Content-Type: multipart/mixed; boundary="-------------0908032129632" This is a multi-part message in MIME format. ---------------0908032129632 Content-Type: text/plain; name="09-132.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="09-132.comments" AMS-Code: 37J40, 37J45, 37J50 ---------------0908032129632 Content-Type: text/plain; name="09-132.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="09-132.keywords" Autonomous Lagrangian Systems, Connecting Orbits, Priori Unstable Type ---------------0908032129632 Content-Type: application/postscript; name="LiCheng-3.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="LiCheng-3.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.94b Copyright 2004 Radical Eye Software %%Title: LiCheng-3.dvi %%CreationDate: Tue Aug 04 09:34:58 2009 %%Pages: 30 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX10 CMR9 CMCSC10 CMMI9 CMR10 CMMI10 CMSY10 MSBM10 %%+ CMMI8 CMR8 CMEX10 EUFM10 CMSY8 EUFM7 CMMI6 CMSY6 CMR6 CMTI10 MSBM7 %%+ MSAM10 rsfs10 rsfs5 CMTI9 CMBX9 CMTT9 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe LiCheng-3.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2009.08.04:0934 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI9 %!PS-AdobeFont-1.1: CMTI9 1.0 %%CreationDate: 1991 Aug 18 21:08:07 % 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 (CMTI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 39 /quoteright put dup 44 /comma put dup 45 /hyphen put dup 49 /one put dup 53 /five put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 79 /O put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-35 -250 1148 750}readonly def /UniqueID 5000827 def currentdict end currentfile eexec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%!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 82 /R put dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: rsfs5 %!PS-AdobeFont-1.0: rsfs5 001.000 %%CreationDate: Sat Mar 21 18:48:20 1998 %%VMusage: 120000 150000 11 dict begin /FontInfo 14 dict dup begin /version (001.001) readonly def /Copyright (Conversion of metafont curves by Metafog (c) 1995 Richard Kinch) readonly def /Notice (Copyright (c) Taco Hoekwater, 1998. All rights reserved.) readonly def /FullName (rsfs5) readonly def /FamilyName (rsfs5) readonly def /ItalicAngle -12 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /Weight (Medium) readonly def end readonly def /FontName /rsfs5 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox {58 -306 1512 731} readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: rsfs10 %!PS-AdobeFont-1.0: rsfs10 001.000 %%CreationDate: Sat Mar 21 18:47:14 1998 %%VMusage: 120000 150000 11 dict begin /FontInfo 14 dict dup begin /version (001.001) readonly def /Copyright (Conversion from mf curves by Metafog (c) 1995 Richard Kinch) readonly def /Notice (Copyright (c) Taco Hoekwater, 1998. All rights reserved.) readonly def /FullName (rsfs10) readonly def /FamilyName (rsfs10) readonly def /ItalicAngle -12 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /Weight (Roman) readonly def end readonly def /FontName /rsfs10 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox {-2 -300 1240 728} readonly def currentdict end currentfile eexec D9D66F633B846A989B9974B0179FC6CC445BCF7C3C3333173232E3FDBFF43949 1DB866C39088C203DC22FDC758584860EC7BB67FDA28CC6208249060E18FAB32 204779B5C03C0493BBBBC95CF02692CC4DEAA8D2EA90B5C2E64374E92BCB8501 429B8FAE4A76C0C6B76D6FF7CF9A7D5EDFBCA0E959541C59BD05B7DE43D25D53 FC3DDA6EF0C2743978A6D03E19CCED4A11F2EA4BCC3110BE8B8D9E2772361969 C19258EFAFDC276CB1ADE9208A941A36D18F9FB1C33DEF76AA315DD8F41F8A25 4AF1DC3456B9DA8108CE7F2380A151DBB59CBB616BB498185CD0D0DE28290A9C 1DB96124F5F796C7A2FC57B079E292DD1A2E5EBB88585421F81BBDC63FE33ABC 2DF705A2BE4F6BE21BED49C3E2FC3D528A348CBEF34339397282607E56471B4E B78EA8C9F24AC14AB83068335F7974D7262C18BEC4AC5FB4CB5E0D7C410A3F08 57FAC1C117ED77399C1E264A853EDDA05A6117D3DFD28F96E44B1B8D3995E42A E0B030F113EC027670E66FF845D9F93808284DC73F51F44B4C210997E261644C 59B3455D7C3D3AFBD65E28D1AA774273FB1B6E9AB4F56464B0D56911EE6CB802 F1C90982836A63EAACD2B2E9D07850C716F7D1291C59433A10F31CB9FB2F2512 1CE1F7F343CC06008D111D235C75DBEB8CE22E88E207F0ADDD18772276176BFC F43EEE4F51613C545A2A49B34E4237BE57CEDEF63179B73879C2674AF40B79AE 024EF39D4D01AA6E8E83BC1D5F393AC1F78D133A004DE64C4AE52897C279CFA1 C87E605E5C1CFC2F4C72A0D0F9ED40A8B4F8E9EFEBDC1D60E353DE792A091B4C 39738379EBB7BDCE4D06844931D77A2BA45D857EC30ECCCB37F85274F283C27B 4C9BDB119847F4422B1FBB4059A5ADEAC44D234520322E2259CFDA894D6148CC 1E0EFAEF5666358F343AA0C6ACCEFCE6F3BFE30C43C85ED1FC565196555E51C1 0395BAF253393AAE179F635A2DC0AF3A9BD67D385A9A96A5009D09BE5AA164FE 8A48F5DFBC3C2C7244F57C91356ADF25AE852673C48C9C3D4A2D1F06BED56256 869E52850D83D2A78DA51010268631BCF5562B70ECF818A741398D63ECA44D23 567F5932BDFDA9520FEC5F94C434FC369F0BE91DFC66ADCCD21A6AD1434BDCDF CAE93B9934C7E24FC7238B6D3A663ED5950C932E0D18A0604526630522F6FCA9 A6998225F6C71FD55A5490C891C2B4F6A8F9B9E454670D51D7F883D0E26253DD 07517660B58ED8E1FE64AC617DBF2FB8C7A3E4B0E9313749E436516518542AB9 2BC049832DCAA3F92F1F53388FFB0B20D278D82A20C029ADD4D9F0C023DA1D3F 876CECD064AAC79B20865677C0F6C6F123E97A46278E51F67A14B5B2C678FF58 35A253D3521F0D6769427993399D380FA3A3EB3BB7E6BABE9D8EE8F4CD2AABE5 3CF57BB6B2CD38516ACE47C604E6C51D6DDBC16F7F4F93CEB861517BD485E6CB 37CD4A245ADAD29370E6E4F79970C021D30B9739D646538ED6283BD390EEA9E1 9B9AF9ECBA8A9DD9DF83672C72D9D4813948668D07A824392AEA77B49B73EB7D 5518CE71D48E3C6057732D80B52C6282E6C964F1A79624A2606174B60BD7C683 CF9EC1A3BC4DB33CD5A34A130031327F3812684E5F262E9C3B7566FF2229E216 B8861601FBADA5010B7EDEE6A3FDCD624C4331DB1E16BFF655FB19CE5E1A2DDA 8B7ECD28388A8915A5D352C6EE5802EADBF694C79942BEE4CBC4DDD12449B7F9 1CF404C1A347BF09F7F859A6242E430D6AE665B45B6542967F3998888469DDDC E0966B6468394DC125D7CEEB7B6AB57459BA6238E72D5622AB74F1BF440716EA 4D2BF7A175CECD4DADFF779CF22FDD3645A10F21359472A96709519E26B67E45 AC47236F9528DC21E7536CC33D8C61DA792DDE69FE2C929FEF7402E0BDD6807A 0B9EB6E89BF89B8795C694A081F3BD46D21B6D0B2ECD3EE5AD5C9DEBBE07877E 6E14ECFD284E2722F7DE17158A17E1FABF85E4FE7A88944795F9BFA7393CE77D A9380C321BB9E395ABFA3A82410E23B2A0C5AE2F269CD89B83FC112D76BBA05C 520069DB1ED50309A2562F7EAF7EFC44206FAC9341D43AC601DF80A80EEC9007 AD0862186139B9F94BFEAD5B0A0457BB8D7B606DBC419225743817957AB44B56 91CD6EA83EBF879D237516 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 52 /four put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 80 /P put dup 83 /S put dup 84 /T put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec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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 1 /Delta put dup 22 /macron put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put dup 111 /o put dup 114 /r put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % 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 (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % 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 (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 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 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 22 /mu put dup 24 /xi put dup 25 /pi put dup 31 /chi put dup 32 /psi put dup 59 /comma put dup 67 /C put dup 71 /G put dup 77 /M put dup 84 /T put dup 86 /V put dup 97 /a put dup 99 /c put dup 101 /e put dup 104 /h put dup 105 /i put dup 106 /j put dup 109 /m put dup 110 /n put dup 116 /t put dup 122 /z put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 4391C9DF440285B8FC159D0E98D4258FC57892DDF0342CA1080743A076089583 6AD6FB2DC4C13F077F17789476E48402796E685107AF60A63FB0DE0266D55CF1 8D0AD65B9342CB686E564758C96164FFA711B11C1CE8C726F3C7BB1044BBD283 9AA4675747DF61E130A55E297CA5F0182A3F12F9085AF2F503481071724077A9 387E27879A9649AD5F186F33500FAC8F7FA26634BDCE1221EC0ED0E359E5EA5E 6166526FEB90C30D30099FBDC1BC2F9B62EFEEC48345160804AA98F8D0AA54B7 A480E715426651865C8E444EDB798C7E11040AF6E5A7ED1888653C6DBF5E6169 70BCD9C063B63B561EF165BF3AF11F8E519F37C6FDA2827685739DE2C48B5ADE EE84F067D704D4511DBFA49E166D543CFD9ECD7417055D8A827F51E087CD2927 BAFC7E6CFBD70B0FE969F890A11149D3D44D422C3370495DA9951AEE7253A49F 3A9444C8CD9158D84117299F7F2332FEB0F94E6ED8BC7AA789A3219BC2F227D3 3B5BC75FB53B55D72AF4A6A7BB613FA235B11BB37D059FD87127CEF73D5B3FBF 9F91ABAD78BD9240BD9525EBA78095EA0BDB25D1A19E876F292882EAD5619D46 D20317A345D931F4FF4EAE6216C27044CBA525E3B917CEA25A04C120466C4B93 FC720E6BA832A06CCA0A3916CEF0968D49085AEBD243C41A448289A6F05CE3F5 79148DC112A3CC7E8FF810B8C1A09E05F496C0F1EBA334E42E05C376C98F5F69 C06C71BFC0A2F3AC9951CFBB143C66FB84F9C4ED27DF70869352D61BD5E11508 0797B87C71BB743A105F7E162050025D82488502354CC2E922F1F6ABEFB44E16 CA35B7EFBC472C31FB98C44D4B3DE16E5D8AA0547C5185632E373E5452B90FDA F34E41C5598ECB262BDC564B8AB5F5C69154A7B0F7988D0DE8F30443975FE15F AA947B94342F1696284734E1D7E1A5C9E24CF4D102AFB05B46ADC517ADCF44C0 350EC2EA84B4308A9D541BD38E7CEE1AA53906B9EA15945B199FA16AAF8A349A 2F79A2F859356791873E26889DFE8ECED2C0659D6B3463A4F827ECA5A8A11DEE BA0AAC43114C380C05CDD20000B21C62B1313F9FC7368438DFBE05A85B44B0CE 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: EUFM7 %!PS-AdobeFont-1.1: EUFM7 2.1 %%CreationDate: 1992 Nov 20 17:36:25 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 77 /M put readonly def /FontBBox{0 -250 1193 750}readonly def /UniqueID 5031992 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163B3DA4F9700DE5807F164169FBB 1458C43CD471029C362871D2FB69E0E5E617BC06F3B8621E3528E4B47E731AE1 0B406A3260424830B3C86A135457A566A1A716189B86883C171DA05829CE2E37 C020AAA2B4E1C2AB63462E7BB571434ED0811B99430919096D08C5F1C659B068 E87DB9366062D68FFD44AAD8A16740DCC214C32CDDDA354637C06EF93BC99352 DF04393C661FFFBF4BA20723ABD9B3E9CA9ABAFEED3C46C638D1A1C726DD7186 51137D23DD526D78AC5284266997E041592776D78B5DD8436571897EFD57A051 F9571E2957B23DF4ADB88D06EAAE9D73ABEC88C934E94205BFDF7115D9E667D6 CD2AFA8DCDA2D49EFC088135417D5D26117B20F923E8F0F5E92E0D8589BE5D93 7D4C3FEEF84B412F8177DE3E3C1711AA06686842AC5A3B3398B098FD06F255F0 2B237EF143FEE4CCC3540EA39B05197D2D9C149A2A71569B8608BA62D982CC7A 4FC2F4170B7A92531E4734A4D1D86FDA1B21AA3683CFC9EA65FB8A91990CEB69 AA62C0F22FDE470FA095FCBE4FB5BD40826BD6FECAA7F8B187412EFDC97E3DAB E2A836F16ED6FFA7F5B5B2714EC5C4A43E4E36D45D9F702CE875B0DFE6E08559 B60BF589C3941BFDAD1392B4D7BCE363E47162D6322FCA12758538A9DB5E4404 56BD008A9DB0E1A2D684F32BDE744F9457C839411F623C0AD633582CAC6FDD79 881E0A4F43DD393738E3CEB1DCAC0CCBAD384E5160BABAA5EF1F46632D05BB7C FC9396D17263BADE1AB2A229367D38F00D7B87124713CD716D9164E64FD1D52C FD6E2226BB350465D3614E241A269ED51FF3283446643AF9F18DCE671611DB2F 5D3EA77EE734D2BED623F973E6840B8DAD1E2C3C2666DD4DD1C1C8384F906A7B AACDF441BB0AAEA2A697E453E61530BEA8FBF4978B098757BF97AE1D85F60821 D9EDD5AAB98230BB618B336A6CE976EF75077BA3134B17684663EAEA983FD2A0 7EE2843DCC31A583D90482A662D31576BC815FB1D27534D579D515C6731CB255 491BF81BE121232CE028246272BB7C6EF85E92081DDD6C47CB43C90508363D90 DEA72BA41AAB889D4A6F2037A28A1BBC786860B175D5A602CAC639CC69CA892E 1224547F50A4D37B74150BCA8F43519AB162FE4089359D7E2087000DA3A09B1C D1091B07FD9948F5B3A72056FA049C0B8C554BB0BEE5D66BE12BFE7F49E1E196 0E58364737F0479F085D6AD1DB9BB6CEE435BD14C6D9D733F1BED2BFB1398610 5A9C87B4A3151C61470B4FA96333FAE1C9DEA88411F7F70E3ACA0B98D95984B9 7B49CD26C845A9DC213B4AC819DF3FDE4059995417EAB7D526039F26DBE15162 2F7AA9B276D1068EA064B1E70C7ECF9D91F34A3D3962D5B81F0D3CDBCFC6B3A0 9211B973192BB244526510EA383256EF9D63B725F5C516DBC322BC46EA0EDDF3 11586B150504FFE42FAF481279A41C776936919E794E66C24336FA4C05F4A636 676FF5F784B9691839B5B0CDF255077722739CA97DFA2FCBEC30A1ED09666343 AE7BB8C8FA68C1143DAF6587FEF79737EA111F381FC25130576F82C8FCA950E2 B57869DECF0EB27E0981ACF0C7B59DD827E76CEFE4CC35ADC92CA0562763556C 4E262996F6611903599FB167A6E404D310D4A3359F0A6B48C982F9D076312C7C DF27F19242B4730557CE8C1825A2BBE8E234543986C715B10D5CD29811BB5800 3F25056B603E4F21E4C91B1E961E3683AF7E1181141E21F8EE46CD379B69BFD3 C2239D0C7A86E46EF06DE8BADB272598105A44FA8E2BA7DE5D20E2A0D9696E45 1431465340E48CB3FC0DDEBB3690288DDA8A5D594537EE051D51CADBB0BAFC65 4AA9861BF12B3BDC5EEB6C04DA99CECA42B984B7D75C783CCFCFBDF5DE550E10 C82CFA1582DBDDE6C3AFD64496CC8D88C037E851FA71EC285D00266AF95D70F6 E7 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 6 /plusminus put dup 7 /minusplus put dup 20 /lessequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 63 /perpendicular put dup 65 /A put dup 110 /backslash put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: EUFM10 %!PS-AdobeFont-1.1: EUFM10 2.1 %%CreationDate: 1992 Nov 20 17:36:20 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 66 /B put dup 67 /C put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 90 /Z put readonly def /FontBBox{-26 -224 1055 741}readonly def /UniqueID 5031986 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163B3DA4F9700DE5807F164169FBB 1458C43CD471029C362871D2FB69E0E5E617BC06F3B8621E3528E4B47E731AE1 0B406A3260424830B3C86A135457A566A1A716189B86883C171DA05829CE2E37 C020AAAD5205DB1A9C1A6A56FFD9008D511B4453C4B33F6B0E977B7C760F8816 4A6F13E41191AA13358515D902E653B81AEF60FE07C4AB673435504C2C376BB2 6BAB038083E5073851178BEB6AA92FE582EFCFA3A22C6795DCAAFAFCA0F3E79D 270BB779E2071AFDF30D3F32082777AEDF90940B6DAA1F34BAC0E9AA08811281 8D4261185EC039EB930024E010615A972BBD0B8EAA09B3931D630F1215D88859 5803A82F87BF0D6B8C64E23358C903C8E022AF457F97845B948D174E063E40F3 FA1F8C2CF0C7C49EC374E22B0A865A90FDC5CE762D643812D9597AC4CA872A23 9CA95A4001607D1E336A8D3C10BBC8BA47B1DD8DAC76B3F7B4E34DA11B11F27E 5CC1B7D0249F7924540CC5E194DD545B39C060A69CFC2145AFED8242A9FEA65F ED03FC2DA7CDD3ADC394D32F22802C8EC818847BA08DE0654277ED11E5C356A7 1E590527E7A231A6AE8B779B88648B69BE95F879361588325BB6F7E01D44C6A1 8422BACAA4C2B6B4802A4452C7A95F232A5B3B9DE2E78780A95F58FF97C276B4 691C2E88AAD4FF004B752120BB3453C12CC6BEDD34165F33D1DACCE965BF81A8 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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 18 /parenleftbigg put dup 19 /parenrightbigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 82 /integraltext put dup 83 /uniontext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 92 /intersectiondisplay put dup 110 /braceleftBig put dup 111 /bracerightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 3 /Lambda put dup 22 /macron put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 88 /X put dup 89 /Y put dup 91 /bracketleft put dup 93 /bracketright put dup 95 /dotaccent put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec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17:46:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 57 /nine put dup 65 /A put dup 67 /C put dup 68 /D put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueID 5000772 def currentdict end currentfile eexec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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 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 17 /eta put dup 22 /mu put dup 23 /nu put dup 25 /pi put dup 32 /psi put dup 59 /comma put dup 62 /greater put dup 68 /D put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 80 /P put dup 84 /T put dup 85 /U put dup 86 /V put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 120 /x put dup 122 /z put dup 123 /dotlessi put dup 126 /vector put readonly def 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BA41D25A802D617AFB6B499156301A 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 63 /emptyset put dup 82 /R put dup 84 /T put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 6 /plusminus put dup 17 /equivalence put dup 18 /reflexsubset put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 27 /propersuperset put dup 33 /arrowright put dup 37 /arrownortheast put dup 38 /arrowsoutheast put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 56 /universal put dup 57 /existential put dup 59 /emptyset put dup 65 /A put dup 67 /C put dup 77 /M put dup 78 /N put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec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1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 37 /rho1 put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 84 /T put dup 85 /U put dup 86 /V put dup 88 /X put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /dotlessi put dup 126 /vector put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 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739713F6AB2ECA148B94E6330A1D3830AD6F7451592463283D 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 3 /Lambda put dup 5 /Pi put dup 6 /Sigma put dup 8 /Phi put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 34 /quotedblright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%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 11 /alpha put dup 99 /c put dup 116 /t put readonly def /FontBBox{-29 -250 1075 750}readonly def /UniqueID 5087384 def currentdict end currentfile eexec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cleartomark %%EndFont %%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 11 /ff put dup 12 /fi put dup 13 /fl put dup 19 /acute put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 123 /endash put dup 126 /tilde put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 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0B50BD4793244912EC0B620AD61A9A9A2D 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 89 /Y put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 119 /w put dup 121 /y put 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begin 39158280 55380996 1000 600 600 (LiCheng-3.dvi) @start /Fa 138[39 3[39 2[39 3[39 1[39 39 1[39 39 39 34[39 17[39 46[{}11 74.7198 /CMTT9 rf /Fb 198[44 44 44 44 44 44 44 44 44 44 6[34 34 40[{}12 74.7198 /CMBX9 rf /Fc 133[31 37 35 51 35 41 25 31 32 35 39 39 43 63 20 35 1[24 39 35 24 35 39 35 35 39 9[77 57 57 55 43 3[59 1[69 48 59 1[30 57 59 50 52 58 55 1[57 6[24 39 3[39 3[39 3[27 24 4[24 25[45 43 47 11[{}52 74.7198 /CMTI9 rf /Fd 173[36 82[{}1 49.8132 /MSBM7 rf /Fe 188[50 67[{}1 49.8132 /rsfs5 rf /Ff 188[44 67[{}1 66.4176 /rsfs10 rf /Fg 188[61 67[{}1 90.9091 /rsfs10 rf /Fh 252[71 3[{}1 90.9091 /MSAM10 rf /Fi 165[44 7[48 82[{}2 66.4176 /MSBM7 rf /Fj 133[37 44 42 60 42 49 30 37 38 42 46 46 51 74 23 42 28 28 46 42 28 42 46 42 42 46 9[91 2[65 51 2[62 3[57 2[35 68 70 59 62 3[68 5[28 28 5[46 1[46 46 2[28 33 28 2[37 37 25[80 53 51 12[{}50 90.9091 /CMTI10 rf /Fk 141[24 2[30 17[18 1[18 29[47 10[30 30 30 4[47 1[24 24 17[30 20[50 1[{}13 49.8132 /CMR6 rf /Fl 201[0 3[42 61 18 14[61 29[32 2[48{}7 49.8132 /CMSY6 rf /Fm 133[30 5[24 5[38 54 2[25 22 36 2[29 1[28 1[34 10[36 1[36 6[58 5[47 3[44 7[19 26[41 38 5[36 28 1[37 4[32 28 26 28 32 13[{}26 49.8132 /CMMI6 rf /Fn 178[72 77[{}1 66.4176 /EUFM7 rf /Fo 145[35 44[56 1[55 12[47 71 19 14[71 12[55 12[55 55 2[35 2[55{}12 66.4176 /CMSY8 rf /Fp 165[55 9[75 75 76 95 9[56 80 66[{}7 90.9091 /EUFM10 rf /Fq 144[61 61 17[101 101 51 1[131 4[76 43 19[81 1[81 1[81 1[81 36[67 67 18[{}14 90.9091 /CMEX10 rf /Fr 160[20 1[20 1[20 1[53 53 2[53 51 39 52 55 1[55 53 65 44 2[25 53 55 46 48 54 51 50 53 3[55 3[35 35 35 35 35 35 35 35 35 35 1[20 24 1[55 1[27 27 17[35 18[49 1[59 44{}43 66.4176 /CMR8 rf /Fs 134[51 1[69 51 51 49 38 50 1[46 53 51 1[43 2[25 51 53 44 46 52 2[51 13[51 2[62 7[68 26[31 45[{}22 83.022 /CMCSC10 rf /Ft 129[35 2[23 33 1[40 3[25 33 32 32 36 1[43 62 1[37 29 24 41 2[33 36 31 30 37 10[41 48 41 3[45 2[68 48 60 1[31 58 3[58 5[55 2[20 26[46 6[41 1[35 42 4[35 1[29 31 36 13[{}40 66.4176 /CMMI8 rf /Fu 165[61 5[61 1[66 18[71 63[{}4 90.9091 /MSBM10 rf 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Fw(c)p Fy(-equiv)-5 b(alen)m(t)30 b(if)f(there)g (exists)g(a)g(con)m(tin)m(uous)456 5216 y(curv)m(e)35 b(\000:)51 b([0)p Fw(;)15 b Fy(1])35 b Fv(!)e Fp(C)i Fy(suc)m(h)g(that)h(\000\(0\))f(=)e Fw(c)p Fy(,)k(\000\(1\))e(=)e Fw(c)2440 5183 y Fo(0)2464 5216 y Fy(,)k Fw(\013)p Fy(\(\000\()p Fw(s)p Fy(\)\))f(k)m(eeps)g(constan)m(t)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 456 251 a Fr(6)1047 b(X.)23 b(LI)h(AND)f(C.-Q.)f (CHENG)456 450 y Fy(for)44 b(all)i Fw(s)j Fv(2)h Fy([0)p Fw(;)15 b Fy(1],)50 b(and)45 b(for)f(eac)m(h)j Fw(s)1816 464 y Fr(0)1904 450 y Fv(2)i Fy([0)p Fw(;)15 b Fy(1])47 b(there)e(exists)h Fw(\016)53 b(>)d Fy(0)45 b(suc)m(h)g(that)456 560 y(\000\()p Fw(s)p Fy(\))20 b Fv(\000)g Fy(\000\()p Fw(s)872 574 y Fr(0)911 560 y Fy(\))26 b Fv(2)f Fw(V)1131 527 y Fo(?)1111 592 y Fr(\000\()p Ft(s)1215 601 y Fk(0)1249 592 y Fr(\))1311 560 y Fy(whenev)m(er)30 b Fw(s)25 b Fv(2)g Fy([0)p Fw(;)15 b Fy(1])32 b(and)e Fv(j)p Fw(s)20 b Fv(\000)g Fw(s)2475 574 y Fr(0)2514 560 y Fv(j)26 b 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Fr(1)2731 1092 y Fy(\()p Fu(T)2827 1059 y Ft(n)p Fo(\000)p Fr(1)2964 1092 y Fw(;)15 b Fu(Z)p Fy(\),)44 b Fw(g)3212 1106 y Ft(i)p Fr(+1)3374 1092 y Fy(=)456 1199 y Fw(')515 1213 y Fo(\003)555 1199 y Fy(\()p Fw(e)632 1213 y Ft(i)660 1199 y Fy(\))g Fv(2)e Fw(H)918 1213 y Fr(1)957 1199 y Fy(\()p Fu(T)1053 1166 y Ft(n)1100 1199 y Fw(;)15 b Fu(Z)p Fy(\).)73 b(Ob)m(viously)-8 b(,)45 b Fw(g)1836 1213 y Ft(i)1905 1199 y Fy(is)c(an)g Fw(n)p Fy(-dimensional)g(in)m(teger)i(v)m (ector)g(for)456 1307 y Fw(i)34 b Fy(=)f(2)p Fw(;)15 b Fy(3)p Fw(;)g Fv(\001)g(\001)g(\001)34 b Fw(;)15 b(n)p Fy(.)56 b(Because)37 b Fw(')f Fy(is)f(injection,)j(there)e(exists)g Fw(g)2594 1321 y Fr(1)2668 1307 y Fv(2)e Fu(Z)2824 1274 y Ft(n)2906 1307 y Fy(suc)m(h)h(that)h(the)456 1415 y Fw(n)20 b Fv(\002)g Fw(n)30 b Fy(matrix)h Fw(G)1074 1429 y Ft(c)1135 1415 y Fy(=)25 b(\()p Fw(g)1309 1429 y Fr(1)1349 1415 y Fw(;)15 b(g)1432 1429 y Fr(2)1473 1415 y Fw(;)g Fv(\001)g(\001)g(\001)31 b Fw(;)15 b(g)1732 1429 y Ft(n)1780 1415 y Fy(\))31 b(is)g(uni-mo)s(dule,)f(i.e.)42 b(det)p Fw(G)2795 1429 y Ft(c)2856 1415 y Fy(=)26 b Fv(\006)p Fy(1.)42 b(\(see)31 b(the)456 1523 y(app)s(endix)e(for)h(the)g (existence)i(pro)s(of)e(of)g Fw(G)1944 1537 y Ft(c)1979 1523 y Fy(.\))555 1721 y(Let)k Fw(q)g Fy(=)d Fw(G)969 1688 y Fo(\000)p Fr(1)969 1743 y Ft(c)1064 1721 y Fw(x)p Fy(,)j(the)g(Lagrangian)1826 1698 y(~)1815 1721 y Fw(L)p Fy(\()p Fw(q)s(;)e Fy(_)-42 b Fw(q)s Fy(\))31 b(=)g Fw(L)p Fy(\()p Fw(G)2376 1735 y Ft(c)2411 1721 y Fw(q)s(;)15 b(G)2566 1735 y Ft(c)2619 1721 y Fy(_)-42 b Fw(q)s Fy(\))34 b(is)f(also)i(2)p Fw(\031)s Fy(-p)s(erio)s(dic)456 1829 y(in)j Fw(q)s Fy(.)65 b(In)37 b(this)i(sense,)i(w)m(e)e(call)g Fw(x)g Fv(!)g Fw(q)j Fy(=)c Fw(G)2080 1796 y Fo(\000)p Fr(1)2080 1851 y Ft(c)2175 1829 y Fw(x)g Fy(co)s(ordinate)i (transformation)e(on)456 1937 y(torus.)44 b(In)31 b Fw(q)s Fy(-co)s(ordinates,)i(let)1593 1914 y(\026)1559 1937 y Fw(M)k Fy(=)27 b Fu(R)21 b Fv(\002)g Fu(T)2022 1904 y Ft(n)p Fo(\000)p Fr(1)2186 1937 y Fy(=)27 b Fv(f)p Fw(q)2370 1951 y Fr(1)2437 1937 y Fv(2)g Fu(R)p Fw(;)15 b Fy(\()p Fw(q)2707 1951 y Fr(2)2746 1937 y Fw(;)g Fv(\001)g(\001)g (\001)32 b Fw(;)15 b(q)3004 1951 y Ft(n)3051 1937 y Fy(\))27 b Fv(2)g Fu(T)3262 1904 y Ft(n)p Fo(\000)p Fr(1)3399 1937 y Fv(g)456 2045 y Fy(b)s(e)e(a)h(co)m(v)m(ering)h(of)f Fu(T)1157 2012 y Ft(n)1204 2045 y Fy(,)h Fw(\031)h Fy(:)1421 2022 y(\026)1387 2045 y Fw(M)35 b Fv(!)25 b Fw(M)36 b Fy(=)24 b Fu(T)1906 2012 y Ft(n)1953 2045 y Fy(.)39 b(The)25 b(lift)i(of)f(\006)2504 2059 y Ft(c)2538 2045 y Fy(,)h Fw(\031)2645 2012 y Fo(\000)p Fr(1)2739 2045 y Fy(\(\006)2840 2059 y Ft(c)2875 2045 y Fy(\))f(has)f(in\014nitely)456 2153 y(man)m(y)30 b(compact)i(comp)s(onen)m(ts)e Fv(f)p Fy(\006)1678 2120 y Ft(i)1678 2175 y(c)1713 2153 y Fv(g)1758 2167 y Ft(i)p Fo(2)p Fi(Z)1878 2153 y Fy(.)41 b(If)30 b Fw(')g Fy(is)h(linear,)g(\006)2560 2120 y Ft(i)2560 2175 y(c)2619 2153 y Fy(=)25 b Fv(f)p Fw(q)2801 2167 y Fr(1)2866 2153 y Fy(=)g(2)p Fw(i\031)s Fv(g)p Fy(.)555 2344 y(\006)621 2358 y Ft(c)699 2344 y Fy(is)44 b(said)g(admissible)g (for)g(some)h(ergo)s(dic)f(comp)s(onen)m(t)g Fw(\026)2716 2358 y Ft(c)2795 2344 y Fy(if)g Fv(9)f Fw(q)51 b Fy(=)c Fw(G)3267 2311 y Fo(\000)p Fr(1)3267 2366 y Ft(c)3362 2344 y Fw(x)p Fy(-)456 2452 y(co)s(ordinates)h(suc)m(h)g(that)h(the)f (\014rst)f(comp)s(onen)m(t)i(of)f(the)g(rotation)i(v)m(ector)f(of)g Fw(\026)3385 2466 y Ft(c)3419 2452 y Fy(,)456 2559 y Fw(!)513 2573 y Fr(1)552 2559 y Fy(\()p Fw(\026)642 2573 y Ft(c)676 2559 y Fy(\))43 b Fw(>)f Fy(0,)i(and)c(the)h(lift)g(of)g (\006)1669 2573 y Ft(c)1744 2559 y Fy(in)f Fu(R)27 b Fv(\002)g Fu(T)2112 2527 y Ft(n)p Fo(\000)p Fr(1)2289 2559 y Fy(has)41 b(in\014nitely)f(man)m(y)h(compact)456 2667 y(comp)s(onen)m(ts.)55 b(\006)1072 2681 y Ft(c)1141 2667 y Fy(is)35 b(said)h(admissible)f(for)g(some)g Fw(c)p Fy(-minimal)h(measure)f(if)g(it)h(is)f(ad-)456 2775 y(missible)30 b(for)g(eac)m(h)i(ergo)s(dic)f(comp)s(onen)m(t)f(of)h(the)f(measure.) 