Minimal subshifts of prescribed mean dimension over general alphabets

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 $[0,1)$.