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A variational principle for the Bowen metric mean dimension of saturated set

Y. Yuan

Abstract

This paper investigates a variational principle for the Bowen metric mean dimension of saturated sets $G_K$, where $K$ is a compact connected subset of the convex combination of finite invariant measures for the systems with g-almost product property. In fact, we prove the variational principle of a saturated set with more information, that is $G_K\cap \{x\in X: C_f(X) \subset ω_f(x)\}$, which reveals that the limit point set of a saturated set contains all structure of the orbits. As an application, we obtain a more general version of multifractal analysis, which is derived independently and can imply partial results of Backes and Rodrigues (2023 IEEE Trans. Inform. Theory. 69 5485-5496).

A variational principle for the Bowen metric mean dimension of saturated set

Abstract

This paper investigates a variational principle for the Bowen metric mean dimension of saturated sets , where is a compact connected subset of the convex combination of finite invariant measures for the systems with g-almost product property. In fact, we prove the variational principle of a saturated set with more information, that is , which reveals that the limit point set of a saturated set contains all structure of the orbits. As an application, we obtain a more general version of multifractal analysis, which is derived independently and can imply partial results of Backes and Rodrigues (2023 IEEE Trans. Inform. Theory. 69 5485-5496).
Paper Structure (15 sections, 27 theorems, 202 equations)

This paper contains 15 sections, 27 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The metric mean dimension
  4. The Bowen metric mean dimension for non-compact subset
  5. Measure theoretic entropy
  6. Entropy formula
  7. g-almost specification
  8. Proof of Theorem \ref{['Maintheorem_G_mu']}
  9. Proof of Theorem \ref{['MainTheorem']}
  10. Upper bound for $\overline{\operatorname{mdim}}^B_{\mathrm{M}}\left(G^C_K, f, d\right)$ and $\underline{\operatorname{mdim}}^B_{\mathrm{M}}\left(G^C_K, f, d\right)$
  11. Lower bound for $\overline{\operatorname{mdim}}^B_{\mathrm{M}}\left(G^C_K, f, d\right)$ and $\underline{\operatorname{mdim}}^B_{\mathrm{M}}\left(G^C_K, f, d\right)$
  12. Proof of Corollary \ref{['cor-maintheorem']}
  13. Applications:Multifractal analysis
  14. Variational principle of level sets
  15. Full Bowen metric mean dimension of irregular sets

Key Result

Theorem 1.1

Let $f$ be a continuous map on a compact metric space and $\mu \in \mathcal{ M}_f(X)$ be ergodic. Then one has and where $\operatorname{diam} \xi$ denotes the diameter of the partition $\xi$ and the infimum is taken over all finite measurable partitions of $X$ satisfying $\operatorname{diam}\xi<\varepsilon$ for any $\mu \in \mathcal{M}_f(X)$.

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Corollary 1.1
  • Corollary 1.2
  • Remark 1.2
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 33 more