555 2966 y(Assume)36 b(\006)964 2980 y Ft(c)1035 2966 y Fy(is)g(admissible)g(for)g(all)h Fw(c)p Fy(-minimal)h(measures.)58 b(Let)40 b(\026)-48 b Fw(\015)2944 2980 y Fr(1)3019 2966 y Fy(denote)37 b(the)456 3074 y(\014rst)20 b(comp)s(onen)m(t)h(of)j (\026)-48 b Fw(\015)30 b Fy(=)25 b(\()s(\026)-48 b Fw(\015)1438 3088 y Fr(1)1478 3074 y Fw(;)15 b Fv(\001)g(\001)g(\001)31 b Fw(;)19 b Fy(\026)-49 b Fw(\015)1741 3088 y Ft(n)1789 3074 y Fy(\))21 b(whic)m(h)f(is)h(the)g(lift)g(of)g(a)g Fw(c)p Fy(-semi-static)j(curv)m(e)d Fw(\015)5 b Fy(,)456 3182 y(Clearly)-8 b(,)32 b(in)f(the)g Fw(q)s Fy(-co)s(ordinate)h (system,)g Fv(j)s Fy(\026)-48 b Fw(\015)1976 3196 y Fr(1)2016 3182 y Fy(\()p Fw(t)p Fy(\))p Fv(j)27 b(!)f(1)31 b Fy(as)g Fw(t)26 b Fv(!)h(\0061)p Fy(.)42 b(If)34 b(\026)-48 b Fw(\015)3067 3196 y Fr(1)3106 3182 y Fy(\()p Fw(t)p Fy(\))27 b Fv(!)f(1)456 3290 y Fy(\()p Fv(!)f(\0001)p Fy(\))k(as)h Fw(t)25 b Fv(!)g(1)p Fy(,)k(then)j(\026)-48 b Fw(\015)1516 3304 y Fr(1)1556 3290 y Fy(\()p Fw(t)p Fy(\))25 b Fv(!)h(\0001)i Fy(\()p Fv(!)e(1)p Fy(\))j(as)g Fw(t)c Fv(!)h(\0001)p Fy(.)40 b(W)-8 b(e)30 b(assume)f(that)459 3398 y(\026)-48 b Fw(\015)503 3412 y Fr(1)542 3398 y Fy(\()p Fw(t)p Fy(\))26 b Fv(!)f(\0061)i Fy(as)i Fw(t)c Fv(!)g(\0061)i Fy(whic)m(h)h(can)g(b)s (e)f(realized)i(b)m(y)f(c)m(ho)s(osing)h Fw(g)2857 3412 y Fr(1)2924 3398 y Fy(or)f Fv(\000)p Fw(g)3147 3412 y Fr(1)3215 3398 y Fy(in)f(the)456 3506 y(matrix)j Fw(G)822 3520 y Ft(c)858 3506 y Fy(.)555 3697 y(In)37 b(the)h(follo)m(wing,)j (for)c(eac)m(h)h Fw(c)g Fv(2)f Fp(C)p Fy(,)i(w)m(e)f(in)m(tro)s(duce)f (suc)m(h)g(a)h(co)s(ordinate)g(trans-)456 3807 y(formation)f(on)f (torus)g Fw(q)i Fy(=)d Fw(G)1506 3774 y Fo(\000)p Fr(1)1506 3829 y Ft(c)1601 3807 y Fw(x)p Fy(,)j(w)m(ork)f(in)f Fw(q)s Fy(-co)s(ordinate)h(system,)i(and)3112 3784 y(~)3102 3807 y Fw(L)d Fy(is)g(still)456 3914 y(denoted)d(b)m(y)h Fw(L)p Fy(.)51 b(When)33 b(w)m(e)i(use)e(the)h(co)m(v)m(ering)i(space) 2436 3892 y(\026)2402 3914 y Fw(M)41 b Fy(=)31 b Fu(R)22 b Fv(\002)g Fu(T)2875 3882 y Ft(n)p Fo(\000)p Fr(1)3012 3914 y Fy(,)35 b(it)f(implies)456 4022 y(that)g(this)g(co)s(ordinate)h (transformation)g(has)e(b)s(een)h(done.)52 b(Note)35 b(that)g(the)f Fw(c)3183 3989 y Fo(0)3207 4022 y Fy(-semi-)456 4130 y(static)i(curv)m(es)g(also)g(in)m(tersects)h(\006)1653 4144 y Ft(c)1722 4130 y Fy(transv)m(ersally)g(if)e Fv(j)p Fw(c)2414 4097 y Fo(0)2461 4130 y Fv(\000)23 b Fw(c)p Fv(j)36 b Fy(is)g(su\016cien)m(tly)g(small,)456 4245 y(b)s(ecause)30 b(of)h(the)f(upp)s(er)e(semi-con)m(tin)m(uit)m(y)33 b(of)2082 4222 y(~)2053 4245 y Fv(N)13 b Fy(\()p Fw(c)p Fy(\))31 b(with)f(resp)s(ect)h(to)g Fw(c)f Fy(\([Ma2)r(]\).)555 4436 y(The)23 b(co)s(dimension)g(one)h(h)m(yp)s(ersurface)e(\006)1996 4403 y Fr(0)1996 4459 y Ft(c)2059 4436 y Fy(separates)2481 4413 y(\026)2447 4436 y Fw(M)34 b Fy(in)m(to)24 b(t)m(w)m(o)h(parts,)f (the)g(up-)456 4544 y(p)s(er)e(part)833 4521 y(\026)799 4544 y Fw(M)897 4511 y Fr(+)980 4544 y Fy(and)h(the)h(lo)m(w)m(er)i (part)1756 4521 y(\026)1722 4544 y Fw(M)1820 4511 y Fo(\000)1879 4544 y Fy(.)39 b(Let)24 b(\006)2165 4511 y Fr(0)2165 4566 y Ft(c)2212 4544 y Fy(+)7 b Fw(\016)27 b Fy(denotes)e(the)f Fw(\016)s Fy(-neigh)m(b)s(orho)s(o)s(d)456 4652 y(of)d(\006)616 4619 y Fr(0)616 4674 y Ft(c)677 4652 y Fy(in)808 4629 y(\026)774 4652 y Fw(M)10 b Fy(,)23 b(w)m(e)f(in)m(tro)s(duce)f(a)h (smo)s(oth)g(function)f Fw(\045)k Fv(2)g Fw(C)2397 4619 y Ft(r)2434 4652 y Fy(\()2503 4629 y(\026)2469 4652 y Fw(M)11 b(;)k Fy([0)p Fw(;)g Fy(1]\))24 b(suc)m(h)d(that)h Fw(\045)j Fy(=)g(0)456 4760 y(if)f Fw(q)k Fv(2)722 4737 y Fy(\026)688 4760 y Fw(M)786 4727 y Fo(\000)845 4760 y Fv(n)p Fy(\(\006)991 4727 y Fr(0)991 4782 y Ft(c)1039 4760 y Fy(+)8 b Fw(\016)s Fy(\),)26 b Fw(\045)f Fy(=)g(1)g(if)f Fw(q)k Fv(2)1751 4737 y Fy(\026)1717 4760 y Fw(M)1815 4727 y Fr(+)1874 4760 y Fv(n)p Fy(\(\006)2020 4727 y Fr(0)2020 4782 y Ft(c)2068 4760 y Fy(+)8 b Fw(\016)s Fy(\).)40 b(Let)24 b Fw(\021)k Fy(and)j(\026)-52 b Fw(\026)24 b Fy(b)s(e)f(closed)i(1-forms)456 4868 y(on)31 b Fw(M)42 b Fy(suc)m(h)32 b(that)g([)p Fw(\021)s Fy(])d(=)f Fw(c)k Fy(and)f([)p Fw(\021)25 b Fy(+)j(\026)-52 b Fw(\026)p Fy(])28 b(=)f Fw(c)2025 4835 y Fo(0)2049 4868 y Fy(.)45 b(These)32 b(forms)f(ha)m(v)m(e)i(natural)f(lifts)g(on)490 4953 y(\026)456 4976 y Fw(M)10 b Fy(.)40 b(Let)31 b Fw(\026)25 b Fy(=)g Fw(\045)7 b Fy(\026)-52 b Fw(\026)30 b Fy(and)1600 5216 y Fw(L)1662 5230 y Ft(\021)r(;\026)1791 5216 y Fy(=)25 b Fw(L)20 b Fv(\000)g Fw(\021)k Fv(\000)c Fw(\026:)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 998 251 a Fr(CONNECTING)31 b(ORBITS)g(F)n(OR)f(A) n(UTONOMOUS)g(SYSTEMS)507 b(7)456 450 y Fy(Let)45 b(\026)-62 b Fw(m;)32 b Fy(\026)-62 b Fw(m)816 417 y Fo(0)867 450 y Fy(b)s(e)27 b(t)m(w)m(o)i(p)s(oin)m(ts)e(in)1565 427 y(\026)1531 450 y Fw(M)10 b Fy(.)40 b(Since)27 b Fw(\013)p Fy(\()p Fw(c)p Fy(\))g(=)e Fw(\013)p Fy(\()p Fw(c)2350 417 y Fo(0)2375 450 y Fy(\))g(=)g(0)j(is)g(assumed,)g(w)m(e)g(de\014ne) 1158 692 y Fw(h)1210 654 y Ft(T)1210 714 y(\021)r(;\026)1314 692 y Fy(\()18 b(\026)-63 b Fw(m;)33 b Fy(\026)-63 b Fw(m)1549 654 y Fo(0)1573 692 y Fy(\))25 b(=)109 b(inf)1729 753 y Fm(\015)s Fk(\()p Fl(\000)p Fm(T)8 b Fk(\)=)13 b(\026)-43 b Fm(m)1742 821 y(\015)s Fk(\()p Fm(T)8 b Fk(\)=)13 b(\026)-43 b Fm(m)1970 800 y Fl(0)2022 568 y Fq(Z)2113 594 y Ft(T)2072 774 y Fo(\000)p Ft(T)2197 692 y Fw(L)2259 706 y Ft(\021)r(;\026)2363 692 y Fy(\()p Fw(d\015)5 b Fy(\()p Fw(t)p Fy(\)\))p Fw(dt:)456 985 y FC(Lemma)24 b(2.1.)35 b Fj(If)24 b(the)g(r)-5 b(otation)27 b(ve)-5 b(ctor)25 b(of)g(e)-5 b(ach)25 b(er)-5 b(go)g(dic)25 b(c)-5 b(omp)g(onent)27 b(of)e(a)f Fw(c)p Fj(-minimal)456 1093 y(me)-5 b(asur)g(e)36 b(has)f(p)-5 b(ositive)36 b(\014rst)f(c)-5 b(omp)g(onent)37 b Fw(!)2007 1107 y Fr(1)2046 1093 y Fy(\()p Fw(\026)2136 1107 y Ft(c)2171 1093 y Fy(\))29 b Fw(>)g Fy(0)p Fj(,)35 b Fv(j)18 b Fy(\026)-63 b Fw(m)22 b Fv(\000)39 b Fy(\026)-63 b Fw(m)2742 1060 y Fo(0)2766 1093 y Fv(j)29 b Fw(>)f Fy(0)35 b Fj(and)h Fv(j)p Fw(c)22 b Fv(\000)g Fw(c)3396 1060 y Fo(0)3419 1093 y Fv(j)456 1201 y Fj(is)32 b(suitably)h(smal)5 b(l,)34 b(then)1535 1378 y Fy(lim)1494 1435 y Fm(T)8 b Fl(!)p Fk(0)p Fm(;)p Fk(or)1515 1486 y Fm(T)g Fl(!1)1718 1378 y Fw(h)1770 1340 y Ft(T)1770 1400 y(\021)r(;\026)1875 1378 y Fy(\()17 b(\026)-62 b Fw(m;)32 b Fy(\026)-62 b Fw(m)2110 1340 y Fo(0)2133 1378 y Fy(\))26 b(=)f Fv(1)p Fw(:)456 1740 y Fj(Pr)-5 b(o)g(of.)43 b Fy(The)30 b(case)h(for)g Fw(T)38 b Fv(!)26 b Fy(0)31 b(is)f(an)h(immediate)g(consequence)h(of)e (the)h(sup)s(er-linear)456 1847 y(gro)m(wth)f(of)h Fw(L)f Fy(in)47 b(_)-42 b Fw(q)s Fy(.)555 2038 y(Let)31 b(\026)-48 b Fw(\015)767 2005 y Ft(T)762 2061 y(\021)r(;\026)866 2038 y Fy(:)39 b([)p Fv(\000)p Fw(T)8 b(;)15 b(T)e Fy(])25 b Fv(!)1393 2015 y Fy(\026)1359 2038 y Fw(M)37 b Fy(b)s(e)27 b(the)g(minimizer)g(of)h Fw(h)2332 2005 y Ft(T)2332 2061 y(\021)r(;\026)2436 2038 y Fy(\()18 b(\026)-63 b Fw(m;)33 b Fy(\026)-63 b Fw(m)2671 2005 y Fo(0)2695 2038 y Fy(\),)28 b(and)e Fw(d\015)3055 2005 y Ft(T)3050 2061 y(\021)r(;\026)3182 2038 y Fy(denote)456 2161 y(the)c(pro)5 b(jection)23 b(of)f Fw(d)s Fy(\026)-48 b Fw(\015)1222 2128 y Ft(T)1217 2183 y(\021)r(;\026)1343 2161 y Fy(to)23 b Fw(T)13 b(M)d Fy(.)38 b(Cho)s(osing)22 b Fw(\026)j Fy(=)g(0,)f(let)f Fw(\027)2500 2175 y Ft(T)2577 2161 y Fy(denote)g(the)f(probabilit)m(y) 456 2283 y(measure)31 b(ev)m(enly)i(distributed)d(along)j Fw(\015)1855 2250 y Ft(T)1850 2307 y(\021)r(;)p Fr(0)1978 2283 y Fy(and)e(let)i Fw(\027)k Fy(b)s(e)31 b(a)h(p)s(oin)m(t)g(of)g (accum)m(ulation)456 2391 y(of)e Fw(\027)604 2405 y Ft(T)659 2391 y Fy(,)h(w)m(e)f(ha)m(v)m(e:)572 2628 y Fw(A)640 2642 y Ft(c)675 2628 y Fy(\()p Fw(\027)6 b Fy(\))25 b(=)917 2504 y Fq(Z)1023 2628 y Fw(L)1085 2642 y Ft(\021)r(;)p Fr(0)1182 2628 y Fw(d\027)31 b Fv(\024)25 b Fy(lim)15 b(inf)1431 2689 y Ft(T)10 b Fo(!1)1668 2504 y Fq(Z)1774 2628 y Fw(L)1836 2642 y Ft(\021)r(;)p Fr(0)1933 2628 y Fw(d\027)2025 2642 y Ft(T)2106 2628 y Fy(=)25 b(lim)15 b(inf)2232 2689 y Ft(T)10 b Fo(!1)2512 2566 y Fy(1)p 2479 2607 112 4 v 2479 2690 a(2)p Fw(T)2615 2504 y Fq(Z)2706 2531 y Ft(T)2666 2710 y Fo(\000)p Ft(T)2791 2628 y Fw(L)2853 2642 y Ft(\021)r(;)p Fr(0)2949 2628 y Fy(\()p Fw(d\015)5 b Fy(\()p Fw(t)p Fy(\)\))p Fw(dt:)456 2867 y Fy(Ob)m(viously)-8 b(,)41 b(there)e(exists)g Fw(C)45 b Fy(suc)m(h)38 b(that)1939 2794 y Fq(R)2015 2867 y Fw(L)2077 2881 y Ft(\021)r(;)p Fr(0)2174 2867 y Fw(d\027)2266 2881 y Ft(T)2360 2867 y Fv(\024)h Fw(C)45 b Fy(for)38 b(all)h Fw(T)52 b Fv(\025)39 b Fy(1.)65 b(Giv)m(en)456 2976 y(an)m(y)39 b(1-form)h Fw(\025)f Fy(on)g(M,)h(the)g(mapping)e Fw(\026)i Fv(!)2085 2903 y Fq(R)2161 2976 y Fw(\025d\026)f Fy(is)h(con)m(tin)m(uous)g(on)f (the)g(set)h(of)456 3085 y(probabilit)m(y)31 b(measure)f Fw(\026)g Fy(suc)m(h)g(that)1762 3012 y Fq(R)1838 3085 y Fw(Ld\026)c Fv(\024)f Fw(C)37 b Fy(\(cf.)k([Ma1)r(]\).)g(Because)49 b(\026)-62 b Fw(m;)33 b Fy(\026)-63 b Fw(m)3269 3052 y Fo(0)3323 3085 y Fy(are)456 3193 y(\014xed,)41 b Fw(!)769 3207 y Fr(1)808 3193 y Fy(\()p Fw(\027)888 3207 y Ft(T)943 3193 y Fy(\))g(=)f(\()18 b(\026)-63 b Fw(m)1245 3160 y Fo(0)1245 3218 y Fr(1)1311 3193 y Fv(\000)43 b Fy(\026)-62 b Fw(m)1488 3207 y Fr(1)1527 3193 y Fy(\))p Fw(=)p Fy(2)42 b Fv(!)e Fy(0)g(as)g Fw(T)53 b Fv(!)40 b(1)p Fy(,)i(where)56 b(\026)-62 b Fw(m)2778 3160 y Fo(0)2778 3218 y Fr(1)2817 3193 y Fw(;)33 b Fy(\026)-63 b Fw(m)2937 3207 y Fr(1)3016 3193 y Fy(denote)40 b(the)456 3301 y(\014rst)g(co)s(ordinate)j(of)58 b(\026)-62 b Fw(m)1307 3268 y Fo(0)1330 3301 y Fw(;)33 b Fy(\026)-63 b Fw(m)42 b Fy(resp)s(ectiv)m(ely)-8 b(.)76 b(Consequen)m(tly)-8 b(,)45 b(the)c(\014rst)g(comp)s(onen)m(t)456 3409 y(of)c(rotation)i(v)m(ector)f(of)g Fw(\027)43 b Fy(is)37 b(zero.)62 b(Because)39 b(of)e Fw(!)2268 3423 y Fr(1)2307 3409 y Fy(\()p Fw(\026)2397 3423 y Ft(c)2432 3409 y Fy(\))g Fw(>)f Fy(0)i(for)f(eac)m(h)i Fw(c)p Fy(-minimal)456 3517 y(measure)30 b Fw(\026)865 3531 y Ft(c)899 3517 y Fy(,)h(w)m(e)g(obtain)g(that)g(the)f(a)m(v)m(erage)j(action)f(of)e Fw(\027)36 b Fy(is)31 b(p)s(ositiv)m(e)g Fw(A)3004 3531 y Ft(c)3039 3517 y Fy(\()p Fw(\027)6 b Fy(\))26 b Fw(>)f Fy(0.)555 3708 y(As)37 b(w)m(e)h(assume)e(the)i(condition)f(of)g(sup)s (er-linear)g(gro)m(wth,)i(there)e(exists)h Fw(D)h(>)d Fy(0)456 3816 y(suc)m(h)c(that)h Fv(j)900 3804 y Fy(_)890 3816 y(\026)-48 b Fw(\015)939 3783 y Ft(T)934 3838 y(\021)r(;\026)1038 3816 y Fv(j)30 b(\024)e Fw(D)36 b Fy(for)c(eac)m(h)i Fw(t)29 b Fv(2)f Fy([)p Fv(\000)p Fw(T)8 b(;)15 b(T)e Fy(])33 b(and)f(all)h Fw(T)42 b(>)29 b Fy(1.)48 b(By)33 b(letting)h Fv(j)p Fw(c)22 b Fv(\000)g Fw(c)3396 3783 y Fo(0)3419 3816 y Fv(j)456 3938 y Fy(b)s(e)36 b(suitably)i(small,)h(w) m(e)f(can)g(c)m(ho)s(ose)g Fw(\026)f Fy(suc)m(h)g(that)h Fv(j)p Fw(L)2416 3952 y Ft(\021)r(;\026)2545 3938 y Fv(\000)24 b Fw(L)2702 3952 y Ft(\021)r(;)p Fr(0)2799 3938 y Fv(j)37 b Fw(<)2979 3902 y Fr(1)p 2979 3917 36 4 v 2979 3970 a(2)3024 3938 y Fw(A)p Fy(\()p Fw(\027)6 b Fy(\))38 b(if)f(w)m(e)456 4053 y(restrict)31 b(them)f(in)g(the)h(region)g Fv(fj)17 b Fy(_)-42 b Fw(q)s Fv(j)26 b(\024)f Fw(D)s Fv(g)20 b(\002)2067 4030 y Fy(\026)2033 4053 y Fw(M)10 b Fy(.)41 b(Th)m(us,)29 b(w)m(e)i(obtain)1092 4294 y Fw(h)1144 4257 y Ft(T)1144 4317 y(\021)r(;\026)1249 4294 y Fy(\()17 b(\026)-62 b Fw(m;)33 b Fy(\026)-63 b Fw(m)1484 4257 y Fo(0)1508 4294 y Fy(\))25 b(=)1654 4171 y Fq(Z)1745 4197 y Ft(T)1705 4377 y Fo(\000)p Ft(T)1830 4294 y Fw(L)1892 4308 y Ft(\021)r(;\026)1995 4294 y Fy(\()p Fw(d)s Fy(\026)-48 b Fw(\015)2129 4257 y Ft(T)2124 4317 y(\021)r(;\026)2229 4294 y Fy(\()p Fw(t)p Fy(\)\))p Fw(dt)-1991 b Fy(\(2.1\))1568 4562 y Fw(>)1654 4438 y Fq(Z)1745 4464 y Ft(T)1705 4644 y Fo(\000)p Ft(T)1830 4562 y Fw(L)1892 4576 y Ft(\021)r(;)p Fr(0)1988 4562 y Fy(\()p Fw(d)s Fy(\026)-48 b Fw(\015)2122 4524 y Ft(T)2117 4584 y(\021)r(;\026)2222 4562 y Fy(\()p Fw(t)p Fy(\)\))p Fw(dt)21 b Fv(\000)f Fw(A)p Fy(\()p Fw(\027)6 b Fy(\))p Fw(T)1568 4829 y Fv(\025)1654 4705 y Fq(Z)1745 4732 y Ft(T)1705 4912 y Fo(\000)p Ft(T)1830 4829 y Fw(L)1892 4843 y Ft(\021)r(;)p Fr(0)1988 4829 y Fy(\()p Fw(d)s Fy(\026)-48 b Fw(\015)2122 4792 y Ft(T)2117 4852 y(\021)r(;)p Fr(0)2215 4829 y Fy(\()p Fw(t)p Fy(\)\))p Fw(dt)21 b Fv(\000)f Fw(A)p Fy(\()p Fw(\027)6 b Fy(\))p Fw(T)1568 5022 y Fv(\025)p Fw(A)p Fy(\()p Fw(\027)g Fy(\))p Fw(T)38 b Fv(!)26 b(1)121 b Fy(as)30 b Fw(T)38 b Fv(!)26 b(1)p Fw(;)456 5198 y Fy(this)k(completes)i(the)e(pro)s(of.)1932 b Fh(\003)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 456 251 a Fr(8)1047 b(X.)23 b(LI)h(AND)f(C.-Q.)f (CHENG)456 450 y FC(De\014nition)35 b(2.1.)42 b Fj(The)33 b(pseudo)g(c)-5 b(onne)g(cting)34 b(orbit)f(set)g(is)f(de\014ne)-5 b(d)34 b(as:)871 626 y Fg(C)932 640 y Ft(\021)r(;\026)1061 626 y Fy(=)1157 525 y Fq(n)1221 626 y Fy(\026)-49 b Fw(\015)31 b Fy(:)25 b Fu(R)g Fv(!)1586 603 y Fy(\026)1552 626 y Fw(M)11 b(;)47 b Fy(for)31 b(all)i Fw(T)38 b(>)25 b Fy(0)33 b(the)e(follo)m(wing)g(holds)1233 739 y Fq(Z)1324 766 y Ft(T)1283 946 y Fo(\000)p Ft(T)1408 863 y Fw(L)1470 877 y Ft(\021)r(;\026)1574 863 y Fy(\()p Fw(d)s Fy(\026)-48 b Fw(\015)6 b Fy(\()p Fw(t)p Fy(\)\))p Fw(dt)26 b Fy(=)52 b(inf)2049 925 y Ft(T)2100 906 y Fl(0)2122 925 y Ft(>)p Fr(0)2228 863 y Fw(h)2280 826 y Ft(T)2331 802 y Fl(0)2280 886 y Ft(\021)r(;\026)2384 863 y Fy(\()s(\026)-48 b Fw(\015)6 b Fy(\()p Fv(\000)p Fw(T)13 b Fy(\))p Fw(;)18 b Fy(\026)-48 b Fw(\015)5 b Fy(\()p Fw(T)13 b Fy(\)\))2942 762 y Fq(o)3004 863 y Fw(:)456 1087 y FC(Prop)s(osition)44 b(2.1.)i Fj(If)39 b Fw(\013)p Fy(\([)p Fw(\021)s Fy(]\))g(=)e Fw(\013)p Fy(\([)p Fw(\021)30 b Fy(+)i(\026)-53 b Fw(\026)p Fy(]\))38 b(=)f(0)p Fj(,)k(the)f(r)-5 b(otation)41 b(ve)-5 b(ctor)40 b(of)f(e)-5 b(ach)456 1195 y(er)g(go)g(dic)30 b Fy([)p Fw(\021)s Fy(])p Fj(-minimal)i(me)-5 b(asur)g(e)31 b(as)f(wel)5 b(l)30 b(as)g(of)g(e)-5 b(ach)30 b(er)-5 b(go)g(dic)31 b Fy([)p Fw(\021)17 b Fy(+)k(\026)-52 b Fw(\026)o Fy(])p Fj(-minimal)31 b(me)-5 b(a-)456 1303 y(sur)g(e)34 b(has)g(p)-5 b(ositive)35 b(\014rst)f(c)-5 b(omp)g(onent,)37 b(then)d(the)g(set)g Fg(C)2375 1317 y Ft(\021)r(;\026)2512 1303 y Fj(is)g(non-empty.)46 b(F)-7 b(or)34 b(e)-5 b(ach)456 1422 y(curve)29 b Fw(\015)5 b Fj(:)41 b Fu(R)25 b Fv(!)g Fw(M)40 b Fj(such)30 b(that)35 b Fy(\026)-48 b Fw(\015)30 b Fv(2)25 b Fg(C)1766 1436 y Ft(\021)r(;\026)1870 1422 y Fj(,)30 b(the)h Fw(!)s Fj(-limit)e(set)i Fw(!)s Fy(\()p Fw(d\015)5 b Fy(\))26 b Fv(\022)2904 1399 y Fy(~)2873 1422 y Fv(A)p Fy(\([)p Fw(\021)18 b Fy(+)k(\026)-52 b Fw(\026)p Fy(]\))30 b Fj(and)456 1540 y(the)j Fw(\013)p Fj(-limit)g(set)f Fw(\013)p Fy(\()p Fw(d\015)5 b Fy(\))27 b Fv(\022)1436 1517 y Fy(~)1405 1540 y Fv(A)p Fy(\([)p Fw(\021)s Fy(]\))p Fj(.)456 1791 y(Pr)-5 b(o)g(of.)43 b Fy(Denote)23 b(b)m(y)j(\026)-49 b Fw(\015)1205 1805 y Ft(\021)r(;\026)1309 1791 y Fy(\()p Fw(t;)33 b Fy(\026)-63 b Fw(m)q(;)32 b Fy(\026)-62 b Fw(m)1618 1758 y Fo(0)1641 1791 y Fy(\))26 b(:)g([)p Fv(\000)p Fw(T)8 b(;)15 b(T)e Fy(])25 b Fv(!)g Fw(M)32 b Fy(the)23 b(minimizer)f(of)h(the)f(quan)m(tit)m(y)456 1910 y(inf)566 1925 y Ft(T)617 1906 y Fl(0)639 1925 y Ft(>)p Fr(0)749 1910 y Fw(h)801 1877 y Ft(T)852 1854 y Fl(0)801 1933 y Ft(\021)r(;\026)905 1910 y Fy(\()c(\026)-63 b Fw(m;)33 b Fy(\026)-63 b Fw(m)1140 1877 y Fo(0)1164 1910 y Fy(\),)31 b(i.e.)45 b(\026)-49 b Fw(\015)1458 1924 y Ft(\021)r(;\026)1563 1910 y Fy(\()p Fv(\000)p Fw(T)8 b(;)32 b Fy(\026)-62 b Fw(m;)32 b Fy(\026)-62 b Fw(m)1970 1877 y Fo(0)1993 1910 y Fy(\))26 b(=)42 b(\026)-62 b Fw(m)o Fy(,)34 b(\026)-48 b Fw(\015)2332 1924 y Ft(\021)r(;\026)2436 1910 y Fy(\()p Fw(T)8 b(;)33 b Fy(\026)-63 b Fw(m;)33 b Fy(\026)-63 b Fw(m)2772 1877 y Fo(0)2796 1910 y Fy(\))25 b(=)43 b(\026)-63 b Fw(m)3032 1877 y Fo(0)3086 1910 y Fy(and)934 2142 y([)p Fw(A)p Fy(\()s(\026)-48 b Fw(\015)1109 2156 y Ft(\021)r(;\026)1214 2142 y Fy(\)])26 b(=)1396 2018 y Fq(Z)1487 2045 y Ft(T)1446 2224 y Fo(\000)p Ft(T)1572 2142 y Fw(L)1634 2156 y Ft(\021)r(;\026)1737 2142 y Fy(\()p Fw(d)s Fy(\026)-48 b Fw(\015)1866 2156 y Ft(\021)r(;\026)1971 2142 y Fy(\()p Fw(t)p Fy(\)\))p Fw(dt)26 b Fy(=)52 b(inf)2311 2204 y Ft(T)2362 2185 y Fl(0)2385 2204 y Ft(>)p Fr(0)2490 2142 y Fw(h)2542 2105 y Ft(T)2593 2081 y Fl(0)2542 2165 y Ft(\021)r(;\026)2646 2142 y Fy(\()18 b(\026)-63 b Fw(m)q(;)32 b Fy(\026)-62 b Fw(m)2882 2105 y Fo(0)2905 2142 y Fy(\))p Fw(:)456 2348 y Fy(Because)40 b(of)e(the)h(lemma)g(2.1,)j(this)c (in\014m)m(um)g(is)g(attained)i(for)e(\014nite)h Fw(T)52 b(>)38 b Fy(0)h(if)56 b(\026)-62 b Fw(m)456 2456 y Fy(and)63 b(\026)-62 b Fw(m)729 2423 y Fo(0)800 2456 y Fy(are)47 b(t)m(w)m(o)i(di\013eren)m(t)f(p)s(oin)m(ts)f(in)1980 2433 y(\026)1946 2456 y Fw(M)10 b Fy(.)91 b(The)47 b(sup)s(er-linear)f (gro)m(wth)i(condi-)456 2564 y(tion)40 b(of)h Fw(L)f Fy(guaran)m(tees)i(that)f Fw(T)54 b Fv(!)42 b(1)e Fy(as)h Fv(\000)16 b Fy(\026)-61 b Fw(m)2182 2578 y Fr(1)2221 2564 y Fw(;)32 b Fy(\026)-62 b Fw(m)2341 2531 y Fo(0)2341 2588 y Fr(1)2422 2564 y Fv(!)42 b(1)p Fy(,)h(where)57 b(\026)-63 b Fw(m)3066 2578 y Fr(1)3146 2564 y Fy(denotes)456 2672 y(the)33 b(\014rst)g(co)s(ordinate)h(of)51 b(\026)-62 b Fw(m)p Fy(.)50 b(Giv)m(en)34 b(an)g(in)m(terv)-5 b(al)35 b([)p Fv(\000)p Fw(T)2395 2686 y Fr(0)2434 2672 y Fw(;)15 b(T)2527 2686 y Fr(0)2567 2672 y Fy(],)35 b(for)e(su\016cien)m(tly)h (large)456 2780 y Fv(\000)16 b Fy(\026)-61 b Fw(m)607 2794 y Fr(1)646 2780 y Fw(;)32 b Fy(\026)-62 b Fw(m)766 2794 y Fr(1)805 2780 y Fy(,)33 b(the)f(set)g Fv(f)s Fy(\026)-48 b Fw(\015)1256 2794 y Ft(c)1291 2780 y Fy(\()p Fw(t;)33 b Fy(\026)-63 b Fw(m)q(;)32 b Fy(\026)-62 b Fw(m)1600 2747 y Fo(0)1623 2780 y Fy(\))p Fv(j)1683 2798 y Fr([)p Fo(\000)p Ft(T)1799 2807 y Fk(0)1834 2798 y Ft(;T)1895 2807 y Fk(0)1929 2798 y Fr(])1953 2780 y Fv(g)32 b Fy(is)g(pre-compact) h(in)e Fw(C)2824 2747 y Fr(1)2863 2780 y Fy(\([)p Fv(\000)p Fw(T)3047 2794 y Fr(0)3087 2780 y Fw(;)15 b(T)3180 2794 y Fr(0)3220 2780 y Fy(])p Fw(;)g(M)10 b Fy(\).)456 2888 y(Let)25 b Fw(T)666 2902 y Fr(0)731 2888 y Fv(!)g(1)p Fy(,)i(b)m(y)e(diagonal)i(extraction)g(argumen)m(t,)g(w)m(e)e(can)h (\014nd)e(a)i(subsequence)e(of)456 2996 y Fv(f)s Fy(\026)-48 b Fw(\015)548 3010 y Ft(c)583 2996 y Fy(\()p Fw(t;)33 b Fy(\026)-63 b Fw(m;)33 b Fy(\026)-63 b Fw(m)891 2963 y Fo(0)915 2996 y Fy(\))p Fv(g)34 b Fy(whic)m(h)f(con)m(v)m(erges)j Fw(C)1775 2963 y Fr(1)1814 2996 y Fy(-uniformly)c(on)i(eac)m(h)g (compact)h(set)f(to)g(a)g Fw(C)3375 2963 y Fr(1)3414 2996 y Fy(-)456 3104 y(curv)m(e)29 b(\026)-48 b Fw(\015)5 b Fy(:)39 b Fu(R)25 b Fv(!)1051 3081 y Fy(\026)1017 3104 y Fw(M)10 b Fy(.)39 b(Ob)m(viously)-8 b(,)27 b(this)f(curv)m(e)h(b)s (elongs)f(to)h Fg(C)2522 3118 y Ft(\021)r(;\026)2625 3104 y Fy(.)39 b(On)26 b(the)g(other)g(hand,)456 3212 y(eac)m(h)f(curv)m(e)i(\026)-48 b Fw(\015)30 b Fv(2)25 b Fg(C)1114 3226 y Ft(\021)r(;\026)1241 3212 y Fy(is)f(an)g(accum)m (ulation)h(p)s(oin)m(t)f(of)g(the)g(set)g Fv(f)s Fy(\026)-48 b Fw(\015)2702 3226 y Ft(c)2737 3212 y Fy(\()p Fw(t;)33 b Fy(\026)-63 b Fw(m)q(;)32 b Fy(\026)-62 b Fw(m)3046 3179 y Fo(0)3069 3212 y Fy(\))p Fv(j)3129 3230 y Fr([)p Fo(\000)p Ft(T)3245 3239 y Fk(0)3280 3230 y Ft(;T)3341 3239 y Fk(0)3375 3230 y Fr(])3399 3212 y Fv(g)456 3325 y Fy(\()p Fw(T)544 3339 y Fr(0)609 3325 y Fv(!)25 b(1)30 b Fy(as)g Fv(\000)17 b Fy(\026)-62 b Fw(m)1108 3339 y Fr(1)1147 3325 y Fw(;)32 b Fy(\026)-62 b Fw(m)1267 3292 y Fo(0)1267 3349 y Fr(1)1332 3325 y Fv(!)25 b(1)p Fy(\).)41 b(In)29 b(fact,)i(giv)m(en)g(a)f(curv)m(e)k(\026)-49 b Fw(\015)31 b Fv(2)24 b Fg(C)2735 3339 y Ft(\021)r(;\026)2839 3325 y Fy(,)30 b(the)g(minimizer)456 3444 y(of)e(inf)667 3459 y Ft(T)718 3440 y Fl(0)741 3459 y Ft(>)p Fr(0)850 3444 y Fw(h)902 3411 y Ft(T)953 3387 y Fl(0)902 3468 y Ft(\021)r(;)p Fr(0)999 3444 y Fy(\()s(\026)-48 b Fw(\015)6 b Fy(\()p Fv(\000)p Fw(T)13 b Fy(\))p Fw(;)19 b Fy(\026)-49 b Fw(\015)6 b Fy(\()p Fw(T)13 b Fy(\)\))28 b(is)k(\026)-49 b Fw(\015)6 b Fv(j)1753 3462 y Fr([)p Fo(\000)p Ft(T)g(;T)k Fr(])1969 3444 y Fy(.)39 b(Th)m(us,)31 b(\026)-48 b Fw(\015)33 b Fy(is)28 b(an)g(accum)m(ulation)i(p)s(oin)m(t)e(of)456 3558 y(the)i(sequence)h Fv(f)s Fy(\026)-48 b Fw(\015)6 b Fv(j)1112 3576 y Fr([)p Fo(\000)p Ft(T)g(;T)k Fr(])1327 3558 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Ft(T)1473 5217 y Fl(0)1495 5235 y Ft(>)p Fr(0)1601 5174 y Fw(h)1653 5136 y Ft(T)1704 5113 y Fl(0)1653 5196 y Ft(\021)r(;)p Fr(0)1750 5174 y Fy(\()s(\026)-48 b Fw(\015)5 b Fy(\()p Fv(\000)p Fw(T)13 b Fy(\))p Fw(;)19 b Fy(\026)-49 b Fw(\015)6 b Fy(\()p Fw(T)13 b Fy(\)\))26 b Fw(<)f(a)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 998 251 a Fr(CONNECTING)31 b(ORBITS)g(F)n(OR)f(A) n(UTONOMOUS)g(SYSTEMS)507 b(9)456 450 y Fy(holds)33 b(for)h(eac)m(h)i (curv)m(e)i(\026)-49 b Fw(\015)37 b Fv(2)31 b Fg(C)1532 464 y Ft(\021)r(;)p Fr(0)1663 450 y Fy(and)j(eac)m(h)h Fw(T)45 b(>)31 b Fy(0.)53 b(In)33 b(fact,)k(w)m(e)d(pic)m(k)h(up)e(the) i(lift)456 561 y(of)c(one)g(semi-static)i(curv)m(e)e Fw(\020)7 b Fy(,)30 b(denoted)h(b)m(y)2007 537 y(\026)1999 561 y Fw(\020)6 b Fy(,)31 b(and)g(join)j(\026)-49 b Fw(\015)5 b Fy(\()p Fv(\000)p Fw(T)13 b Fy(\))31 b(to)h(an)f(in)m(tersection)456 668 y(p)s(oin)m(t)40 b(of)825 645 y(\026)817 668 y Fw(\020)47 b Fy(with)40 b Fv(f)p Fw(q)1207 682 y Fr(1)1289 668 y Fy(=)45 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Fy(and)29 b Fv(j)7 b Fy(\026)-52 b Fw(q)2710 1917 y Ft(i)2759 1950 y Fv(\000)27 b Fy(\026)-52 b Fw(q)2894 1917 y Fo(\003)p Ft(;i)2977 1950 y Fv(j)25 b Fw(<)g(\017)p Fy(.)555 2144 y(F)-8 b(or)32 b(the)g(third)e(step,)i(w)m(e)g(let)1599 2120 y(\026)1591 2144 y Fw(\020)6 b Fv(j)1662 2158 y Ft(I)1693 2168 y Fm(i;j)1802 2144 y Fy(denote)32 b(the)f(segmen)m(t)i(of)2718 2120 y(\026)2710 2144 y Fw(\020)38 b Fy(b)s(et)m(w)m(een)h(\026)-52 b Fw(q)3184 2111 y Fo(\003)p Ft(;i)3298 2144 y Fy(and)462 2263 y(\026)h Fw(q)500 2230 y Fo(\003)p Ft(;j)591 2263 y Fy(,)34 b(\026)-49 b Fw(\015)6 b Fv(j)724 2277 y Ft(I)755 2287 y Fm(i;j)862 2263 y Fy(the)31 b(segmen)m(t)h(of)h(\026)-48 b Fw(\015)35 b Fy(b)s(et)m(w)m(een)j(\026)-51 b Fw(q)1952 2230 y Ft(i)2010 2263 y Fy(and)36 b(\026)-51 b Fw(q)2231 2230 y Ft(j)2267 2263 y Fy(.)41 b(It)30 b(is)h(easy)g(to)g(see)1373 2427 y([)p Fw(A)1466 2441 y Ft(c)1501 2427 y Fy(\()s(\026)-48 b Fw(\015)5 b Fv(j)1613 2441 y Ft(I)1644 2451 y Fm(i;j)1722 2427 y Fy(\)])21 b 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4967 y Ft(i)p Fr(+1)1871 4953 y Fv(\000)22 b Fw(c)2003 4967 y Ft(i)2066 4953 y Fj(and)37 b(smo)-5 b(oth)37 b(functions)f Fw(\045)3002 4967 y Ft(i)3066 4953 y Fj(on)3232 4930 y Fy(\026)3198 4953 y Fw(M)46 b Fj(for)456 5061 y Fw(i)25 b Fy(=)g(0)p Fw(;)15 b Fy(1)p Fw(;)g Fv(\001)g(\001)g(\001)33 b Fw(;)15 b(k)24 b Fv(\000)c Fy(1)p Fj(,)33 b(such)g(that)g(the)g(L)-5 b(agr)g(angian)1561 5216 y Fw(L)1623 5230 y Ft(\021)1658 5240 y Fm(i)1685 5230 y Ft(;\026)1747 5240 y Fm(i)1803 5216 y Fy(=)25 b Fw(L)20 b Fv(\000)f Fw(\021)2116 5230 y Ft(i)2165 5216 y Fv(\000)h Fw(\026)2311 5230 y Ft(i)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 456 251 a Fr(10)1012 b(X.)23 b(LI)h(AND)f(C.-Q.) f(CHENG)456 450 y Fj(with)33 b Fw(\026)708 464 y Ft(i)761 450 y Fy(=)25 b Fw(\045)7 b Fy(\026)-52 b Fw(\026)959 464 y Ft(i)1020 450 y Fj(determines)33 b(the)g(set)g Fg(C)1837 464 y Ft(\021)1872 474 y Fm(i)1898 464 y Ft(;\026)1960 474 y Fm(i)2023 450 y Fj(with)h(the)f(pr)-5 b(op)g(erties:)555 642 y Fy(\(i\))q Fj(,)33 b(e)-5 b(ach)33 b(curve)i Fy(\026)-48 b Fw(\015)31 b Fv(2)24 b Fg(C)1385 656 y Ft(\021)1420 666 y Fm(i)1447 656 y Ft(;\026)1509 666 y Fm(i)1572 642 y Fj(determines)33 b(an)g(orbit)h Fw(d\015)c Fy(=)25 b(\()p Fw(\015)5 b(;)30 b Fy(_)-40 b Fw(\015)6 b Fy(\))33 b Fj(of)g Fw(\036)3012 609 y Ft(t)3012 669 y(L)3064 642 y Fj(;)555 844 y Fy(\(ii\))q Fj(,)41 b(e)-5 b(ach)39 b(of)h(these)f(orbits)h(c)-5 b(onne)g(cts)1968 821 y Fy(~)1937 844 y Fv(A)o Fy(\()p Fw(c)2083 858 y Ft(i)2112 844 y Fy(\))39 b Fj(to)2333 821 y Fy(~)2302 844 y Fv(A)o Fy(\()p Fw(c)2448 858 y Ft(i)p Fr(+1)2567 844 y Fy(\))p Fj(,)i(i.e.,)f(the)f Fw(\013)p Fj(-limit)g(set)456 959 y Fw(\013)p Fy(\()p Fw(d\015)5 b Fy(\))26 b Fv(\022)837 936 y Fy(~)805 959 y Fv(A)p Fy(\()p Fw(c)952 973 y Ft(i)981 959 y Fy(\))33 b Fj(and)g Fw(!)s Fj(-limit)g(set)f Fw(!)s Fy(\()p Fw(d\015)5 b Fy(\))27 b Fv(\022)2058 936 y Fy(~)2027 959 y Fv(A)o Fy(\()p Fw(c)2173 973 y Ft(i)p Fr(+1)2292 959 y Fy(\))p Fj(.)456 1206 y(Pr)-5 b(o)g(of.)43 b Fy(By)37 b(de\014nition,)i(there)e(is)g(a)h(path)f(\000:)54 b([0)p Fw(;)15 b Fy(1])38 b Fv(!)f Fw(H)2528 1173 y Fr(1)2567 1206 y Fy(\()p Fw(M)5 b(;)15 b Fu(R)p Fy(\))38 b(with)f(\000\(0\))h(=)e Fw(c)456 1314 y Fy(and)k(\000\(1\))k(=)f Fw(c)1012 1281 y Fo(0)1076 1314 y Fy(suc)m(h)e(that)h(for)f(eac)m(h)h Fw(s)g Fv(2)h Fy([0)p Fw(;)15 b Fy(1])43 b(there)e(exists)h Fw(\017)h(>)g Fy(0)e(suc)m(h)g(that)456 1424 y(\000\()p Fw(s)591 1391 y Fo(0)614 1424 y Fy(\))22 b Fv(\000)f Fy(\000\()p Fw(s)p Fy(\))30 b Fv(2)e Fw(V)1126 1391 y Fo(?)1105 1455 y Fr(\000\()p Ft(s)p Fr(\))1274 1424 y Fy(whenev)m(er)k Fw(s)1718 1391 y Fo(0)1771 1424 y Fv(2)c Fy([0)p Fw(;)15 b Fy(1])35 b(and)d Fv(j)p Fw(s)22 b Fv(\000)f Fw(s)2479 1391 y Fo(0)2502 1424 y Fv(j)30 b Fw(<)e(\017)p Fy(.)48 b(Th)m(us,)33 b(there)g(exist)456 1548 y(a)k(closed)g(form)44 b(\026)-52 b Fw(\026)1089 1562 y Ft(s)1162 1548 y Fy(and)36 b(a)h(neigh)m(b)s(orho)s(o)s(d)e Fw(U)47 b Fy(of)37 b Fv(N)2298 1567 y Fr(\000\()p Ft(s)p Fr(\))2458 1548 y Fv(\\)24 b Fy(\006)2609 1567 y Fr(\000\()p Ft(s)p Fr(\))2781 1548 y Fy(suc)m(h)36 b(that)i([)7 b(\026)-52 b Fw(\026)3276 1562 y Ft(s)3313 1548 y Fy(])36 b(=)456 1662 y(\000\()p Fw(s)591 1629 y Fo(0)614 1662 y Fy(\))10 b Fv(\000)g Fy(\000\()p Fw(s)p Fy(\))24 b(and)h(supp)5 b(\026)-52 b Fw(\026)1348 1676 y Ft(s)1394 1662 y Fv(\\)10 b Fw(U)35 b Fy(=)25 b Fu(?)p Fy(.)38 b(W)-8 b(e)27 b(in)m(tro)s(duce)e(a)g(co)s (ordinate)h(transformation)456 1774 y Fw(q)39 b Fy(=)c Fw(G)713 1735 y Fo(\000)p Fr(1)713 1807 y(\000\()p Ft(s)p Fr(\))850 1774 y Fw(x)h Fy(on)h(torus)g(as)g(b)s(efore)f(and)h(a)g(smo) s(oth)g(function)f Fw(\045)p Fy(:)2824 1751 y(\026)2790 1774 y Fw(M)46 b Fv(!)36 b Fy([0)p Fw(;)15 b Fy(1])39 b(suc)m(h)456 1907 y(that)26 b Fw(\045)f Fy(=)g(0)g(if)h Fw(q)i Fv(2)1154 1884 y Fy(\026)1120 1907 y Fw(M)1218 1874 y Fo(\000)1277 1907 y Fv(n)p Fy(\(\006)1423 1874 y Fr(0)1423 1939 y(\000\()p Ft(s)p Fr(\))1569 1907 y Fy(+)10 b Fw(\016)s Fy(\),)28 b Fw(\045)d Fy(=)g(1)h(if)f Fw(q)j Fv(2)2287 1884 y Fy(\026)2253 1907 y Fw(M)2351 1874 y Fr(+)2410 1907 y Fv(n)p Fy(\(\006)2556 1874 y Fr(0)2556 1939 y(\000\()p Ft(s)p Fr(\))2702 1907 y Fy(+)10 b Fw(\016)s Fy(\).)40 b(A)26 b(su\016cien)m(tly)456 2027 y(small)k Fw(\016)g(>)25 b Fy(0)30 b(can)h(b)s(e)f(c)m(hosen)h(so)f (that)1374 2177 y(\(\006)1475 2140 y Fr(0)1475 2204 y(\000\()p Ft(s)p Fr(\))1631 2177 y Fy(+)20 b Fw(\016)s Fy(\))h Fv(\\)f Fy(\()p Fv(C)1985 2191 y Ft(\021)r(;)p Fr(0)2102 2177 y Fy(+)g(2)p Fw(\016)s Fy(\))27 b Fv(\032)e Fw(U;)456 2340 y Fy(here)30 b([)p Fw(\021)s Fy(])c(=)f(\000\()p Fw(s)p Fy(\).)41 b(Because)31 b(of)g(upp)s(er)d(semi-con)m(tin)m(uit)m (y)33 b(of)e Fv(C)2609 2354 y Ft(\021)r(;\026)2743 2340 y Fy(on)f Fw(L)2931 2354 y Ft(\021)r(;\026)3035 2340 y Fy(,)h(w)m(e)f(\014nd)456 2498 y(\(2.2\))752 b(\(\006)1494 2460 y Fr(0)1494 2525 y(\000\()p Ft(s)p Fr(\))1650 2498 y Fy(+)20 b Fw(\016)s Fy(\))h Fv(\\)f Fy(\()p Fv(C)2004 2512 y Ft(\021)r(;\026)2129 2498 y Fy(+)f Fw(\016)s Fy(\))27 b Fv(\032)e Fw(U;)456 2668 y 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Fy(\()p Fw(c)p Fy(\))31 b Fv(\032)f Fw(N)45 b Fj(and)36 b(that)h(ther)-5 b(e)36 b(exists)g(an)g(isolate)-5 b(d)37 b(minimal)g(homo)-5 b(clinic)37 b(orbit)f Fw(d\020)42 b Fj(to)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 456 251 a Fr(12)1012 b(X.)23 b(LI)h(AND)f(C.