Measures of maximal entropy for surface diffeomorphisms

We show that C^∞-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.