%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: filej.dvi %%Pages: 19 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o filej.ps filej.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 1999.12.02:1839 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet TeXDict begin 39158280 55380996 1000 600 600 (filej.dvi) @start %DVIPSBitmapFont: Fa cmss10 10 1 /Fa 1 91 df<007FB7FCA55EC8EA03FC15074B5AA24B5A5E153F4B5A5E15FF4A90C7FCA2 4A5A5D14074A5A5D141F4A5AA24A5A5D14FF4990C8FC5C1303495AA2495A5C131F495A5C 137F495AA24890C9FC5B1203485A5B120F485AA2485A5B48B71280B8FCA5293A7BB933> 90 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmss8 7 1 /Fb 1 91 df<007FB512FEA315FCC71201EC03F815F01407EC0FE0EC1FC0A2EC3F80EC7F 00147E14FE495A5C1303495A495AA2495A495A91C7FC5B13FE5B1201485A485AA2485A48 5A5B123F48C8FC90B512FEB6FCA31F287DA726>90 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmsy5 5 2 /Fc 2 49 df0 D48 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmmi5 5 2 /Fd 2 27 df<007F130C48131E120FA2001E133CA2147814F05AEB01E0EB03C0EB078038 781E00137CEA79F0EA7FC048C7FC12F017127C911E>23 D26 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmr5 5 4 /Fe 4 51 df<14E0B0B712C0A3C700E0C7FCB022237C9B2B>43 D48 D<1360EA01E0120F12FF12F1 1201B3A3387FFF80A2111C7B9B1C>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmbx7 7 2 /Ff 2 102 df<003FB612F8A39039F8003FF001C0EB7FE04913FF494813C0007E158000 7C4913005C4A5A00785C4A5A143FC7485A5D4A5A5B495B92C7FC495A130F495A5C494813 78137F495A5C484913F85A4890C7FC491301484814F0001F14034848130749131F484813 7FB7FCA325287CA72E>90 D101 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmbx10 10 23 /Fg 23 120 df46 D65 DI 76 D80 D<003FB91280A4D9F800EBF003D87FC09238007FC049 161F007EC7150FA2007C1707A200781703A400F818E0481701A4C892C7FCB3AE010FB7FC A43B387DB742>84 D97 D<903801FFC0010F13FC017F13FFD9FF8013802603 FE0013C048485AEA0FF8121F13F0123F6E13804848EB7F00151C92C7FC12FFA9127FA27F 123FED01E06C7E15036C6CEB07C06C6C14806C6C131FC69038C07E006DB45A010F13F001 01138023257DA42A>99 DI<903803FF80011F13F0017F13FC3901FF83FE3A03FE007F804848133F4848 14C0001FEC1FE05B003FEC0FF0A2485A16F8150712FFA290B6FCA301E0C8FCA4127FA36C 7E1678121F6C6C14F86D14F000071403D801FFEB0FE06C9038C07FC06DB51200010F13FC 010113E025257DA42C>I<161FD907FEEBFFC090387FFFE348B6EAEFE02607FE07138F26 0FF801131F48486C138F003F15CF4990387FC7C0EEC000007F81A6003F5DA26D13FF001F 5D6C6C4890C7FC3907FE07FE48B512F86D13E0261E07FEC8FC90CAFCA2123E123F7F6C7E 90B512F8EDFF8016E06C15F86C816C815A001F81393FC0000F48C8138048157F5A163FA3 6C157F6C16006D5C6C6C495AD81FF0EB07FCD807FEEB3FF00001B612C06C6C91C7FC0107 13F02B377DA530>103 D<13FFB5FCA412077EAFED7FC0913803FFF8020F13FE91381F03 FFDA3C01138014784A7E4A14C05CA25CA291C7FCB3A3B5D8FC3F13FFA4303A7DB935>I< EA01F0EA07FC487EA2487EA56C5AA26C5AEA01F0C8FCA913FF127FA412077EB3A9B512F8 A4153B7DBA1B>I<13FFB5FCA412077EAF92380FFFE0A4923803FC0016F0ED0FE0ED1F80 4BC7FC157E5DEC03F8EC07E04A5A141FEC7FE04A7E8181A2ECCFFEEC0FFF496C7F806E7F 6E7F82157F6F7E6F7E82150F82B5D8F83F13F8A42D3A7EB932>107 D<13FFB5FCA412077EB3B3ACB512FCA4163A7DB91B>I<01FED97FE0EB0FFC00FF902601 FFFC90383FFF80020701FF90B512E0DA1F81903983F03FF0DA3C00903887801F000749DA CF007F00034914DE6D48D97FFC6D7E4A5CA24A5CA291C75BB3A3B5D8FC1FB50083B512F0 A44C257DA451>I<01FEEB7FC000FF903803FFF8020F13FE91381F03FFDA3C0113800007 13780003497E6D4814C05CA25CA291C7FCB3A3B5D8FC3F13FFA430257DA435>I<903801 FFC0010F13F8017F13FFD9FF807F3A03FE003FE048486D7E48486D7E48486D7EA2003F81 491303007F81A300FF1680A9007F1600A3003F5D6D1307001F5DA26C6C495A6C6C495A6C 6C495A6C6C6CB45A6C6CB5C7FC011F13FC010113C029257DA430>I<9039FF01FF80B500 0F13F0023F13FC9138FE07FFDAF00113800007496C13C06C0180EB7FE091C713F0EE3FF8 A2EE1FFCA3EE0FFEAA17FC161FA217F8163F17F06E137F6E14E06EEBFFC0DAF003138091 39FC07FE0091383FFFF8020F13E0020390C7FC91C9FCACB512FCA42F357EA435>I<9038 FE03F000FFEB0FFEEC3FFF91387C7F809138F8FFC000075B6C6C5A5CA29138807F80ED3F 00150C92C7FC91C8FCB3A2B512FEA422257EA427>114 D<90383FF0383903FFFEF8000F 13FF381FC00F383F0003007E1301007C130012FC15787E7E6D130013FCEBFFE06C13FCEC FF806C14C06C14F06C14F81203C614FC131F9038007FFE140700F0130114007E157E7E15 7C6C14FC6C14F8EB80019038F007F090B512C000F8140038E01FF81F257DA426>I<130F A55BA45BA25B5BA25A1207001FEBFFE0B6FCA3000390C7FCB21578A815F86CEB80F01481 6CEBC3E090383FFFC06D1380903803FE001D357EB425>I119 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmti10 10 24 /Fh 24 119 df12 D<140C140E80EC0380A2EC01C015E0A2140015F0A21578A4157C153CAB157CA7 15FCA215F8A21401A215F0A21403A215E0A21407A215C0140F1580A2141F1500A2143EA2 5CA25CA2495AA2495A5C1307495A91C7FC5B133E133C5B5B485A12035B48C8FC120E5A12 785A12C01E527FBD22>41 D<120EEA3F80127F12FFA31300127E123C0909778819>46 D<0107B612FCEFFF8018C0903B000FF0001FF04BEB07F81703021F15FC17014B14FEA202 3F1400A24B1301A2147F18FC92C7120318F84A140718F04AEC0FE0EF1FC00101ED3F80EF 7F004AEB01FEEE07F849B612E05F9139F80007F0EE01FC01076E7E177F4AEC3F80A2010F 16C0171F5CA2131F173F5CA2133FEF7F805C1800017F5D4C5A91C7485A5F49140FEE1FE0 494A5A00014AB45AB748C7FC16F816C037397BB83A>66 D<0107B512FCA25E9026000FF8 C7FC5D5D141FA25DA2143FA25DA2147FA292C8FCA25CA25CA21301A25CA21303A25CA213 07A25CA2130F170C4A141CA2011F153C17384A1478A2013F157017F04A14E01601017F14 0317C091C71207160F49EC1F80163F4914FF000102071300B8FCA25E2E397BB834>76 D<0107B612F817FF1880903B000FF0003FE04BEB0FF0EF03F8141FEF01FC5DA2023F15FE A25DA2147FEF03FC92C7FCA24A15F817074A15F0EF0FE01301EF1FC04AEC3F80EFFE0001 034A5AEE0FF091B612C04CC7FCD907F8C9FCA25CA2130FA25CA2131FA25CA2133FA25CA2 137FA291CAFCA25BA25B1201B512FCA337397BB838>80 D<14F8EB07FE90381F871C9038 3E03FE137CEBF801120148486C5A485A120FEBC001001F5CA2EA3F801403007F5C1300A2 1407485C5AA2140F5D48ECC1C0A2141F15831680143F1587007C017F1300ECFF076C485B 9038038F8E391F0F079E3907FE03FC3901F000F0222677A42A>97 D<133FEA1FFFA3C67E137EA313FE5BA312015BA312035BA31207EBE0F8EBE7FE9038EF0F 80390FFC07C013F89038F003E013E0D81FC013F0A21380A2123F1300A214075A127EA214 0F12FE4814E0A2141F15C05AEC3F80A215005C147E5C387801F8007C5B383C03E0383E07 C0381E1F80D80FFEC7FCEA01F01C3B77B926>I<147F903803FFC090380FC1E090381F00 70017E13784913383901F801F83803F003120713E0120FD81FC013F091C7FC485AA2127F 90C8FCA35A5AA45AA3153015381578007C14F0007EEB01E0003EEB03C0EC0F806CEB3E00 380F81F83803FFE0C690C7FC1D2677A426>II<147F903803FFC090380FC1E09038 3F00F0017E13785B485A485A485A120F4913F8001F14F0383F8001EC07E0EC1F80397F81 FF00EBFFF891C7FC90C8FC5A5AA55AA21530007C14381578007E14F0003EEB01E0EC03C0 6CEB0F806CEB3E00380781F83803FFE0C690C7FC1D2677A426>II104 D I<150E153F157FA3157E151C1500ABEC1F80EC7FC0ECF1F0EB01C090380380F813071401 130F130E131EEB1C03133C013813F0A2EB0007A215E0A2140FA215C0A2141FA21580A214 3FA21500A25CA2147EA214FEA25CA21301A25CA213035C121C387E07E0A238FE0FC05C49 C7FCEAF83EEA787CEA3FF0EA0FC0204883B619>I109 DI<147F903803FFC090380FC1F090381F00F8017E137C5B4848137E4848133E00 07143F5B120F485AA2485A157F127F90C7FCA215FF5A4814FEA2140115FC5AEC03F8A2EC 07F015E0140F007C14C0007EEB1F80003EEB3F00147E6C13F8380F83F03803FFC0C648C7 FC202677A42A>I<9039078007C090391FE03FF090393CF0787C903938F8E03E9038787F C00170497EECFF00D9F0FE148013E05CEA01E113C15CA2D80003143FA25CA20107147FA2 4A1400A2010F5C5E5C4B5A131F5EEC80035E013F495A6E485A5E6E48C7FC017F133EEC70 FC90387E3FF0EC0F8001FEC9FCA25BA21201A25BA21203A25B1207B512C0A3293580A42A >I<3903C003F0390FF01FFC391E783C0F381C7C703A3C3EE03F8038383FC0EB7F800078 150000701300151CD8F07E90C7FCEAE0FE5BA2120012015BA312035BA312075BA3120F5B A3121F5BA3123F90C9FC120E212679A423>114 D<14FE903807FF8090380F83C090383E 00E04913F00178137001F813F00001130313F0A215E00003EB01C06DC7FC7FEBFFC06C13 F814FE6C7F6D13807F010F13C01300143F141F140F123E127E00FE1480A348EB1F0012E0 6C133E00705B6C5B381E03E06CB45AD801FEC7FC1C267AA422>II<13F8D803FEEB01C0D8078FEB03E0390E0F8007 121E121C0038140F131F007815C01270013F131F00F0130000E015805BD8007E133FA201 FE14005B5D120149137EA215FE120349EBFC0EA20201131E161C15F813E0163CD9F00313 3814070001ECF07091381EF8F03A00F83C78E090393FF03FC090390FC00F00272679A42D >I<01F0130ED803FC133FD8071EEB7F80EA0E1F121C123C0038143F49131F0070140FA2 5BD8F07E140000E08013FEC6485B150E12015B151E0003141C5BA2153C000714385B5DA3 5DA24A5A140300035C6D48C7FC0001130E3800F83CEB7FF8EB0FC0212679A426>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmmi7 7 31 /Fi 31 122 df11 D<137E48B4EB0180000713C0489038E00300481406383E01F0393800700C4813380060EB 181848131CEC0C305AC75B14065DA3EC0780A292C7FCA31406A3140EA2140C141CA45CA3 1430A2142021267E9923>13 D15 D<137001F81338157CA248485BA44848485AA44848485AA44848485AEDC180A3001F9038 0F8300A2141F9038C03786393FE0E7CC9038FFC3FC393E7F00F090C9FC5AA45AA45A5A21 267D9928>22 DI<48B61280000715C0481580481500263C0C06C7FC1270 12C0EB1C0EEA0018A21338A2EB701EA313F013E01201141F120313C0000780A2380F800F A26C486CC7FC221A7D9827>25 D<14FCEB03FF903807878090381E03C0EB3C01017813E0 A213F0000114F013E01203A23907C003E0A4390F8007C0A21580EC0F00EA1F00141E6D5A 6D5A383EE1F0EB7FC0011FC7FC90C8FC5AA45AA45A5A1C267D9922>I<010FB5FC013F14 8049140048B6FC2603F07EC7FC3807C01FEA0F80497E5A123EA2003C5B127CA30078133E 12F8143C0078137C14785C6C485A495A381E0F80D80FFEC8FCEA03F8211A7D9826>I<48 B512F8000714FC4814F84814F0D83C07C7FC1270EAC006130E1200A3131E131CA2133CA3 5BA313F8A3485AA26C5A1E1A7D981F>I<1403A21406A45CA45CA4903807FF80011F13E0 90387C30F0D801F0133C3803C060D80780131ED80F00131F48140F003E13C0A25AA239F8 01801FA3151E903803003E153C157C1578D8780613F0EC01E0003CEB03C0001EEB0F8039 0F0C3E003807FFF8000113E0D8000CC7FC5BA45BA45BA220347CA728>30 D<1238127C12FE12FFA2127F123B1203A31206A3120C121812381270122008127A8614> 59 D<160E163E16FEED03F8ED0FE0ED3F80EDFE00EC03F8EC0FE0EC3F8002FEC7FCEB03 F8EB0FE0EB3F8001FEC8FCEA03F8EA0FE0EA3F8000FEC9FC12F812FEEA3F80EA0FE0EA03 F8EA00FEEB3F80EB0FE0EB03F8EB00FEEC3F80EC0FE0EC03F8EC00FEED3F80ED0FE0ED03 F8ED00FE163E160E27277AA134>II<013FB512F816FF903A01FC001FC04AEB07E0EE03F0010314 01A24A14F8A2130717F04A130317E0010F1407EE0FC04AEB1F80EE7E00011F495A91B512 F0A291388001FC013FEB007E8291C7EA1F80160F4915C0A2137EA213FEEE1F805BEE3F00 0001153E16FE49EB01F84B5A0003EC1FC0B7C7FC15F82D287DA732>66 D<90383FFFF0A2903801FC005CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133F A291C7FCA25BA2137EA213FEA25BA21201A25BA21203B512C0A21C287DA71D>73 D78 D<010FB612C05B9139E0003F800280EB7F00013EC712FE013C495A0138495A49495A4B5A 0160495A01E0495A4949C7FC5D90C75A4A5A4A5A4A5A4A5A4A5A4A5A4AC8FC14FE495A49 5A494813304948137049481360133F4A13E049C75A01FE1301485A4848495A485A484813 074848130F4848013FC7FC484848B4FCB7FC5D2A287CA72D>90 D99 D101 DII<133EEA07FEA2EA007CA213FCA25B A21201A25BA2120314FCEBE3FF9038EF0780D807FC13C0EBF00313E0A2EA0FC014071380 A2121FEC0F801300A248EB1F00A2003E1406143E127EEC7C0C127C151800FCEB3C301570 48EB1FE00070EB0F801F297CA727>I<130E131F5BA2133E131C90C7FCA7EA03E0487EEA 0C78EA187C1230A212605B12C0A2EA01F0A3485AA2485AA2EBC180EA0F81A2381F0300A2 13066C5A131CEA07F06C5A11287DA617>I<133EEA07FEA2EA007CA213FCA25BA21201A2 5BA21203EC07809038E01FC0EC38600007EB61E014C3EBC187EBC307D80FC613C09038CC 038001B8C7FC13E0487E13FEEB3F80EB0FC0486C7E1303003E1460A2127EECC0C0127CEC C18012FC903801E30038F800FE0070137C1B297CA723>107 D<3B07801FC007E03B0FE0 7FF01FF83B18F0E0F8783C3B30F1807CE03E903AFB007D801ED860FEEB3F005B49133E00 C14A133E5B1201A24848495BA35F4848485A1830EE01F0A23C0F8003E003E060A218C093 3801E180271F0007C013E3933800FF00000E6D48137C341B7D993B>109 D<3907801FC0390FE07FF03918F0E0F83930F1807CEBFB00D860FE133C5B5B00C1147C5B 1201A248485BA34A5AEA07C01660EC03E0A23A0F8007C0C0A2EDC180913803C300D81F00 13C7EC01FE000EEB00F8231B7D9929>II<9038F007C03901FC1FF039031E78 780006EBE03C90381FC01C000CEB801E14005B0018141F133E1200137E153E137CA213FC 157C5B1578000114F0A2EC01E0EC03C03903FC07809038FE1F00EBE7FCEBE1F0D807E0C7 FCA25BA2120FA25B121FEAFFF8A22025809922>II<90387C03C03901FF0FF03907079C30390E03B078000CEBF0F8001813E11230 15F0396007C0E015001200A2495AA449C7FC15301238007C1460EAFC3E15C0EAF87E39F0 6F03803970C70700383F83FE381F01F81D1B7D9926>120 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmsy7 7 11 /Fj 11 104 df0 D<1338A50060130C00F8133E00FC137E00FE 13FE383FBBF83807FFC000011300EA007C48B4FC000713C0383FBBF838FE38FE00FC137E 00F8133E0060130C00001300A517197B9A22>3 D<1406140EB3B812E0A3C7000EC8FCB1 B812E0A32B2B7CA834>6 D<160E163E16FEED03F8ED0FE0ED3F80EDFE00EC03F8EC0FE0 EC3F8002FEC7FCEB03F8EB0FE0EB3F8001FEC8FCEA03F8EA0FE0EA3F8000FEC9FC12F812 FEEA3F80EA0FE0EA03F8EA00FEEB3F80EB0FE0EB03F8EB00FEEC3F80EC0FE0EC03F8EC00 FEED3F80ED0FE0ED03F8ED00FE163E160E1600AB007FB612FCB712FEA227357AA734>20 D<176017F01770A217781738173C171C171E83717E717E717EEF00F8BAFC19801900CB12 F8EF01E04D5A4D5A4DC7FC171E171C173C173817781770A217F01760391F7C9D42>33 D<13E0EA01F0EA03F8A3EA07F0A313E0A2120F13C0A3EA1F80A21300A25A123EA35AA312 7812F8A25A12100D1E7D9F13>48 D<017F157F2601FFE0903803FFC0000701F890380FF1 F0260F83FC90381F0038261E00FF013C7F001890263F8078130C4890261FC0E07F007090 260FE1C07F0060EB07E3913803F780486DB4C7EA01806E5A157E157F81824B7E0060DAF7 E0EB0300913801E3F0DBC3F85B6C90260381FC13066C90260F00FE5B001C011E90387F80 3C6C017C90381FE0F82607C7F86DB45A2601FFE0010313C06C6CC86CC7FC391B7C9942> I<49B5FC130F133F01FFC7FCEA01F8EA03E0EA078048C8FC121E121C123C123812781270 A212F05AA2B7FCA300E0C8FCA27E1270A212781238123C121C121E7E6C7EEA03E0EA01F8 6CB4FC013FB5FC130F130120277AA12D>I77 D<147EEB03FEEB0FE0EB1F00133E5BB35BA2485AEA07E0EA FF8000FCC7FCB47EEA07E0EA01F06C7EA2137CB37F7FEB0FE0EB03FEEB007E173B7BAB22 >102 D<12FCB47EEA0FE0EA01F06C7E137CB37FA27FEB0FC0EB03FEEB007EEB03FEEB0F C0EB1F00133EA25BB35B485AEA0FE0EAFF8000FCC7FC173B7BAB22>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk cmmi10 10 58 /Fk 58 123 df<170C171C173E177EA217FE4C7E5EA2EE067F040C7F161C1618EE303F04 607F16E016C0923801801FDB03007F5D15064B130F4B80153815305D4B6D7E14015D4AC7 FC4A14030206815C5C023814010230815C5C010115004A8149C9FC1306010E82010C1780 5B5B0170163F016017C05B485A0003171F90CA13E012065A001FB9FC19F05A5AA2BAFC3C 3C7CBB45>1 D<0103B812F8A34AC8127F010116076EED01F0A26D160081027F16708114 3F81021F1660A281140F8114076F1500140381A21401818082157FA36FC9FC153E15385D 5D4A5A4A5A4AC8120C021E151C023815185C4A1538494815304948157049C9FC010E5E01 3C150101704B5A4915074848150F48484B5A48C9127F000EED0FFF003FB8C7FC5AB9FC5F 3D397BB841>6 D<023FB512F85CA29126003FF0C7FC6F5A5E153FA25EA2157FA2913803 FFE0023F13FED901FEEB7F80D907E0EB0FE0903A1F80FE03F0D97E00EB01FCD9FC016D7E D803F8157ED807F049137FD80FE0ED3F80001F130313C0003F5CEA7F801407D8FF00157F 5DA2020F15005F4B5B4815016C011F5C6C4B5A4B485A6C4B5A261F803F495AD80FC04AC7 FCD807E0EB80FCD803F8EB83F03A00FF7FBFC090261FFFFEC8FC010313E0D9007FC9FC5C A25CA21301A25C497E000FB6FCA331397DB837>8 D11 DII15 D17 DI<133F14C0EB07F06D7E801301A26D7EA314 7FA36E7EA36E7EA36E7EA36E7EA36E7EA36E7EA26E7EA214014A7E5C4A7E91381E3F8014 3C14784A6C7E1301EB03E049486C7EEB0F80EB1F00496D7E137E5B48486D7E485A485A00 0F6E7E485A485A48C87E12FE167F4816800070151F293B7CB930>21 D<017E1438D83FFE147E16FEA2D801FC14FC12000001140116F85BED03F0120315074914 E0150F000715C0ED1F805BED3F00000F147EA2495B4A5A001F495A5D49485A4A5A003F49 C7FC143EEB00F8495A48485AEB0F80D87E3EC8FC13F8EAFFE0138000F8C9FC27257CA429 >23 D<013FB612E090B712F05A120717E0270F807006C7FC391E00600E48140C003813E0 4813C048141CEAC0011200148001035BA213071400A25B1578011E137CA3133E133C137C 157E13FC5B1201157F1203497FA3D801C0131C2C257EA32F>25 D<15FE913803FF809138 0F83E091383E01F091387C00F85C494813FC0103147C4948137E5C130F495AA249C7FC16 FE5B137EA2150113FE4914FCA20001140316F85BED07F01203ED0FE04914C0151F000715 806DEB3F00157E6D5B390FEE01F09038E707E09038C3FF80D9C0FCC7FC001F90C8FCA25B A2123FA290C9FCA25AA2127EA212FEA25AA2127027377EA42B>I<027FB512C00103B612 E0130F5B017F15C09026FF81FEC7FC3901FC007E48487F485A497F484880485AA248C7FC A2127EA2153F00FE92C7FC5AA25D157E5A5DA24A5AA24A5A007C495A5D003C495A003E01 3FC8FC6C137C380F81F83803FFE0C66CC9FC2B257DA32F>I<013FB512FE90B7FC5A5A48 15FE260F801CC7FCEA1E005A00385B5A5A481378C7FC147014F0A4495AA31303A3495AA3 130FA25C131FA3133FA291C8FC131E28257EA324>I<1503A35DA21506A2150EA2150CA2 151CA21518A21538A21530A21570A2EC07FE91383FFFC0903901FCE3F0903907E0E0F890 391F80C03ED93E007FEB7C01D801F8EC0F80D803F0018013C0D807E014071403D80FC015 E0D81F801300A248485AA2007E1306A2020E130F12FE48010C14C0A2021CEB1F80A20218 EB3F00A20238137E007C5D1430007E4A5A003E90387003F06CEC07C09138600F80D80F80 013FC7FC3903E0E0FC3901F8E7F039007FFF80D90FFCC8FCEB01C0A25CA21303A291C9FC A25BA21306A2130EA2130CA22B4B7CB931>30 D34 D<121C127FEAFF80A5EA7F00121C0909798817>58 D<121C127FEAFF80A213C0A3127F12 1C1200A412011380A2120313005A1206120E5A5A5A12600A19798817>II<150C151E153EA2153C157CA21578 15F8A215F01401A215E01403A215C01407A21580140FA215005CA2141E143EA2143C147C A2147814F8A25C1301A25C1303A2495AA25C130FA291C7FC5BA2131E133EA2133C137CA2 137813F8A25B1201A25B1203A25B1207A25B120FA290C8FC5AA2121E123EA2123C127CA2 127812F8A25A12601F537BBD2A>I<126012FCB4FCEA7FC0EA1FF0EA07FCEA01FF38007F C0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FCEC01FF9138007FC0ED1FF0ED07FCED 01FF9238007FC0EE1FF0EE07FCEE01FF9338007F80EF1FC0A2EF7F80933801FF00EE07FC EE1FF0EE7FC04B48C7FCED07FCED1FF0ED7FC04A48C8FCEC07FCEC1FF0EC7FC04948C9FC EB07FCEB1FF0EB7FC04848CAFCEA07FCEA3FF0EA7FC048CBFC12FC1270323279AD41>I< EC03FC91381FFF80027F7F903901F807F0903903C001F890390780007C91C7127E010E80 4980011F1580D93FC0130F17C01607A24A14E0A2011EC7FC90C8FCA5EC0FF0ECFFFC9038 03F00E903907C0078F90381F8001D93E0013CF491300484814FF0003ED7FC05B485A120F 48481580A2485AA2007F160090C8FC167E16FE5A485D15015E1503485D15075E4B5AA200 7E4A5A4BC7FC003E147E003F5C6C6C485A390FC007F03907F01FC06CB5C8FCC613FCEB1F E02B3E7DBB2C>64 D<1760177017F01601A21603A21607160FA24C7EA216331673166316 C3A2ED0183A2ED0303150683150C160115181530A21560A215C014011580DA03007FA202 061300140E140C5C021FB5FC5CA20260C7FC5C83495A8349C8FC1306A25BA25B13385B01 F01680487E000716FFB56C013F13FF5EA2383C7DBB3E>I<0103B77E4916F018FC903B00 07F80003FE4BEB00FFF07F80020FED3FC0181F4B15E0A2141FA25DA2143F19C04B143F19 80027F157F190092C812FE4D5A4A4A5AEF0FF04AEC1FC005FFC7FC49B612FC5F02FCC7B4 FCEF3FC00103ED0FE0717E5C717E1307844A1401A2130F17035CA2131F4D5A5C4D5A133F 4D5A4A4A5A4D5A017F4BC7FC4C5A91C7EA07FC49EC3FF0B812C094C8FC16F83B397DB83F >I<9339FF8001C0030F13E0037F9038F80380913A01FF807E07913A07F8000F0FDA1FE0 EB079FDA3F80903803BF0002FFC76CB4FCD901FC80495A4948157E495A495A4948153E01 7F163C49C9FC5B1201484816385B1207485A1830121F4993C7FCA2485AA3127F5BA312FF 90CCFCA41703A25F1706A26C160E170C171C5F6C7E5F001F5E6D4A5A6C6C4A5A16076C6C 020EC8FC6C6C143C6C6C5C6CB4495A90393FE00FC0010FB5C9FC010313FC9038007FC03A 3D7CBA3B>I<0103B812E05BA290260007F8C7123F4B140FF003C0140F18015DA2141FA2 5D1980143FA25D1760027F14E095C7FC92C75AA24A1301A24A495A16070101141F91B6FC 94C8FCA2903903FC001F824A130EA21307A24A130CA2010F141CA24A90C9FCA2131FA25C A2133FA25CA2137FA291CBFC497EB612C0A33B397DB835>70 DI<0103B5D8F803B512F8495DA290260007F8C73807F8004B5DA2020F150F61 5DA2021F151F615DA2023F153F615DA2027F157F96C7FC92C8FCA24A5D605CA249B7FC60 A202FCC7120101031503605CA201071507605CA2010F150F605CA2011F151F605CA2013F 153F605CA2017F157F95C8FC91C8FC496C4A7EB690B6FCA345397DB845>I<0107B512FC A216F890390007F8005DA2140FA25DA2141FA25DA2143FA25DA2147FA292C7FCA25CA25C A21301A25CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA25CA2137FA291C8 FC497EB6FCA326397DB824>I<0103B500F8903807FFFC5BA290260007F8C813804BEDFC 0019F0020F4B5AF003804B4AC7FC180E021F1538604B5CEF0380023F4AC8FC170E4B133C 1770027F5C4C5ADB0007C9FC160E4A5B167E4A13FE4B7E01015B92380E7F80ECFC1CED38 3F010301E07FECFDC04A486C7EECFF00D907FC6D7E5C4A130783130F707E5C1601011F81 A24A6D7EA2013F6F7EA24A143F84137F717E91C8123F496C81B60107B512C0A26146397D B847>75 D<0103B6FC5B5E90260007FCC8FC5D5D140FA25DA2141FA25DA2143FA25DA214 7FA292C9FCA25CA25CA21301A25CA21303A25CA2130718404A15C0A2010F150118804A14 03A2011F16005F4A1406170E013F151E171C4A143C177C017F5D160391C7120F49EC7FF0 B8FCA25F32397DB839>I<0103B7FC4916E018F8903B0007F80007FC4BEB00FE187F020F ED3F80F01FC05DA2021F16E0A25DA2143FF03FC05DA2027FED7F80A292C8130018FE4A4A 5A604AEC07F04D5A0101ED3FC04CB4C7FC91B612FC17E0D903FCCAFCA25CA21307A25CA2 130FA25CA2131FA25CA2133FA25CA2137FA291CBFC497EB6FCA33B397DB835>80 D<0103B612F849EDFF8018E0903B0007F8001FF84BEB03FCEF00FE020F157FA24BEC3F80 A2021F16C0A25DA2143FF07F805DA2027FEDFF006092C7485A4D5A4A4A5A4D5A4AEC1F80 057FC7FC0101EC07F891B612E094C8FC9139FC000FC00103EC03F0707E4A6D7E83130717 7E5C177F010F5D5F5CA2011F1401A25CA2133F16034A4A1360A2017F17E019C091C71401 496C01011480B61503933900FE0700EF7E0ECAEA1FFCEF07F03B3B7DB83F>82 D<0003B812FEA25A903AF8003FC00101C0913880007E4848163C90C7007F141C121E001C 92C7FCA2485CA200305C007017180060130112E0485CA21403C716005DA21407A25DA214 0FA25DA2141FA25DA2143FA25DA2147FA292C9FCA25CA25CA21301A25CA21303A25CEB0F FC003FB6FC5AA237397EB831>84 D<49B500F890387FFFF095B5FC1AE0D9000301809038 0FFC004BC713E00201ED07804EC7FC6E6C140E606F5C705B606F6C485A4D5A031F91C8FC EEE0065F6F6C5A5F03075B705A16F96FB45A94C9FC6F5AA36F7EA34B7FED037F9238063F C0150E4B6C7E1538ED700F03E07F15C04A486C7EEC0300020613034A805C4A6D7E14704A 1300494880495A49C86C7E130E011E153F017E4B7ED803FF4B7E007F01E0011FEBFFC0B5 FC6144397EB845>88 D<91B712FCA25B9239E00007F84AC7EA0FF0D903F8EC1FE04AEC3F C04AEC7F804A150049485C91C7485A4C5A010E4A5A4C5A010C4A5A011C4A5A01185D167F 4CC7FC90C7485A4B5A4B5A4B5A5E151F4B5A4B5A4BC8FC4A5A4A5A4A5A5D140F4A5A4A5A 4A48130C4AC7FC495A4A141C01031518495A494814384948143049481470495A49C812F0 495D000115014848140348484A5A4848140F4848141F4848EC7F804848EB07FF90B7FCB8 FC94C7FC36397BB839>90 D<1578EC01FEEC07C6EC0F861507EC1E03143E147C1507ECF8 06A2EB01F00103130EECE00C1307A2ECC01C010F1318153890381F80301570156090383F 00E015C01401017F1380EB7E03EC07001406EBFE0E495A5C143000011370495AEBF9C0EB FB8001FFC7FC5B5B485AA25BA4485A120F121DEA39F0127100E1140C0080143C00001470 15E090387801C0EC078090383C1E00EB1FF8EB07E0203C7FBA23>96 D<147E903803FF8090390FC1C38090391F00EFC0017E137F49133F485A4848EB1F801207 5B000F143F48481400A2485A5D007F147E90C7FCA215FE485C5AA214015D48150CA21403 EDF01C16181407007C1538007E010F1330003E131F027B13706C01E113E03A0F83C0F9C0 3A03FF007F80D800FCEB1F0026267DA42C>I<133FEA1FFFA3C67E137EA313FE5BA31201 5BA312035BA31207EBE0FCEBE3FF9038E707C0390FFE03E09038F801F001F013F8EBE000 485A15FC5BA2123F90C7FCA214015A127EA2140312FE4814F8A2140715F05AEC0FE0A215 C0EC1F80143F00781400007C137E5C383C01F86C485A380F07C06CB4C7FCEA01FC1E3B7C B924>II<163FED1FFFA3ED007F167EA216FEA216FCA21501A216F8A21503A216F0A215 07A2027E13E0903803FF8790380FC1CF90381F00EF017EEB7FC049133F485A4848131F00 0715805B000F143F485A1600485A5D127F90C7127EA215FE5A485CA21401A248ECF80CA2 1403161CEDF0181407007C1538007E010F1330003E131F027B13706C01E113E03A0F83C0 F9C03A03FF007F80D800FCEB1F00283B7DB92B>II<16F8ED03FEED0F8792381F 0F80ED3E3F167F157CA215FC1700161C4A48C7FCA414035DA414075DA20107B512F0A390 26000FE0C7FC5DA4141F5DA4143F92C8FCA45C147EA514FE5CA413015CA4495AA45C1307 A25C121E123F387F8F80A200FF90C9FC131E12FEEA7C3CEA7878EA1FF0EA07C0294C7CBA 29>III<14E0EB03F8A21307A314F0EB01C090C7FC AB13F8EA03FEEA070F000E1380121C121812381230EA701F1260133F00E0130012C05BEA 007EA213FE5B1201A25B12035BA20007131813E01438000F133013C01470EB806014E014 C01381EB838038078700EA03FEEA00F815397EB71D>I107 D109 DI<90390F 8003F090391FE00FFC903939F03C1F903A70F8700F80903AE0FDE007C09038C0FF800300 13E00001491303018015F05CEA038113015CA2D800031407A25CA20107140FA24A14E0A2 010F141F17C05CEE3F80131FEE7F004A137E16FE013F5C6E485A4B5A6E485A90397F700F 80DA383FC7FC90387E1FFCEC07E001FEC9FCA25BA21201A25BA21203A25B1207B512C0A3 2C3583A42A>112 D<02FC13C0903803FF0190380F838390383F01C790397E00EF804913 7F485A4848133F000715005B485A001F5C157E485AA2007F14FE90C75AA3481301485CA3 1403485CA314075D140F127C141F007E495A003E137F381F01EF380F839F3903FF1F80EA 