%!PS-Adobe-2.0
%%Creator: dvips(k) 5.94a Copyright 2003 Radical Eye Software
%%Pages: 10
%%PageOrder: Ascend
%%BoundingBox: 0 0 596 842
%%DocumentFonts: Times-Bold Times-Roman Courier MSBM10 CMR7 CMR10 CMMI10
%%+ CMSY10 Times-Italic CMMI7 CMSY7 CMSY9 CMEX10 MSBM7 CMMI5 CMSY5 CMR5
%%+ CMMI12 CMR6 CMMI8 CMMI9 CMR9
%%EndComments
%DVIPSWebPage: (www.radicaleye.com)
%DVIPSParameters: dpi=600, compressed
%DVIPSSource:  TeX output 2005.03.01:2116
%%BeginProcSet: texc.pro 0 0
%!
/TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S
N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72
mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0
0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{
landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize
mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[
matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round
exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{
statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0]
N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin
/FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array
/BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2
array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N
df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A
definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get
}B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub}
B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr
1 add N}if}B/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: 8r.enc 0 0
% File  8r.enc as of 2002-03-12 for PSNFSS 9
%
% This is the encoding vector for Type1 and TrueType fonts to be used
% with TeX.  This file is part of the PSNFSS bundle, version 9
%
% Authors: S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry, W. Schmidt
%
% Idea is to have all the characters normally included in Type 1 fonts
% available for typesetting. This is effectively the characters in Adobe
% Standard Encoding + ISO Latin 1 + extra characters from Lucida + Euro.
%
% Character code assignments were made as follows:
%
% (1) the Windows ANSI characters are almost all in their Windows ANSI
% positions, because some Windows users cannot easily reencode the
% fonts, and it makes no difference on other systems. The only Windows
% ANSI characters not available are those that make no sense for
% typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen
% (173). quotesingle and grave are moved just because it's such an
% irritation not having them in TeX positions.
%
% (2) Remaining characters are assigned arbitrarily to the lower part
% of the range, avoiding 0, 10 and 13 in case we meet dumb software.
%
% (3) Y&Y Lucida Bright includes some extra text characters; in the
% hopes that other PostScript fonts, perhaps created for public
% consumption, will include them, they are included starting at 0x12.
%
% (4) Remaining positions left undefined are for use in (hopefully)
% upward-compatible revisions, if someday more characters are generally
% available.
%
% (5) hyphen appears twice for compatibility with both ASCII and Windows.
%
% (6) /Euro is assigned to 128, as in Windows ANSI
%
/TeXBase1Encoding [
% 0x00 (encoded characters from Adobe Standard not in Windows 3.1)
  /.notdef /dotaccent /fi /fl
  /fraction /hungarumlaut /Lslash /lslash
  /ogonek /ring /.notdef
  /breve /minus /.notdef
% These are the only two remaining unencoded characters, so may as
% well include them.
  /Zcaron /zcaron
% 0x10
 /caron /dotlessi
% (unusual TeX characters available in, e.g., Lucida Bright)
 /dotlessj /ff /ffi /ffl
 /.notdef /.notdef /.notdef /.notdef
 /.notdef /.notdef /.notdef /.notdef
 % very contentious; it's so painful not having quoteleft and quoteright
 % at 96 and 145 that we move the things normally found there down to here.
 /grave /quotesingle
% 0x20 (ASCII begins)
 /space /exclam /quotedbl /numbersign
 /dollar /percent /ampersand /quoteright
 /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash
% 0x30
 /zero /one /two /three /four /five /six /seven
 /eight /nine /colon /semicolon /less /equal /greater /question
% 0x40
 /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O
% 0x50
 /P /Q /R /S /T /U /V /W
 /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore
% 0x60
 /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o
% 0x70
 /p /q /r /s /t /u /v /w
 /x /y /z /braceleft /bar /braceright /asciitilde
 /.notdef % rubout; ASCII ends
% 0x80
 /Euro /.notdef /quotesinglbase /florin
 /quotedblbase /ellipsis /dagger /daggerdbl
 /circumflex /perthousand /Scaron /guilsinglleft
 /OE /.notdef /.notdef /.notdef
% 0x90
 /.notdef /.notdef /.notdef /quotedblleft
 /quotedblright /bullet /endash /emdash
 /tilde /trademark /scaron /guilsinglright
 /oe /.notdef /.notdef /Ydieresis
% 0xA0
 /.notdef % nobreakspace
 /exclamdown /cent /sterling
 /currency /yen /brokenbar /section
 /dieresis /copyright /ordfeminine /guillemotleft
 /logicalnot
 /hyphen % Y&Y (also at 45); Windows' softhyphen
 /registered
 /macron
% 0xD0
 /degree /plusminus /twosuperior /threesuperior
 /acute /mu /paragraph /periodcentered
 /cedilla /onesuperior /ordmasculine /guillemotright
 /onequarter /onehalf /threequarters /questiondown
% 0xC0
 /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla
 /Egrave /Eacute /Ecircumflex /Edieresis
 /Igrave /Iacute /Icircumflex /Idieresis
% 0xD0
 /Eth /Ntilde /Ograve /Oacute
 /Ocircumflex /Otilde /Odieresis /multiply
 /Oslash /Ugrave /Uacute /Ucircumflex
 /Udieresis /Yacute /Thorn /germandbls
% 0xE0
 /agrave /aacute /acircumflex /atilde
 /adieresis /aring /ae /ccedilla
 /egrave /eacute /ecircumflex /edieresis
 /igrave /iacute /icircumflex /idieresis
% 0xF0
 /eth /ntilde /ograve /oacute
 /ocircumflex /otilde /odieresis /divide
 /oslash /ugrave /uacute /ucircumflex
 /udieresis /yacute /thorn /ydieresis
] def

%%EndProcSet
%%BeginProcSet: f7b6d320.enc 0 0
% Thomas Esser, Dec 2002. public domain
%
% Encoding for:
%     cmb10 cmbx10 cmbx12 cmbx5 cmbx6 cmbx7 cmbx8 cmbx9 cmbxsl10
%     cmdunh10 cmr10 cmr12 cmr17cmr6 cmr7 cmr8 cmr9 cmsl10 cmsl12 cmsl8
%     cmsl9 cmss10cmss12 cmss17 cmss8 cmss9 cmssbx10 cmssdc10 cmssi10
%     cmssi12 cmssi17 cmssi8cmssi9 cmssq8 cmssqi8 cmvtt10
%
/TeXf7b6d320Encoding [
/Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega
/ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute /caron /breve
/macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash
/suppress /exclam /quotedblright /numbersign /dollar /percent /ampersand
/quoteright /parenleft /parenright /asterisk /plus /comma /hyphen
/period /slash /zero /one /two /three /four /five /six /seven /eight
/nine /colon /semicolon /exclamdown /equal /questiondown /question /at
/A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X
/Y /Z /bracketleft /quotedblleft /bracketright /circumflex /dotaccent
/quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u
/v /w /x /y /z /endash /emdash /hungarumlaut /tilde /dieresis /suppress
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space
/Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /.notdef
/.notdef /Omega /ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute
/caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE
/OE /Oslash /suppress /dieresis /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
] def

%%EndProcSet
%%BeginProcSet: aae443f0.enc 0 0
% Thomas Esser, Dec 2002. public domain
%
% Encoding for:
%     cmmi10 cmmi12 cmmi5 cmmi6 cmmi7 cmmi8 cmmi9 cmmib10
%
/TeXaae443f0Encoding [
/Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega
/alpha /beta /gamma /delta /epsilon1 /zeta /eta /theta /iota /kappa
/lambda /mu /nu /xi /pi /rho /sigma /tau /upsilon /phi /chi /psi
/omega /epsilon /theta1 /pi1 /rho1 /sigma1 /phi1 /arrowlefttophalf
/arrowleftbothalf /arrowrighttophalf /arrowrightbothalf /arrowhookleft
/arrowhookright /triangleright /triangleleft /zerooldstyle /oneoldstyle
/twooldstyle /threeoldstyle /fouroldstyle /fiveoldstyle /sixoldstyle
/sevenoldstyle /eightoldstyle /nineoldstyle /period /comma /less /slash
/greater /star /partialdiff /A /B /C /D /E /F /G /H /I /J /K /L /M /N
/O /P /Q /R /S /T /U /V /W /X /Y /Z /flat /natural /sharp /slurbelow
/slurabove /lscript /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p
/q /r /s /t /u /v /w /x /y /z /dotlessi /dotlessj /weierstrass /vector
/tie /psi /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/space /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi
/.notdef /.notdef /Omega /alpha /beta /gamma /delta /epsilon1 /zeta /eta
/theta /iota /kappa /lambda /mu /nu /xi /pi /rho /sigma /tau /upsilon
/phi /chi /psi /tie /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef
] def

%%EndProcSet
%%BeginProcSet: bbad153f.enc 0 0
% Thomas Esser, Dec 2002. public domain
%
% Encoding for:
%     cmsy10 cmsy5 cmsy6 cmsy7 cmsy8 cmsy9
%
/TeXbbad153fEncoding [
/minus /periodcentered /multiply /asteriskmath /divide /diamondmath
/plusminus /minusplus /circleplus /circleminus /circlemultiply
/circledivide /circledot /circlecopyrt /openbullet /bullet
/equivasymptotic /equivalence /reflexsubset /reflexsuperset /lessequal
/greaterequal /precedesequal /followsequal /similar /approxequal
/propersubset /propersuperset /lessmuch /greatermuch /precedes /follows
/arrowleft /arrowright /arrowup /arrowdown /arrowboth /arrownortheast
/arrowsoutheast /similarequal /arrowdblleft /arrowdblright /arrowdblup
/arrowdbldown /arrowdblboth /arrownorthwest /arrowsouthwest /proportional
/prime /infinity /element /owner /triangle /triangleinv /negationslash
/mapsto /universal /existential /logicalnot /emptyset /Rfractur /Ifractur
/latticetop /perpendicular /aleph /A /B /C /D /E /F /G /H /I /J /K
/L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /union /intersection
/unionmulti /logicaland /logicalor /turnstileleft /turnstileright
/floorleft /floorright /ceilingleft /ceilingright /braceleft /braceright
/angbracketleft /angbracketright /bar /bardbl /arrowbothv /arrowdblbothv
/backslash /wreathproduct /radical /coproduct /nabla /integral
/unionsq /intersectionsq /subsetsqequal /supersetsqequal /section
/dagger /daggerdbl /paragraph /club /diamond /heart /spade /arrowleft
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/minus /periodcentered /multiply /asteriskmath /divide /diamondmath
/plusminus /minusplus /circleplus /circleminus /.notdef /.notdef
/circlemultiply /circledivide /circledot /circlecopyrt /openbullet
/bullet /equivasymptotic /equivalence /reflexsubset /reflexsuperset
/lessequal /greaterequal /precedesequal /followsequal /similar
/approxequal /propersubset /propersuperset /lessmuch /greatermuch
/precedes /follows /arrowleft /spade /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
] def

%%EndProcSet
%%BeginProcSet: 0ef0afca.enc 0 0
% Thomas Esser, Dec 2002. public domain
%
% Encoding for:
%     cmr5
%
/TeX0ef0afcaEncoding [
/Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega
/arrowup /arrowdown /quotesingle /exclamdown /questiondown /dotlessi
/dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls
/ae /oe /oslash /AE /OE /Oslash /suppress /exclam /quotedblright
/numbersign /dollar /percent /ampersand /quoteright /parenleft
/parenright /asterisk /plus /comma /hyphen /period /slash /zero /one
/two /three /four /five /six /seven /eight /nine /colon /semicolon
/less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K
/L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /quotedblleft
/bracketright /circumflex /dotaccent /quoteleft /a /b /c /d /e /f /g /h
/i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /endash /emdash
/hungarumlaut /tilde /dieresis /suppress /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /space /Gamma /Delta /Theta /Lambda
/Xi /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef /Omega /arrowup
/arrowdown /quotesingle /exclamdown /questiondown /dotlessi /dotlessj
/grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe
/oslash /AE /OE /Oslash /suppress /dieresis /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef
] def

%%EndProcSet
%%BeginProcSet: texps.pro 0 0
%!
TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2
index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll
exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]FontType 0
ne{/Metrics exch def dict begin Encoding{exch dup type/integertype ne{
pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get
div def}ifelse}forall Metrics/Metrics currentdict end def}{{1 index type
/nametype eq{exit}if exch pop}loop}ifelse[2 index currentdict end
definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{dup
sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll
mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[
exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}if}
forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def}def
end

%%EndProcSet
%%BeginFont: CMR9
%!PS-AdobeFont-1.1: CMR9 1.0
%%CreationDate: 1991 Aug 20 16:39:59
% 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 (CMR9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR9 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 /.notdef put
readonly def
/FontBBox{-39 -250 1036 750}readonly def
/UniqueID 5000792 def
currentdict end
currentfile eexec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%%EndFont
%%BeginFont: CMMI9
%!PS-AdobeFont-1.1: CMMI9 1.100
%%CreationDate: 1996 Jul 23 07:53:55
% 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 (CMMI9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI9 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 /.notdef put
readonly def
/FontBBox{-29 -250 1075 750}readonly def
/UniqueID 5087384 def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE
3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B
532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470
B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B
986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE
D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958
9E394A533A081C36D6F5CA5FED4F9AC9ADE41E04F9FC52E758C9F45A92BED935
86F9CFDB57732045913A6422AD4206418610C81D882EE493DE9523CC1BFE1505
DD1390B19BC1947A01B93BC668BE9B2A0E69A968554239B88C00AF9FBDF09CCD
67D3B2094C11A04762FE8CC1E91D020A28B3C122D24BEAACF82313F4604F2FEF
6E176D730A879BE45DD0D4996EF0247AEB1CA0AB08FF374D99F06D47B36F9554
FAD9A2D3CE451B7791C3709D8A1DDDEFBD840C1B42AB824D5A0DFF0E0F15B0B7
22AEEB877FF489581DA6FA8DA64944555101EB16F7AB0B717E148B7B98D8DBFD
730C52937E226545CF8DC3E07C5BA30739BAFCD0F2B44275A6D503F582C0FB4F
449963D0AD2FAFDE33BA3D77BCA9D1DF878DDAFCA2E22CC4BACD542B282164C7
97C2BDE318AF9D501CA21F6E662E7AAB75A5F24D2C182E598D175D44E88AB19A
E7CD59584F95B389183EE21B525BF52A3F23C0FE5383A5565A19361D716F508C
AAB78411CA5A4D27552CC1C435760D5A89D535B71C593E755C616661363308DA
A683F54ED0C23FB2C225A008392B0B719F66F11A946A090B7C00B662A3C69599
B4ECB0CC70C85C4BBBF207E0026F6C7A19F2ACFB7A60804FC98A4BFFD7BFFF2B
9529E6D9D4238002BBC255BC62959D6F3381FE06E0621B879D5FE5B541D45A1E
759A6E7DC32B1D1632368D09A97039DF255B6492B1B2B7E2C1434E8306ECA7D3
5A79B6D614B4979F10988BC76ED53A5F45315CD7DA216221F842FD0F3E050DD2
BAC23C984D506D8F7D614BCB6B244F5F41321549BB0BD041FBF3053307168680
3435E9C9456846433DF6235D79BA455E54E35184F42032331DC8DAA0CA26D710
5892C20FEAED5B92B991D35352CD1CBED1508024C6BD93559098C9CB3496D355
8D1B814C89F5F6B6C6CED48FFDC886CD727885E24C1DD7A311BEB018B021DB50
81C3AF2D949E5B9D6C5C9557BACFD9F48193B2D051EFD6D19CF9A058538EA8E9
D78E842D448C5F56E91EBD0B248346BC180E6FA17AF905588509D2416F221C58
0C0B24738240A617674EB2F78BE53876304CC802CC79629BB4609EE340B68109
895BA07E194B31D9A26C752638BCE8AFEC563F48297611CA699FB5E918D92E8E
E98ADFFEB10D37910EB0BBBAF0C39DFEF8B7E3E221A7FCE35A082238F2B0117F
8FE837326646D87C872B136456C62A28B326CC82B7810D3822ABD3013049670C
8001E93FED95E857D6728C09E1019FFECF32F3606CBFC36C06B386AB6837AB22
235E9E0E6D8767CEF4D5EFE83C2A1D2598751993B24DEEBF451476EF0AB223A5
B84F6E215C349CE6194B0ED8B3996F2F5EF96B5A3715F8956F0664B1603D8BA7
24C1F7E1EE89A37A1D124238CC0E353F11F92E203ED52554B7C3F3724D7E82AD
D69F7E91F6CE2389009EC4F3737BF3CDE72725D6A10B00C8BBE147E81B90F432
3E9BA9BFD7E5463423FA1F23B17B70EB4DD61E3A5DFA8B375DBC65535611AD89
86E1249DD091833612F0701B61D511DDADD74C1E02BC78B3A4EB8DDA06B42886
8A46CD9F4C6F69B2955A493C82A38ACCBCF6001A46602775
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark
%%EndFont
%%BeginFont: CMMI8
%!PS-AdobeFont-1.1: CMMI8 1.100
%%CreationDate: 1996 Jul 23 07:53:54
% 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 (CMMI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI8 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 /.notdef put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
/UniqueID 5087383 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark
%%EndFont
%%BeginFont: CMR6
%!PS-AdobeFont-1.1: CMR6 1.0
%%CreationDate: 1991 Aug 20 16:39:02
% 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 (CMR6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR6 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 /.notdef put
readonly def
/FontBBox{-20 -250 1193 750}readonly def
/UniqueID 5000789 def
currentdict end
currentfile eexec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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 /.notdef put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
/UniqueID 5000820 def
currentdict end
currentfile eexec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%%EndFont
%%BeginFont: CMMI12
%!PS-AdobeFont-1.1: CMMI12 1.100
%%CreationDate: 1996 Jul 27 08:57:55
% 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 (CMMI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI12 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 /.notdef put
readonly def
/FontBBox{-30 -250 1026 750}readonly def
/UniqueID 5087386 def
currentdict end
currentfile eexec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%%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 0 /.notdef put
readonly def
/FontBBox{-341 -250 1304 965}readonly def
