000)217 b Fp(d)370 b Fm(\024)f Fp(q)47923 44624 y Fv(1)48888 44425 y Fm(\024)1102 46019 y Fp(q)1743 45537 y Fg(\003)1695 46349 y Fv(1)2580 46019 y Fq(+)227 b Fp(d)p Fm(g)409 b Fq(or)g(\005)8193 46218 y Fv(2)8789 46019 y Fq(\()p Fp(V)521 b Fm(\\)226 b Fp(R)12723 46218 y Fk(d)13342 46019 y Fq(\))368 b(=)i Fm(f)p Fp(q)16935 45537 y Fg(\003)16887 46349 y Fv(2)17771 46019 y Fm(\000)227 b Fp(d)369 b Fm(\024)h Fp(q)22087 46218 y Fv(2)23051 46019 y Fm(\024)g Fp(q)25095 45537 y Fg(\003)25047 46349 y Fv(2)25931 46019 y Fq(+)227 b Fp(d)p Fm(g)p Fq(,)416 b(where)410 b(\005)34134 46218 y Fk(i)34984 46019 y Fq(is)f(the)f (standard)h(pro)74 b(jection)1102 47613 y(from)481 b Fl(T)5166 47131 y Fv(2)6242 47613 y Fq(to)f(its)g Fp(i)p Fq(-th)f(comp)37 b(onen)-37 b(t)481 b(\(i=1,2\).)703 b(Let)480 b Fp(M)28166 47812 y Fk(d;q)29554 47547 y Fe(\003)30177 47613 y Fq(\()p Fp(S)77 b Fq(\))429 b(=)j Fm(f)p Fp(q)480 b Fq(:)431 b Fp(S)77 b Fq(\()p Fp(q)48 b Fq(\))430 b(=)h(min)43209 47853 y Fk(q)33 b Fg(2)p Fk(R)45245 47994 y Fd(d)45803 47853 y Fv(\()p Fk(q)46738 47587 y Fe(\003)47294 47853 y Fv(\))47997 47613 y Fp(S)77 b Fm(g)p Fq(,)1102 49300 y(w)-37 b(e)444 b(de\014ne)f(a)g(set)g(in)g(the)g(function)f(space)i Fh(F)p Fq(\()p Fp(d;)221 b(q)26064 48817 y Fg(\003)26674 49300 y Fq(\))369 b(=)g Fp(C)30006 48817 y Fv(0)30602 49300 y Fq(\()p Fp(R)32128 49499 y Fk(d)32747 49300 y Fq(\()p Fp(q)33905 48817 y Fg(\003)34515 49300 y Fq(\))p Fp(;)221 b Fl(R)p Fq(\),)3959 53150 y Fh(Z)p Fq(\()p Fp(d;)g(q)7342 52602 y Fg(\003)7953 53150 y Fq(\))369 b(=)10241 51676 y Fi(n)11126 53150 y Fp(S)445 b Fm(2)369 b Fh(F)p Fq(\()p Fp(d;)221 b(q)17039 52602 y Fg(\003)17649 53150 y Fq(\))369 b(:)590 b Fp(M)20783 53349 y Fk(d;q)22171 53084 y Fe(\003)22793 53150 y Fq(\()p Fp(S)77 b Fq(\))221 b(con)-37 b(tains)442 b(a)h(set)g(non-trivial)f(for)i Fp(R)42907 53349 y Fk(d)43526 53150 y Fq(\()p Fp(q)44684 52602 y Fg(\003)45293 53150 y Fq(\))45810 51676 y Fi(o)46696 53150 y Fp(:)1102 56912 y Fq(Let)7203 60431 y Fh(Z)8148 60630 y Fv(1)9113 60431 y Fq(=)10515 58956 y Fi(n)11400 60431 y Fp(S)h Fm(2)369 b Fh(Z)p Fq(\()p Fp(d;)221 b(q)17298 59882 y Fg(\003)17909 60431 y Fq(\))368 b(:)591 b(\005)20750 60630 y Fv(1)21346 60431 y Fq(\()p Fp(M)23152 60630 y Fk(d;q)24540 60364 y Fe(\003)25162 60431 y Fq(\()p Fp(S)77 b Fq(\)\))366 b(=)k Fm(f)p Fp(q)30679 59882 y Fg(\003)30631 60759 y Fv(1)31584 60431 y Fm(\000)296 b Fp(d)369 b Fm(\024)g Fp(q)35968 60630 y Fv(1)36933 60431 y Fm(\024)g Fp(q)38976 59882 y Fg(\003)38928 60759 y Fv(1)39882 60431 y Fq(+)295 b Fp(d)p Fm(g)42565 58956 y Fi(o)43451 60431 y Fp(;)7203 63354 y Fh(Z)8148 63553 y Fv(2)9113 63354 y Fq(=)10515 61879 y Fi(n)11400 63354 y Fp(S)445 b Fm(2)369 b Fh(Z)p Fq(\()p Fp(d;)221 b(q)17298 62805 y Fg(\003)17909 63354 y Fq(\))368 b(:)591 b(\005)20750 63553 y Fv(2)21346 63354 y Fq(\()p Fp(M)23152 63553 y Fk(d;q)24540 63287 y Fe(\003)25162 63354 y Fq(\()p Fp(S)77 b Fq(\)\))366 b(=)k Fm(f)p Fp(q)30679 62805 y Fg(\003)30631 63682 y Fv(2)31584 63354 y Fm(\000)296 b Fp(d)369 b Fm(\024)g Fp(q)35968 63553 y Fv(2)36933 63354 y Fm(\024)g Fp(q)38976 62805 y Fg(\003)38928 63682 y Fv(2)39882 63354 y Fq(+)295 b Fp(d)p Fm(g)42565 61879 y Fi(o)43451 63354 y Fp(;)1102 67131 y Fq(then)19908 68989 y Fh(Z)q Fq(\()p Fp(d;)221 b(q)23292 68440 y Fg(\003)23902 68989 y Fq(\))369 b(=)g Fh(Z)27134 69188 y Fv(1)28025 68989 y Fm([)295 b Fh(Z)30150 69188 y Fv(2)30746 68989 y Fp(:)1102 71777 y Fq(Our)483 b(\014rst)e(task)h(is)f(to)h(sho)-37 b(w)482 b(for)g(eac)-37 b(h)483 b(generating)g(function)f Fp(G)434 b Fm(2)f Fp(C)36631 71295 y Fk(r)37220 71777 y Fq(\()p Fp(M)465 b Fm(\002)322 b Fp(M)71 b(;)221 b Fl(R)p Fq(\))481 b(and)h(eac)-37 b(h)1102 73371 y Fp(\017)538 b(>)h Fq(0,)569 b(there)545 b(is)f(an)g(op)37 b(en)545 b(and)f(dense)h(set)f Fh(H)p Fq(\()p Fp(d;)221 b(q)27701 72889 y Fg(\003)28311 73371 y Fq(\))544 b(of)g Fm(B)31858 73570 y Fk(\017)32368 73371 y Fq(\(0\))537 b Fm(\032)h Fp(C)37218 72889 y Fk(r)37807 73371 y Fq(\()p Fp(R)39333 73570 y Fk(d)39951 73371 y Fq(\()p Fp(q)41109 72889 y Fg(\003)41719 73371 y Fq(\))p Fp(;)221 b Fl(R)p Fq(\),)570 b(for)544 b(eac)-37 b(h)1102 74965 y Fp(G)2146 75164 y Fv(1)3112 74965 y Fm(2)368 b Fh(H)p Fq(\()p Fp(d;)221 b(q)7935 74483 y Fg(\003)8546 74965 y Fq(\),)399 b(the)388 b(image)h(of)g Fp(S)18121 75164 y Fk(\033)19224 74965 y Fq(from)g([)p Fp(A)23675 74483 y Fg(0)24047 74965 y Fp(;)221 b(B)25712 74483 y Fg(0)26085 74965 y Fq(])388 b(to)g Fh(F)g Fq(has)h(no)g(in)-37 b(tersection)389 b(with)f(the)g(set) h Fh(Z)49110 75164 y Fk(i)49552 74965 y Fq(.)p eop %%Page: 48 48 48 47 bop 1102 2239 a Ft(48)2696 5427 y Fq(Ob)-37 b(viously)-111 b(,)385 b(the)369 b(set)h Fh(Z)14298 5626 y Fv(1)15263 5427 y Fq(is)f(a)h(closed)g(set)f(and)g(has)h(in\014nite)f (co-dimensions)h(in)f(the)g(follo)-37 b(wing)1102 7021 y(sense,)542 b(there)522 b(exists)f Fh(N)q Fq(,)541 b(an)521 b(in\014nite)g(dimension)h(subspace)g(of)g Fh(F)p Fq(,)541 b(suc)-37 b(h)521 b(that)g(\()p Fp(S)423 b Fq(+)348 b Fp(F)184 b Fq(\))648 b Fp(=)-812 b Fm(2)500 b Fh(Z)1102 8615 y Fq(for)574 b(all)f Fp(S)663 b Fm(2)586 b Fh(Z)9135 8814 y Fv(1)10304 8615 y Fq(and)574 b Fp(F)771 b Fm(2)586 b Fh(N)p Fm(nf)p Fq(0)p Fm(g)p Fq(.)983 b(In)573 b(fact,)606 b(for)573 b(eac)-37 b(h)575 b(non)e(constan)-37 b(t)573 b(function)g Fp(F)184 b Fq(\()p Fp(q)47336 8814 y Fv(1)47933 8615 y Fq(\))586 b Fm(2)1102 10209 y Fp(C)2146 9727 y Fv(0)2743 10209 y Fq(\([)p Fp(q)4270 9727 y Fg(\003)4222 10539 y Fv(1)5176 10209 y Fm(\000)296 b Fp(d;)221 b(q)8427 9727 y Fg(\003)8379 10539 y Fv(1)9335 10209 y Fq(+)296 b Fp(d)p Fq(])p Fp(;)221 b Fl(R)p Fq(\))445 b(with)f Fp(F)184 b Fq(\()p Fp(q)19458 9727 y Fg(\003)19410 10539 y Fv(1)20069 10209 y Fq(\))371 b(=)h(0)445 b(and)f(eac)-37 b(h)446 b Fp(S)h Fm(2)372 b Fh(Z)32510 10408 y Fv(1)33106 10209 y Fq(,)445 b(w)-37 b(e)445 b(ha)-37 b(v)g(e)445 b Fp(S)372 b Fq(+)297 b Fp(F)704 b(=)-812 b Fm(2)371 b Fh(Z)45070 10408 y Fv(1)45666 10209 y Fq(.)595 b(Th)-37 b(us,)1102 11803 y(w)g(e)581 b(can)e(c)-37 b(ho)37 b(ose)581 b Fh(N)596 b Fq(=)h Fp(C)14692 11321 y Fv(0)15288 11803 y Fq(\([)p Fp(q)16815 11321 y Fg(\003)16767 12133 y Fv(1)17812 11803 y Fm(\000)386 b Fp(d;)221 b(q)21153 11321 y Fg(\003)21105 12133 y Fv(1)22150 11803 y Fq(+)387 b Fp(d)p Fq(])p Fp(;)221 b Fl(R)p Fq(\))p Fp(=)p Fl(R)p Fq(,)614 b(whic)-37 b(h)580 b(w)-37 b(e)580 b(think)f(as)h(the)f(subspace)h(of)1102 13397 y Fp(C)2146 12915 y Fv(0)2743 13397 y Fq(\()p Fp(R)4269 13596 y Fk(d)4887 13397 y Fq(\()p Fp(q)6045 12915 y Fg(\003)6655 13397 y Fq(\))p Fp(;)221 b Fl(R)p Fq(\))443 b(consisting)g(of)g(those)g (con)-37 b(tin)g(uous)443 b(functions)f(indep)37 b(enden)-37 b(t)444 b(of)f Fp(q)42819 13596 y Fv(2)43414 13397 y Fq(.)2696 15658 y(On)580 b(the)f(other)h(hand,)614 b(as)579 b Fp(S)17523 15857 y Fk(\033)18238 15658 y Fq(:)863 b([)p Fp(A)20835 15176 y Fg(0)21208 15658 y Fp(;)221 b(B)22873 15176 y Fg(0)23246 15658 y Fq(])596 b Fm(!)h Fh(F)579 b