Mean dimension of full shifts

TL;DR

This paper investigates the mean dimension of full shifts over finite-dimensional compact metric spaces, establishing a relationship with the dimension of the space and proposing an intriguing problem in infinite dimensional topology.

Contribution

It determines the mean dimension of full shifts based on the dimension of the underlying space and introduces a new problem in infinite dimensional topology.

Findings

Mean dimension equals dim K or dim K - 1 depending on K's type.

Provides a classification of mean dimension for full shifts.

Proposes an open problem in infinite dimensional topology.

Abstract

Let $K$ be a finite dimensional compact metric space and $K^\mathbb{Z}$ the full shift on the alphabet $K$. We prove that its mean dimension is given by $\dim K$ or $\dim K-1$ depending on the "type" of $K$. We propose a problem which seems interesting from the view point of infinite dimensional topology.