00FC1300143F92C7FCA35C147EA314FE5C130190387FFFF0A322357DA425>I<3903E001 F83907F807FE390E3C1E07391C3E381F3A183F703F800038EBE07F0030EBC0FF00705B00 601500EC007E153CD8E07F90C7FCEAC07EA2120013FE5BA312015BA312035BA312075BA3 120F5BA3121F5B0007C9FC21267EA425>I<13F8D803FE1438D8070F147C000E6D13FC12 1C1218003814011230D8701F5C12601503EAE03F00C001005B5BD8007E1307A201FE5C5B 150F1201495CA2151F120349EC80C0A2153F1681EE0180A2ED7F0303FF130012014A5B3A 00F8079F0E90397C0E0F1C90393FFC07F8903907F001F02A267EA430>117 D<01F8EB03C0D803FEEB07E0D8070F130F000E018013F0121C12180038140700301403D8 701F130112601500D8E03F14E000C090C7FC5BEA007E16C013FE5B1501000115805B1503 16001203495B1506150E150C151C151815385D00015C6D485A6C6C485AD97E0FC7FCEB1F FEEB07F024267EA428>I<903907E001F090391FF807FC9039783E0E0F9039E01F1C1FD8 01C09038383F803A03800FF07F0100EBE0FF5A000E4A1300000C157E021F133C001C4AC7 FC1218A2C7123FA292C8FCA25CA2147EA214FEA24A130CA20101141C001E1518003F5BD8 7F81143801835C00FF1560010714E03AFE0E7C01C0D87C1C495A2778383E0FC7FC391FF0 0FFC3907C003F029267EA42F>120 D<13F8D803FE1470D8070F14F8000EEB8001121C12 1800381403003015F0EA701F1260013F130700E0010013E012C05BD8007E130F16C013FE 5B151F000115805BA2153F000315005BA25D157EA315FE5D1401000113033800F8079038 7C1FF8EB3FF9EB0FE1EB00035DA2000E1307D83F805B007F495AA24A5A92C7FCEB003E00 7C5B00705B6C485A381E07C06CB4C8FCEA01FC25367EA429>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl cmsy10 10 24 /Fl 24 113 df<007FB81280B912C0A26C17803204799641>0 D<121C127FEAFF80A5EA 7F00121C0909799917>I<0060150600F8150F6C151F007E153F6C157E6C6C14FC6C6CEB 01F86C6CEB03F06C6CEB07E06C6CEB0FC06C6CEB1F80017EEB3F006D137E6D6C5A90380F C1F8903807E3F0903803F7E06DB45A6D5B6EC7FCA24A7E497F903803F7E0903807E3F090 380FC1F890381F80FC90383F007E017E7F49EB1F804848EB0FC04848EB07E04848EB03F0 4848EB01F84848EB00FC48C8127E007E153F48151F48150F00601506282874A841>I<15 301578B3A6007FB812F8B912FCA26C17F8C80078C8FCB3A3007FB812F8B912FCA26C17F8 36367BB641>6 D14 DI<007FB812F8B912FC A26C17F8CCFCAE007FB812F8B912FCA26C17F8CCFCAE007FB812F8B912FCA26C17F83628 7BA841>17 D20 D<126012F812FEEA7F80EA3FE0EA0FF8EA03FEC66C7EEB3FE0EB0F F8EB03FE903800FF80EC3FE0EC0FF8EC03FE913800FF80ED3FE0ED0FF8ED03FE923800FF 80EE3FE0EE0FF8EE03FE933800FF80EF3FC0171FEF7F80933801FF00EE07FCEE1FF0EE7F C04B48C7FCED07FCED1FF0ED7FC04A48C8FCEC07FCEC1FF0EC7FC04948C9FCEB07FCEB1F F0EB7FC04848CAFCEA07FCEA1FF0EA7FC048CBFC12FC1270CCFCAE007FB81280B912C0A2 6C1780324479B441>I<020FB6128091B712C01303010F1680D91FF8C9FCEB7F8001FECA FCEA01F8485A485A485A5B48CBFCA2123EA25AA2127812F8A25AA87EA21278127CA27EA2 7EA26C7E7F6C7E6C7E6C7EEA00FEEB7F80EB1FF86DB71280010316C01300020F15803232 79AD41>26 D<181EA4181F84A285180785727EA2727E727E85197E85F11F80F10FC0F107 F0007FBA12FCBCFCA26C19FCCCEA07F0F10FC0F11F80F13F00197E61614E5A4E5AA24E5A 61180F96C7FCA260181EA4482C7BAA53>33 D<173CA2173E171E171F8384717E17038471 7E717E187C007FB812FEBAFC856C84CBEA03F0727EF000FEF13F80F11FE0F107F8F101FF A2F107F8F11FE0F13F80F1FE00F001F84E5A007FB912C0BA5A96C7FC6C5FCB127C604D5A 4D5A6017074D5A95C8FC5F171E173E173CA248307BAC53>41 D49 D<91381FFFFE91B6FC1303010F14FED91FF0C7FCEB7F8001FE C8FCEA01F8485A485A485A5B48C9FCA2123EA25AA2127812F8A25AA2B712FE16FFA216FE 00F0C9FCA27EA21278127CA27EA27EA26C7E7F6C7E6C7E6C7EEA00FEEB7F80EB1FF06DB5 12FE010314FF1300021F13FE283279AD37>I54 D<126012F0AD12FCA412F0AD126006207BA400>I<0060161800F0163C6C167CA2007816 78007C16F8A2003C16F0003E1501A26CED03E0A26C16C06D1407A2000716806D140FA26C 6CEC1F00A26CB612FEA36C5D01F8C7127CA2017C5CA2013C5C013E1301A2011E5C011F13 03A26D6C485AA201075CECC00FA2010391C7FC6E5AA2903801F03EA20100133CECF87CA2 EC7878EC7CF8A2EC3FF0A26E5AA36E5AA36E5A6EC8FC2E3C80B92F>I<0238EB07FC02F8 90383FFF80010391B512C0010F010314E0011FEB0F81017B90391E003FF09026E3F07813 1F010349130FECF1E0902607F3C0130714F7DAFF8014E092C7FC18C04A140F49481580EF 1F004A141E5F4A5CEE01E0011F4A5A4A010FC7FC163E9138C001F8ED0FFC013F90383FFF 804AB57E028114F0DA83017F91C7EA3FFC496E7E1607017E6E7E8201FE6E1380A249157F A2173F12015BA21800485AA2177E4848157CA25F48484A5A01C75D019F4A5A261FBF8049 5A496C011EC7FC003F01F0137C9138FC03F0D87E3FB512C0D87C1F91C8FCD8780713F8D8 E00113C0343D7EBA37>66 D77 D102 D<12FCEAFFC0EA07F0EA01FCEA007E7F80131F 80130FB3A7801307806D7E6D7EEB007EEC1FF0EC07F8EC1FF0EC7E00495A495A495A5C13 0F5CB3A7131F5C133F91C7FC137E485AEA07F0EAFFC000FCC8FC1D537ABD2A>I<126012 F0B3B3B3B3A91260045377BD17>106 D<126012F07EA21278127CA2123C123EA2121E12 1FA27E7FA212077FA212037FA212017FA212007FA21378137CA2133C133EA2131E131FA2 7F80A2130780A26D7EA2130180A2130080A21478147CA2143C143EA2141E141FA2801580 A2140715C0A2140315E0A2140115F0A2140015F8A21578157CA2153C153EA2151E150C1F 537BBD2A>110 D112 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmbx12 12 41 /Fm 41 128 df12 D49 DII<163FA25E5E5D5DA25D5D5D5D A25D92B5FCEC01F7EC03E7140715C7EC0F87EC1F07143E147E147C14F8EB01F0EB03E013 0714C0EB0F80EB1F00133E5BA25B485A485A485A120F5B48C7FC123E5A12FCB91280A5C8 000F90C7FCAC027FB61280A531417DC038>I<0007150301E0143F01FFEB07FF91B6FC5E 5E5E5E5E16804BC7FC5D15E092C8FC01C0C9FCAAEC3FF001C1B5FC01C714C001DF14F090 39FFE03FFC9138000FFE01FC6D7E01F06D13804915C0497F6C4815E0C8FC6F13F0A317F8 A4EA0F80EA3FE0487E12FF7FA317F05B5D6C4815E05B007EC74813C0123E003F4A1380D8 1FC0491300D80FF0495AD807FEEBFFFC6CB612F0C65D013F1480010F01FCC7FC010113C0 2D427BC038>I<4AB47E021F13F0027F13FC49B6FC01079038807F8090390FFC001FD93F F014C04948137F4948EBFFE048495A5A1400485A120FA248486D13C0EE7F80EE1E00003F 92C7FCA25B127FA2EC07FC91381FFF8000FF017F13E091B512F89039F9F01FFC9039FBC0 07FE9039FF8003FF17804A6C13C05B6F13E0A24915F0A317F85BA4127FA5123FA217F07F 121FA2000F4A13E0A26C6C15C06D4913806C018014006C6D485A6C9038E01FFC6DB55A01 1F5C010714C0010191C7FC9038003FF02D427BC038>I<121E121F13FC90B712FEA45A17 FC17F817F017E017C0A2481680007EC8EA3F00007C157E5E00785D15014B5A00F84A5A48 4A5A5E151FC848C7FC157E5DA24A5A14035D14074A5AA2141F5D143FA2147F5D14FFA25B A35B92C8FCA35BA55BAA6D5A6D5A6D5A2F447AC238>I I65 DIII< BA1280A419C026003FFEC7121F1701EF007F183F181F180F180719E01803A31801A3EE01 E0F000F0A419001603A31607160F167F91B6FCA59138FE007F160F16071603A31601A693 C9FCAFB712F0A53C447CC346>70 D72 D77 D<923807FFC092B512FE0207ECFFC0021F15F091267F FE0013FC902601FFF0EB1FFF01070180010313C04990C76C7FD91FFC6E6C7E49486F7E49 486F7E01FF8348496F7E48496F1380A248496F13C0A24890C96C13E0A24819F04982003F 19F8A3007F19FC49177FA400FF19FEAD007F19FC6D17FFA3003F19F8A26D5E6C19F0A26E 5D6C19E0A26C6D4B13C06C19806E5D6C6D4B13006C6D4B5A6D6C4B5A6D6C4B5A6D6C4A5B 6D01C001075B6D01F0011F5B010101FE90B5C7FC6D90B65A023F15F8020715C002004AC8 FC030713C047467AC454>79 D82 DI<00 3FBA12E0A59026FE000FEB8003D87FE09338003FF049171F90C71607A2007E1803007C18 01A300781800A400F819F8481978A5C81700B3B3A20107B8FCA545437CC24E>I<903801 FFE0011F13FE017F6D7E48B612E03A03FE007FF84848EB1FFC6D6D7E486C6D7EA26F7FA3 6F7F6C5A6C5AEA00F090C7FCA40203B5FC91B6FC1307013F13F19038FFFC01000313E000 0F1380381FFE00485A5B127F5B12FF5BA35DA26D5B6C6C5B4B13F0D83FFE013EEBFFC03A 1FFF80FC7F0007EBFFF86CECE01FC66CEB8007D90FFCC9FC322F7DAD36>97 D99 DIII104 D<137C48B4FC4813804813C0A24813E0A56C13C0A26C13 806C1300EA007C90C7FCAAEB7FC0EA7FFFA512037EB3AFB6FCA518467CC520>II108 D<90277F8007FEEC0FFCB590263FFFC09038 7FFF8092B5D8F001B512E002816E4880913D87F01FFC0FE03FF8913D8FC00FFE1F801FFC 0003D99F009026FF3E007F6C019E6D013C130F02BC5D02F86D496D7EA24A5D4A5DA34A5D B3A7B60081B60003B512FEA5572D7CAC5E>I<90397F8007FEB590383FFF8092B512E002 8114F8913987F03FFC91388F801F000390399F000FFE6C139E14BC02F86D7E5CA25CA35C B3A7B60083B512FEA5372D7CAC3E>II<90397FC00FF8B590B57E02C314E002CF14F89139DFC03FFC9139FF001FFE00 0301FCEB07FF6C496D13804A15C04A6D13E05C7013F0A2EF7FF8A4EF3FFCACEF7FF8A318 F017FFA24C13E06E15C06E5B6E4913806E4913006E495A9139DFC07FFC02CFB512F002C3 14C002C091C7FCED1FF092C9FCADB67EA536407DAC3E>I<90387F807FB53881FFE00283 13F0028F13F8ED8FFC91389F1FFE000313BE6C13BC14F8A214F0ED0FFC9138E007F8ED01 E092C7FCA35CB3A5B612E0A5272D7DAC2E>114 D<90391FFC038090B51287000314FF12 0F381FF003383FC00049133F48C7121F127E00FE140FA215077EA27F01E090C7FC13FE38 7FFFF014FF6C14C015F06C14FC6C800003806C15806C7E010F14C0EB003F020313E01400 00F0143FA26C141F150FA27EA26C15C06C141FA26DEB3F8001E0EB7F009038F803FE90B5 5A00FC5CD8F03F13E026E007FEC7FC232F7CAD2C>I II120 DI123 D127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn cmex10 10 16 /Fn 16 102 df<1430147014E0EB01C01303EB0780EB0F00A2131E5BA25B13F85B12015B 1203A2485AA3485AA3121F90C7FCA25AA3123EA2127EA6127C12FCB3A2127C127EA6123E A2123FA37EA27F120FA36C7EA36C7EA212017F12007F13787FA27F7FA2EB0780EB03C013 01EB00E0147014301462738226>0 D<12C07E12707E123C7E7EA26C7E6C7EA26C7E7F12 007F1378137CA27FA37FA31480130FA214C0A31307A214E0A6130314F0B3A214E01307A6 14C0A2130FA31480A2131F1400A3133EA35BA2137813F85B12015B485AA2485A48C7FCA2 121E5A12385A5A5A14627C8226>III<12F0B3B3B2043674811C>12 D<151E153E157C15F8EC01F0EC03E01407EC0FC0EC1F8015005C147E5CA2495A495AA249 5AA2495AA2495AA249C7FCA2137EA213FE5B12015BA212035BA21207A25B120FA35B121F A45B123FA548C8FCA912FEB3A8127FA96C7EA5121F7FA4120F7FA312077FA21203A27F12 01A27F12007F137EA27FA26D7EA26D7EA26D7EA26D7EA26D7E6D7EA2147E80801580EC0F C0EC07E01403EC01F0EC00F8157C153E151E1F94718232>16 D<12F07E127C7E7E6C7E7F 6C7E6C7E12017F6C7E137EA27F6D7EA26D7EA26D7EA26D7EA26D7EA26D7EA280147E147F 80A21580141FA215C0A2140F15E0A3140715F0A4140315F8A5EC01FCA9EC00FEB3A8EC01 FCA9EC03F8A515F01407A415E0140FA315C0141FA21580A2143F1500A25C147E14FE5CA2 495AA2495AA2495AA2495AA2495AA249C7FC137EA25B485A5B1203485A485A5B48C8FC12 3E5A5A5A1F947D8232>I<160F161F163E167C16F8ED01F0ED03E0ED07C0150FED1F8016 00153E157E5D4A5A5D14034A5A5D140F4A5AA24AC7FC143E147E5CA2495AA2495AA2495A A2130F5CA2495AA2133F91C8FCA25B137E13FEA25B1201A25B1203A35B1207A35B120FA3 5BA2121FA45B123FA690C9FC5AAA12FEB3AC127FAA7E7FA6121F7FA4120FA27FA312077F A312037FA312017FA212007FA2137E137F7FA280131FA26D7EA2801307A26D7EA26D7EA2 6D7EA2147E143E143F6E7EA26E7E1407816E7E1401816E7E157E153E811680ED0FC01507 ED03E0ED01F0ED00F8167C163E161F160F28C66E823D>I<12F07E127C7E7E6C7E6C7E6C 7E7F6C7E1200137C137E7F6D7E130F806D7E1303806D7EA26D7E147C147E80A26E7EA26E 7EA26E7EA2811403A26E7EA2811400A281157E157FA2811680A2151F16C0A3150F16E0A3 150716F0A31503A216F8A4150116FCA6150016FEAA167FB3AC16FEAA16FC1501A616F815 03A416F0A21507A316E0150FA316C0151FA31680153FA216005DA2157E15FE5DA214015D A24A5AA214075DA24A5AA24A5AA24AC7FCA2147E147C14FC495AA2495A5C1307495A5C13 1F49C8FC137E137C5B1201485A5B485A485A48C9FC123E5A5A5A28C67E823D>I<161E16 7EED01FE1507ED0FF8ED3FE0ED7FC0EDFF80913801FE004A5A4A5A5D140F4A5A5D143F5D 147F92C7FCA25C5CB3B3B3A313015CA3495AA213075C495AA2495A495A137F49C8FC485A 485AEA07F0EA1FE0485AB4C9FC12FCA2B4FCEA3FC06C7EEA07F0EA03FC6C7E6C7E6D7E13 3F6D7E6D7EA26D7E801303A26D7EA3801300B3B3B3A38080A281143F81141F816E7E1407 816E7E6E7E913800FF80ED7FC0ED3FE0ED0FF8ED07FE1501ED007E161E27C675823E>26 D<12F012FCB4FC13C0EA3FE0EA0FF86C7E6C7EC67E6D7E6D7E131F806D7E130780130380 1301A2801300B3B3B3A38080A36E7EA281141F6E7EA26E7E6E7E816E7E6E7EED7F80ED1F C0ED0FF0ED07F8ED01FEED007EA2ED01FEED07F8ED0FF0ED1FC0ED7F80EDFF004A5A4A5A 5D4A5A4A5AA24A5A143F5DA24AC7FCA35C5CB3B3B3A313015CA213035C13075C130F495A 5C133F495A49C8FCEA03FE485A485AEA3FE0B45A90C9FC12FC12F027C675823E>I80 D<167F923801FFC0923803C0F0923807803892380F007892381F01FC151E153E A2157E92387C0070170015FCA44A5AA81403A45DA41407A94A5AAA4A5AA95DA4143FA492 C8FCA7143E147EA4147C123800FE13FC5CA2495A5CEA7803387007C0383C0F80D80FFEC9 FCEA03F82E5C7C7F27>82 D88 D90 D101 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmsy6 6 2 /Fo 2 50 df0 D<01FEEC0FE02603FFC0EB3FF8000F01F0EBFE 3E3B1F0FF801F0073C3C01FC07C003803B3000FE0F00010070D93F1EEB00C00060EB1F9C 00E0D90FF81460485C14076E7E6E7E81020315E00060D9073F14C091390F1F80016C9026 1E0FE01380003890397C07F0073C1C01F003FE1F003B0F8FE001FFFE3B03FF80007FF8C6 48C7EA0FE033177C953D>49 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmmi6 6 1 /Fp 1 111 df<000F13FC381FC3FF3931C707803861EC0301F813C0EAC1F0A213E03903 C00780A3EC0F00EA0780A2EC1E041506D80F00130C143C15181538001EEB1C70EC1FE000 0CEB07801F177D9526>110 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmss9 9 1 /Fq 1 91 df<007FB612FCA416F8C81207ED0FF0ED1FE0A2ED3FC0ED7F80A2EDFF004A5A A24A5A4A5AA24A5A4A5AA24A5A4A5AA24AC7FC495AA2495A495AA2495A495AA2495A495A A249C8FC485AA2485A485AA2485A485AA2485A485A90B612FCB7FCA426347CB32F>90 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmsy9 9 4 /Fr 4 51 df<171C177EEE01FEEE07FCEE1FF0EE7FC0923801FF00ED07FCED1FF0ED7FC0 4A48C7FCEC07FCEC1FF0EC7FC04948C8FCEB07FCEB1FF0EB7FC04848C9FCEA07FCEA1FF0 EA7FC048CAFCA2EA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007F C0EC1FF0EC07FCEC01FF9138007FC0ED1FF0ED07FCED01FF9238007FC0EE1FF0EE07FCEE 01FEEE007E171C1700AC007FB712FCB812FEA26C16FC2F3E7AB03C>20 D<127012FCB4FCEA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007F C0EC1FF0EC07FCEC01FF9138007FC0ED1FF0ED07FCED01FF9238007FC0EE1FF0EE07FCEE 01FEA2EE07FCEE1FF0EE7FC0923801FF00ED07FCED1FF0ED7FC04A48C7FCEC07FCEC1FF0 EC7FC04948C8FCEB07FCEB1FF0EB7FC04848C9FCEA07FCEA1FF0EA7FC048CAFC12FC1270 CBFCAC007FB712FCB812FEA26C16FC2F3E7AB03C>I49 D<91383FFFF849B512FC1307011F14F8D93FE0C7FC01FFC8FCEA01FCEA03F0485A485A5B 48C9FC5A123E5AA21278A212F8A25AB712F816FCA216F800F0C9FC7EA21278A2127CA27E 123F7E6C7E7F6C7E6C7EEA01FC6CB4FCEB3FE06DB512F8010714FC1301D9003F13F8262E 7AA933>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs cmmi9 9 6 /Fs 6 121 df<123C127E12FFA4127E123C08087A8715>58 D<171C177EEE01FEEE07FC EE1FF0EE7FC0923801FF00ED07FCED1FF0ED7FC04A48C7FCEC07FCEC1FF0EC7FC04948C8 FCEB07FCEB1FF0EB7FC04848C9FCEA07FCEA1FF0EA7FC048CAFCA2EA7FC0EA1FF0EA07FC EA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FCEC01FF9138007FC0ED 1FF0ED07FCED01FF9238007FC0EE1FF0EE07FCEE01FEEE007E171C2F2E7AA93C>60 D<010FB612F017FEEFFF80903B003FC0003FE0EF0FF017074B14F81703027F15FCA292C7 FCA25C18F84A140718F00101150F18E04AEC1FC0EF3F800103ED7F00EE01FE4AEB07F891 B612E04915809139F8001FF04AEB03FCEE00FE010F157FA24AEC3F80A2011F16C0A25CA2 133F18804A147FA2017FEDFF005F91C712014C5A494A5A4C5A49EC3FE00001913801FF80 B748C7FC16F816C036337DB23A>66 D97 D 110 D<90391F801F8090397FE07FE09039E0F0E0703A01C0F9C0F83903807D833807007F 000E1403000C15F0001C137E0018EC01C002FEC7FC00385B1210C7FC13015CA31303A25C 1640010714E016C0001C5B007E1401010F148000FE1403011FEB0700011B130E39F839F0 1C397070F878393FE07FE0390F801F8025227EA02C>120 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft cmbx9 9 18 /Ft 18 117 df<120FEA3FC0EA7FE0EAFFF0A6EA7FE0EA3FC0EA0F000C0C7A8B19>46 D48 D<147814F81303131FEA03FFB5FCA3EAFC1F1200B3B2007FB512FEA41F317A B02C>III<151F5D5DA25D5C5C5C5CA25C 143D147D14F9EB01F114E1EB03C1EB0781130FEB1F01133E133C137813F01201EA03E0EA 07C01380EA0F00121E123E5A5AB712FEA4C700031300A80103B512FEA427317EB02C>I< 000C140ED80FE013FE90B5FC5D5D5D5D5D92C7FC14FC14F091C8FC1380A6EB87FE9038BF FFC090B512F09038FC0FF89038E003FE01C07F497E01001480000E6D13C0C8FCA216E0A3 121FEA7F807F487EA316C05B5CD87F801480D87C0014006C5B393F8007FE391FE01FFC00 07B512F06C14C0C691C7FCEB1FF823327CB02C>II<123C123F90B612F8A44815F016E016C01680 16005D007CC7127E00785C4A5A00F8495A48495A4A5A4A5AC7FC4AC7FC147E14FE5C1301 5C1303A2495AA2130FA2131FA25C133FA4137FA96D5AA2010FC8FC25337BB12C>II65 D68 D97 DI<903807FF80013F13F090B512FC3903FE01FE4848487EEA0FF8EA1FF0EA3F E0A2007F6D5A496C5A153000FF91C7FCA9127F7FA2003FEC07807F6C6C130F000FEC1F00 D807FE133E3903FF80FCC6EBFFF8013F13E0010790C7FC21217DA027>I<3901F81F8000 FFEB7FF0ECFFF89038F9E3FC9038FBC7FE380FFF876C1307A213FEEC03FCEC01F8EC0060 491300B1B512F0A41F217EA024>114 D<9038FFE1C0000713FF5A383F803F387E000F14 075A14037EA26C6CC7FC13FCEBFFE06C13FC806CEBFF80000F14C06C14E0C6FC010F13F0 EB007F140F00F0130714037EA26C14E06C13076CEB0FC09038C01F8090B5120000F913FC 38E03FE01C217DA023>I<133CA5137CA313FCA21201A212031207001FB51280B6FCA3D8 07FCC7FCB0EC03C0A79038FE078012033901FF0F006C13FEEB3FFCEB0FF01A2F7EAE22> I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fu cmr9 9 78 /Fu 78 124 df<91393FE00FE0903A01FFF83FF8903A07E01EF83C903A1F800FF07E903A 3F001FE0FE017E133F4914C0485A1738484890381F8000ACB812C0A33B03F0001F8000B3 A7486C497EB50083B5FCA32F357FB42D>11 DI<127812FCA27E7EEA7F80121FEA0F C0EA07E01203EA00F01378133C13080E0E78B326>18 D<137813FCA212011203EA07F813 E0EA0FC0EA1F801300123C5A5A12400E0E71B326>I<003C13F0387E01F838FF03FCA2EB 83FEA2EA7F81383D80F600011306A40003130EEB000CA248131C00061318000E1338000C 1330001C13704813E0387001C00060138017177EB326>34 D<123C127EB4FCA21380A212 7F123D1201A412031300A25A1206120E120C121C5A5A126009177AB315>39 D<14C01301EB0380EB0F00130E5B133C5B5BA2485A485AA212075B120F90C7FC5AA2121E 123EA3123C127CA55AB0127CA5123C123EA3121E121FA27E7F12077F1203A26C7E6C7EA2 13787F131C7F130FEB0380EB01C01300124A79B71E>I<12C07E1270123C121C7E120F6C 7E6C7EA26C7E6C7EA27F1378137C133C133EA2131E131FA37F1480A5EB07C0B0EB0F80A5 14005BA3131E133EA2133C137C137813F85BA2485A485AA2485A48C7FC120E5A123C1270 5A5A124A7CB71E>I<156015F0B3A4007FB812C0B912E0A26C17C0C800F0C8FCB3A41560 33327CAB3C>43 D<123C127EB4FCA21380A2127F123D1201A412031300A25A1206120E12 0C121C5A5A126009177A8715>II<123C127E12FFA4127E123C08 087A8715>I<1530157815F8A215F01401A215E01403A215C01407A21580140FA215005C A2143EA2143C147CA2147814F8A25C1301A25C1303A25C1307A2495AA291C7FC5BA2131E 133EA2133C137CA2137813F8A25B1201A25B1203A2485AA25B120FA290C8FC5AA2121E12 3EA2123C127CA2127812F8A25A12601D4B7CB726>II<13075B5B137FEA07FFB5FC13BFEAF83F1200B3B3A2497E00 7FB51280A319327AB126>IIII<000C14C0380FC00F90B5128015005C5C14F0 14C0D80C18C7FC90C8FCA9EB0FC0EB7FF8EBF07C380FC03F9038001F80EC0FC0120E000C EB07E0A2C713F01403A215F8A41218127E12FEA315F0140712F8006014E01270EC0FC06C 131F003C14806CEB7F00380F80FE3807FFF8000113E038003F801D347CB126>I<14FE90 3807FF80011F13E090383F00F0017C13703901F801F8EBF003EA03E01207EA0FC0EC01F0 4848C7FCA248C8FCA35A127EEB07F0EB1FFC38FE381F9038700F809038E007C039FFC003 E0018013F0EC01F8130015FC1400A24814FEA5127EA4127F6C14FCA26C1301018013F800 0F14F0EBC0030007EB07E03903E00FC03901F81F806CB51200EB3FFCEB0FE01F347DB126 >I<1230123C003FB6FCA34814FEA215FC0070C7123800601430157015E04814C01401EC 0380C7EA07001406140E5C141814385CA25CA2495A1303A3495AA2130FA3131F91C7FCA2 5BA55BA9131C20347CB126>III<123C127E12FFA4127E123C1200B0123C127E12FFA4127E 123C08207A9F15>I<007FB812C0B912E0A26C17C0CCFCAC007FB812C0B912E0A26C17C0 33147C9C3C>61 D<15E0A34A7EA24A7EA34A7EA3EC0DFE140CA2EC187FA34A6C7EA20270 7FEC601FA202E07FECC00FA2D901807F1507A249486C7EA301066D7EA2010E80010FB5FC A249800118C77EA24981163FA2496E7EA3496E7EA20001821607487ED81FF04A7ED8FFFE 49B512E0A333367DB53A>65 DIIIIIIII<017FB5FCA39038003FE0EC1FC0B3B1127EB4FCA4EC3F805A0060140000705B6C13 FE6C485A380F03F03803FFC0C690C7FC20357DB227>IIIII II82 D<90381FE00390387FFC0748B5FC3907 F01FCF390F8003FF48C7FC003E80814880A200788000F880A46C80A27E92C7FC127F13C0 EA3FF013FF6C13F06C13FF6C14C06C14F0C680013F7F01037F9038003FFF140302001380 157F153FED1FC0150F12C0A21507A37EA26CEC0F80A26C15006C5C6C143E6C147E01C05B 39F1FC03F800E0B512E0011F138026C003FEC7FC22377CB42B>I<007FB712FEA3903980 07F001D87C00EC003E0078161E0070160EA20060160600E01607A3481603A6C71500B3AB 4A7E011FB512FCA330337DB237>IIII89 D<003FB612FCA39039F80007F813C090C7EA0FF0003EEC1FE0123C0038EC3FC000 78EC7F801270EDFF004A5AA20060495AA24A5A4A5AC7FC4A5A4A5AA24A5A4AC7FCA2495A 495AA2495A495AA24948130C495AA2495A49C7FCA24848141CA2485A485A1638485A4848 147816F84848130148481307153FB7FCA326337CB22F>I<0003130C48131C000E133848 137000181360003813E0003013C0EA700100601380A2EAE00300C01300A400DE137800FF 13FCEB83FEA2EA7F81A2383F00FC001E1378171774B326>92 D<1320137013F8487EEA03 DEEA078F380F0780381E03C0383C01E0387800F000E0133800401310150C78B326>94 D97 DII<153FEC0FFFA3EC007F81AEEB07F0EB3FFCEBFC0F3901F003BF39 07E001FF48487E48487F8148C7FCA25A127E12FEAA127E127FA27E6C6C5BA26C6C5B6C6C 4813803A03F007BFFC3900F81E3FEB3FFCD90FE0130026357DB32B>III<151F90391FC07F809039FFF8E3C03901F07FC73907E03F033A0FC01F8380 9039800F8000001F80EB00074880A66C5CEB800F000F5CEBC01F6C6C48C7FCEBF07C380E FFF8380C1FC0001CC9FCA3121EA2121F380FFFFEECFFC06C14F06C14FC4880381F000100 3EEB007F4880ED1F8048140FA56C141F007C15006C143E6C5C390FC001F83903F007E0C6 B51280D91FFCC7FC22337EA126>IIIIII<2703F01FE013FF00FF90267FF80313C0903BF1E07C0F03E0 903BF3803E1C01F02807F7003F387FD803FE1470496D486C7EA2495CA2495CB3486C496C 487EB53BC7FFFE3FFFF0A33C217EA041>I<3903F01FC000FFEB7FF09038F1E0FC9038F3 807C3907F7007EEA03FE497FA25BA25BB3486CEB7F80B538C7FFFCA326217EA02B>II<3903F03F8000FFEBFFE09038 F3C0F89038F7007ED807FE7F6C48EB1F804914C049130F16E0ED07F0A3ED03F8A9150716 F0A216E0150F16C06D131F6DEB3F80160001FF13FC9038F381F89038F1FFE0D9F07FC7FC 91C8FCAA487EB512C0A325307EA02B>I<903807F00390383FFC07EBFC0F3901F8038F38 07E001000F14DF48486CB4FC497F123F90C77E5AA25A5AA9127FA36C6C5B121F6D5B000F 5B3907E003BF3903F0073F3800F81EEB3FF8EB0FE090C7FCAAED7F8091380FFFFCA32630 7DA029>I<3803E07C38FFE1FF9038E38F809038E71FC0EA07EEEA03ECA29038FC0F8049 C7FCA35BB2487EB512E0A31A217FA01E>II<1330A51370A313F0A21201A212031207381FFFFEB5FCA23803F000AF1403 A814073801F806A23800FC0EEB7E1CEB1FF8EB07E0182F7FAD1E>IIIII<3A7FFF807FF8A33A07F8001FC00003EC0F800001EC070015066C6C5BA26D131C017E 1318A26D5BA2EC8070011F1360ECC0E0010F5BA2903807E180A214F3010390C7FC14FBEB 01FEA26D5AA31478A21430A25CA214E05CA2495A1278D8FC03C8FCA21306130EEA701CEA 7838EA1FF0EA0FC025307F9F29>I<003FB512F0A2EB000F003C14E00038EB1FC00030EB 3F800070137F1500006013FE495A13035CC6485A495AA2495A495A49C7FC153013FE485A 12035B48481370485A001F14604913E0485A387F000348130F90B5FCA21C207E9F22>I< B712F8A22502809426>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fv cmr6 6 5 /Fv 5 62 df<1438B2B712FEA3C70038C7FCB227277C9F2F>43 D<13FF000313C0380781 E0380F00F0001E137848133CA248131EA400F8131FAD0078131EA2007C133E003C133CA2 6C13786C13F0380781E03803FFC0C6130018227DA01E>48 D<13E01201120712FF12F912 01B3A7487EB512C0A212217AA01E>II61 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fw cmr7 7 13 /Fw 13 116 df<1306130C13181330136013E0EA01C0EA0380A2EA07005A120E121EA212 1C123CA35AA512F85AAB7E1278A57EA3121C121EA2120E120F7EEA0380A2EA01C0EA00E0 136013301318130C13060F3B7AAB1A>40 D<12C012607E7E7E120E7EEA0380A2EA01C013 E0120013F0A213701378A3133CA5133E131EAB133E133CA51378A3137013F0A213E01201 13C0EA0380A2EA0700120E120C5A5A5A5A0F3B7DAB1A>I<140EB3A2B812E0A3C7000EC8 FCB3A22B2B7DA333>43 D48 D<13381378EA01F8121F12FE12E01200B3AB48 7EB512F8A215267BA521>I<13FF000313E0380E03F0381800F848137C48137E00787F12 FC6CEB1F80A4127CC7FC15005C143E147E147C5C495A495A5C495A010EC7FC5B5B903870 018013E0EA0180390300030012065A001FB5FC5A485BB5FCA219267DA521>I61 D91 D93 D108 D<260F81FC137F3BFF8FFF03FFC0903A9C0F8703E03B1FB0 07CC01F0D80FE013D8903AC003F000F8A301805BAF486C486C487E3CFFF83FFE0FFF80A2 311A7E9937>I<380F81FC38FF8FFF90389C0F80391FB007C0EA0FE09038C003E0A31380 AF391FC007F039FFF83FFEA21F1A7E9925>I<3803F840380FFEC0EA3C07EA7803EA7001 EAF000A37E6C1300EA7FC013FC6CB4FC6C1380000713C0C613E0130738C003F013011300 7EA26C13E0130100F813C038EE078038C7FF00EA81FC141C7E9A1A>115 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fx cmr10 10 82 /Fx 82 128 df11 DIII<133C137EA213FE1201EA03FC13F0EA07E0EA0FC0EA1F80EA 1E005A5A5A12C00F0F6FB92A>19 D<001C131C007F137F39FF80FF80A26D13C0A3007F13 7F001C131C00001300A40001130101801380A20003130301001300485B00061306000E13 0E485B485B485B006013601A197DB92A>34 D<121C127FEAFF80A213C0A3127F121C1200 A412011380A2120313005A1206120E5A5A5A12600A1979B917>39 D<146014E0EB01C0EB0380EB0700130E131E5B5BA25B485AA2485AA212075B120F90C7FC A25A121EA2123EA35AA65AB2127CA67EA3121EA2121F7EA27F12077F1203A26C7EA26C7E 1378A27F7F130E7FEB0380EB01C0EB00E01460135278BD20>I<12C07E12707E7E7E120F 6C7E6C7EA26C7E6C7EA21378A2137C133C133E131EA2131F7FA21480A3EB07C0A6EB03E0 