Content-Type: multipart/mixed; boundary="-------------0308140741332" This is a multi-part message in MIME format. ---------------0308140741332 Content-Type: text/plain; name="03-369.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-369.keywords" Anderson model, Lyapunov exponents ---------------0308140741332 Content-Type: application/postscript; name="lyap1.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="lyap1.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: lyap1.dvi %%Pages: 28 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMR10 CMBX10 CMMI10 CMR8 CMBX12 CMMI12 %%+ MSBM10 CMMI8 CMSY8 MSBM7 CMSY10 CMEX10 CMMI6 CMTI12 CMSL12 CMR6 %%+ LASY10 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips lyap1 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.08.14:1428 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def 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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 (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueID 5000829 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 525003F3DBE5BF07B2E83E66B7F97DDD7CE0EEB75A78BD9227BF359D002B6ADB 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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 0 /.notdef put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec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B6BC45A244FB86B470AAB56ED72A745661D2E3BA50BB2E582ADC8F85CB02F0CC 7FB15517147231BF0FFC31FD785A894F5226883017FB0FC180F043F32513FCF7 A1529F26505A6B407C50228293C9B4228E64B6B30969DFC3186DD1875D443452 8DAA55DC67B4D1D77924BBC0C33EB410517A8AC6C5F02C8C0E43E59AEB430868 56137749BF4928BEDD1651177666531AA9E0A6B8A0A30B15AE6A07466DAB3F62 80A5F0FC06AA742F9043E2902EF276AD861A041B8C13A67DDBAE0F5E6C215887 7D04C6F90621DC233491D77549130E4E50B792CA9F9A95A841929FC851ECDCFC A116CFF0F663EBE6C6187CDB5763A659FFF03A34FAB7AAB7E9BFA482CBB61BB6 640D2492E1BFDA75F36632A3BA78F0CA860BB61E91039A7319A836BDE72035AE 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D31FF2B87F97C73D63EECDDA4C49501773468A 27D1663E0B62F461F6E40A5D6676D1D12B51E641C1D4E8E2771864FC104F8CBF 5B78EC1D88228725F1C453A678F58A7E1B7BD7CA700717D288EB8DA1F57C4F09 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5FC379A4091CADD05FAC0A9C79A3C0E4BEAD9B 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C 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: 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 /.notdef put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFBB2A7C1B5D8E7E8AA0 5B10EA43D6A8ED61AF5B23D49920D8F79DAB6A59062134D84AC0100187A6CD1F 80F5DDD9D222ACB1C23326A7656A635C4A241CCD32CBFDF8363206B8AA36E107 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 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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. 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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF4E9D2405B169CD5365D6ECED5D768D66D6C 68618B8C482B341F8CA38E9BB9BAFCFAAD9C2F3FD033B62690986ED43D9C9361 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec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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 0 /.notdef 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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(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. 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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. 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y Fq(2)894 1954 y Fd(,)f Fn(\026)1012 1969 y Fq(11)1119 1954 y Fd(and)g Fn(\026)1367 1969 y Fq(12)1474 1954 y Fd(are)g(outside)h(of)f(the)h(windo)m(w)h Fo([)p Fi(\000)p Fo(2)p Fn(;)17 b Fo(2])32 b Fd(and)g(hence)i(h)m(yp)s (erb)s(olic.)45 b Fo(\(ii\))-24 2074 y Fd(Here)34 b Fn(E)f Fo(=)28 b Fi(\000)p Fo(0)p Fn(:)p Fo(03)p Fd(.)43 b(All)33 b(eigen)m(v)-5 b(alues)35 b(are)d(elliptic.)-24 2359 y Fo(eigen)m(v)-5 b(alues)41 b(are)f(h)m(yp)s(erb)s(olic.)64 b(In)39 b(b)s(et)m(w)m(een)j(one)d(has)h(b)s(oth)e(h)m(yp)s(erb)s(olic) j(and)e(elliptic)h(eigen)m(v)-5 b(alues.)66 b(F)-8 b(or)-24 2480 y(notational)36 b(con)m(v)m(enience,)k(w)m(e)e(will)e(supp)s(ose)i (that)e Fn(\026)1991 2495 y Fl(l)2050 2480 y Fn(<)e Fo(2)i(for)f(all)h Fn(l)r Fo(.)55 b(Hence)37 b(there)g(exists)h Fn(L)3534 2495 y Fl(h)3613 2480 y Fi(\024)3734 2440 y Fl(L)p 3734 2457 48 4 v 3740 2514 a Fq(2)3828 2480 y Fo(suc)m(h)-24 2600 y(that)33 b(that)f Fn(\026)458 2615 y Fl(l)517 2600 y Fo(is)h(h)m(yp)s(erb)s(olic)h(for)e Fn(l)e Fo(=)e(0)p Fn(;)17 b(:)g(:)g(:)e(;)i(L)1737 2615 y Fl(h)1782 2600 y Fo(,)33 b(and)g(elliptic)h(for)e Fn(l)e Fo(=)e Fn(L)2730 2615 y Fl(h)2797 2600 y Fo(+)22 b(1)p Fn(;)17 b(:)g(:)g(:)f(;)h Fo([)3200 2561 y Fl(L)p 3200 2577 V 3206 2634 a Fq(2)3258 2600 y Fo(])32 b(\(here)i([)p Fn(c)p Fo(])e(denotes)-24 2720 y(the)27 b(in)m(teger)f(part)g(of)f Fn(c)j Fi(2)g Fm(R)p Fo(\).)41 b(Moreo)m(v)m(er,)29 b(there)e(is)f(a)g(degeneracy)h Fn(\026)2516 2735 y Fl(L)p Fk(\000)p Fl(l)2672 2720 y Fo(=)h Fn(\026)2835 2735 y Fl(l)2886 2720 y Fo(due)f(to)e(re\015ection) i(symmetry)-24 2841 y(whic)m(h)k(will)f(b)s(e)f(further)h(analysed)g (in)g(Section)g(2.3.)42 b(In)29 b(case)h(there)g(are)f(no)g(h)m(yp)s (erb)s(olic)i(eigen)m(v)-5 b(alues,)32 b(let)d(us)-24 2961 y(set)i Fn(L)192 2976 y Fl(h)265 2961 y Fo(=)d Fi(\000)p Fo(1.)43 b(In)30 b(Fig.)g(1)g(are)h(sho)m(wn)g(examples)i(of)d(\(i\))g (a)g(situation)h(with)g(mixed)g(elliptic)h(and)e(h)m(yp)s(erb)s(olic) -24 3082 y(eigen)m(v)-5 b(alues)35 b(and)e(\(ii\))f(a)g(situation)h (with)h(only)f(elliptic)g(ones.)123 3249 y(F)-8 b(or)34 b(later)i(use,)h(let)f(us)g(set)g Fn(g)g Fo(=)g(diag\()p Fn(g)1616 3264 y Fq(1)1655 3249 y Fn(;)17 b(:)g(:)g(:)e(;)i(g)1920 3264 y Fl(L)1972 3249 y Fo(\))35 b(where)i Fn(g)2377 3264 y Fl(l)2436 3249 y Fo(=)32 b(1)j(if)h Fn(\026)2780 3264 y Fl(l)2841 3249 y Fo(is)g(elliptic)h(and)e Fn(g)3504 3264 y Fl(l)3563 3249 y Fo(=)d Fn({)k Fo(if)f Fn(\026)3889 3264 y Fl(l)3950 3249 y Fo(is)-24 3370 y(h)m(yp)s(erb)s(olic.)45 b(Note)33 b(that)f([)p Fn(d;)17 b(\026)p Fo(])27 b(=)h([)p Fn(d;)17 b(g)t Fo(])26 b(=)i([)p Fn(\026;)17 b(g)t Fo(])26 b(=)i(0.)123 3537 y(In)41 b(order)h(to)e(diagonalize)i(the)g(transfer)f (matrix,)j(let)d(us)h(in)m(tro)s(duce)g(the)g(diagonal)f(complex)i Fn(L)28 b Fi(\002)g Fn(L)-24 3693 y Fo(matrix)33 b Fn(\024)28 b Fo(=)491 3649 y Fl(\026)p 491 3670 43 4 v 495 3728 a Fq(2)565 3693 y Fo(+)663 3573 y Fh(q)p 763 3573 268 4 v 773 3649 a Fl(\026)815 3630 y Fb(2)p 773 3670 78 4 v 794 3728 a Fq(4)882 3693 y Fi(\000)23 b Fo(1)32 b(as)h(w)m(ell)h (as)e(the)h(\(p)s(ossibly)h(complex-v)-5 b(alued\))34 b(matrices)150 4190 y Fn(N)66 b Fo(=)426 3930 y Fh(0)426 4105 y(B)426 4165 y(B)426 4229 y(@)593 4027 y Fn(m)695 3929 y Fh(q)p 794 3929 172 4 v 862 3988 a Fq(1)p 804 4005 152 4 v 804 4068 a Fl(\024)p Fk(\000)913 4041 y Fb(1)p 910 4053 37 3 v 910 4094 a Ff(\024)1124 4027 y Fn(m)1226 3929 y Fh(q)p 1325 3929 172 4 v 1394 3988 a Fq(1)p 1335 4005 152 4 v 1335 4068 a Fl(\024)p Fk(\000)1444 4041 y Fb(1)p 1441 4053 37 3 v 1441 4094 a Ff(\024)554 4331 y Fn(m)656 4233 y Fh(q)p 756 4233 172 4 v 824 4292 a Fq(1)p 766 4308 152 4 v 766 4371 a Fl(\024)p Fk(\000)874 4344 y Fb(1)p 871 4356 37 3 v 871 4398 a Ff(\024)957 4292 y Fq(1)p 954 4308 41 4 v 954 4366 a Fl(\024)1088 4331 y Fn(m)1190 4233 y Fh(q)p 1289 4233 172 4 v 1357 4292 a Fq(1)p 1299 4308 152 4 v 1299 4371 a Fl(\024)p Fk(\000)1407 4344 y Fb(1)p 1405 4356 37 3 v 1405 4398 a Ff(\024)1477 4331 y Fn(\024)1575 3930 y Fh(1)1575 4105 y(C)1575 4165 y(C)1575 4229 y(A)1711 4190 y Fo(,)228 b Fn(N)2054 4149 y Fk(\000)p Fq(1)2204 4190 y Fo(=)2335 3930 y Fh(0)2335 4105 y(B)2335 4165 y(B)2335 4229 y(@)2505 3929 y(q)p 2605 3929 172 4 v 2673 3988 a Fq(1)p 2615 4005 152 4 v 2615 4068 a Fl(\024)p Fk(\000)2723 4041 y Fb(1)p 2720 4053 37 3 v 2720 4094 a Ff(\024)2793 4027 y Fn(\024)17 b(m)2951 3991 y Fk(\003)3114 4027 y Fi(\000)3191 3929 y Fh(q)p 3291 3929 172 4 v 3360 3988 a Fq(1)p 3301 4005 152 4 v 3301 4068 a Fl(\024)p Fk(\000)3410 4041 y Fb(1)p 3407 4053 37 3 v 3407 4094 a Ff(\024)3480 4027 y Fn(m)3565 3991 y Fk(\003)2464 4331 y Fi(\000)2541 4233 y Fh(q)p 2641 4233 172 4 v 2709 4292 a Fq(1)p 2651 4308 152 4 v 2651 4371 a Fl(\024)p Fk(\000)2759 4344 y Fb(1)p 2757 4356 37 3 v 2757 4398 a Ff(\024)2842 4292 y Fq(1)p 2839 4308 41 4 v 2839 4366 a Fl(\024)2906 4331 y Fn(m)2991 4295 y Fk(\003)3153 4233 y Fh(q)p 3253 4233 172 4 v 3321 4292 a Fq(1)p 3263 4308 152 4 v 3263 4371 a Fl(\024)p Fk(\000)3371 4344 y Fb(1)p 3368 4356 37 3 v 3368 4398 a Ff(\024)3441 4331 y Fn(m)3526 4295 y Fk(\003)3646 3930 y Fh(1)3646 4105 y(C)3646 4165 y(C)3646 4229 y(A)3782 4190 y Fo(,)-24 4586 y(where)41 b(here)g(and)f(b)s(elo)m(w)g(all)g(ro)s (ots)f(are)h(tak)m(en)h(on)f(the)g(\014rst)g(branc)m(h.)66 b(Then)41 b(one)f(immediately)i(v)m(eri\014es)-24 4706 y(that)1070 4980 y Fn(N)1158 4939 y Fk(\000)p Fq(1)1286 4839 y Fh(\022)1401 4919 y Fo(\001)1482 4934 y Fl(L)1556 4919 y Fi(\000)23 b Fn(E)89 b Fi(\000)p Fg(1)1539 5039 y(1)264 b Fo(0)1992 4839 y Fh(\023)2098 4980 y Fn(N)66 b Fo(=)2373 4839 y Fh(\022)2488 4919 y Fn(\024)89 b Fo(0)2492 5039 y(0)2640 5000 y Fq(1)p 2637 5016 41 4 v 2637 5074 a Fl(\024)2730 4839 y Fh(\023)2852 4980 y Fo(.)123 5254 y(Finally)25 b(the)g(r.h.s.)42 b(will)26 b(b)s(e)f(transformed)g(in)m (to)h(a)e(normal)h(form)g(whic)m(h)h(is)g(real-v)-5 b(alued)25 b(and)g(symplectic.)1972 5564 y(5)p eop end %%Page: 6 6 TeXDict begin 6 5 bop -24 171 a Fo(F)-8 b(or)38 b(elliptic)i Fn(\026)542 186 y Fl(l)607 171 y Fo(de\014ne)g(0)e Fn(<)g(\021)1144 186 y Fl(l)1208 171 y Fn(<)g(\031)43 b Fo(b)m(y)c Fn(\024)1617 186 y Fl(l)1682 171 y Fo(=)f Fn(e)1841 134 y Fl({\021)1899 146 y Ff(l)1928 171 y Fo(,)i(for)e(a)h(h)m(yp)s(erb)s(olic)h(one)f Fn(\021)2954 186 y Fl(l)3018 171 y Fn(>)f Fo(0)h(b)m(y)g Fn(\024)3417 186 y Fl(l)3482 171 y Fo(=)f Fn(e)3641 134 y Fl(\021)3676 146 y Ff(l)3705 171 y Fo(.)62 b(Then)-24 291 y(in)m(tro)s(duce)34 b(the)f(rotation)f(and)h(h)m(yp)s(erb)s(olic)g (rotation)f(matrices)i(b)m(y)g(\()p Fn(\021)d Fi(2)d Fm(R)p Fo(\):)430 565 y Fn(R)504 580 y Fl(e)541 565 y Fo(\()p Fn(\021)t Fo(\))55 b(=)855 424 y Fh(\022)970 504 y Fo(cos)q(\()p Fn(\021)t Fo(\))83 b Fi(\000)17 b Fo(sin\()p Fn(\021)t Fo(\))976 624 y(sin)q(\()p Fn(\021)t Fo(\))129 b(cos)q(\()p Fn(\021)t Fo(\))1694 424 y Fh(\023)1817 565 y Fo(,)228 b Fn(R)2146 580 y Fl(h)2191 565 y Fo(\()p Fn(\021)t Fo(\))55 b(=)2505 424 y Fh(\022)2620 504 y Fo(cosh)q(\()p Fn(\021)t Fo(\))88 b(sinh)q(\()p Fn(\021)t Fo(\))2625 624 y(sinh)q(\()p Fn(\021)t Fo(\))g(cosh)q(\()p Fn(\021)t Fo(\))3369 424 y Fh(\023)3491 565 y Fo(.)-24 791 y(Setting)825 1094 y Fn(C)62 b Fo(=)1089 948 y Fh(r)p 1188 948 69 4 v 1207 1026 a Fn({)p 1198 1071 49 4 v 1198 1162 a Fo(2)1301 953 y Fh(\022)1416 1033 y Fg(1)122 b Fn({)17 b Fg(1)1416 1153 y(1)83 b Fi(\000)p Fn({)17 b Fg(1)1778 953 y Fh(\023)1900 1094 y Fo(,)228 b Fn(G)55 b Fo(=)2419 923 y Fh( )p 2539 935 105 4 v 2539 1043 a Fn(g)2600 979 y Fb(1)p 2599 991 31 3 v 2599 1032 a(2)2808 1043 y Fo(0)2567 1175 y(0)111 b Fn(g)2778 1139 y Fq(2)2833 1175 y Fn(g)2894 1111 y Fb(1)p 2893 1123 V 2893 1164 a(2)2979 923 y Fh(!)3107 1094 y Fo(,)-24 1350 y(one)33 b(v)m(eri\014es)875 1624 y Fn(G)952 1583 y Fk(\000)p Fq(1)1063 1624 y Fn(C)1140 1583 y Fk(\000)p Fq(1)1267 1483 y Fh(\022)1382 1563 y Fn(\024)89 b Fo(0)1386 1683 y(0)1534 1644 y Fq(1)p 1531 1660 41 4 v 1531 1717 a Fl(\024)1624 1483 y Fh(\023)1730 1624 y Fn(C)24 b(G)55 b Fo(=)g Fn(R)2161 1639 y Fq(1)2201 1624 y Fo(\()p Fn(\021)2287 1639 y Fq(1)2326 1624 y Fo(\))22 b Fi(\010)h Fn(:)17 b(:)g(:)22 b Fi(\010)g Fn(R)2796 1639 y Fl(L)2849 1624 y Fo(\()p Fn(\021)2935 1639 y Fl(L)2987 1624 y Fo(\))32 b(,)-24 1856 y(where)40 b Fn(R)338 1871 y Fl(l)364 1856 y Fo(\()p Fn(\021)450 1871 y Fl(l)477 1856 y Fo(\))e(is)h(either)h Fn(R)1014 1871 y Fl(e)1051 1856 y Fo(\()p Fn(\021)1137 1871 y Fl(l)1163 1856 y Fo(\))e(or)h Fn(R)1439 1871 y Fl(h)1484 1856 y Fo(\()p Fn(\021)1570 1871 y Fl(l)1596 1856 y Fo(\))f(dep)s(ending)i(on) f(whether)h Fn(\026)2728 1871 y Fl(l)2792 1856 y Fo(is)f(elliptic)h(or) e(h)m(yp)s(erb)s(olic,)k(and)-24 1977 y(where)37 b(the)f(direct)h(sum)f (is)g(understo)s(o)s(d)g(suc)m(h)i(that)d Fn(R)2047 1992 y Fl(l)2073 1977 y Fo(\()p Fn(\021)2159 1992 y Fl(l)2185 1977 y Fo(\))h(acts)g(on)f(the)h Fn(l)r Fo(th)g(and)g(\()p Fn(l)27 b Fo(+)d Fn(L)p Fo(\)th)36 b(comp)s(onen)m(t)-24 2097 y(of)c Fm(C)159 2061 y Fq(2)p Fl(L)247 2097 y Fo(,)g(namely)i(the) f(normal)f(form)h Fn(R)28 b Fo(=)g Fn(R)1654 2112 y Fq(1)1693 2097 y Fo(\()p Fn(\021)1779 2112 y Fq(1)1819 2097 y Fo(\))22 b Fi(\010)g Fn(:)17 b(:)g(:)22 b Fi(\010)g Fn(R)2288 2112 y Fl(L)2341 2097 y Fo(\()p Fn(\021)2427 2112 y Fl(L)2479 2097 y Fo(\))32 b(is)h(a)f(real-v)-5 b(alued)33 b(symplectic)i(matrix.) -24 2217 y(In)i(case)g(that)e(there)i(are)f(parab)s(olic)g(eigen)m(v)-5 b(alues,)39 b(the)d(normal)g(form)g(con)m(tains)h(Jordan)e(blo)s(cs)i (just)f(as)g(in)-24 2338 y([SB)q(].)123 2505 y(Let)c(us)i(resume)g(the) f(ab)s(o)m(v)m(e.)44 b(Set)33 b Fn(M)39 b Fo(=)27 b Fn(N)10 b(C)d(G)p Fo(,)33 b(then)1239 2779 y Fn(M)1343 2738 y Fk(\000)p Fq(1)1471 2639 y Fh(\022)1586 2718 y Fo(\001)1667 2733 y Fl(L)1742 2718 y Fi(\000)22 b Fn(E)89 b Fi(\000)p Fg(1)1725 2839 y(1)264 b Fo(0)2177 2639 y Fh(\023)2284 2779 y Fn(M)66 b Fo(=)55 b Fn(R)34 b Fo(.)