Mean dimension and a sharp embedding theorem: extensions of aperiodic subshifts

TL;DR

This paper establishes sharp bounds for embedding aperiodic subshifts with low mean dimension into Euclidean cube shifts, extending previous results and providing precise embedding criteria.

Contribution

It extends embedding theorems for aperiodic subshifts based on mean dimension, including sharp bounds and new embedding conditions for zero-dimensional systems.

Findings

Embedding is possible if mean dimension is less than D/2 for general aperiodic subshifts.

Sharpness of the embedding bounds is demonstrated.

Zero-dimensional extensions embed into ( [0,1]^{D+1} )^{bZ}.

Abstract

We show that if $(X,T)$ is an extension of an aperiodic subshift (a subsystem of $({1,2,...,l}^{\mathbb{Z}},\mathrm{shift})$ for some $l\in\mathbb{N}$) and has mean dimension $mdim(X,T)<\frac{D}{2}$ $(D\in \mathbb{N}$), then it embeds equivariantly in (([0,1]^{D})^{\mathbb{Z}},\mathrm{shift})$. The result is sharp. If $(X,T)$ is an extension of an aperiodic zero-dimensional system then it embeds equivariantly in $(([0,1]^{D+1})^{\mathbb{Z}},\mathrm{shift})$.