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

page revision: 1, last edited: 01 Jun 2011 14:19