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

TL;DR

This paper investigates the properties of directional mean dimension in $Z^k$-actions, revealing discontinuities and bounds under continuum-wise expansiveness, extending classical finite-dimensionality results.

Contribution

It demonstrates the discontinuity of directional mean dimension in certain $Z^2$-actions and establishes bounds for continuum-wise expansive $Z^k$-actions, generalizing Mañé's finite-dimensionality theorem.

Findings

Directional mean dimension can be discontinuous.

Continuum-wise expansive actions have bounded directional mean dimension.

Generalizes classical finite-dimensionality results to multiparameter 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 Ma\~n\'e: Any compact metrizable space admitting an expansive homeomorphism (with respect to a compatible metric) is finite-dimensional.