Homomorphisms from aperiodic subshifts to subshifts with the finite extension property

TL;DR

This paper proves the existence of homomorphisms between certain aperiodic subshifts over groups with polynomial growth, and shows conditions under which one subshift embeds into another, expanding understanding of subshift morphisms.

Contribution

It establishes new conditions for the existence of homomorphisms and embeddings between subshifts over groups with polynomial growth, especially involving the finite extension property.

Findings

Existence of homomorphisms under polynomial growth and FEP conditions

Embedding of subshifts when entropy conditions are met

New insights into subshifts with the finite extension property

Abstract

Given a countable group $G$ and two subshifts $X$ and $Y$ over $G$, a continuous, shift-commuting map $\phi : X \to Y$ is called a homomorphism. Our main result states that if every finitely generated subgroup of $G$ has polynomial growth, $X$ is aperiodic, and $Y$ has the finite extension property (FEP), then there exists a homomorphism $\phi : X \to Y$. By combining this theorem with a previous result of Bland, we obtain that if the same conditions hold, and if additionally the topological entropy of $X$ is less than the topological entropy of $Y$ and $Y$ has no global period, then $X$ embeds into $Y$. We also establish some facts about subshifts with the FEP that may be of independent interest.