Minimal subshifts of prescribed mean dimension over general alphabets

TL;DR

This paper constructs minimal subshifts with prescribed mean dimension over general alphabets, demonstrating the range of mean dimensions achievable within such dynamical systems.

Contribution

It introduces methods to realize any mean dimension value within a certain interval for minimal subshifts over amenable groups.

Findings

Constructed minimal subshifts with any mean dimension in [0, mdim(K^G,σ))

Created a subshift with mean dimension 1 over [0,1]^G

Showed the set of attainable mean dimensions of minimal subsystems is [0,1)

Abstract

Let $G$ be a countable infinite amenable group, $K$ a finite-dimensional compact metrizable space, and $(K^G,\sigma)$ the full $G$-shift on $K^G$. For any $r\in [0,{\rm mdim}(K^G,\sigma))$, we construct a minimal subshift $(X,\sigma)$ of $(K^G,\sigma)$ with mdim$(X,\sigma)=r$. Furthermore, we construct a subshift of $([0,1]^G,\sigma)$ such that its mean dimension is $1$, and that the set of all attainable values of the mean dimension of its minimal subsystems is exactly the interval $[0,1)$.