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.

翻译：从几何视角出发，我们运用度量平均维数研究无穷熵系统中不变测度的泛点集与饱和集。对于具有specification性质的系统，我们基于Kolmogorov-Sinai $ε$-熵建立了饱和集的Bowen与packing度量平均维数的若干变分原理，并证明饱和集的上容量度量平均维数具有满度量平均维数。由此推得：不变测度泛点集的Bowen与packing度量平均维数与平均Rényi信息维数一致，且不变测度泛点集的上容量度量平均维数同样具有满度量平均维数。作为应用，对于具有specification性质的系统，我们给出了水平集的度量平均维数、无穷熵系统中平均Li-Yorke点对集合、以及紧致度量空间上全移位系统中不变测度泛点集的度量平均维数的定性刻画。