Combinatorics, Words and Symbolic Dynamics

497 Pages·2016·4.077 MB·English

by  Valérie Berthé, Michel Rigo

COMBINATORICS, WORDS AND SYMBOLIC DYNAMICS
Internationallyrecognisedresearcherslookatdevelopingtrendsincombinatorics
withapplicationsinthestudyofwordsandinsymbolicdynamics.Theyexplainthe
importantconcepts,providingaclearexpositionofsomerecentresults,and
emphasisetheemergingconnectionsbetweenthesedifferentfields.Topicsinclude
combinatoricsonwords,patternavoidance,graphtheory,tilingsandtheoryof
computation,multidimensionalsubshifts,discretedynamicalsystems,ergodic
theory,numerationsystems,dynamicalarithmetics,automatatheoryand
synchronisedwords,analyticcombinatorics,continuedfractionsandprobabilistic
models.Eachtopicispresentedinawaythatlinksittothemainthemes,butthen
theyarealsoextendedtorepetitionsinwords,similarityrelations,cellular
automata,friezesandDynkindiagrams.
Thebookwillappealtograduatestudents,researchmathematiciansand
computerscientistsworkingincombinatorics,theoryofcomputation,number
theory,symbolicdynamics,tilingsandstringology.Itwillalsointerestbiologists
usingtextalgorithms.
EncyclopediaofMathematicsandItsApplications
Thisseriesisdevotedtosignificanttopicsorthemesthathavewideapplicationin
mathematicsormathematicalscienceandforwhichadetaileddevelopmentofthe
abstracttheoryislessimportantthanathoroughandconcreteexplorationofthe
implicationsandapplications.
BooksintheEncyclopediaofMathematicsandItsApplicationscovertheir
subjectscomprehensively.Lessimportantresultsmaybesummarisedasexercises
attheendsofchapters.Fortechnicalities,readerscanbereferredtothe
bibliography,whichisexpectedtobecomprehensive.Asaresult,volumesare
encyclopedicreferencesormanageableguidestomajorsubjects.
EncyclopediaofMathematicsandItsApplications
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridge
UniversityPress.Foracompleteserieslistingvisit
www.cambridge.org/mathematics.
114 J.BeckCombinatorialGames
115 L.BarreiraandY.PesinNonuniformHyperbolicity
116 D.Z.ArovandH.DymJ-ContractiveMatrixValuedFunctionsandRelatedTopics
117 R.Glowinski,J.-L.LionsandJ.HeExactandApproximateControllabilityforDistributed
ParameterSystems
118 A.A.BorovkovandK.A.BorovkovAsymptoticAnalysisofRandomWalks
119 M.DezaandM.DutourSikiric´GeometryofChemicalGraphs
120 T.NishiuraAbsoluteMeasurableSpaces
121 M.PrestPurity,SpectraandLocalisation
122 S.KhrushchevOrthogonalPolynomialsandContinuedFractions
123 H.NagamochiandT.IbarakiAlgorithmicAspectsofGraphConnectivity
124 F.W.KingHilbertTransformsI
125 F.W.KingHilbertTransformsII
126 O.CalinandD.-C.ChangSub-RiemannianGeometry
127 M.Grabischetal.AggregationFunctions
128 L.W.BeinekeandR.J.Wilson(eds.)withJ.L.GrossandT.W.TuckerTopicsinTopological
GraphTheory
129 J.Berstel,D.PerrinandC.ReutenauerCodesandAutomata
130 T.G.FaticoniModulesoverEndomorphismRings
131 H.MorimotoStochasticControlandMathematicalModeling
132 G.SchmidtRelationalMathematics
133 P.KornerupandD.W.MatulaFinitePrecisionNumberSystemsandArithmetic
134 Y.CramaandP.L.Hammer(eds.)BooleanModelsandMethodsinMathematics,Computer
Science,andEngineering
135 V.BerthéandM.Rigo(eds.)Combinatorics,AutomataandNumberTheory
136 A.Kristály,V.D.Ra˘dulescuandC.VargaVariationalPrinciplesinMathematicalPhysics,
Geometry,andEconomics
137 J.BerstelandC.ReutenauerNoncommutativeRationalSerieswithApplications
138 B.CourcelleandJ.EngelfrietGraphStructureandMonadicSecond-OrderLogic
139 M.FiedlerMatricesandGraphsinGeometry
140 N.VakilRealAnalysisthroughModernInfinitesimals
141 R.B.ParisHadamardExpansionsandHyperasymptoticEvaluation
142 Y.CramaandP.L.HammerBooleanFunctions
143 A.Arapostathis,V.S.BorkarandM.K.GhoshErgodicControlofDiffusionProcesses
144 N.Caspard,B.LeclercandB.MonjardetFiniteOrderedSets
145 D.Z.ArovandH.DymBitangentialDirectandInverseProblemsforSystemsofIntegraland
DifferentialEquations
146 G.DassiosEllipsoidalHarmonics
147 L.W.BeinekeandR.J.Wilson(eds.)withO.R.OellermannTopicsinStructuralGraphTheory
148 L.Berlyand,A.G.KolpakovandA.NovikovIntroductiontotheNetworkApproximationMethod
forMaterialsModeling
149 M.BaakeandU.GrimmAperiodicOrderI:AMathematicalInvitation
150 J.Borweinetal.LatticeSumsThenandNow
151 R.SchneiderConvexBodies:TheBrunn–MinkowskiTheory(SecondEdition)
152 G.DaPratoandJ.ZabczykStochasticEquationsinInfiniteDimensions(SecondEdition)
153 D.Hofmann,G.J.SealandW.Tholen(eds.)MonoidalTopology
154 M.CabreraGarcíaandÁ.RodríguezPalaciosNon-AssociativeNormedAlgebrasI:The
Vidav–PalmerandGelfand–NaimarkTheorems
155 C.F.DunklandY.XuOrthogonalPolynomialsofSeveralVariables(SecondEdition)
156 L.W.BeinekeandR.J.Wilson(eds.)withB.ToftTopicsinChromaticGraphTheory
157 T.MoraSolvingPolynomialEquationSystemsIII:AlgebraicSolving
158 T.MoraSolvingPolynomialEquationSystemsIV:Buchberger’sTheoryandBeyond
159 V.BerthéandM.Rigo(eds.)Combinatorics,WordsandSymbolicDynamics
160 B.RubinIntroductiontoRadonTransforms:WithElementsofFractionalCalculusandHarmonic
Analysis
161 M.GherguandS.D.TaliaferroIsolatedSingularitiesinPartialDifferentialInequalities
162 G.MolicaBisci,V.RadulescuandR.ServadeiVariationalMethodsforNonlocalFractional
Problems
163 S.WagonTheBanach–TarskiParadox(SecondEdition)
Encyclopedia of Mathematics and Its Applications
Combinatorics, Words and
Symbolic Dynamics
Editedby
VALÉRIE BERTHÉ
UniversitéParisDiderot-Paris7,France
MICHEL RIGO
UniversitédeLiège,Belgium
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
education,learningandresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107077027
©CambridgeUniversityPress2016
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2016
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Combinatorics,wordsandsymbolicdynamics/[editedby]ValérieBerthé,
UniversitéParisDiderot,Paris7,MichelRigo,UniversitédeLiège,Belgium.
pagescm.–(Encyclopediaofmathematicsanditsapplications;159)
Includesbibliographicalreferencesandindex.
ISBN978-1-107-07702-7(Hardback)
1. Combinatorialanalysis. 2. Symbolicdynamics. 3. Computerscience.
I. Berthé,V.(Valérie),1957– II. Rigo,Michel.
QA164.C6662015
(cid:2)
511.6–dc23 2015024873
ISBN978-1-107-07702-7Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy
ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication,
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
Contents
Listofcontributors pageix
Preface xi
Acknowledgments xix
1 Preliminaries 1
V. BERTHE´, M.RIGO
1.1 Conventions 1
1.2 Words 1
1.3 Morphisms 4
1.4 Languagesandmachines 5
1.5 Symbolicdynamics 10
2 Expansionsinnon-integerbases 18
M. DEVRIES, V. KOMORNIK
2.1 Introduction 18
2.2 Greedyandlazyexpansions 19
2.3 OnthecardinalityofthesetsEβ(x) 22
2.4 TherandommapKβandinfiniteBernoulliconvolutions 26
2.5 Lexicographiccharacterisations 35
2.6 Univoquebases 39
2.7 Univoquesets 50
2.8 Atwo-dimensionalunivoqueset 55
2.9 Finalremarks 56
2.10 Exercises 57
3 Medieties,end-firstalgorithms,andthecaseofRosencontinuedfrac-
tions 59
B. RITTAUD
3.1 Introduction 59
3.2 Generalities 62
3.3 Examples 68
vi Contents
3.4 End-firstalgorithms 76
3.5 Medietieswithkletters 82
3.6 Anend-firstalgorithmfork-medieties 89
3.7 Exercises 92
3.8 Openproblems 100
4 Repetitionsinwords 101
N. RAMPERSAD, J. SHALLIT
4.1 Introduction 101
4.2 Avoidability 102
4.3 Dejean’stheorem 114
4.4 Avoidingrepetitionsinarithmeticprogressions 120
4.5 Patterns 123
4.6 Abelianrepetitions 123
4.7 Enumeration 134
4.8 Decidabilityforautomaticsequences 143
4.9 Exercises 145
4.10 Notes 146
5 Textredundancies 151
G. BADKOBEH, M.CROCHEMORE, C. S. ILIOPOULOS,M. KUBICA
5.1 Redundancy:aversatilenotion 151
5.2 Avoidingrepetitionsandrepeats 153
5.3 Findingrepetitionsandruns 157
5.4 Findingrepeats 163
5.5 Findingcoversandseeds 167
5.6 Palindromes 171
6 Similarityrelationsonwords 175
V. HALAVA, T.HARJU, T. KA¨RKI
6.1 Introduction 175
6.2 Preliminaries 176
6.3 Coding 181
6.4 Relationalperiods 186
6.5 Repetitionsinrelationalwords 204
6.6 Exercisesandproblems 211
7 Synchronisedautomata 213
M.-P.BE´AL, D. PERRIN
7.1 Introduction 213
7.2 Definitions 214
7.3 Cˇerny´’sconjecture 215
7.4 Roadcolouring 229
Contents vii
8 Cellularautomata,tilingsand(un)computability 241
J. KARI
8.1 Cellularautomata 242
8.2 Tilingsandundecidability 260
8.3 Undecidabilityconcerningcellularautomata 279
8.4 Conclusion 293
8.5 Exercises 293
9 Multidimensionalshiftsoffinitetypeandsoficshifts 296
M. HOCHMAN
9.1 Introduction 296
9.2 Shiftsoffinitetypeandsoficshifts 297
9.3 Basicconstructionsandundecidability 305
9.4 Degreesofcomputability 317
9.5 Slicesandsubdynamicsofsoficshifts 326
9.6 Frequencies,wordgrowthandperiodicpoints 342
10 LinearlyrecursivesequencesandDynkindiagrams 359
C. REUTENAUER
10.1 Introduction 359
10.2 SL -tilingsoftheplane 360
2
10.3 SL -tilingassociatedwithabi-infinitediscretepath 361
2
10.4 ProofofTheorem10.3.1 363
10.5 N-rationalsequences 365
10.6 N-rationalityoftheraysinSL -tilings 369
2
10.7 Friezes 370
10.8 Dynkindiagrams 377
10.9 RationalfriezeimpliesDynkindiagram 382
10.10 RationalityforDynkindiagramsoftypeAandA(cid:2) 385
10.11 FurtherpropertiesofSL -tilings 387
2
10.12 TheotherextendedDynkindiagrams 397
10.13 Problemsandconjectures 397
10.14 Exercises 398
11 Pseudo-randomnessofarandomKroneckersequence.Aninstanceof
dynamicalanalysis 401
E. CESARATTO,B. VALLE´E
11.1 Introduction 401
11.2 FiveparametersforKroneckersequences 404
11.3 Probabilisticmodels 415
11.4 Statementsofthemainresults 417
viii Contents
11.5 Dynamicalanalysis 423
11.6 Balancedcosts 430
11.7 Unbalancedcosts 436
11.8 Summaryoffunctionalanalysis 438
11.9 Conclusionandopenproblems 441
Bibliography 443
Notationindex 464
Generalindex 466