A note on the surjunctivity of algebraic dynamical systems

Let $X$ be a compact metrizable group and $\mathrm{\Gamma }$ a countable group acting on $X$ by continuous group automorphisms. We give sufficient conditions under which the dynamical system $(X,\mathrm{\Gamma })$ is surjunctive, i.e., every injective continuous map $\tau \text{\hspace{0.17em}⁣}:X\to X$ commuting with the action of $\mathrm{\Gamma }$ is surjective.