An Introduction to the Kolmogorov–Bernoulli Equivalence

131 Pages·2019·1.648 MB·English

by  Gabriel Ponce, Régis Varão

SPRINGER BRIEFS IN MATHEMATICS
Gabriel Ponce
Régis Varão
An Introduction
to the
Kolmogorov–
Bernoulli
Equivalence
123
SpringerBriefs in Mathematics
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˜
Gabriel Ponce • Régis Varao
An Introduction to the
Kolmogorov–Bernoulli
Equivalence
123
GabrielPonce RégisVara˜o
IMECC IMECC
UniversityofCampinas-UNICAMP UniversityofCampinas-UNICAMP
Campinas Campinas
Sa˜oPaulo,Brazil Sa˜oPaulo,Brazil
ISSN2191-8198 ISSN2191-8201 (electronic)
SpringerBriefsinMathematics
ISBN978-3-030-27389-7 ISBN978-3-030-27390-3 (eBook)
https://doi.org/10.1007/978-3-030-27390-3
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Preface
Dynamical systems is a broad and active research field in mathematics, part of its
success has made into the general population by the name of “Chaos Theory” or
“The Butterfly Effect.” Ergodic theory is a dynamical system from a probabilistic
pointofview,andfromtheergodictheorypointofviewwemayevenlistsystems
according to their chaotic behavior. On the bottom of this list are the Ergodic
systems,butonthetopofthislistareKolmogorovsystemsandBernoullisystems
(themostchaoticones).Inmanysituations,Kolmogorovsystemsareequivalentto
Bernoullisystems,andwhetherornotaKolmogorovsystemisaBernoullisystem
isaclassicalprobleminergodictheory.
Thisequivalenceproblemisoneofthemostbeautifulchaptersinergodictheory.
It has the quality of combining an incredible amount of techniques and concepts
ofabstractergodictheory(suchasentropytheoryandOrnsteintheory)andsmooth
dynamics (suchasnonuniformlyhyperbolic dynamics andPesin’stheory).Onthe
other hand, what makes this theory so interesting, that is, so many different and
profound beautiful tools, might be a barrier for those willing to enter this area of
research.Therearenobooksthatcouldguideagraduatedynamicalsystemsstudent
into this research area (i.e., the Kolmogorov–Bernoulli equivalence problem from
the smooth dynamical system point of view) although there are plenty of good
books on ergodic theory and smooth dynamics. If one tries to study directly the
Kolmogorov–Bernoulli problem for nonuniformly hyperbolic systems, one could
easily get stuck into many technicalities (e.g., Pesin’s theory) and not grasp the
mainideasbehindtheproblem.
This book has been written with the primary purpose of filling this gap so that
graduate students could feel comfortable with the idea of working on some of the
open problems related to the Kolmogorov–Bernoulli equivalence (or nonequiva-
lence)problem.Thewaywehopewehaveaccomplishedthisgoalinarathersmall
bookisfocusingonthemainideasbehindtheproblem.Wemaysaythatthemost
importantpartofthisbookisChap.3,whereweproveindetailstheKolmogorov–
Bernoulliequivalenceinthecontextofatoymodel(linearAnosovdiffeomorphism).
This will help the reader to understand main ideas. The subsequent chapter deals
with the problem in the context of hyperbolic dynamics. In the last chapter, we
vii
viii Preface
briefly go through some more general contexts (such as nonuniformly hyperbolic
systems)andpresentsomeinterestingrecentideasinthearea.Thefirsttwochapters
aretheintroductionandsomepreliminariesinergodictheory.
Thereaderisassumedtohaveaworkingknowledgeinergodictheoryandhyper-
bolic dynamics so that we can focus on the Kolmogorov–Bernoulli equivalence
problem itself. We hope the reader may find this book very stimulating and feel
interestedindoingresearchontheopenproblemsrelatedtothesubject.
Campinas,Brazil GabrielPonce
Campinas,Brazil RégisVarão
September2019
Acknowledgements
GabrielPoncewaspartiallysupportedbyFAPESP(Grant#2016/05384-0).Régis
VarãowaspartiallysupportedbyFAPESP(Grant#2016/22475-9)andCNPq.
ix
Contents
1 Introduction .................................................................. 1
1.1 GeneralErgodicTheory ................................................ 1
1.2 ChaoticHierarchy....................................................... 2
1.3 KolmogorovandBernoulliSystems ................................... 3
1.4 SmoothErgodicTheoryandHyperbolicStructures................... 5
1.5 TheGoalofThisBook ................................................. 6
References..................................................................... 7
2 PreliminariesinErgodicTheory........................................... 9
2.1 MeasurePreservingDynamicalSystems .............................. 10
2.2 Birkhoff’sErgodicTheoremandtheErgodicProperty............... 12
2.3 OperationswithPartitions.............................................. 15
2.4 MeasureDisintegration................................................. 17
2.4.1 Rokhlin’sDisintegrationTheorem............................. 18
2.5 BasicsonLebesgueSpaces............................................. 19
2.6 SomeResultsonEntropyTheory ...................................... 20
2.6.1 ThePinskerPartitionandSystemswithCompletely
PositiveEntropy ................................................ 22
2.7 TheBernoulliProperty ................................................. 24
2.7.1 BernoulliShifts ................................................. 24
2.7.2 BernoulliPartitions............................................. 27
2.8 TheKolmogorovProperty.............................................. 29
References..................................................................... 32
3 Kolmogorov–BernoulliEquivalenceforErgodicAutomorphisms
ofT2 ........................................................................... 33
3.1 FiniteandVeryWeakBernoulliPartitions............................. 33
3.1.1 Thed-DistanceintheSpaceofFinitePartitions.............. 34
3.1.2 VeryWeakBernoulliPartitionsandOrnsteinTheorems...... 42
3.2 ErgodicAutomorphismsofT2AreKolmogorov...................... 44
xi