Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic group

TL;DR

This paper introduces a unifying framework using Markovian properties to connect algebraic actions, entropy, and homoclinic groups, providing new characterizations for expansive algebraic actions of certain groups.

Contribution

It develops a series of Markovian properties to relate entropy and asymptotic pairs, offering new characterizations of homoclinic groups in algebraic actions for specific groups.

Findings

Established links between entropy and asymptotic pairs in algebraic actions.

Characterized homoclinic groups for expansive algebraic actions of certain groups.

Unified previous results using Markovian properties and independence entropy pairs.

Abstract

We provide a unifying approach which links results on algebraic actions by Lind and Schmidt, Chung and Li, and a topological result by Meyerovitch that relates entropy to the set of asymptotic pairs. In order to do this we introduce a series of Markovian properties and, under the assumption that they are satisfied, we prove several results that relate topological entropy and asymptotic pairs (the homoclinic group in the algebraic case). As new applications of our method, we give a characterization of the homoclinic group of any finitely presented expansive algebraic action of (1) any elementary amenable group with an upper bound on the orders of finite subgroups or (2) any left orderable amenable group, using the language of independence entropy pairs.