Asymptotic pairs in topological actions of amenable groups

Abstract

We provide a definition of a $\prec$-asymptotic pair in a topological action of a countable group $G$, where $\prec$ is an order on $G$ of type $\mathbb{Z}$. We then prove that if $G$ is a countable amenable group and $(X,G)$ is a topological $G$-action of positive entropy, then for every multiorder $(\tilde{\mathcal{O}},\nu ,G)$ and $\nu$-almost every order $\prec \in \tilde{\mathcal{O}}$ there exists a $\prec$-asympotic pair in $X$. This result is a generalization of the Blanchard-Host-Ruette Theorem for classical topological dynamical systems (actions of~$\mathbb{Z}$). We also prove that for every countable amenable group $G$, and every multiorder on $G$ arising from a tiling system, every topological $G$-action of entropy zero has an extension which has no $\prec$-asymptotic pairs for any $\prec$ belonging to this multiorder. Together, these two theorems give a characterization of topological $G$-actions of entropy zero: $(X,G)$ has topological entropy zero if and only if, for any multiorder ${\tilde{\mathcal{O}}}_{\mathsf{T}}$ on $G$ arising from a tiling system of entropy zero, there exists an extension $(Y,G)$ of $(X,G)$, which has no $\prec$-asymptotic pairs for any $\prec \in {\tilde{\mathcal{O}}}_{\mathsf{T}}$, equivalently, there exists a multiorder $(\tilde{\mathcal{O}},\nu ,G)$ on $G$, such that for $\nu$-almost any $\prec \in \tilde{\mathcal{O}}$, there are no $\prec$-asymptotic pairs in $(Y,G)$.

arXiv comment20 pages