A variational principle for the metric mean dimension of level sets

We prove a variational principle for the upper and lower metric mean dimension of level sets $$\{x\in X:\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n}\sum _{j=0}^{n-1}\phi (f^j(x))=\alpha \}$$ associated to continuous potentials $\phi :X\to \mathbb{R}$ and continuous dynamics $f:X\to X$ defined on compact metric spaces and exhibiting the specification property. This result relates the upper and lower metric mean dimension of the above mentioned sets with growth rates of measure-theoretic entropy of partitions decreasing in diameter associated to some special measures. Moreover, we present several examples to which our result may be applied to. Similar results were previously known for the topological entropy and for the topological pressure.