  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.
