Chaotic behavior of group actions

TL;DR

This paper investigates chaotic behaviors in group actions on compact spaces, introducing concepts like local weak mixing and Li-Yorke chaos, and establishing conditions linking entropy, mixing, and chaos for countable discrete groups.

Contribution

It introduces new notions of chaos for group actions and proves their relationships with entropy and group properties, extending classical dynamical systems theory.

Findings

Local weak mixing implies Li-Yorke chaos for infinite groups.

Positive topological entropy implies local weak mixing for amenable groups.

For shifts of finite type, positive entropy correlates with non-trivial homoclinic relations.

Abstract

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.