Ergodic Theory: Independence and Dichotomies

455 Pages·2017·2.623 MB·English

by  David Kerr, Hanfeng Li

Springer Monographs in Mathematics
David Kerr
Hanfeng Li
Ergodic
Theory
Independence and Dichotomies
Springer Monographs in Mathematics
More information about this series at http://www.springer.com/series/3733
David Kerr Hanfeng Li
(cid:129)
Ergodic Theory
Independence and Dichotomies
123
DavidKerr Hanfeng Li
Department ofMathematics Department ofMathematics
Texas A&MUniversity SUNY Buffalo
CollegeStation, TX Buffalo, NY
USA USA
ISSN 1439-7382 ISSN 2196-9922 (electronic)
SpringerMonographs inMathematics
ISBN978-3-319-49845-4 ISBN978-3-319-49847-8 (eBook)
DOI 10.1007/978-3-319-49847-8
LibraryofCongressControlNumber:2016957488
MathematicsSubjectClassification(2010): 37A15,37A20,37A25,37B05,37B40
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Preface
Ergodictheoryinitsbroadestsenseisthestudyofgroupactionsonmeasurespaces.
Historically the discipline has tended to concentrate on the framework of integer
actions, in line with its formal origins in the work of John von Neumann and
George David Birkhoff in the early 1930s. To a considerable extent this continues
toholdtoday,notleastduetoavarietyofdeepinteractionswithnumbertheoryand
smooth dynamics.
Asthesubjectwasmaturinginthe1960s,GeorgeMackeyarguedtheneedfora
broader study of group actions that would also intimately relate to the theory of
unitary representations. This call has been answered in a remarkable way over the
lastfewdecadesnotonlythroughtherigiditytheoryofLiegroupsandtheirlattices
but also through related work in orbit equivalence and its connections to von
Neumann algebras. At the same time there has been a push to broaden the appli-
cation of classical ideas like entropy to actions of more general groups.
Despite the efforts of many mathematicians, a certain cultural division has
persisted in the wake of these various developments, although this has recently
beguntochangewiththechristeningofthesubjectofmeasuredgrouptheory.One
explanationforthedivisionisadifferenceinemphasisbetweentheasymptoticand
theperturbative.Ergodictheoryinitsclassicalsensestudiesasymptoticphenomena
like weak mixing and entropy (although the Rokhlin lemma often makes an
appearance as an indispensible perturbative device and points the way to notions
like amenability as the basis for generalizations to groups other than the integers),
while rigidity and its attendant concepts like amenability and property (T) fall into
the category of the perturbative (although they often appear in conjunction with
asymptotic behaviour like weak mixing and compactness).
Working within the general framework of countable acting groups, one of our
main intentions has been to promote a unified view of ergodic theory that sees the
asymptotic and the perturbative as two sides of the same coin. The common ele-
ment is the notion of independence, which takes on both probabilistic and com-
binatorial forms. Independence plays a crucial role in the celebrated dichotomy
between the asymptotic properties of weak mixing and compactness which
underliesHillelFurstenberg’sproofofSzemerédi’stheoremandleadstoastructure
v
vi Preface
theorem for measure-preserving actions. If we replace compactness by its pertur-
bative counterpart, then this dichotomy breaks down and what emerges at the two
extremes are the group-theoretic concepts of amenability and property (T).
Moreover,theperturbativepropertiesofamenabilityandsoficityformthestructural
basis for entropy, which is the preeminent asymptotic numerical invariant in
dynamics. What tightens this circle of ideas even further is the fact that weak
mixingandpositiveentropyreflectthetwobasicregimeswithwhichindependence
occurs over subsets of orbits. It is this last point which has dictated our division
of the book into two parts. This division is further explained in the introduction,
where the content and organization of the book are mapped out in greater detail.
Upuntilnowtherehasonlybeenonetreatiseonergodictheorywhichadoptsthe
general framework of countable groups, namely Eli Glasner’s Ergodic Theory via
Joinings[104],whichappearedalittlemorethanadecadeago.Eventhen,Glasner
restrictshisdiscussionofentropytheorytointegeractions,inpartbecausethelocal
theoryforactionsofmoregeneralgroupswasnotavailableatthetime.Intheyears
since,asubstantialamountofprogresshasbeenmadenotonlyonthislocaltheory
butalsoinextendingentropytheorybeyondtheamenablecasetotheclassofsofic
groups, a project which was initiated in a breakthrough of Lewis Bowen. At the
same time, Sorin Popa’s cocycle superrigidity theorems have opened up a new
chapterinrigiditytheorywhichiscentredmoreexclusivelyaroundrepresentation-
theoretic techniques. We believe not only that the time is ripe for a textbook
treatment of all of these newer topics, but also that they fit in a basic and integral
way into the picture of abstract ergodic theory that we have sketched above. We
have also aimed to provide a consolidated account of amenability and its ramifi-
cations for dynamics, ranging from the Rokhlin lemma and the pointwise ergodic
theorem to the Connes-Feldman-Weiss and Ornstein-Weiss theorems on orbit
equivalence,especiallysincetheseresultscanonlybefoundinsomewhatscattered
form across the literature.
Our intention has been to make the book flexible enough to serve a variety of
readers. Assuming some rudimentary functional analysis, measure theory, and
topology,webeginourdiscussionofdynamicsfromscratch,andpartsofthebook
canbeused(asbothoftheauthorshavedone)asanintroductorycourseonergodic
theory (for example, Sections 2.1-2.3, 4.1, 4.3-4.7, and 9.1-9.7). While we have
attempted to present the more advanced topics in an accessible way, one of our
goals hasbeen to make everything asself-contained as possible, which means that
we have not shied away from detailed and sometimes technical arguments. We
consequentlyhopethatthebookwillalsobeusefulasareferenceforthoseworking
in ergodic theory and related areas.
In addition to the probability-measure-preserving theory, we have included a
substantial amount of material on topological dynamics, which parallels and
interacts with measure-preserving dynamics in a number offundamental structural
ways, while being an important subject in its own right. The novelty of our treat-
ment is our emphasis on combinatorial independence and how it fits together with
both structure theory and entropy. Although we felt it important to present the
structure theorems of topological dynamics and their ramifications (Sections 7.3
Preface vii
and 8.4), this is the one place in the book where we have not supplied proofs, as
these would have taken us too far afield.
Many sources have helped to shape our understanding of ergodic theory as we
have presented it here. We acknowledge our debt to the classic texts of Halmos
[122], Walters [251], Petersen [208], Cornfeld–Fomin–Sinai [50], and Denker–
Grillenberger–Sigmund [56], to Furstenberg’s book on recurrence and combina-
torialnumbertheory[94],tothebookofGlasnermentionedabove[104],andtothe
books by Schmidt [225], Kechris–Miller [145], Tao [236], and Einsiedler–Ward
[72].Whilereferencesforspecificresultsaresometimesgiveninthemaintext,the
majority are collected together in notes at the end of each chapter, which also
contain supplementary information.
The first author has lectured on material from the book in various settings,
includinggraduate courses atTexas A&M University in 2007 and 2009 and atthe
University of Tokyo in 2013, as well as minicourses at the Fields Institute in
Toronto in2012and2013,attheAGORAmeetingon“Topological Dynamics”at
the Ferme de Courcimont in France in 2012, and in the conference “Dynamics,
Geometry,andOperatorAlgebras”atTexasA&MUniversityin2013.Thesecond
authorusedthebookasabasisforgraduatecoursesatSUNYBuffaloin2014and
2015andatChongqingUniversityin2016.Wethankallofthosewhoattendedfor
their participation and feedback. We would especially like to thank Yuki Arano,
March Boedihardjo, Michael Brannan, Damien Gaboriau, Ben Hayes, Huichi
Huang,XiaojunHuang,AdrianIoana,ZhengxingLian,XinMa,ZhenRong,Song
Shao, Robin Tucker-Drob, and Changrong Zhu for discussions, comments, and
corrections. We also gratefully acknowledge the support of our departments, the
NSF, and the NSFC.
College Station, TX, USA David Kerr
Buffalo, NY, USA Hanfeng Li
Contents
1 General Framework and Notational Conventions.. .... ..... .... 1
1.1 Groups . .... .... ..... .... .... .... .... .... ..... .... 1
1.2 Probability Spaces. ..... .... .... .... .... .... ..... .... 1
1.3 Measure Algebras. ..... .... .... .... .... .... ..... .... 2
1.4 Standard Probability Spaces .. .... .... .... .... ..... .... 3
1.5 Group Actions ... ..... .... .... .... .... .... ..... .... 4
1.6 Measure Conjugacy Versus Measure Algebra Conjugacy. .... 7
1.7 Function Spaces .. ..... .... .... .... .... .... ..... .... 9
1.8 Hilbert Space Operators and Unitary Representations.... .... 10
1.9 The Koopman Representation. .... .... .... .... ..... .... 14
1.10 Conditional Expectations .... .... .... .... .... ..... .... 14
1.11 The Spectral Theorem and the Borel Functional Calculus .... 15
1.12 C(cid:1)-Algebras and von Neumann Algebras.... .... ..... .... 17
Part I Weak Mixing and Compactness
2 Basic Concepts in Ergodic Theory.. .... .... .... .... ..... .... 21
2.1 Ergodicity, Freeness, and Poincaré recurrence .... ..... .... 21
2.2 Mixing, Weak Mixing, and Compactness.... .... ..... .... 26
2.3 Examples ... .... ..... .... .... .... .... .... ..... .... 36
2.3.1 Bernoulli Actions.... .... .... .... .... ..... .... 36
2.3.2 Rotations of the Circle.... .... .... .... ..... .... 40
2.3.3 Skew Transformations of the Torus.. .... ..... .... 41
2.3.4 Odometers..... .... .... .... .... .... ..... .... 42
2.3.5 Actions by Automorphisms of Compact Groups. .... 43
2.3.6 Gaussian Actions.... .... .... .... .... ..... .... 45
2.4 Notes and References... .... .... .... .... .... ..... .... 47
3 Structure Theory for p.m.p. Actions .... .... .... .... ..... .... 49
3.1 Hilbert Modules from Factors of Probability Spaces..... .... 50
3.2 The Furstenberg–Zimmer Structure Theorem . .... ..... .... 56
ix
x Contents
3.3 Multiple Recurrence and Szemerédi’s Theorem ... ..... .... 64
3.3.1 SMR is Preserved Under Weakly Mixing
Extensions..... .... .... .... .... .... ..... .... 65
3.3.2 SMR is Preserved Under Compact Extensions .. .... 69
3.3.3 SMR and Szemerédi’s Theorem .... .... ..... .... 71
3.4 Notes and References... .... .... .... .... .... ..... .... 72
4 Amenability... .... .... ..... .... .... .... .... .... ..... .... 73
4.1 Basic Theory. .... ..... .... .... .... .... .... ..... .... 74
4.2 Amenability and Unitary Representations.... .... ..... .... 80
4.3 Ergodicity, Weak Mixing, and the Mean Ergodic Theorem ... 85
4.4 The Pointwise Ergodic Theorem... .... .... .... ..... .... 87
4.5 Quasitilings and the Subadditivity Theorem .. .... ..... .... 91
4.6 The Ornstein–Weiss Quasitower Theorem ... .... ..... .... 96
4.7 Asymptotic Averages as Infima ... .... .... .... ..... .... 102
4.8 The Connes–Feldman–Weiss Theorem.. .... .... ..... .... 104
4.8.1 P.m.p. Equivalence Relations... .... .... ..... .... 105
4.8.2 Amenability, Hyperfiniteness, and Reiter's
Property.. ..... .... .... .... .... .... ..... .... 108
4.8.3 The Connes–Feldman–Weiss Theorem ... ..... .... 114
4.9 Dye’s Theorem and the Ornstein–Weiss Theorem . ..... .... 121
4.10 Notes and References... .... .... .... .... .... ..... .... 127
5 Property (T)... .... .... ..... .... .... .... .... .... ..... .... 131
5.1 Basic Theory. .... ..... .... .... .... .... .... ..... .... 132
5.2 Characterization in Terms of Isolated Points in the Unitary
Dual ... .... .... ..... .... .... .... .... .... ..... .... 134
5.3 Characterization in Terms of Weak Mixing .. .... ..... .... 138
5.4 Characterization in Terms of Strong Ergodicity ... ..... .... 140
5.5 Generic Weak Mixing and Property (T). .... .... ..... .... 142
5.6 Notes and References... .... .... .... .... .... ..... .... 145
6 Orbit Equivalence Beyond Amenability.. .... .... .... ..... .... 147
6.1 Popa’s Cocycle Superrigidity . .... .... .... .... ..... .... 148
6.2 Bernoulli Actions Over Free Groups.... .... .... ..... .... 157
6.3 Notes and References... .... .... .... .... .... ..... .... 162
7 Topological Dynamics... ..... .... .... .... .... .... ..... .... 163
7.1 Minimality, Topological Transitivity, and Birkhoff
Recurrence .. .... ..... .... .... .... .... .... ..... .... 163
7.2 Weak Mixing and Equicontinuity.. .... .... .... ..... .... 168
7.3 Proximality, Distality, and Structure Theorems.... ..... .... 172
7.4 Notes and References... .... .... .... .... .... ..... .... 177
8 Tameness and Independence .. .... .... .... .... .... ..... .... 179
8.1 Ramsey Theory and a Dichotomy of Rosenthal ... ..... .... 180
8.2 Tameness and IT-Tuples. .... .... .... .... .... ..... .... 182