Joint ergodicity of actions of an abelian group

2013 ◽  
Vol 34 (4) ◽  
pp. 1353-1364
Author(s):  
YOUNGHWAN SON
Keyword(s):  
Abelian Group ◽  
Probability Space ◽  
Countable Abelian Group

AbstractLet $G$ be a countable abelian group and let ${T}^{(1)} , {T}^{(2)} , \ldots , {T}^{(s)} $ be measure preserving $G$-actions on a probability space. We prove that joint ergodicity of ${T}^{(1)} , {T}^{(2)} , \ldots , {T}^{(s)} $ implies total joint ergodicity if each ${T}^{(i)} $ is totally ergodic. We also show that if $G= { \mathbb{Z} }^{d} $, $s\geq d+ 1$ and the actions ${T}^{(1)} , {T}^{(2)} , \ldots , {T}^{(s)} $ commute, then total joint ergodicity of ${T}^{(1)} , {T}^{(2)} , \ldots , {T}^{(s)} $ follows from joint ergodicity. This can be seen as a generalization of Berend’s result for commuting $ \mathbb{Z} $-actions.

scholarly journals Mixing properties of induced random transformations

1997 ◽  
Vol 17 (4) ◽  
pp. 839-847 ◽  
Author(s):  
HANS-OTTO GEORGII
Keyword(s):  
Random Walk ◽  
Abelian Group ◽  
Probability Space ◽  
Return Time ◽  
Skew Product ◽  
Mixing Properties ◽  
Countable Abelian Group ◽  
Measure Preserving Transformations

Let $S(N)$ be a random walk on a countable abelian group $G$ which acts on a probability space $E$ by measure-preserving transformations $(T_v)_{v\in G}$. For any $\Lambda \subset E$ we consider the random return time $\tau$ at which $T_{S(\tau)}\in\Lambda$. We show that the corresponding induced skew product transformation is K-mixing whenever a natural subgroup of $G$ acts ergodically on $E$.

Multiple Mixing and Rank One Group Actions

2000 ◽  
Vol 52 (2) ◽  
pp. 332-347 ◽  
Author(s):  
Andrés del Junco ◽  
Reem Yassawi
Keyword(s):  
Abelian Group ◽  
Large Class ◽  
Probability Space ◽  
Infinite Order ◽  
Group Actions ◽  
Intersection Property ◽  
Product Measure ◽  
Multiple Mixing ◽  
Rank One ◽  
Countable Abelian Group

AbstractSuppose G is a countable, Abelian group with an element of infinite order and let be amixing rank one action of G on a probability space. Suppose further that the Følner sequence {Fn} indexing the towers of satisfies a “bounded intersection property”: there is a constant p such that each {Fn} can intersect no more than p disjoint translates of {Fn}. Then is mixing of all orders. When G = Z, this extends the results of Kalikow and Ryzhikov to a large class of “funny” rank one transformations. We follow Ryzhikov’s joining technique in our proof: the main theorem follows from showing that any pairwise independent joining of k copies of is necessarily product measure. This method generalizes Ryzhikov’s technique.

scholarly journals On uniformly distributed sequences of integers and Poincaré recurrence III

2003 ◽  
Vol 68 (2) ◽  
pp. 345-350
Author(s):  
R. Nair
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Probability Space ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Bohr Compactification ◽  
Poincare Recurrence ◽  
Uniformly Distributed Sequences ◽  
Image Position ◽  
Measure Preserving Transformations

Let S be a semigroup contained in a locally compact Abelian group G. Let Ĝ denote the Bohr compactification of G. We say that a sequence contained in S is Hartman uniform distributed on G iffor any character χ in Ĝ. Suppose that (Tg)g∈s is a semigroup of measurable measure preserving transformations of a probability space (X, β, μ) and B is an element of the σ-algebra β of positive μ measure. For a map T: X → X and a set A ⊆ X let T−1A denote {x ∈ X: Tx ∈ A}. In an earlier paper, the author showed that if k is Hartman uniform distributed thenIn this paper we show that ≥ cannot be replaced by =. A more detailed discussion of this situation ensues.

