On the metric mean dimensions of saturated sets

From a geometric perspective, we employ metric mean dimension to investigate the set of generic points of invariant measures and saturated sets in infinite entropy systems. For systems with the specification property, we establish certain variational principles for the Bowen and packing metric mean dimensions of saturated sets in terms of Kolmogorov-Sinai $ε$-entropy, and prove that the upper capacity metric mean dimension of saturated sets has full metric mean dimension. Consequently, the Bowen and packing metric mean dimensions of the set of generic points of invariant measures coincide with the mean Rényi information dimension, and the upper capacity metric mean dimension of the set of generic points of invariant measures also has full metric mean dimension. As applications, for systems with the specification property, we present the qualitative characterization of the metric mean dimensions of level sets, the set of mean Li-Yorke pairs in infinite-entropy systems, and the set of generic points of invariant measures in full shifts over compact metric spaces.