On the metric mean dimensions of saturated sets

Abstract: Saturated sets 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, we aim to provide some qualitative aspects of saturated sets and the set of generic points from both topological and measure-theoretic perspectives. For systems with the specification property, we establish 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 has full metric mean dimension. As applications, we further present some qualitative aspects of the metric mean dimensions of level sets and the set of mean Li-Yorke pairs in infinite entropy systems.