The Myhill property for hyperbolic homeomorphisms

Suppose that $f$ is an expansive homeomorphism of a compact metrizable space $X$ and that the dynamical system $(X,f)$ is the quotient by a uniformly bounded-to-one factor map of a topologically mixing subshift of finite type. Let $\tau \colon X \to X$ be a continuous map commuting with $f$. We prove that if there is no pair of distinct $f$-homoclinic points in $X$ with the same image under $\tau$ then $\tau$ is surjective. This result extends the Myhill implication in the Garden of Eden theorem of Moore and Myhill for cellular automata and applies in particular to elementary basic sets of Axiom A diffeomorphisms.