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FEDCCE2A3B336C1B58D71D45B7F05D4FFD3386 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 34 /epsilon put readonly def /FontBBox{-29 -250 1075 750}readonly def /UniqueID 5087384 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBXTI10 %!PS-AdobeFont-1.1: CMBXTI10 1.0 %%CreationDate: 1991 Aug 18 17:46:30 % 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 (CMBXTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMBXTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 46 /period put dup 80 /P put dup 83 /S put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 121 /y put readonly def /FontBBox{-29 -250 1274 754}readonly def /UniqueID 5000771 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 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cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put dup 84 /T 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: 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 11 /alpha put dup 12 /beta put dup 20 /kappa put dup 21 /lambda put dup 23 /nu put dup 27 /sigma put dup 29 /upsilon put dup 34 /epsilon put dup 39 /phi1 put dup 59 /comma put dup 61 /slash put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 97 /a put dup 98 /b put dup 101 /e put dup 102 /f put dup 105 /i put dup 106 /j put dup 109 /m put dup 110 /n put dup 111 /o put dup 113 /q put dup 114 /r put dup 115 /s put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 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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 (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 20 /lessequal put dup 48 /prime put dup 106 /bar put dup 112 /radical put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 73 /contintegraldisplay put dup 80 /summationtext put dup 82 /integraltext put dup 83 /uniontext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 112 /radicalbig put dup 113 /radicalBig put dup 114 /radicalbigg put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CAC6A7BEB5D02276E511FFAF2AE11910 DE076F24311D94D07CACC323F360887F1EA11BDDA7927FF3325986FDB0ABDFC8 8E4B40E7988921D551EC0867EBCA44C05657F0DC913E7B3004A5F3E1337B6987 FEBC45F989C8DC6DC0AD577E903F05D0D54208A0AE7F28C734F130C133B48422 BED48639A2B74E4C08F2E710E24A99F347E0F4394CE64EACB549576E89044E52 EABE595BC964156D9D8C2BAB0F49664E951D7C1A3D1789C47F03C7051A63D5E8 DF04FAAC47351E82CAE0794AA9692C6452688A74A7A6A7AD09B8A9783C235EC1 EA2156261B8FB331827145DE315B6EC1B3D8B67B3323F761EAF4C223BB214C4C 6B062D1B281F5041D068319F4911058376D8EFBA59884BA3318C5BC95684F281 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 91 /bracketleft put dup 93 /bracketright put dup 105 /i put dup 110 /n put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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E092B06173AB7F4ABD9D22CF51CCEE000363A8A169B57C5D6344A535C521D086 4A49566E0B759ABDF46D5D2D22AC2F1074704962581A0E8DD4586E9283140140 E8111554E270B960CCABB193712D0C97FAA105150ED700E45ADCD074D7E26414 60D86D2B3AECD62D175CA80D4C37C4E6A2225B6E85DEDDCC3C21256B10969BF1 6A96F78AE8783C41DAAA6C018AC9F78A2E89047FEA5B1B84C85118FB73F9D466 1A3C37B81B9EA1C12D71353001B59C1273C4E4A660216FC5E34E567842377408 C6C0F7D29855D05C324E5938A86D0DE46166BBC17D9D8FBCD88BDF5A3F79C2CF B9B3E312D0BE5716D2DCEA2185A9CDE69874AB2802676453C95486E478594DCE 8F17DCF802592672DC2D971B68F034033A8F6721542F0DC577D2BADA515C49B3 4F9481B232008D77F3DF45580D47C08187B8061C922FBF5EEED915EEBE9E8EC3 B45DEF06D8350340F5E15C2002C6F019F6CFD0B78200160775270EFBF84F95B1 E34DC9E55BB3FEA2081688633F94E40947BF233598A081328BD4D81F5E2F67A2 F4A737AC06995BC5BD0D7F32947417D8A19295C5D8A6F9BB16D008B4396AD59A E341050F1B44D4DB0A4B32D4358CE412EE536E3AE28092F1509BC933C371C145 A94169185514E87A0F99BA47AF3D1CB471F1EF8FD75687B280D5CAA8502588DF EA9EC1F90BB7D30DDF9D5B05B114DC8D5DDE6EE2EDDEFBAE4AF7379D810993BD D1D3E5086511A343F782C9CD99A5B27C8F6A1431BB15CE2E57A47B43C14956E3 08E7927C7CD45C667C1A4F3B7BCBBBCAABE59FFDFEFAA3DD59F46A780FD249E2 9E8B681F88B21E828FEA7C45B2AA4176C0F85776D31DCC6EAF7D811D07F3B09C F33D0074DB9C2E2669AAD40DA5D60565DF2B15E17610100146DE413946C48E72 38ECA6B19561D5CF87FF747BA60ECC790678528759AFC484844DCEC4696FAECE 3DC47C7972F02906AB36D4C8521CC59FBB6ED55A013F12904784C23F4717E6A0 4BA8533F28D923AD68EFFB408F2249473F1A13B8177C1D05508BF745E3FAF86A 2347B40DE8BAD874401D63A70A71D31229ADCC3760B7DF2E69905CF5102B99F2 41C5E83362D01D3F831E31A3D5C84B300C6CCDA15FAF39C4F305F4376CC2F0CC 2EC49F9B24D115AFA5C679EA795D764DB179B12711B6DF30C7948F107FD9AFC8 4C5F8AC7E2268AAB5395B8E5FDB83CCEE01285CDCB0F96877748F1A282357020 C83D5878A0EF5765B57C3DDC1E972874FFF9C3332B98ED35F85E6CBC5D9460C6 1767DB42DAE18AA1497B030AEAD3B6304D32D6870753350E96EE0295758EC96E 73749644BA6468ED0E07B5C7A9F354165C989BBAA55114034089DE8095007D63 E82E200942EDA5D2D61DA671892520203308567254C21B39EAC3F35D6363ECEC C61D899927B129194E57ECD217F2E3A822AD03E8BC27A8111A3F5B7206EEEFAA 1D79FCC407B782DFD1DC91398951E184B2E7A4FD26F598DF4D55A51DD9956E18 2EDC39FF2EF6229593771C3541579FA9280BA24331DD62D756D269DF7369405F DC5C60D287D6ECB3A71DC124E1A93E0BD0C9DFBF1653BFDE36ECBFF324B37B34 4DDA17053AA5A96B87A59B8D16B71E1A5BD990A2C752EA200E54489350E242AF 80A29C3361EE2A4F07F2ACDE1171F884D1D5C4305B6C5F9D0B643CFD4986D0C6 1CDD466BE1DBB88D35D320859F2CDBE3E01334B2891160E0F67CA233735649DB 4CF6046F4B68D485B2241FE60BCEBC014D255764DA72D1B3511ED3F8ECA4F71C A5CBBD9C285641447D5A8BFD7403C914832411B379BEFFAD88536D5B002A274B DE4C9E143A4F9774772AADDA85F260713B497EC26DBB3F514EEFD50B5E8EF315 5167A26BC727795ECCC06B53CB2CE8F2C76C5A88108D92F373912D1C81435F2E A76B66DB152A6B3CFB7D3B3822692923EE463D949D68655376D941315A4EF970 9921FA5D902E191B247E19388466D41CA3DEE0364FAC5492455E87EF3F0C1D2A A38118F73C0083F9EBAA107B7F6503CFFDF1BD7A7EE5716CB9807D0728E7A3A1 36A77EE8B24A46B4AD4D22E2C22A09ED43D087A0A41ED7C4A516C0FBF05811BC 993E9F36AD02AB2800534784078D2E4FEC210E9749C5C80E1B3BA27B4FACAF00 1C9E9789683C27CA81EFEAAD02A5977AE30AE41C32D85A2696ECD49B0F1EBC08 723B827C226135A75DC7802A7EB5E14E759C1A0AACEADE10039B7ED4F766ED57 FE49955731C6561665E79DF6F54D5EF2E9F799D2711BE41721D360FDB4D7FD31 2EA5EE3594F8DBC53FF50D72635724C4AD3E1A131BA537964B6ED633DAD5BF9F 8A4289E62D78DE82F10C315648D25AEB6FA3ECF2EBD3B6A08105B00BF1180C18 F4E32E29B244A5C5C850220C054E0F7DA52072A09FE7C1CFDF39D4E98A5507A7 BF83A8503D6B38EF236B9185699416416B3B3A5D62A97F3342D62D488E350258 FE660E747059C7BF3F57EF4C37E2C44A1423FFEE63B47DCCD9A32AE210C2DE57 B22CF1D2C72D495BA24342F713118A38B91314779A65374A1A719600D4604AB5 2A6E566B94AFCF9DCEF8F7394AB46C8DA89DC6823F4235C4092A43D84BC5ED73 B2194B48480AB654FB405E8543F74F0167E735B18F7DB5644ED3567C1256734F 3DF9634BCD3D0A560FE322C15798563584A9CE0D9D600D36C086F27B756FB4B6 2DC97BB2B372CFF6051CB606A167AE023BE9CE57082627B50392F4D1B5190A8C EE1C17202DB16465D287949EB464CB8B969C1E3A47E8E07209A72FE073C396AC 221684D8538A7E6E464A469464F1AE9F3869C671DC7C4F1E6227F8EF06179A2F 338D06C81DD2A19E8A018F24669EDEA8693E1DB8EC41EEA673F37B47D8159B8A B509F45071549E752004470CB65027407D8EB5314550B3092235F4E3B0E7FB66 DB9D7CA110D52B10F4C17405194782FD18D6571AAA111C6271953C43F7C4EA0A 8671B04BFCB281B167B14FD1444ED53D4B43BB8A9D5CACEF77CEE2B5708359EF 3AAE83E9C9F4A19301D8E71C565DD64A463202DF0E65BAB8F0793B71FB9C87C0 A0CBC4458C106031BFA3DAC711964F92B27957B9BB95E0C759C14E2296F16CC2 52D0183D2FB04A91FF0E2EF1043590C5C3D5FF37DA3D50663D43FFBFD07110AD 14A24E8FD394DAC490E48BF7727FDC49939E36F108197453577AD42B80E1A830 F4E76F6C9CD6F1582E710B68704ADC5F5192BA8030B96A66871267B71E0CF222 2CB38727B63750D3DAF737B7EC9B747AD7E3ECF2A463A99A67C34F84F90534CA A408CA00547C738DB0F8857B1773A7760639205F2A8E96FE620914B8413C23F5 CFBC949F257923397E1B0F389CE046CEA8FBF4A5D7328EF03B738A939ED29183 928F7F9C48E870FF85CF1C79B7C84392134F1C3E328C9597DDDD1C920B59A106 E1858D3C631AA4119E15D37DBE0822432551520A5F2B676A10BE4680F225D1BF 7F6CC08DE7B6BB006FD4BF5FA613D608EE9FEDDA1B624F8AE8E4ED949FDD53A1 82ED2D309337896D98BD3A5A5C300DA95FD970D70862104CA132BF9E2D7B5676 C419B7F8225402E177DCA475E2485540D379822B801E3CE28F5B3DD80CAB8F28 1EB5AFFEFD72E3B0D128558B45ED6AB22FA6A7C8A83835A0B2CE6B90B90DDF92 64C7158FB8812B6A9EEF52123E6CA861 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 dup 116 /fork put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile eexec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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 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 7 /minusplus put dup 10 /circlemultiply put dup 14 /openbullet put dup 20 /lessequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 59 /emptyset put dup 67 /C put dup 68 /D put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 78 /N put dup 82 /R put dup 84 /T put dup 86 /V put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 110 /backslash put dup 112 /radical put dup 114 /nabla put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 29 /upsilon put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon 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 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 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{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 6 /plusminus put dup 7 /minusplus put dup 14 /openbullet put dup 15 /bullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 28 /lessmuch put dup 33 /arrowright put dup 39 /similarequal put dup 41 /arrowdblright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 59 /emptyset put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 16 /dotlessi put dup 18 /grave put dup 19 /acute put dup 34 /quotedblright 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 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 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 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 29 /upsilon put dup 33 /omega put dup 34 /epsilon put dup 37 /rho1 put dup 39 /phi1 put dup 58 /period put dup 59 /comma 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 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 83 /S put dup 84 /T put dup 87 /W put dup 89 /Y 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 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{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /Lambda put dup 22 /macron put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma 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 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 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 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 126 /tilde put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 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 (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 19 /acute put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 52 /four put dup 53 /five put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z 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 dup 127 /dieresis put readonly def /FontBBox{-35 -250 1148 750}readonly def /UniqueID 5000827 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E3948FFB3DF7BFF10C9BDA4EFE5F68A8CB1526990D1357AE6D2F7C2D2EF8496 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 3 /Lambda put dup 5 /Pi put dup 6 /Sigma put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 16 /dotlessi put dup 19 /acute put dup 21 /breve put dup 22 /macron put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMCSC10 %!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 18 /grave 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 54 /six put dup 55 /seven put dup 56 /eight 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 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 88 /X put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueID 5000772 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A30EB76029337 900ECFB1390CA5C0C3A04528044F266BA17BE487C79B94FAC6D6484684C5BFEA 87BCCC77D40AD11552035E95E3007126418ED49B68468B38A14E88E68A267B98 076F1C9769A5AFBC285E5B158EAC9F926F1D6C0B8F1D57D9C31D25AE27123518 9D2CD92E5689E0213089BD268DA5E47525CB8EABAA4B78A15AEA34705889AB3A FFB8953B5B3482E52BFA0940630ADF8C0AC2177D907324299EE980E850F203CD 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 14 /ffi put dup 16 /dotlessi put dup 18 /grave put dup 19 /acute put dup 21 /breve 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 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 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 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 127 /dieresis put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 89 /Y 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 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{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F00F963068B8B731A88D7740B0DDAED1B3F82 7DB9DFB4372D3935C286E39EE7AC9FB6A9B5CE4D2FAE1BC0E55AE02BFC464378 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cleartomark %%EndFont TeXDict begin 40258431 52099146 1000 600 600 (new.dvi) @start /Fa 171[50 84[{}1 74.7198 /MSBM10 rf /Fb 135[39 1[39 39 39 39 39 1[39 2[39 39 3[39 2[39 39 39 1[39 28[39 2[39 39 17[39 39 45[{}19 74.7198 /CMTT9 rf /Fc 221[36 34[{}1 74.7198 /CMMI9 rf /Fd 134[51 4[35 44 46 2[54 59 86 27 2[32 2[36 48 54 48 1[54 13[59 2[72 33[32 46[{}17 90.9091 /CMBXTI10 rf /Fe 165[44 5[44 1[48 82[{}3 66.4176 /MSBM7 rf /Ff 140[29 29 28 1[31 38 54 2[25 22 2[30 29 2[27 34 23[27 1[47 39 45 7[32 1[19 19[40 4[29 4[34 1[36 3[31 1[37 36 7[35 40 11[{}27 49.8132 /CMMI6 rf /Fg 143[50 5[19 57[18 27[48 16[32 2[48{}6 49.8132 /CMSY6 rf /Fh 141[91 91 91 20[101 51 1[131 4[76 43 1[96 6[51 45[68 68 4[48 48 67 67 54 54 3[30 2[53 53 6[42 42{}23 90.9091 /CMEX10 rf /Fi 136[43 1[34 24 24 4[34 4[18 11[18 1[18 35[30 30 30 30 30 30 30 30 4[47 1[24 24 40[{}19 49.8132 /CMR6 rf /Fj 139[61 112[71 3[{}2 90.9091 /MSAM10 rf /Fk 165[61 5[61 1[66 71 2[66 78[{}5 90.9091 /MSBM10 rf /Fl 141[59 1[59 1[35 3[20 2[35 35 9[47 47 4[44 1[39 1[60 3[58 1[49 54 48 39 4[55 37 7[35 4[0 3[47 71 19 14[71 12[55 5[35 3[55 2[55 55 2[35 55 20 55{}33 66.4176 /CMSY8 rf /Fm 133[42 45 52 65 44 52 33 43 41 41 46 44 55 80 27 47 37 31 52 43 45 42 47 39 39 48 38 6[53 1[86 53 62 53 56 69 1[58 69 73 88 62 77 50 40 76 71 58 67 75 65 69 68 48 1[71 45 71 25 25 18[59 1[47 2[42 57 59 57 1[49 40 52 47 52 40 45 55 53 52 1[43 45 2[40 47 51 58 11[{}78 90.9091 /CMMI10 rf /Fn 135[40 5[76 1[76 1[45 3[25 2[45 45 7[61 1[61 61 66 61 65 1[56 1[50 55 77 3[75 109 63 69 62 50 77 54 65 1[70 48 60 73 5[45 1[51 51 0 0 3[61 91 7[91 1[71 5[91 4[91 1[71 4[71 71 2[71 1[45 45 6[71 71 3[71 25 71{}52 90.9091 /CMSY10 rf /Fo 132[46 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 3[28 47 28 3[91 68 1[65 51 66 1[62 70 68 82 57 70 1[35 68 70 59 62 69 65 64 68 6[28 1[46 46 46 46 46 46 46 46 46 1[28 33 28 2[37 37 28 4[47 14[46 46 1[28 1[80 53 51 56 11[{}74 90.9091 /CMTI10 rf /Fp 133[33 35 40 51 34 41 25 33 32 32 36 34 43 62 21 37 29 24 41 1[34 33 36 31 30 37 29 6[41 1[67 2[41 43 2[45 1[56 68 48 60 39 31 1[55 45 52 58 50 53 53 37 1[55 35 1[20 20 18[46 1[36 2[33 44 3[38 1[40 36 41 1[35 42 41 41 1[33 35 2[31 36 40 45 11[{}67 66.4176 /CMMI8 rf /Fq 129[35 6[51 37 39 27 28 28 1[39 35 39 59 20 2[20 39 35 1[31 39 31 39 35 3[20 1[20 4[53 53 51 39 52 2[55 53 65 44 3[53 1[46 48 54 2[53 3[55 3[35 35 35 35 35 35 35 35 35 35 1[20 1[20 55 1[27 27 17[35 18[49 3[{}54 66.4176 /CMR8 rf /Fr 128[39 4[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 6[47 57 1[77 57 57 55 43 56 1[52 59 57 69 48 1[40 30 57 59 50 52 58 55 54 57 6[24 39 39 39 1[39 39 2[39 2[24 27 24 2[31 31 24 19[39 5[45 43 47 11[{}66 74.7198 /CMTI9 rf /Fs 128[45 45 1[91 45 40 48 48 66 48 51 35 36 36 48 51 45 51 76 25 48 28 25 51 45 28 40 51 40 51 45 1[25 45 25 45 25 56 1[68 93 68 68 66 51 67 1[62 71 68 83 57 71 47 33 68 71 59 62 69 66 64 68 3[71 1[25 25 45 45 45 45 45 45 45 45 45 45 45 25 30 25 71 1[35 35 25 4[45 11[45 45 1[45 2[25 1[76 51 51 53 66 71 66 1[66 68 1[63 1[76 57{}97 90.9091 /CMR10 rf /Ft 134[56 56 76 56 56 54 42 55 1[51 58 56 68 47 58 1[27 56 58 49 51 57 54 53 56 12[71 56 73 1[67 77 74 4[37 74 77 65 67 1[71 1[74 9[50 3[50 50 50 2[29 1[29 44[{}42 90.9091 /CMCSC10 rf /Fu 134[46 46 1[46 46 44 34 45 1[42 47 46 56 38 2[23 46 48 40 42 47 44 43 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y(b)s(e)e(used)g(as)h(elemen)m(ts)i(of)e(transition)g(c)m(hains)g (that)h(o)m(v)m(ercome)h(the)e(large)i(gap)456 2501 y(problem.)555 2609 y(The)45 b(second)h(goal)h(\(whic)m(h)f(tak)m(es)h(up)d(the)i (bulk)f(of)g(this)h(pap)s(er\))f(is)g(to)456 2717 y(v)m(erify)25 b(rigorously)g(the)h(existence)g(of)f(these)h(mec)m(hanisms)f(in)g (rather)f(concrete)456 2825 y(systems.)39 b(The)25 b(v)m(eri\014cation) i(will)f(b)s(e)f(rather)g(explicit,)j(and)d(giv)m(en)h(a)g(concrete)456 2932 y(systems)i(there)g(are)g(\014nite)g(calculations)i(whic)m(h)d (establish)i(it.)40 b(In)27 b(particular,)456 3040 y(w)m(e)j(will)h (study)f(a)g(mo)s(del)h(that)g(has)f(b)s(een)f(studied)h(already)h(in)f ([HM82)r(].)555 3148 y(W)-8 b(e)31 b(note)g(that)f(the)g(mo)s(del)g(w)m (e)g(discuss)f(presen)m(ts)h(the)g Fo(lar)-5 b(ge)33 b(gap)g(pr)-5 b(oblem)456 3256 y Fs(\(namely)24 b(that)g(the)f(size)i (of)e(the)h(gaps)g(b)s(et)m(w)m(een)g(the)g(KAM)f(tori)h(is)g(larger)g (than)456 3364 y(the)38 b(size)h(of)f(\014rst)g(order)g(c)m(hange)h(in) f(the)g(\(un\)stable)h(manifolds\))f(that)h(has)456 3472 y(pla)m(y)m(ed)27 b(an)g(imp)s(ortan)m(t)f(role)h(in)g(Arnol'd)f (di\013usion)g(\(an)h(excellen)m(t)h(discussion)456 3580 y(of)j(the)h(problem)f(and,)h(indeed)f(of)h(the)g(problem)f(of)g (di\013usion)g(can)h(b)s(e)f(found)456 3688 y(in)f([Mo)s(e96)r(]\))555 3796 y(Our)36 b(main)g(rigorous)h(result)g(Theorem)f(7)h(establishes)g (that)h(the)f(mec)m(ha-)456 3904 y(nisms)30 b(that)j(w)m(e)f(presen)m (t)f(o)m(v)m(ercome)j(the)e(large)h(gap)f(problem)f(in)g(the)h(mo)s (del)456 4012 y(considered.)39 b(Theorem)28 b(7)h(presen)m(ts)f(sev)m (eral)i(rather)e(explicit)i(su\016cien)m(t)e(con-)456 4120 y(ditions)g(that)i(ensure)e(that)h(a)g(system)g(whic)m(h)f(v)m (eri\014es)h(them)g(has)g(orbits)f(that)456 4228 y(tra)m(v)m(erse)k(a)e (large)i(gap)e(region.)555 4336 y(W)-8 b(e)27 b(b)s(eliev)m(e)f(that)g (the)g(mec)m(hanisms)f(prop)s(osed)g(here)g(ha)m(v)m(e)h(the)g(adv)-5 b(an)m(tage)456 4443 y(that)40 b(they)g(\014t)g(b)s(etter)g(some)g(of)g (the)h(in)m(tuition)f(gathered)h(from)e(n)m(umerical)456 4551 y(and)29 b(real)i(exp)s(erimen)m(ts)g(than)f(the)g(mec)m(hanism)h (of)g([Arn64].)555 4659 y(Note)50 b(that)e(the)g(n)m(umerical,)54 b(exp)s(erimen)m(tal)49 b(and)e(geometric)j(in)m(tuition)456 4767 y(\(see)36 b(e.g.)56 b(the)36 b(classical)h([Chi79)q(])f(or)f ([CSUZ89,)g(Mei92)r(,)h(JVMU99)q(,)g(Las93)q(,)456 4875 y(LHRK02])27 b(among)g(man)m(y)f(others)h(\))g(is)f(that)h(resonances)g (generate)h(di\013usion.)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)789 b(5)456 450 y Fs(On)33 b(the)i(other)g(hand,)f(the)h(mec)m(hanism)g(of)g([Arn64])g(has)f (di\016cult)m(y)h(dealing)456 558 y(with)d(resonances,)i(whic)m(h)e (destro)m(y)h(the)g(primary)f(tori.)48 b(Indeed,)33 b(one)g(of)g(the) 456 666 y(main)25 b(problems)f(to)i(establish)f(rigorously)g(the)h (existence)g(of)f(the)h(mec)m(hanism)456 774 y(of)31 b([Arn64)q(])g(is)h(the)g Fo(lar)-5 b(ge)34 b(gap)g(pr)-5 b(oblem)p Fs(,)34 b(whic)m(h)d(refers)g(to)i(the)e(fact)i(that)f(the) 456 882 y(resonances)22 b(create)i(gaps)e(in)g(the)h(families)g(of)f (primary)f(whisk)m(ered)h(tori)h(whose)456 990 y(size)42 b(is)g(bigger)h(than)e(the)i(angle)g(b)s(et)m(w)m(een)f(the)g(whisk)m (ers)g(of)g(the)g(primary)456 1098 y(whisk)m(ered)34 b(tori.)54 b(In)33 b(other)i(w)m(ords,)h(the)e(mec)m(hanism)h(of)g ([Arn64])g(for)f(di\013u-)456 1206 y(sion)d(is)h(di\016cult)g(to)h(v)m (erify|and)e(can)h(b)s(e)g(presumably)e(false|precisely)j(in)456 1314 y(the)23 b(places)h(where)f(exp)s(erimen)m(tal)i(evidence)f (suggests)g(that)g(di\013usion)f(should)456 1421 y(b)s(e)29 b(most)i(in)m(tense.)555 1529 y(F)-8 b(or)38 b(the)e(mec)m(hanisms)h (that)g(w)m(e)g(prop)s(ose)f(here,)i(w)m(e)f(observ)m(e)g(that)g(reso-) 456 1637 y(nances,)c(ev)m(en)g(if)f(they)h(destro)m(y)g(the)f(primary)g (tori,)h(they)g(create)h(secondary)456 1745 y(tori)f(and)g(tori)g(of)g (lo)m(w)m(er)h(dimension)f(that)g(bridge)g(them,)h(so)f(that)h(the)f (tran-)456 1853 y(sition)38 b(c)m(hains)g(can)g(con)m(tin)m(ue.)64 b(Indeed,)39 b(in)e(agreemen)m(t)j(with)d(the)h(ph)m(ysical)456 1961 y(in)m(tuition,)e(the)f(secondary)g(tori)g(created)g(b)m(y)g (resonances,)h(lead)f(to)g(a)g(larger)456 2069 y(increase)46 b(in)f(action)i(in)f(their)f(elemen)m(ts)i(of)f(the)g(transition)g(c)m (hain.)87 b(\(See)456 2177 y([Hal97)r(,)41 b(Hal99)r(])h(for)f(a)h (discussion)e(of)i(the)g(role)g(of)f(double)g(resonances)h(in)456 2285 y(di\013usion\).)555 2393 y(Of)20 b(course,)j(w)m(e)d(are)h(far)f (from)g(b)s(elieving)h(that)f(the)h(mec)m(hanisms)f(w)m(e)h(presen)m(t) 456 2501 y(here)32 b(are)g(the)g(only)g(mec)m(hanisms)h(for)f (di\013usion.)45 b(Indeed,)32 b(some)g(other)g(geo-)456 2609 y(metric)42 b(mec)m(hanisms)g(ha)m(v)m(e)h(b)s(een)f(rigorously)g (established)g(for)g(other)g(sys-)456 2717 y(tems.)c(F)-8 b(or)24 b(example,)h([DLS00)q(,)e(BT99)q(,)g(DLS01])g(study)f (geometrically)k(a)d(sys-)456 2825 y(tem)f(that)h(has)f(b)s(een)f (studied)h(in)g([Mat95)r(])h(b)m(y)f(using)f(v)-5 b(ariational)24 b(metho)s(ds.)38 b(A)456 2932 y(v)-5 b(ariational)29 b(approac)m(h)e(to)h(Arnol'd)f(di\013usion)f(can)h(b)s(e)g(found)e(in)i ([BB02)r(])g(and)456 3040 y(announcemen)m(ts)36 b(of)g(other)g(v)-5 b(ariational)38 b(metho)s(ds)d(are)h(in)g([Xia98)r(,)g(Mat02)r(].)456 3148 y(The)43 b(pap)s(ers)h([Lla02)q(,)h(Mo)s(e02)q(,)g(EMR01)q(,)f(T) -8 b(re02)q(])45 b(study)e(other)i(geometric)456 3256 y(mec)m(hanisms.)61 b(There)36 b(are)i(heuristic)f(descriptions)g(and)f (n)m(umerical)i(explo-)456 3364 y(rations)30 b(of)h(other)g(geometric)h (mec)m(hanisms)e(in)h([L)-8 b(T83)q(,)30 b(T)-8 b(en82)q(,)31 b(CLSV85].)555 3472 y(In)c(Section)h(2,)h(w)m(e)f(will)g(describ)s(e)f (the)h(prop)s(osed)e(mec)m(hanism)i(in)g(an)f(infor-)456 3580 y(mal)g(and)g(conjectural)i(w)m(a)m(y)-8 b(.)41 b(Hence,)29 b(the)f(reader)f(in)m(terested)h(only)g(in)f(results)456 3688 y(that)k(are)f(rigorously)h(pro)m(v)m(ed)g(can)f(skip)g(this)h (section)g(altogether.)555 3796 y(In)e(Section)i(3,)g(w)m(e)f(presen)m (t)g(a)g(class)h(of)f(mo)s(del)g(systems)g(in)f(whic)m(h)h(w)m(e)h (will)456 3904 y(v)m(erify)24 b(rigorously)h(the)f(mec)m(hanisms)h (describ)s(ed)e(in)h(Section)h(2.)39 b(In)23 b(Section)i(4,)456 4012 y(w)m(e)32 b(will)g(state)h(Theorem)e(7)h(that)g(v)m(eri\014es)g (the)g(mec)m(hanism)g(in)g(the)f(concrete)456 4120 y(mo)s(del)f(in)m (tro)s(duced)g(in)g(Section)h(3.)555 4228 y(The)f(subsequen)m(t)g (sections)h(are)g(dev)m(oted)g(to)g(the)g(pro)s(of)f(of)g(Theorem)g(7.) 555 4336 y(After)j(in)m(tro)s(ducing)f(in)g(Section)i(5)f(some)g (notation,)h(in)e(Section)i(6)f(w)m(e)g(de-)456 4443 y(scrib)s(e)39 b(the)h(geometry)h(of)f(the)g(problem,)i(whic)m(h)d(has) h(sev)m(eral)h(elemen)m(ts)g(in)456 4551 y(common)31 b(with)f(that)i(of)f([DLS00)q(].)42 b(Ev)m(en)31 b(if)g(the)g(presen)m (t)g(pap)s(er)e(is)i(logically)456 4659 y(indep)s(enden)m(t)g(of)i ([DLS00)q(])f(\(w)m(e)i(only)f(use)f(a)h(few)f(of)h(the)g(tec)m(hnical) h(results\),)456 4767 y(w)m(e)f(ha)m(v)m(e)h(included)e(in)g(Section)i (12.2)g(a)f(discussion)f(of)h(the)g(similarities)h(and)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 456 251 a Fq(6)685 b(A.)23 b(Delshams,)g(R.)g(de) h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 450 y Fs(di\013erences)j(b)s (et)m(w)m(een)h(the)f(phenomena)g(considered)f(and)h(the)g(metho)s(ds)g (used)456 558 y(in)k(this)g(pap)s(er)f(and)h(in)g([DLS00)q(].)555 666 y(In)23 b(subsequen)m(t)f(subsections)i(w)m(e)f(will)h(pro)m(v)m(e) g(Theorem)f(7,)j(and)c(w)m(e)i(will)g(de-)456 774 y(v)m(elop)f(to)s (ols)h(to)g(establish)f(that,)i(indeed,)f(our)f(system)g(satis\014es)g (the)g(prop)s(osed)456 882 y(mec)m(hanism.)555 990 y(W)-8 b(e)33 b(note)e(that)h(the)f(metho)s(ds)g(w)m(e)g(presen)m(t)g(are)h (rather)f(constructiv)m(e)h(and)456 1098 y(are)e(capable)h(of)f (establishing)h(\(or)f(at)h(least)g(reduce)f(to)h(a)f(\014nite)g (calculation\))456 1206 y(the)22 b(existence)j(of)d(the)h(mec)m (hanisms)g(in)f(a)h(concrete)i(system)d(giv)m(en)i(b)m(y)f(explicit)456 1314 y(form)m(ulas.)40 b(Indeed,)29 b(in)g(Section)i(13,)g(w)m(e)f (presen)m(t)f(some)h(systems)g(where)f(it)h(is)456 1421 y(p)s(ossible)21 b(to)i(compute)f(the)g(form)m(ulas)h(needed)e(in)h (closed)h(form)e(so)h(that)h(w)m(e)g(can)456 1529 y(ensure)31 b(that)i(these)g(systems)f(ha)m(v)m(e)i(tra)5 b(jectories)34 b(that)f(cross)g(the)f(resonance)456 1637 y(gaps.)555 1745 y(W)-8 b(e)42 b(call)g(atten)m(tion)h(to)f(the)f(fact)g(that)h (the)f(pro)s(of)f(w)m(e)h(presen)m(t)g(is)f(quite)456 1853 y(mo)s(dular)32 b(and)g(consists)h(on)g(w)m(ell)h(de\014ned)e (steps)h(that)g(dep)s(end)f(only)h(on)g(the)456 1961 y(results)k(of)h(the)f(previous)g(ones.)62 b(If)37 b(one)h(is)g (willing)g(to)g(mak)m(e)h(assumptions)456 2069 y(that)e(yield)h(the)f (results)g(of)h(one)f(step,)i(or)f(\014nds)d(a)j(tec)m(hnique)g(that)f (v)m(eri\014es)456 2177 y(the)30 b(steps)g(indep)s(enden)m(tly)-8 b(,)31 b(then)f(the)g(rest)h(of)f(the)h(pro)s(of)e(can)i(b)s(e)f(used.) 555 2285 y(W)-8 b(e)31 b(also)g(note)g(that)f(most)g(of)g(the)g(to)s (ols)h(w)m(e)f(need)g(for)f(the)h(v)m(eri\014cation)i(of)456 2393 y(the)f(mec)m(hanism)h(prop)s(osed)e(here)i(are)g(v)-5 b(arian)m(ts)32 b(of)f(standard)g(tec)m(hniques)h(in)456 2501 y(dynamical)f(systems)g(and)f(sp)s(ecially)i(in)f(di\013usion.)42 b(Most)31 b(of)h(the)f(results)f(w)m(e)456 2609 y(use)d(can)h(b)s(e)f (obtained)g(through)g(readily)h(a)m(v)-5 b(ailable)30 b(tec)m(hniques.)40 b(Hence,)30 b(the)456 2717 y(main)j(no)m(v)m(elt)m (y)i(of)f(this)f(pap)s(er)f(lies)i(in)f(the)g(o)m(v)m(erall)j(strategy) e(and)f(the)g(do)m(v)m(e-)456 2825 y(tailing)e(of)f(di\013eren)m(t)g (geometric)i(structures)e(\(this)g(is)g(wh)m(y)f(w)m(e)i(ha)m(v)m(e)g (decided)456 2932 y(to)g(include)g(the)g(heuristic)h(description)f(of)g (the)g(mec)m(hanism.)43 b(Undoubtedly)-8 b(,)456 3040 y(some)30 b(exp)s(erts)g(will)h(ha)m(v)m(e)h(little)g(trouble)e (\014lling)h(the)f(details)i(themselv)m(es\).)555 3148 y(W)-8 b(e)37 b(hop)s(e)e(that)h(these)f(impro)m(v)m(emen)m(ts)i (\(sharp)s(ening)d(the)i(analytic)h(to)s(ols)456 3256 y(so)42 b(that)g(they)g(b)s(ecome)g(reusable)g(as)g(parts)f(of)h (longer)g(argumen)m(ts\))h(could)456 3364 y(b)s(e)37 b(useful)f(for)i(the)f(study)g(of)h(other)g(mec)m(hanisms)f(for)h (di\013usion.)61 b(W)-8 b(e)39 b(also)456 3472 y(exp)s(ect)25 b(that)g(the)g(ingredien)m(ts)g(w)m(e)g(ha)m(v)m(e)h(studied|mainly)e (normally)h(h)m(yp)s(er-)456 3580 y(b)s(olic)h(in)m(v)-5 b(arian)m(t)27 b(manifolds,)g(secondary)f(tori|can)h(b)s(e)e (rearranged)h(in)g(other)456 3688 y(w)m(a)m(ys)35 b(to)g(pro)m(vide)f (other)h(mec)m(hanisms)f(for)g(di\013usion)g(or)g(that)h(they)g(can)f (b)s(e)456 3796 y(in)m(tegrated)i(with)f(other)g(approac)m(hes,)i (notably)e(v)-5 b(ariational)37 b(metho)s(ds.)54 b(On)456 3904 y(the)29 b(other)h(hand,)e(w)m(e)i(note)g(that)g(con)m(v)m(exit)m (y)i(of)d(the)h(problem)f(do)s(es)g(not)g(pla)m(y)456 4012 y(an)m(y)h(role)h(in)f(our)g(metho)s(ds)g(in)g(con)m(trast)i(with) e(v)-5 b(ariational)32 b(metho)s(d.)555 4120 y(One)40 b(fundamen)m(tal)g(to)s(ol)i(that)f(w)m(e)f(adopted)h(from)f([DLS00)q (])g(is)h(the)f(use)456 4228 y(of)34 b(scattering)h(maps)e(to)i (normally)f(h)m(yp)s(erb)s(olic)f(in)m(v)-5 b(arian)m(t)35 b(manifolds.)50 b(The)456 4336 y(scattering)34 b(map)e(metho)s(d)g (allo)m(ws)i(us)e(to)i(discuss)e(in)g(a)h(geometric)i(w)m(a)m(y)e(het-) 456 4443 y(ero)s(clinic)j(phenomena.)53 b(In)34 b(particular,)j(w)m(e)f (can)f(discuss)f(comfortably)i(the)456 4551 y(in)m(tersection)27 b(of)e(in)m(v)-5 b(arian)m(t)27 b(manifolds)e(of)h(tori)g(of)g (di\013eren)m(t)f(top)s(ologies)j(\(ev)m(en)456 4659 y(of)35 b(di\013eren)m(t)h(dimensions\).)56 b(This)35 b(discussion)g(do)s(es)g(not)h(seem)f(straigh)m(tfor-)456 4767 y(w)m(ard)d(in)g(the)g(usual)g(Melnik)m(o)m(v)i(theory)f(whic)m(h) f(often)h(requires)f(to)h(put)f(b)s(oth)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)789 b(7)456 450 y Fs(ob)5 b(jects)34 b(exp)s(eriencing)g(the)g(homo)s(clinic)g(phenomena)f(in)h (the)g(same)g(system)456 558 y(of)c(co)s(ordinates.)555 666 y(W)-8 b(e)26 b(ha)m(v)m(e)f(c)m(hosen)g(to)g(v)m(erify)g(the)g (results)f(for)g(one)g(parameter)h(families)g(and)456 774 y(form)m(ulate)38 b(our)f(results)g(in)g(suc)m(h)g(a)g(w)m(a)m(y)h (that)g(these)g(results)f(apply)g(for)g(all)456 882 y Fn(j)p Fm(")p Fn(j)26 b Fm(<)f(")712 849 y Fl(\003)752 882 y Fs(.)38 b(Of)23 b(course,)j(the)e Fm(")1428 849 y Fl(\003)1491 882 y Fs(dep)s(ends)e(on)i(the)g(family)g(and)f(can)h (tend)g(to)g(zero)456 990 y(as)43 b(the)g(family)g(considered)f (approac)m(hes)i(a)f(particularly)g(degenerate)i(one)456 1098 y(whic)m(h)27 b(do)s(es)f(not)i(satisfy)f(the)h(assumptions)e(of)i (the)f(theorem.)40 b(W)-8 b(e)29 b(ha)m(v)m(e)f(also)456 1206 y(c)m(hosen)41 b(to)h(form)m(ulate)g(our)e(results)h(b)m(y)f (obtaining)i(computable)f(su\016cien)m(t)456 1314 y(conditions)30 b(on)h(the)f(family)h(for)f(the)h(results)f(to)h(apply)-8 b(.)555 1421 y(F)g(or)48 b(one)f(parameter)h(families,)k(it)c(is)f (customary)g(to)h(classify)g(them|)456 1529 y(follo)m(wing)23 b(the)f(notation)h(of)f([CG94)q(]|)g(in)g Fo(a-priori)k(stable)j Fs(or)22 b Fo(a-priori)k(unsta-)456 1637 y(ble)j Fs(systems)22 b(dep)s(ending)f(on)h(whether)g(the)h(unp)s(erturb)s(ed)18 b(system)23 b(is)f(strongly)456 1745 y(in)m(tegrable)32 b(\(can)f(b)s(e)e(put)h(in)g(action-angle)j(v)-5 b(ariables\))32 b(or)e(not.)555 1853 y(Of)43 b(course,)48 b(this)c(is)g(not)g(the)g (only)g(p)s(ossible)f(p)s(oin)m(t)h(of)g(view)g(and)f(it)i(is)456 1961 y(in)m(teresting)23 b(to)g(compare)g(this)f(form)m(ulation)i(with) e(other)g(form)m(ulations)h(of)g(the)456 2069 y(problem)33 b(of)g(di\013usion.)50 b(F)-8 b(or)34 b(example,)h(the)f(metho)s(d)f (in)g([Arn64])h(considers)456 2177 y(families)d(dep)s(ending)e(on)h(t)m (w)m(o)h(parameters)g(\(the)g(second)f(one)h(exp)s(onen)m(tially)456 2285 y(small)f(with)h(resp)s(ect)f(to)h(the)g(\014rst\).)555 2393 y(It)h(is)g(also)h(natural)f(and)f(customary)h(to)g(mak)m(e)h (assertions)g(for)e(generic)i(or)456 2501 y(for)h(\\cusp)g(residual)h (sets"\(see[Mat02)t(]\).)54 b(Nev)m(ertheless,)38 b(in)c(this)h(pap)s (er)e(w)m(e)456 2609 y(ha)m(v)m(e)g(adopted)e(the)h(concrete)h(v)m (eri\014cation)h(p)s(oin)m(t)d(of)h(view)g(and)f(relegated)i(a)456 2717 y(ten)m(tativ)m(e)g(discussion)d(of)g(genericit)m(y)j(to)e (remarks.)555 2825 y(It)40 b(is)f(imp)s(ortan)m(t)h(to)g(realize)h (that,)i(when)c(one)g(considers)h(t)m(w)m(o)g(parame-)456 2932 y(ter)26 b(families|a)h(fortiori)g(if)e(one)i(consider)f(generic)g (results|the)g(distinction)456 3040 y(b)s(et)m(w)m(een)39 b(a-priori)g(stable)g(systems)g(and)f(a-priori)h(unstable)g(systems)f (do)s(es)456 3148 y(not)g(mak)m(e)g(sense.)63 b(One)38 b(can)g(use)f(the)h(\014rst)f(parameter)h(to)h(mo)m(v)m(e)g(the)f(sys-) 456 3256 y(tem)33 b(a)m(w)m(a)m(y)h(from)e(a-priori)h(stable)g(and)f (then)g(use)g(the)h(second)f(parameter|)456 3364 y(or)46 b(genericit)m(y|applying)j(the)e(metho)s(ds)e(of)i(a-priori)g(unstable) g(systems.)456 3472 y(Hence,)27 b(one)e(can)g(hop)s(e)f(that)i(the)f (metho)s(ds)f(of)h(one)h(parameter)f(families)h(near)456 3580 y(a-priori)32 b(unstable)g(systems)h(apply)f(to)h(obtain)f(t)m(w)m (o)i(parameters)f(or)f(generic)456 3688 y(results.)555 3796 y(The)k(study)e(of)i(one)g(parameter)h(families)f(near)g(a-priori) g(stable)h(systems)456 3904 y(seems)32 b(to)g(require)g(considerations) h(of)f(another)g(nature.)46 b(There)31 b(are)h(indeed)456 4012 y(normal)22 b(forms)g(near)h(resonances)g(similar)g(to)g(the)g (systems)f(w)m(e)h(consider)g(here.)456 4120 y(Nev)m(ertheless)31 b(there)f(app)s(ear)f(exp)s(onen)m(tially)h(small)h(phenomena)e(in)g (the)h(pa-)456 4228 y(rameter)k(and)g(dealing)h(with)f(them)g(without)g (in)m(tro)s(ducing)g(other)g(exp)s(onen-)456 4336 y(tially)c(small)e (parameters)h(or)g(generic)g(p)s(erturbations)e(requires)h(delicate)j (an-)456 4443 y(alytical)h(to)s(ols.)867 4694 y(2.)46 b Ft(Heuristic)34 b(discussion)f(of)g(the)h(mechanism)555 4856 y Fs(In)39 b(this)h(section,)j(w)m(e)d(will)g(describ)s(e)f (heuristically)i(what)f(are)g(the)f(ideas)456 4964 y(that)31 b(lead)f(to)i(the)e(mec)m(hanism)h(of)f(di\013usion)g(prop)s(osed)f(in) h(this)h(pap)s(er.)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 456 251 a Fq(8)685 b(A.)23 b(Delshams,)g(R.)g(de) h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)555 450 y Fs(The)d(discussion)g (in)g(this)h(section)h(will)f(b)s(e)e(completely)k(non-rigorous.)37 b(Nev-)456 558 y(ertheless,)31 b(w)m(e)g(hop)s(e)e(that)i(it)g(can)g (serv)m(e)g(as)g(stim)m(ulus)f(for)g(future)f(w)m(ork.)555 666 y(Of)36 b(course,)j(the)d(reader)h(in)m(terested)h(only)e(in)g (rigorous)h(results)f(can)h(skip)456 774 y(to)30 b(Section)g(3)f(and)g (the)h(follo)m(wing)g(after)g(bro)m(wsing)f(through)g(the)g (de\014nitions)456 882 y(in)m(tro)s(duced)g(in)h(Section)i(2.1.)456 1062 y(2.1.)46 b Fw(In)m(tegrable)33 b(systems,)h(resonances,)g (secondary)g(tori.)46 b Fs(W)-8 b(e)29 b(start)456 1170 y(b)m(y)35 b(collecting)i(some)f(standard)e(de\014nitions)g(and)h (setting)h(the)f(notation)h(w)m(e)456 1278 y(will)30 b(use.)555 1386 y(F)-8 b(or)43 b(our)f(purp)s(oses,)i(w)m(e)f(describ)s (e)f(a)h Fo(str)-5 b(ongly)46 b(inte)-5 b(gr)g(able)44 b(system)51 b Fs(as)42 b(a)456 1494 y(system)36 b(whic)m(h)f(has)h (phase)g(space)g Fk(R)1769 1461 y Fp(d)1833 1494 y Fn(\002)24 b Fk(T)1989 1461 y Fp(d)2029 1494 y Fs(,)38 b(where)d Fk(T)f Fs(=)h Fk(R)p Fm(=)p Fs(2)p Fm(\031)s Fk(Z)p Fs(,)j(and)d(on)456 1602 y(whic)m(h)30 b(the)g(motion)h(is)g(giv)m(en)g(b)m(y:)456 1756 y(\(1\))757 b(\010)1394 1770 y Fp(t)1423 1756 y Fs(\()p Fm(I)7 b(;)15 b(')p Fs(\))27 b(=)e(\()p Fm(I)7 b(;)15 b(')22 b Fs(+)e Fm(!)s Fs(\()p Fm(I)7 b Fs(\))p Fm(t)p Fs(\))456 1911 y(in)30 b(the)g(case)i(of)e(\015o)m(ws)g(or)h(b)m (y:)456 2066 y(\(2\))786 b Fm(F)13 b Fs(\()p Fm(I)7 b(;)15 b(')p Fs(\))27 b(=)e(\()p Fm(I)7 b(;)15 b(')21 b Fs(+)f Fm(!)s Fs(\()p Fm(I)7 b Fs(\)\))456 2221 y(in)30 b(the)g(case)i(of)e (maps.)555 2329 y(According)41 b(to)h(this)e(rather)g(restrictiv)m(e)j (de\014nition,)g(the)e(mathematical)456 2437 y(p)s(endulum,)28 b(a)i(system)h(in)f Fk(R)1457 2404 y Fq(1)1516 2437 y Fn(\002)20 b Fk(T)1668 2404 y Fq(1)1738 2437 y Fs(describ)s(ed)29 b(b)m(y)h(the)h(Hamiltonian)1293 2592 y Fm(P)13 b Fs(\()p Fm(p;)i(q)s Fs(\))26 b(=)f Fm(p)1732 2554 y Fq(2)1772 2592 y Fm(=)p Fs(2)c(+)f(cos)15 b Fm(q)23 b Fn(\000)d Fs(1)p Fm(;)456 2746 y Fs(is)g(not)g(a)h(strongly)g(in)m(tegrable)h (system)e(ev)m(en)h(if)f(it)h(has)f(a)h(conserv)m(ed)g(quan)m(tit)m(y|) 456 2854 y(the)33 b(Hamiltonian)i Fm(P)13 b Fs(|and)33 b(all)h(the)g(motions)g(are)g(quite)g(orderly)-8 b(.)50 b(Some)34 b(of)456 2962 y(the)e(motions)h(consist)g(on)g(librations,)h (some)f(others)f(are)h(rotations)h(and)e(the)456 3070 y(rotations)38 b(and)g(librations)g(are)g(separated)g(b)m(y)f(orbits)h (that)g(start)g(and)g(end)456 3178 y(in)h(the)h(critical)h(p)s(oin)m(t) f Fm(p)g Fs(=)g(0)p Fm(;)15 b(q)44 b Fs(=)d(0.)68 b(Note)41 b(that)f(these)g(orbits)g(\(usually)456 3286 y(called)28 b Fo(sep)-5 b(ar)g(atric)g(es)7 b Fs(\))32 b(separate)d(t)m(w)m(o)g (di\013eren)m(t)f(top)s(ological)i(t)m(yp)s(es)e(of)g(orbits.)456 3394 y(Hence)c(there)g(is)g(no)g(hop)s(e)f(of)h(writing)f(global)i (action-angle)i(co)s(ordinates)d(that)456 3502 y(straddle)30 b(them.)555 3610 y(Hence,)42 b(sometimes,)f(one)e(in)m(tro)s(duces)f (the)g(less)h(stringen)m(t)g(de\014nition)f(of)456 3718 y Fo(inte)-5 b(gr)g(able)29 b(system)34 b Fs(in)26 b(whic)m(h)g(the)g (represen)m(tations)h(\(1\))h(or)e(\(2\))h(hold)f(in)g(op)s(en)456 3826 y(dense)39 b(sets.)70 b(These)40 b(op)s(en)f(sets)h(are)g (delimited)h(b)m(y)f(sp)s(ecial)g(submanifolds)456 3934 y(called)34 b Fo(sep)-5 b(ar)g(atric)g(es)p Fs(.)54 b(These)33 b(separatrices)i(are,)g(at)g(the)f(same)g(time,)h(stable)456 4042 y(and)41 b(unstable)h(manifolds)g(of)g(lo)m(w)m(er)h(dimensional)f (tori)h(in)m(v)-5 b(arian)m(t)43 b(for)f(the)456 4150 y(\015o)m(w.)555 4257 y(F)-8 b(ollo)m(wing)40 b([CG94)q(],)g(it)e(is)g (customary)g(to)g(refer)f(to)i(systems)e(as)h(\(1\))q(,)i(\(2\))456 4365 y(as)45 b Fo(a)h(priori)i(stable)p Fs(.)85 b(Otherwise,)49 b(systems)c(whic)m(h)g(presen)m(t)g(separatrices)456 4473 y(and)29 b(h)m(yp)s(erb)s(olicit)m(y)-8 b(,)31 b(but)e(whic)m(h)g (are)h(strongly)g(in)m(tegrable)i(piecewise)e(in)g(the)456 4581 y(complemen)m(t)h(of)g(the)f(separatrices,)i(are)f(called)g Fo(a)i(priori)h(unstable)p Fs(.)456 4748 y Fw(Remark)43 b(1.)j Fs(Of)37 b(course,)j(there)d(are)h(systems)g(that)g(presen)m(t)g (separatrices)456 4856 y(whic)m(h)g(are)h(not)g(the)f(stable)i(or)e (unstable)h(manifolds)f(of)h(other)g(h)m(yp)s(erb)s(olic)456 4964 y(sets)27 b(\(e.g.,)j(tak)m(e)e(a)g(p)s(endulum)c(with)j(p)s(oten) m(tial)i Fm(V)20 b Fs(\()p Fm(q)s Fs(\))26 b(=)f(\(cos)16 b Fm(q)g Fn(\000)e Fs(1\))2832 4931 y Fq(4)2899 4964 y Fs(so)27 b(that)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)789 b(9)456 450 y Fs(the)35 b(critical)j(p)s(oin)m(ts)d(are)h(not)g(h)m(yp)s(erb)s(olic)e(p)s(oin)m (ts\).)56 b(Suc)m(h)35 b(situations)i(seem)456 558 y(not)30 b(to)i(ha)m(v)m(e)f(receiv)m(ed)h(m)m(uc)m(h)f(atten)m(tion)h(in)f(the) f(literature.)43 b(\(Nev)m(ertheless,)456 666 y(see)31 b([BF02)q(].\))2176 b Fj(\003)555 899 y Fs(A)22 b Fo(quasi-inte)-5 b(gr)g(able)26 b(system)k Fs(is)22 b(a)g(system)g(whic)m(h)g(is)g (close)i(to)e(an)g(in)m(tegrable)456 1007 y(system)39 b(in)g(the)h(top)s(ology)h(of)f(a)g(space)g(of)f(su\016cien)m(tly)i (smo)s(oth)e(functions.)456 1115 y(Sometimes,)k(it)d(will)h(b)s(e)e (con)m(v)m(enien)m(t)j(to)f(consider)e(families)i(indexed)f(b)m(y)f(a) 456 1223 y(parameter)32 b Fm(")g Fs(so)g(that)g(the)g(system)g(for)f Fm(")d Fs(=)f(0)32 b(is)g(in)m(tegrable.)46 b(F)-8 b(or)33 b Fn(j)p Fm(")p Fn(j)f Fs(small)456 1331 y(enough,)e(the)h(system)f (will)h(b)s(e)f(quasi-in)m(tegrable.)555 1439 y(W)-8 b(e)39 b(recall)g(that,)h(under)c(appropriate)h(di\013eren)m(tiabilit)m (y)j(and)d(non-dege-)456 1547 y(neracy)e(conditions,)i(if)f(the)f(unp)s (erturb)s(ed)d(system)j(is)g(strongly)h(in)m(tegrable,)456 1655 y(the)j(KAM)g(theorem)g(sho)m(ws)g(that)h(most)f(of)g(the)h(phase) e(space)i(of)f(a)h(quasi-)456 1763 y(in)m(tegrable)46 b(system)f(is)f(co)m(v)m(ered)j(b)m(y)d(in)m(v)-5 b(arian)m(t)46 b(tori)f(with)g(a)g(Diophan)m(tine)456 1871 y(frequency)29 b Fm(!)s Fs(\()p Fm(I)7 b Fs(\).)555 1979 y(The)30 b(KAM)h(theorem)f (do)s(es)g(not)h(deal)g(with)f(regions)h(on)f(whic)m(h)456 2151 y(\(3\))569 b Fm(!)s Fs(\()p Fm(I)7 b Fs(\))21 b Fn(\001)f Fm(k)29 b Fs(=)1555 2159 y(O)1625 2151 y(\()p Fm(")1702 2114 y Fq(1)p Fp(=)p Fq(2)1813 2151 y Fs(\))p Fm(;)107 b(k)28 b Fn(2)d Fk(Z)2202 2114 y Fp(d)2262 2151 y Fn(n)c(f)p Fs(0)p Fn(g)p Fm(:)456 2333 y Fs(The)35 b(p)s(o)m(w)m(er)h(1)p Fm(=)p Fs(2)i(is)e(optimal)h(as)g(can)f(b)s(e)g (seen)g(in)g(examples)h(suc)m(h)e(as)2999 2297 y Fp(I)3035 2273 y Fi(2)p 2999 2312 71 4 v 3016 2364 a Fq(2)3103 2333 y Fs(+)456 2455 y Fm(")15 b Fs(sin)g Fm(')p Fs(,)32 b(and)e(it)i(w)m(as)f(established)h(in)e([Sv)-5 b(a80)q(,)32 b(Ne)-10 b(\025)-35 b(\02081)q(,)32 b(P\177)-45 b(os82)q(].)43 b(The)2880 2463 y(O)2951 2455 y(\()p Fm(")3028 2422 y Fq(1)p Fp(=)p Fq(2)3139 2455 y Fs(\))456 2563 y(in)28 b(the)h(size)g(that)g(needs)f(to)h(b)s(e)f(excluded)h(is)f(non-uniform) f(in)h Fm(k)k Fs(and)c(go)s(es)h(to)456 2671 y(zero)i(when)e Fn(j)p Fm(k)s Fn(j)d(!)f(1)p Fs(.)555 2779 y(W)-8 b(e)43 b(will)g(call)g Fo(r)-5 b(esonanc)g(es)51 b Fs(the)42 b(v)-5 b(alues)42 b(of)g Fm(I)49 b Fs(for)42 b(whic)m(h)g Fm(!)s Fs(\()p Fm(I)7 b Fs(\))28 b Fn(\001)g Fm(k)48 b Fs(=)c(0,)456 2889 y(for)32 b(some)h Fm(k)g Fn(2)c Fk(Z)1058 2856 y Fp(d)1120 2889 y Fn(n)22 b(f)p Fs(0)p Fn(g)p Fs(.)50 b(W)-8 b(e)34 b(will)f(call)h Fo(r)-5 b(esonant)37 b(r)-5 b(e)g(gions)41 b Fs(the)33 b(regions)h(\(3\))456 2997 y(around)28 b(the)i(resonance)g(that)h(need)e(to)i(b)s(e)e (excluded)g(in)h(the)g(pro)s(of)f(of)h(KAM)456 3105 y(theorem.)555 3213 y(It)k(is)g(kno)m(wn)g(empirically)h(that)g(v)m(ery)f(often,)i(in) d(these)i(resonan)m(t)f(regions)456 3321 y(there)g(are)h(dynamical)g (ob)5 b(jects)36 b(whic)m(h)e(are)h(not)g(presen)m(t)f(in)h(the)f(in)m (tegrable)456 3429 y(system.)66 b(T)m(ypically)-8 b(,)43 b(the)c(resonan)m(t)g(regions)g(con)m(tain)i(p)s(o)s(orly)d(understo)s (o)s(d)456 3537 y(areas)e(called)i Fo(the)g(chaotic)h(se)-5 b(a)44 b Fs(whic)m(h)36 b(includes)f(homo)s(clinic)i(in)m(tersections) 456 3645 y(and,)f(more)g(imp)s(ortan)m(t)g(for)g(us,)g(secondary)g (tori.)58 b(Indeed,)36 b(secondary)g(tori)456 3753 y(are)30 b(the)h(most)g(visible)g(ob)5 b(jects)31 b(in)f(n)m(umerical)h(sim)m (ulations.)42 b(In)29 b(t)m(w)m(o)j(dimen-)456 3860 y(sions,)38 b(when)d(the)i(visualization)i(is)d(v)m(ery)h(clear,)j(these)d (secondary)f(tori)h(are)456 3968 y(the)30 b(tori)h(in)f(the)h Fo(islands)39 b Fs(in)30 b(the)h(c)m(haotic)h(sea.)42 b(\(See)31 b(Figure)f(1.\))555 4076 y(More)h(precisely:)456 4250 y Fw(De\014nition)49 b(2.)f Fo(We)c(say)g(that)h(a)f Fs(\()p Fm(d)29 b Fn(\000)f Fm(k)s Fs(\))p Fo(-dimensional)46 b(torus)f(invariant)456 4358 y(under)d(the)h(\015ow)h(is)e(a)h(se)-5 b(c)g(ondary)45 b(torus)e(when)g(it)f(is)h(r)-5 b(etr)g(actable)44 b(to)f(a)g(set)456 4466 y Fn(f)p Fm(I)7 b Fn(g)21 b(\002)f(f)p Fs(\()p Fm(')844 4480 y Fq(1)884 4466 y Fm(;)15 b(:)g(:)g(:)i(;)e(') 1145 4481 y Fp(l)1172 4466 y Fs(\))p Fn(g)21 b(\002)f Fk(T)1425 4433 y Fp(d)p Fl(\000)p Fp(l)q Fl(\000)p Fp(k)1635 4466 y Fo(.)555 4640 y Fs(The)43 b(fact)i(that)f(resonances)g(generate) h(secondary)e(tori)h(can)g(b)s(e)f(estab-)456 4748 y(lished)34 b(rigorously)g(under)f(suitable)i(non-degeneracy)g(assumptions)e(on)i (the)456 4856 y(in)m(tegrable)40 b(system)f(and)g(the)g(p)s (erturbation.)65 b(W)-8 b(e)41 b(will)e(dev)m(elop)h(one)f(suc)m(h)456 4964 y(pro)s(of)29 b(in)h(Section)h(8.5.)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 456 251 a Fq(10)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)555 450 y Fs(W)-8 b(e)25 b(also)f(p)s(oin)m(t)g(out)g(that)g(b)s(esides)e(the)i Fm(d)p Fs(-dimensional)g(tori,)i(KAM)e(theory)456 558 y(can)33 b(establish)g(the)h(p)s(ersistence)f(of)g Fm(d)22 b Fn(\000)g Fm(k)s Fs(-dimensional)34 b(tori,)g Fm(k)f(<)d(d)p Fs(,)k(when)456 666 y(they)45 b(ha)m(v)m(e)h Fm(k)i Fs(stable)e(and)e (unstable)h(directions.)85 b(\(See)45 b([Gra74)r(,)g(Zeh76].)456 774 y(Optimal)36 b(measure)h(estimates)h(are)f(a)m(v)-5 b(ailable)39 b(from)d([V)-8 b(al00)r(].\))60 b(These)36 b(tori,)456 882 y(called)31 b(whisk)m(ered)f(tori,)h(can)g(b)s(e)f (primary)f(or)i(secondary)-8 b(.)555 990 y(It)21 b(is)f(also)h(kno)m (wn)f(that)h(resonances)f(generate)i(lo)m(w)m(er)g(dimensional)e(whisk) m(ered)456 1098 y(tori.)39 b(\(See,)28 b(for)e(instance,)i([P)m(oi99)r (,)f(Cap)f(V,)g Fn(x)p Fs(81],)j([T)-8 b(re89)q(,)26 b(Nie00)r(,)g(DG01)r(])g(for)456 1206 y(primary)j(tori)i(and)f([L)-10 b(W89)q(])31 b(for)f(secondary)g(tori.\))555 1314 y(In)20 b(our)g(discussion,)h(there)g(will)g(not)f(b)s(e)g(m)m(uc)m(h)g (di\013erence)h(b)s(et)m(w)m(een)g(whisk)m(ered)456 1421 y(tori)36 b(and)f(fully)h(dimensional)g(tori.)58 b(W)-8 b(e)37 b(will)g(use)e(the)h(theory)h(of)f(normally)456 1529 y(h)m(yp)s(erb)s(olic)g(in)m(v)-5 b(arian)m(t)39 b(manifolds)f(to)g(\014nd)e(in)m(v)-5 b(arian)m(t)39 b(submanifolds.)60 b(The)456 1637 y(maximal)37 b(KAM)f(tori)i(in)e (these)h(in)m(v)-5 b(arian)m(t)37 b(submanifolds)e(will)i(corresp)s (ond)456 1745 y(to)c(whisk)m(ered)g(tori)h(for)f(the)g(full)g(system.) 48 b(\(This)33 b(observ)-5 b(ation)34 b(w)m(as)f(already)456 1853 y(done)28 b(in)h([Mo)s(e96)r(,)g(DLS00])g(and)g(it)g(is)g (exploited)h(in)e([Sor02)q(]\).)41 b(On)28 b(the)h(other)456 1961 y(hand,)40 b(distinction)g(b)s(et)m(w)m(een)g(primary)e(and)h (secondary)g(tori)h(will)g(b)s(e)e(v)m(ery)456 2069 y(imp)s(ortan)m(t) 30 b(for)g(us.)456 2266 y(2.2.)46 b Fw(Heuristic)c(description)f(of)h (the)e(mec)m(hanism.)47 b Fs(The)35 b(systems)g(w)m(e)456 2374 y(consider)24 b(will)i(b)s(e)e(p)s(erturbations)g(of)h(in)m (tegrable)i(ones.)39 b(T)-8 b(o)25 b(\014x)f(ideas,)j(w)m(e)f(will)456 2482 y(consider)i(an)h(unp)s(erturb)s(ed)d(system)j(whic)m(h)f(is)h(in) m(tegrable)i(but)d(not)h(strongly)456 2590 y(in)m(tegrable)j(\(i.e.)41 b(it)31 b(is)g(a)g(priori)f(unstable\),)g(and)g(w)m(e)h(will)g(p)s (erturb)d(it.)555 2698 y(Our)g(unp)s(erturb)s(ed)d(system)k(will)g (admit)g(action-angle)j(v)-5 b(ariables)29 b(in)g(op)s(en)456 2806 y(sets)d(divided)g(b)m(y)h Fo(sep)-5 b(ar)g(atric)g(es)36 b Fs(i.e.)41 b(in)m(v)-5 b(arian)m(t)27 b(manifolds)g(that)g(end)e(in)i (lo)m(w)m(er)456 2914 y(dimensional)41 b(tori)g(\(of)h(co)s(dimension)f (one\).)73 b(W)-8 b(e)42 b(will)f(refer)g(to)h(these)f(tori)456 3022 y(presen)m(t)30 b(in)g(the)h(unp)s(erturb)s(ed)26 b(system)31 b(as)g Fo(primary)j(whisker)-5 b(e)g(d)34 b(tori)p Fs(.)555 3129 y(An)23 b(imp)s(ortan)m(t)g(example)g(of)g 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(the)g(resonan)m(t)g(manifold)456 4209 y(agree.)555 4317 y(Note)35 b(also)f(that)f(if)h(w)m(e)f(consider)g(the)h(motion)f(of)h (the)f(in)m(tegrable)i(system)456 4425 y(restricted)42 b(to)h(the)f(resonan)m(t)g(manifold,)j(it)d(is)g(strongly)h(in)m (tegrable.)76 b(The)456 4533 y(whisk)m(ered)37 b(tori)h(of)f(the)h (full)f(system)h(are)g(KAM)f(tori)h(inside)g(the)f(resonan)m(t)456 4640 y(manifold.)555 4748 y(The)20 b(motion)h(restricted)g(to)g(the)f (resonan)m(t)h(manifold)f(presen)m(ts)h(resonances,)456 4856 y(whic)m(h)j(in)g(turn)g(corresp)s(ond)f(to)j(the)e(double)h (resonances)g(of)f(the)h(original)h(sys-)456 4964 y(tem.)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(11)555 450 y Fs(As)24 b(it)h(is)g(w)m(ell)g(kno)m(wn)f(to)h(sp)s(ecialists)g(in)f(Arnol'd)g (di\013usion,)h(adding)f(a)h(p)s(er-)456 558 y(turbation)h(of)i(size)f Fm(")h Fs(preserv)m(es)f(the)g(resonan)m(t)g(manifold)g(and)f(its)i (stable)f(and)456 666 y(unstable)k(in)m(v)-5 b(arian)m(t)33 b(manifolds.)45 b(F)-8 b(or)32 b(t)m(ypical)i(p)s(erturbations)c(the)i (foliation)456 774 y(b)m(y)37 b(tori)h(in)f(the)h(p)s(erturb)s(ed)d (manifold)i(p)s(ersists)g(except)h(for)f(gaps)h(or)f(order)456 891 y(O)526 883 y(\()p Fm(")603 850 y Fq(1)p Fp(=)p Fq(2)714 883 y Fs(\).)69 b(Moreo)m(v)m(er,)44 b(the)c(stable)g(and)g(unstable)f (manifolds)g(mo)m(v)m(e)i(b)m(y)f(an)456 991 y(amoun)m(t)26 b(of)g(order)1114 999 y(O)1185 991 y(\()p Fm(")p Fs(\).)40 b(It)26 b(is)g(rather)g(straigh)m(tforw)m(ard)g(to)h(sho)m(w)e(that)i (when)456 1099 y(there)g(are)h(tori)g(at)g(a)f(distance)1534 1107 y(O)1605 1099 y(\()p Fm(")p Fs(\))h(one)g(can)g(easily)g (construct)g(hetero)s(clinic)456 1206 y(in)m(tersections)g(b)s(et)m(w)m (een)g(the)f(tori)h(and,)f(hence,)h(construct)g(transition)f(c)m(hains) 456 1314 y(along)k(the)g(resonan)m(t)g(manifold.)555 1423 y(Ho)m(w)m(ev)m(er,)k(near)d(the)h(gaps)f(of)h(order)1866 1431 y(O)1937 1423 y(\()p Fm(")2014 1390 y Fq(1)p Fp(=)p Fq(2)2125 1423 y Fs(\),)g(these)g(leading)g(order)e(con-)456 1531 y(siderations)39 b(do)g(not)h(allo)m(w)g(to)g(conclude)g (existence)g(of)g(transition)f(c)m(hains.)456 1639 y(This)29 b(is)i(what)f(is)g(called)i(the)e Fo(lar)-5 b(ge)34 b(gap)f(pr)-5 b(oblem)p Fs(.)555 1747 y(The)34 b(main)h(idea)g(of)f(the)h(metho)s(d)f (prop)s(osed)f(here)i(is)f(to)i(study)d(carefully)456 1855 y(the)e(ob)5 b(jects)33 b(generated)f(b)m(y)g(the)g(double)f (resonance.)45 b(As)31 b(it)i(is)e(more)h(or)g(less)456 1963 y(folklore,)41 b(the)d(resonances)g(destro)m(y)h(the)f(primary)f (whisk)m(ered)g(tori)i(presen)m(t)456 2071 y(in)32 b(the)i(original)g (system)f(but)g(create)h(ob)5 b(jects)34 b(of)g(other)f(t)m(yp)s(es.)49 b(In)33 b(particu-)456 2179 y(lar,)d(they)f(create)i(secondary)f(whisk) m(ered)e(tori)i(and)f(whisk)m(ered)g(tori)h(of)g(lo)m(w)m(er)456 2287 y(dimension.)555 2395 y(The)38 b(k)m(ey)h(p)s(oin)m(t)f(of)g(the)h (prop)s(osed)e(mec)m(hanism)h(is)h(that)f(the)h(secondary)456 2502 y(whisk)m(ered)27 b(tori)i(and)e(the)h(stable)h(and)e(unstable)h (manifolds)f(of)i(the)f(lo)m(w)m(er)h(di-)456 2610 y(mensional)i(tori)g (come)h(v)m(ery)g(close)g(to)g(the)f(region)g(co)m(v)m(ered)i(b)m(y)e (the)g(primary)456 2718 y(tori.)61 b(The)36 b(secondary)i(tori)f Fo(bridge)44 b Fs(the)37 b(resonan)m(t)h(region.)61 b(Hence,)40 b(b)m(y)d(in-)456 2826 y(corp)s(orating)30 b(them)h(in)f(the)g (transition)h(c)m(hains)f(w)m(e)h(can)g(o)m(v)m(ercome)h(the)f(large) 456 2934 y(gap)f(problem.)555 3042 y(The)e(heuristic)g(description)g (of)g(the)g(mec)m(hanism)h(prop)s(osed)e(here)g(to)i(o)m(v)m(er-)456 3150 y(come)i(the)f(large)i(gap)f(problem)e(is:)636 3669 y(1\))42 b(Outside)35 b(the)h(regions)g(of)g(double)f(resonance,)i(KAM) f(theorem)g(ap-)758 3777 y(plies,)23 b(and)d(w)m(e)h(obtain)f(the)h (existence)g(of)g(co)s(dimension)f(one)h(whisk)m(ered)758 3885 y(tori)34 b(whic)m(h)f(are)g(extremely)i(close)f(to)g(eac)m(h)g (other.)49 b(Hence)34 b(one)g(can)758 3993 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b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)636 450 y Fs(3\))42 b(The)20 b(secondary)h(whisk)m(ered)e(tori,)k(the)e(lo)m(w)m(er)g (dimensional)g(whisk)m(ered)758 558 y(tori)33 b(and)d(the)i(primary)f (whisk)m(ered)g(tori)h(lie)h(on)e(a)h(normally)g(h)m(yp)s(er-)758 666 y(b)s(olic)j(in)m(v)-5 b(arian)m(t)36 b(manifold)e(whic)m(h)h(is)f (a)h(con)m(tin)m(uation)i(of)e(the)f(reso-)758 774 y(nan)m(t)d (manifold)f(for)g(the)h(in)m(tegrable)h(system.)636 882 y(4\))42 b(This)37 b(normally)h(h)m(yp)s(erb)s(olic)g(in)m(v)-5 b(arian)m(t)39 b(manifold)e(has)h(stable)g(and)758 990 y(unstable)29 b(in)m(v)-5 b(arian)m(t)30 b(manifolds)e(whic)m(h,)i (under)d(some)i(explicit)h(non-)758 1098 y(degeneracy)23 b(conditions)g(on)e(the)h(p)s(erturbation,)h(in)m(tersect)g(transv)m (er-)758 1206 y(sally)-8 b(.)636 1314 y(5\))42 b(Under)23 b(some)g(explicit)h(non-degeneracy)g(conditions)g(on)f(the)g(p)s (ertur-)758 1421 y(bation)37 b(it)f(is)g(p)s(ossible)f(to)h(pro)s(duce) 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b(manifold)f(thanks)g(to)h(the)f(fact)i(that)e(its)456 3498 y(stable)d(and)g(unstable)f(in)m(v)-5 b(arian)m(t)40 b(manifolds)d(in)m(tersect)j(transv)m(ersally)-8 b(,)41 b(and)456 3606 y(whic)m(h)c(allo)m(ws)j(us)d(to)h(discuss)g(the)g (hetero)s(clinic)h(in)m(tersection)h(of)e(in)m(v)-5 b(arian)m(t)456 3714 y(ob)5 b(jects)31 b(of)f(di\013eren)m(t)h(t)m(yp)s(e.)456 3884 y Fw(Remark)j(3.)42 b Fs(W)-8 b(e)31 b(will)f(see)h(that)f(the)g (resonan)m(t)g(regions,)h(ev)m(en)g(if)f(they)g(ha)m(v)m(e)456 3993 y(a)37 b(size)716 4001 y(O)787 3993 y(\()p Fm(")864 3960 y Fq(1)p Fp(=)p Fq(2)974 3993 y Fs(\),)i(are)e(connected)g(b)m(y)g (only)f(one)h(elemen)m(t)h(in)f(the)f(transition)456 4101 y(c)m(hain,)d(whereas)e(outside)i(of)f(the)g(resonan)m(t)g (regions,)h(one)g(step)f(of)g(the)g(tran-)456 4209 y(sition)f(c)m(hain) f(only)h(mak)m(es)g(a)g(m)m(uc)m(h)f(smaller)h(step)g(in)f(space.)555 4317 y(If)48 b(one)g(mak)m(es)h(the)f(conjecture)h(\(see)g([Chi79],)k (ampli\014ed)48 b(in)f([CV89)q(]\))456 4425 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y([SZF95,)k(Zas02)q(])g(or)g(the)g(exp)s (erimen)m(tal)g(observ)-5 b(ations)41 b(that)f(con\014rm)f(that)456 666 y(di\013usion)34 b(is)i(m)m(uc)m(h)g(more)g(apparen)m(t)f(near)h (resonances)g([Las93)q(,)g(LHRK02],)456 774 y(.)59 b(Of)36 b(course,)j(the)e(curren)m(t)f(state)i(of)f(the)g(art)g(is)g(far)f (from)g(a)h(pro)s(of)f(of)h(this)456 882 y(conjecture,)43 b(In)d(particular,)j(there)d(do)g(not)g(seem)h(to)g(exist)f(tec)m (hniques)h(to)456 990 y(pro)s(duce)24 b(a)i(rigorous)f(an)h (statistical)i(theory)-8 b(.)40 b(Nev)m(ertheless)27 b(w)m(e)f(\014nd)e(encour-)456 1098 y(aging)33 b(that)h(our)e(metho)s (ds)g(of)h(pro)s(of)f(mo)m(v)m(e)j(in)d(the)h(direction)g(suggested)h (b)m(y)456 1206 y(in)m(tuition)27 b(and)g(exp)s(erimen)m(tal)h(observ) -5 b(ations)27 b(and)g(sho)m(w)f(some)i(sem)m(blance)g(of)456 1314 y(agreemen)m(t)k(with)e(them.)1776 b Fj(\003)456 1570 y Fw(Remark)36 b(4.)43 b Fs(In)31 b(man)m(y)h(ph)m(ysical)g (applications,)h(one)f(has)g(to)g(consider)g(sys-)456 1678 y(tems)21 b(whic)m(h)g(consist)h(of)g(man)m(y)f(iden)m(tical)j (systems)d(with)g(a)h(lo)s(cal)h(coupling.)38 b(In)456 1786 y([HL00)q(])30 b(there)g(is)f(an)h(empirical)g(and)f(heuristic)h (study)f(of)h(the)g(abundance)f(of)456 1893 y(secondary)f(tori)g(in)g (these)g(coupled)g(oscillators.)42 b(The)28 b(conclusion)g(of)g(n)m (umer-)456 2001 y(ical)g(exp)s(erimen)m(ts)g(and)f(heuristic)g(argumen) m(ts)h(of)f([HL00)r(])g(is)h(that)g(secondary)456 2109 y(tori)k(are)h(m)m(uc)m(h)f(more)g(abundan)m(t)f(than)h(KAM)g(primary)f (tori)i(in)f(systems)g(of)456 2217 y(coupled)e(oscillators.)1896 b Fj(\003)555 2473 y Fs(F)-8 b(rom)26 b(the)f(mathematical)i(p)s(oin)m (t)d(of)h(view,)i(in)d(this)h(pap)s(er)f(w)m(e)h(will)g(con)m(ten)m(t) 456 2581 y(ourselv)m(es)c(with)g(studying)f(just)g(one)h(mo)s(del)g(c)m (hosen)g(b)s(ecause)g(the)g(v)m(eri\014cation)456 2689 y(of)45 b(the)g(mec)m(hanism)g(is)g(the)h(simplest)f(p)s(ossible)f (among)i(those)f(for)g(whic)m(h)456 2797 y(the)37 b(v)m(eri\014cation)i (is)e(non-trivial)h(and)e(this)h(mo)s(del)g(w)m(as)g(in)m(tro)s(duced)g (in)g(the)456 2905 y(pap)s(er)29 b([HM82)r(],)h(whic)m(h)g(ignored)h (the)g(large)g(gap)g(problem.)555 3013 y(Most)f(of)g(the)f(metho)s(ds)f (that)i(w)m(e)f(emplo)m(y)h(in)f(our)g(v)m(eri\014cation)h(ha)m(v)m(e)h (b)s(een)456 3121 y(standard)g(to)s(ols)j(of)f(the)g(geometric)i (approac)m(h)e(to)g(standard)f(Arnol'd)g(di\013u-)456 3229 y(sion.)60 b(Nev)m(ertheless,)41 b(w)m(e)c(call)h(atten)m(tion)h (to)f(the)f(fact)h(that)g(w)m(e)f(also)h(mak)m(e)456 3337 y(go)s(o)s(d)27 b(use)h(of)g(the)h(p)s(erturbation)e(theory)h(of)g (normally)g(h)m(yp)s(erb)s(olic)g(in)m(v)-5 b(arian)m(t)456 3445 y(manifolds)37 b(to)i(pro)m(vide)f(an)f(sk)m(eleton)j(along)f (whic)m(h)e(di\013usion)g(tak)m(es)j(place.)456 3553 y(\(This)29 b(is)i(a)f(common)h(feature)g(with)f([DLS00)q(,)g(DLS01)q (],)h(whic)m(h)f(consider)g(y)m(et)456 3661 y(another)g(mec)m(hanism)h (of)f(di\013usion)g(and)g(with)g([Mo)s(e96)r(].\))555 3769 y(Indeed,)22 b(the)f(main)g(ob)5 b(ject)22 b(that)f(organizes)h (the)f(mec)m(hanism)g(here)g(is)g(a)g(nor-)456 3878 y(mally)31 b(h)m(yp)s(erb)s(olic)g(in)m(v)-5 b(arian)m(t)32 b(manifold)1920 3855 y(~)1911 3878 y(\003)1974 3892 y Fp(")2042 3878 y Fs(close)g(to)g(the)f(resonances.)43 b(The)456 3986 y(stable)25 b(and)g(unstable)g(manifolds)f(of)i(this)f(manifold)f(will) i(in)m(tersect)g(transv)m(er-)456 4094 y(sally)-8 b(.)45 b(This)31 b(will)h(allo)m(w)h(us)e(to)i(de\014ne)e(t)m(w)m(o)i (dynamics)e(on)g(the)h(manifold)3058 4071 y(~)3049 4094 y(\003)3112 4108 y Fp(")3149 4094 y Fs(.)456 4202 y(One)g(is)g(the)h (dynamics)f(restricted)h(to)h(the)e(manifold,)i(whic)m(h)e(w)m(e)h (called)h(the)456 4310 y Fo(inner)29 b(map)34 b Fs(in)26 b([DLS00)q(].)39 b(F)-8 b(ollo)m(wing)30 b(again)d([DLS00)q(],)h(w)m(e) f(will)g(also)h(de\014ne)d(a)456 4418 y Fo(sc)-5 b(attering)33 b(map)k Fs(in)29 b(Section)i(9.)41 b(Giv)m(en)30 b(an)g(orbit)36 b(~)-51 b Fm(z)t Fs(\()p Fm(t)p Fs(\))31 b(that)f(p)s(erforms)e(a)j (ho-)456 4533 y(mo)s(clinic)e(excursion)f(to)1333 4510 y(~)1324 4533 y(\003)1387 4547 y Fp(")1424 4533 y Fs(,)h(w)m(e)g(can)f (\014nd)f(t)m(w)m(o)j(orbits)e(in)2493 4510 y(~)2484 4533 y(\003)p Fm(")h Fs(that)g(approac)m(h)461 4640 y(~)-50 b Fm(z)t Fs(\()p Fm(t)p Fs(\))29 b(in)f(the)h(future)f(and)g(in)g(the)h (past.)40 b(The)28 b(scattering)i(map)e(asso)s(ciates)j(the)456 4748 y(orbit)d(asymptotic)h(in)f(the)g(past)g(to)h(the)f(orbit)g (asymptotic)i(in)d(the)i(future.)39 b(It)456 4856 y(will)28 b(b)s(e)f(sho)m(wn)g(that,)i(when)e(one)h(considers)g(all)g(the)g (orbits)34 b(~)-51 b Fm(z)t Fs(\()p Fm(t)p Fs(\))29 b(c)m(hosen)f(in)f (a)456 4964 y(homo)s(clinic)k(connection,)g(w)m(e)g(obtain)g(a)g (scattering)g(map)f(whic)m(h)g(is)h(smo)s(oth.)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 456 251 a Fq(14)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)555 450 y Fs(One)38 b(can)g(obtain)h(di\013using)e(pseudo-orbits)g(b)m(y)h(applying)g (alternativ)m(ely)456 558 y(the)g(inner)f(map)g(and)h(the)g(scattering) h(map.)64 b(This)37 b(corresp)s(onds)f(to)j(orbits)456 666 y(that)j(p)s(erform)f(a)h(homo)s(clinic)h(excursion)f(when)f(the)i (p)s(erturbation)e(in)g(fa-)456 776 y(v)m(orable)h(but)e(that,)45 b(otherwise)c(sta)m(y)i(\\park)m(ed")e(near)2438 753 y(~)2429 776 y(\003)2492 790 y Fp(")2529 776 y Fs(.)73 b(If)41 b(the)g(motion)456 884 y(generated)36 b(b)m(y)g(the)g(inner)f (map)g(con)m(tains)i(in)m(v)-5 b(arian)m(t)37 b(ob)5 b(jects)37 b(suc)m(h)e(as)h(tori,)456 991 y(it)h(will)h(b)s(e)f(p)s (ossible)f(to)i(use)f(obstruction)h(prop)s(erties)e(\(see)j(Section)f (11\))g(to)456 1099 y(construct)30 b(orbits)h(that)g(di\013use.)555 1207 y(W)-8 b(e)33 b(note)f(that)g(there)g(is)g(a)g(large)g(literature) h(on)e(obstruction)h(prop)s(erties.)456 1315 y(W)-8 b(e)33 b(will)f(v)m(erify)g(that)g(some)h(metho)s(ds)e(in)g(the)h(literature)h (apply)e(ev)m(en)i(if)f(the)456 1423 y(tori)25 b(ha)m(v)m(e)h 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(considered)g(in)456 3710 y(the)35 b(pap)s(er)g([HM82)r(],)i(but)e(the) h(large)h(gap)f(problem)f(w)m(as)h(not)g(addressed)e(in)456 3818 y(that)d(pap)s(er.)555 3926 y(W)-8 b(e)38 b(also)g(note)f(that)g (the)g(mo)s(del)g(w)m(e)g(consider)g(is)f(a)h(standard)f(mo)s(del)h(of) 456 4034 y(the)30 b(b)s(eha)m(vior)h(near)f(resonances.)555 4142 y(Ev)m(en)38 b(if)g(the)g(analysis)g(in)g(this)f(pap)s(er)g(will)h (require)g(sev)m(eral)h(unduly)d(re-)456 4250 y(strictiv)m(e)g (assumptions,)e(w)m(e)h(hop)s(e)e(that)i(it)g(could)f(la)m(y)h(the)f (foundations)g(for)456 4358 y(further)29 b(progress.)555 4466 y(W)-8 b(e)34 b(will)g(consider)e(a)i(mec)m(hanical)g(system)f (describ)s(ed)f(b)m(y)h(the)g(non-auto-)456 4574 y(nomous)c (Hamiltonian,)j(p)s(erio)s(dic)e(in)g(time)802 4743 y Fm(H)878 4757 y Fp(")914 4743 y Fs(\()p Fm(p;)15 b(q)s(;)g(I)7 b(;)15 b(';)g(t)p Fs(\))28 b(=)d Fm(H)1573 4757 y Fq(0)1612 4743 y Fs(\()p Fm(p;)15 b(q)s(;)g(I)7 b Fs(\))21 b(+)f Fm("h)p Fs(\()p Fm(p;)15 b(q)s(;)g(I)7 b(;)15 b(';)g(t)p Fs(;)g Fm(")p Fs(\))1401 4935 y(=)25 b Fm(P)1555 4949 y Fl(\006)1614 4935 y Fs(\()p Fm(p;)15 b(q)s Fs(\))21 b(+)1936 4874 y(1)p 1936 4914 46 4 v 1936 4997 a(2)1991 4935 y Fm(I)2038 4898 y Fq(2)2098 4935 y Fs(+)f Fm("h)p Fs(\()p Fm(p;)15 b(q)s(;)g(I)7 b(;)15 b(';)g(t)p Fs(;)g Fm(")p Fs(\))456 4859 y(\(4\))p eop end %%Page: 15 15 TeXDict begin 15 14 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(15)456 450 y Fs(where)29 b(w)m(e)i(denote)g(b)m(y)456 634 y(\(5\))729 b Fm(P)1358 648 y Fl(\006)1417 634 y Fs(\()p Fm(p;)15 b(q)s Fs(\))26 b(=)f Fn(\006)p Fs(\()1855 573 y(1)p 1855 613 46 4 v 1855 697 a(2)1910 634 y Fm(p)1956 597 y Fq(2)2016 634 y Fs(+)20 b Fm(V)g Fs(\()p Fm(q)s Fs(\)\))456 811 y(and)29 b(where)h Fm(V)20 b Fs(\()p Fm(q)s Fs(\))31 b(is)g(a)g(2)p Fm(\031)s Fs(-p)s(erio)s(dic)f(function.)555 919 y(W)-8 b(e)31 b(will)e(refer)g(to)h Fm(P)1258 933 y Fl(\006)1317 919 y Fs(\()p Fm(p;)15 b(q)s Fs(\))30 b(as)f(the)g Fo(p)-5 b(endulum)p Fs(.)42 b(\(It)29 b(is)h(a)f(ph)m(ysical)h(p)s(endu-)456 1027 y(lum)25 b(when)h(w)m(e)h(tak)m(e)h Fm(P)1248 1041 y Fq(+)1307 1027 y Fs(,)f(and)f Fm(V)20 b Fs(\()p Fm(q)s Fs(\))26 b(=)f(cos)16 b Fm(q)f Fn(\000)d Fs(1,)28 b(whic)m(h)e(is)g(a)h (go)s(o)s(d)f(example)456 1135 y(to)31 b(k)m(eep)g(in)f(mind.\))555 1245 y(The)j(term)975 1209 y Fq(1)p 975 1224 36 4 v 975 1276 a(2)1020 1245 y Fm(I)1067 1212 y Fq(2)1140 1245 y Fs(of)40 b(\(4\))34 b(describ)s(es)f(a)g(rotator)h(whose)f(frequency) g(c)m(hanges)456 1353 y(with)d(the)g(energy)h(of)f(the)h(oscillation)i (\(equiv)-5 b(alen)m(tly)32 b(with)e(the)h(action\).)555 1461 y(The)42 b(\014nal)h(term)f Fm(h)h Fs(in)g(\(4\))g(describ)s(es)f (a)h(small)g(coupling)g(b)s(et)m(w)m(een)h(the)456 1569 y(rotator)e(and)e(the)g(p)s(endulum)f(that)i(dep)s(ends)e(p)s(erio)s (dically)i(on)f(time.)73 b(W)-8 b(e)456 1676 y(will)42 b(assume|b)s(esides)g(di\013eren)m(tiabilit)m(y)j(prop)s(erties|that)d Fm(V)63 b Fs(reac)m(hes)43 b(a)456 1784 y(maxim)m(um)d(at)i(one)f(p)s (oin)m(t|w)m(e)h(will)f(assume)g(without)f(loss)i(of)f(generalit)m(y) 456 1892 y(that)31 b(it)g(is)g(0|and)g(that)g(the)g(maxim)m(um)g(is)f (non-degenerate)i(\(i.e.)43 b Fm(V)2919 1859 y Fl(00)2961 1892 y Fs(\(0\))27 b Fm(<)456 2000 y Fs(0\).)50 b(W)-8 b(e)35 b(will)f(assume)f(for)g(the)h(sak)m(e)g(of)g(simplicit)m(y)g(of) g(exp)s(osition)g(that)g(0)g(is)456 2108 y(the)27 b(only)h(lo)s(cal)g (maxim)m(um.)40 b(This)26 b(later)j(assumption)d(can)i(b)s(e)f (eliminated)h(at)456 2216 y(the)i(only)h(exp)s(ense)f(of)g (complicating)i(the)f(notation.)456 2378 y Fw(Remark)43 b(5.)i Fs(The)37 b(c)m(hoice)h(of)f(sign)g(in)g Fm(P)1932 2392 y Fl(\006)2029 2378 y Fs(will)g(not)g(mak)m(e)h(an)m(y)f (di\013erence)456 2486 y(in)26 b(our)g(argumen)m(ts)h(whic)m(h)g(are)g (based)f(on)h(h)m(yp)s(erb)s(olicit)m(y)g(and)f(KAM)h(theory)-8 b(.)456 2594 y(Note)34 b(ho)m(w)m(ev)m(er)h(that)f(the)g(Hamiltonian)h (with)e Fm(P)2184 2608 y Fq(+)2277 2594 y Fs(is)h(con)m(v)m(ex)h(for)e (large)i Fm(I)7 b(;)15 b(p)456 2702 y Fs(but)29 b(with)h Fm(P)887 2716 y Fl(\000)977 2702 y Fs(is)g(neither)h(con)m(v)m(ex)h (nor)e(p)s(ositiv)m(e)h(de\014nite.)555 2810 y(The)38 b(assumption)f(of)h(p)s(ositiv)m(e)g(de\014niteness)g(seems)g(to)g(b)s (e)f(v)m(ery)i(imp)s(or-)456 2918 y(tan)m(t)32 b(for)e(v)-5 b(ariational)33 b(metho)s(ds)e([Xia98)r(,)g(Mat02)r(].)43 b(On)30 b(the)h(other)g(hand,)f(w)m(e)456 3026 y(note)h(that)g(the)g (ph)m(ysical)g(in)m(tuition)g([Chi79)q(,)f(TLL80])h(is)g(that)g (systems)f(with)456 3134 y(subsystems)f(of)h(di\013eren)m(t)h (signatures)g(tend)f(to)h(b)s(e)f(more)g(unstable.)252 b Fj(\003)555 3334 y Fs(It)34 b(will)h(also)g(b)s(e)e(con)m(v)m(enien)m (t)j(to)f(consider)e(the)i(system)f(\(4\))h(as)f(describ)s(ed)456 3442 y(b)m(y)28 b(an)g(autonomous)h(Hamiltonian)h(\015o)m(w)f(on)f(\()p Fk(R)17 b Fn(\002)f Fk(T)p Fs(\))2347 3409 y Fq(3)2386 3442 y Fs(|whic)m(h)29 b(w)m(e)g(will)g(call)456 3550 y(the)43 b(symplectic)h(extended)f(phase)g(space|endo)m(w)m(ed)h(with)f (the)h(standard)456 3658 y(symplectic)31 b(structure.)40 b(The)30 b(autonomous)h(Hamiltonian)h(will)e(b)s(e:)717 3783 y(~)694 3806 y Fm(H)770 3820 y Fp(")806 3806 y Fs(\()p Fm(p;)15 b(q)s(;)g(I)7 b(;)15 b(';)g(A;)g(s)p Fs(\))28 b(=)d Fm(A)20 b Fs(+)g Fm(H)1762 3820 y Fp(")1798 3806 y Fs(\()p Fm(p;)15 b(q)s(;)g(I)7 b(;)15 b(';)g(s)p Fs(\))1411 3946 y(=)25 b Fm(A)20 b Fs(+)g Fm(H)1762 3960 y Fq(0)1801 3946 y Fs(\()p Fm(p;)15 b(q)s(;)g(I)7 b Fs(\))21 b(+)f Fm("h)p Fs(\()p Fm(p;)15 b(q)s(;)g(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))1411 4138 y(=)25 b Fm(A)20 b Fs(+)g Fm(P)1744 4152 y Fl(\006)1803 4138 y Fs(\()p Fm(p;)15 b(q)s Fs(\))21 b(+)2125 4076 y(1)p 2125 4117 46 4 v 2125 4200 a(2)2181 4138 y Fm(I)2228 4100 y Fq(2)2287 4138 y Fs(+)f Fm("h)p Fs(\()p Fm(p;)15 b(q)s(;)g(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))p Fm(:)456 3984 y Fs(\(6\))555 4310 y(W)-8 b(e)31 b(will)f(follo)m(w)h(the)f(standard)e(con)m(v)m(en)m (tion)k(of)e(assuming)f(that)h(the)g(pairs)456 4417 y(\()p Fm(p;)15 b(q)s Fs(\))26 b Fn(2)e Fk(R)q Fn(\002)q Fk(T)p Fs(,)e(\()p Fm(I)7 b(;)15 b(')p Fs(\))27 b Fn(2)e Fk(R)q Fn(\002)q Fk(T)p Fs(,)d(and)e(\()p Fm(A;)15 b(s)p Fs(\))26 b Fn(2)f Fk(R)q Fn(\002)q Fk(T)p Fs(,)d(will)f(b)s(e)f(symplectically) 456 4525 y(conjugate)31 b(v)-5 b(ariables.)555 4633 y(Note)31 b(that)g Fm(h)f Fs(do)s(es)f(not)h(in)m(v)m(olv)m(e)i Fm(A)e Fs(so)g(that)g(the)g(equations)g(of)g(motion)h(of)456 4741 y(the)f(pair)g(\()p Fm(A;)15 b(s)p Fs(\))32 b(are)e(just)699 4912 y(_)665 4935 y Fm(A)c Fs(=)f Fn(\000)p Fm(@)974 4949 y Fp(s)1034 4912 y Fs(~)1010 4935 y Fm(H)1086 4949 y Fp(")1123 4935 y Fs(\()p Fm(p;)15 b(q)s(;)g(I)7 b(;)15 b(';)g(s)p Fs(\))27 b(=)e Fn(\000)p Fm(")1838 4874 y(@)5 b(h)p 1838 4914 106 4 v 1843 4997 a(@)g(s)1954 4935 y Fs(\()p Fm(p;)15 b(q)s(;)g(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))p Fm(;)214 b Fs(_)-39 b Fm(s)25 b Fs(=)g(1)p Fm(:)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 456 251 a Fq(16)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 450 y Fs(The)28 b(in)m(tro)s(duction)h(of)g(the)g(extra)h(v)-5 b(ariables)29 b(\()p Fm(A;)15 b(s)p Fs(\))30 b(is)f(a)g(standard)f(device)i(to)456 558 y(form)m(ulate)35 b(p)s(erio)s(dic)f(in)g(time)h(p)s(erturbations)f (as)g(an)h(autonomous)f(system.)456 666 y(The)29 b(extra)i(v)-5 b(ariable)31 b Fm(s)e Fs(mak)m(es)i(the)f(system)g(autonomous)h(and)e (the)h(v)-5 b(ariable)456 774 y Fm(A)37 b Fs(is)g(symplectically)i (conjugate)f(to)g Fm(s)f Fs(to)g(b)s(e)g(able)h(to)f(treat)h(the)g (resulting)456 882 y(system)26 b(as)h(a)g(Hamiltonian)h(one.)40 b(So,)28 b(ev)m(en)f(if)g(the)g(system)f(describ)s(ed)g(b)m(y)i(\(6\)) 456 990 y(is,)g(strictly)h(sp)s(eaking,)g(a)f(three)g(degrees)g(of)g (freedom)g(system,)h(w)m(e)f(refer)g(to)g(it)456 1098 y(as)i(a)h(t)m(w)m(o)h(and)d(a)i(half)g(degrees)f(of)h(freedom)f (system.)555 1206 y(Moreo)m(v)m(er,)35 b(the)e(v)-5 b(ariable)33 b Fm(A)f Fs(do)s(es)g(not)h(pla)m(y)g(an)m(y)g(dynamical)f(role.)48 b(Note)456 1314 y(that)28 b Fm(A)h Fs(do)s(es)f(not)g(app)s(ear)g(in)f (an)m(y)i(of)f(the)h(di\013eren)m(tial)g(equations)g(for)f(an)m(y)h(of) 456 1421 y(the)k(co)s(ordinates,)h(including)f(itself.)49 b(Then,)33 b(one)g(can)g(study)f(the)h(dynamics)456 1531 y(of)26 b(the)h(v)-5 b(ariables)28 b(\()p Fm(p;)15 b(q)s(;)g(I)7 b(;)15 b(';)g(s)p Fs(\),)29 b(and)d(then)2016 1508 y(_)1982 1531 y Fm(A)f Fs(=)g Fn(\000)p Fm(@)2290 1545 y Fp(s)2351 1508 y Fs(~)2327 1531 y Fm(H)2403 1545 y Fp(")2439 1531 y Fs(\()p Fm(p;)15 b(q)s(;)g(I)7 b(;)15 b(';)g(s)p Fs(\))29 b(is)d(just)456 1639 y(a)k(quadrature.)40 b(Hence,)32 b(w)m(e)f(will)f(consider)h(the)f(equations:)456 2324 y(\(7\))1116 1889 y(_)-43 b Fm(p)83 b Fs(=)g Fn(\007)p Fm(V)1525 1856 y Fl(0)1548 1889 y Fs(\()p Fm(q)s Fs(\))g Fn(\000)p Fm(")1868 1828 y(@)5 b(h)p 1868 1868 106 4 v 1872 1952 a(@)g(q)1984 1889 y Fs(\()p Fm(p;)15 b(q)s(;)g(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))1117 2120 y(_)-42 b Fm(q)86 b Fs(=)d Fn(\006)p Fm(p)247 b Fs(+)p Fm(")1868 2059 y(@)5 b(h)p 1868 2100 V 1871 2183 a(@)g(p)1984 2120 y Fs(\()p Fm(p;)15 b(q)s(;)g(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))1118 2329 y(_)1097 2352 y Fm(I)90 b Fs(=)447 b Fn(\000)p Fm(")1872 2290 y(@)5 b(h)p 1868 2331 113 4 v 1868 2414 a(@)g(')1991 2352 y Fs(\()p Fm(p;)15 b(q)s(;)g(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))1109 2583 y(_)-50 b Fm(')84 b Fs(=)f Fm(I)324 b Fs(+)p Fm(")1868 2521 y(@)5 b(h)p 1868 2562 106 4 v 1871 2645 a(@)g(I)1984 2583 y Fs(\()p Fm(p;)15 b(q)s(;)g(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))1115 2799 y(_)-39 b Fm(s)83 b Fs(=)g(1)p Fm(:)456 3013 y Fs(W)-8 b(e)27 b(will)g(denote)h(b)m(y)e Fn(H)1265 3027 y Fp(")1328 3013 y Fs(the)h(v)m(ector)h(\014eld)f(\(7\)) g(generated)h(b)m(y)f(the)f(Hamilton-)456 3121 y(ian)f(of)33 b(\(6\))q(.)39 b(W)-8 b(e)27 b(will)f(denote)g(b)m(y)g(\010)1683 3135 y Fp(t;")1764 3121 y Fs(\()6 b(~)-51 b Fm(x)p Fs(\))26 b(the)g(\015o)m(w)f(generated)i(b)m(y)f(the)f(v)m(ector)456 3232 y(\014eld)30 b Fn(H)730 3246 y Fp(")796 3232 y Fs(in)g(the)h (extended)f(phase)g(space)h(\()p Fk(R)21 b Fn(\002)e Fk(T)p Fs(\))2249 3199 y Fq(2)2309 3232 y Fn(\002)h Fk(T)p Fs(.)456 3414 y Fw(Remark)45 b(6.)h Fs(Notice)41 b(that)e(in)m(tro)s (ducing)f(the)h(extra)h(v)-5 b(ariable)39 b Fm(A)g Fs(mak)m(es)g(a)456 3522 y(certain)34 b(di\013erence)f(in)g(the)g(geometric)j(nature)d(of)g (the)g(ob)5 b(jects.)50 b(F)-8 b(or)34 b(exam-)456 3630 y(ple,)25 b(if)g(w)m(e)f(\014nd)f(a)i(KAM)f(torus)g(in)g(the)h (non-autonomous)f(system,)i(it)f(will)f(b)s(e-)456 3738 y(come)g(a)h(family)f(of)g(tori|indexed)g(b)m(y)f(the)h(v)-5 b(ariable)25 b Fm(A)p Fs(|in)f(the)g(autonomous)456 3845 y(system.)40 b(Hence,)31 b(ev)m(en)f(if)g(the)g(KAM)g(tori)g(in)f(the)h (non-autonomous)g(system)456 3953 y(are,)c(for)f(t)m(ypical)h(p)s (erturbations,)f(a)h(Can)m(tor)f(set)g(of)g(tori,)i(in)e(the)g (autonomous)456 4061 y(v)m(ersion,)31 b(they)f(alw)m(a)m(ys)i(include)e (one)h(parameter)g(families.)578 b Fj(\003)555 4317 y Fs(The)37 b(Hamiltonian)i Fm(H)1355 4331 y Fq(0)1394 4317 y Fs(\()p Fm(p;)15 b(q)s(;)g(I)7 b Fs(\))39 b(in)f(\(4\))g(will)g (b)s(e)f(referred)g(to)i(as)f(the)g(un-)456 4425 y(p)s(erturb)s(ed)23 b(Hamiltonian.)41 b(It)26 b(is)h(customary)f(to)h(describ)s(e)f Fm(H)2584 4439 y Fq(0)2649 4425 y Fs(as)h(in)m(tegrable.)456 4533 y(As)k(w)m(e)h(p)s(oin)m(ted)f(out)g(in)g(Section)h(2.1,)h(the)e (Hamiltonian)i(system)e(of)h(Hamil-)456 4640 y(tonian)i Fm(H)818 4654 y Fq(0)891 4640 y Fs(is)h(a)f(priori)g(unstable.)52 b(Indeed,)35 b(the)f(system)g(describ)s(ed)f(b)m(y)i Fm(P)3115 4654 y Fl(\006)456 4748 y Fs(presen)m(ts)23 b(di\013eren)m(t)h(top)s(ological)i(t)m(yp)s(es)d(of)h(oscillations)h (whic)m(h)f(are)f(separated)456 4856 y(b)m(y)28 b(some)i(sp)s(ecial)f (orbits)g(\(called)i(separatrices\))f(ending)f(with)g(zero)g(v)m(elo)s (cit)m(y)456 4964 y(on)h(the)g(maxim)m(um)h(of)f Fm(V)20 b Fs(.)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(17)1037 450 y Fs(4.)46 b Ft(St)-6 b(a)g(tement)32 b(of)i(rigor)n(ous)f(resul)-6 b(ts)555 612 y Fs(In)28 b(this)h(section,)i(w)m(e)f(will)f(form)m (ulate)h(the)f(main)g(Theorem)g(of)g(this)g(pap)s(er.)555 720 y(Actually)-8 b(,)41 b(w)m(e)d(pro)m(v)m(e)h(somewhat)e(more)h (general)g(results)g(whic)m(h)f(w)m(e)h(for-)456 828 y(m)m(ulate)e(later)g(when)f(w)m(e)g(ha)m(v)m(e)i(in)m(tro)s(duced)e (more)g(notation.)57 b(F)-8 b(or)36 b(example)456 936 y(b)s(esides)i(orbits)i(whic)m(h)f(transv)m(erse)h(the)g(gaps,)j(it)d (follo)m(ws)g(from)g(our)f(pro)s(of)456 1044 y(the)f(existence)i(of)f (a)f(sym)m(b)s(olic)h(dynamics.)65 b(W)-8 b(e)39 b(also)h(use)e(sligh)m (tly)h(w)m(eak)m(er)456 1152 y(h)m(yp)s(otheses)30 b(that)h(tak)m(e)h (adv)-5 b(an)m(tage)32 b(of)f(sev)m(eral)g(homo)s(clinic)g(in)m (tersections.)555 1260 y(W)-8 b(e)37 b(will)g(consider)f(a)g(neigh)m(b) s(orho)s(o)s(d)f Fn(S)41 b(\032)35 b Fk(R)24 b Fn(\002)f Fk(T)36 b Fs(of)g(the)g(separatrix)h(of)456 1367 y(the)30 b(p)s(endulum)e(w)m(e)j(are)g(studying,)f(and)f(consider)i(the)f (compact)i(set)456 1551 y(\(8\))260 b Fn(D)28 b Fs(:=)d Fn(S)i(\002)20 b Fs([)p Fm(I)1288 1565 y Fl(\000)1348 1551 y Fm(;)15 b(I)1428 1565 y Fq(+)1487 1551 y Fs(])21 b Fn(\002)e(f)p Fm(')27 b Fn(2)d Fk(T)p Fn(g)d(\002)f(f)p Fm(s)25 b Fn(2)g Fk(T)p Fn(g)20 b(\002)g Fs([)p Fn(\000)p Fm(")2611 1565 y Fq(0)2651 1551 y Fm(;)15 b(")2733 1565 y Fq(0)2773 1551 y Fs(])456 1734 y(to)31 b(b)s(e)f(the)h(domain)g(of)g (de\014nition)g(of)g(our)f(problem.)41 b(Hence,)33 b(the)e Fm(C)2863 1701 y Fp(r)2931 1734 y Fs(norms)456 1842 y(of)e(functions)g (will)h(refer)f(to)h(sup)e(norms)g(de\014ned)g(on)h(this)h(set.)41 b(Of)28 b(course,)i(in)456 1950 y(the)25 b(case)h(that)g(the)g (functions)f(dep)s(end)e(only)j(on)f(a)g(few)g(of)h(the)f(v)-5 b(ariables)26 b(\(e.g.)456 2058 y Fm(V)20 b Fs(,)31 b(whic)m(h)g(only)h (dep)s(ends)d(on)i Fm(q)s Fs(\),)h(w)m(e)g(can)f(also)h(consider)g (them)f(as)g(function)456 2166 y(of)f(more)h(v)-5 b(ariables)31 b(and)e(de\014ne)h(the)h(norm)e(in)h(the)h(appropriate)f(domain.)555 2274 y(The)g(main)g(rigorous)h(result)f(of)h(this)f(pap)s(er)f(is:)456 2457 y Fw(Theorem)35 b(7.)41 b Fo(Consider)34 b(a)f(system)h(of)e(the)h (form)41 b Fs(\(6\))555 2565 y Fo(Assume:)558 2711 y Fw(H1)p Fs(.)g Fo(The)35 b(terms)h Fm(V)55 b Fo(and)35 b Fm(h)g Fo(in)42 b Fs(\(6\))35 b Fo(ar)-5 b(e)36 b(uniformly)g Fm(C)2493 2678 y Fp(r)2565 2711 y Fo(for)f Fm(r)c Fn(\025)e Fm(r)2925 2725 y Fq(0)2965 2711 y Fo(,)35 b(suf-)758 2819 y(\014ciently)e(lar)-5 b(ge.)558 2927 y Fw(H2)p Fs(.)41 b Fo(The)g(p)-5 b(otential)43 b Fm(V)61 b Fs(:)40 b Fk(T)g Fn(!)g Fk(R)h Fo(has)h(a)f(unique)f(glob)-5 b(al)42 b(maximum)g(at)758 3035 y Fm(q)33 b Fs(=)d(0)35 b Fo(which)h(is)f(non-de)-5 b(gener)g(ate)37 b(\(i.e.,)e Fm(V)2299 3002 y Fl(00)2342 3035 y Fs(\(0\))c Fm(<)e Fs(0)p Fo(\).)50 b(We)35 b(denote)758 3143 y(by)d Fs(\()p Fm(q)952 3157 y Fq(0)992 3143 y Fs(\()p Fm(t)p Fs(\))p Fm(;)15 b(p)1181 3157 y Fq(0)1221 3143 y Fs(\()p Fm(t)p Fs(\)\))33 b Fo(an)g(orbit)f(of)h(the)f(p)-5 b(endulum)34 b Fm(P)2462 3157 y Fl(\006)2521 3143 y Fs(\()p Fm(p;)15 b(q)s Fs(\))33 b Fo(in)39 b Fs(\(5\))q Fo(,)32 b(ho-)758 3251 y(mo)-5 b(clinic)34 b(to)f Fs(\(0)p Fm(;)15 b Fs(0\))p Fo(.)558 3359 y Fw(H3)p Fs(.)41 b Fm(h)33 b Fo(is)g(a)g(trigonometric)h (p)-5 b(olynomial)36 b(in)c Fm(')h Fo(and)h Fm(s)p Fo(:)456 3555 y Fs(\(9\))349 b Fm(h)p Fs(\()p Fm(p;)15 b(q)s(;)g(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))29 b(=)1679 3469 y Fh(X)1648 3667 y Fp(k)r(;l)q Fl(2N)1858 3531 y Fs(^)1857 3555 y Fm(h)1909 3570 y Fp(k)r(;l)1994 3555 y Fs(\()p Fm(p;)15 b(q)s(;)g(I)7 b Fs(;)15 b Fm(")p Fs(\))p Fm(e)2405 3518 y Fp(i)p Fq(\()p Fp(k)r(')p Fq(+)p Fp(l)q(s)p Fq(\))2684 3555 y Fm(;)758 3848 y Fo(wher)-5 b(e)34 b Fn(N)k(\032)25 b Fk(Z)1285 3815 y Fq(2)1357 3848 y Fo(is)32 b(a)h(\014nite)g(set.)558 3956 y Fw(H4)p Fs(.)41 b Fo(Consider)f(the)e(Poinc)-5 b(ar)n(\023)-44 b(e)39 b(function,)h(also)f(c)-5 b(al)5 b(le)-5 b(d)39 b(Melnikov)f(p)-5 b(oten-)758 4064 y(tial,)30 b(asso)-5 b(ciate)g(d)30 b(to)e Fm(h)h Fo(\(and)g(to)f(the)g(homo)-5 b(clinic)30 b(orbit)e Fs(\()p Fm(q)2746 4078 y Fq(0)2785 4064 y Fm(;)15 b(p)2871 4078 y Fq(0)2911 4064 y Fs(\))28 b Fo(men-)758 4172 y(tione)-5 b(d)34 b(in)40 b Fw(H2)p Fo(\):)810 4416 y Fn(L)p Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))26 b(=)20 b Fn(\000)1380 4292 y Fh(Z)1471 4319 y Fq(+)p Fl(1)1431 4499 y(\0001)1601 4315 y Fh(\020)1655 4416 y Fm(h)15 b Fs(\()q Fm(p)1804 4430 y Fq(0)1843 4416 y Fs(\()p Fm(\033)s Fs(\))p Fm(;)g(q)2049 4430 y Fq(0)2089 4416 y Fs(\()p Fm(\033)s Fs(\))p Fm(;)g(I)7 b(;)15 b(')23 b Fs(+)d Fm(I)7 b(\033)n(;)15 b(s)20 b Fs(+)g Fm(\033)s Fs(;)15 b(0\))1475 4645 y Fn(\000)20 b Fm(h)p Fs(\(0)p Fm(;)15 b Fs(0)p Fm(;)g(I)7 b(;)15 b(')24 b Fs(+)19 b Fm(I)7 b(\033)n(;)15 b(s)21 b Fs(+)f Fm(\033)s Fs(;)15 b(0\))2550 4544 y Fh(\021)2621 4645 y Fm(d\033)n(:)456 4515 y Fs(\(10\))758 4856 y Fo(Assume)43 b(that,)j(for)d(any)g(value)g (of)g Fm(I)50 b Fn(2)43 b Fs(\()p Fm(I)2304 4870 y Fl(\000)2364 4856 y Fm(;)15 b(I)2444 4870 y Fq(+)2503 4856 y Fs(\))43 b Fo(ther)-5 b(e)43 b(exists)g(an)758 4964 y(op)-5 b(en)32 b(set)e Fn(J)1171 4978 y Fp(I)1236 4964 y Fn(\032)25 b Fk(T)1393 4931 y Fq(2)1432 4964 y Fo(,)30 b(with)h(the)g(pr)-5 b(op)g(erty)32 b(that)g(when)f Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))27 b Fn(2)d Fm(H)3087 4978 y Fl(\000)3146 4964 y Fo(,)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 456 251 a Fq(18)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)758 450 y Fo(wher)-5 b(e)456 614 y Fs(\(11\))401 b Fm(H)1093 628 y Fl(\000)1177 614 y Fs(=)1383 528 y Fh([)1273 729 y Fp(I)5 b Fl(2)p Fq(\()p Fp(I)1414 738 y Fg(\000)1466 729 y Fp(;I)1517 738 y Fi(+)1567 729 y Fq(\))1595 614 y Fn(f)p Fm(I)i Fn(g)21 b(\002)f(J)1906 628 y Fp(I)1971 614 y Fn(\032)25 b Fs(\()p Fm(I)2142 628 y Fl(\000)2201 614 y Fm(;)15 b(I)2281 628 y Fq(+)2340 614 y Fs(\))21 b Fn(\002)f Fk(T)2548 576 y Fq(2)2587 614 y Fm(;)758 882 y Fo(the)33 b(map)456 1046 y Fs(\(12\))366 b Fm(\034)36 b Fn(2)24 b Fk(R)h Fn(7!)g(L)p Fs(\()p Fm(I)7 b(;)15 b(')22 b Fn(\000)e Fm(I)7 b(\034)e(;)15 b(s)20 b Fn(\000)g Fm(\034)10 b Fs(\))25 b Fn(\021)g Fs(\000\()p Fm(\034)10 b Fs(;)15 b Fm(I)7 b(;)15 b(';)g(s)p Fs(\))758 1210 y Fo(has)29 b(a)f(non-de)-5 b(gener)g(ate)29 b(critic)-5 b(al)28 b(p)-5 b(oint)29 b Fm(\034)37 b Fo(which)29 b(is)e(lo)-5 b(c)g(al)5 b(ly)29 b(given,)f(by)758 1318 y(the)k(implicit)g(function)g (the)-5 b(or)g(em)34 b(in)d(the)h(form)g Fm(\034)j Fs(=)25 b Fm(\034)2637 1285 y Fl(\003)2677 1318 y Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))33 b Fo(with)758 1426 y Fm(\034)808 1393 y Fl(\003)880 1426 y Fo(a)g(smo)-5 b(oth)35 b(function.)858 1534 y(Assume)i(mor)-5 b(e)g(over)40 b(that)e(for)g(every)g Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))35 b Fn(2)f Fm(H)2691 1548 y Fl(\000)2750 1534 y Fo(,)k(the)g(func-)758 1642 y(tion)456 1830 y Fs(\(13\))1382 1769 y Fm(@)5 b Fn(L)p 1382 1810 117 4 v 1384 1893 a Fm(@)g(')1508 1830 y Fs(\()p Fm(I)i(;)15 b(')22 b Fn(\000)e Fm(I)7 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Fm(")p Fn(j)h Fm(<)f(")2335 2630 y Fl(\003)2400 2616 y Fo(and)h(for)f(any)h(interval)456 2723 y Fs([)p Fm(I)528 2690 y Fl(\003)521 2746 y(\000)580 2723 y Fm(;)15 b(I)667 2690 y Fl(\003)660 2746 y Fq(+)719 2723 y Fs(])41 b Fn(\032)f Fs(\()p Fm(I)971 2737 y Fl(\000)1031 2723 y Fm(;)15 b(I)1111 2737 y Fq(+)1170 2723 y Fs(\))p Fo(,)43 b(ther)-5 b(e)42 b(exists)g(a)f(tr)-5 b(aje)g(ctory)49 b Fs(~)-50 b Fm(x)p Fs(\()p Fm(t)p Fs(\))41 b Fo(of)g(the)h(system)49 b Fs(\(6\))456 2831 y Fo(such)32 b(that)i(for)f(some)h Fm(T)k(>)25 b Fs(0)456 2995 y(\(14\))616 b Fm(I)7 b Fs(\()f(~)-51 b Fm(x)p Fs(\(0\)\))27 b Fn(\024)e Fm(I)1686 2958 y Fl(\003)1679 3018 y(\000)1738 2995 y Fs(;)108 b Fm(I)7 b Fs(\()f(~)-51 b Fm(x)q Fs(\()p Fm(T)13 b Fs(\)\))26 b Fn(\025)f Fm(I)2346 2958 y Fl(\003)2339 3018 y Fq(+)456 3159 y Fo(\(r)-5 b(esp)g(e)g(ctively:)456 3323 y Fs(\(15\))586 b Fm(I)7 b Fs(\()f(~)-51 b Fm(x)p Fs(\(0\)\))27 b Fn(\025)e Fm(I)1656 3286 y Fl(\003)1649 3346 y Fq(+)1708 3323 y Fs(;)108 b Fm(I)7 b Fs(\()f(~)-51 b Fm(x)p Fs(\()p Fm(T)13 b Fs(\)\))26 b Fn(\024)f Fm(I)2315 3286 y Fl(\003)2308 3346 y(\000)2367 3323 y Fs(\))p Fm(:)555 3495 y Fs(W)-8 b(e)33 b(will)f(consider)g Fm(I)1282 3509 y Fl(\000)1368 3495 y Fm(<)27 b(I)1506 3509 y Fq(+)1597 3495 y Fs(as)32 b(\014xed)e(and)i(somewhat)g(large.)45 b(In)31 b(partic-)456 3603 y(ular,)e([)p Fm(I)732 3617 y Fl(\000)792 3603 y Fm(;)15 b(I)872 3617 y Fq(+)931 3603 y Fs(])30 b(can)g(con)m(tain)g(all)h(the)f(resonances)g Fm(I)i Fs(=)25 b Fn(\000)p Fm(l)r(=k)s Fs(,)30 b(for)f(\()p Fm(k)s(;)15 b(l)r Fs(\))27 b Fn(2)e(N)13 b Fs(.)456 3711 y(Then,)34 b(the)h(tra)5 b(jectories)36 b(that)g(w)m(e)f(construct)f (cross)h(o)m(v)m(er)h(the)f(resonan)m(t)g(re-)456 3819 y(gions.)555 3927 y(Hence,)25 b(w)m(e)d(o)m(v)m(ercome)j(the)d Fo(lar)-5 b(ge)25 b(gap)h(pr)-5 b(oblem)30 b Fs(b)m(y)22 b(sho)m(wing)g(the)g(existence)456 4035 y(of)32 b(orbits)h(whic)m(h)g (tra)m(v)m(erse)h(regions)f(in)f(whic)m(h)h(primary)e(KAM)i(tori)h(are) f(not)456 4143 y(presen)m(t)c(and)f(indeed,)h(there)g(are)h(no)f (transv)m(erse)g(hetero)s(clinic)h(in)m(tersections)456 4251 y(b)s(et)m(w)m(een)h(them)f(accessible)i(to)f(direct)g(p)s (erturbation)e(theory)-8 b(.)456 4423 y Fw(Remark)41 b(8.)k Fs(Note)38 b(that)e(giv)m(en)h(the)f(fact)h(that)g(w)m(e)f(can)h (\014nd)d(an)i(op)s(en)f(set)456 4531 y Fm(H)532 4545 y Fl(\000)630 4531 y Fs(as)40 b(in)g(\(11\))q(,)j(it)d(will)g(often)h (happ)s(en)d(that)i(w)m(e)h(can)f(\014nd)e(t)m(w)m(o)j(di\013eren)m(t) 456 4640 y(op)s(en)29 b(sets)i Fm(H)936 4602 y Fl(\006)929 4664 y(\000)1025 4640 y Fs(where)f(the)h(sign)f(in)g(\(13\))i(tak)m(e)g (place.)555 4748 y(In)c(a)h(generic)g(situation,)i(the)d (non-degeneracy)i(condition)f(\(11\))h(will)f(only)456 4856 y(fail)k(in)g(a)h(co)s(dimension)f(1)g(manifold.)49 b(Moreo)m(v)m(er,)36 b(generically)-8 b(,)37 b(this)c(critical)456 4964 y(manifold)d(will)h(not)f(b)s(e)g(a)h(lev)m(el)h(set)f(of)f(the)h (action.)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(19)555 450 y Fs(Hence,)47 b(for)c(generic)h(situations)f(w)m(e)g(exp)s(ect)h(that)f(w)m(e)g(will) h(b)s(e)e(able)h(to)456 558 y(\014nd)37 b(o)m(v)m(erlapping)i(in)m (terv)-5 b(als)40 b Fm(I)1561 525 y Fl(\006)1659 558 y Fs(suc)m(h)e(that)h Fm(I)2124 525 y Fl(\006)2209 558 y Fn(\002)25 b Fk(T)2366 525 y Fq(2)2443 558 y Fs(in)m(tersects)40 b Fm(H)2930 572 y Fl(\000)3027 558 y Fs(and)456 666 y(suc)m(h)35 b(that)h(in)f(\()p Fm(I)1061 633 y Fl(\006)1144 666 y Fn(\002)23 b Fk(T)1299 633 y Fq(2)1338 666 y Fs(\))h Fn(\\)f Fm(H)1557 680 y Fl(\000)1651 666 y Fs(w)m(e)35 b(obtain)h(the)g(di\013eren)m(t)g(alternativ)m(es)h(in)456 774 y(\(13\))q(.)49 b(Again,)34 b(w)m(e)g(emphasize)f(that)h(v)m (erifying)g(that)f(this)g(generic)h(situation)456 882 y(happ)s(ens)28 b(in)i(a)h(concrete)h(system,)f(is)f(reduced)g(to)h(a)g (\014nite)f(calculation.)555 990 y(As)k(w)m(e)g(will)f(see)h(in)f (Theorem)h(93)g(this)f(will)h(allo)m(w)h(us)e(to)h(construct)f(v)m(ery) 456 1098 y(complicated)f(orbits.)1889 b Fj(\003)456 1329 y Fw(Remark)30 b(9.)38 b Fs(One)26 b(can)g(obtain)h(more)f(general)h (results)f(in)g(the)g(case)h(that)g(the)456 1436 y(function)1382 1528 y Fm(@)5 b Fn(L)p 1382 1568 117 4 v 1384 1652 a Fm(@)g(')1508 1589 y Fs(\()p Fm(I)i(;)15 b(')22 b Fn(\000)e Fm(I)7 b(\034)1899 1552 y Fl(\003)1938 1589 y Fm(;)15 b(s)20 b Fn(\000)g Fm(\034)2182 1552 y Fl(\003)2222 1589 y Fs(\))456 1779 y(has)36 b(a)h(zero)g(in)g Fn(J)1079 1793 y Fp(I)1118 1779 y Fs(,)h(b)s(ecause)f(in)f(this)h(case)g(w)m(e)h (can)e(obtain)h(subsets)f(where)456 1887 y(it)43 b(is)g(p)s(ositiv)m(e) g(and)g(negativ)m(e.)80 b(This)42 b(allo)m(ws)i(the)f(orbit)48 b(~)-50 b Fm(x)p Fs(\()p Fm(t)p Fs(\))43 b(to)g(p)s(erform)456 1995 y(more)33 b(arbitrary)g(excursions.)49 b(This)33 b(happ)s(ens,)f(for)h(example,)i(if)e(w)m(e)h(c)m(hange)456 2103 y(h)m(yp)s(othesis)c Fw(H4)g Fs(to)h(assuming)f(that)456 2269 y(\(16\))798 b(\()p Fm(I)1489 2283 y Fl(\000)1548 2269 y Fm(;)15 b(I)1628 2283 y Fq(+)1688 2269 y Fs(\))20 b Fn(\002)g Fk(T)1895 2231 y Fq(2)1960 2269 y Fn(\032)25 b Fm(H)2132 2283 y Fl(\000)2190 2269 y Fm(:)456 2434 y Fs(In)f(this)i(case,)h(w)m(e)f(obtain)g(that)g Fn(L)p Fs(\()p Fm(I)7 b(;)15 b(')10 b Fn(\000)g Fm(I)d(\034)1981 2401 y Fl(\003)2022 2434 y Fm(;)15 b(s)10 b Fn(\000)g Fm(\034)2246 2401 y Fl(\003)2286 2434 y Fs(\))26 b(is)g(a)f(p)s(erio)s (dic)g(function)456 2542 y(of)33 b Fm(')g Fs(and)g Fm(s)g Fs(and)f(the)h(fact)h(that)g(its)g(deriv)-5 b(ativ)m(e)34 b(has)f(a)h(zero)f(is)h(guaran)m(teed.)3103 2650 y Fj(\003)555 2881 y Fs(In)25 b(this)g(pap)s(er,)g(w)m(e)h(will)f(not)g(address)g (rigorously)g(the)g(issue)g(of)h(ho)m(w)f(abun-)456 2989 y(dan)m(t)k(is)h(the)g(mec)m(hanism)f(presen)m(ted)h(here.)40 b(Nev)m(ertheless,)32 b(in)d(Section)h(12.3,)456 3097 y(w)m(e)g(will)g(presen)m(t)g(some)h(heuristic)f(remarks)f(on)h(ho)m(w) g(the)g(abundance)g(of)g(the)456 3205 y(h)m(yp)s(otheses.)456 3378 y Fw(Remark)d(10.)36 b Fs(Sev)m(eral)24 b(of)g(the)f(follo)m(wing) i(argumen)m(ts)f(are)f(signi\014can)m(tly)i(eas-)456 3486 y(ier)34 b(if)h(w)m(e)f(assume)h(\(16\))q(.)53 b(The)34 b(reader)g(ma)m(y)h(w)m(an)m(t)h(to)f(c)m(hec)m(k)h(this)e(case)i (\014rst)456 3594 y(to)29 b(gain)f(in)m(tuition.)41 b(F)-8 b(or)29 b(simplicit)m(y)h(of)e(heuristic)g(in)m(tuition,)i(the)e (pictures)g(in)456 3702 y(Figures)36 b(1,)j(2)e(w)m(e)g(presen)m(t)f (to)h(illustrate)h(the)f(results)f(are)h(done)f(under)f(h)m(y-)456 3810 y(p)s(othesis)d(\(16\))r(.)48 b(Note)34 b(that)f(if)g(there)g(is)g (a)g(p)s(oin)m(t)g(\()p Fm(I)2274 3824 y Fq(0)2314 3810 y Fm(;)15 b(')2413 3824 y Fq(0)2453 3810 y Fm(;)g(s)2536 3824 y Fq(0)2575 3810 y Fs(\))34 b(for)e(whic)m(h)h(the)456 3918 y(function)26 b(\000\()p Fm(\034)10 b Fs(;)15 b Fm(I)1030 3932 y Fq(0)1070 3918 y Fm(;)g(')1169 3932 y Fq(0)1209 3918 y Fm(;)g(s)1292 3932 y Fq(0)1331 3918 y Fs(\))27 b(has)f(a)h(non-degenerate)h(critical)g(p)s(oin)m(t)f Fm(\034)2834 3885 y Fl(\003)2873 3918 y Fs(,)g(b)m(y)g(the)456 4026 y(implicit)i(function)f(theorem)g(the)h(same)f(will)h(b)s(e)f (true)g(for)g(all)h(the)f(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))30 b(in)456 4133 y(an)g(op)s(en)g(neigh)m(b)s(orho)s(o)s(d.)555 4241 y(One,)36 b(of)g(course,)g(exp)s(ects)g(that)g(this)f(op)s(en)f (set)i(has)f(a)g(b)s(oundary)e(where)456 4349 y(the)e(critical)j(p)s (oin)m(t)e(b)s(ecomes)f(degenerate.)46 b(So,)32 b(the)g(h)m(yp)s (othesis)f(\(16\))i(holds)456 4457 y(in)i(op)s(en)g(sets)h(of)g(mo)s (dels)g(and)f(it)h(is)g(v)m(eri\014ed)f(in)h(sev)m(eral)h(cases)g(of)e (in)m(terest,)456 4565 y(whic)m(h)30 b(w)m(e)h(will)f(consider)h(in)f (detail)h(in)f(Section)h(13.)555 4673 y(Ev)m(en)40 b(if)47 b(\(16\))41 b(is)f(not)f(generic)i(for)e(one)h(critical)i(p)s(oin)m(t)d (\(it)i(is,)h(ho)m(w)m(ev)m(er,)456 4781 y(op)s(en\),)29 b(as)g(w)m(e)h(will)g(see)f(in)g(Section)h(12.3)h(when)d(w)m(e)i 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Fs([)p Fm(I)528 1062 y Fl(\003)p Fp(;k)521 1129 y Fl(\000)626 1106 y Fm(;)15 b(I)713 1062 y Fl(\003)p Fp(;k)706 1129 y Fq(+)811 1106 y Fs(])30 b(on)g(whic)m(h)g(one)h(can)g(v)m(erify)g (the)f(h)m(yp)s(otheses)g(of)h(Theorem)f(7.)555 1214 y(W)-8 b(e)37 b(will)e(see)h(in)f(Theorem)g(93)i(that)f(if)f(there)g (is)h(a)g(closed)g(in)m(terv)-5 b(al)36 b Fm(I)3030 1181 y Fl(\003)3103 1214 y Fs(=)456 1322 y([)p Fm(I)528 1289 y Fl(\003)521 1344 y(\000)580 1322 y Fm(;)15 b(I)667 1289 y Fl(\003)660 1344 y Fq(+)719 1322 y Fs(])25 b(con)m(tained)g(in)f (the)g(union)f(of)i(the)f(in)m(terior)h(of)f(the)h(in)m(terv)-5 b(als)25 b Fm(I)2879 1337 y Fp(k)2921 1322 y Fs(,)h(then,)456 1430 y(w)m(e)k(also)i(obtain)f(the)f(conclusions)h(of)f(Theorem)h(7)f (for)g(the)h(in)m(terv)-5 b(al)32 b Fm(I)2922 1397 y Fl(\003)2961 1430 y Fs(.)555 1538 y(As)25 b(w)m(e)g(will)h(see)f(in)g (more)g(detail)h(Section)g(12.3)g(this)f(mak)m(es)h(it)f(m)m(uc)m(h)g (easier)456 1646 y(to)41 b(v)m(erify)g(the)f(existence)i(of)f(the)f (mec)m(hanism)h(in)f(concrete)i(mo)s(dels.)70 b(The)456 1754 y(reason)22 b(is)h(that)g(if)f(w)m(e)h(consider)g(just)f(one)h (smo)s(oth)f(curv)m(e)h Fm(\034)2481 1721 y Fl(\003)2543 1754 y Fs(it)g(often)g(happ)s(ens)456 1862 y(that)f(this)g(manifold)g (has)g(co)s(dimension)f(one)i(b)s(oundary)c(in)j(whic)m(h)g(the)g (critical)456 1970 y(p)s(oin)m(t)30 b(b)s(ecomes)h(degenerate.)555 2077 y(If)37 b(the)h(b)s(oundaries)f(for)g(di\013eren)m(t)h Fm(\034)1850 2044 y Fl(\003)1890 2077 y Fs('s)f(do)h(not)g(coincide,)j (the)d(di\013usion)456 2185 y(can)29 b(pro)s(ceed.)40 b(Moreo)m(v)m(er,)32 b(w)m(e)d(will)h(see)g(that)f(if)g(the)h(pro)5 b(jections)29 b(in)g Fm(I)36 b Fs(of)30 b(the)456 2293 y(region)f(where)f(the)h(p)s(ositiv)m(e)h(and)e(negativ)m(e)j (alternativ)m(es)g(of)k(\(13\))c(hold,)e(it)g(is)456 2401 y(p)s(ossible)f(to)h(\014nd)e(orbits)i(whose)f Fm(I)36 b Fs(p)s(erforms)27 b(largely)j(arbitrary)e(excursions.)3103 2509 y Fj(\003)456 2711 y Fw(Remark)j(12.)39 b Fs(F)-8 b(or)27 b Fm(")f Fs(=)e(0,)k(the)f(set)g Fm(p)e Fs(=)g(0)p Fm(;)15 b(q)29 b Fs(=)c(0)i(is)g(a)g(normally)g(h)m(yp)s(erb)s(olic)456 2819 y(in)m(v)-5 b(arian)m(t)26 b(manifold)e(whose)h(stable)h(and)e (unstable)h(manifolds)f(coincide.)40 b(F)-8 b(or)456 2927 y(0)30 b Fm(<)f Fn(j)p Fm(")p Fn(j)i Fm(<)e Fs(1,)34 b(this)f(manifold)g(and)f(its)i(stable)g(and)e(unstable)h(manifolds)f (are)456 3035 y(preserv)m(ed)j(due)g(to)i(the)f(theory)g(of)g(normally) g(h)m(yp)s(erb)s(olic)g(in)m(v)-5 b(arian)m(t)37 b(mani-)456 3143 y(folds.)555 3251 y(Hyp)s(othesis)f Fw(H4)h Fs(means)f(that)h(the) g(\014rst)e(order)h(in)g Fm(")h Fs(p)s(erturbation)e(the-)456 3359 y(ory)j(predicts)h(the)g(transv)m(ersal)g(in)m(tersection)i(of)e (the)f(stable)i(and)e(unstable)456 3467 y(manifolds.)555 3575 y(W)-8 b(e)24 b(ha)m(v)m(e)f(tak)m(en)g(adv)-5 b(an)m(tage)24 b(of)e(the)h(symplectic)g(geometry)g(to)g(express)f(the)456 3683 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y(of)j(co)s(ordinates)h(giv)m(en)g(b)m(y)f(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\).)56 b(Hence,)38 b(w)m(e)d(will)h(not)f (need)g(to)h(distin-)456 1571 y(guish)26 b(v)m(ery)g(explicitly)i (whether)e(w)m(e)h(are)g(talking)h(ab)s(out)e(the)h(three)f(n)m(um)m(b) s(ers)456 1680 y(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))28 b(or)f(the)h(p)s(oin)m(t)f(in)f(the)i(manifold)1917 1657 y(~)1908 1680 y(\003.)39 b(Nev)m(ertheless,)30 b(w)m(e)e(will)f(empha-) 456 1788 y(size)39 b(as)f(m)m(uc)m(h)g(as)h(p)s(ossible)f(the)g (geometric)i(meaning)f(of)f(the)h(calculations)456 1896 y(that)31 b(need)f(to)h(b)s(e)f(carried)g(out)h(in)f(co)s(ordinates.) 