### Abstract

We develop a variational principle between mean dimension theory and rate distortion theory. We consider a minimax problem about the rate distortion dimension with respect to two variables (metrics and measures). We prove that the minimax value is equal to the mean dimension for a dynamical system with the marker property. The proof exhibits a new combination of ergodic theory, rate distortion theory and geometric measure theory. Along the way of the proof, we also show that if a dynamical system has the marker property then it has a metric for which the upper metric mean dimension is equal to the mean dimension.

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

Pages (from-to) | 1048-1109 |

Number of pages | 62 |

Journal | Geometric and Functional Analysis |

Volume | 29 |

Issue number | 4 |

DOIs | |

Publication status | Published - Aug 1 2019 |

Externally published | Yes |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Analysis
- Geometry and Topology

### Cite this

*Geometric and Functional Analysis*,

*29*(4), 1048-1109. https://doi.org/10.1007/s00039-019-00501-8

**Double variational principle for mean dimension.** / Lindenstrauss, Elon; Tsukamoto, Masaki.

Research output: Contribution to journal › Article

*Geometric and Functional Analysis*, vol. 29, no. 4, pp. 1048-1109. https://doi.org/10.1007/s00039-019-00501-8

}

TY - JOUR

T1 - Double variational principle for mean dimension

AU - Lindenstrauss, Elon

AU - Tsukamoto, Masaki

PY - 2019/8/1

Y1 - 2019/8/1

N2 - We develop a variational principle between mean dimension theory and rate distortion theory. We consider a minimax problem about the rate distortion dimension with respect to two variables (metrics and measures). We prove that the minimax value is equal to the mean dimension for a dynamical system with the marker property. The proof exhibits a new combination of ergodic theory, rate distortion theory and geometric measure theory. Along the way of the proof, we also show that if a dynamical system has the marker property then it has a metric for which the upper metric mean dimension is equal to the mean dimension.

AB - We develop a variational principle between mean dimension theory and rate distortion theory. We consider a minimax problem about the rate distortion dimension with respect to two variables (metrics and measures). We prove that the minimax value is equal to the mean dimension for a dynamical system with the marker property. The proof exhibits a new combination of ergodic theory, rate distortion theory and geometric measure theory. Along the way of the proof, we also show that if a dynamical system has the marker property then it has a metric for which the upper metric mean dimension is equal to the mean dimension.

UR - http://www.scopus.com/inward/record.url?scp=85066493351&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85066493351&partnerID=8YFLogxK

U2 - 10.1007/s00039-019-00501-8

DO - 10.1007/s00039-019-00501-8

M3 - Article

AN - SCOPUS:85066493351

VL - 29

SP - 1048

EP - 1109

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

SN - 1016-443X

IS - 4

ER -