%!PS-Adobe-2.0 %%Creator: dvips 5.76 (OzTeX) %%Title: pisafinale.dvi %%Pages: 91 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -M0 -T210mm,297mm -opisafinale.ps %+ pisafinale.dvi %DVIPSParameters: dpi=300 %DVIPSSource: TeX output 1999.12.08:1647 %%BeginProcSet: tex.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{ /sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0] N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]{ch-image}imagemask restore}B /D{/cc X dup type /stringtype ne{]} if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict /eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{dup length product length le{dup length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false} ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false RMat{BDot} imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail{dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M}B /d{ -3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{p 1 w} B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet TeXDict begin 39158280 55380996 1200 300 300 (pisafinale.dvi) @start %DVIPSBitmapFont: Fa msam10 12 1 /Fa 1 4 df 3 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmmi5 6 23 /Fb 23 123 df<07C01C3130196019601AC01AC01CC018403C20CD1F07100B7E8A17>11 D<030007E00C70040006000600030007801DC030C060C060C0C0C0C0C0C0C0408021001E 000C127E9110>14 D<03800C4018603020302060306020C060FFE0C060C040C0C0C080C1 80C300460038000C117E9011>18 D<3C000E0006000600030003000180018000C001C002 E00460187030306038C018801C0E117D9015>21 D<080818181818181818183030303030 32307270B46F1860006000C000C000C0000F107E8A15>I<3FFE7FFE4440844004400840 084018401860306030600F0B7D8A15>25 D<0F803FC0604040007E007E0080008000C0C0 7F803E000A0B7E8A11>34 D<40E060202040408003087D820B>59 D<004000C00180018001800300030003000600060006000C000C000C0018001800180030 0030003000600060006000C000C0000A197D9212>61 D<0FF9FF01C03801C03803807003 80700380700380700700E007FFE00700E00700E00E01C00E01C00E01C00E01C01C0380FF 9FF018117E901C>72 D<0FFFC001C07001C0380380380380380380380380700700E007FF 000700000700000E00000E00000E00000E00001C0000FF800015117F9016>80 D<0FFFC00E0380080700180E00101C00103C0000780000F00000E00001C0000380800701 000E01001C0100380300700E00FFFE0012117D9017>90 D<072018E0306060606060C0C0 C0C0C0C841C862D03C600D0B7E8A13>97 D<07C01820302060207FC0C000C00040006010 20601F800C0B7E8A11>101 D<0C0E0C00000000305898983030606464683007127D910D> 105 D<006000E000400000000000000000038004C008C008C000C0018001800180018003 00030003006300E600EC0078000B1780910F>I<3C000C000C0018001800180018703098 3138363038006F8060C060C860C8C0D0C0600D117E9013>I<7818183030303060606060 C0C0C8C8D06005117E900C>I<70F0F89B090C9C0E0C9C0C0C180C0C3018183018183018 1930183160303260301C180B7D8A1F>I<71F09A189C1898181818303030303032306260 6460380F0B7D8A16>I<07881850303060306030C060C060C06040E061C01EC000C000C0 0180018007E00D107E8A10>113 D<73C05C6098E098C018003000300030003000600060 000B0B7D8A11>I<0E101FE020400080010006000800102020407FC083800C0B7E8A11> 122 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmr5 6 15 /Fc 15 58 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmsl10 12 61 /Fd 61 124 df<0000FE07C00003819C60000E01F0F0001C03E1F0003807E1F0007007C0 E0007003C000007001C00000E003800000E003800000E003800000E003800000E0038000 00E00380003FFFFFF8003FFFFFF80001C007000001C007000001C007000001C007000003 800E000003800E000003800E000003800E000003800E000003800E000007001C00000700 1C000007001C000007001C000007001C000007001C00000F003C00007FE1FFC000FFE3FF C00024237FA21D>11 D<0000FE0000038100000E0080001C03C0003807C0007007C00070 07800070030000E0000000E0000000E0000000E0000000E0000000E000003FFFFF003FFF FF0001C0070001C0070001C0070001C0070003800E0003800E0003800E0003800E000380 0E0003800E0007001C0007001C0007001C0007001C0007001C0007001C000F003C00FFE3 FF80FFE3FF801A237FA21C>I<0000FF60000381E0000E03C0001C07C0003807C0007003 C0007001C0007001C000E0038000E0038000E0038000E0038000E0038000E003803FFFFF 003FFFFF0001C0070001C0070001C0070001C0070003800E0003800E0003800E0003800E 0003800E0003800E0007001C0007001C0007001C0007001C0007001C0007001C000F003C 00FFE3FF80FFE3FF801B237FA21C>I<0000FE03F80003810E04000E00B802001C03F00F 003807E01F007007C01F007007C01E007003C00C00E003800000E003800000E003800000 E003800000E003800000E00380003FFFFFFFFC3FFFFFFFFC01C007003C01C007001C01C0 07001C01C007001C03800E003803800E003803800E003803800E003803800E003803800E 003807001C007007001C007007001C007007001C007007001C007007001C00700F003C00 F0FFE3FF8FFEFFE3FF8FFE28237FA22A>I<01C003C007800F001C003800700060008000 0A0971A219>19 D<000100020004000800100020006000C0018001800300030006000600 0C000C001C00180018003800300030007000700070006000600060006000E000E000E000 E000E000E000600060006000600060003000300030001000180008000C00040002000100 10327BA413>40 D<0080004000200030001000180008000C000C000C0006000600060006 00060007000700070007000700070006000600060006000E000E000E000C000C001C0018 00180038003000300060006000C000C00180018003000600040008001000200040008000 10327FA413>I<1C3E7E7E3A02020404080810204080070F7D840E>44 DI<3078F8787005057C840E>I<0004000C007C07FC0F9C001C00 380038003800380038003800700070007000700070007000E000E000E000E000E000E001 C001C001C001C001C001C003C0FFFEFFFE0F217BA019>49 D<000FC0007FF000E0FC0180 3E02001E04001F07800F0FC00F0FC00F0FC00F0F801F03001E00001E00003C00003C0000 780000F00001E00003C0000700000E00001C0000300000600000C0000180080200080400 080800103000107FFFF0FFFFE0FFFFE018217EA019>I<001F80007FE001C0F803007807 803C07C03C0FC03C0FC03C07803C0300780000780000F00000E00001C000070000FE0000 03800001C00001E00000F00000F00000F00000F81000F87C01F0FC01F0FC01F0FC01E0F8 03E0C007C0600780381F001FFC0007F00016227DA019>I<0000100000300000700000F0 0001F00003700002E00004E00008E00010E00020E00040E00081C00181C00301C00201C0 0401C00801C0100380200380400380C00380FFFFFCFFFFFC000700000700000700000700 000700000700000F0000FFF001FFF016217DA019>I<001FC0007FF000E0780180180200 0C06000C04000C0C000C0E000C0E000C0F001807C03007E06003F1C001FF0000FE00007F 00019FC0060FE00C03E01801F03000F0600070600030C00030C00030C00030C00020E000 606000C07001803C07000FFC0003F00016227DA019>56 D<060F1F1F0E00000000000000 000000003078F8787008157C940E>58 D<030007800F800F800700000000000000000000 0000000000000000000000000018003C007C007C003C0004000400080008000800100020 00200040008000091F7D940E>I<0000040000000006000000000E000000001E00000000 1E000000003E000000003F000000004F000000004F000000008F000000008F000000010F 000000010780000002078000000207800000040780000004078000000807C000000803C0 00001003C000001003C000002003C000003FFFE000007FFFE000004001E000008001E000 008001E000010001E000010000F000020000F000020000F000040000F0001E0000F800FF 800FFF80FF800FFF8021237EA225>65 D<03FFFFE003FFFFF8003C007C003C003E003C00 1F003C001F003C000F0078000F0078000F0078001F0078001F0078003E0078003C00F000 7800F000F000F007C000FFFFC000F001F000F000F801E0007C01E0007C01E0003C01E000 3E01E0003E01E0003E03C0003C03C0007C03C0007C03C000F803C001F003C003E007C00F C0FFFFFF80FFFFFC0020227EA123>I<0000FE010007FF83001F81C6007C006E00F0003E 01E0001E03C0001E0780000E0F00000C1F00000C1E0000043E0000043C0000047C000004 7C000000F8000000F8000000F8000000F8000000F8000000F8000000F0000000F0000000 F0000010F8000010F8000020F8000020780000207C0000403C0000801E0001800F000300 0780060003E0380001FFF000003F800020247AA224>I<03FFFFE00007FFFFF800003C00 7C00003C001E00003C000F00003C000780003C00078000780003C000780003C000780003 E000780003E000780003E000780003E000F00003E000F00003E000F00003E000F00003E0 00F00003E000F00003E001E00003C001E00007C001E00007C001E000078001E0000F8001 E0000F0003C0001F0003C0001E0003C0003C0003C000780003C000F00003C003E00007C0 0FC000FFFFFF0000FFFFF8000023227EA126>I<03FFFFFF8007FFFFFF80003C001F0000 3C000700003C000300003C000300003C0001000078000100007800010000780101000078 0101000078010100007803000000F002000000F00E000000FFFE000000FFFE000000F00E 000000F006000001E004000001E004000001E004010001E004010001E000020001E00002 0003C000020003C000040003C000040003C0000C0003C000180003C000380007C001F800 FFFFFFF000FFFFFFF00021227EA122>I<03FFFFFF8007FFFFFF80003C001F00003C0007 00003C000300003C000300003C0001000078000100007800010000780001000078010100 0078010100007803000000F002000000F006000000F00E000000FFFE000000FFFE000000 F00E000001E004000001E004000001E004000001E004000001E004000001E000000003C0 00000003C000000003C000000003C000000003C000000003C000000007C0000000FFFE00 0000FFFE00000021227EA121>I<0000FE00800007FF8180001F80E300007C00370000F8 001F0001E0000F0003C0000F0007800007000F000006001F000006001E000006003E0000 02003C000002007C000002007C00000000F800000000F800000000F800000000F8000000 00F800000000F800000000F0001FFF80F0001FFF00F000007800F800007800F800007800 780000F000780000F0007C0000F0003C0000F0001E0001F0000F0001F00007C002600003 F01C600000FFF82000003FC0000021247AA227>I<03FFE0FFF803FFE0FFF8003E000F80 003C000F00003C000F00003C000F00003C000F000078001E000078001E000078001E0000 78001E000078001E000078001E0000F0003C0000F0003C0000FFFFFC0000FFFFFC0000F0 003C0000F0003C0001E000780001E000780001E000780001E000780001E000780001E000 780003C000F00003C000F00003C000F00003C000F00003C000F00003C000F00007C001F0 00FFFC3FFF00FFFC3FFF0025227EA125>I<03FFE003FFE0003E00003C00003C00003C00 003C0000780000780000780000780000780000780000F00000F00000F00000F00000F000 00F00001E00001E00001E00001E00001E00001E00003C00003C00003C00003C00003C000 03C00007C000FFFC00FFFC0013227EA112>I<000FFFC0000FFFC000003C0000003C0000 003C0000003C0000003C0000007800000078000000780000007800000078000000780000 00F0000000F0000000F0000000F0000000F0000000F0000001E0000001E0000001E00000 01E0000001E0000001E0003003C0007C03C000FC03C000FC03C000FC078000F807000080 0F0000401C0000307800000FE000001A237EA11A>I<03FFF80003FFF800003E0000003C 0000003C0000003C0000003C000000780000007800000078000000780000007800000078 000000F0000000F0000000F0000000F0000000F0000000F0000001E0000001E0000001E0 001001E0001001E0003001E0002003C0002003C0006003C0006003C000C003C001C003C0 03C007C00FC0FFFFFF80FFFFFF801C227EA11F>76 D<03FC000007FE07FC00000FFC002E 00000FC0002E00001780002E00001780002E00002780002700002780004700004F000047 00008F00004700008F00004700010F00004380010F00004380020F00008380021E000083 80041E000081C0041E000081C0081E000081C0081E000081C0101E000101C0203C000100 E0203C000100E0403C000100E0403C000100E0803C000100E0803C000200710078000200 7100780002007200780002007200780002003C0078000600380078000F003800F800FFE0 301FFF80FFE0301FFF802F227EA12E>I<0001FC00000F0780003C01E0007000F000E000 7803C000380780003C0700001E0F00001E1E00001E1E00001F3C00001F3C00001F7C0000 1F7C00001FF800001FF800001FF800001FF800001FF800001FF800003EF000003EF00000 3EF000007CF000007CF8000078F80000F8780000F0780001E03C0003C03C0007801E000F 000E001E000780380001C0E000007F800020247AA227>79 D<03FFFF800007FFFFF00000 3C00F800003C003C00003C001E00003C001F00003C001F000078001F000078001F000078 001F000078001F000078003E000078003C0000F000780000F000F00000F007C00000FFFE 000000F00E000000F003000001E003800001E001C00001E001C00001E001E00001E001E0 0001E001E00003C003E00003C003E00003C003E00003C003E00003C003E02003C003E040 07C003E040FFFC01F080FFFC00F1800000003E0023237EA125>82 D<000FC040003FF0C000F0398001C00F80018003800300038007000180070001800E0001 000E0001000E0001000F0000000F0000000F80000007F0000007FE000003FFC00001FFF0 00007FF8000007F80000007C0000003C0000001E0000001E0000001E0020001E0040001C 0040001C0060001C00600018006000380070007000F800E000EF03C000C7FF800080FE00 001A247DA21C>I<1FFFFFFF3FFFFFFF3C01E01E3001E0062001E0026001E0024001E002 4003C0024003C0028003C0028003C0028003C0020003C000000780000007800000078000 000780000007800000078000000F0000000F0000000F0000000F0000000F0000000F0000 001E0000001E0000001E0000001E0000001E0000001E0000003E00001FFFF8001FFFF800 20227AA124>I87 D<03FC000E07001F01801F01C01E00C00C00E00001C00001C0003FC003 E1C00F01C01E01C0380380780380F00384F00384F00784F00784F00B887833D01FC1E016 157D9419>97 D<0780003F80003F00000700000700000700000700000700000E00000E00 000E00000E00000E00000E00001C3F001CC1C01D00601E00701C00381C003838003C3800 3C38003C38003C38003C38003C7000787000787000707000F07000E06801C0E80380C60E 0081F80016237BA21C>I<00FF000383800607C00C07C01C0780380300780000700000F0 0000F00000F00000F00000F00000E00000E00000F000807001007001003806001C180007 E00012157C9416>I<00000F00007F00007E00000E00000E00000E00000E00000E00001C 00001C00001C00001C00001C00001C007E3803C1380700B80E00781C0038380038780070 700070F00070F00070F00070F00070E000E0E000E0E000E0F000E07001E07003E03805E0 1C19FC07E1FC18237CA21C>I<00FE000383800701C00C00E01C00E03800E07800E07000 E0FFFFE0F00000F00000F00000F00000E00000E00000F000407000803000803803000E0C 0003F00013157D9416>I<0003E0000E300018780030F80070F800E07000E00000E00001 C00001C00001C00001C00001C00001C0007FFC007FFC0003800003800003800003800007 00000700000700000700000700000700000E00000E00000E00000E00000E00000E00001E 0000FFE000FFE00015237EA20F>I<00000380001F84C00070E9C000E071C001C0380003 C0380003C03800078078000780780007807800078070000380F0000180E00003C3800002 7E00000600000004000000060000000600000007FFC00003FFF00007FFF8001C007C0030 000C0060000C0060000C00C0000C00C0000C004000180060003000300060000C03800003 FE00001A21809519>I<0078000003F8000003F000000070000000700000007000000070 00000070000000E0000000E0000000E0000000E0000000E0000000E0000001C1F80001C6 0C0001D80E0001E0070001E0070001C0070003C00E0003800E0003800E0003800E000380 0E0003800E0007001C0007001C0007001C0007001C0007001C0007001C000F003C00FFE3 FF80FFE3FF8019237FA21C>I<006000F001F001F000E000000000000000000000000000 00000001C00FC00FC003C001C001C0038003800380038003800380070007000700070007 0007000F007FE0FFC00C227FA10E>I<0000C00001E00003E00003E00001C00000000000 00000000000000000000000000000000000000000380003F80003F800007800003800003 80000700000700000700000700000700000700000E00000E00000E00000E00000E00000E 00001C00001C00001C00001C00001C00001C00003800703800F83800F87000F0600061C0 003F0000132C84A10F>I<0078000003F8000003F0000000700000007000000070000000 7000000070000000E0000000E0000000E0000000E0000000E0000000E0000001C07FC001 C07F8001C03C0001C0300001C0400001C080000383000003840000038E000003BE000003 CF00000387000007078000070380000701C0000701C0000700E0000700F0000F00F8007F E3FE00FFE3FE001A237FA21A>I<007803F803F00070007000700070007000E000E000E0 00E000E000E001C001C001C001C001C001C0038003800380038003800380070007000700 0700070007000F007FE0FFE00D237FA20E>I<01C1F807E01FC60C18301FD80E603803E0 07801C01E007801C01C007001C03C00F003803800E003803800E003803800E003803800E 003803800E003807001C007007001C007007001C007007001C007007001C007007001C00 700F003C00F0FFE3FF8FFEFFE3FF8FFE27157F942A>I<01C1F8001FC60C001FD80E0003 E0070001E0070001C0070003C00E0003800E0003800E0003800E0003800E0003800E0007 001C0007001C0007001C0007001C0007001C0007001C000F003C00FFE3FF80FFE3FF8019 157F941C>I<007F0001C1C00700E00E00701C003838003878003C70003CF0003CF0003C F0003CF0003CE00078E00078E00070F000F07000E07001C03803800E0E0003F00016157D 9419>I<00E1F8000FE60E000FE8070000F0038000E003C000E001C001C001E001C001E0 01C001E001C001E001C001E001C001E0038003C0038003C0038007800380078003800F00 03C00E0007403C0007307000070FC0000700000007000000070000000E0000000E000000 0E0000000E0000000E000000FFC00000FFC000001B1F80941C>I<007C10038210070130 0E00F01C00F03800F07800E07800E0F000E0F000E0F000E0F000E0F001C0F001C0F001C0 F001C07003C07005C0380B801C338007C380000380000380000380000700000700000700 000700000700007FE0007FE0141F7C941A>I<01C3E01FC4701FD8F003D0F001E06001C0 0003C0000380000380000380000380000380000700000700000700000700000700000700 000F0000FFF000FFF00014157F9414>I<01FC800603800C018018018018018018008018 01001E00001FE0000FFC0003FE00007E00000700400700400300400300600200600600F0 0400C8180087E00011157E9414>I<0080008000800180010001000300030007000F001F FCFFFC0E000E000E000E001C001C001C001C001C001C0038103810381038103810382038 201C4007800E1F7C9E13>I<0E0038FE03F8FE03F81E00780E00380E00381C00701C0070 1C00701C00701C00701C00703800E03800E03800E03800E03801E03802E01805E01C19FC 07E1FC16157C941C>III<1FF03FC01FF03FC001E01C0000E0100000E0200000704000 0078800000390000001E0000001C0000000E0000001F0000003700000063800000C38000 0181C0000301E0000600E0001E00F000FF03FF00FF03FF001A157F941A>I<0FFC0FF00F FC0FE001E0078000E0020000E0040000E0040000F0080000700800007010000070100000 78200000384000003840000038800000388000001D0000001D0000001E0000001C000000 0C0000000800000008000000100000001000000020000078400000F8400000F8800000F1 000000420000003C0000001C1F80941A>I<07FFF80780380600700C00E00801C0080380 080700100E00001C0000380000700000E00001C02003C0200380200700600E00401C00C0 3801C0700380FFFF8015157F9416>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmex10 12 39 /Fe 39 122 df<000800100020004000C0018003000300060006000C000C001C00180038 003800300070007000700070006000E000E000E000E000E000E000E000E000E000E000E0 00E000E000E000E0006000700070007000700030003800380018001C000C000C00060006 0003000300018000C000400020001000080D3B798117>0 D<800040002000100018000C 0006000600030003000180018001C000C000E000E0006000700070007000700030003800 380038003800380038003800380038003800380038003800380038003000700070007000 70006000E000E000C001C00180018003000300060006000C00180010002000400080000D 3B7E8117>III12 D<0000600000C0000180000300000600000E00001C0000180000380000 700000600000E00001C00001C0000380000380000780000700000700000F00000E00001E 00001E00001C00001C00003C00003C000038000038000078000078000078000078000078 0000700000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F0 0000F00000F00000F00000F00000F00000F00000F00000F0000070000078000078000078 00007800007800003800003800003C00003C00001C00001C00001E00001E00000E00000F 000007000007000007800003800003800001C00001C00000E00000600000700000380000 1800001C00000E000006000003000001800000C0000060135977811E>16 DI<00000180000003000000060000 000C000000180000003000000060000000E0000001C00000018000000380000007000000 0E0000000E0000001C0000001C000000380000007800000070000000F0000000E0000001 E0000001C0000001C0000003C00000038000000780000007800000070000000F0000000F 0000000E0000001E0000001E0000001E0000001C0000003C0000003C0000003C0000003C 000000380000007800000078000000780000007800000078000000780000007800000070 000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0 000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0 000000F0000000F0000000F0000000700000007800000078000000780000007800000078 0000007800000078000000380000003C0000003C0000003C0000003C0000001C0000001E 0000001E0000001E0000000E0000000F0000000F00000007000000078000000780000003 80000003C0000001C0000001C0000001E0000000E0000000F00000007000000078000000 380000001C0000001C0000000E0000000E00000007000000038000000180000001C00000 00E00000006000000030000000180000000C000000060000000300000001801977768125 >IIII<00000700001F00003C0000F80001E00003C0000780000780000F00001E00001E00 001E00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00 003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00 003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00 003C0000780000780000700000F00000E00001C00003C0000780000E00003C0000780000 E000007800003C00000E000007800003C00001C00000E00000F000007000007800007800 003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00 003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00 003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00 001E00001E00001E00000F000007800007800003C00001E00000F800003C00001F000007 18777A8125>26 DI<000003800000078000001F0000003C00000078000000F000 0001E0000003C0000007C00000078000000F8000000F0000001F0000001F0000001E0000 003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000 003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000 003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000 003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000 003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000003C0000 007C0000007C00000078000000F8000000F0000001F0000001E0000003C0000003C00000 078000000F0000001C00000078000000E0000000E0000000780000001C0000000F000000 0780000003C0000003C0000001E0000001F0000000F0000000F8000000780000007C0000 007C0000003C0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000 003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000 003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000 003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000 003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000 003E0000001E0000001F0000001F0000000F0000000F8000000780000007C0000003C000 0001E0000000F0000000780000003C0000001F0000000780000003801994798128>40 D<0000003000000060000000E0000001C000000180000003800000070000000E0000000E 0000001C0000003C000000380000007000000070000000E0000001E0000001C0000003C0 000003C0000007800000078000000F0000000F0000001E0000001E0000001E0000003C00 00003C0000007C0000007800000078000000F8000000F0000000F0000001F0000001F000 0001E0000003E0000003E0000003E0000007C0000007C0000007C0000007C000000FC000 000F8000000F8000000F8000000F8000001F8000001F8000001F0000001F0000001F0000 003F0000003F0000003F0000003F0000003E0000003E0000003E0000003E0000007E0000 007E0000007E0000007E0000007E0000007E0000007E0000007E0000007C0000007C0000 007C000000FC000000FC000000FC000000FC000000FC000000FC000000FC000000FC0000 00FC000000FC000000FC000000FC000000FC000000FC000000FC000000FC000000FC0000 001C5A72802C>48 DIIII<0007000700 070007000700070007000700070007000700070007000700070007000700070007000700 070007000700070007000700070007000700070007000700070007000700070007000700 070007000700070007000700070007000700070007000700070007000700070007000700 070007000700070007000700070007000700070007000700070007000700070007000700 0700070007000700070007000700070007000700070007FFFFFFFFFFFF1059808121>I< 0006001E003C007800F001E003C007C00F800F801F001F003E003E007E007E007C00FC00 FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00 FC00FC00FC00FC00FC00FC00FC00FC00FC000F2D6D7E2C>56 D58 D<003F003F003F003F003F003F003F003F003F003F 003F003F003F003F003F003F003F003F003F003F003F003F003F003F003F003F003F003F 003E007E007E007E007C00FC00F800F801F001E003E003C007800F001E003C007800E000 E00078003C001E000F00078003C003E001E001F000F800F800FC007C007E007E007E003E 003F003F003F003F003F003F003F003F003F003F003F003F003F003F003F003F003F003F 003F003F003F003F003F003F003F003F003F003F105C77802C>60 D62 DII< 000003F0000003F0000003F0000003F0000003F0000003F0000003F0000003F0000003F0 000003F0000003F0000003F0000003F0000003F0000003F0000003F0000003F0000003E0 000003E0000003E0000007E0000007E0000007E0000007E0000007E0000007E0000007E0 000007E0000007C0000007C0000007C0000007C000000FC000000FC000000FC000000FC0 00000F8000000F8000000F8000001F8000001F8000001F0000001F0000001F0000001F00 00003F0000003E0000003E0000003E0000003E0000007C0000007C0000007C0000007800 0000F8000000F8000000F0000000F0000001F0000001E0000001E0000003E0000003C000 0003C000000780000007800000078000000F0000000F0000001E0000001E0000003C0000 003C000000380000007800000070000000E0000000E0000001C0000003C0000003800000 07000000070000000E0000001C00000018000000380000007000000060000000C0000000 1C5A7F822C>I80 D<00000F00000030C000 0060E00000C1E00000C1E00000C0C00001C0000001C00000018000000380000003800000 038000000380000003800000038000000380000007800000078000000780000007800000 0780000007800000070000000F0000000F0000000F0000000F0000000F0000000F000000 0F0000000F0000000F0000000E0000001E0000001E0000001E0000001E0000001E000000 1E0000001C0000001C0000001C0000001C0000001C0000001C0000001C00000018000000 3800000038000060300000F0300000F0600000E0400000618000001F0000001B377D7F18 >82 D88 D<000000000F000000000018800000000039C00000000033E00000000073E00000000063 E000000000E1C000000000E00000000001E00000000001C00000000001C00000000003C0 0000000003C0000000000380000000000380000000000780000000000780000000000780 000000000F00000000000F00000000000F00000000000F00000000001F00000000001E00 000000001E00000000001E00000000003E00000000003E00000000003E00000000003C00 000000007C00000000007C00000000007C00000000007C00000000007C0000000000F800 00000000F80000000000F80000000000F80000000001F80000000001F80000000001F000 00000001F00000000003F00000000003F00000000003F00000000003F00000000003E000 00000007E00000000007E00000000007E00000000007E00000000007E0000000000FC000 0000000FC0000000000FC0000000000FC0000000000FC0000000001F80000000001F8000 0000001F80000000001F80000000001F80000000001F00000000003F00000000003F0000 0000003F00000000003F00000000003E00000000003E00000000007E00000000007E0000 0000007C00000000007C00000000007C00000000007C0000000000F80000000000F80000 000000F80000000000F80000000000F80000000000F00000000001F00000000001F00000 000001F00000000001E00000000001E00000000001E00000000003E00000000003C00000 000003C00000000003C00000000003C00000000007800000000007800000000007800000 00000700000000000700000000000F00000000000E00000000000E00000000001E000000 00001C00000000001C00000000703800000000F83800000000F83000000000F860000000 0070C0000000002180000000001F00000000002B6F7D7F1C>90 D<00001FF800000001FF FF80000007FFFFE000000FFFFFF000003FF00FFC00007F0000FE0000FC00003F0001F800 001F8003E0000007C007C0000003E00F80000001F00F00000000F01F00000000F83E0000 00007C3C000000003C7C000000003E78000000001E78000000001E78000000001EF80000 00001FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF00000 00000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF00000 00000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF00000 00000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF00000 00000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF00000 00000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF00000 00000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF00000 00000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF00000 00000FF0000000000F60000000000630467D7F37>92 D104 DI<0000700001F00003C0000F80001E00003C00003C0000780000F80000F000 00F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F000 00F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F000 00F00000F00001E00001E00003C00003C0000780000F00003C0000780000E00000780000 3C00000F000007800003C00003C00001E00001E00000F00000F00000F00000F00000F000 00F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F000 00F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F800007800 003C00003C00001E00000F800003C00001F000007014597A8121>110 DI<00000000000400000000000C00 000000001800000000001800000000003000000000003000000000006000000000006000 00000000C00000000000C000000000018000000000018000000000030000000000030000 0000000600000000000600000000000600000000000C00000000000C0000000000180000 000000180000000000300000000000300000000000600000000000600000000000C00000 000000C0000000000180000000000180000000000300000200000300000700000600000F 00000600003780000C00006780000C000087800018000003C00018000003C00030000001 E00030000001E00060000000F00060000000F000C0000000F000C0000000780180000000 7801800000003C01800000003C03000000001E03000000001E06000000001E0600000000 0F0C000000000F0C0000000007980000000007980000000003F00000000003F000000000 03E00000000001E00000000001C00000000000C00000002E3C7B8132>I<000000000004 00000000000C00000000000C000000000018000000000018000000000018000000000030 0000000000300000000000300000000000600000000000600000000000600000000000C0 0000000000C00000000000C0000000000180000000000180000000000180000000000300 000000000300000000000300000000000600000000000600000000000600000000000600 000000000C00000000000C00000000000C00000000001800000000001800000000001800 000000003000000000003000000000003000000000006000000000006000000000006000 00000000C00000000000C00000000000C000000000018000000000018000000000018000 0000000300000000000300000000000300000200000300000700000600000F0000060000 1F00000600003780000C00006780000C00004780000C000087800018000003C000180000 03C00018000003C00030000001E00030000001E00030000001E00060000001E000600000 00F00060000000F000C0000000F000C00000007800C00000007801800000007801800000 003C01800000003C01800000003C03000000003C03000000001E03000000001E06000000 001E06000000000F06000000000F0C000000000F0C000000000F0C000000000798000000 0007980000000007980000000003F00000000003F00000000003F00000000003E0000000 0001E00000000001E00000000001C00000000000C00000000000C00000002E5A7B8132> I<0030000030000030000030000030000030000030000030000030000030000030000030 00003000003000003000003000003000003000003000C0300CF0303C3C30F00E31C00733 8001B60000FC00007800007800003000161D7B7F21>121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmsy5 6 3 /Ff 3 49 df0 D<0C000C00CCC0EDC07F800C007F80EDC0 CCC00C000C000A0B7D8B12>3 D<081C1C3838383070706060C0C0060D7E8D0B>48 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg msbm7 8.4 8 /Fg 8 91 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmmi7 8.4 65 /Fh 65 123 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmr7 8.4 28 /Fi 28 127 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmsy7 8.4 25 /Fj 25 113 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fk cmsy10 12 49 /Fk 49 117 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fl cmmi10 12 81 /Fl 81 123 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fm msbm10 12 11 /Fm 11 92 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fn cmr10 14.4 61 /Fn 61 124 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmb10 14.4 67 /Fo 67 124 df<0007FC1FC0003FFF7FF000FE03F8F801F807F1FC03F00FF1FC07F00FE1 FC0FE00FE1FC0FE00FE0F80FE00FE0000FE00FE0000FE00FE0000FE00FE0000FE00FE000 0FE00FE0000FE00FE000FFFFFFFF00FFFFFFFF00FFFFFFFF000FE00FE0000FE00FE0000F E00FE0000FE00FE0000FE00FE0000FE00FE0000FE00FE0000FE00FE0000FE00FE0000FE0 0FE0000FE00FE0000FE00FE0000FE00FE0000FE00FE0000FE00FE0000FE00FE0000FE00F E0000FE00FE0000FE00FE0000FE00FE0000FE00FE000FFFC7FFF00FFFC7FFF00FFFC7FFF 00262A7FA923>11 D<0007FC00003FFF0000FE03C001F803E003F007E007F00FF00FE00F F00FE00FF00FE00FF00FE007E00FE003C00FE000000FE000000FE000000FE000007FFFFF F07FFFFFF07FFFFFF00FE007F00FE007F00FE007F00FE007F00FE007F00FE007F00FE007 F00FE007F00FE007F00FE007F00FE007F00FE007F00FE007F00FE007F00FE007F00FE007 F00FE007F00FE007F00FE007F00FE007F00FE007F0FFFC3FFFFFFC3FFFFFFC3FFF202A80 A921>I<0007F807FC00003FFE3FFF0000FE07FE078001F807F807C003F00FF00FC007F0 1FF01FE00FE01FE01FE00FE01FE01FE00FE01FE01FE00FE00FE00FC00FE00FE007800FE0 0FE000000FE00FE000000FE00FE000000FE00FE00000FFFFFFFFFFE0FFFFFFFFFFE0FFFF FFFFFFE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE0 0FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE0 0FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE0 0FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE0FFFC7FFC7FFEFFFC7FFC7FFEFFFC 7FFC7FFE2F2A7FA932>14 D<018007E007E00FF01FF01FE03FE07F807F00FC00F0006000 0C0C74AA1E>19 D<1C007F007F00FF80FFC0FFC07FC07FC01CC000C000C0018001800180 0300030006000C001C00380020000A157CA911>39 D<000C001C0038007000F000E001C0 03C0078007800F800F001F001F001F003E003E003E007E007E007E007C007C00FC00FC00 FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00FC007C007C007E007E007E003E00 3E003E001F001F001F000F000F800780078003C001C000E000F000700038001C000C0E3C 7BAC17>I<4000E000700038003C001C000E000F000780078007C003C003E003E003E001 F001F001F001F801F801F800F800F800FC00FC00FC00FC00FC00FC00FC00FC00FC00FC00 FC00FC00FC00FC00F800F801F801F801F801F001F001F003E003E003E003C007C0078007 800F000E001C003C0038007000E00040000E3C7DAC17>I<1C007F007F00FF80FFC0FFC0 7FC07FC01CC000C000C00180018001800300030006000C001C00380020000A157C8811> 44 DI<1C003E007F00FF80FF80FF807F003E 001C0009097C8811>I<001C00003C0000FC0007FC00FFFC00FFFC00F9FC0001FC0001FC 0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC 0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC 0001FC0001FC0001FC007FFFF87FFFF87FFFF815277CA61E>49 D<01FE0007FF800FFFE0 1E0FF03807F87C03F8FE03FCFF03FCFF03FEFF01FEFF01FE7E01FE3C01FE0003FE0003FC 0003FC0003F80007F00007F0000FE0000FC0001F80001F00003C0000780000700000E00E 01C00E03800E03001C06001C0C001C1FFFFC1FFFFC3FFFFC7FFFFCFFFFF8FFFFF8FFFFF8 17277DA61E>I<00FF000003FFC00007FFF0000F03F8001F03FC003F81FC003F81FE003F 81FE003F81FE001F81FE000F01FE000001FC000003FC000003F8000003F0000003E00000 07C00000FF000000FFE0000003F8000001FC000001FE000000FE000000FF000000FF0000 00FF800000FF803C00FF807E00FF80FF00FF80FF00FF80FF00FF00FF00FF007E01FE007C 01FC003E03F8001FFFF00007FFE00001FF000019277EA61E>I<00003800000078000000 F8000000F8000001F8000003F8000007F8000007F800000FF800001DF8000039F8000039 F8000071F80000E1F80000E1F80001C1F8000381F8000701F8000701F8000E01F8001C01 F8001801F8003801F8007001F800E001F800FFFFFF80FFFFFF80FFFFFF800003F8000003 F8000003F8000003F8000003F8000003F8000003F8000003F80000FFFF8000FFFF8000FF FF8019277EA61E>I<1000183E00F83FFFF03FFFE03FFFE03FFFC03FFF003FFE003FF800 38000038000038000038000038000038000038FE003FFF803F07E03803F01003F80001F8 0001FC0001FC0001FE0001FE1801FE7C01FEFE01FEFE01FEFE01FEFE01FCFC01FC7801FC 7003F83803F01E0FE00FFFC007FF0001FC0017277DA61E>I<000FE000007FF00000FFF8 0003F83C0007E07E000FC0FE000FC0FE001F80FE003F80FE003F807C007F0000007F0000 007F0000007F000000FF000000FF1F8000FF7FE000FFE1F000FFC0FC00FF807E00FF807E 00FF807F00FF007F00FF007F80FF007F80FF007F80FF007F807F007F807F007F807F007F 803F007F803F007F003F007F001F807E000F80FC0007C1F80003FFF00001FFE000007F80 0019277EA61E>I<380000003E0000003FFFFF803FFFFF803FFFFF807FFFFF007FFFFE00 7FFFFC007FFFFC00700038007000700070007000E000E000E001C000E003800000038000 00070000000F0000000E0000001E0000001E0000003C0000003C0000007C0000007C0000 00FC000000F8000000F8000001F8000001F8000001F8000001F8000003F8000003F80000 03F8000003F8000003F8000003F8000003F8000001F0000000E0000019297DA81E>I<00 7F000001FFE00003FFF0000781F8000F007C001E007C001E003E003E003E003E003E003F 003E003F003E003FC03C003FE07C001FF878001FFCF0000FFFE0000FFFC00007FFE00001 FFF00003FFF80007FFFC000F1FFE001E0FFF003C03FF007C01FF8078007F80F8003F80F8 001F80F8000F80F8000F80F8000F80F8000F007C000F007C001E003E003C001F80FC000F FFF00003FFE00000FF000019277EA61E>I<007F000003FFC00007FFE0000FC1F0001F80 F8003F00FC007F007E007F007E00FF007F00FF007F00FF007F00FF007F00FF007F80FF00 7F80FF007F80FF007F807F007F807F00FF803F00FF803F00FF801F81FF8007C3FF8003FF 7F8000FC7F8000007F8000007F0000007F0000007F0000007F001F00FE003F80FE003F80 FC003F80FC003F81F8003F03F0001E07E0000FFFC00007FF000001FC000019277EA61E> I<00003C000000003C000000003C000000007E000000007E00000000FF00000000FF0000 0000FF00000001FF80000001FF80000001FF80000003FFC0000003BFC0000003BFC00000 071FE00000071FE00000071FE000000E0FF000000E0FF000000E0FF000001C07F800001C 07F800001C07F800003803FC00003803FC00007803FE00007001FE00007FFFFE0000FFFF FF0000FFFFFF0000E000FF0001E0007F8001C0007F8001C0007F800380003FC00380003F C00380003FC00700001FE0FFF803FFFFFFF803FFFFFFF803FFFF28297EA82D>65 DI<0000FF 8030000FFFE070003FFFF8F0007FC07FF001FE000FF003FC0007F007F80003F00FF00001 F00FE00001F01FE00000F03FC00000F03FC00000707FC00000707F800000707F80000070 7F80000000FF80000000FF80000000FF80000000FF80000000FF80000000FF80000000FF 80000000FF80000000FF800000007F800000007F800000007F800000707FC00000703FC0 0000703FC00000701FE00000E00FE00000E00FF00001C007F80001C003FC00038001FE00 0F00007FC03E00003FFFFC00000FFFF0000000FF800024297DA82B>IIII<0000FF8030000FFFE070003FFFF8F0007FC07F F001FE000FF003FC0007F007F80003F00FF00001F00FE00001F01FE00000F03FC00000F0 3FC00000707FC00000707F800000707F800000707F80000000FF80000000FF80000000FF 80000000FF80000000FF80000000FF80000000FF80000000FF80000000FF8007FFFF7F80 07FFFF7F8007FFFF7F80000FF07FC0000FF03FC0000FF03FC0000FF01FE0000FF00FE000 0FF00FF0000FF007F8000FF003FC000FF001FF001FF0007FC07BF0003FFFF1F0000FFFE0 F00000FF803028297DA82F>III< FFFFF01FFF00FFFFF01FFF00FFFFF01FFF0003FC0001E00003FC0003C00003FC00078000 03FC000F000003FC001E000003FC003C000003FC0078000003FC00F0000003FC00E00000 03FC01C0000003FC0380000003FC0700000003FC0E00000003FC1F00000003FC3F000000 03FC7F80000003FCFF80000003FDFFC0000003FF9FE0000003FF1FE0000003FE0FF00000 03FC07F0000003FC07F8000003FC03FC000003FC03FC000003FC01FE000003FC00FF0000 03FC00FF000003FC007F800003FC007F800003FC003FC00003FC001FE00003FC001FE000 03FC000FF00003FC0007F000FFFFF07FFF80FFFFF07FFF80FFFFF07FFF8029297EA82F> 75 DII< FFFC001FFFC0FFFC001FFFC0FFFE001FFFC003FF0000700003FF8000700003FF80007000 03BFC000700003BFE0007000039FE0007000038FF00070000387F80070000387F8007000 0383FC0070000381FE0070000381FE0070000380FF00700003807F80700003807FC07000 03803FC0700003801FE0700003801FF0700003800FF07000038007F87000038003FC7000 038003FC7000038001FE7000038000FF7000038000FF70000380007FF0000380003FF000 0380003FF0000380001FF0000380000FF0000380000FF00003800007F00003800003F000 03800001F00003800001F000FFFE0000F000FFFE00007000FFFE000070002A297EA82F> I<0003FF0000001FFFE000007F03F80000FC00FC0003F8007F0007F0003F800FE0001FC0 0FE0001FC01FC0000FE01FC0000FE03FC0000FF03FC0000FF07F800007F87F800007F87F 800007F8FF800007FCFF800007FCFF800007FCFF800007FCFF800007FCFF800007FCFF80 0007FCFF800007FCFF800007FCFF800007FCFF800007FC7F800007F87F800007F87FC000 0FF83FC0000FF03FC0000FF03FC0000FF01FE0001FE00FE0001FC00FF0003FC007F0003F 8003F8007F0001FC00FE00007F03F800001FFFE0000003FF000026297DA82D>II<0003FF0000001F FFE000007F03F80000FC00FC0003F8007F0007F0003F800FF0003FC00FE0001FC01FE000 1FE01FC0000FE03FC0000FF03FC0000FF07FC0000FF87F800007F87F800007F8FF800007 FCFF800007FCFF800007FCFF800007FCFF800007FCFF800007FCFF800007FCFF800007FC FF800007FCFF800007FCFF800007FC7F800007F87F800007F87F800007F83FC0000FF03F C0000FF03FC0000FF01FE0781FE00FE0FE1FC00FF1873FC007F3033F8003FB01FF0001FF 01FE00007F83F800001FFFE0000003FFE00C000000F00C000000F00C000000F83C000000 FFFC000000FFFC0000007FF80000007FF80000007FF80000003FF00000003FE00000001F C0000000078026357DA82D>II<00FE030003FFC7000FFFEF001F81FF003E007F003E00 1F007C001F007C000F00FC000F00FC000700FC000700FC000700FE000700FF000000FF80 00007FF800007FFF80003FFFF0001FFFF8001FFFFC000FFFFE0003FFFF0000FFFF00001F FF800001FF8000003FC000001FC000001FC060000FC0E0000FC0E0000FC0E0000FC0F000 0FC0F0000F80F8000F80FC001F00FE001E00FFC07E00F7FFFC00E1FFF000C03FC0001A29 7DA821>I<7FFFFFFFF07FFFFFFFF07FFFFFFFF07F01FE07F07C01FE01F07801FE00F070 01FE00707001FE0070F001FE0078F001FE0078E001FE0038E001FE0038E001FE0038E001 FE0038E001FE00380001FE00000001FE00000001FE00000001FE00000001FE00000001FE 00000001FE00000001FE00000001FE00000001FE00000001FE00000001FE00000001FE00 000001FE00000001FE00000001FE00000001FE00000001FE00000001FE00000001FE0000 0001FE00000001FE000001FFFFFE0001FFFFFE0001FFFFFE0025287EA72A>II89 D<3FFFFFFC3FFFFFFC3FFFFFFC3FE007F83F800FF83F000FF03E001FE03C001FE078003F C078007FC078007F807000FF007000FF007001FE007003FE000003FC000007F8000007F8 00000FF000001FF000001FE000003FE000003FC000007F8000007F801C00FF001C01FF00 1C01FE001C03FC001C03FC003C07F8003C0FF8003C0FF000381FE000781FE000F83FC001 F87FC003F87F801FF8FFFFFFF8FFFFFFF8FFFFFFF81E297DA825>I<07FE00001FFF8000 3E07E0007F03F0007F03F8007F01F8003E01FC001C01FC000001FC000001FC000001FC00 003FFC0001FFFC0007F9FC001FC1FC003F81FC007F01FC007E01FC00FE01FC00FE01FC00 FE01FC00FE01FC007E03FC007F03FC003F0EFFC00FFC7FC003F01FC01A1B7E9A1D>97 DI<007FC001 FFF007E0F80FC1FC1F81FC3F81FC3F00F87F00707F0000FF0000FF0000FF0000FF0000FF 0000FF0000FF0000FF0000FF00007F00007F00003F801C3F801C1F80380FC03807E0F001 FFC0007F00161B7E9A1B>I<0001FFC00001FFC00001FFC000001FC000001FC000001FC0 00001FC000001FC000001FC000001FC000001FC000001FC000001FC000001FC000001FC0 007F1FC001FFDFC007E0FFC00FC03FC01F801FC03F801FC03F001FC07F001FC07F001FC0 7F001FC0FF001FC0FF001FC0FF001FC0FF001FC0FF001FC0FF001FC0FF001FC0FF001FC0 7F001FC07F001FC03F001FC03F801FC01F803FC00F807FC007C1FFFC03FFDFFC007E1FFC 1E2A7EA921>I<007F0003FFC007C3E00F81F01F80F83F80FC3F00FC7F007C7F007EFF00 7EFF007EFF007EFFFFFEFFFFFEFF0000FF0000FF00007F00007F00007F00003F000E3F80 0E1F801C0FC03807E0F001FFE0007F80171B7E9A1C>I<001FC0007FF001F8F803F1FC07 F1FC07E1FC0FE1FC0FE0F80FE0000FE0000FE0000FE0000FE0000FE0000FE0007FFF007F FF007FFF000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000F E0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE000FFFF00FF FF00FFFF00162A7FA912>I<01FE070007FF9F801F87FFC03F03F7C07E01FB807E01F900 FE01FC00FE01FC00FE01FC00FE01FC00FE01FC00FE01FC007E01F8007E01F8003F03F000 1F87E0003FFF800031FE0000700000007000000070000000780000007FFFC0007FFFF000 3FFFFC001FFFFE000FFFFF001FFFFF003C003F8078001F80F8000F80F8000F80F8000F80 F8000F8078000F007C001F003E003E001F80FC0007FFF00000FF80001A287E9A1E>II<07000F801FC0 3FE03FE03FE01FC00F8007000000000000000000000000000000FFE0FFE0FFE00FE00FE0 0FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE0 0FE0FFFEFFFEFFFE0F2B7FAA10>I<000E00001F00003F80007FC0007FC0007FC0003F80 001F00000E0000000000000000000000000000000000000000000001FFC001FFC001FFC0 001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0 001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0 001FC0001FC0001FC0001FC07C1FC0FE1FC0FE1F80FE3F00FE3F007C3C003FF8000FE000 123784AA12>IIIII<007F000003FFE00007E3F0000F80F8001F80FC 003F007E003F007E007F007F007F007F00FF007F80FF007F80FF007F80FF007F80FF007F 80FF007F80FF007F80FF007F80FF007F807F007F007F007F007F007F003F007E001F80FC 000F80F80007E3F00003FFE000007F0000191B7E9A1E>II114 D<03F8C01FFFC03C0FC07003 C07001C0F001C0F001C0F801C0FC0000FFC0007FF8007FFE003FFF001FFF800FFFC001FF C0000FE00003E06003E0E001E0E001E0F001E0F001C0F80380FE0700FFFE00C3F800131B 7E9A18>I<01C00001C00001C00001C00001C00003C00003C00003C00007C00007C0000F C0003FFF80FFFF80FFFF801FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001F C0001FC0001FC0001FC0001FC0001FC1C01FC1C01FC1C01FC1C01FC1C01FC1C00FC1C00F C38007E38003FF0000FC0012267FA517>IIII II<7FFFF87FFFF87E07F07807F0700FE0F0 1FE0F01FC0E03F80E03F80E07F00007F0000FE0001FE0001FC0003FC0003F81C07F01C07 F01C0FE01C0FE03C1FC03C3FC0383F80787F00F87F03F8FFFFF8FFFFF8161B7E9A1B>I< FFFFFFF8FFFFFFF81D0280911E>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmti10 12 62 /Fp 62 124 df<00001FC0F8000070718E0000E0F31E0001C0F71E0001C0660C0001800E 000003800E000003800E000003800E000003801C000007001C000007001C000007001C00 0007001C0000FFFFFFC000FFFFFFC0000E003800000E003800000E003800000E00700000 1C007000001C007000001C007000001C007000001C00E000003800E000003800E0000038 00E000003800E000003801C000007001C000007001C000007001C000007001C000007003 800000E003800000E003800000E003000000C007000001C007000071CE060000F19E0C00 00F31E1C0000630C3000003C07E00000272D82A21F>11 D<00000FF00000381C0000E00E 0000C01E0001C01E0003801C000380000003800000038000000700000007000000070000 000700000007000000FFFFF000FFFFF0000E0070000E00E0000E00E0001C00E0001C00E0 001C01C0001C01C0001C01C0003801C00038038000380380003803880038038C00700718 0070071800700718007007100070033000E001E000E0000000E0000000E0000001C00000 01C0000071C00000F1800000F3000000620000003C0000001F2D82A21C>I<00001FF380 0000703F000000E07F000001C07F000001C037000003800E000003800E000003800E0000 03800E000007001C000007001C000007001C000007001C00000700380000FFFFF80000FF FFF800000E003800000E007000000E007000001C007000001C007000001C00E000001C00 E000001C00E000003800E000003801C000003801C000003801C400003801C6000070038C 000070038C000070038C00007003980000F001980000E000F00000E000000000E0000000 00E000000001C000000001C00000007180000000F180000000F30000000062000000003C 00000000212D82A21D>I<60F070703838181C0C0C060A72A219>18 D<0180038007800F000E001C0038007000E000C000090A70A219>I<007000F001F001F0 01F001E001E003E003C003C003C003C00380078007800700070007000E000E000E000C00 0C001C0018001800000000000000000000007000F800F800F000E0000C247BA30F>33 D<0C1E3F3F1D02020204040810204080080F76A20F>39 D<000080000100000200000400 00080000100000300000600000C00000C0000180000380000300000700000600000E0000 0C00001C00001C0000180000380000380000300000700000700000700000600000E00000 E00000E00000E00000C00000C00000C00000C00000C00000C00000C00000C00000C00000 C00000C00000C00000400000600000600000200000300000100000080000113279A414> I<0008000004000006000002000003000003000001000001800001800001800001800001 800001800001800001800001800001800001800001800003800003800003800003800003 00000700000700000700000600000E00000E00000C00001C00001C000018000038000030 0000700000600000E00000C0000180000180000300000600000400000800001000002000 00400000800000113280A414>I<1C3C3C3C3C04040808101020408080060F7D840F>44 DI<70F8F8F0E005057B840F>I<0001000300070006001E00 3E03EE03DC001C001C001C0038003800380038007000700070007000E000E000E000E001 C001C001C001C00380038003800780FFFCFFF810217BA019>49 D<00800800E07000FFE0 01FFC001FF8001FE00018000030000030000030000030000060000067C00068300070380 0C03800C01C00001C00003C00003C00003C00003C00003C0700780F00780F00780E00F00 C00E00C01E00C01C00E0380070F0003FC0001F000015227BA019>53 D<000F80003FE00070E000E07001C0700380380380380700700700700700700780E00780 C007C18003E30001FC0000FC0000FE00033F00061F000C0F801807803003807003806003 80E00380E00380E00300E00700E00600E00E00E01C007878003FE0000F800015227BA019 >56 D<07000F800F800F000E000000000000000000000000000000000000000000000070 00F800F800F000E00009157B940F>58 D<00000180000003800000038000000780000007 C000000FC000000FC000001BC000003BC0000033C0000063C0000063C00000C3C00000C3 C0000183C0000183C0000303C0000703C0000603C0000C03C0000C03C0001803E0001FFF E0003FFFE0003001E0006001E000E001E000C001E0018001E0018001E0030001E0030001 E00F0001E07FE01FFEFFC01FFE1F237EA225>65 D<00FFFFE000FFFFF8000F007C000F00 3E001E001E001E001E001E001F001E001F003C001E003C003E003C003E003C007C007800 78007800F0007801E0007807C000FFFF8000F001E000F000F000F000F801E0007801E000 7801E0007C01E0007C03C0007803C000F803C000F803C001F0078001E0078003E007800F C007801F00FFFFFE00FFFFF00020227DA123>I<00007F01800003FF8300000FC0E70000 3E0067000078003F0000F0001E0001E0001E0003C0001E000780001E000F80000C000F00 000C001F00000C003E00000C003E000018007C000000007C000000007C00000000F80000 0000F800000000F800000000F800000000F800000000F000000000F000006000F0000060 00F00000C000F00000C000F800018000780001800078000300003C000600001E000C0000 1F0038000007C0F0000003FFC0000000FE000000212479A224>I<00FFFFF00000FFFFFC 00000F003E00000F000F00001E000780001E000780001E0003C0001E0003C0003C0003C0 003C0003E0003C0003E0003C0003E000780003E000780003E000780003E000780003E000 F00003C000F00007C000F00007C000F00007C001E0000F8001E0000F8001E0000F0001E0 001F0003C0001E0003C0003C0003C0007C0003C0007800078000F000078001E000078007 800007801F0000FFFFFC0000FFFFF0000023227DA126>I<00FFFFFF8000FFFFFF80000F 000F80000F000780001E000380001E000380001E000300001E000300003C000300003C03 0300003C030300003C0303000078060000007806000000781E0000007FFE000000FFFC00 0000F01C000000F01C000000F01C000001E018000001E0180C0001E0180C0001E0001800 03C000180003C000300003C000300003C00070000780006000078000E000078003C00007 800FC000FFFFFFC000FFFFFF800021227DA122>I<0000FE030003FF86000F81CE003E00 6E0078007E00F0003C01E0003C03C0003C0780001C0F8000180F0000181F0000183E0000 183E0000307C0000007C0000007C000000F8000000F8000000F8000000F8000000F8007F FCF000FFF8F00003C0F00003C0F0000780F0000780F80007807800078078000F003C000F 003C001F001F0037000FC0E60003FFC20000FE0000202478A227>71 D<00FFF87FFC00FFF87FFC000F000780000F000780001E000F00001E000F00001E000F00 001E000F00003C001E00003C001E00003C001E00003C001E000078003C000078003C0000 78003C00007FFFFC0000FFFFF80000F000780000F000780000F000780001E000F00001E0 00F00001E000F00001E000F00003C001E00003C001E00003C001E00003C001E000078003 C000078003C000078003C000078003C000FFF87FFC00FFF87FFC0026227DA125>I<0007 FFC00007FF8000003C0000003C00000078000000780000007800000078000000F0000000 F0000000F0000000F0000001E0000001E0000001E0000001E0000003C0000003C0000003 C0000003C00000078000000780000007800000078000000F0000000F0000380F0000780F 0000F81E0000F81E0000F03C0000403800006070000030E000001F8000001A237DA11A> 74 D<00FFF80FFC00FFF80FFC000F0007C0000F000700001E000E00001E001C00001E00 3800001E006000003C00C000003C018000003C030000003C0E000000781C000000783800 0000787800000078FC000000F1BC000000F33C000000FE1E000000FC1E000001F01F0000 01E00F000001E00F000001E007800003C007800003C007800003C003C00003C003C00007 8003C000078001E000078001E000078001F000FFF80FFE00FFF80FFE0026227DA126>I< 00FFFC0000FFFC00000F0000000F0000001E0000001E0000001E0000001E0000003C0000 003C0000003C0000003C00000078000000780000007800000078000000F0000000F00000 00F0000000F0000001E0000001E000C001E000C001E0018003C0018003C0018003C00300 03C003000780070007800E0007801E0007807E00FFFFFC00FFFFFC001A227DA11F>I<00 FF80000FFC00FF80000FFC000F80001F80000F80003780001B80003F00001B80006F0000 1B80006F00001B8000CF00003380019E00003380019E00003380031E000031C0031E0000 61C0063C000061C0063C000061C00C3C000061C0183C0000C1C018780000C1C030780000 C1C030780000C1C06078000181C0C0F0000181C0C0F0000181C180F0000180E180F00003 00E301E0000300E601E0000300E601E0000300EC01E0000600EC03C0000600F803C00006 00F803C0001F00F003C000FFE0E07FFC00FFE0E07FFC002E227DA12D>I<00FF001FFC00 FF801FFC000F8003C0000F800380001BC00300001BC00300001BC003000019E003000031 E006000031E006000030F006000030F006000060F00C000060780C000060780C00006078 0C0000C03C180000C03C180000C03C180000C01E180001801E300001801E300001800F30 0001800F300003000F6000030007E000030007E000030007E000060007C000060003C000 060003C0001F0003C000FFE0018000FFE001800026227DA125>I<00FFFFE000FFFFF800 0F007C000F001E001E001F001E000F001E000F001E000F003C001F003C001F003C001F00 3C001E0078003E0078003C00780078007800F000F003C000FFFF0000F0000000F0000001 E0000001E0000001E0000001E0000003C0000003C0000003C0000003C000000780000007 8000000780000007800000FFF80000FFF8000020227DA122>80 D<00FFFFC000FFFFF000 0F00F8000F003C001E003E001E001E001E001E001E001E003C003E003C003E003C003E00 3C007C00780078007800F0007801E00078078000FFFC0000F00C0000F00E0000F0070001 E0070001E0078001E0078001E0078003C00F8003C00F8003C00F8003C00F8007801F8007 801F8607801F8607800F8CFFF80F8CFFF80798000001F01F237DA124>82 D<0001F060000FFCC0001E0FC0003807C0007003C000E0038000C0038001C0038001C003 80038003000380030003C0000003C0000003E0000001F8000001FF000000FFE000007FF0 00001FF8000003FC0000007C0000003C0000001E0000001E0000001E0030001C0030001C 0030001C00300018007000380070007000780060007C01C000EF038000C7FF000081FC00 001B247DA21C>I<1FFFFFF81FFFFFF81E03C0783803C038380780383007803860078030 60078030600F0030C00F0030C00F0030C00F0030001E0000001E0000001E0000001E0000 003C0000003C0000003C0000003C00000078000000780000007800000078000000F00000 00F0000000F0000000F0000001E0000001E0000001E0000003E00000FFFF8000FFFF8000 1D2277A124>I<3FFE07FF3FFE07FF03C000F003C000E0078000C0078000C0078000C007 8000C00F0001800F0001800F0001800F0001801E0003001E0003001E0003001E0003003C 0006003C0006003C0006003C00060078000C0078000C0078000C0078000C00F0001800F0 001800F000300070003000700060007000C00038018000380300001E0E00000FFC000003 F00000202377A125>I87 D89 D<00786001C6E00302E00603C00E03C01C01C03C01C0380380 780380780380780380F00700F00700F00710F00718F00E30701E30701E30302E6018C660 0F83C015157C9419>97 D<03C01F803F800380038007000700070007000E000E000E000E 001C001CF81D8C1E0E3C063C073807380F700F700F700F700FE01EE01EE01EE03CE038E0 38607060E031C01F0010237BA217>I<007E0001C3000301800703800E07801C07803C00 00380000780000780000780000F00000F00000F00000F00000F00100700300700600301C 001870000FC00011157B9417>I<00001E0000FC0001FC00001C00001C00003800003800 00380000380000700000700000700000700000E00078E001C6E00302E00603C00E03C01C 01C03C01C0380380780380780380780380F00700F00700F00710F00718F00E30701E3070 1E30302E6018C6600F83C017237CA219>I<00F8038C0E061C063C063806780CF038FFE0 F000F000E000E000E000E000E002E006600C703830E00F800F157A9417>I<00007C0000 CE00019E00039E00038C000380000700000700000700000700000700000E00000E00000E 0000FFE000FFE0001C00001C00001C00001C00001C000038000038000038000038000038 0000700000700000700000700000F00000E00000E00000E00001E00001C00001C00001C0 0001C000038000738000F30000F300006600003C0000172D82A20F>I<001F180031B800 E0F801C0F001C0700380700780700700E00F00E00F00E00F00E01E01C01E01C01E01C01E 01C01E03800E03800E0780060F80061F0001E700000700000700000E00000E00000E0070 1C00F01800F0300060E0003F8000151F7E9417>I<00F00007E0000FE00000E00000E000 01C00001C00001C00001C000038000038000038000038000070000071F0007618007C0C0 0F80E00F00E00F00E00E00E01E01C01C01C01C01C01C01C038038038038038038838070C 700718700618700E10700630E006206003C016237DA219>I<00E000E001E000C0000000 0000000000000000000000000000000E00330023806380C380C700C70007000E000E000E 001C001C001C40386038C038C07080318033001E000B227CA10F>I<0001C00003C00003 C0000180000000000000000000000000000000000000000000000000000000003C000066 0000C700018700010700030700030700000E00000E00000E00000E00001C00001C00001C 00001C0000380000380000380000380000700000700000700000700000E00000E00000E0 0071C000F18000F380006600003C0000122C82A10F>I<00F00007E0000FE00000E00000 E00001C00001C00001C00001C0000380000380000380000380000700000701E007023007 04700E08F00E10F00E20600E40001D80001E00001FC0001C7000383800383800381C2038 1C307038607038607038607018C0E01880600F0014237DA217>I<01E00FC01FC001C001 C0038003800380038007000700070007000E000E000E000E001C001C001C001C00380038 00380038007000700071007180E300E300E300E60066003C000B237CA20D>I<1E07C07C 00331861860063B033030063E03E0380C3C03C0380C3C03C0380C3803803800780780700 0700700700070070070007007007000E00E00E000E00E00E000E00E00E100E00E01C181C 01C01C301C01C038301C01C038601C01C0184038038018801801800F0025157C9429>I< 1C07C0261860672030C74038C78038C78038C700380F00700E00700E00700E00701C00E0 1C00E01C00E21C01C33801C638018638038438018C7001883000F018157C941C>I<003E 0000C1800380C00700E00E00E01C00F03C00F03C00F07800F07800F07800F0F001E0F001 E0F001C0F003C0700380700700700600381C001C380007E00014157C9419>I<01C0F002 631C06740E0C7C0E0C78070C70070C700F00E00F00E00F00E00F00E00F01C01E01C01E01 C01E01C03C03C03803C03803C07003C0E0072180071E000700000700000E00000E00000E 00000E00001C00001C0000FFC000FF8000181F809419>I<00F840018CC00707C00E0780 0E03801C03803C0380380700780700780700780700F00E00F00E00F00E00F00E00F01C00 701C00703C00307C0030F8000F380000380000380000700000700000700000700000E000 00E0000FFE000FFC00121F7B9417>I<1C1F002630C06741C0C7C3C0C783C0C70180C700 000E00000E00000E00000E00001C00001C00001C00001C00003800003800003800003800 0070000030000012157C9415>I<00FC0106020304070C0F0C0F0C040F000FE007F803FC 007C001E000E700EF00CF00CE008601030601F8010157D9414>I<00C001C001C001C001 C003800380038003800700FFF8FFF807000E000E000E000E001C001C001C001C00380038 003810381870307030706070C031801E000D1F7C9E11>I<0F003011807021C07061C0E0 C1C0E0C380E0C380E00381C00701C00701C00701C00E03800E03800E03840E03860E070C 0C070C0E07080E0F1806131003E1E017157C941B>I<0E00C03301E06383E06381E0C380 E0C700E0C700E00700C00E00C00E00C00E00C01C01801C01801C01801C03001C03001C02 001C04000C0C0006180003E00013157C9417>I<0F0030701180707821C070F861C0E078 C1C0E038C380E038C380E0380381C0300701C0300701C0300701C0300E0380600E038060 0E0380600E0380C00E0380C00E0780800E078180060D83000319C60001F07C001D157C94 21>I<03C1C00C6630183C70303CF02038F0603860603800007000007000007000007000 00E00000E00000E02000E03061C060F1C060F1C0C0E3C0804663003C3E0014157D9417> I<0E00303300706380706380E0C380E0C700E0C700E00701C00E01C00E01C00E01C01C03 801C03801C03801C03801C07001C07001C0F001C0F000C3E0003CE00000E00000E00001C 00601C00F03800F03000E06000C0C0006180003E0000141F7C9418>I<00E06003F06007 F8C0060F800C03000C0300000600000C0000180000300000600000C00001800003008006 00C00C01801803803F870071FE0060FC00C0780013157E9414>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmb10 17.28 60 /Fq 60 124 df<00007FC03FC00007FFE1FFF0001FFFFFFFF8007FC0FFF1FC00FF01FFC3 FE01FE03FF83FE03FC03FF83FE03FC03FF03FE07F803FF01FC07F801FF00F807F800FF00 7007F800FF000007F800FF000007F800FF000007F800FF000007F800FF000007F800FF00 0007F800FF0000FFFFFFFFFE00FFFFFFFFFE00FFFFFFFFFE0007F800FF000007F800FF00 0007F800FF000007F800FF000007F800FF000007F800FF000007F800FF000007F800FF00 0007F800FF000007F800FF000007F800FF000007F800FF000007F800FF000007F800FF00 0007F800FF000007F800FF000007F800FF000007F800FF000007F800FF000007F800FF00 0007F800FF000007F800FF000007F800FF000007F800FF000007F800FF000007F800FF00 00FFFF8FFFFC00FFFF8FFFFC00FFFF8FFFFC002F327FB12A>11 D<1F003F807FC0FFE0FF E0FFF0FFF0FFF07FF03FF01F700070007000E000E000E001C001C00380038007000E001E 003C0018000C197CB114>39 D<1F003F807FC0FFE0FFE0FFF0FFF0FFF07FF03FF01F7000 70007000E000E000E001C001C00380038007000E001E003C0018000C197C8A14>44 DI<1F003F807FC0FF E0FFE0FFE0FFE0FFE07FC03F801F000B0B7C8A14>I<001FC00000FFF80003FFFE0007F0 7F000FE03F800FC01F801FC01FC03F800FE03F800FE03F800FE07F800FF07F800FF07F80 0FF07F800FF0FF800FF8FF800FF8FF800FF8FF800FF8FF800FF8FF800FF8FF800FF8FF80 0FF8FF800FF8FF800FF8FF800FF8FF800FF8FF800FF8FF800FF8FF800FF8FF800FF8FF80 0FF8FF800FF87F800FF07F800FF07F800FF07F800FF03F800FE03F800FE03FC01FE01FC0 1FC01FC01FC00FE03F8007F8FF0003FFFE0000FFF800001FC0001D2E7DAD24>48 D<00078000000F8000003F800000FF80001FFF8000FFFF8000FFFF8000E0FF800000FF80 0000FF800000FF800000FF800000FF800000FF800000FF800000FF800000FF800000FF80 0000FF800000FF800000FF800000FF800000FF800000FF800000FF800000FF800000FF80 0000FF800000FF800000FF800000FF800000FF800000FF800000FF800000FF800000FF80 0000FF800000FF800000FF800000FF800000FF800000FF800000FF80007FFFFF807FFFFF 807FFFFF80192E7BAD24>I<007FC00001FFF80007FFFE000F81FF001E007F803C007FC0 7F003FE07F803FF0FFC01FF0FFC01FF0FFC01FF8FFC01FF87F801FF87F801FF83F001FF8 00001FF800001FF000001FF000003FF000003FE000003FC000007F8000007F000000FE00 0001FC000001F8000003F0000007E0000007C000000F8000001F0038003E003800780038 0070007800E0007001C00070038000F007FFFFF00FFFFFF00FFFFFF01FFFFFF03FFFFFF0 7FFFFFE0FFFFFFE0FFFFFFE0FFFFFFE01D2E7DAD24>I<003FC00001FFF80003C1FE0007 00FF000E007F801FC07FC01FE07FC03FE07FE03FF07FE03FF07FE03FF07FE03FF07FE01F E07FC00FC07FC003007FC000007F800000FF000000FE000001FC000003F800007FF00000 7FC000007FF8000001FE0000007F8000007FC000007FE000003FE000003FF000003FF00E 003FF83F803FF87FC03FF87FC03FF8FFE03FF8FFE03FF8FFE03FF0FFE03FF0FFC03FF07F 807FE03F007FC03E007F801FC1FF0007FFFE0003FFF800007FC0001D2E7DAD24>I<0000 03C0000007C0000007C000000FC000001FC000003FC000003FC000007FC00000FFC00001 FFC00001FFC00003FFC00007BFC000073FC0000F3FC0001E3FC0003C3FC000383FC00078 3FC000F03FC000E03FC001E03FC003C03FC007803FC007003FC00F003FC01E003FC01C00 3FC038003FC078003FC0F0003FC0FFFFFFFEFFFFFFFEFFFFFFFE00007FC000007FC00000 7FC000007FC000007FC000007FC000007FC000007FC000007FC0003FFFFE003FFFFE003F FFFE1F2E7EAD24>I<0C0000C01F800FC01FFFFFC01FFFFF801FFFFF001FFFFE001FFFFC 001FFFF8001FFFE0001FFF80001FFC00001E0000001E0000001E0000001E0000001E0000 001E0000001E0000001E1FE0001EFFF8001FFFFE001FE07F001F003F801E003FC00C001F E000001FF000001FF000001FF000001FF800001FF81E001FF83F001FF87F801FF8FF801F F8FFC01FF8FFC01FF8FF801FF0FF801FF07F001FE07C003FE038003FC01E007F800F81FF 0007FFFC0003FFF000007F80001D2E7DAD24>I<0003F800001FFE00003F070000FC0180 01F803C003F00FC007F01FE00FE03FE00FE03FE01FE03FE03FE03FE03FC01FC03FC00F80 7FC000007FC000007FC000007FC10000FFCFF800FFDFFE00FFFFFF00FFF03F80FFE01FC0 FFE01FE0FFE01FE0FFC01FF0FFC01FF0FFC01FF0FFC01FF8FFC01FF8FFC01FF8FFC01FF8 7FC01FF87FC01FF87FC01FF87FC01FF83FC01FF83FC01FF03FC01FF01FC01FE01FC01FE0 0FE01FC007E03F8003F87F0001FFFE00007FF800001FE0001D2E7DAD24>I<380000003E 0000003FFFFFFC3FFFFFFC3FFFFFFC3FFFFFF87FFFFFF07FFFFFE07FFFFFE07FFFFFC07F FFFF8078000F0070001F0070001E00F0003C00E0007C00E000F800E000F0000001E00000 03E0000003C0000007C0000007C000000F8000000F8000001F8000001F8000003F800000 3F8000007F0000007F0000007F0000007F000000FF000000FF000000FF000000FF000001 FF000001FF000001FF000001FF000001FF000001FF000001FF000001FF000000FE000000 FE0000003800001E307CAF24>I<001FE00000FFF80001E07E0003801F0007800F800F00 0FC01F0007C01F0007E01F0007E03F0007E03F8007E03F8007E03FC007E03FF007C03FF8 0FC03FFE0F801FFF1F001FFFFE000FFFFC0007FFF80003FFFC0001FFFF0000FFFF8003FF FFC007FFFFC00F8FFFE01F03FFF03F01FFF07E007FF07E003FF87C000FF8FC0007F8FC00 03F8FC0003F8FC0001F8FC0001F8FC0001F07C0001F07E0001F07E0003E03F0003C01F80 07800FE03F0007FFFE0001FFF800003FE0001D2E7DAD24>I<003FC00000FFF80003F07C 0007E03E000FE03F001FC01F803FC01FC03FC01FC07FC01FE07FC01FE0FFC01FF0FFC01F F0FFC01FF0FFC01FF0FFC01FF0FFC01FF8FFC01FF8FFC01FF8FFC01FF87FC01FF87FC01F F87FC03FF83FC03FF83FC03FF81FC03FF80FE07FF807FFFFF803FFDFF800FF9FF800041F F000001FF000001FF000001FF00F801FE01FC01FE03FE03FE03FE03FC03FE03F803FE03F 803FC07F001F807E001E00FC000F03F80007FFF00003FFC00000FE00001D2E7DAD24>I< 1F003F807FC0FFE0FFE0FFE0FFE0FFE07FC03F801F000000000000000000000000000000 0000000000001F003F807FC0FFE0FFE0FFE0FFE0FFE07FC03F801F000B207C9F14>I<00 0003800000000007C00000000007C0000000000FE0000000000FE0000000001FF0000000 001FF0000000001FF0000000003FF8000000003FF8000000003FF8000000007FFC000000 007FFC000000007FFC00000000FFFE00000000F7FE00000000F7FE00000001F3FF000000 01E3FF00000001E3FF00000003C1FF80000003C1FF80000003C1FF8000000780FFC00000 0780FFC000000F80FFE000000F007FE000000F007FE000001F007FF000001E003FF00000 1E003FF000003E003FF800003C001FF800003FFFFFF800007FFFFFFC00007FFFFFFC0000 78000FFC0000F80007FE0000F00007FE0000F00007FE0001E00003FF0001E00003FF0001 E00003FF0003C00001FF8003C00001FF8007E00001FF80FFFF007FFFFEFFFF007FFFFEFF FF007FFFFE2F317DB036>65 DI<00001FF800600000FFFF 00E00007FFFFC3E0001FFC07E7E0003FE000FFE0007F80007FE001FF00001FE003FE0000 1FE003FC00000FE007FC000007E00FF8000007E01FF8000003E01FF0000003E03FF00000 03E03FF0000001E07FF0000001E07FE0000001E07FE0000001E07FE000000000FFE00000 0000FFE000000000FFE000000000FFE000000000FFE000000000FFE000000000FFE00000 0000FFE000000000FFE000000000FFE000000000FFE0000000007FE0000000007FE00000 00007FE0000000E07FF0000000E03FF0000000E03FF0000000E01FF0000000E01FF80000 01C00FF8000001C007FC0000038003FC0000038003FE0000070001FF00000E00007F8000 1C00003FE0007800001FFC03F0000007FFFFC0000000FFFF000000001FF800002B317CB0 34>II70 D<00001FF000C00001FFFE01C00007FFFF87C000 1FF807CFC0003FE001FFC000FF80007FC001FF00003FC003FE00003FC007FC00001FC007 F800000FC00FF800000FC01FF8000007C01FF0000007C03FF0000007C03FF0000003C07F F0000003C07FE0000003C07FE0000003C07FE000000000FFE000000000FFE000000000FF E000000000FFE000000000FFE000000000FFE000000000FFE000000000FFE000000000FF E000000000FFE001FFFFFEFFE001FFFFFE7FE001FFFFFE7FE00000FFC07FE00000FFC07F F00000FFC03FF00000FFC03FF00000FFC01FF00000FFC01FF80000FFC00FF80000FFC007 FC0000FFC007FC0000FFC003FE0000FFC001FF0000FFC000FF8001FFC0003FE003FFC000 1FFC0FDFC00007FFFF8FC00001FFFE03C000001FF000C02F317CB038>III75 DIII<00007FF000000003FFFE0000000FFFFF8000003FE03F E000007F800FF00001FF0007FC0003FE0003FE0007FC0001FF0007FC0001FF000FF80000 FF801FF80000FFC01FF000007FC03FF000007FE03FF000007FE03FF000007FE07FE00000 3FF07FE000003FF07FE000003FF07FE000003FF0FFE000003FF8FFE000003FF8FFE00000 3FF8FFE000003FF8FFE000003FF8FFE000003FF8FFE000003FF8FFE000003FF8FFE00000 3FF8FFE000003FF8FFE000003FF8FFE000003FF87FE000003FF07FE000003FF07FF00000 7FF03FF000007FE03FF000007FE03FF000007FE01FF80000FFC01FF80000FFC00FF80000 FF8007FC0001FF0007FE0003FF0003FE0003FE0001FF0007FC0000FFC01FF800003FF07F E000000FFFFF80000003FFFE000000007FF000002D317CB036>II<00007FF000000003FFFE0000000FFFFF8000003FF07FE000007FC01FF00001FF 0007FC0003FE0003FE0007FC0001FF0007FC0001FF000FF80000FF801FF80000FFC01FF0 00007FC03FF000007FE03FF000007FE03FF000007FE07FF000007FF07FE000003FF07FE0 00003FF07FE000003FF0FFE000003FF8FFE000003FF8FFE000003FF8FFE000003FF8FFE0 00003FF8FFE000003FF8FFE000003FF8FFE000003FF8FFE000003FF8FFE000003FF8FFE0 00003FF8FFE000003FF87FE000003FF07FE000003FF07FE000003FF03FF000007FE03FF0 00007FE03FF000007FE01FF000007FC01FF80F80FFC00FF83FE0FF8007FC7FF1FF0007FC 7079FF0003FEE03BFE0001FFE01FFC0000FFE01FF800003FF07FE000000FFFFF80000003 FFFF001000007FFF00380000000F80380000000FC0380000000FE0F80000000FFFF80000 000FFFF800000007FFF000000007FFF000000007FFF000000007FFE000000003FFE00000 0003FFC000000001FFC000000000FF80000000003E002D3F7CB036>II<003FC01801FFF83803FFFC780FE03FF81F 800FF81F0003F83F0001F87E0000F87E0000F87E000078FE000078FE000038FE000038FF 000038FF000038FF800000FFC00000FFF800007FFF80007FFFF8003FFFFE003FFFFF801F FFFFC00FFFFFE007FFFFF003FFFFF000FFFFF8003FFFFC0003FFFC00003FFC000007FE00 0003FE000001FE000001FE600000FEE00000FEE00000FEE00000FEE00000FCF00000FCF0 0000FCF80001F8FC0001F8FE0001F0FF8003E0FFF00FC0F1FFFF80E07FFF00C00FF8001F 317CB028>I<3FFFFFFFFF803FFFFFFFFF803FFFFFFFFF803FC07FE07F807F007FE00FC0 7C007FE007C07C007FE003C078007FE003C078007FE003C070007FE001C070007FE001C0 70007FE001C070007FE001C0E0007FE000E0E0007FE000E0E0007FE000E0E0007FE000E0 00007FE0000000007FE0000000007FE0000000007FE0000000007FE0000000007FE00000 00007FE0000000007FE0000000007FE0000000007FE0000000007FE0000000007FE00000 00007FE0000000007FE0000000007FE0000000007FE0000000007FE0000000007FE00000 00007FE0000000007FE0000000007FE0000000007FE0000000007FE0000000007FE00000 00007FE0000000007FE0000000007FE0000000007FE0000000FFFFFFF00000FFFFFFF000 00FFFFFFF0002B307DAF32>I86 D89 D<00FFC00007FFF8000FFFFE001FC0FF003FE07F803FE03FC03FE03FC03FE01FE01FC01F E00F801FE002001FE000001FE000001FE00007FFE0007FFFE003FFDFE007F81FE01FE01F E03FC01FE07F801FE07F801FE0FF001FE0FF001FE0FF001FE0FF001FE0FF003FE07F803F E07F806FE03FC1EFF01FFFC7FF07FF03FF00FC00FF20207E9F23>97 D<01F8000000FFF8000000FFF8000000FFF80000000FF800000007F800000007F8000000 07F800000007F800000007F800000007F800000007F800000007F800000007F800000007 F800000007F800000007F800000007F800000007F83F800007F9FFF00007FBFFFC0007FF 81FE0007FE007F0007FC007F8007F8003F8007F8003FC007F8001FE007F8001FE007F800 1FE007F8001FF007F8001FF007F8001FF007F8001FF007F8001FF007F8001FF007F8001F F007F8001FF007F8001FF007F8001FE007F8001FE007F8001FE007F8003FC007F8003FC0 07F8003F8007FC007F0007FE00FE0007E783FC0007C3FFF8000781FFE00007007F800024 327FB128>I<001FF000007FFE0001FFFF0003F83F8007E07FC00FE07FC01FC07FC03FC0 7FC03FC03F807F801F007F8004007F800000FF800000FF800000FF800000FF800000FF80 0000FF800000FF800000FF8000007F8000007F8000007FC000003FC000E03FC000E01FE0 01C00FE001C007F0038003FC0F0001FFFE00007FFC00001FE0001B207E9F20>I<000000 7E0000003FFE0000003FFE0000003FFE00000003FE00000001FE00000001FE00000001FE 