Measures of maximal entropy for surface diffeomorphisms

By Jérôme Buzzi, S. Crovisier, O. Sarig. Annals of Mathematics, 2018

68 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d48a04bdc1fa998983b0364163a3abf2d3a45f92