Pseudo-Orbit Tracing and Algebraic actions of countable amenable groups

TL;DR

This paper investigates the relationship between expansiveness, positive entropy, and the existence of off-diagonal asymptotic pairs in actions of countable amenable groups, providing new examples and conditions for their occurrence.

Contribution

It demonstrates the near-optimality of Chung and Li's results by providing counterexamples and establishes that expansive actions with positive entropy and pseudo-orbit tracing property necessarily have off-diagonal asymptotic pairs.

Findings

Counterexamples for non-algebraic actions

Counterexamples for non-finitely generated abelian groups

Pseudo-orbit tracing property holds for certain expansive algebraic actions

Abstract

Consider a countable amenable group acting by homeomorphisms on a compact metrizable space. Chung and Li asked if expansiveness and positive entropy of the action imply existence of an off-diagonal asymptotic pair. For algebraic actions of polycyclic-by-finite groups, Chung and Li proved it does. We provide examples showing that Chung and Li's result is near-optimal in the sense that the conclusion fails for some non-algebraic action generated by a single homeomorphism, and for some algebraic actions of non-finitely generated abelian groups. On the other hand, we prove that every expansive action of an amenable group with positive entropy that has the pseudo-orbit tracing property must admit off-diagonal asymptotic pairs. Using Chung and Li's algebraic characterization of expansiveness, we prove the pseudo-orbit tracing property for a class of expansive algebraic actions. This class includes every expansive principal algebraic action of an arbitrary countable group.