00000001FE00000001FE00000001FE00000001FE00000001FE00000001FE00000001FE00 000001FE00000001FE00000001FE00001FE1FE00007FF9FE0001FFFDFE0003FC1FFE0007 F007FE000FE003FE001FC001FE003FC001FE003FC001FE007F8001FE007F8001FE007F80 01FE00FF8001FE00FF8001FE00FF8001FE00FF8001FE00FF8001FE00FF8001FE00FF8001 FE00FF8001FE00FF8001FE007F8001FE007F8001FE003F8001FE003FC001FE001FC001FE 001FE003FE000FE007FE0007F81FFF0001FFFDFFF000FFF1FFF0001FC1FFF024327EB128 >I<001FE00000FFF80001FFFE0007F87F000FE01F801FE00FC01FC00FC03FC00FE03F80 07E07F8007E07F8007F07F8007F0FF8007F0FF8007F0FFFFFFF0FFFFFFF0FFFFFFF0FF80 0000FF800000FF8000007F8000007F8000007F8000003FC000703FC000701FC000F00FE0 00E00FF003C007F80FC001FFFF00007FFC00000FF0001C207E9F21>I<0001FE00000FFF 00003FFF80007F9FC000FE3FE001FC3FE003FC3FE003F83FE007F81FC007F80F8007F807 0007F8000007F8000007F8000007F8000007F8000007F8000007F80000FFFFE000FFFFE0 00FFFFE00007F8000007F8000007F8000007F8000007F8000007F8000007F8000007F800 0007F8000007F8000007F8000007F8000007F8000007F8000007F8000007F8000007F800 0007F8000007F8000007F8000007F8000007F8000007F8000007F8000007F8000007F800 00FFFFE000FFFFE000FFFFE0001B327EB116>I<007FC07C01FFF1FE07FFFFFF0FE0FFBF 1FC07F3F3F803F9E3F803F8C7F803FC07F803FC07F803FC07F803FC07F803FC07F803FC0 7F803FC03F803F803F803F801FC07F000FE0FE000FFFFC001DFFF0001C7FC00038000000 380000003C0000003C0000003E0000003FFFFC003FFFFF801FFFFFE01FFFFFF00FFFFFF8 0FFFFFFC1FFFFFFC3E0003FC7E0000FEFC00007EFC00007EFC00007EFC00007EFC00007E 7E0000FC7E0000FC3F8003F81FE00FF007FFFFC001FFFF00003FF800202F7E9F24>I<01 F8000000FFF8000000FFF8000000FFF80000000FF800000007F800000007F800000007F8 00000007F800000007F800000007F800000007F800000007F800000007F800000007F800 000007F800000007F800000007F800000007F81FC00007F87FF00007F8FFF80007F9E1FC 0007FB81FE0007FF00FE0007FE00FF0007FC00FF0007FC00FF0007FC00FF0007F800FF00 07F800FF0007F800FF0007F800FF0007F800FF0007F800FF0007F800FF0007F800FF0007 F800FF0007F800FF0007F800FF0007F800FF0007F800FF0007F800FF0007F800FF0007F8 00FF0007F800FF0007F800FF0007F800FF00FFFFC7FFF8FFFFC7FFF8FFFFC7FFF825327E B128>I<01C00007F0000FF8000FF8001FFC001FFC001FFC000FF8000FF80007F00001C0 0000000000000000000000000000000000000000000000000001F800FFF800FFF800FFF8 0007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F8 0007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F8 0007F800FFFF80FFFF80FFFF8011337FB213>I<0001C00007F0000FF8000FF8001FFC00 1FFC001FFC000FF8000FF80007F00001C000000000000000000000000000000000000000 00000000000001FC007FFC007FFC007FFC0007FC0003FC0003FC0003FC0003FC0003FC00 03FC0003FC0003FC0003FC0003FC0003FC0003FC0003FC0003FC0003FC0003FC0003FC00 03FC0003FC0003FC0003FC0003FC0003FC0003FC0003FC0003FC0003FC0003FC0003FC00 03FC1C03FC3E03FC7F03FCFF83F8FF87F8FF87F0FF87F07F0FE03FFF801FFF0003F80016 4185B216>I<01F800FFF800FFF800FFF8000FF80007F80007F80007F80007F80007F800 07F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F800 07F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F800 07F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F800 07F800FFFFC0FFFFC0FFFFC012327FB113>108 D<03F00FE000FE0000FFF03FF803FF80 00FFF0FFFC0FFFC000FFF1E0FE1E0FE0000FF380FF380FF00007F7007F7007F00007F600 7FE007F80007FC007FC007F80007FC007FC007F80007FC007FC007F80007F8007F8007F8 0007F8007F8007F80007F8007F8007F80007F8007F8007F80007F8007F8007F80007F800 7F8007F80007F8007F8007F80007F8007F8007F80007F8007F8007F80007F8007F8007F8 0007F8007F8007F80007F8007F8007F80007F8007F8007F80007F8007F8007F80007F800 7F8007F80007F8007F8007F80007F8007F8007F80007F8007F8007F80007F8007F8007F8 00FFFFC7FFFC7FFFC0FFFFC7FFFC7FFFC0FFFFC7FFFC7FFFC03A207E9F3D>I<03F01FC0 00FFF07FF000FFF0FFF800FFF1E1FC000FF381FE0007F700FE0007F600FF0007FC00FF00 07FC00FF0007FC00FF0007F800FF0007F800FF0007F800FF0007F800FF0007F800FF0007 F800FF0007F800FF0007F800FF0007F800FF0007F800FF0007F800FF0007F800FF0007F8 00FF0007F800FF0007F800FF0007F800FF0007F800FF0007F800FF0007F800FF00FFFFC7 FFF8FFFFC7FFF8FFFFC7FFF825207E9F28>I<001FF00000FFFE0001FFFF0007F83FC00F E00FE01FC007F01FC007F03F8003F83F8003F87F8003FC7F8003FC7F8003FCFF8003FEFF 8003FEFF8003FEFF8003FEFF8003FEFF8003FEFF8003FEFF8003FEFF8003FE7F8003FC7F 8003FC7F8003FC3FC007F83FC007F81FC007F00FE00FE007F83FC003FFFF8000FFFE0000 1FF0001F207E9F24>I<01F83F8000FFF9FFF000FFFBFFFC00FFFF81FE0007FE00FF0007 FC007F8007F8007F8007F8003FC007F8003FE007F8003FE007F8001FE007F8001FF007F8 001FF007F8001FF007F8001FF007F8001FF007F8001FF007F8001FF007F8001FF007F800 1FF007F8001FE007F8003FE007F8003FE007F8003FC007F8003FC007F8007F8007FC007F 0007FE00FE0007FF83FC0007FBFFF80007F9FFE00007F87F800007F800000007F8000000 07F800000007F800000007F800000007F800000007F800000007F800000007F800000007 F800000007F8000000FFFFC00000FFFFC00000FFFFC00000242E7F9F28>I<03F07E00FF F1FF80FFF3FFC0FFF78FE00FF61FF007FE1FF007FC1FF007FC1FF007FC0FE007FC07C007 F8010007F8000007F8000007F8000007F8000007F8000007F8000007F8000007F8000007 F8000007F8000007F8000007F8000007F8000007F8000007F8000007F8000007F8000007 F80000FFFFE000FFFFE000FFFFE0001C207F9F1F>114 D<01FE3807FFF81FFFF83E01F8 7C00F8780078F80038F80038F80038FC0038FF0000FFF0007FFF007FFFC03FFFE01FFFF0 0FFFF803FFFC007FFC0003FE0000FE60007EE0003EE0003EF0003EF0003CF8003CFC0078 FF01F8FFFFF0F3FFC0C0FF0017207E9F1C>I<0038000038000038000038000038000078 0000780000780000F80000F80001F80003F80007F8001FF800FFFFF8FFFFF8FFFFF807F8 0007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F8 0007F80007F80007F80007F81C07F81C07F81C07F81C07F81C07F81C07F81C03F81C03FC 3801FC3800FFF0007FE0001F80162E7FAD1C>I<01F8003F00FFF81FFF00FFF81FFF00FF F81FFF000FF801FF0007F800FF0007F800FF0007F800FF0007F800FF0007F800FF0007F8 00FF0007F800FF0007F800FF0007F800FF0007F800FF0007F800FF0007F800FF0007F800 FF0007F800FF0007F800FF0007F800FF0007F800FF0007F800FF0007F800FF0007F801FF 0007F801FF0007F803FF0003F803FF0001FC0EFF8000FFFCFFF8007FF8FFF8001FE0FFF8 25207E9F28>II<7FFF0FFF807F FF0FFF807FFF0FFF8007F801F80003FC01E00001FC01C00001FE03C00000FF078000007F 8F0000007F8E0000003FDE0000001FFC0000001FF80000000FF800000007F800000007F8 00000003FC00000003FE00000007FE00000007FF0000000F7F8000001E3F8000003C3FC0 0000381FE00000780FE00000F00FF00001E007F80001C003F80007E003FC00FFFC1FFFE0 FFFC1FFFE0FFFC1FFFE023207F9F26>120 D<7FFF81FFC07FFF81FFC07FFF81FFC007F8 003E0007F8003C0003FC00380003FC00380001FE00700001FE00700001FE00700000FF00 E00000FF00E00000FF81E000007F81C000007F81C000003FC38000003FC38000003FC780 00001FE70000001FE70000001FFF0000000FFE0000000FFE00000007FC00000007FC0000 0007FC00000003F800000003F800000001F000000001F000000001F000000000E0000000 00E000000001E000000001C000000001C000003C038000007E03800000FF07800000FF07 000000FF0F000000FF1E0000007A7C0000007FF80000003FF00000000FC0000000222E7F 9F26>I<3FFFFFC03FFFFFC03FFFFFC03F807F803E00FF803C01FF003801FE007803FE00 7807FC007007F800700FF800700FF000701FF000003FE000003FC000007FC00000FF8000 00FF01C001FF01C003FE01C003FC01C007FC01C007F803C00FF803C01FF003801FE00780 3FE007807FC01F807F807F80FFFFFF80FFFFFF80FFFFFF801A207E9F20>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmbx10 12 73 /Fr 73 124 df<0007F800007FFC0001FC0E0003F01F0007E03F000FC03F000FC03F000F C03F000FC01E000FC00C000FC000000FC000000FC0FF80FFFFFF80FFFFFF800FC01F800F C01F800FC01F800FC01F800FC01F800FC01F800FC01F800FC01F800FC01F800FC01F800F C01F800FC01F800FC01F800FC01F800FC01F800FC01F800FC01F800FC01F807FF8FFF07F F8FFF01C237FA220>12 D<010007C00FC00FE01FE03FC07F807E00F800E0000B0A74A31D >19 D<3C01E07E03F0FF07F8FF07F8FF87FCFF87FC7F83FC3D81EC01800C01800C030018 0300180700380600300C00601C00E03801C020010016127EA21E>34 D<3C007E00FF00FF00FF80FF807F803D800180018003000300070006000C001C00380020 0009127CA210>39 D<00180030006000C001C00380078007000F001E001E003E003C003C 007C007C007C007800F800F800F800F800F800F800F800F800F800F800F800F800F80078 007C007C007C003C003C003E001E001E000F0007000780038001C000C00060003000180D 317BA416>II<3C007E00FF00FF00FF80FF807F803D800180018003000300070006000C001C0038 00200009127C8710>44 DI<3C7EFFFFFFFF7E3C 08087C8710>I<00FF0003FFC00FC3F01F00F83E007C3E007C7C003E7C003E7C003E7C00 3EFC003FFC003FFC003FFC003FFC003FFC003FFC003FFC003FFC003FFC003FFC003FFC00 3FFC003F7C003E7C003E7E007E3E007C3E007C1F00F80FC3F003FFC000FF0018207E9F1D >48 D<00380000780003F800FFF800FDF80001F80001F80001F80001F80001F80001F800 01F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F800 01F80001F80001F80001F80001F80001F80001F8007FFFF07FFFF014207C9F1D>I<03FC 000FFF803C0FE07007F07C03F8FE01F8FE01FCFE01FCFE01FC7C01FC3801FC0001FC0001 F80003F80003F00007E0000FC0000F80001E00003C0000780000E00C01C00C03800C0300 0C06001C0FFFF81FFFF83FFFF87FFFF8FFFFF8FFFFF816207D9F1D>I<00FF0007FFC00F 03F01E01F83F01F83F01FC3F81FC3F01FC1F01FC0C01F80003F80003F00007E0000FC001 FF0001FF000003E00001F80001FC0000FE0000FE0000FF7C00FF7C00FFFE00FFFE00FFFE 00FE7C01FC7801FC3C03F00FFFE001FF0018207E9F1D>I<0000700000F00001F00001F0 0003F00007F0000FF0001DF00039F00031F00061F000C1F001C1F00381F00701F00601F0 0C01F01801F03801F07001F0E001F0FFFFFFFFFFFF0003F00003F00003F00003F00003F0 0003F00003F0007FFF007FFF18207E9F1D>I<1000301E01F01FFFE01FFFE01FFFC01FFF 001FFE001FF80018000018000018000018000019FE001FFF801E07E01803F01001F00001 F80001F80001FC0001FC7801FCFC01FCFC01FCFC01FCFC01F8F801F86003F03003E01C0F C00FFF0003FC0016207D9F1D>I<001FC000FFF001F03807C0780F80FC1F00FC1F00FC3F 00FC7E00787E00007E0000FE0000FE3FC0FE7FF0FE80F8FF807CFF007EFF007EFE007FFE 007FFE007FFE007F7E007F7E007F7E007F3E007E3E007E1F007C0F00F807C1F003FFE000 FF0018207E9F1D>I<6000007800007FFFFF7FFFFF7FFFFE7FFFFC7FFFF87FFFF0E00060 E00060C000C0C00180C00300000600000E00000C00001C00003800003800007800007800 00F00000F00001F00001F00001F00003F00003F00003F00003F00003F00003F00001E000 00C00018227DA11D>I<00FF0003FFE00701F00E00781C00781C003C3C003C3E003C3F00 3C3FC0383FF0781FF8F01FFFE00FFF8007FFE003FFF007FFF81F3FFC3C0FFE7807FE7801 FFF0007FF0001FF0001FF0000FF0000FF8000E78001C3C001C1F00F80FFFE001FF001820 7E9F1D>I<00FF0007FFC00F83E01F00F03E00F87E007C7E007CFE007EFE007EFE007EFE 007FFE007FFE007FFE007F7E00FF7E00FF3E01FF1F017F0FFE7F03FC7F00007F00007E00 007E1E007E3F00FC3F00FC3F00F83F01F01E03E01C0FC00FFF0003FC0018207E9F1D>I< 3C7EFFFFFFFF7E3C0000000000003C7EFFFFFFFF7E3C08167C9510>I<00003000000000 78000000007800000000FC00000000FC00000000FC00000001FE00000001FE00000003FF 000000037F000000037F000000063F800000063F8000000C3FC000000C1FC000001C1FE0 0000180FE00000180FE000003007F000003007F000007007F800006003F800007FFFF800 00FFFFFC0000C001FC00018001FE00018000FE00038000FF000300007F000300007F0006 00003F800F00003F80FFE007FFFCFFE007FFFC26227EA12B>65 DI<0001FF0040001FFFC1C0007F 80F3C001FC001FC003F0000FC007E00007C00FC00003C01FC00003C03F800001C03F8000 01C07F800000C07F000000C07F000000C0FF00000000FF00000000FF00000000FF000000 00FF00000000FF00000000FF00000000FF000000007F000000007F000000C07F800000C0 3F800000C03F800001C01FC00001800FC000018007E000030003F000060001FC001C0000 7F807800001FFFE0000001FF000022227DA129>IIII<0001FF0020001FFFE0E0007F8079E001FC001FE003F8 0007E007E00003E00FC00001E01FC00001E03F800000E03F800000E07F800000607F0000 00607F00000060FF00000000FF00000000FF00000000FF00000000FF00000000FF000000 00FF0007FFFEFF0007FFFE7F00000FE07F00000FE07F80000FE03F80000FE03F80000FE0 1FC0000FE00FE0000FE007E0000FE003F8000FE001FC001FE0007F8073E0001FFFE1E000 01FF806027227DA12D>III<03FFFF03FFFF0007F00007F00007F00007F00007F00007F00007F0 0007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F0 0007F00007F00007F00007F03807F07C07F0FE07F0FE07F0FE07E07C0FC0381F801FFF00 07F80018227EA11E>IIIII<0007FE00 00003FFFC00000FE07F00003F801FC0007F000FE000FE0007F001FC0003F801F80001F80 3F80001FC03F80001FC07F00000FE07F00000FE07F00000FE0FF00000FF0FF00000FF0FF 00000FF0FF00000FF0FF00000FF0FF00000FF0FF00000FF0FF00000FF0FF00000FF07F00 000FE07F80001FE07F80001FE03F80001FC01FC0003F801FC0003F800FE0007F0007F000 FE0003F801FC0000FE07F000003FFFC0000007FE000024227DA12B>II<0007FE0000003FFFC000 00FE07F00003F801FC0007F000FE000FE0007F001FC0003F801FC0003F803F80001FC03F 80001FC07F80001FE07F00000FE07F00000FE0FF00000FF0FF00000FF0FF00000FF0FF00 000FF0FF00000FF0FF00000FF0FF00000FF0FF00000FF0FF00000FF07F00000FE07F8000 1FE07F80001FE03F80001FC01F80F01F801FC3F83F800FE70C7F0007F606FE0003FE03FC 0000FF07F000003FFFC0000007FFC030000001E030000001F070000001FFF0000001FFF0 000001FFE0000000FFE0000000FFC00000007FC00000003F800000001E00242C7DA12B> II<01FE 020007FFCE001F01FE003C007E003C001E0078000E0078000E00F8000600F8000600FC00 0600FC000000FF000000FFF000007FFF80003FFFE0003FFFF8001FFFFC0007FFFE0003FF FE00003FFF000001FF0000003F8000001F8000001F80C0000F80C0000F80C0000F80E000 0F00E0000F00F0001E00FC001C00FF807800E7FFF000807FC00019227DA120>I<7FFFFF FFC07FFFFFFFC07E03F80FC07803F803C07003F801C06003F800C0E003F800E0E003F800 E0C003F80060C003F80060C003F80060C003F800600003F800000003F800000003F80000 0003F800000003F800000003F800000003F800000003F800000003F800000003F8000000 03F800000003F800000003F800000003F800000003F800000003F800000003F800000003 F800000003F8000001FFFFF00001FFFFF00023217EA028>III89 D91 D<0200100E00701C00E01800C0300180700380600300 600300C00600C00600DE06F0FF07F8FF87FCFF87FC7F83FC7F83FC3F01F81E00F016127B A21E>I I<07FE00001FFF80003F07E0003F03F0003F01F0003F01F8001E01F8000001F8000001F8 00003FF80003FDF8001F81F8003E01F8007C01F800F801F800F801F800F801F800F801F8 007C02F8007E0CF8001FF87F8007E03F8019167E951C>97 DI<00FF8007FFE00F83 F01F03F03E03F07E03F07C01E07C0000FC0000FC0000FC0000FC0000FC0000FC00007C00 007E00007E00003E00181F00300FC06007FFC000FF0015167E9519>I<0001FF000001FF 0000003F0000003F0000003F0000003F0000003F0000003F0000003F0000003F0000003F 0000003F0000003F0000FE3F0007FFBF000FC1FF001F007F003E003F007E003F007C003F 007C003F00FC003F00FC003F00FC003F00FC003F00FC003F00FC003F00FC003F007C003F 007E003F003E003F001F007F000F81FF0007FF3FE001FC3FE01B237EA220>I<00FE0007 FF800F83C01E01E03E00F07E00F07C00F87C0078FC0078FFFFF8FFFFF8FC0000FC0000FC 00007C00007C00003E00183E00181F00300F80E003FFC000FF0015167E951A>I<001F80 00FFE001F1F003E3F007E3F00FC3F00FC1E00FC0000FC0000FC0000FC0000FC0000FC000 FFFE00FFFE000FC0000FC0000FC0000FC0000FC0000FC0000FC0000FC0000FC0000FC000 0FC0000FC0000FC0000FC0000FC0000FC0000FC0000FC0007FFC007FFC0014237EA212> I<00FE0F8003FF9FC00F83E3C01F01F3C01E00F0003E00F8003E00F8003E00F8003E00F8 003E00F8001E00F0001F01F0000F83E0000BFF800008FE000018000000180000001C0000 001FFFE0001FFFFC000FFFFF0007FFFF001FFFFF807C001FC078000FC0F80007C0F80007 C0F80007C07C000F803E001F001F807E000FFFFC0001FFE0001A217F951D>II<1E 003F007F807F807F807F803F001E00000000000000000000000000FF80FF801F801F801F 801F801F801F801F801F801F801F801F801F801F801F801F801F801F801F80FFF0FFF00C 247EA30F>I<003C007E00FF00FF00FF00FF007E003C00000000000000000000000003FF 03FF003F003F003F003F003F003F003F003F003F003F003F003F003F003F003F003F003F 003F003F003F003F003F003F783FFC3FFC3EFC7C78783FF01FC0102E83A312>II< FF80FF801F801F801F801F801F801F801F801F801F801F801F801F801F801F801F801F80 1F801F801F801F801F801F801F801F801F801F801F801F801F801F801F80FFF0FFF00C23 7EA20F>II< FF03F000FF0FFC001F187E001F203E001F403F001F403F001F803F001F803F001F803F00 1F803F001F803F001F803F001F803F001F803F001F803F001F803F001F803F001F803F00 1F803F001F803F00FFF1FFE0FFF1FFE01B167D9520>I<00FF0007FFE00F81F01F00F83E 007C7C003E7C003E7C003EFC003FFC003FFC003FFC003FFC003FFC003FFC003F7C003E7E 007E3E007C1F00F80F81F007FFE000FF0018167E951D>II<00FE030007FF07000FC1CF001F00DF003F007F007E 003F007E003F007C003F00FC003F00FC003F00FC003F00FC003F00FC003F00FC003F00FC 003F007E003F007E003F003E007F001F00FF000FC1FF0007FF3F0000FC3F0000003F0000 003F0000003F0000003F0000003F0000003F0000003F0000003F000001FFE00001FFE01B 207E951E>II<07F9801FFF80380780700380F00180F00180F80000FF0000FFF8007FFE 003FFF001FFF8007FF80003FC0C007C0C003C0E003C0E003C0F00380FC0F00EFFE00C3F8 0012167E9517>I<00C00000C00000C00000C00001C00001C00003C00007C0000FC0001F C000FFFF00FFFF000FC0000FC0000FC0000FC0000FC0000FC0000FC0000FC0000FC0000F C0000FC0000FC1800FC1800FC1800FC1800FC18007C18007E30003FE0000FC0011207F9F 16>IIIIII< 7FFFF07FFFF07C07E0700FC0601FC0E01F80C03F00C07F00C07E0000FC0001FC0003F800 03F03007E0300FE0300FC0701F80703F80603F00E07E03E0FFFFE0FFFFE014167E9519> II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs cmr10 12 100 /Fs 100 128 df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ndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%PaperSize: A4 %%EndSetup %%Page: 0 1 0 0 bop 680 192 a Fs(UNIVERSIT)968 179 y(\022)962 192 y(A)17 b(DI)g(PISA)520 261 y(DIP)l(AR)l(TIMENTO)f(DI)g(MA)l(TEMA)l (TICA)561 449 y(Dottorato)h(di)f(ricerca)g(in)g(matematica)174 755 y Fr(AN)j(INTR)n(ODUCTION)g(TO)g(SMALL)h(DIVISORS)f(PR)n(OBLEMS)916 884 y Fs(b)o(y)781 1013 y(Stefano)d(Marmi)930 2770 y(0)p eop %%Page: 1 2 1 1 bop 57 192 a Fq(Preface)57 379 y Fs(The)11 b(material)f(treated)h (in)g(this)g(b)q(o)q(ok)g(w)o(as)g(brough)o(t)e(together)i(for)g(a)g (PhD)g(course)f(I)h(taugh)o(t)g(at)57 449 y(the)j(Univ)o(ersit)o(y)f (of)h(Pisa)f(in)h(the)g(spring)e(of)i(1999.)21 b(It)14 b(is)g(in)o(tended)f(to)h(b)q(e)g(an)g(in)o(tro)q(duction)e(to)57 519 y(small)f(divisors)g(problems.)19 b(The)13 b(b)q(o)q(ok)g(is)g (divided)e(in)i(t)o(w)o(o)f(parts.)20 b(In)13 b(the)g(\014rst)f(one)g (I)h(discuss)57 589 y(in)k(some)f(detail)h(the)h(theory)f(of)g (linearization)f(of)i(germs)e(of)h(analytic)g(di\013eomorphisms)d(of)57 658 y(one)k(complex)h(v)m(ariable.)28 b(This)18 b(is)g(a)h(part)g(of)g (the)g(theory)g(where)f(man)o(y)g(complete)h(results)57 728 y(are)13 b(kno)o(wn.)21 b(The)14 b(second)f(part)g(is)h(more)f (informal.)20 b(It)14 b(deals)f(with)h(Nash{Moser's)e(implicit)57 798 y(function)h(theorem)f(in)i(F)l(r)o(\023)-24 b(ec)o(het)12 b(spaces)h(and)f(Kolmogoro)o(v{Arnol'd{Mos)o(er)f(theory)l(.)20 b(Man)o(y)57 868 y(results)14 b(\(and)h(ev)o(en)g(some)f(statemen)o (ts\))h(are)g(just)g(brie\015y)g(sk)o(etc)o(hed)f(but)h(I)g(alw)o(a)o (ys)f(refer)h(the)57 937 y(reader)g(to)i(a)g(c)o(hoice)f(of)g(the)h(h)o (uge)f(original)f(literature)h(on)g(the)h(sub)s(ject.)156 1007 y(I)24 b(am)f(particularly)g(fond)g(of)h(the)g(topics)f(describ)q (ed)g(in)g(the)i(\014rst)e(part,)i(esp)q(ecially)57 1077 y(b)q(ecause)17 b(of)g(their)g(in)o(terpla)o(y)f(with)h(complex)g (analysis)f(and)h(n)o(um)o(b)q(er)e(theory)l(.)24 b(The)17 b(second)57 1147 y(part)24 b(is)g(also)g(fascinating)g(b)q(oth)g(b)q (ecause)h(of)g(its)f(generalit)o(y)g(and)g(b)q(ecause)g(it)h(leads)f (to)57 1216 y(applications)11 b(to)i(Hamiltonian)f(systems.)20 b(Both)14 b(are)e(the)i(ob)s(ject)f(of)g(ma)s(jor)f(activ)o(e)h (researc)o(h.)156 1286 y(These)25 b(lectures)g(con)o(tain)f(man)o(y)g (problems)f(\(some)i(of)g(whic)o(h)f(ma)o(y)g(c)o(hallenge)g(the)57 1356 y(reader\))16 b(:)23 b(they)17 b(should)e(b)q(e)j(considered)d(as) i(an)f(essen)o(tial)g(part)g(of)h(the)h(text.)24 b(The)17 b(pro)q(of)f(of)57 1425 y(man)o(y)f(useful)h(and)g(imp)q(ortan)o(t)f (facts)i(is)f(left)h(as)g(an)f(exercise.)156 1495 y(I)c(hop)q(e)g(that) g(the)g(reader)e(will)i(\014nd)e(these)i(notes)g(a)f(useful)g(in)o(tro) q(duction)g(to)h(the)g(sub)s(ject.)57 1565 y(Ho)o(w)o(ev)o(er)18 b(the)h(reason)f(of)h(the)g(long)g(list)f(of)h(references)g(at)g(the)g (end)g(of)g(these)g(notes)f(is)h(m)o(y)57 1635 y(b)q(elief)d(that)g (the)h(b)q(est)f(w)o(a)o(y)f(to)i(learn)e(a)h(sub)s(ject)g(is)f(to)i (study)e(directly)h(the)g(pap)q(ers)f(of)i(those)57 1704 y(who)e(in)o(v)o(en)o(ted)e(it)j(:)21 b(P)o(oincar)o(\023)-24 b(e,)14 b(Siegel,)h(Kolmogoro)o(v,)f(Arnol'd,)g(Moser,)g(Herman,)h(Y)l (o)q(ccoz,)57 1774 y(etc.)156 1844 y(I)j(am)f(v)o(ery)g(grateful)g(to)h (Mariano)d(Giaquin)o(ta)i(for)g(his)g(in)o(vitation)g(to)g(giv)o(e)g (this)g(series)57 1914 y(of)f(lectures.)21 b(I)15 b(also)g(wish)g(to)h (thank)g(Carlo)f(Carminati,)f(whose)h(en)o(th)o(usiasm)e(is)i(also)g (at)h(the)57 1983 y(origin)h(of)i(this)g(pro)s(ject,)g(and)f(whose)g (remarks)g(ha)o(v)o(e)g(b)q(een)h(essen)o(tial)f(in)g(correcting)g (some)57 2053 y(mistak)o(es.)57 2158 y(Udine,)e(Decem)o(b)q(er)g(8,)g (1999.)1502 2228 y Fp(Stefano)i(Marmi)930 2770 y Fs(1)p eop %%Page: 2 3 2 2 bop 663 192 a Fq(T)-6 b(able)25 b(of)f(Con)n(ten)n(ts)57 379 y Fo(P)-5 b(AR)g(T)20 b(I.)f(One{dimensional)i(Small)h(Divisors.)j (Y)-5 b(o)r(ccoz's)17 b(Theorems)80 519 y Fn(1.)29 b(Germs)19 b(of)h(Analytic)e(Di\013eomorphism)o(s.)23 b(Linearization)80 589 y(2.)29 b(T)-5 b(op)r(ological)18 b(Stabilit)n(y)i(vs.)25 b(Analytic)18 b(Linearizabilit)n(y)80 658 y(3.)29 b(The)21 b(Quadratic)h(P)n(olynomial)c(:)29 b(Y)-5 b(o)r(ccoz's)18 b(Pro)r(of)j(of)h(the)f(Siegel)f(Theo-)156 728 y(rem)80 798 y(4.)29 b(Douady{Gh)n(ys')24 b(Theorem.)42 b(Con)n(tin)n(ued)26 b(F)-5 b(ractions)25 b(and)h(the)f(Brjuno)156 868 y(F)-5 b(unction)80 937 y(5.)29 b(Siegel{Brjuno)c(Theorem,)f(Y)-5 b(o)r(ccoz's)21 b(Theorem.)39 b(Some)23 b(Op)r(en)i(Prob-)156 1007 y(lems)80 1077 y(6.)k(Small)18 b(divisors)h(and)h(loss)f(of)h (di\013eren)n(tiabilit)n(y)57 1216 y Fo(P)-5 b(AR)g(T)20 b(I)r(I.)e(Implicit)k(F)-5 b(unction)20 b(Theorems)e(and)j(KAM)f (Theory)80 1356 y Fn(7.)29 b(Hamiltonian)18 b(Systems)f(and)k(In)n (tegrable)e(Systems)80 1425 y(8.)29 b(Quasi{in)n(tegrable)19 b(Hamiltonian)f(Systems)80 1495 y(9.)29 b(Nash{Moser's)18 b(Implicit)f(F)-5 b(unction)20 b(Theorem)50 1565 y(10.)29 b(F)-5 b(rom)19 b(Nash{Moser's)f(Theorem)g(to)i(KAM)g(:)f(Normal)f(F)-5 b(orm)19 b(of)h(V)-5 b(ector)156 1635 y(Fields)19 b(on)h(the)g(T)-5 b(orus)57 1774 y Fo(App)r(endices)35 1914 y Fn(A1.)29 b(Uniformization,)18 b(Distorsion)g(and)j(Quasi{conformal)d(maps)35 1983 y(A2.)29 b(Con)n(tin)n(ued)21 b(F)-5 b(ractions)35 2053 y(A3.)29 b(Distributions,)k(Hyp)r(erfunctions,)f(F)-5 b(ormal)30 b(Series.)59 b(Hyp)r(o)r(elliptici)o(t)n(y)156 2123 y(and)21 b(Diophan)n(tine)e(Conditions)57 2262 y(References)57 2402 y(Analytical)f(index)57 2541 y(List)h(of)h(sym)n(b)r(ols)930 2770 y Fs(2)p eop %%Page: 3 4 3 3 bop 57 192 a Fq(P)n(art)25 b(I.)i(One{Dimensional)h(Small)f (Divisors.)40 b(Y)-6 b(o)r(ccoz's)26 b(The-)57 261 y(orems)57 567 y Fo(1.)g(Germs)19 b(of)h(Analytic)h(Di\013eomorphisms.)k (Linearization)57 672 y Fs(A)18 b Fp(dynamic)m(al)j(system)d Fs(is)f(the)i(action)f(of)g(a)g(group)e(\(or)i(a)g(semigroup\))e(on)i (some)f(space.)26 b(In)57 742 y(lo)q(oking)17 b(for)g(the)h(simplest)e (cases)h(w)o(e)g(are)g(led)h(to)f(ask)h(for)f(the)h(lo)o(w)o(est)e(p)q (ossible)g(dimension)57 812 y(of)j(the)h(am)o(bien)o(t)e(space)g (together)i(with)f(the)h(highest)e(p)q(ossible)h(regularit)o(y)f(of)h (the)h(action.)57 882 y(A)f(remark)m(ably)e(ric)o(h)h(but)g(elemen)o (tary)g(situation)g(is)g(obtained)f(considering)g(the)i(group)e(of)57 951 y(germs)11 b(of)i(holomorphic)e(lo)q(cal)i(di\013eomorphisms)c(of)k Fm(C)25 b Fs(whic)o(h)12 b(lea)o(v)o(e)h(the)g(p)q(oin)o(t)f Fl(z)k Fs(=)e(0)f(\014xed.)57 1021 y(In)g(what)g(follo)o(ws)g(w)o(e)g (will)g(omit)g(the)h(sym)o(b)q(ol)f Fk(\016)g Fs(for)g(the)h(comp)q (osition)e(of)i(t)o(w)o(o)f(germs)f(\(unless)57 1091 y(some)k(confusion)f(ma)o(y)h(b)q(e)h(p)q(ossible\).)156 1161 y(Let)g Fm(C)9 b Fs([[)p Fl(z)r Fs(]])19 b(denote)d(the)h(ring)e (of)h(formal)f(p)q(o)o(w)o(er)g(series)h(and)f Fm(C)9 b Fk(f)p Fl(z)r Fk(g)19 b Fs(denote)e(the)f(ring)f(of)57 1230 y(con)o(v)o(ergen)o(t)g(p)q(o)o(w)o(er)g(series.)156 1300 y(Let)22 b Fl(G)f Fs(denote)g(the)g(group)f(of)h(germs)e(of)i (holomorphic)e(di\013eomorphisms)f(of)j(\()p Fm(C)9 b Fl(;)f Fs(0\))57 1370 y(and)20 b(let)245 1357 y(^)234 1370 y Fl(G)h Fs(denote)f(the)h(group)f(of)h(formal)e(germs)h(of)h (holomorphic)d(di\013eomorphisms)f(of)57 1440 y(\()p Fm(C)9 b Fl(;)f Fs(0\))23 b(:)29 b Fl(G)20 b Fs(=)f Fk(f)p Fl(f)25 b Fk(2)20 b Fl(z)r Fm(C)9 b Fk(f)p Fl(z)s Fk(g)f Fl(;)g(f)679 1421 y Fj(0)696 1440 y Fs(\(0\))20 b Fk(6)p Fs(=)f(0)p Fk(g)p Fs(,)933 1427 y(^)922 1440 y Fl(G)g Fs(=)h Fk(f)1074 1426 y Fs(^)1064 1440 y Fl(f)25 b Fk(2)19 b Fl(z)r Fm(C)10 b Fs([[)p Fl(z)r Fs(]])e Fl(;)1349 1426 y Fs(^)1335 1440 y Fl(f)1359 1447 y Fi(1)1404 1440 y Fk(6)p Fs(=)19 b(0)p Fk(g)p Fs(.)32 b(One)20 b(has)f(the)57 1509 y(trivial)d(\014brations)505 1720 y Fl(G)e Fs(=)f Fk([)643 1727 y Fh(\025)p Fj(2)p Fg(C)716 1717 y Ff(\003)742 1720 y Fl(G)781 1727 y Fh(\025)1089 1707 y Fs(^)1077 1720 y Fl(G)h Fs(=)g Fk([)1216 1727 y Fh(\025)p Fj(2)p Fg(C)1289 1717 y Ff(\003)1326 1707 y Fs(^)1315 1720 y Fl(G)1354 1727 y Fh(\025)604 1867 y(\031)637 1780 y Fe(?)637 1809 y(?)637 1839 y(?)637 1869 y(?)637 1899 y(y)1179 1867 y Fi(^)-23 b Fh(\031)1209 1780 y Fe(?)1209 1809 y(?)1209 1839 y(?)1209 1869 y(?)1209 1899 y(y)627 2013 y Fm(C)660 1995 y Fj(\003)1199 2013 y Fm(C)1232 1995 y Fj(\003)1726 1855 y Fs(\(1)p Fl(:)p Fs(1\))57 2211 y(where)486 2344 y(^)475 2357 y Fl(G)514 2364 y Fh(\025)554 2357 y Fs(=)14 b Fk(f)643 2344 y Fs(^)632 2357 y Fl(f)6 b Fs(\()p Fl(z)r Fs(\))14 b(=)810 2294 y Fj(1)794 2309 y Fe(X)792 2415 y Fh(n)p Fi(=1)886 2344 y Fs(^)875 2357 y Fl(f)899 2364 y Fh(n)927 2357 y Fl(z)952 2336 y Fh(n)993 2357 y Fk(2)g Fm(C)9 b Fs([[)p Fl(z)r Fs(]])f Fl(;)1212 2344 y Fs(^)1201 2357 y Fl(f)1225 2364 y Fi(1)1262 2357 y Fs(=)13 b Fl(\025)p Fk(g)h Fl(;)330 b Fs(\(1)p Fl(:)p Fs(2\))475 2522 y Fl(G)514 2529 y Fh(\025)554 2522 y Fs(=)14 b Fk(f)p Fl(f)5 b Fs(\()p Fl(z)r Fs(\))15 b(=)810 2460 y Fj(1)794 2475 y Fe(X)792 2580 y Fh(n)p Fi(=1)875 2522 y Fl(f)899 2529 y Fh(n)927 2522 y Fl(z)952 2502 y Fh(n)993 2522 y Fk(2)f Fm(C)9 b Fk(f)p Fl(z)s Fk(g)f Fl(;)25 b(f)1220 2529 y Fi(1)1256 2522 y Fs(=)14 b Fl(\025)p Fk(g)g Fl(:)335 b Fs(\(1)p Fl(:)p Fs(3\))930 2770 y(3)p eop %%Page: 4 5 4 4 bop 57 192 a Fo(1.1)19 b(Conjugation,)i(Symmetries)57 298 y Fs(Let)c(Ad)211 305 y Fh(g)242 298 y Fl(f)23 b Fs(denote)16 b(the)h(adjoin)o(t)f(action)g(of)h Fl(g)h Fs(on)e Fl(f)23 b Fs(:)f(Ad)1169 305 y Fh(g)1200 298 y Fl(f)e Fs(=)13 b Fl(g)1322 280 y Fj(\000)p Fi(1)1375 298 y Fl(f)5 b(g)r Fs(.)57 439 y Fr(De\014nition)21 b(1.1)28 b Fd(Let)18 b Fl(f)j Fk(2)16 b Fl(G)i Fd(\(resp.)825 426 y Fs(^)815 439 y Fl(f)j Fk(2)920 427 y Fs(^)908 439 y Fl(G)p Fd(\).)k(W)l(e)18 b(sa)o(y)f(that)h(a)f(germ)g Fl(g)i Fd(\(resp.)24 b(a)17 b(formal)57 509 y(germ)h Fs(^)-27 b Fl(g)16 b Fk(2)281 496 y Fs(^)270 509 y Fl(G)p Fd(\))i(is)f Fs(equiv)m(alen)o(t)g Fd(or)f Fs(conjugate)h Fd(to)h Fl(f)23 b Fd(\(resp.)1186 496 y Fs(^)1176 509 y Fl(f)6 b Fd(\))17 b(if)g(it)h(b)q(elongs)e(to)h(the)h(orbit)e(of)57 579 y Fl(f)22 b Fd(\(resp.)258 566 y Fs(^)247 579 y Fl(f)6 b Fd(\))17 b(under)e(the)i(adjoin)o(t)f(action)g(of)h Fl(G)954 586 y Fi(1)993 579 y Fd(\(resp.)1149 566 y Fs(^)1138 579 y Fl(G)1177 586 y Fi(1)1199 579 y Fd(\))g(:)557 703 y Fl(f)j Fk(\030)13 b Fl(g)29 b Fk(\()-8 b(\))27 b(9)p Fl(h)14 b Fk(2)g Fl(G)982 710 y Fi(1)1026 703 y Fs(:)28 b Fl(g)15 b Fs(=)f Fl(h)1189 683 y Fj(\000)p Fi(1)1242 703 y Fl(f)5 b(h)14 b(;)568 775 y Fs(^)557 788 y Fl(f)20 b Fk(\030)15 b Fs(^)-27 b Fl(g)29 b Fk(\()-8 b(\))27 b(9)854 775 y Fs(^)853 788 y Fl(h)14 b Fk(2)954 776 y Fs(^)943 788 y Fl(G)982 795 y Fi(1)1026 788 y Fs(:)29 b(^)-26 b Fl(g)15 b Fs(=)1160 775 y(^)1160 788 y Fl(h)1189 768 y Fj(\000)p Fi(1)1253 775 y Fs(^)1242 788 y Fl(f)1272 775 y Fs(^)1272 788 y Fl(h)e(:)57 946 y Fs(The)i(set)g(of)h(germs)e (equiv)m(alen)o(t)h(to)h Fl(f)21 b Fs(ob)o(viously)14 b(forms)g(an)h(equiv)m(alence)h(class,)e(the)i Fp(orbit)f Fs(of)57 1016 y Fl(f)22 b Fs(under)15 b(the)i(adjoin)o(t)f(action)g(of) h Fl(G)744 1023 y Fi(1)783 1016 y Fs(:)326 1150 y([)p Fl(f)5 b Fs(])14 b(=)g(Ad)515 1157 y Fh(G)546 1162 y Fc(1)568 1150 y Fl(f)19 b Fs(=)14 b Fk(f)p Fl(g)h Fk(2)f Fl(G)8 b(;)23 b Fk(9)p Fl(h)13 b Fk(2)h Fl(G)1015 1157 y Fi(1)1059 1150 y Fs(:)g Fl(g)h Fs(=)f(Ad)1244 1157 y Fh(h)1270 1150 y Fl(f)19 b Fs(=)14 b Fl(h)1395 1129 y Fj(\000)p Fi(1)1448 1150 y Fl(f)5 b(h)p Fk(g)14 b Fl(:)57 1284 y Fs(The)i(same)g(holds)f(in)h(the)h(formal)f(case.)57 1425 y Fr(De\014nition)25 b(1.2)j Fd(A)20 b(germ)g Fl(g)i Fk(2)f Fl(G)f Fd(is)g(a)h Fs(symmetry)f Fd(of)g Fl(f)26 b Fk(2)21 b Fl(G)g Fd(if)f Fl(g)i Fk(2)f Fd(Cen)o(t)8 b Fs(\()p Fl(f)d Fs(\))p Fd(,)23 b(i.e.)33 b(if)57 1495 y(Ad)122 1502 y Fh(g)145 1495 y Fl(f)19 b Fs(=)14 b Fl(f)5 b Fd(.)23 b(W)l(e)17 b(will)f(denote)g(b)o(y)721 1484 y Fm([)716 1495 y Fd(Cen)o(t)8 b Fs(\()859 1482 y(^)847 1495 y Fl(f)f Fs(\))17 b Fd(the)f(formal)g(analogue)f(of)i(Cen)o(t)8 b Fs(\()p Fl(f)d Fs(\))p Fd(.)57 1634 y Fr(Exercise)23 b(1.3)18 b Fs(Let)h Fl(f)24 b Fk(2)18 b Fl(G)608 1641 y Fh(\025)654 1634 y Fs(\(resp.)816 1621 y(^)805 1634 y Fl(f)24 b Fk(2)915 1622 y Fs(^)904 1634 y Fl(G)943 1641 y Fh(\025)969 1634 y Fs(\))c(and)e(assume)f Fl(g)j Fk(\030)e Fl(f)24 b Fs(\(resp.)31 b(^)-27 b Fl(g)19 b Fk(\030)1693 1621 y Fs(^)1682 1634 y Fl(f)6 b Fs(\),)20 b(i.e.)57 1704 y Fl(f)f Fs(=)14 b Fl(h)182 1686 y Fj(\000)p Fi(1)235 1704 y Fl(g)r(h)i Fs(for)g(some)g Fl(h)d Fk(2)i Fl(G)636 1711 y Fi(1)658 1704 y Fs(.)22 b(Then)16 b(sho)o(w)g(that)68 1775 y(\(1\))25 b Fl(g)16 b Fk(2)e Fl(G)282 1782 y Fh(\025)325 1775 y Fs(\(resp.)23 b(^)-27 b Fl(g)15 b Fk(2)567 1763 y Fs(^)555 1775 y Fl(G)594 1782 y Fh(\025)621 1775 y Fs(\))i(th)o(us)e Fl(f)790 1782 y Fi(1)827 1775 y Fs(=)e Fl(f)908 1757 y Fj(0)923 1775 y Fs(\(0\))i(=)e Fl(\025)k Fs(is)f(in)o(v)m(arian)o(t)f(under)g(conjugation.)68 1847 y(\(2\))25 b(Cen)o(t)8 b(\()p Fl(f)d Fs(\))18 b(is)f(conjugated)f (to)h(Cen)o(t)8 b(\()p Fl(g)r Fs(\),)16 b(i.e.)22 b(Cen)o(t)8 b(\()p Fl(f)d Fs(\))16 b(=)d Fl(h)1286 1828 y Fj(\000)p Fi(1)1339 1847 y Fs(Cen)o(t)c(\()p Fl(g)r Fs(\))p Fl(h)g Fs(;)68 1918 y(\(3\))25 b Fl(f)185 1900 y Fg(Z)226 1918 y Fs(=)13 b Fk(f)p Fl(f)332 1900 y Fh(n)368 1918 y Fl(;)22 b(n)14 b Fk(2)g Fm(Z)-11 b Fk(g)11 b(\032)j Fs(Cen)o(t)8 b(\()p Fl(f)d Fs(\).)57 2094 y Fo(1.2)19 b(Linearization)57 2201 y Fs(Let)14 b Fl(R)181 2208 y Fh(\025)221 2201 y Fs(denote)g(the)g(germ)f Fl(R)622 2208 y Fh(\025)648 2201 y Fs(\()p Fl(z)r Fs(\))i(=)e Fl(\025z)r Fs(.)22 b(This)13 b(is)g(the)h(simplest)f(elemen)o(t)g(of)h Fl(G)1575 2208 y Fh(\025)1602 2201 y Fs(.)21 b(It)14 b(is)f(easy)57 2271 y(to)18 b(c)o(hec)o(k)f(that,)i(if)e Fl(\025)h Fs(is)g(not)g(a)f (ro)q(ot)h(of)g(unit)o(y)l(,)f(its)h(cen)o(tralizer)f(is)g(Cen)o(t)8 b(\()p Fl(R)1497 2278 y Fh(\025)1524 2271 y Fs(\))16 b(=)f Fk(f)p Fl(R)1676 2278 y Fh(\026)1711 2271 y Fl(;)24 b(\026)16 b Fk(2)57 2340 y Fm(C)90 2322 y Fj(\003)116 2340 y Fk(g)p Fs(.)57 2447 y Fr(Exercise)h(1.4)c Fs(Let)i Fl(f)k Fk(2)14 b Fl(G)584 2454 y Fh(\025)625 2447 y Fs(and)f(assume)g (that)h Fl(\025)g Fs(is)g(not)g(a)g(ro)q(ot)g(of)g(unit)o(y)l(.)20 b(The)14 b(morphism)578 2568 y Fl(\026)23 b Fs(:)13 b(Cen)o(t)8 b(\()p Fl(f)d Fs(\))16 b Fk(!)d Fm(C)949 2548 y Fj(\003)813 2653 y Fl(g)j Fk(7!)d Fl(\026)p Fs(\()p Fl(g)r Fs(\))h(:=)g Fl(g)1115 2660 y Fi(1)1151 2653 y Fs(=)f Fl(g)1229 2632 y Fj(0)1243 2653 y Fs(\(0\))930 2770 y(4)p eop %%Page: 5 6 5 5 bop 57 192 a Fs(is)16 b(injectiv)o(e.)22 b([Hin)o(t)16 b(:)22 b(this)15 b(is)h(equiv)m(alen)o(t)h(to)f(sho)o(wing)f(that)i Fl(g)e Fk(2)f Fl(G)1353 199 y Fi(1)1375 192 y Fs(,)j Fl(g)e Fk(2)f Fs(Cen)o(t)8 b(\()p Fl(f)d Fs(\))16 b Fk(\))d Fl(g)j Fs(=)57 261 y(id)8 b(.)24 b(On)17 b(the)h(other)f(hand)g(if)h Fl(g)f Fk(2)e Fl(G)748 268 y Fh(\026)793 261 y Fs(and)i Fl(g)g Fk(2)e Fs(Cen)o(t)8 b(\()p Fl(f)d Fs(\))20 b(one)d(can)g (recursiv)o(ely)f(determine)57 331 y(the)g(p)q(o)o(w)o(er)g(series)f (co)q(e\016cien)o(ts)i(of)f Fl(g)i Fs(:)k(one)17 b(has)57 509 y(\()p Fl(\025)105 488 y Fh(n)136 509 y Fk(\000)t Fl(\025)p Fs(\))p Fl(g)251 516 y Fh(n)292 509 y Fs(=)d(\()p Fl(\026)394 488 y Fh(n)425 509 y Fk(\000)t Fl(\026)p Fs(\))p Fl(f)541 516 y Fh(n)573 509 y Fs(+)616 447 y Fh(n)p Fj(\000)p Fi(1)618 461 y Fe(X)619 568 y Fh(j)r Fi(=2)699 509 y Fl(f)723 516 y Fh(j)821 461 y Fe(X)753 566 y Fh(n)778 571 y Fc(1)797 566 y Fi(+)p Fh(:::)o(n)888 571 y Fb(j)906 566 y Fi(=)p Fh(n)970 509 y Fl(g)994 516 y Fh(n)1019 521 y Fc(1)1048 509 y Fk(\001)8 b(\001)g(\001)h Fl(g)1139 516 y Fh(n)1164 521 y Fb(j)1188 509 y Fk(\000)1231 447 y Fh(n)p Fj(\000)p Fi(1)1233 461 y Fe(X)1234 568 y Fh(j)r Fi(=2)1315 509 y Fl(g)1339 516 y Fh(j)1436 461 y Fe(X)1368 566 y Fh(n)1393 571 y Fc(1)1412 566 y Fi(+)p Fh(:::)o(n)1503 571 y Fb(j)1521 566 y Fi(=)p Fh(n)1585 509 y Fl(f)1609 516 y Fh(n)1634 521 y Fc(1)1664 509 y Fk(\001)f(\001)g(\001)h Fl(f)1755 516 y Fh(n)1780 521 y Fb(j)1815 509 y Fl(;)57 697 y Fs(for)16 b(all)g Fl(n)e Fk(\025)f Fs(2.])57 850 y Fr(De\014nition)33 b(1.5)28 b Fd(A)g(germ)f Fl(f)39 b Fk(2)32 b Fl(G)807 857 y Fh(\025)862 850 y Fd(is)27 b Fs(linearizable)f Fd(if)i(there)g(exists)g Fl(h)1571 857 y Fh(f)1629 850 y Fk(2)33 b Fl(G)1734 857 y Fi(1)1784 850 y Fd(\(a)57 920 y(linearization)15 b(of)j Fl(f)5 b Fd(\))18 b(suc)o(h)e(that)h Fl(h)716 898 y Fj(\000)p Fi(1)716 935 y Fh(f)769 920 y Fl(f)5 b(h)827 927 y Fh(f)868 920 y Fs(=)15 b Fl(R)960 927 y Fh(\025)986 920 y Fd(,)i(i.e.)23 b Fl(f)g Fd(is)17 b(conjugate)g(to)g(\(its)g(linear)g(part\))57 989 y Fl(R)95 996 y Fh(\025)121 989 y Fd(.)k Fl(f)e Fd(is)13 b Fs(formally)f(linearizable)g Fd(if)i(there)f(exists)1001 976 y Fs(^)1000 989 y Fl(h)1029 996 y Fh(f)1069 989 y Fk(2)1127 977 y Fs(^)1116 989 y Fl(G)1155 996 y Fi(1)1191 989 y Fd(suc)o(h)f(that)1405 976 y Fs(^)1405 989 y Fl(h)1434 968 y Fj(\000)p Fi(1)1434 1004 y Fh(f)1487 989 y Fl(f)1517 976 y Fs(^)1516 989 y Fl(h)1545 996 y Fh(f)1585 989 y Fs(=)h Fl(R)1675 996 y Fh(\025)1715 989 y Fd(\(note)57 1059 y(that)20 b(in)g(this)g(case)g(this)g(is)g(a)g(functional)f (equation)h(in)g(the)h(ring)e Fm(C)9 b Fs([[)p Fl(z)r Fs(]])23 b Fd(of)d(formal)g(p)q(o)o(w)o(er)57 1129 y(series\).)57 1277 y Fs(F)l(rom)13 b(Exercise)i(1.4)f(it)h(follo)o(ws)f(that)i(when)e Fl(\025)h Fs(is)g(not)g(a)g(ro)q(ot)g(of)g(unit)o(y)f(the)h (linearization)f(\(if)57 1347 y(it)k(exists\))f(is)h(unique)e(:)24 b(if)18 b Fl(h)593 1354 y Fi(1)633 1347 y Fs(and)f Fl(h)760 1354 y Fi(2)799 1347 y Fs(are)g(t)o(w)o(o)g(linearizations)f(of)i(the)g (same)f Fl(f)k Fk(2)16 b Fl(G)1688 1354 y Fh(\025)1731 1347 y Fs(then)57 1417 y Fl(h)86 1424 y Fi(1)108 1417 y Fl(h)137 1396 y Fj(\000)p Fi(1)137 1430 y(2)204 1417 y Fk(2)e Fs(k)o(er)8 b Fl(\026)p Fs(.)156 1491 y(Our)17 b(\014rst)h(result)f(on)h(the)g(existence)g(of)h(linearizations)d(will) h(concern)h(the)g(case)g(when)57 1561 y Fl(\025)e Fs(is)g(a)h(ro)q(ot)g (of)f(unit)o(y)l(.)57 1714 y Fr(Prop)r(osition)e(1.6)28 b Fd(Assume)12 b Fl(\025)h Fd(is)f(a)h(primitiv)o(e)e(ro)q(ot)i(of)g (unit)o(y)g(of)g(order)e Fl(q)r Fd(.)21 b(A)13 b(germ)f Fl(f)20 b Fk(2)14 b Fl(G)1802 1721 y Fh(\025)57 1784 y Fd(is)i(linearizable)f(if)h(and)g(only)h(if)f Fl(f)696 1766 y Fh(q)733 1784 y Fs(=)d Fd(id)8 b(.)22 b(The)17 b(same)e(holds)h(for)g(a)h(formal)e(germ)1638 1771 y Fs(^)1628 1784 y Fl(f)k Fk(2)1730 1771 y Fs(^)1718 1784 y Fl(G)1757 1791 y Fh(\025)1784 1784 y Fd(.)57 1933 y Fp(Pr)m(o)m(of.)27 b Fs(Assume)c(that)h Fl(f)30 b Fs(is)23 b(linearizable.)43 b(Then)24 b Fl(z)k Fs(=)e Fl(\025)1219 1914 y Fh(q)1241 1933 y Fl(z)j Fs(=)d(\()p Fl(h)1406 1911 y Fj(\000)p Fi(1)1406 1948 y Fh(f)1476 1933 y Fk(\016)16 b Fl(f)21 b Fk(\016)16 b Fl(h)1632 1940 y Fh(f)1658 1933 y Fs(\))1677 1914 y Fh(q)1699 1933 y Fs(\()p Fl(z)r Fs(\))28 b(=)57 2002 y(\()p Fl(h)105 1981 y Fj(\000)p Fi(1)105 2017 y Fh(f)169 2002 y Fk(\016)11 b Fl(f)234 1984 y Fh(q)268 2002 y Fk(\016)g Fl(h)333 2009 y Fh(f)359 2002 y Fs(\)\()p Fl(z)r Fs(\))18 b(from)d(whic)o(h)h(one)g(gets)h Fl(f)942 1984 y Fh(q)965 2002 y Fs(\()p Fl(z)r Fs(\))e(=)e(\()p Fl(h)1143 2009 y Fh(f)1180 2002 y Fk(\016)e Fs(id)19 b Fk(\016)11 b Fl(h)1342 1981 y Fj(\000)p Fi(1)1342 2017 y Fh(f)1395 2002 y Fs(\)\()p Fl(z)r Fs(\))16 b(=)d Fl(z)r Fs(.)156 2077 y(Con)o(v)o(ersely)19 b(if)h Fl(f)490 2058 y Fh(q)532 2077 y Fs(=)f(id)27 b(then)20 b(de\014ning)f Fl(h)997 2055 y Fj(\000)p Fi(1)997 2092 y Fh(f)1069 2077 y Fs(:=)1147 2057 y Fi(1)p 1147 2065 20 2 v 1147 2094 a Fh(q)1181 2039 y Fe(P)1234 2051 y Fh(q)q Fj(\000)p Fi(1)1234 2091 y Fh(j)r Fi(=0)1315 2077 y Fl(\025)1344 2058 y Fj(\000)p Fh(j)1396 2077 y Fl(f)1425 2058 y Fh(j)1467 2077 y Fs(one)g(immediately)57 2153 y(c)o(hec)o(ks)14 b(that)h Fl(h)344 2131 y Fj(\000)p Fi(1)344 2168 y Fh(f)411 2153 y Fk(2)f Fl(G)497 2160 y Fi(1)534 2153 y Fs(if)g Fl(f)20 b Fk(2)14 b Fl(G)707 2160 y Fh(\025)748 2153 y Fs(\(resp.)21 b Fl(h)921 2131 y Fj(\000)p Fi(1)921 2168 y Fh(f)988 2153 y Fk(2)1046 2140 y Fs(^)1035 2153 y Fl(G)1074 2160 y Fi(1)1111 2153 y Fs(if)15 b Fl(f)k Fk(2)1256 2140 y Fs(^)1245 2153 y Fl(G)1284 2160 y Fh(\025)1310 2153 y Fs(\))c(and)f Fl(h)1468 2131 y Fj(\000)p Fi(1)1468 2168 y Fh(f)1529 2153 y Fk(\016)7 b Fl(f)13 b Fk(\016)7 b Fl(h)1659 2160 y Fh(f)1698 2153 y Fs(=)14 b Fl(R)1789 2160 y Fh(\025)1815 2153 y Fs(.)57 2222 y Fa(\003)57 2421 y Fo(1.3)19 b(F)-5 b(ormal)19 b(Conjugacy)h(Classes)57 2531 y Fs(In)g(the)g(formal)f(case,)i(all)f(conjugacy)g(classes)f(of)h (germs)f(whose)g(linear)h(part)f(is)h(a)g(ro)q(ot)g(of)57 2601 y(unit)o(y)c(are)g(w)o(ell)g(kno)o(wn)g(:)930 2770 y(5)p eop %%Page: 6 7 6 6 bop 57 192 a Fr(Prop)r(osition)20 b(1.7)27 b Fd(Let)19 b Fl(\025)e Fd(b)q(e)h(a)g(primitiv)o(e)e(ro)q(ot)i(of)g(unit)o(y)f(of) g(order)g Fl(q)r Fd(.)25 b(Let)1582 178 y Fs(^)1571 192 y Fl(f)c Fk(2)1676 179 y Fs(^)1665 192 y Fl(G)1704 199 y Fh(\025)1748 192 y Fd(and)57 261 y(assume)16 b(that)349 248 y Fs(^)338 261 y Fl(f)367 243 y Fh(q)405 261 y Fk(6)p Fs(=)f Fd(id)8 b(.)25 b(Then)17 b(there)g(exists)g(a)g(unique)g(in)o (teger)g Fl(n)e Fk(\025)g Fs(1)i Fd(and)g(t)o(w)o(o)g(complex)57 331 y(n)o(um)o(b)q(ers)d Fl(a;)8 b(b)14 b Fk(2)g Fm(C)9 b Fd(,)20 b Fl(a)14 b Fk(6)p Fs(=)g(0)p Fd(,)i(suc)o(h)f(that)834 318 y Fs(^)823 331 y Fl(f)23 b Fd(is)16 b(formally)f(conjugated)i(to) 535 477 y Fl(P)567 484 y Fh(n;a;b;\025)692 477 y Fs(\()p Fl(z)r Fs(\))e(=)f Fl(\025z)r Fs(\(1)d(+)g Fl(az)1033 456 y Fh(nq)1092 477 y Fs(+)f Fl(a)1167 456 y Fi(2)1190 477 y Fl(bz)1236 456 y Fi(2)p Fh(nq)1303 477 y Fs(\))k Fl(:)57 664 y Fr(Exercise)g(1.8)24 b Fs(Pro)o(v)o(e)10 b(Prop)q(osition)g(1.7.)20 b(Note)13 b(that)e(if)h(one)f(allo)o(ws)f (to)i(conjugate)g(also)e(with)57 734 y(homoteties)i(then)426 721 y(^)415 734 y Fl(f)19 b Fs(is)12 b(formally)h(conjugated)f(to)i Fl(P)1037 741 y Fh(n;c;\025)1129 734 y Fs(\()p Fl(z)r Fs(\))h(=)e Fl(\025z)r Fs(\(1)t(+)t Fl(z)1429 716 y Fh(nq)1481 734 y Fs(+)t Fl(cz)1571 716 y Fi(2)p Fh(nq)1637 734 y Fs(\).)21 b([Hin)o(t)13 b(:)57 803 y(the)19 b(idea)f(of)i(the)f(pro)q (of)f(is)h(to)g(iterate)g(conjugations)f(b)o(y)h(p)q(olynomials)e Fl(')1484 810 y Fh(j)1505 803 y Fs(\()p Fl(z)r Fs(\))i(=)e Fl(z)e Fs(+)e Fl(\014)1761 810 y Fh(j)1782 803 y Fl(z)1807 785 y Fh(j)57 873 y Fs(with)j Fl(j)g Fk(\025)e Fs(2)i(and)g(suitably)g (c)o(hosen)g Fl(\014)776 880 y Fh(j)797 873 y Fs(.)22 b(See)16 b(also)g([Ar3],)g([Be].])57 984 y(But)h(in)f(the)h(formal)e (case)h(ev)o(erything)h(is)f(v)o(ery)g(simple)f(:)57 1141 y Fr(Prop)r(osition)k(1.9)28 b Fd(Assume)17 b(that)g Fl(\025)h Fd(is)f(not)g(a)h(ro)q(ot)f(of)h(unit)o(y)l(.)24 b(Then)1448 1128 y Fs(^)1436 1141 y Fl(G)1475 1148 y Fh(\025)1519 1141 y Fd(is)17 b(a)g(conjugacy)57 1211 y(class)e(and)282 1198 y Fs(^)270 1211 y Fl(G)309 1218 y Fi(1)348 1211 y Fd(acts)i(freely)g(and)f(transitiv)o(ely)g(on)1026 1198 y Fs(^)1015 1211 y Fl(G)1054 1218 y Fh(\025)1080 1211 y Fd(.)57 1362 y Fp(Pr)m(o)m(of.)21 b Fs(T)l(o)c(see)h(that)g(an)o (y)f Fl(f)k Fk(2)672 1350 y Fs(^)661 1362 y Fl(G)700 1369 y Fh(\025)744 1362 y Fs(is)c(conjugate)g(to)h Fl(R)1121 1369 y Fh(\025)1165 1362 y Fs(w)o(e)f(lo)q(ok)h(for)1426 1349 y(^)1426 1362 y Fl(h)1455 1369 y Fh(f)1496 1362 y Fk(2)1556 1350 y Fs(^)1544 1362 y Fl(G)1583 1369 y Fi(1)1624 1362 y Fs(suc)o(h)e(that)67 1419 y(^)57 1432 y Fl(f)87 1419 y Fs(^)86 1432 y Fl(h)115 1439 y Fh(f)159 1432 y Fs(=)216 1419 y(^)216 1432 y Fl(h)245 1439 y Fh(f)270 1432 y Fl(R)308 1439 y Fh(\025)334 1432 y Fs(.)30 b(W)l(e)19 b(dev)o(elop)g(and)f(solv)o(e)h(this)g(functional)f(equation)h(b)o(y)g (recurrence)f(:)27 b(w)o(e)57 1502 y(get,)17 b(for)f Fl(n)e Fk(\025)f Fs(2)k(\(denoting)592 1489 y(^)592 1502 y Fl(h)621 1509 y Fh(f)646 1502 y Fs(\()p Fl(z)r Fs(\))e(=)777 1464 y Fe(P)829 1477 y Fj(1)829 1517 y Fh(n)p Fi(=1)916 1489 y Fs(^)915 1502 y Fl(h)944 1509 y Fh(n)971 1502 y Fl(z)996 1484 y Fh(n)1024 1502 y Fs(,)1054 1489 y(^)1054 1502 y Fl(h)1083 1509 y Fi(1)1119 1502 y Fs(=)e(1\))482 1667 y(^)481 1680 y Fl(h)510 1687 y Fh(n)551 1680 y Fs(=)670 1646 y(1)p 610 1669 147 2 v 610 1714 a Fl(\025)639 1700 y Fh(n)677 1714 y Fk(\000)e Fl(\025)794 1618 y Fh(n)770 1633 y Fe(X)772 1739 y Fh(j)r Fi(=2)850 1680 y Fl(f)874 1687 y Fh(j)988 1633 y Fe(X)904 1737 y Fh(n)929 1742 y Fc(1)948 1737 y Fi(+)p Fh(:::)o Fi(+)p Fh(n)1070 1742 y Fb(j)1088 1737 y Fi(=)p Fh(n)1152 1667 y Fs(^)1151 1680 y Fl(h)1180 1687 y Fh(n)1205 1692 y Fc(1)1235 1680 y Fk(\001)d(\001)g(\001)1302 1667 y Fs(^)1301 1680 y Fl(h)1330 1687 y Fh(n)1355 1692 y Fb(j)1390 1680 y Fl(:)322 b Fs(\(1)p Fl(:)p Fs(4\))57 1881 y(The)20 b(action)h(of)390 1868 y(^)378 1881 y Fl(G)417 1888 y Fi(1)461 1881 y Fs(on)546 1868 y(^)534 1881 y Fl(G)573 1888 y Fh(\025)621 1881 y Fs(is)f(free.)35 b(This)20 b(follo)o(ws)g(from)g(the)i(fact)f(that)g (the)h(only)e(germ)57 1950 y(tangen)o(t)12 b(to)h(the)f(iden)o(tit)o(y) g(b)q(elonging)g(to)g(the)h(cen)o(tralizer)e(of)1214 1937 y(^)1203 1950 y Fl(f)18 b Fs(is)12 b(the)h(iden)o(tit)o(y)f(\(see) h(Exercise)57 2020 y(1.4\).)21 b(T)l(ransitivit)o(y)12 b(of)h(the)g(action)g(is)f(trivial)h(:)20 b(giv)o(en)13 b(t)o(w)o(o)f(formal)g(germs)1471 2007 y(^)1460 2020 y Fl(f)1484 2027 y Fi(1)1520 2020 y Fs(and)1624 2007 y(^)1613 2020 y Fl(f)1637 2027 y Fi(2)1673 2020 y Fs(b)q(oth)h(in)68 2077 y(^)57 2090 y Fl(G)96 2097 y Fh(\025)139 2090 y Fs(there)k(exist)h(t)o(w)o(o)e(formal)g(linearizations)943 2077 y(^)943 2090 y Fl(h)972 2097 y Fi(1)1011 2090 y Fs(and)1109 2077 y(^)1108 2090 y Fl(h)1137 2097 y Fi(2)1177 2090 y Fs(and)g(clearly)1445 2077 y(^)1434 2090 y Fl(f)1458 2097 y Fi(1)1495 2090 y Fs(=)f(Ad)1615 2097 y Fi(^)1614 2106 y Fh(h)1637 2111 y Fc(2)1657 2097 y Fi(^)1657 2106 y Fh(h)1680 2090 y Ff(\000)p Fc(1)1680 2118 y(1)1729 2090 y Fs(\()1759 2077 y(^)1748 2090 y Fl(f)1772 2097 y Fi(2)1795 2090 y Fs(\).)57 2160 y Fa(\003)57 2291 y Fs(Collecting)g(prop)q(ositions)f(1.6,)i(1.7)f(and)g(1.9)h(together)g (w)o(e)f(ha)o(v)o(e)g(a)h(complete)g(classi\014cation)57 2360 y(of)g(the)h(conjugacy)g(classes)e(of)658 2348 y(^)647 2360 y Fl(G)i Fs(:)75 2436 y(\(I\))25 b(if)17 b Fl(\025)g Fs(is)f(not)g(a)h(ro)q(ot)f(of)h(unit)o(y)f(then)846 2423 y(^)835 2436 y Fl(G)874 2443 y Fh(\025)917 2436 y Fs(is)g(a)g(conjugacy)h(class)9 b(;)55 2511 y(\(I)q(I\))26 b(if)16 b Fl(\025)e Fs(=)g Fl(e)320 2493 y Fi(2)p Fh(\031)q(ip=q)441 2511 y Fs(,)i Fl(q)g Fk(\025)d Fs(1,)j(\()p Fl(p;)8 b(q)r Fs(\))15 b(=)f(1)h(then)h(the)g(conjugacy)f(classes)g(in)1489 2498 y(^)1477 2511 y Fl(G)1516 2518 y Fh(\025)1558 2511 y Fs(are)h([)p Fl(R)1693 2518 y Fh(\025)1718 2511 y Fs(])g(and)156 2581 y Fk(f)p Fs([)p Fl(P)227 2588 y Fh(n;a;b;\025)352 2581 y Fs(])p Fk(g)391 2588 y Fh(a)p Fj(2)p Fg(C)462 2578 y Ff(\003)492 2588 y Fh(;)7 b(b)p Fj(2)p Fg(C)13 b Fh(;)7 b(n)p Fj(\025)p Fi(1)684 2581 y Fs(.)930 2770 y(6)p eop %%Page: 7 8 7 7 bop 57 192 a Fo(1.4)19 b(Ko)r(enigs{P)n(oincar)o(\023)-29 b(e)19 b(Theorem)57 297 y Fs(In)d(the)h(holomorphic)d(case)j(the)g (problem)e(of)i(a)f Fp(c)m(omplete)h Fs(classi\014cation)f(of)h(the)f (conjugacy)57 366 y(classes)25 b(is)g(still)g(op)q(en)h(and,)i(as)d(Y)l (o)q(ccoz)i(sho)o(w)o(ed,)f(p)q(erhaps)f(unreasonable.)47 b(The)26 b(\014rst)57 436 y(imp)q(ortan)o(t)15 b(result)h(in)g(the)h (holomorphic)d(case)i(is)g(the)h(Ko)q(enigs{P)o(oincar)o(\023)-24 b(e)15 b(Theorem)g(:)57 565 y Fr(Theorem)21 b(1.10)g(\(Ko)r(enigs{P)n (oincar)o(\023)-27 b(e\))29 b Fd(If)20 b Fk(j)p Fl(\025)p Fk(j)e(6)p Fs(=)h(1)h Fd(then)g Fl(G)1335 572 y Fh(\025)1381 565 y Fd(is)f(a)h(conjugacy)f(class,)57 635 y(i.e.)i(all)c Fl(f)i Fk(2)14 b Fl(G)341 642 y Fh(\025)384 635 y Fd(are)i (linearizable.)57 764 y Fp(Pr)m(o)m(of.)24 b Fs(Since)d Fl(f)27 b Fs(is)21 b(holomorphic)e(around)h Fl(z)k Fs(=)d(0)g(there)h (exists)f Fl(c)1365 771 y Fi(1)1409 764 y Fl(>)g Fs(1)g(and)g Fl(r)j Fk(2)e Fs(\(0)p Fl(;)8 b Fs(1\))57 834 y(suc)o(h)17 b(that)h Fk(j)p Fl(f)318 841 y Fh(j)339 834 y Fk(j)f(\024)f Fl(c)447 841 y Fi(1)469 834 y Fl(r)492 816 y Fi(1)p Fj(\000)p Fh(j)583 834 y Fs(for)i(all)g Fl(j)h Fk(\025)d Fs(2.)27 b(Since)17 b Fk(j)p Fl(\025)p Fk(j)g(6)p Fs(=)f(1)i(there)g(exists)g Fl(c)1486 841 y Fi(2)1525 834 y Fl(>)e Fs(1)i(suc)o(h)f(that)57 904 y Fk(j)p Fl(\025)100 886 y Fh(n)138 904 y Fk(\000)11 b Fl(\025)p Fk(j)231 886 y Fj(\000)p Fi(1)298 904 y Fk(\024)i Fl(c)372 911 y Fi(2)411 904 y Fs(for)j(all)g Fl(n)e Fk(\025)g Fs(2.)156 973 y(Let)k(\()p Fl(\033)293 980 y Fh(n)321 973 y Fs(\))340 980 y Fh(n)p Fj(\025)p Fi(1)435 973 y Fs(b)q(e)e(the)h(follo)o(wing)f(recursiv)o(ely)f(de\014ned)g(sequence)i (:)489 1108 y Fl(\033)517 1115 y Fi(1)554 1108 y Fs(=)c(1)h Fl(;)36 b(\033)723 1115 y Fh(n)764 1108 y Fs(=)840 1045 y Fh(n)817 1060 y Fe(X)818 1167 y Fh(j)r Fi(=2)981 1060 y Fe(X)897 1165 y Fh(n)922 1170 y Fc(1)941 1165 y Fi(+)p Fh(:::)o Fi(+)p Fh(n)1063 1170 y Fb(j)1081 1165 y Fi(=)p Fh(n)1144 1108 y Fl(\033)1172 1115 y Fh(n)1197 1120 y Fc(1)1228 1108 y Fk(\001)8 b(\001)g(\001)g Fl(\033)1322 1115 y Fh(n)1347 1120 y Fb(j)1382 1108 y Fl(:)330 b Fs(\(1)p Fl(:)p Fs(5\))57 1258 y(The)16 b(generating)g(function)g Fl(\033)r Fs(\()p Fl(z)r Fs(\))f(=)759 1221 y Fe(P)811 1233 y Fj(1)811 1273 y Fh(n)p Fi(=1)897 1258 y Fl(\033)925 1265 y Fh(n)953 1258 y Fl(z)978 1240 y Fh(n)1022 1258 y Fs(satis\014es)g(the)i(functional)f(equation)709 1385 y Fl(\033)r Fs(\()p Fl(z)r Fs(\))f(=)f Fl(z)f Fs(+)994 1352 y Fl(\033)r Fs(\()p Fl(z)r Fs(\))1087 1334 y Fi(2)p 962 1374 181 2 v 962 1420 a Fs(1)e Fk(\000)g Fl(\033)r Fs(\()p Fl(z)r Fs(\))1162 1385 y Fl(;)550 b Fs(\(1)p Fl(:)p Fs(6\))57 1522 y(th)o(us)15 b Fl(\033)r Fs(\()p Fl(z)r Fs(\))g(=)333 1502 y Fi(1+)p Fh(z)q Fj(\000)435 1474 y(p)p 467 1474 162 2 v 467 1502 a Fi(1)p Fj(\000)p Fi(6)p Fh(z)q Fi(+)p Fh(z)609 1492 y Fc(2)p 333 1511 297 2 v 471 1539 a Fi(4)651 1522 y Fs(is)h(analytic)g(in)g(the)h(disk)e Fk(j)p Fl(z)r Fk(j)f Fl(<)g Fs(3)c Fk(\000)g Fs(2)1366 1481 y Fk(p)p 1407 1481 25 2 v 1407 1522 a Fs(2)17 b(and)e(b)q(ounded)g (and)57 1592 y(con)o(tin)o(uous)e(on)j(its)g(closure.)k(By)d(Cauc)o(h)o (y's)d(estimate)i(one)g(has)f Fl(\033)1323 1599 y Fh(n)1365 1592 y Fk(\024)e Fl(c)1439 1599 y Fi(3)1461 1592 y Fs(\(3)d Fk(\000)g Fs(2)1589 1551 y Fk(p)p 1630 1551 V 1630 1592 a Fs(2\))1674 1574 y Fi(1)p Fj(\000)p Fh(n)1769 1592 y Fs(for)57 1662 y(some)16 b Fl(c)204 1669 y Fi(3)239 1662 y Fl(>)e Fs(0.)156 1731 y(Since)j Fl(\025)h Fs(is)e(not)i(a)f(ro)q (ot)h(of)f(unit)o(y)l(,)g Fl(f)23 b Fs(is)17 b(formally)f(linearizable) g(and)h(the)g(p)q(o)o(w)o(er)f(series)57 1801 y(co)q(e\016cien)o(ts)f (of)g(its)h(formal)e(linearization)871 1788 y(^)871 1801 y Fl(h)900 1808 y Fh(f)941 1801 y Fs(satisfy)h(\(1.4\).)22 b(By)16 b(induction)e(one)i(can)f(c)o(hec)o(k)57 1871 y(that)i Fk(j)180 1858 y Fs(^)179 1871 y Fl(h)208 1878 y Fh(n)234 1871 y Fk(j)d(\024)g Fs(\()p Fl(c)356 1878 y Fi(1)378 1871 y Fl(c)400 1878 y Fi(2)422 1871 y Fl(r)445 1853 y Fj(\000)p Fi(1)499 1871 y Fs(\))518 1853 y Fh(n)p Fj(\000)p Fi(1)597 1871 y Fl(\033)625 1878 y Fh(n)652 1871 y Fs(,)j(th)o(us)793 1858 y(^)792 1871 y Fl(h)821 1878 y Fh(f)860 1871 y Fk(2)e Fm(C)9 b Fk(f)p Fl(z)r Fk(g)p Fs(.)760 b Fa(\003)57 1988 y Fr(Remark)12 b(1.11)e Fs(Since)h(the)h(b)q(ound)e Fk(j)p Fl(\025)779 1970 y Fh(n)807 1988 y Fk(\000)q Fl(\025)p Fk(j)890 1970 y Fj(\000)p Fi(1)957 1988 y Fk(\024)j Fl(c)1031 1995 y Fi(2)1065 1988 y Fs(is)e(uniform)f(w.r.t)h Fl(\025)j Fk(2)g Fl(D)q Fs(\()p Fl(\025)1586 1995 y Fi(0)1609 1988 y Fl(;)8 b(\016)r Fs(\),)13 b(where)57 2058 y Fl(\025)86 2065 y Fi(0)122 2058 y Fk(2)h Fm(C)202 2040 y Fj(\003)239 2058 y Fk(n)e Fm(S)306 2040 y Fi(1)342 2058 y Fs(and)k Fl(\016)g(<)e Fs(dist)8 b(\()p Fl(\025)667 2065 y Fi(0)690 2058 y Fl(;)g Fm(S)743 2040 y Fi(1)762 2058 y Fs(\),)17 b(the)g(ab)q(o)o(v)o(e)f(giv) o(en)g(pro)q(of)h(of)g(the)g(P)o(oincar)o(\023)-24 b(e{Ko)q(enigs)57 2127 y(Theorem)15 b(sho)o(ws)g(that)i(the)g(map)765 2218 y Fm(C)798 2198 y Fj(\003)835 2218 y Fk(n)11 b Fm(S)901 2198 y Fi(1)934 2218 y Fk(!)j Fl(G)1037 2225 y Fi(1)892 2303 y Fl(\025)f Fk(7!)h Fl(h)1035 2309 y Fi(~)1027 2318 y Fh(f)1053 2303 y Fs(\()p Fl(\025)p Fs(\))57 2410 y(is)i(analytic)280 2392 y Fi(1)319 2410 y Fs(for)g(all)475 2397 y(~)464 2410 y Fl(f)k Fk(2)14 b Fl(z)580 2392 y Fi(2)602 2410 y Fm(C)9 b Fk(f)p Fl(z)s Fk(g)p Fs(,)19 b(where)d Fl(h)925 2416 y Fi(~)917 2425 y Fh(f)943 2410 y Fs(\()p Fl(\025)p Fs(\))h(is)f(the)h(linearization)e(of)i Fl(\025z)c Fs(+)1631 2397 y(~)1620 2410 y Fl(f)6 b Fs(\()p Fl(z)r Fs(\).)57 2515 y(The)15 b(P)o(oincar)o(\023)-24 b(e{Ko)q(enigs)13 b(Theorem)h(has)g(the)h(follo)o(wing)f(straigh)o(tforw)o(ard)f (generalization)h(:)p 57 2595 600 2 v 109 2632 a Fi(1)156 2650 y Fs(This)19 b(notion)f(needs)h(a)g(little)g(commen)o(t)f(since)h Fm(C)9 b Fk(f)p Fl(z)r Fk(g)22 b Fs(is)d(a)g(rather)f(wild)h(space)f(:) 27 b(it)20 b(is)930 2770 y(7)p eop %%Page: 8 9 8 8 bop 57 192 a Fr(Theorem)33 b(1.12)g(\(Ko)r(enigs{P)n(oincar)o(\023) -27 b(e)34 b(with)i(parameters\))57 b Fd(Let)31 b Fl(r)38 b(>)f Fs(0)p Fd(,)c(let)57 261 y Fl(f)28 b Fs(:)14 b Fm(D)170 243 y Fh(n)170 274 y(r)211 261 y Fk(\002)d Fm(D)294 268 y Fh(r)334 261 y Fk(\032)j Fm(C)420 243 y Fh(n)461 261 y Fk(\002)d Fm(C)26 b Fk(!)14 b Fm(C)9 b Fd(,)20 b Fs(\()p Fl(t;)8 b(z)r Fs(\))16 b Fk(7!)e Fl(f)5 b Fs(\()p Fl(t;)j(z)r Fs(\))16 b(=)e Fl(f)1100 268 y Fh(t)1118 261 y Fs(\()p Fl(z)r Fs(\))k Fd(b)q(e)f(an)f(holomorphic)f(map)h(suc)o (h)57 331 y(that)k Fl(f)192 338 y Fi(0)215 331 y Fs(\()p Fl(z)r Fs(\))g(=)f Fl(\025)p Fs(\(0\))p Fl(z)d Fs(+)d Fd(O)8 b Fs(\()p Fl(z)630 313 y Fi(2)653 331 y Fs(\))p Fd(,)21 b(with)f Fk(j)p Fl(\025)p Fs(\(0\))p Fk(j)g(62)f(f)p Fs(0)p Fl(;)8 b Fs(1)p Fk(g)p Fd(.)32 b(Then)19 b(there)h(exists)f Fl(r)1610 338 y Fi(0)1652 331 y Fk(2)h Fs(\(0)p Fl(;)8 b(r)q Fs(\))p Fd(,)57 401 y(a)19 b(unique)g(holomorphic)e(function)i Fl(z)775 408 y Fi(0)825 401 y Fs(:)f Fm(D)890 383 y Fh(n)890 413 y(r)909 418 y Fc(0)952 401 y Fk(!)h Fm(C)31 b Fd(and)19 b(a)h(unique)e Fl(h)27 b Fs(:)19 b Fm(D)1507 383 y Fh(n)1507 413 y(r)1525 418 y Fc(0)1563 401 y Fk(\002)13 b Fm(D)1648 408 y Fh(r)1666 413 y Fc(0)1710 401 y Fk(!)19 b Fm(C)9 b Fd(,)57 470 y Fs(\()p Fl(t;)f(z)r Fs(\))19 b Fk(7!)f Fl(h)276 477 y Fh(t)293 470 y Fs(\()p Fl(z)r Fs(\))i(=)e Fl(h)p Fs(\()p Fl(t;)8 b(z)r Fs(\))20 b Fd(holomorphic)d(suc)o(h)h (that)h(for)g Fl(t)g Fk(2)f Fm(D)1298 452 y Fh(n)1298 483 y(r)1317 488 y Fc(0)1361 470 y Fd(one)h(has)f(the)i(follo)o(wing)57 540 y(prop)q(erties)15 b(:)79 612 y(\(i\))25 b Fl(f)180 619 y Fh(t)198 612 y Fs(\()p Fl(z)240 619 y Fi(0)263 612 y Fs(\()p Fl(t)p Fs(\)\))15 b(=)f Fl(z)429 619 y Fi(0)451 612 y Fs(\()p Fl(t)p Fs(\))p Fd(,)j Fl(f)562 619 y Fh(t)580 612 y Fs(\()p Fl(z)r Fs(\))e(=)f Fl(\025)p Fs(\()p Fl(t)p Fs(\)\()p Fl(z)g Fk(\000)d Fl(z)925 619 y Fi(0)948 612 y Fs(\()p Fl(t)p Fs(\)\))h(+)f Fd(O)d Fs(\(\()p Fl(z)14 b Fk(\000)d Fl(z)1280 619 y Fi(0)1302 612 y Fs(\()p Fl(t)p Fs(\)\))1377 594 y Fi(2)1401 612 y Fs(\))p Fd(,)16 b Fk(j)p Fl(\025)p Fs(\()p Fl(t)p Fs(\))p Fk(j)f(62)f(f)p Fs(0)p Fl(;)8 b Fs(1)p Fk(g)i Fd(;)65 683 y(\(ii\))25 b Fl(h)185 690 y Fh(t)203 683 y Fs(\(0\))14 b(=)g Fl(z)356 690 y Fi(0)378 683 y Fs(\()p Fl(t)p Fs(\))p Fd(,)j Fl(h)494 665 y Fj(0)494 696 y Fh(t)512 683 y Fs(\(0\))d(=)658 664 y Fh(@)p 648 672 44 2 v 648 701 a(@)r(z)697 683 y Fl(h)726 690 y Fh(t)744 683 y Fk(j)758 690 y Fh(z)q Fi(=0)845 683 y Fs(=)f(1)d Fd(;)51 755 y(\(iii\))25 b Fl(h)185 734 y Fj(\000)p Fi(1)185 767 y Fh(t)250 755 y Fk(\016)11 b Fl(f)16 b Fk(\016)11 b Fl(h)391 762 y Fh(t)422 755 y Fs(=)j Fl(R)513 764 y Fh(\025)p Fi(\()p Fh(t)p Fi(\))585 755 y Fd(.)57 896 y Fp(Pr)m(o)m(of.)j Fs(\(sk)o(etc)o(h\))e(The)f (existence)g(of)g Fl(z)777 903 y Fi(0)814 896 y Fs(and)f(\(i\))i(follo) o(ws)e(easily)h(from)f(the)i(implicit)e(function)57 966 y(theorem)21 b(applied)h(to)g Fl(F)7 b Fs(\()p Fl(t;)h(z)r Fs(\))25 b(=)f Fl(f)5 b Fs(\()p Fl(t;)j(z)r Fs(\))17 b Fk(\000)d Fl(z)25 b Fs(at)e(the)g(p)q(oin)o(t)f(\()p Fl(t;)8 b(z)r Fs(\))24 b(=)g(\(0)p Fl(;)8 b Fs(0\))23 b(\(note)g(that)57 1035 y Fl(F)7 b Fs(\(0)p Fl(;)h Fs(0\))26 b(=)g(0)e(and)466 1016 y Fh(@)p 456 1024 V 456 1052 a(@)r(z)506 1035 y Fl(f)530 1042 y Fh(t)548 1035 y Fs(\()p Fl(z)r Fs(\))p Fk(j)625 1044 y Fi(\()p Fh(t;z)q Fi(\)=\(0)p Fh(;)p Fi(0\))846 1035 y Fs(=)h Fl(\025)p Fs(\(0\))17 b Fk(\000)e Fs(1)26 b Fk(6)p Fs(=)g(0\).)44 b(Therefore)23 b(there)h(exists)f(a)57 1105 y(unique)14 b(\014xed)i(p)q(oin)o(t)f(for) g Fl(f)565 1112 y Fh(t)598 1105 y Fs(close)g(to)h Fl(z)g Fs(=)e(0)h(when)g Fl(t)h Fs(is)f(close)g(to)h(0)f(dep)q(ending)g (analytically)57 1175 y(on)24 b Fl(t)h Fs(as)g Fl(t)g Fs(v)m(aries)g(in)f(a)h(neigh)o(b)q(orho)q(o)q(d)f(of)h(\()p Fl(t;)8 b(z)r Fs(\))29 b(=)f(\(0)p Fl(;)8 b Fs(0\).)48 b(Then)24 b(one)h(can)g(consider)57 1245 y Fl(g)81 1252 y Fh(t)98 1245 y Fs(\()p Fl(z)r Fs(\))g(=)e Fl(f)272 1252 y Fh(t)290 1245 y Fs(\()p Fl(z)18 b Fs(+)d Fl(z)427 1252 y Fi(0)449 1245 y Fs(\()p Fl(t)p Fs(\)\))h Fk(\000)f Fl(z)617 1252 y Fi(0)640 1245 y Fs(\()p Fl(t)p Fs(\))23 b(and)f(apply)g(the)h(pro)q(of)f(giv)o(en)g(ab)q(o)o(v)o(e)g(of)g(the)h (Ko)q(enigs{)57 1314 y(P)o(oincar)o(\023)-24 b(e)14 b(Theorem)g(to)i Fl(g)553 1321 y Fh(t)570 1314 y Fs(\()p Fl(z)r Fs(\).)23 b(It)16 b(is)f(easy)h(to)g(con)o(vince)f(oneself)g(that)h(the)f (linearizing)f(map)57 1384 y(dep)q(ends)h(analytically)i(on)f Fl(t)p Fs(.)1171 b Fa(\003)57 1578 y Fo(1.5)19 b(Cen)n(tralizers)h(and) h(Linearizations)57 1685 y Fs(The)e(study)g(of)g(cen)o(tralizers)f (generalizes)g(the)i(study)f(of)h(linearizabilit)o(y)d(as)i(the)h (follo)o(wing)57 1754 y(exercises)c(sho)o(w)f(:)57 1862 y Fr(Exercise)21 b(1.13)38 b Fs(Pro)o(v)o(e)17 b(that)h(if)g Fl(f)k Fk(2)17 b Fl(G)859 1869 y Fh(\025)903 1862 y Fs(is)g (linearizable)g(and)g Fl(\025)h Fs(is)g(not)g(a)g(ro)q(ot)g(of)g(unit)o (y)57 1931 y(then)f(Cen)o(t)8 b(\()p Fl(f)d Fs(\))18 b Fk(')d Fm(C)455 1913 y Fj(\003)481 1931 y Fs(.)26 b([Hin)o(t)17 b(:)24 b(use)18 b(the)g(fact)g(that)g(the)g(cen)o(tralizer)f(of)h Fl(f)23 b Fs(is)17 b(conjugate)h(to)57 2001 y(the)e(cen)o(tralizer)g (of)h Fl(R)480 2008 y Fh(\025)522 2001 y Fs(whic)o(h)f(is)g(completely) g(kno)o(wn.])57 2108 y Fr(Exercise)f(1.14)c Fs(Pro)o(v)o(e)g(that)h(if) h Fl(g)i Fk(2)f Fs(Cen)o(t)8 b(\()p Fl(f)d Fs(\),)15 b Fl(g)g Fk(2)f Fl(G)1093 2115 y Fh(\026)1132 2108 y Fs(is)e(linearizable)f(and)g Fl(\026)h Fs(is)g(not)g(a)g(ro)q(ot)p 57 2177 600 2 v 57 2232 a(an)17 b(inductiv)o(e)f(limit)h(of)g(Banac)o (h)f(spaces,)h(th)o(us)f(it)h(is)g(a)g(lo)q(cally)h(con)o(v)o(ex)f(top) q(ological)f(v)o(ector)57 2302 y(space)k(and)g(it)g(is)g(complete)h (but)f(it)h(is)f(not)g(metrisable,)g(th)o(us)g(it)g(is)g(not)h(a)f(F)l (r)o(\023)-24 b(ec)o(het)20 b(space)57 2371 y(\(see)15 b(Section)f(9.1\).)21 b(Here)15 b(w)o(e)f(simply)g(mean)f(that)i(if)g Fl(\025)f Fs(v)m(aries)h(in)f(some)f(relativ)o(ely)i(compact)57 2441 y(op)q(en)h(connected)h(subset)f(of)i Fm(C)654 2423 y Fj(\003)691 2441 y Fk(n)11 b Fm(S)758 2423 y Fi(1)794 2441 y Fs(then)17 b Fl(h)945 2447 y Fi(~)937 2457 y Fh(f)962 2441 y Fs(\()p Fl(\025)p Fs(\))h(b)q(elongs)e(to)i(some)e(\014xed)h (Banac)o(h)f(space)57 2511 y(of)24 b(holomorphic)e(functions)i(\(e.g.) 45 b(the)25 b(Hardy)f(space)g Fl(H)1216 2493 y Fj(1)1259 2511 y Fs(\()p Fm(D)1311 2518 y Fh(r)1336 2511 y Fs(\))h(of)f(b)q (ounded)f(analytic)57 2581 y(functions)e(on)h(the)g(disk)g Fm(D)586 2588 y Fh(r)634 2581 y Fs(=)h Fk(f)p Fl(z)i Fk(2)f Fm(C)17 b Fl(;)i Fk(j)p Fl(z)r Fk(j)24 b Fl(<)f(r)q Fk(g)p Fs(,)h(where)d Fl(r)k(>)e Fs(0)f(is)g(\014xed)g(and)g(small)57 2650 y(enough\))16 b(and)g(dep)q(ends)f(analytically)i(on)f Fl(\025)g Fs(in)h(the)f(usual)g(sense.)930 2770 y(8)p eop %%Page: 9 10 9 9 bop 57 192 a Fs(of)13 b(unit)o(y)f(then)h Fl(f)19 b Fs(is)13 b(linearizable.)19 b([Hin)o(t)13 b(:)20 b(use)13 b(that)g Fl(f)20 b Fk(2)14 b Fs(Cen)o(t)8 b(\()p Fl(g)r Fs(\))14 b(=)g Fk(f)p Fl(h)1450 199 y Fh(g)1472 192 y Fl(R)1510 199 y Fh(\027)1535 192 y Fl(h)1564 173 y Fj(\000)p Fi(1)1564 204 y Fh(g)1625 192 y Fl(;)j(\027)g Fk(2)d Fm(C)1778 173 y Fj(\003)1803 192 y Fk(g)57 261 y Fs(and)i(that)g Fl(\027)k Fs(is)c(in)o(v)m(arian)o(t)f(under)g(conjugacy)l(.])57 366 y Fr(Exercise)29 b(1.15)23 b Fs(Pro)o(v)o(e)g(that)i(if)f Fl(f)32 b Fk(2)27 b Fl(G)891 373 y Fh(\025)942 366 y Fs(and)c Fl(\025)i Fs(is)f(not)g(a)g(ro)q(ot)g(of)h(unit)o(y)f(then)g Fl(f)30 b Fs(is)57 436 y(linearizable)15 b(if)h(and)g(only)g(if)h(Cen)o (t)8 b(\()p Fl(f)d Fs(\))15 b Fk(')f Fm(C)897 418 y Fj(\003)923 436 y Fs(.)22 b([Hin)o(t)16 b(:)22 b(apply)16 b(exercises)g(1.13,)g (1.4)g(and)g(the)57 506 y(Ko)q(enigs{P)o(oincar)o(\023)-24 b(e)14 b(Theorem])57 681 y Fo(1.6)19 b(Cremer's)f(Non{Linearizable)j (Germs)57 786 y Fs(When)16 b Fk(j)p Fl(\025)p Fk(j)e Fs(=)f(1)k(and)f Fl(\025)g Fs(is)g(not)h(a)f(ro)q(ot)h(of)g(unit)o(y)f (w)o(e)g(can)g(write)497 887 y Fl(\025)e Fs(=)g Fl(e)616 867 y Fi(2)p Fh(\031)q(i\013)730 887 y Fs(with)g Fl(\013)g Fk(2)g Fm(R)8 b Fk(n)j Fm(Q)h Fk(\\)f Fs(\()p Fk(\000)p Fs(1)p Fl(=)p Fs(2)p Fl(;)d Fs(1)p Fl(=)p Fs(2\))14 b Fl(;)338 b Fs(\(1)p Fl(:)p Fs(7\))57 989 y(and)18 b(whether)h Fl(f)24 b Fk(2)18 b Fl(G)488 996 y Fh(\025)534 989 y Fs(is)g(linearizable)g(or)g(not)i(dep)q(ends)e(crucially)g(on)h(the)g (arithmetical)57 1059 y(prop)q(erties)29 b(of)h Fl(\013)p Fs(.)62 b(Let)31 b Fk(f)p Fl(x)p Fk(g)f Fs(denote)g(the)h(fractional)e (part)g(of)i(a)f(real)f(n)o(um)o(b)q(er)f Fl(x)j Fs(:)57 1128 y Fk(f)p Fl(x)p Fk(g)14 b Fs(=)f Fl(x)f Fk(\000)f Fs([)p Fl(x)p Fs(],)16 b(where)g([)p Fl(x)p Fs(])h(is)f(the)h(in)o (teger)f(part)g(of)h Fl(x)p Fs(.)57 1257 y Fr(Theorem)h(1.16)h (\(Cremer\))28 b Fd(If)34 b Fs(lim)8 b(sup)905 1269 y Fh(n)p Fj(!)p Fi(+)p Fj(1)1050 1257 y Fk(jf)p Fl(n\013)p Fk(gj)1190 1239 y Fj(\000)p Fi(1)p Fh(=n)1304 1257 y Fs(=)15 b(+)p Fk(1)i Fd(then)h(there)f(exists)57 1327 y Fl(f)i Fk(2)14 b Fl(G)186 1336 y Fh(e)205 1326 y Fc(2)p Fb(\031)q(i\013)302 1327 y Fd(whic)o(h)h(is)h(not)h(linearizable.)57 1456 y Fp(Pr)m(o)m(of.)j Fs(First)15 b(of)i(all)f(note)h(that)g(lim)8 b(sup)824 1468 y Fh(n)p Fj(!)p Fi(+)p Fj(1)970 1456 y Fk(jf)p Fl(n\013)p Fk(gj)1110 1438 y Fj(\000)p Fi(1)p Fh(=n)1221 1456 y Fs(=)14 b(+)p Fk(1)i Fs(if)h(and)f(only)g(if)651 1557 y(lim)8 b(sup)659 1597 y Fh(n)p Fj(!)p Fi(+)p Fj(1)811 1557 y Fk(j)p Fl(\025)854 1537 y Fh(n)892 1557 y Fk(\000)j Fs(1)p Fk(j)981 1537 y Fj(\000)p Fi(1)p Fh(=n)1093 1557 y Fs(=)j(+)p Fk(1)57 1676 y Fs(since)456 1746 y Fk(j)p Fl(\025)499 1725 y Fh(n)537 1746 y Fk(\000)d Fs(1)p Fk(j)j Fs(=)f(2)p Fk(j)8 b Fs(sin\()p Fl(\031)r(n\013)p Fs(\))p Fk(j)14 b(2)g Fs(\(2)p Fk(jf)p Fl(n\013)p Fk(gj)p Fl(;)8 b(\031)r Fk(jf)p Fl(n\013)p Fk(gj)p Fs(\))14 b Fl(:)57 1838 y Fs(Then)i(w)o(e)g(construct)f Fl(f)23 b Fs(in)15 b(the)i(follo)o(wing)e(manner)g(:)22 b(for)16 b Fl(n)d Fk(\025)h Fs(2)i(w)o(e)g(tak)o(e)h Fk(j)p Fl(f)1526 1845 y Fh(n)1553 1838 y Fk(j)d Fs(=)g(1)i(and)g(w)o(e)57 1908 y(c)o(ho)q(ose)g(inductiv)o(ely)g(arg)8 b Fl(f)573 1915 y Fh(n)617 1908 y Fs(suc)o(h)15 b(that)487 2044 y(arg)8 b Fl(f)589 2051 y Fh(n)630 2044 y Fs(=)14 b(arg)761 1981 y Fh(n)p Fj(\000)p Fi(1)763 1996 y Fe(X)765 2102 y Fh(j)r Fi(=2)845 2044 y Fl(f)869 2051 y Fh(j)982 1996 y Fe(X)899 2101 y Fh(n)924 2106 y Fc(1)943 2101 y Fi(+)p Fh(:::)o Fi(+)p Fh(n)1065 2106 y Fb(j)1083 2101 y Fi(=)p Fh(n)1147 2031 y Fs(^)1146 2044 y Fl(h)1175 2051 y Fh(n)1200 2056 y Fc(1)1230 2044 y Fk(\001)8 b(\001)g(\001)1297 2031 y Fs(^)1296 2044 y Fl(h)1325 2051 y Fh(n)1350 2056 y Fb(j)1385 2044 y Fl(;)327 b Fs(\(1)p Fl(:)p Fs(8\))57 2190 y(\(recall)14 b(the)g(induction)f(form)o(ula)g(\(1.4\))h(for)g (the)h(co)q(e\016cien)o(ts)e(of)i(the)f(formal)f(linearization)g(of)57 2260 y Fl(f)23 b Fs(and)16 b(note)i(that)f(the)h(r.h.s.)k(of)c(\(1.8\)) f(is)g(a)g(p)q(olynomial)f(in)h Fl(n)12 b Fk(\000)f Fs(2)17 b(v)m(ariables)f Fl(f)1592 2267 y Fi(2)1615 2260 y Fl(;)8 b(:)g(:)g(:)h(;)f(f)1750 2267 y Fh(n)p Fj(\000)p Fi(1)57 2329 y Fs(with)16 b(co)q(e\016cien)o(ts)g(in)g(the)h(\014eld)f Fm(C)9 b Fs(\()p Fl(\025)q Fs(\)\).)25 b(Th)o(us)653 2450 y Fk(j)668 2436 y Fs(^)667 2450 y Fl(h)696 2457 y Fh(n)722 2450 y Fk(j)14 b(\025)854 2416 y(j)p Fl(f)892 2423 y Fh(n)919 2416 y Fk(j)p 809 2438 170 2 v 809 2484 a(j)p Fl(\025)852 2469 y Fh(n)890 2484 y Fk(\000)d Fs(1)p Fk(j)998 2450 y Fs(=)1129 2416 y(1)p 1057 2438 V 1057 2484 a Fk(j)p Fl(\025)1100 2469 y Fh(n)1138 2484 y Fk(\000)g Fs(1)p Fk(j)57 2581 y Fs(and)16 b(lim)8 b(sup)306 2593 y Fh(n)p Fj(!)p Fi(+)p Fj(1)452 2581 y Fk(j)467 2568 y Fs(^)466 2581 y Fl(h)495 2588 y Fh(n)522 2581 y Fk(j)536 2563 y Fi(1)p Fh(=n)617 2581 y Fs(=)15 b(+)p Fk(1)h Fs(:)23 b(the)17 b(formal)f(linearization)1342 2568 y(^)1342 2581 y Fl(h)g Fs(is)h(a)g(div)o(ergen)o(t)f(series.)57 2650 y Fa(\003)930 2770 y Fs(9)p eop %%Page: 10 11 10 10 bop 57 192 a Fr(Exercise)21 b(1.17)16 b Fs(W)l(rite)h(the)g (decimal)g(expansion)f(of)h(an)g(irrational)f(n)o(um)o(b)q(er)f Fl(\013)i Fs(satisfying)57 261 y(the)f(assumption)f(of)i(Cremer's)e (Theorem.)57 366 y Fr(Exercise)k(1.18)c Fs(Sho)o(w)f(that)j(the)f(set)g (of)g(irrational)e(n)o(um)o(b)q(ers)f(satisfying)i(the)h(assumption)57 436 y(of)h(Cremer's)d(Theorem)i(is)g(a)h(dense)f Fl(G)802 443 y Fh(\016)840 436 y Fs(with)h(zero)f(Leb)q(esgue)h(measure)e (\(follo)o(wing)h(Baire,)57 506 y(a)h(set)g(is)g(a)g(dense)g Fl(G)446 513 y Fh(\016)485 506 y Fs(if)g(it)h(is)f(a)g(coun)o(table)f (in)o(tersection)g(of)h(dense)g(op)q(en)g(sets.)24 b(These)17 b(sets)57 576 y(are)f(\\big")g(from)f(the)i(p)q(oin)o(t)f(of)h(view)g (of)g(top)q(ology\).)57 681 y(In)j(the)g(next)h(Chapter)e(w)o(e)h(will) g(con)o(tin)o(ue)f(our)g(study)h(of)g(the)h(problem)d(of)j(the)f (existence)57 751 y(of)f(a)f(linearization)g(of)h(germs)e(of)i (holomorphic)e(di\013eomorphisms)o(.)26 b(T)l(o)18 b(this)h(purp)q(ose) e(the)57 820 y(follo)o(wing)e(\\normalization")f(will)j(b)q(e)f (useful.)57 995 y Fo(1.7)j(Normalized)h(Germs)57 1100 y Fs(Let)d(us)f(note)h(that)f(there)h(is)f(an)g(ob)o(vious)f(action)i (of)g Fm(C)1095 1082 y Fj(\003)1137 1100 y Fs(on)f Fl(G)h Fs(b)o(y)f(homotheties)g(:)518 1195 y(\()p Fl(\026;)8 b(f)d Fs(\))16 b Fk(2)e Fm(C)733 1174 y Fj(\003)770 1195 y Fk(\002)c Fl(G)k Fk(7!)g Fs(Ad)1001 1202 y Fh(R)1031 1207 y Fb(\026)1057 1195 y Fl(f)20 b Fs(=)13 b Fl(R)1191 1174 y Fj(\000)p Fi(1)1191 1207 y Fh(\026)1245 1195 y Fl(f)5 b(R)1312 1202 y Fh(\026)1353 1195 y Fl(:)359 b Fs(\(1)p Fl(:)p Fs(9\))57 1289 y(Note)17 b(that)f(this)g(action)g(lea)o (v)o(es)g(the)g(\014b)q(ers)f Fl(G)931 1296 y Fh(\025)974 1289 y Fs(in)o(v)m(arian)o(t)f(b)o(y)i(Exercise)g(1.3.)22 b(Also,)15 b Fl(f)20 b Fk(2)14 b Fl(G)1802 1296 y Fh(\025)57 1359 y Fs(is)19 b(linearizable)g(if)h(and)g(only)g(if)g(Ad)754 1366 y Fh(R)784 1371 y Fb(\026)810 1359 y Fl(f)26 b Fs(is)20 b(also)f(linearizable)g(for)h(all)g Fl(\026)f Fk(2)h Fm(C)1572 1341 y Fj(\003)1619 1359 y Fs(\(indeed)f(if)57 1429 y Fl(h)86 1436 y Fh(f)129 1429 y Fs(linearizes)d Fl(f)23 b Fs(then)18 b(Ad)573 1436 y Fh(R)603 1441 y Fb(\026)629 1429 y Fl(h)658 1436 y Fh(f)701 1429 y Fs(linearizes)e(Ad) 983 1436 y Fh(R)1013 1441 y Fb(\026)1040 1429 y Fl(f)5 b Fs(\).)26 b(Therefore,)17 b(in)g(order)f(to)i(study)f(the)57 1498 y(problem)k(of)i(the)f(existence)h(of)g(a)g(linearization,)g(it)g (is)f(enough)f(to)i(consider)f Fl(G=)p Fm(C)1701 1480 y Fj(\003)1727 1498 y Fs(,)i(i.e.)57 1568 y(w)o(e)18 b(iden)o(tify)g(t)o(w)o(o)g(germs)f(of)i(holomorphic)d (di\013eomorphisms)f(whic)o(h)j(are)g(conjugate)g(b)o(y)g(a)57 1638 y(homothet)o(y)l(.)156 1708 y(Consider)f(the)h(space)f Fl(S)j Fs(of)e(univ)m(alen)o(t)f(maps)g Fl(F)31 b Fs(:)16 b Fm(D)27 b Fk(!)16 b Fm(C)29 b Fs(suc)o(h)17 b(that)h Fl(F)7 b Fs(\(0\))17 b(=)e(0)j(and)57 1777 y(the)e(pro)s(jection)470 1859 y Fl(G)e Fk(!)g Fl(S)479 1969 y(f)20 b Fk(7!)14 b Fl(F)20 b Fs(=)692 1898 y Fe(\032)738 1937 y Fl(f)173 b Fs(if)17 b Fl(f)22 b Fs(is)16 b(univ)m(alen)o(t)g(in)g Fm(D)738 1997 y Fs(Ad)803 2004 y Fh(R)833 2009 y Fb(r)855 1997 y Fl(f)56 b Fs(if)17 b Fl(f)22 b Fs(is)16 b(univ)m(alen)o(t)g(in)g Fm(D)1384 2004 y Fh(r)57 2080 y Fs(This)10 b(map)h(is)g(clearly)g(on)o (to)g(and)g(t)o(w)o(o)g(germs)f(ha)o(v)o(e)h(the)h(same)e(image)h(only) g(if)h(they)g(coincide)f(or)57 2150 y(if)k(they)g(are)g(conjugate)f(b)o (y)h(some)f(homothet)o(y)l(.)21 b(Th)o(us)13 b(this)i(pro)s(jection)f (induces)g(a)h(bijection)57 2220 y(from)g Fl(G=)p Fm(C)271 2201 y Fj(\003)314 2220 y Fs(on)o(to)h Fl(S)s Fs(.)156 2289 y(In)j(what)h(follo)o(ws)e(w)o(e)i(will)e(alw)o(a)o(ys)h(consider) f(the)h(top)q(ological)h(space)f Fl(S)i Fs(of)f(germs)e(of)57 2359 y(holomorphic)c(di\013eomorphisms)f Fl(f)28 b Fs(:)13 b Fm(D)25 b Fk(!)14 b Fm(C)28 b Fs(suc)o(h)15 b(that)i Fl(f)5 b Fs(\(0\))15 b(=)f(0)i(and)g Fl(f)22 b Fs(is)16 b(univ)m(alen)o(t)g(in)57 2429 y Fm(D)8 b Fs(.)25 b(W)l(e)16 b(will)h(denote)107 2498 y Fk(\017)24 b Fl(S)187 2505 y Fh(\025)230 2498 y Fs(the)16 b(subspace)f(of)i Fl(f)23 b Fs(suc)o(h)15 b(that)i Fl(f)876 2480 y Fj(0)891 2498 y Fs(\(0\))d(=)g Fl(\025)c Fs(;)107 2568 y Fk(\017)24 b Fl(S)187 2575 y Fg(T)229 2568 y Fs(the)17 b(subspace)e(of)i Fl(f)22 b Fs(suc)o(h)16 b(that)g Fk(j)p Fl(f)889 2550 y Fj(0)904 2568 y Fs(\(0\))p Fk(j)e Fs(=)g(1.)156 2638 y(Clearly)i(the)h(pro)s(jection)f(ab)q(o)o(v)o(e)g(induces)f(a)i (bijection)f(b)q(et)o(w)o(een)h Fl(G)1447 2645 y Fh(\025)1473 2638 y Fl(=)p Fm(C)1531 2620 y Fj(\003)1573 2638 y Fs(and)f Fl(S)1701 2645 y