/UniqueID 5000788 def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891
016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171
9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F
D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758
469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8
2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4
87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F
D1F017CE45884D76EF2CB9BC5821FD25365DDEA1F9B0FF4CFF25B8E64D0747A3
7CAD14E0DBA3E3CA95F10F24B7D5D75451845F1FB7221D7794A860756CFBB3E7
704A52A22448C34812C3DBEDD41892577AABA7D555E9298C1A0F7DA638078167
F56E29672683C51CF1C003764A8E7AD9D8ADE77B4983F56FE2D12723AAD8BF36
682CFBB71B1D12210144D39DD841A971F71DB82AC6CD815987CDCF29ABC3CC96
5EEBD5D661F452C6E0C74F9ED8D0C5B3755551A172E0FE31EA02344176E32666
14B6853A1C303A5E818C2E455A6CF8FC9A66DC6E279101D61C523BD9DB8EB82F
EAF4D7FDF6372383C0794C4568D079648689A199D4B65BA646CF95B7647E4BEC
83856C27A8EF177B3A686EDA6354FE9573E123C12EC4BA56A7E8BFB8F9B75147
9DD79A743968F36F7D0D479FA610F0816E6267E5CE327686A5485AB72201525C
FB3B7CA10E1BF26E44C24E1696CB089CB0055BD692C89B237CF269F77A31DC81
0F4B75C8400ABCFDCEC6443CD0E81871CD71AA3064ABDE882C4C52322C27FA8B
41C689F827FB0F8AAF8022CF3C1F41C0B45601190C1328831857CBF9B1E7D1AA
246117E56D6B7938488055F4E63E2A1C8D57C17D213729C68349FEC2C3466F41
171E00413D39DF1F67BC15912F30775AFDF7FB3312587E20A68CF77AD3906040
842D63C45E19278622DD228C18ABDD024DD9613CDC0B109095DB0ADC3A3C0CB5
AB597D490189EA81239E39202CBC7A829EB9B313A8F962F7879D374ADF529BD0
5533EF977142F647AD2F5975BA7E340419116099B19ACCCC37C551226DB28A2A
49F6394324396BF03FEDC34E47427C7B14B81F600732AA1A19FB0A52B6E80FA5
C5281CFA8ADEE3DEECBEFA91BE8A455A4B93364E087A8EDA1EAA0E0E14B621B7
BDBC9CF2476E4DF79FFE7967F12B36403F63B3E9C5D00653975872642043D77F
F178369F138268DD61AFA06EAE4CFED07CF966122596816385AFDC28836247E3
C3F5757E7608292182CFAD5510BE507FB1F760369A2A9D8DD89321B045B89C38
739A1B860E40DAFBB62DA7CA0BD0AED0BB12AF77B8DEC9ABDA40C1712672F3BE
43AF20E1BB228A0AF7CEFB850AD9E4BF4EC373BDF89737C826F92A2FD09DC597
0504D1D0995C22FA958EE2AF7B3DA314F2DB067139AA03392CC74B983A578AD6
0772F1523F22957CD61FE32FA411F8058CE4A7761048D44FA9DCA5FE9282E480
73B8475E07C262FF6E869D40E6166F56F5926CB375064A581A8C9C7B348A907B
0B781FA90718C352B08F20A56C4C3035DFF64D111E1242CF54C2585141CE23EC
92F96E7274A24BD53F700B1D4DD1EAF2531D33A28B524C03CAC60C9FF966C7E2
340719D287EEFB19BAD2D2C9061C74AF6E8E9BEDA9D0EB615FEFA0D1DBD2D772
DE6740409F4B947603305CE2CEC7C5A31C2D0EDB2BA4B91460E08C4B8B78D3A1
B6C46D81620CD32F1B780532ADBBD9BE97DBC21160165907E2BEAF1693810F13
08BCD17CF9D4965EB57FFEBFD96E5F3CE21A8BE43A3B1127DDCBF04669B64029
E9C127D2E9A6566F46BAC76408F9770A9DA2CD6DBE10F3D014D2B741BC7ABCFA
9664ED5385088283F01FC2F5
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 /.notdef put
readonly def
/FontBBox{21 -944 1448 791}readonly def
/UniqueID 5000815 def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964
7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4
A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85
E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A
221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFBAF552B11EFFB6A16C
F03FB920C15AE724EFDF0CCBF00A838D34440FF9FED532F44036AD22561184C5
283722DDFA7285E62754372D716D704AC0E00B2F6AB67154241C7449AA047833
94CEDB08E8C92907FE72A0B05AE36A7B9226ACD6E7890A0B528FDDE84A950FC6
801DE75CF2E739E9121149CCB8B1C87A106822648D84A3D3FBF295EE6C4BF403
BBE9A1C1F6DAEDD1E642ACC486E609703D7612BFFD10C324F5DC710811F7F614
3691B400E3773987424C0D2B0D8A736873C6371DDB2442F05E018A2B5CA9A4AA
17AABB95D09E5890CFFFED5AC01495D89A53D3CF44556D089203F8FC8E7F9119
8496FAB6BED78BAA8A4333579662139575C4339D7716630EA7A09F9CD8CEDD53
7DCD6C852325F0309810A7AB405E10071DF5A6D6E8D386C5EF58E25747EB3A0E
6EB47369D84838AECC135C9CD2D55B6988DF7A0C690DE2F2FEFEBAAF3FAFE74D
39D0796E7ABF125BE85E28BA1CDFBCFDF254A8B1854816097DD2F78E924FCECF
4EFB282B6509B67950DD147BC52099139220A27EF988F2C304EA9AAFB063F726
B144C53D9D8327EF5C4117BAFE672FA8FEF9F593400378DFE157DB05C767D2A6
9BF89CF5AEF8DAB0F247E88AFEE9D0365CDFB790AC783A4C114157F40F55E063
D5E1C88D93E39D7514ECE2DEF5B26A82D6693FFB8326CF8E4BBDEB83CC9C3553
1E3B59F7B12C99F99768CF99E569F4A8824250E5F82A5897855489303A3DFCEF
EF3ABD79184949A68C6D6A7BAE0CE34DDCF39627D7489ADF0504494D5C5AFBB5
1770F7D9E58B639B5F381CE0E38761BDBC24C97D9BA9CC38253BAF39C29B068D
EA3C1B5C1F40C740411D7CFB891DB8AD6CCF8130A36BFA19B65DF0579937B36D
8FBF8495355E0C4BEAD9648920BF6D4D4F1A05A3F901935A61949532DF2D453D
B337C32646835EE3A25BE053FAAF6BE3FCC34EEB8E95F24334435BF711100DE4
A4FD5498023F783D79E04E568E6D408F07B0884CBAB79DB6D7CDBBB9EF66162E
2D20420C62CF0BCD9613F93BE50F3ADC0C8DEE0F6B31C487A415E8AEACBC6DF9
DA8E0B05501E9EF143C52A00D5FEAD3BDDC01534969E64A15D1E7C2BFE17A93D
C69785087E6ABE0F255C2BE29BE371C6B59D2C18A3920796727B39EFDAFFA85B
E5C3AFA6E45D3305BE0D29B09C1FF2BCE1DE0C39485ACE78218EA4103AAFBF85
391CE433DF2C25DD3BDAEF4ED5ABE7DBEA862F49224DAC39377AFC817A015737
002E5CEDDBA5A979F2AD0C3F6154E73F144C1EB2DDC4A31D78D8031B267D72AC
CAD9684268E5F1B48EFB2A9299C882C6B15EDEA770A491520EA3F6D29CE17C1A
95
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 0 /.notdef put
readonly def
/FontBBox{37 -250 1349 750}readonly def
/UniqueID 5087380 def
currentdict end
currentfile eexec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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 90 /Z put
readonly def
/FontBBox{0 -504 2615 1004}readonly def
/UniqueID 5032014 def
currentdict end
currentfile eexec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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 0 /parenleftbig put
dup 1 /parenrightbig put
dup 48 /parenlefttp put
dup 49 /parenrighttp put
dup 64 /parenleftbt put
dup 65 /parenrightbt put
dup 66 /parenleftex put
dup 67 /parenrightex put
dup 88 /summationdisplay put
dup 94 /logicalanddisplay put
dup 112 /radicalbig put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
/UniqueID 5000774 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark
%%EndFont
%%BeginFont: CMSY9
%!PS-AdobeFont-1.1: CMSY9 1.0
%%CreationDate: 1991 Aug 15 07:22:27
% 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 (CMSY9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY9 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 /.notdef put
readonly def
/FontBBox{-30 -958 1146 777}readonly def
/UniqueID 5000819 def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964
7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4
A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85
E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A
221A37D9A807DD01161779DDE7D31FF2B87F97C73D63EECDDA4C49501773468A
27D1663E0B62F461F6E40A5D6676D0037D33F24E2FAC2B0009AD3C8350CDF8CC
65BCA87979C36D14CB552E9A985E48BE4E88ECA16DF418749AF04FDD2B0E1380
D281BB2476BB45FF30946B247DFD7F57305FA87E50CA338121C71CDFDF927A9C
77FF14CB4A1D6D80356FB1171ED38C37702350497B44E42CE31DB2F493807DAA
15B887C671199A54C4C1294BC520F5538C15556BC43C9F62342B121C6DCD6C5F
491DA47FF360201EE21C08A781ED0589A6DF91B99FE118B9B29E4F068672E52F
1A06C514D91C4C937D4E642503392B1CD1B8F5332A2C13884EB17C99A3A4EC0D
3790FC24F3FE52FB340911E79F818459B508C965FC5F2729DCE85519D0AD4BB6
DF3A2B53880D7592FDE696E43B9EEA5D0D651944F9F4181FDEAA6A96209F2626
692273CF7A649D22F746263E5CE945750B68EF63844406EBF0BFF7ECD88C89CA
E64799D58555201302CDCEB7A1613F394BC4EA27F77F0298877B7616E4D33E57
A63F994381FD26CD3582E71B5320400673505DC47932D75FA4050F93D4E8A3B9
5389CC2C22CB9364A4EDC7369FA96AC8D6853AA7C0621D7CFB88008D66D4DA6C
A99118A2A5BA0F9657B10B218E56263BFE8D7A1BB098C858FB421BC113CB67BF
917FB0E990E24467C0501BDF725E8A439D4D5DE1768CE598D2EF8FEFBCCA6970
9F462BED100C101FF26D77B675462F39EC002B68B362B0D332FA5BEABB22D24A
39716C54E2814D491466987A20F162F016F438DBAD7FBBA5223A693C3780680E
08F9AE49C958D9247642E5EC5D9784750656FBB3152FB9750EB9E9A5E43F08A6
365C2E0DA4FCFB607F2CDBB49B9B43D12FF35F8E392F8DF5965C35AA08EDBC63
AB31D79AC72EE37ECE505D35F4453FBF2059C187BEF7473A517DFE130B3681C8
C11184A0A80AA4D382672F4C21207BA4B284E0B44A03175D8DE61B993A44E6A5
E18501A0F15560AA27797B611CD2A437497944702857CAA6EE754A450E9F4A6C
7C71C2CF50887AE674836BEA6743ADCE58E1DD161CD70F7E59DA6B80EE5F700A
6980F6842408BE8DA93F
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 /.notdef put
readonly def
/FontBBox{-15 -951 1252 782}readonly def
/UniqueID 5000817 def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964
7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4
A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85
E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A
221A37D9A807DD01161779DDE7D251491EBF65A98C9FE2B1CF8D725A70281949
8F4AFFE638BBA6B12386C7F32BA350D62EA218D5B24EE612C2C20F43CD3BFD0D
F02B185B692D7B27BEC7290EEFDCF92F95DDEB507068DE0B0B0351E3ECB8E443
E611BE0A41A1F8C89C3BC16B352C3443AB6F665EAC5E0CC4229DECFC58E15765
424C919C273E7FA240BE7B2E951AB789D127625BBCB7033E005050EB2E12B1C8
E5F3AD1F44A71957AD2CC53D917BFD09235601155886EE36D0C3DD6E7AA2EF9C
C402C77FF1549E609A711FC3C211E64E8F263D60A57E9F2B47E3480B978AAF63
868AEA25DA3D5413467B76D2F02F8097D28419D2EEAC108405E4AC2173E4304F
5A8445E1A94C3022676F3B8E366AD06FBA451263D1014AD0ED97FBE558F363DF
DA6BD1699F2B52BCCD8A3F91F6F7AE84A72501DC904505DE87C6F10EB1874721
CA18FC4A7A7937BB915589A460E62278A7C52E2D75D8B1A29458ADE9F21B8596
5D659D3BAB69F6FF11DE44938A28FF578EC12502ECABB442F42FB29C2E76B5EC
9BE7255337BE899A9126EB7CFF1465CCD70877E16863C2A00C2E0BB4F3C8215F
AA0526ECB9D656F2F08E54994170BB60DBADD92ADAE03829D4F9BFCBF4A1CCB9
64B518BD068EDB92E7C3D0F954AF63F051E1D9531424BFD7B47122A986191CE9
D1932DA8F549A439936DC5B41183B8F06FFA04EB737108F86378D7C791038663
19CFEB6E26CAC5A284D2159240543E4EDCC912824B2E8CE6E88A6A6F0E16D8FD
890D5BD39568E7B8064E1DD5234939E027E751FDE1955B64E351614B1521409C
95401B41CA30A09521D7A959F3076230AE125C0FB4925336AD9CEE25FF305585
A146122AD24D7F4698F035456BDAAAAA47E11579FF22F9F0EBFC0D22F02DCF9A
6FA3BFD6C2E097617B0AD28C8D4E4B69075B76E3B9DE4743F801CAA701BECC0C
435E2EFE581C7BFFF3AE2E7C42C1A71D2F9F8529203F9C2DE97562E338B96196
B91B4AB5C420D97F223C140188AB15EB4D888B842D66DCF07674F6B4D217603B
DBD1DFF2F74955C55EA8D836E2C1C6EDFB671469C5EA715500A05D8FEF126031
2D948A30C147E1ED0FA764BBB183D93D34B2C98CED884AC6756654C66B260DC5
41F19219BA7C223A65A89F85137E765CAA234FC8B453C679D6AD1E077FAC7A75
701362C37CE510D3F017A80BF75A84DBE018743499FEEAE80CD6906953026B7A
A9E4D2B748499F3F85
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 0 /.notdef put
readonly def
/FontBBox{0 -250 1171 750}readonly def
/UniqueID 5087382 def
currentdict end
currentfile eexec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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 0 /.notdef put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
/UniqueID 5087385 def
currentdict end
currentfile eexec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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 /.notdef put
readonly def
/FontBBox{-251 -250 1009 969}readonly def
/UniqueID 5000793 def
currentdict end
currentfile eexec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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 /.notdef put
readonly def
/FontBBox{-27 -250 1122 750}readonly def
/UniqueID 5000790 def
currentdict end
currentfile eexec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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 78 /N put
dup 82 /R put
dup 84 /T put
readonly def
/FontBBox{-55 -420 2343 920}readonly def
/UniqueID 5031982 def
currentdict end
currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
cleartomark
%%EndFont
TeXDict begin 39158280 55380996 1000 600 600 (pertenoc.dvi)
@start /Fa 204[38 9[30 30 40[{ TeXf7b6d320Encoding ReEncodeFont }3
74.7198 /CMR9 rf /Fb 172[47 3[58 79[{ TeXaae443f0Encoding ReEncodeFont }
2 74.7198 /CMMI9 rf /Fc 159[29 12[43 60[42 22[{
 TeXaae443f0Encoding ReEncodeFont }3 66.4176 /CMMI8 rf
/Fd 134[33 1[48 33 33 18 26 22 33 33 33 33 52 18 33 1[18
33 1[22 29 33 29 33 29 12[41 37[17 43[37 2[{
 TeXBase1Encoding ReEncodeFont }25 66.4176 /Times-Roman
rf /Fe 204[30 30 30 49[{ TeXf7b6d320Encoding ReEncodeFont }3
49.8132 /CMR6 rf /Ff 205[29 29 49[{ TeXBase1Encoding ReEncodeFont }2
58.1154 /Times-Roman rf /Fg 149[28 106[{
 TeXbbad153fEncoding ReEncodeFont }1 99.6264 /CMSY10
rf /Fh 193[91 1[91 60[{ TeXaae443f0Encoding ReEncodeFont }2
119.552 /CMMI12 rf /Fi 205[28 28 28 48[{
 TeX0ef0afcaEncoding ReEncodeFont }3 41.511 /CMR5 rf
/Fj 145[31 6[31 31 101[45{ TeXbbad153fEncoding ReEncodeFont }4
41.511 /CMSY5 rf /Fk 141[28 13[31 100[{
 TeXaae443f0Encoding ReEncodeFont }2 41.511 /CMMI5 rf
/Fl 165[39 90[{}1 58.1154 /MSBM7 rf /Fm 143[83 17[92
5[120 20[73 73 73 73 14[73 73 46[38 38{}11 83.022 /CMEX10
rf /Fn 145[38 3[21 20[48 85[{ TeXbbad153fEncoding ReEncodeFont }3
74.7198 /CMSY9 rf /Fo 149[20 55[45 46[34 2[52{
 TeXbbad153fEncoding ReEncodeFont }4 58.1154 /CMSY7 rf
/Fp 134[34 5[31 31 1[34 1[41 2[35 27 23 4[35 3[28 12[41
18[35 25[26 3[31 5[31 5[40 22[{ TeXaae443f0Encoding ReEncodeFont }16
58.1154 /CMMI7 rf /Fq 104[74 42 1[46 46 25[37 37 55 37
42 23 32 32 42 42 42 42 60 23 37 1[23 42 42 23 37 42
37 42 42 6[46 2[69 1[60 46 42 51 60 51 60 55 69 46 55
37 28 60 60 51 51 60 55 51 51 6[28 42 42 42 42 1[42 2[42
2[21 28 21 2[28 28 37[42 2[{ TeXBase1Encoding ReEncodeFont }64
83.022 /Times-Italic rf /Fr 105[42 28[42 2[42 42 23 32
28 1[42 42 42 65 23 2[23 42 42 1[37 42 37 42 37 29[55
20[21 46[{ .167 SlantFont TeXBase1Encoding ReEncodeFont }22
83.022 /Times-Roman rf /Fs 143[69 5[23 32 32 42 42 7[55
8[52 4[58 66 1[100 7[44 2[55 66 3[60 60 4[0 0 3[55 16[83
8[65 2[65 65 1[65 3[42 7[65 3[65 23 65{
 TeXbbad153fEncoding ReEncodeFont }29 83.022 /CMSY10
rf /Ft 133[39 41 47 1[40 2[39 37 37 42 1[50 73 25 43
34 29 1[40 41 1[43 36 2[35 7[69 1[48 1[49 51 63 1[53
63 67 1[57 71 46 36 69 3[69 3[44 1[65 1[65 23 23 23[39
52 1[52 49 1[36 47 43 47 2[50 8[43 1[53 11[{
 TeXaae443f0Encoding ReEncodeFont }49 83.022 /CMMI10
rf /Fu 139[32 6[69 3[23 3[37 46 4[23 33[65 1[23 23 42
3[42 42 42 42 42 42 4[65 1[32 32 17[42 12[65 9[{
 TeXf7b6d320Encoding ReEncodeFont }21 83.022 /CMR10 rf
/Fv 194[51 9[33 33 33 33 48[{ TeXf7b6d320Encoding ReEncodeFont }5
58.1154 /CMR7 rf /Fw 171[55 1[60 3[60 78[{}3 83.022 /MSBM10
rf /Fx 141[33 9[50 4[44 1[44 19[89 23[50 2[50 50 50 48[{
 TeXBase1Encoding ReEncodeFont }9 99.6264 /Times-Roman
rf /Fy 133[50 4[50 50 50 4[50 50 50 2[50 50 2[50 3[50
24[50 7[50 17[50 46[{ TeXBase1Encoding ReEncodeFont }14
83.022 /Courier rf /Fz 32[42 42[28 11[28 16[83 42 1[37
37 10[28 13[37 42 42 60 42 42 23 32 28 42 42 42 42 65
23 42 23 23 42 42 28 37 42 37 42 37 3[28 1[28 1[60 60
78 60 60 51 46 55 60 46 60 60 74 51 60 32 28 60 60 46
51 60 55 55 60 1[37 3[23 23 42 42 42 42 42 42 42 42 42
42 23 21 28 21 2[28 28 28 65 2[42 4[28 12[23 28 12[46
46 2[{ TeXBase1Encoding ReEncodeFont }88 83.022 /Times-Roman
rf /FA 32[55 100[44 11[55 83 3[28 3[44 3[50 24[78 72[{
 TeXBase1Encoding ReEncodeFont }8 99.6264 /Times-Bold
rf /FB 166[84 3[84 78 65 84 1[71 90 84 110 78 2[45 90
90 71 78 84 1[78 84 65[{ TeXBase1Encoding ReEncodeFont }18
116.231 /Times-Bold rf /FC 105[42 28[42 1[60 42 46 28
32 37 1[46 42 46 69 23 2[23 46 42 28 37 46 37 46 42 12[55
46 60 1[51 65 60 78 55 65 1[32 3[55 60 60 1[60 7[42 42
42 42 42 42 42 42 42 42 1[21 28 21 41[46 2[{
 TeXBase1Encoding ReEncodeFont }50 83.022 /Times-Bold
rf end
%%EndProlog
%%BeginSetup
%%Feature: *Resolution 600dpi
TeXDict begin
%%PaperSize: A4
 end
%%EndSetup
%%Page: 1 1
TeXDict begin 1 0 bop 1617 -154 a FC()18 b()o()o()j
()f()f()l()i()678 336 y FB(PER)-5
b(TURB)m(A)-11 b(TIONS)27 b(OF)i(INTEGRABLE)e(AND)499
435 y(SUPERINTEGRABLE)f(HAMIL)-11 b(T)n(ONIAN)29 b(SYSTEMS)1512
900 y FA(Heinz)c(Han\337mann)1153 1000 y Fz(Mathematisch)20
b(Instituut,)f(Uni)n(v)o(ersiteit)g(Utrecht)1038 1100
y(Postb)n(us)h(80010,)e(3508)h(T)-8 b(A)21 b(Utrecht,)e(The)h
(Netherlands)1229 1199 y Fy()1587
1465 y Fx(1)25 b(March)g(2005)-107 1910 y FC(Abstract)-71
2010 y Fz(Inte)o(grable)19 b(systems)i(admitting)e(a)j(suf)n
(\002ciently)d(lar)o(ge)h(sym-)-107 2109 y(metry)35 b(group)f(are)i
(considered.)70 b(In)36 b(the)g(non\226de)o(generate)-107
2209 y(case)h(this)f(group)e(is)j(abelian)f(and)f(KAM)h(theory)f
(ensures)-107 2308 y(that)29 b(most)g(of)g(the)g(resulting)g
(Lagrangean)d(tori)j(persist)g(un-)-107 2408 y(der)47
b(small)g(non\226inte)o(grable)d(perturbations.)103 b(F)o(or)47
b(non\226)-107 2508 y(commutati)n(v)o(e)26 b(symmetry)g(groups)h(the)h
(system)g(is)h(superin-)-107 2607 y(te)o(grable,)d(ha)n(ving)f
(additional)g(inte)o(grals)g(of)h(motion)f(that)h(\002-)-107
2707 y(bre)32 b(Lagrangean)d(tori)i(into)h(lo)n(wer)f(dimensional)f(in)
m(v)n(ariant)-107 2807 y(tori.)42 b(This)26 b(simpli\002es)h(the)e
(inte)o(grable)g(dynamics,)h(b)n(ut)f(ren-)-107 2906
y(ders)g(the)f(perturbation)e(analysis)i(more)g(complicated.)36
b(I)25 b(re-)-107 3006 y(vie)n(w)18 b(important)e(cases)j(where)e(it)h
(is)h(possible)f(to)g(\002nd)f(an)h(\223in-)-107 3105
y(termediate\224)32 b(inte)o(grable)f(system)h(that)h(is)h(non\226de)o
(generate)-107 3205 y(and)20 b(approximates)e(the)i(perturbed)e
(dynamics.)-107 3461 y FC(K)n(ey)i(w)o(ords)-71 3561
y Fz(KAM)32 b(theory)-5 b(,)33 b(rami\002ed)e(torus)g(b)n(undle,)i
(perturbed)c(rigid)-107 3660 y(body)-5 b(,)18 b(gyroscopic)g
(stabilization,)i(proper)e(de)o(generac)o(y)-107 3917
y FC(1)83 b(Intr)o(oduction)-71 4016 y Fz(In)51 b(Hamiltonian)e
(dynamics,)57 b(inte)o(grable)49 b(systems)j(are)-107
4116 y(rather)20 b(the)g(e)o(xception)e(than)i(the)g(rule.)25
b(Still,)c(within)f(this)h(cel-)-107 4216 y(ebrated)f(class)i(of)e
(Hamiltonian)g(systems)h(one)f(encounters)f(a)-107 4315
y(whole)d(hierarchy)f(of)i(possibilities.)24 b(An)17
b(important)e(aspect)i(is)-107 4415 y(al)o(w)o(ays)i(ho)n(w)e(the)h
(dynamics)f(beha)n(v)o(e)g(under)f(non\226inte)o(grable)-107
4514 y(perturbations.)-71 4614 y(The)31 b(typical)f(or)h(generic)e
(case)j(\(within)e(the)h(non\226generic)-107 4714 y(class)18
b(of)e(inte)o(grable)f(Hamiltonian)g(systems\))i(is)g(that)g(of)f
(non\226)-107 4813 y(de)o(generate)25 b(inte)o(grable)g(systems)i(;)k
(e)o(xamples)26 b(in)h(three)f(de-)-107 4913 y(grees)31
b(of)f(freedom)f(are)i(easily)g(constructed)e(from)h(a)h(point)-107
