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arXiv · 2108.06308

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

Sebastián Donoso·Lei Jin·Alejandro Maass·Yixiao Qiao
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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.

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Sebastián Donoso, Lei Jin, Alejandro Maass, Yixiao Qiao. 2021-08-13. Directional mean dimension and continuum-wise expansive $\mathbb{Z}^k$-actions. https://arxiv.org/abs/2108.06308

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