<%BANNER%>

Closed Sets, Continuous Functions, and Symbolic Dynamics in the Arithmetic Hierarchy

Permanent Link: http://ufdc.ufl.edu/UFE0045357/00001

Material Information

Title: Closed Sets, Continuous Functions, and Symbolic Dynamics in the Arithmetic Hierarchy
Physical Description: 1 online resource (59 p.)
Language: english
Creator: Wyman, Sebastian P
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: analysis -- capacity -- computablilty -- dynamics -- effective -- symbolic
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: CapacitycompdynamicsIn the study of computable functions on the Cantor space, it is well-known that the image of such a function is an effectively closed set or Pi-0-1 class, and in fact a decidable closed set.  Here a closed subset Q of the Cantor space is decidable if the set of finite strings w which have an extension in Q is a computable set. It was shown recently by Cenzer, Dashti, and King that the set of itineraries of a computable function is also a decidable closed set. Now the set of itineraries of a continuous function is always a subshift, meaning that it is closed under prefixes. It was also shown that any decidable shift is the set of itineraries of some computable function on the Cantor space. We define the new notion of a conservatively approximable function on the Cantor space in order to have a family of functions whose images, and itineraries, can be arbitrary effectively closed sets.  Cenzer, Dashti, and King also showed that the subshifts obtained by forbidding a c.e. set of words are always Pi-0-1. We give a similar characterization of the decidable subshifts. Because the unit interval is a connected set, functions on the unit interval can not produce the subshift correspondences given above through itineraries.  We define a multivalued symbolic dynamics and show that computable Markov functions have decidable itineraries and that many computationally simple subshifts are not itineraries of any computable function. We investigate the connection between measure and capacity for the space C of nonempty closed subsets of the Cantor space.  For any computable measure $\mu^*$, a computable capacity T may be defined by letting T(Q) be the measure of the family of closed sets K which have nonempty intersection with Q. We establish conditions that characterize when the capacity of a random closed set equals zero or is >0. We construct for certain computable measures on the Cantor space, an effectively closed set with positive capacity and with Lebesgue measure zero. We show that for a broad family of computable measures, a real q is upper semi-computable if and only if there is a Pi-0-1 class Q with capacity q.
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 Sebastian P Wyman.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Cenzer, Douglas A.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045357:00001

Permanent Link: http://ufdc.ufl.edu/UFE0045357/00001

Material Information

Title: Closed Sets, Continuous Functions, and Symbolic Dynamics in the Arithmetic Hierarchy
Physical Description: 1 online resource (59 p.)
Language: english
Creator: Wyman, Sebastian P
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: analysis -- capacity -- computablilty -- dynamics -- effective -- symbolic
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: CapacitycompdynamicsIn the study of computable functions on the Cantor space, it is well-known that the image of such a function is an effectively closed set or Pi-0-1 class, and in fact a decidable closed set.  Here a closed subset Q of the Cantor space is decidable if the set of finite strings w which have an extension in Q is a computable set. It was shown recently by Cenzer, Dashti, and King that the set of itineraries of a computable function is also a decidable closed set. Now the set of itineraries of a continuous function is always a subshift, meaning that it is closed under prefixes. It was also shown that any decidable shift is the set of itineraries of some computable function on the Cantor space. We define the new notion of a conservatively approximable function on the Cantor space in order to have a family of functions whose images, and itineraries, can be arbitrary effectively closed sets.  Cenzer, Dashti, and King also showed that the subshifts obtained by forbidding a c.e. set of words are always Pi-0-1. We give a similar characterization of the decidable subshifts. Because the unit interval is a connected set, functions on the unit interval can not produce the subshift correspondences given above through itineraries.  We define a multivalued symbolic dynamics and show that computable Markov functions have decidable itineraries and that many computationally simple subshifts are not itineraries of any computable function. We investigate the connection between measure and capacity for the space C of nonempty closed subsets of the Cantor space.  For any computable measure $\mu^*$, a computable capacity T may be defined by letting T(Q) be the measure of the family of closed sets K which have nonempty intersection with Q. We establish conditions that characterize when the capacity of a random closed set equals zero or is >0. We construct for certain computable measures on the Cantor space, an effectively closed set with positive capacity and with Lebesgue measure zero. We show that for a broad family of computable measures, a real q is upper semi-computable if and only if there is a Pi-0-1 class Q with capacity q.
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 Sebastian P Wyman.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Cenzer, Douglas A.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045357:00001


This item has the following downloads:


Full Text

PAGE 1

CLOSEDSETS,CONTINUOUSFUNCTIONS,ANDSYMBOLICDYNAMICSINTHEARITHMETICHIERARCHYBySEBASTIANWYMANADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

PAGE 2

c2013SebastianWyman 2

PAGE 3

ACKNOWLEDGMENTS Iwishtothankallthosewhohavehelpedmethoughthisprocess.IthankDr.Cenzerandtherestofmycommitteefortheiracademicsupport.IthankthestaffoftheMathematicsDepartmentandoftheUniversityforhelpinnavigatingandsurvivingtheprocess.Ithankmyfriendsformakingitlivable.Ithankmyparentsforalltheirsupport.Finally,Ithankmyanceeforeverything. 3

PAGE 4

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 3 ABSTRACT ......................................... 5 CHAPTER 1BACKGROUND ................................... 7 1.1Introduction ................................... 7 1.2ComputabilityontheNaturalNumbers .................... 11 1.3TreesandtheCantorSpace .......................... 14 1.4ComputableAnalysis .............................. 14 1.4.1SpacesofInterest ........................... 14 1.4.2NamingSystemsandInducedComputability ............ 15 1.4.3ComplexityClasses .......................... 16 1.5SymbolicDynamics .............................. 17 1.6CapacityandRandomClosedSets ...................... 19 1.6.1ThespaceofClosedSets ....................... 19 1.6.2MeasuresonC ............................. 20 1.6.3Capacity ................................. 20 1.6.4AlgorithmicRandomness ....................... 21 2EFFECTIVESYMBOLICDYNAMICSINTHECANTORSPACE ......... 23 2.1WeakeningComputability ........................... 24 2.2DecidableSubshifts .............................. 35 3EFFECTIVECONTINUOUSFUNCTIONSON[0,1] ............... 38 3.1MultivaluedSymbolicDynamics ........................ 38 3.2ConservativelyApproximableFunctions ................... 43 4CAPACITY ...................................... 46 4.1CapacityandRandomness .......................... 46 4.2CapacityofEffectivelyClosedSets ...................... 51 REFERENCES ....................................... 55 BIOGRAPHICALSKETCH ................................ 59 4

PAGE 5

AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyCLOSEDSETS,CONTINUOUSFUNCTIONS,ANDSYMBOLICDYNAMICSINTHEARITHMETICHIERARCHYBySebastianWymanMay2013Chair:DouglasCenzerMajor:MathematicsInthestudyofcomputablefunctionsontheCantorspace2N,itiswell-knownthattheimageofsuchafunctionisaneffectivelyclosedset,or01classandinfactadecidableclosedset.HereaclosedsubsetQoftheCantorspaceisdecidableifthesetofnitestringswwhichhaveanextensioninQisacomputableset.ItwasshownrecentlybyCenzer,Dashti,andKingthatthesetofitinerariesofacomputablefunctionisalsoadecidableclosedset.Nowthesetofitinerariesofacontinuousfunctionisalwaysasubshift,meaningthatitisclosedunderprexes.Itwasalsoshownthatanydecidableshiftisthesetofitinerariesofsomecomputablefunctionon2N.WedenethenewnotionofaconservativelyapproximablefunctionontheCantorspaceinordertohaveafamilyoffunctionswhoseimages,anditineraries,canbearbitrary01classes.Cenzer,Dashti,andKingalsoshowedthatthesubshiftsobtainedbyforbiddingac.e.setofwordsarealways01.Wegiveasimilarcharacterizationofthedecidablesubshifts.Because[0,1]isaconnectedset,functionsontheunitintervalcannotproducethesubshiftcorrespondencesgivenabovethroughitineraries.WedeneamultivaluedsymbolicdynamicsandshowthatcomputableMarkovfunctionshavedecidableitinerariesandthatmanycomputationallysimplesubshiftsarenotitinerariesofanycomputablefunction.WeinvestigatetheconnectionbetweenmeasureandcapacityforthespaceCofnonemptyclosedsubsetsof2N.Foranycomputablemeasure,acomputablecapacity 5

PAGE 6

TmaybedenedbylettingT(Q)bethemeasureofthefamilyofclosedsetsKwhichhavenonemptyintersectionwithQ.Weestablishconditionsthatcharacterizewhenthecapacityofarandomclosedsetequalszerooris>0.Weconstructforcertaincomputablemeasureson2N,aneffectivelyclosedset(01)classwithpositivecapacityandwithLebesguemeasurezero.Weshowthatforabroadfamilyofcomputablemeasures,arealqisuppersemi-computableifandonlyifthereisa01classQwithcapacityq. 6

PAGE 7

CHAPTER1BACKGROUND 1.1IntroductionThereisalonghistoryofinteractionbetweencomputabilityanddynamicalsystems.ATuringmachinemaybeviewedasadynamicalsystemwhichproducesasequenceofcongurationsorwordsbeforepossiblyhalting.Thereversenotionofusinganarbitrarydynamicalsystemforgeneralcomputationhasgeneratedmuchinterestingwork.Seeforexample[ 4 26 ].Thestudyofalgorithmicrandomnesshasbeenanactiveareaofresearchinrecentyears.Thebasicproblemistoquantifytherandomnessofasinglerealnumber.Herewethinkofarealr2[0,1]asaninnitesequenceof0'sand1's,i.easanelementin2N.Therearethreebasicapproachestoalgorithmicrandomness:themeasure-theoreticapproachofMartin-Loftests,theincompressibilityapproachofKolmogorovcomplexity,andthebettingapproachintermsofmartingales.Allthreeapproacheshavebeenshowntoyieldthesamenotionof(algorithmic)randomness.Here,weconsideronlythemeasure-theoreticapproach.ArealxisMartin-LofrandomifforanyeffectivesequenceS1,S2,...ofc.e.opensetswith(Sn)2)]TJ /F11 7.97 Tf 6.59 0 Td[(n,x=2\nSn.Forbackgroundandhistoryofalgorithmicrandomnesswereferto[ 23 33 ].ThestudyofcomputabledynamicalsystemsispartoftheNerodeprogramtostudytheeffectivecontentoftheoremsandconstructionsinanalysis.Weihrauch[ 41 ]hasprovidedacomprehensivefoundationforcomputabilitytheoryonvariousspaces,includingthespaceofcompactsetsandthespaceofcontinuousrealfunctions.Computableanalysisisrelatedaswelltotheso-calledreversemathematicsofFriedmanandSimpson[ 39 ],whereonestudiestheproof-theoreticcontentofvariousmathematicalresults.Thestudyofreversemathematicsisrelatedinturntotheconceptofdegreesofdifculty.HerewesaythatPMQifthereisaTuringcomputablefunctionalFwhichmapsQintoP;thustheproblemofndinganelementofPcan 7

PAGE 8

beuniformlyreducedtothatofndinganelementofQ,sothatPislessdifcultthanQ.SeeMedvedev[ 29 ]andSorbi[ 40 ]fordetails.Thedegreesofdifcultyofeffectivelyclosedsets(alsoknownas01classes)havebeenintensivelyinvestigatedinseveralrecentpapers,forexampleCenzerandHinman[ 14 ]andSimpson[ 38 ].ThecomputabilityofJuliasetsintherealshasbeenstudiedbyCenzer[ 10 ]andKo[ 28 ].ThecomputabilityofcomplexdynamicalsystemshasbeeninvestigatedbyRettingerandWeihrauch[ 35 ]andbyBravermanandYampolsky[ 5 ].Thestudyofthecomputabilityofdynamicalsystemshasreceivedincreasingattentioninrecentyears;seeforexamplepapersofDelvenneetal[ 21 ],Hochman[ 26 ],Miller[ 30 ]andSimpson[ 37 ].ComputablesubshiftsandtheconnectionwitheffectivesymbolicdynamicswereinvestigatedbyCenzer,DashtiandKing[ 13 ]inarecentpaper.Atotal,TuringcomputablefunctionalF:N!Nisalwayscontinuousandthuswillbetermedcomputablycontinuousorjustcomputable.Effectivelyclosedsets(alsoknownas01classes)areacentraltopicincomputabilitytheoryandtheyariseinmanyapplicationsofcomputabilitytheory.Forexample,thesetofzeroesofacomputablefunctionF:N!Nisalwaysa01class,andconverselyevery01classisthesetofzeroesofsomecomputablefunctionF.Nonemptydecidable01classesalwayshaveacomputablememberandareoftenoflessinterestthanarbitrary01classes,buttheyplayanimportantroleineffectivesymbolicdynamics.See[ 16 17 ]formanyotherapplicationsof01classes.Itwasshownin[ 13 ]that,foranycomputablycontinuousfunctionF:N!N,IT[F]isadecidable01classand,conversely,anydecidable01subshiftPisIT[F]forsomecomputablemapF.In[ 11 ],01subshiftsareconstructedin2Nandin2Zwhichhavenocomputableelementsandarenotdecidable.Thusthereisa01subshiftwithnon-trivialMedvedevdegree.J.Miller[ 30 ]showedthatevery01Medvedevdegreecontainsa01subshift.Simpson[ 37 ]studied01subshiftsintwodimensionsandshowed 8