-Q.)f(CHENG)487 428 y Fy(~)456 451 y Fv(A)o Fy(\()p Fw(c)p Fy(\))41 b Fj(with)f Fy([)p Fw(\020)7 b Fy(])39 b Fv(6)p Fy(=)e(0)k Fj(and)f Fw(c)1434 418 y Fo(0)1497 451 y Fj(is)g(su\016ciently)g(close)g(to)h Fw(c)f Fj(with)g Fw(\013)p Fy(\()p Fw(c)p Fy(\))g(=)e Fw(\013)p Fy(\()p Fw(c)3146 418 y Fo(0)3170 451 y Fy(\))p Fj(,)k(then)456 565 y(ther)-5 b(e)33 b(exists)g(some)g(orbit)h Fw(d\015)k Fj(of)32 b Fw(\036)1673 533 y Ft(t)1673 592 y(L)1758 565 y Fj(which)i(c)-5 b(onne)g(cts)2410 543 y Fy(~)2379 565 y Fv(A)o Fy(\()p Fw(c)p Fy(\))34 b Fj(to)2734 543 y Fy(~)2703 565 y Fv(A)o Fy(\()p Fw(c)2849 533 y Fo(0)2873 565 y Fy(\))p Fj(.)456 826 y(Pr)-5 b(o)g(of.)43 b Fy(A)36 b(cohomology)i(class)f Fw(c)f Fy(with)g Fw(!)s Fy(\()p 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Ft(c)3133 1266 y Fy(\))26 b Fv(6)p Fy(=)f(0.)555 1461 y(By)k(condition,)g Fw(!)s Fy(\()p Fw(\026)1268 1475 y Ft(c)1303 1461 y Fy(\))c(=)g(\()p Fw(!)1551 1475 y Fr(1)1590 1461 y Fw(;)15 b Fv(\001)g(\001)g(\001)32 b Fw(;)15 b(!)1864 1475 y Ft(n)p Fo(\000)p Fr(1)2001 1461 y Fw(;)g Fy(0\).)41 b(W)-8 b(e)30 b(can)e(c)m(ho)s(ose)k(^)-48 b Fw(g)2838 1423 y Fo(\000)p Fr(1)2835 1487 y(1)2958 1461 y Fv(2)25 b Fu(Z)3105 1428 y Ft(n)p Fo(\000)p Fr(1)3270 1461 y Fy(suc)m(h)456 1577 y(that)39 b Fv(h)s Fy(^)-48 b Fw(g)742 1539 y Fo(\000)p Fr(1)739 1603 y(1)837 1577 y Fw(;)15 b Fy(\()p Fw(!)969 1591 y Fr(1)1009 1577 y Fw(;)g Fv(\001)g(\001)g(\001)32 b Fw(;)15 b(!)1283 1591 y Ft(n)p Fo(\000)p Fr(1)1420 1577 y Fy(\))p Fv(i)39 b Fw(>)g Fy(0.)65 b(Clearly)-8 b(,)42 b Fv(9)f Fy(^)-48 b Fw(g)2253 1539 y Fo(\000)p Fr(1)2250 1603 y(2)2348 1577 y Fw(;)15 b Fv(\001)g(\001)g(\001)20 b Fy(^)-48 b Fw(g)2556 1539 y Fo(\000)p Fr(1)2553 1603 y Ft(n)p Fo(\000)p Fr(1)2729 1577 y Fv(2)38 b Fu(Z)2889 1544 y Ft(n)p 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y(~)2779 1946 y Fv(A)p Fy(\()p Fw(c)p Fy(\))34 b(as)g(its)g Fw(!)s Fy(-)f(as)456 2054 y(w)m(ell)e(as)g Fw(\013)p Fy(-limit)h(set.)41 b(Here)1458 2030 y(\026)1450 2054 y Fw(\020)36 b Fy(denote)31 b(the)g(lift)g(of)f Fw(\020)37 b Fy(to)31 b(the)g(co)m(v)m(ering)h Fu(R)20 b Fv(\002)g Fu(T)3162 2021 y Ft(n)p Fo(\000)p Fr(1)3299 2054 y Fy(.)555 2245 y(Because)30 b(of)f(the)f(unique)g(ergo)s(dicit)m (y)-8 b(,)31 b(the)e(set)g(of)f(all)i(minimal)e(homo)s(clinic)h(orbits) 456 2353 y(with)40 b(the)h(homology)h([)p Fw(\020)7 b Fy(])40 b(is)h(a)g(Lipsc)m(hitz)g(graph.)71 b(By)41 b(assumption,)i (there)e(exists)456 2461 y(an)34 b(isolated)i(minimal)f(homo)s(clinic)h (orbit)e Fw(\020)41 b Fy(with)35 b([)p Fw(\020)7 b Fy(])32 b Fv(6)p Fy(=)g(0)j(as)g(w)m(ell)h(as)f(an)g(\()p Fw(n)23 b Fv(\000)f Fy(1\)-)456 2569 y(dimensional)44 b(disk)f Fw(D)1251 2536 y Fo(0)1322 2569 y Fv(\032)48 b(f)p Fw(q)1527 2583 y Fr(1)1614 2569 y Fy(=)g Fw(q)1777 2536 y Fr(0)1774 2593 y(1)1816 2569 y Fv(g)d 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Fy(\))26 b(:)f Fu(T)3191 3074 y Ft(n)p Fo(\000)p Fr(1)3353 3107 y Fv(!)456 3215 y Fu(R)522 3182 y Fr(+)624 3215 y Fy(suc)m(h)44 b(that)h(supp)n Fw(w)51 b Fy(=)c Fw(D)1553 3182 y Fo(0)1576 3215 y Fy(,)h Fw(w)j Fv(\021)c Fw(\025)d Fy(for)51 b(^)-52 b Fw(q)51 b Fv(2)d Fw(D)s Fy(,)f(where)d Fw(\025)k(>)g Fy(0,)g Fw(D)j Fv(\032)c Fw(D)3421 3182 y Fo(0)456 3328 y Fy(is)42 b(a)g(smaller)h(disk.)76 b(Let)42 b(\001)p Fw(q)1524 3342 y Fr(1)1608 3328 y Fy(=)i(inf)7 b Fv(fj)p Fw(\020)1944 3342 y Fr(1)1984 3328 y Fv(j)45 b Fy(:)2132 3304 y(^)2124 3328 y Fw(\020)62 b(=)-55 b Fv(2)44 b Fw(D)s Fv(g)e Fy(where)g Fw(\020)49 b Fy(is)42 b(the)g(minimal)456 3436 y(curv)m(e)30 b(of)h Fw(\036)856 3403 y Ft(t)856 3463 y(L)908 3436 y Fy(,)g(homo)s(clinic)g(to)g Fv(A)p Fy(\()p Fw(c)p Fy(\).)42 b(Let)31 b Fw(\045)1991 3403 y Fo(0)2014 3436 y Fy(\()p Fw(q)2090 3450 y Fr(1)2130 3436 y Fy(\))f(b)s(e)g(a)h(smo)s(oth)f (function)h(suc)m(h)f(that)456 3544 y(supp)n Fw(\045)690 3511 y Fo(0)739 3544 y Fv(\032)24 b(fj)p Fw(q)945 3558 y Fr(1)985 3544 y Fv(j)i(\024)f Fw(\016)k(<)c Fy(\001)p Fw(q)1414 3558 y Fr(1)1453 3544 y Fv(g)p Fy(,)31 b Fw(\045)1601 3511 y Fo(0)1624 3544 y Fy(\(0\))26 b Fw(>)f Fy(0,)31 b(0)26 b Fv(\024)f Fw(\045)2176 3511 y Fo(0)2224 3544 y Fv(\024)g Fy(1)31 b(and)f(de\014ne)1593 3705 y Fw(L)1655 3720 y Ft(\021)r(; )1790 3705 y Fy(=)25 b Fw(L)c Fv(\000)e Fw(\021)24 b Fv(\000)c Fw( )s(;)456 3865 y Fy(where)29 b Fw( )g Fy(=)c Fw(\045)949 3832 y Fo(0)972 3865 y Fy(\()p Fw(q)1048 3879 y Fr(1)1088 3865 y Fy(\))p Fw(w)r Fy(\()7 b(^)-52 b Fw(q)t Fy(\))17 b(_)-42 b Fw(q)1346 3879 y Fr(1)1385 3865 y Fy(.)41 b(Recall)32 b Fw(\013)p Fy(\()p Fw(c)p Fy(\))27 b(=)e(0,)31 b(let)1111 4092 y Fw(h)1163 4055 y Ft(T)1163 4115 y(\021)r(; )r(;e)1321 4123 y Fm(n)1369 4092 y Fy(\()p Fw(m;)15 b(m)1604 4055 y Fo(0)1627 4092 y Fy(\))26 b(=)105 b(inf)1784 4160 y Fm(\030)q Fk(\()p Fl(\000)p Fm(T)8 b Fk(\)=)p Fm(m)1797 4227 y(\030)q Fk(\()p Fm(T)g Fk(\)=)p Fm(m)2019 4206 y Fl(0)1830 4284 y Fk([)p Fm(\030)q 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y Fy(when)g(they)h(are)g(restricted)g(in)f Fw(T)13 b(N)d Fy(.)62 b(It)38 b(is)g(not)g(di\016cult)f(to)i(see)f (that)456 4980 y(eac)m(h)g(minimal)f(homo)s(clinic)h(orbit)g(of)f Fw(\036)1885 4947 y Ft(t)1885 5007 y(L)1975 4980 y Fy(to)2124 4957 y(~)2093 4980 y Fv(A)o Fy(\()p Fw(c)p Fy(\))i(is)e(still)h(a)g (homo)s(clinic)g(orbit)f(of)456 5089 y Fw(\036)510 5056 y Ft(t)510 5116 y(L)558 5128 y Fm(\021)r(; )661 5089 y Fy(.)60 b(In)36 b(fact,)j(except)f(for)e Fw(\020)7 b Fy(,)38 b(other)f(minimal)f(homo)s(clinic)i(curv)m(es)e(do)h(not)g (pass)456 5216 y(through)25 b(the)h(supp)s(ort)e(of)i Fw( )s Fy(.)40 b(Note)27 b Fw(\045)1767 5183 y Fo(0)1791 5216 y Fy(\()p Fw(q)1867 5230 y Fr(1)1906 5216 y Fy(\))p Fw(w)r Fy(\()7 b(^)-52 b Fw(q)t Fy(\))p Fw(dq)2211 5230 y Fr(1)2277 5216 y Fy(is)26 b(closed)h(on)e Fv(fj)p Fw(q)2860 5230 y Fr(1)2900 5216 y Fv(j)h(\024)f Fy(\001)p Fw(q)3164 5230 y Fr(1)3202 5216 y Fv(g)11 b(\002)g Fw(D)s Fy(,)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 998 251 a Fr(CONNECTING)31 b(ORBITS)g(F)n(OR)f (A)n(UTONOMOUS)g(SYSTEMS)472 b(13)456 450 y Fy(this)21 b(term)g(has)h(no)f(con)m(tribution)h(to)g(the)g(Euler-Lagrange)g (equation)g(along)h Fw(d\020)7 b Fy(.)37 b(Th)m(us)456 558 y Fw(d\020)g Fy(is)30 b(still)h(a)g(minimal)g(homo)s(clinic)g (orbit)f(of)h Fw(\036)2115 525 y Ft(t)2115 585 y(L)2163 597 y Fm(\021)r(; )2267 558 y Fy(.)555 761 y(Nev)m(ertheless,)43 b(except)e(for)d Fw(\020)7 b Fy(,)41 b(other)f(homo)s(clinic)g(curv)m (es)f(minimal)g(for)g Fw(L)3242 775 y Ft(\021)3323 761 y Fy(are)456 869 y(no)c(longer)h(minimal)g(for)f Fw(L)1430 884 y Ft(\021)r(; )1540 869 y Fy(.)56 b(T)-8 b(o)36 b(see)g(it,)i(let)e (us)f(recall)i(that)f Fw( )h(>)d Fy(0)i(holds)f(on)g(a)456 977 y(segmen)m(t)h(of)g Fw(\020)42 b Fy(and)34 b Fw( )k Fy(=)33 b(0)j(holds)f(along)h(an)m(y)g(other)f(homo)s(clinic)i(curv)m (es)e(minimal)456 1085 y(for)c Fw(L)658 1099 y Ft(\021)700 1085 y Fy(,)h(the)g(quan)m(tit)m(y)h(min)15 b Fw(h)1499 1052 y Fo(1)1499 1112 y Ft(\021)r(; )r(;e)1657 1120 y Fm(n)1704 1085 y Fy(\()p Fw(m;)g(m)p Fy(\))33 b(can)f(only)g(b)s(e)f (ac)m(hiev)m(ed)i(b)m(y)f(the)g(curv)m(e)g Fw(\020)456 1202 y Fy(whic)m(h)24 b(passes)g(through)f(the)i(disk)e Fv(f)p Fw(q)1735 1216 y Fr(1)1800 1202 y Fy(=)i(0)p Fw(;)e Fy(^)-53 b Fw(q)29 b Fv(2)c Fw(D)s Fv(g)p Fy(.)39 b(Th)m(us,)24 b Fw(d\020)31 b Fy(is)24 b(the)h(only)f(minimal)456 1310 y(homo)s(clinic)31 b(orbit)f(with)g(nonzero)h(homology)g(in)g Fw(H)2275 1324 y Fr(1)2314 1310 y Fy(\()p Fw(M)5 b(;)15 b(N)5 b(;)15 b Fu(Z)p Fy(\))31 b(for)f Fw(L)2928 1325 y Ft(\021)r(; )3038 1310 y Fy(.)555 1501 y(Let)38 b Fw(U)46 b Fy(=)36 b Fv(fj)p Fw(q)1051 1515 y Fr(1)1090 1501 y Fv(j)h(\024)f Fy(\001)p Fw(q)1376 1515 y Fr(1)1414 1501 y Fv(g)25 b(\002)g Fw(D)s Fy(.)60 b(A)37 b(closed)h(1-form)44 b(\026)-52 b Fw(\026)36 b Fy(clearly)j(exists)e(suc)m(h)g(that)456 1608 y([)7 b(\026)-52 b Fw(\026)p Fy(])25 b(=)g Fw(c)721 1576 y Fo(0)749 1608 y Fv(\000)t Fw(c)d Fy(and)g(supp)5 b(\026)-52 b Fw(\026)t Fv(\\)1386 1586 y Fy(\026)1365 1608 y Fw(D)28 b Fy(=)d Fu(?)p Fy(.)37 b(W)-8 b(e)24 b(in)m(tro)s(duce)e(a)h(Lagrangian)g Fw(L)2840 1623 y Ft(\021)r(;\026; )3012 1608 y Fy(:)36 b Fw(T)3173 1586 y Fy(\026)3139 1608 y Fw(M)g Fv(!)25 b Fu(R)1380 1759 y Fw(L)1442 1774 y Ft(\021)r(;\026; )1614 1759 y Fy(\()p Fw(q)s(;)32 b Fy(_)-42 b Fw(q)s Fy(\))26 b(=)f Fw(L)20 b Fv(\000)g Fw(\021)k Fv(\000)19 b Fw(\026)h Fv(\000)g Fw( )s(;)456 1909 y Fy(where)35 b Fw(\026)g Fy(=)g Fw(\045)p Fy(\()p Fw(q)1043 1923 y Fr(1)1082 1909 y Fy(\))7 b(\026)-52 b Fw(\026)q Fy(,)37 b Fw(\045)g Fy(is)f(a)h(smo)s(oth)f(function)g (with)g Fw(\045)f Fy(=)g(0)h(for)g Fw(q)2855 1923 y Fr(1)2929 1909 y Fv(\024)f(\000)p Fy(\001)p Fw(q)3223 1923 y Fr(1)3298 1909 y Fy(and)456 2017 y Fw(\045)k Fy(=)f(1)h(for)g Fw(q)924 2031 y Fr(1)1002 2017 y Fv(\025)f Fy(\001)p Fw(q)1228 2031 y Fr(1)1267 2017 y Fy(,)j Fw( )h Fy(=)48 b(\026)-54 b Fw(\045)1591 1984 y Fo(0)1614 2017 y Fy(\()p Fw(q)1690 2031 y Fr(1)1730 2017 y Fy(\))p Fw(w)r Fy(\()7 b(^)-52 b Fw(q)t Fy(\))17 b(_)-42 b Fw(q)1988 2031 y Fr(1)2027 2017 y Fy(,)50 b(\026)-54 b Fw(\045)2140 1984 y Fo(0)2202 2017 y Fy(tak)m(es)40 b(v)-5 b(alue)39 b(of)g(the)g(lift)g(of)g Fw(\045)3274 1984 y Fo(0)3336 2017 y Fy(for)456 2125 y Fv(j)p Fw(q)522 2139 y Fr(1)561 2125 y Fv(j)25 b(\024)g Fw(\031)34 b Fy(and)k(\026)-54 b Fw(\045)1016 2092 y Fo(0)1065 2125 y Fy(=)25 b(0)31 b(for)f Fv(j)p Fw(q)1442 2139 y Fr(1)1481 2125 y Fv(j)c(\025)f Fw(\031)s Fy(.)555 2316 y(Denote)44 b(b)m(y)1044 2293 y(\026)1018 2316 y Fw(N)52 b Fy(the)43 b(lift)g(of)f Fw(N)53 b Fy(in)1861 2293 y(\026)1827 2316 y Fw(M)10 b Fy(,)46 b(let)60 b(\026)-62 b Fw(m;)33 b Fy(\026)-63 b Fw(m)2339 2283 y Fo(0)2408 2316 y Fv(2)2540 2293 y Fy(\026)2514 2316 y Fw(N)52 b Fy(suc)m(h)42 b(that)i Fv(\000)16 b Fy(\026)-61 b Fw(m)3217 2330 y Fr(1)3298 2316 y Fy(and)473 2424 y(\026)f Fw(m)536 2391 y Fo(0)536 2448 y Fr(1)618 2424 y Fy(are)43 b(rather)f(large.)79 b(T)-8 b(o)44 b(eac)m(h)g(curv)m(e)1990 2400 y(\026)1981 2424 y Fw(\030)i Fy(connecting)61 b(\026)-62 b Fw(m)43 b Fy(to)61 b(\026)-63 b Fw(m)2860 2391 y Fo(0)2926 2424 y Fy(w)m(e)43 b(can)g(asso-)456 2532 y(ciate)h(a)f(homology)h(class)f ([)1456 2508 y(\026)1447 2532 y Fw(\030)t Fy(])g(v)-5 b(alued)42 b(with)g(the)h(quan)m(tit)m(y)h([)p Fw(\031)2710 2508 y Fy(\026)2700 2532 y Fw(\030)t Fy(])i Fv(2)f Fw(H)2997 2546 y Fr(1)3036 2532 y Fy(\()p Fw(M)5 b(;)15 b(N)5 b(;)15 b Fu(Z)p Fy(\).)456 2640 y(Let)40 b(\026)-48 b Fw(\015)672 2655 y Ft(\021)r(;\026; )844 2640 y Fy(\()p Fw(t;)15 b(m;)g(m)1152 2607 y Fo(0)1176 2640 y Fw(;)g(e)1258 2654 y Ft(n)1306 2640 y Fy(\):)53 b([)p Fv(\000)p Fw(T)8 b(;)15 b(T)e Fy(])36 b Fv(!)1904 2617 y Fy(\026)1870 2640 y Fw(M)47 b Fy(b)s(e)36 b(the)h(minimizer:)57 b(\026)-48 b Fw(\015)2818 2655 y Ft(\021)r(;\026; )2990 2640 y Fy(\()p Fv(\000)p Fw(T)13 b Fy(\))36 b(=)53 b(\026)-63 b Fw(m)p Fy(,)459 2748 y(\026)-48 b Fw(\015)503 2763 y Ft(\021)r(;\026; )675 2748 y Fy(\()p Fw(T)13 b Fy(\))25 b(=)43 b(\026)-63 b Fw(m)1012 2715 y Fo(0)1036 2748 y Fy(,)30 b([)s(\026)-48 b Fw(\015)1163 2763 y Ft(\021)r(;\026; )1336 2748 y Fy(])25 b(=)g([)p Fw(\020)7 b Fy(])30 b(and)1016 2846 y Fq(Z)1106 2872 y Ft(T)1066 3052 y Fo(\000)p Ft(T)1191 2969 y Fw(L)1253 2984 y Ft(\021)r(;\026; )1425 2969 y Fy(\()p Fw(d)s Fy(\026)-48 b Fw(\015)1554 2984 y Ft(\021)r(;\026; )1727 2969 y Fy(\()p Fw(t)p Fy(\)\))p Fw(dt)26 b Fy(=)52 b(inf)2067 3031 y Ft(T)2118 3012 y Fl(0)2140 3031 y Ft(>)p Fr(0)2246 2969 y Fw(h)2298 2932 y Ft(T)2349 2908 y Fl(0)2298 2993 y Ft(\021)r(;\026; )r(;e)2518 3001 y Fm(n)2565 2969 y Fy(\()18 b(\026)-63 b Fw(m;)33 b Fy(\026)-63 b Fw(m)2800 2932 y Fo(0)2824 2969 y Fy(\))p Fw(;)456 3173 y Fy(where)1027 3334 y Fw(h)1079 3297 y Ft(T)1130 3273 y Fl(0)1079 3357 y Ft(\021)r(;\026; )r(;e)1299 3365 y Fm(n)1346 3334 y Fy(\()18 b(\026)-63 b Fw(m;)33 b Fy(\026)-63 b Fw(m)1581 3297 y Fo(0)1605 3334 y Fy(\))25 b(=)117 b(inf)1768 3394 y Fk(\026)1761 3407 y Fm(\030)r Fk(\()p Fl(\000)p Fm(T)1907 3393 y Fl(0)1930 3407 y Fk(\)=)12 b(\026)-42 b Fm(m)1780 3460 y Fk(\026)1774 3473 y Fm(\030)r Fk(\()p Fm(T)1872 3452 y Fl(0)1895 3473 y Fk(\)=)12 b(\026)-42 b Fm(m)2020 3452 y Fl(0)1819 3535 y Fk([)1843 3522 y(\026)1837 3535 y Fm(\030)r Fk(]=[)p Fm(\020)s Fk(])2070 3211 y Fq(Z)2161 3237 y Ft(T)2212 3213 y Fl(0)2121 3417 y Fo(\000)p Ft(T)2227 3398 y Fl(0)2268 3334 y Fw(L)2330 3349 y Ft(\021)r(;\026; )2502 3334 y Fy(\()p Fw(d)2593 3310 y Fy(\026)2584 3334 y Fw(\030)5 b Fy(\()p Fw(t)p Fy(\)\))p Fw(dt:)456 3652 y Fy(F)-8 b(rom)33 b(the)f(pro)s(of)g(of)h(the)f(lemma)h(2.1,)h(w)m(e)f(can)g (see)g(that)g Fw(T)45 b Fy(is)33 b(\014nite)f(for)g(eac)m(h)i(\014xed) 456 3760 y(\()17 b(\026)-62 b Fw(m;)32 b Fy(\026)-62 b Fw(m)691 3727 y Fo(0)714 3760 y Fy(\))36 b(and)e Fw(T)46 b Fv(!)34 b(1)h Fy(as)g Fv(\000)17 b Fy(\026)-62 b Fw(m)1583 3774 y Fr(1)1622 3760 y Fw(;)33 b Fy(\026)-63 b Fw(m)1742 3727 y Fo(0)1742 3784 y Fr(1)1815 3760 y Fv(!)33 b(1)p Fy(.)55 b(W)-8 b(e)37 b(denote)e(b)m(y)g Fg(C)2761 3775 y Ft(\021)r(;\026; )r(;e)2981 3783 y Fm(n)3063 3760 y Fy(the)g(set)h(of)456 3868 y(all)28 b(accum)m(ulation)h(p)s(oin)m(ts)e (of)g Fv(f)s Fy(\026)-48 b Fw(\015)1596 3883 y Ft(\021)r(;\026; )r(;) 1788 3868 y Fy(\()p Fw(t;)33 b Fy(\026)-63 b Fw(m)q(;)32 b Fy(\026)-62 b Fw(m)2097 3835 y Fo(0)2120 3868 y Fw(;)15 b(e)2202 3882 y Ft(n)2250 3868 y Fy(\))p Fv(j)2310 3886 y Fr([)p Fo(\000)p Ft(T)2426 3896 y Fm(i)2452 3886 y Ft(;T)2513 3896 y Fm(i)2539 3886 y Fr(])2563 3868 y Fv(g)28 b Fy(\()p Fw(T)2724 3882 y Ft(i)2778 3868 y Fv(!)d(1)i Fy(as)g Fw(i)f Fv(!)f(1)p Fy(\).)456 3995 y(F)-8 b(or)46 b(eac)m(h)g(curv)m(e)j(\026)-48 b Fw(\015)56 b Fv(2)49 b Fg(C)1385 4010 y Ft(\021)r(;\026; )r(;e)1605 4018 y Fm(n)1652 3995 y Fy(,)g(let)e Fw(\015)55 b Fy(=)50 b Fw(\031)7 b Fy(\026)-49 b Fw(\015)5 b Fy(,)50 b Fw(d\015)5 b Fy(\()p Fw(t)p Fy(\))46 b(approac)m(hes)g(to)3166 3972 y(~)3135 3995 y Fv(A)p Fy(\()p Fw(c)p Fy(\))g(as)456 4114 y Fw(t)25 b Fv(!)g(\0001)30 b Fy(and)g(approac)m(hes)g(to)1609 4091 y(~)1578 4114 y Fv(A)o Fy(\()p Fw(c)1724 4081 y Fo(0)1748 4114 y Fy(\))h(as)f Fw(t)c Fv(!)f(1)p Fy(.)40 b(Let)1338 4283 y Fv(C)1386 4298 y Ft(\021)r(;\026; )r(;e)1606 4306 y Fm(n)1678 4283 y Fy(=)1910 4197 y Fq([)1776 4392 y Fk(\026)-32 b Fm(\015)s Fl(2)p Fe(C)1901 4409 y Fm(\021)r(;\026; )r (;e)2101 4417 y(n)1910 4460 y(t)p Fl(2)p Fd(R)2166 4283 y Fy(\026)-49 b Fw(\015)5 b Fy(\()p Fw(t)p Fy(\))26 b Fv(\032)2473 4260 y Fy(\026)2439 4283 y Fw(M)10 b(:)456 4593 y Fy(Clearly)-8 b(,)31 b(this)f(set)h(is)g(upp)s(er)d(semi-con)m (tin)m(uous)k(with)e(resp)s(ect)g(to)h(\()p Fw(\021)s(;)15 b(\026;)g( )s Fy(\).)555 4784 y(By)30 b(the)f(c)m(hoice)i(of)e Fw( )s Fy(,)h Fg(C)1401 4799 y Ft(\021)r(;)p Fr(0)p Ft(; )r(;e)1614 4807 y Fm(n)1689 4784 y Fy(con)m(tains)h(only)e(one)g(curv)m(e)h(whic)m (h)f(is)g(a)g(lift)h(of)f(the)456 4892 y(minimal)35 b(homo)s(clinic)h (curv)m(e)f(for)g Fw(L)1729 4906 y Ft(\021)1770 4892 y Fy(,)i(passing)e(through)f(transv)m(ersally)i(the)f(cen)m(ter)456 5000 y(of)i(the)h(disk.)62 b(Due)37 b(to)i(the)e(upp)s(er)f(semi-con)m (tin)m(uit)m(y)k(of)e Fv(C)2527 5015 y Ft(\021)r(;\026; )r(;e)2747 5023 y Fm(n)2831 5000 y Fy(with)f(resp)s(ect)h(to)456 5108 y(\()p Fw(\021)s(;)15 b(\026;)g( )s Fy(\),)44 b(eac)m(h)c(curv)m (e)g(in)f Fg(C)1481 5123 y Ft(\021)r(;\026; )r(;e)1701 5131 y Fm(n)1787 5108 y Fy(passes)g(through)g Fw(U)10 b Fv(n)p Fy(\()p Fv(fj)p Fw(q)2686 5122 y Fr(1)2726 5108 y Fv(j)41 b(\024)f Fy(\001)2979 5122 y Ft(q)3017 5108 y Fv(g)27 b(\002)f Fw(@)5 b(D)s Fy(\))39 b(if)456 5216 y Fv(j)p Fw(c)520 5183 y Fo(0)546 5216 y Fv(\000)s Fw(c)p Fv(j)22 b Fy(is)g(suitably)g(small.)38 b(Th)m(us,)23 b(eac)m(h)g(curv)m(e)i(\026)-48 b Fw(\015)30 b Fv(2)25 b Fg(C)2300 5231 y Ft(\021)r(;\026; )r(;e)2520 5239 y Fm(n)2588 5216 y Fy(determines)d(an)f(orbit)h(of)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 456 251 a Fr(14)1012 b(X.)23 b(LI)h(AND)f(C.-Q.)f(CHENG)456 450 y Fw(\036)510 417 y Ft(t)510 477 y(L)562 450 y Fy(.)37 b(Indeed,)23 b(in)d(the)h(region)h Fv(fj)p Fw(q)1556 464 y Fr(1)1596 450 y Fv(j)j(\025)g Fy(\001)p Fw(q)1859 464 y Fr(1)1898 450 y Fv(g)c Fw( )29 b Fy(=)c(0)c(and)f Fw(\045)p Fy(\()p Fw(q)2504 464 y Fr(1)2544 450 y Fy(\))7 b(\026)-52 b Fw(\026)21 b Fy(is)g(closed,)j (while)d(in)f Fw(U)10 b Fy(,)456 558 y Fw( )33 b Fy(is)e(closed)g(and) 37 b(\026)-52 b Fw(\026)25 b Fy(=)g(0.)41 b(Therefore,)31 b(along)k(\026)-49 b Fw(\015)36 b Fy(these)31 b(terms)f(ha)m(v)m(e)h (no)g(con)m(tribution)456 668 y(to)g(the)f(Lagrangian)i(equation,)f(th) m(us)f Fw(\013)p Fy(\()p Fw(d\015)5 b Fy(\))27 b Fv(\032)2177 645 y Fy(~)2146 668 y Fv(A)o Fy(\()p Fw(c)p Fy(\))32 b(and)e Fw(!)s Fy(\()p Fw(d\015)5 b Fy(\))26 b Fv(\032)2918 645 y Fy(~)2887 668 y Fv(A)o Fy(\()p Fw(c)3033 635 y Fo(0)3057 668 y Fy(\).)257 b Fh(\003)555 915 y Fy(W)-8 b(e)24 b(call)g(the)f(lo)s(cal)i(connecting)f(orbit)e(obtained)i(in)e (the)h(theorem)g(2.2)h Fw(h)p Fy(-t)m(yp)s(e)g(lo)s(cal)456 1023 y(connecting)38 b(orbit.)63 b(Suc)m(h)37 b(connecting)i(orbit)f (has)f(some)h(lo)s(cal)h(minimal)f(prop)s(ert)m(y)456 1131 y(whic)m(h)33 b(will)g(b)s(e)g(used)f(for)h(the)h(construction)g (of)f(global)i(connecting)f(orbits.)49 b(Let)34 b(us)456 1239 y(in)m(tro)s(duce)c(the)g(de\014nition)h(\014rst)e(and)h(giv)m(e)i (explanations)f(afterw)m(ards.)456 1403 y FC(De\014nition)j(2.2.)41 b Fj(We)31 b(assume)i(that)f Fv(A)p Fy(\()p Fw(c)p Fy(\))p Fw(;)15 b Fv(A)p Fy(\()p Fw(c)2181 1370 y Fo(0)2206 1403 y Fy(\))26 b Fv(\032)f Fw(N)10 b Fj(,)31 b Fy([)p Fw(\021)s Fy(])26 b(=)f Fw(c)32 b Fj(and)h Fy([)p Fw(\021)22 b Fy(+)j(\026)-52 b Fw(\026)o Fy(])26 b(=)f Fw(c)3393 1370 y Fo(0)3417 1403 y Fj(.)456 1511 y Fw(d\015)5 b Fj(:)59 b Fw(T)13 b(M)50 b Fv(!)41 b Fu(R)g Fj(is)g(c)-5 b(al)5 b(le)-5 b(d)42 b(lo)-5 b(c)g(al)43 b(minimal)g(orbit)f(of)f Fw(\036)2425 1478 y Ft(t)2425 1538 y(L)2518 1511 y Fj(of)h Fw(h)p Fj(-typ)-5 b(e)42 b(that)g(c)-5 b(onne)g(cts)487 1607 y Fy(~)456 1630 y Fv(A)o Fy(\()p Fw(c)p Fy(\))34 b Fj(to)811 1607 y Fy(~)780 1630 y Fv(A)p Fy(\()p Fw(c)927 1597 y Fo(0)951 1630 y Fy(\))e Fj(if)555 1821 y(1,)40 b Fy(0)c Fv(6)p Fy(=)e([)p Fw(\015)5 b Fy(])36 b Fv(2)f Fw(H)1165 1835 y Fr(1)1204 1821 y Fy(\()p Fw(M)5 b(;)15 b(N)5 b(;)15 b Fu(Z)p Fy(\))p Fj(.)59 b Fw(d\015)5 b Fy(\()p Fw(t)p Fy(\))39 b Fj(is)f(an)h(orbit)f(of)h(the)f(Euler-L)-5 b(agr)g(ange)39 b(\015ow)456 1936 y Fw(\036)510 1903 y Ft(t)510 1963 y(L)562 1936 y Fj(,)32 b(the)h Fw(\013)p Fj(-)g(and)g Fw(!)s Fj(-limit)g(set)g(of)f Fw(d\015)38 b Fj(ar)-5 b(e)34 b(in)2060 1913 y Fy(~)2028 1936 y Fv(A)p Fy(\()p Fw(c)p Fy(\))g Fj(and)2451 1913 y Fy(~)2420 1936 y Fv(A)p Fy(\()p Fw(c)2567 1903 y Fo(0)2591 1936 y Fy(\))f Fj(r)-5 b(esp)g(e)g(ctively.)555 2127 y(2,)35 b(its)f(lift)k Fy(\026)-48 b Fw(\015)33 b Fv(2)28 b Fg(C)1167 2142 y Ft(\021)r(;\026; )r(;e)1387 2150 y Fm(n)1433 2127 y Fj(,)34 b(which)i(is)e(a)g(lo)-5 b(c)g(al)36 b(minimal)g(curve)d(of)i(the)f(L) -5 b(agr)g(angian)456 2238 y Fw(L)518 2253 y Ft(\021)r(;\026; )715 2238 y Fy(:)25 b Fw(T)865 2215 y Fy(\026)831 2238 y Fw(M)35 b Fv(!)26 b Fu(R)31 b Fj(in)g(the)h(fol)5 b(lowing)32 b(sense:)41 b(ther)-5 b(e)32 b(exist)g(two)g Fy(\()p Fw(n)17 b Fv(\000)h Fy(1\))p Fj(dimensional)456 2348 y(disks)35 b Fw(V)754 2315 y Fo(\000)814 2348 y Fj(,)g Fw(V)950 2315 y Fr(+)1039 2348 y Fv(\032)1174 2325 y Fy(\026)1140 2348 y Fw(M)45 b Fj(and)37 b(p)-5 b(ositive)36 b(numb)-5 b(ers)36 b Fw(T)8 b(;)15 b(d)30 b(>)g Fy(0)35 b Fj(such)h(that)k Fy(\026)-49 b Fw(\015)5 b Fy(\()p Fv(\000)p Fw(T)13 b Fy(\))31 b Fv(2)e Fw(V)3357 2315 y Fo(\000)3417 2348 y Fj(,)459 2456 y Fy(\026)-48 b Fw(\015)5 b Fy(\()p Fw(T)13 b Fy(\))25 b Fv(2)g Fw(V)829 2423 y Fr(+)888 2456 y Fj(,)32 b Fw(\031)s(V)1076 2423 y Fo(\000)1161 2456 y Fv(\032)25 b Fw(N)10 b Fv(nA)p Fy(\()p Fw(c)p Fy(\))p Fj(,)33 b Fw(\031)s(V)1756 2423 y Fr(+)1841 2456 y Fv(\032)25 b Fw(N)10 b Fv(nA)p Fy(\()p Fw(c)2212 2423 y Fo(0)2236 2456 y Fy(\))32 b Fj(and)1183 2618 y Fw(h)1235 2581 y Fo(1)1235 2641 y Ft(c)1310 2618 y Fy(\()p Fw(q)1389 2581 y Fo(\000)1449 2618 y Fw(;)15 b(m)1569 2632 y Fr(0)1608 2618 y Fy(\))21 b(+)f Fw(h)1807 2581 y Ft(T)1858 2557 y Fl(0)1807 2642 y Ft(\021)r(;\026; )r(;e)2027 2650 y Fm(n)2074 2618 y Fy(\()e(\026)-63 b Fw(m)2189 2632 y Fr(0)2229 2618 y Fw(;)32 b Fy(\026)-62 b Fw(m)2349 2632 y Fr(1)2388 2618 y Fy(\))21 b(+)f Fw(h)2587 2581 y Fo(1)2587 2642 y Ft(c)2618 2623 y Fl(0)2662 2618 y Fy(\()p Fw(m)2777 2632 y Fr(1)2817 2618 y Fw(;)15 b(q)2901 2581 y Fr(+)2960 2618 y Fy(\))-2539 b(\(2.4\))905 2844 y Fv(\000)48 b Fy(lim)991 2918 y Fm(T)1035 2901 y(i)1027 2942 y Fk(0)1062 2918 y Fl(!1)991 3004 y Fm(T)1035 2983 y(i)1027 3025 y Fk(1)1062 3004 y Fl(!1)1199 2720 y Fq(Z)1289 2747 y Ft(T)1340 2723 y Fm(i)1330 2767 y Fk(1)1249 2927 y Fo(\000)p Ft(T)1355 2904 y Fm(i)1345 2948 y Fk(0)1400 2844 y Fw(L)1462 2859 y Ft(\021)r(;\026; )1634 2844 y Fy(\()p Fw(d\015)5 b Fy(\()p Fw(t)p Fy(\)\))p Fw(dt)22 b Fv(\000)e Fy(\()p Fw(T)2200 2807 y Ft(i)2187 2867 y Fr(0)2249 2844 y Fy(+)g Fw(T)2406 2807 y Ft(i)2393 2867 y Fr(1)2434 2844 y Fy(\))p Fw(\013)p Fy(\()p Fw(c)p Fy(\))1209 3132 y Fw(>)25 b Fy(0)456 3293 y Fj(holds)39 b(for)f(e)-5 b(ach)39 b Fy(\()17 b(\026)-62 b Fw(m)1168 3307 y Fr(0)1208 3293 y Fw(;)32 b Fy(\026)-62 b Fw(m)1328 3307 y Fr(1)1367 3293 y Fw(;)15 b(T)1473 3260 y Fo(0)1497 3293 y Fy(\))35 b Fv(2)f Fw(@)5 b Fy(\()p Fw(V)1824 3260 y Fo(\000)1906 3293 y Fv(\002)24 b Fw(V)2074 3260 y Fr(+)2157 3293 y Fv(\002)g Fy([)p Fw(T)37 b Fv(\000)23 b Fw(d;)15 b(T)38 b Fy(+)23 b Fw(d)p Fy(]\))p Fj(,)40 b Fw(q)2952 3260 y Fo(\000)3046 3293 y Fv(2)34 b(M)p Fy(\()p Fw(c)p Fy(\))25 b Fv(\\)456 3401 y Fw(\031)508 3415 y Ft(q)545 3401 y Fy(\()p Fw(\013)p Fy(\()p Fw(d\015)5 b Fy(\)\))39 b Fj(and)d Fw(q)1104 3368 y Fr(+)1194 3401 y Fv(2)31 b(M)p Fy(\()p Fw(c)1469 3368 y Fo(0)1493 3401 y Fy(\))23 b Fv(\\)f Fw(\031)1686 3415 y Ft(q)1724 3401 y Fy(\()p Fw(!)s Fy(\()p Fw(d\015)5 b Fy(\)\))p Fj(.)54 b(Wher)-5 b(e)36 b Fw(T)2461 3368 y Ft(i)2448 3425 y Fr(0)2520 3401 y Fv(!)31 b(1)36 b Fj(and)h Fw(T)3015 3368 y Ft(i)3002 3425 y Fr(1)3074 3401 y Fv(!)31 b(1)k Fj(ar)-5 b(e)456 3512 y(the)33 b(se)-5 b(quenc)g(es)32 b(such)h(that)h Fw(\015)5 b Fy(\()p Fv(\000)p Fw(T)1630 3479 y Ft(i)1617 3537 y Fr(0)1658 3512 y Fy(\))26 b Fv(!)f Fw(x)1887 3479 y Fo(\000)1978 3512 y Fj(and)34 b Fw(\015)5 b Fy(\()p Fw(T)2308 3479 y Ft(i)2295 3537 y Fr(1)2336 3512 y Fy(\))26 b Fv(!)f Fw(x)2565 3479 y Fr(+)2624 3512 y Fj(.)555 3769 y Fy(Here,)36 b(the)f(term)f Fw(h)1234 3736 y Fo(1)1234 3791 y Ft(c)1310 3769 y Fy(\()p Fw(\030)t(;)15 b(m)1509 3783 y Fr(0)1548 3769 y Fy(\))24 b(+)e Fw(h)1752 3736 y Ft(T)1803 3712 y Fl(0)1752 3797 y Ft(\021)r(;\026; )r(;e)1972 3805 y Fm(n)2020 3769 y Fy(\()17 b(\026)-62 b Fw(m)2135 3783 y Fr(0)2174 3769 y Fw(;)33 b Fy(\026)-63 b Fw(m)2294 3783 y Fr(1)2334 3769 y Fy(\))23 b(+)g Fw(h)2538 3736 y Fo(1)2538 3797 y Ft(c)2569 3778 y Fl(0)2613 3769 y Fy(\()p Fw(m)2728 3783 y Fr(1)2767 3769 y Fw(;)15 b(\020)7 b Fy(\))35 b(measures)f(the) 456 3886 y(smallest)e(action)g(of)f Fw(L)1245 3901 y Ft(\021)r(;\026; )1448 3886 y Fy(along)i(those)e(curv)m(es)g Fv(f)p Fw(\037)p Fv(g)h Fy(whic)m(h)f(satisfy)h(the)f(follo)m(wing)456 3994 y(conditions:)555 4185 y(1,)g(eac)m(h)h Fw(\037)e Fy(joining)48 b(\026)-63 b Fw(m)1328 4199 y Fr(0)1398 4185 y Fy(with)47 b(\026)-62 b Fw(m)1685 4199 y Fr(1)1755 4185 y Fy(with)30 b(time)h(2)p Fw(T)2280 4152 y Fo(0)2334 4185 y Fy(and)e([)p Fw(\037)p Fy(])d Fv(6)p Fy(=)f(0;)555 4376 y(2,)h Fw(x)703 4343 y Fo(\000)785 4376 y Fy(is)d(an)g(accum)m (ulation)i(p)s(oin)m(t)e(of)h Fw(\037)f Fy(as)g Fw(t)i Fv(!)h(\0001)c Fy(and)h Fw(x)2631 4343 y Fr(+)2713 4376 y Fy(is)g(an)h(accum)m(ulation)456 4484 y(p)s(oin)m(t)30 b(of)h Fw(\037)f Fy(as)g Fw(t)25 b Fv(!)h(1)p Fy(.)555 4675 y(By)21 b(requiring)f(those)h(curv)m(es)f(passing)h(through)e Fw(V)2277 4642 y Fo(\000)2357 4675 y Fy(and)h Fw(V)2597 4642 y Fr(+)2676 4675 y Fy(within)g(time)h(in)m(terv)-5 b(al)456 4783 y(2)p Fw(T)567 4750 y Fo(0)617 4783 y Fy(with)26 b Fw(T)886 4750 y Fo(0)935 4783 y Fv(2)f Fy([)p Fw(T)h Fv(\000)13 b Fw(d;)i(T)26 b Fy(+)13 b Fw(d)p Fy(],)28 b(w)m(e)g(obtain)f(a)g(neigh)m(b)s(orho)s(o)s(d)f(of)h(the)g(curv)m(e)g Fw(\015)5 b Fy(.)39 b(Once)456 4891 y(a)j(curv)m(e)h(touc)m(hes)g(the)g (b)s(oundary)d(of)i(this)h(neigh)m(b)s(orho)s(o)s(d,)h(the)e(action)i (of)e Fw(L)3272 4906 y Ft(\021)r(;\026; )456 5000 y Fy(along)e(this)f (curv)m(e)g(will)g(b)s(e)g(larger)g(than)g(the)g(action)i(along)f Fw(\015)5 b Fy(.)66 b(As)39 b Fw(V)3001 4967 y Fo(\000)3060 5000 y Fy(,)j Fw(V)3200 4967 y Fr(+)3298 5000 y Fy(and)456 5108 y(therefore)c Fw(d)g(>)f Fy(0)h(can)g(b)s(e)f(c)m(hosen)h (arbitrarily)g(small,)j(it)d(is)g(reasonable)g(to)h(call)g(it)456 5216 y(lo)s(cally)31 b(minimal.)