constructing center-stable tori andy hammerlindl-匠人.pdf

该【constructing center-stable tori andy hammerlindl-匠人 】是由【刘备文库】上传分享，文档一共【16】页，该文档可以免费在线阅读，需要了解更多关于【constructing center-stable tori andy hammerlindl-匠人 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:..JID:ANIHPCAID:2857/FLA[m3SC+;;Prn:31/08/2017;14:46](1-16)é?AN???(????)???????ate/anihpcConstructingcenter-stabletoriAndyHammerlindlSchoolofMathematicalSciences,MonashUniversity,Victoria3800,AustraliaReceived26October2016;receivedinrevisedform6July2017;accepted31July2017AbstractWeshowthatcertainderived-from-Anosovdiffeomorphismsonthe2-torusmayberealizedasthedynamicsonacenter-stableorcenter-unstabletorusofa3--stableandcenter-unstabletoriinhigherdimensions.?:Partialhyperbolicity;Dynamicalcoherence;[2],buttheyhaveenoughstructuretoallow,insomecases,forthedynamicstobeunderstoodandclassi?ed[5,13].AdiffeomorphismfisstronglypartiallyhyperbolicifthereisasplittingofthetangentbundleintothreeinvariantsubbundlesTM=Eu⊕Ec⊕EssuchthatthederivativeDfexpandsvectorsintheunstablebundleEu,contractsvectorsinstablebundleEs,andthesedominateanyexpansionorcontractioninthecenterdirectionEc.(Seesection2foraprecisede?nition.)[14].Thatis,,however,,thebundlemaynotbeuniquelyintegrablesince,ingeneral,thecenterbundleisonlyH??,bothfandthesplittingcanbetakenassmooth,andthecenterbundleisnotintegrablebecauseitdoesnotsatisfyFrobenius?conditionofinvolutivity[18,20].Suchnon-involutiveexamplesareonlypossibleifthedimensionofthecenterbundleisatleasttwo,andforalongtimeitwasanopenquestionifaone--mailaddress:andy.******@:/~ahammerl/.http://dx./.-1449/?.:..JID:ANIHPCAID:2857/FLA[m3SC+;;Prn:31/08/2017;14:46](1-16).é?AN???(????)???????RodriguezHertz,RodriguezHertz,andUresrecentlyansweredthisquestionbyconstructingacounterexam-ple[17].Theyde?nedapartiallyhyperbolicsystemonthe3-toruswithaone-,thecenterbundleisuniquelyintegrableeverywhereexceptforaninvariantembedded2-torustangenttoEc⊕,,webuildfurtherex-pactsubmanifoldstangenteithertoEc⊕EuorEc⊕Es,[17],thedynamicsonthe2-torustangenttoEc⊕,itisgivenbyahyperboliclinearmaponT2,,thatis,adiffeomorphismg:T2→T2withasplittingoftheformEc⊕EuorEc⊕Es,,onecanaskifaweaklypartiallyhyperbolicsystemwhichisnotAnosovmayberealizedasthedynamicsonaninvariant2-torussittinginsidea3-,infact,thatderived-from-,wesaythatdiffeomorphismsf0andf1ofthe2-torusaredom-isotopicifthereanisotopy{ft}t∈[0,1]:T2→T2beaweaklypartiallyhyperbolicdiffeomorphismwhichisdom-,thereisanembeddingi:T2→T3andastronglypartiallyhyperbolicdiffeomorphismf:T3→T3suchthati(T2)isacenter-stableorcenter-unstabletorusandi?1?f?i=,acenter-stabletorusisanembeddedcopyofTDwithD≥2whichistangenttoEcs:=Ec⊕,acenter-unstabletorusistangenttoEcu:=Ec⊕-torusandcu-⊕Ec,theni(T2)willbeaggcs-⊕Eu,theni(T2)willbeacu-,Eg,wemaybemorespeci?:T2→T2beaweaklypartiallyhyperbolicdiffeomorphismwhichpreservestheorientationofitscenterbundleandisdom-isotopictoalinearAnosovdiffeomorphismA:T2→T2andlet0<<:T3→T3suchthat(1)f(x,t)=(A(x),t)forall(x,t)∈T3with|t|>,(2)f(x,t)=(g0(x),t)forall(x,t)∈T3with|t|<,and2(3)T2×0iseitheracenter-stableorcenter-,:T2→T2isahyperboliclinearautomorphismandforeachi∈{1,...,n}thatgi:T2→T2isaweaklypartiallyhyperbolicdiffeomorphismwhichpreservestheorientationofitscenterbundleandisdom-{t1,...,tn}bea?nitesubsetofthecircle,:T3→T3suchthat(1)f(x,ti)=(gi(x),ti)foreachtiandallx∈T2,and(2)eachT2×tiisacenter-stableorcenter-,theexamplesmaybeconstructedinsuchawaythattheresultingdiffeomorphismf:T3→-,onecouldeasilyconstructasystemwhichhasacsorcu-torusT2×0andhasarobustlytransitiveblenderelsewhereonT3[1].:..JID:ANIHPCAID:2857/FLA[m3SC+;;Prn:31/08/2017;14:46](1-16).é?AN???(????)???????3Thetechnicalassumptionofdom--opensetofdiffeomorphismsofT2withdominatedsplittings,,-stableorcenter-unsta-,RodriguezHertz,andUresonthe3-torusmaybeviewedasaskewproductwithAnosovdynamicsinthe?,theexamplecanbegivenasamapoftheformF(x,v)=(f(x),Av+h(x))wherefisaMorse?Smalediffeomorphismofthecircle,AisthecatmaponT2,andh:S1→?berx0×T2overthissinkgivestheembedded2--,startingfromanydiffeomorphismfofanyclosedmanifoldM,onemayconstructastronglypartiallyhyperbolicdiffeomorphismFofM×TDusingsinksofftoconstructcenter-unstabletoriforFandsourcestoconstructcenter-:M→MbeadiffeomorphismandX?Ma?niteinvariantsetsuchthateveryx∈:M×TD→M×TDoftheformF(x,v)=(f(x),Av+h(x))suchthatfisisotopictof0and,foreachx∈X,thesubmanifoldx×,,RodriguezHertz,RodriguezHertz,andUresshowedthatthe3-manifoldcanonlybeoneofafewpossibilities[16].?berbundleM×,thesametechniquemaybeusedtointroducecenter-stableandcenter-unstabletoriinasystemde?nedonanon-trivial?berbundle,solongasthedynamicsinthe?[9],itmightbepossibletode?neasystemwithacenter-[8]forfurtherconstructions,and[6]forconditionswhichimplythatthe?,justasinthecaseofdimension3,pactcenter-?,section5handleshigher-?Mintotwonon-zerobundlesTM=E⊕E?isdominatedifitisinvariantunderthederivativeoffandthereisk≥1suchthatDfkv<Dfkv?forallx∈andunitvectorsv∈E(x)andv?∈E(x)?.AninvariantbundleE?TMisexpandingifthereisk≥1suchthatDkv>2forallunitvectorsv∈≥1suchthatDkv<1forallunitvectorsv∈=E⊕EwithEcontractingorTM=Es⊕=Es⊕Ec⊕EuwhereEsiscontracting,Euisexpandingandboth(Es⊕Ec)⊕EuandEs⊕(Ec⊕Eu)aredominatedsplittings.:..JID:ANIHPCAID:2857/FLA[m3SC+;;Prn:31/08/2017;14:46](1-16).é?AN???(????)???????-zerovectorv∈TMandn∈Z,letvndenotetheunitvectorDfnvvn=.DfnvOfcourse,vndependsonthediffeomorphismf:M→Mbeingstudied,,,pactinvariantsubsetssuchthat(1)allchainrecurrentpointsoff|YlieinZ,(2)ZhasadominatedsplittingTZM=E⊕E?withd=dimE,and(3)foreveryx∈Y\Z,thereisapointyintheorbitofxandasubspaceVyofdimensiondsuchthatforanynon-zerov∈Vy,eachofthesequencesvnandv?umulatesonavectorinTZM\Easn→+∞.ThenthedominatedsplittingonZextendstoadominatedsplittingonY∪,letNW(f)denotethenon-,TM=Es⊕Ec⊕Euisaninvariantsplitting,andthereisk≥1suchthatTfkvs<Tfkvc<TfkvuandTfkvs<1<Tfkvuforallx∈NW(f)andunitvectorsvu∈Eu(x),vc∈Ec(x),andvs∈Es(x).Then,?étostudyquasi-Anosovsystems[15,],byHirsch,Pugh,Shubinregardstonormallyhyperbolicity[14,],andbyFranksandWilliamsinconstructingnon-transitiveAnosov?ows[7,].,see[10].-isotopicdiffeomorphismsde?nedonT2,thenthereisaC1functiong:T2×[0,1]→T2suchthatg0=g(·,0),g1=g(·,1)andeachg(·,t),wemayassumethereisapiecewiselineardom-isotopyG:T2×[0,1]→T2suchthatG(·,0)=g0andG(·,1)=?neasmoothbijectionα:[0,1]→[0,1](x,t)=G(x,α(t))isthedesiredC1function.IntheC1topology,(see,forinstance,[4,Section2]),,thereisaconstantη>0andafamilyCofconvexconessuchthatif,atapoint(x,t)∈T2×[0,1],thedominatedsplittingisgivenbyTT2=E(x,t)⊕E(x,t),?xthentheconeC(x,t)?TT2satis?esthepropertiesx(1)E(?x,t)?C(x,t),(2)E(x,t)∩C(x,t)=0,and:..JID:ANIHPCAID:2857/FLA[m3SC+;;Prn:31/08/2017;14:46](1-16).é?AN???(????)???????(x,s)∈T2×(e,)anditsforwardorbit(xn,sn):=fn(x,s).Forsimplicity,weassumethesequence{xn}