Expansive multiparameter actions and mean dimension

Mañé proved in 1979 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 the “averaged dimension” of a dynamical system. We prove that if T: ℤ^k ×X → X is expansive and if R: ℤ^{k −1} ×X → 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.