On the asymptotic relation of topological amenable group actions

Wojciech Bułatek, Brunon Kamiński, Jerzy Szymański

DOI: http://dx.doi.org/10.12775/TMNA.2015.086


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 $\A (\Phi)$ and we show that $\A (\Phi)$ 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 $\A (\Phi)$ is dense. These results are generalizations of those of Blanchard, Host and Ruette (\cite{BHR}) that concern the asymptotic relation for $\Z$-actions. We give an example of an expansive $G$-action ($G=\Z^2$) with $\A (\Phi)$ trivial which shows that the Bryant--Walters classical result (\cite{BW}) fails to be true in general case.


Topological G-action; entropy; asymptotic relation; generalized Pinsker formula; Pinsker sigma-algebra

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