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

1981 ◽  
Vol 1 (2) ◽  
pp. 223-236 ◽  
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.

scholarly journals Sofic entropy and amenable groups

2011 ◽  
Vol 32 (2) ◽  
pp. 427-466 ◽  
Author(s):  
LEWIS BOWEN
Keyword(s):  
Group Action ◽  
Amenable Group ◽  
Group Actions ◽  
Amenable Groups ◽  
Orbit Equivalent ◽  
Sofic Entropy ◽  
Classical Entropy

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.

Recurrence in mean for group actions

2019 ◽  
Vol 20 (01) ◽  
pp. 2050006
Author(s):  
Cao Zhao ◽  
Yong Ji
Keyword(s):  
Hausdorff Measure ◽  
Metric Space ◽  
Group Actions ◽  
Finite Measure ◽  
Countable Group ◽  
General Group ◽  
Mean Values ◽  
Measurable Set ◽  
The Mean

In this paper, the mean values of the recurrence are computed for general group actions. Let [Formula: see text] be a metric space with a finite measure [Formula: see text] and [Formula: see text] be a countable group acting on [Formula: see text]. Let [Formula: see text] be a sequence of subsets of [Formula: see text] with [Formula: see text] and put [Formula: see text]. If the Hausdorff measure [Formula: see text] is finite on [Formula: see text] and [Formula: see text] is [Formula: see text]-invariant. We assume that [Formula: see text] and [Formula: see text] are concordant. Then the function [Formula: see text] is [Formula: see text]-integrable and for any [Formula: see text]-measurable set [Formula: see text] we have [Formula: see text] If moreover, [Formula: see text] then [Formula: see text] without the concordance condition for the measure [Formula: see text] and [Formula: see text]

Cocycle and orbit equivalence superrigidity for coinduced actions

2017 ◽  
Vol 38 (7) ◽  
pp. 2644-2665 ◽  
Author(s):  
DANIEL DRIMBE
Keyword(s):  
Probability Measure ◽  
Large Class ◽  
Countable Group ◽  
Rigidity Theory ◽  
Orbit Equivalence ◽  
Amenable Groups ◽  
Property T

We prove a cocycle superrigidity theorem for a large class of coinduced actions. In particular, if $\unicode[STIX]{x1D6EC}$ is a subgroup of a countable group $\unicode[STIX]{x1D6E4}$, we consider a probability measure preserving action $\unicode[STIX]{x1D6EC}\curvearrowright X_{0}$ and let $\unicode[STIX]{x1D6E4}\curvearrowright X$ be the coinduced action. Assume either that $\unicode[STIX]{x1D6E4}$ has property (T) or that $\unicode[STIX]{x1D6EC}$ is amenable and $\unicode[STIX]{x1D6E4}$ is a product of non-amenable groups. Using Popa’s deformation/rigidity theory we prove $\unicode[STIX]{x1D6E4}\curvearrowright X$ is ${\mathcal{U}}_{\text{fin}}$-cocycle superrigid, that is any cocycle for this action to a ${\mathcal{U}}_{\text{fin}}$ (e.g. countable) group ${\mathcal{V}}$ is cohomologous to a homomorphism from $\unicode[STIX]{x1D6E4}$ to ${\mathcal{V}}.$

scholarly journals Approximately transitive flows and ITPFI factors

1985 ◽  
Vol 5 (2) ◽  
pp. 203-236 ◽  
Author(s):  
A. Connes ◽  
E. J. Woods
Keyword(s):  
Lebesgue Measure ◽  
Group Action ◽  
Measure Space ◽  
Finite Measure ◽  
Type Iii ◽  
Ergodic Transformations ◽  
Zero Entropy ◽  
Approximate Transitivity

AbstractWe define a new property of a Borel group action on a Lebesgue measure space, which we call approximate transitivity. Our main results are (i) a type III0 hyperfinite factor is ITPFI if and only if its flow of weights is approximately transitive, and (ii) for ergodic transformations preserving a finite measure, approximate transitivity implies zero entropy.

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

2018 ◽  
Vol 28 (02) ◽  
pp. 1850028 ◽  
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.

On the Integrability of the Ergodic Hilbert Transform for A Class of Functions with Equal Absolute Values

1998 ◽  
Vol 5 (2) ◽  
pp. 101-106
Author(s):  
L. Ephremidze
Keyword(s):  
Hilbert Transform ◽  
Measure Space ◽  
Integrable Function ◽  
Finite Measure ◽  
Ergodic Transformation ◽  
Finite Measure Space ◽  
Class Of Functions

Abstract It is proved that for an arbitrary non-atomic finite measure space with a measure-preserving ergodic transformation there exists an integrable function f such that the ergodic Hilbert transform of any function equal in absolute values to f is non-integrable.

Lacunary ideal convergence in measure for sequences of fuzzy valued functions

2021 ◽  
Vol 40 (3) ◽  
pp. 5517-5526
Author(s):  
Ömer Kişi
Keyword(s):  
Measure Space ◽  
Finite Measure ◽  
Ideal Convergence ◽  
Convergence In Measure ◽  
Ideal Form ◽  
Measurable Functions ◽  
Finite Measure Space ◽  
Convergence Of Sequences

We investigate the concepts of pointwise and uniform I θ -convergence and type of convergence lying between mentioned convergence methods, that is, equi-ideally lacunary convergence of sequences of fuzzy valued functions and acquire several results. We give the lacunary ideal form of Egorov’s theorem for sequences of fuzzy valued measurable functions defined on a finite measure space ( X , M , μ ) . We also introduce the concept of I θ -convergence in measure for sequences of fuzzy valued functions and proved some significant results.

On Unsymmetric Dirichlet Forms

1973 ◽  
Vol 25 (2) ◽  
pp. 252-260 ◽  
Author(s):  
Joanne Elliott
Keyword(s):  
Compact Space ◽  
Measure Space ◽  
Closed Subspace ◽  
Finite Measure ◽  
Dirichlet Forms ◽  
Locally Compact ◽  
Locally Compact Space ◽  
Finite Measure Space

Let F be a linear, but not necessarily closed, subspace of L2[X, dm], where (X,,m) is a σ-finite measure space with the Borel subsets of the locally compact space X. If u and v are measureable functions, then v is called a normalized contraction of u if and Assume that F is stable under normalized contractions, that is, if u ∈ F and v is a normalized contraction of u, then v ∈ F.

A Packing Problem for Measurable Sets

1967 ◽  
Vol 19 ◽  
pp. 749-756
Author(s):  
D. Sankoff ◽  
D. A. Dawson
Keyword(s):  
Probability Measure ◽  
Measure Space ◽  
Packing Problem ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Probability Measure Space

Given a probability measure space (Ω,,P)consider the followingpacking problem.What is the maximum number,b(K,Λ), of sets which may be chosen fromso that each set has measureKand no two sets have intersection of measure larger than Λ <K?In this paper the packing problem is solved for any non-atomic probability measure space. Rather than obtaining the solution explicitly, however, it is convenient to solve the followingminimal paving problem.In a non-atomic a-finite measure space (Ω,,μ)what is the measure,V(b, K,Λ), of the smallest set which is the union of exactlybsubsets of measureKsuch that no subsets have intersection of measure larger than Λ?

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