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Variational principles for metric mean dimension with potential of level sets

Lucas Backes, Chunlin Liu, Fagner B. Rodrigues

TL;DR

The paper develops three variational principles for the upper metric mean dimension with potential of level sets of a continuous map on systems with the specification property, linking geometric size to ergodic quantities via partitions and Katok-type entropy. The authors prove tight upper and lower bounds and obtain a matching variational formula expressed both through $H(μ)$ and through $H_δ^K(μ)$, for measures μ constrained by a fixed level α of a given observable φ. They further extend the framework to suspension flows, establishing flow-analogues and highlighting an Abramov-type relation between base and suspended systems, while showing that equalities may fail in general. The results significantly extend multifractal analysis to systems with infinite entropy and provide tools for analyzing the metric mean dimension of flows, with implications for embedding problems and ergodic optimization in high-entropy regimes.

Abstract

We establish three variational principles for the upper metric mean dimension with potential of level sets of continuous maps in terms of the entropy of partitions and Katok's entropy of the underlying system. Our results hold for dynamical systems exhibiting the specification property. Moreover, we apply our results to study the metric mean dimension of suspension flows.

Variational principles for metric mean dimension with potential of level sets

TL;DR

The paper develops three variational principles for the upper metric mean dimension with potential of level sets of a continuous map on systems with the specification property, linking geometric size to ergodic quantities via partitions and Katok-type entropy. The authors prove tight upper and lower bounds and obtain a matching variational formula expressed both through and through , for measures μ constrained by a fixed level α of a given observable φ. They further extend the framework to suspension flows, establishing flow-analogues and highlighting an Abramov-type relation between base and suspended systems, while showing that equalities may fail in general. The results significantly extend multifractal analysis to systems with infinite entropy and provide tools for analyzing the metric mean dimension of flows, with implications for embedding problems and ergodic optimization in high-entropy regimes.

Abstract

We establish three variational principles for the upper metric mean dimension with potential of level sets of continuous maps in terms of the entropy of partitions and Katok's entropy of the underlying system. Our results hold for dynamical systems exhibiting the specification property. Moreover, we apply our results to study the metric mean dimension of suspension flows.
Paper Structure (22 sections, 13 theorems, 181 equations)

This paper contains 22 sections, 13 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Metric mean dimension with potential
  4. Level sets of a continuous map
  5. A measure-theoretic upper metric mean dimension $H_\delta^K(\mu)$
  6. The measure-theoretic quantities $\mathrm{H_\varphi\overline{mdim}_M}\,(f,\alpha,d,\psi)$ and $H(\mu)$
  7. Specification property
  8. Main result
  9. Proof of Theorem \ref{['thm:main1']}
  10. Upper bounds for $\mathrm{\overline{mdim}_M^B}\,(K_\alpha,f,d,\psi )$
  11. Lower bounds for $\mathrm{\overline{mdim}_M^B}\,(K_\alpha,f,d,\psi )$
  12. Construction of $\mathcal{S}_k$
  13. Construction of $\mathcal{C}_k$
  14. Construction of $\mathcal{T}_k$
  15. Construction of $F$
  16. ...and 7 more sections

Key Result

Theorem 2.2

Suppose $f:X \to X$ is a continuous map with the specification property. Let $\varphi \in C(X,\mathbb R)$ and $\alpha\in\mathbb R$ be such that $K_\alpha\not=\emptyset$. Then, given $\psi \in C(X,\mathbb R)$, for every $\delta \in (0,1)$ we have that

Theorems & Definitions (22)

  • Remark 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3: Lemma 5.1 of TV
  • Lemma 3.4: Lemma 5.2 of TV
  • Lemma 3.5: Lemma 5.3 of TV
  • Lemma 3.6: Lemma 3.10 of MR2765447
  • Lemma 3.7: Lemma 3.17 of MR2765447 and Lemma 3.20 of Tho
  • ...and 12 more