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Effective Symbolic Dynamics

Permanent Link: http://ufdc.ufl.edu/UFE0022083/00001

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Title: Effective Symbolic Dynamics
Physical Description: 1 online resource (87 p.)
Language: english
Creator: Dashti, Ali
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: computability, degrees, dynamics, pi, subshifts
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
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theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
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Notes

Abstract: We investigate computable subshifts and the connection with effective symbolic dynamics. It is shown that a decidable effectively closed class P is a subshift if and only if there is a computable function F mapping Cantor space to Cantor space such that P is the set of itineraries of elements of Cantor space. An effectively closed subshift is constructed which has no computable element. Moreover effectively closed subshifts with higher degrees of difficulty are constructed. We also consider the symbolic dynamics of maps on the unit interval.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ali Dashti.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Cenzer, Douglas A.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-02-28

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Classification: lcc - LD1780 2008
System ID: UFE0022083:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022083/00001

Material Information

Title: Effective Symbolic Dynamics
Physical Description: 1 online resource (87 p.)
Language: english
Creator: Dashti, Ali
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: computability, degrees, dynamics, pi, subshifts
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We investigate computable subshifts and the connection with effective symbolic dynamics. It is shown that a decidable effectively closed class P is a subshift if and only if there is a computable function F mapping Cantor space to Cantor space such that P is the set of itineraries of elements of Cantor space. An effectively closed subshift is constructed which has no computable element. Moreover effectively closed subshifts with higher degrees of difficulty are constructed. We also consider the symbolic dynamics of maps on the unit interval.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ali Dashti.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Cenzer, Douglas A.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-02-28

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022083:00001


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andtomydad 3

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GrandestgratitudegoestothepeopleoftheMathematicsDepartmentoftheUniversityofFloridawhogivetheirtime,skillsandknowledgeselessly.EspeciallyIamthankfultohaveDougCenzer,alsodenotedby asmysupervisor.Itshallremainamaximalhonorformetohavesuchakeenhard-workingworldclassresearcherasmyguidethroughoutsomanyyearsthatewpassbyussoquickly.AlonggoesmyincommensurablerespectandgratitudetoKevinKeating,JLFKing,ScottMcCullough,P.L.Robinson,andJ.Zapletalwhohavebeenutmostlygenerouswiththeirtimeandknowledgetohelpme.AndalonggoesthemostsincerethankstothequeensoftheDepartmentofMathematics:GretchenGarrett,Connie,Julia,MarieHahn,Margaret,SharonandSandyforalltheperfectdetailedcoordinationofallthelittlegearsofthissystemofourdepartment.Andalonggoesthepurestmostheartfeltthankstomycompanionsandmyothergraduatecoworkerswhomadesmooththeroughnessofmanymomentswiththeirsmile,presenceandfriendship.Toallthatgraduatedandwouldgraduate:...MariaHerzog,JamesAscher,...,JustinSmith,RobertStrich,...,Thanos,...Itisalmostattheendofthisveyearlongpathoflife,andhowcouldallhavecometothispointformewithoutthehelpofSteckleyLeewhoseexistenceisatrueblessingforthisworld. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 6 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 8 CHAPTER 1INTRODUCTION .................................. 9 1.1PreliminariesofFormalLanguages ...................... 9 1.2TopologyofNandZ 15 1.3PreliminariesofSymbolicDynamics ...................... 17 1.4PreliminariesofComputabilityTheory .................... 22 1.501Classes .................................... 24 1.6MedvedevandMuchnikDegrees ........................ 27 201SUBSIMILARCLASSES ............................. 29 2.1ForbiddenWords:Simple,ReducedandTotalSets ............. 29 2.2Decidable01SubsimilarClasses ........................ 37 2.3UndecidableSubshiftsof2N 41 2.3.101SubshiftswithDierentDegrees .................. 45 2.3.2Incomparable01Subshifts ....................... 49 2.4UndecidableSubshiftsof2Z 52 2.5Countable01SubsimilarClasses ....................... 53 2.6OpenQuestions ................................. 59 3SYMBOLICDYNAMICSFORFUNCTIONSONNANDZ 64 3.1SymbolicDynamicsforFunctionsonN 64 3.2SymbolicDynamicsforFunctionsonZ 66 4THECOMPUTABILITYOFTHEUNIMODALMAPS ............. 68 4.1Shift-MaximalWords .............................. 68 4.2UnimodalMaps ................................. 73 REFERENCES ....................................... 83 BIOGRAPHICALSKETCH ................................ 87 5

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Table page 4-1Grammaticalhierarchyofthelanguageofkneadingsequences .......... 77 6

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Figure page 1-1Aregulartreethatisnotsubsimilar ........................ 16 1-2Asubsimilartreethatisnotregular ........................ 16 1-3Aninnitenitelybranchingbi-tree ........................ 17 1-4Theniteautomatonthatrecognizesthelanguageofthegoldenmeanshift ... 21 1-5Theniteautomatonthatrecognizesthelanguageoftheevenshift ....... 21 4-1Anexampletoshowtheinclusionisproper .................... 73 4-2AnexampletoshowthatI[F]neednotbeclosed ................. 75 4-3AnexampletoshowthatI[F]canbeclosed .................... 76 4-4Legalinversepath .................................. 80 7

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Weinvestigatecomputablesubshiftsandtheconnectionwitheectivesymbolicdynamics.Itisshownthatadecidable01classPisasubshiftifandonlyifthereisacomputablefunctionFmapping2Nto2NsuchthatPisthesetofitinerariesofelementsof2N.A01subshiftisconstructedwhichhasnocomputableelement.Moreover01subshiftswithhigherdegreesofdicultyareconstructed.Wealsoconsiderthesymbolicdynamicsofmapsontheunitinterval. 8

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Inthisintroductorychapterwereviewthemaindenitionsandfactsthatweshallneedfortherestofthisdissertation.Itrytomakethischapterasself-sucientaspossible,butthisgoalseemsratherambitious.ForthispurposeIdoneedtogooversomefundamentalmaterialsfromsomemainareasofdiscretemathematics.ThiswillincludesubjectslikeCombinatoricsonwords,SymbolicDynamicsandComputabilityTheory. Weadoptthefollowingnotationsanddenitions: Thebinaryalphabet=f0;1gisthemostwidelyusedsetofletters,andthisworkisnotanexceptiontothisfact.Thesetofone-sidedortwo-sidedinnitewordsonthebinaryalphabetistheusualCantorsetdenotedby2N.Theothercommonplacealphabet 9