555 2004 y(The)g(unp)s(erturb)s(ed)d(\015o)m(w)j(on)1592 1981 y(~)1584 2004 y(\003)g(is)g(giv)m(en)i(b)m(y)e(the)h(quasi-p)s (erio)s(dic)f(\015o)m(w:)456 2153 y(\(27\))283 b(\010)965 2167 y Fp(t;)p Fq(0)1049 2153 y Fs(\(0)p Fm(;)15 b Fs(0)p Fm(;)g(I)1294 2167 y Fq(0)1336 2153 y Fm(;)g(')1435 2167 y Fq(0)1475 2153 y Fm(;)g(s)1558 2167 y Fq(0)1598 2153 y Fs(\))25 b(=)g(\(0)p Fm(;)15 b Fs(0)p Fm(;)g(I)1999 2167 y Fq(0)2041 2153 y Fm(;)g(')2140 2167 y Fq(0)2200 2153 y Fs(+)20 b Fm(I)2331 2167 y Fq(0)2370 2153 y Fm(t;)15 b(s)2486 2167 y Fq(0)2546 2153 y Fs(+)20 b Fm(t)p Fs(\))p Fm(;)456 2302 y Fs(and)29 b(w)m(e)i(will)g(b)s(e)f(denote)h(b)m(y)1023 2430 y(~)1019 2454 y Fm(\025)1072 2468 y Fp(t)1102 2454 y Fs(\()p Fm(I)1177 2468 y Fq(0)1216 2454 y Fm(;)15 b(')1315 2468 y Fq(0)1356 2454 y Fm(;)g(s)1439 2468 y Fq(0)1478 2454 y Fs(\))26 b(=)f(\(0)p Fm(;)15 b Fs(0)p Fm(;)g(I)1880 2468 y Fq(0)1921 2454 y Fm(;)g(')2020 2468 y Fq(0)2081 2454 y Fs(+)k Fm(I)2211 2468 y Fq(0)2251 2454 y Fm(t;)c(s)2367 2468 y Fq(0)2426 2454 y Fs(+)20 b Fm(t)p Fs(\))p Fm(:)456 2603 y Fs(the)30 b(orbits)h(in)f(the)g(torus)g Fn(T)1416 2617 y Fp(I)1447 2626 y Fi(0)1485 2603 y Fs(.)555 2711 y(One)g(comp)s(onen)m(t)g(of)g(the)g(stable)h(and)e(unstable)g(in)m(v) -5 b(arian)m(t)31 b(manifolds)f(for)464 2797 y(~)456 2820 y(\003,)g Fm(W)673 2787 y Fq(s)667 2846 y(~)660 2863 y(\003)713 2820 y Fs(,)h Fm(W)868 2787 y Fq(u)862 2846 y(~)855 2863 y(\003)941 2820 y Fs(coincides)g(in)f(a)h(manifold)j (~)-49 b Fm(\015)36 b Fs(of)30 b(orbits)h(homo)s(clinic)g(to)2905 2797 y(~)2896 2820 y(\003)1308 3039 y(~)-48 b Fm(\015)30 b Fn(\032)25 b Fs(\()p Fm(W)1612 3002 y Fq(s)1606 3061 y(~)1599 3078 y(\003)1673 3039 y Fn(n)1747 3016 y Fs(~)1738 3039 y(\003)q(\))20 b Fn(\\)g Fs(\()p Fm(W)2072 3002 y Fq(u)2066 3061 y(~)2059 3078 y(\003)2135 3039 y Fn(n)2210 3016 y Fs(~)2201 3039 y(\003\))p Fm(:)555 3171 y Fs(and)30 b(a)h(parameterization)h(of)i(~)-48 b Fm(\015)35 b Fs(is)c(giv)m(en)g (b)m(y:)456 3320 y(\(28\))51 b(~)-48 b Fm(\015)30 b Fs(:=)25 b Fn(f)p Fs(\()p Fm(p)988 3334 y Fq(0)1028 3320 y Fs(\()p Fm(\034)10 b Fs(\))p Fm(;)15 b(q)1229 3334 y Fq(0)1269 3320 y Fs(\()p Fm(\034)10 b Fs(\))p Fm(;)15 b(I)7 b(;)15 b(';)g(s)p Fs(\))58 b(:)e Fm(I)33 b Fn(2)24 b Fs([)p Fm(I)2055 3334 y Fl(\000)2115 3320 y Fm(;)15 b(I)2195 3334 y Fq(+)2254 3320 y Fs(])p Fm(;)46 b(\034)35 b Fn(2)25 b Fk(R)p Fm(;)30 b Fs(\()p Fm(';)15 b(s)p Fs(\))27 b Fn(2)e Fk(T)3018 3283 y Fq(2)3057 3320 y Fn(g)p Fm(;)456 3469 y Fs(where)32 b(\()p Fm(p)802 3483 y Fq(0)842 3469 y Fm(;)15 b(q)923 3483 y Fq(0)962 3469 y Fs(\))34 b(is,)g(as)g(in)f (\(23\))q(,)h(the)g(c)m(hosen)g(homo)s(clinic)f(orbit)h(of)f(the)h(p)s (en-)456 3577 y(dulum.)555 3685 y(Hence,)46 b(the)c(meaning)g(of)g(the) g(co)s(ordinate)h Fm(\034)52 b Fs(is)42 b(the)g(time)h(of)f(the)g (\015o)m(w)456 3793 y(along)31 b(the)g(unp)s(erturb)s(ed)26 b(separatrix.)41 b(W)-8 b(e)32 b(denote)f(b)m(y)635 3942 y(~)-48 b Fm(\015)679 3956 y Fp(t)709 3942 y Fs(\()p Fm(\034)5 b(;)15 b(I)869 3956 y Fq(0)909 3942 y Fm(;)g(')1008 3956 y Fq(0)1048 3942 y Fm(;)g(s)1131 3956 y Fq(0)1171 3942 y Fs(\))83 b(=)g(\()p Fm(p)1524 3956 y Fq(0)1563 3942 y Fs(\()p Fm(\034)31 b Fs(+)20 b Fm(t)p Fs(\))p Fm(;)15 b(q)1909 3956 y Fq(0)1948 3942 y Fs(\()p Fm(\034)31 b Fs(+)20 b Fm(t)p Fs(\))p Fm(;)15 b(I)2293 3956 y Fq(0)2333 3942 y Fm(;)g(')2432 3956 y Fq(0)2492 3942 y Fs(+)20 b Fm(I)2623 3956 y Fq(0)2663 3942 y Fm(t;)15 b(s)2779 3956 y Fq(0)2838 3942 y Fs(+)20 b Fm(t)p Fs(\))1289 4077 y(=)83 b(\010)1509 4091 y Fp(t;)p Fq(0)1593 4077 y Fs(\()p Fm(p)1674 4091 y Fq(0)1714 4077 y Fs(\()p Fm(\034)10 b Fs(\))p Fm(;)15 b(q)1915 4091 y Fq(0)1955 4077 y Fs(\()p Fm(\034)10 b Fs(\))p Fm(;)15 b(I)2155 4091 y Fq(0)2195 4077 y Fm(;)g(')2294 4091 y Fq(0)2335 4077 y Fm(;)g(s)2418 4091 y Fq(0)2457 4077 y Fs(\))p Fm(;)456 4226 y Fs(the)31 b(unp)s(erturb)s(ed)d(\015o)m(w)j(on)j(~)-48 b Fm(\015)5 b Fs(,)32 b(our)e(c)m(hosen)i(comp)s(onen)m(t)f(of)h(the)f(homo)s (clinic)456 4334 y(manifold.)555 4442 y(W)-8 b(e)32 b(note)f(that,)g (for)f(an)m(y)h Fm(\034)k Fn(2)25 b Fk(R)456 4594 y Fs(\(29\))197 b(dist\()999 4570 y(~)995 4594 y Fm(\025)1048 4608 y Fp(t)1078 4594 y Fs(\()p Fm(I)1153 4608 y Fq(0)1193 4594 y Fm(;)15 b(')1292 4608 y Fq(0)1332 4594 y Fm(;)g(s)1415 4608 y Fq(0)1454 4594 y Fs(\))p Fm(;)20 b Fs(~)-50 b Fm(\015)1576 4608 y Fp(t)1607 4594 y Fs(\()p Fm(\034)5 b(;)15 b(I)1767 4608 y Fq(0)1807 4594 y Fm(;)g(')1906 4608 y Fq(0)1946 4594 y Fm(;)g(s)2029 4608 y Fq(0)2069 4594 y Fs(\)\))26 b Fn(!)f Fs(0)91 b(for)g Fm(t)25 b Fn(!)g(\0061)p Fm(:)555 4742 y Fs(Let)38 b(us)f(note)h(that)g(the)g (unp)s(erturb)s(ed)33 b(system)38 b(is)f(a)h(pro)s(duct,)g(and)f(that) 456 4850 y(\(0)p Fm(;)15 b Fs(0\))25 b(is)e(a)h(critical)h(p)s(oin)m(t) e(of)g(the)h(p)s(endulum)c(with)j(c)m(haracteristic)j(exp)s(onen)m(ts) 456 4964 y Fm(\026)511 4978 y Fl(\006)617 4964 y Fs(=)48 b Fn(\006)p Fm(\026)p Fs(,)f(where)d Fm(\026)k Fs(:=)g(\()p Fn(\000)p Fm(V)1637 4931 y Fl(00)1679 4964 y Fs(\(0\)\))1829 4931 y Fq(1)p Fp(=)p Fq(2)1941 4964 y Fs(.)82 b(Moreo)m(v)m(er,)50 b(the)44 b(exp)s(onen)m(ts)g(of)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 456 251 a Fq(26)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 451 y Fs(con)m(traction) 33 b(in)e(the)g(tangen)m(t)h(direction)g(of)2022 428 y(~)2013 451 y(\003)f(are)h(0)f(\(see)h(\(27\))r(\).)43 b(Then,)31 b(the)456 565 y(stable)41 b(and)g(unstable)f(manifolds)h(of) 1837 543 y(~)1828 565 y(\003)g(are)h(c)m(haracterized)h(as)e(the)g(set) h(of)456 673 y(orbits)f(whose)g(distance)h(to)g(the)g(orbits)f(in)2054 650 y(~)2045 673 y(\003)h(is)f(less)h(than)f Fm(C)21 b Fs(exp\()p Fn(\000)p Fm(\026)15 b Fn(j)p Fm(t)p Fn(j)q Fs(\))456 781 y(resp)s(ectiv)m(ely)31 b(as)g Fm(t)25 b Fn(!)g(\0061)p Fs(,)30 b(and)g(w)m(e)h(ha)m(v)m(e)456 946 y(dist\()642 922 y(~)638 946 y Fm(\025)691 960 y Fp(t)721 946 y Fs(\()p Fm(I)796 960 y Fq(0)835 946 y Fm(;)15 b(')934 960 y Fq(0)975 946 y Fm(;)g(s)1058 960 y Fq(0)1097 946 y Fs(\))p Fm(;)k Fs(~)-49 b Fm(\015)1219 960 y Fp(t)1250 946 y Fs(\()p Fm(\034)5 b(;)15 b(I)1410 960 y Fq(0)1450 946 y Fm(;)g(')1549 960 y Fq(0)1589 946 y Fm(;)g(s)1672 960 y Fq(0)1711 946 y Fs(\)\))26 b Fn(\024)f Fm(C)7 b Fs(\()p Fm(\034)j Fs(\))15 b(exp)q(\()p Fn(\000)p Fm(\026)g Fn(j)p Fm(t)p Fn(j)p Fs(\))91 b(for)g Fm(t)25 b Fn(!)g(\0061)p Fm(:)576 1175 y Fs(7.)46 b Ft(Persistence)33 b(of)g(the)h(normall)-6 b(y)32 b(hyperbolic)h(inv)-8 b(ariant)754 1282 y(manif)n(old)33 b(and)g(its)i(st)-6 b(able)32 b(and)i(unst)-6 b(able)32 b(manif)n(olds)555 1444 y Fs(Since)f(the)g(manifold)1335 1421 y(~)1326 1444 y(\003)g(is)g(normally)g(h)m(yp)s(erb)s(olic,)f(and)g(is)h(lo)s(cally)h (in)m(v)-5 b(ari-)456 1552 y(an)m(t)34 b(for)g(the)g(\015o)m(w)h(\(7\)) g(for)f Fm(")d Fs(=)g(0,)k(b)m(y)f(the)g(theory)g(of)g(normally)h(h)m (yp)s(erb)s(olic)456 1660 y(manifolds)44 b(w)m(e)h(ha)m(v)m(e)h(that)f (the)g(manifold)g(p)s(ersists)f(under)f(small)i(p)s(ertur-)456 1768 y(bations.)58 b(Moreo)m(v)m(er,)40 b(if)d(the)f(system)g(dep)s (ends)f(smo)s(othly)h(on)g(parameters,)456 1876 y(the)d(manifolds|they) g(ma)m(y)g(b)s(e)f(non)h(unique|ma)m(y)f(b)s(e)g(c)m(hosen)i(to)g(dep)s (end)456 1984 y(smo)s(othly)c(on)g(parameters.)555 2092 y(A)37 b(form)m(ulation)g(of)g(the)g(results)f(of)h([F)-8 b(en72)r(,)36 b(F)-8 b(en74)r(,)37 b(F)-8 b(en77)q(])37 b(in)f(the)h(w)m(a)m(y)456 2200 y(that)30 b(w)m(e)h(will)g(use)f(them)g (\(v)m(ery)h(similar)g(to)g(the)f(statemen)m(t)i(of)f(Theorem)f(4.2)456 2308 y(of)g([DLS00)q(]\))h(is:)456 2479 y Fw(Theorem)e(20.)38 b Fo(Consider)29 b(a)g(Hamiltonian)g(as)g(in)35 b Fs(\(6\))q Fo(.)40 b(Assume)28 b(that)i Fm(H)3045 2493 y Fp(")3109 2479 y Fo(is)456 2587 y(uniformly)c Fm(C)935 2554 y Fp(r)972 2587 y Fo(,)h Fm(r)h Fn(\025)d Fs(2)g Fo(in)g(al)5 b(l)26 b(its)f(variables,)j(including)d Fm(")p Fo(,)i(in)e(a)g(neighb)-5 b(orho)g(o)g(d)456 2696 y(of)571 2673 y Fs(~)562 2696 y(\003)33 b Fo(and)k Fs(~)-48 b Fm(\015)5 b Fo(.)555 2804 y(Then,)40 b(ther)-5 b(e)39 b(exists)f Fm(")1355 2771 y Fl(\003)1430 2804 y Fm(>)c Fs(0)39 b Fo(such)f(that)h(for)g Fn(j)p Fm(")p Fn(j)d Fm(<)e(")2448 2771 y Fl(\003)2488 2804 y Fo(,)39 b(ther)-5 b(e)39 b(is)f(a)g Fm(C)3046 2771 y Fp(r)r Fl(\000)p Fq(1)456 2912 y Fo(function)1146 3015 y Fs(~)1122 3038 y Fn(F)c Fs(:)1281 3015 y(~)1272 3038 y(\003)20 b Fn(\002)g Fs(\()p Fn(\000)p Fm(")1594 3000 y Fl(\003)1634 3038 y Fm(;)15 b(")1716 3000 y Fl(\003)1756 3038 y Fs(\))26 b Fn(\000)-16 b(!)26 b Fs(\()p Fk(R)20 b Fn(\002)g Fk(T)p Fs(\))2297 3000 y Fq(2)2356 3038 y Fn(\002)g Fk(T)456 3180 y Fo(such)32 b(that)456 3342 y Fs(\(30\))1474 3319 y(~)1465 3342 y(\003)1528 3356 y Fp(")1590 3342 y Fs(:=)1736 3319 y(~)1712 3342 y Fn(F)9 b Fs(\()1830 3319 y(~)1821 3342 y(\003)21 b Fn(\002)f(f)p Fm(")p Fn(g)p Fs(\))456 3504 y Fo(is)32 b(lo)-5 b(c)g(al)5 b(ly)35 b(invariant)e(for)h(the)f(\015ow)43 b Fs(\(7\))34 b Fo(gener)-5 b(ate)g(d)34 b(by)e(the)i(ve)-5 b(ctor)33 b(\014eld)g Fn(H)3084 3518 y Fp(")3120 3504 y Fo(.)555 3618 y(In)g(p)-5 b(articular,)1126 3595 y Fs(~)1118 3618 y(\003)1181 3632 y Fp(")1250 3618 y Fo(is)33 b Fm(")p Fo(-close)g(to)1764 3595 y Fs(~)1755 3618 y(\003)1818 3632 y Fq(0)1883 3618 y Fs(=)1987 3595 y(~)1979 3618 y(\003)f Fo(in)h(the)g Fm(C)2409 3585 y Fp(r)r Fl(\000)p Fq(2)2569 3618 y Fo(sense.)555 3728 y(Mor)-5 b(e)g(over,)993 3705 y Fs(~)984 3728 y(\003)1047 3742 y Fp(")1118 3728 y Fo(and)35 b(the)f(ve)-5 b(ctor)34 b(\014eld)h Fn(H)1991 3742 y Fp(")2061 3728 y Fo(c)-5 b(an)34 b(b)-5 b(e)34 b(extende)-5 b(d)35 b(so)f(that)h(it)f(is)456 3836 y(a)k(normal)5 b(ly)39 b(hyp)-5 b(erb)g(olic)40 b(invariant)f(manifold)g(for)f(the)g (\015ow)49 b Fs(\(7\))38 b Fo(gener)-5 b(ate)g(d)456 3944 y(by)33 b Fn(H)652 3958 y Fp(")688 3944 y Fo(.)43 b(In)34 b(p)-5 b(articular,)35 b(it)e(is)g(p)-5 b(ossible)35 b(to)e(de\014ne)h(lo)-5 b(c)g(al)35 b(stable)e(and)h(unstable)456 4053 y(manifolds.)68 b(That)43 b(is,)f(we)g(c)-5 b(an)41 b(\014nd)h(a)f Fm(C)1964 4020 y Fp(r)r Fl(\000)p Fq(1)2133 4053 y Fo(function)2523 4031 y Fs(~)2499 4053 y Fn(F)2573 4020 y Fq(s)2646 4053 y Fo(such)g(that)h(the)456 4163 y(\(lo)-5 b(c)g(al\))34 b(stable)g(manifold)g(of)1480 4140 y Fs(~)1471 4163 y(\003)1534 4177 y Fp(")1603 4163 y Fo(takes)f(the)g(form)456 4352 y Fs(\(31\))515 b Fm(W)1230 4308 y Fq(s)p Fp(;)p Fq(lo)r(c)1224 4380 y(~)1217 4397 y(\003)1266 4405 y Ff(")1395 4352 y Fs(=)1515 4329 y(~)1491 4352 y Fn(F)1565 4315 y Fq(s)1612 4251 y Fh(\020)1675 4329 y Fs(~)1667 4352 y(\003)20 b Fn(\002)g Fs(\(0)p Fm(;)15 b Fs(+)p Fn(1)p Fs(\))21 b Fn(\002)f(f)p Fm(")p Fn(g)2402 4251 y Fh(\021)2473 4352 y Fm(:)555 4564 y Fo(If)49 b Fs(~)-51 b Fm(x)44 b Fs(=)896 4541 y(~)871 4564 y Fn(F)10 b Fs(\()p Fm(I)d(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))46 b Fn(2)1486 4541 y Fs(~)1478 4564 y(\003)1541 4578 y Fp(")1577 4564 y Fo(,)g(then)d Fm(W)1962 4520 y Fq(s)p Fp(;)p Fq(lo)r(c)1953 4592 y(~)-39 b Fp(x)2145 4564 y Fs(=)2284 4541 y(~)2260 4564 y Fn(F)2334 4531 y Fq(s)2366 4564 y Fs(\()p Fn(f)p Fm(I)7 b Fn(g)29 b(\002)f(f)p Fm(')p Fn(g)h(\002)e(f)p Fm(s)p Fn(g)h(\002)456 4672 y Fs(\(0)p Fm(;)15 b Fn(1)p Fs(\))21 b Fn(\002)f(f)p Fm(")p Fn(g)p Fs(\))p Fo(.)555 4793 y(In)33 b(p)-5 b(articular,)35 b Fm(W)1217 4749 y Fq(s)p Fp(;)p Fq(lo)r(c)1211 4821 y(~)1204 4838 y(\003)1253 4846 y Ff(")1388 4793 y Fo(is)e Fm(")p Fo(-close)g(to)g Fm(W)1992 4749 y Fq(s)p Fp(;)p Fq(lo)r(c)1986 4821 y(~)1979 4838 y(\003)2164 4793 y Fo(in)g(the)g Fm(C)2499 4760 y Fp(r)r Fl(\000)p Fq(2)2659 4793 y Fo(sense.)555 4945 y(A)n(nalo)-5 b(gous)34 b(r)-5 b(esults)33 b(hold)i(for)e(the)g(\(lo)-5 b(c)g(al\))35 b(unstable)e(manifold)h Fm(W)2873 4900 y Fq(u)p Fp(;)p Fq(lo)r(c)2867 4972 y(~)2860 4989 y(\003)2909 4997 y Ff(")3024 4945 y Fo(.)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(27)555 450 y Fs(The)36 b(pro)s(of)g(of)g(Theorem)g(20)h(is)f(quite)h(standard)f(in)g(the)g (theory)h(of)f(nor-)456 558 y(mally)27 b(h)m(yp)s(erb)s(olic)g(in)m(v) -5 b(arian)m(t)28 b(manifolds.)39 b(It)27 b(is)h(a)f(straigh)m(tforw)m (ard)h(applica-)456 666 y(tion)i(of)g(the)g(theorems)h(in)e([F)-8 b(en74)r(,)30 b(F)-8 b(en77)r(,)30 b(HPS77])h(\(see)g(also)f(App)s (endix)f(A)456 774 y(of)h([DLS00)q(]\).)41 b(A)31 b(mo)s(dern)e(pro)s (of)h(is)g(in)g([Lla00)r(].)555 882 y(A)e(useful)f(observ)-5 b(ation)29 b(is)e(that)i(there)f(is)g(an)f(easy)i(w)m(a)m(y)f(to)h (obtain)f(smo)s(oth)456 990 y(dep)s(endence)i(on)i(parameters)f(from)g (the)h(standard)f(results)g(on)h(p)s(ersistence.)456 1098 y(It)h(su\016ces)h(to)g(consider)g(the)g(system)f(obtained)h(b)m (y)g(taking)g(the)g(pro)s(duct)f(of)456 1206 y(the)i(original)i(system) f(and)f(the)g(iden)m(tit)m(y)i(in)f(the)f(direction)i(of)e(parameters.) 456 1314 y(The)22 b(fact)i(that)f(the)g(in)m(v)-5 b(arian)m(t)25 b(manifolds)d(for)h(the)g(extended)g(system)g(are)g(reg-)456 1421 y(ular,)31 b(giv)m(es)h(the)f(fact)h(that)f(the)h(manifolds)e(of)h (the)g(original)i(system)e(dep)s(end)456 1529 y(regularly)f(on)h (parameters.)555 1639 y(Note)36 b(that)f(the)g(co)s(ordinates)g(along)g (the)g(unp)s(erturb)s(ed)c(manifold)2954 1616 y(~)2945 1639 y(\003)k(un-)456 1747 y(der)29 b(the)i(unp)s(erturb)s(ed)26 b(ev)m(olution)32 b(just)e(rotate)i(or)e(remain)g(in)m(v)-5 b(arian)m(t.)42 b(Since)456 1855 y Fn(jj)p Fm(D)s Fs(\010)650 1869 y Fp(t;)p Fq(0)734 1855 y Fn(j)766 1868 y Fq(~)759 1885 y(\003)812 1855 y Fn(jj)26 b(\024)f Fm(C)d Fn(j)p Fm(t)p Fn(j)30 b Fs(\(see)i(\(27\))q(\),)f(w)m(e)g(ha)m(v)m(e)h(that)f (for)f(ev)m(ery)h Fm(\016)e(>)c Fs(0,)1453 2029 y Fn(jj)p Fs(\010)1569 2043 y Fp(t;)p Fq(0)1654 2029 y Fn(j)1686 2042 y Fq(~)1679 2059 y(\003)1732 2029 y Fn(jj)h(\024)f Fm(C)1969 2044 y Fp(\016)2006 2029 y Fm(e)2048 1992 y Fp(\016)r Fl(j)q Fp(t)p Fl(j)2151 2029 y Fm(:)456 2186 y Fs(This)30 b(sho)m(ws)h(that)g(the)h(tangen)m(tial)h(exp)s(onen)m(ts) e(along)h(the)f(manifold)g(can)g(b)s(e)456 2294 y(tak)m(en)g(as)g (small)g(as)f(w)m(e)h(w)m(an)m(t.)555 2402 y(On)g(the)i(other)f(hand,)f (b)s(ecause)h(of)g(assumption)g Fw(H2)p Fs(,)g(the)h(p)s(oin)m(t)f(\()p Fm(p;)15 b(q)s Fs(\))28 b(=)456 2510 y(\(0)p Fm(;)15 b Fs(0\))26 b(is)e(a)g(h)m(yp)s(erb)s(olic)f(p)s(oin)m(t)h(for)g(the)h (p)s(endulum.)36 b(Since)24 b(the)g(system)g(\(7\))h(for)456 2624 y Fm(")g Fs(=)g(0)k(is)f(a)g(direct)h(pro)s(duct)d(of)j (rotators|along)2205 2602 y(~)2196 2624 y(\003|and)e(a)i(p)s(endulum,)d (the)456 2732 y(h)m(yp)s(erb)s(olic)40 b(directions)i(of)g(the)g(p)s (endulum)c(b)s(ecome)k(the)g(stable/unstable)456 2847 y(bundles)28 b(of)j(the)g(manifold)1435 2824 y(~)1426 2847 y(\003.)555 2955 y(The)d(fact)g(that)h(the)f(tangen)m(tial)i(exp)s (onen)m(ts)e(are)g(arbitrarily)g(small)h(allo)m(ws)456 3063 y(us)f(to)j(conclude)e(that)i(the)e(manifold)h(is)f(as)h(regular)g (as)f(the)h(\015o)m(w)g(when)e(mea-)456 3171 y(sured)20 b(on)i(the)g Fm(C)1028 3138 y Fp(r)r Fl(\000)p Fq(1)1177 3171 y Fs(classes,)j Fm(r)j Fn(2)d Fk(N)p Fs(.)37 b(\(If)22 b(there)g(w)m(as)g(a)g(non-trivial)h(expansion)456 3279 y(exp)s(onen)m(t,)31 b(then)f(the)h(regularit)m(y)h(claimed)f(for)g (the)g(manifold)f(w)m(ould)h(b)s(e)f(the)456 3387 y(in\014m)m(um)d(of)h Fm(r)19 b Fn(\000)c Fs(1)29 b(and)f(a)g(limiting)i(regularit)m(y)f (determined)f(b)m(y)g(the)h(rates)g(of)456 3495 y(expansion)22 b(along)i(the)f(manifold)f(and)g(along)i(the)f(stable/unstable)h (bundles.\))456 3663 y Fw(Remark)36 b(21.)42 b Fs(In)30 b(the)h(general)h(theory)f(of)g(the)g(p)s(ersistence)g(of)g(o)m(v)m (er\015o)m(wing)456 3771 y(lo)s(cally)25 b(in)m(v)-5 b(arian)m(t)25 b(manifolds,)g(the)f(manifold)g(obtained)g(need)g(not)g (b)s(e)f(unique)456 3879 y(since,)j(in)d(principle,)j(it)e(could)g(dep) s(end)f(on)h(some)g(of)h(the)f(c)m(hoices)i(made)e(in)g(the)456 3988 y(pro)s(of.)48 b(Nev)m(ertheless,)36 b(when)c(the)i(manifold)2078 3965 y(~)2069 3988 y(\003)2132 4002 y Fp(")2202 3988 y Fs(is)f(in)m(v)-5 b(arian)m(t)34 b(and)f(not)g(just)456 4096 y(lo)s(cally)i(in)m(v)-5 b(arian)m(t,)36 b(it)e(is)g(unique.)50 b(W)-8 b(e)35 b(will)f(sho)m(w)g(later)h(that)f(the)g(manifold)464 4183 y(~)456 4206 y(\003)519 4220 y Fp(")593 4206 y Fs(is)k(in)m(v)-5 b(arian)m(t)38 b(since)g(the)g(b)s(oundaries)e(are)i(KAM)g(tori.)63 b(Hence,)40 b(for)d(our)456 4314 y(case,)32 b(the)f(manifolds)f(will)i (turn)d(out)i(to)h(b)s(e)e(unique.)41 b(Nev)m(ertheless,)33 b(w)m(e)f(will)456 4422 y(not)d(tak)m(e)i(adv)-5 b(an)m(tage)31 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y(~)g Fp(x)1144 2276 y Fs(where)841 2490 y Fm(W)940 2452 y Fq(s)931 2512 y(~)g Fp(x)996 2490 y Fs(=)25 b Fn(f)6 b Fs(~)-51 b Fm(y)59 b Fs(:)d(dist\(\010)1570 2504 y Fp(t;")1652 2490 y Fs(\()6 b(~)-51 b Fm(x)p Fs(\))p Fm(;)15 b Fs(\010)1880 2504 y Fp(t;")1962 2490 y Fs(\()6 b(~)-51 b Fm(y)t Fs(\)\))26 b Fn(\024)e Fm(C)7 b(e)2351 2452 y Fl(\000)f Fq(~)-41 b Fp(\026)q(t)2478 2490 y Fm(;)15 b(t)26 b(>)f Fs(0)p Fn(g)p Fm(;)456 2680 y Fs(where)37 b(~)-52 b Fm(\026)26 b Fs(=)g Fm(\026)21 b Fs(+)1064 2688 y(O)1135 2680 y(\()p Fm(")p Fs(\).)43 b(Moreo)m(v)m(er)33 b Fm(W)1814 2647 y Fq(s)1805 2706 y(~)-39 b Fp(x)1867 2680 y Fn(\\)20 b Fm(W)2047 2647 y Fq(s)2039 2706 y(~)-40 b Fp(y)2105 2680 y Fs(=)26 b Fn(;)31 b Fs(when)36 b(~)-51 b Fm(x)26 b Fn(6)p Fs(=)32 b(~)-51 b Fm(y)s Fs(,)37 b(~)-51 b Fm(x;)22 b Fs(~)-52 b Fm(y)30 b Fn(2)3058 2657 y Fs(~)3049 2680 y(\003)3112 2694 y Fp(")3149 2680 y Fs(.)456 2796 y(The)j(stable)h(manifolds)f(of)h(the)g(p)s(oin)m(ts)f(are)h(as)g(smo)s 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Fl(\000)p Fq(1)2484 3350 y Fs(,)h Fm(r)25 b Fn(\000)d Fs(1)32 b Fn(2)f Fk(N)p Fs(,)j(then)456 3458 y(the)c(resulting)h Fm(W)1086 3425 y Fq(s)1080 3484 y(~)1073 3501 y(\003)1122 3509 y Ff(")1189 3458 y Fs(will)g(b)s(e)f Fm(C)1557 3425 y Fp(r)r Fl(\000)p Fq(1)1715 3458 y Fs(for)g(small)h Fm(")p Fs(.)555 3581 y(Analogous)h(results)e(hold)g(for)g(the)h(unstable)f(manifold.)555 3689 y(Note)47 b(that)f(the)g(de\014nition)g(of)g(the)f(\(un\)stable)i (manifolds)e(for)g(lo)s(cally)456 3796 y(in)m(v)-5 b(arian)m(t)42 b(manifolds)e(can)i(only)f(b)s(e)f(made)h(in)f(an)h(extended)g(system)g (con-)456 3904 y(structed)30 b(in)g(the)g(pro)s(of)g(for)g(whic)m(h)f (the)i(dynamics)f(is)g(de\014ned)f(for)h(all)h(times.)456 4012 y(In)j(general,)j(these)f(\(un\)stable)f(manifolds)g(dep)s(end)e (on)i(the)g(extended)g(\015o)m(w)456 4120 y(used.)555 4228 y(One)26 b(consequence)h(of)g(that)g(is)f(that)1834 4205 y(~)1825 4228 y(\003)1888 4242 y Fp(")1925 4228 y Fs(,)h Fm(W)2076 4195 y Fq(s)2070 4254 y(~)2063 4271 y(\003)2112 4279 y Ff(")2176 4228 y Fs(ma)m(y)g(fail)g(to)g(b)s(e)e Fm(C)2817 4195 y Fl(1)2918 4228 y Fs(ev)m(en)i(if)456 4351 y(the)32 b(\015o)m(w)f(is)h(analytic.)47 b(Nev)m(ertheless,)34 b(when)d(the)h(manifolds)f(are)h(in)m(v)-5 b(arian)m(t,)456 4459 y(the)21 b(stable)g(and)f(unstable)h(manifolds)f(are)h(uniquely)f (de\014ned.)36 b(In)20 b(our)h(case,)j(as)456 4567 y(stated)g(in)g (Remark)g(21,)i(the)f(KAM)f(tori)g(will)g(pro)s(duce)f(in)m(v)-5 b(arian)m(t)25 b(b)s(oundaries)456 4676 y(for)602 4653 y(~)593 4676 y(\003)656 4690 y Fp(")693 4676 y Fs(,)k(hence)g(stable)g (and)f(unstable)h(manifolds)f(will)h(b)s(e)f(uniquely)g(de\014ned.)456 4784 y(The)33 b(pro)s(ofs)g(w)m(e)h(presen)m(t,)h(ho)m(w)m(ev)m(er,)h (do)e(not)g(tak)m(e)h(adv)-5 b(an)m(tage)36 b(of)e(this)g(fact.)3103 4892 y Fj(\003)p eop end %%Page: 29 29 TeXDict begin 29 28 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(29)456 450 y Fs(7.1.)46 b 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b(\014eld)e Fn(H)1003 1341 y Fp(")1069 1327 y Fs(at)h(a)g(p)s(oin)m(t)g(in)f(the)g(range)h(is)f(tangen)m(t)i (to)f(the)g(range)g(of)3070 1304 y(~)3045 1327 y Fn(F)9 b Fs(.)555 1437 y(Using)31 b(equation)g(\(33\))h(one)f(can)f(\014nd)f (an)h(expansion)h(of)f(the)h(function)3078 1414 y(~)3054 1437 y Fn(F)9 b Fs(.)456 1610 y Fw(Prop)s(osition)31 b(24.)39 b Fo(The)29 b(family)h(of)f(mappings)2171 1587 y Fs(~)2147 1610 y Fn(F)38 b Fo(sp)-5 b(e)g(ci\014e)g(d)30 b(in)f(The)-5 b(or)g(em)31 b(20)456 1718 y(with)i(the)g(normalization) 43 b Fs(\(32\))34 b Fo(admits)g(an)f(exp)-5 b(ansion)900 1856 y Fs(~)875 1879 y Fn(F)35 b Fs(=)1095 1856 y(~)1071 1879 y Fn(F)1136 1893 y Fq(0)1196 1879 y Fs(+)20 b Fm(")1354 1856 y Fs(~)1329 1879 y Fn(F)1394 1893 y Fq(1)1454 1879 y Fs(+)g Fn(\001)15 b(\001)g(\001)21 b Fs(+)f Fm(")1804 1842 y Fp(m)1896 1856 y Fs(~)1871 1879 y Fn(F)1936 1893 y Fp(m)2023 1879 y Fs(+)2114 1887 y(O)2185 1899 y Fp(C)2240 1880 y Ff(r)r Fg(\000)p Ff(m)p Fg(\000)p Fi(2)2459 1879 y Fs(\()p Fm(")2536 1842 y Fp(m)p Fq(+1)2694 1879 y Fs(\))p Fm(;)456 2040 y Fo(wher)-5 b(e)737 2017 y Fs(~)712 2040 y Fn(F)777 2054 y Fq(0)817 2040 y Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))27 b(=)e(\(0)p Fm(;)15 b Fs(0)p Fm(;)g(I)7 b(;)15 b(';)g(s)p Fs(\))p Fo(.)555 2148 y(In)39 b(the)g(c)-5 b(ase)40 b(that)g(the)f(\015ow)h(satis\014es)g(assumptions)48 b Fw(H1)p Fo({)p Fw(H3)p Fo(,)41 b(then)e(the)456 2258 y(functions)881 2235 y Fs(~)857 2258 y Fn(F)922 2272 y Fq(1)961 2258 y Fm(;)15 b(:)g(:)g(:)i(;)1188 2235 y Fs(~)1163 2258 y Fn(F)1228 2272 y Fp(m)1334 2258 y Fo(ar)-5 b(e)39 b(trigonometric)h(p)-5 b(olynomials)42 b(in)d Fm(';)15 b(s)p Fo(,)40 b(and)3105 2235 y Fs(~)3080 2258 y Fn(F)3145 2272 y Fp(i)456 2368 y Fo(ar)-5 b(e)46 b(of)g(class)h Fn(C)1028 2335 y Fp(r)r Fl(\000)p Fq(1)p Fl(\000)p Fp(i)1235 2368 y Fo(.)81 b(Mor)-5 b(e)g(over,)50 b(in)45 b(such)h(a)g(c)-5 b(ase,)50 b Fn(N)13 b Fs(\()2611 2345 y(~)2587 2368 y Fn(F)2652 2382 y Fq(1)2692 2368 y Fs(\))49 b Fn(\032)g(N)13 b Fs(\()p Fm(h)3071 2382 y Fq(1)3111 2368 y Fs(\))p Fo(,)456 2482 y Fn(N)g Fs(\()604 2459 y(~)579 2482 y Fn(F)644 2496 y Fq(2)684 2482 y Fs(\))25 b Fn(\032)g(N)13 b Fs(\()p Fm(h)1015 2496 y Fq(1)1055 2482 y Fs(\))21 b(+)f Fn(N)13 b Fs(\()p Fm(h)1377 2496 y Fq(1)1417 2482 y Fs(\))21 b Fn([)e(N)13 b Fs(\()p Fm(h)1728 2496 y Fq(2)1769 2482 y Fs(\))p Fo(.)456 2702 y(Pr)-5 b(o)g(of)20 b(.)40 b Fs(Once)28 b(w)m(e)g(imp)s(ose)g(normalization)h(\(32\))h(to)2315 2679 y(~)2291 2702 y Fn(F)9 b Fs(,)29 b(w)m(e)f(can)g(compute)3105 2679 y(~)3080 2702 y Fn(F)3145 2716 y Fp(i)456 2810 y Fs(b)m(y)g(matc)m(hing)i(p)s(o)m(w)m(ers)f(of)g Fm(")g Fs(in)g(the)g(equation)h(\(33\))q(.)40 b(W)-8 b(e)31 b(kno)m(w)d(b)m(y)h(Theorem)456 2918 y(20)i(that)g(the)f(expansion)h (exists.)555 3026 y(Equating)37 b(terms)f(in)g(the)h(expansion)f(on)g Fm(")h Fs(of)f(the)h(equation)g(for)f(in)m(v)-5 b(ari-)456 3134 y(ance)31 b(\(33\))h(w)m(e)e(obtain,)i(up)d(to)i(order)f(t)m(w)m (o:)1578 3408 y Fn(H)1655 3422 y Fq(0)1714 3408 y 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3769 y Fq(0)1510 3755 y Fn(\016)1600 3733 y Fs(~)1576 3755 y Fn(F)1641 3769 y Fq(0)1680 3755 y Fs(\))1740 3733 y(~)1715 3755 y Fn(F)1789 3717 y Fl(\012)p Fq(2)1780 3782 y(1)1905 3755 y Fs(+)g(\()p Fm(D)s Fn(H)2186 3769 y Fq(1)2245 3755 y Fn(\016)2335 3733 y Fs(~)2311 3755 y Fn(F)2376 3769 y Fq(0)2416 3755 y Fs(\))2475 3733 y(~)2451 3755 y Fn(F)2516 3769 y Fq(1)2576 3755 y Fs(+)g Fn(H)2744 3769 y Fq(2)2803 3755 y Fn(\016)2893 3733 y Fs(~)2869 3755 y Fn(F)2934 3769 y Fq(0)1910 3937 y Fs(=)25 b Fm(D)2108 3914 y Fs(~)2084 3937 y Fn(F)2149 3951 y Fq(0)2188 3937 y Fn(R)2265 3951 y Fq(2)2325 3937 y Fs(+)20 b Fm(D)2518 3914 y Fs(~)2494 3937 y Fn(F)2559 3951 y Fq(1)2598 3937 y Fn(R)2675 3951 y Fq(1)2735 3937 y Fs(+)g Fn(D)2923 3914 y Fs(~)2899 3937 y Fn(F)2964 3951 y Fq(2)3003 3937 y Fn(R)3080 3951 y Fq(0)3120 3937 y Fm(:)456 3280 y Fs(\(34\))555 4091 y(In)26 b(general,)i(the)f(equation)g(obtained)f (after)h(matc)m(hing)g(the)g(co)s(e\016cien)m(ts)h(of)456 4199 y Fm(")498 4166 y Fp(n)575 4199 y 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Fn(F)3096 4535 y Fp(n)p Fl(\000)p Fq(1)3234 4521 y Fs(,)456 4629 y(and)29 b Fn(R)709 4643 y Fq(0)749 4629 y Fm(;)15 b(:)g(:)g(:)h(;)f Fn(R)1027 4643 y Fp(n)p Fl(\000)p Fq(1)1165 4629 y Fs(.)555 4742 y(Clearly)-8 b(,)25 b(the)e(\014rst)f(equation)h(in)f(\(34\))i(has)f (as)f(solution)2475 4719 y(~)2450 4742 y Fn(F)2515 4756 y Fq(0)2580 4742 y Fs(=)j(\(0)p Fm(;)15 b Fs(0)p Fm(;)g(I)7 b(;)15 b(';)g(s)p Fs(\),)456 4850 y(and)33 b Fn(R)713 4864 y Fq(0)784 4850 y Fs(=)e(\(0)p Fm(;)15 b(I)7 b(;)15 b Fs(1\).)54 b(Therefore,)35 b(if)f(one)g(can)g(dev)m(elop)h(a)f(metho) s(d)g(to)h(solv)m(e)456 4964 y(equations)g(for)1036 4941 y(~)1011 4964 y Fn(F)1076 4978 y Fp(n)1124 4964 y Fs(,)h Fn(R)1262 4978 y Fp(n)1344 4964 y Fs(of)f(the)h(form)e(of)i(the)f (linear)h(Hamiltonian)g(System)p eop end %%Page: 30 30 TeXDict begin 30 29 bop 456 251 a Fq(30)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 450 y Fs(\(35\))36 b(equal)e(to)i(a)e(preassigned)g(righ)m(t)h(hand)f(side,)h(w)m(e)g(can) g(k)m(eep)g(on)f(solving)456 558 y(the)c(hierarc)m(h)m(y)h(of)g (equations)g(\(35\))h(to)f(an)m(y)f(order.)555 666 y(In)44 b(our)f(case,)49 b(since)c(the)g(unp)s(erturb)s(ed)40 b(motion)45 b(due)f(to)h Fn(R)2750 680 y Fq(0)2833 666 y Fs(is)g(quasi-)456 774 y(p)s(erio)s(dic,)26 b(the)h(equations)g(of)g (the)f(form)g(\(35\))i(can)f(b)s(e)e(solv)m(ed)j(quite)e(explicitly)456 882 y(using)37 b(F)-8 b(ourier)38 b(co)s(e\016cien)m(ts.)63 b(\(There)38 b(are)g(more)f(general)i(theories)f([Lla00)r(])456 990 y(that)33 b(allo)m(w)h(to)f(solv)m(e)h(equations)f(of)g(the)f(form) g(\(35\))i(ev)m(en)g(if)e(the)h(motion)g(on)456 1098 y(the)d(base)h(is)f(not)h(quasi-p)s(erio)s(dic.\))555 1206 y(The)k(theory)h(of)g(the)g(equations)h(that)f(w)m(e)g(need)f(is)h (summarized)f(in)h(next)456 1314 y(Lemma)44 b(25.)81 b(Clearly)-8 b(,)49 b(applying)43 b(it)i(recursiv)m(ely)-8 b(,)48 b(w)m(e)c(obtain)h(a)f(pro)s(of)f(of)456 1421 y(Prop)s(osition)30 b(24.)2043 b Fj(\003)456 1626 y Fw(Lemma)41 b(25.)46 b Fo(L)-5 b(et)1199 1603 y Fs(~)1174 1626 y Fn(F)1239 1640 y Fq(0)1279 1626 y Fo(,)40 b Fn(H)1424 1640 y Fq(0)1463 1626 y Fo(,)f Fn(R)1607 1640 y Fq(0)1685 1626 y Fo(b)-5 b(e)38 b(as)h(in)f(Pr)-5 b(op)g(osition)41 b(24.)60 b(Given)38 b(a)h Fm(C)3138 1593 y Fp(s)456 1736 y Fo(function,)34 b Fm(s)28 b Fn(\025)g Fs(1)p Fo(,)35 b Fm(\021)d Fs(:)1261 1713 y(~)1252 1736 y(\003)c Fn(!)h Fk(R)1529 1703 y Fq(5)1568 1736 y Fo(,)34 b(we)h(c)-5 b(an)35 b(\014nd)f(unique)g Fm(C)2485 1703 y Fq(1)2558 1736 y Fo(functions)h Fm(\030)t(;)49 b(\032)29 b Fs(:)464 1822 y(~)456 1845 y(\003)c Fn(!)g Fk(R)726 1812 y Fq(5)798 1845 y Fo(such)33 b(that)456 1993 y Fs(\(36\))519 b(\()p Fm(D)s Fn(H)1325 2007 y Fq(0)1385 1993 y Fn(\016)1475 1970 y Fs(~)1451 1993 y Fn(F)1516 2007 y Fq(0)1555 1993 y Fs(\))p Fm(\030)25 b Fn(\000)20 b Fm(D)s(\030)t Fn(R)1945 2007 y Fq(0)2004 1993 y Fn(\000)g Fm(D)2197 1970 y Fs(~)2173 1993 y Fn(F)2238 2007 y Fq(0)2278 1993 y Fm(\032)25 b Fs(=)g Fm(\021)456 2140 y Fo(and)456 2288 y Fs(\(37\))632 b(\005)1316 2302 y Fp(I)1356 2288 y Fm(\030)29 b Fs(=)c(0)p Fm(;)48 b Fs(\005)1707 2302 y Fp(')1758 2288 y Fm(\030)29 b Fs(=)c(0)p Fm(;)48 b Fs(\005)2109 2302 y Fp(s)2146 2288 y Fm(\030)29 b Fs(=)c(0)p Fm(:)456 2436 y Fo(F)-7 b(urthermor)i(e,)33 b Fm(\030)t Fo(,)d Fm(\032)g Fo(ar)-5 b(e)32 b(of)e(class)h Fm(C)1731 2403 y Fp(s)1798 2436 y Fo(and)g(we)f(have,)h(for)g(a)g(c)-5 b(onstant)32 b Fm(C)37 b Fo(that)456 2544 y(dep)-5 b(ends)34 b(only)f(on)g Fn(H)1199 2558 y Fq(0)1239 2544 y Fm(;)15 b Fn(R)1356 2558 y Fq(0)1395 2544 y Fo(,)1479 2691 y Fn(jj)p Fm(\030)t Fn(jj)1623 2705 y Fp(C)1678 2686 y Ff(s)1741 2691 y Fn(\024)25 b Fm(C)7 b Fn(jj)p Fm(\021)s Fn(jj)2057 2705 y Fp(C)2112 2686 y Ff(s)1465 2826 y Fn(jj)p Fm(\032)p Fn(jj)1612 2840 y Fp(C)1667 2821 y Ff(s)1730 2826 y Fn(\024)25 b Fm(C)7 b Fn(jj)p Fm(\021)s Fn(jj)2046 2840 y Fp(C)2101 2821 y Ff(s)2140 2826 y Fm(:)456 2974 y Fo(Mor)-5 b(e)g(over,)32 b(if)f Fm(\021)j Fo(is)d(a)g(trigonometric)i(p)-5 b(olynomial,)34 b(so)e(ar)-5 b(e)32 b Fm(\030)j Fo(and)d Fm(\032)p Fo(,)f(and)h(we)456 3082 y(have)h Fn(N)13 b Fs(\()p Fm(\030)t Fs(\))26 b Fn(\032)f(N)13 b Fs(\()p Fm(\021)s Fs(\))p Fo(,)33 b Fn(N)13 b Fs(\()p Fm(\032)p Fs(\))26 b Fn(\032)f(N)13 b Fs(\()p Fm(\021)s Fs(\))p Fo(.)456 3291 y(Pr)-5 b(o)g(of)20 b(.)43 b Fs(Note)33 b(that)f(the)f(unp)s(erturb)s(ed)d(\015o)m(w)j(and) g(its)g(di\013eren)m(tial)i(at)2932 3268 y(~)2923 3291 y(\003)e(pre-)456 3399 y(serv)m(e)g(the)f Fm(I)7 b(;)15 b(';)g(s)31 b Fs(directions.)42 b(The)30 b(plane)g Fm(p;)15 b(q)33 b Fs(is)e(in)m(v)-5 b(arian)m(t.)555 3509 y(Moreo)m(v)m(er,)35 b Fm(D)1084 3486 y Fs(~)1060 3509 y Fn(F)1125 3523 y Fq(0)1197 3509 y Fs(in)d(the)g(co)s(ordinates)h(w)m(e)f(are)h(using)e (is)h(a)h(5)22 b Fn(\002)f Fs(3)32 b(matrix.)456 3617 y(It)e(consists)h(of)f(a)h(3)20 b Fn(\002)g Fs(3)31 b(iden)m(tit)m(y)h (along)f(the)f Fm(I)7 b(;)15 b(';)g(s)32 b Fs(directions)e(and)g(0)h (along)456 3725 y(the)f(other)h(directions)g(\()p Fm(p;)15 b(q)s Fs(\).)555 3832 y(The)29 b(t)m(w)m(o)i(observ)-5 b(ations)31 b(ab)s(o)m(v)m(e,)g(immediately)g(giv)m(e)g(us)e(that)h(w)m (e)g(can)g(sat-)456 3940 y(isfy)g(the)g(normalization)i(\(37\))g(in)e (a)h(unique)f(w)m(a)m(y)h(b)m(y)f(setting:)456 4088 y(\(38\))977 b Fm(\032)26 b Fs(=)f(\005)1830 4102 y Fp(I)5 b(;';s)1988 4088 y Fm(\021)456 4236 y Fs(from)30 b(whic)m(h)g(the)g(regularit)m(y)i (claims)f(ab)s(out)f Fm(\032)h Fs(follo)m(w.)555 4344 y(Using)37 b(again)h(the)f(in)m(v)-5 b(ariance)38 b(of)f(the)f(\()p Fm(p;)15 b(q)s Fs(\))38 b(plane)f(it)g(su\016ces)f(to)i(study)456 4451 y(the)30 b(equation)456 4601 y(\(39\))388 b(\()p Fm(D)s Fn(H)1194 4615 y Fq(0)1254 4601 y Fn(\016)1344 4578 y Fs(~)1320 4601 y Fn(F)1385 4615 y Fq(0)1424 4601 y Fs(\))p Fn(j)1484 4615 y Fp(p;q)1578 4601 y Fs(\005)1646 4615 y Fp(p;q)1740 4601 y Fm(\030)24 b Fn(\000)c Fm(D)s Fs(\005)2041 4615 y Fp(p;q)2134 4601 y Fm(\030)t Fn(R)2255 4615 y Fq(0)2319 4601 y Fs(=)25 b(\005)2483 4615 y Fp(p;q)2577 4601 y Fm(\021)555 4748 y Fs(This)35 b(equation)g(\(39\))i(can)e(b)s(e) g(further)f(reduced)g(b)m(y)h(noticing)h(that)g Fm(D)s Fn(H)3135 4762 y Fq(0)456 4856 y Fs(preserv)m(es)27 b(the)g(t)m(w)m(o)i (eigendirections)g(of)e(the)g(equilibrium)g(p)s(oin)m(t)g(of)g(the)h(p) s(en-)456 4964 y(dulum.)72 b(Hence,)45 b(if)d(w)m(e)g(denote)g(b)m(y)f (\005)1865 4931 y Fq(s)1897 4964 y Fs(,)k(\005)2035 4931 y Fq(u)2078 4964 y Fs(,)g(the)c(pro)5 b(jections)42 b(along)h(the)p eop end %%Page: 31 31 TeXDict begin 31 30 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(31)456 450 y Fs(stable)30 b(and)f(the)h(unstable)g(comp)s(onen)m(ts)f(and)h(b)m(y)f Fm(\026)h Fs(the)f(eigen)m(v)-5 b(alue,)32 b(w)m(e)f(ob-)456 558 y(tain)f(that)h(\(39\))h(is)f(equiv)-5 b(alen)m(t)32 b(to:)1292 713 y Fm(\026)p Fs(\005)1415 675 y Fq(u)1459 713 y Fm(\030)24 b Fn(\000)c Fm(D)s Fs(\005)1760 675 y Fq(u)1803 713 y Fm(\030)t Fn(R)1924 727 y Fq(0)2046 713 y Fs(=)25 b(\005)2210 680 y Fq(u)2254 713 y Fm(\021)-1843 b Fs(\(40\))1244 848 y Fn(\000)p Fm(\026)p Fs(\005)1438 810 y Fq(s)1470 848 y Fm(\030)24 b Fn(\000)c Fm(D)s Fs(\005)1771 810 y Fq(s)1803 848 y Fm(\030)t Fn(R)1924 862 y Fq(0)2052 848 y Fs(=)25 b(\005)2216 815 y Fq(s)2248 848 y Fm(\021)555 1002 y Fs(The)35 b(equations)h(\(40\))h(can)e(b)s(e)g(studied)g(easily) -8 b(.)57 b(F)-8 b(or)36 b(example,)h(they)f(can)456 1110 y(b)s(e)24 b(studied)g(noting)h(that)g(the)g(op)s(erator)g(on)g (the)g(linear)g(Hamiltonian)h(system)456 1218 y(of)37 b(\(40\))32 b(is)f(diagonal)g(on)f(F)-8 b(ourier)31 b(series.)41 b(Hence,)32 b(if)1055 1392 y(\005)1123 1354 y Fq(s)1155 1392 y Fm(\021)s Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))28 b(=)1731 1305 y Fh(X)1626 1507 y Fq(\()p Fp(k)r(;l)q Fq(\))p Fl(2N)10 b Fq(\()p Fp(\021)r Fq(\))1989 1392 y Fs(^)-51 b Fm(\021)2028 1407 y Fp(k)r(;l)2112 1392 y Fs(\()p Fm(I)7 b Fs(\))p Fm(e)2271 1354 y Fp(i)p Fq(\()p Fp(k)r(')p Fq(+)p Fp(l)q(s)p Fq(\))2549 1392 y Fm(;)456 1656 y Fs(the)30 b(solution)h(of)38 b(\(40\))32 b(is:)456 1829 y(\(41\))174 b(\005)858 1791 y Fq(s)891 1829 y Fm(\030)t Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))26 b(=)1462 1742 y Fh(X)1356 1944 y Fq(\()p Fp(k)r(;l)q Fq(\))p Fl(2N)10 b Fq(\()p Fp(\021)r Fq(\))1720 1829 y Fs(^)-51 b Fm(\021)1759 1844 y Fp(k)r(;l)1843 1829 y Fs(\()p Fm(I)7 b Fs(\)[)p Fn(\000)p Fm(\026)21 b Fs(+)f Fm(i)p Fs(\()p Fm(k)s(I)28 b Fs(+)20 b Fm(l)r Fs(\)])2587 1791 y Fl(\000)p Fq(1)2682 1829 y Fm(e)2724 1791 y Fp(i)p Fq(\()p Fp(k)r(')p Fq(+)p Fp(l)q(s)p Fq(\))456 2093 y Fs(and)29 b(analogously)j(\(with)f(+)p Fm(\026)f Fs(in)g(place)h(of)f Fn(\000)p Fm(\026)p Fs(\))h(for)f(\005) 2364 2060 y Fq(u)2407 2093 y Fm(\030)t Fs(.)555 2201 y(Note)36 b(that,)h(since)e Fn(\006)p Fm(\026)d Fn(6)p Fs(=)g(0,)k(there)f(are)g(no)f(small)i(denominators)e(in)h(the)456 2309 y(\014nite)c(sum)g(\(41\))r(.)45 b(Then,)32 b(one)g(can)g(b)s (ound)e(the)i(norm)f(of)h Fm(\030)k Fs(b)m(y)c(the)g(norm)f(of)456 2417 y Fm(\021)s Fs(.)41 b(This)29 b(\014nishes)g(the)i(pro)s(of)f(of)g (Lemma)h(25.)1069 b Fj(\003)456 2630 y Fw(Remark)31 b(26.)40 b Fs(Ev)m(en)27 b(if)g(for)g(us,)h(the)f(solution)h(41)g(is)f(useful)g (b)s(ecause)g(it)h(giv)m(es)456 2738 y(the)c(F)-8 b(ourier)25 b(co)s(e\016cien)m(ts)h(that)f(will)g(later)g(b)s(e)e(the)i(base)f(of)h (the)f(study)g(of)g(reso-)456 2846 y(nances,)k(w)m(e)f(note)h(that,)g (in)f(geometric)i(p)s(erturbation)d(theory)-8 b(,)29 b(it)f(is)f(common)456 2954 y(to)k(represen)m(t)f(the)h(solution)g(as)f (in)m(tegral)i(form)m(ulas.)555 3062 y(F)-8 b(or)31 b(example,)h(in)e (our)g(case,)h(w)m(e)g(ha)m(v)m(e:)967 3261 y(\005)1035 3223 y Fq(s)1067 3261 y Fm(\030)t Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))27 b(=)1533 3137 y Fh(Z)1624 3164 y Fl(1)1584 3343 y Fq(0)1714 3261 y Fm(\021)s Fs(\()p Fm(I)7 b(;)15 b(')22 b Fn(\000)e Fm(I)7 b(t;)15 b(s)20 b Fn(\000)g Fm(t)p Fs(\))p Fm(e)2440 3223 y Fl(\000)p Fp(\026t)2582 3261 y Fm(dt)555 3462 y Fs(Out)33 b(of)h(the)g(preceding)f(form)m(ula,) 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b(p)s(erio)s(dic)f(orbits)h(\(see)h(Prop)s(osition)f (66\).)47 b(They)32 b(are)g(arranged)g(in)g(suc)m(h)f(a)456 4748 y(w)m(a)m(y)25 b(that)f(the)g(gaps)h(among)f(them)g(are)g(a)h(p)s (o)m(w)m(er)f(of)g Fm(")g Fs(whic)m(h)g(can)g(b)s(e)g(made)g(as)456 4856 y(large)k(as)g(desired)f(b)m(y)g(assuming)g(enough)g(di\013eren)m (tiabilit)m(y)-8 b(.)42 b(F)-8 b(or)29 b(subsequen)m(t)456 4964 y(dev)m(elopmen)m(ts,)37 b(an)m(y)e(p)s(o)m(w)m(er)f(greater)i (than)f(1)g(will)f(b)s(e)g(enough.)54 b(Hence,)36 b(w)m(e)p eop end %%Page: 32 32 TeXDict begin 32 31 bop 456 251 a Fq(32)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 450 y Fs(ha)m(v)m(e)42 b(dev)m(elop)s(ed)f(all)h(the)g(section)g(trying)f(to)h(obtain)g(only)f (gaps)g(of)g(order)456 559 y Fm(")498 526 y Fq(3)p Fp(=)p Fq(2)608 559 y Fs(.)555 667 y(The)27 b(results)g(are)h(summarized)f(in) g(Figure)h(1,)g(whic)m(h)f(depicts)h(a)f(surface)h(of)456 776 y(section)j(of)g(the)f(manifold)1404 754 y(~)1396 776 y(\003)1459 790 y Fp(")1495 776 y Fs(.)555 884 y(The)22 b(metho)s(d)f(of)h(pro)s(of)f(will)i(b)s(e)e(a)h(sequence)h(of)f (di\013eren)m(t)g(steps.)38 b(Basically)-8 b(,)456 992 y(they)24 b(will)g(b)s(e)f(a)h(com)m(bination)h(of)f(a)m(v)m(eraging)i (metho)s(ds)d(and)g(KAM)h(theorems.)555 1100 y(In)34 b(Section)i(8.1)g(w)m(e)f(study)f(the)h(geometry)h(of)f(the)g(motion)h (restricted)f(to)456 1210 y(the)i(manifold)1011 1187 y(~)1002 1210 y(\003)1065 1224 y Fp(")1102 1210 y Fs(.)61 b(In)37 b(particular,)i(w)m(e)f(will)g(see)g(that)f(the)h(\015o)m(w)f (restricted)456 1320 y(to)574 1297 y(~)565 1320 y(\003)628 1334 y Fp(")694 1320 y Fs(is)28 b(Hamiltonian.)42 b(Moreo)m(v)m(er,)31 b(for)e(con)m(v)m(enience,)i(w)m(e)f(will)f(in)m(tro)s(duce)f(a)456 1428 y(system)i(of)g(co)s(ordinates)h(\()p Fm(J)o(;)15 b(';)g(s)p Fs(\),)32 b(in)e(whic)m(h)g(the)h(symplectic)g(form)f(has)g (the)456 1535 y(standard)f(expression.)555 1643 y(In)c(Section)h(8.2)h 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Fq(2)1763 2082 y Fs(+)g Fn(\001)15 b(\001)g(\001)21 b Fs(+)f Fm(")2114 2049 y Fp(m)p Fl(\000)p Fq(1)2271 2082 y Fm(h)2323 2096 y Fp(m)2411 2082 y Fs(+)2502 2090 y(O)2573 2082 y(\()p Fm(")2650 2049 y Fp(m)2717 2082 y Fs(\))31 b(in)g(\(6\))h(is)e(a)456 2190 y(trigonometric)d(p)s (olynomial)g(with)e(resp)s(ect)h(to)h(\()p Fm(';)15 b(s)p Fs(\),)28 b(so)e(are)h Fm(k)2666 2204 y Fq(0)2706 2190 y Fm(;)15 b(k)2793 2204 y Fq(1)2833 2190 y Fm(;)g(:)g(:)g(:)h(;)f(k) 3081 2204 y Fp(m)3149 2190 y Fs(,)456 2298 y(and)25 b(w)m(e)i(will)g (giv)m(e)h(rather)f(explicit)g(form)m(ulas)g(to)g(compute)g(the)g Fm(k)2733 2312 y Fp(i)2761 2298 y Fs('s)g(in)f(terms)456 2406 y(of)k(the)h Fm(h)768 2420 y Fp(i)796 2406 y Fs('s.)555 2514 y(In)g(Section)i(8.3)g(w)m(e)g(dev)m(elop)g(a)f(global)h(a)m(v)m (eraging)i(pro)s(cedure)30 b(that)j(casts)456 2622 y(the)28 b(reduced)f(Hamiltonian)i Fm(k)s Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))30 b(in)m(to)f(a)f(con)m(v)m(enien)m(t)i (global)f(normal)456 2729 y(form)671 2705 y(\026)669 2729 y Fm(k)s Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(;)g Fm(")p Fs(\))30 b(whic)m(h)f(is)f(giv)m(en)i(b)m(y)e(di\013eren)m(t)i (form)m(ulas)e(in)h(the)f(resonan)m(t)456 2837 y(regions)i(and)g(in)g (the)h(non-resonan)m(t)g(region.)555 2945 y(The)22 b(non-resonan)m(t)h (region)g(is)g(studied)f(in)g(Section)h(8.4.)40 b(Since)22 b(the)h(normal)456 3053 y(form)677 3029 y(\026)675 3053 y Fm(k)s Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(;)g Fm(")p Fs(\))36 b(is)f(v)m(ery)g(close)h(to)f(a)g(strongly)g(in)m (tegrable)h(Hamiltonian,)456 3161 y(a)f(quan)m(titativ)m(e)i(v)m (ersion)e(of)g(the)f(KAM)h(theorem,)h(whic)m(h)f(w)m(e)g(will)g(dev)m (elop,)456 3269 y(will)22 b(sho)m(w)g(that)g(the)g(non-resonan)m(t)h (region)f(con)m(tains)h(KAM)f(tori)h(whic)m(h)e(lea)m(v)m(e)456 3377 y(v)m(ery)30 b(small)h(gaps)g(b)s(et)m(w)m(een)g(them.)555 3485 y(The)h(resonan)m(t)h(region)g(is)g(analyzed)g(in)g(Section)g (8.5.)48 b(In)32 b(this)h(region)g(the)456 3593 y(normal)e(form)984 3569 y(\026)981 3593 y Fm(k)s Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(;)g Fm(")p Fs(\))34 b(is)e(v)m(ery)g(close)g(to)h(a)f (p)s(endulum.)42 b(Note)32 b(that)h(the)456 3701 y(p)s(endulum)e(has)i (rotational)j(and)d(librational)i(motions)f(co)m(v)m(ering)h(op)s(en)e (sets)456 3809 y(as)26 b(w)m(ell)h(as)f(separatrices.)40 b(The)26 b(rotations)h(in)f(the)g(p)s(endulum)d(ha)m(v)m(e)k(the)f (same)456 3917 y(top)s(ology)j(as)f(the)g(primary)e(tori)j(in)e(the)h (in)m(tegrable)h(system.)40 b(The)27 b(librations)456 4025 y(are)k(con)m(tractible)j(to)e(a)g(p)s(erio)s(dic)f(orbit.)44 b(Hence)32 b(the)f(librations)h(corresp)s(ond)456 4133 y(to)46 b(motions)f(with)g(top)s(ologies)i(that)f(are)f(not)h(presen)m (t)f(in)g(the)g(in)m(tegrable)456 4240 y(system.)555 4348 y(The)28 b(heart)g(of)g(the)g(matter)h(is)f(Section)g(8.5.3)i (whic)m(h)e(sho)m(ws)f(that)i(man)m(y)f(of)456 4456 y(these)j (rotational)i(and)e(librational)h(orbits)f(of)g(the)g(p)s(endulum)e(p)s (ersist)h(when)456 4564 y(w)m(e)25 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%27FD09FF7D52FD38FF7D52FD08FF7D2752FD0FFF7DFFFFFF27FD08FFA827 %FD3AFF527DFD08FFF87DFD0FFF52FFFFFF27FD08FF52A8FD3BFF27FD08FF %A852FD0FFF7DFFFFFF27FD08FF27FD3CFF52A8FD07FF7D52FD0FFF7DFFFF %FF27FD08FF52FD3CFF52A8FD07FFA852FD0BFF7D7DA8A87DFFFFFF27FD08 %FF27FD3CFF27FD08FF7D7DFD0BFF7DA852FF7DFFFFFF27FD08FFA827FD3A %FF7D52FD08FFA852FD0AFF7DFD04FF7DFFFFFF27FD09FF5227FD38FF7D27 %FD09FF7D7DFD0AFF7DA8FFFFFF7DFFFFFF27FD0AFF7D27FD36FF7D27FD0A %FFA852FD0FFF7DFFFFFF27FD0BFFA8277DFD32FF7D2752FD0BFF7D7DFD0F %FF52FFFFFF27FD0DFF52277DFD2EFF7D2752FD0DFFA852FD0FFF7DFFFFFF %27FD0FFF7DF8527DFD28FFA8522752FD0FFF7D52FD0FFF7DFFFFFF277DFD %10FFA87D275252A8FD20FFA87D275252FD12FFA852FD0FFF7DFFFFFF2727 %27A8FD11FFA8A85227275252A8A8FD15FF7D7D522727527DFD15FF7D52FD %0FFF7DFFFFFF27FF7D277DFD15FFA8A87D7D27525227277D527D527D527D %527D525227525227527DA8A8FD18FF522752FD0FFF7DFFFFFF27FFFFA827 %F87DFD1EFF7DA87DA87DA8A8FD1FFFA87DF8527D7DFD0FFF7DFFFFFF27FD %04FFA827277DFD1FFF52FD20FFA8522752FFFFA852FD0FFF7DFFFFFF27FD %06FF7D52F852A8FD1BFF7D27FFFFFF7DA87DA8FD16FFA85227F87DA8FFFF %FF7D7DFD0FFF52FFFFFF27FD09FF7D27F87DA8FD18FF7DF8FFFF7D527D7D %FD15FF7D27F852A8FD06FFA852FD0FFF7DFFFFFF27FD0BFFA85227F8527D %FD16FFA8FFFF52A8FD13FFA85227F8527DFD09FF7D52FD0FFF52FFFFFF27 %FD0FFF7D52F82752A8A8FD10FF52F8FD13FF7D7D2727277DA8FD0CFFA852 %FD0EFFA8F87DFFFF27FD12FFA87D5227F827527D7DA8FD0AFFA827FD0DFF %A87D7D2727F85252A8FD10FF7D7DFD0FFF27FFFFFF27FD17FFA8A8525227 %27F8522752527D52A87DA87DA87DA8A8A87D7D527D2752F827277D7DA8FD %15FFA852FD13FF27FD1EFFA8A87D7D527D525252272727527D5252527D7D %A8A8FD1BFF7D7DFD0FFFFD04A827FD28A8FD26FFA852FD13FF27FD4EFF7D %7DFD13FF27FD4EFFA852FD13FF27FD4EFF7D52FD13FF27FD4EFFA852FD13 %FF27FD4EFF7D7DFD13FF27FD4EFFA852FD13FF27FD55FFFF %%EndData %%EndComments %%BeginProlog %%BeginResource: procset Adobe_level2_AI5 1.2 0 %%Title: (Adobe Illustrator (R) Version 5.0 Level 2 Emulation) %%Version: 1.2 0 %%CreationDate: (04/10/93) () %%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved) userdict /Adobe_level2_AI5 26 dict dup begin put /packedarray where not { userdict begin /packedarray { array astore readonly } bind def /setpacking /pop load def /currentpacking false def end 0 } if pop userdict /defaultpacking currentpacking put true setpacking /initialize { Adobe_level2_AI5 begin } bind def /terminate { currentdict Adobe_level2_AI5 eq { end } if } bind def mark /setcustomcolor where not { /findcmykcustomcolor { (AI8_CMYK_CustomColor) 6 packedarray } bind def /findrgbcustomcolor { (AI8_RGB_CustomColor) 5 packedarray } bind def /setcustomcolor { exch aload pop dup (AI8_CMYK_CustomColor) eq { pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } { dup (AI8_RGB_CustomColor) eq { pop pop 3 { 1 exch sub 3 index mul 1 exch sub 3 1 roll } repeat 4 -1 roll pop setrgbcolor } { pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } ifelse } ifelse } def } if /setAIseparationgray { false setoverprint 0 setgray /setseparationgray where{ pop setseparationgray }{ /setcolorspace where{ pop [/Separation (All) /DeviceCMYK {dup dup dup}] setcolorspace 1 exch sub setcolor }{ setgray }ifelse }ifelse } def /gt38? mark {version cvr cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put userdict /level2? systemdict /languagelevel known dup { pop systemdict /languagelevel get 2 ge } if put /level2ScreenFreq { begin 60 HalftoneType 1 eq { pop Frequency } if HalftoneType 2 eq { pop GrayFrequency } if HalftoneType 5 eq { pop Default level2ScreenFreq } if end } bind def userdict /currentScreenFreq level2? {currenthalftone level2ScreenFreq} {currentscreen pop pop} ifelse put level2? not { /setcmykcolor where not { /setcmykcolor { exch .11 mul add exch .59 mul add exch .3 mul add 1 exch sub setgray } def } if /currentcmykcolor where not { /currentcmykcolor { 0 0 0 1 currentgray sub } def } if /setoverprint where not { /setoverprint /pop load def } if /selectfont where not { /selectfont { exch findfont exch dup type /arraytype eq { makefont } { scalefont } ifelse setfont } bind def } if /cshow where not { /cshow { [ 0 0 5 -1 roll aload pop ] cvx bind forall } bind def } if } if cleartomark /anyColor? { add add add 0 ne } bind def /testColor { gsave setcmykcolor currentcmykcolor grestore } bind def /testCMYKColorThrough { testColor anyColor? } bind def userdict /composite? 1 0 0 0 testCMYKColorThrough 0 1 0 0 testCMYKColorThrough 0 0 1 0 testCMYKColorThrough 0 0 0 1 testCMYKColorThrough and and and put composite? not { userdict begin gsave /cyan? 1 0 0 0 testCMYKColorThrough def /magenta? 0 1 0 0 testCMYKColorThrough def /yellow? 0 0 1 0 testCMYKColorThrough def /black? 0 0 0 1 testCMYKColorThrough def grestore /isCMYKSep? cyan? magenta? yellow? black? or or or def /customColor? isCMYKSep? not def end } if end defaultpacking setpacking %%EndResource %%BeginResource: procset Adobe_typography_AI5 1.0 1 %%Title: (Typography Operators) %%Version: 1.0 1 %%CreationDate:(6/10/1996) () %%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_typography_AI5 68 dict dup begin put /initialize { begin begin Adobe_typography_AI5 begin Adobe_typography_AI5 { dup xcheck { bind } if pop pop } forall end end end Adobe_typography_AI5 begin } def /terminate { currentdict Adobe_typography_AI5 eq { end } if } def /modifyEncoding { /_tempEncode exch ddef /_pntr 0 ddef { counttomark -1 roll dup type dup /marktype eq { pop pop exit } { /nametype eq { _tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll put } { /_pntr exch ddef } ifelse } ifelse } loop _tempEncode } def /havefont { systemdict /languagelevel known { /Font resourcestatus dup { exch pop exch pop } if } { systemdict /FontDirectory get 1 index known { pop true } { systemdict /fileposition known { dup length 6 add exch Ss 6 250 getinterval cvs pop Ss exch 0 exch getinterval status { pop pop pop pop true } { false } ifelse } { pop false } ifelse } ifelse } ifelse } def /TE { StandardEncoding 256 array copy modifyEncoding /_nativeEncoding exch def } def /subststring { exch 2 index exch search { exch pop exch dup () eq { pop exch concatstring } { 3 -1 roll exch concatstring concatstring } ifelse exch pop true } { pop pop false } ifelse } def /concatstring { 1 index length 1 index length 1 index add string dup 0 5 index putinterval dup 2 index 4 index putinterval 4 1 roll pop pop pop } def % /TZ { dup type /arraytype eq { /_wv exch def } { /_wv 0 def } ifelse /_useNativeEncoding exch def 2 index havefont { 3 index 255 string cvs dup (_Symbol_) eq { pop 2 index findfont } { 1 index 0 eq { dup length 1 sub 1 exch getinterval cvn findfont } { pop 2 index findfont } ifelse } ifelse } { dup 1 eq { 2 index 64 string cvs dup (-90pv-RKSJ-) (-83pv-RKSJ-) subststring { exch pop dup havefont { findfont false } { pop true } ifelse } { pop dup (-90ms-RKSJ-) (-Ext-RKSJ-) subststring { exch pop dup havefont { findfont false } { pop true } ifelse } { pop pop true } ifelse } ifelse { 1 index 1 eq { /Ryumin-Light-Ext-RKSJ-V havefont {/Ryumin-Light-Ext-RKSJ-V} {/Courier} ifelse } { /Ryumin-Light-83pv-RKSJ-H havefont {/Ryumin-Light-83pv-RKSJ-H} {/Courier} ifelse } ifelse findfont [1 0 0.5 1 0 0] makefont } if } { /Courier findfont } ifelse } ifelse _wv type /arraytype eq { _wv makeblendedfont } if dup length 10 add dict begin mark exch { 1 index /FID ne { def } if cleartomark mark } forall pop /FontScript exch def /FontDirection exch def /FontRequest exch def /FontName exch def counttomark 0 eq { 1 _useNativeEncoding eq { /Encoding _nativeEncoding def } if cleartomark } { /Encoding load 256 array copy modifyEncoding /Encoding exch def } ifelse FontName currentdict end definefont pop } def /tr { _ax _ay 3 2 roll } def /trj { _cx _cy _sp _ax _ay 6 5 roll } def /a0 { /Tx { dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss } ddef /Tj { dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss } ddef } def /a1 { W B } def /e0 { /Tx { tr _psf } ddef /Tj { trj _pjsf } ddef } def /e1 { W F } def /i0 { /Tx { tr sp } ddef /Tj { trj jsp } ddef } def /i1 { W N } def /o0 { /Tx { tr sw rmoveto } ddef /Tj { trj swj rmoveto } ddef } def /r0 { /Tx { tr _ctm _pss } ddef /Tj { trj _ctm _pjss } ddef } def /r1 { W S } def /To { pop _ctm currentmatrix pop } def /TO { iTe _ctm setmatrix newpath } def /Tp { pop _tm astore pop _ctm setmatrix _tDict begin /W { } def /h { } def } def /TP { end iTm 0 0 moveto } def /Tr { _render 3 le { currentpoint newpath moveto } if dup 8 eq { pop 0 } { dup 9 eq { pop 1 } if } ifelse dup /_render exch ddef _renderStart exch get load exec } def /iTm { _ctm setmatrix _tm concat _shift aload pop _lineorientation 1 eq { exch } if translate _scale aload pop _lineorientation 1 eq _yokoorientation 1 eq or { exch } if scale } def /Tm { _tm astore pop iTm 0 0 moveto } def /Td { _mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto } def /iTe { _render -1 eq { } { _renderEnd _render get dup null ne { load exec } { pop } ifelse } ifelse /_render -1 ddef } def /Ta { pop } def /Tf { 1 index type /nametype eq { dup 0.75 mul 1 index 0.25 mul neg } if /_fontDescent exch ddef /_fontAscent exch ddef /_fontSize exch ddef /_fontRotateAdjust _fontAscent _fontDescent add 2 div neg ddef /_fontHeight _fontSize ddef findfont _fontSize scalefont setfont } def /Tl { pop neg 0 exch _leading astore pop } def /Tt { pop } def /TW { 3 npop } def /Tw { /_cx exch ddef } def /TC { 3 npop } def /Tc { /_ax exch ddef } def /Ts { 0 exch _shift astore pop currentpoint iTm moveto } def /Ti { 3 npop } def /Tz { count 1 eq { 100 } if 100 div exch 100 div exch _scale astore pop iTm } def /TA { pop } def /Tq { pop } def /Tg { pop } def /TG { pop } def /Tv { /_lineorientation exch ddef } def /TV { /_charorientation exch ddef } def /Ty { dup /_yokoorientation exch ddef 1 sub neg Tv } def /TY { pop } def /T~ { Tx } def /Th { pop pop pop pop pop } def /TX { pop } def /Tk { _fontSize mul 1000 div _lineorientation 0 eq { neg 0 } { 0 exch } ifelse rmoveto pop } def /TK { 2 npop } def /T* { _leading aload pop _lineorientation 0 ne { exch } if Td } def /T*- { _leading aload pop _lineorientation 0 ne { exch } if exch neg exch neg Td } def /T- { _ax neg 0 rmoveto _lineorientation 1 eq _charorientation 0 eq and { 1 TV _hyphen Tx 0 TV } { _hyphen Tx } ifelse } def /T+ { } def /TR { _ctm currentmatrix pop _tm astore pop iTm 0 0 moveto } def /TS { currentfont 3 1 roll /_Symbol_ findfont _fontSize scalefont setfont 0 eq { Tx } { Tj } ifelse setfont } def /Xb { pop pop } def /Tb /Xb load def /Xe { pop pop pop pop } def /Te /Xe load def /XB { } def /TB /XB load def currentdict readonly pop end setpacking % /X^ { currentfont 5 1 roll dup havefont { findfont _fontSize scalefont setfont } { pop exch } ifelse 2 index 0 eq { Tx } { Tj } ifelse pop pop setfont } def /T^ /X^ load def %%EndResource %%BeginProcSet: Adobe_ColorImage_AI6 1.3 0 userdict /Adobe_ColorImage_AI6 known not { userdict /Adobe_ColorImage_AI6 53 dict put } if userdict /Adobe_ColorImage_AI6 get begin /initialize { Adobe_ColorImage_AI6 begin Adobe_ColorImage_AI6 { dup type /arraytype eq { dup xcheck { bind } if } if pop pop } forall } def /terminate { end } def currentdict /Adobe_ColorImage_AI6_Vars known not { /Adobe_ColorImage_AI6_Vars 41 dict def } if Adobe_ColorImage_AI6_Vars begin /plateindex -1 def /_newproc null def /_proc1 null def /_proc2 null def /sourcearray 4 array def /_ptispace null def /_ptiname null def /_pti0 0 def /_pti1 0 def /_ptiproc null def /_ptiscale 0 def /_pticomps 0 def /_ptibuf 0 string def /_gtigray 0 def /_cticmyk null def /_rtirgb null def /XIEnable true def /XIType 0 def /XIEncoding 0 def /XICompression 0 def /XIChannelCount 0 def /XIBitsPerPixel 0 def /XIImageHeight 0 def /XIImageWidth 0 def /XIImageMatrix null def /XIRowBytes 0 def /XIFile null def /XIBuffer1 null def /XIBuffer2 null def /XIBuffer3 null def /XIDataProc null def /XIColorSpace /DeviceGray def /XIColorValues 0 def /XIPlateList false def end /ci6colorimage /colorimage where {/colorimage get}{null} ifelse def /ci6image systemdict /image get def /ci6curtransfer systemdict /currenttransfer get def /ci6curoverprint /currentoverprint where {/currentoverprint get}{{_of}} ifelse def /ci6foureq { 4 index ne { pop pop pop false }{ 4 index ne { pop pop false }{ 4 index ne { pop false }{ 4 index eq } ifelse } ifelse } ifelse } def /ci6testplate { Adobe_ColorImage_AI6_Vars begin /plateindex -1 def /setcmykcolor where { pop gsave 1 0 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 1 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 1 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 0 1 setcmykcolor systemdict /currentgray get exec 1 exch sub grestore 1 0 0 0 ci6foureq { /plateindex 0 def }{ 0 1 0 0 ci6foureq { /plateindex 1 def }{ 0 0 1 0 ci6foureq { /plateindex 2 def }{ 0 0 0 1 ci6foureq { /plateindex 3 def }{ 0 0 0 0 ci6foureq { /plateindex 5 def } if } ifelse } ifelse } ifelse } ifelse pop pop pop pop } if plateindex end } def /ci6concatprocs { /packedarray where { pop dup type /packedarraytype eq 2 index type /packedarraytype eq or }{ false } ifelse { /_proc2 exch cvlit def /_proc1 exch cvlit def _proc1 aload pop _proc2 aload pop _proc1 length _proc2 length add packedarray cvx }{ /_proc2 exch cvlit def /_proc1 exch cvlit def /_newproc _proc1 length _proc2 length add array def _newproc 0 _proc1 putinterval _newproc _proc1 length _proc2 putinterval _newproc cvx } ifelse } def /ci6istint { type /arraytype eq } def /ci6isspot { dup type /arraytype eq { dup length 1 sub get /Separation eq }{ pop false } ifelse } def /ci6spotname { dup ci6isspot {dup length 2 sub get}{pop ()} ifelse } def /ci6altspace { aload pop pop pop ci6colormake } def /ci6numcomps { dup /DeviceGray eq { pop 1 }{ dup /DeviceRGB eq { pop 3 }{ /DeviceCMYK eq { 4 }{ 1 } ifelse } ifelse } ifelse } def /ci6marksplate { dup /DeviceGray eq { pop plateindex 3 eq }{ dup /DeviceRGB eq { pop plateindex 5 ne }{ dup /DeviceCMYK eq { pop plateindex 5 ne }{ dup ci6isspot { /findcmykcustomcolor where { pop dup length 2 sub get 0.1 0.1 0.1 0.1 5 -1 roll findcmykcustomcolor 1 setcustomcolor systemdict /currentgray get exec 1 ne }{ pop plateindex 5 ne } ifelse }{ pop plateindex 5 ne } ifelse } ifelse } ifelse } ifelse } def /ci6colormake { dup ci6numcomps exch 1 index 2 add 1 roll dup 1 eq {pop}{array astore} ifelse exch } def /ci6colorexpand { dup ci6spotname exch dup ci6istint { ci6altspace exch 4 1 roll }{ 1 3 1 roll } ifelse } def /ci6colortint { dup /DeviceGray eq { 3 1 roll 1 exch sub mul 1 exch sub exch }{ dup /DeviceRGB eq { 3 1 roll {1 exch sub 1 index mul 1 exch sub exch} forall pop 3 array astore exch }{ dup /DeviceCMYK eq { 3 1 roll {1 index mul exch} forall pop 4 array astore exch }{ 3 1 roll mul exch } ifelse } ifelse } ifelse } def /ci6colortocmyk { dup /DeviceGray eq { pop 1 exch sub 0 0 0 4 -1 roll 4 array astore }{ dup /DeviceRGB eq { pop aload pop _rgbtocmyk 4 array astore }{ dup /DeviceCMYK eq { pop }{ ci6altspace ci6colortint ci6colortocmyk } ifelse } ifelse } ifelse } def /ci6makeimagedict { 7 dict begin /ImageType 1 def /Decode exch def /DataSource exch def /ImageMatrix exch def /BitsPerComponent exch def /Height exch def /Width exch def currentdict end } def /ci6stringinvert { 0 1 2 index length 1 sub { dup 2 index exch get 255 exch sub 2 index 3 1 roll put } for } def /ci6stringknockout { 0 1 2 index length 1 sub { 255 2 index 3 1 roll put } for } def /ci6stringapply { 0 1 4 index length 1 sub { dup 4 index exch get 3 index 3 1 roll 3 index exec } for pop exch pop } def /ci6walkrgbstring { 0 3 index dup length 1 sub 0 3 3 -1 roll { 3 getinterval {} forall 5 index exec 3 index } for 5 {pop} repeat } def /ci6walkcmykstring { 0 3 index dup length 1 sub 0 4 3 -1 roll { 4 getinterval {} forall 6 index exec 3 index } for 5 { pop } repeat } def /ci6putrgbtograystr { .11 mul exch .59 mul add exch .3 mul add cvi 3 copy put pop 1 add } def /ci6putcmyktograystr { exch .11 mul add exch .59 mul add exch .3 mul add dup 255 gt { pop 255 } if 255 exch sub cvi 3 copy put pop 1 add } def /ci6rgbtograyproc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec XIBuffer3 dup 3 1 roll /ci6putrgbtograystr load exch ci6walkrgbstring end } def /ci6cmyktograyproc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec XIBuffer3 dup 3 1 roll /ci6putcmyktograystr load exch ci6walkcmykstring end } def /ci6separatecmykproc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec XIBuffer3 0 2 index plateindex 4 2 index length 1 sub { get 255 exch sub 3 copy put pop 1 add 2 index } for pop pop exch pop end } def /ci6compositeimage { dup 1 eq { pop pop image }{ /ci6colorimage load null ne { ci6colorimage }{ 3 1 roll pop sourcearray 0 3 -1 roll put 3 eq {/ci6rgbtograyproc}{/ci6cmyktograyproc} ifelse load image } ifelse } ifelse } def /ci6knockoutimage { gsave 0 ci6curtransfer exec 1 ci6curtransfer exec eq { 0 ci6curtransfer exec 0.5 lt }{ 0 ci6curtransfer exec 1 ci6curtransfer exec gt } ifelse {{pop 0}}{{pop 1}} ifelse systemdict /settransfer get exec ci6compositeimage grestore } def /ci6drawimage { ci6testplate -1 eq { pop ci6compositeimage }{ dup type /arraytype eq { dup length plateindex gt {plateindex get}{pop false} ifelse }{ { true }{ dup 1 eq {plateindex 3 eq}{plateindex 3 le} ifelse } ifelse } ifelse { dup 1 eq { pop pop ci6image }{ dup 3 eq { ci6compositeimage }{ pop pop sourcearray 0 3 -1 roll put /ci6separatecmykproc load ci6image } ifelse } ifelse }{ ci6curoverprint { 7 {pop} repeat }{ ci6knockoutimage } ifelse } ifelse } ifelse } def /ci6proctintimage { /_ptispace exch store /_ptiname exch store /_pti1 exch store /_pti0 exch store /_ptiproc exch store /_pticomps _ptispace ci6numcomps store /_ptiscale _pti1 _pti0 sub store level2? { _ptiname length 0 gt version cvr 2012 ge and { [/Separation _ptiname _ptispace {_ptiproc}] setcolorspace [_pti0 _pti1] ci6makeimagedict ci6image }{ [/Indexed _ptispace 255 {255 div _ptiscale mul _pti0 add _ptiproc}] setcolorspace [0 255] ci6makeimagedict ci6image } ifelse }{ _pticomps 1 eq { { dup { 255 div _ptiscale mul _pti0 add _ptiproc 255 mul cvi put } ci6stringapply } ci6concatprocs ci6image }{ { dup length _pticomps mul dup _ptibuf length ne {/_ptibuf exch string store}{pop} ifelse _ptibuf { exch _pticomps mul exch 255 div _ptiscale mul _pti0 add _ptiproc _pticomps 2 add -2 roll _pticomps 1 sub -1 0 { 1 index add 2 index exch 5 -1 roll 255 mul cvi put } for pop pop } ci6stringapply } ci6concatprocs false _pticomps /ci6colorimage load null eq {7 {pop} repeat}{ci6colorimage} ifelse } ifelse } ifelse } def /ci6graytintimage { /_gtigray 5 -1 roll store {1 _gtigray sub mul 1 exch sub} 4 1 roll /DeviceGray ci6proctintimage } def /ci6cmyktintimage { /_cticmyk 5 -1 roll store {_cticmyk {1 index mul exch} forall pop} 4 1 roll /DeviceCMYK ci6proctintimage } def /ci6rgbtintimage { /_rtirgb 5 -1 roll store {_rtirgb {1 exch sub 1 index mul 1 exch sub exch} forall pop} 4 1 roll /DeviceRGB ci6proctintimage } def /ci6tintimage { ci6testplate -1 eq { ci6colorexpand 3 -1 roll 5 -1 roll {0}{0 exch} ifelse 4 2 roll dup /DeviceGray eq { pop ci6graytintimage }{ dup /DeviceRGB eq { pop ci6rgbtintimage }{ pop ci6cmyktintimage } ifelse } ifelse }{ dup ci6marksplate { plateindex 5 lt { ci6colortocmyk plateindex get dup 0 eq ci6curoverprint and { 7 {pop} repeat }{ 1 exch sub exch {1 0}{0 1} ifelse () ci6graytintimage } ifelse }{ pop exch {0}{0 exch} ifelse 0 3 1 roll () ci6graytintimage } ifelse }{ ci6curoverprint { 8 {pop} repeat }{ pop pop pop {pop 1} 0 1 () /DeviceGray ci6proctintimage } ifelse } ifelse } ifelse } def /XINullImage { } def /XIImageMask { XIImageWidth XIImageHeight false [XIImageWidth 0 0 XIImageHeight neg 0 0] /XIDataProc load imagemask } def /XIImageTint { XIImageWidth XIImageHeight XIBitsPerPixel [XIImageWidth 0 0 XIImageHeight neg 0 0] /XIDataProc load XIType 3 eq XIColorValues XIColorSpace ci6tintimage } def /XIImage { XIImageWidth XIImageHeight XIBitsPerPixel [XIImageWidth 0 0 XIImageHeight neg 0 0] /XIDataProc load false XIChannelCount XIPlateList ci6drawimage } def /XG { pop pop } def /XF { 13 {pop} repeat } def /Xh { Adobe_ColorImage_AI6_Vars begin gsave /XIType exch def /XIImageHeight exch def /XIImageWidth exch def /XIImageMatrix exch def 0 0 moveto XIImageMatrix concat XIImageWidth XIImageHeight scale /_lp /null ddef _fc /_lp /imagemask ddef end } def /XH { Adobe_ColorImage_AI6_Vars begin grestore end } def /XIEnable { Adobe_ColorImage_AI6_Vars /XIEnable 3 -1 roll put } def /XC { Adobe_ColorImage_AI6_Vars begin ci6colormake /XIColorSpace exch def /XIColorValues exch def end } def /XIPlates { Adobe_ColorImage_AI6_Vars begin /XIPlateList exch def end } def /XI { Adobe_ColorImage_AI6_Vars begin gsave /XIType exch def cvi dup 256 idiv /XICompression exch store 256 mod /XIEncoding exch store pop pop /XIChannelCount exch def /XIBitsPerPixel exch def /XIImageHeight exch def /XIImageWidth exch def pop pop pop pop /XIImageMatrix exch def XIBitsPerPixel 1 eq { XIImageWidth 8 div ceiling cvi }{ XIImageWidth XIChannelCount mul } ifelse /XIRowBytes exch def XIEnable { /XIBuffer3 XIImageWidth string def XICompression 0 eq { /XIBuffer1 XIRowBytes string def XIEncoding 0 eq { {currentfile XIBuffer1 readhexstring pop} }{ {currentfile XIBuffer1 readstring pop} } ifelse }{ /XIBuffer1 256 string def /XIBuffer2 XIRowBytes string def {currentfile XIBuffer1 readline pop (%) anchorsearch {pop} if} /ASCII85Decode filter /DCTDecode filter /XIFile exch def {XIFile XIBuffer2 readstring pop} } ifelse /XIDataProc exch def XIType 1 ne { 0 setgray } if XIType 1 eq { XIImageMask }{ XIType 2 eq XIType 3 eq or { XIImageTint }{ XIImage } ifelse } ifelse }{ XINullImage } ifelse /XIPlateList false def grestore end } def end %%EndProcSet %%BeginResource: procset Adobe_Illustrator_AI5 1.3 0 %%Title: (Adobe Illustrator (R) Version 8.0 Full Prolog) %%Version: 1.3 0 %%CreationDate: (3/7/1994) () %%Copyright: ((C) 1987-1998 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_Illustrator_AI5_vars 112 dict dup begin put /_?cmyk false def /_eo false def /_lp /none def /_pf { } def /_ps { } def /_psf { } def /_pss { } def /_pjsf { } def /_pjss { } def /_pola 0 def /_doClip 0 def /cf currentflat def /_lineorientation 0 def /_charorientation 0 def /_yokoorientation 0 def /_tm matrix def /_renderStart [ /e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0 ] def /_renderEnd [ null null null null /i1 /i1 /i1 /i1 ] def /_render -1 def /_shift [0 0] def /_ax 0 def /_ay 0 def /_cx 0 def /_cy 0 def /_leading [ 0 0 ] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fontSize 0 def /_fontAscent 0 def /_fontDescent 0 def /_fontHeight 0 def /_fontRotateAdjust 0 def /Ss 256 string def Ss 0 (fonts/) putinterval /_cnt 0 def /_scale [1 1] def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def /_hfname 100 string def /_hffound false def /Tx { } def /Tj { } def /CRender { } def /_AI3_savepage { } def /_gf null def /_cf 4 array def /_rgbf 3 array def /_if null def /_of false def /_fc { } def /_gs null def /_cs 4 array def /_rgbs 3 array def /_is null def /_os false def /_sc { } def /_pd 1 dict def /_ed 15 dict def /_pm matrix def /_fm null def /_fd null def /_fdd null def /_sm null def /_sd null def /_sdd null def /_i null def /_lobyte 0 def /_hibyte 0 def /_cproc null def /_cscript 0 def /_hvax 0 def /_hvay 0 def /_hvwb 0 def /_hvcx 0 def /_hvcy 0 def /_bitfont null def /_bitlobyte 0 def /_bithibyte 0 def /_bitkey null def /_bitdata null def /_bitindex 0 def /discardSave null def /buffer 256 string def /beginString null def /endString null def /endStringLength null def /layerCnt 1 def /layerCount 1 def /perCent (%) 0 get def /perCentSeen? false def /newBuff null def /newBuffButFirst null def /newBuffLast null def /clipForward? false def end userdict /Adobe_Illustrator_AI5 known not { userdict /Adobe_Illustrator_AI5 100 dict put } if userdict /Adobe_Illustrator_AI5 get begin /initialize { Adobe_Illustrator_AI5 dup begin Adobe_Illustrator_AI5_vars begin /_aicmykps where {pop /_?cmyk _aicmykps def}if discardDict { bind pop pop } forall dup /nc get begin { dup xcheck 1 index type /operatortype ne and { bind } if pop pop } forall end newpath } def /terminate { end end } def /_ null def /ddef { Adobe_Illustrator_AI5_vars 3 1 roll put } def /xput { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop { { pop } repeat } def /hswj { dup stringwidth 3 2 roll { _hvwb eq { exch _hvcx add exch _hvcy add } if exch _hvax add exch _hvay add } cforall } def /vswj { 0 0 3 -1 roll { dup 255 le _charorientation 1 eq and { dup cstring stringwidth 5 2 roll _hvwb eq { exch _hvcy sub exch _hvcx sub } if exch _hvay sub exch _hvax sub 4 -1 roll sub exch 3 -1 roll sub exch } { _hvwb eq { exch _hvcy sub exch _hvcx sub } if exch _hvay sub exch _hvax sub _fontHeight sub } ifelse } cforall } def /swj { 6 1 roll /_hvay exch ddef /_hvax exch ddef /_hvwb exch ddef /_hvcy exch ddef /_hvcx exch ddef _lineorientation 0 eq { hswj } { vswj } ifelse } def /sw { 0 0 0 6 3 roll swj } def /vjss { 4 1 roll { dup cstring dup length 1 eq _charorientation 1 eq and { -90 rotate currentpoint _fontRotateAdjust add moveto gsave false charpath currentpoint 5 index setmatrix stroke grestore _fontRotateAdjust sub moveto _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto 90 rotate } { currentpoint _fontHeight sub 5 index sub 3 index _sp eq { 9 index sub } if currentpoint exch 4 index stringwidth pop 2 div sub exch _fontAscent sub moveto gsave 2 index false charpath 6 index setmatrix stroke grestore moveto pop pop } ifelse } cforall 6 npop } def /hjss { 4 1 roll { dup cstring gsave false charpath currentpoint 5 index setmatrix stroke grestore moveto _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto } cforall 6 npop } def /jss { _lineorientation 0 eq { hjss } { vjss } ifelse } def /ss { 0 0 0 7 3 roll jss } def /vjsp { 4 1 roll { dup cstring dup length 1 eq _charorientation 1 eq and { -90 rotate currentpoint _fontRotateAdjust add moveto false charpath currentpoint _fontRotateAdjust sub moveto _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto 90 rotate } { currentpoint _fontHeight sub 5 index sub 3 index _sp eq { 9 index sub } if currentpoint exch 4 index stringwidth pop 2 div sub exch _fontAscent sub moveto 2 index false charpath moveto pop pop } ifelse } cforall 6 npop } def /hjsp { 4 1 roll { dup cstring false charpath _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto } cforall 6 npop } def /jsp { matrix currentmatrix _lineorientation 0 eq {hjsp} {vjsp} ifelse } def /sp { matrix currentmatrix 0 0 0 7 3 roll _lineorientation 0 eq {hjsp} {vjsp} ifelse } def /pl { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c { curveto } def /C /c load def /v { currentpoint 6 2 roll curveto } def /V /v load def /y { 2 copy curveto } def /Y /y load def /l { lineto } def /L /l load def /m { moveto } def } { /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def } ifelse /d { setdash } def /cf { } def /i { dup 0 eq { pop cf } if setflat } def /j { setlinejoin } def /J { setlinecap } def /M { setmiterlimit } def /w { setlinewidth } def /XR { 0 ne /_eo exch ddef } def /H { } def /h { closepath } def /N { _pola 0 eq { _doClip 1 eq { _eo {eoclip} {clip} ifelse /_doClip 0 ddef } if newpath } { /CRender { N } ddef } ifelse } def /n { N } def /F { _pola 0 eq { _doClip 1 eq { gsave _pf grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf } ifelse } { /CRender { F } ddef } ifelse } def /f { closepath F } def /S { _pola 0 eq { _doClip 1 eq { gsave _ps grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps } ifelse } { /CRender { S } ddef } ifelse } def /s { closepath S } def /B { _pola 0 eq { _doClip 1 eq gsave F grestore { gsave S grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { S } ifelse } { /CRender { B } ddef } ifelse } def /b { closepath B } def /W { /_doClip 1 ddef } def /* { count 0 ne { dup type /stringtype eq { pop } if } if newpath } def /u { } def /U { } def /q { _pola 0 eq { gsave } if } def /Q { _pola 0 eq { grestore } if } def /*u { _pola 1 add /_pola exch ddef } def /*U { _pola 1 sub /_pola exch ddef _pola 0 eq { CRender } if } def /D { pop } def /*w { } def /*W { } def /` { /_i save ddef clipForward? { nulldevice } if 6 1 roll 4 npop concat pop userdict begin /showpage { } def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash /setstrokeadjust where {pop false setstrokeadjust} if newpath 0 setgray false setoverprint } def /~ { end _i restore } def /_rgbtocmyk { 3 { 1 exch sub 3 1 roll } repeat 3 copy 1 4 1 roll 3 { 3 index 2 copy gt { exch } if pop 4 1 roll } repeat pop pop pop 4 1 roll 3 { 3 index sub 3 1 roll } repeat 4 -1 roll } def /setrgbfill { _rgbf astore pop /_fc { _lp /fill ne { _of setoverprint _rgbf aload pop setrgbcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /setrgbstroke { _rgbs astore pop /_sc { _lp /stroke ne { _os setoverprint _rgbs aload pop setrgbcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /O { 0 ne /_of exch ddef /_lp /none ddef } def /R { 0 ne /_os exch ddef /_lp /none ddef } def /g { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /G { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /K { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /Xa { _?cmyk { 3 npop k }{ setrgbfill 4 npop } ifelse } def /XA { _?cmyk { 3 npop K }{ setrgbstroke 4 npop } ifelse } def /Xs { /_gf exch ddef 5 npop /_fc { _lp /fill ne { _of setoverprint _gf setAIseparationgray /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /XS { /_gs exch ddef 5 npop /_sc { _lp /stroke ne { _os setoverprint _gs setAIseparationgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /Xx { exch /_gf exch ddef 0 eq { findcmykcustomcolor }{ _?cmyk {true}{/findrgbcustomcolor where{pop false}{true}ifelse}ifelse { 4 1 roll 3 npop findcmykcustomcolor }{ 8 -4 roll 4 npop findrgbcustomcolor } ifelse } ifelse /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /XX { exch /_gs exch ddef 0 eq { findcmykcustomcolor }{ _?cmyk {true}{/findrgbcustomcolor where{pop false}{true}ifelse}ifelse { 4 1 roll 3 npop findcmykcustomcolor }{ 8 -4 roll 4 npop findrgbcustomcolor } ifelse } ifelse /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /X { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /XK { 3 -1 roll pop 0 eq { 1 exch sub 3 {dup 3 1 roll mul 5 1 roll} repeat mul 4 1 roll K } { 1 exch sub 4 1 roll 3 {1 exch sub 3 index mul 1 exch sub 3 1 roll} repeat 4 -1 roll pop XA } ifelse } def /Xk { 3 -1 roll pop 0 eq { 1 exch sub 3 {dup 3 1 roll mul 5 1 roll} repeat mul 4 1 roll k } { 1 exch sub 4 1 roll 3 {1 exch sub 3 index mul 1 exch sub 3 1 roll} repeat 4 -1 roll pop Xa } ifelse } def /A { pop } def /annotatepage { userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse } def /XT { pop pop } def /Xt { pop } def /discard { save /discardSave exch store discardDict begin /endString exch store gt38? { 2 add } if load stopped pop end discardSave restore } bind def userdict /discardDict 7 dict dup begin put /pre38Initialize { /endStringLength endString length store /newBuff buffer 0 endStringLength getinterval store /newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store /newBuffLast newBuff endStringLength 1 sub 1 getinterval store } def /shiftBuffer { newBuff 0 newBuffButFirst putinterval newBuffLast 0 currentfile read not { stop } if put } def 0 { pre38Initialize mark currentfile newBuff readstring exch pop { { newBuff endString eq { cleartomark stop } if shiftBuffer } loop } { stop } ifelse } def 1 { pre38Initialize /beginString exch store mark currentfile newBuff readstring exch pop { { newBuff beginString eq { /layerCount dup load 1 add store } { newBuff endString eq { /layerCount dup load 1 sub store layerCount 0 eq { cleartomark stop } if } if } ifelse shiftBuffer } loop } if } def 2 { mark { currentfile buffer {readline} stopped { % assume error was due to overfilling the buffer }{ not { stop } if endString eq { cleartomark stop } if }ifelse } loop } def 3 { /beginString exch store /layerCnt 1 store mark { currentfile buffer {readline} stopped { % assume error was due to overfilling the buffer }{ not { stop } if dup beginString eq { pop /layerCnt dup load 1 add store } { endString eq { layerCnt 1 eq { cleartomark stop } { /layerCnt dup load 1 sub store } ifelse } if } ifelse }ifelse } loop } def end userdict /clipRenderOff 15 dict dup begin put { /n /N /s /S /f /F /b /B } { { _doClip 1 eq { /_doClip 0 ddef _eo {eoclip} {clip} ifelse } if newpath } def } forall /Tr /pop load def /Bb {} def /BB /pop load def /Bg {12 npop} def /Bm {6 npop} def /Bc /Bm load def /Bh {4 npop} def end /Lb { 6 npop 7 2 roll 5 npop 0 eq { 0 eq { (%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard } { /clipForward? true def /Tx /pop load def /Tj /pop load def currentdict end clipRenderOff begin begin } ifelse } { 0 eq { save /discardSave exch store } if } ifelse } bind def /LB { discardSave dup null ne { restore } { pop clipForward? { currentdict end end begin /clipForward? false ddef } if } ifelse } bind def /Pb { pop pop 0 (%AI5_EndPalette) discard } bind def /Np { 0 (%AI5_End_NonPrinting--) discard } bind def /Ln /pop load def /Ap /pop load def /Ar { 72 exch div 0 dtransform dup mul exch dup mul add sqrt dup 1 lt { pop 1 } if setflat } def /Mb { q } def /Md { } def /MB { Q } def /nc 4 dict def nc begin /setgray { pop } bind def /setcmykcolor { 4 npop } bind def /setrgbcolor { 3 npop } bind def /setcustomcolor { 2 npop } bind def currentdict readonly pop end /XP { 4 npop } bind def /XD { pop } bind def end setpacking %%EndResource %%BeginResource: procset Adobe_cshow 2.0 8 %%Title: (Writing System Operators) %%Version: 2.0 8 %%CreationDate: (1/23/89) () %%Copyright: ((C) 1992-1996 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_cshow 14 dict dup begin put /initialize { Adobe_cshow begin Adobe_cshow { dup xcheck { bind } if pop pop } forall end Adobe_cshow begin } def /terminate { currentdict Adobe_cshow eq { end } if } def /cforall { /_lobyte 0 ddef /_hibyte 0 ddef /_cproc exch ddef /_cscript currentfont /FontScript known { currentfont /FontScript get } { -1 } ifelse ddef { /_lobyte exch ddef _hibyte 0 eq _cscript 1 eq _lobyte 129 ge _lobyte 159 le and _lobyte 224 ge _lobyte 252 le and or and _cscript 2 eq _lobyte 161 ge _lobyte 254 le and and _cscript 3 eq _lobyte 161 ge _lobyte 254 le and and _cscript 25 eq _lobyte 161 ge _lobyte 254 le and and _cscript -1 eq or or or or and { /_hibyte _lobyte ddef } { _hibyte 256 mul _lobyte add _cproc /_hibyte 0 ddef } ifelse } forall } def /cstring { dup 256 lt { (s) dup 0 4 3 roll put } { dup 256 idiv exch 256 mod (hl) dup dup 0 6 5 roll put 1 4 3 roll put } ifelse } def /clength { 0 exch { 256 lt { 1 } { 2 } ifelse add } cforall } def /hawidthshow { { dup cstring show _hvax _hvay rmoveto _hvwb eq { _hvcx _hvcy rmoveto } if } cforall } def /vawidthshow { { dup 255 le _charorientation 1 eq and { -90 rotate 0 _fontRotateAdjust rmoveto cstring _hvcx _hvcy _hvwb _hvax _hvay 6 -1 roll awidthshow 0 _fontRotateAdjust neg rmoveto 90 rotate } { currentpoint _fontHeight sub exch _hvay sub exch _hvax sub 2 index _hvwb eq { exch _hvcy sub exch _hvcx sub } if 3 2 roll cstring dup stringwidth pop 2 div neg _fontAscent neg rmoveto show moveto } ifelse } cforall } def /hvawidthshow { 6 1 roll /_hvay exch ddef /_hvax exch ddef /_hvwb exch ddef /_hvcy exch ddef /_hvcx exch ddef _lineorientation 0 eq { hawidthshow } { vawidthshow } ifelse } def /hvwidthshow { 0 0 3 -1 roll hvawidthshow } def /hvashow { 0 0 0 6 -3 roll hvawidthshow } def /hvshow { 0 0 0 0 0 6 -1 roll hvawidthshow } def currentdict readonly pop end setpacking %%EndResource %%BeginResource: procset Adobe_shading_AI8 1.0 0 %%Title: (Adobe Illustrator 8 Shading Procset) %%Version: 1.0 0 %%CreationDate: (12/17/97) () %%Copyright: ((C) 1987-1997 Adobe Systems Incorporated All Rights Reserved) userdict /defaultpacking currentpacking put true setpacking userdict /Adobe_shading_AI8 10 dict dup begin put /initialize { Adobe_shading_AI8 begin Adobe_shading_AI8 bdprocs Mesh /initialize get exec } def /terminate { currentdict Adobe_shading_AI8 eq { end } if } def /bdprocs { { dup xcheck 1 index type /arraytype eq and { bind } if pop pop } forall } def /X! {pop} def /X# {pop pop} def /Mesh 40 dict def Mesh begin /initialize { Mesh bdprocs Mesh begin /emulate? /AI8MeshEmulation where { pop AI8MeshEmulation }{ systemdict /shfill known not } ifelse def end } def /bd { shadingdict begin } def /paint { emulate? { end }{ /_lp /none ddef _fc /_lp /none ddef /AIColorSpace AIColorSpace tocolorspace store /ColorSpace AIColorSpace topsspace store version_ge_3010.106 not systemdict /setsmoothness known and { 0.0001 setsmoothness } if composite? { /DataSource getdatasrc def Matrix concat currentdict end shfill }{ AIColorSpace makesmarks AIPlateList markingplate and not isoverprint and { end }{ /ColorSpace /DeviceGray store /Decode [0 1 0 1 0 1] store /DataSource getplatesrc def Matrix concat currentdict end shfill } ifelse } ifelse } ifelse } def /shadingdict 12 dict def shadingdict begin /ShadingType 6 def /BitsPerCoordinate 16 def /BitsPerComponent 8 def /BitsPerFlag 8 def end /datafile null def /databuf 256 string def /dataptr 0 def /srcspace null def /srcchannels 0 def /dstchannels 0 def /dstplate 0 def /srctodstcolor null def /getplatesrc { /srcspace AIColorSpace store /srcchannels AIColorSpace getnchannels store /dstchannels 1 store /dstplate getplateindex store /srctodstcolor srcspace makesmarks { dstplate 4 eq { {1 exch sub} }{ {srcspace tocmyk 3 dstplate sub index 1 exch sub 5 1 roll 4 {pop} repeat} } ifelse }{ {srcchannels {pop} repeat 1} } ifelse store /datafile getdatasrc store /rdpatch168 load DataLength () /SubFileDecode filter } def /getdatasrc { /rdcmntline load /ASCII85Decode filter } def /rdpatch168 { /dataptr 0 store 49 rdcount 4 { dup {pop srcchannels getint8} if dup {pop srctodstcolor dstchannels putint8 true} if } repeat {databuf 0 dataptr getinterval}{()} ifelse } def /rdpatch3216 { /dataptr 0 store 97 rdcount 4 { dup {pop srcchannels getint16} if dup {pop srctodstcolor dstchannels putint16 true} if } repeat {databuf 0 dataptr getinterval}{()} ifelse } def /rdcount { dup 0 gt { datafile databuf dataptr 4 -1 roll getinterval readstring exch length dataptr add /dataptr exch store }{ true } ifelse } def /getint8 { mark true 3 -1 roll { dup {pop datafile read} if dup {pop 255 div true} if } repeat { counttomark 1 add -1 roll pop true }{ cleartomark false } ifelse } def /putint8 { dup dataptr add /dataptr exch store dataptr exch { 1 sub exch 255 mul cvi databuf 2 index 3 -1 roll put } repeat pop } def /getint16 { mark true 3 -1 roll { dup {pop datafile read} if dup {pop 256 mul datafile read} if dup {pop add 65535 div true} if } repeat { counttomark 1 add -1 roll pop true }{ cleartomark false } ifelse } def /putint16 { dup 2 mul dataptr add /dataptr exch store dataptr exch { 2 sub exch 65535 mul cvi dup 256 idiv databuf 3 index 3 -1 roll put 256 mod databuf 2 index 1 add 3 -1 roll put } repeat pop } def /srcbuf 256 string def /rdcmntline { currentfile srcbuf readline pop (%) anchorsearch {pop} if } def /getplateindex { 0 [cyan? magenta? yellow? black? customColor?] {{exit} if 1 add} forall } def /aicsarray 4 array def /aicsaltvals 4 array def /aicsaltcolr aicsaltvals def /tocolorspace { dup type /arraytype eq { mark exch aload pop aicsarray 0 3 -1 roll put aicsarray 1 3 -1 roll put dup aicsarray 2 3 -1 roll put gettintxform aicsarray 3 3 -1 roll put counttomark aicsaltvals 0 3 -1 roll getinterval /aicsaltcolr exch store aicsaltcolr astore pop pop aicsarray } if } def /subtintxform {aicsaltcolr {1 index mul exch} forall pop} def /addtintxform {aicsaltcolr {1 sub 1 index mul 1 add exch} forall pop} def /gettintxform { /DeviceRGB eq {/addtintxform}{/subtintxform} ifelse load } def /getnchannels { dup type /arraytype eq {0 get} if colorspacedict exch get begin Channels end } def /makesmarks { composite? { pop true }{ dup dup type /arraytype eq {0 get} if colorspacedict exch get begin MarksPlate end } ifelse } def /markingplate { composite? { pop true }{ dup type /arraytype eq { dup length getplateindex gt {getplateindex get}{pop false} ifelse } if } ifelse } def /tocmyk { dup dup type /arraytype eq {0 get} if colorspacedict exch get begin ToCMYK end } def /topsspace { dup dup type /arraytype eq {0 get} if colorspacedict exch get begin ToPSSpace end } def /colorspacedict 5 dict dup begin /DeviceGray 4 dict dup begin /Channels 1 def /MarksPlate {pop black?} def /ToCMYK {pop 1 exch sub 0 0 0 4 -1 roll} def /ToPSSpace {} def end def /DeviceRGB 4 dict dup begin /Channels 3 def /MarksPlate {pop isCMYKSep?} def /ToCMYK {pop _rgbtocmyk} def /ToPSSpace {} def end def /DeviceCMYK 4 dict dup begin /Channels 4 def /MarksPlate {pop isCMYKSep?} def /ToCMYK {pop} def /ToPSSpace {} def end def /Separation 4 dict dup begin /Channels 1 def /MarksPlate { /findcmykcustomcolor where { pop dup 1 exch ToCMYK 5 -1 roll 1 get findcmykcustomcolor 1 setcustomcolor systemdict /currentgray get exec 1 ne }{ pop false } ifelse } def /ToCMYK { dup 2 get mark exch 4 2 roll 3 get exec counttomark -1 roll tocmyk 5 -1 roll pop } def /ToPSSpace {} def end def /Process 4 dict dup begin /Channels 1 def /MarksPlate { isCMYKSep? { 1 exch ToCMYK 4 array astore getplateindex get 0 ne }{ pop false } ifelse } def /ToCMYK { dup 2 get mark exch 4 2 roll 3 get exec counttomark -1 roll tocmyk 5 -1 roll pop } def /ToPSSpace { 4 array copy dup 0 /Separation put } def end def end def /isoverprint { /currentoverprint where {pop currentoverprint}{_of} ifelse } def /version_ge_3010.106 { version {cvr} stopped { pop false }{ 3010.106 ge } ifelse } def end end defaultpacking setpacking %%EndResource %%EndProlog %%BeginSetup %%IncludeFont: Symbol userdict /_useSmoothShade false put userdict /_aicmykps true put userdict /_forceToCMYK true put Adobe_level2_AI5 /initialize get exec Adobe_cshow /initialize get exec Adobe_Illustrator_AI5_vars Adobe_Illustrator_AI5 Adobe_typography_AI5 /initialize get exec Adobe_ColorImage_AI6 /initialize get exec Adobe_shading_AI8 /initialize get exec Adobe_Illustrator_AI5 /initialize get exec [ 39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis /Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute /egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde 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1987-1996 Adobe Systems Incorporated All Rights Reserved) userdict /Adobe_level2_AI5 26 dict dup begin put /packedarray where not { userdict begin /packedarray { array astore readonly } bind def /setpacking /pop load def /currentpacking false def end 0 } if pop userdict /defaultpacking currentpacking put true setpacking /initialize { Adobe_level2_AI5 begin } bind def /terminate { currentdict Adobe_level2_AI5 eq { end } if } bind def mark /setcustomcolor where not { /findcmykcustomcolor { (AI8_CMYK_CustomColor) 6 packedarray } bind def /findrgbcustomcolor { (AI8_RGB_CustomColor) 5 packedarray } bind def /setcustomcolor { exch aload pop dup (AI8_CMYK_CustomColor) eq { pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } { dup (AI8_RGB_CustomColor) eq { pop pop 3 { 1 exch sub 3 index mul 1 exch sub 3 1 roll } repeat 4 -1 roll pop setrgbcolor } { pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } ifelse } ifelse } def } if /setAIseparationgray { false setoverprint 0 setgray /setseparationgray where{ pop setseparationgray }{ /setcolorspace where{ pop [/Separation (All) /DeviceCMYK {dup dup dup}] setcolorspace 1 exch sub setcolor }{ setgray }ifelse }ifelse } def /gt38? mark {version cvr cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put userdict /level2? systemdict /languagelevel known dup { pop systemdict /languagelevel get 2 ge } if put /level2ScreenFreq { begin 60 HalftoneType 1 eq { pop Frequency } if HalftoneType 2 eq { pop GrayFrequency } if HalftoneType 5 eq { pop Default level2ScreenFreq } if end } bind def userdict /currentScreenFreq level2? {currenthalftone level2ScreenFreq} {currentscreen pop pop} ifelse put level2? not { /setcmykcolor where not { /setcmykcolor { exch .11 mul add exch .59 mul add exch .3 mul add 1 exch sub setgray } def } if /currentcmykcolor where not { /currentcmykcolor { 0 0 0 1 currentgray sub } def } if /setoverprint where not { /setoverprint /pop load def } if /selectfont where not { /selectfont { exch findfont exch dup type /arraytype eq { makefont } { scalefont } ifelse setfont } bind def } if /cshow where not { /cshow { [ 0 0 5 -1 roll aload pop ] cvx bind forall } bind def } if } if cleartomark /anyColor? { add add add 0 ne } bind def /testColor { gsave setcmykcolor currentcmykcolor grestore } bind def /testCMYKColorThrough { testColor anyColor? } bind def userdict /composite? 1 0 0 0 testCMYKColorThrough 0 1 0 0 testCMYKColorThrough 0 0 1 0 testCMYKColorThrough 0 0 0 1 testCMYKColorThrough and and and put composite? not { userdict begin gsave /cyan? 1 0 0 0 testCMYKColorThrough def /magenta? 0 1 0 0 testCMYKColorThrough def /yellow? 0 0 1 0 testCMYKColorThrough def /black? 