Entropy in Dynamical Systems

404 Pages·2011·2.019 MB·English

by  Tomasz Downarowicz

EntropyinDynamicalSystems
Thiscomprehensivetextonentropycoversthreemajortypesofdynamics:
measure-preservingtransformations;continuousmapsoncompactspaces;and
operatorsonfunctionspaces.
PartIcontainsproofsoftheShannon–McMillan–BreimanTheorem,the
Ornstein–WeissReturnTimeTheorem,theKriegerGeneratorTheorem,theSinaiand
OrnsteinTheorems,andamongthenewestdevelopments,theErgodicLawofSeries.
InPartII,afteranexpandedexpositionofclassicaltopologicalentropy,thebook
addressesSymbolicExtensionEntropy.Itoffersdeepinsightintothetheoryofentropy
structureandexplainstheroleofzero-dimensionaldynamicsasabridgebetween
measurableandtopologicaldynamics.PartIIIexplainshowbothmeasure-theoretic
andtopologicalentropycanbeextendedtooperatorsonrelevantfunctionspaces.
Intuitiveexplanations,examples,exercisesandopenproblemsmakethisanideal
textforagraduatecourseonentropytheory.Moreexperiencedresearcherscanalso
findinspirationforfurtherresearch.
TOMASZDOWNAROWICZisFullProfessorinMathematicsatWroclawUniversityof
Technology,Poland.
NEWMATHEMATICALMONOGRAPHS
EditorialBoard
Be´la Bolloba´s, William Fulton, Anatole Katok, Frances Kirwan, Peter Sarnak, Barry
Simon,BurtTotaro
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridge
UniversityPress.Foracompleteserieslistingvisitwww.cambridge.org/mathematics.
1 M.CabanesandM.EnguehardRepresentationTheoryofFiniteReductiveGroups
2 J.B.GarnettandD.E.MarshallHarmonicMeasure
3 P.CohnFreeIdealRingsandLocalizationinGeneralRings
4 E.BombieriandW.GublerHeightsinDiophantineGeometry
5 Y.J.IoninandM.S.ShrikhandeCombinatoricsofSymmetricDesigns
6 S.Berhanu,P.D.CordaroandJ.HounieAnIntroductiontoInvolutiveStructures
7 A.ShlapentokhHilbert’sTenthProblem
8 G.MichlerTheoryofFiniteSimpleGroupsI
9 A.BakerandG.Wu¨stholzLogarithmicFormsandDiophantineGeometry
10 P.KronheimerandT.MrowkaMonopolesandThree-Manifolds
11 B.Bekka,P.delaHarpeandA.ValetteKazhdan’sProperty(T)
12 J.NeisendorferAlgebraicMethodsinUnstableHomotopyTheory
13 M.GrandisDirectedAlgebraicTopology
14 G.MichlerTheoryofFiniteSimpleGroupsII
15 R.SchertzComplexMultiplication
16 S.BlochLecturesonAlgebraicCycles(2ndEdition)
17 B.Conrad,O.GabberandG.PrasadPseudo-reductiveGroups
Entropy in Dynamical Systems
TOMASZ DOWNAROWICZ
WroclawUniversityofTechnology,Poland
CAMBRIDGEUNIVERSITYPRESS
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T.Downarowicz2011
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Firstpublished2011
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloging-in-PublicationData
Downarowicz,Tomasz,1956–
EntropyinDynamicalSystems/TomaszDownarowicz.
p. cm.–(NewMathematicalMonographs;18)
Includesbibliographicalreferencesandindex.
ISBN978-0-521-88885-1(Hardback)
1. Topologicalentropy–Textbooks. 2. Topologicaldynamics–Textbooks. I. Title.
QA611.5.D6852011
515(cid:2).39–dc22
2010050336
ISBN978-0-521-88885-1Hardback
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To my Parents

Contents
Preface pagexi
Introduction 1
0.1 Theleitmotiv 1
0.2 Afewwordsaboutthehistoryofentropy 3
0.3 Multiplemeaningsofentropy 4
0.4 Conventions 18
PARTI Entropyinergodictheory 21
1 Shannoninformationandentropy 23
1.1 Informationandentropyofprobabilityvectors 23
1.2 Partitionsandsigma-algebras 30
1.3 Informationandstaticentropyofapartition 32
1.4 Conditionalstaticentropy 33
1.5 Conditionalentropyviaprobabilistictools* 35
1.6 Basicpropertiesofstaticentropy 36
1.7 Metricsonthespaceofpartitions 42
1.8 Mutualinformation* 46
1.9 Non-Shannoninequalities* 48
Exercises 51
2 Dynamicalentropyofaprocess 53
2.1 Subadditivity 53
2.2 Preliminariesondynamicalsystems 57
2.3 Dynamicalentropyofaprocess 60
2.4 Propertiesofdynamicalentropy 65
2.5 Affinityofdynamicalentropy 68
2.6 Conditionaldynamicalentropyviadisintegration* 69
viii Contents
2.7 Summaryofthepropertiesofentropy 72
2.8 Combinatorialentropy 73
Exercises 78
3 Entropytheoremsinprocesses 80
3.1 Independenceandε-independence 80
3.2 ThePinskersigma-algebrainaprocess 85
3.3 TheShannon–McMillan–BreimanTheorem 89
3.4 TheOrnstein–WeissReturnTimesTheorem 94
3.5 Horizontaldatacompression 97
Exercises 100
4 Kolmogorov–SinaiEntropy 102
4.1 Entropyofadynamicalsystem 102
4.2 Generators 105
4.3 Thenaturalextension 111
4.4 Joinings 116
4.5 OrnsteinTheory* 120
Exercises 130
5 TheErgodicLawofSeries* 132
5.1 HistoryoftheLawofSeries 132
5.2 Attractingandrepellinginsignalprocesses 135
5.3 Decayofrepellinginpositiveentropy 139
5.4 Typicalityofattractingforlongcylinders 152
PARTII Entropyintopologicaldynamics 157
6 Topologicalentropy 159
6.1 Threedefinitionsoftopologicalentropy 159
6.2 Propertiesoftopologicalentropy 165
6.3 Topologicalconditionalandtailentropies 167
6.4 Propertiesoftopologicalconditionalentropy 171
6.5 Topologicaljoinings 172
6.6 Thesimplexofinvariantmeasures 175
6.7 Topologicalfiberentropy 179
6.8 ThemajorVariationalPrinciples 181
6.9 Determinismintopologicalsystems 190
6.10 Topologicalpreimageentropy* 197
Exercises 199
Contents ix
7 Dynamicsindimensionzero 201
7.1 Zero-dimensionaldynamicalsystems 201
7.2 Topologicalentropyindimensionzero 202
7.3 Theinvariantmeasuresindimensionzero 203
7.4 TheVariationalPrincipleindimensionzero 205
7.5 Tail entropy and asymptotic h-expansiveness in
dimensionzero 206
7.6 Principalzero-dimensionalextensions 212
Exercises 225
8 Theentropystructure 227
8.1 Thetypeofconvergence 227
8.2 U.s.d.a.-sequencesonsimplices 244
8.3 Entropy of a measure with respect to a topological
resolution 254
8.4 Entropystructure 263
Exercises 270
9 Symbolicextensions 272
9.1 Whataresymbolicextensions? 272
9.2 TheSymbolicExtensionEntropyTheorem 274
9.3 Propertiesofsymbolicextensionentropy 287
9.4 Symbolicextensionsofintervalmaps 293
Exercises 301
10 Atouchofsmoothdynamics* 303
10.1 Margulis–RuelleInequalityandPesinEntropyFormula 303
10.2 Tailentropyestimate 307
10.3 Symbolicextensionsofsmoothsystems 308
PARTIII Entropytheoryforoperators 311
11 Measure-theoreticentropyofstochasticoperators 313
11.1 Afewwordsonoperatordynamics 313
11.2 Theaxiomaticmeasure-theoreticdefinition 316
11.3 Anexplicitmeasure-theoreticdefinition 329
11.4 Notsobadpropertiesoftheoperatorentropy 332
Exercises 335
12 TopologicalentropyofaMarkovoperator 336
12.1 Threedefinitions 336
12.2 Propertiesofthetopologicaloperatorentropy 339