PAGE 9

thatevery01Medvedevdegreecontainsa01subshiftofnitetypewhichisastrongerresultthanjustcontaininga01subshift.Cenzer,Dashti,andKingalsoinvestigatedthesymbolicdynamicsoffunctionsontherealsusingkneadingsequences.Noweverynonemptycountable01classcontainsacomputableelement,sothatallcountable01classeshaveMedvedevdegree0,andmanyuncountableclassesalsohavedegree0.Thepaper[ 12 ]considersmorecloselythestructureofcountablesubshifts,usingtheCantor-Bendixsonderivativetocompareandcontrastcountablesubshiftsofniterankwith01subshiftsofniterankaswellaswitharbitrary01classesofniterank.ItisshownthatifQisasubshiftofranktwo,theneverymemberofQiseventuallyperiodic(andthereforecomputable)andfurthermoreifQZ,thenthemembersofranktwoareperiodicandQisadecidableclosedset.However,thereareranktwosubshiftsinNofarbitraryTuringdegreeandranktwo01subshiftsofarbitraryc.e.degree,sothatranktwoundecidable01subshiftsexistinN.01subshiftsofrankthreecontainonlycomputableelements,but01subshiftsofrankfourmaycontainmembersofarbitraryc.e.degree.Foranygiven01classPofranktwo,thereisasubshiftQofrankfoursuchthatthedegreesofthemembersofPandthedegreesofthemembersofQareidentical.Moregenerally,forany01classPN,thereisa01subshiftQNsuchthatthenon-computabledegreesofthemembersofQareidenticalwiththenon-computabledegreesofthememberofP.Inaseriesofrecentpapers[ 2 3 7 8 ],G.Barmpalias,S.Dashti,R.Weberandtheauthorshavedenedanotionof(algorithmic)randomnessforclosedsetsandcontinuousfunctionson2N.Itwasobservedin[ 7 ]thatthereisanaturalisomorphismbetweenthespaceCofnonemptyclosedsubsetsoff0,1gNandthespacef0,1,2gN(withtheproducttopology)whichmapsaclosedsetQf0,1gNtoitscodexQ2f0,1,2gN.AmeasureonthespaceCofclosedsubsetsof2Ncanbedenedasfollows[ 7 ].(X)=(fxQ:Q2Xg) 9

PAGE 10

foranyXCandisthestandardmeasureonf0,1,2gN.Informallythismeansthatgiven2TQ,thereisprobability1 3thatboth_02TQand_12TQand,fori=0,1,thereisprobability1 3thatonly_i2TQ.Inparticular,thismeansthatQ\I()6=;impliesthatfori=0,1,Q\I(_i)6=;withprobability2 3.Brodhead,Cenzer,andDashti[ 7 ]denedaclosedsetQ2Ntobe(Martin-Lof)randomifxQis(Martin-Lof)random.Notethattheequalprobabilityof1 3forthethreecasesofbranchingallowstheapplicationofSchnorr'stheoremthatMartin-Lofrandomnessisequivalenttoprex-freeKolmogorovrandomness.Thenin[ 2 7 ],thefollowingresultsareproved.No01classisrandombutthereisarandom02closedset.Everyrandomclosedsetcontainsarandommemberbutnoteverymemberisrandom.Everyrandomrealbelongstosomerandomclosedset.Everyrandom02closedsetcontainsarandom02member.Everyrandomclosedsetisperfectandcontainsnocomputableelements(infact,itcontainsnon-c.e.elements).Everyrandomclosedsethasmeasure0.Arandomclosedsetisaspecictypeofrandomrecursiveconstruction,asstudiedbyGraf,MauldinandWilliams[ 24 ].McLindenandMauldin[ 19 ]showedthattheHausdorffdimensionofarandomclosedsetislog2(4=3).Justasaneffectivelyclosedsetin2NmaybeviewedasthesetofinnitepathsthroughacomputabletreeTf0,1g,analgorithmicallyrandomclosedsetin2NmaybeviewedasthesetofinnitepathsthroughanalgorithmicallyrandomtreeT.DiamondstoneandKjos-Hanssen[ 22 27 ]giveanalternativedenitionofrandomclosedsetsaccordingtotheGalton-Watsondistributionandshowthatthisdenitionproducesthesamefamilyofalgorithmicallyrandomclosedsets.TheeffectiveHausdorffdimensionofmembersofrandomclosedsetsisstudiedin[ 22 ].In[ 9 ]theauthorsexaminethenotionofcomputablecapacityanditsrelationtocomputablemeasuresonthespaceCofnonemptyclosedsets.AfamilyofcomputablemeasuresonCispresentedanditisshownthattheyinducecapacities.ThenotionofcomputablecapacitydenedandaneffectiveversionofChoquet'stheoremthatevery 10

PAGE 11

capacitycanbeobtainedfromameasureonthespaceofclosedsetsispresented.TheygiveconditionsunderwhichthecapacityT(Q)ofa-randomclosedsetQiseitherequalto0or>0whenisasymmetricmeasure.Inthisrstchapter,wegivebackgroundinformationanddevelopthenotionsneededfortherestofthedocument.Next,inchaptertwo,wenishtheworkstartedbyCenzer,Dashti,andKing[ 13 ]demonstratingcorrespondencesbetweencertainclassesoffunctionsonNorofforbiddenandtheinducedsymbolicdynamicsofdecidableor01subshifts.Todothis,wedenetheconservativelyapproximablefunctions.Inbothcases,wedemonstratethatthecorrespondingclassesmustbemuchmorestrictlydenedthaninthecasesconsideredin[ 13 ].Inthethirdchapter,wedenemultivaluedsymbolicdynamicsandshowthatcomputableMarkovfunctionson[0,1]inducedecidablesubshifts.Correspondingnotionsofconservativelyapproximablefunctionson[0,1]aredenedandinvestigated.Finally,inchapterfour,weextendtheresultof[ 9 ]tonon-symmetricmeasures(alsopublishedin[ 9 ]).Wealsodemonstrateaconnectionbetween01classesandupper-semicomputablerealsthroughcapacity. 1.2ComputabilityontheNaturalNumbers Denition1.2.1(Turingmachine). Letbeanitealphabet(usually=f0,1g)andconsiderabi-innitetapedividedintocountablymanycellsalongwithread/writeheadwhichcanreadandchangethesymbolwritteninacellandmoveleftorrighttoadjacentcells.ATuringMachineisapairM=(Q,),whereQ=fq0,q1,...,qngisanitesetofstatesand:Q!QfL,Rgisapartialfunction.Weintemperateatuple(q,s,q0,s0,X)tomeanthatamachineinstateqwithread/writeheadreadingsymbolswillchangethattosymbols0andmovetostateq0thenmovetheread/writeheadleftorrightifX=LorX=Rrespectively.Thisisexecutingasingleinstruction.Wesupposethatamachinealwaysstartsrunninginstateq1andhaltswhenitreachedthereservedstateq0. 11

PAGE 12

WesaythatMcomputesa(partial)functionf:!iff(w)=uifandonlyifthefollowingoccurs:Givenatapeonwhichwisprinted,withthereadheadattheleftmostcellofw,themachineMhaltswithuprintedonthetapeandtheread/writeheatattheleftmostsymbolofu.(Ineachcasethesymbolsofthewordareprintedonepercell,asingletime,withtherestofthetapeempty.)Inthiscase,wesaythatfis(partial)computable.A(partial)functionf:N!Nis(partial)computableiff0:1n7!1f(n)is.IfMcomputesfandMhaltswithinputnwewritef(n)#.IfMhaltsafterexecutingsinstructions,wesayM(orf)haltsatstagesandwritefs(n)#. Denition1.2.2(RecursiveFunctions). Considerthefollowingfunctions(onN)andfunctionoperations: 1. ThesuccessorfunctionS:x7!x+1. 2. Constantfunctions:foreveryn,m2N,fn,m:x1,x2,...,xn7!m. 3. ProjectionFunctions:foreveryn,m2N,Pn,m:x1,x2,...,xn7!xm. 4. Composition:foreveryn,m2Nandfunctionsg1,...,gm,hFn,m,h,g1,...,gm:x1,...,xn7!h(g1(x1,...,xn),...,gm((x1,...,xn)). 5. Search:foreverynandfunctionhofn+1variables,Sn,h:x1,...,xn7!y[h(x1,...,xn,y)=0^(8zy)(h((x1,...,xn,y)#)].LetC0=fS,fn,m,Pn,m:n,m2Ngandforeachi>0,Ci+1=Ci[fFn,m,g1,...,gn,h,Sn,h:n,m2N,gj,h2Cig.ThenC=TiCiisthecollectionof(partial)recursivefunctions.Ifweremovesearchfromtheabovedenition,wedenetheclassofprimitiverecursivefunctions,whicharealwaystotal.Wecanshowthattheabovedenitionsdenethesameclassoffunctions.Infact,foranymodelofcomputationsofardevised(forexampleFRACTRAN,RealRAMMachines),ithasbeenshownthatthismodelgivesthesameclassoffunctionsalreadydened.Thiswehavethefollowingassumptionwhichweadopt. 12

PAGE 13

Denition1.2.3(Church'sThesis). Theclassofintuitivelycomputablefunctionsis,infact,theclassofrecursive(equivalentlyTuringcomputable)functions.Underthishypothesis,wewillusethewordsrecursiveandcomputableinterchangeablyanddescribeproceduresforcomputingfunctionsmoreinformally.Infactwiththeproliferationofcomputers,itisoftenmosteasytodescribecomputablefunctionsasthosewhichonecouldwriteacomputerprogramtocompute.Inlightofthis,weseethatevery(partial)computablemaybeeffectivelyenumeratedsimplybylistingallpossiblecomputerprogramsinlengthlexicographicorder. Proposition1.2.4. Thereisauniversalpartialcomputablefunction.Thatis,thereisapartialfunctionUsothatU(e,n)='e(n)where'eisthee-thpartialcomputablefunction.Wemaycodethepair(e,n)asasinglenumberhe,niandwriteU(he,ni)=U(e,n)Ontheotherhand, Proposition1.2.5. Thereisnouniversalcomputablefunction. Denition1.2.6. LetAN,AiscomputableifthecharacteristicfunctionAiscomputable.Aiscomputableenumerable(c.e.)ifthereisacomputablefunctionfsothatA=Imf.Aisco-c.e.ifthecomplimentofAisc.e. Proposition1.2.7. LetAN,thefollowingareequivalent: 1. Aisc.e. 2. Aisthedomainofapartialcomputablefunction. 3. A=fn:(9m)(R(n,m))gforsomecomputablerelationR.Everycomputablesetiscomputableenumerable,howevertheclassesarenotequivalent. Proposition1.2.8. Thereisac.e.setwhichinnotcomputable.Atthispointwewillstopdistinguishingbetweenpartialandtotalfunctionsunlessthecontextmakesthingsunclear. 13

PAGE 14

1.3TreesandtheCantorSpace Denition1.3.1. Letbeanitesetofsymbols.DenotethecountablesequencesfrombyN=fX:X:N!g=Qn2Nandthenitesequences(words)by.Letbetheemptyword,thewordoflengthzero.Thelengthofawordwisgivenbyjwj.IfawordwisaprexofasequenceXwewritew
PAGE 15