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Afactorofaword,niteorinnite,isanyconsecutiveblocksoflettersthatoccursomewhereinthatword.Thus,U2oflengthn,isafactorof 1.W2ifthereisi2NsuchthatU=W(i)W(i+1):::W(i+n1). 2.x2Nifthereisi2NsuchthatU=x(i)x(i+1):::x(i+n1). 3.x2Z,ifthereisi2ZsuchthatU=x(i)x(i+1):::x(i+n1). Therearethreespecialtypesoffactors.GivenanydecompositionU1U2U3ofW2,U1;U2andU3areallfactorsofW,butwithdierentnames: 1.U1isaprexoraninitialsegmentoraleftmostfactorofW.Sometimes,assumingjU1j=m,wetakethenotationU1=Wdm.Anotherwidelyusednotationtorefertotheprexofawordis.Soforthiscase,wehaveU1W. 2.U2isaninxofW,insymbolsU2GW.Mostlythroughthisworkweshallassumetheinxeshaveoddlength.ThusgivenawordWoflengthn,aninx,U,ofoddlength2m+1nisdenedasfollows:Letk=bn 3.U3isasuxorarightmostfactorofW.Similartothepreviousitems,givenaword,W,oflengthn,andgivenmn,WemdenotesthesuxoflengthmofW. Thenotionofprexisstillwell-denedforaone-sidedinnitewordx2N.SayWoflengthmisaninitialsegmentofx,Wx,ifW=x(0)x(1):::x(m1)=xdm.Ifxisatwo-sidedinnitewordthenwecantalkaboutitsinxasfollows:Woflength2m+1isaninxofx2Z,insymbolsWGx,ifW=x(m1):::x(1)x(0)x(1):::x(m).Itisforthesakeofsimplicitythatwemainlyconsiderinxesofoddlength.Thisalsowillbesucientforthemostpurposesofourwork. Nowwearepreparedtogivethedenitionofaformallanguage. 10

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ObviouslyH(L)doesnotexist.ButsupposeL:=f0n:n1g,thenU(L;n)=1andH(L)=0.Inalittlewhileweshowthatentropyexistsforaspecialclassofformallanguages,thatofsubsimilartrees. Aswillbecomemoreapparent,thenotionofthesetoffactorsofaformallanguageisimportant.LetL(L)denotethesetofallfactorsofallwordsintheformallanguageL.Also,onecantalkaboutonlythesetofallprexes(allodd-lengthinxesorallsuxes)ofallthewordsofalanguageL,denotedasprex(L)(inx(L)orsux(L)).Clearlywehaveprex(L);inx(L),sux(L)L(L).Inadditionprex(L);sux(L)L[fg. 11

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Givenx2Z,letinx(x):=fU:Uaninxofxg Thelanguageofasetofinnitewordsistheunionofthelanguageofitsmembers. Moreovergivenanyinnitewordx,thelengthdistributionofx,U(x;n),isthedenedasthelengthdistributionofthelanguageofitsfactors,thatis,U(x;n):=U(L(x);n).Forexampleitcanbeseenthatifx=101202:::1n0n:::,U(x;n)/n2andifx=101202:::12n02n:::,U(x;n)/n(1+log2(n)). Nowconsiderthefollowingpropositionthatshowssomeinterestingpropertiesthatcanbeprovedfortheformallanguages,whicharenotobvious.ThisresultisparticularlyimportantforusandisadierentversionofKonig'sLemma. Proof. TogettheusualKonig'slemma,theaboveproposition 1 canbestatedforx2Nasfollows: 12

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Proof. Nowwefocusonsomeespecialformallanguagesthathavecertainclosureproperties,suchasbeingclosedunderprexorsux. Atree,T,overthealphabetisaformallanguagethatisclosedunderprexorinitialsegment,thatisprefix(T)=T.Thisnotionishighlysignicantforourdiscussionsthroughoutthiswork.Atreecanbeniteorinnite,butmostofthetreesthatweconsiderhereareinnite.Atreecanbenitelybranchingorinnitelybranching,butsincethealphabetsforthisworkarenite,allofthetreeswedealwithshallbenitelybranchinginnitetrees.NowwecanrestateProposition 2 intermsoftrees: 3 saysthatifTisinniteandnitebranching,then[T]6=;.EveryW2Tisanodeofthetree.Wisadeadendofthetreeifithasnoextension,thatmeansforalla2,Wa=2T.U2TisanextensionnodeofW2TifthereexistsV2suchthatU=WV.SayW2Tisanextendiblenodeifthereexistsx2[T]suchthatWx.GivenanytreeT,TpdenotesthesetofextendiblenodesofthetreeT. 13

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1.1 .AtreeTiscalledsubsimilarifitisadditionallysux-closed,thatissuffix(T)=T.NotethatifatreeissuxclosedthenitisalsofactorclosedthismeansthatL(T)=T. 1.1 .Intheregulartextbooksandliteratureofsymbolicdynamicsthesetreeshavecloserelativesthatarecalleddynamicallanguages.Theonlybetweenthesetwonotionsisthefactthateverynodeofadynamicallanguageisextendiblewhilesubsimilartreescouldhaveeveninnitelymanydeadends.ThusgivenanysubsimilartreeT,Tpisadynamicallanguage.Aswementionedaboveentropyofalanguagemaynotexist.Eventheentropyofaninnitenitebranchingtreemayfailtoexist.Butifthelanguageisasubsimilartreethenitsentropyalwaysexists:

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ABi-treeisaslightlydierentnotionthanatreeinthesensethatthisoneisaformallanguagethatisclosedunderinxesofoddlength.Wewouldbeinterestedonlyininnitenitelybranchingbi-trees.Usingthisnotiontheproposition 1 impliesthefollowing: 1.1 : Abi-treeissaidtobesubsimilarifitisclosedundersuxesandprexesofoddlength. Thistopologycanalsobeobtainedusingametric,d,denedasfollows:

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Aregulartreethatisnotsubsimilar Figure1-2. Asubsimilartreethatisnotregular 16

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Aninnitenitelybranchingbi-tree ThereisananalogoustopologyforthespaceZ.TaketheZ-producttopologyandgetacountablesetofclopenbasisBforthisspace,suchthatP2BifandonlyifW2suchthatP:=fx2Z:WGxg.Moreoverinparallelwiththespaceofone-sideinnitewords,wehavethefollowing: 17

