Variational relations for metric mean dimension and rate distortion dimension
type: Item
instance of: scholarly article
title: Variational relations for metric mean dimension and rate distortion dimension
zbMATH Open document ID: 1479.37024
author: Yanyan Li
published in: Discrete and Continuous Dynamical Systems
publication date: 2021-07-22T00:00:00Z
review text: \textit{E. Lindenstrauss} and \textit{M. Tsukamoto} [Geom. Funct. Anal. 29, No. 4, 1048--1109 (2019; Zbl 1433.37025)] recently established a connection between mean dimension theory and rate distortion theory by proving that when a dynamical system has the marker property, then \[\mathrm{mdim}(X,T)=\min_{d\in D(X)}\sup_{\mu\in M(X,T)} \mathrm{rdim}(X,T,d,\mu),\] where \(D(X)\) is the collection of metrics compatible with the topology of \(X\) and \(\mu\in M(X,T)\) captures the topological complexity of \((X,T)\). This paper extends this result for compact subsets using the topological entropy introduced by \textit{R. Bowen} [Trans. Am. Math. Soc. 184, 125--136 (1974; Zbl 0274.54030)] and the measure-theoretical lower entropy \(\underline{h}_\mu(T)\) developed by \textit{D.-J. Feng} and \textit{W. Huang} [J. Funct. Anal. 263, No. 8, 2228--2254 (2012; Zbl 1267.37015)]. In fact the author proves that for a topological dynamical system \((X,T)\) with metric \(d\) and any compact subset \(K\) of \(X\) we have \[\mathrm{mdim}^B(K,T,d)=\lim_{\varepsilon\to 0} \sup\left(\sup_{\mu\in M_K(X)}\underline{h}_\mu(T,\varepsilon)/|\log\varepsilon|\right),\] thus establishing the connection between the mean dimension and the lower entropy for subsets. With the help of a redefinition of \(h_\mu(T)\) by \textit{A. Katok} [Publ. Math., Inst. Hautes Étud. Sci. 51, 137--173 (1980; Zbl 0445.58015)] by a quantity \(h^K_\mu(T,\varepsilon)\) and of \textit{C. E. Pfister} and \textit{W. G. Sullivan} [Ergodic Theory Dyn. Syst. 27, No. 3, 929--956 (2007; Zbl 1130.37329)] by \(s(\mu,\varepsilon)\) and their \({L^1}\) versions, a similar connection with the rate distortion dimension is obtained. The upper rate distortion dimension can be given as \[ \mathrm{rdim}(X,T,d,\mu)=\lim_{\varepsilon\to 0}\sup h^K_{\mu,L^1}(T,\varepsilon)/|\log\varepsilon|=\lim_{\varepsilon\to 0}\sup s_{L^1}(\mu,\varepsilon)/|\log \varepsilon|, \] and the \({L^\infty}\)-upper rate distortion dimension as \[ \mathrm{rdim}_…
reviewed by: Piet Borst
Mathematics Subject Classification ID: 37C45
Mathematics Subject Classification ID: 37B40
Mathematics Subject Classification ID: 54F45
zbMATH DE Number: 7373631
zbMATH Keywords: variational relation
zbMATH Keywords: metric mean dimension
zbMATH Keywords: rate distortion dimension
zbMATH Keywords: ergodic measure
MaRDI profile type: Publication
full work available at URL: dcds.2021050
OpenAlex ID: W3137487347
cites work: Topological Entropy
cites work: Topological Entropy for Noncompact Sets
cites work: Q3316247
cites work: Variational principles for topological entropies of subsets
cites work: Topological invariants of dynamical systems and spaces of holomorphic maps. I.
cites work: Lyapunov exponents, entropy and periodic orbits for diffeomorphisms
cites work: From Rate Distortion Theory to Metric Mean Dimension: Variational Principle
cites work: Double variational principle for mean dimension
cites work: Mean topological dimension
cites work: On the topological entropy of saturated sets
cites work: Double variational principle for mean dimension with potential
cites work: Escape of mass and entropy for geodesic flows
cites work: Q3931654
DOI: 10.3934/DCDS.2021050
Recommended article: Double variational principle for mean dimension with potential
Recommended article: Around the variational principle for metric mean dimension
Recommended article: Q3974885
Recommended article: Mean topological dimension
Recommended article: Genericity of continuous maps with positive metric mean dimension
Recommended article: Bowen’s equations for upper metric mean dimension with potential
Recommended article: Mean dimension, small entropy factors and an embedding theorem
Recommended article: Variational principle for metric mean dimension with potential
Recommended article: Mean dimension theory in symbolic dynamics for finitely generated amenable groups
Recommended article: Variational principles for amenable metric mean dimensions
description: scientific article; zbMATH DE number 7373631