-24 3012 y(F)-8 b(urthermore)33 b(the)g(symplectic)i(basis)f(c)m(hange)f Fn(M)39 b Fi(2)17 b Fo(SP\(2)p Fn(L;)g Fm(R)p Fo(\))33 b(is)g(more)g(explicitly)h(giv)m(en)g(b)m(y)109 3446 y Fn(M)66 b Fo(=)400 3245 y Fh(0)400 3425 y(@)729 3324 y Fn(m)17 b(h)439 b Fo(0)529 3565 y Fn(m)641 3526 y Fq(1)p 641 3542 36 4 v 641 3599 a(2)702 3565 y Fo(\()p Fn(\024)23 b Fo(+)929 3526 y Fq(1)p 927 3542 41 4 v 927 3599 a Fl(\024)977 3565 y Fo(\))17 b Fn(h)83 b(m)17 b(h)1329 3529 y Fk(\000)p Fq(1)1440 3565 y Fn(g)1491 3529 y Fq(2)1571 3245 y Fh(1)1571 3425 y(A)1708 3446 y Fo(,)227 b Fn(M)2066 3404 y Fk(\000)p Fq(1)2217 3446 y Fo(=)2348 3245 y Fh(0)2348 3425 y(@)2722 3324 y Fn(h)2778 3288 y Fk(\000)p Fq(1)2889 3324 y Fn(m)2974 3288 y Fk(\003)3470 3324 y Fo(0)2477 3565 y Fi(\000)2564 3526 y Fq(1)p 2564 3542 36 4 v 2564 3599 a(2)2626 3565 y Fn(g)2677 3529 y Fq(2)2732 3565 y Fo(\()p Fn(\024)22 b Fo(+)2959 3526 y Fq(1)p 2956 3542 41 4 v 2956 3599 a Fl(\024)3007 3565 y Fo(\))17 b Fn(h)g(m)3220 3529 y Fk(\003)3342 3565 y Fn(g)3393 3529 y Fq(2)3448 3565 y Fn(h)g(m)3606 3529 y Fk(\003)3688 3245 y Fh(1)3688 3425 y(A)3824 3446 y Fo(,)-24 3778 y(where)273 4224 y Fn(h)56 b Fo(=)516 4052 y Fh(s)p 616 4052 259 4 v 652 4157 a Fo(2)p Fn({)17 b(g)800 4128 y Fq(3)p 626 4201 239 4 v 626 4295 a Fn(\024)22 b Fi(\000)816 4256 y Fq(1)p 814 4272 41 4 v 814 4329 a Fl(\024)929 4224 y Fo(=)56 b(diag\()p Fn(h)1334 4239 y Fq(1)1373 4224 y Fn(;)17 b(:)g(:)g(:)f(;)h(h)1648 4239 y Fl(L)1700 4224 y Fo(\))32 b(,)228 b Fn(h)2081 4239 y Fl(l)2163 4224 y Fo(=)2294 4020 y Fh(8)2294 4109 y(<)2294 4289 y(:)2451 4103 y Fo(sin)q(\()p Fn(\021)2657 4118 y Fl(l)2683 4103 y Fo(\))2721 4066 y Fk(\000)2786 4039 y Fb(1)p 2786 4051 31 3 v 2786 4092 a(2)3074 4103 y Fn(\026)3133 4118 y Fl(l)3219 4103 y Fo(elliptic,)2424 4355 y(sinh)q(\()p Fn(\021)2684 4370 y Fl(l)2710 4355 y Fo(\))2748 4318 y Fk(\000)2813 4291 y Fb(1)p 2813 4303 V 2813 4345 a(2)3051 4355 y Fn(\026)3110 4370 y Fl(l)3196 4355 y Fo(h)m(yp)s(erb)s(olic.)-24 4622 y Fc(2.2)136 b(Calculation)46 b(of)f(the)h(p)t(erturbation)-24 4807 y Fo(First)33 b(note)g(that)745 5081 y Fn(T)69 b Fo(=)1003 4940 y Fh(\022)1117 5020 y Fo(\001)1198 5035 y Fl(L)1273 5020 y Fi(\000)23 b Fn(E)89 b Fi(\000)p Fg(1)1256 5140 y(1)264 b Fo(0)1709 4940 y Fh(\023)44 b(\022)17 b(\022)2031 5020 y Fg(1)87 b Fo(0)2035 5140 y(0)f Fg(1)2268 4940 y Fh(\023)2391 5081 y Fi(\000)39 b Fn(\025)2597 4940 y Fh(\022)2727 5020 y Fo(0)98 b(0)2712 5140 y Fn(V)105 b Fo(0)2964 4940 y Fh(\023)17 b(\023)3176 5081 y Fo(.)-24 5313 y(Hence)34 b Fn(P)41 b Fi(2)17 b Fo(sp\(2)p Fn(L;)g Fm(R)p Fo(\))33 b(in)g(equation)g(\(5\))g(is)g(giv)m(en)g(b)m(y)1972 5564 y(6)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 876 329 a Fn(P)69 b Fo(=)55 b Fn(M)1243 288 y Fk(\000)p Fq(1)1383 189 y Fh(\022)1512 268 y Fo(0)98 b(0)1497 389 y Fn(V)105 b Fo(0)1749 189 y Fh(\023)1839 329 y Fn(M)66 b Fo(=)2131 189 y Fh(\022)2473 263 y Fo(0)311 b(0)2245 394 y Fn(g)2296 357 y Fq(2)2352 394 y Fn(h)17 b(d)2476 357 y Fk(\003)2546 368 y Fo(^)2531 394 y Fn(V)38 b(d)17 b(h)83 b Fo(0)2923 189 y Fh(\023)3046 329 y Fo(.)818 b(\(6\))-24 586 y(Here)28 b(the)g(iden)m(tit)m(y)h Fn(m)f Fo(=)g Fn(f)11 b(d)27 b Fo(w)m(as)h(used)g(as)g(w)m(ell)h(as)e(the)h (de\014nition)2500 561 y(^)2485 586 y Fn(V)50 b Fo(=)27 b Fn(f)2754 550 y Fk(\003)2793 586 y Fn(V)22 b(f)11 b Fo(.)41 b(Actually)29 b Fn(f)38 b Fo(is)27 b(the)h(matrix)-24 707 y(of)41 b(the)g(discrete)h(F)-8 b(ourier)40 b(transform)h(so)g (that)1801 682 y(^)1787 707 y Fn(V)63 b Fo(=)2039 682 y(^)2024 707 y Fn(V)2102 671 y Fk(\003)2183 707 y Fo(is)41 b(the)g(T)-8 b(o)s(eplitz)41 b(matrix)g(asso)s(ciated)h(with)f(the)-24 827 y(F)-8 b(ourier)33 b(transform)f(of)g(the)h(random)g(p)s(oten)m (tial)g(\(at)f(\014xed)i(heigh)m(t)f Fn(n)p Fo(\).)44 b(More)33 b(precisely)-8 b(,)35 b(de\014ne)1220 1136 y(^)-52 b Fn(v)t Fo(\()p Fn(k)s Fo(\))55 b(=)1603 1068 y(1)p 1594 1113 67 4 v 1594 1204 a Fn(L)1763 1011 y Fl(L)1715 1041 y Fh(X)1731 1253 y Fl(l)q Fq(=1)1892 1136 y Fn(v)t Fo(\()p Fn(l)r Fo(\))33 b(exp)2248 995 y Fh(\022)2332 1068 y Fo(2)p Fn(\031)t({)17 b(l)r(k)p 2332 1113 241 4 v 2419 1204 a(L)2582 995 y Fh(\023)2705 1136 y Fo(.)-24 1420 y(One)33 b(has)p 356 1333 181 4 v 37 w(^)-53 b Fn(v)t Fo(\()p Fn(k)s Fo(\))28 b(=)j(^)-53 b Fn(v)t Fo(\()p Fi(\000)p Fn(k)s Fo(\))33 b(and)j(^)-52 b Fn(v)s Fo(\()p Fn(k)25 b Fo(+)d Fn(L)p Fo(\))28 b(=)k(^)-53 b Fn(v)t Fo(\()p Fn(k)s Fo(\).)43 b(Then)950 1831 y(^)935 1856 y Fn(V)77 b Fo(=)1200 1566 y Fh(0)1200 1742 y(B)1200 1801 y(B)1200 1861 y(B)1200 1925 y(@)1464 1633 y Fo(^)-52 b Fn(v)s Fo(\(0\))219 b(^)-53 b Fn(v)t Fo(\(1\))152 b Fn(:)17 b(:)g(:)155 b Fo(^)-53 b Fn(v)t Fo(\()p Fn(L)23 b Fi(\000)f Fo(1\))1426 1795 y(^)-53 b Fn(v)t Fo(\()p Fi(\000)p Fo(1\))1887 1737 y(.)1925 1762 y(.)1963 1787 y(.)2184 1737 y(.)2222 1762 y(.)2261 1787 y(.)2614 1729 y(.)2614 1762 y(.)2614 1795 y(.)1535 1891 y(.)1535 1924 y(.)1535 1957 y(.)1887 1899 y(.)1925 1924 y(.)1963 1949 y(.)2184 1899 y(.)2222 1924 y(.)2261 1949 y(.)2543 1957 y(^)g Fn(v)t Fo(\(1\))1332 2078 y(^)h Fn(v)s Fo(\()p Fi(\000)p Fn(L)23 b Fo(+)f(1\))114 b Fn(:)17 b(:)g(:)116 b Fo(^)-52 b Fn(v)s Fo(\()p Fi(\000)p Fo(1\))181 b(^)-53 b Fn(v)t Fo(\(0\))2850 1566 y Fh(1)2850 1742 y(C)2850 1801 y(C)2850 1861 y(C)2850 1925 y(A)2987 1856 y Fo(.)-24 2344 y Fc(2.3)136 b(Symplectic)45 b(c)l(hannels)-24 2529 y Fo(The)29 b(symplectic)h(c)m(hannels)g(of)d(a)h(symplectic)i(matrix)e (are)g(b)m(y)h(de\014nition)g(the)f(eigenspaces)i(of)d(its)h(\(p)s (ossibly)-24 2649 y(degenerate\))45 b(eigen)m(v)-5 b(alue)45 b(pairs)f(\()p Fn(\025;)1391 2610 y Fq(1)p 1388 2627 42 4 v 1388 2684 a Fl(\025)1439 2649 y Fo(\).)76 b(F)-8 b(or)43 b(the)h(elliptic-h)m(yp)s(erb)s(olic)h(rotation)e Fn(R)k Fo(=)g Fn(R)3455 2664 y Fq(1)3494 2649 y Fo(\()p Fn(\021)3580 2664 y Fq(1)3620 2649 y Fo(\))29 b Fi(\010)h Fn(:)17 b(:)g(:)29 b Fi(\010)-24 2770 y Fn(R)50 2785 y Fl(L)103 2770 y Fo(\()p Fn(\021)189 2785 y Fl(L)241 2770 y Fo(\),)39 b(they)f(are)g(the)g(maximal)h(subspaces)h(of)d Fm(C)1953 2734 y Fq(2)p Fl(L)2078 2770 y Fo(c)m(haracterized)i(b)m(y)g (the)f(prop)s(ert)m(y)g(that)g Fn(R)h Fo(rotates)-24 2890 y(b)m(y)34 b(the)f(same)g(angle.)44 b(F)-8 b(or)32 b(the)h(Anderson)g(mo)s(del)g(on)g(a)f(strip)h(studied)h(here,)g(there) f(are)g Fn(L)3390 2905 y Fl(c)3447 2890 y Fo(+)22 b(1)27 b(=)h([)3762 2851 y Fl(L)p 3762 2867 48 4 v 3768 2925 a Fq(2)3820 2890 y Fo(])22 b(+)g(1)-24 3011 y(c)m(hannels.)75 b(The)43 b(0th)f(c)m(hannel)i(is)f(asso)s(ciated)g(with)g(the)f (fundamen)m(tal)i Fn(\026)2819 3026 y Fl(L)2913 3011 y Fo(and)e(giv)m(en)h(b)m(y)h(the)e(span)h(of)-24 3131 y Fi(f)p Fn(e)71 3146 y Fl(L)124 3131 y Fn(;)17 b(e)213 3146 y Fq(2)p Fl(L)300 3131 y Fi(g)35 b Fo(where)h Fn(e)714 3146 y Fl(l)772 3131 y Fi(2)c Fm(C)942 3095 y Fq(2)p Fl(L)1065 3131 y Fo(has)j(a)g(non-v)-5 b(anshing)35 b(en)m(try)h(equal) g(to)f(1)g(only)g(in)h(the)f Fn(l)r Fo(th)g(comp)s(onen)m(t.)52 b(If)35 b Fn(L)-24 3251 y Fo(is)41 b(ev)m(en,)i(also)e(the)729 3212 y Fl(L)p 729 3228 V 735 3286 a Fq(2)787 3251 y Fo(th)f(c)m(hannel) h(is)g(simple)g(and)f(spanned)i(b)m(y)f Fi(f)p Fn(e)2521 3267 y Fl(L=)p Fq(2)2643 3251 y Fn(;)17 b(e)2732 3267 y Fq(3)p Fl(L=)p Fq(2)2890 3251 y Fi(g)p Fo(.)66 b(Due)40 b(to)g(the)g(degeneracy)-24 3372 y Fn(\026)35 3387 y Fl(l)97 3372 y Fo(=)35 b Fn(\026)267 3387 y Fl(L)p Fk(\000)p Fl(l)433 3372 y Fo(of)h(the)i(sp)s(ectrum)g(of)f(\001)1348 3387 y Fl(L)1401 3372 y Fo(,)h(all)f(other)g(c)m(hannels)i(are)f (doubly)g(degenerate.)58 b(F)-8 b(or)36 b Fn(l)i Fi(6)p Fo(=)e(0)p Fn(;)3757 3333 y Fl(L)p 3757 3349 V 3763 3406 a Fq(2)3815 3372 y Fo(,)i(the)-24 3492 y Fn(l)r Fo(th)29 b(c)m(hannel)h(is)f(spanned)g(b)m(y)h Fi(f)p Fn(e)1175 3507 y Fl(l)1201 3492 y Fn(;)17 b(e)1290 3507 y Fl(L)p Fk(\000)p Fl(l)1419 3492 y Fn(;)g(e)1508 3507 y Fl(l)q Fq(+)p Fl(L)1636 3492 y Fn(;)g(e)1725 3507 y Fq(2)p Fl(L)p Fk(\000)p Fl(l)1889 3492 y Fi(g)p Fo(.)42 b(Let)29 b(us)g(denote)g(the) g(degeneracy)h(of)e(the)h Fn(l)r Fo(th)f(c)m(hannel)-24 3613 y(b)m(y)34 b Fn(\027)160 3628 y Fl(l)186 3613 y Fo(.)43 b(In)33 b(accordance)g(with)g(the)g(ab)s(o)m(v)m(e,)g(the)g Fn(l)r Fo(th)f(c)m(hannel)i(is)f(called)g(elliptic)h(if)e Fi(j)p Fn(\026)3117 3628 y Fl(l)3142 3613 y Fi(j)27 b Fn(<)h Fo(2)k(and)g(h)m(yp)s(erb)s(olic)-24 3733 y(if)41 b Fi(j)p Fn(\026)161 3748 y Fl(l)187 3733 y Fi(j)g Fn(>)h Fo(2.)68 b(With)42 b(these)g(notations,)h Fm(C)1575 3697 y Fq(2)p Fl(L)1704 3733 y Fo(is)e(hence)h(decomp)s(osed)h(in)m(to)e(a)g (direct)h(sum)f(of)g(the)g Fn(L)3800 3748 y Fl(c)3863 3733 y Fo(+)28 b(1)-24 3853 y(c)m(hannels.)123 4021 y(There)38 b(are)f Fn(L)642 4036 y Fl(h)713 4021 y Fo(+)25 b(1)37 b(h)m(yp)s(erb)s(olic)i(c)m(hannels.)59 b(As)38 b(w)m(e)g(supp)s(osed)h (\(for)e(con)m(v)m(enience\))j(that)d Fn(\026)3593 4036 y Fl(l)3655 4021 y Fn(<)e Fo(2,)j(the)-24 4141 y(h)m(yp)s(erb)s(olic)c (c)m(hannels)h(are)d(the)h(\014rst)g(ones,)h(namely)g(c)m(hannels)g(0)p Fn(;)17 b(:)g(:)g(:)f(;)h(L)2689 4156 y Fl(h)2766 4141 y Fo(\(compare)33 b(Fig.)f(1\(i\)\).)123 4309 y(The)27 b(pro)5 b(jection)28 b(on)f(the)g Fn(l)r Fo(th)g(c)m(hannel)h(will)g(b) s(e)e(denoted)i(b)m(y)g Fn(\031)2419 4324 y Fl(l)2445 4309 y Fo(.)42 b(It)27 b(satis\014es)h([)p Fn(\031)3049 4324 y Fl(l)3075 4309 y Fn(;)17 b(J)9 b Fo(])28 b(=)g(0.)41 b(The)28 b(non-zero)-24 4429 y(eigen)m(v)-5 b(alues)33 b(of)d Fn(\031)643 4444 y Fl(l)669 4429 y Fn(R)q(\031)799 4444 y Fl(l)856 4429 y Fo(are)g Fn(e)1061 4393 y Fk(\006)p Fl({\021)1174 4405 y Ff(l)1234 4429 y Fo(if)g(the)h Fn(l)r Fo(th)g(c)m(hannel)h(is)e(elliptic)i(and)f Fn(e)2640 4393 y Fk(\006)p Fl(\021)2730 4405 y Ff(l)2789 4429 y Fo(if)f(it)h(is)g(h)m(yp)s(erb)s(olic.)44 b(Hence)31 b(one)-24 4550 y(can)i(decomp)s(ose)h Fn(\031)701 4565 y Fl(l)760 4550 y Fo(in)m(to)f(the)g(corresp)s(onding)g(eigenspaces)i Fn(\031)2323 4565 y Fl(l)2377 4550 y Fo(=)28 b Fn(\031)2540 4508 y Fq(+)2536 4577 y Fl(l)2621 4550 y Fo(+)22 b Fn(\031)2778 4508 y Fk(\000)2774 4577 y Fl(l)2837 4550 y Fo(.)43 b(Resuming)34 b(all)f(the)g(ab)s(o)m(v)m(e,)927 4858 y Fg(1)983 4877 y Fj(C)1031 4858 y Fb(2)p Ff(L)64 b Fo(=)1330 4734 y Fl(L)1378 4742 y Ff(c)1298 4764 y Fh(X)1314 4976 y Fl(l)q Fq(=0)1486 4858 y Fn(\031)1545 4817 y Fq(+)1541 4886 y Fl(l)1626 4858 y Fo(+)22 b Fn(\031)1783 4817 y Fk(\000)1779 4886 y Fl(l)1875 4858 y Fo(,)228 b Fn(R)28 b(\031)2291 4817 y Fk(\006)2287 4886 y Fl(l)2406 4858 y Fo(=)55 b Fn(e)2582 4817 y Fk(\006)12 b Fl({)p 2684 4780 58 3 v 12 w(g)2718 4829 y Ff(l)2753 4817 y Fl(\021)2788 4829 y Ff(l)2845 4858 y Fn(\031)2904 4817 y Fk(\006)2900 4886 y Fl(l)2995 4858 y Fo(.)869 b(\(7\))1972 5564 y(7)p eop end %%Page: 8 8 TeXDict begin 8 7 bop -24 171 a Fc(2.4)136 b(A)44 b(privileged)i(basis) -24 355 y Fo(F)-8 b(or)37 b(explicit)h(calculations)g(in)f(the)h(next)g (section,)h(it)e(will)h(b)s(e)f(con)m(v)m(enien)m(t)j(to)c(disp)s(ose)j (of)d(a)h(basis)h(of)e Fm(C)3928 319 y Fq(2)p Fl(L)-24 476 y Fo(with)e(the)f(follo)m(wing)f(prop)s(erties:)96 704 y(1.)48 b(the)33 b(basis)h(v)m(ectors)g(are)f(orthonormal)f(and)g (compatible)i(with)f(the)g(symplectic)i(structure;)96 907 y(2.)48 b(the)33 b(basis)h(v)m(ectors)g(are)f(eigen)m(v)m(ectors)i (of)d Fn(R)q Fo(;)96 1111 y(3.)48 b(the)30 b(matrix)g(elemen)m(ts)i(of) c(the)i(p)s(erturbation)g Fn(P)43 b Fo(w.r.t.)f(the)30 b(basis)h(are)e(particularly)h(simple,)i Fd(i.e.)p Fo(,)e(up)220 1231 y(to)i(a)h(constan)m(t,)g(giv)m(en)h(b)m(y)g(the)f(F)-8 b(ourier)32 b(transform)k(^)-52 b Fn(v)36 b Fo(of)c(the)h(p)s(oten)m (tial.)123 1460 y(F)-8 b(or)32 b(this)h(purp)s(ose)h(let)f(us)g(note)g (that,)g(for)g(an)m(y)g Fn(\021)f Fi(2)c Fm(R)p Fo(,)33 