5013 y(mass)20 b(in)g Fw(R)224 4982 y Fv(3)281 5013 y
Fz(mo)o(ving)d(in)j(a)f(separated)g(potential.)24 b(Almost)19
b(all)-107 5112 y(motion)h(is)i(quasi\226periodic)d(with)i(three)g
(frequencies,)e(in)i(ge-)-107 5212 y(ometric)d(language)e(the)i(motion)
f(is)i(con\002ned)e(to)i(in)m(v)n(ariant)d Fu(3)p Fz(\226)-107
5311 y(tori)23 b(in)h(phase)e(space.)34 b(In)23 b(action\226angle)e(v)n
(ariables)i Fu(\()p Ft(x;)14 b(y)s Fu(\))29 b Fs(2)-107
5411 y Fw(T)-52 5381 y Fv(3)-12 5411 y Fs(\002)r Fw(R)115
5381 y Fv(3)167 5411 y Fz(the)16 b(Hamiltonian)e(\(locally\))h(reads)g
Ft(H)30 b Fu(=)23 b Ft(H)7 b Fu(\()p Ft(y)s Fu(\))16
b Fz(and)-107 5511 y(under)j(e.g.)h(the)g(K)m(olmogoro)o(v)d(non\226de)
o(generac)o(y)e(condition)470 5736 y Fu(det)f Ft(D)670
5702 y Fv(2)707 5736 y Ft(H)7 b Fu(\()p Ft(y)s Fu(\))69
b Fs(6)p Fu(=)g(0)479 b Fz(\(1\))1901 1910 y(most)22
b Fu(3)p Fz(\226tori)e(survi)n(v)o(e)g(a)i(suf)n(\002ciently)f(small)h
(non\226inte)o(grable)1901 2010 y(perturbation.)1937
2109 y(In)32 b(case)h(there)e(are)h(more)g(inte)o(grals)f(of)h(motion)f
(than)h(de-)1901 2209 y(grees)f(of)f(freedom)f(almost)i(all)g(motion)f
(is)i(generically)d(\227)1901 2308 y(no)n(w)17 b(within)g(this)h(more)e
(restricted)h(class)h(of)f Fr(superinte)o(grable)1901
2408 y Fz(systems)22 b(\227)g(con\002ned)f(to)g(in)m(v)n(ariant)f
Fu(\()p Ft(d)h Fs(\000)e Fu(1\))p Fz(\226tori,)i(where)g
Ft(d)1901 2508 y Fz(denotes)j(the)i(number)d(of)i(de)o(grees)f(of)h
(freedom.)38 b(Examples)1901 2607 y(are)16 b(gi)n(v)o(en)f(by)g(a)h
(point)g(mass)g(in)g Fw(R)2880 2577 y Fv(3)2934 2607
y Fz(mo)o(ving)e(in)i(a)g(rotationally)1901 2707 y(symmetric)g
(potential)g Ft(V)42 b Fu(=)23 b Ft(V)c Fu(\()p Ft(r)r
Fu(\))f Fz(.)25 b(In)16 b(generalized)g(action\226)1901
2807 y(angle)25 b(v)n(ariables)g Fu(\()p Ft(x;)14 b(y)s(;)g(z)t
Fu(\))33 b Fs(2)h Fw(T)2880 2776 y Fv(2)2940 2807 y Fs(\002)22
b Fw(R)3087 2776 y Fv(2)3146 2807 y Fs(\002)h Fw(R)3294
2776 y Fv(2)3357 2807 y Fz(the)j(Hamil-)1901 2906 y(tonian)j(only)h
(depends)e(on)i Ft(y)j Fz(and)d(one)f(e)o(xpects)h(additional)1901
3006 y(motion)19 b(in)h(the)f Ft(z)t Fz(\226direction)f(already)g
(under)h(inte)o(grable)f(per)n(-)1901 3105 y(turbations.)41
b(Correspondingly)-5 b(,)24 b(the)i(\002rst)g(step)h(in)f(studying)1901
3205 y(non\226inte)o(grable)17 b(perturbations)i(of)i(\(minimally\))e
(superinte-)1901 3305 y(grable)e(systems)i(is)h(to)e(construct)f(an)i
(inte)o(grable)d(approxima-)1901 3404 y(tion)22 b(that)g
Fr(remo)o(v)o(es)f(the)h(de)o(generac)o(y)d Fz(by)j(means)g(of)g
(normal-)1901 3504 y(ization)16 b(with)g(respect)g(to)g(the)g
(unperturbed)d(\003o)n(w)-5 b(.)23 b(The)16 b(result-)1901
3604 y(ing)23 b(Lagrangean)e(tori)i(ha)n(v)o(e)g Ft(d)e
Fs(\000)g Fu(1)i Fz(frequencies)f(of)h(order)f Fu(1)1901
3703 y Fz(and)29 b(one)f(frequenc)o(y)e(of)j(the)g(order)f
Ft(")h Fz(of)g(the)g(perturbation,)1901 3803 y(and)g(a)h(further)e
(application)h(of)g(KAM)h(theory)e(yields)i(such)1901
3902 y(quasi\226periodic)e(motions)i(also)g(in)h(the)f(original)g
(perturbed)1901 4002 y(system.)1937 4102 y(A)e(maximally)f(superinte)o
(grable)d(system)k(in)g Ft(d)g Fz(de)o(grees)f(of)1901
4201 y(freedom)c(\(globally\))f(admits)i Fu(2)p Ft(d)e
Fs(\000)f Fu(1)j Fz(inte)o(grals)g(of)g(motion.)1901
4301 y(An)30 b(e)o(xample)f(is)i(the)g(\(spatial\))e(K)n(epler)h
(system)h(of)f(a)g(point)1901 4401 y(mass)21 b(in)f Fw(R)2233
4370 y Fv(3)2291 4401 y Fz(mo)o(ving)e(in)j(the)f(potential)2507
4677 y Ft(V)f Fu(\()p Ft(r)r Fu(\))71 b(=)e Fs(\000)2957
4621 y Fu(1)p 2957 4658 42 4 v 2958 4734 a Ft(r)3091
4677 y Fu(;)1901 4939 y Fz(after)24 b(re)o(gularization)e(of)j(the)f
(singularity)g(all)h(orbits)f(are)h(pe-)1901 5039 y(riodic.)34
b(Note)23 b(that)h(from)e Ft(d)29 b Fs(\025)g Fu(4)23
b Fz(on)g(there)g(is)h(a)g(whole)f(hier)n(-)1901 5138
y(archy)g(of)h(superinte)o(grable)d(systems)j(between)g(the)g(e)o
(xtreme)1901 5238 y(cases)d(of)f(minimal)f(and)h(maximal)f(superinte)o
(grability)-5 b(.)1937 5338 y(Inte)o(grable)17 b(systems)j(often)f
(admit)g(a)g(symmetry)f(group.)23 b(In-)1901 5437 y(deed,)16
b(by)f(Noether')-5 b(s)16 b(theorem)e(e)n(v)o(ery)h Fu(1)p
Fz(\226parameter)e(symme-)1901 5537 y(try)j(yields)g(an)g(inte)o(gral)f
(of)h(motion.)22 b(T)-7 b(o)17 b(obtain)e Ft(d)i Fz(commuting)1901
5636 y(\002rst)31 b(inte)o(grals)f(in)g(this)h(w)o(ay)f(one)g(needs)g
(a)g Ft(d)p Fz(\226dimensional)1901 5736 y(commutati)n(v)o(e)15
b(symmetry)g(group.)23 b(F)o(or)16 b(these)h(inte)o(grals)f(to)i(be)
1800 6037 y(1)37 b()p eop end
%%Page: 2 2
TeXDict begin 2 1 bop -107 -90 a Fz(independent)16 b(the)i(group)f(has)
i(to)f(act)h(ef)n(fecti)n(v)o(ely)-5 b(.)22 b(More)c(than)-107
9 y Ft(d)h Fz(inte)o(grals)f(in)h Ft(d)g Fz(de)o(grees)f(of)g(freedom)f
(cannot)g(all)i(commute)-107 109 y(with)27 b(each)f(other)g(and)h(a)g
(corresponding)c(symmetry)i(group)-107 209 y(has)c(to)f(be)g
(non\226commutati)n(v)o(e.)-71 308 y(The)33 b(dynamics)f(of)h(inte)o
(grable)e(Hamiltonian)h(systems)i(is)-107 408 y(particularly)20
b(re)o(gular)-5 b(.)29 b(T)-7 b(o)22 b(\002x)g(thoughts)e(we)i
(concentrate)e(on)-107 508 y(compact)d(ener)o(gy)e(shells.)25
b(Then)16 b(all)i(\(b)n(ut)g(some)f(e)o(xceptional\))-107
607 y(motion)24 b(is)i(quasi\226periodic)c(and)j(hence)f(con\002ned)g
(to)h(an)g(in-)-107 707 y(v)n(ariant)19 b(torus.)25 b(These)20
b(tori)g(are)g(the)g(connected)e(components)-107 806
y(of)j(the)h(le)n(v)o(el)f(sets)h(of)f(the)h(inte)o(grals)e(of)h
(motion.)28 b(In)21 b(the)g(non\226)-107 906 y(de)o(generate)e(case)i
(these)f(are)h Ft(d)p Fz(\226tori)f(in)g(the)h Fu(2)p
Ft(d)p Fz(\226dimensional)-107 1006 y(phase)e(space,)h(and)f(in)h(the)f
(superinte)o(grable)e(case)j(the)g Fu(2)p Ft(d)c Fs(\000)g
Ft(r)-107 1105 y Fz(inte)o(grals)33 b(of)g(motion)g(de\002ne)g(in)m(v)n
(ariant)f Ft(r)r Fz(\226tori.)65 b(The)33 b(fol-)-107
1205 y(lo)n(wing)18 b(result)h(formulated)e(in)i([F)o(ass)7
b(\036)-35 b(o,)19 b(2005])e(describes)i(the)-107 1305
y(general)30 b(situation)g(on)g(the)h(re)o(gular)e(part)i
Fs(M)g Fz(of)f(the)h(phase)-107 1404 y(space)20 b Fs(P)28
b Fz(where)19 b(the)h(inte)o(grals)g(are)g(independent.)-107
1569 y FC(Theor)o(em)g(1.1.)40 b Fq(On)35 b(the)g(subset)g
Fs(M)49 b(\022)h(P)42 b Fq(of)35 b(the)f(phase)-107 1669
y(space)g(with)g(symplectic)g(structur)m(e)g Ft(\033)k
Fq(let)83 b Ft(f)57 b Fu(:)48 b Fs(M)g(\000)-14 b(!)-107
1768 y Fw(R)-47 1738 y Fv(2)p Fp(d)p Fo(\000)p Fp(r)168
1768 y Fq(be)26 b(a)f(submer)o(sion)g(with)i(compact)d(and)g(connected)
-107 1868 y(\002br)m(es)35 b(\(hence)o(,)h(a)f(\002br)o(ation\).)66
b(Assume)34 b(that)48 b Fs(f)p Ft(f)1417 1880 y Fp(i)1444
1868 y Ft(;)14 b(f)1522 1880 y Fp(j)1557 1868 y Fs(g)48
b Fu(=)-107 1967 y Ft(P)-54 1979 y Fp(ij)29 1967 y Fs(\016)24
b Ft(f)f(;)k(i;)14 b(j)43 b Fu(=)37 b(1)p Ft(;)14 b(:)g(:)g(:)f(;)h
Fu(2)p Ft(d)24 b Fs(\000)g Ft(r)45 b Fq(and)28 b(that)g(the)g(matrix)g
Ft(P)-107 2067 y Fq(with)g(entries)42 b Ft(P)385 2079
y Fp(ij)480 2067 y Fu(:)36 b Fs(M)f(\000)-14 b(!)36 b
Fw(R)42 b Fq(has)27 b(r)o(ank)g Fu(2)p Ft(d)c Fs(\000)g
Ft(r)31 b Fq(at)c(all)-107 2167 y(points)h(of)g Ft(f)9
b Fu(\()p Fs(M)p Fu(\))28 b Fq(.)49 b(Then)28 b(e)o(very)g(\002br)m(e)f
(of)i Ft(f)37 b Fq(is)29 b(dif)o(feomor)n(-)-107 2266
y(phic)23 b(to)h Fw(T)204 2236 y Fp(r)265 2266 y Fq(and)e(the)i(\002br)
o(ation)e Ft(f)32 b Fq(has)24 b(local)f(trivialisations)-107
2366 y(whic)o(h)d(ar)m(e)g(symplectic.)-107 2531 y Fz(Thus,)f(e)n(v)o
(ery)g(\002bre)g(of)g Ft(f)29 b Fz(has)20 b(a)g(neighbourhood)15
b Fs(U)28 b Fz(with)20 b(co\226)-107 2630 y(ordinates)-65
2876 y Fu(\()p Ft(x;)14 b(y)s(;)g(q)s(;)g(p)p Fu(\))46
b(:)69 b Fs(U)78 b(\000)-15 b(!)70 b Fw(T)809 2842 y
Fp(r)858 2876 y Fs(\002)12 b Fw(R)995 2842 y Fp(r)1043
2876 y Fs(\002)g Fw(R)1180 2842 y Fp(d)p Fo(\000)p Fp(r)1314
2876 y Fs(\002)g Fw(R)1451 2842 y Fp(d)p Fo(\000)p Fp(r)1615
2876 y Fz(\(2\))-107 3122 y(such)20 b(that)g(the)g(le)n(v)o(el)g(sets)h
(of)f Ft(f)29 b Fz(coincide)19 b(with)h(the)g(le)n(v)o(el)g(sets)-107
3221 y(of)g Fu(\()p Ft(y)s(;)14 b(q)s(;)g(p)p Fu(\))21
b Fz(and)28 3549 y Ft(\033)78 3579 y Fn(jU)229 3549 y
Fu(=)406 3445 y Fp(r)362 3470 y Fm(X)369 3647 y Fp(i)p
Fv(=1)496 3549 y Fu(d)p Ft(x)589 3561 y Fp(i)636 3549
y Fs(^)e Fu(d)p Ft(y)797 3561 y Fp(i)866 3549 y Fu(+)972
3445 y Fv(2)p Fp(d)p Fo(\000)p Fp(r)988 3470 y Fm(X)990
3647 y Fp(j)s Fv(=1)1138 3549 y Fu(d)p Ft(q)1221 3561
y Fp(j)1274 3549 y Fs(^)g Fu(d)p Ft(p)1436 3561 y Fp(j)1554
3549 y Ft(:)-107 3878 y Fz(These)54 b(co\226ordinates)d(are)i
(Nekhoroshe)n(v')-5 b(s)51 b(generalized)-107 3977 y(action\226angle)18
b(v)n(ariables.)-71 4077 y(The)58 b(aim)g(of)f(this)i(paper)e(is)h(to)g
(specify)g(ho)n(w)f(\(su-)-107 4177 y(per\)inte)o(grability)40
b(structures)i(the)h(phase)g(space)g(into)g(in-)-107
4276 y(v)n(ariant)36 b(subsets)h(and)f(in)h(ho)n(w)f(f)o(ar)g(this)h
(structure)f(is)h(pre-)-107 4376 y(serv)o(ed)28 b(under)f(small)j
(perturbations.)48 b(In)29 b(the)g(ne)o(xt)f(section)-107
4475 y(the)38 b(non\226de)o(generate)c(inte)o(grable)i(case)i(is)h
(treated.)77 b(Sec-)-107 4575 y(tion)31 b(3)g(starts)h(with)f
(minimally)f(superinte)o(grable)e(systems,)-107 4675
y(where)18 b(the)g(perturbation)d(analysis)k(still)g(goes)f(through)e
(with-)-107 4774 y(out)f(non\226generic)e(assumptions.)22
b(Then)15 b(the)h(e)o(xtreme)e(case)i(of)-107 4874 y(maximally)28
b(superinte)o(grable)d(systems)30 b(is)f(considered,)g(be-)-107
4974 y(fore)e(e)o(x)o(emplyfying)c(the)k(general)f(hierarchy)f(of)i
(superinte-)-107 5073 y(grable)19 b(systems.)-107 5338
y FC(2)83 b(Non\226degenerate)18 b(integrable)h(systems)-71
5437 y Fz(The)27 b(\003o)n(w)h(on)f(the)h(Lagrangean)d(tori)j(of)f(a)h
(Liouville)f(inte-)-107 5537 y(grable)16 b(system)h(is)g(conditionally)
d(periodic.)23 b(Locally)16 b(around)-107 5636 y(such)26
b(a)h(torus)g(the)f(action)g(angle)g(v)n(ariables)g(\(2\))g(simplify)f
(to)-107 5736 y Fu(\()p Ft(x;)14 b(y)s Fu(\))43 b Fs(2)g
Fw(T)281 5706 y Fp(d)347 5736 y Fs(\002)26 b Fw(R)498
5706 y Fp(d)568 5736 y Fz(in)31 b(which)f(the)h(symplectic)f(structure)
1901 -90 y(becomes)d Ft(\033)40 b Fu(=)c(d)p Ft(x)24
b Fs(^)h Fu(d)p Ft(y)31 b Fz(and)26 b(the)i(Hamiltonian)e(function)1901
9 y Ft(H)36 b Fu(=)30 b Ft(H)7 b Fu(\()p Ft(y)s Fu(\))24
b Fz(does)f(not)h(depend)e(on)h(the)h(angles.)35 b(The)24
b(equa-)1901 109 y(tions)c(of)g(motion)f(read)2376 353
y Fu(_)-38 b Ft(x)72 b Fu(=)f Ft(!)s Fu(\()p Ft(y)s Fu(\))e(:=)f
Ft(D)r(H)7 b Fu(\()p Ft(y)s Fu(\))2380 478 y(_)-38 b
Ft(y)74 b Fu(=)d(0)1901 722 y Fz(and)24 b(where)g(the)g(frequenc)o(y)d
(v)o(ector)i Ft(!)28 b Fz(is)d(non\226resonant)c(the)1901
822 y(quasi\226periodic)15 b(\003o)n(w)j(on)g Fw(T)2729
792 y Fp(n)2793 822 y Fz(is)h(dense,)e(e)o(xcluding)f(the)i(e)o(xis-)
1901 922 y(tence)d(of)h(further)e(inte)o(grals)g(of)i(motion.)22
b(W)-7 b(e)17 b(speak)e(of)g(a)h Fr(non\226)1901 1021
y(de)o(generate)f Fz(Liouville)h(inte)o(grable)g(system)h(if)h(almost)f
(all)h(La-)1901 1121 y(grangean)24 b(tori)j(ha)n(v)o(e)f(dense)g
(orbits.)43 b(Suf)n(\002cient)26 b(conditions)1901 1221
y(are)e(the)g(K)m(olmogoro)o(v)d(non\226de)o(generac)o(y)e(condition)j
(\(1\))h(for)1901 1320 y(almost)d(all)h Ft(y)i Fz(or)d(iso\226ener)o
(getic)e(non\226de)o(generac)o(y)-5 b(.)1937 1420 y(The)29
b(Lagrangean)f(tori)h(form)g Ft(d)p Fz(\226parameter)f(f)o(amilies)i
(and)1901 1519 y(the)18 b(singular)g(\002bres)g(of)g(the)h(rami\002ed)e
Ft(d)p Fz(\226torus)h(b)n(undle)f(deter)n(-)1901 1619
y(mine)26 b(ho)n(w)g(these)g(f)o(amilies)g(\002t)h(together)-5
b(.)43 b(At)26 b(the)h Fu(\()p Ft(d)c Fs(\000)g Fu(1\))p
Fz(\226)1901 1719 y(parameter)42 b(f)o(amilies)h(of)g(elliptic)g
Fu(\()p Ft(d)36 b Fs(\000)f Fu(1\))p Fz(\226tori)43 b(the)g(La-)1901
1818 y(grangean)19 b(tori)i(shrink)f(do)n(wn)g(in)h(the)g(same)g(w)o
(ay)g(as)h(periodic)1901 1918 y(orbits)c(shrink)f(do)n(wn)g(to)i
(centres)f(in)g(one)g(de)o(gree)f(of)h(freedom.)1901
2018 y(Dif)n(ferent)23 b(f)o(amilies)i(of)f(Lagrangean)e(tori)j(are)f
(separated)g(by)1901 2117 y Fu(\()p Ft(d)c Fs(\000)g
Fu(1\))p Fz(\226parameter)g(f)o(amilies)i(of)g(hyperbolic)d
Fu(\()p Ft(d)i Fs(\000)e Fu(1\))p Fz(\226tori)1901 2217
y(and)h(their)g(\(un\)stable)e(manifolds.)1937 2316 y(This)h(picture)f
(is)i(repeated)e(in)h(ho)n(w)f(the)h Fu(\()p Ft(d)14
b Fs(\000)g Fu(1\))p Fz(\226tori)j(shrink)1901 2416 y(do)n(wn)h(to)h
Fu(\()p Ft(d)13 b Fs(\000)g Fu(2\))p Fz(\226parameter)j(f)o(amilies)j
(of)f(\(partially\))g(ellip-)1901 2516 y(tic)h Fu(\()p
Ft(d)12 b Fs(\000)g Fu(2\))p Fz(\226tori)17 b(and)h(are)g(separated)f
(by)h Fu(\()p Ft(d)12 b Fs(\000)g Fu(2\))p Fz(\226parameter)1901
2615 y(f)o(amilies)20 b(of)f(\(partially\))e(hyperbolic)g
Fu(\()p Ft(d)e Fs(\000)g Fu(2\))p Fz(\226tori)j(and)h(\(part)1901
2715 y(of\))29 b(their)g(\(un\)stable)f(manifolds.)51
b(Furthermore)27 b(there)i(are)1901 2815 y Fu(\()p Ft(d)k
Fs(\000)e Fu(2\))p Fz(\226parameter)37 b(f)o(amilies)i(of)f(hyperbolic)
e Fu(\()p Ft(d)d Fs(\000)f Fu(2\))p Fz(\226)1901 2914
y(tori)c(with)h(Floquet)f(e)o(xponents)e Fs(\006<)e(\006)g
Fu(i)p Fs(=)29 b Fz(,)i(together)c(with)1901 3014 y(their)18
b(\(un\)stable)f(manifolds)g(these)h(form)g(\223pinched\224)e
Ft(d)p Fz(\226tori.)1901 3113 y(In)40 b(these)g(three)f(w)o(ays)i(we)f
(are)g(led)g(to)g(in)m(v)n(ariant)e(tori)i(of)1901 3213
y(smaller)20 b(and)f(smaller)g(dimension)f(until)i(we)g(end)f(up)g
(with)h Fu(1)p Fz(\226)1901 3313 y(parameter)e(f)o(amilies)i(of)f
(periodic)g(orbits)g(and)g(isolated)g(equi-)1901 3412
y(libria.)1937 3512 y(W)m(ithin)33 b(the)g(f)o(amily)g(of)f(all)i
Fu(\()p Ft(d)28 b Fs(\000)g Fu(1\))p Fz(\226tori)k(we)i(encounter)1901
3612 y(quasi\226periodic)13 b(centre\226saddle)h(and)h(frequenc)o(y)e
(halving)i(bi-)1901 3711 y(furcations)36 b(along)g Fu(\()p
Ft(d)c Fs(\000)e Fu(2\))p Fz(\226parameter)35 b(subf)o(amilies)i(and)
1901 3811 y(more)16 b(generally)g(bifurcations)f(of)i(co\226dimension)d
Ft(k)26 b Fs(\024)d Ft(d)7 b Fs(\000)g Fu(1)1901 3910
y Fz(along)18 b Fu(\()p Ft(d)13 b Fs(\000)g Ft(k)h Fs(\000)f
Fu(1\))p Fz(\226parameter)j(subf)o(amilies.)23 b(Similarly)c(in-)1901
4010 y(v)n(ariant)c Fu(\()p Ft(d)q Fs(\000)q Fu(2\))p
Fz(\226tori)f(under)o(going)e(a)k(quasi\226periodic)d(Hamil-)1901
4110 y(tonian)24 b(Hopf)g(bifurcation)f(form)h Fu(\()p
Ft(d)f Fs(\000)e Fu(3\))p Fz(\226parameter)i(f)o(am-)1901
4209 y(ilies)30 b(and)e(the)g Ft(n)p Fz(\226parameter)e(f)o(amilies)j
(of)f(in)m(v)n(ariant)f Ft(n)p Fz(\226tori)1901 4309
y(ha)n(v)o(e)38 b Fu(\()p Ft(n)32 b Fs(\000)f Ft(k)s
Fu(\))p Fz(\226parameter)37 b(subf)o(amilies)g(where)h(bifurca-)1901
4409 y(tions)22 b(of)g(co\226dimension)e Ft(k)29 b Fs(\024)e
Ft(n)22 b Fz(occur)-5 b(.)31 b(Such)21 b(bifurcations)1901
4508 y(are)j(not)g(restricted)f(to)i(those)e(of)h(semi\226local)g
(type,)g(b)n(ut)g(may)1901 4608 y(also)k(in)m(v)n(olv)o(e)e(coinciding)
g(stable)i(and)f(unstable)g(manifolds)1901 4707 y(of)19
b(dif)n(ferent)f(in)m(v)n(ariant)g(tori.)25 b(F)o(or)19
b(instance,)g(heteroclinic)f(or)n(-)1901 4807 y(bits)28
b(between)f(hyperbolic)f Fu(\()p Ft(d)e Fs(\000)g Fu(1\))p
Fz(\226tori)j(form)g Fu(\(2)p Ft(d)d Fs(\000)g Fu(1\))p
Fz(\226)1901 4907 y(dimensional)19 b(submanifolds)f(of)i(the)g(phase)g
(space)g Fs(P)27 b Fz(.)1901 5138 y FC(2.1)82 b(P)n(erturbation)19
b(analysis)1937 5238 y Fz(T)-7 b(o)31 b(sum)g(up,)h(the)f(dynamics)f
(of)g(a)h(non\226de)o(generate)c(inte-)1901 5338 y(grable)15
b(system)i(mak)o(es)f(the)g(phase)f(space)i Fs(P)23 b
Fz(a)16 b(rami\002ed)g(torus)1901 5437 y(b)n(undle.)46
b(The)27 b(re)o(gular)f(\002bres)h(are)h(the)f(Lagrangean)e(in)m(v)n
(ari-)1901 5537 y(ant)30 b(tori,)h(singular)e(\002bres)h(are)g(in)m(v)n
(ariant)e(tori)h(of)g(lo)n(wer)h(di-)1901 5636 y(mension,)c(together)e
(with)h(their)h(stable)g(and)f(unstable)f(man-)1901 5736
y(ifolds.)54 b(What)31 b(happens)e(to)h(the)g(rami\002ed)f
Ft(d)p Fz(\226torus)h(b)n(undle)
1800 6037 y(2)37 b()p eop end
%%Page: 3 3
TeXDict begin 3 2 bop -107 -90 a Fz(under)22 b(small)i(perturbations)e
(of)h(the)g(Hamiltonian)g(?)35 b(Let)24 b(us)-107 9 y(collect)h(the)g
(partial)f(answers)h(that)g(are)f(already)g(kno)n(wn)f(and)-107
109 y(indicate)d(possible)g(directions)f(of)h(future)f(research.)-71
209 y(Persistence)30 b(of)f(Lagrangean)e(tori)i(is)i(addressed)d(by)h
(clas-)-107 308 y(sical)j(KAM)f(theory)-5 b(.)55 b(Most)31
b(tori)g(survi)n(v)o(e)e(a)j(small)f(pertur)n(-)-107
408 y(bation)26 b(if)h(the)g(K)m(olmogoro)o(v)c(condition)i(\(1\))h(is)
i(satis\002ed)f(\227)-107 508 y(near)20 b(such)f Ft(y)24
b Fz(the)c(relati)n(v)o(e)f(measure)g(of)h(survi)n(ving)e(tori)i(tends)
-107 607 y(to)27 b Fu(1)g Fz(as)h(the)f(perturbation)e(strength)h
(tends)h(to)g(zero.)45 b(These)-107 707 y(tori)38 b(form)g(a)g
(\(Whitne)o(y\)\226smooth)d(Cantor)j(f)o(amily)-5 b(,)42
b(being)-107 806 y(parametrised)21 b(o)o(v)o(er)f(a)j(Cantor)e(set)i
(that)f(has)g(the)g(local)g(struc-)-107 906 y(ture)453
1134 y Fw(R)42 b Fs(\002)f Fr(Cantor)20 b(dust)82 b Ft(:)-107
1362 y Fz(Where)45 b(the)g(ener)o(gy)e(le)n(v)o(el)h(sets)i(are)f
(transv)o(ersal)f(to)h(the)-107 1462 y(continuous)28
b(direction)h(one)g(has)i(persistence)e(of)h(most)g(La-)-107
1561 y(grangean)j(tori)h(on)h(each)f(ener)o(gy)f(shell,)38
b(parametrised)33 b(by)-107 1661 y(Cantor)17 b(dust.)24
b(The)18 b(same)f(result)h(is)h(obtained)d(under)g(the)i(con-)-107
1761 y(dition)33 b(of)g(iso\226ener)o(getic)e(non\226de)o(generac)o(y)
-5 b(,)32 b(which)g(is)j(in-)-107 1860 y(dependent)25
b(of)i(K)m(olmogoro)o(v')-5 b(s)24 b(condition.)43 b(Note)27
b(that)g(it)h(is)-107 1960 y(generic)23 b(for)h(an)g(inte)o(grable)e
(system)i(to)g(satisfy)h(both)e(condi-)-107 2060 y(tions)29
b(almost)g(e)n(v)o(erywhere.)50 b(Ho)n(we)n(v)o(er)m(,)29
b(in)g(applications)f(it)-107 2159 y(is)i(a)g(non\226tri)n(vial)d(task)
j(to)g(actually)e(check)h(this)h(and)f(to)g(de-)-107
2259 y(termine)24 b(the)h(hypersurf)o(aces)e(in)i(action)f(space)h
(where)f(these)-107 2358 y(determinants)19 b(v)n(anish.)-71
2458 y(The)35 b(Cantor)g(set)h(structure)f(is)h(de\002ned)f(by)g
(Diophantine)-107 2558 y(conditions)376 2707 y Fm(^)282
2901 y Fp(k)q Fo(2)p Fl(Z)402 2885 y Fk(d)437 2876 y
Fj(nf)p Fi(0)p Fj(g)658 2786 y Fs(jh)p Ft(k)s(;)14 b(!)s