0 0 0 1 testCMYKColorThrough def grestore /isCMYKSep? cyan? magenta? yellow? black? or or or def /customColor? isCMYKSep? not def end } if end defaultpacking setpacking %%EndResource %%BeginResource: procset Adobe_typography_AI5 1.0 1 %%Title: (Typography Operators) %%Version: 1.0 1 %%CreationDate:(6/10/1996) () %%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_typography_AI5 68 dict dup begin put /initialize { begin begin Adobe_typography_AI5 begin Adobe_typography_AI5 { dup xcheck { bind } if pop pop } forall end end end Adobe_typography_AI5 begin } def /terminate { currentdict Adobe_typography_AI5 eq { end } if } def /modifyEncoding { /_tempEncode exch ddef /_pntr 0 ddef { counttomark -1 roll dup type dup /marktype eq { pop pop exit } { /nametype eq { _tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll put } { /_pntr exch ddef } ifelse } ifelse } loop _tempEncode } def /havefont { systemdict /languagelevel known { /Font resourcestatus dup { exch pop exch pop } if } { systemdict /FontDirectory get 1 index known { pop true } { systemdict /fileposition known { dup length 6 add exch Ss 6 250 getinterval cvs pop Ss exch 0 exch getinterval status { pop pop pop pop true } { false } ifelse } { pop false } ifelse } ifelse } ifelse } def /TE { StandardEncoding 256 array copy modifyEncoding /_nativeEncoding exch def } def /subststring { exch 2 index exch search { exch pop exch dup () eq { pop exch concatstring } { 3 -1 roll exch concatstring concatstring } ifelse exch pop true } { pop pop false } ifelse } def /concatstring { 1 index length 1 index length 1 index add string dup 0 5 index putinterval dup 2 index 4 index putinterval 4 1 roll pop pop pop } def % /TZ { dup type /arraytype eq { /_wv exch def } { /_wv 0 def } ifelse /_useNativeEncoding exch def 2 index havefont { 3 index 255 string cvs dup (_Symbol_) eq { pop 2 index findfont } { 1 index 0 eq { dup length 1 sub 1 exch getinterval cvn findfont } { pop 2 index findfont } ifelse } ifelse } { dup 1 eq { 2 index 64 string cvs dup (-90pv-RKSJ-) (-83pv-RKSJ-) subststring { exch pop dup havefont { findfont false } { pop true } ifelse } { pop dup (-90ms-RKSJ-) (-Ext-RKSJ-) subststring { exch pop dup havefont { findfont false } { pop true } ifelse } { pop pop true } ifelse } ifelse { 1 index 1 eq { /Ryumin-Light-Ext-RKSJ-V havefont {/Ryumin-Light-Ext-RKSJ-V} {/Courier} ifelse } { /Ryumin-Light-83pv-RKSJ-H havefont {/Ryumin-Light-83pv-RKSJ-H} {/Courier} ifelse } ifelse findfont [1 0 0.5 1 0 0] makefont } if } { /Courier findfont } ifelse } ifelse _wv type /arraytype eq { _wv makeblendedfont } if dup length 10 add dict begin mark exch { 1 index /FID ne { def } if cleartomark mark } forall pop /FontScript exch def /FontDirection exch def /FontRequest exch def /FontName exch def counttomark 0 eq { 1 _useNativeEncoding eq { /Encoding _nativeEncoding def } if cleartomark } { /Encoding load 256 array copy modifyEncoding /Encoding exch def } ifelse FontName currentdict end definefont pop } def /tr { _ax _ay 3 2 roll } def /trj { _cx _cy _sp _ax _ay 6 5 roll } def /a0 { /Tx { dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss } ddef /Tj { dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss } ddef } def /a1 { W B } def /e0 { /Tx { tr _psf } ddef /Tj { trj _pjsf } ddef } def /e1 { W F } def /i0 { /Tx { tr sp } ddef /Tj { trj jsp } ddef } def /i1 { W N } def /o0 { /Tx { tr sw rmoveto } ddef /Tj { trj swj rmoveto } ddef } def /r0 { /Tx { tr _ctm _pss } ddef /Tj { trj _ctm _pjss } ddef } def /r1 { W S } def /To { pop _ctm currentmatrix pop } def /TO { iTe _ctm setmatrix newpath } def /Tp { pop _tm astore pop _ctm setmatrix _tDict begin /W { } def /h { } def } def /TP { end iTm 0 0 moveto } def /Tr { _render 3 le { currentpoint newpath moveto } if dup 8 eq { pop 0 } { dup 9 eq { pop 1 } if } ifelse dup /_render exch ddef _renderStart exch get load exec } def /iTm { _ctm setmatrix _tm concat _shift aload pop _lineorientation 1 eq { exch } if translate _scale aload pop _lineorientation 1 eq _yokoorientation 1 eq or { exch } if scale } def /Tm { _tm astore pop iTm 0 0 moveto } def /Td { _mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto } def /iTe { _render -1 eq { } { _renderEnd _render get dup null ne { load exec } { pop } ifelse } ifelse /_render -1 ddef } def /Ta { pop } def /Tf { 1 index type /nametype eq { dup 0.75 mul 1 index 0.25 mul neg } if /_fontDescent exch ddef /_fontAscent exch ddef /_fontSize exch ddef /_fontRotateAdjust _fontAscent _fontDescent add 2 div neg ddef /_fontHeight _fontSize ddef findfont _fontSize scalefont setfont } def /Tl { pop neg 0 exch _leading astore pop } def /Tt { pop } def /TW { 3 npop } def /Tw { /_cx exch ddef } def /TC { 3 npop } def /Tc { /_ax exch ddef } def /Ts { 0 exch _shift astore pop currentpoint iTm moveto } def /Ti { 3 npop } def /Tz { count 1 eq { 100 } if 100 div exch 100 div exch _scale astore pop iTm } def /TA { pop } def /Tq { pop } def /Tg { pop } def /TG { pop } def /Tv { /_lineorientation exch ddef } def /TV { /_charorientation exch ddef } def /Ty { dup /_yokoorientation exch ddef 1 sub neg Tv } def /TY { pop } def /T~ { Tx } def /Th { pop pop pop pop pop } def /TX { pop } def /Tk { _fontSize mul 1000 div _lineorientation 0 eq { neg 0 } { 0 exch } ifelse rmoveto pop } def /TK { 2 npop } def /T* { _leading aload pop _lineorientation 0 ne { exch } if Td } def /T*- { _leading aload pop _lineorientation 0 ne { exch } if exch neg exch neg Td } def /T- { _ax neg 0 rmoveto _lineorientation 1 eq _charorientation 0 eq and { 1 TV _hyphen Tx 0 TV } { _hyphen Tx } ifelse } def /T+ { } def /TR { _ctm currentmatrix pop _tm astore pop iTm 0 0 moveto } def /TS { currentfont 3 1 roll /_Symbol_ findfont _fontSize scalefont setfont 0 eq { Tx } { Tj } ifelse setfont } def /Xb { pop pop } def /Tb /Xb load def /Xe { pop pop pop pop } def /Te /Xe load def /XB { } def /TB /XB load def currentdict readonly pop end setpacking % /X^ { currentfont 5 1 roll dup havefont { findfont _fontSize scalefont setfont } { pop exch } ifelse 2 index 0 eq { Tx } { Tj } ifelse pop pop setfont } def /T^ /X^ load def %%EndResource %%BeginProcSet: Adobe_ColorImage_AI6 1.3 0 userdict /Adobe_ColorImage_AI6 known not { userdict /Adobe_ColorImage_AI6 53 dict put } if userdict /Adobe_ColorImage_AI6 get begin /initialize { Adobe_ColorImage_AI6 begin Adobe_ColorImage_AI6 { dup type /arraytype eq { dup xcheck { bind } if } if pop pop } forall } def /terminate { end } def currentdict /Adobe_ColorImage_AI6_Vars known not { /Adobe_ColorImage_AI6_Vars 41 dict def } if Adobe_ColorImage_AI6_Vars begin /plateindex -1 def /_newproc null def /_proc1 null def /_proc2 null def /sourcearray 4 array def /_ptispace null def /_ptiname null def /_pti0 0 def /_pti1 0 def /_ptiproc null def /_ptiscale 0 def /_pticomps 0 def /_ptibuf 0 string def /_gtigray 0 def /_cticmyk null def /_rtirgb null def /XIEnable true def /XIType 0 def /XIEncoding 0 def /XICompression 0 def /XIChannelCount 0 def /XIBitsPerPixel 0 def /XIImageHeight 0 def /XIImageWidth 0 def /XIImageMatrix null def /XIRowBytes 0 def /XIFile null def /XIBuffer1 null def /XIBuffer2 null def /XIBuffer3 null def /XIDataProc null def /XIColorSpace /DeviceGray def /XIColorValues 0 def /XIPlateList false def end /ci6colorimage /colorimage where {/colorimage get}{null} ifelse def /ci6image systemdict /image get def /ci6curtransfer systemdict /currenttransfer get def /ci6curoverprint /currentoverprint where {/currentoverprint get}{{_of}} ifelse def /ci6foureq { 4 index ne { pop pop pop false }{ 4 index ne { pop pop false }{ 4 index ne { pop false }{ 4 index eq } ifelse } ifelse } ifelse } def /ci6testplate { Adobe_ColorImage_AI6_Vars begin /plateindex -1 def /setcmykcolor where { pop gsave 1 0 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 1 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 1 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 0 1 setcmykcolor systemdict /currentgray get exec 1 exch sub grestore 1 0 0 0 ci6foureq { /plateindex 0 def }{ 0 1 0 0 ci6foureq { /plateindex 1 def }{ 0 0 1 0 ci6foureq { /plateindex 2 def }{ 0 0 0 1 ci6foureq { /plateindex 3 def }{ 0 0 0 0 ci6foureq { /plateindex 5 def } if } ifelse } ifelse } ifelse } ifelse pop pop pop pop } if plateindex end } def /ci6concatprocs { /packedarray where { pop dup type /packedarraytype eq 2 index type /packedarraytype eq or }{ false } ifelse { /_proc2 exch cvlit def /_proc1 exch cvlit def _proc1 aload pop _proc2 aload pop _proc1 length _proc2 length add packedarray cvx }{ /_proc2 exch cvlit def /_proc1 exch cvlit def /_newproc _proc1 length _proc2 length add array def _newproc 0 _proc1 putinterval _newproc _proc1 length _proc2 putinterval _newproc cvx } ifelse } def /ci6istint { type /arraytype eq } def /ci6isspot { dup type /arraytype eq { dup length 1 sub get /Separation eq }{ pop false } ifelse } def /ci6spotname { dup ci6isspot {dup length 2 sub get}{pop ()} ifelse } def /ci6altspace { aload pop pop pop ci6colormake } def /ci6numcomps { dup /DeviceGray eq { pop 1 }{ dup /DeviceRGB eq { pop 3 }{ /DeviceCMYK eq { 4 }{ 1 } ifelse } ifelse } ifelse } def /ci6marksplate { dup /DeviceGray eq { pop plateindex 3 eq }{ dup /DeviceRGB eq { pop plateindex 5 ne }{ dup /DeviceCMYK eq { pop plateindex 5 ne }{ dup ci6isspot { /findcmykcustomcolor where { pop dup length 2 sub get 0.1 0.1 0.1 0.1 5 -1 roll findcmykcustomcolor 1 setcustomcolor systemdict /currentgray get exec 1 ne }{ pop plateindex 5 ne } ifelse }{ pop plateindex 5 ne } ifelse } ifelse } ifelse } ifelse } def /ci6colormake { dup ci6numcomps exch 1 index 2 add 1 roll dup 1 eq {pop}{array astore} ifelse exch } def /ci6colorexpand { dup ci6spotname exch dup ci6istint { ci6altspace exch 4 1 roll }{ 1 3 1 roll } ifelse } def /ci6colortint { dup /DeviceGray eq { 3 1 roll 1 exch sub mul 1 exch sub exch }{ dup /DeviceRGB eq { 3 1 roll {1 exch sub 1 index mul 1 exch sub exch} forall pop 3 array astore exch }{ dup /DeviceCMYK eq { 3 1 roll {1 index mul exch} forall pop 4 array astore exch }{ 3 1 roll mul exch } ifelse } ifelse } ifelse } def /ci6colortocmyk { dup /DeviceGray eq { pop 1 exch sub 0 0 0 4 -1 roll 4 array astore }{ dup /DeviceRGB eq { pop aload pop _rgbtocmyk 4 array astore }{ dup /DeviceCMYK eq { pop }{ ci6altspace ci6colortint ci6colortocmyk } ifelse } ifelse } ifelse } def /ci6makeimagedict { 7 dict begin /ImageType 1 def /Decode exch def /DataSource exch def /ImageMatrix exch def /BitsPerComponent exch def /Height exch def /Width exch def currentdict end } def /ci6stringinvert { 0 1 2 index length 1 sub { dup 2 index exch get 255 exch sub 2 index 3 1 roll put } for } def /ci6stringknockout { 0 1 2 index length 1 sub { 255 2 index 3 1 roll put } for } def /ci6stringapply { 0 1 4 index length 1 sub { dup 4 index exch get 3 index 3 1 roll 3 index exec } for pop exch pop } def /ci6walkrgbstring { 0 3 index dup length 1 sub 0 3 3 -1 roll { 3 getinterval {} forall 5 index exec 3 index } for 5 {pop} repeat } def /ci6walkcmykstring { 0 3 index dup length 1 sub 0 4 3 -1 roll { 4 getinterval {} forall 6 index exec 3 index } for 5 { pop } repeat } def /ci6putrgbtograystr { .11 mul exch .59 mul add exch .3 mul add cvi 3 copy put pop 1 add } def /ci6putcmyktograystr { exch .11 mul add exch .59 mul add exch .3 mul add dup 255 gt { pop 255 } if 255 exch sub cvi 3 copy put pop 1 add } def /ci6rgbtograyproc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec XIBuffer3 dup 3 1 roll /ci6putrgbtograystr load exch ci6walkrgbstring end } def /ci6cmyktograyproc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec XIBuffer3 dup 3 1 roll /ci6putcmyktograystr load exch ci6walkcmykstring end } def /ci6separatecmykproc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec XIBuffer3 0 2 index plateindex 4 2 index length 1 sub { get 255 exch sub 3 copy put pop 1 add 2 index } for pop pop exch pop end } def /ci6compositeimage { dup 1 eq { pop pop image }{ /ci6colorimage load null ne { ci6colorimage }{ 3 1 roll pop sourcearray 0 3 -1 roll put 3 eq {/ci6rgbtograyproc}{/ci6cmyktograyproc} ifelse load image } ifelse } ifelse } def /ci6knockoutimage { gsave 0 ci6curtransfer exec 1 ci6curtransfer exec eq { 0 ci6curtransfer exec 0.5 lt }{ 0 ci6curtransfer exec 1 ci6curtransfer exec gt } ifelse {{pop 0}}{{pop 1}} ifelse systemdict /settransfer get exec ci6compositeimage grestore } def /ci6drawimage { ci6testplate -1 eq { pop ci6compositeimage }{ dup type /arraytype eq { dup length plateindex gt {plateindex get}{pop false} ifelse }{ { true }{ dup 1 eq {plateindex 3 eq}{plateindex 3 le} ifelse } ifelse } ifelse { dup 1 eq { pop pop ci6image }{ dup 3 eq { ci6compositeimage }{ pop pop sourcearray 0 3 -1 roll put /ci6separatecmykproc load ci6image } ifelse } ifelse }{ ci6curoverprint { 7 {pop} repeat }{ ci6knockoutimage } ifelse } ifelse } ifelse } def /ci6proctintimage { /_ptispace exch store /_ptiname exch store /_pti1 exch store /_pti0 exch store /_ptiproc exch store /_pticomps _ptispace ci6numcomps store /_ptiscale _pti1 _pti0 sub store level2? { _ptiname length 0 gt version cvr 2012 ge and { [/Separation _ptiname _ptispace {_ptiproc}] setcolorspace [_pti0 _pti1] ci6makeimagedict ci6image }{ [/Indexed _ptispace 255 {255 div _ptiscale mul _pti0 add _ptiproc}] setcolorspace [0 255] ci6makeimagedict ci6image } ifelse }{ _pticomps 1 eq { { dup { 255 div _ptiscale mul _pti0 add _ptiproc 255 mul cvi put } ci6stringapply } ci6concatprocs ci6image }{ { dup length _pticomps mul dup _ptibuf length ne {/_ptibuf exch string store}{pop} ifelse _ptibuf { exch _pticomps mul exch 255 div _ptiscale mul _pti0 add _ptiproc _pticomps 2 add -2 roll _pticomps 1 sub -1 0 { 1 index add 2 index exch 5 -1 roll 255 mul cvi put } for pop pop } ci6stringapply } ci6concatprocs false _pticomps /ci6colorimage load null eq {7 {pop} repeat}{ci6colorimage} ifelse } ifelse } ifelse } def /ci6graytintimage { /_gtigray 5 -1 roll store {1 _gtigray sub mul 1 exch sub} 4 1 roll /DeviceGray ci6proctintimage } def /ci6cmyktintimage { /_cticmyk 5 -1 roll store {_cticmyk {1 index mul exch} forall pop} 4 1 roll /DeviceCMYK ci6proctintimage } def /ci6rgbtintimage { /_rtirgb 5 -1 roll store {_rtirgb {1 exch sub 1 index mul 1 exch sub exch} forall pop} 4 1 roll /DeviceRGB ci6proctintimage } def /ci6tintimage { ci6testplate -1 eq { ci6colorexpand 3 -1 roll 5 -1 roll {0}{0 exch} ifelse 4 2 roll dup /DeviceGray eq { pop ci6graytintimage }{ dup /DeviceRGB eq { pop ci6rgbtintimage }{ pop ci6cmyktintimage } ifelse } ifelse }{ dup ci6marksplate { plateindex 5 lt { ci6colortocmyk plateindex get dup 0 eq ci6curoverprint and { 7 {pop} repeat }{ 1 exch sub exch {1 0}{0 1} ifelse () ci6graytintimage } ifelse }{ pop exch {0}{0 exch} ifelse 0 3 1 roll () ci6graytintimage } ifelse }{ ci6curoverprint { 8 {pop} repeat }{ pop pop pop {pop 1} 0 1 () /DeviceGray ci6proctintimage } ifelse } ifelse } ifelse } def /XINullImage { } def /XIImageMask { XIImageWidth XIImageHeight false [XIImageWidth 0 0 XIImageHeight neg 0 0] /XIDataProc load imagemask } def /XIImageTint { XIImageWidth XIImageHeight XIBitsPerPixel [XIImageWidth 0 0 XIImageHeight neg 0 0] /XIDataProc load XIType 3 eq XIColorValues XIColorSpace ci6tintimage } def /XIImage { XIImageWidth XIImageHeight XIBitsPerPixel [XIImageWidth 0 0 XIImageHeight neg 0 0] /XIDataProc load false XIChannelCount XIPlateList ci6drawimage } def /XG { pop pop } def /XF { 13 {pop} repeat } def /Xh { Adobe_ColorImage_AI6_Vars begin gsave /XIType exch def /XIImageHeight exch def /XIImageWidth exch def /XIImageMatrix exch def 0 0 moveto XIImageMatrix concat XIImageWidth XIImageHeight scale /_lp /null ddef _fc /_lp /imagemask ddef end } def /XH { Adobe_ColorImage_AI6_Vars begin grestore end } def /XIEnable { Adobe_ColorImage_AI6_Vars /XIEnable 3 -1 roll put } def /XC { Adobe_ColorImage_AI6_Vars begin ci6colormake /XIColorSpace exch def /XIColorValues exch def end } def /XIPlates { Adobe_ColorImage_AI6_Vars begin /XIPlateList exch def end } def /XI { Adobe_ColorImage_AI6_Vars begin gsave /XIType exch def cvi dup 256 idiv /XICompression exch store 256 mod /XIEncoding exch store pop pop /XIChannelCount exch def /XIBitsPerPixel exch def /XIImageHeight exch def /XIImageWidth exch def pop pop pop pop /XIImageMatrix exch def XIBitsPerPixel 1 eq { XIImageWidth 8 div ceiling cvi }{ XIImageWidth XIChannelCount mul } ifelse /XIRowBytes exch def XIEnable { /XIBuffer3 XIImageWidth string def XICompression 0 eq { /XIBuffer1 XIRowBytes string def XIEncoding 0 eq { {currentfile XIBuffer1 readhexstring pop} }{ {currentfile XIBuffer1 readstring pop} } ifelse }{ /XIBuffer1 256 string def /XIBuffer2 XIRowBytes string def {currentfile XIBuffer1 readline pop (%) anchorsearch {pop} if} /ASCII85Decode filter /DCTDecode filter /XIFile exch def {XIFile XIBuffer2 readstring pop} } ifelse /XIDataProc exch def XIType 1 ne { 0 setgray } if XIType 1 eq { XIImageMask }{ XIType 2 eq XIType 3 eq or { XIImageTint }{ XIImage } ifelse } ifelse }{ XINullImage } ifelse /XIPlateList false def grestore end } def end %%EndProcSet %%BeginResource: procset Adobe_Illustrator_AI5 1.3 0 %%Title: (Adobe Illustrator (R) Version 8.0 Full Prolog) %%Version: 1.3 0 %%CreationDate: (3/7/1994) () %%Copyright: ((C) 1987-1998 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_Illustrator_AI5_vars 112 dict dup begin put /_?cmyk false def /_eo false def /_lp /none def /_pf { } def /_ps { } def /_psf { } def /_pss { } def /_pjsf { } def /_pjss { } def /_pola 0 def /_doClip 0 def /cf currentflat def /_lineorientation 0 def /_charorientation 0 def /_yokoorientation 0 def /_tm matrix def /_renderStart [ /e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0 ] def /_renderEnd [ null null null null /i1 /i1 /i1 /i1 ] def /_render -1 def /_shift [0 0] def /_ax 0 def /_ay 0 def /_cx 0 def /_cy 0 def /_leading [ 0 0 ] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fontSize 0 def /_fontAscent 0 def /_fontDescent 0 def /_fontHeight 0 def /_fontRotateAdjust 0 def /Ss 256 string def Ss 0 (fonts/) putinterval /_cnt 0 def /_scale [1 1] def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def /_hfname 100 string def /_hffound false def /Tx { } def /Tj { } def /CRender { } def /_AI3_savepage { } def /_gf null def /_cf 4 array def /_rgbf 3 array def /_if null def /_of false def /_fc { } def /_gs null def /_cs 4 array def /_rgbs 3 array def /_is null def /_os false def /_sc { } def /_pd 1 dict def /_ed 15 dict def /_pm matrix def /_fm null def /_fd null def /_fdd null def /_sm null def /_sd null def /_sdd null def /_i null def /_lobyte 0 def /_hibyte 0 def /_cproc null def /_cscript 0 def /_hvax 0 def /_hvay 0 def /_hvwb 0 def /_hvcx 0 def /_hvcy 0 def /_bitfont null def /_bitlobyte 0 def /_bithibyte 0 def /_bitkey null def /_bitdata null def /_bitindex 0 def /discardSave null def /buffer 256 string def /beginString null def /endString null def /endStringLength null def /layerCnt 1 def /layerCount 1 def /perCent (%) 0 get def /perCentSeen? false def /newBuff null def /newBuffButFirst null def /newBuffLast null def /clipForward? false def end userdict /Adobe_Illustrator_AI5 known not { userdict /Adobe_Illustrator_AI5 100 dict put } if userdict /Adobe_Illustrator_AI5 get begin /initialize { Adobe_Illustrator_AI5 dup begin Adobe_Illustrator_AI5_vars begin /_aicmykps where {pop /_?cmyk _aicmykps def}if discardDict { bind pop pop } forall dup /nc get begin { dup xcheck 1 index type /operatortype ne and { bind } if pop pop } forall end newpath } def /terminate { end end } def /_ null def /ddef { Adobe_Illustrator_AI5_vars 3 1 roll put } def /xput { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop { { pop } repeat } def /hswj { dup stringwidth 3 2 roll { _hvwb eq { exch _hvcx add exch _hvcy add } if exch _hvax add exch _hvay add } cforall } def /vswj { 0 0 3 -1 roll { dup 255 le _charorientation 1 eq and { dup cstring stringwidth 5 2 roll _hvwb eq { exch _hvcy sub exch _hvcx sub } if exch _hvay sub exch _hvax sub 4 -1 roll sub exch 3 -1 roll sub exch } { _hvwb eq { exch _hvcy sub exch _hvcx sub } if exch _hvay sub exch _hvax sub _fontHeight sub } ifelse } cforall } def /swj { 6 1 roll /_hvay exch ddef /_hvax exch ddef /_hvwb exch ddef /_hvcy exch ddef /_hvcx exch ddef _lineorientation 0 eq { hswj } { vswj } ifelse } def /sw { 0 0 0 6 3 roll swj } def /vjss { 4 1 roll { dup cstring dup length 1 eq _charorientation 1 eq and { -90 rotate currentpoint _fontRotateAdjust add moveto gsave false charpath currentpoint 5 index setmatrix stroke grestore _fontRotateAdjust sub moveto _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto 90 rotate } { currentpoint _fontHeight sub 5 index sub 3 index _sp eq { 9 index sub } if currentpoint exch 4 index stringwidth pop 2 div sub exch _fontAscent sub moveto gsave 2 index false charpath 6 index setmatrix stroke grestore moveto pop pop } ifelse } cforall 6 npop } def /hjss { 4 1 roll { dup cstring gsave false charpath currentpoint 5 index setmatrix stroke grestore moveto _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto } cforall 6 npop } def /jss { _lineorientation 0 eq { hjss } { vjss } ifelse } def /ss { 0 0 0 7 3 roll jss } def /vjsp { 4 1 roll { dup cstring dup length 1 eq _charorientation 1 eq and { -90 rotate currentpoint _fontRotateAdjust add moveto false charpath currentpoint _fontRotateAdjust sub moveto _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto 90 rotate } { currentpoint _fontHeight sub 5 index sub 3 index _sp eq { 9 index sub } if currentpoint exch 4 index stringwidth pop 2 div sub exch _fontAscent sub moveto 2 index false charpath moveto pop pop } ifelse } cforall 6 npop } def /hjsp { 4 1 roll { dup cstring false charpath _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto } cforall 6 npop } def /jsp { matrix currentmatrix _lineorientation 0 eq {hjsp} {vjsp} ifelse } def /sp { matrix currentmatrix 0 0 0 7 3 roll _lineorientation 0 eq {hjsp} {vjsp} ifelse } def /pl { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c { curveto } def /C /c load def /v { currentpoint 6 2 roll curveto } def /V /v load def /y { 2 copy curveto } def /Y /y load def /l { lineto } def /L /l load def /m { moveto } def } { /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def } ifelse /d { setdash } def /cf { } def /i { dup 0 eq { pop cf } if setflat } def /j { setlinejoin } def /J { setlinecap } def /M { setmiterlimit } def /w { setlinewidth } def /XR { 0 ne /_eo exch ddef } def /H { } def /h { closepath } def /N { _pola 0 eq { _doClip 1 eq { _eo {eoclip} {clip} ifelse /_doClip 0 ddef } if newpath } { /CRender { N } ddef } ifelse } def /n { N } def /F { _pola 0 eq { _doClip 1 eq { gsave _pf grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf } ifelse } { /CRender { F } ddef } ifelse } def /f { closepath F } def /S { _pola 0 eq { _doClip 1 eq { gsave _ps grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps } ifelse } { /CRender { S } ddef } ifelse } def /s { closepath S } def /B { _pola 0 eq { _doClip 1 eq gsave F grestore { gsave S grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { S } ifelse } { /CRender { B } ddef } ifelse } def /b { closepath B } def /W { /_doClip 1 ddef } def /* { count 0 ne { dup type /stringtype eq { pop } if } if newpath } def /u { } def /U { } def /q { _pola 0 eq { gsave } if } def /Q { _pola 0 eq { grestore } if } def /*u { _pola 1 add /_pola exch ddef } def /*U { _pola 1 sub /_pola exch ddef _pola 0 eq { CRender } if } def /D { pop } def /*w { } def /*W { } def /` { /_i save ddef clipForward? { nulldevice } if 6 1 roll 4 npop concat pop userdict begin /showpage { } def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash /setstrokeadjust where {pop false setstrokeadjust} if newpath 0 setgray false setoverprint } def /~ { end _i restore } def /_rgbtocmyk { 3 { 1 exch sub 3 1 roll } repeat 3 copy 1 4 1 roll 3 { 3 index 2 copy gt { exch } if pop 4 1 roll } repeat pop pop pop 4 1 roll 3 { 3 index sub 3 1 roll } repeat 4 -1 roll } def /setrgbfill { _rgbf astore pop /_fc { _lp /fill ne { _of setoverprint _rgbf aload pop setrgbcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /setrgbstroke { _rgbs astore pop /_sc { _lp /stroke ne { _os setoverprint _rgbs aload pop setrgbcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /O { 0 ne /_of exch ddef /_lp /none ddef } def /R { 0 ne /_os exch ddef /_lp /none ddef } def /g { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /G { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /K { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /Xa { _?cmyk { 3 npop k }{ setrgbfill 4 npop } ifelse } def /XA { _?cmyk { 3 npop K }{ setrgbstroke 4 npop } ifelse } def /Xs { /_gf exch ddef 5 npop /_fc { _lp /fill ne { _of setoverprint _gf setAIseparationgray /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /XS { /_gs exch ddef 5 npop /_sc { _lp /stroke ne { _os setoverprint _gs setAIseparationgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /Xx { exch /_gf exch ddef 0 eq { findcmykcustomcolor }{ _?cmyk {true}{/findrgbcustomcolor where{pop false}{true}ifelse}ifelse { 4 1 roll 3 npop findcmykcustomcolor }{ 8 -4 roll 4 npop findrgbcustomcolor } ifelse } ifelse /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /XX { exch /_gs exch ddef 0 eq { findcmykcustomcolor }{ _?cmyk {true}{/findrgbcustomcolor where{pop false}{true}ifelse}ifelse { 4 1 roll 3 npop findcmykcustomcolor }{ 8 -4 roll 4 npop findrgbcustomcolor } ifelse } ifelse /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /X { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /XK { 3 -1 roll pop 0 eq { 1 exch sub 3 {dup 3 1 roll mul 5 1 roll} repeat mul 4 1 roll K } { 1 exch sub 4 1 roll 3 {1 exch sub 3 index mul 1 exch sub 3 1 roll} repeat 4 -1 roll pop XA } ifelse } def /Xk { 3 -1 roll pop 0 eq { 1 exch sub 3 {dup 3 1 roll mul 5 1 roll} repeat mul 4 1 roll k } { 1 exch sub 4 1 roll 3 {1 exch sub 3 index mul 1 exch sub 3 1 roll} repeat 4 -1 roll pop Xa } ifelse } def /A { pop } def /annotatepage { userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse } def /XT { pop pop } def /Xt { pop } def /discard { save /discardSave exch store discardDict begin /endString exch store gt38? { 2 add } if load stopped pop end discardSave restore } bind def userdict /discardDict 7 dict dup begin put /pre38Initialize { /endStringLength endString length store /newBuff buffer 0 endStringLength getinterval store /newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store /newBuffLast newBuff endStringLength 1 sub 1 getinterval store } def /shiftBuffer { newBuff 0 newBuffButFirst putinterval newBuffLast 0 currentfile read not { stop } if put } def 0 { pre38Initialize mark currentfile newBuff readstring exch pop { { newBuff endString eq { cleartomark stop } if shiftBuffer } loop } { stop } ifelse } def 1 { pre38Initialize /beginString exch store mark currentfile newBuff readstring exch pop { { newBuff beginString eq { /layerCount dup load 1 add store } { newBuff endString eq { /layerCount dup load 1 sub store layerCount 0 eq { cleartomark stop } if } if } ifelse shiftBuffer } loop } if } def 2 { mark { currentfile buffer {readline} stopped { % assume error was due to overfilling the buffer }{ not { stop } if endString eq { cleartomark stop } if }ifelse } loop } def 3 { /beginString exch store /layerCnt 1 store mark { currentfile buffer {readline} stopped { % assume error was due to overfilling the buffer }{ not { stop } if dup beginString eq { pop /layerCnt dup load 1 add store } { endString eq { layerCnt 1 eq { cleartomark stop } { /layerCnt dup load 1 sub store } ifelse } if } ifelse }ifelse } loop } def end userdict /clipRenderOff 15 dict dup begin put { /n /N /s /S /f /F /b /B } { { _doClip 1 eq { /_doClip 0 ddef _eo {eoclip} {clip} ifelse } if newpath } def } forall /Tr /pop load def /Bb {} def /BB /pop load def /Bg {12 npop} def /Bm {6 npop} def /Bc /Bm load def /Bh {4 npop} def end /Lb { 6 npop 7 2 roll 5 npop 0 eq { 0 eq { (%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard } { /clipForward? true def /Tx /pop load def /Tj /pop load def currentdict end clipRenderOff begin begin } ifelse } { 0 eq { save /discardSave exch store } if } ifelse } bind def /LB { discardSave dup null ne { restore } { pop clipForward? { currentdict end end begin /clipForward? false ddef } if } ifelse } bind def /Pb { pop pop 0 (%AI5_EndPalette) discard } bind def /Np { 0 (%AI5_End_NonPrinting--) discard } bind def /Ln /pop load def /Ap /pop load def /Ar { 72 exch div 0 dtransform dup mul exch dup mul add sqrt dup 1 lt { pop 1 } if setflat } def /Mb { q } def /Md { } def /MB { Q } def /nc 4 dict def nc begin /setgray { pop } bind def /setcmykcolor { 4 npop } bind def /setrgbcolor { 3 npop } bind def /setcustomcolor { 2 npop } bind def currentdict readonly pop end /XP { 4 npop } bind def /XD { pop } bind def end setpacking %%EndResource %%BeginResource: procset Adobe_cshow 2.0 8 %%Title: (Writing System Operators) %%Version: 2.0 8 %%CreationDate: (1/23/89) () %%Copyright: ((C) 1992-1996 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_cshow 14 dict dup begin put /initialize { Adobe_cshow begin Adobe_cshow { dup xcheck { bind } if pop pop } forall end Adobe_cshow begin } def /terminate { currentdict Adobe_cshow eq { end } if } def /cforall { /_lobyte 0 ddef /_hibyte 0 ddef /_cproc exch ddef /_cscript currentfont /FontScript known { currentfont /FontScript get } { -1 } ifelse ddef { /_lobyte exch ddef _hibyte 0 eq _cscript 1 eq _lobyte 129 ge _lobyte 159 le and _lobyte 224 ge _lobyte 252 le and or and _cscript 2 eq _lobyte 161 ge _lobyte 254 le and and _cscript 3 eq _lobyte 161 ge _lobyte 254 le and and _cscript 25 eq _lobyte 161 ge _lobyte 254 le and and _cscript -1 eq or or or or and { /_hibyte _lobyte ddef } { _hibyte 256 mul _lobyte add _cproc /_hibyte 0 ddef } ifelse } forall } def /cstring { dup 256 lt { (s) dup 0 4 3 roll put } { dup 256 idiv exch 256 mod (hl) dup dup 0 6 5 roll put 1 4 3 roll put } ifelse } def /clength { 0 exch { 256 lt { 1 } { 2 } ifelse add } cforall } def /hawidthshow { { dup cstring show _hvax _hvay rmoveto _hvwb eq { _hvcx _hvcy rmoveto } if } cforall } def /vawidthshow { { dup 255 le _charorientation 1 eq and { -90 rotate 0 _fontRotateAdjust rmoveto cstring _hvcx _hvcy _hvwb _hvax _hvay 6 -1 roll awidthshow 0 _fontRotateAdjust neg rmoveto 90 rotate } { currentpoint _fontHeight sub exch _hvay sub exch _hvax sub 2 index _hvwb eq { exch _hvcy sub exch _hvcx sub } if 3 2 roll cstring dup stringwidth pop 2 div neg _fontAscent neg rmoveto show moveto } ifelse } cforall } def /hvawidthshow { 6 1 roll /_hvay exch ddef /_hvax exch ddef /_hvwb exch ddef /_hvcy exch ddef /_hvcx exch ddef _lineorientation 0 eq { hawidthshow } { vawidthshow } ifelse } def /hvwidthshow { 0 0 3 -1 roll hvawidthshow } def /hvashow { 0 0 0 6 -3 roll hvawidthshow } def /hvshow { 0 0 0 0 0 6 -1 roll hvawidthshow } def currentdict readonly pop end setpacking %%EndResource %%BeginResource: procset Adobe_shading_AI8 1.0 0 %%Title: (Adobe Illustrator 8 Shading Procset) %%Version: 1.0 0 %%CreationDate: (12/17/97) () %%Copyright: ((C) 1987-1997 Adobe Systems Incorporated All Rights Reserved) userdict /defaultpacking currentpacking put true setpacking userdict /Adobe_shading_AI8 10 dict dup begin put /initialize { Adobe_shading_AI8 begin Adobe_shading_AI8 bdprocs Mesh /initialize get exec } def /terminate { currentdict Adobe_shading_AI8 eq { end } if } def /bdprocs { { dup xcheck 1 index type /arraytype eq and { bind } if pop pop } forall } def /X! {pop} def /X# {pop pop} def /Mesh 40 dict def Mesh begin /initialize { Mesh bdprocs Mesh begin /emulate? /AI8MeshEmulation where { pop AI8MeshEmulation }{ systemdict /shfill known not } ifelse def end } def /bd { shadingdict begin } def /paint { emulate? { end }{ /_lp /none ddef _fc /_lp /none ddef /AIColorSpace AIColorSpace tocolorspace store /ColorSpace AIColorSpace topsspace store version_ge_3010.106 not systemdict /setsmoothness known and { 0.0001 setsmoothness } if composite? { /DataSource getdatasrc def Matrix concat currentdict end shfill }{ AIColorSpace makesmarks AIPlateList markingplate and not isoverprint and { end }{ /ColorSpace /DeviceGray store /Decode [0 1 0 1 0 1] store /DataSource getplatesrc def Matrix concat currentdict end shfill } ifelse } ifelse } ifelse } def /shadingdict 12 dict def shadingdict begin /ShadingType 6 def /BitsPerCoordinate 16 def /BitsPerComponent 8 def /BitsPerFlag 8 def end /datafile null def /databuf 256 string def /dataptr 0 def /srcspace null def /srcchannels 0 def /dstchannels 0 def /dstplate 0 def /srctodstcolor null def /getplatesrc { /srcspace AIColorSpace store /srcchannels AIColorSpace getnchannels store /dstchannels 1 store /dstplate getplateindex store /srctodstcolor srcspace makesmarks { dstplate 4 eq { {1 exch sub} }{ {srcspace tocmyk 3 dstplate sub index 1 exch sub 5 1 roll 4 {pop} repeat} } ifelse }{ {srcchannels {pop} repeat 1} } ifelse store /datafile getdatasrc store /rdpatch168 load DataLength () /SubFileDecode filter } def /getdatasrc { /rdcmntline load /ASCII85Decode filter } def /rdpatch168 { /dataptr 0 store 49 rdcount 4 { dup {pop srcchannels getint8} if dup {pop srctodstcolor dstchannels putint8 true} if } repeat {databuf 0 dataptr getinterval}{()} ifelse } def /rdpatch3216 { /dataptr 0 store 97 rdcount 4 { dup {pop srcchannels getint16} if dup {pop srctodstcolor dstchannels putint16 true} if } repeat {databuf 0 dataptr getinterval}{()} ifelse } def /rdcount { dup 0 gt { datafile databuf dataptr 4 -1 roll getinterval readstring exch length dataptr add /dataptr exch store }{ true } ifelse } def /getint8 { mark true 3 -1 roll { dup {pop datafile read} if dup {pop 255 div true} if } repeat { counttomark 1 add -1 roll pop true }{ cleartomark false } ifelse } def /putint8 { dup dataptr add /dataptr exch store dataptr exch { 1 sub exch 255 mul cvi databuf 2 index 3 -1 roll put } repeat pop } def /getint16 { mark true 3 -1 roll { dup {pop datafile read} if dup {pop 256 mul datafile read} if dup {pop add 65535 div true} if } repeat { counttomark 1 add -1 roll pop true }{ cleartomark false } ifelse } def /putint16 { dup 2 mul dataptr add /dataptr exch store dataptr exch { 2 sub exch 65535 mul cvi dup 256 idiv databuf 3 index 3 -1 roll put 256 mod databuf 2 index 1 add 3 -1 roll put } repeat pop } def /srcbuf 256 string def /rdcmntline { currentfile srcbuf readline pop (%) anchorsearch {pop} if } def /getplateindex { 0 [cyan? magenta? yellow? black? customColor?] {{exit} if 1 add} forall } def /aicsarray 4 array def /aicsaltvals 4 array def /aicsaltcolr aicsaltvals def /tocolorspace { dup type /arraytype eq { mark exch aload pop aicsarray 0 3 -1 roll put aicsarray 1 3 -1 roll put dup aicsarray 2 3 -1 roll put gettintxform aicsarray 3 3 -1 roll put counttomark aicsaltvals 0 3 -1 roll getinterval /aicsaltcolr exch store aicsaltcolr astore pop pop aicsarray } if } def /subtintxform {aicsaltcolr {1 index mul exch} forall pop} def /addtintxform {aicsaltcolr {1 sub 1 index mul 1 add exch} forall pop} def /gettintxform { /DeviceRGB eq {/addtintxform}{/subtintxform} ifelse load } def /getnchannels { dup type /arraytype eq {0 get} if colorspacedict exch get begin Channels end } def /makesmarks { composite? { pop true }{ dup dup type /arraytype eq {0 get} if colorspacedict exch get begin MarksPlate end } ifelse } def /markingplate { composite? { pop true }{ dup type /arraytype eq { dup length getplateindex gt {getplateindex get}{pop false} ifelse } if } ifelse } def /tocmyk { dup dup type /arraytype eq {0 get} if colorspacedict exch get begin ToCMYK end } def /topsspace { dup dup type /arraytype eq {0 get} if colorspacedict exch get begin ToPSSpace end } def /colorspacedict 5 dict dup begin /DeviceGray 4 dict dup begin /Channels 1 def /MarksPlate {pop black?} def /ToCMYK {pop 1 exch sub 0 0 0 4 -1 roll} def /ToPSSpace {} def end def /DeviceRGB 4 dict dup begin /Channels 3 def /MarksPlate {pop isCMYKSep?} def /ToCMYK {pop _rgbtocmyk} def /ToPSSpace {} def end def /DeviceCMYK 4 dict dup begin /Channels 4 def /MarksPlate {pop isCMYKSep?} def /ToCMYK {pop} def /ToPSSpace {} def end def /Separation 4 dict dup begin /Channels 1 def /MarksPlate { /findcmykcustomcolor where { pop dup 1 exch ToCMYK 5 -1 roll 1 get findcmykcustomcolor 1 setcustomcolor systemdict /currentgray get exec 1 ne }{ pop false } ifelse } def /ToCMYK { dup 2 get mark exch 4 2 roll 3 get exec counttomark -1 roll tocmyk 5 -1 roll pop } def /ToPSSpace {} def end def /Process 4 dict dup begin /Channels 1 def /MarksPlate { isCMYKSep? { 1 exch ToCMYK 4 array astore getplateindex get 0 ne }{ pop false } ifelse } def /ToCMYK { dup 2 get mark exch 4 2 roll 3 get exec counttomark -1 roll tocmyk 5 -1 roll pop } def /ToPSSpace { 4 array copy dup 0 /Separation put } def end def end def /isoverprint { /currentoverprint where {pop currentoverprint}{_of} ifelse } def /version_ge_3010.106 { version {cvr} stopped { pop false }{ 3010.106 ge } ifelse } def end end defaultpacking setpacking %%EndResource %%EndProlog %%BeginSetup %%IncludeFont: Symbol userdict /_useSmoothShade false put userdict /_aicmykps true put userdict /_forceToCMYK true put Adobe_level2_AI5 /initialize get exec Adobe_cshow /initialize get exec Adobe_Illustrator_AI5_vars Adobe_Illustrator_AI5 Adobe_typography_AI5 /initialize get exec Adobe_ColorImage_AI6 /initialize get exec Adobe_shading_AI8 /initialize get exec Adobe_Illustrator_AI5 /initialize get exec [ 39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis /Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute /egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde 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/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /Upsilon1 /minute /lessequal /fraction /infinity /florin /club /diamond /heart /spade /arrowboth /arrowleft /arrowup /arrowright /arrowdown /degree /plusminus /second /greaterequal /multiply /proportional /partialdiff /bullet /divide /notequal /equivalence /approxequal /ellipsis /arrowvertex /arrowhorizex /carriagereturn /aleph /Ifraktur /Rfraktur /weierstrass /circlemultiply /circleplus /emptyset /intersection /union /propersuperset /reflexsuperset /notsubset /propersubset /reflexsubset /element /notelement /angle /gradient /registerserif /copyrightserif /trademarkserif /product /radical /dotmath /logicalnot /logicaland /logicalor /arrowdblboth /arrowdblleft /arrowdblup /arrowdblright 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b(';)g(s)p Fs(\))f(+)g Fn(\001)h(\001)g(\001)h Fs(+)e Fm(")2571 1935 y Fp(m)2637 1968 y Fn(J)2699 1982 y Fp(m)2765 1968 y Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\),)456 2083 y(where)30 b Fn(J)781 2097 y Fp(i)839 2083 y Fs(are)h(trigonometric)h(p)s(olynomials)f(in)f(\()p Fm(';)15 b(s)p Fs(\),)32 b Fn(N)13 b Fs(\()p Fn(J)2623 2097 y Fq(2)2662 2083 y Fs(\))26 b Fn(\032)f(N)13 b Fs(\()2967 2060 y(~)2942 2083 y Fn(F)3007 2097 y Fq(1)3048 2083 y Fs(\))20 b(+)456 2198 y Fn(N)13 b Fs(\()604 2175 y(~)579 2198 y Fn(F)644 2212 y Fq(1)684 2198 y Fs(\))25 b Fn(\032)g(N)13 b Fs(\()p Fm(h)1015 2212 y Fq(1)1055 2198 y Fs(\))21 b(+)f Fn(N)13 b Fs(\()p Fm(h)1377 2212 y Fq(1)1417 2198 y Fs(\).)1626 b Fj(\003)456 2400 y Fw(Remark)41 b(29.)k Fs(Notice)37 b(that)f(the)g(explicit)h(form)m(ula)e(\(44\))i(requires)f (that)g Fm(I)7 b Fs(,)456 2508 y Fm(')30 b Fs(are)h(one-dimensional.) 555 2616 y(Nev)m(ertheless,)d(w)m(e)e(p)s(oin)m(t)f(out)g(that)h(a)f (similar)h(result|with)e(a)i(higher)f(loss)456 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Fs(where)42 b(w)m(e)i(ha)m(v)m(e) g(denoted)f(b)m(y)g Fn(I)7 b Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))45 b(the)e(in)m(v)m(erse)h(of)f(the)g(function)456 4748 y Fm(I)e Fn(7!)34 b(J)16 b Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))38 b(giv)m(en)f(in)f(\(44\))q(,)h(whic)m (h)f(is)g(a)g Fn(C)2234 4715 y Fp(r)r Fl(\000)p Fq(2)2398 4748 y Fs(function.)56 b(Since)36 b(the)456 4856 y(function)24 b Fn(J)40 b Fs(has)24 b(an)h(expansion)f(in)g Fm(")h Fs(giv)m(en)g(in)f(Prop)s(osition)g(28)i(the)e(function)456 4964 y Fn(I)36 b Fs(has)30 b(an)h(expansion)f(with)g(similar)h(prop)s (erties.)p eop end %%Page: 38 38 TeXDict begin 38 37 bop 456 251 a Fq(38)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)555 450 y Fs(Therefore,)k(if) g(w)m(e)g(expand)e(in)h Fm(")h Fs(the)g(expression)f(for)g Fm(k)s Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))29 b(giv)m(en)e(in)456 558 y(\(46\))q(,)k(w)m(e)g(obtain)f(the)h(results)f 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Fq(26)1261 3469 y Fs(\))42 b(and)f(to)h(exclude)g(from)g(the)f (resonances)i(a)f(size)g Fm(L)456 3576 y Fs(whic)m(h)30 b(is)g(indep)s(enden)m(t)f(of)i(the)f(resonances)h(and)f(of)g(the)h (parameter)g Fm(")p Fs(.)555 3684 y(Cho)s(osing)d(the)f(order)h(in)f Fm(")h Fs(to)h(b)s(e)e(as)h(high)f(is)h(done)f(to)i(simplify)e(the)h (exp)s(o-)456 3792 y(sition)34 b(later.)51 b(As)33 b(w)m(e)h(will)g (see)g(the)g(only)f(resonances)h(that)g(pro)s(duce)e(e\013ects)456 3900 y(that)e(a\013ect)i(subsequen)m(t)e(argumen)m(ts)g(are)h(those)g (of)f(order)g(1)g(and)g(2)g(\(w)m(e)i(will)456 4008 y(sho)m(w)38 b(that)h(resonances)f(of)h(order)e Fm(j)44 b Fs(pro)s(duce)37 b(gaps)i(among)f(KAM)h(tori)g(of)456 4117 y(size)32 b Fm(")671 4084 y Fp(j)t(=)p Fq(2)811 4117 y Fs(and)f(there)h(is)g(a)h (standard)e(metho)s(d)g(to)i(transv)m(erse)f(those)h(gaps)f(of)456 4225 y(order)26 b(smaller)i(than)e Fm(")p Fs(\).)41 b(Hence,)29 b(an)m(y)e(order)f(of)i(a)m(v)m(eraging)h(greater)f(or)f(equal)456 4333 y(than)g(2)h(w)m(ould)f(ha)m(v)m(e)i(b)s(een)e(enough.)39 b(Nev)m(ertheless,)30 b(w)m(orking)e(harder)e(at)j(the)456 4441 y(a)m(v)m(eraging)41 b(step)d(will)h(mak)m(e)h(other)f(argumen)m (ts)g(in)f(Section)i(8.5.2)g(simpler.)456 4549 y(W)-8 b(e)38 b(decided)f(that)h(this)g(future)e(simpli\014cation)i(w)m(as)g (w)m(orth)f(that)h(the)f(v)m(ery)456 4657 y(sligh)m(t)31 b(complication)h(in)e(the)h(presen)m(t)f(c)m(hapter.)968 b Fj(\003)456 4856 y Fw(Remark)48 b(33.)g Fs(Cho)s(osing)41 b(the)h(width)e(of)i(the)g(resonan)m(t)g(regions)f(to)i(b)s(e)d(a)456 4964 y(\014xed)h(n)m(um)m(b)s(er)f Fm(L)i Fs(is)f(rather)h(w)m (asteful.)76 b(It)41 b(is)h(w)m(ell)h(kno)m(wn)e(that)i(one)f(can)p eop end %%Page: 39 39 TeXDict begin 39 38 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(39)456 450 y Fs(c)m(ho)s(ose)28 b(the)g(width)g(to)g(b)s(e)f(a)i(suitable)f(p)s(o)m(w)m(er)g(of)g Fm(")g Fs(m)m(ultiplied)g(b)m(y)g(the)g(size)h(of)456 558 y(the)h(F)-8 b(ourier)31 b(co)s(e\016cien)m(ts,)h(whic)m(h)f (decrease)g(with)f Fn(j)p Fm(k)s Fn(j)q Fs(.)555 666 y(F)-8 b(or)36 b(the)g(purp)s(oses)d(of)i(this)g(pap)s(er,)h(c)m(ho)s (osing)g(that)g(width)e(to)i(b)s(e)f(a)g(con-)456 774 y(stan)m(t)e(is)f(enough)g(and)f(simpli\014es)h(the)g(exp)s(osition.)46 b(A)32 b(more)h(precise)f(c)m(hoice)456 882 y(w)m(ould)41 b(b)s(e)f(needed)h(to)h(eliminate)h(the)e(h)m(yp)s(othesis)g Fw(H3)g Fs(from)g(Theorem)f(7)456 990 y(that)34 b(requires)g(the)g(p)s (erturbation)f(to)h(b)s(e)g(a)g(trigonometric)i(p)s(olynomial.)52 b(A)456 1098 y(heuristic)30 b(discussion)g(of)g(these)h(c)m(hoices)h (is)f(included)e(in)h(Section)i(12.4.)145 b Fj(\003)456 1333 y Fs(8.3.1.)47 b Fo(The)31 b(in\014nitesimal)g(e)-5 b(quations)31 b(for)g(aver)-5 b(aging.)46 b Fs(No)m(w)29 b(w)m(e)f(turn)f(to)i(the)456 1441 y(implemen)m(tation)37 b(of)f(the)g(ab)s(o)m(v)m(e)g(strategy)-8 b(.)59 b(W)-8 b(e)37 b(\014rst)d(start)j(discussing)e(the)456 1548 y(\014rst)28 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Fm(g)44 b Fs(=)c Fm(K)2720 2210 y Fq(0)2786 2196 y Fs(+)26 b Fm(")p Fs(\()p Fm(K)3037 2210 y Fq(1)3103 2196 y Fs(+)456 2304 y Fn(f)p Fm(K)578 2318 y Fq(0)618 2304 y Fm(;)15 b(G)p Fn(g)p Fs(\))26 b(+)931 2312 y(O)1002 2304 y(\()p Fm(")1079 2271 y Fq(2)1119 2304 y Fs(\).)63 b(Hence)39 b(it)f(is)g(natural)f(to)i(consider)e (equations)i(of)f(the)456 2412 y(form:)456 2688 y(\(48\))827 b Fm(K)26 b Fs(+)20 b Fn(f)p Fm(K)1759 2702 y Fq(0)1799 2688 y Fm(;)15 b(G)p Fn(g)27 b Fs(=)2102 2665 y(\026)2078 2688 y Fm(K)7 b(;)456 2870 y Fs(where)33 b Fm(K)7 b Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(\))35 b(is)f(a)g(giv)m(en)h(Hamiltonian,)i Fm(K)2187 2884 y Fq(0)2257 2870 y Fs(=)31 b Fm(A)23 b Fs(+)2553 2835 y Fp(J)2598 2811 y Fi(2)p 2553 2850 80 4 v 2575 2902 a Fq(2)2642 2870 y Fs(,)35 b(and)f(the)g(un-)456 2978 y(kno)m(wns)28 b(are)h Fm(G)p Fs(,)h(the)f(generator)h(of)f(the)g (in\014nitesimal)h(transformation,)f(and)480 3063 y(\026)456 3086 y Fm(K)6 b Fs(,)37 b(the)e(a)m(v)m(eraged)i(Hamiltonian,)g(whic)m 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Fs(=)d(0,)i(\(49\))f(happ)s(ens)d(when)456 4533 y Fm(J)34 b Fs(=)25 b Fn(\000)p Fm(l)r(=k)s Fs(.)555 4640 y(W)-8 b(e)40 b(will)g(refer)e(to)i(resonances)f(as)g(the)h (places)f Fm(J)49 b Fs(=)39 b Fn(\000)p Fm(l)r(=k)s Fs(,)j(\()p Fm(k)s(;)15 b(l)r Fs(\))41 b Fn(2)e(N)13 b Fs(,)456 4748 y Fm(k)28 b Fn(6)p Fs(=)d(0.)555 4856 y(Notice)36 b(that)e(when)e Fm(J)42 b Fs(is)34 b(not)f(resonan)m(t,)i(w)m(e)f(can)g(reduce)f(the)h (system)f(to)456 4964 y(con)m(tain)e(only)1000 4941 y(\026)976 4964 y Fm(K)1053 4978 y Fq(0)p Fp(;)p Fq(0)1147 4964 y Fs(\()p Fm(J)9 b Fs(\),)32 b(whic)m(h)e(is)g(an)h(in)m(tegrable)h (system.)p eop end %%Page: 40 40 TeXDict begin 40 39 bop 456 251 a Fq(40)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)555 450 y Fs(T)-8 b(o)34 b(preserv)m(e)g(smo)s(othness,)g(w)m(e)f(will)h(ha)m(v)m(e)h (also)f(to)g(k)m(eep)g(a)g(go)s(o)s(d)f(part)h(of)456 558 y(these)d(terms)g(for)g(neigh)m(b)s(oring)g(v)-5 b(alues)31 b(of)g(the)h(action.)44 b(The)30 b(tap)s(ering)h(o\013)h(of) 456 666 y(the)e(elimination)i(in)m(v)m(olv)m(es)g(somewhat)f(arbitrary) f(c)m(hoices.)555 774 y(In)e(the)h(follo)m(wing)h(Lemma)f(34)g(w)m(e)g (in)m(tro)s(duce)f(a)h(sp)s(eci\014c)g(c)m(hoice)h(and)e(pro-)456 882 y(vide)i(estimates)i(for)e(it.)456 1044 y Fw(Lemma)k(34.)42 b Fo(L)-5 b(et)1123 1190 y Fm(K)7 b Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(\))27 b(=)1690 1104 y Fh(X)1631 1305 y Fq(\()p Fp(k)r(;l)q Fq(\))p Fl(2N)1896 1190 y Fm(K)1973 1205 y Fp(k)r(;l)2057 1190 y Fs(\()p Fm(J)9 b Fs(\))p Fm(e)2228 1153 y Fp(i)p Fq(\()p Fp(k)r(')p Fq(+)p Fp(l)q(s)p Fq(\))456 1453 y Fo(b)-5 b(e)29 b(a)h(Hamiltonian,)h(with)f Fn(N)39 b Fs(=)25 b Fn(N)13 b Fs(\()p Fm(K)7 b Fs(\))25 b Fn(\032)g Fk(Z)2018 1420 y Fq(2)2087 1453 y Fo(a)k(\014nite)h(set.)40 b(Assume)30 b(that)g Fm(K)456 1561 y Fo(is)i(of)h(class)g Fn(C)931 1528 y Fp(n)p Fq(+1)1101 1561 y Fo(with)h(r)-5 b(esp)g(e)g(ct)34 b(to)f Fm(J)9 b Fo(.)555 1669 y(Cho)-5 b(ose)51 b Fm(L)d Fo(to)i(b)-5 b(e)49 b(a)g(numb)-5 b(er)49 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Fp(l)2868 2729 y Fi(0)2903 2720 y Fp(s)p Fq(\))1132 2988 y Fs(=:)83 b Fm(K)1388 3002 y Fq(0)p Fp(;)p Fq(0)1482 2988 y Fs(\()p Fm(J)9 b Fs(\))21 b(+)f Fm(U)1785 3003 y Fp(k)1822 3012 y Fi(0)1857 3003 y Fp(;l)1898 3012 y Fi(0)1936 2988 y Fs(\()p Fm(k)2018 3002 y Fq(0)2058 2988 y Fm(')h Fs(+)f Fm(l)2256 3002 y Fq(0)2295 2988 y Fm(s)p Fs(\))p Fm(;)758 3134 y Fo(wher)-5 b(e)34 b Fs(0)26 b Fm(<)f(N)35 b(<)25 b Fn(1)32 b Fo(is)h(such)g(that)h Fs(\()p Fm(tk)2115 3148 y Fq(0)2154 3134 y Fm(;)15 b(tl)2254 3148 y Fq(0)2294 3134 y Fs(\))26 b Fn(2)f(N)13 b Fo(,)32 b(for)h Fn(j)q Fm(t)p Fn(j)25 b(\024)g Fm(N)10 b Fo(.)601 3246 y Fs(\(3\))42 b Fo(The)29 b(function)1318 3223 y Fs(\026)1293 3246 y Fm(K)35 b Fo(veri\014es:)1753 3169 y Fh(\014)1753 3224 y(\014)1807 3223 y Fs(\026)1783 3246 y Fm(K)1867 3169 y Fh(\014)1867 3224 y(\014)1897 3282 y Fl(C)1938 3263 y Ff(n)p Fi(+1)2087 3246 y Fn(\024)25 b Fs(\(1)11 b(+)2422 3210 y Fp(C)p 2366 3225 168 4 v 2366 3280 a(L)2414 3261 y Ff(n)p Fi(+1)2543 3246 y 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Fs(\))32 b(as:)789 1953 y(\(a\))42 b(If)30 b Fm(J)35 b Fn(6)p Fs(=)25 b Fn(\000)1309 1892 y Fm(l)p 1299 1932 51 4 v 1299 2016 a(k)1389 1953 y Fs(then)30 b Fm(G)1667 1968 y Fp(k)r(;l)1751 1953 y Fs(\()p Fm(J)9 b Fs(\))27 b(=)2037 1867 y(\026)2012 1890 y Fm(K)2089 1905 y Fp(k)r(;l)2174 1890 y Fs(\()p Fm(J)9 b Fs(\))21 b Fn(\000)f Fm(K)2492 1905 y Fp(k)r(;l)2576 1890 y Fs(\()p Fm(J)9 b Fs(\))p 2012 1932 694 4 v 2183 2016 a Fm(i)p Fs(\()p Fm(J)g(k)25 b Fs(+)20 b Fm(l)r Fs(\))2716 1953 y(,)784 2193 y(\(b\))41 b Fm(G)1017 2208 y Fp(k)r(;l)1102 2193 y Fs(\()p Fn(\000)p Fm(l)r(=k)s Fs(\))26 b(=)95 b(lim)1489 2257 y Fp(J)6 b Fl(!\000)p Fp(l)q(=k)1804 2106 y Fs(\026)1780 2129 y Fm(K)1857 2144 y Fp(k)r(;l)1941 2129 y Fs(\()p Fm(J)j Fs(\))22 b Fn(\000)d Fm(K)2259 2144 y Fp(k)r(;l)2344 2129 y Fs(\()p Fm(J)9 b Fs(\))p 1780 2172 V 1951 2255 a Fm(i)p Fs(\()p Fm(J)g(k)24 b Fs(+)c Fm(l)r Fs(\))2509 2193 y(=)2615 2116 y Fn(\000)p Fm(K)2770 2083 y Fl(0)2763 2144 y Fp(k)r(;l)2847 2116 y Fs(\()p Fn(\000)p 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Fs(\))2757 3165 y Fp(n)2815 3129 y Fm(:)601 3291 y Fs(\(4\))42 b(If)30 b Fm(L)25 b Fn(\024)g(j)q Fm(J)k Fs(+)20 b Fm(l)1255 3305 y Fq(0)1294 3291 y Fm(=k)1386 3305 y Fq(0)1427 3291 y Fn(j)25 b(\024)g Fs(2)p Fm(L)31 b Fs(then)795 3500 y Fn(j)p Fm(G)891 3515 y Fp(k)928 3524 y Fi(0)963 3515 y Fp(;l)1004 3524 y Fi(0)1042 3500 y Fs(\()p Fm(J)9 b Fs(\))p Fn(j)1198 3532 y Fl(C)1239 3513 y Ff(n)1310 3500 y Fn(\024)25 b Fm(C)1488 3432 y Fn(j)p Fm(K)1590 3447 y Fp(k)1627 3456 y Fi(0)1662 3447 y Fp(;l)1703 3456 y Fi(0)1741 3432 y Fs(\()p Fn(\000)p Fm(l)1874 3446 y Fq(0)1914 3432 y Fm(=k)2006 3446 y Fq(0)2046 3432 y Fs(\))p Fn(j)16 b(j)p Fm( )s Fn(j)2235 3459 y Fl(C)2276 3440 y Ff(n)2342 3432 y Fs(+)k Fn(j)q Fm(K)2536 3447 y Fp(k)2573 3456 y Fi(0)2607 3447 y Fp(;l)2648 3456 y Fi(0)2686 3432 y Fn(j)2712 3464 y Fl(C)2753 3445 y Ff(n)p 1488 3479 1312 4 v 2013 3562 a Fn(j)p Fm(k)2085 3576 y Fq(0)2125 3562 y Fn(j)15 b Fm(L)2227 3536 y Fp(n)2809 3500 y Fm(:)456 3704 y Fs(Then,)30 b Fm(G)p Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(\))34 b(is)d(a)h(trigonometric)h(p)s(olynomial)f(in)f(\() p Fm(';)15 b(s)p Fs(\),)33 b(and)e(of)g(class)456 3812 y Fn(C)509 3779 y Fp(n)586 3812 y Fs(with)f(resp)s(ect)g(to)i Fm(J)9 b Fs(.)1803 b Fj(\003)456 4025 y Fs(8.3.2.)47 b Fo(The)40 b(main)f(aver)-5 b(aging)40 b(r)-5 b(esult,)42 b(The)-5 b(or)g(em)41 b(35.)46 b Fs(Once)37 b(w)m(e)h(kno)m(w)g(ho)m(w) 456 4133 y(to)d(solv)m(e)h(an)m(y)f(homological)i(equation)f(\(48\))q (,)g(w)m(e)g(can)f(pro)s(ceed)f(to)h(obtain)g(a)456 4241 y(suitable)30 b(global)h(normal)f(form)f(of)h(our)f(reduced)g (Hamiltonian)i(b)m(y)f(applying)456 4349 y(rep)s(eatedly)43 b(the)g(pro)s(cedure.)77 b(The)42 b(precise)h(result)g(is)g(form)m (ulated)g(in)g(the)456 4457 y(follo)m(wing)31 b(Theorem)g(35.)456 4621 y Fw(Theorem)51 b(35.)f Fo(L)-5 b(et)46 b Fm(k)s Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))49 b Fo(b)-5 b(e)46 b(a)g Fn(C)2021 4589 y Fp(n)2114 4621 y Fo(Hamiltonian,)51 b Fm(n)d(>)h Fs(1)p Fo(,)h(and)456 4729 y(c)-5 b(onsider)34 b(any)f Fs(1)25 b Fn(\024)g Fm(m)h(<)e(n)p Fo(,)32 b(indep)-5 b(endent)35 b(of)d Fm(")p Fo(.)43 b(Assume)32 b(that)456 4935 y Fs(\(51\))534 b Fm(k)s Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))28 b(=)1717 4874 y Fm(J)1776 4841 y Fq(2)p 1717 4914 99 4 v 1744 4997 a Fs(2)1846 4935 y(+)20 b Fm("k)2029 4898 y Fq(1)2069 4935 y Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))p Fm(:)p eop end %%Page: 42 42 TeXDict begin 42 41 bop 456 251 a Fq(42)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 450 y Fo(L)-5 b(et)30 b Fm(k)657 464 y Fp(i)685 450 y Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(\))32 b Fm(i)25 b Fs(=)g(1)p Fm(;)15 b(:)g(:)g(:)i(;)e(m) 30 b Fo(b)-5 b(e)30 b(the)g(c)-5 b(o)g(e\016cients)31 b(in)e(the)i(T)-7 b(aylor)31 b(exp)-5 b(ansion)456 558 y(with)33 b(r)-5 b(esp)g(e)g(ct)34 b(to)g Fm(")f Fo(of)g Fm(k)1291 525 y Fq(1)1330 558 y Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))p Fo(,)35 b(and)f(assume)f(that)h(the)f Fm(k)2655 572 y Fp(i)2683 558 y Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(\))p Fo(,)35 b Fm(i)25 b Fs(=)456 666 y(1)p Fm(;)15 b(:)g(:)g(:)i(;)e(m)32 b Fo(ar)-5 b(e)34 b(trigonometric)g(p)-5 b(olynomials)36 b(in)c Fm(';)15 b(s)p Fo(.)555 774 y(Then,)33 b(ther)-5 b(e)34 b(exists)f(a)g(\014nite)f(set)h(\(of)g(r)-5 b(esonanc)g(es\))1336 972 y Fn(R)25 b Fs(=)g Fn(R)1611 986 y Fq(1)1671 972 y Fn([)19 b(\001)c(\001)g(\001)22 b([)d(R)2035 986 y Fp(m)2127 972 y Fn(\032)25 b Fk(Q)456 1170 y Fo(\(we)35 b(wil)5 b(l)35 b(give)f(r)-5 b(ather)36 b(explicit)f(expr)-5 b(essions)37 b(for)e Fn(R)2281 1184 y Fp(i)2344 1170 y Fo(involving)g(the)g(supp)-5 b(ort)456 1278 y(of)32 b(the)h(F)-7 b(ourier)34 b(tr)-5 b(ansform)35 b(of)e(the)g Fm(k)1763 1292 y Fp(i)1792 1278 y Fo(\))f(such)h(that:)555 1386 y(Given)38 b Fm(L)f Fo(a)i(numb)-5 b(er,)39 b(indep)-5 b(endent)39 b(of)f Fm(")p Fo(,)i(such)d(that)i(the)g(r)-5 b(e)g(al)39 b(intervals)456 1494 y Fs([)p Fn(\000)p Fm(l)r(=k)23 b Fn(\000)c Fs(2)p Fm(L;)c(l)r(=k)24 b Fs(+)c(2)p Fm(L)p Fs(])33 b Fo(for)g Fn(\000)p Fm(l)r(=k)28 b Fn(2)d(R)p Fo(,)32 b(ar)-5 b(e)34 b(disjoint,)f(ther)-5 b(e)33 b(exists)g(a)g (sym-)456 1602 y(ple)-5 b(ctic)37 b(change)g(of)h(variables,)g(dep)-5 b(ending)39 b(on)e(time,)h Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(\))34 b Fn(7!)f Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(\))p Fo(,)456 1710 y(p)-5 b(erio)g(dic)32 b(in)e Fm(')h Fo(and)g Fm(s)p Fo(,)f(and)h(of)g(class)g Fn(C)1816 1677 y Fp(n)p Fl(\000)p Fq(2)p Fp(m)2016 1710 y Fo(,)f(which)i(is)e Fm(")p Fo(-close)h(to)f(the)h(iden-)456 1818 y(tity)f(in)f(the)h Fn(C)927 1785 y Fp(n)p Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(1)1247 1818 y Fo(sense,)g(such)f(that)i(tr)-5 b(ansforms)33 b(the)d(Hamiltonian)h(sys-)456 1926 y(tem)40 b(asso)-5 b(ciate)g(d)43 b(to)e Fm(k)s Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))42 b Fo(into)f(a)g(Hamiltonian)h(system)e(of)h (Hamil-)456 2034 y(tonian)832 2208 y Fs(\026)830 2232 y Fm(k)s Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(;)g Fm(")p Fs(\))85 b(=)1517 2208 y(\026)1515 2232 y Fm(k)1565 2194 y Fq(0)1604 2232 y Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(;)g Fm(")p Fs(\))22 b(+)e Fm(")2155 2194 y Fp(m)p Fq(+1)2315 2208 y Fs(\026)2312 2232 y Fm(k)2362 2194 y Fq(1)2402 2232 y Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(;)g Fm(")p Fs(\))456 2430 y Fo(wher)-5 b(e)43 b(the)f(function)1252 2406 y Fs(\026)1250 2430 y Fm(k)1300 2397 y Fq(0)1381 2430 y Fo(is)g(of)h(class)g Fn(C)1886 2397 y Fp(n)p Fl(\000)p Fq(2)p Fp(m)p Fq(+2)2175 2430 y Fo(,)i(and)e Fm(")2476 2397 y Fp(m)p Fq(+1)2635 2406 y Fs(\026)2633 2430 y Fm(k)2683 2397 y Fq(1)2765 2430 y Fo(is)f(of)g(class)456 2538 y Fn(C)509 2505 y Fp(n)p Fl(\000)p Fq(2)p Fp(m)741 2538 y Fo(and)33 b(they)h(verify:)601 2691 y Fs(\(1\))42 b Fo(If)f Fn(j)q(B)22 b Fs(+)e Fm(l)r(=k)s Fn(j)42 b(\025)e Fs(2)p Fm(L)i Fo(for)f(any)h Fs(\()p Fm(k)s(;)15 b(l)r Fs(\))42 b Fn(2)f Fk(Z)2242 2658 y Fq(2)2322 2691 y Fo(such)g(that)i Fn(\000)p Fm(l)r(=k)h Fn(2)c(R)p Fo(,)758 2799 y(then)1172 2956 y Fs(\026)1170 2979 y Fm(k)1220 2942 y Fq(0)1259 2979 y Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(;)g Fm(")p Fs(\))28 b(=)1788 2918 y(1)p 1788 2959 46 4 v 1788 3042 a(2)1844 2979 y Fn(B)1907 2942 y Fq(2)1966 2979 y Fs(+)20 b Fm(")2101 2956 y Fs(\026)2099 2979 y Fm(k)2149 2942 y Fq(0)p Fp(;)p Fq(0)2244 2979 y Fs(\()p Fn(B)s Fs(;)15 b Fm(")p Fs(\))758 3197 y Fo(wher)-5 b(e)1017 3173 y Fs(\026)1015 3197 y Fm(k)1065 3164 y Fq(0)p Fp(;)p Fq(0)1160 3197 y Fs(\()p Fn(B)s Fs(;)15 b Fm(")p Fs(\))33 b Fo(is)g(a)g(p)-5 b(olynomial)35 b(of)e(de)-5 b(gr)g(e)g(e)34 b Fm(m)20 b Fn(\000)g Fs(1)33 b Fo(in)f Fm(")p Fo(.)601 3305 y Fs(\(2\))42 b Fo(If)37 b Fn(jB)23 b Fs(+)d Fm(l)1084 3319 y Fq(1)1123 3305 y Fm(=k)1215 3319 y Fq(1)1256 3305 y Fn(j)33 b(\024)f Fm(L)37 b Fo(for)g(some)h Fs(\()p Fm(k)1984 3319 y Fq(1)2024 3305 y Fm(;)15 b(l)2091 3319 y Fq(1)2131 3305 y Fs(\))33 b Fn(2)g Fk(Z)2354 3272 y Fq(2)2393 3305 y Fo(,)k(and)h(for)f(some)h Fs(1)33 b Fn(\024)758 3413 y Fm(i)26 b Fn(\024)f Fm(m)32 b Fo(we)h(have)g(that)h Fn(\000)p Fm(l)1651 3427 y Fq(1)1690 3413 y Fm(=k)1782 3427 y Fq(1)1848 3413 y Fn(2)25 b(R)2011 3427 y Fp(i)2059 3413 y Fn(n)c Fs(\()p Fn(R)2237 3427 y Fq(1)2297 3413 y Fn([)e(\001)c(\001)g(\001)22 b([)e(R)2662 3427 y Fp(i)p Fl(\000)p Fq(1)2780 3413 y Fs(\))p Fo(,)33 b(then)710 3630 y Fs(\026)708 3654 y Fm(k)758 3617 y Fq(0)798 3654 y Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(;)g Fm(")p Fs(\))27 b(=)1327 3593 y(1)p 1327 3634 V 1327 3717 a(2)1382 3654 y Fn(B)1445 3617 y Fq(2)1504 3654 y Fs(+)20 b Fm(")1639 3630 y Fs(\026)1637 3654 y Fm(k)1687 3617 y Fq(0)p Fp(;)p Fq(0)1782 3654 y Fs(\()p Fn(B)s Fs(;)15 b Fm(")p Fs(\))21 b(+)f Fm(")2151 3617 y Fp(i)2180 3654 y Fm(U)2252 3617 y Fp(k)2289 3626 y Fi(1)2323 3617 y Fp(;l)2364 3626 y Fi(1)2402 3654 y Fs(\()p Fm(k)2484 3668 y Fq(1)2525 3654 y Fm(\013)g Fs(+)g Fm(l)2721 3668 y Fq(1)2761 3654 y Fm(s)p Fs(;)15 b Fm(")p Fs(\))758 3894 y Fo(wher)-5 b(e)44 b Fm(U)1097 3861 y Fp(k)1134 3870 y Fi(1)1169 3861 y Fp(;l)1210 3870 y Fi(1)1248 3894 y Fs(\()p Fm(\022)s Fs(;)15 b Fm(")p Fs(\))43 b Fo(is)g(a)g(p)-5 b(olynomial)46 b(in)d Fm(")g Fo(and)h(a)f(trigonometric)758 4002 y(p)-5 b(olynomial)36 b(in)d Fm(\022)s Fo(.)601 4110 y Fs(\(3\))42 b Fo(If)c Fn(j)q(B)22 b Fs(+)e Fm(l)1085 4124 y Fq(0)1125 4110 y Fm(=k)1217 4124 y Fq(0)1257 4110 y Fn(j)36 b(\024)f Fm(L)j Fo(for)h(some)g Fs(\()p Fm(k)1995 4124 y Fq(0)2035 4110 y Fm(;)15 b(l)2102 4124 y Fq(0)2142 4110 y Fs(\))35 b Fn(2)g Fk(Z)2369 4077 y Fq(2)2447 4110 y Fo(and)k Fn(\000)p Fm(l)2727 4124 y Fq(0)2766 4110 y Fm(=k)2858 4124 y Fq(0)2934 4110 y Fn(2)c(R)3107 4124 y Fq(1)3146 4110 y Fo(,)758 4220 y(then)e(the)g(function)g Fm(U)1541 4187 y Fp(k)1578 4196 y Fi(0)1613 4187 y Fp(;l)1654 4196 y Fi(0)1725 4220 y Fo(de\014ne)-5 b(d)33 b(in)g(2)g(is)f(given)g(by:)456 4505 y Fs(\(52\))270 b Fm(U)958 4467 y Fp(k)995 4476 y Fi(0)1030 4467 y Fp(;l)1071 4476 y Fi(0)1109 4505 y Fs(\()p Fm(\022)s Fs(;)15 b Fm(")p Fs(\))26 b(=)1497 4391 y Fp(N)1462 4418 y Fh(X)1429 4615 y Fp(t)p Fq(=)p Fl(\000)p Fp(N)1645 4481 y Fs(^)1642 4505 y Fm(k)1692 4467 y Fq(1)1689 4528 y Fp(tk)1751 4537 y Fi(0)1787 4528 y Fp(;tl)1853 4537 y Fi(0)1891 4505 y Fs(\()p Fn(\000)p Fm(l)2024 4519 y Fq(0)2064 4505 y Fm(=k)2156 4519 y Fq(0)2196 4505 y Fs(;)15 b(0\))p Fm(e)2358 4467 y Fp(it\022)2469 4505 y Fs(+)2559 4513 y(O)2630 4505 y(\()p Fm(")p Fs(\))758 4817 y Fo(wher)-5 b(e)1030 4793 y Fs(^)1028 4817 y Fm(k)1078 4784 y Fq(1)1075 4845 y Fp(k)r(;l)1159 4817 y Fs(\()p Fm(J)9 b Fs(;)15 b Fm(")p Fs(\))48 b Fo(ar)-5 b(e)46 b(the)g(F)-7 b(ourier)46 b(c)-5 b(o)g(e\016cients)47 b(of)f(the)g(function)758 4942 y Fm(k)808 4909 y Fq(1)848 4942 y Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))35 b Fo(with)e(r)-5 b(esp)g(e)g(ct)35 b(to)e Fs(\()p Fm(';)15 b(s)p Fs(\))p Fo(.)p eop end %%Page: 43 43 TeXDict begin 43 42 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(43)456 450 y Fs(8.3.3.)47 b Fo(Pr)-5 b(o)g(of)29 b(of)f(The)-5 b(or)g(em)29 b(35.)47 b Fs(The)24 b(pro)s(of)g(of)h(this)g(Theorem)g(will)g(follo)m(w)h(b)m (y)456 558 y(the)21 b(rep)s(eated)h(application)g(of)g(the)f(follo)m (wing)i(inductiv)m(e)f(Lemma)f(36,)j(that)e(w)m(e)456 666 y(will)k(pro)m(v)m(e)i(using)e(the)g(metho)s(d)g(of)h(Lie)g (transforms)e([Car81)q(,)i(Mey91)q(,)g(Lla01)q(].)555 774 y(The)33 b(h)m(yp)s(othesis)g(of)g(Lemma)g(36)h(will)g(b)s(e)e (that)i(w)m(e)g(ha)m(v)m(e)g(a)g(Hamiltonian)456 882 y(already)21 b(in)g(normal)g(form)f(outside)i(of)f(a)g(set)h(or)f (resonances.)38 b(The)20 b(conclusions)456 990 y(will)38 b(b)s(e)f(that,)j(excluding)e(an)f(sligh)m(tly)i(larger)f(set)g(of)g (resonances|)g(whic)m(h)456 1098 y(w)m(e)c(will)g(giv)m(e)h(rather)e (explicitly|w)m(e)j(can)d(pro)s(duce)g(another)h(Hamiltonian)456 1206 y(whic)m(h)c(is)g(normalized)h(to)g(a)g(higher)f(order)g(in)g Fm(")p Fs(.)456 1377 y Fw(Lemma)k(36.)42 b Fo(Consider)34 b(a)f(Hamiltonian)h(of)f(the)g(form:)456 1539 y Fs(\(53\))287 b Fm(k)950 1553 y Fp(q)988 1539 y Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))28 b(=)d Fm(k)1545 1502 y Fq(0)1542 1562 y Fp(q)1584 1539 y Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))23 b(+)d Fm(")2123 1502 y Fp(q)r Fq(+1)2252 1539 y Fm(k)2302 1502 y Fq(1)2299 1562 y Fp(q)2341 1539 y Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))456 1701 y Fo(wher)-5 b(e)646 1836 y Fs(1.)42 b Fm(k)808 1803 y Fq(0)805 1861 y(0)848 1836 y Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))43 b(=)1395 1801 y Fp(J)1440 1777 y Fi(2)p 1395 1816 80 4 v 1417 1868 a Fq(2)1526 1836 y Fo(and,)h(if)d Fm(q)i Fn(\025)d Fs(1)p Fo(,)k Fm(k)2200 1803 y Fq(0)2197 1859 y Fp(q)2240 1836 y Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))43 b Fo(is)e(a)h Fn(C)2913 1803 y Fp(n)p Fq(+2)p Fl(\000)p Fq(2)p Fp(q)758 1944 y Fo(function)33 b(that)h(veri\014es:)858 2052 y(Ther)-5 b(e)38 b(exist)f(\014nite)g (sets)g Fn(R)1839 2066 y Fp(i)1900 2052 y Fn(\032)c Fk(Q)p Fo(,)k(c)-5 b(al)5 b(le)-5 b(d)38 b(r)-5 b(esonanc)g(es)39 b(of)e(or)-5 b(der)758 2160 y Fm(i)p Fo(,)33 b Fm(i)26 b Fs(=)f(1)15 b Fn(\001)g(\001)g(\001)h Fm(q)s Fo(,)33 b(and)g(a)g(numb)-5 b(er)33 b Fm(L)25 b(>)g Fs(0)33 b Fo(such)g(that:)763 2268 y Fs(1.0.)43 b Fo(The)e(intervals)g Fn(I)1568 2287 y Fl(\000)p Fp(l)q(=k)1760 2268 y Fn(\021)e Fs([)p Fn(\000)p Fm(l)r(=k)29 b Fn(\000)c Fs(2)p Fm(L;)15 b Fn(\000)p Fm(l)r(=k)30 b Fs(+)25 b(2)p Fm(L)p Fs(])p Fo(,)42 b Fn(\000)p Fm(l)r(=k)g Fn(2)946 2394 y(R)1023 2361 y Fq([)p Fl(\024)p Fp(q)r Fq(])1181 2394 y Fn(\021)1277 2326 y Fh(S)1352 2421 y Fp(i)p Fq(=1)p Fp(;:::)o(;q)1618 2394 y Fn(R)1695 2408 y Fp(i)1756 2394 y Fo(ar)-5 b(e)33 b(disjoint.)763 2511 y Fs(1.1.)43 b Fo(If)33 b Fm(J)44 b(=)-55 b Fn(2)1212 2442 y Fh(S)1288 2537 y Fl(\000)p Fp(l)q(=k)r Fl(2R)1546 2518 y Fi([)p Fg(\024)p Ff(q)r Fi(])1682 2511 y Fn(I)1732 2529 y Fl(\000)p Fp(l)q(=k)1887 2511 y Fo(,)32 b(then)1189 2754 y Fm(k)1239 2716 y Fq(0)1236 2776 y Fp(q)1279 2754 y Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))28 b(=)1796 2692 y Fm(J)1855 2659 y Fq(2)p 1796 2733 99 4 v 1822 2816 a Fs(2)1924 2754 y(+)20 b Fm("k)2107 2716 y Fq(0)p Fp(;)p Fq(0)2104 2776 y Fp(q)2202 2754 y Fs(\()p Fm(J)9 b Fs(;)15 b Fm(")p Fs(\))p Fm(;)946 2966 y Fo(wher)-5 b(e)34 b Fm("k)1295 2922 y Fq(0)p Fp(;)p Fq(0)1292 2978 y Fp(q)1390 2966 y Fs(\()p Fm(J)9 b Fs(;)15 b Fm(")p Fs(\))34 b Fo(is)f(a)g(p)-5 b(olynomial)35 b(of)e(de)-5 b(gr)g(e)g(e)34 b Fm(q)h Fo(in)e Fm(")p Fo(.)763 3075 y Fs(1.2.)43 b Fo(If)33 b Fn(j)p Fm(J)c Fs(+)20 b Fm(l)1264 3089 y Fq(1)1304 3075 y Fm(=k)1396 3089 y Fq(1)1436 3075 y Fn(j)26 b(\024)f Fm(L)32 b Fo(for)h(some)g Fs(\()p Fm(k)2136 3089 y Fq(1)2177 3075 y Fm(;)15 b(l)2244 3089 y Fq(1)2284 3075 y Fs(\))25 b Fn(2)g Fk(Z)2491 3042 y Fq(2)2563 3075 y Fo(such)32 b(that)1171 3237 y Fn(\000)p Fm(l)1269 3251 y Fq(1)1308 3237 y Fm(=k)1400 3251 y Fq(1)1466 3237 y Fn(2)25 b(R)1629 3251 y Fp(i)1677 3237 y Fn(n)c Fs(\()p Fn(R)1855 3251 y Fq(1)1915 3237 y Fn([)e(\001)c(\001)g(\001)22 b([)e(R)2280 3251 y Fp(i)p Fl(\000)p Fq(1)2398 3237 y Fs(\))p Fm(;)946 3399 y Fo(for)34 b(some)f Fs(1)26 b Fn(\024)f Fm(i)g Fn(\024)g Fm(q)s Fo(,)32 b(then)740 3617 y Fm(k)790 3579 y Fq(0)787 3639 y Fp(q)829 3617 y Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))28 b(=)1346 3555 y Fm(J)1405 3522 y Fq(2)p 1346 3596 V 1373 3679 a Fs(2)1475 3617 y(+)20 b Fm("k)1658 3579 y Fq(0)p Fp(;)p Fq(0)1655 3639 y Fp(q)1753 3617 y Fs(\()p Fm(J)9 b Fs(;)15 b Fm(")p Fs(\))21 b(+)f Fm(")2118 3579 y Fp(i)2147 3617 y Fm(U)2219 3579 y Fp(k)2256 3588 y Fi(1)2290 3579 y Fp(;l)2331 3588 y Fi(1)2209 3639 y Fp(q)2370 3617 y Fs(\()p Fm(k)2452 3631 y Fq(1)2492 3617 y Fm(')g Fs(+)g Fm(l)2689 3631 y Fq(1)2729 3617 y Fm(s)p Fs(;)15 b Fm(")p Fs(\))946 3832 y Fo(wher)-5 b(e)32 b Fm(U)1273 3788 y Fp(k)1310 3797 y Fi(1)1344 3788 y Fp(;l)1385 3797 y Fi(1)1263 3844 y Fp(q)1424 3832 y Fs(\()p Fm(\022)s Fs(;)15 b Fm(")p Fs(\))30 b Fo(is)h(a)f(p)-5 b(olynomial)34 b(in)c Fm(")h Fo(and)g(a)g(trigonomet-)946 3940 y(ric)i(p)-5 b(olynomial)36 b(in)c Fm(\022)c Fn(\021)c Fm(k)1869 3954 y Fq(1)1909 3940 y Fm(')d Fs(+)f Fm(l)2107 3954 y Fq(1)2146 3940 y Fm(s)p Fo(.)646 4048 y Fs(2.)42 b Fm(k)808 4015 y Fq(1)805 4070 y Fp(q)848 4048 y Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))51 b Fo(is)d(a)g Fn(C)1542 4015 y Fp(n)p Fl(\000)p Fq(2)p Fp(q)1762 4048 y Fo(function)g(whose)i(T)-7 b(aylor)49 b(series)g(c)-5 b(o)g(ef-)758 4156 y(\014cients)49 b(with)g(r)-5 b(esp)g(e)g(ct)50 b(to)f Fm(")g Fo(ar)-5 b(e)49 b(trigonometric)h(p)-5 b(olynomials)52 b(in)758 4264 y Fs(\()p Fm(';)15 b(s)p Fs(\))p Fo(.)555 4399 y(Denote)31 b(by)g Fm(K)h Fs(=)25 b Fm(k)1238 4366 y Fq(1)1235 4421 y Fp(q)1278 4399 y Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g(0\))p Fo(,)33 b(which)f(is)f(the)h(term)g(of)f(the)g(p)-5 b(erturb)g(ation) 456 4507 y(of)32 b(or)-5 b(der)35 b(exactly)e Fm(q)23 b Fs(+)d(1)p Fo(.)42 b(Intr)-5 b(o)g(duc)g(e)35 b(also)f(the)f(set)456 4669 y Fs(\(54\))333 b Fn(R)1026 4683 y Fp(q)r Fq(+1)1179 4669 y Fs(=)25 b Fn(f\000)p Fm(l)r(=k)s(;)49 b Fs(\()p Fm(k)s(;)15 b(l)r Fs(\))26 b Fn(2)f(N)13 b Fs(\()p Fm(k)2063 4631 y Fq(1)2060 4691 y Fp(q)2103 4669 y Fs(\()p Fn(\001)p Fs(;)i(0\)\))p Fm(;)50 b(k)29 b Fn(6)p Fs(=)c(0)p Fn(g)p Fm(:)555 4850 y Fo(Cho)-5 b(ose)36 b Fs(0)30 b Fm(<)1053 4827 y Fs(~)1042 4850 y Fm(L)f(<)f(L)34 b Fo(such)h(that)h(the)f (intervals)g Fs([)p Fn(\000)p Fm(l)r(=k)25 b Fn(\000)c Fs(2)2640 4827 y(~)2629 4850 y Fm(L)q(;)15 b Fn(\000)p Fm(l)r(=k)25 b Fs(+)c(2)3097 4827 y(~)3086 4850 y Fm(L)q Fs(])456 4964 y Fo(ar)-5 b(e)33 b(disjoint)h(when)f Fn(\000)p Fm(l)r(=k)c Fn(2)24 b(R)1553 4931 y Fq([)p Fl(\024)p Fp(q)r Fq(+1])1776 4964 y Fo(.)p eop end %%Page: 44 44 TeXDict begin 44 43 bop 456 251 a Fq(44)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)555 450 y Fo(L)-5 b(et)32 b Fm(G)p Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(\))33 b Fo(b)-5 b(e)32 b(the)g Fn(C)1443 417 y Fp(n)p Fl(\000)p Fq(2)p Fp(q)r Fl(\000)p Fq(1)1736 450 y Fo(function)g(given)f (by)g(L)-5 b(emma)33 b(34,)g(verify-)456 565 y(ing)e(e)-5 b(quation)40 b Fs(\(48\))33 b Fo(with)g Fm(K)f Fs(=)25 b Fm(k)1622 532 y Fq(1)1619 587 y Fp(q)1662 565 y Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g(0\))34 b Fo(and)f(with)f(a)h(distanc)-5 b(e)2894 542 y Fs(~)2883 565 y Fm(L)31 b Fo(away)456 673 y(fr)-5 b(om)34 b(the)f(r)-5 b(esonanc)g(es.)555 781 y(Then,)33 b(the)g Fn(C)1024 748 y Fp(n)p Fl(\000)p Fq(2)p Fp(q)r Fl(\000)p Fq(2)1318 781 y Fo(change)g(of)g(variables)1409 971 y Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(\))27 b(=)e Fm(g)s Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(\))p Fm(;)456 1162 y Fo(given)32 b(by)i(the)g(time)f(one)h(\015ow)h(of)e(the)h (Hamiltonian)h Fm(")2366 1129 y Fp(q)r Fq(+1)2495 1162 y Fm(G)p Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(\))p Fo(,)35 b(tr)-5 b(ans-)456 1270 y(forms)33 b(the)g(Hamiltonian)i Fm(k)1436 1284 y Fp(q)1474 1270 y Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))35 b Fo(into)e(a)g(Hamiltonian)741 1468 y Fm(k)788 1482 y Fp(q)r Fq(+1)916 1468 y Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(;)g Fm(")p Fs(\))27 b(=)e Fm(k)1485 1430 y Fq(0)1482 1490 y Fp(q)r Fq(+1)1611 1468 y Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(;)g Fm(")p Fs(\))22 b(+)e Fm(")2162 1430 y Fp(q)r Fq(+2)2290 1468 y Fm(k)2340 1430 y Fq(1)2337 1490 y Fp(q)r Fq(+1)2466 1468 y Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(;)g Fm(")p Fs(\))p Fm(;)456 1658 y Fo(with)456 1849 y Fs(\(55\))209 b Fm(k)875 1811 y Fq(0)872 1871 y Fp(q)r Fq(+1)1000 1849 y Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(;)g Fm(")p Fs(\))28 b(=)d Fm(k)1570 1811 y Fq(0)1567 1871 y Fp(q)1609 1849 y Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(;)g Fm(")p Fs(\))22 b(+)e Fm(")2160 1811 y Fp(q)r Fq(+1)2291 1825 y Fs(\026)2289 1849 y Fm(k)2339 1811 y Fq(1)2336 1871 y Fp(q)2378 1849 y Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(;)g(0\))p Fm(;)456 2051 y Fo(wher)-5 b(e)728 2027 y Fs(\026)726 2051 y Fm(k)776 2018 y Fq(1)773 2074 y Fp(q)816 2051 y Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(;)g(0\))52 b(=)1412 2028 y(\026)1387 2051 y Fm(K)7 b Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(\))p Fo(,)51 b(given)45 b(in)i(L)-5 b(emma)47 b(34,)k(is)46 b(a)h Fn(C)3003 2018 y Fp(n)p Fl(\000)p Fq(2)p Fp(q)456 2159 y Fo(function.)555 2267 y(Mor)-5 b(e)g(over,)30 b(the)f(Hamiltonian)g Fm(k)1696 2234 y Fq(0)1693 2291 y Fp(q)r Fq(+1)1822 2267 y Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(;)g Fm(")p Fs(\))30 b Fo(veri\014es)e(pr)-5 b(op)g(erties)30 b([1.0],)456 2396 y([1.1],)j([1.2])g(up)f(to)i(or)-5 b(der)34 b Fm(q)23 b Fs(+)d(1)33 b Fo(with)1819 2373 y Fs(~)1808 2396 y Fm(L)g Fo(r)-5 b(eplacing)34 b Fm(L)p Fo(.)555 2504 y(F)-7 b(urthermor)i(e,)32 b Fm(")1147 2471 y Fp(q)r Fq(+1)1276 2504 y Fm(k)1326 2471 y Fq(1)1323 2529 y Fp(q)r Fq(+1)1451 2504 y Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))32 b Fo(is)c(a)i Fn(C)2088 2471 y Fp(n)p Fl(\000)p Fq(2)p Fp(q)r Fl(\000)p Fq(2)2378 2504 y Fo(function)f(whose)h(T)-7 b(ay-)456 2613 y(lor)32 b(series)g(c)-5 b(o)g(e\016cients)33 b(with)g(r)-5 b(esp)g(e)g(ct)34 b(to)e Fm(")g Fo(ar)-5 b(e)33 b(trigonometric)h(p)-5 b(olynomials)456 2721 y(in)32 b Fs(\()p Fm(';)15 b(s)p Fs(\))p Fo(.)456 2991 y(Pr)-5 b(o)g(of)20 b(.)555 3099 y Fs(W)-8 b(e)33 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b(region)g(con)m(tains)h(KAM)f(tori)g(whic)m(h)f(are)h Fm(O)s Fs(\()p Fm(")2537 2316 y Fq(\()p Fp(m)p Fq(+1\))p Fp(=)p Fq(2\))2848 2349 y Fs(\)-closely)456 2457 y(spaced.)456 2634 y Fw(Theorem)38 b(38.)44 b Fo(L)-5 b(et)1243 2610 y Fs(\026)1240 2634 y Fm(k)39 b Fo(b)-5 b(e)36 b(a)g(Hamiltonian)h(of)f (the)g(form)44 b Fs(\(57\))r Fo(,)36 b(wher)-5 b(e)37 b Fm(")f Fo(is)456 2742 y(su\016ciently)c(smal)5 b(l)34 b(and)g(\014xe)-5 b(d.)42 b(Assume)33 b(that)656 2881 y Fs(i\))761 2857 y(\026)758 2881 y Fm(k)46 b Fo(is)c(a)h Fm(C)1119 2848 y Fp(s)p Fq(+)p Fp(\014)1296 2881 y Fo(function)f(and) 1852 2857 y Fs(\026)1850 2881 y Fm(k)1900 2848 y Fq(0)p Fp(;)p Fq(0)2036 2881 y Fo(is)h(a)g Fm(C)2305 2848 y Fp(s)p Fq(+)p Fp(\014)s Fq(+2)2571 2881 y Fo(function)g(of)f(the)758 2989 y(variables)31 b Fm(B)5 b(;)15 b(\013;)g(s)p Fo(,and)31 b(that)g Fn(jj)p Fm(k)1873 2956 y Fq(0)p Fp(;)p Fq(0)1968 2989 y Fn(jj)2018 3011 y Fp(C)2073 2992 y Ff(s)p Fi(+)p Ff(\014)s Fi(+2)2272 2989 y Fm(;)15 b Fn(jj)p 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Fs(1\))p Fo(,)39 b(wher)-5 b(e)38 b Fm(!)h Fo(is)e(a)g(Diophantine)h (numb)-5 b(er)758 4834 y(of)33 b(c)-5 b(onstant)34 b(typ)-5 b(e)34 b(with)f(Markov)g(c)-5 b(onstant)34 b(less)f(that)h Fm(")2706 4801 y Fq(\()p Fp(m)p Fq(+1\))p Fp(=)p Fq(2)3021 4834 y Fo(\(se)-5 b(e)758 4942 y(De\014nition)33 b(42\).)p eop end %%Page: 47 47 TeXDict begin 47 46 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(47)456 450 y Fw(Remark)38 b(39.)44 b Fs(It)33 b(is)g(imp)s(ortan)m(t)g(to)h(note)g(that,)h(for)d (a)i(\014xed)e(v)-5 b(alue)34 b(of)f Fm(")p Fs(,)i(as)456 558 y(long)c(as)f(w)m(e)h(\014x)f Fm(m)25 b(>)g Fs(2,)31 b(it)g(is)f(enough)g(to)h(consider)g(a)f(\014nite)h(n)m(um)m(b)s(er)e (of)h(tori)456 666 y Fn(T)506 680 y Fp(i)573 666 y Fs(to)41 b(ensure)f(that)g(all)h(the)g(p)s(oin)m(ts)f(in)g(the)g(non-resonan)m (t)h(region)f Fn(S)2960 633 y Fp(L)3052 666 y Fs(are)456 782 y(O)526 774 y(\()p Fm(")603 741 y Fq(1+)p Fp(\016)732 774 y Fs(\)-close,)e Fm(\016)f(>)c 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b(ho)m(w)i(separated)f(are)h(the)g(n)m(um)m(b)s(ers)e(with)h(go)s(o)s (d)g(Diophan)m(tine)h(con-)456 558 y(stan)m(ts.)42 b(When)31 b(incorp)s(orated)g(to)g(a)h(KAM)f(theorem)g(this)g(will)g(lead)g(to)h (esti-)456 666 y(mates)j(on)g(the)g(gaps)g(b)s(et)m(w)m(een)g(the)g (KAM)g(tori.)54 b(Later)36 b(on,)f(it)h(will)f(lead)g(to)456 774 y(estimates)d(on)e(ho)m(w)g(close)i(to)f(a)g(resonance)g(w)m(e)g (can)f(get)i(tori.)555 882 y(W)-8 b(e)32 b(start)f(b)m(y)f(recalling)i (the)e(standard)g(de\014nition:)456 1071 y Fw(De\014nition)48 b(42.)g Fo(We)43 b(say)h(that)h(a)e(r)-5 b(e)g(al)45 b(numb)-5 b(er)43 b Fm(!)48 b Fn(2)c(D)s Fs(\()p Fm(\024;)15 b(\027)6 b Fs(\))44 b Fo(when)g(we)456 1179 y(have:)456 1372 y Fs(\(58\))340 b Fn(j)q Fm(!)23 b Fn(\000)d Fm(`=k)s Fn(j)26 b(\025)f Fm(\024)15 b Fn(j)p Fm(k)s Fn(j)1601 1330 y Fl(\000)p Fq(2)p Fl(\000)p Fp(\027)1897 1372 y Fn(8)p Fs(\()p Fm(k)s(;)g(`)p Fs(\))26 b Fn(2)f Fk(Z)2319 1334 y Fq(2)2358 1372 y Fm(;)48 b(k)28 b Fn(6)p Fs(=)d(0)p Fm(:)555 1565 y Fo(We)34 b(wil)5 b(l)35 b(r)-5 b(efer)36 b(to)f Fn(D)s Fs(\()p Fm(\024;)15 b(\027)6 b Fs(\))35 b Fo(as)h(the)f(Diophantine)h(numb)-5 b(ers)35 b(of)g(typ)-5 b(e)36 b Fm(\024;)15 b(\027)6 b Fo(.)456 1673 y(When)41 b Fm(\027)46 b Fs(=)41 b(0)p Fo(,)i(we)f(wil)5 b(l)41 b(r)-5 b(efer)42 b(to)f Fn(D)s Fs(\()p Fm(\024;)15 b Fs(0\))43 b Fo(as)f(c)-5 b(onstant)43 b(typ)-5 b(e)42 b(numb)-5 b(ers)42 b(of)456 1781 y(Markov)33 b(c)-5 b(onstant)34 b Fm(\024)p Fo(.)555 1889 y(We)27 b(wil)5 b(l)28 b(denote)g Fn(D)s Fs(\()p Fm(\027)6 b Fs(\))25 b(=)g Fn([)1543 1903 y Fp(\024>)p Fq(0)1677 1889 y Fn(D)s Fs(\()p Fm(\024;)15 b(\027)6 b Fs(\))28 b Fo(and)h(r)-5 b(efer)27 b(to)h(them)g(as)g (Diophan-)456 1997 y(tine)33 b(numb)-5 b(ers)35 b(of)f(typ)-5 b(e)35 b Fm(\027)6 b Fo(.)45 b(When)35 b Fm(\027)e Fs(=)27 b(0)35 b Fo(we)f(wil)5 b(l)34 b(c)-5 b(al)5 b(l)35 b(them)f(c)-5 b(onstant)36 b(typ)-5 b(e)456 2105 y(numb)g(ers.)555 2294 y Fs(The)31 b(De\014nition)h(42)g(is)f(the)h(same)g(as)f(that)h (used)f(in)g([Her83)q(,)h(p.)f(158].)45 b(W)-8 b(e)456 2402 y(note)31 b(that)g(the)f(inequalit)m(y)j(\(58\))f(is)e(equiv)-5 b(alen)m(t)32 b(to)456 2603 y(\(59\))349 b Fn(j)p Fm(!)s(k)24 b Fn(\000)19 b Fm(`)p Fn(j)1275 2561 y Fl(\000)p Fq(1)1394 2603 y Fn(\024)25 b Fm(C)7 b(\024)1614 2565 y Fl(\000)p Fq(1)1723 2603 y Fn(j)q Fs(\()p Fm(k)s(;)15 b(l)r Fs(\))p Fn(j)1964 2561 y Fq(1+)p Fp(\027)2204 2603 y Fn(8)p Fs(\()p Fm(k)s(;)g(`)p Fs(\))26 b Fn(2)e Fk(Z)2625 2565 y Fq(2)456 2796 y Fs(whic)m(h)30 b(is)g(a)h(form)f(that)h(generalizes)h(to)f (higher)f(dimensions.)555 2904 y(It)c(is)f(w)m(ell)i(kno)m(wn)e(that)h Fn(D)s Fs(\()p Fm(\027)6 b Fs(\))26 b(is)f(of)h(full)f(measure)g(when)g Fm(\027)31 b(>)25 b Fs(0.)39 b(Constan)m(t)456 3012 y(t)m(yp)s(e)31 b(n)m(um)m(b)s(ers)f(are)i(of)g(zero)g(measure)g(but)f(are)h(dense.)44 b(All)32 b(quadratic)g(irra-)456 3120 y(tionals)38 b(are)f(constan)m(t) i(t)m(yp)s(e)e(n)m(um)m(b)s(ers.)60 b(More)38 b(generally)-8 b(,)41 b(it)c(is)g(equiv)-5 b(alen)m(t)456 3228 y(to)33 b(sa)m(y)h(that)f(a)g(n)m(um)m(b)s(er)f(is)g(constan)m(t)i(t)m(yp)s(e)f (and)g(to)g(sa)m(y)h(that)f(its)g(con)m(tin)m(ued)456 3336 y(fraction)e(expansion)f(is)g(b)s(ounded.)39 b(See)31 b(Prop)s(osition)f(44)h(later.)555 3444 y(The)24 b(follo)m(wing)i (statemen)m(t)g(mak)m(es)f(it)g(precise)g(the)f(idea)h(that)g(if)f Fm(\024)h Fs(is)f(small,)456 3551 y(the)31 b(n)m(um)m(b)s(ers)e Fn(D)s Fs(\()p Fm(\024;)15 b(\027)6 b Fs(\),)32 b Fm(\027)f Fn(\025)26 b Fs(0,)31 b(do)g(not)g(ha)m(v)m(e)h(big)e(gaps)h(among)h (them.)41 b(The)456 3659 y(result)30 b(that)h(w)m(e)g(will)f(use)g (later)i(is)e(the)h(case)g(for)f Fm(\027)h Fs(=)25 b(0.)456 3848 y Fw(Lemma)j(43.)37 b Fo(Given)27 b Fm(\027)k Fn(\025)25 b Fs(0)p Fo(,)k(ther)-5 b(e)28 b(exists)g(a)g(c)-5 b(onstant)29 b Fm(K)7 b Fs(\()p Fm(\027)f Fs(\))25 b Fm(>)g Fs(0)j Fo(such)g(that)456 3956 y(in)k(every)h(interval)g Fn(I)39 b Fo(of)33 b(diameter)g Fm(K)7 b Fs(\()p Fm(\027)f Fs(\))p Fm(\024)33 b Fo(ther)-5 b(e)34 b(is)f(a)f(numb)-5 b(er)34 b(in)e Fn(D)s Fs(\()p Fm(\024;)15 b(\027)6 b Fs(\))p Fo(.)456 4231 y(Pr)-5 b(o)g(of)20 b(.)555 4339 y Fs(There)34 b(is)h(a)g(standard)f(pro)s(of)g(for)g Fm(\027)k(>)33 b Fs(0)i(whic)m(h)f(also)i(giv)m(es)g(information)456 4447 y(on)30 b(the)h(measure)g(and)f(also)i(generalizes)h(for)d(higher) g(dimensions.)42 b(W)-8 b(e)32 b(do)e(it)456 4554 y(\014rst.)555 4662 y(W)-8 b(e)32 b(note)f(that)1067 4856 y Fn(D)s Fs(\()p Fm(\024;)15 b(\027)6 b Fs(\))26 b(=)f Fk(R)20 b Fn(n)1780 4769 y Fh([)1627 4972 y Fq(\()p Fp(a;b)p Fq(\))p Fl(2)p Fe(Z)1859 4953 y Fi(2)1895 4972 y Fp(;b)p Fl(6)p Fq(=0)2050 4856 y Fm(B)2119 4875 y Fp(\024b)2190 4856 y Fg(\000)p Fi(2)p Fg(\000)p Ff(\027)12 b Fs(\()p Fm(a=b)p Fs(\))p eop end %%Page: 49 49 TeXDict begin 49 48 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(49)456 450 y Fs(Hence,)25 b(giv)m(en)f(an)f(in)m(terv)-5 b(al)25 b Fn(I)k Fs(and)23 b(denoting)g(b)m(y)g Fn(j)91 b(j)24 b Fs(the)f(Leb)s(esgue)g(measure) 456 558 y(of)30 b(a)h(set,)g(w)m(e)g(ha)m(v)m(e)723 732 y Fn(j)q(I)26 b(\\)20 b(D)s Fs(\()p Fm(\024;)15 b(\027)6 b Fs(\))p Fn(j)26 b(\025)f(jI)7 b(j)20 b(\000)g Fs(2)2097 646 y Fh(X)1617 848 y Fq(\()p Fp(a;b)p Fq(\))p Fl(2)p Fe(Z)1849 829 y Fi(2)1886 848 y Fp(;b)p Fl(6)p Fq(=0)p Fp(;B)2099 874 y Ff(\024)p Fg(j)p Ff(b)p Fg(j)2200 858 y(\000)p Fi(2)p Fg(\000)p Ff(\027)2369 848 y Fq(\()p Fp(a=b)p Fq(\))p Fl(\\I)5 b(6)p Fq(=)p Fl(;)2722 732 y Fm(\024)15 b Fn(j)q Fm(b)p Fn(j)2879 690 y Fl(\000)p Fq(2)p Fl(\000)p Fp(\027)1243 1027 y Fn(\025)25 b(jI)7 b(j)20 b(\000)g Fs(2)1683 940 y Fh(X)1617 1142 y Fp(b)p Fl(2)p Fe(Z)p Fl(nf)p Fq(0)p Fl(g)1895 953 y Fh(\000)1952 1027 y Fn(j)q(I)7 b(j)14 b Fm(\024)h Fn(j)q Fm(b)p Fn(j)2231 985 y Fl(\000)p Fq(1)p Fl(\000)p Fp(\027)2440 1027 y Fs(+)20 b(2)p Fm(\024)15 b Fn(j)q Fm(b)p Fn(j)2733 985 y Fl(\000)p Fq(2)p Fl(\000)p Fp(\027)2936 953 y Fh(\001)1243 1271 y Fn(\025)25 b(jI)7 b(j)15 b Fs(\(1)21 b Fn(\000)f Fm(K)1730 1285 y Fq(1)1770 1271 y Fm(\024)p Fs(\))h Fn(\000)e Fm(\024K)2097 1285 y Fq(2)2138 1271 y Fm(:)456 995 y Fs(\(60\))456 1436 y(The)26 b(second)h(inequalit)m(y)h(in)f(\(60\))h (follo)m(ws)g(from)f(the)g(observ)-5 b(ation)28 b(that,)g(once)456 1544 y(w)m(e)i(\014x)f Fm(b)p Fs(,)h(the)g(n)m(um)m(b)s(er)f(of)g Fm(a)h Fs(suc)m(h)g(that)g Fm(B)1948 1564 y Fp(\024b)2019 1545 y Fg(\000)p Fi(2)p Fg(\000)p Ff(\027)2188 1544 y Fs(\()p Fm(a=b)p Fs(\))20 b Fn(\\)f(I)31 b(6)p Fs(=)25 b Fn(;)30 b Fs(is)g(b)s(ounded)456 1652 y(b)m(y)e Fn(jI)7 b(j)15 b(j)p Fm(b)p Fn(j)h Fs(+)g(2.)40 b(In)27 b(the)i(last)g (inequalit)m(y)g(of)36 b(\(60\))q(,)29 b Fm(K)2260 1666 y Fq(1)2300 1652 y Fm(;)15 b(K)2417 1666 y Fq(2)2485 1652 y Fs(stand)28 b(for)g(p)s(ositiv)m(e)456 1760 y(constan)m(ts)j (that)g(dep)s(end)e(on)h Fm(\027)6 b Fs(.)555 1868 y(W)-8 b(e)34 b(see)f(that)h(for)e Fm(\024)h Fs(small)g(enough)g(and)f(for)g Fn(j)q(I)7 b(j)28 b(\025)h Fm(K)7 b(\024)p Fs(,)34 b(the)f(righ)m(t)g (hand)456 1976 y(side)38 b(of)46 b(\(60\))39 b(is)g(p)s(ositiv)m(e,)j (hence,)e(the)f(measure)f Fn(I)32 b(\\)25 b(D)s Fs(\()p Fm(\024;)15 b(\027)6 b Fs(\))39 b(is)g(p)s(ositiv)m(e,)456 2084 y(whic)m(h)30 b(establishes)h(the)f(claim.)555 2192 y(The)g(ab)s(o)m(v)m(e)g(deriv)-5 b(ation)31 b(uses)e(essen)m(tially)j (that)e Fm(\027)h(>)25 b Fs(0)30 b(since)h(w)m(e)f(use)f(that)456 2232 y Fh(P)552 2327 y Fp(b)601 2300 y Fm(b)640 2267 y Fl(\000)p Fq(1)p Fl(\000)p Fp(\027)859 2300 y Fs(con)m(v)m(erges.)43 b(Indeed,)30 b(it)h(is)f(w)m(ell)i(kno)m(wn)e(that)h Fn(D)s Fs(\()p Fm(\024;)15 b Fs(0\))32 b(is)e(of)h(mea-)456 2408 y(sure)e(0)i(for)f(all)h Fm(\024)p Fs(.)555 2516 y(F)-8 b(or)45 b(the)g(case)g(when)f Fm(\027)54 b Fs(=)48 b(0,)h(w)m(e)c(will)f(use)g(the)h(theory)g(of)f(con)m(tin)m(ued)456 2624 y(fractions.)76 b(W)-8 b(e)43 b(recall)g(the)f(follo)m(wing)i (prop)s(osition)d(whose)h(pro)s(of)f(can)h(b)s(e)456 2732 y(found)29 b(in)h([Her79)q(,)h(p.)f(64].)456 2902 y Fw(Prop)s(osition)36 b(44.)42 b Fo(Given)32 b Fm(L)26 b Fn(2)e Fk(R)1700 2869 y Fq(+)1759 2902 y Fo(,)32 b(denote)i(by)456 3063 y Fs(\(61\))197 b Fn(C)5 b Fs(\()p Fm(L)p Fs(\))26 b(=)f Fn(f)p Fm(x)g Fs(=)g([)p Fm(a)1411 3077 y Fq(1)1451 3063 y Fm(;)15 b(a)1539 3077 y Fq(2)1579 3063 y Fm(;)g(:)g(:)g(:)h(;)f (a)1828 3077 y Fp(n)1876 3063 y Fm(;)g(:)g(:)g(:)q Fs(])26 b(:)58 b Fm(a)2204 3077 y Fp(i)2257 3063 y Fn(2)25 b Fk(N)p Fm(;)48 b Fs(1)26 b Fn(\024)e Fm(a)2696 3077 y Fp(i)2750 3063 y Fn(\024)h Fm(L)p Fn(g)p Fm(:)555 3223 y Fo(With)33 b(this)g(notation,)i(we)d(have)1344 3384 y Fn(C)5 b Fs(\()p Fm(L)p Fs(\))26 b(=)f Fn(D)s Fs(\(1)p Fm(=)p Fs(\()p Fm(L)c Fs(+)f(2\))p Fm(;)15 b Fs(0\))p Fm(:)555 3554 y Fs(Hence,)28 b(to)e(\014nish)e(the)i(pro)s(of)f(of)h (Lemma)g(43,)h(it)g(su\016ces)e(to)h(sho)m(w)g(that)g Fn(I)3097 3568 y Fp(L)3149 3554 y Fs(,)456 3662 y(the)k(largest)i(in)m (terv)-5 b(al)31 b(in)f Fk(R)20 b Fn(n)h(C)5 b Fs(\()p Fm(L)p Fs(\),)31 b(satis\014es)456 3823 y(\(62\))956 b Fn(j)q(I)1648 3837 y Fp(L)1699 3823 y Fn(j)26 b(\024)f Fm(K)q(=L:)555 3983 y Fs(W)-8 b(e)43 b(claim)f(that)g(this)f(in)m(terv) -5 b(al)43 b Fn(I)1762 3997 y Fp(L)1814 3983 y Fs(,)h(whic)m(h)d(is)g (the)h(largest)h(gap)e(among)456 4091 y(n)m(um)m(b)s(ers)29 b(in)h Fn(C)5 b Fs(\()p Fm(L)p Fs(\),)31 b(is)f(precisely:)456 4252 y(\(63\))319 b Fn(I)985 4266 y Fp(L)1062 4252 y Fs(=)25 b(\([2)p Fm(;)15 b(L;)g Fs(1)p Fm(;)g(L;)g Fs(1)p Fm(;)g(L;)g(:)g(:)g(:)5 b Fs(])p Fm(;)15 b Fs([1)p Fm(;)g Fs(1)p Fm(;)g(L;)g Fs(1)p Fm(;)g(L;)g Fs(1)p Fm(;)g(:)g(:)g(:)6 b Fs(]\))555 4412 y(Once)34 b(w)m(e)g(ha)m(v)m(e)i(the)e(claim)h (\(63\))q(,)g(the)f(result)g(in)f(Lemma)h(43)h(follo)m(ws)g(b)s(e-)456 4520 y(cause)c(a)f(direct)h(calculation)i(sho)m(ws)d(that)1022 4681 y([2)p Fm(;)15 b(L;)g Fs(1)p Fm(;)g(L;)g Fs(1)p Fm(;)g(:)g(:)g(:)5 b Fs(])25 b(=)g(1)p Fm(=)p Fs(2)d Fn(\000)e Fs(1)p Fm(=L)h Fs(+)2274 4689 y(O)2344 4681 y(\(1)p Fm(=L)2531 4643 y Fq(2)2572 4681 y Fs(\))456 4841 y(and)967 4964 y([1)p Fm(;)15 b Fs(1)p Fm(;)g(L;)g Fs(1)p Fm(;)g(L;)g Fs(1)p Fm(;)g(:)g(:)g(:)5 b Fs(])26 b(=)f(1)p Fm(=)p Fs(2)c(+)f(1)p Fm(=L)h Fs(+)2304 4972 y(O)2375 4964 y(\(1)p Fm(=L)2562 4927 y Fq(2)2602 4964 y Fs(\))p Fm(:)p eop end %%Page: 50 50 TeXDict begin 50 49 bop 456 251 a Fq(50)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 450 y Fs(Therefore)35 b Fn(jI)948 464 y Fp(L)1000 450 y Fn(j)g Fs(=)g(2)p Fm(=L)24 b Fs(+)1437 458 y(O)1508 450 y(\(1)p Fm(=L)1695 417 y Fq(2)1735 450 y Fs(\).)58 b(Hence,)39 b(using)c(Prop)s(osition)h(44,)j (the)456 558 y(largest)f(gap)g(in)f Fn(D)s Fs(\(1)p Fm(=)p Fs(\()p Fm(L)26 b Fs(+)e(2\))p Fm(;)15 b Fs(0\))39 b(is)f(2)p Fm(=L)25 b Fs(+)2075 566 y(O)2145 558 y(\(1)p Fm(=L)2332 525 y Fq(2)2373 558 y Fs(\).)62 b(That)37 b(is,)i(w)m(e)f(ha)m(v)m(e) 456 666 y(established)j(Lemma)h(43)g(for)g Fm(\024)i Fs(=)f(1)p Fm(=)p Fs(\()p Fm(L)29 b Fs(+)f(2\),)45 b(from)c(whic)m(h)g (the)h(general)456 774 y(result)30 b(follo)m(ws.)555 882 y(Hence,)e(the)e(only)g(thing)g(left)g(for)g(the)g(pro)s(of)f(of)h (Lemma)g(43)g(is)g(to)g(establish)456 990 y(\(63\))q(.)555 1098 y(W)-8 b(e)32 b(recall)f(that)g(giv)m(en)h(t)m(w)m(o)g(n)m(um)m(b) s(ers)1232 1240 y Fm(x)25 b Fs(=)g([)p Fm(a)1478 1254 y Fq(1)1518 1240 y Fm(;)15 b(a)1606 1254 y Fq(2)1646 1240 y Fm(;)g(:)g(:)g(:)i(;)e(a)1896 1254 y Fp(m)1963 1240 y Fm(;)g(b)2042 1254 y Fq(1)2082 1240 y Fm(;)g(b)2161 1254 y Fq(2)2200 1240 y Fm(;)g(:)g(:)g(:)r Fs(])p Fm(;)1232 1380 y(y)28 b Fs(=)d([)p Fm(a)1474 1394 y Fq(1)1514 1380 y Fm(;)15 b(a)1602 1394 y Fq(2)1642 1380 y Fm(;)g(:)g(:)g(:)h(;)f(a) 1891 1394 y Fp(m)1959 1380 y Fm(;)g(c)2038 1394 y Fq(1)2078 1380 y Fm(;)g(c)2157 1394 y Fq(2)2197 1380 y Fm(;)g(:)g(:)g(:)q Fs(])p Fm(;)456 1536 y Fs(and)29 b Fm(b)671 1550 y Fq(1)736 1536 y Fn(6)p Fs(=)c Fm(c)871 1550 y Fq(1)910 1536 y Fs(,)30 b(w)m(e)h(ha)m(v)m(e)g(that)f Fm(x)c(>)f(y)32 b Fs(if)e Fm(b)1877 1550 y Fq(1)1942 1536 y Fm(>)25 b(c)2077 1550 y Fq(1)2146 1536 y Fs(and)k Fm(m)h Fs(is)g(o)s(dd)f(or)h(if)f Fm(b)2934 1550 y Fq(1)2999 1536 y Fm(<)c(c)3134 1550 y Fq(1)456 1644 y Fs(and)k Fm(m)h Fs(is)h(ev)m(en.)555 1752 y(The)42 b(ab)s(o)m(v)m(e)h(observ)-5 b(ation)43 b(allo)m(ws)h(to)f(conclude)f(immediately)i(that)f(the)456 1860 y(n)m(um)m(b)s(er)30 b([2)p Fm(;)15 b(L;)g Fs(1)p Fm(;)g(L;)g Fs(1)p Fm(;)g(L;)g(:)g(:)g(:)5 b Fs(])32 b(is)h(the)f(largest)h(n)m(um)m(b)s(er)e(in)h Fn(C)5 b Fs(\()p Fm(L)p Fs(\))33 b(whose)f(\014rst)456 1968 y(en)m(try)20 b(in)g(the)h(con)m(tin)m(ued)g(fraction)g(is)f(2.)38 b(Similarly)-8 b(,)23 b(the)d(n)m(um)m(b)s(er)f([1)p Fm(;)c Fs(1)p Fm(;)g(L;)g Fs(1)p Fm(;)g(L;)g Fs(1)p Fm(;)g(:)g(:)g(:)6 b Fs(])456 2076 y(is)32 b(the)h(smallest)g(n)m(um)m(b)s(er)e(in)h Fn(C)5 b Fs(\()p Fm(L)p Fs(\))33 b(whose)g(\014rst)e(en)m(try)i(is)f (1.)47 b(This)32 b(mak)m(es)h(it)456 2184 y(clear)e(that)g(there)g(are) f(no)h(p)s(oin)m(ts)f(of)g Fn(C)5 b Fs(\()p Fm(L)p Fs(\))31 b(inside)g Fn(I)2282 2198 y Fp(L)2333 2184 y Fs(.)555 2291 y(The)f(claim)i(\(63\))f(follo)m(ws)h(from)e(the)g(follo)m(wing)i (considerations.)601 2423 y(\(1\))42 b(If)30 b(the)h(in)m(terv)-5 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3142 y(^)1848 3165 y Fn(I)55 b Fs(is)49 b(larger)h(than)e(the)h(in)m(terv)-5 b(al)50 b Fn(I)7 b Fs(.)758 3273 y(Hence)42 b(w)m(e)g(conclude)f(that)g (the)h(in)m(terv)-5 b(al)42 b Fn(I)2332 3287 y Fp(L)2424 3273 y Fs(has)f(to)h(ha)m(v)m(e)g(di\013er-)758 3381 y(en)m(t)31 b(\014rst)f(en)m(tries.)601 3488 y(\(2\))42 b(If)30 b Fm(M)5 b(;)15 b(N)36 b Fn(2)25 b Fk(N)30 b Fs(and)g Fm(N)35 b(>)25 b Fs(1,)31 b(w)m(e)g(ha)m(v)m(e:)543 3644 y([)p Fm(M)f Fs(+)20 b(1)p Fm(;)15 b(b)901 3658 y Fq(2)942 3644 y Fm(;)g(b)1021 3658 y Fq(3)1060 3644 y Fm(;)g(b)1139 3658 y Fq(4)1179 3644 y Fm(;)g(:)g(:)g(:)r Fs(])25 b Fn(2)g Fs(\([)p Fm(M)31 b Fs(+)20 b Fm(N)5 b(;)15 b(b)1889 3658 y Fq(2)1929 3644 y Fm(;)g(b)2008 3658 y Fq(3)2048 3644 y Fm(;)g(b)2127 3658 y Fq(4)2166 3644 y Fm(;)g(:)g(:)g(:)r Fs(])p Fm(;)g Fs([)p Fm(M)5 b(;)15 b(c)2575 3658 y Fq(2)2616 3644 y Fm(;)g(c)2695 3658 y Fq(3)2735 3644 y Fm(;)g(c)2814 3658 y Fq(4)2854 3644 y Fm(;)g(:)g(:)g(:)r Fs(]\))p Fm(:)858 3801 y Fs(Hence,)31 b Fn(I)1203 3815 y Fp(L)1285 3801 y Fs(should)f(b)s(e)g(of)g(the)h (form)907 3957 y Fn(I)957 3971 y Fp(L)1034 3957 y Fs(=)25 b(\([)p Fm(M)31 b Fs(+)20 b(1)p Fm(;)15 b(b)1524 3971 y Fq(2)1564 3957 y Fm(;)g(b)1643 3971 y Fq(3)1683 3957 y Fm(;)g(b)1762 3971 y Fq(4)1802 3957 y Fm(;)g(:)g(:)g(:)q Fs(])p Fm(;)g Fs([)p Fm(M)5 b(;)15 b(c)2210 3971 y Fq(2)2252 3957 y Fm(;)g(c)2331 3971 y Fq(3)2371 3957 y Fm(;)g(c)2450 3971 y Fq(4)2490 3957 y Fm(;)g(:)g(:)g(:)q Fs(]\))p Fm(:)601 4113 y Fs(\(3\))42 b(The)24 b(length)g(of)g(the)g(in)m(terv)-5 b(al)25 b(\([2)p Fm(;)15 b(b)1963 4127 y Fq(2)2004 4113 y Fm(;)g(b)2083 4127 y Fq(3)2123 4113 y Fm(;)g(b)2202 4127 y Fq(4)2242 4113 y Fm(;)g(:)g(:)g(:)q Fs(])p Fm(;)g Fs([1)p Fm(;)g(c)2602 4127 y Fq(2)2644 4113 y Fm(;)g(c)2723 4127 y Fq(3)2763 4113 y Fm(;)g(c)2842 4127 y Fq(4)2882 4113 y Fm(;)g(:)g(:)g(:)q Fs(]\))25 b(is)758 4220 y(larger)c(than)f (that)h(of)g(the)f(in)m(terv)-5 b(al)22 b(\([)p Fm(M)10 b Fs(+1)p Fm(;)15 b(b)2307 4234 y Fq(2)2347 4220 y Fm(;)g(b)2426 4234 y Fq(3)2466 4220 y Fm(;)g(b)2545 4234 y Fq(4)2585 4220 y Fm(;)g(:)g(:)g(:)q Fs(])p Fm(;)g Fs([)p Fm(M)5 b(;)15 b(c)2993 4234 y Fq(2)3035 4220 y Fm(;)g(c)3114 4234 y Fq(3)3154 4220 y Fm(;)g(c)3233 4234 y Fq(4)3273 4220 y Fm(;)g(:)g(:)g(:)q Fs(]\))758 4328 y(for)30 b Fm(M)36 b(>)25 b Fs(1.)858 4436 y(Hence,)31 b Fn(I)1203 4450 y Fp(L)1285 4436 y Fs(should)f(b)s(e)g(of)g(the)h(form)1036 4592 y Fn(I)1086 4606 y Fp(L)1163 4592 y Fs(=)25 b(\([2)p Fm(;)15 b(b)1443 4606 y Fq(2)1483 4592 y Fm(;)g(b)1562 4606 y Fq(3)1602 4592 y Fm(;)g(b)1681 4606 y Fq(4)1721 4592 y Fm(;)g(:)g(:)g(:)r Fs(])p Fm(;)g Fs([1)p Fm(;)g(c)2082 4606 y Fq(2)2123 4592 y Fm(;)g(c)2202 4606 y Fq(3)2242 4592 y Fm(;)g(c)2321 4606 y Fq(4)2362 4592 y Fm(;)g(:)g(:)g(:)q Fs(]\))p Fm(:)601 4748 y Fs(\(4\))42 b(If)i(an)h(in)m(terv)-5 b(al)45 b(of)g(the)g(form)f(\([2)p Fm(;)15 b(b)2048 4762 y Fq(2)2089 4748 y Fm(;)g(b)2168 4762 y Fq(3)2208 4748 y Fm(;)g(b)2287 4762 y Fq(4)2326 4748 y Fm(;)g(:)g(:)g(:)r Fs(])p Fm(;)g Fs([1)p Fm(;)g(c)2687 4762 y Fq(2)2728 4748 y Fm(;)g(c)2807 4762 y Fq(3)2848 4748 y Fm(;)g(c)2927 4762 y Fq(4)2967 4748 y Fm(;)g(:)g(:)g(:)q Fs(]\))758 4856 y(do)s(es)29 b(not)h(con)m(tain)g(an)m(y)g(p)s(oin)m(t)f(in)g Fn(C)5 b Fs(\()p Fm(L)p Fs(\),)30 b(then,)g(it)f(is)h(necessary)f(that) 758 4964 y Fm(b)797 4978 y Fq(2)837 4964 y Fm(;)15 b(b)916 4978 y Fq(3)956 4964 y Fm(;)g(:)g(:)g(:)30 b Fs(are)f(c)m(hosen)g(in)g (suc)m(h)f(a)h(w)m(a)m(y)h(that)f([2)p Fm(;)15 b(b)2479 4978 y Fq(2)2520 4964 y Fm(;)g(b)2599 4978 y Fq(3)2639 4964 y Fm(;)g(b)2718 4978 y Fq(4)2758 4964 y Fm(;)g(:)g(:)g(:)q Fs(])29 b(is)g(the)p eop end %%Page: 51 51 TeXDict begin 51 50 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(51)758 450 y Fs(largest)24 b(n)m(um)m(b)s(er)d(in)h Fn(C)5 b Fs(\()p Fm(L)p Fs(\))23 b(whose)f(\014rst)f(en)m(try)i(in)f(the)g(con)m(tin)m(ued)h(frac-)758 558 y(tion)32 b(expansion)f(is)h(2.)44 b(Similarly)-8 b(,)32 b(w)m(e)g(also)g(need)f(that)h Fm(c)2749 572 y Fq(2)2789 558 y Fm(;)15 b(c)2868 572 y Fq(3)2908 558 y Fm(;)g(c)2987 572 y Fq(4)3027 558 y Fm(;)g(:)g(:)g(:)758 666 y Fs(are)32 b(c)m(hosen)f(in)g(suc)m(h)g(a)h(w)m(a)m(y)g(that)f([1) p Fm(;)15 b(c)2123 680 y Fq(2)2164 666 y Fm(;)g(c)2243 680 y Fq(3)2283 666 y Fm(;)g(c)2362 680 y Fq(4)2403 666 y Fm(;)g(:)g(:)g(:)q Fs(])31 b(is)g(the)h(smallest)758 774 y(n)m(um)m(b)s(er)39 b(in)h Fn(C)5 b Fs(\()p Fm(L)p Fs(\))40 b(whose)g(\014rst)f(en)m(try)i(in)e(the)h(con)m(tin)m(ued)h (fraction)758 882 y(expansion)31 b(is)f(1.)858 990 y(Hence,)h(w)m(e)g (conclude)g(that)g Fn(I)1908 1004 y Fp(L)1990 990 y Fs(is)f(of)h(the)f (form)g(giv)m(en)i(in)e(\(63\))q(.)3103 1122 y Fj(\003)456 1351 y Fs(8.4.2.)47 b Fo(The)32 b(KAM)f(The)-5 b(or)g(em)34 b(for)e(twist)h(maps.)47 b Fs(The)29 b(follo)m(wing)i(result)e(is)h(an) 456 1459 y(easy)37 b(consequence)g(of)g(the)f(Theorem)h(5.4)g(stated)h (in)e([Her83)q(,)j(p.)d(198])i(\(see)456 1567 y(also)31 b(the)f(Theorem)h(5.6)g(in)f([Her83)q(,)h(p.)f(204]\).)456 1735 y Fw(Theorem)39 b(45.)45 b Fo(L)-5 b(et)37 b Fm(F)1301 1749 y Fq(0)1373 1735 y Fs(:)32 b Fk(T)23 b Fn(\002)g Fk(R)36 b Fo(b)-5 b(e)36 b(an)h(inte)-5 b(gr)g(able)38 b(symple)-5 b(ctic)38 b(mapping,)456 1843 y(that)33 b(is:)456 1998 y Fs(\(64\))726 b Fm(F)1400 2012 y Fq(0)1440 1998 y Fs(\()p Fm(\022)s(;)15 b(r)s Fs(\))25 b(=)g(\()p Fm(\022)e Fs(+)d(\001\()p Fm(r)s Fs(\))p Fm(;)15 b(r)s Fs(\))p Fm(:)555 2159 y Fo(Assume)33 b(that)g Fm(F)1138 2173 y Fq(0)1204 2159 y Fn(2)24 b Fm(C)1361 2126 y Fp(n)p Fq(+)p Fp(\014)1506 2159 y Fo(,)32 b Fm(n)25 b Fn(\025)g Fs(3)p Fo(,)33 b Fs(0)26 b Fm(<)f(\014)30 b(<)25 b Fs(1)33 b Fo(and)2473 2123 y Fp(d)p 2456 2138 71 4 v 2456 2190 a(dr)2537 2159 y Fs(\001\()p Fm(r)s Fs(\))25 b Fn(\025)g Fm(M)35 b(>)25 b Fs(0)p Fo(.)555 2267 y(Then)h(we)f(c)-5 b(an)25 b(\014nd)h(a)g(c)-5 b(onstant)26 b Fm(K)32 b Fo(dep)-5 b(ending)26 b(only)g(on)g Fm(n;)15 b(\014)30 b Fo(such)25 b(that)h(for)456 2375 y(any)h Fm(r)665 2389 y Fq(0)730 2375 y Fo(such)g(that)g Fm(!)h Fn(\021)d Fs(\001\()p 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Fq(0)1071 3300 y Fn(jj)1121 3322 y Fp(C)1176 3303 y Ff(n)p Fg(\000)p Fi(1+)p Ff(\014)1411 3300 y Fn(\024)25 b Fm(K)7 b(M)1689 3267 y Fl(\000)p Fq(1)1783 3300 y Fm(\024)1835 3267 y Fl(\000)p Fq(1)1930 3300 y Fm(\016)n(:)641 3408 y Fs(c\))42 b Fo(The)33 b(motion)h(of)f Fm(F)46 b Fo(r)-5 b(estricte)g(d)34 b(to)f Fn(T)55 b Fo(has)34 b(r)-5 b(otation)35 b(numb)-5 b(er)33 b Fm(!)s Fo(.)631 3515 y Fs(d\))41 b Fo(If)33 b(we)g(denote)g(by)f Fm(g)37 b Fo(the)c(map)g(of)g(the)g (torus)h(de\014ne)-5 b(d)33 b(by)1276 3671 y Fm(F)13 b Fs(\()p Fm(\022)s(;)i Fs(\011\()p Fm(\022)s Fs(\)\))25 b(=)g(\()p Fm(g)s Fs(\()p Fm(\022)s Fs(\))p Fm(;)15 b Fs(\011\()p Fm(g)s Fs(\()p Fm(\022)s Fs(\)\))758 3827 y Fo(and)38 b(we)f(denote)h(by)f Fm(h)g Fo(the)h(map)g(that)g(c)-5 b(onjugates)38 b Fm(g)j Fo(to)c(a)g(r)-5 b(otation)758 3935 y(by)33 b Fm(!)s Fo(,)f(i.e.)1434 4053 y Fm(g)24 b Fn(\016)d Fm(h)p Fs(\()p Fm(\022)s Fs(\))k(=)g Fm(h)p Fs(\()p Fm(\022)e Fs(+)d Fm(!)s Fs(\))758 4189 y Fo(normalize)-5 b(d)36 b(to)d Fm(h)p Fs(\(0\))26 b(=)f(0)p Fo(,)33 b(we)g(have)456 4349 y Fs(\(65\))590 b Fn(jj)p Fm(h)21 b Fn(\000)f Fo(Id)p Fn(jj)1551 4371 y Fp(C)1606 4352 y Ff(n)p Fg(\000)p Fi(2+)p Ff(\014)1841 4349 y Fn(\024)25 b Fm(K)7 b(M)2119 4311 y Fl(\000)p Fq(1)2213 4349 y Fm(\024)2265 4311 y Fl(\000)p Fq(1)2360 4349 y Fm(\016)n(:)555 4506 y Fo(Mor)-5 b(e)g(over,)36 b(if)e Fm(n)28 b Fn(\025)g Fs(4)35 b Fo(we)g(have)g(for)g(al)5 b(l)34 b(the)h Fm(F)48 b Fo(in)34 b(a)h Fm(C)2488 4473 y Fp(n)p Fq(+)p Fp(\014)2666 4506 y Fo(neighb)-5 b(orho)g(o)g(d)456 4614 y(of)32 b Fm(F)620 4628 y Fq(0)660 4614 y Fo(:)641 4746 y Fs(e\))42 b Fo(The)30 b(mappings)h(that)g(asso)-5 b(ciate)31 b Fm(F)42 b Fo(to)30 b Fs(\011)p Fo(,)g Fm(h)f Fo(r)-5 b(esp)g(e)g(ctively,)32 b(ar)-5 b(e)30 b(Lips-)758 4856 y(chitz)f(when)g(we)f(give)g Fm(F)41 b Fo(the)29 b Fm(C)1828 4823 y Fp(n)p Fq(+)p Fp(\014)2000 4856 y Fo(top)-5 b(olo)g(gy,)32 b Fs(\011)c Fo(the)h Fm(C)2691 4823 y Fp(n)p Fl(\000)p Fq(2+)p Fp(\014)2953 4856 y Fo(top)-5 b(ol-)758 4964 y(o)g(gy)34 b(and)f Fm(h)g Fo(the)g Fm(C)1403 4931 y Fp(n)p Fl(\000)p Fq(3+)p Fp(\014)1670 4964 y Fo(top)-5 b(olo)g(gy.)p eop end %%Page: 52 52 TeXDict begin 52 51 bop 456 251 a Fq(52)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)647 451 y Fs(f)7 b(\))41 b Fo(De\014ne)32 b Fs(\000)1102 465 y Fp(F)1161 451 y Fs(\()1209 428 y(~)1196 451 y(\011\))h Fo(the)g(gr)-5 b(aph)34 b(tr)-5 b(ansform)35 b(of)2271 428 y Fs(~)2258 451 y(\011)d Fo(by:)1183 619 y Fm(F)13 b Fs(\(Graph\()1591 596 y(~)1578 619 y(\011\)\))26 b(=)f(Graph)o(\(\000)2186 633 y Fp(F)2245 619 y Fs(\()2293 596 y(~)2280 619 y(\011\)\))p Fm(:)758 783 y Fo(Assume)33 b(that)h(for)f(some)g Fm(C)1733 750 y Fp(n)p Fl(\000)p Fq(1+)p Fp(\014)2000 783 y Fo(map)h(we)f(have) 1325 946 y Fn(jj)1388 923 y Fs(~)1375 946 y(\011)20 b Fn(\000)g Fs(\000)1614 960 y Fp(F)1673 946 y Fs(\()1721 923 y(~)1708 946 y(\011\))p Fn(jj)1864 968 y Fp(C)1919 949 y Ff(n)p Fg(\000)p Fi(1+)p Ff(\014)2154 946 y Fn(\024)25 b Fm(\026)758 1115 y Fo(and)34 b(that)1133 1092 y Fs(~)1121 1115 y(\011)e Fo(is)h(in)f(a)h Fm(K)7 b(\024)1648 1082 y Fq(2)1720 1115 y Fo(neighb)-5 b(orho)g(o)g(d)36 b(of)d(a)g(c)-5 b(onstant.)858 1223 y(Then,)45 b(ther)-5 b(e)44 b(is)e(a)h Fm(C)1640 1190 y Fp(n)p Fl(\000)p Fq(1+)p Fp(\014)1917 1223 y Fo(function)f Fs(\011)2355 1190 y Fl(\003)2437 1223 y Fo(whose)i(gr)-5 b(aph)44 b(is)e(an)758 1330 y(invariant)34 b(cir)-5 b(cle)33 b(for)g Fm(F)46 b Fo(such)32 b(that)1270 1494 y Fn(jj)1333 1471 y Fs(~)1320 1494 y(\011)21 b Fn(\000)f Fs(\011)1574 1456 y Fl(\003)1613 1494 y Fn(jj)1663 1516 y Fp(C)1718 1497 y Ff(n)p Fg(\000)p Fi(1+)p Ff(\014)1953 1494 y Fn(\024)25 b Fm(K)7 b(\024)2185 1456 y Fl(\000)p Fq(1)2279 1494 y Fm(\026:)456 1665 y Fw(Remark)43 b(46.)i Fs(F)-8 b(or)38 b(the)f(exp)s(erts)g(in)g(KAM)g(theory)-8 b(,)39 b(w)m(e)f(call)g(atten)m(tion)h(to)456 1773 y(the)32 b(fact)h(that)f(Theorem)g(45)g(allo)m(ws)i(to)e(conclude)h(more)f (regularit)m(y)h(for)f(the)456 1881 y(graph)e(than)g(for)h(the)f (conjugating)i(function.)42 b(Most)31 b(of)g(the)g(v)m(ersions)g(of)g (the)456 1988 y(KAM)h(theorem)g(study)f(the)h(conjugating)h(function,)f (hence,)h(the)f(regularit)m(y)456 2096 y(established)25 b(for)h(the)g(curv)m(e)f(is)h(the)g(same)g(as)g(that)g(for)f(the)h (conjugating)g(func-)456 2204 y(tion.)39 b(\(Some)26 b(theorems)f(that)h(establish)g(that)g(the)f(graphs)g(are)g(more)h (regular)456 2312 y(than)k(the)g(conjugacy)i(are)e([Sal86)r(],)g ([P\177)-45 b(os82)r(].\))1040 b Fj(\003)456 2590 y Fo(Pr)-5 b(o)g(of)20 b(.)555 2698 y Fs(W)-8 b(e)37 b(refer)e(to)h([Her83)r(])f (for)g(the)h(pro)s(of)f(of)g(the)h(result.)56 b(Here)36 b(w)m(e)g(only)f(ex-)456 2806 y(plain)27 b(ho)m(w)h(the)f(statemen)m(t) j(w)m(e)e(ha)m(v)m(e)h(made)e(follo)m(ws)i(from)e(the)h(statemen)m(t)h (in)456 2914 y([Her83)q(].)555 3022 y(W)-8 b(e)32 b(note)g(that)f(the)g (Theorem)f(5.4)i(of)f([Her83)r(])g(is)f(stated)i(as)f(a)g(translated) 456 3130 y(curv)m(e)f(theorem)h(for)f(maps)g(\(not)h(necessarily)g (symplectic\))h(of)f(the)f(form)456 3292 y(\(66\))543 b Fm(F)13 b Fs(\()p Fm(\022)s(;)i(r)s Fs(\))25 b(=)g(\()p Fm(\022)e Fs(+)d Fm(r)i Fs(+)e Fm(\013;)15 b(r)24 b Fs(+)c Fm(')p Fs(\()p Fm(\022)s(;)15 b(r)s Fs(\)\))p Fm(:)555 3453 y Fs(The)39 b(pro)s(of)f(of)h(the)g(translated)g(curv)m(e)g (result)g(in)f([Her83)r(])h(do)s(es)f(not)h(use)456 3561 y(an)m(y)30 b(geometric)j(feature)d(\(e.g.)43 b(that)30 b(the)h(mapping)f(is)g(exact)i(symplectic\))g(of)456 3669 y(the)e(map.)555 3777 y(Note)d(that)f(a)g(t)m(wist)g(mapping)f (can)h(b)s(e)e(alw)m(a)m(ys)j(put)e(in)m(to)h(the)g(form)f(\(66\))i(b)m (y)456 3885 y(a)35 b(c)m(hange)h(of)e(v)-5 b(ariables)36 b(\(p)s(ossibly)e(non-canonical\))i(applying)f(the)g(implicit)456 3993 y(function)40 b(theorem)h(to)g(de\014ne)f Fm(r)j Fs(b)m(y)d(the)h(\014rst)e(comp)s(onen)m(t)i(of)48 b(\(66\))q(.)71 b(The)456 4101 y(map)29 b(of)h(the)f(form)g(\(66\))i(obtained)f (through)f(this)h(pro)s(cedure,)e(has)i(the)f(same)456 4209 y(regularit)m(y)44 b(of)f(the)h(original)g(map.)78 b(This)43 b(c)m(hange)h(of)f(v)-5 b(ariables)44 b(do)s(es)f(not)456 4317 y(c)m(hange)37 b(the)f(regularit)m(y)h(of)g(the)f(in)m(v)-5 b(arian)m(t)37 b(circles)g(or)f(of)h(the)f(conjugacy)h(of)456 4425 y(the)30 b(motion)h(on)g(them)f(to)h(rotations.)555 4533 y(Observ)m(e)22 b(also)h(that)g(an)f(exact)i(symplectic)f(mapping) f(has)g(the)g(in)m(tersection)456 4640 y(prop)s(ert)m(y)k(\(the)j (image)f(of)g(a)g(non)m(trivial)h(circle)f(b)m(y)g(the)g(map)f(has)g (to)h(in)m(tersect)456 4748 y(itself)7 b(\),)29 b(and)d(the)h(in)m (tersection)h(prop)s(ert)m(y)e(of)h(a)g(map)f(is)h(preserv)m(ed)f (under)g(an)m(y)456 4856 y(con)m(tin)m(uous)39 b(c)m(hange)g(of)g(v)-5 b(ariables.)65 b(F)-8 b(or)40 b(an)m(y)e(map)g(with)g(the)h(in)m (tersection)456 4964 y(prop)s(ert)m(y)-8 b(,)30 b(a)h(translated)g (curv)m(e)f(has)h(to)g(b)s(e)e(in)m(v)-5 b(arian)m(t.)p eop end %%Page: 53 53 TeXDict begin 53 52 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(53)555 450 y Fs(The)29 b(pro)s(of)g(of)h(Theorem)f(45)i(presen)m(ted)e(in)g([Her83)r(])h(is)f (extremely)i(simple)456 558 y(since)c(it)g(is)g(not)h(based)e(on)h(a)g (rapidly)g(con)m(v)m(ergen)m(t)i(Nash-Moser)f(metho)s(d)f(but)456 666 y(rather)35 b(on)g(a)h(Sc)m(hauder-T)m(yc)m(hono\013)g(\014xed)e(p) s(oin)m(t)i(Theorem)f(\(when)g Fm(n)e Fn(\025)g Fs(3\))456 774 y(or)d(on)g(a)h(con)m(traction)i(mapping)c(principle)h(\(when)g Fm(n)25 b Fn(\025)g Fs(4\).)555 882 y(The)34 b(fact)g(that)h(the)f(pro) s(of)f(uses)g(a)i(con)m(traction)h(theorem)e(for)f(an)h(appro-)456 990 y(priate)h(map)g(is)h(the)f(reason)g(wh)m(y)g(one)h(has)f(prop)s (erties)f(e\))i(and)f(f)7 b(\).)55 b(Ev)m(en)36 b(if)456 1098 y(w)m(e)27 b(will)g(not)f(use)h(it)g(here)f(\(and,)i(hence)f(not)f (state)i(the)f(results\))g(w)m(e)g(note)g(that)456 1206 y(using)e(the)g(results)h(on)f(comp)s(osition)h(in)f([LO99)q(],)i(it)f (is)g(p)s(ossible)f(to)h(sho)m(w)f(that)456 1316 y(the)31 b(op)s(erator)h(considered)g(in)f([Her83)r(])g(is)h(lo)s(cally)h Fm(C)2306 1283 y Fp(`)2370 1316 y Fs(when)e(the)h Fm(F)45 b Fs(and)31 b(the)456 1424 y(\011)c(are)h(giv)m(en)g(in)g(appropriate)f (top)s(ologies)j(\(and)d Fm(n)g Fs(is)h(large)h(enough\))e(so)h(that) 456 1532 y(one)e(gets)h(that)g(the)f(in)m(v)-5 b(arian)m(t)28 b(tori)e(dep)s(end)f(smo)s(othly)h(on)g Fm(F)39 b Fs(in)26 b(appropriate)456 1640 y(top)s(ologies.)42 b(This)30 b(allo)m(ws)i(to)f(justify)e(formal)i(expansions)f(in)g(parameters.) 