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## Measures of maximal entropy for surface diffeomorphisms

With Sylvain CROVISIER and Omri SARIG, we show that $C^\infty$ surface diffeomorphisms with positive topological entropy have at most finitely many ergodic measures of maximal entropy in general, and at most one in the topologically transitive case. This answers a question of Newhouse, who proved that such measures always exist.

To do this we generalize Smale’s spectral decomposition theorem to non-uniformly hyperbolic surface diffeomorphisms, we introduce homoclinic classes of measures, and we study their properties using codings by irreducible countable state Markov
shifts.

The preprint is on arxiv (you can also download the preprint here).

A measure of maximal entropy of the rational fraction $1/(z^2+d)$ on the Riemann sphere according to Arnaud Chéritat. It describes the distribution of preimages of almost all points.