% % This is a latex2e file % % First part generates two postscript files (num1.ps and num2.ps). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \typeout{The first run will create the following encapsuled Postscript files :} \typeout{num1.ps, num2.ps} \begin{filecontents}{num1.ps} %%!PS-Adobe-2.0 %%Creator: gnuplot 3.5 (pre 3.6) patchlevel beta 340 %%CreationDate: Thu Feb 19 15:08:14 1998 %%DocumentFonts: (atend) %%BoundingBox: 50 50 554 770 %%Orientation: Landscape %%Pages: (atend) %%EndComments /gnudict 120 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /UL { gnulinewidth mul /userlinewidth exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end %%EndProlog %%Page: 1 1 gnudict begin gsave 50 50 translate 0.100 0.100 scale 90 rotate 0 -5040 translate 0 setgray newpath (Helvetica) findfont 140 scalefont setfont LTb 588 350 M 63 0 V 6465 0 R -63 0 V 504 350 M (0) Rshow 588 919 M 63 0 V 6465 0 R -63 0 V 504 919 M (0.1) Rshow 588 1488 M 63 0 V 6465 0 R -63 0 V -6549 0 R (0.2) Rshow 588 2056 M 63 0 V 6465 0 R -63 0 V -6549 0 R (0.3) Rshow 588 2625 M 63 0 V 6465 0 R -63 0 V -6549 0 R (0.4) Rshow 588 3194 M 63 0 V 6465 0 R -63 0 V -6549 0 R (0.5) Rshow 588 3763 M 63 0 V 6465 0 R -63 0 V -6549 0 R (0.6) Rshow 588 4331 M 63 0 V 6465 0 R -63 0 V -6549 0 R (0.7) Rshow 588 4900 M 63 0 V 6465 0 R -63 0 V -6549 0 R (0.8) Rshow 588 350 M 0 63 V 0 4487 R 0 -63 V 588 210 M (3.5) Cshow 1241 350 M 0 63 V 0 4487 R 0 -63 V 0 -4627 R (3.55) Cshow 1894 350 M 0 63 V 0 4487 R 0 -63 V 0 -4627 R (3.6) Cshow 2546 350 M 0 63 V 0 4487 R 0 -63 V 0 -4627 R (3.65) Cshow 3199 350 M 0 63 V 0 4487 R 0 -63 V 0 -4627 R (3.7) Cshow 3852 350 M 0 63 V 0 4487 R 0 -63 V 0 -4627 R (3.75) Cshow 4505 350 M 0 63 V 0 4487 R 0 -63 V 0 -4627 R (3.8) Cshow 5158 350 M 0 63 V 0 4487 R 0 -63 V 0 -4627 R (3.85) Cshow 5810 350 M 0 63 V 0 4487 R 0 -63 V 0 -4627 R (3.9) Cshow 6463 350 M 0 63 V 0 4487 R 0 -63 V 0 -4627 R (3.95) Cshow 7116 350 M 0 63 V 0 4487 R 0 -63 V 0 -4627 R (4) Cshow LTb 588 350 M 6528 0 V 0 4550 V -6528 0 V 588 350 L 1.000 UP 1.000 UL LT0 6465 4767 M (AP's dimension) Rshow 1502 433 Pls 1632 987 Pls 1763 1237 Pls 1894 1476 Pls 2220 1764 Pls 2546 1965 Pls 2873 2278 Pls 3199 2391 Pls 3526 2628 Pls 3852 2647 Pls 4178 2814 Pls 4505 3021 Pls 4831 3122 Pls 5158 2895 Pls 5484 3267 Pls 5810 3555 Pls 6137 3468 Pls 6463 3814 Pls 6790 4049 Pls 6985 4209 Pls 7103 4259 Pls 7116 4346 Pls 6748 4767 Pls 1.000 UL LT1 6465 4627 M (Topological entropy) Rshow 6549 4627 M 399 0 V 588 493 M 41 0 V 41 0 V 41 1 V 41 0 V 41 1 V 41 0 V 41 1 V 41 2 V 42 1 V 41 3 V 41 4 V 41 6 V 41 8 V 41 11 V 41 10 V 41 7 V 41 4 V 41 3 V 41 5 V 41 12 V 41 22 V 41 29 V 41 107 V 41 157 V 41 77 V 41 71 V 42 48 V 41 98 V 41 91 V 41 118 V 41 47 V 41 42 V 41 43 V 41 6 V 41 44 V 41 35 V 41 30 V 41 45 V 41 24 V 41 12 V 41 0 V 41 0 V 41 6 V 41 48 V 42 61 V 41 47 V 41 17 V 41 48 V 41 54 V 41 36 V 41 38 V 41 13 V 41 40 V 41 44 V 41 46 V 41 48 V 41 79 V 41 42 V 41 32 V 41 21 V 41 26 V 42 22 V 41 22 V 41 16 V 41 5 V 41 24 V 41 18 V 41 20 V 41 17 V 41 16 V 41 19 V 41 18 V 41 24 V 41 11 V 41 8 V 41 1 V 41 0 V 41 1 V 41 12 V 42 17 V 41 28 V 41 19 V 41 17 V 41 16 V 41 18 V 41 24 V 41 8 V 41 4 V 41 19 V 41 21 V 41 16 V 41 21 V 41 27 V 41 17 V 41 19 V 41 5 V 41 14 V 42 14 V 41 11 V 41 15 V 41 7 V 41 8 V 41 3 V 41 1 V 41 0 V 41 0 V 41 0 V 41 0 V 41 0 V 41 0 V 41 0 V 41 0 V 41 0 V 41 12 V 42 22 V 41 33 V 41 26 V 41 13 V 41 28 V 41 35 V 41 24 V 41 33 V 41 9 V 41 22 V 41 25 V 41 30 V 41 21 V 41 14 V 41 7 V 41 7 V 41 17 V 41 22 V 42 30 V 41 29 V 41 17 V 41 31 V 41 27 V 41 29 V 41 19 V 41 14 V 41 28 V 41 24 V 41 26 V 41 17 V 41 23 V 41 14 V 41 2 V 41 21 V 41 28 V 41 25 V 42 33 V 41 26 V 41 33 V 41 36 V 41 35 V 41 29 V 41 43 V 41 55 V 41 111 V stroke grestore end showpage \end{filecontents} \begin{filecontents}{num2.ps} %!PS-Adobe-2.0 %%Creator: gnuplot 3.5 (pre 3.6) patchlevel beta 340 %%CreationDate: Wed Feb 18 20:10:49 1998 %%DocumentFonts: (atend) %%BoundingBox: 50 50 554 770 %%Orientation: Landscape %%Pages: (atend) %%EndComments /gnudict 120 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /UL { gnulinewidth mul /userlinewidth exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end %%EndProlog %%Page: 1 1 gnudict begin gsave 50 50 translate 0.100 0.100 scale 90 rotate 0 -5040 translate 0 setgray newpath (Helvetica) findfont 140 scalefont setfont LTb 504 350 M 63 0 V 6549 0 R -63 0 V 420 350 M (0) Rshow 504 1108 M 63 0 V 6549 0 R -63 0 V -6633 0 R (2) Rshow 504 1867 M 63 0 V 6549 0 R -63 0 V -6633 0 R (4) Rshow 504 2625 M 63 0 V 6549 0 R -63 0 V -6633 0 R (6) Rshow 504 3383 M 63 0 V 6549 0 R -63 0 V -6633 0 R (8) Rshow 504 4142 M 63 0 V 6549 0 R -63 0 V -6633 0 R (10) Rshow 504 4900 M 63 0 V 6549 0 R -63 0 V -6633 0 R (12) Rshow 504 350 M 0 63 V 0 4487 R 0 -63 V 504 210 M (0) Cshow 1165 350 M 0 63 V 0 4487 R 0 -63 V 0 -4627 R (5) Cshow 1826 350 M 0 63 V 0 4487 R 0 -63 V 0 -4627 R (10) Cshow 2488 350 M 0 63 V 0 4487 R 0 -63 V 0 -4627 R (15) Cshow 3149 350 M 0 63 V 0 4487 R 0 -63 V 0 -4627 R (20) Cshow 3810 350 M 0 63 V 0 4487 R 0 -63 V 0 -4627 R (25) Cshow 4471 350 M 0 63 V 0 4487 R 0 -63 V 0 -4627 R (30) Cshow 5132 350 M 0 63 V 0 4487 R 0 -63 V 0 -4627 R (35) Cshow 5794 350 M 0 63 V 0 4487 R 0 -63 V 0 -4627 R (40) Cshow 6455 350 M 0 63 V 0 4487 R 0 -63 V 0 -4627 R (45) Cshow 7116 350 M 0 63 V 