3103 1747 y Fj(\003)456 1966 y Fo(Pr)-5 b(o)g(of)34 b(of)f(The)-5 b(or)g(em)34 b(38.)42 b Fs(Note)25 b(that)f(once)g(that)g(w)m(e)g(ha)m (v)m(e)h(Theorem)e(45,)j(The-)456 2074 y(orem)35 b(38)h(follo)m(ws)h (immediately)g(if)e(w)m(e)h(consider)f(the)h(time-2)p Fm(\031)k Fs(map)35 b(of)h(the)456 2182 y(\015o)m(w)e(generated)g(b)m (y)g(the)g(Hamiltonian)i(\(57\))q(.)51 b(Note)36 b(that)e(if)2633 2158 y(\026)2631 2182 y Fm(k)s Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(;)g Fm(")p Fs(\))36 b(is)456 2292 y Fm(C)528 2259 y Fp(s)p Fq(+)p Fp(\014)662 2292 y Fs(,)26 b(the)f(time-2)p Fm(\031)k Fs(map)24 b(is)h Fm(C)1550 2259 y Fp(s)p Fl(\000)p Fq(1+)p Fp(\014)1774 2292 y Fs(.)39 b(Moreo)m(v)m(er)26 b(it)f(is)g(an)g(exact)h(symplectic)456 2400 y(t)m(wist)31 b(map.)555 2508 y(The)20 b(time-2)p Fm(\031)25 b Fs(map)20 b(corresp)s(onding)f(to)i(the)f(in)m(tegrable)i (part)e(of)h(the)f(Hamil-)456 2616 y(tonian)35 b(in)f(\(57\))i(giv)m (es)g(rise)f(to)g(a)h Fm(C)1716 2583 y Fp(s)p Fq(+1+)p Fp(\014)1974 2616 y Fs(strongly)f(in)m(tegrable)i(symplectic)456 2724 y(map.)j(Giv)m(en)29 b(the)h(form)e(of)h(the)h(in)m(tegrable)g (part,)f(the)h(mapping)e(will)h(satisfy)456 2832 y(the)h(t)m(wist)h (condition.)555 2940 y(By)41 b(the)g(standard)f(dep)s(endence)g(on)g (parameters)h(of)g(di\013eren)m(tial)h(equa-)456 3048 y(tions,)28 b(w)m(e)g(obtain)g(that)g(the)g Fm(C)1532 3015 y Fp(s)p Fl(\000)p Fq(1+)p Fp(\014)1784 3048 y Fs(norms)e(of)i (the)g(di\013erence)g(of)f(the)h(time-)456 3156 y(2)p Fm(\031)33 b Fs(maps)d(can)h(b)s(e)f(b)s(ounded)e(b)m(y)i Fm(")1654 3123 y Fp(m)p Fq(+1)1811 3156 y Fs(.)555 3264 y(Then,)25 b(if)f(w)m(e)g(apply)f(Theorem)h(45)h(to)f(constan)m(t)i(t)m (yp)s(e)e(n)m(um)m(b)s(ers)e(of)i(Mark)m(o)m(v)456 3372 y(constan)m(t)47 b Fm(\024)52 b Fs(=)f Fm(K)7 b(")1190 3339 y Fq(\()p Fp(m)p Fq(+1\))p Fp(=)p Fq(2)1518 3372 y Fs(with)45 b Fm(n)51 b Fs(=)g Fm(s)31 b Fn(\000)f Fs(1,)50 b(and)c(tak)m(e)h(in)m(to)g(accoun)m(t)456 3480 y(Lemma)34 b(43)h(to)g(con)m(trol)h(the)f(spacing)f(b)s(et)m(w)m(een)h(suc)m(h)f (n)m(um)m(b)s(ers,)h(w)m(e)f(obtain)456 3588 y(the)c(statemen)m(t)i(of) f(Theorem)f(38.)1466 b Fj(\003)555 3759 y Fs(By)38 b(Remark)f(37,)k (applying)c(Theorem)g(38)h(to)g(the)g(a)m(v)m(eraged)h(Hamilton-)456 3867 y(ian)23 b(\(57\))r(,)h(when)f Fm(r)9 b Fn(\000)d Fs(2)p Fm(m)g Fn(\000)g Fs(2)25 b Fn(\025)g Fs(6,)g(and)e(going)h(bac)m (k)g(to)g(the)f(v)-5 b(ariables)24 b(\()p Fm(J)o(;)15 b(';)g(s)p Fs(\))456 3975 y(using)25 b(the)h(c)m(hange)g(giv)m(en)h(b)m (y)f(Theorem)f(35,)j(w)m(e)e(obtain)g(the)g(follo)m(wing)h(result)456 4083 y(ab)s(out)j(the)g(existence)i(of)f(in)m(v)-5 b(arian)m(t)31 b(tori)g(of)g(Hamiltonian)g(\(45\))r(.)456 4251 y Fw(Prop)s(osition)43 b(47.)j Fo(Assume)38 b Fm(r)h Fn(\025)d Fs(2)p Fm(m)25 b Fs(+)f(8)p Fo(.)60 b(Then,)41 b(for)e Fm(")g Fo(smal)5 b(l)40 b(enough,)456 4359 y(in)g(any)g(c)-5 b(onne)g(cte)g(d)42 b(c)-5 b(omp)g(onent)43 b(of)d(the)g(non-r)-5 b(esonant)43 b(r)-5 b(e)g(gion)41 b Fn(S)2808 4326 y Fp(L)2900 4359 y Fo(de\014ne)-5 b(d)456 4466 y(in)39 b Fs(\(56\))r Fo(,)32 b(ther)-5 b(e)34 b(exists)f(a)g(\014nite)f(set)h(of)g(values)g Fm(J)2156 4480 y Fp(i)2217 4466 y Fo(such)f(that:)636 4613 y Fs(1\))42 b Fm(!)815 4627 y Fp(i)870 4613 y Fs(=)27 b Fm(J)1018 4627 y Fp(i)1068 4613 y Fs(+)21 b Fm(")1212 4577 y Fp(@)1255 4559 y Fq(\026)1253 4577 y Fp(k)1292 4554 y Fi(0)p Ff(;)p Fi(0)p 1212 4592 164 4 v 1251 4644 a Fp(@)t(J)1386 4613 y Fs(\()p Fm(J)1471 4627 y Fp(i)1499 4613 y Fm(;)15 b(")p Fs(\))35 b Fo(\(se)-5 b(e)41 b Fs(\(57\))r Fo(\))33 b(is)h(a)g(Diophantine)h(numb)-5 b(er)35 b(of)758 4748 y(c)-5 b(onstant)35 b(typ)-5 b(e)33 b(and)h(Markov)f(c)-5 b(onstant)34 b Fm(K)7 b(")2315 4685 y Ff(m)p Fi(+1)p 2315 4698 132 3 v 2366 4739 a(2)2461 4748 y Fo(.)636 4856 y Fs(2\))42 b Fo(Ther)-5 b(e)40 b(exists)f(a)g(torus)h Fn(T)1656 4870 y Fp(i)1683 4856 y Fo(,)g(invariant)g(by)f(the)g(\015ow) h(of)f(the)g(Hamil-)758 4964 y(tonian)34 b Fm(k)s Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))35 b Fo(given)d(in)40 b Fs(\(45\))q Fo(,)33 b(such)g(that:)p eop end %%Page: 54 54 TeXDict begin 54 53 bop 456 251 a Fq(54)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)753 450 y Fs(2.1\))43 b Fo(The)d(motion)h(on)f(the)g(torus)g Fn(T)2040 464 y Fp(i)2107 450 y Fo(is)g Fn(C)2265 417 y Fq(1)2304 450 y Fo(-c)-5 b(onjugate)g(d)41 b(to)f(a)g(rigid)946 558 y(tr)-5 b(anslation)36 b(of)d(fr)-5 b(e)g(quencies)32 b Fs(\()p Fm(!)2065 572 y Fp(i)2093 558 y Fm(;)15 b Fs(1\))p Fo(.)753 666 y Fs(2.2\))43 b Fo(The)34 b(torus)g Fn(T)1417 680 y Fp(i)1478 666 y Fo(c)-5 b(an)33 b(b)-5 b(e)34 b(written)g(as)f(a) h(gr)-5 b(aph)35 b(of)e(the)h(variable)g Fm(J)946 774 y Fo(over)f(the)g(angle)g(variables)h Fs(\()p Fm(';)15 b(s)p Fs(\))p Fo(:)961 935 y Fn(T)1011 949 y Fp(i)1064 935 y Fs(=)25 b Fn(f)p Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(\))27 b Fn(2)e(S)1681 897 y Fp(L)1733 935 y Fm(;)48 b(J)34 b Fs(=)25 b Fm(J)2036 949 y Fp(i)2085 935 y Fs(+)20 b Fm(u)2228 949 y Fp(!)2272 959 y Ff(i)2302 935 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))p Fn(g)p Fm(;)946 1090 y Fo(wher)-5 b(e)25 b Fm(u)1246 1104 y Fp(!)1290 1114 y Ff(i)1321 1090 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))26 b Fo(is)d(a)i Fn(C)1853 1057 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(4)p Fl(\000)p Fp(\021)2250 1090 y Fo(function,)h(for)e(any)h Fm(\021)k(>)c Fs(0)p Fo(,)946 1198 y(and)34 b Fn(jj)p Fm(u)1225 1212 y Fp(!)1269 1222 y Ff(i)1300 1198 y Fn(jj)1350 1217 y Fl(C)1391 1198 y Fi(2)g Fn(\024)25 b Fs(cte)p Fm(:)17 b(")p Fo(.)636 1305 y Fs(3\))42 b Fo(Denoting)33 b(by)456 1467 y Fs(\(67\))459 b Fw(B)p Fs(\()p Fn(A)p Fm(;)15 b(\032)p Fs(\))26 b(=)f Fn(f)6 b Fs(~)-51 b Fm(x)26 b Fn(2)1719 1444 y Fs(~)1710 1467 y(\003)1773 1481 y Fp(")1835 1467 y Fs(:)58 b(dist)15 b(\()p Fn(A)p Fm(;)21 b Fs(~)-51 b Fm(x)p Fs(\))26 b Fn(\024)f Fm(\032)p Fn(g)p Fm(;)758 1629 y Fo(for)34 b(any)f Fn(A)25 b(\032)1280 1606 y Fs(~)1272 1629 y(\003)1335 1643 y Fp(")1371 1629 y Fo(,)33 b(one)g(has)g(that)456 1799 y Fs(\(68\))738 b Fn(S)1416 1762 y Fp(L)1493 1799 y Fn(\032)1589 1713 y Fh([)1628 1908 y Fp(i)1705 1799 y Fw(B)p Fs(\()p Fn(T)1864 1813 y Fp(i)1893 1799 y Fm(;)15 b(K)7 b(")2069 1734 y Ff(m)p Fi(+1)p 2069 1747 132 3 v 2120 1788 a(2)2215 1799 y Fs(\))p Fm(:)456 2048 y Fw(Remark)40 b(48.)k Fs(Prop)s(osition)35 b(47)g(giv)m(es)h(the)f(primary)e(KAM)i (tori)g Fn(T)2875 2062 y Fp(i)2937 2048 y Fs(in)g(the)456 2156 y(v)-5 b(ariables)28 b(\()p Fm(J)o(;)15 b(';)g(s)p Fs(\),)29 b(but)e(w)m(e)h(can)f(obtain)h(the)f(tori)h(in)f(the)h (original)g(v)-5 b(ariables)456 2264 y(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))36 b(using)e(the)g(c)m(hange)i(giv)m(en)f(b)m(y)g (Prop)s(osition)f(28,)j(whic)m(h)d(is)g Fm(")2916 2231 y Fq(2)2956 2264 y Fs(-close)456 2372 y(to)e(the)g(iden)m(tit)m(y)-8 b(.)47 b(The)31 b(tori)h(th)m(us)f(obtained)h(are)h(in)m(v)-5 b(arian)m(t)32 b(for)g(the)g(\015o)m(w)h(\(7\))456 2480 y(and)c(are)i(of)g(the)f(form)907 2635 y Fn(T)957 2649 y Fp(i)1010 2635 y Fs(=)25 b Fn(f)p Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))28 b Fn(2)d(I)h(\002)20 b Fk(T)1792 2597 y Fq(2)1831 2635 y Fs(;)15 b Fm(I)33 b Fs(=)25 b Fm(I)2080 2649 y Fp(i)2128 2635 y Fs(+)20 b Fm(U)2281 2649 y Fp(!)2325 2659 y Ff(i)2355 2635 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))p Fn(g)p Fm(;)456 2789 y Fs(where)29 b(the)i(function)f Fm(U)1293 2803 y Fp(!)1337 2813 y Ff(i)1398 2789 y Fs(v)m(eri\014es)h(the)f(same)h(prop)s(erties)f(as)g Fm(u)2677 2803 y Fp(!)2721 2813 y Ff(i)2752 2789 y Fs(.)326 b Fj(\003)456 3003 y Fw(Remark)39 b(49.)44 b Fs(The)34 b(imp)s(ortance)g(of)g(Prop)s(osition)g(47)h(is)f(that)h(in)f(the)g (non-)456 3111 y(resonan)m(t)39 b(region)g Fn(S)1174 3078 y Fp(L)1265 3111 y Fs(w)m(e)h(can)f(\014nd)e(primary)h(KAM)h(tori) g(with)g(extremely)456 3219 y(small)30 b(gaps)h(b)s(et)m(w)m(een)g (them.)555 3327 y(It)40 b(is)f(imp)s(ortan)m(t)g(to)h(note)g(that,)i (for)d(a)h(\014xed)e(v)-5 b(alue)40 b(of)f Fm(")p Fs(,)j(it)e(is)f (enough)456 3435 y(to)48 b(consider)f(a)h(\014nite)f(n)m(um)m(b)s(er)f (of)h(tori)h Fn(T)2004 3449 y Fp(i)2079 3435 y Fs(to)g(ensure)f(that)h (the)f(regions)456 3557 y Fw(B)p Fs(\()p Fn(T)615 3571 y Fp(i)643 3557 y Fm(;)15 b(K)7 b(")819 3493 y Ff(m)p Fi(+1)p 819 3506 V 870 3547 a(2)965 3557 y Fs(\))35 b(co)m(v)m(er)i (all)e(the)g(non-resonan)m(t)g(region)h Fn(S)2454 3524 y Fp(L)2506 3557 y Fs(.)54 b(Of)34 b(course,)i(this)456 3665 y(n)m(um)m(b)s(er)29 b(go)s(es)i(to)g(in\014nit)m(y)f(when)f Fm(")i Fs(decreases)g(to)g(zero.)690 b Fj(\003)456 3879 y Fs(8.5.)46 b Fw(Analyzing)e(the)e(resonances.)k Fs(In)37 b(this)f(section)i(w)m(e)g(study)e(the)h(in-)456 3987 y(v)-5 b(arian)m(t)31 b(sets)g(close)g(to)h(resonan)m(t)e(regions.)555 4095 y(The)46 b(goal)h(is)f(that)g(w)m(e)h(can)f(co)m(v)m(er)h(the)g (whole)f(region)g(with)g(in)m(v)-5 b(arian)m(t)456 4203 y(ob)5 b(jects)39 b(\(either)h(primary)f(tori,)j(secondary)d(tori)h(or) f(p)s(erio)s(dic)f(orbits)h(with)456 4317 y(\(un\)stable)30 b(manifolds\))h(at)g(a)g(distance)g(less)g(than)2259 4325 y(O)2329 4317 y(\()p Fm(")2406 4284 y Fq(3)p Fp(=)p Fq(2)2517 4317 y Fs(\).)555 4425 y(The)25 b(case)h(of)f(resonances)h (of)f(order)g(3)g(and)f(higher)h(will)h(b)s(e)e(studied)h(in)f(Sec-)456 4533 y(tion)33 b(8.5.1.)49 b(It)33 b(will)g(not)g(b)s(e)f(di\013eren)m (t)h(from)f(the)h(non-resonan)m(t)g(region)g(and)456 4640 y(will)25 b(b)s(e)f(enough)g(to)i(apply)e(KAM)h(theorem)g(38)g(to) h(obtain)f(primary)f(tori)h(with)456 4748 y(the)30 b(required)g(gaps.) 555 4856 y(The)37 b(case)g(of)g(resonances)g(or)g(order)g(1)g(and)f(2)h (is)g(signi\014can)m(tly)h(more)f(in-)456 4964 y(v)m(olv)m(ed)31 b(and)f(it)h(will)g(b)s(e)f(done)g(in)g(Section)h(8.5.2,)h(Section)g (8.5.3.)p eop end %%Page: 55 55 TeXDict begin 55 54 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(55)456 450 y Fs(8.5.1.)47 b Fo(R)-5 b(esonanc)g(es)30 b(of)f(or)-5 b(der)30 b Fs(3)e Fo(and)i(higher.)46 b Fs(In)25 b(this)g(section)i(w)m(e)f(study)f(the) 456 558 y(reduced)g(Hamiltonian)j Fm(k)s Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))29 b(giv)m(en)e(in)f(\(45\))r(,)h(in)f (the)h(regions)f(close)i(to)456 666 y(the)36 b(resonances)i(of)e(order) h(3)g(or)f(bigger.)61 b(T)-8 b(o)37 b(this)f(end,)i(w)m(e)f(\014x)f Fm(j)42 b Fn(\025)35 b Fs(3)i(and)456 774 y(de\014ne:)456 882 y(\(69\))456 1008 y Fn(S)518 971 y Fl(R)578 981 y Ff(j)640 1008 y Fs(=)1091 922 y Fh([)736 1123 y Fl(\000)p Fp(l)812 1132 y Fi(1)846 1123 y Fp(=k)918 1132 y Fi(1)953 1123 y Fl(2R)1060 1133 y Ff(j)1093 1123 y Fl(nR)1188 1132 y Fi(1)1223 1123 y Fl([\001\001\001)o([R)1436 1133 y Ff(j)s Fg(\000)p Fi(1)1547 1008 y Fn(f)p Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(\))27 b Fn(2)e Fs([)p Fn(\000)p Fm(l)2129 1022 y Fq(1)2168 1008 y Fm(=k)2260 1022 y Fq(1)2306 1008 y Fn(\000)5 b Fm(L;)15 b Fn(\000)p Fm(l)2582 1022 y Fq(1)2621 1008 y Fm(=k)2713 1022 y Fq(1)2759 1008 y Fs(+)5 b Fm(L)p Fs(])g Fn(\002)g Fk(T)3064 971 y Fq(2)3103 1008 y Fn(g)p Fm(;)456 1284 y Fs(where)29 b Fm(L)i Fs(is)f(the)h(constan)m(t)g(pro)m (vided)f(in)g(Theorem)g(35.)555 1392 y(The)g(connected)h(comp)s(onen)m (ts)g(of)f(this)h(region)g(are)f(the)h(sets)956 1560 y Fn(f)p Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(\))28 b Fn(2)d Fs([)p Fn(\000)p Fm(l)1539 1574 y Fq(1)1578 1560 y Fm(=k)1670 1574 y Fq(1)1730 1560 y Fn(\000)20 b Fm(L;)15 b Fn(\000)p Fm(l)2021 1574 y Fq(1)2061 1560 y Fm(=k)2153 1574 y Fq(1)2213 1560 y Fs(+)20 b Fm(L)p Fs(])h Fn(\002)e Fk(T)2563 1522 y Fq(2)2603 1560 y Fn(g)p Fm(:)456 1728 y Fs(On)39 b(them,)44 b(after)d(the)g(a)m(v)m(eraging)i(pro)s(cedure)d(giv)m(en)h(in)g (Theorem)f(35,)45 b(the)456 1836 y(Hamiltonian)31 b(is)g(a)g Fn(C)1199 1803 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(2)1510 1836 y Fs(function)f(of)g(the)h(form:)456 1943 y(\(70\))516 2050 y Fn(B)579 2017 y Fq(2)p 516 2091 102 4 v 545 2174 a Fs(2)648 2112 y(+)20 b Fm(")783 2088 y Fs(\026)781 2112 y Fm(k)831 2074 y Fq(0)p Fp(;)p Fq(0)926 2112 y Fs(\()p Fn(B)s Fs(;)15 b Fm(")p Fs(\))21 b(+)f Fm(")1295 2074 y Fp(j)1347 2011 y Fh(\020)1402 2112 y Fm(U)1474 2074 y Fp(k)1511 2083 y Fi(1)1545 2074 y Fp(;l)1586 2083 y Fi(1)1624 2112 y Fs(\()p Fm(k)1706 2126 y Fq(1)1747 2112 y Fm(\013)g Fs(+)g Fm(l)1943 2126 y Fq(1)1983 2112 y Fm(s)p Fs(;)15 b Fm(")p Fs(\))21 b(+)f Fm(")2297 2074 y Fp(m)p Fq(+1)p Fl(\000)p Fp(j)2543 2088 y Fs(\026)2541 2112 y Fm(k)2591 2074 y Fq(1)2631 2112 y Fs(\()p Fn(B)s Fm(;)15 b(\013;)g(s)p Fs(;)g Fm(")p Fs(\))3027 2011 y Fh(\021)3098 2112 y Fm(:)456 2311 y Fs(W)-8 b(e)46 b(can)g(apply)f (Theorem)g(38)h(to)g(Hamiltonian)h(\(70\))f(since)g(it)g(is)f(of)h(the) 456 2419 y(form)35 b(\(57\))i(with)f Fm(")1128 2386 y Fp(j)1200 2419 y Fs(instead)g(of)g Fm(")1670 2386 y Fp(m)p Fq(+1)1828 2419 y Fs(.)56 b(When)36 b(w)m(e)g(express)g(the)g(results)f (in)456 2527 y(terms)24 b(of)h(the)g(v)-5 b(ariables)25 b(\()p Fm(J)o(;)15 b(';)g(s)p Fs(\))26 b(using)e(the)h(c)m(hange)g(giv) m(en)h(b)m(y)e(Theorem)h(35,)456 2634 y(w)m(e)30 b(obtain:)456 2809 y Fw(Prop)s(osition)37 b(50.)42 b Fo(Assume)33 b Fm(r)c Fn(\025)e Fs(2)p Fm(m)21 b Fs(+)f(8)p Fo(.)44 b(Then)34 b(for)g(any)g Fs(3)27 b Fn(\024)f Fm(j)33 b Fn(\024)26 b Fm(m)p Fo(,)33 b(in)456 2917 y(any)k(c)-5 b(onne)g(cte)g(d)38 b(c)-5 b(omp)g(onent)39 b(of)d(the)h(r)-5 b(esonant)39 b(r)-5 b(e)g(gion)37 b Fn(S)2482 2884 y Fl(R)2542 2894 y Ff(j)2579 2917 y Fo(,)g(ther)-5 b(e)38 b(exists)e(a)456 3025 y(\014nite)c(set)h(of)g(values)g Fm(J)1261 3039 y Fp(i)1322 3025 y Fo(such)g(that:)636 3172 y Fs(1\))42 b Fm(!)815 3186 y Fp(i)870 3172 y Fs(=)27 b Fm(J)1018 3186 y Fp(i)1068 3172 y Fs(+)21 b Fm(")1212 3137 y Fp(@)1255 3119 y Fq(\026)1253 3137 y Fp(k)1292 3113 y Fi(0)p Ff(;)p Fi(0)p 1212 3152 164 4 v 1251 3204 a Fp(@)t(J)1386 3172 y Fs(\()p Fm(J)1471 3186 y Fp(i)1499 3172 y Fm(;)15 b(")p Fs(\))35 b Fo(\(se)-5 b(e)41 b Fs(\(70\))r Fo(\))33 b(is)h(a)g(Diophantine)h(numb)-5 b(er)35 b(of)758 3295 y(c)-5 b(onstant)35 b(typ)-5 b(e)33 b(and)h(Markov)f(c)-5 b(onstant)34 b Fm(K)7 b(")2305 3262 y Fp(j)t(=)p Fq(2)2412 3295 y Fo(.)636 3403 y Fs(2\))42 b Fo(Ther)-5 b(e)31 b(exists)f(a)g(torus)h Fn(T)1620 3417 y Fp(i)1677 3403 y Fo(invariant)g(by)f(the)g(\015ow)h(of)f(the)g(Hamilton-)758 3511 y(ian)j Fm(k)s Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))35 b Fo(given)d(in)40 b Fs(\(45\))r Fo(,)32 b(such)h(that:)753 3619 y Fs(2.1\))43 b Fo(The)g(motion)i(on)e(the)g (torus)h Fn(T)2057 3633 y Fp(i)2127 3619 y Fo(c)-5 b(an)44 b(b)-5 b(e)43 b Fn(C)2480 3586 y Fq(1)2519 3619 y Fo(-c)-5 b(onjugate)g(d)44 b(to)f(a)946 3727 y(rigid)33 b(tr)-5 b(anslation)36 b(of)d(fr)-5 b(e)g(quencies)32 b Fs(\()p Fm(!)2280 3741 y Fp(i)2308 3727 y Fm(;)15 b Fs(1\))p Fo(.)753 3834 y Fs(2.2\))43 b Fo(The)34 b(torus)g Fn(T)1417 3848 y Fp(i)1478 3834 y Fo(c)-5 b(an)33 b(b)-5 b(e)34 b(written)g(as)f(a)h(gr)-5 b(aph)35 b(of)e(the)h(variable)g Fm(J)946 3942 y Fo(over)f(the)g(angle)g(variables)h Fs(\()p Fm(';)15 b(s)p Fs(\))p Fo(:)951 4116 y Fn(T)1001 4130 y Fp(i)1054 4116 y Fs(=)25 b Fn(f)p Fs(\()p Fm(J)o(;)15 b(';)g(s)p Fs(\))27 b Fn(2)e(S)1671 4079 y Fl(R)1731 4089 y Ff(j)1768 4116 y Fm(;)48 b(J)34 b Fs(=)25 b Fm(J)2071 4130 y Fp(i)2120 4116 y Fs(+)20 b Fm(u)2263 4130 y Fp(!)2307 4140 y Ff(i)2337 4116 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))p Fn(g)946 4284 y Fo(wher)-5 b(e)25 b Fm(u)1246 4298 y Fp(!)1290 4308 y Ff(i)1321 4284 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))26 b Fo(is)d(a)i Fn(C)1853 4251 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(4)p Fl(\000)p Fp(\021)2250 4284 y Fo(function,)h(for)e(any)h Fm(\021)k(>)c Fs(0)p 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4748 y(of)30 b(class)h Fn(C)825 4715 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(2)1136 4748 y Fs(\(see)h(Remark)e(37\).)555 4856 y(No)m(w,)38 b(w)m(e)e(can)g(state)h(explicitly)g(the)f(\014rst)f(part)g(of)h(the)g (non-degeneracy)456 4964 y(conditions)30 b(that)h(constituted)h(Hyp)s (othesis)e Fw(H5)g Fs(of)h(Theorem)f(7:)p eop end %%Page: 57 57 TeXDict begin 57 56 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(57)554 450 y Fw(H5')41 b Fs(F)-8 b(or)21 b(an)m(y)g(\014rst)f(or)g(second)g(order)g (resonance,)j(the)d(function)g Fm(U)2859 417 y Fp(k)2896 426 y Fi(0)2931 417 y Fp(;l)2972 426 y Fi(0)3010 450 y Fs(\()p Fm(\022)s Fs(;)15 b(0\))758 558 y(has)30 b(a)h(global)h (maxim)m(um)e(whic)m(h)g(is)h(non-degenerate.)456 730 y Fw(Remark)40 b(52.)k Fs(By)35 b(Theorem)f(35)i(applied)e(to)i (Hamiltonian)f(\(45\))i(w)m(e)e(ha)m(v)m(e)456 838 y(that,)c(in)f(the)g (case)i(of)e(a)h(\014rst)f(order)g(resonance)h Fn(\000)p Fm(l)2251 852 y Fq(0)2290 838 y Fm(=k)2382 852 y 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Fs(,)g(the)g(h)m(yp)s(othesis)456 3131 y Fw(H5')30 b Fs(holds)g(in)g(a)h Fm(C)1141 3098 y Fp(j)1207 3131 y Fm(j)g Fn(\025)25 b Fs(2)31 b(op)s(en)e(and)h(dense)g(of)h (Hamiltonians.)356 b Fj(\003)555 3359 y Fs(T)-8 b(o)28 b(study)e(Hamiltonian)j(\(72\))g(in)e(the)g(set)2058 3336 y(\026)2037 3359 y Fm(D)s Fs(,)h(w)m(e)g(consider)f(the)g(c)m (hange)i(of)456 3467 y(v)-5 b(ariables)31 b(dep)s(ending)d(on)j(time)g (giv)m(en)g(b)m(y)456 3631 y(\(73\))341 b Fm(b)25 b Fs(=)g Fm(k)1164 3645 y Fq(0)1204 3631 y Fs(\()p Fn(B)e Fs(+)d Fm(l)1440 3645 y Fq(0)1479 3631 y Fm(=k)1571 3645 y Fq(0)1611 3631 y Fs(\))p Fm(;)107 b(\022)27 b Fs(=)e Fm(k)1991 3645 y Fq(0)2031 3631 y Fm(\013)c Fs(+)f Fm(l)2228 3645 y Fq(0)2267 3631 y Fm(s;)106 b(s)25 b Fs(=)g Fm(s:)456 3794 y Fs(The)35 b(c)m(hange)h(\(73\))h(is)f(not)g(a)g(true)f (symplectic)i(c)m(hange)g(of)e(v)-5 b(ariables)36 b(but)f(it)456 3902 y(is)29 b(conformally)h(symplectic,)g(hence)g(the)f(new)f(system)i (of)f(di\013eren)m(tial)h(equa-)456 4010 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1715 y Fs(\()p Fm(")p Fs(\)\))26 b(=)f(\(0)p Fm(;)15 b(\022)1198 1729 y Fq(1)1239 1715 y Fs(\))10 b(+)g Fm(O)s Fs(\()p Fm(")p Fs(\).)39 b(This)25 b(saddle)g(corresp)s (onds)f(to)i(a)g(h)m(yp)s(erb)s(olic)456 1823 y(p)s(erio)s(dic)f(orbit) h(if)g(w)m(e)g(also)h(include)f(the)g Fm(s)f Fs(v)-5 b(ariable)27 b(as)f(the)g(time)h(co)s(ordinate.)456 1931 y(The)i(function)i Fm(b)p Fs(\()p Fm(")p Fs(\))g(is)f(of)h(class)g Fn(C)1642 1898 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(1)1923 1931 y Fs(,)f(and)g Fm(\022)2198 1945 y Fq(1)2237 1931 y Fs(\()p Fm(")p Fs(\))i(is)e(analytic.)555 2039 y(T)-8 b(o)28 b(analyze)h(the)f(b)s(eha)m(vior)g(of)g(this)f(p)s (endulum-lik)m(e)g(system,)h(w)m(e)g(will)g(\014nd)456 2147 y(it)j(con)m(v)m(enien)m(t)h(to)f(mak)m(e)g(the)g(translation)456 2304 y(\(76\))408 b Fm(y)28 b Fs(=)d Fm(b)20 b Fn(\000)g Fm(b)p Fs(\()p Fm(")p Fs(\))p Fm(;)199 b(x)25 b Fs(=)g Fm(\022)d Fn(\000)e Fm(\022)2090 2318 y Fq(1)2129 2304 y Fs(\()p Fm(")p Fs(\))p Fm(;)108 b(s)25 b Fs(=)g 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Fq(0)2698 4941 y Fs(\026)2687 4964 y Fm(L)p Fn(g)p Fm(:)p eop end %%Page: 59 59 TeXDict begin 59 58 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(59)456 450 y Fs(F)-8 b(rom)27 b(no)m(w)g(on,)h(w)m(e)f(w)m(ork)g(in)f(the)i(v)-5 b(ariables)27 b(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(\),)29 b(and,)e(when)f(necessary)-8 b(,)456 558 y(w)m(e)35 b(will)g(reco)m(v)m (er)i(the)e(original)g(Hamiltonian)i(b)m(y)d(p)s(erforming)g(the)h(c)m (hanges)456 666 y(of)30 b(v)-5 b(ariables)31 b(\(73\))r(,)f(\(76\))r(.) 555 774 y(The)35 b(\014rst)g(imp)s(ortan)m(t)h(p)s(oin)m(t)g(is)f (that,)j(up)d(to)h(order)f Fm(")2481 741 y Fp(m)2548 774 y Fs(,)j(the)d(Hamilton-)456 882 y(ian)28 b(\(77\))j(is)e(giv)m(en) g(b)m(y)g(the)g(one)g(degree)g(of)g(freedom)g Fn(C)2347 849 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(1)2656 882 y Fs(Hamiltonian:)456 1035 y(\(82\))540 b Fm(K)1233 1049 y Fq(0)1273 1035 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\))27 b(=)e Fm(h)1740 998 y Fq(0)1779 1035 y Fs(\()p Fm(y)s Fs(;)15 b Fm(")p Fs(\))22 b(+)e Fm(")2134 998 y Fp(j)2171 1035 y Fm(U)10 b Fs(\()p Fm(x)p Fs(;)15 b Fm(")p Fs(\))p Fm(;)456 1185 y Fs(and)30 b(the)h(energy)g(lev)m(el)h Fm(K)1365 1199 y Fq(0)1405 1185 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\))27 b(=)f(0)31 b(consists)g(on)g(the)g(saddle) g(\(0)p Fm(;)15 b Fs(0\))32 b(and)456 1293 y(its)e(separatrices.)456 1456 y Fw(Remark)24 b(53.)35 b Fs(Note)22 b(that)g(the)g(2)p Fm(\031)s(k)1703 1470 y Fq(0)1743 1456 y Fs(-p)s(erio)s(dic)f (Hamiltonians)h(\(74\))h(and)d(\(77\))456 1564 y(are,)28 b(up)e(to)h(order)g Fm(")1142 1531 y Fp(m)1209 1564 y Fs(,)h(2)p Fm(\031)i Fs(p)s(erio)s(dic.)39 b(This)26 b(is)h(the)g(w)m(ell)h(kno)m(wn)f(e\013ect)h(that)g(is)456 1672 y(collo)s(quially)j(describ)s(ed)d(as)i(sa)m(ying)g(that)g(the)g (resonance)g(has)f(\\)p Fm(k)2745 1686 y Fq(0)2814 1672 y Fs(ey)m(es"\(see)456 1780 y(Fig.)41 b(1\).)2347 b Fj(\003)456 1985 y Fs(8.5.3.)47 b Fo(Primary)27 b(and)g(se)-5 b(c)g(ondary)28 b(tori)f(ne)-5 b(ar)27 b(the)f(\014rst)h(and)g(se)-5 b(c)g(ond)27 b(or)-5 b(der)28 b(r)-5 b(es-)456 2093 y(onanc)g(es.)46 b Fs(As)28 b(the)g(Hamiltonian)i Fm(K)1724 2107 y Fq(0)1763 2093 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\))30 b(giv)m(en)f(in)e(\(82\))j(is)e(2)p Fm(\031)j Fs(p)s(erio)s(dic,)456 2201 y(the)f(region)h Fm(D)960 2216 y Fp(k)997 2225 y Fi(0)1066 2201 y Fs(giv)m(en)h(in)e(\(81\))i(can)e(b)s(e)g(seen)g(as)h Fm(k)2248 2215 y Fq(0)2318 2201 y Fs(copies)g(of)g(the)f(region)456 2355 y(\(83\))451 b Fm(D)28 b Fs(=)d Fn(f)p Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(\))26 b Fn(2)f Fk(R)20 b Fn(\002)g Fk(T)1954 2317 y Fq(2)1993 2355 y Fm(;)106 b Fn(j)p Fm(y)s Fn(j)26 b(\024)f Fm(k)2391 2369 y Fq(0)2441 2332 y Fs(\026)2430 2355 y Fm(L)p Fn(g)p Fm(:)456 2504 y Fs(where)k(this)i(Hamiltonian)g (is)g(w)m(ell)g(de\014ned.)555 2612 y(The)36 b(region)i Fm(D)s Fs(|and)d(then)h Fm(D)1667 2627 y Fp(k)1704 2636 y Fi(0)1743 2612 y Fs(|is)h(\014lled)f(b)m(y)h(the)f(energy)h(surfaces) g(of)456 2720 y(the)30 b(Hamiltonian)i Fm(K)1212 2734 y Fq(0)1252 2720 y Fs(,)e(giv)m(en)h(in)g(\(82\))q(:)755 2872 y Fn(T)828 2835 y Fq(0)805 2895 y Fp(E)892 2872 y Fs(=)25 b Fn(f)p Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(\))27 b Fn(2)e Fs([)p Fn(\000)p Fm(k)1582 2886 y Fq(0)1632 2849 y Fs(\026)1622 2872 y Fm(L)o(;)15 b(k)1770 2886 y Fq(0)1821 2849 y Fs(\026)1811 2872 y Fm(L)o Fs(])21 b Fn(\002)f Fk(T)2070 2835 y Fq(2)2134 2872 y Fs(:)41 b Fm(K)2277 2886 y Fq(0)2316 2872 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\))27 b(=)e Fm(E)5 b Fn(g)p Fm(:)456 3022 y Fn(T)528 2989 y Fq(0)506 3048 y Fp(E)598 3022 y Fs(will,)31 b(of)f(course,)h(b)s(e)f(in)m(v)-5 b(arian)m(t)32 b(b)m(y)e(the)g(Hamiltonian)i(\015o)m(w)f(of)f Fm(K)2883 3036 y Fq(0)2923 3022 y Fs(.)555 3129 y(The)35 b(energy)g(surfaces)g (that)h(corresp)s(ond)e(to)i(v)-5 b(alues)36 b(of)f Fm(E)k(>)33 b Fs(0,)k(are)f(t)m(w)m(o)456 3237 y(primary)22 b(tori)h(in)g(the)g (sense)g(of)g(De\014nition)h(2)f(in)g(Section)h(2.1.)39 b(These)23 b(primary)456 3345 y(tori)43 b(can)g(b)s(e)g(written)g(as)g (a)g(graph)g(of)g(the)g(v)-5 b(ariable)44 b Fm(y)h Fs(o)m(v)m(er)g(the) e(angular)456 3453 y(v)-5 b(ariables)31 b(\()p Fm(x;)15 b(s)p Fs(\).)555 3561 y(The)32 b(energy)h(surface)g(corresp)s(onding)e (to)j Fm(E)g Fs(=)28 b(0)33 b(consists)g(of)g(the)g(saddle)456 3669 y(\(0)p Fm(;)15 b Fs(0\))43 b(and)e(the)h(homo)s(clinic)h(orbits)e (to)i(it.)75 b(W)-8 b(e)43 b(will)f(refer)g(to)g Fn(T)2844 3636 y Fq(0)2821 3693 y(0)2925 3669 y Fs(as)g(the)456 3777 y(separatrix)30 b(lo)s(op.)555 3885 y(When)41 b Fm(E)48 b(<)42 b Fs(0)f(the)g(in)m(v)-5 b(arian)m(t)42 b(surfaces)e(\(whic)m(h)h(are)g(con)m(tained)h(inside)456 3993 y(the)37 b(region)h(b)s(ounded)d(b)m(y)i(the)g(separatrix)g(lo)s (op)h Fn(T)2286 3960 y Fq(0)2264 4017 y(0)2326 3993 y Fs(\))f(are)g(tori)h(of)f(di\013eren)m(t)456 4101 y(top)s(ology)25 b(than)g(the)f(primary)g(tori,)i(since)f(they)g(are)g(con)m(tractible)i (to)e(a)g(p)s(oin)m(t.)456 4209 y(They)j(are)g(secondary)h(tori)g(in)f 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b Fs(is)c(small|it)g(con)m (tains)h(a)f(factor)g Fm(")2641 2036 y Fp(m)p Fq(+1)2828 2069 y Fs(and)f Fm(m)g Fs(is)456 2177 y(large|w)m(e)37 b(will)f(see)g(that)g(it)g(remains)f(small)h(when)f(written)h(in)f(the) h(action)456 2285 y(angle-v)-5 b(ariables)37 b(pro)m(vided)d(that)i(w)m (e)g(do)f(not)g(consider)g(it)h(to)s(o)f(close)i(to)f(the)456 2393 y(separatrix.)j(Then,)27 b(w)m(e)g(can)g(apply)f(the)h(KAM)g (theorem)g(38)g(and)f(can)h(obtain)456 2501 y(tori)h(close)h(to)g(a)f (high)g(p)s(o)m(w)m(er)g(of)g Fm(")g Fs(to)h(the)f(separatrix)g(lo)s (op.)40 b(F)-8 b(or)29 b(subsequen)m(t)456 2610 y(purp)s(oses)f (getting)k(tori)f(whic)m(h)f(are)h(at)g(a)g(distance)g Fm(")2308 2577 y Fq(3)p Fp(=)p Fq(2)2449 2610 y Fs(is)f(enough.)456 2779 y Fw(Remark)45 b(54.)i Fs(Similar)38 b(argumen)m(ts)i(for)e (analytic)j(systems)e(w)m(ere)g(used)f(in)456 2887 y([Ne)-10 b(\025)-35 b(\02084)r(].)82 b(In)43 b(the)h(analytic)i(case)f (considered)f(there,)k(the)d(tori)f(are)h(exp)s(o-)456 2995 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b(are)i(at)g(distance)1316 4749 y(O)1386 4741 y(\()p Fm(\016)s Fs(\))h(at)f(all)g(the)f(p)s(oin)m(ts.)39 b(Nev)m(ertheless)28 b(w)m(e)f(see)f(that)456 4848 y Fn(T)506 4863 y Fp(\016)580 4848 y Fs(and)36 b Fn(T)813 4862 y Fq(0)889 4848 y Fs(are)h(t)m(w)m(o)h(tori)g(suc)m(h)e(that)i (for)e Fm(x)g Fs(=)g Fm(\031)s(=)p Fs(2)i(the)f(distance)h(is)2963 4856 y(O)3034 4848 y(\()p Fm(\016)s Fs(\),)456 4964 y(whereas)30 b(at)h Fm(x)25 b Fs(=)g(0)31 b(the)f(distance)h(is)1762 4972 y(O)1833 4964 y(\()1868 4887 y Fn(p)p 1944 4887 44 4 v 77 x Fm(\016)t Fs(\).)p eop end %%Page: 61 61 TeXDict begin 61 60 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(61)555 450 y Fs(F)-8 b(or)38 b(small)g Fm(E)5 b Fs(,)39 b(the)f(same)f(situation)h(p)s (ersists.)61 b(The)37 b(tori)g Fn(T)2696 464 y Fp(E)2793 450 y Fs(and)f Fn(T)3026 465 y Fp(E)t Fq(+)p Fp(\016)456 562 y Fs(are)d(at)h(a)g(distance)1159 570 y(O)1230 562 y(\()p Fm(\016)s Fs(\))h(in)e(the)g(middle)g(and)f(at)i(a)g(distance) 2726 526 y Fp(\016)p 2685 541 115 4 v 2685 551 a Fl(p)p 2744 551 56 3 v 56 x Fp(E)2843 562 y Fs(near)f(the)456 686 y(singularit)m(y)e Fm(x)25 b Fs(=)g(0.)555 794 y(It)32 b(is)g(clear)h(that)g(if)e(w)m(e)i(are)f(considering)g(the)g(tori)g(at) h(distances)f Fm(\016)g Fs(=)c Fm(")3064 761 y Fq(3)p Fp(=)p Fq(2)456 902 y Fs(and)i(are)h(in)m(terested)g(in)g(e\013ects)h (of)e(size)i Fm(")p Fs(,)f(w)m(e)g(ha)m(v)m(e)h(to)f(distinguish)f(the) h(size)456 1010 y(of)38 b(their)h(energies.)65 b(This)38 b(will)h(a\013ect)g(whic)m(h)g(terms)f(in)g(an)g(expansion)g(are)456 1118 y(dominan)m(t.)555 1226 y(The)22 b(c)m(hoices)i(w)m(e)f(ha)m(v)m (e)h(made)f(b)s(elo)m(w)f(are)h(not)g(the)f(only)h(p)s(ossible)f(ones,) j(but)456 1334 y(the)k(ab)s(o)m(v)m(e)i(considerations)g(sho)m(w)e (that)h(there)g(are)g(quan)m(titativ)m(ely)i(di\013eren)m(t)456 1442 y(features)e(in)g(di\013eren)m(t)h(regions.)1538 b Fj(\003)555 1719 y Fm(D)630 1734 y Fp(k)667 1743 y Fi(0)735 1719 y Fs(will)30 b(b)s(e)e(divided)h(in)g(three)g(regions.)41 b Fm(D)2103 1734 y Fp(f)2178 1719 y Fs(is)29 b(the)h(region)g(far)f (from)g(the)456 1827 y(separatrix,)d Fm(D)979 1841 y Fp(o)1042 1827 y Fs(close)g(to)f(the)g(separatrices)h(but)e(out)g(of)h (the)g(region)g(b)s(ounded)456 1935 y(b)m(y)37 b(the)h(separatrix)g(lo) s(op)f(and)g Fm(D)1653 1949 y Fp(in)1762 1935 y Fs(close)i(to)f(the)g (separatrices)g(but)f(inside)456 2043 y(the)30 b(separatrix)h(lo)s(op.) 555 2151 y(The)f(precise)h(de\014nitions)f(w)m(e)h(ha)m(v)m(e)g(found)e (useful)h(are:)559 2346 y Fm(D)634 2361 y Fp(f)763 2346 y Fs(=)82 b Fn(f)p Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(\))27 b Fn(2)e Fm(D)1442 2361 y Fp(k)1479 2370 y Fi(0)1543 2346 y Fs(:)41 b Fm(K)1686 2360 y Fq(0)1725 2346 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\))27 b(=)e Fm(E)5 b(;)31 b(c)2307 2360 y Fq(1)2347 2346 y Fm(")2389 2309 y Fp(j)2451 2346 y Fn(\024)25 b Fm(E)30 b Fn(\024)25 b Fm(c)2779 2360 y Fq(2)2830 2323 y Fs(\026)2819 2346 y Fm(L)p Fn(g)-2470 b Fs(\(84\))566 2491 y Fm(D)641 2505 y Fp(o)763 2491 y Fs(=)82 b Fn(f)p Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(\))27 b Fn(2)e Fm(D)1442 2506 y Fp(k)1479 2515 y Fi(0)1543 2491 y Fs(:)41 b Fm(K)1686 2505 y Fq(0)1725 2491 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\))27 b(=)e Fm(F)s(;)31 b(c)2296 2505 y Fq(3)2335 2491 y Fm(")2377 2453 y Fp(\013)2452 2491 y Fn(\024)25 b Fm(F)39 b Fn(\024)25 b Fm(c)2780 2505 y Fq(1)2820 2491 y Fm(")2862 2453 y Fp(j)2898 2491 y Fn(g)-2487 b Fs(\(85\))533 2631 y Fm(D)608 2645 y Fp(in)763 2631 y Fs(=)82 b Fn(f)p Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(\))27 b Fn(2)e Fm(D)1442 2646 y Fp(k)1479 2655 y Fi(0)1543 2631 y Fs(:)41 b Fm(K)1686 2645 y Fq(0)1725 2631 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\))27 b(=)e Fm(G;)31 b Fn(\000)p Fm(c)2377 2645 y Fq(4)2417 2631 y Fm(")2459 2594 y Fp(j)2521 2631 y Fn(\024)25 b Fm(G)g Fn(\024)g(\000)p Fm(c)2919 2645 y Fq(3)2959 2631 y Fm(")3001 2594 y Fp(\013)3051 2631 y Fn(g)-2640 b Fs(\(86\))456 2826 y(where)31 b Fm(\013)p Fs(,)h(for)f(the)h(time)g(b)s(eing,)g(is)f (arbitrary)g(pro)m(vided)g(that)i Fm(\013)27 b(>)g(j)33 b Fs(=)26 b(1)p Fm(;)15 b Fs(2,)456 2934 y(whic)m(h)42 b(is)g(the)h(order)e(of)i(the)f(resonance,)47 b(and)41 b Fm(c)2216 2948 y Fp(i)2287 2934 y Fs(are)i(suitable)g(constan)m(ts) 456 3042 y(indep)s(enden)m(t)34 b(of)h Fm(")p Fs(.)55 b(\(In)35 b(particular,)i Fm(c)1850 3056 y Fq(4)1923 3042 y Fm(<)c(c)p Fs(,)j(where)f Fn(\000)p Fm(c)g Fs(is)g(the)h(minim)m (um)456 3150 y(of)e Fm(U)10 b Fs(,)35 b(is)g(tak)m(en)g(small)g(enough) f(in)g(order)f(that)i Fm(D)2228 3164 y Fp(in)2334 3150 y Fs(do)s(es)e(not)i(con)m(tain)h(an)m(y)456 3258 y(other)30 b(critical)j(p)s(oin)m(t)d(of)g Fm(K)1414 3272 y Fq(0)1454 3258 y Fs(.\))555 3366 y(As)c(w)m(e)h(will)g(see,)h Fm(\013)f Fs(con)m(trols)g(ho)m(w)f(close)i(to)f(the)g(separatrices)g(w)m(e)g (claim)g(to)456 3474 y(\014nd)20 b(tori.)38 b(Roughly)22 b(sp)s(eaking,)h(w)m(e)f(can)g(tak)m(e)h Fm(\013)f Fs(an)m(y)g(n)m(um)m (b)s(er,)g(pro)m(vided)f(that)456 3582 y(w)m(e)28 b(are)g(willing)h(to) f(p)s(erform)f(enough)h(steps)f(of)h(a)m(v)m(eraging)j(\(this)d(amoun)m (ts)g(to)456 3690 y(tak)m(e)j Fm(m)e Fs(big)g(enough)h(in)f(\(77\))q(,) h(whic)m(h)f(can)h(b)s(e)f(done)g(just)g(b)m(y)g(assuming)g(that)456 3798 y(the)h(original)i(system)e(is)h(di\013eren)m(tiable)g(enough\).) 555 3906 y(Ev)m(en)40 b(if)g(w)m(e)h(state)g(Theorem)e(56)i(for)f (arbitrarily)g Fm(\013)i(>)e(j)46 b Fs(and)39 b Fm(m)h Fs(large)456 4014 y(enough)20 b(with)g(resp)s(ect)g(to)h Fm(\013)p Fs(,)i(w)m(e)e(note)g(that)g(for)f(the)h(subsequen)m(t)f (applications)456 4122 y(in)33 b(this)g(pap)s(er,)g(it)h(will)g (su\016ce)f(to)h(tak)m(e)h(an)m(y)f(v)-5 b(alue)34 b Fm(\013)c(>)g Fs(1)23 b(+)f Fm(j)5 b(=)p Fs(2,)36 b Fm(j)f Fs(=)30 b(1)p Fm(;)15 b Fs(2.)456 4230 y(Later,)33 b(in)g(Corollary)g (57,)h(for)e(the)g(sak)m(e)i(of)f(de\014niteness,)f(w)m(e)h(will)g(tak) m(e)h Fm(\013)29 b Fs(=)456 4337 y(3)p Fm(=)p Fs(2)21 b(+)f Fm(j)5 b(=)p Fs(2,)32 b(and)e Fm(m)25 b Fs(=)g(26.)555 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b(region)i(\(87\))h(not)e(co)m(v)m(ered)i(in)e(the)h(presen)m(t)f (analysis)h(con)m(tains)g(what)f(in)456 558 y(ph)m(ysical)e(language)h (is)f(called)g(the)g(\\c)m(haotic)j(sea".)41 b(In)26 b(Section)j(8.5.5)g(w)m(e)g(will)456 666 y(iden)m(tify)35 b(features,)h(other)f(than)g(in)m(v)-5 b(arian)m(t)35 b(tori,)i(in)d(\(87\))j(namely)-8 b(,)36 b(p)s(erio)s(dic)456 774 y(orbits)30 b(and)g(their)g(\(un\)stable)h(manifolds.)555 882 y(Recall)24 b(that)g Fm(K)1088 896 y Fq(0)1127 882 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\))25 b(as)d(giv)m(en)i (in)e(\(82\))i(describ)s(es)e(a)h(p)s(endulum.)36 b(Since)456 990 y(ha)m(v)m(e)j(assumed)f(that)h(w)m(e)g(do)g(not)f(ha)m(v)m(e)i(an) m(y)f(other)g(critical)h(p)s(oin)m(ts)e(in)h(the)456 1098 y(region)31 b(w)m(e)f(consider,)h(all)g(the)g(orbits)f(w)m(e)h (will)g(study)e(are)i(p)s(erio)s(dic)f(orbits.)555 1206 y(Since)k(w)m(e)g(will)h(b)s(e)e(applying)g(KAM)h(argumen)m(ts,)i(the)e (v)-5 b(alues)34 b(of)g(the)g(fre-)456 1314 y(quency)g Fm(!)s Fs(\()p Fm(E)5 b Fs(\))36 b(and)e(its)i(non-degeneracy)f(prop)s (erties)g(will)g(pla)m(y)h(an)f(imp)s(or-)456 1421 y(tan)m(t)c(role.) 555 1529 y(An)f(orbit)h(of)f Fm(K)1107 1543 y Fq(0)1177 1529 y Fs(of)h(energy)f Fm(E)36 b Fs(has)30 b(frequency)456 1736 y(\(88\))486 b Fm(!)s Fs(\()p Fm(E)5 b Fs(\))26 b(=)1491 1675 y(2)p Fm(\031)p 1436 1715 209 4 v 1436 1799 a(T)13 b Fs(\()p Fm(E)5 b Fs(\))1655 1736 y Fm(;)46 b(T)13 b Fs(\()p Fm(E)5 b Fs(\))26 b(=)2056 1613 y Fh(Z)2106 1819 y Fp(K)2171 1789 y Fg(\000)p Fi(1)2166 1840 y(0)2253 1819 y Fq(\()p Fp(E)t Fq(\))2393 1675 y Fm(dx)p 2393 1715 100 4 v 2418 1799 a(y)2502 1736 y(:)456 1989 y Fw(Theorem)d(56.)34 b Fo(Consider)24 b(the)g Fn(C)1627 1956 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(2)1931 1989 y Fo(r)-5 b(e)g(duc)g(e)g(d)25 b(Hamiltonian)g Fm(K)7 b Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(;)g Fm(")p Fs(\))456 2097 y Fo(as)34 b(in)41 b Fs(\(77\))q Fo(,)34 b(inside)g(the)h(r)-5 b(e)g(gion)35 b Fm(D)1682 2112 y Fp(k)1719 2121 y Fi(0)1791 2097 y Fo(given)e(in)41 b Fs(\(81\))r Fo(.)k(Consider)35 b Fm(\013)28 b(>)g(j)5 b Fo(,)34 b(for)456 2205 y Fm(j)42 b Fs(=)36 b(1)p Fm(;)15 b Fs(2)p Fo(,)41 b Fm(m)36 b Fn(\025)g Fs(max)q(\(11)p Fm(j)31 b Fn(\000)24 b Fs(1)p Fm(;)15 b Fs(14\()p Fm(\013)28 b Fn(\000)c Fm(j)5 b Fs(\))26 b Fn(\000)e Fs(1)h Fn(\000)g Fm(j)5 b(=)p Fs(2\))p Fo(,)41 b(and)f(assume)g(that)456 2312 y Fm(r)27 b Fn(\025)e Fs(2)p Fm(m)c Fs(+)f(8)p Fo(.)42 b(Then,)33 b(for)g Fn(j)p Fm(")p Fn(j)g Fo(smal)5 b(l)34 b(enough,)f(one)g(has:)636 2449 y Fs(1\))42 b Fd(Primary)e(tori)g(far)h (fr)-5 b(om)40 b(r)-5 b(esonanc)g(e.)50 b Fo(Ther)-5 b(e)36 b(exists)g(a)f(set)h(of)758 2557 y(values)d Fm(E)1097 2571 y Fq(1)1162 2557 y Fm(<)25 b Fn(\001)15 b(\001)g(\001)26 b Fm(<)f(E)1552 2572 y Fp(l)1573 2583 y Ff(E)1662 2557 y Fo(verifying)32 b Fm(c)2076 2571 y Fq(1)2116 2557 y Fm(")2158 2524 y Fp(j)2220 2557 y Fn(\024)25 b Fm(E)2383 2571 y Fp(i)2436 2557 y Fn(\024)g Fm(c)2571 2571 y Fq(2)2622 2534 y Fs(\026)2611 2557 y Fm(L)p Fo(,)32 b(such)h(that:)753 2664 y Fs(1.1\))43 b Fo(The)35 b(fr)-5 b(e)g(quencies)34 b Fm(!)s Fs(\()p Fm(E)1761 2678 y Fp(i)1790 2664 y Fs(\))h Fo(ar)-5 b(e)35 b(Diophantine)h(numb)-5 b(ers)35 b(of)g(c)-5 b(on-)946 2795 y(stant)43 b(typ)-5 b(e)42 b(and)g(Markov)g(c)-5 b(onstant)43 b Fm(K)7 b(")2412 2729 y Ff(m)p Fi(+1)p Fg(\000)p Fi(6)p Ff(j)p 2412 2744 239 3 v 2516 2785 a Fi(2)2706 2795 y Fo(\(se)-5 b(e)42 b(De\014ni-)946 2903 y(tion)34 b(42\).)753 3011 y Fs(1.2\))43 b Fo(F)-7 b(or)34 b(any)f(value)g Fm(E)1592 3025 y Fp(i)1620 3011 y Fo(,)g(ther)-5 b(e)33 b(exist)g(two)h(primary)g(invariant)g(tori)946 3120 y Fn(T)1019 3082 y Fl(\006)996 3149 y Fp(E)1048 3159 y Ff(i)1111 3120 y Fo(of)e(Hamiltonian)42 b Fs(\(77\))34 b Fo(c)-5 b(ontaine)g(d)34 b(in)f Fm(D)2537 3135 y Fp(f)2583 3120 y Fo(.)753 3246 y Fs(1.3\))43 b Fo(The)d(motion)i(on)e(the)g(tori) h Fn(T)2007 3208 y Fl(\006)1984 3274 y Fp(E)2036 3284 y Ff(i)2106 3246 y Fo(is)e Fn(C)2263 3213 y Fq(1)2303 3246 y Fo(-c)-5 b(onjugate)g(d)41 b(to)f(a)h(rigid)946 3361 y(tr)-5 b(anslation)36 b(of)d(fr)-5 b(e)g(quencies)32 b Fs(\()p Fm(!)s Fs(\()p Fm(E)2170 3375 y Fp(i)2199 3361 y Fs(\))p Fm(;)15 b Fs(1\))p Fo(.)753 3469 y Fs(1.4\))43 b Fo(These)33 b(tori)h(c)-5 b(an)33 b(b)-5 b(e)32 b(written)i(as:)720 3635 y Fn(T)793 3596 y Fq(+)770 3663 y Fp(E)822 3673 y Ff(i)877 3635 y Fs(=)25 b Fn(f)p Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(\))27 b Fn(2)e Fm(D)1499 3650 y Fp(f)1544 3635 y Fm(;)48 b(K)1694 3649 y Fp(E)1746 3659 y Ff(i)1777 3635 y Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(;)g Fm(")p Fs(\))27 b(=)e Fm(E)2342 3649 y Fp(i)2370 3635 y Fm(;)48 b(y)28 b(>)d Fs(0)p Fn(g)p Fm(;)735 3787 y Fn(T)808 3749 y Fl(\000)785 3816 y Fp(E)837 3826 y Ff(i)892 3787 y Fs(=)g Fn(f)p Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(\))27 b Fn(2)e Fm(D)1514 3802 y Fp(f)1560 3787 y Fm(;)47 b(K)1709 3801 y Fp(E)1761 3811 y Ff(i)1792 3787 y Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(;)g Fm(")p Fs(\))27 b(=)e Fm(E)2357 3801 y Fp(i)2385 3787 y Fm(;)48 b(y)28 b(<)d Fs(0)p Fn(g)p Fm(;)946 3966 y Fo(wher)-5 b(e)47 b Fm(K)1293 3980 y Fp(E)1345 3990 y Ff(i)1375 3966 y Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(;)g Fm(")p Fs(\))48 b Fo(is)d(a)h Fn(C)2053 3933 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(5)p Fl(\000)p Fp(\021)2472 3966 y Fo(function,)j(for)d(any)946 4074 y Fm(\021)29 b(>)c Fs(0)p Fo(,)33 b(given)f(by)456 4261 y Fs(\(89\))285 b Fm(K)978 4275 y Fp(E)1030 4285 y Ff(i)1060 4261 y Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(;)g Fm(")p Fs(\))27 b(=)e Fm(K)1635 4275 y Fq(0)1675 4261 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\))21 b(+)2079 4269 y(O)2150 4281 y Fl(C)2191 4262 y Fi(2)2245 4160 y Fh(\020)2299 4261 y Fm(")2351 4193 y Ff(m)p Fi(+1)p Fg(\000)p Fi(11)p Ff(j)p 2351 4209 269 3 v 2471 4250 a Fi(2)2634 4160 y Fh(\021)2704 4261 y Fm(:)753 4489 y Fs(1.5\))43 b Fm(D)1021 4504 y Fp(f)1092 4489 y Fn(\032)1188 4421 y Fh(S)1264 4516 y Fp(i)1322 4489 y Fm(B)1411 4389 y Fh(\020)1465 4489 y Fn(T)1538 4451 y Fl(\006)1515 4518 y Fp(E)1567 4528 y Ff(i)1597 4489 y Fm(;)15 b(")1689 4423 y 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4978 y Fq(1)2597 4964 y Fm(")2639 4931 y Fp(j)2676 4964 y Fo(,)33 b(such)f(that:)p eop end %%Page: 63 63 TeXDict begin 63 62 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(63)753 450 y Fs(2.1\))43 b Fo(The)36 b(fr)-5 b(e)g(quencies)36 b Fm(!)s Fs(\()p Fm(F)1755 464 y Fp(i)1784 450 y Fs(\))g Fo(ar)-5 b(e)36 b(Diophantine)h(numb)-5 b(ers)37 b(of)f(c)-5 b(on-)946 588 y(stant)34 b(typ)-5 b(e)34 b(and)f(Markov)g(c)-5 b(onstant)34 b Fm(K)7 b(")2368 518 y Ff(m)p Fi(+1)p Fg(\000)p Fi(6\()p Ff(\013)p Fg(\000)p Ff(j)s Fi(\))p Fg(\000)p Ff(j)s(=)p Fi(2)p 2369 537 513 3 v 2610 578 a(2)2896 588 y Fo(.)753 696 y Fs(2.2\))43 b Fo(F)-7 b(or)35 b(any)f(value)g Fm(F)1586 710 y Fp(i)1614 696 y Fo(,)g(ther)-5 b(e)35 b(exist)e(two)i(primary)g(invariant)g(tori)946 805 y Fn(T)1019 767 y Fl(\006)996 834 y Fp(F)1041 844 y Ff(i)1111 805 y Fo(of)d(Hamiltonian)42 b Fs(\(77\))34 b Fo(c)-5 b(ontaine)g(d)34 b(in)f Fm(D)2537 819 y Fp(o)2575 805 y Fo(.)753 931 y Fs(2.3\))43 b Fo(The)d(motion)i(on)e(the)g(tori)h Fn(T)2007 892 y Fl(\006)1984 959 y Fp(F)2029 969 y Ff(i)2106 931 y Fo(is)e Fn(C)2263 898 y Fq(1)2303 931 y Fo(-c)-5 b(onjugate)g(d)41 b(to)f(a)h(rigid)946 1046 y(tr)-5 b(anslation)36 b(of)d(fr)-5 b(e)g(quencies)32 b Fs(\()p Fm(!)s Fs(\()p Fm(F)2161 1060 y Fp(i)2190 1046 y Fs(\))p Fm(;)15 b Fs(1\))p Fo(.)753 1154 y Fs(2.4\))43 b Fo(These)33 b(tori)h(c)-5 b(an)33 b(b)-5 b(e)32 b(written)i(as:)752 1304 y Fn(T)825 1266 y Fq(+)802 1332 y Fp(F)847 1342 y Ff(i)909 1304 y Fs(=)25 b Fn(f)p Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(\))26 b Fn(2)f Fm(D)1530 1318 y Fp(o)1569 1304 y Fm(;)48 b(K)1719 1318 y Fp(F)1764 1328 y Ff(i)1794 1304 y Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(;)g Fm(")p Fs(\))27 b(=)e Fm(F)2350 1318 y Fp(i)2379 1304 y Fm(;)48 b(y)28 b(>)d Fs(0)p Fn(g)752 1456 y(T)825 1418 y Fl(\000)802 1485 y Fp(F)847 1495 y Ff(i)909 1456 y Fs(=)g Fn(f)p Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(\))26 b Fn(2)f Fm(D)1530 1470 y Fp(o)1569 1456 y Fm(;)48 b(K)1719 1470 y Fp(F)1764 1480 y Ff(i)1794 1456 y Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(;)g Fm(")p Fs(\))27 b(=)e Fm(F)2350 1470 y Fp(i)2379 1456 y Fm(;)48 b(y)28 b(<)d Fs(0)p Fn(g)946 1620 y Fo(wher)-5 b(e)48 b Fm(K)1294 1634 y Fp(F)1339 1644 y Ff(i)1369 1620 y Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(;)g Fm(")p Fs(\))49 b Fo(is)d(a)h Fn(C)2050 1587 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(5)p Fl(\000)p Fp(\021)2470 1620 y Fo(function,)j(for)d(any)946 1728 y Fm(\021)29 b(>)c Fs(0)p Fo(,)33 b(given)f(by:)456 1903 y Fs(\(91\))232 b Fm(K)925 1917 y Fp(F)970 1927 y Ff(i)1001 1903 y Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(;)g Fm(")p Fs(\))27 b(=)e Fm(K)1576 1917 y Fq(0)1615 1903 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\))22 b(+)2020 1911 y(O)2091 1922 y Fl(C)2132 1903 y Fi(2)2185 1802 y Fh(\020)2240 1903 y Fm(")2292 1832 y Ff(m)p Fi(+1+)p Ff(j)s(=)p Fi(2)p Fg(\000)p Fi(14\()p Ff(\013)p Fg(\000)p Ff(j)s Fi(\))p 2292 1850 542 3 v 2548 1891 a(2)2848 1802 y Fh(\021)2918 1903 y Fm(:)753 2117 y Fs(2.5\))43 b Fm(D)1021 2131 y Fp(o)1085 2117 y Fn(\032)1181 2049 y Fh(S)1257 2144 y Fp(i)1315 2117 y Fm(B)1404 2016 y Fh(\020)1458 2117 y Fn(T)1531 2079 y Fl(\006)1508 2146 y Fp(F)1553 2156 y Ff(i)1590 2117 y Fm(;)15 b(")1682 2048 y Ff(m)p Fi(+1+)p Ff(j)s(=)p Fi(2)p Fg(\000)p Fi(10\()p Ff(\013)p Fg(\000)p Ff(j)s Fi(\))p 1683 2066 V 1938 2108 a(2)2239 2016 y Fh(\021)2308 2117 y Fm(:)636 2253 y Fs(3\))42 b Fd(Se)-5 b(c)g(ondary)37 b(tori)g(close)g(to)g(r)-5 b(esonanc)g(e.)42 b Fo(Ther)-5 b(e)33 b(exists)g(a)f(set)h(of)758 2361 y(values)38 b Fm(G)1106 2375 y Fq(1)1179 2361 y Fm(<)32 b Fn(\001)15 b(\001)g(\001)35 b Fm(<)e(G)1597 2376 y Fp(l)1618 2387 y Ff(G)1711 2361 y Fo(verifying)j Fn(\000)p Fm(c)2200 2375 y Fq(4)2240 2361 y Fm(")2282 2328 y Fp(j)2352 2361 y Fn(\024)d Fm(G)2527 2375 y Fp(i)2589 2361 y Fn(\024)f(\000)p Fm(c)2802 2375 y Fq(3)2842 2361 y Fm(")2884 2328 y Fp(\013)2934 2361 y Fo(,)38 b(such)758 2469 y(that:)753 2577 y Fs(3.1\))43 b Fo(The)34 b(fr)-5 b(e)g(quencies)34 b Fm(!)s Fs(\()p Fm(G)1764 2591 y Fp(i)1793 2577 y Fs(\))g Fo(ar)-5 b(e)35 b(Diophantine)g(numb)-5 b(ers)35 b(of)f(c)-5 b(on-)946 2714 y(stant)34 b(typ)-5 b(e)34 b(and)f(Markov)g(c)-5 b(onstant)34 b Fm(K)7 b(")2368 2645 y Ff(m)p Fi(+1)p Fg(\000)p Fi(6\()p Ff(\013)p Fg(\000)p Ff(j)s Fi(\))p Fg(\000)p Ff(j)s(=)p Fi(2)p 2369 2664 513 3 v 2610 2705 a(2)2896 2714 y Fo(.)753 2822 y Fs(3.2\))43 b Fo(F)-7 b(or)25 b(any)g(value)f Fm(G)1570 2836 y Fp(i)1598 2822 y Fo(,)i(ther)-5 b(e)24 b(exists)h(a)f(se)-5 b(c)g(ondary)26 b(invariant)f(torus)946 2930 y Fn(T)996 2944 y Fp(G)1051 2954 y Ff(i)1121 2930 y Fo(of)40 b(Hamiltonian)48 b Fs(\(77\))41 b Fo(c)-5 b(ontaine)g(d)41 b(in)f Fm(D)2582 2944 y Fp(in)2653 2930 y Fo(,)h(c)-5 b(ontr)g(actible)946 3038 y(to)34 b(the)f(set)1018 3187 y Fn(f)p Fs(\(0)p Fm(;)15 b(a;)g(s)p Fs(\))p Fm(;)g(a)28 b Fn(2)d Fk(R)p Fm(;)48 b(s)24 b Fn(2)h Fk(R)p Fm(=)p Fs(\(2)p Fm(\031)s(k)2136 3201 y Fq(0)2177 3187 y Fk(Z)p Fs(\))p Fn(g)h(\032)f Fm(D)2515 3201 y Fp(in)2586 3187 y Fm(:)753 3335 y Fs(3.3\))43 b Fo(The)34 b(motion)h(on)g(the)f(torus)g Fn(T)2011 3349 y Fp(G)2066 3359 y Ff(i)2130 3335 y Fo(is)g Fn(C)2282 3302 y Fq(1)2321 3335 y Fo(-c)-5 b(onjugate)g(d)35 b(to)f(a)h(rigid)946 3443 y(tr)-5 b(anslation)36 b(of)d(fr)-5 b(e)g(quencies)32 b Fs(\()p Fm(!)s Fs(\()p Fm(G)2174 3457 y Fp(i)2203 3443 y Fs(\))p Fm(;)15 b Fs(1\))p Fo(.)753 3551 y Fs(3.4\))43 b Fo(This)33 b(torus)h(c)-5 b(an)33 b(b)-5 b(e)33 b(written)g(as:)949 3700 y Fn(T)999 3714 y Fp(G)1054 3724 y Ff(i)1109 3700 y Fs(=)25 b Fn(f)p Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(\))27 b Fn(2)e Fm(D)1731 3714 y Fp(in)1802 3700 y Fm(;)48 b(K)1952 3714 y Fp(G)2007 3724 y Ff(i)2038 3700 y Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(;)g Fm(")p Fs(\))27 b(=)d Fm(G)2606 3714 y Fp(i)2635 3700 y Fn(g)946 3849 y Fo(wher)-5 b(e)47 b Fm(K)1293 3863 y Fp(G)1348 3873 y Ff(i)1378 3849 y Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(;)g Fm(")p Fs(\))47 b Fo(is)e(a)h Fn(C)2055 3816 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(5)p Fl(\000)p Fp(\021)2473 3849 y Fo(function,)j(for)c(any)946 3957 y Fm(\021)29 b(>)c Fs(0)p Fo(,)33 b(given)f(by:)456 4132 y Fs(\(92\))227 b Fm(K)920 4146 y Fp(G)975 4156 y Ff(i)1006 4132 y Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(;)g Fm(")p Fs(\))27 b(=)e Fm(K)1581 4146 y Fq(0)1620 4132 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\))22 b(+)2025 4140 y(O)2096 4151 y Fl(C)2137 4132 y Fi(2)2191 4031 y Fh(\020)2245 4132 y Fm(")2297 4061 y Ff(m)p Fi(+1+)p Ff(j)s(=)p Fi(2)p Fg(\000)p Fi(14\()p Ff(\013)p Fg(\000)p Ff(j)s Fi(\))p 2297 4079 542 3 v 2553 4120 a(2)2853 4031 y Fh(\021)2923 4132 y Fm(:)753 4346 y Fs(3.5\))43 b Fm(D)1021 4360 y Fp(in)1118 4346 y Fn(\032)1214 4278 y Fh(S)1290 4373 y Fp(i)1348 4346 y Fm(B)1437 4246 y Fh(\020)1491 4346 y Fn(T)1541 4360 y Fp(G)1596 4370 y Ff(i)1626 4346 y Fm(;)15 b(")1718 4277 y Ff(m)p Fi(+1+)p Ff(j)s(=)p Fi(2)p Fg(\000)p Fi(10\()p Ff(\013)p Fg(\000)p Ff(j)s Fi(\))p 1719 4295 V 1975 4337 a(2)2275 4246 y Fh(\021)2344 4346 y Fm(:)555 4533 y Fs(Theorem)40 b(56)h(giv)m(es)h(the)e(existence)i(of)e(in)m(v)-5 b(arian)m(t)42 b(tori)e(in)g Fm(D)2753 4548 y Fp(k)2790 4557 y Fi(0)2829 4533 y Fs(.)70 b(In)40 b(the)456 4640 y(follo)m(wing)d(Corollary)f(57,) i(w)m(e)e(will)g(mak)m(e)h(more)f(explicit)h(assertions)f(ab)s(out)456 4748 y(the)44 b(pro)m(ximit)m(y)i(of)f(these)g(tori.)84 b(W)-8 b(e)46 b(will)f(also)h(mak)m(e)f(precise)g(assertions)456 4856 y(of)38 b(their)h(prop)s(erties)f(when)g(expressed)g(as)h(graphs)f (of)g(functions)h(from)f(the)456 4964 y(angle)45 b(v)-5 b(ariables)45 b(to)g(the)f(action)i(v)-5 b(ariable.)83 b(The)43 b(pro)s(of)h(of)g(Corollary)h(57)p eop end %%Page: 64 64 TeXDict begin 64 63 bop 456 251 a Fq(64)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 450 y Fs(from)31 b(Theorem)g(56)h(is)f(just)g(an)h(easy)g(application)h(of)e(the)h (implicit)g(function)456 558 y(theorem.)55 b(Nev)m(ertheless,)38 b(w)m(e)d(state)i(it)f(explicitly)g(b)s(ecause)f(the)g(prop)s(erties) 456 666 y(of)d(the)g(tori)g(as)g(graphs)g(will)g(b)s(e)f(useful)g (later)i(in)f(Section)g(9)h(when)d(w)m(e)j(study)456 774 y(ho)m(w)d(the)h(tori)g(b)s(eha)m(v)m(e)g(under)d(the)j(scattering) h(map.)456 956 y Fw(Corollary)58 b(57.)52 b Fo(Consider)f(the)f(r)-5 b(e)g(duc)g(e)g(d)52 b(Hamiltonian)f Fm(K)7 b Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(;)g Fm(")p Fs(\))51 b Fo(as)456 1064 y(in)35 b Fs(\(77\))r Fo(,)29 b(inside)g(the)g(r)-5 b(e)g(gion)29 b Fm(D)1538 1079 y Fp(k)1575 1088 y Fi(0)1643 1064 y Fo(given)f(in)35 b Fs(\(81\))r Fo(.)40 b(Consider)30 b Fm(\013)c Fs(=)e Fm(j)5 b(=)p Fs(2)11 b(+)g(3)p Fm(=)p Fs(2)p Fo(,)456 1172 y(for)37 b Fm(j)i Fs(=)32 b(1)p Fm(;)15 b Fs(2)p Fo(,)39 b(and)f Fm(m)33 b Fn(\025)g Fs(26)p Fo(.)55 b(Then,)38 b(if)f Fm(r)f Fn(\025)c Fs(2)p Fm(m)24 b Fs(+)f(8)p Fo(,)38 b(the)f(tori)h(de\014ne)-5 b(d)38 b(in)456 1280 y(The)-5 b(or)g(em)34 b(56)g(verify:)601 1424 y Fs(\(1\))42 b Fo(F)-7 b(or)49 b(any)f(value)g Fm(E)1449 1438 y Fp(i)1477 1424 y Fo(,)j(the)d(primary)h(tori)f 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2217 y Fs(=)25 b Fn(f)p Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(\))26 b Fn(2)f Fm(D)1741 2231 y Fp(o)1780 2217 y Fm(;)48 b(y)28 b Fs(=)d Fm(f)2077 2179 y Fl(\006)2067 2246 y Fp(F)2112 2256 y Ff(i)2141 2217 y Fs(\()p Fm(x;)15 b(s)p Fs(;)g Fm(")p Fs(\))p Fn(g)p Fm(:)601 2405 y Fs(\(3\))42 b Fo(Ther)-5 b(e)38 b(exists)g Fm(\032)1325 2419 y Fq(0)1398 2405 y Fm(>)33 b Fs(0)38 b Fo(such)f(that)i(for)f(any)g Fs(0)c Fm(<)f(\032)2546 2419 y Fq(0)2619 2405 y Fn(\024)h Fm(\032)f(<)h(\031)s Fo(,)k(and)758 2513 y(for)c(any)f(value)f Fm(G)1383 2527 y Fp(i)1412 2513 y Fo(,)h(e)-5 b(ach)33 b(of)g(the)g(c)-5 b(omp)g(onents)35 b(of)566 2699 y Fn(T)616 2713 y Fp(G)671 2723 y Ff(i)721 2699 y Fn(\\)20 b(f)p Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(\))27 b(:)58 b Fm(x)25 b Fn(2)g Fm(I)1453 2713 y Fp(\032)1493 2699 y Fn(g)p Fm(;)109 b(I)1712 2713 y Fp(\032)1777 2699 y Fs(=)25 b Fn([)1934 2658 y Fp(k)1971 2667 y Fi(0)2005 2658 y Fl(\000)p Fq(1)1934 2728 y Fp(l)q Fq(=0)2100 2699 y Fs([2)p Fm(\031)s(l)d Fs(+)e Fm(\032;)15 b Fs(2)p Fm(\031)s Fs(\()p Fm(l)24 b Fs(+)c(1\))h Fn(\000)f Fm(\032)p Fs(])p Fn(g)p Fm(;)758 2894 y Fo(that)35 b(we)e(wil)5 b(l)34 b(denote)f(by)g Fn(T)1735 2849 y Fl(\006)p Fp(;\032)1712 2922 y(G)1767 2932 y Ff(i)1850 2894 y Fo(,)g(c)-5 b(an)34 b(b)-5 b(e)33 b(written)h(as)g(a)f(gr)-5 b(aph)35 b(of)e(the)758 3009 y(action)h Fm(y)h Fo(over)e(the)g(angles)g Fs(\()p Fm(x;)15 b(s)p Fs(\))p Fo(:)867 3197 y Fn(T)940 3153 y Fl(\006)p Fp(;\032)917 3226 y(G)972 3236 y Ff(i)1080 3197 y Fs(=)25 b Fn(f)p Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(\))27 b Fn(2)d Fm(D)1701 3211 y Fp(in)1773 3197 y Fm(;)15 b(x)25 b Fn(2)g Fm(I)2016 3211 y Fp(\032)2056 3197 y Fm(;)48 b(y)28 b Fs(=)d Fm(f)2353 3159 y Fl(\006)2343 3226 y Fp(G)2398 3236 y Ff(i)2428 3197 y Fs(\()p Fm(x;)15 b(s)p Fs(;)g Fm(")p Fs(\))p Fn(g)601 3385 y Fs(\(4\))42 b Fo(Ther)-5 b(e)34 b(exists)f(a)g(c)-5 b(onstant)34 b Fm(K)7 b Fo(,)32 b(indep)-5 b(endent)34 b(of)f Fm(")p Fo(,)g(such)g(that)837 3590 y Fn(j)p Fm(E)929 3604 y Fp(i)977 3590 y Fn(\000)20 b Fm(E)1135 3604 y Fp(i)p Fq(+1)1254 3590 y Fn(j)83 b(\024)g Fm(K)7 b(")1642 3553 y Fq(\()p Fp(m)p Fq(+1)p Fl(\000)p Fq(7)p Fp(j)t Fq(\))p Fp(=)p Fq(2)2072 3590 y Fn(\024)25 b Fm(K)7 b(")2304 3526 y Fi(3)p 2304 3538 31 3 v 2304 3579 a(2)2344 3553 y Fq(+)2410 3522 y Ff(j)p 2409 3538 V 2409 3579 a Fi(2)854 3750 y Fn(j)p Fm(F)937 3764 y Fp(i)986 3750 y Fn(\000)20 b Fm(F)1135 3764 y Fp(i)p Fq(+1)1254 3750 y Fn(j)83 b(\024)g Fm(K)7 b(")1642 3712 y Fq(\()p Fp(m)p Fq(+1+)p Fp(j)t(=)p Fq(2)p Fl(\000)p Fq(10\()p Fp(\013)p Fl(\000)p Fp(j)t Fq(\)\))p Fp(=)p Fq(2)2420 3750 y Fn(\024)25 b Fm(K)7 b(")2652 3685 y Fi(3)p 2652 3697 V 2652 3738 a(2)2692 3712 y Fq(+)2758 3681 y Ff(j)p 2757 3697 V 2757 3738 a Fi(2)828 3909 y Fn(j)p Fm(G)924 3923 y Fp(i)973 3909 y Fn(\000)20 b Fm(G)1135 3923 y Fp(i)p Fq(+1)1254 3909 y Fn(j)83 b(\024)g Fm(K)7 b(")1642 3871 y Fq(\()p Fp(m)p Fq(+1+)p Fp(j)t(=)p Fq(2)p Fl(\000)p Fq(10\()p Fp(\013)p Fl(\000)p Fp(j)t Fq(\)\))p Fp(=)p Fq(2)2420 3909 y Fn(\024)25 b Fm(K)7 b(")2652 3844 y Fi(3)p 2652 3856 V 2652 3897 a(2)2692 3871 y Fq(+)2758 3841 y Ff(j)p 2757 3856 V 2757 3897 a Fi(2)601 4090 y Fs(\(5\))753 4295 y Fn(j)p Fm(E)845 4309 y Fq(1)905 4295 y Fn(\000)20 b Fm(F)1054 4310 y Fp(l)1075 4321 y Ff(F)1130 4295 y Fn(j)84 b(\024)e Fm(K)7 b(")1518 4257 y Fq(\()p Fp(m)p Fq(+1)p Fl(\000)p Fq(7)p Fp(j)t Fq(\))p Fp(=)p Fq(2)1949 4295 y Fn(\024)25 b Fm(K)7 b(")2181 4230 y Fi(3)p 2181 4242 V 2181 4283 a(2)2221 4257 y Fq(+)2287 4227 y Ff(j)p 2286 4242 V 2286 4283 a Fi(2)748 4458 y Fn(j)p Fm(F)831 4472 y Fq(1)891 4458 y Fn(\000)20 b Fm(G)1053 4473 y Fp(l)1074 4484 y Ff(G)1130 4458 y Fn(j)84 b(\024)e Fm(K)7 b(")1518 4420 y Fp(\013)1588 4458 y Fs(+)20 b Fm(")1721 4420 y Fq(\()p Fp(m)p Fq(+1+)p Fp(j)t(=)p Fq(2)p Fl(\000)p Fq(10\()p Fp(\013)p Fl(\000)p Fp(j)t Fq(\)\))p Fp(=)p Fq(2)2500 4458 y Fn(\024)25 b Fm(K)7 b(")2732 4393 y Fi(3)p 2732 4405 V 2732 4446 a(2)2772 4420 y Fq(+)2838 4389 y Ff(j)p 2837 4405 V 2837 4446 a Fi(2)601 4640 y Fs(\(6\))42 b Fo(A)n(l)5 b(l)44 b(these)h(functions)g Fm(f)1613 4654 y Fp(\035)1704 4640 y Fs(=)h Fm(f)1876 4607 y Fl(\006)1866 4663 y Fp(\035)1979 4640 y Fo(ar)-5 b(e)45 b(at)h(le)-5 b(ast)45 b(of)g(class)g Fn(C)2887 4607 y Fq(2)2927 4640 y Fo(,)i(and,)758 4748 y(denoting)f(by)e Fm(D)j Fo(the)e(derivatives)g(with)g(r)-5 b(esp)g(e)g(ct)46 b(to)f Fm(x)f Fo(and)i Fm(s)p Fo(,)g(for)758 4856 y Fm(\035)e Fs(=)39 b Fm(E)1028 4870 y Fp(i)1056 4856 y Fo(,)k Fm(i)d Fs(=)g(1)p Fm(;)15 b(:)g(:)g(:)i(;)e(l)1583 4870 y Fp(E)1643 4856 y Fo(,)42 b Fm(\035)i Fs(=)39 b Fm(F)1974 4870 y Fp(i)2003 4856 y Fo(,)j Fm(i)f Fs(=)e(1)p Fm(;)15 b(:)g(:)g(:)i(;)e(l) 2529 4870 y Fp(F)2588 4856 y Fo(,)43 b(and)f Fm(\035)h Fs(=)c Fm(G)3117 4870 y Fp(i)3146 4856 y Fo(,)758 4964 y Fm(i)26 b Fs(=)f(1)p Fm(;)15 b(:)g(:)g(:)i(;)e(l)1185 4978 y Fp(G)1245 4964 y Fo(,)32 b(they)h(verify:)p eop end %%Page: 65 65 TeXDict begin 65 64 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(65)789 450 y Fs(\(a\))42 b Fo(Ther)-5 b(e)32 b(exists)g(a)g(function)f Fn(Y)7 b Fs(\()p Fm(x;)15 b(E)5 b Fs(\))p Fo(,)33 b(given)e(explicitly)h(in)38 b Fs(\(97\))r Fo(,)946 558 y(such)33 b(that:)805 704 y Fn(j)p Fm(f)875 718 y Fp(E)927 728 y Ff(i)977 704 y Fn(\000)20 b(Y)7 b Fs(\()p Fm(x;)15 b(E)1330 718 y Fp(i)1359 704 y Fs(\))p Fn(j)1420 732 y Fl(C)1461 713 y Fi(1)1525 704 y Fn(\024)p Fm(K)7 b(")1722 666 y Fq(\()p Fp(m)p Fq(+1)p Fl(\000)p Fq(12)p Fp(j)t Fq(\))p Fp(=)p Fq(2)2187 704 y Fn(\024)25 b Fm(K)7 b(")2409 666 y Fq(3)p Fp(=)p Fq(2)821 863 y Fn(j)p Fm(f)891 877 y Fp(F)936 887 y Ff(i)986 863 y Fn(\000)20 b(Y)7 b Fs(\()p Fm(x;)15 b(F)1330 877 y Fp(i)1359 863 y Fs(\))p Fn(j)1420 891 y Fl(C)1461 872 y Fi(1)1525 863 y Fn(\024)p Fm(K)7 b(")1722 826 y Fq(\()p Fp(m)p Fq(+1)p Fl(\000)p Fp(j)t(=)p Fq(2)p Fl(\000)p Fq(14\()p Fp(\013)p Fl(\000)p Fp(j)t Fq(\)\))p Fp(=)p Fq(2)2500 863 y Fn(\024)25 b Fm(K)7 b(")2722 826 y Fq(3)p Fp(=)p Fq(2)797 1022 y Fn(j)q Fm(f)868 1036 y Fp(G)923 1046 y Ff(i)973 1022 y Fn(\000)20 b(Y)7 b Fs(\()p 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Fn(j)1550 1546 y Fl(C)1591 1527 y Fi(1)1655 1519 y Fn(\024)1761 1458 y(j)p Fm(\035)24 b Fn(\000)i Fs(\026)-51 b Fm(\035)s Fn(j)p 1761 1498 267 4 v 1819 1585 a Fm(")1861 1559 y Fp(j)t(=)p Fq(2)2062 1519 y Fn(\024)25 b Fm(K)7 b(")2284 1482 y Fq(3)p Fp(=)p Fq(2)2395 1519 y Fm(:)456 1723 y Fw(Remark)32 b(58.)40 b Fs(If)27 b(w)m(e)h(go)g(bac)m(k)h(to)f (the)g(original)h(v)-5 b(ariables)28 b(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))30 b(through)456 1831 y(the)38 b(c)m(hanges)h(giv)m (en)h(b)m(y)e(Prop)s(osition)g(28,)k(Theorem)c(35)h(and)e(the)i(c)m (hanges)456 1946 y(\(73\))q(,)27 b(and)f(\(76\))q(,)i(w)m(e)e(obtain)h (that)f(the)h(tori)f(inside)g(the)h(region)f Fn(S)2713 1913 y Fl(R)2773 1890 y Ff(j)2810 1946 y Fs(,)h Fm(J)35 b Fs(=)25 b(1)p Fm(;)15 b Fs(2)456 2054 y(are)30 b(giv)m(en)i(b)m(y)-8 b(,)1327 2175 y Fm(I)32 b Fs(=)25 b Fn(\000)p Fm(k)1613 2189 y Fq(0)1653 2175 y Fm(=l)1725 2189 y Fq(0)1785 2175 y Fs(+)20 b Fm(U)1948 2137 y Fl(\006)1938 2197 y Fp(\035)2007 2175 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))456 2314 y(where)35 b(the)g(functions)g Fm(U)1354 2281 y Fl(\006)1344 2336 y Fp(\035)1413 2314 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))38 b(v)m(erify)e(the)g(same)f(prop)s(erties)g (than)h(the)456 2422 y(functions)42 b Fm(f)915 2389 y Fl(\006)905 2444 y Fp(\035)973 2422 y Fs(.)79 b(This)42 b(is,)47 b(of)c(course,)k(a)c(re\015ection)h(of)f(the)g(fact)h(that)g (the)456 2530 y(prop)s(erties)22 b(of)h(pro)m(ximit)m(y)h(b)s(et)m(w)m (een)f(the)g(tori)h(are)f(geometric)i(prop)s(erties)d(that)456 2638 y(are)30 b(in)m(v)-5 b(arian)m(t)32 b(under)d(c)m(hanges)i(of)g(v) -5 b(ariables.)1045 b Fj(\003)456 2857 y Fs(8.5.4.)47 b Fo(Pr)-5 b(o)g(of)29 b(of)f(The)-5 b(or)g(em)29 b(56)f(and)g(Cor)-5 b(ol)5 b(lary)31 b(57.)46 b Fs(As)24 b(indicated,)j(the)e(main)456 2965 y(tec)m(hnique)i(will)g(b)s(e)g(to)g(estimate)i(the)e (singularities)h(of)f(the)g(c)m(hange)h(to)g(action)456 3073 y(angle)h(v)-5 b(ariables)30 b(in)e(the)h(curv)m(es)g(that)g(are)g 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Fm(")p Fs(\))29 b(+)f Fm(\016)s(U)10 b Fs(\()p Fm(x)p Fs(;)15 b Fm(")p Fs(\))47 b Fn(')1328 532 y Fp(y)1365 508 y Fi(2)p 1328 552 72 4 v 1346 604 a Fq(2)1438 573 y Fs(+)28 b Fm(")1579 540 y Fp(j)1616 573 y Fm(U)10 b Fs(\()p Fm(x)p Fs(;)15 b Fm(")p Fs(\),)46 b(so)d(that)g(the)f(main)g(term)g(of)h(the)456 681 y(solution)31 b(of)37 b(\(95\))32 b(is)456 856 y(\(96\))710 b Fm(y)28 b Fs(=)d Fn(\006)1566 774 y Fh(p)p 1657 774 621 4 v 82 x Fs(2\()p Fm(E)h Fn(\000)20 b Fm(\016)s(U)10 b Fs(\()p Fm(x)p Fs(;)15 b Fm(")p Fs(\)\))s Fm(:)555 1019 y Fs(Indeed,)46 b(if)d(w)m(e)h(express)f(the)g Fm(y)j Fs(in)d(\(95\))i(as)e(a)h(function)f(of)50 b(\(96\))45 b(equa-)456 1126 y(tion)c(\(95\))i(b)s(ecomes)e(an)g(equation)h(whic)m (h)e(can)i(b)s(e)e(dealt)i(b)m(y)f(the)g(implicit)456 1234 y(function)24 b(theorem.)39 b(Note)26 b(that)f(the)g(energy)g(lev) m(els)h(of)f(the)f(p)s(endulum,)g(when)456 1342 y(expressed)29 b(as)i(a)g(graph,)f(ha)m(v)m(e)h(square)g(ro)s(ot)f(singularities)i(at) f Fm(E)f Fs(=)25 b(0.)555 1450 y(What)i(w)m(e)f(w)m(an)m(t)h(to)f(sho)m (w)g(is)g(that,)h(if)f(w)m(e)h(add)e(extra)h(terms)g(to)h(the)f(Hamil-) 456 1558 y(tonian,)40 b(the)d(lev)m(el)j(sets)e(can)f(b)s(e)g(still)i (expressed)e(as)h(graphs)f(and)g(that)h(the)456 1666 y(singularities)f(are)f(still)h(the)g(square)f(ro)s(ot)g (singularities.)59 b(This)36 b(will)g(b)s(e)g(im-)456 1774 y(p)s(ortan)m(t)i(for)g(us)f(later)j(since)e(the)h(singularities)g (of)f(the)g(graph)g(con)m(trol)i(the)456 1882 y(singularities)31 b(of)f(the)h(action-angle)i(v)-5 b(ariables.)555 1990 y(As)40 b(it)h(is)f(standard)f(in)h(dynamical)h(systems,)i(when)c (dealing)i(with)f(p)s(er-)456 2098 y(turbations)27 b(of)g(systems)h (that)g(in)m(v)m(olv)m(e)h(singularities,)h(it)e(is)f(more)h(con)m(v)m (enien)m(t)456 2206 y(to)34 b(form)m(ulate)g(the)g(problem)f(in)g(a)h (system)g(of)f(co)s(ordinates)i(whic)m(h)e(incorp)s(o-)456 2314 y(rates)25 b(the)h(singularities.)40 b(Hence)25 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Fs(\))47 b(=)f Fn(\000)p Fm(ax)3036 2911 y Fq(2)3103 2944 y Fs(+)758 3052 y Fm(O)s Fs(\()p Fm(x)917 3019 y Fq(3)957 3052 y Fs(\))p Fo(,)33 b(as)g Fm(x)25 b Fn(!)g Fs(0)p Fo(.)601 3160 y Fs(\(3\))42 b Fm(h)p Fs(\()p Fm(y)s Fs(;)15 b Fm(")p Fs(\))35 b Fo(is)d(of)h(the)g(form)41 b Fs(\(80\))q Fo(.)555 3296 y(L)-5 b(et)33 b Fm(\016)c Fn(2)c Fs([)p Fn(\000)p Fm(\016)1003 3310 y Fq(0)1043 3296 y Fm(;)15 b(\016)1123 3310 y Fq(0)1163 3296 y Fs(])33 b Fo(and)g(c)-5 b(onsider)34 b(the)f(e)-5 b(quation)41 b Fs(\(95\))q Fo(.)h(Then)601 3431 y Fs(\(1\))g Fo(F)-7 b(or)34 b Fn(j)p Fm(")p Fn(j)f Fo(smal)5 b(l)33 b(enough,)g(and)g(for)g Fn(\000)p Fm(c\016)c Fn(\024)c Fm(E)30 b Fn(\024)25 b Fm(M)10 b Fo(,)33 b(wher)-5 b(e)33 b Fm(M)42 b Fo(is)32 b(in-)758 3539 y(dep)-5 b(endent)30 b(of)e Fm(")g Fo(and)h Fm(\016)s Fo(,)h(e)-5 b(quation)35 b Fs(\(95\))30 b Fo(de\014nes)e(two) h(functions)f Fm(y)g Fs(=)758 3647 y Fn(Y)819 3661 y Fl(\006)878 3647 y Fs(\()p Fm(x;)15 b(E)5 b Fs(\))28 b Fo(on)g Fn(I)j Fs(=)25 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2815 y Fq(2)p 1414 2889 86 4 v 1434 2972 a Fs(2)1530 2910 y(+)20 b Fm("z)1709 2872 y Fq(2)1750 2886 y Fs(~)1749 2910 y Fm(h)p Fs(\()p Fm(tz)t Fs(;)15 b Fm(")p Fs(\))27 b(=)e(1)p Fm(:)456 3075 y(z)32 b Fs(=)27 b(1)33 b(is)f(a)g(solution)h(when)d Fm(")f Fs(=)e(0.)46 b(The)32 b(implicit)h(function)e(theorem)i(giv)m(es)456 3183 y(us)c(the)i(existence)h(of)e(a)h(solution)g(of)f(the)h(form)1073 3342 y Fm(z)f Fs(=)25 b(1)20 b(+)g Fm("g)s Fs(\()p Fm(t)p Fs(;)15 b Fm(")p Fs(\))28 b(=)d(1)20 b(+)g Fm("b)p Fs(\()p Fm(")p Fs(\))i(+)e Fm(")s Fs(~)-48 b Fm(g)t Fs(\()p Fm(t)p Fs(;)15 b Fm(")p Fs(\))p Fm(;)456 3501 y Fs(where)38 b Fm(b)p Fs(\()p Fm(")p Fs(\))i(=)f Fm(g)s Fs(\(0;)15 b Fm(")p Fs(\),)43 b(so)c(that)g Fm(")s Fs(~)-48 b Fm(g)43 b Fs(is)c Fn(C)1945 3468 y Fp(n)2031 3501 y Fs(function)f(with)g(a)h(b) s(ounded)e Fn(C)3127 3468 y Fp(n)456 3609 y Fs(norm)c(v)m(erifying)38 b(~)-48 b Fm(g)t Fs(\(0)p Fm(;)15 b(")p Fs(\))34 b(=)d(0.)54 b(Therefore,)35 b(the)g(solution)g(of)f(equation)h(\(95\))456 3717 y(is)30 b(giv)m(en)h(b)m(y)941 3875 y Fm(y)d Fs(=)d Fn(\006)p Fm(`)p Fs(\()p Fm(x;)15 b(E)5 b Fs(\)\(1)22 b(+)e Fm("b)p Fs(\))h(+)f Fm("`)p Fs(\()p Fm(x;)15 b(E)5 b Fs(\))s(~)-48 b Fm(g)5 b Fs(\()p Fm(`)p Fs(\()p Fm(x;)15 b(E)5 b Fs(\);)15 b Fm(")p Fs(\))p Fm(:)456 4034 y Fs(W)-8 b(riting)38 b(~)-52 b Fm(y)s Fs(\()p Fm(t)p Fs(\))26 b(=)f Fm(t)s Fs(~)-48 b Fm(g)t Fs(\()p Fm(t)p Fs(;)15 b Fm(")p Fs(\))32 b(w)m(e)f(obtain)g(\(97\))q(.)41 b(Bounds)30 b(\(98\))i(follo)m(w)g(from)e(F)-8 b(aa-)456 4142 y(di-Bruno)29 b(form)m(ulae.)555 4250 y(The)h(pro)s(of)g(of)g(part)g(2.)42 b(of)30 b(the)h(Lemma)f(is)h(analogous.)3103 4358 y Fj(\003)555 4533 y Fs(The)i(next)h(Prop)s(osition)g(61)h(studies)e(the)h (action-angle)j(v)-5 b(ariables)34 b(\()p Fm(A;)15 b( )s Fs(\))456 4640 y(asso)s(ciated)35 b(to)g(the)g(Hamiltonian)g Fm(K)1771 4654 y Fq(0)1811 4640 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\))36 b(in)e(the)g(domain)g Fm(D)2811 4655 y Fp(f)2891 4640 y Fs(de\014ned)456 4748 y(in)c(\(84\))q(.)555 4856 y(In)h(order)g(to)h(simplify)f(the)g(notation,)i(w)m(e)f(note)g (that)g Fm(K)2526 4870 y Fq(0)2597 4856 y Fs(is)f(indep)s(enden)m(t)456 4964 y(of)f Fm(s)g Fs(and)g(that)h(it)g(is)f(2)p Fm(\031)s Fs(-p)s(erio)s(dic)h(in)f Fm(x)g Fs(rather)g(than)g(2)p Fm(k)2433 4978 y Fq(0)2474 4964 y Fm(\031)s Fs(.)p eop end %%Page: 68 68 TeXDict begin 68 67 bop 456 251 a Fq(68)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)555 450 y Fs(Therefore,)31 b(w)m(e)f(will)h(consider)f(the)h(domain)521 629 y Fm(D)599 591 y Fl(\003)596 652 y Fp(f)667 629 y Fs(=)25 b Fn(f)p Fs(\()p Fm(y)s(;)15 b(x)p Fs(\))27 b Fn(2)d Fk(R)c Fn(\002)g Fk(T)p Fm(;)46 b Fn(9)p Fm(l)26 b Fn(2)f(f)p Fs(0)p Fm(;)15 b Fs(1)p Fm(;)g(;)g Fn(\001)g(\001)g(\001)34 b Fm(;)15 b(k)2110 643 y Fq(0)2150 629 y Fn(g)p Fm(;)77 b Fs(\()p Fm(y)s(;)15 b(x)20 b Fs(+)g(2)p Fm(\031)s(l)r(;)15 b(s)p Fs(\))26 b Fn(2)f Fm(D)3017 644 y Fp(f)3063 629 y Fn(g)456 808 y Fs(in)30 b(the)g(v)-5 b(ariables)31 b Fm(y)s(;)15 b(x)p Fs(,)31 b(whic)m(h)f(is)h(2)p Fm(\031)s Fs(-p)s(erio)s(dic.)555 916 y(Roughly)g(sp)s(eaking,)h(what)e(w)m(e)i(do)f(is)g(to)h(restrict)f (the)h(action-angle)h(v)-5 b(ari-)456 1023 y(ables)33 b(in)g(one)g(of)h(the)f(\\ey)m(es)i(of)e(the)g(resonance".)50 b(The)33 b(results)g(obtained)g(in)456 1131 y(one)25 b(ey)m(e)h(extend)f(to)g(the)g(other)g(ey)m(es)h(b)m(y)f(the)g(2)p Fm(\031)s Fs(-p)s(erio)s(dicit)m(y)h(in)e Fm(x)h Fs(of)g Fm(K)2937 1145 y Fq(0)2976 1131 y Fs(,)h(and)456 1239 y(clearly)f(are)g(uniform)f(for)g(all)h Fm(s)f Fs(since)h Fm(s)f Fs(do)s(es)g(not)h(en)m(ter)g(in)g(the)f(Hamiltonian.)456 1420 y Fw(Prop)s(osition)42 b(61.)i Fo(Consider)39 b(a)f(Hamiltonian)g Fm(K)2286 1434 y Fq(0)2326 1420 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\))39 b Fo(as)f(in)44 b Fs(\(82\))39 b Fo(of)456 1528 y(class)33 b Fn(C)727 1495 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(1)1007 1528 y Fo(,)g(with)g Fm(h)1317 1495 y Fq(0)1389 1528 y Fo(verifying)39 b Fs(\(80\))r Fo(,)32 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Fo(,)33 b Fm(i)26 b Fs(=)h(1)p Fm(;)15 b Fs(2)456 2266 y Fo(ar)-5 b(e)33 b(c)-5 b(onstants)35 b(indep)-5 b(endent)34 b(of)f Fm(")p Fo(,)f(such)h(that:)601 2409 y Fs(\(1\))42 b Fm(K)835 2423 y Fq(0)895 2409 y Fn(\016)21 b Fm(\037)1018 2424 y Fp(f)1063 2409 y Fs(\()p Fm(A;)15 b( )s Fs(\))27 b(=)e Fn(G)1480 2424 y Fp(f)1526 2409 y Fs(\()p Fm(A)p Fs(;)15 b Fm(")p Fs(\))p Fm(:)601 2540 y Fs(\(2\))42 b Fn(jj)p Fm(\037)865 2555 y Fp(f)911 2540 y Fn(jj)961 2570 y Fl(C)1002 2552 y Ff(s)1036 2570 y Fq(\()1079 2554 y(~)1063 2570 y Fp(D)1121 2582 y Ff(f)1161 2570 y Fq(\))1218 2540 y Fn(\024)1324 2491 y Fp(M)1392 2503 y Ff(f)p 1324 2520 108 4 v 1331 2575 a Fp(")1364 2556 y Ff(sj)1441 2540 y Fm(;)109 b Fn(jj)p Fm(\037)1682 2502 y Fl(\000)p Fq(1)1682 2570 y Fp(f)1776 2540 y Fn(jj)1826 2559 y Fl(C)1867 2540 y Ff(s)1901 2559 y Fq(\()p Fp(D)1988 2536 y Fg(\003)1986 2583 y Ff(f)2026 2559 y Fq(\))2083 2540 y Fn(\024)2189 2491 y Fp(M)2257 2503 y Ff(f)p 2189 2520 V 2196 2575 a Fp(")2229 2556 y Ff(sj)2306 2540 y Fm(;)48 b Fs(0)26 b Fn(\024)f Fm(s)g Fn(\024)g Fm(r)e Fn(\000)c Fs(2)p Fm(m)i Fn(\000)f Fs(2)p Fm(:)601 2705 y Fs(\(3\))42 b Fn(jjG)862 2720 y Fp(f)908 2705 y Fn(jj)958 2735 y Fl(C)999 2716 y Fi(3)1034 2735 y Fq(\()1078 2718 y(~)1061 2735 y Fp(D)1119 2747 y Ff(f)1159 2735 y Fq(\))1216 2705 y Fn(\024)1332 2656 y Fp(M)1400 2668 y Ff(f)p 1322 2684 128 4 v 1322 2745 a Fp(")1355 2725 y Ff(j)s(=)p Fi(2)1492 2705 y Fo(and)1669 2601 y Fh(\014)1669 2655 y(\014)1669 2710 y(\014)1699 2705 y Fn(G)1758 2672 y Fl(00)1753 2733 y Fp(f)1801 2705 y Fs(\()p Fm(A)p Fs(;)15 b Fm(")p Fs(\))2021 2601 y Fh(\014)2021 2655 y(\014)2021 2710 y(\014)2078 2705 y Fn(\025)25 b Fm(M)2262 2720 y Fp(f)2308 2705 y Fm(:)456 2871 y Fo(wher)-5 b(e)33 b Fm(M)800 2886 y Fp(f)878 2871 y Fo(is)g(a)g(c)-5 b(onstant)34 b(indep)-5 b(endent)35 b(of)d Fm(")p Fo(.)456 3128 y(Pr)-5 b(o)g(of)20 b(.)40 b Fs(In)28 b Fm(D)944 3095 y Fl(\003)941 3155 y Fp(f)1016 3128 y Fs(w)m(e)h(consider)g(the)g(curv)m(es)g Fm(E)i Fs(=)25 b Fm(K)2204 3142 y Fq(0)2244 3128 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\),)31 b(with)d Fm(E)j Fn(\025)25 b Fm(c)3030 3142 y Fq(1)3070 3128 y Fm(")3112 3095 y Fp(j)3149 3128 y Fs(.)456 3240 y(Then)k(the)i(action)g(v)-5 b(ariable)31 b(is)g(giv)m(en)g(b)m(y)f(the)h(w)m(ell)g(kno)m(wn)f(form) m(ula:)1008 3466 y Fm(A)c Fs(=)1208 3405 y Fm(S)5 b Fs(\()p Fm(E)g Fs(\))p 1208 3445 205 4 v 1260 3529 a(2)p Fm(\031)1447 3466 y Fs(=)1553 3405 y Fm(S)g Fs(\()p Fm(E)g Fs(;)15 b Fm(")p Fs(\))p 1553 3445 287 4 v 1646 3529 a(2)p Fm(\031)1875 3466 y Fs(=)2009 3405 y(1)p 1981 3445 101 4 v 1981 3529 a(2)p Fm(\031)2107 3342 y Fh(I)2157 3549 y Fp(K)2222 3519 y Fg(\000)p Fi(1)2217 3570 y(0)2304 3549 y Fq(\()p Fp(E)t Fq(\))2434 3466 y Fm(y)j(dx;)456 3717 y Fs(and)i Fm( )25 b Fs(is)d(the)f(conjugate)i(angle.)39 b(The)21 b(new)g(Hamiltonian)h(will)g(b)s(e)f Fn(G)2811 3732 y Fp(f)2856 3717 y Fs(\()p Fm(A)p Fs(;)15 b Fm(")p Fs(\))27 b(=)456 3829 y Fm(S)517 3796 y Fl(\000)p Fq(1)611 3829 y Fs(\()p Fm(A)p Fs(;)15 b Fm(")p Fs(\).)555 3937 y(T)-8 b(o)36 b(a)m(v)m(oid)g(an)f(unpleasan)m(t)h(t)m(yp)s(ograph)m(y)-8 b(,)37 b(from)e(no)m(w)g(on)g(in)f(this)h(section,)456 4045 y(w)m(e)41 b(will)g(omit)h(the)e(dep)s(endence)g(on)h Fm(")g Fs(of)g(man)m(y)g(of)g(the)g(functions)f(whic)m(h)456 4153 y(app)s(ear)29 b(during)g(the)i(pro)s(of.)555 4261 y(F)-8 b(ollo)m(wing)37 b(standard)d(practice)j(in)d(mec)m(hanics)i(w)m (e)f(\014nd)e(it)j(useful)e(to)h(use)456 4369 y(v)-5 b(ariables)38 b(\()p Fm(x;)15 b(E)5 b Fs(\))38 b(rather)f(than)g(\()p Fm(y)s(;)15 b(x)p Fs(\).)63 b(\(Note)39 b(that)e(w)m(e)h(are)g(in)m (terested)g(in)456 4476 y(the)32 b(curv)m(es)h Fm(E)i Fs(=)29 b Fm(K)1174 4490 y Fq(0)1213 4476 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\).\))50 b(W)-8 b(e)34 b(denote)f(b)m(y)g(\000)f (the)h(map)f(that)i(to)f(\()p Fm(y)s(;)15 b(x)p Fs(\))456 4584 y(asso)s(ciates)32 b(\()p Fm(x;)15 b(E)5 b Fs(\).)42 b(The)30 b(mapping)f(\000)i(satis\014es)f(\000\()p Fm(D)2327 4551 y Fl(\003)2324 4612 y Fp(f)2370 4584 y Fs(\))c Fn(\032)f(J)2589 4599 y Fp(f)2634 4584 y Fs(,)30 b(where)456 4786 y(\(103\))375 b Fn(J)1098 4801 y Fp(f)1168 4786 y Fs(=)25 b Fn(f)p Fs(\()p Fm(x;)15 b(E)5 b Fs(\))p Fm(;)48 b(x)25 b Fn(2)g Fk(T)p Fm(;)45 b(c)1949 4800 y Fq(1)1989 4786 y Fm(")2031 4748 y Fp(j)2093 4786 y Fn(\024)25 b Fm(E)30 b Fn(\024)25 b Fm(c)2421 4800 y Fq(2)2461 4786 y Fm(L)p Fn(g)p Fm(:)456 4964 y Fs(W)-8 b(e)31 b(also)g(note)h(that)e(the)h(mapping)f(\000)g(is) g(lo)s(cally)i(in)m(v)m(ertible.)p eop end %%Page: 69 69 TeXDict begin 69 68 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(69)555 450 y Fs(Using)33 b(part)g(1.)48 b(of)33 b(Lemma)f(60)i(with)e Fm(\016)h Fs(=)c Fm(")2111 417 y Fp(j)2148 450 y Fs(,)34 b(and)e Fm(n)c Fs(=)h Fm(r)c Fn(\000)c Fs(2)p Fm(m)h Fn(\000)f Fs(1,)34 b(w)m(e)456 558 y(de\014ne)29 b(in)h Fm(D)902 525 y Fl(\003)899 586 y Fp(f)945 558 y Fs(:)1241 714 y Fm(E)89 b Fs(=)82 b Fm(K)1627 728 y Fq(0)1667 714 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\))p Fm(;)1028 898 y(\034)10 b Fs(\()p Fm(x;)15 b(E)5 b Fs(\))85 b(=)1550 775 y Fh(Z)1641 801 y Fp(x)1601 981 y Fq(0)2066 837 y Fs(1)p 1710 877 757 4 v 1720 934 a Fp(@)t(K)1821 943 y Fi(0)p 1720 951 136 4 v 1749 1003 a Fp(@)t(y)1865 972 y Fs(\()p Fn(Y)1961 986 y Fl(\006)2021 972 y Fs(\()p Fm(u;)15 b(E)5 b Fs(\))p Fm(;)15 b(u)p Fs(;)g Fm(")p Fs(\))2477 898 y Fm(du:)456 1127 y Fs(where)31 b Fm(E)38 b Fs(is)32 b(the)h(energy)f(of)h(the)f(orbit)g(and)g Fm(\034)42 b Fs(is)33 b(the)f(time)h(along)g(the)g(orbit)456 1235 y(of)e(energy)g Fm(E)37 b Fs(\(w)m(e)32 b(ha)m(v)m(e)g(c)m(hosen)g (the)f(origin)h(of)f(time)h(at)g Fm(x)26 b Fs(=)h(0\).)43 b(With)32 b(this)456 1343 y(c)m(hoice,)39 b(w)m(e)e(ha)m(v)m(e)g(that)g Fm(T)13 b Fs(\()p Fm(E)5 b Fs(\))36 b(=)e Fm(\034)10 b Fs(\(2)p Fm(\031)s(;)15 b(E)5 b Fs(\))38 b(is)f(the)f(p)s(erio)s(d)f (of)h(the)g(p)s(erio)s(dic)456 1450 y(orbit,)30 b(and)g(the)h (action-angle)i(v)-5 b(ariables)31 b(are:)1146 1656 y Fm(A)25 b Fs(=)1345 1594 y Fm(S)5 b Fs(\()p Fm(E)g Fs(\))p 1345 1635 205 4 v 1397 1718 a(2)p Fm(\031)1584 1656 y Fs(=)1718 1594 y(1)p 1690 1635 101 4 v 1690 1718 a(2)p Fm(\031)1816 1532 y Fh(Z)1907 1558 y Fq(2)p Fp(\031)1866 1738 y Fq(0)2004 1656 y Fn(Y)2065 1670 y Fl(\006)2124 1656 y Fs(\()p Fm(x;)15 b(E)5 b Fs(\))p Fm(dx;)1151 1894 y( )29 b Fs(=)1399 1832 y(2)p Fm(\031)p 1345 1873 209 4 v 1345 1956 a(T)13 b Fs(\()p Fm(E)5 b Fs(\))1564 1894 y Fm(\034)10 b Fs(\()p Fm(x;)15 b(E)5 b Fs(\))p Fm(;)456 1770 y Fs(\(104\))456 2096 y(where)29 b Fm(S)779 2063 y Fl(0)803 2096 y Fs(\()p Fm(E)5 b Fs(\))26 b(=)f Fm(T)13 b Fs(\()p Fm(E)5 b Fs(\).)555 2204 y(Although)29 b(region)g Fm(D)1304 2171 y Fl(\003)1301 2232 y Fp(f)1375 2204 y Fs(has)f(t)m(w)m(o)i(connected)g(comp)s(onen)m(ts,)f(w)m(e)g(will)g(do) f(all)456 2316 y(the)i(details)i(only)e(when)f Fm(y)g(>)24 b Fs(0.)555 2424 y(First,)30 b(using)d(implicit)i(deriv)-5 b(ativ)m(es)30 b(in)e(equation)h(\(95\))h(and)e(form)m(ula)g(\(97\))r (,)456 2532 y(w)m(e)i(ha)m(v)m(e:)641 2705 y Fm(\034)10 b Fs(\()p Fm(x;)15 b(E)5 b Fs(\))85 b(=)1164 2582 y Fh(Z)1254 2608 y Fp(x)1214 2788 y Fq(0)1679 2644 y Fs(1)p 1323 2685 757 4 v 1333 2742 a Fp(@)t(K)1434 2751 y Fi(0)p 1333 2758 136 4 v 1362 2810 a Fp(@)t(y)1479 2779 y Fs(\()p Fn(Y)1575 2793 y Fq(+)1634 2779 y Fs(\()p Fm(u;)15 b(E)5 b Fs(\))p Fm(;)15 b(u)p Fs(;)g Fm(")p Fs(\))2090 2705 y Fm(du)25 b Fs(=)2310 2582 y Fh(Z)2401 2608 y Fp(x)2361 2788 y Fq(0)2470 2644 y Fm(@)5 b Fn(Y)2584 2658 y Fq(+)p 2470 2685 174 4 v 2494 2768 a Fm(@)g(E)2653 2705 y Fs(\()p Fm(u;)15 b(E)5 b Fs(\))p Fm(du)1010 2985 y Fs(=)1211 2923 y(1)p 1174 2964 122 4 v 1174 2982 a Fn(p)p 1249 2982 46 4 v 1249 3057 a Fs(2)1320 2861 y Fh(Z)1411 2887 y Fp(x)1370 3067 y Fq(0)1676 2923 y Fs(1)21 b(+)f Fm("b)p 1480 2964 631 4 v 1480 2982 a Fh(p)p 1571 2982 540 4 v 79 x Fm(E)25 b Fn(\000)20 b Fm(")1796 3034 y Fp(j)1833 3061 y Fm(U)10 b Fs(\()p Fm(u)p Fs(;)15 b Fm(")p Fs(\))2121 2985 y Fm(du)20 b Fs(+)g Fm("P)2431 2999 y Fq(1)2471 2985 y Fs(\()p Fm(x;)15 b(E)5 b Fs(\))p Fm(;)-2274 b Fs(\(105\))456 3218 y(where)19 b(the)i(function)f Fm(P)1259 3232 y Fq(1)1299 3218 y Fs(\()p Fm(x;)15 b(E)5 b Fs(\))21 b(is)g(giv)m(en)g(b)m(y)f Fm(P)2037 3232 y Fq(1)2077 3218 y Fs(\()p Fm(x;)15 b(E)5 b Fs(\))26 b(=)2433 3145 y Fh(R)2494 3171 y Fp(x)2476 3250 y Fq(0)2591 3183 y Fp(@)p 2563 3198 97 4 v 2563 3250 a(@)t(E)2688 3195 y Fs(~)2669 3218 y Fn(Y)2730 3232 y Fq(+)2789 3218 y Fs(\()p Fm(`)p Fs(\()p Fm(u;)15 b(E)5 b Fs(\)\))p Fm(du)p Fs(.)456 3327 y(T)-8 b(aking)31 b(in)m(to)g(accoun)m(t)h(that)f(in)f Fm(D)1660 3294 y Fl(\003)1657 3355 y Fp(f)1703 3327 y Fs(,)g(and)g(so)g(in)h Fn(J)2215 3342 y Fp(f)2290 3327 y Fs(\(see)g(\(103\))r(\),)g(one)g(has:)456 3496 y(\(106\))248 b Fm(c)948 3510 y Fq(1)988 3496 y Fm(")1030 3459 y Fp(j)1092 3496 y Fn(\024)25 b Fm(E)31 b Fn(\024)25 b Fm(E)g Fn(\000)20 b Fm(")1607 3459 y Fp(j)1644 3496 y Fm(U)10 b Fs(\()p Fm(x)p Fs(;)15 b Fm(")p Fs(\))27 b Fn(\024)e Fm(E)h Fs(+)19 b Fm(c")2307 3459 y Fp(j)2370 3496 y Fn(\024)25 b Fs(cte)p Fm(:)16 b(E)5 b(;)456 3643 y Fs(b)s(ound)28 b(\(98\))k(giv)m(e)g(us)d (that)456 3831 y(\(107\))48 b Fn(j)p Fm(P)792 3845 y Fq(1)832 3831 y Fn(j)857 3858 y Fl(C)898 3839 y Fi(0)932 3858 y Fq(\()p Fl(J)1007 3870 y Ff(f)1048 3858 y Fq(\))1104 3831 y Fn(\024)25 b Fs(cte)p Fm(:)17 b(;)46 b Fn(j)p Fm(P)1511 3845 y Fq(1)1551 3831 y Fn(j)1576 3858 y Fl(C)1617 3839 y Ff(s)1650 3858 y Fq(\()p Fl(J)1725 3870 y Ff(f)1765 3858 y Fq(\))1822 3831 y Fn(\024)2013 3770 y Fs(cte)p Fm(:)p 1928 3810 328 4 v 1928 3897 a(")1970 3871 y Fp(j)t Fq(\()p Fp(s)p Fl(\000)p Fq(1)p Fp(=)p Fq(2\))2265 3831 y Fm(;)g Fs(1)25 b Fn(\024)g Fm(s)g Fn(\024)g Fm(r)e Fn(\000)d Fs(2)p Fm(m)g Fn(\000)g Fs(2)p Fm(:)555 4013 y Fs(Di\013eren)m(tiating)k(\(105\))f(under)18 b(the)j(in)m(tegral)h (sign,)g(and)e(using)h(\(106\))h(and)e(\(107\))r(,)456 4121 y(w)m(e)30 b(obtain)h(upp)s(er)e(b)s(ounds)f(for)i Fm(\034)10 b Fs(\()p Fm(x;)15 b(E)5 b Fs(\):)456 4308 y(\(108\))327 b Fn(j)q Fm(\034)10 b Fn(j)1089 4335 y Fl(C)1130 4316 y Ff(s)1163 4335 y Fq(\()p Fl(J)1238 4347 y Ff(f)1278 4335 y Fq(\))1335 4308 y Fn(\024)1526 4247 y Fs(cte)p Fm(:)p 1441 4287 V 1441 4375 a(")1483 4348 y Fp(j)t Fq(\()p Fp(s)p Fq(+1)p Fp(=)p Fq(2\))1778 4308 y Fm(;)46 b Fs(0)26 b Fn(\024)f Fm(s)f Fn(\024)h Fm(r)e Fn(\000)d Fs(2)p Fm(m)g Fn(\000)g Fs(2)p Fm(:)456 4492 y Fs(On)29 b(the)i(other)f(hand,)g(using)g(that)h Fm(\034)10 b Fs(\(2)p Fm(\031)s(;)15 b(E)5 b Fs(\))27 b(=)e Fm(S)2205 4459 y Fl(0)2228 4492 y Fs(\()p Fm(E)5 b Fs(\),)32 b(one)e(obtains:)456 4680 y(\(109\))84 b Fn(j)q Fm(S)5 b Fn(j)857 4706 y Fl(C)898 4688 y Fi(0)932 4706 y Fq(\()p Fl(J)1007 4718 y Ff(f)1048 4706 y Fq(\))1104 4680 y Fn(\024)25 b Fs(cte)p Fm(:)17 b(;)46 b Fn(j)p Fm(S)5 b Fn(j)1539 4706 y Fl(C)1580 4688 y Ff(s)1613 4706 y Fq(\()p Fl(J)1688 4718 y Ff(f)1728 4706 y Fq(\))1785 4680 y Fn(\024)1976 4618 y Fs(cte)p Fm(:)p 1891 4659 V 1891 4746 a(")1933 4719 y Fp(j)t Fq(\()p Fp(s)p Fl(\000)p Fq(1)p Fp(=)p Fq(2\))2228 4680 y Fm(;)46 b Fs(1)26 b Fn(\024)f Fm(s)f Fn(\024)h Fm(r)e Fn(\000)d Fs(2)p Fm(m)h Fn(\000)e Fs(2)p Fm(:)456 4856 y Fs(The)44 b(next)g(task)h(is)g(to)g(obtain)g(lo)m(w)m(er)h(b)s(ounds)c(for)i(the) h(\014rst)f(and)g(second)456 4964 y(deriv)-5 b(ativ)m(es)26 b(of)f Fm(S)5 b Fs(.)39 b(W)-8 b(e)26 b(start)g(b)m(y)f(observing)g (that)h(the)f(main)g(term)g(of)g Fm(\034)10 b Fs(\()p Fm(x;)15 b(E)5 b Fs(\))p eop end %%Page: 70 70 TeXDict begin 70 69 bop 456 251 a Fq(70)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 450 y Fs(app)s(ears)34 b(explicitly)j(in)d(\(105\))r(,)j(and)d(w)m(e)i(ha)m(v)m(e)g(dev)m (elop)s(ed)f(in)g(\(107\))i(b)s(ounds)456 558 y(for)27 b(the)g(remainder.)39 b(Using)29 b(\(106\))g(w)m(e)e(can)h(obtain)f (easily)i(lo)m(w)m(er)f(b)s(ounds)d(for)456 666 y(the)30 b(main)g(term.)1216 760 y Fh(\014)1216 814 y(\014)1246 837 y Fm(S)1307 799 y Fl(0)1330 837 y Fs(\()p Fm(E)5 b Fs(\))1472 760 y Fh(\014)1472 814 y(\014)1529 837 y Fs(=)25 b Fn(j)p Fm(T)13 b Fs(\()p Fm(E)5 b Fs(\))p Fn(j)26 b(\025)2028 775 y Fs(cte)p Fm(:)p 2015 816 183 4 v 2015 903 a(E)2087 877 y Fq(1)p Fp(=)p Fq(2)2208 837 y Fm(;)-1777 b Fs(\(110\))1592 979 y Fh(\014)1592 1033 y(\014)1622 1056 y Fm(S)1683 1018 y Fl(00)1726 1056 y Fs(\()p Fm(E)5 b Fs(\))1868 979 y Fh(\014)1868 1033 y(\014)1924 1056 y Fn(\025)2043 994 y Fs(cte)p Fm(:)p 2030 1035 V 2030 1122 a(E)2102 1096 y Fq(3)p Fp(=)p Fq(2)2223 1056 y Fm(:)-1792 b Fs(\(111\))555 1244 y(Once)45 b(w)m(e)g(ha)m(v)m(e)g(estimated)h(the) f(function)f Fm(T)13 b Fs(\()p Fm(E)5 b Fs(\),)49 b(w)m(e)c(can)f(b)s (ound)f(the)456 1352 y(conjugate)d(angle)h Fm( )i Fs(giv)m(en)d(in)g (\(104\))r(,)i(where)d Fm(\034)10 b Fs(\()p Fm(x;)15 b(E)5 b Fs(\))41 b(is)e(giv)m(en)i(b)m(y)g(\(105\))r(.)456 1460 y(First,)29 b(using)g(\(109\))r(,)g(\(110\))i(and)d(F)-8 b(aa-di-Bruno)30 b(form)m(ula)e(for)h(the)f(deriv)-5 b(ativ)m(e)456 1568 y(of)30 b(the)h(comp)s(osition)g(of)f(functions,)g (w)m(e)h(obtain)1091 1639 y Fh(\014)1091 1693 y(\014)1091 1748 y(\014)1091 1803 y(\014)1168 1709 y Fm(@)1221 1676 y Fp(s)p 1132 1750 163 4 v 1132 1833 a Fm(@)5 b(E)1257 1807 y Fp(s)1304 1771 y Fs(\()1431 1709 y(1)p 1349 1750 209 4 v 1349 1833 a Fm(T)13 b Fs(\()p Fm(E)5 b Fs(\))1568 1771 y(\))1603 1639 y Fh(\014)1603 1693 y(\014)1603 1748 y(\014)1603 1803 y(\014)1659 1771 y Fn(\024)1822 1709 y Fs(cte)p Fm(:)p 1765 1750 270 4 v 1765 1837 a(E)1837 1811 y Fp(s)p Fl(\000)p Fq(1)p Fp(=)p Fq(2)2070 1771 y Fn(\024)2261 1709 y Fs(cte)p Fm(:)p 2176 1750 328 4 v 2176 1837 a(")2218 1811 y Fp(j)t Fq(\()p Fp(s)p Fl(\000)p Fq(1)p Fp(=)p Fq(2\))2513 1771 y Fm(;)456 1974 y Fs(and)29 b(then)h(the)h(Leibniz)f(rule)h(giv)m(es)545 2045 y Fh(\014)545 2100 y(\014)545 2154 y(\014)545 2209 y(\014)595 2115 y Fs(1)p 585 2156 66 4 v 585 2239 a Fm(T)661 2177 y(P)719 2191 y Fq(1)758 2045 y Fh(\014)758 2100 y(\014)758 2154 y(\014)758 2209 y(\014)789 2267 y Fl(C)830 2248 y Fi(0)864 2267 y Fq(\()p Fl(J)939 2279 y Ff(f)979 2267 y Fq(\))1036 2177 y Fn(\024)25 b Fs(cte)p Fm(:)17 b(;)1359 2045 y Fh(\014)1359 2100 y(\014)1359 2154 y(\014)1359 2209 y(\014)1410 2115 y Fs(1)p 1400 2156 V 1400 2239 a Fm(T)1475 2177 y(P)1533 2191 y Fq(1)1573 2045 y Fh(\014)1573 2100 y(\014)1573 2154 y(\014)1573 2209 y(\014)1603 2267 y Fl(C)1644 2248 y Ff(s)1678 2267 y Fq(\()p Fl(J)1753 2279 y Ff(f)1793 2267 y Fq(\))1850 2177 y Fn(\024)2005 2115 y Fs(cte)p Fm(:)p 1955 2156 257 4 v 1955 2243 a(")1997 2217 y Fp(j)t Fq(\()p Fp(s)p Fl(\000)p Fq(1\))2222 2177 y Fm(;)46 b Fs(1)25 b Fn(\024)g Fm(s)g Fn(\024)g Fm(r)e Fn(\000)d Fs(2)p Fm(m)g Fn(\000)g Fs(2)p Fm(:)456 2411 y Fs(This)29 b(giv)m(es,)j(using)e(again)h(\(106\))s(:)456 2619 y(\(112\))1026 2487 y Fh(\014)1026 2541 y(\014)1026 2596 y(\014)1026 2650 y(\014)1076 2557 y Fs(1)p 1066 2598 66 4 v 1066 2681 a Fm(T)1142 2619 y(\034)1192 2487 y Fh(\014)1192 2541 y(\014)1192 2596 y(\014)1192 2650 y(\014)1222 2709 y Fl(C)1263 2690 y Ff(s)1296 2709 y Fq(\()p Fl(J)1371 2721 y Ff(f)1411 2709 y Fq(\))1468 2619 y Fn(\024)1574 2557 y Fs(cte)p Fm(:)p 1574 2598 157 4 v 1597 2681 a(")1639 2655 y Fp(j)t(s)1741 2619 y Fm(;)45 b Fs(0)26 b Fn(\024)f Fm(s)g Fn(\024)g Fm(r)e Fn(\000)d Fs(2)p Fm(m)g Fn(\000)g Fs(2)p Fm(:)456 2852 y Fs(Bound)35 b(\(112\))j(together)g(with)d (\(109\))j(giv)m(es)g(the)e(upp)s(er)e(b)s(ound)g(claimed)j(in)456 2981 y(item)24 b(2.)39 b(of)24 b(the)f(Prop)s(osition)h(for)f(the)h Fn(C)1829 2948 y Fp(s)1890 2981 y Fs(norm)f(of)g(the)h(c)m(hange)h Fm(\037)2718 2933 y Fq(\()p Fl(\000)p Fq(1\))2718 3010 y Fp(f)2867 2981 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(\))26 b(=)456 3126 y(\()501 3081 y Fp(S)t Fq(\()p Fp(E)t Fq(\))p 501 3105 V 540 3158 a(2)p Fp(\031)668 3126 y Fm(;)760 3090 y Fq(2)p Fp(\031)p 718 3105 162 4 v 718 3159 a(T)10 b Fq(\()p Fp(E)t Fq(\))889 3126 y Fm(\034)g Fs(\()p Fm(x;)15 b(E)5 b Fs(\)\),)35 b(with)c Fm(E)j Fs(=)27 b Fm(K)1752 3140 y Fq(0)1792 3126 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\),)34 b(for)e(0)d Fn(\024)e Fm(s)h Fn(\024)f Fm(r)d Fn(\000)d Fs(2)p Fm(m)h Fn(\000)f Fs(2)32 b(in)456 3248 y Fm(D)534 3215 y Fl(\003)531 3275 y Fp(f)576 3248 y Fs(.)555 3394 y(Since)37 b(det)16 b Fm(D)s(\037)1076 3346 y Fq(\()p Fl(\000)p Fq(1\))1076 3424 y Fp(f)1225 3394 y Fs(\()p Fm(y)s(;)f(x)p Fs(\))37 b(=)f(1,)j(the)e Fn(C)1904 3361 y Fp(s)1978 3394 y Fs(norm)f(of)i Fm(\037)2390 3409 y Fp(f)2472 3394 y Fs(satis\014es)f(the)g(same)456 3508 y(b)s(ounds.)555 3616 y(Moreo)m(v)m(er,)46 b(taking)41 b(in)m(to)h(accoun)m(t)g(that)f Fn(G)2085 3631 y Fp(f)2173 3616 y Fs(=)h Fm(S)2347 3583 y Fl(\000)p Fq(1)2441 3616 y Fs(,)h(the)e(lo)m(w)m(er)h(b)s(ound)456 3724 y(for)36 b(the)h(second)g(deriv)-5 b(ativ)m(es)38 b(of)f Fn(G)1687 3739 y Fp(f)1769 3724 y Fs(and)f(the)h(upp)s(er)e(b)s(ound)f(for)j(the) g(third)456 3832 y(deriv)-5 b(ativ)m(e)32 b(of)e Fn(G)1030 3847 y Fp(f)1106 3832 y Fs(follo)m(w)h(from)f(\(110-111\).)1125 b Fj(\003)555 3985 y Fs(No)m(w)25 b(that)h(once)f(w)m(e)g(ha)m(v)m(e)g (expressed)f Fm(K)1959 3999 y Fq(0)1999 3985 y Fs(,)i(giv)m(en)f(in)f (\(82\))r(,)i(in)e(action-angle)456 4093 y(v)-5 b(ariables)43 b(in)f Fm(D)1041 4060 y Fl(\003)1038 4121 y Fp(f)1084 4093 y Fs(,)k(the)d(pro)s(of)f(of)g(Theorem)h(56)g(will)g(consist)h(in) e(applying)456 4210 y(Theorem)37 b(45)h(to)g(the)g(time-2)p Fm(\031)s(k)1610 4224 y Fq(0)1688 4210 y Fs(map)f(of)h(the)f(full)g (Hamiltonian)i(\(77\))r(,)g(to)456 4318 y(sho)m(w)c(that)g(it)h(has)f (primary)f(in)m(v)-5 b(arian)m(t)36 b(tori.)56 b(Going)36 b(bac)m(k)g(to)f(the)h(original)456 4426 y(v)-5 b(ariables)30 b(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(\),)30 b(one)g(obtains)f(the)h(result) f(claimed)h(in)f(Theorem)g(56.)42 b(W)-8 b(e)456 4534 y(pro)s(ceed)30 b(to)h(giv)m(e)h(the)e(details.)456 4701 y Fo(Pr)-5 b(o)g(of)34 b(of)f(p)-5 b(art)34 b(1\))f(of)g(The)-5 b(or)g(em)34 b(56.)43 b Fs(The)h(pro)s(of)g(will)i(consist)f(in)g (applying)456 4809 y(Theorem)30 b(45)h(to)g Fm(F)13 b Fs(,)31 b(the)f(time-2)p Fm(\031)s(k)1708 4823 y Fq(0)1780 4809 y Fs(map)g(of)g(the)h(Hamiltonian)456 4964 y(\(113\))970 4941 y(~)946 4964 y Fm(K)7 b Fs(\()p Fm(A;)15 b( )s(;)g(s)p Fs(;)g Fm(")p Fs(\))27 b(=)e Fn(G)1612 4979 y Fp(f)1658 4964 y Fs(\()p Fm(A)p Fs(;)15 b Fm(")p Fs(\))22 b(+)e Fm(")2033 4927 y Fp(m)p Fq(+1)2205 4941 y Fs(~)2190 4964 y Fm(S)5 b Fs(\()p Fm(A;)15 b( )s(;)g(s)p Fs(;)g Fm(")p Fs(\))p Fm(;)p eop end %%Page: 71 71 TeXDict begin 71 70 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(71)456 451 y Fs(where)761 428 y(~)737 451 y Fm(K)6 b Fs(\()p Fm(A;)15 b( )s(;)g(s)p Fs(;)g Fm(")p Fs(\))58 b(=)d Fm(K)39 b Fn(\016)33 b Fm(\037)1660 466 y Fp(f)1705 451 y Fs(\()p Fm(A;)15 b( )s(;)g(s)p Fs(;)g Fm(")p Fs(\),)56 b(and)2401 428 y(~)2385 451 y Fm(S)5 b Fs(\()p Fm(A;)15 b( )s(;)g(s)p Fs(;)g Fm(")p Fs(\))58 b(=)d Fm(S)37 b Fn(\016)456 559 y Fm(\037)513 574 y Fp(f)558 559 y Fs(\()p Fm(A;)15 b( )s(;)g(s)p Fs(;)g Fm(")p Fs(\).)555 678 y(Since)819 655 y(~)795 678 y Fm(K)7 b Fs(,)33 b Fn(G)5 b Fs(,)1070 655 y(~)1055 678 y Fm(S)37 b Fs(are)c Fn(C)1355 645 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(2)1636 678 y Fs(,)h(and)e(w)m(e)h(ha)m(v)m(e)g(assumed)f (in)h(Theorem)f(56)456 788 y(that)f Fm(r)22 b Fn(\000)e Fs(2)p Fm(m)h Fn(\000)f Fs(2)25 b Fn(\025)g Fs(6,)31 b(w)m(e)g(ha)m(v)m(e)h(that)1876 765 y(~)1852 788 y Fm(K)6 b Fs(,)31 b Fn(G)5 b Fs(,)2121 765 y(~)2106 788 y Fm(S)35 b Fs(are)c Fn(C)2402 755 y Fq(6)2442 788 y Fs(.)555 896 y(W)-8 b(e)33 b(denote)e(b)m(y)g Fm(F)1191 910 y Fq(0)1262 896 y Fs(the)g(time-2)p Fm(\031)s(k)1772 910 y Fq(0)1845 896 y Fs(map)g(of)g(Hamiltonian)h Fn(G)2729 911 y Fp(f)2775 896 y Fs(\()p Fm(A)p Fs(;)15 b Fm(")p Fs(\))32 b(and)456 1008 y(w)m(e)e(ha)m(v)m(e)i(that)f Fm(F)43 b Fs(and)30 b Fm(F)1332 1022 y Fq(0)1402 1008 y Fs(are)h Fn(C)1607 975 y Fq(5)1677 1008 y Fs(and)f(w)m(e)h(can)f(b)s(ound)913 1162 y Fn(jj)p Fm(F)k Fn(\000)19 b Fm(F)1203 1176 y Fq(0)1243 1162 y Fn(jj)1293 1182 y Fl(C)1334 1163 y Fi(5)1399 1162 y Fn(\024)25 b Fs(cte)p Fm(:)16 b(")1693 1125 y Fp(m)p Fq(+1)1850 1162 y Fn(jj)p Fm(S)5 b Fn(jj)2011 1182 y Fl(C)2052 1163 y Fi(6)2117 1162 y Fn(\024)25 b Fs(cte)p Fm(:)17 b(")2412 1125 y Fp(m)p Fq(+1)p Fl(\000)p Fq(6)p Fp(j)2692 1162 y Fm(:)456 1317 y Fs(Since)40 b Fn(G)757 1332 y Fp(f)843 1317 y Fs(dep)s(ends)f(only)i(on)f Fm(A)p Fs(,)k Fm(F)1746 1331 y Fq(0)1826 1317 y Fs(is)d(an)f(in)m(tegrable)j (map)d(of)h(the)f(form)456 1425 y(\()p Fm(A;)15 b Fs(\011\))26 b Fn(!)f Fs(\()p Fm(A;)15 b Fs(\011)21 b(+)f(\001\()p Fm(A)p Fs(\).)555 1533 y(F)-8 b(urthermore,)31 b(b)m(y)f(item)h(3.)41 b(of)31 b(Prop)s(osition)f(61,)i(w)m(e)e(ha)m(v)m(e)1232 1681 y Fm(d)p 1198 1722 116 4 v 1198 1805 a(dA)1324 1743 y Fs(\001\()p Fm(A)p Fs(\))25 b(=)1703 1681 y Fm(@)1756 1648 y Fq(2)p 1669 1722 161 4 v 1669 1805 a Fm(@)5 b(A)1790 1779 y Fq(2)1840 1743 y Fn(G)1894 1758 y Fp(f)1940 1743 y Fs(\()p Fm(A)p Fs(;)15 b Fm(")p Fs(\))27 b Fn(\025)e Fm(M)2371 1758 y Fp(f)2416 1743 y Fm(:)456 1923 y Fs(Hence,)38 b(the)f(mapping)e Fm(F)1358 1937 y Fq(0)1434 1923 y Fs(will)h(b)s(e)g (a)h(t)m(wist)g(mapping,)g(and)e(w)m(e)i(can)f(apply)456 2031 y(Theorem)30 b(45)h(with)f Fm(\016)f Fs(=)c Fm(")1375 1998 y Fp(m)p Fq(+1)p Fl(\000)p Fq(6)p Fp(j)1655 2031 y Fs(,)30 b(and)g(w)m(e)h(obtain,)g(if)f Fm(m)25 b(>)g Fs(6)p Fm(j)i Fn(\000)20 b Fs(1:)601 2162 y(\(1\))42 b(There)i(exist)h(a)g(set)g(of)f(v)-5 b(alues)45 b Fm(A)1980 2176 y Fp(i)2008 2162 y Fs(,)j(suc)m(h)c(that)h(the)g(Hamiltonian)758 2270 y Fm(K)27 b Fn(\016)21 b Fm(\037)985 2285 y Fp(f)1060 2270 y Fs(has)31 b(in)m(v)-5 b(arian)m(t)31 b(tori)g(giv)m(en)h(b)m(y) 1404 2425 y Fm(A)25 b Fs(=)g Fm(A)1661 2439 y Fp(i)1710 2425 y Fs(+)20 b Fn(A)1874 2439 y Fp(i)1901 2425 y Fs(\()p Fm( )s(;)15 b(s)p Fs(;)g Fm(")p Fs(\))p Fm(;)758 2579 y Fs(where)20 b Fn(A)1084 2593 y Fp(i)1132 2579 y Fs(are)h Fn(C)1327 2546 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(5)p Fl(\000)p Fp(\021)1721 2579 y Fs(functions,)h(for)e(an)m(y)g Fm(\021)29 b(>)c Fs(0,)e(and)d Fn(jjA)2973 2593 y Fp(i)3001 2579 y Fn(jj)3051 2599 y Fl(C)3092 2580 y Fi(2)3156 2579 y Fn(\024)758 2693 y Fs(cte)p Fm(:)d(")957 2660 y Fq(\()p Fp(m)p Fq(+1)p Fl(\000)p Fq(6)p Fp(j)t Fq(\))p Fp(=)p Fq(2)1362 2693 y Fs(.)601 2801 y(\(2\))42 b(The)36 b(motion)h(on)g (these)g(tori)g(is)f Fn(C)1965 2768 y Fq(1)2005 2801 y Fs(-conjugate)i(to)f(a)g(rigid)f(transla-)758 2909 y(tion)h(of)f(frequencies)g(\()p Fm(!)s Fs(\()p Fm(A)1729 2923 y Fp(i)1758 2909 y Fs(\))p Fm(;)15 b Fs(1\),)40 b(where)35 b Fm(!)s Fs(\()p Fm(A)2409 2923 y Fp(i)2438 2909 y Fs(\))h(is)g(a)h(Diophan)m(tine)758 3040 y(n)m(um)m(b)s(er)31 b(of)g(constan)m(t)i(t)m(yp)s(e)f(and)f(Mark)m(o)m(v)i(constan)m(t)g Fm(K)7 b(")2783 2973 y Ff(m)p Fi(+1)p Fg(\000)p Fi(6)p Ff(j)p 2783 2989 239 3 v 2887 3030 a Fi(2)3036 3040 y Fs(,)32 b(as)758 3147 y(stated)g(in)e(De\014nition)h(42.)601 3255 y(\(3\))42 b(The)34 b(union)f(of)i(neigh)m(b)s(orho)s(o)s(ds)d(of) j(size)g Fm(")2249 3222 y Fq(\()p Fp(m)p Fq(+1)p Fl(\000)p Fq(6)p Fp(j)t Fq(\))p Fp(=)p Fq(2)2688 3255 y Fs(of)g(these)f(tori)758 3370 y(co)m(v)m(er)e(all)g(the)e(region)1570 3347 y(~)1549 3370 y Fm(D)1624 3385 y Fp(f)1670 3370 y Fs(.)555 3501 y(Going)j(bac)m(k)g(to)g(the)f(original)h(v)-5 b(ariables)33 b(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(\))29 b(=)f(\()p Fm(\037)2532 3516 y Fp(f)2578 3501 y Fs(\()p Fm(A;)15 b( )s Fs(\))p Fm(;)g(s)p Fs(\))34 b(,)e(and)456 3614 y(using)j(that)i Fm(K)979 3628 y Fq(0)1019 3614 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\))37 b(=)d Fm(E)41 b Fs(=)35 b Fn(G)5 b Fs(\()p Fm(A)p Fs(;)15 b Fm(")p Fs(\),)40 b(and)35 b(that)i Fn(jjG)2500 3629 y Fp(f)2546 3614 y Fn(jj)2596 3634 y Fl(C)2637 3615 y Fi(3)2711 3614 y Fn(\024)2830 3578 y Fq(cte)p Fp(:)p 2827 3593 128 4 v 2827 3654 a(")2860 3634 y Ff(j)s(=)p Fi(2)2965 3614 y Fs(,)g(and)456 3743 y Fn(jj)p Fm(\037)563 3705 y Fl(\000)p Fq(1)563 3772 y Fp(f)657 3743 y Fn(jj)707 3763 y Fl(C)748 3744 y Fi(2)824 3743 y Fn(\024)940 3707 y Fq(cte)p Fp(:)p 940 3722 122 4 v 953 3778 a(")986 3759 y Fi(2)p Ff(j)1072 3743 y Fs(,)i(one)e (obtains)g(the)g(desired)f(result)h(if)g Fm(m)24 b Fs(+)h(1)g Fn(\000)f Fs(11)p Fm(j)42 b(>)36 b Fs(0,)456 3862 y(calling)31 b Fm(E)810 3876 y Fp(i)864 3862 y Fs(=)25 b Fn(G)5 b Fs(\()p Fm(A)1122 3876 y Fp(i)1151 3862 y Fs(;)15 b Fm(")p Fs(\).)3103 3970 y Fj(\003)555 4139 y Fs(No)m(w)33 b(w)m(e)f(turn)f(to) i(estimate)g(the)f(action-angle)j(v)-5 b(ariables)33 b(in)e(the)h(regions)456 4248 y Fm(D)531 4262 y Fp(o;in)656 4248 y Fs(.)39 b(W)-8 b(e)27 b(use)f(that,)i(in)e Fm(D)1421 4262 y Fp(o;in)1546 4248 y Fs(,)i(the)e(co)s(ordinate)h Fm(y)i Fs(is)e(of)f(size)h Fm(")2665 4215 y Fp(j)t(=)p Fq(2)2799 4248 y Fs(and)f(then,)456 4366 y(it)36 b(is)f(natural)h(to)h (p)s(erform)d(the)i(scaling)g Fm(y)h Fs(=)d Fm(")2132 4333 y Fp(j)t(=)p Fq(2)2240 4366 y Fm(Y)20 b Fs(,)37 b(whic)m(h)e(leads)h(directly)456 4474 y(to)c(the)g(follo)m(wing)i (Lemma.)45 b(This)31 b(transformation)i(is)f(not)g(symplectic,)i(but) 456 4581 y(conformally)28 b(symplectic)h(so)f(that)g(the)g(equations)g (are)h(still)f(in)g(Hamiltonian)456 4689 y(form.)456 4856 y Fw(Lemma)k(62.)40 b Fo(The)31 b(sc)-5 b(aling)31 b Fm(y)d Fs(=)d Fm(")1687 4823 y Fp(j)t(=)p Fq(2)1794 4856 y Fm(Y)51 b Fo(tr)-5 b(ansforms)33 b(the)e(Hamiltonian)h(sys-)456 4964 y(tem)44 b(of)h(Hamiltonian)h Fm(K)7 b Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(;)g Fm(")p Fs(\))45 b Fo(given)f(in)51 b Fs(\(77\))46 b Fo(into)f(a)f(Hamiltonian)p eop end %%Page: 72 72 TeXDict begin 72 71 bop 456 251 a Fq(72)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 450 y Fo(system)33 b(of)g Fn(C)913 417 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(2)1226 450 y Fo(Hamiltonian)456 601 y Fn(K)q Fs(\()p Fm(Y)5 b(;)15 b(x;)g(s)p Fs(;)g Fm(")p Fs(\))85 b(=)e Fm(")1192 564 y Fp(j)t(=)p Fq(2)1299 601 y Fs(\()p Fm(h)1386 615 y Fq(in)n(t)1475 601 y Fs(\()p Fm(Y)21 b Fs(;)15 b Fm(")1666 564 y Fp(j)t(=)p Fq(2)1774 601 y Fs(\))20 b(+)g Fm(U)10 b Fs(\()p Fm(x)p Fs(;)15 b Fm(")p Fs(\)\))22 b(+)e Fm(")2386 564 y Fp(m)p Fq(+1)p Fl(\000)p Fp(j)t(=)p Fq(2)2701 601 y Fm(S)2757 615 y Fq(1)2796 601 y Fs(\()p Fm(Y)5 b(;)15 b(x;)g(s)p Fs(;)g Fm(")3146 564 y Fp(j)t(=)p Fq(2)3254 601 y Fs(\))996 747 y(=)83 b Fm(")1192 709 y Fp(j)t(=)p Fq(2)1299 747 y Fn(K)1368 761 y Fq(in)n(t)1457 747 y Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(;)g Fm(")1724 709 y Fp(j)t(=)p Fq(2)1832 747 y Fs(\))21 b(+)f Fm(")2021 709 y Fp(m)p Fq(+1)p Fl(\000)p Fp(j)t(=)p Fq(2)2336 747 y Fm(S)2392 761 y Fq(1)2431 747 y Fs(\()p Fm(Y)5 b(;)15 b(x;)g(s)p Fs(;)g Fm(")2781 709 y Fp(j)t(=)p Fq(2)2889 747 y Fs(\))-2352 b(\(114\))456 892 y Fo(with)1205 1076 y Fm(h)1257 1090 y Fq(in)n(t)1346 1076 y Fs(\()p Fm(Y)20 b Fs(;)15 b Fm(")1536 1038 y Fp(j)t(=)p Fq(2)1644 1076 y Fs(\))84 b(=)1926 1014 y Fm(Y)1999 981 y Fq(2)p 1926 1055 113 4 v 1960 1138 a Fs(2)2050 1052 y(^)2049 1076 y Fm(h)p Fs(\()p Fm(")2178 1038 y Fp(j)t(=)p Fq(2)2286 1076 y Fm(Y)20 b Fs(;)15 b Fm(")p Fs(\))p Fm(;)1091 1261 y(S)1147 1275 y Fq(1)1186 1261 y Fs(\()p Fm(Y)5 b(;)15 b(x;)g(s)p Fs(;)g Fm(")1536 1223 y Fp(j)t(=)p Fq(2)1644 1261 y Fs(\))84 b(=)e Fm(S)5 b Fs(\()p Fm(")2054 1223 y Fp(j)t(=)p Fq(2)2162 1261 y Fm(Y)g(;)15 b(x;)g(s)p Fs(;)g Fm(")p Fs(\))p Fm(;)456 1417 y Fo(wher)-5 b(e)720 1393 y Fs(^)719 1417 y Fm(h)p Fs(\()p Fm(y)s(;)15 b(")p Fs(\))40 b Fo(is)f(given)f(in)46 b Fs(\(80\))41 b Fo(and,)g(c)-5 b(onse)g(quently,)41 b Fn(K)2523 1431 y Fq(in)n(t)2651 1417 y Fo(is)e(a)g Fn(C)2893 1384 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(1)456 1524 y Fo(function.)555 1632 y(The)33 b(sc)-5 b(aling)33 b(tr)-5 b(ansforms)36 b(the)d(domains)h Fm(D)2082 1646 y Fp(o;in)2240 1632 y Fo(given)e(in)39 b Fs(\(85\))r Fo(,)32 b Fs(\(86\))i Fo(into)552 1792 y Fn(D)622 1806 y Fp(o)744 1792 y Fs(=)82 b Fn(f)p Fs(\()p Fm(Y)5 b(;)15 b(x;)g(s)p Fs(\))27 b(:)e Fn(K)1391 1806 y Fq(in)n(t)1480 1792 y Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(;)g Fm(")1747 1755 y Fp(j)t(=)p Fq(2)1856 1792 y Fs(\))25 b(=)g Fm(F)8 b(=")2165 1755 y Fp(j)2203 1792 y Fm(;)48 b(c)2315 1806 y Fq(3)2354 1792 y Fm(")2396 1755 y Fp(\013)2471 1792 y Fn(\024)25 b Fm(F)39 b Fn(\024)25 b Fm(c)2799 1806 y Fq(1)2838 1792 y Fm(")2880 1755 y Fp(j)2917 1792 y Fn(g)744 1938 y Fs(=)82 b Fn(f)p Fs(\()p Fm(Y)5 b(;)15 b(x;)g(s)p Fs(\))27 b(:)e Fn(K)1391 1952 y Fq(in)n(t)1480 1938 y Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(;)g Fm(")1747 1900 y Fp(j)t(=)p Fq(2)1856 1938 y Fs(\))25 b(=)g Fm(e;)48 b(c)2166 1952 y Fq(3)2206 1938 y Fm(")2248 1900 y Fp(\013)p Fl(\000)p Fp(j)2411 1938 y Fn(\024)25 b Fm(e)g Fn(\024)g Fm(c)2709 1952 y Fq(1)2749 1938 y Fn(g)p Fm(;)-2363 b Fs(\(115\))519 2083 y Fn(D)589 2097 y Fp(in)744 2083 y Fs(=)82 b Fn(f)p Fs(\()p Fm(Y)5 b(;)15 b(x;)g(s)p Fs(\))27 b(:)e Fn(K)1391 2097 y Fq(in)n(t)1480 2083 y Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(;)g Fm(")1747 2046 y Fp(j)t(=)p Fq(2)1856 2083 y Fs(\))25 b(=)g Fm(G=")2170 2046 y Fp(j)2208 2083 y Fm(;)48 b Fn(\000)p Fm(c)2391 2097 y Fq(4)2430 2083 y Fm(")2472 2046 y Fp(j)2535 2083 y Fn(\024)25 b Fm(G)g Fn(\024)g(\000)p Fm(c)2933 2097 y Fq(3)2973 2083 y Fm(")3015 2046 y Fp(\013)3065 2083 y Fn(g)744 2228 y Fs(=)82 b Fn(f)p Fs(\()p Fm(Y)5 b(;)15 b(x;)g(s)p Fs(\))27 b(:)e Fn(K)1391 2242 y Fq(in)n(t)1480 2228 y Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(;)g Fm(")1747 2191 y Fp(j)t(=)p Fq(2)1856 2228 y Fs(\))25 b(=)g Fm(e;)48 b Fn(\000)p Fm(c)2237 2242 y Fq(4)2302 2228 y Fn(\024)25 b Fm(e)h Fn(\024)f(\000)p Fm(c)2672 2242 y Fq(3)2711 2228 y Fm(")2753 2191 y Fp(\013)p Fl(\000)p Fp(j)2890 2228 y Fn(g)p Fm(:)-2504 b Fs(\(116\))555 2373 y(In)27 b(the)h(ab)s(o)m(v)m(e)h(form)m(ulas)f(\()p Fm(x;)15 b(s)p Fs(\))26 b Fn(2)f Fs(\()p Fk(R)p Fm(=)p Fs(\(2)p Fm(\031)s(k)2087 2387 y Fq(0)2128 2373 y Fk(Z)p Fs(\)\))2259 2340 y Fq(2)2299 2373 y Fs(.)40 b(W)-8 b(e)29 b(de\014ne)e(the)h (action)456 2481 y(angle)j(v)-5 b(ariables)31 b(in)456 2589 y(\(117\))537 2697 y Fn(D)610 2659 y Fl(\003)607 2719 y Fp(o)674 2697 y Fs(=)25 b Fn(f)p Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(\))27 b Fn(2)d Fk(R)c Fn(\002)g Fk(T)p Fm(;)46 b Fn(9)p Fm(l)26 b Fn(2)f(f)p Fs(0)p Fm(;)15 b Fs(1)p Fm(;)g(;)g Fn(\001)g(\001)g(\001)34 b Fm(;)15 b(k)2127 2711 y Fq(0)2167 2697 y Fn(g)p Fm(;)46 b Fs(\()p Fm(Y)5 b(;)15 b(x)21 b Fs(+)f(2)p Fm(\031)s(l)r(;)15 b(s)p Fs(\))26 b Fn(2)f(D)3009 2711 y Fp(o)3047 2697 y Fn(g)456 2842 y Fs(b)m(y)39 b(form)m(ulas)h(\(104\))i(as)e(in)f Fn(D)1526 2809 y Fl(\003)1523 2870 y Fp(f)1568 2842 y Fs(.)69 b(The)39 b(only)h(c)m(hange)h(is)e(that,)k Fn(D)2784 2809 y Fl(\003)2781 2865 y Fp(o)2823 2842 y Fs(,)g(instead)456 2960 y(of)37 b(\(106\))c(w)m(e)d(ha)m(v)m(e:)456 3110 y(\(118\))239 b(0)26 b Fn(\024)e Fm(c)1105 3124 y Fq(3)1145 3110 y Fm(")1187 3072 y Fp(\013)p Fl(\000)p Fp(j)1350 3110 y Fn(\024)h Fm(e)g Fn(\024)g Fm(e)c Fn(\000)f Fm(U)10 b Fs(\()p Fm(x)p Fs(;)15 b Fm(")p Fs(\))26 b Fn(\024)f Fm(e)c Fs(+)f Fm(c)25 b Fn(\024)g Fm(c)2514 3124 y Fq(1)2574 3110 y Fs(+)20 b Fm(c:)456 3261 y Fs(T)-8 b(o)20 b(b)s(ound)f (accurately)j(the)e(action-angle)j(v)-5 b(ariables)21 b(for)f(Hamiltonian)i Fn(K)3001 3275 y Fq(in)n(t)3089 3261 y Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(;)g Fm(")3356 3228 y Fp(j)t(=)p Fq(2)3465 3261 y Fs(\))456 3369 y(in)30 b Fn(D)635 3336 y Fl(\003)632 3392 y Fp(o)704 3369 y Fs(w)m(e)h(use)f(the)h(follo)m(wing:)456 3514 y Fw(Lemma)44 b(63.)j 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Fm(a)1109 4978 y Fq(1)1148 4964 y Fm(x)1200 4927 y Fq(2)1265 4964 y Fn(\024)25 b(\000)p Fm(U)10 b Fs(\()p Fm(x)p Fs(;)15 b Fm(")p Fs(\))26 b Fn(\024)f Fm(a)1878 4978 y Fq(2)1917 4964 y Fm(x)1969 4927 y Fq(2)2100 4964 y Fn(8)p Fm(x)g Fn(2)f Fs([)p Fn(\000)p Fm(\032;)15 b(\032)p Fs(])p eop end %%Page: 73 73 TeXDict begin 73 72 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(73)456 450 y Fs(Then,)29 b(w)m(e)i(compute)803 522 y Fh(Z)894 548 y Fp(x)853 728 y Fq(0)1234 584 y Fs(1)p 947 624 620 4 v 947 711 a(\()p Fm(e)21 b Fn(\000)f Fm(U)10 b Fs(\()p Fm(u)p Fs(;)15 b Fm(")p Fs(\)\))1447 685 y Fp(n=)p Fq(2)1577 645 y Fm(du)963 907 y Fs(=)1059 783 y Fh(Z)1150 810 y Fp(\032)1109 989 y Fq(0)1502 846 y Fs(1)p 1215 886 V 1215 973 a(\()p Fm(e)21 b Fn(\000)f Fm(U)10 b Fs(\()p Fm(u)p Fs(;)15 b Fm(")p Fs(\)\))1715 947 y Fp(n=)p Fq(2)1844 907 y Fm(du)21 b Fs(+)2055 783 y Fh(Z)2145 810 y Fp(x)p Fl(\000)p Fp(\032)2105 989 y(\032)2592 846 y Fs(1)p 2305 886 V 2305 973 a(\()p Fm(e)g 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Fs(=)g(1,)k(this)e(in)m(tegral)i(div)m(erges)f(as)f Fm(e)f Fn(!)f Fs(0,)k(but)d(in)m(tegrating)k(explicitly)e(w)m(e)456 2592 y(get)744 2770 y(1)p 691 2811 152 4 v 691 2835 a Fn(p)p 767 2835 77 4 v 59 x Fm(a)815 2908 y Fp(i)868 2708 y Fh(Z)969 2666 y Fg(p)p 1019 2666 60 3 v 28 x Ff(a)1053 2710 y(i)p 969 2719 110 3 v 984 2727 a Fg(p)p 1034 2727 30 3 v 36 x Ff(e)1089 2734 y Fp(\032)918 2914 y Fq(0)1336 2770 y Fs(1)p 1154 2811 410 4 v 1154 2898 a(\(1)21 b(+)f Fm(t)1379 2872 y Fq(2)1418 2898 y Fs(\))1453 2871 y Fq(1)p Fp(=)p Fq(2)1599 2832 y Fs(=)1758 2770 y(1)p 1705 2811 152 4 v 1705 2835 a Fn(p)p 1780 2835 77 4 v 1780 2894 a Fm(a)1828 2908 y Fp(i)1882 2832 y Fs(log)2014 2703 y Fh(\022)2091 2709 y Fn(p)p 2167 2709 V 59 x Fm(a)2215 2782 y Fp(i)p 2091 2811 152 4 v 2108 2829 a Fn(p)p 2184 2829 43 4 v 65 x Fm(e)2253 2832 y(\032)g Fs(+)2411 2699 y Fh(r)p 2502 2699 340 4 v 133 x Fs(1)h(+)2669 2770 y Fm(a)2717 2784 y Fp(i)p 2669 2811 77 4 v 2686 2894 a Fm(e)2755 2832 y(\032)2802 2805 y Fq(2)2841 2703 y Fh(\023)2923 2832 y Fm(:)555 3037 y Fs(If)37 b Fm(n)e Fn(\025)h Fs(1,)j(the)f(in)m (tegral)g(is)f(con)m(v)m(ergen)m(t)i(for)e Fm(e)g Fs(=)e(0.)61 b(This)36 b(giv)m(es)i(us)f(the)456 3145 y(b)s(ounds)28 b(of)i(the)h(Lemma)f(63.)1632 b Fj(\003)555 3296 y Fs(The)20 b(next)g(Prop)s(osition)h(64)g(is)f(dev)m(oted)h(to)g(putting)g (Hamiltonian)g Fn(K)2922 3310 y Fq(in)n(t)3011 3296 y Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(;)g Fm(")3278 3263 y Fp(j)t(=)p Fq(2)3386 3296 y Fs(\))456 3404 y(in)m(to)25 b(action-angle)i(v)-5 b(ariables)25 b(\()p Fm(A;)15 b( )s Fs(\))26 b(in)e(the)h(regions)g Fn(D)2402 3371 y Fl(\003)2399 3430 y Fp(o;in)2523 3404 y Fs(,)h(where)e Fn(D)2904 3371 y Fl(\003)2901 3428 y Fq(0)2967 3404 y Fs(is)h(de-)456 3519 y(\014ned)k(in)h(\(117\))i(and)e Fn(D)1271 3486 y Fl(\003)1268 3544 y Fp(in)1369 3519 y Fs(is)h(de\014ned)e(analogously)-8 b(.)456 3693 y Fw(Prop)s(osition) 24 b(64.)34 b Fo(Consider)25 b(the)e Fn(C)1748 3660 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(1)2052 3693 y Fo(Hamiltonian)i Fm(")2610 3660 y Fp(j)t(=)p Fq(2)2718 3693 y Fn(K)2787 3707 y Fq(in)n(t)2876 3693 y Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(;)g Fm(")3143 3660 y Fp(j)t(=)p Fq(2)3251 3693 y Fs(\))456 3801 y Fo(in)43 b(the)h(r)-5 b(e)g(gions)45 b Fn(D)1132 3768 y Fl(\003)1129 3823 y Fp(o)1171 3801 y Fo(,)h Fn(D)1318 3768 y Fl(\003)1315 3826 y Fp(in)1386 3801 y Fo(.)74 b(Then,)47 b(for)d Fm(\035)k Fs(=)d Fm(o;)15 b(in)p Fo(,)46 b(ther)-5 b(e)44 b(exist)g Fn(C)2893 3768 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(2)456 3908 y