A Garden of Eden theorem for Smale spaces

Abstract

Given a dynamical system $(X,f)$ consisting of a compact metrizable space $X$ and a homeomorphism $f \colon X \to X$, an endomorphism of $(X,f)$ is a continuous map of $X$ into itself which commutes with $f$. One says that a dynamical system $(X,f)$ is surjunctive if every injective endomorphism of $(X,f)$ is surjective. An endomorphism of $(X,f)$ is called pre-injective if its restriction to each $f$-homoclinicity class of $X$ is injective. One says that a dynamical system has the Moore property if every surjective endomorphism of the system is pre-injective and that it has the Myhill property if every pre-injective endomorphism is surjective. One says that a dynamical system satisfies the Garden of Eden theorem if it has both the Moore and the Myhill properties. We prove that every irreducible Smale space satisfies the Garden of Eden theorem and that every non-wandering Smale space is surjunctive and has the Moore property.