Ausefulpropertyofthebasisofdyadicintervalsisthattherearenitelymanyoflengthatmost2)]TJ /F11 7.97 Tf 6.59 0 Td[(nforeachn. Proposition1.4.3. ThecantorspaceNwiththeproducttopologyisacomputabletopologicalspacewithabasisofclopensets=f[w]:w2g,where[w]=fX2N:w
PAGE 16

Theorem1.4.10. LetX22N,thefollowingareequivalent: 1. Xiscomputable. 2. Xisthecharacteristicfunctionofacomputableset.Letx2[0,1],thefollowingareequivelent: 1. xiscomputable. 2. Thereisacomputablesequence)]TJ /F7 11.955 Tf 5.48 -9.69 Td[(qnnsothatforeachn,jqn)]TJ /F7 11.955 Tf 11.96 0 Td[(xj2)]TJ /F11 7.97 Tf 6.58 0 Td[(n. 3. TheupperandlowerDedekindcutsforxarec.e.. 4. TheupperDedekindcutforxiscomputable. 5. ThelowerDedekindcutforxiscomputable.Byanalogy,wemakethefollowingdenition. Denition1.4.11. Letx2[0,1].Wesayxisupper(lower)semi-computableifthereisanincreasing(decreasing)computablesequence)]TJ /F7 11.955 Tf 5.48 -9.69 Td[(qnnofrationalswithlimnqn=x.Equivalently,theupper(lower)Dedekindcutisc.e.. Theorem1.4.12. LetMbeacomputabletopologicalspace,thenF:M!Miscontinuousifanonlyifthereisafunctionf:N!Nsothat: 1. Foreveryx2MandnameAofx,f(A)isanameforF(x),and 2. Foreveryn,n2N,if(n)(m)then(f(n))(f(m)).WecallftheapproximatingfunctionforF. Denition1.4.13. AcontinuousfunctionF:M!Miscomputableiftheapproximatingfunctionfiscomputable. 1.4.3ComplexityClasses Denition1.4.14. LetXbeacomputabletopologicalspaceandQbeaclosedsubsetofX. 1. Qisopenbasiscomputableenumerable(OBc.e.)iffn:(n)\Q6=;gisc.e.. 2. Qisopenbasisco-computablyenumerable(OBco-c.e.)iffn:(n)\Q=;gisc.e.. 16

PAGE 17

3. Qisclosedbasiscomputableenumerable(CBc.e.)iffn: (n)\Q6=;gisc.e.. 4. Qisclosedbasisco-computableenumerable(CBco-c.e.or01)iffn: (n)\Q=;gisc.e.. 5. QisrecursiveifQisbothOBc.e.andCBco-c.e.. 6. Qisopenbasisdecidableiffn:(n)\Q6=;giscomputable. 7. Qisclosedbasisdecidableiffn: (n)\Q6=;giscomputable. 8. QisdecidableifQisopenbasisdecidableandclosedbasisdecidable.InspaceswherethebasissetsareintervalswewillreplacethewordbasiswithintervalandBwithI.SinceNhasabasisofclopensets,manyoftheabovedenitionsareequivalent. Proposition1.4.15. InacantorspaceN,OBc.e.isequivalenttoCBc.e.andOBco-c.e.isequivalenttoCBco-c.e..Additionally,recursive,openbasisdecidable,closedbasisdecidableanddecidableareequivalent. Proposition1.4.16. LetQbeaclosedsubsetofn.Thefollowingareequivalent: 1. Qisa01class. 2. QisCBco-c.e.. 3. TQisac.e.subsetof. 4. Q=[T]forsomecomputabletreeT. 5. Thereisacomputablesequenceofclopensets)]TJ /F7 11.955 Tf 5.48 -9.69 Td[(UnnwithQ=TnUn. Denition1.4.17. AnsetUNisc.e.-openor01ifU{is01equivalentlyU=Sw2A[w]forsomec.e.setA. Proposition1.4.18. Everydecidableclasshascomputableelements. 1.5SymbolicDynamics Denition1.5.1. Adynamicalsystemisapair(X,F),whereXisatopologicalspace,andF:X!Xiscontinuous. 17

PAGE 18

Often,thespaceXisunderstood,andwerefertotheysystembythenameofthefunction.However,asinthenextdenition,thereverseistrue. Denition1.5.2. Let:N!Nbedenedby(X)(n)=X(n+1)forallX2Nandn2N.Wecalltheshiftmap.IfPNisclosedand(P)P,thenP(orthedynamicalsystem(P,))iscalledasubshift. Denition1.5.3. AtreeTissubsimilarifforallwordsuandv,ifuv2Tthenv2T. Denition1.5.4. GivenF:A!AandapartitionfIigofAintoclopensets,theitineraryofX,ITF(X)=a0a1a2...wherean=iifandonlyifFn(X)2IianddeneIT[F]=fITF(X):X2Ag. Proposition1.5.5. LetPN.Thefollowingareequivalent. 1. Pisasubshift. 2. TPissubsimilar. 3. ThereisacontinuousfunctionF:N!NandapartitionofNsothatP=IT[F]. 4. ForsomesetofforbiddenwordsS,P=PS=fX:forallu2S,udoesnotappearasasubsequenceofXg.NotethatwhileTPisuniquelydenedbyP,neitherthefunctionF,northecollectionofforbiddenwordsSneedbeunique. Denition1.5.6. AsubshiftPisasubshiftofnitetypeifthereisanitesetofwordsSsothatP=PS.NoticethatifSisco-nitethenalsoPSisasubshiftofnitetype. Denition1.5.7. LetF:[0,1]![0,1]beacontinuousfunctionifthereisasetP=f0=x0
PAGE 19

Example1.5.8. LetFbeamultimodalmapaswitnessedbyPandsentJi=[xi,xi+1].LetA=fw:(8i
PAGE 20

Proposition1.6.2. ThespaceofclosedsetswiththehitormisstopologyhasacountablebasisofclopensetsofthefromUA=fK2C:TK\f0,1gn=AgfornitetreesA. Proposition1.6.3. Thefunctionf:C!3Ndenedbyf(K)(n)=xK=8>>>>>><>>>>>>:0w_n02TK^w_n1=2TK1w_n0=2TK^w_n12TK2w_n02TK^w_n12TK,wherewnisthenthwordinTKinlengthlexographicorder,isacomputablehomeomor-phism. 1.6.2MeasuresonCWecandeneanumberofmeasuresoncantorsetsintheusualway,bytheCarathedoryprocess. Denition1.6.4. Givenafunctiond:f0,1,2g![0,1]withd()=1and,foranyw2f0,1,2g,d(w)=2Xi=0d(w_i).Thecorrespondingmeasuredonf0,1,2gN,withtheBorelsets,isdenedbylettingd([w])=d(w).Wesaydisregularifthereareconstantsb0,b1,b2withb0+b1+b2=1suchthatforallwandfori2,d(w_i)=bid(w).ThemeasuredonCwiththeBorelsetsisgivenbyd(X)=d(fxK:K2Xg). Denition1.6.5. WesaythatameasureonCiscomputableiftherestrictionoftotheclopensubsetsofCiscomputable. Proposition1.6.6. Foranycomputabled,themeasuredisacomputablemeasureonC. 1.6.3Capacity Denition1.6.7. AcapacityonCisafunctionT:C![0,1]withT(;)=0suchthat 20

PAGE 21

1. Tismonotoneincreasing,thatis,Q1Q2)166(!T(Q1)T(Q2). 2. Thasthealternatingofinniteorderproperty,thatis,forn2andanyQ1,...,Qn2CT(n\i=1Qi)Xf()]TJ /F3 11.955 Tf 9.3 0 Td[(1)jIj+1T([i2IQi):;6=If1,2,...,ngg. 3. IfQ=\nQnandQn+1Qnforalln,thenT(Q)=limn!1T(Qn).Additionally,weassumethatT(2N)=1. Denition1.6.8. AcapacityTiscomputableifitiscomputableontheclopensets. Denition1.6.9. Givenafunctiond:f0,1,2g![0,1]withd()=1and,foranyw2f0,1,2g,d(w)=2Xi=0d(w_i),thecorrespondingcapacityTd(Q)isgivenbyTd(Q)=d(V(Q)).Thus,Td(Q)istheprobabilitythatQhitsarandomlychosenclosedset. Proposition1.6.10. Ifdisa(computable)probabilitymeasureonC,thenTdisa(computable)capacity. Theorem1.6.11(Choquet'sCapacityTheorem). ForeverycapacityT,thereisameasuresothatT=T(Q)=(V(Q)). Theorem1.6.12(EffectiveChoquetCapacityTheorem). IfTisacomputablecapacity,thenthereisacomputablemeasuredonthespaceofclosedsetssuchthatT=Td=Td(Q)=d(V(Q)). 1.6.4AlgorithmicRandomness Denition1.6.13. Letbeameasureonacantorset.AMartin-Loftestisacomputablesequenceofc.e.opensets)]TJ /F7 11.955 Tf 5.48 -9.68 Td[(Unnsothat(UN)2)]TJ /F11 7.97 Tf 6.58 0 Td[(nforeachn.Thesetestscorrespondtostatisticaltests.Thetestis,doesXlieinTnUn?Fromthesetest,wecandeneMartin-Lofrandomnessinthefollowingway. Denition1.6.14. AsequenceXisrandomifforeveryMartin-Loftest)]TJ /F7 11.955 Tf 5.48 -9.69 Td[(Unn,X=2TnUn. 21

PAGE 22

ForaclosedsetinCwedenerandomnessafollows. Denition1.6.15. AclosedsetP2CisrandomifxPisrandom. Theorem1.6.16. Supposethatthemeasuredisdenedbydsuchthat,forallsuf-cientlylong2f0,1g,d(_2)p 2 2d()andd(_0)=d(_1).Then,foranyd-randomclosedsetR,Td(R)=0. Theorem1.6.17. Supposethatb<^bisxedandthatthemeasuredisdenedbydsuchthat,forallsufcientlylong,d(_0)=d(_1)bd().Letdbetheuniformmeasurewithb0=b1=b>0andb2=1)]TJ /F3 11.955 Tf 12.06 0 Td[(2b>0andlet^b=1)]TJ /F9 7.97 Tf 13.25 11.35 Td[(p 2 2andsupposethatb<^b.ThenfR2C:Td(R)>0ghasdmeasureoneandfurthermoreeveryd-randomclosedsethaspositivecapacity. 22

PAGE 23

CHAPTER2EFFECTIVESYMBOLICDYNAMICSINTHECANTORSPACEInthethischapter,weinvestigatethecorrespondencebetweencontinuousfunctionsand01classes.Werstgivedenitionsandpreliminarymaterial.Then,wegivethedenitionforandinvestigatepropertiesofconservativelyapproximablefunctions.Wedemonstratesomepropertiesofthisclassoffunctionsandshowthatthe01classesareexactlytheimagesofthesefunctions.Werenethenotionofconservativelyapproximabletogiveacharacterizationofthe01subshifts.Wegiveagenerallemmawhichaidsinsuchcharacterizationsandinvestigateotherrelatednotions.Finally,weshowthatthedecidablesubshiftsareexactlythosewhoseforbiddenwordshavesubsimilartreeswithnodeadendsascompliments.Letbeanynitealphabet(typicallyaninitialsegmentof!).LetNbethesetofcountablesequencesfromandbethecollectionofnitesequencesorwordsfrom.ForanyX2NletXdn=X(0)X(1)X(n)]TJ /F3 11.955 Tf 12.66 0 Td[(1)andforanywordw2letwdn=wifjwj
PAGE 24

