Symbolic Dynamics for Real Rational Maps

Transcription

1 Symolc Dymcs or Rl Rol Mps João Crl Dprm o Mhmcs o h Azors Uvrsy Po Dlg Porugl ABSTRACT Ths or s mp o suy h ymcs o rl rol mps usg symolc ymcs I s gv mpl h llusrs ho h opologcl ropy c clcul usg g hory Mrov pros hs mly o rol mps Kyors Rol mps symolc ymcs opologcl ropy Mrov pros Th mly o mps INTRODUCTION s lrg sc group o ucos h r smooh h gr zro rom crcl o sl hvg o crcl pos zro y o oo Dpg o h prmrs c orgz hs mly gh r groups s sho gur Ech group hs s o hvor Fgur : Th mly o mps I hs ppr ll suy oly h ymcs o h mp Wh hs rsrcos h Schrz rvv r supr-sl pos hs mp hv o crcl po = S h > > > s lys gv h roos o Th ymcs o s compl u sy o urs h h propr ools Th gol o hs or s o o mor ool o urs usg ools lry o h suy o h logsc mp s symolc ymcs Mrov pros To gv o ou o h comply o h hvor o s ough o s h urco grm sho gur As Ol Jourls jourolcom 9

2 Fgur : Burco grm or h =- So sco usg sr symolc ymcs s rouc mssly ruls o ro squcs ssoc o ll prs h srucur o h s o mssl squcs ulg o g squc r Th chqus o symolc ymcs r s o h Kg Thory or o-msol mulmol mps vlop y Mlor Thurso s [] Rmos JS [] Sc Kg hory s ppropr ool o o opologcl clssco o h o-msol mps hs ors s o plor som rsuls h c cr possly o so o or h suy o rol mps SYMBOLIC DYNAMICS N L h lph H A D L O R E Z h s P h lms o P h N No g h pplco : R \ D E s c s gur A L O R Z N o symolc squcs o y Fgur : Th pros o rl s I o cosr h sh opror : X XX XX X hv h commuv grm As Ol Jourls jourolcom 7

3 h R \ D E N s h symolc spc h lph P W c rouc Ω orr uc lcogrphclly y h orr h rl s R h pry y h survls hr h uco s crsg So hv A L O R Z h h pry s v Z R O L A h h pry s o Do W sy h X Y or X Y X Y l X X l X X X Y hr l X ppr X X X W hv s mpl usg h o RZL : s h umr o ms h h symols A L RZA RZR RZZ TWO METHODS THE SAME ENTROPY VALUE Th or o lm s h s orm y h mgs o h succssv composos or ros o I cosruc corrspoc o ch composo mg o lr o h lph H h s h pro o h rl s hr h mg s clu ch or corrspos o squc Wh som clculus c s h h vlus h l ghorhoo o = h vlus h rgh ghorhoo o = hv h sm squc W ll sg hs pos y p-pos ssoc o P squcs Ω By h sm rso ll sg h vlus h rgh ghorhoo o = h vlus h l ghorhoo o = s s-pos ssoc o S squcs Ω Usg h mp rouc h g PQS hr Q cos h or o h crcl po = P cos h or o h p-pos S cos h or o h s-pos As mpl = = o h ro o P Q S ZARZ ARZR ZARZ ZRZR ZARZ ll hv h g Loog gur c s ll h possl squcs o PQS rough comorl r cosruc usg h orr pry o oc h h mml squc s AZ h mml squc s Z h Bu o ll h squcs r mssl ch mly o mps Fgur : Tr o ll mssl squcs or = Tg = c chrcrz h mssl squcs usg rso mr T I h mr T s h mr hr h ros colums r ll y h lms o h lph H such s As Ol Jourls jourolcom 7

4 I proucs h ollog rsul T Proposo L X X X X mssl squc Th X sss h ollog ruls: Proo T ; X X I X L or X R h X Q h Q h g squc o h crcl po = Wh h co o ch rl rvl chgs s rsrr ohr rvl or h uo o rl rvls Usg h rsuls o Lmpr JP Rmos JS [] o h mp h = cosrg h pros o h rl s ll y h symols A L R Z usg h symol O or = hvg A L O R Z h h pry o s v Z R O L A h h pry o s o c s h smpl clculus h T rprss h ro o h lms o ch pro y ollog urlly h rsuls Fgur : Mrov pro or ZARZ = = Tg proposo h rl s s c s gur ssocg ch = o Q Q hr Q s h corrspog g squc o symolc squc y = o symolc squc P P ; ch hr P s h corrspog g squc o p h =p p-po; ch z = o symolc squc S S hr S s h corrspog g squc o s s-po c ul or = = s gur h rso mr h =s M By Msurcz M Szl W [] c clcul h opologcl ropy h op lm L log hr L s h umr o lps o umr o su-rvls hr s mooo or h log op m M h s h hr M m s h As Ol Jourls jourolcom 7

