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Let $X$ be a compact metrizable space equipped with a continuous action of acountable amenable group $G$. Suppose that the dynamical system $(X,G)$ isexpansive and is the quotient by a uniformly bounded-to-one factor map of astrongly irreducible subshift. Let $\tau \colon X \to X$ be a continuous mapcommuting 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 $\tau$ issurjective.

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