Chaotic Behavior Of Group Actions

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 (and hence Li-Yorke chaos) 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 dense periodic points.