Irregular set and metric mean dimension with potential

TL;DR

This paper investigates the multifractal irregular set in dynamical systems with the specification property, showing it is either empty or has full metric mean dimension with potential, revealing its complex structure.

Contribution

It establishes the full metric mean dimension of the irregular set in systems with the specification property, extending understanding of multifractal analysis in dynamical systems.

Findings

Irregular set is either empty or has full metric mean dimension.

Full metric mean dimension holds with potential for the irregular set.

Results apply to systems with the specification property.

Abstract

Let $(X,f)$ be a dynamical system with the specification property and $\varphi$ be a continuous function. In this paper, we consider the multifractal irregular set \begin{align*} I_{\varphi}=\left\{x\in X:\lim\limits_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}\varphi(f^ix)\ \text{does not exist}\right\} \end{align*} and show that this set is either empty or carries full Bowen upper and lower metric mean dimension with potential.