b(linearly)h(indep)s(enden)m(t)h(eigen)m(v)m(ectors)h(of)c Fn(R)3851 1475 y Fl(e)3888 1460 y Fo(\()p Fn(\021)t Fo(\))-24 1638 y(are)154 1498 y Fh(\022)299 1577 y Fo(1)269 1697 y Fi(\000)p Fn({)419 1498 y Fh(\023)540 1638 y Fo(and)745 1498 y Fh(\022)859 1577 y Fo(1)868 1697 y Fn({)950 1498 y Fh(\023)1023 1638 y Fo(,)52 b(while)c(for)f Fn(R)1609 1653 y Fl(h)1654 1638 y Fo(\()p Fn(\021)t Fo(\))h(one)g(can)f(c)m(ho)s (ose)2542 1498 y Fh(\022)2657 1577 y Fo(1)2657 1697 y(1)2747 1498 y Fh(\023)2868 1638 y Fo(and)3073 1498 y Fh(\022)3226 1577 y Fo(1)3188 1697 y Fi(\000)p Fo(1)3355 1498 y Fh(\023)3429 1638 y Fo(.)88 b(Comparing)-24 1816 y(with)40 b(equation)f(\(6\),)h (the)g(third)f(prop)s(ert)m(y)g(is)g(v)m(eri\014ed)i(if)e(the)g(top)g Fn(L)g Fo(comp)s(onen)m(ts)h(of)e(the)h(basis)h(v)m(ectors)-24 1937 y(comm)m(ute)c(with)f Fn(h)g Fo(and)g(are)f(constructed)i(with)f (the)g(in)m(v)m(erse)i Fn(d)2367 1901 y Fk(\003)2441 1937 y Fo(of)d Fn(d)g Fo(or)g(its)h(quasi-in)m(v)m(erse)i Fn(d)3526 1901 y Fl(t)3556 1937 y Fo(.)49 b(Indeed)36 b(it)-24 2057 y(can)26 b(b)s(e)g(v)m(eri\014ed)i(that)d Fn(dd)915 2021 y Fl(t)972 2057 y Fo(=)i Fi(S)34 b Fo(is)26 b(the)g(re\015ection)h(in)e Fm(C)2015 2021 y Fl(L)2093 2057 y Fo(sending)i(comp)s(onen)m(t)g Fn(l)g Fo(to)f(comp)s(onen)m(t)g (\()p Fn(L)8 b Fi(\000)g Fn(l)r Fo(\).)42 b(In)-24 2178 y(particular,)29 b(the)e(comp)s(onen)m(ts)1153 2138 y Fl(L)p 1153 2155 48 4 v 1159 2212 a Fq(2)1238 2178 y Fo(and)g Fn(L)g Fo(are)g(left)g(in)m(v)-5 b(arian)m(t)28 b(b)m(y)g Fi(S)7 b Fo(.)42 b(Moreo)m(v)m(er,)30 b Fi(S)k Fo(preserv)m(es)c(the)e(c)m(hannels,)-24 2366 y(namely)k([)p Fi(S)7 b Fn(;)17 b(d)p Fo(])28 b(=)g([)p Fi(S)7 b Fn(;)17 b(g)t Fo(])27 b(=)h([)p Fi(S)7 b Fn(;)17 b(h)p Fo(])28 b(=)g(0)i(and)1628 2225 y Fh(\022)1742 2305 y Fi(S)101 b Fo(0)1752 2425 y(0)92 b Fi(S)2003 2225 y Fh(\023)2106 2366 y Fo(comm)m(utes)33 b(with)e Fn(\031)2838 2381 y Fl(l)2864 2366 y Fo(.)43 b(Hence)32 b(one)e(is)h(led)g(to)f(de\014ne) -24 2539 y(the)j(follo)m(wing:)676 2813 y Fn(W)69 b Fo(=)968 2732 y Fh(\000)1013 2813 y Fo(w)1084 2771 y Fq(+)1084 2837 y(1)1143 2813 y Fn(;)17 b(:)g(:)g(:)f(;)h Fo(w)1432 2771 y Fq(+)1432 2840 y Fl(L)1491 2813 y Fn(;)g Fo(w)1605 2771 y Fk(\000)1605 2837 y Fq(1)1664 2813 y Fn(;)g(:)g(:)g(:)f(;)h Fo(w)1953 2771 y Fk(\000)1953 2840 y Fl(L)2012 2732 y Fh(\001)2113 2813 y Fo(=)2296 2745 y(1)p 2254 2790 132 4 v 2254 2810 a Fi(p)p 2337 2810 49 4 v 82 x Fo(2)2440 2672 y Fh(\022)2643 2752 y Fn(d)2694 2715 y Fk(\003)2953 2752 y Fn(d)3004 2715 y Fl(t)2555 2872 y Fi(\000)p Fn({g)k(d)2782 2836 y Fk(\003)2904 2872 y Fn({g)f(d)3053 2836 y Fl(t)3124 2672 y Fh(\023)3246 2813 y Fo(.)618 b(\(8\))-24 3039 y(It)33 b(can)g(readily)g(b)s(e)g(v)m(eri\014ed)i(that)675 3313 y Fn(W)781 3272 y Fk(\003)820 3313 y Fn(W)69 b Fo(=)55 b Fg(1)33 b Fo(,)227 b Fn(W)1561 3272 y Fk(\003)1617 3313 y Fn(J)26 b(W)69 b Fo(=)2008 3246 y Fn({)p 1999 3291 V 1999 3382 a Fo(2)2074 3173 y Fh(\022)2262 3252 y Fn(g)25 b Fo(+)d Fn(g)2483 3216 y Fk(\003)2678 3252 y Fo(\()p Fn(g)2767 3216 y Fk(\003)2827 3252 y Fi(\000)h Fn(g)t Fo(\))p Fi(S)2189 3373 y Fo(\()p Fn(g)i Fi(\000)e Fn(g)2450 3337 y Fk(\003)2489 3373 y Fo(\))p Fi(S)124 b(\000)p Fn(g)26 b Fi(\000)c Fn(g)3011 3337 y Fk(\003)3124 3173 y Fh(\023)3247 3313 y Fo(.)617 b(\(9\))-24 3553 y(Hence)28 b(the)f(v)m(ectors)h Fi(j)p Fo(w)846 3511 y Fq(+)846 3577 y(1)905 3553 y Fi(i)p Fn(;)17 b(:)g(:)g(:)e(;)i Fi(j)p Fo(w)1260 3511 y Fq(+)1260 3580 y Fl(L)1319 3553 y Fi(i)p Fn(;)g Fi(j)p Fo(w)1500 3511 y Fk(\000)1500 3577 y Fq(1)1559 3553 y Fi(i)p Fn(;)g(:)g(:)g(:)e(;)i Fi(j)p Fo(w)1914 3511 y Fk(\000)1914 3580 y Fl(L)1973 3553 y Fi(i)26 b Fo(form)h(an)f(orthonormal)g(basis)i(of)e Fm(C)3355 3517 y Fq(2)p Fl(L)3443 3553 y Fo(.)41 b(Let)27 b(us)g(write)-24 3673 y(out)37 b(examples)i(for)d(a)h(simple)h(and)f(a) g(double)g(eigen)m(v)-5 b(alue)39 b(\()p Fn(\026)2292 3688 y Fl(L)2380 3673 y Fo(and)e Fn(\026)2633 3688 y Fl(l)2659 3673 y Fo(,)h(1)d Fi(\024)g Fn(l)j(<)3108 3634 y Fl(L)p 3108 3650 48 4 v 3114 3707 a Fq(2)3165 3673 y Fo(,)g(resp)s(ectiv)m(ely\))i(more)-24 3793 y(explicitly:)127 4153 y Fi(j)p Fo(w)225 4112 y Fq(+)225 4180 y Fl(L)284 4153 y Fi(i)55 b Fo(=)561 4086 y(1)p 519 4130 132 4 v 519 4150 a Fi(p)p 602 4150 49 4 v 83 x Fo(2)694 4153 y(\()p Fi(j)p Fn(e)805 4168 y Fl(L)857 4153 y Fi(i)22 b(\000)g Fn({g)1095 4168 y Fl(L)1164 4153 y Fi(j)p Fn(e)1237 4168 y Fq(2)p Fl(L)1324 4153 y Fi(i)p Fo(\))49 b(,)228 b Fi(j)p Fo(w)1803 4112 y Fq(+)1803 4181 y Fl(l)1862 4153 y Fi(i)55 b Fo(=)2097 4086 y(1)p 2097 4130 V 2097 4222 a(2)2173 4153 y(\()p Fn({)17 b Fi(j)p Fn(e)2332 4168 y Fl(l)2357 4153 y Fi(i)22 b Fo(+)g Fi(j)p Fn(e)2589 4168 y Fl(L)p Fk(\000)p Fl(l)2718 4153 y Fi(i)g Fo(+)g Fn(g)2924 4168 y Fl(l)2966 4153 y Fi(j)p Fn(e)3039 4168 y Fl(L)p Fq(+)p Fl(l)3168 4153 y Fi(i)g(\000)g Fn({g)3406 4168 y Fl(l)3449 4153 y Fi(j)p Fn(e)3522 4168 y Fq(2)p Fl(L)p Fk(\000)p Fl(l)3686 4153 y Fi(i)p Fo(\))32 b(.)-24 4435 y(Note)47 b(that)f(for)g(an)g(elliptic)i(eigen)m(v)-5 b(alue)48 b Fn(\026)1636 4450 y Fl(l)1661 4435 y Fo(,)i Fi(j)p Fo(w)1836 4394 y Fk(\000)1836 4463 y Fl(l)1895 4435 y Fi(i)c Fo(is)h(the)g(complex)h(conjugate)f(of)e Fi(j)p Fo(w)3349 4394 y Fq(+)3349 4463 y Fl(l)3408 4435 y Fi(i)p Fo(,)k(but)e(if)f Fn(\026)3878 4450 y Fl(l)3950 4435 y Fo(is)-24 4556 y(h)m(yp)s(erb)s(olic,)31 b(this)e(only)g(holds)g (for)f(the)h(\014rst)f Fn(L)h Fo(comp)s(onen)m(ts.)44 b(As)29 b(requested,)i(one)e(reads)g(o\013)f(from)g(\(9\))g(that)1024 4890 y Fn(J)d Fi(j)p Fo(w)1201 4849 y Fk(\006)1201 4918 y Fl(l)1261 4890 y Fi(i)55 b Fo(=)1486 4686 y Fh(8)1486 4775 y(<)1486 4955 y(:)1635 4769 y Fi(\006)17 b Fn({)p Fi(j)p Fo(w)1859 4727 y Fk(\006)1859 4796 y Fl(l)1918 4769 y Fi(i)269 b Fn(\026)2285 4784 y Fl(l)2371 4769 y Fo(elliptic)33 b(,)1616 5009 y Fi(\007)17 b(j)p Fo(w)1808 4968 y Fk(\007)1808 5037 y Fl(L)p Fk(\000)p Fl(l)1937 5009 y Fi(i)249 b Fn(\026)2284 5024 y Fl(l)2370 5009 y Fo(h)m(yp)s(erb)s(olic)34 b(.)3842 4890 y(\(10\))-24 5183 y(Also)f(the)g(second)h(required)g(prop)s(ert)m(y)g(follo)m(ws)f (from)f(\(8\):)1972 5564 y(8)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 1467 291 a Fn(R)18 b Fi(j)p Fo(w)1657 250 y Fk(\006)1657 319 y Fl(l)1716 291 y Fi(i)55 b Fo(=)g Fn(e)1986 250 y Fk(\006)p Fl({)p 2076 212 58 3 v 12 w(g)2110 262 y Ff(l)2146 250 y Fl(\021)2181 262 y Ff(l)2226 291 y Fi(j)p Fo(w)2324 250 y Fk(\006)2324 319 y Fl(l)2383 291 y Fi(i)33 b Fo(.)-24 462 y(Finally)-8 b(,)191 796 y Fn(\031)250 755 y Fk(\006)246 820 y Fq(0)365 796 y Fo(=)55 b Fi(j)p Fo(w)594 755 y Fk(\006)594 823 y Fl(L)653 796 y Fi(ih)p Fo(w)801 755 y Fk(\006)801 823 y Fl(L)860 796 y Fi(j)32 b Fo(,)228 b Fn(\031)1234 755 y Fk(\006)1230 824 y Fl(l)1348 796 y Fo(=)55 b Fi(j)p Fo(w)1578 755 y Fk(\006)1578 824 y Fl(l)1637 796 y Fi(ih)p Fo(w)1784 755 y Fk(\006)1784 824 y Fl(l)1844 796 y Fi(j)21 b Fo(+)h Fi(j)p Fo(w)2089 755 y Fk(\006)2089 824 y Fl(L)p Fk(\000)p Fl(l)2218 796 y Fi(ih)p Fo(w)2366 755 y Fk(\006)2366 824 y Fl(L)p Fk(\000)p Fl(l)2495 796 y Fi(j)32 b Fo(,)228 b Fn(\031)2869 755 y Fk(\006)2875 811 y Ff(L)p 2875 823 43 3 v 2881 865 a Fb(2)2987 796 y Fo(=)55 b Fi(j)p Fo(w)3216 755 y Fk(\006)3226 811 y Ff(L)p 3226 823 V 3232 865 a Fb(2)3283 796 y Fi(ih)p Fo(w)3431 755 y Fk(\006)3440 811 y Ff(L)p 3440 823 V 3446 865 a Fb(2)3497 796 y Fi(j)32 b Fo(.)258 b(\(11\))-24 1094 y Fc(2.5)136 b(Matrix)45 b(elemen)l(ts)i(of)e(the)g(random)g(p)t(erturbation)-24 1279 y Fo(In)d(this)g(section,)i(w)m(e)e(sho)m(w)g(that)f(the)h(third)f (desired)i(prop)s(ert)m(y)e(of)g(the)h(basis)g(stated)f(in)h(Section)g (2.4)e(is)-24 1399 y(ful\014lled)f(and)e(then)h(exploit)g(it)g(in)f (order)h(to)f(calculate)h(the)g(matrix)g(elemen)m(ts)h(of)e(the)h(p)s (erturbation)g(and)-24 1520 y(some)h(exp)s(ectation)g(v)-5 b(alues)39 b(thereof.)59 b(T)-8 b(aking)39 b(in)m(to)f(accoun)m(t)g (that)g([)p Fn(d;)17 b(h)p Fo(])36 b(=)h(0,)i(it)f(no)m(w)g(follo)m(ws) h(from)e(\(6\))-24 1640 y(and)c(\(8\))f(that)17 2017 y Fn(W)123 1976 y Fk(\003)162 2017 y Fn(P)14 b(W)68 b Fo(=)549 1950 y Fn({)p 540 1994 49 4 v 540 2085 a Fo(2)632 1877 y Fh(\022)820 1956 y Fn(g)t(h)942 1931 y Fo(^)927 1956 y Fn(V)21 b(h)228 b(g)t(h)1411 1931 y Fo(^)1396 1956 y Fn(V)21 b(h)p Fi(S)747 2086 y(\000S)7 b Fn(g)t(h)1013 2061 y Fo(^)998 2086 y Fn(V)22 b(h)84 b Fi(\000S)7 b Fn(g)t(h)1483 2061 y Fo(^)1468 2086 y Fn(V)22 b(h)p Fi(S)1712 1877 y Fh(\023)1835 2017 y Fo(,)227 b Fn(W)2195 1976 y Fk(\003)2235 2017 y Fn(P)2312 1976 y Fk(\003)2350 2017 y Fn(W)69 b Fo(=)2661 1950 y Fn({)p 2653 1994 V 2653 2085 a Fo(2)2745 1877 y Fh(\022)2893 1956 y Fi(\000)p Fn(h)3041 1931 y Fo(^)3026 1956 y Fn(V)22 b(hg)3212 1920 y Fk(\003)3402 1956 y Fn(h)3473 1931 y Fo(^)3458 1956 y Fn(V)g(hg)3644 1920 y Fk(\003)3683 1956 y Fi(S)2859 2086 y(\000S)7 b Fn(h)3074 2061 y Fo(^)3059 2086 y Fn(V)23 b(hg)3246 2050 y Fk(\003)3368 2086 y Fi(S)7 b Fn(h)3506 2061 y Fo(^)3491 2086 y Fn(V)23 b(hg)3678 2050 y Fk(\003)3717 2086 y Fi(S)3826 1877 y Fh(\023)3948 2017 y Fo(.)3842 2206 y(\(12\))-24 2326 y(Hence)34 b(one)f(can)g(read)g(o\013,)f(for)g (t)m(w)m(o)h(signs)h Fn(\034)6 b(;)17 b(\033)t Fo(,)129 2659 y Fi(h)p Fo(w)239 2618 y Fl(\034)239 2684 y(l)282 2659 y Fi(j)p Fn(P)d Fi(j)p Fo(w)484 2618 y Fl(\033)484 2684 y(k)531 2659 y Fi(i)55 b Fo(=)g Fn(\034)845 2592 y({)p 836 2637 V 836 2728 a Fo(2)912 2659 y Fn(g)959 2674 y Fl(l)1001 2659 y Fn(h)1057 2674 y Fl(l)1083 2659 y Fn(h)1139 2674 y Fl(k)1202 2659 y Fo(^)-53 b Fn(v)t Fo(\()p Fn(\033)t(k)25 b Fi(\000)e Fn(\034)11 b(l)r Fo(\))33 b(,)228 b Fi(h)p Fo(w)2041 2618 y Fl(\034)2041 2684 y(l)2084 2659 y Fi(j)p Fn(P)2189 2618 y Fk(\003)2228 2659 y Fi(j)p Fo(w)2326 2618 y Fl(\033)2326 2684 y(k)2373 2659 y Fi(i)55 b Fo(=)g Fi(\000)p Fn(\033)2770 2592 y({)p 2761 2637 V 2761 2728 a Fo(2)p 2837 2605 90 4 v 2837 2659 a Fn(g)2884 2674 y Fl(k)2942 2659 y Fn(h)2998 2674 y Fl(l)3025 2659 y Fn(h)3081 2674 y Fl(k)3144 2659 y Fo(^)-53 b Fn(v)t Fo(\()p Fn(\033)t(k)25 b Fi(\000)e Fn(\034)11 b(l)r Fo(\))33 b(.)196 b(\(13\))-24 2898 y(Let)33 b(us)g(collect)h(some)f(useful)h (iden)m(tities.)-24 3118 y Fg(Lemma)39 b(1)49 b Fe(L)-5 b(et)36 b Fn(w)720 3077 y Fk(\006)717 3144 y Fl(j)806 3118 y Fi(2)28 b Fm(C)972 3082 y Fq(2)p Fl(L)1094 3118 y Fe(b)-5 b(e)35 b(unit)g(ve)-5 b(ctors)34 b(satisfying)h Fn(\031)2238 3077 y Fk(\006)2234 3144 y Fl(j)2297 3118 y Fn(w)2370 3077 y Fk(\006)2367 3144 y Fl(j)2456 3118 y Fo(=)27 b Fn(w)2632 3077 y Fk(\006)2629 3144 y Fl(j)2691 3118 y Fe(.)44 b(L)-5 b(et)2956 3093 y Fo(~)2934 3118 y Fn(P)41 b Fo(=)27 b Fn(P)36 b Fo(+)22 b Fn(P)3415 3082 y Fk(\003)3454 3118 y Fe(.)-24 3262 y Fo(\(i\))35 b Fg(E)17 b Fi(h)p Fn(w)s Fi(j)p Fn(P)d Fi(j)p Fn(w)523 3226 y Fk(0)543 3262 y Fi(i)27 b Fo(=)h(0)35 b Fe(for)f(any)h Fn(w)s(;)17 b(w)1329 3226 y Fk(0)1379 3262 y Fi(2)28 b Fm(C)1545 3226 y Fq(2)p Fl(L)1632 3262 y Fe(.)-24 3406 y Fo(\(ii\))35 b Fe(If)g(either)f Fn(k)d Fi(6)p Fo(=)d Fn(l)37 b Fe(or)e Fn(\033)c Fi(6)p Fo(=)d Fn(\034)11 b Fe(,)35 b(then)932 3667 y Fg(E)17 b Fi(h)p Fn(w)1135 3626 y Fl(\034)1132 3691 y(l)1177 3667 y Fi(j)g Fn(P)29 b Fi(j)p Fn(w)1415 3626 y Fl(\033)1412 3691 y(k)1461 3667 y Fi(i)e(h)p Fn(w)1639 3626 y Fl(\033)1636 3691 y(k)1685 3667 y Fi(j)17 b Fn(P)30 b Fi(j)p Fn(w)1924 3626 y Fl(\034)1921 3691 y(l)1966 3667 y Fi(i)55 b Fo(=)g Fi(\000)p Fn(\034)11 b(\033)2441 3599 y Fo(1)p 2408 3644 116 4 v 2408 3735 a(4)p Fn(L)2550 3667 y(g)2597 3682 y Fl(l)2639 3667 y Fn(g)2686 3682 y Fl(k)2756 3667 y Fn(h)2812 3626 y Fq(2)2812 3691 y Fl(l)2851 3667 y Fn(h)2907 3626 y Fq(2)2907 3691 y Fl(k)2985 3667 y Fe(.)-24 3893 y Fo(\(iii\))35 b Fe(If)g(either)f Fn(k)d Fi(6)p Fo(=)d Fn(l)37 b Fe(or)e Fn(\033)c Fi(6)p Fo(=)d Fn(\034)11 b Fe(,)35 b(then)1113 4153 y Fg(E)17 b Fi(h)p Fn(w)1316 4112 y Fl(\034)1313 4178 y(l)1358 4153 y Fi(j)g Fn(P)1480 4112 y Fk(\003)1535 4153 y Fi(j)p Fn(w)1636 4112 y Fl(\033)1633 4178 y(k)1682 4153 y Fi(i)27 b(h)p Fn(w)1860 4112 y Fl(\033)1857 4178 y(k)1906 4153 y Fi(j)17 b Fn(P)29 b Fi(j)p Fn(w)2144 4112 y Fl(\034)2141 4178 y(l)2187 4153 y Fi(i)55 b Fo(=)2455 4086 y(1)p 2422 4130 V 2422 4221 a(4)p Fn(L)2575 4153 y(h)2631 4112 y Fq(2)2631 4178 y Fl(l)2670 4153 y Fn(h)2726 4112 y Fq(2)2726 4178 y Fl(k)2804 4153 y Fe(.)