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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Expansive multiparameter actions and mean dimension


Authors: Tom Meyerovitch and Masaki Tsukamoto
Journal: Trans. Amer. Math. Soc. 371 (2019), 7275-7299
MSC (2010): Primary 37B05, 54F45
DOI: https://doi.org/10.1090/tran/7588
Published electronically: October 2, 2018
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Abstract: 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:\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.


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Additional Information

Tom Meyerovitch
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva 8410501, Israel
Email: mtom@math.bgu.ac.il

Masaki Tsukamoto
Affiliation: Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
Email: tukamoto@math.kyoto-u.ac.jp

DOI: https://doi.org/10.1090/tran/7588
Keywords: Expansive action, mean dimension, topological entropy
Received by editor(s): October 28, 2017
Received by editor(s) in revised form: March 6, 2018
Published electronically: October 2, 2018
Article copyright: © Copyright 2018 American Mathematical Society