Measures of maximal entropy for surface diffeomorphisms

2018/11/06 by Buzzi, Jérôme, Crovisier, Sylvain, Sarig, Omri · 5 citations
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1811.02240

Abstract

We show that C^∞ 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.

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• Critical set for surface diffeomorphisms revisited
• Uniqueness of the measure of maximal entropy for geodesic flows on surfaces
• Metric entropy and homoclinic growth rate
• Construction of equilibrium states revisited
• Unique equilibrium states for some partially hyperbolic diffeomorphisms with dominated splittings

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