Content-Type: multipart/mixed; boundary="-------------9906091439271" This is a multi-part message in MIME format. ---------------9906091439271 Content-Type: text/plain; name="99-222.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-222.keywords" Polygonal billiards, entropy, ergodicity ---------------9906091439271 Content-Type: application/postscript; name="step_0906.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="step_0906.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: step_0906.dvi %%Pages: 27 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips -o step_0906.ps step_0906 %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 1999.06.09:1514 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 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-1 roll pop} {pop dup findfont dup maxlength dict CD dup dup /Encoding exch /Encoding get 256 array copy 7 -1 roll {3 -1 roll dup 4 -2 roll put}forall put definefont} ifelse}bdf /RCJ{4 -1 roll /ourvec xdf 256 string cvs (|______) anchorsearch {pop cvn dup FDFJ exch 1 index eq { _bfh findfont _efh dup maxlength dict CD dup /FontName 3 index put dup /Encoding ourvec put 1 index exch definefont pop } {exch pop} ifelse } {pop} ifelse }bdf /RFJ{ dup FontDirectory exch known {pop 3 -1 roll pop} {RCJ} ifelse }bdf /hasfont { /resourcestatus where { pop /Font resourcestatus { pop pop true } { false } ifelse } { dup FontDirectory exch known {pop true} { 256 string cvs (fonts/) exch MN status {pop pop pop pop true} {false} ifelse } ifelse } ifelse }bdf /FDFJ { dup hasfont not { pop /Ryumin-Light-83pv-RKSJ-H hasfont { /Ryumin-Light-83pv-RKSJ-H } { /Courier } ifelse } if }bdf /FFJ{ _bfh dup 256 string cvs (|______)exch MN cvn dup FontDirectory exch known { exch pop findfont 3 -1 roll pop } { pop FDFJ dup findfont dup maxlength dict CD dup dup /Encoding exch /Encoding get 256 array copy 7 -1 roll { 3 -1 roll dup 4 -2 roll put }forall put definefont } ifelse _efh }bdf /GS { dup hasfont { FFJ curtextmtx makefont setfont exch 5 1 roll ts pop } { pop pop ts } ifelse } bdf /RCK{4 -1 roll /ourvec xdf 256 string cvs (|______) anchorsearch {pop cvn dup FDFK exch 1 index eq { _bfh findfont _efh dup maxlength dict CD dup /FontName 3 index put dup /Encoding ourvec put 1 index exch definefont pop } {exch pop} ifelse } {pop} ifelse }bdf /RFK{ dup FontDirectory exch known {pop 3 -1 roll pop} {RCK} ifelse }bdf /hasfont { /resourcestatus where { pop /Font resourcestatus { pop pop true } { false } ifelse } { dup FontDirectory exch known {pop true} { 256 string cvs (fonts/) exch MN status {pop pop pop pop true} {false} ifelse } ifelse } ifelse }bdf /FDFK { dup hasfont not { pop /JCsm hasfont { /JCsm } { /Courier } ifelse } if }bdf /FFK{ _bfh dup 256 string cvs (|______)exch MN cvn dup FontDirectory exch known { exch pop findfont 3 -1 roll pop } { pop FDFK dup findfont dup maxlength dict CD dup dup /Encoding exch /Encoding get 256 array copy 7 -1 roll { 3 -1 roll dup 4 -2 roll put }forall put definefont } ifelse _efh }bdf /RCTC{4 -1 roll /ourvec xdf 256 string cvs (|______) anchorsearch {pop cvn dup FDFTC exch 1 index eq { _bfh findfont _efh dup maxlength dict CD dup /FontName 3 index put dup /Encoding ourvec put 1 index exch definefont pop } {exch pop} ifelse } {pop} ifelse }bdf /RFTC{ dup FontDirectory exch known {pop 3 -1 roll pop} {RCTC} ifelse }bdf /FDFTC { dup hasfont not { pop /DFMing-Lt-HK-BF hasfont { /DFMing-Lt-HK-BF } { /Courier } ifelse } if }bdf /FFTC{ _bfh dup 256 string cvs (|______)exch MN cvn dup FontDirectory exch known { exch pop findfont 3 -1 roll pop } { pop FDFTC dup findfont dup maxlength dict CD dup dup /Encoding exch /Encoding get 256 array copy 7 -1 roll { 3 -1 roll dup 4 -2 roll put }forall put definefont } ifelse _efh }bdf /fps{ currentflat exch dup 0 le{pop 1}if { dup setflat 3 index stopped {1.3 mul dup 3 index gt{pop setflat pop pop stop}if} {exit} ifelse }loop pop setflat pop pop }bdf /fp{100 currentflat fps}bdf /clipper{clip}bdf /W{/clipper load 100 clipflatness dup setflat fps}bdf /AVec 256 array def AVec 0 /Helvetica findfont /Encoding get 0 256 getinterval putinterval /ANSIPatch[ 16#0/grave 16#1/acute 16#2/circumflex 16#3/tilde 16#4/macron 16#5/breve 16#6/dotaccent 16#7/dieresis 16#8/ring 16#9/cedilla 16#A/hungarumlaut 16#B/ogonek 16#C/caron 16#D/dotlessi 16#27/quotesingle 16#60/grave 16#7C/bar 16#82/quotesinglbase 16#83/florin 16#84/quotedblbase 16#85 /ellipsis 16#86/dagger 16#87/daggerdbl 16#89/perthousand 16#8A/Scaron 16#8B/guilsinglleft 16#8C/OE 16#91/quoteleft 16#92/quoteright 16#93 /quotedblleft 16#94/quotedblright 16#95/bullet 16#96/endash 16#97/emdash 16#99/trademark 16#9A/scaron 16#9B/guilsinglright 16#9C/oe 16#9F/Ydieresis 16#A0/space 16#A4/currency 16#A6/brokenbar 16#A7/section 16#A8/dieresis 16#A9/copyright 16#AA/ordfeminine 16#AB/guillemotleft 16#AC/logicalnot 16#AD/hyphen 16#AE/registered 16#AF/macron 16#B0/degree 16#B1/plusminus 16#B2/twosuperior 16#B3/threesuperior 16#B4/acute 16#B5/mu 16#B6/paragraph 16#B7/periodcentered 16#B8/cedilla 16#B9/onesuperior 16#BA/ordmasculine 16#BB/guillemotright 16#BC/onequarter 16#BD/onehalf 16#BE/threequarters 16#BF/questiondown 16#C0/Agrave 16#C1/Aacute 16#C2/Acircumflex 16#C3/Atilde 16#C4/Adieresis 16#C5/Aring 16#C6/AE 16#C7/Ccedilla 16#C8/Egrave 16#C9/Eacute 16#CA/Ecircumflex 16#CB/Edieresis 16#CC/Igrave 16#CD/Iacute 16#CE/Icircumflex 16#CF/Idieresis 16#D0/Eth 16#D1/Ntilde 16#D2/Ograve 16#D3/Oacute 16#D4/Ocircumflex 16#D5/Otilde 16#D6/Odieresis 16#D7/multiply 16#D8/Oslash 16#D9/Ugrave 16#DA/Uacute 16#DB/Ucircumflex 16#DC/Udieresis 16#DD/Yacute 16#DE/Thorn 16#DF/germandbls 16#E0/agrave 16#E1/aacute 16#E2/acircumflex 16#E3/atilde 16#E4/adieresis 16#E5/aring 16#E6/ae 16#E7/ccedilla 16#E8/egrave 16#E9/eacute 16#EA/ecircumflex 16#EB/edieresis 16#EC/igrave 16#ED/iacute 16#EE/icircumflex 16#EF/idieresis 16#F0/eth 16#F1/ntilde 16#F2/ograve 16#F3/oacute 16#F4/ocircumflex 16#F5/otilde 16#F6/odieresis 16#F7/divide 16#F8/oslash 16#F9/ugrave 16#FA/uacute 16#FB/ucircumflex 16#FC/udieresis 16#FD/yacute 16#FE/thorn 16#FF/ydieresis ] def 127 1 159 { AVec exch/bullet put } for ANSIPatch aload pop ANSIPatch length 2 idiv{AVec 3 1 roll put}repeat /DoPatch { dup /CharStrings known { setfont 0 1 255 { dup currentfont /Encoding get exch get currentfont /CharStrings get exch known {pop} {currentfont /Encoding get exch /bullet put} ifelse } for } {pop} ifelse } def /findheaderfont { AVec 256 array copy /FHT /|______Helvetica dup RF findfont def FHT DoPatch FHT } def end %. AltsysDict AltsysDict begin _bfh / findfont /Encoding get StandardEncoding eq { AVec 256 array copy /f0 /|______Ñc dup RF findfont def f0 DoPatch } { /f0 / findfont def } ifelse _efh end %. AltsysDict AltsysDict begin /onlyk4{false}ndf /ccmyk{dup 5 -1 roll sub 0 max exch}ndf /cmyk2gray{ 4 -1 roll 0.3 mul 4 -1 roll 0.59 mul 4 -1 roll 0.11 mul add add add 1 min neg 1 add }bdf /setcmykcolor{1 exch sub ccmyk ccmyk ccmyk pop setrgbcolor}ndf /maxcolor { max max max } ndf /maxspot { pop } ndf /setcmykcoloroverprint{4{dup -1 eq{pop 0}if 4 1 roll}repeat setcmykcolor}ndf /findcmykcustomcolor{5 packedarray}ndf /setcustomcolor{exch aload pop pop 4{4 index mul 4 1 roll}repeat setcmykcolor pop}ndf /setseparationgray{setgray}ndf /setoverprint{pop}ndf /currentoverprint false ndf /cmykbufs2gray{ 0 1 2 index length 1 sub { 4 index 1 index get 0.3 mul 4 index 2 index get 0.59 mul 4 index 3 index get 0.11 mul 4 index 4 index get add add add cvi 255 min 255 exch sub 2 index 3 1 roll put }for 4 1 roll pop pop pop }bdf /colorimage{ pop pop [ 5 -1 roll/exec cvx 6 -1 roll/exec cvx 7 -1 roll/exec cvx 8 -1 roll/exec cvx /cmykbufs2gray cvx ]cvx image } %. version 47.1 on Linotronic of Postscript defines colorimage incorrectly (rgb model only) version cvr 47.1 le statusdict /product get (Lino) anchorsearch{pop pop true}{pop false}ifelse and{userdict begin bdf end}{ndf}ifelse fhnumcolors 1 ne {/yt save def} if /customcolorimage{ aload pop (_vc_Registration) eq { pop pop pop pop separationimage } { /ik xdf /iy xdf /im xdf /ic xdf ic im iy ik cmyk2gray /xt xdf currenttransfer {dup 1.0 exch sub xt mul add}concatprocs st image } ifelse }ndf fhnumcolors 1 ne {yt restore} if fhnumcolors 3 ne {/yt save def} if /customcolorimage{ aload pop (_vc_Registration) eq { pop pop pop pop separationimage } { /ik xdf /iy xdf /im xdf /ic xdf 1.0 dup ic ik add min sub 1.0 dup im ik add min sub 1.0 dup iy ik add min sub /ic xdf /iy xdf /im xdf currentcolortransfer 4 1 roll {dup 1.0 exch sub ic mul add}concatprocs 4 1 roll {dup 1.0 exch sub iy mul add}concatprocs 4 1 roll {dup 1.0 exch sub im mul add}concatprocs 4 1 roll setcolortransfer {/dummy xdf dummy}concatprocs{dummy}{dummy}true 3 colorimage } ifelse }ndf fhnumcolors 3 ne {yt restore} if fhnumcolors 4 ne {/yt save def} if /customcolorimage{ aload pop (_vc_Registration) eq { pop pop pop pop separationimage } { /ik xdf /iy xdf /im xdf /ic xdf currentcolortransfer {1.0 exch sub ik mul ik sub 1 add}concatprocs 4 1 roll {1.0 exch sub iy mul iy sub 1 add}concatprocs 4 1 roll {1.0 exch sub im mul im sub 1 add}concatprocs 4 1 roll {1.0 exch sub ic mul ic sub 1 add}concatprocs 4 1 roll setcolortransfer {/dummy xdf dummy}concatprocs{dummy}{dummy}{dummy} true 4 colorimage } ifelse }ndf fhnumcolors 4 ne {yt restore} if /separationimage{image}ndf /spotascmyk false ndf /newcmykcustomcolor{6 packedarray}ndf /inkoverprint false ndf /setinkoverprint{pop}ndf /setspotcolor { spots exch get dup 4 get (_vc_Registration) eq {pop 1 exch sub setseparationgray} {0 5 getinterval exch setcustomcolor} ifelse }ndf /currentcolortransfer{currenttransfer dup dup dup}ndf /setcolortransfer{st pop pop pop}ndf /fas{}ndf /sas{}ndf /fhsetspreadsize{pop}ndf /filler{fill}bdf /F{gsave {filler}fp grestore}bdf /f{closepath F}bdf /S{gsave {stroke}fp grestore}bdf /s{closepath S}bdf userdict /islevel2 systemdict /languagelevel known dup { pop systemdict /languagelevel get 2 ge } if put islevel2 not { /currentcmykcolor { 0 0 0 1 currentgray sub } ndf } if /tc { gsave setcmykcolor currentcmykcolor grestore } bind def /testCMYKColorThrough { tc add add add 0 ne } bind def /fhiscomposite where not { userdict /fhiscomposite islevel2 { gsave 1 1 1 1 setcmykcolor currentcmykcolor grestore add add add 4 eq } { 1 0 0 0 testCMYKColorThrough 0 1 0 0 testCMYKColorThrough 0 0 1 0 testCMYKColorThrough 0 0 0 1 testCMYKColorThrough and and and } ifelse put } { pop } ifelse /bc4 [0 0 0 0] def /_lfp4 { 1 pop /yt xdf /xt xdf /ang xdf storerect /taperfcn xdf /k2 xdf /y2 xdf /m2 xdf /c2 xdf /k1 xdf /y1 xdf /m1 xdf /c1 xdf c1 c2 sub abs m1 m2 sub abs y1 y2 sub abs k1 k2 sub abs maxcolor calcgraysteps mul abs round height abs adjnumsteps dup 1 lt {pop 1} if 1 sub /numsteps1 xdf currentflat mark currentflat clipflatness /delta top bottom sub numsteps1 1 add div def /right right left sub def /botsv top delta sub def { { W xt yt translate ang rotate xt neg yt neg translate dup setflat /bottom botsv def 0 1 numsteps1 { numsteps1 dup 0 eq {pop pop 0.5} {div} ifelse taperfcn /frac xdf bc4 0 c2 c1 sub frac mul c1 add put bc4 1 m2 m1 sub frac mul m1 add put bc4 2 y2 y1 sub frac mul y1 add put bc4 3 k2 k1 sub frac mul k1 add put bc4 vc 1 index setflat { mark {newpath left bottom right delta rectfill}stopped {cleartomark exch 1.3 mul dup setflat exch 2 copy gt{stop}if} {cleartomark exit}ifelse }loop /bottom bottom delta sub def }for } gsave stopped grestore {exch pop 2 index exch 1.3 mul dup 100 gt{cleartomark setflat stop}if} {exit}ifelse }loop cleartomark setflat }bdf /bcs [0 0] def /_lfs4 { /yt xdf /xt xdf /ang xdf storerect /taperfcn xdf /tint2 xdf /tint1 xdf bcs exch 1 exch put tint1 tint2 sub abs bcs 1 get maxspot calcgraysteps mul abs round height abs adjnumsteps dup 2 lt {pop 2} if 1 sub /numsteps1 xdf currentflat mark currentflat