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 998 251 a Fr(CONNECTING)31 b(ORBITS)g(F)n(OR)f (A)n(UTONOMOUS)g(SYSTEMS)472 b(15)555 450 y Fy(The)28 b(orbit)g Fw(d\015)33 b Fy(obtained)c(in)e(the)h(theorem)h(2.2)g(is)f (lo)s(cally)h(minimal)g(in)e(this)h(sense.)456 558 y(T)-8 b(o)33 b(see)h(it,)g(let)h(us)d(state)i(a)g(prop)s(ert)m(y)e(of)i (semi-static)h(curv)m(es)e(for)g(autonomous)g(sys-)456 666 y(tems:)456 838 y FC(Prop)s(osition)48 b(2.2.)g Fj(L)-5 b(et)42 b Fw(\015)48 b Fy(:)42 b Fu(R)g Fv(!)h Fw(M)52 b Fj(b)-5 b(e)42 b(a)g Fw(c)p Fj(-semi-static)h(curve)e(of)i(some)g (au-)456 946 y(tonomous)34 b(system.)43 b(If)32 b Fw(\015)38 b Fj(is)32 b(not)i(p)-5 b(erio)g(dic,)34 b(then)1439 1108 y Fy([)p Fw(A)1532 1122 y Ft(c)1567 1108 y Fy(\()p Fw(\015)5 b Fv(j)1679 1127 y Fr([)p Ft(a;b)p Fr(])1810 1108 y Fy(\)])26 b Fw(<)f Fy([)p Fw(A)2085 1122 y Ft(c)2120 1108 y Fy(\()p Fw(\030)t Fv(j)2224 1127 y Fr([)p Ft(a)2281 1108 y Fl(0)2304 1127 y Ft(;b)2354 1108 y Fl(0)2377 1127 y Fr(])2400 1108 y Fy(\)])456 1277 y Fj(holds)43 b(for)g(any)g (absolutely)g(c)-5 b(ontinuous)43 b(curve)e Fw(\030)46 b Fj(if)c Fw(\030)t Fy(\()p Fw(a)2493 1244 y Fo(0)2517 1277 y Fy(\))g(=)g Fw(\015)5 b Fy(\()p Fw(a)p Fy(\))p Fj(,)45 b Fw(\030)t Fy(\()p Fw(b)3068 1244 y Fo(0)3092 1277 y Fy(\))e(=)f Fw(\015)5 b Fy(\()p Fw(b)p Fy(\))456 1384 y Fj(and)36 b Fw(b)22 b Fv(\000)g Fw(a)30 b Fv(6)p Fy(=)g Fw(b)1007 1351 y Fo(0)1052 1384 y Fv(\000)22 b Fw(a)1193 1351 y Fo(0)1217 1384 y Fj(;)36 b(If)f Fw(\015)40 b Fj(is)c(p)-5 b(erio)g(dic)37 b(with)f(le)-5 b(ast)36 b(p)-5 b(erio)g(d)38 b Fw(d)30 b Fv(\025)g Fy(0)p Fj(,)36 b(then)g(the)g(strict)456 1492 y(ine)-5 b(quality)33 b(also)h(holds)g(pr)-5 b(ovide)g(d)35 b Fw(b)20 b Fv(\000)g Fw(a)25 b Fv(6)p Fy(=)g Fw(b)2002 1459 y Fo(0)2046 1492 y Fv(\000)20 b Fw(a)2185 1459 y Fo(0)2241 1492 y Fj(mo)-5 b(d)34 b Fw(d)p Fj(.)456 1756 y(Pr)-5 b(o)g(of.)43 b Fy(Since)37 b(the)h(system)f(is)h(autonomous,)i(w)m(e)d(can)h(assume)f (that)i Fw(a)2985 1723 y Fo(0)3045 1756 y Fy(=)e Fw(a)p Fy(.)62 b(The)456 1864 y(de\014nition)30 b(of)g(semi-static)j(curv)m (es)d(guaran)m(tees)i(that)1450 2026 y([)p Fw(A)1543 2040 y Ft(c)1578 2026 y Fy(\()p Fw(\015)5 b Fv(j)1690 2045 y Fr([)p Ft(a;b)p Fr(])1822 2026 y Fy(\)])26 b Fv(\024)f Fy([)p Fw(A)2097 2040 y Ft(c)2132 2026 y Fy(\()p Fw(\030)t Fv(j)2236 2045 y Fr([)p Ft(a;b)2343 2026 y Fl(0)2365 2045 y Fr(])2389 2026 y Fy(\)])456 2195 y(if)39 b Fw(\030)t Fy(\()p Fw(a)p Fy(\))h(=)f Fw(\015)5 b Fy(\()p Fw(a)p Fy(\),)42 b Fw(\030)t Fy(\()p Fw(b)1215 2162 y Fo(0)1239 2195 y Fy(\))e(=)f Fw(\015)5 b Fy(\()p Fw(b)p Fy(\).)68 b(T)-8 b(o)39 b(sho)m(w)g(that)h(the)f(equalit)m(y)h(can)g(not)f(hold)g (for)456 2303 y(an)m(y)c Fw(b)671 2270 y Fo(0)717 2303 y Fv(\000)23 b Fw(a)33 b Fv(6)p Fy(=)f Fw(b)23 b Fv(\000)g Fw(a)35 b Fy(when)f Fw(\015)40 b Fy(is)35 b(not)g(p)s(erio)s(dic,)g (let)h(us)e(assume)h(the)g(con)m(trary)-8 b(,)37 b(i.e.)456 2410 y([)p Fw(A)549 2424 y Ft(c)584 2410 y Fy(\()p Fw(\015)5 b Fv(j)696 2429 y Fr([)p Ft(a;b)p Fr(])827 2410 y Fy(\)])28 b(=)g([)p Fw(A)1107 2424 y Ft(c)1142 2410 y Fy(\()p Fw(\030)t Fv(j)1246 2429 y Fr([)p Ft(a;b)1353 2410 y Fl(0)1376 2429 y Fr(])1400 2410 y Fy(\)])k(holds)f(for)h(some)g Fw(b)2140 2378 y Fo(0)2185 2410 y Fv(\000)21 b Fw(a)27 b Fv(6)p Fy(=)h Fw(b)21 b Fv(\000)g Fw(a)32 b Fy(and)f Fw(\030)t Fy(:)43 b([)p Fw(a;)15 b(b)3125 2378 y Fo(0)3149 2410 y Fy(])28 b Fv(!)g Fw(M)10 b Fy(.)456 2524 y(The)35 b(prop)s(ert)m(y)f Fw(b)24 b Fv(\000)f Fw(a)34 b Fv(6)p Fy(=)g Fw(b)1407 2491 y Fo(0)1454 2524 y Fv(\000)23 b Fw(a)36 b Fy(and)e(the)i(uniqueness)e(of)i(the)g(solution)g(guaran)m (tee)456 2634 y(that)49 b(_)-39 b Fw(\015)6 b Fy(\()p Fw(a)p Fy(\))34 b Fv(6)p Fy(=)985 2610 y(_)966 2634 y Fw(\030)t Fy(\()p Fw(a)p Fy(\),)51 b(_)-39 b Fw(\015)5 b Fy(\()p Fw(b)p Fy(\))35 b Fv(6)p Fy(=)1509 2610 y(_)1490 2634 y Fw(\030)t Fy(\()p Fw(b)1608 2601 y Fo(0)1631 2634 y Fy(\).)56 b(W)-8 b(e)36 b(join)g(the)f(p)s(oin)m(t)g Fw(\015)5 b Fy(\()p Fw(a)24 b Fv(\000)f Fw(\016)s Fy(\))37 b(with)e(the)g(p)s(oin)m(t)456 2742 y Fw(\030)t Fy(\()p Fw(a)20 b Fy(+)g Fw(\016)s Fy(\))32 b(b)m(y)e(the)g(minimizer)h Fw(\020)7 b Fy(:)40 b([)p Fw(a)21 b Fv(\000)e Fw(\016)n(;)c(a)22 b Fy(+)e Fw(\016)s Fy(])26 b Fv(!)f Fw(M)10 b Fy(:)751 2848 y Fq(Z)842 2874 y Ft(a)p Fr(+)p Ft(\016)802 3054 y(a)p Fo(\000)p Ft(\016)973 2972 y Fy(\()p Fw(L)20 b Fv(\000)g Fw(\021)1226 2986 y Ft(c)1261 2972 y Fy(\)\()p Fw(d\020)7 b Fy(\()p Fw(t)p Fy(\)\))p Fw(dt)27 b Fy(=)189 b(inf)1765 3033 y Fm(\037)p Fk(\()p Fm(a)p Fl(\000)p Fm(\016)r Fk(\)=)p Fm(\015)s Fk(\()p Fm(a)p Fl(\000)p Fm(\016)r Fk(\))1770 3090 y Fm(\037)p Fk(\()p Fm(a)p Fk(+)p Fm(\016)r Fk(\)=)p Fm(\030)q Fk(\()p Fm(a)p Fk(+)p Fm(\016)r Fk(\))2220 2848 y Fq(Z)2311 2874 y Ft(a)p Fr(+)p Ft(\016)2271 3054 y(a)p Fo(\000)p Ft(\016)2441 2972 y Fy(\()p Fw(L)21 b Fv(\000)f Fw(\021)2695 2986 y Ft(c)2729 2972 y Fy(\)\()p Fw(d\037)p Fy(\()p Fw(t)p Fy(\)\))p Fw(dt:)456 3250 y Fy(By)28 b(T)-8 b(onelli)30 b(lemma,)g(the)f (minimizer)f(is)h(a)g Fw(C)2025 3217 y Fr(1)2064 3250 y Fy(-curv)m(e.)41 b(Th)m(us,)28 b(the)g(condition)43 b(_)-38 b Fw(\015)5 b Fy(\()p Fw(a)p Fy(\))26 b Fv(6)p Fy(=)475 3342 y(_)456 3366 y Fw(\030)t Fy(\()p Fw(a)p Fy(\))31 b(implies)g([)p Fw(A)1051 3380 y Ft(c)1086 3366 y Fy(\()p Fw(\015)5 b Fv(j)1198 3385 y Fr([)p Ft(a)p Fo(\000)p Ft(\016)o(;a)p Fr(])1422 3366 y Fy(\)])21 b(+)f([)p Fw(A)1687 3380 y Ft(c)1722 3366 y Fy(\()p Fw(\030)t Fv(j)1826 3385 y Fr([)p Ft(a;a)p Fr(+)p Ft(\016)r Fr(])2053 3366 y Fy(\)])27 b Fw(>)e Fy([)p Fw(A)2329 3380 y Ft(c)2364 3366 y Fy(\()p Fw(\020)7 b Fv(j)2471 3385 y Fr([)p Ft(a)p Fo(\000)p Ft(\016)o(;a)p Fr(+)p Ft(\016)r Fr(])2783 3366 y Fy(\)].)42 b(If)31 b(w)m(e)g(de\014ne)f(a)456 3479 y(curv)m(e)g Fw(\015)750 3446 y Fo(0)774 3479 y Fy(:)40 b Fu(R)25 b Fv(!)g Fw(M)1173 3842 y(\015)1225 3805 y Fo(0)1248 3842 y Fy(\()p Fw(t)p Fy(\))h(=)1473 3574 y Fq(8)1473 3656 y(>)1473 3683 y(>)1473 3710 y(>)1473 3738 y(<)1473 3901 y(>)1473 3928 y(>)1473 3956 y(>)1473 3983 y(:)1554 3651 y Fw(\015)5 b Fy(\()p Fw(t)p Fy(\))61 b(for)30 b Fw(t)25 b Fv(2)g Fy(\()p Fv(\0001)p Fw(;)15 b(a)21 b Fv(\000)f Fw(\016)s Fy(];)1554 3780 y Fw(\020)7 b Fy(\()p Fw(t)p Fy(\))61 b(for)30 b Fw(t)25 b Fv(2)g Fy([)p Fw(a)20 b Fv(\000)g Fw(\016)n(;)15 b(a)22 b Fy(+)d Fw(\016)s Fy(];)1554 3910 y Fw(\030)t Fy(\()p Fw(t)p Fy(\))61 b(for)30 b Fw(t)25 b Fv(2)g Fy([)p Fw(a)c Fy(+)f Fw(\016)n(;)15 b(b)2347 3877 y Fo(0)2371 3910 y Fy(];)1554 4040 y Fw(\015)5 b Fy(\()p Fw(t)21 b Fv(\000)e Fw(b)1824 4007 y Fo(0)1868 4040 y Fy(+)h Fw(b)p Fy(\))31 b(for)f Fw(t)25 b Fv(2)g Fy([)p Fw(b)2411 4007 y Fo(0)2434 4040 y Fw(;)15 b Fv(1)p Fy(\))p Fw(;)456 4211 y Fy(w)m(e)32 b(obtain)h(that)g Fw(\015)5 b Fy(\()p Fw(a)1210 4225 y Fr(1)1250 4211 y Fy(\))29 b(=)g Fw(\015)1466 4178 y Fo(0)1489 4211 y Fy(\()p Fw(a)1572 4225 y Fr(1)1612 4211 y Fy(\))k(for)f Fw(a)1869 4225 y Fr(1)1937 4211 y Fv(\024)c Fw(a)p Fy(,)34 b Fw(\015)2195 4178 y Fo(0)2218 4211 y Fy(\()p Fw(b)2292 4225 y Fr(1)2332 4211 y Fy(\))29 b(=)f Fw(\015)5 b Fy(\()p Fw(b)2621 4225 y Fr(1)2683 4211 y Fv(\000)21 b Fw(b)2814 4178 y Fo(0)2859 4211 y Fy(+)g Fw(b)p Fy(\))33 b(for)f Fw(b)3238 4225 y Fr(1)3306 4211 y Fv(\025)c Fw(b)456 4319 y Fy(and)1268 4444 y([)p Fw(A)1361 4458 y Ft(c)1396 4444 y Fy(\()p Fw(\015)5 b Fv(j)1508 4463 y Fr([)p Ft(a)1565 4472 y Fk(1)1600 4463 y Ft(;b)1650 4472 y Fk(1)1685 4463 y Fr(])1708 4444 y Fy(\)])26 b Fw(>)f Fy([)p Fw(A)1983 4458 y Ft(c)2018 4444 y Fy(\()p Fw(\015)2105 4407 y Fo(0)2129 4444 y Fv(j)2154 4463 y Fr([)p Ft(a)2211 4472 y Fk(1)2246 4463 y Ft(;b)2296 4472 y Fk(1)2330 4463 y Fo(\000)p Ft(b)2415 4444 y Fl(0)2438 4463 y Fr(+)p Ft(b)p Fr(])2547 4444 y Fy(\)])p Fw(:)456 4588 y Fy(This)k(is)i(absurd)d(as)j Fw(\015)k Fy(is)c(assumed)e(to)j(b) s(e)d(a)i Fw(c)p Fy(-semi-static)i(curv)m(e.)555 4779 y(T)-8 b(o)36 b(sho)m(w)g(the)f(prop)s(osition)g(in)g(the)h(case)h (that)f Fw(\015)k Fy(is)c Fw(d)p Fy(-p)s(erio)s(dic,)h(The)e(condition) 469 4890 y(_)-38 b Fw(\015)5 b Fy(\()p Fw(a)p Fy(\))26 b Fv(6)p Fy(=)767 4866 y(_)748 4890 y Fw(\030)t Fy(\()p Fw(a)p Fy(\),)44 b(_)-38 b Fw(\015)5 b Fy(\()p Fw(b)p Fy(\))26 b Fv(6)p Fy(=)1268 4866 y(_)1249 4890 y Fw(\030)t Fy(\()p Fw(b)1367 4857 y Fo(0)1391 4890 y Fy(\))k(holds)g(only)h(when)e Fw(b)20 b Fv(\000)g Fw(a)25 b Fv(6)p Fy(=)g Fw(b)2489 4857 y Fo(0)2533 4890 y Fv(\000)20 b Fw(a)30 b Fy(mo)s(d)g Fw(d)p Fy(.)41 b(W)-8 b(e)31 b(ha)m(v)m(e)870 5053 y([)p Fw(A)963 5067 y Ft(c)999 5053 y Fy(\()p Fw(\015)5 b Fv(j)1111 5071 y Fr([)p Ft(a;b)p Fr(])1242 5053 y Fy(\)])26 b(=)f([)p Fw(A)1517 5067 y Ft(c)1552 5053 y Fy(\()p Fw(\015)5 b Fv(j)1664 5071 y Fr([)p Ft(a;b)1771 5052 y Fl(0)1794 5071 y Fr(])1818 5053 y Fy(\)])p Fw(;)197 b Fv(8)30 b Fw(b)20 b Fv(\000)g Fw(a)26 b Fy(=)f Fw(b)2540 5015 y Fo(0)2583 5053 y Fv(\000)20 b Fw(a)30 b Fy(mo)s(d)g Fw(d:)3374 5216 y Fh(\003)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 456 251 a Fr(16)1012 b(X.)23 b(LI)h(AND)f(C.-Q.)f(CHENG)555 450 y Fy(The)42 b(orbit)g Fw(d\015)48 b Fy(obtained)43 b(in)f(the)g(theorem)h(2.2)g(is)g(lo)s (cally)g(minimal.)77 b(Indeed,)456 558 y(suitably)42 b(small)g(disk)g Fw(\031)s(V)1393 525 y Fo(\000)1494 558 y Fy(and)f Fw(\031)s(V)1811 525 y Fr(+)1912 558 y Fy(can)h(b)s(e)f(c)m(hosen)i(suc)m(h)e(that)i(the)f(minimal)456 666 y(homo)s(clinic)36 b(curv)m(e)g Fw(\020)42 b Fy(in)m(tersects)37 b Fw(\031)s(V)1784 633 y Fo(\000)1879 666 y Fy(as)f(w)m(ell)g(as)g Fw(\031)s(V)2431 633 y Fr(+)2525 666 y Fy(transv)m(ersally)-8 b(.)58 b(Once)36 b(w)m(e)456 774 y(\014xed)e(a)h(lift)916 750 y(\026)908 774 y Fw(\020)7 b Fy(,)36 b Fw(V)1089 741 y Fo(\006)1183 774 y Fy(is)f(then)f(automatically)k(\014xed.)53 b(As)35 b Fw(\020)41 b Fy(is)35 b(an)g(isolated)h(minimal)456 882 y(curv)m(e,)30 b(b)m(y)h(the)f(upp)s(er)e(semi-con)m(tin)m(uit)m(y) 33 b(of)e Fv(C)2055 897 y Ft(\021)r(;\026; )r(;e)2275 905 y Fm(n)2352 882 y Fy(in)f(\()p Fw(\021)s(;)15 b(\026;)g( )s Fy(\),)32 b(w)m(e)f(can)g(see)g(that)456 991 y(no)f(curv)m(e)i(in)e Fg(C)993 1006 y Ft(\021)r(;\026; )r(;e)1213 1014 y Fm(n)1290 991 y Fy(touc)m(hes)i(the)f(b)s(oundary)e(of)i Fw(V)2361 958 y Fo(\000)2451 991 y Fy(as)g(w)m(ell)h(of)f Fw(V)2925 958 y Fr(+)3015 991 y Fy(pro)m(vided)g Fw(\026)456 1099 y Fy(and)k Fw( )40 b Fy(are)d(suitably)f(small.)59 b(Consequen)m(tly)-8 b(,)39 b(\(2.4\))f(holds)e(whenev)m(er)g(\()18 b(\026)-63 b Fw(m)3114 1113 y Fr(0)3154 1099 y Fw(;)32 b Fy(\026)-62 b Fw(m)3274 1113 y Fr(1)3313 1099 y Fy(\))36 b Fv(2)456 1207 y Fw(@)5 b Fy(\()p Fw(V)618 1174 y Fo(\000)697 1207 y Fv(\002)20 b Fw(V)861 1174 y Fr(+)920 1207 y Fy(\),)31 b(regardless)g(what)f(v)-5 b(alue)31 b Fw(T)1960 1174 y Fo(0)2013 1207 y Fy(tak)m(es.)555 1398 y(T)-8 b(o)40 b(see)f(\(2.4\))i(holds)e(for)f(all)i(\()18 b(\026)-63 b Fw(m)1723 1412 y Fr(0)1763 1398 y Fw(;)32 b Fy(\026)-62 b Fw(m)1883 1412 y Fr(1)1922 1398 y Fy(\))40 b Fv(2)g Fw(V)2171 1365 y Fo(\000)2256 1398 y Fv(\002)26 b Fw(V)2426 1365 y Fr(+)2524 1398 y Fy(whenev)m(er)39 b Fw(T)2998 1365 y Fo(0)3060 1398 y Fy(touc)m(h)g(the)456 1506 y(b)s(oundary)28 b(of)i([)p Fw(T)k Fv(\000)20 b Fw(d;)15 b(T)33 b Fy(+)20 b Fw(d)p Fy(],)31 b(b)m(y)g(applying)f(the)g(prop)s(osition)g(2.2)i(w)m (e)f(\014nd)940 1709 y Fw(h)992 1671 y Ft(T)1043 1648 y Fl(0)992 1732 y Ft(\021)r(;\026; )r(;e)1212 1740 y Fm(n)1260 1709 y Fy(\()s(\026)-48 b Fw(\015)5 b Fy(\()p Fv(\000)p Fw(T)13 b Fy(\))p Fw(;)19 b Fy(\026)-49 b Fw(\015)6 b Fy(\(+)p Fw(T)13 b Fy(\)\))26 b Fw(>)f(h)2063 1671 y Ft(T)2063 1732 y(\021)r(;\026; )r(;e)2283 1740 y Fm(n)2330 1709 y Fy(\()s(\026)-48 b Fw(\015)5 b Fy(\()p Fv(\000)p Fw(T)13 b Fy(\))p Fw(;)19 b Fy(\026)-49 b Fw(\015)6 b Fy(\(+)p Fw(T)13 b Fy(\)\))456 1906 y(if)32 b Fw(T)607 1873 y Fo(0)658 1906 y Fv(6)p Fy(=)c Fw(T)45 b Fy(\()p Fw(\015)37 b Fy(is)32 b(clearly)i(not)e(p)s(erio)s(dic)g(in)f(t\).)47 b(Considered)31 b(as)i(the)f(function)g(of)g Fw(T)3396 1873 y Fo(0)3419 1906 y Fy(,)456 2021 y Fw(h)508 1988 y Ft(T)559 1965 y Fl(0)508 2049 y Ft(\021)r(;\026; )r(;e)728 2057 y Fm(n)775 2021 y Fy(\()17 b(\026)-62 b Fw(m)890 2035 y Fr(0)930 2021 y Fw(;)32 b Fy(\026)-62 b Fw(m)1050 2035 y Fr(1)1089 2021 y Fy(\))34 b(is)g(ob)m(viously)g(a)g(Lipsc)m (hitz)g(function.)50 b(Although)34 b(it)g(reac)m(hes)h(its)456 2140 y(minim)m(um)25 b(not)h(necessarily)h(at)f Fw(T)1636 2107 y Fo(0)1685 2140 y Fy(=)e Fw(T)39 b Fy(unless)j(\026)-62 b Fw(m)2216 2154 y Fr(0)2281 2140 y Fy(=)25 b Fw(\015)5 b Fy(\()p Fw(t)11 b Fv(\000)g Fw(T)i Fy(\))26 b(and)43 b(\026)-63 b Fw(m)2969 2154 y Fr(1)3034 2140 y Fy(=)25 b Fw(\015)5 b Fy(\()p Fw(t)11 b Fy(+)g Fw(T)i Fy(\))456 2248 y(for)30 b(some)h Fw(t)p Fy(,)f(w)m(e)h(ha)m(v)m(e)981 2446 y Fw(h)1033 2408 y Ft(T)10 b Fo(\006)p Ft(d)1033 2475 y(\021)r(;\026; )r(;e)1253 2483 y Fm(n)1301 2446 y Fy(\()17 b(\026)-62 b Fw(m)1416 2460 y Fr(0)1455 2446 y Fw(;)33 b Fy(\026)-63 b Fw(m)1575 2460 y Fr(1)1615 2446 y Fy(\))25 b Fw(>)181 b Fy(min)1771 2511 y Ft(T)1822 2492 y Fl(0)1845 2511 y Fo(2)p Fr([)p Ft(T)10 b Fo(\000)p Ft(d;T)g Fr(+)p Ft(d)p Fr(])2250 2446 y Fw(h)2302 2408 y Ft(T)2353 2385 y Fl(0)2302 2469 y Ft(\021)r(;\026; )r(;e)2522 2477 y Fm(n)2569 2446 y Fy(\()18 b(\026)-63 b Fw(m)2684 2460 y Fr(0)2724 2446 y Fw(;)32 b Fy(\026)-62 b Fw(m)2844 2460 y Fr(1)2883 2446 y Fy(\))456 2701 y(for)38 b(eac)m(h)i(\()18 b(\026)-63 b Fw(m)931 2715 y Fr(0)971 2701 y Fw(;)33 b Fy(\026)-63 b Fw(m)1091 2715 y Fr(1)1131 2701 y Fy(\))39 b Fv(2)g Fw(V)1379 2668 y Fo(\000)1464 2701 y Fv(\002)25 b Fw(V)1633 2668 y Fr(+)1692 2701 y Fy(,)42 b(pro)m(vided)c Fw(V)2214 2668 y Fo(\000)2312 2701 y Fy(and)g Fw(V)2571 2668 y Fr(+)2669 2701 y Fy(are)h(c)m(hosen)g(suitably)456 2809 y(small)30 b(for)h(giv)m(en)g Fw(d)26 b(>)f Fy(0.)887 3205 y(3.)46 b Fx(Constr)n(uction)33 b(of)h(global)f(connecting)g (orbits)555 3450 y Fy(Roughly)28 b(sp)s(eaking,)g(global)g(connecting)h (orbits)e(are)g(constructed)h(along)g(a)g(series)456 3558 y(of)d(lo)s(cal)h(minimal)f(connecting)i(orbits.)39 b(A)25 b(su\016cien)m(t)g(condition)h(to)g(construct)f(orbits)456 3668 y(connecting)938 3645 y(~)907 3668 y Fv(A)o Fy(\()p Fw(c)p Fy(\))j(to)1255 3645 y(~)1224 3668 y Fv(A)p Fy(\()p Fw(c)1371 3635 y Fo(0)1395 3668 y Fy(\))f(is)g(the)g(existence)i(of)e (a)g(generalized)i(transition)e(c)m(hain)h(in)456 3776 y Fw(H)539 3743 y Fr(1)578 3776 y Fy(\()p Fw(M)5 b(;)15 b Fu(R)p Fy(\))31 b(that)g(connects)g Fw(c)g Fy(to)g Fw(c)1664 3743 y Fo(0)1688 3776 y Fy(.)456 3964 y FC(De\014nition)41 b(3.1.)46 b Fj(L)-5 b(et)38 b Fw(c)g Fj(and)h Fw(c)1618 3931 y Fo(0)1679 3964 y Fj(b)-5 b(e)38 b(two)h(c)-5 b(ohomolgy)40 b(classes)e(in)g Fw(H)2917 3931 y Fr(1)2957 3964 y Fy(\()p Fw(M)5 b(;)15 b Fu(R)p Fy(\))p Fj(.)58 b(We)456 4072 y(say)42 b(that)g Fw(c)g Fj(is)g(joine)-5 b(d)42 b(with)g Fw(c)1530 4039 y Fo(0)1595 4072 y Fj(by)g(a)f(gener)-5 b(alize)g(d)43 b(tr)-5 b(ansition)44 b(chain)e(if)f(ther)-5 b(e)43 b(is)e(a)456 4179 y(c)-5 b(ontinuous)35 b(curve)g Fy(\000)p Fj(:)45 b Fy([0)p Fw(;)15 b Fy(1])31 b Fv(!)e Fw(H)1707 4146 y Fr(1)1747 4179 y Fy(\()p Fw(M)5 b(;)15 b Fu(R)p Fy(\))35 b Fj(with)h Fy(\000\(0\))30 b(=)e Fw(c)p Fj(,)36 b Fy(\000\(1\))30 b(=)f Fw(c)2996 4146 y Fo(0)3019 4179 y Fj(,)35 b(such)g(that)456 4287 y Fw(\013)p Fy(\(\000\()p Fw(s)p Fy(\)\))i Fj(ke)-5 b(eps)36 b(c)-5 b(onstant)37 b(for)f Fw(s)31 b Fv(2)f Fy([0)p Fw(;)15 b Fy(1])p Fj(,)38 b(and)f(for)f(e)-5 b(ach)37 b Fw(s)30 b Fv(2)h Fy([0)p Fw(;)15 b Fy(1])37 b Fj(at)f(le)-5 b(ast)37 b(one)f(of)456 4395 y(the)d(fol)5 b(lowing)33 b(c)-5 b(ases)34 b(takes)f(plac)-5 b(e:)555 4586 y Fy(\(I\))q Fj(,)28 b Fy(\000\()p Fw(s)p Fy(\))p Fj(-minimal)g(me)-5 b(asur)g(e)29 b(is)e(uniquely)g(er)-5 b(go)g(dic;)30 b(ther)-5 b(e)28 b(is)f(a)h(c)-5 b(o)g(or)g(dinate)30 b(tr)-5 b(ans-)456 4694 y(formation)47 b Fw(x)g Fv(!)g Fw(q)j Fy(=)e Fw(G)1408 4661 y Fo(\000)p Fr(1)1408 4717 y Ft(s)1502 4694 y Fw(x)d Fj(as)h(wel)5 b(l)45 b(as)g Fw(\016)2094 4708 y Ft(s)2179 4694 y Fw(>)i Fy(0)e Fj(such)g(that)h(in) f Fw(q)s Fj(-c)-5 b(o)g(or)g(dinates)490 4782 y Fy(\024)456 4805 y Fw(M)48 b Fy(=)38 b Fu(T)762 4772 y Ft(n)p Fo(\000)p Fr(1)924 4805 y Fv(\002)26 b Fy(\(2)p Fu(T)p Fy(\))p Fj(,)42 b Fv(N)13 b Fy(\(\000\()p Fw(s)p Fy(\)\))26 b(+)f Fw(\016)1757 4819 y Ft(s)1833 4805 y Fv(\032)38 b Fw(N)10 b Fj(,)41 b(and)46 b Fy(\024)-50 b Fw(\031)s Fv(N)13 b Fy(\(\000\()p Fw(s)p Fy(\))p Fw(;)2701 4782 y Fy(\024)2666 4805 y Fw(M)e Fy(\))p Fv(n)p Fy(\()p Fv(N)i Fy(\(\000\()p Fw(s)p Fy(\)\))28 b(+)d Fw(\016)3372 4819 y Ft(s)3409 4805 y Fy(\))456 4913 y Fj(c)-5 b(ontains)34 b(an)f(isolate)-5 b(d)34 b(orbit;)555 5104 y Fy(\(I)s(I\))p Fj(,)41 b Fy(\000\()p Fw(s)p Fy(\))e Fj(is)g Fw(c)p Fj(-e)-5 b(quivalent)39 b(to)h Fy(\000\()p Fw(s)1828 5071 y Fo(0)1851 5104 y Fy(\))g Fj(if)f Fw(s)2064 5071 y Fo(0)2126 5104 y Fj(is)g (su\016ciently)g(close)h(to)g Fw(s)p Fj(;)h Fv(9)e Fy(\006)3309 5122 y Fr(\000\()p Ft(s)p Fr(\))456 5216 y Fj(admissible)33 b(for)h(the)f Fy(\000\()p Fw(s)p Fy(\))p Fj(-minimal)g(me)-5 b(asur)g(e.)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 998 251 a Fr(CONNECTING)31 b(ORBITS)g(F)n(OR)f (A)n(UTONOMOUS)g(SYSTEMS)472 b(17)555 450 y Fy(In)30 b(the)g(item)h(\(I\),)g(the)f(unique)f(ergo)s(dicit)m(y)j(can)e(b)s(e)g (replaced)h(b)m(y)f(that)g(the)h(Aubry)456 558 y(set)h(has)g (\014nitely)g(man)m(y)h(classes.)46 b(This)32 b(assumption)f(is)h(made) g(for)g(brevit)m(y)h(of)f(pro)s(of)456 666 y(and)d(it)i(is)g(enough)f (for)g(the)h(application)g(in)f Fj(a)j(priori)f Fy(unstable)e(systems.) 456 840 y FC(Theorem)53 b(3.1.)f Fj(Supp)-5 b(ose)49 b(that)g(ther)-5 b(e)48 b(is)g(a)g(gener)-5 b(alize)g(d)50 b(tr)-5 b(ansition)49 b(chain)g Fy(\000)j(:)456 948 y([0)p Fw(;)15 b Fy(1])27 b Fv(!)e Fw(H)862 915 y Fr(1)901 948 y Fy(\()p Fw(M)5 b(;)15 b Fu(R)p Fy(\))29 b Fj(joining)f Fw(c)g Fj(to)h Fw(c)1712 915 y Fo(0)1735 948 y Fj(.)41 b(Then)28 b(ther)-5 b(e)29 b(exists)f(an)h(orbit)f(of)g(the)h(L)-5 b(agr)g(ange)456 1063 y(\015ow)34 b Fw(\036)703 1030 y Ft(t)703 1090 y(L)789 1063 y Fw(d\015)5 b Fj(:)44 b Fu(R)26 b Fv(!)h Fw(T)13 b(M)43 b Fj(that)35 b(c)-5 b(onne)g(cts)35 b(the)e(two)i(A)n(ubry)e(sets:)43 b Fw(\013)p Fy(\()p Fw(d\015)5 b Fy(\))29 b Fv(\022)3116 1040 y Fy(~)3084 1063 y Fv(A)p Fy(\()p Fw(c)p Fy(\))34 b Fj(and)456 1182 y Fw(!)s Fy(\()p Fw(d\015)5 b Fy(\))26 b Fv(\022)838 1159 y Fy(~)807 1182 y Fv(A)o Fy(\()p Fw(c)953 1149 y Fo(0)977 1182 y Fy(\))p Fj(.)456 1451 y(Pr)-5 b(o)g(of.)43 b Fy(Roughly)24 b(sp)s(eaking,)i(the)e(global)i(connecting)g(orbits)e (are)h(constructed)g(shad-)456 1559 y(o)m(wing)33 b(a)f(sequence)h(of)g (lo)s(cal)g(connecting)h(orbits.)46 b(According)33 b(to)g(the)g(study)e (in)h(the)456 1669 y(last)40 b(section,)j(if)d Fw(s)1113 1636 y Fo(0)1175 1669 y Fy(is)f(close)i(to)f Fw(s)p Fy(,)i(eac)m(h)f (Aubry)d(set)2420 1646 y(~)2389 1669 y Fv(A)p Fy(\(\000\()p Fw(s)p Fy(\)\))i(can)g(b)s(e)e(connected)456 1784 y(to)31 b(some)827 1761 y(~)796 1784 y Fv(A)o Fy(\(\000\()p Fw(s)1038 1751 y Fo(0)1062 1784 y Fy(\)\))h(b)m(y)f(either)g Fw(h)p Fy(-t)m(yp)s(e,)i(or)e Fw(c)p Fy(-t)m(yp)s(e)g(of)h(lo)s(cal)g(minimal) f(orbits.)43 b(Th)m(us,)456 1892 y(there)30 b(is)h(a)g(sequence)g(0)25 b(=)h Fw(s)1443 1906 y Fr(0)1507 1892 y Fw(<)f(s)1646 1906 y Fr(1)1711 1892 y Fw(<)g Fv(\001)15 b(\001)g(\001)26 b Fw(<)f(s)2077 1907 y Ft(k)2145 1892 y Fy(=)g(1)31 b(suc)m(h)f(that)h (for)f(eac)m(h)i(0)26 b Fv(\024)f Fw(j)31 b(<)25 b(k)487 1978 y Fy(~)456 2001 y Fv(A)o Fy(\(\000\()p Fw(s)698 2015 y Ft(j)735 2001 y Fy(\)\))31 b(is)f(connected)i(to)1494 1978 y(~)1463 2001 y Fv(A)o Fy(\(\000\()p Fw(s)1705 2015 y Ft(j)t Fr(+1)1832 2001 y Fy(\)\))f(b)m(y)f(some)h(lo)s(cal)h(minimal) e(orbits.)555 2192 y(Let)38 b Fw(c)764 2206 y Ft(j)838 2192 y Fy(=)g(\000\()p Fw(s)1082 2206 y Ft(j)1118 2192 y Fy(\),)i Fw(j)j Fy(=)37 b(0)p Fw(;)15 b Fy(1)p Fw(;)g Fv(\001)g(\001)g(\001)33 b Fw(;)15 b(k)s Fy(.)63 b(W)-8 b(e)39 b(divide)e(the)h(set)g Fv(f)p Fy(0)p Fw(;)15 b Fy(1)p Fw(;)g Fv(\001)g(\001)g(\001)33 b Fw(;)15 b(k)s Fv(g)39 b Fy(in)m(to)g Fw(m)456 2300 y Fy(partitions)25 b Fv(f)p Fy(0)p Fw(;)15 b Fy(1)p Fw(;)g Fv(\001)g(\001)g(\001)j Fw(k)s Fv(g)26 b Fy(=)f Fv(f)p Fy(0)p Fw(;)15 b Fy(1)p Fw(;)g Fv(\001)g(\001)g(\001)34 b Fw(;)15 b(i)1845 2314 y Fr(1)1885 2300 y Fw(;)g(i)1956 2314 y Fr(1)2006 2300 y Fy(+)9 b(1)p Fw(;)15 b Fv(\001)g(\001)g(\001)32 b Fw(;)15 b(i)2379 2314 y Fr(2)2419 2300 y Fw(;)g Fv(\001)g(\001)g(\001)32 b Fw(;)15 b(i)2667 2314 y Ft(m)2743 2300 y Fv(\000)9 b Fy(1)p Fw(;)15 b(i)2939 2314 y Ft(m)3032 2300 y Fy(=)25 b Fw(k)s Fv(g)p Fy(.)40 b(The)456 2415 y(rule)25 b(to)i(mak)m(e)g(suc)m (h)f(a)g(division)g(is)g(that)g(for)g(all)h Fw(i)f Fy(=)e Fw(i)2296 2429 y Ft(j)2333 2415 y Fw(;)15 b(i)2404 2429 y Ft(j)2453 2415 y Fy(+)c(1)p Fw(;)k Fv(\001)g(\001)g(\001)32 b Fw(;)15 b(i)2828 2429 y Ft(j)t Fr(+1)2967 2415 y Fv(\000)c Fy(1,)3178 2392 y(~)3147 2415 y Fv(A)o Fy(\()p Fw(c)3293 2429 y Ft(i)3322 2415 y Fy(\))26 b(is)456 2533 y(connected)i(to)1017 2510 y(~)986 2533 y Fv(A)o Fy(\()p Fw(c)1132 2547 y Ft(i)p Fr(+1)1251 2533 y Fy(\))g(b)m(y)g(a)g(lo)s(cal)h(minimal)f(orbit)f(of)h (the)g(same)g(t)m(yp)s(e.)40 b(Let)28 b(\003)3235 2547 y Ft(c)3298 2533 y Fy(and)456 2641 y(\003)519 2656 y Ft(h)587 2641 y Fy(b)s(e)c(the)g(subset)f(of)h Fv(f)p Fw(i)1300 2655 y Fr(1)1340 2641 y Fw(;)15 b(i)1411 2655 y Fr(2)1452 2641 y Fw(;)g Fv(\001)g(\001)g(\001)31 b Fw(;)15 b(i)1699 2655 y Ft(m)1767 2641 y Fv(g)p Fy(,)25 b(\003)1925 2655 y Ft(c)1968 2641 y Fv([)7 b Fy(\003)2099 2656 y Ft(h)2169 2641 y Fy(=)25 b Fv(f)p Fw(i)2341 2655 y Fr(1)2381 2641 y Fw(;)15 b(i)2452 2655 y Fr(2)2492 2641 y Fw(;)g Fv(\001)g(\001)g(\001)i Fw(i)2685 2655 y Ft(m)2752 2641 y Fv(g)p Fy(,)25 b(\003)2910 2655 y Ft(c)2953 2641 y Fv(\\)7 b Fy(\003)3084 2656 y Ft(h)3154 2641 y Fy(=)25 b Fu(?)p Fy(.)38 b(If)456 2756 y Fw(i)487 2770 y Ft(j)549 2756 y Fv(2)25 b Fy(\003)698 2770 y Ft({)725 2756 y Fy(,)j(then)f(for)g(all)i Fw(i)c Fy(=)g Fw(i)1425 2770 y Ft(j)1462 2756 y Fw(;)15 b(i)1533 2770 y Ft(j)1584 2756 y Fy(+)f(1)p Fw(;)h Fv(\001)g(\001)g(\001)33 b Fw(;)15 b(i)1963 2770 y Ft(j)t Fr(+1)2104 2756 y Fv(\000)f Fy(1,)2319 2733 y(~)2288 2756 y Fv(A)o Fy(\()p Fw(c)2434 2770 y Ft(i)2463 2756 y Fy(\))28 b(is)f(connected)h(to)3175 2733 y(~)3143 2756 y Fv(A)p Fy(\()p Fw(c)3290 2770 y Ft(i)p Fr(+1)3409 2756 y Fy(\))456 2864 y(b)m(y)i(a)h(lo)s(cal)g (minimal)g(orbit)f(of)h(the)f Fw({)p Fy(-)h(t)m(yp)s(e)g(\()p Fw({)g Fy(=)p Fw(c)p Fy(,)f(or)h Fw(h)p Fy(\).)555 3062 y(Because)39 b(of)e(the)h(upp)s(er)d(semi-con)m(tin)m(uit)m(y)k Fw(c)e Fv(!)2327 3039 y Fy(~)2298 3062 y Fv(N)13 b Fy(\()p Fw(c;)i(M)10 b Fy(\),)41 b(once)d(the)f(Ma)s(~)-48 b(n)m(\023)-43 b(e)39 b(set)484 3153 y(~)456 3176 y Fv(N)13 b Fy(\()p Fw(\015)5 b Fy(\()p Fw(s)p Fy(\)\))30 b(is)g(in)f(the)h(case)g(I)g (\(or)f(I)s(I\),)h(then)f(for)g(those)h Fw(s)2343 3143 y Fo(0)2396 3176 y Fy(close)g(to)h Fw(s)p Fy(,)2849 3153 y(~)2820 3176 y Fv(N)13 b Fy(\(\000\()p Fw(s)3078 3143 y Fo(0)3102 3176 y Fy(\)\))30 b(is)f(also)456 3284 y(in)38 b(the)h(case)h(I)f(\(or)g(I)s(I\).)g(Th)m(us,)h(for)f(eac)m(h)h Fw(i)1989 3298 y Ft(j)2066 3284 y Fv(2)f Fy(\003)2229 3299 y Ft(h)2274 3284 y Fy(,)i(b)m(y)e(c)m(ho)s(osing)h Fw(c)2892 3298 y Ft(i)2916 3308 y Fm(j)2949 3298 y Fo(\000)p Fr(1)3082 3284 y Fy(and)e Fw(c)3306 3298 y Ft(i)3330 3308 y Fm(j)s Fk(+1)456 3392 y Fy(su\016cien)m(tly)h(close)h(to)f Fw(c)1303 3406 y Ft(i)1327 3416 y Fm(j)1402 3392 y Fy(and)f Fw(c)1626 3406 y Ft(i)1650 3416 y Fm(j)s Fk(+1)1760 3406 y Fo(\000)p Fr(1)1893 3392 y Fy(resp)s(ectiv)m(ely)-8 b(,)42 b(w)m(e)d(can)g(assume)f(that)h(b)s(oth)456 3502 y Fw(c)495 3516 y Ft(i)519 3526 y Fm(j)551 3516 y Fo(\000)p Fr(1)676 3502 y Fy(and)30 b Fw(c)892 3516 y Ft(i)916 3526 y Fm(j)s Fk(+1)1060 3502 y Fy(satisfy)h(the)g(condition)g(of)f (the)h(item)g(\(I\))g(in)f(the)g(de\014nition)g(3.1.)555 3708 y(T)-8 b(o)38 b(eac)m(h)h Fw(c)947 3722 y Ft(i)1013 3708 y Fy(w)m(e)f(asso)s(ciate)i(a)e(co)s(ordinate)g(system)g Fw(x)f Fv(!)g Fw(q)j Fy(=)d Fw(G)2867 3670 y Fo(\000)p Fr(1)2867 3735 y Ft(i)2962 3708 y Fw(x)p Fy(,)i(write)f(the)456 3824 y(in)m(v)m(erse)32 b(of)f Fw(G)933 3838 y Ft(i)993 3824 y Fy(with)h Fw(G)1273 3786 y Fo(\000)p Fr(1)1273 3852 y Ft(i)1394 3824 y Fy(=)27 b([)p Fw(g)1563 3786 y Fo(\000)p Fr(1)1560 3852 y Ft(i;)p Fr(1)1658 3824 y Fw(;)15 b(g)1744 3786 y Fo(\000)p Fr(1)1741 3852 y Ft(i;)p Fr(2)1840 3824 y Fw(;)g Fv(\001)g(\001)g(\001)31 b Fw(;)15 b(g)2102 3786 y Fo(\000)p Fr(1)2099 3852 y Ft(i;n)2198 3824 y Fy(])2223 3791 y Ft(t)2253 3824 y Fy(.)43 b(By)32 b(the)g(assumption,)f(for)g(eac)m(h)456 3936 y Fw(i)26 b Fv(2)f Fy(\003)662 3950 y Ft(c)697 3936 y Fy(,)31 b(w)m(e)h(can)f(c)m (ho)s(ose)h Fw(G)1415 3950 y Ft(i)1474 3936 y Fy(suc)m(h)e(that,)i(in)f Fw(q)s Fy(-co)s(ordinates,)h(the)f(rotation)h(v)m(ector)g(of)456 4044 y(eac)m(h)37 b Fw(c)705 4058 y Ft(i)733 4044 y Fy(-minimal)f (measure)g Fw(\026)1537 4058 y Ft(c)1568 4068 y Fm(i)1633 4044 y Fy(is)g(p)s(ositiv)m(e.)58 b(Note)37 b(that)g(eac)m(h)f (semi-static)i(orbits)456 4152 y(approac)m(hes)45 b(to)g(the)f(supp)s (ort)f(of)i(the)f(minimal)h(measure,)j(and)c(note)h(the)f(upp)s(er)456 4260 y(semi-con)m(tin)m(uit)m(y)32 b(of)f(Ma)s(~)-48 b(n)m(\023)-43 b(e)32 b(set)f(on)f Fw(c)p Fy(,)h(w)m(e)g(can)g(assume)f (that)1254 4427 y Fv(h)p Fw(g)1335 4389 y Fo(\000)p Fr(1)1332 4454 y Ft(j;)p Fr(1)1430 4427 y Fw(;)15 b(!)s Fy(\()p Fw(\026)1620 4441 y Ft(c)1651 4451 y Fm(i)1682 4427 y Fy(\))p Fv(i)26 b Fw(>)f Fy(0)p Fw(;)198 b Fv(8)29 b Fw(j)i Fy(=)25 b Fw(i)c Fv(\000)f Fy(1)p Fw(;)15 b(i)456 4598 y Fy(holds)34 b(for)g(eac)m(h)h(ergo)s(dic)g(comp)s(onen)m(t)g Fw(\026)1893 4612 y Ft(c)1924 4622 y Fm(j)1994 4598 y Fy(if)g Fw(c)2121 4612 y Ft(i)p Fo(\000)p Fr(1)2274 4598 y Fy(is)f(c)m(hosen)h(suitably)g(close)g(to)h Fw(c)3391 4612 y Ft(i)3419 4598 y Fy(.)456 4706 y(Therefore,)30 b Fv(9)g Fw(q)1015 4720 y Ft(i;)p Fr(1)1123 4706 y Fw(>)25 b Fy(0)30 b(suc)m(h)g(that)983 4881 y Fv(h)p Fw(g)1064 4843 y Fo(\000)p Fr(1)1061 4909 y Ft(i;)p Fr(1)1159 4881 y Fw(;)15 b Fy(\001)s(~)-48 b Fw(\015)1322 4895 y Ft(i)1350 4881 y Fv(i)26 b(\025)f Fy(2)p Fw(\031)s(;)107 b Fy(whenev)m(er)60 b Fv(h)p Fw(g)2249 4843 y Fo(\000)p Fr(1)2246 4909 y Ft(i)p Fo(\000)p Fr(1)p Ft(;)p Fr(1)2420 4881 y Fw(;)15 b Fy(\001)s(~)-48 b Fw(\015)2583 4895 y Ft(i)2612 4881 y Fv(i)25 b(\025)g Fw(q)2809 4895 y Ft(i;)p Fr(1)2892 4881 y Fw(;)983 5038 y Fv(h)p Fw(g)1064 5000 y Fo(\000)p Fr(1)1061 5065 y Ft(i)p Fo(\000)p Fr(1)p Ft(;)p Fr(1)1235 5038 y Fw(;)15 b Fy(\001)s(~)-48 b Fw(\015)1398 5052 y Ft(i)1426 5038 y Fv(i)26 b(\025)f Fy(2)p Fw(\031)s(;)107 b Fy(whenev)m(er)60 b Fv(h)p Fw(g)2325 5000 y Fo(\000)p Fr(1)2322 5065 y Ft(i;)p Fr(1)2420 5038 y Fw(;)15 b Fy(\001)s(~)-48 b Fw(\015)2583 5052 y Ft(i)2612 5038 y Fv(i)25 b(\025)g Fw(q)2809 5052 y Ft(i;)p Fr(1)456 5038 y Fy(\(3.1\))456 5216 y(where)33 b(~)-49 b Fw(\015)765 5230 y Ft(i)824 5216 y Fy(is)30 b(an)m(y)h Fw(c)1126 5230 y Ft(i)1155 5216 y Fy(-semi-static)i(curv)m(e,)d(\001)s(~)-48 b Fw(\015)2031 5230 y Ft(i)2085 5216 y Fy(=)28 b(~)-48 b Fw(\015)2228 5230 y Ft(i)2256 5216 y Fy(\()p Fw(t)2324 5183 y Fo(0)2347 5216 y Fy(\))21 b Fv(\000)i Fy(~)-48 b Fw(\015)2541 5230 y Ft(i)2569 5216 y Fy(\()p Fw(t)p Fy(\))31 b(with)f Fw(t)2943 5183 y Fo(0)2991 5216 y Fw(>)25 b(t)p Fy(.)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 456 251 a Fr(18)1012 b(X.)23 b(LI)h(AND)f(C.