On the density of periodic configurations in strongly irreducible subshifts

TL;DR

This paper proves that in certain strongly irreducible subshifts over residually finite groups, periodic configurations are dense, leading to implications for endomorphisms and automorphisms of these subshifts.

Contribution

It establishes the density of periodic configurations in strongly irreducible subshifts over residually finite groups, extending known results and introducing a new class of subshifts.

Findings

Periodic configurations are dense in strongly irreducible subshifts of finite type.

Every injective endomorphism of such subshifts is surjective.

Automorphism groups of these subshifts are residually finite.

Abstract

Let $G$ be a residually finite group and let $A$ be a finite set. We prove that if $X \subset A^G$ is a strongly irreducible subshift of finite type containing a periodic configuration then periodic configurations are dense in $X$. The density of periodic configurations implies in particular that every injective endomorphism of $X$ is surjective and that the group of automorphisms of $X$ is residually finite. We also introduce a class of subshifts $X \subset A^\Z$, including all strongly irreducible subshifts and all irreducible sofic subshifts, in which periodic configurations are dense.