0 4487 R 0 -63 V 0 -4627 R (50) Cshow LTb 504 350 M 6612 0 V 0 4550 V -6612 0 V 504 350 L 1.000 UL LT0 6465 4767 M (Distribution of return times) Rshow 6549 4767 M 399 0 V 504 350 M 636 613 L 768 350 L 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 504 350 L 636 613 L 132 0 V 901 350 L 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 504 350 L 636 613 L 768 767 L 133 0 V 1033 350 L 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 504 350 L 636 613 L 768 767 L 901 960 L 1033 350 L 132 526 V 1297 350 L 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 504 350 L 636 613 L 768 767 L 133 262 V 132 -69 V 132 128 V 1297 767 L 133 0 V 1562 350 L 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 504 350 L 636 613 L 132 0 V 133 475 V 1033 960 L 132 223 V 132 -95 V 133 235 V 132 -100 V 1694 613 L 1826 350 L 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 504 350 L 636 613 L 768 767 L 133 416 V 1033 960 L 132 332 V 132 59 V 133 204 V 132 30 V 132 -497 V 132 171 V 1959 350 L 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 504 350 L 636 613 L 768 767 L 133 321 V 1033 876 L 132 416 V 132 174 V 133 232 V 132 87 V 132 -66 V 132 -55 V 133 -218 V 132 -358 V 2223 876 L 2355 350 L 133 0 V 132 0 V 132 417 V 2884 350 L 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V currentpoint stroke M 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 504 350 L 636 613 L 768 767 L 133 371 V 1033 876 L 132 383 V 132 187 V 133 303 V 132 134 V 132 99 V 132 53 V 133 -126 V 132 -180 V 132 -129 V 132 -417 V 2488 350 L 132 417 V 132 262 V 2884 767 L 3017 613 L 3149 350 L 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 504 350 L 636 613 L 132 0 V 133 525 V 1033 960 L 132 299 V 132 187 V 133 330 V 132 168 V 132 133 V 132 113 V 133 67 V 132 -12 V 132 -149 V 132 -270 V 133 -340 V 132 -227 V 132 118 V 132 -154 V 133 -135 V 132 135 V 3281 350 L 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 504 350 L 636 613 L 768 767 L 133 371 V 1033 960 L 132 263 V 132 223 V 133 312 V 132 192 V 132 192 V 132 144 V 133 137 V 132 82 V 132 -1 V 132 -119 V 133 -220 V 132 -183 V 132 -284 V 132 -71 V 133 -105 V 132 91 V 132 -475 V 3413 960 L 133 128 V 3678 613 L 132 154 V 3942 350 L 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 504 350 L 636 613 L 768 767 L 133 371 V 1033 876 L 132 347 V 132 223 V 133 321 V 132 210 V 132 195 V 132 165 V 133 179 V 132 128 V 132 95 V 132 4 V 133 -82 V 132 -135 V 132 -193 V 132 -218 V 133 -213 V 132 25 V 132 -151 V 132 -57 V 133 -164 V 132 -154 V 132 -50 V 3942 960 L 4074 767 L 4207 350 L 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 504 350 L 636 613 L 768 767 L 133 262 V 1033 876 L 132 347 V 132 223 V 133 321 V 132 220 V 132 194 V 132 178 V 133 204 V 132 161 V 132 143 V 132 89 V 133 50 V 132 -58 V 132 -103 V 132 -192 V 133 -202 V 132 -165 V 132 -99 V 132 -15 V 133 -72 V 132 -252 V 132 -63 V 132 -158 V 132 -203 V 133 -23 V 4339 350 L 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 504 350 L 636 613 L 768 767 L 133 371 V 1033 876 L 132 347 V 132 223 V 133 321 V 132 220 V 132 197 V 132 193 V 133 208 V 132 174 V 132 188 V 132 134 V 133 112 V 132 66 V 132 -31 V 132 -99 V 133 -157 V 132 -187 V 132 -195 V 132 -77 V 133 -18 V 132 -110 V 132 -113 V 132 -126 V 132 -164 V 133 -127 V 132 -454 V 132 -109 V 132 -154 V 4736 960 L 132 128 V 5000 350 L 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 504 350 L 636 613 L 132 0 V 133 525 V 1033 876 L 132 347 V 132 263 V 133 290 V 132 206 V 132 205 V 132 194 V 133 210 V 132 183 V 132 207 V 132 164 V 133 166 V 132 105 V 132 84 V 132 -11 V 133 -59 V 132 -141 V 132 -185 V 132 -176 V 133 -97 V 132 -50 V 132 -43 V 132 -106 V 132 -114 V 133 -159 V 132 -294 V 132 -163 V 132 -69 V 133 -220 V 132 -74 V 132 -510 V 5132 876 L 133 0 V 5397 350 L 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 504 350 L 636 613 L 768 767 L 133 371 V 1033 876 L 132 347 V 132 243 V 133 319 V 132 192 V 132 207 V 132 197 V 133 214 V 132 189 V 132 212 V 132 179 V 133 188 V 132 158 V 132 129 V 132 81 V 133 23 V 132 -45 V 132 -117 V 132 -160 V 133 -180 V 132 -114 V 132 -41 V 132 -40 V 132 -89 V 133 -153 V 132 -155 V 132 -119 V 132 -200 V 133 -192 V currentpoint stroke M 132 -133 V 132 -168 V 132 -235 V 133 -13 V 5397 350 L 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 504 350 L 636 613 L 768 767 L 133 321 V 1033 876 L 132 347 V 132 223 V 133 347 V 132 184 V 132 213 V 132 191 V 133 215 V 132 191 V 132 218 V 132 185 V 133 201 V 132 181 V 132 167 V 132 128 V 133 106 V 132 34 V 132 -28 V 132 -90 V 133 -139 V 132 -167 V 132 -120 V 132 -81 V 132 -27 V 133 -76 V 132 -129 V 132 -137 V 132 -147 V 133 -157 V 132 -173 V 132 -116 V 132 -188 V 133 -70 V 132 -522 V 132 -239 V 132 121 V 133 -434 V 132 0 V 6058 350 L 132 0 V 133 0 V 132 0 V 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 504 350 L 636 613 L 768 767 L 133 321 V 1033 876 L 132 347 V 132 223 V 133 339 V 132 192 V 132 216 V 132 186 V 133 218 V 132 193 V 132 219 V 132 189 V 133 205 V 132 194 V 132 189 V 132 163 V 133 149 V 132 103 V 132 52 V 132 -5 V 133 -69 V 132 -125 V 132 -151 V 132 -134 V 132 -78 V 133 -38 V 132 -66 V 132 -106 V 132 -137 V 133 -128 V 132 -146 V 132 -140 V 132 -137 V 133 -159 V 132 -234 V 132 -264 V 132 20 V 133 -407 V 132 9 V 132 -499 V 132 -76 V 6323 767 L 6455 350 L 132 0 V 132 0 