Fo(changes)33 b(of)g(variables)1430 4061 y Fm(\037)1487 4075 y Fp(\035)1557 4061 y Fs(:)1628 4038 y(~)1607 4061 y Fn(D)1677 4075 y Fp(\035)1805 4061 y Fn(!)83 b(D)2052 4024 y Fl(\003)2049 4084 y Fp(\035)1480 4196 y Fs(\()p Fm(A;)15 b( )s Fs(\))85 b Fn(7!)e Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(\))456 4354 y Fo(wher)-5 b(e)740 4331 y Fs(~)719 4354 y Fm(D)794 4368 y Fp(\035)876 4354 y Fs(=)37 b Fn(f)p Fs(\()p Fm(A;)15 b( )s Fs(\))39 b(:)g(~)-47 b Fm(c)1409 4321 y Fq(1)1409 4377 y Fp(\035)1492 4354 y Fn(\024)36 b Fm(A)i Fn(\024)g Fs(~)-47 b Fm(c)1851 4321 y Fq(2)1851 4377 y Fp(\035)1896 4354 y Fn(g)p Fm(;)40 b Fo(and)i Fs(~)-47 b Fm(c)2228 4321 y Fp(l)2228 4377 y(\035)2273 4354 y Fo(,)41 b Fm(l)e Fs(=)e(1)p Fm(;)15 b Fs(2)p Fo(,)42 b(ar)-5 b(e)40 b(suitable)456 4462 y(c)-5 b(onstants)34 b(indep)-5 b(endent)34 b(of)f Fm(")p Fo(,)g(such)g(that:) 601 4592 y Fs(\(1\))42 b Fm(")800 4559 y Fp(j)t(=)p Fq(2)908 4592 y Fn(K)977 4606 y Fq(in)n(t)1086 4592 y Fn(\016)20 b Fm(\037)1208 4606 y Fp(\035)1253 4592 y Fs(\()p Fm(A;)15 b( )s Fs(\))27 b(=)e Fm(")1658 4559 y Fp(j)t(=)p Fq(2)1766 4592 y Fn(G)1820 4606 y Fp(\035)1864 4592 y Fs(\()p Fm(A)p Fs(;)15 b Fm(")2049 4559 y Fp(j)t(=)p Fq(2)2158 4592 y Fs(\))p Fo(.)601 4700 y Fs(\(2\))42 b Fo(F)-7 b(or)34 b Fs(0)26 b Fn(\024)f Fm(s)g Fn(\024)g Fm(r)d Fn(\000)e Fs(2)p Fm(m)h Fn(\000)f Fs(2)33 b Fo(we)f(have)h(that:)872 4898 y Fn(jj)p Fm(\037)979 4912 y Fp(\035)1024 4898 y Fn(jj)1074 4928 y Fl(C)1115 4909 y Ff(s)1149 4928 y Fq(\()1193 4911 y(~)1176 4928 y Fl(D)1231 4936 y Ff(\035)1271 4928 y Fq(\))1328 4898 y Fn(\024)1501 4837 y Fm(M)1589 4851 y Fp(\035)p 1434 4877 267 4 v 1434 4964 a Fm(")1476 4938 y Fp(s)p Fq(\()p Fp(\013)p Fl(\000)p Fp(j)t Fq(\))1711 4898 y Fm(;)108 b Fn(jj)p Fm(\037)1951 4861 y Fl(\000)p Fq(1)1951 4921 y Fp(\035)2046 4898 y Fn(jj)2096 4917 y Fl(C)2137 4898 y Ff(s)2170 4917 y Fq(\()p Fl(D)2254 4898 y Fg(\003)2252 4933 y Ff(\035)2293 4917 y Fq(\))2350 4898 y Fn(\024)2522 4837 y Fm(M)2610 4851 y Fp(\035)p 2456 4877 V 2456 4964 a Fm(")2498 4938 y Fp(s)p Fq(\()p Fp(\013)p Fl(\000)p Fp(j)t Fq(\))2732 4898 y Fm(:)p eop end %%Page: 74 74 TeXDict begin 74 73 bop 456 251 a Fq(74)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)601 450 y Fs(\(3\))42 b Fn(jjG)862 464 y Fp(\035)908 450 y Fn(jj)958 480 y Fl(C)999 461 y Fi(3)1034 480 y Fq(\()1078 463 y(~)1061 480 y Fl(D)1116 488 y Ff(\035)1156 480 y Fq(\))1213 450 y Fn(\024)1381 414 y Fp(M)1449 422 y Ff(\035)p 1319 429 232 4 v 1319 490 a Fp(")1352 470 y Fi(2\()p Ff(\013)p Fg(\000)p Ff(j)s Fi(\))1560 450 y Fo(,)1621 373 y Fh(\014)1621 427 y(\014)1651 450 y Fn(G)1710 417 y Fl(00)1705 473 y Fp(\035)1753 450 y Fs(\()p Fm(A;)15 b(")1938 417 y Fp(j)t(=)p Fq(2)2046 450 y Fs(\))2081 373 y Fh(\014)2081 427 y(\014)2137 450 y Fn(\025)25 b Fm(M)2321 464 y Fp(\035)2366 450 y Fo(.)456 604 y(wher)-5 b(e)33 b Fm(M)800 618 y Fp(\035)878 604 y Fo(is)f(a)h(c)-5 b(onstant)35 b(indep)-5 b(endent)34 b(of)f Fm(")p Fo(.)456 856 y(Pr)-5 b(o)g(of)20 b(.)37 b Fs(As)21 b(in)f(Prop)s(osition)g(61,)j(w)m(e)e(consider)f(the) h(curv)m(es)f Fm(e)26 b Fs(=)f Fn(K)2705 870 y Fq(in)n(t)2793 856 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")3050 823 y Fp(j)t(=)p Fq(2)3159 856 y Fs(\),)456 964 y(and)29 b(using)h(Lemma)h(60)g(with)f Fm(\016)f Fs(=)c(1,)31 b(w)m(e)g(de\014ne)f(in)g Fm(D)2370 931 y Fl(\003)2367 986 y Fp(o)2409 964 y Fs(:)1199 1153 y Fm(e)84 b Fs(=)e Fn(K)1547 1167 y Fq(in)n(t)1636 1153 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")1893 1115 y Fp(j)t(=)p Fq(2)2001 1153 y Fs(\))p Fm(;)986 1337 y(\034)10 b Fs(\()p Fm(x;)15 b(e)p Fs(\))85 b(=)1478 1213 y Fh(Z)1569 1239 y Fp(x)1529 1419 y Fq(0)2051 1275 y Fs(1)p 1638 1316 871 4 v 1648 1373 a Fp(@)t Fl(K)1743 1384 y Fi(in)n(t)p 1648 1391 173 4 v 1695 1443 a Fp(@)t(y)1831 1412 y Fs(\()p Fn(Y)1927 1426 y Fl(\006)1986 1412 y Fs(\()p Fm(u;)15 b(e)p Fs(\))p Fm(;)g(u)p Fs(;)g Fm(")2364 1386 y Fp(j)t(=)p Fq(2)2474 1412 y Fs(\))2519 1337 y Fm(du:)456 1603 y Fs(Then,)41 b Fm(T)13 b Fs(\()p Fm(e)p Fs(\))42 b(=)f Fm(\034)10 b Fs(\(2)p Fm(\031)s(;)15 b(e)p Fs(\))41 b(is)f(the)g(p)s(erio)s(d)f (of)h(the)g(p)s(erio)s(dic)f(orbit,)j(and)e(the)456 1711 y(action-angle)33 b(v)-5 b(ariables)31 b(are)f(giv)m(en)i(b)m(y:)1110 1948 y Fm(A)83 b Fs(=)1425 1887 y Fm(S)5 b Fs(\()p Fm(e)p Fs(\))p 1425 1927 175 4 v 1462 2011 a(2)p Fm(\031)1634 1948 y Fs(=)1768 1887 y(1)p 1740 1927 101 4 v 1740 2011 a(2)p Fm(\031)1866 1825 y Fh(Z)1957 1851 y Fq(2)p Fp(\031)1916 2031 y Fq(0)2054 1948 y Fn(Y)2115 1962 y Fl(\006)2174 1948 y Fs(\()p Fm(x;)15 b(e)p Fs(\))p Fm(dx;)1116 2186 y( )86 b Fs(=)1464 2125 y(2)p Fm(\031)p 1425 2165 179 4 v 1425 2249 a(T)13 b Fs(\()p Fm(e)p Fs(\))1614 2186 y Fm(\034)d Fs(\()p Fm(x;)15 b(e)p Fs(\))p Fm(;)456 2420 y Fs(where)29 b Fm(S)779 2387 y Fl(0)803 2420 y Fs(\()p Fm(e)p Fs(\))d(=)f Fm(T)13 b Fs(\()p Fm(e)p Fs(\).)555 2528 y(Although)40 b(the)f(region)h Fm(D)1491 2495 y Fl(\003)1488 2551 y Fp(o)1569 2528 y Fs(has)f(t)m(w)m(o)h(connected)g (comp)s(onen)m(ts,)i(w)m(e)e(will)456 2636 y(giv)m(e)22 b(full)g(details)g(only)f(for)h(the)f(comp)s(onen)m(t)h(where)f Fm(y)28 b(>)d Fs(0.)38 b(Let)22 b(us)e(remem)m(b)s(er)456 2744 y(that,)31 b(when)e(\()p Fm(Y)5 b(;)15 b(s)p Fs(\))26 b Fn(2)f(D)1311 2711 y Fl(\003)1308 2766 y Fp(o)1350 2744 y Fs(,)31 b(the)f(v)-5 b(ariables)31 b(\()p Fm(x;)15 b(e)p Fs(\))32 b(are)f(in)f Fn(J)2495 2758 y Fp(o)2532 2744 y Fs(:)456 2927 y(\(121\))402 b Fn(J)1125 2941 y Fp(o)1188 2927 y Fs(=)25 b Fn(f)p Fs(\()p Fm(x;)15 b(e)p Fs(\))p Fm(;)47 b(x)26 b Fn(2)f Fk(T)p Fm(;)45 b(c)1939 2941 y Fq(3)1979 2927 y Fm(")2021 2890 y Fp(\013)p Fl(\000)p Fp(j)2183 2927 y Fn(\024)25 b Fm(e)h Fn(\024)e Fm(c)2481 2941 y Fq(1)2521 2927 y Fn(g)555 3106 y Fs(First,)31 b(using)f(equation)h(\(97\))r(,)f(w)m(e)h(ha)m(v)m(e,)h(as)f(in)f (\(105\))r(:)456 3334 y(\(122\))251 b Fm(\034)10 b Fs(\()p Fm(x;)15 b(e)p Fs(\))26 b(=)1336 3272 y(1)p 1298 3313 122 4 v 1298 3331 a Fn(p)p 1374 3331 46 4 v 75 x Fs(2)1445 3210 y Fh(Z)1536 3236 y Fp(x)1495 3416 y Fq(0)1747 3272 y Fs(1)20 b(+)g Fm("b)p 1605 3313 522 4 v 1605 3331 a Fh(p)p 1696 3331 431 4 v 78 x Fm(e)g Fn(\000)g Fm(U)10 b Fs(\()p Fm(u)p Fs(;)15 b Fm(")p Fs(\))2136 3334 y Fm(du)21 b Fs(+)f Fm("P)2447 3348 y Fq(1)2487 3334 y Fs(\()p Fm(x;)15 b(e)p Fs(\))p Fm(;)456 3599 y Fs(where)29 b Fm(P)776 3613 y Fq(1)816 3599 y Fs(\()p Fm(x;)15 b(e)p Fs(\))26 b(=)1142 3526 y Fh(R)1203 3552 y Fp(x)1185 3631 y Fq(0)1288 3564 y Fp(@)p 1272 3579 74 4 v 1272 3631 a(@)t(e)1375 3576 y Fs(~)1356 3599 y Fn(Y)1417 3613 y Fq(+)1476 3599 y Fs(\()p Fm(`)p Fs(\()p Fm(u;)15 b(e)p Fs(\)\))p Fm(du)p Fs(.)43 b(Bounds)29 b(\(98\))i(and)e(\(118\))j(giv)m(e,)g(in)456 3707 y Fm(D)531 3721 y Fp(o)569 3707 y Fs(,)f(and)e(then,)i(in)f Fn(J)1202 3721 y Fp(o)1240 3707 y Fs(:)533 3921 y Fn(j)p Fm(P)616 3935 y Fq(1)656 3921 y Fn(j)681 3948 y Fl(C)722 3929 y Fi(0)756 3948 y Fq(\()p Fl(J)831 3956 y Ff(o)866 3948 y Fq(\))923 3921 y Fn(\024)25 b Fs(cte)p Fm(:)17 b(;)45 b Fn(j)q Fm(P)1330 3935 y Fq(1)1370 3921 y Fn(j)1395 3948 y Fl(C)1436 3929 y Ff(s)1469 3948 y Fq(\()p Fl(J)1544 3956 y Ff(o)1579 3948 y Fq(\))1636 3921 y Fn(\024)1905 3860 y Fs(cte)p Fm(:)p 1742 3901 483 4 v 1742 3988 a(")1784 3961 y Fq(\()p Fp(\013)p Fl(\000)p Fp(j)t Fq(\)\()p Fp(s)p Fl(\000)p Fq(1)p Fp(=)p Fq(2\))2234 3921 y Fm(;)h Fs(1)25 b Fn(\024)g Fm(s)g Fn(\024)g Fm(r)e Fn(\000)d Fs(2)p Fm(m)g Fn(\000)g Fs(2)p Fm(:)456 4130 y Fs(F)-8 b(rom)32 b(these)h(inequalities)h(and)d(Lemma)i(63,)g(w)m(e)g(obtain)g(upp)s(er) d(b)s(ounds)g(for)456 4238 y Fm(\034)10 b Fs(\()p Fm(x;)15 b(e)p Fs(\):)456 4346 y(\(123\))515 4497 y Fn(j)p Fm(\034)10 b Fn(j)615 4524 y Fl(C)656 4505 y Fi(0)691 4524 y Fq(\()p Fl(J)766 4532 y Ff(o)801 4524 y Fq(\))858 4497 y Fn(\024)25 b Fs(cte)p Fm(:)31 b Fs(log)1335 4435 y(1)p 1268 4476 180 4 v 1268 4559 a Fm(")1310 4533 y Fp(\013)p Fl(\000)p Fp(j)1457 4497 y Fm(;)46 b Fn(j)p Fm(\034)10 b Fn(j)1628 4524 y Fl(C)1669 4505 y Ff(s)1702 4524 y Fq(\()p Fl(J)1777 4532 y Ff(o)1812 4524 y Fq(\))1869 4497 y Fn(\024)2086 4435 y Fs(1)p 1975 4476 267 4 v 1975 4563 a Fm(")2017 4537 y Fq(\()p Fp(\013)p Fl(\000)p Fp(j)t Fq(\))p Fp(s)2252 4497 y Fm(;)45 b Fs(1)26 b Fn(\024)f Fm(s)g Fn(\024)g Fm(r)e Fn(\000)d Fs(2)p Fm(m)g Fn(\000)g Fs(2)p Fm(:)555 4712 y Fs(On)30 b(the)g(other)h(hand,)e(from)h Fm(\034)10 b Fs(\(2)p Fm(\031)s(;)15 b(e)p Fs(\))28 b(=)d Fm(S)2055 4679 y Fl(0)2078 4712 y Fs(\()p Fm(e)p Fs(\),)31 b(w)m(e)g(obtain:)456 4931 y(\(124\))47 b Fn(j)q Fm(S)5 b Fn(j)820 4958 y Fl(C)861 4939 y Fi(0)895 4958 y Fq(\()p Fl(J)970 4966 y Ff(o)1005 4958 y Fq(\))1062 4931 y Fn(\024)25 b Fs(cte)p Fm(:)17 b(;)45 b Fn(j)q Fm(S)5 b Fn(j)1497 4958 y Fl(C)1538 4939 y Ff(s)1571 4958 y Fq(\()p Fl(J)1646 4966 y Ff(o)1681 4958 y Fq(\))1738 4931 y Fn(\024)1971 4870 y Fs(cte)p Fm(:)p 1844 4910 412 4 v 1844 4997 a(")1886 4971 y Fq(\()p Fp(\013)p Fl(\000)p Fp(j)t Fq(\)\()p Fp(s)p Fl(\000)p Fq(1\))2265 4931 y Fm(;)46 b Fs(1)26 b Fn(\024)f Fm(s)g Fn(\024)g Fm(r)d Fn(\000)e Fs(2)p Fm(m)h Fn(\000)f Fs(2)p Fm(;)p eop end %%Page: 75 75 TeXDict begin 75 74 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(75)456 450 y Fs(and,)27 b(using)f(again)i(Lemma)f(63,)i(w)m(e)e(also)h(obtain)f(lo)m(w)m(er)h (b)s(ounds)d(for)i(the)g(\014rst)456 558 y(and)i(second)i(deriv)-5 b(ativ)m(es)32 b(of)e Fm(S)5 b Fs(:)1266 634 y Fh(\014)1266 688 y(\014)1297 711 y Fm(S)1358 674 y Fl(0)1381 711 y Fs(\()p Fm(e)p Fs(\))1493 634 y Fh(\014)1493 688 y(\014)1549 711 y Fs(=)25 b Fn(j)q Fm(T)13 b Fs(\()p Fm(e)p Fs(\))p Fn(j)26 b(\025)f Fs(cte)p Fm(:)32 b Fs(log)r(\(1)p Fm(=e)p Fs(\))p Fm(;)-2056 b Fs(\(125\))1266 818 y Fh(\014)1266 873 y(\014)1297 896 y Fm(S)1358 858 y Fl(00)1400 896 y Fs(\()p Fm(e)p Fs(\))1512 818 y Fh(\014)1512 873 y(\014)1569 896 y Fn(\025)1674 834 y Fs(cte)p Fm(:)p 1674 875 157 4 v 1732 958 a(e)1841 896 y(:)-1410 b Fs(\(126\))456 1090 y(T)-8 b(o)32 b(b)s(ound)d Fm( )h Fs(=)1099 1054 y Fq(2)p Fp(\031)p 1069 1069 139 4 v 1069 1123 a(T)10 b Fq(\()p Fp(e)p Fq(\))1218 1090 y Fm(\034)g Fs(\()p Fm(x;)15 b(e)p Fs(\))32 b(w)m(e)g(use)g(\(125\))r(,)g(\(124\))h(and)e (F)-8 b(aa-di-Bruno)33 b(for-)456 1211 y(m)m(ulas)d(for)g(the)h(comp)s (osition)g(of)g(functions)e(deriv)-5 b(ativ)m(es,)32 b(obtaining:)974 1282 y Fh(\014)974 1336 y(\014)974 1391 y(\014)974 1445 y(\014)1035 1352 y Fm(@)1088 1319 y Fp(s)p 1014 1393 133 4 v 1014 1476 a Fm(@)5 b(e)1109 1450 y Fp(s)1233 1352 y Fs(1)p 1166 1393 179 4 v 1166 1476 a Fm(T)13 b Fs(\()p Fm(e)p Fs(\))1355 1282 y Fh(\014)1355 1336 y(\014)1355 1391 y(\014)1355 1445 y(\014)1410 1414 y Fn(\024)1571 1352 y Fs(cte)p Fm(:)p 1516 1393 267 4 v 1516 1480 a(")1558 1453 y Fq(\()p Fp(\013)p Fl(\000)p Fp(j)t Fq(\))p Fp(s)1793 1414 y Fm(;)46 b Fs(0)25 b Fn(\024)g Fm(s)g Fn(\024)g Fm(r)e Fn(\000)d Fs(2)p Fm(m)g Fn(\000)g Fs(2)p Fm(;)456 1617 y Fs(then,)30 b(the)g(Leibniz)h(rule)f(giv)m(es:) 456 1688 y Fh(\014)456 1742 y(\014)456 1797 y(\014)456 1851 y(\014)506 1758 y Fs(1)p 496 1799 66 4 v 496 1882 a Fm(T)572 1820 y(P)630 1834 y Fq(1)669 1688 y Fh(\014)669 1742 y(\014)669 1797 y(\014)669 1851 y(\014)700 1910 y Fl(C)741 1891 y Fi(0)775 1910 y Fq(\()p Fl(J)850 1918 y Ff(o)885 1910 y Fq(\))942 1820 y Fn(\024)25 b Fs(cte)p Fm(:)17 b(;)1261 1688 y Fh(\014)1261 1742 y(\014)1261 1797 y(\014)1261 1851 y(\014)1312 1758 y Fs(1)p 1302 1799 V 1302 1882 a Fm(T)1377 1820 y(P)1435 1834 y Fq(1)1475 1688 y Fh(\014)1475 1742 y(\014)1475 1797 y(\014)1475 1851 y(\014)1506 1910 y Fl(C)1547 1891 y Ff(s)1580 1910 y Fq(\()p Fl(J)1655 1918 y Ff(o)1690 1910 y Fq(\))1747 1820 y Fn(\024)2015 1758 y Fs(cte)p Fm(:)p 1852 1799 483 4 v 1852 1886 a(")1894 1859 y Fq(\()p Fp(\013)p Fl(\000)p Fp(j)t Fq(\)\()p Fp(s)p Fl(\000)p Fq(1)p Fp(=)p Fq(2\))2345 1820 y Fm(;)42 b Fs(1)25 b Fn(\024)g Fm(s)g Fn(\024)g Fm(r)15 b Fn(\000)e Fs(2)p Fm(m)g Fn(\000)g Fs(2)p Fm(:)456 2048 y Fs(This)29 b(giv)m(es,)j(using)e(again)h(\(118\))s(,)973 2125 y Fh(\014)973 2179 y(\014)973 2234 y(\014)973 2288 y(\014)1024 2195 y Fs(1)p 1014 2236 66 4 v 1014 2319 a Fm(T)1089 2256 y(\034)1139 2125 y Fh(\014)1139 2179 y(\014)1139 2234 y(\014)1139 2288 y(\014)1170 2347 y Fl(C)1211 2328 y Ff(s)1244 2347 y Fq(\()p Fl(J)1319 2355 y Ff(o)1354 2347 y Fq(\))1411 2256 y Fn(\024)1572 2195 y Fs(cte)p Fm(:)p 1517 2236 267 4 v 1517 2323 a(")1559 2296 y Fq(\()p Fp(\013)p Fl(\000)p Fp(j)t Fq(\))p Fp(s)1793 2256 y Fm(;)46 b Fs(0)26 b Fn(\024)f Fm(s)f Fn(\024)h Fm(r)e Fn(\000)d Fs(2)p Fm(m)h Fn(\000)e Fs(2)p Fm(:)456 2485 y Fs(This)44 b(b)s(ound)e(together)47 b(with)d(\(124\))j (establishes)f(the)f(b)s(ound)d(claimed)k(in)456 2593 y(statemen)m(t)34 b(2.)49 b(of)33 b(the)g(prop)s(osition)g(64)h(for)e (the)h Fn(C)2222 2560 y Fp(s)2292 2593 y Fs(norm)f(of)i(the)f(c)m (hange)h(of)456 2701 y(v)-5 b(ariables)1166 2846 y Fm(\037)1223 2809 y Fq(\()p Fl(\000)p Fq(1\))1223 2869 y Fp(\035)1372 2846 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(\))26 b(=)f(\()1749 2785 y Fm(S)5 b Fs(\()p Fm(e)p Fs(\))p 1749 2826 175 4 v 1786 2909 a(2)p Fm(\031)1934 2846 y(;)2023 2785 y Fs(2)p Fm(\031)p 1984 2826 179 4 v 1984 2909 a(T)13 b Fs(\()p Fm(e)p Fs(\))2173 2846 y Fm(\034)d Fs(\()p Fm(x;)15 b(e)p Fs(\)\))456 3049 y(with)30 b Fm(e)25 b Fs(=)g Fm(K)903 3063 y Fq(in)n(t)992 3049 y Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(;)g Fm(")1259 3016 y Fp(j)t(=)p Fq(2)1367 3049 y Fs(\),)31 b(0)26 b Fn(\024)f Fm(s)g Fn(\024)g Fm(r)d Fn(\000)e Fs(2)p Fm(m)h Fn(\000)f Fs(2.)555 3178 y(Since)40 b(det)15 b Fm(D)s(\037)1078 3130 y Fq(\()p Fl(\000)p Fq(1\))1078 3190 y Fp(\035)1227 3178 y Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(\))42 b(=)f(1,)i(the)d Fn(C)1933 3145 y Fp(s)2009 3178 y Fs(norm)g(of)g Fm(\037)f Fs(satis\014es)h(the)g(same)456 3286 y(estimates.)555 3394 y(Moreo)m(v)m(er,)46 b(taking)41 b(in)m(to)h(accoun)m(t)g(that)f Fn(G)2085 3409 y Fp(f)2173 3394 y Fs(=)h Fm(S)2347 3361 y Fl(\000)p Fq(1)2441 3394 y Fs(,)h(the)e(lo)m(w)m(er)h(b)s(ound)456 3502 y(for)36 b(the)h(second)g(deriv)-5 b(ativ)m(es)38 b(of)f Fn(G)1687 3517 y Fp(f)1769 3502 y Fs(and)f(the)h(upp)s(er)e(b)s(ound)f(for)j(the) g(third)456 3609 y(deriv)-5 b(ativ)m(e)32 b(of)e Fn(G)1030 3624 y Fp(f)1106 3609 y Fs(follo)m(w)h(from)f(\(125\))r(,)h(\(124\))r (,)g(\(126\))r(.)555 3717 y(This)f(\014nishes)f(the)i(pro)s(of)e(of)i (Prop)s(osition)f(64)h(for)f(the)h(region)g Fn(D)2832 3684 y Fl(\003)2829 3740 y Fp(o)2871 3717 y Fs(.)555 3825 y(In)f(the)g(region)h Fn(D)1171 3792 y Fl(\003)1168 3851 y Fp(in)1270 3825 y Fs(the)f(action)i(v)-5 b(ariable)31 b(is)g(de\014ned)e(as)456 4034 y(\(127\))580 b Fm(A)26 b Fs(=)1441 3972 y Fm(S)5 b Fs(\()p Fm(e)p Fs(\))p 1441 4013 175 4 v 1477 4096 a(2)p Fm(\031)1650 4034 y Fs(=)1783 3972 y(1)p 1756 4013 101 4 v 1756 4096 a(2)p Fm(\031)1881 3910 y Fh(I)1932 4116 y Fl(K)1986 4087 y Fg(\000)p Fi(1)1986 4139 y(in)n(t)2069 4116 y Fq(\()p Fp(e)p Fq(\))2176 4034 y Fm(Y)35 b(dx;)456 4256 y Fs(and)29 b Fm( )34 b Fs(is)d(the)f (conjugate)i(angle.)555 4364 y(The)26 b(main)h(complication)i(to)e (obtain)g(the)g(b)s(ounds)d(of)j(the)g(c)m(hange)h(of)e(v)-5 b(ari-)456 4487 y(ables)22 b(is)g(that)h(the)f(curv)m(es)h Fn(K)1436 4439 y Fq(\()p Fl(\000)p Fq(1\))1435 4515 y(in)n(t)1585 4487 y Fs(\()p Fm(e)p Fs(\))h(when)d Fm(e)k Fn(\024)g Fs(0,)g(cannot)d(b)s(e)g(parameterized)456 4595 y(as)31 b(graphs)f(of)h(functions)f Fm(Y)46 b Fs(=)26 b Fn(Y)7 b Fs(\()p Fm(x;)15 b(e)p Fs(\))32 b(so)g(that)f(the)g(computation)h(of) 38 b(\(127\))456 4703 y(will)30 b(need)g(to)i(b)s(e)d(divided)h(in)g (di\013eren)m(t)h(pieces.)555 4811 y(Cho)s(osing)f(the)h(origin)g(of)f (time)h(in)f(the)h(section)1261 4964 y(\006)25 b(=)g Fn(f)p Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(\))26 b(:)g Fm(Y)45 b(>)25 b Fs(0)p Fm(;)15 b(x)26 b Fs(=)f Fm(\031)s Fn(g)p Fm(;)p eop end %%Page: 76 76 TeXDict begin 76 75 bop 456 251 a Fq(76)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 450 y Fs(w)m(e)41 b(de\014ne)g Fm(t)p Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(\))41 b(as)h(the)f(time)h(that)g(the)f(tra)5 b(jectory)42 b(starting)g(in)f (\()p Fm(Y)5 b(;)15 b(x)p Fs(\))456 558 y(tak)m(es)31 b(to)h(arriv)m(e)f(to)g(the)f(section)i(\006.)555 666 y(T)-8 b(o)32 b(understand)d(the)i(regularit)m(y)i(prop)s(erties)d(of)i (the)f(function)g Fm(t)p Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(\))31 b(w)m(e)456 774 y(\014nd)d(it)i(con)m(v)m(enien)m(t)j(to)d(p)s (erform)f(di\013eren)m(t)h(argumen)m(ts)g(in)g(di\013eren)m(t)g (regions)456 882 y(of)g(the)h(\()p Fm(Y)5 b(;)15 b(x)p Fs(\))31 b(plane.)555 990 y(Let)g Fm(\032)25 b(>)g Fs(0)31 b(b)s(e)f(the)g(n)m(um)m(b)s(er)f(that)i(app)s(ears)f(in)g(Lemma)h(60,) g(part)f(2.)555 1113 y(If)25 b(\()p Fm(Y)5 b(;)15 b(x)p Fs(\))26 b Fn(2)f(D)1046 1080 y Fl(\003)1043 1139 y Fp(in;\032)1195 1113 y Fn(\021)g(f)p Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(\))27 b Fn(2)d(K)1738 1066 y Fq(\()p Fl(\000)p Fq(1\))1737 1141 y Fp(int)1888 1113 y Fs(\()p Fm(e)p Fs(\))p Fm(;)42 b(Y)j(>)25 b Fs(0)p Fm(;)41 b(\032)25 b(<)g(x)g(<)g Fs(2)p Fm(\031)13 b Fn(\000)d Fm(\032)p Fn(g)p Fs(,)27 b(the)456 1236 y(function)g Fm(t)p Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(\))28 b(is)f(of)h(class)g Fn(C)1542 1203 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(2)1851 1236 y Fs(and)f(its)g Fn(C)2201 1203 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(2)2510 1236 y Fs(norm)f(is)i(b)s(ounded)456 1344 y(indep)s(enden)m (tly)h(of)i Fm(e;)15 b(")p Fs(.)555 1452 y(Outside)22 b(of)h Fn(D)1055 1419 y Fl(\003)1052 1477 y Fp(in;\032)1201 1452 y Fs(the)g(function)f Fm(t)p Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(\))24 b(is)e(more)h(complicated)h(to)f(analyze)456 1562 y(since)28 b(the)h(orbits)f(starting)h(outside)f(of)h Fn(D)1910 1529 y Fl(\003)1907 1588 y Fp(in;\032)2062 1562 y Fs(can)f(pass)g(close)i(to)f(the)f(critical)456 1677 y(p)s(oin)m(t)37 b(\(0)p Fm(;)15 b Fs(0\))39 b(b)s(efore)d(reac)m (hing)i(\006,)h(and)d(the)h(time)h Fm(t)p Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(\))38 b(go)s(es)f(to)h(in\014nit)m(y)456 1785 y(when)29 b Fm(e)i Fs(go)s(es)g(to)g(zero.)555 1893 y(W)-8 b(e)28 b(will)f(explain)g(the)f(case)i(when)e Fm(Y)45 b(>)25 b Fs(0)i(and)f(0)f Fm(<)g(x)g(<)g(\032)p Fs(,)j(and)e(the)h(other)456 2001 y(cases)k(are)g(easy)g(mo)s (di\014cations)f(of)h(this)f(one.)555 2109 y(If)g Fm(x)25 b(<)g(\032)p Fs(,)31 b(and)f Fm(Y)45 b(>)25 b Fs(0,)31 b(w)m(e)g(compute)f(the)h(time)g(as)1266 2280 y Fm(t)p Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(\))26 b(=)f Fm(t)1674 2294 y Fq(1)1713 2280 y Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(\))21 b(+)f Fm(t)2078 2294 y Fq(2)2117 2280 y Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(\))p Fm(;)456 2451 y Fs(where)23 b Fm(t)745 2465 y Fq(1)808 2451 y Fs(is)h(the)h(time)f(to)h(arriv)m(e)g(at)g(the)f (section)h(giv)m(en)g(b)m(y)f Fm(x)h Fs(=)g Fm(\032)f Fs(\(w)m(e)h(denote)456 2565 y(b)m(y)41 b Fm(Y)20 b Fs(\()p Fm(\032)p Fs(\))41 b(the)h(corresp)s(onding)e Fm(Y)61 b Fs(co)s(ordinate:)i Fn(K)2274 2579 y Fp(int)2370 2565 y Fs(\()p Fm(Y)21 b Fs(\()p Fm(\032)p Fs(\))p Fm(;)15 b(x)p Fs(;)g Fm(")2770 2532 y Fp(j)t(=)p Fq(2)2879 2565 y Fs(\))43 b(=)g Fm(e)p Fs(\),)456 2673 y(and)29 b Fm(t)665 2687 y Fq(2)735 2673 y Fs(is)h(the)g(time)h(b)s(et)m(w)m(een)g(the)g (sections)g Fm(x)25 b Fs(=)g Fm(\032)30 b Fs(and)g(\006.)40 b(Note)32 b(that)e Fm(t)3043 2687 y Fq(2)3113 2673 y Fs(is)456 2781 y Fn(C)509 2748 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(2)817 2781 y Fs(with)d(a)i(b)s(ounded)c Fn(C)1516 2748 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(2)1825 2781 y Fs(norm,)i(so)h(w)m(e)g(only)g(need)g(to)g(estimate) 456 2889 y Fm(t)489 2903 y Fq(1)528 2889 y Fs(.)555 2997 y(T)-8 b(o)42 b(b)s(ound)c Fm(t)1023 3011 y Fq(1)1063 2997 y Fs(,)43 b(w)m(e)f(use)e(Lemma)h(60)h(and)e(the)h(Hamiltonian)h (equations)456 3105 y(obtaining)31 b(that)706 3341 y Fm(t)739 3355 y Fq(1)778 3341 y Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(\))84 b(=)f Fn(\000)1322 3218 y Fh(Z)1413 3244 y Fp(Y)15 b Fq(\()p Fp(\032)p Fq(\))1372 3424 y Fp(Y)1977 3280 y Fm(du)p 1590 3320 875 4 v 1600 3377 a Fp(@)t Fl(K)1695 3388 y Fi(in)n(t)p 1600 3395 173 4 v 1646 3448 a Fp(@)t(x)1782 3416 y Fs(\()p Fm(u;)g Fn(X)1974 3430 y Fq(+)2034 3416 y Fs(\()p Fm(u;)g(e)p Fs(\);)g Fm(")2320 3390 y Fp(j)t(=)p Fq(2)2429 3416 y Fs(\))1082 3630 y(=)83 b Fn(\000)1322 3506 y Fh(Z)1413 3533 y Fp(Y)15 b Fq(\()p Fp(\032)p Fq(\))1372 3713 y Fp(Y)1590 3569 y Fm(@)5 b Fn(X)1708 3583 y Fq(+)p 1590 3609 178 4 v 1631 3693 a Fm(@)g(e)1777 3630 y Fs(\()p Fm(u;)15 b(e)p Fs(\))p Fm(du)1082 3895 y Fs(=)1285 3834 y(1)p 1246 3874 124 4 v 1246 3892 a Fn(p)p 1322 3892 49 4 v 66 x Fm(a)1395 3771 y Fh(Z)1486 3798 y Fp(Y)g Fq(\()p Fp(\032)p Fq(\))1445 3978 y Fp(Y)1993 3834 y Fs(1)p 1663 3874 706 4 v 1663 3892 a Fh(p)p 1754 3892 615 4 v 82 x Fm(h)1806 3988 y Fp(int)1903 3974 y Fs(\()p Fm(u)p Fs(;)g Fm(")2072 3948 y Fp(j)t(=)p Fq(2)2180 3974 y Fs(\))20 b Fn(\000)g Fm(e)2379 3895 y(du)g Fs(+)2609 3872 y(~)2589 3895 y Fm(P)2647 3909 y Fq(1)2687 3895 y Fs(\()p Fm(Y)5 b(;)15 b(e)p Fs(\))p Fm(:)456 4138 y Fs(This)39 b(form)m(ula)i(is)g(v)m(ery)f(similar)h(to)h(form)m(ula)e (\(122\))j(and)d(analogous)i(argu-)456 4246 y(men)m(ts)30 b(to)h(the)g(ones)f(used)g(there)h(establish)f(that)488 4435 y Fn(j)q Fm(t)547 4449 y Fq(1)586 4435 y Fn(j)611 4461 y Fl(C)652 4443 y Fi(0)687 4461 y Fq(\()p Fl(J)762 4471 y Ff(i)788 4461 y Fq(\))845 4435 y Fn(\024)25 b Fs(cte)p Fm(:)32 b Fs(log)r(\(1)p Fm(=e)p Fs(\)\))p Fm(;)48 b Fn(j)q Fm(t)1599 4449 y Fq(1)1638 4435 y Fn(j)1663 4461 y Fl(C)1704 4443 y Ff(s)1737 4461 y Fq(\()p Fl(J)1812 4471 y Ff(i)1839 4461 y Fq(\))1896 4435 y Fn(\024)2057 4373 y Fs(cte)p Fm(:)p 2002 4414 267 4 v 2002 4501 a(")2044 4474 y Fq(\()p Fp(\013)p Fl(\000)p Fp(j)t Fq(\))p Fp(s)2278 4435 y Fm(;)e Fs(1)26 b Fn(\024)f Fm(s)f Fn(\024)h Fm(r)e Fn(\000)d Fs(2)p Fm(m)h Fn(\000)e Fs(2)p Fm(:)456 4640 y Fs(where)35 b Fn(J)786 4654 y Fp(i)849 4640 y Fs(is)g(de\014ned)f (analogously)j(to)f Fn(J)1939 4654 y Fp(o)2012 4640 y Fs(in)g(\(121\))h(to)f(b)s(e)f(domain)g(of)h(the)456 4748 y(v)-5 b(ariables)31 b(\()p Fm(x;)15 b(e)p Fs(\).)555 4856 y(Analogous)41 b(argumen)m(ts)g(can)f(b)s(e)f(used)h(in)f(the)h (cases)h(when)e Fm(Y)62 b(<)41 b Fs(0,)i(or)456 4964 y Fm(x)32 b(<)g(\032)p Fs(,)k(obtaining)f(exactly)i(the)d(same)h(kind)f (of)h(b)s(ounds)d(in)j(all)g Fm(D)2825 4931 y Fl(\003)2822 4990 y Fp(in)2893 4964 y Fs(.)54 b(Once)p eop end %%Page: 77 77 TeXDict begin 77 76 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(77)456 450 y Fs(w)m(e)29 b(ha)m(v)m(e)i(de\014ned)d(the)i(function)f Fm(t)p Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(\))29 b(in)g Fm(D)2085 417 y Fl(\003)2082 476 y Fp(in)2154 450 y Fs(,)g(w)m(e)h(ha)m(v)m(e)h(that)f(the)f (action-)456 558 y(angle)i(v)-5 b(ariables)31 b(are)g(giv)m(en)g(b)m(y) 1463 765 y Fm(A)83 b Fs(=)1777 704 y Fm(S)5 b Fs(\()p Fm(e)p Fs(\))p 1777 745 175 4 v 1814 828 a(2)p Fm(\031)1468 993 y( )87 b Fs(=)c(2)p Fm(\031)1878 931 y(t)p Fs(\()p Fm(Y)5 b(;)15 b(x)p Fs(\))p 1878 972 254 4 v 1915 1055 a Fm(T)e Fs(\()p Fm(e)p Fs(\))2142 993 y Fm(;)456 1208 y Fs(where)42 b Fm(T)13 b Fs(\()p Fm(e)p Fs(\))48 b(=)e Fm(S)1135 1175 y Fl(0)1158 1208 y Fs(\()p Fm(e)p Fs(\).)81 b(Pro)s(ceeding)43 b(as)h(w)m(e)f(did)g(in)g Fm(D)2495 1175 y Fl(\003)2492 1230 y Fp(o)2534 1208 y Fs(,)j(w)m(e)e(obtain)g (the)456 1316 y(same)32 b(kind)e(of)i(b)s(ounds)e(in)h(these)h(t)m(w)m (o)h(regions.)45 b(This)31 b(\014nishes)f(the)i(pro)s(of)f(of)456 1424 y(Prop)s(osition)f(64.)2043 b Fj(\003)555 1583 y Fs(Applying)39 b(again)h(Theorem)f(45)g(to)h(the)f(time-2)p Fm(\031)s(k)2369 1597 y Fq(0)2450 1583 y Fs(map)f(of)h(Hamilton-)456 1691 y(ian)27 b(\(77\))i(after)f(it)g(has)f(b)s(een)f(scaled)i(and)f (written)g(in)g(action-angle)j(v)-5 b(ariables)456 1799 y(and)33 b(going)h(bac)m(k)h(to)f(the)g(original)h(v)-5 b(ariables)34 b(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(\))35 b(w)m(e)f(obtain)g(the)g(exis-)456 1907 y(tence)d(of)g(primary)e(tori)i (in)f Fm(D)1499 1874 y Fl(\003)1496 1930 y Fp(o)1569 1907 y Fs(and)f(secondary)i(tori)g(in)f Fm(D)2524 1874 y Fl(\003)2521 1933 y Fp(in)2592 1907 y Fs(.)456 2084 y Fo(Pr)-5 b(o)g(of)34 b(of)f(p)-5 b(arts)34 b(2\))f(and)h(3\))f(of)g (The)-5 b(or)g(em)34 b(56.)43 b Fs(W)-8 b(e)29 b(will)g(pro)m(v)m(e)g (the)g(results)f(in)456 2191 y(b)s(oth)22 b(regions)h Fm(D)1042 2158 y Fl(\003)1039 2214 y Fp(o)1082 2191 y Fs(,)h Fm(D)1209 2158 y Fl(\003)1206 2217 y Fp(in)1300 2191 y Fs(at)g(the)f(same)h(time)f(b)m(y)g(comp)s(osing)g(the)h (Hamiltonian)456 2301 y Fn(K)31 b Fs(with)g(the)f(t)m(w)m(o)i (di\013eren)m(t)f Fn(C)1504 2268 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(2)1815 2301 y Fs(c)m(hanges)g Fm(\037)2208 2315 y Fp(\035)2253 2301 y Fs(,)f Fm(\035)f Fs(=)c Fm(o;)15 b(in)p Fs(.)555 2409 y(Again,)44 b(the)d(pro)s(of)f(will)h(consist)g (in)f(applying)g(Theorem)h(45)g(to)g Fm(F)13 b Fs(,)44 b(the)456 2517 y(time-2)p Fm(\031)s(k)809 2531 y Fq(0)880 2517 y Fs(map)30 b(of)h(the)f(Hamiltonian)456 2682 y(\(128\))772 2659 y(~)754 2682 y Fn(K)823 2696 y Fp(\035)868 2682 y Fs(\()p Fm(A;)15 b( )s(;)g(s)p Fs(;)g Fm(")p Fs(\))28 b(=)d Fm(")1439 2644 y Fp(j)t(=)p Fq(2)1546 2682 y Fn(G)1600 2696 y Fp(\035)1645 2682 y Fs(\()p Fm(A)p Fs(;)15 b Fm(")1830 2644 y Fp(j)t(=)p Fq(2)1939 2682 y Fs(\))20 b(+)g Fm(")2127 2644 y Fp(m)p Fq(+1)p Fl(\000)p Fp(j)t(=)p Fq(2)2457 2659 y Fs(~)2442 2682 y Fm(S)2498 2696 y Fp(\035)2543 2682 y Fs(\()p Fm(A;)15 b( )s(;)g(s)p Fs(;)g Fm(")2913 2644 y Fp(j)t(=)p Fq(2)3022 2682 y Fs(\))p Fm(;)456 2848 y Fs(where)743 2826 y(~)718 2848 y Fm(K)795 2862 y Fp(\035)866 2848 y Fs(=)25 b Fn(K)d(\016)e Fm(\037)1175 2862 y Fp(\035)1250 2848 y Fs(and)1442 2826 y(~)1427 2848 y Fm(S)1483 2862 y Fp(\035)1553 2848 y Fs(=)25 b Fm(S)1705 2862 y Fq(1)1764 2848 y Fn(\016)c Fm(\037)1887 2862 y Fp(\035)1931 2848 y Fs(.)555 2958 y(Since)809 2935 y(~)785 2958 y Fm(K)862 2972 y Fp(\035)907 2958 y Fs(,)j Fn(G)1010 2972 y Fp(\035)1055 2958 y Fs(,)1120 2935 y(~)1104 2958 y Fm(S)1160 2972 y Fp(\035)1227 2958 y Fs(are)f Fn(C)1424 2925 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(2)1705 2958 y Fs(,)i(and)d(w)m(e)h (ha)m(v)m(e)h(assumed)d(in)i(Theorem)f(56)456 3068 y(that)31 b Fm(r)22 b Fn(\000)e Fs(2)p Fm(m)h Fn(\000)f Fs(2)25 b Fn(\025)g Fs(6,)31 b(w)m(e)g(ha)m(v)m(e)h(that)1876 3045 y(~)1852 3068 y Fm(K)1929 3082 y Fp(\035)1974 3068 y Fs(,)e Fn(G)2083 3082 y Fp(\035)2128 3068 y Fs(,)2199 3045 y(~)2184 3068 y Fm(S)2240 3082 y Fp(\035)2315 3068 y Fs(are)g Fn(C)2519 3035 y Fq(6)2559 3068 y Fs(.)555 3177 y(W)-8 b(e)22 b(denote)f(b)m(y)f Fm(F)1159 3191 y Fq(0)1219 3177 y Fs(the)g(time-2)p Fm(\031)s(k)1718 3191 y Fq(0)1780 3177 y Fs(map)g(of)g(Hamiltonian)i Fm(")2620 3144 y Fp(j)t(=)p Fq(2)2727 3177 y Fn(G)2781 3191 y Fp(\035)2826 3177 y Fs(\()p Fm(A)p Fs(;)15 b Fm(")3011 3144 y Fp(j)t(=)p Fq(2)3120 3177 y Fs(\).)456 3285 y(W)-8 b(e)31 b(ha)m(v)m(e)h(that)f Fm(F)43 b Fs(and)30 b Fm(F)1355 3299 y Fq(0)1425 3285 y Fs(are)h Fn(C)1630 3252 y Fq(5)1699 3285 y Fs(and)f(using)g(standard) f(results)i(on)f(dep)s(en-)456 3392 y(dence)g(of)h(solutions)f(of)h(o)s (de's)f(on)h(parameters)658 3558 y Fn(jj)p Fm(F)i Fn(\000)20 b Fm(F)948 3572 y Fq(0)988 3558 y Fn(jj)1038 3577 y Fl(C)1079 3558 y Fi(5)35 b Fn(\024)25 b Fs(cte)p Fm(:)16 b(")1438 3520 y Fp(m)p Fq(+1)p Fl(\000)p Fp(j)t(=)p Fq(2)1753 3558 y Fn(jj)1819 3535 y Fs(~)1803 3558 y Fm(S)1859 3572 y Fp(\035)1904 3558 y Fn(jj)1954 3577 y Fl(C)1995 3558 y Fi(6)35 b Fn(\024)25 b Fs(cte)p Fm(:)16 b(")2354 3520 y Fp(m)p Fq(+1)p Fl(\000)p Fp(j)t(=)p Fq(2)p Fl(\000)p Fq(6\()p Fp(\013)p Fl(\000)p Fp(j)t Fq(\))2947 3558 y Fm(:)456 3717 y Fs(Since)40 b Fn(G)757 3732 y Fp(f)843 3717 y Fs(dep)s(ends)f(only)i(on)f Fm(A)p Fs(,)k Fm(F)1746 3731 y Fq(0)1826 3717 y Fs(is)d(an)f(in)m(tegrable)j(map)d(of)h(the)f (form)456 3825 y(\()p Fm(A;)15 b Fs(\011\))26 b Fn(!)f Fs(\()p Fm(A;)15 b Fs(\011)21 b(+)f(\001\()p Fm(A)p Fs(\).)555 3933 y(F)-8 b(urthermore,)31 b(b)m(y)f(item)h(3.)41 b(of)31 b(Prop)s(osition)f(64,)i(w)m(e)e(ha)m(v)m(e)1030 4087 y Fm(d)p 996 4127 116 4 v 996 4211 a(dA)1121 4148 y Fs(\001\()p Fm(A)p Fs(\))c(=)f Fm(")1499 4111 y Fp(j)t(=)p Fq(2)1650 4087 y Fm(@)1703 4054 y Fq(2)p 1616 4127 161 4 v 1616 4211 a Fm(@)5 b(A)1737 4184 y Fq(2)1787 4148 y Fn(G)1841 4162 y Fp(\035)1886 4148 y Fs(\()p Fm(A)p Fs(;)15 b Fm(")2071 4111 y Fp(j)t(=)p Fq(2)2180 4148 y Fs(\))25 b Fn(\025)g Fm(M)2424 4162 y Fp(\035)2469 4148 y Fm(")2511 4111 y Fp(j)t(=)p Fq(2)2619 4148 y Fm(:)456 4334 y Fs(Hence,)38 b(the)f(mapping)e Fm(F)1358 4348 y Fq(0)1434 4334 y Fs(will)h(b)s(e)g (a)h(t)m(wist)g(mapping,)g(and)e(w)m(e)i(can)f(apply)456 4443 y(Theorem)26 b(45)h(with)f Fm(\016)j Fs(=)c Fm(")1363 4410 y Fp(m)p Fq(+1)p Fl(\000)p Fp(j)t(=)p Fq(2)p Fl(\000)p Fq(6\()p Fp(\013)p Fl(\000)p Fp(j)t Fq(\))1956 4443 y Fs(,)i(and)f(w)m(e)h(obtain,)h(for)e Fm(m)f(>)g Fs(6\()p Fm(\013)13 b Fn(\000)456 4551 y Fm(j)5 b Fs(\))21 b(+)f(3)p Fm(j)5 b(=)p Fs(2)22 b Fn(\000)e Fs(1:)601 4691 y(\(1\))42 b(There)37 b(exist)h(a)g(set)g(of)g(v)-5 b(alues)38 b Fm(A)1939 4706 y Fp(l)1965 4691 y Fs(,)h(suc)m(h)e(that)h(he)g (Hamiltonian)3121 4668 y(~)3103 4691 y Fn(K)758 4799 y Fs(has)30 b(in)m(v)-5 b(arian)m(t)32 b(tori)f(giv)m(en)g(b)m(y)1352 4964 y Fm(A)26 b Fs(=)f Fm(A)1610 4979 y Fp(l)1656 4964 y Fs(+)20 b Fn(A)1820 4979 y Fp(l)1846 4964 y Fs(\()p Fm( )s(;)15 b(s)p Fs(;)g Fm(")2108 4927 y Fp(j)t(=)p Fq(2)2216 4964 y Fs(\))p Fm(;)p eop end %%Page: 78 78 TeXDict begin 78 77 bop 456 251 a Fq(78)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)758 450 y Fs(where)d Fn(A)1084 465 y Fp(l)1130 450 y Fs(are)h Fn(C)1325 417 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(5)p Fl(\000)p Fp(\021)1718 450 y Fs(functions,)h(for)f(an)m(y)f Fm(\021)29 b(>)c Fs(0,)e(and)c Fn(jjA)2970 465 y Fp(l)2996 450 y Fn(jj)3046 470 y Fl(C)3087 451 y Fi(2)3152 450 y Fn(\024)758 588 y Fs(cte)p Fm(:)e(")967 518 y Ff(m)p Fi(+1)p Fg(\000)p Fi(6\()p Ff(\013)p Fg(\000)p Ff(j)s Fi(\))p Fg(\000)p Fi(3)p Ff(j)s(=)p Fi(2)p 967 537 544 3 v 1224 578 a(2)1525 588 y Fs(.)601 696 y(\(2\))42 b(The)24 b(motion)g(on)g(these)g(tori)h (is)e Fn(C)1889 663 y Fq(1)1929 696 y Fs(-conjugate)i(to)g(a)f(rigid)g (translation)758 804 y(of)31 b(frequencies)g(\()p Fm(!)s Fs(\()p Fm(A)1526 819 y Fp(l)1553 804 y Fs(\))p Fm(;)15 b Fs(1\),)33 b(where)d Fm(!)s Fs(\()p Fm(A)2192 819 y Fp(l)2218 804 y Fs(\))h(is)g(a)h(Diophan)m(tine)f(n)m(um-)758 941 y(b)s(er)h(of)i(constan)m(t)g(t)m(yp)s(e)f(and)g(Mark)m(o)m(v)h (constan)m(t)h Fm(K)7 b(")2622 872 y Ff(m)p Fi(+1)p Fg(\000)p Fi(6\()p Ff(\013)p Fg(\000)p Ff(j)s Fi(\))p Fg(\000)p Ff(j)s(=)p Fi(2)p 2622 891 513 3 v 2863 932 a(2)3149 941 y Fs(,)758 1049 y(as)31 b(stated)g(in)f(De\014nition)h(42.)601 1164 y(\(3\))42 b(The)47 b(union)g(of)h(neigh)m(b)s(orho)s(o)s(ds)e(of) h(size)i(cte)p Fm(:)16 b(")2495 1095 y Ff(m)p Fi(+1)p Fg(\000)p Fi(6\()p Ff(\013)p Fg(\000)p Ff(j)s Fi(\))p Fg(\000)p Fi(3)p Ff(j)s(=)p Fi(2)p 2496 1114 544 3 v 2752 1155 a(2)3101 1164 y Fs(of)758 1279 y(these)31 b(tori)g(co)m(v)m (er)h(all)g(the)e(regions)2011 1256 y(~)1990 1279 y Fm(D)2065 1293 y Fp(\035)2110 1279 y Fs(.)555 1412 y(Going)45 b(bac)m(k)g(to)f (the)g(v)-5 b(ariables)45 b(\()p Fm(Y)5 b(;)15 b(x;)g(s)p Fs(\),)48 b(and)43 b(using)h(that)g Fn(jjG)2880 1426 y Fp(\035)2925 1412 y Fn(jj)2975 1431 y Fl(C)3016 1413 y Fi(3)3103 1412 y Fn(\024)456 1526 y Fs(cte)p Fm(:)16 b(")654 1493 y Fl(\000)p Fq(2\()p Fp(\013)p Fl(\000)p Fp(j)t Fq(\))936 1526 y Fs(,)46 b(and)41 b(that)i Fn(jj)p Fm(\037)1511 1493 y Fl(\000)p Fq(1)1511 1548 y Fp(\035)1606 1526 y Fn(jj)1656 1545 y Fl(C)1697 1526 y Fi(2)1732 1545 y Fq(\()p Fl(D)1816 1526 y Fg(\003)1814 1562 y Ff(\035)1854 1545 y Fq(\))1930 1526 y Fn(\024)i Fs(cte)p Fm(:)16 b(")2244 1493 y Fl(\000)p Fq(2\()p Fp(\013)p Fl(\000)p Fp(j)t Fq(\))2527 1526 y Fs(,)45 b(w)m(e)d(obtain,)k(for)456 1638 y Fm(m)25 b(>)g Fs(14\()p Fm(\013)d Fn(\000)d Fm(j)5 b Fs(\))22 b(+)e(3)p Fm(j)5 b(=)p Fs(2)22 b Fn(\000)e Fs(1,)31 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y Ff(m)p Fi(+1+)p Ff(j)s(=)p Fi(2)p Fg(\000)p Fi(14\()p Ff(\013)p Fg(\000)p Ff(j)s Fi(\))p 2020 2229 542 3 v 2276 2270 a(2)2576 2282 y Fs(\))p Fm(:)456 2439 y Fs(No)m(w,)34 b(calling)h Fm(F)1040 2454 y Fp(l)1096 2439 y Fs(=)30 b Fm(")1239 2406 y Fp(j)1275 2439 y Fm(e)1317 2454 y Fp(l)1344 2439 y Fs(,)k(for)f Fm(\035)g Fs(=)c Fm(o)p Fs(,)34 b(and)f Fm(G)2081 2454 y Fp(l)2137 2439 y Fs(=)c Fm(")2279 2406 y Fp(j)2316 2439 y Fm(e)2358 2454 y Fp(l)2385 2439 y Fs(,)k(for)g Fm(\035)g Fs(=)d Fm(i)p Fs(,)k(w)m(e)g(ha)m(v)m(e)456 2547 y(the)c(results)g(claimed)i(in)e(Theorem)g(56.)1263 b Fj(\003)555 2720 y Fs(Before)32 b(pro)m(ving)e(Corollary)h(57,)h(w)m (e)e(analyze)i(the)f(equation)456 2877 y(\(129\))490 b Fm(K)1228 2891 y Fq(0)1268 2877 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\))27 b(=)e Fm(E)g Fs(+)20 b Fm(\027)6 b(g)s Fs(\()p Fm(y)s(;)15 b(x;)g(s;)g(E)5 b Fs(;)15 b Fm(")p Fs(\))p Fm(:)456 3045 y Fw(Lemma)25 b(65.)36 b Fo(L)-5 b(et)26 b(us)f(c)-5 b(onsider)27 b(e)-5 b(quation)33 b Fs(\(129\))28 b Fo(wher)-5 b(e)26 b Fm(K)2520 3059 y Fq(0)2560 3045 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\))27 b Fo(is)e(given)456 3153 y(in)39 b Fs(\(82\))34 b Fo(and)43 b Fs(\(80\))r Fo(,)32 b(and)i Fm(g)s Fs(\()p Fm(y)s(;)15 b(x;)g(s;)g(E)5 b Fs(;)15 b Fm(")p Fs(\))35 b Fo(is)e(at)g(le)-5 b(ast)34 b(of)f(class)g Fn(C)2713 3120 y Fq(2)2753 3153 y Fo(,)f(with)456 3310 y Fs(\(130\))901 b Fn(jj)p Fm(g)s Fn(jj)1708 3330 y Fl(C)1749 3311 y Fi(2)1815 3310 y Fn(\024)25 b Fs(cte)p Fm(:)456 3467 y Fo(for)33 b Fn(j)p Fm(y)s Fn(j)25 b(\024)g Fm(c)859 3481 y Fq(2)899 3467 y Fm(L)33 b Fo(and)636 3600 y Fs(a\))42 b Fm(c)797 3614 y Fq(3)837 3600 y Fm(")879 3567 y Fp(\013)954 3600 y Fn(\024)25 b Fm(E)31 b Fn(\024)25 b Fm(c)1283 3614 y Fq(2)1333 3577 y Fs(\026)1323 3600 y Fm(L)32 b Fo(and)i Fs(\()p Fm(x;)15 b(s)p Fs(\))25 b Fn(2)g Fk(T)1971 3567 y Fq(2)456 3733 y Fo(or)631 3865 y Fs(b\))41 b Fn(\000)p Fm(c)868 3879 y Fq(4)908 3865 y Fm(")950 3832 y Fp(j)1032 3865 y Fn(\024)j Fm(E)50 b Fn(\024)45 b Fs(0)p Fo(,)h Fs(\()p Fm(x;)15 b(s)p Fs(\))45 b Fn(2)g Fk(T)1916 3832 y Fq(2)1955 3865 y Fo(,)h(and)e Fm(x)h Fn(2)f Fs([)p Fm(\032;)15 b Fs(2)p Fm(\031)32 b Fn(\000)c Fm(\032)p Fs(])p Fn(g)p Fo(,)47 b(wher)-5 b(e)758 3973 y Fs(0)26 b Fm(<)f(\032)g(<)g(\031)36 b Fo(is)c(any)i(numb)-5 b(er)33 b(indep)-5 b(endent)34 b(of)f Fm(";)15 b(\027)6 b Fo(.)555 4106 y(Then,)44 b(for)d Fm(\013)g(>)f(j)5 b Fo(,)43 b Fm(j)j Fs(=)41 b(1)p Fm(;)15 b Fs(2)p Fo(,)44 b(and)e(for)f Fm(")g Fo(smal)5 b(l)42 b(enough,)i(ther)-5 b(e)42 b(exists)456 4213 y(some)i Fm(\027)744 4227 y Fq(0)784 4213 y Fo(,)i(indep)-5 b(endent)45 b(of)f Fm(")p Fo(,)j(such)d(that)h(if)f Fm(\027)51 b Fn(\024)46 b Fm(\027)2381 4227 y Fq(0)2420 4213 y Fm(")2462 4180 y Fp(\013)2512 4213 y Fo(,)g(e)-5 b(quation)52 b Fs(\(129\))456 4321 y Fo(de\014nes)34 b(a)g(function)f Fm(y)d Fs(=)d Fm(f)1425 4288 y Fl(\006)1483 4321 y Fs(\()p Fm(x;)15 b(s;)g(E)5 b Fs(;)15 b Fm(";)g(\027)6 b Fs(\))36 b Fo(of)e(class)g Fn(C)2389 4288 y Fq(2)2462 4321 y Fo(in)g(the)g (domains)h(a\))456 4429 y(or)e(b\),)f(such)h(that:)601 4562 y Fs(\(1\))42 b Fm(f)813 4529 y Fl(\006)872 4562 y Fs(\()p Fm(x;)15 b(s;)g(E)5 b Fs(;)15 b Fm(";)g Fs(0\))54 b(=)d Fn(Y)1593 4576 y Fl(\006)1651 4562 y Fs(\()p Fm(x;)15 b(E)5 b Fs(\))p Fo(,)52 b(wher)-5 b(e)48 b Fn(Y)2297 4576 y Fl(\006)2356 4562 y Fs(\()p Fm(x;)15 b(E)5 b Fs(\))48 b Fo(ar)-5 b(e)48 b(the)f(func-)758 4670 y(tions)41 b Fs(\(97\))34 b Fo(intr)-5 b(o)g(duc)g(e)g(d)36 b(in)c(L)-5 b(emma)34 b(60,)f(with)h Fm(\016)29 b Fs(=)c Fm(")2610 4637 y Fp(j)2647 4670 y Fo(.)601 4788 y Fs(\(2\))42 b Fo(If)33 b(we)g(denote)g(by)f Fm(D)s(f)j Fs(=)1661 4747 y Fp(@)t(f)p 1661 4767 83 4 v 1662 4819 a(@)t(x)1753 4788 y Fm(;)1803 4747 y Fp(@)t(f)p 1803 4767 V 1807 4819 a(@)t(s)1896 4788 y Fo(,)d(we)h(have,)g(for)g Fm(f)h Fs(=)25 b Fm(f)2703 4755 y Fl(\006)2762 4788 y Fo(:)456 4964 y Fs(\(131\))338 b Fn(j)q Fm(f)10 b Fn(j)25 b(\024)f Fs(cte)p Fm(:)17 b(;)108 b Fn(j)q Fm(D)s(f)10 b Fn(j)24 b(\024)h Fm(")1861 4927 y Fp(j)t(=)p Fq(2)1969 4964 y Fm(;)2102 4887 y Fh(\014)2102 4942 y(\014)2132 4964 y Fm(D)2210 4927 y Fq(2)2250 4964 y Fm(f)2305 4887 y Fh(\014)2305 4942 y(\014)2359 4964 y Fn(\024)g Fm(")2497 4927 y Fp(j)t(=)p Fq(2)2605 4964 y Fm(;)p eop end %%Page: 79 79 TeXDict begin 79 78 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(79)758 450 y Fo(and:)456 629 y Fs(\(132\))1174 497 y Fh(\014)1174 552 y(\014)1174 606 y(\014)1174 661 y(\014)1223 568 y Fm(@)5 b(f)p 1214 608 126 4 v 1214 692 a(@)g(E)1350 497 y Fh(\014)1350 552 y(\014)1350 606 y(\014)1350 661 y(\014)1405 629 y Fn(\024)25 b Fm(")1543 592 y Fl(\000)p Fp(j)t(=)p Fq(2)1706 629 y Fm(;)1839 497 y Fh(\014)1839 552 y(\014)1839 606 y(\014)1839 661 y(\014)1879 568 y Fm(@)5 b(D)s(f)p 1879 608 186 4 v 1909 692 a(@)g(E)2075 497 y Fh(\014)2075 552 y(\014)2075 606 y(\014)2075 661 y(\014)2130 629 y Fn(\024)25 b Fm(")2268 592 y Fl(\000)p Fp(j)t(=)p Fq(2)2430 629 y Fm(:)456 923 y Fs(\(133\))293 b Fn(j)q Fm(f)29 b Fn(\000)20 b(Y)7 b(j)26 b(\024)f Fm(\027)6 b(")1453 886 y Fl(\000)p Fp(j)t(=)p Fq(2)1615 923 y Fm(;)108 b Fn(j)p Fm(D)s Fs(\()p Fm(f)30 b Fn(\000)20 b(Y)7 b Fs(\))p Fn(j)26 b(\024)f Fm(\027)6 b(")2395 886 y Fl(\000)p Fp(j)t(=)p Fq(2)2557 923 y Fm(;)456 1086 y Fo(Pr)-5 b(o)g(of.)43 b Fs(By)24 b(part)h(1.)39 b(of)25 b(Lemma)g(60)g(with)g Fm(\016)k Fs(=)c Fm(")2115 1053 y Fp(j)2152 1086 y Fs(,)g(for)g Fm(y)j(>)d Fs(0,)h(equation)g(\(129\))456 1194 y(is)k(equiv)-5 b(alen)m(t)32 b(to)f(the)g(equation:)456 1341 y(\(134\))50 b Fm(M)10 b Fs(\()p Fm(y)s(;)15 b(x;)g(s;)g(E)5 b Fs(;)15 b Fm(\027)6 b Fs(\))27 b Fn(\021)e Fm(y)e Fn(\000)d(Y)1648 1355 y Fq(+)1707 1341 y Fs(\()p Fm(x;)15 b(t)p Fs(\))26 b(=)f(0)p Fm(;)107 b(t)25 b Fs(=)g Fm(E)h Fs(+)20 b Fm(\027)6 b(g)s Fs(\()p Fm(y)s(;)15 b(x;)g(s;)g(E)5 b Fs(;)15 b Fm(")p Fs(\))456 1487 y(where)29 b Fn(Y)779 1501 y Fq(+)838 1487 y Fs(\()p Fm(x;)15 b(t)p Fs(\),)32 b(is)e(the)h(function)f(\(97\)) q(.)555 1595 y(Di\013eren)m(tiating)42 b(\(97\))r(,)e(and)e(using)h (\(98\))r(,)h(for)e(cte)p Fm(:)17 b(")2401 1562 y Fp(\013)2489 1595 y Fn(\024)38 b Fm(E)31 b Fn(\000)25 b Fs(cte)p Fm(:)32 b Fn(j)p Fm(\027)6 b Fn(j)38 b(\024)456 1703 y(j)p Fm(t)p Fn(j)j(\024)f Fm(E)32 b Fs(+)26 b(cte)p Fm(:)32 b Fn(j)p Fm(\027)6 b Fn(j)41 b(\024)f Fs(cte)p Fm(:)1479 1680 y Fs(\026)1469 1703 y Fm(L)o Fs(,)j(one)d(has)f(in)g(a\))i(the)f(follo) m(wing)h(b)s(ounds)c(for)456 1811 y Fn(Y)517 1825 y Fq(+)575 1811 y Fs(\()p Fm(x;)15 b(E)5 b Fs(\))755 1881 y Fh(\014)755 1936 y(\014)755 1990 y(\014)755 2045 y(\014)795 1952 y Fm(@)g Fn(Y)909 1966 y Fq(+)p 795 1992 174 4 v 829 2075 a Fm(@)g(x)978 1881 y Fh(\014)978 1936 y(\014)978 1990 y(\014)978 2045 y(\014)1034 2013 y Fn(\024)25 b Fs(cte)p Fm(:)16 b(")1328 1976 y Fp(j)t(=)p Fq(2)1542 1881 y Fh(\014)1542 1936 y(\014)1542 1990 y(\014)1542 2045 y(\014)1582 1952 y Fm(@)5 b Fn(Y)1696 1966 y Fq(+)p 1582 1992 V 1625 2075 a Fm(@)g(t)1765 1881 y Fh(\014)1765 1936 y(\014)1765 1990 y(\014)1765 2045 y(\014)1821 2013 y Fn(\024)24 b Fs(cte)p Fm(:)17 b(")2115 1976 y Fl(\000)p Fp(j)t(=)p Fq(2)715 2140 y Fh(\014)715 2195 y(\014)715 2249 y(\014)715 2304 y(\014)755 2211 y Fm(@)808 2178 y Fq(2)848 2211 y Fn(Y)909 2225 y Fq(+)p 755 2251 213 4 v 789 2334 a Fm(@)5 b(x)894 2308 y Fq(2)978 2140 y Fh(\014)978 2195 y(\014)978 2249 y(\014)978 2304 y(\014)1034 2272 y Fn(\024)25 b Fs(cte)p Fm(:)16 b(")1328 2234 y Fp(j)t(=)p Fq(2)1542 2140 y Fh(\014)1542 2195 y(\014)1542 2249 y(\014)1542 2304 y(\014)1592 2211 y Fm(@)1645 2178 y Fq(2)1684 2211 y Fn(Y)1745 2225 y Fq(+)p 1582 2251 232 4 v 1582 2334 a Fm(@)5 b(x;)15 b(@)5 b(t)1824 2140 y Fh(\014)1824 2195 y(\014)1824 2249 y(\014)1824 2304 y(\014)1879 2272 y Fn(\024)25 b Fs(cte)p Fm(:)17 b(")2174 2234 y Fl(\000)p Fp(j)t(=)p Fq(2)2442 2140 y Fh(\014)2442 2195 y(\014)2442 2249 y(\014)2442 2304 y(\014)2482 2211 y Fm(@)2535 2178 y Fq(2)2575 2211 y Fn(Y)2636 2225 y Fq(+)p 2482 2251 213 4 v 2526 2334 a Fm(@)5 b(t)2612 2308 y Fq(2)2705 2140 y Fh(\014)2705 2195 y(\014)2705 2249 y(\014)2705 2304 y(\014)2761 2272 y Fn(\024)24 b Fm(")2898 2234 y Fl(\000)p Fq(3)p Fp(j)t(=)p Fq(2)3096 2272 y Fm(:)456 2143 y Fs(\(135\))456 2474 y(Using)31 b(\(135\))i(and)d(\(130\))r(,)g(one)h(has:)456 2676 y(\(136\))1341 2544 y Fh(\014)1341 2598 y(\014)1341 2653 y(\014)1341 2708 y(\014)1382 2614 y Fm(@)5 b(M)p 1382 2655 152 4 v 1407 2738 a(@)g(y)1563 2676 y Fn(\000)20 b Fs(1)1699 2544 y Fh(\014)1699 2598 y(\014)1699 2653 y(\014)1699 2708 y(\014)1755 2676 y Fn(\024)25 b Fs(cte)p Fm(:)17 b(\027)6 b(")2101 2638 y Fl(\000)p Fp(j)t(=)p Fq(2)2263 2676 y Fm(;)456 2878 y Fs(so)44 b(that,)k(the)d(implicit)g(function)e (theorem)i(applied)f(to)g(\(134\))s(,)j(giv)m(es)f(the)456 2991 y(existence)29 b(of)e Fm(f)995 2958 y Fq(+)1054 2991 y Fs(\()p Fm(x;)15 b(s;)g(E)5 b Fs(;)15 b Fm(";)g(\027)6 b Fs(\))29 b(if)f Fm(\027)6 b(=")1792 2958 y Fp(j)t(=)p Fq(2)1925 2991 y Fn(\024)24 b Fm(\027)2065 3005 y Fq(0)2105 2991 y Fs(,)k(for)f(some)h Fm(\027)2564 3005 y Fq(0)2631 2991 y Fs(small)g(enough.)555 3099 y(In)21 b(order)h(to)g(b)s(ound)e (the)i(deriv)-5 b(ativ)m(es)23 b(of)f Fm(f)2006 3066 y Fq(+)2064 3099 y Fs(,)i(w)m(e)e(tak)m(e)i(implicit)e(deriv)-5 b(ativ)m(es)456 3207 y(in)35 b(equation)h(\(134\))r(.)55 b(Then,)36 b(taking)g(in)m(to)g(accoun)m(t)h(\(135\))r(,)g(and)d(using) h(that)456 3315 y Fn(j)p Fm(\027)6 b Fn(j)25 b Fm(<)g(\027)723 3329 y Fq(0)762 3315 y Fm(")804 3282 y Fp(\013)879 3315 y Fm(<)g(\027)1020 3329 y Fq(0)1060 3315 y Fm(")1102 3282 y Fp(j)1139 3315 y Fs(,)30 b(w)m(e)h(see)g(that:)753 3385 y Fh(\014)753 3440 y(\014)753 3494 y(\014)753 3549 y(\014)793 3456 y Fm(@)5 b(M)p 793 3496 V 817 3579 a(@)g(\034)954 3385 y Fh(\014)954 3440 y(\014)954 3494 y(\014)954 3549 y(\014)1010 3517 y Fn(\024)25 b Fs(cte)p Fm(:)17 b(")1305 3480 y Fp(j)t(=)p Fq(2)1412 3517 y Fm(;)46 b(\034)35 b Fs(=)25 b Fm(x;)15 b(s;)1980 3385 y Fh(\014)1980 3440 y(\014)1980 3494 y(\014)1980 3549 y(\014)2021 3456 y Fm(@)5 b(M)p 2021 3496 V 2044 3579 a(@)g(\027)2182 3385 y Fh(\014)2182 3440 y(\014)2182 3494 y(\014)2182 3549 y(\014)2237 3517 y Fn(\024)25 b Fs(cte)p Fm(:)17 b(")2532 3480 y Fl(\000)p Fp(j)t(=)p Fq(2)2694 3517 y Fm(;)456 3644 y Fh(\014)456 3699 y(\014)456 3753 y(\014)456 3808 y(\014)506 3714 y Fm(@)559 3681 y Fq(2)599 3714 y Fm(M)p 496 3755 212 4 v 496 3838 a(@)5 b(\034)589 3852 y Fq(1)628 3838 y Fm(\034)668 3852 y Fq(2)718 3644 y Fh(\014)718 3699 y(\014)718 3753 y(\014)718 3808 y(\014)773 3776 y Fn(\024)25 b Fs(cte)p Fm(:)17 b(")1068 3738 y Fp(j)t(=)p Fq(2)1175 3776 y Fm(;)46 b(\034)1286 3790 y Fq(1)1325 3776 y Fm(;)15 b(\034)1405 3790 y Fq(2)1470 3776 y Fs(=)25 b Fm(y)s(;)15 b(x;)g(s;)1980 3644 y Fh(\014)1980 3699 y(\014)1980 3753 y(\014)1980 3808 y(\014)2029 3714 y Fm(@)2082 3681 y Fq(2)2122 3714 y Fm(M)p 2021 3755 208 4 v 2021 3838 a(@)5 b(\027)h(@)f(\034)2238 3644 y Fh(\014)2238 3699 y(\014)2238 3753 y(\014)2238 3808 y(\014)2293 3776 y Fn(\024)25 b Fs(cte)p Fm(:)17 b(")2588 3738 y Fl(\000)p Fp(j)t(=)p Fq(2)2750 3776 y Fm(;)46 b(\034)35 b Fs(=)25 b Fm(y)s(;)15 b(x;)g(s:)456 3978 y Fs(These)30 b(b)s(ounds)e(and)i (\(136\))r(,)g(giv)m(e)i(the)f(desired)f(b)s(ounds)e(\(131\))k(and)e (\(133\))r(.)555 4086 y(Moreo)m(v)m(er,)j(using)e(\(135\))h(w)m(e)f (see)g(that:)1194 4156 y Fh(\014)1194 4210 y(\014)1194 4265 y(\014)1194 4319 y(\014)1234 4226 y Fm(@)5 b(M)p 1234 4267 152 4 v 1247 4350 a(@)g(E)1395 4156 y Fh(\014)1395 4210 y(\014)1395 4265 y(\014)1395 4319 y(\014)1509 4288 y Fn(\024)82 b Fs(cte)p Fm(:)17 b(")1861 4250 y Fl(\000)p Fp(j)t(=)p Fq(2)2023 4288 y Fm(;)1116 4415 y Fh(\014)1116 4469 y(\014)1116 4524 y(\014)1116 4578 y(\014)1175 4485 y Fm(@)1228 4452 y Fq(2)1268 4485 y Fm(M)p 1156 4526 230 4 v 1156 4609 a(@)5 b(\034)10 b(@)5 b(E)1395 4415 y Fh(\014)1395 4469 y(\014)1395 4524 y(\014)1395 4578 y(\014)1509 4547 y Fn(\024)82 b Fs(cte)p Fm(:)17 b(")1861 4509 y Fl(\000)p Fp(j)t(=)p Fq(2)2023 4547 y Fm(;)46 b(\034)35 b Fs(=)25 b Fm(y)s(;)15 b(x;)g(s:)456 4748 y Fs(These)42 b(inequalities)j(and)d(\(136\))j(giv)m(e)f(the)f(desired) g(b)s(ounds)d(\(132\))45 b(in)e(the)456 4856 y(domain)30 b(a\))h(for)f Fm(y)e(>)d Fs(0.)555 4964 y(An)30 b(analogous)i(pro)s(of) e(giv)m(es)h(the)g(b)s(ounds)d(in)i(the)h(domain)f(a\))h(for)f Fm(y)e(<)d Fs(0.)p eop end %%Page: 80 80 TeXDict begin 80 79 bop 456 251 a Fq(80)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)555 450 y Fs(F)-8 b(or)37 b(negativ)m(e)h(v)-5 b(alues)37 b(of)f Fm(E)5 b Fs(|and)36 b(consequen)m(tly)h(of)f Fm(t)p Fs(|,)i(w)m(e)e(note)h (that)456 558 y Fn(Y)7 b Fs(\()p Fm(x;)15 b(t)p Fs(\))44 b(is)g(the)g(comp)s(osition)h(of)f(a)g(regular)g(function)g(with)f(the) h(function)456 675 y Fm(`)p Fs(\()p Fm(x;)15 b(t)p Fs(\))26 b(=)811 600 y Fn(p)p 886 600 46 4 v 886 675 a Fs(2)932 596 y Fh(p)p 1023 596 501 4 v 79 x Fm(t)20 b Fn(\000)g Fm(")1209 649 y Fp(j)1246 675 y Fm(U)10 b Fs(\()p Fm(x;)15 b(")p Fs(\))q(.)555 783 y(W)-8 b(e)30 b(note)e(that)h(in)f(the)g (domain)g(b\)|recall)i(that)e Fm(x)g Fs(is)h(restricted)g(to)f(some)456 891 y(in)m(terv)-5 b(al)42 b([)p Fm(\032;)15 b Fs(2)p Fm(\031)32 b Fn(\000)c Fm(\032)p Fs(]|the)42 b(function)f Fm(`)p Fs(\()p Fm(x;)15 b(t)p Fs(\),)45 b(v)m(eri\014es)d(b)s(ounds)e (\(135\))j(and)456 999 y(therefore)30 b(so)h(do)s(es)f Fn(Y)1215 1013 y Fl(\006)1274 999 y Fs(.)3103 1107 y Fj(\003)456 1279 y Fo(Pr)-5 b(o)g(of)34 b(of)f(Cor)-5 b(ol)5 b(lary)35 b(57.)42 b Fs(F)-8 b(rom)29 b(no)m(w)f(on,)g(w)m(e)h (consider)f Fm(\013)e Fs(=)f Fm(j)5 b(=)p Fs(2)17 b(+)e(3)p Fm(=)p Fs(2,)30 b Fm(j)h Fs(=)456 1387 y(1)p Fm(;)15 b Fs(2)29 b(and)e Fm(m)e Fn(\025)g Fs(26.)41 b(W)-8 b(e)29 b(apply)f(Lemma)g(65)h(to)f(the)g(implicit)h(equations)g(\(89\))r(,)456 1494 y(\(91\))j(and)d(\(92\))r(,)h(of)h(the)g(in)m(v)-5 b(arian)m(t)31 b(tori)g(giv)m(en)g(b)m(y)g(Theorem)f(56.)555 1604 y(The)39 b(equation)h(\(89\))r(,)i(that)e(giv)m(es)g(implicitly)h (the)f(tori)g Fn(T)2632 1566 y Fl(\006)2610 1632 y Fp(E)2662 1642 y Ff(i)2731 1604 y Fs(in)f Fn(D)2916 1619 y Fp(f)2961 1604 y Fs(,)j(is)d(a)456 1734 y(particular)27 b(case)h(of)f(equation)h (\(129\))h(taking)f Fm(E)j Fs(=)25 b Fm(E)2301 1748 y Fp(i)2356 1734 y Fs(and)i Fm(\027)k Fs(=)25 b Fm(")2744 1701 y Fq(\()p Fp(m)p Fq(+1)p Fl(\000)p Fq(7)p Fp(j)t Fq(\))p Fp(=)p Fq(2)3149 1734 y Fs(.)456 1843 y(Analogously)36 b(for)e(equation)i(\(91\))g(whic)m(h)f(giv)m(es)h(implicitly)g(the)f (tori)h Fn(T)3004 1805 y Fl(\006)2982 1872 y Fp(F)3027 1882 y Ff(i)3098 1843 y Fs(in)456 1973 y Fn(D)526 1987 y Fp(o)564 1973 y Fs(,)f(taking)h Fm(E)h Fs(=)31 b Fm(F)1173 1987 y Fp(i)1236 1973 y Fs(and)j Fm(\027)j Fs(=)32 b Fm(")1644 1940 y Fq(\()p Fp(m)p Fq(+1+)p Fp(j)t(=)p Fq(2)p Fl(\000)p Fq(14\()p Fp(\013)p Fl(\000)p Fp(j)t Fq(\)\))p Fp(=)p Fq(2)2431 1973 y Fs(and)i(equation)h(\(92\))r(,)456 2087 y(whic)m(h)20 b(giv)m(es)h(the)g(tori)g Fn(T)1277 2101 y Fp(G)1332 2111 y Ff(i)1382 2087 y Fs(in)f Fn(D)1548 2101 y Fp(in)1619 2087 y Fs(,)i(if)f(w)m(e)f(tak)m(e)i Fm(E)31 b Fs(=)25 b Fm(G)2315 2101 y Fp(i)2364 2087 y Fs(and)19 b Fm(\027)31 b Fs(=)25 b Fm(")2744 2054 y Fq(\()p Fp(m)p Fq(+1+)p Fp(j)t(=)p Fq(2)p Fl(\000)p Fq(14\()p Fp(\013)p Fl(\000)p Fp(j)t Fq(\)\))p Fp(=)p Fq(2)3497 2087 y Fs(.)555 2195 y(F)-8 b(or)25 b Fm(m)g Fn(\025)g Fs(26)g(and)f Fm(\013)i Fs(=)f(3)p Fm(=)p Fs(2)8 b(+)g Fm(j)d(=)p Fs(2,)28 b(the)c(condition)h Fn(j)p Fm(\027)6 b Fn(j)25 b(\024)g Fm(")2593 2162 y Fp(\013)2667 2195 y Fs(of)g(Lemma)f(65)456 2303 y(is)h(v)m(eri\014ed)g(in)h(the)f(three)h (cases)g(of)g(the)f(previous)g(paragraph.)39 b(The)25 b(results)g(of)456 2411 y(Lemma)30 b(65)h(giv)m(e)h(us)e(the)g(items)h (1.,)h(2.,)f(3.,)g(and)f(6.)41 b(a\),)31 b(b\))g(of)f(Corollary)h(57.) 555 2519 y(F)-8 b(rom)31 b(1.5,)h(2.5)f(and)f(3.5)i(of)e(theorem)h(56)g (w)m(e)g(obtain)1155 2694 y Fn(j)p Fm(E)1247 2708 y Fp(i)1296 2694 y Fn(\000)20 b Fm(E)1454 2708 y Fp(i)p Fq(+1)1572 2694 y Fn(j)84 b(\024)e Fm(")1886 2626 y Ff(m)p Fi(+1)p Fg(\000)p Fi(7)p Ff(j)p 1887 2642 239 3 v 1991 2683 a Fi(2)2139 2694 y Fm(;)1172 2861 y Fn(j)q Fm(F)1256 2875 y Fp(i)1305 2861 y Fn(\000)19 b Fm(F)1453 2875 y Fp(i)p Fq(+1)1572 2861 y Fn(j)84 b(\024)e Fm(")1886 2790 y Ff(m)p Fi(+1+)p Ff(j)s(=)p Fi(2)p Fg(\000)p Fi(10\()p Ff(\013)p Fg(\000)p Ff(j)s Fi(\))p 1887 2808 542 3 v 2142 2849 a(2)2443 2861 y Fm(;)1146 3027 y Fn(j)q Fm(G)1243 3041 y Fp(i)1292 3027 y Fn(\000)19 b Fm(G)1453 3041 y Fp(i)p Fq(+1)1572 3027 y Fn(j)84 b(\024)e Fm(")1886 2956 y Ff(m)p Fi(+1+)p Ff(j)s(=)p Fi(2)p Fg(\000)p Fi(10\()p Ff(\013)p Fg(\000)p Ff(j)s Fi(\))p 1887 2975 V 2142 3016 a(2)2443 3027 y Fm(;)456 3184 y Fs(T)-8 b(aking)35 b(in)m(to)h(accoun)m(t)g(the) f(de\014nitions)f(\(84\))r(,)i(\(85\))q(,)g(\(86\))g(of)f Fm(D)2721 3199 y Fp(f)2767 3184 y Fs(,)h Fm(D)2903 3198 y Fp(o)2941 3184 y Fs(,)g Fm(D)3077 3198 y Fp(in)3149 3184 y Fs(,)456 3292 y(w)m(e)30 b(get)1079 3467 y Fn(j)q Fm(E)1172 3481 y Fq(1)1231 3467 y Fn(\000)20 b Fm(F)1380 3482 y Fp(l)1401 3493 y Ff(F)1457 3467 y Fn(j)83 b(\024)g Fm(")1771 3399 y Ff(m)p Fi(+1)p Fg(\000)p Fi(7)p Ff(j)p 1771 3415 239 3 v 1875 3456 a Fi(2)2023 3467 y Fm(;)1074 3637 y Fn(j)p Fm(F)1157 3651 y Fq(1)1218 3637 y Fn(\000)19 b Fm(G)1379 3652 y Fp(l)1400 3663 y Ff(G)1457 3637 y Fn(j)83 b(\024)g Fm(")1761 3599 y Fp(\013)1831 3637 y Fs(+)20 b Fm(")1974 3566 y Ff(m)p Fi(+1+)p Ff(j)s(=)p Fi(2)p Fg(\000)p Fi(10\()p Ff(\013)p Fg(\000)p Ff(j)s Fi(\))p 1974 3584 542 3 v 2230 3625 a(2)2530 3637 y Fm(:)456 3793 y Fs(Since)28 b Fm(\013)e Fs(=)f Fm(j)5 b(=)p Fs(2)17 b(+)f(3)p Fm(=)p Fs(2)29 b(and)f Fm(m)d Fn(\025)g Fs(26,)30 b(all)f(these)f(exp)s(onen)m(ts)h(are)f(bigger)h(than)466 3874 y Fq(3)p 466 3889 36 4 v 466 3941 a(2)536 3909 y Fs(+)644 3868 y Fp(j)p 643 3889 V 643 3941 a Fq(2)688 3909 y Fs(.)65 b(Using)39 b(the)f(b)s(ounds)f(\(132\))j(of)f(Lemma)f (65,)k(and)c(the)g(inequalities)456 4017 y(ab)s(o)m(v)m(e,)31 b(w)m(e)g(obtain)g(the)g(b)s(ounds)d(of)i(6.)41 b(c\).)1161 b Fj(\003)456 4201 y Fs(8.5.5.)47 b Fo(Existenc)-5 b(e)40 b(of)g(stable)g(and)h(unstable)f(manifolds)i(of)e(p)-5 b(erio)g(dic)41 b(orbits.)456 4309 y Fs(W)-8 b(e)41 b(recall)g(that)f (in)g(Section)g(8.5.3)i(w)m(e)e(had)f(sho)m(wn)g(that,)k(in)d (appropriate)456 4417 y(v)-5 b(ariables,)31 b(the)g(motion)h(in)e(the)h (region)g Fn(S)1922 4384 y Fl(R)1982 4394 y Ff(j)2019 4417 y Fs(,)g Fm(j)g Fs(=)26 b(1)p Fm(;)15 b Fs(2,)32 b(is)f(describ)s(ed)e(b)m(y)i(the)456 4524 y Fn(C)509 4491 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(2)820 4524 y Fs(Hamiltonian)h Fm(K)7 b Fs(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(;)g Fm(")p Fs(\))31 b(giv)m(en)h(in)e(\(77\))q(.)555 4632 y(The)46 b(main)f(part)h(of)g(Hamiltonian)h Fm(K)52 b Fs(in)46 b(\(77\))h(is)f(the)g Fn(C)2660 4599 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(1)2987 4632 y Fs(term)456 4740 y Fm(K)533 4754 y Fq(0)572 4740 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\))42 b(in)e(\(82\))q(,)j(whic)m(h)d(has)f(a)i (saddle)f(at)g(\(0)p Fm(;)15 b Fs(0\))42 b(with)e(c)m(haracteristic)456 4856 y(exp)s(onen)m(ts)46 b Fn(\006)970 4779 y Fh(p)p 1061 4779 153 4 v 77 x Fm(c)p Fs(\()p Fm(")p Fs(\))q Fm(")1255 4823 y Fp(j)t(=)p Fq(2)1363 4856 y Fs(,)k(where)c Fm(c)p Fs(\()p Fm(")p Fs(\))54 b(=)e Fn(\000)p Fm(U)2188 4823 y Fl(00)2230 4856 y Fs(\(0;)15 b Fm(")p Fs(\))p Fm(h)2479 4823 y Fl(00)2479 4881 y Fq(0)2523 4856 y Fs(\(0;)g Fm(")p Fs(\),)53 b Fm(c)p Fs(\(0\))h Fn(6)p Fs(=)d(0)456 4964 y(\(see)31 b(\(79\))r(,)f(\(80\))r(\).)p eop end %%Page: 81 81 TeXDict begin 81 80 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(81)555 450 y Fs(The)23 b(stable)h(and)f(unstable)g(manifolds)g(of)h(\(0)p Fm(;)15 b Fs(0\))25 b(coincide)f(along)h(a)e(separa-)456 558 y(trix)f(con)m(tained)h(in)f(the)g(lev)m(el)i(sets)e Fm(K)1716 572 y Fq(0)1756 558 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\))27 b(=)e(0.)38 b(Hence,)25 b(these)d(manifolds)456 666 y(are)30 b Fn(C)660 633 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(1)941 666 y Fs(,)h(the)f(same)h(regularit)m(y)h(as) e(the)h(Hamiltonian)h Fm(K)2664 680 y Fq(0)2703 666 y Fs(.)555 774 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b(they)h(ar)-5 b(e)33 b(given)f(by:)456 3494 y Fs(\(138\))349 b Fm(y)28 b Fs(=)d Fn(Z)1252 3457 y Fq(wu)1245 3517 y Fp(up)1346 3494 y Fs(\()p Fm(x;)15 b(s)p Fs(\))26 b(=)f Fn(Y)1734 3508 y Fq(+)1793 3494 y Fs(\()p Fm(x;)15 b Fs(0\))22 b(+)2112 3502 y(O)2183 3514 y Fl(C)2224 3495 y Fi(1)2263 3494 y Fs(\()p Fm(")2340 3457 y Fp(m)p Fq(+1)p Fl(\000)p Fp(j)2585 3494 y Fs(\))758 3644 y Fo(wher)-5 b(e)47 b Fn(Y)1089 3658 y Fl(\006)1194 3644 y Fo(ar)-5 b(e)47 b(given)f(in)52 b Fs(\(97\))r Fo(.)82 b(A)n(nalo)-5 b(gous)47 b(formulas)h(hold)f(for)758 3752 y Fn(Z)831 3719 y Fq(ws)824 3780 y Fp(dow)r(n)994 3752 y Fo(,)32 b Fn(Z)1127 3719 y Fq(ws)1120 3774 y Fp(up)1211 3752 y Fo(,)g Fn(Z)1344 3719 y Fq(wu)1337 3780 y Fp(dow)r(n)1507 3752 y Fo(.)456 3904 y(Pr)-5 b(o)g(of.)43 b Fs(This)29 b(reduces)h(to)h(Theorems)f(A)h(and)e(C)h(in)g(the)h(pap)s(er)e([FS90a) r(].)555 4018 y(W)-8 b(e)22 b(consider)f(the)f(scaling)i Fm(y)28 b Fs(=)d Fm(")1693 3985 y Fp(j)t(=)p Fq(2)1801 4018 y Fm(Y)20 b 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Fs(\026)813 4856 y Fm(S)869 4870 y Fq(1)953 4856 y Fs(has)46 b(a)g(critical)h(p)s(oin)m (t)f(whic)m(h)f(is)g Fm(")2216 4823 y Fp(m)p Fq(+1)p Fl(\000)p Fp(j)2506 4856 y Fs(close)i(to)g(the)e(ori-)456 4964 y(gin.)62 b(Then,)38 b(applying)f(the)h(results)f(of)g(Theorem)h (C)f(and)f(Prop)s(osition)i(5.1)p eop end %%Page: 82 82 TeXDict begin 82 81 bop 456 251 a Fq(82)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 450 y Fs(in)29 b([FS90a)q(],)i(w)m(e)f(obtain)g(the)f(existence)j(of)d(the)h(p)s(erio) s(dic)f(orbit)h Fm(\025)p Fs(\()p Fm(")p Fs(\),)h(whic)m(h,)456 564 y(after)37 b(the)f(scaling)i Fm(y)g Fs(=)d Fm(")1375 531 y Fp(j)t(=)p Fq(2)1483 564 y Fm(Y)20 b Fs(,)38 b(has)e(the)h Fm(x)f Fs(comp)s(onen)m(t)h(of)f(order)g Fm(")2904 531 y Fp(m)p Fq(+1)p Fl(\000)p Fp(j)3149 564 y Fs(,)456 673 y(and)29 b(the)i Fm(y)i Fs(comp)s(onen)m(t)e(of)f(order)g Fm(")1715 640 y Fp(m)p Fq(+1)p Fl(\000)p Fp(j)t(=)p Fq(2)2030 673 y Fs(.)555 781 y(Hamiltonian)40 b(\(114\))g(is)f(2)p Fm(\031)s(k)1578 795 y Fq(0)1618 781 y Fs(-p)s(erio)s(dic)f(in)g(the)h (v)-5 b(ariable)39 b Fm(s)p Fs(.)64 b(Then,)40 b(one)456 889 y(can)i(consider)h(the)f(time)i(2)p Fm(\031)s(k)1536 903 y Fq(0)1618 889 y Fs(map)e Fm(F)1890 903 y Fp(")1970 889 y Fs(of)h(this)f(Hamiltonian,)47 b(and)42 b(this)456 997 y(map)35 b(has)g(a)g(\014xed)g(p)s(oin)m(t)h Fm(P)1436 1011 y Fp(")1473 997 y Fs(,)g(corresp)s(onding)f(to)h(the)f(p)s(erio)s (dic)g(orbit)h Fm(\025)p Fs(\()p Fm(")p Fs(\).)456 1110 y(W)-8 b(e)32 b(will)f(consider)h(also)g(the)f(time)h(2)p Fm(\031)s(k)1836 1124 y Fq(0)1907 1110 y Fs(map)f Fm(F)2168 1124 y Fq(0)2239 1110 y Fs(of)g(Hamiltonian)i Fm(")2909 1077 y Fp(j)t(=)p Fq(2)3016 1110 y Fn(K)3085 1124 y Fq(in)n(t)456 1218 y Fs(giv)m(en)e(in)f(\(114\))r(.)555 1326 y(These)g(maps)g(v)m (erify:)1169 1490 y Fn(jj)p Fm(F)1277 1504 y Fq(0)1338 1490 y Fn(\000)20 b Fs(Id)o Fn(jj)1562 1510 y Fl(C)1603 1491 y Ff(r)r Fg(\000)p Fi(2)p Ff(m)p Fg(\000)p Fi(3)1936 1490 y Fn(\024)83 b Fm(")2132 1452 y Fp(j)t(=)p Fq(2)1157 1635 y Fn(jj)p Fm(F)1265 1649 y Fp(")1323 1635 y Fn(\000)20 b Fm(F)1472 1649 y Fq(0)1512 1635 y Fn(jj)1562 1655 y Fl(C)1603 1636 y Ff(r)r Fg(\000)p Fi(2)p Ff(m)p Fg(\000)p Fi(4)1936 1635 y Fn(\024)83 b Fm(")2132 1598 y Fp(m)p Fq(+1)p Fl(\000)p Fp(j)t(=)p Fq(2)2447 1635 y Fm(;)456 1799 y Fs(and)39 b(the)i(eigen)m(v)-5 b(alues)42 b(of)f(the)g(\014xed)f (p)s(oin)m(ts)g(are)h(1)27 b(+)g Fm(O)s Fs(\()p Fm(")2564 1766 y Fp(j)t(=)p Fq(2)2671 1799 y Fs(\).)72 b(Then,)42 b(b)m(y)456 1907 y(Theorem)28 b(A)h(in)g([FS90a)r(],)g(w)m(e)h(obtain)f (the)g(existence)i(of)e Fm(W)2547 1863 y Fq(ws)o Fp(;)p Fq(wu)2534 1936 y Fp(P)2579 1944 y Ff(")2739 1907 y Fs(,)h(whic)m(h)e (are)456 2029 y(of)36 b(class)h Fn(C)837 1996 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(3)1118 2029 y Fs(,)i(and)c Fm(")1406 1996 y Fp(m)p Fq(+1)p Fl(\000)p Fp(j)1687 2029 y Fs(close,)40 b(in)c(the)h Fn(C)2266 1996 y Fp(r)r Fl(\000)p Fq(2)p Fp(m)p Fl(\000)p Fq(4)2583 2029 y Fs(sense,)h(to)f(those)456 2137 y(of)c Fm(K)639 2151 y Fq(0)678 2137 y Fs(.)48 b(Moreo)m(v)m(er,) 36 b(in)d([)p Fn(\000)p Fm(\032;)15 b(\032)p Fs(])22 b Fn(\002)g Fk(T)p Fs(,)33 b(using)f(Lemma)h(60)h(and)e(Lemma)h(65for) 456 2245 y Fm(E)d Fs(=)25 b(0,)k Fm(\016)g Fs(=)c Fm(")955 2212 y Fp(j)1020 2245 y Fs(and)i Fm(\027)k Fs(=)25 b Fm(")1408 2212 y Fp(m)p Fq(+1)p Fl(\000)p Fp(j)1652 2245 y Fs(,)k(they)f(can)g(b)s(e)f(written)h(as)g(a)g(graph)g(of)g(the)456 2353 y(v)-5 b(ariable)31 b Fm(Y)50 b Fs(o)m(v)m(er)32 b(the)e(v)-5 b(ariables)31 b(\()p Fm(x;)15 b(s)p Fs(\))31 b(v)m(erifying)i(\(138\))r(.)555 2467 y(Going)49 b(bac)m(k)f(to)h(the)f (v)-5 b(ariables)48 b(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(\))55 b(=)f(\()p Fm(")2317 2434 y Fp(j)t(=)p Fq(2)2425 2467 y Fm(Y)5 b(;)15 b(x;)g(s)p Fs(\),)52 b(w)m(e)c(obtain)456 2575 y(Prop)s(osition)30 b(66)h(for)f Fn(Z)1278 2542 y Fp(\035)1271 2600 y(i)1323 2575 y Fs(.)3103 2683 y Fj(\003)456 2852 y Fw(Remark)51 b(67.)f Fs(Prop)s(osition)45 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b(the)e(manifolds)g Fm(W)1530 2520 y Fq(ws)o Fp(;)p Fq(wu)1517 2598 y Fp(\025)p Fq(\()p Fp(")p Fq(\))1722 2565 y Fs(,)h(when)e(considered)h(as)g(in)m(v)-5 b(arian)m(t)35 b(man-)456 2691 y(ifolds)h(in)f(the)h(whole)g(space)h (are)f(only)g Fo(we)-5 b(ak)47 b Fs(\(un\)stable)36 b(manifolds)g(of)g (the)456 2799 y(p)s(erio)s(dic)g(orbit)h Fm(\025)p Fs(\()p Fm(")p Fs(\).)63 b(W)-8 b(e)38 b(note)g(that)g(in)f([L)-10 b(W95)q(])37 b(one)h(can)f(\014nd)f(a)i(theory)456 2907 y(for)h(these)h(manifolds.)69 b(The)40 b(manifolds)f Fm(W)2039 2862 y Fq(ws)o Fp(;)p Fq(wu)2026 2940 y Fp(\025)p Fq(\()p Fp(")p Fq(\))2271 2907 y Fs(are)h(not)g(the)g(\(un\)stable)456 3028 y(manifolds)30 b(asso)s(ciated)i(in)e(the)g(normal)h(h)m(yp)s(erb) s(olicit)m(y)f(theory)-8 b(.)555 3136 y(F)g(or)38 b(our)e(purp)s(oses,) h(it)h(will)f(b)s(e)f(enough)h(to)g(de\014ne)f(that)i(if)f(w)m(e)g (consider)456 3243 y(the)30 b Fo(total)41 b Fs(\(un\)stable)31 b(manifold)g(of)f(an)g(in)m(v)-5 b(arian)m(t)32 b(ob)5 b(ject)31 b 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Fm(S)30 b Fs(=)25 b(\012)704 1871 y Fq(+)783 1857 y Fn(\016)20 b Fs(\012)914 1819 y Fl(\000)p Fq(1)914 1880 y Fl(\000)1008 1857 y Fs(.)555 1968 y(The)32 b(regularit)m(y)i(of) e(the)h(maps)f Fm(x)d Fn(!)f Fm(W)1964 1924 y Fq(s)p Fp(;)p Fq(u)1951 1980 y Fp(x)2087 1968 y Fs(is)k(studied)g(in)g(great)i (detail)f(in)456 2076 y([HPS77,)j(F)-8 b(en74)r(].)56 b(In)35 b(general,)j(it)e(dep)s(ends)e(on)h(ratios)i(of)f(sev)m(eral)h (rates)f(of)456 2184 y(expansion.)i(In)24 b(our)h(case,)i(ho)m(w)m(ev)m (er,)g(as)f(w)m(e)f(will)g(see)g(in)g(Theorem)f(20,)j(it)f(is)f(as)456 2292 y(smo)s(oth)33 b(as)g(the)g(\015o)m(w)g(b)s(ecause)g(the)h(Lipsc)m (hitz)g(constan)m(t)g(of)f(the)h(\015o)m(w)f(along)456 2400 y(the)d(stable)h(manifold)g(is)f(close)i(to)f(1.)555 2508 y(T)-8 b(o)31 b(c)m(hec)m(k)h(that)f(the)f(scattering)i(map)e(is)g (globally)i(de\014ned,)d(w)m(e)h(just)g(need)456 2616 y(to)44 b(c)m(hec)m(k)h(that)f(if)f(w)m(e)h(con)m(tin)m(ue)h(these)f (lo)s(cal)h(de\014nitions)d(around)h(a)h(lo)s(op)456 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b(this)i(section,)h(w)m(e)f(will)f(sho)m(w)h(that,)h(in)e (the)g(assumptions)g(of)456 3639 y(Theorem)25 b(7,)j(the)e(manifold) 1465 3616 y(~)1457 3639 y(\003)1520 3653 y Fp(")1582 3639 y Fs(constructed)h(in)e(Section)i(7)f(has)g(a)g(scattering)456 3747 y(map)k(and)f(w)m(e)i(will)g(compute)g(the)f(leading)h(order.)555 3856 y(The)c(\014rst)g(di\016cult)m(y)g(to)h(de\014ne)f(the)h (scattering)g(map)f(for)2581 3833 y(~)2573 3856 y(\003)2636 3870 y Fp(")2700 3856 y Fs(in)g(our)g(prob-)456 3964 y(lem)41 b(is)f(that,)45 b(for)40 b Fm(")j Fs(=)f(0,)i(its)d(stable)g (and)g(unstable)f(manifolds)g(coincide:)456 4072 y Fm(W)555 4039 y Fq(s)549 4098 y(~)542 4115 y(\003)630 4072 y Fs(=)c Fm(W)836 4039 y Fq(u)830 4098 y(~)823 4115 y(\003)879 4072 y Fs(.)59 b(W)-8 b(e)38 b(will)f(sho)m(w)g(that)g(under)e(h)m(yp)s (othesis)i Fw(H4)f Fs(of)h(Theorem)f(7)456 4207 y(the)c(stable)g(and)f (unstable)h(in)m(v)-5 b(arian)m(t)33 b(manifolds)e(of)2332 4184 y(~)2323 4207 y(\003)2386 4221 y Fp(")2455 4207 y 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Fs(\()p Fm(\033)s Fs(\))p Fm(;)g(I)7 b(;)15 b(')23 b Fs(+)d Fm(I)7 b(\033)n(;)15 b(s)20 b Fs(+)g Fm(\033)s Fs(\))1277 2863 y Fn(\000)o Fm(h)p Fs(\(0)p Fm(;)15 b Fs(0)p Fm(;)g(I)7 b(;)15 b(')24 b Fs(+)c Fm(I)7 b(\033)n(;)15 b(s)20 b Fs(+)g Fm(\033)s Fs(\))2246 2762 y Fh(\021)2301 2863 y Fm(d\033)n(;)456 3050 y Fs(where)33 b(\()p Fm(p)803 3064 y Fq(0)842 3050 y Fs(\()p Fm(t)p Fs(\))p Fm(;)15 b(q)1026 3064 y Fq(0)1066 3050 y Fs(\()p Fm(t)p Fs(\)\))35 b(is)f(the)g(parameterization)i(\(23\))f(of)f(the)g (separatrix)g(to)456 3158 y(the)i(saddle)h(p)s(oin)m(t)f(\(0)p Fm(;)15 b Fs(0\))39 b(of)d(the)h(p)s(endulum)d Fm(P)2148 3172 y Fl(\006)2207 3158 y Fs(\()p Fm(p;)15 b(q)s Fs(\))36 b(=)f Fn(\006)p Fs(\()p Fm(p)2701 3125 y Fq(2)2741 3158 y Fm(=)p Fs(2)25 b(+)f Fm(V)c Fs(\()p Fm(q)s Fs(\)\))456 3274 y(with)30 b(c)m(haracteristic)j(exp)s(onen)m(t)d Fm(\026)25 b Fs(=)1789 3197 y Fh(p)p 1880 3197 303 4 v 77 x Fn(\000)p Fm(V)2024 3248 y Fl(00)2066 3274 y Fs(\(0\))i Fm(>)e Fs(0.)555 3382 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b(\024)f Fs(cte)p Fm(:)16 b(e)2035 869 y Fl(\000)p Fp(\026)p Fl(j)q Fp(t)p Fl(j)p Fp(=)p Fq(2)2337 906 y Fo(for)66 b Fm(t)25 b Fn(!)g(\0061)p Fm(:)456 1068 y Fo(Mor)-5 b(e)g(over,)45 b(expr)-5 b(essing)44 b(the)f(p)-5 b(oints)49 b Fs(~)-50 b Fm(x)1829 1082 y Fl(\006)1930 1068 y Fs(=)2069 1045 y(~)2044 1068 y Fn(F)9 b Fs(\()p Fm(I)2193 1082 y Fl(\006)2253 1068 y Fm(;)15 b(')2352 1082 y Fl(\006)2412 1068 y Fm(;)g(s)2495 1082 y Fl(\006)2554 1068 y Fs(;)g Fm(")p Fs(\))43 b Fo(in)f(terms)i(of)456 1183 y(the)37 b(p)-5 b(ar)g(ameterization)48 b Fs(\(30\))38 b Fo(of)1635 1160 y Fs(~)1626 1183 y(\003)1689 1197 y Fp(")1763 1183 y Fo(given)e(in)h(The)-5 b(or)g(em)46 b Fs(20)p Fo(,)39 b(the)e(fol)5 b(lowing)456 1291 y(formulas)34 b(hold:)598 1442 y Fm(I)7 b Fs(\()f(~)-51 b Fm(x)732 1456 y Fl(\006)791 1442 y Fs(\))26 b(=)f Fm(I)i Fs(+)1106 1450 y(O)1177 1462 y Fl(C)1218 1443 y Fi(1)1256 1442 y Fs(\()p Fm(")p Fs(\))p Fm(;)203 b(')p Fs(\()6 b(~)-51 b Fm(x)1742 1456 y Fl(\006)1802 1442 y Fs(\))25 b(=)g Fm(')c Fs(+)2129 1450 y(O)2199 1462 y Fl(C)2240 1443 y Fi(1)2279 1442 y Fs(\()p Fm(")p Fs(\))p Fm(;)202 b(s)p Fs(\()6 b(~)-51 b Fm(x)2748 1456 y Fl(\006)2807 1442 y Fs(\))26 b(=)f Fm(s;)456 1594 y Fo(and)783 1770 y Fm(I)7 b Fs(\()f(~)-51 b Fm(x)917 1784 y Fq(+)976 1770 y Fs(\))21 b Fn(\000)f Fm(I)7 b Fs(\()f(~)-51 b Fm(x)1257 1784 y Fl(\000)1316 1770 y Fs(\))26 b(=)f Fm(")1525 1709 y(@)5 b Fn(L)p 1525 1749 117 4 v 1527 1833 a Fm(@)g(')1651 1770 y Fs(\()p Fm(I)i(;)15 b(')22 b Fn(\000)e Fm(I)7 b(\034)e(;)15 b(s)20 b Fn(\000)g Fm(\034)10 b Fs(\))20 b(+)2427 1778 y(O)2498 1790 y Fl(C)2539 1771 y Fi(1)2578 1770 y Fs(\()p Fm(")2655 1733 y Fq(1+)p Fp(\045)2786 1770 y Fs(\))p Fm(;)456 1970 y Fo(wher)-5 b(e)33 b Fm(\034)43 b Fo(is)32 b(given)g(again)i(by)e Fm(\034)j Fs(=)25 b Fm(\034)1715 1937 y Fl(\003)1755 1970 y Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))p Fo(,)34 b(and)g Fm(\045)25 b(>)g Fs(0)p Fo(.)456 2135 y Fw(Remark)38 b(73.)44 b Fs(W)-8 b(e)34 b(recall)h(that)e Fw(H4)h 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Fl(\006)929 2604 y Fs(\()p Fm(p)1010 2618 y Fq(0)1050 2604 y Fs(\()p Fm(\034)10 b Fs(\))p Fm(;)15 b(q)1251 2618 y Fq(0)1291 2604 y Fs(\()p Fm(\034)10 b Fs(\)\)\()5 b(~)-50 b Fm(z)1527 2567 y Fq(u)1592 2604 y Fn(\000)26 b Fs(~)-51 b Fm(z)1729 2567 y Fq(s)1761 2604 y Fs(\))26 b(=)f Fm(")1970 2543 y(@)5 b(L)p 1970 2583 116 4 v 1976 2667 a(@)g(\034)2095 2604 y Fs(\()p Fm(\034)g(;)15 b(I)7 b(;)15 b(';)g(s)p Fs(\))23 b(+)2593 2612 y(O)2663 2624 y Fl(C)2704 2605 y Fi(1)2743 2604 y Fs(\()p Fm(")2820 2567 y Fq(2)2860 2604 y Fs(\))456 2789 y(where)29 b Fm(L)p Fs(\()p Fm(\034)5 b(;)15 b(I)7 b(;)15 b(';)g(s)p Fs(\))33 b(is)d(giv)m(en)i(b)m(y)576 3117 y Fm(L)p Fs(\()p Fm(\034)5 b(;)15 b(I)7 b(;)15 b(';)g(s)p Fs(\))27 b(=)e Fn(\000)1231 2993 y Fh(Z)1322 3020 y Fq(+)p Fl(1)1281 3199 y(\0001)1451 3016 y Fh(\020)1506 3117 y Fm(h)p Fs(\()p Fm(p)1639 3131 y Fq(0)1679 3117 y Fs(\()p Fm(\034)30 b Fs(+)20 b Fm(\033)s Fs(\))p Fm(;)15 b(q)2046 3131 y Fq(0)2086 3117 y Fs(\()p Fm(\034)31 b Fs(+)20 b Fm(\033)s 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Fn(\000)g Fm(\034)10 b Fs(\))26 b(=)f Fn(L)p Fs(\()p Fm(I)7 b(;)15 b(')21 b Fn(\000)f Fm(I)7 b(\034)e(;)15 b(s)20 b Fn(\000)g Fm(\034)10 b Fs(\))p Fm(;)456 4144 y Fs(where)47 b Fn(L)h Fs(is)g(the)g(Melnik)m (o)m(v)i(p)s(oten)m(tial)f(de\014ned)e(in)h(\(10\))q(.)94 b(Then,)51 b(equa-)456 4252 y(tion)30 b(\(143\))j(is)d(equiv)-5 b(alen)m(t)32 b(to)1090 4454 y Fm(")1142 4392 y(@)5 b Fn(L)p 1142 4433 117 4 v 1148 4516 a Fm(@)g(\034)1268 4454 y Fs(\()p Fm(I)i(;)15 b(')22 b Fn(\000)e Fm(I)7 b(\034)e(;)15 b(s)20 b Fn(\000)g Fm(\034)10 b Fs(\))21 b(+)2045 4462 y(O)2115 4473 y Fl(C)2156 4455 y Fi(1)2195 4454 y Fs(\()p Fm(")2272 4416 y Fq(2)2312 4454 y Fs(\))26 b(=)f(0)p Fm(:)456 4634 y Fs(By)38 b(the)g(implicit)h(function)e (Theorem,)j(non-degenerate)f(critical)h(p)s(oin)m(ts)e Fm(\034)456 4742 y Fs(of)31 b(the)h(function)f Fm(\034)37 b Fn(7!)27 b(L)p Fs(\()p Fm(I)7 b(;)15 b(')21 b Fn(\000)g Fm(I)7 b(\034)e(;)15 b(s)21 b Fn(\000)g Fm(\034)10 b Fs(\))31 b(giv)m(e)i(rise,)f(for)f Fm(")h Fs(small)g(enough,)456 4856 y(to)c(transv)m(erse)g(in)m(tersections)h(of)f(the)g(stable)g(and) f(unstable)h(manifolds)f(of)3083 4833 y(~)3074 4856 y(\003)3137 4870 y Fp(")456 4964 y Fs(along)k(p)s(oin)m(ts)36 b(~)-51 b Fm(z)30 b Fs(=)g(~)-50 b Fm(z)t Fs(\()p Fm(\034)5 b(;)15 b(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))33 b(of)e(the)f(form)g (\(141\))r(.)p eop end %%Page: 89 89 TeXDict begin 89 88 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(89)555 450 y Fs(By)22 b(the)g(implicit)g(function)f(Theorem,)j(w)m(e)d(can)h(\014nd)e(a)i (function)f Fm(\034)2830 417 y Fl(\003)2891 450 y Fs(de\014ned)456 558 y(in)36 b(an)i(op)s(en)e(set)i(where)f Fm(\034)1397 525 y Fl(\003)1436 558 y Fs(\()p Fm(I)7 b Fs(;)15 b Fm(';)g(s)p Fs(\))39 b(is)e(a)h(critical)h(p)s(oin)m(t)e(of)g Fm(\034)47 b Fn(7!)36 b(L)p Fs(\()p Fm(I)7 b(;)15 b(')26 b Fn(\000)456 666 y Fm(I)7 b(\034)e(;)15 b(s)20 b Fn(\000)g Fm(\034)10 b Fs(\).)555 774 y(T)-8 b(o)35 b(\014nish)e(the)i(pro)s(of,)g(w)m(e)g (no)m(w)g(consider)f(the)h(expression)g(of)f(the)h(p)s(oin)m(ts)461 884 y(~)-50 b Fm(x)508 898 y Fl(\006)608 884 y Fs(=)745 861 y(~)720 884 y Fn(F)10 b Fs(\()p Fm(I)870 898 y Fl(\006)929 884 y Fm(;)15 b(')1028 898 y Fl(\006)1088 884 y Fm(;)g(s)1171 898 y Fl(\006)1230 884 y Fs(;)g Fm(")p Fs(\))41 b(in)f(terms)g(of)g (the)h(parameterization)h(\(30\))g(of)3083 861 y(~)3074 884 y(\003)3137 898 y Fp(")456 991 y Fs(giv)m(en)29 b(in)e(Theorem)h (20.)41 b(Since)28 b(w)m(e)h(already)f(kno)m(w)g(the)h(existence)g(of) 34 b(~)-50 b Fm(z)32 b Fs(giv)m(en)456 1099 y(in)e(\(141\))i(suc)m(h)e (that)h(\(142\))h(holds,)f(it)f(is)h(clear)g(that)602 1297 y Fm(I)7 b Fs(\()f(~)-51 b Fm(x)736 1311 y Fl(\006)795 1297 y Fs(\))26 b(=)f Fm(I)i Fs(+)1110 1305 y(O)1181 1317 y Fl(C)1222 1298 y Fi(1)1260 1297 y Fs(\()p Fm(")p Fs(\))p Fm(;)199 b(')p Fs(\()6 b(~)-51 b Fm(x)1742 1311 y Fl(\006)1802 1297 y Fs(\))25 b(=)g Fm(')c Fs(+)2129 1305 y(O)2199 1317 y Fl(C)2240 1298 y Fi(1)2279 1297 y Fs(\()p Fm(")p Fs(\))p Fm(;)198 b(s)p Fs(\()6 b(~)-51 b Fm(x)2744 1311 y Fl(\006)2803 1297 y Fs(\))26 b(=)f Fm(s;)456 1495 y Fs(and)k(it)i(only)g(remains)f(to)h(obtain)g(the)f (form)m(ula)h(for)f Fm(I)7 b Fs(\()f(~)-51 b Fm(x)2419 1509 y Fq(+)2479 1495 y Fs(\))20 b Fn(\000)g Fm(I)7 b Fs(\()f(~)-51 b Fm(x)2759 1509 y Fl(\000)2819 1495 y Fs(\).)555 1602 y(W)-8 b(e)32 b(apply)e(no)m(w)g(the)h(F)-8 b(undamen)m(tal)31 b(Theorem)f(of)h(Calculus)f(to)1222 1800 y Fm(t)25 b Fn(7!)g Fm(I)e Fs(\(\010)1560 1814 y Fp(t;")1641 1800 y Fs(\()6 b(~)-51 b Fm(x)1728 1814 y Fl(\006)1788 1800 y Fs(\)\))21 b Fn(\000)f Fm(I)i Fs(\(\010)2133 1814 y Fp(t;")2215 1800 y Fs(\()5 b(~)-50 b Fm(z)t Fs(\)\))16 b Fm(;)456 1998 y Fs(to)31 b(get)589 2254 y Fm(I)7 b Fs(\()f(~)-51 b Fm(x)723 2268 y Fl(\006)783 2254 y Fs(\))20 b Fn(\000)g Fm(I)7 b Fs(\()e(~)-50 b Fm(z)5 b Fs(\))25 b(=)g Fm(")1271 2130 y Fh(Z)1363 2156 y Fq(0)1322 2336 y Fl(\0061)1467 2254 y Fs(\()q Fn(f)p Fm(I)7 b(;)15 b(h)p Fn(g)g Fs(\()r(\010)1850 2268 y Fp(\033)n(;")1945 2254 y Fs(\()6 b(~)-51 b Fm(x)2032 2268 y Fl(\006)2091 2254 y Fs(\)\))21 b Fn(\000)f(f)p Fm(I)7 b(;)15 b(h)p Fn(g)g Fs(\()r(\010)2620 2268 y Fp(\033)n(;")2715 2254 y Fs(\()5 b(~)-50 b Fm(z)t Fs(\)\))q(\))16 b Fm(d\033)n(;)456 2509 y Fs(where)33 b Fn(f)p Fm(I)7 b(;)15 b(h)p Fn(g)32 b Fs(=)f Fm(@)1133 2523 y Fp(')1183 2509 y Fm(I)23 b(@)1294 2523 y Fp(I)1334 2509 y Fm(h)g Fn(\000)f Fm(@)1550 2523 y Fp(I)1590 2509 y Fm(I)g(@)1700 2523 y Fp(')1751 2509 y Fm(h)31 b Fs(=)f Fn(\000)p Fm(@)2054 2523 y Fp(')2104 2509 y Fm(h)k Fs(is)g(the)g(P)m(oisson)g(brac)m(k)m(et)i(of)456 2617 y(the)30 b(functions)g Fm(I)37 b Fs(and)30 b Fm(h)p Fs(.)41 b(Subtracting)30 b(the)h(expressions)f(ab)s(o)m(v)m(e,)i(w)m(e) f(get)620 2870 y Fm(I)7 b Fs(\()f(~)-51 b Fm(x)754 2884 y Fq(+)813 2870 y Fs(\))21 b Fn(\000)f Fm(I)7 b Fs(\()f(~)-51 b Fm(x)1094 2884 y Fl(\000)1153 2870 y Fs(\))26 b(=)f Fm(")1367 2746 y Fh(Z)1458 2773 y Fq(+)p Fl(1)1418 2953 y Fq(0)1616 2809 y Fm(@)5 b(h)p 1613 2849 113 4 v 1613 2933 a(@)g(')1736 2870 y Fs(\(\010)1837 2884 y Fp(\033)n(;")1932 2870 y Fs(\()g(~)-50 b Fm(z)t Fs(\)\))21 b Fn(\000)2209 2809 y Fm(@)5 b(h)p 2205 2849 V 2205 2933 a(@)g(')2328 2870 y Fs(\(\010)2429 2884 y Fp(\033)n(;")2524 2870 y Fs(\()h(~)-51 b Fm(x)2611 2884 y Fq(+)2670 2870 y Fs(\)\))p Fm(d\033)1259 3128 y Fs(+)p Fm(")1387 3004 y Fh(Z)1478 3031 y Fq(0)1438 3211 y Fl(\0001)1596 3067 y Fm(@)5 b(h)p 1593 3107 V 1593 3190 a(@)g(')1715 3128 y Fs(\(\010)1816 3142 y Fp(\033)n(;")1912 3128 y Fs(\()g(~)-50 b Fm(z)t Fs(\)\))21 b Fn(\000)2189 3067 y Fm(@)5 b(h)p 2185 3107 V 2185 3190 a(@)g(')2308 3128 y Fs(\(\010)2409 3142 y Fp(\033)n(;")2504 3128 y Fs(\()h(~)-51 b Fm(x)2591 3142 y Fl(\000)2650 3128 y Fs(\)\))p Fm(d\033)n(:)555 3379 y Fs(W)-8 b(e)32 b(already)f(kno)m(w)f(that)1075 3576 y(\010)1141 3590 y Fp(t;")1222 3576 y Fs(\()5 b(~)-50 b Fm(z)5 b Fs(\))26 b(=)f(\010)1527 3590 y Fp(t;)p Fq(0)1611 3576 y Fs(\()5 b(~)-50 b Fm(z)1688 3590 y Fq(0)1728 3576 y Fs(\))20 b(+)1874 3584 y(O)1945 3596 y Fl(C)1986 3577 y Fi(1)2025 3576 y Fs(\()p Fm(")p Fs(\))p Fm(;)107 b Fn(8)p Fm(t)25 b Fn(2)g Fk(R)p Fm(:)456 3774 y Fs(T)-8 b(aking)22 b Fm(c)789 3788 y Fq(2)850 3774 y Fs(su\016cien)m(tly)g (small)g(\(but)f(indep)s(enden)m(t)f(of)i Fm(")p Fs(\))g(and)f(using)f (Gron)m(w)m(all)456 3882 y(inequalit)m(y)31 b(w)m(e)g(ha)m(v)m(e,)h (for)e Fn(\000)p Fm(c)1495 3896 y Fq(2)1550 3882 y Fn(j)p Fs(log)18 b Fm(")p Fn(j)26 b(\024)e Fm(t)i Fn(\024)f Fm(c)2090 3896 y Fq(2)2144 3882 y Fn(j)q Fs(log)17 b Fm(")p Fn(j)1196 4079 y Fs(\010)1262 4093 y Fp(t;")1344 4079 y Fs(\()6 b(~)-51 b Fm(x)1431 4093 y Fl(\006)1490 4079 y Fs(\))26 b(=)e(\010)1712 4093 y Fp(t;)p Fq(0)1797 4079 y Fs(\()6 b(~)-51 b Fm(x)1884 4093 y Fq(0)1923 4079 y Fs(\))21 b(+)2070 4087 y(O)2141 4099 y Fl(C)2182 4080 y Fi(1)2220 4079 y Fs(\()p Fm(")2297 4042 y Fp(\045)2333 4051 y Fi(1)2373 4079 y Fs(\))p Fm(;)456 4277 y Fs(for)30 b(some)h Fm(\045)870 4291 y Fq(1)934 4277 y Fm(>)25 b Fs(0.)555 4385 y(F)-8 b(rom)22 b(equation)h(\(142\))r(,)h(w)m(e)e (deduce)f(that)i(there)f(exists)g(a)g(constan)m(t)h Fm(c)2942 4399 y Fq(1)3007 4385 y Fm(>)i Fs(0,)554 4532 y Fh(\014)554 4587 y(\014)554 4641 y(\014)554 4696 y(\014)554 4751 y(\014)584 4568 y(Z)675 4594 y Fl(\0061)635 4774 y(\006)p Fp(c)721 4783 y Fi(2)755 4774 y Fl(j)o Fq(log)13 b Fp(")p Fl(j)949 4563 y Fh(\022)1029 4630 y Fm(@)5 b(h)p 1026 4671 V 1026 4754 a(@)g(')1148 4691 y Fs(\(\010)1249 4705 y Fp(\033)n(;")1344 4691 y Fs(\()p Fm(z)t Fs(\)\))22 b Fn(\000)1621 4630 y Fm(@)5 b(h)p 1618 4671 V 1618 4754 a(@)g(')1741 4691 y Fs(\(\010)1842 4705 y Fp(\033)n(;")1937 4691 y Fs(\()p Fm(x)2024 4705 y Fl(\006)2083 4691 y Fs(\)\))2153 4563 y Fh(\023)2236 4691 y Fm(d\033)2338 4532 y Fh(\014)2338 4587 y(\014)2338 4641 y(\014)2338 4696 y(\014)2338 4751 y(\014)1795 4928 y Fn(\024)25 b Fm(c)1930 4942 y Fq(1)1970 4928 y Fm(e)2012 4890 y Fl(\000)p Fp(\026c)2140 4899 y Fi(2)2175 4890 y Fl(j)o Fq(log)13 b Fp(")p Fl(j)p Fp(=)p Fq(2)2450 4928 y Fs(=)2545 4936 y(O)2616 4947 y Fl(C)2657 4929 y Fi(1)2696 4928 y Fs(\()p Fm(")2773 4890 y Fp(\045)2809 4899 y Fi(2)2848 4928 y Fs(\))p Fm(;)p eop end %%Page: 90 90 TeXDict begin 90 89 bop 456 251 a Fq(90)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 450 y Fs(and,)48 b(since)d(for)g(the)g(unp)s(erturb)s(ed)d(system)j(w)m(e)h(ha)m(v)m(e)g (the)f(same)g(kind)g(of)456 558 y(b)s(eha)m(vior,)30 b(w)m(e)h(can)g(conclude)g(\014nally)f(that,)h(for)f(some)h Fm(\045)25 b(>)g Fs(0)539 708 y Fm(I)7 b Fs(\()f(~)-51 b Fm(x)673 722 y Fq(+)733 708 y Fs(\))20 b Fn(\000)g Fm(I)7 b Fs(\()f(~)-51 b Fm(x)1013 722 y Fl(\000)1073 708 y Fs(\))793 990 y(=)25 b Fn(\000)p Fm(")1038 839 y Fq(+)p Fp(c)1124 848 y Fi(2)1158 839 y Fl(j)p Fq(log)12 b Fp(")p Fl(j)1120 866 y Fh(Z)997 1133 y Fl(\000)p Fp(c)1083 1142 y Fi(2)1118 1133 y Fl(j)o Fq(log)h Fp(")p Fl(j)1328 861 y Fh(\022)1408 928 y Fm(@)5 b(h)p 1404 969 113 4 v 1404 1052 a(@)g(')1527 990 y Fs(\()p Fm(q)1603 1004 y Fq(0)1643 990 y Fs(\()p Fm(\034)30 b Fs(+)20 b Fm(\033)s Fs(\))p Fm(;)15 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Fs(\()p Fn(V)889 3445 y Fq(1)928 3431 y Fs(\))23 b(is)f(transv)m(ersal)h(to)g Fn(V)1681 3445 y Fq(2)1743 3431 y Fs(as)f(submanifolds)f(of)2491 3408 y(~)2483 3431 y(\003)2546 3445 y Fp(")2582 3431 y Fs(.)38 b(This)22 b(pro)m(vides)456 3539 y(us)j(with)h(an)f (alternativ)m(e)k(to)d(the)h(customary)f(Melnik)m(o)m(v)i (calculations,)h(whic)m(h)456 3647 y(require)h(common)g(systems)h(of)f (co)s(ordinates)h(for)f(b)s(oth)g(ob)5 b(jects.)555 3755 y(Recall)22 b(that)f(in)f(Section)h(9)g(w)m(e)f(ha)m(v)m(e)i(obtained)f (a)f(explicit)i(expression)e(\(147\))456 3863 y(for)32 b(the)h(scattering)h(map.)47 b(In)32 b(Prop)s(ositions)g(47,)j(50,)f (and)e(Corollary)h(57)g(w)m(e)456 3971 y(ha)m(v)m(e)42 b(obtained)g(explicit)h(expressions)f(\(dep)s(ending)e(on)i(the)f(pro)m (ximit)m(y)i(to)456 4085 y(resonances\))32 b(for)g(the)g(KAM)g(tori,)h (b)s(oth)e(primary)g(or)g(secondary)-8 b(,)33 b(in)2930 4063 y(~)2921 4085 y(\003)2984 4099 y Fp(")3021 4085 y Fs(.)45 b(In)456 4200 y(Prop)s(osition)34 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%52FFFFFF52A8FD08FFA852F82752A8FD17FF7D27FFFFFF7D7DA87DFD12FF %A87DF82752A8FD07FF27FD10FF7DFFFFFF7D7DFD0BFFA87DF82752A8FD14 %FFA87DFFFFA8A8FD12FF7D52F82752A87DFD09FF27FD10FF277DFFFF52A8 %FD0EFFA87D2727275252A8FD0FFF52F8FFFFA87DFD0DFFA87D2727F8527D %A8FFFFFFA8FD09FF27FD10FF277DFFFF7DA8FD13FF7D7D2727F852527DA8 %FD09FFA852FD0BFFA87D7D2727F85252A8A8FD07FFA8FD09FF27FD10FF52 %A8FFFF52A8FD18FF7D7D52522727F8FD042752527D277D527D527DFD0427 %F827277D52A8A8FD0CFFA8FD09FF27FD14FF7DA8FD21FF7DA8A8A87DA8A8 %A87DA8A8A87DFD15FFA8FD09FF27FD0FFFA87DA87DA8277D7DA87DA87DA8 %7DA87DA87DA87DA87DA87DA87DA87DA87DA87DA87DA87DA87DA87DA87DA8 %7DA8A8FD1CFFA8FD09FF27FD14FF7DA8FD43FFA8A852A8FFFFFFA827A827 %FD14FF52A8FD44FF7D27277D527D27277D27FD14FF7D7DFD4DFF27FD14FF %52A8FD4DFF27FD14FF7DA8FD48FF7DA8A8A8FF27FD14FF52A8FD47FF7D52 %A852A8FD5FFF7DFD71FFFF %%EndData %%EndComments %%BeginProlog %%BeginResource: procset Adobe_level2_AI5 1.2 0 %%Title: (Adobe Illustrator (R) Version 5.0 Level 2 Emulation) %%Version: 1.2 0 %%CreationDate: (04/10/93) () %%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved) userdict /Adobe_level2_AI5 26 dict dup begin put /packedarray where not { userdict begin /packedarray { array astore readonly } bind def /setpacking /pop load def /currentpacking false def end 0 } if pop userdict /defaultpacking currentpacking put true setpacking /initialize { Adobe_level2_AI5 begin } bind def /terminate { currentdict Adobe_level2_AI5 eq { end } if } bind def mark /setcustomcolor where not { /findcmykcustomcolor { (AI8_CMYK_CustomColor) 6 packedarray } bind def /findrgbcustomcolor { (AI8_RGB_CustomColor) 5 packedarray } bind def /setcustomcolor { exch aload pop dup (AI8_CMYK_CustomColor) eq { pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } { dup (AI8_RGB_CustomColor) eq { pop pop 3 { 1 exch sub 3 index mul 1 exch sub 3 1 roll } repeat 4 -1 roll pop setrgbcolor } { pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } ifelse } ifelse } def } if /setAIseparationgray { false setoverprint 0 setgray /setseparationgray where{ pop setseparationgray }{ /setcolorspace where{ pop [/Separation (All) /DeviceCMYK {dup dup dup}] setcolorspace 1 exch sub setcolor }{ setgray }ifelse }ifelse } def /gt38? mark {version cvr cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put userdict /level2? systemdict /languagelevel known dup { pop systemdict /languagelevel get 2 ge } if put /level2ScreenFreq { begin 60 HalftoneType 1 eq { pop Frequency } if HalftoneType 2 eq { pop GrayFrequency } if HalftoneType 5 eq { pop Default level2ScreenFreq } if end } bind def userdict /currentScreenFreq level2? {currenthalftone level2ScreenFreq} {currentscreen pop pop} ifelse put level2? not { /setcmykcolor where not { /setcmykcolor { exch .11 mul add exch .59 mul add exch .3 mul add 1 exch sub setgray } def } if /currentcmykcolor where not { /currentcmykcolor { 0 0 0 1 currentgray sub } def } if /setoverprint where not { /setoverprint /pop load def } if /selectfont where not { /selectfont { exch findfont exch dup type /arraytype eq { makefont } { scalefont } ifelse setfont } bind def } if /cshow where not { /cshow { [ 0 0 5 -1 roll aload pop ] cvx bind forall } bind def } if } if cleartomark /anyColor? { add add add 0 ne } bind def /testColor { gsave setcmykcolor currentcmykcolor grestore } bind def /testCMYKColorThrough { testColor anyColor? } bind def userdict /composite? 1 0 0 0 testCMYKColorThrough 0 1 0 0 testCMYKColorThrough 0 0 1 0 testCMYKColorThrough 0 0 0 1 testCMYKColorThrough and and and put composite? not { userdict begin gsave /cyan? 1 0 0 0 testCMYKColorThrough def /magenta? 0 1 0 0 testCMYKColorThrough def /yellow? 0 0 1 0 testCMYKColorThrough def /black? 