Mean dimension theory for infinite dimensional Bedford-McMullen sponges

TL;DR

This paper develops the mean dimension theory for Bedford-McMullen sponge systems in higher dimensions, explicitly computing metric mean dimension and mean Hausdorff dimension, and identifying conditions for their equality.

Contribution

It extends mean dimension theory to higher-dimensional Bedford-McMullen sponge systems and provides explicit formulas for their dimensions.

Findings

Explicit formulas for metric mean dimension and mean Hausdorff dimension.

Identification of conditions where these two dimensions coincide.

Extension of Tsukamoto's results to higher dimensions.

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 $3\leq r\in\mathbb{N}$. 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.