Fq(has)30339 15135 y Fv(1)p 30339 15353 530 54 v 30339 16117 a(2)31028 15658 y Fq(-H\177)-664 b(older)580 b(con)-37 b(tin)g(uit)g(y)-111 b(,)614 b(the)579 b(image)h(is)1102 17252 y(compact)444 b(and)f(its)f(b)37 b(o)-37 b(x)443 b(dimension)g(is)g(not)f(bigger)i(than)f(2)21638 19857 y Fp(D)22738 20056 y Fk(B)23874 19857 y Fq(\()o Fh(F)25349 20056 y Fk(\033)26064 19857 y Fq(\))369 b Fm(\024)g Fq(2)p Fp(:)17806 b Fq(\(6.7\))1102 22461 y(where)530 b Fh(F)5987 22660 y Fk(\033)7214 22461 y Fq(=)511 b Fm(f)p Fp(S)10237 22660 y Fk(\033)11463 22461 y Fq(:)732 b Fp(\033)559 b Fm(2)510 b Fq([)p Fp(A)16643 21979 y Fg(0)17016 22461 y Fp(;)221 b(B)18681 21979 y Fg(0)19053 22461 y Fq(])p Fm(g)p Fq(.)846 b(Clearly)-111 b(,)551 b(this)528 b(set)f(is)h (determined)h(b)-37 b(y)528 b(the)g(generating)1102 24055 y(function)443 b Fp(G)p Fq(.)1102 26317 y Fr(Lemma)509 b(6.1.)764 b Fo(Ther)-68 b(e)475 b(is)h(an)f(op)-68 b(en)475 b(and)g(dense)g(set)h Fh(N)29550 25835 y Fg(\003)30530 26317 y Fm(\032)369 b Fh(N)475 b Fo(such)h(that)f(for)h(al)68 b(l)476 b Fp(F)554 b Fm(2)368 b Fh(N)47439 25835 y Fg(\003)20030 28921 y Fq(\()p Fh(F)21506 29120 y Fk(\033)22516 28921 y Fq(+)295 b Fp(F)184 b Fq(\))296 b Fm(\\)f Fh(Z)369 b Fq(=)g Fl(?)p Fp(:)16198 b Fq(\(6.8\))1102 32193 y Fr(Pro)42 b(of)p Fq(:)563 b(The)389 b(op)37 b(en)388 b(prop)37 b(ert)-37 b(y)388 b(is)f(ob)-37 b(vious.)573 b(If)387 b(there)h(is)g(no)f(densit)-37 b(y)388 b(prop)37 b(ert)-37 b(y)-111 b(,)400 b(there)388 b(w)-37 b(ould)388 b(b)37 b(e)1102 33787 y Fp(n)p Fq(-dimensional)494 b Fp(")p Fq(-ball)f Fm(B)14273 33986 y Fk(")15294 33787 y Fm(\032)452 b Fh(N)493 b Fq(for)g(some)h Fp(")452 b(>)h Fq(0,)505 b(for)493 b(eac)-37 b(h)494 b Fp(F)637 b Fm(2)452 b(B)36957 33986 y Fk(")37526 33787 y Fq(,)505 b(there)493 b(exists)g Fp(S)528 b Fm(2)452 b Fh(F)49206 33986 y Fk(\033)1102 35381 y Fq(suc)-37 b(h)396 b(that)f Fp(F)386 b Fq(+)201 b Fp(S)444 b Fm(2)369 b Fh(Z)12814 35580 y Fv(1)13410 35381 y Fq(.)575 b(F)-111 b(or)396 b(eac)-37 b(h)397 b Fp(S)445 b Fm(2)368 b Fh(F)23105 35580 y Fk(\033)24216 35381 y Fq(there)396 b(is)f(only)h(one)g Fp(F)554 b Fm(2)368 b(B)37645 35580 y Fk(")38610 35381 y Fq(so)395 b(that)g Fp(S)277 b Fq(+)201 b Fp(F)554 b Fm(2)368 b Fh(Z)48956 35580 y Fv(1)49552 35381 y Fq(,)1102 36975 y(for,)614 b(otherwise,)g(there)580 b(w)-37 b(ould)580 b(b)37 b(e)579 b Fp(F)20657 36493 y Fg(0)21627 36975 y Fm(6)p Fq(=)597 b Fp(F)763 b Fq(suc)-37 b(h)580 b(that)f Fp(F)32060 36493 y Fg(0)32819 36975 y Fq(+)386 b Fp(S)673 b Fm(2)596 b Fh(Z)38152 37174 y Fv(1)38748 36975 y Fq(,)613 b(but)579 b(w)-37 b(e)580 b(can)g(write)1102 38569 y Fp(F)2140 38087 y Fg(0)2854 38569 y Fq(+)341 b Fp(S)559 b Fq(=)482 b Fp(F)8155 38087 y Fg(0)8869 38569 y Fm(\000)341 b Fp(F)525 b Fq(+)341 b Fp(F)525 b Fq(+)341 b Fp(S)587 b Fq(where)512 b Fp(F)525 b Fq(+)341 b Fp(S)559 b Fm(2)482 b Fh(Z)27499 38768 y Fv(1)28606 38569 y Fq(and)511 b Fp(F)32295 38087 y Fg(0)33009 38569 y Fm(\000)340 b Fp(F)667 b Fm(2)483 b Fh(N)p Fm(nf)p Fq(0)p Fm(g)p Fq(,)528 b(it)511 b(con)-37 b(tradicts)1102 40163 y(to)575 b(the)g(de\014nition)f(of)h Fh(N)p Fq(.)987 b(Giv)-37 b(en)575 b Fp(F)774 b Fm(2)588 b(B)23571 40362 y Fk(")24140 40163 y Fq(,)608 b(there)575 b(migh)-37 b(t)575 b(b)37 b(e)575 b(more)g(than)g(one)g(elemen)-37 b(t)575 b(in)1102 41757 y Fh(S)2402 41956 y Fk(F)3666 41757 y Fq(=)381 b Fh(S)6380 41956 y Fk(F)7644 41757 y Fq(=)g Fm(f)p Fp(S)458 b Fm(2)380 b Fh(F)13220 41956 y Fk(\033)14316 41757 y Fq(:)h Fp(S)376 b Fq(+)300 b Fp(F)566 b Fm(2)381 b Fh(Z)21221 41956 y Fv(1)21817 41757 y Fm(g)p Fq(.)612 b(Giv)-37 b(en)451 b(an)-37 b(y)450 b(t)-37 b(w)g(o)451 b Fp(F)33203 41956 y Fv(1)33799 41757 y Fp(;)221 b(F)35243 41956 y Fv(2)36220 41757 y Fm(2)381 b(B)38359 41956 y Fk(")38928 41757 y Fq(,)451 b(for)g(an)-37 b(y)450 b Fp(S)45120 41956 y Fv(1)46096 41757 y Fm(2)381 b Fh(S)48663 41956 y Fk(F)49337 42089 y Fj(1)1102 43351 y Fq(and)443 b(an)-37 b(y)443 b Fp(S)7009 43550 y Fv(2)7974 43351 y Fm(2)368 b Fh(S)10528 43550 y Fk(F)11202 43683 y Fj(2)11787 43351 y Fq(,)442 b(w)-37 b(e)444 b(ha)-37 b(v)g(e)7709 45955 y Fp(d)p Fq(\()p Fp(S)9732 46154 y Fv(1)10327 45955 y Fp(;)221 b(S)11732 46154 y Fv(2)12328 45955 y Fq(\))369 b(=)1235 b(max)14468 46874 y Fk(q)33 b Fg(2)p Fk(R)16504 47015 y Fd(d)17062 46874 y Fv(\()p Fk(q)17997 46609 y Fe(\003)18553 46874 y Fv(\))19189 45955 y Fm(j)p Fp(S)20373 46154 y Fv(1)20969 45955 y Fq(\()p Fp(q)48 b Fq(\))294 b Fm(\000)h Fp(S)25081 46154 y Fv(2)25677 45955 y Fq(\()p Fp(q)48 b Fq(\))p Fm(j)13214 48768 y(\025)1448 b Fq(max)14468 49686 y Fg(j)p Fk(q)15270 49819 y Fj(1)15789 49686 y Fg(\000)p Fk(q)17139 49366 y Fe(\003)17106 49975 y Fj(1)17695 49686 y Fg(j\024)p Fk(d)19615 48768 y Fm(j)1577 b Fq(min)20205 49686 y Fg(j)p Fk(q)21007 49819 y Fj(2)21526 49686 y Fg(\000)p Fk(q)22876 49366 y Fe(\003)22843 49975 y Fj(2)23432 49686 y Fg(j\024)p Fk(d)25352 48768 y Fp(S)26167 48967 y Fv(1)26762 48768 y Fq(\()p Fp(q)27872 48967 y Fv(1)28468 48768 y Fp(;)221 b(q)29651 48967 y Fv(2)30247 48768 y Fq(\))295 b Fm(\000)1651 b Fq(min)32387 49686 y Fg(j)p Fk(q)33189 49819 y Fj(2)33708 49686 y Fg(\000)p Fk(q)35058 49366 y Fe(\003)35025 49975 y Fj(2)35614 49686 y Fg(j\024)p Fk(d)37534 48768 y Fp(S)38349 48967 y Fv(2)38944 48768 y Fq(\()p Fp(q)40054 48967 y Fv(1)40650 48768 y Fp(;)221 b(q)41833 48967 y Fv(2)42429 48768 y Fq(\))p Fm(j)13214 51636 y Fq(=)1448 b(max)14468 52555 y Fg(j)p Fk(q)15270 52688 y Fj(1)15789 52555 y Fg(\000)p Fk(q)17139 52234 y Fe(\003)17106 52843 y Fj(1)17695 52555 y Fg(j\024)p Fk(d)19615 51636 y Fm(j)p Fp(F)20838 51835 y Fv(1)21434 51636 y Fq(\()p Fp(q)22544 51835 y Fv(1)23140 51636 y Fq(\))294 b Fm(\000)i Fp(F)26134 51835 y Fv(2)26730 51636 y Fq(\()p Fp(q)27840 51835 y Fv(1)28435 51636 y Fq(\))p Fm(j)13214 54504 y Fq(=)p Fp(d)p Fq(\()p Fp(F)16309 54703 y Fv(1)16905 54504 y Fp(;)221 b(F)18349 54703 y Fv(2)18945 54504 y Fq(\))27729 b(\(6.9\))1102 57260 y(where)320 b Fp(d)p Fq(\()p Fm(\001)p Fp(;)221 b Fm(\001)p Fq(\))318 b(is)g(the)g Fp(C)12608 56778 y Fv(0)13204 57260 y Fq(-metric.)550 b(It)317 b(follo)-37 b(ws)319 b(from)f(\(6.9\))g(and)g(the)g(de\014nition)g(of)h(b)37 b(o)-37 b(x)318 b(dimension)1102 58854 y(that)18553 60448 y Fp(D)19653 60647 y Fk(B)20567 60448 y Fq(\()p Fh(F)22042 60647 y Fk(\033)22758 60448 y Fq(\))368 b Fm(\025)h Fp(D)26145 60647 y Fk(B)27059 60448 y Fq(\()p Fm(B)28448 60647 y Fk(")29017 60448 y Fq(\))f(=)h Fp(n;)1102 62713 y Fq(but)446 b(this)g(is)g(absurd)h(if)f(w)-37 b(e)447 b(c)-37 b(ho)37 b(ose)448 b Fp(n)375 b(>)g Fq(2.)601 b(The)447 b(same)g(argumen)-37 b(t)447 b(can)g(b)37 b(e)446 b(applied)h(to)f(the)g(set)1102 64307 y Fh(Z)2047 64506 y Fv(2)2643 64307 y Fq(.)2165 b Fb(\003)2696 66568 y Fq(As)350 b Fp(C)5610 66086 y Fk(r)6548 66568 y Fq(is)f(dense)h(in)g Fp(C)13821 66086 y Fv(0)14417 66568 y Fq(,)368 b(an)349 b(op)37 b(en)350 b(and)g(dense)g(set)f Fh(H)p Fq(\()p Fp(d;)221 b(q)31591 66086 y Fg(\003)32201 66568 y Fq(\))349 