A note on the surjunctivity of algebraic dynamical systems

TL;DR

This paper investigates conditions under which algebraic dynamical systems on compact metrizable groups are surjunctive, meaning injective equivariant maps are necessarily surjective, contributing to understanding their structural properties.

Contribution

It provides new sufficient conditions for surjunctivity in algebraic dynamical systems involving actions of countable groups on compact metrizable groups.

Findings

Identifies conditions ensuring surjunctivity of algebraic dynamical systems.

Establishes that certain injective equivariant maps are surjective under these conditions.

Enhances understanding of the structure of algebraic dynamical systems.

Abstract

Let $X$ be a compact metrizable group and $\Gamma$ a countable group acting on $X$ by continuous group automorphisms. We give sufficient conditions under which the dynamical system $(X,\Gamma)$ is surjunctive, i.e., every injective continuous map $\tau \colon X \to X$ commuting with the action of $\Gamma$ is surjective.