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1.IfTisasubsimilartree,then[T]Nisasubshift. 2.PNisasubshiftifandonlyifTpisasubsimilartree. 3.IfTisasubsimilarbi-tree,then[T]Zisasubshift. 4.PZisasubshiftifandonlyifTpisasubsimilarbi-tree. Proof. 2.LetPbeasubshift.SincePisclosed,thenthereisatreewithoutdeadendsTpsuchthatP=[Tp].LetW=UV2Tp.ThenUVy2Pforsomey2N.SincePisasubshift,Vy2P.HenceV2TpthusTpissubsimilar.Theconverseistruebytherstclause. 3.LetTbeasubsimilarbi-tree.Then[T]Zisaclosedset.Weneedtoshowitisshift-invariant.Letx2[T],z=1(x),andy=(x).Weneedtoshowz;y2[T].forallnxde2n+3=x(n)x(n+1)yde2n+12T,andxde=zde2n+1x(n1)x(n)2T.SinceTisclosedunderprexesandsuxesofoddlength,wegetyde2n+12Tandzde2n+12Tforalln.Hencey;z2[T]. 4.LetPZbeasubshift.SincePisclosedthereisabi-treeTpsuchthatP=[Tp].LetW=U1VU22Tp,suchthatU1;U2aretwofactorsofoddlength.Thereisx2PsuchthatWGx.WehaveU2Gsigmak(x)andU1Gl(x)forsomel;k1.SincePisshift-invariantandsigmak(x)2Pandsigmal(x)2P,wegetU;V2Tp.HenceTpissubsimilar.Theconversefollowsfromthethirdclause. 18

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Proof. AsanexampleletF=f01g.SF=f1k0N:k0g[f1Ng(2N.Nowletconsideralltheelementsof2ZthatavoidthisF.SimpletoseethatnowSFisacountablesetcontainingelementsofthefollowingform:xm(n)=8><>:0nm1nm1 Alsoy1=:::11111:::andy2=:::00000:::areelementsofthisset.ThusSF=fxm:m0g[fy1;y2g,thatisclosedandshift-invariant.Nowsupposeinthedenitionofthesubshifts,P,ofZ,weallowshift-subinvarianceinlieuofshift-invariance.TheninthiscaseP=fxm:m0g[fy2g(2ZisasubshiftbutP6=SF.HencetherestrictioninthedenitionofsubshiftsofZwarrantsthefollowingsimilarproposition: 19

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Proof. GivenasubshiftPN,considertheformallanguagePrefix(P)=fW:Wisaprexofsomex2Pg.ItisreadilyavailablebythedenitionofasubshiftthatPrefix(P)issux-closedandhenceisalsofactorclosed.ThisimpliesPrefix(P)isasubsimilartreewithnodeadendsanditspathsaretheelementsofP.SimilarlygivenasubshiftPZ,considerthelanguageinfix(P)=fW:Wisaninxofoddlengthofsomex2Pg Asagoodexample,letusconsiderthefamousgoldenmeansubshiftof2N.Theonlyforbiddenwordhereis00.LetP=Sf00g.LetL=L(P)bethelanguageofthissubshift.RecallthatU(L;n)isthelengthdistributionofthelanguageL,andisdenedasUn:=U(L;n)=card(2n\L).ItisnothardtoreasonthatUn+2=Un+1+Un.Thiswillimplythatwouldbeatruncatedbonaccisequence:=<2;3;5;:::>===<1 2)n+2(1p 2)n+2)>

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?>=<89:;11 0// ?>=<89:;00// BCD@GA1 Theniteautomatonthatrecognizesthelanguageofthegoldenmeanshift start// ?>=<89:;11 0// ?>=<89:;01// BCD@GA0 Theniteautomatonthatrecognizesthelanguageoftheevenshift ThustheentropyofP,H(P)=log(1+p 2).Alsoitiscleartheniteautomataingure 1.3 recognizesthelanguageL: NowletusconsidertheevenshiftwhichisdenedbyaninnitesetofforbiddenwordsF:=f102n+11:n0g.SF,iscalledtheevenshiftfortheobviousreasonthatbetweeneverytwooccurrencesofoneineveryfactorofthissubshifttherecanonlybeanevennumberofzeros.Figure 1.3 showstheautomatathatisrecognizingthelanguageofthissubshift. Takeanyfactoroflengthn1,concatenatingthiswith0attheendgivesafactoroflengthnofthesubshift.Thisgivesusallthefactoroflengthnthatendswithzero.Nowhowaboutthosefactorsoflengthnthatendinone.TakeanyfactorWoflengthn2,nowthreedistinctcasescanhappen: Thusthetotalnumberofwordsoflengthn,U(n)followsaFibonacci-likesequence,thatisU(n)=U(n1)+U(n2)+1,whereU(1)=2andU(2)=4.NowitcanbeseenthatU(n)=fn+31.ThusH(SF)=log(1+p 2),sotheentropyoftheevenshiftand 21

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AsetANiscomputablyenumerable,c.e.,ifthereisapartialcomputablefunctionsuchthatA=dom().ItcanbeshownthatthisdenitionforbeingcomputablyenumerableisequivalenttohaveatotalcomputablefunctionfsuchthatA=ran(f).AsetANiscomputableifandonlyifbothAandthecomplementofA,Ac,are 22

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Asagoodexampleoftheapplicationoftheseideasfromcomputabilitytheory,hereisaproofofRice'stheorem.Thistheoremisparticularlyimportantforresearchineectivesymbolicdynamics,sincethereareattemptstocookupsomeRice-styletheoreminthisarea.Tostatethetheoremwealsoneedthenotionofindexsets: 23

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Itfollowsfromthes-m-ntheoremthatfiscomputable.Sincex2K()f(x)2A,wehaveKmA. Thenextimportantconceptforus,isthenotionofrelativecomputation.Givenanypartialcomputablefunction,itcanbeaccompaniedwithanoracleformorecomputationalpower.Anoracleisessentiallyasetofnaturalnumbers.ThusAdenotesapartialcomputablefunctionthatusesasetAisanoracle.AisalsocalledanA-computablefunction.A-computablefunctionscanbecodedusingGodel'snumberinginasimilarmannerastheordinarypartialcomputablefunctionscan.ThuswecanhaveanenumerationofsuchA-computablefunctionsdenotedbyAe.Turingreducibilityistheweakerversionofm-reducibilitythataccompaniesrelativecomputation. 24

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Aninterestingfamilyof01classesisgivenbytheseparatingclasses.LetA;BNbetwodisjointcomputablyenumerablesetsletSep(A;B)=fC:AC;A\C=;gbetheseparatingclassofA;B.SinceA;Baredisjoint,Sep(A;B)isnotempty.Manytimeswemakenodistinctionbetweenasubsetofnaturalnumbersanditscharacteristicfunction,havingsaidthat,wehavethefollowingfact:

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01classesmaybeclassiedintotwomaingroups:decidableandundecidable.Calla01class,Q,decidableifandonlyifTQ,thesetofextendiblenodesofitscomputabletreeTisalsocomputable.Equivalently,QisdecidableifthereisacomputabletreeTQwithnodeadendssuchthatQ=[T].Sinceourclosedsetsareallcompact,beingsubsetsof2N,itcanbeshownthatifacomputabletreeTofa01classQhasatmostnitelymanydeadendsthenTisstilldecidable,andTQwillbecomputable.Andasitisexpecteda01class,Q,isundecidableifandonlyifTQis01butnotcomputable.Wewillalsondthefollowingfactuseful: 26

