Mean dimension theory for infinite dimensional Bedford–McMullen sponges

Tsukamoto (2022) introduced the notion of Bedford–McMullen carpet system, a subsystem of $ ([0, 1]^{ \mathbb{N}}\times[0, 1]^{ \mathbb{N}}, {\rm{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 the Bedford–McMullen sponge system, which is a subsystem of $ (([0, 1]^r)^{ \mathbb{N}}, {\rm{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.