Measures of maximal entropy for surface diffeomorphisms

We show that $C^\infty$-surface diffeomorphisms with positive topological entropy have finitely many ergodic measures of maximal in general, and exactly one the topologically transitive case. This answers a question Newhouse, who proved such always exist. To do this we generalize Smale's spectral decomposition theorem to non-uniformly hyperbolic surface diffeomorphisms, introduce homoclinic classes measures, study their properties using codings by irreducible countable state Markov shifts.