-Q.)f(CHENG)555 450 y Fy(F)-8 b(or)30 b(eac)m(h)h(in)m(teger)g Fw(i)25 b Fv(2)1363 382 y Fq(S)1439 477 y Ft(i)1463 487 y Fm(j)1496 477 y Fo(2)p Fr(\003)1592 489 y Fm(h)1636 450 y Fv(f)p Fw(i)1712 464 y Ft(j)1749 450 y Fw(;)15 b(i)1820 464 y Ft(j)1876 450 y Fy(+)j(1)p Fw(;)d Fv(\001)g(\001)g(\001)32 b Fw(;)15 b(i)2258 464 y Ft(j)t Fr(+1)2404 450 y Fv(\000)j Fy(1)p Fv(g)p Fy(,)31 b(w)m(e)f(\014nd)e(a)h(co)s(ordinate)456 586 y(system)20 b Fw(x)25 b Fv(!)g Fw(q)k Fy(=)c Fw(G)1177 548 y Fo(\000)p Fr(1)1177 613 y Ft(i)1271 586 y Fw(x)c Fy(of)f Fu(T)1498 553 y Ft(n)1565 586 y Fy(suc)m(h)g(that)h(\(3.1\))h(holds)e(and)g Fw(N)35 b Fy(=)25 b Fu(T)2814 553 y Ft(n)p Fo(\000)p Fr(1)2951 586 y Fv(\002fj)p Fw(q)3133 600 y Ft(n)3180 586 y Fv(j)g Fw(<)g(d)p Fv(g)p Fy(.)456 694 y(Let)32 b Fw(U)682 708 y Ft(i)737 694 y Fy(=)27 b Fv(fj)p Fw(q)946 708 y Fr(1)985 694 y Fv(j)h(\024)e Fy(\001)1211 708 y Ft(i)1239 694 y Fv(g)c(\002)e Fw(D)1472 708 y Ft(i)1501 694 y Fy(,)32 b(there)f(exist)i(closed)f(1-forms)f Fw(\021)2649 708 y Ft(i)2709 694 y Fy(and)38 b(\026)-52 b Fw(\026)2942 708 y Ft(i)3001 694 y Fy(on)32 b Fu(T)3190 661 y Ft(n)3268 694 y Fy(with)456 802 y([)p Fw(\021)526 816 y Ft(i)554 802 y Fy(])41 b(=)f Fw(c)770 816 y Ft(i)799 802 y Fy(,)i([)7 b(\026)-52 b Fw(\026)946 816 y Ft(i)974 802 y Fy(])41 b(=)f Fw(c)1190 816 y Ft(i)p Fr(+1)1335 802 y Fv(\000)26 b Fw(c)1471 816 y Ft(i)1539 802 y Fy(and)39 b(supp)6 b(\026)-52 b Fw(\026)1968 816 y Ft(i)2022 802 y Fv(\\)2130 779 y Fy(\026)2109 802 y Fw(D)2184 816 y Ft(i)2253 802 y Fy(=)40 b Fu(?)p Fy(.)68 b(F)-8 b(or)40 b(the)g(co)m(v)m(ering)i (space)490 887 y(\026)456 910 y Fw(M)35 b Fy(=)25 b Fu(R)t Fv(\002)t Fu(T)881 877 y Ft(n)p Fo(\000)p Fr(1)1016 910 y Fy(,)f Fw(\021)1110 924 y Ft(i)1161 910 y Fy(and)k(\026)-52 b Fw(\026)1384 924 y Ft(i)1434 910 y Fy(ha)m(v)m(e)23 b(their)g(natural)f(lift,)i(denoted)f(b)m(y)f(the)g(same)g(sym)m(b)s (ol.)456 1018 y(W)-8 b(e)36 b(also)f(use)g Fw(U)1027 1032 y Ft(i)1090 1018 y Fy(to)g(denote)g Fv(fj)p Fw(q)1613 1032 y Fr(1)1653 1018 y Fv(j)e(\024)f Fy(\001)1890 1032 y Ft(i)1918 1018 y Fv(g)23 b(\002)g Fw(D)2155 1032 y Ft(i)2216 1018 y Fv(\032)2353 995 y Fy(\026)2319 1018 y Fw(M)11 b Fy(.)53 b(There)34 b(is)h(a)g(non-negativ)m(e)456 1126 y(function)f Fw(w)881 1140 y Ft(i)943 1126 y Fy(of)41 b(^)-51 b Fw(q)37 b Fy(with)d(its)h(supp)s(ort)d(con)m(tained)k(in)e(a) h(small)f(neigh)m(b)s(orho)s(o)s(d)f(of)i Fw(D)3416 1140 y Ft(i)456 1234 y Fy(and)d Fw(w)700 1248 y Ft(i)728 1234 y Fv(j)753 1248 y Ft(D)811 1258 y Fm(i)872 1234 y Fy(=)e(constan)m(t)q (,)k(and)f(t)m(w)m(o)h(functions)f Fw(\045)2164 1248 y Ft(i)2192 1234 y Fy(,)43 b(\026)-53 b Fw(\045)2299 1201 y Fo(0)2299 1259 y Ft(i)2360 1234 y Fy(of)33 b Fw(q)2507 1248 y Fr(1)2579 1234 y Fy(suc)m(h)g(that)h Fw(\045)3034 1248 y Ft(i)3092 1234 y Fy(=)k(\026)-53 b Fw(\045)3240 1201 y Fo(0)3240 1259 y Ft(i)3298 1234 y Fy(=)30 b(0)456 1342 y(when)k Fw(q)739 1356 y Fr(1)812 1342 y Fv(2)f Fy(\()p Fv(\0001)p Fw(;)15 b Fv(\000)p Fy(\001)1290 1356 y Ft(i)1318 1342 y Fy(],)37 b(while)f(for)f Fw(q)1833 1356 y Fr(1)1906 1342 y Fv(2)e Fy([\001)2101 1356 y Ft(i)2129 1342 y Fw(;)15 b Fv(1)p Fy(\))36 b(w)m(e)g(ha)m(v)m(e)h Fw(\045)2732 1356 y Ft(i)2793 1342 y Fy(=)d(1)i(and)43 b(\026)-54 b Fw(\045)3207 1309 y Fo(0)3207 1367 y Ft(i)3269 1342 y Fy(=)34 b(0.)456 1450 y(Let)c Fw(\026)673 1464 y Ft(i)727 1450 y Fy(=)25 b Fw(\045)870 1464 y Ft(i)905 1450 y Fy(\026)-52 b Fw(\026)953 1464 y Ft(i)981 1450 y Fy(,)30 b Fw( )1095 1464 y Ft(i)1149 1450 y Fy(=)j(\026)-53 b Fw(\045)1292 1417 y Fo(0)1292 1475 y Ft(i)1320 1450 y Fy(\()p Fw(q)1396 1464 y Fr(1)1435 1450 y Fy(\))p Fw(w)1535 1464 y Ft(i)1564 1450 y Fy(\()7 b(^)-52 b Fw(q)s Fy(\))18 b(_)-43 b Fw(q)1719 1464 y Fr(1)1789 1450 y Fy(and)30 b(de\014ne)f(a)i(Lagrangian)1403 1629 y Fw(L)1465 1644 y Ft(\021)1500 1654 y Fm(i)1527 1644 y Ft(;\026)1589 1654 y Fm(i)1615 1644 y Ft(; )1681 1654 y Fm(i)1737 1629 y Fy(=)25 b Fw(L)20 b Fv(\000)g Fw(\021)2051 1643 y Ft(i)2099 1629 y Fv(\000)g Fw(\026)2245 1643 y Ft(i)2293 1629 y Fv(\000)g Fw( )2443 1643 y Ft(i)2472 1629 y Fw(:)456 1809 y Fy(According)36 b(to)h(the)g(theorem)f(2.2,)j(w)m(e)e(can)f(see) h(that)f(eac)m(h)i(curv)m(e)h(\026)-48 b Fw(\015)2884 1823 y Ft(i)2947 1809 y Fv(2)35 b Fg(C)3104 1824 y Ft(\021)3139 1834 y Fm(i)3165 1824 y Ft(;\026)3227 1834 y Fm(i)3253 1824 y Ft(; )3319 1834 y Fm(i)3345 1824 y Ft(;e)3398 1832 y Fm(n)456 1919 y Fy(determines)k(a)h(lo)s(cal)h(minimal)e(orbit)h (of)g Fw(\036)1995 1886 y Ft(t)1995 1946 y(L)2087 1919 y Fy(of)f Fw(h)p Fy(-t)m(yp)s(e,)k(suc)m(h)c(that)h Fw(d\015)3037 1933 y Ft(i)3106 1919 y Fy(connects)487 2015 y(~)456 2038 y Fv(A)o Fy(\()p Fw(c)602 2052 y Ft(i)631 2038 y Fy(\))i(to)861 2015 y(~)830 2038 y Fv(A)o Fy(\()p Fw(c)976 2052 y Ft(i)p Fr(+1)1095 2038 y Fy(\).)74 b(Note)43 b(that)i(\026)-49 b Fw(\015)1714 2052 y Ft(i)1784 2038 y Fy(is)41 b(a)h(curv)m(e)f Fu(R)i Fv(!)2504 2015 y Fy(\026)2470 2038 y Fw(M)53 b Fy(=)43 b Fu(R)27 b Fv(\002)h Fu(T)2978 2005 y Ft(n)p Fo(\000)p Fr(1)3156 2038 y Fy(passing)456 2146 y(through)h Fw(U)861 2160 y Ft(i)890 2146 y Fv(n)p Fy(\()p Fv(fj)p Fw(q)1081 2160 y Fr(1)1121 2146 y Fv(j)d(\024)f Fy(\001)1344 2160 y Ft(i)1371 2146 y Fv(g)c(\002)f Fw(@)5 b(D)1656 2160 y Ft(i)1685 2146 y Fy(\).)555 2338 y(In)30 b(this)h(case,)h(there) f(exist)h(t)m(w)m(o)g(\()p Fw(n)20 b Fv(\000)g Fy(1\)-dimensional)32 b(disks)e Fw(V)2799 2300 y Fo(\000)2778 2365 y Ft(i)2888 2338 y Fy(and)h Fw(V)3139 2300 y Fr(+)3119 2365 y Ft(i)p Fr(+1)3268 2338 y Fy(with)456 2459 y Fw(\031)s(V)584 2421 y Fo(\000)564 2486 y Ft(i)668 2459 y Fv(\032)25 b Fy(\()p Fv(A)872 2421 y Fo(\000)872 2485 y Fr(0)931 2459 y Fy(\()p Fw(c)1005 2473 y Ft(i)1034 2459 y Fy(\))15 b(+)g Fw(\016)s Fy(\))p Fv(nA)1366 2473 y Fr(0)1406 2459 y Fy(\()p Fw(c)1480 2473 y Ft(i)1509 2459 y Fy(\),)29 b Fw(\031)s(V)1726 2421 y Fr(+)1706 2486 y Ft(i)p Fr(+1)1849 2459 y Fv(\032)c Fy(\()p Fv(A)2053 2421 y Fr(+)2053 2485 y(0)2112 2459 y Fy(\()p Fw(c)2186 2473 y Ft(i)p Fr(+1)2305 2459 y Fy(\))15 b(+)g Fw(\016)s Fy(\))p Fv(nA)2637 2473 y Fr(0)2677 2459 y Fy(\()p Fw(c)2751 2473 y Ft(i)p Fr(+1)2870 2459 y Fy(\),)29 b(big)f(n)m(um)m(b)s(ers)456 2580 y Fw(T)522 2542 y Fr(+)509 2607 y Ft(i)609 2580 y Fw(>)f Fy(0,)34 b(suitably)e(small)g Fw(d)1445 2594 y Ft(i)1502 2580 y Fw(>)c Fy(0)k(and)g(quite)g(small)h Fw(\017)2365 2547 y Fo(\003)2365 2605 y Ft(i)2432 2580 y Fw(>)28 b Fy(0)33 b(suc)m(h)f(that)k(\026)-49 b Fw(\015)3061 2594 y Ft(i)3089 2580 y Fy(\()p Fv(\000)p Fw(T)3261 2542 y Fr(+)3248 2607 y Ft(i)3320 2580 y Fy(\))29 b Fv(2)456 2694 y Fw(V)529 2655 y Fo(\000)509 2721 y Ft(i)588 2694 y Fy(,)34 b(\026)-49 b Fw(\015)690 2708 y Ft(i)719 2694 y Fy(\()p Fw(T)820 2655 y Fr(+)807 2721 y Ft(i)879 2694 y Fy(\))25 b Fv(2)g Fw(V)1099 2655 y Fr(+)1078 2721 y Ft(i)p Fr(+1)1227 2694 y Fy(and)860 2912 y(min)1011 2811 y Fq(n)1072 2912 y Fw(h)1124 2875 y Fo(1)1124 2935 y Ft(c)1199 2912 y Fy(\()p Fw(x)1286 2875 y Fo(\000)1345 2912 y Fw(;)15 b(m)1465 2926 y Fr(0)1505 2912 y Fy(\))20 b(+)g Fw(h)1703 2875 y Ft(T)1703 2935 y(\021)1738 2945 y Fm(i)1765 2935 y Ft(;\026)1827 2945 y Fm(i)1854 2935 y Ft(; )1920 2945 y Fm(i)1946 2935 y Ft(;e)1999 2943 y Fm(n)2045 2912 y Fy(\()d(\026)-62 b Fw(m)2160 2926 y Fr(0)2200 2912 y Fw(;)32 b Fy(\026)-62 b Fw(m)2320 2926 y Fr(1)2359 2912 y Fy(\))21 b(+)f Fw(h)2558 2875 y Fo(1)2558 2936 y Ft(c)2589 2917 y Fl(0)2633 2912 y Fy(\()p Fw(m)2748 2926 y Fr(1)2787 2912 y Fw(;)15 b(x)2879 2875 y Fr(+)2939 2912 y Fy(\))25 b(:)-2568 b(\(3.2\))1011 3111 y(\()18 b(\026)-63 b Fw(m)1126 3125 y Fr(0)1166 3111 y Fw(;)32 b Fy(\026)-62 b Fw(m)1286 3125 y Fr(1)1325 3111 y Fw(;)15 b(T)e Fy(\))26 b Fv(2)f Fw(@)5 b Fy(\()p Fw(V)1740 3073 y Fo(\000)1719 3138 y Ft(i)1819 3111 y Fv(\002)20 b Fw(V)1983 3073 y Fr(+)1963 3138 y Ft(i)p Fr(+1)2102 3111 y Fv(\002)g Fy([)p Fw(T)2284 3073 y Fr(+)2271 3138 y Ft(i)2363 3111 y Fv(\000)g Fw(d)2501 3125 y Ft(i)2529 3111 y Fw(;)15 b(T)2635 3073 y Fr(+)2622 3138 y Ft(i)2715 3111 y Fy(+)20 b Fw(d)2853 3125 y Ft(i)2881 3111 y Fy(]\))2941 3010 y Fq(o)764 3310 y Fv(\025)25 b Fy(min)1011 3209 y Fq(n)1072 3310 y Fw(h)1124 3273 y Fo(1)1124 3333 y Ft(c)1199 3310 y Fy(\()p Fw(x)1286 3273 y Fo(\000)1345 3310 y Fw(;)15 b(m)1465 3324 y Fr(0)1505 3310 y Fy(\))20 b(+)g Fw(h)1703 3273 y Ft(T)1703 3333 y(\021)1738 3343 y Fm(i)1765 3333 y Ft(;\026)1827 3343 y Fm(i)1854 3333 y Ft(; )1920 3343 y Fm(i)1946 3333 y Ft(;e)1999 3341 y Fm(n)2045 3310 y Fy(\()d(\026)-62 b Fw(m)2160 3324 y Fr(0)2200 3310 y Fw(;)32 b Fy(\026)-62 b Fw(m)2320 3324 y Fr(1)2359 3310 y Fy(\))21 b(+)f Fw(h)2558 3273 y Fo(1)2558 3334 y Ft(c)2589 3315 y Fl(0)2633 3310 y Fy(\()p Fw(m)2748 3324 y Fr(1)2787 3310 y Fw(;)15 b(x)2879 3273 y Fr(+)2939 3310 y Fy(\))25 b(:)1011 3509 y(\()18 b(\026)-63 b Fw(m)1126 3523 y Fr(0)1166 3509 y Fw(;)32 b Fy(\026)-62 b Fw(m)1286 3523 y Fr(1)1325 3509 y Fw(;)15 b(T)e Fy(\))26 b Fv(2)f Fw(V)1651 3471 y Fo(\000)1631 3536 y Ft(i)1730 3509 y Fv(\002)20 b Fw(V)1895 3471 y Fr(+)1874 3536 y Ft(i)p Fr(+1)2013 3509 y Fv(\002)g Fy([)p Fw(T)2195 3471 y Fr(+)2182 3536 y Ft(i)2274 3509 y Fv(\000)g Fw(d)2412 3523 y Ft(i)2441 3509 y Fw(;)15 b(T)2547 3471 y Fr(+)2534 3536 y Ft(i)2626 3509 y Fy(+)20 b Fw(d)2764 3523 y Ft(i)2792 3509 y Fy(])2817 3408 y Fq(o)2898 3509 y Fy(+)g(5)p Fw(\017)3071 3472 y Fo(\003)3071 3532 y Ft(i)3111 3509 y Fw(;)456 3727 y Fy(where)30 b Fw(x)771 3694 y Fo(\000)857 3727 y Fv(2)c(M)p Fy(\()p Fw(c)1127 3741 y Ft(i)1156 3727 y Fy(\))21 b Fv(\\)f Fw(\031)1345 3741 y Ft(q)1383 3727 y Fw(\013)p Fy(\()p Fw(d\015)5 b Fy(\),)33 b Fw(x)1720 3694 y Fr(+)1806 3727 y Fv(2)26 b(M)p Fy(\()p Fw(c)2076 3741 y Ft(i)p Fr(+1)2195 3727 y Fy(\))21 b Fv(\\)g Fw(\031)2385 3741 y Ft(q)2422 3727 y Fw(!)s Fy(\()p Fw(d\015)5 b Fy(\).)44 b(The)31 b(disks)g Fw(V)3208 3688 y Fo(\000)3188 3754 y Ft(i)3298 3727 y Fy(and)456 3841 y Fw(V)529 3802 y Fr(+)509 3868 y Ft(i)p Fr(+1)652 3841 y Fy(are)26 b(c)m(hosen)f(so)h(that)j(\026)-49 b Fw(\015)31 b Fy(in)m(tersects)26 b(them)f(transv)m(ersally)-8 b(,)28 b(for)d(eac)m(h)h(\()18 b(\026)-63 b Fw(m)3080 3855 y Fr(0)3120 3841 y Fw(;)33 b Fy(\026)-63 b Fw(m)3240 3855 y Fr(1)3280 3841 y Fw(;)15 b(T)3386 3808 y Fo(0)3409 3841 y Fy(\))456 3961 y Fv(2)24 b Fw(V)615 3923 y Fo(\000)594 3988 y Ft(i)674 3961 y Fv(\002)p Fw(V)818 3923 y Fr(+)798 3988 y Ft(i)p Fr(+1)917 3961 y Fv(\002)p Fy([)p Fw(T)1079 3923 y Fr(+)1066 3988 y Ft(i)1139 3961 y Fv(\000)p Fw(d)1257 3975 y Ft(i)1285 3961 y Fw(;)15 b(T)1391 3923 y Fr(+)1378 3988 y Ft(i)1451 3961 y Fy(+)p Fw(d)1569 3975 y Ft(i)1597 3961 y Fy(],)23 b(the)e(minimizer)g(of)f Fw(h)2377 3928 y Ft(T)2377 3989 y(\021)2412 3999 y Fm(i)2439 3989 y Ft(;\026)2501 3999 y Fm(i)2528 3989 y Ft(; )2594 3999 y Fm(i)2620 3989 y Ft(;e)2673 3997 y Fm(n)2719 3961 y Fy(\()d(\026)-62 b Fw(m)2834 3975 y Fr(0)2873 3961 y Fw(;)33 b Fy(\026)-63 b Fw(m)2993 3975 y Fr(1)3033 3961 y Fy(\),)26 b(\026)-48 b Fw(\015)3163 3975 y Ft(i)3191 3961 y Fy(\()p Fw(t;)33 b Fy(\026)-63 b Fw(m)3379 3975 y Fr(0)3419 3961 y Fw(;)473 4078 y Fy(\026)h Fw(m)536 4092 y Fr(1)575 4078 y Fw(;)15 b(T)e Fy(\))31 b(has)f(the)g(prop)s(ert) m(y)456 4258 y(\(3.3\))550 b(\026)-48 b Fw(\015)1235 4272 y Ft(i)1289 4258 y Fv(2)24 b Fw(U)1436 4272 y Ft(i)1646 4258 y Fy(whenev)m(er)64 b(\026)-48 b Fw(\015)2123 4272 y Ft(i;)p Fr(1)2231 4258 y Fv(2)25 b Fy([)p Fv(\000)p Fy(\001)2489 4272 y Ft(i)2517 4258 y Fw(;)15 b Fy(\001)2633 4272 y Ft(i)2661 4258 y Fy(])p Fw(:)555 4521 y Fy(F)-8 b(or)27 b(eac)m(h)h(in)m(teger)g Fw(i)d Fv(2)1354 4453 y Fq(S)1430 4548 y Ft(i)1454 4558 y Fm(j)1486 4548 y Fo(2)p Fr(\003)1582 4556 y Fm(c)1619 4521 y Fv(f)p Fw(i)1695 4535 y Ft(j)1732 4521 y Fw(;)15 b(i)1803 4535 y Ft(j)1852 4521 y Fy(+)d(1)p Fw(;)j Fv(\001)g(\001)g(\001)33 b Fw(;)15 b(i)2229 4535 y Ft(j)t Fr(+1)2368 4521 y Fv(\000)d Fy(1)p Fv(g)p Fy(,)28 b(there)f(exist)g(an)g(em)m(b)s(ed-)456 4644 y(ded)h(\()p Fw(n)17 b Fv(\000)g Fy(1\)-torus)29 b(\006)1229 4658 y Ft(c)1260 4668 y Fm(i)1290 4644 y Fy(,)g(a)h(neigh)m(b)s(orho)s(o)s(d)d Fw(U)2050 4658 y Ft(i)2107 4644 y Fy(of)i Fv(N)13 b Fy(\()p Fw(c)2371 4658 y Ft(i)2400 4644 y Fy(\))18 b Fv(\\)e Fy(\006)2596 4658 y Ft(c)2627 4668 y Fm(i)2657 4644 y Fy(,)29 b(t)m(w)m(o)i(closed)e (1-forms)456 4752 y Fw(\021)501 4766 y Ft(i)555 4752 y Fy(and)j(\026)-52 b Fw(\026)782 4766 y Ft(i)836 4752 y Fy(on)26 b Fw(M)36 b Fy(with)26 b([)p Fw(\021)1355 4766 y Ft(i)1383 4752 y Fy(])g(=)f Fw(c)1569 4766 y Ft(i)1597 4752 y Fy(,)i([)7 b(\026)-52 b Fw(\026)1729 4766 y Ft(i)1757 4752 y Fy(])26 b(=)f Fw(c)1943 4766 y Ft(i)p Fr(+1)2073 4752 y Fv(\000)11 b Fw(c)2194 4766 y Ft(i)2249 4752 y Fy(and)25 b(supp)6 b(\026)-53 b Fw(\026)2663 4766 y Ft(i)2703 4752 y Fv(\\)11 b Fw(U)2837 4766 y Ft(i)2890 4752 y Fy(=)25 b Fu(?)p Fy(.)39 b(Suitable)456 4864 y(co)s(ordinate)34 b(systems)f Fw(x)d Fv(!)g Fw(q)j Fy(=)d Fw(G)1695 4826 y Fo(\000)p Fr(1)1695 4891 y Ft(i)1790 4864 y Fw(x)j Fy(on)g Fw(M)44 b Fy(is)33 b(c)m(hosen)h(so)f(that)h Fw(\031)2893 4831 y Fo(\000)p Fr(1)2988 4864 y Fy(\006)3054 4878 y Ft(c)3085 4888 y Fm(i)3145 4864 y Fv(\032)3279 4841 y Fy(\026)3245 4864 y Fw(M)41 b Fy(=)456 4988 y Fu(R)20 b Fv(\002)g Fu(T)694 4955 y Ft(n)p Fo(\000)p Fr(1)862 4988 y Fy(has)31 b(in\014nitely)g(man)m(y)h(compact)g(comp)s (onen)m(ts)f(\006)2587 4943 y Ft(j)2587 5000 y(c)2618 5010 y Fm(i)2648 4988 y Fy(.)2750 4965 y(\026)2716 4988 y Fw(M)42 b Fy(is)31 b(separated)g(b)m(y)456 5096 y(\006)522 5063 y Fr(0)522 5119 y Ft(c)553 5129 y Fm(i)614 5096 y Fy(in)m(to)h(upp)s(er)d(part)1292 5073 y(\026)1258 5096 y Fw(M)1356 5058 y Fr(+)1346 5123 y Ft(i)1447 5096 y Fy(and)i(lo)m(w)m(er)h(part)2095 5073 y(\026)2061 5096 y Fw(M)2159 5058 y Fo(\000)2149 5123 y Ft(i)2218 5096 y Fy(.)44 b(A)31 b(smo)s(oth)h(non-negativ)m(e)h(func-)456 5216 y(tion)27 b Fw(\045)686 5230 y Ft(i)740 5216 y Fy(of)g Fw(q)j Fy(is)c(constructed)h(suc)m(h)f(that)i Fw(\045)1927 5230 y Ft(i)1980 5216 y Fy(=)d(0)i(for)f Fw(q)i Fv(2)2472 5193 y Fy(\026)2438 5216 y Fw(M)2536 5177 y Fo(\000)2526 5243 y Ft(i)2595 5216 y Fv(n)p Fy(\(\006)2741 5183 y Fr(0)2741 5238 y Ft(c)2772 5248 y Fm(i)2816 5216 y Fy(+)13 b Fw(\016)2940 5230 y Ft(i)2968 5216 y Fy(\))27 b(and)f Fw(\045)3250 5230 y Ft(i)3303 5216 y Fy(=)f(1)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 998 251 a Fr(CONNECTING)31 b(ORBITS)g(F)n(OR)f (A)n(UTONOMOUS)g(SYSTEMS)472 b(19)456 450 y Fy(for)30 b Fw(q)e Fv(2)784 427 y Fy(\026)750 450 y Fw(M)848 412 y Fr(+)838 477 y Ft(i)907 450 y Fv(n)p Fy(\(\006)1053 417 y Fr(0)1053 473 y Ft(c)1084 483 y Fm(i)1135 450 y Fy(+)19 b Fw(\016)1265 464 y Ft(i)1294 450 y Fy(\).)41 b(Let)31 b Fw(\026)1613 464 y Ft(i)1666 450 y Fy(=)25 b Fw(\045)1809 464 y Ft(i)1844 450 y Fy(\026)-52 b Fw(\026)1892 464 y Ft(i)1951 450 y Fy(and)1548 609 y Fw(L)1610 623 y Ft(\021)1645 633 y Fm(i)1672 623 y Ft(;\026)1734 633 y Fm(i)1790 609 y Fy(=)25 b Fw(L)20 b Fv(\000)g Fw(\021)2104 623 y Ft(i)2152 609 y Fv(\000)g Fw(\026)2298 623 y Ft(i)2326 609 y Fw(:)456 770 y(\016)496 784 y Ft(i)549 770 y Fw(>)25 b Fy(0)k(is)g(so)g(small)g(suc)m(h)f(that)h(\(\006)1654 737 y Fr(0)1654 792 y Ft(c)1685 802 y Fm(i)1732 770 y Fy(+)17 b Fw(\016)1860 784 y Ft(i)1888 770 y Fy(\))g Fv(\\)f Fy(\()p Fv(N)2127 784 y Ft(\021)2162 794 y Fm(i)2189 784 y Ft(;\026)2251 794 y Fm(i)2298 770 y Fy(+)h Fw(\016)2426 784 y Ft(i)2454 770 y Fy(\))26 b Fv(\032)f Fw(U)2673 784 y Ft(i)2701 770 y Fy(,)k(see)g(form)m(ula)g(\(2.2\).)555 961 y(Since)44 b Fw(c)845 975 y Ft(i)917 961 y Fy(is)g(equiv)-5 b(alen)m(t)45 b(to)g Fw(c)1631 975 y Ft(i)p Fr(+1)1793 961 y Fy(and)e(they)h(are)g(closed)h(to)g(eac)m(h)g(other,)i(eac)m(h) 456 1069 y(curv)m(e)34 b(\026)-48 b Fw(\015)31 b Fv(2)26 b Fg(C)925 1083 y Ft(\021)960 1093 y Fm(i)986 1083 y Ft(;\026)1048 1093 y Fm(i)1110 1069 y Fy(determines)k(a)i(lo)s(cal)g (minimal)f(orbit)g Fw(d\015)36 b Fy(of)31 b Fw(\036)2727 1036 y Ft(t)2727 1096 y(L)2810 1069 y Fy(of)g Fw(c)p Fy(-t)m(yp)s(e,)i(whic)m(h)456 1188 y(connects)856 1165 y(~)825 1188 y Fv(A)o Fy(\()p Fw(c)971 1202 y Ft(i)1000 1188 y Fy(\))e(to)1208 1165 y(~)1177 1188 y Fv(A)p Fy(\()p Fw(c)1324 1202 y Ft(i)p Fr(+1)1443 1188 y Fy(\).)555 1379 y(Let)j(\026)-48 b Fw(\015)765 1393 y Ft(i)793 1379 y Fy(\()p Fw(t;)33 b Fy(\026)-63 b Fw(m)q(;)32 b Fy(\026)-62 b Fw(m)1102 1346 y Fo(0)1125 1379 y Fy(\))26 b(:)f([)p Fv(\000)p Fw(T)8 b(;)15 b(T)e Fy(])26 b Fv(!)1700 1356 y Fy(\026)1666 1379 y Fw(M)40 b Fy(b)s(e)30 b(the)h(minimizer)f(of)1287 1549 y Fw(h)1339 1512 y Ft(T)1339 1572 y(\021)1374 1582 y Fm(i)1401 1572 y Ft(;\026)1463 1582 y Fm(i)1493 1549 y Fy(\()18 b(\026)-63 b Fw(m)q(;)32 b Fy(\026)-62 b Fw(m)1729 1512 y Fo(0)1752 1549 y Fy(\))26 b(=)51 b(inf)1909 1611 y Ft(T)1960 1592 y Fl(0)1982 1611 y Ft(>)p Fr(0)2087 1549 y Fw(h)2139 1512 y Ft(T)2190 1488 y Fl(0)2139 1572 y Ft(\021)2174 1582 y Fm(i)2201 1572 y Ft(;\026)2263 1582 y Fm(i)2294 1549 y Fy(\()p Fw(m;)15 b(m)2529 1512 y Fo(0)2553 1549 y Fy(\))p Fw(:)456 1761 y Fy(According)38 b(to)h(the)g(lemma)g(2.1)g(and)e(the)i(corollary)g(2.1)h Fv(9)d Fw(K)2687 1775 y Ft(i)2716 1761 y Fw(;)15 b(T)2822 1723 y Fr(+)2809 1789 y Ft(i)2919 1761 y Fw(>)38 b Fy(0,)j(if)d Fw(K)3307 1775 y Ft(i)3374 1761 y Fv(\024)456 1875 y(\000)16 b Fy(\026)-61 b Fw(m)607 1889 y Fr(1)646 1875 y Fw(;)32 b Fy(\026)-62 b Fw(m)766 1842 y Fo(0)766 1900 y Fr(1)842 1875 y Fv(\024)36 b Fw(K)1026 1889 y Ft(i)1079 1875 y Fy(+)24 b(2)p Fw(\031)s Fy(,)40 b(then)d(there)g(exists)h Fw(T)49 b(<)36 b(T)2325 1837 y Fr(+)2312 1902 y Ft(i)2421 1875 y Fy(suc)m(h)h(that)g Fw(h)2888 1842 y Ft(T)2888 1898 y(\021)2923 1908 y Fm(i)2951 1898 y Ft(;\026)3013 1908 y Fm(i)3043 1875 y Fy(\()18 b(\026)-63 b Fw(m;)33 b Fy(\026)-63 b Fw(m)3278 1842 y Fo(0)3302 1875 y Fy(\))37 b(=)456 2003 y(inf)566 2018 y Ft(T)617 1999 y Fl(0)639 2018 y Ft(>)p Fr(0)749 2003 y Fw(h)801 1970 y Ft(T)852 1947 y Fl(0)801 2025 y Ft(\021)836 2035 y Fm(i)863 2025 y Ft(;\026)925 2035 y Fm(i)956 2003 y Fy(\()17 b(\026)-62 b Fw(m;)32 b Fy(\026)-62 b Fw(m)1191 1970 y Fo(0)1214 2003 y Fy(\))31 b(and)f(its)h(minimizer)i(\026)-48 b Fw(\015)2055 2017 y Ft(i)2083 2003 y Fy(\()p Fw(t;)15 b(m;)g(m)2391 1970 y Fo(0)2415 2003 y Fy(\))26 b(:)f([)p Fv(\000)p Fw(T)8 b(;)15 b(T)e Fy(])26 b Fv(!)2990 1980 y Fy(\026)2956 2003 y Fw(M)40 b Fy(satis\014es)456 2178 y(\(3.4\))222 b(\026)-48 b Fw(\015)907 2192 y Ft(i)935 2178 y Fy(\()p Fw(t;)33 b Fy(\026)-63 b Fw(m;)33 b Fy(\026)-63 b Fw(m)1243 2140 y Fo(0)1267 2178 y Fy(\))25 b Fv(2)g Fw(U)1475 2192 y Ft(i)1503 2178 y Fw(;)198 b Fy(whenev)m(er)64 b(\026)-49 b Fw(\015)2202 2192 y Ft(i)2230 2178 y Fy(\()p Fw(t;)33 b Fy(\026)-63 b Fw(m)q(;)32 b Fy(\026)-62 b Fw(m)2539 2140 y Fo(0)2562 2178 y Fy(\))26 b Fv(2)f Fy(\006)2775 2140 y Fr(0)2775 2200 y Ft(c)2806 2210 y Fm(i)2856 2178 y Fy(+)20 b Fw(\016)2987 2192 y Ft(i)3015 2178 y Fw(:)555 2431 y Fy(T)-8 b(o)32 b(mak)m(e)h(notation)g(simpler,)f(w)m(e)h(let)f Fw( )1967 2445 y Ft(i)2023 2431 y Fy(=)c(0,)k Fw(V)2298 2392 y Fo(\000)2277 2458 y Ft(i)2384 2431 y Fy(=)c Fv(f)p Fw(q)2569 2445 y Fr(1)2636 2431 y Fy(=)f Fv(\000)p Fw(K)2882 2445 y Ft(i)2910 2431 y Fv(g)22 b(\002)e Fu(T)3129 2398 y Ft(n)p Fo(\000)p Fr(1)3298 2431 y Fy(and)456 2544 y Fw(V)529 2506 y Fr(+)509 2572 y Ft(i)p Fr(+1)652 2544 y Fy(=)25 b Fv(f)p Fw(q)834 2558 y Fr(1)899 2544 y Fy(=)g Fw(K)1072 2558 y Ft(i)1100 2544 y Fv(g)11 b(\002)g Fu(T)1299 2511 y Ft(n)p Fo(\000)p Fr(1)1462 2544 y Fy(for)26 b(eac)m(h)h(in)m (teger)g Fw(i)f 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3089 y Fo(\000)p Fr(1)2189 3127 y Fw(V)2263 3088 y Fo(\000)2242 3154 y Ft(i)2347 3127 y Fy(=)25 b Fu(T)2504 3089 y Ft(n)p Fo(\000)p Fr(1)456 3295 y Fy(where)34 b(\026)-50 b Fw(\031)s Fy(:)40 b Fu(R)904 3262 y Ft(n)976 3295 y Fv(!)1126 3272 y Fy(\026)1092 3295 y Fw(M)c Fy(=)25 b Fu(R)19 b Fv(\002)f Fu(T)1547 3262 y Ft(n)p Fo(\000)p Fr(1)1714 3295 y Fy(and)29 b Fw(\031)1942 3309 y Fr(1)1981 3295 y Fy(:)41 b Fu(R)18 b Fv(\002)h Fu(T)2282 3262 y Ft(n)p Fo(\000)p Fr(1)2444 3295 y Fv(!)25 b Fu(T)2621 3262 y Ft(n)p Fo(\000)p Fr(1)2788 3295 y Fy(are)30 b(the)g(standard)456 3403 y(pro)5 b(jections.)41 b(W)-8 b(e)31 b(de\014ne)456 3562 y(\(3.5\))411 b Fw(V)1125 3524 y Fr(+)1105 3589 y Ft(i;)p Fo(\000)1233 3562 y Fy(=)30 b(\026)-50 b Fw(\031)s(G)1455 3524 y Fo(\000)p Fr(1)1455 3589 y Ft(i)p Fo(\000)p Fr(1)1574 3562 y Fw(G)1645 3576 y Ft(i)1687 3539 y Fy(\026)1674 3562 y Fw(V)1747 3524 y Fo(\000)1727 3589 y Ft(i)1806 3562 y Fw(;)197 b(V)2101 3524 y Fo(\000)2081 3589 y Ft(i;)p Fr(+)2209 3562 y Fy(=)30 b(\026)-50 b Fw(\031)s(G)2431 3524 y Fo(\000)p Fr(1)2431 3589 y Ft(i)2526 3562 y Fw(G)2597 3576 y Ft(i)p Fo(\000)p Fr(1)2730 3539 y Fy(\026)2716 3562 y Fw(V)2789 3524 y Fr(+)2769 3589 y Ft(i)456 3739 y Fy(in)33 b(whic)m(h)841 3716 y(\026)828 3739 y Fw(V)901 3701 y Fo(\000)881 3766 y Ft(i)990 3739 y Fy(=)d Fv(f)p Fw(q)j Fv(2)d Fu(R)1367 3706 y Ft(n)1444 3739 y Fy(:)g Fw(q)1540 3753 y Fr(1)1609 3739 y Fy(=)g Fv(\000)p Fw(K)1858 3753 y Ft(i)1886 3739 y Fw(;)49 b(q)2001 3753 y Ft(i)2059 3739 y Fv(2)29 b Fy([0)p Fw(;)15 b Fy(2)p Fw(\031)s Fy(])36 b Fv(8)c Fy(2)f Fv(\024)f Fw(i)g Fv(\024)g Fw(n)p Fv(g)p Fy(,)3015 3716 y(\026)3001 3739 y Fw(V)3074 3701 y Fr(+)3054 3766 y Ft(i)3164 3739 y Fy(=)f Fv(f)p Fw(q)34 b Fv(2)456 3847 y Fu(R)522 3814 y Ft(n)604 3847 y Fy(:)i Fw(q)706 3861 y Fr(1)781 3847 y Fy(=)g Fw(K)965 3861 y Ft(i)p Fo(\000)p Fr(1)1084 3847 y Fw(;)52 b(q)1202 3861 y Ft(i)1265 3847 y Fv(2)36 b Fy([0)p Fw(;)15 b Fy(2)p Fw(\031)s Fy(])39 b Fv(8)d Fy(2)g Fv(\024)g Fw(i)g 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Fo(0)2087 4481 y Fy(.)54 b(W)-8 b(e)37 b(divide)e(the)g(in)m(terv)-5 b(al)36 b([)p Fv(\000)p Fw(K)q(;)15 b(K)3395 4448 y Fo(0)3419 4481 y Fy(])456 4589 y(in)m(to)31 b(2)p Fw(i)716 4603 y Ft(m)803 4589 y Fy(+)20 b(1)31 b(subin)m(terv)-5 b(als)1059 4747 y([)p Fv(\000)p Fw(K)q(;)15 b(K)1357 4710 y Fo(0)1381 4747 y Fy(])25 b(=)g([)p Fw(t)1585 4709 y Fr(+)1585 4773 y(0)1644 4747 y Fw(;)15 b(t)1717 4709 y Fo(\000)1717 4773 y Fr(0)1777 4747 y Fy(])20 b Fv([)g Fy([)p Fw(t)1961 4709 y Fo(\000)1961 4773 y Fr(0)2020 4747 y Fw(;)15 b(t)2093 4709 y Fr(+)2093 4773 y(1)2152 4747 y Fy(])21 b Fv([)f(\001)15 b(\001)g(\001)21 b([)f Fy([)p Fw(t)2544 4709 y Fr(+)2544 4774 y Ft(i)2568 4782 y Fm(m)2631 4747 y Fw(;)15 b(t)2704 4709 y Fo(\000)2704 4774 y Ft(i)2728 4782 y Fm(m)2791 4747 y Fy(])p Fw(;)456 4920 y Fy(where)43 b Fw(t)765 4882 y Fr(+)765 4946 y(0)872 4920 y Fy(=)k Fv(\000)p Fw(K)7 b Fy(,)47 b Fw(t)1250 4882 y Fo(\000)1250 4947 y Ft(i)1274 4955 y Fm(m)1384 4920 y Fy(=)h Fw(K)1587 4887 y Fo(0)1610 4920 y Fy(.)81 b(Corresp)s(ondingly)-8 b(,)46 b(w)m(e)e(divide)g(the)g(curv)m(e)g(in)m(to)456 5042 y(2)p Fw(i)532 5056 y Ft(m)622 5042 y Fy(+)23 b(1)34 b(segmen)m(ts)i Fw(\015)1239 5004 y Fo(\000)1234 5069 y Ft(i)1330 5042 y Fy(=)c Fw(\015)5 b Fv(j)1510 5072 y Fr([)p Ft(t)1555 5043 y Fk(+)1555 5095 y Fm(i)1606 5072 y Ft(;t)1651 5043 y Fl(\000)1651 5095 y Fm(i)1703 5072 y Fr(])1727 5042 y Fy(,)36 b Fw(\015)1840 5004 y Fr(+)1835 5069 y Ft(i)1931 5042 y Fy(=)c Fw(\015)5 b Fv(j)2111 5072 y Fr([)p Ft(t)2156 5043 y Fl(\000)2156 5095 y Fm(i)2209 5072 y Ft(;t)2254 5043 y Fk(+)2254 5095 y Fm(i)p Fk(+1)2357 5072 y Fr(])2415 5042 y Fy(for)35 b Fw(i)d Fy(=)g(0)p Fw(;)15 b Fy(1)p Fw(;)g Fy(2)p Fw(;)g Fv(\001)g(\001)g(\001)34 b Fw(;)15 b(i)3190 5056 y Ft(m)3280 5042 y Fv(\000)23 b Fy(1,)456 5188 y(and)29 b Fw(\015)684 5150 y Fo(\000)679 5215 y Ft(i)703 5223 y Fm(m)792 5188 y Fy(=)24 b Fw(\015)5 b Fv(j)964 5218 y Fr([)p Ft(t)1009 5189 y Fk(+)1009 5240 y Fm(i)1031 5248 y(m)1095 5218 y Ft(;t)1140 5189 y Fl(\000)1140 5240 y Fm(i)1162 5248 y(m)1225 5218 y Fr(])1249 5188 y Fy(.)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 456 251 a Fr(20)1012 b(X.)23 b(LI)h(AND)f(C.-Q.)f(CHENG)555 450 y Fy(W)-8 b(e)25 b(tak)m(e)f(t)m(w)m (o)h(co)m(v)m(erings)g(of)e Fu(T)1601 417 y Ft(n)1648 450 y Fy(,)i(the)e(univ)m(ersal)h(co)m(v)m(ering)h Fu(R)2640 417 y Ft(n)2710 450 y Fy(and)2913 427 y(\026)2879 450 y Fw(M)36 b Fy(=)25 b Fu(R)6 b Fv(\002)g Fu(T)3309 417 y Ft(n)p Fo(\000)p Fr(1)1504 622 y Fu(R)1570 585 y Ft(n)1668 571 y Fr(\026)-39 b Ft(\031)1642 622 y Fv(\000)-64 b(!)25 b Fu(R)20 b Fv(\002)g Fu(T)2003 585 y Ft(n)p Fo(\000)p Fr(1)2187 571 y Ft(\031)2165 622 y Fv(\000)-64 b(!)25 b Fu(T)2349 585 y Ft(n)456 777 y Fy(and)33 b(\014x)h(a)g(lift)k(~)-48 b Fw(\015)39 b Fy(of)34 b Fw(\015)40 b Fy(to)34 b Fu(R)1456 744 y Ft(n)1537 777 y Fy(b)m(y)g(c)m(ho)s(osing)40 b(\026)-50 b Fw(\031)s(G)2166 739 y Fo(\000)p Fr(1)2166 803 y(0)2264 777 y Fy(~)i Fw(\015)5 b Fy(\()p Fw(t)2381 739 y Fo(\000)2381 803 y Fr(0)2440 777 y Fy(\))32 b Fv(2)f Fw(V)2673 739 y Fo(\000)2652 803 y Fr(0)2766 777 y Fy(\()p Fw(G)2872 791 y Ft(i)2935 777 y Fy(determines)j(a)456 890 y(co)s(ordinate)c (transformation)g(on)f Fu(R)1712 857 y Ft(n)1759 890 y Fy(\).)40 b(Corresp)s(ondingly)-8 b(,)29 b(eac)m(h)i Fw(\015)2815 852 y Fo(\006)2810 917 y Ft(i)2903 890 y Fy(has)e(its)h(lift)k(~)-49 b Fw(\015)3385 852 y Fo(\006)3380 917 y Ft(i)456 998 y Fy(to)31 b Fu(R)633 965 y Ft(n)679 998 y Fy(.)555 1189 y(The)f(curv)m(e)h Fw(\015)k Fy(is)c(required)e(to) i(satisfy)g(the)g(conditions:)555 1380 y(1,)g(for)f(eac)m(h)i Fw(i)25 b Fy(=)g(0)p Fw(;)15 b Fy(1)p Fw(;)g Fy(2)p Fw(;)g Fv(\001)g(\001)g(\001)k Fw(i)1562 1394 y Ft(m)1649 1380 y Fv(\000)h Fy(1)31 b(there)g(is)f(some)h Fw(k)2415 1394 y Ft(i)2469 1380 y Fv(2)24 b Fu(Z)30 b Fy(suc)m(h)g(that)1171 1535 y(\026)-50 b Fw(\031)s(G)1292 1497 y Fo(\000)p Fr(1)1292 1562 y Ft(i)1390 1535 y Fy(~)i Fw(\015)1439 1497 y Fr(+)1434 1562 y Ft(i)1498 1535 y Fy(\()p Fw(t)1566 1497 y Fo(\000)1566 1562 y Ft(i)1625 1535 y Fy(\))21 b Fv(\000)f Fy(\(2)p Fw(k)1899 1549 y Ft(i)1928 1535 y Fw(\031)s(;)15 b Fy(0)p Fw(;)g Fv(\001)g(\001)g(\001)33 b Fw(;)15 b Fy(0\))26 b Fv(2)f Fw(V)2551 1497 y Fo(\000)2531 1562 y Ft(i)2610 1535 y Fw(;)1171 1688 y Fy(\026)-50 b Fw(\031)s(G)1292 1650 y Fo(\000)p Fr(1)1292 1715 y Ft(i)1390 1688 y Fy(~)i Fw(\015)1439 1650 y Fr(+)1434 1715 y Ft(i)1498 1688 y Fy(\()p Fw(t)1566 1650 y Fr(+)1566 1715 y Ft(i)p Fr(+1)1685 1688 y Fy(\))20 b Fv(\000)g Fy(\(2)p Fw(k)1958 1702 y Ft(i)1987 1688 y Fw(\031)s(;)15 b Fy(0)p Fw(;)g Fv(\001)g(\001)g(\001) 33 b Fw(;)15 b Fy(0\))27 b Fv(2)d Fw(V)2611 1650 y Fr(+)2590 1715 y Ft(i)p Fr(+1)2709 1688 y Fy(;)-2278 b(\(3.7\))555 1931 y(2,)31 b(for)f Fw(i)c Fv(2)938 1863 y Fq(S)1014 1958 y Ft(i)1038 1968 y Fm(j)1070 1958 y Fo(2)p Fr(\003)1166 1970 y Fm(h)1211 1931 y Fv(f)p Fw(i)1287 1945 y Ft(j)1324 1931 y Fw(;)15 b(i)1395 1945 y Ft(j)1452 1931 y Fy(+)20 b(1)p Fw(;)15 b Fv(\001)g(\001)g(\001)32 b Fw(;)15 b(i)1836 1945 y Ft(j)t Fr(+1)1984 1931 y Fv(\000)20 b Fy(1)p Fv(g)456 2114 y Fy(\(3.8\))745 b Fw(H)1462 2128 y Fr(1)1501 2114 y Fy(\()p Fw(M)5 b(;)15 b(N)5 b(;)15 b Fu(Z)p Fy(\))26 b Fv(3)f Fy([)p Fw(G)2091 2076 y Fo(\000)p Fr(1)2091 2141 y Ft(i)2186 2114 y Fw(\015)2238 2076 y Fr(+)2233 2141 y Ft(i)2297 2114 y Fy(])h Fv(6)p Fy(=)f(0;)555 2352 y(3,)31 b(for)f Fw(i)c Fv(2)938 2284 y Fq(S)1014 2379 y Ft(i)1038 2389 y Fm(j)1070 2379 y Fo(2)p Fr(\003)1166 2387 y Fm(c)1202 2352 y Fv(f)p Fw(i)1278 2366 y Ft(j)1336 2352 y Fy(+)20 b(1)p Fw(;)15 b Fv(\001)g(\001)g(\001)32 b Fw(;)15 b(i)1720 2366 y Ft(j)t Fr(+1)1867 2352 y Fv(\000)20 b Fy(1)p Fv(g)p Fy(,)32 b(b)s(ecause)e(of)h(\(3.1\),)h(w)m(e)f(can)g (require)1105 2536 y(2)p Fw(\031)e Fv(\024)c(h)p Fw(g)1408 2497 y Fo(\000)p Fr(1)1405 2563 y Ft(i;)p Fr(1)1503 2536 y Fw(;)19 b Fy(~)-49 b Fw(\015)1595 2497 y Fo(\000)1590 2563 y Ft(i)1655 2536 y Fy(\()p Fw(t)1723 2497 y Fo(\000)1723 2563 y Ft(i)1782 2536 y Fy(\))20 b Fv(\000)k Fy(~)-49 b Fw(\015)1980 2497 y Fo(\000)1975 2563 y Ft(i)2040 2536 y Fy(\()p Fw(t)2108 2497 y Fr(+)2108 2563 y Ft(i)2167 2536 y Fy(\))p Fv(i)26 b(\024)f Fy(2)p Fw(\031)e Fy(+)d Fw(q)2614 2498 y Fo(\003)2611 2558 y Ft(i;)p Fr(1)1105 2692 y Fy(2)p Fw(\031)29 b Fv(\024)c(h)p Fw(g)1408 2654 y Fo(\000)p Fr(1)1405 2719 y Ft(i)p Fo(\000)p Fr(1)p Ft(;)p Fr(1)1579 2692 y Fw(;)19 b Fy(~)-49 b Fw(\015)1671 2654 y Fo(\000)1666 2719 y Ft(i)1730 2692 y Fy(\()p Fw(t)1798 2654 y Fo(\000)1798 2719 y Ft(i)1858 2692 y Fy(\))20 b Fv(\000)j Fy(~)-48 b Fw(\015)2056 2654 y Fo(\000)2051 2719 y Ft(i)2115 2692 y Fy(\()p Fw(t)2183 2654 y Fr(+)2183 2719 y Ft(i)2242 2692 y Fy(\))p Fv(i)26 b(\024)f Fy(2)p Fw(\031)f Fy(+)c Fw(q)2690 2655 y Fo(\003)2687 