clipflatness /delta top bottom sub numsteps1 1 add div def /right right left sub def /botsv top delta sub def { { W xt yt translate ang rotate xt neg yt neg translate dup setflat /bottom botsv def 0 1 numsteps1 { numsteps1 div taperfcn /frac xdf bcs 0 1.0 tint2 tint1 sub frac mul tint1 add sub put bcs vc 1 index setflat { mark {newpath left bottom right delta rectfill}stopped {cleartomark exch 1.3 mul dup setflat exch 2 copy gt{stop}if} {cleartomark exit}ifelse }loop /bottom bottom delta sub def }for } gsave stopped grestore {exch pop 2 index exch 1.3 mul dup 100 gt{cleartomark setflat stop}if} {exit}ifelse }loop cleartomark setflat }bdf /_rfs6 { /tint2 xdf /tint1 xdf bcs exch 1 exch put /inrad xdf /radius xdf /yt xdf /xt xdf tint1 tint2 sub abs bcs 1 get maxspot calcgraysteps mul abs round radius inrad sub abs adjnumsteps dup 1 lt {pop 1} if 1 sub /numsteps1 xdf radius inrad sub numsteps1 dup 0 eq {pop} {div} ifelse 2 div /halfstep xdf currentflat mark currentflat clipflatness { { dup setflat W 0 1 numsteps1 { dup /radindex xdf numsteps1 dup 0 eq {pop pop 0.5} {div} ifelse /frac xdf bcs 0 tint2 tint1 sub frac mul tint1 add put bcs vc 1 index setflat { newpath mark xt yt radius inrad sub 1 frac sub mul halfstep add inrad add 0 360 { arc radindex numsteps1 ne inrad 0 gt or { xt yt numsteps1 0 eq { inrad } { radindex 1 add numsteps1 div 1 exch sub radius inrad sub mul halfstep add inrad add }ifelse dup xt add yt moveto 360 0 arcn } if fill }stopped {cleartomark exch 1.3 mul dup setflat exch 2 copy gt{stop}if} {cleartomark exit}ifelse }loop }for } gsave stopped grestore {exch pop 2 index exch 1.3 mul dup 100 gt{cleartomark setflat stop}if} {exit}ifelse }loop cleartomark setflat }bdf /_rfp6 { 1 pop /k2 xdf /y2 xdf /m2 xdf /c2 xdf /k1 xdf /y1 xdf /m1 xdf /c1 xdf /inrad xdf /radius xdf /yt xdf /xt xdf c1 c2 sub abs m1 m2 sub abs y1 y2 sub abs k1 k2 sub abs maxcolor calcgraysteps mul abs round radius inrad sub abs adjnumsteps dup 1 lt {pop 1} if 1 sub /numsteps1 xdf radius inrad sub numsteps1 dup 0 eq {pop} {div} ifelse 2 div /halfstep xdf currentflat mark currentflat clipflatness { { dup setflat W 0 1 numsteps1 { dup /radindex xdf numsteps1 dup 0 eq {pop pop 0.5} {div} ifelse /frac xdf bc4 0 c2 c1 sub frac mul c1 add put bc4 1 m2 m1 sub frac mul m1 add put bc4 2 y2 y1 sub frac mul y1 add put bc4 3 k2 k1 sub frac mul k1 add put bc4 vc 1 index setflat { newpath mark xt yt radius inrad sub 1 frac sub mul halfstep add inrad add 0 360 { arc radindex numsteps1 ne inrad 0 gt or { xt yt numsteps1 0 eq { inrad } { radindex 1 add numsteps1 div 1 exch sub radius inrad sub mul halfstep add inrad add }ifelse dup xt add yt moveto 360 0 arcn } if fill }stopped {cleartomark exch 1.3 mul dup setflat exch 2 copy gt{stop}if} {cleartomark exit}ifelse }loop }for } gsave stopped grestore {exch pop 2 index exch 1.3 mul dup 100 gt{cleartomark setflat stop}if} {exit}ifelse }loop cleartomark setflat }bdf /lfp4{_lfp4}ndf /lfs4{_lfs4}ndf /rfs6{_rfs6}ndf /rfp6{_rfp6}ndf /cvc [0 0 0 1] def /vc{ AltsysDict /cvc 2 index put aload length dup 4 eq {pop dup -1 eq{pop setrgbcolor}{setcmykcolor}ifelse} {6 eq {sethexcolor} {setspotcolor} ifelse } ifelse }bdf 0 setseparationgray /imgr {-16128 -16128 16128 16128 } def /bleed 0 def /clpr {-16128 -16128 -144 -144 } def /xs 1 def /ys 1 def /botx 0 def /overlap 0 def /wdist 18 def 0 2 mul fhsetspreadsize 0 0 ne {/df 0 def /clipflatness 0 def} if /maxsteps 256 def /forcemaxsteps false def /minsteps 0 def userdict begin /AGDOrigMtx matrix currentmatrix def end vms -1782 -2093 translate /currentpacking defed{false setpacking}if /spots[ 1 0 0 0 (Process Cyan) false newcmykcustomcolor 0 1 0 0 (Process Magenta) false newcmykcustomcolor 0 0 1 0 (Process Yellow) false newcmykcustomcolor 0 0 0 1 (Process Black) false newcmykcustomcolor ]def /textopf false def /curtextmtx{}def /otw .25 def /msf{dup/curtextmtx xdf makefont setfont}bdf /makesetfont/msf load def /curtextheight{.707104 .707104 curtextmtx dtransform dup mul exch dup mul add sqrt}bdf /ta2{ tempstr 2 index gsave exec grestore cwidth cheight rmoveto 4 index eq{5 index 5 index rmoveto}if 2 index 2 index rmoveto }bdf /ta{exch systemdict/cshow known {{/cheight xdf/cwidth xdf tempstr 0 2 index put ta2}exch cshow} {{tempstr 0 2 index put tempstr stringwidth/cheight xdf/cwidth xdf ta2}forall} ifelse 6{pop}repeat}bdf /sts{/textopf currentoverprint def vc setoverprint /ts{awidthshow}def exec textopf setoverprint}bdf /stol{/xt currentlinewidth def setlinewidth vc newpath /ts{{false charpath stroke}ta}def exec xt setlinewidth}bdf /strk{/textopf currentoverprint def vc setoverprint /ts{{false charpath stroke}ta}def exec textopf setoverprint }bdf n [] 0 d 3.863708 M 1 w 0 j 0 J false setoverprint 0 i false eomode [0 0 0 1]vc vms 1782.5 2189.5 m 1945.5 2189.5 L 1945.5 2188.5 L 1782.5 2188.5 L 1782.5 2189.5 L f S n 1807 2191.5 m 1817.5 2247.5 L 1.1567 w S n 1818.5 2247 m 1867.2064 2240.0696 L S n 1825.5584 2230.2565 m 1827.5 2194 L S n 1807 2191 m 1827.5 2194.5 L S n 1826 2230.5 m 1850.5 2223 L S n 1807 2189.5 m 1795 2138.5 L S n 1791.5 2098 m 1814.5 2108.5 L S n 1815 2108.5 m 1810 2141.5 L S n 1810 2142 m 1828.5 2179 L S n 1828.5 2177.5 m 1845 2140.5 L S n 1844.9048 2140.0305 m 1842 2099 L S n 1842.5 2099 m 1865 2103 L S n 1865.5 2103 m 1858.5 2146.5 L S n 1858 2146.5 m 1848.5 2180.5 L S n 1827.5 2199 m 1855.5 2174 L S n 1856.5 2173.5 m 1873.5 2210 L S n 1873 2209.5 m 1925.9882 2189.2812 L S n 1867.5 2239.5 m 1876 2228 L S n 1877 2228 m 1903 2209 L S n vmrs 1902.5 2209 m 1929 2202.5 L 1.1567 w S n 1929.5 2203 m 1926.499 2189.2812 L S n 1812 2106 m 1813.5 2101.5 1816.125 2099.25 1816.5 2094.5 C 1814.625 2094.25 1813.5 2094 1811.625 2094 C 1806.25 2094 1803.5 2097 1799 2100 C 1 w S n 1864.5 2103 m 1869.75 2103 1871.375 2101.5 1874 2097 C 1873.5 2094.5 L 1868.875 2094.5 1865.625 2094.5 1861 2094.5 C 1855.875 2094.5 1854.125 2097.75 1849.5 2100 C S n 1825.5 2194 m 1825.5 2188 L S n 1825.5 2188 m 1817.5 2187.5 L S n 1927.5 2195 m 1935.5 2195 L S n 1935.5 2195 m 1935.5 2190 L S n 1795.5 2138.5 m 1792 2098.5 L S n 1870.5 2248.5 m 1870.5 2260.9266 1880.5734 2271 1893 2271 C 1905.4266 2271 1915.5 2260.9266 1915.5 2248.5 C 1915.5 2236.0734 1905.4266 2226 1893 2226 C 1880.5734 2226 1870.5 2236.0734 1870.5 2248.5 C s n 1890 2271 m 1898 2250.5 1913.5 2241 1913.5 2241 C S n 1872.1393 2257.2951 m 1880.1393 2236.7951 1895.6393 2227.2951 1895.6393 2227.2951 C S n 1883.5 2248.25 m 1883.5 2252.2541 1886.7459 2255.5 1890.75 2255.5 C 1894.7541 2255.5 1898 2252.2541 1898 2248.25 C 1898 2244.2459 1894.7541 2241 1890.75 2241 C 1886.7459 2241 1883.5 2244.2459 1883.5 2248.25 C s n 1896.5799 2251.2588 m 1892.3232 2255.5156 L 1884.5451 2247.7375 L 1888.8018 2243.4807 L 1896.5799 2251.2588 L n q Q false eomode 1891.3463 2245.977 m 1890.538 2245.1688 L 1887.477 2248.2299 L 1887.8172 2248.5038 1888.2076 2248.6955 1888.6482 2248.8048 C 1889.0887 2248.9142 1889.7805 2248.9518 1890.7234 2248.9178 C 1891.484 2248.8918 1891.9742 2248.8996 1892.194 2248.9413 C 1892.4945 2249.0018 1892.733 2249.1202 1892.9094 2249.2966 C 1893.1044 2249.4916 1893.2022 2249.6938 1893.2026 2249.9029 C 1893.203 2250.1121 1893.1113 2250.3087 1892.9273 2250.4926 C 1892.7454 2250.6745 1892.5457 2250.7643 1892.3282 2250.7618 C 1892.1108 2250.7593 1891.8662 2250.6387 1891.5945 2250.4 C 1890.8097 2251.3589 L 1891.3431 2251.7886 1891.8513 2251.9715 1892.3345 2251.9076 C 1892.8176 2251.8437 1893.2643 2251.6066 1893.6747 2251.1963 C 1894.1243 2250.7466 1894.3561 2250.2718 1894.37 2249.7717 C 1894.384 2249.2717 1894.2102 2248.8409 1893.8487 2248.4794 C 1893.643 2248.2737 1893.4104 2248.1147 1893.1509 2248.0023 C 1892.8914 2247.8899 1892.5801 2247.8117 1892.2169 2247.7677 C 1891.9762 2247.7383 1891.5908 2247.7352 1891.0608 2247.7583 C 1890.5307 2247.7814 1890.1873 2247.7883 1890.0307 2247.7787 C 1889.874 2247.7692 1889.7335 2247.7477 1889.6089 2247.7144 C 1891.3463 2245.977 L [0 0 0 1] vc true setoverprint f false setoverprint n 1950 2189.5 m 1950 2227 L S n 1949.5 2227 m 1980.5 2227 L S n 1980.5 2227 m 1980.5 2213 L S n vmrs 2006 2213 m 2006 2202 L S n 2006 2202 m 2031.5 2202 L S n 2031.5 2202 m 2031.5 2194 L S n 1979.5808 2212.8957 m 2005.0808 2212.8957 L S n 2031.0808 2193.8957 m 2050 2193.8957 L 3.863693 M [4 4 ] 0 d S [] 0 d n 1949.8308 2188.8957 m 2049.5 2188.8957 L 3.863708 M S n 1914 2271 m 1914 2272.6569 1915.567 2274 1917.5 2274 C 1919.433 2274 1921 2272.6569 1921 2271 C 1921 2269.3431 1919.433 2268 1917.5 2268 C 1915.567 2268 1914 2269.3431 1914 2271 C s n 1920 2280 m 1920 2282.2092 1923.3578 2284 1927.5 2284 C 1931.6422 2284 1935 2282.2092 1935 2280 C 1935 2277.7908 1931.6422 2276 1927.5 2276 C 1923.3578 2276 1920 2277.7908 1920 2280 C s n 1934.25 2293.75 m 1934.5 2302.5 1938.375 2309 1952.5 2310.25 C 1956.4843 2310.6026 1957.2522 2310.8903 1961.25 2310.75 C 1975.5 2310.25 1973.127 2299.7083 1984.5 2298 C 1987.1692 2297.5991 1995 2307 2005.5 2307 C 2013.8745 2307 2012 2306.5 2022.5 2300.5 C 2028.2916 2297.1905 2032.375 2302.5 2040 2302.5 C 2045.5 2302.5 2050.1584 2300.8768 2054.5 2297.5 C S n 1934.5 2294.5 m 1935.25 2286.25 1940.0367 2280.1072 1954.1617 2278.8572 C 1958.146 2278.5046 1960.5418 2277.8434 1964.4117 2278.8572 C 1974.5 2281.5 1972.8297 2287.1564 1984.25 2288.5 C 1988.5 2289 1990.25 2280.5 2006.5 2282 C 2014.8389 2282.7697 2011.7058 2282.0081 2022.2058 2288.0081 C 2027.9974 2291.3176 2033.0808 2285.0081 2040.7058 2285.0081 C 2046.2058 2285.0081 2049.5 2286 2053.7058 2290.0081 C 3.863693 M S n 1946 2295.5 m 1961 2286 1971.5 2297.5 1971.5 2297.5 C 3.863708 M S n 1968 2295 m 1959 2305 1950.4184 2292.9999 1950.4184 2292.9999 C S n 1997.0808 2295.6696 m 2007.7316 2288.924 2015.1872 2297.0898 2015.1872 2297.0898 C 3.863693 M S n 2012.702 2295.3146 m 2006.3115 2302.4153 2000.2181 2293.8944 2000.2181 2293.8944 C S n 2031.5808 2295.0364 m 2040.6507 2289.2923 2046.9996 2296.2457 2046.9996 2296.2457 C S n 2044.8833 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y(Orbits)e(starting)g(at)g(p)s(oin)m(ts)h(of)f Fj(E)1541 3644 y Fs(#)1618 3629 y Fo(will)e(nev)m(er)k(collide)c(with)j (an)m(y)g(v)m(ertical)f(side)g(of)141 3749 y Fj(S)6 b Fo(\()p Fj(#)p Fo(\))32 b(\(or)f Fj(P)14 b Fo(\))31 b(and)h(th)m(us,)h (as)e Fj(t)d Fi(!)g Fo(+)p Fi(1)p Fo(,)j(will)e(go)i(to)h(in\014nit)m (y)-8 b(,)31 b(main)m(taining)d(a)k(p)s(ositiv)m(e)141 3870 y(constan)m(t)i Fj(X)8 b Fo(-v)m(elo)s(cit)m(y)-8 b(.)42 b(W)-8 b(e)33 b(call)e(them)h Fk(esc)-5 b(ap)g(e)34 b(orbits)p Fo(.)287 4034 y(It)39 b(turns)h(out)e(that)g(if)g(the)h (\\in\014nite)e(cusp")j(of)e(a)h(step)g(p)s(olygon)f(narro)m(ws)h(do)m (wn)141 4154 y(v)m(ery)34 b(quic)m(kly)-8 b(,)33 b(sev)m(eral)g(in)m (teresting)g(facts)f(can)h(b)s(e)g(sho)m(wn:)141 4359 y Fl(Theorem)k(1)49 b Fk(If)37 b(the)h(heights)g Fi(f)p Fj(p)1420 4374 y Fs(n)1467 4359 y Fi(g)f Fk(of)h(an)g(in\014nite)f (step)h(bil)5 b(liar)-5 b(d)37 b Fj(P)51 b Fk(verify)38 b Fj(p)3131 4374 y Fs(n)p Fh(+1)3302 4359 y Fi(\024)141 4480 y Fj(\025p)247 4495 y Fs(n)294 4480 y Fk(,)d(with)1023 4665 y Fo(0)27 b Fj(<)h(\025)f(<)h(\025)1448 4680 y Fh(0)1515 4665 y Fo(:=)1655 4516 y Fi(p)p 1738 4516 49 4 v 82 x Fo(6)22 b Fi(\000)h Fo(1)p 1655 4642 303 4 v 1782 4734 a(5)1995 4665 y Fi(')29 b Fo(0)p Fj(:)p Fo(290)17 b Fj(:)g(:)g(:)d(;) 141 4862 y Fk(then,)45 b(for)e(almost)g(al)5 b(l)42 b(dir)-5 b(e)g(ctions)43 b Fj(#)g Fi(2)p Fo(]0)p Fj(;)17 b(\031)t(=)p Fo(2[)p Fk(,)45 b(ther)-5 b(e)43 b(exists)f(a)h(subse)-5 b(quenc)g(e)43 b Fi(f)p Fj(n)3293 4877 y Fs(j)3329 4862 y Fi(g)141 5002 y Fk(such)35 b(that)g Fj(n:i:)p Fo(\()p Fj(E)823 4948 y Fh(\()p Fs(n)893 4958 y Fd(j)927 4948 y Fh(\))817 5030 y Fs(#)958 5002 y Fo(\))28 b(=)f(1)p Fk(.)p eop %%Page: 10 10 10 9 bop 515 241 a Fo(10)1160 b Fn(M.)33 b(Degli)d(Esp)s(osti,)j(G.)f (Del)g(Magno)g(and)h(M.)g(Lenci)515 548 