Minimal amenable subshift with full mean dimension

TL;DR

This paper constructs a minimal subshift for an amenable group with mean topological dimension equal to a given finite-dimensional polyhedron, extending previous work for integer actions and answering a specific open question.

Contribution

It introduces a new construction of minimal subshifts with prescribed mean topological dimension for amenable groups, extending prior results from integer actions.

Findings

Constructed minimal subshift with specified mean dimension

Extended results from $bZ$-actions to amenable groups

Provided an answer to D. Dou's open question

Abstract

Let $G$ be an infinite countable amenable group and $P$ a polyhedron with topological dimension $dim(P)<\infty$. We construct a minimal subshift $(X,G)$ such that its mean topological dimension is equal to $dim(P)$. This result answers the question of D. Dou in \cite{DD}, moreover, it is also an extension of the work of L. Jin and Y. Qiao \cite{JQ} for $\mathbb{Z}$-action.