### 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 |
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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 |

### All Science Journal Classification (ASJC) codes

- Analysis
- Geometry and Topology

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## Cite this

Lindenstrauss, E., & Tsukamoto, M. (2019). Double variational principle for mean dimension.

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