Directional mean dimension and continuum-wise expansive ℤ^k-actions

2021/08/13 by Sebastián Donoso, Jin Lei, Donoso, Sebastián +5
Computer Science · Mathematics · #37B05 #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2108.06308

openalex publication_date 2021/08/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study directional mean dimension of ℤ^k-actions (where k is a positive integer). On the one hand, we show that there is a ℤ²-action whose directional mean dimension (considered as a [0,+∞]-valued function on the torus) is not continuous. On the other hand, we prove that if a ℤ^k-action is continuum-wise expansive, then the values of its (k-1)-dimensional directional mean dimension are bounded. This is a generalization (with a view towards Meyerovitch and Tsukamoto's theorem on mean dimension and expansive multiparameter actions) of a classical result due to Mañé: Any compact metrizable space admitting an expansive homeomorphism (with respect to a compatible metric) is finite-dimensional.

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