[Paper Review] Variational principles for amenable metric mean dimensions

Ercai Chen, Dou Dou+1 more|arXiv (Cornell University)|Aug 7, 2017

Advanced Topology and Set Theory4 citations

TL;DR

This paper establishes variational principles linking metric mean dimension and rate distortion functions for countably infinite amenable group actions, extending Lindenstrauss and Tsukamoto's results from $Ζ$-actions to general amenable groups. By leveraging finite tiling properties and Følner sequences, it proves that upper and lower metric mean dimensions equal the limsup/liminf of normalized suprema of $L^p$ and $L^\infty$ rate distortion functions over invariant measures, generalizing classical variational principles to infinite entropy dynamics.

ABSTRACT

In this paper, we prove variational principles between metric mean dimension and rate distortion function for countable discrete amenable group actions which extend recently results by Lindenstrauss and Tsukamoto.

Motivation & Objective

• To generalize Lindenstrauss and Tsukamoto's variational principles for metric mean dimension from $Ζ$-actions to countably infinite amenable group actions.
• To establish a connection between metric mean dimension and rate distortion functions in the context of amenable group dynamics.
• To overcome technical challenges in constructing invariant measures by employing finite tiling results for amenable groups.
• To prove that metric mean dimension equals the asymptotic growth rate of the supremum of rate distortion functions across invariant measures.

Proposed method

• Utilizes the finite tiling result from Downarowicz et al. to construct specific Følner sequences for amenable groups.
• Applies the Misiurewicz method adapted to metric mean dimension via rate distortion functions.
• Employs mutual information and entropy estimation techniques to relate measure-theoretic and topological invariants.
• Uses weak* convergence of pushforward measures along Følner sequences to derive limit expressions.
• Applies data-processing inequality and entropy bounds to control mutual information in the limit.
• Establishes the variational principle by analyzing the asymptotic behavior of rate distortion functions under normalized group actions.

Experimental results

Research questions

• RQ1 Can variational principles for metric mean dimension be extended from $Ζ$-actions to general countably infinite amenable group actions?
• RQ2 Is the metric mean dimension equal to the limsup/liminf of the normalized supremum of $L^p$ and $L^\infty$ rate distortion functions over invariant measures for amenable group actions?
• RQ3 How can one construct invariant measures along Følner sequences in amenable group actions to relate topological and measure-theoretic quantities?
• RQ4 What role do finite tilings and quasi-tilings play in generalizing variational principles to amenable groups?
• RQ5 Does the rate distortion function fully capture the complexity of metric mean dimension in the amenable group setting?

Key findings

• The upper and lower metric mean dimensions of a dynamical system under an amenable group action are equal to the limsup and liminf, respectively, of the normalized supremum of $L^p$ rate distortion functions over all invariant measures.
• For $p>1$, the variational principle holds: $\overline{\rm{mdim}}_M(\mathcal{X},d) = \limsup_{\varepsilon\to 0} \frac{\sup_{\mu\in M(\mathcal{X},T)} R_{\mu,p}(\varepsilon)}{|\log \varepsilon|}$.
• The $L^\infty$ variational principle extends to amenable groups: $\underline{\rm{mdim}}_M(\mathcal{X},d) = \liminf_{\varepsilon\to 0} \frac{\sup_{\mu\in M(\mathcal{X},T)} R_{\mu,\infty}(\varepsilon)}{|\log \varepsilon|}$.
• The proof relies on a finite tiling result for amenable groups to construct Følner sequences that support the necessary measure approximation.
• The authors establish a lower bound on the rate distortion function via mutual information and entropy estimation, showing $R_{\mu,\infty}(\varepsilon) \geq S(\mathcal{X},G,d,12\varepsilon)$ in the limit.
• By taking $\alpha \to 0$, the inequality $R_{\mu,\infty}(\varepsilon) \geq S(\mathcal{X},G,d,12\varepsilon)$ is strengthened to show equality in the asymptotic regime.