An embedding theorem for subshifts over amenable groups with the comparison property

TL;DR

This paper proves an embedding theorem for symbolic dynamical systems over amenable groups with the comparison property, extending classical results to a broader group setting using tiling theory.

Contribution

It extends classical embedding results to subshifts over amenable groups with the comparison property, utilizing tiling and quasi-tiling techniques.

Findings

Embedding theorem for subshifts over amenable groups

Extension of Krieger's classical result to broader groups

Use of tiling theory in dynamical systems

Abstract

We obtain the following embedding theorem for symbolic dynamical systems. Let $G$ be a countable amenable group with the comparison property. Let $X$ be a strongly aperiodic subshift over $G$. Let $Y$ be a strongly irreducible shift of finite type over $G$ which has no global period, meaning that the shift action is faithful on $Y$. If the topological entropy of $X$ is strictly less than that of $Y$, and $Y$ contains at least one factor of $X$, then $X$ embeds into $Y$. This result partially extends the classical result of Krieger when $G = \mathbb{Z}$ and the results of Lightwood when $G = \mathbb{Z}^d$ for $d \geq 2$. The proof relies on recent developments in the theory of tilings and quasi-tilings of amenable groups.