y Fl(Corollary)j(1)48 b Fk(F)-7 b(or)34 b(almost)h(al)5 b(l)34 b(dir)-5 b(e)g(ctions)34 b(ther)-5 b(e)35 b(is)g(exactly)f(one)h(esc)-5 b(ap)g(e)34 b(orbit.)661 794 y Fo(Let)40 b(us)g(name)f Fj(\021)1290 809 y Fs(#)1376 794 y Fo(the)h(\(unique\))f(escap)s(e)i(orbit.)63 b(Also,)41 b(with)e(a)g(certain)h(lac)m(k)f(of)515 914 y(originalit)m(y)-8 b(,)37 b(let)i(us)h(call)d Fk(typic)-5 b(al)50 b Fo(a)39 b(direction)f Fj(#)h Fo(for)g(whic)m(h)h(the)g(ab)s (o)m(v)m(e)g(statemen)m(ts)515 1034 y(hold.)661 1157 y(Theorem)g(1)f(and)h(Corollary)e(1)h(are)h(the)g(analogues)e(of)i (Corollaries)d(3)i(and)h(4)f(of)515 1278 y([DDL)o(],)e(although)e (their)g(pro)s(ofs,)i(whic)m(h)f(are)g(to)g(b)s(e)g(found)g(in)f (Section)h(3,)h(are)f(quite)515 1398 y(di\013eren)m(t.)89 b(In)48 b(our)g(previous)g(article)e(w)m(e)j(exploited)e(thoroughly)g (the)i(arithmetic)515 1518 y(prop)s(erties)39 b(of)h(the)g Fk(exp)-5 b(onential)40 b(bil)5 b(liar)-5 b(d)50 b Fj(p)2186 1533 y Fs(n)2272 1518 y Fo(=)40 b(2)2437 1482 y Fr(\000)p Fs(n)2539 1518 y Fo(,)i(whereas)f(here,)h(to)e(ac)m(hiev)m(e)h(a)515 1639 y(certain)32 b(generalit)m(y)-8 b(,)32 b(w)m(e)h(use)h(rather)e (rough)h(measure-theoretic)f(estimates)g(on)h(some)515 1759 y(sets)h(of)e(directions.)45 b(This)33 b(is)f(wh)m(y)j(w)m(e)f (are)f(prett)m(y)h(con\014den)m(t)g(that)f(the)g(same)g(results)515 1880 y(can)42 b(indeed)g(b)s(e)h(pro)m(v)m(en)g(for)f(a)g(m)m(uc)m(h)g (wider)g(class)h(of)e(billiards)e(\(see)k(the)g(pro)s(of)e(of)515 2000 y(Lemma)31 b(2)h(and)h(in)f(particular)f(estimate)g(\(14\)\).)661 2123 y(The)e(next)f(t)m(w)m(o)h(results)f(are)g(also)f(form)m(ulated)f (for)h(a)h(billiard)c(as)k(in)f(the)h(statemen)m(t)515 2243 y(of)k(Theorem)h(1.)515 2491 y Fl(Lemma)k(1)49 b Fk(F)-7 b(or)34 b(a.a.)44 b Fj(#)p Fk(,)35 b Fj(\021)1573 2506 y Fs(#)1654 2491 y Fk(do)-5 b(es)34 b(not)h(interse)-5 b(ct)35 b(any)g(vertex.)661 2737 y Fo(The)g(ab)s(o)m(v)m(e)f(lemma,)e (to)s(o,)h(is)g(pro)m(v)m(en)j(in)c(Section)i(3.)46 b(Not)34 b(only)f(do)s(es)h(it)f(describ)s(e)515 2857 y(the)g(b)s(eha)m(vior)g (of)g(the)h(escap)s(e)g(orbit,)e(but)i(more)e(imp)s(ortan)m(tly)-8 b(,)31 b(in)i(conjunction)g(with)515 2978 y(Theorem)j(1,)h(yields)e (the)i(follo)m(wing)c(assertions|whic)m(h)k(follo)m(w)d(from)h(the)h (ab)s(o)m(v)m(e)h(in)515 3098 y(the)d(same)f(w)m(a)m(y)i(as)f(the)g(en) m(tire)g(Section)g(3)f(of)g([DDL)o(])h(follo)m(ws)f(from)f(Corollary)g (3)h(and)515 3218 y(Lemma)e(8.)515 3467 y Fl(Prop)s(osition)k(2)49 b Fk(Fixe)-5 b(d)34 b(a)h(typic)-5 b(al)34 b(dir)-5 b(e)g(ction)35 b Fj(#)p Fk(,)600 3682 y(\(i\))49 b Fj(\021)807 3697 y Fs(#)888 3682 y Fk(is)34 b(oscil)5 b(lating)34 b(in)h(the)g(p)-5 b(ast.)570 3898 y(\(ii\))49 b Fj(\021)807 3913 y Fs(#)888 3898 y Fk(is)34 b(c)-5 b(ontaine)g(d)34 b(in)h(the)g Fj(!)t Fk(-limit)e(of)i(every)g(orbit)g(\(exc)-5 b(ept)34 b(for)h(itself)10 b(\).)540 4114 y(\(iii\))49 b Fj(\021)807 4129 y Fs(#)893 4114 y Fk(is)39 b(c)-5 b(ontaine)g(d)39 b(in)g(the)h Fj(\013)q Fk(-limit)f(of)g(every)h(orbit)f(\(exc)-5 b(ept)39 b(for)h(the)g(orbit)f(that)759 4234 y(esc)-5 b(ap)g(es)33 b(in)i(the)g(p)-5 b(ast\).)555 4450 y(\(iv\))49 b(Excluding)32 b(the)h(exc)-5 b(eptions)32 b(mentione)-5 b(d)32 b(ab)-5 b(ove,)33 b(every)g(semi-orbit)f(stays)h(close)759 4570 y(to)i Fj(\021)924 4585 y Fs(#)1005 4570 y Fk(for)f(an)h(arbitr)-5 b(ary)35 b(long)f(time.)585 4786 y(\(v\))49 b(Every)35 b(invariant)f(c)-5 b(ontinuous)34 b(function)h(on)g Fj(S)6 b Fo(\()p Fj(#)p Fo(\))35 b Fk(is)f(c)-5 b(onstant.)555 5002 y(\(vi\))49 b(The)34 b(\015ow)h(is)f(minimal)g(if,)g(and)h(only)f (if,)h Fj(\021)2351 5017 y Fs(#)2432 5002 y Fk(is)g(dense.)p eop %%Page: 11 11 11 10 bop 141 241 a Fn(In\014nite)33 b(Step)g(Billiards)2219 b Fo(11)152 548 y Fk(\(vii\))48 b(The)39 b(closur)-5 b(e)38 b(of)h Fj(\021)1087 563 y Fs(#)1159 548 y Fi(\\)26 b Fj(L)p Fk(,)40 b(which)f(is)g(the)g(\\tr)-5 b(ac)g(e")39 b(of)f(the)i(esc)-5 b(ap)g(e)38 b(orbit)h(on)g(the)385 668 y(usual)c(Poinc)-5 b(ar)n(\023)-47 b(e)33 b(se)-5 b(ction,)34 b(is)h(either)g(the)f(entir)-5 b(e)35 b Fj(L)g Fk(or)g(a)g(Cantor)f(set.)141 924 y Fl(Remark)j(4)49 b Fk(Pr)-5 b(op)g(osition)25 b(2,\(iii\))f(\014xes)h(a)g(minor)g (mistake)g(in)g(Cor)-5 b(ol)5 b(lary)25 b(6)g(of)g([DDL)o(]:)141 1045 y(it)32 b(is)g(obvious)f(that)h(assertion)f(\(i\))g(ther)-5 b(e)32 b(do)-5 b(es)31 b(not)h(work)f(for)h(the)f(\(unique\))h(orbit)g (that)141 1165 y(esc)-5 b(ap)g(es)34 b(in)g(the)h(p)-5 b(ast.)141 1450 y Fb(2.2)135 b(Ergo)t(dic)46 b(Prop)t(erties)141 1634 y Fo(W)-8 b(e)33 b(no)m(w)h(turn)f(to)f(the)h(ergo)s(dic)f(prop)s (erties)h(of)f(our)h(step)h(billiards,)c(and)i(w)m(e)i(start)f(b)m(y) 141 1755 y(presen)m(ting)g(a)g(sligh)m(t)e(enhancemen)m(t)j(of)e(a)h (previous)g(result.)141 1951 y Fl(Theorem)k(2)49 b Fk(Fix)25 b Fj(\013)k Fo(=)e(tan)17 b Fj(#)28 b Fi(62)g Fc(Q)12 b Fk(.)47 b(F)-7 b(or)26 b(every)g(p)-5 b(ositive,)27 b(monotonic)-5 b(al)5 b(ly)24 b(vanishing)141 2072 y(se)-5 b(quenc)g(e)39 b Fi(f)8 b Fo(\026)-57 b Fj(p)641 2087 y Fs(n)687 2072 y Fi(g)36 b(\032)g Fc(Q)12 b Fk(,)46 b(and)39 b(every)g(inte)-5 b(ger)39 b Fj(k)s Fk(,)i(ther)-5 b(e)39 b(exists)g(a)g(de)-5 b(cr)g(e)g(asing)38 b(se)-5 b(quenc)g(e)141 2192 y Fi(f)p Fj(p)240 2207 y Fs(n)287 2192 y Fi(g)27 b(\032)i Fc(Q)11 b Fk(,)41 b(with)1282 2384 y Fj(p)1331 2399 y Fs(n)1405 2384 y Fo(=)36 b(\026)-57 b Fj(p)1558 2399 y Fs(n)1605 2384 y Fj(;)166 b Fo(for)34 b(0)28 b Fi(\024)g Fj(n)g Fi(\024)g Fj(k)s Fo(;)1100 2530 y(0)g Fj(<)f(p)1329 2545 y Fs(n)1404 2530 y Fi(\024)36 b Fo(\026)-57 b Fj(p)1558 2545 y Fs(n)1605 2530 y Fj(;)166 b Fo(for)34 b Fj(n)28 b(>)g(k)s Fo(;)1118 2592 y Fg(X)1168 2801 y Fs(n)1278 2687 y Fj(p)1327 2702 y Fs(n)1402 2687 y Fj(<)f Fi(1)p Fj(:)141 2973 y Fk(such)43 b(that)h(the)g(bil)5 b(liar)-5 b(d)42 b(\015ow)h Fj(\036)1361 2988 y Fs(#;t)1496 2973 y Fk(on)g Fj(S)1710 2937 y Fs(P)1768 2973 y Fo(\()p Fj(#)p Fo(\))p Fk(,)j(for)d Fj(P)57 b Fi(')44 b(f)p Fj(p)2481 2988 y Fs(n)2528 2973 y Fi(g)p Fk(,)h(is)e(er)-5 b(go)g(dic)43 b(\(henc)-5 b(e)141 3094 y(almost)34 b(al)5 b(l)35 b(orbits)g(ar)-5 b(e)34 b(dense\).)287 3290 y Ff(Pr)n(oof.)48 b Fo(Exactly)35 b(the)f(same)g(as)g([DDL)o(],)g(Theorem)h(1,)f(except)h(that)f(the)g (induc-)141 3411 y(tion)e(starts)h(at)f Fj(n)c Fo(=)f Fj(k)s Fo(.)2098 b Fw(Q.E.D.)287 3573 y Fo(Theorem)34 b(2)g(pro)m(vides)g(an)g(example)f(\(man)m(y)-8 b(,)34 b(as)g(a)f(matter)g(of)g(fact\))h(of)f(a)h(billiard)141 3693 y(that)24 b(is)g(ergo)s(dic)g(in)g(one)h(direction.)39 b(One)25 b(w)m(ould)g(lik)m(e)e(to)h(get)h(more:)39 b(a)24 b(billiard)d(ergo)s(dic)141 3814 y(for)36 b(a.a.)g(directions,)h(lik)m (e)e(the)i(rectangular)e(table.)54 b(W)-8 b(e)37 b(adapt)f(to)g(the)h (in\014nite)e(step)141 3934 y(billiards)i(some)i(of)g(the)h(tec)m (hniques)i(presen)m(ted)f(in)e([T)-8 b(r])40 b(to)f(sho)m(w)i(that)e (this)g(is)h(true)141 4054 y(\\in)32 b(general",)g(in)f(the)i(sense)i (of)d([KMS].)287 4175 y(More)43 b(precisely)-8 b(,)44 b(let)e Fi(S)50 b Fo(b)s(e)42 b(the)g(space)h(of)f(all)e(step)j(p)s (olygons)e(with)h(unit)f(area.)141 4295 y(\(This)k(means)g(that,)j(in)c (this)h(section,)j(w)m(e)e(drop)f(the)g(unsubstan)m(tial)g(requiremen)m (t)141 4415 y Fj(p)190 4430 y Fh(0)260 4415 y Fo(=)30 b(1.\))47 b(Since)35 b(eac)m(h)g(step)g(p)s(olygon)e Fj(P)47 b Fo(is)34 b(uniquely)g(determined)g(b)m(y)h(the)f(sequence)141 4536 y(of)i(the)g(heigh)m(ts)g(of)f(its)h(v)m(ertical)f(sides)h Fi(f)p Fj(p)1704 4551 y Fs(n)1751 4536 y Fi(g)1801 4551 y Fs(n)p Fr(2)p Fq(N)1943 4536 y Fo(,)g(w)m(e)h(apply)f(to)f Fi(S)44 b Fo(the)36 b(metric)f(of)h(the)141 4656 y(space)e Fj(l)433 4620 y Fh(1)472 4656 y Fo(:)44 b(giv)m(en)33 b Fj(P)41 b Fi(')28 b(f)p Fj(p)1106 4671 y Fs(n)1153 4656 y Fi(g)p Fo(,)k Fj(Q)c Fi(')g(f)p Fj(q)1565 4671 y Fs(j)1602 4656 y Fi(g)p Fo(,)k(let)1249 4927 y Fj(d)p Fo(\()p Fj(P)s(;)17 b(Q)p Fo(\))27 b(:=)1757 4803 y Fr(1)1720 4833 y Fg(X)1726 5042 y Fs(n)p Fh(=0)1881 4927 y Fi(j)p Fj(p)1958 4942 y Fs(n)2027 4927 y Fi(\000)22 b Fj(q)2169 4942 y Fs(n)2216 4927 y Fi(j)p Fj(:)p eop %%Page: 12 12 12 11 bop 515 241 a Fo(12)1160 b Fn(M.)33 b(Degli)d(Esp)s(osti,)j(G.)f (Del)g(Magno)g(and)h(M.)g(Lenci)515 548 y Fo(The)46 b(metric)e(space)i (\()p Fi(S)7 b Fj(;)17 b(d)p Fo(\))45 b(is)g(complete)g(and)g (separable.)81 b(Note)46 b(that)e Fj(d)p Fo(\()p Fj(P)s(;)17 b(Q)p Fo(\))49 b(=)515 668 y Fj(A)p Fo(\()p Fj(P)14 b Fi(4)p Fj(Q)p Fo(\).)661 793 y(W)-8 b(e)35 b(call)f(a)g(set)i(of)f(a)g (top)s(ological)c(space)36 b Fk(typic)-5 b(al)45 b Fo(if)34 b(it)g(con)m(tains)h(a)g(dense)h Fj(G)3535 808 y Fs(\016)3573 793 y Fo(-set,)515 914 y Fk(me)-5 b(ager)38 b Fo(if)28 b(it)f(is)h(the)h(union)f(of)g(a)h(coun)m(table)f(collection)f(of)h(no) m(where)i(dense)g(sets.)44 b(The)515 1034 y(follo)m(wing)30 b(statemen)m(t)j(is)f(the)h(highligh)m(t)d(of)i(this)g(section:)515 1301 y Fl(Theorem)37 b(3)49 b Fk(A)g(typic)-5 b(al)48 b(step)h(p)-5 b(olygon)48 b Fj(P)62 b Fk(is)48 b(er)-5 b(go)g(dic)48 b(on)g Fj(S)2925 1264 y Fs(P)2984 1301 y Fo(\()p Fj(#)p Fo(\))h Fk(for)f(a.e.)86 b Fj(#)53 b Fi(2)515 1421 y Fo(]0)p Fj(;)17 b(\031)t(=)p Fo(2[)p Fk(.)515 1683 y Fl(Corollary)36 b(2)48 b Fk(Ther)-5 b(e)44 b(ar)-5 b(e)45 b(in\014nite)f(step)g(bil)5 b(liar)-5 b(ds)44 b Fj(P)58 b Fk(er)-5 b(go)g(dic)44 b(on)g Fj(S)3196 1646 y Fs(P)3255 1683 y Fo(\()p Fj(#)p Fo(\))h Fk(for)f(a.e.)515 1803 y Fj(#)28 b Fi(2)p Fo(]0)p Fj(;)17 b(\031)t(=)p Fo(2[)p Fk(.)661 2065 y Fo(Both)31 b(results)h(are)g(pro)m(v)m(en)h(in) d(Section)i(4.)43 b(The)32 b(main)e(di\013erences)j(b)s(et)m(w)m(een)g (The-)515 2185 y(orem)f(3)g(and)h(Theorem)f(5.1)h(of)f([T)-8 b(r])33 b(are:)634 2447 y(1.)49 b(He)33 b(came)f(\014rst.)634 2674 y(2.)49 b(The)37 b(result)g(in)f([T)-8 b(r)q(])36 b(is)h(ab)s(out)f(a)h(broad)f(class)h(of)g(rational)d(in\014nite)i(p)s (olygons,)759 2794 y(while)31 b(ours)i(is)f(restricted)h(to)g(the)g (in\014nite)e(step)j(p)s(olygons.)634 3021 y(3.)49 b(Ev)m(en)30 b(though)f(they)h(use)g(the)f(same)g(ideas,)g(Theorem)g(3)g(cannot)g(b) s(e)g(deriv)m(ed)h(b)m(y)759 3142 y(its)i(analogue)f(in)h([T)-8 b(r].)634 3369 y(4.)49 b(His)32 b(billiards)d(are)k(b)s(ounded;)g(ours) g(can)g(b)s(e)g(un)m(b)s(ounded.)634 3596 y(5.)49 b(The)32 b(metric)e(that)h(w)m(e)h(use)g(is)f(v)m(ery)i(simple)c(and)j(corresp)s (onds)g(to)f(the)h(in)m(tuitiv)m(e)759 3717 y(idea)g(of)g(closeness.) 