B2EB07C0A6EB0F80A31400A25B131EA2133E133C137C1378A25BA2485A485AA2485A48C7 FC120E5A5A5A5A5A13527CBD20>I<15301578B3A6007FB812F8B912FCA26C17F8C80078 C8FCB3A6153036367BAF41>43 D<121C127FEAFF80A213C0A3127F121C1200A412011380 A2120313005A1206120E5A5A5A12600A19798817>II<121C127F EAFF80A5EA7F00121C0909798817>I48 DIII<1538A2157815F8A2140114031407A2140F141F141B1433147314 6314C313011483EB030313071306130C131C131813301370136013C01201EA038013005A 120E120C5A123812305A12E0B712F8A3C73803F800AB4A7E0103B512F8A325397EB82A> I<0006140CD80780133C9038F003F890B5FC5D5D158092C7FC14FC38067FE090C9FCABEB 07F8EB3FFE9038780F803907E007E090388003F0496C7E12066E7EC87EA28181A21680A4 123E127F487EA490C71300485C12E000605C12700030495A00385C6C1303001E495A6C6C 485A3907E03F800001B5C7FC38007FFCEB1FE0213A7CB72A>II<12301238123E003FB612E0A316C05A168016000070C712060060140E5D1518 00E01438485C5D5DC712014A5A92C7FC5C140E140C141C5CA25CA214F0495AA21303A25C 1307A2130FA3495AA3133FA5137FA96DC8FC131E233B7BB82A>III<121C127FEAFF80A5EA7F00121CC7FCB2121C127FEAFF 80A5EA7F00121C092479A317>I<007FB812F8B912FCA26C17F8CCFCAE007FB812F8B912 FCA26C17F836167B9F41>61 D63 D<1538A3157CA315FEA34A7EA34A6C7EA202077FEC063FA202 0E7FEC0C1FA2021C7FEC180FA202387FEC3007A202707FEC6003A202C07F1501A2D90180 7F81A249C77F167FA20106810107B6FCA24981010CC7121FA2496E7EA3496E7EA3496E7E A213E0707E1201486C81D80FFC02071380B56C90B512FEA3373C7DBB3E>65 DI<913A01FF800180020FEBE003027F13F8903A01FF807E07903A03 FC000F0FD90FF0EB039F4948EB01DFD93F80EB00FF49C8127F01FE153F12014848151F48 48150FA248481507A2485A1703123F5B007F1601A35B00FF93C7FCAD127F6DED0180A312 3F7F001F160318006C7E5F6C7E17066C6C150E6C6C5D00001618017F15386D6C5CD91FE0 5C6D6CEB03C0D903FCEB0F80902701FF803FC7FC9039007FFFFC020F13F002011380313D 7BBA3C>III< B812F8A30001903880001F6C90C71201EE00FC177C173C171CA2170CA4170E1706A2ED01 80A21700A41503A21507151F91B5FCA3EC001F15071503A21501A692C8FCAD4813C0B612 C0A32F397DB836>III I<013FB512E0A39039001FFC00EC07F8B3B3A3123FEA7F80EAFFC0A44A5A1380D87F005B 0070131F6C5C6C495A6C49C7FC380781FC3801FFF038007F80233B7DB82B>III< B5933807FFF86E5DA20001F0FC002600DFC0ED1BF8A2D9CFE01533A3D9C7F01563A3D9C3 F815C3A2D9C1FCEC0183A3D9C0FEEC0303A2027F1406A36E6C130CA36E6C1318A26E6C13 30A36E6C1360A26E6C13C0A3913901FC0180A3913900FE0300A2ED7F06A3ED3F8CA2ED1F D8A3ED0FF0A3486C6D5A487ED80FFC6D48497EB500C00203B512F8A2ED018045397DB84C >I III82 D I<003FB812E0A3D9C003EB001F273E0001FE130348EE01F00078160000701770A3006017 30A400E01738481718A4C71600B3B0913807FF80011FB612E0A335397DB83C>II87 D89 D<003FB7FCA39039FC0001 FE01C0130349495A003EC7FC003C4A5A5E0038141F00784A5A12704B5A5E006014FF4A90 C7FCA24A5A5DC712074A5AA24A5A5D143F4A5AA24A5A92C8FC5B495AA2495A5C130F4948 EB0180A2495A5C137F495A16034890C7FC5B1203485AEE0700485A495C001F5D48485C5E 4848495A49130FB8FCA329397BB833>II<3901800180000313033907000700000E130E485B001813180038133800301330 0070137000601360A200E013E0485BA400CE13CE39FF80FF806D13C0A3007F137FA2393F 803F80390E000E001A1974B92A>I I96 DIIIII<147E 903803FF8090380FC1E0EB1F8790383F0FF0137EA213FCA23901F803C091C7FCADB512FC A3D801F8C7FCB3AB487E387FFFF8A31C3B7FBA19>IIIIIII<2703F00FF0EB1FE000 FFD93FFCEB7FF8913AF03F01E07E903BF1C01F83803F3D0FF3800FC7001F802603F70013 CE01FE14DC49D907F8EB0FC0A2495CA3495CB3A3486C496CEB1FE0B500C1B50083B5FCA3 40257EA445>I<3903F00FF000FFEB3FFCECF03F9039F1C01F803A0FF3800FC03803F700 13FE496D7EA25BA35BB3A3486C497EB500C1B51280A329257EA42E>I I<3903F01FE000FFEB7FF89038F1E07E9039F3801F803A0FF7000FC0D803FEEB07E049EB 03F04914F849130116FC150016FEA3167FAA16FEA3ED01FCA26DEB03F816F06D13076DEB 0FE001F614C09039F7803F009038F1E07E9038F0FFF8EC1FC091C8FCAB487EB512C0A328 357EA42E>II<3807E01F00FFEB7FC09038E1E3E09038E387F0380FE707EA03E613EE 9038EC03E09038FC0080491300A45BB3A2487EB512F0A31C257EA421>II<1318A51338A31378A313 F8120112031207001FB5FCB6FCA2D801F8C7FCB215C0A93800FC011580EB7C03017E1300 6D5AEB0FFEEB01F81A347FB220>IIIIII<003FB512FCA2EB8003D83E0013F8003CEB07 F00038EB0FE012300070EB1FC0EC3F800060137F150014FE495AA2C6485A495AA2495A49 5A495AA290387F000613FEA2485A485A0007140E5B4848130C4848131CA24848133C48C7 127C48EB03FC90B5FCA21F247EA325>III<001C131C007F137F39FF80FF80A5397F007F00001C131C190978B72A>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fy cmbx12 14.4 22 /Fy 22 121 df66 D<932601FFFCEC01C0047FD9FFC013030307B600F81307033F03FE131F92B8EA803F0203 DAE003EBC07F020F01FCC7383FF0FF023F01E0EC0FF94A01800203B5FC494848C9FC4901 F8824949824949824949824949824990CA7E494883A2484983485B1B7F485B481A3FA248 49181FA3485B1B0FA25AA298C7FC5CA2B5FCAE7EA280A2F307C07EA36C7FA21B0F6C6D19 80A26C1A1F6C7F1C006C6D606C6D187EA26D6C606D6D4C5A6D6D16036D6D4C5A6D6D4C5A 6D01FC4C5A6D6DEE7F806D6C6C6C4BC7FC6E01E0EC07FE020F01FEEC1FF80203903AFFE0 01FFF0020091B612C0033F93C8FC030715FCDB007F14E0040101FCC9FC525479D261>I< BA7E19FCF1FF801AF01AFCD8000701F0C7000F13FF060014C0071F7F070713F807017F73 7F747E747F747F86747F747F8886888688A2757EA31D8087A21DC0A51DE0A387A963A31D C0A51D80A2631D00A3515AA2646264505B6264505B505B5090C7FCF2FFFE4F5B07075B07 1F5B96B512C0060F91C8FCBB5A1AF01AC007FCC9FC19805B527CD167>I70 D82 D97 D<913801FFF8021FEBFF8091B612F0010315FC010F9038C00FFE903A1FFE0001FFD97FFC 491380D9FFF05B4817C048495B5C5A485BA2486F138091C7FC486F1300705A4892C8FC5B A312FFAD127F7FA27EA2EF03E06C7F17076C6D15C07E6E140F6CEE1F806C6DEC3F006C6D 147ED97FFE5C6D6CEB03F8010F9038E01FF0010390B55A01001580023F49C7FC020113E0 33387CB63C>99 D<4DB47E0407B5FCA5EE001F1707B3A4913801FFE0021F13FC91B6FC01 0315C7010F9038E03FE74990380007F7D97FFC0101B5FC49487F4849143F484980485B83 485B5A91C8FC5AA3485AA412FFAC127FA36C7EA37EA26C7F5F6C6D5C7E6C6D5C6C6D49B5 FC6D6C4914E0D93FFED90FEFEBFF80903A0FFFC07FCF6D90B5128F0101ECFE0FD9003F13 F8020301C049C7FC41547CD24B>I<913803FFC0023F13FC49B6FC010715C04901817F90 3A3FFC007FF849486D7E49486D7E4849130F48496D7E48178048497F18C0488191C7FC48 17E0A248815B18F0A212FFA490B8FCA318E049CAFCA6127FA27F7EA218E06CEE01F06E14 037E6C6DEC07E0A26C6DEC0FC06C6D141F6C6DEC3F806D6CECFF00D91FFEEB03FE903A0F FFC03FF8010390B55A010015C0021F49C7FC020113F034387CB63D>I104 D<137F497E000313E0487FA2487FA76C5BA26C5BC613806DC7FC90C8FCADEB3F F0B5FCA512017EB3B3A6B612E0A51B547BD325>I<157FEDFF80020313E04A13F0A24A13 F8A76E13F0A26E13E002001380ED7F0092C7FCADED1FF891B5FCA51401EC007FB3B3B1EA 0780EA1FE0487E487E486C13FF16F0A216E05C16C04A13806C4848130049485A003F495A 000FB512F06C5C0001148026001FFCC7FC256C87D329>I108 DII<913801FFE0021F13FE91B612C0010315F0010F9038807FFC903A1FFC000FFED97FF8 6D6C7E49486D7F48496D7F48496D7F4A147F48834890C86C7EA24883A248486F7EA3007F 1880A400FF18C0AC007F1880A3003F18006D5DA26C5FA26C5F6E147F6C5F6C6D4A5A6C6D 495B6C6D495B6D6C495BD93FFE011F90C7FC903A0FFF807FFC6D90B55A010015C0023F91 C8FC020113E03A387CB643>I<903A3FF001FFE0B5010F13FE033FEBFFC092B612F002F3 01017F913AF7F8007FFE0003D9FFE0EB1FFFC602806D7F92C76C7F4A824A6E7F4A6E7FA2 717FA285187F85A4721380AC1A0060A36118FFA2615F616E4A5BA26E4A5B6E4A5B6F495B 6F4990C7FC03F0EBFFFC9126FBFE075B02F8B612E06F1480031F01FCC8FC030313C092CB FCB1B612F8A5414D7BB54B>I<90397FE003FEB590380FFF80033F13E04B13F09238FE1F F89139E1F83FFC0003D9E3E013FEC6ECC07FECE78014EF150014EE02FEEB3FFC5CEE1FF8 EE0FF04A90C7FCA55CB3AAB612FCA52F367CB537>114 D<903903FFF00F013FEBFE1F90 B7FC120348EB003FD80FF81307D81FE0130148487F4980127F90C87EA24881A27FA27F01 F091C7FC13FCEBFFC06C13FF15F86C14FF16C06C15F06C816C816C81C681013F1580010F 15C01300020714E0EC003F030713F015010078EC007F00F8153F161F7E160FA27E17E07E 6D141F17C07F6DEC3F8001F8EC7F0001FEEB01FE9039FFC00FFC6DB55AD8FC1F14E0D8F8 07148048C601F8C7FC2C387CB635>I<143EA6147EA414FEA21301A313031307A2130F13 1F133F13FF5A000F90B6FCB8FCA426003FFEC8FCB3A9EE07C0AB011FEC0F8080A26DEC1F 0015806DEBC03E6DEBF0FC6DEBFFF86D6C5B021F5B020313802A4D7ECB34>II<007FB500F090387FFFFEA5C66C48C7000F90C7FC6D6CEC07F86D 6D5C6D6D495A6D4B5A6F495A6D6D91C8FC6D6D137E6D6D5B91387FFE014C5A6E6C485A6E EB8FE06EEBCFC06EEBFF806E91C9FCA26E5B6E5B6F7E6F7EA26F7F834B7F4B7F92B5FCDA 01FD7F03F87F4A486C7E4A486C7E020F7FDA1FC0804A486C7F4A486C7F02FE6D7F4A6D7F 495A49486D7F01076F7E49486E7E49486E7FEBFFF0B500FE49B612C0A542357EB447> 120 D E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 %%EndSetup %%Page: 1 1 1 0 bop 521 -83 5 167 v 3285 -83 V 274 2 167 5 v 3370 2 V 523 407 a Fy(Diophan)l(tine)46 b(Conditions)g(and)e(Real)j(or)e (Complex)523 556 y(Brjuno)f(F)-11 b(unctions)523 851 y Fx(Pierre)26 b(Moussa)1041 821 y Fw(1)1105 851 y Fx(and)h(Stefano)h (Marmi)1807 821 y Fw(2)523 993 y Fv(1)596 1025 y Fu(Service)e(de)f(Ph)n (ysique)g(Th)n(\023)-36 b(eorique,)26 b(CEA/Sacla)n(y)-6 b(,)596 1116 y(F-91191)27 b(Gif)g(sur)f(Yv)n(ette)e(cedex,)i(F)-6 b(rance)523 1176 y Fv(2)596 1207 y Fu(Dipartimen)n(to)25 b(di)g(Matematica)i(\\U.Dini")f(Univ)n(ersit\022)-38 b(a)26 b(di)g(Firenze,)596 1299 y(Viale)g(Morgagni)i(67/A,)f(I-50134)g (Firenze,)f(Italy)596 1390 y(Preprin)n(t)g(T99/116)i(SPhT-CEA/SA)n(CLA) -6 b(Y,)25 b(Septem)n(b)r(er)f(17,)i(1999)523 1677 y Ft(Abstract.)42 b Fu(The)c(con)n(tin)n(ued)e(fraction)j(expansion)e(of) h(the)f(real)i(n)n(um)n(b)r(er)c Fs(x)40 b Fu(=)h Fs(a)3044 1685 y Fv(0)3103 1677 y Fu(+)24 b Fs(x)3231 1685 y Fv(0)3265 1677 y Fu(,)523 1768 y Fs(a)564 1776 y Fv(0)636 1768 y Fr(2)39 b Fq(Z)-31 b(Z)p Fu(,)36 b(is)g(giv)n(en)f(b)n(y)g(0)k Fr(\024)e Fs(x)1478 1776 y Fp(n)1558 1768 y Fs(<)h Fu(1,)e Fs(x)1795 1736 y Fo(\000)p Fv(1)1795 1781 y Fp(n)1915 1768 y Fu(=)i Fs(a)2054 1776 y Fp(n)p Fv(+1)2197 1768 y Fu(+)24 b Fs(x)2325 1776 y Fp(n)p Fv(+1)2444 1768 y Fu(,)36 b Fs(a)2542 1776 y Fp(n)p Fv(+1)2699 1768 y Fr(2)i Fu(I)-13 b(N,)35 b(for)i Fs(n)h Fr(\025)g Fu(0.)523 1859 y(The)28 b(Brjuno)h(function)f(is)h(then)e Fs(B)t Fu(\()p Fs(x)p Fu(\))e(=)1779 1799 y Fn(P)1866 1820 y Fo(1)1866 1886 y Fp(n)p Fv(=0)1999 1859 y Fs(x)2043 1867 y Fv(0)2077 1859 y Fs(x)2121 1867 y Fv(1)2168 1859 y Fs(:)13 b(:)g(:)g(x)2314 1867 y Fp(n)p Fo(\000)p Fv(1)2447 1859 y Fu(ln\()p Fs(x)2585 1828 y Fo(\000)p Fv(1)2585 1872 y Fp(n)2667 1859 y Fu(\),)28 b(and)g(the)g(n)n(um)n(b)r(er)523 1951 y Fs(x)f Fu(satis\014es)i(the)f (Brjuno)g(diophan)n(tine)g(condition)g(whenev)n(er)f Fs(B)t Fu(\()p Fs(x)p Fu(\))g(is)i(b)r(ounded.)e(In)n(v)l(arian)n(t)523 2042 y(circles)f(under)e(a)h(complex)f(rotation)i(p)r(ersist)f(when)g (the)g(map)e(is)j(analytically)g(p)r(erturb)r(ed,)e(if)523 2133 y(and)h(only)f(if)i(the)e(rotation)i(n)n(um)n(b)r(er)d (satis\014es)j(the)e(Brjuno)h(condition,)h(and)e(the)h(same)f(holds)523 2225 y(for)i(in)n(v)l(arian)n(t)f(circles)i(in)e(the)g(semi-standard)g (and)g(standard)g(maps)g(cases.)i(In)e(this)g(lecture,)523 2316 y(w)n(e)19 b(will)i(review)e(some)g(prop)r(erties)h(of)g(the)e (Brjuno)i(function,)f(and)g(giv)n(e)g(some)g(generalisations)523 2407 y(related)29 b(to)g(familiar)h(diophan)n(tine)e(conditions.)i(The) f(Brjuno)g(function)g(is)g(highly)g(singular)523 2499 y(and)23 b(tak)n(es)g(v)l(alue)g(+)p Fr(1)f Fu(on)h(a)g(dense)g(set)h (including)f(rationals.)i(W)-6 b(e)22 b(presen)n(t)h(a)h (regularisation)523 2590 y(leading)29 b(to)g(a)f(complex)g(function)g (holomorphic)h(in)f(the)g(upp)r(er)g(half)h(plane.)g(Its)f(imaginary) 523 2681 y(part)20 b(tends)g(to)h(the)f(Brjuno)h(function)f(on)g(the)g (real)i(axis,)f(the)f(real)h(part)f(remaining)g(b)r(ounded,)523 2773 y(and)25 b(w)n(e)h(also)h(indicate)g(its)f(transformation)g(under) f(the)g(mo)r(dular)g(group.)523 3078 y Fm(1)112 b(Hamiltonian)35 b(Chaos)j(and)h(the)e(Standard)i(Map)523 3307 y Fx(The)32 b(simplest)g(non)g(trivial)f(mo)r(del)h(for)f(Hamiltonian)h(c)n(haos)e (is)i(a)f(t)n(w)n(o)g(dimensional)523 3406 y(real)i(map,)h(called)f (the)h(\\Standard)f(Map".)g(It)h(has)g(b)r(een)g(in)n(tro)r(duced)f (more)g(or)g(less)523 3506 y(indep)r(enden)n(tly)24 b(b)n(y)g(Chirik)n (o)n(v)e(and)h(T)-7 b(a)n(ylor)22 b([1,2].)h(The)h(o)r(ccurrence)f(of)g (c)n(haos)g(w)n(as)f(dis-)523 3606 y(cussed)33 b(b)n(y)g(Greene)g([3])g (who)g(displa)n(y)n(ed)g(man)n(y)f(n)n(umerical)h(results)g(on)g(this)h (mo)r(del,)523 3705 y(whic)n(h)d(describ)r(es)f(a)g(simpli\014ed)i(v)n (ersion)d(of)h(the)i(non{linear)d(coupling)h(of)h(t)n(w)n(o)f(oscil-) 523 3805 y(lators.)37 b(It)h(o)r(ccurs)f(naturally)g(in)h(man)n(y)g (domains)f(of)h(ph)n(ysics,)f(including)h(celestial)523 3905 y(mec)n(hanics,)26 b(classical)f(quasip)r(erio)r(dic)h(systems,)g (quan)n(tum)g(quasicrystals,)f(adiabatic)523 4004 y(resp)r(onse)c(in)i (non{linear)d(mec)n(hanics,)i(magnetic)f(toroidal)g(con\014gurations)f (in)j(plasma)523 4104 y(ph)n(ysics,)k(non{linear)f(electronic)h (devices,)g(and)h(man)n(y)f(others.)648 4209 y(The)f(Standard)f(Map)h (is)g(a)f(map)h(from)g(the)g(cylinder)p 2366 4209 4 52 v 26 w(T)15 b Fl(\002)g Fx(I)-14 b(R)26 b(to)g(itself,)g(de\014ned)h (as)572 4382 y Fn(\000)610 4449 y Fk(\022)r(;)14 b(r)727 4382 y Fn(\001)790 4449 y Fl(7!)896 4382 y Fn(\000)934 4449 y Fk(\022)975 4415 y Fj(0)998 4449 y Fk(;)g(r)1074 4415 y Fj(0)1098 4382 y Fn(\001)1159 4449 y Fx(=)1247 4332 y Fn(\022)1308 4449 y Fk(\022)21 b Fx(+)d Fk(r)j Fx(+)1610 4393 y Fk(K)p 1602 4430 92 4 v 1602 4506 a Fx(2)p Fk(\031)1718 4449 y Fx(sin\(2)p Fk(\031)s(\022)r Fx(\))56 b(\(mo)r(d)28 b(1\))f Fk(;)42 b(r)21 b Fx(+)2617 4393 y Fk(K)p 2609 4430 V 2609 4506 a Fx(2)p Fk(\031)2725 4449 y Fx(sin)o(\(2)p Fk(\031)s(\022)r Fx(\))3023 4332 y Fn(\023)3108 4449 y Fk(:)50 b Fx(\(1\))523 4689 y(Note)28 b(that)h(the)f(second)g(v)-5 b(ariable)27 b(can)h(also)f(b)r(e)i(tak)n (en)e(mo)r(dulo)h(1,)g(in)h(whic)n(h)f(case)f(w)n(e)523 4788 y(get)f(a)f(map)p 928 4788 4 52 v 26 w(T)970 4752 y Fw(2)1030 4788 y Fl(!)p 1154 4788 V 24 w Fx(T)1196 4752 y Fw(2)1234 4788 y Fx(.)h(The)g(ab)r(o)n(v)n(e)e(map)i(can)g(b)r (e)g(written)g(in)h(t)n(w)n(o)e(equiv)-5 b(alen)n(t)25 b(forms:)p eop %%Page: 2 2 2 1 bop 523 232 a Fu(2)237 b(Pierre)27 b(Moussa)g(and)e(Stefano)i (Marmi)581 448 y Fl(\017)41 b Fx(Hamiltonian)28 b(form)1290 647 y Fk(r)1329 613 y Fj(0)1372 647 y Fl(\000)18 b Fk(r)26 b Fx(=)1623 591 y Fk(K)p 1615 628 92 4 v 1615 704 a Fx(2)p Fk(\031)1731 647 y Fx(sin\(2)p Fk(\031)s(\022)r Fx(\))84 b Fk(;)96 b(\022)2274 613 y Fj(0)2316 647 y Fl(\000)18 b Fk(\022)26 b Fx(=)c Fk(r)2590 613 y Fj(0)2638 647 y Fk(;)520 b Fx(\(2\))581 847 y Fl(\017)41 b Fx(Lagrangian)27 b(form,)j(where)f(one)g(considers)f(no)n(w)h(the)h(t)n(wice)g(iterated) f(map,)h(that)664 946 y(is)e(\()p Fk(\022)r(;)14 b(r)r Fx(\))24 b Fl(!)f Fx(\()p Fk(\022)1132 916 y Fj(0)1156 946 y Fk(;)14 b(r)1232 916 y Fj(0)1256 946 y Fx(\))23 b Fl(!)h Fx(\()p Fk(\022)1491 916 y Fj(00)1534 946 y Fk(;)14 b(r)1610 916 y Fj(00)1653 946 y Fx(\),)28 b(so)f(that)1456 1166 y Fk(\022)1497 1132 y Fj(00)1558 1166 y Fl(\000)18 b Fx(2)p Fk(\022)1724 1132 y Fj(0)1766 1166 y Fx(+)g Fk(\022)25 b Fx(=)2018 1110 y Fk(K)p 2011 1147 V 2011 1223 a Fx(2)p Fk(\031)2126 1166 y Fx(sin\(2)p Fk(\031)s(\022)2393 1132 y Fj(0)2417 1166 y Fx(\))e Fk(;)686 b Fx(\(3\))664 1366 y(and)28 b(for)f(the)h Fk(n)p Fx(-th)f(iterated)h(map)f(one)g (gets)1330 1581 y Fk(\022)1369 1593 y Fi(n)p Fw(+1)1517 1581 y Fl(\000)18 b Fx(2)p Fk(\022)1681 1593 y Fi(n)1744 1581 y Fx(+)g Fk(\022)1866 1593 y Fi(n)p Fj(\000)p Fw(1)2019 1581 y Fx(=)2124 1525 y Fk(K)p 2117 1562 V 2117 1638 a Fx(2)p Fk(\031)2232 1581 y Fx(sin\(2)p Fk(\031)s(\022)2497 1593 y Fi(n)2543 1581 y Fx(\))23 b Fk(:)560 b Fx(\(4\))523 1785 y(This)22 b(last)g(equation)g(is)g(sometimes)g(called)g(the)g(F)-7 b(renk)n(el-Kon)n(toro)n(v)i(a)19 b(mo)r(del)j([4])g(whic)n(h)523 1885 y(describ)r(es)j(equilibrium)g(p)r(ositions)g(of)g(a)g(c)n(hain)g (of)g(material)g(p)r(oin)n(ts)g(placed)g(in)g(a)g(p)r(eri-)523 1985 y(o)r(dic)e(p)r(oten)n(tial)f(and)h(submitted)h(to)e(an)h (harmonic)e(elastic)i(force)f(b)r(et)n(w)n(een)g(t)n(w)n(o)g(neigh-)523 2084 y(b)r(ouring)27 b(p)r(oin)n(ts.)648 2184 y(If)h Fk(K)g Fx(=)23 b(0,)k(w)n(e)g(get)h(the)g(so-called)e(\\t)n(wist-map",) g(whic)n(h)i(giv)n(es)787 2361 y Fk(r)824 2373 y Fi(n)p Fw(+1)977 2361 y Fx(=)22 b Fk(r)1101 2373 y Fi(n)1170 2361 y Fx(=)h Fk(r)1295 2373 y Fw(0)1355 2361 y Fx(=)g Fk(\032)g Fx(=)g(constan)n(t)82 b Fk(;)97 b(\022)2146 2373 y Fi(n)p Fw(+1)2298 2361 y Fx(=)23 b Fk(\022)2425 2373 y Fw(0)2480 2361 y Fx(+)18 b Fk(n\032)28 b Fx(\(mo)r(d)g(1\))23 b Fk(;)158 b Fx(\(5\))523 2538 y(after)29 b Fk(n)h Fx(iterations.)f (This)h(map)g(is)f(nothing)h(but)g(a)g(rotation)e(of)i(angle)f(2)p Fk(\031)s(\032)p Fx(|w)n(e)g(sa)n(y)523 2638 y(that)d(the)f(rotation)g (n)n(um)n(b)r(er)g(is)g Fk(\032)p Fx(.)g(The)h(orbits)e(are)h(all)g (transv)n(erse)e(to)i(the)h(axis)e(of)i(the)523 2738 y(cylinder.)f(They)h(are)e(made)i(of)f(a)g(\014nite)h(n)n(um)n(b)r(er)g (of)f(p)r(oin)n(ts)h(\(and)f(therefore)g(discrete\))523 2837 y(if)h Fk(\032)g Fx(is)f(rational,)f(and)h(they)h(are)f(dense)g (on)g(transv)n(erse)f(circles)g(if)i Fk(\032)g Fx(is)f(irrational.)f (The)523 2937 y(cylinder)j(is)g(sliced)g(along)f(orbits)g(at)h (irrational)f(v)-5 b(alues)27 b(of)g Fk(r)r Fx(,)h(whic)n(h)f(are)f(in) n(tert)n(wined)523 3037 y(with)31 b(discrete)e(orbits)h(at)g(rational)f (v)-5 b(alues)29 b(of)i Fk(r)r Fx(.)g(The)f(question)g(is)g(:)g(in)g (whic)n(h)g(sense)523 3136 y(is)e(suc)n(h)f(a)g(pattern)g(stable)h (under)f(p)r(erturbations,)g(that)h(is)f(here)h(when)f Fk(K)i Fl(6)p Fx(=)22 b(0?)648 3236 y(Among)d(the)h(orbits)f(whic)n(h)h (are)f(dense)h(in)g(a)f(curv)n(e)g(wrapp)r(ed)g(around)g(the)h (cylinder,)523 3335 y(particularly)27 b(in)n(teresting)h(are)g(the)g (orbits)g(whic)n(h)h(will)g(p)r(ersist)f(under)g(p)r(erturbation,)523 3435 y(in)39 b(particular)f(b)r(ecause)h(they)h(separate)d(the)j(space) e(in)n(to)h(domains)g(whic)n(h)g(do)g(not)523 3535 y(comm)n(unicate.)31 b(It)h(is)f(kno)n(wn)g(that)g(when)h Fk(K)37 b Fx(is)31 b(large)f(\(for)h(example)g Fk(K)k(>)29 b Fx(4)p Fk(=)p Fx(3,)h(see)523 3634 y([5]\),)d(suc)n(h)g(orbits)g(do)g(not)g(exist,)g (and)g(on)g(the)h(other)f(hand,)g(when)h Fk(K)k Fx(is)27 b(small,)g(some)523 3734 y(of)33 b(the)h(irrational)d(orbits)h(p)r (ersist,)h(dep)r(ending)h(on)f(arithmetical)f(prop)r(erties)g(of)i(the) 523 3834 y(rotation)c(n)n(um)n(b)r(er.)i(F)-7 b(or)31 b(p)r(erturb)r(ed)g(t)n(wist)h(maps,)f(w)n(e)g(de\014ne)h(the)g (rotation)e(n)n(um)n(b)r(er)523 3933 y(as)25 b(lim)738 3945 y Fi(n)p Fj(!1)929 3933 y Fk(n)979 3903 y Fj(\000)p Fw(1)1068 3933 y Fk(\022)1107 3945 y Fi(n)1153 3933 y Fx(,)g(where)g(\()p Fk(\022)1510 3945 y Fi(n)1555 3933 y Fk(;)14 b(r)1629 3945 y Fi(n)1675 3933 y Fx(\))25 b(is)h(the)f Fk(n)p Fx(-th)h(iterated)f(map)g(obtained)g(from)g(\(1\).)648 4033 y(Other)32 b(kinds)h(of)f(in)n(v)-5 b(arian)n(t)32 b(curv)n(es)g(ma)n(y)g(o)r(ccur,)g(attac)n(hed)g(to)h(elliptic)g(p)r (erio)r(dic)523 4132 y(orbits.)j(F)-7 b(or)36 b(instance)g(if)h Fk(K)43 b(>)37 b Fx(0)f(is)g(su\016cien)n(tly)h(small,)f(the)g(p)r(oin) n(t)h(\(1)p Fk(=)p Fx(2)p Fk(;)14 b Fx(0\))35 b(is)h(an)523 4232 y(elliptic)30 b(\014xed)f(p)r(oin)n(t.)g(Due)h(to)e(KAM)i(Theorem) e(\(see)h([6])f(for)h(a)g(review\),)f(there)h(exist)523 4332 y(homotopically)34 b(trivial)i(in)n(v)-5 b(arian)n(t)34 b(curv)n(es)h(on)g(the)h(cylinder)f(winding)h(around)f(this)523 4431 y(\014xed)d(p)r(oin)n(t,)g(whic)n(h)g(form)f(the)h(so{called)f (elliptic)h(islands.)f(W)-7 b(e)33 b(shall)e(not)h(consider)523 4531 y(here)e(the)h(problem)g(of)f(suc)n(h)h(orbits,)f(although)g (there)h(existence)f(is)h(v)n(ery)e(imp)r(ortan)n(t)523 4631 y(in)g(connection)f(with)h(ergo)r(dic)f(theory)-7 b(.)28 b(Indeed)h(one)f(exp)r(ects)h(c)n(haotic)f(b)r(eha)n(viour)f (for)523 4730 y Fk(K)38 b Fx(large,)31 b(but)j(the)f(p)r(ersistence)f (of)g(elliptic)i(islands)e(could)g(prev)n(en)n(t)g(the)h(map)f(from)523 4830 y(b)r(eing)c(ergo)r(dic.)p eop %%Page: 3 3 3 2 bop 1836 232 a Fu(Real)26 b(or)g(Complex)f(Brjuno)i(F)-6 b(unctions)236 b(3)523 448 y Fm(2)112 b(The)38 b(Critical)c(Constan)m (ts)523 653 y Fx(F)-7 b(or)24 b(the)g(standard)f(map,)h(w)n(e)g (consider)f(no)n(w)h(the)g(homotopically)f(non{trivial)g(in)n(v)-5 b(ari-)523 752 y(an)n(t)25 b(curv)n(es)e Fh(i.e.)54 b Fx(wrapp)r(ed)24 b(around)g(the)h(cylinder.)f(A)h(natural)f(w)n(a)n(y)g (to)h(lo)r(ok)f(for)g(their)523 852 y(existence)j(is)h(to)f(replace)g (the)h(angular)e(v)-5 b(ariable)26 b Fk(\022)k Fx(b)n(y)e(the)g(new)f (v)-5 b(ariable)27 b Fk(\036)c Fl(2)h Fg(T)833 1036 y Fk(\022)i Fx(=)c Fk(\036)d Fx(+)f Fk(u)p Fx(\()p Fk(\036;)c(K)q(;)g (\032)p Fx(\))83 b Fk(;)97 b(r)26 b Fx(=)d Fk(\032)18 b Fx(+)g Fk(u)p Fx(\()p Fk(\036;)c(K)q(;)g(\032)p Fx(\))19 b Fl(\000)f Fk(u)p Fx(\()p Fk(\036)h Fl(\000)f Fk(\032;)c(K)q(;)g(\032) p Fx(\))22 b Fk(:)205 b Fx(\(6\))523 1221 y(With)37 b(the)g(condition)f (1)23 b(+)h Fk(@)5 b(u=@)g(\036)36 b(>)h Fx(0,)f(it)h(w)n(ould)e (describ)r(e)h(a)g(curv)n(e)f(around)h(the)523 1321 y(cylinder,)30 b(on)g(whic)n(h)g(the)h(map)f(is)g(expressed)f(as)h Fk(\036)2171 1291 y Fj(0)2222 1321 y Fx(=)d Fk(\036)20 b Fx(+)g Fk(\032)p Fx(,)31 b(when)f Fk(\036)h Fx(describ)r(es)e Fg(T)523 1420 y Fx(for)h Fk(K)36 b Fx(and)30 b Fk(\032)g Fx(\014xed.)g(W)-7 b(e)31 b(sa)n(y)e(that)i(\(6\))f(expresses)f(on)h(the)h(curv)n(e)e(the) i(conjugacy)e(of)523 1520 y(the)e(map)f(to)g(a)g(rotation.)f(The)h (existence)g(of)g(a)g(function)h Fk(u)p Fx(\()p Fk(\036;)14 b(K)q(;)g(\032)p Fx(\),)26 b(analytic)g(in)h(the)523 1620 y(v)-5 b(ariable)27 b Fk(\036)p Fx(,)h(insures)f(the)h(existence)g (of)g(an)f(analytic)g(in)n(v)-5 b(arian)n(t)27 b(curv)n(e)g(with)h (rotation)523 1719 y(n)n(um)n(b)r(er)21 b Fk(\032)p Fx(.)g(W)-7 b(e)22 b(are)e(in)n(terested)h(to)g(determine)g(the)h(critical)f (constan)n(t)f Fk(K)2833 1731 y Fi(c)2866 1719 y Fx(\()p Fk(\032)p Fx(\))i(as)f(b)r(eing)523 1819 y(the)29 b(largest)f(p)r (ossible)h(v)-5 b(alue)29 b(of)g Fk(K)34 b Fx(for)29 b(whic)n(h)g(suc)n(h)g(an)f(analytic)h(function)h Fk(u)e Fx(exists.)523 1918 y(Of)g(course,)e(one)h(could)h(consider)e (regularit)n(y)g(constrain)n(ts)g(w)n(eak)n(er)g(than)i(analyticit)n(y) -7 b(,)523 2018 y(leading)30 b(to)g(other)g(critical)g(constan)n(ts.)g (W)-7 b(e)31 b(lo)r(ok)f(for)g(a)g(p)r(erturbation)g(expansion)g(of)523 2118 y(the)j(function)h Fk(u)p Fx(,)e(and)h(w)n(e)f(follo)n(w)h(the)g (notations)f(of)h([7].)f(F)-7 b(rom)33 b(the)g(standard)f(map)523 2217 y(w)n(e)27 b(get)h(from)f(\(6\))565 2444 y Fk(u)p Fx(\()p Fk(\036)10 b Fx(+)g Fk(\032;)k(K)q(;)g(\032)p Fx(\))c Fl(\000)g Fx(2)p Fk(u)p Fx(\()p Fk(\036;)k(K)q(;)g(\032)p Fx(\))c(+)g Fk(u)p Fx(\()p Fk(\036)g Fl(\000)g Fk(\032;)k(K)q(;)g(\032) p Fx(\))22 b(=)2210 2388 y Fk(K)p 2202 2425 92 4 v 2202 2501 a Fx(2)p Fk(\031)2318 2444 y Fx(sin\(2)p Fk(\031)s Fx(\()p Fk(\036)10 b Fx(+)g Fk(u)p Fx(\()p Fk(\036;)k(K)q(;)g(\032)p Fx(\)\))24 b Fk(:)42 b Fx(\(7\))523 2652 y(F)-7 b(or)27 b Fk(k)f Fl(\025)d Fx(1,)k(w)n(e)g(ha)n(v)n(e)523 2882 y Fk(u)571 2848 y Fw(\()p Fi(k)q Fw(\))663 2882 y Fx(\()p Fk(\036)5 b Fx(+)g Fk(\032;)14 b(\032)p Fx(\))5 b Fl(\000)g Fx(2)p Fk(u)1139 2848 y Fw(\()p Fi(k)q Fw(\))1231 2882 y Fx(\()p Fk(\036;)14 b(\032)p Fx(\))5 b(+)g Fk(u)1547 2848 y Fw(\()p Fi(k)q Fw(\))1639 2882 y Fx(\()p Fk(\036)g Fl(\000)g Fk(\032;)14 b(\032)p Fx(\))24 b(=)2106 2826 y(1)p 2081 2863 V 2081 2939 a(2)p Fk(\031)2197 2882 y Fx(sin)2313 2815 y Fn(\000)2351 2882 y Fx(2)p Fk(\031)s(\036)18 b Fx(+)g(2)p Fk(\031)s(u)p Fx(\()p Fk(\036;)c(K)q(;)g(\032)p Fx(\))3035 2815 y Fn(\001)3073 2762 y(\014)3073 2812 y(\014)3073 2861 y(\014)3073 2911 y(\014)3101 2965 y Fi(k)q Fj(\000)p Fw(1)3264 2882 y Fk(;)3181 3043 y Fx(\(8\))523 3142 y(where)28 b(in)h(the)g(righ)n(t)e(hand)i(side,)f(one)h(k)n(eeps)e (only)h(the)h(terms)g(of)f(order)f Fk(k)22 b Fl(\000)d Fx(1)28 b(in)h(the)523 3242 y(expansion)k(on)g(p)r(o)n(w)n(ers)g(of)g Fk(K)6 b Fx(.)34 b(W)-7 b(e)34 b(use)f(no)n(w)g(the)i(F)-7 b(ourier)32 b(series)h(expansion)g(on)g Fk(\036)p Fx(,)523 3353 y(that)k(is)f Fk(u)852 3323 y Fw(\()p Fi(k)q