-24 4396 y Fo(\(iv\))36 b Fe(L)-5 b(et)35 b(channels)e Fn(l)r(;)17 b(k)38 b Fe(b)-5 b(e)35 b(el)5 b(liptic.)44 b(Then)34 b Fn(\031)1690 4359 y Fl(\033)1686 4421 y(l)1759 4370 y Fo(~)1737 4396 y Fn(P)13 b(\031)1872 4359 y Fl(\033)1868 4421 y(k)1947 4396 y Fo(=)27 b(0)35 b Fe(and)f Fn(\031)2382 4359 y Fl(\033)2378 4421 y(l)2451 4370 y Fo(~)2429 4396 y Fn(P)13 b(\031)2564 4354 y Fk(\000)p Fl(\033)2560 4423 y(k)2694 4396 y Fo(=)27 b(2)17 b Fn(\031)2922 4359 y Fl(\033)2918 4421 y(l)2968 4396 y Fn(P)d(\031)3104 4354 y Fk(\000)p Fl(\033)3100 4423 y(k)3206 4396 y Fe(.)44 b(Mor)-5 b(e)g(over,)411 4656 y Fg(E)17 b Fi(h)p Fn(w)614 4615 y Fl(\033)611 4680 y(l)659 4656 y Fi(j)725 4631 y Fo(~)704 4656 y Fn(P)29 b Fi(j)p Fn(w)897 4615 y Fk(\000)p Fl(\033)894 4684 y(k)998 4656 y Fi(i)f(h)p Fn(w)1177 4615 y Fk(\000)p Fl(\033)1174 4684 y(k)1278 4656 y Fi(j)1344 4631 y Fo(~)1323 4656 y Fn(P)h Fi(j)p Fn(w)1516 4615 y Fl(\033)1513 4680 y(l)1562 4656 y Fi(i)55 b Fo(=)1806 4588 y(1)p 1797 4633 67 4 v 1797 4724 a Fn(L)1901 4656 y(h)1957 4615 y Fq(2)1957 4680 y Fl(l)1997 4656 y Fn(h)2053 4615 y Fq(2)2053 4680 y Fl(k)2151 4656 y Fo(=)g Fg(E)17 b Fi(h)p Fn(w)2485 4615 y Fl(\033)2482 4680 y(l)2530 4656 y Fi(j)2596 4631 y Fo(~)2575 4656 y Fn(P)30 b Fi(j)p Fn(w)2769 4615 y Fk(\000)p Fl(\033)2766 4684 y(k)2869 4656 y Fi(i)e(h)p 2975 4581 120 4 v Fn(w)3048 4621 y Fl(\033)3045 4684 y(l)3094 4656 y Fi(j)3160 4631 y Fo(~)3139 4656 y Fn(P)h Fi(j)p 3259 4562 175 4 v Fn(w)3332 4615 y Fk(\000)p 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Fo(^)-52 b Fn(v)s Fo(\()p Fi(\000)p Fn(\033)t(k)9 b Fi(\000)d Fn(\034)11 b(l)r Fo(\))23 b(+)p 2517 1192 68 4 v 22 w Fn(b)2558 1289 y Fl(l)2585 1274 y Fn(a)2636 1289 y Fl(k)2682 1274 y Fo(^)-52 b Fn(v)s Fo(\()-17 b Fn(\033)t(k)9 b Fo(+)d Fn(\034)11 b(l)r Fo(\))23 b(+)p 3194 1192 V 22 w Fn(b)3235 1289 y Fl(l)3261 1274 y Fn(b)3302 1289 y Fl(k)3349 1274 y Fo(^)-53 b Fn(v)t Fo(\()p Fi(\000)p Fn(\033)t(k)9 b Fo(+)d Fn(\034)11 b(l)r Fo(\))3834 1193 y Fh(\001)3919 1274 y Fi(\001)968 1431 y(\001)1013 1350 y Fh(\000)1058 1431 y Fn(a)1109 1446 y Fl(l)p 1135 1376 94 4 v 1135 1431 a Fn(a)1186 1446 y Fl(k)1233 1431 y Fo(^)-53 b Fn(v)t Fo(\()p Fi(\000)p Fn(\033)t(k)9 b Fo(+)d Fn(\034)11 b(l)r Fo(\))22 b(+)g Fn(a)1889 1446 y Fl(l)p 1915 1350 85 4 v 1915 1431 a Fn(b)1956 1446 y Fl(k)2003 1431 y Fo(^)-53 b Fn(v)t Fo(\()p Fi(\000)-17 b Fn(\033)t(k)9 b Fi(\000)d Fn(\034)11 b(l)r Fo(\))23 b(+)f Fn(b)2634 1446 y Fl(l)p 2661 1376 94 4 v 2661 1431 a Fn(a)2712 1446 y Fl(k)2758 1431 y Fo(^)-52 b Fn(v)s Fo(\()p 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Fo(=)g(1,)h(it)g(follo)m(ws)h(from)f(\(12\))g(that)g Fn(W)2132 2023 y Fk(\003)2193 2034 y Fo(~)2171 2059 y Fn(P)13 b(W)42 b Fo(=)27 b Fn(W)2590 2023 y Fk(\003)2629 2059 y Fo(\()p Fn(P)i Fo(+)15 b Fn(P)2927 2023 y Fk(\003)2966 2059 y Fo(\))p Fn(W)43 b Fo(has)29 b(only)h(o\013-diagonal)-24 2180 y(en)m(tries)37 b(in)d(\()p Fn(\031)499 2195 y Fl(l)549 2180 y Fo(+)23 b Fn(\031)703 2195 y Fl(k)746 2180 y Fo(\))p Fm(C)856 2143 y Fq(2)p Fl(L)978 2180 y Fo(equal)36 b(to)e(t)m(wice)i (those)f(of)f Fn(W)2089 2143 y Fk(\003)2128 2180 y Fn(P)14 b(W)g Fo(.)49 b(The)35 b(\014rst)h(equalit)m(y)g(no)m(w)f(follo)m(ws)g (directly)-24 2300 y(from)e(\(ii\),)f(while)i(the)f(second)h(one)f(is)g (c)m(hec)m(k)m(ed)j(similarly)-8 b(.)123 2444 y(\(v\))33 b(One)g(\014rst)g(v)m(eri\014es)h(as)f(ab)s(o)m(v)m(e)g(that)g(\(also)f (for)g Fn(l)e Fo(=)e Fn(k)35 b Fo(and)e Fn(\033)f Fo(=)27 b Fi(\006)p Fo(\))1142 2704 y Fg(E)17 b Fi(h)p Fn(w)1345 2663 y Fl(\033)1342 2729 y(l)1390 2704 y Fi(j)p Fn(P)1495 2663 y Fk(\003)1533 2704 y Fi(j)p Fo(w)1631 2663 y Fk(\006)1631 2732 y Fl(k)1690 2704 y Fi(ih)p Fo(w)1838 2663 y Fk(\006)1838 2732 y Fl(k)1897 2704 y Fi(j)p Fn(P)d Fi(j)p Fn(w)2103 2663 y Fl(\033)2100 2729 y(l)2149 2704 y Fi(i)55 b Fo(=)2417 2637 y(1)p 2384 2681 116 4 v 2384 2773 a(4)p Fn(L)2537 2704 y(h)2593 2663 y Fq(2)2593 2729 y Fl(k)2652 2704 y Fn(h)2708 2663 y Fq(2)2708 2729 y Fl(l)2780 2704 y Fo(.)-24 2910 y(Hence)34 b(the)f(claim)g(follo)m(ws)h(b)m(y)f (inserting)h(an)e(iden)m(tit)m(y)j(\(7\))d(and)g(summing)i(o)m(v)m(er)g Fn(k)s Fo(.)839 b Fa(2)-24 3242 y Fp(3)161 b(Random)55 b(dynamics)e(of)h(symplectic)e(frames)-24 3490 y Fc(3.1)136 b(Symplectic)45 b(frames)h(and)e(isotropic)i(manifolds)-24 3675 y Fo(The)34 b(space)g Fi(F)510 3690 y Fl(p)581 3675 y Fo(of)e(symplectic)k Fn(p)p Fo(-frames,)c Fn(p)c Fo(=)g(1)p Fn(;)17 b(:)g(:)g(:)e(;)i(L)p Fo(,)33 b(is)g(de\014ned)h(b)m(y)275 4016 y Fi(F)347 4031 y Fl(p)441 4016 y Fo(=)573 3935 y Fh(\010)631 4016 y Fo(\()p Fn(u)725 4031 y Fq(1)764 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Fn(u)2268 1308 y Fq(1)2307 1293 y Fo(\()p Fn(n)p Fo(\))p Fi(ij)2508 1252 y Fq(2)2569 1293 y Fo(+)22 b(\()p Fn(e)2750 1252 y Fk(\000)p Fq(2)p Fl(\021)2875 1264 y Ff(l)2926 1293 y Fi(\000)h Fo(1\))p Fi(jh)p Fo(w)3249 1252 y Fk(\000)3249 1321 y Fl(l)3308 1293 y Fi(j)p Fn(u)3392 1308 y Fq(1)3431 1293 y Fo(\()p Fn(n)p Fo(\))p Fi(ij)3632 1252 y Fq(2)3703 1293 y Fo(,)-24 1592 y(where)36 b(as)f(b)s(efore)g(w)m(e)g(iden)m(ti\014ed)i(indices)f Fn(L)10 b Fh(b)-65 b Fo(=)q(0)34 b(and)h(used)g(that)g Fn(u)2484 1607 y Fq(1)2523 1592 y Fo(\()p Fn(n)p Fo(\))f(is)i (normalized.)50 b(The)36 b(next)f(aim)g(is)-24 1712 y(to)d(pro)m(v)m(e) h(an)f(upp)s(er)g(b)s(ound)h(on)e(this)i(and)f(therefore)g(let)h(us)f (\014rst)h(note)f(that)f Fn(e)2929 1676 y Fk(\000)p Fq(2)p Fl(\021)3054 1688 y Ff(l)3104 1712 y Fi(\000)22 b Fo(1)27 b Fn(<)h Fo(0)j(so)h(that)g(those)-24 1833 y(terms)g(can)g(b)s(e)f (discarded.)44 b(F)-8 b(or)31 b(the)g(remainder,)h(starting)g(from)e (the)i(smallest)g(factor)f Fn(e)3306 1797 y Fq(2)p Fl(\021)3376 1809 y Ff(l)3424 1833 y Fi(\000)19 b Fo(1)31 b(and)g(going)-24 1953 y(iterativ)m(ely)k(to)d(the)h(largest,)g(one)g(gets)g(using)g(eac) m(h)h(time)f Fi(jh)p Fo(w)2283 1912 y Fq(+)2283 1981 y Fl(l)2342 1953 y Fi(j)p Fn(u)2426 1968 y Fq(1)2464 1953 y Fo(\()p Fn(n)p Fo(\))p Fi(ij)2665 1917 y Fq(2)2732 1953 y Fi(\024)28 b Fo(1)22 b Fi(\000)3008 1878 y Fh(P)3113 1905 y Fl(l)q Fk(\000)p Fq(1)3113 1982 y Fl(k)r Fq(=0)3262 1953 y Fi(jh)p Fo(w)3399 1912 y Fq(+)3399 1981 y Fl(k)3458 1953 y Fi(j)p Fn(u)3542 1968 y Fq(1)3581 1953 y Fo(\()p Fn(n)p Fo(\))p Fi(ij)3782 1917 y Fq(2)3821 1953 y Fo(,)661 2353 y Fi(h)p Fn(u)756 2368 y Fq(1)795 2353 y Fo(\()p Fn(n)p Fo(\))p Fi(j)p Fn(R)1032 2312 y Fk(\003)1071 2353 y Fn(R)q Fi(j)p Fn(u)1230 2368 y Fq(1)1269 2353 y Fo(\()p Fn(n)p Fo(\))p Fi(i)83 b(\024)g Fn(e)1730 2312 y Fq(2)p Fl(\021)1800 2323 y Ff(L)1842 2340 y(h)1914 2353 y Fo(+)2012 2225 y Fl(L)2060 2237 y Ff(h)2100 2225 y Fk(\000)p Fq(1)2029 2258 y Fh(X)2045 2470 y Fl(l)q Fq(=0)2207 2353 y Fo(\()p Fn(e)2290 2312 y Fq(2)p Fl(\021)2360 2324 y Ff(l)2412 2353 y Fi(\000)22 b Fn(e)2556 2312 y Fq(2)p Fl(\021)2626 2323 y Ff(L)2668 2340 y(h)2718 2353 y Fo(\))17 b Fi(jh)p Fo(w)2909 2312 y Fq(+)2909 2380 y Fl(l)2968 2353 y Fi(j)p Fn(u)3052 2368 y Fq(1)3091 2353 y Fo(\()p Fn(n)p Fo(\))p Fi(ij)3292 2312 y Fq(2)3842 2588 y Fo(\(22\))1525 2733 y Fi(\024)83 b Fn(e)1730 2692 y Fq(2)p Fl(\021)1800 2701 y Fb(1)1862 2733 y Fo(+)22 b(\()p Fn(e)2043 2692 y Fq(2)p Fl(\021)2113 2703 y Ff(L)2187 2733 y Fi(\000)g Fn(e)2331 2692 y Fq(2)p Fl(\021)2401 2701 y Fb(1)2441 2733 y Fo(\))17 b Fi(jh)p Fo(w)2632 2692 y Fq(+)2632 2760 y Fl(L)2691 2733 y Fi(j)p Fn(u)2775 2748 y Fq(1)2814 2733 y Fo(\()p Fn(n)p Fo(\))p Fi(ij)3015 2692 y Fq(2)3086 2733 y Fo(.)-24 2942 y(Note)33 b(that)f Fn(\021)471 2957 y Fq(1)544 2942 y Fo(is)h(the)g(second)g(largest)g(h)m(yp)s(erb)s(olic)h(angle)f (satisfying)h Fn(\021)2654 2957 y Fq(1)2721 2942 y Fn(<)27 b(\021)2872 2957 y Fl(L)2925 2942 y Fo(.)43 b(Replacing)33 b(in)g(the)g(ab)s(o)m(v)m(e,)613 3224 y Fi(jh)p Fo(w)750 3183 y Fq(+)750 3251 y Fl(L)809 3224 y Fi(j)p Fn(u)893 3239 y Fq(1)932 3224 y Fo(\()p Fn(n)22 b Fo(+)g(1\))p Fi(ij)1302 3183 y Fq(2)1396 3224 y Fi(\025)1747 3157 y Fn(e)1792 3121 y Fq(2\()p Fl(\021)1889 3132 y Ff(L)1936 3121 y Fk(\000)p Fl(\021)2026 3130 y Fb(1)2061 3121 y Fq(\))2109 3157 y Fi(jh)p Fo(w)2246 3115 y Fq(+)2246 3184 y Fl(L)2305 3157 y Fi(j)p Fn(u)2389 3172 y Fq(1)2428 3157 y Fo(\()p Fn(n)p Fo(\))p Fi(ij)2629 3121 y Fq(2)p 1539 3201 1337 4 v 1539 3294 a Fo(1)g(+)g(\()p Fn(e)1791 3266 y Fq(2\()p Fl(\021)1888 3277 y Ff(L)1935 3266 y Fk(\000)p Fl(\021)2025 3275 y Fb(1)2060 3266 y Fq(\))2114 3294 y Fi(\000)h Fo(1\))17 b Fi(jh)p Fo(w)2454 3253 y Fq(+)2454 3322 y Fl(L)2513 3294 y Fi(j)p Fn(u)2597 3309 y Fq(1)2636 3294 y Fo(\()p Fn(n)p Fo(\))p Fi(ij)2837 3266 y Fq(2)2935 3224 y Fo(+)50 b Fi(O)s Fo(\()p Fn(\025)p Fo(\))32 b(.)-24 3460 y(Th)m(us,)i(up)f(to)e(an)h(error,)h Fi(jh)p Fo(w)1046 3419 y Fq(+)1046 3487 y Fl(L)1105 3460 y Fi(j)p Fn(u)1189 3475 y Fq(1)1228 3460 y Fo(\()p Fn(n)21 b Fo(+)g(1\))p Fi(ij)1596 3424 y Fq(2)1667 3460 y Fo(is)33 b(larger)f(than)g(the)g(image)h(of)e Fn(x)d Fo(=)g Fi(jh)p Fo(w)3151 3419 y Fq(+)3151 3487 y Fl(L)3210 3460 y Fi(j)p Fn(u)3294 3475 y Fq(1)3332 3460 y Fo(\()p Fn(n)p Fo(\))p Fi(ij)3533 3424 y Fq(2)3604 3460 y Fo(under)33 b(the)-24 3581 y(function)h Fn(f)407 3596 y Fl(a)449 3581 y Fo(\()p Fn(x)p Fo(\))c(=)842 3541 y Fl(ax)p 725 3558 313 4 v 725 3615 a Fq(1+\()p Fl(a)p Fk(\000)p Fq(1\))p Fl(x)1081 3581 y Fo(where)35 b Fn(a)29 b Fo(=)g Fn(e)1594 3544 y Fq(2\()p Fl(\021)1691 3555 y Ff(L)1739 3544 y Fk(\000)p Fl(\021)1829 3553 y Fb(1)1864 3544 y Fq(\))1925 3581 y Fn(>)g Fo(1.)46 b(As)34 b Fn(n)g Fo(gro)m(ws,)g(this)g(pro)s(cedure)h (is)f(then)g(iterated,)-24 3701 y(giving)39 b(rise)g(to)f(a)g(discrete) i(time)e(dynamics)i(through)f(successiv)m(e)j(application)c(of)g Fn(f)3203 3716 y Fl(a)3245 3701 y Fo(.)60 b(The)39 b(function)g Fn(f)3974 3716 y Fl(a)-24 3821 y Fo(has)j(t)m(w)m(o)h(\014xed)f(p)s (oin)m(ts)g(in)g([0)p Fn(;)17 b Fo(1],)43 b(an)f(unstable)h(one)e(at)h (0)f(and)g(a)h(stable)g(one)g(at)f(1.)70 b(Either)42 b(the)g(initial)-24 3942 y(condition)37 b(is)g(already)f(a)m(w)m(a)m(y) i(from)d(the)i(unstable)g(\014xed)g(p)s(oin)m(t)g(or)e(the)i(random)f (p)s(erturbation)g(leads)h(the)-24 4062 y(discrete)c(time)f(dynamics)h (to)e(lea)m(v)m(e)i(it)e(\(only)g(with)h(exp)s(onen)m(tially)h(small)f (probabilit)m(y)g(one)g(remains)g(in)f(its)-24 4183 y(neigh)m(b)s(orho) s(o)s(d,)f(as)f(an)f(elemen)m(tary)j(argumen)m(t)f(sho)m(ws\).)43 b(As)30 b(it)e(only)i(tak)m(es)g(a)e(\014nite)i(n)m(um)m(b)s(er)g(of)e (iterations)-24 4303 y(to)38 b(get)g(within)h(the)g(neigh)m(b)s(orho)s (o)s(d)e(of)h(the)g(stable)h(\014xed)h(p)s(oin)m(t)e(and)g(the)g (\(random\))g(p)s(erturbation)h(is)f(of)-24 4423 y(order)33 b Fi(O)s Fo(\()p Fn(\025)p Fo(\),)g(one)f(can)h(conclude)h(that,)f(for) f Fn(n)h Fo(large)f(enough,)1379 4664 y Fi(jh)p Fo(w)1516 4623 y Fq(+)1516 4691 y Fl(L)1575 4664 y Fi(j)p Fn(u)1659 4679 y Fq(1)1698 4664 y Fo(\()p Fn(n)p Fo(\))p Fi(ij)1899 4623 y Fq(2)1993 4664 y Fo(=)55 b(1)22 b Fi(\000)h(O)s Fo(\()p Fn(\025)p Fo(\))32 b(.)-24 4834 y(When)42 b(this)e(holds,)j(ho) m(w)m(ev)m(er,)i(the)c(random)f(p)s(erturbation)h(cannot)f(b)s(e)h (linear)f(in)h Fn(\025)f Fo(an)m(ymore,)j(b)s(ecause)-24 4954 y(the)37 b Fi(O)s Fo(\()p Fn(\025)p Fo(\)-term)f(w)m(ould)h(not)f (ha)m(v)m(e)h(de\014nite)g(sign)g(and)f(hence)h(violate)g Fi(jh)p Fo(w)2805 4913 y Fq(+)2805 4981 y Fl(L)2864 4954 y Fi(j)p Fn(u)2948 4969 y Fq(1)2986 4954 y Fo(\()p Fn(n)p Fo(\))p Fi(ij)3187 4918 y Fq(2)3260 4954 y Fi(\024)d Fo(1.)54 b(Indeed,)38 b(it)e(is)-24 5074 y(elemen)m(tary)41 b(to)e(v)m(eryfy)i(also)e(algebraically)h(that)e(the)i(p)s(erturbativ)m (e)g(terms)g(linear)f(in)h Fn(\025)e Fo(v)-5 b(anish)40 b(in)f(\(21\))-24 5195 y(when)34 b(one)g(already)f(kno)m(ws)i Fi(h)p Fo(w)1158 5153 y Fq(+)1158 5222 y Fl(L)1217 5195 y Fi(j)p Fn(u)1301 5210 y Fq(1)1340 5195 y Fo(\()p Fn(n)p Fo(\))p Fi(i)28 b Fo(=)g(1)22 b Fi(\000)h(O)s Fo(\()p Fn(\025)p Fo(\).)45 b(Hence)34 b(one)f(can)g(rep)s(eat)h(the)f(ab)s(o)m (v)m(e)h(argumen)m(t)f(in)-24 5315 y(the)e(neigh)m(b)s(orho)s(o)s(d)f (of)g(the)h(stable)g(\014xed)g(p)s(oin)m(t)g(1,)f(but)h(based)g(on)f (\(21\))g(with)g(an)h(error)f(term)h Fi(O)s Fo(\()p Fn(\025)3679 5279 y Fq(2)3718 5315 y Fo(\).)43 b(This)1947 5564 y(14)p eop end %%Page: 15 15 TeXDict begin 15 14 bop -24 171 a Fo(implies)36 b(the)e(\014rst)g (claim.)48 b(Moreo)m(v)m(er,)36 b(due)f(to)e(normalization,)i Fi(jh)p Fo(w)2529 129 y Fq(+)2529 198 y Fl(l)2588 171 y Fi(j)p Fn(u)2672 186 y Fq(1)2711 171 y Fo(\()p Fn(n)p Fo(\))p Fi(ij)2912 134 y Fq(2)2980 171 y Fo(=)30 b Fi(O)s Fo(\()p Fn(\025)3263 134 y Fq(2)3302 171 y Fo(\))k(for)g(all)f Fn(l)g Fi(6)p Fo(=)c Fn(L)34 b Fo(as)-24 291 y(w)m(ell)g(as)f Fi(jh)p Fo(w)431 250 y Fk(\000)431 319 y Fl(l)490 291 y Fi(j)p Fn(u)574 306 y Fq(1)612 291 y Fo(\()p Fn(n)p Fo(\))p Fi(ij)813 255 y Fq(2)880 291 y Fo(=)28 b Fi(O)s Fo(\()p Fn(\025)1161 255 y Fq(2)1200 291 y Fo(\))k(for)h(an)m(y)g