On the metric mean dimensions of saturated sets

Saturated set and its reduced case, the set of generic points, constitute two significant types of fractal-like sets in multifractal analysis of dynamical systems. In the context of infinite entropy systems, this paper aims to give some qualitative aspects of saturated sets and the set of generic points in both topological and measure-theoretic perspectives. For systems with specification property, we establish the certain variational principles for saturated sets in terms of Bowen and packing metric mean dimensions, and show the upper capacity metric mean dimension of saturated sets have full metric mean dimension. All results are useful for understanding the topological structures of dynamical systems with infinite topological entropy. As applications, we further exhibit some qualitative aspects of metric mean dimensions of level sets and the set of mean Li-Yorke pairs in infinite entropy systems.