### Abstract

This paper contributes to the mean dimension theory of dynamical systems. We introduce a new concept called mean dimension with potential and develop a variational principle for it. This is a mean dimension analogue of the theory of topological pressure. We consider a minimax problem for the sum of rate distortion dimension and the integral of a potential function. We prove that the minimax value is equal to the mean dimension with potential for a dynamical system having the marker property. The basic idea of the proof is a dynamicalization of geometric measure theory.

Original language | English |
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Article number | 106935 |

Journal | Advances in Mathematics |

Volume | 361 |

DOIs | |

Publication status | Published - Feb 12 2020 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

**Double variational principle for mean dimension with potential.** / Tsukamoto, Masaki.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Double variational principle for mean dimension with potential

AU - Tsukamoto, Masaki

PY - 2020/2/12

Y1 - 2020/2/12

N2 - This paper contributes to the mean dimension theory of dynamical systems. We introduce a new concept called mean dimension with potential and develop a variational principle for it. This is a mean dimension analogue of the theory of topological pressure. We consider a minimax problem for the sum of rate distortion dimension and the integral of a potential function. We prove that the minimax value is equal to the mean dimension with potential for a dynamical system having the marker property. The basic idea of the proof is a dynamicalization of geometric measure theory.

AB - This paper contributes to the mean dimension theory of dynamical systems. We introduce a new concept called mean dimension with potential and develop a variational principle for it. This is a mean dimension analogue of the theory of topological pressure. We consider a minimax problem for the sum of rate distortion dimension and the integral of a potential function. We prove that the minimax value is equal to the mean dimension with potential for a dynamical system having the marker property. The basic idea of the proof is a dynamicalization of geometric measure theory.

UR - http://www.scopus.com/inward/record.url?scp=85076246086&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85076246086&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2019.106935

DO - 10.1016/j.aim.2019.106935

M3 - Article

AN - SCOPUS:85076246086

VL - 361

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 106935

ER -