0 0 0 1 testCMYKColorThrough def grestore /isCMYKSep? cyan? magenta? yellow? black? or or or def /customColor? isCMYKSep? not def end } if end defaultpacking setpacking %%EndResource %%BeginResource: procset Adobe_typography_AI5 1.0 1 %%Title: (Typography Operators) %%Version: 1.0 1 %%CreationDate:(6/10/1996) () %%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_typography_AI5 68 dict dup begin put /initialize { begin begin Adobe_typography_AI5 begin Adobe_typography_AI5 { dup xcheck { bind } if pop pop } forall end end end Adobe_typography_AI5 begin } def /terminate { currentdict Adobe_typography_AI5 eq { end } if } def /modifyEncoding { /_tempEncode exch ddef /_pntr 0 ddef { counttomark -1 roll dup type dup /marktype eq { pop pop exit } { /nametype eq { _tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll put } { /_pntr exch ddef } ifelse } ifelse } loop _tempEncode } def /havefont { systemdict /languagelevel known { /Font resourcestatus dup { exch pop exch pop } if } { systemdict /FontDirectory get 1 index known { pop true } { systemdict /fileposition known { dup length 6 add exch Ss 6 250 getinterval cvs pop Ss exch 0 exch getinterval status { pop pop pop pop true } { false } ifelse } { pop false } ifelse } ifelse } ifelse } def /TE { StandardEncoding 256 array copy modifyEncoding /_nativeEncoding exch def } def /subststring { exch 2 index exch search { exch pop exch dup () eq { pop exch concatstring } { 3 -1 roll exch concatstring concatstring } ifelse exch pop true } { pop pop false } ifelse } def /concatstring { 1 index length 1 index length 1 index add string dup 0 5 index putinterval dup 2 index 4 index putinterval 4 1 roll pop pop pop } def % /TZ { dup type /arraytype eq { /_wv exch def } { /_wv 0 def } ifelse /_useNativeEncoding exch def 2 index havefont { 3 index 255 string cvs dup (_Symbol_) eq { pop 2 index findfont } { 1 index 0 eq { dup length 1 sub 1 exch getinterval cvn findfont } { pop 2 index findfont } ifelse } ifelse } { dup 1 eq { 2 index 64 string cvs dup (-90pv-RKSJ-) (-83pv-RKSJ-) subststring { exch pop dup havefont { findfont false } { pop true } ifelse } { pop dup (-90ms-RKSJ-) (-Ext-RKSJ-) subststring { exch pop dup havefont { findfont false } { pop true } ifelse } { pop pop true } ifelse } ifelse { 1 index 1 eq { /Ryumin-Light-Ext-RKSJ-V havefont {/Ryumin-Light-Ext-RKSJ-V} {/Courier} ifelse } { /Ryumin-Light-83pv-RKSJ-H havefont {/Ryumin-Light-83pv-RKSJ-H} {/Courier} ifelse } ifelse findfont [1 0 0.5 1 0 0] makefont } if } { /Courier findfont } ifelse } ifelse _wv type /arraytype eq { _wv makeblendedfont } if dup length 10 add dict begin mark exch { 1 index /FID ne { def } if cleartomark mark } forall pop /FontScript exch def /FontDirection exch def /FontRequest exch def /FontName exch def counttomark 0 eq { 1 _useNativeEncoding eq { /Encoding _nativeEncoding def } if cleartomark } { /Encoding load 256 array copy modifyEncoding /Encoding exch def } ifelse FontName currentdict end definefont pop } def /tr { _ax _ay 3 2 roll } def /trj { _cx _cy _sp _ax _ay 6 5 roll } def /a0 { /Tx { dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss } ddef /Tj { dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss } ddef } def /a1 { W B } def /e0 { /Tx { tr _psf } ddef /Tj { trj _pjsf } ddef } def /e1 { W F } def /i0 { /Tx { tr sp } ddef /Tj { trj jsp } ddef } def /i1 { W N } def /o0 { /Tx { tr sw rmoveto } ddef /Tj { trj swj rmoveto } ddef } def /r0 { /Tx { tr _ctm _pss } ddef /Tj { trj _ctm _pjss } ddef } def /r1 { W S } def /To { pop _ctm currentmatrix pop } def /TO { iTe _ctm setmatrix newpath } def /Tp { pop _tm astore pop _ctm setmatrix _tDict begin /W { } def /h { } def } def /TP { end iTm 0 0 moveto } def /Tr { _render 3 le { currentpoint newpath moveto } if dup 8 eq { pop 0 } { dup 9 eq { pop 1 } if } ifelse dup /_render exch ddef _renderStart exch get load exec } def /iTm { _ctm setmatrix _tm concat _shift aload pop _lineorientation 1 eq { exch } if translate _scale aload pop _lineorientation 1 eq _yokoorientation 1 eq or { exch } if scale } def /Tm { _tm astore pop iTm 0 0 moveto } def /Td { _mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto } def /iTe { _render -1 eq { } { _renderEnd _render get dup null ne { load exec } { pop } ifelse } ifelse /_render -1 ddef } def /Ta { pop } def /Tf { 1 index type /nametype eq { dup 0.75 mul 1 index 0.25 mul neg } if /_fontDescent exch ddef /_fontAscent exch ddef /_fontSize exch ddef /_fontRotateAdjust _fontAscent _fontDescent add 2 div neg ddef /_fontHeight _fontSize ddef findfont _fontSize scalefont setfont } def /Tl { pop neg 0 exch _leading astore pop } def /Tt { pop } def /TW { 3 npop } def /Tw { /_cx exch ddef } def /TC { 3 npop } def /Tc { /_ax exch ddef } def /Ts { 0 exch _shift astore pop currentpoint iTm moveto } def /Ti { 3 npop } def /Tz { count 1 eq { 100 } if 100 div exch 100 div exch _scale astore pop iTm } def /TA { pop } def /Tq { pop } def /Tg { pop } def /TG { pop } def /Tv { /_lineorientation exch ddef } def /TV { /_charorientation exch ddef } def /Ty { dup /_yokoorientation exch ddef 1 sub neg Tv } def /TY { pop } def /T~ { Tx } def /Th { pop pop pop pop pop } def /TX { pop } def /Tk { _fontSize mul 1000 div _lineorientation 0 eq { neg 0 } { 0 exch } ifelse rmoveto pop } def /TK { 2 npop } def /T* { _leading aload pop _lineorientation 0 ne { exch } if Td } def /T*- { _leading aload pop _lineorientation 0 ne { exch } if exch neg exch neg Td } def /T- { _ax neg 0 rmoveto _lineorientation 1 eq _charorientation 0 eq and { 1 TV _hyphen Tx 0 TV } { _hyphen Tx } ifelse } def /T+ { } def /TR { _ctm currentmatrix pop _tm astore pop iTm 0 0 moveto } def /TS { currentfont 3 1 roll /_Symbol_ findfont _fontSize scalefont setfont 0 eq { Tx } { Tj } ifelse setfont } def /Xb { pop pop } def /Tb /Xb load def /Xe { pop pop pop pop } def /Te /Xe load def /XB { } def /TB /XB load def currentdict readonly pop end setpacking % /X^ { currentfont 5 1 roll dup havefont { findfont _fontSize scalefont setfont } { pop exch } ifelse 2 index 0 eq { Tx } { Tj } ifelse pop pop setfont } def /T^ /X^ load def %%EndResource %%BeginProcSet: Adobe_ColorImage_AI6 1.3 0 userdict /Adobe_ColorImage_AI6 known not { userdict /Adobe_ColorImage_AI6 53 dict put } if userdict /Adobe_ColorImage_AI6 get begin /initialize { Adobe_ColorImage_AI6 begin Adobe_ColorImage_AI6 { dup type /arraytype eq { dup xcheck { bind } if } if pop pop } forall } def /terminate { end } def currentdict /Adobe_ColorImage_AI6_Vars known not { /Adobe_ColorImage_AI6_Vars 41 dict def } if Adobe_ColorImage_AI6_Vars begin /plateindex -1 def /_newproc null def /_proc1 null def /_proc2 null def /sourcearray 4 array def /_ptispace null def /_ptiname null def /_pti0 0 def /_pti1 0 def /_ptiproc null def /_ptiscale 0 def /_pticomps 0 def /_ptibuf 0 string def /_gtigray 0 def /_cticmyk null def /_rtirgb null def /XIEnable true def /XIType 0 def /XIEncoding 0 def /XICompression 0 def /XIChannelCount 0 def /XIBitsPerPixel 0 def /XIImageHeight 0 def /XIImageWidth 0 def /XIImageMatrix null def /XIRowBytes 0 def /XIFile null def /XIBuffer1 null def /XIBuffer2 null def /XIBuffer3 null def /XIDataProc null def /XIColorSpace /DeviceGray def /XIColorValues 0 def /XIPlateList false def end /ci6colorimage /colorimage where {/colorimage get}{null} ifelse def /ci6image systemdict /image get def /ci6curtransfer systemdict /currenttransfer get def /ci6curoverprint /currentoverprint where {/currentoverprint get}{{_of}} ifelse def /ci6foureq { 4 index ne { pop pop pop false }{ 4 index ne { pop pop false }{ 4 index ne { pop false }{ 4 index eq } ifelse } ifelse } ifelse } def /ci6testplate { Adobe_ColorImage_AI6_Vars begin /plateindex -1 def /setcmykcolor where { pop gsave 1 0 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 1 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 1 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 0 1 setcmykcolor systemdict /currentgray get exec 1 exch sub grestore 1 0 0 0 ci6foureq { /plateindex 0 def }{ 0 1 0 0 ci6foureq { /plateindex 1 def }{ 0 0 1 0 ci6foureq { /plateindex 2 def }{ 0 0 0 1 ci6foureq { /plateindex 3 def }{ 0 0 0 0 ci6foureq { /plateindex 5 def } if } ifelse } ifelse } ifelse } ifelse pop pop pop pop } if plateindex end } def /ci6concatprocs { /packedarray where { pop dup type /packedarraytype eq 2 index type /packedarraytype eq or }{ false } ifelse { /_proc2 exch cvlit def /_proc1 exch cvlit def _proc1 aload pop _proc2 aload pop _proc1 length _proc2 length add packedarray cvx }{ /_proc2 exch cvlit def /_proc1 exch cvlit def /_newproc _proc1 length _proc2 length add array def _newproc 0 _proc1 putinterval _newproc _proc1 length _proc2 putinterval _newproc cvx } ifelse } def /ci6istint { type /arraytype eq } def /ci6isspot { dup type /arraytype eq { dup length 1 sub get /Separation eq }{ pop false } ifelse } def /ci6spotname { dup ci6isspot {dup length 2 sub get}{pop ()} ifelse } def /ci6altspace { aload pop pop pop ci6colormake } def /ci6numcomps { dup /DeviceGray eq { pop 1 }{ dup /DeviceRGB eq { pop 3 }{ /DeviceCMYK eq { 4 }{ 1 } ifelse } ifelse } ifelse } def /ci6marksplate { dup /DeviceGray eq { pop plateindex 3 eq }{ dup /DeviceRGB eq { pop plateindex 5 ne }{ dup /DeviceCMYK eq { pop plateindex 5 ne }{ dup ci6isspot { /findcmykcustomcolor where { pop dup length 2 sub get 0.1 0.1 0.1 0.1 5 -1 roll findcmykcustomcolor 1 setcustomcolor systemdict /currentgray get exec 1 ne }{ pop plateindex 5 ne } ifelse }{ pop plateindex 5 ne } ifelse } ifelse } ifelse } ifelse } def /ci6colormake { dup ci6numcomps exch 1 index 2 add 1 roll dup 1 eq {pop}{array astore} ifelse exch } def /ci6colorexpand { dup ci6spotname exch dup ci6istint { ci6altspace exch 4 1 roll }{ 1 3 1 roll } ifelse } def /ci6colortint { dup /DeviceGray eq { 3 1 roll 1 exch sub mul 1 exch sub exch }{ dup /DeviceRGB eq { 3 1 roll {1 exch sub 1 index mul 1 exch sub exch} forall pop 3 array astore exch }{ dup /DeviceCMYK eq { 3 1 roll {1 index mul exch} forall pop 4 array astore exch }{ 3 1 roll mul exch } ifelse } ifelse } ifelse } def /ci6colortocmyk { dup /DeviceGray eq { pop 1 exch sub 0 0 0 4 -1 roll 4 array astore }{ dup /DeviceRGB eq { pop aload pop _rgbtocmyk 4 array astore }{ dup /DeviceCMYK eq { pop }{ ci6altspace ci6colortint ci6colortocmyk } ifelse } ifelse } ifelse } def /ci6makeimagedict { 7 dict begin /ImageType 1 def /Decode exch def /DataSource exch def /ImageMatrix exch def /BitsPerComponent exch def /Height exch def /Width exch def currentdict end } def /ci6stringinvert { 0 1 2 index length 1 sub { dup 2 index exch get 255 exch sub 2 index 3 1 roll put } for } def /ci6stringknockout { 0 1 2 index length 1 sub { 255 2 index 3 1 roll put } for } def /ci6stringapply { 0 1 4 index length 1 sub { dup 4 index exch get 3 index 3 1 roll 3 index exec } for pop exch pop } def /ci6walkrgbstring { 0 3 index dup length 1 sub 0 3 3 -1 roll { 3 getinterval {} forall 5 index exec 3 index } for 5 {pop} repeat } def /ci6walkcmykstring { 0 3 index dup length 1 sub 0 4 3 -1 roll { 4 getinterval {} forall 6 index exec 3 index } for 5 { pop } repeat } def /ci6putrgbtograystr { .11 mul exch .59 mul add exch .3 mul add cvi 3 copy put pop 1 add } def /ci6putcmyktograystr { exch .11 mul add exch .59 mul add exch .3 mul add dup 255 gt { pop 255 } if 255 exch sub cvi 3 copy put pop 1 add } def /ci6rgbtograyproc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec XIBuffer3 dup 3 1 roll /ci6putrgbtograystr load exch ci6walkrgbstring end } def /ci6cmyktograyproc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec XIBuffer3 dup 3 1 roll /ci6putcmyktograystr load exch ci6walkcmykstring end } def /ci6separatecmykproc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec XIBuffer3 0 2 index plateindex 4 2 index length 1 sub { get 255 exch sub 3 copy put pop 1 add 2 index } for pop pop exch pop end } def /ci6compositeimage { dup 1 eq { pop pop image }{ /ci6colorimage load null ne { ci6colorimage }{ 3 1 roll pop sourcearray 0 3 -1 roll put 3 eq {/ci6rgbtograyproc}{/ci6cmyktograyproc} ifelse load image } ifelse } ifelse } def /ci6knockoutimage { gsave 0 ci6curtransfer exec 1 ci6curtransfer exec eq { 0 ci6curtransfer exec 0.5 lt }{ 0 ci6curtransfer exec 1 ci6curtransfer exec gt } ifelse {{pop 0}}{{pop 1}} ifelse systemdict /settransfer get exec ci6compositeimage grestore } def /ci6drawimage { ci6testplate -1 eq { pop ci6compositeimage }{ dup type /arraytype eq { dup length plateindex gt {plateindex get}{pop false} ifelse }{ { true }{ dup 1 eq {plateindex 3 eq}{plateindex 3 le} ifelse } ifelse } ifelse { dup 1 eq { pop pop ci6image }{ dup 3 eq { ci6compositeimage }{ pop pop sourcearray 0 3 -1 roll put /ci6separatecmykproc load ci6image } ifelse } ifelse }{ ci6curoverprint { 7 {pop} repeat }{ ci6knockoutimage } ifelse } ifelse } ifelse } def /ci6proctintimage { /_ptispace exch store /_ptiname exch store /_pti1 exch store /_pti0 exch store /_ptiproc exch store /_pticomps _ptispace ci6numcomps store /_ptiscale _pti1 _pti0 sub store level2? { _ptiname length 0 gt version cvr 2012 ge and { [/Separation _ptiname _ptispace {_ptiproc}] setcolorspace [_pti0 _pti1] ci6makeimagedict ci6image }{ [/Indexed _ptispace 255 {255 div _ptiscale mul _pti0 add _ptiproc}] setcolorspace [0 255] ci6makeimagedict ci6image } ifelse }{ _pticomps 1 eq { { dup { 255 div _ptiscale mul _pti0 add _ptiproc 255 mul cvi put } ci6stringapply } ci6concatprocs ci6image }{ { dup length _pticomps mul dup _ptibuf length ne {/_ptibuf exch string store}{pop} ifelse _ptibuf { exch _pticomps mul exch 255 div _ptiscale mul _pti0 add _ptiproc _pticomps 2 add -2 roll _pticomps 1 sub -1 0 { 1 index add 2 index exch 5 -1 roll 255 mul cvi put } for pop pop } ci6stringapply } ci6concatprocs false _pticomps /ci6colorimage load null eq {7 {pop} repeat}{ci6colorimage} ifelse } ifelse } ifelse } def /ci6graytintimage { /_gtigray 5 -1 roll store {1 _gtigray sub mul 1 exch sub} 4 1 roll /DeviceGray ci6proctintimage } def /ci6cmyktintimage { /_cticmyk 5 -1 roll store {_cticmyk {1 index mul exch} forall pop} 4 1 roll /DeviceCMYK ci6proctintimage } def /ci6rgbtintimage { /_rtirgb 5 -1 roll store {_rtirgb {1 exch sub 1 index mul 1 exch sub exch} forall pop} 4 1 roll /DeviceRGB ci6proctintimage } def /ci6tintimage { ci6testplate -1 eq { ci6colorexpand 3 -1 roll 5 -1 roll {0}{0 exch} ifelse 4 2 roll dup /DeviceGray eq { pop ci6graytintimage }{ dup /DeviceRGB eq { pop ci6rgbtintimage }{ pop ci6cmyktintimage } ifelse } ifelse }{ dup ci6marksplate { plateindex 5 lt { ci6colortocmyk plateindex get dup 0 eq ci6curoverprint and { 7 {pop} repeat }{ 1 exch sub exch {1 0}{0 1} ifelse () ci6graytintimage } ifelse }{ pop exch {0}{0 exch} ifelse 0 3 1 roll () ci6graytintimage } ifelse }{ ci6curoverprint { 8 {pop} repeat }{ pop pop pop {pop 1} 0 1 () /DeviceGray ci6proctintimage } ifelse } ifelse } ifelse } def /XINullImage { } def /XIImageMask { XIImageWidth XIImageHeight false [XIImageWidth 0 0 XIImageHeight neg 0 0] /XIDataProc load imagemask } def /XIImageTint { XIImageWidth XIImageHeight XIBitsPerPixel [XIImageWidth 0 0 XIImageHeight neg 0 0] /XIDataProc load XIType 3 eq XIColorValues XIColorSpace ci6tintimage } def /XIImage { XIImageWidth XIImageHeight XIBitsPerPixel [XIImageWidth 0 0 XIImageHeight neg 0 0] /XIDataProc load false XIChannelCount XIPlateList ci6drawimage } def /XG { pop pop } def /XF { 13 {pop} repeat } def /Xh { Adobe_ColorImage_AI6_Vars begin gsave /XIType exch def /XIImageHeight exch def /XIImageWidth exch def /XIImageMatrix exch def 0 0 moveto XIImageMatrix concat XIImageWidth XIImageHeight scale /_lp /null ddef _fc /_lp /imagemask ddef end } def /XH { Adobe_ColorImage_AI6_Vars begin grestore end } def /XIEnable { Adobe_ColorImage_AI6_Vars /XIEnable 3 -1 roll put } def /XC { Adobe_ColorImage_AI6_Vars begin ci6colormake /XIColorSpace exch def /XIColorValues exch def end } def /XIPlates { Adobe_ColorImage_AI6_Vars begin /XIPlateList exch def end } def /XI { Adobe_ColorImage_AI6_Vars begin gsave /XIType exch def cvi dup 256 idiv /XICompression exch store 256 mod /XIEncoding exch store pop pop /XIChannelCount exch def /XIBitsPerPixel exch def /XIImageHeight exch def /XIImageWidth exch def pop pop pop pop /XIImageMatrix exch def XIBitsPerPixel 1 eq { XIImageWidth 8 div ceiling cvi }{ XIImageWidth XIChannelCount mul } ifelse /XIRowBytes exch def XIEnable { /XIBuffer3 XIImageWidth string def XICompression 0 eq { /XIBuffer1 XIRowBytes string def XIEncoding 0 eq { {currentfile XIBuffer1 readhexstring pop} }{ {currentfile XIBuffer1 readstring pop} } ifelse }{ /XIBuffer1 256 string def /XIBuffer2 XIRowBytes string def {currentfile XIBuffer1 readline pop (%) anchorsearch {pop} if} /ASCII85Decode filter /DCTDecode filter /XIFile exch def {XIFile XIBuffer2 readstring pop} } ifelse /XIDataProc exch def XIType 1 ne { 0 setgray } if XIType 1 eq { XIImageMask }{ XIType 2 eq XIType 3 eq or { XIImageTint }{ XIImage } ifelse } ifelse }{ XINullImage } ifelse /XIPlateList false def grestore end } def end %%EndProcSet %%BeginResource: procset Adobe_Illustrator_AI5 1.3 0 %%Title: (Adobe Illustrator (R) Version 8.0 Full Prolog) %%Version: 1.3 0 %%CreationDate: (3/7/1994) () %%Copyright: ((C) 1987-1998 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_Illustrator_AI5_vars 112 dict dup begin put /_?cmyk false def /_eo false def /_lp /none def /_pf { } def /_ps { } def /_psf { } def /_pss { } def /_pjsf { } def /_pjss { } def /_pola 0 def /_doClip 0 def /cf currentflat def /_lineorientation 0 def /_charorientation 0 def /_yokoorientation 0 def /_tm matrix def /_renderStart [ /e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0 ] def /_renderEnd [ null null null null /i1 /i1 /i1 /i1 ] def /_render -1 def /_shift [0 0] def /_ax 0 def /_ay 0 def /_cx 0 def /_cy 0 def /_leading [ 0 0 ] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fontSize 0 def /_fontAscent 0 def /_fontDescent 0 def /_fontHeight 0 def /_fontRotateAdjust 0 def /Ss 256 string def Ss 0 (fonts/) putinterval /_cnt 0 def /_scale [1 1] def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def /_hfname 100 string def /_hffound false def /Tx { } def /Tj { } def /CRender { } def /_AI3_savepage { } def /_gf null def /_cf 4 array def /_rgbf 3 array def /_if null def /_of false def /_fc { } def /_gs null def /_cs 4 array def /_rgbs 3 array def /_is null def /_os false def /_sc { } def /_pd 1 dict def /_ed 15 dict def /_pm matrix def /_fm null def /_fd null def /_fdd null def /_sm null def /_sd null def /_sdd null def /_i null def /_lobyte 0 def /_hibyte 0 def /_cproc null def /_cscript 0 def /_hvax 0 def /_hvay 0 def /_hvwb 0 def /_hvcx 0 def /_hvcy 0 def /_bitfont null def /_bitlobyte 0 def /_bithibyte 0 def /_bitkey null def /_bitdata null def /_bitindex 0 def /discardSave null def /buffer 256 string def /beginString null def /endString null def /endStringLength null def /layerCnt 1 def /layerCount 1 def /perCent (%) 0 get def /perCentSeen? false def /newBuff null def /newBuffButFirst null def /newBuffLast null def /clipForward? false def end userdict /Adobe_Illustrator_AI5 known not { userdict /Adobe_Illustrator_AI5 100 dict put } if userdict /Adobe_Illustrator_AI5 get begin /initialize { Adobe_Illustrator_AI5 dup begin Adobe_Illustrator_AI5_vars begin /_aicmykps where {pop /_?cmyk _aicmykps def}if discardDict { bind pop pop } forall dup /nc get begin { dup xcheck 1 index type /operatortype ne and { bind } if pop pop } forall end newpath } def /terminate { end end } def /_ null def /ddef { Adobe_Illustrator_AI5_vars 3 1 roll put } def /xput { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop { { pop } repeat } def /hswj { dup stringwidth 3 2 roll { _hvwb eq { exch _hvcx add exch _hvcy add } if exch _hvax add exch _hvay add } cforall } def /vswj { 0 0 3 -1 roll { dup 255 le _charorientation 1 eq and { dup cstring stringwidth 5 2 roll _hvwb eq { exch _hvcy sub exch _hvcx sub } if exch _hvay sub exch _hvax sub 4 -1 roll sub exch 3 -1 roll sub exch } { _hvwb eq { exch _hvcy sub exch _hvcx sub } if exch _hvay sub exch _hvax sub _fontHeight sub } ifelse } cforall } def /swj { 6 1 roll /_hvay exch ddef /_hvax exch ddef /_hvwb exch ddef /_hvcy exch ddef /_hvcx exch ddef _lineorientation 0 eq { hswj } { vswj } ifelse } def /sw { 0 0 0 6 3 roll swj } def /vjss { 4 1 roll { dup cstring dup length 1 eq _charorientation 1 eq and { -90 rotate currentpoint _fontRotateAdjust add moveto gsave false charpath currentpoint 5 index setmatrix stroke grestore _fontRotateAdjust sub moveto _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto 90 rotate } { currentpoint _fontHeight sub 5 index sub 3 index _sp eq { 9 index sub } if currentpoint exch 4 index stringwidth pop 2 div sub exch _fontAscent sub moveto gsave 2 index false charpath 6 index setmatrix stroke grestore moveto pop pop } ifelse } cforall 6 npop } def /hjss { 4 1 roll { dup cstring gsave false charpath currentpoint 5 index setmatrix stroke grestore moveto _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto } cforall 6 npop } def /jss { _lineorientation 0 eq { hjss } { vjss } ifelse } def /ss { 0 0 0 7 3 roll jss } def /vjsp { 4 1 roll { dup cstring dup length 1 eq _charorientation 1 eq and { -90 rotate currentpoint _fontRotateAdjust add moveto false charpath currentpoint _fontRotateAdjust sub moveto _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto 90 rotate } { currentpoint _fontHeight sub 5 index sub 3 index _sp eq { 9 index sub } if currentpoint exch 4 index stringwidth pop 2 div sub exch _fontAscent sub moveto 2 index false charpath moveto pop pop } ifelse } cforall 6 npop } def /hjsp { 4 1 roll { dup cstring false charpath _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto } cforall 6 npop } def /jsp { matrix currentmatrix _lineorientation 0 eq {hjsp} {vjsp} ifelse } def /sp { matrix currentmatrix 0 0 0 7 3 roll _lineorientation 0 eq {hjsp} {vjsp} ifelse } def /pl { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c { curveto } def /C /c load def /v { currentpoint 6 2 roll curveto } def /V /v load def /y { 2 copy curveto } def /Y /y load def /l { lineto } def /L /l load def /m { moveto } def } { /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def } ifelse /d { setdash } def /cf { } def /i { dup 0 eq { pop cf } if setflat } def /j { setlinejoin } def /J { setlinecap } def /M { setmiterlimit } def /w { setlinewidth } def /XR { 0 ne /_eo exch ddef } def /H { } def /h { closepath } def /N { _pola 0 eq { _doClip 1 eq { _eo {eoclip} {clip} ifelse /_doClip 0 ddef } if newpath } { /CRender { N } ddef } ifelse } def /n { N } def /F { _pola 0 eq { _doClip 1 eq { gsave _pf grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf } ifelse } { /CRender { F } ddef } ifelse } def /f { closepath F } def /S { _pola 0 eq { _doClip 1 eq { gsave _ps grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps } ifelse } { /CRender { S } ddef } ifelse } def /s { closepath S } def /B { _pola 0 eq { _doClip 1 eq gsave F grestore { gsave S grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { S } ifelse } { /CRender { B } ddef } ifelse } def /b { closepath B } def /W { /_doClip 1 ddef } def /* { count 0 ne { dup type /stringtype eq { pop } if } if newpath } def /u { } def /U { } def /q { _pola 0 eq { gsave } if } def /Q { _pola 0 eq { grestore } if } def /*u { _pola 1 add /_pola exch ddef } def /*U { _pola 1 sub /_pola exch ddef _pola 0 eq { CRender } if } def /D { pop } def /*w { } def /*W { } def /` { /_i save ddef clipForward? { nulldevice } if 6 1 roll 4 npop concat pop userdict begin /showpage { } def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash /setstrokeadjust where {pop false setstrokeadjust} if newpath 0 setgray false setoverprint } def /~ { end _i restore } def /_rgbtocmyk { 3 { 1 exch sub 3 1 roll } repeat 3 copy 1 4 1 roll 3 { 3 index 2 copy gt { exch } if pop 4 1 roll } repeat pop pop pop 4 1 roll 3 { 3 index sub 3 1 roll } repeat 4 -1 roll } def /setrgbfill { _rgbf astore pop /_fc { _lp /fill ne { _of setoverprint _rgbf aload pop setrgbcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /setrgbstroke { _rgbs astore pop /_sc { _lp /stroke ne { _os setoverprint _rgbs aload pop setrgbcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /O { 0 ne /_of exch ddef /_lp /none ddef } def /R { 0 ne /_os exch ddef /_lp /none ddef } def /g { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /G { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /K { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /Xa { _?cmyk { 3 npop k }{ setrgbfill 4 npop } ifelse } def /XA { _?cmyk { 3 npop K }{ setrgbstroke 4 npop } ifelse } def /Xs { /_gf exch ddef 5 npop /_fc { _lp /fill ne { _of setoverprint _gf setAIseparationgray /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /XS { /_gs exch ddef 5 npop /_sc { _lp /stroke ne { _os setoverprint _gs setAIseparationgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /Xx { exch /_gf exch ddef 0 eq { findcmykcustomcolor }{ _?cmyk {true}{/findrgbcustomcolor where{pop false}{true}ifelse}ifelse { 4 1 roll 3 npop findcmykcustomcolor }{ 8 -4 roll 4 npop findrgbcustomcolor } ifelse } ifelse /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /XX { exch /_gs exch ddef 0 eq { findcmykcustomcolor }{ _?cmyk {true}{/findrgbcustomcolor where{pop false}{true}ifelse}ifelse { 4 1 roll 3 npop findcmykcustomcolor }{ 8 -4 roll 4 npop findrgbcustomcolor } ifelse } ifelse /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /X { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /XK { 3 -1 roll pop 0 eq { 1 exch sub 3 {dup 3 1 roll mul 5 1 roll} repeat mul 4 1 roll K } { 1 exch sub 4 1 roll 3 {1 exch sub 3 index mul 1 exch sub 3 1 roll} repeat 4 -1 roll pop XA } ifelse } def /Xk { 3 -1 roll pop 0 eq { 1 exch sub 3 {dup 3 1 roll mul 5 1 roll} repeat mul 4 1 roll k } { 1 exch sub 4 1 roll 3 {1 exch sub 3 index mul 1 exch sub 3 1 roll} repeat 4 -1 roll pop Xa } ifelse } def /A { pop } def /annotatepage { userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse } def /XT { pop pop } def /Xt { pop } def /discard { save /discardSave exch store discardDict begin /endString exch store gt38? { 2 add } if load stopped pop end discardSave restore } bind def userdict /discardDict 7 dict dup begin put /pre38Initialize { /endStringLength endString length store /newBuff buffer 0 endStringLength getinterval store /newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store /newBuffLast newBuff endStringLength 1 sub 1 getinterval store } def /shiftBuffer { newBuff 0 newBuffButFirst putinterval newBuffLast 0 currentfile read not { stop } if put } def 0 { pre38Initialize mark currentfile newBuff readstring exch pop { { newBuff endString eq { cleartomark stop } if shiftBuffer } loop } { stop } ifelse } def 1 { pre38Initialize /beginString exch store mark currentfile newBuff readstring exch pop { { newBuff beginString eq { /layerCount dup load 1 add store } { newBuff endString eq { /layerCount dup load 1 sub store layerCount 0 eq { cleartomark stop } if } if } ifelse shiftBuffer } loop } if } def 2 { mark { currentfile buffer {readline} stopped { % assume error was due to overfilling the buffer }{ not { stop } if endString eq { cleartomark stop } if }ifelse } loop } def 3 { /beginString exch store /layerCnt 1 store mark { currentfile buffer {readline} stopped { % assume error was due to overfilling the buffer }{ not { stop } if dup beginString eq { pop /layerCnt dup load 1 add store } { endString eq { layerCnt 1 eq { cleartomark stop } { /layerCnt dup load 1 sub store } ifelse } if } ifelse }ifelse } loop } def end userdict /clipRenderOff 15 dict dup begin put { /n /N /s /S /f /F /b /B } { { _doClip 1 eq { /_doClip 0 ddef _eo {eoclip} {clip} ifelse } if newpath } def } forall /Tr /pop load def /Bb {} def /BB /pop load def /Bg {12 npop} def /Bm {6 npop} def /Bc /Bm load def /Bh {4 npop} def end /Lb { 6 npop 7 2 roll 5 npop 0 eq { 0 eq { (%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard } { /clipForward? true def /Tx /pop load def /Tj /pop load def currentdict end clipRenderOff begin begin } ifelse } { 0 eq { save /discardSave exch store } if } ifelse } bind def /LB { discardSave dup null ne { restore } { pop clipForward? { currentdict end end begin /clipForward? false ddef } if } ifelse } bind def /Pb { pop pop 0 (%AI5_EndPalette) discard } bind def /Np { 0 (%AI5_End_NonPrinting--) discard } bind def /Ln /pop load def /Ap /pop load def /Ar { 72 exch div 0 dtransform dup mul exch dup mul add sqrt dup 1 lt { pop 1 } if setflat } def /Mb { q } def /Md { } def /MB { Q } def /nc 4 dict def nc begin /setgray { pop } bind def /setcmykcolor { 4 npop } bind def /setrgbcolor { 3 npop } bind def /setcustomcolor { 2 npop } bind def currentdict readonly pop end /XP { 4 npop } bind def /XD { pop } bind def end setpacking %%EndResource %%BeginResource: procset Adobe_cshow 2.0 8 %%Title: (Writing System Operators) %%Version: 2.0 8 %%CreationDate: (1/23/89) () %%Copyright: ((C) 1992-1996 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_cshow 14 dict dup begin put /initialize { Adobe_cshow begin Adobe_cshow { dup xcheck { bind } if pop pop } forall end Adobe_cshow begin } def /terminate { currentdict Adobe_cshow eq { end } if } def /cforall { /_lobyte 0 ddef /_hibyte 0 ddef /_cproc exch ddef /_cscript currentfont /FontScript known { currentfont /FontScript get } { -1 } ifelse ddef { /_lobyte exch ddef _hibyte 0 eq _cscript 1 eq _lobyte 129 ge _lobyte 159 le and _lobyte 224 ge _lobyte 252 le and or and _cscript 2 eq _lobyte 161 ge _lobyte 254 le and and _cscript 3 eq _lobyte 161 ge _lobyte 254 le and and _cscript 25 eq _lobyte 161 ge _lobyte 254 le and and _cscript -1 eq or or or or and { /_hibyte _lobyte ddef } { _hibyte 256 mul _lobyte add _cproc /_hibyte 0 ddef } ifelse } forall } def /cstring { dup 256 lt { (s) dup 0 4 3 roll put } { dup 256 idiv exch 256 mod (hl) dup dup 0 6 5 roll put 1 4 3 roll put } ifelse } def /clength { 0 exch { 256 lt { 1 } { 2 } ifelse add } cforall } def /hawidthshow { { dup cstring show _hvax _hvay rmoveto _hvwb eq { _hvcx _hvcy rmoveto } if } cforall } def /vawidthshow { { dup 255 le _charorientation 1 eq and { -90 rotate 0 _fontRotateAdjust rmoveto cstring _hvcx _hvcy _hvwb _hvax _hvay 6 -1 roll awidthshow 0 _fontRotateAdjust neg rmoveto 90 rotate } { currentpoint _fontHeight sub exch _hvay sub exch _hvax sub 2 index _hvwb eq { exch _hvcy sub exch _hvcx sub } if 3 2 roll cstring dup stringwidth pop 2 div neg _fontAscent neg rmoveto show moveto } ifelse } cforall } def /hvawidthshow { 6 1 roll /_hvay exch ddef /_hvax exch ddef /_hvwb exch ddef /_hvcy exch ddef /_hvcx exch ddef _lineorientation 0 eq { hawidthshow } { vawidthshow } ifelse } def /hvwidthshow { 0 0 3 -1 roll hvawidthshow } def /hvashow { 0 0 0 6 -3 roll hvawidthshow } def /hvshow { 0 0 0 0 0 6 -1 roll hvawidthshow } def currentdict readonly pop end setpacking %%EndResource %%BeginResource: procset Adobe_shading_AI8 1.0 0 %%Title: (Adobe Illustrator 8 Shading Procset) %%Version: 1.0 0 %%CreationDate: (12/17/97) () %%Copyright: ((C) 1987-1997 Adobe Systems Incorporated All Rights Reserved) userdict /defaultpacking currentpacking put true setpacking userdict /Adobe_shading_AI8 10 dict dup begin put /initialize { Adobe_shading_AI8 begin Adobe_shading_AI8 bdprocs Mesh /initialize get exec } def /terminate { currentdict Adobe_shading_AI8 eq { end } if } def /bdprocs { { dup xcheck 1 index type /arraytype eq and { bind } if pop pop } forall } def /X! {pop} def /X# {pop pop} def /Mesh 40 dict def Mesh begin /initialize { Mesh bdprocs Mesh begin /emulate? /AI8MeshEmulation where { pop AI8MeshEmulation }{ systemdict /shfill known not } ifelse def end } def /bd { shadingdict begin } def /paint { emulate? { end }{ /_lp /none ddef _fc /_lp /none ddef /AIColorSpace AIColorSpace tocolorspace store /ColorSpace AIColorSpace topsspace store version_ge_3010.106 not systemdict /setsmoothness known and { 0.0001 setsmoothness } if composite? { /DataSource getdatasrc def Matrix concat currentdict end shfill }{ AIColorSpace makesmarks AIPlateList markingplate and not isoverprint and { end }{ /ColorSpace /DeviceGray store /Decode [0 1 0 1 0 1] store /DataSource getplatesrc def Matrix concat currentdict end shfill } ifelse } ifelse } ifelse } def /shadingdict 12 dict def shadingdict begin /ShadingType 6 def /BitsPerCoordinate 16 def /BitsPerComponent 8 def /BitsPerFlag 8 def end /datafile null def /databuf 256 string def /dataptr 0 def /srcspace null def /srcchannels 0 def /dstchannels 0 def /dstplate 0 def /srctodstcolor null def /getplatesrc { /srcspace AIColorSpace store /srcchannels AIColorSpace getnchannels store /dstchannels 1 store /dstplate getplateindex store /srctodstcolor srcspace makesmarks { dstplate 4 eq { {1 exch sub} }{ {srcspace tocmyk 3 dstplate sub index 1 exch sub 5 1 roll 4 {pop} repeat} } ifelse }{ {srcchannels {pop} repeat 1} } ifelse store /datafile getdatasrc store /rdpatch168 load DataLength () /SubFileDecode filter } def /getdatasrc { /rdcmntline load /ASCII85Decode filter } def /rdpatch168 { /dataptr 0 store 49 rdcount 4 { dup {pop srcchannels getint8} if dup {pop srctodstcolor dstchannels putint8 true} if } repeat {databuf 0 dataptr getinterval}{()} ifelse } def /rdpatch3216 { /dataptr 0 store 97 rdcount 4 { dup {pop srcchannels getint16} if dup {pop srctodstcolor dstchannels putint16 true} if } repeat {databuf 0 dataptr getinterval}{()} ifelse } def /rdcount { dup 0 gt { datafile databuf dataptr 4 -1 roll getinterval readstring exch length dataptr add /dataptr exch store }{ true } ifelse } def /getint8 { mark true 3 -1 roll { dup {pop datafile read} if dup {pop 255 div true} if } repeat { counttomark 1 add -1 roll pop true }{ cleartomark false } ifelse } def /putint8 { dup dataptr add /dataptr exch store dataptr exch { 1 sub exch 255 mul cvi databuf 2 index 3 -1 roll put } repeat pop } def /getint16 { mark true 3 -1 roll { dup {pop datafile read} if dup {pop 256 mul datafile read} if dup {pop add 65535 div true} if } repeat { counttomark 1 add -1 roll pop true }{ cleartomark false } ifelse } def /putint16 { dup 2 mul dataptr add /dataptr exch store dataptr exch { 2 sub exch 65535 mul cvi dup 256 idiv databuf 3 index 3 -1 roll put 256 mod databuf 2 index 1 add 3 -1 roll put } repeat pop } def /srcbuf 256 string def /rdcmntline { currentfile srcbuf readline pop (%) anchorsearch {pop} if } def /getplateindex { 0 [cyan? magenta? yellow? black? customColor?] {{exit} if 1 add} forall } def /aicsarray 4 array def /aicsaltvals 4 array def /aicsaltcolr aicsaltvals def /tocolorspace { dup type /arraytype eq { mark exch aload pop aicsarray 0 3 -1 roll put aicsarray 1 3 -1 roll put dup aicsarray 2 3 -1 roll put gettintxform aicsarray 3 3 -1 roll put counttomark aicsaltvals 0 3 -1 roll getinterval /aicsaltcolr exch store aicsaltcolr astore pop pop aicsarray } if } def /subtintxform {aicsaltcolr {1 index mul exch} forall pop} def /addtintxform {aicsaltcolr {1 sub 1 index mul 1 add exch} forall pop} def /gettintxform { /DeviceRGB eq {/addtintxform}{/subtintxform} ifelse load } def /getnchannels { dup type /arraytype eq {0 get} if colorspacedict exch get begin Channels end } def /makesmarks { composite? { pop true }{ dup dup type /arraytype eq {0 get} if colorspacedict exch get begin MarksPlate end } ifelse } def /markingplate { composite? { pop true }{ dup type /arraytype eq { dup length getplateindex gt {getplateindex get}{pop false} ifelse } if } ifelse } def /tocmyk { dup dup type /arraytype eq {0 get} if colorspacedict exch get begin ToCMYK end } def /topsspace { dup dup type /arraytype eq {0 get} if colorspacedict exch get begin ToPSSpace end } def /colorspacedict 5 dict dup begin /DeviceGray 4 dict dup begin /Channels 1 def /MarksPlate {pop black?} def /ToCMYK {pop 1 exch sub 0 0 0 4 -1 roll} def /ToPSSpace {} def end def /DeviceRGB 4 dict dup begin /Channels 3 def /MarksPlate {pop isCMYKSep?} def /ToCMYK {pop _rgbtocmyk} def /ToPSSpace {} def end def /DeviceCMYK 4 dict dup begin /Channels 4 def /MarksPlate {pop isCMYKSep?} def /ToCMYK {pop} def /ToPSSpace {} def end def /Separation 4 dict dup begin /Channels 1 def /MarksPlate { /findcmykcustomcolor where { pop dup 1 exch ToCMYK 5 -1 roll 1 get findcmykcustomcolor 1 setcustomcolor systemdict /currentgray get exec 1 ne }{ pop false } ifelse } def /ToCMYK { dup 2 get mark exch 4 2 roll 3 get exec counttomark -1 roll tocmyk 5 -1 roll pop } def /ToPSSpace {} def end def /Process 4 dict dup begin /Channels 1 def /MarksPlate { isCMYKSep? { 1 exch ToCMYK 4 array astore getplateindex get 0 ne }{ pop false } ifelse } def /ToCMYK { dup 2 get mark exch 4 2 roll 3 get exec counttomark -1 roll tocmyk 5 -1 roll pop } def /ToPSSpace { 4 array copy dup 0 /Separation put } def end def end def /isoverprint { /currentoverprint where {pop currentoverprint}{_of} ifelse } def /version_ge_3010.106 { version {cvr} stopped { pop false }{ 3010.106 ge } ifelse } def end end defaultpacking setpacking %%EndResource %%EndProlog %%BeginSetup %%IncludeFont: Symbol userdict /_useSmoothShade false put userdict /_aicmykps true put userdict /_forceToCMYK true put Adobe_level2_AI5 /initialize get exec Adobe_cshow /initialize get exec Adobe_Illustrator_AI5_vars Adobe_Illustrator_AI5 Adobe_typography_AI5 /initialize get exec Adobe_ColorImage_AI6 /initialize get exec Adobe_shading_AI8 /initialize get exec Adobe_Illustrator_AI5 /initialize get exec [ 39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis /Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute /egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde /oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex /udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash /.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef /.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash /questiondown/exclamdown/logicalnot/.notdef/florin/.notdef/.notdef /guillemotleft/guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide /.notdef/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright /fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand /Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex /Idieresis/Igrave/Oacute/Ocircumflex/.notdef/Ograve/Uacute/Ucircumflex /Ugrave/dotlessi/circumflex/tilde/macron/breve/dotaccent/ring/cedilla /hungarumlaut/ogonek/caron TE %AI55J_Tsume: None %AI3_BeginEncoding: _Symbol Symbol [/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /exclam /universal /numbersign /existential /percent /ampersand /suchthat /parenleft /parenright /asteriskmath /plus /comma /minus /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /congruent /Alpha /Beta /Chi /Delta /Epsilon /Phi /Gamma /Eta /Iota /theta1 /Kappa /Lambda /Mu /Nu /Omicron /Pi /Theta /Rho /Sigma /Tau /Upsilon /sigma1 /Omega /Xi /Psi /Zeta /bracketleft /therefore /bracketright /perpendicular /underscore /radicalex /alpha /beta /chi /delta /epsilon /phi /gamma /eta /iota /phi1 /kappa /lambda /mu /nu /omicron /pi /theta /rho 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ndData %%EndComments %%BeginProlog %%BeginResource: procset Adobe_level2_AI5 1.2 0 %%Title: (Adobe Illustrator (R) Version 5.0 Level 2 Emulation) %%Version: 1.2 0 %%CreationDate: (04/10/93) () %%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved) userdict /Adobe_level2_AI5 26 dict dup begin put /packedarray where not { userdict begin /packedarray { array astore readonly } bind def /setpacking /pop load def /currentpacking false def end 0 } if pop userdict /defaultpacking currentpacking put true setpacking /initialize { Adobe_level2_AI5 begin } bind def /terminate { currentdict Adobe_level2_AI5 eq { end } if } bind def mark /setcustomcolor where not { /findcmykcustomcolor { (AI8_CMYK_CustomColor) 6 packedarray } bind def /findrgbcustomcolor { (AI8_RGB_CustomColor) 5 packedarray } bind def /setcustomcolor { exch aload pop dup (AI8_CMYK_CustomColor) eq { pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } { dup (AI8_RGB_CustomColor) eq { pop pop 3 { 1 exch sub 3 index mul 1 exch sub 3 1 roll } repeat 4 -1 roll pop setrgbcolor } { pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } ifelse } ifelse } def } if /setAIseparationgray { false setoverprint 0 setgray /setseparationgray where{ pop setseparationgray }{ /setcolorspace where{ pop [/Separation (All) /DeviceCMYK {dup dup dup}] setcolorspace 1 exch sub setcolor }{ setgray }ifelse }ifelse } def /gt38? mark {version cvr cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put userdict /level2? systemdict /languagelevel known dup { pop systemdict /languagelevel get 2 ge } if put /level2ScreenFreq { begin 60 HalftoneType 1 eq { pop Frequency } if HalftoneType 2 eq { pop GrayFrequency } if HalftoneType 5 eq { pop Default level2ScreenFreq } if end } bind def userdict /currentScreenFreq level2? {currenthalftone level2ScreenFreq} {currentscreen pop pop} ifelse put level2? not { /setcmykcolor where not { /setcmykcolor { exch .11 mul add exch .59 mul add exch .3 mul add 1 exch sub setgray } def } if /currentcmykcolor where not { /currentcmykcolor { 0 0 0 1 currentgray sub } def } if /setoverprint where not { /setoverprint /pop load def } if /selectfont where not { /selectfont { exch findfont exch dup type /arraytype eq { makefont } { scalefont } ifelse setfont } bind def } if /cshow where not { /cshow { [ 0 0 5 -1 roll aload pop ] cvx bind forall } bind def } if } if cleartomark /anyColor? { add add add 0 ne } bind def /testColor { gsave setcmykcolor currentcmykcolor grestore } bind def /testCMYKColorThrough { testColor anyColor? } bind def userdict /composite? 1 0 0 0 testCMYKColorThrough 0 1 0 0 testCMYKColorThrough 0 0 1 0 testCMYKColorThrough 0 0 0 1 testCMYKColorThrough and and and put composite? not { userdict begin gsave /cyan? 1 0 0 0 testCMYKColorThrough def /magenta? 0 1 0 0 testCMYKColorThrough def /yellow? 0 0 1 0 testCMYKColorThrough def /black? 0 0 0 1 testCMYKColorThrough def grestore /isCMYKSep? cyan? magenta? yellow? black? or or or def /customColor? isCMYKSep? not def end } if end defaultpacking setpacking %%EndResource %%BeginResource: procset Adobe_typography_AI5 1.0 1 %%Title: (Typography Operators) %%Version: 1.0 1 %%CreationDate:(6/10/1996) () %%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_typography_AI5 68 dict dup begin put /initialize { begin begin Adobe_typography_AI5 begin Adobe_typography_AI5 { dup xcheck { bind } if pop pop } forall end end end Adobe_typography_AI5 begin } def /terminate { currentdict Adobe_typography_AI5 eq { end } if } def /modifyEncoding { /_tempEncode exch ddef /_pntr 0 ddef { counttomark -1 roll dup type dup /marktype eq { pop pop exit } { /nametype eq { _tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll put } { /_pntr exch ddef } ifelse } ifelse } loop _tempEncode } def /havefont { systemdict /languagelevel known { /Font resourcestatus dup { exch pop exch pop } if } { systemdict /FontDirectory get 1 index known { pop true } { systemdict /fileposition known { dup length 6 add exch Ss 6 250 getinterval cvs pop Ss exch 0 exch getinterval status { pop pop pop pop true } { false } ifelse } { pop false } ifelse } ifelse } ifelse } def /TE { StandardEncoding 256 array copy modifyEncoding /_nativeEncoding exch def } def /subststring { exch 2 index exch search { exch pop exch dup () eq { pop exch concatstring } { 3 -1 roll exch concatstring concatstring } ifelse exch pop true } { pop pop false } ifelse } def /concatstring { 1 index length 1 index length 1 index add string dup 0 5 index putinterval dup 2 index 4 index putinterval 4 1 roll pop pop pop } def % /TZ { dup type /arraytype eq { /_wv exch def } { /_wv 0 def } ifelse /_useNativeEncoding exch def 2 index havefont { 3 index 255 string cvs dup (_Symbol_) eq { pop 2 index findfont } { 1 index 0 eq { dup length 1 sub 1 exch getinterval cvn findfont } { pop 2 index findfont } ifelse } ifelse } { dup 1 eq { 2 index 64 string cvs dup (-90pv-RKSJ-) (-83pv-RKSJ-) subststring { exch pop dup havefont { findfont false } { pop true } ifelse } { pop dup (-90ms-RKSJ-) (-Ext-RKSJ-) subststring { exch pop dup havefont { findfont false } { pop true } ifelse } { pop pop true } ifelse } ifelse { 1 index 1 eq { /Ryumin-Light-Ext-RKSJ-V havefont {/Ryumin-Light-Ext-RKSJ-V} {/Courier} ifelse } { /Ryumin-Light-83pv-RKSJ-H havefont {/Ryumin-Light-83pv-RKSJ-H} {/Courier} ifelse } ifelse findfont [1 0 0.5 1 0 0] makefont } if } { /Courier findfont } ifelse } ifelse _wv type /arraytype eq { _wv makeblendedfont } if dup length 10 add dict begin mark exch { 1 index /FID ne { def } if cleartomark mark } forall pop /FontScript exch def /FontDirection exch def /FontRequest exch def /FontName exch def counttomark 0 eq { 1 _useNativeEncoding eq { /Encoding _nativeEncoding def } if cleartomark } { /Encoding load 256 array copy modifyEncoding /Encoding exch def } ifelse FontName currentdict end definefont pop } def /tr { _ax _ay 3 2 roll } def /trj { _cx _cy _sp _ax _ay 6 5 roll } def /a0 { /Tx { dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss } ddef /Tj { dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss } ddef } def /a1 { W B } def /e0 { /Tx { tr _psf } ddef /Tj { trj _pjsf } ddef } def /e1 { W F } def /i0 { /Tx { tr sp } ddef /Tj { trj jsp } ddef } def /i1 { W N } def /o0 { /Tx { tr sw rmoveto } ddef /Tj { trj swj rmoveto } ddef } def /r0 { /Tx { tr _ctm _pss } ddef /Tj { trj _ctm _pjss } ddef } def /r1 { W S } def /To { pop _ctm currentmatrix pop } def /TO { iTe _ctm setmatrix newpath } def /Tp { pop _tm astore pop _ctm setmatrix _tDict begin /W { } def /h { } def } def /TP { end iTm 0 0 moveto } def /Tr { _render 3 le { currentpoint newpath moveto } if dup 8 eq { pop 0 } { dup 9 eq { pop 1 } if } ifelse dup /_render exch ddef _renderStart exch get load exec } def /iTm { _ctm setmatrix _tm concat _shift aload pop _lineorientation 1 eq { exch } if translate _scale aload pop _lineorientation 1 eq _yokoorientation 1 eq or { exch } if scale } def /Tm { _tm astore pop iTm 0 0 moveto } def /Td { _mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto } def /iTe { _render -1 eq { } { _renderEnd _render get dup null ne { load exec } { pop } ifelse } ifelse /_render -1 ddef } def /Ta { pop } def /Tf { 1 index type /nametype eq { dup 0.75 mul 1 index 0.25 mul neg } if /_fontDescent exch ddef /_fontAscent exch ddef /_fontSize exch ddef /_fontRotateAdjust _fontAscent _fontDescent add 2 div neg ddef /_fontHeight _fontSize ddef findfont _fontSize scalefont setfont } def /Tl { pop neg 0 exch _leading astore pop } def /Tt { pop } def /TW { 3 npop } def /Tw { /_cx exch ddef } def /TC { 3 npop } def /Tc { /_ax exch ddef } def /Ts { 0 exch _shift astore pop currentpoint iTm moveto } def /Ti { 3 npop } def /Tz { count 1 eq { 100 } if 100 div exch 100 div exch _scale astore pop iTm } def /TA { pop } def /Tq { pop } def /Tg { pop } def /TG { pop } def /Tv { /_lineorientation exch ddef } def /TV { /_charorientation exch ddef } def /Ty { dup /_yokoorientation exch ddef 1 sub neg Tv } def /TY { pop } def /T~ { Tx } def /Th { pop pop pop pop pop } def /TX { pop } def /Tk { _fontSize mul 1000 div _lineorientation 0 eq { neg 0 } { 0 exch } ifelse rmoveto pop } def /TK { 2 npop } def /T* { _leading aload pop _lineorientation 0 ne { exch } if Td } def /T*- { _leading aload pop _lineorientation 0 ne { exch } if exch neg exch neg Td } def /T- { _ax neg 0 rmoveto _lineorientation 1 eq _charorientation 0 eq and { 1 TV _hyphen Tx 0 TV } { _hyphen Tx } ifelse } def /T+ { } def /TR { _ctm currentmatrix pop _tm astore pop iTm 0 0 moveto } def /TS { currentfont 3 1 roll /_Symbol_ findfont _fontSize scalefont setfont 0 eq { Tx } { Tj } ifelse setfont } def /Xb { pop pop } def /Tb /Xb load def /Xe { pop pop pop pop } def /Te /Xe load def /XB { } def /TB /XB load def currentdict readonly pop end setpacking % /X^ { currentfont 5 1 roll dup havefont { findfont _fontSize scalefont setfont } { pop exch } ifelse 2 index 0 eq { Tx } { Tj } ifelse pop pop setfont } def /T^ /X^ load def %%EndResource %%BeginProcSet: Adobe_ColorImage_AI6 1.3 0 userdict /Adobe_ColorImage_AI6 known not { userdict /Adobe_ColorImage_AI6 53 dict put } if userdict /Adobe_ColorImage_AI6 get begin /initialize { Adobe_ColorImage_AI6 begin Adobe_ColorImage_AI6 { dup type /arraytype eq { dup xcheck { bind } if } if pop pop } forall } def /terminate { end } def currentdict /Adobe_ColorImage_AI6_Vars known not { /Adobe_ColorImage_AI6_Vars 41 dict def } if Adobe_ColorImage_AI6_Vars begin /plateindex -1 def /_newproc null def /_proc1 null def /_proc2 null def /sourcearray 4 array def /_ptispace null def /_ptiname null def /_pti0 0 def /_pti1 0 def /_ptiproc null def /_ptiscale 0 def /_pticomps 0 def /_ptibuf 0 string def /_gtigray 0 def /_cticmyk null def /_rtirgb null def /XIEnable true def /XIType 0 def /XIEncoding 0 def /XICompression 0 def /XIChannelCount 0 def /XIBitsPerPixel 0 def /XIImageHeight 0 def /XIImageWidth 0 def /XIImageMatrix null def /XIRowBytes 0 def /XIFile null def /XIBuffer1 null def /XIBuffer2 null def /XIBuffer3 null def /XIDataProc null def /XIColorSpace /DeviceGray def /XIColorValues 0 def /XIPlateList false def end /ci6colorimage /colorimage where {/colorimage get}{null} ifelse def /ci6image systemdict /image get def /ci6curtransfer systemdict /currenttransfer get def /ci6curoverprint /currentoverprint where {/currentoverprint get}{{_of}} ifelse def /ci6foureq { 4 index ne { pop pop pop false }{ 4 index ne { pop pop false }{ 4 index ne { pop false }{ 4 index eq } ifelse } ifelse } ifelse } def /ci6testplate { Adobe_ColorImage_AI6_Vars begin /plateindex -1 def /setcmykcolor where { pop gsave 1 0 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 1 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 1 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 0 1 setcmykcolor systemdict /currentgray get exec 1 exch sub grestore 1 0 0 0 ci6foureq { /plateindex 0 def }{ 0 1 0 0 ci6foureq { /plateindex 1 def }{ 0 0 1 0 ci6foureq { /plateindex 2 def }{ 0 0 0 1 ci6foureq { /plateindex 3 def }{ 0 0 0 0 ci6foureq { /plateindex 5 def } if } ifelse } ifelse } ifelse } ifelse pop pop pop pop } if plateindex end } def /ci6concatprocs { /packedarray where { pop dup type /packedarraytype eq 2 index type /packedarraytype eq or }{ false } ifelse { /_proc2 exch cvlit def /_proc1 exch cvlit def _proc1 aload pop _proc2 aload pop _proc1 length _proc2 length add packedarray cvx }{ /_proc2 exch cvlit def /_proc1 exch cvlit def /_newproc _proc1 length _proc2 length add array def _newproc 0 _proc1 putinterval _newproc _proc1 length _proc2 putinterval _newproc cvx } ifelse } def /ci6istint { type /arraytype eq } def /ci6isspot { dup type /arraytype eq { dup length 1 sub get /Separation eq }{ pop false } ifelse } def /ci6spotname { dup ci6isspot {dup length 2 sub get}{pop ()} ifelse } def /ci6altspace { aload pop pop pop ci6colormake } def /ci6numcomps { dup /DeviceGray eq { pop 1 }{ dup /DeviceRGB eq { pop 3 }{ /DeviceCMYK eq { 4 }{ 1 } ifelse } ifelse } ifelse } def /ci6marksplate { dup /DeviceGray eq { pop plateindex 3 eq }{ dup /DeviceRGB eq { pop plateindex 5 ne }{ dup /DeviceCMYK eq { pop plateindex 5 ne }{ dup ci6isspot { /findcmykcustomcolor where { pop dup length 2 sub get 0.1 0.1 0.1 0.1 5 -1 roll findcmykcustomcolor 1 setcustomcolor systemdict /currentgray get exec 1 ne }{ pop plateindex 5 ne } ifelse }{ pop plateindex 5 ne } ifelse } ifelse } ifelse } ifelse } def /ci6colormake { dup ci6numcomps exch 1 index 2 add 1 roll dup 1 eq {pop}{array astore} ifelse exch } def /ci6colorexpand { dup ci6spotname exch dup ci6istint { ci6altspace exch 4 1 roll }{ 1 3 1 roll } ifelse } def /ci6colortint { dup /DeviceGray eq { 3 1 roll 1 exch sub mul 1 exch sub exch }{ dup /DeviceRGB eq { 3 1 roll {1 exch sub 1 index mul 1 exch sub exch} forall pop 3 array astore exch }{ dup /DeviceCMYK eq { 3 1 roll {1 index mul exch} forall pop 4 array astore exch }{ 3 1 roll mul exch } ifelse } ifelse } ifelse } def /ci6colortocmyk { dup /DeviceGray eq { pop 1 exch sub 0 0 0 4 -1 roll 4 array astore }{ dup /DeviceRGB eq { pop aload pop _rgbtocmyk 4 array astore }{ dup /DeviceCMYK eq { pop }{ ci6altspace ci6colortint ci6colortocmyk } ifelse } ifelse } ifelse } def /ci6makeimagedict { 7 dict begin /ImageType 1 def /Decode exch def /DataSource exch def /ImageMatrix exch def /BitsPerComponent exch def /Height exch def /Width exch def currentdict end } def /ci6stringinvert { 0 1 2 index length 1 sub { dup 2 index exch get 255 exch sub 2 index 3 1 roll put } for } def /ci6stringknockout { 0 1 2 index length 1 sub { 255 2 index 3 1 roll put } for } def /ci6stringapply { 0 1 4 index length 1 sub { dup 4 index exch get 3 index 3 1 roll 3 index exec } for pop exch pop } def /ci6walkrgbstring { 0 3 index dup length 1 sub 0 3 3 -1 roll { 3 getinterval {} forall 5 index exec 3 index } for 5 {pop} repeat } def /ci6walkcmykstring { 0 3 index dup length 1 sub 0 4 3 -1 roll { 4 getinterval {} forall 6 index exec 3 index } for 5 { pop } repeat } def /ci6putrgbtograystr { .11 mul exch .59 mul add exch .3 mul add cvi 3 copy put pop 1 add } def /ci6putcmyktograystr { exch .11 mul add exch .59 mul add exch .3 mul add dup 255 gt { pop 255 } if 255 exch sub cvi 3 copy put pop 1 add } def /ci6rgbtograyproc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec XIBuffer3 dup 3 1 roll /ci6putrgbtograystr load exch ci6walkrgbstring end } def /ci6cmyktograyproc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec XIBuffer3 dup 3 1 roll /ci6putcmyktograystr load exch ci6walkcmykstring end } def /ci6separatecmykproc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec XIBuffer3 0 2 index plateindex 4 2 index length 1 sub { get 255 exch sub 3 copy put pop 1 add 2 index } for pop pop exch pop end } def /ci6compositeimage { dup 1 eq { pop pop image }{ /ci6colorimage load null ne { ci6colorimage }{ 3 1 roll pop sourcearray 0 3 -1 roll put 3 eq {/ci6rgbtograyproc}{/ci6cmyktograyproc} ifelse load image } ifelse } ifelse } def /ci6knockoutimage { gsave 0 ci6curtransfer exec 1 ci6curtransfer exec eq { 0 ci6curtransfer exec 0.5 lt }{ 0 ci6curtransfer exec 1 ci6curtransfer exec gt } ifelse {{pop 0}}{{pop 1}} ifelse systemdict /settransfer get exec ci6compositeimage grestore } def /ci6drawimage { ci6testplate -1 eq { pop ci6compositeimage }{ dup type /arraytype eq { dup length plateindex gt {plateindex get}{pop false} ifelse }{ { true }{ dup 1 eq {plateindex 3 eq}{plateindex 3 le} ifelse } ifelse } ifelse { dup 1 eq { pop pop ci6image }{ dup 3 eq { ci6compositeimage }{ pop pop sourcearray 0 3 -1 roll put /ci6separatecmykproc load ci6image } ifelse } ifelse }{ ci6curoverprint { 7 {pop} repeat }{ ci6knockoutimage } ifelse } ifelse } ifelse } def /ci6proctintimage { /_ptispace exch store /_ptiname exch store /_pti1 exch store /_pti0 exch store /_ptiproc exch store /_pticomps _ptispace ci6numcomps store /_ptiscale _pti1 _pti0 sub store level2? { _ptiname length 0 gt version cvr 2012 ge and { [/Separation _ptiname _ptispace {_ptiproc}] setcolorspace [_pti0 _pti1] ci6makeimagedict ci6image }{ [/Indexed _ptispace 255 {255 div _ptiscale mul _pti0 add _ptiproc}] setcolorspace [0 255] ci6makeimagedict ci6image } ifelse }{ _pticomps 1 eq { { dup { 255 div _ptiscale mul _pti0 add _ptiproc 255 mul cvi put } ci6stringapply } ci6concatprocs ci6image }{ { dup length _pticomps mul dup _ptibuf length ne {/_ptibuf exch string store}{pop} ifelse _ptibuf { exch _pticomps mul exch 255 div _ptiscale mul _pti0 add _ptiproc _pticomps 2 add -2 roll _pticomps 1 sub -1 0 { 1 index add 2 index exch 5 -1 roll 255 mul cvi put } for pop pop } ci6stringapply } ci6concatprocs false _pticomps /ci6colorimage load null eq {7 {pop} repeat}{ci6colorimage} ifelse } ifelse } ifelse } def /ci6graytintimage { /_gtigray 5 -1 roll store {1 _gtigray sub mul 1 exch sub} 4 1 roll /DeviceGray ci6proctintimage } def /ci6cmyktintimage { /_cticmyk 5 -1 roll store {_cticmyk {1 index mul exch} forall pop} 4 1 roll /DeviceCMYK ci6proctintimage } def /ci6rgbtintimage { /_rtirgb 5 -1 roll store {_rtirgb {1 exch sub 1 index mul 1 exch sub exch} forall pop} 4 1 roll /DeviceRGB ci6proctintimage } def /ci6tintimage { ci6testplate -1 eq { ci6colorexpand 3 -1 roll 5 -1 roll {0}{0 exch} ifelse 4 2 roll dup /DeviceGray eq { pop ci6graytintimage }{ dup /DeviceRGB eq { pop ci6rgbtintimage }{ pop ci6cmyktintimage } ifelse } ifelse }{ dup ci6marksplate { plateindex 5 lt { ci6colortocmyk plateindex get dup 0 eq ci6curoverprint and { 7 {pop} repeat }{ 1 exch sub exch {1 0}{0 1} ifelse () ci6graytintimage } ifelse }{ pop exch {0}{0 exch} ifelse 0 3 1 roll () ci6graytintimage } ifelse }{ ci6curoverprint { 8 {pop} repeat }{ pop pop pop {pop 1} 0 1 () /DeviceGray ci6proctintimage } ifelse } ifelse } ifelse } def /XINullImage { } def /XIImageMask { XIImageWidth XIImageHeight false [XIImageWidth 0 0 XIImageHeight neg 0 0] /XIDataProc load imagemask } def /XIImageTint { XIImageWidth XIImageHeight XIBitsPerPixel [XIImageWidth 0 0 XIImageHeight neg 0 0] /XIDataProc load XIType 3 eq XIColorValues XIColorSpace ci6tintimage } def /XIImage { XIImageWidth XIImageHeight XIBitsPerPixel [XIImageWidth 0 0 XIImageHeight neg 0 0] /XIDataProc load false XIChannelCount XIPlateList ci6drawimage } def /XG { pop pop } def /XF { 13 {pop} repeat } def /Xh { Adobe_ColorImage_AI6_Vars begin gsave /XIType exch def /XIImageHeight exch def /XIImageWidth exch def /XIImageMatrix exch def 0 0 moveto XIImageMatrix concat XIImageWidth XIImageHeight scale /_lp /null ddef _fc /_lp /imagemask ddef end } def /XH { Adobe_ColorImage_AI6_Vars begin grestore end } def /XIEnable { Adobe_ColorImage_AI6_Vars /XIEnable 3 -1 roll put } def /XC { Adobe_ColorImage_AI6_Vars begin ci6colormake /XIColorSpace exch def /XIColorValues exch def end } def /XIPlates { Adobe_ColorImage_AI6_Vars begin /XIPlateList exch def end } def /XI { Adobe_ColorImage_AI6_Vars begin gsave /XIType exch def cvi dup 256 idiv /XICompression exch store 256 mod /XIEncoding exch store pop pop /XIChannelCount exch def /XIBitsPerPixel exch def /XIImageHeight exch def /XIImageWidth exch def pop pop pop pop /XIImageMatrix exch def XIBitsPerPixel 1 eq { XIImageWidth 8 div ceiling cvi }{ XIImageWidth XIChannelCount mul } ifelse /XIRowBytes exch def XIEnable { /XIBuffer3 XIImageWidth string def XICompression 0 eq { /XIBuffer1 XIRowBytes string def XIEncoding 0 eq { {currentfile XIBuffer1 readhexstring pop} }{ {currentfile XIBuffer1 readstring pop} } ifelse }{ /XIBuffer1 256 string def /XIBuffer2 XIRowBytes string def {currentfile XIBuffer1 readline pop (%) anchorsearch {pop} if} /ASCII85Decode filter /DCTDecode filter /XIFile exch def {XIFile XIBuffer2 readstring pop} } ifelse /XIDataProc exch def XIType 1 ne { 0 setgray } if XIType 1 eq { XIImageMask }{ XIType 2 eq XIType 3 eq or { XIImageTint }{ XIImage } ifelse } ifelse }{ XINullImage } ifelse /XIPlateList false def grestore end } def end %%EndProcSet %%BeginResource: procset Adobe_Illustrator_AI5 1.3 0 %%Title: (Adobe Illustrator (R) Version 8.0 Full Prolog) %%Version: 1.3 0 %%CreationDate: (3/7/1994) () %%Copyright: ((C) 1987-1998 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_Illustrator_AI5_vars 112 dict dup begin put /_?cmyk false def /_eo false def /_lp /none def /_pf { } def /_ps { } def /_psf { } def /_pss { } def /_pjsf { } def /_pjss { } def /_pola 0 def /_doClip 0 def /cf currentflat def /_lineorientation 0 def /_charorientation 0 def /_yokoorientation 0 def /_tm matrix def /_renderStart [ /e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0 ] def /_renderEnd [ null null null null /i1 /i1 /i1 /i1 ] def /_render -1 def /_shift [0 0] def /_ax 0 def /_ay 0 def /_cx 0 def /_cy 0 def /_leading [ 0 0 ] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fontSize 0 def /_fontAscent 0 def /_fontDescent 0 def /_fontHeight 0 def /_fontRotateAdjust 0 def /Ss 256 string def Ss 0 (fonts/) putinterval /_cnt 0 def /_scale [1 1] def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def /_hfname 100 string def /_hffound false def /Tx { } def /Tj { } def /CRender { } def /_AI3_savepage { } def /_gf null def /_cf 4 array def /_rgbf 3 array def /_if null def /_of false def /_fc { } def /_gs null def /_cs 4 array def /_rgbs 3 array def /_is null def /_os false def /_sc { } def /_pd 1 dict def /_ed 15 dict def /_pm matrix def /_fm null def /_fd null def /_fdd null def /_sm null def /_sd null def /_sdd null def /_i null def /_lobyte 0 def /_hibyte 0 def /_cproc null def /_cscript 0 def /_hvax 0 def /_hvay 0 def /_hvwb 0 def /_hvcx 0 def /_hvcy 0 def /_bitfont null def /_bitlobyte 0 def /_bithibyte 0 def /_bitkey null def /_bitdata null def /_bitindex 0 def /discardSave null def /buffer 256 string def /beginString null def /endString null def /endStringLength null def /layerCnt 1 def /layerCount 1 def /perCent (%) 0 get def /perCentSeen? false def /newBuff null def /newBuffButFirst null def /newBuffLast null def /clipForward? false def end userdict /Adobe_Illustrator_AI5 known not { userdict /Adobe_Illustrator_AI5 100 dict put } if userdict /Adobe_Illustrator_AI5 get begin /initialize { Adobe_Illustrator_AI5 dup begin Adobe_Illustrator_AI5_vars begin /_aicmykps where {pop /_?cmyk _aicmykps def}if discardDict { bind pop pop } forall dup /nc get begin { dup xcheck 1 index type /operatortype ne and { bind } if pop pop } forall end newpath } def /terminate { end end } def /_ null def /ddef { Adobe_Illustrator_AI5_vars 3 1 roll put } def /xput { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop { { pop } repeat } def /hswj { dup stringwidth 3 2 roll { _hvwb eq { exch _hvcx add exch _hvcy add } if exch _hvax add exch _hvay add } cforall } def /vswj { 0 0 3 -1 roll { dup 255 le _charorientation 1 eq and { dup cstring stringwidth 5 2 roll _hvwb eq { exch _hvcy sub exch _hvcx sub } if exch _hvay sub exch _hvax sub 4 -1 roll sub exch 3 -1 roll sub exch } { _hvwb eq { exch _hvcy sub exch _hvcx sub } if exch _hvay sub exch _hvax sub _fontHeight sub } ifelse } cforall } def /swj { 6 1 roll /_hvay exch ddef /_hvax exch ddef /_hvwb exch ddef /_hvcy exch ddef /_hvcx exch ddef _lineorientation 0 eq { hswj } { vswj } ifelse } def /sw { 0 0 0 6 3 roll swj } def /vjss { 4 1 roll { dup cstring dup length 1 eq _charorientation 1 eq and { -90 rotate currentpoint _fontRotateAdjust add moveto gsave false charpath currentpoint 5 index setmatrix stroke grestore _fontRotateAdjust sub moveto _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto 90 rotate } { currentpoint _fontHeight sub 5 index sub 3 index _sp eq { 9 index sub } if currentpoint exch 4 index stringwidth pop 2 div sub exch _fontAscent sub moveto gsave 2 index false charpath 6 index setmatrix stroke grestore moveto pop pop } ifelse } cforall 6 npop } def /hjss { 4 1 roll { dup cstring gsave false charpath currentpoint 5 index setmatrix stroke grestore moveto _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto } cforall 6 npop } def /jss { _lineorientation 0 eq { hjss } { vjss } ifelse } def /ss { 0 0 0 7 3 roll jss } def /vjsp { 4 1 roll { dup cstring dup length 1 eq _charorientation 1 eq and { -90 rotate currentpoint _fontRotateAdjust add moveto false charpath currentpoint _fontRotateAdjust sub moveto _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto 90 rotate } { currentpoint _fontHeight sub 5 index sub 3 index _sp eq { 9 index sub } if currentpoint exch 4 index stringwidth pop 2 div sub exch _fontAscent sub moveto 2 index false charpath moveto pop pop } ifelse } cforall 6 npop } def /hjsp { 4 1 roll { dup cstring false charpath _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto } cforall 6 npop } def /jsp { matrix currentmatrix _lineorientation 0 eq {hjsp} {vjsp} ifelse } def /sp { matrix currentmatrix 0 0 0 7 3 roll _lineorientation 0 eq {hjsp} {vjsp} ifelse } def /pl { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c { curveto } def /C /c load def /v { currentpoint 6 2 roll curveto } def /V /v load def /y { 2 copy curveto } def /Y /y load def /l { lineto } def /L /l load def /m { moveto } def } { /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def } ifelse /d { setdash } def /cf { } def /i { dup 0 eq { pop cf } if setflat } def /j { setlinejoin } def /J { setlinecap } def /M { setmiterlimit } def /w { setlinewidth } def /XR { 0 ne /_eo exch ddef } def /H { } def /h { closepath } def /N { _pola 0 eq { _doClip 1 eq { _eo {eoclip} {clip} ifelse /_doClip 0 ddef } if newpath } { /CRender { N } ddef } ifelse } def /n { N } def /F { _pola 0 eq { _doClip 1 eq { gsave _pf grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf } ifelse } { /CRender { F } ddef } ifelse } def /f { closepath F } def /S { _pola 0 eq { _doClip 1 eq { gsave _ps grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps } ifelse } { /CRender { S } ddef } ifelse } def /s { closepath S } def /B { _pola 0 eq { _doClip 1 eq gsave F grestore { gsave S grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { S } ifelse } { /CRender { B } ddef } ifelse } def /b { closepath B } def /W { /_doClip 1 ddef } def /* { count 0 ne { dup type /stringtype eq { pop } if } if newpath } def /u { } def /U { } def /q { _pola 0 eq { gsave } if } def /Q { _pola 0 eq { grestore } if } def /*u { _pola 1 add /_pola exch ddef } def /*U { _pola 1 sub /_pola exch ddef _pola 0 eq { CRender } if } def /D { pop } def /*w { } def /*W { } def /` { /_i save ddef clipForward? { nulldevice } if 6 1 roll 4 npop concat pop userdict begin /showpage { } def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash /setstrokeadjust where {pop false setstrokeadjust} if newpath 0 setgray false setoverprint } def /~ { end _i restore } def /_rgbtocmyk { 3 { 1 exch sub 3 1 roll } repeat 3 copy 1 4 1 roll 3 { 3 index 2 copy gt { exch } if pop 4 1 roll } repeat pop pop pop 4 1 roll 3 { 3 index sub 3 1 roll } repeat 4 -1 roll } def /setrgbfill { _rgbf astore pop /_fc { _lp /fill ne { _of setoverprint _rgbf aload pop setrgbcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /setrgbstroke { _rgbs astore pop /_sc { _lp /stroke ne { _os setoverprint _rgbs aload pop setrgbcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /O { 0 ne /_of exch ddef /_lp /none ddef } def /R { 0 ne /_os exch ddef /_lp /none ddef } def /g { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /G { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /K { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /Xa { _?cmyk { 3 npop k }{ setrgbfill 4 npop } ifelse } def /XA { _?cmyk { 3 npop K }{ setrgbstroke 4 npop } ifelse } def /Xs { /_gf exch ddef 5 npop /_fc { _lp /fill ne { _of setoverprint _gf setAIseparationgray /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /XS { /_gs exch ddef 5 npop /_sc { _lp /stroke ne { _os setoverprint _gs setAIseparationgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /Xx { exch /_gf exch ddef 0 eq { findcmykcustomcolor }{ _?cmyk {true}{/findrgbcustomcolor where{pop false}{true}ifelse}ifelse { 4 1 roll 3 npop findcmykcustomcolor }{ 8 -4 roll 4 npop findrgbcustomcolor } ifelse } ifelse /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /XX { exch /_gs exch ddef 0 eq { findcmykcustomcolor }{ _?cmyk {true}{/findrgbcustomcolor where{pop false}{true}ifelse}ifelse { 4 1 roll 3 npop findcmykcustomcolor }{ 8 -4 roll 4 npop findrgbcustomcolor } ifelse } ifelse /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /X { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /XK { 3 -1 roll pop 0 eq { 1 exch sub 3 {dup 3 1 roll mul 5 1 roll} repeat mul 4 1 roll K } { 1 exch sub 4 1 roll 3 {1 exch sub 3 index mul 1 exch sub 3 1 roll} repeat 4 -1 roll pop XA } ifelse } def /Xk { 3 -1 roll pop 0 eq { 1 exch sub 3 {dup 3 1 roll mul 5 1 roll} repeat mul 4 1 roll k } { 1 exch sub 4 1 roll 3 {1 exch sub 3 index mul 1 exch sub 3 1 roll} repeat 4 -1 roll pop Xa } ifelse } def /A { pop } def /annotatepage { userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse } def /XT { pop pop } def /Xt { pop } def /discard { save /discardSave exch store discardDict begin /endString exch store gt38? { 2 add } if load stopped pop end discardSave restore } bind def userdict /discardDict 7 dict dup begin put /pre38Initialize { /endStringLength endString length store /newBuff buffer 0 endStringLength getinterval store /newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store /newBuffLast newBuff endStringLength 1 sub 1 getinterval store } def /shiftBuffer { newBuff 0 newBuffButFirst putinterval newBuffLast 0 currentfile read not { stop } if put } def 0 { pre38Initialize mark currentfile newBuff readstring exch pop { { newBuff endString eq { cleartomark stop } if shiftBuffer } loop } { stop } ifelse } def 1 { pre38Initialize /beginString exch store mark currentfile newBuff readstring exch pop { { newBuff beginString eq { /layerCount dup load 1 add store } { newBuff endString eq { /layerCount dup load 1 sub store layerCount 0 eq { cleartomark stop } if } if } ifelse shiftBuffer } loop } if } def 2 { mark { currentfile buffer {readline} stopped { % assume error was due to overfilling the buffer }{ not { stop } if endString eq { cleartomark stop } if }ifelse } loop } def 3 { /beginString exch store /layerCnt 1 store mark { currentfile buffer {readline} stopped { % assume error was due to overfilling the buffer }{ not { stop } if dup beginString eq { pop /layerCnt dup load 1 add store } { endString eq { layerCnt 1 eq { cleartomark stop } { /layerCnt dup load 1 sub store } ifelse } if } ifelse }ifelse } loop } def end userdict /clipRenderOff 15 dict dup begin put { /n /N /s /S /f /F /b /B } { { _doClip 1 eq { /_doClip 0 ddef _eo {eoclip} {clip} ifelse } if newpath } def } forall /Tr /pop load def /Bb {} def /BB /pop load def /Bg {12 npop} def /Bm {6 npop} def /Bc /Bm load def /Bh {4 npop} def end /Lb { 6 npop 7 2 roll 5 npop 0 eq { 0 eq { (%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard } { /clipForward? true def /Tx /pop load def /Tj /pop load def currentdict end clipRenderOff begin begin } ifelse } { 0 eq { save /discardSave exch store } if } ifelse } bind def /LB { discardSave dup null ne { restore } { pop clipForward? { currentdict end end begin /clipForward? false ddef } if } ifelse } bind def /Pb { pop pop 0 (%AI5_EndPalette) discard } bind def /Np { 0 (%AI5_End_NonPrinting--) discard } bind def /Ln /pop load def /Ap /pop load def /Ar { 72 exch div 0 dtransform dup mul exch dup mul add sqrt dup 1 lt { pop 1 } if setflat } def /Mb { q } def /Md { } def /MB { Q } def /nc 4 dict def nc begin /setgray { pop } bind def /setcmykcolor { 4 npop } bind def /setrgbcolor { 3 npop } bind def /setcustomcolor { 2 npop } bind def currentdict readonly pop end /XP { 4 npop } bind def /XD { pop } bind def end setpacking %%EndResource %%BeginResource: procset Adobe_cshow 2.0 8 %%Title: (Writing System Operators) %%Version: 2.0 8 %%CreationDate: (1/23/89) () %%Copyright: ((C) 1992-1996 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_cshow 14 dict dup begin put /initialize { Adobe_cshow begin Adobe_cshow { dup xcheck { bind } if pop pop } forall end Adobe_cshow begin } def /terminate { currentdict Adobe_cshow eq { end } if } def /cforall { /_lobyte 0 ddef /_hibyte 0 ddef /_cproc exch ddef /_cscript currentfont /FontScript known { currentfont /FontScript get } { -1 } ifelse ddef { /_lobyte exch ddef _hibyte 0 eq _cscript 1 eq _lobyte 129 ge _lobyte 159 le and _lobyte 224 ge _lobyte 252 le and or and _cscript 2 eq _lobyte 161 ge _lobyte 254 le and and _cscript 3 eq _lobyte 161 ge _lobyte 254 le and and _cscript 25 eq _lobyte 161 ge _lobyte 254 le and and _cscript -1 eq or or or or and { /_hibyte _lobyte ddef } { _hibyte 256 mul _lobyte add _cproc /_hibyte 0 ddef } ifelse } forall } def /cstring { dup 256 lt { (s) dup 0 4 3 roll put } { dup 256 idiv exch 256 mod (hl) dup dup 0 6 5 roll put 1 4 3 roll put } ifelse } def /clength { 0 exch { 256 lt { 1 } { 2 } ifelse add } cforall } def /hawidthshow { { dup cstring show _hvax _hvay rmoveto _hvwb eq { _hvcx _hvcy rmoveto } if } cforall } def /vawidthshow { { dup 255 le _charorientation 1 eq and { -90 rotate 0 _fontRotateAdjust rmoveto cstring _hvcx _hvcy _hvwb _hvax _hvay 6 -1 roll awidthshow 0 _fontRotateAdjust neg rmoveto 90 rotate } { currentpoint _fontHeight sub exch _hvay sub exch _hvax sub 2 index _hvwb eq { exch _hvcy sub exch _hvcx sub } if 3 2 roll cstring dup stringwidth pop 2 div neg _fontAscent neg rmoveto show moveto } ifelse } cforall } def /hvawidthshow { 6 1 roll /_hvay exch ddef /_hvax exch ddef /_hvwb exch ddef /_hvcy exch ddef /_hvcx exch ddef _lineorientation 0 eq { hawidthshow } { vawidthshow } ifelse } def /hvwidthshow { 0 0 3 -1 roll hvawidthshow } def /hvashow { 0 0 0 6 -3 roll hvawidthshow } def /hvshow { 0 0 0 0 0 6 -1 roll hvawidthshow } def currentdict readonly pop end setpacking %%EndResource %%BeginResource: procset Adobe_shading_AI8 1.0 0 %%Title: (Adobe Illustrator 8 Shading Procset) %%Version: 1.0 0 %%CreationDate: (12/17/97) () %%Copyright: ((C) 1987-1997 Adobe Systems Incorporated All Rights Reserved) userdict /defaultpacking currentpacking put true setpacking userdict /Adobe_shading_AI8 10 dict dup begin put /initialize { Adobe_shading_AI8 begin Adobe_shading_AI8 bdprocs Mesh /initialize get exec } def /terminate { currentdict Adobe_shading_AI8 eq { end } if } def /bdprocs { { dup xcheck 1 index type /arraytype eq and { bind } if pop pop } forall } def /X! {pop} def /X# {pop pop} def /Mesh 40 dict def Mesh begin /initialize { Mesh bdprocs Mesh begin /emulate? /AI8MeshEmulation where { pop AI8MeshEmulation }{ systemdict /shfill known not } ifelse def end } def /bd { shadingdict begin } def /paint { emulate? { end }{ /_lp /none ddef _fc /_lp /none ddef /AIColorSpace AIColorSpace tocolorspace store /ColorSpace AIColorSpace topsspace store version_ge_3010.106 not systemdict /setsmoothness known and { 0.0001 setsmoothness } if composite? { /DataSource getdatasrc def Matrix concat currentdict end shfill }{ AIColorSpace makesmarks AIPlateList markingplate and not isoverprint and { end }{ /ColorSpace /DeviceGray store /Decode [0 1 0 1 0 1] store /DataSource getplatesrc def Matrix concat currentdict end shfill } ifelse } ifelse } ifelse } def /shadingdict 12 dict def shadingdict begin /ShadingType 6 def /BitsPerCoordinate 16 def /BitsPerComponent 8 def /BitsPerFlag 8 def end /datafile null def /databuf 256 string def /dataptr 0 def /srcspace null def /srcchannels 0 def /dstchannels 0 def /dstplate 0 def /srctodstcolor null def /getplatesrc { /srcspace AIColorSpace store /srcchannels AIColorSpace getnchannels store /dstchannels 1 store /dstplate getplateindex store /srctodstcolor srcspace makesmarks { dstplate 4 eq { {1 exch sub} }{ {srcspace tocmyk 3 dstplate sub index 1 exch sub 5 1 roll 4 {pop} repeat} } ifelse }{ {srcchannels {pop} repeat 1} } ifelse store /datafile getdatasrc store /rdpatch168 load DataLength () /SubFileDecode filter } def /getdatasrc { /rdcmntline load /ASCII85Decode filter } def /rdpatch168 { /dataptr 0 store 49 rdcount 4 { dup {pop srcchannels getint8} if dup {pop srctodstcolor dstchannels putint8 true} if } repeat {databuf 0 dataptr getinterval}{()} ifelse } def /rdpatch3216 { /dataptr 0 store 97 rdcount 4 { dup {pop srcchannels getint16} if dup {pop srctodstcolor dstchannels putint16 true} if } repeat {databuf 0 dataptr getinterval}{()} ifelse } def /rdcount { dup 0 gt { datafile databuf dataptr 4 -1 roll getinterval readstring exch length dataptr add /dataptr exch store }{ true } ifelse } def /getint8 { mark true 3 -1 roll { dup {pop datafile read} if dup {pop 255 div true} if } repeat { counttomark 1 add -1 roll pop true }{ cleartomark false } ifelse } def /putint8 { dup dataptr add /dataptr exch store dataptr exch { 1 sub exch 255 mul cvi databuf 2 index 3 -1 roll put } repeat pop } def /getint16 { mark true 3 -1 roll { dup {pop datafile read} if dup {pop 256 mul datafile read} if dup {pop add 65535 div true} if } repeat { counttomark 1 add -1 roll pop true }{ cleartomark false } ifelse } def /putint16 { dup 2 mul dataptr add /dataptr exch store dataptr exch { 2 sub exch 65535 mul cvi dup 256 idiv databuf 3 index 3 -1 roll put 256 mod databuf 2 index 1 add 3 -1 roll put } repeat pop } def /srcbuf 256 string def /rdcmntline { currentfile srcbuf readline pop (%) anchorsearch {pop} if } def /getplateindex { 0 [cyan? magenta? yellow? black? customColor?] {{exit} if 1 add} forall } def /aicsarray 4 array def /aicsaltvals 4 array def /aicsaltcolr aicsaltvals def /tocolorspace { dup type /arraytype eq { mark exch aload pop aicsarray 0 3 -1 roll put aicsarray 1 3 -1 roll put dup aicsarray 2 3 -1 roll put gettintxform aicsarray 3 3 -1 roll put counttomark aicsaltvals 0 3 -1 roll getinterval /aicsaltcolr exch store aicsaltcolr astore pop pop aicsarray } if } def /subtintxform {aicsaltcolr {1 index mul exch} forall pop} def /addtintxform {aicsaltcolr {1 sub 1 index mul 1 add exch} forall pop} def /gettintxform { /DeviceRGB eq {/addtintxform}{/subtintxform} ifelse load } def /getnchannels { dup type /arraytype eq {0 get} if colorspacedict exch get begin Channels end } def /makesmarks { composite? { pop true }{ dup dup type /arraytype eq {0 get} if colorspacedict exch get begin MarksPlate end } ifelse } def /markingplate { composite? { pop true }{ dup type /arraytype eq { dup length getplateindex gt {getplateindex get}{pop false} ifelse } if } ifelse } def /tocmyk { dup dup type /arraytype eq {0 get} if colorspacedict exch get begin ToCMYK end } def /topsspace { dup dup type /arraytype eq {0 get} if colorspacedict exch get begin ToPSSpace end } def /colorspacedict 5 dict dup begin /DeviceGray 4 dict dup begin /Channels 1 def /MarksPlate {pop black?} def /ToCMYK {pop 1 exch sub 0 0 0 4 -1 roll} def /ToPSSpace {} def end def /DeviceRGB 4 dict dup begin /Channels 3 def /MarksPlate {pop isCMYKSep?} def /ToCMYK {pop _rgbtocmyk} def /ToPSSpace {} def end def /DeviceCMYK 4 dict dup begin /Channels 4 def /MarksPlate {pop isCMYKSep?} def /ToCMYK {pop} def /ToPSSpace {} def end def /Separation 4 dict dup begin /Channels 1 def /MarksPlate { /findcmykcustomcolor where { pop dup 1 exch ToCMYK 5 -1 roll 1 get findcmykcustomcolor 1 setcustomcolor systemdict /currentgray get exec 1 ne }{ pop false } ifelse } def /ToCMYK { dup 2 get mark exch 4 2 roll 3 get exec counttomark -1 roll tocmyk 5 -1 roll pop } def /ToPSSpace {} def end def /Process 4 dict dup begin /Channels 1 def /MarksPlate { isCMYKSep? { 1 exch ToCMYK 4 array astore getplateindex get 0 ne }{ pop false } ifelse } def /ToCMYK { dup 2 get mark exch 4 2 roll 3 get exec counttomark -1 roll tocmyk 5 -1 roll pop } def /ToPSSpace { 4 array copy dup 0 /Separation put } def end def end def /isoverprint { /currentoverprint where {pop currentoverprint}{_of} ifelse } def /version_ge_3010.106 { version {cvr} stopped { pop false }{ 3010.106 ge } ifelse } def end end defaultpacking setpacking %%EndResource %%EndProlog %%BeginSetup %%IncludeFont: Symbol userdict /_useSmoothShade false put userdict /_aicmykps true put userdict /_forceToCMYK true put Adobe_level2_AI5 /initialize get exec Adobe_cshow /initialize get exec Adobe_Illustrator_AI5_vars Adobe_Illustrator_AI5 Adobe_typography_AI5 /initialize get exec Adobe_ColorImage_AI6 /initialize get exec Adobe_shading_AI8 /initialize get exec Adobe_Illustrator_AI5 /initialize get exec [ 39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis /Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute /egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde /oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex /udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash /.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef /.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash /questiondown/exclamdown/logicalnot/.notdef/florin/.notdef/.notdef /guillemotleft/guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide /.notdef/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright /fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand /Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex /Idieresis/Igrave/Oacute/Ocircumflex/.notdef/Ograve/Uacute/Ucircumflex /Ugrave/dotlessi/circumflex/tilde/macron/breve/dotaccent/ring/cedilla /hungarumlaut/ogonek/caron TE %AI55J_Tsume: None %AI3_BeginEncoding: _Symbol Symbol [/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /exclam /universal /numbersign /existential /percent /ampersand /suchthat /parenleft /parenright /asteriskmath /plus /comma /minus /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /congruent /Alpha /Beta /Chi /Delta /Epsilon /Phi /Gamma /Eta /Iota /theta1 /Kappa /Lambda /Mu /Nu /Omicron /Pi /Theta /Rho /Sigma /Tau /Upsilon /sigma1 /Omega /Xi /Psi /Zeta /bracketleft /therefore /bracketright /perpendicular /underscore /radicalex /alpha /beta /chi /delta /epsilon /phi /gamma /eta /iota /phi1 /kappa /lambda /mu /nu /omicron /pi /theta /rho /sigma /tau /upsilon /omega1 /omega /xi /psi /zeta /braceleft /bar /braceright /similar /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /Upsilon1 /minute /lessequal /fraction /infinity /florin /club /diamond /heart /spade /arrowboth /arrowleft /arrowup /arrowright /arrowdown /degree /plusminus /second /greaterequal /multiply /proportional /partialdiff /bullet /divide /notequal /equivalence /approxequal /ellipsis /arrowvertex /arrowhorizex /carriagereturn /aleph /Ifraktur /Rfraktur /weierstrass /circlemultiply /circleplus /emptyset /intersection /union /propersuperset /reflexsuperset /notsubset /propersubset /reflexsubset /element /notelement /angle /gradient /registerserif /copyrightserif /trademarkserif /product /radical /dotmath /logicalnot /logicaland /logicalor /arrowdblboth /arrowdblleft /arrowdblup /arrowdblright /arrowdbldown /lozenge /angleleft /registersans /copyrightsans /trademarksans /summation /parenlefttp /parenleftex /parenleftbt /bracketlefttp /bracketleftex /bracketleftbt /bracelefttp /braceleftmid /braceleftbt /braceex /.notdef /angleright /integral /integraltp /integralex /integralbt /parenrighttp /parenrightex /parenrightbt /bracketrighttp /bracketrightex /bracketrightbt /bracerighttp /bracerightmid /bracerightbt /.notdef /_Symbol/Symbol 0 0 0 TZ %AI3_EndEncoding AdobeType [161/degree 173/notequal 176/infinity/plusminus/lessequal/greaterequal 181/mu/partialdiff/summation/product/pi/integral 189/Omega 195/radical 197/approxequal 198/Delta 214/divide/lozenge 240/apple /_Symbol_/Symbol 0 0 0 TZ %AI5_Begin_NonPrinting Np %AI3_BeginPattern: (Brick) (Brick) 0 0 72 72 [ %AI3_Tile (0 O 0 R 0.3 0.85 0.85 0 k 0.3 0.85 0.85 0 K ) @ ( %AI6_BeginPatternLayer 800 Ar 0 J 0 j 1 w 4 M []0 d %AI3_Note: 0 D 0 XR 0 0 m 0 72 L 72 72 L 72 0 L 0 0 L f %AI6_EndPatternLayer ) & (0 O 0 R 1 g 1 G ) @ ( %AI6_BeginPatternLayer 800 Ar 0 J 0 j 0.3 w 4 M []0 d %AI3_Note: 0 D 0 XR 0 68.4097 m 72 68.4097 l S 0 61.209 m 72 61.209 L S 0 54.0088 m 72 54.0088 L S 0 46.8076 m 72 46.8076 L S 0 39.6084 m 72 39.6084 L S 0 32.4072 m 72 32.4072 L S 0 25.207 m 72 25.207 L S 0 18.0059 m 72 18.0059 L S 0 10.8057 m 72 10.8057 L S 0 3.6064 m 72 3.6064 L S 68.4102 68.4097 m 68.4102 61.2217 l S 54.0098 68.4097 m 54.0098 61.2217 L S 39.6094 68.4097 m 39.6094 61.2217 L S 25.21 68.4097 m 25.21 61.2217 L S 10.8105 68.4097 m 10.8105 61.2217 L S 68.4102 53.9717 m 68.4102 46.7842 l S 54.0098 53.9717 m 54.0098 46.7842 L S 39.6094 53.9717 m 39.6094 46.7842 L S 25.21 53.9717 m 25.21 46.7842 L S 10.8105 53.9717 m 10.8105 46.7842 L S 68.4102 39.5967 m 68.4102 32.4092 l S 54.0098 39.5967 m 54.0098 32.4092 L S 39.6094 39.5967 m 39.6094 32.4092 L S 25.21 39.5967 m 25.21 32.4092 L S 10.8105 39.5967 m 10.8105 32.4092 L S 68.4102 25.2217 m 68.4102 18.0342 l S 54.0098 25.2217 m 54.0098 18.0342 L S 39.6094 25.2217 m 39.6094 18.0342 L S 25.21 25.2217 m 25.21 18.0342 L S 10.8105 25.2217 m 10.8105 18.0342 L S 68.4102 10.7842 m 68.4102 3.5967 l S 54.0098 10.7842 m 54.0098 3.5967 L S 39.6094 10.7842 m 39.6094 3.5967 L S 25.21 10.7842 m 25.21 3.5967 L S 10.8105 10.7842 m 10.8105 3.5967 L S 61.1973 3.5967 m 61.1973 0 L S 46.7969 3.5967 m 46.7969 0 L S 32.3965 3.5967 m 32.3965 0 L S 17.9971 3.5967 m 17.9971 0 L S 3.5967 3.5967 m 3.5967 0 l S 61.1973 18.0342 m 61.1973 10.8467 L S 46.7969 18.0342 m 46.7969 10.8467 L S 32.3965 18.0342 m 32.3965 10.8467 L S 17.9971 18.0342 m 17.9971 10.8467 L S 3.5967 18.0342 m 3.5967 10.8467 l S 61.1973 32.4092 m 61.1973 25.2217 L S 46.7969 32.4092 m 46.7969 25.2217 L S 17.9971 32.4092 m 17.9971 25.2217 L S 3.5967 32.4092 m 3.5967 25.2217 l S 61.1973 46.7842 m 61.1973 39.5967 L S 46.7969 46.7842 m 46.7969 39.5967 L S 32.3965 46.7842 m 32.3965 39.5967 L S 17.9971 46.7842 m 17.9971 39.5967 L S 3.5967 46.7842 m 3.5967 39.5967 l S 61.1973 61.2217 m 61.1973 54.0347 L S 46.7969 61.2217 m 46.7969 54.0347 L S 32.3965 61.2217 m 32.3965 54.0347 L S 17.9971 61.2217 m 17.9971 54.0347 L S 3.5967 61.2217 m 3.5967 54.0347 l S 61.1973 71.959 m 61.1973 68.4717 L S 46.7969 71.959 m 46.7969 68.4717 L S 32.3965 71.959 m 32.3965 68.4717 L S 17.9971 71.959 m 17.9971 68.4717 L S 3.5967 71.959 m 3.5967 68.4717 l S 32.3965 32.4092 m 32.3965 25.2217 L S %AI6_EndPatternLayer ) & ] E %AI3_EndPattern %AI3_BeginPattern: (Confetti) (Confetti) 4.85 3.617 76.85 75.617 [ %AI3_Tile (0 O 0 R 1 g 1 G ) @ ( %AI6_BeginPatternLayer 800 Ar 0 J 0 j 1 w 4 M []0 d %AI3_Note: 0 D 0 XR 4.85 3.617 m 4.85 75.617 L 76.85 75.617 L 76.85 3.617 L 4.85 3.617 L f %AI6_EndPatternLayer ) & (0 O 0 R 0 g 0 G ) @ ( %AI6_BeginPatternLayer 800 Ar 0 J 0 j 0.3 w 4 M []0 d %AI3_Note: 0 D 0 XR 10.6 64.867 m 7.85 62.867 l S 9.1 8.617 m 6.85 6.867 l S 78.1 68.617 m 74.85 67.867 l S 76.85 56.867 m 74.35 55.117 l S 79.6 51.617 m 76.6 51.617 l S 76.35 44.117 m 73.6 45.867 l S 78.6 35.867 m 76.6 34.367 l S 76.1 23.867 m 73.35 26.117 l S 78.1 12.867 m 73.85 13.617 l S 68.35 14.617 m 66.1 12.867 l S 76.6 30.617 m 73.6 30.617 l S 62.85 58.117 m 60.956 60.941 l S 32.85 59.617 m 31.196 62.181 l S 47.891 64.061 m 49.744 66.742 l S 72.814 2.769 m 73.928 5.729 l S 67.976 2.633 m 67.35 5.909 l S 61.85 27.617 m 59.956 30.441 l S 53.504 56.053 m 51.85 58.617 l S 52.762 1.779 m 52.876 4.776 l S 45.391 5.311 m 47.244 7.992 l S 37.062 3.375 m 35.639 5.43 l S 55.165 34.828 m 57.518 37.491 l S 20.795 3.242 m 22.12 5.193 l S 14.097 4.747 m 15.008 8.965 l S 9.736 1.91 m 8.073 4.225 l S 31.891 5.573 m 32.005 8.571 l S 12.1 70.367 m 15.6 68.867 l S 9.35 54.867 m 9.6 58.117 l S 12.85 31.867 m 14.35 28.117 l S 10.1 37.367 m 12.35 41.117 l S 34.1 71.117 m 31.85 68.617 l S 38.35 71.117 m 41.6 68.367 l S 55.1 71.117 m 58.35 69.117 l S 57.35 65.117 m 55.35 61.867 l S 64.35 66.367 m 69.35 68.617 l S 71.85 62.867 m 69.35 61.117 l S 23.6 70.867 m 23.6 67.867 l S 20.6 65.867 m 17.35 65.367 l S 24.85 61.367 m 25.35 58.117 l S 25.85 65.867 m 29.35 66.617 l S 14.1 54.117 m 16.85 56.117 l S 12.35 11.617 m 12.6 15.617 l S 12.1 19.867 m 14.35 22.367 l S 26.1 9.867 m 23.6 13.367 l S 34.6 47.117 m 32.1 45.367 l S 62.6 41.867 m 59.85 43.367 l S 31.6 35.617 m 27.85 36.367 l S 36.35 26.117 m 34.35 24.617 l S 33.85 14.117 m 31.1 16.367 l S 37.1 9.867 m 35.1 11.117 l S 34.35 20.867 m 31.35 20.867 l S 44.6 56.617 m 42.1 54.867 l S 47.35 51.367 m 44.35 51.367 l S 44.1 43.867 m 41.35 45.617 l S 43.35 33.117 m 42.6 30.617 l S 43.85 23.617 m 41.1 25.867 l S 44.35 15.617 m 42.35 16.867 l S 67.823 31.1 m 64.823 31.1 l S 27.1 32.617 m 29.6 30.867 l S 31.85 55.117 m 34.85 55.117 l S 19.6 40.867 m 22.1 39.117 l S 16.85 35.617 m 19.85 35.617 l S 20.1 28.117 m 22.85 29.867 l S 52.1 42.617 m 54.484 44.178 l S 52.437 50.146 m 54.821 48.325 l S 59.572 54.133 m 59.35 51.117 l S 50.185 10.055 m 53.234 9.928 l S 51.187 15.896 m 53.571 14.075 l S 58.322 19.883 m 59.445 16.823 l S 53.1 32.117 m 50.6 30.367 l S 52.85 24.617 m 49.6 25.617 l S 61.85 9.117 m 59.1 10.867 l S 69.35 34.617 m 66.6 36.367 l S 67.1 23.617 m 65.1 22.117 l S 24.435 46.055 m 27.484 45.928 l S 25.437 51.896 m 27.821 50.075 l S 62.6 47.117 m 65.321 46.575 l S 19.85 19.867 m 20.35 16.617 l S 21.85 21.867 m 25.35 22.617 l S 37.6 62.867 m 41.6 62.117 l S 38.323 42.1 m 38.823 38.6 l S 69.35 52.617 m 66.85 53.867 l S 14.85 62.117 m 18.1 59.367 l S 9.6 46.117 m 7.1 44.367 l S 20.6 51.617 m 18.6 50.117 l S 46.141 70.811 m 47.994 73.492 l S 69.391 40.561 m 71.244 43.242 l S 38.641 49.311 m 39.35 52.117 l S 25.141 16.811 m 25.85 19.617 l S 36.6 32.867 m 34.6 31.367 l S 6.1 68.617 m 2.85 67.867 l S 4.85 56.867 m 2.35 55.117 l S 7.6 51.617 m 4.6 51.617 l S 6.6 35.867 m 4.6 34.367 l S 6.1 12.867 m 1.85 13.617 l S 4.6 30.617 m 1.6 30.617 l S 72.814 74.769 m 73.928 77.729 l S 67.976 74.633 m 67.35 77.909 l S 52.762 73.779 m 52.876 76.776 l S 37.062 75.375 m 35.639 77.43 l S 20.795 75.242 m 22.12 77.193 l S 9.736 73.91 m 8.073 76.225 l S 10.1 23.617 m 6.35 24.367 l S 73.217 18.276 m 71.323 21.1 l S 28.823 39.6 m 29.505 42.389 l S 49.6 38.617 m 47.6 37.117 l S 60.323 73.6 m 62.323 76.6 l S 60.323 1.6 m 62.323 4.6 l S %AI6_EndPatternLayer ) & ] E %AI3_EndPattern %AI3_BeginPattern: (Leaves - Fall ) (Leaves - Fall ) 0 0 64.0781 78.9336 [ %AI3_Tile (0 O 0 R 0.05 0.2 1 0 k 0.05 0.2 1 0 K ) @ ( %AI6_BeginPatternLayer 800 Ar 0 J 0 j 1 w 4 M []0 d %AI3_Note: 0 D 0 XR 64.0781 78.9336 m 64.0781 0 L 0 0 L 0 78.9336 L 64.0781 78.9336 L f %AI6_EndPatternLayer ) & (0 O 0 R 0.83 0 1 0 k 0.83 0 1 0 K ) @ ( %AI6_BeginPatternLayer 800 Ar 0 J 0 j 1 w 4 M []0 d %AI3_Note: 1 D 0 XR 29.7578 0.9902 m 30.4346 1.1914 30.7246 1.3428 V 29.2559 4.0547 33.707 8.3359 34.627 9.0762 C 35.2275 8.8506 35.3477 6.3184 34.6699 4.9805 C 35.5137 5.1035 37.7031 3.7256 38.4609 2.4365 C 38.5254 3.125 40.0957 6.0664 40.9219 6.4434 C 40.002 6.8408 39.3359 8.3135 38.5742 9.7617 C 39.5957 9.9287 40.9961 9.0078 42.4668 8.1025 C 42.9814 8.9043 44.3555 9.875 45.6143 10.3916 C 44.5264 11.0781 44.0313 11.8203 43.5352 13.2793 C 42.4922 12.7139 40.3057 12.5645 39.7764 12.8516 C 40.291 13.9648 42.5371 14.5078 43.2676 14.4551 C 43.0137 15.3164 42.8652 17.4697 43.0391 20.0625 C 41.3789 18.7461 39.834 17.4297 38.1738 17.4883 C 38.4434 16.0664 37.8076 14.2607 37.4307 13.7676 C 36.8574 14.5117 36.4463 15.3389 36.8008 17.3164 C 35.3486 17.8008 34.1113 18.3467 32.7373 19.6045 C 32.7373 17.7734 32.166 16.5723 31.2969 15.2959 C 32.5576 14.8076 33.8301 13.6045 33.8252 12.5664 C 32.9775 12.7178 31.2852 13.4619 30.793 14.4551 C 30.0742 13.707 28.3906 12.3984 26.7871 12.3945 C 27.9746 11.5391 28.8945 10.5059 28.9893 8.5938 C 30.2422 9.5645 32.6953 10.1797 34.0752 9.582 C 29.2344 5.3457 29.7031 2.3125 29.7578 0.9902 C f 13.8525 29.9844 m 13.3281 29.5127 13.1309 29.25 V 15.623 27.4326 13.3691 21.6074 12.8555 20.5439 C 12.2168 20.4883 10.8096 23.2285 10.8457 24.7266 C 9.7129 23.9707 8.0488 24.0918 6.4463 24.3779 C 7.0186 23.2891 6.6172 21.3447 5.8164 20.5439 C 6.8184 20.5801 8.1699 19.8652 9.4785 18.8838 C 8.6436 18.0645 6.8164 18.2246 4.9004 18.8838 C 4.9004 17.5107 4.0781 15.7734 3.2412 14.5918 C 4.5576 14.6484 5.7031 13.9629 6.5605 12.9316 C 7.2256 14.5 9.2598 15.6133 10.166 15.5645 C 10.1826 14.1992 8.6094 12.1094 7.5879 11.7109 C 8.1875 11.041 9.207 9.5107 10.166 7.0947 C 10.9648 9.0205 12.1348 10.2627 13.3672 11.1953 C 12.2256 12.7578 12.3994 13.6289 12.7988 15.1074 C 13.541 14.5664 14.5723 14.1338 14.7441 12.1309 C 16.4609 12.416 17.5957 12.3447 19.0938 11.4434 C 18.6387 13.1055 18.6348 14.707 18.9551 16.4063 C 17.1055 16.2666 15.5449 16.4795 14.5156 17.9688 C 15.3457 18.1953 17.6055 18.2549 18.4795 17.3223 C 18.8066 18.3047 19.7012 19.7109 21.1475 20.4043 C 19.707 20.6641 18.7227 21.7637 17.8135 23.4492 C 17.1006 22.0332 14.873 20.3691 13.3711 20.3145 C 15.373 24.3779 15.373 27.2959 13.8525 29.9844 C f 41.2324 26.0742 m 41.5518 26.7021 41.7549 26.959 V 44.1523 25.0176 48.958 28.3262 49.8535 29.0957 C 49.7432 29.7266 47.6182 30.8643 45.9004 29.834 C 46.3408 31.123 45.4395 33.084 44.2402 34.126 C 45.9805 34.0254 48.126 35.3867 48.6484 36.1289 C 48.8701 35.1514 50.0527 33.8809 51.3379 32.8672 C 51.6895 33.8398 50.9941 35.958 50.0781 37.5605 C 51.3125 38.0605 52.4248 38.9912 52.8828 40.25 C 53.3398 38.9336 54.3428 38.2598 55.6875 37.5039 C 54.5273 36.0762 53.7471 33.9023 54.0273 33.0391 C 55.3496 33.374 56.9209 36.0918 57.0439 37.1816 C 57.9189 36.415 59.4727 35.7285 62.0537 35.4219 C 60.3535 34.3438 59.9902 32.3516 59.4063 30.9219 C 58.2588 31.3682 56.0898 31.4277 55.1152 30.8643 C 55.8281 30.2852 57.168 29.7344 59.1777 29.7207 C 59.1777 28.1758 59.6406 27.043 60.8945 25.8281 C 59.1719 25.8418 57.0723 25.3555 55.5762 24.9629 C 55.3281 26.292 54.4844 27.8887 53.3398 28.2891 C 53.334 27.4277 53.5996 25.1797 54.4844 24.5117 C 53.6201 23.9443 52.3672 22.5674 51.9102 20.8496 C 51.2881 22.1758 50.4268 23.4805 48.5645 23.9238 C 49.749 24.9766 50.584 26.9941 50.25 28.4609 C 45.1973 24.4785 42.5215 25.7773 41.2324 26.0742 C f 27.7578 38.7324 m 28.4346 38.9316 28.7246 39.084 V 27.2559 41.7969 31.707 46.0776 32.627 46.8169 C 33.2275 46.5918 33.3477 44.0586 32.6699 42.7227 C 33.5137 42.8457 35.7031 41.4678 36.4609 40.1787 C 36.5254 40.8652 38.0957 43.8066 38.9219 44.1846 C 38.002 44.582 37.3359 46.0547 36.5742 47.5039 C 37.5957 47.6709 38.9961 46.7485 40.4668 45.8438 C 40.9814 46.6445 42.3555 47.6177 43.6143 48.1328 C 42.5264 48.8198 42.0313 49.5615 41.5352 51.0205 C 40.4922 50.4556 38.3057 50.3057 37.7764 50.5938 C 38.291 51.7056 40.5371 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Fp(N)2425 2484 y(i)p Fq(=0)2583 2459 y Fo(of)h Fm(C)2769 2426 y Fq(1)2848 2459 y Fo(p)-5 b(artial)5 b(ly)456 2567 y(hyp)-5 b(erb)g(olic)34 b(tori,)g(invariant)f(by)g Fn(H)g Fo(such)g(that:)656 2693 y Fs(i\))42 b Fo(The)35 b(motion)h(on)f(e)-5 b(ach)36 b(of)f(the)g(tori)g Fn(T)2085 2707 y Fp(i)2148 2693 y Fo(is)f(top)-5 b(olo)g(gic)g(al)5 b(ly)38 b(c)-5 b(onjugate)g(d)758 2801 y(to)34 b(a)f(tr)-5 b(ansitive)33 b(r)-5 b(otation)35 b(on)e(a)g(torus.)631 2909 y Fs(ii\))42 b Fo(Ther)-5 b(e)32 b(exist)e(orbits)i Fm(\015)1527 2923 y Fp(i)1580 2909 y Fn(\032)25 b Fm(W)1775 2876 y Fq(tu)1762 2936 y Fl(T)1801 2946 y Ff(i)p Fg(\000)p Fi(1)1925 2909 y Fn(\\)16 b Fm(W)2101 2876 y Fq(ts)2088 2936 y Fl(T)2127 2946 y Ff(i)2159 2909 y Fo(,)31 b(wher)-5 b(e)32 b Fm(W)2572 2865 y Fq(tu)p Fp(;)p Fq(ts)2559 2938 y Fl(T)2747 2909 y Fo(ar)-5 b(e)32 b(de\014ne)-5 b(d)758 3030 y(in)40 b Fs(\(139\))r Fo(.)575 3138 y Fs(iii\))73 b Fo(The)28 b(interse)-5 b(ction)29 b Fm(W)1527 3105 y Fq(tu)1514 3165 y Fl(T)1553 3175 y Ff(i)p Fg(\000)p Fi(1)1672 3138 y Fn(\\)10 b Fm(W)1842 3105 y Fq(ts)1829 3165 y Fl(T)1868 3175 y Ff(i)1928 3138 y Fo(is)27 b(tr)-5 b(ansversal)30 b(along)f(the)f(orbit)h Fm(\015)3118 3152 y Fp(i)3146 3138 y Fo(.)555 3296 y Fs(W)-8 b(e)43 b(emphasize)g(that)f (w)m(e)h(are)f(not)g(assuming)f(that)i(the)f(tori)h(ha)m(v)m(e)g(the) 456 3404 y(same)30 b(dimension,)g(or)h(that)g(they)f(are)h(homotopic.) 555 3512 y(As)k(the)g(tori)h(are)f(assumed)g(to)g(b)s(e)g(partially)h (h)m(yp)s(erb)s(olic,)f(there)h(exist)f Fm(\016)3145 3526 y Fp(i)456 3620 y Fs(suc)m(h)30 b(that)1154 3753 y(dist\()p Fm(\015)1383 3767 y Fp(i)1412 3753 y Fs(\()p Fm(t)p Fs(\))p Fm(;)15 b Fn(T)1605 3767 y Fp(i)p Fl(\000)p Fq(1)1723 3753 y Fs(\))26 b Fn(\024)f Fm(C)7 b(e)1994 3716 y Fl(\000)p Fp(\016)2080 3726 y Ff(i)2106 3716 y Fl(j)p Fp(t)p Fl(j)2266 3753 y Fm(t)25 b Fn(\024)g Fs(0)p Fm(;)1154 3912 y Fs(dist\()p Fm(\015)1383 3926 y Fp(i)1412 3912 y Fs(\()p Fm(t)p Fs(\))p Fm(;)15 b Fn(T)1605 3926 y Fp(i)1633 3912 y Fs(\))26 b Fn(\024)f Fm(C)7 b(e)1904 3875 y Fl(\000)p Fp(\016)1990 3885 y Ff(i)2016 3875 y Fl(j)p Fp(t)p Fl(j)2176 3912 y Fm(t)25 b Fn(\025)g Fs(0)p Fm(:)456 3830 y Fs(\(148\))456 4058 y(W)-8 b(e)38 b(are)g(not)g (assuming)g(that)g(the)g Fm(\016)1758 4072 y Fp(i)1824 4058 y Fs(are)g(b)s(ounded)d(a)m(w)m(a)m(y)40 b(from)d(zero)h(uni-)456 4166 y(formly)27 b(in)h Fm(i)p Fs(.)40 b(In)27 b(the)h(case)h(that)f Fn(H)g Fs(dep)s(ends)e(on)i(some)g(parameter)g Fm(")p Fs(,)h(w)m(e)f(are)456 4274 y(not)i(assuming)g(the)h Fm(\016)1209 4288 y Fp(i)1268 4274 y Fs(are)f(b)s(ounded)f(a)m(w)m(a)m (y)j(from)e(zero)h(uniformly)e(in)h Fm(")p Fs(.)456 4425 y Fw(Remark)k(77.)42 b Fs(It)30 b(is)g(conceiv)-5 b(able)32 b(that)e(the)g(di\013usion)g(based)f(on)h(ob)5 b(jects)31 b(of)456 4533 y(di\013eren)m(t)f(dimension)g(has)g(di\013eren)m(t)h (quan)m(titativ)m(e)i(prop)s(erties.)555 4640 y(One)39 b(can)h(argue)g(heuristically)h(that)f(the)f(lo)m(w)m(er)i(the)f (dimension)f(of)h(the)456 4748 y(orbit,)34 b(the)f(faster)h(the)f (transition)h(time)g(since)f(the)g(\\ergo)s(dization)j(time")e(is)456 4856 y(smaller)27 b(the)g(smaller)h(the)f(dimension)f(of)h(the)g(tori.) 40 b(In)27 b(this)f(pap)s(er,)h(ho)m(w)m(ev)m(er,)456 4964 y(w)m(e)j(ha)m(v)m(e)i(not)f(in)m(v)m(estigated)i(these)e(quan)m (titativ)m(e)i(prop)s(erties.)473 b Fj(\003)p eop end %%Page: 94 94 TeXDict begin 94 93 bop 456 251 a Fq(94)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 450 y Fs(10.2.)47 b Fw(The)e(scattering)h(map)f(and)h(the)f(transv)m(ersalit)m(y)g(of)h (hetero-)456 558 y(clinic)28 b(in)m(tersections.)47 b Fs(W)-8 b(e)25 b(\014rst)e(turn)g(to)i(pro)m(ving)f(the)g(result)g(ab)s (out)g(c)m(har-)456 666 y(acterizing)h(the)f(existence)h(of)e(transv)m (erse)h(hetero)s(clinic)h(in)m(tersections)g(among)456 776 y(t)m(w)m(o)h(di\013eren)m(t)f(in)m(v)-5 b(arian)m(t)26 b(manifolds)e(of)1864 753 y(~)1855 776 y(\003)1918 790 y Fp(")1979 776 y Fs(in)h(terms)f(of)h(their)g(transv)m(ersalit)m(y)456 884 y(under)g(the)h(scattering)j(map.)39 b(This)26 b(will)h(b)s(e)f(a)h (general)g(result)g(that)g(holds)f(for)456 991 y(normally)31 b(h)m(yp)s(erb)s(olic)f(in)m(v)-5 b(arian)m(t)32 b(manifolds)f(with)f (transv)m(ersal)i(homo)s(clinic)456 1099 y(in)m(tersections.)40 b(W)-8 b(e)24 b(will)g(pro)m(v)m(e)g(it)g(in)f(this)g(generalit)m(y)-8 b(.)41 b(F)-8 b(or)24 b(our)e(applications,)456 1209 y(of)32 b(course,)h(the)f(normally)h(h)m(yp)s(erb)s(olic)e(in)m(v)-5 b(arian)m(t)34 b(manifold)e(will)g(b)s(e)g(the)3083 1186 y(~)3074 1209 y(\003)3137 1223 y Fp(")456 1317 y Fs(pro)s(duced)c(in)i (Section)i(7.)456 1495 y Fw(Lemma)f(78.)41 b Fo(L)-5 b(et)1160 1472 y Fs(~)1151 1495 y(\003)31 b Fo(b)-5 b(e)30 b(a)h(normal)5 b(ly)32 b(hyp)-5 b(erb)g(olic)32 b(invariant)g (manifold.)43 b(As-)456 1603 y(sume)35 b(that)i Fm(W)981 1570 y Fq(s)975 1629 y(~)968 1646 y(\003)1052 1603 y Fj(t)31 b Fm(W)1243 1570 y Fq(u)1237 1629 y(~)1230 1646 y(\003)1322 1603 y Fo(along)36 b(a)h(manifold)j Fs(~)-48 b Fm(\015)5 b Fo(,)37 b(and)f(denote)h(by)f Fm(S)k Fo(the)c(sc)-5 b(at-)456 1717 y(tering)32 b(map)i(asso)-5 b(ciate)g(d)35 b(to)e(this)h(interse)-5 b(ction)37 b Fs(~)-49 b Fm(\015)5 b Fo(.)555 1827 y(L)-5 b(et)27 b Fn(V)762 1841 y Fq(1)801 1827 y Fm(;)15 b Fn(V)897 1841 y Fq(2)962 1827 y Fn(\032)1067 1804 y Fs(~)1058 1827 y(\003)27 b Fo(b)-5 b(e)26 b Fm(C)1325 1794 y Fq(1)1391 1827 y Fo(submanifolds)i(of)2036 1804 y Fs(~)2027 1827 y(\003)p Fo(,)f(and)h(assume)f(that)h Fm(S)5 b Fs(\()p Fn(V)2960 1841 y Fq(1)3000 1827 y Fs(\))25 b Fj(t)3128 1840 y Fq(~)3121 1857 y(\003)456 1935 y Fn(V)512 1949 y Fq(2)551 1935 y Fo(.)41 b(\(In)33 b(p)-5 b(articular)35 b Fn(V)1248 1949 y Fq(1)1308 1935 y Fn(\\)19 b Fm(H)1464 1949 y Fl(\000)1548 1935 y Fn(6)p Fs(=)25 b Fn(;)33 b Fo(and)h Fn(V)1955 1949 y Fq(2)2014 1935 y Fn(\\)20 b Fm(H)2171 1949 y Fq(+)2255 1935 y Fn(6)p Fs(=)25 b Fn(;)p Fo(.\))555 2043 y(Then,)i(ther)-5 b(e)26 b(exists)g(a)f(heter)-5 b(o)g(clinic)27 b(tr)-5 b(aje)g(ctory)28 b Fm(\015)2270 2057 y Fq(1)2334 2043 y Fo(such)e(that)g Fm(\015)2759 2057 y Fq(1)2824 2043 y Fn(\032)f Fm(W)3019 2010 y Fq(u)3006 2070 y Fl(V)3050 2079 y Fi(1)3113 2043 y Fj(t)456 2151 y Fm(W)555 2118 y Fq(s)542 2178 y Fl(V)586 2187 y Fi(2)624 2151 y Fo(.)456 2327 y(Pr)-5 b(o)g(of.)43 b Fs(By)36 b(the)g(de\014nition)f(of)h(the)g(scattering)h(map,)g(w)m(e)f(ha)m(v)m (e)h Fm(W)2839 2294 y Fq(u)2826 2354 y Fl(V)2870 2363 y Fi(1)2932 2327 y Fn(\\)27 b Fs(~)-48 b Fm(\015)39 b Fs(=)456 2435 y Fm(W)555 2402 y Fq(s)542 2467 y Fp(S)t Fq(\()p Fl(V)660 2476 y Fi(1)694 2467 y Fq(\))746 2435 y Fn(\\)23 b Fs(~)-49 b Fm(\015)5 b Fs(.)555 2575 y(By)43 b(the)f(assumption)g(of)h(transv)m(ersalit)m(y)h(of)e Fm(S)5 b Fs(\()p Fn(V)2333 2589 y Fq(1)2373 2575 y Fs(\))42 b(and)g Fn(V)2695 2589 y Fq(2)2776 2575 y Fs(in)2903 2552 y(~)2895 2575 y(\003)g(,)j(w)m(e)456 2683 y(obtain)30 b Fm(W)837 2650 y Fq(s)824 2714 y Fp(S)t Fq(\()p Fl(V)942 2723 y Fi(1)977 2714 y Fq(\))1033 2683 y Fj(t)1094 2713 y Fp(W)1171 2694 y Fi(s)1199 2713 y Fq(\()1233 2696 y(~)1226 2713 y(\003\))1332 2683 y Fm(W)1431 2650 y Fq(s)1418 2710 y Fl(V)1462 2719 y Fi(2)1500 2683 y Fs(,)h(and)f(therefore,)h Fm(W)2239 2650 y Fq(s)2226 2714 y Fp(S)t Fq(\()p Fl(V)2344 2723 y Fi(1)2378 2714 y Fq(\))2435 2683 y Fj(t)2498 2697 y Fq(~)-37 b Fp(\015)2565 2683 y Fm(W)2664 2650 y Fq(s)2651 2710 y Fl(V)2695 2719 y Fi(2)2733 2683 y Fs(.)41 b(Hence,)1556 2869 y Fm(W)1655 2831 y Fq(u)1642 2891 y Fl(V)1686 2900 y Fi(1)1750 2869 y Fj(t)1813 2883 y Fq(~)-37 b Fp(\015)1880 2869 y Fm(W)1979 2831 y Fq(s)1966 2891 y Fl(V)2010 2900 y Fi(2)2048 2869 y Fm(:)456 3040 y Fs(Since)30 b Fm(W)792 3007 y Fq(s)786 3066 y(~)779 3083 y(\003)857 3040 y Fj(t)25 b Fm(W)1042 3007 y Fq(u)1036 3066 y(~)1029 3083 y(\003)1115 3040 y Fs(along)35 b(~)-48 b Fm(\015)5 b Fs(,)30 b(w)m(e)h(obtain)g (the)g(desired)e(result.)508 b Fj(\003)555 3237 y Fs(W)-8 b(e)25 b(no)m(w)e(form)m(ulate)h(and)f(pro)m(v)m(e)h(Lemma)g(81,)h (that)f(will)g(allo)m(w)h(us)d(to)i(v)m(erify)456 3345 y(the)32 b(conditions)g(of)g(Lemma)g(78)h(in)e(the)h(case)h(that)g(the) f(manifolds)f(are)i(close)456 3453 y(to)28 b(lev)m(el)h(sets)e(of)h(a)f (function.)40 b(This)26 b(Lemma)i(will)f(b)s(e)g(useful)f(for)h(us)g (since)h(the)456 3561 y(ob)5 b(jects)35 b(w)m(e)h(ha)m(v)m(e)g (considered)f(b)s(efore)f(\(the)i(primary)e(and)g(secondary)h(tori,)456 3669 y(the)23 b(w)m(eak)h(stable)g(and)f(unstable)g(manifolds)g(of)h(p) s(erio)s(dic)e(orbits\))i(are)g(close)g(to)456 3777 y(b)s(eing)f(lev)m (el)j(sets)f(of)f(the)h(a)m(v)m(eraged)h(Hamiltonian)g(as)e(w)m(e)h(ha) m(v)m(e)h(established)e(in)456 3885 y(Section)k(8.)40 b(\(See)28 b(sp)s(ecially)g(Prop)s(osition)f(47,)i(Prop)s(osition)f (50,)h(Theorem)e(56)456 3993 y(and)i(Prop)s(osition)i(66.