2715 y Ft(i;)p Fr(1)2770 2692 y Fw(;)-2339 b Fy(\(3.9\))456 2858 y(where)41 b Fw(q)774 2825 y Fo(\003)771 2883 y Ft(i;)p Fr(1)897 2858 y Fy(=)j(max)p Fv(f)p Fy(2)p Fw(q)1312 2872 y Ft(i;)p Fr(1)1396 2858 y Fw(;)15 b(q)1480 2825 y Fo(0)1477 2883 y Ft(i;)p Fr(1)1560 2858 y Fv(g)p Fy(,)45 b(\(refer)d(\(3.6\))h(for)f(the)g(de\014nition)f(of)h Fw(q)3053 2825 y Fo(0)3050 2883 y Ft(i;)p Fr(1)3132 2858 y Fy(\).)75 b(This)456 2983 y(condition)23 b(implies)f(that)i Fw(\015)5 b Fy(\()p Fw(t)p Fy(\))23 b(passes)f(through)g Fw(G)2188 2997 y Ft(i)p Fo(\000)p Fr(1)2307 2983 y Fw(\031)s(V)2435 2944 y Fr(+)2415 3010 y Ft(i)2516 2983 y Fy(b)s(efore)h(it)g(passes)f (through)456 3096 y Fw(G)527 3110 y Ft(i)555 3096 y Fw(\031)s(V)684 3058 y Fo(\000)663 3124 y Ft(i)773 3096 y Fy(as)31 b Fw(t)f Fy(increases.)41 b(Let)31 b(us)f(consider)g(the)g(quan)m(tit)m (y)880 3322 y Fw(h)932 3284 y Fr(\001)p Ft(t)932 3344 y(c)963 3354 y Fm(i)1020 3322 y Fy(\()p Fw(z)1101 3284 y Fr(+)1097 3349 y Ft(i)1161 3322 y Fw(;)15 b(z)1247 3284 y Fo(\000)1243 3349 y Ft(i)1307 3322 y Fy(\))25 b(=)300 b(inf)1463 3401 y Fm(\030)q Fk(\()p Fl(\000)p Fk(\001)p Fm(t)p Fk(\)=)p Fm(z)1741 3377 y Fk(+)1739 3427 y Fm(i)1792 3401 y Fl(2)p Fm(G)1881 3417 y(i)p Fl(\000)p Fk(1)1985 3401 y Fm(\031)r(V)2073 3377 y Fk(+)2059 3427 y Fm(i)1525 3490 y(\030)q Fk(\(\001)p Fm(t)p Fk(\)=)p Fm(z)1755 3466 y Fl(\000)1753 3516 y Fm(i)1807 3490 y Fl(2)p Fm(G)1896 3506 y(i)1922 3490 y(\031)r(V)2010 3466 y Fl(\000)1996 3516 y Fm(i)2139 3198 y Fq(Z)2230 3224 y Fr(\001)p Ft(t)2189 3404 y Fo(\000)p Fr(\001)p Ft(t)2332 3322 y Fy(\()p Fw(L)21 b Fv(\000)f Fw(\021)2586 3336 y Ft(i)2614 3322 y Fy(\)\()p Fw(d\030)t Fy(\()p Fw(t)p Fy(\)\))p Fw(dt:)456 3643 y Fy(where)36 b(the)h(in\014m)m(um)f(is)h (tak)m(en)i(for)d(those)i Fw(\030)j Fy(whose)c(lift)h(satisfying)f(the) h(condition)456 3751 y(\(3.9\).)h(In)20 b(view)h(of)g(\(3.1\))i(and)d (the)h(pro)s(of)f(of)h(the)g(lemma)g(2.1,)j(w)m(e)e(ha)m(v)m(e)g Fw(h)2918 3718 y Fr(\001)p Ft(t)2918 3774 y(c)2949 3784 y Fm(i)3006 3751 y Fy(\()p Fw(z)3087 3713 y Fr(+)3083 3778 y Ft(i)3147 3751 y Fw(;)15 b(z)3233 3713 y Fo(\000)3229 3778 y Ft(i)3293 3751 y Fy(\))25 b Fv(!)456 3871 y(1)32 b Fy(as)h(\001)p Fw(t)c Fv(!)g Fy(0)k(or)f Fv(!)d(1)p Fy(.)48 b(Th)m(us,)32 b(if)h(w)m(e)g(tak)m(e)h Fw(T)2171 3832 y Fo(\000)2158 3898 y Ft(i)2259 3871 y Fy(=)29 b Fw(T)2425 3832 y Fo(\000)2412 3898 y Ft(i)2484 3871 y Fy(\()p Fw(z)2565 3832 y Fr(+)2561 3898 y Ft(i)2625 3871 y Fw(;)15 b(z)2711 3832 y Fo(\000)2707 3898 y Ft(i)2770 3871 y Fy(\))33 b(as)g(the)g(quan)m(tit)m(y)456 4021 y(suc)m(h)24 b(that)i Fw(h)899 3968 y Ft(T)950 3938 y Fl(\000)940 3990 y Fm(i)899 4033 y Ft(c)930 4043 y Fm(i)1006 4021 y Fy(\()p Fw(z)1087 3983 y Fr(+)1083 4048 y Ft(i)1147 4021 y Fw(;)15 b(z)1233 3983 y Fo(\000)1229 4048 y Ft(i)1292 4021 y Fy(\))26 b(=)f(min)1601 4035 y Fr(\001)p Ft(t)1704 4021 y Fw(h)1756 3988 y Fr(\001)p Ft(t)1756 4043 y(c)1787 4053 y Fm(i)1845 4021 y Fy(\()p Fw(z)1926 3983 y Fr(+)1922 4048 y Ft(i)1986 4021 y Fw(;)15 b(z)2072 3983 y Fo(\000)2068 4048 y Ft(i)2131 4021 y Fy(\),)27 b(then)d(w)m(e)i(\014nd)d(0)j Fw(<)f(T)2957 3983 y Fo(\000)2944 4048 y Ft(i)3015 4021 y Fy(\()p Fw(x)3102 3983 y Fr(+)3102 4048 y Ft(i)3162 4021 y Fw(;)15 b(x)3254 3983 y Fo(\000)3254 4048 y Ft(i)3313 4021 y Fy(\))26 b Fw(<)456 4148 y Fv(1)p Fy(.)44 b(Since)32 b(b)s(oth)f Fw(\031)s(V)1199 4110 y Fr(+)1179 4175 y Ft(i)1290 4148 y Fy(and)g Fw(\031)s(V)1596 4110 y Fo(\000)1576 4175 y Ft(i)1687 4148 y Fy(are)h(compact,)i(there)e(exist)g(0)c Fw(<)2870 4125 y Fy(^)2852 4148 y Fw(T)2918 4110 y Fo(\000)2905 4175 y Ft(i)3005 4148 y Fw(<)3121 4125 y Fy(\025)3103 4148 y Fw(T)3169 4110 y Fo(\000)3156 4175 y Ft(i)3255 4148 y Fw(<)f Fv(1)456 4270 y Fy(suc)m(h)j(that)875 4247 y(^)858 4270 y Fw(T)924 4231 y Fo(\000)911 4297 y Ft(i)1008 4270 y Fw(<)25 b(T)1170 4231 y Fo(\000)1157 4297 y Ft(i)1228 4270 y Fy(\()p Fw(z)1309 4231 y Fr(+)1305 4297 y Ft(i)1369 4270 y Fw(;)15 b(z)1455 4231 y Fo(\000)1451 4297 y Ft(i)1515 4270 y Fy(\))25 b Fw(<)1689 4247 y Fy(\025)1671 4270 y Fw(T)1737 4231 y Fo(\000)1724 4297 y Ft(i)1826 4270 y Fy(for)31 b(eac)m(h)g Fw(z)2216 4231 y Fr(+)2212 4297 y Ft(i)2301 4270 y Fv(2)25 b Fw(\031)s(V)2515 4231 y Fr(+)2495 4297 y Ft(i)2604 4270 y Fy(and)30 b Fw(z)2827 4231 y Fo(\000)2823 4297 y Ft(i)2912 4270 y Fv(2)24 b Fw(\031)s(V)3126 4231 y Fo(\000)3105 4297 y Ft(i)3185 4270 y Fy(.)40 b(Let)456 4438 y(\(3.10\))920 b(\001)p Fw(T)1748 4400 y Fo(\000)1735 4465 y Ft(i)1832 4438 y Fy(=)25 b([)1971 4415 y(^)1953 4438 y Fw(T)2019 4400 y Fo(\000)2006 4465 y Ft(i)2078 4438 y Fw(;)2136 4415 y Fy(\025)2118 4438 y Fw(T)2184 4400 y Fo(\000)2171 4465 y Ft(i)2243 4438 y Fy(])p Fw(:)555 4680 y Fy(Recall)47 b Fw(\026)899 4694 y Ft(i)972 4680 y Fy(and)e Fw( )1223 4694 y Ft(i)1297 4680 y Fy(are)g(de\014ned)f(on)i Fu(R)29 b Fv(\002)h Fu(T)2190 4647 y Ft(n)p Fo(\000)p Fr(1)2327 4680 y Fy(,)49 b Fw(G)2472 4647 y Fo(\003)2472 4705 y Ft(i)2512 4680 y Fy(\()p Fw(\026)2602 4694 y Ft(i)2661 4680 y Fy(+)29 b Fw( )2820 4694 y Ft(i)2849 4680 y Fy(\)\()p Fw(d)s Fy(~)-48 b Fw(\015)6 b Fy(\))50 b(=)g(\()p Fw(\026)3315 4694 y Ft(i)3374 4680 y Fy(+)456 4795 y Fw( )515 4809 y Ft(i)543 4795 y Fy(\)\()5 b(\026)-50 b Fw(\031)t(G)740 4756 y Fo(\000)p Fr(1)740 4822 y Ft(i)835 4795 y Fw(d)s Fy(~)i Fw(\015)5 b Fy(\))31 b(is)f(w)m(ell)h(de\014ned.)40 b(W)-8 b(e)31 b(in)m(tro)s(duce)g(a)g(translation)1228 4950 y Fw(L)1290 4965 y Ft(\021)1325 4975 y Fm(i)1351 4965 y Ft(;\026)1413 4975 y Fm(i)1440 4965 y Ft(; )1506 4975 y Fm(i)1561 4950 y Fv(!)25 b Fw(L)20 b Fv(\000)g Fw(\021)1895 4964 y Ft(i)1944 4950 y Fv(\000)g Fy(\()p Fw(k)2117 4964 y Ft(i)2146 4950 y Fw(G)2217 4964 y Ft(i)2245 4950 y Fy(\))2280 4912 y Fo(\003)2320 4950 y Fy(\()p Fw(\026)2410 4964 y Ft(i)2459 4950 y Fy(+)g Fw( )2609 4964 y Ft(i)2637 4950 y Fy(\))456 5108 y(where)30 b Fw(k)769 5075 y Fo(\003)766 5133 y Ft(i)840 5108 y Fy(is)h(a)h(translation)g(of) f Fw(q)1615 5122 y Fr(1)1654 5108 y Fy(:)42 b(\()p Fw(k)1803 5122 y Ft(i)1832 5108 y Fy(\))1867 5075 y Fo(\003)1907 5108 y Fw(\036)p Fy(\()p Fw(q)s(;)32 b Fy(_)-42 b Fw(q)s Fy(\))27 b(=)f Fw(\036)p Fy(\()p Fw(q)2413 5122 y Fr(1)2474 5108 y Fv(\000)20 b Fy(2)p Fw(\031)s(k)2712 5122 y Ft(i)2741 5108 y Fw(;)i Fy(^)-52 b Fw(q)s(;)33 b Fy(_)-43 b Fw(q)t Fy(\))31 b(on)g Fw(T)3203 5085 y Fy(\026)3169 5108 y Fw(M)41 b Fy(and)456 5216 y(the)30 b(in)m(teger)i Fw(k)960 5230 y Ft(i)1019 5216 y Fy(is)e(c)m(hosen)h(so)g(that)g(\(3.7\))h (holds.)p eop end %%Page: 21 21 TeXDict begin 21 20 bop 998 251 a Fr(CONNECTING)31 b(ORBITS)g(F)n(OR)f (A)n(UTONOMOUS)g(SYSTEMS)472 b(21)555 450 y Fy(Let)710 433 y Fw(~)716 450 y(t)25 b Fy(=)g(\()p Fw(t)938 412 y Fo(\000)938 476 y Fr(0)997 450 y Fw(;)15 b(t)1070 412 y Fo(\006)1070 476 y Fr(1)1130 450 y Fw(;)g Fv(\001)g(\001)g(\001)31 b Fw(;)15 b(t)1379 412 y Fo(\006)1379 477 y Ft(i)1403 485 y Fm(m)1462 477 y Fo(\000)p Fr(1)1557 450 y Fw(;)g(t)1630 412 y Fr(+)1630 477 y Ft(i)1654 485 y Fm(m)1717 450 y Fy(\),)27 b Fw(~)-43 b(z)29 b Fy(=)c(\()p Fw(z)2054 412 y Fo(\000)2050 476 y Fr(0)2114 450 y Fw(;)15 b(z)2200 412 y Fo(\006)2196 476 y Fr(1)2260 450 y Fw(;)g Fv(\001)g(\001)g(\001) 31 b Fw(;)15 b(z)2522 412 y Fo(\006)2518 477 y Ft(i)2542 485 y Fm(m)2602 477 y Fo(\000)p Fr(1)2696 450 y Fw(;)g(z)2782 412 y Fr(+)2778 477 y Ft(i)2802 485 y Fm(m)2866 450 y Fy(\).)40 b(W)-8 b(e)29 b(consider)456 558 y(the)h(minimal)h(action)611 784 y Fw(h)663 740 y Ft(K)q(;K)809 717 y Fl(0)663 813 y Ft(L)834 784 y Fy(\()p Fw(m;)15 b(m)1069 747 y Fo(0)1092 784 y Fw(;)f(~)-44 b(z)5 b(;)1213 768 y(~)1219 784 y(t)p Fy(\))26 b(=)f(inf)1559 671 y Ft(i)1583 679 y Fm(m)1535 698 y Fq(X)1543 893 y Ft(i)p Fr(=0)1681 661 y Fq(Z)1772 687 y Ft(t)1797 657 y Fl(\000)1797 709 y Fm(i)1732 867 y Ft(t)1757 837 y Fk(+)1757 889 y Fm(i)1854 784 y Fy(\()p Fw(L)20 b Fv(\000)g Fw(\021)2107 798 y Ft(i)2135 784 y Fy(\)\()p Fw(d\015)2304 746 y Fo(\000)2299 812 y Ft(i)2365 784 y Fy(\()p Fw(t)p Fy(\)\))p Fw(dt)1308 1105 y Fy(+)1399 991 y Ft(i)1423 999 y Fm(m)1482 991 y Fo(\000)p Fr(1)1420 1018 y Fq(X)1428 1214 y Ft(i)p Fr(=0)1587 981 y Fq(Z)1678 1007 y Ft(t)1703 977 y Fk(+)1703 1029 y Fm(i)p Fk(+1)1637 1187 y Ft(t)1662 1157 y Fl(\000)1662 1209 y Fm(i)1811 1105 y Fy(\()p Fw(L)g Fv(\000)g Fw(\021)2064 1119 y Ft(i)2113 1105 y Fv(\000)g Fy(\()p Fw(k)2286 1119 y Ft(i)2314 1105 y Fw(G)2385 1119 y Ft(i)2414 1105 y Fy(\))2449 1067 y Fo(\003)2489 1105 y Fy(\()p Fw(\026)2579 1119 y Ft(i)2627 1105 y Fy(+)g Fw( )2777 1119 y Ft(i)2806 1105 y Fy(\)\)\()p Fw(d)s Fy(~)-48 b Fw(\015)3010 1066 y Fr(+)3005 1132 y Ft(i)3070 1105 y Fy(\()p Fw(t)p Fy(\)\))p Fw(dt)-2832 b Fy(\(3.11\))456 1358 y(where)23 b(the)i(in\014m)m(um)e(is)h(tak)m(en) i(o)m(v)m(er)f(all)h Fw(C)1921 1325 y Fr(1)1960 1358 y Fy(-curv)m(es)e Fw(\015)5 b Fy(:)38 b([)p Fv(\000)p Fw(K)q(;)15 b(K)2675 1325 y Fo(0)2699 1358 y Fy(])26 b Fv(!)f Fw(T)13 b(M)34 b Fy(with)24 b(\014xed)456 1470 y(b)s(oundary)30 b(conditions)38 b(\026)-50 b Fw(\031)s(G)1430 1432 y Fo(\000)p Fr(1)1430 1497 y Ft(i)1528 1470 y Fy(~)i Fw(\015)1577 1432 y Fr(+)1572 1497 y Ft(i)1636 1470 y Fy(\()p Fw(t)1704 1432 y Fo(\000)1704 1497 y Ft(i)1763 1470 y Fy(\))22 b Fv(\000)f Fy(\(2)p Fw(k)2039 1484 y Ft(i)2068 1470 y Fw(\031)s(;)15 b Fy(0)p Fw(;)g Fv(\001)g(\001)g(\001) 33 b Fw(;)15 b Fy(0\))30 b(=)e Fw(z)2681 1432 y Fo(\000)2677 1497 y Ft(i)2740 1470 y Fy(,)38 b(\026)-50 b Fw(\031)s(G)2924 1432 y Fo(\000)p Fr(1)2924 1497 y Ft(i)3022 1470 y Fy(~)i Fw(\015)3071 1432 y Fr(+)3066 1497 y Ft(i)3130 1470 y Fy(\()p Fw(t)3198 1432 y Fr(+)3198 1497 y Ft(i)p Fr(+1)3317 1470 y Fy(\))22 b Fv(\000)456 1591 y Fy(\(2)p Fw(k)583 1605 y Ft(i)612 1591 y Fw(\031)s(;)15 b Fy(0)p Fw(;)g Fv(\001)g(\001)g(\001)33 b Fw(;)15 b Fy(0\))26 b(=)f Fw(z)1218 1553 y Fr(+)1214 1618 y Ft(i)p Fr(+1)1359 1591 y Fy(for)h Fw(i)g Fy(=)f(0)p Fw(;)15 b Fy(1)p Fw(;)g Fv(\001)g(\001)g(\001)33 b Fw(;)15 b(i)2026 1605 y Ft(m)2106 1591 y Fv(\000)d Fy(1,)28 b Fw(\015)5 b Fy(\()p Fv(\000)p Fw(K)i Fy(\))25 b(=)g Fw(m)p Fy(,)i Fw(\015)5 b Fy(\()p Fw(K)2988 1558 y Fo(0)3012 1591 y Fy(\))25 b(=)g Fw(m)3248 1558 y Fo(0)3298 1591 y Fy(and)456 1701 y(satisfying)31 b(the)f(conditions)h(\(3.7\),)h(\(3.8\))h(and)c(\(3.9\).)555 1911 y(As)34 b(the)g(system)g(is)g(autonomous,)h(the)f(quan)m(tit)m(y)i Fw(h)2373 1867 y Ft(K)q(;K)2519 1843 y Fl(0)2373 1939 y Ft(L)2544 1911 y Fy(\()p Fw(m;)15 b(m)2779 1878 y Fo(0)2802 1911 y Fw(;)f(~)-44 b(z)5 b(;)2923 1894 y(~)2929 1911 y(t)p Fy(\))34 b(k)m(eeps)h(con-)456 2030 y(stan)m(t)22 b(if)g(\()781 2013 y Fw(~)787 2030 y(t;)15 b(K)q(;)g(K)1062 1997 y Fo(0)1086 2030 y Fy(\))22 b(is)g(sub)5 b(ject)21 b(to)h(a)g(translation.)39 b(Th)m(us,)22 b(it)g(is)g(a)g(function)f(of) h Fw(K)3201 1997 y Fo(0)3226 2030 y Fv(\000)s Fw(t)3333 1991 y Fr(+)3333 2057 y Ft(i)3357 2065 y Fm(m)3419 2030 y Fy(,)456 2155 y Fw(t)489 2117 y Fo(\000)489 2181 y Fr(0)561 2155 y Fy(+)14 b Fw(K)33 b Fy(and)27 b(\001)1000 2138 y Fw(~)1006 2155 y(t)d Fy(=)h Fv(f)p Fy(\001)p Fw(t)1313 2117 y Fr(+)1313 2181 y(0)1372 2155 y Fw(;)15 b Fy(\001)p Fw(t)1521 2117 y Fo(\006)1521 2181 y Fr(1)1580 2155 y Fw(;)g Fv(\001)g(\001)g(\001)32 b Fw(;)15 b Fy(\001)p Fw(t)1906 2117 y Fo(\006)1906 2182 y Ft(i)1930 2190 y Fm(m)1989 2182 y Fo(\000)p Fr(1)2083 2155 y Fv(g)28 b Fy(in)e(whic)m(h)h(\001)p Fw(t)2624 2117 y Fr(+)2624 2182 y Ft(i)2708 2155 y Fy(=)2814 2119 y Fr(1)p 2814 2134 36 4 v 2814 2186 a(2)2859 2155 y Fy(\()p Fw(t)2927 2117 y Fr(+)2927 2182 y Ft(i)p Fr(+1)3059 2155 y Fv(\000)14 b Fw(t)3177 2117 y Fo(\000)3177 2182 y Ft(i)3236 2155 y Fy(\))27 b(and)456 2277 y(\001)p Fw(t)565 2239 y Fo(\000)565 2304 y Ft(i)648 2277 y Fy(=)754 2242 y Fr(1)p 754 2257 V 754 2309 a(2)800 2277 y Fy(\()p Fw(t)868 2239 y Fo(\000)868 2304 y Ft(i)947 2277 y Fv(\000)20 b Fw(t)1071 2239 y Fr(+)1071 2304 y Ft(i)1130 2277 y Fy(\).)555 2482 y(Denote)34 b(b)m(y)e Fw(\015)5 b Fy(\()p Fw(t)p Fy(;)15 b Fw(K)q(;)g(K)1359 2449 y Fo(0)1384 2482 y Fw(;)g(m;)g(m)1624 2449 y Fo(0)1648 2482 y Fw(;)f(~)-44 b(z)t(;)15 b Fy(\001)1844 2465 y Fw(~)1850 2482 y(t)p Fy(\))33 b(the)f(minimizer)g(of)h(\(3.11\).)48 b(The)32 b(purp)s(ose)456 2590 y(of)39 b(the)g(follo)m(wing)i(steps)e (is)g(ob)m(viously)g(to)h(in)m(v)m(estigate)i(when)c(the)i(minimizer)f (is)g(a)456 2698 y(solution)44 b(of)g(the)g(Euler-Lagrange)h(equation)g (determined)e(b)m(y)h Fw(L)p Fy(.)81 b(Indeed,)46 b(It)e(is)456 2819 y(ac)m(hiev)m(ed)d(b)m(y)f(taking)h(the)f(minim)m(um)f(of)h Fw(h)2004 2775 y Ft(K)q(;K)2150 2751 y Fl(0)2004 2848 y Ft(L)2175 2819 y Fy(\()p Fw(m;)15 b(m)2410 2786 y Fo(0)2433 2819 y Fw(;)f(~)-44 b(z)5 b(;)15 b Fy(\001)2630 2802 y Fw(~)2636 2819 y(t)p Fy(\))40 b(in)e Fw(~)-44 b(z)44 b Fy(and)c(\001)3202 2802 y Fw(~)3208 2819 y(t)f Fy(o)m(v)m(er)456 2944 y(some)34 b(set)839 2921 y Fw(~)832 2944 y(V)54 b Fy(and)33 b(\001)1206 2921 y Fw(~)1195 2944 y(T)12 b Fy(,)35 b(in)e(whic)m(h)h(the)1859 2921 y Fw(~)1853 2944 y(V)51 b Fy(=)30 b(\()p Fw(V)2167 2906 y Fo(\000)2146 2970 y Fr(0)2226 2944 y Fw(;)15 b(V)2339 2906 y Fo(\006)2319 2970 y Fr(1)2398 2944 y Fw(;)g Fv(\001)g(\001)g(\001)32 b Fw(;)15 b(V)2689 2906 y Fo(\006)2668 2971 y Ft(i)2692 2979 y Fm(m)2751 2971 y Fo(\000)p Fr(1)2845 2944 y Fw(;)g(V)2959 2906 y Fr(+)2938 2971 y Ft(i)2962 2979 y Fm(m)3026 2944 y Fy(\))34 b(has)f(b)s(een)456 3052 y(w)m(ell)f(de\014ned)e(in)h(the)g (previous)g(demonstration.)44 b(F)-8 b(rom)32 b(no)m(w)f(on,)h(let)g (us)f(de\014ne)f(the)456 3160 y(range)g(where)g(\001)1036 3143 y Fw(~)1042 3160 y(t)g Fy(tak)m(es)i(its)f(v)-5 b(alue.)456 3329 y FC(Prop)s(osition)41 b(3.1.)k Fj(Assume)36 b(ther)-5 b(e)38 b(is)e(a)i(unique)d Fw(c)p Fj(-minimizing)i(me)-5 b(asur)g(e.)56 b(Then)456 3437 y(for)33 b(e)-5 b(ach)33 b Fw(m;)15 b(m)1005 3404 y Fo(0)1054 3437 y Fv(2)25 b Fw(M)42 b Fj(and)34 b(e)-5 b(ach)33 b Fw(x)26 b Fv(2)e(A)p Fy(\()p Fw(c)p Fy(\))p Fj(,)33 b(we)g(have)1229 3596 y Fw(h)1281 3559 y Fo(1)1281 3619 y Ft(c)1356 3596 y Fy(\()p Fw(m;)15 b(x)p Fy(\))21 b(+)f Fw(h)1762 3559 y Fo(1)1762 3619 y Ft(c)1837 3596 y Fy(\()p Fw(x;)15 b(m)2044 3559 y Fo(0)2068 3596 y Fy(\))26 b(=)f Fw(h)2277 3559 y Fo(1)2277 3619 y Ft(c)2352 3596 y Fy(\()p Fw(m;)15 b(m)2587 3559 y Fo(0)2610 3596 y Fy(\))p Fw(:)555 3849 y Fy(Refer)31 b(Corollary)g(5.3.7)h(in)e([F)-8 b(a2)r(])30 b(for)h(the)f(pro)s(of.)555 4040 y(Applying)j(this)f(prop)s(osition)h (to)g Fw(c)1764 4054 y Ft(i)1825 4040 y Fy(with)g Fw(i)c Fv(2)2185 3972 y Fq(S)2261 4067 y Ft(i)2285 4077 y Fm(j)2317 4067 y Fo(2)p Fr(\003)2413 4079 y Fm(h)2458 4040 y Fv(f)p Fw(i)2534 4054 y Ft(j)2571 4040 y Fw(;)15 b(i)2642 4054 y Ft(j)2701 4040 y Fy(+)21 b(1)p Fw(;)15 b Fv(\001)g(\001)g(\001)33 b Fw(;)15 b(i)3087 4054 y Ft(j)t Fr(+1)3236 4040 y Fv(\000)21 b Fy(1)p Fv(g)p Fy(,)456 4173 y(noticing)35 b(the)f(compactness)h(of)f Fw(M)45 b Fy(and)33 b Fw(h)1970 4140 y Fo(1)1970 4196 y Ft(c)2077 4173 y Fy(=)e(lim)2305 4187 y Ft(T)10 b Fo(!1)2517 4173 y Fw(h)2569 4140 y Ft(T)2569 4196 y(c)2658 4173 y Fy(in)34 b(autonomous)g(case)456 4282 y([F)-8 b(a1)r(],)30 b(w)m(e)h(\014nd)e(that,)i(for)f(an)m(y)h Fw(\017)1590 4249 y Fo(\003)1590 4308 y Ft(i)1655 4282 y Fw(>)25 b Fy(0,)31 b(there)f(exists)h Fw(T)2401 4244 y Fo(\000)2388 4309 y Ft(i)2485 4282 y Fy(=)25 b Fw(T)2647 4244 y Fo(\000)2634 4309 y Ft(i)2706 4282 y Fy(\()p Fw(\017)2778 4249 y Fo(\003)2818 4282 y Fy(\))g Fw(>)g Fy(0)31 b(suc)m(h)f(that)456 4452 y(\(3.12\))443 b Fv(j)p Fw(h)1206 4414 y Ft(T)1206 4474 y(c)1237 4484 y Fm(i)1268 4452 y Fy(\()p Fw(m;)15 b(m)1503 4414 y Fo(0)1527 4452 y Fy(\))20 b Fv(\000)g Fw(h)1725 4414 y Fo(1)1725 4474 y Ft(c)1756 4484 y Fm(i)1800 4452 y Fy(\()p Fw(m;)15 b(x)p Fy(\))22 b Fv(\000)d Fw(h)2206 4414 y Fo(1)2206 4474 y Ft(c)2237 4484 y Fm(i)2282 4452 y Fy(\()p Fw(x;)c(m)2489 4414 y Fo(0)2512 4452 y Fy(\))p Fv(j)26 b(\024)f Fw(\017)2731 4414 y Fo(\003)2731 4474 y Ft(i)456 4622 y Fy(holds)k(for)i(eac)m(h)g Fw(T)38 b Fv(\025)25 b Fw(T)1290 4584 y Fo(\000)1277 4650 y Ft(i)1349 4622 y Fy(,)30 b(eac)m(h)i Fw(m;)15 b(m)1809 4589 y Fo(0)1858 4622 y Fv(2)24 b Fw(M)41 b Fy(and)30 b Fw(x)25 b Fv(2)g(A)p Fy(\()p Fw(c)2559 4636 y Ft(i)2587 4622 y Fy(\).)555 4829 y(Denote)44 b(b)m(y)e(\001)1104 4806 y Fw(~)1093 4829 y(T)55 b Fy(the)43 b(domain)f(where)g(\001)2050 4812 y Fw(~)2056 4829 y(t)g Fy(tak)m(es)i(its)f(v)-5 b(alue.)77 b(Its)42 b(de\014nition)g(is)456 4937 y(someho)m(w)30 b(implicit.)42 b(Let)456 5109 y(\(3.13\))81 b(\001)p Fw(T)909 5071 y Fr(+)896 5136 y Ft(i)993 5109 y Fy(=)25 b([)p Fw(T)1180 5071 y Fr(+)1167 5136 y Ft(i)1259 5109 y Fv(\000)20 b Fw(d)1397 5123 y Ft(i)1425 5109 y Fw(;)15 b(T)1531 5071 y Fr(+)1518 5136 y Ft(i)1611 5109 y Fy(+)k Fw(d)1748 5123 y Ft(i)1777 5109 y Fy(])p Fw(;)197 b Fv(8)p Fw(i)26 b Fv(2)2263 5023 y Fq([)2217 5220 y Ft(i)2241 5230 y Fm(j)2274 5220 y Fo(2)p Fr(\003)2370 5232 y Fm(h)2410 5109 y Fv(f)p Fw(i)2486 5123 y Ft(j)2524 5109 y Fw(;)15 b(i)2595 5123 y Ft(j)2652 5109 y Fy(+)20 b(1)p Fw(;)15 b Fv(\001)g(\001)g(\001)32 b Fw(;)15 b(i)3036 5123 y Ft(j)t Fr(+1)3184 5109 y Fv(\000)20 b Fy(1)p Fv(g)p eop end %%Page: 22 22 TeXDict begin 22 21 bop 456 251 a Fr(22)1012 b(X.)23 b(LI)h(AND)f(C.-Q.)f(CHENG)456 453 y Fy(\(3.14\))204 b(\001)p Fw(T)1032 415 y Fr(+)1019 480 y Ft(i)1116 453 y Fy(=)25 b(\(0)p Fw(;)15 b(T)1398 415 y Fr(+)1385 480 y Ft(i)1458 453 y Fy(])p Fw(;)197 b Fy(\001)p Fw(T)1847 415 y Fo(\000)1834 480 y Ft(i)1931 453 y Fy(=)25 b([)2070 430 y(^)2052 453 y Fw(T)2118 415 y Fo(\000)2105 480 y Ft(i)2177 453 y Fw(;)2235 430 y Fy(\025)2217 453 y Fw(T)2283 415 y Fo(\000)2270 480 y Ft(i)2342 453 y Fy(])p Fw(;)198 b Fy(for)30 b(other)h Fw(i)25 b Fv(\024)g Fw(i)3150 467 y Ft(m)3217 453 y Fw(:)456 654 y Fy(See)j(\(3.2\),)i(\(3.4\))g(for)e (the)g(de\014nition)f(of)h Fw(T)1929 616 y Fr(+)1916 681 y Ft(i)2016 654 y Fy(and)f(\(3.10\))j(for)e(the)g(de\014nition)g (of)g(\001)p Fw(T)3386 616 y Fo(\000)3373 681 y Ft(i)456 762 y Fy(resp)s(ectiv)m(ely)-8 b(.)555 959 y(\001)625 942 y Fw(~)631 959 y(t)33 b Fv(2)h Fy(\001)879 936 y Fw(~)868 959 y(T)48 b Fy(implies)35 b(that)h(\001)p Fw(t)1593 920 y Fr(+)1593 986 y Ft(i)1686 959 y Fv(2)d Fy(\001)p Fw(T)1922 920 y Fr(+)1909 986 y Ft(i)2016 959 y Fy(for)i(all)h(0)e Fv(\024)g Fw(i)g(<)f(i)2675 973 y Ft(m)2742 959 y Fy(,)k(\001)p Fw(t)2913 920 y Fo(\000)2913 986 y Ft(i)3005 959 y Fv(2)d Fy(\001)p Fw(T)3242 920 y Fo(\000)3229 986 y Ft(i)3336 959 y Fy(for)456 1072 y Fw(i)25 b Fv(2)598 1004 y Fq(S)674 1099 y Ft(i)698 1109 y Fm(j)730 1099 y Fo(2)p Fr(\003)826 1107 y Fm(c)862 1072 y Fv(f)p Fw(i)938 1086 y Ft(j)989 1072 y Fy(+)13 b(1)p Fw(;)i(i)1189 1086 y Ft(j)1239 1072 y Fy(+)e(2)p Fw(;)i Fv(\001)g(\001)g(\001)32 b Fw(;)15 b(i)1616 1086 y Ft(j)t Fr(+1)1757 1072 y Fv(\000)e Fy(1)p Fv(g)p Fy(,)28 b(for)f(other)g Fw(i)g Fy(\001)p Fw(t)2521 1034 y Fo(\000)2521 1100 y Ft(i)2607 1072 y Fy(satis\014es)g(the)g (follo)m(wing)456 1188 y(restrictions:)725 1383 y Fw(T)791 1345 y Fo(\000)778 1410 y Ft(i)870 1383 y Fy(+)20 b Fw(T)1027 1345 y Fr(+)1014 1410 y Ft(i)1106 1383 y Fy(+)f Fw(T)1262 1345 y Fr(+)1249 1410 y Ft(i)p Fo(\000)p Fr(1)1388 1383 y Fy(+)h Fw(d)1526 1397 y Ft(i)1575 1383 y Fy(+)g Fw(d)1713 1397 y Ft(i)p Fo(\000)p Fr(1)1857 1383 y Fv(\024)25 b Fy(\001)p Fw(t)2062 1345 y Fo(\000)2062 1410 y Ft(i)2140 1383 y Fy(+)20 b(\001)p Fw(t)2340 1345 y Fr(+)2340 1410 y Ft(i)2419 1383 y Fy(+)g(\001)p Fw(t)2619 1345 y Fr(+)2619 1410 y Ft(i)p Fo(\000)p Fr(1)1857 1533 y Fv(\024)25 b Fw(T)2019 1495 y Fo(\000)2006 1560 y Ft(i)2098 1533 y Fy(+)19 b Fw(T)2254 1495 y Fr(+)2241 1560 y Ft(i)2333 1533 y Fy(+)h Fw(T)2490 1495 y Fr(+)2477 1560 y Ft(i)p Fo(\000)p Fr(1)2616 1533 y Fy(+)g(3\()p Fw(d)2834 1547 y Ft(i)2883 1533 y Fy(+)g Fw(d)3021 1547 y Ft(i)p Fo(\000)p Fr(1)3140 1533 y Fy(\))-2719 b(\(3.15\))456 1743 y(holds)29 b(for)i(\001)p Fw(t)942 1705 y Fr(+)942 1770 y Ft(i)1026 1743 y Fv(2)24 b Fy(\001)p Fw(T)1253 1705 y Fr(+)1240 1770 y Ft(i)1312 1743 y Fy(,)30 b(\001)p Fw(t)1476 1705 y Fr(+)1476 1770 y Ft(i)p Fo(\000)p Fr(1)1620 1743 y Fv(2)25 b Fy(\001)p Fw(T)1848 1705 y Fr(+)1835 1770 y Ft(i)p Fo(\000)p Fr(1)1983 1743 y Fy(\(see)31 b(\(3.12\))i(for)d(the)h (de\014nition)f(of)g Fw(T)3297 1705 y Fo(\000)3284 1770 y Ft(i)3356 1743 y Fy(\).)555 1964 y(Let)36 b Fw(h)775 1919 y Ft(K)q(;K)921 1896 y Fl(0)775 1992 y Ft(L)946 1964 y Fy(\()p Fw(m;)15 b(m)1181 1931 y Fo(0)1205 1964 y Fy(\))35 b(b)s(e)g(the)g(minim)m(um)g(of)g Fw(h)2137 1919 y Ft(K)q(;K)2283 1896 y Fl(0)2137 1992 y Ft(L)2308 1964 y Fy(\()p Fw(m;)15 b(m)2543 1931 y Fo(0)2567 1964 y Fw(;)f(~)-44 b(z)5 b(;)2688 1947 y(~)2694 1964 y(t)p Fy(\))35 b(o)m(v)m(er)3004 1941 y Fw(~)2997 1964 y(V)55 b Fy(in)34 b Fw(~)-44 b(z)40 b Fy(and)456 2088 y(o)m(v)m(er)31 b(\001)737 2066 y Fw(~)726 2088 y(T)43 b Fy(in)30 b(\001)998 2072 y Fw(~)1004 2088 y(t)g Fy(resp)s(ectiv)m(ely:)1072 2297 y Fw(h)1124 2252 y Ft(K)q(;K)1270 2229 y Fl(0)1124 2325 y Ft(L)1295 2297 y Fy(\()p Fw(m;)15 b(m)1530 2259 y Fo(0)1554 2297 y Fy(\))25 b(=)150 b(min)1710 2376 y Fr(\001)1765 2364 y Ft(~)1769 2376 y(t)q Fo(2)p Fr(\001)1910 2359 y Ft(~)1901 2376 y(T)9 b(;)o(~)-34 b(z)s Fo(2)2060 2359 y Ft(~)2054 2376 y(V)2126 2297 y Fw(h)2178 2252 y Ft(K)q(;K)2324 2229 y Fl(0)2178 2325 y Ft(L)2349 2297 y Fy(\()p Fw(m;)15 b(m)2584 2259 y Fo(0)2607 2297 y Fw(;)f(~)-44 b(z)5 b(;)2728 2280 y(~)2734 2297 y(t)p Fy(\))p Fw(;)456 2563 y Fy(denote)25 b(the)f(corresp)s(onding)g(minimal)g(curv)m(e)h(b)m (y)g Fw(\015)5 b Fy(\()p Fw(t)p Fy(;)15 b Fw(K)q(;)g(K)2539 2530 y Fo(0)2563 2563 y Fw(;)g(m;)g(m)2803 2530 y Fo(0)2827 2563 y Fy(\),)26 b(w)m(e)f(claim)h(that)456 2671 y Fw(d\015)5 b Fy(\()p Fw(t)p Fy(;)15 b Fw(K)q(;)g(K)865 2638 y Fo(0)890 2671 y Fw(;)g(m;)g(m)1130 2638 y Fo(0)1153 2671 y Fy(\))38 b(is)g(a)g(solution)h(of)e(the)h(Euler-Lagrange)h(equation)g(of)e Fw(L)h Fy(if)g Fw(K)456 2779 y Fy(and)29 b Fw(K)716 2746 y Fo(0)770 2779 y Fy(are)h(su\016cien)m(tly)h(large.)42 b(T)-8 b(o)31 b(v)m(erify)-8 b(,)32 b(w)m(e)e(need)h(to)g(sho)m(w)555 2970 y(1,)42 b Fw(d\015)766 2932 y Fr(+)761 2997 y Ft(i)865 2970 y Fy(=)e Fw(d\015)5 b Fv(j)1100 3000 y Fr(\001)p Ft(t)1184 2970 y Fk(+)1184 3022 y Fm(i)1278 2970 y Fy(solv)m(es)41 b(the)e(Euler-Lagrange)h(equation)g(determined)e(b)m(y)h Fw(L)p Fy(;)456 3109 y(Note)31 b Fw(d\015)774 3071 y Fo(\000)769 3136 y Ft(i)859 3109 y Fy(=)25 b Fw(d\015)5 b Fv(j)1079 3139 y Fr(\001)p Ft(t)1163 3109 y Fl(\000)1163 3161 y Fm(i)1251 3109 y Fy(is)30 b(clearly)i(a)e(solution)h(of)g(the)f (equation;)555 3331 y(2,)49 b Fw(\015)5 b Fy(\()p Fw(t)p Fy(;)15 b Fw(K)q(;)g(K)1036 3298 y Fo(0)1061 3331 y Fw(;)g(m;)g(m)1301 3298 y Fo(0)1325 3331 y Fy(\))45 b(has)f(no)h(corner)g(at)g Fw(G)2211 3345 y Ft(i)2240 3331 y Fw(z)2286 3293 y Fo(\000)2282 3358 y Ft(i)2390 3331 y Fy(and)f Fw(G)2652 3345 y Ft(i)p Fo(\000)p Fr(1)2771 3331 y Fw(z)2817 3293 y Fr(+)2813 3358 y Ft(i)2921 3331 y Fy(for)g(eac)m(h)i Fw(i)k Fy(=)456 3439 y(0)p Fw(;)15 b Fy(1)p Fw(;)g Fv(\001)g(\001)g(\001)j Fw(i)780 3453 y Ft(m)867 3439 y Fv(\000)i Fy(1,)31 b(i.e.)41 b(it)31 b(is)f(smo)s(oth)h(for)f(the)g(whole)h Fw(t)25 b Fv(2)g Fy([)p Fv(\000)p Fw(K)q(;)15 b(K)2714 3406 y Fo(0)2738 3439 y Fy(].)555 3630 y(Indeed,)27 b(if)g Fw(i)f Fv(2)1096 3562 y Fq(S)1172 3657 y Ft(i)1196 3667 y Fm(j)1228 3657 y Fo(2)p Fr(\003)1324 3669 y Fm(h)1369 3630 y Fv(f)p Fw(i)1445 3644 y Ft(j)1482 3630 y Fw(;)15 b(i)1553 3644 y Ft(j)1604 3630 y Fy(+)f(1)p Fw(;)h Fv(\001)g(\001)g(\001)32 b Fw(;)15 b(i)1982 3644 y Ft(j)t Fr(+1)2123 3630 y Fv(\000)f Fy(1)p Fv(g)p Fy(,)28 b(w)m(e)g(obtained)f(from)g(\(3.3\))i(that)1007 3850 y(\026)-48 b Fw(\015)1056 3812 y Fr(+)1051 3877 y Ft(i)1115 3850 y Fy(\()p Fw(t)p Fy(\))26 b Fv(2)f Fw(U)1392 3864 y Ft(i)1602 3850 y Fy(when)63 b(\026)-49 b Fw(\015)1921 3812 y Fr(+)1916 3877 y Ft(i;)p Fr(1)2000 3850 y Fy(\()p Fw(t)p Fy(\))20 b Fv(\000)g Fy(2)p Fw(k)2306 3864 y Ft(i)2335 3850 y Fw(\031)29 b Fv(2)24 b Fy([)p Fv(\000)p Fy(\001)2673 3864 y Ft(i)2701 3850 y Fw(;)15 b Fy(\001)2817 3864 y Ft(i)2846 3850 y Fy(])p Fw(;)456 4063 y Fy(where)30 b(\026)-49 b Fw(\015)767 4025 y Fr(+)762 4091 y Ft(i)852 4063 y Fy(=)29 b(\026)-49 b Fw(\031)s(G)1074 4077 y Ft(i)1106 4063 y Fy(~)g Fw(\015)1154 4025 y Fr(+)1149 4091 y Ft(i)1214 4063 y Fy(.)39 b(Since)27 b(the)h(supp)s(ort)d(of)35 b(\026)-52 b Fw(\026)2155 4077 y Ft(i)2210 4063 y Fy(has)27 b(no)g(in)m(tersection)i(with)e Fw(U)3243 4077 y Ft(i)3298 4063 y Fy(and)456 4177 y Fw( )515 4191 y Ft(i)568 4177 y Fy(is)e(closed)g(in)g Fw(U)1079 4191 y Ft(i)1107 4177 y Fy(,)h(while)f Fw(\026)1445 4191 y Ft(i)1497 4177 y Fy(is)g(closed)h(and)e Fw( )2076 4191 y Ft(i)2130 4177 y Fy(=)h(0)g(in)f(the)h(region)h Fv(f)s Fy(\026)-48 b Fw(\015)2912 4139 y Fr(+)2907 4204 y Ft(i;)p Fr(1)2990 4177 y Fy(\()p Fw(t)p Fy(\))9 b Fv(\000)g Fy(2)p Fw(k)3274 4191 y Ft(i)3303 4177 y Fw(\031)29 b Fv(62)456 4294 y Fy([)p Fv(\000)p Fy(\001)628 4308 y Ft(i)655 4294 y Fw(;)15 b Fy(\001)771 4308 y Ft(i)800 4294 y Fy(])p Fv(g)p Fy(,)32 b(the)f(term)g Fw(\026)1357 4308 y Ft(i)1416 4294 y Fy(and)f Fw( )1652 4308 y Ft(i)1711 4294 y Fy(ha)m(v)m(e)i(no)f(con)m(tribution) h(to)f(the)g(Euler-Lagrange)456 4403 y(equation)d(along)k(\026)-49 b Fw(\015)1115 4365 y Fr(+)1110 4430 y Ft(i)1175 4403 y Fy(.)39 b(F)-8 b(or)28 b(other)g Fw(i)p Fy(,)h(the)e(conclusion)h(is) g(obtained)g(from)f(\(3.4\))i(b)m(y)e(the)456 4511 y(same)j(argumen)m (t.)42 b(This)29 b(pro)m(v)m(es)i(the)g(\014rst)e(p)s(oin)m(t.)555 4702 y(W)-8 b(e)41 b(claim)f(that)h Fw(\015)j Fy(do)s(es)39 b(not)h(touc)m(h)g(the)g(b)s(oundary)d(of)j Fw(G)2682 4716 y Ft(i)2711 4702 y Fw(\031)s(V)2839 4664 y Fo(\000)2819 4729 y Ft(i)2924 4702 y Fv(\002)26 b Fw(G)3092 4716 y Ft(i)3121 4702 y Fw(\031)s(V)3249 4664 y Fr(+)3229 4729 y Ft(i)p Fr(+1)3374 4702 y Fv(\002)456 4823 y Fy([)p Fw(T)547 4785 y Fr(+)534 4850 y Ft(i)621 4823 y Fv(\000)15 b Fw(d)754 4837 y Ft(i)783 4823 y Fw(;)g(T)889 4785 y Fr(+)876 4850 y Ft(i)963 4823 y Fy(+)h Fw(d)1097 4837 y Ft(i)1125 4823 y Fy(])28 b(for)g Fw(i)e Fv(2)1458 4755 y Fq(S)1533 4850 y Ft(i)1557 4860 y Fm(j)1590 4850 y Fo(2)p Fr(\003)1686 4862 y Fm(h)1730 4823 y Fv(f)p Fw(i)1806 4837 y Ft(j)1844 4823 y Fw(;)15 b(i)1915 4837 y Ft(j)1967 4823 y Fy(+)g(1)p Fw(;)g Fv(\001)g(\001)g(\001)33 b Fw(;)15 b(i)2347 4837 y Ft(j)t Fr(+1)2490 4823 y Fv(\000)g Fy(1)p Fv(g)p Fy(.)41 b(Let)28 b(us)f(assume)h(the)456 4956 y(con)m(trary)-8 b(,)33 b(i.e.)44 b(\()p Fw(z)1079 4918 y Fo(\000)1075 4983 y Ft(i)1139 4956 y Fw(;)15 b(z)1225 4918 y Fr(+)1221 4983 y Ft(i)p Fr(+1)1340 4956 y Fw(;)g Fy(\001)p Fw(t)1489 4918 y Fr(+)1489 4983 y Ft(i)1548 4956 y Fy(\))27 b Fv(2)f Fw(@)5 b Fy(\()p Fw(V)1859 4918 y Fo(\000)1838 4983 y Ft(i)1939 4956 y