Fw(\))945 3353 y Fx(\()p Fk(\036;)14 b(\032)p Fx(\))39 b(=)1279 3291 y Fn(P)1367 3378 y Fi(\027)t Fj(2)p Ff(Z)1513 3353 y Fk(u)1561 3310 y Fw(\()p Fi(k)q Fw(\))1561 3363 y Fi(\027)1653 3353 y Fx(\()p Fk(\032)p Fx(\))p Fk(e)1799 3323 y Fw(2)p Fi(i\031)r(\027)t(\036)1978 3353 y Fx(,)e(and)f(w)n(e)g(see)h(that)f (the)h(co)r(e\016cien)n(t)g(of)523 3477 y Fk(e)562 3447 y Fw(2)p Fi(i\031)r(\027)t(\036)773 3477 y Fx(in)c(the)g(left)g(hand)f (side)h(of)f(\(8\))h(is)g(2\(cos)o(\(2)p Fk(\031)s(\027)5 b(\032)p Fx(\))22 b Fl(\000)f Fx(1\))p Fk(u)2544 3434 y Fw(\()p Fi(k)q Fw(\))2544 3487 y Fi(\027)2636 3477 y Fx(\()p Fk(\032)p Fx(\).)33 b(Therefore)f(\(8\))523 3593 y(allo)n(ws)24 b(a)h(recursiv)n(e)f(computation)i(of)f(the)h(F)-7 b(ourier)25 b(co)r(e\016cien)n(ts)g Fk(u)2648 3550 y Fw(\()p Fi(k)q Fw(\))2648 3603 y Fi(\027)2740 3593 y Fx(\()p Fk(\032)p Fx(\),)h(and)g(w)n(e)f(get)523 3693 y(for)30 b Fk(u)701 3663 y Fw(\()p Fi(k)q Fw(\))793 3693 y Fx(\()p Fk(\036;)14 b(\032)p Fx(\))32 b(expressions)c(as)i (trigonometric)f(p)r(olynomials)h(in)h Fk(\036)p Fx(.)g(Ho)n(w)n(ev)n (er)d(terms)523 3792 y(of)34 b(the)h(kind)g(2\(cos)o(\(2)p Fk(\031)s(\027)5 b(\032)p Fx(\))24 b Fl(\000)e Fx(1\))35 b(o)r(ccur)f(in)g(the)h(denominators)e(along)h(the)h(steps)f(of)523 3892 y(the)c(recursion.)f(Suc)n(h)h(factors)f(are)g(called)h(\\small)f (divisors",)f(some)h(of)h(them)h(v)-5 b(anish)523 3992 y(when)27 b Fk(\032)f Fx(is)g(rational,)f(and)i(ma)n(y)e(b)r(ecome)h (arbitrarily)f(small)h(when)g Fk(\027)32 b Fx(b)r(ecomes)26 b(large,)523 4091 y(for)33 b(irrational)f Fk(\032)p Fx(.)i(No)n(w)f (let)1468 4070 y Fn(e)1448 4091 y Fk(K)1519 4103 y Fi(c)1552 4091 y Fx(\()p Fk(\032)p Fx(\))h(b)r(e)g(the)g(minim)n(um)h(o)n(v)n(er) d Fk(\036)i Fx(of)f(the)h(con)n(v)n(ergence)523 4191 y(radius)27 b(of)g(the)h(expansion)1358 4437 y Fk(u)p Fx(\()p Fk(\036;)14 b(K)q(;)g(\032)p Fx(\))24 b(=)1847 4334 y Fj(1)1820 4359 y Fn(X)1819 4537 y Fi(k)q Fw(=1)1954 4437 y Fk(K)2031 4403 y Fi(k)2071 4437 y Fk(u)2119 4403 y Fw(\()p Fi(k)q Fw(\))2211 4437 y Fx(\()p Fk(\036;)14 b(\032)p Fx(\))24 b Fk(:)730 b Fx(\(9\))523 4720 y(F)-7 b(or)34 b Fk(\032)g Fx(rational,)g(\(8\))g(cannot)g(b)r(e)h(solv)n(ed,) f(and)g(w)n(e)g(set)2369 4699 y Fn(e)2349 4720 y Fk(K)2420 4732 y Fi(c)2453 4720 y Fx(\()p Fk(\032)p Fx(\))h(=)g(0.)f(F)-7 b(or)34 b(irrational)523 4830 y(v)-5 b(alues)37 b(of)h Fk(\032)p Fx(,)f(Berretti)g(and)h(Gen)n(tile)g([8])f(w)n(ere)g(able)g (to)g(con)n(trol)2715 4809 y Fn(e)2695 4830 y Fk(K)2766 4842 y Fi(c)2799 4830 y Fx(\()p Fk(\032)p Fx(\))h(using)g(the)p eop %%Page: 4 4 4 3 bop 523 232 a Fu(4)237 b(Pierre)27 b(Moussa)g(and)e(Stefano)i (Marmi)523 448 y Fx(Brjuno)f(function)h Fk(B)t Fx(\()p Fk(\032)p Fx(\))g(whic)n(h)g(is)f(a)g(n)n(um)n(b)r(er)g(theoretic)g (function)i(whic)n(h)e(will)h(de\014ne)523 548 y(in)k(the)g(follo)n (wing)e(Section)h(4.)g(More)g(precisely)-7 b(,)30 b(there)g(exists)g Fk(C)k(>)27 b Fx(0)k(suc)n(h)f(that,)g(for)523 648 y(an)n(y)d (irrational)f Fk(\032)p Fx(,)1334 672 y Fn(\014)1334 722 y(\014)1334 772 y(\014)1361 767 y Fx(ln)1444 675 y Fn(\020)1494 767 y Fx(\()1546 747 y Fn(e)1526 767 y Fk(K)1597 779 y Fi(c)1630 767 y Fx(\()p Fk(\032)p Fx(\)\))1769 733 y Fj(\000)p Fw(1)1859 675 y Fn(\021)1927 767 y Fl(\000)18 b Fx(2)p Fk(B)t Fx(\()p Fk(\032)p Fx(\))2226 672 y Fn(\014)2226 722 y(\014)2226 772 y(\014)2277 767 y Fk(<)23 b(C)29 b(:)663 b Fx(\(10\))523 949 y(The)19 b(functions)1054 928 y Fn(e)1034 949 y Fk(K)1105 961 y Fi(c)1138 949 y Fx(\()p Fk(\032)p Fx(\))g(and)g Fk(e)1456 919 y Fj(\000)p Fw(2)p Fi(B)s Fw(\()p Fi(\032)p Fw(\))1703 949 y Fx(b)r(oth)g(v)-5 b(anish)19 b(on)f(all)g(rationals,)g(but)h(the)g(previous)523 1059 y(equation)29 b(sho)n(ws)f(that)i(the)f(ratio)1650 1038 y Fn(e)1631 1059 y Fk(K)1702 1071 y Fi(c)1735 1059 y Fx(\()p Fk(\032)p Fx(\))p Fk(=e)1923 1029 y Fj(\000)p Fw(2)p Fi(B)s Fw(\()p Fi(\032)p Fw(\))2181 1059 y Fx(remains)f (uniformly)h(b)r(ounded)h(at)523 1159 y(ev)n(ery)18 b(irrationals.)f (The)i(fact)g(that)g(this)g(ratio)f(remains)g(b)r(ounded)h(is)g(in)h (itself)f(amazing,)523 1258 y(but)h(it)f(recalls)f(earlier)g(and)h(no)n (w)g(classical)f(results)g(b)n(y)h(Y)-7 b(o)r(ccoz)19 b([9])g(on)g(the)g(linearisation)523 1358 y(of)k(holomorphic)f(maps.)h (W)-7 b(e)23 b(shall)g(see)g(later)g(that)g(w)n(e)g(ma)n(y)f(ha)n(v)n (e)g(ev)n(en)h(b)r(etter)g(results)523 1458 y(in)28 b(the)g(framew)n (ork)e(of)h(holormorphic)f(maps.)648 1557 y(The)34 b(determination)g (of)g(the)h(radius)e(of)h(con)n(v)n(ergence)e(in)j(\(9\))f(is)g(not)h (the)f(whole)523 1657 y(story)-7 b(.)23 b(It)h(is)f(p)r(ossible)h(that) g(the)g(function)g Fk(u)f Fx(ma)n(y)g(b)r(e)h(analytically)e(con)n(tin) n(ued)i(for)f(real)523 1756 y(v)-5 b(alues)35 b(of)h Fk(\036)g Fx(and)g(for)f Fk(K)42 b(>)1505 1735 y Fn(e)1486 1756 y Fk(K)1557 1768 y Fi(c)1590 1756 y Fx(\()p Fk(\032)p Fx(\))36 b(real.)f(Th)n(us)g(w)n(e)h(w)n(ould)f(ha)n(v)n(e)g(another)g (critical)523 1866 y(constan)n(t)27 b Fk(K)929 1878 y Fi(c)962 1866 y Fx(\()p Fk(\032)p Fx(\))i(suc)n(h)e(that)h(w)n(e)f (still)h(ha)n(v)n(e)f(real)f(analytic)h(curv)n(es)g(for)2820 1845 y Fn(e)2800 1866 y Fk(K)2871 1878 y Fi(c)2904 1866 y Fx(\()p Fk(\032)p Fx(\))d Fk(<)f(K)28 b(<)523 1966 y(K)594 1978 y Fi(c)627 1966 y Fx(\()p Fk(\032)p Fx(\))c(real.)f(The)g (n)n(umerical)g(results)g([10])f(seem)h(to)h(indicate)f(that)h(this)g (is)f(indeed)h(the)523 2066 y(case:)29 b(see)h([11])f(for)h(a)f (detailed)h(discussion)f(of)h(this)h(issue)e(b)r(oth)i(from)e(the)i(n)n (umerical)523 2165 y(and)40 b(the)h(analytical)e(p)r(oin)n(ts)h(of)h (view,)f(whic)n(h)g(also)f(uses)h(results)g(of)g([12,13].)f(The)523 2265 y(de\014nitiv)n(e)d(answ)n(er)d(is)j(not)f(kno)n(wn)g(to)g(us)g (to)r(da)n(y)-7 b(,)35 b(although)g(w)n(e)g(are)f(led)h(to)h(exp)r(ect) 523 2365 y(that)24 b(the)g(function)g Fk(B)t Fx(\()p Fk(\032)p Fx(\))g(pla)n(ys)f(a)g(cen)n(tral)f(role)h(in)h(the)g (determination)f(of)h Fk(K)2984 2377 y Fi(c)3017 2365 y Fx(\()p Fk(\032)p Fx(\))g(\(see)523 2464 y(also)j(Da)n(vie)g([14]\).) 523 2715 y Fm(3)112 b(Complex)37 b(Analytic)e(Maps)523 2898 y Fx(The)28 b(problems)e(of)i(the)f(critical)g(constan)n(t)g(is)g (b)r(etter)h(understo)r(o)r(d)f(in)h(the)g(case)e(of)i(the)523 2998 y(complex)18 b(analytic)h(maps.)f(W)-7 b(e)20 b(ha)n(v)n(e)d (already)h(seen)g(in)h(\(10\))g(that)g(the)g(critical)g(constan)n(t)543 3087 y Fn(e)523 3108 y Fk(K)594 3120 y Fi(c)627 3108 y Fx(\()p Fk(\032)p Fx(\))33 b(of)g(the)g(complexi\014ed)f(v)n(ersion)f (of)i(the)g(standard)e(map)i(is)f(con)n(trolled)f(b)n(y)i(the)523 3208 y(Brjuno)27 b(function.)g(A)h(simpler)f(example)f(is)i(the)f (\\Semi-standard)f(Map",)g(whic)n(h)h(is)g(a)523 3307 y(t)n(w)n(o)21 b(dimensional)g(complex)h(map)g(on)f(the)h(cylinder,)g (closely)f(related)g(to)h(the)g(standard)523 3407 y(map)k(\(1\))h(:)g (to)f(get)g(the)h(semi-standard)e(map,)i(just)g(replace)f(in)g(\(1\))h (the)g(sine)f(function)523 3507 y(sin\(2)p Fk(\031)s(\022)r Fx(\))c(b)n(y)f(its)h(p)r(ositiv)n(e)f(frequency)g(part)g(\(1)p Fk(=)p Fx(2)p Fk(i)p Fx(\))14 b(exp)n(\(2)p Fk(i\031)s(\022)r Fx(\).)22 b(The)f(pro)r(cedure)g(to)g(get)523 3606 y(analytic)28 b(in)n(v)-5 b(arian)n(t)28 b(curv)n(es)g(pro)r(ceeds)h(in)g(a)g (completely)g(similar)f(w)n(a)n(y)g(as)g(Equations)523 3706 y(\(6\))33 b(to)f(\(9\),)h(and)g(it)g(w)n(as)e(pro)n(v)n(en)g (that)i(in)g(this)g(case)f([15,16],)f(the)i(critical)f(constan)n(t)523 3805 y Fk(K)594 3817 y Fw(ssm)705 3805 y Fx(\()p Fk(\032)p Fx(\))c(de\014ned)g(in)g(a)f(same)g(w)n(a)n(y)f(as)h(ab)r(o)n(v)n(e,)g (ful\014ls)h(as)f(in)h(\(10\))1306 3898 y Fn(\014)1306 3948 y(\014)1334 3969 y Fx(ln)1417 3902 y Fn(\000)1455 3969 y Fx(\()p Fk(K)1558 3981 y Fw(ssm)1669 3969 y Fx(\()p Fk(\032)p Fx(\)\))1808 3935 y Fj(\000)p Fw(1)1898 3902 y Fn(\001)1955 3969 y Fl(\000)18 b Fx(2)p Fk(B)t Fx(\()p Fk(\032)p Fx(\))2254 3898 y Fn(\014)2254 3948 y(\014)2304 3969 y Fk(<)23 b(C)29 b(:)636 b Fx(\(11\))523 4132 y(The)25 b(n)n(umerical)e(results)h(\(esp)r(ecially)h(the)g(\014gure)e(16\))h (in)h(ref.)g([16])f(pro)n(vide)f(more.)h(Not)523 4232 y(only)30 b(the)i(ratio)d Fk(K)1127 4244 y Fw(ssm)1238 4232 y Fx(\()p Fk(\032)p Fx(\)\))p Fk(=e)1458 4202 y Fj(\000)p Fw(2)p Fi(B)s Fw(\()p Fi(\032)p Fw(\))1717 4232 y Fx(is)i(b)r(ounded)g(on)g(irrationals,)e(but)i(it)g(is)g (extend-)523 4332 y(able)26 b(to)f(a)h(con)n(tin)n(uous)f(function)h (on)g([0)p Fk(;)14 b Fx(1],)25 b(b)r(ounded)h(b)r(elo)n(w)g(and)f(ab)r (o)n(v)n(e)g(b)n(y)g(p)r(ositiv)n(e)523 4431 y(constan)n(ts.)33 b(This)g(result)h(is)f(amazing)g(if)h(one)f(remem)n(b)r(er)g(that)h(b)r (oth)g Fk(K)2869 4443 y Fw(ssm)2980 4431 y Fx(\()p Fk(\032)p Fx(\)\))g(and)523 4531 y Fk(e)562 4501 y Fj(\000)p Fw(2)p Fi(B)s Fw(\()p Fi(\032)p Fw(\))816 4531 y Fx(v)-5 b(anish)26 b(at)f(all)h(rationals.)e(Therefore)g(the)j(Brjuno)e(function)h Fk(B)t Fx(\()p Fk(\032)p Fx(\))h(is)e(a)h(go)r(o)r(d)523 4631 y(mo)r(del)i(to)f(represen)n(t)g(the)h(singular)e(b)r(eha)n(vior)g (of)i(ln)2204 4563 y Fn(\000)2242 4631 y Fx(\()p Fk(K)2345 4643 y Fw(ssm)2456 4631 y Fx(\()p Fk(\032)p Fx(\)\))2595 4600 y Fj(\000)p Fw(1)2685 4563 y Fn(\001)2723 4631 y Fx(.)648 4730 y(The)d(Brjuno)f(function)i(w)n(as)e(in)n(tro)r(duced)h (b)n(y)g(Y)-7 b(o)r(ccoz)25 b([9])f(in)i(the)f(apparen)n(tly)f(sim-)523 4830 y(pler)j(problem)h(of)f(the)h(linearisation)f(of)g(complex)h (holomorphic)e(maps)h(around)g(their)p eop %%Page: 5 5 5 4 bop 1836 232 a Fu(Real)26 b(or)g(Complex)f(Brjuno)i(F)-6 b(unctions)236 b(5)523 448 y Fx(\014xed)26 b(p)r(oin)n(ts.)f(This)h(is) f(a)h(more)f(than)g(one)h(cen)n(tury)f(old)g(problem)g(\(see)h([17])f (for)g(a)g(nice)523 548 y(review\),)i(whic)n(h)h(w)n(e)f(can)g(state)h (as)f(follo)n(ws.)f(Let)i Fk(f)9 b Fx(\()p Fk(z)t Fx(\))27 b(b)r(e)h(a)f(holomorphic)g(map)g(suc)n(h)523 648 y(that)36 b Fk(f)9 b Fx(\(0\))36 b(=)h(0,)e Fk(f)1155 617 y Fj(0)1178 648 y Fx(\(0\))i(=)f Fk(e)1461 617 y Fw(2)p Fi(i\031)r(\032)1597 648 y Fx(.)g(Is)f(it)i(p)r(ossible)e(to)h(conjugate)f(the)h(map)g Fk(f)44 b Fx(to)36 b(its)523 747 y(linear)26 b(part?)f(This)i(means)e (that)i(w)n(e)f(lo)r(ok)f(for)h(a)g(function)h Fk(h)p Fx(,)f(holomorphic)f(in)h(a)g(disk)523 847 y(of)31 b(radius)f Fk(R)937 859 y Fi(f)980 847 y Fx(,)h(suc)n(h)g(that)g Fk(h)p Fx(\(0\))e(=)f(0,)j Fk(h)1828 817 y Fj(0)1851 847 y Fx(\(0\))e(=)f(1,)j(and)f Fk(f)9 b Fx(\()p Fk(h)p Fx(\()p Fk(z)t Fx(\)\))29 b(=)f Fk(h)p Fx(\()p Fk(z)t(e)2892 817 y Fw(2)p Fi(i\031)r(\032)3027 847 y Fx(\).)k(Note)523 946 y(that)37 b(suc)n(h)f(a)g(function)h Fk(h)p Fx(,)g(if)g(it)g (exists,)f(is)g(unique.)h(In)g(this)g(case,)e(the)i(function)g Fk(f)523 1046 y Fx(is)32 b(said)f(to)h(admit)g(a)f(Siegel)g(disk)h(of)g (radius)f Fk(R)2042 1058 y Fi(f)2085 1046 y Fx(.)h(The)f(Siegel)h(disk) f(is)h(a)f(top)r(ological)523 1146 y(disk)d(with)h(conformal)e(radius)g Fk(R)1586 1158 y Fi(f)1629 1146 y Fx(,)h(since)g(it)h(is)f(the)h(image) e(through)h(the)g(normalised)523 1245 y(conformal)e(map)i Fk(h)f Fx(of)h(the)g(disk)f Fl(j)p Fk(z)t Fl(j)c Fk(<)f(R)1841 1257 y Fi(f)1885 1245 y Fx(.)648 1349 y(W)-7 b(e)24 b(quote)g(no)n(w)g (the)g(classical)f(results)h(on)g(this)g(question)g([17].)g Fh(i\))g Fx(If)h Fk(\032)f Fx(is)g(rational,)523 1448 y(there)30 b(is)g(no)g(disk,)g(that)h(is)f Fk(R)1473 1460 y Fi(f)1543 1448 y Fx(=)e(0.)h Fh(ii\))j Fx(if)e Fk(\032)h Fx(is)f(irrational)e(and)i(satis\014es)g(a)g(\(strong\))523 1548 y(Liouville)f(condition,)g(w)n(e)f(still)i(ha)n(v)n(e)e Fk(R)1797 1560 y Fi(f)1865 1548 y Fx(=)d(0.)k Fh(iii\))i Fx(if)e Fk(\032)g Fx(is)g(a)g(diophan)n(tine)g(irrational)523 1648 y(\(see)40 b(Section)g(5)f(b)r(elo)n(w\),)h(then)g(there)g(exists) f(a)h(Siegel)f(disk)h(and)g Fk(R)2808 1660 y Fi(f)2894 1648 y Fk(>)j Fx(0,)d(more)523 1747 y(precisely)23 b(this)h(happ)r(ens) g(when)f Fk(B)t Fx(\()p Fk(\032)p Fx(\))i(is)e(\014nite.)i Fh(iv\))f Fx(If)g Fk(B)t Fx(\()p Fk(\032)p Fx(\))g(=)f(+)p Fl(1)p Fx(,)g(then)h(that)g(there)523 1847 y(exist)31 b(functions)h Fk(f)40 b Fx(suc)n(h)32 b(that)f Fk(R)1604 1859 y Fi(f)1677 1847 y Fx(=)e(0.)i(Indeed)h(Y)-7 b(o)r(ccoz)31 b([9])g(pro)n(v)n(ed)f(the)i(follo)n(wing)523 1947 y(:)f(de\014ne)f Fk(R)q Fx(\()p Fk(\032)p Fx(\))h(as)f(the)h(smallest)f(radius)f(of)i (the)g(Siegel)f(disks)g Fk(R)2600 1959 y Fi(f)2673 1947 y Fx(obtained)h(when)f Fk(f)523 2046 y Fx(v)-5 b(aries)25 b(in)g(the)h(compact)f(family)h(of)f(all)h(univ)-5 b(alen)n(t)25 b(maps)h(on)f(the)h(unit)g(disk)f(suc)n(h)h(that)523 2146 y Fk(f)9 b Fx(\(0\))23 b(=)f(0,)28 b(and)f Fk(f)1093 2116 y Fj(0)1116 2146 y Fx(\(0\))c(=)g Fk(e)1372 2116 y Fw(2)p Fi(i\031)r(\032)1507 2146 y Fx(.)28 b(Then)g(w)n(e)f(ha)n(v)n (e)1386 2266 y Fn(\014)1386 2316 y(\014)1414 2336 y Fx(ln)1497 2269 y Fn(\000)1535 2336 y Fx(\()p Fk(R)q Fx(\()p Fk(\032)p Fx(\)\))1770 2302 y Fj(\000)p Fw(1)1860 2269 y Fn(\001)1916 2336 y Fl(\000)18 b Fk(B)t Fx(\()p Fk(\032)p Fx(\))2173 2266 y Fn(\014)2173 2316 y(\014)2225 2336 y Fk(<)k(C)30 b(:)715 b Fx(\(12\))523 2527 y(No)n(w)27 b(consider)g(the)i(family)f (of)g(quadratic)e(p)r(olynomial)i Fk(P)2383 2539 y Fw(2)2420 2527 y Fx(\()p Fk(z)t Fx(\))c(=)f Fk(e)2678 2497 y Fw(2)p Fi(i\031)r(\032)2814 2527 y Fx(\()p Fk(z)f Fl(\000)c Fk(z)3033 2497 y Fw(2)3070 2527 y Fx(\),)28 b(and)523 2626 y(call)33 b Fk(R)744 2638 y Fw(2)781 2626 y Fx(\()p Fk(\032)p Fx(\))g(the)h(radius)e(of)g(the)i(Siegel)e(disk)h(asso)r (ciated)f(to)g(it.)i(Observ)n(e)d(\014rst)i(that,)523 2726 y(through)19 b(the)i(rescaling)d Fk(z)27 b Fl(!)c Fk(e)1510 2696 y Fj(\000)p Fw(2)p Fi(i\031)r(\032)1697 2726 y Fk(R)t Fl(\002)s Fk(z)t Fx(,)d(then)g Fk(P)2152 2738 y Fw(2)2210 2726 y Fx(is)g(transformed)f(in)h Fk(e)2871 2696 y Fw(2)p Fi(i\031)r(\032)3007 2726 y Fk(z)7 b Fl(\000)s Fk(R)q(z)3228 2696 y Fw(2)3264 2726 y Fx(.)523 2826 y(In)23 b(the)g(rescaled)e(v)-5 b(ariable,)22 b(w)n(e)g(see)g(that)h Fk(R)1883 2838 y Fw(2)1943 2826 y Fx(is)g(the)g(maxim)n(um)f(v)-5 b(alue)23 b(of)g(the)g(constan)n(t)523 2925 y Fk(R)j Fx(for)f(whic)n(h)g(a)f(circle)h(with)h(conformal)d(radius)i(one)g(is)g (p)r(ersistan)n(t.)f(Therefore)g Fk(R)3142 2937 y Fw(2)3179 2925 y Fx(\()p Fk(\032)p Fx(\))523 3025 y(is)37 b(the)h(critical)f (constan)n(t)f(adapted)h(to)h(the)f(presen)n(t)g(case,)g(and)g(this)g (leads)g(to)h(the)523 3125 y(analogies)26 b(b)r(et)n(w)n(een)h(\(10\),) h(or)e(its)i(equiv)-5 b(alen)n(t)27 b(in)h(the)g(real)f(case,)g(and)g (\(12\).)648 3228 y(Here)39 b(again,)f(the)i(n)n(umerical)e(results)h (\(no)n(w)g(the)h(\014gure)f(6\))g(in)h(ref.)f([16])f(bring)523 3328 y(some)28 b(con)n(tin)n(uit)n(y)g(prop)r(erties.)g(Not)g(only)g (the)h(ratio)f Fk(R)2294 3340 y Fw(2)2331 3328 y Fx(\()p Fk(\032)p Fx(\)\))p Fk(=e)2551 3298 y Fj(\000)p Fi(B)s Fw(\()p Fi(\032)p Fw(\))2775 3328 y Fx(is)g(b)r(ounded)h(on)523 3427 y(irrationals,)h(but)j(it)g(is)f(extendable)h(to)f(a)g(con)n(tin)n (uous)f(function)i(on)f([0)p Fk(;)14 b Fx(1],)31 b(b)r(ounded)523 3527 y(b)r(elo)n(w)e(and)h(ab)r(o)n(v)n(e)e(b)n(y)i(p)r(ositiv)n(e)f (constan)n(ts.)g(Therefore)f(the)j(Brjuno)e(function)h Fk(B)t Fx(\()p Fk(\032)p Fx(\))523 3627 y(is)i(again)e(a)h(go)r(o)r(d)h (mo)r(del)f(to)h(represen)n(t)f(the)h(singular)e(b)r(eha)n(vior)g(of)i (ln)2826 3559 y Fn(\000)2864 3627 y Fx(\()p Fk(R)2959 3639 y Fw(2)2997 3627 y Fx(\()p Fk(\032)p Fx(\)\))3136 3596 y Fj(\000)p Fw(1)3226 3559 y Fn(\001)3264 3627 y Fx(.)523 3726 y(In)h(our)f(w)n(ork)f([18])h(whic)n(h)h(started)f(from)h (these)g(observ)-5 b(ations,)31 b(w)n(e)h(giv)n(e)g(argumen)n(ts)523 3826 y(whic)n(h)k(strongly)f(supp)r(ort)h(the)h(conjecture)f(that)h (the)f(ratio)g Fk(R)2586 3838 y Fw(2)2623 3826 y Fx(\()p Fk(\032)p Fx(\)\))p Fk(=e)2843 3796 y Fj(\000)p Fi(B)s Fw(\()p Fi(\032)p Fw(\))3075 3826 y Fx(is)g(not)523 3925 y(only)22 b(con)n(tin)n(uous)g(but)h(sati\014es)f(a)h(H\177)-42 b(older)22 b(con)n(tin)n(uit)n(y)g(condition)g(with)i(exp)r(onen)n(t)e (1)p Fk(=)p Fx(2.)523 4025 y(More)h(precisely)-7 b(,)23 b(the)h(Brjuno)g(function)g(displa)n(ys)f(the)h(univ)n(ersal)f (singular)f(b)r(eha)n(viour)523 4125 y(\(up)28 b(to)g(some)e(H\177)-42 b(older-)1265 4092 y Fw(1)p 1264 4106 34 4 v 1264 4153 a(2)1335 4125 y Fx(con)n(tin)n(uous)27 b(function\))h(of)f(the)h (critical)f(functions)h(o)r(ccuring)523 4224 y(in)g(small)f(divisors)f (holomorphic)h(problems)g(in)g(dimension)h(one.)648 4328 y(The)37 b(use)h(of)g(the)g(Brjuno)f(function)i(w)n(as)e(somewhat)g (implicit)h(in)g(the)h(w)n(ork)d(of)523 4427 y(Buric)26 b(et)h(al.)f([19],)g(where)g(they)h(attempted)h(to)e(\014nd)h(represen) n(tations)e(of)h(the)i(critical)523 4527 y(constan)n(ts)f(b)n(y)g(what) g(they)h(called)f(mo)r(dular)g(smo)r(othing.)g(Singular)f(functions)i (of)g(the)523 4627 y(same)f(t)n(yp)r(e)h(o)r(ccured)f(in)h(MacKa)n(y)d ([20])i(in)h(relation)f(to)g(the)h(Brjuno)f(condition.)648 4730 y(It)42 b(is)h(nev)n(ertheless)e(useful)h(to)h(recall)e(here)h (brie\015y)g(one)g(of)g(the)h(steps,)f(called)523 4830 y(renormalisation,)34 b(whic)n(h)h(pla)n(ys)g(a)g(sp)r(ecial)h(role)e (in)i(Y)-7 b(o)r(ccoz's)35 b(argumen)n(t,)g(and)h(will)p eop %%Page: 6 6 6 5 bop 523 232 a Fu(6)237 b(Pierre)27 b(Moussa)g(and)e(Stefano)i (Marmi)523 448 y Fx(app)r(ear)f(to)i(b)r(e)g(crucial)e(in)i (understanding)f(the)g(\014ne)h(regularit)n(y)d(prop)r(erties)i(of)g (ratios)523 548 y(of)j(the)g(t)n(yp)r(e)g Fk(R)1017 560 y Fw(2)1054 548 y Fx(\()p Fk(\032)p Fx(\)\))p Fk(=e)1274 518 y Fj(\000)p Fi(B)s Fw(\()p Fi(\032)p Fw(\))1469 548 y Fx(.)g(F)-7 b(or)29 b(this)h(purp)r(ose,)f(w)n(e)h(follo)n(w)f([17],) f(and)i(w)n(e)f(w)n(e)h(con-)523 648 y(sider)39 b(\014rst)g(a)g (rotation)f(of)h(angle)g(2)p Fk(\031)s(\032)p Fx(,)g(with)h(0)i Fl(\024)g Fk(\032)h(<)f Fx(1,)d(that)h(is)f Fk(z)46 b Fl(!)d Fk(e)3086 617 y Fw(2)p Fi(i\031)r(\032)3221 648 y Fk(z)t Fx(,)523 747 y(acting)h(in)g(an)g(op)r(en)g(disk)f(of)h (radius)g Fk(R)1881 759 y Fi(\032)1963 747 y Fx(cen)n(tered)g(at)g(the) g(origin)f(in)h(the)h(com-)523 847 y(plex)f(plane.)f(W)-7 b(e)44 b(need)f(an)h(arbitrary)d(p)r(oin)n(t)j Fk(a)p Fx(,)f(suc)n(h)g(that)h Fl(j)p Fk(a)p Fl(j)50 b Fx(=)f Fk(R)2905 859 y Fi(\032)2943 847 y Fx(,)44 b(and)f(for)523 946 y(simplicit)n(y)h(w)n(e)g(tak)n(e)f(the)h(real)f(p)r(oin)n(t)i Fk(a)50 b Fx(=)g Fk(R)2096 958 y Fi(\032)2134 946 y Fx(.)44 b(Let)h Fk(a)2411 916 y Fj(0)2484 946 y Fx(=)50 b Fk(e)2638 916 y Fw(2)p Fi(i\031)r(\032)2774 946 y Fk(R)2837 958 y Fi(\032)2919 946 y Fx(its)45 b(image.)523 1046 y(No)n(w,)32 b(consider)g(the)i(angular)d(sector)h(b)r(ounded)h(b)n(y)f(the)i(lines) f(0)p Fk(a)f Fx(and)h(0)p Fk(a)2948 1016 y Fj(0)2970 1046 y Fx(,)g(namely)523 1146 y Fk(\001)592 1158 y Fi(\032)672 1146 y Fx(=)42 b Fl(f)f Fk(z)j Fl(j)e Fx(0)f Fl(\024)g Fx(arg)h Fk(z)i(<)e Fx(2)p Fk(\031)s(\032)f(;)55 b Fl(j)p Fk(z)t Fl(j)41 b Fk(<)h(R)2102 1158 y Fi(\032)2140 1146 y Fl(g)p Fx(,)d(the)g(line)g(0)p Fk(a)2652 1116 y Fj(0)2713 1146 y Fx(b)r(eing)g(excluded.)523 1245 y(Consider)33 b(the)i(orbit)e(made)h(of)g(the)g(successiv)n(e)f(iterated)h(p)r(oin)n (ts)g Fk(z)2715 1257 y Fi(n)2759 1245 y Fk(;)48 b(n)34 b Fl(\025)f Fx(1)h(start-)523 1345 y(ing)f(from)g Fk(z)908 1357 y Fw(0)977 1345 y Fl(2)f Fk(\001)1133 1357 y Fi(\032)1172 1345 y Fx(,)h(and)g(let)g Fk(z)1559 1357 y Fi(q)1629 1345 y Fx(the)g(\014rst)g(of)g(these)g(p)r(oin)n(ts)g(whic)n(h)g(also)f (b)r(elong)h(to)523 1445 y Fk(\001)592 1457 y Fi(\032)631 1445 y Fx(.)h(The)g(map)g Fk(z)1095 1457 y Fw(0)1166 1445 y Fl(!)g Fk(z)1322 1457 y Fi(q)1392 1445 y Fx(is)g(th)n(us)h(the)f (\014rst)g(return)g(map)g(in)g(the)g(sector.)g(W)-7 b(e)34 b(ha)n(v)n(e)523 1544 y(\(2)p Fk(\031)s Fx(\))679 1514 y Fj(\000)p Fw(1)769 1544 y Fx(\(arg)23 b Fk(z)980 1556 y Fi(q)1016 1544 y Fx(\))g(=)g(\(2)p Fk(\031)s Fx(\))1315 1514 y Fj(\000)p Fw(1)1404 1544 y Fx(\(arg)g Fk(z)1615 1556 y Fw(0)1652 1544 y Fx(\))11 b(+)g Fk(q)s(\032)g Fl(\000)g Fx(1.)23 b(W)-7 b(e)25 b(no)n(w)e(tak)n(e)g(in)h(the)h (sector)e(the)h(v)-5 b(ari-)523 1662 y(able)22 b Fk(u)f Fx(suc)n(h)h(that)g(its)g(complex)g(conjugate)p 1918 1617 48 4 v 21 w Fk(u)h Fx(=)f Fk(z)2130 1603 y Fe(1)p 2129 1612 31 4 v 2129 1645 a Fd(\032)2173 1662 y Fx(,)g(where)f(no)n(w) h Fk(u)f Fx(b)r(elongs)h(to)g(a)f(disk)523 1793 y(of)27 b(radius)f Fk(R)941 1714 y Fe(1)p 940 1723 V 940 1756 a Fd(\032)929 1802 y Fi(\032)984 1793 y Fx(.)h(F)-7 b(or)27 b(the)g(v)-5 b(alues)26 b Fk(u)1619 1805 y Fw(0)1683 1793 y Fx(and)h Fk(u)1892 1805 y Fi(q)1928 1793 y Fx(,)g(corresp)r (onding)e(to)i Fk(z)2651 1805 y Fw(0)2715 1793 y Fx(and)g Fk(z)2915 1805 y Fi(q)2951 1793 y Fx(,)g(w)n(e)g(ha)n(v)n(e)523 1892 y(\(2)p Fk(\031)s Fx(\))679 1862 y Fj(\000)p Fw(1)769 1892 y Fx(\(arg)c Fk(u)989 1904 y Fi(q)1025 1892 y Fx(\))g(=)g(\(2)p Fk(\031)s Fx(\))1324 1862 y Fj(\000)p Fw(1)1413 1892 y Fx(\(arg)g Fk(u)1633 1904 y Fw(0)1670 1892 y Fx(\))12 b Fl(\000)g Fk(q)i Fx(+)1929 1859 y Fw(1)p 1929 1873 35 4 v 1929 1921 a Fi(\032)1996 1892 y Fx(\(mo)r(d)25 b(1\))46 b(=)22 b(\(2)p Fk(\031)s Fx(\))2575 1862 y Fj(\000)p Fw(1)2665 1892 y Fx(\(arg)h Fk(u)2885 1904 y Fw(0)2922 1892 y Fx(\))12 b(+)3053 1859 y Fw(1)p 3052 1873 V 3052 1921 a Fi(\032)3096 1892 y Fx(.)25 b(The)523 1992 y(original)18 b(map)h(whic)n(h)g(acted)g(in)g(a)g(disk)g(of)g(radius)f Fk(R)2137 2004 y Fi(\032)2176 1992 y Fx(,)h(leads)f(in)i(the)f (new\\renormalised)523 2091 y(v)-5 b(ariable")20 b Fk(u)p Fx(,)g(to)h(a)g(rotation)f(with)h(rotation)f(n)n(um)n(b)r(er)2232 2059 y Fw(1)p 2231 2073 V 2231 2120 a Fi(\032)2276 2091 y Fx(,)h(acting)f(in)i(a)e(disk)h(with)h(radius)523 2191 y Fk(R)586 2206 y Fw(\(1)p Fi(=\032)p Fw(\))744 2191 y Fx(,)27 b(suc)n(h)h(that)g(ln)14 b Fk(R)1308 2203 y Fi(\032)1369 2191 y Fx(=)23 b Fk(\032)14 b Fx(ln)g Fk(R)1660 2206 y Fw(\(1)p Fi(=\032)p Fw(\))1817 2191 y Fx(.)648 2638 y(This)29 b(construction)h(extends)f(to)h(the)h(non)e(linear)h(p)r (erturb)r(ed)g(case,)f(for)g(example)523 2738 y Fk(P)576 2750 y Fw(2)614 2738 y Fx(\()p Fk(\032;)14 b(z)t Fx(\))22 b(=)h Fk(e)950 2708 y Fw(2)p Fi(i\031)r(\032)1086 2738 y Fx(\()p Fk(z)e Fl(\000)d Fk(z)1304 2708 y Fw(2)1341 2738 y Fx(\),)27 b(with)h(a)f(lot)h(of)f(complications.)g(Supp)r(ose)g (that)h(there)f(is)g(a)523 2837 y(Siegel)22 b(disk)g(for)g Fk(P)1096 2849 y Fw(2)1133 2837 y Fx(\()p Fk(\032;)14 b(z)t Fx(\).)22 b(In)h(this)f(disk,)g(there)g(are)f(conformal)g(co)r (ordinates)g(on)h(whic)n(h)523 2937 y(the)f(maps)e(is)i(exactly)e(a)h (rotation)f(of)h(angle)f(2)p Fk(\031)s(\032)p Fx(,)h(and)g(on)g(these)h (co)r(ordinates)d(w)n(e)i(apply)523 3037 y(the)30 b(linear)g (renormalisation.)d(The)j(problem)g(is)g(then)g(to)g(giv)n(e)f(an)h(in) n(terpretation)f(of)523 3136 y(the)24 b(renormalised)d(co)r(ordinates)g Fk(u)i Fx(whic)n(h)g(w)n(e)g(obtain.)g(It)g(app)r(ears)f(that)h(there)g (exist)g(a)523 3236 y(holomorphic)k(map)g(in)h(the)g(v)-5 b(ariable)27 b Fk(u)h Fx(with)g(rotation)f(n)n(um)n(b)r(er)g Fk(\032)2664 3206 y Fj(\000)p Fw(1)2753 3236 y Fx(,)h(whic)n(h)g (admits)523 3335 y(a)f(Siegel)h(disk,)f(with)h(conformal)f(radius)1864 3314 y Fn(e)1849 3335 y Fk(R)1912 3350 y Fw(\(1)p Fi(=\032)p Fw(\))2097 3335 y Fx(suc)n(h)h(that)g(ln)14 b Fk(R)2611 3347 y Fi(\032)2668 3335 y Fl(\000)k Fk(\032)c Fx(ln)2906 3314 y Fn(e)2891 3335 y Fk(R)2954 3350 y Fw(\(1)p Fi(=\032)p Fw(\))3134 3335 y Fx(=)23 b(0.)523 3435 y(Ho)n(w)n(ev)n(er,)35 b(this)h(map)h(is)f(not)g(a)g(p)r(olynomial)g(with)h(degree)e(2.)h (This)h(led)f(Y)-7 b(o)r(ccoz)36 b(to)523 3535 y(extend)41 b(the)f(problem)g(to)g(the)h(compact)f(family)g(of)h(univ)-5 b(alen)n(t)40 b(map)g(on)g(the)h(unit)523 3634 y(disk)34 b(with)h(rotation)e(n)n(um)n(b)r(er)h Fk(\032)p Fx(,)g(and)g(he)h(has)f (considered)f(the)h(minim)n(um)h Fk(R)q