5 spcrl rus o h rso mr M ssoc o h Mrov pro h log op I hs cs hv Aohr rou o clcul h opologcl ropy s rough h g mr s scr Lmpr JP Rmos JS [] usg h chqus vlop y Rmos JS [] As Mlor J Thurso W [] pg 8 h -h g crm o o pcs- I ssoc o h g s mooo mp srcly mooo h rvl h = hr ssoc o h rry o ch urg po s h vr coor o ch symolc squc O h or o Mlor Thurso ch po locl mmum or mmum o h mp W c uc vry sly I I s s h om o couous mp h s crsg I crss I Bu s scouous hs uco s o ru Sc h mooocy o h mp I os o chg cosr I s op rvl c m lso h scous o mp o h urg pos our cs or o Follog h rsuls prs y Lol JL Rmos JS [] or mps h hols c p h rsuls o Mlor Thurso o our mp usg g mr N s [] pg 8 h g rm D D s [] pg 8 hr D s h rm o N hou h colum h cycloomc polyomls h omor corrspos o h sl proc ors o = = Usg h sm mpl o h = = h proucs h Mrov pro sho gur hv h g crm or Q S ZARZ R L Z A R L R P ARZR ZARZ ZRZR ZARZ s: Z Z A R Z L L A Z A R L A Z A R Z R Z A R Z Z Z R Z A R L Z L L Z Z L Z Z L A Z Z A R A R A 7 7 Z Z 8 8 As Ol Jourls jourolcom 7

6 As Ol Jourls jourolcom 7 Rsulg h g mr N Choosg h rs hr p colums hv N sc N h D So y Msurcz M Szl W [] hv 879 * h op hr * s h smlls zro moul o D So clculg h opologcl ropy rough h g mr s scr Lmpr JP Rmos JS [] or rough Mrov pros s scr Msurcz M Szl W [] s h sm gvg us clus h oh mhos ors or rl rol mps or cr slc pos I our prml compuol rsrch ou mor org mpls or oh quvl mhos lg us o h proposo W c proo h oh mhos r quvl Rmos JS [] Proposo Tg h g PQS hr Q cos h or o h crcl po = P cos h or o h p-pos S cos h or o h s-pos o mp h > > > S Q P j or som grs j h o h opologcl ropy vlu * h op h * h smlls zro moul o h g rm D o h g mr N or ch Proo Wh clcul h -h g crm o o pcs-mooo mp srcly mooo h rvl I ssoc o h g ch o o hs vlus r srs Sc or som grs j hv S Q P j h srs hv oly h rs lms s o-commo lms ssurg h ls hr o h colums o h g mr N r lr p As c s Mlor J Thurso W [] r cr rm orr r our cs r j hv por srs commo o Ths s ough o compu h g rm D h opologcl ropy vlu CONCLUSION Ar h co o h vlus h ors or oh mhos scr y proposo hop h our or ops oor o h clculus o h ropy vlu or scou rol rl mps W suspc h oh o hs mhos lso or smll ghorhoos o h lry vlus so our or ll o proc hs rco ryg o y hs rgos I s sy o urs h oh mhos ors h h ors P Q S covrg o h sm or r som ro o h mp u h hpps h hs os o hpps? Wh ll h chgs h mus o o oh mhos orr o us hm o clcul h ropy vlu? Nc qusos h our uur or ll ry o solv REFERENCES [] Lmpr JP Rmos JS Symolc Dymcs o Bmol Mps I Porugl Mhmc vol Fsc 997 [] Lol JL Rmos JS Wgh Kg Thory o Umsol Mps h hols rvorg/s/mh/

7 [] Msurcz M Szl W Eropy o pcs mooo mppgs Su Mh 7 pp - 98 [] Mlor J Thurso W O Ir Mps o h Irvl I JC Alr Procgs Uv Mryl Lc Nos Mh pp - Sprgr-Vrlg rl-n Yor 988 [] Rmos JS Hprolc Burcção Ssms Smólcos PhD Thss Uvrs Lso vll hp://purlp/ As Ol Jourls jourolcom 7