Double variational principle for mean dimension

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 w...

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Bibliographic Details
Published inGeometric and functional analysis Vol. 29; no. 4; pp. 1048 - 1109
Main Authors Lindenstrauss, Elon, Tsukamoto, Masaki
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.08.2019
Springer Nature B.V
Subjects
Analysis
Cameras
Distortion
Dynamical systems
Mathematics
Mathematics and Statistics
Minimax technique
Invariant measure
Geometric measure theory
37B99
Variational principle
37A05
Dynamical system
94A34
Mean dimension
Rate distortion dimension
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Summary: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.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
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content type line 14
ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-019-00501-8