V 133 0 V 132 0 V 132 0 V 504 350 L 636 613 L 132 0 V 133 525 V 1033 960 L 132 263 V 132 243 V 133 310 V 132 206 V 132 211 V 132 186 V 133 218 V 132 193 V 132 220 V 132 191 V 133 211 V 132 197 V 132 197 V 132 184 V 133 178 V 132 147 V 132 116 V 132 67 V 133 10 V 132 -48 V 132 -104 V 132 -139 V 132 -137 V 133 -87 V 132 -52 V 132 -55 V 132 -84 V 133 -118 V 132 -129 V 132 -136 V 132 -133 V 133 -174 V 132 -114 V 132 -196 V 132 -116 V 133 -314 V 132 -112 V 132 -138 V 132 -346 V 133 34 V 132 -781 V 6587 350 L 132 263 V 6852 350 L 132 0 V 132 0 V 504 350 L 636 613 L 768 767 L 133 371 V 1033 876 L 132 347 V 132 243 V 133 301 V 132 205 V 132 224 V 132 185 V 133 215 V 132 194 V 132 221 V 132 192 V 133 212 V 132 200 V 132 203 V 132 192 V 133 194 V 132 174 V 132 157 V 132 122 V 133 82 V 132 23 V 132 -27 V 132 -87 V 132 -124 V 133 -131 V 132 -102 V 132 -53 V 132 -48 V 133 -78 V 132 -101 V 132 -133 V 132 -118 V 133 -142 V 132 -126 V 132 -150 V 132 -155 V 133 -181 V 132 -199 V 132 -107 V 132 -200 V 133 -95 V 132 -484 V 132 -245 V 132 -63 V 133 -262 V 132 0 V 7116 350 L stroke grestore end showpage %%Trailer %%DocumentFonts: Helvetica %%Pages: 1 \end{filecontents} % % Second part : the latex file itself %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \documentclass[12pt]{article} \usepackage{amsthm, amsmath, amsfonts, psfig} \newtheorem{thm}{Theorem}[section] \newtheorem*{TP}{Poincar\'e's theorem} \newtheorem*{LK}{Lemma of Kac} \newtheorem{cor}[thm]{Corollary} \newtheorem{prop}{Proposition}[section] \newtheorem{lem}[thm]{Lemma} \newtheorem{rem}{Remark}[section] \newtheorem*{rem*}{Remark} \newtheorem*{com*}{Comment} \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem*{notation}{Notation} \def \D{\textrm d} \let \mc=\mathcal \def \dimH{\textup{dim}_H} \def \capa{\textup{cap}} \def \htop{h_{\textup{top}} } \def \per{\textup{Per}} \def \dper{\#\textup{Per}} \def \dfix{\#\textup{Fix}} \def \dcyl{\#\textup{Cyl}} \def \bra{\langle} \def \ket{\rangle} \def \ER{\text{E.R.}} \def \lmin{\lambda_{\text{min}}} \def \a{a} \def \Chi{\chi} \def \mpa{m^\Phi_\alpha} \def \mea{m^{\text{exp.}}_\alpha} \def \mla{m^{\text{lin.}}_\alpha} \def \Mpa{M^\Phi_\alpha} \def \Mea{M^{\text{exp.}}_\alpha} \def \Mla{M^{\text{lin.}}_\alpha} \def \MApa{M^{A,\Phi}_\alpha} \def \MOpa{M^{O,\Phi}_\alpha} \def \MCpa{M^{F,\Phi}_\alpha} \def \MFpa{M^{F,\Phi}_\alpha} \def \mOpa{m^{O,\Phi}_\alpha} \def \mCpa{m^{F,\Phi}_\alpha} \def \mFpa{m^{F,\Phi}_\alpha} \def \Rleq{\mc R} %\def \Req{\mc R_=} \def \ROleq{\mc R^O} %\def \ROeq{\mc R^O_=} \def \RCleq{\mc R^F} %\def \RCeq{\mc R^F_=} \def \RFleq{\mc R^F} %\def \RFeq{\mc R^F_=} \def \RAleq{\mc R^A} %\def \RAeq{\mc R^A_=} \def \bor{\partial} \def \epsilon{\varepsilon} \newcommand{\set}[2]{ {\left \{ #1 \Bigm| #2 \right \} }} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def \sizegraph {6.5cm} \begin{document} \font\fifteen=cmbx10 at 15pt \font\twelve=cmbx10 at 12pt \begin{titlepage} \begin{center} \renewcommand{\thefootnote}{\fnsymbol{footnote}} {\twelve Centre de Physique Th\'eorique\footnote{Unit\'e Propre de Recherche 7061 }, CNRS Luminy, Case 907} {\twelve F-13288 Marseille -- Cedex 9} \vspace{1 cm} {\fifteen Fractal and statistical characteristics of recurrence times. \setcounter{footnote}{2} \footnote{talk given by Sandro Vaienti at the conference ``Disorder and Chaos'' (Rome 22-24th sept. 1997), in honour of Giovanni Paladin} } \vspace{0.3 cm} \setcounter{footnote}{0} \renewcommand{\thefootnote}{\arabic{footnote}} {\bf Vincent \textsc{Penn\'e}\footnote{ Centre de Physique Th\'eorique, CNRS Luminy, Case 907, F-13288 Marseille - Cedex 9, FRANCE, and PhyMat, Universit\'e de Toulon et du Var. 83957 La Garde, FRANCE}, Beno\^\i t \textsc{Saussol}$^1$, Sandro \textsc{Vaienti}$^1$. } \begin{center} {\it Dedicated to Giovanni} \end{center} \vspace{2,3 cm} {\bf Abstract. } \end{center} In this paper we introduce and discuss two proprieties related to recurrences in dynamical systems. The first gives the asymptotic law for the return time in a neighborhood, while the second gives a topological index of fractal type to characterize the system or some regions of the system. \vspace{2 cm} \noindent Key-Words: Recurrence, Poisson process, Topological entropy. \bigskip \noindent Number of figures %\footnote{Figures can be obtained via ordinary mail. Please contact: Secr\'etariat du CPT.} : 2 \bigskip \noindent February 1998 \noindent CPT-98/P3623 \bigskip \noindent anonymous ftp : ftp.cpt.univ-mrs.fr \noindent web : www.cpt.univ-mrs.fr \renewcommand{\thefootnote}{\fnsymbol{footnote}} \end{titlepage} \setcounter{footnote}{0} \renewcommand{\thefootnote}{\arabic{footnote}} %\def \proofname{Proof} \section{Introduction} The study of recurrence is at the hearth of ergodic theory. The first rigorous result in this field is probably the famous Poincar\'e's recurrence theorem, which states that given the dynamical system $(X, T, \mu)$ where $T$ is a measurable map on $X$ and $\mu$ a $T-$invariant probability measure, then $\mu-$almost every point in each measurable subset $A\subset X$ comes back to $A$ an infinite number of times. Poincar\'e's result deals with a measurable recurrence propriety in the sense that the measure plays a fundamental role. But there are more genuine topological recurrence proprieties, like , for