Fx(\()p Fk(\032)p Fx(\))g(of)523 3734 y(the)25 b(radius)f(of)h(the)g(Siegel)f(disk)g(tak) n(en)g(o)n(v)n(er)f(this)i(family)g(of)g(maps.)f(The)h(result)f(is)h(t) n(w)n(o)523 3834 y(mo)r(di\014cations)19 b(to)g(the)h(relation)f(ln)14 b Fk(R)1696 3846 y Fi(\032)1737 3834 y Fl(\000)r Fk(\032)g Fx(ln)g Fk(R)2007 3849 y Fw(\(1)p Fi(=\032)p Fw(\))2187 3834 y Fx(=)23 b(0)c(obtained)g(in)h(the)g(linear)f(case.)523 3933 y(First)27 b(due)f(to)h(the)g(minimisation)g(pro)r(cedure,)e(the)i (b)r(est)g(one)g(could)f(get)h(is)f(a)g(p)r(ositiv)n(e)523 4033 y(uniform)32 b(upp)r(er)f(b)r(ound)h(for)f(this)h(expression)e (instead)i(of)f(zero.)g(Second,)g(there)h(is)f(a)523 4132 y(sp)r(ecial)d(di\016cult)n(y)g(when)h Fk(\032)f Fx(go)r(es)f(to)h(zero.)f(In)h(this)h(case)e(the)i(Siegel)e(disk)h(is)g (strongly)523 4232 y(distorted,)d(since)f(there)h(is)g(an)f(other)h (\014xed)f(p)r(oin)n(t)h(whic)n(h)g(tends)g(to)g(zero)f(when)h Fk(\032)g Fx(go)r(es)523 4332 y(to)h(zero.)g(The)g(comparison)f(b)r(et) n(w)n(een)h(the)h(linear)e(and)h(the)h(non)f(linear)g(case)f(b)r (ecomes)523 4431 y(unjusti\014ed)k(in)f(this)h(limit.)g(Y)-7 b(o)r(ccoz)27 b(pro)n(v)n(ed)g(that)h(the)h(result)e(is)h(an)g (additional)g(loga-)523 4531 y(rithmic)23 b(term)f(in)h(the)f (estimate,)h(so)e(that)i(w)n(e)f(only)g(get)g(that)h(ln)14 b Fk(R)2601 4543 y Fi(\032)2647 4531 y Fl(\000)8 b Fk(\032)14 b Fx(ln)g Fk(R)2923 4546 y Fw(\(1)p Fi(=\032)p Fw(\))3088 4531 y Fl(\000)8 b Fx(ln)14 b Fk(\032)523 4631 y Fx(is)25 b(b)r(ounded.)g(It)h(is)e(therefore)g(natural)h(to)f(compare)g(the)h (function)h Fl(\000)14 b Fx(ln)f Fk(R)q Fx(\()p Fk(\032)p Fx(\))26 b(\(as)e(w)n(ell)523 4730 y(as)30 b Fl(\000)14 b Fx(ln)g Fk(R)853 4742 y Fw(2)890 4730 y Fx(\()p Fk(\032)p Fx(\)\))31 b(to)g(the)g(solution)g(of)f(the)h(equation)g Fk(B)t Fx(\()p Fk(\032)p Fx(\))21 b Fl(\000)f Fk(\032)14 b Fx(ln)g Fk(B)t Fx(\(1)p Fk(=\032)p Fx(\))20 b(+)g(ln)14 b Fk(\032)28 b Fx(=)g(0)523 4830 y(whic)n(h)g(w)n(e)f(will)h(see,)f(is) g(nothing)h(else)f(than)h(the)g(Brjuno)f(function.)p eop %%Page: 7 7 7 6 bop 1836 232 a Fu(Real)26 b(or)g(Complex)f(Brjuno)i(F)-6 b(unctions)236 b(7)523 448 y Fm(4)112 b(Con)m(tin)m(ued)37 b(F)-9 b(ractions)37 b(and)h(the)g(Brjuno)f(F)-9 b(unction)523 643 y Fx(W)i(e)26 b(\014rst)g(giv)n(e)f(a)g(somewhat)g(un)n(usual)h (de\014nition)g(of)g(the)g(con)n(tin)n(ued)f(fraction)h(expan-)523 742 y(sion)32 b(sometimes)g(called)g(\\japanese)g(con)n(tin)n(ued)g (fractions")f([21].)h(Let)h Fk(\013)g Fx(b)r(e)g(a)f(\014xed)523 842 y(real)g(n)n(um)n(b)r(er)g(suc)n(h)g(that)1384 809 y Fw(1)p 1384 823 34 4 v 1384 871 a(2)1458 842 y Fl(\024)f Fk(\013)h Fl(\024)f Fx(1.)h(Then,)h(giv)n(en)e(the)i(starting)f(n)n(um) n(b)r(er)g Fk(x)p Fx(,)i(the)523 942 y(co)r(e\016cien)n(ts)27 b Fk(a)987 954 y Fi(n)1060 942 y Fx(and)g Fk(")1260 954 y Fi(n)1333 942 y Fx(are)g(recursiv)n(ely)e(uniquely)j(de\014ned)g(b)n (y)f(the)h(conditions)929 1118 y Fk(x)23 b Fx(=)g Fk(a)1131 1130 y Fw(0)1186 1118 y Fx(+)18 b Fk(")1308 1130 y Fw(0)1345 1118 y Fk(x)1392 1130 y Fw(0)1430 1118 y Fk(;)41 b Fx(and)28 b Fl(8)p Fk(n)22 b Fl(\025)h Fx(0)p Fk(;)41 b(x)2016 1084 y Fj(\000)p Fw(1)2016 1139 y Fi(n)2128 1118 y Fx(=)23 b Fk(a)2260 1130 y Fi(n)p Fw(+1)2408 1118 y Fx(+)18 b Fk(")2530 1130 y Fi(n)p Fw(+1)2659 1118 y Fk(x)2706 1130 y Fi(n)p Fw(+1)2858 1118 y Fk(;)258 b Fx(\(13\))523 1295 y(with)33 b Fl(8)p Fk(n)e Fl(\025)g Fx(0)p Fk(;)46 b(\013)22 b Fl(\000)g Fx(1)31 b Fl(\024)g Fk(")1422 1307 y Fi(n)1467 1295 y Fk(x)1514 1307 y Fi(n)1591 1295 y Fk(<)g(\013)p Fx(.)i(W)-7 b(e)33 b(de\014ne)g(the)g(mo)r(di\014ed)g(in)n(teger)f (part)g([)p Fk(x)p Fx(])3239 1307 y Fi(\013)523 1395 y Fx(and)27 b(the)h(mo)r(di\014ed)g(fractional)f(part)g Fl(f)p Fk(x)p Fl(g)1849 1407 y Fi(\013)1924 1395 y Fx(as)f(follo)n(ws,) 866 1571 y([)p Fk(x)p Fx(])959 1583 y Fi(\013)1030 1571 y Fx(=)c([)p Fk(x)d Fl(\000)f Fk(\013)h Fx(+)f(1])1509 1583 y Fw(1)1629 1571 y Fx(and)83 b Fl(f)p Fk(x)p Fl(g)1977 1583 y Fi(\013)2047 1571 y Fx(=)22 b Fl(f)p Fk(x)d Fl(\000)f Fk(\013)h Fx(+)f(1)p Fl(g)2564 1583 y Fw(1)2618 1571 y Fx(+)g Fk(\013)h Fl(\000)f Fx(1)23 b Fk(;)195 b Fx(\(14\))523 1748 y(where)29 b([)p Fk(x)p Fx(])858 1760 y Fw(1)926 1748 y Fx(and)h Fl(f)p Fk(x)p Fl(g)1221 1760 y Fw(1)1288 1748 y Fx(are)f(the)h(usual)g(in)n(teger)f(and)h(fractional)e(parts)i (of)g Fk(x)g Fx(\(so)g(that)523 1848 y(0)23 b Fl(\024)f(f)p Fk(x)p Fl(g)806 1860 y Fw(1)866 1848 y Fk(<)h Fx(1\).)k(With)i(these)e (notations,)g(w)n(e)g(can)h(rewrite)e(\(13\))i(as)548 2024 y Fk(a)592 2036 y Fw(0)652 2024 y Fx(=)23 b([)p Fk(x)p Fx(])833 2036 y Fi(\013)881 2024 y Fk(;)41 b(")984 2036 y Fw(0)1021 2024 y Fk(x)1068 2036 y Fw(0)1129 2024 y Fx(=)23 b Fl(f)p Fk(x)p Fl(g)1348 2036 y Fi(\013)1395 2024 y Fk(;)41 b Fx(and)p Fk(;)14 b(a)1674 2036 y Fi(n)p Fw(+1)1826 2024 y Fx(=)23 b([)p Fk(x)1984 1990 y Fj(\000)p Fw(1)1984 2045 y Fi(n)2073 2024 y Fx(])2096 2036 y Fi(\013)2144 2024 y Fk(;)41 b(")2247 2036 y Fi(n)p Fw(+1)2376 2024 y Fk(x)2423 2036 y Fi(n)p Fw(+1)2576 2024 y Fx(=)23 b Fl(f)p Fk(x)2753 1990 y Fj(\000)p Fw(1)2753 2045 y Fi(n)2842 2024 y Fl(g)2884 2036 y Fi(\013)2931 2024 y Fk(;)14 b Fl(8)p Fk(n)21 b(>)i Fx(0)g Fk(:)3139 2124 y Fx(\(15\))523 2224 y(Therefore)k(the)i Fk(x)1091 2236 y Fi(n)1165 2224 y Fx(are)e(generated)g(b)n(y)h(iterating)g(the)h(function)g Fk(A)2669 2236 y Fi(\013)2716 2224 y Fx(\()p Fk(x)p Fx(\))d(=)2941 2153 y Fn(\014)2941 2203 y(\014)2969 2224 y Fl(f)p Fk(x)3058 2193 y Fj(\000)p Fw(1)3147 2224 y Fl(g)3189 2236 y Fi(\013)3236 2153 y Fn(\014)3236 2203 y(\014)3264 2224 y Fx(,)523 2332 y(that)32 b(is)f Fl(8)p Fk(n)e Fl(\025)g Fx(0)p Fk(;)77 b(x)1203 2344 y Fi(n)p Fw(+1)1362 2332 y Fx(=)29 b Fk(A)1518 2344 y Fi(\013)1566 2332 y Fx(\()p Fk(x)1645 2344 y Fi(n)1690 2332 y Fx(\))h(=)1846 2261 y Fn(\014)1846 2311 y(\014)1874 2332 y Fl(f)p Fk(x)1963 2301 y Fj(\000)p Fw(1)1963 2352 y Fi(n)2052 2332 y Fl(g)2094 2344 y Fi(\013)2141 2261 y Fn(\014)2141 2311 y(\014)2198 2332 y Fx(=)2293 2261 y Fn(\014)2293 2311 y(\014)2320 2332 y Fk(x)2367 2301 y Fj(\000)p Fw(1)2367 2352 y Fi(n)2475 2332 y Fl(\000)18 b Fx([)p Fk(x)2628 2301 y Fj(\000)p Fw(1)2628 2352 y Fi(n)2718 2332 y Fx(])2741 2344 y Fi(\013)2788 2261 y Fn(\014)2788 2311 y(\014)2816 2332 y Fx(.)32 b(A)f(more)g(de-)523 2431 y(tailed)21 b(description)e(states)i(that)f(the)h(map)g Fk(A)1942 2443 y Fi(\013)2010 2431 y Fx(is)f(made)g(of)h(the)g(follo)n (wing)e(branc)n(hes)852 2642 y(branc)n(h)27 b Fk(k)1173 2608 y Fw(+)1251 2642 y Fx(:)106 b Fk(A)1442 2654 y Fi(\013)1489 2642 y Fx(\()p Fk(x)p Fx(\))26 b(=)1729 2586 y(1)p 1726 2623 48 4 v 1726 2699 a Fk(x)1802 2642 y Fl(\000)18 b Fk(k)86 b Fx(for)2286 2586 y(1)p 2206 2623 201 4 v 2206 2699 a Fk(k)21 b Fx(+)d Fk(\013)2440 2642 y(<)k(x)i Fl(\024)2698 2586 y Fx(1)p 2695 2623 46 4 v 2695 2699 a Fk(k)2793 2642 y(;)282 b Fx(\(16a\))851 2849 y(branc)n(h)27 b Fk(k)1172 2815 y Fj(\000)1251 2849 y Fx(:)106 b Fk(A)1442 2861 y Fi(\013)1489 2849 y Fx(\()p Fk(x)p Fx(\))26 b(=)f Fk(k)c Fl(\000)1876 2793 y Fx(1)p 1873 2830 48 4 v 1873 2906 a Fk(x)2014 2849 y Fx(for)2208 2793 y(1)p 2206 2830 46 4 v 2206 2906 a Fk(k)2285 2849 y(<)h(x)i Fl(\024)2692 2793 y Fx(1)p 2541 2830 344 4 v 2541 2906 a Fk(k)d Fx(+)d Fk(\013)h Fl(\000)f Fx(1)2936 2849 y Fk(:)134 b Fx(\(16b\))523 3068 y(When)776 3036 y Fw(1)p 776 3050 34 4 v 776 3097 a(2)843 3068 y Fk(<)23 b(\013)h Fl(\024)g Fx(1,)k(the)g(function)h Fk(A)1721 3080 y Fi(\013)1796 3068 y Fx(maps)f(the)h(in)n(terv)-5 b(al)27 b([0)p Fk(;)14 b(\013)p Fx(\))29 b(to)f(itself,)g(whereas)523 3168 y(when)22 b Fk(\013)i Fx(=)908 3135 y Fw(1)p 908 3149 V 908 3197 a(2)952 3168 y Fx(,)e(it)g(maps)g(the)h(in)n(terv)-5 b(al)21 b([0)p Fk(;)14 b(\013)p Fx(])23 b(to)f(itself.)g(In)h(b)r(oth)f (cases,)f(it)i(is)f(con)n(v)n(enien)n(t)523 3268 y(to)f(set)h Fk(A)804 3280 y Fi(\013)851 3268 y Fx(\(0\))h(=)g(0,)e(and)g(w)n(e)h (get)f(a)g(map)g(whic)n(h)g(is)h(in\014nitely)g(di\013eren)n(tiable)f (b)n(y)g(pieces,)523 3367 y(and)27 b(the)g(p)r(oin)n(ts)f(where)g(it)h (is)g(not)g(di\013eren)n(tiable)f(accum)n(ulate)g(to)g(0.)h(No)n(w)f Fk(x)h Fx(and)g(the)523 3467 y(reduced)g(fraction)g Fk(p)1181 3479 y Fi(n)1226 3467 y Fk(=q)1305 3479 y Fi(n)1378 3467 y Fx(admit)g(the)h(follo)n(wing)f(represen)n(tation)523 3665 y Fk(x)d Fx(=)e Fk(a)725 3677 y Fw(0)768 3665 y Fx(+)1296 3609 y Fk(")1335 3621 y Fw(0)p 849 3646 971 4 v 849 3748 a Fk(a)893 3760 y Fw(1)948 3748 y Fx(+)1387 3692 y Fk(")1426 3704 y Fw(1)p 1041 3729 768 4 v 1041 3864 a Fk(a)1085 3876 y Fw(2)1141 3864 y Fx(+)1229 3805 y(.)1261 3830 y(.)1293 3856 y(.)1339 3864 y(+)1531 3807 y Fk(")1570 3819 y Fi(n)p Fj(\000)p Fw(1)p 1432 3845 368 4 v 1432 3921 a Fk(a)1476 3933 y Fi(n)1540 3921 y Fx(+)c Fk(")1662 3933 y Fi(n)1707 3921 y Fk(x)1754 3933 y Fi(n)1912 3665 y Fk(;)2042 3609 y(p)2084 3621 y Fi(n)p 2042 3646 87 4 v 2044 3722 a Fk(q)2081 3734 y Fi(n)2162 3665 y Fx(=)23 b Fk(a)2294 3677 y Fw(0)2337 3665 y Fx(+)2765 3609 y Fk(")2804 3621 y Fw(0)p 2417 3646 773 4 v 2417 3748 a Fk(a)2461 3760 y Fw(1)2517 3748 y Fx(+)2857 3692 y Fk(")2896 3704 y Fw(1)p 2610 3729 570 4 v 2610 3864 a Fk(a)2654 3876 y Fw(2)2709 3864 y Fx(+)2797 3805 y(.)2829 3830 y(.)2862 3856 y(.)2908 3864 y(+)3001 3807 y Fk(")3040 3819 y Fi(n)p Fj(\000)p Fw(1)p 3001 3845 169 4 v 3041 3921 a Fk(a)3085 3933 y Fi(n)3242 3665 y Fk(:)3139 4004 y Fx(\(17\))523 4103 y(As)28 b(long)f(as)g(the)h Fk(x)1118 4115 y Fi(n)1163 4103 y Fx('s)g(do)f(not)h(v)-5 b(anish,)27 b(w)n(e)g(ha)n(v)n(e)958 4309 y Fk(x)d Fx(=)1126 4253 y Fk(p)1168 4265 y Fi(n)1232 4253 y Fx(+)18 b Fk(p)1357 4265 y Fi(n)p Fj(\000)p Fw(1)1487 4253 y Fk(")1526 4265 y Fi(n)1571 4253 y Fk(x)1618 4265 y Fi(n)p 1126 4290 537 4 v 1131 4366 a Fk(q)1168 4378 y Fi(n)1232 4366 y Fx(+)g Fk(q)1352 4378 y Fi(n)p Fj(\000)p Fw(1)1482 4366 y Fk(")1521 4378 y Fi(n)1566 4366 y Fk(x)1613 4378 y Fi(n)1756 4309 y Fk(;)97 b(x)1923 4321 y Fi(n)1992 4309 y Fx(=)23 b(\()p Fl(\000)p Fk(")2216 4321 y Fi(n)2260 4309 y Fx(\))2388 4253 y Fk(p)2430 4265 y Fi(n)2493 4253 y Fl(\000)18 b Fk(xq)2660 4265 y Fi(n)p 2303 4290 489 4 v 2303 4366 a Fk(p)2345 4378 y Fi(n)p Fj(\000)p Fw(1)2493 4366 y Fl(\000)g Fk(xq)2660 4378 y Fi(n)p Fj(\000)p Fw(1)2828 4309 y Fk(;)288 b Fx(\(18\))523 4529 y(and)27 b(the)h(recursion)f (relations)1047 4705 y Fk(p)1089 4717 y Fi(n)1158 4705 y Fx(=)e Fk(a)1292 4717 y Fi(n)1337 4705 y Fk(p)1379 4717 y Fi(n)p Fj(\000)p Fw(1)1527 4705 y Fx(+)18 b Fk(")1649 4717 y Fi(n)p Fj(\000)p Fw(1)1779 4705 y Fk(q)1816 4717 y Fi(n)p Fj(\000)p Fw(2)1974 4705 y Fk(;)42 b(p)2081 4717 y Fw(0)2141 4705 y Fx(=)22 b Fk(a)2272 4717 y Fw(0)2337 4705 y Fk(;)42 b(p)2444 4717 y Fj(\000)p Fw(1)2556 4705 y Fx(=)22 b(1)41 b Fk(;)349 b Fx(\(19a\))1051 4830 y Fk(q)1088 4842 y Fi(n)1158 4830 y Fx(=)25 b Fk(a)1292 4842 y Fi(n)1337 4830 y Fk(q)1374 4842 y Fi(n)p Fj(\000)p Fw(1)1523 4830 y Fx(+)18 b Fk(")1645 4842 y Fi(n)p Fj(\000)p Fw(1)1775 4830 y Fk(q)1812 4842 y Fi(n)p Fj(\000)p Fw(2)1969 4830 y Fk(;)42 b(q)2071 4842 y Fw(0)2131 4830 y Fx(=)23 b(1)k Fk(;)42 b(q)2390 4842 y Fj(\000)p Fw(1)2502 4830 y Fx(=)23 b(0)41 b Fk(;)397 b Fx(\(19b\))p eop %%Page: 8 8 8 7 bop 523 232 a Fu(8)237 b(Pierre)27 b(Moussa)g(and)e(Stefano)i (Marmi)523 448 y Fx(so)g(that)h(w)n(e)f(get)g(0)c Fk(<)g(q)1255 460 y Fw(0)1315 448 y Fl(\024)g Fk(q)1440 460 y Fw(1)1500 448 y Fk(<)g(q)1625 460 y Fw(2)1685 448 y Fk(<)g(:)14 b(:)g(:)23 b(<)f(q)2017 460 y Fi(n)2086 448 y Fk(<)g(q)2210 460 y Fi(n)p Fw(+1)2363 448 y Fk(<)g(:)14 b(:)g(:)p Fx(.)28 b(W)-7 b(e)28 b(also)e(de\014ne)1052 618 y Fk(\014)1099 630 y Fi(n)1168 618 y Fx(=)c Fk(x)1302 630 y Fw(0)1340 618 y Fk(x)1387 630 y Fw(1)1438 618 y Fl(\001)14 b(\001)g(\001)g Fk(x)1596 630 y Fi(n)1665 618 y Fx(=)22 b(\()p Fl(\000)p Fx(1\))1923 584 y Fi(n)1968 618 y Fk(")2007 630 y Fw(0)2044 618 y Fk(")2083 630 y Fw(1)2134 618 y Fl(\001)14 b(\001)g(\001)g Fk(")2284 630 y Fi(n)2329 618 y Fx(\()p Fk(q)2398 630 y Fi(n)2443 618 y Fk(x)19 b Fl(\000)f Fk(p)2634 630 y Fi(n)2679 618 y Fx(\))23 b Fk(;)382 b Fx(\(20\))523 788 y(and)27 b(w)n(e)h(ha)n(v)n(e)1170 920 y(1)p 1092 957 197 4 v 1092 1033 a(1)18 b(+)g Fk(\013)1322 976 y Fl(\024)k Fk(\014)1456 988 y Fi(n)1501 976 y Fk(q)1538 988 y Fi(n)p Fw(+1)1691 976 y Fx(=)2052 920 y Fk(q)2089 932 y Fi(n)p Fw(+1)p 1788 957 695 4 v 1788 1033 a Fk(q)1825 1045 y Fi(n)p Fw(+1)1973 1033 y Fx(+)c Fk(")2095 1045 y Fi(n)p Fw(+1)2224 1033 y Fk(q)2261 1045 y Fi(n)2306 1033 y Fk(x)2353 1045 y Fi(n)p Fw(+1)2516 976 y Fl(\024)2620 920 y Fx(1)p 2614 957 54 4 v 2614 1033 a Fk(\013)2705 976 y(:)411 b Fx(\(21\))523 1193 y(No)n(w)27 b(there)g(exist)g Fk(\025)p Fx(\()p Fk(\013)p Fx(\),)i(with)f(0)22 b Fk(<)h(\025)p Fx(\()p Fk(\013)p Fx(\))h Fk(<)f Fx(1,)k(and)g(p)r(ositiv)n(e)g (constan)n(ts)f Fk(C)2941 1205 y Fw(1)3006 1193 y Fx(and)h Fk(C)3226 1205 y Fw(2)3264 1193 y Fx(,)523 1292 y(suc)n(h)g(that[18]) 1246 1392 y Fk(\014)1293 1404 y Fi(n)1361 1392 y Fk(<)c(C)1508 1404 y Fw(1)1545 1392 y Fk(\025)p Fx(\()p Fk(\013)p Fx(\))1710 1358 y Fi(n)1840 1392 y Fk(;)97 b(q)1997 1404 y Fi(n)2065 1392 y Fk(>)23 b(C)2212 1404 y Fw(2)2250 1392 y Fk(\025)p Fx(\()p Fk(\013)p Fx(\))2415 1358 y Fj(\000)p Fi(n)2541 1392 y Fk(:)575 b Fx(\(22\))523 1534 y(Indeed)28 b(w)n(e)f(ha)n(v)n(e) 741 1744 y(for)1017 1619 y Fl(p)p 1086 1619 42 4 v 69 x Fx(5)18 b Fl(\000)g Fx(1)p 1017 1725 254 4 v 1123 1801 a(2)1303 1744 y Fk(<)23 b(\013)g Fl(\024)g Fx(1)k Fk(;)97 b(\025)p Fx(\()p Fk(\013)p Fx(\))26 b(=)f Fk(\025)p Fx(\(1\))e(=)2300 1619 y Fl(p)p 2369 1619 42 4 v 69 x Fx(5)18 b Fl(\000)g Fx(1)p 2300 1725 254 4 v 2406 1801 a(2)2587 1744 y(=)k(0)p Fk(:)p Fx(618)p Fk(:::)40 b(;)101 b Fx(\(23a\))753 1965 y(and)28 b(for)1052 1909 y(1)p 1052 1946 42 4 v 1052 2022 a(2)1126 1965 y Fl(\024)23 b Fk(\013)h Fl(\024)1388 1840 y(p)p 1457 1840 V 69 x Fx(5)18 b Fl(\000)g Fx(1)p 1388 1946 254 4 v 1494 2022 a(2)1680 1965 y Fk(;)41 b(\025)p Fx(\()p Fk(\013)p Fx(\))26 b(=)f Fk(\025)2087 1848 y Fn(\022)2158 1909 y Fx(1)p 2158 1946 42 4 v 2158 2022 a(2)2210 1848 y Fn(\023)2294 1965 y Fx(=)2382 1893 y Fl(p)p 2451 1893 V 72 x Fx(2)18 b Fl(\000)g Fx(1)k(=)h(0)p Fk(:)p Fx(414)p Fk(:::)40 b(:)24 b Fx(\(23b\))523 2185 y(When)34 b Fk(x)818 2197 y Fi(n)864 2185 y Fx(=0)e(for)h(some)f Fk(n)p Fx(,)i(and)f Fk(x)1670 2197 y Fi(m)1766 2185 y Fl(6)p Fx(=)f(0)h(for)f Fk(m)h(<)f(n)p Fx(,)h(then)h(w)n(e)f(ha)n(v)n (e)f Fk(x)h Fx(=)f Fk(p)3118 2197 y Fi(n)3163 2185 y Fk(=q)3242 2197 y Fi(n)523 2284 y Fx(whic)n(h)j(is)g(rational,)e(and)i (w)n(e)g(sa)n(y)e(that)j(the)f(fraction)f(stops)h(at)g(order)e Fk(n)i Fx(\(with)h(our)523 2384 y(con)n(v)n(en)n(tions,)j(w)n(e)g(ha)n (v)n(e)h Fk(x)1393 2396 y Fi(m)1500 2384 y Fx(=)k(0)p Fk(;)14 b Fl(8)p Fk(n)42 b Fl(\025)i Fk(m)p Fx(\).)d(Con)n(v)n(ersely) -7 b(,)38 b(if)j Fk(x)f Fx(is)h(rational,)e(the)523 2484 y(con)n(tin)n(ued)27 b(fraction)g(expansion)g(stops)g(at)h(some)f (\014nite)i(order)d Fk(n)p Fx(.)i(F)-7 b(or)27 b Fk(\013)c Fx(=)g(1,)28 b(w)n(e)f(get)523 2583 y(the)f(classical)e(Gauss)g(con)n (tin)n(ued)h(fraction)g(expansion)f(for)h(whic)n(h)g(all)g(signs)g Fk(")3002 2595 y Fi(n)3070 2583 y Fx(=)e(+1,)523 2683 y(and)k(for)f Fk(\013)e Fx(=)e(1)p Fk(=)p Fx(2,)k(w)n(e)h(ha)n(v)n(e)e (the)j(con)n(tin)n(ued)e(fraction)h(to)f(the)i(nearest)e(in)n(teger.)g (Note)523 2783 y(that)32 b(when)f Fk(\013)f Fl(6)p Fx(=)e(1,)j(the)h (results)f(of)g(equations)f(\(23a\))h(and)g(\(23b\))g(are)f(not)h(ob)n (vious.)523 2882 y(F)-7 b(or)34 b(details,)g(and)h(in)g(particular)e (for)h(the)h(extension)f(to)h(others)e(v)-5 b(alues)35 b(of)f Fk(\013)p Fx(,)h(with)523 2982 y(0)23 b Fl(\024)f Fk(\013)i(<)849 2949 y Fw(1)p 849 2963 34 4 v 849 3010 a(2)892 2982 y Fx(,)k(see)f([18,22].)648 3106 y(Giv)n(en)d(a)g(p)r (ositiv)n(e)g(real)g(function)h Fk(f)33 b Fx(on)24 b(\(0)p Fk(;)14 b Fx(1\),)24 b(the)h(Brjuno)f(series)f Fk(B)2855 3062 y Fw(\()p Fi(\013)p Fw(\))2851 3131 y Fi(f)2955 3106 y Fx(\()p Fk(x)p Fx(\))i(is)g(the)523 3205 y(sum)j(\(whic)n(h)g (can)f(b)r(e)h(in\014nite\))g(of)g(the)g(series)f(with)h(p)r(ositiv)n (e)f(terms)523 3441 y Fk(B)590 3398 y Fw(\()p Fi(\013)p Fw(\))586 3466 y Fi(f)689 3441 y Fx(\()p Fk(x)p Fx(\))d(=)942 3338 y Fj(1)915 3363 y Fn(X)912 3538 y Fi(n)p Fw(=0)1051 3441 y Fk(\014)1098 3453 y Fi(n)p Fj(\000)p Fw(1)1228 3441 y Fk(f)9 b Fx(\()p Fk(x)1357 3453 y Fi(n)1403 3441 y Fx(\))23 b(=)g Fk(f)9 b Fx(\()p Fk(x)1675 3453 y Fw(0)1712 3441 y Fx(\))g(+)g Fk(x)1874 3453 y Fw(0)1911 3441 y Fk(f)g Fx(\()p Fk(x)2040 3453 y Fw(1)2077 3441 y Fx(\))g(+)g Fk(:)14 b(:)g(:)8 b Fx(+)h Fk(x)2418 3453 y Fw(0)2455 3441 y Fk(x)2502 3453 y Fw(1)2553 3441 y Fl(\001)14 b(\001)g(\001)g Fk(x)2711 3453 y Fi(n)p Fj(\000)p Fw(1)2842 3441 y Fk(f)9 b Fx(\()p Fk(x)2971 3453 y Fi(n)3016 3441 y Fx(\))g(+)g Fk(:)14 b(:)g(:)36 b(;)3139 3617 y Fx(\(24\))523 3717 y(where)769 3684 y Fw(1)p 769 3698 V 769 3745 a(2)835 3717 y Fl(\024)22 b Fk(\013)i Fl(\024)f Fx(1,)f(and)h(for)g Fk(k)j Fl(\025)d Fx(0,)f Fk(x)1745 3729 y Fi(k)1810 3717 y Fx(de\014ned)h(in)h(Eq.)f(\(13\))f(or)h(\(15\).)g(As)g(men)n(tioned) 523 3816 y(ab)r(o)n(v)n(e,)h(when)h Fk(x)g Fx(is)g(rational,)e(w)n(e)i (ha)n(v)n(e)f Fk(x)1831 3828 y Fi(n)1899 3816 y Fx(=)f(0)h(for)h(some)f Fk(n)p Fx(,)h(and)g(w)n(e)f(use)h Fk(x)2947 3828 y Fi(m)3033 3816 y Fx(=)e(0)h(for)523 3916 y Fk(m)f Fl(\025)g Fk(n)p Fx(,)k(The)h(follo)n(wing)e(results)h(are)g(easily)g(obtained)g(from)g (the)h(de\014nitions)1070 4097 y Fk(B)1137 4054 y Fw(\()p Fi(\013)p Fw(\))1133 4122 y Fi(f)1236 4097 y Fx(\()p Fk(x)p Fx(\))e(=)f Fk(B)1530 4054 y Fw(\()p Fi(\013)p Fw(\))1526 4122 y Fi(f)1629 4097 y Fx(\()p Fk(x)19 b Fx(+)f(1\))c Fk(;)1177 b Fx(\(25a\))1070 4287 y Fk(B)1137 4244 y Fw(\()p Fi(\013)p Fw(\))1133 4312 y Fi(f)1236 4287 y Fx(\()p Fk(x)p Fx(\))26 b(=)f Fk(xB)1577 4244 y Fw(\()p Fi(\013)p Fw(\))1573 4312 y Fi(f)1690 4170 y Fn(\022)1764 4231 y Fx(1)p 1761 4268 48 4 v 1761 4344 a Fk(x)1819 4170 y Fn(\023)1898 4287 y Fx(+)18 b Fk(f)9 b Fx(\()p Fk(x)p Fx(\))56 b(for)27 b(0)c Fk(<)g(x)g(<)g(\013)14 b(;)367 b Fx(\(25b\))1070 4486 y Fk(B)1137 4443 y Fw(\()p Fi(\013)p Fw(\))1133 4511 y Fi(f)1236 4486 y Fx(\()p Fk(x)p Fx(\))26 b(=)f Fk(B)1530 4443 y Fw(\()p Fi(\013)p Fw(\))1526 4511 y Fi(f)1629 4486 y Fx(\()p Fl(\000)p Fk(x)p Fx(\))56 b(for)27 b(0)c Fk(<)f(x)i Fl(\024)e Fx(1)c Fl(\000)g Fk(\013)d(:)570 b Fx(\(25c\))523 4699 y(In)33 b(particular,)f Fk(B)1111 4650 y Fw(\()1147 4628 y Fe(1)p 1147 4637 29 4 v 1147 4670 a(2)1185 4650 y Fw(\))1107 4724 y Fi(f)1215 4699 y Fx(\()p Fk(x)p Fx(\))i(is)f(an)f(ev)n(en)h (function.)g(More)f(surprising)g(is)h(the)g(follo)n(wing)523 4830 y(result)e([18])g(:)h(in)g(the)g Fk(\013)f Fx(=)e(1)j(case,)f(for) g Fk(B)1875 4787 y Fw(\()p Fj(\006)p Fw(\))1871 4855 y Fi(f)1983 4830 y Fx(\()p Fk(x)p Fx(\))g(=)2229 4797 y Fw(1)p 2229 4811 34 4 v 2229 4858 a(2)2272 4830 y Fx(\()p Fk(B)2371 4787 y Fw(\(1\))2367 4855 y Fi(f)2461 4830 y Fx(\()p Fk(x)p Fx(\))22 b Fl(\006)f Fk(B)2747 4787 y Fw(\(1\))2743 4855 y Fi(f)2836 4830 y Fx(\()p Fl(\000)p Fk(x)p Fx(\)\))33 b(whic)n(h)p eop %%Page: 9 9 9 8 bop 1836 232 a Fu(Real)26 b(or)g(Complex)f(Brjuno)i(F)-6 b(unctions)236 b(9)523 452 y Fx(are)27 b(the)h(ev)n(en)f(and)g(o)r(dd)h (parts)f(of)g Fk(B)1693 409 y Fw(\(1\))1689 477 y Fi(f)1783 452 y Fx(\()p Fk(x)p Fx(\),)h(w)n(e)g(ha)n(v)n(e)e(for)h(0)c Fk(<)f(x)i Fl(\024)2706 419 y Fw(1)p 2706 433 34 4 v 2706 481 a(2)2749 452 y Fx(,)748 713 y Fk(B)815 670 y Fw(\()p Fj(\000)p Fw(\))811 738 y Fi(f)923 713 y Fx(\()p Fk(x)p Fx(\))i(=)1160 657 y(1)p 1160 694 42 4 v 1160 770 a(2)1225 596 y Fn(\022)1286 713 y Fk(f)9 b Fx(\()p Fk(x)p Fx(\))19 b Fl(\000)f Fk(f)9 b Fx(\(1)18 b Fl(\000)g Fk(x)p Fx(\))h Fl(\000)f Fx(\(1)g Fl(\000)g Fk(x)p Fx(\))p Fk(f)2274 596 y Fn(\022)2416 657 y Fk(x)p 2345 694 191 4 v 2345 770 a Fx(1)g Fl(\000)g Fk(x)2545 596 y Fn(\023\023)2695 713 y Fk(;)380 b Fx(\(26a\))749 946 y Fk(B)816 902 y Fw(\(+\))812 971 y Fi(f)923 946 y Fx(\()p Fk(x)p Fx(\))26 b(=)f Fk(xB)1264 902 y Fw(\(+\))1260 971 y Fi(f)1385 828 y Fn(\022)1459 889 y Fx(1)p 1456 926 48 4 v 1456 1002 a Fk(x)1514 828 y Fn(\023)1593 946 y Fx(+)1686 889 y(1)p 1686 926 42 4 v 1686 1002 a(2)1738 946 y Fk(G)p Fx(\()p Fk(x)p Fx(\))14 b Fk(;)98 b Fx(with)883 b(\(26b\))858 1178 y Fk(G)p Fx(\()p Fk(x)p Fx(\))26 b(=)f Fk(f)9 b Fx(\()p Fk(x)p Fx(\))19 b(+)f Fk(f)9 b Fx(\(1)18 b Fl(\000)g Fk(x)p Fx(\))h(+)f(\(1)g Fl(\000)g Fk(x)p Fx(\))p Fk(f)2137 1061 y Fn(\022)2280 1122 y Fk(x)p 2208 1159 191 4 v 2208 1235 a Fx(1)g Fl(\000)g Fk(x)2409 1061 y Fn(\023)2488 1178 y Fx(+)g(2)p Fk(xB)2727 1135 y Fw(\()p Fj(\000)p Fw(\))2723 1203 y Fi(f)2849 1061 y Fn(\022)2923 1122 y Fx(1)p 2920 1159 48 4 v 2920 1235 a Fk(x)2978 1061 y Fn(\023)3066 1178 y Fk(:)13 b Fx(\(26c\))523 1423 y(In)29 b(order)f(to)i(pro)n(v)n(e)d(the)j(previous)e(equations,)g(w)n(e)h(use) g(Equations)g(\(23a{c\))e(and)j(the)523 1523 y(succession)c(of)i (transformations)578 1737 y Fl(\000)p Fk(x)23 b Fl(!)g Fx(1)18 b Fl(\000)g Fk(x)23 b Fl(!)1223 1681 y Fx(1)p 1148 1718 191 4 v 1148 1794 a(1)18 b Fl(\000)g Fk(x)1372 1737 y Fl(!)1562 1681 y Fx(1)p 1488 1718 V 1488 1794 a(1)g Fl(\000)g Fk(x)1707 1737 y Fl(\000)g Fx(1)k(=)2023 1681 y Fk(x)p 1952 1718 V 1952 1794 a Fx(1)c Fl(\000)g Fk(x)2175 1737 y Fl(!)2291 1681 y Fx(1)g Fl(\000)g Fk(x)p 2291 1718 V 2363 1794 a(x)2515 1737 y Fx(=)2615 1681 y(1)p 2613 1718 48 4 v 2613 1794 a Fk(x)2688 1737 y Fl(\000)g Fx(1)23 b Fl(!)2955 1681 y Fx(1)p 2952 1718 V 2952 1794 a Fk(x)3032 1737 y Fl(!)h Fk(x)f(;)523 1967 y Fx(whic)n(h)28 b(pro)n(vides)e(the)i(requested)f(relations)f(b)r(et)n(w)n(een)i Fk(B)t Fx(\()p Fk(x)p Fx(\))g(and)g Fk(B)t Fx(\()p Fl(\000)p Fk(x)p Fx(\).)648 2074 y(No)n(w,)f(it)h(is)f(con)n(v)n(enien)n(t)g(to)g (in)n(tro)r(duce)g(the)h(follo)n(wing)f(sp)r(eci\014c)h(notations:)648 2191 y Fh(i\))e Fx(In)f Fk(B)901 2148 y Fw(\()p Fi(\013)p Fw(\))897 2216 y Fi(f)1001 2191 y Fx(\(x\),)h(when)g Fk(\013)d Fx(=)g(1,)i(w)n(e)g(omit)h(the)g(sup)r(erscript)f(\()p Fk(\013)p Fx(\),)h(and)g(when)g Fk(\013)d Fx(=)3221 2159 y Fw(1)p 3221 2173 34 4 v 3221 2220 a(2)3264 2191 y Fx(,)523 2323 y(w)n(e)31 b(replace)g(the)h(sup)r(erscript)f(\()p Fk(\013)p Fx(\))i(b)n(y)e Fk(e)p Fx(,)h(so)f(that)h Fk(B)2230 2335 y Fi(f)2273 2323 y Fx(\()p Fk(x)p Fx(\))e(=)g Fk(B)2576 2279 y Fw(\(1\))2572 2348 y Fi(f)2665 2323 y Fx(\()p Fk(x)p Fx(\))j(and)e Fk(B)3041 2292 y Fi(e)3037 2346 y(f)3080 2323 y Fx(\()p Fk(x)p Fx(\))g(=)523 2454 y Fk(B)590 2411 y Fw(\(1)p Fi(=)p Fw(2\))586 2479 y Fi(f)746 2454 y Fx(\()p Fk(x)p Fx(\))e(resp)r(ectiv)n(ely)-7 b(.)648 2573 y Fh(ii\))40 b Fx(W)-7 b(e)39 b(omit)g(the)g(subscript)g Fk(f)48 b Fx(when)39 b Fk(f)9 b Fx(\()p Fk(x)p Fx(\))43 b(=)e Fl(\000)14 b Fx(ln\()p Fk(x)p Fx(\))43 b(=)f(ln\()p Fk(x)2837 2543 y Fj(\000)p Fw(1)2927 2573 y Fx(\),)d(so)g(that)523 2688 y Fk(B)590 2658 y Fw(\()p Fi(\013)p Fw(\))689 2688 y Fx(\()p Fk(x)p Fx(\))g(=)e Fk(B)1008 2645 y Fw(\()p Fi(\013)p Fw(\))1004 2714 y Fj(\000)11 b Fw(ln)1126 2688 y Fx(\()p Fk(x)p Fx(\),)38 b Fk(B)t Fx(\()p Fk(x)p Fx(\))g(=)f Fk(B)1683 2645 y Fw(\(1\))1679 2714 y Fj(\000)11 b Fw(ln)1802 2688 y Fx(\()p Fk(x)p Fx(\))37 b(and)f Fk(B)2187 2658 y Fi(e)2223 2688 y Fx(\()p Fk(x)p Fx(\))i(=)f Fk(B)2541 2645 y Fw(\(1)p Fi(=)p Fw(2\))2537 2714 y Fj(\000)11 b Fw(ln)2697 2688 y Fx(\()p Fk(x)p Fx(\))38 b(resp)r(ectiv)n(ely)-7 b(.)523 2788 y(W)g(e)32 b(will)g(call)f Fk(B)t Fx(\()p Fk(x)p Fx(\))i Fh(the)g(Brjuno)h(function)p Fx(,)e(whic)n(h)g(has)f(b)r (een)h(men)n(tioned)f(ab)r(o)n(v)n(e)g(in)523 2888 y(Sects.)d(2)f(and)g (3.)h(W)-7 b(e)28 b(ha)n(v)n(e)523 3129 y Fk(B)t Fx(\()p Fk(x)p Fx(\))9 b(=)801 3025 y Fj(1)773 3050 y Fn(X)770 3226 y Fi(n)p Fw(=0)910 3129 y Fk(\014)957 3141 y Fi(n)p Fj(\000)p Fw(1)1087 3129 y Fx(ln)1157 3062 y Fn(\000)1195 3129 y Fk(x)1242 3095 y Fj(\000)p Fw(1)1242 3150 y Fi(n)1331 3062 y Fn(\001)1365 3129 y Fx(=)g Fl(\000)14 b Fx(ln)o(\()p Fk(x)1665 3141 y Fw(0)1703 3129 y Fx(\))q Fl(\000)q Fk(x)1849 3141 y Fw(0)1901 3129 y Fx(ln\()p Fk(x)2049 3141 y Fw(1)2087 3129 y Fx(\))q(+)q Fk(:)g(:)g(:)r Fl(\000)q Fk(x)2398 3141 y Fw(0)2436 3129 y Fk(x)2483 3141 y Fw(1)2534 3129 y Fl(\001)g(\001)g(\001)g Fk(x)2692 3141 y Fi(n)p Fj(\000)p Fw(1)2836 3129 y Fx(ln)q(\()p Fk(x)2985 3141 y Fi(n)3030 3129 y Fx(\))q Fl(\000)q Fk(:)g(:)g(:)38 b(;)3139 3305 y Fx(\(27\))523 3405 y(where)31 b(the)g Fk(x)960 3417 y Fi(n)1037 3405 y Fx(are)f(obtained)h(from)g(\(13\))g(using)g Fk(\013)e Fx(=)g(1)i(\(Gaussian)f(case\),)h(whereas)523 3504 y Fk(B)590 3474 y Fi(e)626 3504 y Fx(\()p Fk(x)p Fx(\))g(is)e(giv)n(en)g(b)n(y)h(the)g(same)f(equation)h(\(27\))f(with)h Fk(x)2302 3516 y Fi(n)2378 3504 y Fx(obtained)f(from)h(\(13\))f(using) 523 3604 y Fk(\013)35 b Fx(=)720 3571 y Fw(1)p 720 3585 V 720 3632 a(2)797 3604 y Fx(\(con)n(tin)n(ued)g(fraction)e(to)i(the)f (nearest)g(neigh)n(b)r(our\).)g(Both)g(functions)h Fk(B)t Fx(\()p Fk(x)p Fx(\))523 3703 y(and)25 b Fk(B)749 3673 y Fi(e)785 3703 y Fx(\()p Fk(x)p Fx(\))i(are)d(1-p)r(erio)r(dic,)h(and) g(tak)n(e)g(v)-5 b(alue)25 b(+)p Fl(1)g Fx(for)g Fk(x)h Fx(rational.)e(F)-7 b(rom)25 b(\(25a\),)g(the)523 3803 y(o)r(dd)d(part)g(of)h Fk(B)t Fx(\()p Fk(x)p Fx(\))g(is)f(giv)n(en)g (for)g(0)g Fl(\024)h Fk(x)g Fl(\024)1879 3770 y Fw(1)p 1879 3784 V 1879 3832 a(2)1944 3803 y Fx(b)n(y)f Fk(B)2117 3815 y Fj(\000)2173 3803 y Fx(\()p Fk(x)p Fx(\))i(=)2406 3770 y Fw(1)p 2406 3784 V 2406 3832 a(2)2449 3803 y Fk(x)p Fx(\(ln)q(\()p Fk(x)2677 3773 y Fj(\000)p Fw(1)2775 3803 y Fl(\000)8 b Fx(1\)\),)22 b(whic)n(h)g(is)523 3903 y(con)n(tin)n(uous) k(\(and)h(ev)n(en)f(H\177)-42 b(older)26 b(con)n(tin)n(uous)g(for)g(an) n(y)g(exp)r(onen)n(t)h Fk(\033)f(<)d Fx(1\).)k(Moreo)n(v)n(er,)523 4002 y Fk(B)590 3972 y Fi(e)626 4002 y Fx(\()p Fk(x)p Fx(\))f(is)g(ev)n(en,)f(and)g(it)h(has)f(b)r(een)h(pro)n(v)n(en)d([18]) i(that)h(the)g(di\013erence)f Fk(B)2812 3972 y Fi(e)2848 4002 y Fx(\()p Fk(x)p Fx(\))15 b Fl(\000)f Fk(B)3120 3972 y Fw(+)3175 4002 y Fx(\()p Fk(x)p Fx(\))523 4102 y(is)31 b(not)g(only)f(b)r(ounded,)h(but)h(con)n(tin)n(uous,)e(and)h (ev)n(en)f(H\177)-42 b(older)30 b(con)n(tin)n(uous)g(for)g(exp)r(o-)523 4202 y(nen)n(t)728 4169 y Fw(1)p 728 4183 V 728 4230 a(2)772 4202 y Fx(.)36 b(This)g(re\014nes)g(a)g(more)f(general)g (statemen)n(t)h([18,22])f(whic)n(h)h(sa)n(ys)f(that)i(the)523 4325 y(di\013erences)27 b Fk(B)994 4282 y Fw(\()p Fi(\013)p Fw(\))990 4350 y(ln)1112 4325 y Fl(\000)18 b Fk(B)t Fx(\()p Fk(x)p Fx(\))29 b(are)e(b)r(ounded)h(o)n(v)n(er)d(the)j(irrationals.) 