\))555 4101 y(The)39 b(precise)h(application)g(of)g(Lemma)f(78)i(to)f(the)f (case)h(that)g(the)g(man-)456 4209 y(ifolds)j(are)h(\015at|primary)f (KAM)h(tori)g(far)f(from)h(resonance|is)g(done)f(in)456 4317 y(Lemma)34 b(82,)j(and)c(for)h(non-\015at)h(manifolds|primary)e (and)h(secondary)g(tori)456 4425 y(close)39 b(to)f(the)g(resonance)h (or)e(w)m(eak)i(in)m(v)-5 b(arian)m(t)39 b(manifolds)f(of)g(the)g(lo)m (w)m(er)h(di-)456 4533 y(mensional)34 b(tori|is)h(done)f(in)g(Lemma)g (85)h(for)f(the)g(case)h(of)g(a)f(resonance)h(of)456 4640 y(order)29 b(1)i(and)f(in)g(Lemma)h(88)g(in)f(the)g(case)i(of)e(a) h(resonance)g(of)g(order)e(2.)555 4748 y(Lemma)35 b(81)h(is)f(designed) f(to)i(deal)f(in)g(a)g(uni\014ed)e(w)m(a)m(y)j(with)e(the)h(di\013eren) m(t)456 4856 y(t)m(yp)s(e)c(of)g(tori)g(that)g(app)s(ear)f(in)h(our)f (problem.)42 b(That)30 b(is,)i(the)f(rather)f(\015at)h(tori)456 4964 y(that)h(app)s(ear)g(in)f(the)i(non-resonan)m(t)f(region)h(or)f (in)f(the)i(resonan)m(t)f(regions)h(of)p eop end %%Page: 95 95 TeXDict begin 95 94 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(95)456 450 y Fs(order)27 b(3)i(or)f(bigger,)h(and)e(the)i(curv)m(ed)f(tori)g(that)h(app)s(ear)e (near)h(resonances)h(of)456 558 y(order)g(1)i(or)g(2.)555 666 y(As)45 b(w)m(e)g(ha)m(v)m(e)h(argued)f(in)f(Remark)h(55)h(these)f (t)m(w)m(o)h(t)m(yp)s(es)f(of)g(tori)h(ha)m(v)m(e)456 774 y(di\013eren)m(t)37 b(quan)m(titativ)m(e)j(prop)s(erties)d(and)g (this)g(leads)h(to)g(the)f(fact)i(that)f(the)456 882 y(leading)22 b(terms)h(in)e(the)i(asymptotics)g(of)f(the)h(in)m (tersection)g(could)g(b)s(e)e(di\013eren)m(t.)456 990 y(Hence,)31 b(sev)m(eral)h(of)e(the)h(details)g(and)f(the)h(form)m (ulas)f(will)h(b)s(e)f(di\013eren)m(t.)456 1155 y Fw(Remark)51 b(79.)f Fs(The)44 b(case)h(of)f(in)m(tersections)i(of)f(\015at)f(tori)h (is)g(signi\014can)m(tly)456 1263 y(easier)30 b(and)e(can)i(b)s(e)e (dealt)i(with)f(other)h(metho)s(ds.)39 b(Indeed,)29 b(the)g(study)g(of) g(in-)456 1371 y(tersections)e(of)g(\015at)f(tori)h(is)f(signi\014can)m (tly)i(easier)f(that)g(the)f(study)f(of)i([DLS00)q(,)456 1479 y(Lemma)k(4.21].)47 b(In)31 b(the)h(case)h(of)f([DLS00)q(])g(the)g (tori)g(w)m(ere)g(\015at)g(but)f(also)i(pre-)456 1587 y(sen)m(ted)40 b(a)f(phase)h(shift)f(whic)m(h)g(do)s(es)g(not)h(app)s (ear)f(in)g(our)g(case.)70 b(Ho)m(w)m(ev)m(er,)456 1695 y(in)31 b(this)g(pap)s(er,)g(w)m(e)h(will)g(presen)m(t)f(the)h(general) h(approac)m(h)e(that)h(w)m(orks)g(in)f(all)456 1803 y(cases.)3103 1910 y Fj(\003)555 2119 y Fs(Lemma)g(81)h(considers)e(a)i(foliation)g Fn(F)1890 2133 y Fp(F)1980 2119 y Fs(whose)f(lea)m(v)m(es)i(are)e(the)g (lev)m(el)h(sets)456 2227 y(of)e(a)h(function)f Fm(F)13 b Fs(:)849 2379 y Fm(L)911 2341 y Fp(F)911 2401 y(E)996 2379 y Fs(=)25 b Fn(f)p Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))p Fm(;)48 b(F)13 b Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))27 b(=)e Fm(E)5 b Fn(g)p Fm(;)47 b(E)30 b Fn(2)25 b Fs(\()p Fm(E)2558 2393 y Fq(1)2598 2379 y Fm(;)15 b(E)2705 2393 y Fq(2)2745 2379 y Fs(\))456 2530 y(and)29 b(that)i(are)g(also)g(parameterized)h(as)901 2683 y Fm(L)963 2645 y Fp(F)963 2705 y(E)1048 2683 y Fs(=)25 b Fn(f)p Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\)\()p Fm(I)1563 2697 y Fl(\000)1624 2683 y Fm(;)g(I)1704 2697 y Fq(+)1764 2683 y Fs(\))20 b Fn(\002)g Fk(T)1971 2645 y Fq(2)2010 2683 y Fm(;)46 b(I)32 b Fs(=)25 b Fm(\025)2302 2697 y Fp(E)2362 2683 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))p Fn(g)p Fm(;)456 2834 y Fs(and)33 b(it)h(giv)m(es)h (criteria)f(to)h(establish)f(that)g(their)f(lea)m(v)m(es)j(in)m (tersect)f(transv)m(er-)456 2942 y(sally)22 b(their)g(images)h(under)d (the)i(scattering)i(map)d Fm(S)5 b Fs(.)38 b(Note)23 b(that)f(these)h(images)456 3050 y(are)30 b(con)m(tained)i(in)e(the)h (lea)m(v)m(es)h(of)f(the)f(function)g Fm(F)k Fn(\016)20 b Fm(S)2376 3017 y Fl(\000)p Fq(1)2471 3050 y Fs(.)456 3215 y Fw(Remark)37 b(80.)43 b Fs(Giv)m(en)33 b(t)m(w)m(o)h(foliations) f Fn(F)9 b Fs(,)2020 3192 y(~)1996 3215 y Fn(F)g Fs(,)33 b(whic)m(h)f(are)g Fn(C)2596 3182 y Fq(1)2636 3215 y Fs(-close,)j(w)m(e)d(sa)m(y)456 3325 y(that)37 b Fn(F)46 b Fs(in)m(tersects)38 b(transv)m(ersally)1744 3302 y(~)1719 3325 y Fn(F)10 b Fs(|denoted)36 b(as)h Fn(F)45 b Fj(t)2583 3302 y Fs(~)2558 3325 y Fn(F)9 b Fs(|when)36 b(giv)m(en)456 3435 y(an)m(y)42 b(leaf)h(of)g Fn(F)9 b Fs(,)46 b(w)m(e)c(can)h(\014nd) e(another)h(leaf)h(of)2265 3412 y(~)2241 3435 y Fn(F)52 b Fs(for)42 b(whic)m(h)g(there)g(is)g(a)456 3543 y(non-trivial)37 b(in)m(tersection)i(whic)m(h)d(is)h(transv)m(ersal.)60 b(\(There)37 b(could)g(b)s(e)f(other)456 3651 y(non-transv)m(ersal)30 b(in)m(tersections\))555 3758 y(Note)36 b(that)g(our)e(use)g(of)h (\\foliation)i(in)m(tersects)f(transv)m(ersely)g("is)f(at)g(v)-5 b(ari-)456 3866 y(ance)25 b(with)g(standard)f(use)h(in)g(di\013eren)m (tial)h(top)s(ology)g(where)f(it)g(is)g(often)h(tak)m(en)456 3976 y(to)31 b(mean)f(that,)h(giv)m(en)g(a)g(leaf)g(of)f Fn(F)40 b Fs(and)29 b(a)i(leaf)g(of)2270 3953 y(~)2245 3976 y Fn(F)10 b Fs(,)30 b(they)h(either)f(in)m(tersect)456 4084 y(transv)m(ersally)h(or)f(do)h(not)f(in)m(tersect.)1354 b Fj(\003)555 4293 y Fs(T)-8 b(o)33 b(sho)m(w)f(the)h(transv)m(ersalit) m(y)h(b)s(et)m(w)m(een)e(the)h(foliations)h Fn(F)2616 4307 y Fp(F)2707 4293 y Fs(and)e Fn(F)2951 4313 y Fp(F)10 b Fl(\016)p Fp(S)3088 4294 y Fg(\000)p Fi(1)456 4401 y Fs(w)m(e)35 b(only)h(need)f(to)h(obtain)f(lo)m(w)m(er)i(b)s(ounds)c (for)i(the)g(angle)h(b)s(et)m(w)m(een)g(the)g(pa-)456 4509 y(rameterized)31 b(surface)1221 4660 y Fm(S)5 b Fs(\()p Fm(L)1379 4623 y Fp(F)1379 4683 y(E)1438 4660 y Fs(\))26 b(=)f Fn(f)p Fm(S)5 b Fs(\()p Fm(\025)1789 4674 y Fp(E)1849 4660 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))p Fm(;)g(';)g(s)p Fs(\))p Fn(g)456 4812 y Fs(and)29 b(the)i(implicit)g(surface)1103 4964 y Fm(L)1165 4927 y Fp(F)1165 4988 y(E)1221 4969 y Fg(0)1272 4964 y Fs(=)25 b Fn(f)p Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))p Fm(;)48 b(F)13 b Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))27 b(=)e Fm(E)2432 4927 y Fl(0)2456 4964 y Fn(g)p Fm(:)p eop end %%Page: 96 96 TeXDict begin 96 95 bop 456 251 a Fq(96)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 450 y Fs(More)k (precisely)-8 b(,)29 b(w)m(e)f(will)f(compute)h(sin)o(\()p Fm(\013)p Fs(\),)i(where)c Fm(\013)i Fs(is)f(the)g(angle)h(b)s(et)m(w)m (een)456 560 y(the)i(tangen)m(t)i(planes)e(to)h Fm(L)1394 527 y Fp(F)1394 588 y(E)1450 569 y Fg(0)1507 560 y Fs(and)e Fm(S)5 b Fs(\()p Fm(L)1841 527 y Fp(F)1841 586 y(E)1901 560 y Fs(\).)555 674 y(Recalling)42 b(that)e(the)g(normal)f(v)m(ector)i (to)g(the)e(tangen)m(t)j(plane)d(to)h Fm(L)2991 641 y Fp(F)2991 702 y(E)3047 684 y Fg(0)3113 674 y Fs(is)456 805 y(giv)m(en)27 b(b)m(y)847 756 y Fl(r)906 767 y Ff(I)t(;';s)1049 756 y Fp(F)p 822 784 307 4 v 822 856 a Fn(j)847 850 y Fl(r)906 861 y Ff(I)t(;';s)1049 850 y Fp(F)1104 856 y Fn(j)1138 805 y Fs(,)h(and)e(that)h(an)m(y)g(v)m(ector)i(of)e(the)f (tangen)m(t)j(plane)d(to)i Fm(S)5 b Fs(\()p Fm(L)3079 772 y Fp(F)3079 832 y(E)3139 805 y Fs(\))456 955 y(is)30 b(written)g(as)h Fm(D)1052 969 y Fp(v)1093 955 y Fs(\()p Fm(S)26 b Fn(\016)20 b Fs(\()p Fm(\025)1363 969 y Fp(E)1423 955 y Fm(;)15 b Fs(Id)p Fm(;)g Fs(Id\)\))26 b(=)f Fm(D)18 b Fs(\()p Fm(S)25 b Fn(\016)c Fs(\()p Fm(\025)2226 969 y Fp(E)2286 955 y Fm(;)15 b Fs(Id)p Fm(;)g Fs(Id)o(\)\))h Fm(v)s Fs(,)31 b(w)m(e)g(obtain:)456 1239 y(\(149\))111 b(sin)o(\()p Fm(\013)p Fs(\))27 b(=)e(max)1135 1305 y Fp(v)r Fl(2)p Fe(R)1266 1286 y Fi(2)1728 1178 y Fn(j)p Fm(D)1828 1192 y Fp(v)1870 1178 y Fm(F)33 b Fn(\016)21 b Fm(S)k Fn(\016)20 b Fs(\()p Fm(\025)2261 1192 y Fp(E)2321 1178 y Fm(;)15 b Fs(Id)p Fm(;)g Fs(Id\))p Fn(j)p 1328 1218 1702 4 v 1328 1301 a(j)p Fs(\()p Fn(r)1464 1315 y Fp(I)5 b(;';s)1622 1301 y Fm(F)13 b Fs(\))21 b Fn(\016)f Fs(\()p Fm(\025)1902 1315 y Fp(E)1962 1301 y Fm(;)15 b Fs(Id)p Fm(;)g Fs(Id\))p Fn(j)h(j)p Fm(D)2386 1315 y Fp(v)2427 1301 y Fm(S)25 b Fn(\016)c Fs(\()p Fm(\025)2662 1315 y Fp(E)2722 1301 y Fm(;)15 b Fs(Id)o Fm(;)g Fs(Id\))p Fn(j)3040 1239 y Fm(:)456 1438 y Fs(Hence,)39 b(w)m(e)f(will)f(obtain)h (lo)m(w)m(er)g(b)s(ounds)d(for)i(the)g(angles)h(taking)g(v)m(ectors)h Fm(v)456 1546 y Fs(that)31 b(mak)m(e)g(the)g(computations)g(in)f(the)g (righ)m(t)h(hand)f(side)g(of)37 b(\(149\))c(simple.)456 1708 y Fw(Lemma)h(81.)42 b Fo(L)-5 b(et)1083 1853 y Fm(F)38 b Fs(:)26 b Fn(A)f Fs(=)g(\()p Fm(I)1499 1867 y Fq(1)1538 1853 y Fm(;)15 b(I)1618 1867 y Fq(2)1658 1853 y Fs(\))21 b Fn(\002)f(J)36 b(\002)20 b Fs(\()p Fn(\000)p Fm(")2142 1867 y Fq(0)2182 1853 y Fm(;)15 b(")2264 1867 y Fq(0)2304 1853 y Fs(\))26 b Fn(!)f Fk(R)456 1999 y Fo(b)-5 b(e)32 b(a)h Fn(C)699 1966 y Fp(r)770 1999 y Fo(function,)f Fm(r)c Fn(\025)d Fs(2)p Fo(,)33 b(wher)-5 b(e)34 b Fn(J)41 b(\032)25 b Fk(T)1943 1966 y Fq(2)2015 1999 y Fo(is)32 b(an)i(op)-5 b(en)33 b(set.)555 2107 y(Assume)23 b(that)i(for)f(any)g Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))27 b Fn(2)e Fs(\()p Fm(I)1850 2121 y Fq(1)1889 2107 y Fm(;)15 b(I)1969 2121 y Fq(2)2009 2107 y Fs(\))p Fn(\002J)i Fo(,)25 b(the)e(e)-5 b(quation)24 b Fm(F)13 b Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))28 b(=)456 2215 y Fm(E)5 b Fo(,)26 b(for)e Fm(E)31 b Fn(2)25 b Fs(\()p Fm(E)1004 2229 y Fq(1)1044 2215 y Fm(;)15 b(E)1151 2229 y Fq(2)1191 2215 y Fs(\))25 b(=)g Fm(F)13 b Fs(\()p Fn(A)p Fs(\))25 b Fo(de\014nes)f(a)h(smo)-5 b(oth)26 b(surfac)-5 b(e)25 b(given)f(as)g(a)h(gr)-5 b(aph)1514 2361 y Fm(I)32 b Fs(=)25 b Fm(\025)1735 2375 y Fp(E)1795 2361 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))p Fm(:)555 2506 y Fo(L)-5 b(et)31 b Fm(S)j Fo(b)-5 b(e)30 b(the)h(sc)-5 b(attering)31 b(map)g(which)g(has)g(b)-5 b(e)g(en)30 b(c)-5 b(ompute)g(d)32 b(to)e(\014rst)h(or)-5 b(der)456 2614 y(in)42 b Fs(\(147\))c Fo(and)f(assume)g(that)g(ther)-5 b(e)37 b(exists)f(an)g(op)-5 b(en)37 b(set)f Fn(J)2567 2581 y Fl(0)2621 2614 y Fn(\032)31 b(J)16 b Fo(,)37 b Fn(J)2944 2581 y Fl(0)2999 2614 y Fn(6)p Fs(=)31 b Fn(;)p Fo(,)456 2722 y(and)43 b(a)g(c)-5 b(onstant)44 b Fm(C)49 b(>)43 b Fs(0)g Fo(indep)-5 b(endent)44 b(of)e Fm(")h Fo(and)g Fm(E)5 b Fo(,)45 b(such)e(that)g(for)g(any)456 2830 y Fs(\()p Fm(';)15 b(s)p Fs(\))26 b Fn(2)f(J)858 2797 y Fl(0)914 2830 y Fo(one)33 b(has,)456 3029 y Fs(\(150\))1061 2967 y Fn(jr)1162 2981 y Fp(';s)1264 2967 y Fs(\()p Fm(F)h Fn(\016)21 b Fm(S)k Fn(\016)c Fs(\()p Fm(\025)1692 2981 y Fp(E)1752 2967 y Fm(;)15 b Fs(Id)o Fm(;)g Fs(Id\)\()p Fm(';)g(s)p Fs(\)\))p Fn(j)p 1061 3008 1248 4 v 1176 3092 a(jr)1277 3106 y Fp(I)5 b(;';s)1435 3092 y Fm(F)13 b Fs(\()p Fm(\025)1594 3106 y Fp(E)1654 3092 y Fs(\()p Fm(';)i(s)p Fs(;)g Fm(")p Fs(\))p Fm(;)g(';)g(s)p Fs(\))p Fn(j)2343 3029 y(\025)25 b Fm(C)7 b(":)555 3228 y Fo(Then,)33 b(if)f(we)h(denote)h(by)e Fn(F)1517 3242 y Fp(F)1608 3228 y Fo(the)h(foliation)i(given)d(by)627 3375 y Fn(f)p Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))p Fm(;)50 b(F)13 b Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))28 b(=)d Fm(E)5 b(;)48 b(E)31 b Fn(2)25 b Fs(\()p Fm(E)2053 3389 y Fq(1)2093 3375 y Fm(;)15 b(E)2200 3389 y Fq(2)2239 3375 y Fs(\))p Fn(g)27 b Fs(=)d Fn([)2502 3393 y Fp(E)t Fl(2)p Fq(\()p Fp(E)2684 3402 y Fi(1)2718 3393 y Fp(;E)2790 3402 y Fi(2)2824 3393 y Fq(\))2856 3375 y Fm(L)2918 3337 y Fp(F)2918 3397 y(E)2977 3375 y Fm(;)456 3520 y Fo(we)32 b(have:)1540 3628 y Fn(F)1605 3642 y Fp(F)1690 3628 y Fj(t)24 b Fn(F)1840 3648 y Fp(F)10 b Fl(\016)p Fp(S)1977 3629 y Fg(\000)p Fi(1)2064 3628 y Fm(:)555 3755 y Fo(Mor)-5 b(e)42 b(pr)-5 b(e)g(cisely,)44 b(ther)-5 b(e)41 b(exists)h(a)f(c)-5 b(onstant)43 b Fm(C)2226 3722 y Fl(0)2248 3755 y Fo(,)g(indep)-5 b(endent)43 b(of)e Fm(")g Fo(and)456 3863 y Fm(E)5 b Fo(,)40 b(such)g(that)g(the)f(angle)h(b)-5 b(etwe)g(en)39 b(the)h(surfac)-5 b(es)39 b Fm(S)5 b Fs(\()p Fm(L)2410 3830 y Fp(F)2410 3890 y(E)2470 3863 y Fs(\))40 b Fo(and)g Fm(L)2790 3830 y Fp(F)2790 3891 y(E)2846 3872 y Fg(0)2871 3863 y Fo(,)h(at)e(the)456 3971 y(interse)-5 b(ction)33 b(c)-5 b(an)34 b(b)-5 b(e)32 b(b)-5 b(ounde)g(d)34 b(fr)-5 b(om)34 b(b)-5 b(elow)34 b(by)e Fm(C)2221 3938 y Fl(0)2244 3971 y Fm(")p Fo(.)456 4171 y(Pr)-5 b(o)g(of)20 b(.)555 4279 y Fs(T)-8 b(o)30 b(sho)m(w)f(the)h(transv)m(ersalit)m(y)h(b)s(et)m (w)m(een)f(the)g(foliations)h Fn(F)2596 4293 y Fp(F)2684 4279 y Fs(and)e Fn(F)2925 4299 y Fp(F)10 b Fl(\016)p Fp(S)3062 4280 y Fg(\000)p Fi(1)3149 4279 y Fs(,)456 4387 y(w)m(e)43 b(only)f(need)h(to)g(obtain)g(lo)m(w)m(er)h(b)s(ounds)c (for)i(the)h(angle)h(\(149\))g(b)s(et)m(w)m(een)456 4495 y(the)26 b(parameterized)g(surface)g Fm(S)5 b Fs(\()p Fm(\025)1645 4509 y Fp(E)1705 4495 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))p Fm(;)g(';)g(s)p Fs(\))29 b(and)c(the)h(implicit)h(surface)456 4603 y Fm(F)13 b Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))27 b(=)e Fm(E)1103 4570 y Fl(0)1127 4603 y Fs(.)555 4711 y(By)j(form)m(ula)f(\(147\))r(,)h(w)m(e)f(ha)m(v)m(e)i Fm(S)h Fs(=)25 b(Id)13 b(+)h Fm("S)2084 4725 y Fq(1)2136 4711 y Fs(+)2221 4719 y(O)2291 4730 y Fl(C)2332 4711 y Fi(1)9 b Fs(\()p Fm(")2448 4678 y Fq(1+)p Fp(\045)2579 4711 y Fs(\).)40 b(Hence,)29 b(there)456 4818 y(exists)i(a)f(constan)m (t)1170 4796 y(\026)1149 4818 y Fm(C)7 b Fs(,)30 b(indep)s(enden)m(t)f (of)i Fm(")f Fs(suc)m(h)h(that)1290 4964 y Fn(j)p Fm(D)1390 4978 y Fp(v)1431 4964 y Fm(S)25 b Fn(\016)c Fs(\()p Fm(\025)1666 4978 y Fp(E)1726 4964 y Fm(;)15 b Fs(Id)p Fm(;)g Fs(Id)o(\))p Fn(j)26 b(\024)2176 4941 y Fs(\026)2155 4964 y Fm(C)c Fn(j)p Fm(v)s Fn(j)p eop end %%Page: 97 97 TeXDict begin 97 96 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(97)456 450 y Fs(and)41 b(form)m(ula)i(\(149\))h(is)f(b)s(ounded)d(if)49 b(\(150\))c(is)d(v)m (eri\014ed.)77 b(Then,)44 b(in)e(order)456 558 y(to)36 b(obtain)g(that)g(the)g(foliations)h(in)m(tersect)g(transv)m(ersally)f (w)m(e)g(only)g(need)f(to)456 666 y(assume)29 b(condition)i(\(150\))r (,)f(and)g(w)m(e)g(obtain)h(that)g(the)f(angle)h(of)f(in)m(tersection) 456 774 y(is)35 b(b)s(ounded)e(from)i(b)s(elo)m(w)h(b)m(y)f Fm(C)1614 741 y Fl(0)1637 774 y Fm(")p Fs(,)i(where)e Fm(C)2081 741 y Fl(0)2139 774 y Fs(is)h(some)f(suitable)h(constan)m(t.) 3103 882 y Fj(\003)555 1140 y Fs(No)m(w,)45 b(w)m(e)c(apply)g(Lemma)g (81)g(to)h(study)e(the)h(non-resonan)m(t)g(region)h Fn(S)3122 1107 y Fp(L)456 1248 y Fs(and)e(the)h(resonan)m(t)h(regions)f Fn(S)1569 1215 y Fl(R)1629 1225 y Ff(j)1666 1248 y Fs(,)j(for)d Fm(j)49 b Fn(\025)42 b Fs(3,)j(where)40 b(the)i(tori,)i(giv)m(en)e(in) 456 1356 y(Prop)s(ositions)34 b(47)h(and)f(50)h(|see)g(also)g(Remarks)g (48)g(and)f(51|,)i(are)f(rather)456 1464 y(\\\015at".)41 b(That)27 b(is,)i(w)m(e)f(can)g(tak)m(e)h Fm(F)13 b Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))28 b(=)d Fm(I)d Fs(+)2295 1472 y(O)2365 1483 y Fl(C)2406 1464 y Fi(2)8 b Fs(\()p Fm(")p Fs(\))29 b(in)f(the)f(previous)456 1572 y(Lemma)34 b(81,)i(and)d Fm(\025)1168 1586 y Fp(E)1228 1572 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))33 b(=)e Fm(E)d Fs(+)22 b Fm(U)1907 1586 y Fp(E)1967 1572 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\),)37 b(where)d Fn(j)p Fm(U)2677 1586 y Fp(E)2737 1572 y Fn(j)2762 1598 y Fl(C)2803 1580 y Fi(2)2873 1572 y Fn(\024)d Fs(cte)p Fm(:)17 b(")456 1680 y Fs(as)30 b(giv)m(en)i(in)e(Remarks)g(48)h(and)f (51.)456 1906 y Fw(Lemma)g(82.)39 b Fo(Consider)30 b(a)f(foliation)h Fn(F)1873 1920 y Fp(F)1932 1906 y Fo(,)g(c)-5 b(ontaine)g(d)31 b(in)e(a)g(c)-5 b(onne)g(cte)g(d)30 b(c)-5 b(om-)456 2014 y(p)g(onent)35 b(of)g(the)f(non)h(r)-5 b(esonant)36 b(r)-5 b(e)g(gion)35 b Fn(S)1902 1981 y Fp(L)1988 2014 y Fo(de\014ne)-5 b(d)35 b(in)41 b Fs(\(56\))36 b Fo(\(or)f(of)f(the)h (r)-5 b(es-)456 2123 y(onant)33 b(r)-5 b(e)g(gions)34 b Fn(S)1083 2090 y Fl(R)1143 2100 y Ff(j)1180 2123 y Fo(,)e(de\014ne)-5 b(d)34 b(in)40 b Fs(\(69\))q Fo(,)33 b(for)g Fm(j)e Fn(\025)25 b Fs(3)p Fo(\).)555 2231 y(Assume)f(that)h (the)f(function)g Fm(F)36 b Fo(is)24 b(of)g(the)g(form)g Fm(F)13 b Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))28 b(=)d Fm(I)8 b Fs(+)2883 2239 y(O)2953 2251 y Fl(C)2994 2232 y Fi(2)3033 2231 y Fs(\()p Fm(")p Fs(\))p Fo(,)456 2339 y(so)30 b(that)h(e)-5 b(quation)31 b Fm(F)13 b Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))28 b(=)d Fm(E)35 b Fo(de\014nes)30 b(a)h(smo)-5 b(oth)32 b(surfac)-5 b(e)31 b(given)e(as)h(a)456 2447 y(gr)-5 b(aph)34 b Fm(I)e Fs(=)25 b Fm(\025)924 2461 y Fp(E)984 2447 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))27 b(=)e Fm(E)h Fs(+)1585 2455 y(O)1655 2466 y Fl(C)1696 2448 y Fi(2)1735 2447 y Fs(\()p Fm(")p Fs(\))p Fo(,)33 b(for)h Fs(\()p Fm(';)15 b(s)p Fs(\))26 b Fn(2)f Fk(T)2439 2414 y Fq(2)2478 2447 y Fo(.)555 2555 y(Assume)h(also)g(that)h(the)f(r) -5 b(e)g(duc)g(e)g(d)27 b(Poinc)-5 b(ar)n(\023)-44 b(e)27 b(function)e Fn(L)2491 2522 y Fl(\003)2556 2555 y Fo(de\014ne)-5 b(d)26 b(in)33 b Fs(\(145\))456 2664 y Fo(veri\014es,)25 b(for)e(any)h(value)g(of)f Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))27 b Fn(2)e Fm(H)1900 2678 y Fl(\000)1959 2664 y Fn(\\S)2082 2631 y Fp(L)2157 2664 y Fo(\(r)-5 b(esp)g(e)g(ctively)25 b(for)e Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))27 b Fn(2)456 2773 y Fm(H)532 2787 y Fl(\000)610 2773 y Fn(\\)20 b(S)753 2740 y Fl(R)813 2750 y Ff(j)883 2773 y Fo(\))32 b(that)i(the)f (function)456 3080 y Fs(\(151\))858 b Fm(\022)28 b Fn(7!)1716 3018 y Fm(@)5 b Fn(L)1832 2986 y Fl(\003)p 1716 3059 156 4 v 1744 3142 a Fm(@)g(\022)1881 3080 y Fs(\()p Fm(E)g(;)15 b(\022)s Fs(\))456 3365 y Fo(is)32 b(ne)-5 b(gative)33 b(and)h(non-c)-5 b(onstant)34 b(for)f Fm(\022)28 b Fn(2)d(J)2004 3332 y Fl(\003)1988 3392 y Fp(E)2047 3365 y Fo(.)555 3473 y(Then,)33 b(the)g(foliations)h Fn(F)1433 3487 y Fp(F)1525 3473 y Fo(and)f Fn(F)1766 3492 y Fp(F)10 b Fl(\016)p Fp(S)1903 3473 y Fg(\000)p Fi(1)41 b Fo(interse)-5 b(ct)34 b(tr)-5 b(ansversal)5 b(ly.)555 3581 y(Mor)-5 b(e)29 b(pr)-5 b(e)g(cisely,)31 b(ther)-5 b(e)29 b(exist)g(c)-5 b(onstants,)32 b Fm(C)2105 3548 y Fl(0)2127 3581 y Fo(,)e Fm(C)2257 3548 y Fl(00)2299 3581 y Fo(,)f(indep)-5 b(endent)30 b(of)f Fm(")g Fo(and)456 3688 y Fm(E)5 b Fo(,)34 b(such)h(that)g(any)g (surfac)-5 b(e)34 b Fm(S)5 b Fs(\()p Fm(L)1631 3655 y Fp(F)1631 3715 y(E)1691 3688 y Fs(\))35 b Fo(interse)-5 b(cts)35 b(at)f(some)h(p)-5 b(oint)36 b(the)e(surfac)-5 b(e)456 3802 y Fm(L)518 3769 y Fp(F)518 3830 y(E)574 3811 y Fg(0)632 3802 y Fo(such)33 b(that)h Fs(0)25 b Fm(<)g(E)1263 3769 y Fl(0)1307 3802 y Fn(\000)20 b Fm(E)31 b Fn(\024)25 b Fm(C)1664 3769 y Fl(00)1705 3802 y Fm(")p Fo(.)555 3917 y(The)34 b(angle)f(b)-5 b(etwe)g(en)34 b(the)f(surfac)-5 b(es)34 b Fm(S)5 b Fs(\()p Fm(L)1969 3884 y Fp(F)1969 3944 y(E)2029 3917 y Fs(\))34 b Fo(and)g Fm(L)2337 3884 y Fp(F)2337 3945 y(E)2393 3926 y Fg(0)2451 3917 y Fo(at)g(the)g(interse)-5 b(ction)456 4025 y(c)g(an)33 b(b)-5 b(e)32 b(b)-5 b(ounde)g(d)35 b(fr)-5 b(om)33 b(b)-5 b(elow)34 b(by)f Fm(C)1728 3992 y Fl(0)1750 4025 y Fm(")p Fo(.)456 4251 y Fw(Remark)45 b(83.)i Fs(W)-8 b(e)41 b(kno)m(w,)g(b)m(y) f(Prop)s(ositions)f(47)h(and)f(50,)j(that)e(the)g(gaps)456 4360 y(b)s(et)m(w)m(een)23 b(t)m(w)m(o)h(consecutiv)m(e)g(tori)f(are,)i (at)e(least,)j(of)d(order)2442 4337 y(\026)2431 4360 y Fm(I)12 b Fn(\000)5 b Fm(I)31 b Fn(')25 b Fm(")2768 4327 y Fq(3)p Fp(=)p Fq(2)2879 4360 y Fs(.)38 b(Then,)456 4468 y(when)19 b(w)m(e)j(apply)e(Lemma)h(82)h(to)f(these)g(tori)h(w)m (e)f(obtain)g(that)g(the)g(image)h(under)456 4576 y(the)33 b(scattering)h(map)f(of)g(a)g(torus)g(in)f(this)h(region)h(giv)m(en)f (b)m(y)g Fm(I)k Fs(=)29 b Fm(I)2811 4590 y Fq(0)2872 4576 y Fs(+)2965 4584 y(O)3036 4576 y(\()p Fm(")p Fs(\),)456 4683 y(in)m(tersect)43 b(transv)m(ersally)g(another)g(torus)f(giv)m(en) h(b)m(y)f Fm(I)52 b Fs(=)2576 4660 y(\026)2565 4683 y Fm(I)2605 4697 y Fq(0)2673 4683 y Fs(+)2771 4691 y(O)2842 4683 y(\()p Fm(")p Fs(\))43 b(with)467 4768 y(\026)456 4791 y Fm(I)496 4805 y Fq(0)560 4791 y Fm(>)25 b(I)696 4805 y Fq(0)736 4791 y Fs(,)30 b(and)979 4768 y(\026)968 4791 y Fm(I)1008 4805 y Fq(0)1068 4791 y Fn(\000)19 b Fm(I)1198 4805 y Fq(0)1263 4791 y Fs(=)1359 4799 y(O)1430 4791 y(\()p Fm(")p Fs(\).)1536 b Fj(\003)p eop end %%Page: 98 98 TeXDict begin 98 97 bop 456 251 a Fq(98)650 b(A.)23 b(Delshams,)g(R.)g (de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 450 y Fo(Pr)-5 b(o)g(of)20 b(.)69 b Fs(W)-8 b(e)41 b(apply)f(Lemma)g(81,)j(with)d Fm(F)13 b Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))44 b(=)d Fm(I)33 b Fs(+)2697 458 y(O)2767 470 y Fl(C)2808 451 y Fi(2)2847 450 y Fs(\()p Fm(")p Fs(\),)43 b(and)456 558 y Fm(\025)509 572 y Fp(E)568 558 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))53 b(=)d Fm(E)35 b Fs(+)1239 566 y(O)1310 578 y Fl(C)1351 559 y Fi(2)1390 558 y Fs(\()p Fm(")p Fs(\).)87 b(T)-8 b(o)46 b(c)m(hec)m(k)h(inequalit) m(y)h(\(150\))g(w)m(e)e(use)f(for-)456 666 y(m)m(ula)30 b(\(147\))r(,)h(so)g(that)g Fm(S)f Fs(=)25 b(Id)19 b(+)h Fm("S)1726 680 y Fq(1)1786 666 y Fs(+)1877 674 y(O)1948 686 y Fl(C)1989 667 y Fi(1)2027 666 y Fs(\()p Fm(")2104 633 y Fq(1+)p Fp(\045)2235 666 y Fs(\).)41 b(Hence,)32 b(w)m(e)f(compute:)456 830 y Fm(F)i Fn(\016)21 b Fm(S)5 b Fn(\016)p Fs(\()p Fm(\025)807 844 y Fp(E)867 830 y Fm(;)15 b Fs(Id)p Fm(;)g Fs(Id)o(\)\()p Fm(';)g(s)p Fs(\))648 970 y(=)p Fm(F)e Fs(\()p Fm(\025)878 984 y Fp(E)938 970 y Fs(\()p Fm(';)i(s)p Fs(;)g Fm(")p Fs(\))p Fm(;)g(';)g(s)p Fs(;)g Fm(")p Fs(\))739 1110 y(+)20 b Fm(")p Fn(r)948 1124 y Fp(I)5 b(;';s)1106 1110 y Fm(F)13 b Fs(\()p Fm(\025)1265 1124 y Fp(E)1325 1110 y Fs(\()p Fm(';)i(s)p Fs(;)g Fm(")p Fs(\))p Fm(;)g(';)g(s)p Fs(;)g Fm(")p Fs(\))g(\()5 b Fm(S)2029 1124 y Fq(1)2068 1110 y Fs(\()p Fm(\025)2156 1124 y Fp(E)2216 1110 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))p Fm(;)g(';)g(s)p Fs(;)g Fm(")p Fs(\))24 b(+)2924 1118 y(O)2995 1130 y Fl(C)3036 1111 y Fi(1)3074 1110 y Fs(\()p Fm(")3151 1073 y Fp(\045)3192 1110 y Fs(\)\))739 1262 y(+)830 1270 y(O)901 1282 y Fl(C)942 1263 y Fi(1)980 1262 y Fs(\()p Fm(")1057 1225 y Fq(2)1098 1262 y Fs(\))648 1460 y(=)p Fm(E)i Fn(\000)19 b Fm(")954 1398 y(@)5 b Fn(L)1070 1365 y Fl(\003)p 954 1439 156 4 v 983 1522 a Fm(@)g(\022)1120 1460 y Fs(\()p Fm(\025)1208 1474 y Fp(E)1268 1460 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))p Fm(;)g(')23 b Fn(\000)d Fm(\025)1828 1474 y Fp(E)1888 1460 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))p Fm(s)p Fs(\))22 b(+)2373 1468 y(O)2444 1479 y Fl(C)2485 1460 y Fi(1)8 b Fs(\()p Fm(")2600 1422 y Fq(1+)p Fp(\045)2731 1460 y Fs(\))648 1683 y(=)p Fm(E)26 b Fn(\000)19 b Fm(")954 1621 y(@)5 b Fn(L)1070 1588 y Fl(\003)p 954 1662 V 983 1745 a Fm(@)g(\022)1120 1683 y Fs(\()p Fm(E)g(;)15 b(')22 b Fn(\000)e Fm(E)5 b(s)p Fs(\))20 b(+)1700 1691 y(O)1771 1702 y Fl(C)1812 1683 y Fi(1)9 b Fs(\()p Fm(")1928 1645 y Fq(1+)p Fp(\045)2059 1683 y Fs(\))p Fm(;)456 1887 y Fs(and)39 b Fn(jr)743 1901 y Fp(I)5 b(;';s)901 1887 y Fm(F)13 b Fs(\()p Fm(\025)1060 1901 y Fp(E)1120 1887 y Fs(\()p Fm(';)i(s)p Fs(;)g Fm(")p Fs(\))p Fm(;)g(';)g(s)p Fs(;)g Fm(")p Fs(\))p Fn(j)46 b Fs(=)41 b(1)28 b(+)2066 1895 y(O)2137 1906 y Fl(C)2178 1887 y Fi(1)8 b Fs(\()p Fm(")p Fs(\).)72 b(If)39 b(for)h(an)m(y)h(v)-5 b(alue)41 b(of)456 2004 y(\()p Fm(E)5 b(;)15 b(\022)s Fs(\),)26 b(the)f(function)1246 1969 y Fp(@)t Fl(L)1336 1945 y Fg(\003)p 1246 1984 126 4 v 1271 2036 a Fp(@)t(\022)1381 2004 y Fs(\()p Fm(E)5 b(;)15 b(\022)s Fs(\))25 b(is)g(non-constan)m(t,) i(there)d(exists)h(an)g(in)m(terv)-5 b(al)487 2096 y(\026)456 2119 y Fn(J)518 2133 y Fp(E)602 2119 y Fn(\032)25 b(J)776 2086 y Fl(\003)760 2146 y Fp(E)850 2119 y Fs(where)1485 2187 y Fh(\014)1485 2241 y(\014)1485 2296 y(\014)1485 2350 y(\014)1525 2257 y Fm(@)1578 2224 y Fq(2)1618 2257 y Fn(L)1681 2224 y Fl(\003)p 1525 2298 195 4 v 1553 2381 a Fm(@)5 b(\022)1652 2355 y Fq(2)1730 2187 y Fh(\014)1730 2241 y(\014)1730 2296 y(\014)1730 2350 y(\014)1785 2318 y Fn(\025)25 b Fm(C)32 b(>)25 b Fs(0)p Fm(:)456 2534 y Fs(Then,)31 b(h)m(yp)s(othesis)h(\(150\))i(is)e(v)m(eri\014ed)f(in) 1961 2511 y(\026)1929 2534 y Fn(J)1991 2548 y Fp(E)2082 2534 y Fs(and)g(Lemma)h(81)h(applies.)46 b(On)456 2647 y(the)23 b(other)g(hand,)h(as)1197 2612 y Fp(@)t Fl(L)1287 2588 y Fg(\003)p 1197 2627 126 4 v 1222 2679 a Fp(@)t(\022)1332 2647 y Fs(\()p Fm(E)5 b(;)15 b(\022)s Fs(\))26 b Fm(<)f Fs(0,)g(the)f(surface)f Fm(S)5 b Fs(\()p Fm(L)2384 2614 y Fp(F)2384 2674 y(E)2444 2647 y Fs(\))23 b(in)m(tersect)i(surfaces)456 2765 y Fm(L)518 2732 y Fp(F)518 2794 y(E)574 2775 y Fg(0)599 2765 y Fs(,)31 b(for)f Fm(E)866 2732 y Fl(0)915 2765 y Fm(>)25 b(E)5 b Fs(,)31 b(and)f Fm(E)1388 2732 y Fl(0)1431 2765 y Fn(\000)20 b Fm(E)31 b Fs(=)1716 2773 y(O)1787 2765 y(\()p Fm(")p Fs(\).)1179 b Fj(\003)555 2938 y Fs(In)21 b(order)f(to)i(obtain)f(an)g(analogous)h(result)f(to)h(the)f(previous)g (Lemma)g(in)g(the)456 3046 y(regions)26 b(close)h(to)g(the)f (resonances)g(of)g(\014rst)f(order,)h(w)m(e)h(will)f(use)f(the)h(follo) m(wing)456 3154 y(tec)m(hnical)32 b(Lemma:)456 3332 y Fw(Lemma)h(84.)42 b Fo(L)-5 b(et)32 b Fm(a)p Fs(\()p Fm(\022)s Fs(\))p Fo(,)f Fm(b)p Fs(\()p Fm(\022)s Fs(\))h Fo(b)-5 b(e)31 b(functions)h(of)g(class)h Fn(C)2447 3299 y Fp(r)2485 3332 y Fo(,)f Fm(r)c Fn(\025)d Fs(0)p Fo(,)32 b(such)g(that)456 3440 y(the)h(function)f Fm(a)p Fs(\()p Fm(\022)s Fs(\))p Fm(=b)p Fs(\()p Fm(\022)s Fs(\))h Fo(is)g(not)g(c)-5 b(onstant)35 b(in)d(some)h(interval)h Fn(J)16 b Fo(.)555 3555 y(Then,)50 b(ther)-5 b(e)46 b(exists)h(a)f(c)-5 b(onstant)1827 3532 y Fs(~)1806 3555 y Fm(C)56 b(>)49 b Fs(0)d Fo(and)h(two)g(intervals)g Fn(J)2957 3569 y Fq(1)2996 3555 y Fo(,)i Fn(J)3135 3569 y Fq(2)456 3662 y Fo(subsets)35 b(of)h Fn(J)52 b Fo(such)36 b(that,)h(given)e(any)h (value)g Fs(\()p Fm(x;)15 b(y)s Fs(\))32 b Fn(2)e Fk(R)2470 3629 y Fq(2)2509 3662 y Fo(,)36 b(one)g(c)-5 b(an)36 b(cho)-5 b(ose)456 3770 y Fm(\022)34 b Fo(which)g(b)-5 b(elongs)33 b(to)g(one)g(of)g(the)g(two)h(intervals)f Fn(J)2241 3784 y Fq(1)2313 3770 y Fn(J)2375 3784 y Fq(2)2446 3770 y Fo(and)h(such)f(that:)456 3945 y Fs(\(152\))576 b Fn(j)p Fm(a)p Fs(\()p Fm(\022)s Fs(\))p Fm(x)20 b Fs(+)g Fm(b)p Fs(\()p Fm(\022)s Fs(\))p Fm(y)s Fn(j)26 b(\025)1959 3922 y Fs(~)1938 3945 y Fm(C)7 b Fs(\()p Fn(j)q Fm(x)p Fn(j)20 b Fs(+)g Fn(j)p Fm(y)s Fn(j)p Fs(\))456 4123 y Fo(Pr)-5 b(o)g(of.)43 b Fs(If)24 b Fm(a)p Fs(\()p Fm(\022)s Fs(\))p Fm(=b)p Fs(\()p Fm(\022)s Fs(\))h(is)g(not)g(constan)m(t)h(in)f Fn(J)41 b Fs(there)25 b(exist)h(at)f(least)h(t)m(w)m(o)h(v)-5 b(alues)456 4231 y Fm(\022)499 4245 y Fp(i)552 4231 y Fn(2)25 b(J)16 b Fs(,)30 b Fm(i)25 b Fs(=)g(1)p Fm(;)15 b Fs(2)31 b(\(and)e(in)m(terv)-5 b(als)31 b Fn(J)1721 4245 y Fp(i)1778 4231 y Fs(around)e(them\))h(suc)m(h)f(that)h(the)g (matrix)1374 4459 y Fm(A)25 b Fs(=)1563 4330 y Fh(\022)1672 4403 y Fm(a)p Fs(\()p Fm(\022)1798 4417 y Fq(1)1837 4403 y Fs(\))83 b Fm(b)p Fs(\()p Fm(\022)2072 4417 y Fq(1)2112 4403 y Fs(\))1672 4511 y Fm(a)p Fs(\()p Fm(\022)1798 4525 y Fq(2)1837 4511 y Fs(\))g Fm(b)p Fs(\()p Fm(\022)2072 4525 y Fq(2)2112 4511 y Fs(\))2189 4330 y Fh(\023)456 4687 y Fs(is)30 b(in)m(v)m(ertible.)42 b(Then,)30 b(giv)m(en)h(an)m(y)g (v)m(ector)h Fm(z)d Fs(=)c(\()p Fm(x;)15 b(y)s Fs(\),)32 b(w)m(e)f(ha)m(v)m(e)g(that)1555 4912 y Fn(j)p Fm(Az)t Fn(j)26 b(\025)1909 4851 y(j)q Fm(z)t Fn(j)p 1851 4891 214 4 v 1851 4975 a(j)q Fm(A)1945 4948 y Fl(\000)p Fq(1)2039 4975 y Fn(j)p eop end %%Page: 99 99 TeXDict begin 99 98 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)753 b(99)456 450 y Fs(where)34 b Fn(j\001j)h Fs(stands)f(for)g(the)h(sup)e(norm.)53 b(Giv)m(en)35 b Fm(z)t Fs(,)h(w)m(e)f(c)m(ho)s(ose)h Fm(\022)2730 464 y Fp(i)2758 450 y Fs(,)g(for)e Fm(i)f Fs(=)e(1)456 558 y(or)h Fm(i)d Fs(=)f(2,)33 b(suc)m(h)f(that)h Fn(j)p Fm(Az)t Fn(j)c Fs(=)f Fn(j)p Fm(a)p Fs(\()p Fm(\022)1680 572 y Fp(i)1709 558 y Fs(\))p Fm(x)22 b Fs(+)f Fm(b)p Fs(\()p Fm(\022)2027 572 y Fp(i)2055 558 y Fs(\))p Fm(y)s Fn(j)p Fs(,)33 b(and)f(taking)h(in)m(to)g(accoun)m(t)456 666 y(that)e(w)m(e)f(are)h(dealing)g(with)f(con)m(tin)m(uous)h (functions)f(w)m(e)h(obtain)g(that:)1292 868 y Fn(j)p Fm(a)p Fs(\()p Fm(\022)s Fs(\))p Fm(x)20 b Fs(+)g Fm(b)p Fs(\()p Fm(\022)s Fs(\))p Fm(y)s Fn(j)25 b(\025)2087 806 y Fs(1)p 2003 847 214 4 v 2003 930 a Fn(j)p Fm(A)2096 904 y Fl(\000)p Fq(1)2191 930 y Fn(j)2241 868 y(j)p Fm(z)t Fn(j)456 1100 y Fs(in)30 b Fn(J)624 1114 y Fp(i)651 1100 y Fs(.)41 b(The)30 b(pro)s(of)g(\014nishes)f(taking)1762 1077 y(~)1741 1100 y Fm(C)j(<)2013 1065 y Fq(1)p 1944 1080 175 4 v 1944 1134 a Fl(j)o Fp(A)2016 1115 y Fg(\000)p Fi(1)2099 1134 y Fl(j)2128 1100 y Fs(.)950 b Fj(\003)555 1311 y Fs(No)m(w,)32 b(w)m(e)g(will)f(apply)g(Lemma)g(81)h(to)g(study)e (the)h(resonan)m(t)g(regions)h Fn(S)3052 1278 y Fl(R)3112 1288 y Ff(j)3149 1311 y Fs(,)456 1419 y Fm(j)e Fs(=)25 b(1)p Fm(;)15 b Fs(2.)42 b(In)28 b(these)h(regions,)h(the)f(tori)g(and) f(the)h(stable)h(and)e(unstable)h(mani-)456 1527 y(folds)j(of)h(the)g (p)s(erio)s(dic)f(orbit,)i(are)f(not)g(\015at)g(as)g(w)m(e)g(sho)m(w)m (ed)g(in)g(Theorem)f(56)456 1635 y(and)27 b(Prop)s(osition)g(66.)41 b(The)27 b(fact)i(that)f(the)g(tori)h(are)f(not)g(\015at)g(has)f(the)h (conse-)456 1743 y(quence)i(that)g(the)g(dominan)m(t)g(e\013ect)i(of)e (comparing)g(a)g(torus)g(with)f(the)h(torus)456 1851 y(in)g(the)g(image)i(of)e(the)h(scattering)h(map)e(will)h(include)f (some)h(extra)g(terms.)555 1958 y(W)-8 b(e)27 b(recall)g(that)f(in)g (Theorem)f(56)h(w)m(e)g(sho)m(w)m(ed)g(that)g(the)g(in)m(v)-5 b(arian)m(t)27 b(ob)5 b(jects)456 2066 y(in)25 b(the)h(resonan)m(t)h (regions)f Fn(S)1438 2033 y Fl(R)1498 2043 y Ff(j)1535 2066 y Fs(,)h Fm(j)k Fs(=)25 b(1)p Fm(;)15 b Fs(2)27 b(are)f(giv)m(en)h(v)m(ery)g(appro)m(ximately)g(b)m(y)456 2174 y(the)j(lev)m(el)i(sets)f(of)f(the)g(Hamiltonian)i Fm(K)1856 2188 y Fq(0)1896 2174 y Fs(\()p Fm(y)s(;)15 b(x)p Fs(;)g Fm(")p Fs(\))32 b(in)e(\(82\))q(,)h(when)e(written)h(in) 456 2282 y(the)g(a)m(v)m(eraged)j(v)-5 b(ariables)31 b(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(\).)555 2390 y(The)27 b(relation)i(b)s(et)m(w)m(een)e(these)h(v)-5 b(ariables)28 b(and)f(the)g(original)i(ones)e(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))456 2498 y(are)27 b(the)g(c)m(hanges)h(of)f(v)-5 b(ariables)27 b(giv)m(en)h(in)f(Prop)s(osition)g(28,)h(Theorem)f(35)g (and)456 2606 y(in)j(\(73\))q(,)h(\(76\))q(,)g(and)f(is)g(giv)m(en)i (in)e(\014rst)f(order)h(b)m(y)1375 2770 y Fm(y)e Fs(=)d Fm(k)1591 2784 y Fq(0)1630 2770 y Fm(I)j Fs(+)20 b Fm(l)1816 2784 y Fq(0)1875 2770 y Fs(+)1966 2778 y(O)2037 2789 y Fl(C)2078 2770 y Fi(2)9 b Fs(\()p Fm(")p Fs(\))p Fm(:)456 2933 y Fs(Then,)37 b(using)g(\(80\))q(,)h(w)m(e)f(obtain)g(that)g(all)g (these)g(ob)5 b(jects)37 b(are)f(giv)m(en)h(b)m(y)g(the)456 3041 y(lev)m(el)32 b(sets)e(of)h(a)g(function:)456 3242 y(\(153\))98 b Fm(F)13 b Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))28 b(=)1345 3181 y(\()p Fm(k)1427 3195 y Fq(0)1467 3181 y Fm(I)f Fs(+)20 b Fm(l)1652 3195 y Fq(0)1712 3181 y Fs(+)1803 3189 y(O)1873 3201 y Fl(C)1914 3182 y Fi(2)1953 3181 y Fs(\()p Fm(")p Fs(\)\))2100 3148 y Fq(2)p 1345 3222 797 4 v 1720 3305 a Fs(2)2151 3242 y(\(1)h(+)2343 3250 y(O)2413 3262 y Fl(C)2454 3243 y Fi(2)2493 3242 y Fs(\()p Fm(")p Fs(\)\))h(+)2753 3250 y(O)2823 3262 y Fl(C)2864 3243 y Fi(2)2903 3242 y Fs(\()p Fm(")2980 3205 y Fp(j)3017 3242 y Fs(\))p Fm(;)456 3432 y Fs(where)29 b Fm(j)i Fs(=)25 b(1)p Fm(;)15 b Fs(2)32 b(is)e(the)h(order)f(of)g(the)h(resonance.)555 3540 y(Moreo)m(v)m(er,) 51 b(in)44 b(Corollary)h(57)h(and)e(Prop)s(osition)g(66)i(the)e(tori)i (and)e(the)456 3648 y(\(un\)stable)30 b(manifolds)g(of)g(p)s(erio)s (dic)f(orbits)h(w)m(ere)g(written)g(as)g(graphs)f(in)h(the)456 3762 y(v)-5 b(ariables)36 b(\()p Fm(y)s(;)15 b(x;)g(s)p Fs(\))36 b(as)g(functions)f Fm(y)h Fs(=)d Fn(Y)1927 3776 y Fl(\006)1986 3762 y Fs(\()p Fm(x;)15 b(E)5 b Fs(\))25 b(+)2339 3770 y(O)2410 3781 y Fl(C)2451 3762 y Fi(1)9 b Fs(\()p Fm(")2567 3729 y Fq(3)p Fp(=)p Fq(2)2678 3762 y Fs(\),)37 b(and)e(in)g(re-)456 3870 y(marks)d(58)h(and)f(67)h(w)m(e)g (ha)m(v)m(e)g(obtained)g(the)g(corresp)s(onding)e(expressions)h(in)456 3978 y(the)e(original)i(v)-5 b(ariables)31 b(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\).)555 4085 y(The)27 b(dominan)m(t)g(terms)g(in)g Fm(F)40 b Fs(and)26 b(in)h(the)g(expressions)g(of)g(these)h(in)m(v)-5 b(arian)m(t)456 4193 y(ob)5 b(jects)31 b(will)f(b)s(e)g(di\013eren)m(t) g(w)m(ether)h(the)f(resonance)h(is)f(of)h(order)e(1)i(or)f(2.)41 b(The)456 4301 y(h)m(yp)s(othesis)32 b(of)i(the)f(follo)m(wing)i (Lemmas)e(85)h(and)e(88)i(are)g(tailored)g(to)g(apply)456 4409 y(to)d(resonances)f(of)h(order)f(1)h(or)f(2.)456 4581 y Fw(Lemma)25 b(85.)36 b Fo(L)-5 b(et)26 b(us)f(c)-5 b(onsider)27 b(a)f(foliation)h Fn(F)2089 4595 y Fp(F)2173 4581 y Fo(in)f(a)f(c)-5 b(onne)g(cte)g(d)27 b(c)-5 b(omp)g(onent)456 4689 y(of)25 b(the)g(r)-5 b(esonant)27 b(r)-5 b(e)g(gion)26 b Fn(S)1386 4656 y Fl(R)1446 4665 y Fi(1)1485 4689 y Fo(,)g(de\014ne)-5 b(d)26 b(in)32 b Fs(\(69\))q Fo(,)26 b(wher)-5 b(e)26 b Fn(R)2490 4703 y Fq(1)2555 4689 y Fo(is)e(given)h(in)31 b Fs(\(20\))r Fo(.)456 4797 y(Mor)-5 b(e)32 b(pr)-5 b(e)g(cisely,)34 b(c)-5 b(onsider)33 b(a)g(r)-5 b(esonanc)g(e)34 b Fn(\000)p Fm(l)2039 4811 y Fq(0)2078 4797 y Fm(=k)2170 4811 y Fq(0)2235 4797 y Fn(2)25 b(R)2398 4811 y Fq(1)2438 4797 y Fo(,)32 b(and)h(in)f(the)h(r)-5 b(e)g(gion)1028 4964 y Fn(f)p Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))27 b Fn(2)e Fs([)p Fm(l)1537 4978 y Fq(0)1577 4964 y Fm(=k)1669 4978 y Fq(0)1729 4964 y Fn(\000)20 b Fm(L;)15 b(l)1949 4978 y Fq(0)1989 4964 y Fm(=k)2081 4978 y Fq(0)2142 4964 y Fs(+)20 b Fm(L)p Fs(])g Fn(\002)g Fk(T)2492 4927 y Fq(2)2531 4964 y Fn(g)p Fm(;)p eop end %%Page: 100 100 TeXDict begin 100 99 bop 456 251 a Fq(100)615 b(A.)23 b(Delshams,)g(R.)g(de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 450 y Fo(the)33 b(function)f Fm(F)46 b Fo(is)33 b(of)f(the)h(form)1106 686 y Fm(F)13 b Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))28 b(=)1692 625 y(\()p Fm(k)1774 639 y Fq(0)1814 625 y Fm(I)f Fs(+)20 b Fm(l)1999 639 y Fq(0)2039 625 y Fs(\))2074 592 y Fq(2)p 1692 665 422 4 v 1880 748 a Fs(2)2144 686 y(+)2235 694 y(O)2305 706 y Fl(C)2346 687 y Fi(2)2385 686 y Fs(\()p Fm(")p Fs(\))p Fm(;)456 892 y Fo(and,)42 b(for)e(some)h Fs(0)e Fm(<)e(\032)i(<)f(\031)s Fo(,)j(and)g(for)f(some)h(r)-5 b(ange)40 b(of)g(ener)-5 b(gies)40 b Fn(\000)p Fm(c)2983 906 y Fq(4)3023 892 y Fm(")e Fn(\024)456 1000 y Fm(E)31 b Fn(\024)26 b Fm(c)690 1014 y Fq(2)730 1000 y Fm(L)p Fo(,)32 b(the)i(e)-5 b(quation)34 b Fm(F)13 b Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))28 b(=)e Fm(E)5 b Fo(,)33 b(de\014nes)h(two)g(smo)-5 b(oth)35 b(surfac)-5 b(es)456 1110 y(given)32 b(as)h(a)g(gr)-5 b(aph)34 b Fm(I)e Fs(=)25 b Fm(\025)1356 1072 y Fl(\006)1356 1138 y Fp(E)1416 1110 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))35 b Fo(with:)520 1457 y Fm(\025)573 1419 y Fl(\006)573 1486 y Fp(E)632 1457 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))28 b(=)d Fn(\000)1140 1395 y Fm(l)1167 1409 y Fq(0)p 1131 1436 87 4 v 1131 1519 a Fm(k)1178 1533 y Fq(0)1247 1457 y Fs(+)1369 1395 y(1)p 1348 1436 V 1348 1519 a Fm(k)1395 1533 y Fq(0)1445 1457 y Fn(Y)1506 1471 y Fl(\006)1565 1457 y Fs(\()p Fm(\022)s(;)15 b(E)5 b Fs(\))20 b(+)1904 1465 y(O)1975 1477 y Fl(C)2016 1458 y Fi(2)2055 1457 y Fs(\()p Fm(")p Fs(\))954 1693 y(=)25 b Fn(\000)1140 1631 y Fm(l)1167 1645 y Fq(0)p 1131 1672 V 1131 1755 a Fm(k)1178 1769 y Fq(0)1247 1693 y Fn(\006)1369 1631 y Fs(1)p 1348 1672 V 1348 1755 a Fm(k)1395 1769 y Fq(0)1445 1693 y Fs(\(1)c(+)f Fm("b)p Fs(\))p Fm(`)p Fs(\()p Fm(\022)s(;)15 b(E)5 b Fs(\))21 b(+)2163 1631 y Fm(")p 2141 1672 V 2141 1755 a(k)2188 1769 y Fq(0)2257 1670 y Fs(~)2238 1693 y Fn(Y)2299 1707 y Fl(\006)2373 1693 y Fs(\()p Fm(`)p Fs(\()p Fm(\022)s(;)15 b(E)5 b Fs(\)\))21 b(+)2821 1701 y(O)2892 1713 y Fl(C)2933 1694 y Fi(2)2971 1693 y Fs(\()p Fm(")p Fs(\))p Fm(;)456 1287 y Fs(\(154\))456 1935 y Fo(for)36 b Fm(\032)30 b Fn(\024)g Fm(\022)j Fs(:=)d Fm(k)1031 1949 y Fq(0)1071 1935 y Fm(')22 b Fs(+)g Fm(l)1272 1949 y Fq(0)1312 1935 y Fm(s)30 b Fn(\024)g Fs(2)p Fm(\031)c Fn(\000)c Fm(\032)p Fo(,)36 b(wher)-5 b(e)36 b Fm(`)p Fs(\()p Fm(\022)s(;)15 b(E)5 b Fs(\))31 b(=)2470 1857 y Fh(p)p 2561 1857 613 4 v 78 x Fs(2\()p Fm(E)c Fn(\000)19 b Fm("U)10 b Fs(\()p Fm(\022)s Fs(;)15 b Fm(")p Fs(\)\))456 2055 y Fo(with)26 b Fm(U)10 b Fs(\()p Fm(\022)s Fs(;)15 b Fm(")p Fs(\))27 b Fo(de\014ne)-5 b(d)27 b(in)33 b Fs(\(78\))28 b Fo(and)1731 2032 y Fs(~)1712 2055 y Fn(Y)1773 2069 y Fl(\006)1858 2055 y Fo(is)e(given)g(in)f(L)-5 b(emma)28 b(60,)g(with)f Fm(\016)i Fs(=)c Fm(")p Fo(.)555 2162 y(Given)e(the)h(r)-5 b(e)g(duc)g(e)g(d)25 b(Poinc)-5 b(ar)n(\023)-44 b(e)24 b(function)g Fn(L)2049 2129 y Fl(\003)2088 2162 y Fo(,)h(de\014ne)-5 b(d)24 b(in)31 b Fs(\(145\))r Fo(,)25 b(for)f Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))26 b Fn(2)456 2272 y Fm(H)532 2286 y Fl(\000)610 2272 y Fn(\\)20 b(S)753 2239 y Fl(R)813 2248 y Fi(1)852 2272 y Fo(,)32 b(let)h(us)g(assume)g(that)h(the)f (function)456 2499 y Fs(\(155\))871 b Fm(\022)28 b Fn(7!)1729 2437 y Fm(@)5 b Fn(L)1845 2404 y Fl(\003)p 1729 2478 156 4 v 1757 2561 a Fm(@)g(\022)1894 2499 y Fs(\()p Fm(I)i(;)15 b(\022)s Fs(\))456 2706 y Fo(is)32 b(non-c)-5 b(onstant)35 b(and)e(ne)-5 b(gative)33 b(for)g Fm(\022)28 b Fn(2)d(J)2004 2673 y Fl(\003)1988 2733 y Fp(I)2044 2706 y Fo(.)555 2814 y(Assume,)j(mor)-5 b(e)g(over,)30 b(the)e(fol)5 b(lowing)28 b(hyp)-5 b(othesis,)31 b(which)d(is)35 b Fw(H5")27 b Fo(in)g(The-)456 2922 y(or)-5 b(em)33 b(7.)555 3030 y(The)g(function)456 3296 y Fs(\(156\))220 b Fm(\022)27 b Fn(7!)1077 3220 y Fm(k)1124 3234 y Fq(0)1164 3220 y Fm(U)1236 3187 y Fl(0)1259 3220 y Fs(\()p Fm(\022)s Fs(;)15 b(0\))1470 3184 y Fp(@)t Fl(L)1560 3160 y Fg(\003)p 1471 3199 126 4 v 1496 3251 a Fp(@)t(\022)1607 3220 y Fs(\()p Fn(\000)1731 3182 y Fp(l)1752 3191 y Fi(0)p 1723 3199 72 4 v 1723 3251 a Fp(k)1760 3260 y Fi(0)1804 3220 y Fm(;)1873 3184 y Fp(\022)p 1854 3199 V 1854 3251 a(k)1891 3260 y Fi(0)1936 3220 y Fs(\))20 b(+)g(2)p Fm(U)10 b Fs(\()p Fm(\022)s Fs(;)15 b(0\))2410 3184 y Fp(@)2451 3160 y Fi(2)2487 3184 y Fl(L)2536 3160 y Fg(\003)p 2411 3199 161 4 v 2436 3253 a Fp(@)t(\022)2512 3234 y Fi(2)2581 3220 y Fs(\()p Fn(\000)2705 3182 y Fp(l)2726 3191 y Fi(0)p 2698 3199 72 4 v 2698 3251 a Fp(k)2735 3260 y Fi(0)2779 3220 y Fm(;)2847 3184 y Fp(\022)p 2829 3199 V 2829 3251 a(k)2866 3260 y Fi(0)2910 3220 y Fs(\))p 1077 3275 1869 4 v 1717 3372 a(2)1772 3337 y Fp(@)1813 3318 y Fi(2)1848 3337 y Fl(L)1897 3318 y Fg(\003)p 1772 3352 161 4 v 1797 3406 a Fp(@)t(\022)1873 3387 y Fi(2)1942 3372 y Fs(\()p Fn(\000)2066 3335 y Fp(l)2087 3344 y Fi(0)p 2058 3352 72 4 v 2058 3404 a Fp(k)2095 3413 y Fi(0)2140 3372 y Fm(;)2208 3337 y Fp(\022)p 2190 3352 V 2190 3404 a(k)2227 3413 y Fi(0)2271 3372 y Fs(\))456 3554 y Fo(is)32 b(non-c)-5 b(onstant.)555 3661 y(Then,)33 b(the)g(foliations)h Fn(F)1433 3675 y Fp(F)1525 3661 y Fo(and)f Fn(F)1766 3681 y Fp(F)10 b Fl(\016)p Fp(S)1903 3662 y Fg(\000)p Fi(1)2022 3661 y Fo(interse)-5 b(ct)34 b(tr)-5 b(ansversal)5 b(ly.)555 3769 y(Mor)-5 b(e)29 b(pr)-5 b(e)g(cisely,)31 b(ther)-5 b(e)29 b(exist)g(c)-5 b(onstants,)32 b Fm(C)2105 3736 y Fl(0)2127 3769 y Fo(,)e Fm(C)2257 3736 y Fl(00)2299 3769 y Fo(,)f(indep)-5 b(endent)30 b(of)f Fm(")g Fo(and)456 3877 y Fm(E)5 b Fo(,)32 b(such)h(that:)601 4029 y Fs(\(1\))42 b Fo(A)n(ny)25 b(surfac)-5 b(e)26 b Fm(S)5 b Fs(\()p Fm(L)1405 3985 y Fp(F)r(;)p Fl(\000)1405 4058 y Fp(E)1531 4029 y Fs(\))25 b Fo(interse)-5 b(cts)27 b(at)f(some)g(p)-5 b(oint)27 b(the)f(surfac)-5 b(e)26 b Fm(L)3049 3985 y Fp(F)r(;)p Fl(\000)3049 4059 y Fp(E)3105 4040 y Fg(0)758 4159 y Fo(such)33 b(that)h Fm(E)1223 4126 y Fl(0)1272 4159 y Fm(<)25 b(E)38 b Fo(and)33 b Fm(E)26 b Fn(\000)20 b Fm(E)1905 4126 y Fl(0)1954 4159 y Fn(\024)25 b Fm(C)2122 4126 y Fl(00)2163 4159 y Fm(")15 b Fs(max)h(\()p Fn(j)q Fm(E)5 b Fn(j)2564 4117 y Fq(1)p Fp(=)p Fq(2)2689 4159 y Fm(;)15 b(")2771 4126 y Fq(1)p Fp(=)p Fq(2)2882 4159 y Fs(\))601 4280 y(\(2\))42 b Fo(A)n(ny)25 b(surfac)-5 b(e)26 b Fm(S)5 b Fs(\()p Fm(L)1405 4235 y Fp(F)r(;)p Fq(+)1405 4308 y Fp(E)1531 4280 y Fs(\))25 b Fo(interse)-5 b(cts)27 b(at)f(some)g(p)-5 b(oint)27 b(the)f(surfac)-5 b(e)26 b Fm(L)3049 4235 y Fp(F)r(;)p Fq(+)3049 4310 y Fp(E)3105 4291 y Fg(0)758 4410 y Fo(such)33 b(that)h Fm(E)1223 4377 y Fl(0)1272 4410 y Fm(>)25 b(E)38 b Fo(and)33 b Fm(E)1721 4377 y Fl(0)1765 4410 y Fn(\000)20 b Fm(E)31 b Fn(\024)25 b Fm(C)2122 4377 y Fl(00)2163 4410 y Fm(")15 b Fs(max)h(\()p Fn(j)q Fm(E)5 b Fn(j)2564 4368 y Fq(1)p Fp(=)p Fq(2)2689 4410 y Fm(;)15 b(")2771 4377 y Fq(1)p Fp(=)p Fq(2)2882 4410 y Fs(\))555 4567 y Fo(The)35 b(angle)g(b)-5 b(etwe)g(en)35 b(the)g(surfac)-5 b(es)35 b Fm(S)5 b Fs(\()p Fm(L)1976 4522 y Fp(F)r(;)p Fl(\006)1976 4595 y Fp(E)2102 4567 y Fs(\))35 b Fo(and)g Fm(L)2412 4522 y Fp(F)r(;)p Fl(\006)2412 4596 y Fp(E)2468 4577 y Fg(0)2572 4567 y Fo(at)g(the)g(interse)-5 b(c-)456 4675 y(tion)33 b(c)-5 b(an)33 b(b)-5 b(e)33 b(b)-5 b(ounde)g(d)34 b(fr)-5 b(om)34 b(b)-5 b(elow)33 b(by)g Fm(C)1916 4642 y Fl(0)1938 4675 y Fm(")p Fo(.)456 4856 y Fw(Remark)h(86.)42 b Fs(W)-8 b(e)31 b(kno)m(w,)f(b)m(y)g(Theorem)f(56)i(that)g(all)f(the)g(tori)h (in)e(the)h(reso-)456 4964 y(nan)m(t)h(region)i(are)f(giv)m(en)g(b)m(y) g(form)m(ulas)g(\(154\))h(for)f Fm(E)h Fs(=)27 b Fm(E)2475 4978 y Fp(i)2503 4964 y Fm(;)15 b(F)2601 4978 y Fp(i)2630 4964 y Fm(;)g(G)32 b Fs(and)f(v)m(erify)p eop end %%Page: 101 101 TeXDict begin 101 100 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)718 b(101)456 450 y Fs(that)960 890 y Fn(j)p Fm(E)1052 904 y Fp(i)1100 890 y Fn(\000)20 b Fm(E)1258 904 y Fp(i)p Fq(+1)1377 890 y Fn(j)83 b(\024)g Fm(")1691 826 y Fi(3)p 1691 838 31 3 v 1691 879 a(2)1731 853 y Fq(+)1796 826 y Fi(1)p 1796 838 V 1796 879 a(2)1866 890 y Fn(\024)25 b Fs(max\()2166 807 y Fh(p)p 2257 807 96 4 v 83 x Fm(E)2324 904 y Fp(i)2353 890 y Fm(;)15 b(")2435 853 y Fq(1)p Fp(=)p Fq(2)2546 890 y Fs(\))p Fm(;)998 1045 y Fn(j)q Fm(E)1091 1060 y Fp(l)1112 1071 y Ff(E)1188 1045 y Fn(\000)20 b Fm(F)1337 1059 y Fq(1)1377 1045 y Fn(j)83 b(\024)g Fm(")1691 980 y Fi(3)p 1691 992 31 3 v 1691 1034 a(2)1731 1008 y Fq(+)1796 980 y Fi(1)p 1796 992 V 1796 1034 a(2)1866 1045 y Fn(\024)25 b Fs(max\()2166 968 y Fh(p)p 2257 968 144 4 v 77 x Fm(E)2324 1060 y Fp(l)2345 1071 y Ff(E)2401 1045 y Fm(;)15 b(")2483 1008 y Fq(1)p Fp(=)p Fq(2)2594 1045 y Fs(\))p Fm(;)977 1205 y Fn(j)p Fm(F)1060 1219 y Fp(i)1109 1205 y Fn(\000)20 b Fm(F)1258 1219 y Fp(i)p Fq(+1)1377 1205 y Fn(j)83 b(\024)g Fm(")1691 1140 y Fi(3)p 1691 1152 31 3 v 1691 1194 a(2)1731 1168 y Fq(+)1796 1140 y Fi(1)p 1796 1152 V 1796 1194 a(2)1866 1205 y Fn(\024)25 b Fs(max\()2166 1121 y Fh(p)p 2257 1121 87 4 v 84 x Fm(F)2315 1219 y Fp(i)2344 1205 y Fm(;)15 b(")2426 1168 y Fq(1)p Fp(=)p Fq(2)2537 1205 y Fs(\))1034 1360 y Fn(j)q Fm(F)1118 1375 y Fp(l)1139 1386 y Ff(F)1214 1360 y Fn(\000)20 b Fm(G)p Fn(j)84 b(\024)f Fm(")1691 1295 y Fi(3)p 1691 1307 31 3 v 1691 1348 a(2)1731 1322 y Fq(+)1796 1295 y Fi(1)p 1796 1307 V 1796 1348 a(2)1866 1360 y Fn(\024)25 b Fs(max\()2166 1282 y Fh(p)p 2257 1282 135 4 v 78 x Fm(F)2315 1375 y Fp(l)2336 1386 y Ff(F)2392 1360 y Fm(;)15 b(")2474 1322 y Fq(1)p Fp(=)p Fq(2)2585 1360 y Fs(\))p Fm(:)456 1800 y Fs(and)37 b(that)h(b)m(y)g(and)g (Corollary)g(57)h(and)e(Remark)h(58)h(that,)h(when)d(they)h(are)456 1908 y(written)25 b(as)g(graphs,)h(the)f(distance)g(b)s(et)m(w)m(een)h (to)g(consecutiv)m(e)h(tori)e(is)g(of)g(order)456 2025 y(O)526 2017 y(\()p Fm(")603 1984 y Fq(3)p Fp(=)p Fq(2)714 2017 y Fs(\).)555 2125 y(Then,)33 b(when)f(w)m(e)i(apply)f(Lemma)g(85)h (to)g(these)f(tori)h(w)m(e)f(obtain)h(that)g(the)456 2233 y(image)42 b(under)d(the)i(scattering)h(map)e(of)h(a)g(torus)g(in) f(this)h(region)g(in)m(tersect)456 2341 y(transv)m(ersally)31 b(another)g(torus.)1551 b Fj(\003)456 3003 y Fo(Pr)-5 b(o)g(of.)43 b Fs(It)30 b(su\016ces)g(to)h(apply)f(Lemma)h(81)g(to)g (the)g(function)1106 3498 y Fm(F)13 b Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))28 b(=)1692 3437 y(\()p Fm(k)1774 3451 y Fq(0)1814 3437 y Fm(I)f Fs(+)20 b Fm(l)1999 3451 y Fq(0)2039 3437 y Fs(\))2074 3404 y Fq(2)p 1692 3477 422 4 v 1880 3561 a Fs(2)2144 3498 y(+)2235 3506 y(O)2305 3518 y Fl(C)2346 3499 y Fi(2)2385 3498 y Fs(\()p Fm(")p Fs(\))p Fm(;)456 3980 y Fs(and)29 b Fm(\025)685 3941 y Fl(\006)685 4008 y Fp(E)775 3980 y Fs(giv)m(en)j(b)m(y)f(\(154\))r(.)555 4088 y(Note)e(that)f(to)g(study)f (transv)m(ersalit)m(y)i(of)f(the)f(foliations,)j(w)m(e)e(can)g (consider)456 4195 y Fm(E)35 b Fs(\014xed)30 b(since)h Fm(E)k Fs(is)c(only)f(a)h(lab)s(el)g(for)f(the)g(lea)m(v)m(es.)555 4303 y(Since)25 b(w)m(e)h(consider)f Fm(E)30 b Fs(\014xed,)c(the)f (analysis)h(of)f(dominan)m(t)g(terms)g(will)h(only)456 4411 y(in)m(v)m(olv)m(e)32 b(estimating)g(deriv)-5 b(ativ)m(es)32 b(with)e(resp)s(ect)g(to)h(the)g(angle)g(v)-5 b(ariables.)555 4519 y(W)d(e)28 b(will)f(also)g(see)g(that)g(the)g(angle)g(v)-5 b(ariables)27 b(will)g(en)m(ter)g(in)f(the)h(dominan)m(t)456 4627 y(terms)j(only)g(trough)h Fm(\022)c Fs(=)e Fm(k)1415 4641 y Fq(0)1455 4627 y Fm(')20 b Fs(+)g Fm(l)1652 4641 y Fq(0)1692 4627 y Fm(s)30 b Fs(\(the)h(resonan)m(t)g(angle\).)555 4735 y(In)37 b(order)h(to)h(c)m(hec)m(k)g(inequalit)m(y)i(\(150\))f(w)m (e)e(use)g(form)m(ula)g(\(147\))r(,)i(so)e(that)456 4845 y Fm(S)30 b Fs(=)25 b(Id)s(+)t Fm("S)898 4859 y Fq(1)942 4845 y Fs(+)1017 4853 y(O)1087 4864 y Fl(C)1128 4845 y Fi(1)9 b Fs(\()p Fm(")1244 4812 y Fq(1+)p Fp(\045)1375 4845 y Fs(\),)24 b(and)e(that,)j(b)m(y)e(\(98\))r(,)h Fm(\025)2227 4806 y Fl(\006)2227 4873 y Fp(E)2309 4845 y Fs(is)e(a)h(b)s(ounded)d(function,)456 4964 y(and)33 b(its)h(deriv)-5 b(ativ)m(es)35 b(with)e(resp)s(ect)g(to)i Fm(';)15 b(s)34 b Fs(are)g(of)g(order)2540 4972 y(O)2610 4964 y(\()p Fm(")2687 4931 y Fq(1)p Fp(=)p Fq(2)2798 4964 y Fs(\).)51 b(Hence,)p eop end %%Page: 102 102 TeXDict begin 102 101 bop 456 251 a Fq(102)615 b(A.)23 b(Delshams,)g(R.)g(de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 450 y Fs(w)m(e)30 b(can)h(compute:)456 635 y Fm(F)i Fn(\016)21 b Fm(S)5 b Fn(\016)p Fs(\()p Fm(\025)807 596 y Fl(\006)807 663 y Fp(E)867 635 y Fm(;)15 b Fs(Id)p Fm(;)g Fs(Id)o(\)\()p Fm(';)g(s)p Fs(\))744 785 y(=)25 b Fm(F)13 b Fs(\()p Fm(\025)999 747 y Fl(\006)999 813 y Fp(E)1059 785 y Fs(\()p Fm(';)i(s)p Fs(;)g Fm(")p Fs(\))p Fm(;)g(';)g(s)p Fs(;)g Fm(")p Fs(\))24 b(+)c Fm(")p Fn(r)1885 799 y Fp(I)5 b(;';s)2043 785 y Fm(F)13 b Fs(\()p Fm(\025)2202 747 y Fl(\006)2202 813 y Fp(E)2262 785 y Fs(\()p Fm(';)i(s)p Fs(;)g Fm(")p Fs(\))p Fm(;)g(';)g(s)p Fs(;)g Fm(")p Fs(\))p 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1747 a Fm(k)2208 1761 y Fq(0)2278 1684 y Fs(+)2399 1623 y(1)p 2379 1663 V 2379 1747 a Fm(k)2426 1761 y Fq(0)2475 1684 y Fn(Y)2536 1698 y Fl(\006)2595 1684 y Fs(\()p Fm(\022)s(;)15 b(E)5 b Fs(\))2823 1556 y Fh(\023)2906 1684 y Fm(s)2949 1556 y Fh(\023)3036 1684 y Fs(+)3126 1692 y(O)3197 1704 y Fl(C)3238 1685 y Fi(1)3277 1684 y Fs(\()p Fm(")3354 1647 y Fp(\045)3395 1684 y Fs(\))3430 1556 y Fh(\023)648 1886 y Fs(+)734 1894 y(O)805 1906 y Fl(C)846 1887 y Fi(1)885 1886 y Fs(\()p Fm(")962 1849 y Fq(2)1002 1886 y Fs(\))744 2026 y(=)25 b Fm(E)h Fn(\007)20 b Fm("k)1113 2040 y Fq(0)1153 2026 y Fs(\(1)h(+)f Fm("b)p Fs(\))p Fm(`)p Fs(\()p Fm(\022)s(;)15 b(E)5 b Fs(\))p Fn(\001)734 2102 y Fh(\022)811 2168 y Fm(@)g Fn(L)927 2135 y Fl(\003)p 811 2209 156 4 v 839 2292 a Fm(@)g(\022)992 2102 y Fh(\022)1059 2230 y Fn(\000)1149 2168 y Fm(l)1176 2182 y Fq(0)p 1140 2209 87 4 v 1140 2292 a Fm(k)1187 2306 y Fq(0)1256 2230 y Fn(\006)1357 2168 y Fs(\(1)21 b(+)f Fm("b)p Fs(\))p 1357 2209 309 4 v 1468 2292 a Fm(k)1515 2306 y Fq(0)1676 2230 y Fm(`)p Fs(\()p Fm(\022)s(;)15 b(E)5 b Fs(\))p Fm(;)15 b(')21 b Fn(\000)2153 2102 y Fh(\022)2220 2230 y Fn(\000)2311 2168 y Fm(l)2338 2182 y Fq(0)p 2301 2209 87 4 v 2301 2292 a Fm(k)2348 2306 y Fq(0)2418 2230 y Fn(\006)2519 2168 y Fs(\(1)g(+)e Fm("b)p Fs(\))p 2519 2209 309 4 v 2629 2292 a Fm(k)2676 2306 y Fq(0)2837 2230 y Fm(`)p Fs(\()p Fm(\022)s(;)c(E)5 b Fs(\))3103 2102 y Fh(\023)3186 2230 y Fm(s)3229 2102 y Fh(\023)739 2456 y Fs(+)830 2464 y(O)901 2476 y Fl(C)942 2457 y Fi(1)980 2456 y Fs(\()p Fm(")1057 2418 y Fp(\045)1099 2456 y Fs(\))1134 2355 y Fh(\021)1208 2456 y Fs(+)1299 2464 y(O)1370 2476 y Fl(C)1411 2457 y Fi(1)1450 2456 y Fs(\()p Fm(")1527 2418 y Fq(2)1567 2456 y Fs(\))719 2631 y Fm(E)26 b Fn(\007)19 b Fm(")p Fn(M)p Fs(\()p Fm(\022)s Fs(;)c Fm(")p Fs(\))22 b(+)1364 2639 y(O)1435 2650 y Fl(C)1476 2632 y Fi(1)1514 2631 y Fs(\()p Fm(")1591 2593 y Fq(1+)p Fp(\045)1722 2631 y Fs(\))555 2819 y(No)m(w,)32 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4302 y Fn(M)p Fs(\()p Fm(\022)s Fs(;)15 b Fm(")p Fs(\))26 b(=)795 4505 y(=)p Fm(k)913 4519 y Fq(0)953 4423 y Fh(p)p 1044 4423 616 4 v 82 x Fs(2\()p Fm(E)g Fn(\000)20 b Fm("U)10 b Fs(\()p Fm(\022)s Fs(;)15 b(0\)\))1675 4377 y Fh(\022)1752 4444 y Fm(@)5 b Fn(L)1868 4411 y Fl(\003)p 1752 4484 156 4 v 1780 4568 a Fm(@)g(\022)1917 4505 y Fs(\()p Fn(\000)2043 4444 y Fm(l)2070 4458 y Fq(0)p 2033 4484 87 4 v 2033 4568 a Fm(k)2080 4582 y Fq(0)2130 4505 y Fm(;)2201 4444 y(\022)p 2180 4484 V 2180 4568 a(k)2227 4582 y Fq(0)2277 4505 y Fs(\))20 b(+)2423 4513 y(O)2494 4525 y Fl(C)2535 4506 y Fi(1)2574 4505 y Fs(\()p Fm(")2651 4468 y Fp(\027)t(=)p Fq(2)2765 4505 y Fs(\))2800 4377 y Fh(\023)2883 4505 y Fm(:)758 4748 y Fs(T)-8 b(o)37 b(apply)f(Lemma)g(81)h(w)m(e)g(need)f(to)h(c)m(hec)m(k)g(\(150\))r(,)h (that)f(is,)h(it)e(suf-)758 4856 y(\014ces)29 b(to)h(sho)m(w)f(that)h (w)m(e)f(can)g(b)s(ound)e(from)i(b)s(elo)m(w)g(the)g(deriv)-5 b(ativ)m(e)31 b(of)758 4964 y(this)39 b(function)f(divided)f(b)m(y)i Fm(k)1818 4978 y Fq(0)1858 4964 y Fm(\025)1911 4926 y Fl(\006)1911 4993 y Fp(E)1970 4964 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))28 b(+)d Fm(l)2415 4978 y Fq(0)2454 4964 y Fs(.)65 b(Computing)38 b(this)p eop end %%Page: 103 103 TeXDict begin 103 102 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)718 b(103)758 450 y Fs(deriv)-5 b(ativ)m(e,)33 b(w)m(e)d(obtain:)610 601 y Fm(@)p 587 641 99 4 v 587 725 a(@)5 b(\022)711 662 y Fs(\()p Fn(M)p Fs(\()p Fm(\022)s Fs(;)15 b Fm(")p Fs(\)\))27 b(=)625 897 y(=)1016 836 y Fm(k)1063 850 y Fq(0)p 706 876 707 4 v 706 894 a Fh(p)p 797 894 616 4 v 78 x Fs(2\()p Fm(E)f Fn(\000)20 b Fm("U)10 b Fs(\()p Fm(\022)s Fs(;)15 b(0\)\))1438 769 y Fh(\022)1505 897 y Fs(2)p Fm(E)1655 836 y(@)1708 803 y Fq(2)p 1632 876 138 4 v 1632 959 a Fm(@)5 b(\022)1731 933 y Fq(2)1780 897 y Fn(L)1843 860 y Fl(\003)1882 897 y Fs(\()p Fn(\000)2009 836 y Fm(l)2036 850 y Fq(0)p 1998 876 87 4 v 1998 959 a Fm(k)2045 973 y Fq(0)2095 897 y Fm(;)2166 836 y(\022)p 2146 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Fq(0)2763 2317 y Fm(;)2835 2256 y(\022)p 2814 2297 V 2814 2380 a(k)2861 2394 y Fq(0)2910 2317 y Fs(\))2945 2189 y Fh(\025)q(\023)720 2526 y Fs(+)806 2534 y(O)876 2546 y Fl(C)917 2527 y Fi(0)956 2526 y Fs(\()p Fm(")1033 2489 y Fp(\027)t(=)p Fq(2)1147 2526 y Fs(\))p Fm(;)858 2696 y Fs(No)m(w,)31 b(w)m(e)g(apply)f(Lemma)h (84,)g(with)f Fm(x)25 b Fs(=)g Fn(\000)p Fm(")p Fs(,)31 b Fm(y)d Fs(=)d Fm(E)5 b Fs(,)31 b(and)617 2921 y Fm(a)p Fs(\()p Fm(\022)s Fs(\))83 b(=)g Fm(k)1065 2935 y Fq(0)1105 2921 y Fm(U)1177 2883 y Fl(0)1200 2921 y Fs(\()p Fm(\022)s Fs(;)15 b(0\))1411 2859 y Fm(@)5 b Fn(L)1527 2826 y Fl(\003)p 1411 2900 156 4 v 1440 2983 a Fm(@)g(\022)1577 2921 y Fs(\()p Fn(\000)1703 2859 y Fm(l)1730 2873 y Fq(0)p 1693 2900 87 4 v 1693 2983 a Fm(k)1740 2997 y Fq(0)1790 2921 y Fm(;)1861 2859 y(\022)p 1840 2900 V 1840 2983 a(k)1887 2997 y Fq(0)1937 2921 y Fs(\))20 b(+)g(2)p Fm(U)10 b Fs(\()p Fm(\022)s Fs(;)15 b(0\))2411 2859 y Fm(@)2464 2826 y Fq(2)2505 2859 y Fn(L)2568 2826 y Fl(\003)p 2411 2900 195 4 v 2441 2983 a Fm(@)5 b(\022)2540 2957 y Fq(2)2617 2921 y Fs(\()p Fn(\000)2743 2859 y Fm(l)2770 2873 y Fq(0)p 2733 2900 87 4 v 2733 2983 a Fm(k)2780 2997 y Fq(0)2830 2921 y Fm(;)2901 2859 y(\022)p 2880 2900 V 2880 2983 a(k)2927 2997 y Fq(0)2977 2921 y Fs(\))626 3169 y Fm(b)p Fs(\()p Fm(\022)s Fs(\))83 b(=)g(2)1073 3108 y Fm(@)1126 3075 y Fq(2)1166 3108 y Fn(L)1229 3075 y Fl(\003)p 1073 3148 195 4 v 1102 3231 a Fm(@)5 b(\022)1201 3205 y Fq(2)1278 3169 y Fs(\()p Fn(\000)1404 3108 y Fm(l)1431 3122 y Fq(0)p 1394 3148 87 4 v 1394 3231 a Fm(k)1441 3245 y Fq(0)1491 3169 y Fm(;)1562 3108 y(\022)p 1541 3148 V 1541 3231 a(k)1588 3245 y Fq(0)1638 3169 y Fs(\))p Fm(:)758 3378 y Fs(Using)31 b(Lemma)f(84,)i(and)e(that)1131 3547 y Fn(j)q Fs(2\()p Fm(E)c Fn(\000)20 b Fm("U)10 b Fs(\()p Fm(\022)s(;)15 b Fs(0\)\))p Fn(j)27 b(\024)e Fs(cte)p Fm(:)16 b Fs(\()p Fn(j)q Fm(E)5 b Fn(j)21 b Fs(+)f Fn(j)p Fm(")p Fn(j)q Fs(\))p Fm(;)758 3717 y Fs(w)m(e)40 b(obtain)h(that)f (for)f Fm(\022)j Fs(either)e(in)f Fn(J)2077 3731 y Fq(1)2116 3717 y Fs(,)k Fn(J)2246 3731 y Fq(2)2285 3717 y Fs(,)f(where)d Fn(J)2686 3731 y Fp(i)2754 3717 y Fn(\032)i(J)2944 3684 y Fl(\003)2928 3748 y(\000)p Fp(l)3004 3757 y Fi(0)3038 3748 y Fp(=k)3110 3757 y Fi(0)3149 3717 y Fs(,)758 3833 y Fm(i)26 b Fs(=)f(1)p Fm(;)15 b Fs(2)1329 3883 y Fh(\014)1329 3937 y(\014)1329 3992 y(\014)1329 4046 y(\014)1329 4101 y(\014)1488 3939 y Fp(@)p 1471 3954 76 4 v 1471 4006 a(@)t(\022)1572 3975 y Fs(\()p Fn(M)p Fs(\()p Fm(\022)s Fs(;)g Fm(")p Fs(\)\))p 1369 4021 674 4 v 1369 4110 a Fm(k)1416 4124 y Fq(0)1456 4110 y Fm(\025)1509 4072 y Fl(\006)1509 4139 y Fp(E)1569 4110 y Fs(\()p Fm(';)g(s)p Fs(;)g Fm(")p Fs(\))22 b(+)e Fm(l)2003 4124 y Fq(0)2052 3883 y Fh(\014)2052 3937 y(\014)2052 3992 y(\014)2052 4046 y(\014)2052 4101 y(\014)2108 4042 y Fn(\025)2224 4019 y Fs(~)2204 4042 y Fm(C)6 b(:)758 4269 y Fs(Consequen)m(tly)-8 b(,)32 b(the)f(angle)h(of)e(in)m(tersection)j(can)e(b)s(e)f(b)s(ounded) e(from)758 4377 y(b)s(elo)m(w)j(b)m(y)f Fm(C)1214 4344 y Fl(0)1237 4377 y Fm(")p Fs(,)h(for)f(some)h(suitable)g(constan)m(t)g (indep)s(enden)m(t)f(of)g Fm(")p Fs(.)858 4493 y(T)-8 b(o)21 b(iden)m(tify)f(more)h(precisely)g(whic)m(h)f(surfaces)g(will)g (in)m(tersect)i Fm(S)5 b Fs(\()p Fm(L)3140 4449 y Fp(F)r(;)p Fl(\006)3140 4521 y Fp(E)3266 4493 y Fs(\),)758 4601 y(w)m(e)25 b(only)f(need)g(to)h(observ)m(e)g(that)g(the)f(function)g Fn(M)p Fs(\()p Fm(\022)s Fs(;)15 b Fm(")p Fs(\))25 b(is)f(giv)m(en)i (ap-)758 4709 y(pro)m(ximately)32 b(b)m(y)1120 4921 y Fm(k)1167 4935 y Fq(0)1207 4839 y Fh(p)p 1298 4839 616 4 v 82 x Fs(2\()p Fm(E)26 b Fn(\000)20 b Fm("U)10 b Fs(\()p Fm(\022)s Fs(;)15 b(0\)\))1924 4860 y Fm(@)5 b Fn(L)2040 4827 y Fl(\003)p 1924 4901 156 4 v 1952 4984 a Fm(@)g(\022)2089 4921 y Fs(\()p Fn(\000)2215 4860 y Fm(l)2242 4874 y Fq(0)p 2205 4901 87 4 v 2205 4984 a Fm(k)2252 4998 y Fq(0)2302 4921 y Fm(;)2373 4860 y(\022)p 2352 4901 V 2352 4984 a(k)2399 4998 y Fq(0)2449 4921 y Fs(\))p Fm(;)p eop end %%Page: 104 104 TeXDict begin 104 103 bop 456 251 a Fq(104)615 b(A.)23 b(Delshams,)g(R.)g(de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)758 450 y Fs(and)i(then)f(it)i(is)e(a)i(negativ)m(e)h(function.)38 b(On)24 b(the)h(other)h(hand)d(w)m(e)j(ha)m(v)m(e)758 558 y(that:)1001 704 y Fm(F)33 b Fn(\016)21 b Fm(S)k Fn(\016)c Fs(\()p Fm(\025)1393 666 y Fl(\006)1393 732 y Fp(E)1453 704 y Fm(;)15 b Fs(Id)o Fm(;)g Fs(Id\)\()p Fm(';)g(s)p Fs(\))27 b Fn(')e Fm(E)g Fn(\007)20 b Fm(")p Fn(M)p Fs(\()p Fm(\022)s Fs(;)15 b Fm(")p Fs(\))p Fm(;)758 868 y Fs(then)38 b(the)g(surface)f Fm(S)5 b Fs(\()p Fm(L)1608 824 y Fp(F)r(;)p Fl(\000)1608 897 y Fp(E)1734 868 y Fs(\))38 b(in)m(tersect)h(surfaces)f Fm(L)2593 824 y Fp(F)r(;)p Fl(\000)2593 898 y Fp(E)2649 879 y Fg(0)2756 868 y Fs(with)f Fm(E)3042 835 y Fl(0)3103 868 y Fm(<)758 985 y(E)5 b Fs(.)81 b(\(Note)45 b(that)g(if)e Fm(E)1583 952 y Fl(0)1654 985 y Fm(<)k(E)i Fs(then)43 b Fm(\025)2161 946 y Fl(\000)2161 1013 y Fp(E)2221 985 y Fs(\()p Fm(';)15 b(s)p Fs(;)g 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Fp(\027)1942 1789 y Fn(\024)25 b Fm(E)30 b Fn(\024)25 b Fm(c)2270 1803 y Fq(2)2310 1789 y Fm(L)p Fs(.)858 1897 y(This)j(case)j(is)e(analogous)h(to)g(the)f(non)g(resonan)m(t)h (region,)g(b)s(ecause)758 2004 y(in)g(this)h(case,)764 2209 y Fm(`)p Fs(\()p Fm(\022)s(;)15 b(s)p Fs(\))82 b(=)1237 2126 y Fh(p)p 1328 2126 613 4 v 83 x Fs(2\()p Fm(E)26 b Fn(\000)20 b Fm("U)10 b Fs(\()p Fm(\022)s Fs(;)15 b Fm(")p Fs(\)\))27 b(=)2062 2127 y Fn(p)p 2138 2127 118 4 v 82 x Fs(2)p Fm(E)2256 2075 y Fh(r)p 2346 2075 520 4 v 2346 2209 a Fs(1)21 b Fn(\000)2528 2147 y Fm(")p 2513 2188 73 4 v 2513 2271 a(E)2595 2209 y(U)10 b Fs(\()p Fm(\022)s Fs(;)15 b Fm(")p Fs(\))1083 2414 y(=)1237 2333 y Fn(p)p 1313 2333 118 4 v 81 x Fs(2)p Fm(E)6 b Fs(\(1)21 b(+)1623 2422 y(O)1693 2434 y Fl(C)1734 2415 y Fi(2)1773 2414 y Fs(\()p Fm(")1850 2376 y Fq(1)p Fl(\000)p Fp(\027)1984 2414 y Fs(\)\))p Fm(;)758 2560 y Fs(consequen)m(tly)-8 b(,)32 b(the)f(function)f Fn(M)h Fs(b)s(ecomes)637 2705 y Fn(M)p Fs(\()p Fm(\022)s Fs(;)15 b Fm(")p Fs(\))637 2895 y(=)25 b Fm(k)780 2909 y Fq(0)819 2814 y Fn(p)p 895 2814 V 81 x Fs(2)p Fm(E)1023 2834 y(@)5 b Fn(L)1139 2801 y Fl(\003)p 1023 2874 156 4 v 1051 2958 a Fm(@)g(\022)1203 2767 y Fh(\022)1270 2895 y Fn(\000)1361 2834 y Fm(l)1388 2848 y Fq(0)p 1351 2874 87 4 v 1351 2958 a Fm(k)1398 2972 y Fq(0)1468 2895 y Fs(+)1559 2814 y Fn(p)p 1635 2814 118 4 v 81 x Fs(2)p Fm(E)g(;)15 b(')22 b Fn(\000)d Fs(\()p Fn(\000)2090 2834 y Fm(l)2117 2848 y Fq(0)p 2079 2874 87 4 v 2079 2958 a Fm(k)2126 2972 y Fq(0)2196 2895 y Fs(+)2287 2814 y Fn(p)p 2363 2814 118 4 v 81 x Fs(2)p Fm(E)6 b Fs(\))p Fm(s)2559 2767 y Fh(\023)2646 2895 y Fs(+)2737 2903 y(O)2808 2915 y Fl(C)2849 2896 y Fi(2)2887 2895 y Fs(\()p Fm(")2964 2858 y Fq(1)p Fl(\000)p Fp(\027)3098 2895 y Fs(\))p Fm(:)758 3096 y Fs(Then,)37 b(if)e(the)h(function)g Fn(L)1703 3063 y Fl(\003)1742 3096 y Fs(\()p Fm(I)7 b(;)15 b(\022)s Fs(\))36 b(is)g(not)f(constan)m(t,)k(w)m(e)d(can)g(apply)758 3204 y(Lemma)31 b(81,)g(to)g(get)h(the)f(desired)e(result.)3103 3331 y Fj(\003)456 3485 y Fw(Remark)38 b(87.)44 b Fs(W)-8 b(e)34 b(observ)m(e)g(that)g(the)f(transv)m(ersalit)m(y)i(lemmas)e(82)h (and)f(85)456 3593 y(use)27 b(some)h(non-degeneracy)g(h)m(yp)s (othesis.)40 b(Hyp)s(othesis)27 b(\(151\))j(and)d(\(155\))j(re-)456 3701 y(fer)20 b(to)h(the)f(reduced)g(P)m(oincar)m(\023)-43 b(e)22 b(function)e Fn(L)1927 3668 y Fl(\003)1966 3701 y Fs(,)j(and)c(therefore)i(to)g Fn(L)f Fs(\(see)h(\(145\))r(\),)456 3809 y(and)h(b)s(oth)h(are)h(con)m(tained)h(in)e(Hyp)s(othesis)g Fw(H4)h Fs(of)f(Theorem)h(7.)38 b(The)23 b(h)m(yp)s(oth-)456 3917 y(esis)k(that)g(the)g(function)f(\(156\))j(is)d(non-constan)m(t)i (also)g(in)m(v)m(olv)m(es)g(the)f(function)456 4025 y Fm(U)10 b Fs(\()p Fm(\022)s Fs(;)15 b Fm(")p Fs(\),)27 b(whic)m(h)f(comes)h(from)f(the)g(expression)g(of)h(the)f(Hamiltonian)i (reduced)456 4140 y(to)574 4117 y(~)565 4140 y(\003)628 4154 y Fp(")694 4140 y Fs(near)h(the)g(resonance)h Fm(I)i Fs(=)25 b 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b(and)h(in)f(the)h(r)-5 b(e)g(gion)1028 4964 y Fn(f)p Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))27 b Fn(2)e Fs([)p Fm(l)1537 4978 y Fq(0)1577 4964 y Fm(=k)1669 4978 y Fq(0)1729 4964 y Fn(\000)20 b Fm(L;)15 b(l)1949 4978 y Fq(0)1989 4964 y Fm(=k)2081 4978 y Fq(0)2142 4964 y Fs(+)20 b Fm(L)p Fs(])g Fn(\002)g Fk(T)2492 4927 y Fq(2)2531 4964 y Fn(g)p Fm(;)p eop end %%Page: 105 105 TeXDict begin 105 104 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)718 b(105)456 450 y Fo(the)33 b(function)f Fm(F)46 b Fo(is)33 b(of)f(the)h(form)1008 659 y Fm(F)13 b Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))28 b(=)1594 597 y Fm(y)1642 564 y Fq(2)p 1594 638 88 4 v 1615 721 a Fs(2)1691 659 y(\(1)21 b(+)1883 667 y(O)1954 678 y Fl(C)1995 660 y Fi(2)2034 659 y Fs(\()p Fm(")p Fs(\)\))g(+)2293 667 y(O)2364 678 y Fl(C)2405 660 y Fi(2)2443 659 y Fs(\()p Fm(")2520 621 y Fq(2)2561 659 y Fs(\))p Fm(;)456 843 y Fo(wher)-5 b(e)36 b Fm(y)e Fs(=)d Fm(y)s Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(;)g 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2640 y Fi(2)2628 2639 y Fs(\()p Fm(")2705 2601 y Fq(3)p Fp(=)p Fq(2)2816 2639 y Fs(\))p Fm(;)456 2557 y Fs(\(159\))456 2814 y Fo(for)33 b Fm(\032)27 b Fn(\024)f Fm(\022)i Fn(\024)f Fs(2)p Fm(\031)d Fn(\000)c Fm(\032)p Fo(,)33 b(wher)-5 b(e)35 b Fm(`)p Fs(\()p Fm(x;)15 b(E)5 b Fs(\))27 b(=)1915 2736 y Fh(p)p 2006 2736 659 4 v 78 x Fs(2\()p Fm(E)g Fn(\000)20 b Fm(")2313 2788 y Fq(2)2352 2814 y Fm(U)10 b Fs(\()p Fm(x)p Fs(;)15 b Fm(")p Fs(\)\))35 b Fo(with)f Fm(U)10 b Fs(\()p Fm(x)p Fs(;)15 b Fm(")p Fs(\))456 2934 y Fo(de\014ne)-5 b(d)33 b(in)40 b Fs(\(78\))34 b Fo(and)1270 2911 y Fs(~)1251 2934 y Fn(Y)1312 2948 y Fl(\006)1404 2934 y Fo(is)e(given)g(in)h(L)-5 b(emma)34 b(60)f(with)h Fm(\016)29 b Fs(=)c Fm(")2702 2901 y Fq(2)2742 2934 y Fo(.)555 3042 y(Given)e(the)h(r)-5 b(e)g(duc)g(e)g(d)25 b(Poinc)-5 b(ar)n(\023)-44 b(e)24 b(function)g Fn(L)2049 3009 y Fl(\003)2088 3042 y Fo(,)h(de\014ne)-5 b(d)24 b(in)31 b Fs(\(145\))r Fo(,)25 b(for)f Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))26 b Fn(2)456 3151 y Fm(H)532 3165 y Fl(\000)610 3151 y Fn(\\)20 b(S)753 3118 y Fl(R)813 3127 y Fi(2)852 3151 y Fo(,)32 b(let)h(us)g(assume)g(that)h(the)f (function)456 3351 y Fs(\(160\))871 b Fm(\022)28 b Fn(7!)1729 3289 y Fm(@)5 b Fn(L)1845 3256 y Fl(\003)p 1729 3330 156 4 v 1757 3413 a Fm(@)g(\022)1894 3351 y Fs(\()p Fm(I)i(;)15 b(\022)s Fs(\))456 3531 y Fo(is)32 b(non-c)-5 b(onstant)35 b(and)e(ne)-5 b(gative)33 b(for)g Fm(\022)28 b Fn(2)d(J)2004 3498 y Fl(\003)1988 3557 y Fp(I)2044 3531 y Fo(.)555 3638 y(Assume)31 b(mor)-5 b(e)g(over)32 b(the)f(fol)5 b(lowing)32 b(hyp)-5 b(othesis,)33 b(which)e(is)38 b Fw(H5"')31 b Fo(in)f(The-)456 3746 y(or)-5 b(em)33 b(7.)555 3854 y(Ther)-5 b(e)36 b(exists)f(a)h(c)-5 b(onstant)37 b Fm(C)7 b Fo(,)34 b(indep)-5 b(endent)37 b(of)e Fm(E)40 b Fo(and)c Fm(")p Fo(,)g(and)g(thr)-5 b(e)g(e)36 b(in-)456 3962 y(tervals)43 b Fn(J)819 3976 y Fp(i)890 3962 y Fn(\032)g(J)1083 3929 y Fl(\003)1066 3994 y(\000)p Fp(l)1142 4003 y Fi(0)1176 3994 y Fp(=k)1248 4003 y Fi(0)1287 3962 y Fo(,)i Fm(i)f Fs(=)f(1)p Fm(;)15 b Fs(2)p Fm(;)g Fs(3)p Fo(,)47 b(such)c(that,)j (given)c(any)h Fm(E)5 b(;)15 b(")44 b Fo(in)e(this)456 4094 y(r)-5 b(e)g(gion)36 b(\()p Fn(\000)p Fm(c)876 4108 y Fq(4)915 4094 y Fm(")957 4061 y Fq(2)1027 4094 y Fn(\024)29 b Fm(E)35 b Fn(\024)29 b Fm(c)1368 4108 y Fq(2)1408 4094 y Fm(L)p Fo(\),)36 b(one)f(c)-5 b(an)36 b(cho)-5 b(ose)36 b Fm(i)30 b Fs(=)f(1)p Fm(;)15 b Fs(2)p Fm(;)g Fs(3)37 b Fo(in)e(such)h(a)f(way)456 4202 y(that)769 4316 y Fh(\014)769 4370 y(\014)1093 4332 y Fm(k)1140 4346 y Fq(0)p 809 4372 656 4 v 809 4456 a Fs(2\()p Fm(E)27 b Fn(\000)19 b Fm(")1115 4429 y Fq(2)1155 4456 y Fm(U)10 b Fs(\()p Fm(\022)s Fs(;)15 b(0\)\))1490 4265 y Fh(\022)1556 4393 y Fs(2)p Fm(E)1684 4332 y(@)1737 4299 y Fq(2)1777 4332 y Fn(L)1840 4299 y Fl(\003)p 1684 4372 195 4 v 1713 4456 a Fm(@)5 b(\022)1812 4429 y Fq(2)1889 4393 y Fs(\()p Fn(\000)2015 4332 y Fm(l)2042 4346 y Fq(0)p 2005 4372 87 4 v 2005 4456 a Fm(k)2052 4470 y Fq(0)2102 4393 y Fm(;)2173 4332 y(\022)p 2152 4372 V 2152 4456 a(k)2199 4470 y Fq(0)2249 4393 y Fs(\))789 4652 y Fn(\000)20 b Fm(")922 4615 y Fq(2)977 4524 y Fh(\024)1025 4652 y Fm(k)1072 4666 y Fq(0)1112 4652 y Fm(U)1184 4615 y Fl(0)1207 4652 y Fs(\()p Fm(\022)s(;)15 b Fs(0\))1418 4591 y Fm(@)5 b Fn(L)1534 4558 y Fl(\003)p 1418 4631 156 4 v 1447 4715 a Fm(@)g(\022)1584 4652 y Fs(\()p Fn(\000)1710 4591 y Fm(l)1737 4605 y Fq(0)p 1700 4631 87 4 v 1700 4715 a Fm(k)1747 4729 y Fq(0)1797 4652 y Fm(;)1868 4591 y(\022)p 1847 4631 V 1847 4715 a(k)1894 4729 y Fq(0)1944 4652 y Fs(\))21 b(+)e(2)p Fm(U)10 b Fs(\()p Fm(\022)s Fs(;)15 b(0\))2418 4591 y Fm(@)2471 4558 y Fq(2)2512 4591 y Fn(L)2575 4558 y Fl(\003)p 2418 4631 195 4 v 2448 4715 a Fm(@)5 b(\022)2547 4688 y Fq(2)2625 4652 y Fs(\()p Fn(\000)2751 4591 y Fm(l)2778 4605 y Fq(0)p 2741 4631 87 4 v 2741 4715 a Fm(k)2788 4729 y Fq(0)2837 4652 y Fm(;)2908 4591 y(\022)p 2888 4631 V 2888 4715 a(k)2935 4729 y Fq(0)2984 4652 y Fs(\))3019 4524 y Fh(\025)794 4911 y Fn(\006)p Fm(")907 4828 y Fh(p)p 998 4828 656 4 v 83 x Fs(2\()p Fm(E)26 b Fn(\000)20 b Fm(")1304 4885 y Fq(2)1344 4911 y Fm(U)10 b Fs(\()p Fm(\022)s Fs(;)15 b(0\)\))1663 4850 y Fm(@)1716 4817 y Fq(2)1756 4850 y Fn(L)1819 4817 y Fl(\003)p 1663 4890 195 4 v 1692 4973 a Fm(@)5 b(\022)1791 4947 y Fq(2)1868 4911 y Fs(\()p Fn(\000)1994 4850 y Fm(l)2021 4864 y Fq(0)p 1984 4890 87 4 v 1984 4973 a Fm(k)2031 4987 y Fq(0)2081 4911 y Fm(;)2152 4850 y(\022)p 2131 4890 V 2131 4973 a(k)2178 4987 y Fq(0)2228 4911 y Fs(\))2263 4783 y Fh(\023)2345 4834 y(\014)2345 4888 y(\014)2401 4911 y Fn(\025)25 b Fm(C)456 4650 y Fs(\(161\))p eop end %%Page: 106 106 TeXDict begin 106 105 bop 456 251 a Fq(106)615 b(A.)23 b(Delshams,)g(R.)g(de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)555 450 y Fo(Then,)33 b(the)g(foliations)h Fn(F)1433 464 y Fp(F)1525 450 y Fo(and)f Fn(F)1766 470 y Fp(F)10 b Fl(\016)p Fp(S)1903 451 y Fg(\000)p Fi(1)2022 450 y Fo(interse)-5 b(ct)34 b(tr)-5 b(ansversal)5 b(ly.)555 558 y(Mor)-5 b(e)29 b(pr)-5 b(e)g(cisely,)31 b(ther)-5 b(e)29 b(exist)g(c)-5 b(onstants,)32 b Fm(C)2105 525 y Fl(0)2127 558 y Fo(,)e Fm(C)2257 525 y Fl(00)2299 558 y Fo(,)f(indep)-5 b(endent)30 b(of)f Fm(")g Fo(and)456 666 y Fm(E)5 b Fo(,)32 b(such)h(that:)601 826 y Fs(\(1\))42 b Fo(Given)33 b(a)g(surfac)-5 b(e)33 b Fm(S)5 b Fs(\()p Fm(L)1571 781 y Fp(F)r(;)p Fl(\000)1571 854 y Fp(E)1696 826 y Fs(\))33 b Fo(we)g(have)g(one)g(of)g(the)g(fol)5 b(lowing)34 b(c)-5 b(ases:)789 952 y Fs(\(a\))42 b Fm(S)5 b Fs(\()p Fm(L)1104 908 y Fp(F)r(;)p Fl(\000)1104 981 y Fp(E)1230 952 y Fs(\))36 b Fo(interse)-5 b(cts)36 b(at)h(some)f(p)-5 b(oint)37 b(the)f(surfac)-5 b(e)37 b Fm(L)2820 908 y Fp(F)r(;)p Fl(\000)2820 982 y Fp(E)2876 963 y Fg(0)2945 952 y Fo(,)f(with)946 1082 y Fm(E)1018 1049 y Fl(0)1067 1082 y Fm(<)25 b(E)5 b Fo(,)33 b(and)h Fm(E)25 b Fn(\000)20 b Fm(E)1728 1049 y Fl(0)1777 1082 y Fn(\024)25 b Fm(C)1945 1049 y Fl(00)1987 1082 y Fm(")15 b Fs(max)h(\()p Fn(j)p Fm(E)5 b Fn(j)2387 1040 y Fq(1)p Fp(=)p Fq(2)2512 1082 y Fm(;)15 b(")p Fs(\))q Fo(,)784 1203 y Fs(\(b\))41 b Fm(S)5 b Fs(\()p Fm(L)1104 1158 y Fp(F)r(;)p Fl(\000)1104 1231 y Fp(E)1230 1203 y Fs(\))36 b Fo(interse)-5 b(cts)36 b(at)h(some)f(p)-5 b(oint)37 b(the)f(surfac)-5 b(e)37 b Fm(L)2820 1158 y Fp(F)r(;)p Fq(+)2820 1233 y Fp(E)2876 1214 y Fg(0)2945 1203 y Fo(,)f(with)946 1333 y Fm(E)1018 1300 y Fl(0)1067 1333 y Fn(\025)25 b Fm(E)5 b Fo(,)33 b(and)h Fm(E)1545 1300 y Fl(0)1588 1333 y Fn(\000)20 b Fm(E)31 b Fn(\024)25 b Fm(C)1945 1300 y Fl(00)1987 1333 y Fm(")15 b Fs(max)h(\()p Fn(j)p Fm(E)5 b Fn(j)2387 1291 y Fq(1)p Fp(=)p Fq(2)2512 1333 y Fm(;)15 b(")p Fs(\))q Fo(.)601 1453 y Fs(\(2\))42 b Fo(A)n(ny)23 b(surfac)-5 b(e)24 b Fm(S)5 b Fs(\()p Fm(L)1401 1409 y Fp(F)r(;)p Fq(+)1401 1482 y Fp(E)1526 1453 y Fs(\))24 b Fo(interse)-5 b(cts)24 b(at)g(some)g(p)-5 b(oint)25 b(the)e(surfac)-5 b(e)24 b Fm(L)3029 1409 y Fp(F)r(;)p Fq(+)3029 1483 y Fp(E)3085 1464 y Fg(0)3154 1453 y Fo(,)758 1583 y(with)34 b Fm(E)1028 1550 y Fl(0)1077 1583 y Fn(\025)25 b Fm(E)5 b Fo(,)32 b(and)i Fm(E)1554 1550 y Fl(0)1598 1583 y Fn(\000)20 b Fm(E)30 b Fn(\024)25 b Fm(C)1954 1550 y Fl(00)1996 1583 y Fm(")15 b Fs(max)h(\()p Fn(j)q Fm(E)5 b Fn(j)2396 1541 y Fq(1)p Fp(=)p Fq(2)2521 1583 y Fm(;)15 b(")p Fs(\))r Fo(.)555 1735 y(In)30 b(al)5 b(l)30 b(the)g(c)-5 b(ases)31 b(the)f(angle)g(b)-5 b(etwe)g(en)30 b(these)g(surfac)-5 b(es)30 b(at)h(the)f(interse)-5 b(ction)456 1843 y(c)g(an)33 b(b)-5 b(e)32 b(b)-5 b(ounde)g(d)35 b(fr)-5 b(om)33 b(b)-5 b(elow)34 b(by)f Fm(C)1728 1810 y Fl(0)1750 1843 y Fm(")p Fo(.)456 2034 y Fw(Remark)h(89.)42 b Fs(W)-8 b(e)31 b(kno)m(w,)f(b)m(y) g(Theorem)f(56)i(that)g(all)f(the)g(tori)h(in)e(the)h(reso-)456 2142 y(nan)m(t)h(region)i(are)f(giv)m(en)g(b)m(y)g(form)m(ulas)g (\(154\))h(for)f Fm(E)h Fs(=)27 b Fm(E)2475 2156 y Fp(i)2503 2142 y Fm(;)15 b(F)2601 2156 y Fp(i)2630 2142 y Fm(;)g(G)32 b Fs(and)f(v)m(erify)456 2250 y(that)1022 2446 y Fn(j)p Fm(E)1114 2460 y Fp(i)1163 2446 y Fn(\000)20 b Fm(E)1321 2460 y Fp(i)p Fq(+1)1439 2446 y Fn(j)83 b(\024)g Fm(")1753 2381 y Fi(3)p 1754 2393 31 3 v 1754 2434 a(2)1794 2408 y Fq(+1)1914 2446 y Fn(\024)24 b Fs(max)q(\()2214 2362 y Fh(p)p 2305 2362 96 4 v 84 x Fm(E)2372 2460 y Fp(i)2400 2446 y Fm(;)15 b(")p Fs(\))p Fm(;)1061 2601 y Fn(j)p Fm(E)1153 2616 y Fp(l)1174 2627 y Ff(E)1250 2601 y Fn(\000)20 b Fm(F)1399 2615 y Fq(1)1439 2601 y Fn(j)83 b(\024)g Fm(")1753 2536 y Fi(3)p 1754 2548 31 3 v 1754 2589 a(2)1794 2563 y Fq(+1)1914 2601 y Fn(\024)24 b Fs(max)q(\()2214 2523 y Fh(p)p 2305 2523 144 4 v 78 x Fm(E)2372 2616 y Fp(l)2393 2627 y Ff(E)2449 2601 y Fm(;)15 b(")p Fs(\))p Fm(;)1039 2761 y Fn(j)q Fm(F)1123 2775 y Fp(i)1171 2761 y Fn(\000)20 b Fm(F)1320 2775 y Fp(i)p Fq(+1)1439 2761 y Fn(j)83 b(\024)g Fm(")1753 2696 y Fi(3)p 1754 2708 31 3 v 1754 2749 a(2)1794 2723 y Fq(+1)1914 2761 y Fn(\024)24 b Fs(max)q(\()2214 2677 y Fh(p)p 2305 2677 87 4 v 84 x Fm(F)2363 2775 y Fp(i)2392 2761 y Fm(;)15 b(")p Fs(\))1097 2915 y Fn(j)p Fm(F)1180 2930 y Fp(l)1201 2941 y Ff(F)1277 2915 y Fn(\000)20 b Fm(G)p Fn(j)83 b(\024)g Fm(")1753 2851 y Fi(3)p 1754 2863 31 3 v 1754 2904 a(2)1794 2878 y Fq(+1)1914 2915 y Fn(\024)24 b Fs(max)q(\()2214 2838 y Fh(p)p 2305 2838 135 4 v 77 x Fm(F)2363 2930 y Fp(l)2384 2941 y Ff(F)2439 2915 y Fm(;)15 b(")p Fs(\))p Fm(:)456 3112 y Fs(and)37 b(that)h(b)m(y)g(and)g(Corollary)g(57)h(and)e(Remark)h (58)h(that,)h(when)d(they)h(are)456 3220 y(written)25 b(as)g(graphs,)h(the)f(distance)g(b)s(et)m(w)m(een)h(to)g(consecutiv)m (e)h(tori)e(is)g(of)g(order)456 3336 y(O)526 3328 y(\()p Fm(")603 3295 y Fq(3)p Fp(=)p Fq(2)714 3328 y Fs(\).)555 3436 y(Then,)33 b(when)f(w)m(e)i(apply)f(Lemma)g(88)h(to)g(these)f (tori)h(w)m(e)f(obtain)h(that)g(the)456 3544 y(image)42 b(under)d(the)i(scattering)h(map)e(of)h(a)g(torus)g(in)f(this)h(region) g(in)m(tersect)456 3652 y(transv)m(ersally)31 b(another)g(torus.)1551 b Fj(\003)456 3931 y Fo(Pr)-5 b(o)g(of.)43 b Fs(It)30 b(su\016ces)g(to)h(apply)f(Lemma)h(81)g(to)g(the)g(function)1090 4183 y Fm(F)13 b Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))27 b(=)1592 4121 y Fm(y)1640 4088 y Fq(2)p 1592 4162 88 4 v 1613 4245 a Fs(2)1690 4183 y(\(1)21 b(+)1882 4191 y(O)1952 4183 y(\()p Fm(")p Fs(\)\))h(+)2212 4191 y(O)2283 4202 y Fl(C)2324 4183 y Fi(2)8 b Fs(\()p Fm(")2439 4145 y Fq(2)2479 4183 y Fs(\))p Fm(;)456 4420 y Fs(and)29 b Fm(\025)685 4382 y Fl(\006)685 4449 y Fp(E)775 4420 y Fs(giv)m(en)j(b)m(y)f(\(158\))i(and)c(\(159\))r(.)555 4528 y(In)37 b(order)h(to)h(c)m(hec)m(k)g(inequalit)m(y)i(\(150\))f(w)m (e)e(use)g(form)m(ula)g(\(147\))r(,)i(so)e(that)456 4637 y Fm(S)30 b Fs(=)25 b(Id)6 b(+)h Fm("S)904 4651 y Fq(1)951 4637 y Fs(+)1029 4645 y(O)1099 4657 y Fl(C)1140 4638 y Fi(1)1179 4637 y Fs(\()p Fm(")1256 4604 y Fq(1+)p Fp(\045)1387 4637 y Fs(\),)26 b(and)d(that,)j(b)m(y)f(\(98\))r(,)g Fm(\025)2246 4599 y Fl(\006)2246 4666 y Fp(E)2329 4637 y Fs(is)f(a)h(b)s(ounded)c(function)456 4745 y(and)29 b(its)i(deriv)-5 b(ativ)m(es)32 b(with)e(resp)s(ect)g(to)h Fm(';)15 b(s)31 b Fs(are)g(of)g(order)2507 4753 y(O)2578 4745 y(\()p Fm(")p Fs(\).)555 4853 y(W)-8 b(e)22 b(will)e(also)i(use)e (that)g Fn(r)1445 4867 y Fp(I)5 b(;';s)1603 4853 y Fm(F)13 b Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\))28 b(=)d Fm(y)s Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(;)g Fm(")p Fs(\)\(1+)2759 4861 y(O)2832 4873 y Fl(C)2873 4854 y Fi(2)2912 4853 y Fs(\()p Fm(")p Fs(\)\)\()p Fm(k)3141 4867 y Fq(0)3182 4853 y Fm(;)g Fs(0)p Fm(;)g Fs(0\)+)456 4972 y(O)526 4984 y Fl(C)567 4965 y Fi(2)606 4964 y Fs(\()p Fm(")683 4931 y Fq(2)723 4964 y Fs(\),)47 b(and)c(analogous)i (estimates)g(for)f(the)f(second)h(deriv)-5 b(ativ)m(es)45 b(of)f Fm(F)13 b Fs(.)p eop end %%Page: 107 107 TeXDict begin 107 106 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)718 b(107)456 450 y Fs(Hence,)31 b(w)m(e)g(can)g(compute,)g(as)f(in)g(Lemma)h(85:)456 644 y Fm(F)i Fn(\016)21 b Fm(S)5 b Fn(\016)p Fs(\()p Fm(\025)807 606 y Fl(\006)807 672 y Fp(E)867 644 y Fm(;)15 b Fs(Id)p Fm(;)g Fs(Id)o(\)\()p Fm(';)g(s)p Fs(\))744 794 y(=)25 b Fm(E)h Fn(\000)20 b Fm(")p 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b(s)p Fs(;)g Fm(")p Fs(\))p Fm(;)g(')23 b Fn(\000)d Fm(\025)1967 1229 y Fl(\006)1967 1296 y Fp(E)2027 1268 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))p Fm(s)p Fs(\))22 b(+)2512 1276 y(O)2582 1287 y Fl(C)2623 1268 y Fi(1)9 b Fs(\()p Fm(")2739 1230 y Fp(\045)2780 1268 y Fs(\))2815 1139 y Fh(\023)2882 1162 y Fq(2)2942 1268 y Fs(+)3033 1276 y(O)3104 1287 y Fl(C)3145 1268 y Fi(1)f Fs(\()p Fm(")3260 1230 y Fq(3)3300 1268 y Fs(\))744 1457 y(=)25 b Fm(E)h Fn(\000)20 b Fm("k)1113 1471 y Fq(0)1153 1457 y Fn(Y)1214 1471 y Fl(\006)1273 1457 y Fs(\()p Fm(\022)s(;)15 b(E)5 b Fs(\))p Fn(\001)734 1533 y Fh(\022)811 1600 y Fm(@)g Fn(L)927 1567 y Fl(\003)p 811 1640 V 839 1723 a Fm(@)g(\022)992 1533 y Fh(\022)1059 1661 y Fn(\000)1149 1600 y Fm(l)1176 1614 y Fq(0)p 1140 1640 87 4 v 1140 1723 a Fm(k)1187 1737 y Fq(0)1256 1661 y Fs(+)20 b Fn(Y)1408 1675 y Fl(\006)1467 1661 y Fs(\()p Fm(\022)s(;)15 b(E)5 b Fs(\))p Fm(;)15 b(')22 b Fn(\000)1906 1533 y Fh(\022)1973 1661 y Fn(\000)2064 1600 y Fm(l)2091 1614 y Fq(0)p 2054 1640 V 2054 1723 a Fm(k)2101 1737 y Fq(0)2171 1661 y Fs(+)e Fn(Y)2323 1675 y Fl(\006)2382 1661 y Fs(\()p Fm(\022)s(;)15 b(E)5 b Fs(\))2610 1533 y Fh(\023)2692 1661 y Fm(s)2735 1533 y Fh(\023)2822 1661 y Fs(+)2913 1669 y(O)2984 1681 y Fl(C)3025 1662 y Fi(1)3063 1661 y Fs(\()p Fm(")3140 1624 y Fp(\045)3181 1661 y Fs(\))3216 1533 y Fh(\023)739 1931 y Fs(+)20 b Fm(k)877 1945 y Fq(0)917 1931 y Fm(")959 1894 y Fq(2)1014 1803 y Fh(\022)1091 1870 y Fm(@)5 b Fn(L)1207 1837 y Fl(\003)p 1091 1910 156 4 v 1119 1993 a Fm(@)g(\022)1271 1803 y Fh(\022)1338 1931 y Fn(\000)1429 1870 y Fm(l)1456 1884 y Fq(0)p 1419 1910 87 4 v 1419 1993 a Fm(k)1466 2007 y Fq(0)1536 1931 y Fs(+)20 b Fn(Y)1688 1945 y Fl(\006)1747 1931 y Fs(\()p Fm(\022)s(;)15 b(E)5 b Fs(\))p Fm(;)15 b(')21 b Fn(\000)2186 1803 y Fh(\022)2253 1931 y Fn(\000)2344 1870 y Fm(l)2371 1884 y Fq(0)p 2334 1910 V 2334 1993 a Fm(k)2381 2007 y Fq(0)2451 1931 y Fs(+)f Fn(Y)2603 1945 y Fl(\006)2661 1931 y Fs(\()p Fm(\022)s(;)15 b(E)5 b Fs(\))2889 1803 y Fh(\023)2972 1931 y Fm(s)3015 1803 y Fh(\023\023)3149 1826 y Fq(2)739 2140 y Fs(+)830 2148 y(O)901 2160 y Fl(C)942 2141 y Fi(1)980 2140 y Fs(\()p Fm(")1057 2103 y Fq(5)p Fp(=)p Fq(2)1168 2140 y Fm(;)15 b(")1250 2103 y Fq(2+)p Fp(\045)1381 2140 y Fs(\))744 2299 y(=)25 b Fm(E)h Fn(\000)20 b Fm(")p Fn(M)1175 2313 y Fl(\006)1234 2299 y Fs(\()p Fm(\022)s Fs(;)15 b Fm(")p Fs(\))21 b(+)1544 2307 y(O)1615 2319 y Fl(C)1656 2300 y Fi(1)1694 2299 y Fs(\()p Fm(")1771 2262 y Fq(5)p Fp(=)p Fq(2)1882 2299 y Fm(;)15 b(")1964 2262 y Fq(2+)p Fp(\045)2095 2299 y Fs(\))555 2497 y(No)m(w,)32 b(w)m(e)e(compute)h(the)g(main)f (part)g(of)h(the)f(functions)g Fn(M)2632 2511 y Fl(\006)622 2694 y Fn(M)731 2708 y Fl(\006)790 2694 y Fs(\()p Fm(\022)s Fs(;)15 b Fm(")p Fs(\))26 b(=)f Fm(k)1157 2708 y Fq(0)1197 2694 y Fn(Y)1258 2708 y Fl(\006)1316 2694 y Fs(\()p Fm(\022)s(;)15 b(E)5 b Fs(\))21 b Fn(\001)622 2756 y Fh(\022)699 2822 y Fm(@)5 b Fn(L)815 2789 y Fl(\003)p 699 2863 156 4 v 727 2946 a Fm(@)g(\022)879 2756 y Fh(\022)946 2884 y Fn(\000)1037 2822 y Fm(l)1064 2836 y Fq(0)p 1027 2863 87 4 v 1027 2946 a Fm(k)1074 2960 y Fq(0)1144 2884 y Fs(+)1265 2822 y(1)p 1244 2863 V 1244 2946 a Fm(k)1291 2960 y Fq(0)1341 2884 y Fn(Y)1402 2898 y Fl(\006)1461 2884 y Fs(\()p Fm(\022)s(;)15 b(E)5 b Fs(\))p Fm(;)15 b(')22 b Fn(\000)1901 2756 y Fh(\022)1968 2884 y Fn(\000)2058 2822 y Fm(l)2085 2836 y Fq(0)p 2049 2863 V 2049 2946 a Fm(k)2096 2960 y Fq(0)2165 2884 y Fs(+)2287 2822 y(1)p 2266 2863 V 2266 2946 a Fm(k)2313 2960 y Fq(0)2363 2884 y Fn(Y)2424 2898 y Fl(\006)2483 2884 y Fs(\()p Fm(\022)s(;)15 b(E)5 b Fs(\))2711 2756 y Fh(\023)2793 2884 y Fm(s)2836 2756 y Fh(\023)622 3110 y Fs(+)708 3118 y(O)778 3130 y Fl(C)819 3111 y Fi(1)858 3110 y Fs(\()p Fm(")935 3072 y Fp(\045)976 3110 y Fs(\))1011 3009 y Fh(\021)622 3353 y Fn(\000)p Fm("k)785 3315 y Fq(2)782 3375 y(0)840 3225 y Fh(\022)916 3291 y Fm(@)g Fn(L)1032 3258 y Fl(\003)p 916 3332 156 4 v 945 3415 a Fm(@)g(\022)1097 3225 y Fh(\022)1164 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b(that)h(1)25 b Fn(\024)g Fm(\027)31 b(<)25 b Fs(2,)30 b(and)f(w)m(e)h(will)g(consider)f(the)g(follo)m(wing) 456 3924 y(t)m(w)m(o)i(cases:)659 4076 y Fw(Close)k(to)f(the)h (resonance:)42 b Fn(\000)p Fm(c)1875 4090 y Fq(4)1915 4076 y Fm(")1957 4043 y Fq(2)2022 4076 y Fn(\024)25 b Fm(E)30 b Fn(\024)d Fs(~)-47 b Fm(c)q(")2393 4043 y Fp(\027)2436 4076 y Fs(.)858 4184 y(In)30 b(this)g(region,)h(w)m(e)g(ha)m(v)m(e)h (that)f Fn(j)p Fm(`)p Fs(\()p Fn(\001)p Fm(;)15 b(E)5 b Fs(\))p Fn(j)2285 4211 y Fl(C)2326 4192 y Fi(1)2390 4184 y Fn(\024)25 b Fm(")2528 4151 y Fp(\027)t(=)p Fq(2)2642 4184 y Fs(,)30 b(and)931 4393 y Fn(Y)992 4407 y Fl(\006)1051 4393 y Fs(\()p Fm(\022)s(;)15 b(E)5 b Fs(\))26 b(=)f Fn(\006)p Fs(\(1)c(+)f Fm("b)p Fs(\))p Fm(`)p Fs(\()p Fm(\022)s(;)15 b(E)5 b Fs(\))21 b(+)f Fm(")2220 4370 y Fs(~)2200 4393 y Fn(Y)2261 4407 y Fl(\006)2341 4393 y Fn(\016)g Fm(`)p Fs(\()p Fm(\022)s(;)15 b(E)5 b Fs(\))p Fm(:)758 4597 y Fs(By)28 b(Lemma)g(60)h(w)m(e)f(kno)m(w)g(that)1918 4574 y(~)1899 4597 y Fn(Y)1960 4611 y Fl(\006)2046 4597 y Fs(ha)m(v)m(e)h Fn(C)2306 4564 y Fq(2)2373 4597 y Fs(norm)e(b)s (ounded)f(inde-)758 4709 y(p)s(enden)m(tly)k(of)h Fm(")p Fs(,)g Fm(E)k Fs(and)1680 4686 y(~)1661 4709 y Fn(Y)7 b Fs(\(0\))27 b(=)1986 4686 y(~)1967 4709 y Fn(Y)2035 4676 y Fl(0)2058 4709 y Fs(\(0\))g(=)d(0,)31 b(so)g(that)1388 4830 y Fh(\014)1388 4884 y(\014)1388 4939 y(\014)1418 4934 y Fm(")1480 4911 y Fs(~)1460 4934 y Fn(Y)d(\016)21 b Fm(`)1653 4830 y Fh(\014)1653 4884 y(\014)1653 4939 y(\014)1683 4997 y Fl(C)1724 4979 y Fi(1)1788 4934 y Fn(\024)k Fs(cte)p Fm(:)16 b(")2082 4897 y Fq(1+)p Fp(\027)2216 4934 y Fm(:)p eop end %%Page: 108 108 TeXDict begin 108 107 bop 456 251 a Fq(108)615 b(A.)23 b(Delshams,)g(R.)g(de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)758 450 y Fs(Then,)30 b(the)h(main)f(terms)g(in)g Fn(M)1873 464 y Fl(\006)1963 450 y Fs(are)592 645 y Fn(M)701 659 y Fl(\006)760 645 y Fs(\()p Fm(\022)s Fs(;)15 b Fm(")p Fs(\))26 b(=)521 865 y(=)20 b Fn(\006)g Fm(k)750 879 y Fq(0)790 782 y Fh(p)p 881 782 656 4 v 83 x Fs(2\()p Fm(E)26 b Fn(\000)20 b Fm(")1187 838 y Fq(2)1227 865 y Fm(U)10 b Fs(\()p Fm(\022)s Fs(;)15 b(0\)\))1546 803 y Fm(@)5 b Fn(L)1662 770 y Fl(\003)p 1546 844 156 4 v 1574 927 a Fm(@)g(\022)1711 865 y Fs(\()p Fn(\000)1837 803 y Fm(l)1864 817 y Fq(0)p 1827 844 87 4 v 1827 927 a Fm(k)1874 941 y Fq(0)1924 865 y Fm(;)1995 803 y(\022)p 1974 844 V 1974 927 a(k)2021 941 y Fq(0)2071 865 y Fs(\))21 b Fn(\000)e Fm("k)2309 827 y Fq(2)2306 887 y(0)2365 737 y Fh(\022)2441 803 y Fm(@)5 b Fn(L)2557 770 y Fl(\003)p 2441 844 156 4 v 2470 927 a Fm(@)g(\022)2607 865 y Fs(\()p Fn(\000)2733 803 y Fm(l)2760 817 y Fq(0)p 2723 844 87 4 v 2723 927 a Fm(k)2770 941 y Fq(0)2820 865 y Fm(;)2891 803 y(\022)p 2870 844 V 2870 927 a(k)2917 941 y Fq(0)2967 865 y Fs(\))3002 737 y Fh(\023)3069 759 y Fq(2)521 1074 y Fs(+)15 b Fn(j)p Fm(`)p Fn(j)710 1082 y Fs(O)781 1093 y Fl(C)822 1075 y Fi(1)861 1074 y Fs(\()p Fm(")938 1036 y Fp(\027)t(=)p Fq(2)1052 1074 y Fm(;)g(")1134 1036 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Fm(")1100 3618 y Fq(2)1140 3644 y Fm(U)10 b Fs(\()p Fm(\022)s Fs(;)15 b(0\)\))1474 3454 y Fh(\022)1541 3582 y Fs(2)p Fm(E)1691 3521 y(@)1744 3488 y Fq(2)p 1669 3561 138 4 v 1669 3644 a Fm(@)5 b(\022)1768 3618 y Fq(2)1816 3582 y Fn(L)1879 3544 y Fl(\003)1919 3582 y Fs(\()p Fn(\000)2045 3521 y Fm(l)2072 3535 y Fq(0)p 2035 3561 87 4 v 2035 3644 a Fm(k)2082 3658 y Fq(0)2131 3582 y Fm(;)2202 3521 y(\022)p 2182 3561 V 2182 3644 a(k)2229 3658 y Fq(0)2278 3582 y Fs(\))794 3841 y Fn(\000)p Fm(")922 3713 y Fh(\024)970 3841 y Fm(k)1017 3855 y Fq(0)1057 3841 y Fm(U)1129 3803 y Fl(0)1152 3841 y Fs(\()p Fm(\022)s Fs(;)15 b(0\))1363 3779 y Fm(@)5 b Fn(L)1479 3746 y Fl(\003)p 1363 3820 156 4 v 1392 3903 a Fm(@)g(\022)1529 3841 y Fs(\()p Fn(\000)1655 3779 y Fm(l)1682 3793 y Fq(0)p 1645 3820 87 4 v 1645 3903 a Fm(k)1692 3917 y Fq(0)1742 3841 y Fm(;)1813 3779 y(\022)p 1792 3820 V 1792 3903 a(k)1839 3917 y Fq(0)1889 3841 y Fs(\))20 b(+)g(2)p Fm(U)10 b Fs(\()p Fm(\022)s Fs(;)15 b(0\))2387 3779 y Fm(@)2440 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Fn(\000)2732 4038 y Fm(l)2759 4052 y Fq(0)p 2722 4079 87 4 v 2722 4162 a Fm(k)2769 4176 y Fq(0)2818 4100 y Fm(;)2889 4038 y(\022)p 2869 4079 V 2869 4162 a(k)2916 4176 y Fq(0)2965 4100 y Fs(\))3000 3972 y Fh(\023)784 4309 y Fs(+)870 4317 y(O)940 4329 y Fl(C)981 4310 y Fi(0)1020 4309 y Fs(\()p Fm(")1097 4271 y Fp(\027)t(=)p Fq(2)1211 4309 y Fm(;)15 b(")1293 4271 y Fp(\045)1334 4309 y Fs(\))p Fm(:)858 4514 y Fs(By)37 b(h)m(yp)s(othesis)g(\(161\))i(w)m(e)f(kno)m (w)f(that)h(the)f(main)g(term)h(of)f(this)758 4622 y(expression)k(is)f (b)s(ounded)e(a)m(w)m(a)m(y)43 b(from)d(zero)h(b)m(y)g(a)f(constan)m(t) i Fm(C)47 b Fs(for)758 4730 y Fm(\022)i Fn(2)e(J)1020 4744 y Fp(i)1094 4730 y Fn(\032)g(J)1290 4697 y Fl(\003)1274 4762 y(\000)p Fp(l)1350 4771 y Fi(0)1384 4762 y Fp(=k)1456 4771 y Fi(0)1495 4730 y Fs(,)f Fm(i)h Fs(=)g(1)p Fm(;)15 b Fs(2)p Fm(;)g Fs(3.)81 b(Consequen)m(tly)-8 b(,)48 b(the)43 b(angle)i(of)758 4856 y(in)m(tersection)h(can)e(b)s(e)f(b)s (ounded)e(again)k(from)e(b)s(elo)m(w)h(b)m(y)g Fm(C)2928 4823 y Fl(0)2951 4856 y Fm(")p Fs(,)j(for)758 4964 y(some)31 b(suitable)g(constan)m(t)h(indep)s(enden)m(t)d(of)h Fm(")p Fs(.)p eop end %%Page: 109 109 TeXDict begin 109 108 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)718 b(109)858 457 y Fs(In)39 b(order)g(to)i(see)f(whic)m(h)g(surfaces)g(will)g(in)m(tersect)h Fm(S)5 b Fs(\()p Fm(L)2842 412 y Fp(F)r(;)p Fl(\006)2842 485 y Fp(E)2968 457 y Fs(\),)42 b(w)m(e)758 565 y(observ)m(e)27 b(that)f(the)f(function)h Fn(M)1882 579 y Fl(\006)1941 565 y Fs(\()p Fm(\022)s Fs(;)15 b Fm(")p Fs(\))26 b(is)g(giv)m(en)g (appro)m(ximately)h(b)m(y)598 846 y Fm(k)645 860 y Fq(0)695 785 y Fm(@)5 b Fn(L)811 752 y Fl(\003)p 695 825 156 4 v 723 909 a Fm(@)g(\022)860 846 y Fs(\()p Fn(\000)986 785 y Fm(l)1013 799 y Fq(0)p 976 825 87 4 v 976 909 a Fm(k)1023 923 y Fq(0)1073 846 y Fm(;)1144 785 y(\022)p 1123 825 V 1123 909 a(k)1170 923 y Fq(0)1220 846 y Fs(\))1270 718 y Fh(\022)1337 846 y Fn(\006)1408 764 y Fh(p)p 1499 764 656 4 v 82 x Fs(2\()p Fm(E)26 b Fn(\000)20 b Fm(")1805 820 y Fq(2)1845 846 y Fm(U)10 b Fs(\()p Fm(\022)s Fs(;)15 b(0\)\))21 b Fn(\000)f Fm("k)2354 860 y Fq(0)2404 785 y Fm(@)5 b Fn(L)2520 752 y Fl(\003)p 2404 825 156 4 v 2433 909 a Fm(@)g(\022)2570 846 y Fs(\()p Fn(\000)2696 785 y Fm(l)2723 799 y Fq(0)p 2686 825 87 4 v 2686 909 a Fm(k)2733 923 y Fq(0)2782 846 y Fm(;)2854 785 y(\022)p 2833 825 V 2833 909 a(k)2880 923 y Fq(0)2929 846 y Fs(\))2964 718 y Fh(\023)758 1123 y Fs(and)26 b(w)m(e)i(will)f(ha)m(v)m(e)h (di\013eren)m(t)f(b)s(eha)m(viors)g(dep)s(ending)e(on)i(the)f(branc)m (h)758 1239 y Fm(S)5 b Fs(\()p Fm(L)916 1195 y Fp(F)r(;)p Fl(\006)916 1267 y Fp(E)1042 1239 y Fs(\).)858 1365 y(If)42 b(w)m(e)g(fo)s(cus)g(in)g(the)g(case)i(of)e Fm(S)5 b Fs(\()p Fm(L)2115 1321 y Fp(F)r(;)p Fl(\000)2115 1394 y Fp(E)2241 1365 y Fs(\),)45 b(whic)m(h)d(corresp)s(ond)f(to)758 1473 y(the)c(lo)m(w)m(er)h(branc)m(h,)f(the)g(function)f Fn(M)2132 1487 y Fl(\000)2228 1473 y Fs(can)g(ha)m(v)m(e)i(di\013eren)m (t)f(signs)758 1587 y(dep)s(ending)29 b(on)h(the)h(size)g(of)1754 1509 y Fh(p)p 1845 1509 656 4 v 78 x Fs(2\()p Fm(E)26 b Fn(\000)20 b Fm(")2151 1561 y Fq(2)2191 1587 y Fm(U)10 b Fs(\()p Fm(\022)s Fs(;)15 b(0\)\))q(.)858 1709 y(If)941 1630 y Fh(p)p 1032 1630 V 79 x Fs(2\()p Fm(E)27 b Fn(\000)20 b Fm(")1339 1682 y Fq(2)1378 1709 y Fm(U)10 b Fs(\()p Fm(\022)s Fs(;)15 b(0\)\))24 b(is)f(bigger)h(than)f Fn(\000)p Fm("k)2425 1723 y Fq(0)2474 1673 y Fp(@)t Fl(L)2564 1649 y Fg(\003)p 2474 1688 126 4 v 2499 1740 a Fp(@)t(\022)2610 1709 y Fs(\()p Fn(\000)2734 1671 y Fp(l)2755 1680 y Fi(0)p 2726 1688 72 4 v 2726 1740 a Fp(k)2763 1749 y Fi(0)2807 1709 y Fm(;)2876 1673 y Fp(\022)p 2858 1688 V 2858 1740 a(k)2895 1749 y Fi(0)2939 1709 y Fs(\))g(then)758 1847 y Fn(M)867 1861 y Fl(\000)927 1847 y Fs(\()p Fm(\022)s Fs(;)15 b Fm(")p Fs(\))38 b Fm(>)f Fs(0)h(and)f Fm(S)5 b Fs(\()p Fm(L)1696 1803 y Fp(F)r(;)p Fl(\000)1696 1876 y Fp(E)1822 1847 y Fs(\))38 b(will)g(in)m(tersect)h(the)f(surfaces)g Fm(L)3024 1803 y Fp(F)r(;)p Fl(\000)3024 1877 y Fp(E)3080 1858 y Fg(0)3149 1847 y Fs(,)758 1955 y(with)e Fm(E)1043 1922 y Fl(0)1101 1955 y Fm(<)f(E)5 b Fs(.)58 b(This)35 b(happ)s(ens)f(for)i(p)s(ositiv)m(e)h(v)-5 b(alues)36 b(of)g Fm(E)5 b Fs(,)38 b(whic)m(h)758 2063 y(corresp)s(ond)24 b(to)i(the)f(primary)f(tori)i(under)d(the)j(separatrix)f(lo)s(op,)h (and)758 2171 y(for)34 b(negativ)m(e)i(v)-5 b(alues)34 b(of)g Fm(E)5 b Fs(,)36 b(whic)m(h)d(corresp)s(ond)g(to)i(the)f (secondary)758 2279 y(tori)d(inside)f(the)h(separatrix)g(lo)s(op.)858 2390 y(If)939 2312 y Fh(p)p 1030 2312 656 4 v 78 x Fs(2\()p Fm(E)26 b Fn(\000)20 b Fm(")1336 2364 y Fq(2)1376 2390 y Fm(U)10 b Fs(\()p Fm(\022)s Fs(;)15 b(0\)\))21 b(is)f(smaller)h(than) f Fn(\000)p Fm("k)2452 2404 y Fq(0)2502 2354 y Fp(@)t Fl(L)2592 2331 y Fg(\003)p 2502 2369 126 4 v 2527 2422 a Fp(@)t(\022)2638 2390 y Fs(\()p Fn(\000)2761 2353 y Fp(l)2782 2362 y Fi(0)p 2754 2369 72 4 v 2754 2422 a Fp(k)2791 2431 y Fi(0)2835 2390 y Fm(;)2904 2354 y Fp(\022)p 2885 2369 V 2885 2422 a(k)2922 2431 y Fi(0)2967 2390 y Fs(\))g(then)758 2529 y Fn(M)867 2543 y Fl(\000)927 2529 y Fs(\()p Fm(\022)s Fs(;)15 b Fm(")p Fs(\))28 b Fm(<)f Fs(0,)33 b(w)m(e)f(obtain)g(that)h Fm(S)5 b Fs(\()p Fm(L)2130 2484 y Fp(F)r(;)p Fl(\000)2130 2557 y Fp(E)2255 2529 y Fs(\))32 b(will)g(in)m(tersect)i(the)e(sur-)758 2637 y(faces)e(with)e Fm(E)1254 2604 y Fl(0)1303 2637 y Fm(>)d(E)5 b Fs(.)41 b(This)28 b(means)g(that,)i(due)e(to)i(the)f (fact)g(that)h(the)758 2745 y(Melnik)m(o)m(v)43 b(function)c(is)i(big,) i(this)d(surface)g(in)m(tersect)i(the)e(surfaces)758 2860 y Fm(L)820 2816 y Fp(F)r(;)p Fq(+)820 2890 y Fp(E)876 2871 y Fg(0)945 2860 y Fs(.)55 b(Then,)35 b(in)f(the)h(case)h(of)f(a)h (resonance)f(of)g(second)g(order)f(it)i(is)758 2968 y(p)s(ossible)26 b(that)i(only)e(with)h(one)g(application)g(of)g Fm(S)32 b Fs(w)m(e)27 b(cross)f(the)h(gap.)858 3076 y(Once)35 b(w)m(e)h(ha)m(v)m(e)h(cross)e(the)h(separatrix)g(lo)s(op,)h(that)f (is,)h(when)d(w)m(e)758 3192 y(consider)e Fm(S)5 b Fs(\()p Fm(L)1272 3147 y Fp(F)r(;)p Fq(+)1272 3220 y Fp(E)1398 3192 y Fs(\),)33 b(w)m(e)f(ha)m(v)m(e)h(that)g Fn(M)2145 3206 y Fq(+)2204 3192 y Fs(\()p Fm(\022)s Fs(;)15 b Fm(")p Fs(\))33 b(is)f(alw)m(a)m(ys)h(negativ)m(e,)758 3318 y(so)e(w)m(e)g(cross)f(the)h(surfaces)f Fm(L)1788 3274 y Fp(F)r(;)p Fq(+)1788 3348 y Fp(E)1844 3329 y Fg(0)1913 3318 y Fs(,)h(for)f Fm(E)2180 3285 y Fl(0)2229 3318 y Fm(>)25 b(E)5 b Fs(.)858 3426 y(In)32 b(all)i(these)g(cases)g(w)m(e)f (ha)m(v)m(e)i(that)e(there)g(exists)h(some)g(constan)m(t)758 3534 y Fm(C)830 3501 y Fl(00)903 3534 y Fs(suc)m(h)c(that)674 3761 y(max)547 3822 y Fp(\022)r Fl(2J)677 3831 y Fi(1)711 3822 y Fl([J)806 3831 y Fi(2)841 3822 y Fl([J)936 3831 y Fi(3)986 3761 y Fn(jM)p Fs(\()p Fm(\025)1208 3775 y Fp(E)1268 3761 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))p Fm(;)g(')23 b Fn(\000)d Fm(\025)1828 3775 y Fp(E)1888 3761 y Fs(\()p Fm(';)15 b(s)p Fs(;)g Fm(")p Fs(\))p Fm(s)p Fs(\))p Fn(j)27 b(\025)e Fm(C)2480 3723 y Fl(00)2537 3761 y Fs(max)15 b(\()p Fm(E)2828 3723 y Fq(1)p Fp(=)p Fq(2)2939 3761 y Fm(;)g(")p Fs(\))q Fm(;)758 4031 y Fs(and)30 b(applying)g(Lemma)h(150)h(w)m(e)e(obtain)h(the)g(desired)f(result.)659 4258 y Fw(F)-9 b(ar)35 b(from)g(the)f(resonance:)45 b Fs(~)-47 b Fm(c")1873 4225 y Fp(\027)1942 4258 y Fn(\024)25 b Fm(E)30 b Fn(\024)25 b Fm(c)2270 4272 y Fq(2)2310 4258 y Fm(L)p Fs(.)858 4366 y(This)j(case)j(is)e(analogous)h(to)g(the)f(non) g(resonan)m(t)h(region,)g(b)s(ecause)758 4474 y(in)g(this)h(case,)707 4774 y Fn(j)p Fm(`)p Fs(\()p Fm(\022)s(;)15 b(s)p Fs(\))p Fn(j)83 b Fs(=)1231 4691 y Fh(p)p 1322 4691 659 4 v 83 x Fs(2\()p Fm(E)26 b Fn(\000)20 b Fm(")1628 4748 y Fq(2)1668 4774 y Fm(U)10 b Fs(\()p Fm(x)p Fs(;)15 b Fm(")p Fs(\)\))27 b(=)2102 4692 y Fn(p)p 2178 4692 118 4 v 82 x Fs(2)p Fm(E)2296 4626 y Fh(r)p 2387 4626 536 4 v 148 x Fs(1)20 b Fn(\000)2553 4712 y Fm(")2595 4686 y Fq(2)p 2553 4753 82 4 v 2558 4836 a Fm(E)2645 4774 y(U)10 b Fs(\()p Fm(x)p Fs(;)15 b Fm(")p Fs(\))1077 4964 y(=)1231 4883 y Fn(p)p 1307 4883 118 4 v 81 x Fs(2)p Fm(E)6 b Fs(\(1)21 b(+)1617 4972 y(O)1687 4984 y Fl(C)1728 4965 y Fi(2)1767 4964 y Fs(\()p Fm(")1844 4927 y Fq(2)p Fl(\000)p Fp(\027)1978 4964 y Fs(\)\))p Fm(;)p eop end %%Page: 110 110 TeXDict begin 110 109 bop 456 251 a Fq(110)615 b(A.)23 b(Delshams,)g(R.)g(de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)758 451 y Fs(consequen)m(tly)-8 b(,)25 b(as)1408 374 y Fn(p)p 1484 374 118 4 v 77 x Fs(2)p Fm(E)31 b Fn(\025)25 b Fm(")1765 418 y Fp(\027)t(=)p Fq(2)1904 451 y Fm(>>)g(")p Fs(,)e(the)e(functions) f Fn(M)2799 465 y Fl(\006)2878 451 y Fs(b)s(ecome)637 597 y Fn(M)746 611 y Fl(\006)805 597 y Fs(\()p Fm(\022)s Fs(;)15 b Fm(")p Fs(\))637 787 y(=)25 b Fm(k)780 801 y Fq(0)819 706 y Fn(p)p 895 706 V 81 x Fs(2)p Fm(E)1023 726 y(@)5 b Fn(L)1139 693 y Fl(\003)p 1023 766 156 4 v 1051 850 a Fm(@)g(\022)1203 659 y Fh(\022)1270 787 y Fn(\000)1361 726 y Fm(l)1388 740 y Fq(0)p 1351 766 87 4 v 1351 850 a Fm(k)1398 864 y Fq(0)1468 787 y Fs(+)1559 706 y Fn(p)p 1635 706 118 4 v 81 x Fs(2)p Fm(E)g(;)15 b(')22 b Fn(\000)d Fs(\()p Fn(\000)2090 726 y Fm(l)2117 740 y Fq(0)p 2079 766 87 4 v 2079 850 a Fm(k)2126 864 y Fq(0)2196 787 y Fs(+)2287 706 y Fn(p)p 2363 706 118 4 v 81 x Fs(2)p Fm(E)6 b Fs(\))p Fm(s)2559 659 y Fh(\023)2646 787 y Fs(+)2737 795 y(O)2808 807 y Fl(C)2849 788 y Fi(2)2887 787 y Fs(\()p Fm(")2964 750 y Fq(2)p Fl(\000)p Fp(\027)3098 787 y Fs(\))p Fm(:)758 1002 y Fs(Then,)37 b(if)f(the)h(function)1651 967 y Fp(@)t Fl(L)1741 943 y Fg(\003)p 1651 982 126 4 v 1676 1034 a Fp(@)t(\022)1787 1002 y Fs(\()p Fm(I)7 b(;)15 b(\022)s Fs(\))36 b(is)g(not)h(constan)m(t)g(and)f(negativ)m(e) 758 1110 y(as)27 b(a)g(function)f(of)g Fm(\022)s Fs(,)h(w)m(e)f(can)h (apply)f(Lemma)g(81,)i(to)f(get)h(the)e(desired)758 1218 y(result.)3103 1345 y Fj(\003)555 1503 y Fs(Hyp)s(othesis)46 b Fw(H5"')f Fs(in)g(\(161\))i(amoun)m(ts)f(to)g(the)g(existence)h(of)f 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b(recall)g(that)g(in)e(Section)i(8,)g(b)m(y)f(analyzing)h(the)f(inner)f (map,)i(for)f(an)m(y)456 3408 y Fn(j)p Fm(")p Fn(j)26 b(\034)f Fs(1)30 b(w)m(e)g(ha)m(v)m(e)h(pro)s(duced)e(a)h(collection)i Fn(fT)2077 3422 y Fp(i)2105 3408 y Fn(g)2150 3360 y Fp(N)7 b Fq(\()p Fp(")p Fq(\))2150 3435 y Fp(i)p Fq(=1)2335 3408 y Fs(consisting)31 b(of)f(primary)456 3516 y(KAM)f(tori,)h (secondary)f(KAM)g(tori)g(or)g(p)s(erio)s(dic)f(orbits)h(suc)m(h)f (that)i(primary)456 3624 y(tori,)35 b(the)f(secondary)g(tori)h(and)e (the)h(lo)s(cal)h(stable)g(and)e(unstable)h(manifolds)456 3734 y(of)26 b(the)g(p)s(erio)s(dic)g(orbits)g(are)h(at)g(a)f(distance) h(less)g(or)f(equal)h(than)f Fm(C)7 b(")2811 3701 y Fq(3)p Fp(=)p Fq(2)2947 3734 y Fs(in)3058 3711 y(~)3049 3734 y(\003)3112 3748 y Fp(")3149 3734 y Fs(.)456 3842 y(W)-8 b(e)28 b(will)f(assume)g(without)g(loss)h(of)f(generalit)m(y)i(that)f (the)f Fn(T)2506 3856 y Fp(i)2561 3842 y Fs(are)h(di\013eren)m(t)f(for) 456 3949 y(di\013eren)m(t)j Fm(i)p Fs(.)555 4057 y(W)-8 b(e)28 b(will)e(refer)g(to)h(this)f(collection)j(as)d(an)1990 4065 y(O)2061 4057 y(\()p Fm(")2138 4024 y Fq(3)p Fp(=)p Fq(2)2249 4057 y Fs(\))g(sca\013olding.)41 b(Note)27 b(that)456 4165 y(these)i(sca\013oldings)g(consist)g(of)g(ob)5 b(jects)30 b(of)e(di\013eren)m(t)h(dimensions)f(\(the)h(p)s(eri-)456 4273 y(o)s(dic)c(orbits)g(are)h(of)g(dimension)f(1)g(and)g(the)h(the)f (KAM)h(tori)g(are)g(of)f(dimension)456 4381 y(2\))31 b(and,)f(of)g(course,)h(di\013eren)m(t)g(top)s(ologies.)555 4489 y(W)-8 b(e)43 b(will)f(\014nd)f(it)h(con)m(v)m(enien)m(t)i(to)e (in)m(tro)s(duced)g(the)f(notion)i(of)f(lea)m(v)m(es)i(of)456 4597 y(the)d(sca\013olding.)73 b(If)40 b Fn(T)1288 4611 y Fp(i)1356 4597 y Fs(is)h(a)g(KAM)g(tori,)j(either)d(primary)f(or)h (secondary)-8 b(,)456 4705 y(w)m(e)44 b(denote)f(the)h(leaf)g(corresp)s (onding)f(to)h Fn(T)2032 4719 y Fp(i)2103 4705 y Fs(b)m(y)f Fm(L)2304 4719 y Fp(i)2332 4705 y Fs(.)80 b(If)43 b Fn(T)2591 4719 y Fp(i)2662 4705 y Fs(is)h(a)f(p)s(erio)s(dic)456 4821 y(orbit,)37 b(w)m(e)e(will)h(asso)s(ciate)h(t)m(w)m(o)g(lea)m(v)m (es)g(of)f(the)f(sca\013olding)i Fm(L)2639 4788 y Fq(ws)2639 4847 y Fp(i)2755 4821 y Fs(=)c Fm(W)2958 4777 y Fq(ws)p Fp(;)p Fq(lo)r(c)2945 4850 y Fl(T)2984 4860 y Ff(i)3149 4821 y Fs(,)456 4959 y Fm(L)518 4926 y Fq(wu)518 4984 y Fp(i)637 4959 y Fs(=)25 b Fm(W)832 4914 y Fq(wu)p Fp(;)p Fq(lo)r(c)819 4987 y Fl(T)858 4997 y Ff(i)p eop end %%Page: 112 112 TeXDict begin 112 111 bop 456 251 a Fq(112)615 b(A.)23 b(Delshams,)g(R.)g(de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 450 y Fw(Remark)48 b(91.)g Fs(F)-8 b(or)42 b(man)m(y)g(purp)s(oses,)h (w)m(e)f(will)g(not)g(need)f(to)i(distinguish)456 558 y(b)s(et)m(w)m(een)26 b Fm(L)864 525 y Fq(ws)864 583 y Fp(i)946 558 y Fs(,)h Fm(L)1060 525 y Fq(wu)1060 583 y Fp(i)1179 558 y Fs(since)f(w)m(e)g(ha)m(v)m(e)g(sho)m(wn)f(that)h (they)f(are)h(at)g(a)g(distance)g Fm(")3107 525 y Fp(m)456 675 y Fs(in)571 653 y(~)562 675 y(\003)625 689 y Fp(")662 675 y Fs(,)31 b(whic)m(h)f(is)h(m)m(uc)m(h)g(smaller)g(than)g(the)g (e\013ects)h(w)m(e)f(will)g(b)s(e)f(considering.)456 783 y(Hence,)40 b(when)d(w)m(e)h(sa)m(y)g(that)h Fm(L)1577 797 y Fp(i)1642 783 y Fj(t)1710 796 y Fq(~)1703 813 y(\003)1752 821 y Ff(")1826 783 y Fm(L)1888 799 y Fp(i)1912 780 y Fg(0)1976 783 y Fs(and)f Fn(T)2211 797 y Fp(i)2276 783 y Fs(is)f(a)i(p)s(erio)s(dic)e(orbit,)i(w)m(e)456 893 y(will)i(mean)h(that)g Fm(L)1162 860 y Fq(ws)1162 919 y Fp(i)1288 893 y Fj(t)1356 906 y Fq(~)1349 923 y(\003)1398 931 y Ff(")1479 893 y Fm(L)1541 908 y Fp(i)1565 890 y Fg(0)1632 893 y Fs(and)f(that)h Fm(L)2090 860 y Fq(wu)2090 919 y Fp(i)2228 893 y Fj(t)2296 906 y Fq(~)2289 923 y(\003)2338 931 y Ff(")2418 893 y Fm(L)2480 908 y Fp(i)2504 890 y Fg(0)2572 893 y Fs(in)f(the)g(sense)h(of)456 1003 y(in)m(tersection)32 b(of)e(the)h(manifolds.)1492 b Fj(\003)555 1215 y Fs(Note)25 b(that)f(the)g(lea)m(v)m(es)h(of)f(the)g(sca\013olding)g(are)g(all)g (of)g(dimension)f(2)h(and)e(w)m(e)456 1323 y(ha)m(v)m(e)36 b(already)f(sho)m(wn)g(that)g(they)g(are)g(v)m(ery)h(close)g(to)g(the)f (lev)m(el)h(sets)g(of)f(the)456 1431 y(function)j(giv)m(en)i(b)m(y)f (the)g(a)m(v)m(eraged)i(hamiltonian.)67 b(In)39 b(particular,)i(w)m(e)f (can)456 1539 y(assume)e(that)i(the)f(a)m(v)m(eraged)i(energy)e(ev)-5 b(aluated)40 b(in)f(di\013eren)m(t)g(lea)m(v)m(es)i(do)s(es)456 1646 y(not)36 b(ha)m(v)m(e)i(an)m(y)f(common)f(v)-5 b(alue.)60 b(This)35 b(allo)m(ws)j(us)e(to)h(order)f(the)g(lea)m(v)m(es)j(b)m(y) 456 1754 y(the)27 b(v)-5 b(alues)27 b(of)h(the)f(a)m(v)m(erage)j (energy)d(on)h(them.)39 b(W)-8 b(e)29 b(can)e(arrange)h(that)f(order) 456 1862 y(in)h Fm(i)i Fs(is)f(the)g(order)g(of)g(increasing)h(a)m(v)m (eraged)h(energy)e(an)g(refer)g(to)h(them)f(as)g(an)456 1970 y(ordered)g(sca\013olding.)555 2078 y(W)-8 b(e)37 b(also)f(recall)h(that)f(in)f(Section)h(9)g(w)m(e)f(ha)m(v)m(e)i (computed)e(and)g(analyzed)456 2186 y(the)e(scattering)i(map.)48 b(W)-8 b(e)34 b(ha)m(v)m(e)h(sho)m(wn)d(in)h(Lemmas)g(82,)i(85)f(and)e (88)i(that,)456 2294 y(under)20 b(the)h(transv)m(ersalit)m(y)j (conditions)e(for)f(a)h(homo)s(clinic)g(in)m(tersection)i(made)456 2402 y(in)34 b(Theorem)h(7,)h(an)m(y)f(leaf)h(of)f(the)g(sca\013olding) h(in)m(tersects)g(transv)m(ersally)g(all)456 2510 y(the)c(lea)m(v)m(es) j(of)e(the)f(sca\013olding)i(at)f(a)g(distance)2138 2518 y(O)2209 2510 y(\()p Fm(")p Fs(\))g(whic)m(h)f(is)h(in)f(the)g(same)456 2618 y(direction.)555 2726 y(It)k(is)f(w)m(orth)g(emphasizing)g(that)h (the)g(sca\013olding)g(dep)s(ends)d(only)j(on)f(the)456 2834 y(inner)c(map)h(and,)h(therefore)g(is)g(indep)s(enden)m(t)e(of)i (the)f(the)h(in)m(tersection)h(that)456 2942 y(w)m(e)h(are)h (considering.)55 b(In)34 b(particular,)j(the)f(constan)m(t)g Fm(C)42 b Fs(is)35 b(indep)s(enden)m(t)f(of)456 3050 y(the)43 b(in)m(tersection.)81 b(On)43 b(the)g(other)h(hand,)i(the)d (scattering)i(map)e(and)g(the)456 3157 y(in)m(tersections)32 b(do)e(dep)s(end)f(on)h(the)g(homo)s(clinic)h(manifold)g(considered.) 