515 4035 y Fb(2.3)135 b(Metric)45 b(En)l(trop)l(y)515 4229 y Fo(Let)35 b Fj(P)46 b Fi(')34 b(f)p Fj(p)1011 4244 y Fs(n)1057 4229 y Fi(g)i Fo(b)s(e)f(a)g(giv)m(en)h(in\014nite)f (step)h(billiard.)49 b(W)-8 b(e)36 b(denote)g(b)m(y)g Fi(V)41 b Fo(=)33 b Fi(f)p Fj(V)3490 4244 y Fs(k)3532 4229 y Fi(g)i Fo(the)515 4350 y(set)f(of)f(non-remo)m(v)-5 b(able)32 b(\(singular\))g(v)m(ertices,)j(de\014ned)g(at)e(the)h(b)s (eginning)e(of)h(Section)515 4470 y(2.)46 b(One)34 b(can)f(pro)m(v)m(e) i(the)f(follo)m(wing)d(prop)s(ert)m(y)j(of)f(non-p)s(erio)s(dic)f (orbits,)h(whic)m(h)h(turns)515 4590 y(out)e(to)g(b)s(e)h(v)m(ery)h (useful)f(in)f(certain)g(en)m(trop)m(y)i(estimates.)515 4857 y Fl(Prop)s(osition)h(3)49 b Fk(The)30 b(closur)-5 b(e)31 b(of)g(the)g(forwar)-5 b(d)30 b(and)g(b)-5 b(ackwar)g(d)30 b(semi-orbit)g(of)h(every)515 4977 y(non-p)-5 b(erio)g(dic)33 b(p)-5 b(oint)34 b(interse)-5 b(cts)35 b Fi(V)30 b([)23 b(f1g)p Fk(.)p eop %%Page: 13 13 13 12 bop 141 241 a Fn(In\014nite)33 b(Step)g(Billiards)2219 b Fo(13)141 548 y Fl(Remark)37 b(5)49 b Fk(This)33 b(r)-5 b(esult)35 b(c)-5 b(an)34 b(b)-5 b(e)34 b(c)-5 b(onsider)g(e)g(d)33 b(the)h(adapte)-5 b(d)33 b(version)h(of)g(Pr)-5 b(op)g(osition)141 668 y(4.1)34 b(in)h([T)-7 b(r)o(].)45 b(Notic)-5 b(e)35 b(that)g(in)f(our)h(c)-5 b(ase)34 b(the)h(bil)5 b(liar)-5 b(d)34 b(is)g(always)h(r)-5 b(ational)34 b(\(or)h(we)-5 b(akly)141 789 y(r)g(ational)37 b(ac)-5 b(c)g(or)g(ding)36 b(to)h([T)-7 b(r],)37 b(De\014nition)f(4.3\))h(and)g Fi(V)45 b Fk(is)37 b(only)g(a)g(subset)h(of)f(al)5 b(l)36 b(ver-)141 909 y(tic)-5 b(es.)86 b(A)n(lso,)51 b(it)e(is)f(imme)-5 b(diate)48 b(to)h(che)-5 b(ck)47 b(that)i(for)g(bil)5 b(liar)-5 b(ds)47 b(as)h(in)h(Se)-5 b(ction)48 b(2.2,)141 1029 y(Pr)-5 b(op)g(osition)34 b(3)h(is)g(c)-5 b(ontaine)g(d)33 b(in)i(Pr)-5 b(op)g(osition)34 b(2.)287 1305 y Ff(Pr)n(oof)40 b(of)h(Pr)n(oposition)e(3.)96 b Fo(If)34 b(a)g(semi-orbit)e Fj(\015)39 b Fo(\(either)34 b(forw)m(ard)h(or)e(bac)m(k-)141 1425 y(w)m(ard\))k(of)f(a)g(non-p)s(erio)s(dic)e(p)s(oin)m(t)i(is)g(b)s (ounded,)i(then)f(it)e(is)h(en)m(tirely)g(con)m(tained)h(in)e(a)141 1546 y(\014nite)30 b(step)h(billiard)c Fj(P)1003 1510 y Fh(\()p Fs(n)p Fh(\))1105 1546 y Fo(,)j(for)g Fj(n)h Fo(large)e(enough.)43 b(According)30 b(to)g([GKT],)h(the)g(closure)141 1666 y(of)i Fj(\015)38 b Fo(m)m(ust)c(con)m(tain)f(a)g(singular)f(v)m (ertex)j(of)e Fj(P)1867 1630 y Fh(\()p Fs(n)p Fh(\))1968 1666 y Fo(,)h(whic)m(h)g(clearly)e(is)h(in)g Fi(V)8 b Fo(.)46 b(If)33 b(instead)141 1787 y Fj(\015)38 b Fo(is)32 b(un)m(b)s(ounded,)i(then)f(the)g(set)g(of)f(its)g(limiting)d(p)s(oin)m (ts)j(con)m(tains)g Fi(f1g)p Fo(.)187 b Fw(Q.E.D.)287 1977 y Fo(T)-8 b(ak)m(e)45 b Fj(n)h Fi(\025)g Fo(1.)75 b(In)43 b(Fig.)f(2,)j(let)e Fj(L)1593 1992 y Fs(n)1686 1977 y Fo(:=)j Fi(f)p Fj(n)p Fi(g)29 b(\002)h Fo([)p Fj(p)2205 1992 y Fs(n)2252 1977 y Fj(;)17 b(p)2345 1992 y Fs(n)p Fr(\000)p Fh(1)2481 1977 y Fo([)44 b(and)f Fj(L)2818 1992 y Fr(\000)p Fs(n)2966 1977 y Fo(:=)i Fi(f)p Fj(n)p Fi(g)30 b(\002)141 2097 y Fo([)p Fi(\000)p Fj(p)294 2112 y Fs(n)p Fr(\000)p Fh(1)432 2097 y Fj(;)17 b Fi(\000)p Fj(p)602 2112 y Fs(n)649 2097 y Fo([)38 b(b)s(e)h(the)g(t)m(w)m(o)g (copies)g(of)f(the)h Fj(n)p Fo(-th)f(v)m(ertical)g(side.)61 b(W)-8 b(e)39 b(iden)m(tify)f(them)141 2218 y(with)c Fj(d)416 2233 y Fs(n)492 2218 y Fo(:=)c([)p Fj(p)701 2233 y Fs(n)749 2218 y Fj(;)17 b(p)842 2233 y Fs(n)p Fr(\000)p Fh(1)978 2218 y Fo([)34 b(and)g Fj(d)1281 2233 y Fr(\000)p Fs(n)1413 2218 y Fo(:=)c([)p Fi(\000)p Fj(p)1699 2233 y Fs(n)p Fr(\000)p Fh(1)1837 2218 y Fj(;)17 b Fi(\000)p Fj(p)2007 2233 y Fs(n)2054 2218 y Fo([.)48 b(The)35 b(family)c(of)j (these)h(in)m(terv)-5 b(als)141 2338 y(partitions)35 b Fj(I)41 b Fo(:=)34 b([)p Fi(\000)p Fo(1)p Fj(;)17 b Fo(0[)p Fi([)p Fo(]0)p Fj(;)g Fo(1[.)53 b(Th)m(us)38 b(it)d(mak)m(es)i(sense)g(to)f(de\014ne)h Fj(f)2784 2353 y Fs(#)2831 2338 y Fo(,)g(as)f(the)g(i.e.t.)141 2459 y(induced)d(b)m(y)h Fj(\036)698 2474 y Fs(#;t)821 2459 y Fo(on)f Fj(I)8 b Fo(.)287 2586 y(F)-8 b(or)32 b(an)m(y)h Fj(f)694 2601 y Fs(#)741 2586 y Fo(-in)m(v)-5 b(arian)m(t)30 b(Borel)i(probabilit)m(y)f(measure)i Fj(\027)6 b Fo(,)33 b(w)m(e)g(set)1078 2896 y Fj(H)1159 2911 y Fs(\027)1202 2896 y Fo(\()p Fj(P)14 b Fo(\))27 b(:=)g Fi(\000)1643 2772 y Fr(1)1606 2802 y Fg(X)1612 3011 y Fs(n)p Fh(=1)1784 2896 y Fj(\027)6 b Fo(\()p Fj(d)1927 2911 y Fs(n)1974 2896 y Fo(\))33 b(log)16 b Fj(\027)6 b Fo(\()p Fj(d)2330 2911 y Fs(n)2377 2896 y Fo(\))p Fj(:)141 3246 y Fo(Call)378 3221 y Fg(e)349 3246 y Fi(X)420 3261 y Fs(#)498 3246 y Fo(:=)31 b Fj(L)24 b Fi(n)795 3171 y Fg(T)878 3198 y Fr(1)878 3275 y Fs(n)p Fh(=0)1032 3246 y Fi(P)1109 3205 y Fr(\000)p Fs(n)1101 3274 y(#)1212 3246 y Fj(E)1284 3261 y Fs(#)1365 3246 y Fo(the)35 b(set)g(of)f(the)i(p)s(oin)m(ts)e(in) g Fj(L)h Fo(whose)h(forw)m(ard)f(orbits)141 3367 y(k)m(eep)43 b(returning)d(there;)46 b(and)41 b(denote)h(b)m(y)f Fi(X)1832 3382 y Fs(#)1919 3367 y Fo(the)h(corresp)s(onding)f(set)g(in)f Fj(I)8 b Fo(.)69 b(More)141 3487 y(precisely)-8 b(,)1428 3649 y Fi(X)1499 3664 y Fs(#)1573 3649 y Fo(:=)27 b Fj(g)1750 3664 y Fs(#)1813 3539 y Fg(\020)1901 3624 y(e)1872 3649 y Fi(X)1943 3664 y Fs(#)1989 3539 y Fg(\021)2065 3649 y Fj(;)141 3897 y Fo(where)36 b Fj(g)472 3912 y Fs(#)564 3897 y Fo(:)668 3872 y Fg(e)639 3897 y Fi(X)710 3912 y Fs(#)786 3897 y Fi(\000)-16 b(!)30 b Fj(I)42 b Fo(is)34 b(giv)m(en)g(b)m(y)h Fj(g)1602 3912 y Fs(#)1648 3897 y Fo(\()p Fj(x)p Fo(\))c(=)f Fj(\036)1974 3912 y Fs(#;t)2061 3921 y Fe(1)2100 3897 y Fo(\()p Fj(x)p Fo(\))k(and)g Fj(t)2491 3912 y Fh(1)2565 3897 y Fo(is)g(the)g(\014rst)h(collision)141 4031 y(time)c(at)i(a)f(v)m(ertical)g(w)m(all.)42 b(F)-8 b(or)32 b(an)m(y)h Fj(x)28 b Fi(2)1717 4005 y Fg(e)1688 4031 y Fi(X)1759 4046 y Fs(#)1805 4031 y Fo(,)33 b(let)640 4271 y Fg(e)-57 b Fj(\031)693 4286 y Fs(#)739 4271 y Fo(\()p Fj(x)p Fo(\))28 b(:=)g Fi(f)p Fj(:)17 b(:)g(:)f(;)h(L)1320 4286 y Fs(!)1364 4295 y Fa(\000)p Fe(1)1450 4271 y Fj(;)g(L;)g(L)1670 4286 y Fs(!)1714 4295 y Fe(0)1753 4271 y Fj(;)g(L;)g(L)1973 4286 y Fs(!)2017 4295 y Fe(1)2056 4271 y Fj(;)g(L;)g(:)g(:)g(:)f(;)h (L;)g(L)2561 4286 y Fs(!)2605 4298 y Fd(k)2647 4271 y Fj(;)g(:)g(:)g(:)o Fi(g)p Fj(;)141 4511 y Fo(b)s(e)42 b(the)h(sequence)i(of)c(v)m(ertical)g(sides)i(that)f(the)g(orbit)f(of)g Fj(x)i Fo(crosses)h(\(adopting)d(the)141 4631 y(con)m(v)m(en)m(tion)35 b(that)e Fj(L)908 4646 y Fs(!)952 4655 y Fe(0)1024 4631 y Fo(is)g(the)h(\014rst)g(v)m(ertical)e(side)i(encoun)m(tered)h(in)e (the)h(past\).)46 b(Then)141 4762 y(w)m(e)34 b(de\014ne)g(the)f Fk(c)-5 b(o)g(ding)40 b Fj(\031)1095 4777 y Fs(#)1186 4762 y Fo(:)1286 4737 y Fg(e)1257 4762 y Fi(X)1328 4777 y Fs(#)1402 4762 y Fi(\000)-16 b(!)27 b Fo(\()p Fc(Z)1697 4726 y Fr(\003)1734 4762 y Fo(\))1772 4726 y Fq(Z)1853 4762 y Fo(b)m(y)1127 5002 y Fj(\031)1182 5017 y Fs(#)1228 5002 y Fo(\()p Fj(x)p Fo(\))h(:=)g Fi(f)p Fj(:)17 b(:)g(:)f(;)h(!)1804 5017 y Fr(\000)p Fh(1)1898 5002 y Fj(;)g(!)2003 5017 y Fh(0)2041 5002 y Fj(;)g(!)2146 5017 y Fh(1)2185 5002 y Fj(;)g(:)g(:)g(:)p Fi(g)p eop %%Page: 14 14 14 13 bop 515 241 a Fo(14)1160 b Fn(M.)33 b(Degli)d(Esp)s(osti,)j(G.)f (Del)g(Magno)g(and)h(M.)g(Lenci)515 548 y Fo(W)-8 b(e)35 b(equip)h Fj(C)1021 563 y Fs(#)1098 548 y Fo(:=)c Fj(\031)1288 563 y Fs(#)1335 548 y Fo(\()p Fi(X)1444 563 y Fs(#)1490 548 y Fo(\))j(with)g(the)g(pro)s(duct)g(top)s(ology)f(and)h(w)m(e)h (denote)g(b)m(y)g Fj(\033)j Fo(the)515 668 y(left)31 b(shift.)43 b(The)34 b(follo)m(wing)c(diagram)g(comm)m(utes:)1765 858 y Fj(f)1813 873 y Fs(#)1887 858 y Fo(:)e Fi(X)2013 873 y Fs(#)2152 858 y Fi(\000)-16 b(!)83 b(X)2467 873 y Fs(#)1952 979 y Fi(")184 b Fj(g)2233 994 y Fs(#)2430 979 y Fi(")1754 1112 y(P)1823 1127 y Fs(#)1897 1112 y Fo(:)1981 1087 y Fg(e)1952 1112 y Fi(X)2023 1127 y Fs(#)2152 1112 y Fi(\000)-16 b(!)2425 1087 y Fg(e)2396 1112 y Fi(X)2467 1127 y Fs(#)1952 1232 y Fi(#)180 b Fj(\031)2237 1247 y Fs(#)2430 1232 y Fi(#)1783 1353 y Fj(\033)32 b Fo(:)c Fj(C)1995 1368 y Fs(#)2152 1353 y Fi(\000)-16 b(!)84 b Fj(C)2467 1368 y Fs(#)515 1550 y Fl(Prop)s(osition)35 b(4)49 b Fk(L)-5 b(et)29 b Fj(\027)35 b Fk(b)-5 b(e)28 b(any)g Fj(f)1811 1565 y Fs(#)1858 1550 y Fk(-invariant,)g(Bor)-5 b(el,)29 b(er)-5 b(go)g(dic)28 b(pr)-5 b(ob)g(ability)28 b(me)-5 b(asur)g(e)515 1670 y(on)34 b Fi(X)725 1685 y Fs(#)771 1670 y Fk(.)45 b(If)34 b Fj(H)1029 1685 y Fs(\027)1072 1670 y Fo(\()p Fj(P)14 b Fo(\))27 b Fj(<)h Fi(1)p Fk(,)34 b(then)h Fj(h)1793 1685 y Fs(\027)1864 1670 y Fo(=)28 b(0)p Fk(.)515 1874 y Fl(Remark)37 b(6)48 b Fk(F)-7 b(or)38 b(instanc)-5 b(e,)39 b(if)f Fj(p)1793 1889 y Fs(n)1878 1874 y Fk(de)-5 b(c)g(ays)38 b(exp)-5 b(onential)5 b(ly)38 b(or)g(faster,)h(as)f(in)h(the)f(ex-)515 1994 y(amples)k(that)i(we)f (have)g(r)-5 b(e)g(c)g(al)5 b(le)-5 b(d)43 b(e)-5 b(arlier,)45 b(then)e Fj(H)2448 2009 y Fs(`)2481 1994 y Fo(\()p Fj(P)14 b Fo(\))43 b Fk(is)g(\014nite)g(for)h(the)f(L)-5 b(eb)g(esgue)515 2115 y(me)g(asur)g(e)34 b Fj(`)p Fk(.)661 2318 y Ff(Pr)n(oof)46 b(of)h(Pr)n(oposition)e(4.)132 b Fo(First)39 b(of)g(all,)h(notice)g (that)g(the)g(assertion)g(is)515 2439 y(ob)m(vious)c(if)f Fj(\027)43 b Fo(is)35 b(atomic.)53 b(W)-8 b(e)36 b(claim)e(that)i(the)h (partition)d Fj(\013)g Fo(:=)g Fi(f)p Fj(\013)3102 2454 y Fh(1)3141 2439 y Fj(;)17 b(\013)3247 2454 y Fh(2)3286 2439 y Fj(;)g(:)g(:)g(:)o Fi(g)36 b Fo(giv)m(en)515 2559 y(b)m(y)1613 2680 y Fj(\013)1675 2695 y Fs(j)1739 2680 y Fo(:=)28 b Fi(f)p Fj(!)j Fi(2)d Fj(C)2176 2695 y Fs(#)2244 2680 y Fi(j)22 b Fj(!)2355 2695 y Fh(1)2438 2680 y Fo(=)44 b Fj(j)6 b Fi(g)515 2845 y Fo(is)28 b(a)g(one-side)g(generator.)42 b(This)29 b(implies)d(b)m(y)j(standard)g(results)f(\(e.g.,)i([R],)f (11.3\))f(that)515 2965 y Fj(h)571 2980 y Fs(\025)664 2965 y Fo(=)48 b(0)c(for)g(an)m(y)i Fj(\033)t Fo(-in)m(v)-5 b(arian)m(t)42 b(Borel)i(probabilit)m(y)f(measure)h Fj(\025)h Fo(on)f Fj(C)3243 2980 y Fs(#)3289 2965 y Fo(.)80 b(No)m(w,)48 b Fj(g)3707 2980 y Fs(#)515 3086 y Fo(is)36 b(bijectiv)m(e)h(b)m(y)h (construction,)g(therefore)g(the)f(pull-bac)m(k)g Fj(g)2801 3101 y Fs(#)p Fr(\003)2882 3086 y Fj(\027)43 b Fo(is)37 b(also)f(non-atomic)515 3206 y(and)48 b(ergo)s(dic.)88 b(Hence,)53 b(there)48 b(is)g(a)f Fj(Y)75 b Fi(\022)2244 3181 y Fg(e)2215 3206 y Fi(X)2286 3221 y Fs(#)2380 3206 y Fo(suc)m(h)49 b(that)f Fj(\031)2897 3221 y Fs(#)2943 3206 y Fi(j)2971 3221 y Fs(Y)3079 3206 y Fo(is)g(injectiv)m(e)f(and)515 3326 y Fj(g)562 3341 y Fs(#)p Fr(\003)643 3326 y Fj(\027)6 b Fo(\()p Fj(Y)22 b Fo(\))27 b(=)h(1.)43 b(W)-8 b(e)33 b(conclude)g(that)f Fj(h)1938 3341 y Fs(\027)1982 3326 y Fo(\()p Fj(f)2068 3341 y Fs(#)2114 3326 y Fo(\))27 b(=)h Fj(h)2339 3341 y Fs(\031)2382 3318 y Fa(\003)2380 3365 y Fd(#)2421 3341 y Fs(g)2455 3353 y Fd(#)p Fa(\003)2528 3341 y Fs(\027)2571 3326 y Fo(\()p Fj(\033)t Fo(\))g(=)f(0.)661 3447 y(It)40 b(remains)f(to)g(pro)m(v)m(e)j(the)e(initial)c(claim.)63 b(This)40 b(is)f(done)i(if)e(w)m(e)h(sho)m(w)h(that)f(the)515 3567 y(sequence)47 b(of)d(sides)h(an)g(orbit)f(visits)g(in)g(the)h (future)g(determines)g(the)g(sequence)i(of)515 3687 y(sides)34 b(visited)f(in)g(the)h(past.)46 b(Clearly)-8 b(,)33 b(if)f(the)i(orbit) f(is)g(p)s(erio)s(dic,)f(there)i(is)f(nothing)g(to)515 3808 y(pro)m(v)m(e.)62 b(In)39 b(the)f(non-p)s(erio)s(dic)f(case)i(w)m (e)g(can)g(use)g(Prop)s(osition)e(3)h(to)g(conclude)h(that)515 3928 y(an)m(y)28 b(parallel)c(strip)j(of)f(orbits)h(m)m(ust)g(ev)m(en)m (tually)h(\\split")d(at)i(some)g(v)m(ertex)i Fj(V)3341 3943 y Fs(k)3411 3928 y Fo(and)e(this)515 4049 y(of)35 b(course)i(implies)c(that)j(t)m(w)m(o)h(distinct)e(orbits)g(cannot)h (hit)f(the)h(same)g(v)m(ertical)f(sides)515 4169 y(in)d(the)h(future.) 