Fn(l)r Fo(.)123 459 y(The)28 b(remaining)h(estimates)g(are)f(pro)m(v)m (en)h(b)m(y)f(recurrence)i(o)m(v)m(er)e Fn(k)s Fo(.)42 b(After)28 b(ha)m(ving)g(exploited)i(orthogonal)-24 579 y(and)c(symplecting)i(blo)s(c)m(king,)h(the)d(basic)h(argumen)m(t)f(is) h(as)f(b)s(efore)g(and)g(therefore)g(some)h(calculatory)g(details)-24 699 y(are)d(suppressed)i(and)e(left)g(to)f(the)h(reader.)41 b(Hence)25 b(let)f(us)g(supp)s(ose)h(that)e Fi(jh)p Fo(w)2781 658 y Fq(+)2781 727 y Fl(l)2843 699 y Fi(^)t Fo(w)2984 658 y Fq(+)2984 727 y Fl(L)p Fk(\000)p Fl(l)3113 699 y Fi(j)p Fn(u)3197 714 y Fq(2)p Fl(l)3257 699 y Fo(\()p Fn(n)p Fo(\))t Fi(^)t Fn(u)3521 714 y Fq(2)p Fl(l)q Fq(+1)3672 699 y Fo(\()p Fn(n)p Fo(\))p Fi(ij)3873 663 y Fq(2)3940 699 y Fo(=)-24 820 y(1)j Fi(\000)h(O)s Fo(\()p Fn(\025)332 784 y Fq(2)371 820 y Fo(\))39 b(for)f(all)g Fn(l)i(<)e(k)s Fo(.)61 b(As)39 b Fi(h)p Fn(u)1314 835 y Fq(2)p Fl(k)1392 820 y Fo(\()p Fn(n)p Fo(\))p Fi(j)p Fn(u)1610 835 y Fl(m)1675 820 y Fo(\()p Fn(n)p Fo(\))p Fi(i)f Fo(=)g Fn(\016)2043 835 y Fq(2)p Fl(k)r(;m)2241 820 y Fo(and)h Fi(h)p Fn(u)2532 835 y Fq(2)p Fl(k)2609 820 y Fo(\()p Fn(n)p Fo(\))p Fi(j)p Fn(J)9 b Fi(j)p Fn(u)2918 835 y Fl(m)2984 820 y Fo(\()p Fn(n)p Fo(\))p Fi(i)37 b Fo(=)h(0,)i(one)f(concludes)-24 940 y(that)32 b Fi(h)p Fn(u)282 955 y Fq(2)p Fl(k)359 940 y Fo(\()p Fn(n)p Fo(\))p Fi(j)p Fo(w)591 904 y Fq(+)591 965 y Fl(m)658 940 y Fi(i)27 b Fo(=)h Fi(O)s Fo(\()p Fn(\025)p Fo(\))j(and)h Fi(h)p Fn(u)1358 955 y Fq(2)p Fl(k)1436 940 y Fo(\()p Fn(n)p Fo(\))p Fi(j)p Fn(J)9 b Fi(j)p Fo(w)1759 904 y Fq(+)1759 965 y Fl(m)1825 940 y Fi(i)28 b Fo(=)f Fi(O)s Fo(\()p Fn(\025)p Fo(\))32 b(for)f Fn(m)d Fo(=)f(1)p Fn(;)17 b(:)g(:)g(:)f(;)h(k)23 b Fi(\000)e Fo(1)p Fn(;)c(L)k Fi(\000)g Fn(k)i Fo(+)d(1)p Fn(;)d(:)g(:)g(:)f(L)p Fo(.)44 b(But)-24 1061 y(for)34 b(these)h(h)m(yp)s(erb)s(olic)h(c)m(hannels)g Fn(J)9 b Fi(j)p Fo(w)1413 1024 y Fq(+)1413 1085 y Fl(m)1480 1061 y Fi(i)30 b Fo(=)g Fi(j)p Fo(w)1753 1024 y Fk(\000)1753 1085 y Fl(m)1820 1061 y Fi(i)p Fo(.)48 b(As)34 b(the)h(same)g(holds)g (for)e Fn(u)2958 1076 y Fq(2)p Fl(k)r Fq(+1)3126 1061 y Fo(\()p Fn(n)p Fo(\),)i(w)m(e)g(can)f(conclude)-24 1181 y(that)259 1516 y Fi(h)p Fo(w)368 1475 y Fl(\033)368 1541 y(m)457 1516 y Fi(^)23 b Fo(w)616 1475 y Fl(\034)616 1541 y(l)659 1516 y Fi(j)p Fn(u)743 1531 y Fq(2)p Fl(k)820 1516 y Fo(\()p Fn(n)p Fo(\))g Fi(^)f Fn(u)1121 1531 y Fq(2)p Fl(k)r Fq(+1)1289 1516 y Fo(\()p Fn(n)p Fo(\))p Fi(i)55 b Fo(=)g Fi(O)s Fo(\()p Fn(\025)p Fo(\))33 b(,)227 b Fn(l)r(;)17 b(m)28 b Fo(=)g(1)p Fn(;)17 b(:)g(:)g(:)f(;)h(k)24 b Fi(\000)f Fo(1)p Fn(;)17 b(L)22 b Fi(\000)h Fn(k)i Fo(+)d(1)p Fn(;)17 b(:)g(:)g(:)e(L)33 b Fo(.)-24 1731 y(Using)h(this,)f(a)f(short)h(p)s(erturbativ)m(e)h(calculation)f (starting)g(from)f(\(15\))g(sho)m(ws)-24 2108 y Fi(jh)p Fo(w)113 2067 y Fq(+)113 2136 y Fl(k)181 2108 y Fi(^)8 b Fo(w)326 2067 y Fq(+)326 2136 y Fl(L)p Fk(\000)p Fl(k)471 2108 y Fi(j)p Fn(u)555 2123 y Fq(2)p Fl(k)632 2108 y Fo(\()p Fn(n)g Fo(+)g(1\))g Fi(^)g Fn(u)1045 2123 y Fq(2)p Fl(k)r Fq(+1)1214 2108 y Fo(\()p Fn(n)g Fo(+)g(1\))p Fi(ij)1556 2067 y Fq(2)1640 2108 y Fo(=)1932 2035 y Fn(e)1977 1999 y Fq(4)p Fl(\021)2047 2011 y Ff(k)2117 2035 y Fi(jh)p Fo(w)2254 1994 y Fq(+)2254 2063 y Fl(k)2336 2035 y Fi(^)22 b Fo(w)2495 1994 y Fq(+)2495 2063 y Fl(L)p Fk(\000)p Fl(k)2640 2035 y Fi(j)p Fn(u)2724 2050 y Fq(2)p Fl(k)2801 2035 y Fo(\()p Fn(n)p Fo(\))g Fi(^)h Fn(u)3102 2050 y Fq(2)p Fl(k)r Fq(+1)3270 2035 y Fo(\()p Fn(n)p Fo(\))p Fi(ij)3471 1999 y Fq(2)p 1770 2086 1902 4 v 1770 2177 a Fi(h)p Fn(u)1865 2192 y Fq(2)p Fl(k)1942 2177 y Fo(\()p Fn(n)p Fo(\))f Fi(^)h Fn(u)2243 2192 y Fq(2)p Fl(k)r Fq(+1)2410 2177 y Fo(\()p Fn(n)p Fo(\))p Fi(j)p Fo(\003)2640 2148 y Fq(2)2679 2177 y Fn(R)2754 2148 y Fk(\003)2794 2177 y Fn(R)q Fi(j)p Fn(u)2953 2192 y Fq(2)p Fl(k)3030 2177 y Fo(\()p Fn(n)p Fo(\))f Fi(^)h Fn(u)3331 2192 y Fq(2)p Fl(k)r Fq(+1)3499 2177 y Fo(\()p Fn(n)p Fo(\))p 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Fo(,)k(one)h(can)g(use)h(the)f(same)g (function)g Fn(f)1952 2815 y Fl(a)2027 2800 y Fo(as)f(ab)s(o)m(v)m(e)i (with)f Fn(a)28 b Fo(=)g Fn(e)2873 2764 y Fq(4\()p Fl(\021)2970 2776 y Ff(k)3009 2764 y Fk(\000)p Fl(\021)3099 2776 y Ff(k)q Fb(+1)3214 2764 y Fq(\))3279 2800 y Fo(and)k(complete)i(the)-24 2921 y(\(t)m(w)m(o-stepp)s(ed\))g(argumen)m(t)g(as)e(ab)s(o)m(v)m(e.) 2518 b Fa(2)-24 3300 y Fp(4)161 b(Ly)l(apuno)l(v)53 b(exp)t(onen)l(ts) -24 3548 y Fc(4.1)136 b(Calculating)46 b(Ly)l(apuno)l(v)f(exp)t(onen)l (ts)h(with)f(symplectic)g(frames)-24 3733 y Fo(In)32 b(the)f(de\014nition)h(\(23\))e(of)g(the)i(Ly)m(apuno)m(v)g(exp)s(onen) m(ts)h(app)s(ears)e(the)g(op)s(erator)f(norm.)43 b(Instead,)33 b(one)e(ma)m(y)-24 3853 y(use)45 b(symplectic)h(planes)e(as)g(initial)g (condition)g(if)f(an)g(a)m(v)m(eraging)i(o)m(v)m(er)f(them)g(is)g (done.)77 b(This)45 b(is)e(brie\015y)-24 3974 y(discussed)36 b(in)d(this)g(section.)123 4141 y(Imp)s(ortan)m(t)26 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Fb(2)p Ff(L)2823 587 y Fo(\))49 b(,)-24 838 y(where)31 b(here)f(the)g Fg(E)f Fo(is)h(only)g(an)g(a)m(v) m(erage)g(o)m(v)m(er)h(the)f(single)g(site)g(transfer)g(matrix)g Fn(T)14 b Fo(.)42 b(Similar)30 b(form)m(ulas)g(can)-24 959 y(b)s(e)j(found)g(in)g([CL,)g(Section)g(IV.6].)-24 1245 y Fc(4.2)136 b(Basic)45 b(p)t(erturbativ)l(e)h(form)l(ula:)61 b(only)45 b(symplectic)h(c)l(hannels)-24 1430 y Fo(As)36 b(it)g(is)f(considerably)j(more)d(transparen)m(t,)i(let)f(us)g(\014rst) g(p)s(erform)f(the)h(p)s(erturbativ)m(e)g(calculation)g(of)f(the)-24 1550 y(Ly)m(apuno)m(v)29 b(exp)s(onen)m(ts)h(in)d(the)h(case)h(where)f (there)h(are)e(no)g(h)m(yp)s(erb)s(olic)i(c)m(hannels)h(\(as)d(in)h (Fig.)f(1\(ii\)\).)41 b(Hence)-24 1670 y(w)m(e)34 b(assume)g Fn(R)g Fo(to)e(b)s(e)h(orthogonal.)123 1838 y(Let)24 b(us)g(insert)h Fg(1)j Fo(=)f(\003)923 1802 y Fl(p)963 1838 y Fn(M)10 b Fo(\003)1135 1802 y Fl(p)1175 1838 y Fn(M)1279 1802 y Fk(\000)p Fq(1)1398 1838 y Fo(in)24 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b(Ly)m(apuno)m(v)g(exp)s (onen)m(t.)47 b(This)34 b(will)g(then)f(giv)m(e)h(\(26\).)44 b(As)34 b(it)f(do)s(es)g(not)g(tak)m(e)h(more)f(e\013ort)g(at)g(this)h (p)s(oin)m(t)-24 2724 y(and)f(will)h(b)s(e)f(needed)i(b)s(elo)m(w,)e(w) m(e)h(will)g(ho)m(w)m(ev)m(er)h(not)e(supp)s(ose)h(all)f(c)m(hannels)i (to)d(b)s(e)h(elliptic,)i(but)e(only)g(the)-24 2845 y(exterior)h(frame) e(v)m(ector)i Fn(u)959 2860 y Fl(p)1031 2845 y Fo(to)e(b)s(e)h (elliptic,)h Fd(i.e.)44 b Fn(p)27 b(>)h Fo(2)p Fn(L)2093 2860 y Fl(h)2160 2845 y Fo(+)22 b(1.)-24 3061 y Fg(Lemma)39 b(3)49 b Fe(L)-5 b(et)36 b Fn(p)27 b(>)h Fo(2)p Fn(L)942 3076 y Fl(h)1009 3061 y Fo(+)22 b(1)34 b Fe(and)h(supp)-5 b(ose)34 b(that)h(the)g(Main)g(hyp)-5 b(othesis)34 b(holds.)-24 3205 y Fo(\(i\))550 3351 y(1)p 530 3395 89 4 v 530 3487 a Fn(N)645 3294 y Fl(N)7 b Fk(\000)p Fq(1)649 3324 y Fh(X)655 3533 y Fl(n)p Fq(=0)842 3418 y Fg(E)17 b Fi(h)p Fn(u)1028 3433 y Fl(p)1066 3418 y Fo(\()p Fn(n)p Fo(\))p Fi(j)g(j)p Fn(R)q(P)d Fi(j)1453 3377 y 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3488 y(den)m(t)40 b(of)d Fn(w)391 3452 y Fl(\033)388 3514 y(l)476 3488 y Fo(and)h(giv)m(en)i(b)m(y)f (Lemma)g(1\(v\).)61 b(This)39 b(directly)h(leads)f(to)f(the)h(\014rst)g (claim)f(b)s(ecause)i(the)f(sum)-24 3609 y(can)33 b(again)f(b)s(e)h (extended)i(to)d Fn(l)e Fo(=)d(0)p Fn(;)17 b(:)g(:)g(:)f(;)h(L)1578 3624 y Fl(c)1645 3609 y Fo(b)m(y)34 b(Prop)s(osition)f(3.)123 3776 y(\(ii\))f(One)h(has)g(for)f Fn(p;)17 b(q)31 b(>)d Fo(2)p Fn(L)1200 3791 y Fl(h)1267 3776 y Fo(+)22 b(1)99 4080 y Fi(h)p Fn(u)194 4095 y Fl(p)233 4080 y Fo(\()p Fn(n)p Fo(\))p Fi(j)417 4055 y Fo(~)395 4080 y Fn(P)13 b Fi(j)p Fn(u)555 4095 y Fl(q)593 4080 y Fo(\()p Fn(n)p Fo(\))p Fi(i)766 4039 y Fq(2)860 4080 y Fo(=)1104 3956 y Fl(L)1152 3964 y Ff(c)1072 3986 y Fh(X)992 4198 y Fl(k)r(;l)q(;m;j)t Fq(=0)1516 3986 y Fh(X)1340 4195 y Fl(\033)1380 4207 y Ff(k)1418 4195 y Fl(;\033)1478 4207 y Ff(l)1502 4195 y Fl(;\033)1562 4203 y Ff(m)1621 4195 y Fl(;\033)1681 4205 y Ff(j)1714 4195 y Fq(=)e Fk(\006)1863 4080 y Fi(h)p Fn(u)1958 4095 y Fl(p)1997 4080 y Fo(\()p Fn(n)p Fo(\))p Fi(j)p Fn(\031)2218 4034 y Fl(\033)2258 4046 y Ff(k)2214 4108 y Fl(k)2322 4055 y Fo(~)2300 4080 y Fn(P)i(\031)2435 4034 y Fl(\033)2475 4046 y Ff(l)2431 4108 y Fl(l)2504 4080 y Fi(j)p Fn(u)2588 4095 y Fl(q)2625 4080 y Fo(\()p Fn(n)p Fo(\))p Fi(i)28 b(h)p Fn(u)2921 4095 y Fl(q)2958 4080 y Fo(\()p Fn(n)p Fo(\))p Fi(j)p Fn(\031)3179 4039 y Fl(\033)3219 4047 y Ff(m)3175 4105 y Fl(m)3304 4055 y Fo(~)3282 4080 y Fn(P)13 b(\031)3417 4026 y Fl(\033)3457 4036 y Ff(j)3413 4106 y Fl(j)3494 4080 y Fi(j)p Fn(u)3578 4095 y Fl(p)3617 4080 y Fo(\()p Fn(n)p Fo(\))p Fi(i)32 b Fo(.)-24 4393 y(F)-8 b(or)34 b(the)i(same)f(reason)g(as)g(ab)s(o)m(v) m(e,)i(the)e(sum)h(can)f(b)s(e)g(restricted)h(to)e(elliptic)i(c)m (hannels)h Fn(k)s(;)17 b(l)r(;)g(m;)g(j)37 b(>)31 b(L)3827 4408 y Fl(h)3907 4393 y Fo(up)-24 4513 y(to)i(errors)g(of)g(order)g Fi(O)s Fo(\()p Fn(\025)p Fo(\).)44 b(F)-8 b(rom)32 b(the)h(16)g(signs,) h(Lemma)f(1\(iv\))g(eliminates)i(half,)e(forcing)f Fn(\033)3491 4528 y Fl(k)3563 4513 y Fo(=)c Fi(\000)p Fn(\033)3799 4528 y Fl(l)3859 4513 y Fo(and)-24 4634 y Fn(\033)31 4649 y Fl(m)132 4634 y Fo(=)33 b Fi(\000)p Fn(\033)373 4649 y Fl(j)411 4634 y Fo(.)53 b(T)-8 b(o)36 b(eac)m(h)h(of)e(the)i (\014nite)f(n)m(um)m(b)s(er)i(of)d(remaining)i(summands)h(an)e (oscilatory)g(sum)h(argumen)m(t)-24 4754 y(will)d(no)m(w)f(b)s(e)g (applied.)44 b(Set)583 5063 y Fn(J)9 b Fo(\()p Fn(N)h Fo(\))56 b(=)1027 4995 y(1)p 1007 5040 89 4 v 1007 5131 a Fn(N)1122 4938 y Fl(N)7 b Fk(\000)p Fq(1)1127 4968 y Fh(X)1132 5177 y Fl(n)p Fq(=0)1320 5063 y Fg(E)27 b Fi(h)p Fn(u)1516 5078 y Fl(p)1555 5063 y Fo(\()p Fn(n)p Fo(\))p Fi(j)p Fn(\031)1776 5022 y Fk(\000)p Fl(\033)1772 5091 y(k)1899 5038 y Fo(~)1877 5063 y Fn(P)14 b(\031)2013 5022 y Fl(\033)2009 5087 y(l)2059 5063 y Fi(j)p Fn(u)2143 5078 y Fl(q)2181 5063 y Fo(\()p Fn(n)p Fo(\))p Fi(i)27 b(h)p Fn(u)2476 5078 y Fl(q)2513 5063 y Fo(\()p Fn(n)p Fo(\))p Fi(j)p Fn(\031)2734 5022 y Fk(\000)p Fl(\034)2730 5087 y(m)2854 5038 y Fo(~)2832 5063 y Fn(P)14 b(\031)2968 5022 y Fl(\034)2964 5087 y(j)3011 5063 y Fi(j)p Fn(u)3095 5078 y Fl(p)3133 5063 y Fo(\()p Fn(n)p Fo(\))p Fi(i)33 b Fo(.)-24 5315 y(Pro)s(ceeding)h(as)f(ab)s(o)m(v)m(e)g(sho)m(ws)1947 5564 y(18)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 928 291 a Fn(J)9 b Fo(\()p Fn(N)h Fo(\))56 b(=)f Fn(e)1387 250 y Fl({\033)r Fq(\()p Fl(\021)1514 262 y Ff(k)1554 250 y Fq(+)p Fl(\021)1644 262 y Ff(l)1669 250 y Fq(\))1717 291 y Fn(e)1762 250 y Fl({\034)8 b Fq(\()p Fl(\021)1886 258 y Ff(m)1946 250 y Fq(+)p Fl(\021)2036 260 y Ff(j)2069 250 y Fq(\))2117 291 y Fn(J)h Fo(\()p Fn(N)h Fo(\))50 b(+)g Fi(O)s Fo(\()p Fn(N)2728 250 y Fk(\000)p Fq(1)2823 291 y Fn(;)17 b(\025)p Fo(\))32 b(.)