Maximal Entropy Measures

Hyperbolic (and hyperbolic like) dynamical systems have plenty of invariant measures, some of which are more equal than others: the physical measures (often coinciding with the Sinai-Ruelle-Bowen ones) and the measures of maximal entropy. In the best situations, measures of maximal entropy are those reflecting the periodic points and represent the richness of the dynamics.

In the smooth setting

Existence results are known under quite general assumptions: C infinity smoothness, central dimension at most 1. But basic problems remain open.

  • Find a C^1 (or better C^2) diffeomorphim of a compact surface without such a measure (examples, due to Misiurewicz are known only in dimension 4 and more).

Uniqueness (or finiteness) is harder to come by:

  • Find a C^2 diffeomophism of a compact surface with infinitely many (ergodic, invariant probability) measures of maximal entropy
  • Show that a C^infinity diffeomorphism of a compact surface has finitely many (ergodic, invariant probability) measures of maximal entropy
  • Show that a C^2 diffeomorphism of a compact manifold has at most countably many hyperbolic (ergodic, invariant probability) measures of maximal entropy

Properties of such measures are also of interest:

  • Is invariance by stable and unstable holonomies characteristic of such measures beyond the uniform case?

(J.Buzzi)

In the symbolic setting