Fh(\025)1727 2638 y Fs(.)918 2770 y(10)p eop %%Page: 11 12 11 11 bop 57 192 a Fq(2.)31 b(T)-6 b(op)r(ological)26 b(Stabilit)n(y)g(vs.)32 b(Analytic)25 b(Linearizabil)q(i)q(t)n(y)57 300 y Fs(The)19 b(purp)q(ose)g(of)g(this)h(Chapter)f(is)g(to)h(connect) g(the)g(study)f(of)h(the)g(conjugacy)g(classes)e(of)57 370 y(germs)11 b(of)i(holomorphic)e(di\013eomorphisms)e(to)14 b(the)f(theory)f(of)i(one{dimensional)c(conformal)57 440 y(dynamical)i(systems)g(and)h(in)g(particular)e(to)j(the)g(notion)e (of)i(stabilit)o(y)e(of)i(a)f(\014xed)g(p)q(oin)o(t.)21 b(The)57 510 y(extremely)g(remark)m(able)e(fact)i(is)g(that)g(stabilit) o(y)l(,)g(whic)o(h)e(is)h(a)h(top)q(ological)f(prop)q(ert)o(y)l(,)h (will)57 579 y(turn)16 b(out)g(to)h(b)q(e)g(equiv)m(alen)o(t)f(to)h (linearizabilit)o(y)l(,)e(whic)o(h)g(is)h(an)h(analytic)f(prop)q(ert)o (y)l(.)57 758 y Fo(2.1)j(Dynamics)g(of)h(Rational)i(Maps)57 867 y Fs(Let)e(us)e(\014rst)h(of)g(all)g(recall)f(the)i(notion)e(of)i (normal)d(family)i(on)g(an)g(op)q(en)g(subset)f Fl(U)25 b Fs(of)19 b(the)57 937 y(Riemann)h(sphere)p 433 896 36 2 v 20 w Fm(C)35 b Fs(=)22 b Fm(C)k Fk([)15 b(f1g)p Fs(.)37 b(T)l(o)21 b(this)h(purp)q(ose)e(w)o(e)h(recall)h(the)f(usual)g (system)g(of)57 1006 y(co)q(ordinates)16 b(on)p 391 966 V 16 w Fm(C)29 b Fs(determined)16 b(b)o(y)h(the)g(stereographic)e(pro)s (jection)h(:)23 b Fl(z)i Fs(:)p 1517 966 V 15 w Fm(C)e Fk(n)11 b(f1g)j(!)h Fm(C)9 b Fs(,)57 1076 y Fl(z)r Fs(\(0\))15 b(=)e(0)h(and)f Fl(w)23 b Fs(:)p 432 1036 V 14 w Fm(C)18 b Fk(n)6 b(f)p Fs(0)p Fk(g)13 b(!)h Fm(C)9 b Fs(,)17 b Fl(w)q Fs(\()p Fk(1)p Fs(\))d(=)g(0,)g(related)g(b)o(y)f Fl(z)r(w)i Fs(=)f(1.)21 b(The)14 b Fp(spheric)m(al)j(metric)57 1146 y Fs(on)p 126 1106 V 16 w Fm(C)28 b Fs(is)17 b(de\014ned)e(as)h (follo)o(ws)g(:)580 1313 y Fl(ds)p 629 1300 28 2 v 17 x Fg(C)671 1313 y Fs(=)724 1228 y Fe(\()793 1248 y Fi(2)p Fj(j)p Fh(dz)q Fj(j)p 778 1261 115 2 v 778 1289 a Fi(1+)p Fj(j)p Fh(z)q Fj(j)873 1279 y Fc(2)957 1272 y Fs(in)g(the)h Fl(z)r Fs({c)o(hart)9 b(;)793 1329 y Fi(2)p Fj(j)p Fh(dw)q Fj(j)p 778 1342 124 2 v 778 1370 a Fi(1+)p Fj(j)p Fh(w)q Fj(j)883 1361 y Fc(2)957 1353 y Fs(in)16 b(the)h Fl(w)q Fs({c)o(hart)9 b(;)1726 1313 y(\(2)p Fl(:)p Fs(1\))57 1499 y(Let)18 b Fl(U)i Fk(\032)p 255 1459 36 2 v 15 w Fm(C)30 b Fs(b)q(e)17 b(op)q(en)h(and)e Fk(F)632 1506 y Fh(U)681 1499 y Fs(=)f Fk(f)p Fl(f)29 b Fs(:)15 b Fl(U)21 b Fk(!)p 962 1459 V 15 w Fm(C)f Fl(;)j(f)e Fs(meromorphic)n Fk(g)p Fs(.)j(W)l(e)18 b(endo)o(w)p 1678 1459 V 16 w Fm(C)29 b Fs(with)57 1569 y(the)14 b(spherical)e(metric)h(and)g Fk(F)627 1576 y Fh(U)674 1569 y Fs(with)g(the)h(top)q(ology)g(of)g (uniform)e(con)o(v)o(ergence)g(on)i(compact)57 1639 y(subsets)j(of)h Fl(U)5 b Fs(.)27 b(It)19 b(is)e(a)h(classical)g(result)f(of)h(W)l (eierstrass)f(that)h(the)h(limit)e(of)i(a)f(con)o(v)o(ergen)o(t)57 1709 y(sequence)j(in)g Fk(F)367 1716 y Fh(U)422 1709 y Fs(still)g(b)q(elongs)g(to)h Fk(F)810 1716 y Fh(U)865 1709 y Fs(\(note)g(that)g(the)g(constan)o(t)f(function)g Fl(f)28 b Fk(\021)22 b(1)g Fs(is)57 1778 y(considered)15 b(meromorphic\).)57 1928 y Fr(De\014nition)20 b(2.1)28 b Fd(A)17 b(family)f Fk(F)j(\032)14 b(F)776 1935 y Fh(U)825 1928 y Fd(is)j Fs(normal)e Fd(if)h(it)h(is)f(relativ)o(ely)h(compact)f (in)g Fk(F)1701 1935 y Fh(U)1734 1928 y Fd(,)h(i.e.)57 1998 y(an)o(y)f(sequence)h Fk(f)p Fl(f)407 2005 y Fh(n)435 1998 y Fk(g)d(\032)h(F)22 b Fd(con)o(tains)16 b(a)h(subsequence)f(whic) o(h)g(con)o(v)o(erges)g(uniformly)f(in)i(the)57 2068 y(spherical)e(metric)h(on)g(compact)g(subsets)g(of)g Fl(U)5 b Fd(.)57 2214 y Fp(Warning)10 b(!)49 b Fs(If)26 b Fk(f)p Fl(f)429 2221 y Fh(n)456 2214 y Fk(g)g Fs(is)e(a)i(normal)e (family)h(then)g Fk(f)p Fl(f)1131 2196 y Fj(0)1126 2226 y Fh(n)1153 2214 y Fk(g)h Fs(needs)e(not)i(b)q(e)f(normal)f(:)40 b(e.g.)57 2284 y Fl(f)81 2291 y Fh(n)108 2284 y Fs(\()p Fl(z)r Fs(\))15 b(=)f Fl(n)p Fs(\()p Fl(z)313 2266 y Fi(2)347 2284 y Fk(\000)d Fl(n)p Fs(\))16 b(on)h Fm(C)9 b Fs(.)156 2357 y(By)17 b(means)f(of)h(the)g(Ascoli{Arzel\022)-25 b(a)16 b(theorem)f(one)i(gets)f(:)57 2507 y Fr(Prop)r(osition)i(2.2)75 2581 y Fd(\(I\))25 b(A)20 b(family)f(of)g(meromorphic)e(functions)h(on) h Fl(U)25 b Fd(is)18 b(normal)g(on)h Fl(U)24 b Fd(if)c(and)e(only)h(if) g(it)h(is)156 2650 y(equicon)o(tin)o(uous)15 b(on)h(ev)o(ery)g(compact) h(subset)e(of)i Fl(U)f Fd(;)918 2770 y Fs(11)p eop %%Page: 12 13 12 12 bop 55 192 a Fd(\(I)q(I\))26 b(A)17 b(family)f(of)h(analytic)f (functions)g(on)g Fl(U)21 b Fd(is)16 b(normal)f(on)h Fl(U)22 b Fd(if)16 b(and)g(only)g(if)h(it)f(is)g(lo)q(cally)57 261 y(uniformly)f(b)q(ounded)g(\(i.e.)22 b(uniformly)15 b(b)q(ounded)h(on)g(ev)o(ery)h(compact)f(subset)g(of)g Fl(U)5 b Fd(\).)57 395 y Fp(Pr)m(o)m(of.)21 b Fs(The)c(\014rst)g (statemen)o(t)g(is)g(ob)o(vious)f(since)h(the)h(compactness)e(of)p 1435 354 36 2 v 18 w Fm(C)29 b Fs(guaran)o(tees)16 b(that)57 464 y(the)22 b(family)f(is)g(uniformly)g(b)q(ounded.)36 b(The)22 b(second)f(statemen)o(t)g(follo)o(ws)g(from)g(Cauc)o(h)o(y's) 57 534 y(in)o(tegral)15 b(theorem.)1360 b Fa(\003)57 653 y Fs(The)24 b(notion)f(of)i(normal)e(family)g(allo)o(ws)h(us)f(to)i (in)o(tro)q(duce)e(the)h(basic)g(notions)f(of)i(one{)57 723 y(dimensional)i(holomorphic)f(dynamics.)59 b(Here)29 b(w)o(e)g(are)g(in)o(terested)f(in)h(studying)f(the)57 793 y(dynamics)19 b(of)i(a)g(discrete)f(dynamical)g(system)g(\(i.e.)35 b(an)21 b(action)f(of)h Fm(N)p Fs(\))h(on)f(the)g(Riemann)57 863 y(sphere)p 210 822 V 13 w Fm(C)27 b Fs(generated)14 b(b)o(y)h(a)f(holomorphic)f(transformation)f Fl(R)23 b Fs(:)p 1302 822 V 13 w Fm(C)j Fk(!)p 1416 822 V 14 w Fm(C)12 b Fs(,)j(i.e.)21 b(and)14 b(elemen)o(t)57 932 y(of)i(End)8 b(\()p 230 892 V Fm(C)k Fs(\).)156 1002 y(Let)19 b Fl(d)f Fs(denote)g(the)g(top)q(ological)g(degree)f(of)i Fl(R)p Fs(.)26 b(W)l(e)18 b(will)g(assume)e Fl(d)h Fk(\025)f Fs(2)i(th)o(us)f Fl(R)h Fs(is)f(a)57 1072 y Fl(d)p Fs({fold)h(branc)o (hed)f(co)o(v)o(ering)g(of)i(the)g(Riemann)e(sphere)h(and)g(can)g(b)q (e)h(written)g(in)f(a)h(unique)57 1142 y(w)o(a)o(y)f(in)h(the)g(form)f Fl(R)p Fs(\()p Fl(z)r Fs(\))i(=)613 1117 y Fh(P)5 b Fi(\()p Fh(z)q Fi(\))p 612 1130 83 2 v 612 1159 a Fh(Q)p Fi(\()p Fh(z)q Fi(\))701 1142 y Fs(,)20 b(where)e Fl(P)7 b Fs(\()p Fl(z)r Fs(\))19 b Fk(2)f Fm(C)9 b Fs([)p Fl(z)s Fs(],)22 b Fl(Q)p Fs(\()p Fl(z)r Fs(\))d Fk(2)g Fm(C)9 b Fs([)p Fl(z)r Fs(])22 b(ha)o(v)o(e)c(no)h(common)57 1211 y(factors)12 b(and)g Fl(d)i Fs(=)f(max\(deg)8 b Fl(P)q(;)g Fs(deg)h Fl(Q)p Fs(\).)21 b(In)12 b(fact)i(ev)o(ery)e Fl(d)i Fs(:)g(1)e (conformal)f(branc)o(hed)g(co)o(v)o(ering)57 1281 y(of)p 113 1241 36 2 v 16 w Fm(C)29 b Fs(comes)16 b(from)g(some)g(suc)o(h)f (rational)g(function)i(and)131 1403 y(End)8 b(\()p 248 1363 V Fm(C)k Fs(\))i(=)g Fk(f)p Fl(R)22 b Fs(:)p 483 1363 V 14 w Fm(C)j Fk(!)p 596 1363 V 14 w Fm(C)20 b Fs(holomorphic)n Fk(g)14 b Fs(=)f Fk(f)p Fl(R)23 b Fs(:)p 1113 1363 V 13 w Fm(C)j Fk(!)p 1227 1363 V 14 w Fm(C)20 b Fl(;)c(R)p Fs(\()p Fl(z)r Fs(\))g(=)d Fl(P)7 b Fs(\()p Fl(z)r Fs(\))p Fl(=Q)p Fs(\()p Fl(z)r Fs(\))p Fk(g)16 b Fl(:)57 1525 y Fs(Note)h(that)539 1649 y Fl(R)p Fs(\()p Fl(z)r Fs(\))f(=)708 1532 y Fe(8)708 1577 y(>)708 1592 y(<)708 1681 y(>)708 1696 y(:)767 1556 y Fh(P)5 b Fi(\()p Fh(z)q Fi(\))p 767 1569 83 2 v 767 1598 a Fh(Q)p Fi(\()p Fh(z)q Fi(\))1085 1581 y Fs(if)17 b Fl(Q)p Fs(\()p Fl(z)r Fs(\))e Fk(6)p Fs(=)f(0,)761 1649 y Fk(1)274 b Fs(if)17 b Fl(Q)p Fs(\()p Fl(z)r Fs(\))e(=)f(0,)761 1717 y(lim)830 1724 y Fh(z)q Fj(!1)947 1692 y Fh(P)5 b Fi(\()p Fh(z)q Fi(\))p 947 1705 V 947 1734 a Fh(Q)p Fi(\()p Fh(z)q Fi(\))1085 1717 y Fs(if)17 b Fl(z)f Fs(=)e Fk(1)p Fs(.)57 1820 y(W)l(e)g(de\014ne)f (the)h(iterates)g Fl(R)577 1802 y Fh(n)618 1820 y Fs(of)g Fl(R)g Fs(as)f(usual)g(:)20 b Fl(R)982 1802 y Fh(n)1024 1820 y Fs(=)13 b Fl(R)6 b Fk(\016)g Fl(R)1189 1802 y Fh(n)p Fj(\000)p Fi(1)1267 1820 y Fs(.)21 b(Note)14 b(that)h Fl(R)1563 1802 y Fh(n)1604 1820 y Fs(has)e(degree)57 1889 y Fl(d)83 1871 y Fh(n)110 1889 y Fs(.)156 1959 y(Giv)o(en)h(a)f(p) q(oin)o(t)h Fl(z)487 1966 y Fi(0)523 1959 y Fk(2)p 570 1919 36 2 v 14 w Fm(C)26 b Fs(the)14 b(sequence)f(of)h(p)q(oin)o(ts)f Fk(f)p Fl(z)1155 1966 y Fh(n)1182 1959 y Fk(g)1207 1966 y Fh(n)p Fj(\025)p Fi(0)1299 1959 y Fs(de\014ned)g(b)o(y)g Fl(z)1557 1966 y Fh(n)p Fi(+1)1649 1959 y Fs(=)g Fl(R)p Fs(\()p Fl(z)1781 1966 y Fh(n)1809 1959 y Fs(\))57 2029 y(is)19 b(called)g(the)h(orbit)e(of)i Fl(z)550 2036 y Fi(0)573 2029 y Fs(.)30 b(A)20 b(p)q(oin)o(t)f Fl(z)830 2036 y Fi(0)872 2029 y Fs(is)h(a)f Fp(\014xe)m(d)i(p)m(oint)g Fs(of)f Fl(R)g Fs(if)f Fl(R)p Fs(\()p Fl(z)1461 2036 y Fi(0)1484 2029 y Fs(\))h(=)e Fl(z)1603 2036 y Fi(0)1626 2029 y Fs(,)i Fp(p)m(erio)m(dic)57 2099 y Fs(if)26 b Fl(z)135 2106 y Fh(n)192 2099 y Fs(=)k Fl(R)299 2080 y Fh(n)326 2099 y Fs(\()p Fl(z)368 2106 y Fi(0)391 2099 y Fs(\))h(=)f Fl(z)533 2106 y Fi(0)581 2099 y Fs(for)c(some)g Fl(n)g Fs(\(the)h(minimal)d Fl(n)j Fs(is)e(the)i(p)q(erio)q(d\).)51 b(The)26 b(orbit)57 2168 y Fk(f)p Fl(z)105 2175 y Fi(1)127 2168 y Fl(;)8 b(:)g(:)g(:)h(;)f(z)261 2175 y Fh(n)314 2168 y Fs(=)25 b Fl(z)401 2175 y Fi(0)424 2168 y Fk(g)f Fs(is)f(called)g(a)h Fp(cycle)p Fs(.)43 b(The)23 b(p)q(oin)o(t)h Fl(z)1160 2175 y Fi(0)1206 2168 y Fs(is)f(called)g Fp(pr)m(ep)m(erio)m (d)q(ic)k Fs(if)c Fl(z)1746 2175 y Fh(k)1795 2168 y Fs(is)57 2238 y(p)q(erio)q(dic)16 b(for)g(some)g Fl(k)f(>)f Fs(0.)156 2308 y(The)25 b(fundamen)o(tal)d(dic)o(hotom)o(y)i(of)p 877 2267 V 24 w Fm(C)37 b Fs(asso)q(ciated)24 b(to)h(the)f(dynamics)g (of)g Fl(R)h Fs(is)f(the)57 2378 y(follo)o(wing)15 b(:)57 2511 y Fr(De\014nition)24 b(2.3)k Fd(The)19 b Fs(F)l(atou)g(set)h Fl(F)7 b Fs(\()p Fl(R)p Fs(\))20 b Fd(of)g Fl(R)g Fd(is)f(the)h(set)g (of)g(p)q(oin)o(ts)e Fl(z)1472 2518 y Fi(0)1514 2511 y Fk(2)p 1566 2471 V 19 w Fm(C)32 b Fd(suc)o(h)18 b(that)57 2581 y Fk(f)p Fl(R)120 2563 y Fh(n)147 2581 y Fk(g)172 2588 y Fh(n)p Fj(\025)p Fi(0)270 2581 y Fd(is)i(a)g(normal)e(family)i (in)g(some)f(disk)g Fl(D)q Fs(\()p Fl(z)1078 2588 y Fi(0)1102 2581 y Fl(;)8 b(r)q Fs(\))21 b Fd(\(w.r.t.)33 b(the)20 b(spherical)f(metric\).)57 2650 y(The)d(complemen)o(t)f(of)i(the)g(F)l (atou)f(set)g(is)g(the)h Fs(Julia)f(set)g Fl(J)5 b Fs(\()p Fl(R)p Fs(\))p Fd(.)918 2770 y Fs(12)p eop %%Page: 13 14 13 13 bop 57 192 a Fr(Exercise)20 b(2.4)c Fs(Sho)o(w)g(that)h Fl(J)5 b Fs(\()p Fl(R)p Fs(\))15 b(=)f Fl(J)5 b Fs(\()p Fl(R)881 173 y Fh(n)908 192 y Fs(\))18 b(for)e(all)h Fl(n)d Fk(\025)g Fs(1)j(and)f(that)i Fl(J)5 b Fs(\()p Fl(R)p Fs(\))17 b(is)f(nonempt)o(y)57 261 y(and)i(closed.)31 b([Hin)o(t)19 b(:)28 b(if)19 b Fl(J)5 b Fs(\()p Fl(R)p Fs(\))19 b(=)g Fk(;)g Fs(then)g Fk(f)p Fl(R)961 243 y Fh(n)989 261 y Fk(g)1014 268 y Fh(n)p Fj(\025)p Fi(0)1111 261 y Fs(is)g(a)h(normal)e(family)h(on)g(all)p 1680 221 36 2 v 19 w Fm(C)31 b Fs(th)o(us)57 331 y Fl(R)95 313 y Fh(n)120 318 y Fb(j)154 331 y Fk(!)14 b Fl(S)i Fk(2)e Fs(End)8 b(\()p 429 291 V Fm(C)k Fs(\).)23 b(Compare)15 b(degrees.])57 506 y Fo(2.2)k(Stabilit)n(y)57 634 y Fr(De\014nition)c (2.5)28 b Fd(A)12 b(p)q(oin)o(t)g Fl(z)619 641 y Fi(0)655 634 y Fk(2)p 702 593 V 14 w Fm(C)24 b Fd(is)11 b Fs(stable)h Fd(if)g(for)g(all)f Fl(\016)16 b(>)e Fs(0)d Fd(there)h(exists)g(a)g (neigh)o(b)q(orho)q(o)q(d)57 703 y Fl(W)25 b Fd(of)19 b Fl(z)211 710 y Fi(0)252 703 y Fd(suc)o(h)e(that)h(for)g(all)g Fl(z)i Fk(2)d Fl(W)25 b Fd(and)18 b(for)g(all)g Fl(n)e Fk(\025)h Fs(0)h Fd(one)g(has)g Fl(d)p Fs(\()p Fl(R)1449 685 y Fh(n)1477 703 y Fs(\()p Fl(z)r Fs(\))p Fl(;)8 b(R)1600 685 y Fh(n)1628 703 y Fs(\()p Fl(z)1670 710 y Fi(0)1693 703 y Fs(\)\))18 b Fk(\024)e Fl(\016)57 773 y Fd(\(here,)g(as)g(usual,) f Fl(d)i Fd(denotes)f(the)h(spherical)e(metric\).)57 901 y Fr(Exercise)i(2.6)d Fs(Sho)o(w)f(that)h(a)g(p)q(oin)o(t)g(is)g (stable)g(if)g(and)g(only)g(if)g(it)g(b)q(elongs)g(to)g(the)h(F)l(atou) e(set.)57 1006 y Fr(Exercise)k(2.7)d Fs(Let)h Fl(R)f Fk(2)g Fs(End)7 b(\()p 671 966 V Fm(C)13 b Fs(\))h(and)g(assume)f(that) h Fl(R)p Fs(\(0\))h(=)e(0.)22 b(If)14 b Fl(R)h Fs(is)e(linearizable)g (and)57 1076 y Fl(R)95 1058 y Fj(0)109 1076 y Fs(\(0\))h(=)g Fl(\025)p Fs(,)i Fk(j)p Fl(\025)p Fk(j)e(\024)g Fs(1)i(then)h(0)f(is)g (a)h(stable)f(\014xed)g(p)q(oin)o(t.)57 1181 y(If)24 b(w)o(e)f(consider)f(the)i(more)f(general)g(situation)g(of)h(a)f(germ)g Fl(f)31 b Fk(2)26 b Fl(S)s Fs(,)f(i.e.)43 b Fl(f)d Fs(:)25 b Fm(D)37 b Fk(!)25 b Fm(C)57 1251 y Fs(injectiv)o(ely)h(and)f (holomorphically)l(,)g Fl(f)5 b Fs(\(0\))30 b(=)e(0,)g(the)e (de\014nition)e(of)i(stabilit)o(y)f(m)o(ust)f(b)q(e)57 1321 y(sligh)o(tly)18 b(generalized)g(so)h(as)f(to)i(tak)o(e)f(in)o(to) g(accoun)o(t)f(the)i(fact)g(that)f(the)g(iterates)h(of)f Fl(f)25 b Fs(are)57 1390 y(not)16 b(necesaarily)g(de\014ned)f(for)i (all)f Fl(n)p Fs(.)57 1518 y Fr(De\014nition)k(2.8)28 b Fs(0)16 b Fd(is)h Fs(stable)f Fd(if)g(and)g(only)g(if)h(there)g (exists)f(a)h(neigh)o(b)q(orho)q(o)q(d)d Fl(U)22 b Fd(of)17 b Fs(0)g Fd(suc)o(h)57 1588 y(that)23 b Fl(f)200 1570 y Fh(n)252 1588 y Fd(is)g(de\014ned)f(on)h Fl(U)29 b Fd(for)23 b(all)g Fl(n)i Fk(\025)g Fs(0)e Fd(and)g(for)g(all)g Fl(z)28 b Fk(2)d Fl(U)k Fd(and)23 b Fl(n)i Fk(\025)g Fs(0)e Fd(one)g(has)57 1658 y Fk(j)p Fl(f)100 1640 y Fh(n)127 1658 y Fs(\()p Fl(z)r Fs(\))p Fk(j)15 b Fl(<)f Fs(1)p Fd(.)57 1785 y Fr(Exercise)i(2.9)d Fs(Sho)o(w)f(that)i(if)g Fl(f)19 b Fs(is)13 b(a)h(rational)e(map)h(with)g(a)h(\014xed)f(p)q(oin) o(t)g(0)g(then)h(de\014nitions)57 1855 y(2.8)i(and)g(2.5)g(are)g(equiv) m(alen)o(t.)57 1960 y Fr(Exercise)k(2.10)15 b Fs(If)i Fl(f)481 1942 y Fj(0)496 1960 y Fs(\(0\))d(=)g Fl(\025)i Fs(and)g Fk(j)p Fl(\025)p Fk(j)e Fl(>)f Fs(1)k(then)f(0)h(is)f(not)h (stable.)57 2066 y(T)l(o)f(eac)o(h)f(germ)h Fl(f)j Fk(2)14 b Fl(S)s Fs(,)i Fk(j)p Fl(f)563 2047 y Fj(0)577 2066 y Fs(\(0\))p Fk(j)f(\024)e Fs(1,)j(one)g(can)g(asso)q(ciate)g(a)g (natural)f Fl(f)5 b Fs({in)o(v)m(arian)o(t)16 b(compact)57 2135 y(set)688 2205 y(0)e Fk(2)g Fl(K)816 2212 y Fh(f)856 2205 y Fs(:=)933 2158 y Fe(\\)922 2265 y Fh(n)p Fj(\025)p Fi(0)1006 2205 y Fl(f)1035 2184 y Fj(\000)p Fh(n)1094 2205 y Fs(\()p Fm(D)9 b Fs(\))17 b Fl(:)529 b Fs(\(2)p Fl(:)p Fs(2\))57 2336 y(Let)20 b Fl(U)183 2343 y Fh(f)228 2336 y Fs(denote)f(the)h(connected)f(comp)q(onen)o(t)f(of)i(the)f(in)o (terior)f(of)i Fl(K)1401 2343 y Fh(f)1446 2336 y Fs(whic)o(h)e(con)o (tains)h(0.)57 2406 y(Then)d(0)i(is)f(stable)f(if)i(and)e(only)h(if)h Fl(U)760 2413 y Fh(f)801 2406 y Fk(6)p Fs(=)c Fk(;)p Fs(,)k(i.e.)24 b(if)17 b(and)g(only)g(if)g(0)g(b)q(elongs)g(to)g(the)h (in)o(terior)57 2475 y(of)e Fl(K)155 2482 y Fh(f)182 2475 y Fs(.)57 2581 y Fr(Exercise)f(2.11)d Fs(Sho)o(w)g(that)h(if)g Fl(f)19 b Fk(2)14 b Fl(S)i Fs(and)c Fk(j)p Fl(f)941 2563 y Fj(0)955 2581 y Fs(\(0\))p Fk(j)j Fl(<)e Fs(1)g(then)g(0)g(is)f (stable.)20 b([Hin)o(t)13 b(:)20 b(consider)57 2650 y(a)c(small)g(disk) g(around)f(0)h(on)g(whic)o(h)g(the)h(inequalit)o(y)f Fk(j)p Fl(f)5 b Fs(\()p Fl(z)r Fs(\))p Fk(j)15 b(\024)f Fl(\032)p Fk(j)p Fl(z)r Fk(j)i Fs(with)g Fl(\032)e(<)g Fs(1)i(holds.])918 2770 y(13)p eop %%Page: 14 15 14 14 bop 57 297 a Fo(2.3)19 b(Stabilit)n(y)k(vs.)i(Linearizabilit)n(y) 57 402 y Fs(The)18 b(main)g(result)g(of)g(this)h(section)f(is)g(the)h (equiv)m(alence)g(\(for)g Fk(j)p Fl(f)1301 384 y Fj(0)1315 402 y Fs(\(0\))p Fk(j)f(\024)f Fs(1\))i(of)g(stabilit)o(y)f(\(a)57 472 y(top)q(ological)e(notion\))g(and)g(linearizabilit)o(y)f(\(an)i (analytical)f(notion\))g(:)57 606 y Fr(Theorem)h(2.12)27 b Fd(Let)18 b Fl(f)h Fk(2)14 b Fl(S)s Fd(,)i Fk(j)p Fl(f)717 588 y Fj(0)732 606 y Fs(\(0\))p Fk(j)e(\024)g Fs(1)p Fd(.)22 b Fs(0)16 b Fd(is)g(stable)g(if)h(and)f(only)g(if)h Fl(f)22 b Fd(is)16 b(linearizable.)57 741 y Fp(Pr)m(o)m(of.)23 b Fs(The)d(statemen)o(t)g(is)g(non{trivial)f(only)h(if)g Fl(\025)g Fs(=)g Fl(f)1160 723 y Fj(0)1175 741 y Fs(\(0\))h(has)e(unit) h(mo)q(dulus.)31 b(If)21 b Fl(f)26 b Fs(is)57 811 y(linearizable)10 b(then)i(the)g(linearization)e Fl(h)814 818 y Fh(f)852 811 y Fs(maps)g(a)i(small)f(disk)g Fm(D)1272 818 y Fh(r)1309 811 y Fs(around)f(zero)i(conformally)57 880 y(in)o(to)g Fm(D)c Fs(.)24 b(Since)13 b Fl(h)381 887 y Fh(f)407 880 y Fs(\(0\))h(=)g(0)f(and)g Fk(j)p Fl(f)712 862 y Fh(n)739 880 y Fs(\()p Fl(z)r Fs(\))p Fk(j)i Fl(<)f Fs(1)f(for)g(all)g Fl(z)j Fk(2)e Fl(h)1176 887 y Fh(f)1202 880 y Fs(\()p Fm(D)1254 887 y Fh(r)1279 880 y Fs(\))g(one)f(sees)g(that)h(0)f(is)g (stable.)156 950 y(Con)o(v)o(ersely)j(assume)f(no)o(w)h(that)h(0)g(is)f (stable.)22 b(Then)16 b Fl(U)1213 957 y Fh(f)1253 950 y Fk(6)p Fs(=)e Fk(;)i Fs(and)g(one)h(can)f(easily)g(see)57 1020 y(that)h(it)g(m)o(ust)f(also)g(b)q(e)i(simply)d(connected)i (\(otherwise,)g(if)g(it)g(had)f(a)h(hole)g Fl(V)11 b Fs(,)17 b(surroundin)o(g)57 1089 y(it)j(with)f(some)g(closed)g(curv)o (e)g Fl(\015)j Fs(con)o(tained)d(in)g Fl(U)1012 1096 y Fh(f)1057 1089 y Fs(since)g Fk(j)p Fl(f)1225 1071 y Fh(n)1253 1089 y Fs(\()p Fl(z)r Fs(\))p Fk(j)h Fl(<)f Fs(1)g(for)g(all)h Fl(z)h Fk(2)e Fl(\015)j Fs(and)57 1159 y Fl(n)13 b Fk(\025)h Fs(0)i(the)g(maxim)o(um)f(principle)f(leads) h(to)i(the)f(same)g(conclusion)e(for)i(all)g(the)g(p)q(oin)o(ts)g(in)f Fl(V)57 1229 y Fs(th)o(us)j Fl(V)28 b Fk(\032)18 b Fl(U)317 1236 y Fh(f)343 1229 y Fs(\).)29 b(Applying)18 b(the)h(Riemann)e (mapping)g(theorem)i(to)g Fl(U)1425 1236 y Fh(f)1469 1229 y Fs(one)g(sees)g(that)g(b)o(y)57 1299 y(conjugation)e(with)g(the) h(Riemann)e(map)h Fl(f)24 b Fs(induces)16 b(a)i(univ)m(alen)o(t)f(map)g Fl(g)i Fs(of)f(the)g(disk)f(in)o(to)57 1368 y(itself)g(with)g(the)h (same)e(linear)g(part)h Fl(\025)p Fs(.)24 b(By)18 b(Sc)o(h)o(w)o(arz')d (Lemma)i(one)g(m)o(ust)f(ha)o(v)o(e)h Fl(g)r Fs(\()p Fl(z)r Fs(\))e(=)g Fl(\025z)57 1438 y Fs(th)o(us)g Fl(f)23 b Fs(is)16 b(analytically)g(linearizable.)997 b Fa(\003)57 1558 y Fs(When)22 b Fl(\025)h Fs(=)h Fl(f)352 1540 y Fj(0)367 1558 y Fs(\(0\))f(has)e(mo)q(dulus)g(one,)j(is)e(not)g(a)g(ro) q(ot)h(of)f(unit)o(y)g(and)g(0)g(is)g(stable)g(then)57 1628 y Fl(U)91 1635 y Fh(f)138 1628 y Fs(is)f(conformally)f(equiv)m (alen)o(t)h(to)h(a)f(disk)f(and)h(is)g(called)g(the)g Fp(Sie)m(gel)h(disk)h Fs(of)e Fl(f)27 b Fs(\(at)22 b(0\).)57 1698 y(Th)o(us)g(the)j(Siegel)e(disk)h(of)g Fl(f)30 b Fs(is)24 b(the)g(maximal)f(connected)h(op)q(en)g(set)g(con)o(taining)f (0)h(on)57 1767 y(whic)o(h)18 b Fl(f)25 b Fs(is)19 b(conjugated)g(to)g Fl(R)661 1774 y Fh(\025)687 1767 y Fs(.)30 b(The)19 b(conformal)f (represen)o(tation)1399 1754 y(~)1399 1767 y Fl(h)1428 1774 y Fh(f)1480 1767 y Fs(:)g Fm(D)1545 1776 y Fh(c)p Fi(\()p Fh(f)t Fi(\))1641 1767 y Fk(!)g Fl(U)1743 1774 y Fh(f)1788 1767 y Fs(of)57 1837 y Fl(U)91 1844 y Fh(f)136 1837 y Fs(whic)o(h)g(satis\014es)468 1824 y(~)468 1837 y Fl(h)497 1844 y Fh(f)522 1837 y Fs(\(0\))i(=)e(0,)722 1824 y(~)721 1837 y Fl(h)750 1819 y Fj(0)750 1851 y Fh(f)776 1837 y Fs(\(0\))h(=)f(1)i(linearizes)e Fl(f)25 b Fs(th)o(us)19 b(the)g(p)q(o)o(w)o(er)g(series)f(of)1774 1824 y(~)1774 1837 y Fl(h)1803 1844 y Fh(f)57 1907 y Fs(and)e Fl(h)183 1914 y Fh(f)225 1907 y Fs(coincide.)22 b(If)c Fl(r)q Fs(\()p Fl(f)5 b Fs(\))19 b(denotes)d(the)h(radius)e(of)i(con)o(v)o (ergence)f(of)h(the)g(linearization)f Fl(h)1803 1914 y Fh(f)57 1977 y Fs(\(whose)j(p)q(o)o(w)o(er)f(series)g(co)q(e\016cien) o(ts)h(are)g(recursiv)o(ely)f(determined)g(as)h(in)g(\(1.4\)\),)i (recalling)57 2046 y(the)16 b(de\014nition)g(of)h(conformal)e(capacit)o (y)h(\(Exercise)h(A1.4,)f(App)q(endix)h(1\))g(w)o(e)f(see)g(that)h(:)79 2116 y(\(i\))25 b(if)17 b Fl(r)q Fs(\()p Fl(f)5 b Fs(\))16 b Fl(>)e Fs(0)i(then)h(0)c Fl(<)h(c)p Fs(\()p Fl(U)682 2123 y Fh(f)708 2116 y Fl(;)8 b Fs(0\))14 b(=)g Fl(c)p Fs(\()p Fl(f)5 b Fs(\))15 b Fk(\024)e Fl(r)q Fs(\()p Fl(f)5 b Fs(\))12 b(;)65 2186 y(\(ii\))25 b(if)17 b Fl(r)q Fs(\()p Fl(f)5 b Fs(\))16 b(=)e(0)i(then)h Fl(c)p Fs(\()p Fl(f)5 b Fs(\))15 b(=)e Fl(r)q Fs(\()p Fl(f)5 b Fs(\))16 b(=)e(0.)57 2291 y Fr(Exercise)21 b(2.13)16 b Fs(Sho)o(w)g(that)i(the)g (map)f Fl(c)23 b Fs(:)15 b Fl(S)949 2298 y Fg(T)990 2291 y Fk(!)g Fs([0)p Fl(;)8 b Fs(1])17 b(whic)o(h)f(asso)q(ciates)h(to)h (eac)o(h)f(germ)57 2361 y Fl(f)22 b Fs(the)17 b(conformal)e(capacit)o (y)i(of)f Fl(U)707 2368 y Fh(f)750 2361 y Fs(w.r.t.)g(0)g(is)g(upp)q (er)g(semicon)o(tin)o(uous.)57 2466 y(When)g(0)g(is)h(not)f(stable)g (and)g Fl(\025)h Fs(is)f(not)g(a)h(ro)q(ot)g(of)f(unit)o(y)g Fl(K)1184 2473 y Fh(f)1227 2466 y Fs(is)g(called)g(a)h Fp(he)m(dgeho)m(g)p Fs(.)156 2536 y(W)l(e)29 b(conclude)g(this)f(in)o (tro)q(duction)g(to)h(Siegel)f(disks)g(with)h(t)o(w)o(o)g(results)e(on) i(their)57 2605 y(conformal)15 b(capacit)o(y)l(.)918 2770 y(14)p eop %%Page: 15 16 15 15 bop 57 192 a Fr(Prop)r(osition)22 b(2.14)27 b Fd(One)19 b(has)g Fl(c)p Fs(\()p Fl(f)5 b Fs(\))20 b(=)f Fl(r)q Fs(\()p Fl(f)5 b Fs(\))21 b Fd(when)f(at)f(least)h(one)f(of)h(the)g(t)o (w)o(o)f(follo)o(wing)57 261 y(conditions)c(is)h(satis\014ed)g(:)79 331 y(\(i\))25 b Fl(U)190 338 y Fh(f)233 331 y Fd(is)16 b(relativ)o(ely)g(compact)g(in)h Fm(D)k Fd(;)65 401 y(\(ii\))k(eac)o(h) 16 b(p)q(oin)o(t)g(of)h Fm(S)486 383 y Fi(1)522 401 y Fd(is)f(a)g(singularit)o(y)f(of)i Fl(f)5 b Fd(.)57 566 y Fr(Prop)r(osition)18 b(2.15)28 b Fd(Let)17 b Fl(\025)d Fk(2)g Fm(T)9 b Fd(and)16 b(assume)f(that)i Fl(\025)f Fd(is)h(not)f(a)h(ro)q(ot)f(of)h(unit)o(y)l(.)k(Then)717 695 y Fs(inf)698 728 y Fh(f)t Fj(2)p Fh(S)772 733 y Fb(\025)804 695 y Fl(c)p Fs(\()p Fl(f)5 b Fs(\))15 b(=)32 b(inf)961 728 y Fh(f)t Fj(2)p Fh(S)1035 733 y Fb(\025)1067 695 y Fl(r)q Fs(\()p Fl(f)5 b Fs(\))16 b Fl(:)57 877 y Fs(F)l(or)f(the)i (pro)q(ofs)f(see)g([Y)l(o2,)h(p.19].)918 2770 y(15)p eop %%Page: 16 17 16 16 bop 57 192 a Fq(3.)43 b(The)28 b(Quadratic)g(P)n(olynomial.)44 b(Y)-6 b(o)r(ccoz's)28 b(Pro)r(of)f(of)h(Siegel)57 261 y(Theorem)57 368 y Fs(In)16 b(this)g(Chapter)g(w)o(e)g(will)g(sho)o(w)g (the)h(sp)q(ecial)f(role)g(pla)o(y)o(ed)f(b)o(y)h(the)h(quadratic)f(p)q (olynomial)702 521 y Fl(P)734 528 y Fh(\025)760 521 y Fs(\()p Fl(z)r Fs(\))f(=)f Fl(\025)928 451 y Fe(\022)965 521 y Fl(z)f Fk(\000)1057 488 y Fl(z)1082 470 y Fi(2)p 1057 510 48 2 v 1068 556 a Fs(2)1110 451 y Fe(\023)1169 521 y Fl(:)543 b Fs(\(3)p Fl(:)p Fs(1\))57 673 y(Indeed)10 b Fl(P)245 680 y Fh(\025)283 673 y Fs(is)g(the)i(\\w)o(orst)e(p)q (ossible)g(p)q(erturbation)g(of)h(the)g(linear)g(part)f Fl(R)1436 680 y Fh(\025)1462 673 y Fs(")h(as)g(the)h(follo)o(wing)57 743 y(theorem)k(sho)o(ws)57 884 y Fr(Theorem)22 b(3.1)g(\(Y)-5 b(o)r(ccoz\))30 b Fd(Let)22 b Fl(\025)e Fs(=)h Fl(e)874 866 y Fi(2)p Fh(\031)q(i\013)961 884 y Fd(,)g Fl(\013)g Fk(2)g Fm(R)10 b Fk(n)k Fm(Q)p Fd(.)35 b(If)21 b Fl(P)1364 891 y Fh(\025)1410 884 y Fd(is)f(linearizable)f(then)57 954 y(ev)o(ery)d(germ)g Fl(f)k Fk(2)14 b Fl(G)443 961 y Fh(\025)485 954 y Fd(is)j(also)e(linearizable.)57 1093 y Fp(Pr)m(o)m(of.)32 b Fs(Let)e Fl(f)41 b Fk(2)36 b Fl(G)493 1100 y Fh(\025)519 1093 y Fs(,)d Fl(f)5 b Fs(\()p Fl(z)r Fs(\))37 b(=)f Fl(\025z)22 b Fs(+)902 1056 y Fe(P)955 1068 y Fj(1)955 1108 y Fh(n)p Fi(=2)1041 1093 y Fl(f)1065 1100 y Fh(n)1092 1093 y Fl(z)1117 1075 y Fh(n)1145 1093 y Fs(.)61 b(By)30 b(conjugating)f(with)g(some)57 1163 y(homothet)o(y)18 b(one)g(has)g Fk(j)p Fl(f)530 1170 y Fh(n)558 1163 y Fk(j)f(\024)g Fs(10)695 1145 y Fj(\000)p Fi(3)749 1163 y Fs(4)774 1145 y Fj(\000)p Fh(n)832 1163 y Fs(.)28 b(W)l(e)19 b(no)o(w)f(consider)g(the)h(one{parameter)e (family)57 1233 y Fl(f)81 1240 y Fh(b)101 1233 y Fs(\()p Fl(z)r Fs(\))e(=)f Fl(\025z)s Fs(+)q Fl(bz)373 1215 y Fi(2)397 1233 y Fs(+)437 1196 y Fe(P)489 1208 y Fj(1)489 1248 y Fh(n)p Fi(=2)575 1233 y Fl(f)599 1240 y Fh(n)626 1233 y Fl(z)651 1215 y Fh(n)679 1233 y Fs(.)20 b(Note)12 b(that)g Fl(f)955 1240 y Fi(0)992 1233 y Fs(=)i Fl(f)5 b Fs(.)21 b(Since)11 b Fl(\025)g Fs(is)g(not)h(a)f(ro)q(ot)h(of)f(unit) o(y)h(there)57 1303 y(exists)19 b(a)h(unique)f(formal)g(germ)696 1290 y(^)695 1303 y Fl(h)724 1310 y Fh(b)763 1303 y Fk(2)827 1290 y Fs(^)816 1303 y Fl(G)855 1310 y Fi(1)897 1303 y Fs(suc)o(h)f(that)1124 1290 y(^)1124 1303 y Fl(h)1153 1281 y Fj(\000)p Fi(1)1153 1318 y Fh(b)1206 1303 y Fl(f)1230 1310 y Fh(b)1251 1290 y Fs(^)1250 1303 y Fl(h)1279 1310 y Fh(b)1318 1303 y Fs(=)h Fl(R)1414 1310 y Fh(\025)1440 1303 y Fs(.)32 b(Its)20 b(p)q(o)o(w)o(er)e(series)57 1372 y(expansion)c(is)337 1359 y(^)337 1372 y Fl(h)366 1379 y Fh(b)386 1372 y Fs(\()p Fl(z)r Fs(\))h(=)e Fl(z)f Fs(+)600 1335 y Fe(P)652 1347 y Fj(1)652 1387 y Fh(n)p Fi(=2)738 1372 y Fl(h)767 1379 y Fh(n)794 1372 y Fs(\()p Fl(b)p Fs(\))p Fl(z)878 1354 y Fh(n)923 1372 y Fs(with)k Fl(h)1065 1379 y Fh(n)1091 1372 y Fs(\()p Fl(b)p Fs(\))f Fk(2)f Fm(C)9 b Fs([)p Fl(b)q Fs(].)24 b(Th)o(us)15 b(b)o(y)g(the)h (maxim)o(um)57 1442 y(principle)j(one)i(has)f Fk(j)p Fl(h)498 1449 y Fh(n)525 1442 y Fs(\(0\))p Fk(j)i(\024)f Fs(max)777 1451 y Fj(j)p Fh(b)p Fj(j)p Fi(=1)p Fh(=)p Fi(2)920 1442 y Fk(j)p Fl(h)963 1449 y Fh(n)989 1442 y Fs(\()p Fl(b)p Fs(\))p Fk(j)p Fs(.)37 b(If)21 b Fk(j)p Fl(b)p Fk(j)h Fs(=)f(1)p Fl(=)p Fs(2)g(then)g(\(p)q(ossibly)f(after)57 1512 y(conjugation)i(with)h(a)g(rotation\))f Fl(f)741 1519 y Fh(b)762 1512 y Fs(\()p Fl(z)r Fs(\))j(=)g Fl(P)946 1519 y Fh(\025)972 1512 y Fs(\()p Fl(z)r Fs(\))16 b(+)1105 1475 y Fe(P)1158 1487 y Fj(1)1158 1527 y Fh(n)p Fi(=2)1244 1512 y Fl(f)1268 1519 y Fh(n)1295 1512 y Fl(z)1320 1494 y Fh(n)1372 1512 y Fs(=)24 b Fl(P)1467 1519 y Fh(\025)1493 1512 y Fs(\()p Fl(z)r Fs(\))17 b(+)e Fl( )r Fs(\()p Fl(z)r Fs(\))24 b(and)57 1582 y(it)d(is)g(immediate)g(to)g(c)o(hec)o(k)g(that) h(sup)807 1594 y Fh(z)q Fj(2)p Fg(D)877 1599 y Fc(3)910 1582 y Fk(j)p Fl( )r Fs(\()p Fl(z)r Fs(\))p Fk(j)h Fl(<)e Fs(10)1168 1564 y Fj(\000)p Fi(2)1222 1582 y Fs(.)36 b(F)l(rom)20 b(Douady{Hubbard's)57 1651 y(theorem)15 b(on)h(the)g(stabilit)o(y)f(of)h(the)g(quadratic)g(p)q(olynomial)f (\(App)q(endix)h(1\))g(it)g(follo)o(ws)f(that)57 1721 y Fl(f)81 1728 y Fh(b)119 1721 y Fs(is)j(quasiconformally)e(conjugated) h(to)i Fl(P)905 1728 y Fh(\025)931 1721 y Fs(.)26 b(If)18 b Fl(P)1054 1728 y Fh(\025)1098 1721 y Fs(is)f(linearizable)g(then)h(0) g(is)f(stable)h(for)57 1791 y Fl(P)89 1798 y Fh(\025)131 1791 y Fs(th)o(us)d(also)h(for)f Fl(f)439 1798 y Fh(b)476 1791 y Fs(since)h(a)g(quasiconformal)e(conjugacy)i(is)f(in)h (particular)f(a)h(top)q(ological)57 1861 y(conjugacy)l(.)23 b(But)18 b(w)o(e)f(kno)o(w)f(that)i(this)e(implies)g(that)i Fl(f)1119 1868 y Fh(b)1156 1861 y Fs(is)f(linearizable.)22 b(Therefore)17 b(there)57 1930 y(exists)c(t)o(w)o(o)f(p)q(ositiv)o(e)h (constan)o(ts)f Fl(C)k Fs(and)c Fl(r)j Fs(suc)o(h)d(that)h Fk(j)p Fl(h)1117 1937 y Fh(n)1144 1930 y Fs(\()p Fl(b)p Fs(\))p Fk(j)i(\024)e Fl(C)t(r)1347 1912 y Fj(\000)p Fh(n)1418 1930 y Fs(for)g(all)g Fl(b)g Fs(of)g(mo)q(dulus)57 2000 y(1)p Fl(=)p Fs(2,)j(th)o(us)f Fk(j)p Fl(h)314 2007 y Fh(n)341 2000 y Fs(\(0\))p Fk(j)f(\024)g Fl(C)t(r)548 1982 y Fj(\000)p Fh(n)606 2000 y Fs(.)22 b(Then)773 1987 y(^)772 2000 y Fl(h)801 2007 y Fi(0)840 2000 y Fs(con)o(v)o(erges)15 b(and)h Fl(f)1184 2007 y Fi(0)1221 2000 y Fs(=)d Fl(f)23 b Fs(is)16 b(linearizable.)158 b Fa(\003)57 2193 y Fo(3.1)19 b(Y)-5 b(o)r(ccoz's)17 b(Linearization)22 b(Theorem)d(for)g(the)i (Quadratic)f(P)n(olynomial)57 2300 y Fs(Once)e(one)g(has)g(established) f(that)h(the)h(linearizabilit)o(y)e(of)h(the)h(quadratic)f(p)q (olynomial)f(for)57 2370 y(a)e(certain)h Fl(\025)f Fs(implies)g(that)h Fl(G)622 2377 y Fh(\025)664 2370 y Fs(is)f(a)h(conjugacy)f(class)g(the) h(follo)o(wing)f(remark)m(able)f(theorem)57 2439 y(of)i(Y)l(o)q(ccoz)i (sho)o(ws)d(that)i Fl(G)571 2446 y Fh(\025)614 2439 y Fs(is)f(a)g(conjugacy)h(class)f(for)g(almost)g(all)g Fl(\025)e Fk(2)g Fm(T)-5 b Fs(.)57 2581 y Fr(Theorem)22 b(3.2)28 b Fd(Let)21 b Fl(\025)h Fs(=)e Fl(e)633 2563 y Fi(2)p Fh(\031)q(i\013)721 2581 y Fd(,)h Fl(\013)h Fk(2)f Fm(R)11 b Fk(n)j Fm(Q)p Fd(.)36 b(F)l(or)19 b(almost)i(all)f Fl(\025)h Fk(2)h Fm(T)13 b Fd(the)21 b(quadratic)57 2650 y(p)q(olynomial)15 b Fl(P)346 2657 y Fh(\025)389 2650 y Fd(is)h(linearizable.)918 2770 y Fs(16)p eop %%Page: 17 18 17 17 bop 57 192 a Fs(This)16 b(statemen)o(t)h(deserv)o(es)f(a)h (commen)o(t.)23 b(As)17 b(w)o(e)g(will)g(see)g(in)g(Chapter)g(5)g(this) g(theorem)f(of)57 261 y(Y)l(o)q(ccoz)h(is)g(indeed)f(w)o(eak)o(er)g (than)g(the)h(Siegel)g([S])f(and)g(Brjuno)g([Br])h(theorems)e(whic)o(h) h(date)57 331 y(resp)q(ectiv)o(ely)i(to)g(1942)g(and)f(1970.)27 b(What)18 b(is)g Fp(very)h(r)m(emarkable)g Fs(is)e(the)h(pro)q(of)g(of) g(Theorem)57 401 y(3.2)23 b(whic)o(h)f(do)q(es)i(not)f(need)g(an)o(y)h (subtle)e(estimate)i(on)f(the)h(gro)o(wth)e(of)i(the)g(co)q(e\016cien)o (ts)57 470 y(of)19 b(the)h(formal)e(linearization)g(as)h(pro)o(vided)f (b)o(y)h(\(1.4\))h(:)27 b(compare)19 b(with)g(the)h(pro)q(of)f(of)g (the)57 540 y(Siegel{Brjuno)c(Theorem)g(giv)o(en)i(in)f(Section)g(5.1.) 156 612 y(Let)g(us)f(note)g(that)g Fl(P)554 619 y Fh(\025)596 612 y Fs(has)f(a)h(unique)f(critical)h(p)q(oin)o(t)g Fl(c)e Fs(=)h(1)h(\(apart)g(from)f Fl(z)i Fs(=)e Fk(1)p Fs(\))h(and)57 682 y(that)i(the)h(corresp)q(onding)d(critical)h(v)m (alue)i(is)f Fl(v)944 689 y Fh(\025)985 682 y Fs(=)d Fl(P)1070 689 y Fh(\025)1097 682 y Fs(\()p Fl(c)p Fs(\))h(=)f Fl(\025=)p Fs(2.)24 b(If)18 b Fk(j)p Fl(\025)p Fk(j)c Fl(<)h Fs(1)i(b)o(y)g(Ko)q(enigs{)57 752 y(P)o(oincar)o(\023)-24 b(e)14 b(theorem)i(w)o(e)g(kno)o(w)f(that)i(there)f(exists)g(a)g (unique)g(analytic)g(linearization)f Fl(H)1746 759 y Fh(\025)1788 752 y Fs(of)57 822 y Fl(P)89 829 y Fh(\025)134 822 y Fs(and)j(that)h(it)g(dep)q(ends)f(analytically)h(on)f Fl(\025)h Fs(as)g Fl(\025)g Fs(v)m(aries)f(in)g Fm(D)8 b Fs(.)32 b(Let)20 b Fl(r)1487 829 y Fi(2)1510 822 y Fs(\()p Fl(\025)p Fs(\))f(denote)g(the)57 891 y(radius)c(of)h(con)o(v)o (ergence)g(of)h Fl(H)637 898 y Fh(\025)663 891 y Fs(.)22 b(One)16 b(has)g(the)h(follo)o(wing)57 1036 y Fr(Prop)r(osition)h(3.3) 28 b Fd(Let)17 b Fl(\025)d Fk(2)g Fm(D)8 b Fd(.)25 b(Then)16 b(:)68 1108 y(\(1\))25 b Fl(r)178 1115 y Fi(2)201 1108 y Fs(\()p Fl(\025)p Fs(\))15 b Fl(>)e Fs(0)d Fd(;)68 1181 y(\(2\))25 b Fl(r)178 1188 y Fi(2)201 1181 y Fs(\()p Fl(\025)p Fs(\))15 b Fl(<)e Fs(+)p Fk(1)i Fd(and)f Fl(H)575 1188 y Fh(\025)616 1181 y Fd(has)g(a)g(con)o(tin)o(uous)f(extension)h (to)p 1267 1140 132 2 v 15 w Fm(D)1300 1190 y Fh(r)1318 1195 y Fc(2)1341 1190 y Fi(\()p Fh(\025)p Fi(\))1398 1181 y Fd(.)21 b(Moreo)o(v)o(er)13 b(the)i(map)156 1250 y Fl(H)197 1257 y Fh(\025)246 1250 y Fs(:)p 274 1210 V 14 w Fm(D)307 1259 y Fh(r)325 1264 y Fc(2)348 1259 y Fi(\()p Fh(\025)p Fi(\))419 1250 y Fk(!)e Fm(C)29 b Fd(is)16 b(conformal)f(and)h(v)o(eri\014es)g Fl(P)1112 1257 y Fh(\025)1149 1250 y Fk(\016)11 b Fl(H)1226 1257 y Fh(\025)1266 1250 y Fs(=)j Fl(H)1360 1257 y Fh(\025)1397 1250 y Fk(\016)d Fl(R)1471 1257 y Fh(\025)1497 1250 y Fd(.)68 1323 y(\(3\))25 b(On)16 b(its)h(circle)f(of)h(con)o(v)o (ergence)f Fk(f)p Fl(z)11 b(;)16 b Fk(j)p Fl(z)r Fk(j)e Fs(=)g Fl(r)1003 1330 y Fi(2)1026 1323 y Fs(\()p Fl(\025)p Fs(\))p Fk(g)p Fd(,)j Fl(H)1190 1330 y Fh(\025)1233 1323 y Fd(has)f(a)h(unique)f(singular)f(p)q(oin)o(t)156 1392 y(whic)o(h)h(will)g(b)q(e)h(denoted)f Fl(u)p Fs(\()p Fl(\025)p Fs(\))p Fd(.)68 1465 y(\(4\))25 b Fl(H)197 1472 y Fh(\025)224 1465 y Fs(\()p Fl(u)p Fs(\()p Fl(\025)p Fs(\)\))15 b(=)e(1)k Fd(and)f Fs(\()p Fl(H)624 1472 y Fh(\025)651 1465 y Fs(\()p Fl(z)r Fs(\))c Fk(\000)f Fs(1\))820 1446 y Fi(2)859 1465 y Fd(is)16 b(holomorphic)e(in)i Fl(z)g Fs(=)e Fl(u)p Fs(\()p Fl(\025)p Fs(\))p Fd(.)57 1607 y Fp(Pr)m(o)m(of.)20 b Fs(The)c(\014rst)g(assertion)f(is)h(just)h (a)f(consequence)g(of)h(Ko)q(enigs{P)o(oincar)o(\023)-24 b(e)15 b(theorem.)156 1679 y(The)20 b(functional)e(equation)i Fl(P)737 1686 y Fh(\025)763 1679 y Fs(\()p Fl(H)823 1686 y Fh(\025)850 1679 y Fs(\()p Fl(z)r Fs(\)\))g(=)e Fl(H)1050 1686 y Fh(\025)1076 1679 y Fs(\()p Fl(\025z)r Fs(\))j(is)e(satis\014ed) f(for)i(all)f Fl(z)i Fk(2)d Fm(D)1716 1688 y Fh(r)1735 1693 y Fc(2)1757 1688 y Fi(\()p Fh(\025)p Fi(\))1815 1679 y Fs(.)57 1749 y(Moreo)o(v)o(er)c Fl(H)316 1756 y Fh(\025)365 1749 y Fs(:)g Fm(D)426 1758 y Fh(r)444 1763 y Fc(2)467 1758 y Fi(\()p Fh(\025)p Fi(\))538 1749 y Fk(!)f Fm(C)29 b Fs(is)15 b(univ)m(alen)o(t)h(\(if)h(one)f(had)f Fl(H)1212 1756 y Fh(\025)1239 1749 y Fs(\()p Fl(z)1281 1756 y Fi(1)1304 1749 y Fs(\))f(=)g Fl(H)1431 1756 y Fh(\025)1457 1749 y Fs(\()p Fl(z)1499 1756 y Fi(2)1522 1749 y Fs(\))j(with)f Fl(z)1694 1756 y Fi(1)1730 1749 y Fk(6)p Fs(=)e Fl(z)1806 1756 y Fi(2)57 1819 y Fs(and)k Fl(z)179 1826 y Fi(1)201 1819 y Fl(;)8 b(z)246 1826 y Fi(2)286 1819 y Fk(2)18 b Fm(D)370 1828 y Fh(r)388 1833 y Fc(2)411 1828 y Fi(\()p Fh(\025)p Fi(\))487 1819 y Fs(one)g(w)o(ould)g(ha)o(v)o(e)g Fl(H)886 1826 y Fh(\025)912 1819 y Fs(\()p Fl(\025)960 1801 y Fh(n)988 1819 y Fl(z)1011 1826 y Fi(1)1033 1819 y Fs(\))g(=)f Fl(H)1167 1826 y Fh(\025)1194 1819 y Fs(\()p Fl(\025)1242 1801 y Fh(n)1270 1819 y Fl(z)1293 1826 y Fi(2)1315 1819 y Fs(\))i(for)g(all)f Fl(n)f Fk(\025)g Fs(0)i(whic)o(h)f(is)57 1888 y(imp)q(ossible)e(since)h Fk(j)p Fl(\025)p Fk(j)f Fl(<)f Fs(1)j(and)f Fl(H)737 1870 y Fj(0)733 1903 y Fh(\025)760 1888 y Fs(\(0\))g(=)e(1\).)26 b(Th)o(us)17 b Fl(r)1128 1895 y Fi(2)1151 1888 y Fs(\()p Fl(\025)p Fs(\))f Fl(<)g Fs(+)p Fk(1)p Fs(.)26 b(On)17 b(the)h(other)f(hand)57 1958 y(if)k Fl(H)148 1965 y Fh(\025)195 1958 y Fs(is)g(holomorphic)e(in)h Fm(D)635 1965 y Fh(r)681 1958 y Fs(for)h(some)f Fl(r)k(>)d Fs(0)g(and)f(the)h(critical)g(v)m (alue)g Fl(v)1562 1965 y Fh(\025)1610 1958 y Fk(62)g Fl(H)1705 1965 y Fh(\025)1732 1958 y Fs(\()p Fm(D)1784 1965 y Fh(r)1809 1958 y Fs(\))57 2028 y(the)i(functional)f(equation)h (allo)o(ws)f(to)h(con)o(tin)o(ue)f(analytically)h Fl(H)1343 2035 y Fh(\025)1393 2028 y Fs(to)g(the)g(disk)g Fm(D)1696 2037 y Fj(j)p Fh(\025)p Fj(j)1744 2027 y Ff(\000)p Fc(1)1792 2037 y Fh(r)1815 2028 y Fs(.)57 2098 y(Therefore)d(there)g(exists)h Fl(u)p Fs(\()p Fl(\025)p Fs(\))f Fk(2)h Fm(C)33 b Fs(suc)o(h)19 b(that)i Fk(j)p Fl(u)p Fs(\()p Fl(\025)p Fs(\))p Fk(j)f Fs(=)h Fl(r)1241 2105 y Fi(2)1264 2098 y Fs(\()p Fl(\025)p Fs(\))g(and)f Fl(H)1494 2105 y Fh(\025)1520 2098 y Fs(\()p Fl(\025u)p Fs(\()p Fl(\025)p Fs(\)\))i(=)e Fl(v)1788 2105 y Fh(\025)1815 2098 y Fs(.)57 2167 y(Suc)o(h)j(a)i Fl(u)p Fs(\()p Fl(\025)p Fs(\))g(is)f(unique)g(since)g Fl(H)755 2174 y Fh(\025)806 2167 y Fs(is)g(injectiv)o(e)h(on)f Fm(D)1180 2176 y Fh(r)1199 2181 y Fc(2)1221 2176 y Fi(\()p Fh(\025)p Fi(\))1278 2167 y Fs(.)46 b(If)25 b Fk(j)p Fl(w)q Fk(j)i Fs(=)g Fk(j)p Fl(\025)p Fk(j)p Fl(r)1633 2174 y Fi(2)1656 2167 y Fs(\()p Fl(\025)p Fs(\))e(and)57 2237 y Fl(w)15 b Fk(6)p Fs(=)e Fl(\025u)p Fs(\()p Fl(\025)p Fs(\))k(one)f(has)g Fl(H)523 2244 y Fh(\025)550 2237 y Fs(\()p Fl(w)q Fs(\))e(=)g Fl(P)724 2244 y Fh(\025)750 2237 y Fs(\()p Fl(H)810 2244 y Fh(\025)837 2237 y Fs(\()p Fl(\025)885 2219 y Fj(\000)p Fi(1)939 2237 y Fl(w)q Fs(\)\))j(and)560 2374 y Fl(H)601 2381 y Fh(\025)628 2374 y Fs(\()p Fl(\025)676 2353 y Fj(\000)p Fi(1)730 2374 y Fl(w)q Fs(\))d(=)g(1)d Fk(\000)938 2329 y Fe(p)p 988 2329 337 2 v 45 x Fs(1)g Fk(\000)g Fs(2)p Fl(\025)1128 2360 y Fj(\000)p Fi(1)1181 2374 y Fl(H)1222 2381 y Fh(\025)1249 2374 y Fs(\()p Fl(w)q Fs(\))402 b(\(3)p Fl(:)p Fs(2\))57 2511 y(whic)o(h)28 b(sho)o(ws)f(ho)o(w)h(to)i(extend)f(con)o(tin)o(uously)e(and)h (injectiv)o(ely)i Fl(H)1434 2518 y Fh(\025)1489 2511 y Fs(to)p 1562 2471 132 2 v 29 w Fm(D)1595 2520 y Fh(r)1614 2525 y Fc(2)1636 2520 y Fi(\()p Fh(\025)p Fi(\))1694 2511 y Fs(.)59 b(By)57 2581 y(construction)14 b(the)h(functional)g (equation)g(is)g(trivially)g(v)o(eri\014ed.)20 b(This)15 b(completes)f(the)i(pro)q(of)57 2650 y(of)g(\(2\).)918 2770 y(17)p eop %%Page: 18 19 18 18 bop 156 192 a Fs(T)l(o)21 b(pro)o(v)o(e)f(\(3\))i(and)e(\(4\))i (note)f(that)g(from)g Fl(H)1035 199 y Fh(\025)1061 192 y Fs(\()p Fl(\025u)p Fs(\()p Fl(\025)p Fs(\)\))i(=)e Fl(P)1339 199 y Fh(\025)1365 192 y Fs(\()p Fl(H)1425 199 y Fh(\025)1452 192 y Fs(\()p Fl(u)p Fs(\()p Fl(\025)p Fs(\)\)\))h(it)f(follo)o(ws)57 261 y(that)f Fl(H)209 268 y Fh(\025)236 261 y Fs(\()p Fl(u)p Fs(\()p Fl(\025)p Fs(\)\))h(=)e(1.)33 b(F)l(orm)o(ula)18 b(\(3.2\))i(sho)o(ws)f(that)i (all)f(p)q(oin)o(ts)f Fl(z)j Fk(2)e Fm(C)9 b Fs(,)24 b Fk(j)p Fl(z)r Fk(j)c Fs(=)g Fl(r)1651 268 y Fi(2)1674 261 y Fs(\()p Fl(\025)p Fs(\))h(are)57 331 y(regular)12 b(except)i(for)g Fl(z)i Fs(=)d Fl(u)p Fs(\()p Fl(\025)p Fs(\).)22 b(Finally)12 b(one)i(has)e(\()p Fl(H)1073 338 y Fh(\025)1100 331 y Fs(\()p Fl(z)r Fs(\))5 b Fk(\000)g Fs(1\))1256 313 y Fi(2)1294 331 y Fs(=)14 b(1)5 b Fk(\000)g Fs(2)p Fl(\025)1475 313 y Fj(\000)p Fi(1)1528 331 y Fl(H)1569 338 y Fh(\025)1596 331 y Fs(\()p Fl(\025z)r Fs(\))14 b(whic)o(h)57 401 y(is)i(holomorphic)e(also)i(at)h Fl(z)f Fs(=)e Fl(u)p Fs(\()p Fl(\025)p Fs(\).)1035 b Fa(\003)57 518 y Fs(The)21 b(fact)i(that)f Fl(H)422 525 y Fh(\025)470 518 y Fs(is)f(injectiv)o(e)h(on)p 802 478 36 2 v 21 w Fm(D)838 527 y Fh(r)856 532 y Fc(2)876 527 y Fi(\()p Fh(\025)p Fi(\))955 518 y Fs(implies)e(that)i Fl(r)1264 525 y Fi(2)1287 518 y Fs(\()p Fl(\025)p Fs(\))h Fl(<)g Fs(+)p Fk(1)e Fs(\(otherwise)g(it)57 588 y(w)o(ould)h(b)q(e)j(a)f (biholomorphis)o(m)d(of)j Fm(C)36 b Fs(th)o(us)23 b(an)h(a\016ne)f (map\).)44 b(A)25 b(more)e(precise)g(upp)q(er)57 658 y(b)q(ound)15 b(is)h(pro)o(vided)f(b)o(y)h(the)h(follo)o(wing)57 787 y Fr(Lemma)g(3.4)h(\(a)h(priori)h(estimate)f(of)f Fl(r)897 794 y Fi(2)920 787 y Fs(\()p Fl(\025)p Fs(\))p Fr(\).)52 b Fl(r)1099 794 y Fi(2)1122 787 y Fs(\()p Fl(\025)p Fs(\))14 b Fk(\024)g Fs(2)p Fd(.)57 916 y Fp(Pr)m(o)m(of.)23 b Fs(It)e(is)g(an)f(easy)g(consequence)h(of)f(Ko)q(eb)q(e)h(1)p Fl(=)p Fs(4{Theorem.)33 b(Indeed)20 b(if)1581 903 y(~)1570 916 y Fl(f)27 b Fk(2)21 b Fl(S)1706 923 y Fi(1)1748 916 y Fs(and)57 986 y Fl(t)e(>)f Fs(0)i(then)f Fl(f)25 b Fs(=)19 b Fl(R)457 965 y Fj(\000)p Fi(1)457 998 y Fh(t)521 973 y Fs(~)510 986 y Fl(f)6 b(R)578 993 y Fh(t)623 986 y Fs(:)18 b Fm(D)688 995 y Fh(t)703 985 y Ff(\000)p Fc(1)774 986 y Fk(!)h Fm(C)32 b Fs(is)19 b(univ)m(alen)o(t)g(and)f Fl(f)5 b Fs(\()p Fm(D)1353 995 y Fh(t)1368 985 y Ff(\000)p Fc(1)1419 986 y Fs(\))20 b(=)e Fl(R)1553 995 y Fh(t)1568 985 y Ff(\000)p Fc(1)1628 973 y Fs(~)1617 986 y Fl(f)6 b Fs(\()p Fm(D)i Fs(\))q(.)34 b(By)57 1056 y(Ko)q(eb)q(e)19 b(1)p Fl(=)p Fs(4{Theorem)e(one)h(has)g Fm(D)743 1065 y Fi(1)p Fh(=)p Fi(4)825 1056 y Fk(\032)892 1043 y Fs(~)881 1056 y Fl(f)6 b Fs(\()p Fm(D)i Fs(\))22 b(th)o(us)c Fm(D)1149 1065 y Fh(t)1164 1055 y Ff(\000)p Fc(1)1213 1065 y Fh(=)p Fi(4)1273 1056 y Fk(\032)f Fl(f)5 b Fs(\()p Fm(D)1411 1065 y Fh(t)1426 1055 y Ff(\000)p Fc(1)1478 1056 y Fs(\).)28 b(But)19 b(w)o(e)f(kno)o(w)57 1125 y(that)f Fl(v)189 1132 y Fh(\025)229 1125 y Fk(62)d Fl(H)317 1132 y Fh(\025)343 1125 y Fs(\()p Fm(D)396 1134 y Fj(j)p Fh(\025)p Fj(j)p Fh(r)462 1139 y Fc(2)484 1134 y Fi(\()p Fh(\025)p Fi(\))541 1125 y Fs(\))j(th)o(us)f Fk(j)p Fl(v)725 1132 y Fh(\025)751 1125 y Fk(j)d Fs(=)837 1101 y Fj(j)p Fh(\025)p Fj(j)p 837 1114 48 2 v 851 1143 a Fi(2)904 1125 y Fk(\025)963 1101 y Fj(j)p Fh(\025)p Fj(j)p Fh(r)1029 1106 y Fc(2)1048 1101 y Fi(\()p Fh(\025)p Fi(\))p 963 1114 140 2 v 1023 1143 a(4)1109 1125 y Fs(.)667 b Fa(\003)57 1243 y Fr(Exercise)20 b(3.5)c Fs(Sho)o(w)f(that)i Fl(r)635 1250 y Fi(2)658 1243 y Fs(\()p Fl(\025)p Fs(\))d Fk(\024)g Fs(8)p Fl(=)p Fs(7.)22 b([Hin)o(t)16 b(:)22 b(apply)16 b(\(A1.1\))h(to)g(the)g (function)568 1364 y(~)555 1377 y Fl(H)596 1384 y Fh(\025)622 1377 y Fs(\()p Fl(z)r Fs(\))693 1306 y Fe(\022)732 1377 y Fs(1)11 b(+)823 1343 y(2)p Fl(r)870 1350 y Fi(2)893 1343 y Fs(\()p Fl(\025)p Fs(\))p 823 1365 138 2 v 878 1411 a Fl(\025)980 1364 y Fs(~)967 1377 y Fl(H)1008 1384 y Fh(\025)1034 1377 y Fs(\()p Fl(z)r Fs(\))1097 1306 y Fe(\023)1135 1317 y Fj(\000)p Fi(1)1202 1377 y Fk(2)j Fl(S)1280 1384 y Fi(1)1316 1377 y Fl(;)57 1513 y Fs(where)214 1501 y(~)201 1513 y Fl(H)242 1520 y Fh(\025)268 1513 y Fs(\()p Fl(z)r Fs(\))h(=)405 1489 y Fh(H)438 1494 y Fb(\025)461 1489 y Fi(\()p Fh(r)495 1494 y Fc(2)514 1489 y Fi(\()p Fh(\025)p Fi(\))p Fh(z)q Fi(\))p 405 1502 201 2 v 459 1531 a Fh(r)477 1536 y Fc(2)497 1531 y Fi(\()p Fh(\025)p Fi(\))611 1513 y Fs(.])57 1619 y Fr(Exercise)20 b(3.6)c Fs(Sho)o(w)f(that)i(the)g(image)f(b)o(y)h Fl(H)954 1626 y Fh(\025)997 1619 y Fs(of)g(its)f(circle)h(of)g(con)o(v)o (ergence)e(is)h(a)h(Jordan)57 1688 y(curv)o(e,)f(analytic)g(except)h (at)g Fl(H)651 1695 y Fh(\025)678 1688 y Fs(\()p Fl(u)p Fs(\()p Fl(\025)p Fs(\)\))e(=)e(1)k(where)f(it)h(has)f(a)g(righ)o(t)g (angle.)57 1818 y Fr(Prop)r(osition)j(3.7)28 b Fl(u)23 b Fs(:)15 b Fm(D)583 1800 y Fj(\003)624 1818 y Fk(!)g Fm(C)29 b Fd(has)17 b(a)g(b)q(ounded)g(analytic)g(extension)g(to)h Fm(D)8 b Fd(.)27 b(Moreo)o(v)o(er)57 1887 y(it)22 b(is)g(the)g(limit)g (of)g(the)h(sequence)f(of)g(p)q(olynomials)f Fl(u)1124 1894 y Fh(n)1151 1887 y Fs(\()p Fl(\025)p Fs(\))j(=)f Fl(\025)1333 1869 y Fj(\000)p Fh(n)1391 1887 y Fl(P)1430 1869 y Fh(n)1423 1902 y(\025)1457 1887 y Fs(\(1\))g Fd(uniformly)e(on) 57 1957 y(compact)16 b(subsets)f(of)i Fm(D)8 b Fd(.)25 b(One)16 b(has)g Fl(u)p Fs(\(0\))e(=)g(1)p Fl(=)p Fs(2)p Fd(.)57 2087 y Fp(Pr)m(o)m(of.)23 b Fs(By)e(Prop)q(osition)e(3.3)h(one) g(has)f Fl(P)873 2068 y Fh(n)866 2101 y(\025)900 2087 y Fs(\(1\))i(=)e Fl(H)1083 2094 y Fh(\025)1110 2087 y Fs(\()p Fl(\025)1158 2068 y Fh(n)1185 2087 y Fl(u)p Fs(\()p Fl(\025)p Fs(\)\).)34 b(F)l(rom)19 b(Lemma)g(3.4)h(and)57 2156 y(Ko)q(eb)q(e's)g(distorsion)e(estimates)i(\(sp)q(eci\014cally)g (\(A1.4\))h(\))g(applied)e(to)1433 2144 y(~)1420 2156 y Fl(H)1461 2163 y Fh(\025)1488 2156 y Fs(\()p Fl(z)r Fs(\))i(=)1636 2132 y Fh(H)1669 2137 y Fb(\025)1692 2132 y Fi(\()p Fh(u)p Fi(\()p Fh(\025)p Fi(\))p Fh(z)q Fi(\))p 1636 2145 186 2 v 1690 2173 a Fh(u)p Fi(\()p Fh(\025)p Fi(\))57 2226 y Fs(one)16 b(has)296 2336 y Fk(j)p Fl(P)349 2316 y Fh(n)342 2349 y(\025)376 2336 y Fs(\(1\))p Fk(j)e Fs(=)f Fk(j)p Fl(u)p Fs(\()p Fl(\025)p Fs(\))643 2324 y(~)629 2336 y Fl(H)670 2343 y Fh(\025)697 2336 y Fs(\()p Fl(\025)745 2316 y Fh(n)773 2336 y Fs(\))p Fk(j)h(\024)f Fl(r)894 2343 y Fi(2)917 2336 y Fs(\()p Fl(\025)p Fs(\))1064 2303 y Fk(j)p Fl(\025)p Fk(j)1121 2284 y Fh(n)p 991 2325 231 2 v 991 2370 a Fs(\(1)e Fk(\000)g(j)p Fl(\025)p Fk(j)1153 2356 y Fh(n)1180 2370 y Fs(\))1199 2356 y Fi(2)1241 2336 y Fk(\024)j Fs(2)1398 2303 y Fk(j)p Fl(\025)p Fk(j)1455 2284 y Fh(n)p 1325 2325 V 1325 2370 a Fs(\(1)d Fk(\000)g(j)p Fl(\025)p Fk(j)1487 2356 y Fh(n)1514 2370 y Fs(\))1533 2356 y Fi(2)1575 2336 y Fl(;)57 2463 y Fs(th)o(us)23 b Fk(j)p Fl(u)217 2470 y Fh(n)244 2463 y Fs(\()p Fl(\025)p Fs(\))p Fk(j)k(\024)g Fs(2\(1)16 b Fk(\000)g(j)p Fl(\025)p Fk(j)p Fs(\))634 2444 y Fj(\000)p Fi(2)712 2463 y Fs(for)24 b(all)g Fl(\025)j Fk(2)g Fm(D)36 b Fs(and)23 b(the)i(p)q(olynomials)e Fl(u)1562 2470 y Fh(n)1613 2463 y Fs(v)o(erify)h(the)57 2532 y(recurrence)15 b(relation)446 2636 y Fl(u)475 2643 y Fi(0)497 2636 y Fs(\()p Fl(\025)p Fs(\))g(=)e(1)h Fl(;)50 b(u)763 2643 y Fh(n)p Fi(+1)840 2636 y Fs(\()p Fl(\025)p Fs(\))14 b(=)g Fl(u)1003 2643 y Fh(n)1030 2636 y Fs(\()p Fl(\025)p Fs(\))e Fk(\000)1164 2602 y Fl(\025)1193 2584 y Fh(n)p 1164 2625 57 2 v 1180 2670 a Fs(2)1227 2636 y(\()p Fl(u)1275 2643 y Fh(n)1302 2636 y Fs(\()p Fl(\025)p Fs(\)\))1388 2616 y Fi(2)1425 2636 y Fl(:)287 b Fs(\(3)p Fl(:)p Fs(3\))918 2770 y(18)p eop %%Page: 19 20 19 19 bop 57 192 a Fs(This)16 b(sho)o(ws)g(that)i Fl(u)453 199 y Fh(n)497 192 y Fs(con)o(v)o(erges)f(uniformly)f(on)h(compact)g (subsets)f(of)i Fm(D)8 b Fs(.)28 b(The)17 b(limit)g(is)g Fl(u)57 261 y Fs(since)469 331 y(lim)436 361 y Fh(n)p Fj(!)p Fi(+)p Fj(1)579 331 y Fl(u)608 338 y Fh(n)635 331 y Fs(\()p Fl(\025)p Fs(\))e(=)46 b(lim)769 361 y Fh(n)p Fj(!)p Fi(+)p Fj(1)912 331 y Fl(\025)941 310 y Fj(\000)p Fh(n)999 331 y Fl(H)1040 338 y Fh(\025)1067 331 y Fs(\()p Fl(\025)1115 310 y Fh(n)1143 331 y Fl(u)p Fs(\()p Fl(\025)p Fs(\)\))14 b(=)g Fl(u)p Fs(\()p Fl(\025)p Fs(\))g Fl(:)57 435 y Fs(Finally)h(from)h Fk(j)p Fl(u)p Fs(\()p Fl(\025)p Fs(\))p Fk(j)e Fs(=)f Fl(r)556 442 y Fi(2)579 435 y Fs(\()p Fl(\025)p Fs(\))18 b(and)d(Lemma)h(3.4)h(one)f (has)g Fk(j)p Fl(u)p Fs(\()p Fl(\025)p Fs(\))p Fk(j)d(\024)h Fs(2)j(on)f Fm(D)8 b Fs(.)243 b Fa(\003)57 552 y Fs(The)16 b(function)g Fl(u)22 b Fs(:)13 b Fm(D)25 b Fk(!)14 b Fm(C)28 b Fs(will)16 b(b)q(e)h(called)f Fp(Y)l(o)m(c)m(c)m(oz's)21 b(function)p Fs(.)h(It)17 b(has)e(man)o(y)h(remark)m(able)57 622 y(prop)q(erties)f(and)h(it)h(is)f(the)h(ob)s(ject)f(of)h(v)m (arious)f(conjectures)g(\(see)h(Section)f(5.3\).)57 727 y Fr(Exercise)k(3.8)c Fs(Chec)o(k)g(that)h(:)87 797 y(1\))25 b Fl(u)p Fs(\()p Fl(\025)p Fs(\))12 b Fk(\000)f Fl(u)343 804 y Fh(n)369 797 y Fs(\()p Fl(\025)p Fs(\))k(=)e(O)8 b(\()p Fl(\025)598 779 y Fh(n)626 797 y Fs(\))i(;)87 866 y(2\))25 b Fl(u)p Fs(\()p Fl(\025)p Fs(\))14 b(=)325 847 y Fi(1)p 325 855 20 2 v 325 884 a(2)354 866 y Fk(\000)402 847 y Fh(\025)p 402 855 24 2 v 404 884 a Fi(8)435 866 y Fk(\000)483 847 y Fh(\025)507 832 y Fc(2)p 483 855 44 2 v 495 884 a Fi(8)535 866 y Fk(\000)584 847 y Fh(\025)608 832 y Fc(3)p 584 855 V 586 884 a Fi(16)636 866 y Fk(\000)684 847 y Fi(9)p Fh(\025)728 832 y Fc(4)p 684 855 63 2 v 686 884 a Fi(128)756 866 y Fk(\000)812 847 y Fh(\025)836 832 y Fc(5)p 804 855 60 2 v 804 884 a Fi(128)873 866 y Fk(\000)921 847 y Fi(7)p Fh(\025)965 832 y Fc(6)p 921 855 63 2 v 923 884 a Fi(128)993 866 y Fs(+)1041 847 y Fi(3)p Fh(\025)1085 832 y Fc(7)p 1041 855 V 1043 884 a Fi(256)1113 866 y Fk(\000)1161 847 y Fi(29)p Fh(\025)1225 832 y Fc(8)p 1161 855 83 2 v 1163 884 a Fi(1024)1254 866 y Fk(\000)1310 847 y Fh(\025)1334 832 y Fc(9)p 1302 855 60 2 v 1302 884 a Fi(256)1370 866 y Fs(+)1418 847 y Fi(25)p Fh(\025)1482 832 y Fc(10)p 1418 855 100 2 v 1429 884 a Fi(2048)1527 866 y Fs(+)1576 847 y Fi(559)p Fh(\025)1660 832 y Fc(11)p 1576 855 120 2 v 1586 884 a Fi(32768)1704 866 y Fs(+)s Fl(:)8 b(:)g(:)j Fs(;)87 936 y(3\))25 b Fl(u)p Fs(\()p Fl(\025)p Fs(\))14 b Fk(2)h Fm(Q)p Fk(f)p Fl(\025)p Fk(g)i Fs(and)f(all)g(the)h(denominators)d(are) i(a)h(p)q(o)o(w)o(er)e(of)i(2.)57 1006 y(W)l(rite)i(a)h(computer)e (program)f(to)j(calculate)g(the)f(p)q(o)o(w)o(er)g(series)f(expansion)g (of)i Fl(u)f Fs(and)g(use)57 1076 y(it)25 b(to)h(design)e(the)h(lev)o (el)g(sets)g(of)g(log)9 b Fk(j)p Fl(u)p Fk(j)24 b Fs(and)h(arg)8 b Fl(u)p Fs(.)47 b(T)l(ry)25 b(to)g(compute)g(the)g(graph)f(of)57 1145 y Fl(\022)g Fk(7!)f Fs(arg)8 b Fl(u)p Fs(\()p Fl(r)q(e)349 1127 y Fi(2)p Fh(\031)q(i\022)431 1145 y Fs(\))23 b(as)e Fl(r)k Fk(!)e Fs(1)p Fk(\000)p Fs(.)38 b(Y)l(ou)22 b(ma)o(y)f(use)h (some)f(form)o(ulas)f(giv)o(en)h(in)h([Y)l(o2,)h(pp.)57 1215 y(70{71])16 b(and)h(to)h(compare)e(with)h([MMY2].)23 b(If)18 b(y)o(ou)f(get)g(nice)h(pictures)e(I)h(w)o(ould)f(lik)o(e)h(to) h(get)57 1285 y(a)e(cop)o(y)g(of)h(them.)57 1460 y Fo(3.2)i(Radial)j (Limits)f(of)f(Y)-5 b(o)r(ccoz's)17 b(F)-5 b(unction.)27 b(Conclusion)21 b(of)f(the)g(Pro)r(of)57 1589 y Fr(Prop)r(osition)i (3.9)28 b Fd(Let)21 b Fl(\025)594 1596 y Fi(0)635 1589 y Fk(2)f Fm(T)12 b Fd(and)19 b(assume)g(that)h Fl(\025)1157 1596 y Fi(0)1199 1589 y Fd(is)g(not)g(a)g(ro)q(ot)g(of)g(unit)o(y)l(.) 31 b(Then)57 1658 y Fl(r)79 1665 y Fi(2)102 1658 y Fs(\()p Fl(\025)150 1665 y Fi(0)172 1658 y Fs(\))14 b Fk(\025)g Fs(lim)8 b(sup)411 1671 y Fg(D)t Fj(3)p Fh(\025)p Fj(!)p Fh(\025)548 1676 y Fc(0)580 1658 y Fk(j)p Fl(u)p Fs(\()p Fl(\025)p Fs(\))p Fk(j)p Fd(.)57 1787 y Fp(Pr)m(o)m(of.)23 b Fs(Let)e Fl(r)h Fs(=)e(lim)8 b(sup)558 1800 y Fg(D)t Fj(3)p Fh(\025)p Fj(!)p Fh(\025)695 1805 y Fc(0)727 1787 y Fk(j)p Fl(u)p Fs(\()p Fl(\025)p Fs(\))p Fk(j)p Fs(.)34 b(It)20 b(is)g(not)h(restrictiv)o(e)f(to)g(assume)f Fl(r)j(>)e Fs(0.)34 b(Let)57 1857 y(\()p Fl(\025)105 1864 y Fh(n)132 1857 y Fs(\))151 1864 y Fh(n)p Fj(\025)p Fi(1)248 1857 y Fk(\032)19 b Fm(D)30 b Fs(suc)o(h)18 b(that)i Fl(\025)616 1864 y Fh(n)662 1857 y Fk(!)e Fl(\025)759 1864 y Fi(0)801 1857 y Fs(and)h Fk(j)p Fl(u)p Fs(\()p Fl(\025)992 1864 y Fh(n)1019 1857 y Fs(\))p Fk(j)f(!)h Fl(r)q Fs(.)31 b(Since)19 b(the)h(linearizations)d Fl(H)1777 1864 y Fh(\025)1801 1869 y Fb(n)57 1927 y Fs(are)f(univ)m(alen)o(t)f(on)h (their)g(disks)f(of)i(con)o(v)o(ergence)e Fm(D)1031 1936 y Fh(r)1050 1941 y Fc(2)1072 1936 y Fi(\()p Fh(\025)1112 1941 y Fb(n)1136 1936 y Fi(\))1170 1927 y Fs(one)h(can)g(extract)h(a)g (subsequence)57 1997 y(uniformly)d(con)o(v)o(ergen)o(t)g(on)h(the)h (compact)f(susb)q(ets)g(of)g Fm(D)1143 2004 y Fh(r)1168 1997 y Fs(.)22 b(The)15 b(limit)g(function)g Fl(H)20 b Fs(v)o(eri\014es)57 2066 y Fl(H)t Fs(\(0\))e(=)f(0,)i Fl(H)342 2048 y Fj(0)357 2066 y Fs(\(0\))f(=)f(1)i(and)f Fl(H)t Fs(\()p Fl(\025)730 2073 y Fi(0)753 2066 y Fl(z)r Fs(\))g(=)f Fl(P)903 2073 y Fh(\025)927 2078 y Fc(0)949 2066 y Fs(\()p Fl(H)t Fs(\()p Fl(z)r Fs(\)\))k(\(this)d(is)g(immediate) g(b)o(y)g(taking)h(the)57 2136 y(limit)d(of)g(the)h(corresp)q(onding)d (equations)i(for)h Fl(\025)967 2143 y Fh(n)994 2136 y Fs(\).)22 b(Th)o(us)15 b Fl(H)1216 2143 y Fh(\025)1240 2148 y Fc(0)1276 2136 y Fs(=)f Fl(H)21 b Fs(and)16 b Fl(r)1510 2143 y Fi(2)1533 2136 y Fs(\()p Fl(\025)1581 2143 y Fi(0)1603 2136 y Fs(\))f Fk(\025)e Fl(r)q Fs(.)64 b Fa(\003)57 2253 y Fs(Y)l(o)q(ccoz)17 b(has)f(indeed)g(pro)o(v)o(ed)f (the)i(follo)o(wing)e(stronger)g(result)h([Y)l(o2,)g(pp.)22 b(65-69])57 2382 y Fr(Theorem)d(3.10)28 b Fd(F)l(or)17 b(all)h Fl(\025)624 2389 y Fi(0)664 2382 y Fk(2)g Fm(T)-5 b Fd(,)16 b Fk(j)p Fl(u)p Fs(\()p Fl(\025)p Fs(\))p Fk(j)i Fd(has)g(a)h(non{tangen)o(tial)e(limit)h(in)g Fl(\025)1609 2389 y Fi(0)1650 2382 y Fd(whic)o(h)g(is)57 2452 y(equal)e(to)h(the)g (radius)e(of)h(con)o(v)o(ergence)g Fl(r)839 2459 y Fi(2)862 2452 y Fs(\()p Fl(\025)910 2459 y Fi(0)932 2452 y Fs(\))h Fd(of)g Fl(H)1066 2459 y Fh(\025)1090 2464 y Fc(0)1112 2452 y Fd(.)57 2581 y Fs(Of)h(course,)h(if)g Fl(\025)376 2588 y Fi(0)417 2581 y Fs(is)f(a)h(ro)q(ot)g(of)g(unit)o(y)f(then)h Fl(P)961 2588 y Fh(\025)985 2593 y Fc(0)1025 2581 y Fs(is)f(not)h(ev)o (en)g(formally)f(linearizable)f(and)57 2650 y(one)f(p)q(oses)g Fl(r)302 2657 y Fi(2)325 2650 y Fs(\()p Fl(\025)373 2657 y Fi(0)396 2650 y Fs(\))e(=)f(0.)918 2770 y(19)p eop %%Page: 20 21 20 20 bop 156 192 a Fs(Collecting)17 b(Prop)q(ositions)e(3.3,)i(3.7)g (and)f(3.9)h(together)g(one)g(can)g(\014nally)f(pro)o(v)o(e)g(Theo-)57 261 y(rem)f(3.2.)57 366 y Fp(Pr)m(o)m(of.)22 b Fs(of)d(Theorem)f(3.2.) 29 b(Applying)18 b(F)l(atou's)g(Theorem)g(to)h Fl(u)25 b Fs(:)18 b Fm(D)29 b Fk(!)18 b Fm(C)31 b Fs(one)18 b(\014nds)g(that)57 436 y(there)g(exists)h Fl(u)355 418 y Fj(\003)394 436 y Fk(2)f Fl(L)479 418 y Fj(1)521 436 y Fs(\()p Fm(T)-5 b Fl(;)8 b Fm(C)f Fs(\))21 b(suc)o(h)d(that)h(for)f(almost)g(all)g Fl(\025)1235 443 y Fi(0)1275 436 y Fk(2)f Fm(T)11 b Fs(one)18 b(has)g Fk(j)p Fl(u)1604 418 y Fj(\003)1626 436 y Fs(\()p Fl(\025)1674 443 y Fi(0)1697 436 y Fs(\))p Fk(j)g Fl(>)e Fs(0)57 506 y(and)c Fl(u)p Fs(\()p Fl(\025)p Fs(\))i Fk(!)g Fl(u)p Fs(\()p Fl(\025)401 513 y Fi(0)423 506 y Fs(\))f(as)g Fl(\025)h Fk(!)f Fl(\025)648 513 y Fi(0)684 506 y Fs(non)f(tangen)o(tially)l(.)20 b(F)l(rom)11 b(Prop)q(osition)g (3.7)i(one)g(concludes)57 576 y(that)k(for)f(almost)g(all)g Fl(\025)500 583 y Fi(0)536 576 y Fk(2)e Fm(T)9 b Fs(one)16 b(has)g Fl(r)835 583 y Fi(2)858 576 y Fs(\()p Fl(\025)906 583 y Fi(0)929 576 y Fs(\))e Fl(>)f Fs(0.)737 b Fa(\003)57 696 y Fr(Remark)26 b(3.11)d Fs(Con)o(tin)o(uing)e(the)j(ab)q(o)o(v)o(e) g(argumen)o(t)e(of)i(Y)l(o)q(ccoz,)i(L.)e(Carleson)f(and)g(P)l(.)57 766 y(Jones)15 b(pro)o(v)o(e)f(that)j(for)e(almost)h(all)f Fl(\025)f Fk(2)g Fm(T)8 b Fs(the)17 b(critical)e(p)q(oin)o(t)h Fl(z)g Fs(=)d(1)j(of)h Fl(P)1479 773 y Fh(\025)1521 766 y Fs(b)q(elongs)e(to)h(the)57 835 y(b)q(oundary)i(of)i(the)g(Siegel)f (disk)h(\(see,)g(for)g(example,)f([CG]\).)h(This)f(has)g(also)g(b)q (een)h(pro)o(v)o(ed)57 905 y(directly)i(b)o(y)f(M.)h(Herman)f(under)g (the)i(assumption)d(that)i Fl(\013)g Fs(is)g(diophan)o(tine)e([He3].)39 b(M.)57 975 y(Herman)13 b(has)h(also)g(sho)o(wn)f(that)i(there)f(are)g Fl(\025)p Fs('s)g(for)g(whic)o(h)g(the)h(critical)f(p)q(oin)o(t)g(is)g (not)g(on)g(the)57 1044 y(b)q(oundary)h(of)i(the)g(Siegel)f(disk)g (\(ev)o(en)h(though)e(the)i(b)q(oundary)e(is)i(a)f(quasicircle\))g ([Do].)918 2770 y(20)p eop %%Page: 21 22 21 21 bop 57 192 a Fq(4.)62 b(Douady{Gh)n(ys')33 b(Theorem.)61 b(Con)n(tin)n(ued)34 b(F)-6 b(ractions)35 b(and)57 261 y(the)24 b(Brjuno)f(F)-6 b(unction)57 366 y Fs(F)l(rom)17 b(Y)l(o)q(ccoz's)i(theorem)e(it)i(follo)o(ws)f(that)h Fl(G)955 373 y Fh(\025)981 366 y Fs(,)g Fl(\025)f Fs(=)e Fl(e)1139 348 y Fi(2)p Fh(\031)q(i\013)1227 366 y Fs(,)j Fl(\013)e Fk(2)g Fm(R)10 b Fk(n)i Fm(Q)p Fs(,)20 b(is)e(a)g(conjugacy) 57 436 y(class)c(for)g(almost)g(all)g(v)m(alues)g(of)h Fl(\013)p Fs(.)21 b(Let)16 b Fk(Y)j Fs(denote)14 b(the)h(set)g(of)g Fl(\013)f Fk(2)g Fm(R)7 b Fk(n)g Fm(Q)12 b Fs(suc)o(h)i(that)h Fl(G)1730 446 y Fh(e)1749 436 y Fc(2)p Fb(\031)q(i\013)57 506 y Fs(is)i(a)g(conjugacy)h(class.)24 b(Then)17 b(w)o(e)h(already)f (kno)o(w)g(that)h Fk(Y)k Fs(has)17 b(full)g(measure)f(but)i(that)f(its) 57 576 y(complemen)o(t)d(in)i Fm(R)8 b Fk(n)i Fm(Q)17 b Fs(is)f(a)g Fl(G)659 583 y Fh(\016)681 576 y Fs({dense)f(\(Exercise)h (1.18\).)22 b(The)16 b(goal)g(of)h(this)e(Section)h(is)g(to)57 645 y(pro)o(v)o(e)e(a)h(result)f(due)h(to)g(Douady)g(and)f(Gh)o(ys)h (on)g(the)g(structure)f(of)i Fk(Y)j Fs(\(Section)c(4.1\))h(and)e(to)57 715 y(in)o(tro)q(duce)g(v)m(arious)h(sets)g(of)h(irrational)e(n)o(um)o (b)q(ers)f(\(Sections)j(4.3)f(and)g(4.4\))h(whic)o(h)e(ha)o(v)o(e)h (the)57 785 y(same)k(prop)q(erties)f(of)i Fk(Y)t Fs(.)32 b(Our)18 b(main)h(to)q(ol)h(will)f(b)q(e)h(the)g(use)f(of)h(con)o(tin)o (ued)e(fractions)h(\(see)57 855 y(App)q(endix)d(2)h(for)f(a)g(short)g (in)o(tro)q(duction\).)57 1030 y Fo(4.1)j(Douady{Gh)n(ys')g(Theorem)57 1154 y Fs(W)l(e)27 b(recall)f(that)h(SL)8 b(\(2)p Fl(;)g Fm(Z)-10 b Fs(\))24 b(is)i(the)h(group)f(of)h(matrices)f Fl(g)32 b Fs(=)1338 1084 y Fe(\022)1383 1124 y Fl(a)52 b(b)1385 1184 y(c)g(d)1493 1084 y Fe(\023)1556 1154 y Fs(with)27 b(in)o(teger)57 1243 y(co)q(e\016cien)o(ts)16 b Fl(a;)8 b(b;)g(c;)g(d)17 b Fs(suc)o(h)f(that)h Fl(ad)11 b Fk(\000)g Fl(bc)j Fs(=)f(1.)22 b(It)17 b(acts)g(on)f Fm(R)8 b Fk([)k(f1g)k Fs(\(th)o(us)g(on)g Fk(Y)21 b Fs(to)q(o\))d(as)57 1333 y(usual)e(:)25 b Fl(g)13 b Fk(\001)f Fl(\013)k Fs(=)398 1314 y Fh(a\013)p Fi(+)p Fh(b)p 398 1322 96 2 v 398 1350 a(c\013)p Fi(+)p Fh(d)500 1333 y Fs(.)25 b(SL)8 b(\(2)p Fl(;)g Fm(Z)-10 b Fs(\))15 b(is)i(generated)h(b)o(y)f Fl(T)23 b Fs(=)1199 1263 y Fe(\022)1244 1303 y Fs(1)50 b(1)1244 1363 y(0)g(1)1352 1263 y Fe(\023)1389 1333 y Fs(,)18 b Fl(T)h Fk(\001)11 b Fl(\013)16 b Fs(=)g Fl(\013)c Fs(+)f(1,)18 b(and)57 1458 y Fl(U)h Fs(=)163 1387 y Fe(\022)208 1428 y Fs(0)49 b Fk(\000)p Fs(1)208 1488 y(1)69 b(0)354 1387 y Fe(\023)391 1458 y Fs(,)16 b Fl(U)h Fk(\001)11 b Fl(\013)j Fs(=)f Fk(\000)p Fs(1)p Fl(=\013)p Fs(.)156 1545 y(F)l(urther)i(information)g(on)h(the)h(structure)f(of)g Fk(Y)21 b Fs(is)c(pro)o(vided)d(b)o(y)j(the)f(follo)o(wing)57 1679 y Fr(Theorem)h(4.1)h(\(Douady{Gh)n(ys\))28 b Fk(Y)21 b Fd(is)c(SL)8 b Fs(\(2)p Fl(;)g Fm(Z)-11 b Fs(\))p Fd({in)o(v)m(ari)o (an)o(t.)57 1814 y Fp(Pr)m(o)m(of.)20 b Fs(\(sk)o(etc)o(h\).)j Fk(Y)e Fs(is)16 b(clearly)h(in)o(v)m(arian)o(t)e(under)g Fl(T)7 b Fs(,)17 b(th)o(us)f(w)o(e)g(only)h(need)f(to)i(sho)o(w)d(that) i(if)57 1883 y Fl(\013)d Fk(2)g(Y)21 b Fs(then)16 b(also)g Fl(U)g Fk(\001)11 b Fl(\013)j Fs(=)g Fk(\000)p Fs(1)p Fl(=\013)f Fk(2)h(Y)t Fs(.)156 1953 y(Let)k Fl(f)h Fk(2)14 b Fl(S)367 1962 y Fh(e)386 1952 y Fc(2)p Fb(\031)q(i\013)482 1953 y Fs(and)i(consider)f(a)h(domain)f Fl(V)1032 1935 y Fj(0)1062 1953 y Fs(b)q(ounded)h(b)o(y)87 2023 y(1\))25 b(a)17 b(segmen)o(t)e Fl(l)j Fs(joining)e(0)g(to)h Fl(z)713 2030 y Fi(0)749 2023 y Fk(2)d Fm(D)829 2005 y Fj(\003)855 2023 y Fs(,)j Fl(l)d Fk(\032)g Fm(D)21 b Fs(;)87 2092 y(2\))k(its)17 b(image)f Fl(f)5 b Fs(\()p Fl(l)q Fs(\))11 b(;)87 2162 y(3\))25 b(a)17 b(curv)o(e)f Fl(l)347 2144 y Fj(0)377 2162 y Fs(joining)g Fl(z)565 2169 y Fi(0)604 2162 y Fs(to)h Fl(f)5 b Fs(\()p Fl(z)736 2169 y Fi(0)760 2162 y Fs(\).)57 2232 y(W)l(e)17 b(c)o(ho)q(ose)g Fl(l)318 2214 y Fj(0)349 2232 y Fs(and)f Fl(z)469 2239 y Fi(0)509 2232 y Fs(\(su\016cien)o(tly)g(close)h(to)h(0\))f(so)g(that)h Fl(l)q Fs(,)f Fl(l)1255 2214 y Fj(0)1286 2232 y Fs(and)f Fl(f)5 b Fs(\()p Fl(l)q Fs(\))19 b(do)e(not)g(in)o(tersect)57 2302 y(except)24 b(at)h(their)e(extremities.)44 b(Note)24 b(that)g Fl(l)h Fs(and)e Fl(f)5 b Fs(\()p Fl(l)q Fs(\))26 b(form)d(an)g(angle)g(of)h(2)p Fl(\031)r(\013)g Fs(at)h(0.)57 2371 y(Then)i(glueing)f Fl(l)j Fs(to)e Fl(f)5 b Fs(\()p Fl(l)q Fs(\))29 b(one)e(obtains)g(a)g(top)q(ological)h(manifold)p 1429 2331 41 2 v 26 w Fl(V)38 b Fs(with)28 b(b)q(oundary)57 2441 y(whic)o(h)17 b(is)h(homeomorphic)d(to)p 652 2401 36 2 v 19 w Fm(D)c Fs(.)27 b(With)18 b(the)h(induced)e(complex)h (structure)f(its)h(in)o(terior)f(is)57 2511 y(biholomorphic)10 b(to)j Fm(D)8 b Fs(.)23 b(Let)14 b(us)e(no)o(w)g(consider)f(the)i Fp(\014rst)h(r)m(eturn)g(map)f Fl(g)1401 2518 y Fh(V)1433 2508 y Ff(0)1462 2511 y Fs(to)g(the)g(domain)e Fl(V)1814 2493 y Fj(0)57 2581 y Fs(\(this)g(is)h(w)o(ell)f(de\014ned)g(if)h Fl(z)i Fs(is)d(c)o(ho)q(osen)g(with)h Fk(j)p Fl(z)r Fk(j)g Fs(small)e(enough\))h(:)20 b(if)12 b Fl(z)k Fk(2)e Fl(V)1467 2563 y Fj(0)1493 2581 y Fs(\(and)d Fk(j)p Fl(z)r Fk(j)h Fs(is)g(small)57 2650 y(enough\))17 b(w)o(e)h(de\014ne)g Fl(g)494 2657 y Fh(V)525 2648 y Ff(0)541 2650 y Fs(\()p Fl(z)r Fs(\))g(=)e Fl(f)706 2632 y Fh(n)734 2650 y Fs(\()p Fl(z)r Fs(\))j(where)f Fl(n)g Fs(\(dep)q(ends)f(on)h Fl(z)r Fs(\))h(is)e(de\014ned)h(asking)f(that)918 2770 y(21)p eop %%Page: 22 23 22 22 bop 57 192 a Fl(f)5 b Fs(\()p Fl(z)r Fs(\))p Fl(;)j(:)g(:)g(:)i (;)e(f)290 173 y Fh(n)p Fj(\000)p Fi(1)369 192 y Fs(\()p Fl(z)r Fs(\))18 b Fk(62)e Fl(V)539 173 y Fj(0)571 192 y Fs(and)i Fl(f)699 173 y Fh(n)727 192 y Fs(\()p Fl(z)r Fs(\))f Fk(2)g Fl(V)897 173 y Fj(0)911 192 y Fs(,)h(i.e.)26 b Fl(n)17 b Fs(=)f(inf)s Fk(f)p Fl(k)i Fk(2)f Fm(N)8 b Fl(;)18 b(k)f Fk(\025)g Fs(1)8 b Fl(;)16 b(f)1580 173 y Fh(k)1605 192 y Fs(\()p Fl(z)r Fs(\))i Fk(2)e Fl(V)1776 173 y Fj(0)1790 192 y Fk(g)p Fs(.)57 261 y(Then)h(it)i(is)e(easy)i(to)f (c)o(hec)o(k)g(that)g Fl(n)f Fs(=)812 221 y Fe(\002)842 242 y Fi(1)p 839 250 26 2 v 839 278 a Fh(\013)871 221 y Fe(\003)910 261 y Fs(or)g Fl(n)g Fs(=)1074 221 y Fe(\002)1104 242 y Fi(1)p 1101 250 V 1101 278 a Fh(\013)1132 221 y Fe(\003)1165 261 y Fs(+)12 b(1.)27 b(The)18 b(\014rst)f(return)g(map)h Fl(g)1781 268 y Fh(V)1812 259 y Ff(0)57 331 y Fs(induces)13 b(a)h(map)f Fl(g)p 403 318 35 2 v 17 x Fh(V)451 331 y Fs(on)h(a)g(neigh)o(b)q(orho)q(o)q(d)f(of)h(0)g Fk(2)p 1008 291 41 2 v 14 w Fl(V)25 b Fs(and)13 b(\014nally)h(a)g(germ)f Fl(g)j Fs(of)e(holomorphic)57 401 y(di\013eomorphism)f(at)18 b(0)c Fk(2)g Fm(D)28 b Fs(\()p Fl(g)r(p)14 b Fs(=)g Fl(pg)p 794 388 35 2 v 16 x Fh(V)829 401 y Fs(,)i(where)h Fl(p)f Fs(is)h(the)g(pro)s(jection)f(from)p 1537 361 41 2 v 16 w Fl(V)28 b Fs(to)17 b(the)g(disk)57 470 y Fm(D)8 b Fs(\).)37 b(It)20 b(is)g(easy)h(to)g(c)o(hec)o(k)f(that)h Fl(g)r Fs(\()p Fl(z)r Fs(\))f(=)g Fl(e)891 452 y Fj(\000)p Fi(2)p Fh(\031)q(i=\013)1030 470 y Fl(z)c Fs(+)d(O\()p Fl(z)1204 452 y Fi(2)1227 470 y Fs(\))21 b(\(note)g(that)g(in)f(the)h (passage)57 540 y(from)15 b Fl(V)215 522 y Fj(0)245 540 y Fs(to)i Fm(D)28 b Fs(through)p 547 500 V 15 w Fl(V)g Fs(the)16 b(angle)h(2)p Fl(\031)r(\013)f Fs(at)h(the)g(origin)e(is)h (mapp)q(ed)g(in)g(2)p Fl(\031)r Fs(\).)156 613 y(T)l(o)h(eac)o(h)f (orbit)g(of)g Fl(f)23 b Fs(near)16 b(0)g(corresp)q(onds)e(an)j(orbit)f (of)g Fl(g)i Fs(near)e(0.)22 b(In)16 b(particular)107 685 y Fk(\017)24 b Fl(f)f Fs(is)16 b(linearizable)f(if)h(and)g(only)h (if)f Fl(g)i Fs(is)f(linearizable)8 b(;)107 757 y Fk(\017)24 b Fs(if)17 b Fl(f)22 b Fs(has)16 b(a)h(p)q(erio)q(dic)f(orbit)g(near)g (0)g(then)g(also)g Fl(g)j Fs(has)c(a)i(p)q(erio)q(dic)f(orbit)9 b(;)107 830 y Fk(\017)24 b Fs(if)16 b Fl(f)22 b Fs(has)14 b(a)i(p)q(oin)o(t)f(of)h(instabilit)o(y)f(\(i.e.)22 b(a)15 b(p)q(oin)o(t)g(whic)o(h)g(do)q(es)g(not)h(b)q(elong)f(to)h Fl(K)1670 837 y Fh(f)1696 830 y Fs(\))g(then)156 899 y(also)g Fl(g)h Fs(has)f(a)g(p)q(oin)o(t)f(of)h(instabilit)o(y)f (\(whic)o(h,)h(after)g(ha)o(ving)f(normalized)f Fl(g)k Fs(so)d(as)h(to)g(b)q(e)156 969 y(univ)m(alen)o(t)g(on)g Fm(D)8 b Fs(,)20 b(will)c(lea)o(v)o(e)g(the)h(unit)f(disk)g(ev)o(en)g (more)g(rapidly\).)156 1042 y(In)h(particular)e(these)h(statemen)o(ts)g (sho)o(w)g(that)g Fl(\013)e Fk(2)g(Y)21 b Fs(if)c(and)f(only)g(if)h Fk(\000)p Fs(1)p Fl(=\013)c Fk(2)h(Y)t Fs(.)55 b Fa(\003)57 1237 y Fo(4.2)19 b(SL\(2,Z\))h(and)g(Con)n(tin)n(ued)i(F)-5 b(ractions)57 1345 y Fs(T)l(o)29 b(b)q(etter)i(understand)c(the)j (action)g(of)g(SL)8 b(\(2)p Fl(;)g Fm(Z)-10 b Fs(\))27 b(on)i Fm(R)17 b Fk(n)j Fm(Q)31 b Fs(w)o(e)e(can)h(in)o(tro)q(duce)e(a) 57 1414 y(fundamen)o(tal)17 b(domain)h([0)p Fl(;)8 b Fs(1\))19 b(for)g(one)f(of)i(the)f(t)o(w)o(o)f(generators)g(\(the)i (translation)e Fl(T)7 b Fs(\))19 b(and)57 1484 y(restrict)e(our)g (atten)o(tion)h(to)h(the)f(in)o(v)o(ersion)e Fl(\013)g Fk(7!)g Fs(1)p Fl(=\013)i Fs(restricted)f(to)i([0)p Fl(;)8 b Fs(1\).)26 b(This)18 b(giv)o(es)f(us)57 1554 y(a)h(\\microscop)q(e")e (since)h Fl(\013)f Fk(7!)g Fs(1)p Fl(=\013)i Fs(is)f(expanding)g(on)h ([0)p Fl(;)8 b Fs(1\),)18 b(i.e.)26 b(its)17 b(deriv)m(ativ)o(e)h(is)g (alw)o(a)o(ys)57 1623 y(greater)d(than)g(1.)21 b(Our)14 b(microscop)q(e)g(magni\014es)g(more)g(and)h(more)f(as)h Fl(\013)f Fk(!)g Fs(0+)h(and)g(leads)f(to)57 1693 y(the)i(in)o(tro)q (duction)g(of)g(con)o(tin)o(ued)f(fractions)h(discussed)f(in)h(App)q (endix)g(A2.)57 1801 y Fr(Exercise)24 b(4.2)c Fs(Sho)o(w)g(that)h(giv)o (en)f(an)o(y)h(pair)e Fl(x;)8 b(y)24 b Fk(2)d Fm(R)11 b Fk(n)i Fm(Q)22 b Fs(there)e(exists)h Fl(g)h Fk(2)f Fs(SL)8 b(\(2)p Fl(;)g Fm(Z)-10 b Fs(\))57 1871 y(suc)o(h)28 b(that)h Fl(x)35 b Fs(=)f Fl(g)21 b Fk(\001)f Fl(y)31 b Fp(if)f(and)g(only)g(if)f Fl(x)36 b Fs(=)e([)p Fl(a)1096 1878 y Fi(0)1118 1871 y Fl(;)8 b(a)1166 1878 y Fi(1)1189 1871 y Fl(;)g(:)g(:)g(:)h(;)f(a)1326 1878 y Fh(m)1364 1871 y Fl(;)g(c)1408 1878 y Fi(0)1430 1871 y Fl(;)g(c)1474 1878 y Fi(1)1496 1871 y Fl(;)g(:)g(:)g(:)p Fs(])29 b(and)g Fl(y)37 b Fs(=)57 1940 y([)p Fl(b)92 1947 y Fi(0)114 1940 y Fl(;)8 b(b)157 1947 y Fi(1)180 1940 y Fl(;)g(:)g(:)g(:)h(;)f(b) 312 1947 y Fh(n)339 1940 y Fl(;)g(c)383 1947 y Fi(0)405 1940 y Fl(;)g(c)449 1947 y Fi(1)471 1940 y Fl(;)g(:)g(:)g(:)q Fs(].)22 b([Hin)o(t)17 b(:)22 b(it)17 b(is)g(easy)g(to)g(c)o(hec)o(k)f (that)h(the)g(condition)f(is)h(su\016cien)o(t)57 2010 y(for)f(ha)o(ving)f Fl(x)g Fs(=)e Fl(g)g Fk(\001)e Fl(y)h Fs(;)k(necessit)o(y)h(is)f(more)f(tric)o(ky)l(,)i(see)f([HW])h(pp.)k (141{143.])57 2118 y Fr(Exercise)g(4.3)c Fs(Sho)o(w)g(that)h(if)g Fl(x)g Fs(is)f(a)h(quadratic)f(irrational,)g(i.e.)25 b Fl(x)16 b Fk(2)g Fm(R)9 b Fk(n)j Fm(Q)19 b Fs(is)e(a)h(zero)f(of)57 2188 y(a)j(monic)g(quadratic)g(p)q(olynomial)f(equation)i(with)f(co)q (e\016cien)o(ts)g(in)g Fm(Q)p Fs(,)i(then)f(there)f(exists)57 2258 y Fl(N)f Fk(2)14 b Fm(N)k Fs(suc)o(h)d(that)i(the)g(partial)e (fractions)h Fl(a)912 2265 y Fh(n)956 2258 y Fs(of)h Fl(x)g Fs(are)f(b)q(ounded)g Fl(a)1371 2265 y Fh(n)1412 2258 y Fk(\024)d Fl(N)22 b Fs(for)17 b(all)f Fl(n)e Fk(\025)f Fs(0.)57 2365 y(The)21 b(t)o(w)o(o)g(main)f(results)h(whic)o(h)f(mak)o (e)h(con)o(tin)o(ued)f(fractions)h(so)g(useful)f(in)h(the)h(study)f(of) 57 2435 y(one{dimensional)14 b(small)h(divisors)g(problems)f(are)j(the) f(follo)o(wing)57 2581 y Fr(Theorem)j(4.4)i(\(Best)g(appro)n (ximation\))29 b Fd(Let)19 b Fl(x)e Fk(2)h Fm(R)9 b Fk(n)j Fm(Q)19 b Fd(and)f(let)h Fl(p)1493 2588 y Fh(n)1520 2581 y Fl(=q)1567 2588 y Fh(n)1613 2581 y Fd(denote)f(its)57 2650 y Fl(n)p Fd({th)i(con)o(v)o(ergen)o(t.)31 b(If)20 b Fs(0)g Fl(<)f(q)j(<)d(q)738 2657 y Fh(n)p Fi(+1)836 2650 y Fd(then)h Fk(j)p Fl(q)r(x)13 b Fk(\000)h Fl(p)p Fk(j)19 b(\025)h(j)p Fl(q)1238 2657 y Fh(n)1265 2650 y Fl(x)14 b Fk(\000)f Fl(p)1384 2657 y Fh(n)1411 2650 y Fk(j)20 b Fd(for)g(all)f Fl(p)h Fk(2)g Fm(Z)6 b Fd(and)918 2770 y Fs(22)p eop %%Page: 23 24 23 23 bop 57 192 a Fd(equalit)o(y)16 b(can)h(o)q(ccur)f(only)g(if)h Fl(q)f Fs(=)d Fl(q)739 199 y Fh(n)767 192 y Fd(,)j Fl(p)e Fs(=)g Fl(p)914 199 y Fh(n)941 192 y Fd(.)57 364 y Fr(Theorem)j(4.5)28 b Fd(If)452 306 y Fe(\014)452 336 y(\014)452 366 y(\014)468 364 y Fl(x)12 b Fk(\000)564 341 y Fh(p)p 564 352 21 2 v 565 381 a(q)590 306 y Fe(\014)590 336 y(\014)590 366 y(\014)620 364 y Fl(<)699 344 y Fi(1)p 679 352 59 2 v 679 381 a(2)p Fh(q)718 371 y Fc(2)760 364 y Fd(then)880 341 y Fh(p)p 880 352 21 2 v 881 381 a(q)923 364 y Fd(is)k(a)h(con)o(v)o (ergen)o(t)e(of)i Fl(x)p Fd(.)57 507 y Fs(F)l(or)e(the)i(pro)q(ofs)f (see)g([HW],)h(resp)q(ectiv)o(ely)g(Theorems)e(182,)h(p.)21 b(151)c(and)f(184,)g(p.)21 b(153.)57 683 y Fo(4.3)e(Classical)i (Diophan)n(tine)h(Conditions)57 790 y Fs(Let)17 b Fl(\015)f(>)e Fs(0)i(and)g Fl(\034)k Fk(\025)13 b Fs(0)k(b)q(e)f(t)o(w)o(o)h(real)f (n)o(um)o(b)q(ers.)57 930 y Fr(De\014nition)24 b(4.6)47 b Fl(x)20 b Fk(2)f Fm(R)10 b Fk(n)j Fm(Q)20 b Fd(is)f Fs(diophan)o(tine)f Fd(of)i(exp)q(onen)o(t)f Fl(\034)25 b Fd(and)19 b(constan)o(t)g Fl(\015)j Fd(if)e(and)57 1004 y(only)c(if)h(for)f(all)g Fl(p;)8 b(q)16 b Fk(2)e Fm(Z)-11 b Fd(,)14 b Fl(q)i(>)d Fs(0)p Fd(,)k(one)f(has)879 947 y Fe(\014)879 977 y(\014)879 1007 y(\014)895 1004 y Fl(x)c Fk(\000)991 