# Ergodic Actions Of Abelian Groups And Properties Of Their Joint Actions

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### Workshop on von Neumann algebras and group actions Copenhagen

then the groups are isomorphic and the actions are conjugate. Paul Jolissaint (Universit´e de Neuchatel) : Mixing MASAs in group factors In 1954, J. Dixmier identiﬁed three types of maximal abelian -subalgebras in ﬁnite factors in terms of the size of their normalizer in the ambient factor. Since then, interested sub-

### PROCEEDINGS OF SYMPOSIA

Actions of non-abelian groups and invariant T R. LIMA Cohomological invariants for groups of outer automorphisms algebras COLIN E. SUTHERLAND Automorphism groups and invariant states ERLING ST0RMER Compact ergodic groups of automorphisms MAGNUS B. LANDSTAD Actions of discrete groups on factors V. F. R. JONES Ergodic theory and von Neumann algebras

### arXiv:0806.0430v1 [math.DS] 3 Jun 2008

applications in the study of orbit equivalence of actions see Epstein [E]. For further potential applications of this method, it seems that one should have a better understanding of the connection of ergodic properties between a,bas above. We show, for example, that if b 0 is free, mixing and a 0 is ergodic, then: ais mixing ⇒ bis mixing.

### INVARIANT MEASURES FOR MULTIPARAMETER DIAGONALIZABLE

as actions on tori, and actions on totally disconnected groups. I also do not cover my own work on arithmetic quantum unique ergodicity, which is closely related to the topics I survey here; the interested reader can consult [17] or the expository papers [20, 19]. 2. More general algebraic actions

### Motivation and statement of the main result

A from countably in nite abelian groups to freem ergodic actions of satisfying: (1) A˘=A0if and only if A ˘ A0, and (2) if Ais a collection of abelian groups such that f A: A2Agare pairwise not conjugate but are all OE, then Ais countable. With this, it follows that there is a countable-to-one (Borel) reduction from isomorphism of abelian groups

### KURT W VINHAGE

Joint with Zhenqi Jenny Wang COCYCLE RIGIDITY OF PARTIALLY HYEPERBOLIC ABELIAN ACTIONS WITH ALMOST RANK ONE FACTORS Ergodic Theory Dynam. Systems 39 (2019), no. 7, 2006-2016. ON THE NON-EQUIVALENCE OF THE BERNOULLI AND K PROPERTIES IN DIMENSION FOUR J. Mod. Dyn. 13 (2018), 221-250. Joint with Adam Kanigowski and Federico Rodriguez-Hertz ON THE

### Ergodic Theory and Number Theory - univie.ac.at

measures on T which are invariant and ergodic under the joint action of T pand T q with (p;q) = 1.7 Theorem 3 ([29, Theorem 4.9]). Let p;q>1 be two relatively prime integers, and let be a nonatomic probability measure on T which is invariant and ergodic under the joint action of T pand T q. If h (T p) >0 then =

### MEASURE RIGIDITY BEYOND UNIFORM HYPERBOLICITY: RESULTS AND

Algebraic actions Goal: every ergodic Borel probability invariant measure for an action of a higher rank abelian group is either essentially of algebraic nature or comes from a rank one factor on an algebraic invariant set. This problem was discussed in the talk by Elon Lindenstrauss primarily for homogeneous actions.

### TITLES AND ABSTRACTS David Aulicino

smoothness and singularities. Joint work with Tobias Hurth, Jonathan Mattingly and Sean Lawley. Aaron Brown (Chicago) Title: Ergodic theory of lattice actions in low dimensions Abstract: Given a lattice in a higher-rank semisimple Lie group G, we consider C2 actions of the lattice on a manifold M. We show that if the

### Mathematisches Forschungsinstitut Oberwolfach

), asymptotic geometry of ﬁnitely generated groups and Lie groups (E. Breuillard, G. Margulis). Interestingly enough, quasi-morphisms a group theoretical notion which was

### AFFINE ISOMETRIC ACTIONS OF DISCRETE GROUPS

Asymptotic median structure of mapping class groups, applications to homomor-phisms. Median spaces are non-discrete versions of CAT(0) cubical complexes. Both Kazh-dan and Haagerup properties can be formulated in terms of actions of groups on median spaces. Moreover, it turns out that every asymptotic cone of a mapping class

### GLOBAL RIGIDITY OF HIGHER RANK ANOSOV ACTIONS ON TORI AND

not adapt easily to the case of actions of higher rank abelian groups. Indeed because of the delicate cutting and pasting arguments used in their constructions, it would be hard to guarantee that di erent elements continue to commute. As a consequence of Theorem 1.1, a positive answer to Question 1.4 can only occur for an action where

### CURRICULUM VITAECURRICULUM VITAE

Ergodic Theory : Nonsingular groups actions and their properties 2 Estimation of angular and hidden angular densities using the kernel method (joint work with

### Benjamin Weiss: Ergodic theory beyond amenable groups

Benjamin Weiss: Ergodic theory beyond amenable groups In the last few years there has been great progress in extending the classical aspects of ergodic theory to actions of non-amenable groups. I will survey a part of this activity and in particular present a new proof of Kolmogorov s theorem that isomorphic Bernoulli shifts have the same

### PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Volume 38, Part 1

Actions of non-abelian groups and invariant T R. LIMA Cohomological invariants for groups of outer automorphisms algebras COLIN E. SUTHERLAND Automorphism groups and invariant states ERLING ST0RMER Compact ergodic groups of automorphisms MAGNUS B. LANDSTAD Actions of discrete groups on factors V. F. R. JONES Ergodic theory and von Neumann algebras

### A Thesis Submitted for the Degree of PhD at the - Warwick

1All actions are assumed to be actions by continuous automorphisms, that is they are given by a homomorphism F-+ Aut(X) where Aut(X) is the group of continuous automorphisms of X. Such actions are convenient for ergodic theory because they are automatically measure-preserving actions on the probability space (X, Borel sets, Haar measure).

### ON THE WORK OF RODRIGUEZ HERTZ ON RIGIDITY IN DYNAMICS

of this seemingly algebraic fact uses ideas from dynamics and ergodic theory. The presence of higher rank abelian subgroups in ¡ consisting of semisimple elements is key. Later, R. J. Zimmer formulated the program of analyzing smooth actions of higher rank semisimple Lie groups and their lattices on manifolds [67]. We refer

### On the Ext-group of an AF algebra 23 (1978), 251-267.

33. S. Popa, A. Wassermann: Actions of compact Lie groups on von Neumann algebras, Comptes Rendus de l Academie de Sciences de Paris, 315 (1992), 421-425, 34. S. Popa: On the classi cation of actions of amenable groups on subfactors, Comptes Ren. Acad. Sci. Paris, 315 (1992), 295-299 35. S.

### Subequivalence Relations and Positive-Deﬁnite Functions

applications in the study of orbit equivalence of actions see Epstein [E]. For further potential applications of this method, it seems that one should have a better understanding of the connection of ergodic properties between a,bas above. We show, for example, that if b 0 is free, mixing and a 0 is ergodic, then: ais mixing ⇒ bis mixing.

### 2020 Vision for Dynamics - IM PAN

Measure rigidity and projective actions of lattices Aaron Brown I will discuss ariousv rigidity properties of projective actions of higher-rank lattices. oT establish the main results, we show a measure classi cation theorem for certain actions of higher-rank abelian groups; this closely follows work of A. Katok and his collaborators.

### ENDOMORPHISMS OF MEASURED EQUIVALENCE RELATIONS, LOCALLY

(ii) extensions of amenable l.c.s.c. groups by (i)-type groups. Generic properties of cocycles with values in these groups are discussed. The last Section 3 is organized like §§5,6 from [Da3]. We start with the lifting theory in orbital setting. Many of the results here are only slight modiﬁcations of their invertible analogues from

### GLOBAL RIGIDITY OF HIGHER RANK ANOSOV ACTIONS ON TORI AND

GLOBAL RIGIDITY OF HIGHER RANK ANOSOV ACTIONS ON TORI AND NILMANIFOLDS DAVID FISHER, BORIS KALININ, RALF SPATZIER (WITH AN APPENDIX BY JAMES F. DAVIS ) Abstract. We show that su ciently irreducible Anosov actions of higher rank abelian groups on tori and nilmanifolds are C1-conjugate to a ne actions. 1. Introduction

### Diagonalizable ﬂows on locally homogeneous spaces and number

classiﬁcation of invariant measures, and their applications. A basic invariant in ergodic theory is the ergodic theoretic entropy introduced by A. Kolmogorov and Ya. Sinai. This invariant plays a surprisingly big role in the study of actions of diagonalizable groups on locally homogeneous spaces as well as in the applications.

### DYNAMICAL AND ALGEBRAIC PROPERTIES OF ALGEBRAIC ACTIONS

DYNAMICAL AND ALGEBRAIC PROPERTIES OF ALGEBRAIC ACTIONS NHAN-PHU CHUNG Actions of countable discrete groups on compact (metrizable) groups Xby (con-tinuous) automorphisms are a rich class of dynamical systems, and have drawn much attention since the beginning of ergodic theory. The fact that Z[Zd] is a commu-

### Abstracts of the Talks - Institute for Research in

Operator Algebras and their Applications, January 6-9, 2020 School of Mathematics, IPM, Tehran, Iran Mehrdad Kalantar (University of Houston, USA) Rigidity Phenomena in Non-Commutative Ergodic Theory Abstract: We study dynamical and ergodic properties of groups, from the point of view of their associated C* and von Neumann algebras.

### Abstracts

force their degrees to be bounded, i. e. locally simplify the triangulation. Lei Chen California Institute of Technology Actions of Homeo and Di eo groups on manifolds Coauthors: Kathryn Mann In this talk, I discuss the general question of how to obstruct and construct group actions on manifolds.

### Thematic lectures Andrei Lerner Bar-Ilan University, Israel

Joint work with Parasar Mohanty and Saurabh Srivastava. Guixiang Hong Wuhan University, P. R. China Title: Some progresses on noncommutative ergodic theory Abstract: In this talk, I shall rst give a historical review on the ergodic theory associated to group actions. Then I shall present our recent works on noncommutative ergodic theory.

### Rigidity, Dynamics, and Group Actions

M. Handel explained his joint work with Franks on ﬁxed points for actions of higher rank abelian groups on R2 and S2. Higher rank abelian actions have been prominent in recent years, due to the discovery of many rigidity properties. The work of Franks and Handel again shows that such actions are very special.

### JOINT DYNAMICS

3. new and exciting results regarding mixing properties of actions on zero dimensional compact abelian groups by commuting automorphisms were presented, with applications to rigidity. Partial list of problems from problem session. A fruitful problem session was held. Among the problems presented were the following: 1.

### Emmanuel Breuillard

systems, Abelian (or, more generally, Moore group) actions, and measure preserving systems. This is a joint work with Benjy Weiss. Vadim Kaimanovich Title: Random walks with the same Poisson boundary (joint with Behrang Forghani) Abstract: I will show that Markov stopping times (and their randomizations) of ran-

### Abstracts - UH

11. A. Katok, V. Nit˘ic a and A. T or ok. Nonabelian cohomology of abelian Anosov actions, Ergodic Theory and Dynamical Systems 20 (2000), 259{288. We develop a new technique for calculating the rst cohomology of certain classes of actions of higher{rank abelian groups (Zk and Rk, k 2) with values in a linear Lie group.

### Banach Algebras and Abstract Harmonic Analysis

Ergodic theory for quantum (semi)groups We shall describe a generalization of recent work on aspects of ergodic theory of semigroup actions on von Neumann algebras to the context of quantum semigroups. These results give a Jacobs-de Leeuw-Glicksberg splitting at the von Neumann algebra level. DILIAN YANG, University of Windsor

### GROUP ACTIONS ON INJECTIVE FACTORS

factors of type III in Section 3. This is a joint work with M. Takesaki. Sutherland and Takesaki classiﬁed discrete amenable group actions on injective factors of type III λ,0<λ<1,38 and these two and the author classiﬁed discrete abelian group actions on injective factors of type III1.26 Since abelian groups are amenable, we

### SRB Measures for Anosov actions

exist and they prove that this holds for certain algebraic actions. Another work in that direction of classifying measures with positive entropy for actions is the result of Kalinin-Katok-Rodriguez Hertz [KKRH11] showing that for a locally free Anosov abelian action (not necessarily algebraic) with dimM= 2 + 1 with 2, an invariant ergodic mea-

### Krieger s Finite Generator Theorem for Ergodic Actions of

Rokhlin s theorem was generalized to actions of abelian groups by Conze [12] in 1972 and was just recently extended to amenable groups by Seward and Tucker-Drob [48]. Speci cally, if Gy (X; ) is a free ergodic p:m:p:action of an amenable group then the entropy h G(X; ) is equal to the in mum of H( ) over all countable generating partitions

### Invited talks - UniPD

Entropy of actions of amenable groups Antongiulio Fornasiero Hebrew University of Jerusalem Let G be an amenable group (or more generally, cancellative monoid). We describe the entropy of the action of G on various kind of spaces For actions on Abelian groups, we have the so-called algebraic entropy.

### Diagonalizable ﬂows on locally homogeneous spaces and number

their applications. A basic invariant in ergodic theory is the ergodic theoretic entropy introduced by A. Kolmogorov andYa. Sinai. This invariant plays a surprisingly big role in the study of actions of diagonalizable groups on locally homogeneous spaces as well as in the applications. We discuss entropy and how it naturally arises in several

### Titles and abstracts of talks - math.ucsd.edu

These groups were introduced by Baumslag and Solitar to provide the rst examples of nitely presented non-Hop an groups. We examine both the group von Neumann algebra of the Baumslag-Solitar groups as the group measure space construction of some of their actions. In the case of the group von Neumann algebra, the rational number jn=mjis an invariant.