Fs(ij)47 b(\025)1116 2730 y Ft(\015)p 1073 2767 134 4
v 1073 2843 a Fs(j)p Ft(k)s Fs(j)1165 2819 y Fp(\034)1300
2786 y Ft(:)292 b Fz(\(3\))-107 3107 y(and)35 b(this)h(can)g(be)f(used)
h(to)g(weak)o(en)e(the)i(necessary)f(non\226)-107 3207
y(de)o(generac)o(y)26 b(condition.)50 b(Indeed,)29 b(since)g(the)g
(gaps)f(are)h(de-)-107 3307 y(\002ned)g(by)f(linear)h(inequalities)g
(the)g(conditions)e(on)i(the)g(\002rst)-107 3406 y(deri)n(v)n(ati)n(v)o
(es)i(of)h(the)h(frequenc)o(y)c(mapping)45 b Ft(y)j Fs(7!)e
Ft(!)s Fu(\()p Ft(y)s Fu(\))f(=)-107 3506 y Ft(D)r(H)7
b Fu(\()p Ft(y)s Fu(\))33 b Fz(can)18 b(be)g(replaced)f(by)h
(conditions)f(on)g(the)i(curv)n(ature)-107 3606 y(or)f(e)n(v)o(en)f
(higher)f(deri)n(v)n(ati)n(v)o(es.)23 b(Such)17 b(R)7
b(\250)-35 b(ussmann\226lik)o(e)17 b(condi-)-107 3705
y(tions)j(still)g(guarantee)e(that)h(the)h(relati)n(v)o(e)e(measure)h
(of)g(survi)n(v-)-107 3805 y(ing)k(tori)f(tends)h(to)f
Fu(1)h Fz(as)h(the)e(perturbation)e(strength)i(tends)g(to)-107
3904 y(zero,)e(b)n(ut)g(at)h(a)g(price.)k(F)o(or)20 b(instance,)g(the)h
(highest)f(deri)n(v)n(ati)n(v)o(e)-107 4004 y Ft(L)j
Fs(2)g Fw(N)e Fz(needed)e(in)247 4274 y Fh(<)371 4218
y Ft(@)420 4188 y Fo(j)p Fp(`)p Fo(j)491 4218 y Ft(!)p
371 4255 175 4 v 412 4331 a(@)5 b(y)611 4274 y Fg(j)55
b Fs(j)p Ft(`)p Fs(j)23 b(\024)g Ft(L)f Fh(>)70 b Fu(=)e
Fw(R)1319 4240 y Fp(d)1615 4274 y Fz(\(4\))-107 4541
y(enters)41 b(the)f(Diophantine)f(conditions)g(on)i(the)f(frequenc)o(y)
-107 4640 y(v)o(ector)24 b(by)h(means)g(of)g(the)g(inequality)e
Ft(\034)42 b(>)32 b(dL)22 b Fs(\000)g Fu(1)j Fz(on)g(the)-107
4740 y(Diophantine)i(constant)h Ft(\034)38 b Fz(.)51
b(F)o(or)28 b(more)g(details)h(the)g(reader)-107 4839
y(is)43 b(referred)e(to)i([Broer)m(,)j(Huitema)c(and)g(Se)n(vryuk,)k
(1996;)-107 4939 y(R)7 b(\250)-35 b(ussmann,)20 b(2001])e(and)i
(references)f(therein.)-71 5039 y(F)o(or)25 b(hyperbolic)e
Ft(n)p Fz(\226tori)h(the)h(abo)o(v)o(e)f(criteria)h(remain)f(v)n(alid)
-107 5138 y(almost)32 b(v)o(erbatim)e(;)38 b(the)32 b(k)o(e)o(y)f(step)
h(is)g(to)g(pass)g(to)g(a)g(centre)-107 5238 y(manifold)c(\(and)g(to)i
(replace)e Ft(d)i Fz(by)f Ft(n)h Fz(in)f(the)h(formulas\).)50
b(A)-107 5338 y(technical)31 b(dif)n(\002culty)f(is)j(that)e(e)n(v)o
(en)g(for)f(analytic)h(Hamilto-)-107 5437 y(nians)26
b(centre)f(manifolds)g(may)g(only)g(be)h(of)g(\002nite)g(dif)n(feren-)
-107 5537 y(tiability)-5 b(.)50 b(KAM\226theorems)27
b(remain)h(true)g(in)h(this)g(conte)o(xt,)-107 5636 y(during)22
b(the)i(proof)e(one)h(has)h(to)g(intersperse)f(an)h(analytic)f(ap-)-107
5736 y(proximation)28 b(at)j(each)f(iteration)g(step.)57
b(Still,)34 b(the)c(analytic)1901 -90 y(conte)o(xt)25
b(has)h(its)h(adv)n(antages)e(\227)i(for)e(instance)h(\(4\))g(is)h
(satis-)1901 9 y(\002ed)i(for)f(some)g Ft(L)39 b Fs(2)g
Fw(N)29 b Fz(for)f(an)g(analytic)g(frequenc)o(y)e(map-)1901
109 y(ping)f Ft(!)j Fz(if)e(and)f(only)g(if)h Fu(im)8
b Ft(!)29 b Fz(does)c(not)h(lie)g(within)f(a)h(linear)1901
209 y(hyperplane.)d(An)e(alternati)n(v)o(e)e(is)i(therefore)e(to)i(pro)
o(v)o(e)e(persis-)1901 308 y(tence)h(of)g(hyperbolic)e(tori)i(directly)
-5 b(,)19 b(see)i([R)7 b(\250)-35 b(ussmann,)19 b(2001;)1901
408 y(Rudne)n(v)-5 b(,)16 b(2003])g(and)h(references)f(therein.)23
b(This)17 b(also)h(gi)n(v)o(es)f(a)1901 508 y(more)e(direct)h(hold)f
(on)h(their)g(stable)g(and)g(unstable)f(manifolds.)1937
607 y(Elliptic)20 b Fu(\()p Ft(d)f Fs(\000)f Fu(1\))p
Fz(\226tori)i(need)f(one)h(e)o(xtra)f(parameter)g(to)h(con-)1901
707 y(trol)26 b(the)g(normal)e(frequenc)o(y)f(as)k(well.)43
b(Similar)26 b(to)g(the)g(iso\226)1901 806 y(ener)o(getic)17
b(case)j(one)e(can)h(use)h(time)f(re\226parametrisation)d(and)1901
906 y(obtain)31 b(Cantor)g(f)o(amilies)g(of)g(persistent)g(elliptic)h
Fu(\()p Ft(d)27 b Fs(\000)g Fu(1\))p Fz(\226)1901 1006
y(tori)c(parametrised)f(by)h(Cantor)f(dust)h(without)g(the)g(use)h(of)f
(an)1901 1105 y(e)o(xternal)16 b(parameter)-5 b(.)22
b(Where)17 b(there)f(are)h(more)f(than)g(one)g(nor)n(-)1901
1205 y(mal)h(frequenc)o(y)d(to)j(control)f(this)h(can)g(no)g(longer)e
(be)i(done)f(in)h(a)1901 1305 y(linear)f(w)o(ay)g(;)i(a)e(problem)e
(solv)o(ed)i(by)f(R)7 b(\250)-35 b(ussmann\226lik)o(e)15
b(condi-)1901 1404 y(tions)22 b(on)f(the)h(higher)f(deri)n(v)n(ati)n(v)
o(es)f(of)h(the)h(frequenc)o(y)d(v)o(ector)m(,)1901 1504
y(see)35 b([Broer)m(,)h(Huitema)d(and)h(Se)n(vryuk,)i(1996;)j(R)7
b(\250)-35 b(ussmann,)1901 1603 y(2001])25 b(and)i(references)f
(therein.)44 b(In)27 b(case)g(the)h(mapping)d(of)1901
1703 y(internal)h(frequencies)f(satis\002es)k(K)m(olmogoro)o(v')-5
b(s)24 b(condition,)1901 1803 y(the)f(higher)e(order)g(deri)n(v)n(ati)n
(v)o(es)g(are)i(only)e(needed)g(of)i(normal)1901 1902
y(frequencies.)f(No)n(w)17 b(normal)f(frequencies)f Ft(\013)3212
1914 y Fp(j)3265 1902 y Fz(enter)i(the)g(Dio-)1901 2002
y(phantine)i(conditions)2326 2260 y Fs(j)p Fu(2)p Ft(\031)s
Fs(h)p Ft(k)s(;)14 b(!)s Fs(i)32 b Fu(+)g Fs(h)p Ft(`;)14
b(\013)p Fs(ij)47 b(\025)3194 2204 y Ft(\015)p 3151 2241
134 4 v 3151 2317 a Fs(j)p Ft(k)s Fs(j)3243 2293 y Fp(\034)3623
2260 y Fz(\(5\))1901 2548 y(only)22 b(as)h(combinations)d
Fs(h)p Ft(`;)14 b(\013)p Fs(i)24 b Fz(with)e Fs(j)p Ft(`)p
Fs(j)27 b(\024)g Fu(2)22 b Fz(.)33 b(This)22 b(allo)n(ws)1901
2648 y(to)d(e)o(xtend)e(the)i(result)g(to)g(\002nite\226dimensional)e
(elliptic)i(tori)f(in)1901 2747 y(in\002nitely)g(man)o(y)g(de)o(grees)f
(of)i(freedom,)e(cf.)i([P)7 b(\250)-35 b(oschel,)18 b(1989;)1901
2847 y(K)o(uksin,)36 b(1993].)64 b(F)o(or)34 b(hypo\226elliptic)d(tori)
j(one)f(may)h(deal)1901 2946 y(with)20 b(the)g(hyperbolic)d(part)i(by)g
(means)h(of)f(a)h(centre)f(manifold)1901 3046 y(or)32
b(use)g(a)h(direct)e(approach,)i(cf.)f([Huitema,)i(1988;)i(Broer)m(,)
1901 3146 y(Huitema)20 b(and)f(T)-7 b(ak)o(ens,)20 b(1990;)f(R)7
b(\250)-35 b(ussmann,)20 b(2001].)1937 3245 y(Where)32
b(\(lo)n(wer)n(\226dimensional\))c Ft(n)p Fz(\226tori)j(under)o(go)d(a)
33 b(semi\226)1901 3345 y(local)24 b(bifurcation)e(the)i
Ft(n)h Fz(actions)f Ft(y)j Fz(conjugate)22 b(to)j(the)f(toral)1901
3445 y(angles)38 b Ft(x)h Fz(\002rst)g(of)f(all)h(ha)n(v)o(e)e(to)i(v)o
(ersally)e(unfold)g(the)h(bi-)1901 3544 y(furcation)i(scenario.)88
b(It)42 b(is)g(generic)f(for)g(the)g(inte)o(grable)1901
3644 y(Hamiltonian)34 b Ft(H)43 b Fz(that)35 b(the)h
Ft(n)p Fz(\226parameter)d(f)o(amilies)i(of)h Ft(n)p Fz(\226)1901
3743 y(tori,)30 b Fu(1)38 b Fs(\024)f Ft(n)h Fs(\024)g
Ft(d)25 b Fs(\000)f Fu(1)k Fz(,)j(do)d(not)g(encounter)e(bifurcations)
1901 3843 y(of)34 b(co\226dimension)e(higher)g(than)i
Ft(n)h Fz(,)j(so)c(this)h(is)g(possible.)1901 3943 y(The)29
b(curv)n(ature)e(of)h(the)h(frequenc)o(y)d(mapping)h(is)j(then)f(used)
1901 4042 y(to)35 b(ensure)e(Diophanticity)g(of)h(most)h(bifurcating)d
(tori,)37 b(i.e.)1901 4142 y(a)30 b(R)7 b(\250)-35 b(ussmann\226lik)o
(e)29 b(condition)f(with)i Ft(L)40 b Fu(=)h(2)30 b Fz(is)g(suf)n
(\002cient,)1901 4242 y(cf.)35 b([Broer)m(,)j(Han\337mann)c(and)g(Y)-9
b(ou,)39 b(2003;)j(2004;)f(Han\337-)1901 4341 y(mann,)19
b(2003;)g(2004].)1937 4441 y(While)31 b(the)f(proof)e(in)i(the)g(abo)o
(v)o(e)f(papers)g(is)i(k)o(ept)f(as)g(sim-)1901 4541
y(ple)h(as)i(possible,)g(restricting)e(to)g Ft(n)44 b
Fu(=)f Ft(d)27 b Fs(\000)f Fu(1)32 b Fz(,)i(it)f(should)1901
4640 y(be)19 b(feasible)f(to)i(include)d(additional)h(elliptic)h(and)f
(hyperbolic)1901 4740 y(normal)h(directions.)26 b(On)20
b(the)h(other)f(hand,)f(additional)g(viola-)1901 4839
y(tions)f(of)g(\(5\))g(pose)g(a)g(much)g(harder)f(problem,)f(as)j(in)g
(this)f(situ-)1901 4939 y(ation)k(e)n(v)o(en)g(the)g(corresponding)e
(bifurcations)g(of)j(equilibria)1901 5039 y(ha)n(v)o(e)29
b(yet)g(to)g(be)g(understood.)49 b(Thus,)31 b(if)e(we)h(e)o(xplicitly)e
(re-)1901 5138 y(quire)16 b(that)h(the)g(bifurcation)e(results)i(from)f
(violating)g(\(5\))g(with)1901 5238 y(a)22 b(single)f
(normal\226internal)e(resonance,)h(the)h(quasi\226periodic)1901
5338 y(bifurcation)27 b(scenario)h(should)g(persist)h(for)f(all)h
Ft(n)p Fz(\226tori)f(with)1901 5437 y Fu(2)23 b Fs(\024)h
Ft(n)g Fs(\024)f Ft(d)c Fs(\000)f Fu(1)j Fz(and)f(in)h(f)o(act)g(also)f
(in)h(in\002nite\226dimensional)1901 5537 y(Hamiltonian)37
b(systems.)78 b(Recall)39 b(that)f(the)g(maximal)f(co\226)1901
5636 y(dimension)17 b(of)i(occurring)e(bifurcations)g(is)j(the)f
(dimension)e Ft(n)1901 5736 y Fz(of)27 b(the)f(bifurcating)f(torus)h
(and)h(not)f(related)h(to)f(the)h(number)
1800 6037 y(3)37 b()p eop end
%%Page: 4 4
TeXDict begin 4 3 bop -107 -90 a Fz(of)32 b(de)o(grees)f(of)h(freedom.)
60 b(F)o(or)32 b(instance,)i(the)e(abo)o(v)o(e)f(cur)n(-)-107
9 y(v)n(ature)d(requirement)e(is)k(not)f(necessary)f(for)g
Fu(2)p Fz(\226tori)g(;)33 b(these)-107 109 y(may)27 b(under)o(go)d(the)
j(quasi\226periodic)d(analogues)i(of)g(the)h(co\226)-107
209 y(dimension)e(one)h(bifurcations)f(of)i(periodic)e(orbits)h
(detailed)-107 308 y(in)c([Me)o(yer)m(,)d(1970;)i(1975].)27
b(Indeed,)19 b(co\226dimension)g(tw)o(o)j(bi-)-107 408
y(furcations)h(are)i(isolated)f(within)g(these)h Fu(2)p
Fz(\226parameter)d(f)o(ami-)-107 508 y(lies)h(and)f(cannot)e(be)i(pre)n
(v)o(ented)e(to)i(disappear)f(in)h(resonance)-107 607
y(gaps.)-71 719 y(Let)31 b(an)f Fu(\()p Ft(n)c Fs(\000)g
Ft(k)s Fu(\))p Fz(\226parameter)i(f)o(amily)i(of)g Ft(n)p
Fz(\226tori)g(that)g(un-)-107 818 y(der)o(go)37 b(a)j(bifurcation)d(of)
h(co\226dimension)f Ft(k)42 b Fz(ha)n(v)o(e)d Ft(m)g
Fz(ad-)-107 918 y(ditional)28 b(pairs)h(of)f(purely)g(imaginary)f
(Floquet)h(e)o(xponents.)-107 1018 y(Then)g(e)o(xcitation)g(of)h
(normal)f(modes,)i(cf.)f([Jorba)f(and)g(V)-5 b(il-)-107
1117 y(lanue)n(v)n(a,)21 b(1997;)g(Se)n(vryuk,)f(1997],)g(leads)i(for)f
Ft(l)28 b Fu(=)d(1)p Ft(;)14 b(:)g(:)g(:)f(;)h(m)-107
1217 y Fz(to)k Fu(\()p Ft(n)10 b Fu(+)g Ft(l)i Fs(\000)e
Ft(k)s Fu(\))p Fz(\226parameter)15 b(f)o(amilies)j(of)f
Fu(\()p Ft(n)10 b Fu(+)g Ft(l)r Fu(\))p Fz(\226tori)17
b(under)n(-)-107 1317 y(going)g(that)i(co\226dimension)c
Ft(k)22 b Fz(bifurcation)16 b(in)j(the)f(inte)o(grable)-107
1416 y(system.)63 b(This)33 b(whole)f(structure)f(should)h(persist)h
(under)e(a)-107 1516 y(\(suf)n(\002ciently)17 b(small\))i(non\226inte)o
(grable)14 b(perturbation)i(on)i(per)n(-)-107 1615 y(tinent)36
b(Cantor)g(sets.)74 b(Additional)35 b(hyperbolic)f(directions)-107
1715 y(augment)d(the)i(dimension)e(of)h(stable)h(and)f(unstable)g
(mani-)-107 1815 y(folds.)-71 1926 y(Up)h(to)g(no)n(w)f(the)g(reported)
f(changes)h(of)g(the)g(rami\002ed)g Ft(d)p Fz(\226)-107
2026 y(torus)18 b(b)n(undle)e(under)h(a)h(small)g(perturbation)d(of)j
(the)f(Hamilto-)-107 2126 y(nian)j(were)h(of)f(the)g(form)g
Fq(\223Diophantine)e(tori)j(per)o(sist\224)h Fz(lead-)-107
2225 y(ing)h(to)h(a)f(\223Cantori\002cation\224)f(of)h(the)g
(rami\002ed)g Ft(d)p Fz(\226torus)g(b)n(un-)-107 2325
y(dle)30 b(\227)h(the)f(strati\002cation)g(of)g(the)g(action)f(space)i
(into)e(v)n(ari-)-107 2424 y(ous)j(subf)o(amilies)g(parametrising)e
(the)i(tori)g(is)h(replaced)e(by)-107 2524 y(a)c(Cantor)e
(strati\002cation.)42 b(Of)26 b(equal)g(importance)e(are)i(those)-107
2624 y(changes)35 b(that)h(mak)o(e)f(sure)h(that)g(the)g(non\226inte)o
(grable)c(per)n(-)-107 2723 y(turbed)24 b(dynamics)g(is)i(indeed)e
(qualitati)n(v)o(ely)g(dif)n(ferent)g(from)-107 2823
y(the)18 b(inte)o(grable)e(unperturbed)e(dynamics.)23
b(While)18 b(the)g(former)-107 2923 y(persistence)k(results)g(are)g
(obtained)f(upon)f(genericity)h(condi-)-107 3022 y(tions)i(on)g(the)g
(unperturbed)d(system,)j(such)g(changes)f(require)-107
3122 y(the)e(perturbation)e(to)i(be)g(generic.)-71 3233
y(One)29 b(of)f(the)h(ef)n(fects)f(of)h(a)g(small)g(generic)f
(perturbation)e(is)-107 3333 y(that)33 b(stable)g(and)f(unstable)g
(manifolds)f(of)i(hyperbolic)d(tori)-107 3433 y(no)22
b(longer)e(coincide,)h(b)n(ut)h(split)h(and)e(intersect)h(transv)o
(ersely)-5 b(,)-107 3532 y(cf.)31 b([Robinson,)h(1970a;)j(1970b;)g
(Delshams,)e(de)e(la)h(Lla)n(v)o(e)-107 3632 y(and)26
b(Seara,)h(2003a;)h(2003b].)41 b(Where)26 b(this)h(concerns)e(hete-)
-107 3732 y(roclinic)19 b(orbits)g(between)g(tw)o(o)h(dif)n(ferent)e(f)
o(amilies)i(of)f(hyper)n(-)-107 3831 y(bolic)e(tori)g(this)g(leads)h
(to)f(drastic)g(changes)f(of)h(the)g(connection)-107
3931 y(bifurcation)27 b(scenario.)51 b(Indeed,)30 b(heteroclinic)e
(orbits)g(e)o(xist)-107 4030 y(in)21 b(the)g(inte)o(grable)f(system)h
(only)f(at)i Ft(\026)i Fu(=)h(0)c Fz(for)f(an)h(appropri-)-107
4130 y(ately)c(chosen)e(transv)o(ersal)g(parameter)g
Ft(\026)i Fz(.)25 b(F)o(or)16 b(a)h(suf)n(\002ciently)-107
4230 y(small)26 b(generic)e(perturbation)f(there)i(is)h(a)g(whole)e
(interv)n(al)h(of)-107 4329 y Ft(\026)p Fz(\226v)n(alues)30
b(containing)e(a)j(Cantor)e(subset)i(of)f(relati)n(v)o(e)f(mea-)-107
4429 y(sure)d(near)g Fu(1)h Fz(for)e(which)h(there)g(are)g
(heteroclinic)f(orbits)h(be-)-107 4529 y(tween)h(survi)n(ving)e
(hyperbolic)f(tori.)45 b(Similar)27 b(observ)n(ations)-107
4628 y(apply)17 b(to)i(stable)f(and)g(unstable)f(manifolds)g(of)h
(parabolic)e(and)-107 4728 y(other)k(bifurcating)e(tori.)-71
4839 y(Completely)28 b(ne)n(w)g(phenomena)e(are)j(also)g(to)g(be)f(e)o
(xpected)-107 4939 y(in)d(the)h(gaps)e(of)h(the)g(Cantor)f(sets)j
(parametrising)c(persistent)-107 5039 y(tori.)55 b(Disinte)o(grating)28
b(Lagrangean)g(tori)i(lead)g(to)g(in)m(v)n(ariant)-107
5138 y Ft(n)p Fz(\226tori,)22 b(where)f Ft(d)f Fs(\000)g
Ft(n)i Fz(is)h(the)f(number)e(of)i(independent)e(res-)-107
5238 y(onances)h Fs(h)p Ft(k)s(;)14 b(!)s Fs(i)25 b Fu(=)g(0)c
Fz(of)g(the)h(\(internal\))e(frequencies.)26 b(Most)-107
5338 y(of)16 b(these)g(lo)n(wer)g(dimensional)e(tori)i(will)h(be)f
(elliptic)g(or)f(hyper)n(-)-107 5437 y(bolic,)28 b(cf.)e([T)m(reshch)5
b(\250)-33 b(ev)-5 b(,)26 b(1991].)42 b(The)27 b(ne)n(w)f(hyperbolic)e
(tori)-107 5537 y(lie)19 b(at)h(the)e(basis)i(of)e(the)h(e)o(xample)e
(in)i([Arnol')l(d,)e(1964])g(of)h(dy-)-107 5636 y(namical)f
(instablility)-5 b(.)23 b(This)17 b(approach)e(to)i(Arnol')l(d)e(dif)n
(fusion)-107 5736 y(relies)27 b(on)f(the)g(splitting)g(of)g
(separatrices)f(which)h(also)g(leads)1901 -90 y(to)21
b(transv)o(ersal)e(intersections)g(of)h(stable)h(and)f(unstable)f(man-)
1901 9 y(ifolds)25 b(of)g(neighbouring)c(hyperbolic)h(tori)j(in)h(the)f
(same)g(en-)1901 109 y(er)o(gy)19 b(shell.)25 b(These)20
b(hyperbolic)e(tori)i(form)f(a)i(Cantor)f(f)o(amily)-5
b(,)1901 209 y(and)23 b(one)h(of)g(the)g(main)f(problems)g(is)i(to)f
(mak)o(e)g(sure)f(that)h(the)1901 308 y(transition)16
b(chain)g(of)h(hyperbolic)d(tori)j(and)f(their)h(heteroclinic)1901
408 y(connections)23 b(bridges)g(the)i(occuring)e(gaps,)i(cf.)f
([Delshams,)1901 508 y(de)29 b(la)h(Lla)n(v)o(e)g(and)f(Seara,)i
(2003a;)i(2003b])27 b(and)i(references)1901 607 y(therein.)1937
707 y(The)18 b(dynamics)g(in)g(the)h(gaps)f(of)g(Cantor)g(f)o(amilies)h
(of)f(hyper)n(-)1901 806 y(bolic)d(tori)h(can)f(already)f(be)i(studied)
f(in)h(the)f(perturbation)e(near)1901 906 y(resonant)19
b(singular)g(\002bres)g(of)h(the)g(rami\002ed)f Ft(d)p
Fz(\226torus)g(b)n(undle.)1901 1006 y(On)24 b(the)f(centre)g(manifold)f
(these)i(become)e(again)g(\(resonant\))1901 1105 y(re)o(gular)28
b(\002bres,)k(b)n(ut)e(the)f(full)h(perturbed)d(motion)i(is)h(super)n
(-)1901 1205 y(posed)j(by)h(the)g(hyperbolic)d(dynamics)i(in)h(the)g
(symplectic)1901 1305 y(normal)23 b(directions.)37 b(In)24
b(particular)m(,)g(secondary)e(hyperbolic)1901 1404 y(tori)32
b(\227)h(maximal)f(tori)g(on)g(the)h(centre)f(manifold)e(that)j(ap-)
1901 1504 y(pear)d(in)h(the)g(resonance)e(gap)h(\227)h(are)g(used)f(in)
h([Delshams,)1901 1603 y(de)20 b(la)g(Lla)n(v)o(e)g(and)g(Seara,)f
(2003a])f(together)h(with)h(hyperbolic)1901 1703 y(tori)j(of)f(e)n(v)o
(en)g(lo)n(wer)g(dimension)g(to)h(continuate)e(a)i(transition)1901
1803 y(chain)d(through)e(the)i(resonance)f(gap.)1937
1902 y(A)d(Lagrangean)d(torus)i(with)h Ft(d)q Fs(\000)q
Fu(1)f Fz(independent)e(resonances)1901 2002 y(consists)28
b(of)e(periodic)g(orbits.)45 b(When)27 b(the)g(torus)g(breaks)f(up)1901
2102 y(under)c(the)i(perturbation,)e(only)g(\002nitely)i(man)o(y)e(of)i
(these)g(are)1901 2201 y(e)o(xpected)h(to)i(survi)n(v)o(e.)42
b(At)27 b(the)g(same)g(time)f(the)h(tri)n(vial)f(nor)n(-)1901
2301 y(mal)d(beha)n(viour)d(of)j(these)f(periodic)g(orbits)g(changes,)g
(result-)1901 2400 y(ing)j(in)g(hyperbolic)e(and)h(elliptic)i(periodic)
e(orbits.)39 b(The)25 b(lat-)1901 2500 y(ter)i(can)f(serv)o(e)g(as)h
(starting)f(points)g(for)g(the)g(construction)f(of)1901
2600 y(solenoids,)33 b(cf.)e([Markus)f(and)h(Me)o(yer)m(,)i(1980].)56
b(This)31 b(con-)1901 2699 y(struction)f(should)g(carry)g(o)o(v)o(er)g
(to)h(elliptic)h(tori,)h(where)d(the)1901 2799 y(\223encircling\224)14
b(tori)h(emer)o(ge)f(from)g(the)i(normal\226internal)c(reso-)1901
2899 y(nances)18 b(studied)f(in)i([Broer)m(,)e(Han\337mann,)f(Jorba,)i
(V)-5 b(illanue)n(v)n(a)1901 2998 y(and)15 b(W)-7 b(agener)m(,)15
b(2003].)22 b(This)15 b(might)g(also)g(result)h(in)f(solenoids)1901
3098 y(that)20 b(are)g(limits)h(of)f(tori)g(with)h(v)n(arying)d
(dimension.)1937 3197 y(The)29 b(nature)e(of)i(the)g(gaps)f(where)g
(\(5\))g(is)i(not)f(satis\002ed)g(for)1901 3297 y(elliptic)h(tori)g(is)
h(tw)o(ofold.)54 b(Internal)29 b(resonances)g Fs(h)p
Ft(k)s(;)14 b(!)s Fs(i)41 b Fu(=)1901 3397 y(0)i Fz(lead)f(again)g(to)g
(the)h(destruction)e(of)i(the)f(torus.)92 b(The)1901
3496 y(study)32 b([Broer)m(,)i(Han\337mann,)g(Jorba,)g(V)-5
b(illanue)n(v)n(a)32 b(and)g(W)-7 b(a-)1901 3596 y(gener)m(,)52
b(2003])45 b(of)h(normal\226internal)e(resonances)h(relates)1901
3696 y(boundary)16 b(points)i(of)h(the)g(resulting)f(gaps)g(to)h
(quasi\226periodic)1901 3795 y(bifurcations.)k(In)d(particular)f
(resonance)g(gaps)2330 4044 y Fs(j)k Fu(2)p Ft(\031)s
Fs(h)p Ft(k)s(;)14 b(!)s Fs(i)41 b Fu(+)g(2)p Ft(\013)23
b Fs(j)46 b Ft(<)3191 3988 y(\015)p 3148 4025 134 4 v
3148 4101 a Fs(j)p Ft(k)s Fs(j)3240 4077 y Fp(\034)1901
4317 y Fz(are)22 b(completely)f(\002lled)h(by)g(hyperbolic)e(tori)i
(\(in)g(accordance)1901 4417 y(with)k([Bour)o(gain,)d(1994;)k(1997;)g
(Xu)e(and)g(Y)-9 b(ou,)26 b(2001]\))d(that)1901 4517
y(terminate)g(in)i(frequenc)o(y)c(halving)i(bifurcations.)36
b(One)24 b(may)1901 4616 y(speculate)c(that)g(resonance)f(gaps)2236