-24 448 y(Again)41 b(in)m(v)m(oking)i(the)e(main)g(h)m(yp)s(othesis,)k Fn(J)9 b Fo(\()p Fn(N)h Fo(\))43 b(=)e Fi(O)s Fo(\(1\))g(is)g (therefore)h(p)s(ossible)g(only)f(if)g Fn(\034)54 b Fo(=)41 b Fi(\000)p Fn(\033)46 b Fo(and)-24 568 y Fi(f)p Fn(k)s(;)17 b(l)r Fi(g)28 b Fo(=)f Fi(f)p Fn(m;)17 b(j)6 b Fi(g)p Fo(.)43 b(Hence)34 b(up)f(to)f Fi(O)s Fo(\()p Fn(N)1439 532 y Fk(\000)p Fq(1)1534 568 y Fn(;)17 b(\025)p Fo(\),)259 869 y(1)p 239 913 89 4 v 239 1004 a Fn(N)354 812 y Fl(N)7 b Fk(\000)p Fq(1)359 841 y Fh(X)364 1051 y Fl(n)p Fq(=0)552 936 y Fg(E)17 b Fi(h)p Fn(u)738 951 y Fl(p)776 936 y Fo(\()p Fn(n)p Fo(\))p Fi(j)959 911 y Fo(~)938 936 y Fn(P)c Fi(j)p Fn(u)1098 951 y Fl(q)1135 936 y Fo(\()p Fn(n)p Fo(\))p Fi(i)1308 895 y Fq(2)478 1274 y Fo(=)639 1206 y(1)p 619 1251 V 619 1342 a Fn(N)734 1149 y Fl(N)7 b Fk(\000)p Fq(1)739 1179 y Fh(X)744 1388 y Fl(n)p Fq(=0)1003 1149 y Fl(L)1051 1157 y Ff(c)971 1179 y Fh(X)932 1391 y Fl(k)r(;l)q(>L)1116 1403 y Ff(h)1192 1179 y Fh(X)1188 1388 y Fl(\033)r Fq(=)p Fk(\006)1385 1274 y Fg(E)17 b Fi(h)p Fn(u)1571 1289 y Fl(p)1609 1274 y Fo(\()p Fn(n)p Fo(\))p Fi(j)p Fn(\031)1830 1232 y Fk(\000)p Fl(\033)1826 1301 y(k)1953 1249 y Fo(~)1931 1274 y Fn(P)d(\031)2067 1233 y Fl(\033)2063 1298 y(l)2114 1274 y Fi(j)p Fn(u)2198 1289 y Fl(q)2235 1274 y Fo(\()p Fn(n)p Fo(\))p Fi(i)27 b(h)p Fn(u)2530 1289 y Fl(q)2568 1274 y Fo(\()p Fn(n)p Fo(\))p Fi(j)p Fn(\031)2789 1233 y Fl(\033)2785 1298 y(l)2857 1249 y Fo(~)2835 1274 y Fn(P)14 b(\031)2971 1232 y Fk(\000)p Fl(\033)2967 1301 y(k)3072 1274 y Fi(j)p Fn(u)3156 1289 y Fl(p)3195 1274 y Fo(\()p Fn(n)p Fo(\))p Fi(i)474 b Fo(\(28\))616 1627 y(+)750 1560 y(1)p 730 1604 V 730 1696 a Fn(N)845 1503 y Fl(N)7 b Fk(\000)p Fq(1)850 1533 y Fh(X)855 1742 y Fl(n)p Fq(=0)1187 1503 y Fl(L)1235 1511 y Ff(c)1155 1533 y Fh(X)1043 1745 y Fl(k)r(;l)q(>L)1227 1757 y Ff(h)1266 1745 y Fl(;)k(k)r Fk(6)p Fq(=)p Fl(l)1450 1533 y Fh(X)1445 1742 y Fl(\033)r Fq(=)p Fk(\006)1642 1627 y Fg(E)17 b Fi(h)p Fn(u)1828 1642 y Fl(p)1866 1627 y Fo(\()p Fn(n)p Fo(\))p Fi(j)p Fn(\031)2087 1586 y Fk(\000)p Fl(\033)2083 1655 y(k)2211 1602 y Fo(~)2189 1627 y Fn(P)c(\031)2324 1586 y Fl(\033)2320 1652 y(l)2371 1627 y Fi(j)p Fn(u)2455 1642 y Fl(q)2492 1627 y Fo(\()p Fn(n)p Fo(\))p Fi(i)28 b(h)p Fn(u)2788 1642 y Fl(q)2825 1627 y Fo(\()p Fn(n)p Fo(\))p Fi(j)p Fn(\031)3046 1586 y Fl(\033)3042 1652 y(k)3115 1602 y Fo(~)3093 1627 y Fn(P)13 b(\031)3228 1586 y Fk(\000)p Fl(\033)3224 1655 y(l)3330 1627 y Fi(j)p Fn(u)3414 1642 y Fl(p)3453 1627 y Fo(\()p Fn(n)p Fo(\))p Fi(i)27 b Fn(:)162 b Fo(\(29\))-24 1910 y(Normalizing)33 b(the)g(pro)5 b(jections)34 b(of)e(the)h(frame)g(v)m(ectors)h(and)f(then)g(applying)g(Lemma)h (1\(iv\))e(giv)m(es)847 2218 y(\(28\))55 b(=)1250 2151 y(1)p 1217 2195 116 4 v 1217 2287 a(2)p Fn(L)1458 2094 y Fl(L)1506 2102 y Ff(c)1426 2124 y Fh(X)1386 2336 y Fl(k)r(;l)q(>L)1570 2348 y Ff(h)1626 2218 y Fn(h)1682 2177 y Fq(2)1682 2243 y Fl(l)1738 2218 y Fn(h)1794 2177 y Fq(2)1794 2243 y Fl(k)1881 2048 y Fh( )1990 2151 y Fo(1)p 1970 2195 89 4 v 1970 2287 a Fn(N)2085 2094 y Fl(N)7 b Fk(\000)p Fq(1)2090 2124 y Fh(X)2095 2333 y Fl(n)p Fq(=0)2283 2218 y Fg(E)27 b Fn(\032)2434 2233 y Fl(p;k)2532 2218 y Fo(\()p Fn(n)p Fo(\))17 b Fn(\032)2733 2233 y Fl(q)r(;l)2813 2218 y Fo(\()p Fn(n)p Fo(\))2947 2048 y Fh(!)3075 2218 y Fo(.)-24 2478 y(The)29 b(con)m(tribution)f (\(29\))f(can)h(only)g(b)s(e)f(treated)h(similarly)h(if)e Fn(q)32 b Fo(=)27 b Fn(p)p Fo(.)42 b(Supp)s(osing)28 b(this,)h(the)f(second)h(iden)m(tit)m(y)-24 2599 y(in)k(Lemma)g (1\(iv\))g(sho)m(ws)773 2907 y(\(29\))55 b(=)1176 2840 y(1)p 1143 2884 116 4 v 1143 2976 a(2)p Fn(L)1457 2783 y Fl(L)1505 2791 y Ff(c)1425 2813 y Fh(X)1312 3025 y Fl(k)r(;l)q(>L)1496 3037 y Ff(h)1535 3025 y Fl(;)12 b(k)r Fk(6)p Fq(=)p Fl(l)1699 2907 y Fn(h)1755 2866 y Fq(2)1755 2932 y Fl(l)1811 2907 y Fn(h)1867 2866 y Fq(2)1867 2932 y Fl(k)1954 2737 y Fh( )2063 2840 y Fo(1)p 2043 2884 89 4 v 2043 2976 a Fn(N)2158 2783 y Fl(N)7 b Fk(\000)p Fq(1)2162 2813 y Fh(X)2168 3022 y Fl(n)p Fq(=0)2355 2907 y Fg(E)28 b Fn(\032)2507 2922 y Fl(p;k)2605 2907 y Fo(\()p Fn(n)p Fo(\))17 b Fn(\032)2806 2922 y Fl(p;l)2887 2907 y Fo(\()p Fn(n)p Fo(\))3021 2737 y Fh(!)3149 2907 y Fo(.)-24 3167 y(The)34 b(sum)f(of)f(the)h(latter)g(t)m(w)m(o)g(con)m(tributions) h(is)f(giv)m(en)h(in)f(\(ii\).)123 3335 y(\(iii\))47 b(One)g(no)m(w)h(has)f(to)g(sum)h(\(28\))e(and)h(\(29\))g(o)m(v)m(er)h Fn(q)56 b Fo(=)c(2)p Fn(L)2518 3350 y Fl(h)2595 3335 y Fo(+)32 b(2)p Fn(;)17 b(:)g(:)g(:)e(;)i(L)p Fo(,)51 b(namely)d(precisely)h(the)-24 3455 y(elliptic)38 b(frame)e(v)m (ectors.)55 b(But)36 b(b)s(ecause)h Fn(k)i Fo(and)e Fn(l)h Fo(only)e(corresp)s(ond)h(to)f(elliptic)h(c)m(hannels,)i(the)d(sum)h (ma)m(y)-24 3576 y(b)s(e)g(extended)h(to)d Fn(q)j Fo(=)33 b(1)p Fn(;)17 b(:)g(:)g(:)f(;)h(L)36 b Fo(b)s(ecause)i(the)e(w)m(eigh)m (t)h(of)f(the)h(h)m(yp)s(erb)s(olic)g(frame)f(v)m(ectors)i(in)e(the)h (elliptic)-24 3696 y(c)m(hannels)28 b(is)e(of)g(order)g Fi(O)s Fo(\()p Fn(\025)p Fo(\))f(b)m(y)i(Prop)s(osition)f(3.)41 b(It)26 b(no)m(w)g(follo)m(ws)h(that)f(the)g(con)m(tribution)h(of)e (\(29\))g(v)-5 b(anishes.)-24 3817 y(In)29 b(order)f(to)f(sho)m(w)i (this,)h(decomp)s(ose)f Fn(\031)1454 3780 y Fl(\033)1450 3842 y(k)1529 3817 y Fo(and)f Fn(\031)1773 3780 y Fl(\033)1769 3842 y(l)1847 3817 y Fo(therein)h(using)g(\(11\))e(and)h(note)g(that)f (the)i(directions)g Fi(j)p Fo(w)3930 3780 y Fl(\033)3930 3842 y(k)3977 3817 y Fi(i)-24 3937 y Fo(and)38 b Fi(j)p Fo(w)269 3901 y Fl(\033)269 3963 y(l)316 3937 y Fi(i)f Fo(on)h(whic)m(h)h(they)f(pro)5 b(ject)39 b(satisfy)f(the)g(h)m(yp)s (othesis)i(of)d(Lemma)h(2)f(due)i(to)e(the)h(iden)m(tities)h(\(10\).) -24 4057 y(Therefore)1121 4241 y Fl(L)1073 4271 y Fh(X)1083 4480 y Fl(q)r Fq(=1)1261 4366 y Fn(\031)1320 4325 y Fl(\033)1316 4391 y(k)1367 4366 y Fi(j)p Fn(u)1451 4381 y Fl(q)1488 4366 y Fo(\()p Fn(n)p Fo(\))p Fi(i)27 b(h)p Fn(u)1783 4381 y Fl(q)1821 4366 y Fo(\()p Fn(n)p Fo(\))p Fi(j)p Fn(\031)2042 4325 y Fl(\033)2038 4391 y(l)2144 4366 y Fo(=)55 b(0)32 b(,)228 b Fn(k)31 b Fi(6)p Fo(=)c Fn(l)35 b Fo(,)955 b(\(30\))-24 4623 y(implying)45 b(the)f(claim.)77 b(The)44 b(sum)h(of)e(\(28\))g(o)m(v)m(er)h Fn(q)50 b Fo(=)c(1)p Fn(;)17 b(:)g(:)g(:)f(;)h(L)43 b Fo(can)h(easily)h(b)s(e)f (carried)g(out)f(using)h(the)-24 4743 y(iden)m(tit)m(y)35 b(\(18\):)817 4927 y Fl(L)769 4957 y Fh(X)779 5167 y Fl(q)r Fq(=1)941 5052 y Fo(\(28\))55 b(=)1344 4985 y(1)p 1311 5029 116 4 v 1311 5120 a(2)p Fn(L)1523 4927 y Fl(L)1571 4935 y Ff(c)1491 4957 y Fh(X)1480 5169 y Fl(l)q(>L)1605 5181 y Ff(h)1662 5052 y Fn(h)1718 5011 y Fq(2)1718 5077 y Fl(l)1774 5052 y Fn(\027)1822 5067 y Fl(l)1943 4927 y(L)1991 4935 y Ff(c)1911 4957 y Fh(X)1893 5169 y Fl(k)r(>L)2035 5181 y Ff(h)2091 5052 y Fn(h)2147 5011 y Fq(2)2147 5077 y Fl(k)2223 4882 y Fh( )2332 4985 y Fo(1)p 2312 5029 89 4 v 2312 5120 a Fn(N)2427 4927 y Fl(N)7 b Fk(\000)p Fq(1)2431 4957 y Fh(X)2437 5167 y Fl(n)p Fq(=0)2624 5052 y Fg(E)28 b Fn(\032)2776 5067 y Fl(p;k)2874 5052 y Fo(\()p Fn(n)p Fo(\))3025 4882 y Fh(!)3152 5052 y Fo(.)-24 5315 y(Because)34 b Fn(u)405 5330 y Fl(p)477 5315 y Fo(is)f(elliptic,)h(the) f(sum)h(ma)m(y)f(carry)g(o)m(v)m(er)h Fn(k)c Fo(=)e(0)p Fn(;)17 b(:)g(:)g(:)f(;)h(L)2488 5330 y Fl(c)2555 5315 y Fo(b)s(ecause)34 b(the)f(error)f(is)h Fi(O)s Fo(\()p Fn(\025)3597 5279 y Fq(2)3637 5315 y Fo(\).)239 b Fa(2)1947 5564 y Fo(19)p eop end %%Page: 20 20 TeXDict begin 20 19 bop -24 171 a Fc(4.4)136 b(Sum)44 b(of)h(Ly)l(apuno)l(v)g(exp)t(onen)l(ts)g(near)h(band)e(cen)l(ter)-24 355 y Fo(Let)34 b(us)h(again)e(supp)s(ose)j(in)e(this)g(section)h(that) f(there)h(are)f(only)g(elliptic)h(c)m(hannels.)49 b(Then)35 b(it)f(follo)m(ws)h(from)-24 476 y(\(24\))d(and)h(Lemma)g(3)f(that)71 759 y Fl(L)23 789 y Fh(X)39 1001 y Fl(l)q Fq(=1)200 884 y Fn(\015)251 899 y Fl(l)360 884 y Fo(=)529 816 y Fn(\025)586 780 y Fq(2)p 529 861 97 4 v 553 952 a Fo(2)703 884 y(lim)668 946 y Fl(N)7 b Fk(!1)936 816 y Fo(1)p 916 861 89 4 v 916 952 a Fn(N)1059 759 y Fl(N)g Fk(\000)p Fq(1)1063 789 y Fh(X)1069 998 y Fl(n)p Fq(=0)1293 759 y Fl(L)1245 789 y Fh(X)1261 1001 y Fl(l)q Fq(=1)1422 884 y Fg(E)1513 743 y Fh(\022)1586 884 y Fi(h)p Fn(u)1681 899 y Fl(l)1706 884 y Fo(\()p Fn(n)p Fo(\))p Fi(j)17 b(j)p Fn(P)d Fi(j)2018 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1217 y Fq(3)2229 1258 y Fo(\))-24 1568 y(Using)34 b(\(18\),)e(one)g(therefore)i(gets:)-24 1796 y Fg(Theorem)k(2)49 b Fe(Supp)-5 b(ose)34 b(that)i(the)f(Main)f (hyp)-5 b(othesis)35 b(holds)f(and)g(al)5 b(l)34 b(channels)g(ar)-5 b(e)34 b(el)5 b(liptic.)45 b(Then)1375 1980 y Fl(L)1327 2010 y Fh(X)1343 2222 y Fl(l)q Fq(=1)1488 2105 y Fn(\015)1539 2120 y Fl(l)1620 2105 y Fo(=)1761 2037 y Fn(L)17 b(\025)1901 2001 y Fq(2)p 1761 2082 180 4 v 1826 2173 a Fo(8)1995 2024 y Fh(\000)2040 2105 y Fn(h)2096 2064 y Fq(2)2096 2129 y Fb(a)n(v)2161 2024 y Fh(\001)2207 2046 y Fq(2)2269 2105 y Fo(+)22 b Fi(O)s Fo(\()p Fn(\025)2544 2064 y Fq(3)2583 2105 y Fo(\))17 b Fn(:)1177 b Fo(\(31\))123 2475 y(Via)41 b(the)h(Thouless)i(form)m(ula)e([CS)q(],)i(this)e(theorem)g(also)g (allo)m(ws)h(to)e(deduce)i(a)f Fi(O)s Fo(\()p Fn(\025)3346 2438 y Fq(2)3385 2475 y Fo(\))g(correction)g(to)-24 2595 y(the)36 b(densit)m(y)h(of)e(states.)53 b(Ho)m(w)m(ev)m(er,)39 b(the)c(term)h(linear)g(in)f Fn(\025)h Fo(has)f(to)g(b)s(e)h 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1044 1883 a Fl(N)1134 1774 y Fh(P)1239 1800 y Fl(N)7 b Fk(\000)p Fq(1)1239 1878 y Fl(n)p Fq(=0)1430 1848 y Fi(jh)p Fo(w)1567 1812 y Fl(\033)1567 1874 y(l)1614 1848 y Fi(j)p Fn(u)1698 1863 y Fl(p)1737 1848 y Fo(\()p Fn(n)p Fo(\))p Fi(ij)1938 1812 y Fq(2)1976 1848 y Fo(.)67 b(Because)42 b Fg(E)17 b Fn(P)53 b Fo(=)41 b(0,)h(one)e(gets)h(b)m(y)g(going)f(bac)m(k)h(in)-24 1969 y(history)34 b(once)340 2377 y Fn(J)403 2336 y Fk(\000)462 2377 y Fo(\()p Fn(N)10 b Fo(\))83 b(=)g Fn(e)913 2336 y Fk(\000)p Fq(2)p Fl(\021)1038 2348 y Ff(l)1084 2377 y Fn(J)1147 2336 y Fk(\000)1206 2377 y Fo(\()p Fn(N)10 b Fo(\))23 b(+)f Fn(\025)1548 2336 y Fq(2)1604 2377 y Fn(e)1649 2336 y Fk(\000)p Fq(2)p Fl(\021)1774 2348 y Ff(l)1819 2377 y Fg(E)1939 2310 y Fo(1)p 1919 2354 89 4 v 1919 2445 a Fn(N)2035 2253 y Fl(N)7 b Fk(\000)p Fq(1)2039 2282 y Fh(X)2045 2492 y Fl(n)p Fq(=0)2221 2377 y Fi(jh)p Fo(w)2358 2336 y Fk(\000)2358 2402 y Fl(q)2417 2377 y Fi(j)p Fn(P)14 b Fi(j)p Fn(u)2606 2392 y Fl(p)2644 2377 y Fo(\()p 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y(The)h(app)s(earing)e(oscillatory)h(sum)g(can)g(b)s(e)f(treated)h (b)m(y)g(the)g(same)g(argumen)m(t)g(as)g(in)f(the)h(pro)s(of)e(of)h (Lemma)-24 3137 y(3\(i\).)44 b(This)33 b(giv)m(es)712 3445 y Fn(J)775 3404 y Fk(\000)834 3445 y Fo(\()p Fn(N)10 b Fo(\))56 b(=)f Fn(\025)1242 3404 y Fq(2)1341 3378 y Fo(1)p 1308 3422 116 4 v 1308 3514 a(4)p Fn(L)1593 3378 y Fo(1)p 1460 3422 315 4 v 1460 3514 a Fn(e)1505 3485 y Fq(2)p Fl(\021)1575 3497 y Ff(l)1626 3514 y Fi(\000)23 b Fo(1)1812 3445 y Fn(h)1868 3404 y Fq(2)1868 3470 y Fl(l)1973 3321 y(L)2021 3329 y Ff(c)1941 3351 y Fh(X)1949 3563 y Fl(k)r Fq(=0)2101 3445 y Fn(h)2157 3404 y Fq(2)2157 3470 y Fl(k)2228 3445 y Fi(h)p Fn(\032)2317 3460 y Fl(p;k)2415 3445 y Fi(i)2454 3460 y Fl(N)2571 3445 y Fo(+)49 b Fi(O)s Fo(\()p Fn(N)2904 3404 y Fk(\000)p Fq(1)2999 3445 y Fn(;)17 b(\025)3100 3404 y Fq(3)3139 3445 y Fo(\))33 b(.)-24 3711 y(This)47 b(argumen)m(t)e(can)g(b)s(e)h(rep)s(eated)f(for)g Fn(J)1632 3675 y Fq(+)1691 3711 y Fo(\()p Fn(N)10 b Fo(\))45 b(using)h(\(34\))e(instead)i(of)f(\(33\).)80 b(One)45 b(\014nds)h Fn(J)3667 3675 y Fq(+)3727 3711 y Fo(\()p Fn(N)10 b Fo(\))49 b(=)-24 3832 y Fn(J)39 3796 y Fk(\000)99 3832 y Fo(\()p Fn(N)10 b Fo(\))17 b(+)g Fi(O)s Fo(\()p Fn(N)581 3796 y Fk(\000)p Fq(1)676 3832 y Fn(;)g(\025)777 3796 y Fq(3)816 3832 y Fo(\).)43 b(No)m(w)31 b(the)f(whole)h(argumen)m (t)g(can)f(b)s(e)h(rep)s(eated)g(for)e(w)2937 3796 y Fl(\033)2937 3858 y(L)p Fk(\000)p Fl(l)3066 3832 y Fo(.)42 b(Finally)31 b(summing)g(the)-24 3952 y(con)m(tributions)j(of)e(w)752 3916 y Fl(\033)752 3978 y(l)831 3952 y Fo(and)h(w)1091 3916 y Fl(\033)1091 3978 y(L)p Fk(\000)p Fl(l)1253 3952 y Fo(allo)m(ws)g(to)g(conclude)h(the)f(pro)s(of.)1459 b Fa(2)-24 4241 y Fc(4.6)136 b(P)l(erturbativ)l(e)46 b(form)l(ula)g(for)f(b)t(ottom)h(Ly)l(apuno)l(v)f(exp)t(onen)l(t)-24 4426 y Fo(The)36 b(aim)f(is)h(no)m(w)f(to)g(generalize)h(the)f(p)s (erturbativ)m(e)h(calculation)g(of)e(the)h(Ly)m(apuno)m(v)i(exp)s(onen) m(ts)f(giv)m(en)g(in)-24 4546 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Fo(^)-55 b Fn(u)2827 5252 y Fl(l)2853 5237 y Fo(\()p Fn(n)p Fo(\))p Fi(i)3026 5196 y Fq(2)3093 5067 y Fh(#)3201 5237 y Fo(+)49 b Fi(O)s Fo(\()p Fn(\025)3503 5196 y Fq(3)3543 5237 y Fo(\))17 b Fn(:)217 b Fo(\(39\))1947 5564 y(23)p eop end %%Page: 24 24 TeXDict begin 24 23 bop 123 171 a Fo(No)m(w)38 b(the)g(terms)g(\(35\))f (to)h(\(39\),)g(eac)m(h)h(b)m(y)f(de\014nition)h(con)m(taining)f(the)g (a)m(v)m(erage)3208 131 y Fq(1)p 3194 148 64 4 v 3194 205 a Fl(N)3284 96 y Fh(P)3389 122 y Fl(N)7 b Fk(\000)p Fq(1)3389 200 y Fl(n)p Fq(=0)3580 171 y Fg(E)37 b Fo(but)h(not)-24 291 y(the)27 b(limit)h Fn(N)38 b Fi(!)27 b(1)p Fo(,)h(will)f(b)s(e)g (treated)g(sep)s(erately)-8 b(.)43 b(Inserting)28 b(\(7\))e(and)h (using)g(the)g(normalization)g(prop)s(ert)m(y)-24 411 y(\(18\),)-24 821 y(\(35\))55 b(=)346 754 y(1)p 346 798 49 4 v 346 890 a(2)481 697 y Fl(L)529 705 y Ff(c)449 727 y Fh(X)465 939 y Fl(l)q Fq(=0)638 711 y Fh(\020)697 821 y Fn(e)742 780 y Fq(\(1)p Fk(\000)p Fl(g)895 757 y Fb(2)893 803 y 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Fo(is)e(actually)f (restricted)h(to)f(the)g(h)m(yp)s(erb)s(olic)h(c)m(hannels.)62 b(But)38 b(for)f(h)m(yp)s(erb)s(olic)i(a)-24 1953 y(c)m(hannel)33 b Fn(k)s Fo(,)e(one)h(has)f Fi(h)p Fn(u)887 1968 y Fl(L)939 1953 y Fo(\()p Fn(n)p Fo(\))p Fi(j)p Fn(\031)1160 1917 y Fl(\033)1156 1979 y(k)1206 1953 y Fi(j)6 b Fo(^)-55 b Fn(u)1290 1968 y Fl(l)1315 1953 y Fo(\()p Fn(n)p Fo(\))p Fi(i)28 b Fo(=)f Fi(O)s Fo(\()p Fn(\025)1796 1917 y Fq(2)1836 1953 y Fo(\))k(unless)i Fn(\033)e Fo(=)d(+)j(and)g Fn(l)f Fo(=)d(2)p Fn(k)s(;)17 b Fo(2)p Fn(k)22 b Fo(+)d(1)31 b(b)m(y)h(Prop)s(osition)f(3.)