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arXiv · 1710.09647

Expansive multiparameter actions and mean dimension

Tom Meyerovitch·Masaki Tsukamoto
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Abstract

Ma\~n\'e (1979) proved that if a compact metric space admits an expansive homeomorphism then it is finite dimensional. We generalize this theorem to multiparameter actions. The generalization involves mean dimension theory, which counts "averaged dimension" of a dynamical system. We prove that if $T:\mathbb{Z}^k\times X\to X$ is expansive and if $R:\mathbb{Z}^{k-1}\times X\to X$ commutes with $T$ then $R$ has finite mean dimension. When $k=1$, this statement reduces to Ma\~{n}\'{e}'s theorem. We also study several related issues, especially the connection with entropy theory.

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Tom Meyerovitch, Masaki Tsukamoto. 2017-10-26. Expansive multiparameter actions and mean dimension. https://arxiv.org/abs/1710.09647

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