4865 y Fs(j)k Fu(2)p Ft(\031)s Fs(h)p Ft(k)s(;)14 b(!)s
Fs(i)41 b Fu(+)g Ft(\013)2776 4877 y Fv(1)2832 4865 y
Fu(+)18 b Ft(\013)2968 4877 y Fv(2)3029 4865 y Fs(j)46
b Ft(<)3285 4809 y(\015)p 3242 4846 V 3242 4922 a Fs(j)p
Ft(k)s Fs(j)3334 4898 y Fp(\034)1901 5138 y Fz(are)16
b(similarly)f(\002lled)i(by)e(hyperbolic)e(tori)j(obtained)e(in)i
(quasi\226)1901 5238 y(periodic)30 b(Hamiltonian)g(Hopf)g(bifurcations)
f(generated)h(by)1901 5338 y(the)20 b(perturbation.)1937
5437 y(The)42 b(results)g(in)h([Broer)m(,)j(Han\337mann)40
b(and)i(Y)-9 b(ou,)47 b(2003;)1901 5537 y(2004;)29 b(Han\337mann,)e
(2003;)i(2004])d(address)g(persistence)h(of)1901 5636
y(Diophantine)21 b(tori)i(in)m(v)n(olv)o(ed)e(in)j(a)f(bifurcation)e
(and)i(the)g(cor)n(-)1901 5736 y(responding)39 b(gaps)h(trigger)g
(again)h(ne)n(w)g(phenomena.)85 b(A)
1800 6037 y(4)37 b()p eop end
%%Page: 5 5
TeXDict begin 5 4 bop -107 -90 a Fz(\002rst)32 b(step)g(has)g(been)f
(made)g(in)h([Litv)n(ak\226Hinenzon,)e(2001;)-107 9 y(Litv)n
(ak\226Hinenzon)50 b(and)k(Rom\226K)n(edar)m(,)59 b(2002a;)69
b(2002b;)-107 109 y(2004])40 b(where)g(\(internally\))g(resonant)g
(parabolic)g(tori)h(in-)-107 209 y(v)n(olv)o(ed)15 b(in)h(a)g
(quasi\226periodic)d(Hamiltonian)i(pitchfork)f(bifur)n(-)-107
308 y(cation)20 b(are)h(considered.)k(This)c(may)f(result)h(in)g(lar)o
(ge)e(dynam-)-107 408 y(ical)34 b(instabilities,)k(especially)33
b(where)g(multiple)g(parabolic)-107 508 y(resonances)25
b(are)g(encountered.)39 b(The)26 b(ef)n(fect)f(is)i(further)d(am-)-107
607 y(pli\002ed)19 b(for)g(tangent)f(\(or)g(\003at\))i(parabolic)d
(resonances,)h(which)-107 707 y(f)o(ail)k(to)f(satisfy)h(the)f
(iso\226ener)o(getic)e(non\226de)o(generac)o(y)e(condi-)-107
806 y(tion.)-107 1036 y FC(2.2)82 b(The)21 b(Lagrange)e(top)-71
1135 y Fz(The)h(rigid)f(body)g(with)i(a)f(\002x)o(ed)g(point)f(is)i(a)g
(mechanical)d(sys-)-107 1235 y(tem)35 b(with)g(three)g(de)o(grees)e(of)
i(freedom,)h(the)f(phase)g(space)-107 1334 y Fs(P)47
b Fu(=)40 b Ft(T)164 1304 y Fo(\003)201 1334 y Ft(S)5
b(O)r Fu(\(3\))469 1312 y Fs(\030)469 1339 y Fu(=)573
1334 y Ft(S)g(O)r Fu(\(3\))26 b Fs(\002)f Fw(R)976 1304
y Fv(3)1043 1334 y Fz(being)k(the)g(cotangent)-107 1434
y(b)n(undle)22 b(of)g(the)h(group)e Ft(S)5 b(O)r Fu(\(3\))24
b Fz(of)f(three\226dimensional)c(rota-)-107 1534 y(tions)28
b Ft(g)k Fz(.)49 b(An)29 b(e)o(xample)e(of)g(a)i(non\226de)o(generate)
24 b(inte)o(grable)-107 1633 y(system)j(on)g Fs(P)34
b Fz(is)28 b(the)e(Lagrange)f(top,)j(an)f(axially)f(symmet-)-107
1733 y(ric)d(rigid)f(body)f(subject)h(to)h(a)g(constant)f(v)o(ertical)g
(force)g(\002eld,)-107 1833 y(cf.)e([Cushman)f(and)h(Bates,)h(1997].)i
(Ne)o(xt)d(to)g(the)h(ener)o(gy)93 2116 y Ft(H)7 b Fu(\()p
Ft(\032;)14 b(`)p Fu(\))69 b(=)g Ft(I)587 2128 y Fv(1)634
2059 y Ft(`)669 2029 y Fv(2)669 2080 y(1)725 2059 y Fu(+)18
b Ft(`)843 2029 y Fv(2)843 2080 y(2)p 634 2096 246 4
v 736 2172 a Fu(2)931 2116 y(+)41 b Ft(I)1073 2128 y
Fv(3)1121 2059 y Ft(`)1156 2029 y Fv(2)1156 2080 y(3)p
1121 2096 72 4 v 1136 2172 a Fu(2)1244 2116 y(+)g Ft(\037g)1442
2128 y Fv(33)-107 2371 y Fz(both)17 b(the)h(component)d
Ft(`)601 2383 y Fv(3)656 2371 y Fz(of)j(the)g(angular)e(momentum)g
(along)-107 2471 y(the)k(\002gure)g(axis)g(and)g(the)g(component)279
2704 y Ft(\026)329 2716 y Fv(3)412 2704 y Fu(=)46 b Ft(g)563
2716 y Fv(31)633 2704 y Ft(`)668 2716 y Fv(1)737 2704
y Fu(+)32 b Ft(g)874 2716 y Fv(32)944 2704 y Ft(`)979
2716 y Fv(2)1048 2704 y Fu(+)g Ft(g)1185 2716 y Fv(33)1255
2704 y Ft(`)1290 2716 y Fv(3)-107 2936 y Fz(of)f(the)g(angular)e
(momentum)g(along)h(the)h(v)o(ertical)f(axis)i(are)-107
3036 y(\(commuting\))22 b(inte)o(grals)j(of)g(motion.)38
b(These)25 b(tw)o(o)g(inte)o(grals)-107 3135 y(generate)18
b(the)i(rotations)e(about)h(the)g(\002gure)g(axis)h(and)f(the)g(v)o(er)
n(-)-107 3235 y(tical)29 b(axis,)i(respecti)n(v)o(ely)-5
b(.)47 b(When)28 b(the)h(top)f(is)h(standing)e(up-)-107
3335 y(right)19 b(or)f(hanging)f(upside)i(do)n(wn)e(these)j(tw)o(o)f
Ft(S)1273 3304 y Fv(1)1310 3335 y Fz(\226actions)f(co-)-107
3434 y(incide)h(and)g(correspondingly)c(the)20 b(motion)e(is)i
(periodic)1541 3404 y Ff(1)1592 3434 y Fz(and)-107 3534
y(consists)30 b(of)f(rotation)f(about)h(that)g(common)e(axis.)53
b(In)29 b(case)-107 3633 y Ft(X)-52 3662 y(H)50 3633
y Fz(lies)23 b(within)e(the)h(plane)f(spanned)f(by)h
Ft(X)1202 3645 y(\026)1252 3657 y Fv(3)1315 3633 y Fz(and)h
Ft(X)1513 3663 y(`)1548 3675 y Fv(3)1611 3633 y Fz(the)-107
3733 y(motion)i(is)i(called)f(re)o(gular)e(precession)h(\227)h(a)h
(superposition)-107 3833 y(of)c(the)g(rotation)f(about)h(the)g
(\002gure)f(axis)i(and)e(the)i(precession)-107 3932 y(of)k(the)h
(\002gure)f(axis)h(about)e(the)i(v)o(ertical)e(axis)i(\227)g(and)f(tak)
o(es)-107 4032 y(place)f(on)g(a)h Fu(2)p Fz(\226torus.)42
b(F)o(or)26 b(re)o(gular)f(v)n(alues)h(of)g(the)g(ener)o(gy\226)-107
4132 y(momentum)c(mapping)g Fs(E)-16 b(M)30 b Fu(=)g(\()p
Ft(H)r(;)14 b(`)1039 4144 y Fv(3)1076 4132 y Ft(;)g(\026)1163
4144 y Fv(3)1200 4132 y Fu(\))25 b Fz(we)f(obtain)g(the)-107
4231 y(Lagrangean)c Fu(3)p Fz(\226tori)h(as)i(the)f(\002gure)f(axis)i
(starts)g(to)f(nutate)g(up)-107 4331 y(and)e(do)n(wn)f(as)i(well.)-71
4430 y(T)-7 b(o)17 b(complete)f(this)i(description)d(of)i(the)g
(rami\002ed)f(torus)h(b)n(un-)-107 4530 y(dle)i(note)f(that)h(unstable)
f(rotations)g(about)f(the)i(upright)e(stand-)-107 4630
y(ing)37 b(\002gure)g(axis)g(are)g(accompanied)e(by)i(asymptotic)f(mo-)
-107 4729 y(tions)f(forming)d(the)j(\(un\)stable)e(manifold,)j(this)f
(turns)f(the)-107 4829 y(le)n(v)o(el)i(set)i(of)e Fs(E)-16
b(M)37 b Fz(into)g(a)g(pinched)e Fu(3)p Fz(\226torus.)73
b(When)37 b(the)-107 4929 y(magnitude)e(of)i Ft(`)423
4941 y Fv(3)514 4929 y Fu(=)54 b Ft(\026)683 4941 y Fv(3)757
4929 y Fz(increases)37 b(these)g(periodic)f(or)n(-)-107
5028 y(bits)h(get)g(gyroscopically)d(stabilized)i(through)f(a)i
(periodic)-107 5128 y(Hamiltonian)30 b(Hopf)h(bifurcation.)57
b(Corresponding)29 b(to)j(the)-107 5228 y(tw)o(o)22 b(rotation)f
(senses)i(tw)o(o)f(such)f(bifurcations)g(tak)o(e)g(place)h(at)-107
5327 y Ft(`)-72 5339 y Fv(3)-12 5327 y Fu(=)h Ft(\026)126
5339 y Fv(3)186 5327 y Fu(=)g Fs(\006)p Fu(2)381 5265
y Fs(p)p 449 5265 126 4 v 449 5327 a Ft(I)485 5339 y
Fv(1)523 5327 y Ft(\037)d Fz(.)p -107 5514 300 3 v -22
5630 a Fe(1)13 5653 y Fd(There)h(is)f(a)g(whole)i Fc(S)531
5630 y Fe(1)585 5653 y Fd(of)e(equilibria)k(when)d(furthermore)h
Fc(\026)1478 5662 y Fe(3)1533 5653 y Fd(and)f Fc(`)1678
5662 y Fe(3)-107 5736 y Fd(v)n(anish.)1937 -90 y Fz(While)h(the)f
(periodic)f(orbits)h(survi)n(v)o(e)f(a)i(small)g(perturbation)1901
9 y(by)f(means)f(of)h(the)g(implicit)g(mapping)e(theorem,)g(the)i(tw)o
(o)h(bi-)1901 109 y(furcations)i(serv)o(e)h(as)h(or)o(ganizing)21
b(centres)k(for)g(the)g(Cantori-)1901 209 y(\002cation)31
b(of)f(the)h(f)o(amily)f(of)h(in)m(v)n(ariant)e Fu(2)p
Fz(\226tori,)j(see)g([P)o(acha,)1901 308 y(2002].)114
b(Furthermore,)56 b(the)51 b(monodromy)c(around)h(the)1901
408 y(pinched)40 b Fu(3)p Fz(\226tori)g(ensures)g(that)i(the)f(K)m
(olmogoro)o(v)d(condi-)1901 508 y(tion)20 b(\(1\))g(is)h(satis\002ed)g
(almost)f(e)n(v)o(erywhere.)1937 617 y(Let)g(us)g(discuss)g(tw)o(o)f
(procedures)f(to)h(generalize)f(this)i(result)1901 717
y(to)26 b(more)f(de)o(grees)g(of)h(freedom.)40 b(A)26
b(weak)g(coupling)e(with)i(a)1901 817 y(quasi\226peirodic)14
b(oscillator)j(of)f Ft(n)h Fz(frequencies)f(is)h(considered)1901
916 y(in)h([Hoo,)g(2005;)f(Broer)m(,)h(Han\337mann)e(and)h(Hoo,)h
(2004].)k(This)1901 1016 y(turns)k(the)h(Lagrangean)e(tori)h(into)h
Fu(\()p Ft(n)c Fu(+)g(3\))p Fz(\226tori,)28 b(the)f(ellip-)1901
1115 y(tic)k(tori)g(into)f Fu(\()p Ft(n)d Fu(+)e(2\))p
Fz(\226tori)30 b(and)g(the)h(periodic)e(orbits)i(into)1901
1215 y Fu(\()p Ft(n)c Fu(+)g(1\))p Fz(\226tori.)59 b(Again)32
b(the)f(quasi\226periodic)f(Hamiltonian)1901 1315 y(Hopf)g
(bifurcations)g(serv)o(e)g(as)i(or)o(ganizing)c(centres)j(for)f(the)
1901 1414 y(Cantori\002cation)22 b(of)g(the)h(rami\002ed)f(torus)h(b)n
(undle,)f(see)i([Hoo,)1901 1514 y(2005;)31 b(Broer)m(,)f(Han\337mann)c
(and)i(Hoo,)i(2004].)48 b(In)29 b(particu-)1901 1614
y(lar)e(the)g Fu(\()p Ft(n)d Fu(+)f(1\))p Fz(\226parameter)h(f)o
(amilies)k(of)e(elliptic)h(and)g(hy-)1901 1713 y(perbolic)36
b(tori)h(persist)h(on)f(Cantor)g(sets,)43 b(as)38 b(does)f(the)h
Ft(n)p Fz(\226)1901 1813 y(parameter)19 b(f)o(amily)g(of)h
Fu(\()p Ft(n)f Fu(+)f(1\))p Fz(\226tori)h(in)i Fu(1)p
Fz(:)p Fs(\000)p Fu(1)e Fz(resonance.)1937 1923 y(One)24
b(may)f(also)h(weakly)f(couple)g(tw)o(o)h(\(or)f(e)n(v)o(en)g(more\))g
(La-)1901 2022 y(grange)32 b(tops.)64 b(Where)33 b Fu(3)p
Fz(\226periodic)d(motion)i(of)h(one)g(body)1901 2122
y(is)g(superposed)d(with)j(the)f(rami\002ed)g(torus)f(b)n(undle)h
(de\002ned)1901 2221 y(by)i(the)g(other)g(body)e(this)j(resembles)f
(the)g(weak)g(coupling)1901 2321 y(with)17 b(a)g Fu(3)p
Fz(\226dimensional)d(oscillator)i(and)g(the)h(results)g(of)g([Hoo,)1901
2421 y(2005;)f(Broer)m(,)g(Han\337mann)d(and)j(Hoo,)g(2004])e(still)j
(apply)-5 b(.)22 b(Su-)1901 2520 y(perposition)30 b(with)i(elliptic)h
Fu(2)p Fz(\226periodic)c(motion)i(yields)h Fu(5)p Fz(\226)1901
2620 y(tori,)i Fu(4)p Fz(\226tori)d(and)g Fu(3)p Fz(\226tori.)58
b(Persistence)31 b(of)h(the)f(elliptic)h Fu(5)p Fz(\226)1901
2720 y(tori)24 b(and)f Fu(4)p Fz(\226tori)g(follo)n(ws)h(from)f
([Huitema,)g(1988;)i(P)7 b(\250)-35 b(oschel,)1901 2819
y(1989;)36 b(Broer)m(,)d(Huitema)e(and)g(T)-7 b(ak)o(ens,)34
b(1990;)h(R)7 b(\250)-35 b(ussmann,)1901 2919 y(2001],)23
b(and)g(the)h(same)g(holds)f(true)g(for)h(the)f(elliptic)h(and)g(hy-)
1901 3018 y(perbolic)33 b Fu(3)p Fz(\226tori.)68 b(T)-7
b(o)35 b(pro)o(v)o(e)d(persistence)j(of)f(a)h(normally)1901
3118 y(elliptic)i(quasi\226periodic)d(Hamiltonian)i(Hopf)g(bifurcation)
1901 3218 y(one)20 b(w)o(ould)h(ha)n(v)o(e)f(to)h(drag)f(the)h
(Diophantine)e(conditions)g(\(5\))1901 3317 y(through)g(the)i
(computations)e(in)j([Hoo,)e(2005;)g(Broer)m(,)g(Han\337-)1901
3417 y(mann)f(and)h(Hoo,)f(2004].)1937 3527 y(The)j(abo)o(v)o(e)f
(applies)i(mutatis)f(mutandi)g(for)f(the)i(superposi-)1901
3626 y(tion)f(with)g(elliptic)g(or)g(hyperbolic)d(rotations)i(about)g
(the)h(v)o(er)n(-)1901 3726 y(tical)i(\002gure)g(axis.)36
b(The)23 b(coupling)f(of)i(tw)o(o)g(periodic)f(Hamil-)1901
3826 y(tonian)18 b(Hopf)g(bifurcations)f(is)j(a)g(much)e(more)g(dif)n
(\002cult)g(prob-)1901 3925 y(lem.)43 b(Still,)28 b(one)e(may)f(tak)o
(e)i([Hoo,)f(2005;)i(Broer)m(,)e(Hoo)g(and)1901 4025
y(Naudot,)d(2004])f(as)i(a)g(starting)f(point)g(where)g(persistence)g
(of)1901 4125 y(the)17 b(resulting)f Fu(2)p Fz(\226tori)g(in)h(normal)e
Fu(1)p Fz(:)p Fu(1)p Fz(:)p Fs(\000)p Fu(1)p Fz(:)p Fs(\000)p
Fu(1)g Fz(resonance)g(has)1901 4224 y(been)20 b(pro)o(v)o(en.)1901
4530 y FC(3)83 b(P)n(erturbations)19 b(of)h(superintegrable)g(systems)
1937 4640 y Fz(W)-7 b(e)20 b(speak)f(of)g(a)h(superinte)o(grable)c
(system)k(if)f(the)g(re)o(gular)f(\002-)1901 4740 y(bres)29
b(of)g(the)g(rami\002ed)f(torus)h(b)n(undle)f(are)h(isotropic)f(tori)h
(of)1901 4839 y(dimension)c Ft(<)34 b(d)14 b Fu(\(=)2572
4807 y Fv(1)p 2572 4821 34 4 v 2572 4868 a(2)2629 4839
y Fu(dim)g Fs(P)7 b Fu(\))27 b Fz(.)44 b(Determined)25
b(by)h(the)h(di-)1901 4939 y(mension)f(of)h(these)h(\223maximal\224)e
(tori)h(this)h(de\002nes)f(a)h(whole)1901 5039 y(hierarchy)-5
b(,)26 b(starting)h(with)g(the)g(minimally)f(superinte)o(grable)1901
5138 y(systems)e(where)g(the)g(re)o(gular)e(\002bres)i(are)g
Fu(\()p Ft(d)e Fs(\000)e Fu(1\))p Fz(\226tori)j(\(and)1901
5238 y(almost)18 b(all)g(of)g(them)g(ha)n(v)o(e)f(dense)g
(quasi\226periodic)f(orbits\))h(up)1901 5338 y(to)i(maximally)f
(superinte)o(grable)f(systems)i(where)g(almost)g(all)1901
5437 y(orbits)30 b(are)g(periodic.)53 b(In)29 b(the)h(case)h(of)f
Ft(d)41 b Fu(=)g(2)30 b Fz(de)o(grees)f(of)1901 5537
y(freedom)21 b(all)i(these)f(notions)g(coincide.)30 b(According)21
b(to)h(The-)1901 5636 y(orem)i(1.1)h(the)g(Hamiltonian)f(of)h(a)h
(superinte)o(grable)c(system)1901 5736 y(only)c(depends)f(on)i(the)f
Ft(r)23 b Fz(actions)18 b(conjugate)f(to)i(the)f(toral)h(an-)
1800 6037 y(5)37 b()p eop end
%%Page: 6 6
TeXDict begin 6 5 bop -107 -90 a Fz(gles,)21 b(so)h(the)f
Fu(2\()p Ft(d)d Fs(\000)h Ft(r)r Fu(\))k Fz(\223e)o(xtra)d(inte)o
(grals\224)g(are)g(mute)h(param-)-107 9 y(eters)h(and)e(a)h(f)o(amily)g
(of)g Ft(n)p Fz(\226tori)f(still)i(encounters)d(only)i(bifur)n(-)-107
109 y(cations)g(up)f(to)h(co\226dimension)d Ft(n)j Fz(\227)h(although)c
(these)j(are)g(no)-107 209 y(longer)e(isolated)h(b)n(ut)g(form)f
Fu(2\()p Ft(d)g Fs(\000)f Ft(r)r Fu(\))p Fz(\226parameter)h(f)o
(amilies.)-71 308 y(Again)i(we)h(w)o(ant)g(to)f(kno)n(w)g(what)h
(happens)e(to)i(the)f(rami\002ed)-107 408 y Ft(r)r Fz(\226torus)e(b)n
(undle)e(under)g(small)j(perturbations)c(of)i(the)h(Hamil-)-107
508 y(tonian.)k(The)18 b(strate)o(gy)f(is)h(to)g(\002nd)g(an)f
(\223intermediate\224)f(system)-107 607 y(that)k(is)i(also)e(inte)o
(grable,)e(b)n(ut)j(non\226de)o(generately)15 b(so.)-107
771 y FC(De\002nition)21 b(3.1.)40 b Fq(The)d(perturbation)f
Ft("P)50 b Fq(of)38 b(a)f(superinte-)-107 871 y(gr)o(able)28
b(Hamiltonian)f Ft(N)38 b Fr(remo)o(v)o(es)27 b(the)i(de)o(generac)o(y)
c Fq(if)30 b(the)-107 971 y(perturbed)18 b(Hamiltonian)f
Ft(H)30 b Fu(=)23 b Ft(N)f Fu(+)13 b Ft("P)31 b Fq(can)18
b(be)h(written)g(in)-107 1070 y(the)h(form)358 1314 y
Ft(H)76 b Fu(=)69 b Ft(N)50 b Fu(+)42 b Ft(")5 b(S)45
b Fu(+)c Ft(")1146 1280 y Fv(2)1183 1314 y Ft(R)-107
1559 y Fq(wher)m(e)24 b Ft(N)30 b Fu(+)20 b Ft("S)29
b Fq(is)24 b(a)g(non\226de)m(g)o(ener)o(ate)19 b(inte)m(gr)o(able)j
(Hamil-)-107 1658 y(tonian.)-107 1822 y Fz(Let)45 b Fu(\()p
Ft(x;)14 b(q)s(;)g(y)s(;)g(p)p Fu(\))46 b Fz(complete)e(the)h(action)g
(angle)f(v)n(ariables)-107 1922 y Fu(\()p Ft(x;)14 b(y)s
Fu(\))43 b Fz(of)e Ft(N)71 b Fu(=)62 b Ft(N)9 b Fu(\()p
Ft(y)s Fu(\))42 b Fz(to)g(action)e(angle)h(v)n(ariables)g(of)-107
2022 y Ft(N)9 b Fu(\()p Ft(y)s Fu(\))h(+)g Ft("S)5 b
Fu(\()p Ft(y)s(;)14 b(p)p Fu(\))j Fz(.)24 b(If)18 b Ft(N)27
b Fz(satis\002es)19 b(\223its\224)g(K)m(olmogoro)o(v)c(con-)-107
2121 y(dition)20 b Fu(det)14 b Ft(D)308 2091 y Fv(2)345
2121 y Ft(N)9 b Fu(\()p Ft(y)s Fu(\))23 b Fs(6)p Fu(=)g(0)d
Fz(for)g(almost)g(all)h Ft(y)i Fz(then)149 2531 y Fu(det)279
2314 y Fm(0)279 2460 y(B)279 2510 y(B)279 2563 y(@)460
2347 y Fp(@)499 2322 y Fi(2)531 2347 y Fp(S)p 460 2361
116 4 v 465 2410 a(@)t(p)538 2390 y Fi(2)538 2428 y(1)697
2379 y Fs(\001)14 b(\001)g(\001)915 2347 y Fp(@)954 2322
y Fi(2)986 2347 y Fp(S)p 828 2361 289 4 v 828 2408 a(@)t(p)901
2416 y Fi(1)934 2408 y Fp(@)t(p)1007 2417 y Fk(d)p Fj(\000)p
Fk(r)507 2487 y Fz(.)507 2520 y(.)507 2553 y(.)705 2495
y(.)735 2520 y(.)765 2545 y(.)962 2487 y(.)962 2520 y(.)962
2553 y(.)460 2635 y Fp(@)499 2610 y Fi(2)531 2635 y Fp(S)p
374 2649 V 374 2696 a(@)t(p)447 2704 y Fi(1)479 2696
y Fp(@)t(p)552 2705 y Fk(d)p Fj(\000)p Fk(r)697 2667
y Fs(\001)g(\001)g(\001)915 2635 y Fp(@)954 2610 y Fi(2)986
2635 y Fp(S)p 881 2649 183 4 v 881 2698 a(@)t(p)954 2678
y Fi(2)954 2719 y Fk(d)p Fj(\000)p Fk(r)1139 2314 y Fm(1)1139
2460 y(C)1139 2510 y(C)1139 2563 y(A)1281 2531 y Fs(6)p
Fu(=)68 b(0)-107 2935 y Fz(ensures)17 b(that)h(the)f(inte)o(grable)f
(Hamiltonian)g Ft(N)h Fu(+)8 b Ft("S)22 b Fz(is)c(non\226)-107
3035 y(de)o(generate.)23 b(Under)c(this)i(condition)d(most)j(in)m(v)n
(ariant)d(tori)385 3279 y Fw(T)440 3245 y Fp(r)518 3279
y Fs(\002)41 b Fw(T)679 3245 y Fp(d)p Fo(\000)p Fp(r)844
3279 y Fs(\002)h(f)p Fu(\()p Ft(y)s(;)14 b(p)p Fu(\))p
Fs(g)-107 3523 y Fz(persist)27 b(under)d(the)i(perturbation)e(of)i(the)
g(intermediate)e(sys-)-107 3623 y(tem)j(by)f Ft(")194
3593 y Fv(2)231 3623 y Ft(R)q Fu(\()p Ft(x;)14 b(q)s(;)g(y)s(;)g(p)p
Fu(\))28 b Fz(,)h(cf.)d([Arnol')l(d,)g(1963b].)43 b(If)27
b Ft(N)36 b Fz(is)-107 3722 y(iso\226ener)o(getically)26
b(non\226de)o(generate)f(in)30 b Ft(y)i Fz(then)d(this)g(holds)-107
3822 y(true)20 b(on)g(e)n(v)o(ery)f(ener)o(gy)f(shell.)-107
4054 y FC(3.1)82 b(Minimally)21 b(superintegrable)f(systems)-71
4153 y Fz(One)d(w)o(ay)g(to)g(put)g Ft("P)12 b Fu(\()p
Ft(x;)i(q)s(;)g(y)s(;)g(p)p Fu(\))j Fz(into)g(the)g(form)f
Ft("S)5 b Fu(\()p Ft(y)s(;)14 b(p)p Fu(\))6 b(+)-107
4253 y Ft(")-68 4223 y Fv(2)-31 4253 y Ft(R)q Fu(\()p
Ft(x;)14 b(q)s(;)g(y)s(;)g(p)p Fu(\))19 b Fz(is)g(to)g(compute)e(a)i
(normal)e(form)h(of)g Ft("P)30 b Fz(with)-107 4352 y(respect)21
b(to)g Ft(N)30 b Fz(.)e(This)21 b(results)h(in)f(an)g(intermediate)e
(Hamilto-)-107 4452 y(nian)80 4431 y Fu(\026)59 4452
y Ft(H)32 b Fu(=)26 b Ft(N)j Fu(+)19 b Ft(")488 4431
y Fu(\026)470 4452 y Ft(P)34 b Fz(where)801 4431 y Fu(\026)782
4452 y Ft(P)g Fz(is)23 b(the)f(a)n(v)o(erage)f(of)g Ft(P)35
b Fz(along)-107 4552 y(the)27 b(\002bres)f(of)h(the)f(rami\002ed)g
(torus)g(b)n(undle)g(de\002ned)f(by)h Ft(N)36 b Fz(.)-107
4651 y(On)31 b(the)g(re)o(gular)e(part)i(of)f(this)i(b)n(undle)d(this)j
(de\002nes)e(a)h Fw(T)1633 4621 y Fp(r)1671 4651 y Fz(\226)-107
4751 y(symmetry)d(and)g(re)o(gular)g(reduction)f(mak)o(es)1270
4730 y Fu(\026)1252 4751 y Ft(P)41 b Fz(a)29 b(Hamilto-)-107
4851 y(nian)20 b(in)g Ft(d)f Fs(\000)f Ft(r)23 b Fz(de)o(grees)c(of)h
(freedom.)-71 4950 y(In)j(the)h(minimally)e(superinte)o(grable)f(case)j
Ft(r)31 b Fu(=)e Ft(d)21 b Fs(\000)g Fu(1)i Fz(this)-107
5050 y(is)31 b(a)g(one\226de)o(gree\226of\226freed)o(om)24
b(system)30 b(and)g(al)o(w)o(ays)h(inte-)-107 5149 y(grable.)43
b(Furthermore,)26 b(it)h(is)g(generic)f(for)1200 5128
y Fu(\026)1182 5149 y Ft(P)38 b Fz(to)27 b(ha)n(v)o(e)f(non\226)-107
5249 y(tri)n(vial)33 b(dynamics)e(in)i(one)f(de)o(gree)g(of)g(freedom,)
i(so)f Ft(P)45 b Fz(re-)-107 5349 y(mo)o(v)o(es)28 b(the)h(de)o
(generac)o(y)d(with)j Ft(S)44 b Fu(=)1068 5328 y(\026)1049
5349 y Ft(P)e Fz(.)52 b(The)28 b(remainder)-107 5448
y(term)20 b Ft(")105 5418 y Fv(2)142 5448 y Ft(R)h Fz(is)h(gi)n(v)o(en)
d(by)383 5729 y Ft(R)70 b Fu(=)660 5673 y(1)p 660 5710
42 4 v 661 5786 a Ft(")725 5662 y Fm(\000)763 5729 y
Ft(P)31 b Fs(\016)18 b Fu(\011)41 b Fs(\000)1137 5708
y Fu(\026)1119 5729 y Ft(P)1184 5662 y Fm(\001)1901 -90
y Fz(where)25 b Fu(\011)h Fz(is)g(the)f(normalizing)f(transformation.)