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Proof. Anewertechniquetoconstructaspecial01classistouseacomputablyenumerablesetofforbiddenwords.Thiswillbecoveredinfulldetailinthenextchapters. 27

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1.Itisclearthat[NN]isthebottomofthislattice.Indeedevery01classthathasacomputableelementbelongstothebottomdegreeofthelattice.Thetopdegreehassomeveryinterestingclasses.ForexampleDNC2isamemberofthetopdegree. 2.Givenanytwodegree[P]and[Q],let[P]_[Q]:=[PQ],wherePQ=fxy:x2P;y2Qg.Itcanbeshownthat[P]_[Q]isthejoin,theleastupperbound,ofthesetwodegrees.InparticularwegetPQMQP.NotethatgiventwosubshiftsPandQ,PQmaynotbeasubshift.ButPQ[QPisasubshift. 3.Givenanytwodegree[P]and[Q],let[P]^[Q]:=[PQ],wherePQ=f0x:x2Pg[f1y:y2Qg.[P]^[Q]isthemeet,greatestlowerbound,ofthesetwodegrees.InparticularPQMQP.NotethatgivenanytwosubshiftsPandQ,PQ[QPmaynotbeasubshift. 4.Given[P],[Q]and[R],[P]_([Q]^[R])M([P]_[Q])^([P]_[R]),and[P]^([Q]_[R])M([P]^[Q])_([P]^[R]) MuchnikdegreesareaweakerversionofMedvedevdegrees.GivenmassproblemsPandQ,PisMuchnikreducibletoQ,PWQ,ifandonlyifeveryx2Qcomputessomey2P,thatisyTx.ThisentailsthatifPMQthenPWQ.HencetheMedvedevdegreeofPiscontainedintheMuchnikdegreeofP.ThesetofMuchnikdegreesformadistributivelatticeinthesamefashionthattheMedevedevdegreesdo. 28

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Inthischapterwestudythecomputabilityaspectsofsubshifts.Recallthataone-sidedsubshiftP2Nisaclosedsubsetwhichisshiftsubinvariantmeaning:(P)P.Weknoweveryclosedsubset,P,of2NcanberegardedastheinnitepathsthroughatreeTP2.ButPbeingasubshiftgivesanotherpropertytothistree.suffix(Tp)Tp,thatistosaythatthetreeofasubshiftisclosedundersux.Thiscanbeunderstoodaswhateverstructurecanbeseenaboveanynodeofthetreecanalsobeseenfromtherootofthetree.Becauseofthisobservationwetendtocall01subshifts,01subsimilarclasses.Theyarenotquiteself-similar,sinceitisnotthecasethatthestructureabovetherootisidenticaltothatofanyothernodeofthetree.Thiscouldonlyhappenforthefoursets;,2N,f0Ngandf1Ng.Themostusefulcharacterizationofsubshiftsisinthetermofforbiddenwords. 29

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1.AsetFadmitsfactorialreductionifthereexistW;U2FsuchthatUisafactorofW.LetWbethesmallestwordthathasafactorinF.Thendeneaone-stepfactorial-reducedderivativeofF,FD(F),asFD(F)=FfWg.IfFdoesnotadmitfactorreductionletFD(F)=F. 2.AsetFadmitsrightsimplicationifUa2F,wherea2isaletter,andforallotherlettersb6=athereexistsasuxZofUsuchthatZb2F.LetUabethesmallestwordthatinducesarightsimplication.Thendeneaone-stepsimplicationderivativeofF,SD(F),asSD(F)=(FfUag)[fUg.IfFdoesnotadmitanysimplicationletSD(F)=F. Proof. ConverselysupposeFdoesnotadmitrightsimplication,butTF=fW22:WhasnofactorinFghasadeadend.LetUbethesmallestdeadendsuchthatUhasnoimmediateextensioninTF.ThusUa2Fforalllettersa2.HenceFadmitsrightsimplication,whichisacontradiction. Moreoverthefollowingusefulfactcanbeexpressed:

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Proof. 31

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Proof. 16 ,2F.SinceFdoesnotallowfactorialreduction,F=fg. Proof. 16 Nowwewouldliketoknowwhetherwecanstartwithanarbitrarysetofforbiddenwords,F,andderivebothasimplesetFS,andareducedset,FRbyasequenceofone-stepreductionssuchthat:[TF]=[TFR]=[TFS]orSF=SFR=SFS.Firstwegivesapositiveanswerforanitesetofforbiddenwords,andthenweinvestigateinnitesetsofforbiddenwords. Onecanhaveachainofderivatives,thatis,iterationsoftheformFDn+1(F)=FD(FDn(F))orSDn+1(F)=SD(SDn(F)). Proof. 15 ,SF=SFS.StartingfromFS,recursivelycomputeFDj+1(FS)forj0.SinceCard(FS)isnite,forsomemCard(FS)wearrivedatthereducedsetFR=FD(FR)=FDm(FS).Oncemorebyproposition 15 ,haveSFS=SFR. 32

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Proof. 3 ,SFisemptyifandonlyif2FR. Considerthefollowingexample: 0 1 0 1 0 1 1 1 Proof. 33

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Ingeneralgivenanycomputablesetofforbiddenwords,itsreducedsetofforbiddenwordsmaynotbecomputable.Furthermorewehavethefollowingundecidabilityoftheemptinessproblem: Proof. Thesituationisevenmorecomplicated,inthatthereducedsetFRmaynotbecomputablyenumerableevenwhenFiscomputable. Wehavethefollowingbasicfacts. 1.SF=SFR=SFT.MoreoverF;FRFT. 2.FT=fW:WhasafactorinFRg. Proof. ThefactthatSF=SFR=SFSarisesasacorollarytoProposition 15 .Bylemma 5 ,SF=SFTandFFT.SupposeW2FR,thenW=2SFR=SF,henceW2FT. 2. ThisisanimmediatecorollarytoProposition 16 34

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ConverselysupposeW=2SnFn;T.Thenforeachn,W2L(SFn),sothereexistsxn2SFnwithfactorW.SinceSFnisshift-subinvariant,withoutlossofgenerality,wecanassumethatWxn.BycompactnesshasalimitpointxandWx.Itremainstoshowthatx2SF.Ifnot,thenxhassomefactorU2F.SoU2Fnforallsucientlylargen.NowUafactorofxdkforsomekandforsucientlylargen,xndk=xdk,whichwouldmakeUafactorofxn,contradictingxn2SFn. 1.IfFisnite,thenFTiscomputable. 2.IfFiscomputablyenumerablethenFTisalsocomputablyenumerable. Proof. 17 ,givenanyW22,ifWhasafactorinFRthenW2FT,andotherwiseW=2FT.HenceFTiscomputable. 2. 35