982 y Fh(p)p 991 993 V 992 1021 a(q)1017 947 y Fe(\014)1017 977 y(\014)1017 1007 y(\014)1048 1004 y Fk(\025)h Fl(\015)s(q)1153 986 y Fj(\000)p Fi(2)p Fj(\000)p Fh(\034)1260 1004 y Fd(.)57 1143 y Fr(Remark)24 b(4.7)d Fs(Note)i(that)f(Theorem)f(4.5)g(implies)g(that)h(giv)o(en)g (an)o(y)f(irrational)g(n)o(um)o(b)q(er)57 1213 y(there)f(are)f (in\014nitely)h(man)o(y)f(solutions)g(to)902 1155 y Fe(\014)902 1185 y(\014)902 1215 y(\014)919 1213 y Fl(x)11 b Fk(\000)1014 1190 y Fh(p)p 1014 1201 V 1015 1230 a(q)1040 1155 y Fe(\014)1040 1185 y(\014)1040 1215 y(\014)1077 1213 y Fl(<)1151 1193 y Fi(1)p 1141 1201 39 2 v 1141 1230 a Fh(q)1160 1220 y Fc(2)1206 1213 y Fs(with)20 b Fl(p)g Fs(and)g Fl(q)i Fs(coprime.)31 b(This)57 1282 y(explains)15 b(wh)o(y)h(the)h(previous)f (de\014nition)f(w)o(ould)g(nev)o(er)h(b)q(e)h(satis\014ed)f(if)g Fl(\034)j(<)14 b Fs(0.)57 1389 y(W)l(e)26 b(denote)g(CD)8 b(\()p Fl(\015)s(;)g(\034)e Fs(\))26 b(the)g(set)g(of)h(all)e (irrationals)f Fl(x)j Fs(suc)o(h)e(that)1416 1331 y Fe(\014)1416 1361 y(\014)1416 1391 y(\014)1432 1389 y Fl(x)12 b Fk(\000)1528 1366 y Fh(p)p 1528 1377 21 2 v 1529 1406 a(q)1554 1331 y Fe(\014)1554 1361 y(\014)1554 1391 y(\014)1600 1389 y Fk(\025)30 b Fl(\015)s(q)1722 1371 y Fj(\000)p Fi(2)p Fj(\000)p Fh(\034)57 1463 y Fs(for)e(all)g Fl(p;)8 b(q)37 b Fk(2)d Fm(Z)-11 b Fs(,)29 b Fl(q)36 b(>)e Fs(0.)58 b(CD)8 b(\()p Fl(\034)e Fs(\))29 b(will)f(denote)h(the)g(union)e Fk([)1443 1470 y Fh(\015)r(>)p Fi(0)1520 1463 y Fs(CD)9 b(\()p Fl(\015)s(;)f(\034)e Fs(\))28 b(and)57 1533 y(CD)22 b(=)13 b Fk([)238 1540 y Fh(\034)t Fj(\025)p Fi(0)315 1533 y Fs(CD)8 b(\()p Fl(\034)e Fs(\).)57 1640 y Fr(Exercise)20 b(4.8)c Fs(Sho)o(w)f(that)137 1751 y(CD)8 b(\()p Fl(\034)e Fs(\))14 b(=)f Fk(f)p Fl(x)i Fk(2)f Fm(R)8 b Fk(n)j Fm(Q)j Fk(j)28 b Fl(q)665 1758 y Fh(n)p Fi(+1)756 1751 y Fs(=)14 b(O\()p Fl(q)891 1730 y Fi(1+)p Fh(\034)889 1763 y(n)966 1751 y Fs(\))p Fk(g)h Fs(=)e Fk(f)p Fl(x)h Fk(2)g Fm(R)8 b Fk(n)j Fm(Q)k Fk(j)28 b Fl(a)1395 1758 y Fh(n)p Fi(+1)1486 1751 y Fs(=)14 b(O\()p Fl(q)1621 1730 y Fh(\034)1619 1763 y(n)1646 1751 y Fs(\))p Fk(g)299 1836 y Fs(=)f Fk(f)p Fl(x)i Fk(2)f Fm(R)8 b Fk(n)j Fm(Q)j Fk(j)28 b Fl(x)671 1815 y Fj(\000)p Fi(1)671 1848 y Fh(n)739 1836 y Fs(=)13 b(O\()p Fl(\014)880 1814 y Fj(\000)p Fh(\034)877 1849 y(n)p Fj(\000)p Fi(1)956 1836 y Fs(\))p Fk(g)h Fs(=)f Fk(f)p Fl(x)h Fk(2)h Fm(R)8 b Fk(n)j Fm(Q)j Fk(j)28 b Fl(\014)1389 1815 y Fj(\000)p Fi(1)1386 1848 y Fh(n)1456 1836 y Fs(=)13 b(O\()p Fl(\014)1597 1814 y Fj(\000)p Fi(1)p Fj(\000)p Fh(\034)1594 1849 y(n)p Fj(\000)p Fi(1)1704 1836 y Fs(\))p Fk(g)57 1993 y Fr(Exercise)33 b(4.9)26 b Fs(Sho)o(w)h(that)h(if)g Fl(x)g Fs(is)f(an)h(algebraic)e(n)o(um)o(b)q (er)g(of)i(degree)f Fl(n)32 b Fk(\025)g Fs(2,)f(i.e.)57 2062 y Fl(x)26 b Fk(2)h Fm(R)13 b Fk(n)i Fm(Q)25 b Fs(is)f(a)g(zero)f (of)h(a)g(monic)f(p)q(olynomial)g(with)h(co)q(e\016cien)o(ts)f(in)h Fm(Q)h Fs(and)e(degree)57 2132 y Fl(n)p Fs(,)18 b(then)g Fl(x)f Fk(2)f Fs(CD)9 b(\()p Fl(n)j Fk(\000)g Fs(2\))18 b(\(Liouville's)g(theorem\).)26 b(Th)o(ue)17 b(impro)o(v)o(ed)f(this)i (result)f(in)h(1909)57 2202 y(sho)o(wing)d(that)j Fl(x)e Fk(2)f Fs(CD)9 b(\()p Fl(\034)17 b Fk(\000)12 b Fs(1)f(+)g Fl(n=)p Fs(2\))18 b(for)f(all)g Fl(\034)k(>)15 b Fs(0)i(\(see)h([ST],)f (Chapter)f(V,)i(for)f(a)g(v)o(ery)57 2271 y(nice)f(discussion)e(of)i (the)g(pro)q(of)g(in)g(the)g(cubic)g(case\).)22 b(Actually)17 b(one)f(can)f(pro)o(v)o(e)g(that)i(if)f Fl(x)h Fs(is)57 2341 y(algebraic)12 b(then)i Fl(x)h Fk(2)f Fs(CD)8 b(\()p Fl(\034)e Fs(\))14 b(for)g(all)f Fl(\034)19 b(>)14 b Fs(0)g(regardless)e(of)i(the)g(degree,)g(but)f(this)h(is)f(di\016cult) 57 2411 y(\(Roth's)j(theorem\).)57 2517 y Fr(Exercise)k(4.10)15 b Fs(Using)h(the)h(fact)g(that)g(the)g(con)o(tin)o(ued)e(fraction)h(of) h Fl(e)d Fs(=)1477 2480 y Fe(P)1530 2492 y Fj(1)1530 2532 y Fh(n)p Fi(=0)1630 2498 y(1)p 1622 2506 36 2 v 1622 2534 a Fh(n)p Fi(!)1680 2517 y Fs(is)534 2650 y([2)p Fl(;)8 b Fs(1)p Fl(;)g Fs(2)p Fl(;)g Fs(1)p Fl(;)g Fs(1)p Fl(;)g Fs(4)p Fl(;)g Fs(1)p Fl(;)g Fs(1)p Fl(;)g Fs(6)p Fl(;)g Fs(1)p Fl(;)g Fs(1)p Fl(;)g Fs(8)p Fl(;)g Fs(1)p Fl(;)g Fs(1)p Fl(;)g Fs(10)p Fl(;)g(:)g(:)g(:)q Fs(])918 2770 y(23)p eop %%Page: 24 25 24 24 bop 57 192 a Fs(sho)o(w)21 b(that)i Fl(e)h Fk(2)h(\\)438 199 y Fh(\034)t(>)p Fi(0)514 192 y Fs(CD)8 b(\()p Fl(\034)e Fs(\).)41 b(A)23 b(pro)q(of)f(of)h(the)g(con)o(tin)o(ued)e(fraction)h (expansion)f(of)i Fl(e)p Fs(,)57 261 y(whic)o(h)14 b(is)h(due)g(to)g (L.)h(Euler,)e(can)h(b)q(e)h(found)e(in)h([L],)g(Chapter)g(V.)g(P)o (erhaps)f(y)o(ou)g(ma)o(y)h(lik)o(e)g(to)57 331 y(try)j(to)g(obtain)f (it)h(y)o(ourself)f(starting)g(from)g(the)h(kno)o(wledge)f(of)h(the)g (con)o(tin)o(ued)e(fraction)h(of)63 381 y Fh(e)p Fi(+1)p 63 389 70 2 v 63 418 a Fh(e)p Fj(\000)p Fi(1)152 401 y Fs(=)d([2)p Fl(;)8 b Fs(6)p Fl(;)g Fs(10)p Fl(;)g Fs(14)p Fl(;)g(:)g(:)g(:)p Fs(].)57 506 y Fr(Exercise)30 b(4.11)25 b Fs(Use)h(the)h(result)e(of)h(Exercise)f(4.9)h(to)g(exhibit)g (explicit)g(examples)f(of)57 576 y(trascenden)o(tal)15 b(n)o(um)o(b)q(ers,)f(e.g.)22 b Fl(x)14 b Fs(=)770 538 y Fe(P)822 551 y Fj(1)822 591 y Fh(n)p Fi(=0)908 576 y Fs(10)958 558 y Fj(\000)p Fh(n)p Fi(!)1027 576 y Fs(.)57 681 y(The)i(complemen)o(t)f(in)h Fm(R)8 b Fk(n)j Fm(Q)18 b Fs(of)e(CD)h(is)f(called)g(the)h(set)g(of)g(Liouville)f(n)o(um)o(b)q (ers.)57 786 y Fr(Exercise)k(4.12)15 b Fs(Sho)o(w)h(that)h(CD)8 b(\()p Fl(\034)e Fs(\))17 b(and)f(CD)g(are)g(b)q(oth)h(SL)8 b(\(2)p Fl(;)g Fm(Z)-10 b Fs(\){in)n(v)m(arian)n(t.)57 921 y Fr(Prop)r(osition)38 b(4.13)27 b Fd(F)l(or)33 b(all)g Fl(\015)45 b(>)d Fs(0)33 b Fd(and)g Fl(\034)47 b(>)42 b Fs(0)34 b Fd(the)g(Leb)q(esgue)f(measure)f(of)57 990 y(CD)8 b Fs(\()p Fl(\015)s(;)g(\034)e Fs(\)\()p Fd(mo)q(d)i Fs(1\))22 b Fd(is)f(at)g(least)h Fs(1)14 b Fk(\000)g Fs(2)p Fl(\015)s(\020)t Fs(\(1)f(+)h Fl(\034)6 b Fs(\))p Fd(,)23 b(where)d Fl(\020)25 b Fd(denotes)c(the)h(Riemann)e(zeta)57 1060 y(function.)57 1194 y Fp(Pr)m(o)m(of.)g Fs(The)c(complemen)o(t)f (of)i(CD)8 b(\()p Fl(\015)s(;)g(\034)e Fs(\)\(mo)q(d)i(1\))17 b(is)f(con)o(tained)g(in)508 1341 y Fk([)541 1350 y Fh(p=q)q Fj(2)p Fg(Q)6 b Fj(\\)p Fi([0)p Fh(;)p Fi(1])767 1271 y Fe(\022)810 1308 y Fl(p)p 810 1330 26 2 v 811 1376 a(q)852 1341 y Fk(\000)11 b Fl(\015)s(q)955 1321 y Fj(\000)p Fi(2)p Fj(\000)p Fh(\034)1061 1341 y Fl(;)1089 1308 y(p)p 1089 1330 V 1090 1376 a(q)1131 1341 y Fs(+)g Fl(\015)s(q)1234 1321 y Fj(\000)p Fi(2)p Fj(\000)p Fh(\034)1341 1271 y Fe(\023)57 1485 y Fs(whose)16 b(Leb)q(esgue)g(measure)f(is)h(b)q (ounded)g(b)o(y)603 1580 y Fj(1)586 1595 y Fe(X)587 1700 y Fh(q)q Fi(=1)693 1577 y Fh(q)667 1595 y Fe(X)667 1700 y Fh(p)p Fi(=1)747 1642 y Fs(2)p Fl(\015)s(q)825 1621 y Fj(\000)p Fi(2)p Fj(\000)p Fh(\034)945 1642 y Fk(\024)e Fs(2)p Fl(\015)1076 1580 y Fj(1)1060 1595 y Fe(X)1061 1700 y Fh(q)q Fi(=1)1140 1642 y Fl(q)1164 1621 y Fj(\000)p Fi(1)p Fj(\000)p Fh(\034)1285 1642 y Fl(:)1790 1808 y Fa(\003)57 1928 y Fs(F)l(rom)h(the)h(p)q(oin)o(t)g(of)h(view)g(of)g (dimension)d(one)j(has)e(\(see)i([F)l(a],)f(p.)21 b(142)c(for)f(a)g (pro)q(of)t(\))57 2062 y Fr(Theorem)11 b(4.14)h(\(Jarnik\))30 b Fd(Let)12 b Fl(\034)19 b(>)14 b Fs(0)d Fd(and)g(let)h Fl(F)1047 2069 y Fh(\034)1083 2062 y Fd(b)q(e)g(the)g(set)g(of)f(real)g (n)o(um)o(b)q(ers)e Fl(x)15 b Fk(2)f Fs([0)p Fl(;)8 b Fs(1])57 2132 y Fd(suc)o(h)20 b(that)h Fk(f)p Fl(q)r(x)p Fk(g)h(\024)g Fl(q)495 2114 y Fj(\000)p Fi(1)p Fj(\000)p Fh(\034)623 2132 y Fd(for)f(in\014nitely)g(man)o(y)f(p)q(ositiv)o(e)h (in)o(tegers)f Fl(q)r Fd(.)36 b(The)22 b(Hausdor\013)57 2202 y(dimension)14 b(of)j Fl(F)381 2209 y Fh(\034)422 2202 y Fd(is)g Fs(2)p Fl(=)p Fs(\(2)11 b(+)g Fl(\034)6 b Fs(\))p Fd(.)57 2336 y Fr(Exercise)22 b(4.15)c Fs(The)g(set)h(CD)8 b(\(0\))20 b(is)e(also)g(called)g(the)h(set)g(of)f(n)o(um)o(b)q(ers)e (of)j Fp(c)m(onstant)i(typ)m(e)57 2406 y Fs(since)f Fl(x)h Fk(2)g Fs(CD)8 b(\(0\))21 b(if)g(and)f(only)g(if)h(the)g(sequence)g(of) f(its)h(partial)f(fractions)g(is)g(b)q(ounded.)57 2475 y(Sho)o(w)15 b(that)i(CD)8 b(\(0\))18 b(has)d(Hausdor\013)h(dimension)e (1)j(and)f(zero)g(Leb)q(esgue)g(measure.)57 2581 y Fr(Exercise)g(4.16)c Fs(Sho)o(w)g(that)i(the)g(set)f(of)h(Liouville)e(n)o(um)o(b)q(ers)f (has)i(zero)g(Leb)q(esgue)g(measure,)57 2650 y(zero)j(Hausdor\013)f (dimension)g(but)h(it)h(is)f(a)h(dense)e Fl(G)1032 2657 y Fh(\016)1054 2650 y Fs({set)918 2770 y(24)p eop %%Page: 25 26 25 25 bop 57 297 a Fo(4.4)19 b(Brjuno)h(Num)n(b)r(ers)f(and)h(the)g (Brjuno)g(F)-5 b(unction)57 402 y Fs(Let)13 b Fl(x)h Fk(2)g Fm(R)s Fk(n)s Fm(Q)p Fs(,)e(let)433 347 y Fe(\020)468 380 y Fh(p)489 385 y Fb(n)p 468 390 45 2 v 469 419 a Fh(q)487 424 y Fb(n)519 347 y Fe(\021)549 437 y Fh(n)p Fj(\025)p Fi(0)640 402 y Fs(denote)g(the)h(sequence)g(of)g(its)f(con)o (v)o(ergen)o(ts)f(and)h(let)h(\()p Fl(\014)1672 409 y Fh(n)1700 402 y Fs(\))1719 409 y Fh(n)p Fj(\025\000)p Fi(1)57 486 y Fs(b)q(e)j(de\014ned)g(as)g(in)g(\(A2.14\).)57 619 y Fr(De\014nition)27 b(4.17)g Fl(x)c Fd(is)f(a)g Fs(Brjuno)g(n)o(um)o(b)q(er)e Fd(if)i Fl(B)r Fs(\()p Fl(x)p Fs(\))j(:=)1233 581 y Fe(P)1285 594 y Fj(1)1285 634 y Fh(n)p Fi(=0)1371 619 y Fl(\014)1399 626 y Fh(n)p Fj(\000)p Fi(1)1486 619 y Fs(log)9 b Fl(x)1587 601 y Fj(\000)p Fi(1)1587 631 y Fh(n)1664 619 y Fl(<)23 b Fs(+)p Fk(1)p Fd(.)57 688 y(The)16 b(function)g Fl(B)25 b Fs(:)13 b Fm(R)8 b Fk(n)j Fm(Q)k Fk(!)f Fs(\(0)p Fl(;)8 b Fs(+)p Fk(1)p Fs(])16 b Fd(is)g(called)g(the)h Fs(Brjuno)f(function)p Fd(.)57 822 y Fr(Exercise)k(4.18)15 b Fs(Sho)o(w)h(that)h(all)f (diophan)o(tine)e(n)o(um)o(b)q(ers)g(are)j(Brjuno)e(n)o(um)o(b)q(ers.) 57 927 y Fr(Exercise)21 b(4.19)16 b Fs(Sho)o(w)g(that)i(there)f(exists) h Fl(C)g(>)d Fs(0)j(suc)o(h)e(tah)o(t)h(for)g(all)g(Brjuno)g(n)o(um)o (b)q(ers)e Fl(x)57 997 y Fs(one)h(has)641 1002 y Fe(\014)641 1031 y(\014)641 1061 y(\014)641 1091 y(\014)641 1121 y(\014)658 1089 y Fl(B)r Fs(\()p Fl(x)p Fs(\))c Fk(\000)844 1026 y Fj(1)828 1041 y Fe(X)826 1147 y Fh(n)p Fi(=0)916 1055 y Fs(log)c Fl(q)1010 1062 y Fh(n)p Fi(+1)p 916 1077 173 2 v 977 1123 a Fl(q)999 1130 y Fh(n)1094 1002 y Fe(\014)1094 1031 y(\014)1094 1061 y(\014)1094 1091 y(\014)1094 1121 y(\014)1124 1089 y Fk(\024)14 b Fl(C)j(:)57 1261 y Fr(Exercise)j(4.20)e (\(see)h([MMY]\))f Fs(Sho)o(w)d(that)i(the)g(Brjuno)f(function)g (satis\014es)424 1372 y Fl(B)r Fs(\()p Fl(x)p Fs(\))f(=)f Fl(B)r Fs(\()p Fl(x)e Fs(+)f(1\))j Fl(;)50 b Fk(8)p Fl(x)13 b Fk(2)h Fm(R)8 b Fk(n)j Fm(Q)424 1481 y Fl(B)r Fs(\()p Fl(x)p Fs(\))k(=)f Fk(\000)8 b Fs(log)h Fl(x)i Fs(+)g Fl(xB)884 1411 y Fe(\022)929 1448 y Fs(1)p 927 1470 29 2 v 927 1515 a Fl(x)961 1411 y Fe(\023)1020 1481 y Fl(;)50 b(x)14 b Fk(2)g Fm(R)8 b Fk(n)j Fm(Q)h Fk(\\)f Fs(\(0)p Fl(;)d Fs(1\))1726 1441 y(\(4)p Fl(:)p Fs(1\))57 1622 y(Deduce)18 b(from)g(this)g(that)h(the)g(set)g(of)g(Brjuno)f(n)o(um)o (b)q(ers)e(is)i(SL)9 b(\(2)p Fl(;)f Fm(Z)-11 b Fs(\){in)o(v)m(ari)o(an) o(t.)25 b(Use)19 b(the)57 1707 y(ab)q(o)o(v)o(e)d(giv)o(en)g (functional)g(equation)g(to)h(compute)f Fl(B)r Fs(\()p Fl(x)1115 1714 y Fh(p)1139 1707 y Fs(\),)h(where)f Fl(x)1361 1714 y Fh(p)1398 1707 y Fs(=)1457 1644 y Fk(p)p 1498 1644 91 2 v 1498 1678 a Fh(p)1519 1668 y Fc(2)1538 1678 y Fi(+4)p Fj(\000)p Fh(p)p 1457 1695 184 2 v 1539 1724 a Fi(2)1647 1707 y Fs(,)g Fl(p)e Fk(2)g Fm(N)p Fs(.)57 1812 y Fr(Exercise)22 b(4.21)d(\(see)j([MMY]\))d Fs(Sho)o(w)e(that)i (the)f(linear)g(op)q(erator)f(\()p Fl(T)7 b(f)e Fs(\)\()p Fl(x)p Fs(\))19 b(=)e Fl(xf)1734 1772 y Fe(\000)1764 1792 y Fi(1)p 1763 1801 23 2 v 1763 1829 a Fh(x)1792 1772 y Fe(\001)1815 1812 y Fs(,)57 1892 y Fl(x)25 b Fk(2)g Fs(\(0)p Fl(;)8 b Fs(1\),)26 b(acting)d(on)g(p)q(erio)q(dic)g (functions)f(whic)o(h)g(b)q(elong)h(to)h Fl(L)1386 1874 y Fh(p)1417 1836 y Fe(\020)1447 1892 y Fm(T)-5 b Fl(;)1581 1872 y Fh(dx)p 1508 1880 190 2 v 1508 1909 a Fi(\(1+)p Fh(x)p Fi(\))5 b(log)i(2)1703 1836 y Fe(\021)1756 1892 y Fs(has)57 1981 y(sp)q(ectral)27 b(radius)f(b)q(ounded)h(b)o(y)g Fl(g)34 b Fs(=)847 1933 y Fj(p)p 879 1933 20 2 v 879 1962 a Fi(5)p Fj(\000)p Fi(1)p 847 1970 104 2 v 888 1998 a(2)956 1981 y Fs(.)56 b(Conclude)27 b(that)h(the)g(Brjuno)f(function) 57 2051 y Fl(B)16 b Fk(2)e(\\)191 2058 y Fh(p)p Fj(\025)p Fi(1)265 2051 y Fl(L)299 2033 y Fh(p)322 2051 y Fs(\()p Fm(T)l Fs(\).)19 b(Note)e(that)g Fl(B)f Fk(62)e Fl(L)793 2033 y Fj(1)836 2051 y Fs(\()p Fm(T)-5 b Fs(\).)57 2156 y Fr(Exercise)18 b(4.22)d Fs(W)l(rite)g(the)h(con)o(tin)o(ued)d (fraction)i(expansion)g(of)g(a)g(Brjuno)g(n)o(um)o(b)q(er)e(whic)o(h)57 2226 y(is)h(not)h(a)g(diophan)o(tine)e(n)o(um)o(b)q(er.)20 b(The)14 b(same)h(for)f(the)i(decimal)e(expansion.)20 b(Is)1571 2189 y Fe(P)1623 2201 y Fj(1)1623 2241 y Fh(n)p Fi(=0)1709 2226 y Fs(10)1759 2208 y Fj(\000)p Fh(n)p Fi(!)57 2296 y Fs(a)c(Brjuno)g(n)o(um)o(b)q(er)8 b(?)22 b(What)16 b(ab)q(out)768 2258 y Fe(P)820 2271 y Fj(1)820 2311 y Fh(n)p Fi(=0)906 2296 y Fs(10)956 2278 y Fj(\000)p Fi(10)1027 2262 y Fb(n)p Fc(!)1074 2296 y Fs(?)57 2401 y Fr(Exercise)k(4.23)15 b Fs(Let)i Fl(\033)f(>)e Fs(0.)22 b(Use)16 b(the)h(results)f(of)g(Exercise)g(4.21)g(to)h(study)f(the)h (functions)377 2513 y Fl(B)417 2493 y Fi(\()p Fh(\033)q Fi(\))475 2513 y Fs(\()p Fl(x)p Fs(\))e(=)f Fl(B)649 2493 y Fi(\()p Fh(\033)q Fi(\))707 2513 y Fs(\()p Fl(x)e Fs(+)f(1\))j Fl(;)50 b Fk(8)p Fl(x)13 b Fk(2)h Fm(R)8 b Fk(n)j Fm(Q)377 2623 y Fl(B)417 2602 y Fi(\()p Fh(\033)q Fi(\))475 2623 y Fs(\()p Fl(x)p Fs(\))k(=)f Fl(x)637 2602 y Fj(\000)p Fi(1)p Fh(=\033)747 2623 y Fs(+)c Fl(xB)864 2602 y Fi(\()p Fh(\033)q Fi(\))931 2553 y Fe(\022)976 2589 y Fs(1)p 974 2612 29 2 v 974 2657 a Fl(x)1008 2553 y Fe(\023)1067 2623 y Fl(;)50 b(x)14 b Fk(2)g Fm(R)8 b Fk(n)j Fm(Q)h Fk(\\)f Fs(\(0)p Fl(;)d Fs(1\))918 2770 y(25)p eop %%Page: 26 27 26 26 bop 57 192 a Fs(Sho)o(w)21 b(that)i(if)f Fl(B)399 173 y Fi(\()p Fh(\033)q Fi(\))458 192 y Fs(\()p Fl(x)p Fs(\))j Fl(<)e Fs(+)p Fk(1)f Fs(then)g Fl(x)j Fk(2)e Fs(CD)9 b(\()p Fl(\033)r Fs(\).)40 b(Vicev)o(ersa,)24 b(if)e Fl(x)j Fk(2)e Fs(CD)9 b(\()p Fl(\034)d Fs(\))22 b(then)57 261 y Fl(B)97 243 y Fi(\()p Fh(\033)q Fi(\))155 261 y Fs(\()p Fl(x)p Fs(\))15 b Fl(<)f Fs(+)p Fk(1)i Fs(for)g(all)g Fl(\033)g(>)e(\034)6 b Fs(.)918 2770 y(26)p eop %%Page: 27 28 27 27 bop 57 192 a Fq(5.)31 b(Siegel{Brjuno)24 b(Theorem,)d(Y)-6 b(o)r(ccoz's)22 b(Theorem)g(and)f(Some)57 261 y(Op)r(en)j(Problems)156 372 y Fs(Recall)19 b(that)h Fk(Y)j Fs(denotes)c(the)g(set)h(of)f Fl(\013)f Fk(2)h Fm(R)10 b Fk(n)i Fm(Q)20 b Fs(suc)o(h)e(that)i Fl(G)1398 381 y Fh(e)1417 371 y Fc(2)p Fb(\031)q(i\013)1516 372 y Fs(is)e(a)h(conjugacy)57 442 y(class.)i(Here)c(w)o(e)f(list)g (what)h(w)o(e)f(already)g(kno)o(w)g(ab)q(out)h(it)107 514 y Fk(\017)24 b(Y)d Fs(has)16 b(full)g(measure)f(\(Chapter)i(3\))10 b(;)107 586 y Fk(\017)24 b Fs(the)17 b(complemen)o(t)e(of)i Fk(Y)k Fs(in)16 b Fm(R)8 b Fk(n)j Fm(Q)17 b Fs(is)g(a)f Fl(G)958 593 y Fh(\016)980 586 y Fs({dense)g(\(Exercise)g(1.18\))10 b(;)107 659 y Fk(\017)24 b(Y)18 b Fs(is)12 b(in)o(v)m(arian)o(t)g (under)g(the)h(action)g(of)g(SL)8 b(\(2)p Fl(;)g Fm(Z)-10 b Fs(\))11 b(\(Douady{Gh)o(ys')h(Theorem,)g(Chapter)156 728 y(4\).)156 801 y(The)17 b(purp)q(ose)e(of)i(this)g(Chapter)f(is)g (to)h(pro)o(v)o(e)f(the)h(classical)f(results)g(of)g(Siegel)h([S])f (and)57 871 y(Brjuno)g([Br])g(whic)o(h)f(sho)o(w)h(that)h(:)156 943 y Fp(al)s(l)h(Brjuno)g(numb)m(ers)g(b)m(elong)g(to)g Fk(Y)156 1015 y Fs(Moreo)o(v)o(er)h(w)o(e)i(will)f(state)h(the)g (Theorems)f(of)h(Y)l(o)q(ccoz)g([Y)l(o2])g(and)f(in)g(particular)f(his) 57 1085 y(celebrated)d(result)g(:)156 1158 y Fk(Y)23 b Fp(is)18 b(e)m(qual)g(to)g(the)g(set)f(of)i(Brjuno)f(numb)m(ers.)156 1230 y Fs(W)l(e)f(will)f(also)g(men)o(tion)f(sev)o(eral)h(op)q(en)h (problems.)156 1302 y(F)l(or)h(the)h(sak)o(e)f(of)h(brevit)o(y)l(,)f (starting)g(with)h(this)f(section)g(all)h(the)g(pro)q(ofs)f(will)g(b)q (e)h(only)57 1372 y(sk)o(etc)o(hed)13 b(:)21 b(the)15 b(reader)e(can)h(try)h(autonomously)d(to)j(\014ll)e(in)h(the)h(details) f(but)g(w)o(e)g(will)g(alw)o(a)o(ys)57 1442 y(refer)i(to)h(the)f (original)g(literature)f(where)h(complete)h(pro)q(ofs)f(are)g(giv)o (en.)57 1619 y Fo(5.1)j(Siegel{Brjuno)i(Theorem)57 1727 y Fs(The)15 b(theorem)g(of)h(Siegel)f(and)f(Brjuno)h(sa)o(ys)g(that)h (the)f(set)h(of)g(Brjuno)f(n)o(um)o(b)q(ers)e(is)i(a)g(subset)57 1797 y(of)f Fk(Y)t Fs(.)21 b(Indeed)14 b(in)f(1942)h(C.L.)g(Siegel)f(w) o(as)h(the)g(\014rst)f(to)h(sho)o(w)f(that)i Fk(Y)j Fs(is)13 b(not)h(empt)o(y)g(sho)o(wing)57 1867 y(that)j(CD)22 b Fk(\032)13 b(Y)t Fs(.)156 1939 y(W)l(e)25 b(will)g(sk)o(etc)o(h)f (the)h(pro)q(of)f(of)h(a)g(more)f(precise)g(result)g(whic)o(h)f(follo)o (ws)h(from)g(the)57 2009 y(Theorem)17 b(of)i(Y)l(o)q(ccoz)h(whic)o(h)d (w)o(e)i(will)f(discuss)f(in)h(the)h(next)g(section)g(but)f(whic)o(h)g (can)g(also)57 2079 y(b)q(e)e(pro)o(v)o(ed)f(follo)o(wing)g(the)h (classical)f(ma)s(joran)o(t)g(series)g(metho)q(d)g(\(see)i([CM]\).)f (Let)g(us)g(recall)57 2148 y(that)e Fl(S)193 2155 y Fh(\025)233 2148 y Fs(denotes)g(the)g(top)q(ological)g(space)f(of)h(all)g(germs)f (of)h(holomorphic)e(di\013eomorphisms)57 2218 y Fl(f)31 b Fs(:)16 b Fm(D)28 b Fk(!)17 b Fm(C)30 b Fs(suc)o(h)17 b(that)i Fl(f)5 b Fs(\(0\))18 b(=)e(0,)j Fl(f)24 b Fs(is)18 b(univ)m(alen)o(t)g(on)g Fm(D)29 b Fs(and)18 b Fl(f)1334 2200 y Fj(0)1348 2218 y Fs(\(0\))g(=)e Fl(\025)p Fs(.)28 b(By)18 b(Theorem)57 2288 y(A1.19)e(it)h(is)f(a)h(compact)f(space.)21 b(Giv)o(en)c(a)f(germ)g Fl(f)j Fk(2)c Fl(S)1125 2295 y Fh(\025)1167 2288 y Fs(w)o(e)h(let)h Fl(r)q Fs(\()p Fl(f)5 b Fs(\))19 b(indicate)d(the)h(radius)57 2358 y(of)f(con)o(v)o (ergence)g(of)h(the)f(linearization)g Fl(h)846 2365 y Fh(f)888 2358 y Fs(of)g Fl(f)23 b Fs(and)16 b(w)o(e)g(set)712 2495 y Fl(r)q Fs(\()p Fl(\013)p Fs(\))g(=)69 b(inf)873 2528 y Fh(f)t Fj(2)p Fh(S)947 2540 y Fb(e)964 2533 y Fc(2)p Fb(\031)q(i\013)1053 2495 y Fl(r)q Fs(\()p Fl(f)5 b Fs(\))16 b Fl(:)553 b Fs(\(5)p Fl(:)p Fs(1\))918 2770 y(27)p eop %%Page: 28 29 28 28 bop 57 192 a Fr(Theorem)17 b(5.1)h(\(Y)-5 b(o)r(ccoz's)20 b(lo)n(w)n(er)h(b)r(ound\))701 314 y Fs(log)8 b Fl(r)q Fs(\()p Fl(\013)p Fs(\))16 b Fk(\025)d(\000)p Fl(B)r Fs(\()p Fl(\013)p Fs(\))f Fk(\000)f Fl(C)545 b Fs(\(5)p Fl(:)p Fs(2\))57 435 y Fd(where)23 b Fl(C)28 b(>)d Fs(0)e Fd(is)g(a)g(univ)o(ersal)f(constan)o(t)g(\(indep)q(enden)o(t)h(of)g Fl(\013)p Fd(\))h(and)f Fl(B)j Fd(is)c(the)i(Brjuno)57 505 y(function.)57 639 y Fs(Before)e(of)g(sk)o(etc)o(hing)e(the)i(pro)q (of)g(let)g(us)f(brie\015y)g(men)o(tion)g(what)g(is)h(the)g(main)e (di\016cult)o(y)57 708 y(whic)o(h)k(w)o(as)h(\014rst)f(o)o(v)o(ercome)h (b)o(y)g(Siegel)g(and)f(whic)o(h)h(w)o(as)f(clearly)i(w)o(ell{kno)o(wn) d(among)57 778 y(mathematicians)16 b(at)j(the)f(end)g(of)h(the)f(19th)g (and)g(at)g(the)h(b)q(eginning)e(of)h(the)h(20th)f(cen)o(tury)57 848 y(\(in)k(1919)g(Gaston)f(Julia)g(ev)o(en)i(claimed,)f(in)g(an)g (incorrect)f(pap)q(er,)i(to)f(dispro)o(v)o(e)e(Siegel's)57 918 y(theorem\).)156 987 y(Assume)e(that)i Fl(\013)d Fk(2)h Fs(CD)9 b(\()p Fl(\034)d Fs(\))19 b(for)f(some)g Fl(\034)24 b Fk(\025)17 b Fs(0.)29 b(Recalling)18 b(the)h(recurrence)e (\(1.2\))j(for)57 1057 y(the)c(p)q(o)o(w)o(er)g(series)f(co)q (e\016cien)o(ts)i(of)f(the)h(linearization)e Fl(h)1130 1064 y Fh(f)1156 1057 y Fs(\()p Fl(z)r Fs(\))g(=)1286 1020 y Fe(P)1339 1032 y Fj(1)1339 1072 y Fh(n)p Fi(=1)1425 1057 y Fl(h)1454 1064 y Fh(n)1481 1057 y Fl(z)1506 1039 y Fh(n)297 1211 y Fl(h)326 1218 y Fi(1)362 1211 y Fs(=)f(1)f Fl(;)50 b(h)546 1218 y Fh(n)587 1211 y Fs(=)706 1178 y(1)p 645 1200 147 2 v 645 1246 a Fl(\025)674 1231 y Fh(n)712 1246 y Fk(\000)11 b Fl(\025)829 1149 y Fh(n)806 1164 y Fe(X)807 1270 y Fh(j)r Fi(=2)886 1211 y Fl(f)910 1218 y Fh(j)1023 1164 y Fe(X)940 1269 y Fh(n)965 1274 y Fc(1)984 1269 y Fi(+)p Fh(:::)o Fi(+)p Fh(n)1106 1274 y Fb(j)1123 1269 y Fi(=)p Fh(n)1187 1211 y Fl(h)1216 1218 y Fh(n)1241 1223 y Fc(1)1271 1211 y Fk(\001)d(\001)g(\001)g Fl(h)1366 1218 y Fh(n)1391 1223 y Fb(j)1439 1211 y Fl(n)14 b Fk(\025)f Fs(2)166 b(\(5)p Fl(:)p Fs(3\))57 1375 y(one)17 b(sees)f(that)i Fl(h)387 1382 y Fh(n)431 1375 y Fs(is)e(a)h(p)q (olynomial)f(in)h Fl(f)864 1382 y Fi(2)887 1375 y Fl(;)8 b(:)g(:)g(:)g(;)g(f)1021 1382 y Fh(n)1066 1375 y Fs(with)17 b(co)q(e\016cien)o(ts)g(whic)o(h)f(are)g(rational)57 1445 y(functions)f(of)i Fl(\025)g Fs(:)22 b Fl(h)439 1452 y Fh(n)479 1445 y Fk(2)15 b Fm(C)8 b Fs(\()q Fl(\025)p Fs(\)[)p Fl(f)666 1452 y Fi(2)691 1445 y Fl(;)g(:)g(:)g(:)g(;)g(f)825 1452 y Fh(n)853 1445 y Fs(])17 b(for)f(all)g Fl(n)e Fk(\025)f Fs(2.)156 1515 y(Let)18 b(us)d(compute)i(explicitely)g(the)f(\014rst)g (few)h(terms)f(of)h(the)f(recurrence)301 1625 y Fl(h)330 1632 y Fi(2)366 1625 y Fs(=)d(\()p Fl(\025)466 1604 y Fi(2)500 1625 y Fk(\000)e Fl(\025)p Fs(\))598 1604 y Fj(\000)p Fi(1)652 1625 y Fl(f)676 1632 y Fi(2)713 1625 y Fl(;)301 1709 y(h)330 1716 y Fi(3)366 1709 y Fs(=)i(\()p Fl(\025)466 1689 y Fi(3)500 1709 y Fk(\000)e Fl(\025)p Fs(\))598 1689 y Fj(\000)p Fi(1)652 1709 y Fs([)p Fl(f)690 1716 y Fi(3)724 1709 y Fs(+)f(2)p Fl(f)827 1689 y Fi(2)822 1722 y(2)850 1709 y Fs(\()p Fl(\025)898 1689 y Fi(2)932 1709 y Fk(\000)h Fl(\025)p Fs(\))1030 1689 y Fj(\000)p Fi(1)1084 1709 y Fs(])j Fl(;)301 1794 y(h)330 1801 y Fi(4)366 1794 y Fs(=)f(\()p Fl(\025)466 1774 y Fi(4)500 1794 y Fk(\000)e Fl(\025)p Fs(\))598 1774 y Fj(\000)p Fi(1)652 1794 y Fs([)p Fl(f)690 1801 y Fi(4)724 1794 y Fs(+)f(3)p Fl(f)822 1801 y Fi(3)845 1794 y Fl(f)869 1801 y Fi(2)892 1794 y Fs(\()p Fl(\025)940 1774 y Fi(2)974 1794 y Fk(\000)g Fl(\025)p Fs(\))1071 1774 y Fj(\000)p Fi(1)1136 1794 y Fs(+)h(2)p Fl(f)1235 1801 y Fi(2)1258 1794 y Fl(f)1282 1801 y Fi(3)1305 1794 y Fs(\()p Fl(\025)1353 1774 y Fi(3)1386 1794 y Fk(\000)g Fl(\025)p Fs(\))1484 1774 y Fj(\000)p Fi(1)652 1879 y Fs(4)p Fl(f)706 1858 y Fi(3)701 1891 y(2)729 1879 y Fs(\()p Fl(\025)777 1858 y Fi(3)811 1879 y Fk(\000)g Fl(\025)p Fs(\))909 1858 y Fj(\000)p Fi(1)963 1879 y Fs(\()p Fl(\025)1011 1858 y Fi(2)1044 1879 y Fk(\000)g Fl(\025)p Fs(\))1142 1858 y Fj(\000)p Fi(1)1207 1879 y Fs(+)g Fl(f)1286 1858 y Fi(3)1281 1891 y(2)1309 1879 y Fs(\()p Fl(\025)1357 1858 y Fi(2)1391 1879 y Fk(\000)g Fl(\025)p Fs(\))1489 1858 y Fj(\000)p Fi(2)1543 1879 y Fs(])i Fl(;)1726 1751 y Fs(\(5)p Fl(:)p Fs(4\))57 1988 y(and)g(so)h(on.)20 b(It)15 b(is)e(not)h(di\016cult)f(to)i(see)f(that)g(among)f(all)g(con)o (tributes)g(to)h Fl(h)1472 1995 y Fh(n)1513 1988 y Fs(there)g(is)g(alw) o(a)o(ys)57 2058 y(a)i(term)g(of)g(the)h(form)e(2)501 2040 y Fh(n)p Fj(\000)p Fi(2)579 2058 y Fl(f)608 2036 y Fh(n)p Fj(\000)p Fi(1)603 2071 y(2)687 2058 y Fs([\()p Fl(\025)749 2040 y Fh(n)787 2058 y Fk(\000)10 b Fl(\025)p Fs(\))e Fl(:)g(:)g(:)h Fs(\()p Fl(\025)1007 2040 y Fi(3)1040 2058 y Fk(\000)h Fl(\025)p Fs(\)\()p Fl(\025)1185 2040 y Fi(2)1219 2058 y Fk(\000)g Fl(\025)p Fs(\)])1330 2040 y Fj(\000)p Fi(1)1384 2058 y Fs(.)22 b(If)16 b(one)g(then)g(tries)g(to) 57 2128 y(estimate)e Fk(j)p Fl(h)297 2135 y Fh(n)324 2128 y Fk(j)h Fs(b)o(y)f(simply)f(summing)g(up)h(the)h(absolute)f(v)m (alues)g(of)h(eac)o(h)g(con)o(tribution)d(then)57 2197 y(one)k(term)g(will)g(b)q(e)82 2319 y(2)107 2299 y Fh(n)p Fj(\000)p Fi(2)185 2319 y Fk(j)p Fl(f)223 2326 y Fi(2)245 2319 y Fk(j)259 2299 y Fh(n)p Fj(\000)p Fi(1)337 2319 y Fs([)p Fk(j)p Fl(\025)394 2299 y Fh(n)423 2319 y Fk(\000)r Fl(\025)p Fk(j)8 b Fl(:)g(:)g(:)h Fk(j)p Fl(\025)625 2299 y Fi(3)649 2319 y Fk(\000)r Fl(\025)p Fk(jj)p Fl(\025)776 2299 y Fi(2)801 2319 y Fk(\000)r Fl(\025)p Fk(j)p Fs(])899 2299 y Fj(\000)p Fi(1)966 2319 y Fk(\024)k Fs(2)1043 2299 y Fh(n)p Fj(\000)p Fi(2)1121 2319 y Fk(j)p Fl(f)1159 2326 y Fi(2)1182 2319 y Fk(j)1196 2299 y Fh(n)p Fj(\000)p Fi(1)1274 2319 y Fs(\(2)p Fl(\015)s Fs(\))1366 2299 y Fi(\()p Fh(n)p Fj(\000)p Fi(1\))p Fh(\034)1498 2319 y Fs([\()p Fl(n)r Fk(\000)r Fs(1\)!])1676 2299 y Fh(\034)1726 2319 y Fs(\(5)p Fl(:)p Fs(5\))57 2441 y(if)k Fl(\013)e Fk(2)g Fs(CD)8 b(\()p Fl(\015)s(;)g(\034)e Fs(\))17 b(and)g(one)g (obtains)f(a)h(div)o(ergen)o(t)f(b)q(ound.)23 b(Note)18 b(the)g(di\013erence)e(with)h(the)57 2511 y(case)f Fk(j)p Fl(\025)p Fk(j)e(6)p Fs(=)f(1)k(:)22 b(in)16 b(this)g(case)g(the)h(b)q (ound)e(w)o(ould)g(b)q(e)i Fk(j)p Fl(\025)p Fk(j)1132 2493 y Fj(\000)p Fi(\()p Fh(n)p Fj(\000)p Fi(1\))1272 2511 y Fs(2)1297 2493 y Fh(n)p Fj(\000)p Fi(2)1375 2511 y Fk(j)p Fl(f)1413 2518 y Fi(2)1436 2511 y Fk(j)1450 2493 y Fh(n)p Fj(\000)p Fi(1)1528 2511 y Fl(c)1550 2493 y Fh(n)p Fj(\000)p Fi(1)1644 2511 y Fs(for)f(some)57 2581 y(p)q(ositiv)o(e)g(constan)o(t)g Fl(c)h Fs(indep)q(enden)o(t)e(of) i Fl(f)5 b Fs(.)24 b(Th)o(us)15 b(one)i(m)o(ust)f(use)g(a)h(more)f (subtle)g(ma)s(joran)o(t)57 2650 y(series)f(metho)q(d.)918 2770 y(28)p eop %%Page: 29 30 29 29 bop 156 192 a Fs(The)17 b(k)o(ey)g(p)q(oin)o(t)f(is)g(that)h(the) g(estimate)f(\(5.5\))h(is)f(far)g(to)q(o)i(p)q(essimistic)d(:)57 301 y Fr(Exercise)31 b(5.2)25 b Fs(Sho)o(w)g(that)h(the)g(series)890 263 y Fe(P)943 276 y Fj(1)943 316 y Fh(n)p Fi(=1)1148 281 y Fh(z)1168 266 y Fb(n)p 1035 289 272 2 v 1035 318 a Fi(\()p Fh(\025)1075 308 y Fb(n)1098 318 y Fj(\000)p Fi(1\))p Fh(:::)o Fi(\()p Fh(\025)p Fj(\000)p Fi(1\))1312 301 y Fs(,)i(with)e Fl(\025)k Fs(=)f Fl(e)1627 283 y Fi(2)p Fh(\031)q(i\013)1714 301 y Fs(,)f(has)57 381 y(p)q(ositiv)o(e)19 b(radius)g(of)h(con)o(v)o(ergence)f(whenev)o(er)g(lim)8 b(sup)1111 393 y Fh(n)p Fj(!1)1232 356 y Fi(log)f Fh(q)1308 361 y Fb(k)p Fc(+1)p 1232 369 141 2 v 1282 398 a Fh(q)1300 403 y Fb(k)1398 381 y Fl(<)19 b Fs(+)p Fk(1)p Fs(.)32 b([Hin)o(t)20 b(:)28 b(see)57 450 y([HL])16 b(for)h(a)f(pro)q(of.])57 560 y(Indeed)e(when)g(a)h(small)f(divisor)g(is)g(really)h(small)e (then,)i(for)g(a)g(certain)f(time,)h(all)g(other)g(small)57 629 y(divisors)h(cannot)h(b)q(e)h(to)q(o)g(small.)25 b(This)16 b(v)m(ague)i(idea)g(is)f(made)g(clear)g(b)o(y)g(the)h(t)o(w)o (o)g(follo)o(wing)57 699 y(lemmas)h(of)i(A.M.)g(Da)o(vie)g([Da])f(whic) o(h)g(extend)h(and)g(impro)o(v)o(e)e(previous)g(results)h(of)h(A.D.)57 769 y(Brjuno.)156 843 y(Let)d Fl(x)c Fk(2)g Fm(R)p Fs(,)f Fl(x)i Fk(6)p Fs(=)e(1)p Fl(=)p Fs(2,)j(w)o(e)h(denote)f Fk(k)p Fl(x)p Fk(k)913 850 y Fg(Z)953 843 y Fs(=)e(min)1089 850 y Fh(p)p Fj(2)p Fg(Z)1170 843 y Fk(j)p Fl(x)d Fs(+)g Fl(p)p Fk(j)p Fs(.)57 994 y Fr(Lemma)21 b(5.3)27 b Fd(Let)21 b Fl(\013)e Fk(2)h Fm(R)10 b Fk(n)j Fm(Q)p Fd(,)21 b Fs(\()p Fl(p)766 1001 y Fh(j)788 994 y Fl(=q)835 1001 y Fh(j)856 994 y Fs(\))875 1001 y Fh(j)r Fj(\025)p Fi(0)967 994 y Fd(denote)f(the)g(sequence)g(of)g(its)f(con)o(v)o(ergen)o(ts,)57 1064 y Fl(k)d Fk(2)f Fm(N)p Fd(,)k Fl(n)14 b Fk(2)i Fm(N)p Fd(,)i Fl(n)d Fk(6)p Fs(=)f(0)p Fd(,)k(and)e(assume)g(that)h Fk(k)p Fl(n\013)p Fk(k)1020 1071 y Fg(Z)1061 1064 y Fk(\024)e Fs(1)p Fl(=)p Fs(\(4)p Fl(q)1231 1071 y Fh(k)1255 1064 y Fs(\))p Fd(.)25 b(Then)17 b Fl(n)d Fk(\025)h Fl(q)1564 1071 y Fh(k)1606 1064 y Fd(and)i(either)57 1134 y Fl(q)79 1141 y Fh(k)120 1134 y Fd(divides)f Fl(n)g Fd(or)g Fl(n)e Fk(\025)g Fl(q)514 1141 y Fh(k)q Fi(+1)589 1134 y Fl(=)p Fs(4)p Fd(.)57 1281 y Fp(Pr)m(o)m(of.)23 b Fs(F)l(rom)18 b(Theorem)h(4.5)h(it)g(follo)o(ws)f(that)h(if)h Fl(r)g Fs(is)f(an)g(in)o(teger)f(and)g(0)h Fl(<)f(r)i(<)f(q)1687 1288 y Fh(k)1731 1281 y Fs(then)57 1351 y Fk(k)p Fl(r)q(\013)p Fk(k)162 1358 y Fg(Z)212 1351 y Fk(\025)j Fs(\(2)p Fl(q)340 1358 y Fh(k)365 1351 y Fs(\))384 1332 y Fj(\000)p Fi(1)438 1351 y Fs(.)40 b(Th)o(us)21 b Fl(n)j Fk(\025)g Fl(q)763 1358 y Fh(k)787 1351 y Fs(.)40 b(Assume)22 b(that)h Fl(q)1168 1358 y Fh(k)1215 1351 y Fs(do)q(es)f(not)h(divide)f Fl(n)g Fs(and)g(that)57 1420 y Fl(n)d(<)g(q)186 1427 y Fh(k)q Fi(+1)261 1420 y Fl(=)p Fs(4.)32 b(Then)19 b Fl(n)h Fs(=)f Fl(mq)664 1427 y Fh(k)702 1420 y Fs(+)13 b Fl(r)21 b Fs(where)f(0)f Fl(<)g(r)i(<)e(q)1170 1427 y Fh(k)1215 1420 y Fs(and)g Fl(m)g(<)g(q)1458 1427 y Fh(k)q Fi(+1)1534 1420 y Fl(=)p Fs(\(4)p Fl(q)1625 1427 y Fh(k)1650 1420 y Fs(\).)32 b(Since)57 1490 y Fk(k)p Fl(q)104 1497 y Fh(k)128 1490 y Fl(\013)p Fk(k)185 1497 y Fg(Z)225 1490 y Fk(\024)14 b Fl(q)302 1469 y Fj(\000)p Fi(1)300 1505 y Fh(k)q Fi(+1)391 1490 y Fs(one)j(gets)g Fk(k)p Fl(mq)677 1497 y Fh(k)701 1490 y Fl(\013)p Fk(k)758 1497 y Fg(Z)798 1490 y Fk(\024)c Fl(mq)918 1469 y Fj(\000)p Fi(1)916 1505 y Fh(k)q Fi(+1)1005 1490 y Fl(<)h Fs(\(4)p Fl(q)1124 1497 y Fh(k)1149 1490 y Fs(\))1168 1472 y Fj(\000)p Fi(1)1222 1490 y Fs(.)22 b(But)17 b Fk(k)p Fl(r)q(\013)p Fk(k)1462 1497 y Fg(Z)1502 1490 y Fk(\025)d Fs(\(2)p Fl(q)1621 1497 y Fh(k)1646 1490 y Fs(\))1665 1472 y Fj(\000)p Fi(1)1735 1490 y Fs(th)o(us)57 1560 y Fk(k)p Fl(n\013)p Fk(k)169 1567 y Fg(Z)208 1560 y Fl(>)g Fs(\(4)p Fl(q)327 1567 y Fh(k)352 1560 y Fs(\))371 1542 y Fj(\000)p Fi(1)424 1560 y Fs(.)1352 b Fa(\003)57 1688 y Fs(Using)19 b(this)h(information)e (on)i(the)g(sequence)g(\()p Fk(k)p Fl(n\013)p Fk(k)1077 1695 y Fg(Z)1103 1688 y Fs(\))1122 1695 y Fh(n)p Fj(\025)p Fi(0)1221 1688 y Fs(Da)o(vie)g(sho)o(ws)e(the)j(follo)o(wing)e(:)57 1764 y(Let)f Fl(A)184 1771 y Fh(k)223 1764 y Fs(=)277 1709 y Fe(n)310 1764 y Fl(n)c Fk(\025)g Fs(0)f Fk(j)h(k)p Fl(n\013)p Fk(k)g(\024)677 1745 y Fi(1)p 657 1753 60 2 v 657 1781 a(8)p Fh(q)695 1786 y Fb(k)723 1709 y Fe(o)756 1764 y Fs(,)k Fl(E)825 1771 y Fh(k)864 1764 y Fs(=)c(max)8 b(\()q Fl(q)1060 1771 y Fh(k)1085 1764 y Fl(;)g(q)1129 1771 y Fh(k)q Fi(+1)1204 1764 y Fl(=)p Fs(4\))17 b(and)g Fl(\021)1413 1771 y Fh(k)1452 1764 y Fs(=)e Fl(q)1528 1771 y Fh(k)1552 1764 y Fl(=E)1614 1771 y Fh(k)1639 1764 y Fs(.)24 b(Let)17 b Fl(A)1803 1746 y Fj(\003)1803 1778 y Fh(k)57 1834 y Fs(b)q(e)e(the)h(set)g(of)f(non)g(negativ)o(e)g(in)o (tegers)f Fl(j)k Fs(suc)o(h)c(that)i(either)f Fl(j)i Fk(2)d Fl(A)1333 1841 y Fh(k)1373 1834 y Fs(or)h(for)g(some)f Fl(j)1652 1841 y Fi(1)1690 1834 y Fs(and)g Fl(j)1806 1841 y Fi(2)57 1904 y Fs(in)i Fl(A)152 1911 y Fh(k)177 1904 y Fs(,)h(with)g Fl(j)343 1911 y Fi(2)376 1904 y Fk(\000)11 b Fl(j)447 1911 y Fi(1)483 1904 y Fl(<)j(E)573 1911 y Fh(k)598 1904 y Fs(,)i(one)h(has)f Fl(j)830 1911 y Fi(1)867 1904 y Fl(<)e(j)i(<)f(j)1032 1911 y Fi(2)1070 1904 y Fs(and)i Fl(q)1190 1911 y Fh(k)1231 1904 y Fs(divides)f Fl(j)e Fk(\000)d Fl(j)1505 1911 y Fi(1)1527 1904 y Fs(.)23 b(F)l(or)16 b(an)o(y)g(non)57 1973 y(negativ)o(e)g(in)o(teger)g Fl(n)g Fs(de\014ne)g(:)399 2132 y Fl(l)10 b Fs(\()p Fl(n)p Fs(\))k(=)g(max)660 2062 y Fe(\032)697 2132 y Fs(\(1)d(+)g Fl(\021)827 2139 y Fh(k)851 2132 y Fs(\))894 2099 y Fl(n)p 885 2121 47 2 v 885 2166 a(q)907 2173 y Fh(k)949 2132 y Fk(\000)g Fs(2)p Fl(;)d Fs(\()p Fl(m)1109 2139 y Fh(n)1136 2132 y Fl(\021)1161 2139 y Fh(k)1196 2132 y Fs(+)j Fl(n)p Fs(\))1321 2099 y(1)p 1310 2121 V 1310 2166 a Fl(q)1332 2173 y Fh(k)1373 2132 y Fk(\000)g Fs(1)1448 2062 y Fe(\033)1726 2132 y Fs(\(5)p Fl(:)p Fs(6\))57 2294 y(where)16 b Fl(m)245 2301 y Fh(n)286 2294 y Fs(=)d(max)p Fk(f)p Fl(j)j Fk(j)e Fs(0)g Fk(\024)g Fl(j)i Fk(\024)e Fl(n;)8 b(j)16 b Fk(2)e Fl(A)876 2276 y Fj(\003)876 2308 y Fh(k)901 2294 y Fk(g)p Fs(.)22 b(W)l(e)17 b(then)g(de\014ne)f(a)g(function)g Fl(h)1571 2301 y Fh(k)1618 2294 y Fs(:)e Fm(N)h Fk(!)e Fm(R)1798 2301 y Fi(+)57 2364 y Fs(as)j(follo)o(ws)484 2455 y Fl(h)513 2462 y Fh(k)546 2455 y Fs(\()p Fl(n)p Fs(\))e(=)681 2385 y Fe(\032)732 2402 y Fh(m)767 2407 y Fb(n)792 2402 y Fi(+)p Fh(\021)844 2407 y Fb(k)865 2402 y Fh(n)p 732 2413 158 2 v 791 2442 a(q)809 2447 y Fb(k)907 2425 y Fk(\000)d Fs(1)49 b(if)17 b Fl(m)1121 2432 y Fh(n)1159 2425 y Fs(+)11 b Fl(q)1231 2432 y Fh(k)1269 2425 y Fk(2)j Fl(A)1353 2407 y Fj(\003)1353 2439 y Fh(k)726 2491 y Fl(l)c Fs(\()p Fl(n)p Fs(\))212 b(if)17 b Fl(m)1121 2498 y Fh(n)1159 2491 y Fs(+)11 b Fl(q)1231 2498 y Fh(k)1269 2491 y Fk(62)j Fl(A)1353 2473 y Fj(\003)1353 2505 y Fh(k)1726 2455 y Fs(\(5)p Fl(:)p Fs(7\))156 2598 y(The)g(function)g Fl(h)478 2605 y Fh(k)510 2598 y Fs(\()q Fl(n)p Fs(\))g(has)g(some)f (prop)q(erties)g(collected)h(in)g(the)g(follo)o(wing)f(prop)q(osition) 918 2770 y(29)p eop %%Page: 30 31 30 30 bop 57 192 a Fr(Prop)r(osition)18 b(5.4)45 b Fd(The)16 b(function)g Fl(h)811 199 y Fh(k)844 192 y Fs(\()p Fl(n)p Fs(\))h Fd(v)o(eri\014es)68 261 y(\(1\))162 237 y Fi(\(1+)p Fh(\021)250 242 y Fb(k)271 237 y Fi(\))o Fh(n)p 162 250 149 2 v 217 278 a(q)235 283 y Fb(k)328 261 y Fk(\000)11 b Fs(2)i Fk(\024)h Fl(h)498 268 y Fh(k)530 261 y Fs(\()q Fl(n)p Fs(\))g Fk(\024)672 237 y Fi(\()o(1+)p Fh(\021)759 242 y Fb(k)780 237 y Fi(\))o Fh(n)p 672 250 V 726 278 a(q)744 283 y Fb(k)837 261 y Fk(\000)d Fs(1)16 b Fd(for)h(all)f Fl(n)p Fd(.)68 331 y(\(2\))25 b(If)17 b Fl(n)d(>)f Fs(0)k Fd(and)f Fl(n)e Fk(2)g Fl(A)569 313 y Fj(\003)569 345 y Fh(k)610 331 y Fd(then)j Fl(h)753 338 y Fh(k)785 331 y Fs(\()p Fl(n)p Fs(\))e Fk(\025)e Fl(h)949 338 y Fh(k)982 331 y Fs(\()p Fl(n)e Fk(\000)g Fs(1\))g(+)g(1)p Fd(.)68 401 y(\(3\))25 b Fl(h)185 408 y Fh(k)218 401 y Fs(\()p Fl(n)p Fs(\))14 b Fk(\025)g Fl(h)382 408 y Fh(k)414 401 y Fs(\()q Fl(n)d Fk(\000)g Fs(1\))16 b Fd(for)h(all)f Fl(n)e(>)f Fs(0)p Fd(.)68 470 y(\(4\))25 b Fl(h)185 477 y Fh(k)218 470 y Fs(\()p Fl(n)11 b Fs(+)g Fl(q)350 477 y Fh(k)375 470 y Fs(\))j Fk(\025)g Fl(h)490 477 y Fh(k)522 470 y Fs(\()p Fl(n)p Fs(\))e(+)f(1)16 b Fd(for)g(all)g Fl(n)p Fd(.)57 603 y Fs(No)o(w)g(w)o(e)g(set)h Fl(g)345 610 y Fh(k)378 603 y Fs(\()p Fl(n)p Fs(\))d(=)g(max)614 548 y Fe(\020)644 603 y Fl(h)673 610 y Fh(k)705 603 y Fs(\()q Fl(n)p Fs(\))8 b Fl(;)804 548 y Fe(h)841 584 y Fh(n)p 834 592 41 2 v 834 621 a(q)852 626 y Fb(k)880 548 y Fe(i)o(\021)950 603 y Fs(and)15 b(w)o(e)i(state)g(the)g(follo)o (wing)e(prop)q(osition)57 731 y Fr(Prop)r(osition)j(5.5)45 b Fd(The)16 b(function)g Fl(g)806 738 y Fh(k)847 731 y Fd(is)g(non)g(negativ)o(e)h(and)f(v)o(eri\014es)f(:)68 801 y(\(1\))25 b Fl(g)180 808 y Fh(k)205 801 y Fs(\(0\))14 b(=)g(0)c Fd(;)68 871 y(\(2\))25 b Fl(g)180 878 y Fh(k)213 871 y Fs(\()p Fl(n)p Fs(\))14 b Fk(\024)354 847 y Fi(\()o(1+)p Fh(\021)441 852 y Fb(k)462 847 y Fi(\))p Fh(n)p 354 859 149 2 v 408 888 a(q)426 893 y Fb(k)525 871 y Fd(for)i(all)g Fl(n)10 b Fd(;)68 941 y(\(3\))25 b Fl(g)180 948 y Fh(k)213 941 y Fs(\()p Fl(n)262 948 y Fi(1)285 941 y Fs(\))11 b(+)g Fl(g)389 948 y Fh(k)421 941 y Fs(\()q Fl(n)471 948 y Fi(2)493 941 y Fs(\))j Fk(\024)g Fl(g)603 948 y Fh(k)635 941 y Fs(\()q Fl(n)685 948 y Fi(1)718 941 y Fs(+)d Fl(n)798 948 y Fi(2)820 941 y Fs(\))17 b Fd(for)f(all)g Fl(n)1031 948 y Fi(1)1070 941 y Fd(and)g Fl(n)1197 948 y Fi(2)1229 941 y Fd(;)68 1010 y(\(4\))25 b(if)17 b Fl(n)d Fk(2)g Fl(A)330 1017 y Fh(k)371 1010 y Fd(and)i Fl(n)e(>)f Fs(0)k Fd(then)f Fl(g)743 1017 y Fh(k)776 1010 y Fs(\()p Fl(n)p Fs(\))e Fk(\025)g Fl(g)935 1017 y Fh(k)968 1010 y Fs(\()p Fl(n)d Fk(\000)g Fs(1\))g(+)g(1)p Fd(.)57 1138 y Fs(The)16 b(pro)q(of)g(of)h(these)g(prop)q(ositions)d(can)j(b)q(e)f (found)g(in)g([Da].)156 1208 y(Let)22 b Fl(k)r Fs(\()p Fl(n)p Fs(\))g(b)q(e)g(de\014ned)e(b)o(y)i(the)f(condition)g Fl(q)1028 1217 y Fh(k)q Fi(\()p Fh(n)p Fi(\))1130 1208 y Fk(\024)h Fl(n)g(<)g(q)1326 1217 y Fh(k)q Fi(\()p Fh(n)p Fi(\)+1)1457 1208 y Fs(.)37 b(Note)22 b(that)g Fl(k)h Fs(is)57 1278 y(non{decreasing.)76 1406 y Fr(Lemma)17 b(5.6)h(\(Da)n(vie's)j(lemma\))27 b Fd(Let)522 1549 y Fl(K)t Fs(\()p Fl(n)p Fs(\))14 b(=)g Fl(n)8 b Fs(log)g(2)j(+)899 1482 y Fh(k)q Fi(\()p Fh(n)p Fi(\))902 1502 y Fe(X)902 1609 y Fh(k)q Fi(=0)985 1549 y Fl(g)1009 1556 y Fh(k)1034 1549 y Fs(\()p Fl(n)p Fs(\))d(log)q(\(2)p Fl(q)1241 1556 y Fh(k)q Fi(+1)1316 1549 y Fs(\))15 b Fl(:)362 b Fs(\(5)p Fl(:)p Fs(8\))57 1684 y Fd(The)16 b(function)g Fl(K)c Fs(\()q Fl(n)p Fs(\))17 b Fd(v)o(eri\014es)e(:)68 1754 y(\(a\))25 b(There)16 b(exists)h(a)f(univ)o(ersal)f(constan)o(t)h Fl(c)913 1761 y Fi(0)949 1754 y Fl(>)d Fs(0)k Fd(suc)o(h)e(that)635 1901 y Fl(K)t Fs(\()p Fl(n)p Fs(\))f Fk(\024)g Fl(n)854 1801 y Fe(0)854 1891 y(@)898 1835 y Fh(k)q Fi(\()p Fh(n)p Fi(\))900 1854 y Fe(X)900 1961 y Fh(k)q Fi(=0)990 1868 y Fs(log)8 b Fl(q)1084 1875 y Fh(k)q Fi(+1)p 990 1890 170 2 v 1051 1935 a Fl(q)1073 1942 y Fh(k)1177 1901 y Fs(+)i Fl(c)1248 1908 y Fi(0)1270 1801 y Fe(1)1270 1891 y(A)1336 1901 y Fs(;)376 b(\(5)p Fl(:)p Fs(9\))65 2049 y Fd(\(b\))25 b Fl(K)t Fs(\()p Fl(n)251 2056 y Fi(1)274 2049 y Fs(\))11 b(+)g Fl(K)t Fs(\()p Fl(n)449 2056 y Fi(2)472 2049 y Fs(\))j Fk(\024)f Fl(K)t Fs(\()p Fl(n)652 2056 y Fi(1)686 2049 y Fs(+)e Fl(n)766 2056 y Fi(2)788 2049 y Fs(\))17 b Fd(for)f(all)g Fl(n)999 2056 y Fi(1)1038 2049 y Fd(and)g Fl(n)1165 2056 y Fi(2)1197 2049 y Fd(;)71 2118 y(\(c\))25 b Fk(\000)8 b Fs(log)h Fk(j)p Fl(\025)319 2100 y Fh(n)357 2118 y Fk(\000)i Fs(1)p Fk(j)i(\024)h Fl(K)t Fs(\()p Fl(n)p Fs(\))e Fk(\000)e Fl(K)t Fs(\()p Fl(n)i Fk(\000)e Fs(1\))p Fd(.)57 2246 y Fp(Pr)m(o)m(of.)20 b Fs(W)l(e)c(will)h(apply)f(Prop)q(osition)f(5.4.)21 b(By)d(\(2\))f(w)o(e)f(ha)o(v)o(e)134 2394 y Fl(K)t Fs(\()p Fl(n)p Fs(\))e Fk(\024)f Fl(n)352 2294 y Fe(2)352 2383 y(4)386 2394 y Fs(log)8 b(2)j(+)544 2327 y Fh(k)q Fi(\()p Fh(n)p Fi(\))547 2347 y Fe(X)547 2454 y Fh(k)q Fi(=0)636 2360 y Fs(\(1)h(+)e Fl(\021)766 2367 y Fh(k)791 2360 y Fs(\))p 636 2382 174 2 v 700 2428 a Fl(q)722 2435 y Fh(k)824 2394 y Fs(log)q(\(2)p Fl(q)955 2401 y Fh(k)q Fi(+1)1030 2394 y Fs(\))1049 2294 y Fe(3)1049 2383 y(5)262 2593 y Fk(\024)j Fl(n)352 2493 y Fe(2)352 2583 y(4)386 2527 y Fh(k)q Fi(\()p Fh(n)p Fi(\))389 2546 y Fe(X)389 2653 y Fh(k)q Fi(=0)478 2559 y Fs(log)c Fl(q)573 2566 y Fh(k)q Fi(+1)p 478 2582 170 2 v 539 2627 a Fl(q)561 2634 y Fh(k)665 2593 y Fs(+)i(log)d(2)j(+)g(log)e(2)995 2531 y Fj(1)979 2546 y Fe(X)979 2653 y Fh(k)q Fi(=0)1060 2523 y Fe(\022)1113 2559 y Fs(1)p 1102 2582 47 2 v 1102 2627 a Fl(q)1124 2634 y Fh(k)1166 2593 y Fs(+)1258 2559 y(4)p 1222 2582 98 2 v 1222 2627 a Fl(q)1244 2634 y Fh(k)q Fi(+1)1325 2523 y Fe(\023)1373 2593 y Fs(+)i(4)1472 2531 y Fj(1)1456 2546 y Fe(X)1456 2653 y Fh(k)q Fi(=0)1543 2559 y Fs(log)d Fl(q)1637 2566 y Fh(k)q Fi(+1)p 1543 2582 170 2 v 1579 2627 a Fl(q)1601 2634 y Fh(k)q Fi(+1)1718 2493 y Fe(3)1718 2583 y(5)918 2770 y Fs(30)p eop %%Page: 31 32 31 31 bop 57 192 a Fs(since)16 b Fl(\021)204 199 y Fh(k)242 192 y Fk(\024)d Fs(4)p Fl(q)341 199 y Fh(k)366 192 y Fl(q)390 170 y Fj(\000)p Fi(1)388 207 y Fh(k)q Fi(+1)463 192 y Fs(.)156 261 y(By)19 b(Remark)f(A2.4)g(the)h(series)769 224 y Fe(P)835 236 y Fi(log)8 b Fh(q)912 241 y Fb(k)p Fc(+1)p 835 250 141 2 v 865 278 a Fh(q)883 283 y Fb(k)p Fc(+1)1001 261 y Fs(and)1100 224 y Fe(P)1160 261 y Fl(q)1184 240 y Fj(\000)p Fi(1)1182 276 y Fh(k)1256 261 y Fs(are)19 b(uniformly)d(b)q(ounded)i(b)o(y)57 331 y(some)e(constan)o(t)f(indep)q (enden)o(t)h(of)g Fl(\013)p Fs(,)h(th)o(us)f(\(a\))h(follo)o(ws.)156 401 y(\(b\))22 b(is)f(an)g(immediate)f(consequence)h(of)h(Prop)q (osition)e(5.5,)i(\(3\),)h(and)d(the)i(fact)g(that)57 470 y Fl(k)r Fs(\()p Fl(n)p Fs(\))16 b(is)h(not)f(decreasing.)156 540 y(Finally)g(recall)g(that)234 666 y Fk(\000)8 b Fs(log)h Fk(j)p Fl(\025)397 645 y Fh(n)435 666 y Fk(\000)i Fs(1)p Fk(j)i Fs(=)h Fk(\000)8 b Fs(log)h(2)p Fk(j)f Fs(sin)f Fl(\031)r(n\013)p Fk(j)14 b(2)g Fs(\()p Fk(\000)8 b Fs(log)h Fl(\031)r Fk(k)p Fl(n\013)p Fk(k)1274 673 y Fg(Z)1300 666 y Fl(;)f Fk(\000)g Fs(log)g(2)p Fk(k)p Fl(n\013)p Fk(k)1578 673 y Fg(Z)1604 666 y Fs(\))14 b Fl(:)57 791 y Fs(F)l(or)k(all)h Fl(n)g Fs(w)o(e)g(ha)o(v)o(e)f(either)h Fk(k)p Fl(n\013)p Fk(k)719 798 y Fg(Z)763 791 y Fl(>)f Fs(1)p Fl(=)p Fs(4)h(or)g(there)g(exists)g(some)g(non{negativ)o(e)f(in) o(teger)57 861 y Fl(k)27 b Fs(suc)o(h)e(that)h(\(8)p Fl(q)415 868 y Fh(k)440 861 y Fs(\))459 843 y Fj(\000)p Fi(1)542 861 y Fl(>)j Fk(k)p Fl(n\013)p Fk(k)722 868 y Fg(Z)777 861 y Fk(\025)g Fs(\(8)p Fl(q)911 868 y Fh(k)q Fi(+1)986 861 y Fs(\))1005 843 y Fj(\000)p Fi(1)1059 861 y Fs(,)f(th)o(us)d Fl(n)k Fk(2)h Fl(A)1379 868 y Fh(k)1404 861 y Fs(,)e(whic)o(h)c(implies)h(b)o(y)57 931 y(Prop)q(osition)d(5.5,)i(\(4\),)h(that)f Fl(g)675 938 y Fh(k)699 931 y Fs(\()p Fl(n)p Fs(\))i Fk(\025)f Fl(g)881 938 y Fh(k)905 931 y Fs(\()p Fl(n)16 b Fk(\000)f Fs(1\))h(+)f(1,)25 b(and)d Fk(\000)8 b Fs(log)h Fk(j)p Fl(\025)1468 912 y Fh(n)1511 931 y Fk(\000)15 b Fs(1)p Fk(j)25 b(\024)f Fs(2)p Fl(q)1739 938 y Fh(k)q Fi(+1)1815 931 y Fs(.)57 1000 y(Com)o(bining)14 b(these)i(facts)h(together)g(one)f (gets)h(\(c\).)770 b Fa(\003)57 1120 y Fs(F)l(ollo)o(wing)14 b([CM])i(w)o(e)g(can)h(no)o(w)f(pro)o(v)o(e)f(Theorem)g(5.1.)57 1190 y Fp(Pr)m(o)m(of.)22 b(of)f(The)m(or)m(em)g(5.1.)61 b Fs(Let)19 b Fl(s)8 b Fs(\()q Fl(z)r Fs(\))19 b(=)881 1152 y Fe(P)933 1205 y Fh(n)p Fj(\025)p Fi(1)1020 1190 y Fl(s)1043 1197 y Fh(n)1070 1190 y Fl(z)1095 1172 y Fh(n)1142 1190 y Fs(b)q(e)g(the)g(unique)f(solution)g(analytic)57 1271 y(at)f Fl(z)h Fs(=)d(0)i(of)h(the)g(equation)f Fl(s)8 b Fs(\()q Fl(z)r Fs(\))16 b(=)f Fl(z)f Fs(+)d Fl(\033)g Fs(\()p Fl(s)d Fs(\()q Fl(z)r Fs(\))q(\),)18 b(where)f Fl(\033)r Fs(\()p Fl(z)r Fs(\))f(=)1375 1247 y Fh(z)1395 1232 y Fc(2)1415 1247 y Fi(\(2)p Fj(\000)p Fh(z)q Fi(\))p 1375 1260 143 2 v 1386 1289 a(\(1)p Fj(\000)p Fh(z)q Fi(\))1489 1279 y Fc(2)1539 1271 y Fs(=)1593 1234 y Fe(P)1646 1286 y Fh(n)p Fj(\025)p Fi(2)1732 1271 y Fl(nz)1787 1253 y Fh(n)1815 1271 y Fs(.)57 1341 y(The)g(co)q(e\016cien)o(ts)g(satisfy) 400 1494 y Fl(s)423 1501 y Fi(1)460 1494 y Fs(=)d(1)h Fl(;)22 b(s)610 1501 y Fh(n)651 1494 y Fs(=)735 1431 y Fh(n)711 1446 y Fe(X)704 1552 y Fh(m)p Fi(=2)798 1494 y Fl(m)999 1446 y Fe(X)850 1553 y Fh(n)875 1558 y Fc(1)894 1553 y Fi(+)p Fh(:::)o Fi(+)p Fh(n)1016 1558 y Fb(m)1049 1553 y Fi(=)p Fh(n)6 b(;)g(n)1154 1558 y Fb(i)1170 1553 y Fj(\025)p Fi(1)1229 1494 y Fl(s)1252 1501 y Fh(n)1277 1506 y Fc(1)1307 1494 y Fl(:)i(:)g(:)h(s)1397 1501 y Fh(n)1422 1506 y Fb(m)1471 1494 y Fl(:)216 b Fs(\(5)p Fl(:)p Fs(10\))57 1660 y(Clearly)16 b(there)g(exist)h(t)o(w)o(o)f(p)q (ositiv)o(e)h(constan)o(ts)e Fl(c)996 1667 y Fi(1)1018 1660 y Fl(;)8 b(c)1062 1667 y Fi(2)1101 1660 y Fs(suc)o(h)15 b(that)810 1786 y Fk(j)p Fl(s)847 1793 y Fh(n)874 1786 y Fk(j)f(\024)g Fl(c)977 1793 y Fi(1)999 1786 y Fl(c)1021 1765 y Fh(n)1021 1798 y Fi(2)1061 1786 y Fl(:)57 1911 y Fs(F)l(rom)g(the)i(recurrence)e(relation)h(and)g(Bieb)q(erbac)o(h{De) f(Branges's)h(b)q(ound)f Fk(j)p Fl(f)1547 1918 y Fh(n)1575 1911 y Fk(j)f(\024)h Fl(n)h Fs(for)h(all)57 1981 y Fl(n)d Fk(\025)h Fs(2)j(w)o(e)f(obtain)347 2131 y Fk(j)p Fl(h)390 2138 y Fh(n)417 2131 y Fk(j)d(\024)577 2097 y Fs(1)p 503 2119 174 2 v 503 2165 a Fk(j)p Fl(\025)546 2150 y Fh(n)584 2165 y Fk(\000)e Fl(\025)p Fk(j)722 2068 y Fh(n)698 2083 y Fe(X)691 2189 y Fh(m)p Fi(=2)785 2131 y Fl(m)986 2083 y Fe(X)837 2190 y Fh(n)862 2195 y Fc(1)881 2190 y Fi(+)p Fh(:::)o Fi(+)p Fh(n)1003 2195 y Fb(m)1036 2190 y Fi(=)p Fh(n)6 b(;)g(n)1141 2195 y Fb(i)1157 2190 y Fj(\025)p Fi(1)1216 2131 y Fk(j)p Fl(h)1259 2138 y Fh(n)1284 2143 y Fc(1)1305 2131 y Fk(j)i Fl(:)g(:)g(:)h Fk(j)p Fl(h)1437 2138 y Fh(n)1462 2143 y Fb(m)1496 2131 y Fk(j)14 b Fl(:)57 2307 y Fs(W)l(e)25 b(no)o(w)g(deduce)g(b)o(y)g(induction)g (on)g Fl(n)h Fs(that)f Fk(j)p Fl(h)1041 2314 y Fh(n)1068 2307 y Fk(j)k(\024)f Fl(s)1201 2314 y Fh(n)1229 2307 y Fl(e)1252 2289 y Fh(K)r Fi(\()p Fh(n)p Fj(\000)p Fi(1\))1423 2307 y Fs(for)d Fl(n)k Fk(\025)f Fs(1,)g(where)57 2377 y Fl(K)37 b Fs(:)25 b Fm(N)h Fk(!)f Fm(R)c Fs(is)h(de\014ned)h(in)g (\(5.8\).)42 b(If)24 b(w)o(e)f(assume)f(this)h(holds)f(for)h(all)g Fl(n)1575 2359 y Fj(0)1614 2377 y Fl(<)i(n)e Fs(then)57 2446 y(the)16 b(ab)q(o)o(v)o(e)h(inequalit)o(y)f(giv)o(es)165 2599 y Fk(j)p Fl(h)208 2606 y Fh(n)235 2599 y Fk(j)e(\024)396 2565 y Fs(1)p 321 2587 V 321 2633 a Fk(j)p Fl(\025)364 2619 y Fh(n)402 2633 y Fk(\000)d Fl(\025)p Fk(j)540 2537 y Fh(n)516 2552 y Fe(X)509 2657 y Fh(m)p Fi(=2)603 2599 y Fl(m)805 2552 y Fe(X)655 2659 y Fh(n)680 2664 y Fc(1)700 2659 y Fi(+)p Fh(:::)o Fi(+)p Fh(n)822 2664 y Fb(m)854 2659 y Fi(=)p Fh(n)6 b(;)h(n)960 2664 y Fb(i)975 2659 y Fj(\025)p Fi(1)1034 2599 y Fl(s)1057 2606 y Fh(n)1082 2611 y Fc(1)1112 2599 y Fl(:)h(:)g(:)h(s)1202 2606 y Fh(n)1227 2611 y Fb(m)1262 2599 y Fl(e)1285 2578 y Fh(K)r Fi(\()p Fh(n)1361 2583 y Fc(1)1381 2578 y Fj(\000)p Fi(1\)+)p Fh(:::)o(K)r Fi(\()p Fh(n)1590 2583 y Fb(m)1623 2578 y Fj(\000)p Fi(1\))1706 2599 y Fl(:)918 2770 y Fs(31)p eop %%Page: 32 33 32 32 bop 57 192 a Fs(But)21 b Fl(K)t Fs(\()p Fl(n)255 199 y Fi(1)291 192 y Fk(\000)13 b Fs(1\))h(+)f Fl(:)8 b(:)g(:)h(K)t Fs(\()p Fl(n)615 199 y Fh(m)666 192 y Fk(\000)14 b Fs(1\))20 b Fk(\024)g Fl(K)t Fs(\()p Fl(n)14 b Fk(\000)g Fs(2\))20 b Fk(\024)g Fl(K)t Fs(\()p Fl(n)14 b Fk(\000)f Fs(1\))h(+)g(log)8 b Fk(j)p Fl(\025)1514 173 y Fh(n)1555 192 y Fk(\000)14 b Fl(\025)p Fk(j)20 b Fs(and)g(w)o(e)57 261 y(deduce)c(that)246 409 y Fk(j)p Fl(h)289 416 y Fh(n)316 409 y Fk(j)e(\024)f Fl(e)419 388 y Fh(K)r Fi(\()p Fh(n)p Fj(\000)p Fi(1\))603 346 y Fh(n)580 361 y Fe(X)573 467 y Fh(m)p Fi(=2)667 409 y Fl(m)868 361 y Fe(X)719 468 y Fh(n)744 473 y Fc(1)763 468 y Fi(+)p Fh(:::)o Fi(+)p Fh(n)885 473 y Fb(m)917 468 y Fi(=)p Fh(n)6 b(;)h(n)1023 473 y Fb(i)1038 468 y Fj(\025)p Fi(1)1098 409 y Fl(s)1121 416 y Fh(n)1146 421 y Fc(1)1176 409 y Fl(:)h(:)g(:)g(s)1265 416 y Fh(n)1290 421 y Fb(m)1340 409 y Fs(=)13 b Fl(s)1415 416 y Fh(n)1443 409 y Fl(e)1466 388 y Fh(K)r Fi(\()p Fh(n)p Fj(\000)p Fi(1\))1625 409 y Fl(;)57 585 y Fs(as)18 b(required.)28 b(Theorem)17 b(5.1)i(then)f(follo)o(ws)g(from)g(the)h (fact)h(that)f Fl(n)1367 567 y Fj(\000)p Fi(1)1420 585 y Fl(K)t Fs(\()p Fl(n)p Fs(\))f Fk(\024)g Fl(B)r Fs(\()p Fl(!)r Fs(\))13 b(+)f Fl(c)1806 592 y Fi(0)57 655 y Fs(for)k(some)g (univ)o(ersal)e(constan)o(t)i Fl(c)691 662 y Fi(0)727 655 y Fl(>)e Fs(0)i(\(Da)o(vie's)h(lemma\).)597 b Fa(\003)57 775 y Fr(Exercise)27 b(5.7)22 b Fs(Consider)g(the)h(quadratic)f(p)q (olynomial)g Fl(P)1219 782 y Fh(\025)1245 775 y Fs(\()p Fl(z)r Fs(\))k(=)e Fl(\025)p Fs(\()p Fl(z)19 b Fk(\000)c Fl(z)1566 757 y Fi(2)1588 775 y Fs(\))24 b(\(w)o(e)f(ha)o(v)o(e)57 845 y(conjugated)12 b(\(3.1\))i(b)o(y)e(an)h(homotheh)o(t)o(y)e(so)i (as)f(to)h(eliminate)f(a)h(factor)g(1)p Fl(=)p Fs(2)g(in)f(what)h (follo)o(ws\).)57 915 y(Its)j(formal)g(linearization)f Fl(H)613 922 y Fh(\025)640 915 y Fs(\()p Fl(z)r Fs(\))f(=)770 878 y Fe(P)823 890 y Fj(1)823 930 y Fh(n)p Fi(=1)908 915 y Fl(H)949 922 y Fh(n)977 915 y Fs(\()p Fl(\025)p Fs(\))p Fl(z)1069 897 y Fh(n)1114 915 y Fs(is)i(giv)o(en)g(b)o(y)g(the) h(recurrence)346 1045 y Fl(H)387 1052 y Fi(1)410 1045 y Fs(\()p Fl(\025)p Fs(\))d(=)g(1)g Fl(;)36 b(H)674 1052 y Fh(n)701 1045 y Fs(\()p Fl(\025)p Fs(\))15 b(=)e(\(1)f Fk(\000)e Fl(\025)969 1024 y Fh(n)p Fj(\000)p Fi(1)1048 1045 y Fs(\))1067 1024 y Fj(\000)p Fi(1)1152 997 y Fe(X)1129 1103 y Fh(i)p Fi(+)p Fh(j)r Fi(=)p Fh(n)1255 1045 y Fl(H)1296 1052 y Fh(i)1313 1045 y Fs(\()p Fl(\025)p Fs(\))p Fl(H)1421 1052 y Fh(j)1444 1045 y Fs(\()p Fl(\025)p Fs(\))k Fl(:)57 1217 y Fs(De\014ne)i(the)h(sequence)f(of)h(p)q(ositiv)o(e)f(real)g(n)o (um)o(b)q(ers)e(\()p Fl(h)1090 1224 y Fh(n)1117 1217 y Fs(\()p Fl(\025)p Fs(\)\))1203 1224 y Fh(n)p Fj(\025)p Fi(1)1299 1217 y Fs(b)o(y)j(the)f(recurrence)377 1347 y Fl(h)406 1354 y Fi(1)428 1347 y Fs(\()p Fl(\025)p Fs(\))f(=)e(1)h Fl(;)36 b(h)680 1354 y Fh(n)707 1347 y Fs(\()p Fl(\025)p Fs(\))14 b(=)g Fk(j)p Fs(1)d Fk(\000)f Fl(\025)969 1327 y Fh(n)p Fj(\000)p Fi(1)1048 1347 y Fk(j)1062 1327 y Fj(\000)p Fi(1)1146 1300 y Fe(X)1123 1406 y Fh(i)p Fi(+)p Fh(j)r Fi(=)p Fh(n)1250 1347 y Fl(h)1279 1354 y Fh(i)1295 1347 y Fs(\()p Fl(\025)p Fs(\))p Fl(h)1391 1354 y Fh(j)1413 1347 y Fs(\()p Fl(\025)p Fs(\))k Fl(:)57 1524 y Fs(Clearly)e Fk(j)p Fl(H)281 1531 y Fh(n)308 1524 y Fs(\()p Fl(\025)p Fs(\))p Fk(j)j(\024)e Fl(h)485 1531 y Fh(n)512 1524 y Fs(\()p Fl(\025)p Fs(\).)22 b(Sho)o(w)11 b(that)j(if)e Fl(\025)i Fs(=)g Fl(e)1007 1506 y Fi(2)p Fh(\031)q(i\013)1094 1524 y Fs(,)f Fl(\013)h Fk(2)g Fm(R)s Fk(n)s Fm(Q)e Fs(is)g Fp(not)h Fs(a)f(Brjuno)h(n)o(um)o(b)q(er)57 1594 y(then)23 b(lim)8 b(sup)330 1606 y Fh(n)p Fj(!1)445 1594 y Fl(n)475 1576 y Fj(\000)p Fi(1)536 1594 y Fs(log)h Fl(h)638 1601 y Fh(n)665 1594 y Fs(\()p Fl(\025)p Fs(\))26 b(=)f(+)p Fk(1)p Fs(,)g(i.e.)43 b(the)24 b(ma)s(joran)o(t)e(series)1513 1556 y Fe(P)1566 1609 y Fh(n)p Fj(\025)p Fi(1)1652 1594 y Fl(h)1681 1601 y Fh(n)1708 1594 y Fs(\()p Fl(\025)p Fs(\))p Fl(z)1800 1576 y Fh(n)57 1663 y Fs(is)f(div)o(ergen)o(t.)35 b(Under)20 b(what)i(assumptions)d(on)h Fl(\013)i Fs(one)f(can)g(sho)o (w)f(that)i(there)f(exist)h(t)o(w)o(o)57 1733 y(p)q(ositiv)o(e)e (constan)o(ts)g Fl(c)492 1740 y Fi(0)514 1733 y Fl(;)8 b(c)558 1740 y Fi(1)601 1733 y Fs(and)21 b Fl(s)g(>)g Fs(0)g(suc)o(h)f(that)h Fl(h)1111 1740 y Fh(n)1138 1733 y Fs(\()p Fl(\025)p Fs(\))h Fk(\024)f Fl(c)1309 1740 y Fi(0)1331 1733 y Fl(c)1353 1715 y Fh(n)1353 1745 y Fi(1)1380 1733 y Fs(\()p Fl(n)p Fs(!\))1462 1715 y Fh(s)1505 1733 y Fs(for)f(all)h Fl(n)g Fk(\025)g Fs(1)10 b(?)57 1803 y([Hin)o(t)25 b(:)40 b(First)25 b(sho)o(w)f(that)i(the)g(set)f Fk(f)p Fl(n)k Fk(\025)g Fs(0)f Fk(j)h Fl(q)1078 1810 y Fh(n)p Fi(+1)1185 1803 y Fk(\025)f Fs(\()p Fl(q)1293 1810 y Fh(n)1338 1803 y Fs(+)17 b(1\))1438 1785 y Fi(2)1461 1803 y Fk(g)25 b Fs(is)g(in\014nite)g(and)57 1873 y(denote)16 b(\()p Fl(q)260 1854 y Fj(0)258 1886 y Fh(i)275 1873 y Fs(\))294 1880 y Fh(i)p Fj(\025)p Fi(0)379 1873 y Fs(its)g(elemen)o (ts.)21 b(Then)16 b(sho)o(w)g(that)h(the)g(sequence)f Fl(h)1354 1880 y Fh(n)1381 1873 y Fs(\()p Fl(\025)p Fs(\))h(is)f (increasing)f(and)57 1987 y Fl(h)86 1996 y Fh(q)105 1984 y Ff(0)104 2008 y Fb(r)q Fc(+1)166 1996 y Fi(+1)219 1987 y Fs(\()p Fl(\025)p Fs(\))g Fk(\025)e(j)p Fs(1)t Fk(\000)t Fl(\025)468 1968 y Fh(q)487 1953 y Ff(0)486 1978 y Fb(r)q Fc(+1)551 1987 y Fk(j)565 1969 y Fj(\000)p Fi(1)618 1987 y Fs(\()p Fl(h)666 1994 y Fh(q)685 1985 y Ff(0)684 2005 y Fb(r)705 1994 y Fi(+1)758 1987 y Fs(\()p Fl(\025)p Fs(\)\))844 1894 y Fe(h)874 1915 y Fb(q)891 1905 y Ff(0)890 1930 y Fb(r)q Fc(+1)p 874 1938 80 2 v 874 1962 a Fb(q)891 1954 y Ff(0)890 1973 y Fb(r)911 1962 y Fc(+1)959 1894 y Fe(i)986 1987 y Fs(.)20 b(One)13 b(can)g(also)f(consult)g([Y)l(o2,)i (App)q(endice)57 2057 y(2,)i(pp.)21 b(83{85])16 b(and)g([CM].])57 2162 y Fr(Exercise)j(5.8)d Fs(\(Linearization)f(and)h(Gevrey)g (classes,)f(see)i([CM])e(for)h(solutions)f(and)g(more)57 2232 y(information.)40 b(\))i(Bet)o(w)o(een)23 b Fm(C)9 b Fs([[)p Fl(z)r Fs(]])26 b(and)d Fm(C)9 b Fk(f)p Fl(z)r Fk(g)26 b Fs(one)d(has)f(man)o(y)h(imp)q(ortan)o(t)e(algebras)h(of)57 2302 y(\\ultradi\013eren)o(tiable")11 b(p)q(o)o(w)o(er)i(series)g (\(i.e.)21 b(asymptotic)13 b(expansions)g(at)h Fl(z)i Fs(=)d(0)h(of)g(functions)57 2371 y(whic)o(h)e(are)i(\\b)q(et)o(w)o (een")f Fk(C)545 2353 y Fj(1)602 2371 y Fs(and)g Fm(C)c Fk(f)p Fl(z)r Fk(g)p Fs(\).)24 b(Consider)13 b(t)o(w)o(o)g(subalgebras) e Fl(A)1461 2378 y Fi(1)1498 2371 y Fk(\032)i Fl(A)1587 2378 y Fi(2)1624 2371 y Fs(of)h Fl(z)r Fm(C)k Fs([)r([)p Fl(z)r Fs(])q(])57 2441 y(closed)i(with)i(resp)q(ect)f(to)g(the)h(comp) q(osition)e(of)h(formal)g(series.)35 b(F)l(or)20 b(example)h(Gevrey{)p Fl(s)57 2511 y Fs(classes,)16 b Fl(s)g(>)f Fs(0)j(\(i.e.)25 b(series)16 b Fl(F)7 b Fs(\()p Fl(z)r Fs(\))17 b(=)781 2474 y Fe(P)833 2526 y Fh(n)p Fj(\025)p Fi(0)920 2511 y Fl(f)944 2518 y Fh(n)971 2511 y Fl(z)996 2493 y Fh(n)1041 2511 y Fs(suc)o(h)g(that)g(there)h(exist)g Fl(c)1533 2518 y Fi(1)1555 2511 y Fl(;)8 b(c)1599 2518 y Fi(2)1636 2511 y Fl(>)15 b Fs(0)j(suc)o(h)57 2581 y(that)g Fk(j)p Fl(f)204 2588 y Fh(n)231 2581 y Fk(j)d(\024)g Fl(c)336 2588 y Fi(1)358 2581 y Fl(c)380 2563 y Fh(n)380 2593 y Fi(2)407 2581 y Fs(\()p Fl(n)p Fs(!\))489 2563 y Fh(s)528 2581 y Fs(for)j(all)f Fl(n)e Fk(\025)g Fs(0\).)25 b(Let)19 b Fl(f)i Fk(2)15 b Fl(A)1079 2588 y Fi(1)1120 2581 y Fs(b)q(eing)i(suc)o(h)f(that)i Fl(f)1506 2563 y Fj(0)1529 2581 y Fs(\()q(0\))d(=)g Fl(\025)h Fk(2)g Fm(C)1789 2563 y Fj(\003)1815 2581 y Fs(.)57 2650 y(W)l(e)f(sa)o(y)g(that)h Fl(f)21 b Fp(is)c(line)m(arizable)h(in)32 b Fl(A)801 2657 y Fi(2)840 2650 y Fs(if)15 b(there)g(exists)g Fl(h)1175 2657 y Fh(f)1215 2650 y Fk(2)f Fl(A)1299 2657 y Fi(2)1337 2650 y Fs(tangen)o(t)h(to)g(the)h(iden)o(tit)o(y)918 2770 y(32)p eop %%Page: 33 34 33 33 bop 57 192 a Fs(and)20 b(suc)o(h)h(that)h Fl(f)e Fk(\016)14 b Fl(h)501 199 y Fh(f)548 192 y Fs(=)22 b Fl(h)638 199 y Fh(f)678 192 y Fk(\016)14 b Fl(R)755 199 y Fh(\025)781 192 y Fs(.)37 b(Sho)o(w)20 b(that)i(if)g(one)f(requires)f Fl(A)1457 199 y Fi(2)1502 192 y Fs(=)h Fl(A)1599 199 y Fi(1)1622 192 y Fs(,)i(i.e.)36 b(the)57 261 y(linearization)23 b Fl(h)379 268 y Fh(f)429 261 y Fs(to)i(b)q(e)f(as)g(regular)f(as)h (the)h(giv)o(en)f(germ)f Fl(f)5 b Fs(,)27 b(once)e(again)e(the)i (Brjuno)57 331 y(condition)18 b(is)h(su\016cien)o(t.)30 b(It)20 b(is)f(quite)h(in)o(teresting)e(to)i(notice)g(that)g(giv)o(en)f (an)o(y)g(algebra)f(of)57 401 y(formal)c(p)q(o)o(w)o(er)g(series)h (whic)o(h)f(is)h(closed)g(under)f(comp)q(osition)g(\(as)i(it)f(should)f (if)i(one)f(whishes)57 470 y(to)21 b(study)g(conjugacy)g(problems\))e (a)i(germ)f(in)h(the)g(algebra)f(is)g(linearizable)g Fp(in)i(the)g(same)57 540 y(algebr)m(a)g Fs(if)f(the)g(Brjuno)f (condition)g(is)h(satis\014ed.)34 b(If)21 b(the)g(linearization)f(is)g (allo)o(w)o(ed)g(to)h(b)q(e)57 610 y(less)c(regular)f(than)h(the)h(giv) o(en)f(germ)g(\(i.e.)26 b Fl(A)927 617 y Fi(1)967 610 y Fs(is)17 b(a)h(prop)q(er)e(subset)h(of)h Fl(A)1470 617 y Fi(2)1493 610 y Fs(\))g(one)f(\014nds)g(new)57 680 y(arithmetical)22 b(conditions,)h(w)o(eak)o(er)f(than)g(the)h (Brjuno)g(condition.)40 b(Let)23 b(\()p Fl(M)1581 687 y Fh(n)1609 680 y Fs(\))1628 687 y Fh(n)p Fj(\025)p Fi(1)1729 680 y Fs(b)q(e)g(a)57 749 y(sequence)16 b(of)h(p)q(ositiv)o(e)f(real)g (n)o(um)o(b)q(ers)e(suc)o(h)i(that)h(:)93 819 y(0.)24 b(inf)217 826 y Fh(n)p Fj(\025)p Fi(1)303 819 y Fl(M)356 793 y Fi(1)p Fh(=n)351 825 y(n)438 819 y Fl(>)14 b Fs(0)c(;)93 889 y(1.)24 b(There)16 b(exists)h Fl(C)474 896 y Fi(1)510 889 y Fl(>)c Fs(0)k(suc)o(h)e(that)i Fl(M)872 896 y Fh(n)p Fi(+1)964 889 y Fk(\024)c Fl(C)1056 868 y Fh(n)p Fi(+1)1052 902 y(1)1133 889 y Fl(M)1181 896 y Fh(n)1225 889 y Fs(for)j(all)h Fl(n)c Fk(\025)h Fs(1)c(;)93 959 y(2.)24 b(The)17 b(sequence)f(\()p Fl(M)532 966 y Fh(n)560 959 y Fs(\))579 966 y Fh(n)p Fj(\025)p Fi(1)674 959 y Fs(is)g(logarithmically)f(con)o(v)o(ex)10 b(;)93 1028 y(3.)24 b Fl(M)204 1035 y Fh(n)232 1028 y Fl(M)280 1035 y Fh(m)332 1028 y Fk(\024)13 b Fl(M)432 1035 y Fh(m)p Fi(+)p Fh(n)p Fj(\000)p Fi(1)593 1028 y Fs(for)k(all)f Fl(m;)8 b(n)13 b Fk(\025)h Fs(1.)57 1098 y(Let)j Fl(f)k Fs(=)244 1061 y Fe(P)297 1113 y Fh(n)p Fj(\025)p Fi(1)383 1098 y Fl(f)407 1105 y Fh(n)435 1098 y Fl(z)460 1080 y Fh(n)502 1098 y Fk(2)15 b Fl(z)r Fm(C)i Fs([)s([)p Fl(z)r Fs(]])10 b(;)17 b Fl(f)23 b Fs(b)q(elongs)16 b(to)h(the)h(algebra)e Fl(z)r Fm(C)h Fs([)s([)p Fl(z)r Fs(]])1440 1113 y Fi(\()p Fh(M)1494 1118 y Fb(n)1518 1113 y Fi(\))1553 1098 y Fs(if)g(there)g(exist)57 1168 y(t)o(w)o(o)f(p)q(ositiv)o(e)g(constan)o(ts)g Fl(c)578 1175 y Fi(1)600 1168 y Fl(;)8 b(c)644 1175 y Fi(2)682 1168 y Fs(suc)o(h)16 b(that)626 1296 y Fk(j)p Fl(f)664 1303 y Fh(n)691 1296 y Fk(j)e(\024)g Fl(c)794 1303 y Fi(1)816 1296 y Fl(c)838 1275 y Fh(n)838 1308 y Fi(2)864 1296 y Fl(M)912 1303 y Fh(n)967 1296 y Fs(for)j(all)c Fl(n)h Fk(\025)f Fs(1)h Fl(:)57 1424 y Fs(Sho)o(w)e(that)h(the)g (condition)f(3)h(ab)q(o)o(v)o(e)f(implies)g(that)h Fl(z)r Fm(C)18 b Fs([)r([)p Fl(z)r Fs(])q(])1181 1439 y Fi(\()p Fh(M)1235 1444 y Fb(n)1259 1439 y Fi(\))1290 1424 y Fs(is)13 b(closed)f(for)h(comp)q(osition.)57 1493 y(Sho)o(w)i(that)i(if)g Fl(f)i Fk(2)14 b Fl(z)r Fm(C)k Fs([)s([)p Fl(z)r Fs(]])583 1508 y Fi(\()p Fh(N)631 1513 y Fb(n)654 1508 y Fi(\))672 1493 y Fs(,)f Fl(f)727 1500 y Fi(1)763 1493 y Fs(=)d Fl(e)839 1475 y Fi(2)p Fh(\031)q(i\013)943 1493 y Fs(and)h Fl(\013)i Fs(v)o(eri\014es)472 1682 y(lim)8 b(sup)481 1722 y Fh(n)p Fj(!)p Fi(+)p Fj(1)632 1582 y Fe(0)632 1672 y(@)676 1616 y Fh(k)q Fi(\()p Fh(n)p Fi(\))679 1635 y Fe(X)679 1742 y Fh(k)q Fi(=0)768 1649 y Fs(log)h Fl(q)863 1656 y Fh(k)q Fi(+1)p 768 1671 170 2 v 830 1717 a Fl(q)852 1724 y Fh(k)955 1682 y Fk(\000)1013 1649 y Fs(1)p 1011 1671 30 2 v 1011 1717 a Fl(n)1055 1682 y Fs(log)1133 1649 y Fl(M)1181 1656 y Fh(n)p 1133 1671 76 2 v 1137 1717 a Fl(N)1177 1724 y Fh(n)1215 1582 y Fe(1)1215 1672 y(A)1272 1682 y Fl(<)14 b Fs(+)p Fk(1)57 1860 y Fs(where)i Fl(k)r Fs(\()p Fl(n)p Fs(\))h(is)f(de\014ned)g(b)o(y)g(the)h(condition) f Fl(q)932 1869 y Fh(k)q Fi(\()p Fh(n)p Fi(\))1027 1860 y Fk(\024)e Fl(n)g(<)g(q)1199 1869 y Fh(k)q Fi(\()p Fh(n)p Fi(\)+1)1329 1860 y Fs(,)j(then)g(the)g(linearization)57 1930 y Fl(h)86 1937 y Fh(f)134 1930 y Fk(2)22 b Fl(z)r Fm(C)c Fs([)s([)p Fl(z)r Fs(]])340 1945 y Fi(\()p Fh(M)394 1950 y Fb(n)418 1945 y Fi(\))436 1930 y Fs(.)37 b(\(W)l(e)23 b(of)f(course)f(assume)f(that)i(the)g(sequence)g(\()p Fl(N)1470 1937 y Fh(n)1497 1930 y Fs(\))1516 1937 y Fh(n)p Fj(\025)p Fi(0)1616 1930 y Fs(is)g(asymp-)57 2000 y(totically)e(b)q (ounded)f(b)o(y)g(the)i(sequence)e(\()p Fl(M)900 2007 y Fh(n)928 2000 y Fs(\),)i(i.e.)32 b Fl(M)1126 2007 y Fh(n)1173 2000 y Fk(\025)19 b Fl(N)1271 2007 y Fh(n)1318 2000 y Fs(for)h(all)f(su\016cien)o(tly)g(large)57 2069 y Fl(n)p Fs(\).)57 2244 y Fo(5.2)g(Y)-5 b(o)r(ccoz's)17 b(Theorem)57 2349 y Fs(The)f(main)g(result)f(of)i(Y)l(o)q(ccoz)h(can)e (b)q(e)h(v)o(ery)f(simply)g(stated)g(as)378 2477 y Fk(Y)i Fs(=)c Fk(f)p Fl(\013)f Fk(2)h Fm(R)8 b Fk(n)j Fm(Q)k Fk(j)22 b Fl(B)r Fs(\()p Fl(\013)p Fs(\))15 b Fl(<)e Fs(+)p Fk(1g)h Fs(=)f(Brjuno)j(n)o(um)o(b)q(ers)11 b Fl(;)57 2605 y Fs(but)16 b(he)g(pro)o(v)o(es)f(m)o(uc)o(h)g(more)h (than)g(the)h(ab)q(o)o(v)o(e)f(:)918 2770 y(33)p eop %%Page: 34 35 34 34 bop 57 192 a Fr(Theorem)17 b(5.9)68 261 y Fd(\(a\))25 b(If)17 b Fl(B)r Fs(\()p Fl(\013)p Fs(\))e(=)f(+)p Fk(1)i Fd(there)g(exists)h(a)f(non{linearizable)e(germ)i Fl(f)k Fk(2)14 b Fl(S)1410 271 y Fh(e)1429 261 y Fc(2)p Fb(\031)q(i\013)g Fd(;)65 331 y(\(b\))25 b(If)17 b Fl(B)r Fs(\()p Fl(\013)p Fs(\))e Fl(<)f Fs(+)p Fk(1)i Fd(then)g Fl(r)q Fs(\()p Fl(\013)p Fs(\))g Fl(>)d Fs(0)k Fd(and)738 419 y Fk(j)8 b Fs(log)h Fl(r)q Fs(\()p Fl(\013)p Fs(\))k(+)d Fl(B)r Fs(\()p Fl(\013)p Fs(\))p Fk(j)15 b(\024)f Fl(C)j(;)454 b Fs(\(5)p Fl(:)p Fs(11\))156 508 y Fd(where)16 b Fl(C)k Fd(is)c(a)h(univ)o(ersal)d(constan)o(t)i(\(i.e.)23 b(indep)q(enden)o(t) 15 b(of)i Fl(\013)p Fd(\))10 b(;)71 577 y(\(c\))25 b(F)l(or)16 b(all)g Fl(")e(>)g Fs(0)i Fd(there)g(exists)h Fl(C)747 584 y Fh(")782 577 y Fl(>)c Fs(0)k Fd(suc)o(h)e(that)i(for)f(all)h (Brjuno)e(n)o(um)o(b)q(ers)g Fl(\013)h Fd(one)g(has)460 666 y Fk(\000)p Fl(B)r Fs(\()p Fl(\013)p Fs(\))c Fk(\000)f Fl(C)17 b Fk(\024)d Fs(log)8 b Fl(r)q Fs(\()p Fl(P)923 675 y Fh(e)942 665 y Fc(2)p Fb(\031)q(i\013)1023 666 y Fs(\))14 b Fk(\024)g(\000)p Fs(\(1)d Fk(\000)g Fl(")p Fs(\))p Fl(B)r Fs(\()p Fl(\013)p Fs(\))h(+)f Fl(C)1503 673 y Fh(")1701 666 y Fs(\(5)p Fl(:)p Fs(12\))57 754 y Fd(where)16 b Fl(C)k Fd(is)c(a)g(univ)o(ersal)f(constan)o(t)h(\(i.e.) 22 b(indep)q(enden)o(t)15 b(of)i Fl(\013)g Fd(and)e Fl(")p Fd(.)57 880 y Fs(The)i(remark)m(able)f(consequence)i(of)f(\(5.11\))h (and)f(\(5.12\))h(is)f(that)h(the)g(Brjuno)f(function)g(not)57 950 y(only)22 b(iden)o(ti\014es)f(the)i(set)g Fk(Y)k Fs(but)22 b(also)g(giv)o(es)g(a)h(rather)f(precise)g(estimate)g(of)h (the)g(size)f(of)57 1019 y(the)h(Siegel)g(disks.)41 b(When)23 b Fl(\013)h Fs(is)f(not)g(a)g(Brjuno)g(n)o(um)o(b)q(er)e(the)i(problem) f(of)h(a)h(complete)57 1089 y(classi\014cation)13 b(of)i(the)f (conjugacy)h(classes)e(of)i(germs)f(in)g Fl(G)1164 1099 y Fh(e)1183 1089 y Fc(2)p Fb(\031)q(i\013)19 b Fs(is)14 b(op)q(en)g(and)g(quite)h(di\016cult)57 1159 y(\(p)q(erhaps)g (unreasonable\))g(as)h(the)h(follo)o(wing)e(result)h(of)h(Y)l(o)q(ccoz) g(sho)o(ws)e(:)57 1285 y Fr(Theorem)k(5.10)27 b Fd(Let)19 b Fl(\013)e Fk(2)f Fm(R)9 b Fk(n)k Fm(Q)p Fd(,)19 b Fl(B)r Fs(\()p Fl(\013)p Fs(\))e(=)g(+)p Fk(1)p Fd(.)26 b(There)17 b(exists)i(a)f(set)g(with)g(the)h(p)q(o)o(w)o(er)57 1355 y(of)e(the)h(con)o(tin)o(uum)d(of)j(conjugacy)f(classes)g(of)h(germs)e (of)i Fl(G)1199 1364 y Fh(e)1218 1354 y Fc(2)p Fb(\031)q(i\013)1297 1355 y Fd(,)g(eac)o(h)f(of)h(whic)o(h)e(do)q(es)h(not)57 1424 y(con)o(tain)e(an)i(en)o(tire)f(function.)57 1550 y Fs(The)24 b(pro)q(of)h(of)f(the)h(Theorem)f(of)h(Y)l(o)q(ccoz)g (\(5.9)g(ab)q(o)o(v)o(e\))g(uses)f(a)g(metho)q(d,)i(in)o(v)o(en)o(ted)e (b)o(y)57 1620 y(Y)l(o)q(ccoz)f(himself,)f(kno)o(wn)g(as)g Fp(ge)m(ometric)h(r)m(enormalization)p Fs(.)41 b(Roughly)21 b(sp)q(eaking)h(it)h(is)e(a)57 1690 y(quan)o(titativ)o(e)16 b(v)o(ersion)g(of)g(the)h(top)q(ological)g(construction)e(of)i (Douady{Gh)o(ys)e(describ)q(ed)h(in)57 1759 y(Chapter)k(4)h(and)g(whic) o(h)f(sho)o(ws)g(that)i(the)f(set)h Fk(Y)j Fs(is)c(SL)8 b(\(2)p Fl(;)g Fm(Z)-10 b Fs(\){in)n(v)m(arian)n(t.)34 b(Whereas)20 b(the)57 1829 y(construction)12 b(of)i(non{linearizable)e (germs)g Fl(f)20 b Fk(2)14 b Fl(G)1030 1839 y Fh(e)1049 1829 y Fc(2)p Fb(\031)q(i\013)19 b Fs(when)13 b Fl(B)r Fs(\()p Fl(\013)p Fs(\))i(=)e(+)p Fk(1)h Fs(and)f(Y)l(o)q(ccoz's)57 1899 y(upp)q(er)i(b)q(ound)720 1969 y(log)9 b Fl(r)q Fs(\()p Fl(\013)p Fs(\))15 b Fk(\024)f Fl(C)g Fk(\000)d Fl(B)r Fs(\()p Fl(\013)p Fs(\))57 2055 y(go)17 b(far)g(b)q(ey)o(ond)h (the)f(scop)q(e)h(of)g(these)f(lectures,)h(it)g(is)f(not)g(to)q(o)i (di\016cult)d(to)i(giv)o(e)g(an)f(idea)g(of)57 2125 y(ho)o(w)e(Y)l(o)q (ccoz)j(pro)o(v)o(es)d(Theorem)g(5.1,)h(i.e.)22 b(the)17 b(lo)o(w)o(er)e(b)q(ound)687 2213 y(log)9 b Fl(r)q Fs(\()p Fl(\013)p Fs(\))15 b Fk(\025)f(\000)p Fl(C)g Fk(\000)c Fl(B)r Fs(\()p Fl(\013)p Fs(\))15 b Fl(:)57 2301 y Fs(Let)h Fl(f)k Fk(2)14 b Fl(S)267 2311 y Fh(e)286 2301 y Fc(2)p Fb(\031)q(i\013)20 b Fs(and)15 b(let)i Fl(E)25 b Fs(:)13 b Fm(H)24 b Fk(!)13 b Fm(D)787 2283 y Fj(\003)829 2301 y Fs(b)q(e)k(the)f(exp)q(onen)o(tial)f(map)g Fl(E)s Fs(\()p Fl(z)r Fs(\))g(=)f Fl(e)1552 2283 y Fi(2)p Fh(\031)q(iz)1633 2301 y Fs(.)22 b(Then)16 b Fl(f)57 2371 y Fs(lifts)g(to)h(a)g(map)e Fl(F)29 b Fs(:)14 b Fm(H)24 b Fk(!)13 b Fm(C)29 b Fs(suc)o(h)15 b(that)167 2459 y Fl(E)f Fk(\016)d Fl(F)21 b Fs(=)13 b Fl(f)k Fk(\016)11 b Fl(E)17 b(;)1197 b Fs(\(5)p Fl(:)p Fs(13\))166 2544 y Fl(F)29 b Fs(:)13 b Fm(H)24 b Fk(!)14 b Fm(C)25 b Fs(is)16 b(univ)m(alen)o(t)e Fl(;)1003 b Fs(\(5)p Fl(:)p Fs(14\))190 2629 y Fl(F)7 b Fs(\()p Fl(z)r Fs(\))15 b(=)e Fl(z)h Fs(+)d Fl(\013)g Fs(+)g Fl(')p Fs(\()p Fl(z)r Fs(\))j(where)8 b Fl(')13 b Fs(is)8 b Fm(Z)m Fk(\000)j(\000)p Fs(p)q(erio)q(dic)16 b(and)82 b(lim)1294 2661 y Fj(=)p Fh(m)7 b(z)q Fj(!)p Fi(+)p Fj(1)1504 2629 y Fl(')p Fs(\()p Fl(z)r Fs(\))14 b(=)g(0)g Fl(:)-19 b Fs(\(5)p Fl(:)p Fs(15\))918 2770 y(34)p eop %%Page: 35 36 35 35 bop 57 192 a Fs(W)l(e)13 b(will)h(denote)f Fl(S)s Fs(\()p Fl(\013)p Fs(\))h(the)g(space)f(of)h(univ)m(alen)o(t)f (functions)f Fl(F)21 b Fs(v)o(erifying)13 b(\(5.13\),)h(\(5.14\))g(and) 57 261 y(\(5.15\).)57 366 y Fr(Exercise)24 b(5.11)18 b Fs(Sho)o(w)h(that)h Fl(S)s Fs(\()p Fl(\013)p Fs(\))g(is)g(compact)f (and)g(that)i(it)f(is)f(the)h(univ)o(ersal)e(co)o(v)o(er)h(of)57 436 y Fl(S)88 446 y Fh(e)107 436 y Fc(2)p Fb(\031)q(i\013)t Fs(.)57 541 y Fr(Exercise)c(5.12)d Fs(Sho)o(w)g(that)h(if)g Fl(f)19 b Fk(2)14 b Fl(S)789 548 y Fi(1)824 541 y Fs(then)f(one)f(has)g (the)h(follo)o(wing)f(distorsion)f(estimate)i(:)551 609 y Fe(\014)551 639 y(\014)551 669 y(\014)551 699 y(\014)568 681 y Fl(z)599 648 y(f)628 630 y Fj(0)643 648 y Fs(\()p Fl(z)r Fs(\))p 599 670 108 2 v 606 716 a Fl(f)5 b Fs(\()p Fl(z)r Fs(\))724 681 y Fk(\000)11 b Fs(1)799 609 y Fe(\014)799 639 y(\014)799 669 y(\014)799 699 y(\014)829 681 y Fk(\024)918 648 y Fs(2)p Fk(j)p Fl(z)r Fk(j)p 888 670 139 2 v 888 716 a Fs(1)g Fk(\000)g(j)p Fl(z)r Fk(j)1047 681 y Fs(for)16 b(all)8 b Fl(z)16 b Fk(2)e Fm(D)25 b Fl(:)367 b Fs(\(5)p Fl(:)p Fs(16\))57 857 y Fr(Exercise)23 b(5.13)18 b Fs(Use)h(the)g (result)f(of)i(the)f(previous)f(exercise)h(to)g(sho)o(w)f(that)i(if)f Fl(')g Fs(is)f(as)h(in)57 927 y(\(5.15\))e(and)e Fl(z)i Fk(2)d Fm(H)26 b Fs(then)522 1068 y Fk(j)p Fl(')569 1048 y Fj(0)582 1068 y Fs(\()p Fl(z)r Fs(\))p Fk(j)15 b(\024)759 1035 y Fs(2)8 b(exp\()p Fk(\000)p Fs(2)p Fl(\031)j Fk(=)p Fl(m)d(z)r Fs(\))p 733 1057 417 2 v 733 1103 a(1)j Fk(\000)g Fs(exp\()p Fk(\000)p Fs(2)p Fl(\031)f Fk(=)p Fl(m)e(z)r Fs(\))1169 1068 y Fl(;)518 b Fs(\(5)p Fl(:)p Fs(17\))536 1201 y Fk(j)p Fl(')p Fs(\()p Fl(z)r Fs(\))p Fk(j)14 b(\024)g(\000)774 1167 y Fs(1)p 772 1189 31 2 v 772 1235 a Fl(\031)816 1201 y Fs(log\(1)e Fk(\000)f Fs(exp\()p Fk(\000)p Fs(2)p Fl(\031)f Fk(=)p Fl(m)e(z)r Fs(\)\))15 b Fl(:)338 b Fs(\(5)p Fl(:)p Fs(18\))57 1366 y(Let)20 b Fl(r)i(>)d Fs(0,)i Fm(H)347 1373 y Fh(r)392 1366 y Fs(=)e Fm(H)k Fs(+)13 b Fl(ir)q Fs(.)34 b(It)20 b(is)g(clear)g(that)g(if)g Fl(F)27 b Fk(2)20 b Fl(S)s Fs(\()p Fl(\013)p Fs(\))g(and)g Fl(r)i Fs(is)d(su\016cien)o(tly)g(large)57 1436 y(then)g Fl(F)26 b Fs(is)19 b(v)o(ery)h(close)f(to)h(the)f(translation)g Fl(z)i Fk(7!)d Fl(z)d Fs(+)e Fl(\013)20 b Fs(for)f Fl(z)i Fk(2)e Fm(H)1391 1443 y Fh(r)1416 1436 y Fs(.)31 b(Indeed)19 b(using)f(the)57 1505 y(compactness)d(of)i Fl(S)s Fs(\()p Fl(\013)p Fs(\))g(and)e(Exercise)i(5.13)f(one)g(can)g(pro)o(v)o(e)g (the)h(follo)o(wing)e(:)57 1610 y Fr(Exercise)j(5.14)c Fs(Let)i Fl(\013)e Fk(6)p Fs(=)g(0.)21 b(Sho)o(w)14 b(that)i(there)f (exists)g(a)h(univ)o(ersal)d(constan)o(t)i Fl(c)1617 1617 y Fi(0)1653 1610 y Fl(>)e Fs(0)i(\(i.e.)57 1680 y(indep)q(enden)o(t)g(of)i Fl(\013)p Fs(\))g(suc)o(h)e(that)i(for)f (all)g Fl(F)21 b Fk(2)14 b Fl(S)s Fs(\()p Fl(\013)p Fs(\))j(and)f(for)g (all)g Fl(z)h Fk(2)d Fm(H)1414 1689 y Fh(t)p Fi(\()p