The Myhill property for strongly irreducible subshifts over amenable groups

TL;DR

This paper proves that strongly irreducible subshifts over amenable groups possess the Myhill property, ensuring pre-injective cellular automata are surjective, thus extending classical results to a broader algebraic setting.

Contribution

It establishes the Myhill property for strongly irreducible subshifts over amenable groups, a significant generalization in symbolic dynamics and cellular automata theory.

Findings

Strongly irreducible subshifts over amenable groups have the Myhill property.

Pre-injective cellular automata on these subshifts are necessarily surjective.

The result extends classical Garden of Eden theorems to a broader class of groups and subshifts.

Abstract

Let $G$ be an amenable group and let $A$ be a finite set. We prove that if $X \subset A^G$ is a strongly irreducible subshift then $X$ has the Myhill property, that is, every pre-injective cellular automaton $\tau \colon X \to X$ is surjective.