38 b(Note)25 b(that)1901 9 y(the)41 b(dynamics)f(de\002ned)g(by)g
Ft(N)51 b Fz(is)42 b(f)o(ast)f(with)h(respect)e(to)1901
109 y(the)27 b Ft(")p Fz(\226slo)n(w)f(one\226de)o(gree\226of\226freed)
o(om)21 b(dynamics)26 b(de\002ned)1901 209 y(by)20 b
Ft(")2062 188 y Fu(\026)2044 209 y Ft(P)32 b Fz(.)1937
319 y(From)21 b(the)h(rami\002ed)f Fu(\()p Ft(d)g Fs(\000)e
Fu(1\))p Fz(\226torus)i(b)n(undle)f(and)i(the)g(one\226)1901
418 y(de)o(gree\226of\226freedom)36 b(dynamics)j(we)j(no)n(w)e
(construct)g(the)1901 518 y(rami\002ed)c Ft(d)p Fz(\226torus)g(b)n
(undle)f(de\002ned)h(by)g(the)h(intermediate)1901 618
y(system.)74 b(The)36 b(Lagrangean)e(tori)i(consist)h(of)f(the)g(re)o
(gular)1901 717 y Fu(\()p Ft(d)27 b Fs(\000)e Fu(1\))p
Fz(\226tori)30 b(superposed)e(with)j(the)f(slo)n(w)h(periodic)e
(one\226)1901 817 y(de)o(gree-of\226freedom)13 b(dynamics.)23
b(T)-7 b(o)18 b(obtain)f(singular)g(\002bres)1901 916
y(we)k(can)f(proceed)e(in)i(tw)o(o)h(dif)n(ferent)d(w)o(ays.)1937
1026 y(One)i(the)h(one)e(hand,)h(the)g(\(relati)n(v)o(e\))f(equilibria)
g(of)h(the)h(one\226)1901 1126 y(de)o(gree\226of\226freedom)12
b(system)18 b(lead)g(to)f(singular)g(\002bres.)24 b(This)1901
1226 y(is)19 b(already)d(true)h(for)g(the)h Fu(\()p Ft(d)9
b Fs(\000)g Fu(1\))p Fz(\226tori)16 b(of)i(the)f(\223f)o(ast\224)h
(rami\002ed)1901 1325 y(torus)28 b(b)n(undle)g(and)g(e)n(v)o(en)g(more)
f(so)i(for)f(its)i(singular)e(\002bres.)1901 1425 y(A)e(further)e
(hierarchical)g(structure)h(is)i(imposed)d(by)h(the)h(co\226)1901
1525 y(dimensions)31 b(of)h(the)g(v)n(arious)f(equilibria)g(of)h(the)g
Fu(\()p Ft(d)c Fs(\000)f Fu(1\))p Fz(\226)1901 1624 y(parameter)15
b(f)o(amily)h Ft(S)2534 1636 y Fp(y)2597 1624 y Fu(=)2703
1603 y(\026)2684 1624 y Ft(P)c Fu(\()p Ft(y)s(;)i(::)p
Fu(\))j Fz(,)h(starting)e(at)h(saddles)f(and)1901 1724
y(centres)26 b(of)h(co\226dimension)d(zero)i(and)g(generically)f
(ranging)1901 1823 y(to)20 b(bifurcations)f(up)h(to)g(co\226dimension)d
Ft(d)i Fs(\000)f Fu(1)i Fz(.)1937 1933 y(On)25 b(the)g(other)e(hand,)i
(superposing)e(singular)g(\002bres)i(of)f(the)1901 2033
y(\(f)o(ast\))19 b(rami\002ed)f Fu(\()p Ft(d)13 b Fs(\000)g
Fu(1\))p Fz(\226torus)18 b(b)n(undle)f(with)i(the)g(slo)n(w)g(peri-)
1901 2133 y(odic)h(one\226de)o(gree\226of\226freed)o(om)14
b(dynamics)19 b(leads)i(to)f(singu-)1901 2232 y(lar)f(\002bres)g(of)g
(the)g(rami\002ed)g Ft(d)p Fz(\226torus)f(b)n(undle)g(as)i(well.)25
b(In)19 b(this)1901 2332 y(w)o(ay)g(the)h(intermediate)d(system)j(has)f
(four)f(kinds)h(of)g(motion)f(:)1916 2540 y Fu(\()p Ft(i)p
Fu(\))42 b Fz(The)23 b(re)o(gular)f(\002bres)h(correspond)e(to)j
(conditionally)d(pe-)2051 2639 y(riodic)e(motions)f(with)h
Ft(d)d Fs(\000)f Fu(1)k Fz(f)o(ast)h(frequencies)d(and)i(one)2051
2739 y(slo)n(w)i(frequenc)o(y)-5 b(.)1888 2838 y Fu(\()p
Ft(ii)p Fu(\))41 b Fz(Singular)31 b(\002bres)h(with)g(periodic)f(slo)n
(w)h(motion)f(corre-)2051 2938 y(spond)22 b(on)g(resulting)g
Fu(\()p Ft(n)e Fu(+)g(1\))p Fz(\226tori,)i Fu(0)27 b
Fs(\024)g Ft(n)h Fs(\024)f Ft(d)21 b Fs(\000)e Fu(2)k
Fz(,)2051 3038 y(to)k(conditionally)d(periodic)h(motions)h(with)h
Ft(n)g Fz(f)o(ast)g(and)2051 3137 y(one)d(slo)n(w)g(frequencies.)35
b(The)24 b(symplectic)f(normal)g(be-)2051 3237 y(ha)n(viour)e(is)h(f)o
(ast)h(as)g(well,)f(this)h(is)g(in)f(particular)f(true)g(for)2051
3337 y(the)f(asymptotic)e(motion)g(on)h(e)o(xisting)g(\(un\)stable)f
(man-)2051 3436 y(ifolds.)1859 3536 y Fu(\()p Ft(iii)p
Fu(\))41 b Fz(Singular)24 b(\002bres)g(constructed)f(from)h(re)o(gular)
f Fu(\()p Ft(d)f Fs(\000)g Fu(1\))p Fz(\226)2051 3635
y(tori)35 b(ha)n(v)o(e)g(f)o(ast)h(conditionally)d(periodic)h(motion)g
(and)2051 3735 y(slo)n(w)29 b(symplectic)f(normal)f(beha)n(viour)-5
b(.)49 b(In)28 b(particular)m(,)2051 3835 y(the)d(motion)f(on)g(e)o
(xisting)g(\(un\)stable)g(manifolds)f(com-)2051 3934
y(bines)d(a)g(f)o(ast)h(rotational)d(motion)h(with)h(a)g(slo)n(w)h
(approxi-)2051 4034 y(mation)e(of)h(the)h(in)m(v)n(ariant)d
Fu(\()p Ft(d)h Fs(\000)f Fu(1\))p Fz(\226torus.)1873
4134 y Fu(\()p Ft(iv)s Fu(\))42 b Fz(The)70 b(superposition)f(of)h
(one\226de)o(gree\226of\226fre)o(edo)o(m)2051 4233 y(equilibria)29
b(and)g(singular)g(f)o(ast)h(\002bres)g(leads)g(to)g(condi-)2051
4333 y(tionally)g(periodic)f(motion)h(with)h Ft(n)g Fz(f)o(ast)h
(frequencies,)2051 4433 y Fu(0)27 b Fs(\024)g Ft(n)h
Fs(\024)f Ft(d)21 b Fs(\000)f Fu(2)i Fz(,)i(while)f(the)f(symplectic)g
(normal)g(be-)2051 4532 y(ha)n(viour)c(is)j(a)f(combination)d(of)i
Ft(d)d Fs(\000)g Ft(n)g Fs(\000)g Fu(1)j Fz(f)o(ast)i(de)o(grees)2051
4632 y(of)f(freedom)e(and)i(one)g(slo)n(w)g(de)o(gree)f(of)h(freedom.)
1901 4839 y(It)25 b(remains)f(to)g(understand)f(what)h(happens)f(to)i
(the)f(rami\002ed)1901 4939 y Ft(d)p Fz(\226torus)29
b(b)n(undle)f(de\002ned)h(by)2845 4918 y Fu(\026)2824
4939 y Ft(H)47 b Fu(=)39 b Ft(N)34 b Fu(+)25 b Ft(")3293
4918 y Fu(\026)3274 4939 y Ft(P)42 b Fz(under)28 b(per)n(-)1901
5039 y(turbation)i(by)i Ft(")2388 5009 y Fv(2)2425 5039
y Ft(R)h Fz(.)62 b(F)o(or)31 b(the)h(re)o(gular)f(\002bres)h
Fu(\()p Ft(i)p Fu(\))h Fz(the)f(re-)1901 5138 y(sult)g(in)f([Arnol')l
(d,)h(1963b])d(yields)i(persistence)g(of)g(a)h(Can-)1901
5238 y(tor)23 b(f)o(amily)g(of)g(Lagrangean)e(tori.)34
b(The)23 b(proof)f(relies)i(on)f(ini-)1901 5338 y(tial)29
b(normalizing)e(transformations,)h(using)h(the)f(ultra)n(violet)1901
5437 y(cut\226of)n(f)33 b(introduced)f(in)i([Arnol')l(d,)i(1963a].)66
b(The)34 b(lo)n(wer)n(\226)1901 5537 y(dimensional)24
b(tori)h Fu(\()p Ft(ii)p Fu(\))h Fz(also)g(ha)n(v)o(e)f(the)g(slo)n(w)h
(dynamics)e(en-)1901 5636 y(coded)17 b(in)h(one)f(of)g(the)h(internal)f
(frequencies,)f(the)i(symplectic)1901 5736 y(normal)k(beha)n(viour)g
(is)i(of)f(the)h(same)g(magnitude)d(as)j(the)g(f)o(ast)
1800 6037 y(6)37 b()p eop end
%%Page: 7 7
TeXDict begin 7 6 bop -107 -90 a Fz(frequencies.)46 b(This)28
b(should)f(allo)n(w)g(to)h(obtain)f(their)h(persis-)-107
9 y(tence)20 b(along)f(the)i(same)f(lines.)-71 130 y(F)o(or)26
b(the)h Fu(\()p Ft(d)d Fs(\000)e Fu(1\))p Fz(\226tori)k
Fu(\()p Ft(iii)p Fu(\))g Fz(the)h(tw)o(o)g(time)g(scales)g(distin-)-107
229 y(guish)j(the)h(internal)f(from)g(the)g(normal)g(dynamics.)55
b(In)31 b(the)-107 329 y(hyperbolic)f(case)i(the)g(normal)f
(hyperbolicity)e(of)j(the)g(cen-)-107 429 y(tre)h(manifold)e(is)j(of)e
(order)f Ft(")i Fz(and)f(thus)h(suf)n(\002ciently)f(lar)o(ge)-107
528 y(with)k(respect)f(to)h(the)g(perturbation)d(strength)i
Ft(")1372 498 y Fv(2)1445 528 y Fz(to)h(yield)-107 628
y(persistence.)43 b(During)25 b(local)h(bifurcations)e(in)j(the)f(slo)n
(w)g(dy-)-107 727 y(namics)c(one)g(has)g(the)g(alternati)n(v)o(e)f
(between)h(a)g(scaling)g(ar)o(gu-)-107 827 y(ment)15
b([Broer)m(,)h(Han\337mann)d(and)j(Y)-9 b(ou,)15 b(2003])f(and)h(a)i
(direct)e(in-)-107 927 y(corporation)23 b(in)i(the)h(KAM\226iteration)e
([Han\337mann,)g(1998].)-107 1026 y(In)16 b(the)h(elliptic)f(case)h
(\(treated)f(in)h([Lieberman,)d(1971;)j(1972]\))-107
1126 y(the)30 b(Diophantine)d(conditions)h(again)g(in)m(v)n(olv)o(e)g
(both)g(the)i Ft(")p Fz(\226)-107 1226 y(small)21 b(normal)e(and)g(the)
i(\223lar)o(ge\224)e(internal)g(frequencies.)-71 1346
y(The)24 b(f)o(ast\226slo)n(w)g(dynamics)e(is)j(contained)e(in)h(the)g
(symplec-)-107 1446 y(tic)k(normal)f(beha)n(viour)e(for)i(lo)n(wer)n
(\226dimensional)e(tori)i Fu(\()p Ft(iv)s Fu(\))p Fz(.)-107
1545 y(Again)e(the)g(hypo\226elliptic)e(case)j(does)f(not)g(pose)g(ne)n
(w)g(prob-)-107 1645 y(lems.)53 b(When)29 b(incorporating)d(additional)
i(elliptic)i(and)f(hy-)-107 1744 y(perbolic)24 b(directions)f(into)i
(the)f(KAM)h(iteration)f(one)h(should)-107 1844 y(furthermore)13
b(be)k(able)f(to)g(let)h(one)e(of)h(them)g(be)g(slo)n(w)g(\227)h(where)
-107 1944 y(the)28 b(bifurcation)d(scenario)i(is)h(de)n(v)o(eloping)d
(in)i(the)h(slo)n(w)f(dy-)-107 2043 y(namics)17 b(the)g(abo)o(v)o(e)f
(alternati)n(v)o(es)g(still)i(apply)-5 b(.)23 b(More)16
b(interest-)-107 2143 y(ing)26 b(is)g(the)g(combination)e(of)h(tw)o(o)i
(bifurcations)d(in)i(both)f(the)-107 2243 y(f)o(ast)31
b(and)e(the)h(slo)n(w)g(\(symplectic)f(normal\))f(dynamics.)53
b(In-)-107 2342 y(deed,)25 b(with)f(tw)o(o)h(dif)n(ferent)e(time)h
(scales)i(e.g.)d(the)i(dynamics)-107 2442 y(triggered)16
b(by)h(tw)o(o)h(simultaneous)e(violations)h(of)h(\(5\))f(appears)-107
2541 y(to)i(be)g(of)g Fu(\(1)13 b(+)g(1\))p Fz(\226de)o
(gree\226of\226freedo)o(m)g(rather)18 b(than)h(ha)n(ving)-107
2641 y(truly)25 b Fu(2)g Fz(de)o(grees)f(of)g(freedom.)38
b(This)25 b(might)g(help)f(to)i(obtain)-107 2741 y(more)20
b(detailed)f(results.)-107 3035 y FC(3.2)82 b(Systems)21
b(in)g(thr)o(ee)e(degr)o(ees)h(of)g(fr)o(eedom)-71 3155
y Fz(F)o(or)28 b(superinte)o(grable)c(systems)29 b(with)e(re)o(gular)g
Ft(r)r Fz(\226tori)g(with)-107 3255 y Ft(r)45 b Fs(\024)e
Ft(d)26 b Fs(\000)g Fu(2)31 b Fz(it)g(is)h(no)e(longer)f(automatic)h
(for)g(the)h(a)n(v)o(erage)-107 3355 y Ft(")-50 3334
y Fu(\026)-68 3355 y Ft(P)i Fz(of)21 b(the)g(perturbation)d
Ft("P)33 b Fz(to)21 b(reduce)f(to)h(an)g Ft(r)r Fz(\226parameter)-107
3454 y(f)o(amily)h(of)f(inte)o(grable)g(systems,)h(the)g(remaining)f
(number)f(of)-107 3554 y(de)o(grees)j(of)h(freedom)f(being)g
Ft(d)e Fs(\000)g Ft(r)34 b Fs(\025)c Fu(2)24 b Fz(.)37
b(It)25 b(seems)f(there-)-107 3654 y(fore)d(unlik)o(ely)g(that)h(a)g
(gi)n(v)o(en)f(perturbation)e(remo)o(v)o(es)h(the)i(de-)-107
3753 y(generac)o(y)-5 b(,)28 b(b)n(ut)g(see)h([Arnol')l(d,)g(1963b])d
(for)i(a)g(treatment)g(of)-107 3853 y(the)20 b(planetary)e(system)i(as)
h(a)f(perturbation)e(of)h(the)h(superinte-)-107 3952
y(grable)26 b(superposition)e(of)i Fu(9)g Fz(K)n(eplerian)g(systems)g
(that)h(does)-107 4052 y(remo)o(v)o(e)22 b(the)h(de)o(generac)o(y)-5
b(.)32 b(The)23 b(ensuing)g(problems)f(can)h(al-)-107
4152 y(ready)c(be)h(illustrated)g(in)h(three)e(de)o(grees)g(of)h
(freedom.)-71 4272 y(The)28 b(Euler)n(\226Poinsot)f(system)h(is)h(a)g
(\223free\224)e(rigid)h(body)f(not)-107 4372 y(subject)21
b(to)g(an)o(y)f(e)o(xternal)g(force)g(or)h(torque.)26
b(This)c(mak)o(es)f(the)-107 4471 y(spatial)28 b(components)e
Ft(\026)618 4483 y Fv(1)655 4471 y Ft(;)14 b(\026)742
4483 y Fv(2)779 4471 y Ft(;)g(\026)866 4483 y Fv(3)931
4471 y Fz(of)28 b(the)f(angular)f(momen-)-107 4571 y(tum)g(three)f
(non\226commuting)c(inte)o(grals)k(of)h(motion)e(ne)o(xt)h(to)-107
4671 y(the)20 b(\(kinetic\))f(ener)o(gy)313 5001 y Ft(N)78
b Fu(=)623 4945 y Ft(`)658 4957 y Fv(1)p 602 4982 116
4 v 602 5058 a Fu(2)p Ft(I)680 5070 y Fv(1)768 5001 y
Fu(+)906 4945 y Ft(`)941 4957 y Fv(2)p 884 4982 V 884
5058 a Fu(2)p Ft(I)962 5070 y Fv(2)1051 5001 y Fu(+)1189
4945 y Ft(`)1224 4957 y Fv(3)p 1167 4982 V 1167 5058
a Fu(2)p Ft(I)1245 5070 y Fv(3)-107 5338 y Fz(and)19
b(replacing)e(one)h(of)h(them)f(by)h(the)g(sum)32 b Ft(\026)1219
5307 y Fv(2)1280 5338 y Fu(=)22 b Ft(\026)1417 5307 y
Fv(2)1417 5358 y(1)1468 5338 y Fu(+)14 b Ft(\026)1597
5307 y Fv(2)1597 5358 y(2)1647 5338 y Fu(+)-107 5437
y Ft(\026)-57 5407 y Fv(2)-57 5458 y(3)27 5437 y Fz(of)32
b(their)g(squares)g(yields)g(a)h(second)e(inte)o(gral)g(\(ne)o(xt)g(to)
-107 5537 y(the)26 b(ener)o(gy\))d(that)i(commutes)g(with)g(all)h
(other)f(inte)o(grals.)40 b(In)-107 5636 y(this)19 b(w)o(ay)f(the)g
(phase)g(space)g Fs(P)26 b Fz(becomes)17 b(a)i(rami\002ed)e
Fu(2)p Fz(\226torus)-107 5736 y(b)n(undle)j(with)h(a)h(complicated)d
(singular)h(set)i(at)g Ft(\026)1329 5706 y Fv(2)1391
5736 y Fu(=)i(0)d Fz(.)28 b(Re-)1901 -90 y(de\002ning)2388
132 y Fs(P)75 b Fu(=)69 b Ft(T)2716 97 y Fo(\003)2753
132 y Ft(S)5 b(O)r Fu(\(3\))2980 100 y Fn(n)p Fb(S)t(O)r
Fa(\(3\))1901 339 y Fz(by)38 b(taking)f(out)h(the)h(zero)e(section)h
(\(where)g(no)g(dynamics)1901 439 y(tak)o(es)f(place,)i
Ft(S)5 b(O)r Fu(\(3\))38 b Fz(consists)e(of)g(equilibria\))f
(simpli\002es)1901 539 y(this)30 b(situation)f(and)g(also)g(allo)n(ws)h
(to)f(replace)g Ft(\026)3339 508 y Fv(2)3406 539 y Fz(by)g
Fs(j)p Ft(\026)p Fs(j)40 b Fu(=)1901 574 y Fm(p)p 1984
574 88 4 v 73 x Ft(\026)2034 623 y Fv(2)2092 647 y Fz(.)1937
747 y(In)30 b(the)h(dynamically)e(symmetric)h(case)h
Ft(I)3204 759 y Fv(1)3284 747 y Fu(=)42 b Ft(I)3427 759
y Fv(2)3496 747 y Fz(of)30 b(tw)o(o)1901 846 y(equal)39
b(moments)g(of)g(inertia)h(the)f(conditionally)f(periodic)1901
946 y(motion)30 b(along)h(the)g(re)o(gular)f Fu(2)p Fz(\226tori)h
(becomes)f(particularly)1901 1046 y(transparent.)40 b(Indeed,)26
b(for)f(such)g(an)h(Euler)f(top)g(the)h(preces-)1901
1145 y(sion)d(of)g(the)g(\002gure)g(axis)g(about)g(the)g(angular)f
(momentum)f(is)1901 1245 y(superposed)k(by)i(a)h(rotation)e(of)h(the)h
(body)e(about)g(the)h(\002gure)1901 1345 y(axis.)k(At)22
b Ft(`)2227 1357 y Fv(3)2290 1345 y Fu(=)k(0)c Fz(the)g
Fu(2)p Fz(\226tori)f(become)g(\(internally\))f Fu(1)p
Fz(:)p Fu(0)h Fz(res-)1901 1444 y(onant)30 b(as)j(the)e(precession)f
(consists)i(of)f(the)h(body)e(rotating)1901 1544 y(about)d(an)o(y)g
(axis)i(perpendicular)c(to)j(the)g(\002gure)g(axis.)49
b(Note)1901 1643 y(that)20 b(the)h(Hamiltonian)2333 1884
y Ft(N)78 b Fu(=)2622 1828 y Fs(j)p Ft(\026)p Fs(j)2718
1798 y Fv(2)p 2622 1865 134 4 v 2631 1941 a Fu(2)p Ft(I)2709
1953 y Fv(1)2807 1884 y Fs(\000)2923 1828 y Ft(I)2959
1840 y Fv(3)3015 1828 y Fs(\000)18 b Ft(I)3134 1840 y
Fv(1)p 2923 1865 249 4 v 2974 1941 a Ft(I)3010 1953 y
Fv(1)3047 1941 y Ft(I)3083 1953 y Fv(3)3206 1828 y Ft(`)3241
1798 y Fv(2)3241 1849 y(3)p 3206 1865 72 4 v 3221 1941
a Fu(2)1901 2132 y Fz(can)30 b(be)g(e)o(xpressed)f(as)h(a)h(function)d
(of)i Fs(j)p Ft(\026)p Fs(j)g Fz(and)g Ft(`)3399 2144
y Fv(3)3466 2132 y Fz(whence)1901 2232 y(the)36 b(additional)f(inte)o
(gral)h Ft(`)2733 2244 y Fv(3)2807 2232 y Fz(does)g(not)g(lead)g(to)g
(topologi-)1901 2331 y(cal)h(changes)f(of)g(the)h(re)o(gular)e
(\002bres)i(of)f(the)h(rami\002ed)f Fu(2)p Fz(\226)1901
2431 y(torus)21 b(b)n(undle.)26 b(This)21 b(will)h(change)d(belo)n(w)i
(when)f(we)h(replace)1901 2531 y(the)26 b(additional)e
Ft(S)2443 2501 y Fv(1)2480 2531 y Fz(\226symmetry)f(by)i(an)h
(additional)e Ft(S)5 b(O)r Fu(\(3\))p Fz(\226)1901 2630
y(symmetry)-5 b(.)1937 2730 y(F)o(or)33 b(the)h(general)f(free)g(rigid)
g(body)f(with)i(three)f(dif)n(ferent)1901 2830 y(moments)19
b(of)h(inertia)g(the)g(abo)o(v)o(e)f Fu(1)p Fz(:)p Fu(0)g
Fz(resonant)g Fu(2)p Fz(\226tori)h(break)1901 2929 y(up)29
b(and)f(the)h(\(un\)stable)f(manifolds)g(of)g(the)i(rotation)d(about)
1901 3029 y(the)k(\223middle\224)f(axis)h(of)g(inertia)f(separate)h
(four)f(f)o(amilies)h(of)1901 3128 y(re)o(gular)26 b
Fu(2)p Fz(\226tori.)45 b(Depending)25 b(on)i(which)f(f)o(amily)h(a)h
Fu(2)p Fz(\226torus)1901 3228 y(belongs)37 b(to,)42 b(the)37
b(r)7 b(\210)-35 b(ole)38 b(of)f(the)h(\002gure)f(axis)h(is)g(played)f
(by)1901 3328 y(the)32 b(\223longest\224)f(or)h(the)g(\223shortest\224)
f(axis)h(of)g(inertia,)i(which)1901 3427 y(still)29 b(precesses)e(re)o
(gularly)f(about)g(the)i(direction)e(of)h(the)h(an-)1901
3527 y(gular)h(momentum.)52 b(Ho)n(we)n(v)o(er)m(,)30
b(the)g(rotational)e(motion)h(of)1901 3627 y(the)20 b(body)e(about)h
(this)i(\002gure)e(axis)h(looks)f(more)g(complicated)1901
3726 y(as)k(the)g(sines)h(and)e(cosines)g(of)h(the)g(dynamically)d
(symmetric)1901 3826 y(case)h(ha)n(v)o(e)f(to)h(be)f(replaced)g(by)g
(elliptic)h(functions.)k(The)20 b(free)1901 3926 y(rigid)g(body)e(is)k
(a)e(minimally)f(superinte)o(grable)f(system.)1937 4025
y(The)g(torque)f(e)o(x)o(erted)f(by)i(a)h(perturbing)c(e)o(xternal)i
(force)g(\002eld)1901 4125 y(causes)g(the)g(angular)e(momentum)f(to)j
(slo)n(wly)g(mo)o(v)o(e)e(in)i(space.)1901 4224 y(F)o(or)i(the)f
(intermediate)g(system)h(this)g(motion)f(is)i(periodic)d(and)1901
4324 y(superposed)29 b(to)h(the)h(f)o(ast)g(precessional\226rotational)
c(motion)1901 4424 y(of)18 b(the)h(free)f(rigid)g(body)-5
b(.)23 b(In)18 b([Mazzocco,)f(1997])g(persistence)1901
4523 y(of)25 b(the)h(resulting)f Fu(3)p Fz(\226tori)f(is)j(e)o
(xplicitly)d(pro)o(v)o(en.)39 b(The)25 b(struc-)1901
4623 y(ture)17 b(of)g(the)h(rami\002ed)e Fu(3)p Fz(\226torus)h(b)n
(undle)f(de\002ned)g(by)h(the)h(inter)n(-)1901 4723 y(mediate)g(system)
g(depends)f(on)h(the)g(precise)g(form)g(of)g(\(the)g(a)n(v-)1901
4822 y(erage)j(of\))f(the)i(perturbation.)j(The)c(case)h(of)f(an)g(af)n
(\002ne)3495 4792 y Ff(2)3549 4822 y Fz(force)1901 4922
y(\002eld)f(is)h(detailed)e(in)h([Han\337mann,)d(1995;)i(1997])f(for)h
(the)h(dy-)1901 5021 y(namically)f(symmetric)g(case.)1937
5121 y(A)d(free)f(rigid)g(body)f(with)h(three)g(equal)g(moments)f
Ft(I)3410 5133 y Fv(1)3471 5121 y Fu(=)23 b Ft(I)3595
5133 y Fv(2)3655 5121 y Fu(=)1901 5221 y Ft(I)1937 5233
y Fv(3)1993 5221 y Fz(of)16 b(inertia)h(has)h(only)e(periodic)g
(motions)g(\(we)i(still)g(e)o(xclude)1901 5320 y(the)h(zero)f(section)g
Ft(S)5 b(O)r Fu(\(3\))20 b Fz(from)e(the)g(phase)h(space\))f(since)h(e)
n(v-)1901 5420 y(ery)28 b(axis)i(through)c(the)j(\002x)o(ed)f(point)g
(is)i(a)f(principal)f(axis)h(of)p 1901 5598 300 3 v 1986
5713 a Fe(2)2021 5736 y Fd(The)16 b(linear)h(part)g(is)f(needed)i(to)e
(break)h(the)g(rotational)i(symmetry)e(of)f(the)1901
5819 y(constant)j(part)f(of)f(the)h(force)g(\002eld.)
