Existence of asymptotic pairs in positive entropy group actions

TL;DR

This paper generalizes the Blanchard-Host-Ruette Theorem by proving that positive entropy actions of countable amenable groups almost surely contain asymptotic pairs with respect to a multiorder, extending classical results to broader group actions.

Contribution

It introduces a new definition of asymptotic pairs for group actions and proves their almost sure existence in positive entropy systems for amenable groups, generalizing classical theorems.

Findings

Existence of asymptotic pairs in positive entropy group actions

Generalization of classical theorems to amenable group actions

Almost sure presence of asymptotic pairs for a multiorder on the group

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 $\mathbb Z$).