example, the density of all the orbits for irrational rotations. In this paper we present and discuss two problems which are related to these two aspects of recurrence, the measurable and the topological ones. We will get some precise statistics of the return time of typical orbits in a given neighborhood, while in the second case we will get some global informations on the asymptotic distribution of orbits through the definition of a ``dimension''. The construction of this dimension uses explicitly the shortest return time in a given set and is a reminiscent of the way of constructing Haussdorf's dimension : this explains the attribute fractal in the title of our note, but probably there are deeper reasons. We hope that this dimension could be used as a statistical indicator of chaos and complexity. We now define the fundamental quantities investigated in this paper. Take $U$ a subset of $X$ and define for each $x\in U$ the first return time into $U$ as : \[\tau_U(x)=\inf\set{k>0}{T^kx\in U}.\] The Poincar\'e recurrence of a \emph{point} $\tau_U(x)$ as defined above leads to the first return time of a \emph{set} : it is the infimum over all return times of the points of the set, and we denote it \[\tau(U)= \inf_{x\in U} \tau_U(x).\] \section{Poisson statistics for the return time} We will consider as above a dynamical system $(X, T, \mu)$ where $X$ is a (not necessarily compact) metric space, $T$ a measurable application on $X$ and $\mu$ a probability $T-$invariant Borel measure. A refinement of Poincar\'e's theorem can be found if we assume that $\mu$ is $T-$ergodic. Under this hypothesis, take $U$ a measurable subset of $X$. Then the function $\tau_U: U\rightarrow \mathbb N$ is measurable. If we introduce the induced probability measure $\mu_U$ defined by $\mu_U=\frac {\mu_{|U}} {\mu(U)}$, then the Kac Lemma \cite{petersen} states that \[ \int_U \tau_U(x) \text{d}\mu_U = \frac 1 {\mu(U)}, \] whenever $\mu(U)>0$. Suppose now that $U_\epsilon(z)$ is a neighborhood of $z\in X$ with diameter $\epsilon$. Define the random variable $\mu(U_\epsilon)\tau_{U_\epsilon(z)}$ (we will call it the normalized return time). Next results assert that, under very general conditions, the normalized return time converges in law, with respect to the measure $\mu_\epsilon\equiv\mu_{U_\epsilon(z)}$, to a mean one exponential random variable, and that for $\mu-$almost every $z\in X$, precisely : \begin{equation} \label{eq poisson} \mu_\epsilon\set{x\in U_\epsilon}{\mu(U_\epsilon)\tau_{U_\epsilon}(x)>t} \xrightarrow[\epsilon\rightarrow 0]{} e^{-t}. \end{equation} This kind of result was first proved for Axiom-A systems in a series of independent papers by Pitskel \cite{pitskel}, Hirata \cite{hirata1}, Collet \cite{collet lect}. Actually, these authors proved a stronger result : in fact they consider the sequence of successive normalized return times in $\tau_{U_\epsilon}$ and proved that this sequence converges to the Poisson point process in finite-dimensional distribution when $\epsilon\rightarrow 0$. We want here to give a heuristic proof which shows how naturally the exponential law for the statistic of the first return time arises. \begin{proof}[Heuristic proof] Let $t>0$ be fixed, $U\subset X$, $\tau=\tau(U)$ and $n=n(U)>0$ which will be fixed later. We want to estimate the quantity \begin{equation}\label{eqt} \mu_U \set{x\in U}{n(U)\tau_U(x)> t}. \end{equation} This can be rewritten as $$ \mu_U ( T^{-1}U^c \cap\cdots\cap T^{-t/n}U^c ) = \mu_U ( T^{-\tau}U^c \cap\cdots\cap T^{-t/n}U^c ). $$ Now we suppose (H1) that $\tau$ is big enough, and that (H2) the events $T^{-\tau}U^c,...,$ $T^{-t/c}U^c$ are nearly independent, which yields $$ (\ref{eqt}) \approx \mu(T^{-\tau}U^c\cap\cdots\cap T^{-t/n}U^c) \approx \left( 1-\mu(U) \right)^{(t/n)-\tau}. $$ we assume now (H3) that $t/n(U)-\tau(U)$ tends to infinity as $\mu(U)$ goes to zero, $$ (\ref{eqt}) \approx \exp \left( -\frac {\mu(U)}{n(U)} t + \mu(U)\tau(U) \right). $$ The exponential law appears now if we take the normalization $n(U)=\mu(U)$, and (H3) now reads $\mu(U)\tau(U)\to 0$ as $\mu(U)$ vanishes. \end{proof} Besides the statistics of return times one could equivalently study the process of the successive entrance times into a region $\Omega$ and prove under some conditions that, when times are correctly rescaled, the process converges in law to a Poisson point process \cite{collet galves}. In some cases it is even possible to estimate the rate of this convergence \cite{galves schmitt}, and it is a promising field for further researches. The Poisson statistics for the first return times was successively extended by Hirata \cite{hirata2} to a large class of systems verifying what he called the ``self-mixing'' conditions. Unfortunately these conditions do not hold for non-uniformly hyperbolic dynamical systems. Nevertheless the techniques introduced in \cite{hirata2} can be adapted to this kind of dynamical system. In particular we will consider the well known one parameter family of one-dimensional intermittent maps : \[ T(x) = \begin{cases} x(1+2^\alpha x^\alpha);& \qquad \forall x\in [0,1/2 [, \\ 2x-1;& \qquad \forall x\in [1/2,1]. \end{cases} \] When the parameter $\alpha>1$, $T$ has a $\sigma-$finite absolutely continuous invariant measure : in these cases it is possible to