2394 b Fw(Q.E.D.)661 4333 y Fo(W)-8 b(e)33 b(no)m(w)g(turn)g(to)f(the)h (pro)s(ofs)f(of)h(the)g(other)f(results.)515 4663 y Fm(3)161 b(Escap)t(e)53 b(orbits)515 4882 y Fo(In)g(this)f(section)g(w)m(e)i (denote)f(directions)f(b)m(y)h Fj(\013)63 b Fo(=)e(tan)17 b Fj(#)62 b Fi(2)p Fo(]0)p Fj(;)17 b Fo(+)p Fi(1)p Fo([,)57 b(using)52 b(the)515 5002 y(Leb)s(esgue)44 b(measure)f(there.)75 b(All)41 b(surfaces)j Fj(S)6 b Fo(\()p Fj(\013)q Fo(\))42 b(will)f(b)s(e)i(iden)m(ti\014ed)f(with)h(the)g(set)p eop %%Page: 15 15 15 14 bop 141 241 a Fn(In\014nite)33 b(Step)g(Billiards)2219 b Fo(15)141 548 y Fj(S)37 b Fo(of)31 b(Fig.)f(2)i(on)f(whic)m(h)h(w)m (e)h(use)f(the)g(co)s(ordinates)f(\()p Fj(X)r(;)17 b(Y)k Fo(\).)43 b(F)-8 b(or)31 b(instance,)h Fj(\015)3015 563 y Fs(k)3057 548 y Fo(\()p Fj(\013)q Fo(\))f(will)141 668 y(denote)43 b(the)g(orbit)f(on)g Fj(S)6 b Fo(\()p Fj(\013)q Fo(\))42 b(starting)f(at)h(p)s(oin)m(t)g(the)h(\()p Fj(k)s(;)17 b Fi(\000)p Fj(p)2491 683 y Fs(k)2534 668 y Fo(\))44 b Fi(')h Fj(V)2795 683 y Fs(k)2838 668 y Fo(|this)d(is)g (the)141 789 y(same)33 b(as)f(the)h(orbit)f(on)g Fj(S)39 b Fo(starting)31 b(at)h(\()p Fj(k)s(;)17 b Fi(\000)p Fj(p)1890 804 y Fs(k)1933 789 y Fo(\))33 b(with)f(slop)s(e)g Fj(\013)q Fo(.)287 950 y Ff(Pr)n(oof)50 b(of)g(Theorem)g(1.)152 b Fo(Let)44 b Fj(k)49 b(<)d(m)e Fo(b)s(e)g(t)m(w)m(o)g(natural)e(n)m (um)m(b)s(ers.)78 b(W)-8 b(e)141 1070 y(in)m(tro)s(duce)33 b(the)g(follo)m(wing)d(sets:)837 1257 y Fj(A)910 1272 y Fs(k)r(;m)1117 1257 y Fo(:=)83 b Fi(f)p Fj(\013)23 b Fi(j)f Fj(\015)1539 1272 y Fs(k)1581 1257 y Fo(\()p Fj(\013)q Fo(\))32 b(reac)m(hes)i(directly)e Fj(G)2523 1272 y Fs(m)2590 1257 y Fi(g)p Fo(;)587 b(\(1\))894 1496 y Fj(B)968 1511 y Fs(m)1117 1496 y Fo(:=)1303 1371 y Fs(m)p Fr(\000)p Fh(1)1324 1401 y Fg([)1315 1613 y Fs(k)r Fh(=1)1473 1496 y Fj(A)1546 1511 y Fs(k)r(;m)1670 1496 y Fj(:)1557 b Fo(\(2\))141 1768 y(Therefore,)391 1955 y Fj(B)470 1914 y Fs(c)465 1980 y(m)560 1955 y Fo(=)27 b Fi(f)p Fj(\013)c Fi(j)f Fo(no)32 b Fj(\015)1034 1970 y Fs(k)1076 1955 y Fo(\()p Fj(\013)q Fo(\))p Fj(;)49 b Fo(with)32 b(1)c Fi(\024)g Fj(k)j Fi(\024)d Fj(m)22 b 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b(The)29 b(only)f(thing)g(w)m(e)h(need)g(to)g(kno)m(w)g(ab)s(out)f (this)g(set)h(is)f(that)g(\()p Fj(A)2764 2540 y Fs(k)r(;m)2903 2525 y Fi(\\)14 b Fj(I)8 b Fo(\))27 b Fi(\022)h Fj(D)3285 2540 y Fs(m)3352 2525 y Fo(,)141 2646 y(whic)m(h)37 b(should)e(b)s(e)h (clear)g(b)m(y)g(construction)g(\(see)h(also)e(the)h(previous)h (remark\).)53 b(This)141 2766 y(fact)33 b(and)f(the)h(rep)s(eated)h (use)f(of)f(\(8\),)g(yield)1176 3068 y Fi(j)p Fj(A)1277 3083 y Fs(k)r(;m)1424 3068 y Fi(\\)22 b Fj(I)8 b Fi(j)28 b(\024)g(j)p Fj(D)1833 3083 y Fs(k)r Fh(+1)1965 3068 y Fi(j)2037 2943 y Fs(m)p Fr(\000)p Fh(1)2050 2973 y Fg(Y)2010 3185 y Fs(i)p Fh(=)p Fs(k)r Fh(+1)2234 3068 y Fj(\014)2289 3083 y Fs(i)2317 3068 y Fj(:)910 b Fo(\(9\))287 3377 y(Let)39 b(us)f(consider)h(our)f(sp)s(eci\014c)h(case:)55 b Fj(p)1812 3392 y Fs(n)p Fh(+1)1986 3377 y Fi(\024)38 b Fj(\025p)2207 3392 y Fs(n)2254 3377 y Fo(.)60 b(F)-8 b(rom)36 b(de\014nition)h(\(8\),)j(one)141 3497 y(v)m(eri\014es)34 b(that:)860 3739 y Fj(\014)915 3754 y Fs(n)990 3739 y Fo(=)1103 3671 y(2\(2\()p Fj(p)1326 3686 y Fs(n)p Fh(+1)1463 3671 y Fj(=p)1561 3686 y Fs(n)1607 3671 y Fo(\))22 b(+)g(1\))p 1103 3716 750 4 v 1152 3807 a(\(\()p Fj(p)1277 3822 y Fs(n)1324 3807 y Fj(=p)1422 3822 y Fs(n)p Fh(+1)1559 3807 y Fo(\))g(+)g(1\))1890 3739 y Fi(\024)2005 3671 y Fo(2\(2)p Fj(\025)f Fo(+)h(1\))p 2005 3716 400 4 v 2045 3807 a Fj(\025)2102 3778 y Fr(\000)p Fh(1)2218 3807 y Fo(+)g(1)2442 3739 y(=:)27 b Fj(\014)6 b(:)546 b Fo(\(10\))141 4001 y(It)44 b(is)f(going)g(to)g(b)s(e)h(crucial)f(later)g(that)h Fj(\014)49 b Fo(b)s(e)44 b(less)g(than)g(1.)77 b(F)-8 b(or)43 b Fj(\025)h Fo(p)s(ositiv)m(e,)i(this)141 4121 y(amoun)m(ts)33 b(to)f(4)p Fj(\025)760 4085 y Fh(2)821 4121 y Fo(+)22 b Fj(\025)g Fi(\000)h Fo(1)k Fj(<)h Fo(0,)k(whic)m(h)h (is)f(easily)g(solv)m(ed)h(b)m(y)998 4398 y(0)28 b Fj(<)f(\025)h(<)f (\025)1423 4413 y Fh(1)1490 4398 y Fo(:=)1631 4248 y Fi(p)p 1714 4248 98 4 v 83 x Fo(17)22 b Fi(\000)g Fo(1)p 1631 4375 352 4 v 1782 4467 a(8)2020 4398 y Fi(')28 b Fo(0)p Fj(:)p Fo(390)17 b Fj(:)g(:)g(:)e(;)684 b Fo(\(11\))141 4641 y(Going)29 b(bac)m(k)i(to)f(the)g(de\014nition)f(of)h Fj(D)1555 4656 y Fs(k)r Fh(+1)1688 4641 y Fo(,)g(and)h(to)e(\(9\),)i(w) m(e)g(see)g(that)f(it)f(is)h(p)s(ossible)f(to)141 4761 y(giv)m(e)36 b(an)f(estimate)g(of)g(the)h(measure)g(of)f Fj(D)1740 4776 y Fs(k)r Fh(+1)1909 4761 y Fo(in)f Fj(I)8 b Fo(,)37 b(for)e(large)f Fj(k)s Fo(.)53 b(In)36 b(fact,)g(when)h Fi(j)p Fj(I)8 b Fi(j)141 4882 y Fo(is)38 b(m)m(uc)m(h)h(bigger)e(than)i (2)p Fj(p)1141 4897 y Fs(k)1183 4882 y Fo(,)h(then)f(it)e(is)h(clear)g (\(from)f(Fig.)g(6,)i(sa)m(y\))h(that)e Fj(I)46 b Fo(includes)141 5002 y(v)m(ery)38 b(man)m(y)f(cones)g(of)f(size)h(2)p Fj(p)1295 5017 y Fs(k)r Fh(+1)1428 5002 y Fo(,)g(placed)g(on)f(a)g(2)p Fj(p)2122 5017 y Fs(k)2165 5002 y Fo(-p)s(erio)s(dic)e(arra)m(y)-8 b(.)56 b(As)37 b Fj(k)i Fo(gro)m(ws,)p eop %%Page: 20 20 20 19 bop 515 241 a Fo(20)1160 b Fn(M.)33 b(Degli)d(Esp)s(osti,)j(G.)f (Del)g(Magno)g(and)h(M.)g(Lenci)515 548 y Fo(the)e(densit)m(y)i(of)d (these)j(cones)f(in)e Fj(I)39 b Fo(can)32 b(b)s(e)f(made)g(arbitrarily) e(close)i(to)g Fj(p)3263 563 y Fs(k)r Fh(+1)3395 548 y Fj(=p)3493 563 y Fs(k)3563 548 y Fi(\024)e Fj(\025)p Fo(.)515 668 y(The)35 b(precise)g(statemen)m(t)g(then)g(is:)47 b(giv)m(en)34 b(an)m(y)i Fj(")30 b(>)h Fo(0,)j(there)i(exists)f(a)f Fj(q)h Fo(=)30 b Fj(q)t Fo(\()p Fj(")p Fo(\))g Fi(2)i Fc(N)515 789 y Fo(suc)m(h)i(that)1494 909 y Fi(8)p Fj(k)d Fi(\025)d Fj(q)t(;)114 b Fi(j)p Fj(D)2033 924 y Fs(k)r Fh(+1)2166 909 y Fi(j)27 b Fj(<)h Fo(\()p Fj(\025)22 b Fo(+)g Fj(")p Fo(\))17 b Fi(j)p Fj(I)8 b Fi(j)p Fj(:)515 1078 y Fo(Plugging)31 b(this)h(in)m(to)g(\(9\),)g(w)m(e)h(obtain)1293 1286 y Fi(8)p Fj(m)28 b(>)f(k)k Fi(\025)d Fj(q)102 b Fi(j)p Fj(A)1997 1301 y Fs(k)r(;m)2121 1286 y Fi(j)2149 1301 y Fs(I)2217 1286 y Fi(\024)28 b Fo(\()p Fj(\025)22 b Fo(+)g Fj(")p Fo(\))17 b Fj(\014)2699 1245 y Fs(m)p Fr(\000)p Fs(k)r Fr(\000)p Fh(1)2948 1286 y Fj(;)604 b Fo(\(12\))515 1495 y(ha)m(ving)33 b(used)i(\(10\))e(as)h(w)m(ell.)46 b(F)-8 b(or)33 b(the)h(other)f(v)-5 b(alues)34 b(of)f Fj(k)s Fo(,)h(w)m(e)h(ha)m(v)m(e)g(no)f(con)m(trol)e(o)m(v)m(er)515 1615 y Fj(C)585 1630 y Fs(k)655 1615 y Fo(:=)c Fi(j)p Fj(D)895 1630 y Fs(k)r Fh(+1)1027 1615 y Fi(j)p Fj(=)p Fi(j)p Fj(I)8 b Fi(j)31 b Fo(and)i(w)m(e)g(just)g(write)1430 1823 y Fi(8)p Fj(m;)17 b(q)32 b(>)c(k)100 b Fi(j)p Fj(A)2045 1838 y Fs(k)r(;m)2170 1823 y Fi(j)2198 1838 y Fs(I)2265 1823 y Fi(\024)28 b Fj(C)2440 1838 y Fs(k)2499 1823 y Fj(\014)2560 1782 y Fs(m)p Fr(\000)p Fs(k)r Fr(\000)p Fh(1)2810 1823 y Fj(:)742 b Fo(\(13\))515 2031 y(In)37 b(the)g(case)h Fj(A)1097 2046 y Fs(k)r(;k)r Fh(+1)1313 2031 y Fi(\\)26 b Fj(I)42 b Fo(=)35 b Fi(;)p Fo(,)j(whic)m(h)f(w)m(e)h (did)e(not)h(consider)g(b)s(efore,)h(\(12\)-\(13\))e(are)515 2152 y(trivial)30 b(consequences)36 b(of)c(the)h(fact)f(that)h Fj(A)2150 2167 y Fs(k)r(;m)2297 2152 y Fi(\\)22 b Fj(I)36 b Fo(=)27 b Fi(;)p Fo(,)33 b(for)f(all)e Fj(m)e(>)g(k)s Fo(.)661 2272 y(W)-8 b(e)34 b(mo)m(v)m(e)g(on)f(to)g(the)h(\014nal)e (estimation.)44 b(T)-8 b(ak)m(e)35 b Fj(m)29 b(>)g(q)t Fo(:)45 b(from)32 b(de\014nition)h(\(2\))f(w)m(e)515 2392 y(ha)m(v)m(e,)i(using)e(\(12\))g(and)h(\(13\),)1078 2695 y Fi(j)p Fj(B)1180 2710 y Fs(m)1246 2695 y Fi(j)1274 2710 y Fs(I)1397 2695 y Fi(\024)1567 2566 y Fs(q)r Fr(\000)p Fh(1)1558 2600 y Fg(X)1565 2812 y Fs(k)r Fh(=1)1718 2695 y Fi(j)p Fj(A)1819 2710 y Fs(k)r(;m)1944 2695 y Fi(j)1972 2710 y Fs(I)2033 2695 y Fo(+)2131 2570 y Fs(m)p Fr(\000)p Fh(1)2136 2600 y Fg(X)2144 2812 y Fs(k)r Fh(=)p Fs(q)2300 2695 y Fi(j)p Fj(A)2401 2710 y Fs(k)r(;m)2526 2695 y Fi(j)2554 2710 y Fs(I)2621 2695 y Fi(\024)1397 3050 y(\024)1567 2921 y Fs(q)r Fr(\000)p Fh(1)1558 2955 y Fg(X)1565 3167 y Fs(k)r Fh(=1)1718 3050 y Fj(C)1788 3065 y Fs(k)1847 3050 y Fj(\014)1908 3008 y Fs(m)p Fr(\000)p Fs(k)r Fr(\000)p Fh(1)2180 3050 y Fo(+)2278 2925 y Fs(m)p Fr(\000)p Fh(1)2282 2955 y Fg(X)2291 3167 y Fs(k)r Fh(=)p Fs(q)2431 3050 y Fo(\()p Fj(\025)22 b Fo(+)g Fj(")p Fo(\))17 b Fj(\014)2808 3008 y Fs(m)p Fr(\000)p Fs(k)r Fr(\000)p Fh(1)3084 3050 y Fi(\024)418 b Fo(\(14\))1397 3360 y Fi(\024)84 b Fj(o)p Fo(\(1\))21 b(+)1863 3292 y Fj(\025)h Fo(+)g Fj(")p 1859 3337 232 4 v 1859 3428 a Fo(1)g Fi(\000)h Fj(\014)2100 3360 y(;)515 3613 y Fo(as)37 b Fj(m)f Fi(!)f(1)p Fo(.)56 b(In)37 b(the)h(last)e(inequalit)m(y)g(w)m(e)i(ha)m(v)m(e)h(used)f(t)m (wice)f(the)h(fact)f(that)g Fj(\014)k(<)35 b Fo(1.)515 3733 y(W)-8 b(e)33 b(imp)s(ose)e(the)i(condition)1641 3923 y Fj(\025)p 1554 3968 V 1554 4059 a Fo(1)22 b Fi(\000)g Fj(\014)1823 3991 y Fo(=)2042 3923 y Fj(\025)p Fo(\()p Fj(\025)g Fo(+)g(1\))p 1936 3968 571 4 v 1936 4059 a Fi(\000)p Fo(4)p Fj(\025)2119 4030 y Fh(2)2181 4059 y Fi(\000)h Fj(\025)f Fo(+)g(1)2544 3991 y Fj(<)28 b Fo(1)p Fj(:)855 b Fo(\(15\))515 4249 y(F)-8 b(or)30 b Fj(\025)i Fo(as)f(in)g(\(11\),)g(the)h(denominator)e(is)h(p)s(ositiv)m(e,)g (hence)i(\(15\))e(can)h(b)s(e)g(rewritten)f(as)515 4369 y(5)p Fj(\025)621 4333 y Fh(2)682 4369 y Fo(+)22 b(2)p Fj(\025)g Fi(\000)g Fo(1)28 b Fj(<)f Fo(0,)33 b(whose)h(solutions)d (are)1547 4643 y(0)d Fj(<)f(\025)h(<)1925 4493 y Fi(p)p 2008 4493 49 4 v 82 x Fo(6)22 b Fi(\000)h Fo(1)p 1925 4620 303 4 v 2052 4711 a(5)2265 4643 y(=)28 b Fj(\025)2426 4658 y Fh(0)2493 4643 y Fj(<)g(\025)2654 4658 y Fh(1)2693 4643 y Fj(;)859 b Fo(\(16\))515 4882 y(as)37 b(in)f(the)i(statemen)m(t) f(of)g(the)g(lemma.)55 b(F)-8 b(or)36 b(these)j(v)-5 b(alues)37 b(of)f Fj(\025)p Fo(,)i(b)m(y)g(virtue)f(of)g(\(11\),)515 5002 y(1)12 b Fi(\000)g Fj(\014)34 b Fo(k)m(eeps)29 b(a)m(w)m(a)m(y)g (from)e(0.)41 b(Therefore)29 b(\(15\))e(implies)f(that,)i(for)f Fj(")h Fo(small)d(enough,)k(the)p eop %%Page: 21 21 21 20 bop 141 241 a Fn(In\014nite)33 b(Step)g(Billiards)2219 b Fo(21)141 548 y(last)31 b(term)f(in)h(\(14\))f(can)i(b)s(e)f(tak)m (en)h(less)g(than)f(a)g(certain)g Fj(\016)h(<)27 b Fo(1,)k(whence)i (the)f(pro)s(of)e(of)141 668 y(Lemma)h(2.)2536 b Fw(Q.E.D.)287 827 y Ff(Pr)n(oof)41 b(of)g(Lemma)f(1.)100 b Fo(Consider)36 b(a)e(v)m(ertex)j Fj(V)56 b Fo(of)34 b Fj(P)2436 842 y Fs(\013)2521 827 y Fo(and)g(let)h Fj(\015)2907 842 y Fs(V)2967 827 y Fo(\()p Fj(\013)q Fo(\))f(b)s(e)h(its)141 948 y(forw)m(ard)i(semi-orbit.)52 b(F)-8 b(or)35 b(a)h(\014xed)i (\014nite)d(sequence)k Fj(S)h Fo(:=)34 b(\()p Fj(S)2511 963 y Fh(1)2550 948 y Fj(;)17 b(S)2654 963 y Fh(2)2694 948 y Fj(;)g(:)g(:)g(:)e(S)2928 963 y Fs(`)2961 948 y Fo(\))37 b(of)e(sides,)141 1068 y(w)m(e)f(de\014ne)208 1249 y Fj(A)281 1264 y Fs(V)t(;m)412 1249 y Fo(\()p Fj(S)6 b Fo(\))27 b(:=)h Fi(f)p Fj(\013)23 b Fi(j)e Fj(\015)947 1264 y Fs(V)1008 1249 y Fo(\()p Fj(\013)q Fo(\))32 b(hits)g Fj(S)1429 1264 y Fh(1)1469 1249 y Fj(;)17 b(:)g(:)g(:)e(;)i(S)1747 1264 y Fs(`)1813 1249 y Fo(and)32 b(then)h(reac)m(hes)i(directly)d Fj(G)2996 1264 y Fs(m)3062 1249 y Fi(g)p Fj(:)67 b Fo(\(17\))141 1429 y(This)34 b(means)h(that)f(these)h(tra)5 b(jectories)34 b(do)g(not)h(hit)e(an)m(y)i(v)m(ertical)e(w)m(all)g(after)h(lea)m(ving) 141 1549 y Fj(S)201 1564 y Fs(`)268 1549 y Fo(and)f(b)s(efore)h(reac)m (hing)g Fj(G)1218 1564 y Fs(m)1284 1549 y Fo(.)47 b(Notice)33 b(the)h(similarities)29 b(with)k(de\014nition)g(\(1\).)46 b(As)34 b(a)141 1670 y(matter)21 b(of)h(fact,)i(if)d Fj(V)49 b Fo(=)28 b Fj(V)1108 1685 y Fs(k)1173 1670 y Fo(for)21 b(some)h Fj(k)s Fo(,)i(then)f Fj(A)1935 1685 y Fs(V)1976 1697 y Fd(k)2014 1685 y Fs(;m)2101 1670 y Fo(\()p Fi(;)p Fo(\))k(=)h Fj(A)2431 1685 y Fs(k)r(;m)2555 1670 y Fo(.)40 b(Ho)m(w)m(ev)m(er,)27 b(for)22 b(most)141 1790 y(sequences)33 b Fj(S)6 b Fo(,)30 b(\(17\))f(de\014nes)i(the)f (empt)m(y)h(set.)43 b(F)-8 b(or)29 b(example,)g Fj(S)36 b Fo(can)30 b(b)s(e)f Fk(inc)-5 b(omp)g(atible)141 1911 y Fo(in)28 b(the)h(sense)i(that)d(no)h(orbit)e(can)i(go)f(from)g Fj(S)1823 1926 y Fs(i)1879 1911 y Fo(to)h Fj(S)2055 1926 y Fs(i)p Fh(+1)2202 1911 y Fo(without)f(crossing)h(other)f(sides)141 2031 y(in)43 b(the)g(mean)m(time.)74 b(But,)46 b(ev)m(en)f(for)e (compatible)e(sequences,)49 b(if)42 b Fj(G)2757 2046 y Fs(m)2867 2031 y Fo(do)s(es)h(not)g(lie)141 2151 y(to)37 b(the)g(righ)m(t)f(of)h Fj(S)853 2166 y Fs(`)886 2151 y Fo(,)h(ob)m(viously)f Fj(A)1460 2166 y Fs(V)t(;m)1591 2151 y Fo(\()p Fj(S)6 b Fo(\))35 b(=)g Fi(;)p Fo(.)56 b(T)-8 b(o)37 b(a)m(v)m(oid)g(this)f(latter)g(case,)j(w)m(e)f(\014x)141 2272 y Fj(m)226 2287 y Fs(o)308 2272 y Fo(=)43 b Fj(m)512 2287 y Fs(o)551 2272 y Fo(\()p Fj(V)5 b(;)17 b(S)6 b Fo(\))41 b(bigger)g(than)h(the)h(largest)e Fj(Y)21 b Fo(-co)s(ordinate)40 b(in)h Fj(S)2663 2287 y Fs(`)2696 2272 y Fo(.)72 b(F)-8 b(or)41 b Fj(m)i Fi(\025)h Fj(m)3313 2287 y Fs(o)3352 2272 y Fo(,)141 2392 y Fj(A)214 2407 y Fs(V)t(;m)p Fh(+1)436 2392 y Fo(\()p Fj(S)6 b Fo(\))27 b Fi(\032)h Fj(A)783 2407 y Fs(V)t(;m)914 2392 y Fo(\()p Fj(S)6 b Fo(\).)44 b(Let)32 b(us)i(then)f(de\014ne)358 2573 y Fj(B)432 2588 y Fs(V)493 2573 y Fo(\()p Fj(S)6 b Fo(\))83 b(:=)g Fi(f)p Fj(\013)23 b Fi(j)f Fj(\015)1140 2588 y Fs(V)1200 2573 y Fo(\()p Fj(\013)q Fo(\))32 b(hits)g Fj(S)1621 2588 y Fh(1)1661 2573 y Fj(;)17 b(:)g(:)g(:)f(;)h(S)1940 2588 y Fs(`)2005 2573 y Fo(and)32 b(then)i(escap)s(es)g(to)e Fi(1g)27 b Fo(=)718 2794 y(:=)976 2669 y Fr(1)956 2699 y Fg(\\)904 2908 y Fs(m)p Fh(=)p Fs(m)1083 2916 y Fd(o)1135 2794 y Fj(A)1208 2809 y Fs(V)t(;m)1339 2794 y Fo(\()p Fj(S)6 b Fo(\))p Fj(;)1698 b Fo(\(18\))141 3064 y(W)-8 b(orking)27 