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In this paper we study chaotic behavior of actions of a countable discrete group acting on a compact metric space by self-homeomorphisms. For actions of a countable discrete group G, we introduce local weak mixing and Li-Yorke chaos; and prove that local weak mixing implies Li-Yorke chaos if G is infinite, and positive topological entropy implies local weak mixing if G is an infinite countable discrete amenable group. Moreover, when considering a shift of finite type for actions of an infinite countable amenable group G, if the action has positive topological entropy then its homoclinic equivalence relation is non-trivial, and the converse holds true if additionally G is residually finite and the action contains a dense set of periodic points.
Let $H_3(Bbb R)$ denote the 3-dimensional real Heisenberg group. Given a family of lattices $Gamma_1supsetGamma_2supsetcdots$ in it, let $T$ stand for the associated uniquely ergodic $H_3(Bbb R)$-{it odometer}, i.e. the inverse limit of the $H_3(Bbb
An textit{algebraic} action of a discrete group $Gamma $ is a homomorphism from $Gamma $ to the group of continuous automorphisms of a compact abelian group $X$. By duality, such an action of $Gamma $ is determined by a module $M=widehat{X}$ over the
The paper offers a thorough study of multiorders and their applications to measure-preserving actions of countable amenable groups. By a multiorder on a countable group we mean any probability measure $ u$ on the collection $mathcal O$ of linear orde
We consider a minimal equicontinuous action of a finitely generated group $G$ on a Cantor set $X$ with invariant probability measure $mu$, and stabilizers of points for such an action. We give sufficient conditions under which there exists a subgroup
In this work, we investigate the dynamical and geometric properties of weak solenoids, as part of the development of a calculus of group chains associated to Cantor minimal actions. The study of the properties of group chains was initiated in the wor