648 4431 y(The)38 b(n)n(umerical)f(computation)g(of)h Fk(B)t Fx(\()p Fk(x)p Fx(\))h(and)f Fk(B)2275 4401 y Fi(e)2311 4431 y Fx(\()p Fk(x)p Fx(\))h(is)f(delicate,)g(due)g(to)g (the)523 4531 y(instabilities)30 b(of)g(the)g(con)n(tin)n(ued)g (fraction)f(expansion.)g(Ho)n(w)n(ev)n(er,)f(it)i(is)g(v)n(ery)f(easy)g (to)523 4631 y(compute)c(their)g(v)-5 b(alues)25 b(when)g(the)g(con)n (tin)n(ued)g(fraction)f(expansion)g(is)h(p)r(erio)r(dic,)g(that)523 4730 y(is)j(when)h Fk(x)f Fx(is)g(an)g(irrational)f(quadratic)g(n)n(um) n(b)r(er.)h(This)g(applies)g(to)g(noble)g(n)n(um)n(b)r(ers,)523 4830 y(in)g(whic)n(h)f(case)g(the)h Fk(x)1223 4842 y Fi(n)1296 4830 y Fx(are)f(constan)n(t)g(after)g(a)g(certain)g(order.)p eop %%Page: 10 10 10 9 bop 523 232 a Fu(10)199 b(Pierre)27 b(Moussa)g(and)e(Stefano)i (Marmi)523 448 y Fm(5)112 b(The)38 b(Brjuno)g(Series)f(and)h(Diophan)m (tine)f(conditions)523 615 y Fx(A)d(real)f(n)n(um)n(b)r(er)h(is)g(said) g(to)f(b)r(e)i(a)e Fh(Brjuno)j(numb)l(er)e Fx(if)g(and)g(only)g(if)g Fk(B)t Fx(\()p Fk(x)p Fx(\))h(is)f(\014nite,)523 715 y(and)j(w)n(e)g(also)f(sa)n(y)g(that)h Fk(x)h Fx(satis\014es)e Fh(the)j(Brjuno)g(diophantine)i(c)l(ondition)p Fx(.)d(Brjuno)523 814 y(n)n(um)n(b)r(ers)24 b(are)g(irrationals)f(and)i(real)f(n)n(um)n (b)r(ers)g(satisfying)g(the)i(classical)d(diophan)n(tine)523 914 y(conditions)g(\(whic)n(h)g(w)n(e)g(recall)f(b)r(elo)n(w\))h(are)f (Brjuno)h(n)n(um)n(b)r(ers.)g(In)g([18],)g(w)n(e)f(sho)n(w)h(that)523 1013 y(for)665 981 y Fw(1)p 665 995 34 4 v 665 1042 a(2)739 1013 y Fl(\024)31 b Fk(\013)h Fl(\024)e Fx(1,)i Fk(B)1179 983 y Fw(\()p Fi(\013)p Fw(\))1279 1013 y Fx(\()p Fk(x)p Fx(\))h(is)g(\014nite)g(if)g(and)f(only)g(if)h Fk(x)g Fx(is)f(a)h(Brjuno)f(n)n(um)n(b)r(er.)g(More)523 1124 y(precisely)-7 b(,)20 b(the)h(pro)r(of)g(sa)n(ys)e(that)i(for)f(an)n(y) g Fk(\013)k Fl(2)f Fx([0)p Fk(;)2098 1091 y Fw(1)p 2098 1105 V 2098 1153 a(2)2141 1124 y Fx(],)e(the)g(di\013erence)g Fl(j)p Fk(B)2799 1094 y Fw(\()p Fi(\013)p Fw(\))2898 1124 y Fx(\()p Fk(x)p Fx(\))5 b Fl(\000)g Fk(B)t Fx(\()p Fk(x)p Fx(\))p Fl(j)523 1224 y Fx(is)38 b(b)r(ounded)g(o)n(v)n(er)e (irrational)g(v)-5 b(alues)37 b(of)h Fk(x)p Fx(.)g(W)-7 b(e)38 b(also)e(sho)n(w)h(that)h(for)f Fk(\013)k Fx(=)e(1,)f(the)523 1323 y(di\013erence)g Fl(j)p Fk(B)t Fx(\()p Fk(x)p Fx(\))26 b Fl(\000)1222 1261 y Fn(P)1310 1282 y Fj(1)1310 1348 y Fi(n)p Fw(=0)1453 1323 y Fk(q)1493 1293 y Fj(\000)p Fw(1)1490 1344 y Fi(n)1596 1323 y Fx(ln\()p Fk(q)1734 1335 y Fi(n)p Fw(+1)1864 1323 y Fx(\))p Fl(j)38 b Fx(is)f(b)r(ounded)i (o)n(v)n(er)d(irrational)g(v)-5 b(alues)37 b(of)523 1423 y Fk(x)p Fx(,)e(so)f(that)g(w)n(e)g(reco)n(v)n(er)e(the)j(original)d (de\014nition)j(of)f(the)h(Brjuno)f(n)n(um)n(b)r(ers)g([23]:)f Fk(x)523 1523 y Fx(is)k(a)g(Brjuno)g(n)n(um)n(b)r(er)g(if)h(and)g(only) f(if)1830 1460 y Fn(P)1918 1481 y Fj(1)1918 1548 y Fi(n)p Fw(=0)2061 1523 y Fk(q)2101 1492 y Fj(\000)p Fw(1)2098 1543 y Fi(n)2204 1523 y Fx(ln\()p Fk(q)2342 1535 y Fi(n)p Fw(+1)2472 1523 y Fx(\))h(is)f(b)r(ounded)h(o)n(v)n(er)e(the)523 1622 y(irrational.)28 b(One)h(can)g(see)g([18,22])f(that)i(suc)n(h)f(a) g(de\014nition)h(of)f(the)h(Brjuno)f(n)n(um)n(b)r(ers)523 1722 y(do)r(es)e(not)h(dep)r(end)g(of)g(the)g(particular)e(v)-5 b(alue)27 b(of)h Fk(\013)g Fx(used)g(to)f(compute)h(the)g Fk(q)2965 1734 y Fi(n)3010 1722 y Fx(.)648 1822 y(W)-7 b(e)38 b(no)n(w)f(rep)r(ort)h(the)g(usual)g(de\014nition)g([6])g(of)g (the)g(diophan)n(tine)g(conditions)g(:)523 1921 y(w)n(e)33 b(sa)n(y)g(that)g Fk(x)i Fx(is)e(an)g(irrational)f(diophan)n(tine)h(n)n (um)n(b)r(er)h(of)f(order)f Fk(\034)43 b Fl(\025)32 b Fx(0)i(\(and)f(w)n(e)523 2021 y(write)j Fk(x)h Fl(2)g Fx(C\()p Fk(\034)9 b Fx(\)\),)38 b(if)e(there)g(exists)g Fk(c)g(>)h Fx(0)e(suc)n(h)h(that)g(for)f(an)n(y)h(in)n(tegers)e Fk(p)i Fx(and)g Fk(q)s Fx(,)523 2120 y(suc)n(h)26 b(that)h Fk(q)f(>)d Fx(0,)j(w)n(e)g(ha)n(v)n(e)f Fl(j)q Fk(x)18 b Fl(\000)g Fk(p=q)s Fl(j)23 b(\025)g Fk(cq)1947 2090 y Fj(\000)p Fw(2)p Fj(\000)p Fi(\034)2125 2120 y Fx(.)k(Some)f (classical)f(facts)h(need)h(to)f(b)r(e)523 2220 y(recalled)33 b(here)g([24].)g(First,)g(for)g(an)n(y)g Fk(p)h Fx(and)f Fk(q)k Fx(suc)n(h)c(that)h(0)e Fk(<)h(q)j(<)d(q)2809 2232 y Fi(n)p Fw(+1)2938 2220 y Fx(,)h(w)n(e)f(ha)n(v)n(e)523 2320 y Fl(j)p Fk(q)s(x)24 b Fl(\000)f Fk(p)p Fl(j)35 b(\025)g(j)p Fk(q)1005 2332 y Fi(n)1051 2320 y Fk(x)23 b Fl(\000)g Fk(p)1251 2332 y Fi(n)1296 2320 y Fl(j)p Fx(,)36 b(where)e Fk(p)1667 2332 y Fi(n)1712 2320 y Fk(=q)1791 2332 y Fi(n)1871 2320 y Fx(is)h(the)g(Gaussian)f(reduced)h(fraction)f (to)h Fk(x)p Fx(.)523 2419 y(Therefore,)e(in)i(order)e(to)h(ha)n(v)n(e) f Fk(x)i Fl(2)g Fx(C\()p Fk(\034)9 b Fx(\),)36 b(it)f(is)f(su\016cien)n (t)g(to)h(c)n(hec)n(k)e(that)i(for)f(an)n(y)523 2519 y Fk(n)j(>)f Fx(0,)g Fl(j)p Fk(x)19 b Fl(\000)f Fk(p)1026 2531 y Fi(n)1071 2519 y Fk(=q)1150 2531 y Fi(n)1195 2519 y Fl(j)37 b(\025)f Fk(cq)1432 2489 y Fj(\000)p Fw(2)p Fj(\000)p Fi(\034)1429 2539 y(n)1610 2519 y Fx(.)g(Second,)g (Liouville's)f(classical)g(theorem)g(asserts)523 2619 y(that)27 b(algebraic)d(n)n(um)n(b)r(ers)i(of)g(degree)f Fk(n)i Fx(b)r(elong)f(to)g Fk(x)d Fl(2)h Fx(C\()p Fk(n)16 b Fl(\000)f Fx(2\).)26 b(Moreo)n(v)n(er)e(Roth's)523 2718 y(theorem)39 b(sho)n(ws)f(that)i(all)f(algebraic)f(n)n(um)n(b)r (ers)h(b)r(elong)g(to)g(C\()p Fk(\034)9 b Fx(\),)41 b(for)e(all)g Fk(\034)53 b(>)42 b Fx(0.)523 2818 y(Finally)-7 b(,)35 b(for)f(an)g(arbitrary)f(irrational,)g(and)i(an)n(y)f Fk(n)g(>)h Fx(0,)f(w)n(e)h(ha)n(v)n(e)e(\()p Fk(q)2854 2830 y Fi(n)2900 2818 y Fk(q)2937 2830 y Fi(n)p Fw(+1)3066 2818 y Fx(\))3098 2788 y Fj(\000)p Fw(1)3222 2818 y Fl(\024)523 2917 y(j)p Fk(x)19 b Fl(\000)f Fk(p)737 2929 y Fi(n)782 2917 y Fk(=q)861 2929 y Fi(n)906 2917 y Fl(j)23 b(\024)g Fk(q)1080 2887 y Fj(\000)p Fw(2)1077 2938 y Fi(n)1169 2917 y Fx(.)h(Using)h(\(21\))f(for)g Fk(\013)g Fx(=)f(1,)h(w)n(e)g(get) h(an)f(equiv)-5 b(alen)n(t)24 b(caracterisation)523 3017 y(of)j(the)g(diophan)n(tine)g(conditions:)f Fk(x)e Fl(2)f Fx(C\()p Fk(\034)9 b Fx(\))28 b(if)g(and)e(only)h(if)g(there)g(exists)f (a)h(constan)n(t)523 3117 y Fk(c)c(>)g Fx(0)k(suc)n(h)g(that)h Fk(\014)1153 3129 y Fi(n)1221 3117 y Fl(\025)23 b Fk(c)14 b(\014)1410 3081 y Fw(1+)p Fi(\034)1406 3139 y(n)p Fj(\000)p Fw(1)1563 3117 y Fx(for)27 b(an)n(y)g Fk(n)c(>)g Fx(0.)648 3216 y(No)n(w)h(w)n(e)h(in)n(tro)r(duce)g(for)f Fk(\027)29 b(>)22 b Fx(0,)j(the)h(Brjuno)e(series)g Fk(B)2391 3231 y Fj(f)p Fi(\027)t Fj(g)2501 3216 y Fx(\()p Fk(x)p Fx(\))g Fl(\021)e Fk(B)2786 3232 y Fi(x)2824 3215 y Fc(\000)p Fd(\027)2910 3216 y Fx(\()p Fk(x)p Fx(\))k(for)f(the)523 3316 y(fonctions)i Fk(f)9 b Fx(\()p Fk(x)p Fx(\))24 b(=)f Fk(x)1196 3286 y Fj(\000)p Fi(\027)1317 3316 y Fx(\(still)28 b(using)f Fk(\013)d Fx(=)e(1\),)587 3524 y Fk(B)650 3539 y Fj(f)p Fi(\027)t Fj(g)760 3524 y Fx(\()p Fk(x)p Fx(\))i(=)1012 3421 y Fj(1)985 3445 y Fn(X)982 3621 y Fi(n)p Fw(=0)1121 3524 y Fk(\014)1168 3536 y Fi(n)p Fj(\000)p Fw(1)1298 3524 y Fx(\()p Fk(x)p Fx(\))p Fk(x)1456 3490 y Fj(\000)p Fi(\027)1456 3545 y(n)1574 3524 y Fx(=)1691 3421 y Fj(1)1664 3445 y Fn(X)1662 3621 y Fi(n)p Fw(=0)1801 3524 y Fk(\014)1848 3536 y Fi(n)p Fj(\000)p Fw(1)1992 3407 y Fn(\022)2105 3468 y Fk(\014)2152 3480 y Fi(n)p 2063 3505 178 4 v 2063 3581 a Fk(\014)2110 3593 y Fi(n)p Fj(\000)p Fw(1)2250 3407 y Fn(\023)2311 3424 y Fj(\000)p Fi(\027)2400 3524 y Fx(=)2517 3421 y Fj(1)2490 3445 y Fn(X)2487 3621 y Fi(n)p Fw(=0)2626 3524 y Fk(\014)2677 3489 y Fw(1+)p Fi(\027)2673 3546 y(n)p Fj(\000)p Fw(1)2803 3524 y Fx(\()p Fk(x)p Fx(\))p Fk(\014)2965 3490 y Fj(\000)p Fi(\027)2961 3545 y(n)3060 3524 y Fx(\()p Fk(x)p Fx(\))29 b Fk(:)3139 3700 y Fx(\(28\))523 3800 y(Using)e(\(21\))h(one)f(gets)664 4008 y(2)706 3974 y Fj(\000)p Fi(\027)842 3904 y Fj(1)815 3929 y Fn(X)813 4105 y Fi(n)p Fw(=0)951 4008 y Fk(q)991 3974 y Fj(\000)p Fw(1)p Fj(\000)p Fi(\027)988 4029 y(n)1170 4008 y Fl(j)p Fk(q)1230 4020 y Fi(n)1275 4008 y Fk(x)19 b Fl(\000)f Fk(p)1466 4020 y Fi(n)1511 4008 y Fl(j)1534 3974 y Fj(\000)p Fi(\027)1650 4008 y Fl(\024)23 b Fk(B)1801 4023 y Fj(f)p Fi(\027)t Fj(g)1910 4008 y Fx(\()p Fk(x)p Fx(\))h Fl(\024)2162 3904 y Fj(1)2135 3929 y Fn(X)2133 4105 y Fi(n)p Fw(=0)2272 4008 y Fk(q)2312 3974 y Fj(\000)p Fw(1)p Fj(\000)p Fi(\027)2309 4029 y(n)2490 4008 y Fl(j)p Fk(q)2550 4020 y Fi(n)2595 4008 y Fk(x)19 b Fl(\000)f Fk(p)2786 4020 y Fi(n)2831 4008 y Fl(j)2854 3974 y Fj(\000)p Fi(\027)2975 4008 y Fk(:)141 b Fx(\(29\))523 4232 y(The)22 b(series)f Fk(B)968 4247 y Fj(f)p Fi(\027)t Fj(g)1077 4232 y Fx(\()p Fk(x)p Fx(\))i(con)n(v)n(erges)c(if)j(and)g(only)f(if)i (the)f(series)2404 4169 y Fn(P)2491 4190 y Fj(1)2491 4257 y Fi(n)p Fw(=0)2634 4232 y Fk(q)2674 4201 y Fj(\000)p Fw(1)p Fj(\000)p Fi(\027)2671 4252 y(n)2853 4232 y Fl(j)p Fk(q)2913 4244 y Fi(n)2958 4232 y Fk(x)7 b Fl(\000)g Fk(p)3126 4244 y Fi(n)3171 4232 y Fl(j)3194 4201 y Fj(\000)p Fi(\027)523 4347 y Fx(con)n(v)n(erges,)30 b(that)k(is)e(if)i(the)f (series)1655 4285 y Fn(P)1743 4305 y Fj(1)1743 4372 y Fi(n)p Fw(=0)1886 4347 y Fk(q)1926 4317 y Fj(\000)p Fw(1)p Fj(\000)p Fw(2)p Fi(\027)1923 4368 y(n)2151 4347 y Fl(j)p Fk(x)19 b Fl(\000)f Fx(\()p Fk(p)2397 4359 y Fi(n)2442 4347 y Fk(=q)2521 4359 y Fi(n)2566 4347 y Fx(\))p Fl(j)2621 4305 y Fj(\000)p Fi(\027)2748 4347 y Fx(also)31 b(con)n(v)n(erges.)523 4458 y(As)e(a)f(consequence,)g(if)h Fk(B)1350 4473 y Fj(f)p Fi(\027)t Fj(g)1459 4458 y Fx(\()p Fk(x)p Fx(\))d Fk(<)e Fl(1)p Fx(,)29 b(then)g Fk(q)2050 4428 y Fj(\000)p Fw(1)p Fj(\000)p Fw(2)p Fi(\027)2047 4479 y(n)2276 4458 y Fl(j)p Fk(x)19 b Fl(\000)f Fx(\()p Fk(p)2522 4470 y Fi(n)2567 4458 y Fk(=q)2646 4470 y Fi(n)2691 4458 y Fx(\))p Fl(j)2746 4416 y Fj(\000)p Fi(\027)2868 4458 y Fx(is)28 b(b)r(ounded,)523 4558 y(and)f Fk(x)d Fl(2)f Fx(C\(1)p Fk(=\027)5 b Fx(\).)28 b(Con)n(v)n(ersely)-7 b(,)26 b(assume)g Fk(\034)33 b Fl(\025)23 b Fx(0,)k(and)g Fk(x)d Fl(2)f Fx(C\()p Fk(\034)9 b Fx(\),)29 b(then)f(w)n(e)g(ha)n(v)n(e)1407 4766 y Fk(B)1470 4781 y Fj(f)p Fi(\027)t Fj(g)1579 4766 y Fx(\()p Fk(x)p Fx(\))c Fl(\024)f Fk(c)1838 4732 y Fj(\000)p Fi(\027)1988 4662 y Fj(1)1961 4687 y Fn(X)1959 4863 y Fi(n)p Fw(=0)2098 4766 y Fk(q)2138 4732 y Fj(\000)p Fw(1+)p Fi(\034)7 b(\027)2135 4787 y(n)2380 4766 y Fk(:)736 b Fx(\(30\))p eop %%Page: 11 11 11 10 bop 1836 232 a Fu(Real)26 b(or)g(Complex)f(Brjuno)i(F)-6 b(unctions)198 b(11)523 448 y Fx(Using)30 b(b)r(ounds)h(in)f(\(22\),)g (w)n(e)g(get)g(the)h(follo)n(wing)f(statemen)n(t:)g(If)h Fk(x)d Fl(2)g Fx(C\()p Fk(\034)9 b Fx(\),)32 b(then)e(for)523 548 y(an)n(y)21 b Fk(\027)27 b Fx(suc)n(h)21 b(that)g Fk(\034)33 b(<)23 b(\027)1299 518 y Fj(\000)p Fw(1)1388 548 y Fx(,)f Fk(B)1496 563 y Fj(f)p Fi(\027)t Fj(g)1605 548 y Fx(\()p Fk(x)p Fx(\))i Fk(<)f Fl(1)p Fx(.)e(Therefore,)g(there)g (is)g(a)g(relation)g(b)r(et)n(w)n(een)523 648 y(the)i(diophan)n(tine)g (conditions)f(C\()p Fk(\034)9 b Fx(\),)24 b(and)f(the)g(con)n(v)n (ergence)d(of)j(the)g(Brjuno)g(series)e(for)523 747 y Fk(f)9 b Fx(\()p Fk(x)p Fx(\))24 b(=)e Fk(x)842 717 y Fj(\000)p Fi(\027)936 747 y Fx(:)h(the)h(set)f(of)g(irrationals)e Fk(x)i Fx(suc)n(h)g(that)g Fk(B)2220 762 y Fj(f)p Fi(\027)t Fj(g)2329 747 y Fx(\()p Fk(x)p Fx(\))h Fl(\021)f Fk(B)2615 763 y Fi(x)2653 746 y Fc(\000)p Fd(\027)2738 747 y Fx(\()p Fk(x)p Fx(\))i(is)d(b)r(ounded,)523 847 y(is)37 b(con)n(tained)g(in)h (C\(1)p Fk(=\027)5 b Fx(\),)38 b(and)f(con)n(tains)f(C\()p Fl(\000)p Fk(")25 b Fx(+)g(1)p Fk(=\027)5 b Fx(\),)37 b(for)g(an)n(y)g(0)i Fk(<)g(")g Fl(\024)g Fx(1)p Fk(=\027)5 b Fx(.)523 946 y(In)33 b(some)g(sense,)g(the)g(Brjuno)g(conditions)g (is)g(related)g(to)g(the)g(limiting)h(case)e Fk(\027)38 b Fx(=)32 b(0,)523 1046 y(and)38 b(in)g(particular,)f Fk(x)k Fl(2)g Fx(C\()p Fk(\034)9 b Fx(\))39 b(for)f Fk(\034)50 b(>)40 b Fx(0)e(implies)g Fk(B)t Fx(\()p Fk(x)p Fx(\))k Fk(<)e Fl(1)p Fx(,)e(that)h(is)f Fk(x)g Fx(is)g(a)523 1146 y(Brjuno)j(n)n(um)n(b)r(er.)f(Using)h(a)f(more)h(general)e (function)j Fk(f)9 b Fx(,)40 b(p)r(ositiv)n(e)h(on)f(\(0)p Fk(;)14 b Fx(1\),)41 b(and)523 1245 y(monotoneously)34 b(decreasing)f(in)i(the)h(vicinit)n(y)f(of)g(zero,)f(w)n(e)h(can)f(in)n (tro)r(duce)h(a)g(wide)523 1345 y(family)28 b(of)f(conditions)g Fk(B)1331 1357 y Fi(f)1374 1345 y Fx(\()p Fk(x)p Fx(\))d Fk(<)f Fx(+)p Fl(1)p Fx(.)k(The)g(diophan)n(tine)h(conditions)f (obtained)g(will)523 1445 y(b)r(e)41 b(mainly)g(go)n(v)n(erned)e(b)n(y) h(the)i(singular)d(b)r(eha)n(vior)h(of)h Fk(f)49 b Fx(around)40 b(zero.)g(A)h(p)r(o)n(w)n(er)523 1544 y(la)n(w)36 b(b)r(eha)n(viour)f (w)n(ould)i(sim)n(ulate)f(the)h(usual)f(conditions,)g(whereas)g(a)g (logarithmic)523 1644 y(b)r(eha)n(viour)24 b(w)n(ould)h(generate)e(a)i (condition)g(similar)f(to)h(the)h(Brjuno)e(condition.)h(Other)523 1743 y(in)n(teresting)35 b(examples)g(w)n(ould)g(b)r(e)h(obtained)f(b)n (y)h(taking)f(functions)g Fk(f)45 b Fx(of)35 b(the)h(form)523 1843 y Fk(x)570 1813 y Fj(\000)p Fi(\027)664 1843 y Fl(j)14 b Fx(log\()p Fk(x)p Fx(\))p Fl(j)942 1813 y Fi(\026)987 1843 y Fx(,)28 b Fk(x)1085 1813 y Fj(\000)p Fi(\027)1179 1843 y Fl(j)14 b Fx(log\()p Fk(x)p Fx(\))p Fl(j)1457 1813 y Fi(\026)1503 1843 y Fl(j)g Fx(log\()p Fl(j)g Fx(log)g Fk(x)p Fl(j)p Fx(\))p Fl(j)1962 1813 y Fi(\033)2008 1843 y Fx(,)27 b(and)h(so)f(one.)523 2118 y Fm(6)112 b(The)38 b(Brjuno)g(Op)s(erator)523 2326 y Fx(W)-7 b(e)31 b(will)f(in)n(tro)r (duce)g(no)n(w)g(some)f(functional)i(analysis)e(in)h(order)f(to)h(solv) n(e)f(Equations)523 2426 y(\(25a{c\).)k(F)-7 b(or)33 b(\014xed)1220 2393 y Fw(1)p 1220 2407 34 4 v 1220 2454 a(2)1297 2426 y Fl(\024)g Fk(\013)h Fl(\024)f Fx(1,)h(let)g(us)g (consider)f(the)i(op)r(erator)d Fk(T)2788 2441 y Fw(\()p Fi(\013)p Fw(\))2887 2426 y Fx(,)i(acting)f(on)523 2525 y(lo)r(cally)27 b(Leb)r(esgue)g(in)n(tegrable)f(functions)i Fk(f)37 b Fx(on)27 b(the)h(real)f(line,)g(whic)n(h)h(v)n(erify)1066 2711 y Fk(f)9 b Fx(\()p Fk(x)p Fx(\))25 b(=)g Fk(f)9 b Fx(\()p Fk(x)18 b Fx(+)g(1\))23 b(for)k(almost)g(ev)n(ery)13 b Fk(x)23 b Fl(2)h Fx(I)-14 b(R)14 b Fk(;)565 b Fx(\(31a\))1066 2836 y Fk(f)9 b Fx(\()p Fk(x)p Fx(\))25 b(=)g Fk(f)9 b Fx(\()p Fl(\000)p Fk(x)p Fx(\))51 b(for)27 b(almost)g(ev)n(ery)12 b Fk(x)24 b Fl(2)f Fx(\(0)p Fk(;)14 b Fx(1)k Fl(\000)g Fk(\013)p Fx(\))c Fk(:)349 b Fx(\(31b\))523 3022 y(The)28 b(op)r(erator)e(is)h(de\014ned)h(b)n(y)1198 3254 y(\()p Fk(T)1279 3269 y Fw(\()p Fi(\013)p Fw(\))1378 3254 y Fk(f)9 b Fx(\)\()p Fk(x)p Fx(\))24 b(=)e Fk(xf)1793 3137 y Fn(\022)1867 3197 y Fx(1)p 1864 3234 48 4 v 1864 3311 a Fk(x)1922 3137 y Fn(\023)2020 3254 y Fk(;)69 b Fx(if)56 b Fk(x)23 b Fl(2)h Fx(\(0)p Fk(;)14 b(\013)p Fx(\))28 b Fk(:)527 b Fx(\(32\))523 3490 y(It)23 b(is)f(understo)r(o)r(d)f(that) i(the)f(function)h Fk(T)1795 3505 y Fw(\()p Fi(\013)p Fw(\))1894 3490 y Fk(f)31 b Fx(is)22 b(completed)g(outside)g(\(0)p Fk(;)14 b(\013)p Fx(\))22 b(b)n(y)g(imp)r(os-)523 3589 y(ing)27 b(on)g Fk(T)825 3604 y Fw(\()p Fi(\013)p Fw(\))923 3589 y Fk(f)36 b Fx(the)28 b(same)e(parit)n(y)g(and)h(p)r(erio)r(dicit) n(y)g(conditions)g(whic)n(h)g(are)f(expressed)523 3689 y(for)k Fk(f)39 b Fx(in)31 b(the)g(ab)r(o)n(v)n(e)f(equations)f (\(31a{b\).)h(The)h(functional)f(equations)g(\(25a{c\))f(can)523 3788 y(then)f(b)r(e)g(written)g(in)g(the)g(form)1528 3907 y Fn(\000)1566 3975 y Fx(1)18 b Fl(\000)g Fk(T)1758 3990 y Fw(\()p Fi(\013)p Fw(\))1857 3907 y Fn(\001)1909 3975 y Fk(B)1976 3932 y Fw(\()p Fi(\013)p Fw(\))1972 4000 y Fi(f)2098 3975 y Fx(=)23 b Fk(f)32 b(:)857 b Fx(\(33\))523 4161 y(This)28 b(suggest)e(to)i(study)f(the)h(op)r(erator)e Fk(T)1861 4176 y Fw(\()p Fi(\013)p Fw(\))1988 4161 y Fx(on)h(the)h(Banac)n(h)e(spaces)904 4347 y Fk(X)973 4359 y Fi(\013;p)1097 4347 y Fx(=)d Fl(f)o Fk(f)32 b Fx(:)23 b(I)-14 b(R)24 b Fl(!)f Fx(I)-14 b(R)23 b Fl(j)g Fk(f)g Fx(v)n(eri\014es)j(\(31a{b\))c Fk(;)60 b(f)31 b Fl(2)24 b Fk(L)2630 4313 y Fi(p)2668 4347 y Fx(\(0)p Fk(;)14 b(\013)p Fx(\))p Fl(g)233 b Fx(\(34\))523 4533 y(endo)n(w)n(ed)27 b(with)h(the)g(norm)f(of)g Fk(L)1560 4503 y Fi(p)1598 4533 y Fx(\(0)p Fk(;)14 b(\013)p Fx(\),)28 b(namely)1337 4793 y Fl(jj)p Fk(f)9 b Fl(jj)1479 4805 y Fi(\013;p)1603 4793 y Fx(=)1691 4675 y Fn(\022)1752 4679 y(Z)1835 4700 y Fi(\013)1798 4868 y Fw(0)1896 4793 y Fl(j)p Fk(f)g Fx(\()p Fk(x)p Fx(\))p Fl(j)2103 4758 y Fi(p)2156 4793 y Fk(dx)2246 4675 y Fn(\023)2308 4693 y Fw(1)p Fi(=p)2450 4793 y Fk(;)666 b Fx(\(35\))p eop %%Page: 12 12 12 11 bop 523 232 a Fu(12)199 b(Pierre)27 b(Moussa)g(and)e(Stefano)i (Marmi)523 448 y Fx(for)f Fk(p)d Fl(2)g Fx([1)p Fk(;)14 b Fl(1)p Fx(].)26 b(Note)g(that)h(one)f(could)g(also)f(use)h Fk(L)2162 418 y Fi(p)2200 448 y Fx(\(0)p Fk(;)14 b Fx(1\),)26 b(instead)g(of)g Fk(L)2869 418 y Fi(p)2907 448 y Fx(\(0)p Fk(;)14 b(\013)p Fx(\),)27 b(and)523 548 y(that)35 b(if)f Fk(p)g(<)g(p)1009 518 y Fj(0)1067 548 y Fx(one)f(has)h(the)h(ob)n (vious)e(inclusion)h Fk(X)2259 560 y Fi(\013;p)2356 544 y Fc(0)2417 548 y Fl(\032)f Fk(X)2584 560 y Fi(\013;p)2685 548 y Fx(.)i(If)f(\(1)23 b Fl(\000)g Fk(T)3066 563 y Fw(\()p Fi(\013)p Fw(\))3164 548 y Fx(\))35 b(is)523 648 y(in)n(v)n(ertible)f(in)h(the)g(considered)e(space,)h(then)i (\(25a{c\))d(ha)n(v)n(e)g(a)h(unique)h(solution)f(for)523 763 y Fk(B)590 720 y Fw(\()p Fi(\013)p Fw(\))586 788 y Fi(f)689 763 y Fx(,)28 b(pro)n(vided)e(that)h(the)h Fk(f)35 b Fx(in)28 b(the)f(righ)n(t)g(hand)g(side)g(of)g(\(25b\))g (also)f(b)r(elongs)g(to)i(the)523 863 y(space.)36 b(The)h(in)n(v)n (ertibilit)n(y)g(prop)r(ert)n(y)e(is)i(giv)n(en)f(b)n(y)h(the)h(follo)n (wing)d(theorem,)i(whic)n(h)523 963 y(states)26 b(in)g(particular)f (that)h(the)h(sp)r(ectral)e(radius)g(of)h Fk(T)2255 978 y Fw(\()p Fi(\013)p Fw(\))2380 963 y Fx(is)g(strictly)g(smaller)f(than) h(1.)523 1062 y Fg(Theorem.)36 b Fk(T)1010 1077 y Fw(\()p Fi(\013)p Fw(\))1147 1062 y Fx(is)i(a)g(linear)f(b)r(ounded)h(op)r (erator)f(from)g Fk(X)2532 1074 y Fi(\013;p)2671 1062 y Fx(in)n(to)h(itself)h(for)e(all)523 1170 y Fk(\013)32 b Fl(2)f Fx([)727 1137 y Fw(1)p 727 1151 34 4 v 727 1199 a(2)770 1170 y Fk(;)14 b Fx(1])32 b(and)h(for)e(all)i Fk(p)e Fl(2)g Fx([1)p Fk(;)14 b Fl(1)p Fx(].)32 b(Its)h(sp)r(ectral)f (radius)f(on)h Fk(X)2633 1182 y Fi(\013;p)2767 1170 y Fx(is)g(b)r(ounded)h(b)n(y)523 1270 y(the)28 b(constan)n(t)f Fk(\025)p Fx(\()p Fk(\013)p Fx(\))i(of)f(Equation)e(\(22\),)h(and)h (therefore)f(1)18 b Fl(\000)g Fk(T)2549 1285 y Fw(\()p Fi(\013)p Fw(\))2675 1270 y Fx(is)27 b(in)n(v)n(ertible.)523 1369 y(F)-7 b(or)24 b(the)i(pro)r(of,)e(see)g([18].)h(W)-7 b(e)25 b(will)g(just)g(observ)n(e)e(here)i(that)g(the)g(result)g(is)g (immediate)523 1469 y(in)j(the)g Fk(p)23 b Fx(=)f Fl(1)28 b Fx(case.)f(Indeed,)985 1611 y(\()p Fk(T)1078 1576 y Fi(n)1066 1633 y Fw(\()p Fi(\013)p Fw(\))1165 1611 y Fk(f)9 b Fx(\)\()p Fk(x)p Fx(\))24 b(=)e Fk(\014)1516 1623 y Fi(n)p Fj(\000)p Fw(1)1646 1611 y Fx(\()p Fk(x)p Fx(\))p Fk(f)9 b Fx(\()p Fk(x)1886 1623 y Fi(n)1933 1611 y Fx(\))23 b(=)g Fk(\014)2123 1623 y Fi(n)p Fj(\000)p Fw(1)2253 1611 y Fx(\()p Fk(x)p Fx(\)\()p Fk(f)28 b Fl(\016)18 b Fk(A)2587 1576 y Fi(n)2587 1631 y(\013)2635 1611 y Fx(\)\()p Fk(x)p Fx(\))24 b Fk(;)314 b Fx(\(36\))523 1752 y(where)27 b(the)h(map)g Fk(A)1153 1764 y Fi(\013)1228 1752 y Fx(is)f(de\014ned)h(ab)r(o)n(v)n(e)f(\(see)g(\(16a{b\)\).)g (Therefore)913 1894 y Fl(jj)p Fk(T)1020 1860 y Fi(n)1008 1917 y Fw(\()p Fi(\013)p Fw(\))1107 1894 y Fk(f)9 b Fl(jj)1203 1906 y Fi(\013;)p Fj(1)1359 1894 y Fl(\024)23 b Fx(sup)1572 1914 y Fi(x)1613 1894 y Fx(\()p Fk(\014)1692 1906 y Fi(n)p Fj(\000)p Fw(1)1823 1894 y Fx(\()p Fk(x)p Fx(\)\))p Fl(jj)p Fk(f)9 b Fl(jj)2108 1906 y Fi(\013;p)2233 1894 y Fl(\024)23 b Fk(c\025)p Fx(\()p Fk(\013)p Fx(\))2522 1860 y Fi(n)p Fj(\000)p Fw(1)2654 1894 y Fl(jj)p Fk(f)9 b Fl(jj)2796 1906 y Fi(\013;p)3139 1894 y Fx(\(37\))523 2036 y(and)30 b(one)g(gets)f(the)i(theorem)e(\(for)h Fk(p)g Fx(in\014nite\))h(b)n(y)f (taking)g(the)g(1)p Fk(=n)p Fx({th)f(ro)r(ot)g(of)h(b)r(oth)523 2135 y(sides.)37 b(F)-7 b(or)37 b(the)h(other)e(v)-5 b(alues)37 b(of)h Fk(p)p Fx(,)f(it)h(is)f(con)n(v)n(enien)n(t)f(to)i (mak)n(e)e(use)h(of)h(the)g(mea-)523 2235 y(sure)28 b(whic)n(h)h(is)g (in)n(v)-5 b(arian)n(t)28 b(under)h(transformation)f(b)n(y)g(the)i(map) f Fk(A)2688 2247 y Fi(\013)2735 2235 y Fx(,)h(instead)e(of)i(the)523 2335 y(Leb)r(esgue)g(measure.)f(An)i(immediate)g(consequence)e(of)h (the)h(theorem,)f(is)g(that)g(if)h(w)n(e)523 2434 y(tak)n(e)24 b Fk(f)9 b Fx(\()p Fk(x)p Fx(\))24 b(=)f(ln\()p Fk(x)p Fx(\),)j(for)e(0)f Fk(<)f(x)i(<)f(\013)p Fx(,)i(then)g Fk(f)32 b Fl(2)24 b Fk(X)2145 2446 y Fi(\013;p)2246 2434 y Fx(,)h(for)f(all)h Fh(\014nite)f Fk(p)h Fx(and)g(therefore)523 2534 y(w)n(e)i(also)g(ha)n(v)n(e)f Fk(B)1070 2504 y Fw(\()p Fi(\013)p Fw(\))1193 