b(in)g(the)h(unfolded-billiard)23 b(plane)k(and)h(iden)m (tifying)e(directions)h(with)g(orbits,)141 3185 y(it)46 b(is)h(not)g(hard)g(to)g(realize)f(that)h Fj(A)1547 3200 y Fs(V)t(;m)1674 3208 y Fd(o)1713 3185 y Fo(\()p Fj(S)6 b Fo(\))47 b(is)g(made)g(up)g(of)g(a)g(\014nite)g(n)m(um)m(b)s(er)g(of) 141 3305 y(in)m(terv)-5 b(als/cones,)37 b(eac)m(h)g(of)e(whic)m(h)h (reac)m(hes)i(a)d(cop)m(y)i(of)e Fj(G)2308 3320 y Fs(m)2370 3328 y Fd(o)2445 3305 y Fo(after)h(hitting)e(a)h(certain)141 3425 y(sequence)i(of)c(sides)i(\()p Fj(S)993 3440 y Fh(1)1032 3425 y Fj(;)17 b(:)g(:)g(:)f(S)1267 3440 y Fs(`)1300 3425 y Fj(;)h(S)1404 3440 y Fs(`)p Fh(+1)1527 3425 y Fj(;)g(:)g(:)g(:)e(S)1761 3440 y Fs(n)1808 3425 y Fo(\))34 b(\(the)g(\014rst)h Fj(`)f Fo(sides)g(are)g(common)e(to)i(all)141 3546 y(cones)45 b(and)e(the)h(others)g(can)f(only)g(b)s(e)h(horizon)m (tal\).)74 b(It)43 b(is)g(p)s(ossible)g(that)g(some)g(of)141 3666 y(these)37 b(b)s(eams)e(of)g(tra)5 b(jectories)35 b(in)m(tersect)h(the)g(corresp)s(onding)f(cop)m(y)i(of)d Fj(G)2944 3681 y Fs(m)3006 3689 y Fd(o)3081 3666 y Fo(only)h(in)141 3787 y(a)f(prop)s(er)g(sub-segmen)m(t.)50 b(Let)34 b(us)h(\014x)f(this) g(situation)f(b)m(y)i(enlarging)d(an)m(y)j(suc)m(h)h(b)s(eam)141 3907 y(un)m(til)c(it)h(co)m(v)m(ers)i(the)f(whole)g(segmen)m(t.)46 b(W)-8 b(e)34 b(call)e Fj(D)g Fi(\023)e Fj(A)2268 3922 y Fs(V)t(;m)2395 3930 y Fd(o)2434 3907 y Fo(\()p Fj(S)6 b Fo(\))33 b(the)h(union)f(of)g(these)141 4027 y(new)h(cones.)287 4148 y(Pro)s(ceeding)e(exactly)f(as)h(in)e(the)i(pro)s(of)e(of)h(Lemma) f(2)h(\(see)h(in)f(particular)e(\(9\))i(and)141 4268 y(\(13\)\))h(w)m(e)i(get,)e(for)g Fj(m)c(>)g(m)1169 4283 y Fs(o)1207 4268 y Fo(,)1262 4449 y Fi(j)p Fj(A)1363 4464 y Fs(V)t(;m)1494 4449 y Fo(\()p Fj(S)6 b Fo(\))p Fi(j)27 b(\024)h(j)p Fj(D)s Fi(j)17 b Fj(\014)2014 4407 y Fs(m)p Fr(\000)p Fs(m)2193 4415 y Fd(o)2231 4449 y Fj(;)141 4629 y Fo(with)34 b Fj(\014)i(<)31 b Fo(1.)48 b(Hence,)37 b(for)c Fj(m)e Fi(!)g(1)p Fo(,)j Fi(j)p Fj(A)1666 4644 y Fs(V)t(;m)1797 4629 y Fo(\()p Fj(S)6 b Fo(\))p Fi(j)30 b(!)g Fo(0.)49 b(By)35 b(\(18\),)f Fi(j)p Fj(B)2744 4644 y Fs(V)2805 4629 y Fo(\()p Fj(S)6 b Fo(\))p Fi(j)30 b Fo(=)g(0,)35 b(and)141 4749 y(the)e(set)1424 4775 y Fg([)1454 4975 y Fd(V)1396 5032 y Fe(v)n(ertex)1659 4775 y Fg([)1609 4976 y Fd(S)i Fe(\014nite)1598 5032 y(sequence)1857 4870 y Fj(B)1931 4885 y Fs(V)1992 4870 y Fo(\()p Fj(S)6 b Fo(\))p eop %%Page: 22 22 22 21 bop 515 241 a Fo(22)1160 b Fn(M.)33 b(Degli)d(Esp)s(osti,)j(G.)f (Del)g(Magno)g(and)h(M.)g(Lenci)515 548 y Fo(has)42 b(measure)f(zero.) 71 b(This)41 b(is)g(the)h(set)g(of)f(directions)g(as)h(in)e(the)i (statemen)m(t)g(of)f(the)515 668 y(Lemma)31 b(2,)h(whic)m(h)h(w)m(e)h (ha)m(v)m(e)g(no)m(w)f(pro)m(v)m(ed.)1343 b Fw(Q.E.D.)515 1049 y Fm(4)161 b(Generic)53 b(ergo)t(dicit)l(y)515 1268 y Fo(W)-8 b(e)35 b(b)s(egin)e(this)i(section)f(b)m(y)i(giving)d(some)h (de\014nitions)g(and)h(a)f(tec)m(hnical)g(lemma.)47 b(In)515 1388 y Fc(R)581 1352 y Fh(2)626 1388 y Fo(,)33 b(let)e Fj(\032)i Fo(b)s(e)g(the)g(Euclidean)f(metric)f(and)i Fj(A)f Fo(the)h(area.)44 b(Also,)32 b(recall)f(the)i(de\014nition)515 1509 y(of)k Fi(S)46 b Fo(from)36 b(Section)i(2.2;)i(later)d(on)h(w)m(e) h(will)d(need)j(to)e(use)i Fi(S)2815 1524 y Fh(0)2855 1509 y Fo(,)g(the)f(collection)e(of)i(all)515 1629 y(\014nite)32 b Fj(P)41 b Fi(2)28 b(S)41 b Fo(with)32 b(rational)e(heigh)m(ts.)515 1852 y Fl(De\014nition)36 b(1)49 b Fk(Given)38 b Fj(P)s(;)17 b(Q)35 b Fi(2)h(S)7 b Fk(,)40 b Fj(#)c Fi(2)p Fo(]0)p Fj(;)17 b(\031)t(=)p Fo(2[)p Fk(,)39 b Fj(x)d Fi(2)f Fj(S)2682 1815 y Fs(P)2741 1852 y Fo(\()p Fj(#)p Fo(\))25 b Fi(\\)h Fj(S)3057 1815 y Fs(Q)3116 1852 y Fo(\()p Fj(#)p Fo(\))40 b Fk(and)e Fj(")d(>)g Fo(0)p Fk(,)515 1972 y(let)853 2187 y Fj(I)8 b Fo(\()p Fj(P)s(;)17 b(Q;)g(x;)g(")p Fo(\))26 b(:=)i Fi(f)p Fj(t)f Fi(2)i Fo([0)p Fj(;)17 b Fo(1)p Fj(=")p Fo(])k Fi(j)g Fj(\032)p Fo(\()p Fj(\036)2227 2146 y Fs(P)2227 2212 y(t)2286 2187 y Fo(\()p Fj(x)p Fo(\))p Fj(;)c(\036)2519 2140 y Fs(Q)2519 2209 y(t)2578 2187 y Fo(\()p Fj(x)p Fo(\)\))28 b Fj(>)g(")p Fi(g)p Fj(;)577 b Fo(\(19\))853 2332 y Fj(G)p Fo(\()p Fj(P)s(;)17 b(Q;)g(#;)g(")p Fo(\))26 b(:=)i Fi(f)p Fj(x)g Fi(2)g Fj(S)1834 2291 y Fs(P)1892 2332 y Fo(\()p Fj(#)p Fo(\))23 b Fi(\\)f Fj(S)2202 2291 y Fs(Q)2262 2332 y Fo(\()p Fj(#)p Fo(\))g Fi(j)g(j)p Fj(I)8 b Fo(\()p Fj(P)s(;)17 b(Q;)g(x;)g(")p Fo(\))p Fi(j)26 b Fj(<)i(")p Fi(g)p Fj(;)300 b Fo(\(20\))853 2478 y Fj(E)6 b Fo(\()p Fj(P)s(;)17 b(Q;)g(")p Fo(\))26 b(:=)1441 2397 y Fg(\010)1499 2478 y Fj(#)i Fi(2)p Fo(]0)p Fj(;)17 b(\031)t(=)p Fo(2[)22 b Fi(j)g Fj(A)p Fo(\()p Fj(G)p Fo(\()p Fj(P)s(;)17 b(Q;)g(#;)g(")p Fo(\)\))26 b Fi(\025)i Fj(A)p Fo(\()p Fj(S)3014 2436 y Fs(P)r(;Q)3140 2478 y Fo(\()p Fj(#)p Fo(\)\))23 b Fi(\000)f Fj(")3479 2397 y Fg(\011)3554 2478 y Fj(:)n Fo(\(21\))515 2700 y Fl(Lemma)37 b(3)49 b Fk(F)-7 b(or)34 b(any)g Fj(P)41 b Fi(2)29 b(S)42 b Fk(and)35 b Fj(")27 b(>)g Fo(0)p Fk(,)35 b(we)f(have)1711 2915 y Fo(lim)1689 2977 y Fs(Q)p Fr(!)p Fs(P)1886 2915 y Fi(j)p Fj(E)6 b Fo(\()p Fj(P)s(;)17 b(Q;)g(")p Fo(\))p Fi(j)26 b Fo(=)i(1)p Fj(:)661 3180 y Ff(Pr)n(oof)51 b(of)g(Lemma)f(3.)156 b Fo(It)44 b(is)g(enough)h(to)f (pro)m(v)m(e)h(the)g(statemen)m(t)f(for)g(an)m(y)515 3300 y(sequence)h(con)m(v)m(erging)d(to)g Fj(P)14 b Fo(.)72 b(Let)42 b Fi(f)p Fj(P)2025 3315 y Fs(n)2072 3300 y Fi(g)f Fo(b)s(e)i(suc)m(h)g(a)f(sequence.)75 b(Fix)42 b Fj(#)i Fi(2)p Fo(]0)p Fj(;)17 b(\031)t(=)p Fo(2[)515 3421 y(and)36 b Fj(")e(>)f Fo(0.)55 b(Only)35 b(coun)m(tably)i(man)m(y)f(orbits)g(of) f Fj(S)2456 3384 y Fs(P)2515 3421 y Fo(\()p Fj(#)p Fo(\))h(con)m(tain)g (a)g(singular)f(v)m(ertex)515 3541 y Fj(V)21 b Fo(.)49 b(Let)35 b Fj(x)c Fi(2)h Fj(S)1096 3505 y Fs(P)1189 3541 y Fo(b)s(e)i(a)h(p)s(oin)m(t)e(that)i(do)s(es)f(not)h(b)s(elong)e(to)h (an)m(y)i(of)e(these)h(orbits.)49 b(In)35 b(a)515 3661 y(\014nite)e(in)m(terv)-5 b(al)32 b(of)h(time,)f(its)h(tra)5 b(jectory)34 b(can)g(get)f(close)h(only)f(to)g(a)g(\014nite)g(n)m(um)m (b)s(er)h(of)515 3782 y(singular)28 b(v)m(ertices.)43 b(Therefore)31 b Fj(\016)1752 3797 y Fs(")1789 3782 y Fo(,)f(the)f(distance)h(b)s(et)m(w)m(een)h(the)f Fi([)2991 3797 y Fh(0)p Fr(\024)p Fs(t)p Fr(\024)p Fh(1)p Fs(=")3286 3782 y Fj(\036)3344 3746 y Fs(P)3344 3808 y(#;t)3435 3782 y Fo(\()p Fj(x)p Fo(\))f(and)515 3902 y Fi(V)8 b Fo(,)30 b(is)g(p)s(ositiv)m(e.)42 b(Let)30 b Fj(N)1385 3917 y Fs(")1452 3902 y Fo(b)s(e)g(the)g(n)m(um)m(b)s(er)g(of)g (collisions)d(of)j Fi([)2793 3918 y Fh(0)p Fr(\024)p Fs(t)p Fr(\024)p Fh(1)p Fs(=")3087 3902 y Fj(\036)3145 3866 y Fs(P)3145 3928 y(#;t)3236 3902 y Fo(\()p Fj(x)p Fo(\))h(with)e(the)515 4022 y(horizon)m(tal)i(sides)i(of)f Fj(S)1384 3986 y Fs(P)1442 4022 y Fo(\()p Fj(#)p Fo(\).)661 4143 y(The)c(t)m(w)m(o)g(surfaces)h Fj(S)1463 4107 y Fs(P)1521 4143 y Fo(\()p Fj(#)p Fo(\))f(and)f Fj(S)1932 4107 y Fs(P)1977 4115 y Fd(n)2023 4143 y Fo(\()p Fj(#)p Fo(\))h(o)m(v)m(erlap)f(\(as)h(subsets)h(of)e Fc(R)3174 4107 y Fh(2)3219 4143 y Fo(\).)42 b(If)27 b Fj(x)h Fi(2)g Fj(S)3661 4107 y Fs(P)3706 4115 y Fd(n)515 4263 y Fo(as)g(w)m(ell,)g (then)g(the)g(set)g Fi([)1444 4279 y Fh(0)p Fr(\024)p Fs(t)p Fr(\024)p Fh(1)p Fs(=")1739 4263 y Fj(\036)1797 4221 y Fs(P)1842 4229 y Fd(n)1797 4291 y Fs(#;t)1888 4263 y Fo(\()p Fj(x)p Fo(\))g(represen)m(ts)i(a)d(\014nite)h(piece)g (of)f(the)h(tra)5 b(jectory)515 4400 y(of)27 b Fj(x)i Fo(in)e(the)i(surface)g Fj(S)1368 4364 y Fs(P)1413 4372 y Fd(n)1459 4400 y Fo(\()p Fj(#)p Fo(\).)42 b(W)-8 b(e)29 b(w)m(an)m(t)g(to)f(estimate)f Fj(\032)p Fo(\()p Fj(\036)2706 4364 y Fs(P)2706 4425 y(t)2765 4400 y Fo(\()p Fj(x)p Fo(\))p Fj(;)17 b(\036)2998 4358 y Fs(P)3043 4366 y Fd(n)2998 4423 y Fs(t)3089 4400 y Fo(\()p Fj(x)p Fo(\)\))28 b(for)g(0)f Fi(\024)h Fj(t)g Fi(\024)515 4521 y Fo(1)p Fj(=")p Fo(.)66 b(Since)40 b(there)h(is)f(a)g(natural)g(corresp)s(ondence)i(b)s(et)m(w) m(een)g(the)f(sides)g(of)f(the)h(t)m(w)m(o)515 4641 y(surfaces,)27 b(when)e(w)m(e)h(sa)m(y)f(that)f(the)g(t)m(w)m(o)h(tra)5 b(jectories)24 b(hit)g(the)g(same)g(sequence)j(of)d(sides,)515 4761 y(w)m(e)42 b(mean)e(corresp)s(onding)g(sides.)69 b(Notice)40 b(that)h Fj(\032)p Fo(\()p Fj(\036)2546 4725 y Fs(P)2546 4786 y(t)2605 4761 y Fo(\()p Fj(x)p Fo(\))p Fj(;)17 b(\036)2838 4719 y Fs(P)2883 4727 y Fd(n)2838 4784 y Fs(t)2929 4761 y Fo(\()p Fj(x)p Fo(\)\))41 b(sta)m(ys)h(constan) m(t)515 4882 y(when)36 b(neither)f(orbit)f(crosses)i(an)m(y)g(sides.)51 b(Also,)35 b(as)g(long)f(as)h Fj(\036)2901 4846 y Fs(P)2901 4906 y(t)2960 4882 y Fo(\()p Fj(x)p Fo(\))g(and)g Fj(\036)3376 4839 y Fs(P)3421 4847 y Fd(n)3376 4904 y Fs(t)3467 4882 y Fo(\()p Fj(x)p Fo(\))g(hit)515 5002 y(the)e(same)h(sequence)i(of)d (sides,)h(ev)m(ery)h(time)d(that)h(there)h(is)f(collision)d(at)j(a)g (horizon)m(tal)p eop %%Page: 23 23 23 22 bop 141 241 a Fn(In\014nite)33 b(Step)g(Billiards)2219 b Fo(23)141 548 y(side,)30 b(the)f(distance)g Fj(\032)p Fo(\()p Fj(\036)1047 512 y Fs(P)1047 573 y(t)1105 548 y Fo(\()p Fj(x)p Fo(\))p Fj(;)17 b(\036)1338 505 y Fs(P)1383 513 y Fd(n)1338 570 y Fs(t)1430 548 y Fo(\()p Fj(x)p Fo(\)\))28 b(increases)i(b)m(y)g(a)e(term)g Fj(h)g Fi(\024)g Fo(2)p Fj(d)p Fo(\()p Fj(P)s(;)17 b(P)2969 563 y Fs(n)3015 548 y Fo(\).)42 b(Hence)141 668 y(the)24 b(maxim)m(um)e(distance)i(b)s (et)m(w)m(een)i(the)e(t)m(w)m(o)h(tra)5 b(jectories)23 b(in)g(the)i(in)m(terv)-5 b(al)22 b Fj(t)28 b Fi(2)g Fo([0)p Fj(;)17 b Fo(1)p Fj(=")p Fo(])141 789 y(is)34 b Fi(\024)e Fo(2)p Fj(d)p Fo(\()p Fj(P)s(;)17 b(P)661 804 y Fs(n)707 789 y Fo(\))p Fj(N)823 804 y Fs(")860 789 y Fo(.)50 b(Therefore,)36 b(b)m(y)g(de\014nition)d(of)i Fj(\016)2138 804 y Fs(")2175 789 y Fo(,)g(if)f Fj(d)p Fo(\()p Fj(P)s(;)17 b(P)2591 804 y Fs(n)2636 789 y Fo(\))32 b Fj(<)f(\016)2856 804 y Fs(")2893 789 y Fj(=)p Fo(\(2)p Fj(N)3107 804 y Fs(")3143 789 y Fo(\),)36 b(the)141 909 y(tra)5 b(jectories)33 b(encoun)m(ter)h(the)f(same)f(sequence)k(of)c (sides)h(for)f(0)27 b Fi(\024)h Fj(t)g Fi(\024)h Fo(1)p Fj(=")p Fo(.)287 1030 y(Since)34 b Fj(d)p Fo(\()p Fj(P)s(;)17 b(P)805 1045 y Fs(n)851 1030 y Fo(\))30 b Fi(!)f Fo(0)k(as)h Fj(n)c Fi(!)f(1)p Fo(,)34 b(w)m(e)h(can)f(\014nd)g(an)f Fj(m)p Fo(\()p Fj(x;)17 b(")p Fo(\))30 b Fj(>)f Fo(0)34 b(suc)m(h)h(that)f(the)141 1150 y(previous)g(inequalit)m(y)f(is)g (satis\014ed)h(for)e(all)g Fj(n)d(>)h(m)p Fo(\()p Fj(x;)17 b(")p Fo(\).)46 b(It)33 b(is)g(clear)g(no)m(w)h(that,)g(as)g Fj(n)141 1271 y Fo(gro)m(ws)k(larger,)g(the)g(distance)g(b)s(et)m(w)m (een)i(the)e(t)m(w)m(o)g(tra)5 b(jectories)38 b(decreases.)61 b(W)-8 b(e)38 b(con-)141 1391 y(clude)33 b(that,)f(for)g(p)s(oin)m(ts)h Fj(x)28 b Fi(2)g Fj(S)1320 1355 y Fs(P)1378 1391 y Fo(\()p Fj(#)p Fo(\))33 b(with)f(non-singular)f(p)s(ositiv)m(e)h(semi-tra)5 b(jectory)-8 b(,)1078 1613 y(lim)1054 1673 y Fs(n)p Fr(!1)1317 1613 y Fo(max)1271 1679 y Fh(0)p Fr(\024)p Fs(t)p Fr(\024)p Fh(1)p Fs(=")1561 1613 y Fj(\032)p Fo(\()p Fj(\036)1707 1572 y Fs(P)1707 1637 y(t)1766 1613 y Fo(\()p Fj(x)p Fo(\))p Fj(;)17 b(\036)1999 1572 y Fs(P)2044 1580 y Fd(n)1999 1637 y Fs(t)2090 1613 y Fo(\()p Fj(x)p Fo(\)\))28 b(=)f(0)p Fj(:)141 1897 y Fo(As)40 b(a)e(consequence)k(w)m(e)e(ha)m(v)m(e)g(that) f(a.e.)62 b Fj(x)39 b Fi(2)g Fj(S)2008 1861 y Fs(P)2067 1897 y Fo(\()p Fj(#)p Fo(\))g(b)s(elongs)f(to)h Fj(G)p Fo(\()p Fj(P)s(;)17 b(P)3009 1912 y Fs(n)3055 1897 y Fj(;)g(#;)g(")p Fo(\))38 b(if)141 2017 y Fj(n)p Fo(,)i(whic)m(h)e(dep)s (ends)h(on)f Fj(x)p Fo(,)i(is)d(su\016cien)m(tly)i(large.)58 b(Therefore)39 b Fj(A)p Fo(\()p Fj(G)p Fo(\()p Fj(P)s(;)17 b(P)2930 2032 y Fs(n)2976 2017 y Fj(;)g(#;)g(")p Fo(\)\))36 b Fi(!)141 2138 y Fj(A)p Fo(\()p Fj(S)318 2102 y Fs(P)377 2138 y Fo(\()p Fj(#)p Fo(\)\))j(as)g Fj(n)f Fi(!)g(1)p Fo(.)62 b(Being)39 b(this)f(true)h(for)g(all)e Fj(#)h Fi(2)p Fo(]0)p Fj(;)17 b(\031)t(=)p Fo(2[,)40 b(w)m(e)g(\014nally)e (obtain)141 2258 y Fi(j)p Fj(E)6 b Fo(\()p Fj(P)s(;)17 