b(of)h Fm(B)35359 66767 y Fk(\017)36237 66568 y Fm(\032)370 b Fp(C)38684 66086 y Fk(r)39272 66568 y Fq(\()p Fp(R)40798 66767 y Fk(d)41417 66568 y Fq(\()p Fp(q)42575 66086 y Fg(\003)43185 66568 y Fq(\))p Fp(;)221 b Fl(R)p Fq(\))349 b(clearly)1102 68162 y(exists)434 b(suc)-37 b(h)434 b(that)f(for)h(eac)-37 b(h)435 b(p)37 b(erturbation)434 b(of)f(generating)i(function)f Fp(G)37635 68361 y Fv(1)38600 68162 y Fm(2)368 b Fh(H)p Fq(\()p Fp(d;)221 b(q)43423 67680 y Fg(\003)44034 68162 y Fq(\),)435 b(w)-37 b(e)434 b(ha)-37 b(v)g(e)16297 70766 y Fh(F)17256 70965 y Fk(\033)18266 70766 y Fm(\\)295 b Fh(Z)p Fq(\()p Fp(d;)221 b(q)22830 70218 y Fg(\003)23441 70766 y Fq(\))368 b(=)h Fl(?)p Fp(;)2879 b Fm(8)442 b Fp(\033)417 b Fm(2)369 b Fl(S)p Fp(:)1102 73371 y Fq(where)500 b(b)-37 b(y)498 b(abuse)h(of)g(terminology)g(w)-37 b(e)499 b(con)-37 b(tin)g(ue)499 b(to)f(denote)g Fp(S)33832 73570 y Fk(\033)35046 73371 y Fq(and)g(its)g(restriction)h Fp(R)47018 73570 y Fk(d)47637 73371 y Fq(\()p Fp(q)48795 72889 y Fg(\003)49405 73371 y Fq(\))1102 74965 y(b)-37 b(y)443 b(the)g(same)g(sym)-37 b(b)37 b(ol.)p eop %%Page: 49 49 49 48 bop 48792 2239 a Ft(49)2696 5427 y Fq(Recall)574 b(w)-37 b(e)574 b(ha)-37 b(v)g(e)574 b(de\014ned)f(the)g(set)g Fp(U)731 b Fq(=)586 b Fl(T)382 b Fm(\002)g Fq([)p Fp(a;)221 b Fq(2)p Fp(\031)430 b Fm(\000)382 b Fp(a)p Fq(])574 b(b)37 b(efore.)981 b(Let)574 b Fp(M)42643 5626 y Fk(U)43532 5427 y Fq(\()p Fp(S)77 b Fq(\))584 b(=)i Fm(f)p Fp(q)634 b Fq(:)1102 7021 y Fp(S)77 b Fq(\()p Fp(q)48 b Fq(\))367 b(=)i(min)7652 7220 y Fk(q)33 b Fg(2)p Fk(U)10000 7021 y Fp(S)77 b Fm(g)442 b Fq(and)9018 10822 y Fh(Z)370 b Fq(=)11734 9347 y Fi(n)12619 10822 y Fp(S)445 b Fm(2)369 b Fp(C)16178 10273 y Fv(0)16774 10822 y Fq(\()p Fp(U)m(;)221 b Fl(R)p Fq(\))369 b(:)591 b Fp(M)22879 11021 y Fk(U)23768 10822 y Fq(\()p Fp(S)77 b Fq(\))221 b(consists)442 b(of)g(discrete)i(p) 37 b(oin)-37 b(ts)40751 9347 y Fi(o)41636 10822 y Fp(:)1102 14574 y Fq(Giv)g(en)559 b Fp(d)5755 14773 y Fk(i)6758 14574 y Fp(>)i Fq(0,)586 b(there)559 b(are)f(\014nite)g Fp(q)19920 14773 y Fk(ij)21413 14574 y Fq(suc)-37 b(h)558 b(that)f Fm([)28403 14773 y Fk(j)28963 14574 y Fp(R)29972 14773 y Fk(d)30525 14906 y Fd(i)31011 14574 y Fq(\()p Fp(q)32121 14773 y Fk(ij)33057 14574 y Fq(\))j Fm(\033)g Fp(U)145 b Fq(.)935 b(Th)-37 b(us,)587 b(there)558 b(exists)g(a)1102 16168 y(sequence)445 b Fp(d)7299 16367 y Fk(i)8110 16168 y Fm(!)370 b Fq(0)443 b(and)f(a)h(coun)-37 b(table)444 b(set)f Fm(f)p Fp(q)23950 16367 y Fk(ij)24885 16168 y Fm(g)g Fq(suc)-37 b(h)443 b(that)16576 18494 y Fi(0)16576 20885 y(@)19146 19504 y Fg(1)18937 19902 y Fi(\\)17738 22732 y Fk(i)p Fv(=1)p Fk(;j)54 b Fv(=1)21834 21164 y Fh(H)p Fq(\()p Fp(d)24172 21363 y Fk(i)24614 21164 y Fp(;)221 b(q)25797 21363 y Fk(ij)26733 21164 y Fq(\))27250 18494 y Fi(1)27250 20885 y(A)28633 19902 y(\\)30330 21164 y Fh(Z)370 b Fq(=)f Fl(?)p Fp(:)1102 26113 y Fq(Therefore)498 b(there)e(is)g(generic)h(set)f(in)g Fm(B)21177 26312 y Fk(\017)22144 26113 y Fm(\032)457 b Fp(C)24678 25631 y Fk(r)25267 26113 y Fq(\()p Fp(U)m(;)221 b Fl(R)p Fq(\),)509 b(the)495 b(h)-37 b(yp)37 b(othesis)496 b(\(H1\))f(holds)h(for)g(eac) -37 b(h)1102 27707 y Fp(G)2146 27906 y Fv(1)3263 27707 y Fq(in)521 b(this)f(generic)h(set.)824 b(Note)521 b Fp(U)665 b Fq(is)520 b(an)h(ann)-37 b(ulus,)540 b(w)-37 b(e)521 b(can)g(write)g Fp(G)37608 27906 y Fv(1)38703 27707 y Fq(=)498 b Fp(G)41278 27225 y Fg(0)41278 28036 y Fv(1)42222 27707 y Fq(+)347 b Fp(G)44646 27225 y Fg(\003)44646 28036 y Fv(1)45777 27707 y Fq(so)521 b(that)1102 29301 y(b)37 b(oth)443 b Fp(G)5283 28819 y Fg(0)5283 29630 y Fv(1)6322 29301 y Fq(and)g Fp(G)9949 28819 y Fg(\003)9949 29630 y Fv(1)11003 29301 y Fq(ha)-37 b(v)g(e)443 b(simply)g(connected)h (supp)37 b(ort.)2696 31858 y(The)419 b(p)37 b(erturbation)419 b(to)f(the)g(generating)i(function)e Fp(G)h Fq(can)g(b)37 b(e)418 b(ac)-37 b(hiev)g(ed)420 b(b)-37 b(y)419 b(p)37 b(erturbing)418 b(the)1102 33452 y(Hamiltonian)537 b(function)e Fp(H)633 b Fm(!)524 b Fp(H)18930 32970 y Fg(0)19827 33452 y Fq(=)h Fp(H)465 b Fq(+)358 b Fp(\016)50 b(H)108 b Fq(.)870 b(Let)536 b(\010)30868 32970 y Fg(0)31777 33452 y Fq(b)37 b(e)536 b(the)g(map)g(determined)h(b)-37 b(y)536 b(the)1102 35046 y(generating)498 b(function)f Fp(G)331 b Fq(+)g Fp(\024G)17459 34564 y Fg(0)17459 35376 y Fv(1)18056 35046 y Fq(,)510 b(the)497 b(symplectic)g(di\013eomorphism)g(\011)459 b(=)g(\010)41212 34564 y Fg(0)41916 35046 y Fm(\016)332 b Fq(\010)43871 34564 y Fg(\000)p Fv(1)45794 35046 y Fq(is)496 b(close)1102 36640 y(to)518 b(iden)-37 b(tit)g(y)-111 b(.)815 b(W)-111 b(e)519 b(c)-37 b(ho)37 b(ose)519 b(a)f(smo)37 b(oth)517 b(function)h Fp(\032)p Fq(\()p Fp(s)p Fq(\))f(with)g Fp(\032)p Fq(\(0\))493 b(=)i(0)517 b(and)h Fp(\032)p Fq(\(1\))494 b(=)g(1,)536 b(let)518 b(\010)49353 36158 y Fg(0)49353 36968 y Fk(s)1102 38234 y Fq(b)37 b(e)452 b(the)g(symplectic)g(map)f(determined)i(b)-37 b(y)451 b Fp(G)302 b Fq(+)f Fp(\032)p Fq(\()p Fp(s)p Fq(\))p Fp(\024G)30359 37752 y Fg(0)30359 38564 y Fv(1)31405 38234 y Fq(and)452 b(let)f(\011)36957 38433 y Fk(s)37909 38234 y Fq(=)384 b(\010)40285 37752 y Fg(0)40285 38562 y Fk(s)41154 38234 y Fm(\016)301 b Fq(\010)43078 37752 y Fg(\000)p Fv(1)44504 38234 y Fq(.)617 b(Clearly)-111 b(,)1102 39828 y(\011)2135 40027 y Fk(s)3112 39828 y Fq(de\014nes)409 b(a)f(symplectic)h(isotop)-37 b(y)409 b(b)37 b(et)-37 b(w)g(een)409 b(iden)-37 b(tit)g(y)408 b(map)h(and)f(\011.)579 b(Th)-37 b(us)409 b(there)g(is)f(a)h(unique) 1102 41422 y(family)443 b(of)g(symplectic)h(v)-37 b(ector)443 b(\014eld)g Fp(X)21180 41621 y Fk(s)21748 41422 y Fq(:)590 b Fp(T)23667 40940 y Fg(\003)24278 41422 y Fp(M)514 b Fm(!)369 b Fp(T)184 b(T)29698 40940 y Fg(\003)30310 41422 y Fp(M)588 b Fq(suc)-37 b(h)443 b(that)21034 44496 y Fp(d)p 20722 45089 1315 54 v 20722 46305 a(ds)22196 45394 y Fq(\011)23229 45593 y Fk(s)24166 45394 y Fq(=)369 b Fp(X)26669 45593 y Fk(s)27531 45394 y Fm(\016)296 b Fq(\011)29524 45593 y Fk(s)30092 45394 y Fp(:)1102 49121 y Fq(By)427 b(the)f(c)-37 b(hoice)428 b(of)e(p)37 b(erturbation,)430 b(there)d(is)f(a)h(simply)f(connected)h(and)g(compact)g(domain)f Fp(D)48897 49320 y Fk(K)1102 50715 y Fq(suc)-37 b(h)524 b(that)e(\011)8170 50914 y Fk(s)8738 50715 y Fm(j)9107 50955 y Fk(T)9877 50689 y Fe(\003)10433 50955 y Fk(M)100 b Fg(n)p Fk(D)12959 51089 y Fd(K)14395 50715 y Fq(=)503 b Fp(id)p Fq(.)831 b(It)522 b(follo)-37 b(ws)524 b(that)e(there)i(is)f (a)g(Hamiltonian)g Fp(H)42103 50914 y Fv(1)42699 50715 y Fq(\()p Fp(p;)221 b(q)48 b(;)221 b(s)p Fq(\))523 b(suc)-37 b(h)1102 52316 y(that)492 b Fp(dH)5825 52515 y Fv(1)6422 52316 y Fq(\()p Fp(Y)295 b Fq(\))451 b(=)h Fp(dp)329 b Fm(^)e Fp(dq)48 b Fq(\()p Fp(X)16309 52515 y Fk(s)16876 52316 y Fp(;)221 b(Y)296 b Fq(\))492 b(holds)g(for)h(an)-37 b(y)493 b(v)-37 b(ector)493 b(\014eld)g Fp(Y)295 b Fq(.)739 b(Re-parametrizing)494 b Fp(s)e Fq(b)-37 b(y)1102 53910 y Fp(t)503 b Fq(w)-37 b(e)503 