Multifractal level sets and metric mean dimension with potential

Abstract

Let $(X,f)$ be a dynamical system with the specification property and $\ddot{I}†$ 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 $K_{\hat{I}}\pm :=\{x\in X:\underset{n\to \infty }{\lim }\frac{1}{n}\sum _{i=0}^{n-1}\ddot{I}†(f^{i}x)=\hat{I}\pm \}.$

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