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Proof. 16 ,W2FTifandonlyifWhasafactorinF.WeonlyneedtocheckthisforthenitesetoffactorsofW.ThusFTiscomputable. Conversely,supposeFTiscomputable.WeknowFTkF.OrdertheelementsofFTrstinlengthandthenlexicographically.Enumeratetheshortestword,W1,intoF.GivenW1;:::;Wn2F,letWn+1bethesmallestwordofwhichnoneisafactor.WeclaimisreducedandhenceequalsF.Clearlydoesnotadmitfactorreduction.NowsupposeUa2andletUa=Wn.Supposeforallb6=a,thereisasuxzofUsuchthatZb2.TheneachUaandZbisinFT,andthusbyProposition 15 ,U2FT.NowUahadnoproperfactorin,soU=2,butalsoUhasnoproperfactorinsothealgorithmwouldhaveputUintothelist.Thiscontradictionshowsthatdoesnotadmitrightsimplication. 36

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Proof. 20 .Since,FTisalsocomputablyenumerable,thenL(SF)isco.ce.Hence,SFisa01class,thesetofinnitepaththroughthetreeL(SF). Moreover 37

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Proof. ButonecanhaveacomputablereducedsetofforbiddenwordsFsuchthatSFisadecidable01subsimilarclassandhasnoperiodicelement.Beforegivingsuchanexample,Letusintroducesometerminologythatareusedinthesection.xisrecurrentifanyW2L(x)occursinnitelyofteninx.Aninnitewordx2Nisuniformlyrecurrent,ifthereexistsarecurrencefunctionh:!!!suchthatifw;v2L(x)thenwisafactorofvifjvjh(jwj). 1.Foralln,L(x)=L(n(x)) 2.IngeneralgivenanyS2N,ifyisalimitpointofS,thenL(y)L(S).IndeedletybealimitpathofS.ForanyfactorUofy,Uisafactorofsomeydn.Thentheremustbex2Swithxdn=ydnbydenitionoflimitpoint.HenceU2L(x),soU2L(S).ThusinparticularL(y)L(x).LetW2L(x).ThereexistsawordU2L(y)suchthatjUjh(jWj).HenceWisafactorofU,andthusbelongstoL(y) 38

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Notethatifforalln,L(x)=L(n(x)),thisdoesnotentailthatxisuniformlyrecurrent.Asacounterexamplewehavethefollowing: 1.xisuniformlyrecurrent. 2. Proof. O(x),forbiddinganyotherwordtogetapropernonemptysubclasswouldnotbepossible.Supposethatxisnotuniformlyrecurrent.ThenforsomeW2L(x),xhasarbitrarilylargefactorsUnwithjUnjnnothavingWasafactor.LetL=L(fU1;U2;:::g)(L(x).Thenbyproposition 1 ,thereexistsysuchthatL(y)LanddoesnotcontainW.Itfollowsthat Becauseoftheabovefacts,wehave: 1.Forallx2P,xisuniformlyrecurrent. 2.L(P)=L(x),forallx2P 39

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38 ]: O(f)hasnopropersubshift.Thiswouldimmediatelyentailthatnoelementof FRiscomputable,andSFR= Proof. 1.Ifw=2TpthenthereexistsssuchthatwhasnoextensioninTs(bycompactness). 2.Ifw2Tp,thenthereexiststsuchthatwisasubwordofallwordsoflengthtinT.Thatis,wemusthaveWafactorofeveryx2P,orelsewecouldforbidWtogetapropersubshift.Henceeveryx2Pbelongstosomeopenset[UWV].Againusingcompactness,nitelymanyofthesecoverP.IfSisthemaximumlengthofUWVfortheseopensets,theneverywordinTp\ShasWasafactor.Thosewordsin 40

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Thusitisanitetesttoseewhetherw2Tpornot.Hencedenetherecurrencefunctionasfollowsh(m)=maxft:w2Tp;jwj=m;wappearsasasubwordofallwordsoflengthtg 21 ,Fwouldbecomputable.Contradiction.EventhoughSFisanundecidable01class,butithasacomputableelement,namely10N. Proof.

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Itisimportanttonoticethatgivenanywordwoflength2k,whasatmostk+1distinctfactorsoflengthk.Sincethereare2kwordsoflengthk,forklargeenoughsothat2k>k+1,therearewordsoflengthkthatdonotappearasafactorofw.Withthisinmind,weconstructrecursivelytwosequencesofwordsn2Nandn2Nsuchthat,foralln: 1.jAnj=jBnj=ln; 2.An6=Bn; 3.AnandBnarenotfactorsofwn;thisispossibleforn=0sincejw0j=6sow0hasatmost4distinctfactorsoflength3. 4.An+1andBn+1aretakenfromfAn;Bng,haveAnasaprex,andhavelengthm=22n+4=ln+1=ln.Thisispossiblesincethereare2m1suchwords,butthereareatmostln+1+1factorsoflengthln+1inwn+1and2m1=ln+1+1+2.Tocheckthis,notethatln+1+3=3(2n2+3n)+32n2+3n+22m1,sincem1=22n+4122n+3n2+3n+2foralln. NowletA=limnAn.Thisexistssinceeach[An][An+1]andbycompactness\n0An=fAg Notethatthesequenceofnaturalnumbersthatweconstructedisnotunique.Infactanyothersequencethatdominatesthissequencealsosatisfythelemma 9 42

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Proof. 9 9 ,weclaimxalsoavoidsF.Supposenot,thenthereisawordUmthatappearsasafactorofx.SinceWmisaninitialsegmentofUmitensuesthatWmisafactorofx,contradiction. Notethatinfactanysubsequenceof6(2k(k+3))dominates6(2k(k+3)),sosatisesthelemma 10 Proof. 9 .Let0;1;:::;e;:::beanenumerationofthepartialcomputablefunctions.Nowdenethepartialrecursivefunctionby(k)=8>><>>:kdlk;ifk(i)#foralli
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9 ,weshowedthereexistsx2fA0;B0gNthatbelongstoSF0.SincewehavetwodierentpairsforA0;B0,wegetatleasttwodierentelementsx;y2SF.Herex2f101;111gNandy2f010;011gN.LetF1=f00g[F0.Clearlyx2SF1,buty=2SF1.NowtoaccomplishthisforanyK,Partitionthesequenceofnumbersofthelemma 9 ,intoanitesequenceandtherestofthenumbers.Usethenewinnitesetofnumbersfortheconstructionofthe01class,andusethearbitrarynitewordsfW0;:::;Wk1gsuchthatjWij=litogetanitedescendingchain.FortheeaseofcomputationwecanassumeW0=0l0,W1=1l1,W2=(01)l2,...,Wk=Ulk1,whereUisthekthnitewordintheordinarylength-lexicographicorderingoftheelementsof2. Therearesomeimmediateusefulcorollariestothetheorem 27 Proof. 27 hasone,sayz.