b(can)h(mak)-37 b(e)503 b Fp(H)11228 54109 y Fv(1)12327 53910 y Fq(smo)37 b(othly)503 b(and)g(p)37 b(erio)g(dically)504 b(dep)37 b(end)503 b(on)g Fp(t)p Fq(.)770 b(T)-111 b(o)503 b(see)h(that)e Fp(dH)45205 54109 y Fv(1)46304 53910 y Fq(is)h(also)1102 55504 y(small,)435 b(let)e(us)g(mak)-37 b(e)434 b(use)f(of)g(a)g (theorem)h(of)f(W)-111 b(eistain)433 b([W].)g(A)h(neigh)-37 b(b)37 b(orho)g(o)g(d)434 b(of)f(the)f(iden)-37 b(tit)g(y)1102 57098 y(in)415 b(the)g(symplectic)h(di\013eomorphism)f(group)h(of)f(a)g (compact)h(symplectic)f(manifold)g Fr(M)g Fq(can)h(b)37 b(e)1102 58692 y(iden)-37 b(ti\014ed)512 b(with)f(a)h(neigh)-37 b(b)37 b(orho)g(o)g(d)513 b(of)e(the)g(zero)i(in)f(the)f(v)-37 b(ector)512 b(space)h(of)e(closed)h(1-forms)g(on)1102 60286 y Fr(M)p Fq(.)k(Since)g(Hamiltomorphism)g(is)f(a)h(subgroup)g(of) f(symplectic)h(di\013eomorphism,)535 b(there)516 b(is)f(a)1102 61880 y(function)443 b Fp(H)7517 61398 y Fg(0)7890 61880 y Fq(,)g(su\016cien)-37 b(tly)443 b(close)g(to)g Fp(H)108 b Fq(,)443 b(suc)-37 b(h)443 b(that)f(\010)29012 62079 y Fk(H)29882 62212 y Fj(1)30763 61880 y Fm(\016)295 b Fq(\010)32681 62079 y Fk(H)34058 61880 y Fq(=)369 b(\010)36419 61398 y Fk(t)36419 62246 y(H)37360 61981 y Fe(0)37786 61880 y Fm(j)38155 62079 y Fk(t)p Fv(=1)39968 61880 y Fq(.)2696 64438 y(Th)-37 b(us)444 b(the)e(densit)-37 b(y)443 b(of)g(\(H1\))f(is)h(pro)-37 b(v)g(ed.)2696 66995 y(F)-111 b(or)571 b(the)e(h)-37 b(yp)37 b(othesis)569 b(\(H2\))g(let)g(us)g(consider)i(the)e(t)-37 b(wist)569 b(map)g(on)h(the)f(cylinder.)971 b(In)569 b(this)1102 68589 y(case,)550 b(eac)-37 b(h)528 b(co-homology)h(class)f(corresp)37 b(onds)529 b(to)e(a)g(unique)h(rotation)g(n)-37 b(um)g(b)37 b(er.)845 b(Giv)-37 b(en)528 b(an)-37 b(y)1102 70183 y(rational)446 b(n)-37 b(um)g(b)37 b(er)445 b Fp(p=q)420 b Fm(2)372 b Fl(Q)p Fq(,)446 b(it)e(is)h(ob)-37 b(vious)445 b(that)f(there)i(is)f(a)g(op)37 b(en)445 b(dense)g(set)g(in)g(the)f (space)i(of)1102 71777 y(area-preserving)527 b(t)-37 b(wist)523 b(map)h(suc)-37 b(h)524 b(that)f(there)i(is)f(only)g(one)g (minimal)g(\()p Fp(p;)221 b(q)48 b Fq(\)-p)37 b(erio)g(dic)524 b(orbit)1102 73371 y(without)350 b(homo)37 b(clinic)350 b(lo)37 b(op.)559 b(T)-111 b(ak)-37 b(e)351 b(the)f(in)-37 b(tersection)350 b(of)g(coun)-37 b(tably)350 b(op)37 b(en)349 b(dense)i(set)e(w)-37 b(e)350 b(obtain)1102 74965 y(that)442 b(\(H2\))h(is)g(a)g(generic)h(prop)37 b(ert)-37 b(y)-111 b(.)p eop %%Page: 50 50 50 49 bop 1102 2239 a Ft(50)2696 5427 y Fq(T)-111 b(o)375 b(v)-37 b(erify)375 b(the)f(\(H3\),)387 b(let)374 b(us)g(consider)h(an) f(in)-37 b(v)-74 b(arian)-37 b(t)375 b(circle)g(\000)33987 5626 y Fk(\033)35076 5427 y Fq(on)f(\006.)568 b(There)376 b(is)d(an)i(in)-37 b(terv)-74 b(al)1102 7021 y Fp(I)104 b Fq(\()p Fp(c)2882 7220 y Fv(1)3478 7021 y Fq(\))458 b(=)g Fm(f)p Fp(c)g Fq(=)g(\()p Fp(c)10224 7220 y Fv(1)10819 7021 y Fp(;)221 b(c)11984 7220 y Fv(2)12581 7021 y Fq(\))457 b Fm(2)h Fl(R)15858 6539 y Fv(2)16912 7021 y Fq(:)g Fp(a)p Fq(\()p Fp(c)19533 7220 y Fv(1)20129 7021 y Fq(\))f Fm(\024)i Fp(c)23170 7220 y Fv(2)24224 7021 y Fm(\024)f Fp(b)p Fq(\()p Fp(c)27377 7220 y Fv(1)27972 7021 y Fq(\))p Fm(g)496 b Fq(suc)-37 b(h)497 b(that)e(supp\()p Fm(M)p Fq(\()p Fp(c)p Fq(\))p Fm(j)42458 7220 y Fk(t)p Fv(=0)44270 7021 y Fq(\))457 b Fm(\022)i Fq(\000)47566 7220 y Fk(\033)48777 7021 y Fq(i\013)1102 8615 y Fp(c)377 b Fm(2)f Fp(I)104 b Fq(\()p Fp(c)5096 8814 y Fv(1)5692 8615 y Fq(\).)604 b(Let)447 b Fp(U)592 b Fq(b)37 b(e)448 b(a)g(small)f(neigh)-37 b(b)37 b(orho)g(o)g(d)449 b(of)e Fp(\031)48 b Fq(\(\000)29487 8814 y Fk(\033)30202 8615 y Fq(\).)604 b(Under)448 b(the)g(h)-37 b(yp)37 b(othesis)447 b(\(H1\),)h(the)1102 10209 y(set)477 b Fm(f)p Fp(B)4949 9727 y Fg(\003)4882 10537 y Fk(c)5984 10209 y Fq(=)425 b(0)p Fm(gn)p Fp(U)622 b Fq(is)476 b(homotopically)h (trivial)g(for)g Fp(c)425 b Fq(=)g(\()p Fp(c)30566 10408 y Fv(1)31161 10209 y Fp(;)221 b(a)32453 10408 y Fv(1)33050 10209 y Fq(\()p Fp(c)34142 10408 y Fv(1)34737 10209 y Fq(\)\))476 b(and)g(for)h Fp(c)425 b Fq(=)g(\()p Fp(c)44480 10408 y Fv(1)45075 10209 y Fp(;)221 b(b)46235 10408 y Fv(1)46832 10209 y Fq(\()p Fp(c)47924 10408 y Fv(1)48519 10209 y Fq(\)\).)1102 11897 y(By)439 b(the)g(upp)37 b(er)440 b(semi-con)-37 b(tin)g(uit)g(y)439 b(of)g(Ma)37 b(~)-701 b(n)-37 b(\023)-627 b(e)439 b(sets)g Fp(c)369 b Fm(!)29385 11561 y Fq(~)28964 11897 y Fm(N)195 b Fq(\()p Fp(c)p Fq(\),)439 b(the)g(set)g Fm(N)196 b Fq(\()p Fp(c)39398 11415 y Fg(0)39770 11897 y Fq(\))p Fm(j)40656 12096 y Fk(t)p Fv(=0)42907 11897 y Fq(is)439 b(in)g(a)g(small)1102 13491 y(neigh)-37 b(b)37 b(orho)g(o)g(d)446 b(of)e Fm(f)p Fp(B)12700 13009 y Fg(\003)12633 13819 y Fk(c)13681 13491 y Fq(=)372 b(0)p Fm(g)444 b Fq(if)g Fp(c)18652 13009 y Fg(0)19396 13491 y Fq(=)371 b(\()p Fp(c)21892 13690 y Fv(1)22488 13491 y Fp(;)221 b(b)p Fq(\()p Fp(c)24740 13690 y Fv(1)25336 13491 y Fq(\))295 b(+)i Fp(\016)50 b Fq(\))444 b(with)g Fp(\016)422 b(>)371 b Fq(0)445 b(su\016cien)-37 b(tly)444 b(small.)595 b(Let)444 b(us)1102 15179 y(consider)479 b(suc)-37 b(h)478 b(a)g(minimal)f(measure)21549 14843 y(~)20899 15179 y Fm(M)p Fq(\()p Fp(c)23586 14697 y Fg(0)23958 15179 y Fq(\).)695 b(Let)477 b Fp(\026)h Fq(b)37 b(e)478 b(an)g(ergo)37 b(dic)479 b(comp)37 b(onen)-37 b(t)478 b(of)46626 14843 y(~)45976 15179 y Fm(M)q Fq(\()p Fp(c)48664 14697 y Fg(0)49036 15179 y Fq(\),)1102 16867 y(there)446 b(exists)g Fp(\017)8713 16385 y Fg(\003)9697 16867 y Fp(>)373 b Fq(0)445 b(suc)-37 b(h)446 b(that)f(dist\(supp)o Fp(\026;)24968 16531 y Fq(~)24884 16867 y(\000)25714 17066 y Fk(\033)26430 16867 y Fq(\))373 b Fm(\025)g Fq(3)p Fp(\017)29929 16385 y Fg(\003)30986 16867 y Fq(for)445 b(all)h Fp(\033)421 b Fm(2)372 b Fl(S)p Fq(.)598 b(F)-111 b(or)446 b(an)-37 b(y)446 b Fp(\017)373 b(>)h Fq(0)445 b(with)1102 18461 y Fp(\017)370 b Fm(\024)f Fp(\017)3952 17979 y Fg(\003)5000 18461 y Fq(w)-37 b(e)438 b(can)g(de\014ne)g(a)g Fp(C)15358 17979 y Fk(r)15947 18461 y Fq(-smo)37 b(oth)437 b(function)g Fp(L)27179 17979 y Fk(\033)27179 18837 y(k)22 b(;\017)28591 18461 y Fq(:)587 b Fp(T)184 b(M)430 b Fm(\002)285 b Fl(T)368 b Fm(!)i Fl(R)437 b Fq(so)h(that)f Fp(L)43295 17979 y Fk(\033)43295 18837 y(k)22 b(;\017)44707 18461 y Fq(\()p Fp(z)58 b(;)221 b(t)p Fq(\))368 b(=)h(0)1102 20278 y(if)534 b(\()p Fp(z)58 b(;)221 b(t)p Fq(\))520 b Fm(2)h Fq(supp)p Fp(\026)356 b Fq(+)g(2)13065 19796 y Fg(\000)p Fk(k)22 b Fg(\000)p Fv(1)15909 20278 y Fp(\017)p Fq(,)557 b Fp(L)18278 19796 y Fk(\033)18278 20654 y(k)22 b(;\017)20210 20278 y Fq(=)521 b(2)22428 19796 y Fg(\000)p Fk(k)23912 20278 y Fp(\017)24451 19796 y Fk(r)29 b Fv(+1)26918 20278 y Fq(if)534 b(\()p Fp(z)58 b(;)221 b(t)p Fq(\))668 b Fp(=)-812 b Fm(2)521 b Fq(supp)p Fp(\026)356 b Fq(+)g(2)38881 19796 y Fg(\000)p Fk(k)40365 20278 y Fp(\017)534 b Fq(and)g Fp(L)45016 19796 y Fk(\033)45016 20654 y(k)22 b(;\017)46962 20278 y Fq(tak)-37 b(es)1102 22096 y(the)649 b(v)-74 b(alue)649 b(b)37 b(et)-37 b(w)g(een)649 