prove that the sequence of successive (suitably normalized) entrance times in a small neighborhood of the neutral fixed point converges to a Poisson point process provided the system is equipped with an absolutely continuous distribution with density of bounded variations. \cite{campanino, collet galves} We instead consider the case $0<\alpha<1$, for which we construct in the paper \cite{liverani saussol vaienti} an absolute continuous invariant probability measure by using a new technique based on a stochastic perturbation of the Perron-Frobenius operator. We also proved polynomial decay of correlations for H\"older continuous observables. The case $0<\alpha<1$ is quite different if compared with the case of the $\sigma-$finite measure, especially for the techniques used to control the distortion of the application. Moreover, strictly speaking, our analysis will be local around almost any point and not only restricted to a neighborhood of the neutral fixed point. Let us now consider the infinite Markov partition $\xi$ generated by the left preimages of $1$ and consider the dynamical partitions $\xi_n\equiv \bigvee_{i=0}^{n-1}T^{-1}\xi$. If we now fix a point $z\in[0,1]$ and denote by $U_n$ the member of $\xi_n$ containing $z$ and by $\mu_n\equiv \frac {\mu_{U_n}} {\mu(U_n)}$, we could ask if the propriety (\ref{eq poisson}) follows. This is just the content of the next theorem. \begin{thm}[\cite{hirata saussol vaienti}] For $\mu-$almost every $z\in [0,1]$ and $\forall t>0$ we have : \[ \mu_n\set{x\in U_n}{\mu(U_n)\tau_{U_n}>t} \xrightarrow[n\rightarrow \infty]{} e^{-t}. \] \end{thm} The proof of this theorem heavily relies on the techniques developed in \cite{liverani saussol vaienti} for decay of correlations. (Laplace transform technique are used in the rigorous proof). Following similar arguments, it is also possible to prove a related result, that, although expected, was not known to be true. \begin{thm}[\cite{hirata saussol vaienti}] $\xi$ is a weak-Bernoulli partition for $T$, i.e. $$ \sup_{n,l \geq 0} \sum_{\substack{A \in \xi^n\\ B\in T^{-(n+t)}\xi^l} } | \mu(A\cap B) - \mu(A)\mu(B) | \xrightarrow[t\to\infty]{} 0. $$ \end{thm} This propriety implies a stronger one, that the system $(X,T,\mu)$ is in fact Bernoulli. \section{Dimensional characteristic for Poincar\'e recurrence} In this section, we present a new approach originally introduced by Afraimovich \cite{afr1} for minimal sets, in order to characterize topological recurrence. We will present a series of preliminary results, some of which apply to general dynamical systems, others to specific class of hyperbolic systems : the main result is the possibility to define a ``dimension'' (in the fractal language) which turns out to be an invariant for topological conjugation. This dimension reveals to be a good indicator to distinguish systems of zero topological entropy. On the contrary, in the case of some positive entropy systems, it coincides with topological entropy (for subshifts of finite type and $\beta-$shifts for example). It is a matter of investigation whether the identification with topological entropy persists for more complicated chaotic systems. We will present some numerical evidence in this direction at the end of this chapter. \subsection{Construction of the dimension} We are using the well known Caratheodory's construction \cite{rogers, pesin}, for which Haussdorf's measures are special cases. We will work on a compact metric space $(X,d)$ together with a continuous transformation $T$, which form a dynamical system $(X, T)$. % Obviously $\tau(U)$ is a monotonic function, i.e. : $A\subset B \Rightarrow %\tau(A)\geq\tau(B)$. For any $A\subset X$, we define $\mc R^A(A, \epsilon)$ the collection of all countable covers of $A$ by subsets of $X$ with diameters less than $\epsilon$ (the superscript $A$ above $\mc R$ stands for \emph{arbitrary} type of set that form the cover %, i.e. they need not to be open, closed nor even to be in the topology of $X$ ). In the same way, we denote by $\ROleq$ and $\RCleq$ the restriction of the precedent collection to cover with respectively \emph{open} and \emph{closed} sets. Then we define a pre-measure (or gauge function) $\Phi(U) : 2^X \rightarrow \mathbb R^+$ with the propriety that $\Phi(\emptyset)=0$. Now we define \footnote{a slightly more general definition would be to, instead of put $\Phi^\alpha$ in the sum, rather use a one parameter family of pre-measure $\Phi_\alpha$.} \begin{equation}\label{sum} \Mpa(A, \epsilon) = \inf_{R\in\mc R(A,\epsilon)} \sum_{U\in R} \Phi^\alpha(U). \end{equation} (we do not precise here if we use covers by arbitrary sets, open sets or closed sets.) %We will use the convention that %$\{ \emptyset \} \in \mc R(A,\epsilon)$, so that %$\Mpa(\emptyset)=0$. It is easy to show that $\Mpa$ is a family of outer measure with the parameter $\alpha$. The idea of Afraimovich was to apply this construction in the case where $\Phi(U)$ is a decreasing function of $\tau(U)$, i.e. $\Phi(U) = g\circ \tau(U)$ where $g : \mathbb N\rightarrow \mathbb R$ is decreasing and converge to zero. We will also call this function a gauge function. Typically, we will set $\Phi(U)=e^{-\tau(U)}$ or $\Phi(U)=\frac 1 {\tau(U)}$, the choice being determined by the type of growth rate of Poincar\'e recurrence