Expansive multiparameter actions and mean dimension

2017/10/26 by Tom Meyerovitch, Meyerovitch, Tom, Masaki Tsukamoto +1 · 1 citation
Mathematics · #37B05 #54F45 #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics #math.DS #msc:37B05 #msc:54F45

paper · pdf · doi:10.48550/arxiv.1710.09647

27 pages

arxiv created 2017/10/26 · openalex publication_date 2017/10/26 · arxiv updated 2017/10/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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: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.

Citations

• Mean topological dimension

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