Irregular set and metric mean dimension with potential

Abstract

Let $(X,f)$ be a dynamical system with the specification property and $\ddot{I}†$ be a continuous function. In this paper, we consider the multifractal irregular set \begin{align*} I_φ=\left\{x\in X:\lim\limits_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}φ(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.

23pages. arXiv admin note: substantial text overlap with arXiv:2407.15027