Fv(\002)20 b Fw(V)2104 4918 y Fr(+)2083 4983 y Ft(i)p Fr(+1)2223 4956 y Fv(\002)g Fy([)p Fw(T)2405 4918 y Fr(+)2392 4983 y Ft(i)2485 4956 y Fv(\000)g Fw(d)2623 4970 y Ft(i)2652 4956 y Fw(;)15 b(T)2758 4918 y Fr(+)2745 4983 y Ft(i)2838 4956 y Fy(+)20 b Fw(d)2976 4970 y Ft(i)3005 4956 y Fy(]\))32 b(holds)f(for)456 5080 y(some)36 b Fw(i)e Fv(2)849 5011 y Fq(S)924 5106 y Ft(i)948 5116 y Fm(j)981 5106 y Fo(2)p Fr(\003)1077 5118 y Fm(h)1121 5080 y Fv(f)p Fw(i)1197 5094 y Ft(j)1235 5080 y Fw(;)15 b(i)1306 5094 y Ft(j)1367 5080 y Fy(+)23 b(1)p Fw(;)15 b Fv(\001)g(\001)g(\001)32 b Fw(;)15 b(i)1754 5094 y Ft(j)t Fr(+1)1905 5080 y Fv(\000)24 b Fy(1)p Fv(g)36 b Fy(b)m(y)g(noticing)g Fw(z)2657 5041 y Fo(\000)2653 5107 y Ft(i)2751 5080 y Fy(=)i(\026)-50 b Fw(\031)s(G)2981 5041 y Fo(\000)p Fr(1)2981 5107 y Ft(i)3080 5080 y Fy(~)h Fw(\015)3128 5041 y Fr(+)3123 5107 y Ft(i)3187 5080 y Fy(\()p Fw(t)3255 5041 y Fo(\000)3255 5107 y Ft(i)3315 5080 y Fy(\))24 b Fv(\000)456 5216 y Fy(\(2)p Fw(k)583 5230 y Ft(i)612 5216 y Fw(\031)s(;)15 b Fy(0)p Fw(;)g Fv(\001)g(\001)g(\001)33 b Fw(;)15 b Fy(0\))45 b(and)e Fw(z)1331 5177 y Fr(+)1327 5243 y Ft(i)1438 5216 y Fy(=)53 b(\026)-50 b Fw(\031)s(G)1683 5177 y Fo(\000)p Fr(1)1683 5243 y Ft(i)1781 5216 y Fy(~)i Fw(\015)1830 5177 y Fr(+)1825 5243 y Ft(i)1889 5216 y Fy(\()p Fw(t)1957 5177 y Fr(+)1957 5243 y Ft(i)2016 5216 y Fy(\))30 b Fv(\000)f Fy(\(2)p Fw(k)2308 5230 y Ft(i)2337 5216 y Fw(\031)s(;)15 b Fy(0)p Fw(;)g Fv(\001)g(\001)g(\001)33 b Fw(;)15 b Fy(0\).)83 b(Then,)46 b(in)e Fw(q)51 b Fy(=)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 998 251 a Fr(CONNECTING)31 b(ORBITS)g(F)n(OR)f (A)n(UTONOMOUS)g(SYSTEMS)472 b(23)456 450 y Fw(G)527 412 y Fo(\000)p Fr(1)527 477 y Ft(i)621 450 y Fw(x)p Fy(-co)s(ordinates,)32 b(w)m(e)f(obtain)g(from)f(\(3.2\))i(and)e (\(3.12\))i(that)899 649 y Fw(h)951 596 y Fr(\001)p Ft(t)1035 566 y Fl(\000)1035 618 y Fm(i)951 661 y Ft(c)982 671 y Fm(i)1092 649 y Fy(\()p Fw(G)1198 611 y Fo(\000)p Fr(1)1198 676 y Ft(i)1293 649 y Fw(G)1364 663 y Ft(i)p Fo(\000)p Fr(1)1483 649 y Fw(\031)s(z)1584 611 y Fr(+)1580 676 y Ft(i)1643 649 y Fw(;)15 b(\031)s(z)1784 611 y Fo(\000)1780 676 y Ft(i)1844 649 y Fy(\))21 b(+)f Fw(h)2043 596 y Fr(\001)p Ft(t)2127 566 y Fk(+)2127 618 y Fm(i)2043 678 y Ft(\021)2078 688 y Fm(i)2105 678 y Ft(;\026)2167 688 y Fm(i)2193 678 y Ft(; )2259 688 y Fm(i)2285 678 y Ft(;e)2338 686 y Fm(n)2384 649 y Fy(\()p Fw(z)2465 611 y Fo(\000)2461 676 y Ft(i)2525 649 y Fw(;)15 b(z)2611 611 y Fr(+)2607 676 y Ft(i)p Fr(+1)2726 649 y Fy(\))919 849 y(+)20 b Fw(h)1062 788 y Fr(\001)p Ft(t)1146 758 y Fl(\000)1146 810 y Fm(i)p Fk(+1)1062 860 y Ft(c)1093 870 y Fm(i)p Fk(+1)1254 849 y Fy(\()p Fw(\031)s(z)1390 810 y Fr(+)1386 876 y Ft(i)p Fr(+1)1505 849 y Fw(;)15 b(G)1616 810 y Fo(\000)p Fr(1)1616 876 y Ft(i)1712 849 y Fw(G)1783 863 y Ft(i)p Fr(+1)1901 849 y Fw(\031)s(z)2002 810 y Fo(\000)1998 876 y Ft(i)p Fr(+1)2117 849 y Fy(\))828 1035 y Fv(\025)p Fw(h)951 998 y Fo(1)951 1058 y Ft(c)982 1068 y Fm(i)1026 1035 y Fy(\()p Fw(\030)t(;)g(\031)s(z)1246 997 y Fo(\000)1242 1063 y Ft(i)1306 1035 y Fy(\))21 b(+)f Fw(h)1505 982 y Fr(\001)p Ft(t)1589 952 y Fk(+)1589 1004 y Fm(i)1505 1065 y Ft(\021)1540 1075 y Fm(i)1567 1065 y Ft(;\026)1629 1075 y Fm(i)1655 1065 y Ft(; )1721 1075 y Fm(i)1747 1065 y Ft(;e)1800 1073 y Fm(n)1846 1035 y Fy(\()p Fw(z)1927 997 y Fo(\000)1923 1063 y Ft(i)1987 1035 y Fw(;)15 b(z)2073 997 y Fr(+)2069 1063 y Ft(i)p Fr(+1)2188 1035 y Fy(\))21 b(+)e Fw(h)2386 998 y Fo(1)2386 1058 y Ft(c)2417 1068 y Fm(i)p Fk(+1)2525 1035 y Fy(\()p Fw(\031)s(z)2661 997 y Fr(+)2657 1063 y Ft(i)p Fr(+1)2776 1035 y Fw(;)c(\020)7 b Fy(\))919 1194 y(+)20 b Fw(h)1062 1156 y Fo(1)1062 1216 y Ft(c)1093 1226 y Fm(i)1137 1194 y Fy(\()p Fw(G)1243 1156 y Fo(\000)p Fr(1)1243 1221 y Ft(i)1338 1194 y Fw(G)1409 1208 y Ft(i)p Fo(\000)p Fr(1)1528 1194 y Fw(\031)s(z)1629 1156 y Fr(+)1625 1221 y Ft(i)1689 1194 y Fw(;)15 b(\030)t Fy(\))21 b(+)e Fw(h)1971 1156 y Fo(1)1971 1216 y Ft(c)2002 1226 y Fm(i)p Fk(+1)2110 1194 y Fy(\()p Fw(\020)7 b(;)15 b(G)2303 1156 y Fo(\000)p Fr(1)2303 1221 y Ft(i)2398 1194 y Fw(G)2469 1208 y Ft(i)p Fr(+1)2588 1194 y Fw(\031)s(z)2689 1156 y Fo(\000)2685 1221 y Ft(i)p Fr(+1)2803 1194 y Fy(\))21 b Fv(\000)f Fy(2)p Fw(\017)3032 1156 y Fo(\003)3032 1216 y Ft(i)828 1384 y Fv(\025)p Fw(h)951 1346 y Fo(1)951 1406 y Ft(c)982 1416 y Fm(i)1026 1384 y Fy(\()p Fw(\030)t(;)15 b(\031)9 b Fy(\026)-51 b Fw(x)1252 1346 y Fo(\000)1252 1411 y Ft(i)1312 1384 y Fy(\))20 b(+)g Fw(h)1510 1330 y Ft(T)1561 1301 y Fk(+)1551 1353 y Fm(i)1510 1413 y Ft(\021)1545 1423 y Fm(i)1572 1413 y Ft(;\026)1634 1423 y Fm(i)1661 1413 y Ft(; )1727 1423 y Fm(i)1753 1413 y Ft(;e)1806 1421 y Fm(n)1852 1384 y Fy(\()6 b(\026)-51 b Fw(x)1939 1346 y Fo(\000)1939 1411 y Ft(i)1998 1384 y Fw(;)21 b Fy(\026)-51 b Fw(x)2090 1346 y Fr(+)2090 1411 y Ft(i)p Fr(+1)2209 1384 y Fy(\))21 b(+)e Fw(h)2407 1346 y Fo(1)2407 1406 y Ft(c)2438 1416 y Fm(i)p Fk(+1)2546 1384 y Fy(\()p Fw(\031)9 b Fy(\026)-51 b Fw(x)2688 1346 y Fr(+)2688 1411 y Ft(i)p Fr(+1)2807 1384 y Fw(;)15 b(\020)7 b Fy(\))919 1542 y(+)20 b Fw(h)1062 1505 y Fo(1)1062 1565 y Ft(c)1093 1575 y Fm(i)1137 1542 y Fy(\()p Fw(G)1243 1504 y Fo(\000)p Fr(1)1243 1570 y Ft(i)1338 1542 y Fw(G)1409 1556 y Ft(i)p Fo(\000)p Fr(1)1528 1542 y Fw(\031)s(z)1629 1504 y Fr(+)1625 1570 y Ft(i)1689 1542 y Fw(;)15 b(\030)t Fy(\))21 b(+)e Fw(h)1971 1505 y Fo(1)1971 1565 y Ft(c)2002 1575 y Fm(i)p Fk(+1)2110 1542 y Fy(\()p Fw(\020)7 b(;)15 b(G)2303 1504 y Fo(\000)p Fr(1)2303 1570 y Ft(i)2398 1542 y Fw(G)2469 1556 y Ft(i)p Fr(+1)2588 1542 y Fw(\031)s(z)2689 1504 y Fo(\000)2685 1570 y Ft(i)p Fr(+1)2803 1542 y Fy(\))21 b(+)f(3)p Fw(\017)3032 1505 y Fo(\003)3032 1565 y Ft(i)828 1732 y Fv(\025)p Fw(h)951 1679 y Fr(\001)p Ft(t)1035 1649 y Fl(\000)1035 1701 y Fm(i)951 1744 y Ft(c)982 1754 y Fm(i)1092 1732 y Fy(\()p Fw(G)1198 1694 y Fo(\000)p Fr(1)1198 1759 y Ft(i)1293 1732 y Fw(G)1364 1746 y Ft(i)p Fo(\000)p Fr(1)1483 1732 y Fw(\031)s(z)1584 1694 y Fr(+)1580 1759 y Ft(i)1643 1732 y Fw(;)15 b(\031)9 b Fy(\026)-51 b Fw(x)1790 1694 y Fo(\000)1790 1759 y Ft(i)1850 1732 y Fy(\))20 b(+)g Fw(h)2048 1679 y Ft(T)2099 1649 y Fk(+)2089 1701 y Fm(i)2048 1762 y Ft(\021)2083 1772 y Fm(i)2110 1762 y Ft(;\026)2172 1772 y Fm(i)2199 1762 y Ft(; )2265 1772 y Fm(i)2291 1762 y Ft(;e)2344 1770 y Fm(n)2390 1732 y Fy(\()6 b(\026)-51 b Fw(x)2477 1694 y Fo(\000)2477 1759 y Ft(i)2536 1732 y Fw(;)21 b Fy(\026)-51 b Fw(x)2628 1694 y Fr(+)2628 1759 y Ft(i)p Fr(+1)2747 1732 y Fy(\))919 1932 y(+)20 b Fw(h)1062 1871 y Fr(\001)p Ft(t)1146 1842 y Fl(\000)1146 1893 y Fm(i)p Fk(+1)1062 1944 y Ft(c)1093 1954 y Fm(i)p Fk(+1)1254 1932 y Fy(\()p Fw(\031)9 b Fy(\026)-51 b Fw(x)1396 1893 y Fr(+)1396 1959 y Ft(i)p Fr(+1)1515 1932 y Fw(;)15 b(G)1626 1893 y Fo(\000)p Fr(1)1626 1959 y Ft(i)1721 1932 y Fw(G)1792 1946 y Ft(i)p Fr(+1)1911 1932 y Fw(\031)s(z)2012 1893 y Fo(\000)2008 1959 y Ft(i)p Fr(+1)2127 1932 y Fy(\))20 b(+)g Fw(\017)2310 1894 y Fo(\003)2310 1954 y Ft(i)456 2103 y Fy(where)38 b(\026)-51 b Fw(x)773 2065 y Fo(\000)773 2131 y Ft(i)865 2103 y Fy(and)39 b(\026)-51 b Fw(x)1097 2065 y Fr(+)1097 2131 y Ft(i)p Fr(+1)1248 2103 y Fy(are)34 b(the)f(in)m(tersection)i(p)s(oin) m(t)e(of)h(a)f(curv)m(e)h(in)f Fg(C)2894 2118 y Ft(\021)2929 2128 y Fm(i)2955 2118 y Ft(;\026)3017 2128 y Fm(i)3043 2118 y Ft(; )3109 2128 y Fm(i)3136 2118 y Ft(;e)3189 2126 y Fm(n)3268 2103 y Fy(with)456 2224 y Fw(V)529 2186 y Fo(\000)509 2251 y Ft(i)619 2224 y Fy(and)e(with)h Fw(V)1079 2186 y Fr(+)1059 2251 y Ft(i)p Fr(+1)1208 2224 y Fy(resp)s(ectiv)m(ely)-8 b(,)34 b Fw(\030)d Fv(2)c(M)p Fy(\()p Fw(c)2065 2238 y Ft(i)p Fo(\000)p Fr(1)2184 2224 y Fy(\))32 b(and)f Fw(\020)j Fv(2)27 b(M)p Fy(\()p Fw(c)2774 2238 y Ft(i)2803 2224 y Fy(\).)44 b(It)32 b(con)m(tradicts)456 2332 y(the)e(minimalit)m(y)i(of)e Fw(\015)5 b Fy(,)31 b(th)m(us)f(v)m(eri\014es)h(our)f(claim.)555 2523 y(T)-8 b(o)30 b(see)g(the)f(curv)m(e)35 b(\026)-50 b Fw(\031)s(G)1357 2485 y Fo(\000)p Fr(1)1357 2550 y Ft(i)1455 2523 y Fy(~)i Fw(\015)23 b Fv(\000)17 b Fy(\(2)p Fw(k)1737 2537 y Ft(i)1767 2523 y Fw(\031)s(;)e Fy(0)p Fw(;)g Fv(\001)g(\001)g(\001)32 b Fw(;)15 b Fy(0\))31 b(is)e(smo)s(oth)g(at)h Fw(z)2801 2485 y Fo(\000)2797 2550 y Ft(i)2860 2523 y Fy(,)g(let)g(us)f(assume) 456 2640 y(the)36 b(con)m(trary)h(again.)59 b(Let)37 b Fw(z)1498 2607 y Fo(0)1557 2640 y Fy(=)i(\026)-49 b Fw(\031)s(G)1789 2601 y Fo(\000)p Fr(1)1789 2667 y Ft(i)1887 2640 y Fy(~)g Fw(\015)1935 2601 y Fo(\000)1930 2667 y Ft(i)1995 2640 y Fy(\()p Fw(t)2063 2601 y Fo(\000)2063 2667 y Ft(i)2146 2640 y Fv(\000)24 b Fw(\016)s Fy(\))h Fv(\000)f Fy(\(2)p Fw(k)2566 2654 y Ft(i)2595 2640 y Fw(\031)s(;)15 b Fy(0)p Fw(;)g Fv(\001)g(\001)g(\001)33 b Fw(;)15 b Fy(0\))37 b(and)f Fw(z)3299 2607 y Fo(\003)3374 2640 y Fy(=)460 2756 y(\026)-49 b Fw(\031)s(G)582 2718 y Fo(\000)p Fr(1)582 2783 y Ft(i)680 2756 y Fy(~)g Fw(\015)728 2718 y Fr(+)723 2783 y Ft(i)788 2756 y Fy(\()p Fw(t)856 2718 y Fo(\000)856 2783 y Ft(i)932 2756 y Fy(+)17 b Fw(\016)s Fy(\))h Fv(\000)f Fy(\(2)p Fw(k)1331 2770 y Ft(i)1360 2756 y Fw(\031)s(;)e Fy(0)p Fw(;)g Fv(\001)g(\001)g(\001)33 b Fw(;)15 b Fy(0\),)30 b(here)f Fw(\016)j Fy(is)d(c)m(hosen)h(so)f (small)g(that)g(\001)p Fw(t)3151 2718 y Fr(+)3151 2783 y Ft(i)3227 2756 y Fv(\006)17 b Fw(\016)29 b Fv(2)456 2870 y Fy([)p Fw(T)547 2832 y Fr(+)534 2897 y Ft(i)628 2870 y Fv(\000)23 b Fw(d)769 2884 y Ft(i)797 2870 y Fw(;)15 b(T)903 2832 y Fr(+)890 2897 y Ft(i)985 2870 y Fy(+)22 b Fw(d)1125 2884 y Ft(i)1154 2870 y Fy(].)52 b(This)33 b(is)h(p)s(ossible)g(since)g(\()p Fw(z)2215 2832 y Fo(\000)2211 2897 y Ft(i)2275 2870 y Fw(;)15 b(z)2361 2832 y Fr(+)2357 2897 y Ft(i)p Fr(+1)2476 2870 y Fw(;)g Fy(\001)p Fw(t)2625 2832 y Fr(+)2625 2897 y Ft(i)2684 2870 y Fy(\))32 b Fv(62)f Fw(@)5 b Fy(\()p Fw(V)3005 2832 y Fo(\000)2984 2897 y Ft(i)3086 2870 y Fv(\002)23 b Fw(V)3253 2832 y Fr(+)3233 2897 y Ft(i)p Fr(+1)3374 2870 y Fv(\002)456 2991 y Fy([)p Fw(T)547 2952 y Fr(+)534 3018 y Ft(i)628 2991 y Fv(\000)f Fw(d)768 3005 y Ft(i)797 2991 y Fw(;)15 b(T)903 2952 y Fr(+)890 3018 y Ft(i)984 2991 y Fy(+)22 b Fw(d)1124 3005 y Ft(i)1153 2991 y Fy(]\))34 b(implies)g(that)g Fw(T)1825 2952 y Fr(+)1812 3018 y Ft(i)1907 2991 y Fv(\000)22 b Fw(d)2047 3005 y Ft(i)2106 2991 y Fw(<)31 b Fy(\001)p Fw(t)2317 2952 y Fr(+)2317 3018 y Ft(i)2406 2991 y Fw(<)g(T)2574 2952 y Fr(+)2561 3018 y Ft(i)2655 2991 y Fy(+)22 b Fw(d)2795 3005 y Ft(i)2824 2991 y Fy(.)50 b(W)-8 b(e)35 b(join)f(these)456 3098 y(t)m(w)m(o)d(p)s(oin)m(ts)g(b)m(y)f(a)h(minimizer)f Fw(\030)f Fy(:)d([)p Fv(\000)p Fw(\016)n(;)15 b(\016)s Fy(])27 b Fv(!)e Fw(M)41 b Fy(with)30 b Fw(\030)t Fy(\()p Fv(\000)p Fw(\016)s Fy(\))c(=)f Fw(z)2764 3065 y Fo(0)2818 3098 y Fy(and)30 b Fw(\030)t Fy(\()p Fw(\016)s Fy(\))d(=)e Fw(z)3321 3065 y Fo(\003)1077 3322 y Fy([)p Fw(A)1170 3336 y Ft(c)1201 3346 y Fm(i)1231 3322 y Fy(\()p Fw(\030)t Fv(j)1335 3340 y Fr([)p Fo(\000)p Ft(\016)o(;\016)r Fr(])1517 3322 y Fy(\)])h(=)99 b(inf)1699 3387 y Fm(\020)s Fk(\()p Fl(\000)p Fm(\016)r Fk(\)=)p Fm(z)1935 3373 y Fl(0)1716 3451 y Fm(\020)s Fk(\()p Fm(\016)r Fk(\)=)p Fm(z)1904 3430 y Fl(\003)1972 3198 y Fq(Z)2063 3224 y Ft(\016)2023 3404 y Fo(\000)p Ft(\016)2116 3322 y Fy(\()p Fw(L)20 b Fv(\000)g Fw(\021)2369 3336 y Ft(i)2397 3322 y Fy(\)\()p Fw(d\030)t Fy(\()p Fw(s)p Fy(\)\))p Fw(ds:)456 3608 y Fy(If)35 b Fw(\030)40 b Fy(passes)c(through)f Fw(V)1334 3569 y Fo(\000)1314 3635 y Ft(i)1394 3608 y Fy(,)i(w)m(e)g(obtain)f(a)g (curv)m(e)h Fw(\015)2266 3575 y Fo(0)2325 3608 y Fy(b)m(y)f(replacing)h (the)f(segmen)m(t)h(of)456 3721 y(the)32 b(minimizer)g Fw(\015)5 b Fv(j)1117 3752 y Fr([)p Ft(t)1162 3722 y Fl(\000)1162 3774 y Fm(i)1215 3752 y Fo(\000)p Ft(\016)o(;t)1345 3722 y Fl(\000)1345 3774 y Fm(i)1397 3752 y Fr(+)p Ft(\016)r Fr(])1541 3721 y Fy(with)32 b Fw(G)1821 3735 y Ft(i)1849 3721 y Fw(\030)h Fy(:)28 b([)p Fv(\000)p Fw(\016)n(;)15 b(\016)s Fy(])30 b Fv(!)e Fw(M)10 b Fy(.)46 b(Let)33 b Fw(t)2733 3683 y Fo(0\000)2733 3749 y Ft(i)2843 3721 y Fy(b)s(e)e(the)i(time)g(for)456 3861 y Fw(\015)508 3828 y Fo(0)570 3861 y Fy(passing)39 b(through)f Fw(G)1321 3875 y Ft(i)1349 3861 y Fw(\031)s(V)1478 3823 y Fo(\000)1457 3888 y Ft(i)1537 3861 y Fy(,)j(then)1828 3825 y Fr(1)p 1828 3840 36 4 v 1828 3892 a(2)1874 3861 y Fy(\()p Fw(t)1942 3823 y Fr(+)1942 3888 y Ft(i)p Fr(+1)2086 3861 y Fv(\000)26 b Fw(t)2216 3823 y Fo(0\000)2216 3888 y Ft(i)2294 3861 y Fy(\))39 b Fv(2)g Fy(\001)p Fw(T)2610 3823 y Fr(+)2597 3888 y Ft(i)2708 3861 y Fy(and)2903 3825 y Fr(1)p 2903 3840 V 2903 3892 a(2)2948 3861 y Fy(\()p Fw(t)3016 3823 y Fo(0\000)3016 3888 y Ft(i)3121 3861 y Fv(\000)25 b Fw(t)3250 3823 y Fr(+)3250 3888 y Ft(i)3309 3861 y Fy(\))40 b Fv(2)456 3982 y Fy(\001)p Fw(T)598 3943 y Fo(\000)585 4009 y Ft(i)656 3982 y Fy(.)k(Consequen)m(tly)-8 b(,)33 b(w)m(e)f(w)m(ould)f(get)h(an)g(absolutely)g(con)m(tin)m(uous)g(curv)m (e)g(whic)m(h)f(is)456 4089 y(admissible)k(for)g(eac)m(h)h(condition)g (\(see)g(\(3.13\))h(and)e(\(3.15\)\).)57 b(Along)36 b(this)f(curv)m(e)h (w)m(e)456 4216 y(w)m(ould)24 b(get)i(ev)m(en)f(smaller)g Fw(h)1422 4172 y Ft(K)q(;K)1568 4148 y Fl(0)1422 4244 y Ft(L)1593 4216 y Fy(\()p Fw(m;)15 b(m)1828 4183 y Fo(0)1852 4216 y Fy(\),)26 b(but)e(this)h(is)f(absurd.)37 b(So,)26 b(w)m(e)f(only)g(need)g(to)456 4331 y(sho)m(w)g Fw(\030)30 b Fy(passes)c(through)f Fw(G)1426 4345 y Ft(i)1455 4331 y Fw(V)1528 4293 y Fo(\000)1508 4358 y Ft(i)1587 4331 y Fy(.)40 b(Indeed,)26 b(as)g Fw(V)2149 4293 y Fo(\000)2129 4358 y Ft(i)2234 4331 y Fy(is)g(c)m(hosen)h(small)f(and)g(transv)m (ersal)456 4445 y(to)45 b(the)f(lo)s(cal)h(connecting)h(curv)m(e)e(in)g Fg(C)1885 4460 y Ft(\021)1920 4470 y Fm(i)1946 4460 y Ft(;\026)2008 4470 y Fm(i)2035 4460 y Ft(; )2101 4470 y Fm(i)2127 4460 y Ft(;e)2180 4468 y Fm(n)2226 4445 y Fy(,)k Fw(\015)2351 4407 y Fo(\000)2346 4472 y Ft(i)2410 4445 y Fy(\()p Fw(t)p Fy(\))c(and)g Fw(\015)2800 4407 y Fr(+)2795 4472 y Ft(i)2859 4445 y Fy(\()p Fw(t)p Fy(\))h(approac)m (hes)456 4559 y Fw(G)527 4573 y Ft(i)555 4559 y Fw(V)628 4521 y Fo(\000)608 4586 y Ft(i)724 4559 y Fy(from)36 b(di\013eren)m(t)g(sides)h(as)f Fw(t)f Fv(&)g Fw(t)1878 4521 y Fo(\000)1878 4586 y Ft(i)1973 4559 y Fy(and)h Fw(t)f Fv(\045)g Fw(t)2383 4521 y Fo(\000)2383 4586 y Ft(i)2478 4559 y Fy(resp)s(ectiv)m(ely)-8 b(.)60 b(Otherwise,)456 4667 y(the)32 b(minimalit)m(y)h(of)f Fw(\015)37 b Fy(will)32 b(b)s(e)f(violated.)47 b(One)32 b(refers)f(to)i([BCV])f(for)g(the)g (details)h(of)456 4777 y(argumen)m(t.)41 b(The)30 b(smo)s(othness)f(at) j Fw(G)1751 4791 y Ft(i)1779 4777 y Fw(z)1825 4738 y Fr(+)1821 4804 y Ft(i)p Fr(+1)1970 4777 y Fy(can)f(b)s(e)f(pro)m(v)m (ed)g(similarly)-8 b(.)555 4980 y(T)g(o)26 b(sho)m(w)f(the)h(smo)s (othness)f(of)h Fw(\015)k Fy(at)c Fw(t)f Fy(=)g Fw(t)2004 4942 y Fo(\006)2004 5007 y Ft(i)2089 4980 y Fy(for)g Fw(i)g Fv(2)2365 4912 y Fq(S)2441 5007 y Ft(i)2465 5017 y Fm(j)2498 5007 y Fo(2)p Fr(\003)2594 5015 y Fm(c)2630 4980 y Fv(f)p Fw(i)2706 4994 y Ft(j)2743 4980 y Fw(;)15 b(i)2814 4994 y Ft(j)2862 4980 y Fy(+)10 b(1)p Fw(;)15 b Fv(\001)g(\001)g(\001)32 b Fw(;)15 b(i)3236 4994 y Ft(j)t Fr(+1)3374 4980 y Fv(\000)456 5104 y Fy(1)p Fv(g)p Fy(,)35 b(let)f(us)f(consider)g(\014rst)f(the)i(case)g(that)g Fw(c)2001 5118 y Ft(i)p Fo(\000)p Fr(1)2153 5104 y Fy(is)f(connected)i (to)f Fw(c)2828 5118 y Ft(i)2889 5104 y Fy(also)h(b)m(y)e(an)g(or-)456 5216 y(bit)k(of)g Fw(c)p Fy(-t)m(yp)s(e.)62 b(In)37 b(the)g(co)s (ordinate)i(system)e Fw(q)j Fy(=)c Fw(G)2350 5177 y Fo(\000)p Fr(1)2350 5243 y Ft(i)2445 5216 y Fw(x)h Fy(w)m(e)h(consider)f(a)g(co)m (v)m(ering)p eop end %%Page: 24 24 TeXDict begin 24 23 bop 456 251 a Fr(24)1012 b(X.)23 b(LI)h(AND)f(C.-Q.)f(CHENG)456 450 y Fy(space)743 427 y(\026)709 450 y Fw(M)53 b Fy(=)43 b Fu(R)27 b Fv(\002)g Fu(T)1216 417 y Ft(n)p Fo(\000)p Fr(1)1352 450 y Fy(.)73 b(In)40 b Fw(q)46 b Fy(=)c Fw(G)1845 412 y Fo(\000)p Fr(1)1845 477 y Ft(i)1940 450 y Fw(x)p Fy(-co)s(ordinates,)j(b)s(oth)40 b Fw(G)2840 412 y Fo(\000)p Fr(1)2840 477 y Ft(i)2935 450 y Fw(G)3006 464 y Ft(i)p Fo(\000)p Fr(1)3125 450 y Fw(V)3198 412 y Fr(+)3178 477 y Ft(i)3298 450 y Fy(and)456 567 y Fw(G)527 528 y Fo(\000)p Fr(1)527 594 y Ft(i)621 567 y Fw(G)692 581 y Ft(i)p Fr(+1)811 567 y Fw(V)884 528 y Fo(\000)864 594 y Ft(i)p Fr(+1)1014 567 y Fy(are)31 b(co)s(dimension-1)g(sub-manifold)f(whic)m(h)h(ma)m(y)h(extend)f(to)h Fv(\0001)e Fy(as)456 690 y(w)m(ell)24 b(as)g Fv(1)f Fy(in)g Fw(q)992 704 y Fr(1)1032 690 y Fy(-direction.)39 b(By)24 b(construction)g(ho)m(w)m(ev)m(er,)31 b(\026)-50 b Fw(\031)s(G)2618 652 y Fo(\000)p Fr(1)2618 717 y Ft(i)2717 690 y Fy(~)h Fw(\015)12 b Fv(\000)7 b Fy(\(2)p Fw(k)2977 704 y Ft(i)3005 690 y Fw(\031)s(;)15 b Fy(0)p Fw(;)g Fv(\001)g(\001)g(\001)33 b Fw(;)15 b Fy(0\))456 804 y(passes)26 b(through)g Fw(V)1139 766 y Fo(\000)1119 831 y Ft(i;)p Fr(+)1222 804 y Fy(,)h Fw(V)1347 766 y Fo(\000)1327 831 y Ft(i)1407 804 y Fy(,)g Fw(V)1532 766 y Fr(+)1512 831 y Ft(i)p Fr(+1)1657 804 y Fy(and)f Fw(V)1903 766 y Fr(+)1883 831 y Ft(i)p Fr(+1)p Ft(;)p 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1283 y Ft(i)999 1256 y Fy(\))21 b Fv(\000)f Fy(\(2)p Fw(k)1273 1270 y Ft(i)1302 1256 y Fw(\031)s(;)15 b Fy(0)p Fw(;)g Fv(\001)g(\001)g(\001)33 b Fw(;)15 b Fy(0\))26 b Fv(2)f(f)p Fw(q)1938 1270 y Fr(1)2003 1256 y Fy(=)g Fv(\000)p Fw(K)2247 1270 y Ft(i)2275 1256 y Fv(g)p Fw(;)604 1409 y Fy(\026)-50 b Fw(\031)s(G)725 1371 y Fo(\000)p Fr(1)725 1436 y Ft(i)823 1409 y Fy(~)i Fw(\015)5 b Fy(\()p Fw(t)940 1371 y Fr(+)940 1436 y Ft(i)p Fr(+1)1059 1409 y Fy(\))20 b Fv(\000)g Fy(\(2)p Fw(k)1332 1423 y Ft(i)1361 1409 y Fw(\031)s(;)15 b Fy(0)p Fw(;)g Fv(\001)g(\001)g(\001)33 b Fw(;)15 b Fy(0\))27 b Fv(2)d(f)p Fw(q)1997 1423 y Fr(1)2062 1409 y Fy(=)h Fw(K)2235 1423 y Ft(i)2263 1409 y Fv(g)p Fw(;)604 1562 y Fy(\026)-50 b Fw(\031)s(G)725 1524 y Fo(\000)p Fr(1)725 1589 y Ft(i)823 1562 y Fy(~)i Fw(\015)5 b Fy(\()p Fw(t)940 1524 y Fo(\000)940 1589 y Ft(i)p Fr(+1)1059 1562 y Fy(\))20 b Fv(\000)g Fy(\(2)p Fw(k)1332 1576 y Ft(i)1361 1562 y Fw(\031)s(;)15 b Fy(0)p Fw(;)g Fv(\001)g(\001)g(\001) 33 b Fw(;)15 b Fy(0\))27 b Fv(2)d(f)p Fw(K)2033 1576 y Ft(i)2082 1562 y Fy(+)c(2)p Fw(\031)29 b Fv(\024)c Fw(q)2436 1576 y Fr(1)2500 1562 y Fv(\024)g Fw(K)2673 1576 y Ft(i)2722 1562 y Fy(+)20 b(2)p Fw(\031)k Fy(+)19 b Fw(q)3068 1524 y Fo(\003)3065 1584 y Ft(i;)p Fr(1)3148 1562 y Fv(g)p Fw(:)456 1748 y Fy(If)38 b Fw(\015)43 b Fy(had)38 b(corner)g(at)h Fw(\015)5 b Fy(\()p Fw(t)1355 1710 y Fo(\000)1355 1775 y Ft(i)1415 1748 y Fy(\),)41 b(w)m(e)e(w)m(ould)f(get)h(a)g(minimizer)g Fw(\030)t Fy(:)56 b([)p Fw(t)2790 1710 y Fr(+)2790 1775 y Ft(i)2850 1748 y Fw(;)15 b(t)2923 1710 y Fr(+)2923 1775 y Ft(i)p Fr(+1)3041 1748 y Fy(])39 b Fv(!)3269 1725 y Fy(\026)3235 1748 y Fw(M)48 b Fy(of)456 1872 y Fw(L)29 b Fv(\000)h Fw(\021)693 1886 y Ft(i)751 1872 y Fv(\000)g Fw(k)902 1839 y Fo(\003)899 1897 y Ft(i)942 1872 y Fw(\026)997 1886 y Ft(i)1070 1872 y Fy(in)1224 1849 y(\026)1190 1872 y Fw(M)56 b Fy(connecting)46 b(from)j(\026)-50 b Fw(\031)s(G)2158 1833 y Fo(\000)p Fr(1)2158 1899 y Ft(i)2257 1872 y Fy(~)h Fw(\015)5 b Fy(\()p Fw(t)2373 1833 y Fr(+)2373 1899 y Ft(i)2433 1872 y Fy(\))45 b(to)50 b(\026)-49 b Fw(\031)s(G)2765 1833 y Fo(\000)p Fr(1)2765 1899 y Ft(i)2863 1872 y Fy(~)h Fw(\015)5 b Fy(\()p Fw(t)2980 1833 y Fr(+)2980 1899 y Ft(i)p Fr(+1)3098 1872 y Fy(\).)85 b(As)45 b Fw(\015)456 1992 y Fy(is)e(assumed)g(not)h(smo)s(oth)f(at)i Fw(\015)5 b Fy(\()p Fw(t)1692 1954 y Fo(\000)1692 2019 y Ft(i)1751 1992 y Fy(\),)48 b(the)43 b(action)i(of)f Fw(L)29 b Fv(\000)g Fw(\021)2667 2006 y Ft(i)2724 1992 y Fv(\000)g Fw(k)2874 1959 y Fo(\003)2871 2018 y Ft(i)2914 1992 y Fw(\026)2969 2006 y Ft(i)3040 1992 y Fy(along)45 b Fw(\030)i Fy(is)456 2109 y(smaller)30 b(than)f(the)h(action)h(along)36 b(\026)-50 b Fw(\031)s(G)1776 2071 y Fo(\000)p Fr(1)1776 2136 y Ft(i)1874 2109 y Fy(~)i Fw(\015)5 b Fy(\()p Fw(t)1991 2071 y Fr(+)1991 2136 y Ft(i)2050 2109 y Fy(\))p Fv(j)2110 2139 y Fr([)p Ft(t)2155 2109 y Fk(+)2155 2161 y Fm(i)2207 2139 y Ft(;t)2252 2109 y Fk(+)2252 2161 y Fm(i)p Fk(+1)2355 2139 y Fr(])2379 2109 y Fy(.)41 b(Replacing)30 b(the)g(segmen)m(t)h(of) 460 2257 y(\026)-49 b Fw(\031)s(G)582 2219 y Fo(\000)p Fr(1)582 2285 y Ft(i)680 2257 y Fy(~)g Fw(\015)6 b Fv(j)754 2288 y Fr([)p Ft(t)799 2258 y Fk(+)799 2310 y Fm(i)850 2288 y Ft(;t)895 2258 y Fk(+)895 2310 y Fm(i)p Fk(+1)998 2288 y Fr(])1047 2257 y Fy(b)m(y)25 b Fw(\030)t Fy(,)h(w)m(e)g(w)m (ould)f(obtain)g(a)h(curv)m(e)f Fw(\015)2289 2224 y Fo(0)2337 2257 y Fy(with)g(smaller)h(action.)40 b(Indeed,)456 2403 y(let)d Fw(t)626 2365 y Fo(0\000)626 2431 y Ft(i)741 2403 y Fy(b)s(e)f(the)i(time)f(when)f Fw(\030)41 b Fy(passes)36 b(through)h Fv(f)p Fw(q)2290 2417 y Fr(1)2365 2403 y Fy(=)f Fv(\000)p Fw(K)2620 2417 y Ft(i)2648 2403 y Fv(g)p Fy(,)j(w)m(e)f(can)f(see)g(easily)456 2517 y(from)h(the)g(de\014nition) g(of)h(\001)p Fw(T)1509 2479 y Fo(\006)1496 2544 y Ft(i)1606 2517 y Fy(that)g Fw(t)1844 2479 y Fo(0\000)1844 2544 y Ft(i)1947 2517 y Fv(\000)25 b Fw(t)2076 2479 y Fr(+)2076 2544 y Ft(i)2174 2517 y Fv(2)38 b Fy(\001)p Fw(T)2415 2479 y Fo(\000)2402 2544 y Ft(i)2512 2517 y Fy(and)g Fw(t)2730 2479 y Fr(+)2730 2544 y Ft(i)p Fr(+1)2873 2517 y Fv(\000)26 b Fw(t)3003 2479 y Fo(0\000)3003 2544 y Ft(i)3119 2517 y Fv(2)39 b Fy(\001)p Fw(T)3361 2479 y Fr(+)3348 2544 y Ft(i)3419 2517 y Fy(.)456 2638 y(This)33 b(absurdit)m(y)i(implies)f(the)h(smo)s(othness)f(of)h Fw(\015)40 b Fy(at)35 b Fw(\015)5 b Fy(\()p Fw(t)2468 2600 y Fo(\000)2468 2665 y Ft(i)2528 2638 y Fy(\).)53 b(In)34 b(the)h(same)g(w)m(a)m(y)h(w)m(e)456 2752 y(can)30 b(sho)m(w)h(the)f(smo)s(othness)g(of)g Fw(\015)36 b Fy(at)31 b Fw(\015)5 b Fy(\()p Fw(t)1905 2713 y Fr(+)1905 2779 y Ft(i)p Fr(+1)2024 2752 y Fy(\).)555 2943 y(If)35 b Fw(c)690 2957 y Ft(i)p Fo(\000)p Fr(1)844 2943 y Fy(is)g(connected)h (to)g Fw(c)1524 2957 y Ft(i)1587 2943 y Fy(b)m(y)f(an)g(orbit)g(of)h Fw(h)p Fy(-t)m(yp)s(e,)h(our)e(assumptions)f(ensure)456 3051 y(the)f(uniqueness)f(of)h Fw(c)1221 3065 y Ft(i)p Fo(\000)p Fr(1)1340 3051 y Fy(-minimal)g(measure.)49 b(One)33 b(obtains)g(the)g(smo)s(othness)g(of)k(~)-49 b Fw(\015)456 3160 y Fy(at)39 b Fw(G)646 3122 y Fo(\000)p Fr(1)646 3188 y Ft(i)p Fo(\000)p Fr(1)765 3160 y Fw(V)838 3122 y Fr(+)818 3188 y Ft(i)936 3160 y Fy(from)g(the)g(argumen)m(ts)g (for)g Fw(h)p Fy(-t)m(yp)s(e)g(and)g(the)g(smo)s(othness)f(at)i Fw(G)3253 3122 y Fo(\000)3253 3188 y Ft(i)3312 3160 y Fw(V)3385 3122 y Fo(\000)3365 3188 y Ft(i)456 3268 y Fy(from)30 b(the)g(argumen)m(ts)h(for)f Fw(c)p Fy(-t)m(yp)s(e.)555 3459 y(As)h(the)f(system)h(is)f(autonomous,)h(the)f(follo)m(wing)i (limit)g(exists)1246 3644 y Fw(h)1298 3606 y Fo(1)1298 3666 y Ft(L)1373 3644 y Fy(\()p Fw(m;)15 b(m)1608 3606 y Fo(0)1632 3644 y Fy(\))26 b(=)115 b(lim)1789 3706 y Ft(K)q(;K)1935 3687 y Fl(0)1955 3706 y Fo(!1)2111 3644 y Fw(h)2163 3600 y Ft(K)q(;K)2309 3576 y Fl(0)2163 3673 y Ft(L)2334 3644 y Fy(\()p Fw(m;)15 b(m)2569 3606 y Fo(0)2593 3644 y Fy(\))p Fw(:)456 3869 y Fy(W)-8 b(e)28 b(pic)m(k)g(out)g(a)g (sequence)g(of)f Fw(\015)5 b Fy(\()p Fw(t)p Fy(;)15 b Fw(K)q(;)g(K)1866 3836 y Fo(0)1891 3869 y Fw(;)g(m;)g(m)2131 3836 y Fo(0)2155 3869 y Fy(\))27 b(for)h(large)g Fw(K)34 b Fy(and)27 b Fw(K)2942 3836 y Fo(0)2965 3869 y Fy(.)40 b(Ob)m(viously)-8 b(,)456 3977 y(the)24 b(set)h Fv(f)p Fw(\015)5 b Fy(\()p Fw(t)p Fy(;)15 b Fw(K)q(;)g(K)1149 3944 y Fo(0)1174 3977 y Fw(;)g(m;)g(m)1414 3944 y Fo(0)1438 3977 y Fy(\))p Fv(g)25 b Fy(has)f(at)h(least)h(one)e(accum)m(ulation)i (p)s(oin)m(t)f Fw(\015)3002 3991 y Fo(1)3077 3977 y Fy(:)37 b Fu(R)25 b Fv(!)g Fw(M)456 4092 y Fy(with)30 b(the)h(prop)s(ert)m(y)f Fw(\013)p Fy(\()p Fw(d\015)1379 4106 y Fo(1)1454 4092 y Fy(\))c Fv(\022)1643 4069 y Fy(~)1612 4092 y Fv(A)o Fy(\()p Fw(c)p Fy(\))32 b(and)e Fw(!)s Fy(\()p Fw(d\015)2191 4106 y Fo(1)2266 4092 y Fy(\))c Fv(\022)2454 4069 y Fy(~)2423 4092 y Fv(A)p Fy(\()p Fw(c)2570 4059 y Fo(0)2594 4092 y Fy(\).)41 b(As)31 b(w)m(e)g(ha)m(v)m(e)h(sho)m(wn,)456 4200 y(it)f(is)f(an)g(orbit)h(of)f Fw(\036)1144 4167 y Ft(t)1144 4227 y(L)1197 4200 y Fy(.)2152 b Fh(\003)916 4539 y Fy(4.)46 b Fx(Applica)-6 b(tion)33 b(to)h(a)g(priori)h(Unst)-6 b(able)32 b(Systems)555 4784 y Fy(Arnold)42 b(di\013usion)g(in)g (time-dep)s(enden)m(t)g(systems)h(has)f(b)s(een)f(studied)h(in)g(man)m (y)456 4892 y(pap)s(ers,)47 b(for)e(instance,)50 b(see)45 b([Ar1)q(],)k([CY1],)h([CY2],)f([DLS],)g([T)-8 b(r])45 b(and)g([Be2)q(].)85 b(It)456 5000 y(has)37 b(b)s(een)g(sho)m(wn)h(a)g (generic)h(phenomenon)d(in)i([CY1],)j([CY2].)63 b(F)-8 b(or)39 b(autonomous)456 5108 y(systems,)29 b(there)f(are)h(some)g (examples)g(of)f(Arnold)g(di\013usion,)h(for)f(instance,)h(see)g([Bs)q (],)456 5216 y([KL)o(])d(and)e([Zh)o(].)39 b(In)25 b(this)f(section,)k (w)m(e)d(apply)g(the)g(result)g(in)g(the)g(last)h(section)g(to)f(sho)m (w)p eop end %%Page: 25 25 TeXDict begin 25 24 bop 998 251 a Fr(CONNECTING)31 b(ORBITS)g(F)n(OR)f (A)n(UTONOMOUS)g(SYSTEMS)472 b(25)456 450 y Fy(that,)29 b(in)g(autonomous)f(Hamiltonian)i(systems)f(of)f Fj(a)k(priori)d Fy(unstable)g(t)m(yp)s(e,)g(Arnold)456 558 y(di\013usion)g(is)i (generic)g(phenomenon.)555 749 y(W)-8 b(e)32 b(consider)e(the)h (autonomous)f(Hamiltonian)i(of)f Fj(a)i(priori)e Fy(unstable)f(t)m(yp)s (e)456 926 y(\(4.1\))413 b Fw(H)1130 940 y Ft(\017)1162 926 y Fy(\()p Fw(p;)15 b(q)s(;)g(x;)g(y)s Fy(\))26 b(=)f Fw(R)q Fy(\()p Fw(p)p Fy(\))c(+)f Fw(P)13 b Fy(\()p Fw(x;)i(y)s Fy(\))21 b(+)f Fw(h)2407 940 y Ft(\017)2440 926 y Fy(\()p Fw(p;)15 b(q)s(;)g(x;)g(y)s Fy(\))p Fw(:)456 1106 y Fy(Here,)36 b(\()p Fw(q)s(;)15 b(p)p Fy(\))32 b Fv(2)g Fu(T)1087 1073 y Fr(2)1149 1106 y Fv(\002)23 b Fu(R)1309 1073 y Fr(2)1348 1106 y Fy(,)35 b Fw(x)d Fv(2)g Fu(T)1646 1073 y Ft(k)1711 1106 y Fv(\002)23 b Fu(R)1871 1073 y Ft(k)1947 1106 y Fy(and)34 b Fw(H)2204 1120 y Ft(\017)2271 1106 y Fy(is)g(con)m(v)m(ex)i(in)e(\()p Fw(p;)15 b(y)s Fy(\),)36 b Fw(h)3095 1120 y Ft(\017)3160 1106 y Fv(!)c Fy(0)j(as)456 1214 y Fw(\017)c Fv(!)h Fy(0.)53 b(Let)35 b(\010)1003 1181 y Ft(t)1003 1241 y(H)1061 1249 y Fm(\017)1130 1214 y 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(b)m(y)j(a)g(Denjo)m(y)g(set,)g(or)g(a)456 5216 y(closed)g(orbit.)