Expansive actions of countable amenable groups, homoclinic pairs, and the Myhill property

TL;DR

This paper investigates expansive actions of countable amenable groups on compact spaces, establishing conditions under which certain continuous maps are surjective, extending the understanding of the Myhill property in dynamical systems.

Contribution

It proves that for expansive systems with specific factor structures, the absence of distinct homoclinic pairs with the same image implies surjectivity of commuting maps, extending prior results on the Myhill property.

Findings

If no pair of distinct G-homoclinic points share the same image under τ, then τ is surjective.

The result applies to systems that are quotients of strongly irreducible subshifts by uniformly bounded-to-one factor maps.

The work extends the class of systems known to satisfy the Myhill property.

Abstract

Let $X$ be a compact metrizable space equipped with a continuous action of a countable amenable group $G$. Suppose that the dynamical system $(X,G)$ is expansive and is the quotient by a uniformly bounded-to-one factor map of a strongly irreducible subshift. Let $\tau \colon X \to X$ be a continuous map commuting with the action of $G$. We prove that if there is no pair of distinct $G$-homoclinic points in $X$ having the same image under $\tau$, then $\tau$ is surjective.