b(2)13231 21614 y Fg(\000)p Fk(k)14715 22096 y Fp(\017)15254 21614 y Fk(r)29 b Fv(+1)17836 22096 y Fq(and)648 b(0)g(elsewhere.)1209 b(Ob)-37 b(viously)-111 b(,)700 b Fp(\026)649 b Fq(is)f(the)g(unique)h (ergo)37 b(dic)1102 23690 y(comp)g(onen)-37 b(t)444 b(of)f Fp(c)9980 23208 y Fg(0)10352 23690 y Fq(-minimal)g(measure)h(of)e(the)h (Lagrangian)19545 27792 y Fp(L)20449 27243 y Fk(\033)20449 28120 y(\017)21534 27792 y Fq(=)369 b Fp(L)295 b Fq(+)25793 26893 y(1)p 25623 27486 1006 54 v 25623 28703 a Fp(r)37 b Fq(!)27445 26131 y Fg(1)27016 26530 y Fi(X)27009 29389 y Fk(k)22 b Fv(=1)29162 27792 y Fp(L)30066 27243 y Fk(\033)30066 28120 y(k)g(;\017)1102 32168 y Fq(and)431 b Fm(k)p Fp(L)5241 31686 y Fk(\033)5241 32497 y(\017)6226 32168 y Fm(\000)270 b Fp(L)p Fm(k)9097 32367 y Fk(C)9924 32102 y Fd(r)10895 32168 y Fm(\024)369 b Fp(\017)p Fq(.)586 b(Since)431 b(\(H3\))e(is)h(required)i(only)e(for)g(coun)-37 b(table)431 b Fp(\033)417 b Fm(2)368 b Fl(S)p Fq(,)433 b(w)-37 b(e)431 b(can)f(c)-37 b(ho)37 b(ose)1102 33762 y(ev)-37 b(en)444 b(smaller)g Fp(\017)9254 33961 y Fk(\033)10412 33762 y Fq(so)f(that)f(the)h(supp)37 b(orts)442 b(of)h(these)g Fp(L)28470 33280 y Fk(\033)28470 34091 y(\017)28913 34224 y Fd(\033)29906 33762 y Fm(\000)295 b Fp(L)443 b Fq(ha)-37 b(v)g(e)444 b(no)f(in)-37 b(tersection.)2696 36136 y(Note)610 b(the)g(p)37 b(erturbation)610 b(w)-37 b(e)610 b(in)-37 b(tro)37 b(duced)611 b(for)f(\(H1\))f(has)h(compact)g(supp)37 b(ort)610 b(whic)-37 b(h)610 b(has)1102 37730 y(no)471 b(in)-37 b(tersection)470 b(with)g(the)g(cylinder,)478 b(the)470 b(p)37 b(erturbation)470 b(w)-37 b(e)471 b(in)-37 b(tro)37 b(duced)471 b(for)f(\(H3\))g(do)37 b(es)471 b(not)1102 39324 y(touc)-37 b(h)358 b(the)g(set)g Fm(f)p Fp(B)10601 38842 y Fg(\003)10534 39752 y Fk(c)p Fv(\()p Fk(\033)33 b Fv(\))12922 39324 y Fq(=)370 b(0)p Fm(g)358 b Fq(for)g(all)g Fp(\033)416 b Fm(2)369 b Fl(S)p Fq(,)374 b(there)359 b(is)e(a)h(dense)h(set)e(for)h Fp(P)542 b Fq(suc)-37 b(h)358 b(that)f(\(H1\),)375 b(\(H2\))1102 41113 y(and)489 b(\(H3\))f(hold.)728 b(Th)-37 b(us)489 b(w)-37 b(e)490 b(obtain)e(the)h(densit)-37 b(y)488 b(of)h(the)g(p)37 b(erturbation.)728 b(Since)489 b(the)f(time)h(for)1102 42708 y(eac)-37 b(h)401 b(orbit)f(drifts)g(from)g Fp(p)14495 42907 y Fv(1)15460 42708 y Fp(<)369 b(A)400 b Fq(to)g Fp(p)20507 42907 y Fv(1)21472 42708 y Fp(>)369 b(B)466 b Fq(is)400 b(\014nite,)409 b(the)399 b(smo)37 b(oth)400 b(dep)37 b(endence)402 b(of)d(solutions)1102 44302 y(of)443 b(ODE's)g(on)g(parameter)h(guaran)-37 b(tees)444 b(the)e(op)37 b(enness.)2696 46675 y(Therefore,)445 b(the)e(pro)37 b(of)443 b(of)g(the)g(theorem)g(1.1)g(is)g(completed.)1102 49844 y Fr(App)42 b(endix)2696 52218 y Fq(In)418 b(this)g(app)37 b(endix)418 b(w)-37 b(e)419 b(presen)-37 b(t)418 b(the)g(pro)37 b(of)419 b(of)f(the)g(lemma)g(2.6,)423 b(giv)-37 b(en)419 b(b)-37 b(y)418 b(Bernard)h(in)f([Be],)1102 53812 y(for)444 b(the)e(completeness)i(sak)-37 b(e.)1102 56206 y Fr(Lemma)567 b(2.6.)764 b Fo(If)11808 55870 y Fq(~)11158 56206 y Fm(M)p Fq(\()p Fp(c)p Fq(\))521 b Fo(is)h(minimal)h(in)f(the)g(sense)g(of)g (top)-68 b(olo)g(gic)g(al)523 b(dynamics)f(and)f(if)i(ther)-68 b(e)1102 57800 y(exists)575 b(a)e(se)-68 b(quenc)g(e)573 b Fp(\015)12273 57999 y Fk(n)13569 57800 y Fo(of)h Fp(n)p Fo(-p)-68 b(erio)g(dic)574 b(curves)g(such)f Fp(A)29910 57999 y Fk(c)30451 57800 y Fq(\()p Fp(\015)31656 57999 y Fk(n)32379 57800 y Fq(\))546 b Fm(!)h Fq(0)573 b Fo(as)g Fp(n)547 b Fm(!)g(1)p Fo(,)598 b(then)573 b Fp(L)47856 57999 y Fk(c)48971 57800 y Fo(is)1102 59488 y(r)-68 b(e)g(gular,)477 b(henc)-68 b(e)10046 59152 y Fq(~)9589 59488 y Fm(A)p Fq(\()p Fp(c)p Fq(\))368 b(=)14450 59152 y(~)14029 59488 y Fm(N)195 b Fq(\()p Fp(c)p Fq(\))368 b(=)18944 59152 y(~)18694 59488 y Fm(G)79 b Fq(\()p Fp(c)p Fq(\))p Fo(.)1102 61862 y Fr(Pro)42 b(of)p Fq(:)552 b(As)366 b(the)f(\014rst)g(step)h(w) -37 b(e)366 b(sho)-37 b(w)366 b(that)f(the)g(follo)-37 b(wing)366 b(limit)f(exists)g(for)h(all)g(\()p Fp(x;)221 b(t)p Fq(\))368 b Fm(2)h(M)140 b(\002)g Fl(T)p Fq(:)18527 64973 y(lim)18063 65770 y Fk(n)p Fg(!1)21059 64973 y Fp(F)21913 65172 y Fk(c)22454 64973 y Fq(\()p Fp(x;)221 b(x;)g(t;)g(t)296 b Fq(+)f Fp(n)p Fq(\))369 b(=)g(0)p Fp(:)13898 b Fq(\(A.1\))1102 68475 y(By)401 b(the)g(condition,)409 b(w)-37 b(e)401 b(can)g(supp)37 b(ose)401 b(these)g Fp(n)p Fq(-p)37 b(erio)g(dic)402 b(curv)-37 b(es)402 b Fp(\015)35186 68674 y Fk(n)36309 68475 y Fq(are)f(minimizers,)410 b(their)401 b Fp(n)p Fq(-)1102 70069 y(p)37 b(erio)g(dic)377 b(orbits)f Fp(X)10940 70268 y Fk(n)11663 70069 y Fq(\()p Fp(t)p Fq(\))368 b(=)h(\()p Fp(d\015)16843 70268 y Fk(n)17566 70069 y Fq(\()p Fp(t)p Fq(\))p Fp(;)221 b(t)p Fq(\))374 b(is)i(a)g(compact)h(subset)f(of)g Fp(T)184 b(M)307 b Fm(\002)162 b Fl(T)p Fq(.)567 b(Eac)-37 b(h)377 b(subsequence)1102 71663 y(of)382 b Fp(X)3655 71862 y Fk(n)4760 71663 y Fq(has)g(a)g(con)-37 b(v)g(ergen)g(t)384 b(subsequence)f(in)f(the)g (sense)h(of)f(Hausdor\013)g(top)37 b(ology)-111 b(.)571 b(The)383 b(limit)e(set)1102 73277 y(of)426 b(suc)-37 b(h)427 b(a)f(sequence)h(is)f(ob)-37 b(viously)427 b(an)f(in)-37 b(v)-74 b(arian)-37 b(t)427 b(subset)e(of)32922 72941 y(~)32272 73277 y Fm(M)q Fq(\()p Fp(c)p Fq(\).)583 b(Since)40531 72941 y(~)39881 73277 y Fm(M)p Fq(\()p Fp(c)p Fq(\))425 b(is)h(minimal,)1102 74965 y(this)c(limit)g(set)g(has)h(to)f(b)37 b(e)15267 74629 y(~)14617 74965 y Fm(M)q Fq(\()p Fp(c)p Fq(\))421 b(itself.)583 b(Therefore,)429 b(the)422 b(sequence)i(of)e (subsets)g Fp(X)43267 75164 y Fk(n)44412 74965 y Fq(con)-37 b(v)g(erges)p eop %%Page: 51 51 51 50 bop 48792 2239 a Ft(51)1102 5427 y Fq(to)3441 5091 y(~)2791 5427 y Fm(M)p Fq(\()p Fp(c)p Fq(\))507 b(in)h(the)f (Hausdor\013)i(top)37 b(ology)-111 b(.)785 b(It)508 b(follo)-37 b(ws)508 b(that)f(eac)-37 b(h)509 b(p)37 b(oin)-37 b(t)507 b(\()p Fp(x;)221 b(s)p Fq(\))477 b Fm(2)43615 5091 y Fq(~)42965 5427 y Fm(M)p Fq(\()p Fp(c)p Fq(\))507 b(is)g(the)1102 7021 y(limit)445 b(of)h(a)g(sequence)h(\()p Fp(\015)74 b Fq(\()p Fp(t)14688 7220 y Fk(n)15409 7021 y Fq(\))p Fp(;)221 b(s)p Fq(\))445 b(with)g Fp(t)21609 7220 y Fk(n)22705 7021 y Fq(=)374 b Fp(s)445 b Fq(mo)37 b(d)446 b(1)g(for)f(eac)-37 b(h)447 b Fp(n)p Fq(.)599 b(As)445 b Fp(F)38893 7220 y Fk(c)39880 7021 y Fq(is)h(of)f(Lipsc)-37 b(hitz)446 b(w)-37 b(e)1102 8615 y(ha)g(v)g(e)8427 11235 y(lim)221 b(sup)9072 12291 y Fk(n)p Fg(!1)12714 11235 y Fp(F)13568 11434 y Fk(c)14110 11235 y Fq(\()p Fp(x;)g(x;)g(t;)g(t)295 b Fq(+)h Fp(n)p Fq(\))369 b(=)221 b(lim)g(sup)24082 12291 y Fk(n)p Fg(!1)27724 11235 y Fp(F)28578 11434 y Fk(c)29119 11235 y Fq(\()p Fp(\015)30324 11434 y Fk(n)31046 11235 y Fq(\()p Fp(t)32043 11434 y Fk(n)32766 11235 y Fq(\))p Fp(;)g(\015)34561 11434 y Fk(n)35283 11235 y Fq(\()p Fp(t)36280 11434 y Fk(n)37003 11235 y Fq(\))p Fp(;)g(t;)g(t)295 b Fq(+)g Fp(n)p Fq(\))22182 13952 y(=)221 b(lim)g(sup)24082 15007 y Fk(n)p Fg(!1)27724 13952 y Fp(A)28720 14151 y Fk(c)29261 13952 