2.1WeakeningComputabilityInthissection,x=f0,1,...,n)]TJ /F3 11.955 Tf 11.96 0 Td[(1g. Denition2.1.1. AfunctionF:N!Nisconservativelyapproximableifthereisacomputablefunctionf:!andafunctiong:N!NsothatForeveryX2N,)]TJ /F2 11.955 Tf 7.47 -9.69 Td[(jf(Xdn)jisunboundedandlimnf(Xdn)=F(X).1.Ifwn.Also,notethatifjwjg(m)thenjf(w)jm.ForletY=w.Thensincejf(Ydn)jisunbounded,thereisn>jwjwithjf(Ydn)j>mbutby(2),f(w)dm=f(Ydn)dm. Proposition2.1.2. IfFisconservativelyapproximablethenFiscontinuous. Proof. Letfandgbeasindenition 2.1.1 ,w2andX2F)]TJ /F5 7.97 Tf 6.59 0 Td[(1(I[w]).Thenifu=Xdg(jwj),f(u)djwj=whenceifY=Xdg(jwj)thenF(Y)=w.ThusF(I[Xdg(jwj)])I[w]soX2I[Xdg(jwj)]F)]TJ /F5 7.97 Tf 6.59 0 Td[(1(I[w])henceFiscontinuous. Proposition2.1.3. IfFisconservativelyapproximablethenImFisa01class. Proof. Letfandgbeasindenition 2.1.1 andletUn=SfI[f(w)dn]:w2ng.Then(Un)isacomputablesequenceofclopensetsandsoP=TnUnisa01class.WewillshowthatImF=P.SupposeX2P.Thenthereisasequence(wn)withjwnj=nandf(wn)dnnandsolimnF(Yn)=X.SinceFiscontinuousandNiscompact,ImFiscompact.Inparticular,ImFisclosed,thusX2ImF.SuposeX2ImFsothereisaYwithF(Y)=Xandsolimnf(Ydn)=X.Now,givenk,f(Ydg(k))dk=Xdk.Weshowbyinductionn)]TJ /F7 11.955 Tf 11.99 0 Td[(kthatforw2withjwj=n>kthereisau2withjuj=kandf(u)
PAGE 25

Theinductionproceedsasfollows:Ifn)]TJ /F7 11.955 Tf 12.78 0 Td[(k=1,thenthestatementisclearbyproperty(3)indenition 2.1.1 .Ifn)]TJ /F7 11.955 Tf 12.12 0 Td[(k=i+1andthestatementholdswhenn)]TJ /F7 11.955 Tf 12.13 0 Td[(k=i,thenthereisauwithjuj=k+1andf(u)><>>:LPXLLPXPX6LLPissuchafunction.Clearly,ImF=PsinceF(X)=XforX2PandLP,XP2PforeveryX.ToseethatFisconservativelyapproximable,wemustndfandgsatisfyingthethreepropertiesindenition 2.1.1 .FirstletTbeacomputabletreesothatP=[T]anddenefasfollows.Givenw,ifpossible,ndu2T\jwjnearesttowsothatalsouLw;f(w)=u.Otherwisef(w)istheleftmostinT\jwj.Usingtheabovenotation,wemightwritethisasf(w)=8>><>>:lTdjwjwLlTdjwjwTdjwjw6LlTdjwj.Thenfiscomputable,sinceTiscomputable.Also,letg(n)betheleastksothatanyuoflengthkwhichhasanextensionoflengthninThasaninniteextensionin[T]. 25

PAGE 26

1. ForanyX,)]TJ /F2 11.955 Tf 7.47 -9.69 Td[(jf(Xdn)jisunboundedsincejf(Xdn)j=n.Additionally,F(X)=limnf(Xdn). 2. Ifw><>>:LQiXLLQiXQiX6LLQiwhereQi=fX:i_X2Qgisa01classsinceQis.SinceQ\I[i]6=;foreveryi2,thefunctionG(i_X)=Gi(X)hasG(X)(0)=X(0)andsatisesalltheaboveproperties 26

PAGE 27

so,byLemma 2.1.5 ,F=GhasITF=Q.FisalsoconservativelyapproximablewithapproximationsfF(w)=(fG(w))andgF(n)=gG(n+1).Whentryingtoshowtheconverse,theproofrunsintotroubleinthesameplaceasitdoeswhentryingtodemonstrateclosureundercomposition. Example2.1.7. ThereisaconservativelyapproximablefunctionFsothatImF2isnota01class.ThusF2isnotconservativelyapproximable. Proof. Let)]TJ /F7 11.955 Tf 5.48 -9.69 Td[(Teebeacomputableenumerationofprimitiverecursivetreessothat)]TJ /F3 11.955 Tf 5.48 -9.69 Td[([Te]eenumeratesall01classes(see[ 15 ]).Wewillfocusonprexesoffourtypesofsequences:He=0e+110!,Ie=10e+110!,Je=110e+110!,andXe=110e+1110!.OurgoalisthatifXe62[Te]thenF(Ie)=JeandF(Je)=XehenceXe2ImF2;otherwiseXe62ImF2.WeuseHetoseeminglyremoveXefromImF2whilepreservingtheconservativeofFandtheabilitytoaddXetoImF2whenitleaves[Te].PrexesofHeareneverinImf,however,theyalwaysmaptoprexesofXe.Wedenefinductivelyonwordsoflengthn.Firstdenef(0)=f(1)=1.Ifjw_ij=n,denef(w_i)asfollows:f(w_i)=8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:Xednw_i=HednJednw_i=Iedn_w_i=XednJednw_i=Jedn^(Xdn2Te\f0,1gn_e+5>n)Xednw_i=Jedn^Xdn62Te\f0,1gn^e+5nf(w)_iotherwise 27

PAGE 28

Werstcheckthatthereisagsatisfying(2)in 2.1.1 .WeclaimthatforeveryXandnthereisanMX,nsothatifu=XdMX,nthenf(u)dn=f(XdMX,n)dn. Subproof. LetXandnbegivenandnotethatjf(u)j=juj.Thereseveralcases: 1. IfX=wwherew2f0,10,111,110k1,110e+1111,110e+110k1,110e+1111,110e+1110:e,k2Ng.Thenifu=v=wwehavef(u)=f(v).Hence,ifM=maxfn,jwjg,wehavethatu=XdMimpliesf(u)dn=f(XdM)dn. 2. IfX=JeandXe2[Te],thenforeverym,f(Jedm)=Jedm.SoforM=maxfn,e+5gwehaveu=v=XdMthenf(u)=f(v).Hence,wehavethatu=XdMimpliesf(u)dn=f(XdM)dn. 3. IfX=JeandXe62[Te]thenforsomek,Xdk62Te\f0,1gk.SoforM=maxfk,e+5,ngwehaveu=v=XdMthenf(u)=f(v).Hence,wehavethatu=XdMimpliesf(u)dn=f(XdM)dn. 4. IfX=limeXe=limeJe=110!thenforeverymandfore>mXdm=Xedm=Jedmsof(Xdm)=Xdm.LetM=n.Ifu=XdMeitherug(n)andu=w,wehaveu=w=XidMXi,nforsomeihencef(u)dn=f(w)dn=f(XidMXi,n)dn.Itisclearthatfsatises(1)and(3)indenition 2.1.1 andiscomputable.ThusFdenedbyF(X)=limnf(Xdn)isconservativelyapproximable.Weneedtoshowthat[Te]6=ImF2foranyesoImF2isnota01classandF2isnotconservativelyapproximable.Toseethis,supposeXe62[Te].ThenforsomeN,f(Iedn)=Jednandf(Je)dn)=Xednforeveryn>N.ThusF2(Ie)=F(Je)=XesoXe2ImF2.OntheotherhandifXe2[Te],thenF(Y)=XeonlyifY=HeandHe62ImFsoXe62ImF2. Example2.1.8. Thereisadecidable01classPandconservativelyapproximablefunctionFsothatF[P]isnot01. 28

PAGE 29

Proof. Inthenotationofthepreviousexample,letFbeasabove,andletP=fJe:e2Ng.ThenXe2F[P]ifandonlyife=2[Te]andsoF[P]6=[Te]foranyehenceisnota01class.Theclassofconservativelyapproximablefunctionsisnotclosedundercomposition;however,itisclosedunderpost-compositionwithcomputablefunctions. Proposition2.1.9. IfFisconservativelyapproximableandHiscomputable,thenF0=HFisconservativelyapproximable. Proof. LetfandgbeapproximationfunctionsforFasindenition 2.1.1 andhbeanapproximationfunctionforH.Wedenef0andg0forF0byf0=hfandg0=gh0whereh0(n)=minfm:(8w2m)(jh(w)jn)g.Notethath0iswelldenedbycompactness.Weneedtoshowthatf0andg0satisfydenition 2.1.1 2. Letu=wandnbegiven.Wherejwjg0(n).Sincejwjg(h0(n)),wehavev=f(w)dh0(n)=f(u)dh0(n).So,jh(v)jn,h(f(w))=h(v)andh(f(u))=h(v).Thus,h(f(w))dn=h(v)dn=h(f(u))dn. 1. Clearly)]TJ /F2 11.955 Tf 7.47 -9.68 Td[(jf0(Xdn)jnisunboundedasboth)]TJ /F2 11.955 Tf 7.47 -9.68 Td[(jf(Xdn)jnand)]TJ /F2 11.955 Tf 7.47 -9.68 Td[(jh(Xdn)jnare.Further,(2)showsthat)]TJ /F2 11.955 Tf 7.48 -9.68 Td[(jf0(Xdn)jnconvergesandthecontinuityofFandHshowsthatthelimitisF0(X). 3. Letw_ibegiven.Thereisausothatf(u)
PAGE 30

Denefinductivelyonwordsoflengthn.Ifn2thenf(w)=(w).Ifjw_ij=n,denefby:f(w_i)=8>>>>>>>>>><>>>>>>>>>>:(Yedn)w_i=Xedn^(Xedn2Te\f0,1gn_e+3n)(Xedn)w_i=Xedn^Xdn62Te\f0,1gn^e+30,minimalf(w)_iotherwiseToshowthatthereisagsatisfying(2)indenition 2.1.1 ,weclaim,asinthepreviousproof,thatforeveryXandnthereisanMX,nsothatifu=XdMX,nthenf(u)dn=f(XdMX,n)dn. Subproof. LetXandnbegiven.Notethatforallu,jf(u)j=juj)]TJ /F3 11.955 Tf 19.54 0 Td[(1.Therearetwocases. 1. IfXdn+16=Xedn+1foralle
PAGE 31

4. IfX=w0k+11andw2f0n+11m+1,1n+2,10e+11m+1,n,m2N,Xe62[Te]g,thenF(X)=(w0!)andsoITF(X)=w0!.Ineachcase,wecancalculateITF(X)bynotingthatforeveryX,F(X)fallsundertherstcase.Thusforn>2,wehaveFn(X)=n)]TJ /F5 7.97 Tf 6.59 0 Td[(1(F(X)).FromthiswecanseethatXe2IT[F]ifandonlyifXe62[Te].Since[Te]isthee-th01class,IT[F]isnota01class.Intheexampleabove,wehavethatFn+1(X)=n(F(X)).Becauseofthis,manyimmediatestrengtheningsofthenotionofconservativelyapproximablearenotviabletoclassify01subshift.Forexample,assumingthatthesequence)]TJ /F7 11.955 Tf 5.48 -9.68 Td[(Fnnisuniformlyconservativelyapproximable.However,thereisnotmuchroomforfurtherstrengthening. Denition2.1.11. AfunctionF:N!NisstronglyconservativelyapproximableifthefunctionFdenedbyF(X)=hX,F(X),F2(X),F3(X),...iisconservativelyapproximable. Proposition2.1.12. IfFisstronglyconservativelyapproximable,thenFiscomputable. Proof. SinceFisconservativelyapproximable,F0:X7!hX,F(X)iisconservativelyapproximable.Thus,ImF0isa01class.However,ImF0=GraphFhence,Fiscomputable.Asanintermediatestep,considerthefollowing: Proposition2.1.13. SupposeFisaconservativelyapproximablefunctionsuchthat 1. F2=HFforsomecomputablefunctionHand 2. Fw(X)=F(w_X)isconservativelyapproximableforeachw2where[w]\ImF6=;.ThenIT[F]isa01subshift. Proof. LetIibeapartitionofNintonitelymanyclopensetsandletAithethecorrespondingpartitionofmforsomeappropriatem.WerstnotethatFn+1=HFn 31

