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 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 $E$.

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.

A Family of Generalized Riesz Products

1996 ◽  
Vol 48 (2) ◽  
pp. 302-315 ◽  
Author(s):  
A. H. Dooley ◽  
S. J. Eigen
Keyword(s):  
Lebesgue Measure ◽  
Spectral Measure ◽  
Probability Space ◽  
Rank One ◽  
Almost All ◽  
Riesz Products

AbstractGeneralized Riesz products similar to the type which arise as the spectral measure for a rank-one transformation are studied. A condition for the mutual singularity of two such measures is given. As an application, a probability space of transformations is presented in which almost all transformations are singular with respect to Lebesgue measure.

scholarly journals On abelian group actions with TNI-centralizers

2019 ◽  
Vol 47 (7) ◽  
pp. 3003-3006
Author(s):  
Gülin Ercan ◽  
İsmail Ş. Güloğlu
Keyword(s):  
Abelian Group ◽  
Group Actions ◽  
Abelian Group Actions

On Intermediate Subalgebras of C*-simple Group Actions

2019 ◽  
Author(s):  
Tattwamasi Amrutam
Keyword(s):  
Simple Group ◽  
Large Class ◽  
Hyperbolic Group ◽  
Group Actions ◽  
Simple Groups ◽  
C Algebras ◽  
Amenable Radical

Abstract We show that for a large class of actions $\Gamma \curvearrowright \mathcal{A}$ of $C^*$-simple groups $\Gamma $ on unital $C^*$-algebras $\mathcal{A}$, including any non-faithful action of a hyperbolic group with trivial amenable radical, every intermediate $C^*$-subalgebra $\mathcal{B}$, $C_{\lambda }^*(\Gamma )\subseteq \mathcal{B} \subseteq \mathcal{A}\rtimes _{r}\Gamma $, is of the form $\mathcal{A}_1\rtimes _{r}\Gamma $, where $\mathcal{A}_1$ is a unital $\Gamma $-$C^*$-subalgebra of $\mathcal{A}$.

scholarly journals SKEW CATEGORY ALGEBRAS ASSOCIATED WITH PARTIALLY DEFINED DYNAMICAL SYSTEMS

2012 ◽  
Vol 23 (04) ◽  
pp. 1250040 ◽  
Author(s):  
PATRIK LUNDSTRÖM ◽  
JOHAN ÖINERT
Keyword(s):  
Dynamical Systems ◽  
Topological Space ◽  
Large Class ◽  
Additive Group ◽  
Nonzero Ideal ◽  
Intersection Property ◽  
The Other ◽  
The One ◽  
Category Algebras ◽  
The Ideal

We introduce partially defined dynamical systems defined on a topological space. To each such system we associate a functor s from a category G to Topop and show that it defines what we call a skew category algebra A ⋊σ G. We study the connection between topological freeness of s and, on the one hand, ideal properties of A ⋊σ G and, on the other hand, maximal commutativity of A in A ⋊σ G. In particular, we show that if G is a groupoid and for each e ∈ ob (G) the group of all morphisms e → e is countable and the topological space s(e) is Tychonoff and Baire. Then the following assertions are equivalent: (i) s is topologically free; (ii) A has the ideal intersection property, i.e. if I is a nonzero ideal of A ⋊σ G, then I ∩ A ≠ {0}; (iii) the ring A is a maximal abelian complex subalgebra of A ⋊σ G. Thereby, we generalize a result by Svensson, Silvestrov and de Jeu from the additive group of integers to a large class of groupoids.

scholarly journals Subnomality in soluble minimax groups

1974 ◽  
Vol 17 (1) ◽  
pp. 113-128 ◽  
Author(s):  
D. J. McCaughan
Keyword(s):  
Large Class ◽  
Upper Bound ◽  
Nilpotent Groups ◽  
Subnormal Subgroup ◽  
Wreath Products ◽  
Minimal Length ◽  
Intersection Property ◽  
Finite Chain ◽  
Subnormal Subgroups

A subgroup H of a group G is said to be subnormal in G if there is a finite chain of subgroups, each normal in its successor, connecting H to G. If such chains exist there is one of minimal length; the number of strict inclusions in this chain is called the subnormal index, or defect, of H in G. The rather large class of groups which have an upper bound for the subnormal indices of their subnormal subgroups has been inverstigated to same extent, mainly with a restriction to solublegroups — for instance, in [10] McDougall considered soluble p-groups in this class. Robinson, in [14], restricted his attention to wreath products of nilpotent groups but extended his investigations to the strictly larger class of groups in which the intersection of any family of subnormal subgroups is a subnormal subgroup. These groups are said to have the subnormal intersection property.

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