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