### 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: 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.

Original language | English |
---|---|

Pages (from-to) | 7275-7299 |

Number of pages | 25 |

Journal | Transactions of the American Mathematical Society |

Volume | 371 |

Issue number | 10 |

DOIs | |

Publication status | Published - Jan 1 2019 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*371*(10), 7275-7299. https://doi.org/10.1090/tran/7588

**Expansive multiparameter actions and mean dimension.** / Meyerovitch, Tom; Tsukamoto, Masaki.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 371, no. 10, pp. 7275-7299. https://doi.org/10.1090/tran/7588

}

TY - JOUR

T1 - Expansive multiparameter actions and mean dimension

AU - Meyerovitch, Tom

AU - Tsukamoto, Masaki

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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: ℤk ×X → X is expansive and if R: ℤk −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.

AB - 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: ℤk ×X → X is expansive and if R: ℤk −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.

UR - http://www.scopus.com/inward/record.url?scp=85066473842&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85066473842&partnerID=8YFLogxK

U2 - 10.1090/tran/7588

DO - 10.1090/tran/7588

M3 - Article

AN - SCOPUS:85066473842

VL - 371

SP - 7275

EP - 7299

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 10

ER -