Multifractal level sets and metric mean dimension with potential

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