2534 y Fl(2)d Fk(X)1340 2546 y Fi(\013;p)1469 2534 y Fx(for)k(all)g(\014nite)h Fk(p)p Fx(.)648 2634 y(Ho)n(w)n(ev)n(er,)k(w)n(e)j(ha)n(v)n(e)e(a)h(stronger)f(prop)r(ert)n (y)h(in)h(the)g Fk(\013)g Fx(=)2545 2601 y Fw(1)p 2545 2615 V 2545 2662 a(2)2623 2634 y Fx(case.)f(Here,)h(w)n(e)f(set)523 2733 y(again)d Fk(T)798 2745 y Fi(e)863 2733 y Fl(\021)f Fk(T)1007 2748 y Fw(\(1)p Fi(=)p Fw(2\))1163 2733 y Fx(.)i(In)g(this)g (case,)f(the)h(logarithm)f(b)r(elongs)g(also)g(to)h(the)g(set)g Fk(X)3154 2745 y Fj(\003)3222 2733 y Fl(\032)523 2833 y Fk(X)592 2848 y Fi(\013)p Fw(=1)p Fi(=)p Fw(2)p Fi(;p)p Fw(=1)928 2833 y Fx(,)j(made)g(of)g(ev)n(en,)f(p)r(erio)r(dic)h(\(with) g(p)r(erio)r(d)g(1\))g(functions)g(b)r(elonging)f(to)523 2932 y(the)e(so-called)f(\\BMO-space".)f(In)i(this)g(space,)f Fk(f)41 b Fx(has)32 b(b)r(ounded)g(mean)g(oscillation,)523 3032 y(more)27 b(precisely)g(the)h(follo)n(wing)e(semi-norm)h Fl(jj)p Fk(f)9 b Fl(jj)2113 3044 y Fj(\003)2178 3032 y Fx(is)28 b(b)r(ounded,)g(with)1371 3216 y Fl(jj)p Fk(f)9 b Fl(jj)1513 3228 y Fj(\003)1574 3216 y Fx(=)23 b(sup)1707 3286 y Fi(I)1834 3160 y Fx(1)p 1811 3197 90 4 v 1811 3273 a Fl(j)p Fk(I)7 b Fl(j)1923 3103 y Fn(Z)1970 3292 y Fi(I)2022 3216 y Fl(j)p Fk(f)27 b Fl(\000)18 b Fk(f)2237 3228 y Fi(I)2274 3216 y Fl(j)p Fk(dx)29 b(;)700 b Fx(\(38\))523 3412 y(where)34 b(the)i(mean)e(v)-5 b(alue)35 b(of)g Fk(f)44 b Fx(o)n(v)n(er)33 b Fk(I)42 b Fx(is)34 b Fk(f)1952 3424 y Fi(I)2025 3412 y Fx(=)h Fl(j)p Fk(I)7 b Fl(j)2214 3382 y Fj(\000)p Fw(1)2317 3345 y Fn(R)2357 3442 y Fi(I)2408 3412 y Fl(j)p Fk(f)32 b Fl(\000)23 b Fk(f)2633 3424 y Fi(I)2671 3412 y Fl(j)p Fk(dx)p Fx(,)36 b(and)f(the)g(sup)523 3512 y(is)f(tak)n(en)g(o)n(v)n(er)e(all)i(p)r(ossible)g(in)n(terv)-5 b(als)34 b Fk(I)41 b Fx(with)35 b(length)g Fl(j)p Fk(I)7 b Fl(j)34 b Fx(smaller)f(than)i(one.)f(The)523 3611 y(BMO)29 b(space)f(has)h(remark)-5 b(able)28 b(prop)r(erties.)g(First)h(it)h(is) f(con)n(tained)f(in)i(all)f Fk(L)2993 3623 y Fi(p)3060 3611 y Fx(spaces)523 3711 y(for)e Fk(p)g Fx(\014nite,)g(and)h(it)f(con) n(tains)f(the)i Fk(L)1722 3723 y Fj(1)1819 3711 y Fx(space,)e(second)h (it)h(is)f(the)g(space)g(adequate)f(to)523 3811 y(describ)r(e)d (functions)h(ha)n(ving)e(singular)g(b)r(eha)n(viour)g(not)h(w)n(orse)f (than)h(logarithmic,)f(but)523 3910 y(around)f(ev)n(ery)h(p)r(oin)n(t)g (in)h(a)f(dense)g(set)g(of)h(the)g(real)e(line,)i(and)f(third,)h(it)g (has)e(remark)-5 b(able)523 4010 y(prop)r(erties)27 b(connected)g(to)h (the)g(harmonic)e(conjugacy)h(transformation)f([25,18].)648 4110 y(In)f([18,26],)e(w)n(e)i(ha)n(v)n(e)f(sho)n(wn)g(that)i(1)13 b Fl(\000)g Fk(T)1950 4122 y Fi(e)2008 4110 y Fl(\021)22 b Fx(1)13 b Fl(\000)g Fk(T)2277 4125 y Fw(\(1)p Fi(=)p Fw(2\))2458 4110 y Fx(is)25 b(in)n(v)n(ertible)g(in)g Fk(X)3067 4122 y Fj(\003)3105 4110 y Fx(,)g(and)523 4221 y(therefore)h(that)h Fk(B)1117 4191 y Fi(e)1175 4221 y Fl(\021)c Fk(B)1330 4191 y Fw(\(1)p Fi(=)p Fw(2\))1510 4221 y Fl(2)g Fk(X)1657 4233 y Fj(\003)1695 4221 y Fx(.)k(Since)f Fl(j)p Fk(B)2050 4191 y Fi(\013)2114 4221 y Fl(\000)16 b Fk(B)2262 4191 y Fi(e)2298 4221 y Fl(j)27 b Fx(is)f(b)r(ounded)h(for) 2904 4188 y Fw(1)p 2904 4202 34 4 v 2904 4250 a(2)2970 4221 y Fl(\024)c Fk(\013)h Fl(\024)e Fx(1,)523 4332 y(w)n(e)29 b(ha)n(v)n(e)g(the)g(unexp)r(ected)h(consequence)f(that)h(all)f Fk(B)2262 4302 y Fw(\()p Fi(\013)p Fw(\))2391 4332 y Fx(for)2530 4299 y Fw(1)p 2530 4313 V 2530 4360 a(2)2599 4332 y Fl(\024)d Fk(\013)h Fl(\024)f Fx(1,)j(also)f(ha)n(v)n(e)523 4431 y(the)21 b(Bounded)g(Mean)f(Oscillation)g(prop)r(ert)n(y)-7 b(,)20 b(although)g(it)h(cannot)f(b)r(e)h(sho)n(wn)f(directly)523 4531 y(through)27 b(the)h(prop)r(erties)f(of)g(1)18 b Fl(\000)g Fk(T)1657 4546 y Fw(\()p Fi(\013)p Fw(\))1756 4531 y Fx(.)648 4631 y(The)23 b(BMO)h(prop)r(ert)n(y)f(obtained)g(for)h (the)g(Brjuno)f(function)i(in)f(the)g(real)f(case,)g(w)n(as)523 4730 y(one)29 b(of)g(our)g(motiv)-5 b(ations)29 b(to)g(consider)f(the)i (complexi\014cation)f(pro)r(cedure)f(whic)n(h)h(w)n(e)523 4830 y(will)f(describ)r(e)f(in)h(the)g(last)f(Section)h(of)f(this)h (pap)r(er)f([27].)p eop %%Page: 13 13 13 12 bop 1836 232 a Fu(Real)26 b(or)g(Complex)f(Brjuno)i(F)-6 b(unctions)198 b(13)523 448 y Fm(7)112 b(Application)35 b(to)i(H\177)-56 b(older{con)m(tin)m(uous)36 b(F)-9 b(unctions)523 641 y Fx(In)32 b(this)g(Section,)g(w)n(e)f(will)h(consider)f(only)g (the)h(case)f Fk(\013)g Fx(=)2431 608 y Fw(1)p 2431 622 34 4 v 2431 670 a(2)2474 641 y Fx(.)h(In)g(this)g(case)f(the)h(map)523 741 y Fk(A)585 753 y Fi(\013)663 741 y Fl(\021)d Fk(A)819 756 y Fw(1)p Fi(=)p Fw(2)956 741 y Fx(is)i(con)n(tin)n(uous)g(on)h(the) g(in)n(terv)-5 b(al)31 b(\(0)p Fk(;)2155 708 y Fw(1)p 2155 722 V 2155 769 a(2)2198 741 y Fx(].)h(The)f(functional)h(equation) f(for)523 840 y(the)d(Brjuno)f(function)h Fk(B)1336 810 y Fi(e)1400 840 y Fx(for)f Fk(\013)c Fx(=)g(1)p Fk(=)p Fx(2)j(is)1427 1014 y([\(1)18 b Fl(\000)g Fk(T)1674 1026 y Fi(e)1709 1014 y Fx(\))p Fk(B)1808 980 y Fi(e)1844 1014 y Fx(]\()p Fk(x)p Fx(\))24 b(=)f Fl(\000)14 b Fx(log)g Fk(x)23 b(;)756 b Fx(\(39\))523 1189 y(for)34 b(all)g Fk(x)g Fl(2)h Fx(\(0)p Fk(;)14 b Fx(1)p Fk(=)p Fx(2\),)32 b(complemen)n(ted)j(with)f(the)h(condition)f(that)g Fk(B)2797 1159 y Fi(e)2867 1189 y Fx(is)g(ev)n(en)g(and)523 1288 y(p)r(erio)r(dic.)d(W)-7 b(e)31 b(supp)r(ose)g(that)g(the)g(righ)n(t)f (hand)h(side)g(of)g(this)g(equation)f(is)h(p)r(ertub)r(ed,)523 1388 y(b)n(y)24 b(an)g(additional)f(term)h Fk(f)9 b Fx(,)24 b(whic)n(h)g(is)g(less)g(singular)f(than)h(the)g(logarithmic)f (function,)523 1488 y(and)g(w)n(e)g(w)n(an)n(t)f(to)i(study)f(the)g (singular)f(prop)r(erties)h(of)g(the)g(p)r(erturb)r(ed)h(solution.)f (Since)523 1587 y(the)34 b(equation)g(is)g(linear,)f(w)n(e)h(only)f (need)i(to)e(consider)g(the)i(action)e(on)h Fk(f)43 b Fx(of)34 b Fk(T)3084 1599 y Fi(e)3153 1587 y Fx(and)523 1687 y(\(1)9 b Fl(\000)g Fk(T)729 1699 y Fi(e)764 1687 y Fx(\))796 1657 y Fj(\000)p Fw(1)886 1687 y Fx(,)23 b(whic)n(h)g(w)n(e)f(will)i(con)n(v)n(enien)n(tly)d(recall)h(the)i (Brjuno)e(op)r(erator)g Fg(B)2935 1699 y Ff(e)2973 1687 y Fx(.)h(W)-7 b(e)24 b(will)523 1787 y(consider)29 b(ev)n(en)h(and)f(p) r(erio)r(dic)h(functions)h Fk(f)38 b Fx(whic)n(h)30 b(are)f Fh(c)l(ontinuous)p Fx(.)h(It)g(is)g(su\016cien)n(t)523 1886 y(to)e(kno)n(w)g(the)g(v)-5 b(alue)28 b(of)g Fk(f)37 b Fx(on)28 b([0)p Fk(;)14 b Fx(1)p Fk(=)p Fx(2],)27 b(so)g(w)n(e)h (assume)g Fk(f)k Fl(2)25 b Fk(C)2522 1856 y Fw(0)2516 1913 y([0)p Fi(;)p Fw(1)p Fi(=)p Fw(2])2711 1886 y Fx(.)j(One)g(can)g (c)n(hec)n(k)523 1998 y(that)22 b Fk(T)12 b(f)30 b Fx(is)22 b(also)f(con)n(tin)n(uous)g(pro)n(vided)g(w)n(e)g(set)h Fk(T)12 b(f)d Fx(\(0\))22 b(=)h(0.)e(W)-7 b(e)22 b(need)h(no)n(w)e(the) h(usual)523 2098 y(H\177)-42 b(older's)27 b(t)n(yp)r(e)h(semi-norms)e (for)h(con)n(tin)n(uous)g(functions)h(:)g(let)g Fk(f)k Fl(2)23 b Fk(C)2763 2068 y Fw(0)2757 2124 y([0)p Fi(;)p Fw(1)p Fi(=)p Fw(2])2952 2098 y Fx(,)28 b(then)g(w)n(e)523 2197 y(de\014ne)g(the)g(H\177)-42 b(older's)27 b Fk(\015)5 b Fx(-norm)26 b(as)1323 2415 y Fl(j)p Fk(f)9 b Fl(j)1419 2427 y Fi(\015)1485 2415 y Fx(=)141 b(sup)1572 2488 y Fw(0)p Fj(\024)p Fi(x)g Fx(\(2)2701 3481 y Fi(\015)2767 3511 y Fl(\000)23 b Fx(2)2897 3481 y Fj(\000)p Fi(\015)2991 3511 y Fx(\))3023 3481 y Fj(\000)p Fw(1)3113 3511 y Fx(,)36 b(the)523 3610 y(norm)e(of)g Fk(T)897 3622 y Fi(e)967 3610 y Fx(corresp)r(onding)e(to)i(the)h(norm)f (\(41\))g(satis\014es)f Fl(jj)p Fk(T)2578 3622 y Fi(e)2613 3610 y Fl(jj)2659 3622 y Fi(\015)2736 3610 y Fl(\024)h Fx(2)2877 3580 y Fw(\(2)p Fi(\015)t Fj(\000)p Fw(1\))3123 3610 y Fl(\024)g Fx(1.)523 3710 y(Therefore)26 b(for)h(0)c Fk(<)g(\015)k(<)c Fx(1)p Fk(=)p Fx(2,)j Fk(T)1561 3722 y Fi(e)1624 3710 y Fx(is)i(a)f(con)n(traction,)f(and)h(1)18 b Fl(\000)g Fk(T)2589 3722 y Fi(e)2652 3710 y Fx(is)28 b(in)n(v)n(ertible.)648 3810 y(W)-7 b(e)28 b(need)f(the)h(follo)n(wing) f(Lemma)523 3909 y Fg(Lemma.)38 b Fx(Let)i(0)k Fk(<)f(y)k(<)d(x)g Fl(\024)g Fx(1)p Fk(=)p Fx(2,)39 b(and)h(de\014ne)g Fk(x)2310 3921 y Fw(1)2388 3909 y Fx(and)g Fk(y)2603 3921 y Fw(1)2680 3909 y Fx(b)n(y)g(the)h(follo)n(wing)523 4009 y(conditions)1358 4127 y Fk(y)25 b Fx(=)1616 4071 y(1)p 1522 4108 230 4 v 1522 4184 a Fk(n)18 b Fx(+)g Fk(y)1714 4196 y Fw(1)1844 4127 y Fk(;)97 b(x)24 b Fx(=)2241 4071 y(1)p 2132 4108 260 4 v 2132 4184 a Fk(m)19 b Fx(+)f Fk(x)2354 4196 y Fw(1)2429 4127 y Fk(;)687 b Fx(\(42\))523 4315 y(with)27 b Fk(n)c Fl(\025)g Fx(2)j(and)g Fk(m)d Fl(\025)g Fx(2,)j(and)g Fl(\000)p Fx(1)p Fk(=)p Fx(2)21 b Fl(\024)i Fk(x)1882 4327 y Fw(1)1943 4315 y Fk(<)f Fx(1)p Fk(=)p Fx(2)j(and)i Fl(\000)p Fx(1)p Fk(=)p Fx(2)21 b Fl(\024)h Fk(y)2682 4327 y Fw(1)2742 4315 y Fk(<)h Fx(1)p Fk(=)p Fx(2,)i(then)i(w)n(e)523 4415 y(ha)n(v)n(e)1494 4542 y Fl(jj)p Fk(x)1587 4554 y Fw(1)1624 4542 y Fl(j)19 b(\000)f(j)p Fk(y)1813 4554 y Fw(1)1850 4542 y Fl(jj)23 b(\024)2017 4486 y(j)p Fk(x)c Fl(\000)f Fk(y)s Fl(j)p 2017 4523 239 4 v 2044 4599 a(j)p Fk(x)p Fl(jj)p Fk(y)s Fl(j)2293 4542 y Fk(:)823 b Fx(\(43\))523 4730 y Fh(Pr)l(o)l(of)32 b(of)f(the)g(L)l(emma.)e Fx(Since)f Fk(y)g(<)c(x)p Fx(,)29 b(w)n(e)f(ha)n(v)n(e)f Fk(n)19 b Fl(\000)f Fk(m)25 b(>)f(x)2477 4742 y Fw(1)2533 4730 y Fl(\000)19 b Fk(y)2658 4742 y Fw(1)2719 4730 y Fk(>)24 b Fl(\000)p Fx(1.)k(therefore)523 4830 y Fk(n)23 b Fl(\025)g Fk(m)p Fx(.)h(Let)g Fk(n)12 b Fl(\000)g Fk(m)22 b Fx(=)h Fk(p)g Fl(\025)f Fx(0.)i(W)-7 b(e)25 b(ha)n(v)n(e)e Fk(x)12 b Fl(\000)g Fk(y)25 b Fx(=)e Fk(xy)s Fx(\()p Fk(p)12 b Fx(+)g Fk(y)2425 4842 y Fw(1)2473 4830 y Fl(\000)g Fk(x)2597 4842 y Fw(1)2634 4830 y Fx(\).)24 b(So)g(that)h(w)n(e)f(need) p eop %%Page: 14 14 14 13 bop 523 232 a Fu(14)199 b(Pierre)27 b(Moussa)g(and)e(Stefano)i (Marmi)523 448 y Fx(to)h(pro)n(v)n(e)e Fl(jj)p Fk(x)942 460 y Fw(1)980 448 y Fl(j)18 b(\000)h(j)p Fk(y)1169 460 y Fw(1)1206 448 y Fl(jj)k(\024)g(j)p Fk(p)c Fx(+)f Fk(y)1571 460 y Fw(1)1627 448 y Fl(\000)g Fk(x)1757 460 y Fw(1)1795 448 y Fl(j)p Fx(.)28 b(This)f(is)h(ob)n(vious)f(when)h Fk(p)23 b Fx(=)g(0.)28 b(W)-7 b(e)28 b(alw)n(a)n(ys)523 548 y(ha)n(v)n(e)f Fl(jj)p Fk(x)808 560 y Fw(1)846 548 y Fl(j)18 b(\000)h(j)p Fk(y)1035 560 y Fw(1)1072 548 y Fl(jj)24 b(\024)f Fx(1)p Fk(=)p Fx(2,)j(so)i(the)g(required)f (inequalit)n(y)h(also)f(holds)g(when)h Fk(p)c Fl(\025)f Fx(2.)28 b(In)523 648 y(the)f(remaining)f(case)h Fk(p)22 b Fx(=)h(1,)k(w)n(e)f(set)h Fk(\021)g Fx(=)22 b(sign\()p Fk(y)2094 660 y Fw(1)2131 648 y Fx(\))27 b(and)g Fk(\017)c Fx(=)f(sign\()p Fk(x)2718 660 y Fw(1)2756 648 y Fx(\),)27 b(and)g(w)n(e)g(need)523 747 y(to)22 b(c)n(hec)n(k)e(that)i Fl(jj)p Fk(x)1103 759 y Fw(1)1141 747 y Fl(j)7 b(\000)g(j)p Fk(y)1307 759 y Fw(1)1343 747 y Fl(jj)24 b(\024)e(j)p Fx(1)7 b(+)g Fk(\021)s Fl(j)p Fk(y)1752 759 y Fw(1)1788 747 y Fl(j)g(\000)g Fk(\017)p Fl(j)p Fk(x)1994 759 y Fw(1)2030 747 y Fl(jj)p Fx(.)22 b(Still)h(b)r(ecause)e(the)h(left)h (hand)e(side)h(is)523 847 y(smaller)j(or)h(equal)g(to)g(1)p Fk(=)p Fx(2,)f(this)i(last)f(inequalit)n(y)g(is)g(not)g(ob)n(vious)f (only)h(when)h Fk(\021)f Fx(=)d Fl(\000)p Fx(1)523 946 y(and)k Fk(\017)22 b Fx(=)h(+1.)j(It)h(therefore)f(remains)g(to)h(sho)n (w)f(that)h Fl(jj)p Fk(x)2308 958 y Fw(1)2346 946 y Fl(j)16 b(\000)h(j)p Fk(y)2531 958 y Fw(1)2568 946 y Fl(jj)23 b(\024)g(j)p Fx(1)16 b Fl(\000)h(j)p Fk(y)2952 958 y Fw(1)2989 946 y Fl(j)f(\000)h(j)p Fk(x)3180 958 y Fw(1)3218 946 y Fl(jj)p Fx(.)523 1046 y(Setting)33 b Fk(u)d Fx(=)h(1)p Fk(=)p Fx(2)20 b Fl(\000)h(j)p Fk(x)1290 1058 y Fw(1)1328 1046 y Fl(j)33 b Fx(and)f Fk(v)i Fx(=)d(1)p Fk(=)p Fx(2)20 b Fl(\000)h(j)p Fk(y)2016 1058 y Fw(1)2053 1046 y Fl(j)p Fx(,)33 b(the)g(last)f(inequalit)n(y)g(is)g(equiv)-5 b(alen)n(t)523 1146 y(to)29 b Fl(j)p Fx(1)19 b Fl(\000)g Fk(v)s(=u)p Fl(j)25 b(\024)g(j)p Fx(1)20 b(+)f Fk(v)s(=u)p Fl(j)p Fx(,)28 b(whic)n(h)h(is)g(readily)g(c)n(hec)n(k)n(ed)f(since)h Fk(u=v)i Fx(is)e(real)f(and)h(non-)523 1245 y(negativ)n(e.)523 1345 y Fh(Pr)l(o)l(of)39 b(of)g(the)f(Pr)l(op)l(osition.)h Fx(Let)e(0)h Fk(<)f(y)k(<)d(x)h Fl(\024)e Fx(1)p Fk(=)p Fx(2,)f(and)g Fk(x)2595 1357 y Fw(1)2669 1345 y Fx(and)h Fk(y)2881 1357 y Fw(1)2955 1345 y Fx(as)f(in)h(the)523 1445 y(preceding)27 b(lemma.)g(W)-7 b(e)28 b(ha)n(v)n(e)745 1592 y Fl(j)p Fk(T)817 1604 y Fi(e)852 1592 y Fk(f)9 b Fx(\()p Fk(x)p Fx(\))19 b Fl(\000)f Fk(T)1164 1604 y Fi(e)1199 1592 y Fk(f)9 b Fx(\()p Fk(y)s Fx(\))p Fl(j)25 b Fx(=)f Fl(j)p Fk(xf)9 b Fx(\(1)p Fk(=x)p Fx(\))19 b Fl(\000)f Fk(y)s(f)9 b Fx(\(1)p Fk(=y)s Fx(\))p Fl(j)22 b Fx(=)g Fl(j)p Fk(xf)9 b Fx(\()p Fl(j)p Fk(x)2551 1604 y Fw(1)2590 1592 y Fl(j)p Fx(\))18 b Fl(\000)g Fk(y)s(f)9 b Fx(\()p Fl(j)p Fk(y)2936 1604 y Fw(1)2973 1592 y Fl(j)p Fx(\))p Fl(j)47 b Fx(\(44a\))1405 1717 y Fl(\024)24 b(j)p Fk(x)19 b Fl(\000)f Fk(y)s Fl(jj)p Fk(f)9 b Fx(\()p Fl(j)p Fk(x)1908 1729 y Fw(1)1946 1717 y Fl(j)p Fx(\))p Fl(j)19 b Fx(+)f Fl(j)p Fk(y)s Fl(jj)p Fk(f)9 b Fx(\()p Fl(j)p Fk(x)2391 1729 y Fw(1)2428 1717 y Fl(j)p Fx(\))19 b Fl(\000)f Fk(f)9 b Fx(\()p Fl(j)p Fk(y)2731 1729 y Fw(1)2768 1717 y Fl(j)p Fx(\))p Fl(j)247 b Fx(\(44b\))1405 1841 y Fl(\024)24 b(j)p Fk(x)19 b Fl(\000)f Fk(y)s Fl(jj)p Fk(f)9 b Fl(j)1829 1853 y Fj(1)1918 1841 y Fx(+)18 b Fl(j)p Fk(y)s Fl(jj)p Fk(f)9 b Fl(j)2187 1853 y Fi(\015)2229 1841 y Fl(jj)p Fk(x)2322 1853 y Fw(1)2360 1841 y Fl(j)18 b(\000)g(j)p Fk(y)2548 1853 y Fw(1)2585 1841 y Fl(jj)2631 1807 y Fi(\015)3102 1841 y Fx(\(44c\))1405 2017 y Fl(\024)24 b(j)p Fk(x)19 b Fl(\000)f Fk(y)s Fl(jj)p Fk(f)9 b Fl(j)1829 2029 y Fj(1)1918 2017 y Fx(+)18 b Fl(j)p Fk(f)9 b Fl(j)2097 2029 y Fi(\015)2185 1961 y Fl(j)p Fk(x)19 b Fl(\000)f Fk(y)s Fl(j)2424 1931 y Fi(\015)p 2149 1998 354 4 v 2149 2074 a Fl(j)p Fk(x)p Fl(j)2242 2050 y Fi(\015)2285 2074 y Fl(j)p Fk(y)s Fl(j)2375 2050 y Fi(\015)t Fj(\000)p Fw(1)2526 2017 y Fk(;)544 b Fx(\(44d\))523 2209 y(where)27 b(w)n(e)g(ha)n(v)n(e)g(used)g Fk(f)32 b Fl(2)23 b Fk(C)1482 2179 y Fi(\015)1525 2209 y Fx(,)28 b(and)f(the)h(Lemma.)g(Therefore)866 2350 y Fl(j)p Fk(T)938 2362 y Fi(e)973 2350 y Fk(f)9 b Fx(\()p Fk(x)p Fx(\))19 b Fl(\000)f Fk(T)1285 2362 y Fi(e)1320 2350 y Fk(f)9 b Fx(\()p Fk(y)s Fx(\))p Fl(j)p 866 2387 636 4 v 1043 2463 a(j)p Fk(x)19 b Fl(\000)f Fk(y)s Fl(j)1282 2439 y Fi(\015)1536 2406 y Fl(\024)24 b(j)p Fk(x)19 b Fl(\000)f Fk(y)s Fl(j)1864 2372 y Fw(1)p Fj(\000)p Fi(\015)1992 2406 y Fl(j)p Fk(f)9 b Fl(j)2088 2418 y Fj(1)2176 2406 y Fx(+)2259 2289 y Fn(\022)2332 2350 y Fl(j)p Fk(y)s Fl(j)p 2330 2387 94 4 v 2330 2463 a(j)p Fk(x)p Fl(j)2434 2289 y Fn(\023)2495 2306 y Fi(\015)2551 2406 y Fl(j)p Fk(y)s Fl(j)2641 2372 y Fw(1)p Fj(\000)p Fw(2)p Fi(\015)2802 2406 y Fl(j)p Fk(f)g Fl(j)2898 2418 y Fi(\015)3098 2406 y Fx(\(45a\))1536 2590 y Fl(\024)24 b Fx(\(1)p Fk(=)p Fx(2\))1815 2556 y Fw(1)p Fj(\000)p Fi(\015)1942 2590 y Fl(j)p Fk(f)9 b Fl(j)2038 2602 y Fj(1)2127 2590 y Fx(+)18 b(\(1)p Fk(=)p Fx(2\))2400 2556 y Fw(1)p Fj(\000)p Fw(2)p Fi(\015)2559 2590 y Fl(j)p Fk(f)9 b Fl(j)2655 2602 y Fi(\015)2712 2590 y Fk(;)358 b Fx(\(45b\))523 2738 y(since)30 b(0)d Fk(<)g(y)k(<)c(x)h Fl(\024)f Fx(1)p Fk(=)p Fx(2,)i(and)h Fk(\015)i Fl(\024)27 b Fx(1)p Fk(=)p Fx(2.)i(F)-7 b(or)30 b Fk(y)g Fx(=)d(0,)j(the)h(righ)n (t)e(hand)h(side)h(can)e(b)r(e)523 2837 y(replaced)e(b)n(y)g(its)h (\014rst)f(term)h(\(1)p Fk(=)p Fx(2\))1642 2807 y Fw(1)p Fj(\000)p Fi(\015)1769 2837 y Fl(j)p Fk(f)9 b Fl(j)1865 2849 y Fj(1)1935 2837 y Fx(,)27 b(and)h(the)g(ab)r(o)n(v)n(e)e (inequalit)n(y)h(extends)h(to)523 2937 y(the)g(case)f(where)g Fk(y)j Fx(v)-5 b(anishes,)27 b(so)g(that)1155 3085 y Fl(j)p Fk(T)1227 3097 y Fi(e)1263 3085 y Fk(f)9 b Fl(j)1336 3097 y Fi(\015)1401 3085 y Fl(\024)23 b Fk(K)1560 3097 y Fi(\015)1602 3085 y Fx(\()p Fk(f)9 b Fx(\))23 b(=)g(2)1869 3051 y Fi(\015)t Fj(\000)p Fw(1)1996 3085 y Fl(j)p Fk(f)9 b Fl(j)2092 3097 y Fj(1)2180 3085 y Fx(+)18 b(2)2305 3051 y Fw(2)p Fi(\015)t Fj(\000)p Fw(1)2465 3085 y Fl(j)p Fk(f)9 b Fl(j)2561 3097 y Fi(\015)2631 3085 y Fk(:)485 b Fx(\(46\))523 3233 y(F)-7 b(or)27 b(the)h(norm,)f(w)n(e)g(get)1004 3381 y Fl(jj)p Fk(T)1099 3393 y Fi(e)1135 3381 y Fk(f)9 b Fl(jj)1231 3393 y Fi(\015)1298 3381 y Fx(=)24 b Fk(A)p Fl(j)p Fk(T)1521 3393 y Fi(e)1557 3381 y Fk(f)9 b Fl(j)1630 3393 y Fi(\015)1690 3381 y Fx(+)18 b Fk(B)t Fl(j)p Fk(T)1912 3393 y Fi(e)1948 3381 y Fk(f)9 b Fl(j)2021 3393 y Fj(1)3098 3381 y Fx(\(47a\))1298 3505 y Fl(\024)24 b Fx(2)1429 3471 y Fw(2)p Fi(\015)t Fj(\000)p Fw(1)1603 3438 y Fn(\002)1638 3505 y Fk(A)p Fl(j)p Fk(f)9 b Fl(j)1796 3517 y Fi(\015)1857 3505 y Fx(+)18 b(\(2)2014 3471 y Fj(\000)p Fw(2)p Fi(\015)2141 3505 y Fk(B)23 b Fx(+)18 b(2)2352 3471 y Fj(\000)p Fi(\015)2446 3505 y Fk(A)p Fx(\))p Fl(j)p Fk(f)9 b Fl(j)2636 3517 y Fj(1)2706 3438 y Fn(\003)2769 3505 y Fk(;)301 b Fx(\(47b\))1298 3645 y Fl(\024)24 b Fx(2)1429 3611 y Fw(\(2)p Fi(\015)t Fj(\000)p Fw(1\))1641 3645 y Fl(jj)p Fk(f)9 b Fl(jj)1783 3657 y Fi(\015)1840 3645 y Fk(;)1239 b Fx(\(47c\))523 3793 y(pro)n(vided)28 b(2)908 3763 y Fj(\000)p Fw(2)p Fi(\015)1035 3793 y Fk(B)c Fx(+)19 b(2)1248 3763 y Fj(\000)p Fi(\015)1342 3793 y Fk(A)26 b Fl(\024)g Fk(B)t Fx(,)j(that)h(is)f Fk(A=B)h Fl(\024)25 b Fx(2)2236 3763 y Fi(\015)2298 3793 y Fl(\000)19 b Fx(2)2424 3763 y Fj(\000)p Fi(\015)2547 3793 y Fx(whic)n(h)30 b(completes)f(the)523 3893 y(pro)r(of.)e(The)h (ab)r(o)n(v)n(e)e(prop)r(osition)h(has)g(t)n(w)n(o)f(ob)n(vious)h (consequences)581 4024 y Fl(\017)41 b Fx(Since)28 b Fk(C)946 3994 y Fi(\015)1012 4024 y Fl(\032)23 b Fk(C)1165 3994 y Fi(\015)1204 3969 y Fc(0)1258 4024 y Fx(whenev)n(er)j Fk(\015)1670 3994 y Fj(0)1716 4024 y Fl(\024)d Fk(\015)5 b Fx(,)27 b(w)n(e)g(ha)n(v)n(e)922 4172 y Fk(f)33 b Fl(2)24 b Fk(C)1140 4137 y Fi(\015)1266 4172 y Fx(and)82 b Fk(\015)28 b Fl(\025)23 b Fx(1)p Fk(=)p Fx(2)187 b(=)-14 b Fl(\))106 b Fk(T)2243 4184 y Fi(e)2278 4172 y Fk(f)32 b Fl(2)23 b Fk(C)2494 4137 y Fw(1)p Fi(=)p Fw(2)3098 4172 y Fx(\(48a\))922 4296 y Fk(f)33 b Fl(2)24 b Fk(C)1140 4262 y Fi(\015)1266 4296 y Fx(and)82 b Fk(\015)28 b Fl(\025)23 b Fk(\015)1684 4308 y Fw(0)1804 4296 y Fk(;)97 b(\015)1967 4308 y Fw(0)2027 4296 y Fl(\024)23 b Fx(1)p Fk(=)p Fx(2)187 b(=)-14 b Fl(\))106 b Fk(T)2717 4308 y Fi(e)2752 4296 y Fk(f)32 b Fl(2)24 b Fk(C)2969 4262 y Fi(\015)3004 4270 y Fe(0)3093 4296 y Fx(\(48b\))581 4437 y Fl(\017)41 b Fx(When)30 b Fk(\015)f(<)c Fx(1)p Fk(=)p Fx(2,)i Fk(T)1295 4449 y Fi(e)1359 4437 y Fx(is)i(a)f(con)n(traction)g(on)g Fk(C)2133 4407 y Fi(\015)2176 4437 y Fx(.)h(Therefore)e(1)19 b Fl(\000)g Fk(T)2799 4449 y Fi(e)2863 4437 y Fx(is)29 b(in)n(v)n(ertible)664 4537 y(and)f Fg(B)894 4549 y Ff(e)955 4537 y Fx(=)1043 4474 y Fn(P)1130 4495 y Fj(1)1130 4562 y Fw(0)1201 4537 y Fx(\()p Fk(T)1282 4549 y Fi(e)1317 4537 y Fx(\))1349 4507 y Fi(n)1418 4537 y Fx(=)22 b(\(1)d Fl(\000)f Fk(T)1730 4549 y Fi(e)1765 4537 y Fx(\))1797 4507 y Fj(\000)p Fw(1)1914 4537 y Fx(preserv)n(es)25 b Fk(C)2335 4507 y Fi(\015)2378 4537 y Fx(,)j(and)g(w)n(e)f(ha)n(v)n(e) 723 4685 y Fk(f)34 b Fl(2)23 b Fk(C)941 4650 y Fi(\015)1067 4685 y Fx(and)83 b(0)22 b Fk(<)h(\015)28 b(<)22 b Fx(1)p Fk(=)p Fx(2)188 b(=)-14 b Fl(\))106 b Fg(B)2216 4697 y Ff(e)2254 4685 y Fk(f)32 b Fl(2)23 b Fk(C)2470 4650 y Fi(\015)3098 4685 y Fx(\(49a\))723 4809 y Fk(f)34 b Fl(2)23 b Fk(C)941 4775 y Fw(1)p Fi(=)p Fw(2)1235 4809 y Fx(=)-14 b Fl(\))106 b Fg(B)1543 4821 y Ff(e)1581 4809 y Fk(f)32 b Fl(2)23 b Fk(C)1797 4775 y Fi(\015)1923 4809 y Fk(;)97 b Fl(8)p Fk(\015)31 b Fx(suc)n(h)d(that)83 b(0)23 b Fk(<)f(\015)28 b(<)22 b Fx(1)p Fk(=)p Fx(2)27 b Fk(:)20 b Fx(\(49b\))p eop %%Page: 15 15 15 14 bop 1836 232 a Fu(Real)26 b(or)g(Complex)f(Brjuno)i(F)-6 b(unctions)198 b(15)523 448 y Fx(W)-7 b(e)20 b(ha)n(v)n(e)e(repro)r (duced)g(here)h(the)g(pro)r(of)g(of)g([26],)f(b)r(ecause)h(it)h(is)f (essen)n(tially)f(elemen)n(tary)-7 b(.)523 548 y(In)32 b(fact)g(w)n(e)f(ha)n(v)n(e)f(a)h(sligh)n(tly)g(b)r(etter)h(result)g (for)f Fg(B)2174 560 y Fi(e)2241 548 y Fx(than)h(for)f Fk(T)2619 560 y Fi(e)2654 548 y Fx(,)h(as)f(sho)n(wn)g(in)h(the)523 648 y(next)d(prop)r(osition,)f(whic)n(h)h(sho)n(ws)f(that)h(the)g Fk(C)2041 617 y Fw(1)p Fi(=)p Fw(2)2175 648 y Fx(prop)r(ert)n(y)f(is)g (e\013ectiv)n(ely)h(reac)n(hed.)523 747 y(Its)f(pro)r(of)f([18,26])f (is)h(to)r(o)g(di\016cult)i(to)e(b)r(e)h(repro)r(duced)f(here.)g(W)-7 b(e)28 b(ha)n(v)n(e)523 847 y Fg(Prop)s(osition.)e Fx(If)i Fk(f)j Fl(2)24 b Fk(C)1361 817 y Fi(\015)1431 847 y Fx(,)k(and)f Fk(\015)h(>)23 b Fx(1)p Fk(=)p Fx(2,)j(then)i Fg(B)2234 859 y Fi(e)2270 847 y Fk(f)j Fl(2)24 b Fk(C)2486 817 y Fw(1)p Fi(=)p Fw(2)2590 847 y Fx(.)648 946 y(The)34 b(Brjuno)g(function)g Fk(B)1508 916 y Fi(e)1578 946 y Fx(whic)n(h)h(w)n(e)e(ha)n(v)n(e)g(studied)i(in)g(the)f(previous)f (section)523 1046 y(is)c(nothing)f(else)g(than)h Fg(B)1334 1058 y Fi(e)1369 1046 y Fk(`)p Fx(,)g(where)f Fk(`)g Fx(is)g(equal)h(to)f(min)n(us)h(the)g(logarithmic)e(fonction)523 1146 y(restricted)40 b(to)h([0)p Fk(;)14 b Fx(1)p Fk(=)p Fx(2].)39 b(When)j(made)e(ev)n(en)h(and)g(p)r(erio)r(dic,)f(this)i (function)f(is)g(not)523 1245 y(con)n(tin)n(uous.)26 b(Supp)r(ose)i(that)f(w)n(e)g(p)r(erturb)h Fk(`)f Fx(b)n(y)g(a)f (function)i(with)g(enough)f(regularit)n(y)523 1345 y(prop)r(erties)f (\(for)g(example)g Fk(C)1460 1315 y Fw(1)1524 1345 y Fx(or)g Fk(C)1690 1315 y Fw(1)p Fi(=)p Fw(2+)p Fi(\017)1873 1345 y Fx(\),)h(the)g(c)n(hange)f(in)g Fg(B)2534 1357 y Fi(e)2570 1345 y Fk(`)g Fx(will)h(b)r(e)g(con)n(tin)n(uous)523 1445 y(and)19 b(ev)n(en)f(in)h Fk(C)1009 1414 y Fw(1)p Fi(=)p Fw(2)1114 1445 y Fx(,)g(that)g(is)f(H\177)-42 b(older-)1682 1412 y Fw(1)p 1681 1426 34 4 v 1681 1473 a(2)1743 1445 y Fx(con)n(tin)n(uous.)18 b(In)h(this)g(sense,)f(the)i (`most)e(singular)523 1544 y(part')24 b(of)f(the)i(Brjuno)e(function)i (is)f(stable)f(or)g(`univ)n(ersal',)g(roughly)g(sp)r(eaking)g(mo)r (dulo)523 1644 y(H\177)-42 b(older-)804 1611 y Fw(1)p 803 1625 V 803 1672 a(2)871 1644 y Fx(con)n(tin)n(uous)24 b(con)n(tributions.)g(As)h(noticed)f(at)h(the)g(end)g(of)g(Section)g(5) f(ab)r(o)n(v)n(e,)523 1754 y(w)n(e)i(can)f(use)h(either)g Fk(B)1237 1724 y Fi(e)1296 1754 y Fl(\021)d Fk(B)1451 1724 y Fw(\(1)p Fi(=)p Fw(2\))1633 1754 y Fx(or)j Fk(B)h Fl(\021)22 b Fk(B)1978 1724 y Fw(\(1\))2094 1754 y Fx(since)k(a)f (similar)h(argumen)n(t)f(starting)523 1854 y(from)i(\(26a{c\))f(sho)n (ws)h(that)h(their)f(di\013erence)h(is)f(also)g(H\177)-42 b(older-)2534 1821 y Fw(1)p 2533 1835 V 2533 1883 a(2)2604 1854 y Fx(con)n(tin)n(uous)26 b([18].)648 1954 y(This)35 b(pro)n(vides)e(a)i(frame)g(to)g(understand)g(the)g(prop)r(erties)f(of) i(the)f(critical)g(con-)523 2053 y(stan)n(ts)40 b Fk(K)45 b Fx(of)40 b(the)g(Sections)g(2)g(and)g(3)f(ab)r(o)n(v)n(e.)g(W)-7 b(e)40 b(assume)g(that)g(the)g(singularit)n(y)523 2153 y(comes)30 b(from)h(the)g(renormalisation)d(equation)j(\(1)20 b Fl(\000)g Fk(T)2294 2123 y Fi(e)2329 2153 y Fx(\))p Fk(K)34 b Fx(=)28 b Fk(`)20 b Fx(+)g Fk(f)9 b Fx(,)31 b(and)f(not)h(from)523 2253 y(additional)39 b(singular)g(b)r(eha)n (viour)g(coming)g(from)h Fk(f)49 b Fx(in)40 b(the)g(righ)n(t-hand)f (side.)h(If)g(it)523 2352 y(w)n(ould)34 b(exist,)g(suc)n(h)g(an)f (additional)h(singular)f(b)r(eha)n(viour)g(w)n(ould)g(require)g(a)h (further)523 2452 y(ph)n(ysical)23 b(in)n(terpretation.)g(This)h (argumen)n(t,)f(whic)n(h)h(is)g(usual)g(in)g(the)g(renormalisation)523 2551 y(analysis)38 b(of)g(singularities,)g(w)n(as)g(one)g(of)h(the)g (motiv)-5 b(ation)39 b(for)f(our)h(previous)e(w)n(ork)523 2651 y([18].)f(The)h(renormalisation)d(equation)i(allo)n(ws)g (naturally)g(to)g(conjecture)g(that)i(the)523 2751 y(di\013erence)27 