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Metric mean dimension of irregular sets for maps with shadowing

Magdalena Foryś-Krawiec, Piotr Oprocha

TL;DR

The paper addresses how irregular points, defined via divergent Birkhoff averages for a continuous observable $\Phi$, manifest dynamical complexity in systems with the shadowing property. It develops lower bounds for topological entropy and metric mean dimension of the $\Phi$-irregular set restricted to a neighborhood of a chain recurrent class $Y$, under the assumption that $Y$ supports ergodic measures with distinct $\Phi$-means. The method blends a simplified Pressure Distribution Principle with a Katok-type counting framework, constructing a recurrent, shadowing subset inside $B(Y,\varepsilon)$ whose growth rates bound the complexity from below by the corresponding invariants of $Y$. The results extend understanding of irregular sets beyond full space considerations, highlighting how local recurrence structure and shadowing enforce nontrivial complexity, and they raise open questions about when these lower bounds are actually equal to the invariants of $Y$.

Abstract

We study the metric mean dimension of $Φ$-irregular set $I_Φ(f)$ in dynamical systems with the shadowing property. In particular we prove that for dynamical systems with shadowing containing a chain recurrent class $Y$, the values of topological entropy together with the values of lower and upper metric mean dimension of the set $I_Φ(f)\cap B(Y,\varepsilon)\cap CR(f)$ are bounded from below by the respective values for class $Y$.

Metric mean dimension of irregular sets for maps with shadowing

TL;DR

The paper addresses how irregular points, defined via divergent Birkhoff averages for a continuous observable , manifest dynamical complexity in systems with the shadowing property. It develops lower bounds for topological entropy and metric mean dimension of the -irregular set restricted to a neighborhood of a chain recurrent class , under the assumption that supports ergodic measures with distinct -means. The method blends a simplified Pressure Distribution Principle with a Katok-type counting framework, constructing a recurrent, shadowing subset inside whose growth rates bound the complexity from below by the corresponding invariants of . The results extend understanding of irregular sets beyond full space considerations, highlighting how local recurrence structure and shadowing enforce nontrivial complexity, and they raise open questions about when these lower bounds are actually equal to the invariants of .

Abstract

We study the metric mean dimension of -irregular set in dynamical systems with the shadowing property. In particular we prove that for dynamical systems with shadowing containing a chain recurrent class , the values of topological entropy together with the values of lower and upper metric mean dimension of the set are bounded from below by the respective values for class .
Paper Structure (6 sections, 6 theorems, 19 equations)

This paper contains 6 sections, 6 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notions and definitions
  4. Measures
  5. Topological entropy and metric mean dimension
  6. Metric mean dimension of irregular sets

Key Result

Theorem 1.1

LimVar Let $f:X\to X$ be a continuous map with the gluing orbit property on compact metric space $X$ and let $\Phi:X\to \mathbb{R}^d$ be a continuous observable. Assume that $I_{\Phi}(f)\neq \emptyset$. Then $I_{\Phi}(f)$ carries full topological pressure and full metric mean dimension.

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 3.1
  • Theorem 3.2
  • ...and 2 more