Expansive multiparameter actions and mean dimension

Abstract

Mañé (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 Mañé's theorem. We also study several related issues, especially the connection with entropy theory.

27 pages

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