Saturated sets and its reduced case the set of generic points are two types of important 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 generic points in both topological and measure-theoretic situations. 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.

翻译：饱和集及其简化情形——通有点集是动力系统多重分形分析中两类重要的分形结构。在无限熵系统的背景下，本文旨在从拓扑和测度论两个角度描述饱和集与通有点集的若干定性性质。对于具有指定性质的系统，我们建立了基于Bowen度量平均维数与填充度量平均维数的饱和集变分原理，并证明饱和集的上容量度量平均维数等于系统的全度量平均维数。所有结果对于理解具有无限拓扑熵的动力系统的拓扑结构具有重要意义。作为应用，我们进一步展示了无限熵系统中水平集的度量平均维数与平均Li-Yorke点集度量平均维数的若干定性特征。