Content-Type: multipart/mixed; boundary="-------------0506010304395" This is a multi-part message in MIME format. ---------------0506010304395 Content-Type: text/plain; name="05-198.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-198.keywords" dynamical zeta function, periodic reflecting rays, analytic singularity ---------------0506010304395 Content-Type: application/postscript; name="zeta5.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="zeta5.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: zeta5.dvi %%Pages: 12 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX10 CMR9 CMCSC10 CMMI9 CMMI6 CMR6 CMSY9 CMR10 CMSY10 %%+ MSBM10 CMMI8 CMMI10 CMSY8 CMR8 CMTI10 CMEX10 CMBX8 CMTI9 CMBX9 CMTT9 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o zeta5.ps zeta5 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2005.05.24:1055 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX9 %!PS-AdobeFont-1.1: CMBX9 1.0 %%CreationDate: 1991 Aug 20 16:36:25 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX8 %!PS-AdobeFont-1.1: CMBX8 1.0 %%CreationDate: 1991 Aug 20 16:36:07 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-59 -250 1235 750}readonly def /UniqueID 5000766 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 16 /parenleftBig put dup 17 /parenrightBig put dup 40 /braceleftBigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 104 /bracketleftBig put dup 105 /bracketrightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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%%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-30 -958 1146 777}readonly def /UniqueID 5000819 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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%%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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Fi(T)1448 1130 y Fj(\015)1518 1116 y Fl(\024)25 b Fi(m)p Fl(k)p Fi(f)10 b Fl(k)1839 1130 y Fh(1)1944 1116 y Fn(and)30 b(w)m(e)g(get)1021 1262 y Fi(S)1077 1276 y Fj(m)1143 1262 y Fi(h)p Fn(\()p Fi(\021)s Fn(\))d Fl(\024)e Fi(m)p Fn(\()p Fi(k)e Fn(+)d(1\))p Fi(k)s(\017s)1922 1277 y Fj(k)1965 1262 y Fl(k)p Fi(f)10 b Fl(k)2110 1276 y Fh(1)2211 1262 y Fl(\024)25 b Fn(\()p Fi(k)e Fn(+)d(1\))p Fi(k)s(\017s)2713 1277 y Fj(k)2756 1262 y Fi(T)2809 1276 y Fj(\015)2854 1262 y Fi(:)0 1408 y Fn(F)-8 b(rom)31 b(the)f(equalit)m(y)i Fi(S)797 1422 y Fj(m)864 1408 y Fn(\()p Fl(\000)p Fi(k)s(s)1063 1423 y Fj(k)1105 1408 y Fi(f)e Fn(+)20 b Fi(k)s(g)s Fn(\)\()p Fi(\021)s Fn(\))27 b(=)e Fi(S)1699 1422 y Fj(m)1765 1408 y Fi(h)p Fn(\()p Fi(\021)s Fn(\))p Fi(;)33 b Fn(w)m(e)d(deduce)1231 1554 y Fl(\000)p Fi(s)1345 1569 y Fj(k)1387 1554 y Fi(T)1440 1568 y Fj(\015)1505 1554 y Fn(+)20 b Fi(\016)1636 1568 y Fj(\015)1706 1554 y Fl(\024)25 b Fn(\()p Fi(k)e Fn(+)d(1\))p Fi(\017s)2158 1569 y Fj(k)2201 1554 y Fi(T)2254 1568 y Fj(\015)2299 1554 y Fi(;)35 b Fl(8)p Fi(\015)30 b Fl(2)25 b(P)7 b Fi(:)0 1700 y Fn(No)m(w)31 b(let)g(0)26 b Fl(\024)f Fi(u)g Fl(\024)738 1665 y Fj(\017)p 688 1680 129 4 v 688 1732 a(k)r Fg(+1)827 1700 y Fi(:)30 b Fn(Then)1103 1853 y Fl(\000)p Fi(s)1217 1868 y Fj(k)1259 1853 y Fn(\(1)21 b(+)f Fi(u)p Fn(\))p Fi(T)1591 1867 y Fj(\015)1656 1853 y Fn(+)g Fi(\016)1787 1867 y Fj(\015)1857 1853 y Fl(\024)25 b Fn(\()p Fi(k)f Fn(+)c(1\))p Fi(\017s)2310 1868 y Fj(k)2353 1853 y Fi(T)2406 1867 y Fj(\015)2471 1853 y Fl(\000)g Fi(s)2605 1868 y Fj(k)2647 1853 y Fi(uT)2752 1867 y Fj(\015)1260 2031 y Fl(\024)1356 1930 y Fe(\020)1410 2031 y Fn(\()p Fi(k)k