# Measures of maximal entropy for surface diffeomorphisms

### Abstract

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.

## Authors

Jérôme Buzzi

Laboratoire de Mathématiques d'Orsay, CNRS - UMR 8628, Université Paris-Saclay, Orsay 91405, France

Sylvain Crovisier

Laboratoire de Mathématiques d'Orsay, CNRS - UMR 8628, Université Paris-Saclay, Orsay 91405, France

Omri Sarig

Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot 7610001, Israel