y Fq(\()p Fp(\015)30466 14151 y Fk(n)31188 13952 y Fq(\))22182 16602 y(=0)p Fp(;)1102 19288 y Fq(whic)-37 b(h)444 b(implies)f(\(A.1\).)2696 21553 y(Next,)362 b(w)-37 b(e)343 b(claim)e(that)g(\(A.1\))g(implies)h(that)e Fp(L)92 b Fm(\000)g Fp(\021)27672 21752 y Fk(c)28555 21553 y Fq(is)342 b(regular,)363 b(i.e.)557 b(for)341 b(an)-37 b(y)342 b(\()p Fp(x;)221 b(s)p Fq(\),)361 b(\()p Fp(x)46192 21071 y Fg(0)46564 21553 y Fp(;)221 b(s)47777 21071 y Fg(0)48150 21553 y Fq(\))369 b Fm(2)1102 23147 y Fp(M)440 b Fm(\002)295 b Fl(T)p Fq(,)443 b Fp(\017)369 b(>)g Fq(0,)443 b(there)g(exists)g Fp(T)628 b Fq(suc)-37 b(h)443 b(that)16167 25768 y Fp(F)17021 25967 y Fk(c)17562 25768 y Fq(\()p Fp(x;)221 b(x)20187 25219 y Fg(0)20560 25768 y Fp(;)g(t;)g(t)22700 25219 y Fg(0)23073 25768 y Fq(\))368 b Fm(\024)h Fp(h)26125 25967 y Fk(c)26667 25768 y Fq(\()p Fp(x;)221 b(x)29292 25219 y Fg(0)29665 25768 y Fp(;)g(t;)g(t)31805 25219 y Fg(0)32178 25768 y Fq(\))294 b(+)i Fp(\017)1102 28388 y Fq(if)494 b Fp(t)f Fq(and)g Fp(t)6457 27906 y Fg(0)7323 28388 y Fq(satisfy)g Fp(t)453 b Fq(=)h Fp(s)493 b Fq(mo)37 b(d)493 b(1,)507 b Fp(t)20116 27906 y Fg(0)20941 28388 y Fq(=)454 b Fp(s)23051 27906 y Fg(0)23917 28388 y Fq(mo)37 b(d)493 b(1)h(and)f Fp(t)31227 27906 y Fg(0)32053 28388 y Fm(\025)453 b Fp(t)329 b Fq(+)g Fp(T)184 b Fq(.)743 b(Indeed,)507 b(let)493 b Fp(K)589 b Fq(b)37 b(e)493 b(the)1102 29982 y(common)381 b(Lipsc)-37 b(hitz)381 b(constan)-37 b(t)380 b(of)g(all)h(functions)f Fp(F)27008 30181 y Fk(c)27549 29982 y Fq(\()p Fm(\001)p Fp(;)221 b Fm(\001)p Fp(;)g(t;)g(t)31534 29500 y Fg(0)31907 29982 y Fq(\))380 b(with)g Fp(t)36247 29500 y Fg(0)36988 29982 y Fm(\025)369 b Fp(t)170 b Fq(+)g(1,)393 b(let)381 b Fp(t)44006 30181 y Fv(0)44970 29982 y Fq(=)369 b Fp(s)380 b Fq(mo)37 b(d)1102 31576 y(1,)480 b Fp(t)3095 31094 y Fg(0)3095 31906 y Fv(0)4109 31576 y Fq(=)418 b Fp(s)6183 31094 y Fg(0)7028 31576 y Fq(mo)37 b(d)472 b(1,)480 b(let)472 b Fp(\015)74 b Fq(:)649 b([)p Fp(t)16136 31775 y Fv(0)16731 31576 y Fp(;)221 b(t)17801 31094 y Fg(0)17801 31906 y Fv(0)18397 31576 y Fq(])418 b Fm(!)h Fp(M)616 b Fq(b)37 b(e)473 b(a)f(minimizer)h(with)f Fp(\015)74 b Fq(\()p Fp(t)36856 31775 y Fv(0)37451 31576 y Fq(\))418 b(=)g Fp(x)472 b Fq(and)h Fp(\015)74 b Fq(\()p Fp(t)45440 31094 y Fg(0)45440 31906 y Fv(0)46035 31576 y Fq(\))417 b(=)h Fp(x)49179 31094 y Fg(0)49552 31576 y Fq(,)1102 33170 y(i.e.)667 b Fp(A)4462 33369 y Fk(c)5003 33170 y Fq(\()p Fp(\015)74 b Fq(\))410 b(=)h Fp(F)9507 33369 y Fk(c)10048 33170 y Fq(\()p Fp(x;)221 b(x)12673 32688 y Fg(0)13046 33170 y Fp(;)g(t)14116 33369 y Fv(0)14712 33170 y Fp(;)g(t)15782 32688 y Fg(0)15782 33500 y Fv(0)16378 33170 y Fq(\).)665 b(W)-111 b(e)469 b(can)g(mak)-37 b(e)468 b Fp(t)26676 32688 y Fg(0)26676 33500 y Fv(0)27584 33170 y Fm(\000)312 b Fp(t)29409 33369 y Fv(0)30473 33170 y Fq(is)468 b(su\016cien)-37 b(tly)468 b(large)h(so)f(that)f Fm(9)h Fp(t)48029 33369 y Fv(1)49036 33170 y Fm(2)1102 34764 y Fq([)p Fp(t)1951 34963 y Fv(0)2547 34764 y Fp(;)221 b(t)3617 34282 y Fg(0)3617 35094 y Fv(0)4213 34764 y Fq(])446 b(suc)-37 b(h)447 b(that)f(dist\()p Fp(\015)74 b Fq(\()p Fp(t)15334 34963 y Fv(1)15928 34764 y Fq(\))p Fp(;)221 b(y)48 b Fq(\))375 b Fm(\024)g Fp(\017=)p Fq(4)p Fp(K)542 b Fq(for)447 b(some)g Fp(y)423 b Fm(2)375 b(M)p Fq(\()p Fp(c)p Fq(\))p Fm(j)34848 34963 y Fk(t)p Fv(=)p Fk(t)36465 35096 y Fj(1)37050 34764 y Fq(,)447 b(in)g(virtue)g(of)f(standard)1102 36358 y(argumen)-37 b(t)591 b(of)f(top)37 b(ological)591 b(dynamics.)1033 b(Since)591 b Fp(h)26968 36557 y Fk(c)27510 36358 y Fq(\()p Fp(x;)221 b(x)30135 35876 y Fg(0)30508 36358 y Fp(;)g(s;)g(s)32934 35876 y Fg(0)33307 36358 y Fq(\))614 b(=)h(lim)221 b(inf)325 b Fp(F)40844 36557 y Fk(c)41385 36358 y Fq(\()p Fp(x;)221 b(x)44010 35876 y Fg(0)44383 36358 y Fp(;)g(t;)g(t)46523 35876 y Fg(0)46896 36358 y Fq(\),)626 b(w)-37 b(e)1102 37952 y(can)444 b(supp)37 b(ose)443 b(in)g(addition)f(that)15208 40840 y Fp(F)16062 41039 y Fk(c)16603 40840 y Fq(\()p Fp(x;)221 b(x)19228 40291 y Fg(0)19601 40840 y Fp(;)g(t)20671 41039 y Fv(0)21267 40840 y Fp(;)g(t)22337 40291 y Fg(0)22337 41168 y Fv(0)22933 40840 y Fq(\))369 b Fm(\024)g Fp(h)25986 41039 y Fk(c)26527 40840 y Fq(\()p Fp(x;)221 b(x)29152 40291 y Fg(0)29525 40840 y Fp(;)g(s;)g(s)31951 40291 y Fg(0)32324 40840 y Fq(\))295 b(+)34686 39941 y Fp(\017)p 34623 40534 665 54 v 34623 41751 a Fq(2)35447 40840 y Fp(:)1102 43907 y Fq(Let)443 b Fp(x)4241 44106 y Fv(1)5206 43907 y Fq(=)370 b Fp(\015)74 b Fq(\()p Fp(t)8368 44106 y Fv(1)8962 43907 y Fq(\))443 b(w)-37 b(e)443 b(ha)-37 b(v)g(e)10858 46527 y Fp(F)11712 46726 y Fk(c)12253 46527 y Fq(\()p Fp(x;)221 b(x)14878 45978 y Fg(0)15251 46527 y Fp(;)g(t)16321 46726 y Fv(0)16917 46527 y Fp(;)g(t)17987 45978 y Fg(0)17987 46855 y Fv(0)18583 46527 y Fq(\))368 b(=)i Fp(F)21725 46726 y Fk(c)22266 46527 y Fq(\()p Fp(x;)221 b(x)24891 46726 y Fv(1)25487 46527 y Fp(;)g(t)26557 46726 y Fv(0)27153 46527 y Fp(;)g(t)28223 46726 y Fv(1)28819 46527 y Fq(\))295 b(+)g Fp(F)31813 46726 y Fk(c)32355 46527 y Fq(\()p Fp(x)33631 46726 y Fv(1)34226 46527 y Fp(;)221 b(x)35575 45978 y Fg(0)35949 46527 y Fp(;)g(t)37019 46726 y Fv(1)37614 46527 y Fp(;)g(t)38684 45978 y Fg(0)38684 46855 y Fv(0)39280 46527 y Fq(\))p Fp(;)1102 49147 y Fq(It)442 b(follo)-37 b(ws)444 b(that)9842 51777 y Fm(j)p Fp(F)11065 51976 y Fk(c)11606 51777 y Fq(\()p Fp(x;)221 b(x)14231 51228 y Fg(0)14604 51777 y Fp(;)g(t)15674 51976 y Fv(0)16270 51777 y Fp(;)g(t)17340 51228 y Fg(0)17340 52105 y Fv(0)17936 51777 y Fq(\))295 b Fm(\000)g Fp(F)20930 51976 y Fk(c)21471 51777 y Fq(\()p Fp(x;)221 b(y)48 b(;)221 b(t)25106 51976 y Fv(0)25702 51777 y Fp(;)g(t)26772 51976 y Fv(1)27368 51777 y Fq(\))295 b Fm(\000)g Fp(F)30362 51976 y Fk(c)30904 51777 y Fq(\()p Fp(y)48 b(;)221 b(x)33469 51228 y Fg(0)33841 51777 y Fp(;)g(t)34911 51976 y Fv(1)35507 51777 y Fp(;)g(t)36577 51228 y Fg(0)36577 52105 y Fv(0)37173 51777 y Fq(\))p Fm(j)369 b(\024)40052 50878 y Fp(\017)p 39990 51471 V 39990 52688 a Fq(2)40813 51777 y Fp(;)1102 54770 y Fq(th)-37 b(us)10743 56364 y Fp(F)11597 56563 y Fk(c)12139 56364 y Fq(\()p Fp(x;)221 b(y)48 b(;)221 b(t)15774 56563 y Fv(0)16370 56364 y Fp(;)g(t)17440 56563 y Fv(1)18036 56364 y Fq(\))295 b(+)g Fp(F)21030 56563 y Fk(c)21571 56364 y Fq(\()p Fp(y)48 b(;)221 b(x)24136 55815 y Fg(0)24509 56364 y Fp(;)g(t)25579 56563 y Fv(1)26175 56364 y Fp(;)g(t)27245 55815 y Fg(0)27245 56692 y Fv(0)27841 56364 y Fq(\))369 b Fm(\024)g Fp(h)30894 56563 y Fk(c)31435 56364 y Fq(\()p Fp(x;)221 b(x)34060 55815 y Fg(0)34433 56364 y Fp(;)g(s;)g(s)36859 55815 y Fg(0)37232 56364 y Fq(\))295 b(+)g Fp(\017:)1102 58635 y Fq(By)448 b(the)f(c)-37 b(hoice)449 b(of)f Fp(t)f Fq(and)h Fp(t)14950 58153 y Fg(0)15769 58635 y Fq(w)-37 b(e)449 b(kno)-37 b(w)448 b(that)f Fm(9)g Fp(n)377 b Fm(2)g Fl(N)448 b Fq(suc)-37 b(h)447 b(that)g Fp(t)35478 58153 y Fg(0)36149 58635 y Fm(\000)298 b Fp(t)377 b Fq(=)g Fp(t)40227 58153 y Fg(0)40227 