TOPOLOGICAL ENTROPY OF NONAUTONOMOUS PIECEWISE MON

TOPOLOGICALENTROPYOFNONAUTONOMOUSPIECEWISEMONOTONEDYNAMICALSYSTEMSONTHEINTERVALSergiKolyada,MihalMisiurewizandL’ubomrSnohaDediatedtothememoryofWieslawSzlenkAbstrat.Topologialentropyofanonautonomousdynamialsystemgivenbyasequeneofompatmetrispaes(Xi)1i=1andasequeneofontinuousmaps(fi)1i=1,fi:Xi!Xi+1,isdened.Ifallthespaesareompatrealintervalsandallthemapsarepieewisemonotonethen,undersomeadditionalassumptions,aformulafortheentropyofthesystemisobtainedintermsofthenumberofpieesofmonotoniityoffnÆÆf2Æf1.Asanappliationweonstrutalargelassofsmoothtriangularmapsofthesquareoftype21andpositivetopologialentropy.1.IntrodutionandmainresultsLetX1;1:=(Xi)1i=1beasequeneofompatmetrispaesandf1;1:=(fi)1i=1asequeneofontinuousmaps,wherefiisamapfromXitoXi+1.Foranypositiveintegersi;nsetfni=fi+(n1)ÆÆfi+1Æfiandadditionallyf0i=idXi.Wewillalsowritefni=(fni)1(thisnotationwillbeappliedtosets;wedonotassumethatthemapsfiareinvertible).Wewillall(X1;1;f1;1)anonautonomousdisretedynamialsystem.Thetrajetoryofapointx2X1willbethesequene(fn1(x))1n=0.Wewillspeakofthen-thiterate(X[n℄1;1;f[n℄1;1)ofthesystem,whereX[n℄i=X(i1)n+1andf[n℄i=fn(i1)n+1.Topologialentropyh(f)ofanautonomousdynamialsystem(X;f)wasintro-duedbyAdler,KonheimandMAndrew[AKM℄andequivalentdenitionsweregivenbyBowen[B℄andDinaburg[D℄.Topologialentropyh(f1;1)ofanonau-tonomousdynamialsystem(X1;1;f1;1)wasstudiedin[KS℄undertheadditionalassumptionthatallthespaesXioinide.Inthepresentpaperwegeneralizethesedenitionstothesystem(X1;1;f1;1).Themainaimofthepaperistoproveanaloguesofthefollowingresult.1991MathematisSubjetClassiation.Primary58F03;Seondary58F08,54H20.Keywordsandphrases.Nonautonomousdynamialsystem,topologialentropy,triangularmaps,pieewisemonotonemaps,C1maps.TypesetbyAMS-TEX12S.KOLYADA,M.MISIUREWICZANDL’.SNOHATheorem1.1([MS℄).Iffisapieewisemonotoneontinuousselfmapofaom-patintervalandndenotesthenumberofpieesofmonotoniityoffnthen(1.1)h(f)=limn!11nlogn:Wewillonsideradynamialsystem(I1;1;f1;1),whereIiisaompatrealintervalforanyi.Moreover,weassumethateveryfiispieewisemonotone.BythiswemeanthatthereisanitepartitionofIiintointervalssuhthatfiismonotone(notneessarilystritly)oneahelementofthispartition.Thenfn1isalsopieewisemonotone.Alapofapieewisemonotonemapisanymaximal(withrespettoinlusion)intervalonwhihitismonotone.Wedenotethenumberoflapsoffn1by1;n.Wewanttondonditionsunderwhihtheformula(1.2)h(f1;1)=limsupn!11nlog1;n;analogousto(1.1),holds.Wepresentthreesetsofsuhonditions.InTheoremA,theassumptionsaretheweakestandeasytostate.However,theyarediÆulttoverifydiretly.ThereforewestrengthenthemtogetweakerTheoremsBandC,whoseassumptionsareofteneasytoverify.SinetheappliationfromSetion5usesTheoremB,thistheoremanbeper-eivedasthemainresultofthepaper.Intheautonomousaseompatnessofthespaeplaysanimportantroleinthetheory.Inordernottolosethisadvantage,wehavetoputsomerestritionsonthebehaviorofoursequeneofspaes.Denition1.2.Adynamialsystem(I1;1;f1;1)issaidtobeboundedifthelengthsoftheintervalsareuniformlyboundedfromabove.TostateTheoremB,weneedthefollowingdenitions.Denition1.3.Let(I1;1;f1;1)beadynamialsystem.ItissaidtohaveMarkovpropertyifthereexistsÆ0andasequeneC1;1ofnitesubsetsCiIisuhthatforeveryi1(a)theendpointsofIibelongtoCi,(b)eitherfiismonotoneonthewholeintervalIiorthelengthofeveryom-ponentofIinCiisatleastÆ,()foreveryomponentJofIinCithemapfiismonotoneonJ,(d)fi(Ci)Ci+1.Weusethename\Markovbeauseoftheproperties(a),()and(d).However,property(b)isalsoveryimportant.WhilewedonotwanttheelementsoftheMarkovpartitionstobetooshort(thisholdsautomatiallyintheautonomousase),weadmitexeptionsiffiisgloballymonotone.Denition1.4.Aboundeddynamialsystem(I1;1;f1;1)issaidtobeaMarkovsystemifithasMarkovpropertyandthemapsfi,i=1;2;3;:::,areequiontinu-ous.TOPOLOGICALENTROPYOFNONAUTONOMOUSDYNAMICALSYSTEMS3TheoremB.If(I1;1;f1;1)isaMarkovdynamialsystemthen(1.2)holds.Anotherimportantasewhen(1.2)holdsisthefollowingone.Denition1.5.Aboundeddynamialsystem(I1;1;f1;1)issaidtobeanitepieewisemonotonesystemifthesetofmapsffi:i2Nghasonlynitelymanydistintelementsandeahofthemispieewisemonotone.TheoremC.If(I1;1;f1;1)isanitepieewisemonotonesystemthen(1.2)holds.NotethatTheorem1.1isaspeialaseofTheoremC.WededueTheoremsBandCfrommoregeneral,butmoretehnialresult.Denition1.6.Asystem(I1;1;f1;1),whereallfiarepieewisemonotone,willbealledlong-lappedifthereexistsÆ0suhthatforeveryieitherfiismonotoneoreverylapoffihaslengthatleastÆ.Itwillbealledtotallylong-lappedifitsn-thiterateislong-lappedforeveryn1.TheoremA.If(I1;1;f1;1)isaboundedtotallylong-lappeddynamialsystemthen(1.2)holds.TheoremAwillbeprovedinSetion3.InSetion4weshowthatMarkovandnitesystemsaretotallylong-lapped,soTheoremsBandCfollowfromTheoremA.Wepresentalsosomerelatedexamples.InSetion5weapplyTheoremBtotriangularmaps.AontinuousmapFofthesquareIIintoitselfisalledtriangular(seee.g.[Kl℄,[Ko℄)ifitisoftheformF(x;y)=(f(x);gx(y)):Itissaidtobeoftype21ifithasperiodipointsofperiods2nforeveryn0andofnootherperiods.Similarlyasintheaseofontinuousintervalmaps,ifanyotherperiodispresent,topologialentropyisp