Double variational principle for mean dimension

Elon Lindenstrauss, Masaki Tsukamoto*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

26 Scopus citations

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 languageAmerican English
Pages (from-to)1048-1109
Number of pages62
JournalGeometric and Functional Analysis
Volume29
Issue number4
DOIs
StatePublished - 1 Aug 2019

Bibliographical note

Publisher Copyright:
© 2019, Springer Nature Switzerland AG.

Keywords

  • Dynamical system
  • Geometric measure theory
  • Invariant measure
  • Mean dimension
  • Rate distortion dimension
  • Variational principle

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