-24 2073 y(Hence)869 2385 y(\(36\))54 b(=)i Fi(\000)1316 2317 y Fo(1)p 1316 2362 49 4 v 1316 2453 a(2)1447 2257 y Fl(L)1495 2269 y Ff(h)1419 2290 y Fh(X)1427 2502 y Fl(k)r Fq(=0)1607 2304 y Fh(\000)1653 2385 y Fn(e)1698 2344 y Fq(2)p Fl(\021)1768 2356 y Ff(k)1833 2385 y Fi(\000)23 b Fo(1)1982 2304 y Fh(\001)2027 2326 y Fq(2)2083 2385 y Fn(e)2128 2344 y Fk(\000)p Fq(2)p Fl(\021)2253 2356 y Ff(k)2296 2385 y Fi(h)p 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Fi(h)p Fn(u)851 3539 y Fl(p)890 3524 y Fo(\()p Fn(n)p Fo(\))p Fi(j)p Fo(\()p Fn(R)1165 3482 y Fk(\003)1204 3524 y Fn(R)q(P)36 b Fo(+)22 b Fn(P)1553 3482 y Fk(\003)1592 3524 y Fn(R)1667 3482 y Fk(\003)1706 3524 y Fn(R)q Fo(\))p Fi(j)6 b Fo(^)-55 b Fn(u)1903 3539 y Fl(l)1928 3524 y Fo(\()p Fn(n)p Fo(\))p Fi(i)2101 3482 y Fq(2)2196 3524 y Fo(=)2337 3456 y(1)p 2337 3501 49 4 v 2337 3592 a(2)2412 3524 y Fn(h)2468 3482 y Fq(2)2468 3548 y Fb(a)n(v)2610 3399 y Fl(L)2658 3407 y Ff(c)2578 3429 y Fh(X)2594 3641 y Fl(l)q Fq(=0)2738 3524 y Fn(h)2794 3482 y Fq(2)2794 3548 y Fl(l)2862 3524 y Fi(h)p Fn(\032)2951 3539 y Fl(p;l)3032 3524 y Fi(i)3071 3539 y Fl(N)3188 3524 y Fo(+)49 b Fi(O)s Fo(\()p Fn(N)3521 3482 y Fk(\000)p Fq(1)3616 3524 y Fn(;)17 b(\025)p Fo(\))27 b Fn(:)-24 3848 y Fg(Pro)s(of.)44 b Fo(Let)32 b(us)i(call)e(the)h(l.h.s.)45 b Fn(J)9 b Fo(\()p Fn(N)h Fo(\).)44 b(Because)34 b Fn(R)1960 3812 y Fk(\003)1999 3848 y Fn(R)q Fi(j)p Fn(u)2158 3863 y Fl(k)2200 3848 y Fo(\()p Fn(n)p Fo(\))p 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b(of)f(the)h(last)g(factor)f (is)-24 1611 y(b)m(y)32 b(Lemma)f(1)f(indep)s(enden)m(t)j(of)d Fn(w)1268 1575 y Fl(\033)1265 1636 y(m)1332 1611 y Fo(.)42 b(No)m(w)31 b Fi(h)p Fn(\032)1710 1575 y Fq(+)1710 1636 y Fl(p;m)1832 1611 y Fi(i)1871 1626 y Fl(N)1966 1611 y Fo(=)c Fi(h)p Fn(\032)2158 1575 y Fk(\000)2158 1636 y Fl(p;m)2280 1611 y Fi(i)2319 1626 y Fl(N)2404 1611 y Fo(+)18 b Fi(O)s Fo(\()p Fn(\025)p Fo(\))31 b(b)m(y)g(Prop)s(osition) g(4)f(and)h(\(19\).)42 b(Of)-24 1732 y(the)36 b(four)e(terms,)j(the)e (\()p Fn(P)s(;)17 b(P)d Fo(\))33 b(and)i(\()p Fn(P)1427 1696 y Fk(\003)1466 1732 y Fn(;)17 b(P)1587 1696 y Fk(\003)1625 1732 y Fo(\))35 b(pairs)g(v)-5 b(anish)36 b(after)f(summing)h(o)m(v)m (er)g Fn(\033)i Fo(b)s(ecause)f(of)d(the)h(sign)-24 1852 y(in)e(Lemma)g(1\(ii\).)44 b(The)33 b(remaining)g(terms)h(\()p Fn(P)s(;)17 b(P)1840 1816 y Fk(\003)1878 1852 y Fo(\))32 b(and)h(\()p Fn(P)2253 1816 y Fk(\003)2292 1852 y Fn(;)17 b(P)d Fo(\))32 b(are)g(giv)m(en)i(b)m(y)f(Lemma)g(1\(iii\))g(so)g(that) 546 2164 y Fn(J)600 2179 y Fl(h)645 2164 y Fo(\()p Fn(N)10 b Fo(\))56 b(=)1006 2096 y(1)p 1006 2141 49 4 v 1006 2232 a(2)1145 2039 y Fl(L)1193 2047 y Ff(c)1113 2069 y Fh(X)1109 2278 y Fl(m)p Fq(=0)1295 2164 y Fn(h)1351 2122 y Fq(2)1351 2188 y Fl(m)1434 2164 y Fi(h)p Fn(\032)1523 2122 y Fl(\033)1523 2188 y(p;m)1645 2164 y Fi(i)1684 2179 y Fl(N)1797 2096 y Fo(1)p 1788 2141 67 4 v 1788 2232 a Fn(L)1926 2036 y Fl(L)1974 2048 y Ff(h)1898 2069 y Fh(X)1906 2281 y Fl(k)r Fq(=0)2086 2164 y Fn(h)2142 2122 y Fq(2)2142 2188 y Fl(k)2202 2164 y Fn(\027)2250 2179 y Fl(k)2326 2164 y Fo(cosh)q(\(2)p Fn(\021)2646 2179 y Fl(k)2688 2164 y Fo(\))50 b(+)f Fi(O)s Fo(\()p Fn(N)3109 2122 y Fk(\000)p Fq(1)3204 2164 y Fn(;)17 b(\025)p Fo(\))32 b(.)123 2445 y(Finally)g(for)f(the)h Fn(l)i Fo(in)f(the)f(sum)h(of)e Fn(J)1483 2460 y Fl(e)1520 2445 y Fo(\()p Fn(h)p Fo(\),)h(one)g(has)g Fn(e)2112 2408 y Fq(^)-40 b Fl(\021)2142 2420 y Ff(l)2171 2445 y Fn(P)34 b Fo(+)20 b Fn(e)2409 2408 y Fk(\000)5 b Fq(^)-40 b Fl(\021)2499 2420 y Ff(l)2528 2445 y Fn(P)2605 2408 y Fk(\003)2672 2445 y Fo(=)2797 2419 y(~)2775 2445 y Fn(P)14 b Fo(.)43 b(Therefore,)33 b(this)g(is)f(actually)-24 2565 y(the)h(term)g(treated) g(in)g(Lemma)g(3\(iii\).)44 b(Com)m(bining)34 b(pro)m(v)m(es)g(the)f (lemma.)1216 b Fa(2)123 2733 y Fo(Therefore,)38 b(\(37\))e(and)g (\(39\))f(comp)s(ensate)j(to)e(leading)h(order)f(just)g(as)h(do)f (\(35\))f(to)h(\(36\).)54 b(The)37 b(leading)-24 2853 y(order)28 b(con)m(tribution)h(to)e Fn(\015)942 2868 y Fl(L)1022 2853 y Fo(is)h(th)m(us)h(solely)g(giv)m(en)g(b)m(y)g (\(38\).)41 b(Noting)28 b(that)f Fn(R)2811 2817 y Fk(\003)2851 2853 y Fn(R)q Fi(j)p Fn(u)3010 2868 y Fl(L)3061 2853 y Fo(\()p Fn(n)p Fo(\))p Fi(i)h Fo(=)f Fi(j)p Fn(u)3449 2868 y Fl(L)3500 2853 y Fo(\()p Fn(n)p Fo(\))p Fi(i)12 b Fo(+)g Fi(O)s Fo(\()p Fn(\025)p Fo(\),)-24 2973 y(the)33 b(con)m(tribution)h(\(38\))e(w)m(as)h(already)g(dealt)g(with,)h(in)e (Lemma)i(3\(ii\))e(and)h(w)m(e)g(ha)m(v)m(e)h(pro)m(v)m(en:)-24 3249 y Fg(Theorem)k(3)49 b Fe(Supp)-5 b(ose)30 b(that)i Fn(E)h Fi(2)28 b Fm(R)j Fe(is)g(in)f(the)h(sp)-5 b(e)g(ctrum)31 b(of)f Fn(H)2392 3264 y Fl(L)2444 3249 y Fo(\(0\))h Fe(and)f (satis\014es)g(the)h(Main)f(hyp)-5 b(othesis.)-24 3369 y(Then)787 3678 y Fn(\015)838 3693 y Fl(L)945 3678 y Fo(=)90 b(lim)1076 3740 y Fl(N)7 b Fk(!1)1344 3611 y Fn(\025)1401 3574 y Fq(2)p 1335 3655 116 4 v 1335 3746 a Fo(8)p Fn(L)1525 3553 y Fl(L)1573 3561 y Ff(c)1493 3583 y Fh(X)1476 3795 y Fl(j;k)r Fq(=0)1670 3678 y Fn(h)1726 3637 y Fq(2)1726 3703 y Fl(j)1765 3678 y Fn(h)1821 3637 y Fq(2)1821 3703 y Fl(k)1864 3678 y Fo(\(2)22 b Fi(\000)h Fn(\016)2116 3693 y Fl(j;k)2206 3678 y Fo(\))28 b Fi(h)p Fn(\032)2361 3693 y Fl(L;j)2465 3678 y Fn(\032)2515 3693 y Fl(L;k)2626 3678 y Fi(i)2665 3693 y Fl(N)2782 3678 y Fo(+)49 b Fi(O)s Fo(\()p Fn(\025)3084 3637 y Fq(3)3124 3678 y Fo(\))17 b Fn(:)636 b Fo(\(40\))123 4051 y(The)38 b(presen)m(ted)i(tec)m(hniques)g(also)e(allo)m(w)g(to)f(write)h(out)f (form)m(ulas)h(for)f(the)h(top)f(Ly)m(apuno)m(v)i(exp)s(onen)m(t)-24 4171 y(and)33 b(the)g(sum)h(of)e(the)h(p)s(ositiv)m(e)h(Ly)m(apuno)m(v) g(exp)s(onen)m(ts.)-24 4339 y Fg(Pro)s(of)39 b Fo(of)g(Theorem)i(1.)63 b(\(i\))40 b(follo)m(ws)g(immediately)h(from)e Fn(h)2288 4303 y Fq(2)2288 4365 y Fl(k)2370 4339 y Fi(\025)h Fo(1)f(and)g(the)h (fact)f(that)g Fi(h)p Fn(\032)3452 4354 y Fl(L;j)3556 4339 y Fn(\032)3606 4354 y Fl(L;k)3717 4339 y Fi(i)3756 4354 y Fl(N)3862 4339 y Fo(is)h(a)-24 4459 y(probabilit)m(y)i (distribution.)69 b(\(ii\))40 b(F)-8 b(or)40 b(h)m(yp)s(erb)s(olic)i(c) m(hannels)h Fn(j;)17 b(k)45 b Fo(=)c(0)p Fn(;)17 b(:)g(:)g(:)e(L)2903 4474 y Fl(h)2949 4459 y Fo(,)42 b(the)f(w)m(eigh)m(ts)i(in)e(\(40\))f (are)-24 4580 y Fi(O)s Fo(\()p Fn(\025)p Fo(\))30 b(b)m(y)h(Prop)s (osition)f(3)g(so)g(that)f(they)i(can)f(b)s(e)g(neglected.)44 b(F)-8 b(or)29 b(the)i(remaining)f(elliptic)h(c)m(hannels)h Fn(j)6 b Fo(,)30 b(one)-24 4730 y(has)40 b Fn(h)213 4694 y Fq(2)213 4755 y Fl(j)293 4730 y Fi(\025)g Fn(h)466 4694 y Fq(2)466 4755 y Fl(L)514 4763 y Ff(c)550 4730 y Fo(.)65 b(But)40 b Fn(h)899 4694 y Fq(2)899 4755 y Fl(L)947 4763 y Ff(c)1023 4730 y Fo(=)f(1)p Fn(=)17 b Fo(sin\()p Fn(\021)1458 4745 y Fl(L)1506 4753 y Ff(c)1542 4730 y Fo(\))40 b(=)g(\(1)26 b Fi(\000)1964 4668 y Fl(\026)2006 4645 y Fb(2)2006 4691 y Ff(L)2048 4699 y(c)p 1964 4707 121 4 v 2006 4765 a Fq(4)2094 4730 y Fo(\))2132 4694 y Fk(\000)p Fq(1)p Fl(=)p Fq(2)2337 4730 y Fo(=)39 b(\()p Fn(\017)28 b Fi(\000)2671 4691 y Fl(\017)2700 4668 y Fb(2)p 2671 4707 64 4 v 2685 4765 a Fq(4)2744 4730 y Fo(\))2782 4694 y Fk(\000)p Fq(1)p Fl(=)p Fq(2)2987 4730 y Fi(\025)40 b Fn(\017)3143 4694 y Fk(\000)p Fq(1)p Fl(=)p Fq(2)3308 4730 y Fo(.)65 b(Replacing)40 b(this)-24 4851 y(concludes)35 b(the)e(pro)s(of.)3107 b Fa(2)1947 5564 y Fo(25)p eop end %%Page: 26 26 TeXDict begin 26 25 bop -24 171 a Fp(5)161 b(More)54 b(insigh)l(ts)d(on)i(the)g(c)l(hannel)f(w)l(eigh)l(ts)-24 390 y Fo(This)29 b(short)f(section)g(do)s(es)g(not)f(con)m(tain)h (rigorous)g(results.)43 b(The)28 b(aim)g(is)g(to)f(get)g(a)g(b)s(etter) h(understanding)h(of)-24 510 y(the)f(a)m(v)m(eraged)h(c)m(hannel)g(w)m (eigh)m(ts)g(en)m(tering)g(in)e(the)h(p)s(erturbativ)m(e)h(form)m(ulas) f(ab)s(o)m(v)m(e.)43 b(Again,)28 b(for)f(simplicit)m(y)-8 b(,)-24 630 y(let)30 b(us)g(restrict)g(ourselv)m(es)i(to)d(the)g (situation)h(where)g(there)g(are)g(only)f(elliptic)i(c)m(hannels.)44 b(W)-8 b(e)30 b(\014rst)f(fo)s(cus)h(on)-24 751 y(the)j(w)m(eigh)m(ts)i (of)d(the)h(\014rst)g(frame)g(v)m(ector)g Fn(u)1594 766 y Fq(1)1633 751 y Fo(.)44 b(It)32 b(follo)m(ws)i(from)e(\(14\))g(that,) g(up)h(to)f Fi(O)s Fo(\()p Fn(\025)3241 715 y Fq(3)3281 751 y Fo(\),)38 1089 y Fg(E)17 b Fn(\032)179 1104 y Fq(1)p Fl(;k)276 1089 y Fo(\()p Fn(n)22 b Fo(+)g(1\))g Fi(\000)h Fn(\032)751 1104 y Fq(1)p Fl(;k)849 1089 y Fo(\()p Fn(n)p Fo(\))83 b(=)g Fn(\025)1282 1048 y Fq(2)1349 1089 y Fg(E)1423 978 y Fh(h)1469 1089 y Fi(h)p Fn(u)1564 1104 y Fq(1)1603 1089 y Fi(j)p Fn(P)1708 1048 y Fk(\003)1746 1089 y Fn(\031)1801 1104 y Fl(k)1844 1089 y Fn(P)14 b Fi(j)p Fn(u)2005 1104 y Fq(1)2044 1089 y Fi(i)21 b(\000)i(h)p Fn(u)2299 1104 y Fq(1)2338 1089 y Fi(j)p Fn(\031)2421 1104 y Fl(k)2463 1089 y Fi(j)p Fn(u)2547 1104 y Fq(1)2586 1089 y Fi(ih)p Fn(u)2720 1104 y Fq(1)2758 1089 y Fi(j)17 b(j)p Fn(P)d Fi(j)2936 1048 y Fq(2)2991 1089 y Fi(j)p Fn(u)3075 1104 y Fq(1)3113 1089 y Fi(i)1446 1301 y Fo(+)26 b Fi(h)p Fn(u)1643 1316 y Fq(1)1682 1301 y Fi(j)p Fn(\031)1765 1316 y Fl(k)1808 1301 y Fi(j)p Fn(u)1892 1316 y Fq(1)1930 1301 y Fi(ih)p Fn(u)2064 1316 y Fq(1)2103 1301 y Fi(j)2153 1276 y Fo(~)2131 1301 y Fn(P)13 b Fi(j)p Fn(u)2291 1316 y Fq(1)2330 1301 y Fi(i)2369 1260 y Fq(2)2430 1301 y Fi(\000)23 b(h)p Fn(u)2625 1316 y Fq(1)2663 1301 y Fi(j)p Fo(\()p Fn(P)2806 1260 y Fk(\003)2845 1301 y Fn(\031)2900 1316 y Fl(k)2965 1301 y Fo(+)f Fn(\031)3118 1316 y Fl(k)3161 1301 y Fn(P)14 b Fo(\))p Fi(j)p Fn(u)3360 1316 y Fq(1)3398 1301 y Fi(ih)p Fn(u)3532 1316 y Fq(1)3570 1301 y Fi(j)3620 1276 y Fo(~)3598 1301 y Fn(P)f Fi(j)p Fn(u)3758 1316 y Fq(1)3797 1301 y Fi(i)3836 1191 y Fh(i)3927 1301 y Fn(;)-24 1546 y Fo(where)30 b(the)e(index)i Fn(n)e Fo(is)h(left)f(out)g (on)g(the)h(r.h.s.)43 b(and)28 b(the)h(exp)s(ectation)g(v)-5 b(alues)29 b(is)g(o)m(v)m(er)g Fn(P)14 b Fo(\()p Fn(n)f Fo(+)g(1\))28 b(only)-8 b(.)42 b(No)m(w)-24 1667 y(let)d(us)g(a)m(v)m (erage)h Fg(E)724 1628 y Fq(1)p 710 1644 64 4 v 710 1701 a Fl(N)799 1592 y Fh(P)905 1618 y Fl(N)7 b Fk(\000)p Fq(1)905 1696 y Fl(n)p Fq(=0)1100 1667 y Fo(and)39 b(supp)s(ose)h(that) e(the)h(limit)g(exists.)62 b(Then)40 b(the)f(l.h.s.)62 b(v)-5 b(anishes.)62 b(Hence)-24 1787 y(the)42 b(co)s(e\016cien)m(t)h (of)d Fn(\025)793 1751 y Fq(2)874 1787 y Fo(on)h(the)g(r.h.s.)70 b(has)42 b(to)e(v)-5 b(anish)42 b(as)g(w)m(ell,)i(up)e(to)e Fi(O)s Fo(\()p Fn(\025)p Fo(\).)69 b(Calculating)42 b(it)f(with)h(an) -24 1908 y(oscillatory)34 b(sum)f(argumen)m(t)g(as)g(in)g(Section)g (4.3)g(sho)m(ws)h(that)e(for)g(all)h Fn(k)d Fo(=)e(0)p Fn(;)17 b(:)g(:)g(:)f(;)h(L)3116 1923 y Fl(c)3150 1908 y Fo(:)289 2313 y(0)83 b(=)623 2246 y(1)p 590 2290 116 4 v 590 2382 a(2)p Fn(L)791 2189 y Fl(L)839 2197 y Ff(c)759 2219 y Fh(X)775 2431 y Fl(l)q Fq(=0)920 2313 y Fn(\027)968 2328 y Fl(k)1011 2313 y Fn(h)1067 2272 y Fq(2)1067 2338 y Fl(k)1109 2313 y Fn(h)1165 2272 y Fq(2)1165 2338 y Fl(l)1233 2313 y Fi(h)p Fn(\032)1322 2328 y Fq(1)p Fl(;l)1403 2313 y Fi(i)21 b(\000)1606 2246 y Fo(1)p 1573 2290 V 1573 2382 a(2)p Fn(L)1799 2189 y Fl(L)1847 2197 y Ff(c)1767 2219 y Fh(X)1742 2431 y Fl(l)q(;m)p Fq(=0)1953 2313 y Fn(\027)2001 2328 y Fl(m)2068 2313 y Fn(h)2124 2272 y Fq(2)2124 2338 y Fl(m)2191 2313 y Fn(h)2247 2272 y Fq(2)2247 2338 y Fl(l)2286 2313 y Fi(h)p Fn(\032)2375 2328 y Fq(1)p Fl(;l)2456 2313 y Fn(\032)2506 2328 y Fq(1)p Fl(;k)2604 2313 y Fi(i)580 2667 y Fo(+)726 2600 y(1)p 693 2644 V 693 2735 a(2)p Fn(L)892 2542 y Fl(L)940 2550 y Ff(c)860 2572 y Fh(X)835 2784 y Fl(l)q(;m)p Fq(=0)1046 2667 y Fn(h)1102 2626 y Fq(2)1102 2692 y Fl(l)1141 2667 y Fn(h)1197 2626 y Fq(2)1197 2692 y Fl(m)1264 2667 y Fo(\(2)h Fi(\000)g Fn(\016)1515 2682 y Fl(l)q(;m)1624 2667 y Fo(\))27 b Fi(h)p Fn(\032)1778 2682 y Fq(1)p Fl(;l)1859 2667 y Fn(\032)1909 2682 y Fq(1)p Fl(;m)2031 2667 y Fn(\032)2081 2682 y Fq(1)p Fl(;k)2179 2667 y Fi(i)49 b(\000)2438 2600 y Fo(1)p 2405 2644 V 2405 2735 a(2)p Fn(L)2578 2542 y Fl(L)2626 2550 y Ff(c)2546 2572 y Fh(X)2562 2784 y Fl(l)q Fq(=0)2707 2667 y Fn(h)2763 2626 y Fq(2)2763 2692 y Fl(l)2803 2667 y Fn(h)2859 2626 y Fq(2)2859 2692 y Fl(k)2901 2667 y Fo(\(2)22 b Fi(\000)h Fn(\016)3153 2682 y Fl(k)r(;l)3237 2667 y Fo(\))28 b Fi(h)p Fn(\032)3392 2682 y Fq(1)p Fl(;l)3473 2667 y Fn(\032)3523 2682 y Fq(1)p Fl(;k)3621 2667 y Fi(i)17 b Fn(:)-24 2982 y Fo(These)45 b(equations)f(giv)m(e)g(relations)g(b)s (et)m(w)m(een)h(the)e(a)m(v)m(eraged)h(\014rst,)i(second)e(and)f(third) g(momen)m(ts)i(of)d(the)-24 