A Garden of Eden theorem for Smale spaces

TL;DR

This paper proves that all irreducible Smale spaces satisfy the Garden of Eden theorem, establishing a fundamental link between surjunctivity, Moore, and Myhill properties in these complex dynamical systems.

Contribution

It demonstrates that irreducible Smale spaces satisfy the Garden of Eden theorem, extending classical results to a broad class of hyperbolic dynamical systems.

Findings

Irreducible Smale spaces satisfy the Garden of Eden theorem.

Non-wandering Smale spaces are surjunctive.

Non-wandering Smale spaces have the Moore property.

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.