b(P)458 2273 y Fs(n)504 2258 y Fj(;)g(")p Fo(\))p Fi(j)27 b(!)g Fo(1)32 b(as)h Fj(n)28 b Fi(!)f(1)p Fo(,)32 b(for)g(an)m(y)i Fj(")27 b(>)h Fo(0.)1127 b Fw(Q.E.D.)287 2430 y Fo(W)-8 b(e)33 b(are)g(no)m(w)g(in)f(p)s(osition)f(to)h(attac)m(k)h(the)g(main) e(pro)s(of)h(of)g(this)g(section:)287 2602 y Ff(Pr)n(oof)50 b(of)g(Theorem)f(3.)150 b Fo(Cho)s(ose)44 b Fj(")1944 2617 y Fs(n)2037 2602 y Fj(>)h Fo(0)e(suc)m(h)i(that)e(lim)2838 2617 y Fs(n)p Fr(!1)3043 2602 y Fj(")3089 2617 y Fs(n)3181 2602 y Fo(=)j(0.)141 2722 y(Let)37 b Fi(f)p Fj(f)418 2737 y Fs(i)446 2722 y Fi(g)496 2737 y Fs(i)p Fr(2)p Fq(N)656 2722 y Fo(b)s(e)g(a)g(coun)m(table)f(collection)f(of)i(con)m (tin)m(uous)g(functions)g(with)f(compact)141 2843 y(supp)s(ort)d(and)f (let)g(it)f(b)s(e)i(dense)h(in)d Fj(L)1507 2806 y Fh(1)1547 2843 y Fo(\()p Fc(R)1651 2806 y Fh(2)1696 2843 y Fo(\).)44 b(F)-8 b(or)31 b(eac)m(h)j(step)f(p)s(olygon)e Fj(P)14 b Fo(,)32 b Fj(f)2968 2806 y Fs(P)2957 2867 y(i)3059 2843 y Fo(denotes)141 2963 y(the)d(restriction)f(of)h Fj(f)924 2978 y Fs(i)981 2963 y Fo(to)g Fj(S)1163 2927 y Fs(P)1221 2963 y Fo(\()p Fj(#)p Fo(\),)h(corrected)g(to)f(ob)s(ey)g (the)g(iden)m(ti\014cations)f(on)h Fj(S)3160 2927 y Fs(P)3219 2963 y Fo(\()p Fj(#)p Fo(\).)141 3083 y(These)39 b(corrections)d(o)s (ccur)h(on)g(a)f(set)h(of)g(zero)g(Leb)s(esgue)g(measure)g(in)f Fc(R)2892 3047 y Fh(2)2938 3083 y Fo(,)h(therefore)141 3204 y Fi(f)p Fj(f)250 3168 y Fs(P)239 3228 y(i)308 3204 y Fi(g)358 3219 y Fs(i)p Fr(2)p Fq(N)514 3204 y Fo(is)32 b(dense)i(in)e Fj(L)1058 3168 y Fh(1)1098 3204 y Fo(\()p Fj(S)1202 3168 y Fs(P)1260 3204 y Fo(\()p Fj(#)p Fo(\)\))h(for)f(eac)m (h)h Fj(#)c Fi(2)p Fo(]0)p Fj(;)17 b(\031)t(=)p Fo(2[.)287 3325 y(Giv)m(en)33 b Fj(P)41 b Fi(2)28 b(S)7 b Fo(,)34 b Fj(x)28 b Fi(2)g Fj(S)1139 3289 y Fs(P)1197 3325 y Fo(\()p Fj(#)p Fo(\))33 b(and)g Fj(i)28 b Fi(2)g Fc(N)9 b Fo(,)38 b(let)32 b(us)h(in)m(tro)s(duce)474 3614 y Fj(B)553 3573 y Fs(P)548 3639 y(T)612 3614 y Fo(\()p Fj(#;)17 b(i;)g(x)p Fo(\))28 b(:=)1079 3469 y Fg(\014)1079 3529 y(\014)1079 3589 y(\014)1079 3649 y(\014)1133 3547 y Fo(1)p 1122 3591 71 4 v 1122 3682 a Fj(T)1220 3478 y Fg(Z)1319 3505 y Fs(T)1275 3704 y Fh(0)1391 3614 y Fj(f)1450 3573 y Fs(P)1439 3639 y(i)1531 3614 y Fi(\016)22 b Fj(\036)1661 3573 y Fs(P)1661 3639 y(t)1719 3614 y Fo(\()p Fj(x)p Fo(\))p Fj(dt)g Fi(\000)2247 3547 y Fo(1)p 2068 3591 407 4 v 2068 3682 a Fj(A)p Fo(\()p Fj(S)2245 3653 y Fs(P)2303 3682 y Fo(\()p Fj(#)p Fo(\)\))2501 3478 y Fg(Z)2557 3704 y Fs(S)2604 3685 y Fd(P)2654 3704 y Fh(\()p Fs(#)p Fh(\))2772 3614 y Fj(f)2831 3573 y Fs(P)2820 3639 y(i)2889 3614 y Fj(dA)3013 3469 y Fg(\014)3013 3529 y(\014)3013 3589 y(\014)3013 3649 y(\014)141 3899 y Fo(and)548 4021 y Fj(C)625 3980 y Fs(P)618 4046 y(T)683 4021 y Fo(\()p Fj(#;)17 b(n)p Fo(\))28 b(:=)1077 3941 y Fg(\010)1135 4021 y Fj(x)g Fi(2)g Fj(S)1378 3980 y Fs(P)1437 4021 y Fo(\()p Fj(#)p Fo(\))22 b Fi(j)g Fj(B)1721 3980 y Fs(P)1716 4046 y(T)1780 4021 y Fo(\()p Fj(#;)17 b(i;)g(x)p Fo(\))28 b Fi(\024)g Fj(")2268 4036 y Fs(n)2315 4021 y Fj(;)39 b(i)28 b Fo(=)f(1)p Fj(;)17 b(:)g(:)g(:)f(;)h(n)2871 3941 y Fg(\011)2945 4021 y Fj(:)141 4198 y Fo(If)37 b Fj(P)48 b Fi(2)35 b(S)515 4213 y Fh(0)555 4198 y Fo(,)j(the)f(billiard) c(\015o)m(w)k Fj(\036)1397 4161 y Fs(P)1397 4223 y(#;t)1525 4198 y Fo(is)f(uniquely)h(ergo)s(dic)f(for)g(all)f(irrational)e Fj(#)k Fo(\(i.e.,)141 4318 y(tan)17 b Fj(#)33 b Fi(62)h Fc(Q)11 b Fo(\).)59 b(Th)m(us)37 b(for)e(ev)m(ery)j Fj(n)33 b(>)g Fo(0)j(and)f(ev)m(ery)j(irrational)32 b Fj(#)i Fi(2)p Fo(]0)p Fj(;)17 b(\031)t(=)p Fo(2[,)36 b(w)m(e)h(ha)m(v)m(e)141 4438 y(that)32 b(lim)488 4453 y Fs(T)10 b Fr(!)p Fh(+)p Fr(1)756 4438 y Fj(A)p Fo(\()p Fj(C)944 4402 y Fs(P)937 4464 y(T)1002 4438 y Fo(\()p Fj(#;)17 b(n)p Fo(\)\))28 b(=)g Fj(A)p Fo(\()p Fj(S)1584 4402 y Fs(P)1642 4438 y Fo(\()p Fj(#)p Fo(\)\).)44 b(Let)562 4660 y Fj(D)646 4619 y Fs(P)643 4685 y(T)705 4660 y Fo(\()p Fj(n)p Fo(\))28 b(:=)997 4579 y Fg(\010)1055 4660 y Fj(#)g Fi(2)p Fo(]0)p Fj(;)17 b(\031)t(=)p Fo(2[)22 b Fi(j)g Fj(A)p Fo(\()p Fj(C)1770 4619 y Fs(P)1763 4685 y(T)1829 4660 y Fo(\()p Fj(#;)17 b(n)p Fo(\)\))28 b Fi(\025)g Fj(A)p Fo(\()p Fj(S)2412 4619 y Fs(P)2470 4660 y Fo(\()p Fj(#)p Fo(\)\))23 b Fi(\000)f Fj(")2809 4675 y Fs(n)2856 4579 y Fg(\011)2931 4660 y Fj(:)141 4882 y Fo(Then)38 b(w)m(e)g(can)f(c)m(ho)s(ose)h(a)f Fj(T)1188 4897 y Fs(n)1235 4882 y Fo(\()p Fj(P)14 b Fo(\))34 b Fi(\025)i Fj(n)h Fo(suc)m(h)h(that)f Fi(j)p Fj(D)2182 4846 y Fs(P)2179 4907 y(T)2240 4882 y Fo(\()p Fj(n)p Fo(\))p Fi(j)e Fj(>)g Fo(1)25 b Fi(\000)g Fj(")2770 4897 y Fs(n)2854 4882 y Fo(for)36 b(an)m(y)i Fj(T)49 b Fi(\025)141 5002 y Fj(T)198 5017 y Fs(n)245 5002 y Fo(\()p Fj(P)14 b Fo(\).)41 b(Since)26 b Fj(f)762 5017 y Fh(1)802 5002 y Fj(;)17 b(f)894 5017 y Fh(2)933 5002 y Fj(;)g(:)g(:)g(:)f(;)h(f)1200 5017 y Fs(n)1273 5002 y Fo(are)26 b(uniformly)e(con)m(tin)m(uous)j(on)f Fc(R)2544 4966 y Fh(2)2590 5002 y Fo(,)h(there)g(is)f(an)g Fj(r)3152 5017 y Fs(n)3227 5002 y Fj(>)h Fo(0)p eop %%Page: 24 24 24 23 bop 515 241 a Fo(24)1160 b Fn(M.)33 b(Degli)d(Esp)s(osti,)j(G.)f (Del)g(Magno)g(and)h(M.)g(Lenci)515 548 y Fo(for)f(whic)m(h)h Fi(j)p Fj(f)1019 563 y Fs(i)1047 548 y Fo(\()p Fj(x)p Fo(\))22 b Fi(\000)h Fj(f)1348 563 y Fs(i)1376 548 y Fo(\()p Fj(y)t Fo(\))p Fi(j)j(\024)j Fj(")1710 563 y Fs(n)1789 548 y Fo(whenev)m(er)35 b Fj(\032)p Fo(\()p Fj(x;)17 b(y)t Fo(\))27 b Fi(\024)h Fj(r)2670 563 y Fs(n)2750 548 y Fo(and)k Fj(i)c Fo(=)g(1)p Fj(;)17 b(:)g(:)g(:)e(;)i(n)p Fo(.)44 b(Let)1131 839 y Fj(\016)1174 854 y Fs(n)1221 839 y Fo(\()p Fj(P)14 b Fo(\))27 b(:=)h(min)1711 698 y Fg(\032)1900 771 y Fo(1)p 1796 816 257 4 v 1796 907 a Fj(T)1853 922 y Fs(n)1900 907 y Fo(\()p Fj(P)14 b Fo(\))2062 839 y Fj(;)j(")2152 854 y 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Fs(n)3675 1250 y Fi(\025)515 1370 y Fo(1)i Fi(\000)g Fj(")731 1385 y Fs(n)778 1370 y Fo(.)661 1495 y(Let)42 b Fj(E)917 1510 y Fs(n)1008 1495 y Fo(:=)h Fj(E)6 b Fo(\()p Fj(P)s(;)17 b(Q;)g(\016)1544 1510 y Fs(n)1590 1495 y Fo(\))42 b(and)g Fj(I)1912 1510 y Fs(n)1959 1495 y Fo(\()p Fj(x)p Fo(\),)i Fj(G)2238 1510 y Fs(n)2285 1495 y Fo(\()p Fj(#)p Fo(\))e(b)s(e)g(the)h (sets)g(\(19\),)g(\(20\))e(used)i(in)515 1616 y(the)i(de\014nition)f (of)g Fj(E)6 b Fo(\()p Fj(P)s(;)17 b(Q;)g(\016)1654 1631 y Fs(n)1700 1616 y Fo(\).)80 b(These)46 b(sets)g(dep)s(end)g(on)f Fj(P)14 b Fo(.)79 b(F)-8 b(or)44 b Fj(#)49 b Fi(2)g Fj(E)3504 1631 y Fs(n)3595 1616 y Fo(and)515 1736 y Fj(x)28 b Fi(2)g Fj(G)769 1751 y Fs(n)816 1736 y Fo(\()p Fj(#)p Fo(\),)h Fj(A)p Fo(\()p Fj(G)1193 1751 y Fs(n)1240 1736 y Fo(\()p Fj(#)p Fo(\)\))f Fi(\025)g Fj(A)p Fo(\()p Fj(S)1721 1700 y Fs(P)r(;Q)1847 1736 y Fo(\()p Fj(#)p Fo(\)\))13 b Fi(\000)g Fj(\016)2164 1751 y Fs(n)2239 1736 y Fi(\025)28 b Fj(A)p Fo(\()p Fj(S)2521 1700 y 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Fj(C)1868 3312 y Fs(P)1861 3372 y(\034)1892 3380 y Fd(n)1939 3348 y Fo(\()p Fj(#;)17 b(n)p Fo(\))22 b Fi(\\)h Fj(G)2362 3363 y Fs(n)2409 3348 y Fo(\()p Fj(#)p Fo(\).)44 b(Then)1124 3502 y Fg(\014)1124 3562 y(\014)1124 3622 y(\014)1124 3681 y(\014)1187 3579 y Fo(1)p 1167 3624 90 4 v 1167 3715 a Fj(\034)1209 3730 y Fs(n)1283 3511 y Fg(Z)1382 3537 y Fs(\034)1413 3545 y Fd(n)1338 3736 y Fh(0)1477 3647 y Fj(f)1536 3599 y Fs(Q)1525 3672 y(i)1617 3647 y Fi(\016)22 b Fj(\036)1747 3599 y Fs(Q)1747 3669 y(t)1807 3647 y Fo(\()p Fj(x)p Fo(\))p Fj(dt)g Fi(\000)2269 3579 y Fo(1)p 2156 3624 275 4 v 2156 3715 a Fj(A)p Fo(\()p Fj(S)2333 3686 y Fs(Q)2392 3715 y Fo(\))2457 3511 y Fg(Z)2512 3736 y Fs(S)2559 3718 y Fd(Q)2632 3647 y Fj(f)2691 3599 y Fs(Q)2680 3672 y(i)2750 3647 y Fj(dA)2874 3502 y Fg(\014)2874 3562 y(\014)2874 3622 y(\014)2874 3681 y(\014)1319 3912 y Fi(\024)1454 3845 y Fo(1)p 1434 3889 90 4 v 1434 3981 a Fj(\034)1476 3996 y Fs(n)1550 3777 y Fg(Z)1649 3803 y Fs(\034)1680 3811 y Fd(n)1605 4002 y Fh(0)1744 3798 y Fg(\014)1744 3858 y(\014)1744 3917 y(\014)1777 3912 y Fj(f)1836 3865 y Fs(Q)1825 3938 y(i)1918 3912 y Fi(\016)g Fj(\036)2048 3865 y Fs(Q)2048 3935 y(t)2107 3912 y Fo(\()p Fj(x)p Fo(\))g Fi(\000)h Fj(f)2419 3871 y Fs(P)2408 3937 y(i)2500 3912 y Fi(\016)f Fj(\036)2630 3871 y Fs(P)2630 3937 y(t)2688 3912 y Fo(\()p Fj(x)p Fo(\))2819 3798 y Fg(\014)2819 3858 y(\014)2819 3917 y(\014)2869 3912 y Fj(dt)g Fo(+)1319 4180 y(+)1412 4035 y Fg(\014)1412 4095 y(\014)1412 4155 y(\014)1412 4215 y(\014)1475 4113 y Fo(1)p 1454 4157 V 1454 4248 a Fj(\034)1496 4263 y Fs(n)1570 4044 y Fg(Z)1670 4071 y Fs(\034)1701 4079 y Fd(n)1626 4270 y Fh(0)1764 4180 y Fj(f)1823 4139 y Fs(P)1812 4205 y(i)1904 4180 y Fi(\016)g Fj(\036)2034 4139 y Fs(P)2034 4205 y(t)2092 4180 y Fo(\()p Fj(x)p Fo(\))p Fj(dt)h Fi(\000)2554 4113 y Fo(1)p 2441 4157 274 4 v 2441 4248 a Fj(A)p Fo(\()p Fj(S)2618 4219 y Fs(P)2677 4248 y Fo(\))2741 4044 y Fg(Z)2797 4270 y Fs(S)2844 4251 y Fd(P)2915 4180 y Fj(f)2974 4139 y Fs(P)2963 4205 y(i)3033 4180 y Fj(dA)3157 4035 y Fg(\014)3157 4095 y(\014)3157 4155 y(\014)3157 4215 y(\014)3212 4180 y Fo(+)1319 4452 y(+)1412 4308 y Fg(\014)1412 4368 y(\014)1412 4427 y(\014)1412 4487 y(\014)1567 4385 y Fo(1)p 1454 4429 V 1454 4521 a Fj(A)p Fo(\()p Fj(S)1631 4492 y Fs(P)1690 4521 y Fo(\))1754 4317 y Fg(Z)1810 4542 y Fs(S)1857 4523 y Fd(P)1928 4452 y Fj(f)1987 4411 y Fs(P)1976 4477 y(i)2046 4452 y Fj(dA)f Fi(\000)2414 4385 y Fo(1)p 2301 4429 275 4 v 2301 4521 a Fj(A)p Fo(\()p Fj(S)2478 4492 y Fs(Q)2538 4521 y Fo(\))2602 4317 y Fg(Z)2658 4542 y Fs(S)2705 4523 y Fd(Q)2777 4452 y Fj(f)2836 4405 y Fs(Q)2825 4478 y(i)2896 4452 y Fj(dA)3020 4308 y Fg(\014)3020 4368 y(\014)3020 4427 y(\014)3020 4487 y(\014)1319 4648 y Fo(=:)27 b Fj(I)j Fo(+)22 b Fj(I)8 b(I)30 b Fo(+)22 b Fj(I)8 b(I)g(I)g(:)515 4882 y Fo(W)-8 b(e)43 b(ha)m(v)m(e)h Fj(I)52 b Fi(\024)45 b Fo(2)17 b Fj(\016)1258 4846 y Fh(2)1254 4906 y Fs(n)1301 4882 y Fi(k)p Fj(f)1399 4897 y Fs(i)1427 4882 y Fi(k)1477 4897 y Fr(1)1581 4882 y Fo(+)28 b Fj(")1731 4897 y Fs(n)1823 4882 y Fi(\024)45 b Fo(3)p Fj(")2040 4897 y Fs(n)2086 4882 y Fo(|sa)m(y|for)e Fj(n)f Fo(large)g(enough.)74 b(Moreo)m(v)m(er,)515 5002 y Fj(x)46 b Fi(2)g Fj(C)805 4966 y Fs(P)798 5027 y(\034)829 5035 y Fd(n)876 5002 y Fo(\()p Fj(#;)17 b(n)p Fo(\))43 b(implies)e Fj(I)8 b(I)54 b Fi(\024)46 b Fj(")1813 5017 y Fs(n)1860 5002 y Fo(.)75 b(Finally)-8 b(,)43 b Fj(I)8 b(I)g(I)53 b Fi(\024)47 b Fj(")2694 5017 y Fs(n)2784 5002 y Fo(b)m(y)d(\(22\).)74 b(W)-8 b(e)44 b(conclude)p eop %%Page: 25 25 25 24 bop 141 241 a Fn(In\014nite)33 b(Step)g(Billiards)2219 b Fo(25)141 548 y(that)39 b Fj(I)c Fo(+)27 b Fj(I)8 b(I)35 b Fo(+)26 b Fj(I)8 b(I)g(I)47 b Fi(\024)40 b Fo(5)p Fj(")1175 563 y Fs(n)1261 548 y Fo(for)f Fj(#)h Fi(2)g Fj(D)1704 512 y Fs(P)1701 573 y(\034)1732 581 y Fd(n)1779 548 y Fo(\()p Fj(n)p Fo(\))27 b Fi(\\)g Fj(E)2105 563 y Fs(n)2153 548 y Fo(,)41 b Fj(x)f Fi(2)g Fj(C)2499 512 y Fs(P)2492 573 y(\034)2523 581 y Fd(n)2570 548 y Fo(\()p Fj(#;)17 b(n)p Fo(\))27 b Fi(\\)g Fj(G)3002 563 y Fs(n)3049 548 y Fo(\()p Fj(#)p Fo(\))40 b(and)141 668 y Fj(i)28 b Fo(=)g(1)p Fj(;)17 b(:)g(:)g(:)e(;)i(n)p Fo(.)44 b(By)33 b(de\014nition)e(of)h Fj(D)1483 632 y Fs(P)1480 693 y(\034)1511 701 y Fd(n)1558 668 y Fo(\()p Fj(n)p Fo(\))h(and)g Fj(E)1987 683 y Fs(n)2034 668 y Fo(,)f(w)m(e)i(ha)m(v)m(e:)1236 913 y Fi(j)p Fj(D)1348 872 y Fs(P)1345 938 y(\034)1376 946 y Fd(n)1422 913 y Fo(\()p Fj(n)p Fo(\))23 b Fi(\\)f Fj(E)1739 928 y Fs(n)1786 913 y Fi(j)28 b Fj(>)f Fo(1)22 b Fi(\000)h Fo(2)p Fj(")2211 928 y Fs(n)2257 913 y Fj(:)922 b Fo(\(23\))141 1158 y(F)-8 b(rom)45 b Fj(A)p Fo(\()p Fj(C)599 1122 y Fs(P)592 1183 y(\034)623 1191 y Fd(n)670 1158 y Fo(\()p Fj(#;)17 b(n)p Fo(\)\))51 b Fi(\025)g Fj(A)p Fo(\()p Fj(S)1299 1122 y Fs(P)1358 1158 y Fo(\()p Fj(#)p Fo(\)\))32 b Fi(\000)g Fj(")1716 1173 y Fs(n)1808 1158 y Fo(and)47 b Fj(A)p Fo(\()p Fj(G)2200 1173 y Fs(n)2247 1158 y Fo(\()p Fj(#)p Fo(\)\))k