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

Donoso, Sebastián; Jin, Lei; Maass, Alejandro; Qiao, Yixiao

doi:10.1090/proc/16027

Directional mean dimension and continuum-wise expansive $\mathbb {Z}^k$-actions
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by Sebastián Donoso, Lei Jin, Alejandro Maass and Yixiao Qiao

Proc. Amer. Math. Soc. 150 (2022), 4841-4853

DOI: https://doi.org/10.1090/proc/16027

Published electronically: July 29, 2022

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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 Mañé: Any compact metrizable space admitting an expansive homeomorphism (with respect to a compatible metric) is finite-dimensional.

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