3102 y(w)m(eigh)m(ts)32 b Fn(\032)371 3117 y Fq(1)p Fl(;l)452 3102 y Fo(.)43 b(Analogously)-8 b(,)31 b(one)g(can)f(write)h(out)f(equations)h(for)f Fi(h)p Fn(\032)2533 3117 y Fq(1)p Fl(;l)2614 3102 y Fn(\032)2664 3117 y Fq(1)p Fl(;k)2762 3102 y Fi(i)f Fo(whic)m(h)j(then)f(in)m(v)m (ok)m(e)h(up)f(to)e(the)-24 3223 y(a)m(v)m(eraged)34 b(sixth)g(momen)m(ts)h(of)d(the)h(c)m(hannel)h(w)m(eigh)m(ts,)h(and)e (so)g(on.)44 b(This)34 b(giv)m(es)g(a)f(hierarc)m(h)m(y)h(of)e (equations)-24 3343 y(for)37 b(the)g(c)m(hannel)h(w)m(eigh)m(ts.)59 b(It)37 b(results)h(that)f(the)g(w)m(eigh)m(ts)i(are)e(indep)s(enden)m (t)i(of)d Fn(\025)h Fo(and)g(only)h(dep)s(end)g(on)-24 3464 y(energy)c Fn(E)39 b Fo(\(through)32 b(the)h(the)g(frequencies)i Fn(\021)t Fo(\).)123 3631 y(It)43 b(order)g(to)g(analyse)h(these)g (equations,)j(let)d(us)f(close)i(them)e(already)h(at)f(\014rst)g(order) g(b)m(y)h(assuming)-24 3752 y(factorization)i Fi(h)p Fn(\032)645 3767 y Fq(1)p Fl(;l)726 3752 y Fn(\032)776 3767 y Fq(1)p Fl(;k)874 3752 y Fi(i)51 b Fo(=)g Fi(h)p Fn(\032)1180 3767 y Fq(1)p Fl(;l)1261 3752 y Fi(i)17 b(h)p Fn(\032)1406 3767 y Fq(1)p Fl(;k)1503 3752 y Fi(i)46 b Fo(and)g Fi(h)p Fn(\032)1880 3767 y Fq(1)p Fl(;l)1961 3752 y Fn(\032)2011 3767 y Fq(1)p Fl(;m)2133 3752 y Fn(\032)2183 3767 y Fq(1)p Fl(;k)2281 3752 y Fi(i)51 b Fo(=)g Fi(h)p Fn(\032)2587 3767 y Fq(1)p Fl(;l)2668 3752 y Fi(i)17 b(h)p Fn(\032)2813 3767 y Fq(1)p Fl(;m)2934 3752 y Fi(i)g(h)p Fn(\032)3079 3767 y Fq(1)p Fl(;k)3176 3752 y Fi(i)p Fo(.)84 b(F)-8 b(urthermore)47 b(w)m(e)-24 3872 y(neglect)39 b(the)f Fn(\016)528 3887 y Fl(m;l)674 3872 y Fo(and)f(supp)s(ose)i Fn(\027)1288 3887 y Fl(k)1367 3872 y Fo(=)e(2,)h(b)s(oth)g(appro)m (ximations)g(whic)m(h)h(are)f Fi(O)s Fo(\(1)p Fn(=L)p Fo(\))g(w.r.t.)59 b(the)38 b(other)-24 3992 y(terms.)45 b(No)m(w)33 b(the)g(sum)g(o)m(v)m(er)h Fn(l)h Fo(factors)d(and)h(one)g (obtains:)680 4301 y(0)55 b(=)h Fn(h)972 4260 y Fq(2)972 4326 y Fl(k)1064 4301 y Fi(\000)50 b(h)p Fn(\032)1280 4316 y Fq(1)p Fl(;k)1378 4301 y Fi(i)1486 4176 y Fl(L)1534 4184 y Ff(c)1454 4206 y Fh(X)1450 4416 y Fl(m)p Fq(=0)1619 4301 y Fn(h)1675 4260 y Fq(2)1675 4326 y Fl(m)1792 4301 y Fo(+)f Fi(h)p Fn(\032)2006 4316 y Fq(1)p Fl(;k)2104 4301 y Fi(i)2196 4176 y Fl(L)2244 4184 y Ff(c)2164 4206 y Fh(X)2160 4416 y Fl(m)p Fq(=0)2329 4301 y Fn(h)2385 4260 y Fq(2)2385 4326 y Fl(m)2479 4301 y Fi(h)p Fn(\032)2568 4316 y Fq(1)p Fl(;m)2689 4301 y Fi(i)h(\000)g Fn(h)2961 4260 y Fq(2)2961 4326 y Fl(k)3032 4301 y Fi(h)p Fn(\032)3121 4316 y Fq(1)p Fl(;k)3218 4301 y Fi(i)28 b Fn(:)-24 4566 y Fo(These)35 b(equations)f(ha)m(v)m(e)f(the)g(unique)h(solution)g (\(recall)f Fn(h)2129 4530 y Fq(2)2129 4592 y Fl(k)2199 4566 y Fo(=)28 b(1)p Fn(=)17 b Fo(sin\()p Fn(\021)2623 4581 y Fl(k)2665 4566 y Fo(\)\))1453 4826 y Fi(h)p Fn(\032)1542 4841 y Fq(1)p Fl(;k)1639 4826 y Fi(i)56 b Fo(=)2132 4759 y(1)p 1875 4803 563 4 v 1875 4894 a(1)22 b(+)g Fn(Z)40 b Fo(sin)q(\()p Fn(\021)2357 4909 y Fl(k)2399 4894 y Fo(\))2480 4826 y(,)-24 5074 y(where)j Fn(Z)50 b Fi(\025)43 b Fo(0)e(is)h(suc)m(h)h(that)e(normalization)1776 5000 y Fh(P)1881 5026 y Fl(L)1929 5034 y Ff(c)1881 5103 y Fl(k)r Fq(=0)2014 5074 y Fi(h)p Fn(\032)2103 5089 y Fq(1)p Fl(;k)2201 5074 y Fi(i)h Fo(=)h(1)e(is)h(assured.)71 b(One)42 b(easily)g(v)m(eri\014es)i(that)-24 5195 y Fn(Z)e Fi(\030)36 b Fn(L)p Fo(.)57 b(F)-8 b(or)36 b(small)i Fn(k)i Fo(\(and)d(large)g(ones)h Fn(L)25 b Fi(\000)h Fn(k)40 b Fo(as)d(w)m(ell\))h(one)f(has)h(sin\()p Fn(\021)2791 5210 y Fl(k)2834 5195 y Fo(\))d Fi(\030)h Fn(k)s(=L)p Fo(.)57 b(Hence)38 b(the)f(w)m(eigh)m(t)-24 5315 y(on)e(these)h(c)m (hannels)g(is)f(of)f(order)h(of)f(unit)m(y)-8 b(,)36 b(while)g(it)e(is)h(of)g(order)f(1)p Fn(=L)h Fo(on)f(the)h(others.)50 b(But)35 b(the)g(c)m(hannels)1947 5564 y(26)p eop end %%Page: 27 27 TeXDict begin 27 26 bop -24 171 a Fo(with)35 b(small)g(and)f(large)g Fn(k)j Fo(are)d(precisely)j(those)d(near)h(the)f(band)g(edges)i(in)e (Fig.)g(1\(ii\))f(where)j(the)e(rotation)-24 291 y(frequency)k(is)d (small.)53 b(Hence)36 b(the)g(w)m(eigh)m(t)g(of)f Fn(u)1767 306 y Fq(1)1841 291 y Fo(is)h(concen)m(trated)h(on)e(the)h(slo)m(wly)h (rotating)d(c)m(hannels)j(for)-24 411 y(whic)m(h)d Fn(h)311 375 y Fq(2)311 437 y Fl(k)382 411 y Fo(=)27 b Fi(O)s Fo(\()p Fn(L)p Fo(\))33 b(\(they)g(can)f(b)s(e)h(considered)h(to)e(b)s (e)g(most)h(similar)g(to)e(h)m(yp)s(erb)s(olic)j(c)m(hannels\).)45 b(Hence)34 b(it)-24 532 y(is)f(exp)s(ected)h(from)e(\(32\))g(that)g Fn(\015)1178 547 y Fq(1)1244 532 y Fo(=)c Fn(c)17 b(\025)1464 496 y Fq(2)1524 532 y Fo(+)22 b Fi(O)s Fo(\()p Fn(\025)1799 496 y Fq(2)1838 532 y Fo(\))32 b(where)h Fn(c)28 b Fo(=)g Fi(O)s Fo(\(1\))j(as)i Fn(L)28 b Fi(!)f(1)p Fo(.)43 b(Presumably)-8 b(,)35 b(only)d(the)-24 652 y(\014rst)h(few)h(exp)s(onen)m(ts)g(are)f (considerably)h(larger)e(than)h Fn(\015)2091 667 y Fl(L)2143 652 y Fo(.)123 820 y(Due)j(to)g(symplectic)j(blo)s(c)m(king,)f(the)f(w) m(eigh)m(t)h(of)e Fn(u)2013 835 y Fq(2)2088 820 y Fo(has)h(to)f(b)s(e)g (cen)m(tered)j(on)d(sligh)m(tly)i(faster)e(rotating)-24 940 y(c)m(hannels,)h(ecc)m(\022)-46 b(etera.)50 b(In)35 b(conclusion,)h(the)f(w)m(eigh)m(t)g(of)f(last)h(frame)f(v)m(ector)i Fn(u)2847 955 y Fl(L)2933 940 y Fo(is)e(exp)s(ected)j(to)c(b)s(e)i (concen-)-24 1061 y(trated)k(on)g(the)g(c)m(hannels)h(whic)m(h)g (rotate)f(the)g(fastest)g(and)g(hence)h(corresp)s(ond)f(to)f(the)h (band)g(cen)m(ter.)63 b(In)-24 1181 y(these)36 b(c)m(hannels,)h Fn(h)707 1145 y Fq(2)707 1207 y Fl(k)780 1181 y Fo(=)31 b Fi(O)s Fo(\(1\).)48 b(Therefore,)36 b(the)f(lo)m(w)m(er)g(b)s(ound)g Fn(h)2425 1145 y Fq(2)2425 1207 y Fl(k)2498 1181 y Fi(\025)c Fo(1)j(whic)m(h)i(allo)m(w)m(ed)g(to)e(deduce)h(Theo-)-24 1301 y(rem)h(1)e(from)h(Theorem)h(3)e(is)i(presumably)g(not)f(so)g(bad) g(b)s(ecause)h(the)f(w)m(eigh)m(ts)i(of)d Fn(u)3155 1316 y Fl(L)3242 1301 y Fo(en)m(ter)i(in)m(to)f(form)m(ula)-24 1422 y(\(40\).)48 b(In)35 b(the)f(case)h(of)f(mixed)i(elliptic)f(and)f (h)m(yp)s(erb)s(olic)i(c)m(hannels,)g(w)m(e)g(exp)s(ect)f(the)g(ab)s(o) m(v)m(e)g(arguemen)m(t)g(to)-24 1542 y(hold)e(within)h(the)f(elliptic)h (part)e(of)h Fm(C)1380 1506 y Fq(2)p Fl(L)1467 1542 y Fo(,)g(namely)g(for)g(the)g(w)m(eigh)m(t)h(v)m(ectors)g(not)f(alined)g (to)f(the)h(h)m(yp)s(erb)s(olic)-24 1662 y(c)m(hannels)i(b)m(y)e(Prop)s (osition)g(3.)-24 2042 y Fp(References)-24 2261 y Fo([AALR])49 b(E.)24 b(Abrahams,)i(P)-8 b(.)24 b(W.)g(Anderson,)j(D.)c(C.)h (Licciardello,)i(T.)e(V.)g(Ramakrishnan,)j Fd(Scaling)d(Theory)177 2382 y(of)35 b(Lo)s(calization:)50 b(Absence)38 b(of)d(Quan)m(tum)h (Di\013usion)g(in)g(Tw)m(o)h(Dimensions)p Fo(,)h(Ph)m(ys.)f(Rev.)g (Lett.)f Fg(42)p Fo(,)177 2502 y(673-677)31 b(\(1979\).)-24 2706 y([APW])50 b(L.)34 b(Arnold,)h(G.)e(P)m(apanicolaou,)i(V.)f (Wihstutz,)i Fd(Asymptotic)f(analysis)g(of)f(the)g(Ly)m(apuno)m(v)i (exp)s(o-)177 2826 y(nen)m(t)31 b(and)f(rotation)f(n)m(um)m(b)s(er)j (of)d(the)i(random)f(oscillator)g(and)g(applications)p Fo(,)i(SIAM)f(J.)f(Appl.)g(Math.)177 2946 y Fg(46)p Fo(,)j(427-450)d (\(1986\).)-24 3150 y([Ben])50 b(C.)32 b(W.)g(J.)f(Beenakk)m(er,)j Fd(Random-matrix)e(theory)g(of)e(quan)m(tum)j(transp)s(ort)p Fo(,)f(Rev.)g(Mo)s(d.)f(Ph)m(ys.)j Fg(69)p Fo(,)177 3270 y(731-808)d(\(1997\).)-24 3474 y([BK])49 b(A.)i(Bo)m(vier,)56 b(A.)51 b(Klein,)56 b Fd(W)-8 b(eak)51 b(disorder)h(expansion)g(of)e (the)h(in)m(v)-5 b(arian)m(t)52 b(measure)g(for)e(the)h(one-)177 3594 y(dimensional)34 b(Anderson)g(mo)s(del)p Fo(,)f(J.)f(Stat.)h(Ph)m (ys.)h Fg(51)p Fo(,)f(501-517)e(\(1988\).)-24 3797 y([BL])49 b(P)-8 b(.)36 b(Bougerol,)h(J.)f(Lacroix,)h Fd(Pro)s(ducts)f(of)f (Random)h(Matrices)h(with)f(Applications)h(to)e(Sc)m(hr\177)-49 b(odinger)177 3918 y(Op)s(erators)p Fo(,)32 b(\(Birkh\177)-49 b(auser,)35 b(Boston,)d(1985\).)-24 4121 y([CL])50 b(R.)29 b(Carmona,)g(J.)g(Lacroix,)i Fd(Sp)s(ectral)e(Theory)h(of)e(Random)h (Sc)m(hr\177)-49 b(odinger)31 b(Op)s(erators)p Fo(,)e(\(Birkh\177)-49 b(auser,)177 4242 y(Boston,)33 b(1990\).)-24 4445 y([CS])50 b(W.)25 b(Craig,)h(B.)f(Simon,)h Fd(Log)e(H\177)-49 b(older)25 b(Con)m(tin)m(uit)m(y)i(of)d(the)h(In)m(tegrated)h(Densit)m(y)f(of)f (States)i(for)e(Sto)s(c)m(has-)177 4565 y(tic)33 b(Jacobi)g(Matrices)p Fo(,)h(Comm)m(un.)g(Math.)f(Ph)m(ys.)h Fg(90)p Fo(,)f(207-218)d (\(1983\).)-24 4769 y([DK])49 b(H.)26 b(v)m(on)g(Dreifus;)j(A.)d (Klein,)i Fd(A)e(new)h(pro)s(of)e(of)g(lo)s(calization)h(in)g(the)g (Anderson)i(tigh)m(t)e(binding)g(mo)s(del)p Fo(,)177 4889 y(Comm)m(un.)34 b(Math.)f(Ph)m(ys.)h Fg(124)p Fo(,)f(285-299)e (\(1989\).)-24 5093 y([FS])49 b(J.)29 b(F)-8 b(r\177)-49 b(ohlic)m(h,)29 b(T.)g(Sp)s(encer,)i Fd(Absence)f(of)e(di\013usion)h (in)g(the)f(Anderson)i(tigh)m(t)e(binding)h(mo)s(del)g(for)f(large)177 5213 y(disorder)33 b(or)g(lo)m(w)g(energy)p Fo(,)g(Comm)m(un.)i(Math.)d (Ph)m(ys.)j Fg(88)p Fo(,)e(151-184)d(\(1983\).)1947 5564 y(27)p eop end %%Page: 28 28 TeXDict begin 28 27 bop -24 171 a Fo([F)-8 b(ur])49 b(H.)30 b(F)-8 b(ursten)m(b)s(erg,)31 b Fd(Noncomm)m(uting)f(random)g(pro)s (ducts)p Fo(,)h(T)-8 b(rans.)30 b(Amer.)g(Math.)g(So)s(c.)f Fg(108)p Fo(,)i(377-428)177 291 y(\(1963\).)-24 494 y([GK])49 b(F.)25 b(Germinet,)j(A.)d(Klein,)j Fd(Bo)s(otstrap)d(Multiscale)i (Analysis)h(and)d(Lo)s(calization)h(in)f(Random)h(Media)p Fo(,)177 615 y(Comm)m(un.)34 b(Math.)f(Ph)m(ys.)h Fg(222)p Fo(,)f(415-448)e(\(2001\).)-24 818 y([GM])49 b(Y)-8 b(a.)34 b(I.)h(Goldsheid,)h(A.)e(G.)g(Margulis,)i Fd(Ly)m(apuno)m(v)g(exp)s (onen)m(ts)g(of)e(a)g(pro)s(duct)g(of)g(random)h(matrices)177 939 y Fo(Russian)e(Math.)g(Surv)m(eys)i Fg(44)p Fo(,)e(11-71)e (\(1989\).)-24 1142 y([GR])49 b(Y.)j(Guiv)-5 b(arc'h,)57 b(A.)c(Raugi,)j Fd(Propri)m(\023)-46 b(et)m(\023)g(es)54 b(de)f(con)m(traction)f(d'un)h(semi-group)s(e)g(de)g(matrices)g(in-)177 1262 y(v)m(ersibles.)30 b(Co)s(e\016cien)m(ts)g(de)e(Liapuno\013)g (d'un)g(pro)s(duit)g(de)g(matrices)h(alatoires)f(ind)m(\023)-46 b(ep)s(endan)m(tes.)30 b Fo(Israel)177 1383 y(J.)j(Math.)g Fg(65)p Fo(,)f(165-196)f(\(1989\).)-24 1586 y([JSS])50 b(S.)39 b(Jitomirsk)-5 b(a)m(y)m(a,)43 b(H.)c(Sc)m(h)m(ulz-Baldes,)44 b(G.)39 b(Stolz,)i Fd(Delo)s(calization)f(in)f(random)h(p)s(olymer)g (mo)s(dels)p Fo(,)177 1706 y(Comm)m(un.)34 b(Math.)f(Ph)m(ys.)h Fg(233)p Fo(,)f(27-48)e(\(2003\).)-24 1910 y([KW])49 b(M.)41 b(Kappus,)h(F.)e(W)-8 b(egner,)42 b Fd(Anomaly)f(in)f(the)g (band)g(cen)m(tre)i(of)d(the)h(one-dimensional)h(Anderson)177 2030 y(mo)s(del)p Fo(,)33 b(Z.)f(Ph)m(ys.)j Fg(B45)p Fo(,)d(15-21)f(\(1981\).)-24 2234 y([KLS])49 b(A.)c(Klein,)i(J.)e (Lacroix,)i(A.)e(Sp)s(eis,)j Fd(Lo)s(calization)c(for)g(the)g(Anderson) i(Mo)s(del)e(on)h(a)f(Strip)g(with)177 2354 y(Singular)33 b(P)m(oten)m(tials)p Fo(,)h(J.)f(F)-8 b(unct.)32 b(Anal.)h Fg(94)p Fo(,)g(135-155)e(\(1990\).)-24 2557 y([Klo])49 b(F.)31 b(Klopp,)i Fd(W)-8 b(eak)32 b(disorder)h(lo)s(calization)e(and) h(Lifshitz)h(tails)p Fo(,)f(Comm)m(un.)h(Math.)f(Ph)m(ys.)i Fg(232)p Fo(,)e(125-)177 2678 y(155)g(\(2002\).)-24 2881 y([Mir])50 b(A.)26 b(Mirlin,)j Fd(Statistics)e(of)f(energy)i(lev)m(els) g(and)f(eigenfunctions)h(in)f(disordered)h(and)e(c)m(haotic)h(systems:) 177 3002 y(Sup)s(ersymmetry)35 b(approac)m(h)p Fo(,)e(Ph)m(ys.)i(Rep.)e Fg(326)p Fo(,)f(259-382)f(\(2000\).)-24 3205 y([PF])49 b(L.)g(P)m(astur,)k(A.)c(Figotin,)j Fd(Sp)s(ectra)c(of)g(Random)h(and)f (Almost-P)m(erio)s(dic)i(Op)s(erators)p Fo(,)i(\(Springer,)177 3325 y(Berlin,)33 b(1992\).)-24 3529 y([SSW])50 b(W.)41 b(Sc)m(hlag,)j(C.)d(Sh)m(ubin,)k(T.)c(W)-8 b(ol\013,)43 b Fd(F)-8 b(requency)43 b(concen)m(tration)f(and)f(lo)s(calization)f (lengths)i(for)177 3649 y(the)33 b(Anderson)h(mo)s(del)f(at)f(small)h (disorders)p Fo(,)h(Journal)f(d'Analyse)h(Math.)f Fg(88)p Fo(,)g(173-220)d(\(2002\).)-24 3853 y([SB])50 b(H.)28 b(Sc)m(h)m(ulz-Baldes,)j Fd(Lifshitz)e(tails)f(for)g(the)g(1D)g (Bernoulli-Anderson)h(mo)s(del)p Fo(,)h(to)d(app)s(ear)h(in)g(Mark)m(o) m(v)177 3973 y(Chains)34 b(and)e(Rel.)h(Fields..)-24 4176 y([T)-8 b(ei])50 b(H.)28 b(T)-8 b(eic)m(hert,)30 b Fd(Asymptotic)g(analysis)f(of)e(the)h(Ly)m(apuno)m(v)h(exp)s(onen)m (ts)h(of)d(certain)h(linear)g(Hamiltonian)177 4297 y(systems)35 b(with)e(small)g(noise)p Fo(,)h(Math.)f(Nac)m(hr.)g Fg(172)p Fo(,)g(291-311)d(\(1995\).)-24 4500 y([Tho])50 b(D.)30 b(J.)g(Thouless,)j Fd(Maxim)m(um)f(Metallic)g(Resistance)g(in)f(Thin)g (Wires)p Fo(,)h(Ph)m(ys.)g(Rev.)f(Lett.)g Fg(39)p Fo(,)g(1167-)177 4621 y(1169)h(\(1977\).)1947 5564 y(28)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0308140741332--