One expects measure maximizing the entropy (m.m.e.) to be especially “interesting”, especially for dynamics with “some hyperbolicity”. For instance, under some (hyperbolicity) assumptions, one expects them to determine all the aperiodic invariant probability measure (see this expository paper). By a theorem of Newhouse (1987) based on Yomdin’s theory, smoothness ensures the existence of some m.m.e.

Finite multiplicity is a harder question – often it can be solved only after a thorough understanding of the dynamics. It is a classical result for uniformly hyperbolic diffeomorphisms. I proved it in my thesis for interval maps with nonzero entropy (finite smoothness is not enough, even though Ruelle’s inequality shows that all ergodic measures with lower bounded entropy have lower bounded Lyapunov exponents).

Here, at the School and Conference on Dynamical Systems at ICTP, I presented the following answer to a long standing question of Newhouse:

**Theorem** (B-Crovisier-Sarig). *A smooth diffeomorphism of a compact surface with nonzero topological entropy has finitely many ergodic measures maximizing the entropy.*

You can see the slides here.