Multifractal level sets and metric mean dimension with potential

TL;DR

This paper investigates the multifractal structure of dynamical systems with the specification property by establishing variational principles for metric mean dimension with potential on level sets defined by Birkhoff averages.

Contribution

It introduces conditional variational principles for the upper and lower Bowen/packing metric mean dimension with potential on multifractal level sets in systems with the specification property.

Findings

Established variational principles for metric mean dimension with potential.

Analyzed the structure of multifractal level sets in dynamical systems.

Extended the understanding of dimension theory in dynamical systems.

Abstract

Let $(X,f)$ be a dynamical system with the specification property and $\varphi$ be continuous functions. In this paper, we establish some conditional variational principles for the upper and lower Bowen/packing metric mean dimension with potential of multifractal level set $K_\alpha:=\{x\in X:\lim\limits_{n\to\infty}\dfrac{1}{n}\sum\limits_{i=0}^{n-1}\varphi(f^ix)=\alpha\}.$