On the asymptotic relation of topological amenable group actions

Abstract

For a topological action $\Phi$ of a countable amenable orderable group $G$ on a compact metric space we introduce a concept of the asymptotic relation $\mathbf{A} (\Phi)$ and we show that $\mathbf{A} (\Phi)$ is non-trivial if the topological entropy $h(\Phi)$ is positive. It is also proved that if the Pinsker $\sigma$-algebra $\pi_{\mu}(\Phi)$ is trivial, where $\mu$ is an invariant measure with full support, then $\mathbf{A} (\Phi)$ is dense. These results are generalizations of those of Blanchard, Host and Ruette ([B.F. Bryant and P. Walters, Asymptotic properties of expansive homeomorphisms, Math. System Theory 3 (1969), 60-66]) that concern the asymptotic relation for $\mathbb{Z}$-actions.

We give an example of an expansive $G$-action ($G=\mathbb{Z}^2$) with $\mathbf{A} (\Phi)$ trivial which shows that the Bryant-Walters classical result ([B.F. Bryant and P. Walters, Asymptotic properties of expansive homeomorphisms, Math. System Theory 3 (1969), 60-66]) fails to be true in general case.