58965 y Fv(0)41121 58635 y Fm(\000)298 b Fp(t)42932 58834 y Fv(0)43826 58635 y Fq(+)h Fp(n)p Fq(,)449 b(so)e(w)-37 b(e)1102 60229 y(ha)g(v)g(e)11298 62850 y Fp(F)12152 63049 y Fk(c)12694 62850 y Fq(\()p Fp(x;)221 b(x)15319 62301 y Fg(0)15692 62850 y Fp(;)g(t;)g(t)17832 62301 y Fg(0)18205 62850 y Fq(\))368 b(=)p Fp(F)20977 63049 y Fk(c)21519 62850 y Fq(\()p Fp(x;)221 b(x)24144 62301 y Fg(0)24517 62850 y Fp(;)g(t)25587 63049 y Fv(0)26183 62850 y Fp(;)g(t)27253 63049 y Fv(0)28144 62850 y Fq(+)295 b Fp(n)p Fq(\))19090 64909 y Fm(\024)p Fp(F)20977 65108 y Fk(c)21519 64909 y Fq(\()p Fp(x;)221 b(y)48 b(;)221 b(t)25154 65108 y Fv(0)25750 64909 y Fp(;)g(t)26820 65108 y Fv(1)27416 64909 y Fq(\))294 b(+)i Fp(F)30410 65108 y Fk(c)30951 64909 y Fq(\()p Fp(y)48 b(;)221 b(y)48 b(;)221 b(t)34526 65108 y Fv(1)35122 64909 y Fp(;)g(t)36192 65108 y Fv(1)37083 64909 y Fq(+)295 b Fp(n)p Fq(\))20418 67037 y(+)h Fp(F)22601 67236 y Fk(c)23142 67037 y Fq(\()p Fp(y)48 b(;)221 b(x)25707 66489 y Fg(0)26080 67037 y Fp(;)g(t)27150 67236 y Fv(1)28041 67037 y Fq(+)295 b Fp(n;)221 b(t)31236 66489 y Fg(0)31236 67366 y Fv(0)32128 67037 y Fq(+)295 b Fp(n)p Fq(\))p Fp(:)1102 69724 y Fq(Let)443 b Fp(n)370 b Fm(!)f(1)p Fq(,)443 b(thanks)g(to)f(\(A.1\),)h(w)-37 b(e)443 b(obtain)13695 72344 y(lim)221 b(sup)h Fp(F)18837 72543 y Fk(c)19378 72344 y Fq(\()p Fp(x;)f(x)22003 71796 y Fg(0)22376 72344 y Fp(;)g(t;)g(t)24516 71796 y Fg(0)24889 72344 y Fq(\))369 b Fm(\024)g Fp(h)27942 72543 y Fk(c)28483 72344 y Fq(\()p Fp(x;)221 b(x)31108 71796 y Fg(0)31481 72344 y Fp(;)g(s;)g(s)33907 71796 y Fg(0)34280 72344 y Fq(\))295 b(+)g Fp(\017:)1102 74965 y Fq(As)443 b(this)g(holds)g(for) g(arbitrary)g Fp(\017)369 b(>)h Fq(0,)442 b(w)-37 b(e)444 b(see)g(that)e Fp(L)h Fq(is)f(regular.)p eop %%Page: 52 52 52 51 bop 1102 2239 a Ft(52)2696 5427 y Fq(As)307 b(the)f(third)g (step,)334 b(w)-37 b(e)307 b(claim)g(that)e Fp(L)h Fq(is)h(regular)g (implies)g(that)35046 5091 y(~)34796 5427 y Fm(G)448 b Fq(=)37858 5091 y(~)37436 5427 y Fm(N)196 b Fq(.)545 b(Let)306 b Fp(\015)443 b Fm(2)368 b Fp(C)45308 4945 y Fv(1)45905 5427 y Fq(\()p Fl(R)p Fp(;)221 b(M)145 b Fq(\))1102 7021 y(b)37 b(e)508 b(a)f(minimizing)g(curv)-37 b(e,)524 b(let)507 b Fp(t)17531 7220 y Fk(k)18661 7021 y Fm(!)476 b(\0001)508 b Fq(b)37 b(e)507 b(a)g(sequence)i(suc)-37 b(h)507 b(that)f Fp(s)476 b Fq(=)g Fp(t)41037 7220 y Fk(k)42198 7021 y Fq(mo)37 b(d)507 b(1)g(for)g(all)1102 8615 y Fp(k)422 b Fm(2)379 b Fl(Z)449 b Fq(and)g(suc)-37 b(h)450 b(that)e Fp(\013)385 b Fq(=)380 b(lim)221 b Fp(\015)74 b Fq(\()p Fp(t)19765 8814 y Fk(k)20417 8615 y Fq(\),)451 b(let)e Fp(t)24159 8133 y Fg(0)24159 8991 y Fk(k)25192 8615 y Fm(!)380 b(1)450 b Fq(b)37 b(e)449 b(a)h(sequence)g(suc)-37 b(h)450 b(that)f Fp(s)43628 8133 y Fg(0)44380 8615 y Fq(=)380 b Fp(t)46273 8133 y Fg(0)46273 8991 y Fk(k)47375 8615 y Fq(mo)37 b(d)1102 10209 y(1)443 b(and)g(suc)-37 b(h)443 b(that)g Fp(!)416 b Fq(=)369 b(lim)221 b Fp(\015)74 b Fq(\()p Fp(t)17137 9727 y Fg(0)17137 10585 y Fk(k)17790 10209 y Fq(\).)590 b(In)442 b(this)h(case)10914 13439 y Fp(A)p Fq(\()p Fp(\015)74 b Fm(j)13558 13679 y Fv([)p Fk(t)14260 13820 y Fd(k)14840 13679 y Fk(;t)15556 13359 y Fe(0)15556 14000 y Fd(k)16137 13679 y Fv(])16505 13439 y Fq(\))368 b(=)h Fp(F)184 b Fq(\()p Fp(\015)74 b Fq(\()p Fp(t)22106 13638 y Fk(k)22759 13439 y Fq(\))p Fp(;)221 b(\015)74 b Fq(\()p Fp(t)25625 12891 y Fg(0)25625 13767 y Fk(k)26277 13439 y Fq(\))p Fp(;)221 b(t)27864 13638 y Fk(k)28517 13439 y Fp(;)g(t)29587 12891 y Fg(0)29587 13767 y Fk(k)30241 13439 y Fq(\))369 b Fm(!)g Fp(h)p Fq(\()p Fp(\013)5 b(;)221 b(!)48 b(;)221 b(s;)g(s)38852 12891 y Fg(0)39224 13439 y Fq(\))p Fp(:)1102 16768 y Fq(Let)408 b(us)f(consider)i(a)f(compact)g(in)-37 b(terv)-74 b(al)408 b(of)g(times)f([)p Fp(a;)221 b(b)p Fq(],)415 b(where)409 b Fp(s)34178 16286 y Fg(0)34919 16768 y Fq(=)369 b Fp(a)408 b Fq(mo)37 b(d)408 b(1)f(and)h Fp(s)369 b Fq(=)g Fp(b)407 b Fq(mo)37 b(d)1102 18362 y(1.)591 b(F)-111 b(or)444 b Fp(k)484 b Fq(su\016cien)-37 b(tly)443 b(large)h(w)-37 b(e)443 b(ha)-37 b(v)g(e)10442 21592 y Fp(A)11438 21791 y Fk(c)11979 21592 y Fq(\()p Fp(\015)74 b Fm(j)13627 21832 y Fv([)p Fk(a;b)p Fv(])15653 21592 y Fq(\))369 b(=)g Fp(A)18937 21791 y Fk(c)19478 21592 y Fq(\()p Fp(\015)74 b Fm(j)21126 21832 y Fv([)p Fk(t)21828 21973 y Fd(k)22408 21832 y Fk(;t)23124 21511 y Fe(0)23124 22153 y Fd(k)23706 21832 y Fv(])24073 21592 y Fq(\))295 b Fm(\000)g Fp(A)27209 21791 y Fk(c)27750 21592 y Fq(\()p Fp(\015)74 b Fm(j)29398 21832 y Fv([)p Fk(t)30100 21973 y Fd(k)30680 21832 y Fk(;a)p Fv(])31939 21592 y Fq(\))295 b Fm(\000)g Fp(A)35075 21791 y Fk(c)35616 21592 y Fq(\()p Fp(\015)74 b Fm(j)37264 21832 y Fv([)p Fk(b;t)38748 21511 y Fe(0)38748 22153 y Fd(k)39329 21832 y Fv(])39696 21592 y Fq(\))p Fp(;)1102 24845 y Fq(taking)443 b(the)g(limit)f(w)-37 b(e)444 b(obtain)7201 28074 y Fp(A)8197 28273 y Fk(c)8739 28074 y Fq(\()p Fp(\015)74 b Fm(j)10387 28314 y Fv([)p Fk(a;b)p Fv(])12412 28074 y Fq(\))369 b(=)g Fp(h)15465 28273 y Fk(c)16007 28074 y Fq(\()p Fp(\013)5 b(;)221 b(!)48 b(;)221 b(s;)g(s)21270 27526 y Fg(0)21641 28074 y Fq(\))295 b Fm(\000)h Fp(h)24547 28273 y Fk(c)25088 28074 y Fq(\()p Fp(\013)5 b(;)221 b(\015)74 b Fq(\()p Fp(a)p Fq(\))p Fp(;)221 b(s;)g(s)p Fq(\))293 b Fm(\000)j Fp(h)34878 28273 y Fk(c)35419 28074 y Fq(\()p Fp(\015)74 b Fq(\()p Fp(b)p Fq(\))p Fp(;)221 b(!)48 b(;)221 b(s)40980 27526 y Fg(0)41351 28074 y Fp(;)g(s)42564 27526 y Fg(0)42937 28074 y Fq(\))p Fp(:)1102 31304 y Fq(On)443 b(the)g(other)g(hand,)g(w)-37 b(e)444 b(observ)-37 b(e)444 b(that)e(if)h Fp(L)f Fq(is)h(regular)h(then)4857 34534 y Fp(h)5622 34733 y Fk(c)6164 34534 y Fq(\()p Fp(\013)5 b(;)221 b(!)48 b(;)221 b(s;)g(s)11427 33986 y Fg(0)11799 34534 y Fq(\))368 b Fm(\024)h Fp(h)14851 34733 y Fk(c)15393 34534 y Fq(\()p Fp(\013)5 b(;)221 b(\015)74 b Fq(\()p Fp(a)p Fq(\))p Fp(;)221 b(s;)g(s)p Fq(\))293 b(+)j(\010)25377 34733 y Fk(c)25918 34534 y Fq(\()p Fp(\015)74 b Fq(\()p Fp(a)p Fq(\))p Fp(;)221 b(\015)74 b Fq(\()p Fp(b)p Fq(\))p Fp(;)221 b(s;)g(s)34315 33986 y Fg(0)34685 34534 y Fq(\))295 b(+)h Fp(h)37591 34733 y Fk(c)38132 34534 y Fq(\()p Fp(\015)74 b Fq(\()p Fp(b)p Fq(\))p Fp(;)221 b(!)48 b(;)221 b(s)43693 33986 y Fg(0)44064 34534 y Fp(;)g(s)45277 33986 y Fg(0)45650 34534 y Fq(\))1102 37764 y(it)443 b(follo)-37 b(ws)443 b(that)16456 39400 y Fp(A)p Fq(\()p Fp(\015)74 b Fm(j)19100 39640 y Fv([)p Fk(a;b)p Fv(])21126 39400 y Fq(\))369 b Fm(\024)g Fq(\010)24373 39599 y Fk(c)24914 39400 y Fq(\()p Fp(\015)74 b Fq(\()p Fp(a)p Fq(\))p Fp(;)221 b(\015)74 b Fq(\()p Fp(b)p Fq(\))p Fp(;)221 b(s;)g(s)33311 38852 y Fg(0)33682 39400 y Fq(\))p Fp(;)1102 42200 y Fq(hence)444 b Fp(\015)517 b Fq(is)442 b(semi-static.)590 b(It)442 b(has)h(b)37 b(een)444 b(sho)-37 b(wn)443 b(in)g([Ma4])g(that) 34055 41864 y(~)33634 42200 y Fm(N)195 b Fq(\()p Fp(c)p Fq(\))368 b(=)38755 41864 y(~)38298 42200 y Fm(A)p Fq(\()p Fp(c)p Fq(\).)2164 b Fb(\003)1102 