456 3324 y Fw(Prop)s(osition)24 b(92.)34 b Fo(Consider)25 b(a)e(Hamiltonian)i(system)f(of)g(Hamiltonian)32 b Fs(\(6\))q Fo(,)456 3432 y(and)43 b(c)-5 b(onsider)44 b(any)f(interval)g Fs(\()p Fm(I)1613 3446 y Fl(\000)1673 3432 y Fm(;)15 b(I)1753 3446 y Fq(+)1812 3432 y Fs(\))43 b Fo(verifying)f(hyp)-5 b(othesis)53 b Fw(H1)p Fo(,.)14 b(.)g(.)g(,)p Fw(H5)456 3540 y Fo(state)-5 b(d)34 b(in)e(The)-5 b(or)g(em)35 b(7.)42 b(Assume)32 b(that)i Fs(0)26 b Fm(<)f Fn(j)p Fm(")p Fn(j)h Fm(<<)f Fs(1)33 b Fo(is)f(given.)555 3668 y(L)-5 b(et)24 b Fn(fT)798 3682 y Fp(i)826 3668 y Fn(g)871 3620 y Fp(N)7 b Fq(\()p Fp(")p Fq(\))871 3696 y Fp(i)p Fq(=1)1049 3668 y Fo(b)-5 b(e)24 b(a)f Fm(C)7 b(")1335 3635 y Fq(3)p Fp(=)p Fq(2)1468 3668 y Fo(or)-5 b(der)g(e)g(d)26 b(sc)-5 b(a\013olding)25 b(as)f(pr)-5 b(o)g(duc)g(e)g(d)26 b(in)e(Se)-5 b(ction)31 b Fs(8)p Fo(.)555 3802 y(Then,)i Fn(fT)915 3816 y Fp(i)943 3802 y Fn(g)988 3754 y Fp(N)7 b Fq(\()p Fp(")p Fq(\))988 3829 y Fp(i)p Fq(=1)1176 3802 y Fo(is)32 b(a)h(tr)-5 b(ansition)35 b(chain.)555 3935 y(If)23 b(in)31 b Fs(\(13\))24 b Fo(we)g(take)f(the)h(p)-5 b(ositive)24 b(alternative,)j(we)c(c)-5 b(onclude)24 b(that)h Fn(fT)2934 3953 y Fp(N)7 b Fq(\()p Fp(")p Fq(\)+1)p Fl(\000)p Fp(i)3258 3935 y Fn(g)3303 3887 y Fp(N)g Fq(\()p Fp(")p Fq(\))3303 3962 y Fp(i)p Fq(=1)3458 3935 y Fo(.)555 4101 y Fs(The)46 b(Prop)s(osition)h(is)g(an)f(ob)m(vious)h(consequence) h(of)e(the)h(Lemmas)g(82,)456 4209 y(85)e(whic)m(h)f(establish)h(that,) k(under)43 b(the)i(h)m(yp)s(othesis)f(of)h(transv)m(ersalit)m(y)h(in) 456 4317 y(Theorem)i(7)i(and)e(the)i(pro)m(ximit)m(y)g(assumptions)e (in)h(Prop)s(osition)g(92)h(for)456 4425 y Fm(L)518 4443 y Fp(\033)r Fq(\()p Fp(i)p Fq(\))643 4425 y Fm(;)15 b(L)745 4443 y Fp(\033)r Fq(\()p Fp(i)p Fq(+1\))961 4425 y Fs(,)31 b(w)m(e)g(ha)m(v)m(e)g(that)g Fm(S)5 b Fs(\()p Fm(L)1715 4443 y Fp(\033)r Fq(\()p Fp(i)p Fq(\))1841 4425 y Fs(\))26 b Fj(t)1970 4438 y Fq(~)1963 4455 y(\003)2012 4463 y Ff(")2074 4425 y Fm(L)2136 4443 y Fp(\033)r Fq(\()p Fp(i)p Fq(+1\))2352 4425 y Fs(.)555 4535 y(By)31 b(Lemma)f(78)i(this)e(means)g (that)1471 4688 y Fm(W)1570 4651 y Fq(u)1557 4711 y Fp(L)1605 4725 y Ff(\033)r Fi(\()p Ff(i)p Fi(\))1746 4688 y Fj(t)25 b Fm(W)1931 4651 y Fq(s)1918 4711 y Fp(L)1966 4725 y Ff(\033)r Fi(\()p Ff(i)p Fi(+1\))555 4854 y Fs(When)35 b(b)s(oth)e Fm(L)1105 4873 y Fp(\033)r Fq(\()p Fp(i)p Fq(\))1265 4854 y Fm(L)1327 4873 y Fp(\033)r Fq(\()p Fp(i)p Fq(+1\))1578 4854 y Fs(are)h(KAM)h(tori,)h(this)f(implies)g (immediately)456 4962 y(that)c Fn(T)703 4981 y Fp(\033)r Fq(\()p Fp(i)p Fq(\))858 4962 y Fn(T)908 4981 y Fp(\033)r Fq(\()p Fp(i)p Fq(+1\))1154 4962 y Fs(are)g(elemen)m(ts)g(of)g(a)g (transition)g(c)m(hain.)p eop end %%Page: 113 113 TeXDict begin 113 112 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)718 b(113)555 450 y Fs(When)37 b(either)h(of)f Fn(T)1252 469 y Fp(\033)r Fq(\()p Fp(i)p Fq(\))1414 450 y Fn(T)1464 469 y Fp(\033)r Fq(\()p Fp(i)p Fq(+1\))1717 450 y Fs(are)g(p)s(erio)s(dic)g(orbits,)i(w)m(e)e(tak)m(e) i(in)m(to)f(ac-)456 558 y(coun)m(t)31 b(the)f(observ)-5 b(ations)31 b(in)f(Remark)h(91.)555 666 y(If)42 b Fn(T)708 684 y Fp(\033)r Fq(\()p Fp(i)p Fq(\))876 666 y Fs(is)g(a)h(p)s(erio)s (dic)e(orbit)i(w)m(e)f(observ)m(e)h(that)g(w)m(e)g(ha)m(v)m(e)h(sho)m (wn)d(that)456 774 y Fm(W)555 741 y Fq(wu)542 805 y Fp(\033)r Fq(\()p Fp(i)p Fq(\))692 774 y Fj(t)760 787 y Fq(~)753 804 y(\003)802 812 y Ff(")865 774 y Fm(L)927 792 y Fp(\033)r Fq(\()p Fp(i)p Fq(+1\))1143 774 y Fs(.)e(If)26 b Fn(T)1344 792 y Fp(\033)r Fq(\()p Fp(i)p Fq(\))1496 774 y Fs(is)h(a)g(p)s(erio)s (dic)g(orbit)f(w)m(e)i(observ)m(e)f(that)g(w)m(e)h(ha)m(v)m(e)456 898 y(sho)m(wn)37 b(that)i Fm(W)1043 865 y Fq(ws)1030 930 y Fp(\033)r Fq(\()p Fp(i)p Fq(+1\))1284 898 y Fj(t)1352 911 y Fq(~)1345 928 y(\003)1394 936 y Ff(")1470 898 y Fm(L)1532 917 y Fp(\033)r Fq(\()p Fp(i)p Fq(\))1658 898 y Fs(.)64 b(\(In)38 b(this)g(pap)s(er,)h(the)g(case)g(when)f(b)s(oth) 456 1018 y Fn(T)506 1036 y Fp(\033)r Fq(\()p Fp(i)p Fq(\))631 1018 y Fs(,)28 b Fn(T)734 1036 y Fp(\033)r Fq(\()p Fp(i)p Fq(+1\))977 1018 y Fs(are)g(p)s(erio)s(dic)e(orbits)i(do)s(es)f(not)h (app)s(ear,)f(but)g(the)g(conclusions)456 1129 y(remain)j(the)g (same\).)555 1237 y(In)g(either)h(case,)g(w)m(e)g(obtain)g(that)g Fm(W)1857 1204 y Fq(u)1844 1264 y Fl(T)1883 1278 y Ff(\033)r Fi(\()p Ff(i)p Fi(\))2024 1237 y Fj(t)25 b Fm(W)2209 1204 y Fq(s)2196 1264 y Fl(T)2235 1278 y Ff(\033)r Fi(\()p Ff(i)p Fi(+1\))2428 1237 y Fs(.)650 b Fj(\003)555 1363 y Fs(The)28 b(meaning)h(of)g(Prop)s(osition)g(92)g(is)g(that)g(w)m(e)g (can)g(use)f(the)h(ob)5 b(jects)29 b(pro-)456 1471 y(duced)h(in)h(the)g (sca\013olding)h(to)g(pro)s(duce)e(transition)i(c)m(hains.)43 b(The)31 b(only)g(con-)456 1578 y(strain)m(ts)j(is)h(that)f(the)h (steps)f(can)g(not)h(b)s(e)e(longer)i(that)2416 1586 y(O)2486 1578 y(\()p Fm(")p Fs(\))h({)e(the)g(amoun)m(t)456 1686 y(mo)m(v)m(ed)29 b(b)m(y)e(a)i(hetero)s(clinic)g(excursion)f({)g (that)h(the)f(ob)5 b(jects)29 b(are)f(ordered)f(and)456 1794 y(that)d(w)m(e)g(do)g(not)f(step)h(out)g(of)g(the)g(region)g (where)f(w)m(e)h(ha)m(v)m(e)h(the)f(transv)m(ersalit)m(y)456 1902 y(conditions.)555 2010 y(W)-8 b(e)32 b(emphasize)f(that)g(the)f (conclusion)456 2195 y(\(167\))932 b Fm(W)1692 2158 y Fq(u)1679 2218 y Fl(T)1718 2228 y Ff(i)1773 2195 y Fj(t)25 b Fm(W)1958 2158 y Fq(s)1945 2218 y Fl(T)1984 2237 y Ff(i)2006 2223 y Fg(0)456 2386 y Fs(is)44 b(indep)s(enden)m(t)f(of)i (the)g(scattering)h(map)e(that)h(w)m(as)g(used)f(in)g(the)g(pro)s(of.) 456 2494 y(Hence,)31 b(if)f(there)g(are)g(di\013eren)m(t)g(in)m (tersections)i(b)s(et)m(w)m(een)e Fm(W)2548 2461 y Fq(s)2542 2520 y(~)2535 2537 y(\003)2584 2545 y Ff(")2622 2494 y Fs(,)g Fm(W)2776 2461 y Fq(u)2770 2520 y(~)2763 2537 y(\003)2812 2545 y Ff(")2849 2494 y Fs(,)g(w)m(e)h(can)456 2616 y(construct)f(more)h(complicated)h(transitions.)555 2724 y(Recall|see)f(remark)d(11|that)j(there)d(is)h(a)g(P)m(oincar)m (\023)-43 b(e)31 b(function)d Fn(L)p Fs(\()p Fm(I)7 b(;)15 b(')j Fn(\000)456 2832 y Fm(I)7 b(\034)e(;)15 b(s)24 b Fn(\000)g Fm(\034)10 b Fs(\),)39 b(where)d Fn(L)h Fs(is)g(giv)m(en)g (in)g(\(10\))q(,)i(for)d(eac)m(h)i(parameterization)h(\(23\))456 2940 y(c)m(hosen.)555 3048 y(Moreo)m(v)m(er,)34 b(the)d(function)g Fm(\034)37 b Fn(7!)26 b(L)p Fs(\()p Fm(I)7 b(;)15 b(')22 b Fn(\000)e Fm(I)7 b(\034)e(;)15 b(s)21 b Fn(\000)g Fm(\034)10 b Fs(\))31 b(will)h(t)m(ypically)h(ha)m(v)m(e)456 3156 y(sev)m(eral)h(critical)i(p)s(oin)m(ts)d Fm(\034)1390 3123 y Fl(\003)1429 3156 y Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\).)52 b(\(If)33 b(it)h(is)g(not)f(constan)m(t,)j(w)m(e)e(will)g(ha) m(v)m(e)456 3264 y(at)d(least)g(a)g(minim)m(um)f(and)f(a)i(maxim)m (um.\))555 3338 y Fq(1)555 3481 y Fs(F)-8 b(urthermore,)35 b(as)e(observ)m(ed)h(in)f(Remark)h(8,)h(t)m(ypically)-8 b(,)37 b(giv)m(en)d(an)f(homo-)456 3589 y(clinic)d(in)m(tersection,)i (there)f(will)f(b)s(e)f(t)m(w)m(o)i(di\013eren)m(t)g(op)s(en)e(sets)h Fm(H)2741 3551 y Fq(+)2734 3612 y Fl(\000)2800 3589 y Fs(,)g Fm(H)2938 3551 y Fq(+)2931 3612 y Fl(\000)3027 3589 y Fs(sat-)456 3697 y(isfying)f(the)h(h)m(yp)s(othesis)f Fw(H4)h Fs(but)f(taking)i(the)e(p)s(ositiv)m(e)i(and)e(negativ)m(e)j (signs)456 3805 y(in)e(\(13\))q(.)555 3913 y(By)d(Prop)s(osition)f(72,) h(an)m(y)g(of)f(them)g(giv)m(es)h(rise)f(to)h(a)f(homo)s(clinic)h (manifold)456 4022 y(to)22 b(the)f(manifold)1082 3999 y(~)1073 4022 y(\003)1136 4036 y Fp(")1173 4022 y Fs(.)37 b(Eac)m(h)22 b(of)g(these)g(homo)s(clinic)g(manifolds)f(can)h(b)s(e)f (used)f(to)456 4130 y(construct)28 b(a)g(scattering)h(map)e Fm(S)5 b Fs(.)40 b(Of)27 b(course,)i(the)f(results)f(of)h(transv)m (ersalit)m(y)456 4238 y(of)38 b(the)g(hetero)s(clinic)h(in)m (tersections)g(of)f(Lemma)g(78,)j(are)d(v)-5 b(alid)38 b(for)g(eac)m(h)h(of)456 4346 y(the)30 b(scattering)i(maps)e (constructed.)p 456 4500 499 4 v 555 4570 a Fq(1)591 4599 y Fv(Moreo)n(v)n(er,)j(as)e(it)g(is)g(w)n(ell)h(kno)n(wn)e(to)h (dynamicists,)i(the)d(fact)i(that)e(one)h(in)n(tersection)456 4690 y(exists)j(implies)i(that)e(their)g(images)h(are)g(also)h(in)n (tersections)f(and)f(that)g(there)g(are)h(other)456 4782 y(secondary)19 b(in)n(tersections)h(\(See)f(e.g.)34 b([Mos73)r(])19 b(Theorem)i(3.8\))f(whic)n(h)f(cannot)g(b)r(e)h(con)n(tin)n(ued)456 4873 y(till)28 b(the)g(case)g Fc(")d Fv(=)f(0.)42 b(Since)27 b(in)h(this)g(pap)r(er,)g(w)n(e)h(are)f(making)g(statemen)n(ts)g(that)g (are)g(v)l(alid)456 4964 y(for)e(all)h Fc(")e Fv(su\016cien)n(tly)g (small,)j(w)n(e)e(will)h(not)e(use)h(these)g(secondary)g(in)n (tersections.)p eop end %%Page: 114 114 TeXDict begin 114 113 bop 456 251 a Fq(114)615 b(A.)23 b(Delshams,)g(R.)g(de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)555 450 y Fs(In)h(the)g(follo)m(wing)i(result,)g(w)m(e)f(form)m(ulate)g(a)g (result)f(that)h(tak)m(es)g(in)m(to)h(acoun)m(t)456 558 y(all)36 b(these)h(p)s(ossibilities.)57 b(Note)37 b(that,)h(in)e (particular,)i(w)m(e)e(ha)m(v)m(e)h(that,)h(when)456 666 y(w)m(e)30 b(ha)m(v)m(e)i(\(16\))r(,)e(w)m(e)h(ha)m(v)m(e)h(a)e (sym)m(b)s(olic)h(dynamics)f(in)g(the)h(transition)g(c)m(hains.)456 831 y Fw(Theorem)43 b(93.)k Fo(Consider)41 b(a)f(Hamiltonian)h(system)g (of)e(Hamiltonian)49 b Fs(\(6\))456 938 y Fo(satisfying)41 b Fw(H1)p Fo(,)32 b Fw(H2)p Fo(,)h Fw(H3)f Fo(in)h(The)-5 b(or)g(em)35 b(7.)555 1048 y(Assume)29 b(that)h(the)f(stable)h(and)g (unstable)f(manifolds)i(of)2514 1025 y Fs(~)2505 1048 y(\003)2568 1062 y Fp(")2634 1048 y Fo(have)f Fm(K)35 b Fo(inter-)456 1156 y(se)-5 b(ctions)33 b(which)h(verify)40 b Fw(H4)p Fo(,)33 b Fw(H5)f Fo(in)h(The)-5 b(or)g(em)34 b(7.)555 1264 y(Denote)44 b(by)f Fm(I)1048 1279 y Fp(k)1136 1264 y Fs(=)h(\()p Fm(I)1333 1231 y Fp(k)1326 1286 y Fl(\000)1386 1264 y Fm(;)15 b(I)1473 1231 y Fp(k)1466 1286 y Fq(+)1525 1264 y Fs(\))p Fo(,)47 b Fm(k)h Fs(=)c(1)p Fm(;)15 b(:)g(:)g(:)j(K)50 b Fo(the)44 b(intervals)g(in)50 b Fw(H4)44 b Fo(and)456 1372 y(by)37 b Fm(\015)626 1387 y Fp(k)702 1372 y Fs(=)c Fn(\006)k Fo(b)-5 b(e)37 b(the)h(signs)f (taken)h(by)f(the)h(function)f(in)44 b Fs(\(13\))r Fo(.)56 b(Assume)37 b(that)456 1412 y Fh(S)531 1507 y Fp(k)589 1480 y Fm(I)629 1495 y Fp(k)697 1480 y Fs(=)25 b Fm(I)840 1447 y Fl(\003)905 1480 y Fn(\021)g Fs(\()p Fm(I)1083 1447 y Fl(\003)1076 1502 y(\000)1135 1480 y Fm(;)15 b(I)1222 1447 y Fl(\003)1215 1502 y Fq(+)1275 1480 y Fs(\))p Fo(.)555 1588 y(Ther)-5 b(e)37 b(is)f(a)h(c)-5 b(onstant)38 b Fm(C)h(>)31 b Fs(0)37 b Fo(and)g Fm(")1880 1555 y Fl(\003)1952 1588 y Fm(>)31 b Fs(0)37 b Fo(such)f(that)i(for)f Fs(0)32 b Fm(<)g Fn(j)p Fm(")p Fn(j)h Fm(<)e(")3134 1555 y Fl(\003)456 1696 y Fo(we)h(have)h(the)g(fol)5 b(lowing.)555 1819 y(L)-5 b(et)33 b Fn(fT)807 1833 y Fp(i)835 1819 y Fn(g)880 1772 y Fp(N)7 b Fq(\()p Fp(")p Fq(\))880 1847 y Fp(i)p Fq(=1)1068 1819 y Fo(b)-5 b(e)32 b(a)h Fm(C)7 b(")1372 1786 y Fq(3)p Fp(=)p Fq(2)1515 1819 y Fo(sc)-5 b(a\013olding)34 b(as)f(pr)-5 b(o)g(duc)g(e)g(d)36 b(in)c(Se)-5 b(ction)40 b Fs(8)p Fo(.)555 1927 y(Given)33 b(a)g(map)g Fm(\033)c Fs(:)c Fk(N)g Fn(!)h(f)p Fs(1)p Fm(;)15 b Fs(2)p Fm(;)g(:)g(:)g(:)j(N) 10 b Fs(\()p Fm(")p Fs(\))p Fn(g)34 b Fo(such)f(that:)671 2057 y Fn(\017)1181 2169 y(T)1231 2188 y Fp(\033)r Fq(\()p Fp(i)p Fq(\))1382 2169 y Fn(\032)25 b Fs(\()p Fm(I)1560 2132 y Fl(\003)1553 2192 y(\000)1632 2169 y Fs(+)20 b Fm(C)7 b(";)15 b(I)1924 2132 y Fl(\003)1917 2192 y Fq(+)1997 2169 y Fn(\000)20 b Fm(C)7 b(")p Fs(\))20 b Fn(\002)g Fk(T)2409 2132 y Fq(2)671 2301 y Fn(\017)1296 2413 y Fs(dist)1450 2426 y Fq(~)1443 2443 y(\003)1492 2451 y Ff(")1530 2413 y Fs(\()p Fm(L)1627 2432 y Fp(\033)r Fq(\()p Fp(i)p Fq(\))1753 2413 y Fm(;)15 b(L)1855 2432 y Fp(\033)r Fq(\()p Fp(i)p Fq(+1\)\))2124 2413 y Fn(\024)24 b Fm(C)7 b(")671 2545 y Fn(\017)1456 2658 y Fm(L)1518 2676 y Fp(\033)r Fq(\()p Fp(i)p Fq(\))1644 2658 y Fm(;)15 b(L)1746 2676 y Fp(\033)r Fq(\()p Fp(i)p Fq(+1\))1987 2658 y Fn(\032)25 b Fm(I)2130 2620 y Fp(k)758 2789 y Fo(and)k(the)f(or)-5 b(dering)29 b(of)e Fm(L)1594 2808 y Fp(\033)r Fq(\()p Fp(i)p Fq(\))1720 2789 y Fm(;)15 b(L)1822 2808 y Fp(\033)r Fq(\()p Fp(i)p Fq(+1\))2065 2789 y Fo(is)28 b(c)-5 b(omp)g(atible)29 b(with)g(the)f(or)-5 b(der-)758 2897 y(ing)33 b(of)g(the)g(interval.) 555 3026 y(Then,)g(the)g(se)-5 b(quenc)g(e)32 b Fn(fT)1435 3045 y Fp(\033)r Fq(\()p Fp(i)p Fq(\))1561 3026 y Fn(g)1606 2993 y Fl(1)1606 3052 y Fp(i)p Fq(=1)1758 3026 y Fo(is)g(a)h(tr)-5 b(ansition)35 b(chain.)555 3202 y Fs(Note)f(that,)h(in)d(particular,)i (if)f(w)m(e)g(ha)m(v)m(e)h(\(16\))h(w)m(e)e(can)g(tak)m(e)h Fm(I)2739 3169 y Fq(1)2808 3202 y Fs(=)29 b Fm(I)2955 3169 y Fq(2)3027 3202 y Fs(and)456 3310 y(b)s(oth)g(equal)i(to)h(the)e (in)m(terv)-5 b(al,)32 b(but)e(w)m(e)h(tak)m(e)h(a)f(di\013eren)m(t)g (c)m(hoices)h(of)f(signs)f(in)456 3418 y(\(13\))q(.)41 b(Hence,)31 b(w)m(e)g(can)g(obtain)g(man)m(y)f(di\013eren)m(t)h (transition)g(c)m(hains.)555 3526 y(In)42 b(general,)k(when)c(it)g(is)h (p)s(ossible)f(to)h(mak)m(e)g(a)g(circuit)g(along)g(orien)m(ted)456 3634 y(in)m(terv)-5 b(als,)31 b(w)m(e)g(ha)m(v)m(e)h(an)e(uncoun)m (table)h(c)m(hoice)h(for)e(transition)h(c)m(hains.)480 3831 y(11.)47 b Ft(Orbits)33 b(shado)n(wing)h(the)f(transition)h (chains)f(and)h(pr)n(oof)f(of)1658 3939 y(theorem)f(7)555 4101 y Fs(In)38 b(this)h(section,)j(w)m(e)d(just)f(state)i(the)e(w)m (ell)i(kno)m(wn)e(result)h(that)g(giv)m(en)g(a)456 4209 y(transition)31 b(c)m(hain,)g(w)m(e)g(can)g(\014nd)e(an)h(orbit)h (visiting)g(all)g(the)g(elemen)m(ts)h(of)f(the)456 4317 y(c)m(hain.)555 4425 y(There)j(are)g(man)m(y)h(pro)s(ofs)e(of)h (similar)h(results)f(in)f(the)i(literature.)53 b(In)33 b(our)456 4533 y(case,)45 b(ho)m(w)m(ev)m(er,)g(w)m(e)c(ha)m(v)m(e)i (to)e(mak)m(e)h(sure)f(that)g(the)g(pro)s(ofs)g(remain)g(v)-5 b(alid)456 4640 y(for)36 b(transition)h(c)m(hains)h(that)f(incorp)s (orate)g(ob)5 b(jects)38 b(with)e(di\013eren)m(t)i(top)s(olo-)456 4748 y(gies)23 b(and)g(indeed)f(di\013eren)m(t)h(dimensions.)38 b(In)22 b(our)h(form)m(ulation)g(also)h(w)m(e)f(allo)m(w)456 4856 y(the)f(transition)g(c)m(hains)h(to)g(ha)m(v)m(e)g(in\014nite)f (man)m(y)g(transition)g(tori.)39 b(This)21 b(mak)m(es)456 4964 y(certain)29 b(w)m(ell)g(kno)m(wn)f(pro)s(ofs)g(in)g(the)g (literature)i(not)f(applicable.)41 b(Hence,)29 b(w)m(e)p eop end %%Page: 115 115 TeXDict begin 115 114 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)718 b(115)456 450 y Fs(presen)m(t)33 b(full)h(details,)h(follo)m(wing)g(the)f(exp)s(osition)g(in)g([DLS00)q (].)50 b(An)34 b(exp)s(osi-)456 558 y(tion)c(of)h(related)g(results)f (along)i(similar)f(lines)f(can)h(b)s(e)f(found)f(in)h([FM03)r(].)456 736 y Fw(Lemma)k(94.)42 b Fo(L)-5 b(et)33 b Fn(fT)1253 750 y Fp(i)1281 736 y Fn(g)1326 703 y Fp(N)1326 762 y(i)p Fq(=0)1478 736 y Fo(b)-5 b(e)32 b(a)h(tr)-5 b(ansition)35 b(chain.)555 844 y(Given)27 b Fs(\()p Fm(\016)890 858 y Fp(i)919 844 y Fs(\))954 858 y Fp(i)p Fq(=0)p Fp(;:::)o(;N)1260 844 y Fo(b)-5 b(e)27 b(a)g(se)-5 b(quenc)g(e)27 b(of)g(strictly)h(p)-5 b(ositive)28 b(numb)-5 b(ers,)28 b(we)f(c)-5 b(an)456 955 y(\014nd)33 b(a)f(p)-5 b(oint)40 b Fs(~)-51 b Fm(x)25 b Fn(2)g Fs(\()p Fk(R)20 b Fn(\002)f Fk(T)p Fs(\))1415 922 y Fq(2)1475 955 y Fn(\002)g Fk(T)p Fo(,)32 b(and)h(a)g(incr)-5 b(e)g(asing)33 b(se)-5 b(quenc)g(e)33 b(of)f(numb)-5 b(ers)456 1063 y Fs(0)25 b(=)g Fm(t)655 1077 y Fq(0)720 1063 y Fm(<)g(:)15 b(:)g(:)26 b(<)f(t)1076 1077 y Fp(N)1176 1063 y Fo(such)32 b(that)1455 1212 y Fs(~)1445 1235 y(\010)1511 1249 y Fp(t)1536 1259 y Ff(i)1563 1249 y Fp(;")1619 1235 y Fs(\()6 b(~)-51 b Fm(x)p Fs(\))26 b Fn(2)f Fw(B)p Fs(\()p Fn(T)2012 1249 y Fp(i)2040 1235 y Fm(;)15 b(\016)2120 1249 y Fp(i)2149 1235 y Fs(\))456 1407 y Fo(wher)-5 b(e)33 b Fw(B)p Fs(\()p Fn(T)871 1421 y Fp(i)900 1407 y Fm(;)15 b(\016)980 1421 y Fp(i)1009 1407 y Fs(\))33 b Fo(is)f(neighb)-5 b(orho)g(o)g(d)36 b(of)d(size)g Fm(\016)2038 1421 y Fp(i)2099 1407 y Fo(of)g(the)g(torus)g Fn(T)2640 1421 y Fp(i)2668 1407 y Fo(.)555 1584 y Fs(A)e(particular)f(case)i(of)e(the)h(results)f (of)h([FM00)q(])g(is:)456 1760 y Fw(Lemma)h(95.)40 b Fo(L)-5 b(et)32 b Fm(f)40 b Fo(b)-5 b(e)30 b(a)h Fn(C)1477 1727 y Fq(2)1548 1760 y Fo(symple)-5 b(ctic)32 b(mapping)g(in)e(a)i (symple)-5 b(ctic)31 b(man-)456 1868 y(ifold.)43 b(Assume)32 b(that)i(the)f(map)h(le)-5 b(aves)34 b(invariant)g Fn(C)2282 1835 y Fq(1)2354 1868 y Fo(torus)f Fn(T)56 b Fo(and)33 b(that)h(the)456 1976 y(motion)41 b(on)f(the)g(torus)h(is)e(an)i(irr)-5 b(ational)42 b(r)-5 b(otation.)66 b(L)-5 b(et)40 b Fs(\000)f Fo(b)-5 b(e)40 b(a)g(manifold)456 2084 y(interse)-5 b(cting)33 b Fm(W)1044 2051 y Fq(u)1031 2111 y Fl(T)1123 2084 y Fo(tr)-5 b(ansversal)5 b(ly.)45 b(Then,)1484 2297 y Fm(W)1583 2259 y Fq(s)1570 2319 y Fl(T)1655 2297 y Fn(\032)p 1751 2191 395 4 v 1758 2211 a Fh([)1751 2406 y Fp(i>)p Fq(0)1881 2297 y Fm(f)1936 2271 y Fl(\000)p Fp(i)2018 2297 y Fs(\(\000\))555 2565 y(W)-8 b(e)31 b(emphasize)g(that)f(since)h(the)f(pro)s(of)f(in)h ([FM00)r(])g(only)g(mak)m(es)h(assump-)456 2673 y(tions)g(ab)s(out)g 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y Fp(i)572 3713 y Fn(g)617 3680 y Fl(1)617 3738 y Fp(i)p Fq(=1)765 3713 y Fo(a)30 b(se)-5 b(quenc)g(e)30 b(of)f(strictly)i(p)-5 b(ositive)31 b(numb)-5 b(ers,)30 b(we)g(c)-5 b(an)31 b(\014nd)f(a)g(p)-5 b(oint)31 b Fm(P)456 3821 y Fo(and)i(a)g(incr)-5 b(e)g(asing)34 b(se)-5 b(quenc)g(e)32 b(of)h(numb)-5 b(ers)33 b Fm(T)2030 3835 y Fp(i)2091 3821 y Fo(such)g(that)1477 3992 y Fs(\010)1543 4006 y Fp(T)1584 4016 y Ff(i)1615 3992 y Fs(\()p Fm(P)13 b Fs(\))26 b Fn(2)f Fm(N)1941 4006 y Fp(")1974 4016 y Ff(i)2004 3992 y Fs(\()p Fn(T)2089 4006 y Fp(i)2117 3992 y Fs(\))456 4164 y Fo(wher)-5 b(e)33 b Fm(N)785 4178 y Fp(")818 4188 y Ff(i)848 4164 y Fs(\()p Fn(T)933 4178 y Fp(i)961 4164 y Fs(\))g Fo(is)g(a)g(neighb)-5 b(orho)g(o)g(d)36 b(of)d(size)g Fm(")2072 4178 y Fp(i)2133 4164 y Fo(of)f(the)h(torus)h Fn(T)2674 4178 y Fp(i)2702 4164 y Fo(.)456 4341 y(Pr)-5 b(o)g(of.)43 b Fs(Let)26 b Fm(x)f Fn(2)g Fm(W)1157 4308 y Fq(s)1144 4368 y Fl(T)1183 4377 y Fi(1)1222 4341 y Fs(.)39 b(W)-8 b(e)27 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Fs(suc)m(h)i(that,)h(b)s(esides)e(satisfying)j(\(168\):)1427 716 y(\010)1493 730 y Fp(T)1534 739 y Fi(2)1573 716 y Fs(\()p Fm(B)1677 730 y Fq(2)1716 716 y Fs(\))26 b Fn(\032)f Fm(N)1946 730 y Fp(")1979 739 y Fi(2)2017 716 y Fs(\()p Fn(T)2102 730 y Fq(2)2142 716 y Fs(\))p Fm(:)555 868 y Fs(Pro)s(ceeding)31 b(b)m(y)f(induction,)g(w)m(e)h(can)g(\014nd)e(a)i (sequence)f(of)h(closed)g(balls)1359 1021 y Fm(B)1428 1035 y Fp(i)1481 1021 y Fn(\032)25 b Fm(B)1646 1035 y Fp(i)p Fl(\000)p Fq(1)1790 1021 y Fn(\032)g(\001)15 b(\001)g(\001)26 b(\032)f Fm(B)2182 1035 y Fq(1)1359 1156 y Fs(\010)1425 1170 y Fp(T)1466 1180 y Ff(j)1502 1156 y Fs(\()p Fm(B)1606 1170 y Fp(i)1635 1156 y Fs(\))g Fn(\032)g Fm(N)1864 1170 y Fp(")1897 1180 y Ff(j)1933 1156 y Fs(\()p Fn(T)2018 1170 y Fp(j)2055 1156 y Fs(\))p Fm(;)107 b(i)25 b Fn(\024)g Fm(j:)555 1311 y Fs(Since)e(the)g(balls)g(are)g(compact,)j Fn(\\)p Fm(B)1796 1325 y Fp(i)1849 1311 y Fn(6)p Fs(=)f Fn(;)p Fs(.)39 b(A)23 b(p)s(oin)m(t)g Fm(P)36 b Fs(in)22 b(the)h(in)m(tersection)456 1419 y(satis\014es)30 b(the)h(required)f (prop)s(ert)m(y)-8 b(.)1440 b Fj(\003)456 1586 y Fo(End)32 b(of)h(the)g(pr)-5 b(o)g(of)35 b(of)e(The)-5 b(or)g(em)34 b(7.)42 b Fs(In)28 b(order)h(to)g(pro)m(v)m(e)h(the)f(existence)h(of)f (an)456 1696 y(orbit)g(~)-51 b Fm(x)p Fs(\()p Fm(t)p Fs(\))26 b(=)958 1673 y(~)948 1696 y(\010)1014 1710 y Fp(t;")1095 1696 y Fs(\()6 b(~)-51 b Fm(x)q Fs(\))23 b(of)h(system)g(\(6\))g(v)m(erifying)h(\(14\))g(w)m(e)f(only)g(need)f (to)h(apply)456 1804 y(Lemma)30 b(94)h(to)g(the)g(transition)g(c)m (hains)g(obtained)f(in)g(prop)s(osition)g(92.)555 1912 y(Then,)g(Theorem)g(7)h(is)f(pro)m(v)m(ed.)3103 2020 y Fj(\003)1141 2223 y Fs(12.)47 b Ft(Conclusions)33 b(and)g(remarks)456 2385 y Fs(12.1.)47 b Fw(The)40 b(role)h(of)f(secondary)i(tori)e(and)g (the)g(sp)s(eed)h(of)f(di\013usion.)456 2493 y Fs(W)-8 b(e)40 b(ha)m(v)m(e)g(sho)m(wn)e(that)i(the)f(secondary)g(whisk)m(ered) f(tori)i(as)f(w)m(ell)g(as)h(lo)m(w)m(er)456 2601 y(dimensional)28 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b(manifolds)g Fm(W)1468 4709 y Fq(s)1462 4768 y(~)1455 4785 y(\003)1544 4742 y Fs(and)f Fm(W)1825 4709 y Fq(u)1819 4768 y(~)1812 4785 y(\003)1904 4742 y Fs(considered)h(in)g (this)g(pap)s(er)f(do)h(not)758 4856 y(in)m(tersect)j(transv)m(ersely)f (in)f(the)g(unp)s(erturb)s(ed)d(case.)62 b(This)36 b(is)h(wh)m(y)758 4964 y(w)m(e)25 b(refer)f(to)h(the)f(system)h(here)f(as)g(a)h(priori)f (unstable,)h(but)f(w)m(e)g(coined)p eop end %%Page: 117 117 TeXDict begin 117 116 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)718 b(117)758 450 y Fs(the)34 b(name)g Fo(a)i(priori)g(chaotic)k Fs(for)33 b(the)h(system)g(in)f ([Mat95)r(,)h(DLS00)q(,)758 558 y(BT99)q(,)k(DLS01)q(].)63 b(This)37 b(has)g(made)h(it)g(necessary)g(for)f(us)g(to)i(study)758 666 y(the)31 b(transv)m(erse)g(in)m(tersection)h(through)e(a)g(p)s (erturbation)g(theory)-8 b(.)671 774 y Fn(\017)42 b Fs(The)21 b(homo)s(clinic)h(connections)g(of)f(the)g(mo)s(del)g(here)g(do)g(not)h (include)e(a)758 882 y(phase)28 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y(mec)m(hanism)35 b(is)h(geared)g(to)m(w)m(ards)g (the)f(explicit)i(v)m(eri\014cation)g(of)e(the)h(mec)m(ha-)456 3345 y(nism)29 b(in)h(concrete)i(systems.)555 3453 y(This)21 b(p)s(oin)m(t)g(of)g(view)h(is)f(motiv)-5 b(ated)23 b(b)s(ecause)e(in)g (applications)h(or)f(in)g(mathe-)456 3561 y(matics,)29 b(one)f(has)g(to)g(deal)h(with)e(concrete)i(systems)f({)g(e.g.)41 b(the)28 b(solar)h(system)456 3669 y(or)42 b(the)g(systems)h(app)s (earing)f(in)g(tec)m(hnology)i({)f(whic)m(h)f(ha)m(v)m(e)i(v)m(ery)e (sp)s(ecial)456 3777 y(prop)s(erties.)555 3885 y(Nev)m(ertheless,)31 b(sev)m(eral)f(colleagues)g(ha)m(v)m(e)g(ask)m(ed)f(ab)s(out)f(the)g (genericit)m(y)j(of)456 3993 y(the)f(mec)m(hanism)h(that)g(w)m(e)g(ha)m (v)m(e)g(discussed)f(here.)555 4101 y(In)25 b(this)h(section,)i(w)m(e)f (will)f(discuss)f(brie\015y)g(and)h(informally)g(the)g(genericit)m(y) 456 4209 y(h)m(yp)s(othesis)k(of)g(Theorem)g(7)h(and)f(of)g(Theorem)g (93.)555 4317 y(First)j(note)h(that)f(the)g(h)m(yp)s(othesis)g(on)f 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g(w)m(e)f(note)h(that)f(the)g(mapping)456 558 y(the)45 b(sends)f(the)h(p)s(erturbation)g(to)g(the)h(Melnik)m(o)m(v)h(in)m (tegral)g(is)e(linear)g(and)456 666 y(non)m(trivial.)c(In)29 b(this)h(case,)h(generic)g(prop)s(erties)f(on)f(the)h(Melnik)m(o)m(v)i (p)s(oten)m(tial)456 774 y(are)j(implied)g(b)m(y)g(generic)h (assumptions)e(on)h(the)g(p)s(erturbation.)53 b(Moreo)m(v)m(er,)456 882 y(since)32 b Fm(h)h Fs(is)f(assumed)f(to)i(b)s(e)f(a)g(p)s (olynomial,)i(w)m(e)e(do)g(not)h(need)f(to)h(discuss)e(in)456 990 y(whic)m(h)f Fm(C)788 957 y Fp(r)855 990 y Fs(genericit)m(y)j(prop) s(erties)d(hold.)555 1098 y(W)-8 b(e)31 b(note)g(that)f(it)g(is)g (generic)h(that)f(there)g(are)g(op)s(en)f(and)h(non-empt)m(y)f(sets)456 1207 y(for)h(whic)m(h)g(w)m(e)h(ha)m(v)m(e)g Fw(H4)g Fs(whic)m(h)f(include)g(the)g(resonances)h(in)2658 1184 y(~)2649 1207 y(\003)2712 1221 y Fp(")2749 1207 y Fs(.)555 1315 y(This)39 b(follo)m(ws)i(b)s(ecause)f(the)g(existence)i(if)e(suc)m (h)f(an)h(op)s(en)f(set)i(is)f(clearly)456 1423 y(op)s(en)27 b(in)h(the)h(space)g(of)f Fm(h)p Fs('s.)41 b(The)27 b(densit)m(y)i (follo)m(ws)g(b)s(ecause)g(the)f(negation)i(of)456 1533 y(the)h(conclusion)h(is)f(that)g(for)g(all)h(p)s(oin)m(ts)f(in)2000 1510 y(~)1991 1533 y(\003)2054 1547 y Fp(")2122 1533 y Fs(in)g(a)g(resonance)h Fn(f)p Fm(k)s(=l)r Fn(g)22 b(\002)f Fk(T)3110 1500 y Fq(2)3149 1533 y Fs(,)456 1641 y(the)27 b(function)h(\000)f(has)h(only)f(degenerate)j(critical)f(p)s (oin)m(ts.)40 b(This)27 b(can)h(b)s(e)f(easily)456 1749 y(destro)m(y)m(ed)k(b)m(y)f(arbitrarily)h(small)g(p)s(erturbations.)555 1857 y(Hence,)44 b(w)m(e)d(conclude)g(that)g(the)f(set)h(of)g(mo)s (dels)f(of)g(the)h(form)f(\(6\))h(that)456 1965 y(o)m(v)m(ercome)32 b(the)f(large)g(problem)f(in)g(one)h(resonance)g(is)f(op)s(en)g(and)g (dense.)555 2073 y(W)-8 b(e)31 b(can)e(also)h(w)m(onder)f(ab)s(out)g (the)h(genericit)m(y)h(of)e(the)h(existence)g(of)g(orbits)456 2180 y(that)38 b(transv)m(erse)g(all)h(the)f(resonances.)64 b(The)37 b(follo)m(wing)j(discussion,)f(based)456 2288 y(simply)31 b(on)g(the)h(coun)m(ting)h(of)f(parameters,)g(presen)m(ts)g (some)g(plausibilit)m(y)h(ar-)456 2396 y(gumen)m(ts)d(for)g(what)h(one) f(w)m(ould)g(exp)s(ect.)555 2504 y(Notice)39 b(that)e(giv)m(en)h(a)f (critical)h(p)s(oin)m(t)f(for)f(\000)h(w)m(e)g(exp)s(ect)g(that)g(it)h (will)e(b)s(e)456 2612 y(non-degenerate)e(except)f(for)g(a)g(set)h(of)f Fm(I)7 b(;)15 b(';)g(s)33 b Fs(of)g(co)s(dimension)g(1.)48 b(In)32 b(these)456 2720 y(sets,)c(w)m(e)f(will)f(ha)m(v)m(e)i(that)f (it)g(is)g(generically)i(p)s(ossible)d(to)h(\014nd)e(sets)i Fm(H)2856 2682 y Fl(\006)2849 2743 y(\000)2941 2720 y Fs(where)456 2828 y(w)m(e)h(ha)m(v)m(e)i(di\013eren)m(t)e(signs)g(in)g (\(13\))q(.)40 b(Generically)-8 b(,)31 b(these)e(sets)f(will)h(ha)m(v)m (e)g(o)m(v)m(er-)456 2936 y(lapping)36 b(pro)5 b(jections)38 b(on)f(the)g Fm(I)44 b Fs(direction,)39 b(hence,)g(it)f(will)f(b)s(e)g (p)s(ossible)f(to)456 3044 y(obtain)30 b(the)h(existence)h(of)e(a)h (sym)m(b)s(olic)g(dynamics.)555 3152 y(If)d(w)m(e)i(consider)e(t)m(w)m (o)i(critical)h(p)s(oin)m(ts,)e(w)m(e)g(exp)s(ect)g(that)g(the)g(sets)g (of)g Fm(I)7 b(;)15 b(';)g(s)456 3260 y Fs(for)30 b(whic)m(h)g(one)h (of)f(them)g(are)h(non-degenerate)h(is)e(a)h(set)g(of)f(co)s(dimension) h(2.)555 3368 y(F)-8 b(urthermore,)37 b(if)e(w)m(e)g(consider)g (systems)g(for)g(whic)m(h)f(the)i(p)s(endulum)c(has)456 3476 y(t)m(w)m(o)44 b(homo)s(clinics,)j(w)m(e)d(ha)m(v)m(e)g(more)f (critical)i(p)s(oin)m(ts)e(to)h(study)-8 b(.)78 b(Once)43 b(w)m(e)456 3585 y(ha)m(v)m(e)36 b(4)f(or)g(more)g(w)m(e)g(exp)s(ect)h (that)f(all)h(the)f(p)s(oin)m(ts)g(in)2419 3562 y(~)2410 3585 y(\003)2473 3599 y Fp(")2545 3585 y Fs(corresp)s(ond)f(to)h(a)456 3693 y(non-degenerate)c(critical)h(p)s(oin)m(t.)555 3801 y(Hence,)46 b(w)m(e)c(exp)s(ect)h(that)f(the)g(in)m(terv)-5 b(als)43 b(corresp)s(onding)e(to)i(a)f(p)s(ositiv)m(e)456 3909 y(sign)34 b(in)g(\(13\))i(o)m(v)m(erlap)h(and)c(their)i(union)f (co)m(v)m(ers)i(all)f(the)g(resonan)m(t)g(regions.)456 4017 y(Similarly)-8 b(,)46 b(the)c(in)m(terv)-5 b(als)44 b(the)e(negativ)m(e)j(signs)d(in)g(\(13\))i(will)f(o)m(v)m(erlap)g(and) 456 4125 y(co)m(v)m(er)i(the)f(resonances.)80 b(Hence,)48 b(w)m(e)c(exp)s(ect)g(that,)k(for)43 b(the)h(systems)f(w)m(e)456 4233 y(ha)m(v)m(e)30 b(consider)f(it)h(will)g(b)s(e)f(generic)h(to)g (ha)m(v)m(e)h(tra)5 b(jectories)31 b(that)f(cross)f(all)i(the)456 4341 y(resonances.)555 4449 y(W)-8 b(e)35 b(p)s(ostp)s(one)f(a)g (detailed)h(v)m(eri\014cation)h(of)e(the)g(results)g(for)g(generic)h (sys-)456 4557 y(tems.)73 b(W)-8 b(e)42 b(hop)s(e)f(that)h(a)f(generic) h(v)m(eri\014cation)h(of)f(the)f(system)g(could)g(b)s(e)456 4665 y(simpler)30 b(and)f(applicable)j(without)e Fw(H3)p Fs(.)555 4772 y(Note)41 b(that)e(all)h(our)f(results)g(on)g(existence)h (of)f(orbits)g(c)m(hanging)i(the)e(ac-)456 4880 y(tion)j(are)h(form)m (ulated)g(for)f(all)h Fm(")f Fs(with)g Fn(j)p Fm(")p Fn(j)k(\024)f Fm(")2178 4847 y Fl(\003)2260 4880 y Fs(with)d Fm(")2521 4847 y Fl(\003)2606 4880 y Fm(>)i Fs(0.)77 b(The)42 b Fm(")3109 4847 y Fl(\003)3149 4880 y Fs(,)p eop end %%Page: 119 119 TeXDict begin 119 118 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)718 b(119)456 450 y Fs(ho)m(w)m(ev)m(er,) 42 b(dep)s(ends)37 b(on)i Fm(h)g Fs(and)f(is)h(only)g(de\014ned)e(for)i Fm(h)g Fs(that)g(satisfy)g(some)456 558 y(non-degeneracy)k(conditions.) 78 b(As)42 b Fm(h)h Fs(approac)m(hes)g(the)g(degenerate)h Fm(h)e Fs(w)m(e)456 666 y(ha)m(v)m(e)31 b Fm(")707 633 y Fl(\003)747 666 y Fs(\()p Fm(h)p Fs(\))g(approac)m(hes)g(zero.)555 774 y(If)g(w)m(e)h(do)f(not)g(w)m(an)m(t)h(to)g(consider)f(parameters,) h(w)m(e)g(are)f(lead)h(to)g(consider)456 882 y(the)e(set)h(of)g Fm("h)g Fs(for)f(whic)m(h)g(w)m(e)h(can)g(establish)g(di\013usion)f (across)h(the)f(biggaps.)456 990 y(This)f(set)h(con)m(tains)h(in)m (terv)-5 b(als)30 b(in)g Fm(")p Fs(,)g(but)f(the)h(size)h(of)f(the)g (in)m(terv)-5 b(al)31 b(can)f(go)g(to)456 1098 y(zero)h(as)f(w)m(e)h (approac)m(h)g(the)f(degenerate)i(sets.)555 1206 y(This)20 b(is)g(v)m(ery)h(similar)f(to)h(the)g Fo(\\cusp)i(r)-5 b(esidual")28 b Fs(sets)20 b(considered)h(in)f([Mat02)r(].)456 1314 y(The)27 b(main)h(di\013erence)g(is)g(that)g(in)g(cusp)f(residual) g(sets)i(one)f(do)s(es)f(not)h(require)456 1421 y(that)38 b(there)g(are)g(whole)g(in)m(terv)-5 b(als)38 b(in)g(the)g(v)-5 b(ariables)38 b Fm(")g Fs(but)f(allo)m(ws)i(to)f(tak)m(e)456 1529 y(a)m(w)m(a)m(y)32 b(some)f(exceptional)h(v)-5 b(alues.)456 1700 y(12.4.)47 b Fw(The)i(h)m(yp)s(othesis)i(of)f(p)s(olynomial)h(p)s (erturbations.)45 b Fs(It)f(seems)456 1808 y(quite)36 b(p)s(ossible)f(that)h Fw(H3)g Fs(can)g(b)s(e)f(a)m(v)m(oided)i(at)g (the)f(price)g(of)g(just)f(carrying)456 1916 y(out)28 b(more)h(delicate)h(estimates.)42 b(A)m(t)29 b(the)f(momen)m(t,)i(w)m (e)f(though)m(t)g(it)g(b)s(etter)f(to)456 2024 y(a)m(v)m(oid)36 b(this)f(complication.)57 b(Nev)m(ertheless)36 b(in)f(the)g(follo)m (wing,)j(w)m(e)e(outline)f(a)456 2132 y(sk)m(etc)m(h)28 b(of)e(a)h(heuristic)g(argumen)m(t)g(for)g(resonances)g(of)g(order)f (1.)40 b(F)-8 b(or)27 b(the)g(sak)m(e)456 2240 y(of)j(clarit)m(y)-8 b(,)33 b(w)m(e)e(ignore)g(constan)m(ts.)555 2348 y(W)-8 b(e)31 b(note)g(that,)g(b)s(ecause)f(the)g(scattering)i(map)d(mo)m(v)m (es)j(b)m(y)d(an)h(amoun)m(t)h(of)456 2456 y(order)23 b Fm(")p Fs(,)i(w)m(e)g(only)e(need)h(to)g(consider)g(resonan)m(t)g (regions)g(whose)g(size)h(is)e(bigger)456 2564 y(or)30 b(equal)h(than)f Fm(")1061 2531 y Fq(1+)p Fp(\016)1220 2564 y Fs(for)g(an)g(arbitrary)g(p)s(ositiv)m(e)i Fm(\016)s Fs(.)555 2672 y(A)c(more)f(careful)h(analysis)f(sho)m(ws)h(that)f(the)h (size)g(of)f(a)h(resonance)g(of)f(order)456 2781 y(1)g(and)g (denominator)h Fm(k)i Fs(can)e(b)s(e)f(b)s(ounded)e(from)i(ab)s(o)m(v)m (e)h(b)m(y)g(\()p Fm(")p Fn(j)p Fm(a)2700 2796 y Fp(k)2743 2781 y Fn(j)p Fs(\))2803 2748 y Fq(1)p Fp(=)p Fq(2)2941 2781 y Fs(where)456 2889 y Fm(a)504 2904 y Fp(k)577 2889 y Fs(is)i(the)h(co)s(e\016cien)m(t)h(of)e(the)h(F)-8 b(ourier)31 b(co)s(e\016cien)m(t)h(of)f(the)f(p)s(erturbation.)555 2996 y(If)c(the)h(p)s(erturbation)f(is)g Fm(C)1484 2963 y Fp(`)1517 2996 y Fs(,)h(w)m(e)h(ha)m(v)m(e)f(that)h Fn(j)p Fm(a)2172 3011 y Fp(k)2215 2996 y Fn(j)d(\024)g Fm(k)2411 2963 y Fl(\000)p Fp(`)2499 2996 y Fs(.)39 b(So)27 b(that)g(the)g(size)456 3110 y(of)j(a)h(resonan)m(t)g(region)g(can)f(b) s(e)g(b)s(ounded)e(from)i(ab)s(o)m(v)m(e)i(b)m(y)e Fm(")2577 3077 y Fq(1)p Fp(=)p Fq(2)2688 3110 y Fn(j)p Fm(k)s Fn(j)2788 3077 y Fl(\000)p Fp(`=)p Fq(2)2947 3110 y Fs(.)555 3218 y(Hence,)i(it)f(is)g(clear)h(that)f(it)g(is)g(only)f(necessary)h(to)h (consider)e(resonan)m(t)i(re-)456 3327 y(gions)20 b(with)g(denominator) h Fm(k)28 b Fn(\024)d Fm(k)1614 3341 y Fs(max)1808 3327 y(where)20 b Fm(k)2108 3341 y Fs(max)2302 3327 y(is)g(suc)m(h)g(that)h Fm(")2807 3294 y Fq(1)p Fp(=)p Fq(2)2917 3327 y Fn(j)p Fm(k)2989 3341 y Fs(max)3163 3327 y Fn(j)3188 3294 y Fl(\000)p Fp(`=)p Fq(2)3372 3327 y Fs(=)456 3437 y Fm(")498 3404 y Fq(1+)p Fp(\016)626 3437 y Fs(.)41 b(That)30 b(is,)1424 3555 y Fn(j)p Fm(k)1496 3569 y Fs(max)1670 3555 y Fn(j)c Fs(=)f Fm(")1859 3517 y Fl(\000)p Fq(1)p Fp(=`)p Fq(+2)p Fp(\016)r(=`)555 3686 y Fs(W)-8 b(e)26 b(note)f(that)g(the)g(minim)m (um)f(distance)h(b)s(et)m(w)m(een)h(t)m(w)m(o)f(rational)h(n)m(um)m(b)s (ers)456 3794 y(of)44 b(denominator)h(smaller)g(that)g Fm(k)1704 3808 y Fs(max)1922 3794 y({)g(and)e(therefore,)49 b(the)c(minim)m(um)456 3902 y(distance)31 b(b)s(et)m(w)m(een)g(the)f (resonances)h(that)g(need)f(to)h(b)s(e)f(considered)g(is:)1404 4053 y Fn(j)p Fm(k)1476 4067 y Fs(max)1650 4053 y Fn(j)1675 4015 y Fl(\000)p Fq(2)1795 4053 y Fs(=)25 b Fm(")1933 4015 y Fq(2)p Fp(=`)p Fq(+4)p Fp(\016)r(=`)555 4209 y Fs(The)30 b(maxim)m(um)g(width)g(of)g(an)m(y)h(of)g(these)g(resonan)m (t)f(regions)h(is)g Fm(")2845 4176 y Fq(1)p Fp(=)p Fq(2)2955 4209 y Fs(.)555 4317 y(Hence,)k(w)m(e)e(see)g(that)h(a)f(condition)h (that)f(ensures)f(that)h(the)g(resonan)m(t)h(re-)456 4425 y(gions)c(are)h(separated)g(is)f Fm(`)c(>)f Fs(4.)555 4533 y(A)31 b(similar)f(analysis)h(w)m(orks)g(for)f(the)g(resonances)h (of)g(order)f(2.)555 4640 y(Once)d(that)g(w)m(e)g(ha)m(v)m(e)h(that)f (the)g(resonan)m(t)g(regions)g(are)g(separated,)h(w)m(e)g(can)456 4748 y(p)s(erform)37 b(a)i(v)m(ery)g(similar)g(analysis)g(than)f(the)h (one)g(p)s(erformed)e(here.)65 b(The)456 4856 y(main)42 b(di\013erence)h(is)g(that)g(rather)f(than)g(c)m(ho)s(osing)i Fm(L)e Fs(as)h(w)m(e)g(ha)m(v)m(e)g(c)m(hosen)456 4964 y(here,)32 b(one)f(w)m(ould)h(need)f(to)h(tak)m(e)h Fm(L)f Fs(dep)s(ending)e(on)h Fm(")h Fs(and)f Fm(k)k Fs(in)c(suc)m(h)g(a)h(w)m (a)m(y)p eop end %%Page: 120 120 TeXDict begin 120 119 bop 456 251 a Fq(120)615 b(A.)23 b(Delshams,)g(R.)g(de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 450 y Fs(that)i Fm(L)f Fs(is)h(m)m(uc)m(h)g(larger)g(than)g(the)f(size) i(of)f(the)g(resonan)m(t)g(region.)39 b(If)25 b Fm(`)f Fs(is)h(large)456 559 y(enough,)30 b(this)g(can)h(b)s(e)f(accomplished) h(e.g.)41 b(b)m(y)31 b(taking)g Fm(L)25 b Fs(=)g Fm(")2626 526 y Fq(1)p Fp(=)p Fq(4)2737 559 y Fn(j)p Fm(k)s Fn(j)2837 526 y Fl(\000)p Fp(`=)p Fq(4)2996 559 y Fs(.)555 667 y(The)31 b(dep)s(endence)g(of)g(the)h(a)m(v)m(eraging)i(results)d(on)h Fm(L)f Fs(can)h(b)s(e)e(w)m(ork)m(ed)i(out.)456 775 y(It)41 b(seems)h(that)g(they)f(can)h(also)h(b)s(e)d(made)i(small)g(for)f (su\016cien)m(tly)h(smo)s(oth)456 883 y(systems.)49 b(On)32 b(the)h(other)h(hand,)f(w)m(e)g(remark)g(that)h(the)f(in)m(tuition)h (gathered)456 991 y(in)g([Chi79)q(],)i(it)g(seems)f(that)h(when)e(the)h (resonances)g(o)m(v)m(erlap,)j(the)e(di\013usion)456 1099 y(should)29 b(b)s(e)h(more)g(in)m(tense.)456 1344 y(12.5.)47 b Fw(In)m(v)m(olving)29 b(other)f(ob)6 b(jects.)46 b Fs(W)-8 b(e)26 b(also)g(note)f(that)g(it)h(should)d(b)s(e)h(p)s(os-) 456 1452 y(sible)h(to)h(consider)f(other)h(orbits)f(b)s(esides)f(the)i (whisk)m(ered)e(tori)i(w)m(e)g(ha)m(v)m(e)g(used.)456 1560 y(In)h(particular,)j(giv)m(en)g(that)f(in)f(a)h(generic)h(system)e (there)h(are)g(h)m(yp)s(erb)s(olic)f(p)s(e-)456 1668 y(rio)s(dic)f(orbits)h(appro)m(ximating)h(the)f(KAM)g(tori)h(\(See)f (e.g.)41 b([FL92)r(]\))28 b(and)f(that)456 1776 y(the)e(stable)g(and)g (unstable)g(manifolds)f(are)i(close)g(to)g(the)f(torus,)h(it)f(seems)g (p)s(os-)456 1883 y(sible)h(that,)h(at)g(least)g(for)f(generic)h (systems)f(one)g(could)g(construct)h(h)m(yp)s(erb)s(olic)456 1991 y(orbits)j(that)h(connect.)555 2099 y(It)e(also)g(seems)f (plausible)g(that)h(one)g(could)f(adapt)g(the)h(metho)s(d)f(of)g ([Lla02)r(])456 2207 y(using)h(normally)i(h)m(yp)s(erb)s(olic)f (laminations)h(to)g(discuss)f(the)g(problem.)555 2315 y(More)25 b(ten)m(tativ)m(ely)-8 b(,)30 b(it)25 b(lo)s(oks)g(plausible) f(that)h(Aubry-Mather)g(Can)m(tor)f(sets)456 2423 y(and)29 b(their)i(in)m(v)-5 b(arian)m(t)31 b(manifolds)f(could)h(b)s(e)f(used)f (as)i(transition)g(elemen)m(ts.)456 2668 y(12.6.)47 b Fw(V)-9 b(ariational)32 b(metho)s(ds.)46 b Fs(There)27 b(is)i(an)f(uncann)m(y)f(relation)i(b)s(et)m(w)m(een)456 2776 y(the)39 b(geometric)i(metho)s(ds)d(and)h(v)-5 b(ariational)41 b(metho)s(ds.)66 b(W)-8 b(e)40 b(think)f(that)g(it)456 2884 y(w)m(ould)h(b)s(e)h(v)m(ery)g(useful)f(to)i(pursue)d(the)j(study) e(of)h(the)g(parallels)h(b)s(et)m(w)m(een)456 2992 y(the)28 b(t)m(w)m(o)i(v)m(ery)f(di\013eren)m(t)g(metho)s(ds.)39 b(It)29 b(seems)g(plausible)f(that)h(if)g(one)g(pro)m(v)m(ed)456 3100 y(connecting)h(lemmas)g(based)g(on)f(v)-5 b(ariational)32 b(metho)s(ds,)d(one)h(could)g(also)g(use)456 3208 y(Aubry-Mather)20 b(sets)h(in)f(place)h(of)g(the)f(tori.)38 b(Of)20 b(course,)j(v)-5 b(ariational)23 b(metho)s(ds)456 3316 y(seem)38 b(to)g(ha)m(v)m(e)h(to) f(use)f(p)s(ositivit)m(y)i(or)e(con)m(v)m(exit)m(y)k(prop)s(erties)c (that)h(are)g(not)456 3424 y(presen)m(t)30 b(in)g(our)g(metho)s(ds.)555 3532 y(W)-8 b(e)23 b(note)g(that)f(it)h(is)e(quite)i(p)s(ossible)e (that)i(one)f(can)g(use)f(a)i(mixed)e(approac)m(h.)456 3640 y(Once)34 b(one)g(iden)m(ti\014es)g(the)g(relev)-5 b(an)m(t)36 b(geometric)g(ob)5 b(jects)34 b(and)g(pro)s(duce)e(het-)456 3748 y(ero)s(clinic)h(connections)g(among)g(them,)g(v)-5 b(ariational)35 b(metho)s(ds)c(can)i(pro)s(duce)456 3856 y(v)m(ery)d(e\013ectiv)m(e)j(shado)m(wing)e(orbits)f([Mat93)r(,)h (CP02,)g(RS02].)555 3963 y(Some)h(implemen)m(tations)h(of)f(these)g (mixed)f(metho)s(ds)g(happ)s(en)f(in)h([Bes96)r(,)456 4071 y(BCV01)q(])456 4317 y(12.7.)47 b Fw(Di\013usion)40 b(times.)45 b Fs(W)-8 b(e)35 b(notice)g(that,)h(during)c(this)i(pap)s (er,)g(nothing)456 4425 y(is)41 b(said)g(ab)s(out)g(the)h(di\013usion)f (time)h(since)f(the)h(shado)m(wing)f(Lemma)h(94)g(is)456 4533 y(only)28 b(based)h(in)f(top)s(ological)j(metho)s(ds)d(and)g(do)s (es)h(not)f(pro)m(vide)h(quan)m(titativ)m(e)456 4640 y(estimates)34 b(ab)s(out)f(the)g(ergo)s(dization)i(time)e(as)h(other)f (metho)s(ds)f(do)h(\([T)-8 b(re02)r(,)456 4748 y(CG03)q(,)30 b(BB02)r(]\).)555 4856 y(W)-8 b(e)30 b(also)g(note)f(the)g(metho)s(ds)e (of)i([Lla02)r(])g(using)f(h)m(yp)s(erb)s(olic)g(laminations,)456 4964 y(that)j(yield)f(v)m(ery)h(go)s(o)s(d)f(times.)p eop end %%Page: 121 121 TeXDict begin 121 120 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)718 b(121)1463 450 y Fs(13.)46 b Ft(An)35 b(example)555 612 y Fs(Consider)30 b(the)g(Hamiltonian)456 815 y(\(169\))90 b Fm(H)827 829 y Fp(")863 815 y Fs(\()p Fm(p;)15 b(q)s(;)g(I)7 b(;)15 b(';)g(t)p Fs(\))27 b(=)e Fn(\006)1531 687 y Fh(\022)1608 754 y Fm(p)1654 721 y Fq(2)p 1608 794 86 4 v 1628 877 a Fs(2)1723 815 y(+)20 b(cos)c Fm(q)23 b Fn(\000)d Fs(1)2151 687 y Fh(\023)2239 815 y Fs(+)2340 754 y Fm(I)2387 721 y Fq(2)p 2340 794 87 4 v 2360 877 a Fs(2)2456 815 y(+)g Fm(")15 b Fs(cos)i Fm(q)28 b(g)s Fs(\()p Fm(';)15 b(t)p Fs(\))p Fm(;)456 1030 y Fs(where)829 1196 y Fm(g)s Fs(\()p Fm(';)g(t)p Fs(\))27 b(=)1259 1110 y Fh(X)1200 1311 y Fq(\()p Fp(k)r(;l)q Fq(\))p Fl(2N)1464 1196 y Fm(a)1512 1211 y Fp(k)r(;l)1612 1196 y Fs(cos)q(\()p Fm(k)s(')21 b Fs(+)f Fm(l)r(t)p Fs(\))g(+)g Fm(b)2237 1211 y Fp(k)r(;l)2336 1196 y Fs(sin\()p Fm(k)s(')h Fs(+)f Fm(l)r(t)p Fs(\))456 1471 y(is)40 b(a)h (trigonometric)i(p)s(olynomial)e(in)f(the)h(angles)h Fm(')p Fs(,)i Fm(t)c Fs(\()p Fn(N)54 b Fs(is)41 b(a)g(\014nite)f(set) 456 1578 y(of)e(indexes\).)63 b(The)38 b(Hamiltonian)h(of)f(one)h (degree)f(of)g(freedom)g Fm(P)2805 1592 y Fl(\006)2865 1578 y Fs(\()p Fm(p;)15 b(q)s Fs(\))38 b(=)456 1687 y Fn(\006)542 1613 y Fh(\000)583 1687 y Fm(p)629 1654 y Fq(2)668 1687 y Fm(=)p Fs(2)21 b(+)f(cos)c Fm(q)23 b Fn(\000)d Fs(1)1207 1613 y Fh(\001)1286 1687 y Fs(is)38 b(the)f(standard)f(p)s(endulum)f(when)h(w)m(e)i(c)m(ho)s(ose)g(the)456 1795 y(+)30 b(sign,)g(and)g(its)h(separatrix)g(for)f(p)s(ositiv)m(e)h Fm(p)f Fs(is)h(giv)m(en)g(b)m(y)g(\(24\))1046 1962 y Fm(q)1087 1976 y Fq(0)1126 1962 y Fs(\()p Fm(t)p Fs(\))26 b(=)f(4)15 b(arctan)h Fm(e)1721 1924 y Fl(\006)p Fp(t)1806 1962 y Fm(;)106 b(p)1983 1976 y Fq(0)2023 1962 y Fs(\()p Fm(t)p Fs(\))25 b(=)g(2)p Fm(=)p Fs(cosh)17 b Fm(t)o(:)555 2127 y Fs(An)22 b(imp)s(ortan)m(t)h(feature)g(of)f(the)h(Hamiltonian)h (\(169\))g(is)f(that)g(the)f(3-dimen-)456 2235 y(sional)31 b(h)m(yp)s(erb)s(olic)e(in)m(v)-5 b(arian)m(t)32 b(manifold)1068 2380 y(~)1059 2403 y(\003)25 b(=)g Fn(f)p Fs(\(0)p Fm(;)15 b Fs(0)p Fm(;)g(I)7 b(;)15 b(';)g(s)p Fs(\))29 b(:)d(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))27 b Fn(2)e Fk(R)19 b Fn(\002)h Fk(T)2486 2365 y Fq(2)2525 2403 y Fn(g)456 2568 y Fs(is)31 b Fo(pr)-5 b(eserve)g(d)42 b Fs(for)31 b Fm(")c Fn(6)p Fs(=)g(0:)43 b Fm(p)26 b Fs(=)h Fm(q)i Fs(=)e(0)g Fn(\))45 b Fs(_)-43 b Fm(p)26 b Fs(=)44 b(_)-42 b Fm(q)29 b Fs(=)e(0.)44 b(Ho)m(w)m(ev)m(er,)34 b(in)c(con)m(trast)456 2676 y(with)35 b(the)h(example)h(in)f([Arn63b],)i(the)e(p)s (erturbation)f(do)s(es)g(not)i(v)-5 b(anish)35 b(on)464 2768 y(~)456 2791 y(\003.)50 b(Indeed,)34 b(restricted)h(to)1451 2768 y(~)1442 2791 y(\003,)g(the)f(reduced)f(Hamiltonian)i(tak)m(es)g (the)f(form)456 2899 y Fm(I)503 2866 y Fq(2)542 2899 y Fm(=)p Fs(2)21 b(+)f Fm("g)s Fs(\()p Fm(';)15 b(t)p Fs(\).)43 b(Hence,)32 b(2-dimensional)f(whisk)m(ered)f(tori)1166 3067 y Fn(T)1239 3030 y Fq(0)1216 3090 y Fp(I)1304 3067 y Fs(=)25 b Fn(f)p Fs(\(0)p Fm(;)15 b Fs(0)p Fm(;)g(I)7 b(;)15 b(';)g(s)p Fs(\))29 b(:)c(\()p Fm(';)15 b(s)p Fs(\))27 b Fn(2)e Fk(T)2379 3030 y Fq(2)2418 3067 y Fn(g)456 3233 y Fs(are)h(not)g(preserv)m(ed,)h(and)f(primary)f(resonances)h(tak) m(e)i(place)f(at)g Fm(I)32 b Fs(=)25 b Fn(\000)p Fm(l)r(=k)k Fs(for)456 3341 y(eac)m(h)35 b(\()p Fm(k)s(;)15 b(l)r Fs(\))33 b Fn(2)e(N)13 b Fm(;)i(k)35 b Fn(6)p Fs(=)d(0)i(suc)m(h)g (that)h Fm(a)1828 3356 y Fp(k)r(;l)1944 3341 y Fn(6)p Fs(=)c(0.)53 b(Therefore,)35 b(\(169\))h(presen)m(ts)456 3449 y(the)30 b(large)i(gap)e(problem.)555 3557 y(The)g(Melnik)m(o)m(v) j(p)s(oten)m(tial)e(\(10\))h(of)f(the)f(Hamiltonian)i(\(169\))g(is)f (giv)m(en)g(b)m(y)950 3771 y Fn(L)p Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))27 b(=)1445 3710 y(1)p 1445 3750 46 4 v 1445 3834 a(2)1516 3648 y Fh(Z)1606 3674 y Fl(1)1566 3854 y(\0001)1711 3771 y Fm(p)1757 3734 y Fq(2)1757 3794 y(0)1796 3771 y Fs(\()p Fm(\033)s Fs(\))p Fm(g)s Fs(\()p Fm(')c Fs(+)c Fm(I)7 b(\033)n(;)15 b(s)21 b Fs(+)f Fm(\033)s Fs(\))p Fm(d\033)n(;)456 3990 y Fs(and)29 b(computing)i(the)f(in)m (tegrals)i(b)m(y)f(the)f(residue)g(theorem,)h(w)m(e)g(obtain:)616 4169 y Fn(L)p Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))27 b(=)1160 4082 y Fh(X)1101 4284 y Fq(\()p Fp(k)r(;l)q Fq(\))p Fl(2N)1366 4169 y Fm(A)1434 4184 y Fp(k)r(;l)1518 4169 y Fs(\()p Fm(I)7 b Fs(\))15 b(cos)r(\()p Fm(k)s(')21 b Fs(+)f Fm(l)r(t)p Fs(\))g(+)g Fm(B)2306 4184 y Fp(k)r(;l)2390 4169 y Fs(\()p Fm(I)7 b Fs(\))15 b(sin)q(\()p Fm(k)s(')21 b Fs(+)f Fm(l)r(t)p Fs(\))p Fm(;)456 4438 y Fs(with)666 4634 y Fm(A)734 4649 y Fp(k)r(;l)844 4634 y Fs(=)25 b(2)p Fm(\031)1170 4572 y Fs(\()p Fm(k)s(I)j Fs(+)20 b Fm(l)r Fs(\))p 1050 4613 549 4 v 1050 4696 a(sinh)1238 4660 y Fp(\031)p 1238 4675 43 4 v 1242 4727 a Fq(2)1290 4696 y Fs(\()p Fm(k)s(I)28 b Fs(+)20 b Fm(l)r Fs(\))1608 4634 y Fm(a)1656 4649 y Fp(k)r(;l)1741 4634 y Fm(;)106 b(B)1941 4649 y Fp(k)r(;l)2050 4634 y Fs(=)25 b(2)p Fm(\031)2377 4572 y Fs(\()p Fm(k)s(I)j Fs(+)20 b Fm(l)r Fs(\))p 2256 4613 549 4 v 2256 4696 a(sinh)2444 4660 y Fp(\031)p 2444 4675 43 4 v 2448 4727 a Fq(2)2497 4696 y Fs(\()p Fm(k)s(I)28 b Fs(+)20 b Fm(l)r Fs(\))2815 4634 y Fm(b)2854 4649 y Fp(k)r(;l)2938 4634 y Fm(:)555 4856 y Fs(As)28 b(w)m(e)h(will)f(v)m (erify)h(in)e(concrete)j(examples,)f(for)f(a)g(v)m(ery)h(general)g(c)m (hoice)h(of)456 4964 y(the)g(co)s(e\016cien)m(ts)i Fm(a)1120 4979 y Fp(k)r(;l)1205 4964 y Fm(;)15 b(b)1284 4979 y Fp(k)r(;l)1398 4964 y Fs(w)m(e)31 b(can)g(\014nd)e(op)s(en)h(sets)h(of) f(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))27 b Fn(2)e Fs([)p Fm(I)2859 4978 y Fl(\000)2918 4964 y Fm(;)15 b(I)2998 4978 y Fq(+)3058 4964 y Fs(])20 b Fn(\002)p eop end %%Page: 122 122 TeXDict begin 122 121 bop 456 251 a Fq(122)615 b(A.)23 b(Delshams,)g(R.)g(de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 450 y Fk(T)517 417 y Fq(2)556 450 y Fs(,)44 b(suc)m(h)d(that)h(the)g (function)f Fm(\034)53 b Fn(2)44 b Fk(R)f Fn(7!)h(L)p Fs(\()p Fm(I)7 b(;)15 b(')28 b Fn(\000)f Fm(I)7 b(\034)e(;)15 b(s)28 b Fn(\000)f Fm(\034)10 b Fs(\))42 b(has)f(non-)456 558 y(degenerate)29 b(critical)g(p)s(oin)m(ts)f(at)g(a)g Fm(\034)35 b Fs(=)25 b Fm(\034)1881 525 y Fl(\003)1921 558 y Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))29 b(whic)m(h)e(v)m(erify) h(h)m(yp)s(othesis)456 666 y Fw(H4)p Fs(.)555 774 y(F)-8 b(or)29 b(instance,)g(let)f(us)f(consider)h(the)g(case)g(of)g(a)g (function)f Fm(g)k Fs(with)d(only)g(t)m(w)m(o)456 882 y(harmonics)456 1034 y(\(170\))486 b Fm(g)s Fs(\()p Fm(';)15 b(t)p Fs(\))27 b(=)e Fm(a)1566 1048 y Fq(0)1621 1034 y Fs(cos)q(\()p Fm(')p Fs(\))c(+)f Fm(a)2032 1048 y Fq(1)2087 1034 y Fs(cos\()p Fm(')h Fn(\000)f Fm(t)p Fs(\))456 1187 y(whic)m(h)37 b(giv)m(es)h(rise)g(to)g(t)m(w)m(o)g(\\large)h(gaps")f (asso)s(ciated)h(to)f(the)f(t)m(w)m(o)i(primary)456 1294 y(resonances)27 b(\(20\))i Fm(I)j Fs(=)25 b(0)p Fm(;)15 b Fs(1,)29 b(as)e(w)m(ell)i(as)e(to)h(the)f(\\big)h(gap")f(asso)s (ciated)i(to)f(the)456 1402 y(secondary)i(resonance)h(\(21\))h Fm(I)g Fs(=)25 b(1)p Fm(=)p Fs(2.)42 b(Assuming)30 b(that)456 1558 y(\(171\))805 b Fm(a)1514 1572 y Fq(0)1553 1558 y Fm(a)1601 1572 y Fq(1)1641 1558 y Fs(\()p Fm(a)1724 1521 y Fq(2)1724 1581 y(0)1784 1558 y Fn(\000)19 b Fm(a)1922 1521 y Fq(2)1922 1581 y(1)1962 1558 y Fs(\))26 b Fn(6)p Fs(=)f(0)456 1710 y(and)39 b(c)m(ho)s(osing,)44 b(for)c(instance,)k([)p Fm(I)1655 1724 y Fl(\000)1714 1710 y Fm(;)15 b(I)1794 1724 y Fq(+)1854 1710 y Fs(])42 b(=)f([)p Fn(\000)p Fs(1)p Fm(=)p Fs(2)p Fm(;)15 b Fs(3)p Fm(=)p Fs(2],)46 b(w)m(e)41 b(are)g(going)g(to)456 1818 y(c)m(hec)m(k)i(that)f(for)f(all)h(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))46 b Fn(2)d Fs([)p Fm(I)1721 1832 y Fl(\000)1780 1818 y Fm(;)15 b(I)1860 1832 y Fq(+)1920 1818 y Fs(])28 b Fn(\002)f Fk(T)2132 1785 y Fq(2)2171 1818 y Fs(,)45 b(the)c(function)g Fm(\034)54 b Fn(2)44 b Fk(R)f Fn(7!)456 1926 y(L)p Fs(\()p Fm(I)7 b(;)15 b(')21 b Fn(\000)f Fm(I)7 b(\034)e(;)15 b(s)20 b Fn(\000)g Fm(\034)10 b Fs(\))31 b(has)f(non-degenerate)h(critical)i(p)s(oin)m(ts.)555 2034 y(First,)e(notice)h(that)456 2187 y Fn(L)p Fs(\()p Fm(\034)10 b Fs(\))25 b(:=)h Fn(L)p Fs(\()p Fm(I)7 b(;)15 b(')10 b Fn(\000)g Fm(I)d(\034)e(;)15 b(s)10 b Fn(\000)g Fm(\034)g Fs(\))25 b(=)g Fm(A)1661 2201 y Fq(0)1716 2187 y Fs(cos)q(\()p Fm(')10 b Fn(\000)g Fm(I)d(\034)j Fs(\))g(+)g Fm(A)2314 2201 y Fq(1)2369 2187 y Fs(cos)q(\()p Fm(')g Fn(\000)g Fm(s)g Fn(\000)g Fs(\()p Fm(I)17 b Fn(\000)10 b Fs(1\))p Fm(\034)g Fs(\))p Fm(;)456 2339 y Fs(with)552 2521 y Fm(A)620 2535 y Fq(0)685 2521 y Fs(=)25 b Fm(A)849 2535 y Fq(0)889 2521 y Fs(\()p Fm(I)7 b Fs(\))26 b(=)f(2)p Fm(\031)1358 2460 y(I)p 1238 2500 288 4 v 1238 2584 a Fs(sinh)1425 2548 y Fp(\031)p 1425 2563 43 4 v 1429 2615 a Fq(2)1478 2584 y Fm(I)1535 2521 y(a)1583 2535 y Fq(0)1623 2521 y Fm(;)106 b(A)1822 2535 y Fq(1)1887 2521 y Fs(=)25 b Fm(A)2051 2535 y Fq(1)2091 2521 y Fs(\()p Fm(I)7 b Fs(\))26 b(=)f(2)p Fm(\031)2560 2460 y Fs(\()p Fm(I)j Fn(\000)20 b Fs(1\))p 2440 2500 515 4 v 2440 2584 a(sinh)2627 2548 y Fp(\031)p 2627 2563 43 4 v 2631 2615 a Fq(2)2680 2584 y Fs(\()p Fm(I)28 b Fn(\000)20 b Fs(1\))2965 2521 y Fm(a)3013 2535 y Fq(1)3052 2521 y Fm(;)456 2736 y Fs(so,)25 b(\014xed)f(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\),)27 b(w)m(e)e(only)f (need)f(to)i(study)e(the)i(ev)m(olution)g(of)g(the)f(Melnik)m(o)m(v)456 2844 y(p)s(oten)m(tial)456 2996 y(\(172\))382 b Fn(L)p Fs(\()p Fm(I)7 b(;)15 b( )1287 3010 y Fq(0)1327 2996 y Fm(;)g( )1426 3010 y Fq(1)1466 2996 y Fs(\))26 b(=)f Fm(A)1691 3010 y Fq(0)1745 2996 y Fs(cos)q(\()p Fm( )1961 3010 y Fq(0)2001 2996 y Fs(\))c(+)f Fm(A)2216 3010 y Fq(1)2270 2996 y Fs(cos)q(\()p Fm( )2486 3010 y Fq(1)2526 2996 y Fs(\))p Fm(;)456 3149 y Fs(expressed)45 b(in)h(the)g(v)-5 b(ariables)46 b(\()p Fm( )1659 3163 y Fq(0)1699 3149 y Fm(;)15 b( )1798 3163 y Fq(1)1838 3149 y Fs(\))52 b(=)f(\()p Fm(';)15 b(')32 b Fn(\000)f Fm(s)p Fs(\))51 b Fn(2)g Fk(T)2676 3116 y Fq(2)2715 3149 y Fs(,)f(along)d(the)456 3256 y(straigh)m(t)31 b(lines)g Fm(R)26 b Fs(=)f Fm(R)q Fs(\()p Fm(I)7 b(;)15 b(';)g(s)p Fs(\))31 b(on)g(the)f(torus:)456 3409 y(\(173\))282 b Fm( )1002 3423 y Fq(0)1041 3409 y Fs(\()p Fm(\034)10 b Fs(\))26 b(=)f Fm(')c Fn(\000)f Fm(I)7 b(\034)e(;)106 b( )1736 3423 y Fq(1)1776 3409 y Fs(\()p Fm(\034)10 b Fs(\))26 b(=)f Fm(')c Fn(\000)e Fm(s)h Fn(\000)g Fs(\()p Fm(I)28 b Fn(\000)20 b Fs(1\))p Fm(\034)5 b(:)555 3561 y Fs(As)23 b(long)h(as)f Fm(a)1024 3575 y Fq(0)1063 3561 y Fm(a)1111 3575 y Fq(1)1176 3561 y Fn(6)p Fs(=)i(0)e(\(and)g(therefore)g Fm(A)1987 3575 y Fq(0)2027 3561 y Fm(A)2095 3575 y Fq(1)2160 3561 y Fn(6)p Fs(=)i(0\),)g(the)e(Melnik)m(o)m(v)i(p)s(oten-)456 3669 y(tial)31 b(\(172\))i(p)s(ossesses)d(four)g(non-degenerate)i (critical)h(p)s(oin)m(ts:)41 b(a)31 b(maxim)m(um,)456 3777 y(a)k(minim)m(um)f(and)g(t)m(w)m(o)i(saddles.)54 b(Around)33 b(the)i(t)m(w)m(o)h(extrem)m(um)f(p)s(oin)m(ts,)h(its)456 3885 y(lev)m(el)41 b(curv)m(es)f(are)g(closed)g(\(and)g(indeed)f(con)m (v)m(ex\))j(curv)m(es)e(whic)m(h)f(\014ll)h(out)f(a)456 3993 y(basin)29 b(ending)h(at)i(the)e(lev)m(el)i(curv)m(e)f(of)f(one)h (of)g(the)f(saddle)g(p)s(oin)m(ts.)555 4101 y(Therefore,)g(an)m(y)f (straigh)m(t)i(line)e(\(173\))i(that)f(en)m(ters)g(in)m(to)g(some)g (extrem)m(um)456 4209 y(basin)24 b(is)g(tangen)m(t)i(to)f(one)g(of)g (the)g(con)m(v)m(ex)h(closed)f(lev)m(el)h(curv)m(es,)g(giving)g(rise)e (to)456 4317 y(a)k(non-degenerate)i(extrem)m(um)f(of)f Fn(L)p Fs(\()p Fm(\034)10 b Fs(\).)41 b(So,)29 b(degenerate)h(extrema)f (of)f Fn(L)p Fs(\()p Fm(\034)10 b Fs(\))456 4425 y(can)39 b(only)h(exist)h(for)e(straigh)m(t)i(lines)f(\(173\))h(that)f Fo(never)50 b Fs(en)m(ter)40 b(inside)f(suc)m(h)456 4533 y(extrem)m(um)33 b(basins.)49 b(In)33 b(particular,)i(this)e(could)g (only)h(happ)s(en)d(for)j(rational)456 4640 y(v)-5 b(alues)42 b(of)f Fm(I)7 b Fs(,)45 b(since)d(an)g(irrational)h(v)-5 b(alue)42 b(of)g Fm(I)48 b Fs(implies)42 b(a)g(dense)g(straigh)m(t)456 4748 y(line)30 b(\(173\))j(\(and)d(in\014nite)g(non-degenerate)h (extrema)h(for)e Fn(L)p Fs(\()p Fm(\034)10 b Fs(\)\).)555 4856 y(A)45 b(closer)h(lo)s(ok)g(at)g(the)f(Melnik)m(o)m(v)i(p)s(oten)m (tial)g(\(172\))g(sho)m(ws)e(that)h(b)s(oth)456 4964 y(extrem)m(um)36 b(basins)g(con)m(tain)h(a)g(v)m(ertical)h (1-dimensional)g(torus)e(\(for)g Fn(j)p Fm(A)3003 4978 y Fq(0)3043 4964 y Fn(j)f(\025)p eop end %%Page: 123 123 TeXDict begin 123 122 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)718 b(123)116 367 y gsave currentpoint currentpoint translate -90 neg rotate neg exch neg exch translate 116 367 a @beginspecial 50 @llx 50 @lly 554 @urx 770 @ury 2880 @rwi @setspecial %%BeginDocument: fig6.eps %!PS-Adobe-2.0 %%Title: gapsexample.ps %%Creator: gnuplot 3.7 patchlevel 3 %%CreationDate: Mon Apr 14 18:08:32 2003 %%DocumentFonts: (atend) %%BoundingBox: 50 50 554 770 %%Orientation: Landscape %%Pages: (atend) %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid true def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def /Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall currentdict end definefont pop /MFshow {{dup dup 0 get findfont exch 1 get scalefont setfont [ currentpoint ] exch dup 2 get 0 exch rmoveto dup dup 5 get exch 4 get {show} {stringwidth pop 0 rmoveto}ifelse dup 3 get {2 get neg 0 exch rmoveto pop} {pop aload pop moveto}ifelse} forall} bind def /MFwidth {0 exch {dup 3 get{dup dup 0 get findfont exch 1 get scalefont setfont 5 get stringwidth pop add} {pop} ifelse} forall} bind def /MLshow { currentpoint stroke M 0 exch R MFshow } bind def /MRshow { currentpoint stroke M exch dup MFwidth neg 3 -1 roll R MFshow } def 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21 V 86 31 V 85 41 V 86 48 V 85 56 V 86 59 V 85 62 V 86 63 V 85 61 V 86 58 V 85 52 V 86 45 V 85 36 V 86 26 V 85 15 V 86 3 V 85 -10 V 1788 3907 M 86 -23 V 85 -35 V 86 -48 V 85 -58 V 85 -69 V 86 -78 V 85 -85 V 86 -91 V 85 -94 V 86 -95 V 85 -95 V 86 -93 V 85 -88 V 86 -81 V 85 -74 V 86 -64 V 85 -53 V 86 -41 V 85 -30 V 86 -16 V 85 -4 V 86 9 V 85 20 V 86 32 V 85 40 V 86 49 V 85 55 V 86 60 V 85 62 V 86 63 V 85 61 V 86 58 V 85 52 V 86 45 V 85 36 V 86 26 V 85 15 V 86 3 V 85 -11 V 1836 3949 M 86 -22 V 85 -36 V 86 -47 V 85 -59 V 86 -69 V 85 -78 V 86 -85 V 85 -90 V 86 -94 V 85 -96 V 86 -95 V 85 -92 V 86 -88 V 85 -82 V 86 -73 V 85 -64 V 86 -53 V 85 -42 V 86 -29 V 85 -16 V 85 -4 V 86 9 V 85 20 V 86 31 V 85 41 V 86 49 V 85 55 V 86 60 V 85 62 V 86 63 V 85 61 V 86 58 V 85 52 V 86 45 V 85 36 V 86 26 V 85 14 V 86 3 V 85 -10 V 1884 3986 M 86 -23 V 85 -35 V 86 -47 V 85 -59 V 86 -69 V 85 -78 V 86 -85 V 85 -90 V 86 -94 V 85 -96 V 86 -95 V 85 -92 V 86 -88 V 85 -82 V 86 -74 V 85 -64 V 86 -53 V 85 -41 V 86 -29 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86 -82 V 85 -73 V 86 -64 V 85 -54 V 86 -41 V 85 -29 V 85 -17 V 86 -3 V 85 8 V 86 21 V 85 31 V 86 41 V 85 48 V 86 56 V 85 59 V 86 63 V 85 62 V 86 62 V 85 57 V 86 52 V 85 45 V 86 37 V 85 26 V 86 14 V 85 3 V 86 -10 V 2077 4073 M 85 -23 V 86 -35 V 85 -47 V 86 -59 V 85 -69 V 86 -78 V 85 -85 V 86 -90 V 85 -94 V 86 -96 V 85 -95 V 86 -92 V 85 -88 V 86 -82 V 85 -74 V 86 -64 V 85 -53 V 86 -41 V 85 -29 V 86 -17 V 85 -3 V 86 8 V 85 21 V 86 31 V 85 41 V 86 48 V 85 56 V 86 59 V 85 63 V 86 62 V 85 61 V 86 58 V 85 52 V 86 45 V 85 36 V 85 26 V 86 15 V 85 3 V 86 -10 V 2125 4081 M 85 -23 V 86 -35 V 85 -48 V 86 -59 V 85 -69 V 86 -77 V 85 -85 V 86 -91 V 85 -94 V 86 -95 V 85 -95 V 86 -93 V 85 -88 V 86 -82 V 85 -73 V 86 -64 V 85 -53 V 86 -42 V 85 -29 V 86 -16 V 85 -4 V 86 9 V 85 20 V 86 31 V 85 41 V 86 49 V 85 55 V 86 60 V 85 62 V 86 63 V 85 61 V 86 58 V 85 52 V 86 45 V 85 36 V 86 26 V 85 15 V 86 2 V 85 -10 V 2173 4083 M 86 -23 V 85 -35 V 86 -47 V 85 -59 V 85 -69 V 86 -78 V 85 -85 V 86 -90 V 85 -94 V 86 -96 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V 86 -35 V 85 -48 V 86 -58 V 85 -69 V 86 -78 V 85 -85 V 86 -91 V 85 -94 V 86 -95 V 85 -95 V 86 -93 V 85 -88 V 86 -81 V 85 -74 V 86 -64 V 85 -53 V 86 -42 V 85 -29 V 86 -16 V 85 -4 V 86 9 V 85 20 V 86 32 V 85 40 V 86 49 V 85 55 V 86 60 V 85 62 V 86 63 V 85 61 V 86 58 V 85 52 V 86 45 V 85 36 V 85 26 V 86 15 V 85 2 V 86 -10 V 2510 4027 M 85 -23 V 86 -35 V 85 -48 V 86 -58 V 85 -69 V 86 -78 V 85 -85 V 86 -91 V 85 -94 V 86 -95 V 85 -95 V 86 -93 V 85 -88 V 86 -81 V 85 -74 V 86 -64 V 85 -53 V 86 -41 V 85 -30 V 86 -16 V 85 -4 V 86 9 V 85 20 V 86 32 V 85 40 V 86 49 V 85 55 V 86 60 V 85 62 V 86 63 V 85 61 V 86 58 V 85 52 V 86 45 V 85 36 V 86 26 V 85 15 V 86 3 V 85 -11 V 2558 4019 M 86 -23 V 85 -35 V 86 -48 V 85 -58 V 85 -69 V 86 -78 V 85 -85 V 86 -91 V 85 -94 V 86 -95 V 85 -95 V 86 -93 V 85 -88 V 86 -81 V 85 -74 V 86 -64 V 85 -53 V 86 -41 V 85 -30 V 86 -16 V 85 -4 V 86 9 V 85 20 V 86 32 V 85 40 V 86 49 V 85 55 V 86 60 V 85 62 V 86 63 V 85 61 V 86 58 V 85 52 V 86 45 V 85 36 V 86 26 V 85 15 V 86 3 V 85 -11 V 2606 4014 M 86 -23 V 85 -35 V 86 -48 V 85 -59 V 86 -69 V 85 -77 V 86 -85 V 85 -91 V 86 -94 V 85 -95 V 86 -95 V 85 -93 V 86 -88 V 85 -81 V 86 -74 V 85 -64 V 86 -53 V 85 -42 V 86 -29 V 85 -16 V 85 -4 V 86 9 V 85 20 V 86 31 V 85 41 V 86 49 V 85 55 V 86 60 V 85 62 V 86 63 V 85 61 V 86 58 V 85 52 V 86 45 V 85 36 V 86 26 V 85 15 V 86 2 V 85 -10 V 2654 4012 M 86 -23 V 85 -35 V 86 -48 V 85 -58 V 86 -69 V 85 -78 V 86 -85 V 85 -90 V 86 -94 V 85 -96 V 86 -95 V 85 -93 V 86 -88 V 85 -81 V 86 -74 V 85 -64 V 86 -53 V 85 -41 V 86 -29 V 85 -17 V 86 -4 V 85 9 V 86 21 V 85 31 V 86 41 V 85 48 V 86 55 V 85 60 V 86 62 V 85 63 V 86 61 V 85 58 V 86 52 V 85 45 V 86 36 V 85 26 V 85 15 V 86 3 V 85 -10 V 2702 4015 M 86 -23 V 85 -35 V 86 -48 V 85 -59 V 86 -69 V 85 -77 V 86 -85 V 85 -91 V 86 -94 V 85 -95 V 86 -95 V 85 -93 V 86 -88 V 85 -82 V 86 -73 V 85 -64 V 86 -53 V 85 -42 V 86 -29 V 85 -16 V 86 -4 V 85 9 V 86 20 V 85 31 V 86 41 V 85 49 V 86 55 V 85 60 V 86 62 V 85 63 V 86 61 V 85 58 V 86 52 V 85 45 V 86 36 V 85 26 V 86 15 V 85 2 V 86 -10 V 2751 4022 M 85 -23 V 86 -35 V 85 -47 V 85 -59 V 86 -69 V 85 -78 V 86 -85 V 85 -90 V 86 -94 V 85 -96 V 86 -95 V 85 -92 V 86 -88 V 85 -82 V 86 -74 V 85 -64 V 86 -53 V 85 -41 V 86 -29 V 85 -17 V 86 -3 V 85 8 V 86 21 V 85 31 V 86 41 V 85 48 V 86 56 V 85 59 V 86 63 V 85 62 V 86 62 V 85 57 V 86 52 V 85 45 V 86 37 V 85 25 V 86 15 V 85 3 V 86 -10 V 2799 4035 M 85 -23 V 86 -35 V 85 -47 V 86 -59 V 85 -69 V 86 -78 V 85 -85 V 86 -90 V 85 -94 V 86 -96 V 85 -95 V 86 -93 V 85 -88 V 86 -81 V 85 -74 V 86 -64 V 85 -53 V 86 -41 V 85 -29 V 85 -17 V 86 -4 V 85 9 V 86 21 V 85 31 V 86 41 V 85 48 V 86 56 V 85 59 V 86 62 V 85 63 V 86 61 V 85 58 V 86 52 V 85 45 V 86 36 V 85 26 V 86 15 V 85 3 V 86 -10 V 2847 4054 M 85 -23 V 86 -35 V 85 -48 V 86 -59 V 85 -68 V 86 -78 V 85 -85 V 86 -91 V 85 -94 V 86 -95 V 85 -95 V 86 -93 V 85 -88 V 86 -81 V 85 -74 V 86 -64 V 85 -53 V 86 -42 V 85 -29 V 86 -16 V 85 -4 V 86 9 V 85 20 V 86 31 V 85 41 V 86 49 V 85 55 V 86 60 V 85 62 V 86 63 V 85 61 V 86 58 V 85 52 V 86 45 V 85 36 V 85 26 V 86 15 V 85 2 V 86 -10 V currentpoint stroke M 2895 4078 M 85 -22 V 86 -36 V 85 -47 V 86 -59 V 85 -69 V 86 -78 V 85 -85 V 86 -90 V 85 -94 V 86 -96 V 85 -95 V 86 -92 V 85 -88 V 86 -82 V 85 -73 V 86 -64 V 85 -54 V 86 -41 V 85 -29 V 86 -16 V 85 -4 V 86 9 V 85 20 V 86 31 V 85 41 V 86 49 V 85 55 V 86 60 V 85 62 V 86 63 V 85 61 V 86 57 V 85 53 V 86 45 V 85 36 V 86 26 V 85 14 V 86 3 V 85 -10 V 970 2971 M 49 31 V 50 38 V 49 44 V 49 50 V 50 54 V 49 60 V 49 62 V 50 66 V 49 67 V 49 68 V 50 68 V 49 66 V 50 64 V 49 61 V 49 57 V 50 52 V 49 47 V 49 41 V 50 35 V 49 28 V 49 22 V 50 16 V 49 10 V 50 5 V 49 0 V 49 -5 V 50 -7 V 49 -10 V 49 -11 V 50 -11 V 49 -10 V 49 -9 V 50 -6 V 49 -2 V 49 2 V 50 7 V 49 13 V 50 19 V 49 25 V 1053 2949 M 50 31 V 49 38 V 49 44 V 50 50 V 49 54 V 49 59 V 50 63 V 49 66 V 49 67 V 50 68 V 49 68 V 50 66 V 49 64 V 49 61 V 50 57 V 49 52 V 49 47 V 50 41 V 49 35 V 49 28 V 50 22 V 49 16 V 50 10 V 49 5 V 49 0 V 50 -5 V 49 -7 V 49 -10 V 50 -11 V 49 -11 V 49 -10 V 50 -9 V 49 -6 V 49 -2 V 50 2 V 49 7 V 50 13 V 49 19 V 49 25 V 1137 2915 M 49 31 V 49 38 V 50 44 V 49 50 V 49 54 V 50 59 V 49 63 V 49 66 V 50 67 V 49 68 V 50 68 V 49 66 V 49 64 V 50 61 V 49 57 V 49 52 V 50 47 V 49 41 V 49 35 V 50 28 V 49 22 V 50 16 V 49 10 V 49 5 V 50 0 V 49 -5 V 49 -7 V 50 -10 V 49 -11 V 49 -11 V 50 -10 V 49 -9 V 49 -6 V 50 -2 V 49 2 V 50 7 V 49 13 V 49 19 V 50 25 V 1220 2869 M 49 32 V 50 38 V 49 44 V 49 49 V 50 55 V 49 59 V 49 63 V 50 65 V 49 67 V 50 68 V 49 68 V 49 66 V 50 65 V 49 61 V 49 57 V 50 52 V 49 47 V 49 41 V 50 34 V 49 29 V 50 22 V 49 16 V 49 10 V 50 4 V 49 0 V 49 -4 V 50 -8 V 49 -9 V 49 -11 V 50 -11 V 49 -11 V 49 -8 V 50 -6 V 49 -3 V 50 3 V 49 7 V 49 13 V 50 19 V 49 25 V 1303 2813 M 50 31 V 49 38 V 49 44 V 50 50 V 49 54 V 49 60 V 50 62 V 49 66 V 50 67 V 49 68 V 49 68 V 50 66 V 49 64 V 49 61 V 50 57 V 49 52 V 49 47 V 50 41 V 49 35 V 50 28 V 49 22 V 49 16 V 50 10 V 49 5 V 49 0 V 50 -5 V 49 -7 V 49 -10 V 50 -10 V 49 -12 V 49 -10 V 50 -9 V 49 -6 V 50 -2 V 49 2 V 49 7 V 50 13 V 49 19 V 49 25 V 1387 2747 M 49 31 V 49 38 V 50 44 V 49 50 V 49 54 V 50 59 V 49 63 V 50 66 V 49 67 V 49 68 V 50 67 V 49 67 V 49 64 V 50 61 V 49 57 V 49 52 V 50 47 V 49 41 V 49 35 V 50 28 V 49 22 V 50 16 V 49 10 V 49 5 V 50 0 V 49 -5 V 49 -7 V 50 -10 V 49 -11 V 49 -11 V 50 -10 V 49 -9 V 50 -6 V 49 -2 V 49 2 V 50 7 V 49 13 V 49 19 V 50 25 V 1470 2672 M 49 32 V 50 37 V 49 44 V 49 50 V 50 55 V 49 59 V 50 62 V 49 66 V 49 67 V 50 68 V 49 68 V 49 66 V 50 64 V 49 61 V 49 57 V 50 53 V 49 46 V 49 41 V 50 35 V 49 28 V 50 22 V 49 16 V 49 10 V 50 5 V 49 0 V 49 -4 V 50 -8 V 49 -9 V 49 -11 V 50 -12 V 49 -10 V 50 -9 V 49 -6 V 49 -2 V 50 2 V 49 8 V 49 12 V 50 19 V 49 26 V 1553 2590 M 50 32 V 49 38 V 49 43 V 50 50 V 49 55 V 50 59 V 49 63 V 49 65 V 50 67 V 49 68 V 49 68 V 50 66 V 49 65 V 49 61 V 50 56 V 49 53 V 49 46 V 50 41 V 49 35 V 50 29 V 49 22 V 49 15 V 50 10 V 49 5 V 49 0 V 50 -4 V 49 -8 V 49 -9 V 50 -11 V 49 -11 V 50 -11 V 49 -8 V 49 -6 V 50 -3 V 49 2 V 49 8 V 50 13 V 49 19 V 49 25 V 1637 2503 M 49 31 V 49 38 V 50 44 V 49 50 V 50 54 V 49 59 V 49 63 V 50 66 V 49 67 V 49 68 V 50 68 V 49 66 V 49 64 V 50 61 V 49 57 V 49 52 V 50 47 V 49 41 V 50 35 V 49 28 V 49 22 V 50 16 V 49 10 V 49 5 V 50 0 V 49 -5 V 49 -7 V 50 -10 V 49 -11 V 50 -11 V 49 -10 V 49 -9 V 50 -6 V 49 -2 V 49 2 V 50 7 V 49 13 V 49 19 V 50 25 V currentpoint stroke M 1720 2412 M 49 31 V 50 38 V 49 44 V 50 50 V 49 54 V 49 59 V 50 63 V 49 65 V 49 68 V 50 68 V 49 67 V 49 67 V 50 64 V 49 61 V 49 57 V 50 52 V 49 47 V 50 41 V 49 35 V 49 28 V 50 22 V 49 16 V 49 10 V 50 5 V 49 -1 V 49 -4 V 50 -7 V 49 -10 V 50 -11 V 49 -11 V 49 -10 V 50 -9 V 49 -6 V 49 -2 V 50 2 V 49 7 V 49 13 V 50 19 V 49 25 V 1803 2319 M 50 31 V 49 38 V 50 44 V 49 50 V 49 54 V 50 59 V 49 63 V 49 65 V 50 68 V 49 68 V 49 67 V 50 67 V 49 64 V 49 61 V 50 57 V 49 52 V 50 47 V 49 41 V 49 35 V 50 28 V 49 22 V 49 16 V 50 10 V 49 5 V 49 -1 V 50 -4 V 49 -7 V 50 -10 V 49 -11 V 49 -11 V 50 -10 V 49 -9 V 49 -6 V 50 -2 V 49 2 V 49 7 V 50 13 V 49 19 V 49 25 V 1887 2226 M 49 31 V 50 38 V 49 44 V 49 50 V 50 54 V 49 59 V 49 63 V 50 65 V 49 68 V 49 68 V 50 67 V 49 67 V 49 64 V 50 61 V 49 57 V 50 52 V 49 47 V 49 41 V 50 35 V 49 28 V 49 22 V 50 16 V 49 10 V 49 5 V 50 -1 V 49 -4 V 49 -7 V 50 -10 V 49 -11 V 50 -11 V 49 -10 V 49 -9 V 50 -6 V 49 -2 V 49 2 V 50 7 V 49 13 V 49 19 V 50 25 V 1970 2135 M 50 31 V 49 38 V 49 44 V 50 49 V 49 55 V 49 59 V 50 63 V 49 65 V 49 68 V 50 68 V 49 67 V 49 67 V 50 64 V 49 61 V 50 57 V 49 52 V 49 47 V 50 41 V 49 35 V 49 28 V 50 22 V 49 16 V 49 10 V 50 5 V 49 -1 V 49 -4 V 50 -7 V 49 -10 V 50 -11 V 49 -11 V 49 -10 V 50 -9 V 49 -6 V 49 -2 V 50 2 V 49 7 V 49 13 V 50 19 V 49 25 V 2054 2047 M 49 32 V 49 38 V 50 44 V 49 49 V 49 55 V 50 59 V 49 63 V 49 65 V 50 67 V 49 68 V 49 68 V 50 66 V 49 65 V 50 61 V 49 57 V 49 52 V 50 47 V 49 41 V 49 34 V 50 29 V 49 22 V 49 16 V 50 10 V 49 4 V 49 0 V 50 -4 V 49 -8 V 50 -9 V 49 -11 V 49 -11 V 50 -11 V 49 -8 V 49 -6 V 50 -3 V 49 3 V 49 7 V 50 13 V 49 19 V 50 25 V 2137 1965 M 49 32 V 50 38 V 49 44 V 49 49 V 50 55 V 49 59 V 49 63 V 50 65 V 49 67 V 49 68 V 50 68 V 49 67 V 50 64 V 49 61 V 49 57 V 50 52 V 49 47 V 49 41 V 50 34 V 49 29 V 49 22 V 50 16 V 49 10 V 49 4 V 50 0 V 49 -4 V 50 -7 V 49 -10 V 49 -11 V 50 -11 V 49 -11 V 49 -8 V 50 -6 V 49 -2 V 49 2 V 50 7 V 49 13 V 50 19 V 49 25 V 2220 1891 M 50 31 V 49 38 V 49 44 V 50 50 V 49 54 V 49 59 V 50 63 V 49 65 V 49 68 V 50 68 V 49 67 V 50 67 V 49 64 V 49 61 V 50 57 V 49 52 V 49 47 V 50 41 V 49 35 V 49 28 V 50 22 V 49 16 V 49 10 V 50 5 V 49 -1 V 50 -4 V 49 -7 V 49 -10 V 50 -11 V 49 -11 V 49 -10 V 50 -9 V 49 -6 V 49 -2 V 50 2 V 49 7 V 50 13 V 49 19 V 49 25 V 2304 1824 M 49 32 V 49 38 V 50 44 V 49 49 V 49 55 V 50 59 V 49 63 V 49 65 V 50 68 V 49 67 V 50 68 V 49 67 V 49 64 V 50 61 V 49 57 V 49 52 V 50 47 V 49 41 V 49 35 V 50 28 V 49 22 V 49 16 V 50 10 V 49 4 V 50 0 V 49 -4 V 49 -7 V 50 -10 V 49 -11 V 49 -11 V 50 -10 V 49 -9 V 49 -6 V 50 -2 V 49 2 V 50 7 V 49 13 V 49 19 V 50 25 V 2387 1768 M 49 32 V 50 38 V 49 43 V 49 50 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Fm(;)15 b(a)631 4654 y Fq(1)698 4640 y Fo(ar)-5 b(e)29 b(such)e(that)i(they)f(ar)-5 b(e)28 b(not)h(in)e(the)h(c)-5 b(o)g(dimension)30 b Fs(1)d Fo(surfac)-5 b(e)28 b(of)g(e)-5 b(qua-)456 4748 y(tion)36 b Fm(a)695 4762 y Fq(0)735 4748 y Fm(a)783 4762 y Fq(1)822 4748 y Fs(\()p Fm(a)905 4715 y Fq(2)905 4773 y(0)967 4748 y Fn(\000)23 b Fm(a)1109 4715 y Fq(2)1109 4773 y(1)1148 4748 y Fs(\))32 b(=)f(0)p Fo(,)37 b(and)f Fn(j)q Fm(")p Fn(j)c(\024)e Fm(")1874 4715 y Fl(\003)1914 4748 y Fs(\()p Fm(a)1997 4762 y Fq(0)2037 4748 y Fm(;)15 b(a)2125 4762 y Fq(1)2165 4748 y Fs(\))p Fo(,)37 b(admits)g(orbits)g(fol)5 b(lowing)456 4856 y(the)40 b(me)-5 b(chanism)42 b(describ)-5 b(e)g(d)41 b(in)f(this)h(p)-5 b(ap)g(er)42 b(and)f(such)f(that)i Fm(I)7 b Fs(\(0\))40 b Fn(\024)e(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(,)456 4964 y Fm(I)7 b Fs(\()p Fm(T)13 b Fs(\))25 b Fn(\025)g Fs(3)p Fm(=)p Fs(2)34 b Fo(for)f(some)h Fm(T)k(>)25 b Fs(0)p Fo(.)p eop end %%Page: 125 125 TeXDict begin 125 124 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)718 b(125)555 450 y Fs(W)-8 b(e)37 b(note)f(that)g(Prop)s(osition)g(97)g(is)g(extremely)h(conserv) -5 b(ativ)m(e.)58 b(W)-8 b(e)37 b(ha)m(v)m(e)456 558 y(only)25 b(used)g(the)h(critical)h(p)s(oin)m(ts)f(close)g(to)h(the)f (extrema)g(of)g Fn(L)p Fs(.)38 b(It)26 b(is)g(clear)g(that)456 666 y(there)k(are)h(man)m(y)f(other)h(critical)h(p)s(oin)m(ts.)555 774 y(The)c(example)i(\(169\))g(is)f(somewhat)g(non-generic)g(b)s (ecause)g(it)g(has)f(a)h(sym-)456 882 y(metry)38 b(that)h(causes)g (that)g(the)f(t)m(w)m(o)i(homo)s(clinic)f(orbits)f(of)h(the)f(p)s (endulum)456 990 y(giv)m(e)g(the)f(same)h(Melnik)m(o)m(v)h(function.)61 b(Nev)m(ertheless,)40 b(in)d(spite)g(of)h(the)f(fact)456 1098 y(that)f(the)g(system)f(is)h(v)m(ery)g(non-generic,)i(w)m(e)e(can) g(v)m(erify)g(easily)h(the)f(condi-)456 1206 y(tions)30 b(of)h(our)f(theorem.)1308 1395 y(14.)46 b Ft(A)m(ckno)n(wledgments)555 1557 y Fs(W)-8 b(e)38 b(thank)f(J.)g(V)-8 b(aaler)38 b(for)f(sev)m(eral)h(discussions)e(and)g(for)h(help)f(with)h(the)456 1665 y(pro)s(of)29 b(of)i(Lemma)f(43.)555 1773 y(This)44 b(w)m(ork)h(has)g(b)s(een)f(supp)s(orted)f(b)m(y)i(the)g Fo(Comisi\023)-46 b(on)47 b(Conjunta)g(His-)456 1881 y(p)-5 b(ano)37 b(Norte)-5 b(americ)g(ana)39 b(de)e(Co)-5 b(op)g(er)g(aci\023)-46 b(on)39 b(Cient)-9 b(\023)-37 b(\020\014c)-5 b(a)37 b(y)f(T)-7 b(e)i(cnol\023)-46 b(ogic)-5 b(a)p Fs(.)53 b(The)456 1989 y(\014nal)47 b(v)m(ersion)h(w)m(as)h (prepared)d(while)i(R.L.)g(w)m(as)h(enjo)m(ying)f(a)h Fo(C\023)-46 b(ate)-5 b(dr)g(a)51 b(de)456 2097 y(la)42 b(F)-7 b(undaci\023)-46 b(on)44 b(FBBV)p Fs(,)d(and)f(A.D.)i(w)m(as)f (visiting)h(the)f Fo(Centr)-5 b(e)42 b(de)h(R)-5 b(e)g(c)g(er)g(c)g(a) 456 2205 y(Matem\022)-46 b(atic)-5 b(a)p Fs(,)31 b(for)f(whose)g (hospitalit)m(y)i(he)e(is)g(v)m(ery)g(grateful.)42 b(A.D.)31 b(and)e(T.S.)456 2313 y(ha)m(v)m(e)24 b(also)g(b)s(een)e(partially)i (supp)s(orted)e(b)m(y)h(the)g(Catalan)h(gran)m(t)g(2001SGR-70,)456 2421 y(the)30 b(Spanish)f(gran)m(t)i(BFM2000-0805-C02)36 b(and)30 b(the)h(INT)-8 b(AS)30 b(gran)m(t)h(00-221,)456 2529 y(and)e(R.L.)i(b)m(y)f(NSF)h(gran)m(ts.)1550 2719 y Ft(References)456 2864 y Fv([AA67])92 b(V.I.)24 b(Arnold)g(and)g(A.)g (Av)n(ez.)g Fr(Er)l(go)l(dic)k(pr)l(oblems)f(of)f(classic)l(al)g(me)l (chanics)p Fv(.)g(Ben-)782 2955 y(jamin,)h(New)f(Y)-6 b(ork,)25 b(1967.)456 3046 y([AKN88])38 b(V.I.)29 b(Arnold,)h(V.V.)f (Kozlo)n(v,)i(and)d(A.I.)h(Neish)n(tadt.)g Fr(Dynamic)l(al)i(Systems)h (III)p Fv(,)782 3138 y(v)n(olume)26 b(3)g(of)g Fr(Encyclop)l(ae)l(dia)j (Math.)f(Sci.)d Fv(Springer,)h(Berlin,)h(1988.)456 3229 y([Arn63a])39 b(V.)31 b(I.)g(Arnol'd.)h(Pro)r(of)g(of)g(a)g(theorem)f (of)h(A.)f(N.)g(Kolmogoro)n(v)i(on)e(the)f(in)n(v)l(ari-)782 3320 y(ance)18 b(of)h(quasi-p)r(erio)r(dic)f(motions)h(under)e(small)i (p)r(erturbations.)f Fr(R)n(ussian)j(Math.)782 3412 y(Surveys)p Fv(,)28 b(18\(5\):9{36,)h(1963.)456 3503 y([Arn63b])38 b(V.)20 b(I.)g(Arnol)1144 3471 y Fg(0)1167 3503 y Fv(d.)g(Small)g (denominators)h(and)f(problems)g(of)h(stabilit)n(y)g(of)f(motion)h(in) 782 3594 y(classical)j(and)d(celestial)j(mec)n(hanics.)e Fr(Usp)l(ehi)j(Mat.)e(Nauk)p Fv(,)h(18\(6)e(\(114\)\):91{192,)782 3686 y(1963.)456 3777 y([Arn64])77 b(V.I.)36 b(Arnold.)f(Instabilit)n (y)g(of)i(dynamical)f(systems)g(with)f(sev)n(eral)i(degrees)f(of)782 3868 y(freedom.)27 b Fr(Sov.)h(Math.)f(Doklady)p Fv(,)g(5:581{585,)i (1964.)456 3960 y([BB02])100 b(M.)29 b(Berti)g(and)f(P)-6 b(.)28 b(Bolle.)i(A)e(functional)h(analysis)g(approac)n(h)g(to)f (Arnold)g(di\013u-)782 4051 y(sion.)f Fr(A)n(nn.)g(Inst.)h(H.)f(Poinc)l (ar)n(\023)-37 b(e)28 b(A)n(nal.)f(Non)h(Lin)n(\023)-37 b(eair)l(e)p Fv(,)26 b(19\(4\):395{450,)k(2002.)456 4142 y([BCV01])41 b(Ugo)d(Bessi,)43 b(Luigi)38 b(Chierc)n(hia,)k(and)c (Enrico)g(V)-6 b(aldino)r(ci.)39 b(Upp)r(er)e(b)r(ounds)g(on)782 4234 y(Arnold)f(di\013usion)h(times)g(via)g(Mather)g(theory)-6 b(.)36 b Fr(J.)h(Math.)h(Pur)l(es)h(Appl.)e(\(9\))p Fv(,)782 4325 y(80\(1\):105{129,)30 b(2001.)456 4416 y([Bes96])90 b(Ugo)31 b(Bessi.)h(An)d(approac)n(h)i(to)f(Arnol)1923 4385 y Fg(0)1946 4416 y Fv(d's)g(di\013usion)h(through)f(the)g (calculus)h(of)782 4508 y(v)l(ariations.)c Fr(Nonline)l(ar)h(A)n(nal.)p Fv(,)d(26\(6\):1115{1135,)31 b(1996.)456 4599 y([BF02])104 b(I.)57 b(Baldom\022)-38 b(a)58 b(and)e(E.)h(F)-6 b(on)n(tic)n(h.)56 b(Exp)r(onen)n(tially)g(small)i(splitting)f(of)g(in-)782 4690 y(v)l(arian)n(t)63 b(manifolds)i(of)g(parab)r(olic)f(p)r(oin)n (ts.)g(Preprin)n(t)g(02-166,)74 b Fb(mp)p 2993 4690 24 4 v 29 w(arc@)782 4782 y(math.utexas.edu)p Fv(,)29 b(2002.)456 4873 y([BT99])99 b(S.)37 b(Bolotin)h(and)f(D.)g(T)-6 b(resc)n(hev.)37 b(Un)n(b)r(ounded)e(gro)n(wth)i(of)h(energy)f(in)g (nonau-)782 4964 y(tonomous)26 b(Hamiltonian)h(systems.)f Fr(Nonline)l(arity)p Fv(,)h(12\(2\):365{388,)j(1999.)p eop end %%Page: 126 126 TeXDict begin 126 125 bop 456 251 a Fq(126)615 b(A.)23 b(Delshams,)g(R.)g(de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 450 y Fv([Car81])85 b(John)30 b(R.)f(Cary)-6 b(.)30 b(Lie)g(transform)h (p)r(erturbation)e(theory)g(for)i(Hamiltonian)f(sys-)782 541 y(tems.)c Fr(Phys.)i(R)l(ep.)p Fv(,)e(79\(2\):129{159,)k(1981.)456 633 y([CFdlL03])40 b(Xa)n(vier)24 b(Cabr)n(\023)-36 b(e,)25 b(Ernest)f(F)-6 b(on)n(tic)n(h,)25 b(and)e(Rafael)i(de)f(la)h(Lla)n(v)n (e.)f(The)h(parameter-)782 724 y(ization)32 b(metho)r(d)g(for)g(in)n(v) l(arian)n(t)f(manifolds:)47 b(I,)31 b(I)r(I.)g Fr(Indiana)i(Univ.)f (Math.)h(J.)p Fv(,)782 815 y(2003.)28 b(T)-6 b(o)26 b(app)r(ear.)456 907 y([CG94])93 b(L.)23 b(Chierc)n(hia)i(and)d(G.)i(Galla)n(v)n(otti.)h (Drift)e(and)g(di\013usion)g(in)g(phase)g(space.)h Fr(A)n(nn.)782 998 y(Inst.)k(H.)f(Poinc)l(ar)n(\023)-37 b(e)28 b(Phys.)g(Th)n(\023)-37 b(eor.)p Fv(,)26 b(60\(1\):144,)j(1994.)456 1089 y([CG98])93 b(L.)35 b(Chierc)n(hia)h(and)f(G.)g(Galla)n(v)n(otti.)h(Erratum)f (drift)h(and)e(di\013usion)h(in)g(phase)782 1181 y(space.)27 b Fr(A)n(nn.)g(Inst.)g(H.)g(Poinc)l(ar)n(\023)-37 b(e)29 b(Phys.)f(Th)n(\023)-37 b(eor.)p Fv(,)26 b(68:135,)i(1998.)456 1272 y([CG03])93 b(Jac)n(ky)29 b(Cresson)h(and)f(Christophe)g(Guillet.) h(P)n(erio)r(dic)g(orbits)g(and)e(Arnold)h(dif-)782 1363 y(fusion.)e Fr(Discr)l(ete)i(Contin.)e(Dyn.)g(Syst.)p Fv(,)g(9\(2\):451{470,)j(2003.)456 1455 y([Chi79])89 b(B.V.)38 b(Chirik)n(o)n(v.)h(A)e(univ)n(ersal)h(instabilit)n(y)g(of)g (man)n(y-dimensional)g(oscillator)782 1546 y(systems.)26 b Fr(Phys.)i(R)l(ep.)p Fv(,)e(52\(5\):264{379,)k(1979.)456 1637 y([CLSV85])39 b(B.)29 b(V.)f(Chirik)n(o)n(v,)h(M.)g(A.)f(Lieb)r (erman,)h(D.)g(L.)f(Shep)r(ely)n(ansky)-6 b(,)27 b(and)h(F.)h(M.)f(Vi-) 782 1729 y(v)l(aldi.)23 b(A)f(theory)g(of)h(mo)r(dulational)h (di\013usion.)f Fr(Phys.)i(D)p Fv(,)e(14\(3\):289{304,)k(1985.)456 1820 y([CP02])101 b(Gonzalo)22 b(Con)n(treras)f(and)e(Gabriel)j(P)-6 b(.)20 b(P)n(aternain.)h(Connecting)f(orbits)h(b)r(et)n(w)n(een)782 1911 y(static)30 b(classes)i(for)e(generic)g(Lagrangian)i(systems.)e Fr(T)-6 b(op)l(olo)l(gy)p Fv(,)32 b(41\(4\):645{666,)782 2003 y(2002.)456 2094 y([CSUZ89])39 b(A.)30 b(A.)f(Chernik)n(o)n(v,)h (R.)g(Z.)f(Sagdeev,)i(D.)f(A.)f(Usik)n(o)n(v,)h(and)g(G.)g(M.)g(Zasla)n (vsky)-6 b(.)782 2185 y Fr(We)l(ak)29 b(chaos)h(and)g(structur)l(es)p Fv(,)h(v)n(olume)c(8)g(of)h Fr(Soviet)i(Scienti\014c)f(R)l(eviews,)h (Se)l(c-)782 2277 y(tion)g(C:)e(Mathematic)l(al)j(Physics)f(R)l(eviews) p Fv(.)e(Harw)n(o)r(o)r(d)h(Academic)f(Publishers,)782 2368 y(Ch)n(ur,)e(1989.)456 2459 y([CV89])95 b(B.)31 b(V.)f(Chirik)n(o)n(v)g(and)g(V.)f(V.)h(V)-6 b(ec)n(hesla)n(v)n(o)n(v.) 30 b(Ho)n(w)g(fast)h(is)g(the)f(Arnold's)g(di\013u-)782 2551 y(sion??)35 b(T)-6 b(ec)n(hnical)27 b(Rep)r(ort)e(98-72,)i(Inst.)f (Plasma)h(Ph)n(ys.,)f(No)n(v)n(osibirsk,)g(1989.)456 2642 y([DG96])89 b(A.)30 b(Delshams)g(and)f(P)-6 b(.)30 b(Guti)n(\023)-36 b(errez.)31 b(E\013ectiv)n(e)e(stabilit)n(y)h(and)g (KAM)f(theory)-6 b(.)29 b Fr(J.)782 2733 y(Di\013er)l(ential)f (Equations)p Fv(,)f(128\(2\):415{490,)k(1996.)456 2825 y([DG00])89 b(A.)37 b(Delshams)g(and)f(P)-6 b(.)37 b(Guti)n(\023)-36 b(errez.)37 b(Splitting)g(p)r(oten)n(tial)g(and)f(the)g(Poincar)n(\023) -36 b(e-)782 2916 y(Melnik)n(o)n(v)24 b(metho)r(d)f(for)i(whisk)n(ered) f(tori)g(in)g(Hamiltonian)g(systems.)g Fr(J.)i(Nonlin-)782 3007 y(e)l(ar)j(Sci.)p Fv(,)c(10\(4\):433{476,)30 b(2000.)456 3099 y([DG01])89 b(A.)35 b(Delshams)g(and)g(P)-6 b(.)35 b(Guti)n(\023)-36 b(errez.)36 b(Homo)r(clinic)g(orbits)f(to)g(in)n(v)l (arian)n(t)g(tori)g(in)782 3190 y(Hamiltonian)30 b(systems.)g(In)f (Christopher)h(K.)f(R.)g(T.)h(Jones)g(and)f(Alexander)g(I.)782 3281 y(Khibnik,)35 b(editors,)i Fr(Multiple-time-sc)l(ale)g(dynamic)l (al)e(systems)i(\(Minne)l(ap)l(olis,)782 3372 y(MN,)27 b(1997\))p Fv(,)g(pages)g(1{27.)g(Springer,)f(New)g(Y)-6 b(ork,)25 b(2001.)456 3464 y([DLC83])46 b(Rapha)n(\177)-36 b(el)20 b(Douady)g(and)g(P)n(atrice)h(Le)f(Calv)n(ez.)i(Exemple)e(de)g (p)r(oin)n(t)g(\014xe)g(elliptique)782 3555 y(non)i(top)r(ologiquemen)n (t)i(stable)f(en)f(dimension)h(4.)g Fr(C.)h(R.)h(A)l(c)l(ad.)g(Sci.)f (Paris)h(S)n(\023)-37 b(er.)782 3646 y(I)27 b(Math.)p Fv(,)f(296\(21\):895{898,)31 b(1983.)456 3738 y([DLS00])58 b(A.)34 b(Delshams,)i(R.)e(de)f(la)h(Lla)n(v)n(e,)j(and)c(T.M.)i (Seara.)g(A)e(geometric)i(approac)n(h)782 3829 y(to)h(the)f(existence)g (of)h(orbits)g(with)g(un)n(b)r(ounded)d(energy)j(in)f(generic)h(p)r (erio)r(dic)782 3920 y(p)r(erturbations)h(b)n(y)g(a)g(p)r(oten)n(tial)h (of)g(generic)g(geo)r(desic)g(\015o)n(ws)g(of)g Fa(T)2838 3889 y Fi(2)2872 3920 y Fv(.)g Fr(Comm.)782 4012 y(Math.)28 b(Phys.)p Fv(,)e(209\(2\):353{392,)k(2000.)456 4103 y([DLS01])58 b(A.)37 b(Delshams,)k(R.)c(de)g(la)h(Lla)n(v)n(e,)i(and)d(T.M.)i (Seara.)f(Orbits)f(of)h(un)n(b)r(ounded)782 4194 y(energy)20 b(in)g(generic)h(quasip)r(erio)r(dic)h(p)r(erturbations)e(of)h(geo)r (desic)h(\015o)n(ws)e(of)h(certain)782 4286 y(manifolds,)28 b(2001.)f(preprin)n(t.)456 4377 y([Dou88])68 b(R.)23 b(Douady)-6 b(.)22 b(Stabilit)n(\023)-36 b(e)23 b(ou)g(instabilit)n (\023)-36 b(e)24 b(des)f(p)r(oin)n(ts)g(\014xes)g(elliptiques.)h Fr(A)n(nn.)g(Sci.)793 4456 y(\023)782 4475 y(Ec)l(ole)k(Norm.)f(Sup.)h (\(4\))p Fv(,)f(21\(1\):1{46,)h(1988.)456 4566 y([EMR01])40 b(R.)33 b(W.)g(Easton,)j(J.)e(D.)g(Meiss,)i(and)d(G.)h(Rob)r(erts.)g (Drift)f(b)n(y)g(coupling)g(to)h(an)782 4657 y(an)n(ti-in)n(tegrable)26 b(limit.)h Fr(Phys.)h(D)p Fv(,)e(156\(3-4\):201{218,)k(2001.)456 4749 y([F)-6 b(en72])87 b(Neil)40 b(F)-6 b(enic)n(hel.)39 b(P)n(ersistence)h(and)f(smo)r(othness)i(of)f(in)n(v)l(arian)n(t)e (manifolds)j(for)782 4840 y(\015o)n(ws.)27 b Fr(Indiana)g(Univ.)g (Math.)h(J.)p Fv(,)d(21:193{226,)30 b(1971/1972.)p eop end %%Page: 127 127 TeXDict begin 127 126 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)718 b(127)456 450 y Fv([F)-6 b(en77])87 b(N.)22 b(F)-6 b(enic)n(hel.)22 b(Asymptotic)f(stabilit)n(y) h(with)g(rate)g(conditions.)g(II.)g Fr(Indiana)h(Univ.)782 541 y(Math.)28 b(J.)p Fv(,)d(26\(1\):81{93,)30 b(1977.)456 633 y([F)-6 b(en74])87 b(N.)34 b(F)-6 b(enic)n(hel.)34 b(Asymptotic)g(stabilit)n(y)h(with)f(rate)g(conditions.)h Fr(Indiana)g(Univ.)782 724 y(Math.)28 b(J.)p Fv(,)d(23:1109{1137,)31 b(1973/74.)456 815 y([FL92])110 b(C.)45 b(F)-6 b(alcolini)46 b(and)f(R.)f(de)g(la)h(Lla)n(v)n(e.)g(A)f(rigorous)i(partial)g (justi\014cation)f(of)782 907 y(Greene's)27 b(criterion.)g Fr(J.)g(Statist.)i(Phys.)p Fv(,)d(67\(3-4\):609{643,)k(1992.)456 998 y([FM00])88 b(E.)21 b(F)-6 b(on)n(tic)n(h)19 b(and)g(P)-6 b(.)20 b(Mart)-9 b(\023)-30 b(\020n.)22 b(Di\013eren)n(tiable)e(in)n(v) l(arian)n(t)g(manifolds)h(for)g(partially)782 1089 y(h)n(yp)r(erb)r (olic)38 b(tori)h(and)f(a)g(lam)n(b)r(da)h(lemma.)g Fr(Nonline)l(arity) p Fv(,)k(13\(5\):1561{1593,)782 1181 y(2000.)456 1272 y([FM01])88 b(Ernest)36 b(F)-6 b(on)n(tic)n(h)34 b(and)h(P)n(au)h(Mart) -9 b(\023)-30 b(\020n.)37 b(Arnold)e(di\013usion)g(in)h(p)r (erturbations)f(of)782 1363 y(analytic)24 b(in)n(tegrable)g (Hamiltonian)h(systems.)f Fr(Discr)l(ete)i(Contin.)g(Dynam.)f(Sys-)782 1455 y(tems)p Fv(,)i(7\(1\):61{84,)i(2001.)456 1546 y([FM03])88 b(Ernest)29 b(F)-6 b(on)n(tic)n(h)29 b(and)g(P)n(au)g(Mart)-9 b(\023)-30 b(\020n.)31 b(Hamiltonian)f(systems)f(with)h(orbits)f(co)n (v-)782 1637 y(ering)49 b(densely)g(submanifolds)g(of)h(small)g(co)r (dimension.)g Fr(Nonline)l(ar)f(A)n(nal.)p Fv(,)782 1729 y(52\(1\):315{327,)30 b(2003.)456 1820 y([FS90a])77 b(E.)29 b(F)-6 b(on)n(tic)n(h)28 b(and)h(C.)g(Sim\023)-38 b(o.)29 b(In)n(v)l(arian)n(t)f(manifolds)i(for)f(near)g(iden)n(tit)n(y)f (di\013eren-)782 1911 y(tiable)i(maps)f(and)f(splitting)i(of)g (separatrices.)h Fr(Er)l(go)l(dic)g(The)l(ory)h(Dynam.)e(Sys-)782 2003 y(tems)p Fv(,)d(10\(2\):319{346,)j(1990.)456 2094 y([FS90b])72 b(E.)23 b(F)-6 b(on)n(tic)n(h)23 b(and)f(C.)i(Sim\023)-38 b(o.)23 b(The)g(splitting)h(of)f(separatrices)i(for)f(analytic)f (di\013eo-)782 2185 y(morphisms.)k Fr(Er)l(go)l(dic)h(The)l(ory)h (Dynam.)e(Systems)p Fv(,)h(10\(2\):295{318,)i(1990.)456 2277 y([Gal94])89 b(G.)30 b(Galla)n(v)n(otti.)h(Twistless)g(KAM)e (tori,)i(quasi)e(\015at)g(homo)r(clinic)i(in)n(tersections,)782 2368 y(and)26 b(other)h(cancellations)h(in)e(the)g(p)r(erturbation)h (series)g(of)g(certain)g(completely)782 2459 y(in)n(tegrable)g (Hamiltonian)g(systems.)g(A)e(review.)i Fr(R)l(ev.)h(Math.)g(Phys.)p Fv(,)e(6\(3\):343{)782 2551 y(411,)h(1994.)456 2642 y([Gal99])89 b(G.)35 b(Galla)n(v)n(otti.)g(Arnold's)f(di\013usion)g(in)g(iso)r(c)n (hronous)h(systems.)f Fr(Math.)i(Phys.)782 2733 y(A)n(nal.)27 b(Ge)l(om.)p Fv(,)f(1\(4\):295{312,)k(1998/99.)456 2825 y([Gra74])80 b(Sam)n(uel)32 b(M.)h(Gra\013.)f(On)g(the)f(conserv)l (ation)h(of)h(h)n(yp)r(erb)r(olic)f(in)n(v)l(arian)n(t)g(tori)g(for)782 2916 y(Hamiltonian)27 b(systems.)f Fr(J.)h(Di\013er)l(ential)h (Equations)p Fv(,)f(15:1{69,)i(1974.)456 3007 y([Hal97])91 b(G.)26 b(Haller.)g(Univ)n(ersal)f(homo)r(clinic)h(bifurcations)g(and)e (c)n(haos)i(near)f(double)g(res-)782 3099 y(onances.)i Fr(J.)g(Statist.)i(Phys.)p Fv(,)d(86\(5-6\):1011{1051,)31 b(1997.)456 3190 y([Hal99])91 b(G.)26 b(Haller.)h Fr(Chaos)h(ne)l(ar)h (r)l(esonanc)l(e)p Fv(.)f(Springer-V)-6 b(erlag,)26 b(New)g(Y)-6 b(ork,)25 b(1999.)456 3281 y([Her79])86 b(M.-R.)32 b(Herman.)g(Sur)e (la)j(conjugaison)g(di\013)n(\023)-36 b(eren)n(tiable)32 b(des)g(di\013)n(\023)-36 b(eomorphismes)782 3379 y(du)26 b(cercle)h(\022)-38 b(a)26 b(des)g(rotations.)i Fr(Inst.)g(Hautes)2101 3360 y(\023)2090 3379 y(Etudes)i(Sci.)d(Publ.)h(Math.)p Fv(,)e(\(49\):5{)782 3470 y(233,)h(1979.)456 3562 y([Her83])86 b(M.-R.)26 b(Herman.)g Fr(Sur)i(les)f(c)l(ourb)l(es)j(invariantes)f(p)l (ar)f(les)g(di\013)n(\023)-37 b(eomorphismes)28 b(de)782 3653 y(l'anne)l(au.)34 b(Vol.)g(1)p Fv(,)i(v)n(olume)d(103)h(of)g Fr(Ast)n(\023)-37 b(erisque)p Fv(.)36 b(So)r(ci)n(\023)-36 b(et)n(\023)g(e)34 b(Math)n(\023)-36 b(ematique)33 b(de)782 3744 y(F)-6 b(rance,)26 b(P)n(aris,)h(1983.)456 3836 y([HL00])102 b(A.)26 b(Haro)g(and)f(R.)h(de)f(la)i(Lla)n(v)n(e.)f(New)g (mec)n(hanisms)g(for)h(lac)n(k)f(of)g(equipartion)g(of)782 3927 y(energy)-6 b(.)26 b Fr(Phys.)h(R)l(ev.)h(L)l(ett.)p Fv(,)f(89\(7\):1859{1862,)k(2000.)456 4018 y([HM82])80 b(P)-6 b(.J.)21 b(Holmes)f(and)g(J.E.)h(Marsden.)g(Melnik)n(o)n(v's)f (metho)r(d)g(and)f(Arnol'd)h(di\013usion)782 4110 y(for)37 b(p)r(erturbations)f(of)h(in)n(tegrable)g(Hamiltonian)g(systems.)g Fr(J.)g(Math.)g(Phys.)p Fv(,)782 4201 y(23\(4\):669{675,)30 b(1982.)456 4292 y([HPS77])55 b(M.W.)35 b(Hirsc)n(h,)i(C.C.)f(Pugh,)h (and)c(M.)i(Sh)n(ub.)f Fr(Invariant)i(manifolds)p Fv(,)g(v)n(olume)782 4383 y(583)27 b(of)f Fr(L)l(e)l(ctur)l(e)k(Notes)f(in)e(Math.)f Fv(Springer-V)-6 b(erlag,)26 b(Berlin,)h(1977.)456 4475 y([JVMU99])39 b(G.)32 b(H.)f(F.)g(Dierc)n(ksen)g(J.)g(V)-6 b(on)30 b(Milczewski)j(and)e(T.)g(Uzer.)g(The)g(arnold)h(w)n(eb)782 4566 y(in)26 b(atomic)g(ph)n(ysics.)g(In)e Fr(Hamiltonian)j(Systems)i (with)f(Thr)l(e)l(e)g(or)g(Mor)l(e)g(De)l(gr)l(e)l(es)782 4657 y(of)i(F)-6 b(r)l(e)l(e)l(dom)31 b(\(S'A)l(gar\023)-39 b(o,)31 b(1995\))p Fv(,)g(pages)e(499{503.)i(Klu)n(w)n(er)d(Acad.)h (Publ.,)g(Dor-)782 4749 y(drec)n(h)n(t,)c(1999.)456 4840 y([Las93])92 b(Jacques)53 b(Lask)l(ar.)f(F)-6 b(requency)51 b(analysis)i(for)f(m)n(ulti-dimensional)h(systems.)782 4931 y(Global)27 b(dynamics)f(and)f(di\013usion.)h Fr(Phys.)i(D)p Fv(,)e(67\(1-3\):257{281,)k(1993.)p eop end %%Page: 128 128 TeXDict begin 128 127 bop 456 251 a Fq(128)615 b(A.)23 b(Delshams,)g(R.)g(de)h(la)f(Lla)n(v)n(e,)h(T.)f(M.)g(Seara)456 450 y Fv([LHRK02])38 b(Anna)29 b(Litv)l(ak-Hinenzon)e(and)i(V)-6 b(ered)29 b(Rom-Kedar.)g(Resonan)n(t)g(tori)g(and)g(in-)782 541 y(stabilities)24 b(in)f(Hamiltonian)g(systems.)h Fr(Nonline)l(arity)p Fv(,)g(15\(4\):1149{1177,)k(2002.)456 633 y([Lla00])101 b(R.)26 b(de)f(la)h(Lla)n(v)n(e.)g(P)n(ersistence)h (of)g(h)n(yp)r(erb)r(olic)e(manifolds,)j(2000.)456 724 y([Lla01])101 b(R.)22 b(de)g(la)h(Lla)n(v)n(e.)g(A)f(tutorial)h(on)f (KAM)g(theory)-6 b(.)22 b(In)f Fr(Smo)l(oth)26 b(er)l(go)l(dic)f(the)l (ory)h(and)782 815 y(its)i(applic)l(ations)h(\(Se)l(attle,)g(W)-6 b(A,)26 b(1999\))p Fv(,)h(pages)g(175{292.)i(Amer.)d(Math.)g(So)r(c.,) 782 907 y(Pro)n(vidence,)g(RI,)f(2001.)456 998 y([Lla02])101 b(R.)23 b(Lla)n(v)n(e.)g(Orbits)f(of)i(un)n(b)r(ounded)d(energy)h(in)h (p)r(erturbations)f(of)i(geo)r(desic)g(\015o)n(ws)782 1089 y(b)n(y)h(p)r(erio)r(dic)h(p)r(oten)n(tials.)h(a)f(simple)h (construction.)f(Preprin)n(t,)g(2002.)456 1181 y([LM88])90 b(P)-6 b(.)24 b(Lo)r(c)n(hak)f(and)g(C.)i(Meunier.)f Fr(Multiphase)i(A)n(ver)l(aging)h(for)e(Classic)l(al)h(Systems)p Fv(,)782 1272 y(v)n(olume)g(72)g(of)h Fr(Appl.)g(Math.)g(Sci.)e Fv(Springer,)h(New)g(Y)-6 b(ork,)25 b(1988.)456 1363 y([LO99])100 b(R.)24 b(de)g(la)h(Lla)n(v)n(e)f(and)g(R.)g(Oba)n(y)n(a.) g(Regularit)n(y)g(of)h(the)f(comp)r(osition)i(op)r(erator)f(in)782 1455 y(spaces)e(of)f(H\177)-38 b(older)22 b(functions.)h Fr(Discr)l(ete)i(Contin.)f(Dynam.)f(Systems)p Fv(,)h(5\(1\):157{)782 1546 y(184,)j(1999.)456 1637 y([L)-6 b(T83])111 b(M.)28 b(A.)g(Lieb)r(erman)g(and)f(Je\013rey)h(L.)g(T)-6 b(enn)n(yson.)27 b(Chaotic)i(motion)f(along)g(reso-)782 1729 y(nance)c(la)n(y)n(ers)g (in)g(near-in)n(tegrable)g(Hamiltonian)h(systems)f(with)g(three)f(or)h (more)782 1820 y(degrees)c(of)f(freedom.)h(In)e Fr(L)l(ong-time)23 b(pr)l(e)l(diction)f(in)f(dynamics)h(\(L)l(akeway,)h(T)-6 b(ex.,)782 1911 y(1981\))p Fv(,)27 b(pages)g(179{211.)h(Wiley)-6 b(,)26 b(New)g(Y)-6 b(ork,)25 b(1983.)456 2003 y([L)-9 b(W89])90 b(R.)28 b(de)g(la)h(Lla)n(v)n(e)f(and)g(C.E.)i(W)-6 b(a)n(yne.)27 b(Whisk)n(ered)h(and)g(lo)n(w)n(er)h(dimensional)g(tori) 782 2094 y(in)d(nearly)f(in)n(tegrable)i(Hamiltonian)f(systems.)h (Preprin)n(t,)f(1989.)456 2185 y([L)-9 b(W95])90 b(R.)29 b(de)g(la)g(Lla)n(v)n(e)g(and)g(C.)g(E.)h(W)-6 b(a)n(yne.)28 b(On)g(Irwin's)i(pro)r(of)g(of)g(the)e(pseudostable)782 2277 y(manifold)f(theorem.)f Fr(Math.)i(Z.)p Fv(,)d(219\(2\):301{321,) 31 b(1995.)456 2368 y([Mat93])70 b(John)38 b(N.)f(Mather.)h(V)-6 b(ariational)38 b(construction)g(of)g(connecting)g(orbits.)g Fr(A)n(nn.)782 2459 y(Inst.)28 b(F)-6 b(ourier)28 b(\(Gr)l(enoble\))p Fv(,)g(43\(5\):1349{1386,)j(1993.)456 2551 y([Mat95])70 b(J.N.)19 b(Mather.)g(Graduate)f(course)h(at)f(Princeton,)j(95{96,)h (and)17 b(Lectures)i(at)f(Penn)782 2642 y(State,)26 b(Spring)f(96,)i (Paris,)g(Summer)e(96,)i(Austin,)f(Fall)g(96,)h(1995.)456 2733 y([Mat02])70 b(J.)36 b(N.)e(Mather.)i(Arnold)f(di\013usion)g(I:)f (Announcemen)n(t)g(of)h(results.)h Fr(Pr)l(eprint)p Fv(,)782 2825 y(2002.)456 2916 y([Mei92])83 b(J.)20 b(D.)g(Meiss.)h(Symplectic)e (maps,)j(v)l(ariational)f(principles,)g(and)f(transp)r(ort.)g Fr(R)l(ev.)782 3007 y(Mo)l(dern)29 b(Phys.)p Fv(,)d(64\(3\):795{848,)k (1992.)456 3099 y([Mey91])63 b(K.)32 b(R.)g(Mey)n(er.)h(Lie)g (transform)g(tutorial.)h(II.)e(In)f Fr(Computer)k(A)n(ide)l(d)f(Pr)l(o) l(ofs)g(in)782 3190 y(A)n(nalysis)e(\(Cincinnati,)g(OH,)f(1989\))p Fv(,)h(pages)f(190{210.)i(Springer,)e(New)f(Y)-6 b(ork,)782 3281 y(1991.)456 3372 y([Mo)r(e96])64 b(Ric)n(hard)26 b(Mo)r(ec)n(k)n(el.)h(T)-6 b(ransition)27 b(tori)g(in)f(the)f(\014v)n (e-b)r(o)r(dy)g(problem.)h Fr(J.)i(Di\013er)l(en-)782 3464 y(tial)f(Equations)p Fv(,)h(129\(2\):290{314,)i(1996.)456 3555 y([Mo)r(e02])64 b(Ric)n(hard)20 b(Mo)r(ec)n(k)n(el.)h(Generic)g (drift)f(on)h(Can)n(tor)f(sets)h(of)g(ann)n(uli.)f(In)g Fr(Celestial)i(me-)782 3646 y(chanics)29 b(\(Evanston,)h(IL,)d(1999\))p Fv(,)h(pages)f(163{171.)j(Amer.)c(Math.)h(So)r(c.,)g(Pro)n(v-)782 3738 y(idence,)f(RI,)f(2002.)456 3829 y([Mos73])70 b(J.)30 b(Moser.)g Fr(Stable)h(and)g(r)l(andom)g(motions)f(in)g(dynamic)l(al)h (systems)p Fv(.)f(Princeton)782 3920 y(Univ)n(ersit)n(y)e(Press,)j (Princeton,)f(N.)f(J.,)i(1973.)g(With)d(sp)r(ecial)j(emphasis)e(on)g (ce-)782 4012 y(lestial)i(mec)n(hanics,)g(Hermann)e(W)-6 b(eyl)29 b(Lectures,)i(the)e(Institute)f(for)i(Adv)l(anced)782 4103 y(Study)-6 b(,)24 b(Princeton,)i(N.)g(J,)g(Annals)g(of)g (Mathematics)h(Studies,)f(No.)g(77.)456 4194 y([Ne)-9 b(\025)-30 b(\02081])96 b(A.)27 b(I.)g(Ne)-9 b(\025)-30 b(\020sh)n(tadt.)27 b(Estimates)h(in)f(the)f(Kolmogoro)n(v)j(theorem)e (on)f(conserv)l(ation)782 4286 y(of)31 b(conditionally)f(p)r(erio)r (dic)h(motions.)g Fr(Prikl.)g(Mat.)g(Mekh.)p Fv(,)g(45\(6\):1016{1025,) 782 4377 y(1981.)456 4468 y([Ne)-9 b(\025)-30 b(\02084])96 b(A.)28 b(I.)g(Ne)-9 b(\025)-30 b(\020sh)n(tadt.)29 b(The)g(separation) g(of)g(motions)g(in)f(systems)g(with)h(rapidly)f(ro-)782 4560 y(tating)e(phase.)g Fr(Prikl.)h(Mat.)h(Mekh.)p Fv(,)e (48\(2\):197{204,)k(1984.)456 4651 y([Nie00])95 b(L.)44 b(Niederman.)g(Dynamics)g(around)f(simple)i(resonan)n(t)f(tori)g(in)g (nearly)f(in-)782 4742 y(tegrable)37 b(Hamiltonian)h(systems.)f Fr(J.)g(Di\013er)l(ential)h(Equations)p Fv(,)i(161\(1\):1{41,)782 4834 y(2000.)p eop end %%Page: 129 129 TeXDict begin 129 128 bop 1280 251 a Fq(Ov)n(ercoming)24 b(the)g(large)g(gap)g(problem)718 b(129)456 450 y Fv([P)n(oi99])99 b(H.)26 b(P)n(oincar)n(\023)-36 b(e.)27 b Fr(L)l(es)h(m)n(\023)-37 b(etho)l(des)29 b(nouvel)t(les)f(de)g(la)f(m)n(\023)-37 b(ec)l(anique)28 b(c)n(\023)-37 b(eleste)p Fv(,)28 b(v)n(olume)d(1,)782 541 y(2,)h(3.)h(Gauthier-Villars,)g(P)n(aris,)g(1892{1899.)456 633 y([P\177)-38 b(os82])88 b(J.)24 b(P\177)-38 b(osc)n(hel.)24 b(In)n(tegrabilit)n(y)f(of)g(Hamiltonian)h(systems)f(on)g(Can)n(tor)g (sets.)h Fr(Comm.)782 724 y(Pur)l(e)k(Appl.)f(Math.)p Fv(,)g(35\(5\):653{696,)j(1982.)456 815 y([RS02])108 b(P)n(aul)31 b(H.)f(Rabino)n(witz)h(and)e(Edw)n(ard)i(W.)f (Stredulinsky)-6 b(.)29 b(A)h(v)l(ariational)h(shad-)782 907 y(o)n(wing)22 b(metho)r(d.)f(In)f Fr(Celestial)k(me)l(chanics)g (\(Evanston,)h(IL,)e(1999\))p Fv(,)g(v)n(olume)e(292)782 998 y(of)28 b Fr(Contemp.)i(Math.)p Fv(,)e(pages)h(185{197.)h(Amer.)e (Math.)g(So)r(c.,)h(Pro)n(vidence,)f(RI,)782 1089 y(2002.)456 1181 y([Sal86])106 b(D.)30 b(Salamon.)g(The)g(Kolmogoro)n (v-Arnold-Moser)h(theorem.)f Fr(Z)q(\177)-40 b(urich)32 b(pr)l(eprint)p Fv(,)782 1272 y(1986.)456 1363 y([Sor02])97 b(A.)31 b(Sorren)n(tino.)g Fr(Sul)t(le)g(soluzioni)h(quasi-p)l(erio)l (diche)j(di)c(sistemi)i(Hamiltoniani)782 1455 y(di\013er)l(enziabili)p Fv(.)26 b(PhD)f(thesis,)i(Univ.)e(di)g(Roma)h(T)-6 b(re,)26 b(2002.)456 1546 y([Sv)l(a80])90 b(N.)26 b(V.)f(Sv)l(anidze.)h(Small)g (p)r(erturbations)g(of)g(an)g(in)n(tegrable)h(dynamical)f(system)782 1637 y(with)g(an)g(in)n(tegral)h(in)n(v)l(arian)n(t.)f Fr(T)-6 b(rudy)29 b(Mat.)f(Inst.)g(Steklov.)p Fv(,)f(147:124{146,)j (204,)782 1729 y(1980.)e(English)e(translation:)36 b Fr(Pr)l(o)l(c.)28 b(Steklov)h(Inst.)e(Math.)p Fv(,)f(1981,)i(no.)e(2.) 456 1820 y([SZF95])68 b(Mic)n(hael)32 b(F.)e(Shlesinger,)j(George)f(M.) f(Zasla)n(vsky)-6 b(,)31 b(and)f(Uriel)h(F)-6 b(risc)n(h,)32 b(editors.)782 1911 y Fr(L)n(\023)-37 b(evy)28 b(\015ights)f(and)g(r)l (elate)l(d)i(topics)e(in)f(physics)p Fv(,)h(Berlin,)f(1995.)h (Springer-V)-6 b(erlag.)456 2003 y([T)g(en82])82 b(Je\013rey)27 b(T)-6 b(enn)n(yson.)27 b(Resonance)g(transp)r(ort)h(in)f(near-in)n (tegrable)g(systems)h(with)782 2094 y(man)n(y)d(degrees)i(of)f (freedom.)h Fr(Phys.)h(D)p Fv(,)e(5\(1\):123{135,)j(1982.)456 2185 y([TLL80])57 b(J.)40 b(L.)f(T)-6 b(enn)n(yson,)41 b(M.)f(A.)e(Lieb)r(erman,)43 b(and)c(A.)f(J.)i(Lic)n(h)n(ten)n(b)r (erg.)f(Di\013usion)782 2277 y(in)47 b(near-in)n(tegrable)h (Hamiltonian)g(systems)g(with)f(three)g(degrees)h(of)h(free-)782 2368 y(dom.)30 b(In)g Fr(Nonline)l(ar)h(dynamics)h(and)g(the)g(b)l(e)l (am-b)l(e)l(am)i(inter)l(action)f(\(Symp)l(os.,)782 2459 y(Br)l(o)l(okhaven)39 b(Nat.)d(L)l(ab.,)i(New)f(Y)-6 b(ork,)39 b(1979\))p Fv(,)g(pages)d(272{301.)j(Amer.)c(Inst.)782 2551 y(Ph)n(ysics,)26 b(New)g(Y)-6 b(ork,)25 b(1980.)456 2642 y([T)-6 b(re89])95 b(D.)28 b(V.)f(T)-6 b(reshc)n(h)n(\177)-36 b(ev.)28 b(A)f(mec)n(hanism)h(for)h(the)e(destruction)h(of)g(resonance) h(tori)f(in)782 2733 y(Hamiltonian)f(systems.)f Fr(Mat.)h(Sb.)p Fv(,)f(180\(10\):1325{1346,)32 b(1439,)27 b(1989.)456 2825 y([T)-6 b(re02])95 b(D.)19 b(T)-6 b(resc)n(hev.)20 b(Multidimensional)h(symplectic)f(separatrix)g(maps.)g Fr(J.)h(Nonline)l(ar)782 2916 y(Sci.)p Fv(,)26 b(12\(1\):27{58,)j (2002.)456 3007 y([V)-6 b(al00])97 b(Enrico)25 b(V)-6 b(aldino)r(ci.)26 b(F)-6 b(amilies)26 b(of)f(whisk)n(ered)g(tori)g(for) g(a-priori)h(stable/unstable)782 3099 y(Hamiltonian)31 b(systems)f(and)g(construction)g(of)h(unstable)f(orbits.)h Fr(Math.)h(Phys.)782 3190 y(Ele)l(ctr)l(on.)d(J.)p Fv(,)c(6:P)n(ap)r (er)i(2,)g(31)f(pp.)f(\(electronic\),)i(2000.)456 3281 y([Xia98])91 b(Zhihong)18 b(Xia.)h(Arnold)f(di\013usion:)31 b(a)19 b(v)l(ariational)h(construction.)f(In)f Fr(Pr)l(o)l(c)l(e)l(e)l (dings)782 3372 y(of)23 b(the)h(International)g(Congr)l(ess)h(of)e (Mathematicians,)i(V)-6 b(ol.)23 b(II)f(\(Berlin,)h(1998\))p Fv(,)782 3464 y(n)n(um)n(b)r(er)i(Extra)g(V)-6 b(ol.)26 b(I)r(I,)g(pages)g(867{877)j(\(electronic\),)e(1998.)456 3555 y([Zas02])93 b(G.)31 b(M.)g(Zasla)n(vsky)-6 b(.)30 b(Chaos,)j(fractional)f(kinetics,)g(and)e(anomalous)i(transp)r(ort.)782 3646 y Fr(Phys.)c(R)l(ep.)p Fv(,)e(371\(6\):461{580,)k(2002.)456 3738 y([Zeh76])84 b(E.)23 b(Zehnder.)e(Generalized)j(implicit)f (function)f(theorems)h(with)f(applications)i(to)782 3829 y(some)33 b(small)g(divisor)f(problems/I)r(I.)g Fr(Comm.)g(Pur)l(e)i (Appl.)e(Math.)p Fv(,)i(29:49{111,)782 3920 y(1976.)555 4097 y Fu(Dep)-5 b(ar)g(t)g(ament)26 b(de)h(Ma)-5 b(tem)1456 4091 y(\022)1454 4097 y(atica)26 b(Aplicad)n(a)h(I,)g(Universit)-5 b(a)g(t)26 b(Polit)2813 4091 y(\022)2813 4097 y(ecnica)i(de)456 4188 y(Ca)-5 b(t)g(aluny)g(a,)27 b(Dia)n(gonal)h(647,)h(08028)h(Bar)n (celona,)e(Sp)-5 b(ain)555 4280 y Fr(E-mail)27 b(addr)l(ess)p Fv(,)h(AD:)d Fb(Amadeu.Delshams@upc.es)555 4439 y Fu(Dep)-5 b(ar)g(tment)22 b(of)h(Ma)-5 b(thema)g(tics,)23 b(University)e(of)j (Texas,)f(A)n(ustin,)g(TX)f(78712-)456 4531 y(1802)555 4622 y Fr(E-mail)27 b(addr)l(ess)p Fv(,)h(RL:)e Fb (llave@math.utexas.edu)555 4782 y Fu(Dep)-5 b(ar)g(t)g(ament)26 b(de)h(Ma)-5 b(tem)1456 4776 y(\022)1454 4782 y(atica)26 b(Aplicad)n(a)h(I,)g(Universit)-5 b(a)g(t)26 b(Polit)2813 4776 y(\022)2813 4782 y(ecnica)i(de)456 4873 y(Ca)-5 b(t)g(aluny)g(a,)27 b(Dia)n(gonal)h(647,)h(08028)h(Bar)n(celona,)e(Sp) -5 b(ain)555 4964 y Fr(E-mail)27 b(addr)l(ess)p Fv(,)h(TS:)e Fb(tere.m-seara@upc.es)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF