Semi-Conjugacies to Tent Maps, Transitivity and Patterns, and Inverse Limit Spaces

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Semi-Conjugacies to Tent Maps, Transitivity and Patterns, and Inverse Limit Spaces
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english
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Ledis,Dennis Joel
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Doctorate ( Ph.D.)
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University of Florida
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Mathematics
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Block, Louis S
Committee Co-Chair:
Keesling, James E
Committee Members:
Boyland, Philip L
Pilyugin, Sergei
Gilland, David R

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conjugacy -- conjugate -- dynamical -- dynamics -- inverse -- knaster -- limit -- map -- pattern -- semi -- solenoid -- space -- systems -- tent -- transitive -- transitivity
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Abstract:
Let f be a continuous map of the interval to itself. We prove that if f has a k-horseshoe, then f is topologically semi-conjugate to a tent map with slope plus or minus k. Moreover, we prove that if f is topologically transitive, exhibits a particular pattern as a piecewise linear map L, and h(f)=h(L), then f is topologically conjugate to L. Using the first result, we prove that if f has positive topological entropy, and n is greater than or equal to 2, then there is a continuous surjective map from the inverse limit space (I,f) to the Knaster continuum K sub n. In particular, for m and n greater than or equal to 2, there is a continuous surjective map from K sub m to K sub n. On the other hand, we prove that the analogous statement does not hold for solenoids.
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by Dennis Joel Ledis.
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Thesis (Ph.D.)--University of Florida, 2011.
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Adviser: Block, Louis S.
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Co-adviser: Keesling, James E.

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SEMI-CONJUGACIESTOTENTMAPS,TRANSITIVITYANDPATTERNS,ANDINVERSELIMITSPACESByDENNISJOELLEDISADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011DennisJoelLedis 2

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Idedicatethistomyparents,whohavealwayssupportedmeineverythingI'vedone. 3

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ACKNOWLEDGMENTS Iwouldliketothankmyadviser,Dr.LouisBlock,forallhishelp,encouragement,ideas,anduncountableamountoftimehespentwithme.Iwouldalsoliketothanktheco-chairofmycommittee,Dr.JamesKeesling,forallofhishelpandnewoutlooksontheproject.Finally,Iwouldliketothanktheothermembersofmycommittee,Dr.SergeiPilyugin,Dr.PhilipBoyland,andDr.DavidGilland,formakingthispossible. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 6 ABSTRACT ......................................... 7 CHAPTER 1INTRODUCTION ................................... 8 2DEFINITIONSANDPRELIMINARIES ....................... 11 3RESULTSONSEMI-CONJUGACIESTOTENTMAPS ............. 15 4ARESULTONTOPOLOGICALCONJUGACY,TRANSITIVITY,ANDPATTERNS 27 5INVERSELIMITS .................................. 35 6SEMI-CONJUGACIESANDINVERSELIMITSPACES .............. 39 7ARESULTONTOPOLOGICALSEMI-CONJUGACIESFORMAPSOFTHECIRCLE ........................................ 43 REFERENCES ....................................... 46 BIOGRAPHICALSKETCH ................................ 48 5

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LISTOFFIGURES Figure page 3-1Graphoffwithlabels ................................ 19 3-2Graphofsemi-conjugacyT2)]TJ /F1 11.955 Tf 10.08 -.3 Td[(fromftoT2 ..................... 19 3-3Graphoff ....................................... 20 3-4Graphofsemi-conjugacyfromftoT2 ...................... 20 3-5Graphofsemi-conjugacy1 ............................. 21 3-6Graphofsemi-conjugacy2 ............................. 21 4-1Graphoff ....................................... 34 4-2GraphofLP ...................................... 34 6

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophySEMI-CONJUGACIESTOTENTMAPS,TRANSITIVITYANDPATTERNS,ANDINVERSELIMITSPACESByDennisJoelLedisAugust2011Chair:LouisBlockMajor:MathematicsLetfbeacontinuousmapoftheintervaltoitself.Weprovethatiffhasak-horseshoe,thenfistopologicallysemi-conjugatetoatentmapwithslopek.Moreover,weprovethatiffistopologicallytransitive,exhibitsaparticularpatternasapiecewiselinearmapL,andh(f)=h(L),thenfistopologicallyconjugatetoL.Usingtherstresult,weprovethatiffhaspositivetopologicalentropy,andn>2,thenthereisacontinuoussurjectivemapfromtheinverselimitspace(I,f)totheKnastercontinuumKn.Inparticular,form,n>2,thereisacontinuoussurjectivemapfromKmtoKn.Ontheotherhand,weprovethattheanalogousstatementdoesnotholdforsolenoids. 7

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CHAPTER1INTRODUCTIONInstudyingthedynamicsofcontinuousmaps,thenotionoftopologicalconjugacyplaysafundamentalrole.Letf:X!Xandg:Y!Ybecontinuousfunctions.Wesaythatfandgaretopologicallyconjugateifthereexistsahomeomorphism:X!Ysuchthatthefollowingdiagramcommutes.Xf)305()223()305(!X??y??yYg)305()223()305(!YIffandgaretopologicallyconjugate,wethinkofthedynamicalpropertiesofthetwomapsasbeingthesame.Arelatedideaisthenotionoftopologicalsemi-conjugacy.Wesaythatfistopologicallysemi-conjugatetogifthereexistsacontinuous,surjectivefunction:X!Ysuchthatthediagramabovecommutes.Note,whenwesayfissemi-conjugatetog,wemeanthatfisthetopmapandgisthebottommapinthediagram.Sometimes,intheliterature,thephrasefisanextensionofgisusedwiththesamemeaning.Iffistopologicallysemi-conjugatetogwethinkofthedynamicsoffasbeingatleastascomplicatedasthedynamicsofg.Manyresultsinthetheoryofdynamicalsystemsinvolvetopologicalsemi-conjugacy.AsourrstexamplewementionaresultofM.Shub[ 20 ,Theorem2,p.192].AsnotedbyShubintheremarkonpage184,aparticularcaseofthisresultimpliesthatifgisacontinuousmapofthecircleontoitselfofdegreenwithjnj>1,thengistopologicallysemi-conjugatetothemapofthecircletoitselfgivenbyz!zn.Asimpleproofofthisresultappearsin[ 8 ].Similarresultsmayalsobefoundin[ 9 ].AsecondexampleisaresultofBowen[ 7 ,Theorem28,p.741].TheresultisthatiffistherestrictionofadiffeomorphismsatisfyingS.Smale'sAxiomAtoabasicset,then 8

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thereexistsasubshiftofnitetypehavingthesametopologicalentropyasfwhichistopologicallysemi-conjugatetof.OurnalexampleisaresultofMilnorandThurston[ 14 ](comparewiththeresultofParry[ 16 ]).Theresultisthatiffisapiecewisemonotonecontinuousmapoftheintervaltoitselfwithtopologicalentropylogs,ands>1,thenfistopologicallysemi-conjugatetoapiecewiselinearmapwithslopeeverywhereequaltos.Also,thetopologicalsemi-conjugacyismonotone.Inthisdissertation,weobtainadifferentresultdealingwithtopologicalsemi-conjugaciesformapsoftheintervaltoitself.WeletTnandTn)]TJ /F1 11.955 Tf 10.07 -.3 Td[(denotethepiecewiselinearmapsof[0,1]ontoitselfwithslopesalternatingbetween+nand)]TJ /F3 11.955 Tf 9.3 0 Td[(n(seeDenition 1 ).Weprovethefollowingresultintherstpartofthedissertation. Theorem 3.6 Letf:[0,1]![0,1]becontinuousandhaveann-horseshoe.Thenfistopologicallysemi-conjugatetoTnorTn)]TJ /F13 11.955 Tf 6.76 -.3 Td[(.Weremarkthatthetopologicalsemi-conjugacyobtainedinTheorem 3.6 isnotingeneralmonotone.Infact,wegiveanexample(seeProposition 3.5 )wherethetopologicalsemi-conjugacyisnowheredifferentiableandadensesetofpointshaveuncountablymanyinverseimages.Furthermore,weprovethefollowingtheoremaboutwhenamapfexhibitsacertainpattern(seeDenition 24 ). Theorem 4.6 Let(P,)beapatternandletLbethelinearizationofthepattern.Supposef:I!Iistopologicallytransitiveandfexhibitsthepattern(P,).Supposeh(f)=h(L).ThenfistopologicallyconjugatetoL.Thelastpartexploitstheconnectionbetweentopologicalsemi-conjugaciesandinverselimitspaces.Foreachintegerm>1,weletKmdenotethecorrespondingKnastercontinuum(seeDenition 26 ).Also,givenf:[0,1]![0,1]welet(I,f)denotethecorrespondinginverselimitspace.Weprovethefollowingresult: 9

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Theorem 6.1 Letf:I!Ibeacontinuousfunctionwithpositivetopologicalentropy.Thenforanym>2,thereisacontinuoussurjectivemapfrom(I,f)toKm.Weobtaintwocorollariestothistheorem.Therstisawell-knownresultofBargeandMartin[ 3 ].AstrongerresulthasrecentlybeenobtainedbyMouron[ 15 ]. Corollary 5 Letf:I!Ibeacontinuousfunctionwithpositivetopologicalentropy.Then(I,f)hasanindecomposablesubcontinuum. Corollary 6 Foranym,n>2,thereisacontinuoussurjectivemapfromKmtoKn.WealsoshowthattheanalogousstatementtoCorollary 6 doesnotholdforsolenoids.Weletndenotethen-solenoidandprovethefollowing: Theorem 6.4 Supposem,narepositiveintegers,m,n>2,suchthatthereisaprimepwithpjnbutp-m.Thentheredoesnotexistacontinuoussurjectivemapfrommton.Weconcludethedissertationwiththefollowingresult: Theorem 7.2 Supposefandgaremapsofthecircletoitselfwithdeg(f)=m>0,deg(g)=n>1,andm6=n.Thenfisnottopologicallysemi-conjugatetog. 10

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CHAPTER2DEFINITIONSANDPRELIMINARIES Denition1. Thefulltentmap,T:I!I,whereIistheunitinterval[0,1],isdenedasT(x)=8><>:2xifx61=2)]TJ /F8 11.955 Tf 9.3 0 Td[(2(x)]TJ /F8 11.955 Tf 11.95 0 Td[(1)ifx>1=2Ingeneral,forapositiveintegern>2,letTn:I!Ibepiecewisemonotoneandlinearwithslopendenedasfollows:Tn(k=n)=8><>:0ifkisaneveninteger,06k6n1ifkisanoddinteger,06k6nandTnislinearbetweenthesepoints.ThemapsTnaresometimescalledsawtoothmaps.Inparticular,thefulltentmapisT2,butwhenweonlywriteT,wewillmeanthefulltentmap,T2.WealsodeneTn)]TJ /F8 11.955 Tf 10.85 -.3 Td[(:I!IsimilartoTnexceptwithslopesreversed,i.e.Tn)]TJ /F8 11.955 Tf 6.75 -.3 Td[((k=n)=8><>:1ifkisaneveninteger,06k6n0ifkisanoddinteger,06k6n Remark1. Forneven,TnandTn)]TJ /F13 11.955 Tf 10.07 -.3 Td[(aretopologicallyconjugatevia(x)=1)]TJ /F3 11.955 Tf 11.95 0 Td[(x. Remark2. Fornisodd,TnandTn)]TJ /F13 11.955 Tf 10.07 -.3 Td[(arenottopologicallyconjugate;simplyconsiderthedynamicsoftheendpoints. Denition2. Letf:I!Ibecontinuous.ThemapfisturbulentifthereexisttwoclosedsubintervalsofI,A1andA2,withdisjointinteriors,suchthat(A1[A2)(f(A1)\f(A2)).Ingeneral,wesayacontinuousmapoftheintervalhasann-horseshoeifthereexistnclosedsubintervalsofI,A1,...,An,withpairwisedisjointinteriors,suchthat(A1[...[An)(f(A1)\...\f(An)). 11

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Turbulenceisthensimplyhavinga2-horseshoe. Denition3. LetXbeacompactmetricspaceandf:X!Xbeacontinuousmapofthisspaceintoitself.Wesaythatfistopologicallytransitiveifforeverypairofnon-emptyopensetsUandVinX,thereisapositiveintegerksuchthatfk(U)\Visnonempty.Inthisdissertation,wewilloftenusethetermtransitiveinplaceoftopologicallytransitive.Thefollowingresults,Propositions 2.1 through 2.3 ,abouttopologicaltransitivitymaybefoundin[ 6 ]andwillbepresentedwithoutproof. Proposition2.1. Acontinuousmapf:X!Xofacompactmetricspaceintoitselfistransitiveifandonlyifthereisapointx2Xsuchthattheset!(x,f)=Tn2N ffk(x):k>ng=X. Proposition2.2. Letf:I!Ibeatransitivecontinuousmap.Thenexactlyoneofthefollowingholds: (1) fsistransitiveforeverypositiveintegers, (2) thereexistnon-degenerateclosedintervalsJ,KwithJ[K=IandJ\K=fyg,whereyisaxedpointoff,suchthatf(J)=Kandf(K)=J.Moreover,inthiscase,(f2jJ)sand(f2jK)saretopologicallytransitiveforeachpositiveintegers. Proposition2.3. Letf:I!Ibeacontinuousmap.Thenf2istransitiveifandonlyif,foreveryopensubintervalJandeveryclosedsubintervalHwhichdoesnotcontainanendpointofI,thereisapositiveintegerNsuchthatHfn(J)foreveryn>N.Ofparticularinterestwillbetheconceptoftopologicalentropy.Topologicalentropygivesusawaytomeasurethecomplexityofagivenmapf.Thefollowing,Denitions 4 through 7 andPropositions 2.4 through 2.7 ,arewell-knownandwillbestatedwithoutproof,butmaybefoundin[ 6 ]. Denition4. LetXbeacompacttopologicalspace.AnopencoverofXisacollectionofopensetswhoseunionisX.Anopencoverissaidtobearenementofanopencover(notation:<),ifeveryopensetofiscontainedinsomeopensetof.Wesaythatisasubcoverofifeveryopensetofactuallyisanopensetof. 12

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Ifandaretwoopensubcovers,theirjoin_istheopencoverconsistingofallsetsA\BwithA2andB2.Thus_isarenementofbothand. Denition5. SinceXiscompact,everyopencoverhasanitesubcover.TheentropyofanopencoverisdenedtobeH()=logN(),whereN()istheminimumnumberofopensetsinanynitesubcover.Theentropyoffrelativetothecoverisdenedtobeh(f,)=limn!1H(_f)]TJ /F4 7.97 Tf 6.58 0 Td[(1_..._f)]TJ /F7 7.97 Tf 6.58 0 Td[(n+1)=n,wheref)]TJ /F7 7.97 Tf 6.59 0 Td[(kareallthesetsf)]TJ /F7 7.97 Tf 6.58 0 Td[(k(A)withA2. Denition6. Thetopologicalentropyofacontinuousmapf:X!Xisdenedtobeh(f)=suph(f,),wherethesupremumistakenoverallopencovers.Ingeneral,though,thisisnotalwaysthebestwaytocalculatethetopologicalentropyofamap.Wewillutilizethefollowingresultsinthispaper. Proposition2.4. Iff:X!Xisacontinuousmapthen,foranypositiveintegerk,h(fk)=kh(f). Proposition2.5. Iffistopologicallysemi-conjugatetog,thenh(f)>h(g).Iftheyaretopologicallyconjugate,wehaveequality. Denition7. LetJ1,J2,...,Jnbenon-emptysubintervalsofIwithpairwisedisjointinteriorsandacontinuousfunctionf:I!I.TheadjacencymatrixunderfassociatedtothiscollectionofsubintervalsisannnmatrixA=(aik)denedbyaik=1iff(Ji)Jkandaik=0otherwise. Proposition2.6. Letf:I!Ibeacontinuousmap.LetJ1,...,Jnbeclosedintervalswithpairwisedisjointinteriors.LetAbetheadjacencymatrixforthiscollectionunderf.Thenh(f)>log,whereisthemaximaleigenvalueofA. Proposition2.7. Letf:I!IbeacontinuousmapandsupposeI=J1[...[Jn,whereJ1,...,Jnareclosedintervalswithnon-emptyandpairwisedisjointinteriorssuchthatfismonotoniconeachintervalJiandmapsthesetofendpointsofallintervalsJiintoitself.LetAbetheadjacencymatrixofthiscollectionunderfandletbethe 13

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maximaleigenvalueofA.Thenh(f)=max(0,log).Moreover,iffisnotconstantoneachintervalJi,i=1,...,n,then>1andh(f)=log. 14

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CHAPTER3RESULTSONSEMI-CONJUGACIESTOTENTMAPSOurmaingoalinthissectionistoproveTheorem 3.6 .WewillrstdealwiththecaseofthetentmapT=T2.Webeginwithasimple,basicfactaboutthetentmap.Aproofmaybefoundin[ 18 ,page385]. Proposition3.1. GivenasequenceB=b0,b1,b2,...ofL'sandR's,thereexistsauniquey2[0,1]suchthatforthetentmapT,thefollowingholds:Foreachnonnegativeintegerk,bk=LimpliesTk(y)2[0,1=2]andbk=RimpliesTk(y)2[1=2,1]. Denition8. LetBbeasequenceofL'sandR's.LetybetheuniquepointgiveninProposition 3.1 .WewillcallythepointwhoseitineraryunderTisB.WewishtouseDenition 8 todeneamapwhichisapotentialsemi-conjugacyfromamapftothetentmapT.Withthisinmind,wemakethefollowingthreedenitions: Denition9. Alabeledpartitionof[0,1]withrespecttofisanorderedpair(C,)suchthat (1) Cisanonemptynitesubsetoftheopeninterval(0,1), (2) isafunctionwhosedomainisthesetofconnectedcomponentsof[0,1])]TJ /F3 11.955 Tf 12.07 0 Td[(CandwhoserangeisthesetcontainingthetwosymbolsLandR,and (3) alternatesbetweenLandRonconsecutiveconnectedcomponents. Denition10. Theitineraryofxwithrespecttofand(C,)isthesequenceB=b0,b1,b2,...ofL'sandR'ssuchthatbk=Riffk(x)liesinsomecomponentJof[0,1])]TJ /F3 11.955 Tf 9.77 0 Td[(Cwith(J)=R,andbk=Lotherwise.Notethatintheabovedenition,bk=Liffk(x)2C.ThechoiceofLisarbitrary.Wewillseelaterthatthischoicedoesnotmatter. Denition11. Givenalabeledpartition(C,)withrespecttofwedeneamap:[0,1]![0,1]asfollows: 15

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FormtheitineraryofxunderftoobtainasequenceBofL'sandR's.Set(x)=ywhereyisthepointwhoseitineraryunderTisB. Denition12. LetAandBbenitedisjointsubsetsofI.BythedistancebetweenAandB,wemeantheminimumvalueofja)]TJ /F3 11.955 Tf 11.96 0 Td[(bjwherea2A,b2B. Proposition3.2. Supposef:[0,1]![0,1]iscontinuous,(C,)isalabeledpartitionwithrespecttof,andistheassociatedmap.Thenforallpointsxsuchthatfn(x)=2Cforallnonnegativeintegersn,iscontinuousatx. Proof. Letx2Ibesuchthatfn(x)=2Cforallnonnegativeintegersn.Sincefiscontinuous,allofitsiteratesarecontinuousaswell.Let>0.ChooseNlargeenoughsothat2)]TJ /F4 7.97 Tf 6.59 0 Td[((N+1)<.Let0denotethedistancebetweenthesetsfx,f(x),...,fN(x)gandC.Thereisa>0suchthatify2Iwithjx)]TJ /F3 11.955 Tf 12.14 0 Td[(yj<,thenjfk(x))]TJ /F3 11.955 Tf 12.13 0 Td[(fk(y)j<0foreachk=0,...,N.Supposey2Iwithjx)]TJ /F3 11.955 Tf 12.04 0 Td[(yj<.ThentheitinerariesofxandyunderfagreeontherstN+1places.Hencetheitinerariesof(x)and(y)underTagreeontherstN+1places.Hencej(x))]TJ /F6 11.955 Tf 11.95 0 Td[((y)j62)]TJ /F4 7.97 Tf 6.59 0 Td[((N+1)<. Proposition3.3. f=T. Proof. Put(f)(x)=y.ThenyisthepointthatsatisesI f(f(x))=I T(y),whereI fandI TaretheitinerariesofthepointunderfandT,respectively.IfI f(x)=b0b1b2...,thenyhasitineraryI T(y)=b1b2....Nowletz=(x).ThenzhastheitineraryI T(z)=b0b1b2....ApplyingTtoz,wehaveI T(T(z))=b1b2....SinceyandT(z)havethesameitineraryunderT,y=T(z). Proposition3.4. LetBandB0besequencesofL'sandR's.Letxandx0bethecorrespondingpointswhoseitinerariesareBandB0underT,respectively.LetAbeanitesequenceofL'sandR'swithlengthn.Letyandy0bethecorrespondingpointswhoseitinerariesareABandAB0underT,respectively.Ifjx)]TJ /F3 11.955 Tf 11.62 0 Td[(x0j<,thenjy)]TJ /F3 11.955 Tf 11.62 0 Td[(y0j< 2n. 16

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Proof. LetussupposethatA=L.Thenx=2yandx0=2y0.Sojy)]TJ /F3 11.955 Tf 12.07 0 Td[(y0j=jx)]TJ /F7 7.97 Tf 6.59 0 Td[(x0 2j< 2.ItisclearthatthisholdsforA=Raswell.Formallybyinduction,weseebyconcatenatingonebyone,wehavethedesiredresult. Theorem3.1. Supposef:[0,1]![0,1]iscontinuous,(C,)isalabeledpartitionwithrespecttof,andistheassociatedmap.Thenthefollowingareequivalent: (1) iscontinuous. (2) iscontinuousateachz2C. (3) (z)=1=2forallz2C. (4) Eachz2ChasitineraryLR L,where LdenotesthesequenceLLL....Moreover,ifiscontinuous,thenisalsosurjective,andhencefistopologicallysemi-conjugatetoT. Proof. (1))(2)isclear.(2))(1):ByProposition 3.2 ,itsufcestoshowthatiscontinuousonthepointsxsuchthatx=2Cbutfn(x)2Cforsomepositiveintegern.Letxbesuchthatx=2Cbutfn(x)2C,wherenisthesmallestsuchpositiveinteger.Callfn(x)=yandlet>0.Sinceiscontinuousaty,thereisa0>0suchthatifjy)]TJ /F3 11.955 Tf 12.93 0 Td[(y0j<0,thenj(y))]TJ /F6 11.955 Tf 11.95 0 Td[((y0)j<.Letdenotethedistancebetweenthesetsfx,f(x),...,fn)]TJ /F4 7.97 Tf 6.58 0 Td[(1(x)gandC.Thereexists>0suchthatifx02Iandjx)]TJ /F3 11.955 Tf 11.11 0 Td[(x0j<,thenjfn(x))]TJ /F3 11.955 Tf 11.1 0 Td[(fn(x0)j<0andjfk(x))]TJ /F3 11.955 Tf 11.1 0 Td[(fk(x0)jN.Since(zn)>1=2foralln,andis 17

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continuousatz,(zn)!(z)>1=2,acontradictionsince(z)wasassumedtobestrictlylessthan1=2.Asimilarproofworksintheothercases.(3))(2):TherstthingweobserveisthatthepointthatcorrespondstotheitinerariesB=LRLLL...andB0=RRLLL...underTis1=2.TheniteitineraryLRLLL...Lwithncharactersdenespointsintheinterval[1=2)]TJ /F8 11.955 Tf 13.09 0 Td[(1=2n,1=2]andtheniteitineraryRRLLL...Lwithncharactersdenespointsintheinterval[1=2,1=2+1=2n].Letz2Candsuppose(z)=1=2.Let>0.Takentobesuchthat2)]TJ /F7 7.97 Tf 6.59 0 Td[(n<.Byhypothesis(3),eachelementofChasitineraryLR L.Thus,ifz2C,wehavefk(z)2I)]TJ /F3 11.955 Tf 12.46 0 Td[(Cforeachpositiveintegerk.InthesamemannerasProposition 3.2 ,wemaynda>0suchthatifjz)]TJ /F3 11.955 Tf 12.23 0 Td[(z0j<,zandz0shareitinerariesstartingwith,atworst,thesecondcharacteruptothe(n+1)stcharacter.Sincezismappedto1=2by,theimagesofbothpointsmustlieintheinterval[1=2)]TJ /F8 11.955 Tf 12.19 0 Td[(1=2n+1,1=2+1=2n+1].Soj(z))]TJ /F6 11.955 Tf 12.19 0 Td[((z0)j<2)]TJ /F7 7.97 Tf 6.58 0 Td[(n<.(3),(4)isclear.Moreover,ifiscontinuous,thenissurjective:Letz2C.Then(z)=1=2and(f(z))=1and(f2(z))=0.Henceissurjective.ItfollowsfromProposition 3.3 thatfistopologicallysemi-conjugatetoT. WenowuseTheorem 3.1 toobtainthedesiredresultforturbulentmaps.WewillusethefollowinglemmafromBlockandCoppel[ 6 ,page26]. Lemma1. Letfbeaturbulentcontinuousmapoftheinterval.Thenthereexistpointsa,bandcsuchthatf(a)=a,f(b)=a,andf(c)=bfora
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Theorem3.3. Iffistopologicallysemi-conjugatetothetentmap,T,thenfneednotbeturbulent. Proof. Letfbethefunctionwithf(0)=1=2,f(1=4)=1,f(1=2)=1=2,f(3=4)=0,f(1)=1=2,andlinearbetweenthesepoints(seeFigure 3-1 ).Itiseasytoseethismapisnotturbulentsincetheredonotexistpointsa,b,andcwiththepropertyf(a)=a,f(b)=a,f(c)=bwithcinbetweenaandb.However,ifwedenealabeledpartitionwithC=f1=4,3=4gwiththelabelinginFigure 3-1 ,weseetheitinerariesof1=4and3=4areLR L,hencebyTheorem 3.1 ,fistopologicallysemi-conjugatetoT. Remark3. Thetopologicalsemi-conjugacycanbeveriedtobe(x)=T2)]TJ /F13 11.955 Tf 6.75 -.3 Td[(,asshowninFigure 3-2 Figure3-1. Graphoffwithlabels. Figure3-2. Graphofsemi-conjugacyT2)]TJ /F1 11.955 Tf -149.19 -14.75 Td[(fromftoT2. Remark4. ItfollowsfromTheorem 3.2 thatforn>2,TnandTn)]TJ /F13 11.955 Tf 10.07 -.3 Td[(aresemi-conjugatetothetentmap,T,sinceitiseasytoseethatTnandTn)]TJ /F13 11.955 Tf 10.08 -.3 Td[(areturbulent. Remark5. Iffissemi-conjugatetothetentmap,T,aturningpointoffneednotmaptotheturningpointofT.Moreover,apointthatisnotaturningpointoffmaymaptotheturningpointofT. 19

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Toseethis,considerthemapfwithf(0)=0,f(3=8)=1,f(3=4)=0,f(1)=1=2,andflinearbetweenthesepoints,asshowninFigure 3-3 .Weseethatfisturbulentwitha=0,b=3=4,c=9=32.WedeneasintheproofofofTheorem 3.2 .Itisclearthat(c)=1=2althoughcisnotaturningpointoff.Byfollowingtheitineraryoftheturningpointoff,3=8,weseethat(3=8)6=1=2.ThegraphofisshowninFigure 3-4 Figure3-3. Graphoff. Figure3-4. Graphofsemi-conjugacyfromftoT2. Remark6. Wealsonotethatfmaybeturbulentbutnotonto.Sowecanhaveasemi-conjugacytoTwithoutfbeingontoI. Remark7. Iffissemi-conjugatetoT,thenisnotnecessarilyunique. Proof. Letf=T4.ThenwemaydeneonelabeledpartitionwithC=f1=4g,Ltotheleftof1=4,andRtotherightof1=4andanotherlabeledpartitionwithC=f3=4g,Ltotheleftof3=4,andRtotherightof3=4.Theseeachyieldtwodifferentsemi-conjugacies.Wecalltherst1andthesecond2.Figure 3-5 belowisthegraphof1,andFigure 3-6 isthegraphof2. Proposition3.5. Thetopologicalsemi-conjugacies1and2fromtheprevioustwoguresarenowheredifferentiable,andfori=1,2,thereisadensesetDsuchthatforeachx2D,)]TJ /F4 7.97 Tf 6.59 0 Td[(1i(x)isuncountable. 20

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Figure3-5. Graphofsemi-conjugacy1. Figure3-6. Graphofsemi-conjugacy2. Proof. Wewillprovethepropositiononlyfor1,astheproofissimilarfor2.LetA=a0,a1,a2,...beanysequencewhereeachai2f1,2,3,4g.Thereisauniquex2[0,1]suchthatifan=k,thenTn4(x)2[k)]TJ /F4 7.97 Tf 6.58 0 Td[(1 4,k 4].Letx2[0,1]correspondtosuchasequenceAunderT4.Wecanview1asobtaininganewsequenceA0=b0,b1,b2,...,wherebi=Lifai=1andbi=Rifai2f2,3,4g,thenmappingxtothepointwhoseitineraryunderT2isA0.SinceT2istransitive,thereisapointywithadenseorbit.LetD=fy,T2(y),T22(y),...g.ForeachpointinD,theitineraryunderT2hasinnitelymanyR's.ForeachRthatappears,thereare3pre-imagesinT4,since2,3,4areallconvertedtoRunder1.ThereforeeverypointinDhasuncountablymanypre-imagesunder1.Now,weshowthat1isnotdifferentiableanywhere.Ifwexn,wemayconsideracanonicalpartitionoftheintervalas[0,1=4n],[1=4n,2=4n],andsoon.EachintervalJinthispartitioncorrespondstoauniquenitesequencefa1,a2,...,angwhereeachai2f1,2,3,4g.Considerthesequencefb1,b2,...,bngwherebi=Lifai=1andbi=Rifai2f2,3,4g.ThereisauniqueintervalKoflength1=2ncorrespondingtofb1,b2,...,bng.Then1(J)=K. 21

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Now,letx2[0,1]andxanintegern>1.ThepointxliesinoneofthesecanonicallychosenintervalsJoflength1=4n,with1(J)=K,asabove.LetxnbeapointinJsuchthat1(xn)isanendpointofKandj1(x))]TJ /F6 11.955 Tf 12.48 0 Td[(1(xn)jisaslargeaspossible.ThisdistanceisatleasthalfthetotallengthofK.Thereforej1(x))]TJ /F11 7.97 Tf 6.59 0 Td[(1(xn)j jx)]TJ /F7 7.97 Tf 6.59 0 Td[(xnj>2n 2=2n)]TJ /F4 7.97 Tf 6.58 0 Td[(1(halftheexpansionrate).Asn!1,weseethat2n)]TJ /F4 7.97 Tf 6.59 0 Td[(1!1andxn!x.Hence,1isnowheredifferentiable. Theorem3.4. Letf:[0,1]![0,1]becontinuous.Supposefhasann-horseshoe.(1)Ifniseven,thenthereexistpointsc0,c1,...,cnsuchthatf(c2i)=c0andf(c2i+1)=cnfori=0,...,n=2andeitherc0y1and[y1,x2]decreasesonaveragecoveringA1,...,An.Continuinginthismanner,wendthatthereareatleastnintervalsthatalternateincreasinganddecreasing(onaverage)suchthattheimageunderfofeachoneoftheseintervalscontainstheinterval[v1,wn].LetuscalltheseintervalsB1,B2,...,Bn,orderingthemasbefore.SupposethatnisevenandthatB1increasesonaverage.Letc0bethesmallestxedpointinB1.NowinB2,takethelargestpoint,callitc2,thatmapstoc0.Wecontinuendingpointsc2iinthismannerthatmaptoc0untilwendthelastoneinBn.Wemaynowndapointc1inB1suchthatc1mapstocnandc0
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thismanneragain,wendthenecessaryc2i+1.Thuswehavethedesiredresultwithc0<...
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Proof. Let>0.ChooseNsuchthatn)]TJ /F7 7.97 Tf 6.59 0 Td[(N<.LetdenotethedistancebetweenthesetsCandfx,f(x),...,fN(x)g.Ifjx)]TJ /F3 11.955 Tf 10.33 0 Td[(yj<,thenyisinthesameconnectedcomponentofxonatleasttherstNiteratesoff.Therefore,j(x))]TJ /F6 11.955 Tf 11.96 0 Td[((y)j
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Proof. Thereareactually4casestoprove;whennisevenoroddandwhetheritistoTnorTn)]TJ /F1 11.955 Tf 6.75 -.3 Td[(.ThefollowingistheprooffornevenandTn.Theother3casesarevirtuallyidentical.Letc0,c1,...,cnbethepointsthatsatisfyTheorem 3.4 andsupposec0<...h(d)=log(2).Moreover,ifgistransitive,thengislocallyeventuallyonto.Also,ifisdegree1,thenisunique.Weseethatthetentmapontheintervalisananalogytotheangle-doublingmaponthecirclesincebothmaphalfofthespacetothewholespace.Moreover,theangle-doublingmapdissemi-conjugatetoT.However,bylookingattheexampleshowninFigures 3-1 and 3-2 ,theentropyofthemapislog2andthesemi-conjugacyhasdisconnectedpointinverses.WehavealsoseenthatthemapinFigure1is 25

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transitive.Noneoftheaboveresultsthatholdonthecircleholdforitsequivalent,thetentmap,ontheinterval. 26

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CHAPTER4ARESULTONTOPOLOGICALCONJUGACY,TRANSITIVITY,ANDPATTERNSForthischapter,wewillpresentsomefactsfromgraphtheoryandlinearalgebra.Thedenitionsandremarksthatappearwithoutproofcanbefoundin[ 13 ].Wewillmentionhowitreadilyrelatestodynamicalsystems. Denition17. AgraphGconsistsofanitesetV=V(G)ofverticestogetherwithanitesetE=E(G)ofedges.Eachedgee2E(G)startsatavertexdenotedbyi(e)2V(G)andterminatesatavertext(e)2V(E)(whichcanbethesameasi(e)).Theremaybemorethanoneedgebetweenagiveninitialstateandterminalstate.(Thisissometimescalledadirectedgraph,ordigraph.) Denition18. LetGbeagraphwithvertexsetV=fJ1,...,Jng.ForverticesJi,Jk2V,letai,kdenotethenumberofedgesinGwithinitialstateJiandterminalstateJk.ThentheadjacencymatrixofGisA=(ai,k)andisdenotedAG.Withthisdenition,weseethattherelationshipbetweentheadjacencymatrixforthecollectionJ1,...,JnunderfisthesameastheadjacencymatrixdenitiongivenforgraphtheoryifwethinkofformingagraphGwithverticesJ1,...,JnandanedgeJi!Jkifandonlyiff(Ji)Jk.Thisgivesusawaytorelatethefollowingresultstodynamics. Denition19. LetGbeagraphwithedgesetEandadjacencymatrixA.TheedgeshiftXGistheshiftspaceoverEdenedbyXG=f=(i)i2Z2EZ:t(i)=i(i+1)foralli2Zg.TheshiftmaponXGiscalledtheedgeshiftmapandisdenotedG. Denition20. LetGbeagraphwithvertexsetVandedgesetE.FixavertexI2V.LetEIbethesetofedgesinEstartingatI.PartitionEIintodisjointsubsetsE1IandE2I.WeconstructanewgraphHbasedonthispartitionasfollows:TheverticesofHarethoseofG,exceptthevertexIisreplacedbytwoverticescalledI1andI2,soW=V(H)=(V)-267(fIg)[fI1,I2g.Foreache2EiI,putanedgeinHfromIitot(e)havingthesamenamee.Foreachf2EstartingatJandendingatI,puttwoedgesf1andf2inH,wheref1goesfromJtoI1andf2goesfromJtoI2.Therestofthe 27

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initialandterminalstateswecopyovertoH.WecallthisprocesssplittingandwecallagraphHobtainedinthiswayasplittingofG. Denition21. Similarlyasfordynamics,wemaydenetheentropyofagraphGaslog,whereisthelargesteigenvalueoftheadjacencymatrixAG. Theorem4.1. IfagraphHisasplittingofagraphG,thentheedgeshiftsXGandXHaretopologicallyconjugate.ThisimpliesthattheentropiesofXGandXHarethesame.Whilethistheoremisstatedasagraphtheoreticresult,wewillbeabletouseitforourpurposes,sincetheedgeshiftsarethesameassubshiftsofnitetypeandanabstractgraphwillcontaintheinformationofadynamicalsystem;wewillusegraphsthatyieldthesameadjacencymatrix. Denition22. AnonnegativematrixAisirreducibleifforeachorderedpairofindicesI,J,thereexistsanonnegativeintegernsuchthatAnI,J>0. Remark10. Thesplittingofanirreduciblegraph(viaapartitionwhoseelementsareallnon-empty)isirreducible. Theorem4.2. LetAbeanirreduciblematrix,06B6Aand(bk,l)<(ak,l)forapairk,l.ThenB
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Denition24. LetP=fx10.ThenthereexistsoneandonlyonemapF:I!IsuchthatFispiecewiselinearwithslopeequaltoeverywhereandfistopologicallysemi-conjugatetoFviaatopologicalsemi-conjugacythatismonotone(andofcoursecontinuousandonto).Moreover,iffistopologicallytransitive,thenfistopologicallyconjugatetoF(themoreoverpartwasnotoriginallyprovedin[ 14 ],butlaterin[ 1 ].) Theorem4.5(BlockandCoven). If(P,)and(Q, )areequivalent,thentheirlineariza-tionsaretopologicallyconjugate.Wearenowreadytostateandprovethemaintheoremofthischapter. Theorem4.6. Let(P,)beapatternandletLbethelinearizationofthepattern.Supposef:I!Iistopologicallytransitiveandfexhibitsthepattern(P,).Supposeh(f)=h(L).ThenfistopologicallyconjugatetoL.Wewillprovethistheoreminseveralparts.FirstwewillassumethatListransitiveandthatfandf2aretransitive.Thenwewillmovetothecasethatf2isnottransitive.Finally,wewilldothesamewhenLisassumedtonotbetransitive. 29

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Part1.SupposeListransitive. ProofofPart1. WemayassumethatthecardinalityofPisatleastfour.IfPcontainedtwopoints,Lwouldnotbetransitive(itwouldbeahomeomorphismontheconvexhullofP.)IfthecardinalityofPisthree,wemayinsteadconsiderthesetL)]TJ /F4 7.97 Tf 6.59 0 Td[(1(P),whichwouldhaveatleastfourpointsandwemayndthesamenumberofpointsunderfthatexhibit(L)]TJ /F4 7.97 Tf 6.58 0 Td[(1(P),LjL)]TJ /F19 5.978 Tf 5.75 0 Td[(1(P)).(Wewillusethisfactlateron.)First,letussupposethatf2istopologicallytransitive.LetP=fx01<...ni)]TJ /F4 7.97 Tf 6.59 0 Td[(1foreachpositiveintegeri.Hence,lettingAki=[xki,xki+1]fori=1,...,nk)]TJ /F8 11.955 Tf 10.84 0 Td[(1,theadjacencymatrixoftheAki'sunderLk+1haslargesteigenvalueLk+1=eh(Lk+1).Alsonotethisadjacencymatrixisirreducible(see[ 4 ]).NowletQ=fy01<...L(andlogf6h(f)).Wemaychoosefy11
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([y01,c1][[c2,y0n0]))suchthatf(y1k)=y0jifandonlyifL(x1k)=x0j.ThentheadjacencymatrixformedbyB1i=[y1i,y1i+1],i=1,2,...,n1)]TJ /F8 11.955 Tf 12.18 0 Td[(1,underf2haslargesteigenvaluef2withh(f2)>logf2>logL2=h(L2).Formally,byinduction,wemaydothistoobtainanadjacencymatrixformformedbyBm)]TJ /F4 7.97 Tf 6.59 0 Td[(1i,i=1,2,...,nm)]TJ /F4 7.97 Tf 6.58 0 Td[(1)]TJ /F8 11.955 Tf 12.29 0 Td[(1,underfmwithlargesteigenvaluefmwithh(fm)>logfm>logLm=h(Lm).Byconstruction,wehavethepoints(andordering)ym)]TJ /F4 7.97 Tf 6.59 0 Td[(11<...q.Thenfm([ym)]TJ /F4 7.97 Tf 6.58 0 Td[(1i,ym)]TJ /F4 7.97 Tf 6.58 0 Td[(1i+1])coversatleast[y0p)]TJ /F4 7.97 Tf 6.59 0 Td[(1,y0q]or[y0p,y0q+1],whichwouldaddaonetoanirreduciblematrix,sobyTheorem 4.2 ,thiswouldraisetheentropy,acontradiction.Case3.Supposefm(d1)=y0lwherep
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conjugatetoamapF1viaaconjugacyh1asinTheorem 4.4 .Also,byTheorem 4.4 ,ListopologicallyconjugatetoamapF2viaaconjugacyh2asinTheorem 4.4 .Set^Q=h1(Q)and^P=h2(P).Set 1=F1j^Qand 2=F2j^P.Thepatterns(^Q, 1)and(^P, 2)areequivalent.ObservethateachturningpointofF1isanelementof^Q.ItfollowsthatF1isthelinearizationof(^Q, 1).Similarly,F2isthelinearizationof(^P, 2).ByTheorem 4.5 ,F1andF2aretopologicallyconjugate.Hence,fandLaretopologicallyconjugate,sincetopologicalconjugacyisanequivalencerelation. Claim2. Iff2isnottransitive,westillobtainthesameresult. ProofofClaim. Iffistransitivebutf2isnot,thenbyProposition 2.2 therearetwoclosedintervalsJ,KcoveringIwhichintersectatexactlyaxedpointewithf(J)=Kandf(K)=J.NoticethatL2cannotbetransitiveeither,sothereareJ0andK0withthesamepropertiesunderL.Forthepattern(P,),justasbefore,wemayconsiderapatternthatcontainsmorepoints.WemaytakethispatterntocontainatleastfourpointsinJ0andatleastfourpointsinK0(hencealsoinJandKunderf),andwedothissothatatleastoneoftheB0k'soneachsidedoesnotcontaintheendpointsofIorthexedpointe.NowsupposethatfisnotmonotoneononeoftheB0i's,callitB0j.WemayassumeB0jJ.Thenwemayndd1msuchthatfn([d1,d2])coversB0k,wherethisisanintervalthatdoesnotcontaintheendpointsofIorthexedpointe.NotethatProposition 2.3 appliessince(f2jJ)2istransitivebyProposition 2.2 .Therestoftheproofisthesameasthepreviouscase. ThiscompletestheproofofPart1. Part2.SupposeLisnottransitive.Thenh(f)>h(L). ProofofPart2. Wemayassumethath(L)6=0,sinceifh(L)=0,thenwearedoneash(f)>0becausefistransitive[ 4 ].Now,formtheadjacencymatricesforLnandfnasbefore,choosingapatternforftomimicthemodelmap,L.LetAbetheadjacency 32

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matrixformedfromL)]TJ /F7 7.97 Tf 6.59 0 Td[(k(P)sothatwehaveatleastfourpoints,asbefore(andwewillstillrefertothismodiedpatternas(P,)).ThereisanirreduciblesubmatrixAithatyieldsthesamelargesteigenvalueasA(henceyieldstheentropyofL).SinceLisnottransitive,theremustbeanintervaloutsideofwhatcorrespondstotheirreduciblepart,Ai.LetJbeanintervalthatcorrespondstosubmatrixAijoiningadjacentpointsofPthatdoesnotcontaintheendpointsofI.LetKbeanintervaloutsideofAithatdoesnotcontaintheendpointsofI.AsinPart1,thereisanintegermsuchthatfm(J)Kandfm(K)J.Justasbefore,thisimpliestheentropyoffmustbegreaterthantheentropyofL. Thiscompletestheproofofthetheorem. Fromthis,wemaypresentthefollowingcorollariesthatrelatetotentmaps. Corollary2. Supposef:I!Iisturbulent,topologicallytransitive,andh(f)=log2.ThenfistopologicallyconjugatetoT2. Proof. Sincefisturbulent,itexhibitsthepattern(P,)withP=fa,b,cgand(a)=a,(c)=a,and(b)=cwitheithera
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h(f)=h(LP),i.e.,thesupremumisamaximum.Ingeneral,itispossiblethatfisnottopologicallyconjugatetoLP,butstillh(f)=h(LP).Hereisanexample.Letfbethefunctionf(x)=x2 2for06x62andlinearfrom(2,2)to(3,4)and(3,4)to(4,1),shownbelowinFigure 4-1 .Let(P,)bethepatternwithP=f0,1,2g,(0)=0,(1)=2,and(2)=0,showninFigure 4-2 .Thenh(f)=h(LP)butfandLParenottopologicallyconjugate. Figure4-1. Graphoff. Figure4-2. GraphofLP. However,byTheorem 4.6 ,wehavethefollowing: Corollary4. Supposef:I!IistopologicallytransitiveandexhibitsapatternP.Thenh(f)=h(LP)ifandonlyiffistopologicallyconjugatetoLP. 34

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CHAPTER5INVERSELIMITS Denition25. LetfXi,dig1i=0beacollectionofcompactmetricspaceseachwithametricdiboundedby1,andsuchthatforeachi,fi:Xi+1!Xiisacontinuousmap.TheinverselimitspaceoftheinverselimitsystemfXi,dig1i=0isthesetlim fXi,fig1i=0=fx=(x0,x1,...)jx21Yi=0Xi,fi(xi+1)=xi,i2Ng,withametricdgivenbyd(x,y)=1Xi=0di(xi,yi) 2i.IfXi=Xandfi=fforalli,theinverselimitspaceisdenoted(X,f).Themap:(X,f)!(X,f)denedby(x0,x1,...)=(f(x0),x0,x1,...)iscalledtheshifthomeomorphism,ortheinducedhomeomorphism. Denition26. Ofparticularinterest,wewillconsiderlaterthe-adicsolenoid,denoted,andthe-adicKnastercontinuum,denotedK.Let=(a0,a1,a2,...)beasequenceofintegerswithak>1forallk>0.Deneastheinverselimitspaceofmappingsz7!zak,k=1,2,...onthecircleinthecomplexplaneS1=fz2Cjjzj=1g.Then=lim fS1,zakg1k=0.DeneKasthequotientbytherelationwherexyifandonlyifx=yorx=y)]TJ /F4 7.97 Tf 6.58 0 Td[(1.Fornotation,if=(n,n,n,...),wewilldenotethespacesnandKn.Boththeoremsinthischapterarewell-known,andsomeinterestingresultsrelatingtothemmaybefoundinYe[ 21 ].Thefollowingtheoremestablishesaconnectionbetweentopologicalsemi-conjugaciesandinverselimitspaces. 35

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Theorem5.1. Supposef:X!Xandg:Y!Y,arecontinuousmapsoncompactspacesXandY.Supposealsothatfistopologicallysemi-conjugatetog,withsemi-conjugacy.Then^iscontinuousandsurjectiveandthefollowingdiagramcommutes:(X,f)^f)305()223()305(!(X,f)^??y??y^(Y,g)^g)305()223()305(!(Y,g)where^((x0,x1,...))=((x0),(x1),...)and^fand^garetheshifthomeomorphismsontheirrespectivespaces. Proof. Westartbyshowing^indeedsendspointsto(Y,g).Let(x0,x1,...)2(X,f).Sincef(xn)=xn)]TJ /F4 7.97 Tf 6.59 0 Td[(1,(f(xn))=(xn)]TJ /F4 7.97 Tf 6.58 0 Td[(1)and(f(xn))=g((xn))bythesemi-conjugacy,g((xn))=(xn)]TJ /F4 7.97 Tf 6.59 0 Td[(1),so^sendspointsto(Y,g).Clearly,^iscontinuous.Nowweshowthat^issurjective.Let(y0,y1,y2,...)beapointof(Y,g).Fixapositiveintegern.Letxnbesuchthat(xn)=yn.Wehavethatf(xn)=g(xn)=g(yn)=yn)]TJ /F4 7.97 Tf 6.59 0 Td[(1,sof(xn)2)]TJ /F4 7.97 Tf 6.58 0 Td[(1(yn)]TJ /F4 7.97 Tf 6.59 0 Td[(1).Wemaychoosexn)]TJ /F4 7.97 Tf 6.59 0 Td[(12)]TJ /F4 7.97 Tf 6.59 0 Td[(1(yn)]TJ /F4 7.97 Tf 6.59 0 Td[(1)suchthatf(xn)=xn)]TJ /F4 7.97 Tf 6.59 0 Td[(1.LetL=T1j=0fj(X).Forapoint(x0,x1,x2,...)=x2(X,f),letn(x)betheprojectionmapontothecoordinatexn.Noticethatf(L)=L,soforallx2L,thereexistsanx2(X,f)with0(x)=x.Now,xy=(y0,y1,y2,...)2(Y,g)andxanintegerk>0. Claim3. Thereisaw2(X,f)withk(^(w))=yk. Proof. LetE0=)]TJ /F4 7.97 Tf 6.59 0 Td[(1(fykg),E1=)]TJ /F4 7.97 Tf 6.59 0 Td[(1(fyk+1g),andsoon.WeseethateachEjisnon-empty,compact,andE0E1E2....Sothereisanx2T1k=0Ek.Thenx2L,sothereisanxwith0(x)=x.Nowsetw=^fk(x).Thenk(^(w))=ykbecausek(w)2)]TJ /F4 7.97 Tf 6.58 0 Td[(1(fykg).Thisprovestheclaim. 36

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SetAk=fxjk(^(x))=ykg.WeseethatA0A1...witheachcompactandnonempty,sothereisanxintheintersectionaswell.Then^(x)=y.Hence,^issurjective.Finally,wejustcheckthatthediagramcommutes.^(^f((x0,x1,x2...)))=^((f(x0),x0,x1,...))=((f(x0)),(x0),(x1),...)=(g((x0)),(x0),(x1),...)=^g(^((x0,x1,...))).Thus^fistopologicallysemi-conjugateto^g. Denition27. WesaythatXisacontinuumifXisanonempty,compact,connectedmetricspace.IfXisnottheunionofanytwopropersubcontinua,wesayXisindecom-posable. Theorem5.2. LetXandYbecontinua.SupposethatYisindecomposableandf:X!Yiscontinuousandsurjective.ThenXhasanindecomposablesubcontinuum. Proof. LetSdenotethesetofallcontinuaKXsuchthatf(K)=Y.WehavethatSisnonemptysinceX2S.WemayorderSbyreverseinclusion.Sispartiallyordered.IfwetakeachainXK1K2...,thishasanupperbound,namelyTK.SobyZorn'slemma,thereisamaximalelementinS.(Noticethatsinceweareorderingbyreverseinclusion,maximalmeanssmallest.)CallthiselementK02S. Claim4. K0isindecomposable.Bywayofcontradiction,supposethatK0isnotindecomposable.ThenK0=V[WwhereVandWarepropersubcontinuaofK0.Now,f(V)[f(W)=Y,butYisindecomposable,soitmustbethecasethatoneiscontainedtheother,sayf(V)f(W).Therefore,f(W)=Y,butthenWisapropersubsetofK0andW2S,acontradiction.SoK0isindecomposable. Thefollowingremarksarewell-knownandwillbestatedwithoutproof.Someoftheseresults,andothers,maybefoundin[ 12 ],[ 2 ]. Remark12. Forn>1,theinverselimitspaces(I,f)and(I,fn)arehomeomorphic. 37

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Remark13. Forn>2,theinverselimitspace(I,Tn)ishomeomorphictotheKnastercontinuumKn.NotethatsinceT2n)]TJ /F13 11.955 Tf 10.07 1.4 Td[(andT2naretopologicallyconjugate(fornodd,theyareactuallyequal),wehavethat(I,Tn)]TJ /F8 11.955 Tf 6.76 -.3 Td[()ishomeomorphicto(I,Tn)bythepreviousremark. Remark14. IfXisanindecomposablecontinuum,thenXisnotpathconnected. 38

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CHAPTER6SEMI-CONJUGACIESANDINVERSELIMITSPACESWenowuseourpreviousresultstoproveTheorem 6.1 andtwocorollaries. Theorem6.1. Letf:I!Ibeacontinuousfunctionwithpositivetopologicalentropy.Thenforanym>2,thereisacontinuoussurjectivemapfrom(I,f)toKm. Proof. Letkbesuchthatfkhasanm-horseshoe[ 6 ,TheoremVIII.29,p.215][ 1 ,Theorem4.3.5,p.207].Thenfkissemi-conjugatetoTmorTm)]TJ /F1 11.955 Tf 6.75 -.3 Td[(.SobyTheorem 5.1 ,thereisacontinuoussurjectivemapfrom(I,fk)to(I,Tm)or(I,Tm)]TJ /F8 11.955 Tf 6.75 -.3 Td[().Since(I,fk)'(I,f),(I,Tm)'Km,and(I,Tm)]TJ /F8 11.955 Tf 6.75 -.3 Td[()'Km,wehavethedesiredresult. Corollary5. Letf:I!Ibeacontinuousfunctionwithpositivetopologicalentropy.Then(I,f)hasanindecomposablesubcontinuum. Proof. ByTheorem 6.1 ,thereisacontinuoussurjectivemapfrom(I,f)toK2.SinceK2isindecomposable,theconclusionfollowsfromTheorem 5.2 Corollary6. Foranym,n>2,thereisacontinuoussurjectivemapfromKmtoKn. Proof. Setf=Tm.Then(I,f)ishomeomorphictoKm.Thetopologicalentropyoffislogm.ByTheorem 6.1 ,thereisacontinuoussurjectivemapfrom(I,f)toKn. GiventheconnectionbetweensolenoidsandKnastercontinua,onemightexpectthattheanalogousresulttoCorollary 6 holdsformandn.Itturnsoutthatthisisnotthecase.Toseethis,wewillusesomeresultsabouttopologicalgroups. Denition28. LetGbealocallycompactabeliangroup.AcharacterofGisacontinu-ousgrouphomomorphismfromGtothecirclegroup,S1. Denition29. Thecharactergroup,alsocalledthePontryagindual,ofalocallycompactabeliangroupGisthesetofallcharactersonG. Remark15. ThePontryagindual,G0,ofalocallycompactabeliangroupGisalsoalocallycompactabeliangroupwiththecompact-opentopology.G00,thePontryagindual 39

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ofG0,isnaturallyisomorphictoG,hencethereasonforusingthetermdual.Forfurtherreading,wemayconsult[ 17 ]. Theorem6.2. ThePontryagindualofthesolenoidnistheadditivegroupQn=fj a1n1a2n2...aknkgwherea1...akarethedistinctprimesintheprimefactorizationofn,n1,...nkarepositiveintegers,andj2Z.Ifthereisaone-to-onecontinuoushomomor-phismfromQmtoQn,thenthisinducesacontinuoussurjectivehomomorphismfromntom.Dually,ifthereisacontinuoussurjectivehomomorphismfromntom,thenthisinducesaone-to-onecontinuoushomomorphismfromQmtoQn.Thesetheoremsaredetailedin[ 11 ,p.392,p.402].Inamoregeneralsetting,thesetheoremsarestatedashavingdenseinplaceofsurjective.However,sincenandmarecompactandconnected,wehavethestatedstrongerresult. Theorem6.3. Letm,nbeintegers,m,n>2.Supposethereisacontinuoushomomor-phismf:Qn!QmthatdoesnotmapeveryelementofQntotheidentity.Thenfisone-to-one. Proof. Suppose,bywayofcontradiction,thatf(a)=0forsomea6=0.Wemayassumeaisanintegersinceiff(c)=0andc=a=bwhereaandbareintegers,then0=bf(a=b)=f(a).Then,a=n2Qn.So,0=f(a)=nf(a=n)=naf(1=n),sof(1=n)=0.Similarly,f(1=n2)=0.Byinduction,f(1=nk)=0fork=1,2,3,....Sincethesetf1=n,1=n2,...ggeneratesQn,fmustsendeverythingto0,acontradiction.Hence,fmustbeone-to-one. Corollary7. Iff:m!nisacontinuoushomomorphismthatdoesnottakeeveryelementtotheidentity,thenfissurjective. Proof. Thisfollowsbydualitysincefinducesamapf:Qn!Qmwhichdoesnottakeeveryelementtotheidentity. 40

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Denition30. Letgbeanelementofagroup(G,).Givenapositiveintegerp,wesayghasinnitepthrootsifforanypositiveintegerk,thereisanelementh2Gsuchthathh...h| {z }pktimes=g. Lemma2. Supposemandnarepositiveintegerssuchthatthereisaprimepwithpjnbutp-m.Thentheredoesnotexistacontinuousone-to-onehomomorphismfromQntoQm. Proof. WerstnotethattheonlyelementinQmwithinnitepthrootsis0.Nowsupposethereisacontinuousone-to-onehomomorphismf:Qn!Qm.Thenf(1)=bforsomeb6=0.Sob=f(1)=pkf(1=pk)foranypositiveintegerk.Butthismeansbhasinnitepthroots,sob=0,acontradiction. WearenowreadytoprovethattheanalogousresulttoCorollary 6 doesnotholdforsolenoids. Theorem6.4. Supposem,narepositiveintegers,m,n>2,suchthatthereisaprimepwithpjnbutp-m.Thentheredoesnotexistacontinuoussurjectivemapfrommton. Proof. Suppose,bywayofcontradiction,thatthereexistsacontinuoussurjectivemapg:m!n.ByScheffer'stheorem[ 19 ],thereisauniquecontinuoushomomorphism^g:m!nsuchthatgishomotopicto^g.Sincenisnotpathconnected,^gdoesnotmapeveryelementofmtotheidentityofn.ByCorollary 7 ,^gissurjective.ByTheorem 6.2 ,^ginducesacontinuous,one-to-onehomomorphismfromQmtoQn.ThiscontradictsLemma 2 Corollary8. Supposemandnarepositiveintegerssuchthatthereisaprimepwithpjnbutp-m.Letfandgbethemapsofthecircletoitselfdenedbyf(z)=zmandg(z)=zn.Thenfisnottopologicallysemi-conjugatetog. Proof. Suppose,bywayofcontradiction,thatfistopologicallysemi-conjugatetogwithsemi-conjugacy.ByTheorem 5.1 ,thereexistsacontinuoussurjectivemap^:m!n.ThiscontradictsTheorem 6.4 41

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Whilethiscorollaryfollowsimmediately,wewillproveamoregeneralresultinthenextchapter. 42

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CHAPTER7ARESULTONTOPOLOGICALSEMI-CONJUGACIESFORMAPSOFTHECIRCLE Denition31. Letp:~X!Xbeamapoftopologicalspaces.Aliftofamapf:Y!X,Yatopologicalspace,isamap~f:Y!~Xsuchthatp~f=f. Denition32. AcoveringspaceofatopologicalspaceXisatopologicalspace~Xtogetherwithamapp:~X!Xsatisfying:ThereexistsanopencoverfUgofXsuchthatforeach,p)]TJ /F4 7.97 Tf 6.58 0 Td[(1(U)isadisjointunionofopensetsin~X,eachofwhichismappedbyphomeomorphicallyontoU. Theorem7.1(TheHomotopyLiftingProperty). Givenacoveringspacep:~X!X,ahomotopyft:Y!X,andamap~f0:Y!~Xliftingf0,thenthereexistsauniquehomotopy~ft:Y!~Xof~f0thatliftsft. Remark16. Thecircleandthemapp:S1!S1,p(z)=zn,isacoveringspacewhennisanypositiveinteger. Remark17. Letfandgbemapsofthecircletoitself.Letdeg(f)denotethedegreeoff.Wehavethefollowingresults:Themapsfandgarehomotopicifandonlyifdeg(f)=deg(g).Also,deg(fg)=deg(f)deg(g).Finally,deg(f)=0ifandonlyiffishomotopictoaconstantmap.Thepreviousresultsarewell-known.TheformulationsofthedenitionsaswellastheproofstotheremarkscanbefoundinHatcher'stext[ 10 ]. Lemma3. Supposefandgaremapsofthecircletoitself,withdeg(f)=m>0,deg(g)=n>1,andm6=n.Iffistopologicallysemi-conjugatetog,thenfistopologicallysemi-conjugatetothemapp(z)=zn.Moreover,thistopologicalsemi-conjugacyhasdegree0. Proof. Byhypothesis,fistopologicallysemi-conjugatetogviaasemi-conjugacy1.Also,bytheTheoremofShub[ 20 ]mentionedintheintroduction,gistopologically 43

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semi-conjugatetopviaasemi-conjugacy2.Wehavethefollowingdiagram:S1f )223()305()]TJ /F3 11.955 Tf 35.86 0 Td[(S11??y??y1S1g )223()305()]TJ /F3 11.955 Tf 35.86 0 Td[(S12??y??y2S1p )223()305()]TJ /F3 11.955 Tf 35.86 0 Td[(S1Thenfissemi-conjugatetopvia21.Also,byRemark 17 ,1hasdegree0,so21alsohasdegree0. Wearenowinpositiontoproveournalresult. Theorem7.2. Supposefandgaremapsofthecircletoitselfwithdeg(f)=m>0,deg(g)=n>1,andm6=n.Thenfisnottopologicallysemi-conjugatetog. Proof. Suppose,bywayofcontradiction,thatwiththeabovehypotheses,fistopologicallysemi-conjugatetog.Step1.ByLemma 3 ,thereisatopologicalsemi-conjugacyhfromftothemapp(z)=zn.Therearehomotopies(h0)tand(h1)tsuchthat(h0)tf=p(h1)tforallt,where(hk)0=hand(hk)1isaconstantmap,k=0,1.ProofofStep1.ByLemma 3 ,wehavedeg(h)=0.Thismeansthathishomotopictoaconstantmapc.Let(h0)tbeahomotopywith(h0)0=hand(h0)1=c.Sincepisacoveringmap,wemayapplyTheorem 7.1 .Thereadermightwanttotakealookattherstboxinthediagrambelowatthispoint.Considerthehomotopy(h0)tf.Wehave(h0)0f=hfand(h0)1f=c.Thereisamapthatlifts(h0)0f,namelyh.Thus,byTheorem 7.1 ,thereexistsahomotopy(h1)twith(h1)0=hsuchthat(h0)tf=p(h1)tforallt.Soatt=1,wehavecf=c=p(h1)1.Sincethepre-imageofasinglepointunderpisadiscretesetofnpointsand(h1)1isacontinuousfunctionofaconnectedspace,(h1)1isaconstantmapaswell. 44

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Step2.Thereexistsahomotopy(h1)t:(S1,f)!n,where(h1)0=^h,theinducedmapfromh,and(h1)1isaconstantmap.ProofofStep2.ByapplyingTheorem 7.1 inductivelyweobtainahomotopy(hn)tforeachpositiveintegern,suchthat(hn)0=h,(hn)1isaconstantmap,andthediagrambelowiscommutative.S1f )223()305()]TJ /F3 11.955 Tf 35.86 0 Td[(S1f )222()306()]TJ /F3 11.955 Tf 35.87 0 Td[(S1f )222()306()]TJ /F8 11.955 Tf 35.86 0 Td[(...(h0)t??y(h1)t??y(h2)t??yS1p )223()305()]TJ /F3 11.955 Tf 35.86 0 Td[(S1p )222()306()]TJ /F3 11.955 Tf 35.87 0 Td[(S1p )222()306()]TJ /F8 11.955 Tf 35.86 0 Td[(...Dene(h1)t:(S1,f)!nby(h1)t(x0,x1,x2,...)=((h0)t(x0),(h1)t(x1),(h2)t(x2),...).Then(h1)0=^hand(h1)1isaconstantmap.Step3.Therecannotbeahomotopyhtfromatopologicalspacetonfromasurjectivemaptoaconstant.Hence,wecometoacontradiction,sofcannotbetopologicallysemi-conjugatetog.ProofofStep3.Ifthereisahomotopyfromasurjectivemaptoaconstantmap,itwouldimplythatnispath-connected,butnisnotpath-connected.Wehavereachedthedesiredcontradiction. 45

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REFERENCES [1] L.Alseda,J.Llibre,M.Misiurewicz,Combinatorialdynamicsandentropyindimensionone,volume5ofAdvancedSeriesinNonlinearDynamics,WorldScienticPublishingCo.Inc.,RiverEdge,NJ,secondedition,2000. [2] M.Barge,ThetopologicalentropyofhomeomorphismsofKnastercontinua,HoustonJ.Math.13(1987)465. [3] M.Barge,J.Martin,Chaos,periodicity,andsnakelikecontinua,Trans.Amer.Math.Soc.289(1985)355. [4] L.Block,E.M.Coven,Topologicalconjugacyandtransitivityforaclassofpiecewisemonotonemapsoftheinterval,Trans.Amer.Math.Soc.300(1987)297. [5] L.Block,E.M.Coven,Approximatingentropyofmapsoftheinterval,in:Dynamicalsystemsandergodictheory(Warsaw,1986),volume23ofBanachCenterPubl.,PWN,Warsaw,1989,pp.237. [6] L.S.Block,W.A.Coppel,Dynamicsinonedimension,volume1513ofLectureNotesinMathematics,Springer-Verlag,Berlin,1992. [7] R.Bowen,MarkovpartitionsforAxiomAdiffeomorphisms,Amer.J.Math.92(1970)725. [8] P.Boyland,Semiconjugaciestoangle-doubling,Proc.Amer.Math.Soc.134(2006)1299(electronic). [9] J.Franks,Anosovdiffeomorphisms,in:GlobalAnalysis(Proc.Sympos.PureMath.,Vol.XIV,Berkeley,Calif.,1968),Amer.Math.Soc.,Providence,R.I.,1970,pp.61. [10] A.Hatcher,Algebraictopology,CambridgeUniversityPress,Cambridge,2002. [11] E.Hewitt,K.A.Ross,Abstractharmonicanalysis.Vol.I,volume115ofGrundlehrenderMathematischenWissenschaften[FundamentalPrinciplesofMathematicalSciences],Springer-Verlag,Berlin,secondedition,1979.Structureoftopologicalgroups,integrationtheory,grouprepresentations. [12] J.Keesling,V.A.Ssembatya,OnxedpointsofKnastercontinua,TopologyAppl.153(2005)318. [13] D.Lind,B.Marcus,Anintroductiontosymbolicdynamicsandcoding,CambridgeUniversityPress,Cambridge,1995. [14] J.Milnor,W.Thurston,Oniteratedmapsoftheinterval,in:Dynamicalsystems(CollegePark,MD,1986),volume1342ofLectureNotesinMath.,Springer,Berlin,1988,pp.465. 46

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[15] C.Mouron,Positiveentropyhomeomorphismsofchainablecontinuaandindecomposablesubcontinua,Proc.Amer.Math.Soc.139(2011)2783. [16] W.Parry,Symbolicdynamicsandtransformationsoftheunitinterval,Trans.Amer.Math.Soc.122(1966)368. [17] L.S.Pontryagin,Topologicalgroups,TranslatedfromthesecondRussianeditionbyArlenBrown,GordonandBreachSciencePublishers,Inc.,NewYork,1966. [18] R.C.Robinson,Anintroductiontodynamicalsystems:continuousanddiscrete,PearsonPrenticeHall,UpperSaddleRiver,NJ,2004. [19] W.Scheffer,Mapsbetweentopologicalgroupsthatarehomotopictohomomorphisms,Proc.Amer.Math.Soc.33(1972)562. [20] M.Shub,Endomorphismsofcompactdifferentiablemanifolds,Amer.J.Math.91(1969)175. [21] X.D.Ye,Topologicalentropyoftheinducedmapsoftheinverselimitswithbondingmaps,TopologyAppl.67(1995)113. 47

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BIOGRAPHICALSKETCH DennisLediswasborninSouthFloridain1981.PriortostudyingmathematicsattheUniversityofFlorida,heearnedhisB.S.inmathematicsatFloridaInternationalUniversityinMiami,Florida. 48