PAGE 32

sobyinduction,Fn+1=HnF.SoIT[F]=[i[w2Ai[w]\ImF6=;i_ITH[ImFw].Now,sinceHiscomputable,soisITHandsinceFwisconservativelyapproximable,ImFwisa01class.Thus,ITH[ImFw]isa01classandsotheaboveniteunionisalsoa01class.Conversely: Proposition2.1.14. FixthepartionIi=[i]of2NandletQbe01subshiftQsothat[i]\Q6=;foreachi.ThereisaconservativelyapproximablefunctionFsothatF2=HFforsomecomputablefunctionHandFw(X)=F(w_X)isconservativelyapproximableforeachw2. Proof. ByLemma 2.1.5 andProposition 2.1.9 weneedonlyndasuitablefunctionGwithImG=QandGwconservativelyapproximableforeachw.Now,letQ=[T]forsomecomputabletreeTanddenef(w)inductivelyasfollows:f(w)(0)=w(0)f(w)(n+1)=8>><>>:w(n+1)(9v2T\2jwj)(f(w)dn+1_w(n+1)
PAGE 33

ToshowthatGwisconservativelyapproximable,wedenefw(u)inasimilarway.First,ndvw2ext(T),jvwj=jwj:vw(0)=w(0)vw(n+1)=8>><>>:w(n+1)(9v2ext(T)\2jwj)(vwdn+1_w(n+1)><>>:u(n)9v2[vw]\T\2jwj(fw(u)dn+jwj_u(n)m,f(wu)=vwhencefw(u)=f(wu)andsoFw(X)=F(wX)=limnf(wXdn)=limnfw(Xdn)andFw(X)isconservativesince,ifvwu2Twehavefw(u)=vwu.Notethattherstconditionisnotenoughbyexample 2.1.10 andthatthesecondpropertyisnottrueofallconservativelyapproximablefunctionsasthefollowingexampleshows. Example2.1.15. ThereisaconservativelyapproximablefunctionFsothatF0(X):=F(0_X)isnotconservativelyapproximable. Proof. LetXe=1e+10e+11!.WeconstructFsothatXe2ImF0ifandonlyife2;0.Thus,;01TImF0soF0isnotconservativelyapproximable,asitsimageisnota01class.Todothis,letAnbeacomputableapproximationof;0and,forju_ij=n,denef(u_i)=8>>>>>><>>>>>>:Xednu_i<0_Xe^e2An(u_i)1
PAGE 34

Thus,F(X)=limnf(Xdn)isconservativelyapproximableandhasF(0_1e+10e+1X)=8>><>>:Xee2;01!otherwise,whileF(00X)=1!andF(1X)=XforeveryXin2N.Thus,Fhasthepropertydesired.NorcanweaskthatFwbeapproximableuniformly,forthenFiscomputable. Proposition2.1.16. IfFisconservativelyapproximableandFw(X)=F(wX)isalsoconservativelyapproximableviafw(u)=f(wu)foreveryw,thenFiscomputable. Proof. Byassumption,wehavethatforeverywandeveryu_i,thereisavwithjuj=jvjandfw(v)
PAGE 35

ToseethatFis02then,notethatforsg(juj)andus)(jfs(w)j>n^fs0(w)=f(w)].NotethatgiswelldenedsinceNiscompact.Clearly,ifjwjg(n)thenf0(u)dn=f0(w)dnforeveryu=w.Finally,givenw_i,thereisau,juj=jwj,sothatfjwj(u)
PAGE 36

Thisisthebestclassicationofdecidablesubshiftsbyforbiddenwordsthatwecanget.ThenexttwoexamplesshowthatPSneednotbedecidableifS{ismerelyacomputablesubsimilartreeoracomputabletreewithnodeadends. Example2.2.3. ThereisanSwhichiscomputableandS{isasubsimilartreebutPSisnotdecidable. Proof. LetPbeanundecidable01subshift.ThenP=[T]forsomecomputabletreeT.WewilltrimTintoacomputablesubsimilartreeT0.DeneT0=fgandT0n=fw:jwj=n^w2T^(8u,v22
PAGE 37

ends.Fixw02Tsothatw062ext(T)andifu
PAGE 38

CHAPTER3EFFECTIVECONTINUOUSFUNCTIONSON[0,1]Becausethereisadearthofclopensetsintherealnumbers,partitioningintoclopensetsisimpossible.Cenzer,Dashti,andKing[ 13 ]usedapartitionI0=[0,1 2),I1=(1 2,1],andIC=f1 2gof[0,1]totrytogetaroundthis.However,theywere,then,onlyabletoworkwiththemorecomplicatednotionsofkneadingsequence,shiftmaximalsequence,andadmissiblesequence.Ourstrategywillbetoinsteaduse(necessarily)overlappingcoveringsbyclosedsets.Inthischapter,wedenethenotionofthemultivaluedsymbolicdynamicsofafunction.WeshowthatcomputableMarkovfunctionshavedecidableitinerariesandgiveevidencethatstrengtheningthetheoremmaynotbepossible.Asapartialconverse,weshowthatmanydecidablesubshiftsarenotitinerariesofanycomputablefunction.Finally,webeginthedevelopmentofthenotionofconservativelyapproximableontherealswithafewdenitions. 3.1MultivaluedSymbolicDynamics Denition3.1.1. LetIibeacoveringof[0,1]byclosedsetsandF:[0,1]![0,1]beacontinuousfunction.Thesetofitinerariesofx2[0,1]underFisITF(x)=fX:(8n)(Fn(x)2IX(n))gandthesetofitinerariesofF,IT[F]=SxITF(x).Noticethathere,wecannotdeneanitineraryfunctionasinthecaseoftheCantorspacebecausetheclosedsetsmustoverlap.However,wecouldhaveamultivaluedfunctionITFwithGraphITF=f(x,X):(8n)(fn(x)2IX(n))gwherewerecoverthatIT[F]=ImITF. Proposition3.1.2. ForeverycontinuousfunctionF,IT[F]isasubshift. Proof. LetIibeacoveringof[0,1]byclosedsets.WemustrstshowthatIT[F]isclosed.Supposethat)]TJ /F7 11.955 Tf 5.48 -9.69 Td[(XnnisasequencefromIT[F]whichconvergestoX.SincelimnXn=X,foreveryMthereisanNsothatXNdM=XdM.Additionally,sinceXN2IT[F]thereisanxMsothatXN2ITF(xM)andsoforeveryn
PAGE 39

continuous.LetFNbeniteandletM)]TJ /F3 11.955 Tf 12.09 0 Td[(1bethemaximuminF,thenxM2Anforalln><>>:2xx1 25 4)]TJ /F5 7.97 Tf 13.15 4.71 Td[(1 2xx>1 2G(x)=8>><>>:3 4+1 2xx1 22)]TJ /F3 11.955 Tf 11.96 0 Td[(2xx>1 2ThenIT[F]=Sf10gandIT[G]=Sf00g.However,manyquitesimplesubshiftscannotbegeneratedasitinerariesofcontinuousfunctionsinparticular,nitesubshifts. Proposition3.1.4. LetP=IT[F]forsomecontinuousfunctionFunderthecoverf[0,1 2],[1 2,1]gandX,Y2P.IfX(n)6=Y(n)forinnitelymanyn,thenforeveryk,k(P)isinnite. Proof. WewillshowthattherearedistinctznsothatFkn(zn)=1 2henceTPhasabranchingaboveeverylevel.Let)]TJ /F7 11.955 Tf 5.48 -9.68 Td[(mnnbeanincreasingsequencesuchthatX(mn)6=Y(mn).Thenforeachn,wehavexn,ynsothatFmn(xn)1 2andFmn(yn)1 2.Bytheintermediatevaluetheorem,then,thereisaznsothatFmn(zn)=1 2.Now,ifzi=zjfori
PAGE 40

v_n1_Xn,v_n0_Xn2ITF(zn)IT[F].Andso,TPhasabranchingaboveeverylevelasdoesk(TP)foreveryk.Thisshowsthatevensimplesubshiftsmaynotevenbetheshiftoftheitineraryforacontinuousfunction. Example3.1.5. TherearesubshiftsofnitetypeP,whicharenottheitineraryofanycontinuousfunctionon[0,1]usingthecoverf[0,1 2],[1 2,1]g. Proof. LetP=S01,10=f0!,1!g.Pisniteand0!(n)6=1!foranyn.Inthefollowing,letJ0=[0,1 2]andJ1=[1 2,0],ourusualcoveringoftheunitinterval.Additionally,letGRIT[F]=f(x,Y):Y2ITF(x)g,thegraphofITFviewedasamultivaluedfunction. Proposition3.1.6. IfFisacomputablefunction,thenGRIT[F]isclosedintervalco-c.e. Proof. Toseethis,wemustshowthat[p,q][w]\GRIT[F]=;isac.e.statement.Or,equivalently,(8x2[p,q])(9n)(Fn(x)=2Jw(n))isc.e.Wewillshowthatthisisequivalentto(9k)(8i2k)]TJ /F3 11.955 Tf 11.95 0 Td[(1)(9n0:Inthiscase,givenacomputablesequenceIhin,niwithTnIhin,ni=fxgthereisannkandIk2fI0,I1gsothatfk(Ihi,ni)IkandsoinfactFk(x)2Ik.Thus,todecidewhetherw2ITF(x)rstcheckthatx2Iw(0)thenforeachk
PAGE 41

3. Fk(x)=1 2forsomek(x6=1 2):InthiscaseITF(x)=x_ITF(1 2)forsomewoflengthksoisdecidable. Proposition3.1.8. IfFiscomputablewithnitelymanyextrema,thenGRIT[F]isopenintervalc.e. Proof. Letc1,...,cnbethecriticalpoints,thelocationsoftheextrema.Firstnotethattheextremaareisolatedandeachciiscomputable.Toseethis,rstndarationalinterval[a,b]sothat[a,b]\fc1,c2,...,cng=fcig.Bytheintermediatevaluetheorem,F([a,b])=[p,q].Now,since[a,b]isdecidable,sois[p,q]andso,bothpandqarecomputable.ThusF)]TJ /F5 7.97 Tf 6.58 0 Td[(1(F(ci))isa01classandsoisF)]TJ /F5 7.97 Tf 6.59 0 Td[(1(F(ci))\[a,b]=fcig.Thusasasingletonina01class,ciiscomputable.Weshowthatfp,q,w:(p,q)[w]\GRIT[F]6=;gc.e.bygivingarecursiveprocedurewhichhaltsexactlywhen(p,q)[w]\IT[F]6=;.First,assumewearegivenfanapproximatingfunctionforFalongwithiand1 2whichdecidewhetheragivenwordisinITF(ci)orITF(1 2)respectively.Theprocedureisdenedbythefollowingsteps,givenrationalinterval(p,q)andnitewordw: 1. Foreachiwithci2(p,q)andfor1 2if1 2orci2(p,q)determineifwisintherespectiveitinerary.Ifsohalt,otherwiseproceedtothenextstep. 2. Checkthat(p,q)\Iw(0)6=;ifso,proceedtothenextstep,otherwiseloopforever. 3. Ifjwj=1thenhalt. 4. Ifjwj>1thensearchforarationalinterval(p0,q0)F((p,q)\Iw(0))sothat(p0,q0)[w(1)w(2)...w(n)]TJ /F3 11.955 Tf 11.95 0 Td[(1)]\GRIT[F]6=;andhaltiffound.Checkingthatarationalinterval(p0,q0)F((p,q)\Iw(0))alsohasarecursiveprocedure.SearchforintervalsA1,A2(p,q)\Iw(0)sothatf(A1)isentirelybelowp0andf(A2)isentirelyabovep0.Weclaimthattheaboveprocedurehaltsifandonlyif(p,q)[w]\GRIT[F]6=;.Iftheprocedurehaltsatstep(1)or(3)thenthisisclear.Ifithaltsatstep(4)thentherearerationalintervals(p,q)=A0,A1,...Ajwj)]TJ /F5 7.97 Tf 8.94 0 Td[(1sothat 41

PAGE 42

Ai+1F(Ai\Iw(i)).Hence,thereisanintervalA=Tjwj)]TJ /F5 7.97 Tf 8.94 0 Td[(1i=0F)]TJ /F11 7.97 Tf 6.59 0 Td[(i(Ai)(p,q)withfi(A)Iw(i)andso(p,q)[w]\GRIT[F]6=;.Ontheotherhand,if(p,q)[w]\GRIT[F]6=;,thenbyinductiononthelengthofw,theprocedurehalts.Ifjwj=1thisisclear.Supposethatforjuj
PAGE 43

Infact,thepreviousproofshowsthatGRIT[F]isclosedintervalco-c.e.uniformlyovertheapproximatingfunctionfofF.Ontheotherhand,thereisnoalgorithmwhichshowsthatGRIT[F]isopenintervalc.e.whichisuniformevenovertuples(f,1,...,n)where1,...,narenamesforthenitelymanylocationsofextremaofafunctionapproximatedbyf. Proposition3.1.12. GRIT[F]isnotopenintervalc.e.inAf=hf,1,...,niuniformlyforallfsuchthatfapproximatesafunctionFwithnitelymanyextremaatpointsin[0,1]withnames1,...,n. Proof. Supposewehadanalgorithm,T,asdescribedabove;which,whengiven(f,1,...,n),p,q,w,haltsexactlywhen(p,q)[w]\GRIT[F]6=;.ConsiderthefunctionFconsistingofthelinesegmentsfrom(0,1)to(1 2,3 4thento(3 4),1 2)andnallyto(1,1).ThenITF(3 4)3(10)!.Thus,givensomeintervalI33 4,I[10]\GRIT[F]6=;andso,givenIand01,TwillhaltwithAfasanoracle.Inthiscase,letKbetheuseofAfintheabovecomputation,sothatTdoesnotuseinformationaboutf(Ihi,ki)fork>K.Then,wecanndanotherfunctionF2withapproximatingfunctionf2sothatf2(Ihi,ki)=f(Ihi,ki)forkKandsothattheonlyminimumofF2isat3 4butF2(x)>1 2forallx.Thus,infactI[10]\GRIT[F2]=;howeverTwillhaltgivenIand01withAf2asoracle,asAf2dK=AfdK. 3.2ConservativelyApproximableFunctionsAsin[ 41 ],wedenea-nameofarealnumberxtobeasequenceofrationalswhichconvergetoxsothatjx)]TJ /F7 11.955 Tf 11.96 0 Td[(qnj2)]TJ /F11 7.97 Tf 6.59 0 Td[(n.AndafunctionfromF:R!Riscomputable,ifthereisacomputablef:QN!QNwhichmaps-namesofxto-namesofF(x).Noticethattheclassoffunctionsdescribedin[ 42 ]isnottheclassdescribedhereasthatclassincludesnon-continuousfunctions. Denition3.2.1. AfunctionF:R!Risconservativelyapproximableifthereisaconservativelyapproximablef:QN!QNwhichmaps-namesofxto-namesofF(x). Proposition3.2.2. IfFisconservativelyapproximablethenFcontinuous. 43

PAGE 44

Proof. Let">0begivenandletgbethemodulousofconvergenceoftheapproximateingfunctionftoFgiveninDenition 2.1.1 .ChooseNsothat2)]TJ /F11 7.97 Tf 6.59 0 Td[(N<"andMsothatg(M)N+1.Thenifjx)]TJ /F7 11.955 Tf 11.95 0 Td[(yj<2)]TJ /F11 7.97 Tf 6.58 0 Td[(M,thereisarationalqMsothatmaxfjx)]TJ /F7 11.955 Tf 11.96 0 Td[(qMj,jy)]TJ /F7 11.955 Tf 11.95 0 Td[(qMjg<2M.Thus,forany-names(xn)and(yn)ofxandy,(x0n)and(y0n)arealso-namesofxandywherex0n=8>><>>:qMnMxnn>Mandy0n=8>><>>:qMnMynn>M.Butsinceg(M)N+1,f((x0n))dN+1=f((y0n))dN+1inparticular,f((x0n))N+1=f((y0n))N+1=:qhencemaxfjF(x))]TJ /F7 11.955 Tf 11.95 0 Td[(qj,jF(y))]TJ /F7 11.955 Tf 11.95 0 Td[(qjg<2)]TJ /F5 7.97 Tf 6.59 0 Td[((N+1)andsojF(x))]TJ /F7 11.955 Tf 11.95 0 Td[(F(y)j<2)]TJ /F11 7.97 Tf 6.59 0 Td[(N<". Denition3.2.3. AfunctionF:[0,1]![0,1]isconservativelyapproximableifthereisacomputablerepresentingfunctionf:N!Nandafunctiong:N!Nsothat: 1. Ifthereisasequence(jn)withTnIhn,jni=fxgthenTkIf(hg(k),jg(k)i)=fF(x)g. 2. Foreachkandi,thereisanlsothatIhk,liIf(hg(k),ii)andadditionally,ifIhg(k),iiIhn,i0ithenIf(hg(k),ii)If(hn,i0i). 3. Foreachn,i,jwithIhn,iiIhn+1,jithereisaksothatIf(hn,ki)If(hn+1,ji).RecallthatifJnisaneffectiveenumerationofopenintervalsin[0,1]withrationalendpoints,thenasetKisco-c.e.closediffn: Jn\K=;gisc.e. Proposition3.2.4. IfFisaconservativelyapproximablefunctionon[0,1]thenImFisco-c.eclosed. 44

PAGE 45

Proof. Since JnandIhm,iiareintervalswithrationalendpoints,Ihm,ii\ Jn=;isacomputablerelation.ThusA:=fn:(9m)(80i<2m+1)]TJ /F3 11.955 Tf 11.95 0 Td[(1)(If(hm,ii)\ Jn=;)gisc.e.WemustseethatA=fn: Jn\ImF=;g.Clearly,ifn2Athen Jn\ImF=;by(1)and(3)ofdenition 3.2.3 .Toseethereverseimplication,supposeImF\ Jn=;.ThenthereisanopensetUImFsothatU\ Jn=;.Andbycompactness,thereisan">0sothatforx2ImFandanyy,jx)]TJ /F7 11.955 Tf 11.95 0 Td[(yj<"forcesy2U.Pickksothat2)]TJ /F11 7.97 Tf 6.59 0 Td[(k<"andletI=fi:0i<2k+1)]TJ /F5 7.97 Tf 6.59 0 Td[(1^ImF\Ihk,ii6=;g.ThenSi2IIhk,iiUhencedisjointfrom Jn.Andso,by(3)indenition 3.2.3 wehaveSi<2g(k)+1)]TJ /F5 7.97 Tf 6.58 0 Td[(1If(hg(k),ii)Si2IIhk,iiwhichimpliesthatforevery0i<2g(k)+1)]TJ /F3 11.955 Tf 11.95 0 Td[(1,wehaveIf(hg(k),ii)\ Jn=;.Hencen2A. 45

PAGE 46

CHAPTER4CAPACITYInthischapter,wewillexaminethenotionofcomputablecapacityanditsrelationtocomputablemeasuresonthespaceCofnonemptyclosedsets.WegiveconditionsunderwhichthecapacityT(Q)ofa-randomclosedsetQiseitherequalto0or>0whenisanonsymmetricmeasure.Wealsoconstructa01classwithLebesguemeasurezerobutwithpositivecapacity,foreachcapacityofacertaintype. 4.1CapacityandRandomnessIncontrasttopreviouswork,weconsidernon-symmetricmeasures,whered(_0)doesnotnecessarilyequald(_1). Theorem4.1.1. Fixbandletdbeameasuredenedbydwhered(_i)=bid()withb0+b1=2b>0andb2=1)]TJ /F3 11.955 Tf 11.95 0 Td[(2b>0andlet^b=1)]TJ /F9 7.97 Tf 13.15 11.35 Td[(p 2 2.Then 1. Ifb^bandkb0)]TJ /F7 11.955 Tf 12.62 0 Td[(b1kp 8b)]TJ /F3 11.955 Tf 11.95 0 Td[(4b2)]TJ /F3 11.955 Tf 11.95 0 Td[(2,thenforanyd-randomclosedsetR,Td(R)=0. 2. Ifb<^borkb0)]TJ /F7 11.955 Tf 12.1 0 Td[(b1k>p 8b)]TJ /F3 11.955 Tf 11.95 0 Td[(4b2)]TJ /F3 11.955 Tf 11.96 0 Td[(2,thenthereisad-randomclosedsetRwithTd(R)>0. Proof. Forconveniencelet=dandlet2=betheusualproductmeasureontheproductspaceCC.WewillcomputetheprobabilityP=2(f(Q,K):Q\K6=;g).LetQn=SfI(w):jwj=n^Q\I(w)6=;gforanyclosedsetQandletpn=2(QnKn6=;)betheprobabilitythatQnKn6=;forarbitraryclosedsetsQandK,thensincefQnKn6=;gdecreasestofQK6=;g,pndecreasestoP.Clearly,p1=1)]TJ /F3 11.955 Tf 12.17 0 Td[(2b0b1sinceQ1\K1=;onlywhenQ1=I(i)andK1=I(1)]TJ /F7 11.955 Tf 12.18 0 Td[(i).Now,thereisaquadraticfunctionfsothatpn+1=f(pn).InorderthatQn+1\Kn+16=;,theremustbeacommonnodeinQ1\K1andthenheighttreesabovethatnodeinQandKmustalsointersect.Thelaterhasprobabilitypn.Theformercanhappen7ways(of9possibilitiesforKandQ)whichcanbedividedinto3distinctcases. 1. TherearetwowaysQ1=K1=I(i),eachwithprobabilityb2i.InthiscasewendthatQn+1\Kn+16=;withprobability(b20+b21)pn. 46

PAGE 47

2. Thereare4waysQ1=2NandK1=I(i)orviceversa,eachwithprobabilitybi(1)]TJ /F3 11.955 Tf 12.27 0 Td[(2b)foratotalprobability(2b0+2b1)(1)]TJ /F3 11.955 Tf 12.27 0 Td[(2b)=4b(1)]TJ /F3 11.955 Tf 12.27 0 Td[(2b).InthiscasewendthatQn+1\Kn+16=;withprobability4b(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2b)pn. 3. ThereisonewayQ1=K1)]TJ /F3 11.955 Tf 10.39 0 Td[(2N,whichhasprobability(1)]TJ /F3 11.955 Tf 10.39 0 Td[(2b)2.Here,Qn+1\Kn+1=;onlyifI(i)\Qn+1\Kn+1=;fori=0,1.i.e.neitherofthetreesabove0or1forQorKintersect.Thishappenswithprobability1)]TJ /F3 11.955 Tf 11.43 0 Td[((1)]TJ /F7 11.955 Tf 11.43 0 Td[(pn)2=2pn)]TJ /F7 11.955 Tf 11.43 0 Td[(p2n.InthiscasewendthatQn+1\Kn+16=;withprobability(1)]TJ /F3 11.955 Tf 11.95 0 Td[(2b)2(2pn)]TJ /F7 11.955 Tf 11.95 0 Td[(p2n).Intotal,wehavepn+1=(b20+b21+4b(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2b))pn+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(2b)2(2pn)]TJ /F7 11.955 Tf 11.95 0 Td[(p2n)=(2b20)]TJ /F3 11.955 Tf 11.95 0 Td[(4bb0+4b2)]TJ /F3 11.955 Tf 11.96 0 Td[(4b+2)pn)]TJ /F3 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(2b)2p2nNext,weinvestigatelimnpn.Letf(p)=(2b20)]TJ /F3 11.955 Tf 11.95 0 Td[(4bb0+4b2)]TJ /F3 11.955 Tf 11.96 0 Td[(4b+2)p)]TJ /F3 11.955 Tf 11.95 0 Td[((1)]TJ /F3 11.955 Tf 11.95 0 Td[(2b)2p2Thisfunctionhasxedpointsp=0andp=2b20)]TJ /F5 7.97 Tf 6.59 0 Td[(4bb0+4b2)]TJ /F5 7.97 Tf 6.59 0 Td[(4b+1 (1)]TJ /F5 7.97 Tf 6.58 0 Td[(2b)2.Notethatwemusthaveb<1 2so(1)]TJ /F3 11.955 Tf 11.95 0 Td[(2b)2>0.Nowconsiderthefunctiong(a)=2a2)]TJ /F3 11.955 Tf 12.95 0 Td[(4ba+4b2)]TJ /F3 11.955 Tf 12.95 0 Td[(4b+1whichhasrootsa=bq )]TJ /F7 11.955 Tf 9.3 0 Td[(b2+2b)]TJ /F5 7.97 Tf 13.15 4.7 Td[(1 2.Additionally,h(b)=)]TJ /F7 11.955 Tf 9.3 0 Td[(b2+2b)]TJ /F5 7.97 Tf 13.25 4.7 Td[(1 2=)]TJ /F3 11.955 Tf 9.3 0 Td[((x)]TJ /F3 11.955 Tf 12.05 0 Td[(1)2+1 2hasroot^b.Thereare(3)casestoconsiderwhencomparingband^b. 1. Ifb>^banda)]TJ /F2 11.955 Tf 11.35 1.8 Td[(b0a+,theng(b0)<0andhencetheotherxedpointoffisnegative.Since(pn)isdecreasingwithlowerbound0thesequenceconvergestoanon-negativexedpointoff.HenceP=limnpn=0. 2. Ifb=^bandb0=borifb0=atheng(b0)=0andsop=0istheonlyxedpointoffhenceP=limnpn=0. 3. Ifb<^borb062[a+,a)]TJ /F3 11.955 Tf 7.08 1.79 Td[(],theng(b0)>0andsofhaspositivexedpointmb,b0=2b20)]TJ /F5 7.97 Tf 6.59 0 Td[(4bb0+4b2)]TJ /F5 7.97 Tf 6.59 0 Td[(4b+1 (1)]TJ /F5 7.97 Tf 6.58 0 Td[(2b)2.Furthermore,fhasitsmaximumatp=b0)]TJ /F5 7.97 Tf 6.59 0 Td[(2bb0+2b2+2b+1 (1)]TJ /F5 7.97 Tf 6.58 0 Td[(2b)2>1(since2b>2bb0).Thusfisincreasingforp<1,soifp>mb,b0,thenf(p)>f(mb,b0)=mb,b0.Hence,sincep0=1,(pn)isboundedbellowbymb,bandso,P=limnpn=mb,b0>0. 47

PAGE 48

Proofof(1). Inthesetwocases,wecandeneaMartin-LoftesttoprovethatTd(R)=0forany-randomclosedsetR.Foreachm,n2!,letBm=f(K,Q):Km\Qm6=;g,sothat(Bm)=pmandletAm,n=fQ:(fK:Km\Qm6=;g)2)]TJ /F11 7.97 Tf 6.59 0 Td[(ng. Claim4.1.2. Foreachmandn,(Am,n)2npm.ProofofClaim 4.1.2 .DenetheBorelmeasurablefunctionFm:CC:!f0,1gtobethecharacteristicfunctionofBm.Thenpm=2(Bm)=ZQ2CZK2CF(Q,K)dKdQ.NowforxedQ,(fK:Km\Qm6=;g)=ZK2CF(Q,K)dK,sothatforQ2Am,n,wehaveRK2CF(Q,K)dK2)]TJ /F11 7.97 Tf 6.59 0 Td[(n.Itfollowsthatpm=ZQ2CZK2CF(Q,K)dKdQZQ2Am,nZK2CF(Q,K)dKdQZQ2Am,n2)]TJ /F11 7.97 Tf 6.58 0 Td[(ndQ=2)]TJ /F11 7.97 Tf 6.59 0 Td[(n(Am,n).Multiplyingbothsidesby2ncompletestheproofofClaim 4.1.2 .Sincethecomputablesequence)]TJ /F7 11.955 Tf 5.48 -9.69 Td[(pnn2!convergesto0,theremustbeacomputablesubsequencem0,m1,...suchthatpmn<2)]TJ /F5 7.97 Tf 6.59 0 Td[(2n)]TJ /F5 7.97 Tf 6.59 0 Td[(1foralln.WecannowdeneourMartin-Loftest.LetSr=Amr,r 48

PAGE 49

andletVn=[r>nSr.Itfollowsthat(An)2n+1(Bmn)<2n+12)]TJ /F5 7.97 Tf 6.59 0 Td[(2n)]TJ /F5 7.97 Tf 6.58 0 Td[(1=2)]TJ /F11 7.97 Tf 6.59 0 Td[(nandtherefore(Vn)Xr>n2)]TJ /F11 7.97 Tf 6.58 0 Td[(r=2)]TJ /F11 7.97 Tf 6.59 0 Td[(nNowsupposethatRisarandomclosedset.Thesequence)]TJ /F7 11.955 Tf 5.48 -9.69 Td[(Vnn2!isacomputablesequenceofc.e.opensetswithmeasure2)]TJ /F11 7.97 Tf 6.59 0 Td[(n,sothatthereissomensuchthatR=2Sn.Thusforallr>n,(fK:Kmr\Rmr6=;g)<2)]TJ /F11 7.97 Tf 6.59 0 Td[(randitfollowsthat(fK:K\R6=;g)=limn(fK:Kmn\Rmn6=;g)=0.ThusTd(R)=0,asdesired. Proofof(2). NowB=f(Q,K):Q\K6=;g=TnBnistheintersectionofadecreasingsequenceofsetsandhence2(B)=limnpn=mb>0. Claim4.1.3. (fQ:(fK:K\Q6=;g)>0g)mb.ProofofClaim 4.1.3 .LetB=f(K,Q):K\Q6=;,letA=fQ:(fK:K\Q6=;g)>0gandsupposethat(A)
PAGE 50

ProofofClaim 4.1.4 .RecallthatTd(Q)=(fK:Q\K6=;g).LetB=f(K,Q):K\Q6=;,letA=fQ:Td(Q)mbgandsupposethat(A)=0.AsintheproofofClaim 4.1.2 ,wehavemb=2(B)=ZQ2CTd(Q)dQ.Since(A)=0,itfollowsthatforanyBC,wehaveZQ2BTd(Q)dQmb(B).Furthermore,Td(Q)0.Thenwehavemb=ZQ2CTd(Q)dQ=ZQ2ETd(Q)dQ+ZQ=2ETd(Q)dQ(mb)]TJ /F8 11.955 Tf 11.95 0 Td[()+(1)]TJ /F8 11.955 Tf 11.95 0 Td[()mb=mb)]TJ /F8 11.955 Tf 11.95 0 Td[(0withprobabilityone.Thatis,letpbetheprobabilitythatTd(R)=0.ThenbyconsideringtherstlevelofR,wecanseethatp=2bp+(1)]TJ /F3 11.955 Tf 12.49 0 Td[(2b)p2andhenceeitherp=0orp=1.Sinceweknowthatp>0,itfollowsthatp=1.Sincethesetof-randomclosedsetshasmeasureone,theremustbearandomclosedsetRsuchthatTd(R)mbandfurthermore,almostevery-randomclosedsethaspositivecapacity.Furthermore,wecanconstructaMartin-Loftestasfollows.Firstobservethatforanycomputableq,fQ:Td(Q)
PAGE 51

Nowleth(p)betheprobabilitythatTd(Q)a+b>0.Forany01classQ=[T],QistheeffectiveintersectionofthedecreasingsequenceQn=[Tn]ofclopensets,whereTn=T\f0,1gn.ThusforacomputablemeasureTd,thecapacityTd(Q)isthelimitofacomputable,decreasingsequenceandisthereforeanuppersemi-computablereal.Wewillshowthatforeveryuppersemi-computablerealq2[0,1],thereexistsa01classQwithTd(Q)=q. Lemma4.2.1. LetQ=0_Q0[1_Q1.Then,T(Q)=(1)]TJ /F7 11.955 Tf 10.76 0 Td[(b)x+(1)]TJ /F7 11.955 Tf 10.75 0 Td[(a)y)]TJ /F3 11.955 Tf 10.76 0 Td[((1)]TJ /F3 11.955 Tf 10.76 0 Td[((a+b))xy,wherex=T(Q0)andy=T(Q1). Proof. ForaclosedsetK,K\Q6=;ifandonlyifoneofthefollowingholds: 1. K=0_K0andQ0\K06=;(whichhasprobabilityaT(Q0)),or 2. K=1_K1andQ1\K16=;(whichhasprobabilitybT(Q1)),or 3. K=0_K0[1_K1andeitherQ0\K06=;orQ1\K16=;(whichhasprobability(1)]TJ /F3 11.955 Tf 11.95 0 Td[((a+b))(1)]TJ /F3 11.955 Tf 11.96 0 Td[((1)-222(T(Q0)(1)-222(T(Q1))). 51

PAGE 52

Thus,T(Q)=ax+by+(1)]TJ /F3 11.955 Tf 11.95 0 Td[((a+b))(1)]TJ /F3 11.955 Tf 11.96 0 Td[((1)]TJ /F7 11.955 Tf 11.96 0 Td[(x)(1)]TJ /F7 11.955 Tf 11.95 0 Td[(y))=(1)]TJ /F7 11.955 Tf 11.95 0 Td[(b)x+(1)]TJ /F7 11.955 Tf 11.95 0 Td[(a)y)]TJ /F3 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[((a+b))xyasdesired. Lemma4.2.2. LetQ=Sk=nk=0I[k].Thenforeachik,T(Q))-175(T(QnI[i])(1)]TJ /F7 11.955 Tf 11.4 0 Td[(b)jij. Proof. Byinductiononjij.Letx=T(Q0)andy=T(Q1).Basecase: Ifi=0,thenT(Q)=(1)]TJ /F7 11.955 Tf 12.19 0 Td[(b)+byandT(QnI[0])=(1)]TJ /F7 11.955 Tf 12.19 0 Td[(a)y.Then,T(Q))]TJ /F7 11.955 Tf 11.96 0 Td[(T(QnI[0])=(1)]TJ /F7 11.955 Tf 11.95 0 Td[(b))]TJ /F3 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[((a+b))y(1)]TJ /F7 11.955 Tf 11.96 0 Td[(b).Ifi=1,thenT(Q)=(1)]TJ /F7 11.955 Tf 11.95 0 Td[(a)+axandT(QnI[1]=(1)]TJ /F7 11.955 Tf 11.96 0 Td[(b)x.Then,T(Q))]TJ /F7 11.955 Tf 11.96 0 Td[(T(QnI[0])=(1)]TJ /F7 11.955 Tf 11.96 0 Td[(a))]TJ /F3 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[((a+b))x(1)]TJ /F7 11.955 Tf 11.96 0 Td[(b).Inductionstep: Letjij=n.Firstconsiderthecasethati=0_forsome.Letx0=T(Q0nI[]).Then,T(Q))-222(T(QnI[i])=(1)]TJ /F7 11.955 Tf 11.96 0 Td[(b)(x)]TJ /F7 11.955 Tf 11.97 0 Td[(x0))]TJ /F3 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[((a+b)y(x)]TJ /F7 11.955 Tf 11.96 0 Td[(x0)(1)]TJ /F7 11.955 Tf 12.66 0 Td[(b)(x)]TJ /F7 11.955 Tf 12.66 0 Td[(x0)(1)]TJ /F7 11.955 Tf 12.66 0 Td[(b)(1)]TJ /F7 11.955 Tf 12.66 0 Td[(b)n)]TJ /F5 7.97 Tf 6.58 0 Td[(1wherethelastinequalityholdsbytheinductionhypothesis.Next,leti=1_forsome.Lety0=T(Q1nI[]).ThensimilartothepreviouscasewegetT(Q))-246(T(QnI[i])(1)]TJ /F7 11.955 Tf 12.25 0 Td[(a)(y)]TJ /F7 11.955 Tf 12.25 0 Td[(y0)(1)]TJ /F7 11.955 Tf 12.25 0 Td[(b)(1)]TJ /F7 11.955 Tf 12.25 0 Td[(b)n)]TJ /F5 7.97 Tf 6.59 0 Td[(1wherethelastinequalityfollowsfromtheinductionhypothesisandab. Theorem4.2.3. Letqbeuppersemi-computable,i.e.thereisacomputable,decreasingsequence(qn)suchthatlimqn=q.Thenthereexistsa01classQsuchthatT(Q)=q.Moreover,QcanbewrittenasTnQnwhere(Qn)isacomputablesequenceofclopensetswithqn+1T(Qn)qn. Proof. ThroughouttheproofQnwillbeaunionofintervalsandwewillwriteQn=Sk=nkk=0I[k]forsomenitenkwherejij=jjj=lnforalli,jnk.Additionaly,letQn,i=Sk=nkk=iI[k].Also,withoutlossofgeneralitywemayassumethatq0=1.Wewillconstruct(Qn)byinduction. 52

PAGE 53

Basecase: LetQ0=2N.Inductionstep: SupposewehaveconstructedQn)]TJ /F5 7.97 Tf 6.58 0 Td[(1.Computelnsothat(1)]TJ /F7 11.955 Tf 11.99 0 Td[(b)ln+1qn)]TJ /F7 11.955 Tf 13.21 0 Td[(qn+1=:.Thenforanyi,T(Qn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,i))-327(T(Qn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,i+1byLemma 4.2.2 .IfT(Qn)]TJ /F5 7.97 Tf 6.59 0 Td[(1)=qn,setQn=Qn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,otherwise,T(Qn)]TJ /F5 7.97 Tf 6.58 0 Td[(1)>qn.LettbetheleastsuchthatT(Qn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,t)qn.ThenT(Qn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,t)]TJ /F5 7.97 Tf 6.59 0 Td[(1)>qnandT(Qn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,t)T(Qn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)]TJ /F8 11.955 Tf 12 0 Td[(>qn)]TJ /F8 11.955 Tf 12 0 Td[(qn+1.SosetQn=Qn)]TJ /F5 7.97 Tf 6.58 0 Td[(1,t.Thus,wehave(Qn)acomputablesequenceofclopensetsandforQ=TnQn,wehaveT(Q)=limnT(Qn)=q. Theorem4.2.4. Fortheregularmeasuredwithb=b1=b2,thereisa01classQwithLebesguemeasurezeroandpositivecapacityTd(Q). Proof. FirstletuscomputethecapacityofXn=fx:x(n)=0g.Forn=0,wehaveTd(X0)=1)]TJ /F7 11.955 Tf 10.61 0 Td[(b.Thatis,QmeetsX0ifandonlyifQ0=I(0)(whichoccurswithprobabilityb),orQ0=2N(whichoccurswithprobability1)]TJ /F3 11.955 Tf 12.36 0 Td[(2b.NowtheprobabilityTd(Xn+1)thatanarbitraryclosedsetKmeetsXn+1maybecalculatedintwodistinctcases.AsintheproofofTheorem 4.1.1 ,letKn=[fI():2f0,1gn&K\I()6=;gCaseIIfK0=2N,thenTd(Xn+1)=1)]TJ /F3 11.955 Tf 11.96 0 Td[((1)-222(Td(Xn))2.CaseIIIfK0=I((i))forsomei<2,thenTd(Xn+1)=Td(Xn).ItfollowsthatTd(Xn+1)=2bTd(Xn)+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2b)(2Td(Xn))]TJ /F3 11.955 Tf 11.95 0 Td[((Td(Xn))2)=(2)]TJ /F3 11.955 Tf 11.96 0 Td[(2b)Td(Xn))]TJ /F3 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(2b)(Td(Xn))2Nowconsiderthefunctionf(p)=(2)]TJ /F3 11.955 Tf 12.35 0 Td[(2b)p)]TJ /F3 11.955 Tf 12.34 0 Td[((1)]TJ /F3 11.955 Tf 12.35 0 Td[(2b)p2,where0pfor0
PAGE 54

Td(Xn+1)=f(Td(Xn)),itfollowsthatlimnTd(Xn)=1andisthelimitofacomputablesequence.Forany=(n0,n1,...,nk),withn0
PAGE 55

REFERENCES [1] L.M.AXON,Algorithmicallyrandomclosedsetsandprobability,PhDthesis,UniversityofNotreDame,2010. [2] G.BARMPALIAS,P.BRODHEAD,D.CENZER,S.DASHTI,ANDR.WEBER,Algo-rithmicrandomnessofclosedsets,JournalofLogicandComputation,17(2007),p.1041. [3] GEORGEBARMPALIAS,PAULBRODHEAD,DOUGLASCENZER,JEFFREYB.REM-MEL,ANDREBECCAWEBER,Algorithmicrandomnessofcontinuousfunctions,Arch.Math.Logic,46(2008),pp.533. [4] OLIVIERBOURNEZANDMICHELCOSNARD,Onthecomputationalpowerofdynamicalsystemsandhybridsystems,TheoreticalComputerScience,168(1996),pp.417459. [5] M.BRAVERMANANDM.YAMPOLSKY,Non-computableJuliasets,J.Amer.Math.Soc.,19(2006),pp.551(electronic). [6] P.BRODHEADANDD.CENZER,Effectivecapacityandrandomnessofclosedsets,ComputabilityandComplexityinAnalysis,CCA2010,(2009),pp.67. [7] P.BRODHEAD,D.CENZER,ANDS.DASHTI,Randomclosedsets,inLogicalApproachestoComputationalBarriers,etallA.Beckmann,U.Berger,ed.,vol.3988,Springer-Verlag,2006,pp.55. [8] PAULBRODHEAD,DOUGLASCENZER,ANDJEFFREYB.REMMEL,Randomcontinuousfunctions,inProceedingsoftheThirdInternationalConferenceonComputabilityandComplexityinAnalysis(CCA2006),vol.167ofElectron.NotesTheor.Comput.Sci.,Amsterdam,2007,Elsevier,pp.275(electronic). [9] PAULBRODHEAD,DOUGLASCENZER,FERITTOSKA,ANDSEBASTIANWYMAN,Algorithmicrandomnessandcapacityofclosedsets,Log.MethodsComput.Sci.,Specialissue:7thInternationalConferenceonComputabilityandComplexityinAnalysis(CCA2010)(2011),pp.3:16,16. [10] DOUGLASCENZER,Effectivedynamics,inLogicalMethods:InHonorofAnilNerode'sSixtiethBirthday,JohnN.CrossleyandetallCherniavsky,eds.,Birkhauser,1994. [11] DOUGLASCENZER,ALIDASHTI,FERITTOSKA,ANDSEBASTIANWYMAN,Com-putabilityofcountablesubshifts,inPrograms,proofs,processes,vol.6158ofLectureNotesinComput.Sci.,Springer,Berlin,2010,pp.88. [12] ,Computabilityofcountablesubshiftsinonedimension,Theor.Comp.Sys.,51(2012),pp.352. 55

PAGE 56

[13] DOUGLASCENZER,S.ALIDASHTI,ANDJONATHANL.F.KING,Computablesymbolicdynamics,MLQMath.Log.Q.,54(2008),pp.460. [14] DOUGLASCENZERANDPETERG.HINMAN,Degreesofdifcultyofgeneralizedr.e.separatingclasses,Arch.Math.Logic,46(2008),pp.629. [15] DOUGLASCENZERANDJEFFREYREMMEL,Indexsetsfor01classes,AnnalsofPureandAppliedLogic,93(1998),pp.361.ComputabilityTheory. [16] DOUGLASCENZERANDJEFFREYB.REMMEL,01classes,inHandbookofRecursiveMathematicsVolume2:RecursiveAlgebra,AnalysisandCombinatorics,etall.Yu.L.Ershov,S.S.Goncharov,ed.,vol.139ofStudiesinLogicandtheFoundationsofMathematics,Elsevier,1998,pp.viixlvi. [17] DOUGLASCENZERANDJEFFREYB.REMMEL,EffectivelyClosedSets,PerspectivesinMathematicalLogic,CambridgeUniversityPress,2013. [18] G.CHOQUET,Theoryofcapacities,inAnnalesdel'institutFourier,vol.5,1953,p.87. [19] R.DANIELMAULDINANDALEXANDERP.MCLINDEN,Randomclosedsetsviewedasrandomrecursions,Arch.Math.Logic,48(2009),pp.257. [20] C.DELLACHERIE,Lesderivationsentheoriedescriptivedesensemblesetletheoremedelaborne,inSeminairedeProbabilitesXI,C.Dellacherie,P.A.Meyer,andM.Weil,eds.,vol.581ofLectureNotesinMathematics,SpringerBerlinHeidelberg,1977,pp.34. [21] JEAN-CHARLESDELVENNE,PETRKURKA,ANDVINCENTBLONDEL,Decidabil-ityanduniversalityinsymbolicdynamicalsystems,Fund.Inform.,74(2006),pp.463. [22] DAVIDDIAMONDSTONEANDBJRNKJOS-HANSSEN,Membersofrandomclosedsets,inMathematicaltheoryandcomputationalpractice,vol.5635ofLectureNotesinComput.Sci.,Springer,Berlin,2009,pp.144. [23] RODNEYG.DOWNEYANDDENISR.HIRSCHFELDT,Algorithmicrandomnessandcomplexity,TheoryandApplicationsofComputability,Springer,NewYork,2010. [24] SIEGFRIEDGRAF,R.DANIELMAULDIN,ANDS.C.WILLIAMS,TheexactHausdorffdimensioninrandomrecursiveconstructions,Mem.Amer.Math.Soc.,71(1988),pp.x+121. [25] P.R.HALMOS,Measuretheory,vol.1950,Springer-VerlagNewYork;,1974. [26] MICHAELHOCHMAN,Onthedynamicsandrecursivepropertiesofmultidimensionalsymbolicsystems,Invent.Math.,176(2009),pp.131. 56

PAGE 57

[27] BJRNKJOS-HANSSEN,Innitesubsetsofrandomsetsofintegers,Math.Res.Lett.,16(2009),pp.103. [28] KER-I.KO,Onthecomputabilityoffractaldimensionsandhausdorffmeasure,AnnalsofPureandAppliedLogic,93(1998),pp.195216. [29] YU.T.MEDVEDEV,Degreesofdifcultyofthemassproblem,Dokl.Akad.NaukSSSR(N.S.),104(1955),pp.501. [30] JOSEPHS.MILLER,Twonotesonsubshifts,Proc.Amer.Math.Soc.,140(2012),pp.1617. [31] I.MOLCHANOV,Theoryofrandomsets,Springer,2005. [32] HUNGT.NGUYEN,Anintroductiontorandomsets,Chapman&Hall/CRC,BocaRaton,FL,2006. [33] ANDRENIES,Computabilityandrandomness,vol.51ofOxfordLogicGuides,OxfordUniversityPress,Oxford,2009. [34] J.REIMANNANDT.A.SLAMAN,Measuresandtheirrandomreals,arXivpreprintarXiv:0802.2705,(2008). [35] ROBERTRETTINGERANDKLAUSWEIHRAUCH,ThecomputationalcomplexityofsomeJuliasets,inProceedingsoftheThirty-FifthAnnualACMSymposiumonTheoryofComputing,NewYork,2003,ACM,pp.177(electronic). [36] W.RUDIN,Principlesofmathematicalanalysis,vol.3,McGraw-HillNewYork,1964. [37] STEPHENG.SIMPSON,Medvedevdegreesoftwo-dimensionalsubshiftsofnitetype,ErgodicTheoryandDynamicalSystems,FirstView,pp.1. [38] ,Massproblemsandrandomness,Bull.SymbolicLogic,11(2005),pp.1. [39] ,Subsystemsofsecondorderarithmetic,PerspectivesinLogic,CambridgeUniversityPress,Cambridge,seconded.,2009. [40] ANDREASORBI,TheMedvedevlatticeofdegreesofdifculty,inComputability,enumerability,unsolvability,vol.224ofLondonMath.Soc.LectureNoteSer.,CambridgeUniv.Press,Cambridge,1996,pp.289. [41] KLAUSWEIHRAUCH,Computableanalysis,TextsinTheoreticalComputerScience.AnEATCSSeries,Springer-Verlag,Berlin,2000.Anintroduction. [42] KLAUSWEIHRAUCHANDXIZHONGZHENG,Computabilityoncontinuous,lowersemi-continuousanduppersemi-continuousrealfunctions,TheoreticalComputerScience,234(2000),pp.109133. 57

PAGE 58

[43] SEBASTIANWYMAN,Conservativelyapproximablefunctions,inLogicalFoundationsofComputerScience,SergeiArtemovandAnilNerode,eds.,vol.7734ofLectureNotesinComputerScience,SpringerBerlinHeidelberg,2013,pp.387. 58

PAGE 59

BIOGRAPHICALSKETCH SebastianwasborninColoradoandraisedinthefoothillsoftheRockyMountainsat8800feet.HemajoredinmathematicsandphysicsatCaseWesternReserveUniversityandreceivedhismastersandPhDfromtheUniversityofFlorida.Heenjoysteachingmathematicsandteachingstudentsstrongreasoningandproblemsolvingskills.Outsideofacademia,heenjoysUltimateFrisbee,ballroomandsocialdancing.Hemarriedhisgirlfriendof6yearsonedayaftergraduation. 59