Existence of asymptotic pairs in positive entropy group actions

Abstract

We provide a definition of a $\prec$-asymptotic pair in a topological action of a countable amenable group $G$, where $\prec$ is an order on $G$ of type $\mathbb{Z}$. We then prove that if $(\tilde{\mathcal{O}},\nu ,G)$ is a multiorder on $G$, then for every topological $G$-action of positive entropy there exists an $\prec$-asympotic pair for almost every order $\prec \in \tilde{\mathcal{O}}$. This result is a generalization of the Blanchard-Host-Ruette Theorem for classical topological dynamical systems (actions of $\mathbb{Z}$).

The content of the work with necessary corrections was included in a more extensive article: Asymptotic pairs in topological actions of amenable groups (arXiv:2303.12923)

References in corpus (1)

• Multiorders in amenable group actions