Topological entropy of continuous actions of compactly generated groups

Abstract

We introduce a notion of topological entropy for continuous actions of compactly generated topological groups on compact Hausdorff spaces. It is shown that any continuous action of a compactly generated topological group on a compact Hausdorff space with vanishing topological entropy is amenable. Given an arbitrary compactly generated locally compact Hausdorff topological group $G$, we consider the canonical action of $G$ on the closed unit ball of $L^{1}(G)' \cong L^{\infty}(G)$ endowed with the corresponding weak-$^{\ast}$ topology. We prove that this action has vanishing topological entropy if and only if $G$ is compact. Furthermore, we show that the considered action has infinite topological entropy if $G$ is almost connected and non-compact.