Metric mean dimension of irregular sets for maps with shadowing

TL;DR

This paper investigates the metric mean dimension of irregular sets in dynamical systems with shadowing, establishing bounds related to chain recurrent classes and their entropy and dimension values.

Contribution

It introduces bounds for the metric mean dimension of irregular sets in systems with shadowing, linking them to chain recurrent classes and their entropy.

Findings

Lower bounds for metric mean dimension of irregular sets

Connection between irregular sets and chain recurrent classes

Bounds involving topological entropy and dimension values

Abstract

We study the metric mean dimension of $\Phi$-irregular set $I_{\Phi}(f)$ 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 $I_{\Phi}(f)\cap B(Y,\varepsilon)\cap CR(f)$ are bounded from below by the respective values for class $Y$.