1800 6037 y(7)37 b()p eop end
%%Page: 8 8
TeXDict begin 8 7 bop -107 -90 a Fz(inertia.)37 b(Correspondingly)-5
b(,)21 b(one)j(has)g(\002)n(v)o(e)g(independent)e(in-)-107
9 y(te)o(grals)e(of)g(motion)f(by)h(choosing)e(ne)o(xt)i(to)g(the)g
(ener)o(gy)587 316 y Ft(N)78 b Fu(=)875 260 y Fs(j)p
Ft(\026)p Fs(j)971 230 y Fv(2)p 875 297 134 4 v 884 373
a Fu(2)p Ft(I)962 385 y Fv(1)-107 608 y Fz(tw)o(o)31
b(of)f(the)h(three)f(components)e Ft(`)945 620 y Fv(1)982
608 y Ft(;)14 b(`)1054 620 y Fv(2)1091 608 y Ft(;)g(`)1163
620 y Fv(3)1230 608 y Fz(of)31 b(the)f(angular)-107 708
y(momentum)18 b(about)h(a)h(body)f(set)h(of)g(ax)o(es)g(and)f(tw)o(o)h
(out)g(of)g(the)-107 807 y Ft(\026)-57 819 y Fv(1)-20
807 y Ft(;)14 b(\026)67 819 y Fv(2)105 807 y Ft(;)g(\026)192
819 y Fv(3)252 807 y Fz(in)23 b(a)h(spatial)f(frame.)32
b(The)23 b(free)g(rigid)f(body)f(with)-107 907 y(tri)n(vial)k(tensor)g
(of)g(inertia)g(is)h(a)g(maximally)d(superinte)o(grable)-107
1006 y(system.)-71 1110 y(The)28 b(ef)n(fect)f(of)h(the)g(torque)e(of)i
(the)g(a)n(v)o(erage)f(of)g(a)i(perturb-)-107 1209 y(ing)23
b(e)o(xternal)g(force)f(\002eld)i(is)g(no)n(w)f(that)h(the)f(direction)
f(of)i(the)-107 1309 y(angular)i(momentum)g(mo)o(v)o(es)g(both)h(in)h
(the)g(spatial)g(and)f(the)-107 1408 y(body)d(frame.)41
b(Fixing)25 b Fs(j)p Ft(\026)p Fs(j)h Fz(,)h(re)o(gular)d(reduction)g
(of)h(the)h Ft(S)1634 1378 y Fv(1)1671 1408 y Fz(\226)-107
1508 y(symmetry)34 b(generated)f(by)h Fs(j)p Ft(\026)p
Fs(j)i Fz(yields)e(a)i(tw)o(o\226de)o(gree\226of\226)-107
1608 y(freedom)25 b(system)i(on)f Ft(S)621 1578 y Fv(2)616
1634 y Fo(j)p Fp(\026)p Fo(j)724 1608 y Fs(\002)c Ft(S)867
1578 y Fv(2)862 1634 y Fo(j)p Fp(\026)p Fo(j)974 1608
y Fz(.)45 b(This)27 b(system)g(may)f(of)-107 1725 y(course)19
b(be)g(inte)o(grable,)f(e.g.)h(because)g(the)g(force)g(\002eld)g(is)i
Ft(S)1634 1695 y Fv(1)1671 1725 y Fz(\226)-107 1825 y(symmetric.)63
b(Note)33 b(that)g(the)h(e)o(xternal)e(force)g(\002eld)h(has)g(to)-107
1924 y(\223detect\224)d(the)g(asymetries)g(of)g(the)h(rigid)e(body)-5
b(,)31 b(whence)f(an)-107 2024 y(af)n(\002ne)19 b(force)g(\002eld)g(is)
i(no)e(longer)f(suf)n(\002ciently)h(general,)f(lead-)-107
2123 y(ing)k(to)g(an)g Ft(S)266 2093 y Fv(1)303 2123
y Fz(\226symmetric)e(system.)30 b(But)23 b(already)d(a)j(generic)-107
2223 y(quadratic)17 b(force)h(\002eld)g(has)h(an)f(a)n(v)o(erage)g
(that)g(cannot)f(be)i(used)-107 2323 y(to)i(remo)o(v)o(e)d(the)i(de)o
(generac)o(y)-5 b(.)-71 2426 y(An)23 b(interesting)f(phenomenon)e
(appears)i(in)h(some)g(applica-)-107 2525 y(tions)e(in)m(v)n(olving)e
(the)j(K)n(epler)e(system.)28 b(Indeed,)20 b(the)h(re)o(gular)n(-)-107
2625 y(ized)26 b(spatial)g(K)n(epler)f(problem)f(is)i(a)h(maximally)d
(superinte-)-107 2725 y(grable)18 b(three\226de)o(gree\226of\226freed)o
(om)12 b(system)19 b(with)g(Hamilto-)-107 2824 y(nian)h
Ft(N)9 b Fu(\()p Ft(K)d Fu(\))23 b(=)f Ft(K)27 b Fz(and)19
b(has)i(a)f(\223\002rst\224)h(normal)e(form)409 3060
y Fu(\026)387 3081 y Ft(H)76 b Fu(=)69 b Ft(K)46 b Fu(+)c
Ft(")5 b(S)g Fu(\()p Ft(K)q(;)14 b(L)p Fu(\))-107 3337
y Fz(for)19 b(the)h(lunar)f(problem,)f(the)i(Rydber)o(g)e(\(or)h
(hydrogen\))d(atom)-107 3436 y(in)42 b(crossed)f(\002elds)h(and)f(the)g
(problem)f(of)h(orbiting)f(dust,)-107 3536 y(cf.)32 b([v)n(an)f(der)h
(Meer)g(and)f(Cushman,)j(1986;)j(1987;)g(Cush-)-107 3636
y(man,)32 b(1992;)h(Cushman)c(and)h(Sado)o(vski)n(\020)-26
b(\021,)31 b(2000;)i(Sommer)m(,)-107 3735 y(2003;)g(Efstathiou,)d
(2004].)50 b(Here)29 b Ft(L)g Fz(is)i(the)e(third)f(compo-)-107
3835 y(nent)e(of)g(the)h(angular)e(momentum)f(and)i(in)g(the)h(tw)o(o)f
(former)-107 3934 y(cases)f Ft(S)5 b Fu(\()p Ft(K)q(;)14
b(L)p Fu(\))25 b Fz(is)g(a)g(multiple)f(of)g Ft(K)j Fs(\001)22
b Ft(L)i Fz(,)i(while)e(it)h(can)g(be)-107 4034 y(brought)c(into)j
(this)f(form)g(by)g(an)g(additional)f(transformation)-107
4134 y(in)16 b(the)g(latter)g(case)h(as)f(well.)24 b(A)17
b(second)e(normalization)f(yields)p 90 4323 76 4 v 90
4340 V 90 4406 a Ft(H)76 b Fu(=)69 b Ft(K)47 b Fu(+)41
b Ft(")5 b(S)g Fu(\()p Ft(K)q(;)14 b(L)p Fu(\))40 b(+)h
Ft(")1108 4372 y Fv(2)1145 4406 y Ft(T)12 b Fu(\()p Ft(K)q(;)i(L;)g(I)7
b Fu(\))-107 4663 y Fz(with)21 b(an)f(appropriately)d(chosen)i(third)h
(action)g Ft(I)27 b Fz(.)-71 4766 y(In)33 b(all)h(these)g(cases)g(the)g
(conditionally)d(periodic)h(motion)-107 4882 y(de\002ned)25
b(by)p 272 4799 V 272 4815 V 24 w Ft(H)33 b Fz(has)26
b(three)f(time)g(scales,)i(while)f(the)f(rates)h(of)-107
4982 y(change)c(of)g(these)h(frequencies)e(are)i(only)f(of)h(the)f(tw)o
(o)i(orders)-107 5081 y Ft(")e Fz(and)f Ft(")135 5051
y Fv(2)195 5081 y Fz(of)g(magnitude.)28 b(This)21 b(is)i(not)f(a)g
(coincidence)d(since)-107 5181 y(for)25 b(a)h(maximally)f(superinte)o
(grable)e(system)i(with)h(no)n(where)-107 5281 y(v)n(anishing)g
(periodic)g(\003o)n(w)i(the)f(\002rst)h(action)f(can)h(al)o(w)o(ays)g
(be)-107 5380 y(chosen)k(to)i(be)f(the)g(unperturbed)d(Hamiltonian.)63
b(While)33 b(a)-107 5480 y(perturbation)-107 5736 y Ft(H)-38
5748 y Fp(")-2 5736 y Fu(\()p Ft(J)o(;)14 b(\036)p Fu(\))70
b(=)e Ft(J)442 5748 y Fv(1)512 5736 y Fu(+)31 b Ft(")-5
b(S)5 b Fu(\()p Ft(J)776 5748 y Fv(1)813 5736 y Ft(;)14
b(J)896 5748 y Fv(2)933 5736 y Fu(\))33 b(+)e Ft(")1133
5702 y Fv(2)1170 5736 y Ft(T)12 b Fu(\()p Ft(J)o(;)i(")p
Fu(\))31 b(+)h Fr(h.o.t.)1901 -90 y Fz(does)d(not)g(ful\002l)g
(De\002nition)g(3.1,)i(it)f(could)e(nonetheless)h(be)1901
9 y(sho)n(wn)15 b(in)g([Sommer)m(,)g(2003])e(that)j(most)f(re)o(gular)f
(\002bres)h(of)g(the)1901 126 y(rami\002ed)g Fu(3)p Fz(\226torus)g(b)n
(undle)g(de\002ned)g(by)p 3061 42 V 3061 59 V 15 w Ft(H)24
b Fz(persists)16 b(as)h(a)g(Can-)1901 225 y(tor)30 b(f)o(amily)-5
b(,)32 b(pro)o(vided)c(that)i(appropriate)e(non\226de)o(generac)o(y)
1901 325 y(conditions)19 b(hold)g(true.)1901 557 y FC(3.3)82
b(The)21 b(hierar)o(ch)o(y)e(of)h(superintegrable)g(systems)1937
657 y Fz(One)26 b(still)i(obtains)d(a)i(minimally)e(superinte)o(grable)
e(system)1901 756 y(when)32 b(coupling)f(an)i(Euler)f(top)h(with)g(one)
f(or)h(se)n(v)o(eral)f(La-)1901 856 y(grange)c(tops.)52
b(Note)29 b(that)g(the)h(Euler)n(\226Poinsot)d(system)j(can)1901
956 y(be)22 b(realized)g(e)n(v)o(en)f(in)i(the)f(presence)f(of)h
(constant)g(gra)n(vity)f(by)1901 1055 y(letting)e(the)h(\002x)o(ed)f
(point)f(coincide)h(with)g(the)h(centre)f(of)g(mass.)1901
1155 y(Similarly)-5 b(,)29 b(the)g(weak)f(coupling)e(of)i
Ft(m)h Fz(tops)f(appropriately)1901 1255 y(chosen)22
b(among)g(the)g(Lagrange)f(top,)i(the)g(Euler)g(top)f(and)g(the)1901
1354 y(dynamically)k(spherically)h(symmetric)h(Euler)f(top)h(yields)g
(a)1901 1454 y(superinte)o(grable)20 b(system)i(in)h
Ft(d)k Fu(=)g(3)p Ft(m)c Fz(de)o(grees)e(of)h(freedom)1901
1553 y(with)f(the)g(dimension)e Ft(r)24 b Fz(of)c(the)h(re)o(gular)e
(\002bres)i(of)f(the)h(result-)1901 1653 y(ing)f(rami\002ed)f(torus)h
(b)n(undle)f(satisfying)h Ft(m)j Fs(\024)f Ft(r)k Fs(\024)d
Fu(3)p Ft(m)d Fz(.)1937 1753 y(A)28 b(class)g(of)f(e)o(xamples)f(where)
h(the)g(dimension)f Ft(r)k Fz(may)d(as-)1901 1852 y(sume)j(an)o(y)g
(number)f(between)h(the)g(number)f(of)h(de)o(grees)g(of)1901
1952 y(freedom)17 b Ft(r)26 b Fu(=)d Ft(d)d Fz(and)f(the)g(maximally)f
(superinte)o(grable)e(case)1901 2052 y Ft(r)36 b Fu(=)c(1)26
b Fz(is)h(gi)n(v)o(en)d(by)h(the)h(C.Neumann)e(system,)j(cf.)e([Dullin)
1901 2151 y(and)e(Han\337mann,)e(2005].)32 b(A)23 b(point)f(mo)o(v)o
(es)g(on)h(a)g(sphere)g Ft(S)3682 2121 y Fp(d)1901 2251
y Fz(under)31 b(the)h(in\003uence)f(of)h(a)g(linear)g(force)f(\002eld.)
61 b(Only)31 b(the)1901 2350 y(dif)n(ferences)c(between)h(the)h(coef)n
(\002cients)e(of)i(the)g(force)e(\002eld)1901 2450 y(ha)n(v)o(e)e
(dynamical)f(consequences.)39 b(In)26 b(particular)m(,)f(when)g(all)
1901 2550 y(coef)n(\002cients)16 b(are)g(equal)g(to)g(each)g(other)g
(the)g(C.Neumann)f(sys-)1901 2649 y(tem)26 b(becomes)f(the)h(geodesic)f
(\003o)n(w)-5 b(,)27 b(a)f(maximally)f(superin-)1901
2749 y(te)o(grable)e(system)i(with)g(symmetry)e(group)g
Ft(S)5 b(O)r Fu(\()p Ft(d)22 b Fu(+)g(1\))i Fz(.)39 b(A)1901
2849 y(subgroup)18 b(of)i Ft(S)5 b(O)r Fu(\()p Ft(d)19
b Fu(+)f(1\))j Fz(of)e(product)g(form)2365 3097 y Ft(S)5
b(O)r Fu(\()p Ft(m)2591 3109 y Fv(0)2629 3097 y Fu(\))19
b Fs(\002)f(\001)c(\001)g(\001)k(\002)g Ft(S)5 b(O)r
Fu(\()p Ft(m)3187 3109 y Fp(s)3223 3097 y Fu(\))1901
3345 y Fz(is)30 b(the)f(symmetry)f(group)g(of)h(the)g(de)o(generate)e
(C.Neumann)1901 3445 y(system)17 b(with)f Ft(s)t Fu(+)t(1)g
Fz(groups)f(of)h Ft(m)2885 3457 y Fp(&)2936 3445 y Fz(equal)g(coef)n
(\002cients.)23 b(Note)1901 3544 y(that)28 b Ft(m)2127
3556 y Fp(&)2197 3544 y Fu(=)36 b(2)28 b Fz(equal)f(coef)n(\002cients)g
(do)g(not)g(yet)g(lead)h(to)f(su-)1901 3644 y(perinte)o(grability)-5
b(.)28 b(Indeed,)21 b(the)h(angular)f(momentum)f(de\002n-)1901
3743 y(ing)32 b(the)f(corresponding)e Ft(S)5 b(O)r Fu(\(2\))p
Fz(\226action)31 b(merely)g(replaces)1901 3843 y(one)23
b(of)f(the)i(Uhlenbeck)d(inte)o(grals,)i(see)g([Dullin)g(and)f
(Han\337-)1901 3943 y(mann,)k(2005])e(for)h(more)g(details.)41
b(F)o(or)26 b Ft(m)3167 3955 y Fp(&)3234 3943 y Fs(\025)33
b Fu(3)25 b Fz(the)h(f)o(actor)1901 4042 y Ft(S)5 b(O)r
Fu(\()p Ft(m)2127 4054 y Fp(&)2162 4042 y Fu(\))28 b
Fz(is)g(non\226commutati)n(v)o(e)23 b(and)j(does)h(lead)g(to)g(super)n
(-)1901 4142 y(inte)o(grability)-5 b(.)29 b(Thus,)22
b(the)g(de)o(generate)e(C.Neumann)g(system)1901 4242
y(is)j(minimally)e(superinte)o(grable)e(if)j(and)f(only)g(if)h(there)f
(is)i(one)1901 4341 y(group)28 b(of)h(three)g(equal)g(coef)n
(\002cients)g(and)g(all)i(other)d(coef)n(\002-)1901 4441
y(cients)20 b(equal)g(at)h(most)f(one)g(more)f(coef)n(\002cient.)1937
4541 y(As)34 b(formulated)d(in)i(Theorem)f(1.1,)k(the)d(re)o(gular)e
(part)i Fs(M)1901 4640 y Fz(of)h(the)h(rami\002ed)f(torus)g(b)n(undle)g
(determined)f(by)h(a)h(super)n(-)1901 4740 y(inte)o(grable)29
b(system)i(is)h(a)g(\002bration)e(by)g(isotropic)g(in)m(v)n(ariant)1901
4839 y(tori)18 b Fw(T)2090 4809 y Fp(r)2146 4839 y Fz(.)25
b(Since)19 b(the)f(Poisson)h(brack)o(et)e(of)h(tw)o(o)h(\002rst)g(inte)
o(grals)1901 4939 y Fs(f)p Ft(f)1984 4951 y Fp(i)2011
4939 y Ft(;)14 b(f)2089 4951 y Fp(j)2123 4939 y Fs(g)23
b Fu(=)g Ft(P)2329 4951 y Fp(ij)2405 4939 y Fs(\016)16
b Ft(f)30 b Fz(is)20 b(again)f(a)h(\002rst)h(inte)o(gral)e(this)h
(isotropic)1901 5039 y(\002bration)e(admits)h(a)h(polar)e(foliation)g
(which)h(is)h(co\226isotropic.)1901 5138 y(In)35 b(the)g(local)g
(co\226ordinates)e(\(2\))h(the)h(co\226isotropic)e(lea)n(v)o(es)1901
5238 y(are)27 b(co\226ordinized)e(by)i Fu(\()p Ft(x;)14
b(q)s(;)g(p)p Fu(\))28 b Fz(while)g Ft(y)i Fz(is)f(\002x)o(ed.)46
b(Where)1901 5338 y(superinte)o(grability)24 b(is)j(coming)e(from)h(a)h
(non\226commutati)n(v)o(e)1901 5437 y(symetry)17 b(group)g(the)h
(quotient)f(of)g(a)i(co\226isotropic)d(leaf)i(by)g Fw(T)3683
5407 y Fp(r)1901 5537 y Fz(is)25 b(the)f(co\226adjoint)e(orbit,)i(with)
g(local)g(co\226ordinates)d Fu(\()p Ft(q)s(;)14 b(p)p
Fu(\))25 b Fz(.)1901 5636 y(On)17 b(the)g(other)f(hand,)g(for)h(a)g
(non\226de)o(generate)c(inte)o(grable)i(sys-)1901 5736
y(tem)j(both)e(the)i(isotropic)f(\002bration)f(and)h(the)h
(co\226isotropic)d(fo-)
1800 6037 y(8)37 b()p eop end
%%Page: 9 9
TeXDict begin 9 8 bop -107 -90 a Fz(liation)20 b(coincide)f(with)i(the)
f(\002bration)f(by)h(Lagrangean)d(tori.)-71 21 y(A)25
b(superinte)o(grable)d(Hamiltonian)h Ft(N)34 b Fz(depends)23
b(only)g(on)h Ft(y)-107 121 y Fz(and)f(in)g(case)h(the)f
(co\226isotropic)e(foliation)i(is)h(in)f(f)o(act)h(a)f(\002bra-)-107
221 y(tion)42 b Ft(c)c Fu(:)g Fs(M)g(\000)-15 b(!)38
b(A)43 b Fz(\227)29 b(as)g(holds)f(true)g(in)g(all)h(e)o(xamples)-107
320 y(in)g(the)g(present)f(paper)f(\227)j(one)e(may)g(write)h
Ft(N)34 b Fs(\016)24 b Ft(c)29 b Fz(for)f(this)-107 420
y(Hamiltonian.)49 b(Similarly)-5 b(,)30 b(a)f(normal)e(form)1276
399 y Fu(\026)1257 420 y Ft(P)41 b Fz(of)28 b(the)h(per)n(-)-107
519 y(turbation)21 b Ft(P)35 b Fz(with)22 b(respect)h(to)f
Ft(N)32 b Fz(may)22 b(be)g(written)h(as)1555 498 y Fu(\026)1537
519 y Ft(P)32 b Fs(\016)19 b Ft(i)-107 619 y Fz(where)42
b Ft(i)c Fu(:)g Fs(M)g(\000)-14 b(!)38 b(B)45 b Fz(denotes)28
b(the)g(isotropic)g(\002bration.)-107 719 y(W)-7 b(e)21
b(ha)n(v)o(e)e(already)f(seen)i(that)33 b Ft(N)25 b Fs(\016)15
b Ft(c)g Fu(+)h Ft(")1124 698 y Fu(\026)1106 719 y Ft(P)27
b Fs(\016)15 b Ft(i)34 b Fz(is)20 b(inte)o(grable)-107
818 y(and)e(may)g(serv)o(e)f(as)i(an)g(intermediate)e(system)h(for)g
(minimally)-107 918 y(superinte)o(grable)g Ft(N)29 b
Fz(.)-71 1029 y(But)19 b(e)n(v)o(en)e(when)h(not)g(inte)o(grable)e
(\(and)h(thus)h(not)g(helpful)f(for)-107 1129 y(KAM\226lik)o(e)i
(results\))h(normal)e(forms)32 b Ft(N)25 b Fs(\016)15
b Ft(c)h Fu(+)f Ft(")1357 1108 y Fu(\026)1339 1129 y
Ft(P)27 b Fs(\016)15 b Ft(i)34 b Fz(sho)n(w)-107 1229
y(that)18 b(the)g(perturbed)d(motion)i(has)h(three)f(time)h(scales.)25
b(Ne)o(xt)17 b(to)-107 1328 y(the)j(f)o(ast)h(motion)460
1625 y Fu(_)-37 b Ft(x)69 b Fu(=)706 1568 y Ft(@)5 b(N)p
706 1606 125 4 v 722 1682 a(@)g(y)882 1625 y Fu(+)41
b Fs(O)r Fu(\()p Ft(")p Fu(\))-107 1936 y Fz(there)26
b(is)i(an)e Ft(")p Fz(\226slo)n(w)g(motion)g(in)h Ft(q)j
Fz(and)c Ft(p)h Fz(.)45 b(Moreo)o(v)o(er)m(,)25 b(the)-107
2035 y(motion)j(in)h Ft(y)j Fz(can)c(be)h(made)f(v)o(ery)g(slo)n(w)h
(by)f(considering)f(a)-107 2135 y(high)e(order)g(of)g(the)h(normal)e
(form.)41 b(Along)25 b(these)h(lines)g(one)-107 2235
y(may)c(obtain)g(Nekhoroshe)n(v\226lik)o(e)e(results,)j(for)f(more)g
(details)-107 2334 y(see)f([F)o(ass)7 b(\036)-35 b(o,)20
b(2005])e(and)i(references)f(therein.)-107 2648 y FC(Refer)o(ences)-65
