A variational principle for the Bowen metric mean dimension of saturated set

For dynamical systems with infinite topological entropy, the classical entropy fails to quantify their complexity effectively, while the metric mean dimension provides a natural extension in this context. In this paper, we study the complexity of saturated sets from the perspective of Bowen upper and lower metric mean dimensions. We show that if a dynamical system $(X,f)$ satisfies the $g$-almost product property, then for any compact connected non-empty subset $K$ of a set of the convex combination of finitely many invariant measures, the saturated set $G_K$ satisfies $$\overline{\operatorname{mdim}}^B_{M}\left(G_K, f, d\right)=\limsup_{\varepsilon \rightarrow 0} \frac{1}{|\ln \varepsilon|} \inf_{\mu \in K} \inf _{\operatorname{diam}(\xi)<\varepsilon} h_\mu(f, \xi),$$ $$\underline{\operatorname{mdim}}^B_{M}\left(G_K , f, d\right)=\liminf_{\varepsilon \rightarrow 0} \frac{1}{|\ln \varepsilon|} \inf_{\mu \in K} \inf_{\operatorname{diam}(\xi)<\varepsilon} h_\mu(f, \xi),$$ where $\overline{\operatorname{mdim}}^B_{M}$ and $\underline{\operatorname{mdim}}^B_{M}$ denote Bowen upper and lower metric mean dimensions of $f$ on $G_K$, respectively, and $h_\mu(f,\xi)$ is the measure-theoretic entropy of the measure $\mu$ with respect to the partition $\xi$. As an application, we give an abstract framework for multifractal analysis of general continuous functions, which extends the prior work of Backes (2023, Trans. Inform. Theory, 69, 5485-5496) and Liu (2024, J. Math. Anal. Appl., 534, 128043).