Variational relations for metric mean dimension and rate distortion dimension
\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 \[ \mathrm{rdim}_{L^\infty}(X,T,d,\mu)=\lim_{\varepsilon\to 0}\sup h^K_{\mu}(T,\varepsilon)/|\log\varepsilon|=\lim_{\varepsilon\to 0}\sup s(\mu,\varepsilon)/|\log \varepsilon|. \] When the space has a tame growth of covering numbers, \(\mathrm{rdim}(X,T,d,\mu)=\mathrm{rdim}_{L^\infty}(X,T,d,\mu)\). In that case the following is interesting. For a set \(G_\mu\) of \(\mu\)-generic points there holds \[ \mathrm{mdim}^B(G_\mu,T,d)\leq\mathrm{rdim} (X,T,d,\mu)\leq\mathrm{mdim}^B(\mathrm{cl}(G_\mu),T,d). \] In the appendix another proof of the equivalence between the Bowen topological entropy and the weighted Bowen topological entropy is provided. The paper gives a clear introduction in the progress made in this field also adding interesting new results about the dimension of the involved subsets.
scientific article; zbMATH DE number 7373631
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