Mean dimension theory for infinite dimensional Bedford-McMullen sponges

Abstract

Tsukamoto (2022) introduced the notion of Bedford-McMullen carpet system, a subsystem of $([0,1{]}^{\mathbb{N}}\times [0,1{]}^{\mathbb{N}},shift)$ whose metric mean dimension and mean Hausdorff dimension does not coincide in general. The aim of this paper is to develop the mean dimension theory for Bedford-McMullen sponge system, which is a subsystem of $(([0,1{]}^r{)}^{\mathbb{N}},shift)$ with arbitrary $3\le r\in \mathbb{N}$. In particular, we compute the metric mean dimension and mean Hausdorff dimension of such topological dynamical systems explicitly, extending the results by Tsukamoto. The metric mean dimension is a weighted combination of the standard topological entropy, whereas the mean Hausdorff dimension is expressed in terms of weighted topological entropy. We also exhibit a special situation for which the metric mean dimension and the mean Hausdorff dimension of a sponge system coincide.

V2: Accepted for publication in Discrete and Continuous Dynamical Systems