Fn(+)c(1\))p Fi(\017)h Fl(\000)1930 1970 y Fi(\017)p 1845 2010 207 4 v 1845 2094 a(k)j Fn(+)c(1)2062 1930 y Fe(\021)2116 2031 y Fi(s)2159 2046 y Fj(k)2202 2031 y Fi(T)2255 2045 y Fj(\015)2325 2031 y Fl(\024)25 b Fi(\017s)2501 2046 y Fj(k)2543 2031 y Fi(T)2596 2045 y Fj(\015)0 2192 y Fn(and)30 b(w)m(e)h(get)g(the)g(lo)m(w)m(er)g(b)s(ound)744 2376 y(1)26 b Fi(>)f Fn(1)c Fl(\000)f Fi(e)1110 2338 y Fh(\000)p Fj(s)1198 2350 y Fq(k)1236 2338 y Fg(\(1+)p Fj(u)p Fg(\))p Fj(T)1462 2346 y Fq(\015)1502 2338 y Fg(+)p Fj(\016)1588 2346 y Fq(\015)1658 2376 y Fl(\025)25 b Fn(1)20 b Fl(\000)g Fi(e)1952 2338 y Fj(\017s)2014 2350 y Fq(k)2052 2338 y Fj(T)2093 2346 y Fq(\015)2162 2376 y Fl(\025)25 b Fn(1)c Fl(\000)f Fi(e)2457 2338 y Fg(2)p Fj(s)2525 2350 y Fq(k)2563 2338 y Fj(\017d)2628 2347 y Fp(0)2692 2376 y Fn(=)2853 2315 y(1)p 2798 2355 156 4 v 2798 2438 a Fi(C)2863 2453 y Fj(\017;k)2989 2376 y Fi(>)25 b Fn(0)p Fi(:)0 2577 y Fn(Th)m(us)k(for)h(0)c Fl(\024)f Fi(u)g Fl(\024)769 2541 y Fj(\017)p 719 2556 129 4 v 719 2608 a(k)r Fg(+1)888 2577 y Fn(the)31 b(series)554 2699 y Fh(1)523 2727 y Fe(X)466 2924 y Fj(m)p Fg(=)p Fj(k)r Fg(+1)727 2813 y Fi(e)769 2775 y Fj(m)p Fg(\()p Fh(\000)p Fj(s)946 2787 y Fq(k)985 2775 y Fg(\(1+)p Fj(u)p Fg(\))p Fj(T)1211 2783 y Fq(\015)1252 2775 y Fg(+)p Fj(\016)1338 2783 y Fq(\015)1378 2775 y Fg(\))1434 2813 y 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4719 a(k)r Fg(+1)1692 4688 y Fn(.)150 4904 y(P)m(assing)31 b(to)g(the)g(case)g Fi(s)981 4919 y Fj(k)1049 4904 y Fi(>)25 b Fn(0,)31 b(c)m(ho)s(ose)g Fi(\017)25 b(>)g Fn(0)31 b(to)g(arrange)g(the)f(inequalities)1345 5050 y(sup)1320 5144 y Fj(\021)r Fh(2)p Fg(\006)1455 5114 y Fp(+)1455 5167 y Fq(A)1522 5050 y Fi(h)p Fn(\()p Fi(\021)s Fn(\))c Fl(\024)f(\000)p Fn(\()p Fi(k)f Fn(+)c(1\))p Fi(k)s(\017s)2292 5065 y Fj(k)2335 5050 y Fl(k)p Fi(f)10 b Fl(k)2480 5064 y Fh(1)2555 5050 y Fi(;)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 0 66 a Fg(8)1714 b(V.)23 b(PETK)n(O)n(V)1173 266 y Fl(\000)p Fi(s)1287 281 y Fj(k)1329 266 y Fi(T)1382 280 y Fj(\015)1447 266 y Fn(+)c Fi(\016)1577 280 y Fj(\015)1648 266 y Fl(\024)25 b(\000)p Fn(\()p Fi(k)e Fn(+)d(1\))p Fi(\017s)2171 281 y Fj(k)2214 266 y Fi(T)2267 280 y Fj(\015)2311 266 y Fi(;)36 b Fl(8)p Fi(T)2476 280 y Fj(\015)2545 266 y Fl(2)25 b(P)7 b Fi(:)0 396 y Fn(F)-8 b(or)31 b(0)26 b Fl(\024)f Fi(u)g Fl(\024)563 361 y Fj(\017)p 513 376 129 4 v 513 428 a(k)r 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y(satis\014es)g(the)f(estimate)1700 5012 y Fi(c)1739 5026 y Fg(0)1804 5012 y Fl(\025)25 b Fi(s)1943 5026 y Fg(1)2002 5012 y Fl(\000)20 b Fi(h)2145 5026 y Fj(t)2175 5012 y Fi(;)0 5143 y Fn(but)30 b(it)h(is)f(di\016cult) g(to)h(compare)g Fi(c)1196 5157 y Fg(0)1266 5143 y Fn(with)f Fi(s)1516 5158 y Fj(k)1559 5143 y Fn(.)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 539 66 a Fg(D)n(YNAMICAL)30 b(ZET)-6 b(A)31 b(FUNCTION)f(F)n(OR)h(SEVERAL)f(STRICTL)-6 b(Y)31 b(CONVEX)f(OBST)-6 b(A)n(CLES)505 b(9)1135 266 y Fn(4.)46 b Fm(Singularities)33 b(on)h(the)g(line)g Fl(<)p Fi(s)24 b Fn(=)h Fi(s)2726 280 y Fg(2)150 428 y Fn(Consider)30 b(the)g(Diric)m(hlet)i(series)1459 596 y Fi(P)1517 610 y Fg(2)1557 596 y Fn(\()p Fi(s)p Fn(\))26 b(=)1801 535 y(1)p 1801 575 46 4 v 1801 659 a(2)1877 510 y Fe(X)1872 707 y Fj(\015)t Fh(2P)2029 596 y Fi(e)2071 559 y Fh(\000)p Fg(2)p Fj(sT)2235 567 y Fq(\015)2275 559 y Fg(+2)p Fj(\016)2396 567 y Fq(\015)0 859 y Fn(with)32 b(p)s(ositiv)m(e)h(co)s(e\016cien)m (ts.)47 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