with respect to the diameter in our system. From now on, we will implicitly use one of these pre-measure. \begin{thm}[\cite{nous1}] \label{thm open closed} The outer measure for Poincar\'e recurrence $\MApa$ constructed with arbitrary covers is concentrated on periodic points. The outer measures for Poincar\'e recurrence $\MOpa$ and $\MCpa$ constructed respectively with open and closed covers, coincide on \emph{closed sets}. \end{thm} \begin{rem*} We recall that in the case of Haussdorf's measures (obtained when we set $\Phi(U)=|U|$), the three constructions (arbitrary, open and closed covers) coincide on any borelian sets \cite{rogers}. \end{rem*} The first part of this theorem results from a very simple construction, in which we construct covers of the space minus its periodic points with sets that all have infinite Poincar\'e recurrence \footnote{ To give an idea : suppose that the transformation $T$ is invertible, then we can construct a set $U$ by taking one point (and no more) of each non-periodic orbit of the system (remark that we need to use the Axiom of Choice to do that). One can check that $\tau(U)=\infty$, but even more : $\forall k, \tau(T^k U)=\infty$ . Thus, the countable family of set $U_k \equiv T^k U$ is a cover of the space minus the periodic points, whose members have all infinite Poincar\'e recurrence. Because of Poincar\'e's recurrence theorem, these sets cannot have positive measure for any invariant measure, and thus for any invariant non-atomic probability measure they cannot be all measurable . } ! (but usually these sets cannot be borelian\ldots) The second part is due to a nice propriety of the pre-measure $\Phi$ and comes from a more general theorem that we prove in \cite{nous1}. This indicate that the choice of open or closed covers is natural, since otherwise we would obtain trivial results. %\begin{proof}[Proof of the first part] %We define an equivalence %class for the points of $X$ : for any $x, y\in X$, we will say that they are in %the same orbit if there is $k_x>0$ and $k_y>0$ for which $T^{k_x} x = T^{k_y} y$. %We now define a slice of an orbit : it is a set $S$ such that for any %$x, y\in S$, there is $k>0$ such $T^k x = T^k y$. It is obviously a subset of %an orbit. Note that a slice of orbit might be a periodic point, %and that an orbit might contain only periodic points. % %Then we construct a set $U$ by taking, for each orbit one (and no more) %slice of orbit %(we need to use the Axiom of Choice to do that). %If $F$ denotes the set of all periodic points, %one can check that $\tau(U\setminus F)=\infty$, but even more : $\forall k, %\tau(T^k U\setminus F)=\infty$ . %Thus, the countable family of set $U_k \equiv T^k U \setminus F$ is %a cover of $X\setminus F$ whose members have all infinite Poincar\'e %recurrence. %\end{proof} We now take the limit $\epsilon\rightarrow 0$ : \begin{equation} \mpa(A) = \lim_{\epsilon\rightarrow 0} \Mpa(A, \epsilon). \end{equation} The set function $\mpa(A)$ is a family of \emph{borelian measure} (as shown in \cite{falconer}). If $\Phi(U)$ goes uniformly to zero when $|U|$ goes to zero \footnote{more precisely if $ \forall\epsilon >0, \quad \exists\delta \quad \text{ such that } \quad |U|<\delta\Rightarrow\Phi(U)<\epsilon. $ } then we meet all the conditions of the classical Carath\'eodory's construction. It is well known, then, that there exists a unique transition exponent $\alpha^\Phi_c(A)$ such that \[ \mpa(A) = \begin{cases} \infty & \text{ if }\alpha<\alpha^\Phi_c(A) \\ 0 & \text{ if }\alpha>\alpha^\Phi_c(A) \end{cases} \] It is true with Haussdorf's measures, and Afraimovich has proved the same result with Poincar\'e recurrence if the system is minimal\footnote{a dynamical system is minimal if each orbit is dense in the space.} \cite{afr1}. \begin{figure} \begin{center} \leavevmode \begin{picture} (320,90) (-15,-10) %% Premier dessin : infini -> zero \put (-12,68){$\infty$} \put (0,70){\line(1,0){50}} \put (50,70){\line(0,-1){60}} \put (50,10){\line(1,0){50}} \put (102,7){$0$} \put (-10,0){\thicklines \vector(1,0){120}} \put (50,-2){\line(0,1){4}} \put (45,-8){$\alpha_c$} %% Second dessin : infini -> fini \put (188,68){$\infty$} \put (200,70){\line(1,0){50}} \multiput (250,-2)(0,8) {9}{\line(0,1){4}} %\put (250,0){\dashbox{5}(0,100)} %\put (250,-2){\line(0,1){4}} \qbezier (250,70)(250,10)(300,10) \put (302,7){$<\infty$} \put (190,0){\thicklines \vector(1,0){120}} \put (245,-8){$\alpha_c$} \end{picture} \end{center} \quote{\small{{\bf Figure 1.} On the left, the transition is net, on the right it is not, however we are able to define a unique critical exponent $\alpha_c$ in both cases.}} \end{figure} However, we will consider cases where the transition point is not so net, and still we can define a critical exponent without any ambiguity (see Figure 1). This is possible because the set function $\mpa(A)$ is non-increasing with $\alpha$. So, we define the critical exponent of a set $A\subset X$ as \begin{equation} \alpha^\Phi_c(A) = \sup\set{\alpha>0}{\mpa(A)=\infty}. \end{equation} It is always well defined and positive if we adopt the convention that $\sup \emptyset = 0$. We will call this dimension-like characteristic either dimension for Poincar\'e recurrence, either Afraimovich-Pesin (AP)'s dimension. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Some proprieties of Afraimovich-Pesin's dimension} \begin{thm} \label{THM top inv} The borelian measure $\mpa$ and the dimension $\alpha^\Phi_c$, constructed with open or closed covers, are invariant under topological conjugation, i.e. if $(X,d,T)$ and $(X',d',T')$ are two continuous dynamical system on metric spaces and if there exists a homeomorphism $h : X\rightarrow X'$ such that $T=h^{-1}\circ T'\circ h$, then for any $A\subset X$, $\mpa(A) = m^{'\Phi}_\alpha(h(A))$. \end{thm} %Before we write the proof, We would like to remark that this result is important since it allows us to say that AP's dimension is a \emph{topological propriety}, i.e. two dynamical systems that are similar from a certain (topological) point of view will have the same AP's dimension. Topological entropy is a tool that allows us to classify systems that are similar. The problem is that it cannot be used to distinguish systems with zero topological entropy although they may have very different behavior, %a large class of dynamical systems has %zero topological entropy (so it does not distinguish between systems of this %large class), which show the need to find other tools. AP's dimension with a non-exponential gauge function (e.g. $\Phi(U)= \frac 1 {\tau(U)}$) is such a tool. It is interesting to note that for some classes of minimal sets with zero topological entropy, some topological invariant numbers have been recently proposed : for example the symbolic (or topological) complexity \cite{ferenczi}, and the covering number \cite{chabarchina}. It would be interesting to compare them to AP's dimension. \begin{proof}[Proof of Theorem \ref{THM top inv}] We recall that we supposed from the beginning that $X$ is compact, so that $h$ is uniformly continuous. We write the uniform continuity \[ \begin{split} \forall \delta >0, \exists \epsilon(\delta) \text{ such that } \forall x,y\in X, \text{ if } d(x,y)&<\epsilon(\delta) \\ \text{ then } d'(h(x),h(y))&<\delta. \end{split} \] Let $A\subset X$ and $A'=h(A)\subset X'$. Let $\mc R'(A',\delta)$ be the set of all covers $R'$ of $A'$ with sets of diameter less than $\delta$. Let $h(\mc R(A,\epsilon(\delta)))$ be the set of all transformed covers $h(R) \equiv \{ h(U), U\in R\}$ of $A$ with sets of diameter less than $\epsilon(\delta)$. Then $\mc R'(A',\delta)$ contains $h(\mc R(A, \epsilon(\delta)))$. Moreover, topological conjugation implies obviously that $\tau(U)=\tau'(h(U))$ for any set $U \subset X$, thus we have $\Phi(U)= \Phi'(h(U))$. This shows that \[ \Mpa(A, \epsilon(\delta)) = \inf_{R\in\mc R(A, \epsilon(\delta))}\sum_{U\in R} \Phi(U)^\alpha \geq \inf_{R'\in\mc R'(A', \delta)}\sum_{U'\in R'} \Phi'(U')^\alpha = M^{'\Phi'}_\alpha(A',\delta). \] Then, taking the limit $\delta\rightarrow 0$ (hence $\epsilon\rightarrow 0$), we obtain $ \mpa(A) \geq m^{'\Phi'}_\alpha(A')$. Now, by reversing $A$ and $A'$'s rules, one can apply the same idea to obtain the opposite inequality, which yields \[ \mpa(A) = m^{'\Phi}_\alpha(A'). \] It is then obvious that $\alpha^\Phi_c(A) = \alpha^{'\Phi'}_c(A')$. \end{proof} The next theorem establishes other proprieties of AP's dimension, the most important being the one that says that AP's dimension over $X$ coincide with AP's dimension restricted to the set of non-wandering points\footnote{a point $x$ is non-wandering if any open neighborhood $V$ has a finite Poincar\'e recurrence, i.e. $\tau(V)<\infty$.}, that is exactly what happens for the topological entropy. %An other propriety %is to see what happens when we consider some power of the initial application %$T'=T^k=\underbrace{T\circ T\circ\ldots\circ T}_{\text{k times}}$ (we recall %that topological entropy is then multiplied by $k$). \begin{thm} \label{thm non wander} Dimension for Poincar\'e recurrence has the following proprieties : \begin{enumerate} \item if we use the pre-measure $\Phi(U) = e^{-\tau(U)}$, then for any $k>0$, we have $\alpha^\Phi_c(T^k, X) \leq k \alpha^\Phi_c(T, X)$, \item if we use the pre-measure $\Phi(U) = \frac 1 {\tau(U)}$, then for any $k>0$, we have $\alpha^\Phi_c(T^k, X) \leq \alpha^\Phi_c(T, X)$, \item if $T$ is invertible, then $\mpa$ is an invariant measure. \item $\alpha_c^\Phi(T, X) = \alpha_c^\Phi(T, NW) = \alpha_c^\Phi(T_{|NW}, NW)$, where $NW$ denotes the set of non-wandering points. \item if we use the pre-measure $\Phi(U)=e^{-\tau(U)}$, there is the lower-bound \[ \alpha_c^\Phi(X) \geq \overline\lim_{k\rightarrow\infty}\frac 1 k \log \dper(k), \] where $\dper(k)$ denotes the number of periodic points with smallest period $k$. \end{enumerate} \end{thm} \begin{rem*} We point out that these results are true with open and closed covers. We proved it in \cite{nous1}. There are examples of diffeomorphisms of the unit disk where strict inequality holds in the two first points of this theorem. However, these constructions depend heavily on the combinatorics of the periodic points, and these maps are somewhat unnatural. That is why we conjecture that for a large class of dynamical systems the equality holds. We recall that for topological entropy, the following equality holds : \[ \htop(T^k) = k \htop(T). \] The last point gives a lower-bound to AP's dimension with the periodic points. We recall that there exist a similar lower-bound for topological entropy with expansive map (\cite{walters}, p178) : \[ \htop\geq\overline{\lim}_{k\rightarrow\infty}\frac 1 k \log\dfix(k), \] where $\dfix(k)$ denotes the number of fixed point of $T^k$. \end{rem*} \subsection{Application of AP's dimension to classical dynamical systems} \subsubsection{Systems with positive topological entropy} We now state some results about AP's dimension in simple cases, as subshifts of finite type and $\beta-$shift. For these systems, AP's dimension with exponential gauge function ($\Phi(U)=e^{-\tau(U)}$) and topological entropy are equal. These systems are symbolic systems. We will work on the space $\Omega = \{0, \ldots, p-1\}^{\mathbb N}$ of all semi-infinite sequences $\omega = \omega_1 \omega_2 \ldots$, with the product topology. We consider %\begin{itemize} %\item The first is the shift to the left $\sigma$ such that $\sigma\omega_1 \omega_2 \omega_3 \ldots = \omega_2 \omega_3 \ldots$. Subshifts of finite type and $\beta-$shift are restrictions of the shift on some invariant subsets of $\Omega$. See \cite{parry1,parry2} for a complete description of these systems. Many dynamical systems are topologically conjugate to subshifts of finite type, whereas the $\beta-$transformation $T_\beta(x)=\beta x \mod 1$ is conjugate to the $\beta-$shift \cite{parry1}. %\item The second transformation is the addic transformation $T$ on $\Omega$ %which we define as %\[ (T\omega)_i = \begin{cases} \omega_i+1 \mod p; & \qquad %\text{if either $i=1$, either $\forall 1\leq j0} N_\epsilon(k)$, then it is possible to prove that the critical exponent $\alpha_c$ is bounded by \[ \overline{\lim}_{k\rightarrow\infty} \frac 1 k \log\dper(k) \leq \alpha_c \leq \overline{\lim}_{k\rightarrow\infty} \frac 1 k \log N(k).\] The first inequality is actually always verified (provided $T$ is continuous). Thus the numerical experimentation shows that for the logistic map $N(k) \propto \dper(k)$ and therefore that AP's dimension and topological entropy coincide. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{Acknowledgments} We want to thanks M. Hirata who is actually the coauthor of section two's results and V. Afraimovich who introduced us to the recurrence dimension. We also thank S. Kolyada for remarks and stimulating discussions. S. Vaienti acknowledges the support of the ISI-Turin. \begin{thebibliography}{99} \bibitem{afr1} V. Afraimovich, Chaos {\bf 7} (1997) \bibitem{afr2} V. Afraimovich, G.M. Zaslavsky, ``Sticky Orbits of Chaotic Hamiltonian Dynamics'', Preprint 1998 \bibitem{ALM} L. Alsed\`a, J. Llibre and M. Misiurewicz, ``Combinatorial Dynamics and Entropy in Dimension One'', (1993, World Scientific Publ., Singapore) \bibitem{auslander and katznelson} J. Auslander, Y. Katznelson, Israel J. Math, {\bf 32}, 375-381, (1979) \bibitem{campanino} M. Campanino, S. Isola, Forum Mathematicum, {\bf 7}, 331--348 (1995) \bibitem{chabarchina} N. Chabarchina, ``Nombres de recouvrement'', Th\`ese - I.M.L. Marseille (1997) \bibitem{collet lect} P. Collet, Lectures given at the C.I.M.P.A. - Temuco (1991) \bibitem{collet galves} P. Collet, A. Galves, Journal of Statistical Physics, {\bf 72}, 459--478 (1993) \bibitem{collet galves schmitt} P. Collet, A. Galves, B. Schmitt, ``Fluctuation of repetition times for Gibbsian sources'', Preprint 1997 \bibitem{falconer} K.J. Falconer, ``The geometry of fractal sets'', (1985, Cambridge University Press) \bibitem{ferenczi} S. Ferenczi, to appear in Israel Journal of Mathematics (1997) \bibitem{galves schmitt} A. Galves, B. Schmitt, Random Comput. Dyn. to appear (1997) \bibitem{hirata1} M. Hirata, Ergodic Th. \& Dynamical Systems, {\bf 13}, 533--556 (1993) \bibitem{hirata2} M. Hirata, ``Poisson law for the dynamical systems with the "self mixing" conditions'', preprint TMU (1996) \bibitem{hirata saussol vaienti} M. Hirata, B. Saussol, S. Vaienti, ``Poisson law for the return time in an intermittent map'', in preparation \bibitem{knig} J.L. Knig, J. Analyse Math, {\bf 51}, 182--227 (1988) \bibitem{liverani saussol vaienti} C. Liverani, B. Saussol, S. Vaienti, to appear in Ergodic Th. \& Dynamical Systems (1997) \bibitem{parry1} W. Parry, Acta Math. Acad. Sci. Hung. {\bf 11}, 401--416 (1960) \bibitem{parry2} W. Parry, Trans. A.M.S., {\bf 122}, 368--378 (1966) \bibitem{pesin} Y. B. Pesin, ``Dimension Theory in Dynamical Systems'', (1997, The University of Chicago Press, Chicago and London) \bibitem{petersen} K. Petersen, ``Ergodic Theory'', (1983, Cambridge University Press). \bibitem{nous1} V. Penn\'e, B. Saussol, S. Vaienti, ``A topological dynamical number constructed with recurrence times'', preprint (1998) \bibitem{pitskel} B. Pitskel, Ergodic Th. \& Dynamical Systems, {\bf 11}, 501--533 (1991) \bibitem{rogers} C.A. Rogers, ``Hausdorff measures'', (1970, Cambridge) \bibitem{walters} P. Walters, ``Ergodic Theory - Introductory Lectures'', (1975, Springer-Verlag) %\bibitem{book} Name F., Title (Publisher, Publisher's location, year %of publication) pp. 134-137. %\bibitem{article} Name A. and Name B., J. Phys. A {\bf 99} (1999) 9-17 %\bibitem{conference} Name A. and Name B. "Title of the paper", %Title of the conference, Location and date of the Conference, C.Name and %D. Name Eds. (Publisher, Publisher's location, year %of publications) pp. 544-549 \end{thebibliography} \end{document}