Garden of Eden and weakly periodic points for certain expansive actions of groups

TL;DR

This paper explores the properties of expansive group actions on compact spaces, establishing the Garden of Eden theorem for certain actions and demonstrating the existence and density of weakly periodic points under specific conditions.

Contribution

It extends the Garden of Eden theorem to expansive actions of countable amenable groups with weak specification and topological Markov properties, and generalizes results on weakly periodic points.

Findings

Establishes the Moore and Myhill properties for specific expansive group actions.

Proves the existence of weakly periodic points for expansive actions of groups with at least two ends.

Shows the density of weakly periodic points when the weak specification property is satisfied.

Abstract

We present several applications of the weak specification property and certain topological Markov properties, recently introduced by S. Barbieri, F. Garc\'{i}a-Ramos and H. Li, and implied by the pseudo-orbit tracing property, for general expansive group actions on compact spaces. First we show that any expansive action of a countable amenable group on a compact metrizable space satisfying the weak specification and strong topological Markov properties satisfies the Moore property, i.e. every surjective automorphism of such dynamical system is pre-injective. This together with an earlier result of H. Li (where the strong topological Markov property is not needed) of the Myhill property, which we also re-prove here, establishes the Garden of Eden theorem for all expansive actions of countable amenable groups on compact metrizable spaces satisfying the weak specification and strong topological Markov properties. We hint how to easily generalize this result even for uncountable amenable groups and general compact, not necessarily metrizable, spaces. Second, we generalize the recent result of D. B. Cohen that any subshift of finite type of a finitely generated group having at least two ends has weakly periodic points. We show that every expansive action of such a group having a certain Markov topological property, again implied by the pseudo-orbit tracing property, has a weakly periodic point. If it has additionally the weak specification property, the set of such points is dense.