  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.
