Expansive multiparameter actions and mean dimension

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}$^{k-1}\times X\to X$commuteswith$T$then$R$hasfinitemeandimension\mathrm{.}When$k=1$, this statement reduces to Ma~{n}'{e}'s theorem. We also study several related issues, especially the connection with entropy theory.