Directional mean dimension and continuum-wise expansive ${\mathbb{Z}}^k$-actions

Abstract

We study directional mean dimension of ${\mathbb{Z}}^k$-actions (where $k$ is a positive integer). On the one hand, we show that there is a ${\mathbb{Z}}^2$-action whose directional mean dimension (considered as a $[0,+\infty ]$-valued function on the torus) is not continuous. On the other hand, we prove that if a ${\mathbb{Z}}^k$-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.

Comments welcome!