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2n(P).Thus(P)2n1 2n(P),hence(P)=0. Asaresultofthis: 27 weget:

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Thenthereexistsa01subsimilarclassQsuchthatP=SFMQandQMP. Proof. 10 ,weknowPisanonempty01subsimilarclass.Weneedtondanx2PandconstructQsuchthatxdoesnotcomputeanyelementinQ.ThatistosayFQforbidsallxe.Theconstructionproceedsatthestages,0,1,...,s+1,...Atstages+1,wehavesomeapproximationxs+1ofxandanapproximationofFs+1QofFQ.Asitisgoingtobeseenshortly,foralls+11wehavethefollowing: 9 ,wechoosetwoarbitrarydistinctwords,As,Bs,oflengthjUsj Asitmightbeexpectedxwearetryingtondislimxs=As.AndalsosincealwaysanewforbiddenwordenumeratedinFhaslongerlengththanallthepreviouswords,wehavexsxs+1.SincejWejn2eandjUejn2e+1,itfollowsfromlemma8thatF1=F[fW0;W1;:::gisavoidable.LetQ=SF1.Supposethatforsomeewehave 46

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Proof. ToachieveourgoalweconstructtwocomputablyenumerablesetofforbiddenwordsFPandFQsuchthat Webreakdowntheseconditionsintoaninnitelistofrequirements: 9 asthelimitofAn. 47

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Atstages=0,F0P=;=F0Q.Weenterthestages+1withFsP,FsQ,andAs.WesayReneedsattentionifedl<2e;n>#forsomen1.WesaySeneedsattentionifAsedl<2e+1;n>#forsomen1.Whenarequirementneedsattention,wetakesomeactiontosatisfyit.Butbeforeweexplainhowtheactionstakeplace,letusexplainsomepriorityconnectionamongtherequirements. 9 thatthisrestraintcanberespected. 48

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2.ThesequenceofwordsAs0;Bs0;:::;Asn;Bsn;:::convergestoA0;B0;A1;B1;:::;An;Bn;:::.ThusthesequenceAssconvergestoalimitA=limnAn.Thisfollowsfrom(1)sinceAsnonlychangesforrequirementSjwithj>n. 3.Foreache,ifeistotalthentherequirementRiiseventuallysatised,andthereforePhasnocomputableelement.Itfollowsfrom(1)thatRiwilleventuallybethehighestpriorityrequirementneedingattention. 4.Foreache,ifAeistotal,thentherequirementSeiseventuallypermanentlysatised,andthereforeAe=2Qforanye.Asin(3),Sewilleventuallybethehighestpriorityrequirementneedingattention.OnceallhigherpriorityRihaveceasedreceivingattention,therewillbeaxedniterestraintontheuseofAinAe.OnceeAdl<2e+1;n>convergesforsomenewnwithusecontainedinAs,thenwearefreetotakeactiononSe. 49

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Proof. Againthroughoutthisproofwewouldusethesequenceofnaturalnumbers.Moreoverviaapairingfunctionwebreakthissequencedownintoacountableunionofinnitesequences.Letbeanenumerationofpartialcomputableoraclefunctions. Webreakdowntheseconditionsintoaninnitelistofrequirements: ThepathsAandCwouldbeapproximatedsimilartowhathasbeendoneallalongintheproofofthelemma 9 50

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Weenterthestages+1withFsP,FsQ,As,andCs.WesayRerequiresattentionifCsedl#.WesaySerequiresattentionifAsedl#.Whenarequirementneedsattention,wetakesomeactiontosatisfyit.Butbeforeweexplainhowtheactionstakeplace,letusexplainsomepriorityconnectionamongtherequirements. AgainPrioritygoesas:R0;S0;R1;S1;:::,therequirementswithhigherprioritycomerstinthelist.Re-requirementsareindependentofeachother.ThisisalsotrueamongSe-requirements.ButtherearesomeneconsiderationsaboutthewayReandSeinteract: Thustheconstructionatthestages+1canbesummarizedasfollows: 51

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Proof. 52

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Proof. Proof. Sincetheset,S,of(k)sisacomputableenumerablesetofforbiddenwordswhoseelementshavelengthswhichcomposeasubsetofn2!,itfollowsthatP=SFisanonemptysubsimilar01classsuchthatnoelementofPhasanyword(k)asafactor.Thenforanycomputableelementkof2Z,wk=kd[lk=2;lk=2]isdenedandisnotafactorofanyx2P,andhencek=2P.

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TherankofP,rank(P)isdenedastheleastordinalsuchthatD(P)=D+1(P).Thereisasimilardenitionofrankforelementsofaclosedset,P,thatarenotinitskernel,K(P).Takez2PK(P).Therankrank(z)istheleastordinalsuchthatz2D(P)D+1(P).Wewillshowthatanycountablesubshiftcontainsaneventuallyperiodicelement.Firstweconsideraveryusefulgaptheoremforthelength-distributionfunctionU(x;n),wherex2Nandn2N. 1.xiseventuallyperiodic. 2.U(x;n)isbounded. 3.U(x;n)
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ThisisagaptheoremsinceitsaysthereisnoU(x;n)whichisnoteventuallyconstantbutgrowsatmostasfastasn. Proof. Proof. O(x)isinniteandhencecontainsalimitpathy6=xifori=1;:::;n Proof. 55

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Proof. 36 therankof 1. ifx2(D(P)),then 2. Ifx2D((P)),then Nowx=limnxnwhere(P)andxn=ynwhereyn2P. Thenbycompactness,hasasubsequencethatconvergestosomez2D(P). 56

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Proof. TherankofaclosedPisneveralimitordinal: Proof. 57

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SincePisasubshift,wehaveP.Bycompactness,getasubsequencethatconvergestoy=WU1CN,whereU1;C22andWU1.Notethatify=x,thenU6=.Otherwise,U1==UandC=W.Thisisnotpossible,sinceaswesaidifU=thenforallnVn6=,andthusU16=. Nowgivenanym1,P.Hencezmnk!Wm1y.Notethatforallm,tm=Wm1yarealldistinctandhaaverankatleast.tm!WN=z.ThiscompletestheproofthattherankofPisgreaterthan+1. Wewouldliketoconstructacountablesubshiftthathasonlyeventuallyperiodicelementsanditsrankisnotnite.Firstweseehowtoachievethisforarbitraryniteranks.Thenwegivethenalconstruction. Proof.