45383 y Fr(Ac)-42 b(kno)g(wledgemen)g (t)411 b Fq(W)-111 b(e)412 b(w)-37 b(ould)412 b(lik)-37 b(e)413 b(to)e(thank)h(Professor)g(John)g(Mather)f(for)h(v)-37 b(ery)413 b(helpful)1102 46977 y(discussion)394 b(and)f(useful)h (suggestions.)574 b(W)-111 b(e)394 b(also)g(thank)f(Jian)g(Cheng)h(for) g(listening)f(to)g(our)h(oral)1102 48571 y(presen)-37 b(tation)436 b(and)e(for)h(thorough)g(discussions.)588 b(The)436 b(\014rst)e(author)h(thanks)g(the)f(mathematics)1102 50165 y(departmen)-37 b(t)299 b(of)f(Harv)-74 b(ard)299 b(Univ)-37 b(ersit)g(y)-111 b(,)328 b(where)299 b(the)f(partial)g(w)-37 b(ork)299 b(w)-37 b(as)298 b(done,)328 b(for)298 b(its)f(hospitalit)-37 b(y)-111 b(.)1102 51759 y(This)331 b(w)-37 b(ork)330 b(is)g(under)h(the)e(supp)37 b(ort)330 b(of)g(the)g(state)f(basic)i (researc)-37 b(h)331 b(pro)74 b(ject)331 b(of)f(China)g(\\Nonlinear) 1102 53353 y(Sciences")445 b(\(G2000077303\).)1102 56536 y Fr(References)2736 58969 y Fq([Ar].)664 b(Arnold)419 b(V.I.,)j Fo(Instabilityof)455 b(dynamic)-68 b(al)453 b(systems)g(with)g(sever)-68 b(al)453 b(de)-68 b(gr)g(e)g(es)453 b(of)f(fr)-68 b(e)g(e)g(dom)p Fq(,)6023 60563 y(So)-37 b(viet)443 b(Math.)590 b(Dokl.)443 b Fr(5)f Fq(\(1964\),)h(581-585.) 2094 62171 y([BM].)663 b(Ball)419 b(J.)g(and)g(Mizel)g(V.,)424 b Fo(One)454 b(dimensional)g(variational)h(pr)-68 b(oblems)454 b(whose)f(minimiz-)6023 63765 y(ers)501 b(do)f(not)h(satisfy)g(the)g (Euler-L)-68 b(agr)g(ange)501 b(e)-68 b(quation)p Fq(,)478 b(Arc)-37 b(h.)673 b(Ration.)f(Mec)-37 b(h.)673 b(Anal.)6023 65359 y Fr(90)443 b Fq(\(1985\),)f(325-388.)2647 66967 y([Ba].)664 b(Bangert)479 b(V.,)487 b Fo(Ge)-68 b(o)g(desic)508 b(r)-68 b(ays,)516 b(Busemann)508 b(functions)g(and)g(monotone)g(twist) g(maps)p Fq(,)6023 68561 y(Calc.)592 b(V)-111 b(ar.)443 b Fr(2)g Fq(\(1994\),)g(49-63.)2721 70169 y([Be].)664 b(Bernard)482 b(P)-111 b(.,)490 b Fo(Conne)-68 b(cting)509 b(orbits)i(of)f(time)h(dep)-68 b(endent)510 b(L)-68 b(agr)g(angian)510 b(systems)p Fq(,)491 b(Ann.)6023 71763 y(Inst.)590 b(F)-111 b(ourier)444 b Fr(52)e Fq(\(2002\),)h(1533-1568.)2787 73371 y([Bs].)664 b(Bessi)617 b(U.,)659 b Fo(A)-34 b(n)635 b(appr)-68 b(o)g(ach)635 b(to)g(A)-34 b(rnold)636 b(di\013usion)f(thr) -68 b(ough)635 b(c)-68 b(alculus)635 b(of)g(variations)p Fq(,)6023 74965 y(Nonlinear)444 b(Analysis)f Fr(.)680 b(26)442 b Fq(\(1996\),)h(1115-1135.)p eop %%Page: 53 53 53 52 bop 48792 2239 a Ft(53)1356 5427 y Fq([BCV].)664 b(Bessi)495 b(U.,)508 b(Chierc)-37 b(hia)497 b(L.)e(and)g(V)-111 b(aldino)37 b(ci)496 b(E.,)508 b Fo(Upp)-68 b(er)523 b(b)-68 b(ounds)523 b(on)g(A)-34 b(rnold)524 b(di\013usion)6023 7021 y(time)476 b(via)g(Mather)g(the)-68 b(ory)p Fq(,)443 b(J.)f(Math.)590 b(Pure.)h(Appl.)443 b Fr(80)f Fq(\(2001\),)h(105-129.) 2647 8615 y([Bo].)664 b(Bolotin)654 b(S.,)706 b Fo(Homo)-68 b(clonic)671 b(orbits)f(to)g(invariant)g(tori)g(for)f(Hamiltonian)i (systems)p Fq(,)6023 10209 y(Amer.)592 b(Math.)d(So)37 b(c.)591 b(T)-111 b(ransl.\(2\))443 b Fr(168)f Fq(\(1995\),)h(21-90.) 1669 11803 y([DLS].)664 b(Delshama)635 b(A.,)682 b(de)635 b(la)f(Lla)-37 b(v)g(e)635 b(R.)g(and)f(Seara)h(T.M.,)682 b Fo(Ge)-68 b(ometric)652 b(me)-68 b(chanism)653 b(for)6023 13397 y(di\013usion)352 b(in)g(Hamiltonian)h(systems)g(over)-68 b(c)g(oming)353 b(the)f(lar)-68 b(ge)352 b(gap)g(pr)-68 b(oblem:)550 b(heuristic)6023 14991 y(and)475 b(rigor)-68 b(ous)476 b(veri\014c)-68 b(ation)476 b(of)g(a)f(mo)-68 b(del)p Fq(,)443 b(preprin)-37 b(t)444 b(\(2003\).)2167 16585 y([FM].)664 b(F)-111 b(athi)446 b(A.)g(&)g(Mather)f(J.,)h Fo(F)-102 b(ailur)-68 b(e)479 b(of)g(c)-68 b(onver)g(genc)g(e)478 b(of)g(the)g(L)-68 b(ax-Oleinik)480 b(semi-gr)-68 b(oup)6023 18179 y(in)476 b(the)f(time-p)-68 b(erio)g(dic)478 b(c)-68 b(ase)p Fq(,)443 b(Bull.)590 b(So)37 b(c.)591 b(Math.)e(F)-111 b(rance)445 b Fr(128)d Fq(\(2000\),)h(473-483.)1614 19773 y([HPS].)664 b(Hirsc)-37 b(h)313 b(M.W.,)338 b(Pugh)311 b(C.C.)i(&)f(Sh)-37 b(ub)311 b(M.,)338 b Fo(Invariant)355 b(Manifolds)p Fq(,)339 b(Lect.)547 b(Notes)312 b(Math.)6023 21367 y Fr(583)442 b Fq(\(1977\).)1706 22961 y([Ma1].)664 b(Mather)609 b(J.,)649 b Fo(Existenc)-68 b(e)629 b(of)f(quasip)-68 b(erio)g(dic)629 b(orbits)f(for)h(twist)f(home)-68 b(omorphisms)629 b(of)6023 24555 y(annulus)p Fq(,)443 b(T)-111 b(op)37 b(ology)444 b Fr(21)e Fq(\(1982\),)h(457-467.)1706 26149 y([Ma2].)664 b(Mather)561 b(J.,)590 b Fo(Di\013er)-68 b(entiablity)585 b(of)f(the)g(minimal)h(aver)-68 b(age)584 b(action)g(as)g(a)g(function)g(of)6023 27743 y(the)476 b(r)-68 b(otation)476 b(numb)-68 b(er)p Fq(,)443 b(Bol.)590 b(So)37 b(c.)591 b(Bras.)g(Mat.)442 b Fr(21)g Fq(\(1990\),)h(59-70.) 1706 29337 y([Ma3].)664 b(Mather)576 b(J.,)608 b Fo(A)-68 b(ction)598 b(minimizing)i(invariant)e(me)-68 b(asur)g(es)598 b(for)g(p)-68 b(ositive)598 b(de\014nite)g(L)-68 b(a-)6023 30931 y(gr)g(angian)476 b(systems)p Fq(,)443 b(Math.)590 b(Z.)443 b Fr(207)f Fq(\(1991\),)h(169-207.)1706 32525 y([Ma4].)664 b(Mather)362 b(J.,)377 b Fo(V)-102 b(ariational)402 b(c)-68 b(onstruction)402 b(of)e(c)-68 b(onne)g(cting)401 b(orbits)p Fq(,)379 b(Ann.)563 b(Inst.)g(F)-111 b(ourier,)6023 34119 y Fr(43)443 b Fq(\(1993\),)f(1349-1386.)1706 35713 y([Ma5].)664 b(Mather)443 b(J.,)f Fo(A)-34 b(rnold)476 b(di\013usion,)g(I:)g(A)-34 b(nnounc)-68 b(ement)476 b(of)f(r)-68 b(esults)p Fq(,)444 b(preprin)-37 b(t)443 b(\(2002\).)2370 37307 y([Mo].)664 b(Moser)448 b(J.K.,)h Fo(On)480 b(the)g(volume)h(elements)f(on)g(manifold)p Fq(,)450 b(T)-111 b(rans.)606 b(AMS)449 b Fr(120)f Fq(\(1966\),)6023 38901 y(280-296.)3274 40495 y([R].)664 b(Rademac)-37 b(her)422 b(H.,)15985 40160 y Fo(\177)15747 40495 y(Ub)-68 b(er)455 b(p)-68 b(artiel)68 b(le)456 b(und)f(totale)h(Di\013er)-68 b(entierb)g(arkeit)456 b(I.)p Fq(,)426 b(Math.)582 b(Ann.)6023 42089 y Fr(79)443 b Fq(\(1919\),)f(340-359.)2887 43684 y([W].)664 b(W)-111 b(eistein)521 b(A.,)540 b Fo(Symple)-68 b(ctic)549 b(manifolds)e(and)g(their)g(L)-68 b(agr)g(angian)547 b(submanifolds)p Fq(,)541 b(Ad-)6023 45278 y(v)-74 b(ances)444 b(in)f(Math.)f Fr(6)h Fq(\(1971\),)f(329-346.)2592 46872 y([X1].)664 b(Xia)538 b(Z.,)560 b Fo(A)-34 b(rnold)563 b(di\013usion:)785 b(a)561 b(variational)j(c)-68 b(onstruction.)p Fq(,)561 b(Do)37 b(c)538 b(Math.)872 b(J.)537 b(DMV)6023 48466 y(Extra)443 b(V)-111 b(olume)444 b(ICM)e Fr(I)42 b(I)444 b Fq(\(1998\),)f(867-877.)2592 50060 y([X2].)664 b(Xia)405 b(Z.,)412 b Fo(A)-34 b(rnold)440 b(di\013usion)g(and)g (instabilities)i(in)e(Hamiltonian)h(dynamics)p Fq(,)413 b(preprin)-37 b(t)6023 51654 y(\(2002\).)1102 54822 y(1,)443 b(Departmen)-37 b(t)443 b(of)g(Mathematics,)g(Nanjing)g(Univ)-37 b(ersit)g(y)-111 b(,)444 b(Nanjing)f(210093,)h(China)2696 56416 y Fo(e-mail)477 b(addr)-68 b(ess)p Fq(:)591 b Fw (chengcq@nju.edu.cn)1102 59585 y Fq(2,)443 b(Institute)f(of)h (Mathematics,)f(F)-111 b(udan)444 b(Univ)-37 b(ersit)g(y)-111 b(,)443 b(Shanghai)h(200433,)f(China)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF