Double variational principle for mean dimension

Elon Lindenstrauss, Masaki Tsukamoto

Research output: Contribution to journalArticle

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 languageEnglish
Pages (from-to)1048-1109
Number of pages62
JournalGeometric and Functional Analysis
Volume29
Issue number4
DOIs
Publication statusPublished - Aug 1 2019
Externally publishedYes

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Geometric Measure Theory
Invariant Measure
Variational Principle
Rate-distortion
Dynamical system
Metric
Dimension Theory
Minimax Problems
Ergodic Theory
Minimax

All Science Journal Classification (ASJC) codes

  • Analysis
  • Geometry and Topology

Cite this

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

In: Geometric and Functional Analysis, Vol. 29, No. 4, 01.08.2019, p. 1048-1109.

Research output: Contribution to journalArticle

Lindenstrauss, Elon ; Tsukamoto, Masaki. / Double variational principle for mean dimension. In: Geometric and Functional Analysis. 2019 ; Vol. 29, No. 4. pp. 1048-1109.
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