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Proof. 1. 2. 3. TheelementsofPm+1areallthelimitpointsforthePm. 4. Alltheelementsoff0m10N:m0ghaverank!+1. 6. 0Nhasthehighestrankof!+2 Incomputabilitytheory01classesmaybeclassiedintermsoftheirsizeasMinimal,Smallorthin. 59

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Proof. Wecanweakenthenotionofminimalityfora01subsimilarclassasfollows: Proof. 60

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40 ].ForamorerecentexpositionofthisconstructionIalsorecommend[?]or[?]. ThenextmajorquestionwouldbetheK-TrivialityofSubsimilarClasses. LetQ2Nbeaclosedset.Qcanbecodedasanelementof3Nasfollows: 61

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ThenxP=2122121221221212212::::.Itmayseemcomplicated,butthisisexactlythebonacciinniteword,andcanbesimplydescribedasfollows: 62

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Notethat(iii)implies(ii)bycompactness. Proof.

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Foranycomputablex2N,theitineraryIt(x)iscomputable. (b) ThesetIT[F]ofitinerariesisadecidable,subsimilar01class. Proof. 12 Nextweprovetheconverse.NotethatF0(x)=xforallx2NandthereforeIT[F]meetseveryUi.NotethatifQisasubshiftandQdoesnotmeetJ[i],thenQf0;1;:::;i1;i+1;:::;kgN. Proof. Foranyx2Q,wehaveF(x)=(x)and(x)2Q,sinceQisasubshift.HenceFn(x)=n(x),sothatFn(x)(0)=x(n),sothatFn(x)belongstothesetUx(n).ThustheitineraryI(x)=x.ThisshowsthatQIT[F]. Nextconsideranyx2N.WewillshowbyinductionthatFn(x)=n(G(x))foralln>0.Forn=1,thisisthedenition.ThenFn+1(x)=(G(Fn(x)))=(G(n(G(x))));

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Proof. NotethatTiscomputablesincetherewillbeaboundonthelengthofthewi.ItfollowsfromthedenitionofTthaty2[T]ifandonlyif,foreachn,thereexistsnitesequence(wn;:::;w1;w0;w1;:::;wn)suchthatf(wi)wi+1andwi(0)=y(i)i2fn;:::;n1g.CertainlyITZ(F)[T].TheotherinclusionfollowsfromthecompactnessofXZ.Thatis,xy2[T]andletKnbethesetofbi-innitesequences(:::;x1;x0;x1;:::)2XZsuchthatforni
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Proof. 45 .WedeneG:Z!QsuchthatG(x)isthenearestelementinQtox.Thatis,theapproximatingfunctiongmapsxd[n;n]toxd[n;n]aslongasxd[n;n]2TQandwhennistheleastsuchthatxd[n1;n+1]=2TQ,g([nk1;n+k+1])isthelexicographicallyleastextensionofxd[n;n]whichisinTQ.ThenweletF(x)=(G(x)).Onceagainforx2Q,wehaveF(x)=(x),sothat(:::;1(x);x;(x);2(x);:::)hasitineraryxandthereforeQITZ(F).Ontheotherhand,F(x)2Qforallx,sothatforanybi-innitesequencez=(:::;x1;x0;x1;:::)withF(xi)=xi+1foralli2Z,wemusthaveeachxi2Qandthereforetheitineraryofzisx02Q.ThusITZ(F)Qaswell. ThefunctiongiveninTheorem 47 aboveisingeneralnotone-to-oneoronto,andwedon'tknowifaone-to-oneand/orontofunctioncanbegiven. 67

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Thischapterhastwosections.Intherstsectionwediscusstheshift-maximalwordsandadmissiblesequences.Thesecondsectionthatishighlydependentontherstoneosaboutthecomputabilityaspectsofsymbolicdynamicsformappingsontheunitinterval. (i) Ifw1w2,thenw1w2andviceversa. (ii) Otherwise,letubethelargestcommonprexofw1andw2andletjuj=m.Ifuiseven,thenw1w2(m). Thisparitylexicographicorderingisalinearorderingbutnotwell-ordering. Theparitylexicographicorderingcanalsobedenedforinnitewords.Giventwodistinctinnitewordsx;y2N,letWbetheirlargestcommonprexoflengthm,sayx
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62 ].Takeforexamplethebinaryalphabetf0;1g.LetS1=100andSn+1=SnCnwhereCn=11orCn=101.EachSnisashiftmaximalword.Nowtakeanys2S1f11;101gN,clearlys=limSnforsomesequenceoffSn:SnSn+1foralln1g,andthusisshiftmaximal. Thereisadierentwayofshowingthereareuncountablymanyshift-maximalwords:GivenanyL2Nf1g.LetbyanysubsequenceofNf1;2;:::;Lg,then101L01n001n101n20:::1nk0:::isshift-maximal.Thus10120130140:::1n0:::and10120(130)Nareshift-maximal. Letx2Nbeanyinniteword.Denethesetofadmissibleelementswithrespecttox,Adm(x),asfollows:Adm(x):=fy22N:i(y)xforallig Wehavethefollowingfact Proof.

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Mostlyweareinterestedintheadmissiblesequenceswithrespecttoshiftmaximalinnitewords: 5. 70

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ThenweshowthatifW22hasthepropertyP,theneitherW0orW1hasthisproperty.SupposeW0doesnothavethepropertyP.Thusforsome0jjW0j1=jWj,V0=j(W0)>,whereVisanonemptyword.NotethatifVwereempty,then0>,whichisnotpossible.BecauseV,haveV.VandV0>togetherimplythatVisodd,and(jVj+1)=1or(jVj+1)=c.Inotherwords=V1xor=VcxforsomeoddwordVandsomex2N.NotethatnowW=UV 1.Visasuxofs.Thatiss=tV.Thus=Vax=tVbywherea2f1;cgandb2f0;cg.a=c=bcannothappensincejtVj>jVjandtV22.SobVax=,contradictingthefactthatisshift-maximal. 2.sisasuxofV.ThatisV=ts.Thus=tsax=sby,wherea2f1;cgandb2f0;cg.a=c=bcannothappensincejtsj>jsjandts22.Sobsby=,whichisacontradiction,sinceisshift-maximal. HenceeitherW0orW1hasthepropertyP.ThuswegetachainofwordsW0=WW1W2::::suchthatforalliWihasthepropertyP.Thusx=limWiisadmissiblewithrespectto,andWisafactorofit.ThusW2L(Adm()).Thiscompletestheproof. Notethatsomeoftheclausesinthispropositionfailingeneral.Forexample,Letx=1011100ywherey22N.W=101110satisestheconditionthati(w)xforall0ijWj1.ButW0andW1donothavethisproperty.Since4(W0)=100>x

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Nowwearereadytodiscussthecomputabilitysideofthestory. Proof. Proof. Thefollowingexampleshowsthattheinclusioninthetheoremcanbeproper:LetQ:=f1n0N:n0gSf1Ng,clearlyQisacountablesubsimilar01class.(Q)=Q.Takex=10N,thenAdm(x)=2N,andclearlyQ(2N

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Anexampletoshowtheinclusionisproper Itisnotthecasethateverydecidable,subsimilar01classQequalsAdm(x)forthemaximalelementxofQ.Forexample,ifx=10!,thenAdm(x)=2N.However,forF=f111gandQ=SF,wehavex2QandthusxisthemaximalelementofQ,butcertainlyQ6=Adm(x)=2N. Wealsoneedtorecallsomedenitionsandfactsofthedynamicsoftheunimodalmapsthedetailsofwhichcanbefoundinreference[ 43 ]. 1.F(c)istheuniqueabsolutemaximumofF. 2.Fisstrictlyincreasingovertheinterval[0;c)andisstrictlydecreasingover(c;1].

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2. TheQuadraticfamily:F(x)=x(1x)where04 1.Thetentfamily:F(x)=sx0x1 2;s(1x)1 2x1; ItiswellknownthatF=fF:[0;1]![0;1];Fcontinuousgwithuniformconvergencetopologyisapolishspace.ItcanbeseenthatthecollectionofunimodalmapsasdenedaboveisaGsubset.Indeed,givenany0f(q4)gisanopensubsetofF.HenceF:=funimodalF:[0;1]![0;1]g=\q1>0\1 2>q2>q1\q3>1 2\1>q4>q3F(q1;q2;q3;q4) isaGsubset. 43 ]isoneoftheearliestworksthatsetsforthsomecombinatorialandsymbolicdynamicsapproachtowardsthestudyofunimodalmaps.Thisapproachassignsanaddressandanitinerarytoanypointof[0;1]asfollows: 74

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AnexampletoshowthatI[F]neednotbeclosed Givenz2[0;1],theaddressofz,A(z),isdenedasensues:A(z)=8>>>><>>>>:1z>1 2;cz=1 2;0z<1 2 Hencethespaceofitineraries,IT[F],oftheelementsoftheinterval[0;1]isasymbolicsubspaceofX=f0;1;2g!.Wealsoareinterestedinthesubsetofitinerariesin2N,thatis,I[F]=IT[F]\2N 62 ]andshowsthatI[F]neednotbeclosed. 75

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AnexampletoshowthatI[F]canbeclosed Indeeds=010Nisnotanitineraryofanypointintheunitinterval.Butontheotherhand,givenanyn2NthereisapointzsuchthatIt(z)=010nywherey22N. Anothergoodexampleisthefollowingfunction: NotethatIt(F(1=2))isagain10N.ButitcanbeshownthatI[F]=2N,andthusclosedinthiscase.AsitcanbeseenthelowestcomplexityofI[F]dependsprimarilyonthefunctionathand.But,inthegeneralcase,itcanbeseenthatI[F]isananalyticset.IndeedletB:=fx2[0;1]:fn(x)=cforsomeng.Biscountable,soisF,andthusA=[0;1]BisaGset.Leth:A!2Nbedenedasx7!It(x).Notethathisacontinuousfunction,andsinceAisapolishspace,bydenitionh(A)=I[F]isanalytic. ThemostimportantitineraryofaunimodalmapFisitskneadingsequence,KS(F),whichisIt(F(1 2)).Notethatifc2L(KS),thenKSisaperiodicelementofN=f0;1;cgN.ThereisaconnectionbetweenthekneadingsequenceandthesetI[F]bywayoflexicographic-paritylinearorderingonf0;1;cg. GivenanyunimodalmapF,wehavetheusefulfactthatforallx;y2[0;F(c)],It(x)It(y).Formoredetailssee[ 45 ].Thispropertyentailsthefollowingtworesults 76

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Grammaticalhierarchyofthelanguageofkneadingsequences ThelanguageofaunimodalmapFisdenedasfollows: MoreonL(KS): Moreoveritcanbeshownthat[ 62 ] 77

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52 ,thereexistssn2I0[F],suchthatsndn=sdn.Hencelimsn=s. 62 ].ForacontinuousfunctionF,letAdm(F)denoteAdm(KS(F)). Proof. 2gandR(x)=fn:Fn(x)>1 2arebothc.e.sets.SupposerstthatFn(x)6=1 2foranyn.ThenL(x)andR(x)arecomplementsandhencebothsetsarecomputable.ThenI(x)(n)=0()n2L(x),sothatI(x)iscomputable. Fortheothercase,werstconsiderthekneadingsequence.IfFn(1 2)=1 2forsomen,thenKS(F)isperiodicandcertainlycomputable.ThusKS(F)iscomputableinanycase.Nowforarbitraryx2[0;1]suchthatFn(x)=1 2forsomen,thenI(x)=I(x)dn+1_KS(F)andisthereforecomputablesinceKS(F)iscomputable. WehavethefollowingcorollarytoTheorems 50 and 55 78

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Fortherestofthesection,weconnethediscussionofunimodalmapstothequadraticmapsF(x)=x(1x).Theseformaso-calledfullfamily,sothat,by[ 62 ],wehave 2)>1 2,sothatthereexistpointsx02(0;1 2)andx12(1 2;1)suchthatF(x0)=1 2=F(x1)andtherefore0CI(x0)and1CI(x1).Ingeneral,Fmayhavekthorderinversesof1 2forallk. ForthesurjectivequadraticmapF4(x)=4x(1x),itisclearthatGhas2kk-thorderinversesof1 2forallk.Forexample:00c,10c,11c,01c,theseareallinversepathoflength2.(Considerthefollowinggraph) Henceforanyw20;1,thereexistsxsuchthatwCI(x);wewillsaythatthisxisthecoordinateofthepathwC.ForthesurjectivequadraticmapF=F4,wesaythatwCisalegalinversepath(l.i.p.)ifthecoordinater2[0;1]ofthepathisthegreatestnumericalvalueofanypointonthepath,thatis,ifFn(r))nmforallm>k.Henceisamonotoneandboundedsequencesoitconverges. 79

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Legalinversepath 2.xhasnitelymany0's.Againletmbetheplacewherethelastzeroinxoccurs.Letnk=m+kfor1k.ThusHencexn_kc=W1kcforsomeW2f0;1g.Observethatnk+2alwaysfalls"strictly"betweennkandnk+1.Henceconverges. 3.xhasinnitelymanyonesandzeros.Writexas0l010l110l21:::.Letxnk=0l010l11:::0lk110lk1.Thenobservethatasituationsimilartothepreviouscasehappens.Indeednk+2falls"strictly"betweennkandnk+1.Henceconverges. Metropolisetal[ 43 ]providesthefollowingcrucialfact.

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Wenextgiveaconditionforanitewordw2f0;1gwhichwillimplythatwCisshift-maximalifwisshift-maximal.Supposethatwisshift-maximalbutwCisnotshift-maximal.Thensomei(wc)>wc;leti(wc)=uCandletv=wdi,sothatw=vu.ThenwCn
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56 ,wemayassumethatn+2isbetweennandn+1,sothatthelimitisalsocomputable. Someopenquestionstoconsider: 82

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