The classical variational principle of entropy has sparked extensive studies on maximal entropy measures. Motivated by the Brin–Katok formula, Feng and Huang introduced measure-theoretic entropy for arbitrary Borel probability measures and established a corresponding variational principle — one links the topological entropy of an analytic set to the measure-theoretic entropy of measures supported on it. Against this background, this talk focuses on a key question: when do analytic sets admit maximal entropy measures? Our main results are as follows:

1. In any positive-entropy system, there exist two types of analytic sets: one admitting maximal entropy measures, and the other lacking them.
2. We construct a smooth surface system that contains a smooth curve supporting no Borel probability measure of maximal entropy.
3. For all h-expansive systems, we fully characterize the analytic sets carrying Borel probability measures of maximal entropy by introducing a new concept: the increasing countable entropy slice. As a consequence, in a Euclidean space, any analytic set with positive Hausdorff dimension either supports a measure of full lower Hausdorff dimension, or can be partitioned into a countable union of analytic sets with smaller Hausdorff dimension.

This talk is based on joint work with Xulei Wang.