Abstract

We study directional mean dimension of Zk-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 Zk-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.