Fi(\025)g Fj(A)p Fo(\()p Fj(S)2774 1122 y Fs(P)2833 1158 y Fo(\()p Fj(#)p Fo(\)\))32 b Fi(\000)g Fj(")3191 1173 y Fs(n)3237 1158 y Fo(,)50 b(it)141 1279 y(follo)m(ws)31 b(that)494 1524 y Fj(A)p Fo(\()p Fj(C)682 1483 y Fs(P)675 1548 y(\034)706 1556 y Fd(n)753 1524 y Fo(\()p Fj(#;)17 b(n)p Fo(\))22 b Fi(\\)h Fj(G)1176 1539 y Fs(n)1223 1524 y Fo(\()p Fj(#)p Fo(\)\))28 b Fi(\025)g Fj(A)p Fo(\()p Fj(S)1704 1483 y Fs(P)1762 1524 y Fo(\()p Fj(#)p Fo(\)\))23 b Fi(\000)f Fo(2)p Fj(")2150 1539 y Fs(n)2225 1524 y Fo(=)27 b Fj(A)p Fo(\()p Fj(S)2505 1483 y Fs(Q)2565 1524 y Fo(\()p Fj(#)p Fo(\)\))22 b Fi(\000)h Fo(2)p Fj(")2953 1539 y Fs(n)2999 1524 y Fj(:)180 b Fo(\(24\))287 1777 y(Let)33 b Fi(f)p Fj(P)575 1792 y Fs(j)611 1777 y Fi(g)661 1792 y Fs(j)t Fr(2)p Fq(N)825 1777 y Fo(b)s(e)g(an)g(en)m(umeration)e (of)i Fi(S)1826 1792 y Fh(0)1898 1777 y Fo(and)1245 2101 y Fj(H)i Fo(:=)1523 1976 y Fr(1)1503 2006 y Fg(\\)1491 2215 y Fs(n)p Fh(=1)1671 1976 y Fr(1)1651 2006 y Fg([)1641 2216 y Fs(j)t Fh(=)p Fs(n)1788 2101 y Fj(U)1854 2116 y Fs(\033)1894 2124 y Fd(n)1937 2116 y Fh(\()p Fs(P)2009 2126 y Fd(j)2042 2116 y Fh(\))2073 2101 y Fo(\()p Fj(P)2174 2116 y Fs(j)2210 2101 y Fo(\))p Fj(:)141 2441 y Fo(It)46 b(is)f(easy)i(to)e(see)i(that)f Fj(H)53 b Fo(is)45 b(a)h(dense)h Fj(G)1823 2456 y Fs(\016)1861 2441 y Fo(-subset)g(of)e Fi(S)7 b Fo(.)84 b(If)46 b Fj(Q)k Fi(2)g Fj(H)8 b Fo(,)49 b(then)d(for)141 2562 y(ev)m(ery)54 b Fj(n)62 b(>)g Fo(0)52 b(there)h(is)f(a)g Fj(j)1305 2577 y Fs(n)1405 2562 y Fo(for)g(whic)m(h)h Fj(Q)62 b Fi(2)g Fj(U)2206 2577 y Fs(\033)2246 2585 y Fd(n)2289 2577 y Fh(\()p Fs(P)2361 2587 y Fd(j)2386 2595 y(n)2433 2577 y Fh(\))2465 2562 y Fo(\()p Fj(P)2566 2577 y Fs(j)2595 2585 y Fd(n)2641 2562 y Fo(\).)103 b(De\014ne)53 b Fj(D)64 b Fo(:=)141 2630 y Fg(T)224 2656 y Fr(1)224 2734 y Fs(m)p Fh(=1)398 2630 y Fg(S)481 2734 y Fs(n)p Fr(\025)p Fs(m)661 2705 y Fj(D)745 2651 y Fs(P)790 2661 y Fd(j)815 2669 y(n)742 2715 y Fs(\034)773 2723 y Fd(n)866 2705 y Fo(\()p Fj(n)p Fo(\))31 b Fi(\\)f Fj(E)1199 2720 y Fs(n)1246 2705 y Fo(.)80 b(By)45 b(\(23\),)i Fi(j)p Fj(D)s Fi(j)g Fo(=)h(1.)79 b(This)45 b(means)g(that,)i(for)d(eac)m(h)141 2869 y Fj(#)28 b Fi(2)g Fj(D)s Fo(,)k(there)g(is)g(a)f(subsequence)k Fi(f)p Fj(n)1547 2884 y Fs(k)1590 2869 y Fi(g)d Fo(suc)m(h)h(that)e Fj(#)d Fi(2)g Fj(D)2364 2799 y Fs(P)2409 2809 y Fd(j)2434 2817 y(n)2472 2835 y(k)2361 2879 y Fs(\034)2392 2887 y Fd(n)2430 2905 y(k)2523 2869 y Fo(\()p Fj(n)2619 2884 y Fs(k)2662 2869 y Fo(\))k(for)f(all)e Fj(k)s Fo(.)44 b(In)32 b(or-)141 2989 y(der)g(to)e(a)m(v)m(oid)h(hea)m(vy)h(notation,) e(let)h(us)g(denote)h(suc)m(h)g(a)f(sequence)j(b)m(y)d Fi(f)p Fj(n)p Fi(g)p Fo(.)43 b(No)m(w)32 b(call)141 3122 y Fj(C)7 b Fo(\()p Fj(#)p Fo(\))28 b(:=)510 3047 y Fg(T)593 3073 y Fr(1)593 3151 y Fs(m)p Fh(=1)766 3047 y Fg(S)849 3151 y Fs(n)p Fr(\025)p Fs(m)1030 3122 y Fj(C)1107 3068 y Fs(P)1152 3078 y Fd(j)1177 3086 y(n)1100 3132 y Fs(\034)1131 3140 y Fd(n)1228 3122 y Fo(\()p Fj(#;)17 b(n)p Fo(\))22 b Fi(\\)h Fj(G)1651 3137 y Fs(n)1698 3122 y Fo(\()p Fj(#)p Fo(\).)44 b(F)-8 b(rom)31 b(\(24\),)h Fj(A)p Fo(\()p Fj(C)7 b Fo(\()p Fj(#)p Fo(\)\))28 b(=)f Fj(A)p Fo(\()p Fj(S)3058 3086 y Fs(Q)3118 3122 y Fo(\()p Fj(#)p Fo(\)\).)287 3260 y(So,)37 b(for)e(eac)m(h)i Fj(#)d Fi(2)f Fj(D)38 b Fo(and)e Fj(x)e Fi(2)f Fj(C)7 b Fo(\()p Fj(#)p Fo(\))36 b(\(i.e.,)h(a.e.)f Fj(#)g Fo(and)g(a.e.)g Fj(x)d Fi(2)h Fj(S)2896 3224 y Fs(Q)2955 3260 y Fo(\()p Fj(#)p Fo(\)\))i(there)141 3381 y(exists)d(a)g(subsequence)j Fi(f)p Fj(n)1151 3396 y Fs(k)1194 3381 y Fi(g)c Fo(suc)m(h)i(that)e(lim)1843 3396 y Fs(k)r Fr(!1)2043 3381 y Fj(\034)2085 3396 y Fs(n)2128 3408 y Fd(k)2198 3381 y Fo(=)c(+)p Fi(1)k Fo(and)769 3678 y(lim)747 3741 y Fs(k)r Fr(!1)993 3611 y Fo(1)p 953 3655 128 4 v 953 3747 a Fj(\034)995 3762 y Fs(n)1038 3774 y Fd(k)1107 3543 y Fg(Z)1207 3569 y Fs(\034)1238 3577 y Fd(n)1276 3595 y(k)1163 3768 y Fh(0)1340 3678 y Fj(f)1399 3631 y Fs(Q)1388 3704 y(i)1480 3678 y Fi(\016)22 b Fj(\036)1610 3631 y Fs(Q)1610 3701 y(t)1669 3678 y Fo(\()p Fj(x)p Fo(\))p Fj(dt)28 b Fo(=)2141 3611 y(1)p 2028 3655 275 4 v 2028 3747 a Fj(A)p Fo(\()p Fj(S)2205 3718 y Fs(Q)2264 3747 y Fo(\))2329 3543 y Fg(Z)2384 3768 y Fs(S)2431 3749 y Fd(Q)2504 3678 y Fj(f)2563 3631 y Fs(Q)2552 3704 y(i)2622 3678 y Fj(dA;)141 3999 y Fo(for)h(all)e Fj(i)h Fi(2)h Fc(N)8 b Fo(.)49 b(Since)30 b Fi(f)p Fj(f)1077 3952 y Fs(Q)1066 4024 y(i)1136 3999 y Fi(g)1186 4014 y Fs(i)p Fr(2)p Fs(N)1353 3999 y Fo(is)f(dense)i(in)e Fj(L)1888 3963 y Fh(1)1928 3999 y Fo(\()p Fj(S)2032 3963 y Fs(Q)2091 3999 y Fo(\()p Fj(#)p Fo(\)\),)i(using)e(a)g(standard)h (appro)m(x-)141 4119 y(imation)f(argumen)m(t,)j(and)h(Birkho\013)7 b('s)32 b(Theorem,)g(w)m(e)i(conclude)e(that)g Fj(\036)2852 4072 y Fs(Q)2852 4147 y(#;t)2975 4119 y Fo(is)g(ergo)s(dic)141 4240 y(for)g(a.e.)44 b Fj(#)28 b Fi(2)p Fo(]0)p Fj(;)17 b(\031)t(=)p Fo(2[.)2139 b Fw(Q.E.D.)287 4435 y Ff(Pr)n(oof)52 b(of)f(Cor)n(ollar)-7 b(y)52 b(2.)158 b Fo(Let)44 b Fj(F)1916 4399 y Fs(N)2031 4435 y Fi(\032)49 b(S)j Fo(b)s(e)45 b(the)g(collection)d(of)i(\014nite)141 4555 y(step)c(p)s(olygons)f (with)g Fj(N)49 b Fo(sides.)64 b(Eac)m(h)40 b Fj(F)1742 4519 y Fs(N)1848 4555 y Fo(is)f(a)g(no)m(where)h(dense)h(set)f(in)e Fi(S)7 b Fo(,)42 b(so)e(the)141 4675 y(subset)h(of)e(all)e(\014nite)i (step)h(p)s(olygons)f Fj(F)53 b Fo(:=)1848 4601 y Fg(S)1931 4705 y Fs(N)7 b Fr(\025)p Fh(4)2105 4675 y Fj(F)2182 4639 y Fs(N)2288 4675 y Fo(is)39 b(a)g(meager)f(set)i(of)f Fi(S)7 b Fo(.)64 b(By)141 4796 y(Baire's)41 b(Theorem,)j Fj(H)49 b Fo(is)41 b(a)g(set)h(of)f(second)i(category)-8 b(,)43 b(therefore)f(it)f(in)m(tersects)i(the)141 4916 y(complemen)m(t)31 b(of)f Fj(F)14 b Fo(.)43 b(In)32 b(other)f(w)m (ords,)i Fj(H)39 b Fo(con)m(tains)31 b(in\014nite)f(step)j(p)s (olygons.)42 b Fw(Q.E.D.)p eop %%Page: 26 26 26 25 bop 515 241 a Fo(26)1160 b Fn(M.)33 b(Degli)d(Esp)s(osti,)j(G.)f (Del)g(Magno)g(and)h(M.)g(Lenci)515 548 y Fm(References)515 774 y Fo([Ar])148 b Ff(P.)35 b(Arnoux)p Fo(,)d Fk(Er)-5 b(go)g(dicit)n(\023)-47 b(e)32 b(generique)h(des)f(bil)5 b(lar)-5 b(ds)33 b(p)-5 b(olygonaux)p Fo(,)30 b(Seminaire)828 894 y(Bourbaki)i Fl(696)g Fo(\(1988\),)g(203-221.)515 1114 y([B])190 b Ff(M.)28 b(Boshernitzan)p Fo(,)e Fk(Bil)5 b(liar)-5 b(ds)26 b(and)g(r)-5 b(ational)27 b(p)-5 b(erio)g(dic)26 b(dir)-5 b(e)g(ctions)27 b(in)g(p)-5 b(oly-)828 1235 y(gons)p Fo(,)32 b(Amer.)g(Math.)h(Mon)m(thly)h Fl(99)p Fo(\(6\))e(\(1992\),)g(522-529.)515 1455 y([DDL])48 b Ff(M.)41 b(Degli)g(Esposti,)h(G.)g(Del)f(Ma)n(gno)h(and)g(M.)f(Lenci)p Fo(,)d Fk(A)n(n)g(in\014nite)828 1575 y(step)c(bil)5 b(liar)-5 b(d)p Fo(,)32 b(Nonlinearit)m(y)g Fl(11)p Fo(\(4\))h (\(1998\),)e(991-1013.)515 1796 y([FK])119 b Ff(R.)37 b(H.)i(F)m(o)n(x)f(and)h(R.)f(B.)f(Kershner)p Fo(,)c Fk(Conc)-5 b(erning)34 b(the)h(tr)-5 b(ansitive)35 b(pr)-5 b(op-)828 1916 y(erties)37 b(of)h(ge)-5 b(o)g(desic)37 b(on)g(a)h(r)-5 b(ational)38 b(p)-5 b(olyhe)g(dr)g(on)p Fo(,)36 b(Duk)m(e)g(Math.)h(J.)f Fl(2)g Fo(\(1936\),)828 2036 y(147-150.)515 2257 y([EFV])56 b Ff(B.)e(Eckhardt,)60 b(J.)55 b(F)m(ord)g(and)h(F.)f(Viv)-9 b(aldi)p Fo(,)53 b Fk(A)n(nalytic)-5 b(al)5 b(ly)50 b(solvable)828 2377 y(dynamic)-5 b(al)38 b(systems)h(which)g(ar)-5 b(e)39 b(not)g(inte)-5 b(gr)g(able.)p Fo(,)38 b(Ph)m(ys.)h Fl(D)k(13)p Fo(\(3\))38 b(\(1984\),)828 2497 y(339-356.)515 2718 y([G1])133 b Ff(E.)45 b(Gutkin)p Fo(,)d Fk(Bil)5 b(liar)-5 b(d)41 b(\015ows)g(on)g(almost)g(inte)-5 b(gr)g(able)41 b(p)-5 b(olyhe)g(dr)g(al)41 b(surfac)-5 b(es)p Fo(,)828 2838 y(Ergo)s(d.)32 b(Th.)h(&)g(Dyn.)f(Sys.)i Fl(4)f Fo(\(1984\),)e(569-584.)515 3058 y([G2])133 b Ff(E.)37 b(Gutkin)p Fo(,)c Fk(Bil)5 b(liar)-5 b(ds)34 b(in)g(p)-5 b(olygons)p Fo(,)32 b(Ph)m(ysica)h Fl(D)38 b(19)32 b Fo(\(1986\))g(311-333.)515 3278 y([G3])133 b Ff(E.)44 b(Gutkin)p Fo(,)d Fk(Bil)5 b(liar)-5 b(ds)39 b(in)i(p)-5 b(olygons:)56 b(survey)41 b(of)f(r)-5 b(e)g(c)g(ent)41 b(r)-5 b(esults)p Fo(,)41 b(J.)e(Stat.)828 3399 y(Ph)m(ys.)34 b Fl(83)f Fo(\(1996\),)e(7-26.)515 3619 y([GKT])49 b Ff(G.)25 b(Galperin,)j(T.)e(Kr)1786 3611 y(\177)1783 3619 y(uger)g(and)g(S.)g(Tr)n(oubetzk)n(o)n(y)p Fo(,)e Fk(L)-5 b(o)g(c)g(al)25 b(instability)828 3739 y(of)44 b(orbits)g(in)g(p)-5 b(olygonal)43 b(and)h(p)-5 b(olyhe)g(dr)g(al)43 b(bil)5 b(liar)-5 b(ds)p Fo(,)45 b(Comm.)d(Math.)h(Ph)m(ys.)828 3860 y Fl(169)32 b Fo(\(1995\),)g(463-473.)515 4080 y([GU])109 b Ff(M.-J.)37 b(Giannoni)j(and)e(D.)g(Ullmo)p Fo(,)d Fk(Co)-5 b(ding)34 b(chaotic)h(bil)5 b(liar)-5 b(ds.)35 b(I.)g(Non-)828 4200 y(c)-5 b(omp)g(act)43 b(bil)5 b(liar)-5 b(ds)43 b(on)h(a)g(ne)-5 b(gative)43 b(curvatur)-5 b(e)45 b(manifold)p Fo(,)f(Ph)m(ys.)g Fl(D)49 b(41)p Fo(\(3\))828 4321 y(\(1990\),)31 b(371-390.)515 4541 y([KMS])49 b Ff(S.Ker)n(ckhoff,)65 b(H.Masur)58 b(and)i(J.Smillie)p Fo(,)e Fk(Er)-5 b(go)g(dicity)53 b(of)g(bil)5 b(liar)-5 b(d)828 4661 y(\015ows)34 b(and)h(quadr)-5 b(atic)34 b(di\013er)-5 b(entials)p Fo(,)32 b(Ann.)h(Math.)g Fl(124)f Fo(\(1986\),)g(293-311.)515 4882 y([KZ])123 b Ff(A.)37 b(Ka)-7 b(tok)38 b(and)f(A.)g(Zeml)-7 b(y)g(ak)n(o)n(v)p Fo(,)33 b Fk(T)-7 b(op)i(olo)g(gic)g(al)33 b(tr)-5 b(ansitivity)35 b(of)f(bil)5 b(liar)-5 b(ds)828 5002 y(in)34 b(p)-5 b(olygons)p Fo(,)32 b(Math)h(Notes)g Fl(18)g Fo(\(1975\),)e(760-764.)p eop %%Page: 27 27 27 26 bop 141 241 a Fn(In\014nite)33 b(Step)g(Billiards)2219 b Fo(27)141 548 y([Le])155 b Ff(M.)44 b(Lenci)p Fo(,)d Fk(Esc)-5 b(ap)g(e)40 b(orbits)h(for)g(non-c)-5 b(omp)g(act)40 b(\015at)h(bil)5 b(liar)-5 b(ds)p Fo(,)40 b(Chaos)g Fl(6)p Fo(\(3\))454 668 y(\(1996\),)32 b(428-431.)141 872 y([L])198 b Ff(A.)42 b(M.)f(Leonto)n(vich)p Fo(,)d Fk(The)f(existenc)-5 b(e)38 b(of)g(unb)-5 b(ounde)g(d)38 b(oscil)5 b(lating)37 b(tr)-5 b(aje)g(c-)454 992 y(tories)38 b(in)f(a)g(pr)-5 b(oblem)37 b(of)g(bil)5 b(liar)-5 b(ds)p Fo(,)35 b(So)m(v.)i(Acad.)f (Sci.,)g(doklady)f(Math.)h Fl(3)p Fo(\(4\))454 1112 y(\(1962\),)c (1049-1052.)141 1316 y([M])170 b Ff(H.)50 b(Masur)p Fo(,)c Fk(Close)-5 b(d)44 b(tr)-5 b(aje)g(ctories)45 b(of)g(a)g(quadr)-5 b(atic)45 b(di\013er)-5 b(ential)p Fo(,)45 b(Duk)m(e)g(J.)454 1436 y(Math.)33 b Fl(53)g Fo(\(1986\),)e(307-314.)141 1640 y([PP])127 b Ff(Y)-9 b(a.)29 b(Pesin)e(and)i(B.)f(Pitskel)p Fo(,)d Fk(T)-7 b(op)i(olo)g(gic)g(al)25 b(pr)-5 b(essur)g(e)27 b(and)f(the)h(variational)454 1760 y(principle)32 b(for)i(nonc)-5 b(omap)g(ct)32 b(sets)p Fo(,)f(F)-8 b(unc.)31 b(Anal.)g(Appl.)f Fl(18)h Fo(\(1984\),)g(307-318.)141 1963 y([R])187 b Ff(V.)78 b(A.)g(R)m(okhlin)p Fo(,)j Fk(L)-5 b(e)g(ctur)g(es)70 b(on)g(the)f(entr)-5 b(opy)70 b(the)-5 b(ory)70 b(of)g(me)-5 b(asur)g(e-)454 2084 y(pr)g(eserving)34 b(tr)-5 b(ansformations)p Fo(,)31 b(Russ.)j(Math.)f(Surv.)g Fl(22)p Fo(\(5\))g(\(1967\),)e(1-52.) 141 2287 y([S])205 b Ff(Y)-9 b(a.)27 b(G.)f(Sinai)h(\(ed.\))p Fo(,)d Fk(Dynamic)-5 b(al)24 b(systems)h(II)g(\(Er)-5 b(go)g(dic)24 b(the)-5 b(ory)26 b(with)f(appli-)454 2408 y(c)-5 b(ations)32 b(to)h(dynamic)-5 b(al)32 b(systems)g(and)h (statistic)-5 b(al)32 b(me)-5 b(chanics\))p Fo(,)30 b(Encyclopae-)454 2528 y(dia)i(of)g(Mathematical)e(Sciences,)k(2.)f(Springer-V)-8 b(erlag,)30 b(Berlin,)i(1989.)141 2731 y([T])189 b Ff(S.)35 b(T)-9 b(aba)n(chnik)n(o)n(v)p Fo(,)31 b Fk(Bil)5 b(liar)-5 b(ds)p Fo(,)29 b(So)s(ci)m(\023)-46 b(et)m(\023)g(e)30 b(math)m(\023)-46 b(ematique)29 b(de)h(F)-8 b(rance,)31 b(P)m(aris,)454 2852 y(1995.)141 3055 y([T)-8 b(r])159 b Ff(S.)56 b(Tr)n(oubetzk)n(o)n(y)p Fo(,)f Fk(Bil)5 b(liar)-5 b(ds)50 b(in)h(in\014nite)f(p)-5 b(olygons)p Fo(,)54 b(Nonlinearit)m(y)c Fl(12)454 3176 y Fo(\(1999\),)32 b(293-311.)141 3379 y([T)-8 b(r2])110 b Ff(S.)47 b(Tr)n(oubetzk)n(o)n (y)p Fo(,)c Fk(T)-7 b(op)i(olo)g(gic)g(al)40 b(tr)-5 b(ansitivity)44 b(of)e(in\014nite)g(step)h(bil)5 b(liar)-5 b(ds)p Fo(,)454 3499 y(preprin)m(t)33 b(\(1999\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9906091439271--