scholarly journals Plünnecke inequalities for countable abelian groups

2017 ◽  
Vol 2017 (730) ◽  
Author(s):  
Michael Björklund ◽  
Alexander Fish
Keyword(s):  
Abelian Group ◽  
Probability Measure ◽  
Lower Bounds ◽  
Correspondence Principle ◽  
Additive Basis ◽  
Banach Density ◽  
New Form ◽  
Countable Abelian Group ◽  
Product Sets ◽  
Measure Preserving Actions

AbstractWe establish in this paper a new form of Plünnecke-type inequalities for ergodic probability measure-preserving actions of any countable abelian group. Using a correspondence principle for product sets, this allows us to deduce lower bounds on the upper and lower Banach densities of any product set in terms of the upper Banach density of an iterated product set of one of its addends. These bounds are new already in the case of the integers.We also introduce the notion of an ergodic basis, which is parallel, but significantly weaker than the analogous notion of an additive basis, and deduce Plünnecke bounds on their impact functions with respect to both the upper and lower Banach densities on any countable abelian group.

Frames and Single Wavelets for Unitary Groups

2002 ◽  
Vol 54 (3) ◽  
pp. 634-647 ◽  
Author(s):  
Eric Weber
Keyword(s):  
Hilbert Space ◽  
Abelian Group ◽  
Multiresolution Analysis ◽  
Separable Hilbert Space ◽  
Unitary Groups ◽  
Analytic Method ◽  
Generalized Frame ◽  
Countable Abelian Group ◽  
Frame Multiresolution Analysis ◽  
Stone’S Theorem

AbstractWe consider a unitary representation of a discrete countable abelian group on a separable Hilbert space which is associated to a cyclic generalized frame multiresolution analysis. We extend Robertson’s theorem to apply to frames generated by the action of the group. Within this setup we use Stone’s theorem and the theory of projection valued measures to analyze wandering frame collections. This yields a functional analytic method of constructing a wavelet from a generalized frame multiresolution analysis in terms of the frame scaling vectors. We then explicitly apply our results to the action of the integers given by translations on L2(ℝ).

Purely Infinite Simple C*-Crossed Products II

1996 ◽  
Vol 39 (2) ◽  
pp. 203-210 ◽  
Author(s):  
JA A. Jeong ◽  
Kazunori Kodaka ◽  
Hiroyuki Osaka
Keyword(s):  
Abelian Group ◽  
Finite Groups ◽  
Crossed Product ◽  
Crossed Products ◽  
Outer Action ◽  
Countable Abelian Group

AbstractWe study the pure infiniteness of C* -crossed products by endomorphisms and automorphisms. Let A be a purely infinité simple unital C*-algebra. At first we show that a crossed product A × p N by a corner endomorphism p is purely infinite if it is simple. From this observation we prove that any simple C*-crossed products A ×αZ by an automorphism α is purely infinite. Combining this with the result in [Je] on pure infiniteness of crossed products by finite groups, one sees that if α is an outer action by a countable abelian group G then the simple C*-algebra A ×α G is purely infinite.

Geometric K-homology with coefficients I: ℤ/kℤ-cycles and Bockstein sequence

2011 ◽  
Vol 9 (3) ◽  
pp. 537-564 ◽  
Author(s):  
Robin J. Deeley
Keyword(s):  
Abelian Group ◽  
Index Theorem ◽  
Geometric Object ◽  
Type Model ◽  
Inductive Limits ◽  
Geometric Models ◽  
Countable Abelian Group ◽  
The Relationship

AbstractWe construct a Baum-Douglas type model for K-homology with coefficients in ℤ/kℤ. The basic geometric object in a cycle is a spinc ℤ/kℤ-manifold. The relationship between these cycles and the topological side of the Freed-Melrose index theorem is discussed in detail. Finally, using inductive limits, we construct geometric models for K-homology with coefficients in any countable abelian group.

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