b(b)r(et)n(w)n(een)g(the)h(Brjuno)e(function)i Fk(B)2026 2721 y Fi(e)2062 2751 y Fx(,)f(and)g(the)h(logarithm)e(of)h(the)g(v)-5 b(arious)523 2850 y(critical)20 b(constan)n(ts)g(\(m)n(ultiplied)i(b)n (y)f(a)g(suitable)g(co)r(e\016cien)n(t\),)g(is)g(con)n(tin)n(uous)f (and)g(ev)n(en)523 2950 y(H\177)-42 b(older-)804 2917 y Fw(1)p 803 2931 V 803 2979 a(2)869 2950 y Fx(con)n(tin)n(uous.)22 b(As)h(an)f(example,)g(w)n(e)h(conjecture)f(that)h(the)g(ratio)f(of)h Fk(e)3022 2920 y Fj(\000)p Fi(B)3153 2950 y Fx(and)523 3050 y(the)i(radius)f(of)h(the)g(Siegel)f(disk)g(of)h(the)g(quadratic)f (p)r(olynomial,)g(is)g(an)h(H\177)-42 b(older-)3068 3017 y Fw(1)p 3067 3031 V 3067 3078 a(2)3135 3050 y Fx(con-)523 3149 y(tin)n(uous)26 b(function)g(of)g(the)h(rotation)e(n)n(um)n(b)r (er.)h(These)g(conjectures)f(are)g(in)h(agreemen)n(t)523 3249 y(with)i(the)g(n)n(umerical)f(results)g([16,10].)523 3503 y Fm(8)150 b(The)37 b(Complexi\014cation)e(of)j(the)f(Brjuno)h(F) -9 b(unction)523 3691 y Fx(W)i(e)32 b(consider)e(here)h(the)h(case)f Fk(\013)f Fx(=)f(1,)i(and)g(w)n(e)g(w)n(an)n(t)g(to)g(asso)r(ciate)f (to)h(the)h(function)523 3791 y Fk(f)50 b Fx(in)42 b Fk(X)794 3803 y Fw(1)p Fi(;)p Fw(2)884 3791 y Fx(,)f(a)g(function)h Fk(\010)p Fx(,)g(holomorphic)e(in)i(the)f(upp)r(er)h(half)f(plane,)h (suc)n(h)f(that)523 3890 y(Im)14 b Fk(\010)40 b Fl(!)g Fk(B)917 3902 y Fi(f)998 3890 y Fx(when)d(z)h(go)r(es)f(to)g(the)h (real)f(axis.)g(Since)g(Re)14 b Fk(\010)38 b Fx(is)g(asso)r(ciated)e (to)i(the)523 3990 y(harmonic)c(conjugate)h(of)h Fk(f)9 b Fx(,)35 b(w)n(e)g(exp)r(ect)g(to)h(\014nd)g(b)r(etter)f(b)r (oundedness)h(prop)r(erties)523 4090 y(when)h Fk(f)44 b Fx(has)36 b(the)h(BMO)f(prop)r(ert)n(y)-7 b(.)35 b(W)-7 b(e)37 b(will)f(here)g(describ)r(e)g(our)g(pro)r(cedure,)f(and)523 4189 y(rep)r(ort)27 b(the)h(results)f([27].)648 4289 y(W)-7 b(e)26 b(asso)r(ciate)f(to)h Fk(f)34 b Fx(a)26 b(function)h Fk(F)12 b Fx(\()p Fk(z)t Fx(\))26 b(holomorphic)e(in)p 2495 4289 4 52 v 27 w(C)p Fl(n)p Fx([0)p Fk(;)14 b Fx(1],)24 b(and)i(v)-5 b(anishing)523 4388 y(at)28 b(in\014nit)n(y)-7 b(,)28 b(as)f(follo)n(ws)1471 4543 y Fk(F)12 b Fx(\()p Fk(z)t Fx(\))23 b(=)1775 4487 y Fk(i)p 1764 4524 51 4 v 1764 4600 a(\031)1838 4430 y Fn(Z)1921 4450 y Fw(1)1884 4618 y Fi(o)1997 4487 y Fk(f)9 b Fx(\()p Fk(x)p Fx(\))p 1982 4524 192 4 v 1982 4600 a Fk(x)19 b Fl(\000)f Fk(z)2197 4543 y(dx)28 b(:)801 b Fx(\(50\))523 4730 y(F)-7 b(or)33 b(x)h(real,)f(w)n(e)g(ha)n(v)n(e)g(Im)14 b Fk(F)e Fx(\()p Fk(x)23 b Fl(\006)f Fk(i")p Fx(\))33 b(=)g Fl(\006)p Fk(f)9 b Fx(\()p Fk(x)p Fx(\))34 b(for)f Fk(x)h Fl(2)g Fx([0)p Fk(;)14 b Fx(1],)33 b(and)g(Im)14 b Fk(F)e Fx(\()p Fk(x)p Fx(\))34 b(=)f(0)523 4830 y(for)d Fk(x)e Fl(62)g Fx([0)p Fk(;)14 b Fx(1].)29 b(W)-7 b(e)31 b(will)g(b)r(e)f (particularly)f(in)n(terested)h(in)h(the)f(case)g Fk(f)9 b Fx(\()p Fk(z)t Fx(\))27 b(=)g(ln\()p Fk(z)t Fx(\),)k(in)p eop %%Page: 16 16 16 15 bop 523 232 a Fu(16)199 b(Pierre)27 b(Moussa)g(and)e(Stefano)i (Marmi)523 448 y Fx(whic)n(h)c(case)f(w)n(e)g(get)g Fk(F)12 b Fx(\()p Fk(z)t Fx(\))23 b(=)g Fl(\000)p Fk(\031)1575 418 y Fj(\000)p Fw(1)1664 448 y Fx(Li)1739 460 y Fw(2)1776 448 y Fx(\(1)p Fk(=z)t Fx(\),)f(where)g(Li)2322 460 y Fw(2)2382 448 y Fx(is)h(the)g(classical)e(dilogaritm)523 548 y(function)28 b([28].)f(No)n(w)g(w)n(e)g(set)1412 811 y Fk(\010)p Fx(\()p Fk(z)t Fx(\))c(=)61 b(lim)1685 865 y Fi(N)6 b Fj(!1)1895 708 y Fw(+)p Fi(N)1890 733 y Fn(X)1895 911 y Fj(\000)p Fi(N)2024 811 y Fk(F)12 b Fx(\()p Fk(z)22 b Fx(+)c Fk(n)p Fx(\))28 b Fk(;)741 b Fx(\(51\))523 1074 y(and)32 b(w)n(e)f(get)h(a)f(function)i Fk(\010)f Fx(holomorphic)f(in)h(the)g(upp)r(er-half)g(plane)f(I)-14 b(H)2886 1086 y Fw(+)2941 1074 y Fx(,)32 b(p)r(erio)r(dic)523 1174 y(with)20 b(a)f(real)f(p)r(erio)r(d)h(equal)g(to)g(one,)g(and)g (suc)n(h)g(that)h(for)f(x)g(real,)f(Im)c Fk(F)e Fx(\()p Fk(x)r Fl(\006)r Fk(i")p Fx(\))23 b(=)g Fl(\006)p Fk(f)9 b Fx(\()p Fk(x)p Fx(\).)523 1274 y(In)38 b(fact)g(the)g(previous)e (equation)h(de\014nes)h(a)f(pair)g(on)h(functions)g Fk(\010)2745 1286 y Fj(\006)2801 1274 y Fx(,)g(resp)r(ectiv)n(ely)523 1373 y(holomorphic)26 b(in)i(the)g(upp)r(er)g(or)f(lo)n(w)n(er)f(half)i (plane)f(I)-14 b(H)2259 1385 y Fj(\006)2315 1373 y Fx(,)27 b(so)g(that)h(the)g(natural)f(frame)523 1473 y(in)k(whic)n(h)f(our)g (pro)r(cedure)g(tak)n(es)f(place)i(is)f(the)h(frame)f(of)h(complex)f(h) n(yp)r(erfunctions,)523 1573 y(whic)n(h)e(w)n(e)f(will)h(not)f (consider)g(here)g([27].)648 1672 y(W)-7 b(e)30 b(consider)f(no)n(w)g (the)h(action)g(of)f Fk(T)12 b Fx(,)29 b(with)i(\()p Fk(T)12 b(f)d Fx(\)\()p Fk(x)p Fx(\))27 b(=)f Fk(xf)9 b Fx(\(1)p Fk(=x)p Fx(\))30 b(if)h(0)26 b Fl(\024)h Fk(x)g(<)f Fx(1,)523 1772 y Fk(f)34 b Fx(and)26 b Fk(T)12 b(f)33 b Fx(b)r(eing)26 b(complemen)n(ted)f(using)h(p)r(erio)r(dicit)n(y)-7 b(.)25 b(Using)h(the)g(ab)r(o)n(v)n(e)e(corresp)r(on-)523 1871 y(dence)30 b Fk(f)35 b Fl(7!)27 b Fk(T)12 b(f)d Fx(,)28 b(a)h(corresp)r(ondence)f Fk(F)39 b Fl(7!)26 b Fk(T)12 b(F)41 b Fx(is)30 b(induced)g(on)f(holomorphic)g(func-)523 1971 y(tions)e(in)p 847 1971 4 52 v 28 w(C)p Fl(n)p Fx([0)p Fk(;)14 b Fx(1],)27 b(v)-5 b(anishing)27 b(at)g(in\014nit)n(y)-7 b(.)28 b(W)-7 b(e)28 b(get)725 2220 y(\()p Fk(T)12 b(F)g Fx(\)\()p Fk(z)t Fx(\))23 b(=)f Fl(\000)p Fk(z)1291 2116 y Fj(1)1264 2141 y Fn(X)1253 2317 y Fi(m)p Fw(=1)1410 2103 y Fn(\022)1471 2220 y Fk(F)1550 2103 y Fn(\022)1621 2164 y Fx(1)p 1621 2201 43 4 v 1621 2277 a Fk(z)1691 2220 y Fl(\000)c Fk(m)1847 2103 y Fn(\023)1927 2220 y Fl(\000)g Fk(F)12 b Fx(\()p Fl(\000)p Fk(m)p Fx(\))2277 2103 y Fn(\023)2356 2220 y Fx(+)2478 2116 y Fj(1)2451 2141 y Fn(X)2439 2317 y Fi(m)p Fw(=1)2596 2220 y Fk(F)2661 2186 y Fj(0)2684 2220 y Fx(\()p Fl(\000)p Fk(m)p Fx(\))28 b Fk(:)202 b Fx(\(52\))523 2487 y(In)39 b(fact,)g Fk(T)12 b(F)49 b Fx(is)39 b(essen)n(tially)e Fl(\000)p Fk(z)1623 2424 y Fn(P)1711 2445 y Fj(1)1711 2511 y Fi(m)p Fw(=1)1872 2487 y Fk(F)1950 2419 y Fn(\000)1988 2487 y Fk(z)2031 2456 y Fj(\000)p Fw(1)2138 2487 y Fl(\000)18 b Fk(m)2294 2419 y Fn(\001)2332 2487 y Fx(,)39 b(up)g(to)f(an)h(a\016ne)f(additiv)n (e)523 2586 y(correction,)24 b(whic)n(h)i(could)f(b)r(e)h(determined)g (b)n(y)g(the)g(v)-5 b(anishing)25 b(condition)h(at)f(in\014nit)n(y)-7 b(.)648 2686 y(If)24 b Fk(f)34 b Fx(is)24 b(asso)r(ciated)f(to)h Fk(F)37 b Fx(as)24 b(ab)r(o)n(v)n(e,)f(the)i(solution)f Fk(B)2330 2698 y Fi(f)2373 2686 y Fx(\()p Fk(x)p Fx(\))h(\(for)f Fk(\013)g Fx(=)f(1\))h(of)g(\(25a{c\))523 2785 y(is)k(asso)r(ciated)e (to)h(the)h(series)1417 3013 y Fl(B)1472 3025 y Fi(f)1515 3013 y Fx(\()p Fk(z)t Fx(\))23 b(=)1733 2935 y Fn(X)1767 3113 y Fb(Z)-25 b(Z)1905 2910 y Fj(1)1878 2935 y Fn(X)1866 3110 y Fi(m)p Fw(=0)2009 3013 y Fx(\()p Fk(T)2102 2979 y Fi(n)2147 3013 y Fk(F)12 b Fx(\()p Fk(z)t Fx(\)\))18 b Fk(;)747 b Fx(\(53\))523 3284 y(where)38 b(w)n(e)h(use)f(the)h (notation)1559 3222 y Fn(P)1647 3309 y Fb(Z)-25 b(Z)1716 3284 y Fk(F)50 b Fx(for)38 b Fk(\006)2027 3248 y Fw(+)p Fj(1)2022 3305 y Fi(n)p Fw(=)p Fj(\0001)2236 3284 y Fk(F)12 b Fx(\()p Fk(z)30 b Fx(+)25 b Fk(n)p Fx(\))39 b(understo)r(o)r(d)f(as)h (the)523 3384 y(symmetric)27 b(summation)h(\(51\))f(to)g(insure)h(con)n (v)n(ergence.)648 3483 y(It)35 b(is)g(no)n(w)g(in)n(teresting)f(to)h (displa)n(y)g(the)g(link)h(b)r(et)n(w)n(een)f(\(53\))g(and)g(the)g(mo)r (dular)523 3633 y(group)j Fk(GL)p Fx(\(2)p Fk(;)14 b Fa(Z)-33 b(Z)n Fx(\).)39 b(Let)g Fk(g)44 b Fx(=)1514 3516 y Fn(\022)1588 3582 y Fk(a)29 b(b)1592 3682 y(c)g(d)1712 3516 y Fn(\023)1815 3633 y Fl(2)42 b Fk(GL)p Fx(\(2)p Fk(;)14 b Fa(Z)-33 b(Z)o Fx(\),)39 b(whic)n(h)f(mean)h Fk(a;)14 b(b;)g(c;)g(d)41 b Fl(2)g Fa(Z)-33 b(Z)p Fx(,)523 3777 y Fk(")562 3789 y Fi(g)623 3777 y Fx(=)23 b Fk(ad)13 b Fl(\000)g Fk(bc)22 b Fx(=)h Fl(\006)p Fx(1.)h(T)-7 b(o)24 b Fk(g)k Fx(w)n(e)c(asso)r(ciate)f(the)i(follo)n(wing)f(group)g (action)g(on)h(functions)523 3877 y(holomorphic)h(on)p 1138 3877 4 52 v 28 w(C)p Fl(n)p Fx([0)p Fk(;)14 b Fx(1],)26 b(that)i(is)f Fk(g)f Fl(7!)d Fk(L)1924 3889 y Fi(g)1962 3877 y Fk(F)12 b Fx(,)28 b(with)651 4045 y Fn(\000)689 4112 y Fk(L)746 4124 y Fi(g)784 4112 y Fk(F)12 b Fx(\)\()p Fk(z)t Fx(\))24 b(=)e(\()p Fk(a)d Fl(\000)f Fk(cz)t Fx(\))1402 3995 y Fn(\032)1463 4112 y Fk(F)1542 3995 y Fn(\022)1614 4056 y Fk(dz)k Fl(\000)c Fk(b)p 1613 4093 224 4 v 1613 4169 a(a)h Fl(\000)f Fk(cz)1847 3995 y Fn(\023)1926 4112 y Fl(\000)g Fk(F)2088 3995 y Fn(\022)2149 4112 y Fl(\000)2224 4056 y Fk(d)p 2224 4093 44 4 v 2228 4169 a(c)2277 3995 y Fn(\023\033)2419 4112 y Fl(\000)2512 4056 y Fk(")2551 4068 y Fi(g)p 2512 4093 78 4 v 2532 4169 a Fk(c)2599 4112 y(F)2664 4078 y Fj(0)2701 3995 y Fn(\022)2762 4112 y Fl(\000)2837 4056 y Fk(d)p 2837 4093 44 4 v 2841 4169 a(c)2890 3995 y Fn(\023)2988 4112 y Fk(:)128 b Fx(\(54\))523 4344 y(Let)38 b Fl(M)782 4356 y Fw(+)878 4344 y Fl(\032)i Fk(GL)p Fx(\(2)p Fk(;)14 b Fa(Z)-33 b(Z)o Fx(\))38 b(b)r(e)h(the)f(m)n (ultiplicativ)n(e)g(monoid)g(generated)f(b)n(y)h(the)g(unit)523 4494 y(matrix,)g(and)g(the)h(set)g(of)f(matrices)1743 4377 y Fn(\022)1817 4443 y Fx(0)i(1)1817 4543 y(1)24 b Fk(m)1968 4377 y Fn(\023)2029 4494 y Fx(,)39 b(for)f Fk(m)j Fl(\025)g Fx(1)d(in)n(teger.)g(The)g(monoid)523 4643 y Fl(M)623 4655 y Fw(+)701 4643 y Fl(\032)23 b Fk(GL)p Fx(\(2)p Fk(;)14 b Fa(Z)-33 b(Z)n Fx(\))26 b(can)g(also)e(b)r(e)i (de\014ned)g(as)g(the)g(set)f(of)h(matrices)f(including)h(iden)n(tit)n (y)523 4793 y(and)35 b(the)h(matrices)1183 4675 y Fn(\022)1257 4742 y Fk(a)28 b(b)1261 4841 y(c)g(d)1381 4675 y Fn(\023)1478 4793 y Fl(2)36 b Fk(GL)p Fx(\(2)p Fk(;)14 b Fa(Z)-33 b(Z)o Fx(\))35 b(suc)n(h)g(that)h(\014rst,)f Fk(d)h Fl(\025)f Fk(c)h Fl(\025)f Fk(a)h Fl(\025)f Fx(0,)g(and)p eop %%Page: 17 17 17 16 bop 1836 232 a Fu(Real)26 b(or)g(Complex)f(Brjuno)i(F)-6 b(unctions)198 b(17)523 448 y Fx(second,)24 b Fk(d)f Fl(\025)g Fk(b)g Fl(\025)f Fk(a)h Fl(\025)g Fx(0.)h(In)h Fk(GL)p Fx(\(2)p Fk(;)14 b Fa(Z)-33 b(Z)n Fx(\))25 b(there)g(is)f(a)g (unique)h(pro)r(duct)f(decomp)r(osition,)523 548 y(namely)32 b Fl(8)p Fk(g)h Fl(2)f Fk(GL)p Fx(\(2)p Fk(;)14 b Fa(Z)-33 b(Z)o Fx(\))33 b(there)f(exist)h(a)f(unique)h(set)f(of)h(three)f (matrices)g Fk(k)s Fx(,)h Fk(m)f Fx(and)523 648 y Fk(h)p Fx(,)c(with)g Fk(g)e Fx(=)d Fk(k)s(mh)p Fx(,)28 b(and)g Fk(m)23 b Fl(2)h(M)1620 617 y Fw(+)1675 648 y Fx(,)k Fk(k)e Fl(2)e Fk(Z)6 b Fx(,)27 b(where)h(Z)f(is)h(the)g(translation)f (subgroup)523 792 y(of)k(matrices)957 675 y Fn(\022)1031 742 y Fx(1)24 b Fk(n)1031 841 y Fx(0)k(1)1159 675 y Fn(\023)1221 792 y Fx(,)j Fk(n)d Fl(2)h Fa(Z)-33 b(Z)p Fx(,)31 b(and)g Fk(h)d Fl(2)h Fk(H)7 b Fx(,)31 b(where)g Fk(h)g Fx(is)f(the)i(order)e (eigh)n(t)g(sugroup)523 1000 y(of)k Fk(GL)p Fx(\(2)p Fk(;)14 b Fa(Z)-33 b(Z)o Fx(\))35 b(made)f(of)g(the)h(matrices)1811 883 y Fn(\022)1885 949 y Fk(")i Fx(0)1884 1049 y(0)24 b Fk(")1989 1019 y Fj(0)2025 883 y Fn(\023)2086 1000 y Fx(,)35 b(and)2312 883 y Fn(\022)2396 949 y Fx(0)h Fk(")2385 1049 y(")2424 1019 y Fj(0)2472 1049 y Fx(0)2526 883 y Fn(\023)2587 1000 y Fx(,)f(with)g Fk(")f Fx(=)g Fl(\006)p Fx(1)f(and)523 1150 y Fk(")562 1119 y Fj(0)608 1150 y Fx(=)23 b Fl(\006)p Fx(1.)648 1249 y(No)n(w)k(\(53\))g(is)g (rewritten)h(as)1317 1438 y Fl(B)1372 1450 y Fi(f)1415 1438 y Fx(\()p Fk(z)t Fx(\))22 b(=)1637 1359 y Fn(X)1632 1538 y Fi(k)q Fj(2)p Fi(Z)1819 1359 y Fn(X)1777 1542 y Fi(g)r Fj(2M)1934 1525 y Fe(+)1995 1371 y Fn(\000)2033 1438 y Fk(L)2090 1453 y Fw(\()p Fi(hg)r Fw(\))2218 1438 y Fk(F)2283 1371 y Fn(\001)2335 1438 y Fx(\()p Fk(z)t Fx(\))28 b Fk(:)646 b Fx(\(55\))523 1700 y(The)20 b(double)g(sum)h(o)n (v)n(er)d Fk(g)23 b Fx(and)d Fk(k)j Fx(amoun)n(ts)c(to)h(a)g(sum)g(o)n (v)n(er)f(a)g(part)h(of)g(the)h(full)f(mo)r(dular)523 1800 y(group)30 b(\(here)h(one)f(o)n(v)n(er)g(eigh)n(t\).)h(The)g(con)n (tribution)f(o)n(v)n(er)f(the)j(sev)n(en)e(other)g(p)r(ossible)523 1899 y(parts)d(w)n(ould)g(b)r(e)h Fl(\000B)1211 1911 y Fi(f)1253 1899 y Fx(\()p Fk(z)t Fx(\),)g Fl(\006B)1531 1911 y Fi(f)1573 1899 y Fx(\()p Fk(z)1648 1869 y Fj(\000)p Fw(1)1737 1899 y Fx(\),)g Fl(\006B)1940 1911 y Fi(f)1982 1899 y Fx(\()p Fl(\000)p Fk(z)t Fx(\),)f(and)g Fl(B)2420 1911 y Fi(f)2463 1899 y Fx(\()p Fl(\000)p Fk(z)2603 1869 y Fj(\000)p Fw(1)2691 1899 y Fx(\).)648 1999 y(W)-7 b(e)28 b(will)f(no)n(w)g(summarize)g(the)h(results:)523 2098 y(i\))35 b(The)f(sums)h(in)g(\(55\))f(con)n(v)n(erge)e(in)j(the)f(op)r (en)h(upp)r(er)g(half)f(plane)g(as)g(long)g(as)g Fk(f)43 b Fx(is)523 2198 y(in)31 b Fk(L)680 2210 y Fw(1)717 2198 y Fx(\(0)p Fk(;)14 b Fx(1\))30 b(whic)n(h)h(insures)f(that)h Fk(F)43 b Fx(is)30 b(holomorphic)g(in)p 2422 2198 4 52 v 31 w(C)p Fl(n)p Fx([0)p Fk(;)14 b Fx(1],)29 b(and)i(v)-5 b(anishes)30 b(at)523 2298 y(in\014nit)n(y)-7 b(.)523 2397 y(ii\))28 b(When)g Fk(f)37 b Fx(is)27 b(in)h Fk(L)1186 2409 y Fi(p)1224 2397 y Fx(\(0)p Fk(;)14 b Fx(1\),)27 b Fk(p)g Fx(\014nite,)i(then)f Fl(B)2008 2409 y Fi(f)2078 2397 y Fx(is)f(in)h(the)g(Hardy)f(I)-14 b(H)2734 2409 y Fi(p)2800 2397 y Fx(space.)523 2497 y(iii\))28 b(If)g Fk(f)36 b Fx(is)28 b(suc)n(h)f(that)h Fk(F)40 b Fx(has)27 b(b)r(ounded)h(real)e(part,)i(then)g(the)g(same)f(holds)g(for)g Fl(B)3147 2509 y Fi(f)3190 2497 y Fx(.)523 2597 y(iv\))37 b(F)-7 b(or)36 b Fk(f)9 b Fx(\()p Fk(x)p Fx(\))38 b(=)f(ln)q(\()p Fk(x)p Fx(\),)g(and)f Fk(F)12 b Fx(\()p Fk(z)t Fx(\))38 b(=)f Fl(\000)p Fk(\031)1956 2566 y Fj(\000)p Fw(1)2045 2597 y Fx(Li)2120 2609 y Fw(2)2158 2597 y Fx(\(1)p Fk(=z)t Fx(\),)e(w)n(e)h(get)g(the)h(complexi\014ed)523 2696 y(Brjuno)27 b(function,)i Fl(B)s Fx(,)d(holomorphic)h(in)h(the)g(upp)r (er)g(half)f(plane,)h(v)-5 b(anishing)27 b(at)h(+)p Fk(i)p Fl(1)p Fx(.)523 2796 y(When)d Fk(z)i Fx(go)r(es)d(to)g(a)g(real)f(n)n (um)n(b)r(er)h(in)h(a)e(non-tangen)n(tial)g(w)n(a)n(y)-7 b(,)23 b(w)n(e)h(ha)n(v)n(e)f(the)i(follo)n(wing)523 2895 y(limits)e(when)g Fk(")f(>)h Fx(0)f(go)r(es)f(to)i(zero)14 b(:)21 b(the)i(real)e(part)h(Re)14 b Fl(B)s Fx(\()p Fk(x)8 b Fx(+)g Fk(i")p Fx(\))21 b(has)h(a)g(b)r(ounded)h(limit)523 2995 y(for)i(an)n(y)f(real)g Fk(x)p Fx(.)i(This)f(limit)h(is)f(con)n (tin)n(uous)f(at)h(all)g(irrationals)e(and)i(has)g(a)g(decreasing)523 3095 y(jump)j(of)g Fk(\031)s(=q)i Fx(at)e(eac)n(h)f(rational)f Fk(p=q)s Fx(.)h(When)h Fk(x)g Fx(is)g(a)f(Brjuno)g(n)n(um)n(b)r(er,)g (Im)14 b Fl(B)s Fx(\()p Fk(x)k Fx(+)g Fk(i")p Fx(\))523 3194 y(go)r(es)27 b(to)g(the)h(Brjuno)f(function)h Fk(B)t Fx(\()p Fk(x)p Fx(\).)648 3294 y(The)g(limit)i(prop)r(erties)e(of)g (the)i(complex)e(Brjuno)g(function)h(on)g(the)g(real)f(axis)g(are)523 3394 y(c)n(haracteristic)d(of)h(functions)h Fk(f)35 b Fx(ha)n(ving)26 b(a)g(logarithmic)g(singularit)n(y)f(aroud)g(zero.)h (Al-)523 3493 y(though)31 b(the)g(b)r(oundedness)g(of)f(the)i(real)e (part)g(reminds)g(the)i(BMO)e(prop)r(ert)n(y)g(of)h(the)523 3593 y(real)h(Brjuno)g(function,)h(it)g(is)g(in)f(fact)h(a)f(stronger)f (prop)r(ert)n(y)-7 b(.)32 b(This)g(is)h(one)f(more)g(re-)523 3692 y(mark)-5 b(able)30 b(feature)h(of)g(this)g(function.)h(W)-7 b(e)31 b(are)f(con)n(vinced)g(that)i(the)f(in)n(terpretation)523 3792 y(of)36 b(the)g(prop)r(erties)f(of)h(the)g(Brjuno)g(function)g(in) g(terms)g(of)g(the)g(mo)r(dular)f(group)g(is)523 3892 y(promising.)29 b(On)h(the)g(other)g(hand,)g(w)n(e)g(can)f(hop)r(e)h (to)g(\014nd)h(an)e(in)n(terpretation)h(of)g(the)523 3991 y(complexi\014ed)f(v)n(ersion)e(of)i(rotation)f(n)n(um)n(b)r(ers)g (analogous)f(with)i(the)g(usual)g(in)n(terpre-)523 4091 y(tation)38 b(of)g(complex)g(frequencies)g(in)h(terms)f(of)g(damp)r(ed) g(oscillations,)g(but)g(this)h(is)523 4191 y(another)27 b(story)-7 b(.)523 4431 y Fg(Ac)m(kno)m(wledgemen)m(ts.)35 b Fx(The)21 b(\014rst)h(author)f(thanks)h(the)g(CNRS)g(and)g(the)g (organizors)523 4531 y(of)31 b(the)775 4510 y(\023)768 4531 y(Ecole)f(Thematique)h(held)g(at)g(La)f(Chap)r(elle)i(des)f(Bois,) f(with)h(sp)r(ecial)g(thanks)523 4631 y(for)22 b(Mic)n(hel)h(Planat.)f (This)h(w)n(ork)e(b)r(egun)i(during)f(a)h(visit)g(of)f(the)i(second)e (author)g(at)h(the)523 4730 y(S.Ph.T.{CEA)30 b(Sacla)n(y)g(and)h(at)h (the)f(Dept.)h(of)f(Mathematics)g(of)h(Orsa)n(y)d(during)i(the)523 4830 y(academic)36 b(y)n(ear)f(1993{1994.)d(This)37 b(researc)n(h)d (has)i(b)r(een)h(supp)r(orted)f(b)n(y)h(the)f(CNR,)p eop %%Page: 18 18 18 17 bop 523 232 a Fu(18)199 b(Pierre)27 b(Moussa)g(and)e(Stefano)i (Marmi)523 448 y Fx(CNRS,)39 b(INFN,)f(MURST)h(and)f(a)f(EEC)g(gran)n (t.)g(W)-7 b(e)39 b(thank)f(J.-C.)f(Y)-7 b(o)r(ccoz)37 b(for)h(his)523 548 y(constan)n(t)29 b(help)h(and)f(w)n(arm)f (encouragemen)n(ts,)g(and)h(also)g(for)g(allo)n(wing)f(us)h(to)h(rep)r (ort)523 648 y(here)d(results)g(obtained)h(in)f(collab)r(oration)f (with)i(him.)523 910 y Fm(References)523 1098 y Fu(1.)42 b(Chirik)n(o)n(v,)50 b(B.)26 b(\(1979\))g(A)f(univ)n(ersal)g (instabilit)n(y)g(of)h(man)n(y-)d(dimensional)i(oscillator)i(sys-)624 1190 y(tems.)f(Ph)n(ys.)f(Rep)r(orts,)h Ft(52)p Fu(,)g(263{279)523 1280 y(2.)42 b(Escande,)25 b(D.)g(\(1985\))g(Sto)r(c)n(hasticit)n(y)g (in)f(classical)j(Hamiltonian)d(systems:)g(univ)n(ersal)h(as-)624 1372 y(p)r(ects.)h(Ph)n(ys.)g(Rep)r(orts,)g Ft(121)p Fu(,)h(165{261)523 1462 y(3.)42 b(Greene,)23 b(J.)f(M.)g(\(1979\))h(A)e (metho)r(d)f(for)j(determining)d(a)i(sto)r(c)n(hastic)h(transition.)g (J.)f(Math.)624 1554 y(Ph)n(ys.)k Ft(20)p Fu(,)g(1183{1201)523 1644 y(4.)42 b(Aubry)25 b(S.)h(and)g(Le)h(Daeron,)g(P)-6 b(.)26 b(\(1983\))i(The)e(discrete)h(F)-6 b(renk)n(el{Kon)n(toro)n(v)l (a)26 b(mo)r(del)g(and)624 1736 y(its)g(extensions.)h(Ph)n(ysica)f Ft(8D)p Fu(,)g(381{422)523 1826 y(5.)42 b(Mather)29 b(J.)h(N.)e (\(1984\))i(Non)e(existence)h(of)h(in)n(v)l(arian)n(t)e(circles.)i (Ergo)r(d.)g(Theor.)g(and)e(Dy-)624 1918 y(nam.)d(Sys.)h Ft(4)p Fu(,)g(301{309)523 2008 y(6.)42 b(Y)-6 b(o)r(ccoz)25 b(J.-C.)g(\(1992\))g(An)f(in)n(tro)r(duction)g(to)g(small)h(divisors)f (problems,)g(in:)h(F)-6 b(rom)23 b(Num-)624 2099 y(b)r(er)29 b(Theory)f(to)h(Ph)n(ysics,)g(W)-6 b(aldsc)n(hmidt)28 b(M.,)h(Moussa)h(P)-6 b(.,)29 b(Luc)n(k)f(J.-M.,)h(and)g(Itzykson)624 2191 y(C.)e(editors,)f(Springer-V)-6 b(erlag,)26 b(Berlin,)h(pp.)e (659{679)523 2281 y(7.)42 b(Berretti)22 b(A.)f(and)f(Gen)n(tile)i(G.)f (\(1998\))h(Scaling)g(prop)r(erties)g(of)f(the)g(radius)g(of)h(con)n(v) n(ergence)624 2373 y(of)27 b(the)e(Lindstedt)g(series)i(:)f(the)f (standard)h(map.)f(Univ)n(ersit)n(y)f(of)j(Roma,)e(Italy)-6 b(,)26 b(preprin)n(t)523 2463 y(8.)42 b(Berretti)26 b(A.)e(and)h(Gen)n (tile)g(G.)h(\(1998\))g(Bryuno)e(function)h(and)f(the)h(standard)g (map.)f(Uni-)624 2555 y(v)n(ersit)n(y)h(of)i(Roma,)e(Italy)-6 b(,)25 b(preprin)n(t)523 2645 y(9.)42 b(Y)-6 b(o)r(ccoz)18 b(J.-C.)g(\(1995\))h(Th)n(\023)-36 b(eor)n(\022)g(eme)17 b(de)g(Siegel,)h(nom)n(bres)e(de)h(Bruno)h(et)f(p)r(olyn^)-38 b(omes)17 b(quadra-)624 2737 y(tiques.)26 b(Ast)n(\023)-36 b(erisque,)26 b Ft(231)p Fu(,)g(3{88,)i(\(app)r(eared)e(\014rst)f(as)h (a)g(preprin)n(t)f(in)h(1987\).)523 2827 y(10.)43 b(Marmi)27 b(S.)h(and)f(Stark)g(J.)i(\(1992\))g(On)e(the)g(standard)g(map)g (critical)i(function.)f(Nonlin-)624 2919 y(earit)n(y)e Ft(5)p Fu(,)g(743{761)523 3009 y(11.)43 b(Carletti)25 b(T.)f(and)f(Lask)l(ar)h(J.)g(\(1999\))h(Scaling)g(la)n(w)f(in)g(the)f (standard)h(map)e(critical)j(func-)624 3100 y(tion,)34 b(in)n(terp)r(olating)g(hamiltonian)f(and)g(frequency)f(analysis.)j (\(Preprin)n(t,)e(Bureau)h(des)624 3192 y(Longitudes,)27 b(P)n(aris,)g(in)e(preparation\))523 3282 y(12.)43 b(T)-6 b(reshev)31 b(D.)g(and)g(Zub)r(elevitc)n(h)g(O.)g(\(1998\))i(In)n(v)l (arian)n(t)d(tori)i(in)f(Hamiltonian)g(systems)624 3374 y(with)36 b(t)n(w)n(o)h(degrees)g(of)f(freedom)g(in)g(a)g(neigh)n(b)r (orho)r(o)r(d)h(of)f(a)h(resonance.)g(Regular)f(and)624 3465 y(Chaotic)27 b(dynamics,)e Ft(3)p Fu(,)h(73{81)523 3556 y(13.)43 b(Gelfreic)n(h)38 b(G.)g(V.)g(\(1999\))g(A)f(pro)r(of)i (of)f(exp)r(onen)n(tially)f(small)h(transv)n(ersalit)n(y)g(of)g(the)624 3647 y(separatrices)28 b(for)e(the)f(standard)h(map.)f(Comm)n(un.)f (Math.)i(Ph)n(ys.)g Ft(201)p Fu(,)g(155{216)523 3738 y(14.)43 b(Da)n(vie)31 b(A.)g(M.)h(\(1995\))h(Renormalisation)f(for)h (analytic)f(area)g(preserving)g(maps.)f(Uni-)624 3829 y(v)n(ersit)n(y)25 b(of)i(Edin)n(burgh)e(preprin)n(t)523 3919 y(15.)43 b(Da)n(vie)19 b(A.)g(M.)h(\(1994\))g(the)f(critical)i (function)e(for)h(the)f(semistandard)g(map.)g(Nonlinearit)n(y)624 4011 y Ft(7)p Fu(,)26 b(219{229)523 4101 y(16.)43 b(Marmi)19 b(S.)h(\(1990\))g(Critical)i(functions)e(for)g(complex)f(analytic)h (maps)f(function.)h(J.)g(Ph)n(ys.)624 4193 y(A:)26 b(Math.Gen.)g Ft(23)p Fu(,)h(3447{3474)523 4283 y(17.)43 b(P)n(erez-Marco)20 b(R.)f(\(1992\))h(Solution)f(compl)n(\022)-36 b(ete)19 b(du)f(probl)n(\022)-36 b(eme)18 b(de)h(Siegel)h(de)f(lin)n(\023)-36 b(earisation)624 4375 y(d'une)18 b(application)h(holomorphe)f(autour)g (d'un)f(p)r(oin)n(t)h(\014xe.)g(S)n(\023)-36 b(eminaire)18 b(Bourbaki)g(nr.753,)624 4466 y(Ast)n(\023)-36 b(erisque,)26 b Ft(206)p Fu(,)h(273{310)523 4557 y(18.)43 b(Marmi)27 b(S.,)g(Moussa)h(P)-6 b(.,)27 b(and)g(Y)-6 b(o)r(ccoz)27 b(J.-C.)h(\(1997\))h(The)e(Brjuno)g(function)g(and)g(their)624 4648 y(regularit)n(y)f(prop)r(erties.)h(Comm)n(un.)d(Math.)i(Ph)n(ys.)g Ft(186)p Fu(,)g(265-293)523 4739 y(19.)43 b(Buric)29 b(N.,)h(P)n(erciv)l(al)g(I.)f(C.,)i(and)d(Viv)l(aldi)i(F.)f(\(1990\))i (Critical)g(function)e(and)g(mo)r(dular)624 4830 y(smo)r(othing,)d (Nonlinearit)n(y)g Ft(3)p Fu(,)g(21{37)p eop %%Page: 19 19 19 18 bop 1836 232 a Fu(Real)26 b(or)g(Complex)f(Brjuno)i(F)-6 b(unctions)198 b(19)523 448 y(20.)43 b(MacKa)n(y)23 b(R.)f(S.)h (\(1988\))h(Exact)e(results)i(for)f(an)g(appro)n(ximate)f (renormalisation)i(sc)n(heme)624 540 y(and)k(some)g(predictions)h(for)g (the)f(breakup)f(of)i(in)n(v)l(arian)n(t)f(tori,)h(Ph)n(ysica)g Ft(33D)p Fu(,)g(240{265,)624 631 y(and)d(Erratum)e(\(1989\))j(Ph)n (ysica)f Ft(36D)p Fu(,)h(358{265)523 722 y(21.)43 b(Sc)n(h)n(w)n(eiger) 18 b(F.)h(\(1995\))h(Ergo)r(dic)f(theory)f(of)h(\014b)r(ered)f(systems) f(and)h(metric)g(n)n(um)n(b)r(er)e(thory)-6 b(,)624 814 y(Clarendon)27 b(Press,)g(Oxford,)523 905 y(22.)43 b(Moussa)35 b(P)-6 b(.,)35 b(Cassa)h(A.,)f(and)f(Marmi)g(S.)h(\(1999\))g(Con)n(tin) n(ued)f(fractions)i(and)e(Brjuno)624 996 y(functions,)27 b(J.)f(Comput.)f(Appl.)g(Math.)h Ft(105)h Fu(403{415)523 1088 y(23.)43 b(Brjuno.)27 b(A.)f(D.)g(\(1971\))h(Analytical)g(form)f (of)h(di\013eren)n(tial)f(equations,)h(T)-6 b(rans.)27 b(Mosco)n(w)624 1179 y(Math.)g(So)r(c.)f Ft(25)g Fu(131{288,)j(and,)c (\(1972\),)i Ft(26)f Fu(199{239)523 1270 y(24.)43 b(Hardy)25 b(G.)h(H.,)g(and)f(W)-6 b(righ)n(t)25 b(E.)h(M.)h(\(1938\))g(An)d(in)n (tro)r(duction)i(to)g(the)f(theory)h(of)g(n)n(um-)624 1362 y(b)r(ers,)g(Clarendon)h(Press,)g(Oxford,)f(c)n(hapter)f(11,)i (\014fth)e(edition)h(1979)523 1453 y(25.)43 b(Garnett)24 b(J.)i(B.)f(\(1981\))h(Bounded)e(Analytic)g(functions,)i(Academic)e (Press,)i(New)e(Y)-6 b(ork.)523 1544 y(26.)43 b(Marmi)33 b(S.,)h(Moussa)g(P)-6 b(.,)34 b(and)f(Y)-6 b(o)r(ccoz)34 b(J.-C.)g(\(1995\))g(D)n(\023)-36 b(ev)n(elopp)r(emen)n(ts)32 b(en)h(fractions)624 1636 y(con)n(tin)n(ues,)21 b(fonctions)h(de)f (Brjuno)g(et)g(espaces)g(BMO,)h(CEA/Sacla)n(y)-6 b(,)21 b(Note)g(CEA-N-2788)523 1727 y(27.)43 b(Marmi)c(S.,)g(Moussa)h(P)-6 b(.,)39 b(and)g(Y)-6 b(o)r(ccoz)39 b(J.-C.)h(\(1999\))g(Complex)f (Brjuno)g(functions,)624 1818 y(Preprin)n(t)26 b(SPhT/CEA)g(Sacla)n(y)g (T99/066,)j(71)d(p.)523 1910 y(28.)43 b(Lewin)22 b(L,)h(\(1981\))g(P)n (olylogarithms)h(and)d(Asso)r(ciated)j(F)-6 b(unctions,)22 b(Elsevier)h(North)f(Hol-)624 2001 y(land,)k(New)g(Y)-6 b(ork.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF