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

Abstract

We study directional mean dimension of Z k \mathbb {Z}^k -actions (where k k is a positive integer). On the one hand, we show that there is a Z 2 \mathbb {Z}^2 -action whose directional mean dimension (considered as a [ 0 , + ∞ ] [0,+\infty ] -valued function on the torus) is not continuous. On the other hand, we prove that if a Z k \mathbb {Z}^k -action is continuum-wise expansive, then the values of its ( k − 1 ) (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.