Abstract
We show that $C^\infty$-surface diffeomorphisms with positive topological entropy have finitely many ergodic measures of maximal entropy in general, and exactly 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.
Citation
Jérôme Buzzi. Sylvain Crovisier. Omri Sarig. "Measures of maximal entropy for surface diffeomorphisms." Ann. of Math. (2) 195 (2) 421 - 508, March 2022. https://doi.org/10.4007/annals.2022.195.2.2
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