2024/12/03 by Qiang Huo, Huo, Qiang
Engineering · Mathematics · #28D20 #37A35 #37C45 #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Rings, Modules, and Algebras #graph theory and CDMA systems #primary: 28A80 #secondary: 37B40
paper · pdf · doi:10.48550/arxiv.2412.02278
openalex publication_date 2024/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Tsukamoto (2022) introduced the notion of Bedford-McMullen carpet system, a subsystem of ([0,1]ℕ×[0,1]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)ℕ,shift) with arbitrary 3≤ r∈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.