2747 y Fz(Arnol')l(d,)e(V)-11 b(.I.)19 b(\(1963\).)d(Proof)i(of)h(a)g
(theorem)f(of)g(A.)i(N.)f(K)m(ol-)-65 2847 y(mogoro)o(v)f(on)i(the)h
(in)m(v)n(ariance)e(of)h(quasi\226periodic)e(motions)-65
2946 y(under)29 b(small)i(perturbations)d(of)i(the)h(Hamiltonian.)d
Fq(Russ.)-65 3046 y(Math.)19 b(Surv)-6 b(.)20 b FC(18)p
Fz(\(5\),)e(pp.)i(9)g(\226)g(36)-65 3146 y(Arnol')l(d,)31
b(V)-11 b(.I.)31 b(\(1963\).)e(Small)i(denominators)e(and)h(prob-)-65
3245 y(lems)g(of)f(stability)h(of)f(motion)g(in)g(classical)i(and)e
(celestial)-65 3345 y(mechanics.)19 b Fq(Russ.)h(Math.)g(Surv)-6
b(.)19 b FC(18)p Fz(\(6\),)g(pp.)g(85)h(\226)g(191)-65
3445 y(Arnol')l(d,)e(V)-11 b(.I.)19 b(\(1964\).)f(Instability)h(of)h
(dynamical)e(systems)-65 3544 y(with)25 b(se)n(v)o(eral)f(de)o(grees)g
(of)h(freedom.)e Fq(So)o(v)-6 b(.)24 b(Math.)h(Dokl.)g
FC(5)p Fz(,)-65 3644 y(pp.)19 b(581)g(\226)i(585)-65
3743 y(Bour)o(gain,)36 b(J.)f(\(1994\).)e(Construction)g(of)i
(quasi-periodic)-65 3843 y(solutions)52 b(for)h(Hamiltonian)f
(perturbations)f(of)i(linear)-65 3943 y(equations)43
b(and)h(applications)f(to)h(nonlinear)f(PDE.)h Fq(Int.)-65
4042 y(Math.)19 b(Res.)i(Notices)g FC(1994)p Fz(\(11\),)c(pp.)i(475)g
(\226)i(497)-65 4142 y(Bour)o(gain,)c(J.)k(\(1997\).)d(On)i(Melnik)o(o)
o(v')-5 b(s)19 b(persistenc)o(y)g(prob-)-65 4242 y(lem.)h
Fq(Math.)f(Res.)i(Lett.)g FC(4)p Fz(,)f(pp.)f(445)g(\226)i(458)-65
4341 y(Broer)m(,H.W)-8 b(.,)61 b(Han\337mann,)g(H.)54
b(and)g(Hoo,)62 b(J.)55 b(\(2004\))-65 4441 y(The)46
b(quasi\226periodic)e(Hamiltonian)h(Hopf)h(bifurcation.)-65
4541 y(Preprint,)19 b(Rijksuni)n(v)o(ersiteit)g(Groningen)-65
4640 y(Broer)m(,H.W)-8 b(.,)29 b(Han\337mann,)e(H.,)j(Jorba,)1159
4622 y(\036)1143 4640 y(A.,)g(V)-5 b(illanue)n(v)n(a)27
b(J.)-65 4740 y(and)36 b(W)-7 b(agener)m(,)39 b(F)-7
b(.O.O.)37 b(\(2003\))d(Normal\226internal)g(reso-)-65
4839 y(nances)k(in)h(quasi\226periodically)d(forced)h(oscillators)i(:)
63 b(a)-65 4939 y(conserv)n(ati)n(v)o(e)30 b(approach.)h
Fq(Nonlinearity)h FC(16)p Fz(,)j(pp.)d(1751)g(\226)-65
5039 y(1791)-65 5138 y(Broer)m(,)g(H.W)-8 b(.,)33 b(Han\337mann,)d(H.)h
(and)f(Y)-9 b(ou,)32 b(J.)f(\(2003\))d(Bi-)-65 5238 y(furcations)39
b(of)g(Normally)h(P)o(arabolic)f(T)-7 b(ori)40 b(in)g(Hamilto-)-65
5338 y(nian)24 b(Systems.)g(Preprint,)g(Inst.)g(Reine)h(&)f(Ange)n(w)-5
b(.)23 b(Math.,)-65 5437 y(R)-5 b(WTH)21 b(Aachen)-65
5537 y(Broer)m(,)k(H.W)-8 b(.,)27 b(Han\337mann,)d(H.)i(and)e(Y)-9
b(ou,)26 b(J.)g(\(2004\))d(Um-)-65 5636 y(bilical)38
b(T)-7 b(orus)38 b(Bifurcations)g(in)g(Hamiltonian)g(Systems.)-65
5736 y(Preprint,)16 b(Inst.)g(Reine)h(&)g(Ange)n(w)-5
b(.)16 b(Math.,)g(R)-5 b(WTH)19 b(Aachen)1943 -90 y(Broer)m(,)42
b(H.W)-8 b(.,)43 b(Hoo,)f(J.)d(and)f(Naudot,)k(V)-11
b(.)39 b(\(2004\))d(Nor)n(-)1943 9 y(mal)20 b(Linear)h(Stability)f(of)h
(Quasi\226Periodic)e(T)-7 b(ori.)21 b(Preprint,)1943
109 y(Rijksuni)n(v)o(ersiteit)e(Groningen)1943 209 y(Broer)m(,)55
b(H.W)-8 b(.,)57 b(Huitema,)f(G.B.)50 b(and)f(Se)n(vryuk,)54
b(M.B.)1943 308 y(\(1996\))35 b Fq(Quasi\226P)-7 b(eriodic)36
b(Motions)h(in)g(F)-6 b(amilies)37 b(of)g(Dy-)1943 408
y(namical)24 b(Systems)i(:)37 b(Or)m(der)26 b(amidst)f(Chaos)p
Fz(.)g(LNM)h FC(1645)p Fz(,)1943 508 y(Springer)1943
607 y(Broer)m(,)15 b(H.W)-8 b(.,)17 b(Huitema,)e(G.B.)h(and)f(T)-7
b(ak)o(ens,)16 b(F)-7 b(.)16 b(\(1990\))d(Un-)1943 707
y(foldings)25 b(of)i(quasi\226periodic)e(tori.)h Fq(Mem.)i(AMS)f
FC(83)f Fz(#421,)1943 806 y(pp.)19 b(1)h(\226)h(82)1943
906 y(Cushman,)15 b(R.)i(\(1992\))d(A)i(Surv)o(e)o(y)e(of)i
(Normalization)e(T)-6 b(ech-)1943 1006 y(niques)25 b(Applied)g(to)h
(Perturbed)e(K)n(eplerian)h(Systems.)g Fq(Dy-)1943 1105
y(namics)20 b(Rep.,)f(ne)o(w)h(ser)-9 b(.)21 b FC(1)p
Fz(,)f(pp.)f(54)h(\226)g(112)g(\(1992\))1943 1205 y(Cushman,)j(R.)h
(and)f(Bates,)i(L.M.)e(\(1997\))f Fq(Global)g(Aspects)1943
1305 y(of)e(Classical)h(Inte)m(gr)o(able)d(Systems)p
Fz(.)i(Birkh)5 b(\250)-33 b(auser)1943 1404 y(Cushman,)16
b(R.)g(and)g(Sado)o(vski)n(\020)-26 b(\021,)16 b(D.A.)g(\(2000\))e
(Monodromy)1943 1504 y(in)k(the)h(hydrogen)c(atom)j(in)g(crossed)g
(\002elds.)h Fq(Physica)f(D)g FC(142)1943 1603 y Fz(pp.)h(166)g(\226)i
(196)1943 1703 y(Delshams,)g(A.,)h(de)g(la)g(Lla)n(v)o(e,)f(R.)h(and)f
(Seara,)h(T)-6 b(.M.)21 b(\(2003\))1943 1803 y(A)41 b(geometric)e
(mechanism)h(for)g(dif)n(fusion)f(in)i(Hamilto-)1943
1902 y(nian)21 b(systems)h(o)o(v)o(ercoming)d(the)j(lar)o(ge)f(gap)g
(problem)f(:)29 b(an-)1943 2002 y(nouncement)20 b(of)k(results.)f
Fq(Electr)l(on.)g(Res.)h(Announc.)d(AMS)1943 2102 y FC(9)p
Fz(,)f(pp.)f(125)g(\226)i(134)1943 2201 y(Delshams,)g(A.,)h(de)g(la)g
(Lla)n(v)o(e,)f(R.)h(and)f(Seara,)h(T)-6 b(.M.)21 b(\(2003\))1943
2301 y(A)h(geometric)e(mechanism)g(for)h(dif)n(fusion)f(in)i
(Hamiltonian)1943 2400 y(systems)27 b(o)o(v)o(ercoming)d(the)j(lar)o
(ge)f(gap)g(problem)f(:)39 b(heuris-)1943 2500 y(tics)d(and)f(rigorous)
f(v)o(eri\002cation)f(on)i(a)h(model.)f(Preprint,)1943
2600 y(Uni)n(v)o(ersitat)19 b(de)h(Barcelona)1943 2699
y(Dullin,)i(H.R.)h(and)g(Han\337mann,)e(H.)h(\(2005\))f(The)h(de)o
(gener)n(-)1943 2799 y(ate)29 b(C.)g(Neumann)e(system)i(I)g(:)h
(symmetry)d(reduction)g(and)1943 2899 y(con)m(v)o(e)o(xity)17
b(\227)k Fq(in)f(pr)m(epar)o(ation)1943 2998 y Fz(Efstathiou,)e(K.)h
(\(2004\))e Fq(Metamorphoses)h(of)h(Hamiltonian)1943
3098 y(systems)36 b(with)f(symmetry)p Fz(.)h(Ph.D.)f(thesis,)k(Uni)n(v)
o(ersit)5 b(\264)-33 b(e)34 b(du)1943 3197 y(Littoral)19
b(C)7 b(\210)-35 b(ote)21 b(d'Opale)1943 3297 y(F)o(ass)7
b(\036)-35 b(o,)46 b(F)-7 b(.)42 b(\(2005\))d(Superinte)o(grable)f
(Hamiltonian)i(sys-)1943 3397 y(tems)26 b(:)38 b(Geometry)25
b(and)g(Perturbation.)f(Preprint,)j(Uni)n(v)o(er)n(-)1943
3496 y(sit)5 b(\036)-33 b(a)21 b(di)f(P)o(ado)o(v)n(a)1943
3596 y(Han\337mann,)32 b(H.)f(\(1995\))f Fq(Quasi\226periodic)f
(Motions)j(of)f(a)1943 3696 y(Rigid)e(Body)h(\227)h(A)f(case)h(study)f
(on)g(perturbations)f(of)h(su-)1943 3795 y(perinte)m(gr)o(able)24
b(systems)p Fz(.)k(Ph.D.)e(thesis,)j(Rijksuni)n(v)o(ersiteit)1943
3895 y(Groningen)1943 3994 y(Han\337mann,)49 b(H.)d(\(1997\))d
(Quasi\226periodic)h(Motions)h(of)1943 4094 y(a)38 b(Rigid)g(Body)f(I)h
(\227)g(Quadratic)f(Hamiltonians)f(on)h(the)1943 4194
y(Sphere)28 b(with)i(a)g(Distinguished)e(P)o(arameter)-5
b(.)29 b Fq(Re)m(g)o(.)f(Chaot.)1943 4293 y(Dyn.)19 b
FC(2)p Fz(\(2\),)g(pp.)h(41)g(\226)g(57)1943 4393 y(Han\337mann,)37
b(H.)f(\(1998\))e(The)i(Quasi\226Periodic)e(Centre\226)1943
4493 y(Saddle)19 b(Bifurcation.)g Fq(J)n(.)h(Dif)o(f)o(.)g(Eq.)g
FC(142)p Fz(\(2\),)e(pp.)h(305)h(\226)g(370)1943 4592
y(Han\337mann,)43 b(H.)e(\(2003\))e(Hamiltonian)g(T)-7
b(orus)40 b(Bifurca-)1943 4692 y(tions)30 b(Related)h(to)f(Simple)g
(Singularities.)g(Preprint,)i(Inst.)1943 4791 y(Reine)20
b(&)g(Ange)n(w)-5 b(.)19 b(Math.,)h(R)-5 b(WTH)21 b(Aachen)1943
4891 y(Han\337mann,)15 b(H.)j(\(2004\))d(On)i(Hamiltonian)f
(bifurcations)g(of)1943 4991 y(in)m(v)n(ariant)25 b(tori)h(with)h(a)h
(Floquet)e(multiplier)f Fs(\000)p Fu(1)p Fz(.)i(Preprint,)1943
5090 y(Inst.)20 b(Reine)g(&)g(Ange)n(w)-5 b(.)19 b(Math.,)h(R)-5
b(WTH)21 b(Aachen)1943 5190 y(Hoo,)c(J.)h(\(2005\))d(Quasi\226periodic)
g(bifurcations)h(in)i(a)f(strong)1943 5290 y(resonance)31
b(:)51 b(combination)31 b(tones)i(in)g(gyroscopic)d(stabi-)1943
5389 y(lization.)19 b(Ph.D.)h(thesis,)h(Rijksuni)n(v)o(ersiteit)e
(Groningen)1943 5489 y(Huitema,)28 b(G.B.)f(\(1988\))e
Fq(Unfoldings)h(of)h(Quasi\226P)-7 b(eriodic)1943 5588
y(T)f(ori)p Fz(.)20 b(Ph.D.)g(thesis,)h(Rijksuni)n(v)o(ersiteit)e
(Groningen)1943 5688 y(Jorba,)2180 5670 y(\036)2163 5688
y(A.)h(and)f(V)-5 b(illanue)n(v)n(a,)19 b(J.)h(\(1997\))d(On)j(the)g
(normal)e(be-)
1800 6037 y(9)37 b()p eop end
%%Page: 10 10
TeXDict begin 10 9 bop -65 -90 a Fz(ha)n(viour)20 b(of)h(partially)f
(elliptic)i(lo)n(wer)n(\226dimensional)d(tori)i(of)-65
9 y(Hamiltonian)e(systems.)h Fq(Nonlinearity)f FC(10)p
Fz(,)h(pp.)g(783)f(\226)h(822)-65 109 y(K)o(uksin,)70
b(S.B.)62 b(\(1993\))d Fq(Nearly)i(inte)m(gr)o(able)f(in\002nite\226)
-65 209 y(dimensional)74 b(Hamiltonian)g(systems)p Fz(.)j(LNM)f
FC(1556)p Fz(,)-65 308 y(Springer)-65 408 y(Lieberman,)25
b(B.B.)i(\(1971\))c(Existence)i(of)h(Quasi-Periodic)-65
508 y(Solutions)d(to)h(the)h(Three\226Body)c(Problem.)i
Fq(Celest.)i(Mec)o(h.)-65 607 y FC(3)p Fz(,)20 b(pp.)f(408)h(\226)g
(426)-65 707 y(Lieberman,)27 b(B.B.)h(\(1972\))e(Quasi-Periodic)g
(Solutions)h(of)-65 806 y(Hamiltonian)19 b(Systems.)h
Fq(J)n(.)g(Dif)o(f)o(.)g(Eq.)g FC(11)p Fz(,)f(pp.)h(109)f(\226)h(137)
-65 906 y(Litv)n(ak\226Hinenzon,)15 b(A.)j(\(2001\))f
Fq(P)-7 b(ar)o(abolic)17 b(Resonances)f(in)-65 1006 y(Hamiltonian)d
(Systems)p Fz(.)j(Ph.D.)f(thesis,)h(The)f(W)-7 b(eizmann)15
b(In-)-65 1105 y(stitute)20 b(of)g(Science,)g(Reho)o(v)n(ot)-65
1205 y(Litv)n(ak\226Hinenzon,)53 b(A.)c(and)g(Rom\226K)n(edar)m(,)55
b(V)-11 b(.)50 b(\(2002\))-65 1305 y(P)o(arabolic)33
b(resonances)h(in)g Fu(3)h Fz(de)o(gree)e(of)i(freedom)d(near)n(\226)
-65 1404 y(inte)o(grable)26 b(Hamiltonian)i(systems.)g
Fq(Physica)g(D)h FC(164)p Fz(,)g(pp.)-65 1504 y(213)19
b(\226)h(250)-65 1603 y(Litv)n(ak\226Hinenzon,)14 b(A.)19
b(and)e(Rom\226K)n(edar)m(,)f(V)-11 b(.)19 b(\(2002\))d(Res-)-65
1703 y(onant)33 b(tori)i(and)f(instabilities)h(in)g(Hamiltonian)e
(systems.)-65 1803 y Fq(Nonlinearity)19 b FC(15)p Fz(,)h(pp.)f(1149)g
(\226)h(1177)-65 1902 y(Litv)n(ak\226Hinenzon,)k(A.)i(and)g(Rom\226K)n
(edar)m(,)f(V)-11 b(.)27 b(\(2004\))d(On)-65 2002 y(ener)o(gy)16
b(surf)o(aces)i(and)g(the)g(resonance)f(web)m(.)g Fq(SIAM)h(J)n(.)g
(Appl.)-65 2102 y(Dynam.)h(Syst.)h FC(3)p Fz(\(4\),)f(pp.)h(525)f(\226)
h(573)-65 2201 y(Markus,)34 b(L.)e(and)f(Me)o(yer)m(,)j(K.R.)f
(\(1980\))d(Periodic)h(orbits)-65 2301 y(and)22 b(solenoids)g(in)h
(generic)f(Hamiltonian)f(dynamical)g(sys-)-65 2400 y(tems.)f
Fq(Am.)g(J)n(.)g(Math.)g FC(102)p Fz(\(1\),)e(pp.)i(25)f(\226)i(92)-65
2500 y(Mazzocco,)k(M.)h(\(1997\))e(KAM)i(theorem)f(for)g(generic)g(an-)
-65 2600 y(alytic)35 b(perturbations)d(of)j(the)f(Euler)h(system.)f
Fq(Z.)i(Ang)o(e)o(w)-6 b(.)-65 2699 y(Math.)19 b(Phys.)h
FC(48)p Fz(,)g(pp.)f(193)h(\226)g(219)-65 2799 y(v)n(an)37
b(der)g(Meer)m(,)k(J.C.)d(and)f(Cushman,)k(R.)e(\(1986\))c(Con-)-65
2899 y(strained)h(normalization)e(of)j(Hamiltonian)e(systems)i(and)-65
2998 y(perturbed)29 b(K)n(eplerian)i(motion.)f Fq(Z.)j(Ang)o(e)o(w)-6
b(.)31 b(Math.)g(Phys.)-65 3098 y FC(37)p Fz(\(3\),)18
b(pp.)i(402)f(\226)h(424)-65 3197 y(v)n(an)j(der)h(Meer)m(,)g(J.C.)h
(and)e(Cushman,)h(R.)h(\(1987\))d(Orbiting)-65 3297 y(dust)d(under)f
(radiation)g(pressure.)g(In)h Fq(Dif)o(fer)m(ential)g(Geomet-)-65
3397 y(ric)28 b(Methods)f(in)g(Theor)m(etical)g(Physics,)i(Clausthal,)f
(1986)-65 3496 y Fz(\(ed.)19 b(H)i(Doebner\))d(W)-7 b(orld)20
b(Scienti\002c.)h(pp.)e(403)g(\226)i(414)-65 3596 y(Me)o(yer)m(,)33
b(K.R.)f(\(1970\))e(Generic)i(bifurcation)d(of)j(periodic)-65
3696 y(points.)19 b Fq(T)-5 b(r)o(ans.)21 b(AMS)f FC(149)p
Fz(,)f(pp.)g(95)h(\226)g(107)-65 3795 y(Me)o(yer)m(,)26
b(K.R.)g(\(1975\))e(Generic)h(Bifurcation)g(in)h(Hamilto-)-65
3895 y(nian)15 b(Systems.)h(In)g Fq(Dynamical)e(Systems)j(\227)f(W)-8
b(arwic)n(k)17 b(1974)-65 3994 y Fz(\(ed.)i(A.)i(Manning\))d(LNM)i
FC(468)p Fz(,)f(Springer)-5 b(.)19 b(pp.)h(62)f(\226)i(70)-65
4094 y(P)o(acha,)36 b(J.R.)e(\(2002\))e Fq(On)h(the)h
(quasi\226periodic)d(Hamilto-)-65 4194 y(nian)f(Andr)l(ono)o(v\226Hopf)
e(bifur)m(cation)p Fz(.)i(Ph.D.)h(thesis,)j(Uni-)-65
4293 y(v)o(ersitat)20 b(Polit)5 b(\036)-33 b(ecnica)19
b(de)h(Catalun)o(ya)-65 4393 y(P)7 b(\250)-35 b(oschel,)42
b(J.)e(\(1989\))c(On)i(Elliptic)h(Lo)n(wer)f(Dimensional)-65
4493 y(T)-7 b(ori)27 b(in)h(Hamiltonian)e(Systems.)i
Fq(Math.)f(Z.)h FC(202)p Fz(,)h(pp.)e(559)-65 4592 y(\226)20
b(608)-65 4692 y(Robinson,)25 b(R.C.)i(\(1970\))c(Generic)i(properties)
f(of)h(conser)n(-)-65 4791 y(v)n(ati)n(v)o(e)19 b(systems.)h
Fq(Am.)h(J)n(.)f(Math.)g FC(92)p Fz(,)f(pp.)h(562)f(\226)h(603)-65
4891 y(Robinson,)25 b(R.C.)i(\(1970\))c(Generic)i(properties)f(of)h
(conser)n(-)-65 4991 y(v)n(ati)n(v)o(e)19 b(systems)i(II.)e
Fq(Am.)i(J)n(.)f(Math.)g FC(92)p Fz(,)f(pp.)h(897)f(\226)h(906)-65
5090 y(Rudne)n(v)-5 b(,)41 b(M.)e(\(2003\))d(Hamilton\226Jacobi)h
(method)g(for)h(a)-65 5190 y(simple)20 b(resonance.)e(Preprint,)h(Uni)n
(v)o(ersity)g(of)h(Bristol)-65 5290 y(R)7 b(\250)-35
b(ussmann,)17 b(H.)h(\(2001\))e(In)m(v)n(ariant)f(tori)j(in)g
(non\226de)o(generate)-65 5389 y(nearly)32 b(inte)o(grable)g
(Hamiltonian)h(systems.)h Fq(Re)m(g)o(.)e(Chaot.)-65
5489 y(Dyn.)20 b FC(6)p Fz(\(2\),)e(pp.)i(119)f(\226)h(204)-65
5588 y(Se)n(vryuk,)41 b(M.B.)e(\(1997\))d(In)m(v)n(ariant)h(T)-7
b(ori)38 b(of)h(Intermedi-)-65 5688 y(ate)22 b(Dimensions)g(in)h
(Hamiltonian)e(Systems.)h Fq(Re)m(g)o(.)f(Chaot.)1943
-90 y(Dyn.)e FC(2)p Fz(\(3/4\),)g(pp.)g(30)h(\226)g(40)1943
9 y(Sommer)m(,)e(B.S.)i(\(2003\))e Fq(A)i(KAM)g(Theor)m(em)g(for)g(the)
g(Spatial)1943 109 y(Lunar)f(Pr)l(oblem)p Fz(.)h(Ph.D.)g(thesis,)h(R)-5
b(WTH)21 b(Aachen)1943 209 y(T)m(reshch)5 b(\250)-33
b(ev)-5 b(,)14 b(D.V)-11 b(.)17 b(\(1991\))d(The)i(mechanism)f(of)h
(destruction)1943 308 y(of)f(resonant)f(tori)i(of)f(Hamiltonian)f
(systems.)i Fq(Math.)f(USSR\226)1943 408 y(Sb)m(.)k FC(68)p
Fz(,)g(pp.)h(181)f(\226)h(203)1943 508 y(Xu,)52 b(J.)46
b(and)f(Y)-9 b(ou,)52 b(J.)46 b(\(2001\))e(Persistence)i(of)f(lo)n(wer)
n(\226)1943 607 y(dimensional)h(tori)j(under)e(the)h(\002rst)i(Melnik)o
(o)o(v')-5 b(s)47 b(non\226)1943 707 y(resonance)18 b(condition.)g
Fq(J)n(.)j(Math.)f(Pur)m(es)g(Appl.)f FC(80)p Fz(\(10\),)f(pp.)1943
806 y(1045)g(\226)j(1067)
1800 6037 y(10)37 b()p eop end
%%Trailer

userdict /end-hook known{end-hook}if
%%EOF