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 998 251 a Fr(CONNECTING)31 b(ORBITS)g(F)n(OR)f (A)n(UTONOMOUS)g(SYSTEMS)472 b(27)555 450 y Fy(3,)31 b(for)f(an)m(y)h(lo)s(op)g Fw(\015)k Fy(with)30 b(0)c Fv(6)p Fy(=)f([)p Fw(\015)5 b Fy(])26 b Fv(2)e Fw(H)1912 464 y Fr(1)1951 450 y Fy(\()p Fu(T)2047 417 y Fr(2+)p Ft(k)2180 450 y Fw(;)15 b Fu(T)2281 417 y Fr(2)2321 450 y Fw(;)g Fu(Z)p Fy(\),)31 b(w)m(e)f(ha)m(v)m(e)456 644 y(\(4.3\))1330 520 y Fq(Z)1421 644 y Fy(\()p Fw(L)20 b Fv(\000)g Fw(\021)1674 658 y Ft(c)1729 644 y Fy(+)g Fw(\013)p Fy(\()p Fw(c)p Fy(\)\)\()p Fw(d\015)5 b Fy(\()p Fw(t)p Fy(\)\))p Fw(dt)29 b(>)c Fy(0)p Fw(:)555 915 y Fy(T)-8 b(o)30 b(v)m(erify)f(the)g(existence)h(of)g(a)f(generalized)h (transition)g(c)m(hain,)g(w)m(e)f(consider)g(\014rst)456 1023 y(the)23 b(case)i(that)f(there)f(is)h(no)f(in)m(v)-5 b(arian)m(t)25 b(curv)m(e)f(in)f(the)g(cylinder)h(for)f(the)g(corresp)s (onding)456 1131 y Fw(c)i Fv(2)g Fy(\000)663 1145 y Ft(\017;e)757 1131 y Fv(\002)9 b Fw(O)906 1145 y Ft(\017;x)998 1131 y Fy(.)39 b(F)-8 b(or)25 b(these)h(cohomology)g(classes,)i(w)m(e)d(can) g(c)m(ho)s(ose)h(a)f(v)m(ector)h(\()p Fw(a)3241 1145 y Fr(1)3281 1131 y Fw(;)15 b(a)3369 1145 y Fr(2)3409 1131 y Fy(\))456 1239 y(from)38 b(the)h(three)g(candidates)h Fv(f)p Fy(\(1)p Fw(;)15 b Fy(0\))p Fw(;)g Fy(\(1)p Fw(;)g Fy(1\))p Fw(;)g Fy(\(0)p Fw(;)g Fy(1)q(\))p Fv(g)45 b Fy(and)38 b(c)m(ho)s(ose)i(\006)2908 1253 y Ft(c)2982 1239 y Fy(=)f Fv(f)p Fy(\()p Fw(q)s(;)15 b(x)p Fy(\))41 b Fv(2)456 1349 y Fu(T)517 1316 y Ft(k)r Fr(+2)676 1349 y Fy(:)28 b Fw(a)777 1363 y Fr(1)816 1349 y Fw(q)857 1363 y Fr(1)917 1349 y Fy(+)21 b Fw(a)1057 1363 y Fr(2)1096 1349 y Fw(q)1137 1363 y Fr(2)1203 1349 y Fy(=)27 b(0)p Fv(g)33 b Fy(suc)m(h)e(that)h Fv(N)13 b Fy(\()p Fw(c)p Fy(\))22 b Fv(\\)f Fy(\006)2195 1363 y Ft(c)2260 1349 y Fy(is)32 b(totally)h(disconnected.)45 b(Th)m(us,)456 1457 y(for)32 b(eac)m(h)i Fw(c)f Fy(under)e(consideration,)j(one)f(of)f (these)h(three)g Fw(k)25 b Fy(+)c(1)33 b(dimensional)g(tori)g(is)456 1565 y(admissible,)d(eac)m(h)i Fw(c)e Fy(is)h(equiv)-5 b(alen)m(t)32 b(to)f(an)m(y)g Fw(c)2041 1532 y Fo(0)2094 1565 y Fy(nearb)m(y)-8 b(.)555 1756 y(If)31 b(there)h(is)f(an)h(in)m(v) -5 b(arian)m(t)32 b(curv)m(e)g(for)f Fw(c)p Fy(,)h(w)m(e)g(consider)g (a)f(\014nite)h(co)m(v)m(ering)h(of)f Fu(T)3287 1723 y Fr(2+)p Ft(k)3419 1756 y Fy(,)490 1843 y(~)456 1866 y Fw(M)j Fy(=)25 b Fu(T)736 1833 y Fr(1+)p Ft(k)883 1866 y Fv(\002)15 b Fy(2)p Fu(T)p Fy(.)40 b(The)27 b(condition)h(\(4.3\))i (implies)e(that)g Fw(\031)2487 1880 y Fr(1)2526 1866 y Fv(N)13 b Fy(\()p Fw(c;)2763 1843 y Fy(~)2728 1866 y Fw(M)e Fy(\))p Fv(n)p Fy(\()p Fv(A)p Fy(\()p Fw(c;)k(M)10 b Fy(\))15 b(+)g Fw(\016)s Fy(\))456 1973 y(is)31 b(not)g(empt)m(y)-8 b(.)44 b(Therefore,)32 b(for)f(the)g(existence)i(of)e(a)h(generalized)h (transition)f(c)m(hain,)456 2081 y(w)m(e)27 b(need)g(to)h(v)m(erify)g (that)g(there)f(is)g(an)h(isolated)g(minimal)g(homo)s(clinic)g(curv)m (e)f(to)h(eac)m(h)456 2189 y(in)m(v)-5 b(arian)m(t)31 b(curv)m(e.)555 2380 y(T)-8 b(o)m(w)m(ards)33 b(that)g(goal,)h(w)m(e)f (fully)f(in)m(v)m(ok)m(e)i(the)f(strength)f(of)h(the)f(tec)m(hnique)h (used)f(in)456 2488 y([CY1].)61 b(Let)38 b(us)e(brie\015y)g(describ)s (e)h(it)g(for)g(the)g(time-p)s(erio)s(dic)h(case.)61 b(Let)38 b Fp(Z)f Fy(b)s(e)f(the)456 2596 y(set)j(of)g(those)g (functions)f(whose)h(minimal)g(p)s(oin)m(ts)g(are)g(not)g(totally)i (disconnected,)456 2704 y(this)30 b(set)i(has)f(\\in\014nite)g(co)s (dimensions")g(in)f Fw(C)2071 2671 y Fr(0)2110 2704 y Fy(:)42 b(there)31 b(is)g(a)h(subspace)e(with)h(in\014nite)456 2812 y(dimensions)41 b Fp(N)46 b Fv(\032)f Fw(C)1243 2779 y Ft(r)1323 2812 y Fy(suc)m(h)d(that)i Fw(U)1812 2826 y Fr(1)1879 2812 y Fy(+)28 b Fw(U)2040 2826 y Fr(2)2136 2812 y Fw(=)-55 b Fv(2)45 b Fp(Z)d Fy(holds)g(for)h(eac)m(h)g Fw(U)3009 2826 y Fr(1)3094 2812 y Fv(2)j Fp(Z)c Fy(and)456 2920 y(eac)m(h)37 b Fw(U)728 2934 y Fr(2)803 2920 y Fv(2)e Fp(N)p Fv(nf)p Fy(0)p Fv(g)p Fy(.)61 b(If)36 b(the)h(set)g(of)g (barrier)f(functions)g Fp(B)f Fy(=)h Fv([)p Fw(B)2819 2887 y Fo(\003)2814 2942 y Ft(c)2894 2920 y Fy(has)g(\014nite)g(b)s(o)m (x)456 3028 y(dimensions,)28 b(one)i(can)f(exp)s(ect)g(that)h (arbitrarily)f(small)g(translation)h Fp(B)c Fv(!)f Fp(B)17 b Fy(+)g Fw(u)26 b Fy(=)456 3136 y Fv([)p Fy(\()p Fw(B)626 3103 y Fo(\003)621 3158 y Ft(c)686 3136 y Fy(+)20 b Fw(u)p Fy(\))32 b(exists)h(suc)m(h)e(that)h Fp(Z)21 b Fv(\\)g Fy(\()p Fp(B)h Fy(+)e Fw(u)p Fy(\))28 b(=)f Fu(?)p Fy(,)32 b(i.e.)46 b(the)32 b(set)g(of)g(minimal)f(p)s(oin)m(ts)456 3244 y(of)f(eac)m(h)i Fw(B)838 3211 y Fo(\003)833 3266 y Ft(c)897 3244 y Fy(+)20 b Fw(u)30 b Fy(is)h(totally)h(disconnected.) 555 3435 y(The)25 b(\014niteness)f(of)h(the)g(b)s(o)m(x)g(dimension)f (is)h(obtained)g(b)m(y)g(studying)f(the)h(regularit)m(y)456 3543 y(of)i(the)h(barrier)e(functions)h(in)g Fw(c)p Fy(.)40 b(If)27 b Fw(B)1763 3510 y Fo(\003)1758 3565 y Ft(c)1830 3543 y Fy(is)g Fw(\013)p Fy(-H\177)-45 b(older)28 b(con)m(tin)m(uous)g (in)g Fw(c)f Fy(and)g(if)g Fw(c)h Fy(tak)m(es)456 3650 y(its)g(v)-5 b(alue)29 b(on)f(a)g(smo)s(oth)h(curv)m(e,)g(the)f(set)h (of)f(barrier)g(function)g Fv(f)p Fw(B)2761 3617 y Fo(\003)2756 3673 y Ft(c)2800 3650 y Fv(g)h Fy(in)e Fw(C)3049 3617 y Fr(0)3088 3650 y Fy(-function)456 3760 y(space)k(has)g(b)s(o)m (x-dimension)g(not)h(bigger)g(than)f Fw(\013)2175 3727 y Fo(\000)p Fr(1)2270 3760 y Fy(.)43 b(W)-8 b(e)32 b(\014nd)2688 3724 y Fr(1)p 2688 3739 36 4 v 2688 3791 a(2)2733 3760 y Fy(-H\177)-45 b(older)32 b(regularit)m(y)456 3868 y(on)g(those)g (cohomology)i(classes)f(where)f(the)g(in)m(v)-5 b(arian)m(t)34 b(curv)m(es)e(exist.)46 b(The)32 b(pro)s(of)f(is)456 3976 y(the)f(same)h(as)g(in)f([CY1],)h(w)m(e)g(omit)g(the)g(details)g (here.)555 4167 y(F)-8 b(or)22 b(autonomous)f(systems,)i(the)e(minimal) g(p)s(oin)m(t)g(of)g Fw(B)2452 4134 y Fo(\003)2447 4189 y Ft(c)2491 4167 y Fv(j)2516 4185 y Ft(M)7 b Fo(nA)p Fr(\()p Ft(c;M)g Fr(\)+)p Ft(\016)2976 4167 y Fy(corresp)s(onds)456 4294 y(to)39 b(the)f(minimal)g(homo)s(clinic)h(curv)m(es)g(in)f Fw(\031)2021 4261 y Fo(0)2044 4294 y Fv(N)13 b Fy(\()p Fw(c;)2281 4271 y Fy(~)2246 4294 y Fw(M)e Fy(\))p Fv(n)p Fy(\()p Fv(A)p Fy(\()p Fw(c;)k(M)10 b Fy(\))28 b(+)d Fw(\016)s Fy(\).)66 b(Since)38 b(the)456 4402 y(system)44 b(is)h(autonomous,)k(the)c(barrier)f(function)g(reac)m(hes)i(its)f (minim)m(um)f(on)g(the)456 4510 y(whole)36 b(curv)m(e.)60 b(What)38 b(w)m(e)f(need)f(is)h(that)g(the)g(set)g(of)g(the)g(minimal)g (p)s(oin)m(ts)f(of)h(eac)m(h)456 4618 y(barrier)j(function)h(is)h (totally)h(disconnected)f(when)f(it)h(is)f(restricted)h(on)f(some)h (co-)456 4726 y(dimension)29 b(one)i(section.)555 4917 y(In)c(virtue)h(of)f(the)h(sp)s(ecial)g(form)f(of)h(the)f(Hamiltonian)i (\(4.1\),)h(there)e(exist)g Fw(m)g Fy(pairs)456 5025 y(of)i(op)s(en)g(disks)g Fw(U)1067 5039 y Fr(1)1131 5025 y Fv(\032)25 b Fw(U)1299 4992 y Fo(0)1289 5049 y Fr(1)1329 5025 y Fw(;)15 b(U)1431 5039 y Fr(2)1496 5025 y Fv(\032)25 b Fw(U)1664 4992 y Fo(0)1654 5049 y Fr(2)1693 5025 y Fw(;)15 b Fv(\001)g(\001)g(\001)i Fw(U)1917 5039 y Ft(m)2009 5025 y Fv(\032)25 b Fw(U)2177 4992 y Fo(0)2167 5047 y Ft(m)2259 5025 y Fv(\032)g Fu(R)2421 4992 y Fr(2)2490 5025 y Fy(suc)m(h)30 b(that)555 5216 y(1,)h Fw(U)718 5230 y Fr(1)778 5216 y Fv([)20 b Fw(U)921 5230 y Fr(2)980 5216 y Fv([)g(\001)15 b(\001)g(\001)22 b([)d Fw(U)1330 5230 y Ft(m)1422 5216 y Fv(\033)25 b Fy(\000)1575 5230 y Ft(e)1612 5216 y Fy(;)p eop end %%Page: 28 28 TeXDict begin 28 27 bop 456 251 a Fr(28)1012 b(X.)23 b(LI)h(AND)f(C.-Q.)f(CHENG)555 450 y Fy(2,)36 b(Restricted)f(on)g(eac)m (h)g Fw(U)1512 417 y Fo(0)1502 476 y Ft(i)1558 450 y Fv(\002)23 b Fu(R)1718 417 y Ft(k)1783 450 y Fv(\002)f Fu(T)1937 417 y Ft(k)r Fr(+2)2070 450 y Fy(,)35 b(there)f(exist)h Fw(j)j Fv(2)31 b(f)p Fy(1)p Fw(;)15 b Fy(2)p Fv(g)37 b Fy(and)c Fw(d)3236 464 y Ft(i)3297 450 y Fw(>)e Fy(0)456 558 y(suc)m(h)f(that)h Fw(@)906 572 y Ft(p)942 582 y Fm(j)978 558 y Fw(H)h Fv(\025)25 b Fw(d)1229 572 y Ft(i)1288 558 y Fy(or)30 b Fw(@)1447 572 y Ft(p)1483 582 y Fm(j)1520 558 y Fw(H)i Fv(\024)25 b(\000)p Fw(d)1842 572 y Ft(i)1870 558 y Fy(.)555 762 y(In)45 b(the)h(case)g(that)g Fw(j)56 b Fy(=)50 b(1)c(in)f(the)h(region)g Fw(U)2223 776 y Ft(i)2282 762 y Fv(\002)30 b Fu(R)2449 729 y Ft(k)2521 762 y Fv(\002)g Fu(T)2683 729 y Ft(k)r Fr(+2)2816 762 y Fy(,)49 b(i.e.)104 b(_)-42 b Fw(q)3133 776 y Fr(1)3223 762 y Fv(\025)50 b Fw(d)3391 776 y Ft(i)3419 762 y Fy(.)456 870 y(Let)36 b Fw(K)7 b Fy(\()p Fw(p)789 884 y Fr(2)828 870 y Fw(;)15 b(x;)g(y)s(;)g(T)8 b(;)15 b(q)1190 884 y Fr(2)1230 870 y Fw(;)g(e)p Fy(\))37 b(b)s(e)d(the)i(solution)g(of)g Fw(H)7 b Fy(\()p Fw(p)2296 884 y Fr(1)2335 870 y Fw(;)15 b(p)2421 884 y Fr(2)2461 870 y Fw(;)g Fv(\000)p Fw(T)8 b(;)15 b(q)2714 884 y Fr(2)2753 870 y Fw(;)g(x;)g(y)s Fy(\))35 b(=)f Fw(e)p Fy(,)j(where)456 979 y Fw(T)k Fy(=)28 b Fv(\000)p Fw(q)761 993 y Fr(1)800 979 y Fy(,)33 b(then)f(the)h (Hamiltonian)h(\015o)m(w)e(\010)2008 946 y Ft(T)2008 1006 y(K)2109 979 y Fy(is)g(equiv)-5 b(alen)m(t)34 b(to)f(\010)2815 946 y Ft(t)2815 1006 y(H)2914 979 y Fy(in)f(this)h(region)456 1087 y([Ar2].)61 b(By)37 b(simple)g(calculation,)k(w)m(e)d(see)f(that)h Fw(K)43 b Fy(is)38 b(p)s(ositiv)m(e)f(de\014nite)g(in)g(\()p Fw(p)3256 1101 y Fr(2)3296 1087 y Fw(;)15 b(y)s Fy(\).)456 1195 y(Clearly)-8 b(,)38 b Fw(K)43 b Fy(is)36 b(p)s(erio)s(dic)f(in)h Fw(T)47 b Fy(=)34 b Fv(\000)p Fw(q)1796 1209 y Fr(1)1835 1195 y Fy(.)57 b(Because)37 b(of)f(the)h(sp)s(ecial)f(form)g(of)g Fw(H)7 b Fy(,)37 b(the)456 1303 y(v)-5 b(ariation)30 b(of)g Fw(p)f Fy(is)g(small)h(along)g(eac)m(h)h(orbit)e(in)g(the)h(Ma)s (~)-48 b(n)m(\023)-43 b(e)31 b(sets,)f(along)g(p)s(ositiv)m(e)g(half) 456 1411 y(of)38 b(the)h(forw)m(ard)e(semi-static)k(orbits)d(and)g (along)h(negativ)m(e)i(half)d(of)g(the)h(bac)m(kw)m(ard)456 1519 y(semi-static)g(orbits.)60 b(By)37 b(re\014ning)f(the)h(division)g Fw(U)2283 1533 y Fr(1)2323 1519 y Fw(;)15 b(U)2425 1533 y Fr(2)2464 1519 y Fw(;)g Fv(\001)g(\001)g(\001)32 b Fw(;)15 b(U)2743 1533 y Ft(m)2847 1519 y Fy(w)m(e)37 b(can)h(assume)456 1627 y(that)d(these)h(orbits)f(en)m(tirely)h(sta)m (y)h(in)e Fw(U)1872 1594 y Fo(0)1862 1652 y Ft(i)1930 1627 y Fy(if)g(they)g(start)h(from)f Fw(U)2732 1641 y Ft(i)2760 1627 y Fy(.)55 b(These)35 b(orbits)g(of)456 1735 y Fw(K)h Fy(exactly)31 b(corresp)s(ond)e(to)h(those)h(for)e Fw(H)7 b Fy(.)41 b(Denote)31 b(b)m(y)f Fw(B)2485 1749 y Ft(K)q(;c)2629 1735 y Fy(the)g(barrier)f(functions)456 1843 y(corresp)s(onding)20 b(to)i Fw(K)7 b Fy(.)37 b(Clearly)-8 b(,)24 b(eac)m(h)e(minimal)g(p)s(oin)m(t)f(of)g Fw(B)2536 1857 y Ft(K)q(;c)2672 1843 y Fy(exactly)i(corresp)s(onds)456 1951 y(to)29 b(a)h(minimal)f(p)s(oin)m(t)g(of)g(the)g(barrier)g (function)f Fw(B)2206 1965 y Ft(c)2270 1951 y Fy(for)h Fw(H)36 b Fy(in)28 b(the)h(section)i Fv(f)p Fw(q)3168 1965 y Fr(1)3232 1951 y Fy(=)25 b(0)p Fv(g)p Fy(.)456 2059 y(According)31 b(to)h(the)f(study)f(in)h([CY1)q(])g(and)f([CY2)q (]\),)i(outside)f(a)g(small)h(neigh)m(b)s(orho)s(o)s(d)456 2166 y(of)c(Aubry)e(set,)j(an)m(y)f(small)h(p)s(erturbation)d Fw(B)2006 2180 y Ft(K)q(;c)2146 2166 y Fv(!)f Fw(B)2331 2180 y Ft(K)q(;c)2461 2166 y Fv(\000)15 b Fw(u)2599 2180 y Ft(\017)2659 2166 y Fy(can)28 b(b)s(e)f(realized)i(b)m(y)f(a)456 2274 y(Hamiltonian)33 b(p)s(erturbation)f Fw(K)j Fv(!)29 b Fw(K)f Fy(+)21 b Fw(K)2022 2288 y Ft(\017)2055 2274 y Fy(.)47 b(Note)34 b Fw(@)2397 2288 y Ft(p)2433 2297 y Fk(1)2471 2274 y Fw(H)i Fv(\025)28 b Fw(d)2729 2288 y Ft(i)2786 2274 y Fw(>)h Fy(0,)k(there)g(exists)456 2382 y Fw(h)508 2396 y Ft(\017)572 2382 y Fy(suc)m(h)e(that)h Fw(K)27 b Fy(+)21 b Fw(K)1249 2396 y Ft(\017)1313 2382 y Fy(is)31 b(the)g(solution)h(of)g Fw(H)27 b Fy(+)21 b Fw(h)2259 2396 y Ft(\017)2319 2382 y Fy(=)26 b Fw(e)p Fy(.)44 b(Th)m(us,)30 b(b)m(y)h(the)h(argumen)m(t)456 2490 y(in)f([CY1)q(])h(and)g([CY2],)h Fv(9)f Fy(an)g(op)s(en)f(and)h (dense)g(set)g Fp(O)2370 2504 y Ft(i;\017;K)2559 2490 y Fv(\032)c Fp(B)2738 2504 y Ft(i;\017;K)2930 2490 y Fy(suc)m(h)k(that)h(for)456 2605 y(eac)m(h)27 b Fw(h)708 2619 y Ft(\017)767 2605 y Fv(2)e Fp(O)928 2619 y Ft(i;\017;K)1088 2605 y Fy(,)i(the)g(set)h(of)e(minimal)h(homo)s(clinic)h(curv)m(es)e (in)h Fw(U)2774 2619 y Ft(i)2815 2605 y Fv(\002)13 b Fu(R)2965 2572 y Ft(k)3019 2605 y Fv(\002)g Fu(T)3164 2572 y Ft(k)r Fr(+2)3323 2605 y Fy(are)456 2713 y(totally)32 b(disconnected)e(for)g(eac)m(h)h(related)f(cohomology)i(class.)42 b(Here,)30 b Fp(B)2991 2727 y Ft(i;\017;K)3182 2713 y Fy(denote)456 2827 y(the)22 b(set)h(of)g(Hamiltonians)h(de\014ned)d(on) h(\()p Fw(U)1915 2794 y Fo(0)1905 2851 y Fr(1)1950 2827 y Fv([)t Fw(U)2087 2794 y Fo(0)2077 2851 y Fr(2)2121 2827 y Fv([)t(\001)15 b(\001)g(\001)6 b([)t Fw(U)2434 2794 y Fo(0)2424 2849 y Ft(m)2490 2827 y Fy(\))t Fv(\002)t Fu(T)2665 2794 y Ft(k)r Fr(+2)2803 2827 y Fv(\002)t Fy([)p Fv(\000)p Fw(K)q(;)15 b(K)7 b Fy(])3201 2794 y Ft(k)3268 2827 y Fy(with)456 2939 y(the)35 b(top)s(ology)i(induced)d(b)m(y)h(the) g(standard)f Fw(C)2087 2906 y Ft(r)2125 2939 y Fy(-norm)h(on)g Fw(U)2601 2906 y Fo(0)2591 2964 y Ft(i)2647 2939 y Fv(\002)24 b Fu(T)2803 2906 y Ft(k)r Fr(+2)2958 2939 y Fv(\002)f Fy([)p Fv(\000)p Fw(K)q(;)15 b(K)7 b Fy(])3375 2906 y Ft(k)3419 2939 y Fy(.)456 3052 y(Ob)m(viously)-8 b(,)25 b(the)e(set)g Fv(\\)1239 3019 y Ft(m)1239 3077 y(i)p Fr(=1)1357 3052 y Fp(O)1432 3066 y Ft(i;\017;K)1618 3052 y Fv(\032)i Fw(C)1786 3019 y Ft(r)1823 3052 y Fy(\(\()p Fw(U)1965 3019 y Fo(0)1955 3076 y Fr(1)2001 3052 y Fv([)5 b Fw(U)2139 3019 y Fo(0)2129 3076 y Fr(2)2173 3052 y Fv([)g(\001)15 b(\001)g(\001)6 b([)f Fw(U)2488 3019 y Fo(0)2478 3074 y Ft(m)2545 3052 y Fy(\))g Fv(\002)g Fu(T)2722 3019 y Ft(k)r Fr(+2)2860 3052 y Fv(\002)g Fy([)p Fv(\000)p Fw(K)q(;)15 b(K)7 b Fy(])3259 3019 y Ft(k)3303 3052 y Fw(;)15 b Fu(R)p Fy(\))456 3160 y(is)30 b(op)s(en)g(and)f(dense.)555 3351 y(Therefore,)k(giv)m(en)g(an)m(y)g(t)m(w)m(o)g(p)s(oin)m(ts)g(in)f (\000)2022 3365 y Ft(\017;e)2128 3351 y Fv(\002)21 b Fw(O)2289 3365 y Ft(\017;x)2381 3351 y Fy(,)33 b(generically)-8 b(,)36 b(there)c(exists)h(a)456 3458 y(generalized)40 b(transition)g(c)m(hain)g(joining)f(these)h(t)m(w)m(o)g(p)s(oin)m(ts.) 67 b(This)39 b(completes)h(the)456 3566 y(pro)s(of.)2685 b Fh(\003)1681 3850 y Fy(5.)46 b Fx(Appendix)456 4094 y FC(Prop)s(osition)51 b(5.1.)f Fj(L)-5 b(et)45 b(a)h(smo)-5 b(oth)47 b(inje)-5 b(ction)46 b Fw(')p Fj(:)67 b Fu(T)2432 4061 y Ft(n)p Fo(\000)p Fr(1)2617 4094 y Fv(!)48 b Fu(T)2817 4061 y Ft(n)2908 4094 y Fj(such)e(that)g(the)456 4202 y(induc)-5 b(e)g(d)30 b(map)g Fw(')1035 4216 y Fo(\003)1075 4202 y Fj(:)40 b Fw(H)1219 4216 y Fr(1)1258 4202 y Fy(\()p Fu(T)1354 4169 y Ft(n)p Fo(\000)p Fr(1)1491 4202 y Fw(;)15 b Fu(Z)p Fy(\))26 b Fv(!)f Fw(H)1845 4216 y Fr(1)1884 4202 y Fy(\()p Fu(T)1980 4169 y Ft(n)2027 4202 y Fw(;)15 b Fu(Z)p Fy(\))29 b Fj(is)g(an)h(inje)-5 b(ction.)41 b(L)-5 b(et)30 b Fv(f)p Fw(e)3064 4216 y Ft(i)3093 4202 y Fv(g)3138 4216 y Fr(1)p Fo(\024)p Ft(i)p Fo(\024)p Ft(n)p Fo(\000)p Fr(1)456 4311 y Fj(b)-5 b(e)26 b(the)h(standar)-5 b(d)29 b(b)-5 b(asis)28 b(of)e Fw(H)1459 4325 y Fr(1)1498 4311 y Fy(\()p Fu(T)1594 4278 y Ft(n)p Fo(\000)p Fr(1)1732 4311 y Fw(;)15 b Fu(Z)p Fy(\))p Fj(,)27 b Fw(g)1966 4325 y Ft(i)p Fr(+1)2110 4311 y Fy(=)e Fw(')2265 4325 y Fo(\003)2305 4311 y Fy(\()p Fw(e)2382 4325 y Ft(i)2411 4311 y Fy(\))h Fv(2)f Fw(H)2634 4325 y Fr(1)2673 4311 y Fy(\()p Fu(T)2769 4278 y Ft(n)2816 4311 y Fw(;)15 b Fu(Z)p Fy(\))p Fj(.)40 b(Then)26 b(ther)-5 b(e)456 4419 y(exists)35 b Fw(g)750 4433 y Fr(1)820 4419 y Fv(2)29 b Fu(Z)971 4386 y Ft(n)1052 4419 y Fj(such)36 b(that)g(the)f(matrix)i Fw(G)30 b Fy(=)f(\()p Fw(g)2177 4433 y Fr(1)2218 4419 y Fw(;)15 b(g)2301 4433 y Fr(2)2341 4419 y Fw(;)g Fv(\001)g(\001)g(\001)32 b Fw(;)15 b(g)2601 4433 y Ft(n)2648 4419 y Fy(\))36 b Fj(is)f(uni-mo)-5 b(dule,)36 b(i.e.)456 4527 y Fy(det)p Fw(G)26 b Fy(=)f Fv(\006)p Fy(1)p Fj(.)456 4776 y(Pr)-5 b(o)g(of.)43 b Fy(:)g(F)-8 b(or)33 b 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Fw(x)1566 5120 y Ft(k)1609 5105 y Fy(\)\),)39 b(where)d(\()p Fw(a)p Fy(\))i(denotes)f(the)f(decimal)i (part)e(of)h(the)456 5216 y(n)m(um)m(b)s(er)29 b Fw(a)p Fy(.)40 b(W)-8 b(e)32 b(de\014ne)e(the)g(pro)5 b(jection)31 b(map)f Fw(\031)s Fy(:)41 b Fu(R)2297 5183 y Ft(k)2365 5216 y Fv(!)25 b Fw(I)2528 5183 y Ft(k)2601 5216 y Fy(suc)m(h)30 b(that)h Fw(\031)s(x)25 b Fy(=)g(\()p Fw(x)p Fy(\).)p eop end %%Page: 29 29 TeXDict begin 29 28 bop 998 251 a Fr(CONNECTING)31 b(ORBITS)g(F)n(OR)f (A)n(UTONOMOUS)g(SYSTEMS)472 b(29)555 450 y Fy(W)-8 b(e)24 b(assume)f(that)g Fw(\036)p Fy(:)37 b Fu(T)1377 417 y Ft(n)p Fo(\000)p Fr(1)1539 450 y Fv(!)25 b Fu(T)1716 417 y Ft(n)1785 450 y Fy(is)e(a)g(linear)h(injection.)39 b(Denote)24 b(b)m(y)3021 426 y(\026)3009 450 y Fw(\036)q Fy(:)36 b Fu(R)3191 417 y Ft(n)p Fo(\000)p Fr(1)3353 450 y Fv(!)456 558 y Fu(R)522 525 y Ft(n)600 558 y Fy(the)31 b(lift)i(to)f(the)g(univ)m(ersal)f(co)m(v)m(ering)j(with)2136 534 y(\026)2124 558 y Fw(\036)p Fy(\(0\))29 b(=)e(0.)44 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Fw(\036)p Fy(\()p Fw(I)1589 957 y Ft(n)p Fo(\000)p Fr(1)1727 990 y Fy(\))31 b(=)f Fv([)p Fw(D)2030 1004 y Ft(j)2066 990 y Fw(;)15 b(j)37 b Fy(=)30 b(1)p Fw(;)15 b Fy(2)p Fw(;)g Fv(\001)g(\001)g(\001)33 b Fw(;)15 b(J)9 b Fy(,)35 b(where)e(the)h(in)m(teger)456 1098 y Fw(J)49 b Fy(dep)s(ends)39 b(on)i Fv(f)p Fw(g)1142 1112 y Ft(i)1171 1098 y Fv(g)1216 1112 y Ft(i)p Fo(\024)p Ft(n)p Fo(\000)p Fr(1)1432 1098 y Fy(.)72 b(W)-8 b(e)42 b(use)e Fw(D)1938 1112 y Fr(1)2019 1098 y Fy(to)h(denote)g(the)g(piece) h(whic)m(h)e(con)m(tains)456 1207 y(the)33 b(origin.)51 b(Let)33 b(\006)g(b)s(e)g(the)h(h)m(yp)s(erplane)e(con)m(taining)2396 1183 y(\026)2384 1207 y Fw(\036)p Fy(\()p Fw(I)2520 1174 y Ft(n)p Fo(\000)p Fr(1)2658 1207 y Fy(\),)j(w)m(e)f(consider)f(some) 456 1315 y(the)45 b(parallel)h(shift)f(of)h(\006:)69 b(\006)1530 1329 y Ft(j)1567 1315 y Fw(;)15 b(j)56 b Fy(=)49 b(2)p Fw(;)15 b Fv(\001)g(\001)g(\001)32 b Fw(;)15 b(J)9 b Fy(.)86 b(This)45 b(shift)f(is)i(made)f(suc)m(h)g(that)456 1422 y(\006)522 1436 y Ft(j)586 1422 y Fv(\033)28 b Fw(D)760 1436 y Ft(j)829 1422 y Fy(for)j(eac)m(h)j Fw(j)g Fv(\024)27 b Fw(J)9 b Fy(.)46 b(Clearly)-8 b(,)34 b(eac)m(h)f(\006)2083 1436 y Ft(j)2152 1422 y Fy(passes)f(some)g(p)s(oin)m(ts)g(in)g Fu(Z)3101 1389 y Ft(n)3148 1422 y Fy(.)45 b(Being)456 1530 y(a)m(w)m(are)30 b(of)f(co)s(dimension)f(one,)i(w)m(e)f(see)g (that)g(there)g(is)g(some)g(\006)2606 1544 y Ft(j)2635 1553 y Fk(0)2701 1530 y Fy(suc)m(h)g(that)g(the)g(la)m(y)m(er)456 1638 y(b)s(ounded)34 b(b)m(y)i(\006)h(and)f(\006)1317 1652 y Ft(j)1346 1661 y Fk(0)1420 1638 y Fy(do)s(es)g(not)h(con)m(tain) h(an)m(y)f(other)g(\006)2614 1652 y Ft(j)2650 1638 y Fy(.)59 b(Let)37 b Fw(g)2946 1652 y Ft(n)3029 1638 y Fv(2)e Fu(Z)3186 1605 y Ft(n)3269 1638 y Fy(b)s(e)h(a)456 1746 y(p)s(oin)m(t)e(in)g(\006)873 1760 y Ft(j)902 1769 y Fk(0)940 1746 y Fy(,)h(denote)g(b)m(y)f Fw(V)55 b Fy(the)34 b Fw(n)g Fy(dimensional)g(parallel)i(p)s(olyhedron,)e(spanned)456 1857 y(b)m(y)i Fw(g)631 1871 y Ft(i)696 1857 y Fy(=)814 1833 y(\026)802 1857 y Fw(\036)p Fy(\()p Fw(e)933 1871 y Ft(i)962 1857 y Fy(\))g Fv(2)g Fu(Z)1191 1824 y Ft(n)1274 1857 y Fy(\()p Fw(i)g Fy(=)g(1)p Fw(;)15 b Fy(2)p Fw(;)g Fv(\001)g(\001)g(\001)33 b Fw(;)15 b(n)p Fy(\).)60 b(W)-8 b(e)38 b(pro)5 b(ject)37 b(this)g(parallel)h(p)s(olyhedron)456 1965 y(in)m(to)31 b(the)g(unit)f(cub)s(e)g Fw(\031)f Fy(:)d Fw(V)45 b Fv(!)26 b Fw(I)1598 1932 y Ft(n)1645 1965 y Fy(.)41 b(Clearly)-8 b(,)32 b(this)f(map)f(is)g(on)m(to,)j(and)d (one)g(to)i(one)f(in)456 2073 y Fw(I)503 2040 y Ft(k)545 2073 y Fv(n)20 b([)f Fw(D)765 2087 y Ft(j)802 2073 y Fy(.)40 b(Let)31 b Fw(G)26 b Fy(=)e(\()p Fw(g)1300 2087 y Fr(1)1341 2073 y Fw(;)15 b(g)1424 2087 y Fr(2)1464 2073 y Fw(;)g Fv(\001)g(\001)g(\001)32 b Fw(;)15 b(g)1724 2087 y Ft(n)1771 2073 y Fy(\))31 b(b)s(e)e(an)h Fw(n)19 b Fv(\002)g Fw(n)29 b Fy(matrix,)i(the)f(ab)s(o)m(v)m(e)h(argumen)m(t) 456 2180 y(pro)m(v)m(es)g(that)g(det)p Fw(G)26 b Fy(=)f Fv(\006)p Fy(1)p Fw(:)1980 b Fh(\003)456 2423 y FC(Ac)m(kno)m (wledgemen)m(t)44 b Fy(This)f(w)m(ork)h(is)g(partially)h(supp)s(orted)d (b)m(y)h(National)j(Basic)456 2530 y(Researc)m(h)27 b(Program)f(of)g (China)f(\(973)j(Program,)g(2007CB814800\),)k(NNSF)26 b(of)g(China)456 2638 y(\(Gran)m(t)j(10531050\))k(and)28 b(Basic)i(Researc)m(h)f(Program)g(of)f(Jiangsu)h(Pro)m(vince,)h(China.) 456 2746 y(The)i(authors)g(thank)h(the)g(referees)g(for)g(their)g (commen)m(ts)h(and)e(suggestions)i(for)e(the)456 2854 y(revise)e(of)h(the)g(pap)s(er.)1685 3123 y Fx(References)456 3298 y FB([Ar1])66 b(Arnol'd)38 b(V.)g(I.,)j Fc(Instability)e(of)g (dynamic)l(al)g(systems)h(with)f(sever)l(al)i(de)l(gr)l(e)l(es)g(of)d (fr)l(e)l(e)l(dom,)690 3389 y FB(\(Russian,)h(English\))e(So)n(v.)f (Math.,)k(Dokl.,)f Fb(5)p FB(\(1964\),)h(581-585;)k(translation)37 b(from)g(Dokl.)690 3480 y(Ak)l(ad.)25 b(Nauk)g(SSSR,)f Fb(156)p FB(\(1964\),)k(9-12.)456 3572 y([Ar2])66 b(Arnol'd)26 b(V.)f(I.,)h Fc(Mathematic)l(al)j(Metho)l(ds)g(of)e(Classic)l(al)h(Me)l (chanics)e FB(\(1978\))h(Springer.)456 3663 y([Be0])66 b(Bernard)25 b(P)-6 b(.,)26 b Fc(Homo)l(clinic)g(orbits)i(to)g (invariant)f(sets)h(of)f(quasi-inte)l(gr)l(able)i(exact)g(maps)p FB(,)d(Er-)690 3754 y(go)r(d.)h(Theory)f(Dynam.)f(Syst.)g Fb(20)h FB(\(2000\),)h(1583-1601.)456 3846 y([Be1])66 b(Bernard)32 b(P)-6 b(.,)34 b Fc(Conne)l(cting)g(orbits)h(of)d(time)h (dep)l(endent)j(L)l(agr)l(angian)e(systems)p FB(,)i(Ann.)31 b(Inst.)690 3937 y(F)-6 b(ourier,)27 b Fb(52)f FB(\(2002\),)h (1533-1568.)456 4028 y([Be2])66 b(Bernard)18 b(P)-6 b(.,)19 b Fc(The)h(dynamics)g(of)g(pseudo)l(gr)l(aphs)j(in)c(c)l(onvex)i (Hamiltonian)e(systems)p FB(,)i(J.)d(Amer.)690 4120 y(Math.)26 b(So)r(c.)h Fb(21)f FB(\(2008\))h(615-669.)456 4211 y([Bs])108 b(Bessi)27 b(U.)f Fc(A)n(rnold's)i(example)g(with)f(thr)l(e)l(e)j(r)l (otators)p FB(,)e(Nonlinearit)n(y)e Fb(10)g FB(\(1997\))h(763-781.)456 4302 y([BCV])39 b(Bessi)25 b(U.,)f(Chierc)n(hia)g(L.)f(&)g(V)-6 b(aldino)r(ci)24 b(E.,)g Fc(Upp)l(er)j(b)l(ounds)f(on)g(A)n(rnold)f (di\013usion)h(times)f(via)690 4394 y(Mather)k(the)l(ory)p FB(,)f(J.)e(Math.)g(Pure)g(Appl.)f Fb(80)h FB(\(2001\))h(105-129.)456 4485 y([Ca])99 b(Carneiro)19 b(M.J.D.,)p Fc(On)j(minimizing)d(me)l (asur)l(es)j(of)d(the)i(action)g(of)f(autonomous)i(L)l(agr)l(angians)p FB(,)690 4576 y(Nonlinearit)n(y)k Fb(6)g FB(\(1996\))h(1077-1085.)456 4668 y([CY1])41 b(Cheng)18 b(C.-Q.)g(&)g(Y)-6 b(an)16 b(J.,)21 b Fc(Existenc)l(e)h(of)d(di\013usion)i(orbits)g(in)f(a)g (priori)g(unstable)i(Hamiltonian)690 4759 y(systems,)27 b FB(J.)g(Di\013eren)n(tial)f(Geometry)-6 b(,)26 b Fb(67)g FB(\(2004\),)h(457-517.)456 4850 y([CY2])41 b(Cheng)24 b(C.-Q.)g(&)f(Y)-6 b(an)23 b(J.,)i Fc(A)n(rnold)h(di\013usion)g(in)f (Hamiltonian)h(systems:)35 b(a)26 b(priori)g(unstable)690 4942 y(c)l(ase)p FB(,)h(J.)g(Di\013eren)n(tial)f(Geometry)-6 b(,)26 b Fb(82)g FB(\(2009\),)h(229-277.)456 5033 y([DLS])42 b(Delshams)27 b(A.,)g(de)f(la)h(Lla)n(v)n(e)g(R.)f(and)g(Seara)h(T.)g (M.,)h Fc(Ge)l(ometric)h(me)l(chanism)g(for)f(di\013usion)690 5124 y(in)i(Hamiltonian)f(systems)j(over)l(c)l(oming)g(the)e(lar)l(ge)h (gap)g(pr)l(oblem:)40 b(heuristic)31 b(and)g(rigor)l(ous)690 5216 y(veri\014c)l(ation)e(of)e(a)h(mo)l(del)p FB(,)e(Memoirs)h(Amer.)f (Math.)g(So)r(c.)g Fb(179)p FB(\(844\),)i(\(2006\).)p eop end %%Page: 30 30 TeXDict begin 30 29 bop 456 251 a Fr(30)1012 b(X.)23 b(LI)h(AND)f(C.-Q.)f(CHENG)456 450 y FB([F)-6 b(a1])72 b(F)-6 b(athi)23 b(A.,)h Fc(Sur)i(la)f(c)l(onver)l(genc)l(e)j(du)e (semi-gr)l(oup)l(e)h(de)f(L)l(ax-Oleinik)p FB(,)e(C.)g(R.)f(Acad.)h (Sci.)f(P)n(aris)690 541 y(S)n(\023)-36 b(er.)26 b(I)f(Math.)i Fb(324)f FB(\(1998\))h(267-270.)456 633 y([F)-6 b(a2])72 b(F)-6 b(athi)34 b(A.,)j Fc(We)l(ak)f(KAM)f(The)l(or)l(em)h(in)f(L)l (agr)l(angian)i(Dynamics)e FB(\(Cam)n(bridge)g(Studies)e(in)690 724 y(Adv)l(anced)24 b(Mathematics\),)k(\(2009\))f(Cam)n(bridge)f(Univ) n(ersit)n(y)f(Press.)456 815 y([FM])72 b(F)-6 b(athi)31 b(A.)g(and)g(Mather)g(J.,)i Fc(F)-6 b(ailur)l(e)33 b(of)f(c)l(onver)l (genc)l(e)k(of)c(the)h(L)l(ax-Oleinik)g(semi-gr)l(oup)h(in)690 907 y(the)29 b(time-p)l(erio)l(dic)f(c)l(ase,)f FB(Bull.)f(So)r(c.)h (Math.)f(F)-6 b(rance,)26 b Fb(128\(3\))p FB(\(2000\),)j(473{483,)456 998 y([HPS])39 b(Hirsc)n(h)27 b(M.W.)h(Pugh)f(C.C.)h(&)f(Sh)n(ub)e(M.,) j Fc(Invariant)i(Manifolds)p FB(,)d(Lect.)g(Notes)g(Math.)h Fb(583)690 1089 y FB(\(1977\),)g(Springer-V)-6 b(erlag.)456 1181 y([It])134 b(Iturriaga)28 b(R.,)f Fc(Minimizing)g(me)l(asur)l(es)k (for)e(time)f(dep)l(endent)j(L)l(agr)l(angians)p FB(,)e(Pro)r(c.)g (London)690 1272 y(Math.)d(So)r(c.)h Fb(73)f FB(\(1996\))h(216-240.)456 1363 y([KL])84 b(Kaloshin)29 b(V.)e(and)h(Levi)g(M.,)h Fc(A)n(rnold)h(di\013usion)g(for)f(ne)l(ar-inte)l(gr)l(able)k (Hamiltonians)p FB(,)28 b(Bul-)690 1455 y(letin)e(AMS)f Fb(45)h FB(\(2008\))h(409-427.)456 1546 y([Mas])54 b(Massart)35 b(D.,)f Fc(On)g(A)n(ubry)h(sets)h(and)e(Mather's)h(action)g(functional) p FB(,)g(Israel)f(J.)f(Math.)h Fb(134)690 1637 y FB(\(2003\),)28 b(157-171.)456 1729 y([Ma1])46 b(Mather)29 b(J.,)i Fc(A)l(ction)g (minimizing)d(invariant)j(me)l(asur)l(es)h(for)f(p)l(ositive)g (de\014nite)g(L)l(agr)l(angian)690 1820 y(systems,)c FB(Math.)f(Z.,)h Fb(207\(2\))p FB(\(1991\),)h(169{207.)456 1911 y([Ma2])46 b(Mather)19 b(J.,)i Fc(V)-6 b(ariational)22 b(c)l(onstruction)i(of)c(c)l(onne)l(cting)j(orbits,)e FB(Ann.)d(Inst.)g(F)-6 b(ourier)19 b(\(Greno-)690 2003 y(ble\),)26 b Fb(43\(5\))p FB(\(1993\),)j(1349{1386.)456 2094 y([Me1])50 b(Ma)r(~)-41 b(n)n(\023)-36 b(e)19 b(R.,)h Fc(L)l(agr)l(angian)h(\015ows:)32 b(the)21 b(dynamics)g(of)f(glob)l(al) t(ly)g(minimizing)e(orbits)p FB(,)j(Pro)r(ceedings)690 2185 y(In)n(t.)g(Congress)j(in)e(Dynamical)g(Systems)f(\()p Fc(Montevide)l(o)26 b(1995)p FB(\),)d(Pitman)f(Researc)n(h)g(Notes)g (in)690 2277 y(Math.)k Fb(362)h FB(\(1996\),)g(120-131.)456 2368 y([Me2])50 b(Ma)r(~)-41 b(n)n(\023)-36 b(e)30 b(R.,)g Fc(Generic)h(pr)l(op)l(erties)h(and)f(pr)l(oblems)g(of)f(minimizing)e (me)l(asur)l(es)k(of)e(L)l(agr)l(angian)690 2459 y(systems)p FB(,)e(Nonlinearit)n(y)-6 b(,)26 b Fb(9)g FB(\(1996\),169-207.)456 2551 y([T)-6 b(r])113 b(T)-6 b(resc)n(hev)31 b(D.V.,)i Fc(Evolution)h(of)e(slow)h(variables)g(in)g(a)g(priori)f(unstable)j (Hamiltonian)d(sys-)690 2642 y(tems)p FB(,)27 b(Nonlinearit)n(y)-6 b(,)26 b Fb(17)g FB(\(2004\),)h(1803-1841.)456 2733 y([Zh])102 b(Zheng)26 b(Y.,)g(Arnold)f(di\013usion)h(in)f(planar)h(\014v)n(e-b)r (o)r(dy)e(problem,)i(\(2009\))h(preprin)n(t.)555 2987 y FA(Dep)-5 b(ar)g(tment)27 b(of)i(Ma)-5 b(thema)g(tics,)27 b(Nanjing)g(University,)g(Nanjing)f(210093,)k(China,)555 3162 y Fc(E-mail)d(addr)l(ess)6 b FB(:)37 b Fa(chengcq@nju.edu.cn)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0908032129632--