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arXiv · 2410.18795

Homomorphisms from aperiodic subshifts to subshifts with the finite extension property

Robert Bland·Kevin McGoff
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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.

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Robert Bland, Kevin McGoff. 2024-10-24. Homomorphisms from aperiodic subshifts to subshifts with the finite extension property. https://doi.org/10.1017/etds.2025.16

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