Fh(\013)p Fi(\))1509 1680 y Fs(where)682 1822 y Fl(t)p Fs(\()p Fl(\013)p Fs(\))h(=)858 1788 y(1)p 843 1810 56 2 v 843 1856 a(2)p Fl(\031)912 1822 y Fs(log)9 b Fl(\013)1017 1801 y Fj(\000)p Fi(1)1082 1822 y Fs(+)i Fl(c)1154 1829 y Fi(0)1189 1822 y Fl(;)498 b Fs(\(5)p Fl(:)p Fs(19\))57 1945 y(one)16 b(has)719 2015 y Fk(j)p Fl(F)7 b Fs(\()p Fl(z)r Fs(\))k Fk(\000)g Fl(z)j Fk(\000)d Fl(\013)p Fk(j)i(\024)1101 1981 y Fl(\013)p 1101 2003 33 2 v 1105 2049 a Fs(4)1153 2015 y Fl(:)534 b Fs(\(5)p Fl(:)p Fs(20\))57 2127 y([Hin)o(t)26 b(:)41 b(Let)27 b Fl(')p Fs(\()p Fl(z)r Fs(\))k(=)f Fl(F)7 b Fs(\()p Fl(z)r Fs(\))19 b Fk(\000)e Fl(z)j Fk(\000)d Fl(\013)31 b Fs(=)952 2089 y Fe(P)1004 2102 y Fj(1)1004 2142 y Fh(n)p Fi(=1)1090 2127 y Fl(')1123 2134 y Fh(n)1150 2127 y Fl(e)1173 2109 y Fi(2)p Fh(\031)q(inz)1279 2127 y Fs(.)51 b(If)27 b Fk(=)p Fl(m)8 b(z)32 b(>)e(t)p Fs(\()p Fl(\013)p Fs(\))d(then)57 2197 y Fk(j)p Fl(F)7 b Fs(\()p Fl(z)r Fs(\))12 b Fk(\000)e Fl(\013)i Fk(\000)e Fl(z)r Fk(j)k(\024)433 2159 y Fe(P)486 2171 y Fj(1)486 2211 y Fh(n)p Fi(=1)571 2197 y Fk(j)p Fl(')618 2204 y Fh(n)645 2197 y Fk(j)p Fl(\013)691 2178 y Fh(n)718 2197 y Fl(e)741 2178 y Fj(\000)p Fi(2)p Fh(\031)q(nc)859 2183 y Fc(0)898 2197 y Fs(th)o(us)h Fl(:)8 b(:)g(:)p Fs(.])57 2302 y(Giv)o(en)16 b Fl(F)7 b Fs(,)17 b(the)h(lo)o(w)o(est)e(admissible)e(v)m(alue)k Fl(t)p Fs(\()p Fl(F)q(;)8 b(\013)p Fs(\))18 b(of)f Fl(t)p Fs(\()p Fl(\013)p Fs(\))h(suc)o(h)e(that)h(\(5.20\))g(holds)f(for)h (all)57 2371 y Fl(z)k Fk(2)e Fm(H)188 2380 y Fh(t)p Fi(\()p Fh(F)q(;\013)p Fi(\))324 2371 y Fs(represen)o(ts)e(the)j(heigh)o(t)f (in)g(the)h(upp)q(er)e(half)h(plane)g Fm(H)29 b Fs(at)20 b(whic)o(h)f(the)g(strong)57 2441 y(nonlinearities)14 b(of)j Fl(F)24 b Fs(manifest)16 b(themselv)o(es.)22 b(When)16 b Fk(=)p Fl(m)8 b(z)17 b(>)d(t)p Fs(\()p Fl(F)q(;)8 b(\013)p Fs(\),)18 b Fl(F)24 b Fs(is)16 b(v)o(ery)h(close)f(to)57 2511 y(the)h(translation)f Fl(z)h Fk(7!)d Fl(z)g Fs(+)d Fl(\013)p Fs(.)23 b(An)18 b(example)e(of)h(strong)f(nonlinearit)o(y)g (is)g(of)h(course)g(a)g(\014xed)57 2581 y(p)q(oin)o(t)g(:)25 b(if)19 b Fl(F)7 b Fs(\()p Fl(z)r Fs(\))17 b(=)f Fl(z)f Fs(+)d Fl(\013)g Fs(+)658 2561 y Fi(1)p 639 2569 59 2 v 639 2598 a(2)p Fh(\031)q(i)703 2581 y Fl(e)726 2563 y Fi(2)p Fh(\031)q(iz)808 2581 y Fs(,)18 b Fl(\013)e(>)h Fs(0,)h(then)g Fl(z)h Fs(=)d Fk(\000)1258 2561 y Fi(1)p 1258 2569 20 2 v 1258 2598 a(4)1296 2581 y Fs(+)1368 2561 y Fh(i)p 1352 2569 45 2 v 1352 2598 a Fi(2)p Fh(\031)1411 2581 y Fs(log\(2)p Fl(\031)r(\013)p Fs(\))1600 2563 y Fj(\000)p Fi(1)1673 2581 y Fs(is)i(\014xed)57 2650 y(and)e Fl(t)p Fs(\()p Fl(F)q(;)8 b(\013)p Fs(\))15 b Fk(\025)383 2631 y Fi(1)p 370 2639 V 370 2668 a(2)p Fh(\031)429 2650 y Fs(log\(2)p Fl(\031)r(\013)p Fs(\))618 2632 y Fj(\000)p Fi(1)673 2650 y Fs(.)918 2770 y(35)p eop %%Page: 36 37 36 36 bop 156 192 a Fs(The)20 b(estimates)g(\(5.19\))g(and)f(\(5.20\))i (are)e(the)h(fundamen)o(tal)f(ingredien)o(t)f(of)i(Y)l(o)q(ccoz's)57 261 y(pro)q(of)f(of)h(the)g(lo)o(w)o(er)e(b)q(ound)g(\(5.2\))i (together)g(with)g(Prop)q(osition)e(2.15)h(and)g(the)h(follo)o(wing)57 331 y(elemen)o(tary)15 b(prop)q(erties)h(of)h(the)f(conformal)g (capacit)o(y)l(.)57 436 y Fr(Exercise)g(5.15)d Fs(Let)h Fl(U)19 b Fk(\032)14 b Fl(C)i Fs(b)q(e)e(a)g(simply)e(connected)i(op)q (en)f(set,)h Fl(U)19 b Fk(6)p Fs(=)14 b Fm(C)9 b Fs(,)17 b(and)c(let)h Fl(z)1692 443 y Fi(0)1728 436 y Fk(2)g Fl(U)5 b Fs(.)57 506 y(Let)15 b Fl(d)f Fs(b)q(e)g(the)h(distance)e(of)h Fl(z)601 513 y Fi(0)638 506 y Fs(from)g(the)g(complemen)o(t)f(of)h Fl(U)20 b Fs(in)14 b Fm(C)26 b Fs(and)13 b(let)i Fl(C)t Fs(\()p Fl(U;)8 b(z)1629 513 y Fi(0)1651 506 y Fs(\))14 b(denote)57 576 y(the)i(conformal)g(capacit)o(y)g(of)h Fl(U)22 b Fs(w.r.t.)f Fl(z)844 583 y Fi(0)867 576 y Fs(.)h(Then)734 680 y Fl(d)14 b Fk(\024)f Fl(C)t Fs(\()p Fl(U;)8 b(z)964 687 y Fi(0)987 680 y Fs(\))14 b Fk(\024)f Fs(4)p Fl(d)h(:)550 b Fs(\(5)p Fl(:)p Fs(21\))57 821 y(As)11 b(in)g(the)h(pro)q(of)f(of)h (Douady{Gh)o(ys)e(theorem)h(w)o(e)g(can)g(no)o(w)g(construct)g(the)g (\014rst)g(return)f(map)57 890 y(in)i(the)g(strip)f Fl(B)k Fs(delimited)c(b)o(y)h Fl(l)j Fs(=)e([)p Fl(it)p Fs(\()p Fl(\013)p Fs(\))p Fl(;)8 b Fs(+)p Fl(i)p Fk(1)p Fs([,)14 b Fl(F)7 b Fs(\()p Fl(l)q Fs(\))13 b(and)e(the)i(segmen)o(t)e([)p Fl(it)p Fs(\()p Fl(\013)p Fs(\))p Fl(;)d(F)f Fs(\()p Fl(it)p Fs(\()p Fl(\013)p Fs(\)\)].)57 960 y(Giv)o(en)18 b Fl(z)j Fs(in)d Fl(B)j Fs(w)o(e)e(can)f(iterate)h Fl(F)26 b Fs(un)o(til)17 b Fk(<)p Fl(e)9 b(F)979 942 y Fh(n)1006 960 y Fs(\()p Fl(z)r Fs(\))18 b Fl(>)f Fs(1.)29 b(If)18 b Fk(=)p Fl(m)8 b(z)20 b Fk(\025)d Fl(t)p Fs(\()p Fl(\013)p Fs(\))c(+)g Fl(c)18 b Fs(for)g(some)57 1030 y Fl(c)h(>)g Fs(0)g(then)h Fl(z)342 1012 y Fj(0)376 1030 y Fs(=)f Fl(F)473 1012 y Fh(n)500 1030 y Fs(\()p Fl(z)r Fs(\))14 b Fk(\000)f Fs(1)20 b Fk(2)f Fl(B)j Fs(and)d Fl(z)j Fk(7!)d Fl(z)1025 1012 y Fj(0)1059 1030 y Fs(is)h(the)g(\014rst)f(return)g(map) g(in)g(the)h(strip)57 1100 y Fl(B)r Fs(.)30 b(Glueing)19 b Fl(l)h Fs(and)f Fl(F)7 b Fs(\()p Fl(l)q Fs(\))20 b(b)o(y)f Fl(F)26 b Fs(one)19 b(obtains)g(a)g(Riemann)f(surface)g Fl(S)k Fs(corresp)q(onding)17 b(to)57 1169 y(in)o(t)7 b Fl(B)24 b Fs(and)d(biholomorphic)e(to)j Fm(D)719 1151 y Fj(\003)745 1169 y Fs(.)37 b(This)21 b(induces)g(a)g(map)g Fl(g)j Fk(2)f Fl(S)1395 1181 y Fh(e)1414 1171 y Fc(2)p Fb(\031)q(i=\013)1533 1169 y Fs(whic)o(h)d(lifts)i(to)57 1239 y Fl(G)14 b Fk(2)g Fl(S)s Fs(\()p Fl(\013)242 1221 y Fj(\000)p Fi(1)295 1239 y Fs(\).)22 b(One)17 b(can)f(then)g(sho)o(w)g (the)h(follo)o(wing)e(\(see)i([Y)l(o2],)f(pp.)21 b(32{33\))57 1369 y Fr(Prop)r(osition)13 b(5.16)28 b Fd(Let)12 b Fl(\013)i Fk(2)g Fs(\(0)p Fl(;)8 b Fs(1\))p Fd(,)14 b Fl(F)21 b Fk(2)14 b Fl(S)s Fs(\()p Fl(\013)p Fs(\))e Fd(and)f Fl(t)p Fs(\()p Fl(\013)p Fs(\))k Fl(>)e Fs(0)f Fd(suc)o(h)f(that)i(if)f Fk(=)p Fl(m)c(z)16 b Fk(\025)e Fl(t)p Fs(\()p Fl(\013)p Fs(\))57 1438 y Fd(then)d Fk(j)p Fl(F)c Fs(\()p Fl(z)r Fs(\))q Fk(\000)q Fl(z)s Fk(\000)q Fl(\013)p Fk(j)14 b(\024)g Fl(\013=)p Fs(4)p Fd(.)20 b(There)11 b(exists)g Fl(G)j Fk(2)g Fl(S)s Fs(\()p Fl(\013)1073 1420 y Fj(\000)p Fi(1)1127 1438 y Fs(\))e Fd(suc)o(h)e(that)i(if)f Fl(z)17 b Fk(2)d Fm(H)6 b Fd(,)16 b Fk(=)p Fl(m)8 b(z)16 b Fk(\025)e Fl(t)p Fs(\()p Fl(\013)p Fs(\))57 1508 y Fd(and)j Fl(F)194 1490 y Fh(i)211 1508 y Fs(\()p Fl(z)r Fs(\))h Fk(2)f Fm(H)28 b Fd(for)18 b(all)g Fl(i)e Fs(=)h(0)p Fl(;)8 b Fs(1)p Fl(;)g(:)g(:)g(:)g(;)g(n)13 b Fk(\000)f Fs(1)18 b Fd(but)g Fl(F)1088 1490 y Fh(n)1115 1508 y Fs(\()p Fl(z)r Fs(\))g Fk(62)f Fm(H)28 b Fd(then)18 b(there)g(exists)g Fl(z)1711 1490 y Fj(0)1742 1508 y Fk(2)f Fm(C)57 1578 y Fd(suc)o(h)e(that)93 1648 y(1.)24 b Fk(=)p Fl(m)8 b(z)269 1629 y Fj(0)298 1648 y Fk(\025)13 b Fl(\013)382 1629 y Fj(\000)p Fi(1)436 1648 y Fs(\()p Fk(=)p Fl(m)8 b(z)13 b Fk(\000)e Fl(t)p Fs(\()p Fl(\013)p Fs(\))h Fk(\000)f Fl(c)801 1655 y Fi(1)823 1648 y Fs(\))p Fd(,)17 b(where)f Fl(c)1039 1655 y Fi(1)1074 1648 y Fl(>)e Fs(0)i Fd(is)h(a)f(univ)o (ersal)f(constan)o(t)9 b(;)93 1717 y(2.)24 b(There)16 b(exists)h(an)f(in)o(teger)g Fl(m)g Fd(suc)o(h)g(that)h Fs(0)c Fk(\024)h Fl(m)f(<)h(n)i Fd(and)g Fl(G)1336 1699 y Fh(m)1374 1717 y Fs(\()p Fl(z)1418 1699 y Fj(0)1433 1717 y Fs(\))e Fk(62)g Fm(H)7 b Fd(.)57 1847 y Fs(F)l(rom)18 b(this)h(Prop)q(osition)g(one)h(can)f(conclude)h(the)g(pro)q(of)f(as)h (follo)o(ws.)31 b(Let)20 b(us)g(recall)f(that)57 1917 y Fl(K)99 1924 y Fh(f)149 1917 y Fs(=)24 b Fk(\\)245 1924 y Fh(n)p Fj(\025)p Fi(0)324 1917 y Fl(f)353 1898 y Fj(\000)p Fh(n)412 1917 y Fs(\()p Fm(D)8 b Fs(\))26 b(is)d(the)g(maximal)f(compact)g Fl(f)5 b Fs({in)o(v)m(arian)o(t)22 b(set)h(con)o(taining)f(0.)41 b(Let)57 1986 y Fl(F)20 b Fk(2)15 b Fl(S)s Fs(\()p Fl(\013)p Fs(\))h(b)q(e)g(the)g(lift)h(of)f Fl(f)k Fk(2)14 b Fl(S)686 1996 y Fh(e)705 1986 y Fc(2)p Fb(\031)q(i\013)20 b Fs(and)15 b(let)i Fl(K)1010 1993 y Fh(F)1057 1986 y Fk(\032)c Fm(C)28 b Fs(b)q(e)17 b(de\014ned)e(as)g (the)i(co)o(v)o(er)e(of)h Fl(K)1772 1993 y Fh(f)1815 1986 y Fs(:)57 2056 y Fl(K)99 2063 y Fh(F)146 2056 y Fs(=)e Fl(E)239 2038 y Fj(\000)p Fi(1)292 2056 y Fs(\()p Fl(K)353 2063 y Fh(f)379 2056 y Fs(\).)23 b(It)16 b(is)h(immediate)e (to)i(c)o(hec)o(k)f(that)306 2176 y Fl(d)332 2183 y Fh(F)379 2176 y Fs(=)d(sup)o Fk(f=)p Fl(m)8 b(z)25 b Fk(j)d Fl(z)16 b Fk(2)e Fm(C)23 b Fk(n)11 b Fl(K)914 2183 y Fh(F)948 2176 y Fk(g)i Fs(=)h Fk(\000)1099 2142 y Fs(1)p 1084 2165 56 2 v 1084 2210 a(2)p Fl(\031)1153 2176 y Fs(log)9 b(dist)e(\(0)p Fl(;)h Fm(C)24 b Fk(n)11 b Fl(K)1506 2183 y Fh(f)1532 2176 y Fs(\))j Fl(:)57 2288 y Fs(Th)o(us)h(b)o(y)h (\(5.21\))h(one)f(gets)481 2392 y(exp\()p Fk(\000)p Fs(2)p Fl(\031)r(d)696 2399 y Fh(F)730 2392 y Fs(\))e Fk(\024)g Fl(C)t Fs(\()p Fl(K)917 2399 y Fh(f)942 2392 y Fl(;)8 b Fs(0\))14 b Fk(\024)g Fs(4)8 b(exp\()p Fk(\000)p Fs(2)p Fl(\031)r(d)1323 2399 y Fh(F)1357 2392 y Fs(\))14 b Fl(:)57 2497 y Fs(Theorem)h(5.1)h(is)g(therefore)h(equiv)m(alen)o(t)f(to)h(the) g(lo)o(w)o(er)e(b)q(ound)699 2612 y(sup)666 2657 y Fh(F)5 b Fj(2)p Fh(S)r Fi(\()p Fh(\013)p Fi(\))815 2612 y Fl(d)841 2619 y Fh(F)888 2612 y Fk(\024)962 2579 y Fs(1)p 947 2601 V 947 2647 a(2)p Fl(\031)1008 2612 y(B)r Fs(\()p Fl(\013)p Fs(\))12 b(+)f Fl(C)485 b Fs(\(5)p Fl(:)p Fs(22\))918 2770 y(36)p eop %%Page: 37 38 37 37 bop 57 192 a Fs(for)16 b(some)g(univ)o(ersal)e(constan)o(t)i Fl(C)h(>)d Fs(0.)156 261 y(Assume)i(that)h(\(5.22\))f(is)g(not)h(true)f (and)g(that)g(there)h(exist)g Fl(\013)c Fk(2)h Fm(R)8 b Fk(n)j Fm(Q)g Fk(\\)h Fs(\(0)p Fl(;)c Fs(1)p Fl(=)p Fs(2\))16 b(with)57 331 y Fl(B)r Fs(\()p Fl(\013)p Fs(\))f Fl(<)e Fs(+)p Fk(1)j Fs(,)h Fl(F)k Fk(2)14 b Fl(S)s Fs(\()p Fl(\013)p Fs(\),)i Fl(z)g Fk(2)f Fm(H)26 b Fs(and)16 b Fl(n)d(>)h Fs(0)j(suc)o(h)e(that)647 451 y Fk(=)p Fl(m)8 b(F)774 431 y Fh(n)801 451 y Fs(\()p Fl(z)r Fs(\))15 b Fk(\024)f Fs(0)g Fl(;)752 554 y Fk(=)p Fl(m)8 b(z)16 b Fk(\025)953 521 y Fs(1)p 938 543 56 2 v 938 588 a(2)p Fl(\031)999 554 y(B)r Fs(\()p Fl(\013)p Fs(\))c(+)f Fl(C)17 b(:)156 688 y Fs(Let)j(us)f(c)o(ho)q(ose)g Fl(\013)p Fs(,)g Fl(F)27 b Fs(and)18 b Fl(z)k Fs(so)d(that)g Fl(n)h Fs(is)e(as)h(small)f(as)h(p)q(ossible.)29 b(By)20 b(Prop)q(osition)57 758 y(5.16,)c(if)g Fl(C)h(>)d(c)349 765 y Fi(0)371 758 y Fs(,)i(one)h(gets)382 876 y Fk(=)p Fl(m)8 b(z)495 855 y Fj(0)523 876 y Fk(\025)13 b Fl(\013)607 855 y Fj(\000)p Fi(1)661 876 y Fs([)p Fk(=)p Fl(m)8 b(z)13 b Fk(\000)e Fl(t)p Fs(\()p Fl(\013)p Fs(\))h Fk(\000)f Fl(c)1021 883 y Fi(1)1043 876 y Fs(])523 986 y Fk(\025)i Fl(\013)607 965 y Fj(\000)p Fi(1)669 915 y Fe(\024)717 952 y Fs(1)p 702 974 V 702 1020 a(2)p Fl(\031)763 986 y Fs(\()p Fl(B)r Fs(\()p Fl(\013)p Fs(\))f Fk(\000)f Fs(log)e Fl(\013)1059 965 y Fj(\000)p Fi(1)1112 986 y Fs(\))j(+)e Fl(C)15 b Fk(\000)10 b Fl(c)1314 993 y Fi(0)1347 986 y Fk(\000)h Fl(c)1419 993 y Fi(1)1441 915 y Fe(\025)1490 986 y Fl(:)57 1133 y Fs(By)17 b(the)g(functional)f(equation)g(of)h Fl(B)i Fs(one)d(gets)319 1273 y Fk(=)p Fl(m)8 b(z)432 1253 y Fj(0)461 1273 y Fk(\025)534 1240 y Fs(1)p 519 1262 V 519 1307 a(2)p Fl(\031)580 1273 y(B)r Fs(\()p Fl(\013)671 1253 y Fj(\000)p Fi(1)725 1273 y Fs(\))k(+)f Fl(\013)838 1253 y Fj(\000)p Fi(1)891 1273 y Fs([)p Fl(C)j Fk(\000)d Fl(c)1027 1280 y Fi(0)1060 1273 y Fk(\000)g Fl(c)1132 1280 y Fi(1)1154 1273 y Fs(])i Fk(\025)1255 1240 y Fs(1)p 1240 1262 V 1240 1307 a(2)p Fl(\031)1301 1273 y(B)r Fs(\()p Fl(\013)1392 1253 y Fj(\000)p Fi(1)1446 1273 y Fs(\))f(+)e Fl(C)57 1410 y Fs(pro)o(vided)17 b(that)j Fl(C)i Fk(\025)c Fs(2\()p Fl(c)556 1417 y Fi(0)592 1410 y Fs(+)12 b Fl(c)665 1417 y Fi(1)687 1410 y Fs(\).)31 b(But)20 b(Prop)q(osition)e(5.16)h(sho)o(ws)f(that)i(this)f(con)o (tradicts)57 1480 y(the)d(minimalit)o(y)f(of)i Fl(n)g Fs(and)e(w)o(e)i(m)o(ust)e(therefore)h(conclude)g(that)h(\(5.22\))g (holds.)204 b Fa(\003)57 1600 y Fs(A)19 b(nice)g(description)f(of)i (the)f(pro)q(of)g(of)g(the)h(upp)q(er)e(b)q(ound)g(log)9 b Fl(r)q Fs(\()p Fl(\013)p Fs(\))20 b Fk(\024)e Fl(C)e Fk(\000)d Fl(B)r Fs(\()p Fl(\013)p Fs(\))20 b(can)f(b)q(e)57 1670 y(found)c(in)i(the)f(Bourbaki)g(seminar)f(of)i(Ricardo)e(P)o (erez{Marco)g([PM1].)57 1845 y Fo(5.3)k(Some)h(Op)r(en)f(Problems)57 1950 y Fs(The)c(\014rst)g(op)q(en)h(problem)d(w)o(e)j(w)o(an)o(t)f(to)h (address)e(is)h(whether)g(or)g(not)h(the)g(in\014m)o(um)d(in)i(\(5.1\)) 57 2020 y(is)h(attained)g(b)o(y)g(the)h(quadratic)f(p)q(olynomial)g Fl(P)971 2027 y Fh(\025)997 2020 y Fs(\()p Fl(z)r Fs(\))f(=)e Fl(\025z)1190 1980 y Fe(\000)1213 2020 y Fs(1)e Fk(\000)1304 2000 y Fh(z)p 1304 2008 21 2 v 1304 2037 a Fi(2)1331 1980 y Fe(\001)1370 2020 y Fs(:)57 2125 y Fr(Question)29 b(5.17)24 b Fs(Do)q(es)i Fl(r)q Fs(\()p Fl(\013)p Fs(\))k(=)e(inf)818 2132 y Fh(f)t Fj(2)p Fh(S)892 2145 y Fb(e)909 2138 y Fc(2)p Fb(\031)q(i\013)1000 2125 y Fl(r)q Fs(\()p Fl(f)5 b Fs(\))31 b(=)d Fl(r)1210 2132 y Fi(2)1233 2125 y Fs(\()p Fl(e)1275 2107 y Fi(2)p Fh(\031)q(i\013)1362 2125 y Fs(\),)g(i.e.)48 b(the)26 b(radius)d(of)57 2195 y(con)o(v)o(ergence)15 b(of)i(the)g(quadratic)f(p)q(olynomial)8 b(?)57 2300 y(If)24 b(this)g(w)o(ere)f(true)h(then)g(the)h(v)o(ery)f(precise)f(b)q (ound)g(\(5.11\))i(w)o(ould)e(hold)g(also)g(for)h Fl(P)1764 2307 y Fh(\025)1815 2300 y Fs(:)57 2370 y(recalling)16 b(that)i Fl(r)388 2377 y Fi(2)411 2370 y Fs(\()p Fl(\025)p Fs(\))f(=)e Fk(j)p Fl(u)p Fs(\()p Fl(\025)p Fs(\))p Fk(j)p Fs(,)j(where)g Fl(u)23 b Fs(:)16 b Fm(D)27 b Fk(!)15 b Fm(C)30 b Fs(is)17 b(the)h(function)g(de\014ned)e(in)i(Section)57 2440 y(3.1,)e(one)g(can)g(ask)57 2545 y Fr(Question)j(5.18)d Fs(Do)q(es)g(the)h(function)f Fl(\013)e Fk(7!)g Fs(log)8 b Fk(j)p Fl(u)p Fs(\()p Fl(e)1087 2527 y Fi(2)p Fh(\031)q(i\013)1174 2545 y Fs(\))p Fk(j)j Fs(+)g Fl(B)r Fs(\()p Fl(\013)p Fs(\))k Fk(2)f Fl(L)1474 2527 y Fj(1)1516 2545 y Fs(\()p Fm(S)1566 2527 y Fi(1)1586 2545 y Fs(\))c(?)57 2650 y(Indeed)16 b(there)g(is)g(a)h(go)q(o)q(d)f(n)o(umerical)f(evidence)h(that)h(m)o (uc)o(h)e(more)h(could)g(b)q(e)h(true)f(:)918 2770 y(37)p eop %%Page: 38 39 38 38 bop 57 192 a Fr(Conjecture)16 b(5.19)d Fs(The)g(function)h Fl(\013)g Fk(7!)f Fs(log)c Fk(j)p Fl(u)p Fs(\()p Fl(e)1023 173 y Fi(2)p Fh(\031)q(i\013)1110 192 y Fs(\))p Fk(j)d Fs(+)g Fl(B)r Fs(\()p Fl(\013)p Fs(\))14 b(extends)g(to)g(a)g(H\177)-25 b(older)14 b(1)p Fl(=)p Fs(2)57 261 y(function.)57 366 y(W)l(e)26 b(refer)g(to)h([Ma1])f(and)g(to)g([MMY2])g(and)g(references) g(therein)f(for)i(a)f(discussion)e(of)57 436 y(Conjecture)12 b(5.19.)20 b(The)12 b(next)h(Exercises)e(giv)o(e)i(an)f(idea)f(of)i(ho) o(w)e(to)i(compute)f(appro)o(ximately)57 506 y(but)17 b(e\013ectiv)o(ely)h(the)f(function)g Fl(\013)e Fk(7!)g Fs(log)9 b Fk(j)p Fl(u)p Fs(\()p Fl(e)934 488 y Fi(2)p Fh(\031)q(i\013)1020 506 y Fs(\))p Fk(j)18 b Fs(on)f(a)g(computer.)23 b(More)16 b(informations)57 576 y(can)f(b)q(e)h(found)f(in)h([He4])g (\(where)g(one)f(can)h(also)f(\014nd)g(man)o(y)g(problems,)f(most)h(of) h(whic)o(h)f(are)57 645 y(still)h(op)q(en\))h(and)e([Ma1].)57 751 y Fr(Exercise)27 b(5.20)c Fs(Let)h Fl(f)30 b Fk(2)25 b Fl(G)664 758 y Fh(\025)714 751 y Fs(b)q(e)f(linearizable,)f Fl(\025)i Fs(=)g Fl(e)1215 733 y Fi(2)p Fh(\031)q(i\013)1302 751 y Fs(.)42 b(Let)24 b Fl(U)1488 758 y Fh(f)1537 751 y Fs(b)q(e)f(the)h(Siegel)57 820 y(disk)d(of)h Fl(f)5 b Fs(,)25 b Fl(h)325 827 y Fh(f)372 820 y Fs(b)q(e)d(the)h (linearization)d(of)j Fl(f)5 b Fs(,)24 b Fl(z)h Fk(2)e Fl(U)1095 827 y Fh(f)1121 820 y Fs(,)h Fl(z)h Fs(=)e Fl(h)1298 827 y Fh(f)1323 820 y Fs(\()p Fl(w)q Fs(\),)h(where)e Fl(w)i Fk(2)f Fm(D)1735 829 y Fh(r)r Fi(\()p Fh(f)t Fi(\))1815 820 y Fs(,)57 890 y Fk(j)p Fl(w)q Fk(j)13 b Fs(=)h Fl(r)h(<)f(r)q Fs(\()p Fl(f)5 b Fs(\).)24 b(Sho)o(w)16 b(that)635 1057 y(lim)597 1087 y Fh(m)p Fj(!)p Fi(+)p Fj(1)766 1023 y Fs(1)p 757 1045 44 2 v 757 1091 a Fl(m)815 995 y Fh(m)p Fj(\000)p Fi(1)822 1009 y Fe(X)824 1116 y Fh(j)r Fi(=0)910 1057 y Fs(log)8 b Fk(j)p Fl(f)1025 1036 y Fh(j)1047 1057 y Fs(\()p Fl(z)r Fs(\))p Fk(j)15 b Fs(=)e(log)c Fl(r)415 b Fs(\(5)p Fl(:)p Fs(23\))57 1233 y([Solution)14 b(:)21 b(Since)15 b Fl(h)459 1240 y Fh(f)500 1233 y Fs(conjugates)g Fl(f)21 b Fs(to)16 b Fl(R)887 1240 y Fh(\025)928 1233 y Fs(one)f(has)g Fl(f)1135 1215 y Fh(j)1156 1233 y Fs(\()p Fl(z)r Fs(\))g(=)f Fl(f)1316 1215 y Fh(j)1338 1233 y Fs(\()p Fl(h)1386 1240 y Fh(f)1411 1233 y Fs(\()p Fl(w)q Fs(\)\))h(=)f Fl(h)1602 1240 y Fh(f)1628 1233 y Fs(\()p Fl(\025)1676 1215 y Fh(j)1697 1233 y Fl(w)q Fs(\))i(for)57 1303 y(all)g Fl(j)g Fk(\025)e Fs(0)i(and)g Fl(w)f Fk(2)f Fm(D)485 1312 y Fh(r)q Fi(\()p Fh(f)t Fi(\))564 1303 y Fs(,)j(th)o(us)507 1441 y(1)p 498 1463 V 498 1509 a Fl(m)556 1413 y Fh(m)p Fj(\000)p Fi(1)563 1427 y Fe(X)564 1534 y Fh(j)r Fi(=0)650 1475 y Fs(log)9 b Fk(j)p Fl(f)766 1454 y Fh(j)788 1475 y Fs(\()p Fl(z)r Fs(\))p Fk(j)14 b Fs(=)947 1441 y(1)p 938 1463 V 938 1509 a Fl(m)996 1413 y Fh(m)p Fj(\000)p Fi(1)1003 1427 y Fe(X)1005 1534 y Fh(j)r Fi(=0)1091 1475 y Fs(log)8 b Fk(j)p Fl(h)1206 1482 y Fh(f)1232 1475 y Fs(\()p Fl(\025)1280 1454 y Fh(j)1301 1475 y Fl(w)q Fs(\))p Fk(j)g Fl(:)57 1644 y(h)86 1651 y Fh(f)128 1644 y Fs(has)16 b(neither)h(p)q(oles)f(nor)g(zeros)h(but)f Fl(w)g Fs(=)e(0)j(th)o(us)f(b)o(y)h(the)g(mean)f(prop)q(ert)o(y)g(of)h (harmonic)57 1714 y(functions)g(one)g(has)455 1674 y Fe(R)488 1686 y Fi(1)478 1732 y(0)519 1714 y Fs(log)8 b Fk(j)p Fl(h)634 1721 y Fh(f)660 1714 y Fs(\()p Fl(r)q(e)725 1696 y Fi(2)p Fh(\031)q(ix)810 1714 y Fs(\))p Fk(j)p Fl(dx)16 b Fs(=)f(log)9 b Fl(r)20 b Fs(for)d(all)g Fl(r)h Fk(\024)d Fl(r)q Fs(\()p Fl(f)5 b Fs(\).)28 b(Finally)16 b(note)i(that)57 1784 y Fl(w)d Fk(7!)e Fl(\025w)h Fs(is)f(uniquely)f (ergo)q(dic)g(on)h Fk(j)p Fl(w)q Fk(j)g Fs(=)h Fl(r)q Fs(,)g(and)e(in)h(this)f(case)h(Birkho\013)t('s)f(ergo)q(dic)g(theorem) 57 1854 y(holds)j(for)h(all)g(initial)g(p)q(oin)o(ts,)g(th)o(us)362 2018 y(lim)324 2048 y Fh(m)p Fj(!)p Fi(+)p Fj(1)493 1984 y Fs(1)p 483 2006 V 483 2052 a Fl(m)541 1955 y Fh(m)p Fj(\000)p Fi(1)548 1970 y Fe(X)550 2076 y Fh(j)r Fi(=0)636 2018 y Fs(log)8 b Fk(j)p Fl(f)751 1997 y Fh(j)773 2018 y Fs(\()p Fl(z)r Fs(\))p Fk(j)15 b Fs(=)51 b(lim)917 2048 y Fh(m)p Fj(!)p Fi(+)p Fj(1)1086 1984 y Fs(1)p 1077 2006 V 1077 2052 a Fl(m)1135 1955 y Fh(m)p Fj(\000)p Fi(1)1142 1970 y Fe(X)1144 2076 y Fh(j)r Fi(=0)1230 2018 y Fs(log)8 b Fk(j)p Fl(h)1345 2025 y Fh(f)1371 2018 y Fs(\()p Fl(\025)1419 1997 y Fh(j)1440 2018 y Fl(w)q Fs(\))p Fk(j)865 2186 y Fs(=)917 2118 y Fe(Z)967 2131 y Fi(1)945 2231 y(0)998 2186 y Fs(log)g Fk(j)p Fl(h)1113 2193 y Fh(f)1139 2186 y Fs(\()p Fl(r)q(e)1204 2166 y Fi(2)p Fh(\031)q(ix)1289 2186 y Fs(\))p Fk(j)p Fl(dx)14 b Fs(=)g(log)9 b Fl(r)h(:)57 2366 y Fr(Exercise)21 b(5.21)16 b Fs(Deduce)h(from)g(the) g(previous)f(exercise)i(that)f(for)g(almost)g(ev)o(ery)g Fl(z)h Fk(2)d Fl(@)s(U)1802 2373 y Fh(f)57 2436 y Fs(with)h(resp)q(ect) h(to)g(the)f(harmonic)f(measure)g(one)i(has)587 2600 y(lim)549 2630 y Fh(m)p Fj(!)p Fi(+)p Fj(1)718 2566 y Fs(1)p 709 2588 V 709 2634 a Fl(m)767 2538 y Fh(m)p Fj(\000)p Fi(1)774 2552 y Fe(X)776 2659 y Fh(j)r Fi(=0)862 2600 y Fs(log)8 b Fk(j)p Fl(f)977 2579 y Fh(j)999 2600 y Fs(\()p Fl(z)r Fs(\))p Fk(j)15 b Fs(=)e(log)c Fl(r)q Fs(\()p Fl(f)c Fs(\))16 b Fl(:)365 b Fs(\(5)p Fl(:)p Fs(24\))918 2770 y(38)p eop %%Page: 39 40 39 39 bop 57 192 a Fs(Let)17 b(us)e(no)o(w)h(consider)f(the)h (quadratic)g(p)q(olynomial)f Fl(P)1104 199 y Fh(\025)1147 192 y Fs(once)h(more.)21 b(According)15 b(to)i(\(5.24\))57 261 y(to)f(compute)g Fk(j)p Fl(u)p Fs(\()p Fl(\025)p Fs(\))p Fk(j)h Fs(one)f(needs)f(to)i(kno)o(w)f(that)h(some)e(p)q(oin)o (t)h(b)q(elongs)g(to)h(the)f(b)q(oundary)f(of)57 331 y(the)f(Siegel)f(disk)g(of)h Fl(P)465 338 y Fh(\025)505 331 y Fs(\(and)g(hop)q(e)f Fl(:)8 b(:)g(:)q Fs(\).)21 b(The)14 b(critical)f(p)q(oin)o(t)g(cannot)h(b)q(e)g(con)o(tained)f(in) g Fl(U)1777 338 y Fh(P)1802 343 y Fb(\025)57 401 y Fs(b)q(ecause)20 b Fl(f)5 b Fk(j)288 408 y Fh(U)315 413 y Fb(P)337 422 y(\025)386 401 y Fs(is)19 b(injectiv)o(e,)i(and)f(from)f(the)h (classical)g(theory)g(of)g(F)l(atou)f(and)h(Julia)f(one)57 470 y(kno)o(ws)c(that)i Fl(@)s(U)377 477 y Fh(P)402 482 y Fb(\025)444 470 y Fs(is)f(con)o(tained)f(in)h(the)g(closure)g(of)g (the)g(forw)o(ard)f(orbit)h Fk(f)p Fl(P)1539 452 y Fh(k)1532 485 y(\025)1563 470 y Fs(\(1\))23 b Fk(j)f Fl(k)15 b Fk(\025)f Fs(0)p Fk(g)57 540 y Fs(of)20 b(the)h(critical)f(p)q(oin)o(t) g Fl(z)i Fs(=)e(1.)33 b(Finally)20 b(Herman)f(pro)o(v)o(ed)g(if)i Fl(\013)f Fs(v)o(eri\014es)f(an)h(arithmetical)57 610 y(condition)13 b Fk(H)p Fs(,)i(w)o(eak)o(er)e(than)h(the)g(Diophan)o (tine)f(condition)g(but)h(stronger)f(than)h(the)g(Brjuno)57 680 y(condition)21 b(\(see,)j(for)d(example,)i([Y)l(o1])f(for)g(its)g (precise)f(form)o(ulation\))g(the)h(critical)g(p)q(oin)o(t)57 749 y(b)q(elongs)15 b(to)i Fl(@)s(U)359 756 y Fh(P)384 761 y Fb(\025)411 749 y Fs(.)57 855 y Fr(Exercise)j(5.22)15 b Fs(If)i Fl(\013)d Fk(2)23 b Fs(CD)8 b(\(0\))18 b(Herman)d(has)h(also) g(pro)o(v)o(ed)f(that)j Fl(@)s(U)1429 862 y Fh(P)1454 867 y Fb(\025)1497 855 y Fs(is)e(a)h Fd(quasicircle)p Fs(,)57 924 y(that)23 b(is)g(the)g(image)g(of)g Fm(S)565 906 y Fi(1)607 924 y Fs(under)f(a)h(quasiconformal)f(homeomorphism)o(.) 39 b(In)23 b(this)g(case)57 994 y Fl(h)86 1001 y Fh(P)111 1006 y Fb(\025)160 994 y Fs(admits)e(a)i(quasiconformal)e(extension)h (to)i Fk(j)p Fl(w)q Fk(j)g Fs(=)g Fl(r)1195 1001 y Fi(2)1217 994 y Fs(\()p Fl(\025)p Fs(\))g(and)e(therefore)h(is)f(H\177)-25 b(older)57 1064 y(con)o(tin)o(uous)14 b([P)o(o])i(:)534 1134 y Fk(j)p Fl(h)577 1141 y Fh(P)602 1146 y Fb(\025)628 1134 y Fs(\()p Fl(w)683 1141 y Fi(1)706 1134 y Fs(\))11 b Fk(\000)g Fl(h)815 1141 y Fh(P)840 1146 y Fb(\025)866 1134 y Fs(\()p Fl(w)921 1141 y Fi(2)943 1134 y Fs(\))p Fk(j)j(\024)g Fs(4)p Fk(j)p Fl(w)1118 1141 y Fi(1)1151 1134 y Fk(\000)c Fl(w)1236 1141 y Fi(2)1258 1134 y Fk(j)1272 1113 y Fi(1)p Fj(\000)p Fh(\037)1701 1134 y Fs(\(5)p Fl(:)p Fs(25\))57 1238 y(for)18 b(all)g Fl(w)242 1245 y Fi(1)272 1238 y Fl(;)8 b(w)330 1245 y Fi(2)370 1238 y Fk(2)17 b Fl(@)s Fm(D)482 1247 y Fh(r)501 1252 y Fc(2)523 1247 y Fi(\()p Fh(\025)p Fi(\))580 1238 y Fs(,)i(where)f Fl(\037)g Fk(2)f Fs([0)p Fl(;)8 b Fs(1[)18 b(dep)q(ends)g(on)g Fl(\025)h Fs(is)f(the)h(so{called)e(Grunsky)57 1308 y(norm)i([P)o(o])h (asso)q(ciated)g(with)h(the)g(univ)m(alen)o(t)f(function)h Fl(g)r Fs(\()p Fl(z)r Fs(\))g(=)g Fl(r)1358 1315 y Fi(2)1381 1308 y Fs(\()p Fl(\025)p Fs(\))p Fl(=h)1502 1315 y Fh(P)1527 1320 y Fb(\025)1553 1308 y Fs(\()p Fl(r)1594 1315 y Fi(2)1617 1308 y Fs(\()p Fl(\025)p Fs(\))p Fl(=z)r Fs(\))i(on)57 1378 y Fk(j)p Fl(z)r Fk(j)14 b Fl(>)f Fs(1.)22 b(Using)16 b(this)g(information)f(sho)o(w)h(that)409 1547 y Fk(j)440 1513 y Fs(1)p 429 1535 47 2 v 429 1581 a Fl(q)451 1588 y Fh(k)490 1482 y(q)508 1487 y Fb(k)530 1482 y Fj(\000)p Fi(1)499 1500 y Fe(X)501 1606 y Fh(j)r Fi(=0)589 1547 y Fs(log)9 b Fk(j)p Fl(P)715 1523 y Fh(j)708 1562 y(\025)735 1547 y Fs(\()p Fl(z)r Fs(\))p Fk(j)j(\000)f Fs(log)e Fl(r)969 1554 y Fi(2)992 1547 y Fs(\()p Fl(\025)p Fs(\))p Fk(j)14 b(\024)1190 1513 y Fs(8)p 1146 1535 113 2 v 1146 1581 a Fl(r)1168 1588 y Fi(2)1191 1581 y Fs(\()p Fl(\025)p Fs(\))1264 1547 y(\()1289 1513 y(2)p Fl(\031)p 1289 1535 56 2 v 1293 1581 a(q)1315 1588 y Fh(k)1351 1547 y Fs(\))1370 1526 y Fi(1)p Fj(\000)p Fh(\037)1462 1547 y Fl(;)225 b Fs(\(5)p Fl(:)p Fs(26\))57 1720 y(where)18 b Fl(p)228 1727 y Fh(k)252 1720 y Fl(=q)299 1727 y Fh(k)342 1720 y Fs(is)g(a)h(con)o(v)o(ergen)o(t)e(of)h(the)h(con)o(tin)o(ued)e (fraction)h(expansion)f(of)h Fl(\013)p Fs(.)28 b(Note)19 b(that)57 1789 y(\(5.26\))f(implies)e(con)o(v)o(ergence)h(to)h(log)8 b Fl(r)805 1796 y Fi(2)828 1789 y Fs(\()p Fl(\025)p Fs(\))19 b Fp(for)h(al)s(l)d Fl(z)h Fk(2)e Fl(@)s(U)1219 1796 y Fh(P)1244 1801 y Fb(\025)1271 1789 y Fs(,)h(th)o(us)g(also)g(for)h (the)g(critical)57 1859 y(p)q(oin)o(t)e Fl(z)g Fs(=)e(1.)918 2770 y(39)p eop %%Page: 40 41 40 40 bop 57 192 a Fq(6.)31 b(Small)25 b(divisors)g(and)f(loss)h(of)f (di\013eren)n(tiabili)q(t)n(y)57 300 y Fs(In)13 b(this)h(Chapter)f(w)o (e)g(will)g(\(v)o(ery)e(!\))21 b(brie\015y)13 b(illustrate)g(other)g(t) o(w)o(o)h(completely)f(di\013eren)o(t)g(ap-)57 369 y(proac)o(hes)g(to)j (the)g(problem)e(of)h(linearization)f(of)i(germs)e(of)i(holomorphic)d (di\013eomorphisms)57 439 y(with)j(an)g(indi\013eren)o(t)f(\014xed)i(p) q(oin)o(t.)156 512 y(In)j(the)f(previous)g(c)o(hapter)f(w)o(e)h(sa)o(w) g(ho)o(w)g(the)h(optimal)e(su\016cien)o(t)g(condition)h(can)g(b)q(e)57 582 y(obtained)h(b)o(y)h(the)h(classical)e(ma)s(joran)o(t)g(series)g (metho)q(d)h(as)g(Siegel)g(and)g(Brjuno)g(did)f(and)57 651 y(that)c(Y)l(o)q(ccoz)i(w)o(as)d(able)h(to)h(sho)o(w)e(that)i(it)g (is)f(also)g(necessary)f(with)h(his)g(ingenious)f(creation)57 721 y(of)21 b(\\geometric)g(renormalization".)34 b(Here)22 b(w)o(e)f(will)g(giv)o(e)g(an)g(idea)g(of)h(t)o(w)o(o)f(pro)q(ofs)g(of) g(the)57 791 y(Siegel)f(theorem,)g(one)h(due)f(to)h(Herman)e([He1,)j (He2])f(and)f(the)g(other)g(due)h(essen)o(tially)e(to)57 861 y(Kolmogoro)o(v)14 b([K])i(\(see)h(also)f(Arnol'd)g([Ar3])g(and)f (Zehnder)h([Ze2]\).)156 933 y(Herman's)11 b(metho)q(d)g(is)g(far)g (from)g(giving)g(the)h(optimal)e(n)o(um)o(b)q(er{theoretical)f (condition,)57 1003 y(the)18 b(idea)f(is)h(simply)f(so)g(original)g (and)g(b)q(eautiful)h(that)g(it)g(deserv)o(es)f(b)q(eing)g(kno)o(wn.)26 b(It)18 b(also)57 1073 y(illustrates)f(ho)o(w)g(in)h(one{dimensional)e (small)h(divisor)g(problems)f(the)j(problem)d(kno)o(wn)i(as)57 1143 y(\\loss)j(of)i(di\013eren)o(tiabilit)o(y")d(do)q(es)j(not)f(prev) o(en)o(t)g(from)f(the)i(application)e(of)i(simple)e(to)q(ols)57 1212 y(lik)o(e)d(the)g(con)o(traction)f(principle.)26 b(Herman's)16 b(metho)q(d)i(can)g(also)g(b)q(e)g(extended)h(to)f(\(and) g(it)57 1282 y(is)j(actually)h(describ)q(ed)f(b)o(y)g(him)g(for\))h (the)g(problem)e(of)i(lo)q(cal)g(conjugacy)g(to)g(rotation)f(of)57 1352 y(smo)q(oth)16 b(orien)o(tation{preserving)d(di\013eomorphisms)g (of)k(the)g(circle.)156 1425 y(The)i(idea)g(of)g(Kolmogoro)o(v)f(is)g (do)h(adapt)g(Newton's)g(metho)q(d)g(for)g(\014nding)e(the)j(ro)q(ots) 57 1494 y(of)f(algebraic)f(equations)h(so)g(as)f(to)i(apply)e(it)i(for) f(\014nding)e(the)j(solution)e(of)h(the)g(conjugacy)57 1564 y(equation.)24 b(This)17 b(metho)q(d)g(has)f(b)q(een)i(sho)o(wn)e (b)o(y)h(R)q(\177)-26 b(ussmann)14 b([R)q(\177)-26 b(u])17 b(to)h(b)q(e)f(adaptable)g(so)g(as)57 1634 y(to)e(pro)o(v)o(e)e(the)i (su\016ciency)f(of)h(a)g(condition)f(\(sligh)o(tly)g(stronger)g(than\)) h(Brjuno's.)20 b(The)14 b(main)57 1704 y(reason)d(for)h(sk)o(etc)o (hing)f(Kolmogoro)o(v's)f(argumen)o(t)h(in)h(this)g(rather)f(limited)h (setting)g(is)g(that)h(in)57 1773 y(the)f(second)f(part)g(of)h(this)f (monograph)f(w)o(e)h(will)g(illustrate)g(Nash{Moser's)f(implicit)h (function)57 1843 y(theorem)i(whic)o(h)h(is)g(essen)o(tially)f(the)h (abstract)g(and)g(\015exible)g(form)o(ulation)e(of)j(Kolmogoro)o(v's)57 1913 y(idea.)57 2091 y Fo(6.1)k(Hardy{Sob)r(olev)g(spaces)g(and)h(loss) g(of)g(di\013eren)n(tiabilit)o(y)57 2199 y Fs(Let)d Fl(k)e Fk(2)f Fm(N)p Fs(,)k Fl(r)d(>)f Fs(0.)22 b(F)l(ollo)o(wing)14 b([He2],)j(w)o(e)f(in)o(tro)q(duce)g(the)h(Hardy{Sob)q(olev)f(spaces)82 2368 y Fk(O)123 2347 y Fh(k)q(;)p Fi(2)122 2380 y Fh(r)193 2368 y Fs(=)d Fk(f)p Fl(f)5 b Fs(\()p Fl(z)r Fs(\))16 b(=)448 2305 y Fj(1)432 2320 y Fe(X)430 2426 y Fh(n)p Fi(=0)514 2368 y Fl(f)538 2375 y Fh(n)565 2368 y Fl(z)590 2347 y Fh(n)640 2368 y Fk(j)22 b(k)p Fl(f)5 b Fk(k)755 2385 y Fj(O)788 2368 y Fb(k;)p Fc(2)787 2391 y Fb(r)855 2368 y Fs(=)13 b(\()p Fk(j)p Fl(f)964 2375 y Fi(0)987 2368 y Fk(j)1001 2347 y Fi(2)1033 2368 y Fs(+)1099 2305 y Fj(1)1083 2320 y Fe(X)1082 2426 y Fh(n)p Fi(=1)1165 2368 y Fl(n)1195 2347 y Fi(2)p Fh(k)1239 2368 y Fk(j)p Fl(f)1277 2375 y Fh(n)1304 2368 y Fk(j)1318 2347 y Fi(2)1341 2368 y Fl(r)1364 2347 y Fi(2)p Fh(n)1412 2368 y Fs(\))1431 2347 y Fi(1)p Fh(=)p Fi(2)1507 2368 y Fl(<)h Fs(+)p Fk(1g)f Fl(:)25 b Fs(\(6)p Fl(:)p Fs(1\))57 2578 y Fr(Exercise)20 b(6.1)68 2650 y Fs(\(a\))25 b(Sho)o(w)16 b(that)h Fk(O)437 2632 y Fh(k)q(;)p Fi(2)436 2663 y Fh(r)510 2650 y Fs(is)f(a)g(Hilb)q (ert)h(space.)918 2770 y(40)p eop %%Page: 41 42 41 41 bop 65 192 a Fs(\(b\))25 b(Sho)o(w)16 b(if)g Fl(f)k Fk(2)14 b(O)465 173 y Fh(k)q(;)p Fi(2)464 204 y Fh(r)521 192 y Fs(,)j Fl(f)5 b Fs(\(0\))15 b(=)f(0,)i(one)g(has)677 321 y(sup)667 365 y Fj(j)p Fh(z)q Fj(j\024)p Fh(r)770 321 y Fk(j)p Fl(f)5 b Fs(\()p Fl(z)r Fs(\))p Fk(j)16 b(\024)958 276 y Fe(p)p 1008 276 117 2 v 45 x Fl(\020)t Fs(\(2)p Fl(k)r Fs(\))p Fk(k)p Fl(f)5 b Fk(k)1204 338 y Fj(O)1237 321 y Fb(k;)p Fc(2)1236 345 y Fb(r)1304 321 y Fl(;)156 478 y Fs(where)16 b Fl(\020)k Fs(denotes)d(Riemann's)d(zeta) j(function.)71 548 y(\(c\))25 b(Sho)o(w)16 b(that)h(if)f Fl(k)g Fk(\025)d Fs(1)k(then)f Fk(O)731 530 y Fh(k)q(;)p Fi(2)730 560 y Fh(r)804 548 y Fs(is)g(a)h(Banac)o(h)f(algebra.)65 618 y(\(d\))25 b(Sho)o(w)15 b(that)i(if)f Fl(f)k Fk(2)14 b(O)572 600 y Fh(k)q(;)p Fi(2)571 630 y Fh(r)644 618 y Fs(and)i Fl(\036)g Fs(is)f(holomorphic)f(in)i(a)g(neigh)o(b)q(orho)q (o)q(d)e(of)j Fl(f)5 b Fs(\()p Fm(D)1671 625 y Fh(r)1696 618 y Fs(\))16 b(then)156 687 y Fl(\036)9 b Fk(\016)f Fl(f)20 b Fk(2)14 b(O)360 669 y Fh(k)q(;)p Fi(2)359 700 y Fh(r)431 687 y Fs(and)h(on)g(a)g(su\016cien)o(tly)f(small)g(neigh)o (b)q(orho)q(o)q(d)g Fl(V)27 b Fs(of)15 b Fl(f)21 b Fs(in)15 b Fk(O)1578 669 y Fh(k)q(;)p Fi(2)1577 700 y Fh(r)1650 687 y Fs(the)g(map)156 757 y Fl( )h Fk(2)e Fl(V)25 b Fk(7!)14 b Fl(\036)d Fk(\016)g Fl( )16 b Fk(2)e(O)582 739 y Fh(k)q(;)p Fi(2)581 769 y Fh(r)655 757 y Fs(is)i(holomorphic.)57 862 y(The)d(follo)o(wing)e(v)o(ery)i(elemen)o(tary)g(prop)q(osition)e (w)o(ell)i(illustrates)f(the)h(phenomenon)e(of)i(\\loss)57 932 y(of)19 b(di\013eren)o(tiabilit)o(y")e(due)i(to)g(the)h(small)e (divisors)f(whic)o(h)h(already)h(arises)e(at)j(the)f(lev)o(el)g(of)57 1002 y(the)d(linearized)g(conjugacy)g(equation)h(\(6.2\).)57 1137 y Fr(Prop)r(osition)23 b(6.2)k Fd(Let)21 b Fs(0)f Fk(\024)g Fl(\034)25 b Fk(\024)19 b Fl(\034)796 1144 y Fi(0)818 1137 y Fd(,)i Fl(\034)875 1144 y Fi(0)917 1137 y Fk(2)f Fm(N)p Fd(,)i Fl(k)g Fk(\025)d Fs(1)14 b(+)f Fl(\034)1260 1144 y Fi(0)1282 1137 y Fd(,)21 b Fl(f)k Fk(2)20 b(O)1460 1119 y Fh(k)q(;)p Fi(2)1459 1149 y Fh(r)1517 1137 y Fd(,)h Fl(f)5 b Fs(\(0\))21 b(=)e(0)p Fd(.)33 b(If)57 1207 y Fl(\013)14 b Fk(2)g Fd(CD)8 b Fs(\()p Fl(\034)e Fs(\))17 b Fd(then)f(the)h(unique)f(solution)f Fl(g)h Fs(:=)d Fl(D)1012 1185 y Fj(\000)p Fi(1)1011 1222 y Fh(\025)1066 1207 y Fl(f)23 b Fd(v)o(erifying)16 b Fl(g)r Fs(\(0\))e(=)f(0)k Fd(of)769 1336 y Fl(g)c Fk(\016)e Fl(R)880 1343 y Fh(\025)917 1336 y Fk(\000)g Fl(g)k Fs(=)e Fl(f)20 b(;)610 b Fs(\(6)p Fl(:)p Fs(2\))57 1465 y Fd(b)q(elongs)15 b(to)h Fk(O)336 1447 y Fh(k)q Fj(\000)p Fi(1)p Fj(\000)p Fh(\034)458 1452 y Fc(0)478 1447 y Fh(;)p Fi(2)335 1478 y Fh(r)512 1465 y Fd(.)21 b(Moreo)o(v)o(er)15 b(there)g(exists)h(a)f (univ)o(ersal)f(constan)o(t)h Fl(C)i(>)d Fs(0)i Fd(suc)o(h)e(that)587 1618 y Fk(k)618 1585 y Fl(d)644 1567 y Fh(k)q Fj(\000)p Fi(1)p Fj(\000)p Fh(\034)766 1572 y Fc(0)788 1585 y Fl(g)p 618 1607 196 2 v 690 1653 a(dz)820 1618 y Fk(k)845 1634 y Fj(O)878 1617 y Fc(0)p Fb(;)p Fc(2)877 1641 y Fb(r)942 1618 y Fk(\024)1000 1585 y Fl(C)p 1000 1607 40 2 v 1005 1653 a(\015)1045 1618 y Fk(k)1076 1585 y Fl(d)1102 1567 y Fh(k)1127 1585 y Fl(f)p 1076 1607 81 2 v 1091 1653 a(dz)1162 1618 y Fk(k)1187 1634 y Fj(O)1220 1617 y Fc(0)p Fb(;)p Fc(2)1219 1641 y Fb(r)1284 1618 y Fl(;)57 1768 y Fd(where)i Fl(\015)g Fs(=)e(inf)356 1775 y Fh(n)p Fj(\025)p Fi(1)443 1768 y Fl(n)473 1750 y Fi(1+)p Fh(\034)548 1768 y Fk(j)p Fl(\025)591 1750 y Fh(n)629 1768 y Fk(\000)d Fs(1)p Fk(j)p Fd(.)57 1903 y Fp(Pr)m(o)m(of.)29 b Fs(It)f(is)e(a)h (straigh)o(tforw)o(ard)d(computation)i(starting)g(from)g(the)h(iden)o (tit)o(y)g Fl(g)r Fs(\()p Fl(z)r Fs(\))32 b(=)57 1935 y Fe(P)109 1947 y Fj(1)109 1987 y Fh(n)p Fi(=1)187 1972 y Fs(\()p Fl(\025)235 1954 y Fh(n)273 1972 y Fk(\000)11 b Fs(1\))367 1954 y Fj(\000)p Fi(1)421 1972 y Fl(f)445 1979 y Fh(n)473 1972 y Fs(.)1303 b Fa(\003)57 2092 y Fs(This)19 b(Prop)q(osition)g(sho)o(ws)g(that)i(solving)f(the)g(linear) g(equation)g(\(6.2\))h(the)g(small)e(divisors)57 2162 y(cause)k(the)i(loss)e(of)h(1)16 b(+)g Fl(\034)575 2169 y Fi(0)621 2162 y Fs(deriv)m(ativ)o(es.)44 b(This)23 b(loss)g(of)h(di\013eren)o(tiabilit)o(y)e(phenomenon)57 2232 y(is)j(t)o(ypical)h(of)h(small)e(divisors)g(problems)f(and)h(it)i (will)f(b)q(e)g(crucial)g(in)g(the)g(discussions)57 2302 y(in)21 b(the)h(second)e(part)h(of)h(this)f(monograph.)35 b(The)21 b(most)g(anno)o(ying)f(consequence)h(of)h(this)57 2371 y(phenomenon)14 b(is)i(the)g(imp)q(ossibilit)o(y)e(of)i(using)f (\014xed)h(p)q(oin)o(ts)g(metho)q(ds)f(to)i(solv)o(e)e(conjugacy)57 2441 y(equations,)23 b(simply)e(b)q(ecause)h(the)g(op)q(erator)f Fl(D)994 2448 y Fh(\025)1043 2441 y Fs(is)h Fp(unb)m(ounde)m(d)i Fs(if)e(regarded)e(on)i(a)g Fp(\014xe)m(d)57 2511 y Fs(Hardy{Sob)q (olev)15 b(space.)22 b(Ho)o(w)o(ev)o(er)15 b(under)g(some)g (restriction)h(on)f Fl(\034)1349 2518 y Fi(0)1388 2511 y Fs(one)h(can)g(actually)g(use)57 2581 y(the)i(con)o(traction)f (principle)f(to)j(solv)o(e)e(the)h(conjugacy)g(problem,)f(thanks)g(to)h (an)g(ingenious)57 2650 y(idea)e(of)h(Herman)e(w)o(e)i(will)f(shortly)g (describ)q(e)f(in)h(the)h(next)g(section.)918 2770 y(41)p eop %%Page: 42 43 42 42 bop 57 297 a Fo(6.2)19 b(Herman's)f(Sc)n(h)n(w)n(arzian)k(deriv)m (ativ)n(e)e(tric)n(k)57 402 y Fs(Let)d(\012)g(b)q(e)f(a)h(region)e(in)i (the)f(complex)g(plane)g(and)g Fl(f)28 b Fs(:)14 b(\012)f Fk(!)h Fm(C)29 b Fs(b)q(e)17 b(holomorphic.)57 535 y Fr(De\014nition)j(6.3)28 b Fd(The)16 b Fs(Sc)o(h)o(w)o(arzian)e(deriv)m (ativ)o(e)j Fl(S)s Fs(\()p Fl(f)5 b Fs(\))17 b Fd(of)g Fl(f)22 b Fd(is)390 688 y Fl(S)s Fs(\()p Fl(f)5 b Fs(\))15 b(:=)e(\(log)c Fl(f)693 667 y Fj(0)708 688 y Fs(\))727 667 y Fj(00)764 688 y Fk(\000)820 654 y Fs(1)p 820 676 25 2 v 820 722 a(2)851 688 y(\(\(log)g Fl(f)991 667 y Fj(0)1006 688 y Fs(\))1025 667 y Fj(0)1039 688 y Fs(\))1058 667 y Fi(2)1095 688 y Fs(=)1153 654 y Fl(f)1182 636 y Fj(000)p 1153 676 67 2 v 1165 722 a Fl(f)1194 708 y Fj(0)1237 688 y Fk(\000)1293 654 y Fs(3)p 1293 676 25 2 v 1293 722 a(2)1332 618 y Fe(\022)1375 654 y Fl(f)1404 636 y Fj(00)p 1375 676 56 2 v 1381 722 a Fl(f)1410 708 y Fj(0)1436 618 y Fe(\023)1472 628 y Fi(2)506 834 y Fs(=)14 b Fk(\000)p Fs(2)623 789 y Fe(p)p 672 789 44 2 v 672 834 a Fl(f)701 820 y Fj(0)724 764 y Fe(\022)797 801 y Fs(1)p 767 823 86 2 v 767 833 a Fk(p)p 808 833 44 2 v 808 871 a Fl(f)837 856 y Fj(0)858 764 y Fe(\023)895 775 y Fj(00)1726 756 y Fs(\(6)p Fl(:)p Fs(3\))57 1009 y Fr(Exercise)20 b(6.4)68 1079 y Fs(\(a\))25 b(Pro)o(v)o(e)13 b(that)i(the)f(follo)o(wing)f(\\c)o (hain)g(rule")g(holds)g(:)21 b Fl(S)s Fs(\()p Fl(f)12 b Fk(\016)6 b Fl(g)r Fs(\))14 b(=)f(\()p Fl(S)s Fs(\()p Fl(f)5 b Fs(\))h Fk(\016)g Fl(g)r Fs(\)\()p Fl(g)1608 1061 y Fj(0)1624 1079 y Fs(\))1643 1061 y Fi(2)1672 1079 y Fs(+)g Fl(S)s Fs(\()p Fl(g)r Fs(\).)65 1149 y(\(b\))25 b(Sho)o(w)16 b(that)h Fl(S)s Fs(\()p Fl(f)5 b Fs(\))14 b Fk(\021)g Fs(0)j(if)f(and)g(only)g(if)h Fl(f)22 b Fs(is)16 b(a)h(M\177)-25 b(obius)15 b(map.)57 1254 y(The)c(idea)g(of)g(Herman)f (is)h(to)h(apply)e(the)i(Sc)o(h)o(w)o(arzian)c(deriv)m(ativ)o(e)j(to)h (the)f(conjugacy)h(equation)57 1324 y Fl(f)17 b Fk(\016)10 b Fl(h)k Fs(=)g Fl(h)c Fk(\016)h Fl(R)342 1331 y Fh(\025)385 1324 y Fs(:)22 b(one)16 b(obtains)504 1446 y Fl(\025)533 1426 y Fi(2)555 1446 y Fs(\()p Fl(S)s Fs(\()p Fl(h)p Fs(\))11 b Fk(\016)g Fl(R)760 1453 y Fh(\025)786 1446 y Fs(\))h Fk(\000)e Fl(S)s Fs(\()p Fl(h)p Fs(\))k(=)g(\()p Fl(S)s Fs(\()p Fl(f)5 b Fs(\))12 b Fk(\016)f Fl(h)p Fs(\)\()p Fl(h)1298 1426 y Fj(0)1312 1446 y Fs(\))1331 1426 y Fi(2)1368 1446 y Fl(:)344 b Fs(\(6)p Fl(:)p Fs(4\))57 1569 y(A)o(t)14 b(the)h(r.h.s.)20 b(app)q(ears)13 b Fl(h)551 1551 y Fj(0)564 1569 y Fs(,)i(th)o(us)e(one)h(has)g(already)g(lost)g(one)g(deriv)m (ativ)o(e)g(and)f(this)h(do)q(es)g(not)57 1638 y(seem)k(to)g(lead)g(to) h(an)o(ything)e(go)q(o)q(d.)27 b(Assuming)17 b(the)i(r.h.s.)25 b(as)18 b(giv)o(en)g(one)g(could)g(solv)o(e)g(for)57 1708 y Fl(S)s Fs(\()p Fl(h)p Fs(\))d(but)h(this)f(w)o(ould)f(cost)i(1)9 b(+)g Fl(\034)710 1715 y Fi(0)747 1708 y Fs(deriv)m(ativ)o(es)15 b(according)g(to)h(Prop)q(osition)e(6.2.)21 b(Ho)o(w)o(ev)o(er)57 1778 y(if)e Fl(\034)127 1785 y Fi(0)168 1778 y Fs(=)f(1)h(\(whic)o(h)g (is)g(true)g(for)g(almost)f(all)h Fl(\013)h Fs(as)f(w)o(e)g(sa)o(w)g (in)f(Section)i(4.2\))f(the)h(total)f(loss)57 1848 y(of)e(deriv)m(ativ) o(es)h(is)f(three.)25 b(The)17 b(idea)g(no)o(w)g(is)g(that)h(applying)f Fl(S)1278 1830 y Fj(\000)p Fi(1)1348 1848 y Fs(one)h(should)e(recup)q (erate)57 1917 y(three)g(deriv)m(ativ)o(es)g(and)g(this)g(w)o(ould)g (imply)f(that)i(the)g(map)545 2040 y Fk(O)586 2019 y Fh(k)q(;)p Fi(2)585 2052 y Fh(r)656 2040 y Fk(3)d Fl(h)g Fk(7!)f Fl(S)843 2019 y Fj(\000)p Fi(1)896 2040 y Fl(D)938 2018 y Fj(\000)p Fi(1)937 2055 y Fh(\025)992 2040 y Fs([\()p Fl(S)s Fs(\()p Fl(f)5 b Fs(\))12 b Fk(\016)f Fl(h)p Fs(\)\()p Fl(h)1270 2019 y Fj(0)1285 2040 y Fs(\))1304 2019 y Fi(2)1326 2040 y Fs(])386 b(\(6)p Fl(:)p Fs(5\))57 2162 y(tak)o(es)17 b(its)f(v)m(alues)h(in)g Fk(O)503 2144 y Fh(k)q(;)p Fi(2)502 2175 y Fh(r)576 2162 y Fs(to)q(o.)24 b(Note)17 b(that)h(w)o(e)e(ha)o(v) o(e)h(sligh)o(tly)f(mo)q(di\014ed)f(the)i(de\014nition)f(of)57 2232 y Fl(D)98 2239 y Fh(\025)143 2232 y Fs(with)j(resp)q(ect)g(to)g (the)h(previous)d(Section)i(:)27 b(here)18 b Fl(D)1153 2239 y Fh(\025)1180 2232 y Fl(f)23 b Fs(=)18 b Fl(\025)1313 2214 y Fi(2)1335 2232 y Fl(f)h Fk(\016)12 b Fl(R)1453 2239 y Fh(\025)1492 2232 y Fk(\000)g Fl(f)5 b Fs(.)30 b(This)19 b(do)q(es)57 2302 y(not)d(c)o(hange)g(the)h(conclusions)d(of) j(Section)f(6.1.)156 2371 y(On)i(a)g(disk)f Fm(D)422 2378 y Fh(r)465 2371 y Fs(of)h(su\016cien)o(tly)f(small)g(radius)f Fl(f)24 b Fs(is)17 b(close)h(to)g Fl(R)1377 2378 y Fh(\025)1421 2371 y Fs(th)o(us)f Fl(S)s Fs(\()p Fl(f)5 b Fs(\))19 b(m)o(ust)e(b)q(e)57 2441 y(small)f(and)g(one)h(can)g(hop)q(e)h(to)f (conclude)g(using)f(a)h(\014xed)g(p)q(oin)o(t)g(theorem)f(\(the)i(con)o (traction)57 2511 y(principle,)c(sa)o(y\).)22 b(This)16 b(strategy)h(indeed)e(w)o(orks)h(:)22 b(see)17 b([He1,)f(He2])h(for)f (the)h(details.)156 2581 y(The)24 b(in)o(v)o(ersion)d(of)j Fl(S)i Fs(is)d(ac)o(hiev)o(ed)g(as)g(follo)o(ws.)43 b(First)22 b(of)i(all)f(note)h(that)g(it)g(is)f(not)57 2650 y(restrictiv)o(e)13 b(to)i(assume)e Fl(f)545 2632 y Fj(00)571 2650 y Fs(\(0\))i(=)e(0,)i (so)f(that)g Fl(h)948 2632 y Fj(00)974 2650 y Fs(\(0\))g(=)g(0)g(\(one) g(can)g(preliminarly)e(conjugate)918 2770 y(42)p eop %%Page: 43 44 43 43 bop 57 195 a Fl(f)25 b Fs(b)o(y)20 b(the)g(p)q(olynomial)e Fl(z)e Fk(\000)650 171 y Fh(f)673 156 y Ff(00)698 171 y Fi(\(0\))p 625 183 149 2 v 625 212 a(2\()p Fh(\025)p Fj(\000)p Fh(\025)740 202 y Fc(2)758 212 y Fi(\))780 195 y Fl(z)805 177 y Fi(2)828 195 y Fs(\))k(:)28 b(this)20 b(implies)e([\()p Fl(S)s Fs(\()p Fl(f)5 b Fs(\))14 b Fk(\016)g Fl(h)p Fs(\)\()p Fl(h)1465 177 y Fj(0)1479 195 y Fs(\))1498 177 y Fi(2)1520 195 y Fs(])1534 202 y Fh(z)q Fi(=0)1627 195 y Fs(=)19 b(0.)32 b(Let)57 264 y Fl( )25 b Fs(=)d Fl(D)217 243 y Fj(\000)p Fi(1)216 280 y Fh(\025)271 264 y Fs([\()p Fl(S)s Fs(\()p Fl(f)5 b Fs(\))16 b Fk(\016)f Fl(h)p Fs(\)\()p Fl(h)557 246 y Fj(0)571 264 y Fs(\))590 246 y Fi(2)613 264 y Fs(].)38 b(If)22 b Fl( )i Fs(is)e(small)e(enough)h(\(this)h(is)g(alw)o(a)o(ys)f (the)h(case)g(if)g(one)57 334 y(considers)14 b(a)j(su\016cien)o(tly)e (small)h(disk\))g(then)h(one)f(can)g(write)h Fl( )h Fs(uniquely)f(in)f (the)g(form)788 469 y Fl( )g Fs(=)d Fl( )922 449 y Fj(0)920 482 y Fi(1)954 469 y Fk(\000)1010 436 y Fs(1)p 1010 458 25 2 v 1010 504 a(2)1041 469 y Fl( )1075 449 y Fi(2)1073 482 y(1)57 603 y Fs(with)i Fl( )201 610 y Fi(1)224 603 y Fs(\(0\))f(=)g(0.)21 b(No)o(w)16 b(one)f(can)g(easily)g(solv)o(e)g (the)h(problem)d Fl(S)s Fs(\()p Fl(h)1323 610 y Fi(1)1345 603 y Fs(\))h(=)g Fl( )i Fs(=)e Fl( )1566 585 y Fj(0)1564 615 y Fi(1)1595 603 y Fk(\000)1649 583 y Fi(1)p 1649 591 20 2 v 1649 620 a(2)1674 603 y Fl( )1708 585 y Fi(2)1706 615 y(1)1746 603 y Fs(just)57 672 y(lo)q(oking)i(for)g Fl(h)336 679 y Fi(1)375 672 y Fs(suc)o(h)f(that)i(\(log)9 b Fl(h)716 654 y Fj(0)716 685 y Fi(1)738 672 y Fs(\))757 654 y Fj(0)785 672 y Fs(=)14 b Fl( )870 679 y Fi(1)893 672 y Fs(.)22 b(This)16 b(is)g(ac)o(hiev)o(ed)f(in)h(three)h(steps)f(:) 467 784 y Fl( )501 764 y Fj(0)499 796 y Fi(2)536 784 y Fs(=)d Fl( )620 791 y Fi(1)657 784 y Fl(;)467 869 y( )499 876 y Fi(3)536 869 y Fs(=)g Fl(e)611 848 y Fh(c)p Fi(+)p Fh( )686 853 y Fc(2)722 869 y Fs(where)8 b Fl(c)16 b Fs(is)g(c)o(hosen)f(s.t.)8 b Fl( )1212 876 y Fi(3)1235 869 y Fs(\(0\))15 b(=)e(1)h Fl(;)471 977 y(h)500 984 y Fi(1)536 977 y Fs(=)588 909 y Fe(Z)638 921 y Fh(z)616 1022 y Fi(0)669 977 y Fl( )701 984 y Fi(3)724 977 y Fs(\()p Fl(\020)t Fs(\))p Fl(d\020)k(:)57 1116 y Fs(Then)e(it)h(is)f(immediate) f(to)i(c)o(hec)o(k)f(that)h Fl(S)s Fs(\()p Fl(h)915 1123 y Fi(1)937 1116 y Fs(\))d(=)g Fl( )r Fs(.)57 1222 y Fr(Exercise)i(6.5)d Fs(Let)h Fl(r)h(>)f Fs(0,)g Fl( )h Fs(and)e Fl(h)766 1229 y Fi(1)801 1222 y Fs(as)g(ab)q(o)o(v)o(e.)20 b(Sho)o(w)13 b(that)g(if)h Fl( )i Fk(2)e(O)1430 1204 y Fi(0)p Fh(;)p Fi(2)1429 1234 y Fh(r)1497 1222 y Fs(then)g Fl(h)1637 1229 y Fi(1)1672 1222 y Fk(2)g(O)1760 1204 y Fi(3)p Fh(;)p Fi(2)1759 1234 y Fh(r)1815 1222 y Fs(.)57 1396 y Fo(6.3)19 b(Kolmogoro)n(v's)f(mo)r(di\014ed)j(Newton)f(metho)r(d)57 1502 y Fs(Here)f(w)o(e)g(follo)o(w)f(quite)i(closely)f([St],)g(V)l (olume)f(I)q(I,)h(Chapter)g(I)q(I)q(I,)h(Section)e(7.)30 b(W)l(e)19 b(suggest)57 1571 y(ho)o(w)o(ev)o(er)c(the)i(reader)e(to)i (lo)q(ok)g(also)f(at)h([Ze2])f(for)h(a)f(complete)g(pro)q(of.)156 1641 y(Let)i Fl(f)h Fk(2)14 b Fl(S)367 1648 y Fh(\025)393 1641 y Fs(,)j Fl(\025)d Fs(=)f Fl(e)542 1623 y Fi(2)p Fh(\031)q(i\013)646 1641 y Fs(and)j(assume)f(that)i Fl(\013)g Fs(is)f(a)h(diophan)o(tine)d(n)o(um)o(b)q(er.)20 b(W)l(e)d(w)o(an)o(t) 57 1711 y(to)c(construct)g Fl(h)g Fs(tangen)o(t)g(to)g(the)h(iden)o (tit)o(y)e(suc)o(h)g(that)i Fl(R)1122 1718 y Fh(\025)1152 1711 y Fk(\000)t Fl(h)1224 1693 y Fj(\000)p Fi(1)1282 1711 y Fk(\016)t Fl(f)c Fk(\016)t Fl(h)k Fs(=)g(0.)21 b(Let)1616 1698 y(~)1616 1711 y Fl(h)13 b Fs(=)h Fl(h)t Fk(\000)t Fs(id)57 1781 y(and)i(let)h(us)e(de\014ne)h(the)h(comp)q (osition)f(la)o(w)f Fk(\014)i Fs(as)547 1902 y(\(id)11 b(+)669 1889 y(~)669 1902 y Fl(h)698 1909 y Fi(1)720 1902 y Fs(\))g Fk(\016)g Fs(\(id)g(+)909 1889 y(~)908 1902 y Fl(h)937 1909 y Fi(2)959 1902 y Fs(\))j(=)g(id)c(+)1148 1889 y(~)1147 1902 y Fl(h)1176 1909 y Fi(1)1209 1902 y Fk(\014)1260 1889 y Fs(~)1259 1902 y Fl(h)1288 1909 y Fi(2)1324 1902 y Fl(:)388 b Fs(\(6)p Fl(:)p Fs(6\))57 2024 y(Clearly)16 b(one)g(exp)q(ects)i(that)530 2133 y(~)530 2146 y Fl(h)559 2153 y Fi(1)592 2146 y Fk(\014)642 2133 y Fs(~)641 2146 y Fl(h)670 2153 y Fi(2)706 2146 y Fs(=)759 2133 y(~)759 2146 y Fl(h)788 2153 y Fi(1)821 2146 y Fs(+)871 2133 y(~)871 2146 y Fl(h)900 2153 y Fi(2)933 2146 y Fs(+)11 b(quadratic)16 b(terms)d Fl(:)57 2267 y Fs(Let)157 2254 y(~)146 2267 y Fl(f)20 b Fs(=)13 b Fl(f)k Fk(\000)11 b Fl(R)371 2274 y Fh(\025)413 2267 y Fs(and)16 b(let)h(us)f(de\014ne)g(a)g(second)g(comp)q(osition)g(la)o (w)g Fk(\012)g Fs(as)433 2389 y Fl(R)471 2396 y Fh(\025)508 2389 y Fs(+)558 2376 y(~)558 2389 y Fl(h)11 b Fk(\012)658 2376 y Fs(~)648 2389 y Fl(f)19 b Fs(=)14 b(\(id)d(+)866 2376 y(~)865 2389 y Fl(h)p Fs(\))913 2369 y Fj(\000)p Fi(1)978 2389 y Fk(\016)g Fs(\()p Fl(R)1071 2396 y Fh(\025)1108 2389 y Fs(+)1169 2376 y(~)1158 2389 y Fl(f)6 b Fs(\))11 b Fk(\016)g Fs(\(id)g(+)1377 2376 y(~)1376 2389 y Fl(h)p Fs(\))j Fl(:)274 b Fs(\(6)p Fl(:)p Fs(7\))57 2511 y(Of)22 b(course)g(one)h(needs)532 2498 y(~)531 2511 y Fl(h)g Fs(b)q(e)g(small)e(so)h(as)h(to)g(assure)e(the)i(existence)g(of)g(the)g (in)o(v)o(erse)e(in)57 2581 y(\(6.7\))k(but)g(this)g(is)g(not)g (di\016cult)g(to)g(obtain)g(considering)e(a)j(small)e(enough)g(disk)h (since)57 2637 y(~)57 2650 y Fl(h)13 b Fs(=)h Fl(h)181 2657 y Fi(2)203 2650 y Fl(z)228 2632 y Fi(2)262 2650 y Fs(+)c Fl(:)e(:)g(:)q Fs(.)918 2770 y(43)p eop %%Page: 44 45 44 44 bop 57 192 a Fr(Exercise)24 b(6.6)19 b Fs(Recall)h(Lagrange's)e (Theorem)h(on)h(the)g(in)o(v)o(ersion)e(of)i(analytic)g(functions)57 261 y(\(see)c([Di],)g(p.)22 b(250\))16 b(:)22 b(if)17 b Fl(h)22 b Fs(:)13 b Fm(D)633 268 y Fh(r)672 261 y Fk(!)h Fm(C)28 b Fs(is)16 b(holomorphic)e(and)i(tangen)o(t)g(to)h(the)g(iden)o (tit)o(y)e(then)57 331 y(c)o(ho)q(osing)i Fl(r)j Fs(small)d(enough)h (there)g(exists)g(a)h(unique)e(solution)h Fl(z)h Fs(=)d Fl(\024)p Fs(\()p Fl(w)q Fs(\))j(of)g(the)g(equation)57 401 y Fl(w)k Fs(=)e Fl(h)p Fs(\()p Fl(z)r Fs(\).)37 b(Moreo)o(v)o(er)20 b Fl(\024)h Fs(is)g(holomorphic)e(in)i(a)g(neigh)o(b)q(orho)q(o)q(d)f (of)h(0)h(and)e(is)h(explicitly)57 470 y(giv)o(en)16 b(b)o(y)569 571 y Fl(\024)p Fs(\()p Fl(w)q Fs(\))e(=)758 508 y Fj(1)742 523 y Fe(X)740 629 y Fh(n)p Fi(=1)829 537 y Fs(\()p Fk(\000)p Fs(1\))931 519 y Fh(n)p 829 559 130 2 v 872 605 a Fl(n)p Fs(!)989 537 y Fl(d)1015 519 y Fh(n)p Fj(\000)p Fi(1)p 971 559 142 2 v 971 605 a Fl(dw)1034 590 y Fh(n)p Fj(\000)p Fi(1)1118 571 y Fs(\()p Fl(h)p Fs(\()p Fl(w)q Fs(\)\))1260 550 y Fh(n)1302 571 y Fl(:)57 711 y Fs(Get)21 b(some)f(precise)g(estimate)h(on)f(the)h(size)g(of)g (the)g(domain)e(and)h(of)h(the)g(norm)e(of)i Fl(\024)g Fs(if)g Fl(h)57 781 y Fs(b)q(elongs)15 b(to)i(some)f(Hardy{Sob)q(olev)g (space.)57 887 y(W)l(e)g(then)h(de\014ne)439 1020 y Fk(R)p Fs(\()501 1007 y(~)500 1020 y Fl(h;)562 1007 y Fs(~)551 1020 y Fl(f)6 b Fs(\))14 b(=)g(\(id)d(+)789 1007 y(~)788 1020 y Fl(h)p Fs(\))h Fk(\016)f Fl(R)922 1027 y Fh(\025)959 1020 y Fk(\000)f Fs(\()p Fl(R)1065 1027 y Fh(\025)1103 1020 y Fs(+)1163 1007 y(~)1153 1020 y Fl(f)5 b Fs(\))12 b Fk(\016)f Fs(\(id)g(+)1371 1007 y(~)1371 1020 y Fl(h)o Fs(\))j Fl(:)280 b Fs(\(6)p Fl(:)p Fs(8\))57 1153 y(Clearly)16 b Fk(R)p Fs(\(0)p Fl(;)8 b Fs(0\))15 b(=)e(0,)j Fk(R)p Fs(\(0)p Fl(;)624 1139 y Fs(~)612 1153 y Fl(f)7 b Fs(\))14 b(=)g Fk(\000)778 1139 y Fs(~)768 1153 y Fl(f)22 b Fs(and)83 1285 y Fk(R)p Fs(\()145 1272 y(~)144 1285 y Fl(h)173 1292 y Fi(1)207 1285 y Fk(\014)257 1272 y Fs(~)257 1285 y Fl(h)286 1292 y Fi(2)308 1285 y Fl(;)341 1272 y Fs(~)330 1285 y Fl(f)6 b Fs(\))14 b(=)g(\(id)c(+)568 1272 y(~)567 1285 y Fl(h)596 1292 y Fi(1)618 1285 y Fs(\))i Fk(\016)f Fs(\(id)g(+)807 1272 y(~)807 1285 y Fl(h)836 1292 y Fi(2)858 1285 y Fs(\))g Fk(\016)g Fl(R)962 1292 y Fh(\025)999 1285 y Fk(\000)g Fs(\()p Fl(R)1106 1292 y Fh(\025)1143 1285 y Fs(+)1204 1272 y(~)1193 1285 y Fl(f)6 b Fs(\))11 b Fk(\016)g Fs(\(id)g(+)1411 1272 y(~)1411 1285 y Fl(h)1440 1292 y Fi(1)1462 1285 y Fs(\))g Fk(\016)g Fs(\(id)g(+)1651 1272 y(~)1650 1285 y Fl(h)1679 1292 y Fi(2)1701 1285 y Fs(\))j Fl(;)-22 b Fs(\(6)p Fl(:)p Fs(9\))83 1370 y Fk(R)p Fs(\()145 1357 y(~)144 1370 y Fl(h)173 1377 y Fi(2)196 1370 y Fl(;)219 1357 y Fs(~)218 1370 y Fl(h)247 1377 y Fi(1)280 1370 y Fk(\012)341 1357 y Fs(~)330 1370 y Fl(f)6 b Fs(\))14 b(=)g(\(id)c(+)568 1357 y(~)567 1370 y Fl(h)596 1377 y Fi(2)618 1370 y Fs(\))i Fk(\016)f Fl(R)723 1377 y Fh(\025)760 1370 y Fk(\000)g Fs(\(id)g(+)932 1357 y(~)931 1370 y Fl(h)960 1377 y Fi(1)982 1370 y Fs(\))1001 1349 y Fj(\000)p Fi(1)1066 1370 y Fk(\016)g Fs(\()p Fl(R)1159 1377 y Fh(\025)1197 1370 y Fs(+)1257 1357 y(~)1246 1370 y Fl(f)6 b Fs(\))12 b Fk(\016)f Fs(\(id)g(+)1465 1357 y(~)1464 1370 y Fl(h)1493 1377 y Fi(1)1515 1370 y Fs(\))h Fk(\016)f Fs(\(id)g(+)1704 1357 y(~)1704 1370 y Fl(h)1733 1377 y Fi(2)1755 1370 y Fs(\))j Fl(:)-101 b Fs(\(6)p Fl(:)p Fs(10\))57 1503 y(Comparing)14 b(\(6.9\))j(with)g(\(6.10\))g(w)o (e)f(ha)o(v)o(e)g(\(for)g Fl(z)j Fs(small)d(enough\))57 1635 y(\(inf)c Fk(j)p Fs(1+)224 1622 y(~)223 1635 y Fl(h)252 1615 y Fj(0)252 1648 y Fi(1)274 1635 y Fk(j)p Fs(\))p Fk(jR)p Fs(\()383 1622 y(~)382 1635 y Fl(h)411 1642 y Fi(2)433 1635 y Fl(;)456 1622 y Fs(~)455 1635 y Fl(h)484 1642 y Fi(1)506 1635 y Fk(\012)p Fl(f)5 b Fs(\)\()p Fl(z)r Fs(\))p Fk(j)16 b(\024)e(jR)p Fs(\()815 1622 y(~)814 1635 y Fl(h)843 1642 y Fi(1)865 1635 y Fk(\014)905 1622 y Fs(~)904 1635 y Fl(h)933 1642 y Fi(2)955 1635 y Fl(;)988 1622 y Fs(~)977 1635 y Fl(f)6 b Fs(\)\()p Fl(z)r Fs(\))p Fk(j)15 b(\024)f Fs(\(sup)7 b Fk(j)p Fs(1+)1352 1622 y(~)1351 1635 y Fl(h)1380 1615 y Fj(0)1380 1648 y Fi(1)1402 1635 y Fk(j)p Fs(\))p Fk(jR)p Fs(\()1511 1622 y(~)1510 1635 y Fl(h)1539 1642 y Fi(2)1562 1635 y Fl(;)1584 1622 y Fs(~)1584 1635 y Fl(h)1613 1642 y Fi(1)1635 1635 y Fk(\012)p Fl(f)e Fs(\)\()p Fl(z)r Fs(\))p Fk(j)16 b Fl(;)57 1768 y Fs(th)o(us)f(one)i(should)e(get)214 1901 y Fl(C)254 1880 y Fj(\000)p Fi(1)307 1901 y Fk(kR)p Fs(\()394 1887 y(~)393 1901 y Fl(h)422 1908 y Fi(2)444 1901 y Fl(;)467 1887 y Fs(~)466 1901 y Fl(h)495 1908 y Fi(1)529 1901 y Fk(\012)c Fl(f)5 b Fs(\))p Fk(k)14 b(\024)g(kR)p Fs(\()806 1887 y(~)805 1901 y Fl(h)834 1908 y Fi(1)868 1901 y Fk(\014)918 1887 y Fs(~)917 1901 y Fl(h)946 1908 y Fi(2)968 1901 y Fl(;)1001 1887 y Fs(~)990 1901 y Fl(f)6 b Fs(\))p Fk(k)14 b(\024)g Fl(C)t Fk(kR)p Fs(\()1258 1887 y(~)1257 1901 y Fl(h)1286 1908 y Fi(2)1308 1901 y Fl(;)1330 1887 y Fs(~)1330 1901 y Fl(h)1359 1908 y Fi(1)1392 1901 y Fk(\012)d Fl(f)5 b Fs(\))p Fk(k)15 b Fl(:)157 b Fs(\(6)p Fl(:)p Fs(11\))57 2033 y(for)16 b(a)g(suitably)g(c)o(hosen)g(norm)f Fk(k)f(k)i Fs(and)g(some)g Fl(C)h(>)c Fs(0)k(\(see)g(Exercise)f(6.6\).) 156 2104 y(Let)23 b(us)d(no)o(w)h(try)h(to)g(solv)o(e)f(the)g(equation) h Fk(R)p Fs(\()1069 2091 y(~)1068 2104 y Fl(h;)1130 2091 y Fs(~)1119 2104 y Fl(f)6 b Fs(\))22 b(=)g(0)g(b)o(y)f(taking)g(a)h (sequence)f(of)57 2174 y(appro)o(ximations)14 b(de\014ned)h(as)h(follo) o(ws)g(:)68 2245 y(\(0\))25 b(Let)246 2231 y(~)246 2245 y Fl(h)275 2252 y Fi(0)310 2245 y Fs(=)16 b(~)-27 b Fl(g)387 2252 y Fi(0)423 2245 y Fs(=)14 b(0,)542 2231 y(~)531 2245 y Fl(f)555 2252 y Fi(0)591 2245 y Fs(=)655 2231 y(~)644 2245 y Fl(f)22 b Fs(:)g(th)o(us)16 b Fk(R)p Fs(\()898 2231 y(~)897 2245 y Fl(h)926 2252 y Fi(0)949 2245 y Fl(;)981 2231 y Fs(~)971 2245 y Fl(f)995 2252 y Fi(0)1017 2245 y Fs(\))f(=)e Fk(\000)1153 2231 y Fs(~)1142 2245 y Fl(f)1166 2252 y Fi(0)1199 2245 y Fs(;)68 2315 y(\(1\))25 b(Let)254 2302 y(~)244 2315 y Fl(f)268 2322 y Fi(1)304 2315 y Fs(=)16 b(~)-27 b Fl(g)381 2322 y Fi(0)411 2315 y Fk(\012)467 2302 y Fs(~)457 2315 y Fl(f)481 2322 y Fi(0)517 2315 y Fs(=)581 2302 y(~)570 2315 y Fl(f)594 2322 y Fi(0)617 2315 y Fs(.)21 b(Cho)q(ose)16 b(~)-27 b Fl(g)847 2322 y Fi(1)884 2315 y Fs(to)15 b(b)q(e)g(the)g(solution)f(of)h(the)g (linearized)f(equation)654 2448 y Fl(@)680 2455 y Fi(1)703 2448 y Fk(R)p Fs(\(0)p Fl(;)8 b Fs(0\))r(~)-27 b Fl(g)879 2455 y Fi(1)913 2448 y Fs(+)11 b Fl(@)989 2455 y Fi(2)1012 2448 y Fk(R)p Fs(\(0)p Fl(;)d Fs(0\))1175 2435 y(~)1164 2448 y Fl(f)1188 2455 y Fi(1)1225 2448 y Fs(=)14 b(0)g Fl(;)348 b Fs(\(6)p Fl(:)p Fs(12\))1806 2455 y Fi(1)156 2581 y Fs(where)17 b Fl(@)327 2588 y Fh(j)365 2581 y Fs(denotes)f(the)h(partial)f(deriv)m(ativ)o(e)h(w.r.t.)22 b(the)c Fl(j)s Fs({th)e(argumen)o(t.)21 b(Finally)16 b(w)o(e)156 2650 y(set)235 2637 y(~)234 2650 y Fl(h)263 2657 y Fi(1)299 2650 y Fs(=)352 2637 y(~)352 2650 y Fl(h)381 2657 y Fi(0)414 2650 y Fk(\014)c Fs(~)-27 b Fl(g)487 2657 y Fi(1)510 2650 y Fs(.)918 2770 y(44)p eop %%Page: 45 46 45 45 bop 15 192 a Fs(\(i+1\))25 b(W)l(e)17 b(c)o(ho)q(ose)411 178 y(~)400 192 y Fl(f)424 199 y Fh(i)p Fi(+1)505 192 y Fs(=)f(~)-27 b Fl(g)582 199 y Fh(i)609 192 y Fk(\012)670 178 y Fs(~)659 192 y Fl(f)683 199 y Fh(i)717 192 y Fs(and)17 b(~)-26 b Fl(g)838 199 y Fh(i)p Fi(+1)921 192 y Fs(to)17 b(b)q(e)g(the)f(solution)g(of)610 322 y Fl(@)636 329 y Fi(1)658 322 y Fk(R)p Fs(\(0)p Fl(;)8 b Fs(0\))r(~)-27 b Fl(g)834 329 y Fh(i)p Fi(+1)913 322 y Fs(+)11 b Fl(@)989 329 y Fi(2)1012 322 y Fk(R)p Fs(\(0)p Fl(;)d Fs(0\))1175 309 y(~)1164 322 y Fl(f)1188 329 y Fh(i)p Fi(+1)1270 322 y Fs(=)13 b(0)h Fl(;)259 b Fs(\(6)p Fl(:)p Fs(12\))1761 329 y Fh(i)p Fi(+1)156 453 y Fs(and)16 b(w)o(e)h(set)405 439 y(~)404 453 y Fl(h)433 460 y Fh(i)p Fi(+1)514 453 y Fs(=)567 439 y(~)566 453 y Fl(h)595 460 y Fh(i)623 453 y Fk(\014)12 b Fs(~)-26 b Fl(g)697 460 y Fh(i)p Fi(+1)763 453 y Fs(.)156 523 y(It)17 b(is)f(immediate)g(to)h(c)o(hec)o(k)f(that)h (the)g(linearized)e(equations)h(\(6.12\))1467 530 y Fh(i)1501 523 y Fs(ha)o(v)o(e)f(the)i(form)695 653 y(~)-27 b Fl(g)717 660 y Fh(i)745 653 y Fk(\016)11 b Fl(R)819 660 y Fh(\025)856 653 y Fk(\000)f Fl(R)943 660 y Fh(\025)980 653 y Fk(\016)j Fs(~)-27 b Fl(g)1040 660 y Fh(i)1071 653 y Fs(=)1134 640 y(~)1123 653 y Fl(f)1147 660 y Fh(i)1178 653 y Fl(;)509 b Fs(\(6)p Fl(:)p Fs(13\))57 784 y(whic)o(h)15 b(w)o(e)i(studied)f(in)g (Section)g(6.2.)23 b(Note)17 b(that)g(w)o(e)g(do)f Fp(not)h Fs(linearize)f(at)h(the)g(p)q(oin)o(t)f(\(0)p Fl(;)8 b(f)d Fs(\))57 853 y(since)16 b(w)o(e)g(w)o(ould)f(get)i(a)g (di\013erence)f(equation)g Fp(without)h Fs(constan)o(t)f(co)q (e\016cien)o(ts)g(:)232 984 y Fl(@)258 991 y Fi(1)280 984 y Fk(R)p Fs(\(0)p Fl(;)400 971 y Fs(~)388 984 y Fl(f)412 991 y Fh(i)p Fj(\000)p Fi(1)481 984 y Fs(\))r(~)-27 b Fl(g)524 991 y Fh(i)552 984 y Fs(+)11 b Fl(@)628 991 y Fi(2)650 984 y Fk(R)p Fs(\(0)p Fl(;)770 971 y Fs(~)758 984 y Fl(f)782 991 y Fh(i)p Fj(\000)p Fi(1)851 984 y Fs(\))881 971 y(~)870 984 y Fl(f)894 991 y Fh(i)925 984 y Fs(=)k(~)-26 b Fl(g)1002 991 y Fh(i)1029 984 y Fk(\016)11 b Fl(R)1103 991 y Fh(\025)1140 984 y Fk(\000)g Fl(R)1228 991 y Fh(\025)1265 984 y Fk(\016)i Fs(~)-27 b Fl(g)1325 991 y Fh(i)1352 984 y Fk(\000)1413 971 y Fs(~)1402 984 y Fl(f)1431 963 y Fj(0)1426 996 y Fh(i)p Fj(\000)p Fi(1)1496 984 y Fs(~)g Fl(g)1518 991 y Fh(i)1548 984 y Fs(=)14 b(0)g Fl(:)57 1114 y Fs(If)j(one)f(could)g(solv)o(e)g(\(6.13\))h(at)f (eac)o(h)h(step)f(with)g(a)h(b)q(ound)785 1245 y Fk(k)r Fs(~)-27 b Fl(g)834 1252 y Fh(i)851 1245 y Fk(k)13 b(\024)h Fl(C)t Fk(k)1017 1232 y Fs(~)1007 1245 y Fl(f)1031 1252 y Fh(i)1047 1245 y Fk(k)g Fl(;)601 b Fs(\(6)p Fl(:)p Fs(14\))57 1375 y(b)o(y)16 b(\(6.11\))h(and)f(\(6.14\))h(one)f(w)o (ould)f(ha)o(v)o(e)207 1506 y Fk(kR)p Fs(\()r(~)-27 b Fl(g)317 1513 y Fh(i)334 1506 y Fl(;)366 1493 y Fs(~)356 1506 y Fl(f)380 1513 y Fh(i)397 1506 y Fs(\))p Fk(k)14 b(\024)f Fs(sup)8 b Fk(k)p Fl(d)642 1485 y Fi(2)664 1506 y Fk(Rk)p Fs(\()p Fk(k)r Fs(~)-27 b Fl(g)799 1513 y Fh(i)816 1506 y Fk(k)841 1485 y Fi(2)874 1506 y Fs(+)11 b Fk(k)959 1493 y Fs(~)949 1506 y Fl(f)973 1513 y Fh(i)990 1506 y Fk(k)1015 1485 y Fi(2)1051 1506 y Fk(\024)i Fl(C)1143 1485 y Fi(3)1165 1506 y Fk(kR)p Fs(\()r(~)-27 b Fl(g)1275 1513 y Fh(i)p Fj(\000)p Fi(1)1343 1506 y Fl(;)1375 1493 y Fs(~)1365 1506 y Fl(f)1389 1513 y Fh(i)p Fj(\000)p Fi(1)1457 1506 y Fs(\))p Fk(k)1501 1485 y Fi(2)1537 1506 y Fl(:)150 b Fs(\(6)p Fl(:)p Fs(15\))57 1636 y(This)11 b(w)o(ould)g(imply)g(the)h(con)o(v)o(ergence)f(of)h(the)g(iterativ)o(e) g(sc)o(heme)f(to)h(a)g(solution)f(of)h Fk(R)p Fs(\()1676 1623 y(~)1675 1636 y Fl(h)q(;)1737 1623 y Fs(~)1727 1636 y Fl(f)5 b Fs(\))15 b(=)57 1706 y(0)k(pro)o(vided)e(that)i(one)g(c)o (ho)q(oses)f Fk(k)727 1693 y Fs(~)717 1706 y Fl(f)5 b Fk(k)19 b Fs(small)f(enough)g(\(i.e.)29 b(one)18 b(considers)f(the)i (restriction)57 1776 y(of)d Fl(f)23 b Fs(to)17 b(a)f(small)g(enough)f (disk)h Fm(D)701 1783 y Fh(r)726 1776 y Fs(\))h(:)22 b(indeed)16 b(iterating)g(\(6.15\))h(one)f(gets)580 1906 y Fk(kR)p Fs(\()r(~)-27 b Fl(g)690 1913 y Fh(i)707 1906 y Fl(;)740 1893 y Fs(~)729 1906 y Fl(f)753 1913 y Fh(i)770 1906 y Fs(\))p Fk(k)14 b(\024)g Fs(\()p Fl(C)940 1886 y Fi(3)p Fh(=)p Fi(2)1002 1906 y Fk(kR)p Fs(\()r(~)-27 b Fl(g)1112 1913 y Fi(0)1134 1906 y Fl(;)1167 1893 y Fs(~)1156 1906 y Fl(f)1180 1913 y Fi(0)1203 1906 y Fs(\))p Fk(k)p Fs(\))1266 1886 y Fi(2)1286 1871 y Fb(i)57 2037 y Fs(th)o(us)15 b(again)h(b)o(y)g(\(6.11\))h(one)g(has)498 2161 y Fk(kR)p Fs(\()585 2148 y(~)584 2161 y Fl(h)613 2168 y Fh(i)630 2161 y Fl(;)662 2148 y Fs(~)652 2161 y Fl(f)5 b Fs(\))p Fk(k)15 b Fs(=)e Fk(kR)p Fs(\()r(~)-27 b Fl(g)902 2168 y Fi(0)936 2161 y Fk(\014)12 b Fs(~)-26 b Fl(g)1010 2168 y Fi(1)1043 2161 y Fk(\014)11 b Fl(:)d(:)g(:)j Fk(\014)h Fs(~)-26 b Fl(g)1236 2168 y Fh(i)1252 2161 y Fl(;)1285 2148 y Fs(~)1274 2161 y Fl(f)6 b Fs(\))p Fk(k)740 2246 y(\024)13 b Fl(C)t Fk(kR)p Fs(\()r(~)-27 b Fl(g)942 2253 y Fi(1)975 2246 y Fk(\014)11 b Fl(:)d(:)g(:)j Fk(\014)h Fs(~)-26 b Fl(g)1168 2253 y Fh(i)1184 2246 y Fl(;)10 b Fs(~)-27 b Fl(g)1230 2253 y Fi(0)1263 2246 y Fk(\012)1324 2233 y Fs(~)1313 2246 y Fl(f)6 b Fs(\))p Fk(k)740 2331 y(\024)13 b Fl(C)832 2310 y Fh(i)848 2331 y Fk(kR)p Fs(\()r(~)-27 b Fl(g)958 2338 y Fh(i)975 2331 y Fl(;)1008 2317 y Fs(~)997 2331 y Fl(f)1021 2338 y Fh(i)1038 2331 y Fs(\))p Fk(k)740 2415 y(\024)13 b Fs(\()p Fl(C)851 2395 y Fi(2)873 2415 y Fk(kR)p Fs(\()r(~)-27 b Fl(g)983 2422 y Fi(0)1006 2415 y Fl(;)1038 2402 y Fs(~)1028 2415 y Fl(f)1052 2422 y Fi(0)1074 2415 y Fs(\))p Fk(k)p Fs(\))1137 2395 y Fi(2)1157 2380 y Fb(i)57 2581 y Fr(Exercise)17 b(6.7)d Fs(Assuming)f(the)i(estimates)f(ab)q(o)o(v)o(e)g(sho)o(w)f (that)i(the)g(sequence)1552 2568 y(~)1551 2581 y Fl(h)1580 2588 y Fh(n)1622 2581 y Fs(con)o(v)o(erges)57 2650 y(th)o(us)g(b)o(y)h (con)o(tin)o(uit)o(y)g(of)g Fk(R)h Fs(one)g(gets)f(the)h(desired)e (result.)918 2770 y(45)p eop %%Page: 46 47 46 46 bop 57 192 a Fr(Exercise)29 b(6.8)23 b Fs(Use)i(the)f(ab)q(o)o(v) o(e)g(sc)o(heme)g(to)g(giv)o(e)h(an)f(alternativ)o(e)g(pro)q(of)g(of)g (Ko)q(enigs{)57 261 y(P)o(oincar)o(\023)-24 b(e)15 b(theorem.)57 366 y(The)29 b(ab)q(o)o(v)o(e)g(discussion)f(sho)o(ws)g(ho)o(w)h(to)h (pro)o(v)o(e)e(the)i(existence)g(of)g(the)g(linearization)57 436 y(disregarding)19 b(the)j(problem)e(of)i(loss)f(of)h(di\013eren)o (tiabilit)o(y)e(due)h(to)h(small)f(divisors.)36 b(This)57 506 y(mak)o(es)18 b(imp)q(ossible)g(to)i(get)g(an)f(estimate)g(lik)o(e) h(\(6.14\))g Fp(unless)g(one)h(r)m(e)m(gularizes)h(the)f(r.h.s.)p Fs(.)57 576 y(The)14 b(simplest)g(metho)q(d)g(of)h(regolarization,)e (whic)o(h)h(is)g(adapted)g(to)h(the)g(analytic)g(case,)g(is)f(to)57 645 y Fp(c)m(onsider)19 b(r)m(estrictions)f Fs(of)f(the)g(domains)d(:) 57 751 y Fr(Exercise)20 b(6.9)c Fs(Sho)o(w)f(that)i(if)g Fl(f)i Fk(2)14 b(O)790 733 y Fi(0)p Fh(;)p Fi(2)789 763 y Fh(r)844 751 y Fs(,)j Fl(f)5 b Fs(\(0\))15 b(=)f(0,)i Fl(k)f Fk(2)f Fm(N)p Fs(,)k(for)e(all)g Fl(\016)g(>)e Fs(0)i(one)g(has)603 912 y Fk(k)p Fl(f)5 b Fk(k)682 929 y Fj(O)715 912 y Fb(k;)p Fc(2)714 950 y Fb(r)q(e)749 943 y Ff(\000)p Fb(\016)814 912 y Fk(\024)867 842 y Fe(\022)910 878 y Fl(k)p 910 900 28 2 v 912 946 a(\016)943 842 y Fe(\023)980 852 y Fh(k)1013 912 y Fl(e)1036 891 y Fj(\000)p Fh(k)1091 912 y Fk(k)p Fl(f)g Fk(k)1170 928 y Fj(O)1203 911 y Fc(0)p Fb(;)p Fc(2)1202 934 y Fb(r)1268 912 y Fl(:)419 b Fs(\(6)p Fl(:)p Fs(16\))57 1097 y(Com)o(bining)18 b(the)i(ab)q(o)o(v) o(e)g(giv)o(en)g(discussion)e(with)i(a)h(suitable)e(c)o(hoice)h(of)g (restrictions)f(\(i.e.)57 1167 y(a)24 b(sequence)h(\()p Fl(\016)362 1174 y Fh(n)390 1167 y Fs(\))409 1174 y Fh(n)p Fj(\025)p Fi(0)512 1167 y Fs(suc)o(h)f(that)749 1129 y Fe(P)801 1141 y Fj(1)801 1181 y Fh(n)p Fi(=0)887 1167 y Fl(\016)909 1174 y Fh(n)964 1167 y Fl(<)j Fs(+)p Fk(1)p Fs(\))e(one)f(can)h(indeed)f(pro)o(v)o(e)f(Siegel's)57 1236 y(Theorem)15 b(follo)o(wing)g(the)i(iteration)f(metho)q(d.)918 2770 y(46)p eop %%Page: 47 48 47 47 bop 72 192 a Fq(P)n(art)24 b(I)r(I.)g(Implicit)i(F)-6 b(unction)24 b(Theorems)f(and)h(KAM)g(Theory)57 501 y(7.)31 b(Hamiltonian)25 b(Systems)f(and)f(In)n(tegrable)j(Systems)57 609 y Fs(In)20 b(this)g(Chapter)g(w)o(e)g(will)g(v)o(ery)g(brie\015y)g (recall)g(some)g(w)o(ell{kno)o(wn)f(facts)i(on)f(symplectic)57 678 y(manifolds)15 b(and)g(Hamiltonian)h(systems.)21 b(V)l(ery)c(go)q(o)q(d)g(references)e(are)i([AKN])f(and)g([AM].)57 856 y Fo(7.1)j(Symplectic)h(Manifolds)h(and)g(Hamiltonian)h(Systems)57 1003 y Fr(De\014nition)g(7.1)45 b Fd(A)18 b Fk(C)529 985 y Fj(1)589 1003 y Fs(symplectic)g(manifold)e Fd(is)h(a)h Fs(2)p Fl(l)q Fd({dimensional)d Fk(C)1507 985 y Fj(1)1567 1003 y Fd(manifold)i Fl(M)57 1073 y Fd(equipp)q(ed)11 b(with)h(a)g(non{degenerate)f Fk(C)790 1055 y Fj(1)845 1073 y Fd(t)o(w)o(o{form)f(\(the)j(symplectic)e(form\))h Fl(!)r Fd(.)20 b(A)13 b Fk(C)1680 1055 y Fj(1)1734 1073 y Fd(map)57 1143 y Fl(f)33 b Fs(:)18 b Fl(U)24 b Fk(!)19 b Fl(M)326 1125 y Fj(0)360 1143 y Fd(where)g Fl(U)24 b Fk(\032)19 b Fl(M)25 b Fd(is)19 b(op)q(en)h(and)f Fl(M)1026 1125 y Fj(0)1060 1143 y Fd(is)g(also)g(symplectic)g(\(with)h (symplectic)57 1213 y(form)15 b Fl(!)207 1195 y Fj(0)221 1213 y Fd(\))i(is)f Fs(symplectic)g Fd(\(or)h Fs(canonical)p Fd(\))f(if)h Fl(f)946 1195 y Fj(\003)969 1213 y Fl(!)1002 1195 y Fj(0)1030 1213 y Fs(=)c Fl(!)r Fd(.)57 1357 y Fs(The)j(simplest)f(\(but)i(imp)q(ortan)o(t\))f(examples)g(of)g (symplectic)g(manifolds)f(are)h(:)107 1429 y Fk(\017)24 b Fl(M)j Fs(=)21 b Fm(R)330 1411 y Fi(2)p Fh(l)383 1429 y Fk(3)g Fs(\()p Fl(p)481 1436 y Fi(1)504 1429 y Fl(;)8 b(:)g(:)g(:)g(;)g(p)639 1436 y Fh(l)655 1429 y Fl(;)g(q)699 1436 y Fi(1)722 1429 y Fl(;)g(:)g(:)g(:)g(;)g(q)854 1436 y Fh(l)870 1429 y Fs(\),)22 b Fl(!)h Fs(=)1038 1392 y Fe(P)1091 1404 y Fh(l)1091 1444 y(i)p Fi(=1)1166 1429 y Fl(dp)1217 1436 y Fh(i)1248 1429 y Fk(^)14 b Fl(dq)1343 1436 y Fh(i)1381 1429 y Fs(\()p Fp(standar)m(d)24 b(symple)m(ctic)156 1499 y(structur)m(e)p Fs(\).)32 b(If)20 b Fl(U)26 b Fs(and)19 b Fl(V)31 b Fs(are)20 b(t)o(w)o(o)f(op)q(en)h(sets)g(in)f Fm(R)1196 1481 y Fi(2)p Fh(l)1249 1499 y Fs(and)g Fl(f)33 b Fs(:)20 b Fl(U)25 b Fk(!)19 b Fl(V)31 b Fs(then)20 b Fl(f)26 b Fs(is)156 1569 y(symplectic)18 b(if)f(and)g(only)h(if)g (its)f(Jacobian)g(matrix)g Fl(J)1176 1576 y Fh(f)1217 1569 y Fk(2)f Fs(Sp)8 b(\()p Fl(l)q(;)g Fm(R)p Fs(\),)15 b(the)j(Lie)g(group)e(of)156 1660 y(2)p Fl(l)c Fk(\002)f Fs(2)p Fl(l)17 b Fs(real)f(matrices)g Fl(A)h Fs(suc)o(h)e(that)i Fl(A)923 1642 y Fh(T)955 1660 y Fk(I)t Fl(A)d Fs(=)g Fk(I)t Fs(,)i(where)g Fk(I)h Fs(=)1392 1590 y Fe(\022)1437 1630 y Fs(0)50 b Fk(\000)p Fs(1)1437 1690 y(1)69 b(0)1584 1590 y Fe(\023)1620 1660 y Fs(.)107 1750 y Fk(\017)24 b Fl(M)c Fs(=)13 b Fl(T)312 1732 y Fj(\003)335 1750 y Fl(N)k Fs(where)11 b Fl(N)16 b Fs(is)11 b(a)g Fk(C)697 1732 y Fj(1)750 1750 y Fs(Riemannian)e(manifold.)19 b(This)10 b(is)h(the)g(t)o(ypical)g(situation)156 1820 y(in)g(classical)g(mec)o (hanics.)19 b(If)11 b(\()p Fl(q)736 1827 y Fi(1)759 1820 y Fl(;)d(:)g(:)g(:)h(;)f(q)892 1827 y Fh(l)908 1820 y Fs(\))k(are)f(lo)q(cal)g(co)q(ordinates)g(in)g Fl(N)17 b Fs(and)11 b(\()p Fl(p)1635 1827 y Fi(1)1658 1820 y Fl(;)d(:)g(:)g(:)g(;)g(p)1793 1827 y Fh(l)1809 1820 y Fs(\))156 1890 y(are)k(the)h(corresp)q(onding)d(lo)q(cal)j(co)q (ordinates)e(in)i(the)f(cotangen)o(t)h(space)f(at)g(a)h(p)q(oin)o(t,)f (then)156 1959 y Fl(!)k Fs(=)256 1922 y Fe(P)308 1934 y Fh(l)308 1974 y(i)p Fi(=1)383 1959 y Fl(dp)434 1966 y Fh(i)462 1959 y Fk(^)11 b Fl(dq)554 1966 y Fh(i)571 1959 y Fs(.)107 2032 y Fk(\017)24 b Fl(M)c Fs(=)13 b Fm(T)313 2014 y Fi(2)p Fh(l)345 2032 y Fs(,)j Fl(!)g Fs(=)475 1995 y Fe(P)527 2007 y Fh(l)527 2047 y(i)p Fi(=1)602 2032 y Fl(d\022)651 2039 y Fh(i)679 2032 y Fk(^)c Fl(d\022)773 2039 y Fh(i)p Fi(+)p Fh(l)833 2032 y Fs(.)57 2179 y Fr(Theorem)f(7.2)h (\(Darb)r(oux\))27 b Fd(Eac)o(h)11 b(symplectic)f(manifold)g Fl(M)17 b Fd(has)10 b(an)h(atlas)g Fs(\()p Fl(U)1610 2186 y Fh(\013)1639 2179 y Fl(;)d(')1694 2186 y Fh(\013)1722 2179 y Fs(\))1741 2186 y Fh(\013)p Fj(2A)57 2249 y Fd(suc)o(h)j(that)h (on)g Fl(')366 2256 y Fh(\013)394 2249 y Fs(\()p Fl(U)447 2256 y Fh(\013)476 2249 y Fs(\))i Fk(\032)g Fm(R)601 2231 y Fi(2)o Fh(l)645 2249 y Fd(one)e(has)f Fl(!)16 b Fs(=)e Fl(')949 2231 y Fj(\003)949 2261 y Fh(\013)985 2212 y Fe(P)1038 2224 y Fh(l)1038 2264 y(i)p Fi(=1)1113 2249 y Fl(dp)1164 2256 y Fh(i)1183 2249 y Fk(^)r Fl(dq)1266 2256 y Fh(i)1295 2249 y Fd(\(the)f(standard)e(symplectic)57 2319 y(structure)k(on)h Fm(R)378 2301 y Fi(2)p Fh(l)410 2319 y Fd(\).)22 b(The)16 b(transition)f(maps)g Fl(')957 2326 y Fh(\013)996 2319 y Fk(\016)10 b Fl(')1064 2297 y Fj(\000)p Fi(1)1064 2334 y Fh(\014)1133 2319 y Fd(are)16 b(symplectic)g(di\013eomorphisms,)57 2388 y(i.e.)21 b(their)c (Jacobians)e Fl(J)518 2395 y Fh(\013\014)570 2388 y Fs(\()p Fl(x)p Fs(\))g Fk(2)f Fd(Sp)8 b Fs(\()p Fl(l)q(;)g Fm(R)p Fs(\))14 b Fd(for)i(all)g Fl(x)f Fk(2)f Fl(')1159 2395 y Fh(\014)1185 2388 y Fs(\()p Fl(U)1238 2395 y Fh(\013)1278 2388 y Fk(\\)d Fl(U)1356 2395 y Fh(\014)1383 2388 y Fs(\))p Fd(.)57 2533 y Fs(The)19 b(atlas)h(giv)o(en)f(b)o(y)g(Darb)q(oux's)g (Theorem)f(and)h(the)h(corresp)q(onding)e(lo)q(cal)h(co)q(ordinates)57 2602 y(are)d(called)g(symplectic.)918 2770 y(47)p eop %%Page: 48 49 48 48 bop 57 192 a Fr(De\014nition)28 b(7.3)g Fd(A)c Fs(Hamiltonian)e(function)h Fd(on)f(a)i(symplectic)f(manifold)f Fs(\()p Fl(M)s(;)8 b(!)r Fs(\))24 b Fd(is)e(a)57 261 y(function)g Fl(H)28 b Fk(2)23 b(C)412 243 y Fj(1)455 261 y Fs(\()p Fl(M)s(;)8 b Fm(R)p Fs(\))p Fd(.)37 b(The)22 b Fs(Hamiltonian)f(v)o(ector)i(\014eld)f Fd(asso)q(ciated)g(to)g Fl(H)27 b Fd(is)22 b(the)57 331 y(unique)16 b Fl(X)260 338 y Fh(H)311 331 y Fk(2)f(C)388 313 y Fj(1)430 331 y Fs(\()p Fl(M)s(;)8 b(T)f(M)e Fs(\))18 b Fd(suc)o(h)d(that)i Fl(i)885 338 y Fh(X)918 343 y Fb(H)953 331 y Fl(!)e Fs(=)f Fl(dH)t Fd(.)57 473 y Fs(Note)g(that)f(in)g(symplectic)f(lo)q(cal)h(co) q(ordinates)g(a)f(Hamiltonian)g(v)o(ector)h(\014eld)g(tak)o(es)g(the)g (form)598 646 y Fl(X)639 653 y Fh(H)691 646 y Fs(=)773 584 y Fh(l)743 599 y Fe(X)747 705 y Fh(i)p Fi(=1)824 646 y Fk(\000)869 612 y Fl(@)s(H)p 869 635 75 2 v 872 680 a(@)s(q)923 687 y Fh(i)976 612 y Fl(@)p 955 635 71 2 v 955 680 a(@)s(p)1009 687 y Fh(i)1043 646 y Fs(+)1099 612 y Fl(@)s(H)p 1099 635 75 2 v 1101 680 a(@)s(p)1155 687 y Fh(i)1205 612 y Fl(@)p 1185 635 69 2 v 1185 680 a(@)s(q)1236 687 y Fh(i)1273 646 y Fl(;)439 b Fs(\(7)p Fl(:)p Fs(1\))57 816 y(and)25 b(the)g(asso)q(ciated)h(ordinary)e (di\013eren)o(tial)g(equations)h(are)g(the)h(classical)e(Hamilton's)57 886 y(equations)e(of)h(the)g(motion)f(of)h(a)g(conserv)m(ativ)o(e)g (mec)o(hanical)e(system)i(with)f Fl(l)i Fs(degrees)e(of)57 956 y(freedom)15 b(:)571 1032 y(_)-28 b Fl(p)582 1039 y Fh(i)612 1032 y Fs(=)14 b Fk(\000)710 999 y Fl(@)s(H)p 710 1021 75 2 v 713 1067 a(@)s(q)764 1074 y Fh(i)804 1032 y Fl(;)49 b Fs(_)-27 b Fl(q)876 1039 y Fh(i)907 1032 y Fs(=)965 999 y Fl(@)s(H)p 965 1021 V 967 1067 a(@)s(p)1021 1074 y Fh(i)1060 1032 y Fl(;)36 b Fs(1)13 b Fk(\024)h Fl(i)g Fk(\024)g Fl(l)g(:)398 b Fs(\(7)p Fl(:)p Fs(2\))57 1160 y(Clearly)20 b(the)g(Hamiltonian)g(function)g(is) g(a)g(\014rst)g(in)o(tegral)f(of)i(\(7.2\).)34 b(The)20 b(co)q(ordinates)g Fl(q)1812 1167 y Fh(i)57 1230 y Fs(are)e(also)h (called)g(\\generalized)f(co)q(ordinates")g(and)g(the)h Fl(p)1178 1237 y Fh(i)1214 1230 y Fs(their)g(\\conjugate)g(momen)o (ta".)57 1300 y(In)14 b(man)o(y)g(problems)f(arising)h(from)g (celestial)h(mec)o(hanics)e(the)i(\015o)o(w)f(is)h(not)g(complete)g (due)f(to)57 1370 y(the)19 b(una)o(v)o(oidable)e(o)q(ccurance)h(of)h (collisions,)f(but)h(w)o(e)g(will)f(alw)o(a)o(ys)g(assume)g (completeness)57 1439 y(of)e(the)h(Hamiltonian)f(\015o)o(w.)57 1584 y Fr(De\014nition)23 b(7.4)48 b Fd(The)19 b Fs(P)o(oisson)f(brac)o (k)o(et)g Fd(of)i(t)o(w)o(o)f(functions)g Fl(f)s(;)8 b(g)20 b Fk(2)f(C)1464 1566 y Fj(1)1506 1584 y Fs(\()p Fl(M)s(;)8 b Fm(R)p Fs(\))17 b Fd(de\014ned)57 1654 y(on)j(an)h(op)q (en)g(subset)f(of)h Fs(\()p Fl(M)s(;)8 b(!)r Fs(\))22 b Fd(is)f Fk(f)p Fl(F)q(;)8 b(G)p Fk(g)22 b Fs(:=)e Fl(X)1048 1661 y Fh(G)1082 1654 y Fl(F)29 b Fs(=)21 b Fl(!)r Fs(\()p Fl(X)1296 1661 y Fh(F)1329 1654 y Fl(;)8 b(X)1392 1661 y Fh(G)1426 1654 y Fs(\))22 b(=)f Fk(\000)p Fl(X)1607 1661 y Fh(F)1640 1654 y Fl(G)g(;)g Fd(th)o(us)57 1723 y Fl(X)98 1732 y Fj(f)p Fh(F)q(;G)p Fj(g)230 1723 y Fs(=)e Fk(\000)p Fs([)p Fl(X)382 1730 y Fh(F)415 1723 y Fl(;)8 b(X)478 1730 y Fh(G)512 1723 y Fs(])20 b Fl(:)g Fd(Tw)o(o)f(functions)g Fl(F)q(;)8 b(G)21 b Fd(are)f Fs(in)f(in)o(v)o(olution)f Fd(if)j Fk(f)p Fl(F)q(;)8 b(G)p Fk(g)20 b Fs(=)f(0)p Fd(,)i(i.)e(e.)57 1793 y(when)d(their)g(hamiltonian)f(\015o)o(ws)g (comm)o(ute.)57 1935 y Fr(Exercise)23 b(7.5)18 b Fs(Sho)o(w)g(that)h (the)g Fp(Hamiltonian)i(\015ow)f Fs(\010)26 b(:)18 b Fm(R)10 b Fk(\002)i Fl(M)24 b Fk(!)18 b Fl(M)24 b Fs(is)19 b(symplectic)g(:)57 2005 y(for)c(all)h Fl(t)e Fk(2)g Fm(R)f Fs(one)i(has)h(\010\()p Fl(t;)8 b Fk(\001)p Fs(\))639 1987 y Fj(\003)662 2005 y Fl(!)16 b Fs(=)d Fl(!)r Fs(.)22 b([Hin)o(t)16 b(:)21 b(use)16 b(Cartan's)f(form)o(ula)1475 1985 y Fh(d)p 1468 1993 36 2 v 1468 2022 a(dt)1510 2005 y Fk(j)1524 2012 y Fh(t)p Fi(=0)1591 2005 y Fs(\010\()p Fl(t;)8 b Fk(\001)p Fs(\))1719 1987 y Fj(\003)1743 2005 y Fl(!)16 b Fs(=)57 2075 y Fl(d)p Fs(\()p Fl(i)119 2082 y Fh(X)152 2087 y Fb(H)186 2075 y Fl(!)r Fs(\))j(+)f Fl(i)331 2082 y Fh(X)364 2087 y Fb(H)398 2075 y Fl(d!)r Fs(,)30 b(where)c Fl(X)696 2082 y Fh(H)766 2075 y Fs(=)850 2055 y Fh(d)p 843 2063 V 843 2092 a(dt)884 2075 y Fk(j)898 2082 y Fh(t)p Fi(=0)966 2075 y Fs(\010\()p Fl(t;)8 b Fk(\001)p Fs(\))28 b(is)f(the)h(Hamiltonian)e(v)o(ector)h(\014eld)57 2144 y(asso)q(ciated)16 b(to)h(\010.])57 2252 y(The)22 b(imp)q(ortance)g(of)h(Exercise)f(7.5)h(is)f(that)h(to)g(mak)o(e)f (symplectic)g(co)q(ordinate)h(c)o(hanges)57 2322 y(of)d(a)g (Hamiltonian)f(v)o(ector)h(\014eld)f(it)h(is)g(su\016cien)o(t)e(to)i(c) o(hange)g(the)g(v)m(arables)f(in)g(the)i(corre-)57 2391 y(sp)q(onding)e(Hamiltonian)h(function.)36 b(This)20 b(is)h(a)g(simpler)e(op)q(eration,)j(b)q(oth)f(conceptually)57 2461 y(and)16 b(computationally)l(.)156 2533 y(As)h(w)o(e)g(will)f(see) h(in)g(the)g(next)g(Section,)g(among)f(the)h(p)q(ossible)f(orbits)f(of) j(Hamiltonian)57 2603 y(systems,)d Fp(quasip)m(erio)m(d)q(ic)k Fs(orbits)d(are)g(of)h(sp)q(ecial)f(in)o(terest.)918 2770 y(48)p eop %%Page: 49 50 49 49 bop 57 192 a Fr(De\014nition)22 b(7.6)28 b Fd(A)18 b(con)o(tin)o(uous)e(function)h Fl(F)31 b Fs(:)16 b Fm(R)d Fk(!)j Fm(R)f Fd(is)i Fs(quasip)q(erio)q(dic)g Fd(if)h(there)g(exist)57 261 y Fl(n)13 b Fk(\025)h Fs(2)p Fd(,)i Fl(f)28 b Fs(:)14 b Fm(T)324 243 y Fh(n)362 261 y Fk(!)g Fm(R)f Fd(con)o(tin)o(uous)i (and)h Fl(\027)g Fk(2)e Fm(R)952 243 y Fh(n)987 261 y Fk(n)d(f)p Fs(0)p Fk(g)17 b Fd(suc)o(h)e(that)i Fl(F)7 b Fs(\()p Fl(t)p Fs(\))14 b(=)g Fl(f)5 b Fs(\()p Fl(\027)1570 268 y Fi(1)1593 261 y Fl(t;)j(:)g(:)g(:)h(\027)1725 268 y Fh(n)1751 261 y Fl(t)p Fs(\))p Fd(.)57 395 y Fs(Let)18 b Fk(M)c Fs(=)h Fk(f)p Fl(k)h Fk(2)f Fm(Z)427 377 y Fh(n)474 395 y Fk(j)23 b Fl(k)13 b Fk(\001)e Fl(\027)18 b Fs(=)d(0)p Fk(g)p Fs(.)23 b(Note)18 b(that)g Fk(M)f Fs(is)g(a)g Fm(Z)-11 b Fs({mo)q(du)o(le.)21 b(If)d(dim)7 b Fk(M)15 b Fs(=)f Fl(n)j Fs(then)57 465 y Fl(\027)f Fs(=)e(0,)i(if)h(dim)7 b Fk(M)14 b Fs(=)f Fl(n)e Fk(\000)f Fs(1)17 b(then)f(there)g(exists)g Fl(\013)e Fk(2)g Fm(R)g Fs(and)h Fl(k)h Fk(2)e Fm(Z)1345 447 y Fh(n)1386 465 y Fs(suc)o(h)h(that)h Fl(\027)h Fs(=)d Fl(\013k)r Fs(.)21 b(If)57 535 y(dim)7 b Fk(M)14 b Fs(=)f(0)k(then)f Fl(\027)k Fs(is)c(called)g Fp(non{r)m(esonant)p Fs(.)57 640 y Fr(Exercise)j(7.7)c Fs(Sho)o(w)g(that)h(the)h(closure)e(of)h(an)o (y)f(orbit)g(of)i(the)f(linear)f(\015o)o(w)1525 627 y(_)1515 640 y Fl(\022)h Fs(=)d Fl(\027)19 b Fs(on)d Fm(T)1755 622 y Fh(n)1795 640 y Fs(is)57 710 y(di\013eomorphic)e(to)j(the)g (torus)f Fm(T)679 691 y Fh(n)o Fj(\000)p Fi(dim)t Fj(M)854 710 y Fs(.)57 815 y Fr(Exercise)j(7.8)d Fs(Sho)o(w)f(that)h(if)h(dim)7 b Fk(M)14 b(2)g(f)p Fs(1)p Fl(;)8 b(:)g(:)g(:)g(n)i Fk(\000)h Fs(1)p Fk(g)16 b Fs(there)g(exists)g Fl(A)e Fk(2)g Fs(SL)8 b(\()p Fl(n;)g Fm(Z)-10 b Fs(\))14 b(suc)o(h)57 884 y(that)k(p)q(osing) e Fl(')f Fs(=)h Fl(A\022)j Fs(the)f(linear)f(\015o)o(w)846 871 y(_)836 884 y Fl(\022)h Fs(=)d Fl(\027)20 b Fs(on)d Fm(T)1083 866 y Fh(n)1124 884 y Fs(b)q(ecomes)31 b(_)-28 b Fl(')1356 891 y Fh(i)1388 884 y Fs(=)15 b(0)j(for)f Fl(i)f Fs(=)f(1)p Fl(;)8 b(:)g(:)g(:)g(;)g(m)57 954 y Fs(and)29 b(_)-27 b Fl(')187 961 y Fh(i)217 954 y Fs(=)13 b Fl(\027)297 936 y Fj(0)294 967 y Fh(i)327 954 y Fs(for)k Fl(i)d Fs(=)f Fl(m)e Fs(+)g(1)p Fl(;)d(:)g(:)g(:)g(;)g(n)17 b Fs(with)g(\()p Fl(\027)935 936 y Fj(0)932 967 y Fh(m)p Fi(+1)1020 954 y Fl(;)8 b(:)g(:)g(:)g(;)g(\027)1158 936 y Fj(0)1155 967 y Fh(n)1182 954 y Fs(\))14 b Fk(2)g Fm(R)1301 936 y Fh(n)p Fj(\000)p Fh(m)1409 954 y Fs(non{resonan)o(t.)57 1059 y Fr(Exercise)26 b(7.9)21 b Fs(Sho)o(w)f(that)i(if)g Fl(\027)j Fs(is)c(non{resonan)o(t)e(then)j(the)g(Haar)f(measure)f(on)i Fm(T)1749 1041 y Fh(n)1795 1059 y Fs(is)57 1129 y(uniquely)16 b(ergo)q(dic)g(\(see)h([Mn])e(for)i(its)f(de\014nition\))g(for)g(the)h (linear)e(\015o)o(w)1450 1116 y(_)1440 1129 y Fl(\022)h Fs(=)d Fl(\027)20 b Fs(on)c Fm(T)1681 1111 y Fh(n)1705 1129 y Fs(.)57 1304 y Fo(7.2)j(In)n(tegrable)i(Systems)57 1409 y Fs(An)13 b(esp)q(ecially)g(in)o(teresting)f(example)h(of)g (symplectic)g(manifold)e(is)i Fl(M)20 b Fs(=)13 b Fm(R)1493 1391 y Fh(l)1510 1409 y Fk(\002)t Fm(T)1589 1391 y Fh(l)1615 1409 y Fs(whic)o(h)f(can)57 1479 y(b)q(e)h(iden)o(ti\014ed)e(with)i (the)g(cotangen)o(t)g(bundle)f(of)h(the)g Fl(l)q Fs({dimensional)d (torus)i Fm(T)1517 1461 y Fh(l)1543 1479 y Fs(=)i Fm(R)1635 1461 y Fh(l)1647 1479 y Fl(=)p Fs(\(2)p Fl(\031)r Fm(Z)-10 b Fs(\))1802 1461 y Fh(l)1815 1479 y Fs(.)57 1549 y(This)21 b(manifold)g(has)h(a)g(natural)f(symplectic)h(structure)g(de\014ned)f (b)o(y)h(the)h(closed)e(2{form)57 1618 y Fl(!)15 b Fs(=)156 1581 y Fe(P)208 1593 y Fh(l)208 1633 y(i)p Fi(=1)284 1618 y Fl(dJ)338 1625 y Fh(i)365 1618 y Fk(^)c Fl(d#)464 1625 y Fh(i)498 1618 y Fs(where)16 b(\()p Fl(J)689 1625 y Fi(1)711 1618 y Fl(;)8 b(:)g(:)g(:)h(J)828 1625 y Fh(l)843 1618 y Fl(;)f(#)894 1625 y Fi(1)917 1618 y Fl(;)g(:)g(:)g(:)g(#)1034 1625 y Fh(l)1050 1618 y Fs(\))17 b(are)f(co)q(ordinates)g(on)g Fm(R)1542 1600 y Fh(l)1565 1618 y Fk(\002)11 b Fm(T)1651 1600 y Fh(l)1664 1618 y Fs(.)57 1752 y Fr(De\014nition)26 b(7.10)h Fd(Let)22 b Fl(U)27 b Fd(denote)21 b(an)g(op)q(en)g(connected) h(subset)e(of)i Fm(R)1474 1734 y Fh(l)1487 1752 y Fd(.)36 b(Whenev)o(er)21 b(an)57 1822 y(Hamiltonian)e(system)h(can)h(b)q(e)f (reduced)g(b)o(y)g(a)g(symplectic)h(c)o(hange)e(of)i(co)q(ordinates)f (to)g(a)57 1892 y(function)d Fl(H)29 b Fs(:)15 b Fl(U)j Fk(\002)11 b Fm(T)491 1874 y Fh(l)519 1892 y Fk(!)16 b Fm(R)e Fd(whic)o(h)j Fs(do)q(es)h(not)g(dep)q(end)f(on)g(the)h (angular)e(v)m(ariables)h Fl(#)i Fd(one)57 1962 y(sa)o(ys)d(that)i(the) g(system)f(is)g Fs(completely)h(canonically)e(in)o(tegrable)h Fd(and)f(the)i(v)m(ariables)f Fl(J)22 b Fd(are)57 2031 y(called)16 b Fs(action)g(v)m(ariables)p Fd(.)57 2165 y Fs(Note)e(that)g(in)g(this)f(case)h(Hamilton's)f(equations)g(\(7.2\)) h(tak)o(e)g(the)g(particularly)f(simple)f(form)499 2293 y(_)484 2305 y Fl(J)512 2312 y Fh(i)542 2305 y Fs(=)i Fk(\000)640 2272 y Fl(@)s(H)p 640 2294 76 2 v 640 2340 a(@)s(#)698 2347 y Fh(i)734 2305 y Fs(=)g(0)g Fl(;)906 2292 y Fs(_)889 2305 y Fl(#)918 2312 y Fh(i)949 2305 y Fs(=)1008 2272 y Fl(@)s(H)p 1008 2294 75 2 v 1009 2340 a(@)s(J)1066 2347 y Fh(i)1102 2305 y Fl(;)50 b(i)14 b Fs(=)g(1)p Fl(;)8 b(:)g(:)g(:)g(;)g(l)57 2441 y Fs(and)17 b(the)i(\015o)o(w)f(lea)o(v)o(es)g(in)o(v)m(arian)o(t)f(the)h Fl(l)q Fs({dimensional)e(torus)i Fl(J)j Fs(=)e(constan)o(t.)27 b(The)18 b(motion)57 2511 y(is)e(therefore)g(b)q(ounded)f(and)h(quasip) q(erio)q(dic)g(\(or)g(p)q(erio)q(dic\).)156 2581 y(Being)i(completely)g (canonically)g(in)o(tegrable)e(is)i(a)g(stronger)f(requiremen)o(t)f (than)i(in)o(te-)57 2650 y(grabilit)o(y)12 b(b)o(y)i(quadratures)e(or)h (complete)g(in)o(tegrabilit)o(y)g(\(see)h([AKN])g(for)f(their)g (discussion\).)918 2770 y(49)p eop %%Page: 50 51 50 50 bop 57 192 a Fs(In)14 b(the)g(latter)g(case)g(one)g(requires)f (the)h(existence)h(of)f Fl(l)h Fs(indep)q(enden)o(t)e(\014rst)g(in)o (tegrals)g(in)h(in)o(v)o(o-)57 261 y(lution)i(but)i(their)f(join)o(t)g (lev)o(el{set)g(ma)o(y)g(w)o(ell)g(b)q(e)g(non)g(compact)g(\(this)g(is) g(already)g(the)h(case)57 331 y(in)h(the)h(t)o(w)o(o)g(b)q(o)q(dy)g (problem)e(for)i(non)f(negativ)o(e)h(energy)f(v)m(alues\))h(and)f(the)i (\015o)o(w)e(do)q(es)g(not)57 401 y(need)d(to)h(b)q(e)g(quasip)q(erio)q (dic)e(\(scattering)i(states\).)156 473 y(The)g(main)e(risult)g(in)h (the)h(theory)f(of)h(completely)f(canonically)g(in)o(tegrable)f (systems)h(is)57 543 y(the)g(celebrated)57 651 y Fr(Theorem)35 b(7.11)g(\(Arnol'd{Liouvill)q(e\))q Fp(L)m(et)g Fl(H)44 b Fk(2)c(C)1223 633 y Fj(1)1265 651 y Fs(\()p Fl(M)s(;)8 b Fm(R)p Fs(\))30 b Fp(and)j(assume)f(that)57 721 y Fl(F)89 728 y Fi(1)111 721 y Fl(;)8 b(:)g(:)g(:)h(;)f(F)254 728 y Fh(l)284 721 y Fk(2)15 b(C)361 703 y Fj(1)403 721 y Fs(\()p Fl(M)s(;)8 b Fm(R)p Fs(\))17 b Fp(ar)m(e)i Fl(l)g Fp(\014rst)g(inte)m(gr)m(als)g(in)g(involution)g(for)g(the)f (Hamiltonian)i(\015ow)57 790 y(asso)m(ciate)m(d)26 b(to)d Fl(H)t Fp(.)38 b(L)m(et)23 b Fl(a)h Fk(2)f Fm(R)694 772 y Fh(l)729 790 y Fp(b)m(e)g(such)g(that)g Fl(M)1068 797 y Fh(a)1115 790 y Fs(=)g Fk(f)p Fl(m)g Fk(2)g Fl(M)37 b Fk(j)31 b Fl(F)1487 797 y Fh(i)1504 790 y Fs(\()p Fl(m)p Fs(\))23 b(=)g Fl(a)1697 797 y Fh(i)1722 790 y Fk(8)p Fl(i)f Fs(=)57 860 y(1)p Fl(;)8 b(:)g(:)g(:)g(;)g(l)q Fk(g)16 b Fp(is)h(not)g(empty)f(and)h(assume)g(that)g(the)f Fl(l)h Fp(functions)g Fl(F)1263 867 y Fi(1)1285 860 y Fl(;)8 b(:)g(:)g(:)h(;)f(F)1428 867 y Fh(l)1460 860 y Fp(ar)m(e)17 b(indep)m(endent)1805 842 y Fi(1)57 930 y Fp(in)j(a)h(neighb)m(orho)m(o)n(d)j(of)d Fl(M)574 937 y Fh(a)598 930 y Fp(.)31 b(Then)20 b(if)h Fl(M)875 937 y Fh(a)920 930 y Fp(is)g(c)m(omp)m(act)h(and)f(c)m(onne)m(cte)m(d)h(it) e(is)h(di\013e)m(omor-)57 1000 y(phic)d(to)f(the)g Fl(l)q Fp({torus.)24 b(Mor)m(e)m(over)18 b(ther)m(e)f(exists)h(an)g(invariant) g(op)m(en)g(subset)f Fl(V)28 b Fp(of)18 b Fl(M)23 b Fp(which)57 1069 y(c)m(ontains)f Fl(M)305 1076 y Fh(a)351 1069 y Fp(and)g(is)f(symple)m(ctic)m(al)s(ly)h(di\013e)m(omorphic)i(to)d Fl(U)e Fk(\002)13 b Fm(T)1344 1051 y Fh(l)1356 1069 y Fp(,)22 b(wher)m(e)g Fl(U)27 b Fp(is)22 b(an)f(op)m(en)57 1139 y(subset)c(of)i Fm(R)302 1121 y Fh(l)314 1139 y Fp(.)57 1247 y Fs(Arnol'd{Liouville's)e(Theorem)h(th)o(us)g(assures)f (that)j(the)f(existence)h(of)f(su\016cien)o(tly)f(man)o(y)57 1317 y(\014rst)f(in)o(tegrals)f(together)i(with)g(the)g(compactness)f (and)g(connectedness)g(of)h(their)f(lev)o(el)h(set)57 1386 y(guaran)o(tees)d(complete)h(canonical)g(in)o(tegrabilit)o(y)l(.) 57 1564 y Fo(7.3)j(Examples)h(of)g(completely)g(canonically)h(in)n (tegrable)h(systems)57 1672 y Fs(In)i(this)f(section)h(w)o(e)g(will)g (brie\015y)f(describ)q(e)g(some)h(examples)f(of)h(completely)g (canonical)57 1742 y(in)o(tegrable)15 b(systems.)57 1850 y Fr(Example)28 b(7.12)f(:)44 b(Harmonic)28 b(oscillators.)48 b Fs(Let)26 b Fl(M)33 b Fs(=)27 b Fm(R)1364 1832 y Fi(2)p Fh(l)1421 1850 y Fs(with)e(the)g(standard)57 1919 y(symplectic)15 b(structure,)f Fl(S)j Fk(2)d Fs(GL)8 b(\(2)p Fl(l)q(;)g Fm(R)p Fs(\))13 b(b)q(e)j(symmetric)f(and)g(p)q(ositiv)o(e)g (de\014nite.)21 b(Consider)57 1989 y(the)f(Hamiltonian)f(system)h Fl(H)t Fs(\()p Fl(x)p Fs(\))i(=)802 1969 y Fi(1)p 802 1978 20 2 v 802 2006 a(2)828 1989 y Fl(x)856 1971 y Fh(T)888 1989 y Fl(S)s(x)p Fs(.)32 b(This)20 b(is)g(completely)g(in)o(tegrable.) 31 b(Indeed)57 2059 y(if)19 b Fl(J)25 b Fs(is)19 b(a)g(symplectic)h (matrix)f(whic)o(h)f(diagonalizes)h Fl(S)s Fs(,)g(in)h(the)f(v)m (ariables)g Fl(y)i Fs(=)e Fl(J)1658 2041 y Fj(\000)p Fi(1)1711 2059 y Fl(x)h Fs(the)57 2129 y(Hamiltonian)15 b(will)h(b)q(e)630 2233 y Fl(H)t Fs(\()p Fl(y)r Fs(\))f(=)826 2171 y Fi(2)p Fh(l)807 2186 y Fe(X)810 2292 y Fh(i)p Fi(=1)893 2196 y Fl(\025)922 2203 y Fh(i)939 2196 y Fl(y)965 2177 y Fi(2)963 2208 y Fh(i)998 2196 y Fs(+)c Fl(\025)1077 2203 y Fh(i)p Fi(+)p Fh(l)1137 2196 y Fl(y)1163 2177 y Fi(2)1161 2210 y Fh(i)p Fi(+)p Fh(l)p 893 2222 329 2 v 1045 2267 a Fs(2)1241 2233 y Fl(;)57 2380 y Fs(where)26 b Fl(\025)240 2387 y Fh(i)265 2380 y Fl(;)17 b(i)32 b Fs(=)f(1)p Fl(;)8 b(:)g(:)g(:)h(;)f Fs(2)p Fl(l)28 b Fs(are)f(the)g(eigen)o(v)m(alues)g(of)g Fl(S)s Fs(.)54 b(Then)26 b(the)i(functions)e Fl(F)1741 2387 y Fh(i)1790 2380 y Fs(=)63 2425 y Fh(\025)87 2430 y Fb(i)102 2425 y Fh(y)123 2410 y Fc(2)122 2435 y Fb(i)143 2425 y Fi(+)p Fh(\025)198 2430 y Fb(i)p Fc(+)p Fb(l)251 2425 y Fh(y)272 2410 y Fc(2)271 2436 y Fb(i)p Fc(+)p Fb(l)p 63 2444 262 2 v 183 2472 a Fi(2)330 2455 y Fs(,)21 b Fl(i)e Fs(=)g(1)p Fl(;)8 b(:)g(:)g(:)g(;)g(l)q Fs(,)21 b(are)e(indep)q(enden)o(t)g (\014rst)g(in)o(tegrals)f(in)i(in)o(v)o(olution)e(and)h(their)57 2525 y(common)12 b(lev)o(el)h(set)g(is)g(compact)g(and)f(connected)h (\(since)g Fl(\025)1169 2532 y Fh(i)1200 2525 y Fl(>)g Fs(0)h(for)f(all)f Fl(i)p Fs(\).)22 b(The)13 b(symplectic)p 57 2595 600 2 v 109 2632 a Fi(1)156 2650 y Fs(As)k(usual)e Fl(F)392 2657 y Fi(1)415 2650 y Fl(;)8 b(:)g(:)g(:)g(;)g(F)557 2657 y Fh(l)589 2650 y Fs(are)16 b(indep)q(enden)o(t)g(if)g Fl(dF)1058 2657 y Fi(1)1092 2650 y Fk(^)11 b Fl(:)d(:)g(:)j Fk(^)g Fl(dF)1307 2657 y Fh(l)1337 2650 y Fk(6)p Fs(=)i(0.)918 2770 y(50)p eop %%Page: 51 52 51 51 bop 57 192 a Fs(transformation)14 b(to)j(action{angle)f(v)m (ariables)g(is)148 335 y Fl(y)172 342 y Fh(i)203 335 y Fs(=)255 269 y Fe(q)p 305 269 279 2 v 66 x Fs(2)p Fl(J)358 342 y Fh(i)374 290 y Fe(p)p 424 290 160 2 v 45 x Fl(\025)453 342 y Fh(i)p Fi(+)p Fh(l)513 335 y Fl(=\025)567 342 y Fh(i)592 335 y Fs(cos)8 b Fl(\037)698 342 y Fh(i)729 335 y Fl(;)49 b(y)816 342 y Fh(i)p Fi(+)p Fh(l)890 335 y Fs(=)943 269 y Fe(q)p 993 269 279 2 v 66 x Fs(2)p Fl(J)1046 342 y Fh(i)1062 290 y Fe(p)p 1112 290 160 2 v 45 x Fl(\025)1141 342 y Fh(i)1157 335 y Fl(=\025)1211 342 y Fh(i)p Fi(+)p Fh(l)1280 335 y Fs(sin)7 b Fl(\037)1380 342 y Fh(i)1411 335 y Fl(;)49 b(i)14 b Fs(=)g(1)p Fl(;)8 b(:)g(:)g(:)g(;)g(l)15 b(:)57 499 y Fr(Example)d(7.13)h(:)22 b(The)13 b(t)n(w)n(o)i(b)r(o)r (dy)d(problem.)19 b Fs(The)11 b(Hamiltonian)g Fk(H)22 b Fs(:)14 b Fl(T)1543 481 y Fj(\003)1566 499 y Fs(\()p Fm(R)1624 481 y Fi(3)1645 499 y Fk(n)q(f)p Fs(0)p Fk(g)p Fs(\))g Fk(7!)57 569 y Fm(R)c Fs(of)j(the)h(t)o(w)o(o-b)q(o)q(dy)e (problem)g(in)h(the)g(cen)o(ter)g(of)g(mass)f(frame)h(is)g(\(w)o(e)g (ha)o(v)o(e)g(assumed)e Fl(G)j Fs(=)g(1,)57 638 y(where)i Fl(G)h Fs(is)f(the)g(univ)o(ersal)f(gra)o(vitational)h(constan)o(t\)) 660 781 y Fk(H)p Fs(\()p Fl(p;)8 b(q)r Fs(\))15 b(=)900 747 y(1)p 885 769 55 2 v 885 815 a(2)p Fl(\026)945 781 y Fk(k)p Fl(p)p Fk(k)1020 760 y Fi(2)1054 781 y Fk(\000)1110 747 y Fl(m)1154 754 y Fi(0)1176 747 y Fl(m)p 1110 769 110 2 v 1128 815 a Fk(k)p Fl(q)r Fk(k)57 928 y Fs(where)h Fl(\026)e Fs(=)f Fl(m)341 935 y Fi(0)363 928 y Fl(m=)p Fs(\()p Fl(m)495 935 y Fi(0)528 928 y Fs(+)e Fl(m)p Fs(\))17 b(is)f(the)h(reduced)e(mass)h(of)g(the)h(system.)156 998 y(It)j(is)f(w)o(ell-kno)o(wn)f(that)i(for)f(negativ)o(e)g(energy)g (the)h(solutions)e(are)h(ellipses)f(with)i(one)57 1068 y(fo)q(cus)13 b(at)g(the)h(origin)e(\(i.)h(e.)21 b(the)13 b(cen)o(ter)g(of)g(mass\).)20 b(These)13 b(are)g(called)g Fp(keplerian)i(orbits)p Fs(.)22 b(The)57 1138 y(shap)q(e)16 b(and)g(the)h(p)q(osition)g(of)g(the)g(ellipse)f(in)g(space)h(are)f (determined)g(from)g(the)h(kno)o(wledge)57 1207 y(of)h(the)h(ma)s(jor)e (semiaxis)h Fl(a)p Fs(,)h(the)g(eccen)o(tricit)o(y)f Fl(e)p Fs(,)h(the)g(angle)f(of)g(inclination)f Fl(i)i Fs(of)g(its)f(plane)57 1277 y(w.r.t.)j(the)15 b(horizon)o(tal)f(plane)g Fl(q)665 1284 y Fi(3)701 1277 y Fs(=)g(0,)h(the)g(argumen)o(t)f(of)h(p) q(erihelion)f Fl(!)j Fs(and)d(the)h(longitude)57 1347 y(of)e(the)h(ascending)f(no)q(de)g(\012.)21 b(The)13 b(p)q(osition)g(of)h(the)g(planet)f(along)g(the)h(ellipse)f(is)g (determined)57 1417 y(b)o(y)18 b(the)i(mean)e(anomaly)g Fl(l)q Fs(,)h(whic)o(h)f(is)h(prop)q(ortional)e(to)i(the)g(area)g(sw)o (ept)f(b)o(y)h(the)g(p)q(osition)57 1486 y(v)o(ector)d Fl(q)j Fs(of)e(the)f(planet)h(starting)e(from)h(the)h(p)q(erihelion.) 156 1556 y(The)k(systems)f(admits)g(5)h Fp(indep)m(endent)i Fs(\014rst)d(in)o(tegrals)f(:)31 b(the)22 b(total)f(energy)g Fk(H)p Fs(,)h(the)57 1626 y(three)c(comp)q(onen)o(ts)e(of)j(the)f (angular)e(momen)o(tum)g Fl(q)e Fk(^)f Fl(p)18 b Fs(and)f(one)h(of)g (the)g(comp)q(onen)o(ts)f(of)57 1696 y(the)e(Laplace{Runge{Lenz)f(v)o (ector)i Fl(A)e Fs(=)f Fl(p)c Fk(^)g Fl(q)i Fk(^)d Fl(p)h Fk(\000)1105 1673 y Fh(m)1140 1678 y Fc(0)1159 1673 y Fh(mq)p 1105 1684 V 1129 1713 a Fj(k)p Fh(q)q Fj(k)1220 1696 y Fs(.)22 b(Among)14 b(these)i(in)o(tegrals)e(one)57 1765 y(can)21 b(c)o(ho)q(ose)g(three)g(in)o(tegrals)f(in)g(in)o(v)o (olution)g(and)h(construct)g(the)g(completely)g(canonical)57 1835 y(transformation)15 b(to)i(action{angle)g(v)m(ariables.)22 b(The)17 b(other)g(t)o(w)o(o)g(in)o(tegrals)f(are)g(resp)q(onsible)57 1905 y(for)23 b(the)g(prop)q(er)g(complete)g(degeneration)f(of)i(the)g (Kepler)e(problem)g(:)36 b(one)23 b(can)g(c)o(ho)q(ose)57 1975 y(action{angle)13 b(v)m(ariables)f(so)h(that)h(the)g(Hamiltonian)f (dep)q(ends)f(only)i(on)f(one)g(of)h(the)g(actions.)57 2044 y(Indeed)25 b(the)h(Delauna)o(y)g(action{angle)f(v)m(ariables)g (\()p Fl(L;)8 b(G;)g Fs(\002)p Fl(;)g(l)q(;)g(g)r(;)g(\022)q Fs(\))27 b(are)f(related)g(to)g(the)57 2114 y(orbital)15 b(elemen)o(ts)h(as)g(follo)o(ws)g(:)145 2243 y Fl(L)e Fs(=)f Fl(\026)275 2198 y Fe(p)p 325 2198 236 2 v 45 x Fs(\()p Fl(m)388 2250 y Fi(0)422 2243 y Fs(+)d Fl(m)p Fs(\))p Fl(a)15 b(;)36 b(G)14 b Fs(=)f Fl(L)764 2194 y Fe(p)p 814 2194 132 2 v 49 x Fs(1)e Fk(\000)g Fl(e)923 2228 y Fi(2)959 2243 y Fl(;)36 b Fs(\002)13 b(=)h Fl(G)8 b Fs(cos)g Fl(i)14 b(;)36 b(l)15 b(;)36 b(g)15 b Fs(=)f Fl(!)h(;)36 b(\022)16 b Fs(=)d(\012)h Fl(:)57 2371 y Fs(Note)k(that)g Fl(G)f Fs(is)g(the)h(mo)q(dulus)e(of)h(angular)f (momen)o(tum)f Fl(q)f Fk(^)e Fl(p)p Fs(,)17 b(th)o(us)g(\002)g(is)g (its)g(pro)s(jection)57 2441 y(along)e(the)g Fl(q)295 2448 y Fi(3)318 2441 y Fs({axis.)21 b(One)15 b(has)g(the)h(ob)o(vious)e (limitation)h Fk(j)p Fs(\002)p Fk(j)e(\024)h Fl(G)p Fs(.)22 b(The)15 b(new)h(Hamiltonian)57 2511 y(reads)f Fk(H)g Fs(=)e Fk(\000)341 2487 y Fh(\026)365 2472 y Fc(3)385 2487 y Fi(\()p Fh(m)436 2492 y Fc(0)455 2487 y Fi(+)p Fh(m)p Fi(\))537 2472 y Fc(2)p 341 2499 215 2 v 415 2528 a Fi(2)p Fh(L)462 2518 y Fc(2)562 2511 y Fs(.)156 2581 y(The)j(relation)f(among)g(Delauna)o(y)f(v)m(ariables)h(and)g(the)h (original)e(momen)o(tum{p)q(osition)57 2650 y(\()p Fl(p;)8 b(q)r Fs(\))17 b(v)m(ariables)f(is)g(m)o(uc)o(h)f(more)g(subtle)h(and)g (will)g(not)h(b)q(e)g(discussed)d(here.)918 2770 y(51)p eop %%Page: 52 53 52 52 bop 156 192 a Fs(The)17 b(t)o(w)o(o{b)q(o)q(dy)f(problem)g(is)g (the)h(mo)q(delization)f(of)h(the)g(motion)f(of)h(a)g(planet)f(around) 57 261 y(the)k(Sun.)32 b(But)21 b(the)f(Delauna)o(y)f(v)m(ariables)h (are)f(not)i(suitable)e(for)h(the)g(description)f(of)h(the)57 331 y(orbits)15 b(of)i(the)g(planets)f(of)h(the)g(solar)e(system)i (since)f(they)h(are)f(singular)f(for)h(circular)f(orbits)57 401 y(\()p Fl(e)j Fs(=)f(0,)i(th)o(us)f Fl(L)f Fs(=)h Fl(G)h Fs(anf)f(the)h(argumen)o(t)f(of)h(the)g(p)q(erihelion)e Fl(g)j Fs(is)f(not)f(de\014ned\))h(and)f(for)57 470 y(horizon)o(tal)e (orbits)g(\()p Fl(i)g Fs(=)f(0)j(or)f Fl(i)e Fs(=)h Fl(\031)r Fs(,)h(th)o(us)g Fl(G)f Fs(=)f(\002)i(and)g(the)h(longitude)e(of)i(the) g(ascending)57 540 y(no)q(de)c Fl(\022)i Fs(is)e(not)g(de\014ned\).)21 b(But)15 b(all)f(the)g(planets)g(of)h(the)f(solar)f(system)i(ha)o(v)o (e)e(almost)h(circular)57 610 y(orbits)h(\(with)i(the)g(exception)g(of) f(Mercury)g(and)g(Mars\))f(and)h(small)g(inclinations.)156 680 y(P)o(oincar)o(\023)-24 b(e)28 b(solv)o(ed)f(the)i(problem)e (\014rst)h(in)o(tro)q(ducing)f(a)h(new)h(set)f(of)h(action{angle)57 749 y(v)m(ariables)21 b(\(\003)p Fl(;)8 b(H)q(;)g(Z)q(;)g(\025;)g(h;)g (\020)t Fs(\))23 b(:)34 b(\003)23 b(=)h Fl(L)p Fs(,)g Fl(H)k Fs(=)23 b Fl(L)15 b Fk(\000)g Fl(G)p Fs(,)24 b Fl(Z)j Fs(=)c Fl(G)15 b Fk(\000)g Fs(\002,)24 b Fl(\025)f Fs(=)h Fl(l)16 b Fs(+)e Fl(g)j Fs(+)e Fl(\022)q Fs(,)57 819 y Fl(h)24 b Fs(=)h Fk(\000)p Fl(g)17 b Fk(\000)e Fl(\022)q Fs(,)26 b Fl(\020)j Fs(=)24 b Fk(\000)p Fl(\022)h Fs(\()p Fl(\025)f Fs(is)f(called)g(the)g(mean)g(longitude,)h Fk(\000)p Fl(h)f Fs(is)f(the)i(longitude)e(of)57 889 y(the)16 b(p)q(erihelion\))g(then)g(considering)f(the)h(couples)g(\()p Fl(H)q(;)8 b(h)p Fs(\))18 b(and)d(\()p Fl(Z)q(;)8 b(\020)t Fs(\))17 b(as)f(p)q(olar)g(symplectic)57 959 y(co)q(ordinates)f(:)130 1088 y Fl(\030)152 1095 y Fi(1)188 1088 y Fs(=)240 1044 y Fk(p)p 282 1044 71 2 v 44 x Fs(2)p Fl(H)d Fs(cos)c Fl(h)14 b(;)50 b(\021)567 1095 y Fi(1)602 1088 y Fs(=)655 1044 y Fk(p)p 697 1044 V 44 x Fs(2)p Fl(H)12 b Fs(sin)c Fl(h)13 b(;)105 b(\030)1028 1095 y Fi(2)1064 1088 y Fs(=)1117 1044 y Fk(p)p 1158 1044 63 2 v 1158 1088 a Fs(2)p Fl(Z)12 b Fs(cos)c Fl(\020)17 b(;)50 b(\021)1432 1095 y Fi(2)1468 1088 y Fs(=)1521 1044 y Fk(p)p 1562 1044 V 1562 1088 a Fs(2)p Fl(Z)12 b Fs(sin)7 b Fl(\020)18 b(:)57 1218 y Fs(The)e(v)m(ariables)g(\(\003)p Fl(;)8 b(\030)r(;)g(\025;)g(\021)r Fs(\))17 b(are)f(called)g Fp(Poinc)m(ar)o(\023)-24 b(e)19 b(variables)p Fs(.)k(They)16 b(are)h(w)o(ell)f(de\014ned)f(also)57 1287 y(in)h(the)h(case)f(of)h(circular)e(\()p Fl(H)k Fs(=)13 b(0\))k(or)f(horizon)o(tal)f(\()p Fl(Z)j Fs(=)13 b(0\))k(orbits.)57 1393 y Fr(Example)f(7.14)f(:)25 b(Motion)17 b(of)f(a)g(\\hea)n(vy")i(particle)h(on)d(a)g(surface)g(of)g(rev)n (olution.)57 1462 y Fs(Let)h Fl(S)f Fk(\032)e Fm(R)285 1444 y Fi(3)321 1462 y Fs(b)q(e)i(a)h(surface)f(of)g(rev)o(olution)f (with)i(the)g(Riemannian)d(metric)i(induced)f(b)o(y)h(its)57 1532 y(em)o(b)q(edding)g(in)o(to)i Fm(R)452 1514 y Fi(3)489 1532 y Fs(and)g(assume)f(that)h Fl(x)899 1539 y Fi(3)940 1532 y Fs(is)g(its)g(symmetry)g(axis.)26 b(Let)19 b Fl(f)k Fk(2)16 b(C)1639 1514 y Fj(1)1682 1532 y Fs(\()p Fm(R)p Fl(;)8 b Fm(R)m Fs(\).)57 1602 y(If)23 b(the)h(surface)e(nev)o(er)h (meets)g(the)h Fl(x)789 1609 y Fi(3)835 1602 y Fs(axis)f(then)g(it)h (is)f(di\013eomorphic)e(to)j(the)f(cylinder)57 1672 y Fl(S)16 b Fk(\031)e Fm(R)p Fk(\002)q Fm(T)269 1653 y Fi(1)300 1672 y Fs(and)c(its)i(cotangen)o(t)f(bundle)f(will)h(b)q(e)g Fm(R)1027 1653 y Fi(3)1047 1672 y Fk(\002)q Fm(T)1123 1653 y Fi(1)1142 1672 y Fs(.)21 b(If)11 b(\()p Fl(p)1265 1679 y Fi(1)1288 1672 y Fl(;)d(p)1335 1679 y Fi(2)1358 1672 y Fl(;)g(q)1402 1679 y Fi(1)1424 1672 y Fl(;)g(q)1468 1679 y Fi(2)1491 1672 y Fs(\))k(are)f(symplectic)57 1741 y(co)q(ordinates)k(the)h(Hamiltonian)f(of)i(a)f(\(hea)o(vy\))h(p)q(oin) o(t)e(mass)g(constrained)g(to)h(mo)o(v)o(e)g(on)f Fl(S)k Fs(is)57 1811 y Fl(H)t Fs(\()p Fl(p;)8 b(q)r Fs(\))21 b(=)297 1791 y Fi(1)p 297 1800 20 2 v 297 1828 a(2)323 1811 y Fk(k)p Fl(p)p Fk(k)398 1793 y Fi(2)434 1811 y Fs(+)13 b Fl(f)5 b Fs(\()p Fl(q)556 1818 y Fi(1)580 1811 y Fs(\).)33 b(Here)21 b Fl(f)26 b Fs(is)20 b(the)g(\\w)o(eigh)o(t")g (and)f Fl(p)1298 1818 y Fi(2)1341 1811 y Fs(\(whic)o(h)h(corresp)q (onds)e(to)57 1881 y(the)g(pro)s(jection)g(of)h(the)f(angular)f(momen)o (tum)g(of)h(the)h(particle)f(along)g(the)g Fl(x)1553 1888 y Fi(3)1576 1881 y Fs({axis\))h(is)f(an)57 1951 y(indep)q(enden)o(t)d(in)o(tegral)g(of)h(the)g(motion.)22 b(Complete)15 b(in)o(tegrabilit)o(y)g(is)h(assured)e(if)j(the)f(curv)o (e)57 2020 y Fk(f)p Fs(\()p Fl(p)126 2027 y Fi(1)148 2020 y Fl(;)8 b(q)192 2027 y Fi(1)215 2020 y Fs(\))14 b Fk(2)g Fm(R)334 2002 y Fi(2)381 2020 y Fk(j)28 b Fl(H)t Fs(\()p Fl(p)512 2027 y Fi(1)535 2020 y Fl(;)8 b(a;)g(q)627 2027 y Fi(1)650 2020 y Fl(;)g(q)694 2027 y Fi(2)717 2020 y Fs(\))14 b(=)g Fl(E)s Fk(g)i Fs(is)g(closed)g(for)g(some)g(v)m(alue)h (of)f Fl(a)h Fs(and)f Fl(E)s Fs(.)918 2770 y(52)p eop %%Page: 53 54 53 53 bop 57 192 a Fq(8.)31 b(Quasi{in)n(tegrable)c(Hamiltonian)e (Systems)57 297 y Fs(The)17 b(imp)q(ortance)g(of)h(completely)g (canonically)f(in)o(tegrable)g(Hamiltonian)f(systems)h(is)h(due)57 366 y(b)q(oth)j(to)h(the)g(fact)h(that)f(their)f(\015o)o(ws)g(can)g(b)q (e)h(studied)f(in)g(great)h(detail)f(and)g(that)h(man)o(y)57 436 y(problems)12 b(in)i(mathematical)g(ph)o(ysics)f(can)i(b)q(e)g (considered)e(as)h Fp(p)m(erturb)m(ations)i Fs(of)f(in)o(tegrable)57 506 y(systems.)27 b(The)18 b(most)g(famous)g(example)g(is)g(giv)o(en)g (b)o(y)g(the)h(motion)f(of)g(the)h(planets)f(in)g(the)57 576 y(Solar)c(System)h(\(see)g([Ma2])g(and)f(references)h(therein)f (for)h(an)g(in)o(tro)q(duction\).)21 b(If)15 b(the)h(\(w)o(eak\))57 645 y(m)o(utual)24 b(attraction)i(b)q(et)o(w)o(een)f(the)h(planets)f (is)g(neglected)g(the)h(system)f(decouples)g(in)o(to)57 715 y(sev)o(eral)14 b(indep)q(enden)o(t)g(Kepler)h(problems)e(and)i(it) h(is)f(completely)g(in)o(tegrable.)21 b(Exactly)16 b(this)57 785 y(problem)g(ga)o(v)o(e)i(origin)e(in)i(the)g(18th)g(cen)o(tury)f (to)i(\\p)q(erturbation)d(theory")i(whose)g(mo)q(dern)57 855 y(form)o(ulation)i(is)h(mainly)g(due)h(to)g(the)g(mon)o(umen)o(tal) e(w)o(ork)h(of)h(Henri)g(P)o(oincar)o(\023)-24 b(e)20 b([P].)i(The)57 924 y(goal)14 b(of)g(p)q(ertubation)g(theory)g(is)g(to) h(understand)d(the)i(dynamics)g(of)g(a)g(\\p)q(erturb)q(ed")g(system)57 994 y(whic)o(h)h(is)h(close)h(to)f(a)h(w)o(ell{understo)q(o)q(d)e(one)h (\(usually)g(an)g(in)o(tegrable)f(system\).)57 1169 y Fo(8.1)k(Quasi{in)n(tegrable)j(Systems)57 1274 y Fs(F)l(ollo)o(wing)10 b(P)o(oincar)o(\023)-24 b(e)11 b([P],)h(the)g Fp(fundamental)i(pr)m (oblem)h(of)g(dynamics)e Fs(is)f(the)h(study)f(of)g(quasi{)57 1344 y(integrable)j(Hamiltonian)h(systems)g(:)22 b(let)17 b Fl(")892 1351 y Fi(0)928 1344 y Fl(>)d Fs(0,)57 1474 y Fr(De\014nition)38 b(8.1)28 b Fd(A)k Fs(quasi{in)o(tegrable)e Fd(Hamiltonian)h(system)h(is)f(a)h(function)g Fk(H)40 b(2)57 1543 y(C)86 1525 y Fj(1)128 1543 y Fs(\(\()p Fk(\000)p Fl(")228 1550 y Fi(0)251 1543 y Fl(;)8 b(")296 1550 y Fi(0)319 1543 y Fs(\))k Fk(\002)g Fl(M)s(;)c Fm(R)p Fs(\))16 b Fd(suc)o(h)g(that)j(the)f(Hamiltonian)f(function)h Fl(H)i Fs(=)c Fk(H)p Fs(\(0)p Fl(;)8 b Fk(\001)p Fs(\))26 b(:)16 b Fl(M)22 b Fk(!)16 b Fm(R)57 1613 y Fd(is)g(completely)g (canonically)g(in)o(tegrable.)57 1743 y Fs(Using)d(the)i(canonical)e (transformation)f(to)j(action{angle)e(v)m(ariables)g(asso)q(ciated)h (to)g Fk(H)p Fs(\(0)p Fl(;)8 b Fk(\001)p Fs(\),)57 1812 y(Hamiltonians)15 b Fk(H)22 b Fs(:)14 b(\()p Fk(\000)p Fl(")536 1819 y Fi(0)559 1812 y Fl(;)8 b(")604 1819 y Fi(0)627 1812 y Fs(\))j Fk(\002)g Fl(U)16 b Fk(\002)11 b Fm(T)843 1794 y Fh(l)869 1812 y Fk(7!)j Fm(R)g Fs(\(smo)q(oth)i(or)g (analytic\))h(of)g(the)g(form)615 1918 y Fk(H)p Fs(\()p Fl(";)8 b(J)o(;)g(\037)p Fs(\))15 b(=)f Fl(h)917 1925 y Fi(0)939 1918 y Fs(\()p Fl(J)5 b Fs(\))11 b(+)g Fl("f)5 b Fs(\()p Fl(J)o(;)j(\037)p Fs(\))16 b Fl(;)455 b Fs(\(8)p Fl(:)p Fs(1\))57 2023 y(where)19 b Fl(f)24 b Fk(2)18 b(C)332 2005 y Fj(1)375 2023 y Fs(\()p Fl(U)h Fk(\002)12 b Fm(T)534 2005 y Fh(l)547 2023 y Fl(;)c Fm(R)p Fs(\),)17 b(are)i(t)o(ypical)g(examples)g(of)g(quasi{in)o(tegrable)f(Hamiltonian) 57 2092 y(systems.)156 2162 y(The)d(most)g(am)o(bitious)e(program)h(w)o (ould)g(b)q(e)h(to)h(pro)o(v)o(e)e(that)h(quasi{in)o(tegrable)e(Hamil-) 57 2232 y(tonian)24 b(systems)h(are)g(indeed)f(in)o(tegrable)g(:)40 b(i.e.)48 b(to)25 b(sho)o(w)f(that)i(there)f(exists)g(a)h(one{)57 2302 y(parameter)20 b(family)i Fl(V)487 2309 y Fh(")531 2302 y Fs(of)g(op)q(en)g(connected)g(in)o(v)m(arian)o(t)e(subsets)h(of) i Fl(M)28 b Fs(whic)o(h)21 b(are)g(sym-)57 2371 y(plectically)e (di\013eomorphic)e(to)i Fl(U)707 2378 y Fh(")741 2371 y Fk(\002)13 b Fm(T)829 2353 y Fh(l)861 2371 y Fs(where)18 b Fl(U)1041 2378 y Fh(")1081 2371 y Fk(\032)g Fm(R)1177 2353 y Fh(l)1208 2371 y Fs(is)h(op)q(en,)g(connected)g(and)g(suc)o(h)57 2441 y(that)e(if)f(\()242 2429 y(~)229 2441 y Fl(J)5 b(;)14 b Fs(~)-31 b Fl(\037)p Fs(\))17 b(are)f(the)h(co)q(ordinates)f (in)g Fl(U)877 2448 y Fh(")909 2441 y Fk(\002)11 b Fm(T)995 2423 y Fh(l)1024 2441 y Fs(one)16 b(has)g Fk(H)p Fs(\()p Fl(";)8 b Fk(\001)p Fl(;)g Fk(\001)p Fs(\))15 b Fk(j)1408 2448 y Fh(V)1432 2453 y Fb(")1454 2441 y Fs(=)e Fl(h)1535 2448 y Fh(")1556 2441 y Fs(\()1588 2429 y(~)1575 2441 y Fl(J)5 b Fs(\))17 b(for)f(some)57 2511 y(smo)q(oth)g(one{parameter)f (family)h(of)h(smo)q(oth)f(function)g Fl(h)1180 2518 y Fh(")1223 2511 y Fs(:)d Fl(U)1284 2518 y Fh(")1317 2511 y Fk(\002)e Fm(T)1403 2493 y Fh(l)1429 2511 y Fk(!)j Fm(R)p Fs(.)156 2581 y(In)h(general)f(this)g(is)h(asking)f(to)q(o)h(m)o (uc)o(h)f(:)21 b(a)15 b(result)f(of)h(P)o(oincar)o(\023)-24 b(e)13 b(sho)o(ws)h(that)h(in)f(general)57 2650 y(quasi{in)o(tegrable)j (Hamiltonian)h(systems)g(are)h(not)g(completely)g(in)o(tegrable)f(\(in) h(addition)918 2770 y(53)p eop %%Page: 54 55 54 54 bop 57 192 a Fs(to)17 b([P],)g(T)l(ome)f(I,)h(Chapitre)g(V,)g (see)g([BF)o(GG])g(for)f(a)i(nice)f(discussion)d(of)k(the)f (consequences)57 261 y(of)f(this)h(problem)d(and)i(a)h(related)f (result)g(of)g(F)l(ermi\).)57 393 y Fr(Theorem)h(8.2)h(\(P)n(oincar)o (\023)-27 b(e\))29 b Fd(Consider)15 b(a)h(quasi{in)o(tegrable)e (Hamiltonian)h(of)i(the)f(form)57 463 y(\(8.1\),)i Fl(l)e Fk(\025)f Fs(2)p Fd(.)24 b(Assume)17 b(that)g(the)h(t)o(w)o(o)f(follo)o (wing)f(genericit)o(y)h(assumptions)e(are)i(satis\014ed)g(:)57 538 y(\(1\))i Fs(non{degeneracy)e Fd(:)26 b Fs(det)620 482 y Fe(\020)677 517 y Fh(@)700 502 y Fc(2)720 517 y Fh(h)743 522 y Fc(0)p 655 526 129 2 v 655 555 a Fh(@)r(J)700 560 y Fb(i)717 555 y Fh(@)r(J)762 560 y Fb(k)790 482 y Fe(\021)837 538 y Fk(6)p Fs(=)17 b(0)h Fd(on)h Fl(U)c Fd(;)k(\(2\))h Fs(generic)e(p)q(erturbations)f Fd(:)26 b(for)18 b(all)57 623 y Fl(J)i Fk(2)15 b Fl(U)23 b Fd(and)17 b(for)h(all)f Fl(k)g Fk(2)e Fm(Z)584 605 y Fh(l)608 623 y Fk(n)c(f)p Fs(0)p Fk(g)18 b Fd(either)f(the)h Fl(k)r Fd({th)f(F)l(ourier)e(co)q(e\016cien)o(t)1501 610 y Fs(^)1490 623 y Fl(f)1514 630 y Fh(k)1539 623 y Fs(\()p Fl(J)5 b Fs(\))17 b Fd(of)h Fl(f)24 b Fd(do)q(es)57 693 y(not)13 b(v)m(anish)g(or)g(there)h(exists)f Fl(k)638 674 y Fj(0)666 693 y Fk(2)h Fm(Z)749 674 y Fh(l)766 693 y Fk(n)5 b(f)p Fs(0)p Fk(g)14 b Fd(parallel)e(to)i Fl(k)h Fd(suc)o(h)d(that)1383 679 y Fs(^)1372 693 y Fl(f)1396 700 y Fh(k)1418 690 y Ff(0)1435 693 y Fs(\()p Fl(J)5 b Fs(\))14 b Fk(6)p Fs(=)f(0)p Fd(.)21 b(Then)13 b(the)57 762 y(system)h(is)g(not)g(a)h(smo)q(oth)f (one{parameter)f(family)h(of)h(completely)f(canonically)g(in)o (tegrable)57 832 y(Hamiltonians.)57 964 y Fs(One)i(can)g(also)g(recall) g(the)h(follo)o(wing)e(theorem)h(of)h(Markus)e(and)h(Mey)o(er)g([MM])57 1096 y Fr(Theorem)24 b(8.3)k Fd(Generically)22 b(hamiltonian)e(systems) i(are)g(neither)g(completely)h(canoni-)57 1166 y(cally)16 b(in)o(tegrable)g(nor)f(ergo)q(dic)57 1298 y Fr(Exercise)h(8.4)c Fs(Pro)o(v)o(e)g(P)o(oincar)o(\023)-24 b(e's)11 b(Theorem)h(follo)o (wing)g(these)h(lines.)20 b(Using)13 b(the)g(notations)57 1367 y(in)o(tro)q(duced)18 b(ab)q(o)o(v)o(e,)i(if)g(the)g(system)f(w)o (ere)g(completely)h(canonically)f(in)o(tegrable)g(then)g(the)57 1437 y(new)12 b(actions)332 1424 y(~)320 1437 y Fl(J)17 b Fs(w)o(ould)11 b(b)q(e)i(a)g(system)f(of)h Fl(l)g Fs(indep)q(enden)o (t)e(\014rst)h(in)o(tegrals)f(of)i(the)g(Hamiltonian)57 1507 y(\015o)o(w)d(of)h Fk(H)h Fs(in)e(in)o(v)o(olution.)19 b(W)l(riting)10 b(them)h(explicitly)g(in)g(terms)f(of)h(the)h(old)e(lo) q(cal)h(co)q(ordinates)57 1577 y(\()p Fl(J)o(;)d(\037)p Fs(\))17 b(one)g(has)655 1634 y(~)643 1646 y Fl(J)i Fs(=)13 b Fl(J)j Fs(+)11 b Fl(")871 1634 y Fs(~)859 1646 y Fl(J)887 1653 y Fi(1)909 1646 y Fs(\()p Fl(J)o(;)d(\037)p Fs(\))k(+)f Fk(O)q Fs(\()p Fl(")1172 1626 y Fi(2)1195 1646 y Fs(\))j Fl(;)484 b Fs(\(8)p Fl(:)p Fs(2\))57 1745 y(for)17 b(some)g(smo)q(oth)g (function)645 1732 y(~)633 1745 y Fl(J)661 1752 y Fi(1)707 1745 y Fs(:)24 b Fl(V)774 1752 y Fh(")811 1745 y Fk(!)16 b Fl(U)911 1752 y Fh(")932 1745 y Fs(.)26 b(Imp)q(osing)16 b(that)i Fk(f)1335 1732 y Fs(~)1323 1745 y Fl(J)t(;)8 b Fk(Hg)17 b Fs(=)e Fk(O)q Fs(\()p Fl(")1598 1727 y Fi(2)1621 1745 y Fs(\))j(leads)f(to)57 1814 y(the)f(system)h(of)f(linear)g (partial)g(di\013eren)o(tial)f(equations)585 1904 y Fh(l)555 1919 y Fe(X)559 2025 y Fh(i)p Fi(=1)642 1933 y Fl(@)s(h)700 1940 y Fi(0)p 642 1955 81 2 v 645 2000 a Fl(@)s(J)702 2007 y Fh(i)734 1933 y Fl(@)775 1920 y Fs(~)763 1933 y Fl(J)791 1940 y Fi(1)p Fh(j)p 734 1955 98 2 v 744 2000 a Fl(@)s(\037)804 2007 y Fh(i)851 1966 y Fs(=)921 1933 y Fl(@)s(f)p 910 1955 82 2 v 910 2000 a(@)s(\037)970 2007 y Fh(j)1011 1966 y Fl(;)50 b(j)16 b Fs(=)d(1)p Fl(;)8 b(:)g(:)g(:)h(;)f(l)q(:)396 b Fs(\(8)p Fl(:)p Fs(3\))57 2115 y(Using)16 b(F)l(ourier)e(series)i(try)h(to)f(\014nd)g(a)h(smo)q (oth)f(solution)f(to)i(these)g(equations)f Fl(:)8 b(:)g(:)p Fs(.)57 2290 y Fo(8.2)17 b(Constan)n(t)j(Co)r(e\016cien)n(ts)e(Linear)h (PDE)f(on)g Fm(T)1175 2272 y Fh(n)1218 2290 y Fo(and)h(Loss)e(of)h (Di\013eren)n(tia-)57 2360 y(bilit)n(y)-5 b(.)57 2465 y Fs(The)16 b(\(v)o(ery\))h(short)e(sk)o(etc)o(h)h(of)g(the)h(pro)q(of) f(of)g(P)o(oincar)o(\023)-24 b(e's)15 b(Theorem)g(led)h(us)g(to)g (consider)f(the)57 2535 y(general)g(constan)o(t)h(co)q(e\016cien)o(ts)g (linear)g(partial)g(di\013eren)o(tial)f(equation)h(on)g Fm(T)1539 2517 y Fh(n)732 2650 y Fl(D)773 2657 y Fh(\026)800 2650 y Fl(u)e Fs(:=)f Fl(\026)e Fk(\001)g Fl(@)s(u)j Fs(=)f Fl(v)j(;)573 b Fs(\(8)p Fl(:)p Fs(4\))918 2770 y(54)p eop %%Page: 55 56 55 55 bop 57 192 a Fs(where)18 b Fl(\026)g Fk(2)g Fm(R)341 173 y Fh(n)365 192 y Fs(,)i Fl(@)s(u)e Fs(=)f(\()p Fl(@)576 199 y Fi(1)599 192 y Fl(u;)8 b(:)g(:)g(:)g(;)g(@)764 199 y Fh(n)792 192 y Fl(u)p Fs(\),)20 b(is)e(the)h(gradien)o(t)f(of)h Fl(u)p Fs(,)g Fl(v)h Fk(2)e(C)1457 173 y Fi(0)p Fh(;)p Fj(1)1531 192 y Fs(\()p Fm(T)1587 173 y Fh(n)1611 192 y Fl(;)8 b Fm(R)1672 173 y Fh(m)1707 192 y Fs(\))19 b(\(i.e.)57 261 y Fl(v)d Fk(2)e(C)173 243 y Fj(1)215 261 y Fs(\()p Fm(T)271 243 y Fh(n)295 261 y Fl(;)8 b Fm(R)356 243 y Fh(m)391 261 y Fs(\))17 b Fp(and)523 221 y Fe(R)546 279 y Fg(T)567 269 y Fb(n)605 261 y Fl(v)r Fs(\()p Fl(\022)q Fs(\))p Fl(d\022)g Fs(=)d(0\).)22 b(Indeed)17 b(for)f(all)g(\014xed)h (v)m(alue)g(of)f Fl(J)22 b Fs(the)17 b(equation)57 331 y(\(8.3\))g(is)f(a)g(sp)q(ecial)g(case)h(of)g(\(8.4\))g(with)f Fl(n)e Fs(=)f Fl(m)h Fs(=)g Fl(l)q Fs(.)156 403 y(It)23 b(is)g(easy)f(to)h(c)o(hec)o(k)g(\(see)g(App)q(endix)f(A3)h(for)f(a)h (detailed)f(discussion)f(of)i(the)g(case)57 473 y Fl(n)13 b Fs(=)h(2\))i(that)h Fl(D)362 480 y Fh(\026)405 473 y Fs(is)f(h)o(yp)q(o)q(elliptic)725 454 y Fi(1)764 473 y Fs(if)g(and)g(only)g(if)g Fl(\026)g Fs(is)g(a)g(diophan)o(tine)e(v)o (ector,)i(i.e.)22 b(there)57 542 y(exist)17 b(t)o(w)o(o)f(constan)o(ts) f Fl(\015)i(>)c Fs(0)k(and)f Fl(\034)j Fk(\025)13 b Fl(n)e Fk(\000)g Fs(1)17 b(suc)o(h)e(that)609 678 y Fk(j)p Fl(\026)c Fk(\001)g Fl(k)r Fk(j)i(\025)h Fl(\015)s Fk(j)p Fl(k)r Fk(j)882 657 y Fj(\000)p Fh(\034)950 678 y Fk(8)p Fl(k)g Fk(2)g Fm(Z)1102 657 y Fh(n)1138 678 y Fk(n)d(f)p Fs(0)p Fk(g)i Fl(;)450 b Fs(\(8)p Fl(:)p Fs(5\))57 813 y(where)16 b Fl(k)f Fs(=)f(\()p Fl(k)340 820 y Fi(1)362 813 y Fl(;)8 b(:)g(:)g(:)h(k)477 820 y Fh(n)504 813 y Fs(\),)17 b Fk(j)p Fl(k)r Fk(j)c Fs(=)g Fk(j)p Fl(k)715 820 y Fi(1)737 813 y Fk(j)e Fs(+)g Fl(:)d(:)g(:)j Fs(+)g Fk(j)p Fl(k)971 820 y Fh(n)998 813 y Fk(j)p Fs(.)57 921 y Fr(Exercise)19 b(8.5)c Fs(Pro)o(v)o(e)g(that)h(almost)f(all)g Fl(\026)f Fk(2)g Fm(R)977 902 y Fh(n)1017 921 y Fs(is)i(diophan)o(tine)d(of)j (exp)q(onen)o(t)g Fl(\034)j(>)14 b(n)9 b Fk(\000)h Fs(1.)57 1028 y Fr(Exercise)17 b(8.6)c Fs(Ha)o(v)o(e)h(a)g(lo)q(ok)h(to)f(the)h (b)q(o)q(ok)f(of)h(Y.)f(Mey)o(er)f([Me].)21 b(Among)13 b(man)o(y)h(in)o(teresting)57 1097 y(things)24 b(one)h(\014nds)f(the)i (follo)o(wing)e(theorem)h(\(Prop)q(osition)f(2,)k(p.)48 b(16\))25 b(:)40 b(Let)26 b Fk(R)g Fs(b)q(e)f(a)57 1167 y(real)e(algebraic)g(n)o(um)o(b)q(er)f(\014eld)h(and)g(let)h Fl(n)g Fs(b)q(e)g(its)g(degree)g(o)o(v)o(er)f Fm(Q)p Fs(.)45 b(Let)24 b Fl(\033)i Fs(b)q(e)f(the)f Fm(Q)p Fs({)57 1237 y(isomorphism)15 b(of)k Fk(R)h Fs(suc)o(h)d(that)j Fl(\033)r Fs(\()p Fk(R)p Fs(\))f Fk(\032)e Fm(R)f Fs(and)i(let)i Fl(\026)1140 1244 y Fi(1)1162 1237 y Fl(;)8 b(:)g(:)g(:)h(\026)1281 1244 y Fh(n)1327 1237 y Fs(b)q(e)19 b(an)o(y)f(basis)g(of)h Fk(R)g Fs(o)o(v)o(er)57 1307 y Fm(Q)p Fs(.)30 b(Then)18 b(\()p Fl(\033)r Fs(\()p Fl(\026)369 1314 y Fi(1)393 1307 y Fs(\))p Fl(;)8 b(:)g(:)g(:)h(;)f(\033)r Fs(\()p Fl(\026)602 1314 y Fh(n)629 1307 y Fs(\)\))19 b Fk(2)f Fm(R)776 1289 y Fh(n)819 1307 y Fs(is)h(diophan)o(tine)e(of)i(exp)q (onen)o(t)g Fl(\034)k Fs(=)18 b Fl(n)13 b Fk(\000)f Fs(1.)29 b(T)l(ry)19 b(to)57 1376 y(pro)o(v)o(e)d(it)j(if)f(y)o(ou)f(remem)o(b)q (er)g(a)g(tin)o(y)h(bit)g(of)g(Galois)g(theory)l(.)26 b(Apply)17 b(it)i(to)f(\(1)p Fl(;)1565 1335 y Fk(p)p 1607 1335 25 2 v 41 x Fs(2)p Fl(;)1654 1335 y Fk(p)p 1696 1335 V 41 x Fs(3)o Fl(;)1742 1335 y Fk(p)p 1784 1335 V 41 x Fs(6\))57 1446 y(and)g(\(1)p Fl(;)8 b Fs(2)247 1428 y Fi(1)p Fh(=)p Fi(3)310 1446 y Fl(;)g Fs(2)357 1428 y Fi(2)p Fh(=)p Fi(3)420 1446 y Fs(\).)31 b(There)18 b(exist)i(also)f(higher)f(dimensional)f(generalizations)h(of)h(Roth's) 57 1516 y(Theorem)d(quoted)h(in)f(Exercise)h(4.9)f(:)23 b(see,)17 b(for)g(example,)f(the)h(Subspace)f(Theorem)g([Sc)o(h1,)57 1586 y(Sc)o(h2].)57 1693 y(In)i(addition)f(to)i(kno)o(wing)e(that)i Fl(u)e Fk(2)g(C)815 1675 y Fj(1)857 1693 y Fs(\()p Fm(T)913 1675 y Fh(n)937 1693 y Fl(;)8 b Fm(R)998 1675 y Fh(m)1033 1693 y Fs(\))19 b(one)f(has)g(the)g(follo)o(wing)g(more)f(precise)57 1763 y(estimate)f(:)57 1906 y Fr(Prop)r(osition)f(8.7)28 b Fd(Let)15 b Fk(k)f(k)616 1913 y Fh(k)654 1906 y Fd(denote)g(the)g Fk(C)924 1888 y Fh(k)962 1906 y Fd(norm.)20 b(If)14 b Fl(\026)g Fd(is)g(diophan)o(tine)e(with)h(exp)q(onen)o(t)57 1975 y Fl(\034)23 b Fd(then)c(for)e(all)h Fl(r)h(>)d(\034)i Fs(+)12 b Fl(n)g Fk(\000)g Fs(1)18 b Fd(and)g(for)f(all)h Fl(i)f Fk(2)g Fm(N)i Fd(there)f(exists)h(a)f(p)q(ositiv)o(e)g(constan)o (t)f Fl(A)1811 1982 y Fh(i)57 2045 y Fd(suc)o(h)e(that)750 2121 y Fk(k)p Fl(u)p Fk(k)829 2128 y Fh(i)859 2121 y Fk(\024)e Fl(A)948 2128 y Fh(i)965 2121 y Fk(k)p Fl(v)r Fk(k)1041 2128 y Fh(i)p Fi(+)p Fh(r)1121 2121 y Fl(:)591 b Fs(\(8)p Fl(:)p Fs(6\))57 2269 y Fp(Pr)m(o)m(of.)20 b Fs(Let)d Fl(u)p Fs(\()p Fl(\022)q Fs(\))e(=)453 2232 y Fe(P)506 2284 y Fh(k)q Fj(2)p Fg(Z)575 2274 y Fb(n)616 2269 y Fs(^)-28 b Fl(u)642 2276 y Fh(k)666 2269 y Fl(e)689 2251 y Fi(2)p Fh(\031)q(ik)q Fj(\001)p Fh(\022)804 2269 y Fs(,)17 b(where)f(ob)o(v)o(oiusly)f(one)h(has)673 2425 y(^)-28 b Fl(u)699 2432 y Fh(k)737 2425 y Fs(=)789 2357 y Fe(Z)817 2470 y Fg(T)837 2461 y Fb(n)876 2425 y Fl(u)p Fs(\()p Fl(\022)q Fs(\))p Fl(e)990 2405 y Fj(\000)p Fi(2)p Fh(\031)q(ik)q Fj(\001)p Fh(\022)1137 2425 y Fl(d\022)15 b(:)p 57 2525 600 2 v 109 2563 a Fi(1)156 2581 y Fs(A)j(constan)o(t)f (co)q(e\016cien)o(ts)g(linear)g(partial)f(di\013eren)o(tial)h(op)q (erator)g Fl(P)24 b Fs(is)17 b(h)o(yp)q(o)q(elliptic)g(if)57 2650 y(all)f Fl(u)g Fs(suc)o(h)f(that)i Fl(P)7 b(u)14 b Fs(=)f Fl(v)19 b Fs(are)d Fk(C)680 2632 y Fj(1)739 2650 y Fs(on)g(all)g(op)q(en)h(sets)f(where)g Fl(v)j Fs(is)d Fk(C)1361 2632 y Fj(1)1420 2650 y Fs(\(see)g([H1],)h(p.109\).) 918 2770 y(55)p eop %%Page: 56 57 56 56 bop 57 192 a Fs(Then)16 b(the)i Fk(C)303 173 y Fh(k)327 192 y Fs({norm)e(can)h(b)q(e)g(equiv)m(alen)o(tly)g(giv)o(en)g (in)f(terms)h(of)g(F)l(ourier)e(co)q(e\016cien)o(ts)i(:)23 b(for)57 261 y(all)16 b Fl(i)e Fk(2)g Fm(N)k Fs(there)e(exists)g(a)h(p) q(ositiv)o(e)f(constan)o(t)g Fl(B)986 268 y Fh(i)1019 261 y Fs(suc)o(h)g(that)278 380 y Fl(B)318 359 y Fj(\000)p Fi(1)316 394 y Fh(i)391 380 y Fs(sup)380 423 y Fh(k)q Fj(2)p Fg(Z)450 413 y Fb(n)477 380 y Fs([\(1)11 b(+)g Fk(j)p Fl(k)r Fk(j)p Fs(\))671 360 y Fh(i)687 380 y Fk(j)s Fs(^)-28 b Fl(u)730 387 y Fh(k)754 380 y Fk(j)p Fs(])14 b Fk(\024)f(k)p Fl(u)p Fk(k)927 387 y Fh(i)957 380 y Fk(\024)g Fl(B)1047 387 y Fh(i)1083 380 y Fs(sup)1072 423 y Fh(k)q Fj(2)p Fg(Z)1141 413 y Fb(n)1169 380 y Fs([\(1)e(+)g Fk(j)p Fl(k)r Fk(j)p Fs(\))1363 360 y Fh(i)p Fi(+)p Fh(n)p Fi(+1)1484 380 y Fk(j)s Fs(^)-28 b Fl(u)1527 387 y Fh(k)1551 380 y Fk(j)p Fs(])14 b Fl(:)119 b Fs(\(8)p Fl(:)p Fs(7\))57 519 y(Comparing)14 b(the)j(F)l(ourier)e(co)q(e\016cien)o(ts)h(of)g Fl(u)g Fs(with)h(those)f(of)h Fl(v)i Fs(one)d(has)613 651 y(^)-28 b Fl(u)639 658 y Fh(k)677 651 y Fs(=)796 618 y(^)h Fl(v)818 625 y Fh(k)p 736 640 166 2 v 736 686 a Fs(2)p Fl(\031)r(ik)13 b Fk(\001)d Fl(\026)949 651 y Fk(8)p Fl(k)k Fk(2)g Fm(Z)1101 631 y Fh(n)1136 651 y Fk(n)d(f)p Fs(0)p Fk(g)j Fl(;)451 b Fs(\(8)p Fl(:)p Fs(8\))57 787 y(The)17 b(desired)f(estimates)h(are)g(an)g(easy)g (consequence)g(of)h(\(8.7\),)g(the)f(assumption)f(that)h Fl(\026)h Fs(is)57 857 y(diophan)o(tine)c(and)i(of)h(the)g(elemen)o (tary)e(fact)922 819 y Fe(P)974 872 y Fh(k)q Fj(2)p Fg(Z)1043 862 y Fb(n)1071 872 y Fj(nf)p Fi(0)p Fj(g)1163 857 y Fk(j)p Fl(k)r Fk(j)1219 839 y Fj(\000)p Fh(\016)1285 857 y Fl(<)e Fs(+)p Fk(1)j Fs(for)h(all)f Fl(\016)g(>)d(n)p Fs(.)68 b Fa(\003)57 976 y Fs(The)22 b(fact)h(that)g(one)g(needs)f Fl(r)i Fs(more)e(deriv)m(ativ)o(es)g(to)h(b)q(ound)e(the)i(norms)e(of)i Fl(u)f Fs(in)g(terms)57 1045 y(of)d(those)g(of)g Fl(v)i Fs(is)e(what)g(is)g(called)f(the)i(\\loss)e(of)h(di\013eren)o(tiabilit) o(y".)28 b(As)19 b(w)o(e)g(ha)o(v)o(e)f(already)57 1115 y(seen)f(in)g(Chapter)f(6)i(this)e(is)h(a)h(t)o(ypical)f(phenomenon)e (asso)q(ciated)i(to)h(small)e(divisors.)23 b(The)57 1185 y(analogue)15 b(in)i(the)g(analytic)g(case)f(w)o(ould)g(b)q(e)h(the)g (necessary)f(restriction)g(of)h(the)g(domain)e(to)57 1254 y(con)o(trol)g(the)j(maxim)o(um)c(norm)i(of)h Fl(u)f Fs(in)g(terms)g(of)h Fl(v)i Fs(b)o(y)d(means)g(of)h(Cauc)o(h)o(y's)e (estimates)i(as)57 1324 y(w)o(e)f(did)g(in)g(Section)g(6.3.)156 1394 y(In)k(b)q(oth)f(cases)g(\(smo)q(oth)h(and)e(analytic\))i(these)g (are)f(not)h(artefacts)g(of)f(the)h(metho)q(ds)57 1464 y(used)h(but)h(a)g(concrete)h(manifestation)e(of)h(the)h(un)o(b)q (oundedness)c(of)j(the)h(linear)e(op)q(erator)57 1533 y Fl(D)99 1515 y Fj(\000)p Fi(1)98 1546 y Fh(\026)153 1533 y Fs(.)35 b(The)21 b(main)f(consequence)h(of)g(this)f(fact)i(is)f (that)g(one)g(cannot)f(use)h(Banac)o(h)f(spaces)57 1603 y(tec)o(hniques)e(to)h(study)g(semilinear)e(equations)i(lik)o(e)g Fl(D)1105 1610 y Fh(\026)1132 1603 y Fl(u)e Fs(=)h Fl(v)c Fs(+)f Fl("f)5 b Fs(\()p Fl(u)p Fs(\),)21 b(where)d Fl(")i Fs(is)e(some)57 1673 y(small)e(parameter.)24 b(These)17 b(semilinear)e(equations)i(are)h(ho)o(w)o(ev)o(er)e(t)o(ypical)h(of)h (p)q(erturbation)57 1743 y(theory)l(.)57 1918 y Fo(8.3)h(KAM)h(Theory) -5 b(,)18 b(Nekhoroshev)g(Theorem,)g(Arnol'd)i(Di\013usion)57 2023 y Fs(Despite)12 b(Theorem)g(8.2,)h(most)g(results)f(on)g(quasi{in) o(tegrable)f(systems)h(ha)o(v)o(e)g(b)q(een)h(obtained)57 2092 y(under)21 b(the)i(assumption)d(of)j(non{degeneracy)e(\(i.e.)40 b(the)23 b(hessian)e(matrix)h(of)g Fl(h)1647 2099 y Fi(0)1692 2092 y Fs(is)g(non)57 2162 y(degenerate)28 b(th)o(us)h(the)g(frequency) g(map)f Fl(J)39 b Fk(7!)c Fl(\027)1077 2169 y Fi(0)1099 2162 y Fs(\()p Fl(J)5 b Fs(\))35 b(=)1284 2142 y Fh(@)r(h)1330 2147 y Fc(0)p 1284 2151 67 2 v 1293 2179 a Fh(@)r(J)1356 2162 y Fs(\()p Fl(J)5 b Fs(\))35 b Fk(2)g Fm(R)1569 2144 y Fh(l)1611 2162 y Fs(is)28 b(a)h(lo)q(cal)57 2232 y (di\013eomorphism\))15 b(but)i(accepting)h(the)g(fact)g(that)g(one)f (cannot)h(hop)q(e)f(for)h(in)o(tegrabilit)o(y)e(on)57 2302 y(op)q(en)g(sets.)156 2371 y(The)f(general)f(picture)h(is)f(pro)o (vided)f(b)o(y)i(KAM)g([Ar1,Ga,)f(Bo,)i(Y)l(o1])f(and)f(Nekhoroshev)57 2441 y([Ne,)e(Lo])g(theorems)e(:)20 b(if)11 b Fl(")h Fs(is)f(su\016cien)o(tly)f(small,)h(most)g(initial)g(conditions)f (\(w.r.t.)21 b(Leb)q(esgue)57 2511 y(measure\))k(lie)h(on)g(in)o(v)m (arian)o(t)f Fl(l)q Fs({dimensional)f(lagrangian)g(tori)i(carrying)g (quasip)q(erio)q(dic)57 2581 y(motions)11 b(with)g(Diophan)o(tine)g (frequencies.)20 b(The)12 b(action)f(v)m(ariables)g(corresp)q(onding)f (to)i(these)57 2650 y(orbits)20 b(will)i(remain)e Fl(\017)p Fs({close)h(to)h(their)g(initial)f(v)m(alues)g(for)g(all)h(times.)36 b(The)22 b(complemen)o(t)918 2770 y(56)p eop %%Page: 57 58 57 57 bop 57 192 a Fs(of)28 b(this)h(set)f(is)h(op)q(en)f(and)g(dense)g (and)g(it)g(is)h(connected)f(if)h Fl(l)34 b Fk(\025)g Fs(3.)58 b(It)29 b(con)o(tains)e(a)57 261 y(connected)13 b(\()p Fl(l)i Fk(\025)f Fs(3\))f(w)o(eb)g Fk(R)h Fs(of)g(resonan)o(t)d (zones)i(corresp)q(onding)f(to)h Fm(Z)1391 243 y Fh(l)1404 261 y Fs({linearly)f(dep)q(enden)o(t)57 331 y(frequencies)19 b(:)29 b Fk([)391 341 y Fh(k)q Fj(2)p Fg(Z)460 331 y Fb(l)8 b Fk(f)p Fl(J)24 b Fk(2)c Fl(U)14 b(;)8 b(\027)705 338 y Fi(0)727 331 y Fs(\()p Fl(J)d Fs(\))13 b Fk(\001)h Fl(k)21 b Fs(=)e(0)p Fk(g)13 b(\002)g Fm(T)1096 313 y Fh(l)1108 331 y Fs(.)32 b(Motion)20 b(along)f(these)h(resonances)57 401 y(cannot)c(b)q(e)i(excluded)e(\(see)h([Ar2])g(for)f(an)h(explicit)g (example\),)g(resulting)e(in)i(a)g(v)m(ariation)f(of)57 470 y Fk(O)q Fs(\(1\))21 b(of)f(the)h(actions)f(in)g(a)g(\014nite)g (time)840 452 y Fi(2)862 470 y Fs(.)34 b(But)20 b(if)h(the)f (hamiltonian)f(is)h Fp(analytic)h Fs(and)e Fl(h)1806 477 y Fi(0)57 540 y Fs(is)e Fp(ste)m(ep)h Fs(\(for)f(example)g(con)o(v) o(ex)h(or)f(quasi)f(con)o(v)o(ex\))i(then)f(this)g(v)m(ariation)g(is)g (v)o(ery)h(slo)o(w)e(:)24 b(it)57 610 y(tak)o(es)c(a)h(time)g(at)g (least)g Fk(O)587 570 y Fe(\000)610 610 y Fs(exp)694 570 y Fe(\000)733 590 y Fi(1)p 723 599 41 2 v 723 627 a Fh(")742 617 y Fb(a)770 570 y Fe(\001)o(\001)836 610 y Fs(to)g(c)o(hange)f(the)h(actions)f(of)h Fk(O)q Fs(\()p Fl(")1477 592 y Fh(b)1498 610 y Fs(\),)h(where)e Fl(a)h Fs(and)57 680 y Fl(b)f Fs(are)f(t)o(w)o(o)g(p)q(ositiv)o(e)g(constan)o (ts.)30 b(Moreo)o(v)o(er)17 b(eac)o(h)i(in)o(v)m(arian)o(t)f(torus)h (has)g(a)g(neigh)o(b)q(orho)q(o)q(d)57 749 y(\014lled)13 b(in)g(with)h(tra)s(jectiories)f(whic)o(h)g(remain)f(close)i(to)g(it)g (for)f(an)h(ev)o(en)f(longer)g(time.)21 b(Indeed,)57 819 y(if)13 b Fl(h)128 826 y Fi(0)164 819 y Fs(is)g(quasi{con)o(v)o(ex) f(one)i(can)f(pro)o(v)o(e)f([GM])h(that)h(all)f(tra)s(jectories)g (starting)g(at)h(a)f(distance)57 889 y(of)18 b(order)f Fl(\032)f(<)g(\032)370 871 y Fj(\003)411 889 y Fs(from)i(a)g(Diophan)o (tine)e Fl(l)q Fs({torus)h(of)i(exp)q(onen)o(t)f Fl(\034)23 b Fs(will)18 b(remain)f(close)h(to)g(it)57 976 y(for)e(a)g(time)h Fk(O)337 906 y Fe(\022)374 976 y Fs(exp)458 906 y Fe(\022)495 976 y Fs(exp)579 921 y Fe(\020)615 954 y Fh(\032)636 939 y Ff(\003)p 615 964 42 2 v 626 993 a Fh(\032)663 921 y Fe(\021)692 931 y Fi(1)p Fh(=\034)t Fi(+1)808 906 y Fe(\023\023)881 976 y Fs(.)156 1065 y(One)c(of)h(the)f(consequences)g (of)g(KAM)g(theorem)g([P\177)-25 b(o])12 b(is)h(the)h(existence,)g(for) f(su\016cien)o(tly)57 1135 y(small)j(v)m(alues)i(of)g Fl(")p Fs(,)h(of)f(a)g Fp(Cantor)h(set)f Fl(N)842 1142 y Fh(")882 1135 y Fs(of)g(v)m(alues)g(of)g(the)g(frequencies)g Fl(\027)i Fs(for)e(whic)o(h)f(the)57 1205 y(Hamiltonian)j(system)h (\(8.1\))g(has)g(smo)q(oth)f(in)o(v)m(arian)o(t)g(tori)h(with)g(linear) f(\015o)o(w.)35 b(Moreo)o(v)o(er)57 1275 y(there)21 b(exists)g(a)h (homeomorphis)o(m)c Fl(F)788 1282 y Fh(")840 1275 y Fs(:)k Fl(N)916 1282 y Fh(")951 1275 y Fk(\002)14 b Fm(T)1040 1257 y Fh(l)1074 1275 y Fk(!)22 b Fl(U)e Fk(\002)14 b Fm(T)1289 1257 y Fh(l)1323 1275 y Fl(")p Fs({close)21 b(to)h(the)f(iden)o(tit)o(y)l(,)57 1344 y(Whitney)j(smo)q(oth)f(w.r.t.) 43 b(the)24 b(\014rst)f(factor)h(and)f(analytic)h(w.r.t.)43 b(the)24 b(second)f(\(if)h(the)57 1414 y(Hamiltonian)16 b(\(8.1\))h(is)g(analytic\))g(whic)o(h)f(transforms)f(Hamilton's)h (equations)g(in)o(to)25 b(_)-22 b Fl(\027)17 b Fs(=)e(0,)70 1484 y(_)-27 b Fl(')13 b Fs(=)h Fl(\027)s Fs(.)20 b(This)13 b(foliation)g(in)o(to)g(in)o(v)m(arian)o(t)e(tori)i(is)g(th)o(us)g (parametrized)e(o)o(v)o(er)i(a)g(Can)o(tor)g(set)g(and)57 1554 y(hence)g(no)o(where)e(dense.)21 b(It)13 b(exhibits)g(the)g (phenomenon)e(of)j(\\anisotropic)d(di\013eren)o(tiabilit)o(y")57 1623 y(since)k(it)g(is)g(m)o(uc)o(h)f(more)h(regular)f(tangen)o(tially) h(to)g(these)h(tori)f(than)g(transv)o(ersally)f(to)i(them)57 1693 y(\(see)h(also)e([BHS]\).)p 57 2525 600 2 v 109 2563 a Fi(2)156 2581 y Fs(It)f(is)e(conjectured)h([AKN,)g(p.)20 b(189])13 b(that)g(generically)g(quasi{in)o(tegrable)e(hamiltonians)57 2650 y(with)16 b(more)g(than)g(t)o(w)o(o)g(degrees)g(of)h(freedom)e (are)i(top)q(ologically)f(unstable)918 2770 y(57)p eop %%Page: 58 59 58 58 bop 57 192 a Fq(9.)31 b(The)24 b(In)n(v)n(erse)h(F)-6 b(unction)25 b(Theorem)e(of)g(Nash)h(and)g(Moser)57 302 y Fs(The)19 b(In)o(v)o(erse)f(F)l(unction)g(Theorem)h(for)g(Banac)o(h)f (spaces)h(is)g(one)g(of)h(the)f(extremely)h(useful)57 372 y(standard)15 b(to)q(ols)i(in)f(the)h(study)g(of)g(a)g(v)m(ariet)o (y)g(of)g(non{linear)e(problems,)g(ranging)g(from)h(the)57 442 y(go)q(o)q(d)j(p)q(osition)g(of)g(the)h(Cauc)o(h)o(y)e(problem)f (for)i(ordinary)f(di\013eren)o(tial)g(equations)h(to)g(non{)57 511 y(linear)i(elliptic)h(equations.)40 b(Unfortunately)22 b(the)h(\\loss)f(of)g(di\013eren)o(tiabilit)o(y")f(t)o(ypical)h(of)57 581 y(small)c(divisors)h(problems)f(prev)o(en)o(ts)h(from)g(its)h(use)f (\(with)i(some)e(remark)m(able)g(exceptions)57 651 y(ho)o(w)o(ev)o(er,) e(see)h(Section)g(6.2)g(and)f([He2]\).)28 b(In)18 b(the)g(analytic)h (case,)f(Kolmogoro)o(v)e(suggested)57 721 y(the)27 b(use)f(of)h(a)f(mo) q(di\014ed)g(Newton)h(metho)q(d)f(to)h(o)o(v)o(ercome)f(this)g (di\016cult)o(y)g(but)g(in)g(the)57 790 y(di\013eren)o(tiable)c(case)i (the)g(need)f(of)h(an)g(In)o(v)o(erse)f(F)l(unction)f(Theorem)h(in)g(F) l(r)o(\023)-24 b(ec)o(het)23 b(spaces)57 860 y(has)i(also)h(other)g (sources)f(:)42 b(its)26 b(origin)f(is)h(the)h(solution)e(of)i(the)f (em)o(b)q(edding)f(problem)57 930 y(for)f(Riemannian)f(manifolds)g(b)o (y)h(Nash)h([N].)f(Later)h(Moser)f(disco)o(v)o(ered)f(ho)o(w)h(to)h (adapt)57 1000 y(Kolmogoro)o(v's)9 b(idea)i(to)h(the)g(di\013eren)o (tiable)e(case)i(creating)f(a)g(theory)h(with)f(a)h(wide)f(sp)q(ectrum) 57 1069 y(of)16 b(applications)f([A)o(G,)h(Gr,)g(H2,)h(Ha,)f(Ni,)h (Ser,)e(St,)i(SZ,)f(Ze1])h(:)22 b(to)16 b(geometry)l(,)g(to)h(the)g (study)57 1139 y(of)k(foliations)g(and)g(deformations)f(of)i(complex)f (and)g(CR)g(structures,)g(to)h(free)g(b)q(oundary)57 1209 y(problems,)j(etc..)48 b(In)25 b(all)g(these)g(cases)g(a)g (non{linear)f(partial)g(di\013eren)o(tial)g(equation)h(is)57 1279 y(solv)o(ed)e(using)h(a)g(rapidly)f(con)o(v)o(ergen)o(t)g (iterativ)o(e)i(algorithm)e(in)o(tro)q(ducing)g(at)i(eac)o(h)f(step)57 1348 y(of)16 b(the)h(iteration)f(a)h(smo)q(othing)e(of)i(the)g(appro)o (ximate)e(solution.)156 1423 y(In)i(this)f(Chapter)g(w)o(e)g(will)g (follo)o(w)g(the)h(presen)o(tation)e(of)i([Ha])f(v)o(ery)h(closely)l(.) 57 1604 y Fo(9.1)i(Calculus)i(in)g(F)-5 b(r)o(\023)-29 b(ec)n(het)19 b(Spaces)57 1761 y Fr(De\014nition)c(9.1)28 b Fd(A)12 b Fs(F)l(r)o(\023)-24 b(ec)o(het)11 b(space)h Fd(is)g(a)g(lo)q(cally)g(con)o(v)o(ex)g(top)q(ological)g(v)o(ector)g (space)f(\(lctvs\))57 1830 y(whic)o(h)k(is)h(complete,)g(Hausdor\013)g (and)g(metrizable.)57 1982 y Fr(Exercise)h(9.2)c Fs(Sho)o(w)g(that)h(a) g(lctvs)g Fl(X)k Fs(is)13 b(Hausdor\013)g(if)h(and)f(only)h(if)g Fl(x)g Fk(2)g Fl(X)t Fs(,)g Fk(k)p Fl(x)p Fk(k)1614 1989 y Fh(i)1645 1982 y Fs(=)g(0)f Fk(8)p Fl(i)g Fk(2)57 2052 y(I)20 b Fs(then)d Fl(x)d Fs(=)g(0)j(\(where)f(\()p Fk(k)c(\001)f(k)624 2059 y Fh(i)640 2052 y Fs(\))659 2059 y Fh(i)p Fj(2I)744 2052 y Fs(is)17 b(the)g(collection)f(of)h(seminorms)d(giving)i(the)h (top)q(ology)57 2121 y(of)f Fl(X)t Fs(\).)23 b(Sho)o(w)15 b(that)i Fl(X)k Fs(is)16 b(metrizable)g(if)h(and)e(only)i(if)f Fk(I)21 b Fs(is)16 b(coun)o(table.)57 2232 y Fr(Exercise)26 b(9.3)21 b Fs(Sho)o(w)f(that)i Fm(R)673 2214 y Fj(1)734 2232 y Fs(\(space)g(of)g(all)f(sequences)h(of)g(real)f(n)o(um)o(b)q (ers\),)g Fk(C)1694 2214 y Fj(1)1736 2232 y Fs(\()p Fl(M)5 b Fs(\))57 2302 y(\(where)22 b Fl(M)28 b Fs(is)22 b(a)g(smo)q(oth)g (compact)f(manifold\),)i Fk(A)p Fs(\()p Fm(C)9 b Fs(\))26 b(\(en)o(tire)c(functions\))g(are)f(F)l(r)o(\023)-24 b(ec)o(het)57 2371 y(spaces)14 b(\(th)o(us)g(the)h(exercise)f(asks)h(y) o(ou)f(to)h(de\014ne)f(suitable)g(seminorms\).)19 b(Sho)o(w)14 b(that)h Fk(C)1731 2378 y Fi(0)1754 2371 y Fs(\()p Fm(R)p Fs(\))57 2441 y(\(con)o(tin)o(uous)k(functions)i(with)g(compact)g(supp) q(ort\))g(with)g(the)h(usual)e(top)q(ology)h(\()p Fl(f)1677 2448 y Fh(n)1727 2441 y Fk(!)h Fl(f)57 2511 y Fs(if)i(and)f(only)g(if)h (there)g(exists)g(a)f(compact)h(in)o(terv)m(al)f Fl(I)28 b Fs(suc)o(h)22 b(that)j(supp)7 b Fl(f)1525 2518 y Fh(n)1578 2511 y Fk(\032)26 b Fl(I)h Fs(for)d(all)57 2581 y(su\016cien)o(tly)18 b(large)g Fl(n)p Fs(,)h(supp)7 b Fl(f)24 b Fk(\032)18 b Fl(I)23 b Fs(and)18 b Fl(f)880 2588 y Fh(n)927 2581 y Fs(con)o(v)o(erges)g(uniformly)f(to)j Fl(f)k Fs(on)19 b Fl(I)t Fs(\))g(is)g(a)g(lctvs)57 2650 y(but)d(it)h(is)f(not)g(a)h(F)l (r)o(\023)-24 b(ec)o(het)16 b(space)g(since)g(it)h(is)f(not)g (metrizable.)918 2770 y(58)p eop %%Page: 59 60 59 59 bop 57 192 a Fr(Exercise)22 b(9.4)d Fs(Pro)o(v)o(e)f(that)h (Hahn{Banac)o(h)e(Theorem)h(holds)g(in)g(F)l(r)o(\023)-24 b(ec)o(het)18 b(spaces)g(:)27 b(if)19 b Fl(X)57 261 y Fs(is)d(a)h(F)l(r)o(\023)-24 b(ec)o(het)16 b(space)g(and)g Fl(x)h Fs(is)g(a)g(non{zero)e(v)o(ector)i(in)g Fl(X)k Fs(then)16 b(there)h(exists)g(a)g(con)o(tin)o(uous)57 331 y(linear)h(functional)h Fl(l)28 b Fs(:)19 b Fl(X)k Fk(!)c Fm(R)8 b Fs(\(or)d Fm(C)k Fs(\))23 b(suc)o(h)c(that)h Fl(l)q Fs(\()p Fl(x)p Fs(\))g(=)e(1.)31 b(This)19 b(allo)o(ws)f(to)i (in)o(tro)q(duce)57 401 y(quite)d(straigh)o(tforw)o(ardly)d Fl(X)t Fs({v)m(alued)i(analytic)h(functions)f([V)l(a].)22 b(A)17 b(function)g Fl(x)22 b Fs(:)14 b(\012)g Fk(!)g Fl(X)t Fs(,)57 470 y(where)h(\012)i(is)e(a)i(region)e(in)h Fm(C)9 b Fs(,)19 b(is)d(analytic)g(if)g(and)g(only)g(if)g(for)g(all)g Fl(l)f Fk(2)f Fl(X)1419 452 y Fj(\003)1458 470 y Fs(the)i(function)g Fl(l)c Fk(\016)e Fl(x)57 540 y Fs(is)19 b(analytic.)33 b(Sho)o(w)19 b(that)i(this)f(is)f(equiv)m(alen)o(t)i(to)f(asking)g (that,)h(for)f(all)g Fl(z)1496 547 y Fi(0)1538 540 y Fk(2)g Fs(\012,)h Fl(x)g Fs(has)e(a)57 610 y(con)o(v)o(ergen)o(t)c(p)q (o)o(w)o(er)g(series)h(expansion)f(at)i Fl(z)900 617 y Fi(0)939 610 y Fs(:)22 b Fl(x)p Fs(\()p Fl(z)r Fs(\))16 b(=)1134 573 y Fe(P)1187 585 y Fj(1)1187 625 y Fh(n)p Fi(=0)1264 610 y Fs(\()p Fl(z)e Fk(\000)d Fl(z)1393 617 y Fi(0)1415 610 y Fs(\))1434 592 y Fh(n)1462 610 y Fl(x)1490 617 y Fh(n)1517 610 y Fs(.)57 718 y Fr(Exercise)23 b(9.5)c Fs(Extend)g(the)h(theory)f(of)g(Riemann's)f(in)o(tegration,)g (including)g(the)i(funda-)57 788 y(men)o(tal)15 b(theorem)h(of)h (calculus,)e(to)i(con)o(tin)o(uous)d Fl(X)t Fs({v)m(alued)j(functions)f (on)g([)p Fl(a;)8 b(b)p Fs(])14 b Fk(\032)g Fm(R)p Fs(.)57 933 y Fr(De\014nition)21 b(9.6)28 b Fd(Let)18 b Fl(X)q(;)8 b(Y)28 b Fd(b)q(e)18 b(t)o(w)o(o)e(F)l(r)o(\023)-24 b(ec)o(het)17 b(spaces,)f Fl(U)k Fk(\032)15 b Fl(X)21 b Fd(b)q(e)d(op)q(en,)e Fl(f)29 b Fs(:)15 b Fl(U)20 b Fk(!)15 b Fl(Y)28 b Fd(b)q(e)57 1003 y(con)o(tin)o(uous.)19 b(The)e Fs(deriv)m(ativ)o(e)f Fd(of)h Fl(f)22 b Fd(at)17 b Fl(x)e Fk(2)f Fl(U)22 b Fd(in)16 b(the)h(direction)e(of)i Fl(h)c Fk(2)i Fl(X)20 b Fd(is)558 1157 y Fl(D)q(f)5 b Fs(\()p Fl(x)p Fs(\))13 b Fk(\001)e Fl(h)j Fs(:=)i(lim)842 1188 y Fh(t)p Fj(!)p Fi(0)931 1123 y Fl(f)5 b Fs(\()p Fl(x)13 b Fs(+)e Fl(th)p Fs(\))g Fk(\000)g Fl(f)5 b Fs(\()p Fl(x)p Fs(\))p 931 1145 363 2 v 1103 1191 a Fl(t)1313 1157 y(:)399 b Fs(\(9)p Fl(:)p Fs(1\))57 1306 y Fl(f)30 b Fd(is)24 b Fk(C)198 1288 y Fi(1)245 1306 y Fd(on)g Fl(U)30 b Fd(if)24 b(and)g(only)g(if)g Fl(D)q(f)31 b Fd(exists)25 b(for)f(all)g Fl(x)j Fk(2)g Fl(U)j Fd(and)24 b(for)g(all)g Fl(h)i Fk(2)h Fl(X)i Fs(and)57 1376 y Fl(D)q(f)f Fs(:)14 b Fl(U)i Fk(\002)11 b Fl(X)18 b Fk(!)c Fl(Y)27 b Fd(is)17 b(con)o(tin)o(uous.)57 1519 y Fr(Remark)f(9.7)e Fs(In)h(the)h(case)f(of)g(Banac)o(h)g(spaces)f (this)h(de\014nition)f(of)h Fk(C)1394 1501 y Fi(1)1432 1519 y Fs(is)g(w)o(eak)o(er)f(than)h(the)57 1589 y(usual)g(one.)57 1696 y Fr(Exercise)i(9.8)c Fs(Pro)o(v)o(e)h(that)g(the)g(comp)q (osition)f(of)i Fk(C)1055 1678 y Fi(1)1091 1696 y Fs(maps)e(is)h Fk(C)1296 1678 y Fi(1)1333 1696 y Fs(and)f(that)i(the)f(c)o(hain)f (rule)57 1766 y(holds)i(:)22 b Fl(D)q Fs(\()p Fl(g)14 b Fk(\016)d Fl(f)5 b Fs(\)\()p Fl(x)p Fs(\))13 b Fk(\001)e Fl(h)i Fs(=)h Fl(D)q(g)r Fs(\()p Fl(f)5 b Fs(\()p Fl(x)p Fs(\)\))14 b Fk(\001)d Fs(\()p Fl(D)q(f)5 b Fs(\()p Fl(x)p Fs(\))14 b Fk(\001)d Fl(h)p Fs(\).)57 1874 y Fr(Exercise)k(9.9)d Fs(De\014ne)g(higher)f(order)g(deriv)m(ativ)o(es)h(and)g Fk(C)1155 1856 y Fh(k)1192 1874 y Fs(maps)f(b)q(et)o(w)o(een)h(F)l(r)o (\023)-24 b(ec)o(het)11 b(spaces.)57 1982 y Fr(Exercise)23 b(9.10)18 b Fs(Let)i Fl(f)33 b Fs(:)19 b Fk(C)619 1964 y Fj(1)661 1982 y Fs(\([)p Fl(a;)8 b(b)p Fs(]\))20 b Fk(!)e(C)913 1964 y Fj(1)956 1982 y Fs(\([)p Fl(a;)8 b(b)p Fs(]\),)21 b Fl(f)5 b Fs(\()p Fl(x)p Fs(\))20 b(=)f Fl(P)7 b Fs(\()p Fl(x;)h(x)1435 1964 y Fj(0)1450 1982 y Fl(;)g(:)g(:)g(:)h(;)f(x)1589 1964 y Fi(\()p Fh(n)p Fi(\))1648 1982 y Fs(\),)20 b(where)57 2052 y Fl(P)g Fk(2)14 b Fm(R)p Fs([)p Fl(X)250 2059 y Fi(0)270 2052 y Fl(;)8 b(:)g(:)g(:)g(X)399 2059 y Fh(n)427 2052 y Fs(],)15 b(is)f Fk(C)547 2034 y Fj(1)589 2052 y Fs(.)21 b(Is)15 b(there)f(a)h(nice)f(form)o(ula)f(for)i Fl(D)q(f)5 b Fs(\()p Fl(x)p Fs(\))i Fk(\001)g Fl(h)12 b Fs(?)21 b([Hin)o(t)15 b(:)21 b(start)14 b(from)57 2121 y(monomials)g(lik)o(e)i(\()p Fl(x)444 2103 y Fi(\()p Fh(i)p Fi(\))493 2121 y Fs(\))512 2103 y Fh(k)537 2121 y Fs(.])57 2229 y(The)j(follo)o(wing)f(examples)g (sho)o(w)g(wh)o(y)h(the)h(extension)f(of)g(the)h(in)o(v)o(erse)d (function)i(theorem)57 2299 y(to)i(F)l(r)o(\023)-24 b(ec)o(het)20 b(spaces)f(is)i(not)f(a)h(straigh)o(tforw)o(ard)d(generalization)i(of)h (the)g(in)o(v)o(erse)e(function)57 2369 y(theorem)d(in)g(Banac)o(h)f (spaces)h(but)h(needs)e(some)h(extra)h(assumption.)156 2441 y(The)f(map)f Fl(x)f Fk(7!)f Fl(f)5 b Fs(\()p Fl(x)p Fs(\))16 b(=)e(sin)7 b Fl(x)p Fs(,)16 b(where)g Fl(x)e Fk(2)g Fl(X)k Fs(=)13 b Fl(L)1142 2423 y Fi(2)1165 2441 y Fs(\([0)p Fl(;)8 b Fs(1]\),)16 b(is)f(of)h(class)f Fk(C)1583 2423 y Fi(1)1621 2441 y Fs(according)57 2511 y(to)g(De\014nition)g(9.6.)21 b(Its)16 b(deriv)m(ativ)o(e)f Fl(D)q(f)5 b Fs(\(0\))16 b(=iden)o(tit)o(y)f(but)g Fl(f)21 b Fs(is)15 b(not)g(in)o(v)o(ertible)f(:)22 b Fl(f)5 b Fs(\(0\))15 b(=)e(0)57 2581 y(and)19 b(the)i(functions)f Fl(x)494 2588 y Fh(n)521 2581 y Fs(\()p Fl(\030)r Fs(\))h(=)f Fl(\031)r(\037)724 2590 y Fi([0)p Fh(;)p Fi(1)p Fh(=n)p Fi(])846 2581 y Fs(\()p Fl(\030)r Fs(\),)i(where)e Fl(\037)1123 2590 y Fi([0)p Fh(;)p Fi(1)p Fh(=n)p Fi(])1265 2581 y Fs(denotes)g(the)h(c)o(haracteristic)57 2650 y(function)16 b(of)h(the)f(in)o(terv)m(al)g([0)p Fl(;)8 b Fs(1)p Fl(=n)p Fs(],)16 b(con)o(v)o(erge)g(to)h Fl(x)d Fs(=)g(0)i(but)g Fl(f)5 b Fs(\()p Fl(x)1327 2657 y Fh(n)1356 2650 y Fs(\))14 b(=)g(0)i(for)h(all)f Fl(n)p Fs(.)918 2770 y(59)p eop %%Page: 60 61 60 60 bop 156 192 a Fs(Another)12 b(example)g(is)f(obtained)g(taking)i Fl(X)18 b Fs(=)13 b Fk(C)1073 173 y Fj(1)1115 192 y Fs(\([)p Fk(\000)p Fs(1)p Fl(;)8 b Fs(1]\))13 b(and)e(considering)f(the)i(map)57 261 y Fl(f)30 b Fs(:)17 b Fl(X)j Fk(!)d Fl(X)22 b Fs(de\014ned)17 b(as)h Fl(f)5 b Fs(\()p Fl(x)p Fs(\)\()p Fl(\030)r Fs(\))19 b(=)d Fl(x)p Fs(\()p Fl(\030)r Fs(\))e Fk(\000)e Fl(\030)r(x)p Fs(\()p Fl(\030)r Fs(\))p Fl(x)1097 243 y Fj(0)1113 261 y Fs(\()p Fl(\030)r Fs(\))19 b(for)f(all)f Fl(\030)i Fk(2)e Fs([)p Fk(\000)p Fs(1)p Fl(;)8 b Fs(1].)26 b(Then)17 b(it)i(is)57 331 y(immediate)12 b(to)h(c)o(hec)o(k)f(that)h Fl(f)19 b Fs(is)13 b(smo)q(oth)f(and)g Fl(D)q(f)5 b Fs(\()p Fl(x)p Fs(\))t Fk(\001)t Fl(h)15 b Fs(=)f Fl(h)t Fk(\000)t Fl(\030)r(x)1327 313 y Fj(0)1340 331 y Fl(h)t Fk(\000)t Fl(\030)r(xh)1497 313 y Fj(0)1510 331 y Fs(,)g(th)o(us)e Fl(f)5 b Fs(\(0\))15 b(=)e(0)57 401 y(and)f Fl(D)q(f)5 b Fs(\(0\))16 b(=iden)o(tit)o(y)l(.)k(But)14 b Fl(f)19 b Fs(is)13 b(not)g(in)o(v)o(ertible)f(:)21 b(the)13 b(sequence)h Fl(x)1374 408 y Fh(n)1401 401 y Fs(\()p Fl(\030)r Fs(\))h(=)1539 381 y Fi(1)p 1537 389 25 2 v 1537 418 a Fh(n)1572 401 y Fs(+)1621 379 y Fh(\030)1640 364 y Fb(n)p 1621 389 44 2 v 1625 418 a Fh(n)p Fi(!)1685 401 y Fk(!)e Fs(0)h(in)57 470 y Fk(C)86 452 y Fj(1)128 470 y Fs(\([)p Fk(\000)p Fs(1)p Fl(;)8 b Fs(1]\))j(as)g Fl(n)j Fk(!)f Fs(+)p Fk(1)e Fs(but)g(one)g(can)g(c)o(hec)o(k)f(that)i(it)f(do)q(es)g(not)g(b)q (elong)g(to)g Fl(f)5 b Fs(\()p Fk(C)1588 452 y Fj(1)1632 470 y Fs(\([)p Fk(\000)p Fs(1)p Fl(;)j Fs(1]\)\))57 540 y(for)17 b(all)g Fl(n)f Fk(\025)f Fs(1.)26 b([Hin)o(t)17 b(:)24 b(use)18 b(the)f(fact)i(that)f(if)g Fl(x)e Fk(2)g(C)1087 522 y Fj(1)1129 540 y Fs(\([)p Fk(\000)p Fs(1)p Fl(;)8 b Fs(1]\))18 b(one)f(can)h(tak)o(e)g(its)f(T)l(a)o(ylor)57 610 y(series)e(at)i(0)g(at)f(an)o(y)h(\014nite)f(order)f(and)h(apply)g Fl(f)5 b Fs(.)23 b(])156 680 y(An)18 b(ev)o(en)f(more)g(in)o(teresting) f(coun)o(terexample)g(\(see)i([Ha])f(for)g(details\))h(is)f(the)g (follo)o(w-)57 749 y(ing)j(:)31 b(let)21 b Fl(M)27 b Fs(b)q(e)21 b(a)g(compact)g(manifold,)f Fl(X)26 b Fs(=)20 b Fk(C)1042 731 y Fj(1)1085 749 y Fs(\()p Fl(M)s(;)8 b(T)f(M)e Fs(\))22 b(b)q(e)g(the)f(F)l(r)o(\023)-24 b(ec)o(het)20 b(space)g(of)57 819 y(smo)q(oth)g(v)o(ector)h(\014elds)g(on)f Fl(M)5 b Fs(,)23 b(Di\013)768 797 y Fj(1)811 819 y Fs(\()p Fl(M)5 b Fs(\))22 b(b)q(e)g(the)f(group)f(of)h(smo)q(oth)g (di\013eomorphisms)57 889 y(of)d Fl(M)25 b Fs(\(it)19 b(is)e(a)i(F)l(r)o(\023)-24 b(ec)o(het)17 b(manifold,)h(it's)g(not)g(v) o(ery)g(hard)g(to)g(\014gure)g(out)g(what)h(this)f(means,)57 959 y(otherwise)e(lo)q(ok)g(in)h([Ha]\).)22 b(Then)16 b(the)h(usual)e(exp)q(onen)o(tial)h(map)596 1058 y(exp)23 b(:)13 b Fk(C)751 1037 y Fj(1)794 1058 y Fs(\()p Fl(M)s(;)8 b(T)f(M)e Fs(\))15 b Fk(!)f Fs(Di\013)1154 1036 y Fj(1)1196 1058 y Fs(\()p Fl(M)5 b Fs(\))969 1143 y Fl(v)16 b Fk(7!)e Fs(exp\()p Fl(v)r Fs(\))57 1242 y(clearly)j(v)o(eri\014es)g(exp\(0\))f (=)f(id)636 1249 y Fh(M)698 1242 y Fs(and)i Fl(D)10 b Fs(exp\(0\))16 b(=iden)o(tit)o(y)l(,)h(but)g(the)h(exp)q(onen)o(tial)f (map)g(is)57 1312 y(not)11 b(in)o(v)o(ertible)f(in)h(general.)19 b(Note)13 b(that)e(this)g(w)o(ould)f(ha)o(v)o(e)h(mean)o(t)g(that)h(an) o(y)e(di\013eomorphism)57 1381 y(extends)16 b(to)h(a)g(one)f(parameter) f(\015o)o(w.)156 1451 y(F)l(or)j(example)h(a)h(di\013eomorphism)c(of)j Fm(S)934 1433 y Fi(1)973 1451 y Fs(without)g(\014xed)g(p)q(oin)o(ts)g (is)f(the)i(exp)q(onen)o(tial)57 1521 y(of)26 b(a)h(v)o(ector)f (\014eld)g(only)g(if)h(it)f(is)g(conjugate)h(to)f(a)h(rotation.)51 b(But)27 b(there)f(exist)h([Y)l(o3])57 1591 y(di\013eomorphisms)19 b(of)j Fm(S)526 1572 y Fi(1)568 1591 y Fs(arbitrarily)f(close)h(to)h (the)f(iden)o(tit)o(y)g(whic)o(h)g(are)g(not)g(conjugate)57 1660 y(to)17 b(a)f(rotation.)156 1730 y(What)h(go)q(es)f(wrong)g(in)g (all)g(these)g(examples)g(is)g(that)g(although)g(the)g(deriv)m(ativ)o (e)h(of)f(the)57 1800 y(map)c(is)h(the)h(iden)o(tit)o(y)f(at)h(the)g (origin)e(it)i(fails)f(to)h(b)q(e)g(in)o(v)o(ertible)e(at)h(nearb)o(y)g (p)q(oin)o(ts.)20 b(Indeed)13 b(in)57 1869 y(the)g(second)f(example)g (ab)q(o)o(v)o(e)g(one)h(has)f Fl(D)q(f)5 b Fs(\(1)p Fl(=n)p Fs(\))s Fk(\001)s Fl(\030)1030 1851 y Fh(k)1071 1869 y Fs(=)1123 1829 y Fe(\000)1146 1869 y Fs(1)11 b Fk(\000)1239 1850 y Fh(k)p 1238 1858 25 2 v 1238 1887 a(n)1268 1829 y Fe(\001)1299 1869 y Fl(\030)1323 1851 y Fh(k)1348 1869 y Fs(,)i(th)o(us)f Fl(D)q(f)5 b Fs(\(1)p Fl(=n)p Fs(\))p Fl(\030)1694 1851 y Fh(n)1737 1869 y Fs(=)14 b(0.)156 1939 y(Th)o(us)19 b Fp(one)j(has)g(to)f(r)m(e)m(quir)m(e)h(the)f (invertibility)g(of)h Fl(D)q(f)28 b Fp(on)21 b(a)h(neighb)m(orho)m(o)n (d)i(explicitly)57 2009 y Fs(and)e(this)g(is)g(usually)f(di\016cult)h (to)h(b)q(e)g(c)o(hec)o(k)o(ed.)39 b(In)22 b(a)h(Banac)o(h)f(space)g (\(with)h(the)g(usual)57 2079 y(de\014nition)15 b(of)i(deriv)m(ativ)o (e)f(of)h(a)g(map)e(instead)h(of)h(De\014nition)f(9.6\))g(this)g(is)g (not)h(needed.)57 2254 y Fo(9.2)i(T)-5 b(ame)19 b(Maps)h(and)g(T)-5 b(ame)19 b(Spaces)57 2384 y Fr(De\014nition)j(9.11)28 b Fd(A)19 b Fs(graded)e(F)l(r)o(\023)-24 b(ec)o(het)18 b(space)g Fl(X)23 b Fd(is)18 b(a)g(F)l(r)o(\023)-24 b(ec)o(het)18 b(space)g(with)g(a)h(collection)57 2454 y(of)d(seminorms)e Fs(\()p Fk(k)h(k)p Fs(\))463 2461 y Fh(n)p Fj(2)p Fg(N)559 2454 y Fd(whic)o(h)g(de\014ne)h(the)h(top)q(ology)g(and)e(are)i (increasing)d(in)j(strength)550 2563 y Fk(k)p Fl(x)p Fk(k)628 2570 y Fi(0)664 2563 y Fk(\024)d(k)p Fl(x)p Fk(k)795 2570 y Fi(1)831 2563 y Fk(\024)g(k)p Fl(x)p Fk(k)962 2570 y Fi(2)999 2563 y Fk(\024)f Fl(:)8 b(:)g(:)36 b Fk(8)p Fl(x)13 b Fk(2)h Fl(X)k(:)918 2770 y Fs(60)p eop %%Page: 61 62 61 61 bop 57 192 a Fr(De\014nition)17 b(9.12)28 b Fd(Let)14 b Fl(X)q(;)8 b(Y)26 b Fd(b)q(e)14 b(graded)e(F)l(r)o(\023)-24 b(ec)o(het)13 b(spaces,)h Fl(U)19 b Fk(\032)13 b Fl(X)19 b Fd(op)q(en,)14 b Fl(P)29 b Fs(:)13 b Fl(U)20 b Fk(!)13 b Fl(Y)25 b Fd(b)q(e)57 261 y(a)16 b(con)o(tin)o(uous)e(map.)21 b Fl(P)i Fd(is)16 b Fs(tame)g Fd(if)h(for)f(all)g Fl(x)925 268 y Fi(0)961 261 y Fk(2)e Fl(U)22 b Fd(there)16 b(exists)h(a)f(neigh) o(b)q(orho)q(o)q(d)e Fl(V)25 b Fk(\032)14 b Fl(U)57 331 y Fd(of)i Fl(x)141 338 y Fi(0)180 331 y Fd(and)f(a)i(non)e(negativ)o(e) h(in)o(teger)f Fl(r)j Fd(suc)o(h)d(that)i(for)f(all)g Fl(i)e Fk(2)g Fm(N)j Fd(there)f(exists)g Fl(C)1609 338 y Fh(i)1639 331 y Fl(>)d Fs(0)k Fd(suc)o(h)57 401 y(that)556 470 y Fk(k)p Fl(P)7 b Fs(\()p Fl(x)p Fs(\))p Fk(k)711 477 y Fh(i)742 470 y Fk(\024)14 b Fl(C)831 477 y Fh(i)847 470 y Fs(\(1)d(+)g Fk(k)p Fl(x)p Fk(k)1030 477 y Fh(i)p Fi(+)p Fh(r)1097 470 y Fs(\))28 b Fk(8)p Fl(x)13 b Fk(2)h Fl(V)25 b(:)397 b Fs(\(9)p Fl(:)p Fs(2\))57 565 y Fd(A)17 b Fk(C)140 547 y Fh(k)181 565 y Fd(tame)f(map)g(is)g(a)h Fk(C)537 547 y Fh(k)578 565 y Fd(map)f Fl(P)23 b Fd(suc)o(h)16 b(that)g Fl(D)1006 547 y Fh(j)1028 565 y Fl(P)24 b Fd(is)16 b(tame)g(for)g(all)h Fs(0)c Fk(\024)h Fl(j)i Fk(\024)e Fl(k)r Fd(.)57 695 y Fs(The)j(most)g(t)o(ypical)g(example)g(of)g(a)h (tame)f(op)q(erator)g(b)q(et)o(w)o(een)g(F)l(r)o(\023)-24 b(ec)o(het)16 b(spaces)h(is)g(giv)o(en)g(b)o(y)57 765 y(nonlinear)g(partial)i(di\013eren)o(tial)f(op)q(erators)g(on)h (compact)g(manifolds.)29 b(If)20 b Fl(P)34 b Fs(:)18 b Fk(C)1625 747 y Fj(1)1667 765 y Fs(\()p Fl(M)5 b Fs(\))21 b Fk(!)57 835 y(C)86 817 y Fj(1)128 835 y Fs(\()p Fl(M)5 b Fs(\))20 b(is)e(a)h(smo)q(oth)f(function)g(of)h Fl(x)f Fk(2)f(C)894 817 y Fj(1)937 835 y Fs(\()p Fl(M)5 b Fs(\))20 b(and)e(its)g(partial)g(deriv)m(ativ)o(es)g(of)h(degree)57 904 y(at)27 b(most)f Fl(r)i Fs(then)e(w)o(e)h(sa)o(y)f(that)h(the)f (degree)g(of)h Fl(P)33 b Fs(is)27 b Fl(r)h Fs(and)e(this)g(will)g(b)q (e)h(the)f(\\loss)57 974 y(of)e(di\013eren)o(tiabilit)o(y")f(in)i (\(9.2\).)46 b(The)25 b(pro)q(of)f(of)h(this)f(fact)h(is)f(giv)o(en)g (in)h([Ha])f(and)g(uses)57 1044 y(Hadamard's)c(inequalities)i(for)g (functions)g Fl(x)i Fk(2)g(C)1050 1026 y Fj(1)1092 1044 y Fs(\()p Fl(M)5 b Fs(\))24 b(:)34 b(for)22 b(all)g Fl(n)i Fk(2)g Fl(N)k Fs(and)22 b(for)g(all)57 1114 y(in)o(teger)15 b Fl(k)k Fs(suc)o(h)c(that)i(0)d Fk(\024)f Fl(k)j Fk(\024)d Fl(n)k Fs(there)f(exists)h Fl(C)1019 1121 y Fh(k)q(;n)1093 1114 y Fl(>)d Fs(0)i(suc)o(h)f(that)428 1221 y Fk(k)p Fl(x)p Fk(k)506 1228 y Fh(k)545 1221 y Fk(\024)e Fl(C)633 1228 y Fh(k)q(;n)694 1221 y Fk(k)p Fl(x)p Fk(k)772 1200 y Fh(k)q(=n)772 1233 y(n)842 1221 y Fk(k)p Fl(x)p Fk(k)920 1195 y Fi(1)p Fj(\000)p Fh(k)q(=n)920 1234 y Fi(0)1054 1221 y Fk(8)p Fl(x)g Fk(2)h(C)1200 1200 y Fj(1)1243 1221 y Fs(\()p Fl(M)5 b Fs(\))15 b(and)e Fl(:)269 b Fs(\(9)p Fl(:)p Fs(3\))57 1363 y Fr(Exercise)25 b(9.13)20 b Fs(Pro)o(v)o(e)h (that)h(the)f(comp)q(osition)f(of)i(t)o(w)o(o)f Fk(C)1234 1345 y Fh(k)1280 1363 y Fs(tame)g(maps)f(is)h(a)g Fk(C)1674 1345 y Fh(k)1720 1363 y Fs(tame)57 1433 y(map.)57 1563 y Fr(De\014nition)33 b(9.14)28 b Fd(A)g(graded)e(F)l(r)o(\023)-24 b(ec)o(het)27 b(space)h Fl(X)k Fd(is)27 b Fs(tame)h Fd(if)g(it)g (admits)f Fs(smo)q(othing)57 1633 y(op)q(erators)p Fd(,)16 b(i.e.)24 b(a)17 b(one{parameter)f(family)g Fl(S)s Fs(\()p Fl(t)p Fs(\))24 b(:)15 b Fl(X)k Fk(!)c Fl(X)t Fd(,)i Fl(t)e Fk(2)g Fs([1)p Fl(;)8 b Fs(+)p Fk(1)p Fs(\))p Fd(,)17 b(of)h(con)o(tin)o(uous)57 1703 y(linear)f(op)q(erators)h(suc)o (h)f(that)i(there)f(exists)h(a)f(non)g(negativ)o(e)h(in)o(teger)e Fl(r)k Fd(and)d(p)q(ositiv)o(e)g(real)57 1772 y(constan)o(ts)f Fs(\()p Fl(C)334 1779 y Fh(n;k)395 1772 y Fs(\))414 1779 y Fh(n;k)q Fj(2)p Fg(N)546 1772 y Fd(suc)o(h)g(that)i(for)f(all)g Fl(x)g Fk(2)f Fl(X)22 b Fd(and)c(for)g(all)g Fl(t)f Fk(2)h Fs([1)p Fl(;)8 b Fs(+)p Fk(1)p Fs(\))18 b Fd(and)g(for)g(all)57 1842 y Fl(k)d Fk(2)f(f)p Fs(0)p Fl(;)8 b Fs(1)p Fl(;)g(:)g(:)g(:)g(;)g (n)p Fk(g)17 b Fd(one)f(has)705 1942 y Fk(k)p Fl(S)s Fs(\()p Fl(t)p Fs(\))p Fl(x)p Fk(k)873 1949 y Fh(n)914 1942 y Fk(\024)e Fl(C)1003 1949 y Fh(n;k)1063 1942 y Fl(t)1081 1921 y Fh(n)p Fj(\000)p Fh(k)1162 1942 y Fk(k)p Fl(x)p Fk(k)1240 1949 y Fh(k)618 2027 y Fk(k)p Fl(x)e Fk(\000)e Fl(S)s Fs(\()p Fl(t)p Fs(\))p Fl(x)p Fk(k)875 2034 y Fh(k)914 2027 y Fk(\024)k Fl(C)1003 2034 y Fh(k)q(;n)1063 2027 y Fl(t)1081 2006 y Fh(k)q Fj(\000)p Fh(n)1162 2027 y Fk(k)p Fl(x)p Fk(k)1240 2034 y Fh(n)1726 1983 y Fs(\(9)p Fl(:)p Fs(4\))57 2159 y Fr(Exercise)28 b(9.15)e(\(con)n(v)n(olution)j (with)g(regularizing)f(k)n(ernels\))d Fs(Let)g Fl( )i Fk(2)f(C)1684 2141 y Fj(1)1681 2171 y Fi(0)1727 2159 y Fs(\()p Fm(R)1785 2141 y Fh(n)1809 2159 y Fs(\))57 2229 y(and)20 b(assume)g(that)h Fl( )j Fk(\025)d Fs(0,)h Fl( )i Fk(\021)d Fs(1)h(near)e(0.)36 b(Let)22 b Fl(')e Fs(b)q(e)i(the)f(F)l(ourier)e(transform)h(of)h Fl( )j Fs(:)57 2299 y Fl(')p Fs(\()p Fl(\030)r Fs(\))d(=)232 2258 y Fe(R)256 2316 y Fg(R)278 2306 y Fb(n)316 2299 y Fl( )r Fs(\()p Fl(\021)r Fs(\))p Fl(e)438 2281 y Fj(\000)p Fi(2)p Fh(\031)q(i\030)q(\021)572 2299 y Fl(d\021)r Fs(.)34 b(Let)21 b Fl(t)f Fk(\025)h Fs(1,)g Fl(')957 2306 y Fh(t)974 2299 y Fs(\()p Fl(\030)r Fs(\))h(=)e Fl(t)1135 2281 y Fh(n)1162 2299 y Fl(')p Fs(\()p Fl(t\030)r Fs(\).)35 b(De\014ne)20 b Fl(S)s Fs(\()p Fl(t)p Fs(\))p Fl(f)27 b Fs(=)20 b Fl(')1715 2306 y Fh(t)1746 2299 y Fl(?)14 b(f)5 b Fs(,)57 2368 y(where)19 b Fl(f)26 b Fk(2)21 b(C)337 2350 y Fj(1)379 2368 y Fs(\()p Fm(T)434 2350 y Fh(n)459 2368 y Fs(\).)33 b(Sho)o(w)19 b(that)i(\()p Fl(S)s Fs(\()p Fl(t)p Fs(\)\))900 2375 y Fh(t)p Fj(\025)p Fi(1)990 2368 y Fs(is)f(a)g(family)g(of)g(smo)q(othing)f(op)q(erators)h(on)57 2438 y Fk(C)86 2420 y Fj(1)128 2438 y Fs(\()p Fm(T)184 2420 y Fh(n)208 2438 y Fs(\).)57 2543 y Fr(Exercise)e(9.16)13 b Fs(Pro)o(v)o(e)h(that)h(in)g(a)g(tame)f(F)l(r)o(\023)-24 b(ec)o(het)14 b(space)h(Hadamard's)d(inequalities)j(hold)f(:)274 2650 y Fk(k)p Fl(x)p Fk(k)352 2657 y Fh(l)381 2650 y Fk(\024)g Fl(C)t Fs(\()p Fl(k)r(;)8 b(n)p Fs(\))p Fk(k)p Fl(x)p Fk(k)670 2629 y Fi(1)p Fj(\000)p Fh(\013)670 2665 y(k)749 2650 y Fk(k)p Fl(x)p Fk(k)827 2630 y Fh(\013)827 2663 y(n)883 2650 y Fk(8)p Fl(k)14 b Fk(\024)g Fl(l)g Fk(\024)g Fl(n)28 b(;)35 b(l)15 b Fs(=)f(\(1)d Fk(\000)g Fl(\013)p Fs(\))p Fl(k)i Fs(+)e Fl(\013n)i(:)918 2770 y Fs(61)p eop %%Page: 62 63 62 62 bop 57 194 a Fs([Hin)o(t)16 b(:)22 b(use)16 b(\(9.4\))h(with)f Fl(t)e Fs(=)g Fk(k)p Fl(x)p Fk(k)701 168 y Fi(1)p Fh(=)p Fi(\()p Fh(n)p Fj(\000)p Fh(k)q Fi(\))701 200 y Fh(n)853 194 y Fk(k)p Fl(x)p Fk(k)931 168 y Fj(\000)p Fi(1)p Fh(=)p Fi(\()p Fh(n)p Fj(\000)p Fh(k)q Fi(\))931 209 y Fh(k)1114 194 y Fs(.])57 391 y Fo(9.3)19 b(The)h(Nash{Moser)e(Theorem)57 519 y Fs(W)l(e)e(can)h(\014nally)f(state)h(Nash{Moser's)d([N,M])i (implicit)g(and)g(in)o(v)o(erse)f(function)h(theorems.)57 743 y Fr(Theorem)g(9.17)i(\(implicit)j(function\))46 b Fd(Let)16 b Fl(X)q(;)8 b(Y)s(;)g(Z)20 b Fd(b)q(e)c(three)g(tame)g(F)l (r)o(\023)-24 b(ec)o(het)15 b(spaces,)57 813 y Fl(U)25 b Fk(\032)20 b Fl(X)e Fk(\002)13 b Fl(Y)32 b Fd(op)q(en,)21 b Fs(\010)28 b(:)20 b Fl(U)26 b Fk(!)20 b Fl(Z)k Fd(a)c(tame)g Fk(C)974 795 y Fh(r)1017 813 y Fd(map,)g Fs(2)g Fk(\024)g Fl(r)i Fk(\024)e(1)p Fd(.)33 b(Let)21 b Fs(\()p Fl(x)1590 820 y Fi(0)1613 813 y Fl(;)8 b(y)1659 820 y Fi(0)1682 813 y Fs(\))21 b Fk(2)f Fl(U)5 b Fd(.)57 883 y(Assume)15 b(that)i(there)g(exists)f(a)h(neigh)o(b)q(orho)q(o)q(d)e Fl(V)998 890 y Fi(0)1037 883 y Fd(of)h Fs(\()p Fl(x)1140 890 y Fi(0)1164 883 y Fl(;)8 b(y)1210 890 y Fi(0)1232 883 y Fs(\))17 b Fd(and)f(a)h(con)o(tin)o(uous)d Fl(z)r Fd({linear)57 952 y(tame)j(map)g Fl(L)24 b Fs(:)16 b Fl(V)411 959 y Fi(0)445 952 y Fk(\002)11 b Fl(Z)20 b Fk(!)15 b Fl(Y)c Fd(,)18 b Fs(\(\()p Fl(x;)8 b(y)r Fs(\))p Fl(;)g(z)r Fs(\))18 b Fk(7!)e Fl(L)p Fs(\()p Fl(x;)8 b(y)r Fs(\))13 b Fk(\001)f Fl(z)r Fd(,)18 b(suc)o(h)e(that)i(if)g Fs(\()p Fl(x;)8 b(y)r Fs(\))17 b Fk(2)f Fl(V)1691 959 y Fi(0)1731 952 y Fd(then)57 1022 y Fl(D)98 1029 y Fh(y)122 1022 y Fs(\010\()p Fl(x;)8 b(y)r Fs(\))21 b Fd(is)e(in)o(v)o(ertible)f (with)h(in)o(v)o(erse)f Fl(L)p Fs(\()p Fl(x;)8 b(y)r Fs(\))p Fd(.)32 b(Then)19 b Fl(x)1208 1029 y Fi(0)1250 1022 y Fd(has)g(a)h(neigh)o(b)q(orho)q(o)q(d)d Fl(W)27 b Fd(on)57 1092 y(whic)o(h)17 b Fs(\011)g Fk(2)g(C)336 1074 y Fh(r)358 1092 y Fs(\()p Fl(W)o(;)8 b(Y)j Fs(\))19 b Fd(is)f(de\014ned)g(and)f(suc)o(h)h(that)g Fs(\011\()p Fl(x)1157 1099 y Fi(0)1180 1092 y Fs(\))g(=)e Fl(y)1296 1099 y Fi(0)1338 1092 y Fd(and)h(for)h(all)h Fl(x)e Fk(2)g Fl(W)26 b Fd(one)57 1162 y(has)16 b Fs(\()p Fl(x;)8 b Fs(\011\()p Fl(x)p Fs(\)\))16 b Fk(2)e Fl(U)22 b Fd(and)16 b Fs(\010\()p Fl(x;)8 b Fs(\011\()p Fl(x)p Fs(\)\))16 b(=)d(\010\()p Fl(x)935 1169 y Fi(0)958 1162 y Fl(;)8 b(y)1004 1169 y Fi(0)1027 1162 y Fs(\))p Fd(.)57 1461 y Fr(Theorem)28 b(9.18)g(\(in)n(v)n(erse)k(function\))55 b Fd(Let)26 b Fl(X)q(;)8 b(Y)37 b Fd(b)q(e)26 b(t)o(w)o(o)g(tame)g(F)l (r)o(\023)-24 b(ec)o(het)24 b(spaces,)57 1530 y Fl(U)e Fk(\032)16 b Fl(X)23 b Fd(op)q(en,)18 b Fs(\010)25 b(:)16 b Fl(U)23 b Fk(!)16 b Fl(Y)29 b Fd(a)19 b(tame)f Fk(C)839 1512 y Fh(r)879 1530 y Fd(map,)g Fs(2)f Fk(\024)f Fl(r)i Fk(\024)f(1)p Fd(.)27 b(Let)19 b Fl(x)1409 1537 y Fi(0)1448 1530 y Fk(2)e Fl(U)5 b Fd(,)19 b Fl(y)1594 1537 y Fi(0)1634 1530 y Fs(=)d(\010\()p Fl(x)1772 1537 y Fi(0)1795 1530 y Fs(\))p Fd(.)57 1600 y(Assume)e(that)h(there)h(exists)f(a)g(neigh)o (b)q(orho)q(o)q(d)e Fl(V)989 1607 y Fi(0)1027 1600 y Fd(of)i Fl(x)1110 1607 y Fi(0)1149 1600 y Fd(and)f(a)h(con)o(tin)o (uous)e Fl(y)r Fd({linear)h(tame)57 1670 y(map)e Fl(L)22 b Fs(:)13 b Fl(V)276 1677 y Fi(0)302 1670 y Fk(\002)s Fl(Y)25 b Fk(!)14 b Fl(X)t Fd(,)f Fs(\()p Fl(x;)8 b(y)r Fs(\))16 b Fk(7!)d Fl(L)p Fs(\()p Fl(x)p Fs(\))s Fk(\001)s Fl(y)r Fd(,)i(suc)o(h)d(that)h(if)g Fl(x)h Fk(2)g Fl(V)1275 1677 y Fi(0)1310 1670 y Fd(then)f Fl(D)q Fs(\010\()p Fl(x)p Fs(\))i Fd(is)d(in)o(v)o(ertible)57 1740 y(with)j(in)o(v)o(erse) e Fl(L)p Fs(\()p Fl(x)p Fs(\))p Fd(.)23 b(Then)14 b Fl(x)625 1747 y Fi(0)663 1740 y Fd(has)g(a)h(neigh)o(b)q(orho)q(o)q(d)e Fl(V)25 b Fk(\032)14 b Fl(V)1237 1747 y Fi(0)1274 1740 y Fd(and)g Fl(y)1393 1747 y Fi(0)1431 1740 y Fd(has)g(a)h(neigb)q(orho) q(o)q(d)57 1809 y Fl(W)23 b Fd(suc)o(h)16 b(that)h Fs(\010)22 b(:)13 b Fl(V)25 b Fk(!)14 b Fl(W)24 b Fd(is)16 b(a)g(tame)h Fk(C)867 1791 y Fh(r)905 1809 y Fd(di\013eomorphism.)57 2012 y Fr(Exercise)j(9.19)15 b Fs(Sho)o(w)h(that)h(the)f(t)o(w)o(o)g (previous)g(theorems)f(are)h(equiv)m(alen)o(t.)57 2139 y(W)l(e)i(refer)g(the)h(reader)e(to)h([Ha])h(for)f(the)g(pro)q(ofs)g (of)g(Theorems)f(9.17)h(and)g(9.18.)26 b(The)18 b(main)57 2209 y(idea)d(of)i(the)f(pro)q(of)g(is)f(to)i(use)e(a)h(mo)q(di\014ed)f (Newton's)h(metho)q(d)g(for)g(\014nding)e(the)j(ro)q(ot)f(of)g(the)57 2279 y(equation)k(\010\()p Fl(x)p Fs(\))h(=)f Fl(y)r Fs(.)34 b(It)20 b(mak)o(es)g(use)g(of)g(the)h(smo)q(othing)e(op)q (erators)g Fl(S)s Fs(\()p Fl(t)p Fs(\))i(to)g(guaran)o(tee)57 2348 y(con)o(v)o(ergence.)f(Here)15 b(w)o(e)f(will)g(con)o(ten)o(t)g (ourselv)o(es)f(with)h(a)h(brief)f(sk)o(etc)o(h)o(y)g(description)e(of) j(the)57 2418 y(argumen)o(t.)156 2510 y(Without)22 b(loss)f(of)g (generalit)o(y)g(w)o(e)h(can)f(assume)f Fl(x)1138 2517 y Fi(0)1183 2510 y Fs(=)i Fl(y)1268 2517 y Fi(0)1313 2510 y Fs(=)g(0.)37 b(An)22 b(algorithm)e(for)57 2580 y(constructing)11 b(a)h(sequence)g Fl(x)605 2587 y Fh(j)640 2580 y Fk(2)i Fl(X)i Fs(whic)o(h)11 b(will)h(con)o(v)o(erge)f(to)h(a)h (solution)e Fl(x)h Fs(of)g(\010\()p Fl(x)p Fs(\))j(=)f Fl(y)g Fs(\(for)57 2650 y(small)i(enough)g Fl(y)r Fs(\))i(is)f(the)h (follo)o(wing)e(:)23 b(\014x)17 b(a)h(sequence)f Fl(t)1148 2657 y Fh(j)1184 2650 y Fs(=)e Fl(e)1261 2632 y Fi(\(3)p Fh(=)p Fi(2\))1353 2617 y Fb(j)1373 2650 y Fs(,)j(so)f(that)h Fl(t)1594 2657 y Fh(j)r Fi(+1)1680 2650 y Fs(=)d Fl(t)1752 2624 y Fi(3)p Fh(=)p Fi(2)1752 2664 y Fh(j)1815 2650 y Fs(,)918 2770 y(62)p eop %%Page: 63 64 63 63 bop 57 192 a Fs(and)16 b(let)534 239 y Fl(x)562 246 y Fi(0)598 239 y Fs(=)e(0)69 b(\(initial)16 b(guess)8 b(\))14 b Fl(;)535 324 y(x)563 331 y Fh(j)598 324 y Fs(=)g Fl(:)8 b(:)g(:)78 b Fs(\()p Fl(j)s Fs({th)16 b(guess)7 b(\))14 b Fl(;)540 408 y(z)563 415 y Fh(j)598 408 y Fs(=)g Fl(y)f Fk(\000)e Fs(\010\()p Fl(x)821 415 y Fh(j)843 408 y Fs(\))69 b(\()p Fl(j)s Fs({th)17 b(error)7 b(\))14 b Fl(;)494 493 y Fs(\001)p Fl(x)564 500 y Fh(j)598 493 y Fs(=)g Fl(S)s Fs(\()p Fl(t)722 500 y Fh(j)743 493 y Fs(\))p Fl(L)p Fs(\()p Fl(x)843 500 y Fh(j)865 493 y Fs(\))p Fl(z)907 500 y Fh(j)998 493 y Fs(\()p Fl(j)s Fs({th)i(correction)7 b(\))15 b Fl(;)485 578 y(x)513 585 y Fh(j)r Fi(+1)598 578 y Fs(=)f Fl(x)679 585 y Fh(j)712 578 y Fs(+)c(\001)p Fl(x)831 585 y Fh(j)921 578 y Fs(\()p Fl(j)k Fs(+)d(1{th)17 b(guess)7 b(\))14 b Fl(:)57 669 y Fs(The)i(idea)g(to)h(sho)o(w)f(con)o(v)o(ergence)f(of)i(this)f (algorithm)f(is)h(the)h(follo)o(wing)e(:)22 b(let)572 822 y Fl(R)p Fs(\()p Fl(x;)8 b(h)p Fs(\))15 b(=)795 754 y Fe(Z)844 766 y Fi(1)822 867 y(0)875 822 y Fl(D)917 801 y Fi(2)940 822 y Fs(\010\()p Fl(x)d Fs(+)f Fl(th)p Fs(\)\()p Fl(h;)d(h)p Fs(\))p Fl(dt)57 967 y Fs(denote)16 b(the)h(quadratic)f(in)o(tegral)f(remainder)g(in)h(T)l(a)o(ylor's)f (form)o(ula.)20 b(Since)517 1096 y(\010\()p Fl(x)12 b Fs(+)f Fl(h)p Fs(\))j(=)f(\010\()p Fl(x)p Fs(\))g(+)d Fl(D)q Fs(\010\()p Fl(x)p Fs(\))j Fk(\001)e Fl(h)g Fs(+)g Fl(R)p Fs(\()p Fl(x;)d(h)p Fs(\))57 1226 y(one)16 b(gets)423 1283 y Fl(z)446 1290 y Fh(j)r Fi(+1)532 1283 y Fs(=)d Fl(y)h Fk(\000)c Fs(\010\()p Fl(x)754 1290 y Fh(j)r Fi(+1)827 1283 y Fs(\))k(=)f Fl(y)h Fk(\000)d Fs(\010\()p Fl(x)1083 1290 y Fh(j)1115 1283 y Fs(+)g(\001)p Fl(x)1235 1290 y Fh(j)1256 1283 y Fs(\))532 1367 y(=)i Fl(z)607 1374 y Fh(j)640 1367 y Fk(\000)d Fl(D)q Fs(\010\()p Fl(x)814 1374 y Fh(j)837 1367 y Fs(\))p Fl(S)s Fs(\()p Fl(t)927 1374 y Fh(j)948 1367 y Fs(\))p Fl(L)p Fs(\()p Fl(x)1048 1374 y Fh(j)1070 1367 y Fs(\))p Fl(z)1112 1374 y Fh(j)1145 1367 y Fk(\000)g Fl(R)p Fs(\()p Fl(x)1279 1374 y Fh(j)1302 1367 y Fl(;)e Fs(\001)p Fl(x)1394 1374 y Fh(j)1415 1367 y Fs(\))14 b Fl(:)57 1462 y Fs(Using)i(the)h(iden)o(tit)o(y)f Fl(z)490 1469 y Fh(j)525 1462 y Fs(=)d Fl(D)q Fs(\010\()p Fl(x)702 1469 y Fh(j)725 1462 y Fs(\))p Fl(L)p Fs(\()p Fl(x)825 1469 y Fh(j)847 1462 y Fs(\))p Fl(z)889 1469 y Fh(j)927 1462 y Fs(w)o(e)j(\014nd)419 1591 y Fl(z)442 1598 y Fh(j)r Fi(+1)527 1591 y Fs(=)e Fl(D)q Fs(\010\()p Fl(x)705 1598 y Fh(j)727 1591 y Fs(\)[)p Fl(I)h Fk(\000)c Fl(S)s Fs(\()p Fl(t)918 1598 y Fh(j)939 1591 y Fs(\)])p Fl(L)p Fs(\()p Fl(x)1053 1598 y Fh(j)1075 1591 y Fs(\))p Fl(z)1117 1598 y Fh(j)1149 1591 y Fs(+)g Fl(R)p Fs(\()p Fl(x)1284 1598 y Fh(j)1306 1591 y Fl(;)d Fs(\001)p Fl(x)1398 1598 y Fh(j)1419 1591 y Fs(\))15 b Fl(:)57 1721 y Fs(The)20 b(\014rst)g(term)h(tends)f(to)h(zero)f(v)o(ery)h(rapidly)e(since)i Fl(S)s Fs(\()p Fl(t)1191 1728 y Fh(j)1211 1721 y Fs(\))h Fk(!)e Fl(I)25 b Fs(as)20 b Fl(j)j Fk(!)e Fs(+)p Fk(1)f Fs(and)g(the)57 1791 y(second)15 b(term)i(is)f(quadratic.)156 1860 y(This)e(short)g(description)e(of)j(the)g(idea)f(of)h(the)f(pro)q (of)g(mak)o(es)g(also)g(clear)g(wh)o(y)g(one)g(needs)57 1930 y(the)i(assumption)f(\010)i(at)f(least)h(of)g(class)e Fk(C)838 1912 y Fi(2)878 1930 y Fs(\(in)h(Banac)o(h)g(spaces)g Fk(C)1315 1912 y Fi(1)1354 1930 y Fs(is)g(enough\).)918 2770 y(63)p eop %%Page: 64 65 64 64 bop 57 192 a Fq(10.)70 b(F)-6 b(rom)36 b(Nash{Moser's)h(Theorem)f (to)h(KAM)g(:)g(Normal)57 261 y(F)-6 b(orm)22 b(of)i(V)-6 b(ector)24 b(Fields)i(on)e(the)g(T)-6 b(orus)57 366 y Fs(F)l(ollo)o(wing)12 b(Herman)i(w)o(e)h(will)f(pro)o(v)o(e)g(in)g (this)g(Chapter)g(a)h(normal)e(form)h(theorem)g(for)g(v)o(ector)57 436 y(\014elds)j(on)g(the)h(torus)f(whic)o(h)g(can)g(b)q(e)i (considered)d(as)h(the)h(basic)f(KAM)h(theorem)f(in)h(higher)57 506 y(dimension)c(\(without)j(taking)f(the)h(symplectic)f(structure)g (in)o(to)g(accoun)o(t\).)21 b(The)c(pro)q(of)f(will)57 576 y(b)q(e)k(an)f(application)g(of)h(Nash{Moser's)e(Theorem.)31 b(F)l(or)18 b(a)i(pro)q(of)g(of)g(KAM)f(theorem)h(see,)57 645 y(for)c(example,)g([Bo].)156 715 y(Let)21 b(Di\013)330 693 y Fj(1)372 715 y Fs(\()p Fm(T)428 697 y Fh(l)440 715 y Fl(;)8 b Fs(0\))20 b(denote)g(the)h(group)d(of)j Fk(C)1014 697 y Fj(1)1076 715 y Fs(di\013eomorphisms)c Fl(f)25 b Fs(of)20 b(the)h(torus)e Fm(T)1816 697 y Fh(l)57 785 y Fs(homotopic)c(to)h(the)g(iden)o(tit)o(y)f(and)h(suc)o(h)e(that)j Fl(f)5 b Fs(\(0\))15 b(=)e(0.)22 b(This)15 b(space)h(can)f(b)q(e)h (iden)o(ti\014ed)f(to)57 855 y(an)e(op)q(en)h(subset)f(of)g(the)h(tame) g(F)l(r)o(\023)-24 b(ec)o(het)13 b(space)g Fk(C)982 837 y Fj(1)1024 855 y Fs(\()p Fm(T)1080 837 y Fh(l)1092 855 y Fl(;)8 b Fm(R)1153 837 y Fh(l)1166 855 y Fl(;)g Fs(0\))14 b(=)g Fk(f)p Fl(u)f Fk(2)h(C)1442 837 y Fj(1)1484 855 y Fs(\()p Fm(T)1540 837 y Fh(l)1552 855 y Fl(;)8 b Fm(R)1613 837 y Fh(l)1625 855 y Fs(\))g Fl(;)18 b(u)p Fs(\(0\))c(=)57 924 y(0)p Fk(g)20 b Fs(:)31 b Fl(u)20 b Fs(corresp)q(onds)f(to)i(a)g (di\013eomorphism)d Fl(f)27 b Fs(if)21 b(and)f(only)h(if)g(for)f(all)h Fl(\037)g Fk(2)g Fm(T)1627 906 y Fh(l)1660 924 y Fs(one)g(has)57 994 y(id)10 b(+)h Fl(@)s(u)p Fs(\()p Fl(\037)p Fs(\))k Fk(2)f Fs(GL)8 b(\()p Fl(l)q(;)g Fm(R)p Fs(\).)20 b(In)c(this)g(case)h (one)f(has)g Fl(f)j Fs(=)14 b(id)1157 1004 y Fg(T)1178 994 y Fb(l)19 b Fs(+)11 b Fl(u)p Fs(.)156 1064 y(Since)18 b(the)h(tangen)o(t)f(bundle)f(of)i Fm(T)819 1046 y Fh(l)850 1064 y Fs(is)f(canonically)f(isomorphic)f(to)j Fm(T)1515 1046 y Fh(l)1540 1064 y Fk(\002)12 b Fm(R)1630 1046 y Fh(l)1660 1064 y Fs(one)19 b(can)57 1134 y(also)d(iden)o(tify)g(the)h (space)f(of)g Fk(C)642 1116 y Fj(1)701 1134 y Fs(v)o(ector)h(\014elds)f (on)g(the)h(torus)e(with)i Fk(C)1404 1116 y Fj(1)1446 1134 y Fs(\()p Fm(T)1502 1116 y Fh(l)1514 1134 y Fl(;)8 b Fm(R)1575 1116 y Fh(l)1587 1134 y Fs(\).)156 1203 y(Let)18 b Fl(\026)e Fk(2)f Fm(R)379 1185 y Fh(l)409 1203 y Fs(b)q(e)j(Diophan)o (tine)e(with)h(exp)q(onen)o(t)g Fl(\034)23 b Fs(and)17 b(constan)o(t)g Fl(\015)s Fs(.)24 b(W)l(e)17 b(will)g(denote)57 1273 y Fl(R)95 1280 y Fh(\026)137 1273 y Fs(the)f(translation)f(b)o(y)h Fl(\026)g Fs(on)f(the)i(torus)e Fm(T)906 1255 y Fh(l)934 1273 y Fs(:)22 b Fl(R)1008 1280 y Fh(\026)1034 1273 y Fs(\()p Fl(\037)1084 1280 y Fi(1)1107 1273 y Fl(;)8 b(:)g(:)g(:)h(;)f (\037)1249 1280 y Fh(l)1265 1273 y Fs(\))14 b(=)f(\()p Fl(\037)1400 1280 y Fi(1)1433 1273 y Fs(+)d Fl(\026)1512 1280 y Fi(1)1534 1273 y Fl(;)e(:)g(:)g(:)h(;)f(\037)1676 1280 y Fh(l)1701 1273 y Fs(+)i Fl(\026)1780 1280 y Fh(l)1795 1273 y Fs(\).)57 1378 y Fr(Exercise)20 b(10.1)15 b Fs(Sho)o(w)h(that)h (the)f(map)609 1483 y Fl(I)25 b Fs(:)14 b(Di\013)765 1460 y Fj(1)807 1483 y Fs(\()p Fm(T)863 1462 y Fh(l)875 1483 y Fl(;)8 b Fs(0\))14 b Fk(!)g Fs(Di\013)1100 1460 y Fj(1)1142 1483 y Fs(\()p Fm(T)1198 1462 y Fh(l)1210 1483 y Fl(;)8 b Fs(0\))912 1567 y Fl(f)19 b Fk(7!)14 b Fl(I)t Fs(\()p Fl(f)5 b Fs(\))15 b(=)f Fl(f)1209 1547 y Fj(\000)p Fi(1)57 1669 y Fs(is)i(a)g(tame)h Fk(C)302 1651 y Fj(1)361 1669 y Fs(map.)k(Its)16 b(deriv)m(ativ)o(e)h(is)588 1784 y Fl(D)q(I)t Fs(\()p Fl(f)5 b Fs(\))13 b Fk(\001)e Fl(h)j Fs(=)f Fk(\000)p Fs([\()p Fl(@)s(f)5 b Fs(\))1005 1763 y Fj(\000)p Fi(1)1071 1784 y Fk(\001)11 b Fl(h)p Fs(])g Fk(\016)g Fl(f)1215 1763 y Fj(\000)p Fi(1)1283 1784 y Fl(:)404 b Fs(\(10)p Fl(:)p Fs(1\))57 1934 y(The)22 b(follo)o(wing)g(statemen)o(ts)g(\(and)g(pro)q(of)t(\))h(are)f(tak)o (en)h(from)f(\([Bo],)j(pp.)40 b(139{141\))22 b(and)57 2003 y([He5].)57 2109 y Fr(Theorem)17 b(10.2)p Fp(L)m(et)g Fl(\026)d Fk(2)g Fm(R)618 2090 y Fh(l)630 2109 y Fp(.)23 b(The)18 b(map)457 2213 y Fs(\010)493 2220 y Fh(\026)542 2213 y Fs(:)c Fp(Di\013)653 2191 y Fj(1)695 2213 y Fs(\()p Fm(T)751 2192 y Fh(l)763 2213 y Fl(;)8 b Fs(0\))k Fk(\002)e Fm(R)929 2192 y Fh(l)955 2213 y Fk(!)k Fp(Di\013)1103 2191 y Fj(1)1145 2213 y Fs(\()p Fm(T)1200 2192 y Fh(l)1213 2213 y Fs(\))g Fl(;)826 2298 y Fs(\()p Fl(f)s(;)8 b(\027)s Fs(\))14 b Fk(7!)g Fl(R)1057 2305 y Fh(\027)1093 2298 y Fk(\016)d Fl(f)17 b Fk(\016)11 b Fl(R)1244 2305 y Fh(\026)1281 2298 y Fk(\016)g Fl(f)1346 2277 y Fj(\000)p Fi(1)1414 2298 y Fl(;)1701 2254 y Fs(\(10)p Fl(:)p Fs(2\))57 2405 y Fp(is)17 b(a)g(tame)f Fk(C)302 2387 y Fj(1)361 2405 y Fp(map.)23 b(Mor)m(e)m(over,)18 b(if)f Fl(\026)g Fp(is)g(a)g (diophantine)1166 2387 y Fi(1)1207 2405 y Fp(ve)m(ctor)f(then)h Fs(\010)1498 2412 y Fh(\026)1541 2405 y Fp(is)g(a)g(tame)f Fk(C)1786 2387 y Fj(1)57 2475 y Fp(lo)m(c)m(al)j(di\013e)m(omorphism)h (ne)m(ar)f Fl(f)g Fs(=)14 b Fp(id)768 2485 y Fg(T)788 2475 y Fb(l)8 b Fp(,)18 b Fl(\027)f Fs(=)c(0)p Fp(.)p 57 2525 600 2 v 109 2563 a Fi(1)156 2581 y Fs(In)j(this)f(situation)g Fl(\026)g Fs(is)g(diophan)o(tine)f(if)h(there)h(exist)g(t)o(w)o(o)f (constan)o(ts)f Fl(\015)j(>)c Fs(0)j(and)f Fl(\034)k Fk(\025)14 b Fl(l)57 2650 y Fs(suc)o(h)h(that)i Fk(j)p Fl(\026)11 b Fk(\001)g Fl(k)h Fs(+)f Fl(p)p Fk(j)j(\025)g Fl(\015)s Fk(j)p Fl(k)r Fk(j)636 2632 y Fj(\000)p Fh(\034)707 2650 y Fs(for)i(all)g Fl(k)g Fk(2)e Fm(Z)977 2632 y Fh(l)1000 2650 y Fk(n)d(f)p Fs(0)p Fk(g)17 b Fs(and)e(for)i(all)f Fl(p)e Fk(2)g Fm(Z)-11 b Fs(.)918 2770 y(64)p eop %%Page: 65 66 65 65 bop 57 192 a Fs(The)13 b(meaning)f(of)i(the)g(second)f(part)g(is) g(that)h(when)f Fl(\026)h Fs(is)f(diophan)o(tine)f(the)i (di\013eomorphisms)57 261 y(of)25 b(the)g(torus)e Fm(T)388 243 y Fh(l)425 261 y Fs(conjugate)h(to)h(the)g(translation)f Fl(R)1120 268 y Fh(\026)1171 261 y Fs(b)o(y)g(a)h(di\013eomorphism)d (close)i(to)57 331 y(the)d(iden)o(tit)o(y)f(form)g(a)h(F)l(r)o(\023)-24 b(ec)o(het)20 b(submanifold)f(of)i(co)q(dimension)e Fl(l)j Fs(of)f(Di\013)1498 309 y Fj(1)1540 331 y Fs(\()p Fm(T)1595 313 y Fh(l)1608 331 y Fs(\))g(whic)o(h)f(is)57 401 y(transv)o(erse)15 b(in)h Fl(\026)g Fs(to)h(the)g(space)f Fm(R)716 383 y Fh(l)745 401 y Fs(of)h(the)g(translations)e(on)h(the)h(torus.)57 509 y Fr(Exercise)26 b(10.3)21 b Fs(Guess)h(the)g(statemen)o(t)g(of)g (for)g(v)o(ector)h(\014elds)e(equiv)m(alen)o(t)h(to)g(Theorem)57 579 y(10.2.)57 688 y(Here)16 b(is)h(the)f(solution)g(:)57 796 y Fr(Theorem)h(10.3)p Fp(L)m(et)g Fl(\026)d Fk(2)g Fm(R)618 778 y Fh(l)630 796 y Fp(.)23 b(The)18 b(map)358 925 y Fs(\011)397 932 y Fh(\026)446 925 y Fs(:)13 b Fp(Di\013)557 903 y Fj(1)599 925 y Fs(\()p Fm(T)655 905 y Fh(l)667 925 y Fl(;)8 b Fs(0\))j Fk(\002)g Fm(R)833 905 y Fh(l)859 925 y Fk(!)j(C)952 905 y Fj(1)994 925 y Fs(\()p Fm(T)1050 905 y Fh(l)1062 925 y Fl(;)8 b Fm(R)1123 905 y Fh(l)1136 925 y Fs(\))14 b Fl(;)730 1010 y Fs(\()p Fl(f)s(;)8 b(\027)s Fs(\))14 b Fk(7!)g Fl(\027)g Fs(+)d Fl(f)1036 1017 y Fj(\003)1059 1010 y Fl(\026)j Fs(=)f Fl(\027)h Fs(+)d Fl(@)s(f)17 b Fk(\016)11 b Fl(f)1379 990 y Fj(\000)p Fi(1)1444 1010 y Fk(\001)g Fl(\026)j(;)1701 967 y Fs(\(10)p Fl(:)p Fs(3\))57 1140 y Fp(is)k(a)g(tame)f Fk(C)305 1122 y Fj(1)364 1140 y Fp(map.)24 b(Mor)m(e)m(over,)19 b(if)f Fl(\026)f Fp(is)h(a)g(diophantine)i(ve)m(ctor)d(\(se)m(e)i(\(8.5\))g (\))f(then)f Fs(\011)1749 1147 y Fh(\026)1793 1140 y Fp(is)57 1209 y(a)h(tame)f Fk(C)252 1191 y Fj(1)313 1209 y Fp(lo)m(c)m(al)i(di\013e)m(omorphism)h(ne)m(ar)f Fl(f)g Fs(=)14 b Fp(id)1024 1220 y Fg(T)1044 1210 y Fb(l)1064 1209 y Fp(,)k Fl(\027)f Fs(=)c(0)p Fp(.)57 1318 y(Pr)m(o)m(of.)21 b Fs(First)d(of)g(all)g(note)h(that)g(\011)722 1325 y Fh(\026)748 1318 y Fs(\(id)809 1328 y Fg(T)829 1318 y Fb(l)8 b Fl(;)g Fs(0\))17 b(=)g Fl(\026)p Fs(.)28 b(The)18 b(\014rst)f(assertion)g(is)h(an)g(immediate)57 1388 y(consequence)e(of) h(Exercises)f(9.13)h(and)f(10.1.)23 b(Moreo)o(v)o(er,)15 b(using)h(\(10.1\),)h(one)f(easily)h(c)o(hec)o(ks)57 1457 y(that)171 1575 y Fl(D)q Fs(\011)252 1582 y Fh(\026)279 1575 y Fs(\()p Fl(f)s(;)8 b(\027)s Fs(\))k Fk(\001)f Fs(\(\001)p Fl(f)s(;)d Fs(\001)p Fl(\027)s Fs(\))13 b(=)h(\001)p Fl(\027)f Fs(+)e(\()p Fl(@)s Fs(\001)p Fl(f)5 b Fs(\))12 b Fk(\016)f Fl(f)1041 1555 y Fj(\000)p Fi(1)1107 1575 y Fk(\001)g Fl(\026)641 1660 y Fs(+)f Fl(@)719 1639 y Fi(2)742 1660 y Fl(f)17 b Fk(\016)11 b Fl(f)848 1639 y Fj(\000)p Fi(1)913 1660 y Fk(\001)g Fs(\()p Fk(\000)p Fs(\()p Fl(@)s(f)5 b Fs(\))1092 1639 y Fj(\000)p Fi(1)1158 1660 y Fk(\016)11 b Fl(f)1223 1639 y Fj(\000)p Fi(1)1289 1660 y Fk(\001)g Fs(\001)p Fl(f)16 b Fk(\016)11 b Fl(f)1461 1639 y Fj(\000)p Fi(1)1515 1660 y Fl(;)d(\026)p Fs(\))643 1745 y(=)14 b(\001)p Fl(\027)f Fs(+)e([\()p Fl(@)s Fs(\001)p Fl(f)5 b Fs(\))12 b Fk(\001)f Fl(\026)641 1829 y Fs(+)f Fl(@)719 1809 y Fi(2)742 1829 y Fl(f)17 b Fk(\001)11 b Fs(\()p Fk(\000)p Fs(\()p Fl(@)s(f)5 b Fs(\))962 1809 y Fj(\000)p Fi(1)1028 1829 y Fk(\001)11 b Fs(\001)p Fl(f)s(;)d(\026)p Fs(\)])j Fk(\016)g Fl(f)1283 1809 y Fj(\000)p Fi(1)1701 1702 y Fs(\(10)p Fl(:)p Fs(4\))57 1964 y(\(w)o(e)16 b(recall)g(that)h (here)f(one)h(has)f(\001)p Fl(f)j Fk(2)14 b(C)840 1946 y Fj(1)882 1964 y Fs(\()p Fm(T)938 1946 y Fh(l)950 1964 y Fl(;)8 b Fm(R)1011 1946 y Fh(l)1023 1964 y Fl(;)g Fs(0\),)17 b(\001)p Fl(\027)f Fk(2)e Fm(R)1289 1946 y Fh(l)1302 1964 y Fs(\).)156 2037 y(If)j(one)f(in)o(tro)q(duces)g Fl(u)p Fs(,)g(writing)f(\001)p Fl(f)k Fs(=)14 b Fl(@)s(f)j Fk(\001)11 b Fl(u)16 b Fs(then)h(one)f(gets)417 2177 y(\()p Fl(@)s Fs(\001)p Fl(f)5 b Fs(\))12 b Fk(\016)f Fl(f)632 2156 y Fj(\000)p Fi(1)697 2177 y Fk(\001)g Fl(\026)j Fs(=)g([)p Fl(@)862 2156 y Fi(2)884 2177 y Fl(f)j Fk(\001)11 b Fl(u)g Fk(\001)f Fl(\026)i Fs(+)e Fl(@)s(f)17 b Fk(\001)11 b Fl(@)s(u)g Fk(\001)g Fl(\026)p Fs(])g Fk(\016)g Fl(f)1414 2156 y Fj(\000)p Fi(1)57 2316 y Fs(and)243 2396 y Fl(@)272 2376 y Fi(2)295 2396 y Fl(f)17 b Fk(\016)10 b Fl(f)400 2376 y Fj(\000)p Fi(1)466 2396 y Fk(\001)h Fs(\()p Fk(\000)p Fs(\()p Fl(@)s(f)5 b Fs(\))645 2376 y Fj(\000)p Fi(1)711 2396 y Fk(\016)11 b Fl(f)776 2376 y Fj(\000)p Fi(1)841 2396 y Fk(\001)g Fs(\001)p Fl(f)17 b Fk(\016)11 b Fl(f)1014 2376 y Fj(\000)p Fi(1)1068 2396 y Fl(;)d(\026)p Fs(\))14 b(=)g Fk(\000)p Fs([)p Fl(@)1288 2376 y Fi(2)1310 2396 y Fl(f)j Fk(\001)11 b Fl(u)f Fk(\001)h Fl(\026)p Fs(])g Fk(\016)g Fl(f)1560 2376 y Fj(\000)p Fi(1)1628 2396 y Fl(:)57 2511 y Fs(Therefore)16 b(\(10.4\))g(simpli\014es)f (considerably)g(and)h(b)q(ecomes)380 2650 y Fl(D)q Fs(\011)461 2657 y Fh(\026)488 2650 y Fs(\()p Fl(f)s(;)8 b(\027)s Fs(\))k Fk(\001)f Fs(\()p Fl(@)s(f)17 b Fk(\001)11 b Fl(u;)d Fs(\001)p Fl(\027)s Fs(\))13 b(=)h(\001)p Fl(\027)f Fs(+)e(\()p Fl(@)s(f)17 b Fk(\001)11 b Fl(@)s(u)g Fk(\001)g Fl(\026)p Fs(\))g Fk(\016)g Fl(f)1423 2630 y Fj(\000)p Fi(1)1491 2650 y Fl(:)196 b Fs(\(10)p Fl(:)p Fs(5\))918 2770 y(65)p eop %%Page: 66 67 66 66 bop 57 192 a Fs(T)l(o)13 b(pro)o(v)o(e)g(the)h(second)g (assertion)e(w)o(e)i(will)g(apply)f(Theorem)g(9.18)h(to)g(\010)g(=)f (\011)1513 199 y Fh(\026)1554 192 y Fs(at)h(the)g(p)q(oin)o(ts)57 261 y Fl(x)85 268 y Fi(0)121 261 y Fs(=)g(\(id)235 271 y Fg(T)255 261 y Fb(l)8 b Fl(;)g Fs(0\))17 b(and)f Fl(y)479 268 y Fi(0)516 261 y Fs(=)d Fl(\026)p Fs(.)23 b(W)l(e)16 b(m)o(ust)g(just)g(c)o(hec)o(k)h(that)g Fl(D)q Fs(\011)1266 268 y Fh(\026)1293 261 y Fs(\()p Fl(f)s(;)8 b(\027)s Fs(\))17 b(is)f(in)o(v)o(ertible)f Fp(for)k(al)s(l)57 331 y Fs(\()p Fl(f)s(;)8 b(\027)s Fs(\))17 b(in)f(a)h(neigh)o(b)q(orho) q(o)q(d)d(of)j(\(id)719 341 y Fg(T)739 331 y Fb(l)8 b Fl(;)g Fs(0\).)23 b(This)15 b(leads)h(us)g(to)h(the)g(equation)618 461 y(\001)p Fl(\027)c Fs(+)e(\()p Fl(@)s(f)18 b Fk(\001)11 b Fl(@)s(u)f Fk(\001)h Fl(\026)p Fs(\))h Fk(\016)f Fl(f)1082 440 y Fj(\000)p Fi(1)1150 461 y Fs(=)i Fl(w)i(:)434 b Fs(\(10)p Fl(:)p Fs(6\))57 590 y(Comp)q(osing)15 b(on)h(the)h(righ)o(t) e(with)i Fl(f)22 b Fs(and)16 b(m)o(ultiplying)f(b)q(oth)h(sides)g(b)o (y)g(\()p Fl(@)s(f)5 b Fs(\))1516 572 y Fj(\000)p Fi(1)1588 590 y Fs(one)16 b(gets)604 720 y Fl(\026)11 b Fk(\001)g Fl(@)s(u)j Fs(=)f(\()p Fl(@)s(f)5 b Fs(\))890 699 y Fj(\000)p Fi(1)957 720 y Fk(\001)11 b Fs([)p Fl(w)g Fk(\016)g Fl(f)17 b Fk(\000)11 b Fs(\001)p Fl(\027)s Fs(])i Fl(;)420 b Fs(\(10)p Fl(:)p Fs(7\))57 849 y(i.e.)14 b(an)g(equation)h(of)f(the)h (form)f(\(8.4\))h(with)g Fl(v)g Fs(=)f(\()p Fl(@)s(f)5 b Fs(\))1075 831 y Fj(\000)p Fi(1)1137 849 y Fk(\001)i Fs([)p Fl(w)h Fk(\016)f Fl(f)13 b Fk(\000)7 b Fs(\001)p Fl(\027)s Fs(].)21 b(This)13 b(clari\014es)h(wh)o(y)57 919 y(one)k(needs)f(the)i(term)f Fl(\027)j Fs(in)d(the)g(de\014nition)g (\(10.3\))g(of)h(\011)1156 926 y Fh(\026)1200 919 y Fs(:)26 b(indeed)17 b(one)i(\014xes)f(it)g(so)g(as)g(to)57 989 y(assure)c(that)h Fl(v)h Fk(2)e(C)428 970 y Fi(0)p Fh(;)p Fj(1)502 989 y Fs(\()p Fm(T)558 970 y Fh(l)570 989 y Fl(;)8 b Fm(R)631 970 y Fh(l)643 989 y Fs(\),)16 b(i.e.)f(it)g(has)g (zero)g(a)o(v)o(erage)f(on)h(the)h(torus)e Fm(T)1503 970 y Fh(l)1516 989 y Fs(.)21 b(One)15 b(can)g(also)57 1058 y(c)o(hec)o(k)h(that)h(the)g(map)e(\()p Fl(f)s(;)8 b(w)q Fs(\))15 b Fk(7!)f Fs(\001)p Fl(\027)19 b Fs(is)d(tame.)156 1128 y(Prop)q(osition)g(8.7)i(allo)o(ws)e(to)i(conclude)f(since)g(it)h (sho)o(ws)e(that)i(the)f(map)g(\()p Fl(f)s(;)8 b(\027)s Fs(\))16 b Fk(7!)g Fl(u)f Fs(=)57 1198 y Fl(D)99 1180 y Fj(\000)p Fi(1)98 1210 y Fh(\026)153 1198 y Fl(v)j Fs(is)e(tame.)1423 b Fa(\003)918 2770 y Fs(66)p eop %%Page: 67 68 67 67 bop 761 192 a Fq(App)r(endices)57 499 y Fo(A1.)26 b(Uniformization,)21 b(Distorsion)f(and)g(Quasi{conformal)g(maps)156 608 y Fs(In)h(this)g(app)q(endix)f(w)o(e)h(recall)f(some)g(elemen)o (tary)g(and)g(less)h(elemen)o(tary)f(facts)h(from)57 678 y(the)16 b(theory)h(of)g(conformal)e(and)h(quasi{conformal)e(maps)i (of)g(one)h(complex)f(v)m(ariable.)57 854 y Fr(A1.1)g Fs(A)h(nonempt)o(y)e(connected)i(op)q(en)f(set)h(is)f(called)g(a)h Fp(r)m(e)m(gion)p Fs(.)57 996 y Fr(Theorem)29 b(A1.1)i(\(The)g(Maxim)n (um)g(Principle\))g Fd(If)d Fl(f)5 b Fs(\()p Fl(z)r Fs(\))29 b Fd(is)e(analytic)g(and)f(non{)57 1066 y(constan)o(t)d(in)g(a)h (region)e Fs(\012)i Fd(of)g(the)g(complex)f(plane)g Fm(C)9 b Fd(,)29 b(then)23 b(its)h(absolute)f(v)m(alue)h Fk(j)p Fl(f)5 b Fs(\()p Fl(z)r Fs(\))p Fk(j)57 1136 y Fd(has)16 b(no)g(maxim)o(um)e(in)i Fs(\012)p Fd(.)57 1276 y Fp(Pr)m(o)m(of.)21 b Fs(It)d(is)g(an)f(easy)h(consequence)g(of)g(the)g(fact)h(that)f (non{constan)o(t)e(analytic)i(functions)57 1346 y(map)d(op)q(en)i(sets) f(on)o(to)g(op)q(en)h(sets.)1077 b Fa(\003)57 1470 y Fs(The)16 b(maxim)o(um)f(principle)g(implies)g(that)i(if)g Fl(f)22 b Fs(is)17 b(de\014ned)e(and)h(con)o(tin)o(uous)f(on)h(a)h (compact)57 1539 y(set)24 b Fl(K)j Fs(and)c(analytic)h(in)f(the)h(in)o (terior)e(of)i Fl(K)k Fs(then)23 b(the)h(maxim)o(um)e(of)i Fk(j)p Fl(f)5 b Fs(\()p Fl(z)r Fs(\))p Fk(j)25 b Fs(on)e Fl(K)28 b Fs(is)57 1609 y(assumed)14 b(on)j(the)f(b)q(oundary)g(of)h Fl(K)t Fs(.)k(Another)c(easy)f(consequence)g(is)g(the)h(follo)o(wing)57 1716 y Fr(Exercise)h(A1.2)g(\(Sc)n(h)n(w)n(arz's)h(Lemma,)d (automorphism)o(s)e(of)j(the)h(disk\))e Fp(Schwarz's)57 1786 y(L)m(emma)k Fs(:)29 b(If)20 b Fk(j)p Fl(f)5 b Fs(\()p Fl(z)r Fs(\))p Fk(j)21 b Fs(is)f(analytic)f(for)h Fk(j)p Fl(z)r Fk(j)f Fl(<)g Fs(1)h(and)f(satis\014es)g(the)h(conditions)f Fk(j)p Fl(f)5 b Fs(\()p Fl(z)r Fs(\))p Fk(j)21 b(\024)e Fs(1,)57 1855 y Fl(f)5 b Fs(\(0\))20 b(=)f(0,)h(then)g Fk(j)p Fl(f)5 b Fs(\()p Fl(z)r Fs(\))p Fk(j)21 b(\024)d(j)p Fl(z)r Fk(j)i Fs(and)f Fk(j)p Fl(f)817 1837 y Fj(0)832 1855 y Fs(\(0\))p Fk(j)g(\024)g Fs(1.)32 b(If)20 b Fk(j)p Fl(f)5 b Fs(\()p Fl(z)r Fs(\))p Fk(j)20 b Fs(=)f Fk(j)p Fl(z)r Fk(j)h Fs(for)f(some)g Fl(z)j Fk(6)p Fs(=)c(0)i(or)f(if)57 1925 y Fk(j)p Fl(f)100 1907 y Fj(0)114 1925 y Fs(\(0\))p Fk(j)c Fs(=)e(1)k(then)g Fl(f)5 b Fs(\()p Fl(z)r Fs(\))15 b(=)f Fl(cz)19 b Fs(with)d Fl(c)e Fk(2)g Fm(C)9 b Fs(,)20 b Fk(j)p Fl(c)p Fk(j)13 b Fs(=)h(1.)22 b Fp(A)o(utomorphisms)d(of)g (the)f(disk)f Fs(:)22 b(Sho)o(w)57 1995 y(that)c(if)g Fl(f)31 b Fs(:)16 b Fm(D)27 b Fk(!)17 b Fm(D)29 b Fs(is)17 b(an)h(automorphism)d(of)k(the)f(disk)f(and)h Fl(f)5 b Fs(\(0\))18 b(=)e(0)i(then)g Fk(j)p Fl(f)1640 1977 y Fj(0)1654 1995 y Fs(\(0\))p Fk(j)f Fs(=)f(1)57 2065 y(and)d Fl(f)20 b Fs(is)13 b(a)h(rotation.)21 b(Deduce)14 b(from)f(this)g(that)i(the)f(group)f(of)h(automorphisms)c(of)15 b(the)f(unit)57 2134 y(disk)i Fm(D)27 b Fs(is)295 2269 y(Aut)8 b(\()p Fm(D)h Fs(\))17 b(=)d Fk(f)p Fl(z)i Fk(7!)d Fl(T)7 b Fs(\()p Fl(z)r Fs(\))15 b(=)829 2236 y Fl(az)f Fs(+)c Fl(b)p 829 2258 134 2 v 829 2268 22 2 v 829 2308 a(bz)k Fs(+)p 936 2281 27 2 v 10 w Fl(a)977 2269 y(;)22 b(a;)8 b(b)15 b Fk(2)f Fm(C)j Fl(;)25 b Fk(j)p Fl(a)p Fk(j)1278 2249 y Fi(2)1311 2269 y Fk(\000)11 b(j)p Fl(b)p Fk(j)1410 2249 y Fi(2)1446 2269 y Fs(=)j(1)p Fk(g)p Fs(])g Fl(:)57 2511 y Fr(A1.2)f Fs(A)g(mapping)f Fl(f)19 b Fs(of)14 b(a)f(region)g(\012)g(in)o(to)g Fm(C)25 b Fs(is)13 b(called)g Fp(c)m(onformal)i Fs(if)f(it)f(is)g(holomorphic)e(and)57 2581 y(injectiv)o(e.)25 b(Suc)o(h)16 b(maps)g(are)h(also)g(called)g Fp(univalent)p Fs(.)24 b(Since)17 b(an)g(analytic)h(map)e(is)h (injectiv)o(e)57 2650 y(if)g(and)f(only)g(if)h Fl(f)384 2632 y Fj(0)413 2650 y Fk(6)p Fs(=)d(0)j(if)g Fl(f)22 b Fs(is)17 b(univ)m(alen)o(t)f(in)g(\012)h(its)g(deriv)m(ativ)o(e)f (nev)o(er)h(v)m(anishes,)f(i.e.)22 b(it)17 b(has)918 2770 y(67)p eop %%Page: 68 69 68 68 bop 57 192 a Fs(no)16 b(critical)h(p)q(oin)o(ts)f(inside)f(\012.) 23 b(The)17 b(most)f(imp)q(ortan)o(t)g(result)g(of)h(the)g(theory)g(of) g(conformal)57 261 y(maps)e(is)h(certainly)g(the)57 414 y Fr(Theorem)g(A1.3)j(\(Riemann)f(Mapping)h(Theorem\))27 b Fd(Giv)o(en)16 b(an)o(y)f(simply)g(connected)57 484 y(region)i Fs(\012)i Fd(whic)o(h)e(is)h(not)h(the)f(whole)g(plane,)g (and)g(a)h(p)q(oin)o(t)f Fl(z)1227 491 y Fi(0)1266 484 y Fk(2)f Fs(\012)i Fd(there)f(exists)h(a)f(unique)57 554 y(conformal)c(map)h(\(the)i(Riemann)d(map\))i Fl(f)28 b Fs(:)13 b Fm(D)25 b Fk(!)14 b Fs(\012)i Fd(suc)o(h)f(that)h Fl(f)22 b Fd(is)15 b(on)o(to,)h Fl(f)5 b Fs(\(0\))15 b(=)f Fl(z)1710 561 y Fi(0)1748 554 y Fd(and)57 624 y Fl(f)86 606 y Fj(0)100 624 y Fs(\(0\))h Fl(>)e Fs(0)p Fd(.)57 772 y Fr(Exercise)23 b(A1.4)c Fs(Drop)g(the)g(requiremen)o(t)f Fl(f)950 754 y Fj(0)965 772 y Fs(\(0\))h Fl(>)f Fs(0.)31 b(Then)19 b Fl(f)25 b Fs(is)19 b(not)g(unique)g(but)g(the)57 842 y(n)o(um)o(b)q(er)14 b Fk(j)p Fl(f)281 824 y Fj(0)296 842 y Fs(\(0\))p Fk(j)j Fs(do)q(es)g(not)f(dep)q(end)g(on)h Fl(f)5 b Fs(.)23 b(It)17 b(is)f(called)h([Ah1])f(the)h Fp(c)m(onformal)i(c)m(ap)m(acity)g Fs(of)57 912 y(\012)d(with)h(resp)q (ect)f(to)h Fl(z)477 919 y Fi(0)516 912 y Fs(and)f(it)h(will)f(b)q(e)h (denoted)f Fl(C)t Fs(\(\012)p Fl(;)8 b(z)1153 919 y Fi(0)1175 912 y Fs(\).)23 b([Hin)o(t)16 b(:)22 b(use)16 b(Exercise)g(A1.3])57 1021 y(One)h(should)e(not)i(think)h(to)f(an)g(arbitrary)f(simply)g (connected)h(region)f(as)h(the)h(\\p)q(otato")f(of)57 1091 y(PDEs)d(but)h(as)g(a)g(rather)g(irregular)e(ob)s(ject.)22 b(F)l(or)14 b(example)g(the)i(b)q(oundary)e(needs)g(not)i(to)f(b)q(e)57 1161 y(lo)q(cally)h(connected.)57 1340 y Fr(A1.3)g Fs(Assume)g(that)h (the)g(region)f(\012)h(is)f(b)q(ounded)g(and)g Fl(@)s Fs(\012)h(is)g(a)g(closed)f(Jordan)f(curv)o(e)h(\(i.e.)57 1410 y Fl(@)s Fs(\012)f(=)g Fl(\015)s Fs(\([0)p Fl(;)8 b Fs(1]\),)18 b(where)f Fl(\015)26 b Fs(:)e([0)p Fl(;)8 b Fs(1])15 b Fk(!)g Fm(C)30 b Fs(is)17 b(con)o(tin)o(uous,)e Fl(\015)s Fs(\(0\))h(=)f Fl(\015)s Fs(\(1\))j(and)e Fl(\015)s Fs(\()p Fl(t)1609 1417 y Fi(1)1632 1410 y Fs(\))g(=)f Fl(\015)s Fs(\()p Fl(t)1787 1417 y Fi(2)1809 1410 y Fs(\))57 1480 y(if)h(and)g(only)g(if)h Fl(t)372 1487 y Fi(1)408 1480 y Fs(=)d Fl(t)479 1487 y Fi(2)517 1480 y Fs(or)i Fl(t)596 1487 y Fi(1)633 1480 y Fs(=)d(0,)j Fl(t)758 1487 y Fi(2)795 1480 y Fs(=)d(1\).)22 b(In)17 b(this)f(case)g(the)h (Riemann)e(map)g Fl(f)28 b Fs(:)14 b Fm(D)25 b Fk(!)13 b Fs(\012)57 1550 y(has)j(a)g(nice)g(b)q(oundary)g(b)q(eha)o(viour)f(:) 57 1703 y Fr(Theorem)22 b(A1.5)h(\(Caratheo)r(dory\))49 b Fd(A)21 b(Riemann)f(map)g Fl(f)35 b Fs(:)21 b Fm(D)32 b Fk(!)20 b Fs(\012)h Fd(extends)g(to)g(a)57 1773 y(homeomorphism)13 b(of)p 487 1732 36 2 v 16 w Fm(D)28 b Fd(on)o(to)p 652 1733 V 16 w Fs(\012)17 b Fd(if)f(and)g(only)h(if)f Fl(@)s Fs(\012)h Fd(is)f(a)h(closed)f(Jordan)e(curv)o(e.)57 1921 y Fs(The)19 b(topic)g(of)g(the)h(b)q(oundary)e(b)q(eha)o(viour)g (of)h(conformal)f(maps)g(is)h(v)o(ery)g(ric)o(h)f(and)h(it's)g(an)57 1991 y(activ)o(e)12 b(researc)o(h)e(area)i(:)19 b(w)o(e)12 b(refer)g(to)g([P)o(o])f(for)h(more)f(informations)f(and)h(references.) 20 b(W)l(e)12 b(will)57 2061 y(only)18 b(need)g(t)o(w)o(o)f(other)h (results)f(:)26 b(the)18 b(\014rst)g(extends)g(Caratheo)q(dory's)f (theorem)g(dropping)57 2130 y(the)f(assumption)f(that)i(the)g (restriction)e(of)i Fl(f)22 b Fs(to)17 b(the)g(b)q(oundary)e(of)i(the)g (disk)f(is)g(injectiv)o(e.)57 2284 y Fr(Theorem)c(A1.6)28 b Fd(Let)13 b Fl(f)28 b Fs(:)14 b Fm(D)25 b Fk(!)13 b Fs(\012)g Fd(b)q(e)f(a)g(Riemann)f(map.)20 b(The)12 b(follo)o(wing)f (four)h(conditions)57 2353 y(are)k(equiv)m(alen)o(t)g(:)79 2428 y(\(i\))25 b Fl(f)e Fd(has)16 b(a)g(con)o(tin)o(uous)e(extension)j (to)p 864 2387 V 17 w Fm(D)j Fd(;)65 2502 y(\(ii\))25 b Fl(@)s Fs(\012)17 b Fd(is)f(a)h(con)o(tin)o(uous)d(curv)o(e,)i(i.e.) 22 b Fl(@)s Fs(\012)14 b(=)f Fk(f)p Fl(')p Fs(\()p Fl(\020)t Fs(\))8 b Fl(;)g(\020)18 b Fk(2)c Fm(T)-5 b Fk(g)13 b Fd(with)k Fl(')f Fd(con)o(tin)o(uous)8 b(;)51 2576 y(\(iii\))25 b Fl(@)s Fs(\012)17 b Fd(is)f(lo)q(cally)h(connected)10 b(;)53 2650 y(\(iv\))25 b Fm(C)e Fk(n)11 b Fs(\012)17 b Fd(is)f(lo)q(cally)h(connected.)918 2770 y Fs(68)p eop %%Page: 69 70 69 69 bop 57 192 a Fs(Our)11 b(second)g(result,)h(due)g(to)g(F)l(atou,) h(applies)e(to)h(all)g Fl(f)28 b Fs(:)13 b Fm(D)25 b Fk(!)14 b Fm(C)24 b Fs(holomorphic)10 b Fp(and)k(b)m(ounde)m(d)57 261 y Fs(\(th)o(us)i(w)o(e)g(are)g(dropping)f(the)h(assumption)f(of)i Fl(f)22 b Fs(b)q(eing)16 b(injectiv)o(e)h(in)f Fm(D)8 b Fs(\).)57 414 y Fr(Theorem)18 b(A1.7)h(\(F)-5 b(atou\))29 b Fd(Let)18 b Fl(f)29 b Fs(:)14 b Fm(D)26 b Fk(!)15 b Fm(C)29 b Fd(b)q(e)17 b(holomorphic)e(and)h(b)q(ounded.)23 b(Then)17 b Fl(f)57 483 y Fd(has)g(a)g(non{tangen)o(tial)f(limit)h(at)h (almost)f(all)g(p)q(oin)o(ts)g Fl(\020)i Fk(2)d Fm(T)8 b Fs(=)15 b Fl(@)s Fm(D)8 b Fd(.)28 b(Moreo)o(v)o(er)16 b(if)i Fl(f)23 b Fd(is)17 b(not)57 553 y(iden)o(tically)f(zero)h(then)h Fl(')p Fs(\()p Fl(\020)t Fs(\))d(=)g(lim)754 560 y Fh(r)q Fj(!)p Fi(1)p Fj(\000)875 553 y Fl(f)5 b Fs(\()p Fl(r)q(\020)t Fs(\))19 b Fd(\(whic)o(h)e(b)q(elongs)g(to)g Fl(L)1448 535 y Fj(1)1491 553 y Fs(\()p Fm(T)-5 b Fs(\))p Fd(\))15 b(is)i(not)h(zero)57 623 y(almost)d(ev)o(erywhere.)57 841 y Fr(A1.4)h Fs(Another)g(fundamen)o(tal)f(result)h(is)g(the)h (celebrated)57 993 y Fr(Theorem)h(A1.8)i(\(Uniformization)g(Theorem\)) 27 b Fd(The)17 b(only)h(simply)e(connected)h(Rie-)57 1063 y(mann)k(surfaces,)i(up)f(to)g(biholomorphic)e(equiv)m(alence,)k (are)e(the)h(Riemann)e(sphere)p 1730 1022 36 2 v 21 w Fm(C)36 b Fs(=)57 1132 y Fm(C)23 b Fk([)11 b(f1g)p Fd(,)16 b(the)h(complex)f(plane)g Fm(C)28 b Fd(and)16 b(the)h(unit)f(disk)g Fm(D)8 b Fd(.)57 1280 y Fr(Exercise)19 b(A1.9)c Fs(Pro)o(v)o(e)f(that)i (the)g(group)e(of)i(automorphisms)c(of)k(the)g(Riemann)e(sphere)g(is)57 1367 y(the)h(group)e(PGL)c(\(2)p Fl(;)f Fm(C)h Fs(\))18 b(acting)d(b)o(y)f(homographies)f(:)21 b(if)15 b Fl(g)g Fs(=)1230 1297 y Fe(\022)1275 1337 y Fl(a)52 b(b)1277 1397 y(c)g(d)1385 1297 y Fe(\023)1435 1367 y Fk(2)14 b Fs(PGL)9 b(\(2)p Fl(;)f Fm(C)h Fs(\))18 b(then)57 1463 y Fl(z)e Fk(7!)e Fl(g)9 b Fk(\001)f Fl(z)16 b Fs(=)313 1444 y Fh(az)q Fi(+)p Fh(b)p 313 1452 91 2 v 313 1481 a(cz)q Fi(+)p Fh(d)409 1463 y Fs(.)21 b(The)15 b(group)f(of)h (automorphisms)d(of)j(the)h(complex)e(plane)h(is)f(simply)g(the)57 1533 y(a\016ne)i(group.)57 1712 y Fr(A1.5)i Fs(One)h(can)g(consider)e (univ)m(alen)o(t)i(functions)f Fl(f)25 b Fs(on)18 b(regions)g(of)p 1377 1672 36 2 v 19 w Fm(C)31 b Fs(with)19 b(v)m(alues)g(in)p 1759 1672 V 18 w Fm(C)32 b Fs(:)57 1782 y(in)22 b(this)h(case)g Fl(f)29 b Fs(m)o(ust)23 b(b)q(e)g(meromorphic)e(and)h(injectiv)o(e.)42 b(Here)24 b(are)f(some)f(elemen)o(tary)57 1852 y(prop)q(erties)15 b(:)57 1961 y Fr(Exercise)20 b(A1.10)68 2035 y Fs(\(a\))25 b(If)20 b Fl(f)25 b Fs(is)19 b(univ)m(alen)o(t)g(on)g(a)g(region)g (\012)f Fk(\032)p 910 1995 V 18 w Fm(C)32 b Fs(then)19 b Fl(f)25 b Fs(is)19 b(analytic)h(except)g(for)f(at)h(most)e(a)156 2105 y(single)e(simple)f(p)q(ole)i(and)f Fl(f)683 2087 y Fj(0)714 2105 y Fs(nev)o(er)g(v)m(anishes.)65 2179 y(\(b\))25 b(If)17 b Fl(f)28 b Fs(:)14 b(\012)g Fk(!)f Fs(\012)435 2161 y Fj(0)466 2179 y Fs(is)j(on)o(to)g(and)g(univ)m(alen) o(t)g(then)g Fl(f)1083 2161 y Fj(\000)p Fi(1)1159 2179 y Fs(:)e(\012)1223 2161 y Fj(0)1251 2179 y Fk(!)g Fs(\012)i(is)g(also)g (univ)m(alen)o(t.)71 2253 y(\(c\))25 b(A)17 b(univ)m(alen)o(t)f(map)g (is)g(a)g(homeomorphism.)65 2327 y(\(d\))25 b(A)14 b(univ)m(alen)o(t)f (map)g(preserv)o(es)f(angles)g(b)q(et)o(w)o(een)h(curv)o(es)g(and)g (their)g(orien)o(tation)f(\(that's)156 2397 y(wh)o(y)k(they're)h (called)f(conformal)9 b(!\).)71 2471 y(\(e\))25 b(The)17 b(comp)q(osition)f(of)h(univ)m(alen)o(t)f(maps)g(is)h(univ)m(alen)o(t)9 b(;)17 b Fl(f)23 b Fs(is)16 b(univ)m(alen)o(t)g(if)h(and)g(only)f(if) 156 2541 y(1)p Fl(=f)22 b Fs(is)17 b(univ)m(alen)o(t.)57 2650 y Fr(Exercise)k(A1.11)d Fs(Pro)o(v)o(e)f(that)h(if)g Fl(f)30 b Fs(:)16 b(\012)g Fk(!)p 949 2610 V 16 w Fm(C)30 b Fs(is)18 b(univ)m(alen)o(t)f(and)g Fl(A)g Fk(\032)f Fs(\012)i(is)f(measurable)918 2770 y(69)p eop %%Page: 70 71 70 70 bop 57 192 a Fs(then)547 268 y(Area)8 b(\()p Fl(f)d Fs(\()p Fl(A)p Fs(\)\))17 b(=)870 200 y Fe(Z)898 313 y Fh(A)939 268 y Fk(j)p Fl(f)982 247 y Fj(0)996 268 y Fs(\()p Fl(x)12 b Fs(+)f Fl(iy)r Fs(\))p Fk(j)1181 247 y Fi(2)1204 268 y Fl(dxdy)16 b(:)57 434 y Fr(Exercise)24 b(A1.12)e(\(The)i(Area)f(F)-5 b(orm)n(ula\))20 b Fs(Sho)o(w)f(that)i (if)f Fl(f)34 b Fs(:)20 b Fm(D)31 b Fk(!)19 b Fm(C)33 b Fs(is)19 b(univ)m(alen)o(t,)57 504 y(letting)d Fl(f)5 b Fs(\()p Fl(z)r Fs(\))16 b(=)375 466 y Fe(P)427 479 y Fj(1)427 519 y Fh(n)p Fi(=0)513 504 y Fl(f)537 511 y Fh(n)565 504 y Fl(z)590 486 y Fh(n)617 504 y Fs(,)h(one)f(has)641 672 y(Area)8 b(\()p Fl(f)d Fs(\()p Fm(D)10 b Fs(\)\))17 b(=)d Fl(\031)1019 610 y Fj(1)1003 624 y Fe(X)1001 730 y Fh(n)p Fi(=1)1085 672 y Fl(n)p Fk(j)p Fl(f)1153 679 y Fh(n)1180 672 y Fk(j)1194 651 y Fi(2)1230 672 y Fl(:)57 843 y Fs([Hin)o(t)28 b(:)46 b(First)28 b(consider)f(the)i(disk)f Fm(D)843 850 y Fh(r)897 843 y Fs(of)h(radius)e Fl(r)36 b(<)d Fs(1.)59 b(If)29 b Fl(f)39 b Fs(=)34 b Fl(u)19 b Fs(+)g Fl(iv)30 b Fs(then)57 913 y(Area)8 b(\()p Fl(f)d Fs(\()p Fm(D)k Fs(\))q(\))17 b(=)379 873 y Fe(R)402 931 y Fh(@)r Fg(D)448 936 y Fb(r)482 913 y Fl(udv)e Fs(=)637 893 y Fh(i)p 634 902 20 2 v 634 930 a Fi(2)668 873 y Fe(R)692 931 y Fh(@)r Fg(D)738 936 y Fb(r)771 913 y Fl(f)5 b(d)p 826 872 30 2 v(f)i Fs(.)22 b(Then)16 b(let)h Fl(r)f Fk(!)d Fs(1)p Fk(\000)p Fs(.])57 1020 y(Let)j Fl(S)176 1027 y Fi(1)213 1020 y Fs(denote)g(the)f(collection)h(of)f(functions)g Fl(f)21 b Fs(univ)m(alen)o(t)15 b(in)g Fm(D)27 b Fs(and)15 b(suc)o(h)f(that)i Fl(f)5 b Fs(\(0\))15 b(=)f(0,)57 1090 y Fl(f)86 1072 y Fj(0)100 1090 y Fs(\(0\))26 b(=)f(1,)g(th)o(us)d Fl(f)5 b Fs(\()p Fl(z)r Fs(\))27 b(=)e Fl(z)17 b Fs(+)f Fl(f)735 1097 y Fi(2)757 1090 y Fl(z)782 1072 y Fi(2)821 1090 y Fs(+)f Fl(:)8 b(:)g(:)p Fs(.)42 b(With)24 b(\006)1161 1097 y Fi(1)1206 1090 y Fs(w)o(e)f(will)g(denote)g(all)g(functions)57 1160 y Fl(g)r Fs(\()p Fl(\020)t Fs(\))e(=)g Fl(\020)d Fs(+)c Fl(g)345 1167 y Fi(0)381 1160 y Fs(+)g Fl(g)458 1167 y Fi(1)480 1160 y Fl(\020)506 1142 y Fj(\000)p Fi(1)573 1160 y Fs(+)g Fl(:)8 b(:)g(:)21 b Fs(univ)m(alen)o(t)g(in)f(the)i (outer)e(disk)h Fm(E)29 b Fs(=)21 b Fk(f)p Fl(\020)k Fk(2)p 1563 1119 36 2 v 21 w Fm(C)c Fl(;)8 b Fk(j)p Fl(\020)t Fk(j)20 b Fl(>)i Fs(1)p Fk(g)p Fs(.)57 1230 y(Clearly)d(if)g Fl(f)24 b Fk(2)19 b Fl(S)412 1237 y Fi(1)454 1230 y Fs(then)g Fl(g)r Fs(\()p Fl(\020)t Fs(\))f(=)g(1)p Fl(=f)5 b Fs(\()p Fl(\020)859 1211 y Fj(\000)p Fi(1)913 1230 y Fs(\))20 b(b)q(elongs)f(to)g(\006)1233 1237 y Fi(1)1275 1230 y Fs(and)g(omits)g(0.)30 b(Con)o(v)o(ersely)l(,)57 1299 y(if)20 b Fl(g)g Fk(2)g Fs(\006)239 1306 y Fi(1)281 1299 y Fs(and)f Fl(g)r Fs(\()p Fl(\020)t Fs(\))g Fk(6)p Fs(=)g(0)h(for)f (all)h Fl(\020)i Fk(2)e Fm(E)27 b Fs(then)19 b Fl(f)5 b Fs(\()p Fl(z)r Fs(\))21 b(=)e(1)p Fl(=g)r Fs(\()p Fl(z)1302 1281 y Fj(\000)p Fi(1)1356 1299 y Fs(\))h(b)q(elongs)f(to)h Fl(S)1672 1306 y Fi(1)1694 1299 y Fs(.)32 b(One)57 1369 y(of)16 b(the)g(most)f(imp)q(ortan)o(t)g(results)f(on)i(univ)m(alen)o (t)f(functions)g(is)g(the)h(ob)s(ject)h(of)f(the)g(follo)o(wing)57 1439 y(exercise)g(:)57 1546 y Fr(Exercise)f(A1.13)f(\(Area)h(Theorem\)) c Fs(If)i Fl(g)i Fk(2)f Fs(\006)1024 1553 y Fi(1)1059 1546 y Fs(then)f Fk(j)p Fl(g)1207 1553 y Fi(1)1229 1546 y Fk(j)h(\024)1309 1509 y Fe(P)1362 1521 y Fj(1)1362 1561 y Fh(n)p Fi(=1)1448 1546 y Fl(n)p Fk(j)p Fl(g)1516 1553 y Fh(n)1542 1546 y Fk(j)1556 1528 y Fi(2)1592 1546 y Fk(\024)g Fs(1.)20 b(Pro)o(v)o(e)57 1616 y(that)15 b(equalit)o(y)g(holds)f(if)h(and)f(only)h(if)g Fl(g)r Fs(\()p Fl(\020)t Fs(\))f(=)f Fl(\020)f Fs(+)c Fl(g)1032 1623 y Fi(0)1061 1616 y Fs(+)g Fl(g)1132 1623 y Fi(1)1154 1616 y Fl(\020)1180 1598 y Fj(\000)p Fi(1)1248 1616 y Fs(with)15 b Fk(j)p Fl(g)1398 1623 y Fi(1)1420 1616 y Fk(j)f Fs(=)f(1.)22 b([Hin)o(t)14 b(:)22 b(sho)o(w)57 1685 y(that)17 b(area\()p Fm(C)23 b Fk(n)10 b Fl(g)r Fs(\()p Fm(E)5 b Fs(\)\))17 b(=)d Fl(\031)r Fs(\(1)d Fk(\000)677 1648 y Fe(P)730 1660 y Fj(1)730 1700 y Fh(n)p Fi(=1)816 1685 y Fl(n)p Fk(j)p Fl(g)884 1692 y Fh(n)910 1685 y Fk(j)924 1667 y Fi(2)946 1685 y Fs(\).])57 1793 y(One)22 b(of)h(the)g(main)f(consequences)g(is)h(the)g(follo)o(wing)f (apparen)o(tly)f(inno)q(cen)o(t)i(b)q(ound)f(:)34 b(if)57 1862 y Fl(f)19 b Fk(2)14 b Fl(S)178 1869 y Fi(1)217 1862 y Fs(then)846 1938 y Fk(j)p Fl(f)884 1945 y Fi(2)906 1938 y Fk(j)g(\024)g Fs(2)f Fl(;)650 b Fs(\()p Fl(A)p Fs(1)p Fl(:)p Fs(1\))57 2049 y(as)16 b(one)g(can)h(easily)f(c)o(hec)o (k)h(applying)e(the)i(Area)g(Theorem)e(to)i Fl(g)r Fs(\()p Fl(\020)t Fs(\))d(=)1411 2007 y Fe(p)p 1460 2007 148 2 v 1460 2049 a Fl(f)5 b Fs(\()p Fl(\020)1534 2035 y Fj(\000)p Fi(2)1589 2049 y Fs(\).)22 b(Ho)o(w)o(ev)o(er)57 2119 y(this)13 b(estimate)g(will)g(ha)o(v)o(e)g(man)o(y)f(imp)q(ortan)o (t)g(consequences)h(as)g(w)o(e)g(will)g(see)g(so)q(on.)21 b(A)14 b(\(m)o(uc)o(h)57 2189 y(harder)h(and)h(for)g(a)g(long)g(time)h (conjectural\))f(result)g(is)g(the)h(celebrated)f([DeB])57 2332 y Fr(Theorem)j(A1.14)h(\(Bieb)r(erbac)n(h{De)i(Branges\))46 b Fd(If)19 b Fl(f)j Fk(2)17 b Fl(S)1342 2339 y Fi(1)1383 2332 y Fd(then)h Fk(j)p Fl(f)1536 2339 y Fh(n)1563 2332 y Fk(j)f(\024)f Fl(n)j Fd(for)f(all)57 2402 y Fl(n)p Fd(.)57 2543 y Fs(Using)h(\(A1.1\))i(one)e(can)h(easily)f(sho)o(w)g (that)h(the)g(image)f(of)h Fm(D)31 b Fs(through)18 b(a)i(univ)m(alen)o (t)f(map)57 2613 y(cannot)d(b)q(e)h(to)q(o)g(small)e(:)918 2770 y(70)p eop %%Page: 71 72 71 71 bop 57 192 a Fr(Theorem)17 b(A1.15)h(\(Ko)r(eb)r(e)h Fs(1)p Fl(=)p Fs(4)p Fr({Theorem\))26 b Fd(If)17 b Fl(f)i Fk(2)c Fl(S)1217 199 y Fi(1)1255 192 y Fd(then)h Fm(D)1401 201 y Fi(1)p Fh(=)q Fi(4)1481 192 y Fk(\032)d Fl(f)5 b Fs(\()p Fm(D)10 b Fs(\))p Fd(.)57 324 y Fp(Pr)m(o)m(of.)20 b Fs(Let)d Fl(w)e Fk(2)f Fm(D)27 b Fs(and)16 b(assume)f Fl(w)21 b(=)-31 b Fk(2)14 b Fl(f)5 b Fs(\()p Fm(D)k Fs(\))q(.)25 b(Then)493 449 y(~)482 462 y Fl(f)6 b Fs(\()p Fl(z)r Fs(\))15 b(=)679 428 y Fl(w)q(f)5 b Fs(\()p Fl(z)r Fs(\))p 649 451 192 2 v 649 496 a Fl(w)12 b Fk(\000)f Fl(f)5 b Fs(\()p Fl(z)r Fs(\))860 462 y(=)14 b Fl(z)f Fs(+)e(\()p Fl(f)1042 469 y Fi(2)1076 462 y Fs(+)g Fl(w)1163 442 y Fj(\000)p Fi(1)1217 462 y Fs(\))p Fl(z)1261 442 y Fi(2)1295 462 y Fs(+)f Fl(:)e(:)g(:)57 609 y Fs(b)q(elongs)j(to)h Fl(S)318 616 y Fi(1)340 609 y Fs(.)20 b(Applying)12 b(\(A1.1\))g(to)h (b)q(oth)f Fl(f)17 b Fs(and)1051 596 y(~)1041 609 y Fl(f)g Fs(one)12 b(gets)g Fk(j)p Fl(w)q Fk(j)1332 591 y Fj(\000)p Fi(1)1399 609 y Fk(\024)i(j)p Fl(f)1490 616 y Fi(2)1512 609 y Fk(j)r Fs(+)r Fk(j)p Fl(f)1607 616 y Fi(2)1631 609 y Fs(+)r Fl(w)1709 591 y Fj(\000)p Fi(1)1762 609 y Fk(j)g(\024)57 679 y Fs(4.)1694 b Fa(\003)57 798 y Fs(The)27 b Fp(Ko)m(eb)m(e)i(function)f Fl(f)5 b Fs(\()p Fl(z)r Fs(\))34 b(=)e Fl(z)r Fs(\(1)19 b Fk(\000)f Fl(z)r Fs(\))917 780 y Fj(\000)p Fi(2)1004 798 y Fs(=)1075 761 y Fe(P)1127 813 y Fh(n)p Fi(=1)1213 798 y Fl(nz)1268 780 y Fh(n)1323 798 y Fs(maps)26 b(the)i(unit)f(disk)g Fm(D)57 868 y Fs(conformally)19 b(on)o(to)h Fm(C)26 b Fk(n)14 b Fs(\()p Fk(\0001)p Fl(;)8 b Fk(\000)p Fs(1)p Fl(=)p Fs(4\).)33 b(Therefore)20 b(Bieb)q(erbac)o(h{De)g(Branges')f (Theorem)57 938 y(and)f(Ko)q(eb)q(e)i(1)p Fl(=)p Fs(4{Theorem)d(are)i (optimal.)28 b(If)20 b Fl(f)k Fs(is)19 b(univ)m(alen)o(t)f(and)h (analytic)g(in)f Fm(D)8 b Fs(,)23 b(giv)o(en)57 1007 y(an)o(y)16 b Fl(z)174 1014 y Fi(0)210 1007 y Fk(2)e Fm(D)28 b Fs(the)17 b Fp(Ko)m(eb)m(e)i(tr)m(ansform)36 b Fs(of)16 b Fl(f)23 b Fs(at)16 b Fl(z)973 1014 y Fi(0)422 1180 y Fl(K)464 1187 y Fh(z)483 1192 y Fc(0)503 1187 y Fh(;f)541 1180 y Fs(\()p Fl(z)r Fs(\))f(=)677 1129 y Fl(f)715 1073 y Fe(\020)761 1108 y Fh(z)q Fi(+)p Fh(z)831 1113 y Fc(0)p 751 1117 110 2 v 751 1146 a Fi(1+)p 802 1125 39 2 v Fh(z)821 1151 y Fc(0)840 1146 y Fh(z)866 1073 y Fe(\021)907 1129 y Fk(\000)c Fl(f)5 b Fs(\()p Fl(z)1028 1136 y Fi(0)1052 1129 y Fs(\))p 677 1169 394 2 v 700 1215 a(\(1)11 b Fk(\000)g(j)p Fl(z)842 1222 y Fi(0)864 1215 y Fk(j)p Fs(\))897 1200 y Fi(2)920 1215 y Fl(f)949 1200 y Fj(0)964 1215 y Fs(\()p Fl(z)1006 1222 y Fi(0)1029 1215 y Fs(\))619 1321 y(=)i Fl(z)h Fs(+)757 1251 y Fe(\024)790 1288 y Fs(\(1)d Fk(\000)g(j)p Fl(z)932 1295 y Fi(0)954 1288 y Fk(j)p Fs(\))987 1269 y Fi(2)1010 1288 y Fl(f)1039 1269 y Fj(00)1065 1288 y Fs(\()p Fl(z)1107 1295 y Fi(0)1130 1288 y Fs(\))p 790 1310 360 2 v 893 1355 a(2)p Fl(f)947 1341 y Fj(0)962 1355 y Fs(\()p Fl(z)1004 1362 y Fi(0)1026 1355 y Fs(\))1166 1321 y Fk(\000)p 1216 1294 46 2 v 11 w Fl(z)1239 1328 y Fi(0)1262 1251 y Fe(\025)1296 1321 y Fl(z)1321 1301 y Fi(2)1355 1321 y Fs(+)g Fl(:)d(:)g(:)1689 1232 y Fs(\()p Fl(A)p Fs(1)p Fl(:)p Fs(2\))57 1457 y(b)q(elongs)16 b(to)h Fl(S)328 1464 y Fi(1)350 1457 y Fs(.)24 b(This)16 b(is)h(a)g(v)o(ery)g(useful)f(to)q(ol)h(in)g(order)f(to)h(transfer)f (the)i(information)d(at)i(0)57 1527 y(to)i(information)e(at)i(an)o(y)g (p)q(oin)o(t)f(of)h(the)g(disk.)28 b(Applying)18 b(systematically)h (this)f(idea,)h(from)57 1597 y(\(A1.1\))e(one)f(deduces)g(the)h(follo)o (wing)e(imp)q(ortan)o(t)h(distorsion)e(estimates)i(:)57 1702 y Fr(Exercise)21 b(A1.16)f(\(Ko)r(eb)r(e)f(distortion)i (theorems\))c Fs(If)h Fl(f)23 b Fs(maps)17 b Fm(D)28 b Fs(conformally)17 b(in)o(to)57 1772 y Fm(C)28 b Fs(then)17 b Fk(8)p Fl(z)e Fk(2)f Fm(D)27 b Fs(one)17 b(has)e(:)648 1828 y Fe(\014)648 1858 y(\014)648 1888 y(\014)648 1918 y(\014)664 1900 y Fs(\(1)c Fk(\000)g(j)p Fl(z)r Fk(j)822 1880 y Fi(2)845 1900 y Fs(\))870 1867 y Fl(f)899 1849 y Fj(00)925 1867 y Fs(\()p Fl(z)r Fs(\))p 870 1889 120 2 v 876 1934 a Fl(f)905 1920 y Fj(0)919 1934 y Fs(\()p Fl(z)r Fs(\))1006 1900 y Fk(\000)g Fs(2)p 1081 1873 26 2 v Fl(z)1106 1828 y Fe(\014)1106 1858 y(\014)1106 1888 y(\014)1106 1918 y(\014)1137 1900 y Fk(\024)i Fs(4)h Fl(;)447 b Fs(\()p Fl(A)p Fs(1)p Fl(:)p Fs(3\))434 2039 y Fk(j)p Fl(f)477 2018 y Fj(0)491 2039 y Fs(\(0\))p Fk(j)648 2005 y(j)p Fl(z)r Fk(j)p 574 2027 200 2 v 574 2073 a Fs(\(1)12 b(+)f Fk(j)p Fl(z)r Fk(j)p Fs(\))752 2058 y Fi(2)795 2039 y Fk(\024)i(j)p Fl(f)5 b Fs(\()p Fl(z)r Fs(\))13 b Fk(\000)e Fl(f)5 b Fs(\(0\))p Fk(j)15 b(\024)e(j)p Fl(f)1232 2018 y Fj(0)1247 2039 y Fs(\(0\))p Fk(j)1404 2005 y(j)p Fl(z)r Fk(j)p 1330 2027 V 1330 2073 a Fs(\(1)f Fk(\000)f(j)p Fl(z)r Fk(j)p Fs(\))1508 2058 y Fi(2)1550 2039 y Fl(;)125 b Fs(\()p Fl(A)p Fs(1)p Fl(:)p Fs(4\))574 2176 y Fk(j)p Fl(f)617 2156 y Fj(0)632 2176 y Fs(\(0\))p Fk(j)746 2142 y Fs(1)11 b Fk(\000)g(j)p Fl(z)r Fk(j)p 715 2165 V 715 2210 a Fs(\(1)h(+)e Fk(j)p Fl(z)r Fk(j)p Fs(\))892 2196 y Fi(3)935 2176 y Fk(\024)k(j)p Fl(f)1031 2156 y Fj(0)1045 2176 y Fs(\()p Fl(z)r Fs(\))p Fk(j)h(\024)e(j)p Fl(f)1232 2156 y Fj(0)1247 2176 y Fs(\(0\))p Fk(j)1361 2142 y Fs(1)e(+)g Fk(j)p Fl(z)r Fk(j)p 1330 2165 V 1330 2210 a Fs(\(1)h Fk(\000)f(j)p Fl(z)r Fk(j)p Fs(\))1508 2196 y Fi(3)1550 2176 y Fl(;)125 b Fs(\()p Fl(A)p Fs(1)p Fl(:)p Fs(5\))313 2275 y(1)p 313 2297 25 2 v 313 2343 a(4)344 2309 y(\(1)11 b Fk(\000)g(j)p Fl(z)r Fk(j)502 2288 y Fi(2)525 2309 y Fs(\))p Fk(j)p Fl(f)587 2288 y Fj(0)601 2309 y Fs(\()p Fl(z)r Fs(\))p Fk(j)k(\024)f Fs(dist)8 b(\()p Fl(f)d Fs(\()p Fl(z)r Fs(\))p Fl(;)j(@)s(f)d Fs(\()p Fm(D)11 b Fs(\))q(\))17 b Fk(\024)c Fs(\(1)f Fk(\000)f(j)p Fl(z)r Fk(j)1348 2288 y Fi(2)1370 2309 y Fs(\))p Fk(j)p Fl(f)1432 2288 y Fj(0)1447 2309 y Fs(\()p Fl(z)r Fs(\))p Fk(j)k Fl(:)136 b Fs(\()p Fl(A)p Fs(1)p Fl(:)p Fs(6\))57 2463 y(These)18 b(estimates)g(sho)o(w)f(that)i(the)g (gro)o(wth)e(of)i Fl(f)24 b Fs(as)19 b Fl(z)h Fs(approac)o(hes)d Fl(@)s Fm(D)30 b Fs(cannot)18 b(b)q(e)h(faster)57 2533 y(than)14 b(\(1)7 b Fk(\000)g Fl(r)q Fs(\))310 2515 y Fj(\000)p Fi(2)364 2533 y Fs(,)15 b(where)e Fl(r)j Fs(=)e Fk(j)p Fl(z)r Fk(j)p Fs(.)21 b(The)14 b(next)h(theorem)f(\(see)h([P)o (o]\))e(for)i(a)f(pro)q(of)t(\))g(sho)o(ws)f(that)57 2603 y(the)j(a)o(v)o(erage)g(gro)o(wth)g(is)g(m)o(uc)o(h)f(lo)o(w)o(er) g(than)h(\(1)11 b Fk(\000)g Fl(r)q Fs(\))1065 2584 y Fj(\000)p Fi(2)1120 2603 y Fs(.)918 2770 y(71)p eop %%Page: 72 73 72 72 bop 57 192 a Fr(Theorem)16 b(A1.17)28 b Fd(Let)16 b Fl(f)21 b Fd(map)15 b Fm(D)26 b Fd(conformally)15 b(in)o(to)g Fm(C)9 b Fd(.)24 b(Then)15 b Fl(f)5 b Fs(\()p Fl(\020)t Fs(\))15 b(=)f(lim)1568 199 y Fh(r)q Fj(!)p Fi(1)1658 192 y Fl(f)5 b Fs(\()p Fl(r)q(\020)t Fs(\))16 b Fk(6)p Fs(=)57 261 y Fk(1)g Fd(exists)h(for)f(almost)g(all)g Fl(\020)h Fk(2)d Fm(T)6 b Fs(=)14 b Fl(@)s Fm(D)28 b Fd(and)15 b(for)i Fs(0)c Fk(\024)h Fl(r)i(<)d Fs(1)k Fd(one)f(has)504 366 y Fs(1)p 489 388 56 2 v 489 434 a(2)p Fl(\031)559 332 y Fe(Z)609 344 y Fi(2)p Fh(\031)586 445 y Fi(0)664 400 y Fk(j)p Fl(f)5 b Fs(\()p Fl(r)q(e)772 379 y Fh(i\022)821 400 y Fk(\000)11 b Fl(f)5 b Fs(\(0\))p Fk(j)977 379 y Fi(2)p Fh(=)p Fi(5)1041 400 y Fl(dt)14 b Fk(\024)f Fs(5)p Fk(j)p Fl(f)1219 379 y Fj(0)1234 400 y Fs(\(0\))p Fk(j)1311 379 y Fi(2)p Fh(=)p Fi(5)1388 400 y Fl(:)287 b Fs(\()p Fl(A)p Fs(1)p Fl(:)p Fs(7\))57 620 y Fr(A1.6)16 b Fs(W)l(e)g(conclude)g(our)f(brief)h(in)o(tro)q (duction)f(to)i(univ)m(alen)o(t)f(functions)f(with)i(the)f(pro)q(of)g (of)57 690 y(a)g(fundamen)o(tal)f(prop)q(ert)o(y)h(of)g Fl(S)677 697 y Fi(1)699 690 y Fs(.)22 b(In)17 b(order)e(to)i(do)f(this) g(w)o(e)g(recall)g(\([Re],)h(p.)k(163\))57 822 y Fr(Lemma)h(A1.18)i (\(Hurwitz\))30 b Fd(If)22 b(a)f(sequence)g Fs(\()p Fl(f)1068 829 y Fh(n)1096 822 y Fs(\))1115 829 y Fh(n)p Fj(2)p Fg(N)1216 822 y Fd(of)g(functions)g(holomorphic)e(in)57 892 y(a)f(region)g Fs(\012)g Fk(\032)f Fm(C)31 b Fd(con)o(v)o(erges)17 b(uniformly)g(on)h(compact)h(subsets)e(of)i Fs(\012)g Fd(to)g(a)g(non{constan)o(t)57 961 y(holomorphic)14 b(function)i Fl(f)28 b Fs(:)14 b(\012)f Fk(!)h Fm(C)29 b Fd(then)16 b(the)h(follo)o(wing)e(statemen)o(ts)h(hold)g(:)68 1031 y(\(a\))25 b(if)17 b(all)f(the)h(images)f Fl(f)545 1038 y Fh(n)572 1031 y Fs(\(\012\))h Fd(are)f(con)o(tained)g(in)g(a)h (\014xed)f(set)h Fl(A)g Fd(then)f Fl(f)5 b Fs(\(\012\))16 b Fk(\032)d Fl(A)d Fd(;)65 1101 y(\(b\))25 b(if)17 b(all)f(the)h(maps)e Fl(f)511 1108 y Fh(n)561 1101 y Fs(:)f(\012)f Fk(!)h Fm(C)29 b Fd(are)16 b(injectiv)o(e)h(then)f(so)g(is)g Fl(f)28 b Fs(:)14 b(\012)g Fk(!)f Fm(C)22 b Fd(;)71 1170 y(\(c\))j(if)17 b(all)f(the)h(maps)e Fl(f)511 1177 y Fh(n)561 1170 y Fs(:)f(\012)f Fk(!)h Fm(C)29 b Fd(are)16 b(lo)q(cally)h(biholomorphic,)c(then)k(so)f(is)g Fl(f)28 b Fs(:)13 b(\012)h Fk(!)g Fm(C)9 b Fd(.)57 1302 y Fs(This)15 b(is)i(the)f(ingredien)o(t)f(w)o(e)i(missed)e(for)h(the)h(pro)q(of)f (of)h(the)f(follo)o(wing)57 1434 y Fr(Theorem)30 b(A1.19)e Fl(S)518 1441 y Fi(1)568 1434 y Fd(endo)o(w)o(ed)f(with)h(the)h(top)q (ology)f(of)h(uniform)e(con)o(v)o(ergence)g(on)57 1504 y(compact)16 b(subsets)f(of)i Fm(D)28 b Fd(is)16 b(a)g(compact)h(top)q (ological)f(space.)57 1614 y Fp(Pr)m(o)m(of.)22 b Fs(An)o(y)e(sequence) f(\()p Fl(f)570 1621 y Fh(n)598 1614 y Fs(\))617 1621 y Fh(n)p Fj(2)p Fg(N)715 1614 y Fk(\032)f Fl(S)803 1621 y Fi(1)845 1614 y Fs(is)h(equicon)o(tin)o(uous)e(and)i(uniformly)e(b)q (ounded)i(on)57 1684 y(compact)h(subsets)g(of)h Fm(D)32 b Fs(b)o(y)21 b(Ko)q(eb)q(e)g(distortion)f(theorems.)34 b(Limit)20 b(functions)g(are)h(in)f Fl(S)1806 1691 y Fi(1)57 1753 y Fs(b)q(ecause)d(they)g(are)g(univ)m(alen)o(t)f(b)o(y)h (Hurwitz's)f(lemma)g(and)h(the)g(normalisation)e Fk(j)p Fl(f)1652 1735 y Fj(0)1647 1766 y Fh(n)1674 1753 y Fs(\(0\))p Fk(j)j Fs(for)57 1823 y(all)e Fl(n)e Fk(2)g Fm(N)p Fs(.)1524 b Fa(\003)57 1942 y Fr(Exercise)25 b(A1.20)c Fs(Pro)o(v)o(e)f(that)i (Theorem)f(A1.19)g(is)g(equiv)m(alen)o(t)g(to)h(the)g(follo)o(wing)e (\(see)57 2011 y([Mc]\))k(:)38 b(the)25 b(space)f(of)h Fp(al)s(l)f Fs(univ)m(alen)o(t)g(maps)g Fl(f)41 b Fs(:)27 b Fm(D)38 b Fk(!)p 1235 1971 36 2 v 27 w Fm(C)f Fs(is)24 b(compact)g(up)h(to)f(p)q(ost{)57 2081 y(comp)q(osition)13 b(with)i(automorphisms)d(of)p 848 2041 V 15 w Fm(C)g Fs(.)22 b(This)14 b(precisely)g(means)g(that)h(an)o(y)f(sequence)h(of) 57 2151 y(univ)m(alen)o(t)i(maps)g(con)o(tains)f(a)i(subsequence)f Fl(f)952 2158 y Fh(n)1004 2151 y Fs(:)e Fm(D)27 b Fk(!)p 1151 2111 V 16 w Fm(C)j Fs(suc)o(h)17 b(that)h Fl(M)1476 2158 y Fh(n)1515 2151 y Fk(\016)12 b Fl(f)1576 2158 y Fh(n)1622 2151 y Fs(con)o(v)o(erges)57 2221 y(to)20 b(a)f(univ)m(alen)o (t)g(map)g Fl(f)5 b Fs(,)21 b(uniformly)d(on)h(compact)g(subsets)g(of)g Fm(D)8 b Fs(,)24 b(for)19 b(some)g(sequence)g(of)57 2290 y(M\177)-25 b(obius)15 b(transforms)f Fl(M)529 2297 y Fh(n)570 2290 y Fk(2)h Fs(PGL)8 b(\(2)p Fl(;)p 796 2250 V 8 w Fm(C)k Fs(\).)57 2465 y Fr(A1.7)k Fs(Let)i Fl(f)29 b Fs(:)15 b Fm(C)27 b Fk(!)15 b Fm(C)29 b Fs(b)q(e)17 b(a)g Fk(C)670 2447 y Fi(1)710 2465 y Fs(orien)o(tation{preserving)d (di\013eomorphism.)21 b(Then)c(giv)o(en)57 2535 y(an)o(y)f(p)q(oin)o(t) g Fl(z)304 2542 y Fi(0)340 2535 y Fk(2)e Fm(C)29 b Fs(one)16 b(has)213 2650 y Fl(f)5 b Fs(\()p Fl(z)r Fs(\))15 b(=)f Fl(f)5 b Fs(\()p Fl(z)444 2657 y Fi(0)467 2650 y Fs(\))12 b(+)f Fl(f)572 2657 y Fh(z)595 2650 y Fs(\()p Fl(z)637 2657 y Fi(0)660 2650 y Fs(\)\()p Fl(z)j Fk(\000)d Fl(z)808 2657 y Fi(0)830 2650 y Fs(\))h(+)f Fl(f)937 2657 y Fi(\026)-22 b Fh(z)958 2650 y Fs(\()p Fl(z)1000 2657 y Fi(0)1023 2650 y Fs(\)\()s(\026)-28 b Fl(z)14 b Fk(\000)g Fs(\026)-28 b Fl(z)1171 2657 y Fi(0)1193 2650 y Fs(\))12 b(+)f(o)d(\()p Fk(j)p Fl(z)14 b Fk(\000)c Fl(z)1449 2657 y Fi(0)1472 2650 y Fk(j)p Fs(\))k Fl(;)156 b Fs(\()p Fl(A)p Fs(1)p Fl(:)p Fs(8\))918 2770 y(72)p eop %%Page: 73 74 73 73 bop 57 192 a Fs(where)189 330 y Fl(f)213 337 y Fh(z)250 330 y Fs(=)309 296 y(1)p 309 318 25 2 v 309 364 a(2)348 260 y Fe(\022)391 296 y Fl(@)s(f)p 391 318 59 2 v 392 364 a(@)s(x)467 330 y Fk(\000)11 b Fl(i)540 296 y(@)s(f)p 540 318 V 542 364 a(@)s(y)605 260 y Fe(\023)664 330 y Fl(;)49 b(f)753 337 y Fi(\026)-22 b Fh(z)788 330 y Fs(=)847 296 y(1)p 847 318 25 2 v 847 364 a(2)886 260 y Fe(\022)929 296 y Fl(@)s(f)p 929 318 59 2 v 930 364 a(@)s(x)1005 330 y Fs(+)11 b Fl(i)1078 296 y(@)s(f)p 1078 318 V 1080 364 a(@)s(y)1143 260 y Fe(\023)1201 330 y Fl(;)50 b Fs(\()p Fl(z)17 b Fs(=)c Fl(x)f Fs(+)f Fl(iy)r Fs(\))j Fl(:)133 b Fs(\()p Fl(A)p Fs(1)p Fl(:)p Fs(9\))57 481 y(Note)16 b(that)g(if)g Fl(f)21 b Fs(is)15 b(analytic)h(in)f Fl(z)691 488 y Fi(0)729 481 y Fs(then)g Fl(f)867 488 y Fi(\026)-22 b Fh(z)889 481 y Fs(\()p Fl(z)931 488 y Fi(0)953 481 y Fs(\))15 b(=)e(0)j(\(Cauc)o(h)o(y{Riemann\).)j(The)d (Jacobian)57 550 y(determinan)o(t)i(of)i Fl(f)26 b Fs(is)19 b Fl(J)24 b Fs(=)19 b Fk(j)p Fl(f)652 557 y Fh(z)675 550 y Fk(j)689 532 y Fi(2)725 550 y Fk(\000)13 b(j)p Fl(f)817 557 y Fi(\026)-22 b Fh(z)838 550 y Fk(j)852 532 y Fi(2)874 550 y Fs(.)32 b(Since)19 b Fl(f)26 b Fs(is)20 b(orien)o(tation{preserving)c(one)k(has)57 620 y Fl(J)e(>)c Fs(0,)i(th)o(us)f Fk(j)p Fl(f)358 627 y Fh(z)382 620 y Fk(j)e Fl(>)h Fk(j)p Fl(f)502 627 y Fi(\026)-22 b Fh(z)523 620 y Fk(j)p Fs(.)57 757 y Fr(De\014nition)20 b(A1.21)28 b Fd(The)16 b Fs(dilatation)g Fd(of)h Fl(f)22 b Fd(in)17 b Fl(z)1012 764 y Fi(0)1051 757 y Fd(is)572 904 y Fl(D)613 911 y Fh(f)639 904 y Fs(\()p Fl(z)681 911 y Fi(0)703 904 y Fs(\))e(:=)809 870 y Fk(j)p Fl(f)847 877 y Fh(z)870 870 y Fs(\()p Fl(z)912 877 y Fi(0)935 870 y Fs(\))p Fk(j)c Fs(+)g Fk(j)p Fl(f)1069 877 y Fi(\026)-22 b Fh(z)1090 870 y Fs(\()p Fl(z)1132 877 y Fi(0)1155 870 y Fs(\))p Fk(j)p 809 892 380 2 v 809 938 a(j)p Fl(f)847 945 y Fh(z)870 938 y Fs(\()p Fl(z)912 945 y Fi(0)935 938 y Fs(\))p Fk(j)11 b(\000)g(j)p Fl(f)1069 945 y Fi(\026)-22 b Fh(z)1090 938 y Fs(\()p Fl(z)1132 945 y Fi(0)1155 938 y Fs(\))p Fk(j)1208 904 y(\025)14 b Fs(1)g Fl(:)350 b Fs(\()p Fl(A)p Fs(1)p Fl(:)p Fs(10\))57 1087 y(Note)17 b(that)g(if)g Fl(f)22 b Fs(is)16 b(conformal)f(then)i Fl(D)813 1094 y Fh(f)853 1087 y Fs(=)c(1.)156 1157 y(Here)25 b(is)f(a)g(geometric)g (in)o(terpretation)f(of)i(the)f(meaning)f(of)i(the)g(dilatation)e(:)38 b(the)57 1227 y(di\013eren)o(tial)25 b Fl(d)-8 b(f)5 b Fs(\()p Fl(z)405 1234 y Fi(0)428 1227 y Fs(\))27 b(maps)e(a)i(circle) f(in)g(the)h(tangen)o(t)f(space)g Fl(T)1333 1234 y Fh(z)1352 1239 y Fc(0)1374 1227 y Fm(C)39 b Fs(in)o(to)26 b(an)g(ellipse)g(in)57 1297 y Fl(T)86 1306 y Fh(f)t Fi(\()p Fh(z)144 1311 y Fc(0)163 1306 y Fi(\))181 1297 y Fm(C)9 b Fs(.)29 b(The)18 b(dilatation)f(measures)f(the)j(distorsion)d(since)h(it)h(is)g(the)g (ratio)f(of)h(the)h(ma)s(jor)57 1367 y(semiaxis)12 b(amd)g(the)i(minor) e(semiaxis.)20 b(Indeed)12 b(applying)h(\(A1.8\))h(to)f(an)g (in\014nitesimal)f(circle)57 1436 y(\001)p Fl(z)17 b Fs(=)f Fl("e)240 1418 y Fh(i\022)295 1436 y Fs(cen)o(tered)h(at)h Fl(z)579 1443 y Fi(0)620 1436 y Fs(one)f(\014nds)g(an)g (in\014nitesimal)f(ellipse)h(cen)o(tered)g(at)i Fl(f)5 b Fs(\()p Fl(z)1671 1443 y Fi(0)1694 1436 y Fs(\))18 b(with)57 1506 y(ma)s(jor)d(semiaxis)h([)p Fk(j)p Fl(f)453 1513 y Fh(z)476 1506 y Fk(j)10 b(\000)h(j)p Fl(f)590 1513 y Fi(\026)-22 b Fh(z)611 1506 y Fk(j)p Fs(])639 1488 y Fj(\000)p Fi(1)693 1506 y Fl(")16 b Fs(and)g(minor)f(semiaxis)h ([)p Fk(j)p Fl(f)1224 1513 y Fh(z)1247 1506 y Fk(j)11 b Fs(+)g Fk(j)p Fl(f)1362 1513 y Fi(\026)-22 b Fh(z)1383 1506 y Fk(j)p Fs(])1411 1488 y Fj(\000)p Fi(1)1464 1506 y Fl(")p Fs(.)156 1576 y(The)19 b Fp(maximal)i(dilatation)g Fs(of)e Fl(f)25 b Fs(on)19 b Fm(C)32 b Fs(is)18 b Fl(D)1020 1583 y Fh(f)1065 1576 y Fs(=)g(sup)1197 1588 y Fh(z)q Fj(2)p Fg(C)1280 1576 y Fl(D)1321 1583 y Fh(f)1347 1576 y Fs(\()p Fl(z)r Fs(\).)32 b(If)19 b Fl(D)1549 1583 y Fh(f)1593 1576 y Fl(<)g Fs(+)p Fk(1)p Fs(,)g(let)57 1646 y Fl(\024)86 1653 y Fh(f)125 1646 y Fs(=)14 b(\()p Fl(D)238 1653 y Fh(f)275 1646 y Fk(\000)d Fs(1\))p Fl(=)p Fs(\()p Fl(D)454 1653 y Fh(f)492 1646 y Fs(+)f(1\).)23 b(Then)16 b(one)g(has)938 1622 y Fj(j)p Fh(f)971 1627 y Fc(\026)-19 b Fb(z)990 1622 y Fj(j)p 938 1635 64 2 v 938 1663 a(j)p Fh(f)969 1668 y Fb(z)990 1663 y Fj(j)1021 1646 y Fk(\024)14 b Fl(\024)1103 1653 y Fh(f)1142 1646 y Fl(<)g Fs(1.)57 1783 y Fr(De\014nition)20 b(A1.22)28 b Fl(f)22 b Fd(is)16 b Fs(quasiconformal)f Fd(if)i Fl(D)1025 1790 y Fh(f)1065 1783 y Fl(<)c Fs(+)p Fk(1)p Fd(,)j(i.e.)22 b Fl(\024)1351 1790 y Fh(f)1390 1783 y Fl(<)14 b Fs(1)p Fd(.)57 1919 y Fs(Clearly)i(if)g Fl(f)23 b Fs(is)16 b(conformal)f(then)i Fl(D)758 1926 y Fh(f)797 1919 y Fs(=)d(1,)i Fl(\024)934 1926 y Fh(f)974 1919 y Fs(=)d(0.)156 1989 y(W)l(e)25 b(w)o(an)o(t)e(no)o(w)g(to)h(extend)h(the)f(notion)f(of)h (quasiconformal)e(map)h(to)i(homeomor-)57 2059 y(phisms.)20 b(W)l(e)c(will)h(follo)o(w)e(the)i(geometric)f(approac)o(h)f(outlined)h (in)g([Ah2].)57 2165 y Fr(Exercise)k(A1.23)15 b Fs(Giv)o(en)h(t)o(w)o (o)g(rectangles)g Fl(R)954 2172 y Fi(1)992 2165 y Fs(and)g Fl(R)1127 2172 y Fi(2)1165 2165 y Fs(resp)q(ectiv)o(ely)h(with)f(sides) g Fl(a)1696 2172 y Fi(1)1732 2165 y Fk(\024)e Fl(b)1806 2172 y Fi(1)57 2235 y Fs(and)j Fl(a)181 2242 y Fi(2)220 2235 y Fk(\024)f Fl(b)296 2242 y Fi(2)319 2235 y Fs(,)i(sho)o(w)f(that) i(there)f(exists)g(a)g(conformal)f(map)g(of)h Fl(R)1338 2242 y Fi(1)1379 2235 y Fs(on)o(to)f Fl(R)1530 2242 y Fi(2)1571 2235 y Fs(whic)o(h)g(maps)57 2304 y(v)o(ertices)f(on)g(v)o (ertices)g(if)h(and)f(only)g(if)790 2284 y Fh(a)812 2289 y Fc(1)p 790 2293 42 2 v 792 2321 a Fh(b)810 2326 y Fc(1)851 2304 y Fs(=)909 2284 y Fh(a)931 2289 y Fc(2)p 909 2293 V 911 2321 a Fh(b)929 2326 y Fc(2)956 2304 y Fs(.)57 2441 y Fr(De\014nition)k(A1.24)27 b Fd(A)17 b Fs(quadrilateral)d Fl(Q)p Fs(\()p Fl(z)926 2448 y Fi(1)950 2441 y Fl(;)8 b(z)995 2448 y Fi(2)1017 2441 y Fl(;)g(z)1062 2448 y Fi(3)1085 2441 y Fl(;)g(z)1130 2448 y Fi(4)1153 2441 y Fs(\))16 b Fd(is)g(a)g(Jordan)e(domain)h(in)g Fm(C)28 b Fd(with)57 2511 y(four)19 b(distinguished)f(b)q(oundary)i(p)q(oin)o (ts)f Fl(z)875 2518 y Fi(1)898 2511 y Fl(;)8 b(z)943 2518 y Fi(2)965 2511 y Fl(;)g(z)1010 2518 y Fi(3)1033 2511 y Fl(;)g(z)1078 2518 y Fi(4)1101 2511 y Fd(.)33 b(Its)20 b Fs(mo)q(dulus)f Fl(M)5 b Fs(\()p Fl(Q)p Fs(\))22 b Fd(is)e(the)h(ratio)57 2581 y Fl(a=b)f Fd(of)h(the)f(lengths)f Fl(a)h(<)f(b)i Fd(of)f(the)g(sides)f(of)h(an)o(y)g(rectangle)g Fl(R)g Fd(whic)o(h)f(is)g(the)i(conformal)57 2650 y(image)16 b(of)g Fl(Q)h Fd(and)f(whose)g(v)o(ertices)g(are)h(image)e(of)i(the)g (distinguished)d(p)q(oin)o(ts.)918 2770 y Fs(73)p eop %%Page: 74 75 74 74 bop 57 192 a Fs(Note)18 b(that)h(the)f(mo)q(dulus)e(of)i(a)g (quadrilateral)e(is)i(a)f(conformal)g(in)o(v)m(arian)o(t.)24 b(Th)o(us)17 b(one)h(can)57 261 y(use)f(its)i(v)m(ariation)f(under)f(a) h(homeomorphism)d(to)j(measure)f(the)i(lac)o(k)f(of)g(conformalit)o(y)f (of)57 331 y(a)f(map.)57 477 y Fr(De\014nition)i(A1.25)27 b Fd(Let)15 b Fl(f)28 b Fs(:)14 b Fm(C)26 b Fk(!)13 b Fm(C)27 b Fd(b)q(e)14 b(an)g(orien)o(tation{preserving)d (homeomorphism.)57 547 y(Its)28 b Fs(maximal)f(dilatation)h Fd(is)f Fl(D)698 554 y Fh(f)758 547 y Fs(:=)33 b(sup)919 564 y Fh(Q)p Fj(\032)p Fg(C)14 b Fh(;)7 b(Q)g Fd(quadrilateral)1365 523 y Fh(M)t Fi(\()p Fh(f)t Fi(\()p Fh(Q)p Fi(\)\))p 1365 535 159 2 v 1392 564 a Fh(M)t Fi(\()p Fh(Q)p Fi(\))1530 547 y Fd(.)57 b(Let)29 b Fl(\024)1731 554 y Fh(f)1790 547 y Fs(=)57 616 y(\()p Fl(D)117 623 y Fh(f)151 616 y Fk(\000)8 b Fs(1\))p Fl(=)p Fs(\()p Fl(D)327 623 y Fh(f)360 616 y Fs(+)g(1\))p Fd(.)21 b Fl(f)g Fd(is)14 b(a)h Fs(quasiconformal)e(homeomorphis)o(m)f Fd(if)j Fl(D)1418 623 y Fh(f)1444 616 y Fl(;)8 b Fs(+)p Fk(1)p Fd(,)14 b(i.e.)22 b Fl(\024)1698 623 y Fh(f)1737 616 y Fl(<)14 b Fs(1)p Fd(.)57 760 y Fr(Exercise)20 b(A1.26)15 b Fs(Pro)o(v)o(e)h(that)h Fl(f)22 b Fs(is)16 b(conformal)g(if)g(and)g (only)g(if)h Fl(D)1360 767 y Fh(f)1400 760 y Fs(=)d(1.)57 906 y Fr(Theorem)k(A1.27)28 b Fd(A)19 b(quasiconformal)c(homeomorphism) g Fl(f)23 b Fd(with)18 b(maximal)f(dilatation)57 975 y Fl(D)98 982 y Fh(f)135 975 y Fd(is)11 b(almost)g(ev)o(erywhere)g (di\013eren)o(tiable)f(and)h(at)h(eac)o(h)f(p)q(oin)o(t)g Fl(z)1287 982 y Fi(0)1321 975 y Fd(where)g Fl(f)17 b Fd(is)11 b(di\013eren)o(tiable)57 1045 y(one)16 b(has)789 1096 y Fk(j)p Fl(f)829 1103 y Fi(\026)-22 b Fh(z)850 1096 y Fs(\()p Fl(z)892 1103 y Fi(0)915 1096 y Fs(\))p Fk(j)p 789 1118 160 2 v 789 1164 a(j)p Fl(f)827 1171 y Fh(z)850 1164 y Fs(\()p Fl(z)892 1171 y Fi(0)915 1164 y Fs(\))p Fk(j)968 1129 y(\024)13 b Fl(\024)1049 1136 y Fh(f)1089 1129 y Fl(:)57 1297 y Fs(P)o(erhaps)c(the)j(most)e(useful)h (result)f(in)h(the)g(theory)h(of)f(quasiconformal)e(maps)h(is)h(the)h (follo)o(wing)57 1367 y(existence)20 b(theorem)g(also)g(kno)o(wn)f(as)h (Measurable)f(Riemann)g(Mapping)f(Theorem)i([Ah2,)57 1436 y(p.98])57 1582 y Fr(Theorem)11 b(A1.28)28 b Fd(Let)12 b Fl(\026)g Fd(b)q(e)f(a)h(complex{v)m(alued)e(measurable)g(function)h (with)g Fk(k)p Fl(\026)p Fk(k)1681 1589 y Fj(1)1737 1582 y Fl(<)j Fs(1)p Fd(.)57 1652 y(There)i(exists)g(a)h(quasiconformal)d (mapping)h Fl(f)23 b Fd(suc)o(h)15 b(that)606 1790 y Fl(f)632 1797 y Fi(\026)-22 b Fh(z)668 1790 y Fs(=)13 b Fl(\026)p Fs(\()p Fl(z)r Fs(\))p Fl(f)837 1797 y Fh(z)875 1790 y Fd(almost)j(ev)o(erywhere)385 b Fs(\()p Fl(A)p Fs(1)p Fl(:)p Fs(11\))57 1927 y Fd(and)16 b Fl(f)22 b Fd(lea)o(v)o(es)16 b(the)g(p)q(oin)o(ts)g Fs(0)p Fl(;)8 b Fs(1)p Fl(;)g Fk(1)17 b Fd(\014xed.)57 2071 y Fs(The)f(equation)g (\(A1.11\))i(is)e(also)g(kno)o(wn)f(as)i Fp(Beltr)m(ami)h(e)m(quation)p Fs(.)156 2143 y(Quasiconformal)j(maps)h(ha)o(v)o(e)h(b)q(een)h(in)o (tro)q(duced)e(in)g(the)i(sub)s(ject)f(of)h(holomorphic)57 2213 y(dynamics)g(b)o(y)g(Dennis)g(Sulliv)m(an)g(and)h(Adrien)f(Douady) h(and)f(ha)o(v)o(e)h(rapidly)e(b)q(ecome)i(a)57 2283 y(standard)15 b(to)q(ol.)22 b(What)17 b(w)o(e)f(will)g(need)g(in)h (Chapter)e(3)i(is)f(the)h(follo)o(wing)57 2429 y Fr(Theorem)g(A1.29)i (\(Douady{Hubbard)e(:)27 b(stabilit)n(y)c(of)18 b(the)i(quadratic)h(p)r (olyno-)57 2503 y(mial\))29 b Fd(Let)12 b Fl(P)331 2510 y Fh(\025)357 2503 y Fs(\()p Fl(z)r Fs(\))j(=)e Fl(\025)524 2448 y Fe(\020)554 2503 y Fl(z)h Fk(\000)647 2484 y Fh(z)667 2469 y Fc(2)p 647 2492 40 2 v 657 2520 a Fi(2)693 2448 y Fe(\021)733 2503 y Fd(and)d(let)g Fl(F)c Fs(\()p Fl(z)r Fs(\))15 b(=)f Fl(P)1093 2510 y Fh(\025)1119 2503 y Fs(\()p Fl(z)r Fs(\)+)p Fl( )r Fs(\()p Fl(z)r Fs(\))f Fd(where)e Fl( )i Fd(is)e(holomorphic)57 2581 y(and)16 b(b)q(ounded)g(in)g(the)h (disk)f Fm(D)639 2588 y Fi(3)665 2581 y Fd(,)h Fl( )r Fs(\()p Fl(z)r Fs(\))e(=)861 2543 y Fe(P)914 2556 y Fj(1)914 2596 y Fh(n)p Fi(=2)1000 2581 y Fl( )1032 2588 y Fh(n)1059 2581 y Fl(z)1084 2563 y Fh(n)1129 2581 y Fd(\(i.e.)23 b Fl( )r Fs(\(0\))15 b(=)f Fl( )1434 2563 y Fj(0)1448 2581 y Fs(\(0\))h(=)f(0)p Fd(\).)23 b(Assume)57 2650 y(that)18 b Fs(sup)241 2663 y Fh(z)q Fj(2)p Fg(D)310 2668 y Fc(3)344 2650 y Fk(j)p Fl( )r Fs(\()p Fl(z)r Fs(\))p Fk(j)f Fl(<)f Fs(10)591 2632 y Fj(\000)p Fi(2)644 2650 y Fd(.)26 b(Then)17 b(there)h(exists)g(a)g(quasiconformal)e (homeomorphism)918 2770 y Fs(74)p eop %%Page: 75 76 75 75 bop 57 192 a Fl(h)18 b Fd(suc)o(h)f(that)h(on)g(the)g(disk)g Fm(D)624 199 y Fi(2)668 192 y Fd(one)g(has)f Fl(h)880 173 y Fj(\000)p Fi(1)933 192 y Fl(F)7 b(h)17 b Fs(=)f Fl(P)1105 199 y Fh(\025)1131 192 y Fd(.)27 b(If)18 b Fl( )j Fd(is)d(small)f(enough)g(then)h Fl(h)g Fd(is)57 261 y(near)e(the)g(iden)o(tit)o(y)g(in)g(the)h Fk(C)610 243 y Fi(0)649 261 y Fd(top)q(ology)l(.)57 655 y Fo(A2.)26 b(Con)n(tin)n(ued)c(F)-5 b(ractions)57 761 y Fs(In)26 b(this)g(app)q(endix)f(w)o(e)i(recall)e(some)h(elemen)o(tary)g(facts)g (on)g(standard)f(real)h(con)o(tin)o(ued)57 830 y(fractions)17 b(\(w)o(e)i(refer)f(to)h([MMY],)e(and)h(references)g(therein,)g(for)g (more)f(general)h(con)o(tin)o(ued)57 900 y(fractions\).)156 970 y(W)l(e)f(will)f(consider)f(the)i(iteration)f(of)h(the)g(Gauss)e (map)745 1095 y Fl(A)g Fs(:)e(\(0)p Fl(;)8 b Fs(1\))15 b Fk(7!)e Fs([0)p Fl(;)8 b Fs(1])14 b Fl(;)549 b Fs(\()p Fl(A)p Fs(2)p Fl(:)p Fs(1\))57 1221 y(de\014ned)15 b(b)o(y)696 1308 y Fl(A)p Fs(\()p Fl(x)p Fs(\))g(=)889 1274 y(1)p 873 1297 59 2 v 887 1342 a Fl(x)948 1308 y Fk(\000)998 1238 y Fe(\024)1063 1274 y Fs(1)p 1046 1297 V 1061 1342 a Fl(x)1127 1238 y Fe(\025)1176 1308 y Fl(:)499 b Fs(\()p Fl(A)p Fs(2)p Fl(:)p Fs(2\))57 1428 y Fl(A)17 b Fs(is)f(piecewise)g (analytic)g(with)h(branc)o(hes)478 1565 y Fl(A)p Fs(\()p Fl(x)p Fs(\))e(=)f Fl(x)677 1544 y Fj(\000)p Fi(1)742 1565 y Fk(\000)c Fl(n)28 b Fs(if)943 1531 y(1)p 898 1553 116 2 v 898 1599 a Fl(n)11 b Fs(+)g(1)1033 1565 y Fl(<)j(x)g Fk(\024)1189 1531 y Fs(1)p 1187 1553 30 2 v 1187 1599 a Fl(n)1236 1565 y(;)8 b(n)14 b Fk(\025)g Fs(1)g Fl(:)57 1740 y Fr(Exercise)22 b(A2.1)c Fs(Pro)o(v)o(e)g(that)h Fl(A)711 1722 y Fj(\003)734 1740 y Fs(\()p Fl(\032)p Fs(\()p Fl(x)p Fs(\))p Fl(dx)p Fs(\))g(=)e Fl(\032)p Fs(\()p Fl(x)p Fs(\))p Fl(dx)j Fs(where)f Fl(\032)p Fs(\()p Fl(x)p Fs(\))f(=)f([\(1)12 b(+)g Fl(x)p Fs(\))c(log)i(2])1761 1722 y Fj(\000)p Fi(1)1815 1740 y Fs(,)57 1810 y(i.e.)21 b Fl(\032)c Fs(is)f(an)g(in)o(v)m(arian)o(t)f(probabilit)o(y)g(densit)o (y)h(for)g(the)h(Gauss)e(map.)57 1915 y(Let)541 1999 y Fl(G)f Fs(=)652 1924 y Fk(p)p 694 1924 25 2 v 41 x Fs(5)d(+)f(1)p 652 1987 153 2 v 716 2033 a(2)824 1999 y Fl(;)22 b(g)15 b Fs(=)f Fl(G)991 1978 y Fj(\000)p Fi(1)1059 1999 y Fs(=)1117 1924 y Fk(p)p 1159 1924 25 2 v 41 x Fs(5)d Fk(\000)f Fs(1)p 1117 1987 153 2 v 1181 2033 a(2)1289 1999 y Fl(:)57 2108 y Fs(T)l(o)17 b(eac)o(h)h Fl(x)e Fk(2)g Fm(R)9 b Fk(n)j Fm(Q)19 b Fs(w)o(e)f(asso)q(ciate)f(a)h(con)o (tin)o(ued)e(fraction)i(expansion)f(b)o(y)g(iterating)h Fl(A)g Fs(as)57 2178 y(follo)o(ws.)j(Let)797 2226 y Fl(x)825 2233 y Fi(0)862 2226 y Fs(=)14 b Fl(x)d Fk(\000)g Fs([)p Fl(x)p Fs(])j Fl(;)799 2310 y(a)825 2317 y Fi(0)862 2310 y Fs(=)g([)p Fl(x)p Fs(])g Fl(;)1689 2268 y Fs(\()p Fl(A)p Fs(2)p Fl(:)p Fs(3\))57 2400 y(then)i(one)g(ob)o(viously)g(has)g Fl(x)e Fs(=)f Fl(a)691 2407 y Fi(0)725 2400 y Fs(+)e Fl(x)803 2407 y Fi(0)826 2400 y Fs(.)22 b(W)l(e)17 b(no)o(w)e(de\014ne) h(inductiv)o(ely)g(for)h(all)f Fl(n)d Fk(\025)h Fs(0)737 2513 y Fl(x)765 2520 y Fh(n)p Fi(+1)857 2513 y Fs(=)f Fl(A)p Fs(\()p Fl(x)993 2520 y Fh(n)1021 2513 y Fs(\))i Fl(;)739 2623 y(a)765 2630 y Fh(n)p Fi(+1)857 2623 y Fs(=)909 2553 y Fe(\024)957 2589 y Fs(1)p 941 2612 56 2 v 941 2657 a Fl(x)969 2664 y Fh(n)1003 2553 y Fe(\025)1043 2623 y Fk(\025)f Fs(1)f Fl(;)1689 2583 y Fs(\()p Fl(A)p Fs(2)p Fl(:)p Fs(4\))918 2770 y(75)p eop %%Page: 76 77 76 76 bop 57 192 a Fs(th)o(us)719 261 y Fl(x)747 241 y Fj(\000)p Fi(1)747 274 y Fh(n)815 261 y Fs(=)13 b Fl(a)893 268 y Fh(n)p Fi(+1)982 261 y Fs(+)e Fl(x)1060 268 y Fh(n)p Fi(+1)1152 261 y Fl(:)523 b Fs(\()p Fl(A)p Fs(2)p Fl(:)p Fs(5\))57 353 y(Therefore)16 b(w)o(e)g(ha)o(v)o(e)135 452 y Fl(x)e Fs(=)g Fl(a)256 459 y Fi(0)289 452 y Fs(+)d Fl(x)367 459 y Fi(0)404 452 y Fs(=)i Fl(a)482 459 y Fi(0)516 452 y Fs(+)640 418 y(1)p 572 441 161 2 v 572 486 a Fl(a)598 493 y Fi(1)632 486 y Fs(+)e Fl(x)710 493 y Fi(1)752 452 y Fs(=)j Fl(:)8 b(:)g(:)14 b Fs(=)f Fl(a)955 459 y Fi(0)989 452 y Fs(+)1298 418 y(1)p 1045 441 532 2 v 1045 512 a Fl(a)1071 519 y Fi(1)1105 512 y Fs(+)1353 479 y(1)p 1160 501 411 2 v 1160 582 a Fl(a)1186 589 y Fi(2)1220 582 y Fs(+)1273 547 y(.)1292 562 y(.)1311 577 y(.)1339 582 y(+)1467 548 y(1)p 1395 570 170 2 v 1395 616 a Fl(a)1421 623 y Fh(n)1459 616 y Fs(+)e Fl(x)1537 623 y Fh(n)1597 452 y Fl(;)78 b Fs(\()p Fl(A)p Fs(2)p Fl(:)p Fs(6\))57 693 y(and)16 b(w)o(e)g(will)g(write)685 762 y Fl(x)f Fs(=)e([)p Fl(a)820 769 y Fi(0)843 762 y Fl(;)8 b(a)891 769 y Fi(1)914 762 y Fl(;)g(:)g(:)g(:)g(;)g(a)1050 769 y Fh(n)1078 762 y Fl(;)g(:)g(:)g(:)p Fs(])14 b Fl(:)489 b Fs(\()p Fl(A)p Fs(2)p Fl(:)p Fs(7\))57 854 y(The)16 b(n)o(th-con)o(v)o(ergen)o(t)e(is)i(de\014ned)g(b)o(y)412 930 y Fl(p)437 937 y Fh(n)p 412 952 53 2 v 413 998 a Fl(q)435 1005 y Fh(n)484 964 y Fs(=)e([)p Fl(a)577 971 y Fi(0)599 964 y Fl(;)8 b(a)647 971 y Fi(1)670 964 y Fl(;)g(:)g(:)g(:)g(;)g(a)806 971 y Fh(n)834 964 y Fs(])14 b(=)f Fl(a)940 971 y Fi(0)974 964 y Fs(+)1225 930 y(1)p 1030 952 416 2 v 1030 1024 a Fl(a)1056 1031 y Fi(1)1090 1024 y Fs(+)1280 991 y(1)p 1145 1013 294 2 v 1145 1094 a Fl(a)1171 1101 y Fi(2)1205 1094 y Fs(+)1258 1059 y(.)1277 1074 y(.)1296 1089 y(.)1324 1094 y(+)1394 1060 y(1)p 1380 1082 54 2 v 1380 1128 a Fl(a)1406 1135 y Fh(n)1465 964 y Fl(:)210 b Fs(\()p Fl(A)p Fs(2)p Fl(:)p Fs(8\))57 1240 y Fr(Exercise)16 b(A2.2)e Fs(Sho)o(w)e(that)i(the)g(n)o(umerators) d Fl(p)1011 1247 y Fh(n)1052 1240 y Fs(and)i(denominators)e Fl(q)1475 1247 y Fh(n)1516 1240 y Fs(are)j(recursiv)o(ely)57 1310 y(determined)h(b)o(y)564 1380 y Fl(p)589 1387 y Fj(\000)p Fi(1)656 1380 y Fs(=)f Fl(q)731 1387 y Fj(\000)p Fi(2)799 1380 y Fs(=)f(1)28 b Fl(;)49 b(p)992 1387 y Fj(\000)p Fi(2)1060 1380 y Fs(=)13 b Fl(q)1134 1387 y Fj(\000)p Fi(1)1202 1380 y Fs(=)h(0)27 b Fl(;)368 b Fs(\()p Fl(A)p Fs(2)p Fl(:)p Fs(9\))57 1471 y(and)16 b(for)g(all)g Fl(n)e Fk(\025)f Fs(0)k(one)f(has)709 1525 y Fl(p)734 1532 y Fh(n)775 1525 y Fs(=)e Fl(a)854 1532 y Fh(n)881 1525 y Fl(p)906 1532 y Fh(n)p Fj(\000)p Fi(1)995 1525 y Fs(+)d Fl(p)1070 1532 y Fh(n)p Fj(\000)p Fi(2)1162 1525 y Fl(;)712 1610 y(q)734 1617 y Fh(n)775 1610 y Fs(=)j Fl(a)854 1617 y Fh(n)881 1610 y Fl(q)903 1617 y Fh(n)p Fj(\000)p Fi(1)993 1610 y Fs(+)c Fl(q)1064 1617 y Fh(n)p Fj(\000)p Fi(2)1156 1610 y Fl(:)1664 1567 y Fs(\()p Fl(A)p Fs(2)p Fl(:)p Fs(10\))57 1724 y Fr(Exercise)20 b(A2.3)c Fs(Sho)o(w)f(that)i(for)f(all)h Fl(n)c Fk(\025)h Fs(0)i(one)h(has)878 1823 y Fl(x)d Fs(=)979 1789 y Fl(p)1004 1796 y Fh(n)1042 1789 y Fs(+)d Fl(p)1117 1796 y Fh(n)p Fj(\000)p Fi(1)1195 1789 y Fl(x)1223 1796 y Fh(n)p 979 1812 272 2 v 982 1857 a Fl(q)1004 1864 y Fh(n)1042 1857 y Fs(+)g Fl(q)1114 1864 y Fh(n)p Fj(\000)p Fi(1)1192 1857 y Fl(x)1220 1864 y Fh(n)1271 1823 y Fl(;)379 b Fs(\()p Fl(A)p Fs(2)p Fl(:)p Fs(11\))851 1951 y Fl(x)879 1958 y Fh(n)920 1951 y Fs(=)14 b Fk(\000)1069 1918 y Fl(q)1091 1925 y Fh(n)1118 1918 y Fl(x)e Fk(\000)e Fl(p)1232 1925 y Fh(n)p 1018 1940 293 2 v 1018 1986 a Fl(q)1040 1993 y Fh(n)p Fj(\000)p Fi(1)1118 1986 y Fl(x)i Fk(\000)e Fl(p)1232 1993 y Fh(n)p Fj(\000)p Fi(1)1330 1951 y Fl(;)320 b Fs(\()p Fl(A)p Fs(2)p Fl(:)p Fs(12\))541 2055 y Fl(q)563 2062 y Fh(n)590 2055 y Fl(p)615 2062 y Fh(n)p Fj(\000)p Fi(1)704 2055 y Fk(\000)11 b Fl(p)779 2062 y Fh(n)806 2055 y Fl(q)828 2062 y Fh(n)p Fj(\000)p Fi(1)920 2055 y Fs(=)j(\()p Fk(\000)p Fs(1\))1075 2034 y Fh(n)1130 2055 y Fl(:)520 b Fs(\()p Fl(A)p Fs(2)p Fl(:)p Fs(13\))57 2189 y(Note)15 b(that)f Fl(q)302 2196 y Fh(n)p Fi(+1)393 2189 y Fl(>)g(q)468 2196 y Fh(n)509 2189 y Fl(>)g Fs(0)g(and)f(that)h(the)h(sequence)f(of)g (the)g(n)o(umerators)d Fl(p)1509 2196 y Fh(n)1551 2189 y Fs(has)i(the)h(same)57 2259 y(constan)o(t)19 b(sign)g(of)h Fl(x)p Fs(.)33 b(Equation)20 b(\(A2.13\))h(implies)d(also)i(that)g(for) g(all)f Fl(k)i Fk(\025)f Fs(0)g(and)f(for)h(all)57 2329 y Fl(x)14 b Fk(2)g Fm(R)8 b Fk(n)j Fm(Q)17 b Fs(one)g(has)471 2307 y Fh(p)492 2312 y Fc(2)p Fb(k)p 471 2317 60 2 v 472 2346 a Fh(q)490 2351 y Fc(2)p Fb(k)550 2329 y Fl(<)c(x)i(<)703 2304 y Fh(p)724 2309 y Fc(2)p Fb(k)p Fc(+1)p 703 2317 102 2 v 704 2346 a Fh(q)722 2351 y Fc(2)p Fb(k)p Fc(+1)811 2329 y Fs(.)156 2399 y(Let)197 2498 y Fl(\014)225 2505 y Fh(n)266 2498 y Fs(=)e(\005)355 2478 y Fh(n)355 2510 y(i)p Fi(=0)423 2498 y Fl(x)451 2505 y Fh(i)482 2498 y Fs(=)g(\()p Fk(\000)p Fs(1\))636 2478 y Fh(n)664 2498 y Fs(\()p Fl(q)705 2505 y Fh(n)732 2498 y Fl(x)f Fk(\000)f Fl(p)847 2505 y Fh(n)874 2498 y Fs(\))50 b(for)16 b Fl(n)e Fk(\025)g Fs(0)p Fl(;)57 b Fs(and)16 b Fl(\014)1337 2505 y Fj(\000)p Fi(1)1405 2498 y Fs(=)d(1)28 b Fl(:)140 b Fs(\()p Fl(A)p Fs(2)p Fl(:)p Fs(14\))57 2598 y(Then)16 b Fl(x)215 2605 y Fh(n)256 2598 y Fs(=)e Fl(\014)337 2605 y Fh(n)364 2598 y Fl(\014)395 2576 y Fj(\000)p Fi(1)392 2611 y Fh(n)p Fj(\000)p Fi(1)487 2598 y Fs(and)i Fl(\014)612 2605 y Fh(n)p Fj(\000)p Fi(2)704 2598 y Fs(=)d Fl(a)782 2605 y Fh(n)810 2598 y Fl(\014)838 2605 y Fh(n)p Fj(\000)p Fi(1)927 2598 y Fs(+)e Fl(\014)1005 2605 y Fh(n)1032 2598 y Fs(.)918 2770 y(76)p eop %%Page: 77 78 77 77 bop 57 192 a Fr(Prop)r(osition)18 b(A2.4)45 b Fd(F)l(or)16 b(all)g Fl(x)e Fk(2)g Fm(R)8 b Fk(n)j Fm(Q)17 b Fd(and)f(for)h(all)f Fl(n)d Fk(\025)h Fs(1)j Fd(one)f(has)79 261 y(\(i\))125 b Fk(j)p Fl(q)292 268 y Fh(n)319 261 y Fl(x)12 b Fk(\000)e Fl(p)433 268 y Fh(n)461 261 y Fk(j)j Fs(=)692 242 y(1)p 547 250 317 2 v 547 281 a Fl(q)569 288 y Fh(n)p Fi(+1)658 281 y Fs(+)e Fl(q)730 288 y Fh(n)757 281 y Fl(x)785 288 y Fh(n)p Fi(+1)869 261 y Fd(,)16 b(so)h(that)1074 242 y Fs(1)p 1074 250 25 2 v 1074 284 a(2)1119 261 y Fl(<)d(\014)1200 268 y Fh(n)1227 261 y Fl(q)1249 268 y Fh(n)p Fi(+1)1341 261 y Fl(<)f Fs(1)d Fd(;)65 350 y(\(ii\))25 b Fl(\014)184 357 y Fh(n)225 350 y Fk(\024)14 b Fl(g)304 331 y Fh(n)347 350 y Fd(and)i Fl(q)466 357 y Fh(n)507 350 y Fk(\025)566 330 y Fs(1)p 566 338 V 566 372 a(2)597 350 y Fl(G)636 331 y Fh(n)p Fj(\000)p Fi(1)731 350 y Fd(.)57 485 y Fp(Pr)m(o)m(of.)k Fs(Using)c(\(A2.11\))h(one)f(has)300 635 y Fk(j)p Fl(q)336 642 y Fh(n)363 635 y Fl(x)c Fk(\000)f Fl(p)478 642 y Fh(n)505 635 y Fk(j)i Fs(=)585 563 y Fe(\014)585 593 y(\014)585 622 y(\014)585 652 y(\014)602 635 y Fl(q)624 642 y Fh(n)657 601 y Fl(p)682 608 y Fh(n)p Fi(+1)771 601 y Fs(+)d Fl(p)845 608 y Fh(n)873 601 y Fl(x)901 608 y Fh(n)p Fi(+1)p 657 623 322 2 v 660 669 a Fl(q)682 676 y Fh(n)p Fi(+1)771 669 y Fs(+)g Fl(q)842 676 y Fh(n)870 669 y Fl(x)898 676 y Fh(n)p Fi(+1)996 635 y Fk(\000)g Fl(p)1070 642 y Fh(n)1098 563 y Fe(\014)1098 593 y(\014)1098 622 y(\014)1098 652 y(\014)1128 635 y Fs(=)1187 601 y Fk(j)p Fl(q)1223 608 y Fh(n)1250 601 y Fl(p)1275 608 y Fh(n)p Fi(+1)1363 601 y Fk(\000)h Fl(p)1438 608 y Fh(n)1465 601 y Fl(q)1487 608 y Fh(n)p Fi(+1)1565 601 y Fk(j)p 1187 623 393 2 v 1225 669 a Fl(q)1247 676 y Fh(n)p Fi(+1)1336 669 y Fs(+)f Fl(q)1407 676 y Fh(n)1435 669 y Fl(x)1463 676 y Fh(n)p Fi(+1)532 768 y Fs(=)736 734 y(1)p 591 757 317 2 v 591 802 a Fl(q)613 809 y Fh(n)p Fi(+1)702 802 y Fs(+)h Fl(q)774 809 y Fh(n)801 802 y Fl(x)829 809 y Fh(n)p Fi(+1)57 916 y Fs(b)o(y)16 b(\(A2.13\).)23 b(This)15 b(pro)o(v)o(es)g(\(i\).)156 986 y(Let)g(us)e(no)o(w)h(consider)e Fl(\014)625 993 y Fh(n)666 986 y Fs(=)i Fl(x)747 993 y Fi(0)770 986 y Fl(x)798 993 y Fi(1)829 986 y Fl(:)8 b(:)g(:)g(x)923 993 y Fh(n)951 986 y Fs(.)21 b(If)14 b Fl(x)1061 993 y Fh(k)1100 986 y Fk(\025)g Fl(g)h Fs(for)f(some)f Fl(k)j Fk(2)e(f)p Fs(0)p Fl(;)8 b Fs(1)p Fl(;)g(:)g(:)g(:)g(;)g(n)e Fk(\000)g Fs(1)p Fk(g)p Fs(,)57 1056 y(then,)16 b(letting)h Fl(m)c Fs(=)h Fl(x)480 1035 y Fj(\000)p Fi(1)480 1071 y Fh(k)545 1056 y Fk(\000)d Fl(x)623 1063 y Fh(k)q Fi(+1)712 1056 y Fk(\025)j Fs(1,)477 1186 y Fl(x)505 1193 y Fh(k)530 1186 y Fl(x)558 1193 y Fh(k)q Fi(+1)648 1186 y Fs(=)f(1)e Fk(\000)g Fl(mx)858 1193 y Fh(k)897 1186 y Fk(\024)i Fs(1)e Fk(\000)g Fl(x)1063 1193 y Fh(k)1102 1186 y Fk(\024)i Fs(1)e Fk(\000)g Fl(g)k Fs(=)f Fl(g)1358 1165 y Fi(2)1394 1186 y Fl(:)57 1316 y Fs(This)h(pro)o(v)o(es)g(\(ii\).)1386 b Fa(\003)57 1436 y Fr(Remark)21 b(A2.5)e Fs(Note)h(that)g(from)e Fp(\(ii\))j Fs(it)f(follo)o(ws)e(that)1189 1399 y Fe(P)1242 1411 y Fj(1)1242 1451 y Fh(k)q Fi(=0)1331 1414 y(log)7 b Fh(q)1407 1419 y Fb(k)p 1331 1425 99 2 v 1360 1454 a Fh(q)1378 1459 y Fb(k)1455 1436 y Fs(and)1554 1399 y Fe(P)1607 1411 y Fj(1)1607 1451 y Fh(k)q Fi(=0)1706 1417 y(1)p 1696 1425 41 2 v 1696 1454 a Fh(q)1714 1459 y Fb(k)1762 1436 y Fs(are)57 1506 y(alw)o(a)o(ys)15 b(con)o(v)o(ergen)o (t)g(and)h(their)g(sum)g(is)g(uniformly)f(b)q(ounded.)57 1611 y(F)l(or)d(all)g(in)o(tegers)g Fl(k)k Fk(\025)d Fs(1,)h(the)f(iteration)g(of)g(the)g(Gauss)f(map)g Fl(k)j Fs(times)e(leads)f(to)h(the)h(follo)o(wing)57 1681 y(partition)h(of)i (\(0)p Fl(;)8 b Fs(1\))j(;)16 b Fk(t)505 1688 y Fh(a)527 1693 y Fc(1)546 1688 y Fh(;:::;a)628 1693 y Fb(k)652 1681 y Fl(I)t Fs(\()p Fl(a)723 1688 y Fi(1)745 1681 y Fl(;)8 b(:)g(:)g(:)h(;)f(a)882 1688 y Fh(k)907 1681 y Fs(\),)17 b(where)f Fl(a)1127 1688 y Fh(i)1158 1681 y Fk(2)e Fm(N)p Fs(,)j Fl(i)d Fs(=)g(1)p Fl(;)8 b(:)g(:)g(:)g(;)g(k)r Fs(,)16 b(and)446 1865 y Fl(I)t Fs(\()p Fl(a)517 1872 y Fi(1)540 1865 y Fl(;)8 b(:)g(:)g(:)g(;)g(a)676 1872 y Fh(k)701 1865 y Fs(\))14 b(=)787 1763 y Fe(8)787 1808 y(<)787 1898 y(:)840 1763 y(\020)875 1796 y Fh(p)896 1801 y Fb(k)p 875 1806 43 2 v 876 1835 a Fh(q)894 1840 y Fb(k)924 1818 y Fl(;)952 1793 y Fh(p)973 1798 y Fb(k)994 1793 y Fi(+)p Fh(p)1046 1798 y Fb(k)p Ff(\000)p Fc(1)p 952 1806 160 2 v 954 1835 a Fh(q)972 1840 y Fb(k)994 1835 y Fi(+)p Fh(q)1043 1840 y Fb(k)p Ff(\000)p Fc(1)1117 1763 y Fe(\021)1196 1818 y Fs(if)j Fl(k)h Fs(is)e(ev)o(en)840 1857 y Fe(\020)875 1888 y Fh(p)896 1893 y Fb(k)918 1888 y Fi(+)p Fh(p)970 1893 y Fb(k)p Ff(\000)p Fc(1)p 875 1901 V 877 1930 a Fh(q)895 1935 y Fb(k)918 1930 y Fi(+)p Fh(q)967 1935 y Fb(k)p Ff(\000)p Fc(1)1040 1913 y Fl(;)1069 1890 y Fh(p)1090 1895 y Fb(k)p 1069 1901 43 2 v 1070 1930 a Fh(q)1088 1935 y Fb(k)1117 1857 y Fe(\021)1196 1913 y Fs(if)h Fl(k)h Fs(is)e(o)q(dd)57 2052 y(is)h(the)h(branc)o(h)d (of)j Fl(A)455 2034 y Fh(k)498 2052 y Fs(determined)e(b)o(y)h(the)h (fact)g(that)g(all)f(p)q(oin)o(ts)g Fl(x)f Fk(2)g Fl(I)t Fs(\()p Fl(a)1509 2059 y Fi(1)1532 2052 y Fl(;)8 b(:)g(:)g(:)g(;)g(a) 1668 2059 y Fh(k)1693 2052 y Fs(\))18 b(ha)o(v)o(e)57 2122 y(the)e(\014rst)g Fl(k)d Fs(+)e(1)16 b(partial)g(quotien)o(ts)g (exactly)i(equal)e(to)h Fk(f)p Fs(0)p Fl(;)8 b(a)1214 2129 y Fi(1)1237 2122 y Fl(;)g(:)g(:)g(:)g(;)g(a)1373 2129 y Fh(k)1398 2122 y Fk(g)p Fs(.)22 b(Th)o(us)289 2272 y Fl(I)t Fs(\()p Fl(a)360 2279 y Fi(1)383 2272 y Fl(;)8 b(:)g(:)g(:)g(;)g(a)519 2279 y Fh(k)544 2272 y Fs(\))14 b(=)630 2202 y Fe(\032)667 2272 y Fl(x)h Fk(2)f Fs(\(0)p Fl(;)8 b Fs(1\))23 b Fk(j)k Fl(x)14 b Fs(=)1032 2238 y Fl(p)1057 2245 y Fh(k)1093 2238 y Fs(+)c Fl(p)1167 2245 y Fh(k)q Fj(\000)p Fi(1)1243 2238 y Fl(y)p 1032 2261 238 2 v 1035 2306 a(q)1057 2313 y Fh(k)1093 2306 y Fs(+)g Fl(q)1164 2313 y Fh(k)q Fj(\000)p Fi(1)1240 2306 y Fl(y)1289 2272 y(;)22 b(y)16 b Fk(2)e Fs(\(0)p Fl(;)8 b Fs(1\))1522 2202 y Fe(\033)1582 2272 y Fl(:)57 2441 y Fs(Note)14 b(that)284 2422 y Fh(dx)p 284 2430 44 2 v 285 2458 a(dy)347 2441 y Fs(=)467 2417 y Fi(\()p Fj(\000)p Fi(1\))550 2402 y Fb(k)p 406 2430 227 2 v 406 2458 a Fi(\()p Fh(q)440 2463 y Fb(k)461 2458 y Fi(+)p Fh(q)510 2463 y Fb(k)p Ff(\000)p Fc(1)576 2458 y Fh(y)q Fi(\))613 2448 y Fc(2)651 2441 y Fs(is)f(p)q(ositiv)o(e)g(\(negativ)o (e\))g(if)g Fl(k)i Fs(is)d(ev)o(en)h(\(o)q(dd\).)21 b(It)14 b(is)e(immediate)57 2511 y(to)23 b(c)o(hec)o(k)g(that)g(an)o(y)f (rational)g(n)o(um)o(b)q(er)f Fl(p=q)27 b Fk(2)d Fs(\(0)p Fl(;)8 b Fs(1\),)26 b(\()p Fl(p;)8 b(q)r Fs(\))25 b(=)f(1,)h(is)d(the)h (endp)q(oin)o(t)f(of)57 2581 y(exactly)f(t)o(w)o(o)g(branc)o(hes)e(of)h (the)h(iterated)g(Gauss)f(map.)34 b(Indeed)20 b Fl(p=q)j Fs(can)d(b)q(e)h(written)g(as)57 2650 y Fl(p=q)16 b Fs(=)d([)q(\026)-26 b Fl(a)237 2657 y Fi(1)260 2650 y Fl(;)8 b(:)g(:)g(:)g(;)h Fs(\026)-26 b Fl(a)396 2657 y Fh(k)421 2650 y Fs(])16 b(with)h Fl(k)e Fk(\025)f Fs(1)i(and)g(\026)-25 b Fl(a)823 2657 y Fh(k)861 2650 y Fk(\025)14 b Fs(2)i(in)g(a)g(unique)g(w)o(a)o(y) g(and)g(it)g(is)g(the)h(left)g(\(righ)o(t\))918 2770 y(77)p eop %%Page: 78 79 78 78 bop 57 192 a Fs(endp)q(oin)o(t)17 b(of)i Fl(I)t Fs(\()q(\026)-26 b Fl(a)396 199 y Fi(1)419 192 y Fl(;)8 b(:)g(:)g(:)g(;)h Fs(\026)-26 b Fl(a)555 199 y Fh(k)580 192 y Fs(\))19 b(and)f(the)h(righ)o(t)f(\(left\))i(endp)q(oin)o(t)d(of) i Fl(I)t Fs(\()q(\026)-26 b Fl(a)1395 199 y Fi(1)1418 192 y Fl(;)8 b(:)g(:)g(:)h(;)f Fs(\026)-25 b Fl(a)1555 199 y Fh(k)1592 192 y Fk(\000)12 b Fs(1)p Fl(;)c Fs(1\))19 b(if)g Fl(k)57 261 y Fs(is)d(ev)o(en)g(\(o)q(dd\).)57 567 y Fo(A3.)38 b(Distributions,)26 b(Hyp)r(erfunctions,)e(F)-5 b(ormal)23 b(Series.)38 b(Hyp)r(o)r(ellipticit)n(y)57 637 y(and)20 b(Diophan)n(tine)i(Conditions.)57 742 y Fr(A3.1)16 b Fs(W)l(e)h(follo)o(w)f(here)g([H1],)g(Chapter)g(9)h(but)f (w)o(e)h(also)f(recommend)f([Ph],)h(esp)q(ecially)g(the)57 812 y(\014rst)d(few)h(c)o(hapters,)f(for)h(a)g(nice)g(in)o(tro)q (duction)e(to)j(h)o(yp)q(erfunctions)d(and)i(their)f(applications.)156 882 y(Let)f Fl(K)j Fs(b)q(e)d(a)f(non)f(empt)o(y)h(compact)g(subset)g (of)g Fm(R)p Fs(.)17 b(A)12 b Fp(hyp)m(erfunction)i(with)g(supp)m(ort)g (in)d Fl(K)57 951 y Fs(is)j(a)h(linear)f(functional)g Fl(u)g Fs(on)g(the)h(space)g Fk(O)q Fs(\()p Fl(K)t Fs(\))g(of)g (functions)f(analytic)h(in)f(a)h(neigh)o(b)q(orho)q(o)q(d)57 1021 y(of)h Fl(K)k Fs(suc)o(h)c(that)g(for)g(all)g(neigh)o(b)q(orho)q (o)q(d)f Fl(V)27 b Fs(of)17 b Fl(K)j Fs(there)c(is)g(a)g(constan)o(t)g Fl(C)1483 1028 y Fh(V)1531 1021 y Fl(>)e Fs(0)i(suc)o(h)f(that)561 1130 y Fk(j)p Fl(u)p Fs(\()p Fl(')p Fs(\))p Fk(j)f(\024)f Fl(C)791 1137 y Fh(V)834 1130 y Fs(sup)855 1172 y Fh(V)917 1130 y Fk(j)p Fl(')p Fk(j)g Fl(;)50 b Fk(8)p Fl(')12 b Fk(2)i(O)q Fs(\()p Fl(V)e Fs(\))i Fl(:)57 1259 y Fs(W)l(e)f(denote)g (b)o(y)f Fl(A)398 1241 y Fj(0)412 1259 y Fs(\()p Fl(K)t Fs(\))i(the)f(space)f(of)h(h)o(yp)q(erfunctions)f(with)g(supp)q(ort)g (in)g Fl(K)t Fs(.)21 b(It)13 b(is)g(a)f(F)l(r)o(\023)-24 b(ec)o(het)57 1329 y(space)16 b(:)22 b(a)16 b(seminorm)f(is)h(asso)q (ciated)g(to)h(eac)o(h)f(neigh)o(b)q(orho)q(o)q(d)f Fl(V)27 b Fs(of)17 b Fl(K)t Fs(.)156 1398 y(Let)h Fk(O)287 1380 y Fi(1)309 1398 y Fs(\()p 328 1358 36 2 v Fm(C)24 b Fk(n)11 b Fl(K)t Fs(\))17 b(denote)f(the)h(complex)g(v)o(ector)f(space)h(of)g (functions)e(holomorphic)g(on)57 1468 y(\()p 76 1428 V Fm(C)23 b Fk(n)11 b Fl(K)t Fs(\))17 b(and)f(v)m(anishing)f(at)i (in\014nit)o(y)l(.)k(One)16 b(has)g(the)h(follo)o(wing)57 1599 y Fr(Prop)r(osition)e(A3.1)41 b Fd(The)13 b(spaces)g Fl(A)807 1580 y Fj(0)821 1599 y Fs(\()p Fl(K)t Fs(\))h Fd(and)e Fk(O)1053 1580 y Fi(1)1076 1599 y Fs(\()p 1095 1558 V Fm(C)17 b Fk(n)t Fl(K)t Fs(\))d Fd(are)e(canonically)h (isomorphic.)57 1668 y(T)l(o)j(eac)o(h)g Fl(u)d Fk(2)i Fl(A)369 1650 y Fj(0)383 1668 y Fs(\()p Fl(K)t Fs(\))i Fd(corresp)q(onds)d Fl(')g Fk(2)g(O)892 1650 y Fi(1)914 1668 y Fs(\()p 933 1628 V Fm(C)24 b Fk(n)11 b Fl(K)t Fs(\))17 b Fd(giv)o(en)f(b)o(y)645 1777 y Fl(')p Fs(\()p Fl(z)r Fs(\))e(=)g Fl(u)p Fs(\()p Fl(c)878 1784 y Fh(z)900 1777 y Fs(\))h Fl(;)22 b Fk(8)p Fl(z)14 b Fk(2)h Fm(C)23 b Fk(n)11 b Fl(K)17 b(;)57 1886 y Fd(where)26 b Fl(c)233 1893 y Fh(z)256 1886 y Fs(\()p Fl(x)p Fs(\))33 b(=)434 1867 y Fi(1)p 431 1875 25 2 v 431 1904 a Fh(\031)495 1867 y Fi(1)p 468 1875 75 2 v 468 1904 a Fh(x)p Fj(\000)p Fh(z)548 1886 y Fd(.)54 b(Con)o(v)o(ersely)26 b(to)i(eac)o(h)e Fl(')32 b Fk(2)f(O)1243 1868 y Fi(1)1266 1886 y Fs(\()p 1285 1846 36 2 v Fm(C)f Fk(n)18 b Fl(K)t Fs(\))28 b Fd(corresp)q(onds)d (the)57 1956 y(h)o(yp)q(erfunction)562 2039 y Fl(u)p Fs(\()p Fl( )r Fs(\))14 b(=)755 2005 y Fl(i)p 736 2027 56 2 v 736 2073 a Fs(2)p Fl(\031)805 1971 y Fe(Z)833 2084 y Fh(\015)867 2039 y Fl(')p Fs(\()p Fl(z)r Fs(\))p Fl( )r Fs(\()p Fl(z)r Fs(\))p Fl(dz)k(;)k Fk(8)p Fl( )15 b Fk(2)f Fl(A)57 2162 y Fd(where)h Fl(\015)k Fd(is)d(an)o(y)f (piecewise)h Fk(C)633 2144 y Fi(1)671 2162 y Fd(path)g(winding)f (around)f Fl(K)20 b Fd(in)c(the)g(p)q(ositiv)o(e)g(direction.)21 b(W)l(e)57 2232 y(will)16 b(also)g(use)g(the)h(notation)587 2341 y Fl(u)p Fs(\()p Fl(x)p Fs(\))e(=)764 2307 y(1)p 756 2330 43 2 v 756 2375 a(2)p Fl(i)804 2341 y Fs([)p Fl(')p Fs(\()p Fl(x)c Fs(+)g Fl(i)p Fs(0\))g Fk(\000)g Fl(')p Fs(\()p Fl(x)h Fk(\000)e Fl(i)p Fs(0\)])57 2450 y Fd(for)16 b(short.)57 2581 y Fp(Pr)m(o)m(of.)30 b Fs(It)e(is)g(v)o (ery)g(easy)g(:)44 b(note)28 b(that)h(the)f(function)f Fl(x)33 b Fk(7!)g Fl(c)1354 2588 y Fh(z)1376 2581 y Fs(\()p Fl(x)p Fs(\))d(is)d(analytic)h(in)f(a)57 2650 y(neigh)o(b)q(orho)q(o)q (d)11 b(of)j Fl(K)j Fs(for)c(all)f Fl(z)22 b(=)-30 b Fk(2)14 b Fl(K)t Fs(.)20 b(Then)13 b(it)h(is)e(immediate)h(to)g(c)o (hec)o(k)g(applying)f(Cauc)o(h)o(y's)918 2770 y(78)p eop %%Page: 79 80 79 79 bop 57 192 a Fs(form)o(ula)16 b(that)j(these)g(t)o(w)o(o)f (corresp)q(ondences)e(are)i(surjectiv)o(e)g(and)g(are)g(the)h(in)o(v)o (erse)e(of)i(one)57 261 y(another.)1552 b Fa(\003)57 453 y Fr(A3.2)17 b Fs(Let)g Fm(T)317 435 y Fi(1)351 453 y Fs(=)e Fm(R)p Fl(=)p Fm(Z)n Fk(\032)g Fm(C)9 b Fl(=)p Fm(Z)-8 b Fs(.)21 b(A)d Fp(hyp)m(erfunction)i(on)e Fm(T)9 b Fs(is)17 b(a)g(linear)f(fun)o(tional)g Fl(U)23 b Fs(on)17 b(the)57 522 y(space)g Fk(O)q Fs(\()p Fm(T)287 504 y Fi(1)306 522 y Fs(\))h(of)g(functions)e(analytic)h(in)g(a)g(complex)g (neigh)o(b)q(orho)q(o)q(d)f(of)h Fm(T)1510 504 y Fi(1)1547 522 y Fs(suc)o(h)f(that)i(for)57 592 y(all)e(neigh)o(b)q(orho)q(o)q(d)f Fl(V)27 b Fs(of)17 b Fm(T)9 b Fs(there)16 b(exists)h Fl(C)903 599 y Fh(V)951 592 y Fl(>)d Fs(0)i(suc)o(h)f(that)553 724 y Fk(j)p Fl(U)5 b Fs(\(\010\))p Fk(j)14 b(\024)g Fl(C)797 731 y Fh(V)839 724 y Fs(sup)861 766 y Fh(V)923 724 y Fk(j)p Fl(')p Fk(j)f Fl(;)49 b Fk(8)p Fs(\010)13 b Fk(2)h(O)q Fs(\()p Fl(V)d Fs(\))k Fl(:)57 878 y Fs(W)l(e)21 b(will)f(denote)g Fl(A)446 860 y Fj(0)460 878 y Fs(\()p Fm(T)516 860 y Fi(1)535 878 y Fs(\).)35 b(the)21 b(F)l(r)o(\023)-24 b(ec)o(het)19 b(space)h(of)h(h)o(yp)q(erfunctions)e(with)i(supp)q(ort)e (in)h Fm(T)l Fs(.)57 948 y(F)l(or)15 b Fl(U)k Fk(2)14 b Fl(A)283 930 y Fj(0)298 948 y Fs(\()p Fm(T)-5 b Fs(\),)14 b(let)481 935 y(^)472 948 y Fl(U)6 b Fs(\()p Fl(n)p Fs(\))14 b(:=)f Fl(U)5 b Fs(\()p Fl(e)741 955 y Fj(\000)p Fh(n)801 948 y Fs(\))17 b(with)f Fl(e)973 955 y Fh(n)1000 948 y Fs(\()p Fl(z)r Fs(\))f(=)f Fl(e)1154 930 y Fi(2)p Fh(\031)q(inz)1260 948 y Fs(.)57 1054 y Fr(Exercise)20 b(A3.2)c Fs(Sho)o(w)f(that)i(the)g (doubly)f(in\014nite)f(sequence)i(\()1310 1041 y(^)1301 1054 y Fl(U)6 b Fs(\()p Fl(n)p Fs(\)\))1428 1061 y Fh(n)p Fj(2)p Fg(Z)1523 1054 y Fs(satis\014es)731 1186 y Fk(j)753 1173 y Fs(^)745 1186 y Fl(U)f Fs(\()p Fl(n)p Fs(\))p Fk(j)14 b Fl(<)g(C)969 1193 y Fh(")990 1186 y Fl(e)1013 1165 y Fi(2)p Fh(\031)q Fj(j)p Fh(n)p Fj(j)p Fh(")1141 1186 y Fl(:)57 1317 y Fs(for)19 b(all)f Fl(")h(>)f Fs(0)h(and)f(for)h (all)g Fl(n)f Fk(2)g Fm(Z)6 b Fs(with)19 b(a)g(suitably)f(c)o(hosen)g Fl(C)1302 1324 y Fh(")1341 1317 y Fl(>)g Fs(0.)30 b(Con)o(v)o(ersely)18 b(sho)o(w)57 1387 y(that)i(an)o(y)f(suc)o(h)f(sequence)h(is)g(the)h(F)l (ourier)e(expansion)g(of)i(a)f(unique)g(h)o(yp)q(erfunction)f(with)57 1457 y(supp)q(ort)d(in)h Fm(T)-5 b Fs(.)57 1563 y(Let)16 b Fk(O)185 1570 y Fi(\006)231 1563 y Fs(denote)f(the)g(complex)g(v)o (ector)h(space)e(of)i(holomorphic)d(functions)h(\010)22 b(:)14 b Fm(C)21 b Fk(n)8 b Fm(R)j Fk(!)j Fm(C)9 b Fs(,)57 1633 y(1{p)q(erio)q(dic,)21 b(b)q(ounded)f(at)h Fk(\006)p Fl(i)p Fk(1)g Fs(and)g(suc)o(h)f(that)h(\010\()p Fk(\006)p Fl(i)p Fk(1)p Fs(\))h(:=)f(lim)1393 1640 y Fj(=)p Fh(m)7 b(z)q Fj(!\0061)1606 1633 y Fs(\010\()p Fl(z)r Fs(\))22 b(exist)57 1702 y(and)16 b(v)o(erify)g(\010\(+)p Fl(i)p Fk(1)p Fs(\))e(=)g Fk(\000)p Fs(\010\()p Fk(\000)p Fl(i)p Fk(1)p Fs(\).)57 1808 y Fr(Exercise)19 b(A3.3)d Fs(Sho)o(w)f(that)i (the)f(spaces)g Fl(A)930 1790 y Fj(0)944 1808 y Fs(\()p Fm(T)999 1790 y Fi(1)1019 1808 y Fs(\))g(and)g Fk(O)1191 1815 y Fi(\006)1238 1808 y Fs(are)f(canonically)h(isomorphic.)57 1878 y(Indeed)g(to)g(eac)o(h)h Fl(U)i Fk(2)14 b Fl(A)529 1860 y Fj(0)543 1878 y Fs(\()p Fm(T)599 1860 y Fi(1)618 1878 y Fs(\))j(corresp)q(onds)e(\010)e Fk(2)h(O)1064 1885 y Fi(\006)1112 1878 y Fs(giv)o(en)i(b)o(y)631 2010 y(\010\()p Fl(z)r Fs(\))f(=)e Fl(U)5 b Fs(\()p Fl(C)891 2017 y Fh(z)915 2010 y Fs(\))14 b Fl(;)22 b Fk(8)p Fl(z)15 b Fk(2)f Fm(C)23 b Fk(n)11 b Fl(K)17 b(;)57 2141 y Fs(where)25 b Fl(C)246 2148 y Fh(z)269 2141 y Fs(\()p Fl(x)p Fs(\))31 b(=)f(cotg)8 b Fl(\031)r Fs(\()p Fl(x)19 b Fk(\000)e Fl(z)r Fs(\).)52 b(Con)o(v)o(ersely)25 b(to)h(eac)o(h)g(\010)k Fk(2)g(O)1419 2148 y Fi(\006)1476 2141 y Fs(corresp)q(onds)24 b(the)57 2211 y(h)o(yp)q(erfunction)515 2296 y Fl(U)5 b Fs(\(\011\))15 b(=)708 2262 y Fl(i)p 705 2284 25 2 v 705 2330 a Fs(2)744 2228 y Fe(Z)771 2341 y Fi(\000)807 2296 y Fs(\010\()p Fl(z)r Fs(\)\011\()p Fl(z)r Fs(\))p Fl(dz)j(;)k Fk(8)p Fs(\011)12 b Fk(2)i Fl(A)p Fs(\()p Fm(T)1331 2275 y Fi(1)1350 2296 y Fs(\))57 2424 y(where)g(\000)h(is)f (an)o(y)g(piecewise)g Fk(C)628 2406 y Fi(1)665 2424 y Fs(path)g(winding)g(around)f(a)h(closed)g(in)o(terv)m(al)g Fl(I)k Fk(\032)13 b Fm(R)f Fs(of)j(length)57 2494 y(1)h(in)g(the)h(p)q (ositiv)o(e)f(direction.)21 b(W)l(e)c(will)f(also)g(use)g(the)h (notation)579 2636 y Fl(U)5 b Fs(\()p Fl(x)p Fs(\))15 b(=)766 2602 y(1)p 758 2625 43 2 v 758 2670 a(2)p Fl(i)806 2636 y Fs([\010\()p Fl(x)c Fs(+)g Fl(i)p Fs(0\))h Fk(\000)f Fs(\010\()p Fl(x)g Fk(\000)g Fl(i)p Fs(0\)])918 2770 y(79)p eop %%Page: 80 81 80 80 bop 57 192 a Fs(for)16 b(short.)57 299 y(The)g(nice)g(fact)i(is)e (that)h(the)g(follo)o(wing)e(diagram)g(comm)o(utes)g(:)570 524 y Fl(A)607 506 y Fj(0)622 524 y Fs(\([0)p Fl(;)8 b Fs(1]\))50 b Fk(\000)-21 b(\000)-11 b(\000)f(\000)h(\000)-20 b(!)50 b(O)1071 506 y Fi(1)1093 524 y Fs(\()p 1112 483 36 2 v Fm(C)24 b Fk(n)11 b Fs([0)p Fl(;)d Fs(1]\))557 640 y Fe(P)610 692 y Fb(Z)646 586 y Fe(?)646 616 y(?)646 646 y(?)646 676 y(?)646 706 y(y)1153 586 y(?)1153 616 y(?)1153 646 y(?)1153 676 y(?)1153 706 y(y)1200 640 y(P)1253 692 y Fb(Z)592 826 y Fl(A)629 808 y Fj(0)644 826 y Fs(\()p Fm(T)699 808 y Fi(1)719 826 y Fs(\))72 b Fk(\000)-21 b(\000)-11 b(\000)f(\000)h(\000)-20 b(!)157 b(O)1177 833 y Fi(\006)57 1047 y Fs(the)19 b(horizon)o(tal)e(lines)h(are)h(the)g (ab)q(o)o(v)o(e)f(men)o(tioned)g(isomorphism)o(s,)1386 1009 y Fe(P)1439 1062 y Fh(Z)1490 1047 y Fs(is)g(the)i(sum)d(o)o(v)o (er)57 1116 y(in)o(teger)e(translates)h(:)22 b(\()504 1079 y Fe(P)557 1131 y Fg(Z)591 1116 y Fl(')p Fs(\)\()p Fl(z)r Fs(\))15 b(=)774 1079 y Fe(P)826 1131 y Fh(n)p Fj(2)p Fg(Z)912 1116 y Fl(')p Fs(\()p Fl(z)e Fk(\000)e Fl(n)p Fs(\).)57 1293 y Fr(A3.3)26 b Fs(As)g(w)o(e)g(ha)o(v)o(e)g(seen) g(in)g(A3.2)h(p)q(erio)q(dic)f(distributions)e(and)i(h)o(yp)q (erfunctions)f(are)57 1363 y(naturally)16 b(iden)o(ti\014ed)f(with)i (the)g(t)o(w)o(o)g(follo)o(wing)f(subspaces)f(of)i(the)g(complex)g(v)o (ector)g(space)57 1433 y(of)f Fp(formal)i Fs(F)l(ourier)c(series)170 1603 y Fl(')f Fk(2)h(D)302 1582 y Fj(0)317 1603 y Fs(\()p Fm(T)l Fs(\))d Fk(,)j Fl(')p Fs(\()p Fl(\022)q Fs(\))h(=)630 1540 y Fi(+)p Fj(1)629 1555 y Fe(X)630 1660 y Fj(\0001)717 1603 y Fs(^)-33 b Fl(')p Fs(\()p Fl(n)p Fs(\))p Fl(e)833 1582 y Fi(2)p Fh(\031)q(in\022)953 1603 y Fs(and)16 b(there)h(exists)p Fl(M)i(>)14 b Fs(0)8 b Fl(;)22 b(r)16 b(>)d Fs(0)439 1731 y(suc)o(h)i(that)9 b Fk(j)f Fs(^)-33 b Fl(')o Fs(\()p Fl(n)p Fs(\))p Fk(j)15 b(\024)e Fl(M)5 b Fk(j)p Fl(n)p Fk(j)957 1711 y Fi(+)p Fh(r)1024 1731 y Fk(8)p Fl(n)12 b Fk(2)j Fm(Z)1179 1711 y Fj(\003)1212 1731 y Fl(;)170 1858 y(')e Fk(2)h(A)303 1837 y Fj(0)317 1858 y Fs(\()p Fm(T)l Fs(\))d Fk(,)j Fl(')p Fs(\()p Fl(\022)q Fs(\))h(=)630 1795 y Fi(+)p Fj(1)629 1810 y Fe(X)630 1915 y Fj(\0001)717 1858 y Fs(^)-33 b Fl(')p Fs(\()p Fl(n)p Fs(\))p Fl(e)833 1837 y Fi(2)p Fh(\031)q(in\022)953 1858 y Fs(and)16 b(for)g(all)8 b Fl(")14 b(>)g Fs(0)i(there)h(exists)p Fl(C)1603 1865 y Fh(")1638 1858 y Fl(>)c Fs(0)439 1986 y(suc)o(h)i(that)9 b Fk(j)f Fs(^)-33 b Fl(')o Fs(\()p Fl(n)p Fs(\))p Fk(j)15 b(\024)e Fl(C)882 1993 y Fh(")911 1986 y Fs(exp)q(\(2)p Fl(\031)r Fk(j)p Fl(n)p Fk(j)p Fl(")p Fs(\))h Fk(8)p Fl(n)e Fk(2)j Fm(Z)p Fl(:)57 2116 y Fs(Let)i(us)f(no)o(w)g(consider)f (the)i(follo)o(wing)e(linear)g(\014rst{order)g(di\013erence)h(equation) g(on)g Fm(T)1716 2097 y Fi(1)677 2252 y Fl(f)701 2259 y Fh(g)724 2252 y Fs(\()p Fl(\022)e Fs(+)c Fl(\013)p Fs(\))i Fk(\000)f Fl(f)966 2259 y Fh(g)989 2252 y Fs(\()p Fl(\022)q Fs(\))16 b(=)d Fl(g)r Fs(\()p Fl(\022)q Fs(\))57 2388 y(where)20 b Fl(\013)h Fk(2)h Fm(R)11 b Fk(n)i Fm(Q)p Fs(.)36 b(A)22 b(necessary)e(condition)g(for)g(the)h(existence)h(of)f (a)g(solution)f(is)g(that)57 2417 y Fe(R)90 2430 y Fi(2)p Fh(\031)80 2475 y Fi(0)145 2457 y Fl(g)r Fs(\()p Fl(\022)q Fs(\))p Fl(d\022)d Fs(=)c(0.)22 b(Th)o(us)15 b(w)o(e)i(in)o(tro)q(duce) e(the)i(zero{mean)e(Dirac)h(delta)h(function)f(on)g Fm(T)1726 2439 y Fi(1)685 2599 y Fl(\016)707 2606 y Fg(T)r Fh(;)p Fi(0)765 2599 y Fs(\()p Fl(\022)q Fs(\))f(=)943 2551 y Fe(X)894 2659 y Fh(n)p Fj(2)p Fg(Z)d Fh(;n)p Fj(6)p Fi(=0)1071 2599 y Fl(e)1094 2578 y Fi(2)p Fh(\031)q(in\022)918 2770 y Fs(80)p eop %%Page: 81 82 81 81 bop 57 192 a Fs(and)19 b(w)o(e)h(note)g(that)h(the)f(corresp)q (onding)e Fl(f)897 199 y Fh(\016)939 192 y Fs(pla)o(ys)h(the)h(role)g (of)g(a)g(fundamen)o(tal)e(solution)57 261 y(since)227 415 y Fl(f)251 422 y Fh(g)289 415 y Fs(=)13 b Fl(f)365 422 y Fh(\016)399 415 y Fk(\014)e Fl(g)k Fs(=)568 353 y Fi(+)p Fj(1)568 368 y Fe(X)541 472 y Fh(n)p Fi(=)p Fj(\0001)686 402 y Fs(^)675 415 y Fl(f)699 422 y Fh(\016)721 415 y Fs(\()p Fl(n)p Fs(\))r(^)-27 b Fl(g)r Fs(\()p Fl(n)p Fs(\))p Fl(e)906 394 y Fi(2)p Fh(\031)q(in\022)1027 415 y Fs(=)1100 381 y(1)p 1085 403 56 2 v 1085 449 a(2)p Fl(\031)1155 347 y Fe(Z)1204 359 y Fi(2)p Fh(\031)1182 460 y Fi(0)1260 415 y Fl(f)1284 422 y Fh(\016)1306 415 y Fs(\()p Fl(\022)13 b Fk(\000)e Fl(\022)1434 422 y Fi(1)1457 415 y Fs(\))p Fl(g)r Fs(\()p Fl(\022)1544 422 y Fi(1)1567 415 y Fs(\))p Fl(d\022)1635 422 y Fi(1)57 581 y Fs(when)16 b(the)h(in)o(tegral)e(mak)o(es)h(sense.)156 650 y(On)g(the)h(other)f (hand)g(one)g(clearly)g(has)697 801 y Fl(f)721 808 y Fh(\016)743 801 y Fs(\()p Fl(\022)q Fs(\))f(=)874 753 y Fe(X)873 861 y Fh(n)p Fj(6)p Fi(=0)1008 767 y Fl(e)1031 749 y Fi(2)p Fh(\031)q(in\022)p 962 789 221 2 v 962 835 a Fl(e)985 821 y Fi(2)p Fh(\031)q(in\013)1108 835 y Fk(\000)c Fs(1)57 972 y(as)j(a)g(formal)g(p)q(o)o(w)o(er)f(series.)20 b(W)l(e)15 b(ha)o(v)o(e)f(the)h(follo)o(wing)e(elemen)o(tary)h(Prop)q (osition)f(whic)o(h)h(can)57 1042 y(also)i(b)q(e)g(tak)o(en)h(as)f(an)g (equiv)m(alen)o(t)h(de\014nition)e(of)i(diophan)o(tine)e(n)o(um)o(b)q (ers)57 1177 y Fr(Prop)r(osition)33 b(A3.4)61 b Fl(f)583 1184 y Fh(\016)635 1177 y Fd(is)28 b(a)h(distribution)f(if)h(and)f (only)h(if)g Fl(\013)35 b Fk(2)g Fd(CD.)59 b Fl(f)1689 1184 y Fh(\016)1741 1177 y Fd(is)28 b(a)57 1246 y(h)o(yp)q(erfunction) 19 b(if)i(and)f(only)h(if)g(the)g(denominators)d Fl(q)1120 1253 y Fh(n)1169 1246 y Fd(of)i(the)i(con)o(v)o(ergen)o(ts)d(of)i Fl(\013)f Fd(v)o(erify)57 1316 y Fs(lim)126 1323 y Fh(n)p Fj(!)p Fi(+)p Fj(1)277 1292 y Fh(log)q(q)348 1297 y Fb(n)p Fc(+1)p 277 1305 139 2 v 325 1333 a Fh(q)343 1338 y Fb(n)435 1316 y Fs(=)14 b(0)p Fd(.)57 1451 y Fs(The)i(pro)q(of)g(is)g(immediate) g(and)g(it)h(is)f(left)h(as)f(an)g(Exercise.)156 1521 y(Note)k(that)g(the)g(ab)q(o)o(v)o(e)e(discussion)f(carries)h(o)o(v)o (er)h(easily)g(to)g(the)h(linear)e(PDE)h(on)g(the)57 1591 y(t)o(w)o(o{dimensional)13 b(torus)j Fm(T)597 1573 y Fi(2)748 1660 y Fs(\()p Fl(@)793 1667 y Fh(\022)812 1672 y Fc(1)846 1660 y Fs(+)11 b Fl(\013@)954 1667 y Fh(\022)973 1672 y Fc(2)995 1660 y Fs(\))p Fl(f)20 b Fs(=)14 b Fl(g)57 1765 y Fs(\(whic)o(h)26 b(is)h(asso)q(ciated)g(to)h (the)g(linear)e(\015o)o(w)987 1752 y(_)971 1765 y Fl(\022)994 1772 y Fi(1)1048 1765 y Fs(=)32 b(1,)1204 1752 y(_)1188 1765 y Fl(\022)1211 1772 y Fi(2)1266 1765 y Fs(=)f Fl(\013)p Fs(\).)55 b(In)27 b(this)g(case)h(one)57 1835 y(has)f Fl(\016)179 1844 y Fg(T)199 1834 y Fc(2)222 1844 y Fh(;)p Fi(0)289 1835 y Fs(=)361 1797 y Fe(P)413 1850 y Fh(n)p Fj(2)p Fg(Z)485 1840 y Fc(2)507 1850 y Fh(;n)p Fj(6)p Fi(=0)605 1835 y Fl(e)628 1817 y Fi(2)p Fh(\031)q(i)p Fi(\()p Fh(n)727 1822 y Fc(1)746 1817 y Fh(\022)765 1822 y Fc(1)785 1817 y Fi(+)p Fh(n)841 1822 y Fc(2)859 1817 y Fh(\022)878 1822 y Fc(2)898 1817 y Fi(\))944 1835 y Fs(and)g(the)i(fundamen)o(tal)d(solution)h(is)g Fl(f)1734 1842 y Fh(\016)1790 1835 y Fs(=)57 1877 y Fe(P)109 1930 y Fh(n)p Fj(2)p Fg(Z)181 1920 y Fc(2)203 1930 y Fh(;n)p Fj(6)p Fi(=0)307 1895 y Fh(e)326 1880 y Fc(2)p Fb(\031)q(i)p Fc(\()p Fb(n)413 1887 y Fc(1)433 1880 y Fb(\022)450 1887 y Fc(1)469 1880 y(+)p Fb(n)517 1887 y Fc(2)536 1880 y Fb(\022)553 1887 y Fc(2)572 1880 y(\))p 307 1903 282 2 v 331 1932 a Fi(2)p Fh(\031)q(i)p Fi(\()p Fh(n)430 1937 y Fc(1)449 1932 y Fi(+)p Fh(n)505 1937 y Fc(2)523 1932 y Fh(\013)p Fi(\))594 1915 y Fs(.)c(Then)17 b(Prop)q(osition)e (A3.4)i(holds)f(also)g(in)h(this)f(case,)h(sho)o(wing)57 1985 y(that)g(the)f(op)q(erator)g Fl(@)477 1992 y Fh(\022)496 1997 y Fc(1)530 1985 y Fs(+)11 b Fl(\013@)638 1992 y Fh(\022)657 1997 y Fc(2)696 1985 y Fs(is)16 b(h)o(yp)q(o)q(elliptic)g (if)h(and)e(only)i(if)f Fl(\013)h Fs(is)f(diophan)o(tine.)918 2770 y(81)p eop %%Page: 82 83 82 82 bop 57 192 a Fq(References)29 300 y Fs([A)o(G])24 b(S.)15 b(Alinhac,)g(P)l(.)g(G)o(\023)-24 b(erard)14 b(\\Op)o(\023)-24 b(erateurs)13 b(pseudo{di\013)o(\023)-24 b(eren)o(tiels)12 b(et)k(th)o(\022)-24 b(eor)o(\023)g(eme)15 b(de)g(Nash{)156 370 y(Moser")h(Sa)o(v)o(oirs)e(Actuels,)j(CNRS)f (Editions)g(\(1991\))14 442 y([Ah1])24 b(L.V.)17 b(Ahlfors)f (\\Conformal)f(In)o(v)m(arian)o(ts)g(:)22 b(T)l(opics)15 b(in)h(Geometric)h(F)l(unction)e(Theory")156 512 y(McGra)o(w{Hill)h (\(1973\))14 585 y([Ah2])24 b(L.V.)17 b(Ahlfors)f(\\Lectures)g(on)g (Quasiconformal)e(Mappings")h(V)l(an)h(Nostrand)g(\(1966\))21 658 y([AM])24 b(R.)17 b(Abraham,)g(J.)g(Marsden)f(\\F)l(oundations)f (of)j(Mec)o(hanics")e(Benjamin)h(Cummings,)156 727 y(New)g(Y)l(ork)g (\(1978\))22 800 y([Ar1])24 b(V.)g(I.)f(Arnol'd)f(\\Small)g (denominators)f(and)h(problems)f(of)i(stabilit)o(y)g(of)g(motion)g(in) 156 870 y(classical)16 b(celestial)g(mec)o(hanics")f(Russ.)21 b(Math.)g(Surv.)g Fr(18)16 b Fs(\(1963\),)h(85{193.)22 943 y([Ar2])24 b(V.)31 b(I.)f(Arnol'd)f(\\Instabilit)o(y)g(of)i (dynamical)e(systems)g(with)h(sev)o(eral)g(degrees)f(of)156 1013 y(freedom")16 b(So)o(v.)21 b(Math.)h(Dokl.)g Fr(5)16 b Fs(\(1964\),)h(581{585.)22 1085 y([Ar3])24 b(V.)c(I.)g(Arnol'd)f (\\Geometrical)g(Metho)q(ds)g(in)h(the)g(Theory)f(of)h(Ordinary)e (Di\013eren)o(tial)156 1155 y(Equations")e(Springer{V)l(erlag)e (\(1983\))-10 1228 y([AKN])25 b(V.)d(I.)f(Arnol'd,)g(V.)g(V.)g(Kozlo)o (v)g(and)f(A.)i(I.)f(Neish)o(tadt)f(\\Dynamical)g(Systems)h(I)q(I)q (I",)156 1298 y(Springer{V)l(erlag)14 b(\(1988\).)46 1370 y([Be])25 b(A.)17 b(Beardon)f(\\Iteration)g(of)h(Rational)e(F)l (unctions")g(Springer{V)l(erlag)f(\(1991\))-41 1443 y([BF)o(GG])24 b(G.)30 b(Benettin,)k(G.)29 b(F)l(errari,)i(L.)f(Galgani)f(and)g(A.)h (Giorgilli)e(\\An)i(Extension)f(of)156 1513 y(the)20 b(P)o(oincar)o(\023)-24 b(e{F)l(ermi)16 b(Theorem)i(on)g(the)i (Nonexistence)f(of)h(In)o(v)m(arian)o(t)d(Manifolds)h(in)156 1583 y(Nearly)f(In)o(tegrable)e(Hamiltonian)h(Systems")g(Il)g(Nuo)o(v)o (o)g(Cimen)o(to)g Fr(72B)f Fs(\(1982\))i(137)3 1656 y([BHS])25 b(H.)17 b(W.)f(Bro)q(er,)g(G.)g(B.)h(Huitema)f(and)f(M.)h(B.)h(Sevryuk) f(\\Quasi{P)o(erio)q(dic)e(Motions)h(in)156 1725 y(F)l(amilies)g(of)i (Dynamical)e(Systems")h(Springer{V)l(erlag)e(\(1996\))44 1798 y([Bo])24 b(J.)d(B.)g(Bost)g(\\T)l(ores)f(in)o(v)m(arian)o(ts)f (des)h(syst)o(\022)-24 b(emes)20 b(dynamiques)g(hamiltoniens")f(S)o (\023)-24 b(emi-)156 1868 y(naire)16 b(Bourbaki)g Fr(639)p Fs(,)f(Ast)o(\023)-24 b(erisque)16 b Fr(133{134)e Fs(\(1986\),)j (113{157.)49 1941 y([Br])24 b(A.)12 b(D.)g(Brjuno)f(\\Analytical)g (form)g(of)h(di\013eren)o(tial)e(equations")h Fp(T)l(r)m(ans.)23 b(Mosc)m(ow)15 b(Math.)156 2010 y(So)m(c.)24 b Fr(25)15 b Fs(\(1971\),)i(131-288)9 b(;)17 b Fr(26)f Fs(\(1972\),)g(199-239.)29 2083 y([CG])24 b(L.)f(Carleson)e(and)h(T.)g(Gamelin)f(\\Complex)h (Dynamics")g(Univ)o(ersitext,)i(Springer{)156 2153 y(V)l(erlag,)16 b(Berlin)g(Heidelb)q(er)g(New)i(Y)l(ork)e(\(1993\))22 2226 y([CM])24 b(T.)13 b(Carletti)g(and)f(S.)g(Marmi)f(\\Linearization) g(of)i(Analytic)g(and)f(Non{Analytic)h(Germs)156 2296 y(of)j(Di\013eomorphisms)c(of)k(\()p Fm(C)9 b Fl(;)g Fs(0\)")18 b(Bulletin)d(de)h(la)f(So)q(ciet)o(\023)-24 b(e)16 b(Math)o(\023)-24 b(ematique)15 b(de)h(F)l(rance)156 2365 y(\(1999\))41 2438 y([Da])24 b(A.M.)16 b(Da)o(vie)h(\\The)f (critical)g(function)g(for)g(the)h(semistandard)c(map")j Fp(Nonline)m(arity)i Fr(7)156 2508 y Fs(\(1994\),)f(219)f(-)h(229.)8 2581 y([DeB])25 b(L.)13 b(de)g(Branges)e(\\A)i(pro)q(of)g(of)f(the)h (Bieb)q(erbac)o(h)f(conjecture")h Fp(A)m(cta)h(Math.)21 b Fr(154)12 b Fs(\(1985\),)156 2650 y(137{152.)918 2770 y(82)p eop %%Page: 83 84 83 83 bop 52 192 a Fs([Di])24 b(J.)16 b(Dieudonn)o(\023)-24 b(e)16 b(\\Calcul)g(In\014nit)o(\023)-24 b(esimal")14 b(Hermann,)h(P)o(aris)g(\(1980\))41 263 y([Do])24 b(A.)15 b(Douady)f(\\Disques)g(de)g(Siegel)g(et)h(anneaux)f(de)g(Herman")g(S)o (\023)-24 b(eminaire)12 b(Bourbaki)i(n.)156 332 y(677,)i Fp(Ast)o(\023)-24 b(erisque)17 b Fr(152{153)d Fs(\(1987\),)j(151{172)50 404 y([F)l(a])24 b(K.)11 b(F)l(alconer)f(\\F)l(ractal)g(Geometry)l(.)20 b(Mathematical)11 b(F)l(oundations)e(and)i(Applications")156 473 y(John)16 b(Wiley)h(and)f(Sons)f(\(1990\))40 544 y([Ga])24 b(G.)i(Galla)o(v)o(otti)f(\\Quasi{In)o(tegrable)f(Mec)o (hanical)g(Systems")h(in)g(Ph)o(\023)-24 b(enom)o(\022)g(enes)24 b(cri-)156 614 y(tiques,)d(syst)o(\022)-24 b(emes)20 b(al)o(\023)-24 b(eatoires,)20 b(th)o(\023)-24 b(eories)20 b(de)g(jauge,)h(P)o(art)f(I,)g(I)q(I,)g(Les)g(Houc)o(hes)g(1984,)156 684 y(North{Holland,)c(Amsterdam)f(\(1986\))i(539{624)19 755 y([GM])24 b(A.)d(Giorgilli)d(and)i(A.)g(Morbidelli)e(\\In)o(v)m (arian)o(t)g(KAM)i(tori)g(and)f(global)g(stabilit)o(y)h(for)156 825 y(Hamiltonian)c(systems")g(ZAMP)g Fr(48)g Fs(\(1997\),)h(102{134.) 45 896 y([Gr])24 b(M.L.)h(Gromo)o(v)g(\\Smo)q(othing)f(and)g(In)o(v)o (ersion)g(of)h(Di\013eren)o(tial)g(Op)q(erators")f Fp(Math.)156 966 y(USSR)19 b(Sb)m(ornik)e Fr(17)f Fs(\(1972\),)h(381{434)41 1037 y([Ha])25 b(R.S.)e(Hamilton)g(\\The)g(In)o(v)o(erse)f(F)l(unction) h(Theorem)f(of)i(Nash)f(and)g(Moser")f Fp(Bul)s(l.)156 1106 y(A.M.S.)17 b Fr(7)g Fs(\(1982\),)g(65{222.)41 1178 y([H1])25 b(L.)k(H\177)-25 b(ormander)26 b(\\The)j(Analysis)e(of)i (Linear)e(P)o(artial)h(Di\013eren)o(tial)f(Op)q(erators)g(I")156 1247 y(Grundlehren)13 b(der)h(mathematisc)o(hen)f(Wissensc)o(haften)g Fr(256)p Fs(,)h(Springer{V)l(erlag,)f(Ber-)156 1317 y(lin,)j(Heidelb)q (erg,)g(New)h(Y)l(ork,)g(T)l(oky)o(o)f(\(1983\))41 1388 y([H2])25 b(L.)12 b(H\177)-25 b(ormander)11 b(\\The)g(b)q(oundary)g (problem)g(of)h(ph)o(ysical)f(geo)q(desy")h Fp(A)o(r)m(ch.)22 b(R)m(at.)h(Me)m(ch.)156 1458 y(A)o(nal.)g Fr(62)16 b Fs(\(1976\),)g(1{52)19 1529 y([He1])25 b(M.)17 b(R.)f(Herman)g (\\Examples)g(de)g(fractions)g(rationelles)g(a)o(y)o(an)o(t)g(une)g (orbite)g(dense)h(sur)156 1599 y(la)i(sphere)g(de)g(Riemann")f (Bulletin)h(de)g(la)g(So)q(ciet)o(\023)-24 b(e)20 b(Math)o(\023)-24 b(ematique)19 b(de)g(F)l(rance)f Fr(112)156 1669 y Fs(\(1984\),)f (93{142)19 1740 y([He2])25 b(M.)12 b(R.)f(Herman)g(\\Simple)f(pro)q (ofs)h(of)h(lo)q(cal)f(conjugacy)h(theorems)e(for)i(di\013eomorphisms) 156 1809 y(of)18 b(the)g(circle)f(with)g(almost)g(ev)o(ery)h(rotation)f (n)o(um)o(b)q(ers")e(Bull.)24 b(So)q(c.)h(Bras.)f(Mat.)h Fr(16)156 1879 y Fs(\(1985\))17 b(45{83)19 1950 y([He3])25 b(M.)30 b(R.)f(Herman)g(\\Are)h(there)f(critical)h(p)q(oin)o(ts)e(one)i (the)g(b)q(oundary)e(of)i(singular)156 2020 y(domains)8 b(?")22 b(Comm)o(un.)e(Math.)h(Ph)o(ys.)h Fr(99)15 b Fs(\(1985\))i(593{612.)19 2091 y([He4])25 b(M.)15 b(R.)g(Herman)g (\\Recen)o(t)g(results)f(and)h(some)g(op)q(en)g(questions)f(on)h (Siegel's)g(lineariza-)156 2161 y(tion)f(theorem)f(of)h(germs)e(of)i (complex)f(analytic)h(di\013eomorphisms)c(of)k Fr(C)1537 2143 y Fh(n)1578 2161 y Fs(near)f(a)h(\014xed)156 2231 y(p)q(oin)o(t")j(Pro)q(c.)26 b(VI)q(I)q(I)19 b(In)o(t.)25 b(Conf.)g(Math.)g(Ph)o(ys.)g(Mebkhout)17 b(and)g(Seneor)g(eds.)25 b(\(Sin-)156 2300 y(gap)q(ore)16 b(:)22 b(W)l(orld)16 b(Scien)o(ti\014c\))g(\(1986\),)h(138{184.)19 2371 y([He5])25 b(M.)17 b(R.)g(Herman)f(\\D)o(\023)-24 b(emonstration)16 b(du)h(th)o(\023)-24 b(eor)o(\022)g(eme)16 b(des)h(courb)q(es)f (translat)o(\023)-24 b(ees)17 b(par)f(dif-)156 2441 y(f)o(\023)-24 b(eomorphismes)12 b(de)i(l'anneau)9 b(;)15 b(d)o(\023)-24 b(emonstration)12 b(du)i(th)o(\023)-24 b(eor)o(\022)g(eme)14 b(des)f(tores)h(in)o(v)m(arian)o(ts")156 2511 y(man)o(uscripts)28 b(\(1980\))i(and)g(\\Abstract)g(metho)q(ds)g(in)f(small)g(divisors)g(:) 49 b(implicit)156 2581 y(function)17 b(theorems)e(in)i(F)l(r)o(\023)-24 b(ec)o(het)15 b(spaces",)h(lectures)g(giv)o(en)g(at)h(the)g(CIME)g (conference)156 2650 y(on)g(Dynamical)e(Systems)h(and)g(Small)f (Divisors,)h(Cetraro)f(1998)918 2770 y(83)p eop %%Page: 84 85 84 84 bop 35 192 a Fs([HL])25 b(G.)17 b(H.)h(Hardy)e(and)h(J.)f(E.)h (Littlew)o(o)q(o)q(d)g(\\Notes)h(on)e(the)i(theory)f(of)g(series)f (\(XXIV\))j(:)d(a)156 261 y(curious)f(p)q(o)o(w)o(er)h(series")f(Pro)q (c.)22 b(Cam)o(bridge)14 b(Phil.)22 b(So)q(c.)g Fr(42)16 b Fs(\(1946\),)g(85{90)15 337 y([HW])25 b(G.H.)d(Hardy)e(and)h(E.M.)f (W)l(righ)o(t)g(\\An)h(in)o(tro)q(duction)f(to)h(the)h(theory)f(of)g(n) o(um)o(b)q(ers")156 407 y(Fifth)c(Edition,)e(Oxford)h(Science)g (Publications)f(\(1990\))65 482 y([K])24 b(A.)12 b(N.)g(Kolmogoro)o(v)e (\\On)h(the)h(p)q(ersistence)f(of)g(conditionally)g(p)q(erio)q(dic)g (motions)g(under)156 552 y(a)18 b(small)e(p)q(erturbation)g(of)h(the)g (Hamilton)g(function")g(Dokl.)24 b(Ak)m(ad.)h(Nauk)17 b(SSSR)f Fr(98)156 622 y Fs(\(1954\))21 b(527{530)e(\(in)h(Russian)e(:) 29 b(English)19 b(translation)g(in)g(G.)h(Casati)g(and)g(J.)f(F)l(ord,) 156 691 y(editors,)14 b(Sto)q(c)o(hastic)h(Beha)o(vior)f(in)h (Classical)e(and)h(Quan)o(tum)f(Hamiltonian)h(Systems,)156 761 y(Lecture)j(Notes)g(in)f(Ph)o(ysics)g Fr(93)f Fs(\(1979\))i(51{56)g (Springer{V)l(erlag\).)73 837 y([L])24 b(S.)34 b(Lang)g(\\In)o(tro)q (duction)f(to)i(Diophan)o(tine)d(Appro)o(ximation")h(Addison{W)l(esley) 156 906 y(\(1966\))48 982 y([Lo])24 b(P)l(.)13 b(Lo)q(c)o(hak)h (\\Canonical)e(p)q(erturbation)g(theory)i(via)f(sim)o(ultaneous)e (appro)o(ximations",)156 1052 y(Russ.)21 b(Math.)h(Surv.)f Fr(47)16 b Fs(\(1992\),)g(57{133.)8 1127 y([Ma1])24 b(S.)12 b(Marmi)f(\\Critical)h(F)l(unctions)f(for)i(Complex)f(Analytic)h(Maps") e Fp(J.)k(Phys.)22 b(A)15 b(:)f(Math.)156 1197 y(Gen.)22 b Fr(23)16 b Fs(\(1990\),)h(3447{3474)8 1273 y([Ma2])24 b(S.)g(Marmi)e(\\Chaotic)i(Beha)o(viour)e(in)i(the)g(Solar)e(System)i (\(F)l(ollo)o(wing)e(J.)h(Lask)m(ar\)")156 1342 y(S)o(\023)-24 b(eminaire)15 b(Bourbaki)h(n.)21 b(854,)c(No)o(v)o(em)o(b)q(er)e(1998,) h(to)h(app)q(ear)f(in)g(Ast)o(\023)-24 b(erisque)36 1418 y([Me])24 b(Y.)13 b(Mey)o(er)f(\\Algebraic)g(Num)o(b)q(ers)f(and)h (Harmonic)g(Analysis")g(North{Holland)f(Math-)156 1488 y(ematical)17 b(Library)e Fr(2)h Fs(\(1972\))12 1563 y([MM])24 b(L.)29 b(Markus)f(and)g(K.)h(R.)f(Mey)o(er)g(\\Generic)h (Hamiltonian)e(Systems)i(are)f(neither)156 1633 y(in)o(tegrable)16 b(nor)f(ergo)q(dic")h(Memoirs)f(of)i(the)g(A.M.S.)e Fr(144)h Fs(\(1974\))-25 1709 y([MMY])24 b(S.)h(Marmi,)g(P)l(.)g(Moussa)f(and)g (J.{C.)g(Y)l(o)q(ccoz)i(\\The)f(Brjuno)f(functions)g(and)h(their)156 1778 y(regularit)o(y)16 b(prop)q(erties")f Fp(Commun.)22 b(Math.)i(Phys.)f Fr(186)15 b Fs(\(1997\),)i(265-293)-50 1854 y([MMY2])24 b(S.)14 b(Marmi,)f(P)l(.)g(Moussa)g(and)g(J.{C.)g(Y)l (o)q(ccoz)i(\\Complex)e(Brjuno)g(F)l(unctions")f(preprin)o(t)156 1924 y(SPhT)k(Sacla)o(y)l(,)g(F)l(rance,)f(71)h(pages)g(\(1999\))36 1999 y([Mc])24 b(C.)19 b(T.)g(McMullen)f(\\Complex)g(Dynamics)g(and)g (Renormalization")f(Ann.)29 b(of)19 b(Math.)156 2069 y(Studies,)d(Princeton)g(Univ)o(ersit)o(y)f(Press)h(\(1994\))30 2145 y([Mn])24 b(R.)j(Ma)q(~)-26 b(ne)27 b(\\Ergo)q(dic)g(Theory)g(and) g(Di\013eren)o(tiable)f(Dynamics")h(Springer{V)l(erlag)156 2214 y(\(1987\))58 2290 y([M])d(J.)d(Moser)e(\\A)i(rapidly)f(con)o(v)o (ergen)o(t)f(iteration)h(metho)q(d)h(and)f(nonlinear)f(di\013eren)o (tial)156 2360 y(equations")d Fp(A)o(nn.)23 b(Scuola)18 b(Norm.)24 b(Sup.)f(Pisa)18 b Fr(20)e Fs(\(1966\))h(499-535)66 2435 y([N])25 b(J.)16 b(Nash)g(\\The)g(em)o(b)q(edding)e(problem)h(for) g(Riemannian)g(manifolds")f Fp(A)o(nn.)23 b(of)18 b(Math.)156 2505 y Fr(63)e Fs(\(1956\))h(20{63)44 2581 y([Ne])25 b(N.N.)14 b(Nekhoroshev)g(\\An)f(exp)q(onen)o(tial)g(estimate)h(for)f (the)h(time)f(of)h(stabilit)o(y)f(of)h(nearly)156 2650 y(in)o(tegrable)i(Hamiltonian)f(systems")h Fp(R)o(uss.)23 b(Math.)h(Surveys)17 b Fr(32)e Fs(\(1977\),)i(1{65.)918 2770 y(84)p eop %%Page: 85 86 85 85 bop 53 192 a Fs([Ni])24 b(L.)h(Nirem)o(b)q(erg)e(\\An)h(abstract) h(form)e(of)i(the)g(non{linear)d(Cauc)o(h)o(y{Ko)o(w)o(alewsk)m(a)o(y)o (a)156 261 y(theorem")16 b Fp(J.)i(Di\013.)23 b(Ge)m(om.)f Fr(6)17 b Fs(\(1972\))g(561{576)-1 332 y([PM1])24 b(R.)e(P)o(\023)-24 b(erez{Marco)21 b(\\Solution)g(compl)o(\022)-24 b(ete)22 b(au)f(probl)o(\022)-24 b(eme)21 b(de)h(Siegel)f(de)h(lin)o(\023)-24 b(earisation)156 402 y(d'une)25 b(application)g(holomorphe)e(au)j(v)o (oisinage)f(d'un)f(p)q(oin)o(t)i(\014xe)g(\(d'apr)o(\022)-24 b(es)24 b(J.{C.)156 472 y(Y)l(o)q(ccoz\)")18 b(S)o(\023)-24 b(eminaire)15 b(Bourbaki)h(n.)21 b(753,)16 b Fp(Ast)o(\023)-24 b(erisque)17 b Fr(206)f Fs(\(1992\),)g(273{310)42 542 y([Ph])24 b(F.)c(Pham)f(\(Editor\))h(\\Hyp)q(erfunctions)g(and)f (Theoretical)h(Ph)o(ysics")f(Lecture)h(Notes)156 612 y(in)c(Mathematics)g Fr(449)g Fs(Springer{V)l(erlag)e(\(1975\))70 683 y([P])24 b(H.)18 b(P)o(oincar)o(\023)-24 b(e)16 b(\\Les)i(M)o(\023) -24 b(etho)q(des)17 b(Nouv)o(elles)g(de)g(la)h(M)o(\023)-24 b(ecanique)17 b(Celeste",)h(tomes)f(I{I)q(I)q(I)156 753 y(P)o(aris)e(Gauthier{Villars)g(\(1892,)h(1893,)h(1899\).)46 824 y([P)o(o])24 b(Ch.)38 b(P)o(ommerenk)o(e)20 b(\\Boundary)h(Beha)o (viour)g(of)h(Conformal)e(Maps")h(Grundlehren)156 893 y(der)16 b(Mathematisc)o(hen)o(t)f(Wissensc)o(haften)g Fr(299)p Fs(,)h(Springer{V)l(erlag)e(\(1992\))45 964 y([P\177)-25 b(o])24 b(J.)12 b(P\177)-25 b(osc)o(hel)10 b(\\In)o(tegrabilit)o(y)h(of)h(Hamiltonian)f(systems)g(on)g(Can)o(tor)g (sets")h(Comm.)19 b(Pure)156 1034 y(Appl.)j(Math.)f Fr(35)16 b Fs(653{696)g(\(1982\))45 1105 y([Re])24 b(R.)15 b(Remmert)e (\\Classical)h(T)l(opics)g(in)g(Complex)g(F)l(unction)g(Theory")g (Graduate)g(T)l(exts)156 1175 y(in)i(Mathematics)g Fr(172)p Fs(,)g(Springer{V)l(erlag)e(\(1998\))39 1245 y([R)q(\177)-26 b(u])24 b(H.)18 b(R)q(\177)-26 b(ussmann)15 b(\\Kleine)i(Nenner)f(I)q (I)i(:)g(Bemerkungen)e(zur)h(Newtonsc)o(hen)f(Metho)q(de")156 1315 y(Nac)o(hr.)22 b(Ak)m(ad.)g(Wiss.)f(G\177)-25 b(ottingen)17 b(Math.)k(Ph)o(ys.)h(Kl)16 b(\(1972\))h(1{20)76 1386 y([S])24 b(C.L.)12 b(Siegel)g(\\Iteration)f(of)h(analytic)g(functions") f(Annals)h(of)g(Mathematics)f Fr(43)g Fs(\(1942\),)156 1456 y(807-812.)3 1527 y([Sc)o(h1])23 b(W.M.)14 b(Sc)o(hmidt)f (\\Diophan)o(tine)g(Appro)o(ximation")f(Lecture)j(Notes)f(in)g (Mathematics,)156 1596 y Fr(785)p Fs(,)i(Springer{V)l(erlag)e(\(1980\)) 3 1667 y([Sc)o(h2])23 b(W.M.)g(Sc)o(hmidt)f(\\Diophan)o(tine)g(Appro)o (ximations)f(and)h(Diophan)o(tine)g(Equations")156 1737 y(Lecture)17 b(Notes)g(in)f(Mathematics,)g Fr(1467)p Fs(,)f(Springer{V)l(erlag)f(\(1991\))34 1808 y([Ser])24 b(F.)d(Sergeraert)g(\\Un)h(th)o(\023)-24 b(eor)o(\022)g(eme)21 b(de)g(fonctions)g(implicites)g(sur)g(certains)g(espaces)g(de)156 1878 y(F)l(r)o(\023)-24 b(ec)o(het)19 b(et)h(quelques)g(applications")e Fp(A)o(nn.)31 b(Scient.)1234 1865 y(\022)1226 1878 y(Ec.)h(Norm.)g (Sup.)g Fr(5)19 b Fs(\(1972\),)156 1947 y(599{660.)40 2018 y([ST])24 b(J.)14 b(Silv)o(erman)e(and)h(J.)g(T)l(ate)i (\\Rational)e(P)o(oin)o(ts)f(on)i(Elliptic)f(Curv)o(es")g (Undergraduate)156 2088 y(T)l(exts)k(in)f(Mathematics,)g(Springer{V)l (erlag)e(\(1992\))46 2159 y([SZ])24 b(D.)c(Salomon)f(and)g(E.)h (Zehnder)g(\\KAM)g(theory)g(in)f(confuguration)g(space")g Fp(Comm.)156 2229 y(Math.)24 b(Helvetici)16 b Fr(64)p Fs(,)f(\(1989\),)i(84{132)57 2299 y([St])24 b(S.)14 b(Stern)o(b)q(erg)g (\\Celestial)g(Mec)o(hanics")f(\(t)o(w)o(o)h(v)o(olumes\))g(W.A.)g (Benjamin,)g(New)h(Y)l(ork)156 2369 y(\(1969\).)46 2440 y([V)l(a])24 b(F.H.)16 b(V)l(asilescu)g(\\Analytic)h(F)l(unctional)e (Calculus")g(D.)i(Reidel)f(Publ.)21 b(Co.)h(\(1982\))21 2511 y([Y)l(o1])i(J.{C.)15 b(Y)l(o)q(ccoz)i(\\An)f(in)o(tro)q(duction)e (to)i(small)f(divisors)f(problems")f(in)j(\\F)l(rom)e(n)o(um)o(b)q(er) 156 2581 y(theory)k(to)g(ph)o(ysics",)f(M.)g(W)l(aldsc)o(hmidt,)f(P)l (.)h(Moussa,)g(J.M.)g(Luc)o(k)g(and)g(C.)h(Itzykson)156 2650 y(\(editors\))f(Springer{V)l(erlag)d(\(1992\),)j(659{679)918 2770 y(85)p eop %%Page: 86 87 86 86 bop 21 192 a Fs([Y)l(o2])24 b(J.{C.)17 b(Y)l(o)q(ccoz)h(\\Th)o (\023)-24 b(eor)o(\022)g(eme)16 b(de)h(Siegel,)g(nom)o(bres)e(de)i (Bruno)f(et)i(p)q(olyn^)-25 b(omes)16 b(quadra-)156 261 y(tiques")h Fp(Ast)o(\023)-24 b(erisque)16 b Fr(231)g Fs(\(1995\),)h(3-88.)21 331 y([Y)l(o3])24 b(J.{C.)17 b(Y)l(o)q(ccoz,)i(lectures)e(giv)o(en)h(at)g(the)g(CIME)f(conference)h (on)f(Dynamical)g(Systems)156 401 y(and)d(Small)f(Divisors,)g(Cetraro)g (1998,)h(to)h(app)q(ear)e(in)h(Lecture)g(Notes)h(in)e(Mathematics)26 470 y([Ze1])25 b(E.)d(Zehnder)g(\\Generalized)f(Implicit)g(F)l(unction) g(Theorems)g(with)h(Applications)f(to)156 540 y(some)f(Small)g(Divisor) f(Problems)g(\(I)i(and)f(I)q(I\)")h Fp(Commun.)34 b(Pur)m(e)21 b(Appl.)35 b(Math.)g Fr(28)156 610 y Fs(\(1975\))17 b(91{140,)f Fr(29)g Fs(\(1976\))h(49{113.)26 680 y([Ze2])25 b(E.)15 b(Zehnder)f(\\A)i(simple)d(pro)q(of)i(of)g(a)g(generalization)f(of)h(a) g(Theorem)f(b)o(y)g(C.)h(L.)g(Siegel")156 749 y(Lecture)i(Notes)g(in)f (Mathematics)g Fr(597)f Fs(\(1977\))i(855{866.)918 2770 y(86)p eop %%Page: 87 88 87 87 bop 57 192 a Fq(Analytical)26 b(index)156 298 y Fs(action{angle)16 b(v)m(ariables)g(pp.49{53)156 368 y(adjoin)o(t)g(action)h(p.)k(4)156 438 y(area)16 b(form)o(ula)f(p.)22 b(70)156 508 y(area)16 b(theorem)g(p.)22 b(70)156 578 y(Arnol'd{Liouville)15 b(Theorem)h(p.)21 b(50)156 684 y(Beltrami)16 b(equation)h(p.)k(74)156 754 y(Best)c(Appro)o(ximation)e (Theorem)h(p.)21 b(22)156 824 y(Bieb)q(erbac)o(h{De)16 b(Branges)g(Theorem)f(pp.)21 b(31,)c(70,)f(71)156 894 y(Brjuno)g(n)o(um)o(b)q(er)f(pp.)21 b(25,)16 b(27,)g(32,)h(33,)f(34)156 965 y(Brjuno)g(function)g(p.)22 b(25,)16 b(28,)g(34)156 1035 y(Brjuno)g(Theorem)g(p.)21 b(17,)c(27)156 1141 y(Caratheo)q (dory's)f(theorem)f(p.)22 b(68)156 1211 y(cen)o(tralizer)16 b(p.)21 b(4,)c(8)156 1281 y(completely)g(canonically)f(in)o(tegrable)f (pp.)21 b(49,)16 b(50,)h(51,)f(53,)g(54)156 1351 y(conformal)g(map)f (p.)22 b(14,)16 b(17,)g(67,)h(73,)f(74)156 1421 y(conformal)g(capacit)o (y)g(pp.)21 b(14,)c(36,)f(68)156 1491 y(conjugate)h(p.)22 b(4)156 1561 y(con)o(tin)o(ued)15 b(fractions)h(pp.)21 b(21,)c(22,)f(23,)g(25,)g(39,)h(75)156 1632 y(Cremer's)e(Theorem)h(p.) 21 b(9)156 1702 y(critical)c(p)q(oin)o(t)f(\(on)g(the)h(b)q(oundary\))f (pp.)21 b(20,)16 b(39)156 1772 y(cycle)h(p.)22 b(12)156 1878 y(Darb)q(oux's)16 b(theorem)g(p.)21 b(47)156 1948 y(Da)o(vie's)16 b(lemmas)f(pp.)22 b(29,)16 b(30,)g(32)156 2018 y(dilatation)g(pp.)22 b(73,)16 b(74)156 2088 y(diophan)o(tine)h(n) o(um)o(b)q(er,)f(condition,)i(v)o(ector)h(pp.)27 b(20,)18 b(23,)h(25,)g(39,)f(43,)h(55{57,)f(64,)h(65,)156 2158 y(81)156 2228 y(Douady{Gh)o(ys')c(Theorem)h(pp.)21 b(21,)c(27,)f(36)156 2298 y(Douady{Hubbard's)e(theorem)i(pp.)21 b(16,)c(74)156 2404 y(F)l(atou)f(set)h(pp.)k(12,)16 b(13)156 2474 y(F)l(atou's)g (theorem)f(pp.)22 b(20,)16 b(69)156 2544 y(F)l(r)o(\023)-24 b(ec)o(het)16 b(space)g(pp.)21 b(58{62,)16 b(64)156 2650 y(Gauss)g(map)g(pp.)21 b(75,)16 b(77)918 2770 y(87)p eop %%Page: 88 89 88 88 bop 156 192 a Fl(G)195 199 y Fh(\016)234 192 y Fs(set)17 b(p.)k(9)156 262 y(geometric)16 b(renormalization)f(pp.)21 b(34,)16 b(40)156 332 y(germ)g(p.)22 b(3)156 402 y(Gevrey)17 b(class)f(p.)22 b(32)156 472 y(Grunsky)16 b(norm)f(p.)22 b(39)156 578 y(Hamiltonian)16 b(pp.)21 b(48,)16 b(53,)h(57)156 648 y(Hardy{Sob)q(olev)f(spaces)g(pp.)21 b(40,)c(44)156 718 y(hedgehog)f(p.)22 b(14)156 788 y(h)o(yp)q(erfunction)16 b(pp.)21 b(78{80)156 858 y(h)o(yp)q(o)q(elliptic)c(pp.)k(55,)16 b(81)156 928 y(Hurwitz')h(Lemma)e(p.)22 b(72)156 1034 y(Jarnik's)15 b(theorem)h(p.)22 b(24)156 1104 y(Julia)16 b(set)h(p.)k(12)156 1210 y(KAM)c(Theorem)e(pp.)21 b(56,)c(57,)f(64)156 1280 y(Ko)q(eb)q(e)h(1)p Fl(=)p Fs(4{Theorem)e(p.)22 b(71)156 1350 y(Ko)q(eb)q(e)17 b(distorsion)e(theorems)g(p.)22 b(71)156 1420 y(Ko)q(eb)q(e)17 b(function)f(p.)22 b(71)156 1490 y(Ko)q(eb)q(e)17 b(transform)e(p.)22 b(71)156 1561 y(Ko)q(enigs{P)o(oincar)o(\023)-24 b(e)15 b(Theorem)g(pp.)21 b(7-9)156 1666 y(Lagrange's)15 b(in)o(v)o(ersion)g(theorem)h(p.)21 b(44)156 1737 y(linearizable,)d(linearization)g(pp.)29 b(5{10,)19 b(13{14,)h(16,)f(19,)g(22,)h(27{28,)f(32{34,)g(38,)h(40,)156 1806 y(46)156 1876 y(Liouville)c(n)o(um)o(b)q(er)f(p.)21 b(24)156 1946 y(Liouville's)16 b(Theorem)f(p.)22 b(23)156 2017 y(loss)16 b(of)h(di\013eren)o(tiabilit)o(y)e(pp.)21 b(40{41,)16 b(46,)g(56,)g(58,)h(61)156 2122 y(Maxim)o(um)e(principle)g (p.)22 b(67)156 2193 y(Measurable)15 b(Riemann)g(Mapping)g(Theorem)g (p.)22 b(74)156 2298 y(Nash{Moser)15 b(Theorem)h(pp.)21 b(58,)16 b(62{64)156 2369 y(Nekhoroshev)g(Theorem)g(pp.)21 b(56{57)156 2439 y(normal)15 b(family)h(p.)22 b(11{12)156 2545 y(orbit)16 b(p.)22 b(4)156 2650 y(P)o(oincar)o(\023)-24 b(e)16 b(Theorem)f(p.)22 b(54)918 2770 y(88)p eop %%Page: 89 90 89 89 bop 156 192 a Fs(P)o(oisson)15 b(brac)o(k)o(et)h(p.)22 b(48)156 297 y(quadratic)16 b(p)q(olynomial)g(pp.)21 b(16,)16 b(37)156 366 y(quadrilateral)f(pp.)22 b(73{74)156 436 y(quasicircle)16 b(p.)22 b(39)156 506 y(quasiconformal)15 b(map)h(pp.)21 b(16,)16 b(39,)g(73{74)156 576 y(quasi{in)o(tegrable)f (system)h(pp.)21 b(53{54,)16 b(56)156 645 y(quasip)q(erio)q(dic)g (function)g(p.)22 b(49)156 751 y(rational)16 b(map)g(pp.)21 b(11{12)156 820 y(region)16 b(p.)22 b(67)156 890 y(Riemann)15 b(mapping)g(theorem)h(p.)22 b(68)156 960 y(Roth's)16 b(Theorem)f(pp.)22 b(23{55)156 1065 y(Sc)o(h)o(w)o(arzian)14 b(deriv)m(ativ)o(e)j(p.)k(42)156 1135 y(Sc)o(h)o(w)o(arz's)14 b(Lemma)i(p.)22 b(67)156 1204 y(Siegel{Brjuno)16 b(Theorem)f(pp.)21 b(17,)c(27)156 1274 y(Siegel)f(disk)h(p.)k(14)156 1344 y(smo)q(othing)16 b(op)q(erators)f(pp.)22 b(58,)16 b(61{62)156 1414 y(spherical)f(metric)h(p.)22 b(11)156 1483 y(stable)17 b(p)q(oin)o(t)f(pp.)21 b(13{14)156 1553 y(symmetry)16 b(p.)22 b(4)156 1623 y(symplectic)17 b(manifold)e(p.)21 b(47)156 1728 y(tame)c(map)f(pp.)21 b(61{62,)16 b(64{66)156 1798 y(tame)h(F)l(r)o(\023)-24 b(ec)o(het)15 b(space)i(pp.)k(61{62)156 1903 y(ultradi\013eren)o(tiable)15 b(p)q(o)o(w)o(er)g(series)h(p.)21 b(32)156 1973 y(Uniformization)16 b(theorem)f(p.)22 b(69)156 2042 y(uniquely)16 b(ergo)q(dic)g(pp.)22 b(38,)16 b(49)156 2112 y(univ)m(alen)o(t)g(function)g(pp.)22 b(10,39,)16 b(67,)g(69{72)156 2217 y(Y)l(o)q(ccoz's)h(lo)o(w)o(er)e(b)q(ound)h(p.) 22 b(28)156 2287 y(Y)l(o)q(ccoz's)17 b(pro)q(of)f(of)h(Siegel's)f (Theorem)f(pp.)22 b(16{20)156 2357 y(Y)l(o)q(ccoz's)17 b Fl(u)p Fs(\()p Fl(\025)p Fs(\))g(function)f(pp.)22 b(17{20,)16 b(37{39)156 2427 y(Y)l(o)q(ccoz's)h(Theorem)f(pp.)21 b(33{34)918 2770 y(89)p eop %%Page: 90 91 90 90 bop 57 192 a Fo(List)20 b(of)g(sym)n(b)r(ols)156 297 y Fs(Ad)d(:)22 b(adjoin)o(t)16 b(action)156 366 y Fl(c)p Fs(\(\012)p Fl(;)8 b(z)278 373 y Fi(0)301 366 y Fs(\))17 b(:)22 b(conformal)15 b(capacit)o(y)i(of)f(\012)h(w.r.t.)k Fl(z)1071 373 y Fi(0)156 436 y Fm(C)29 b Fs(:)22 b(complex)16 b(plane)156 506 y Fm(C)189 488 y Fj(\003)232 506 y Fs(:)22 b Fm(C)h Fk(n)11 b(f)p Fs(0)p Fk(g)p 156 535 36 2 v 156 576 a Fm(C)29 b Fs(:)22 b(Riemann)15 b(sphere)156 645 y Fm(C)9 b Fk(f)p Fl(z)s Fk(g)19 b Fs(:)j(ring)16 b(of)g(con)o(v)o (ergen)o(t)g(p)q(o)o(w)o(er)f(series)g(in)i(one)f(complex)g(v)m (ariable)156 715 y Fm(C)9 b Fs([[)p Fl(z)r Fs(]])20 b(:)i(ring)15 b(of)i(formal)e(p)q(o)o(w)o(er)h(series)f(in)h(one)h(complex)f(v)m (ariable)156 785 y(Cen)o(t)h(:)22 b(cen)o(tralizer)161 844 y Fm([)156 855 y Fs(Cen)o(t)17 b(:)22 b(formal)15 b(cen)o(tralizer)156 924 y Fl(D)197 931 y Fh(f)223 924 y Fs(\()p Fl(z)265 931 y Fi(0)288 924 y Fs(\),)i Fl(D)379 931 y Fh(f)422 924 y Fs(:)22 b(dilatiation)15 b(of)i(f)g(\(maximal\)) 156 994 y Fm(D)28 b Fs(:)22 b(op)q(en)16 b(unit)g(disk)g Fk(fj)p Fl(z)r Fk(j)e Fl(<)g Fs(1)p Fk(g)156 1064 y Fm(D)189 1071 y Fh(r)231 1064 y Fs(:)22 b(op)q(en)16 b(disk)g Fk(fj)p Fl(z)r Fk(j)e Fl(<)g(r)q Fk(g)j Fs(of)g(radius)e Fl(r)g(>)f Fs(0.)156 1134 y Fm(E)24 b Fs(:)e(outer)16 b(disk)g Fk(fj)p Fl(z)r Fk(j)e Fl(>)g Fs(1)p Fk(g)156 1203 y Fs([)p Fl(f)5 b Fs(])17 b(:)22 b(orbit)16 b(of)h Fl(f)156 1273 y(F)7 b Fs(\()p Fl(R)p Fs(\))18 b(:)k(F)l(atou)16 b(set)g(of)h Fl(R)156 1343 y(G)g Fs(:)22 b(group)15 b(of)i(germs)e(of)i (holomorphic)d(di\013eomorphisms)f(of)k(\()p Fm(C)10 b Fl(;)e Fs(0\))168 1400 y(^)156 1413 y Fl(G)17 b Fs(:)22 b(formal)16 b(analogue)f(of)i Fl(G)156 1482 y(G)195 1489 y Fh(\025)238 1482 y Fs(:)22 b(elemen)o(ts)16 b(of)h Fl(G)f Fs(with)h(linear)e(part)h Fl(\025)168 1539 y Fs(^)156 1552 y Fl(G)195 1559 y Fh(\025)238 1552 y Fs(:)22 b(formal)16 b(analogue)f(of)i Fl(G)734 1559 y Fh(\025)156 1622 y Fl(J)5 b Fs(\()p Fl(R)p Fs(\))17 b(:)22 b(Julia)16 b(set)g(of)h Fl(R)156 1691 y Fm(N)h Fs(:)k(non{negativ)o(e)16 b(in)o(tegers)156 1761 y(\012)h(:)22 b(a)16 b(region)g(of)h Fm(C)156 1831 y(Q)h Fs(:)k(rational)15 b(in)o(tegers)156 1901 y Fl(R)194 1908 y Fh(\025)237 1901 y Fs(:)22 b(the)17 b(germ)e Fl(R)521 1908 y Fh(\025)547 1901 y Fs(\()p Fl(z)r Fs(\))g(=)f Fl(\025z)156 1970 y(S)19 b Fs(:)j(univ)m(alen)o(t)16 b(maps)g(on)g Fm(D)156 2040 y Fl(S)187 2047 y Fh(\025)230 2040 y Fs(:)22 b(elemen)o(ts)15 b(of)i Fl(S)i Fs(with)d(linear)g(part)g Fl(R)974 2047 y Fh(\025)156 2110 y Fl(S)187 2119 y Fg(S)205 2109 y Fc(1)245 2110 y Fs(:)22 b(elemen)o(ts)15 b(of)i Fl(S)i Fs(with)e(linear)e(part)h(of)h(unit)f(mo)q(dulus)156 2180 y Fm(S)187 2162 y Fi(1)223 2180 y Fs(:)22 b(unit)16 b(circle)g Fk(fj)p Fl(z)r Fk(j)e Fs(=)g(1)p Fk(g)156 2249 y Fl(u)i Fs(Y)l(o)q(ccoz's)h(function,)f(see)h(Section)f(3.1)156 2319 y Fk(Y)21 b Fs(:)h(see)17 b(Chapter)f(4)156 2389 y Fm(Z)s Fs(:)22 b(in)o(tegers)57 2459 y(Stefano)c(Marmi,)e(Dipartimen) o(to)h(di)h(Matematica)f(e)i(Informatica,)e(Univ)o(ersit\022)-25 b(a)17 b(di)h(Udine,)57 2528 y(Via)e(delle)h(Scienze)f(206,)g(Lo)q (calit\022)-25 b(a)17 b(Rizzi,)f(33100)g(Udine,)g(Italy)11 b(;)16 b(marmi@dimi.un)o(iud.)o(it)918 2770 y(90)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF