Sofic entropy and amenable groups

AbstractIn previous work, the author introduced a measure-conjugacy invariant for sofic group actions called sofic entropy. Here, it is proven that the sofic entropy of an amenable group action equals its classical entropy. The proof uses a new measure-conjugacy invariant called upper-sofic entropy and a theorem of Rudolph and Weiss for the entropy of orbit-equivalent actions relative to the orbit changeσ-algebra.

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Mean Proximality, Mean Sensitivity and Mean Li–Yorke Chaos for Amenable Group Actions

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127418500281 ◽

2018 ◽

Vol 28 (02) ◽

pp. 1850028 ◽

Cited By ~ 1

Author(s):

Kesong Yan ◽

Fanping Zeng

Keyword(s):

Abelian Group ◽

Periodic Point ◽

Group Action ◽

Amenable Group ◽

Group Actions ◽

Amenable Groups ◽

Transitive Point ◽

Infinite Abelian Group ◽

Mean Sensitivity

We consider mean proximality and mean Li–Yorke chaos for [Formula: see text]-systems, where [Formula: see text] is a countable discrete infinite amenable group. We prove that if a countable discrete infinite abelian group action is mean sensitive and there is a mean proximal pair consisting of a transitive point and a periodic point, then it is mean Li–Yorke chaotic. Moreover, we give some characterizations of mean proximal systems for general countable discrete infinite amenable groups.

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Amenability, Kazhdan's property T, strong ergodicity and invariant means for ergodic group-actions

Ergodic Theory and Dynamical Systems ◽

10.1017/s014338570000924x ◽

1981 ◽

Vol 1 (2) ◽

pp. 223-236 ◽

Cited By ~ 43

Author(s):

Klaus Schmidt

Keyword(s):

Group Action ◽

Measure Space ◽

Group Actions ◽

Finite Measure ◽

Countable Group ◽

Invariant Sets ◽

Amenable Groups ◽

Strong Ergodicity ◽

Property T ◽

Invariant Means

AbstractThis paper discusses the relations between the following properties o finite measure preserving ergodic actions of a countable group G: strong ergodicity (i.e. the non-existence of almost invariant sets), uniqueness of G-invariant means on the measure space carrying the group action, and certain cohomological properties. Using these properties one can characterize all actions of amenable groups and of groups with Kazhdan's property T. For groups which fall in between these two definations these notions lead to some interesting examples.

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Weak amenability of free products of hyperbolic and amenable groups

Glasgow Mathematical Journal ◽

10.1017/s0017089521000458 ◽

2022 ◽

pp. 1-4

Author(s):

Ignacio Vergara

Keyword(s):

Free Product ◽

Hyperbolic Group ◽

Amenable Group ◽

Free Products ◽

Amenable Groups ◽

Weak Amenability ◽

Orbit Equivalent

Abstract We show that if G is an amenable group and H is a hyperbolic group, then the free product $G\ast H$ is weakly amenable. A key ingredient in the proof is the fact that $G\ast H$ is orbit equivalent to $\mathbb{Z}\ast H$ .

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Examples in the entropy theory of countable group actions

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2019.18 ◽

2019 ◽

Vol 40 (10) ◽

pp. 2593-2680 ◽

Cited By ~ 2

Author(s):

LEWIS BOWEN

Keyword(s):

Classical Theory ◽

Group Actions ◽

Countable Group ◽

Classification Theory ◽

Entropy Theory ◽

Survey Article ◽

Sofic Entropy ◽

Factor Maps ◽

Measure Preserving Actions ◽

Classical Entropy

Kolmogorov–Sinai entropy is an invariant of measure-preserving actions of the group of integers that is central to classification theory. There are two recently developed invariants, sofic entropy and Rokhlin entropy, that generalize classical entropy to actions of countable groups. These new theories have counterintuitive properties such as factor maps that increase entropy. This survey article focusses on examples, many of which have not appeared before, that highlight the differences and similarities with classical theory.

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Local Entropy, Metric Entropy and Topological Entropy for Countable Discrete Amenable Group Actions

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416501108 ◽

2016 ◽

Vol 26 (07) ◽

pp. 1650110

Author(s):

Xiankun Ren ◽

Wenxiang Sun

Keyword(s):

Topological Entropy ◽

Group Action ◽

Metric Space ◽

Amenable Group ◽

Group Actions ◽

Entropy Function ◽

Metric Entropy ◽

Local Entropy ◽

Compact Metric Space ◽

The Difference

Let [Formula: see text] be a compact metric space and [Formula: see text] a countable infinite discrete amenable group acting on [Formula: see text]. Like in the [Formula: see text]-action cases we define the notion of local entropy and by it we bound the difference between metric entropy and that of a partition, and bound the difference between topological entropy and that of a separated set, which generalize Theorems 1(1) and 1(2) in [Newhouse, 1989] from [Formula: see text]-actions to amenable group actions. We further prove that the entropy function [Formula: see text] is upper semi-continuous on [Formula: see text] for an asymptotic entropy expansive amenable group action.

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On a construction of unitary cocycles and the representation theory of amenable groups

Ergodic Theory and Dynamical Systems ◽

10.1017/s014338570000184x ◽

1983 ◽

Vol 3 (1) ◽

pp. 129-133 ◽

Cited By ~ 1

Author(s):

Colin E. Sutherland

Keyword(s):

Representation Theory ◽

Probability Space ◽

Amenable Group ◽

Unitary Representations ◽

Intertwining Operators ◽

Type I ◽

Amenable Groups

AbstractIf K is a countable amenable group acting freely and ergodically on a probability space (Γ, μ), and G is an arbitrary countable amenable group, we construct an injection of the space of unitary representations of G into the space of unitary 1-cocyles for K on (Γ, μ); this injection preserves intertwining operators. We apply this to show that for many of the standard non-type-I amenable groups H, the representation theory of H contains that of every countable amenable group.

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