Bernoulli equilibrium states for surface diffeomorphisms

Suppose f: M → M is a C^1+α (α > 0) diffeomorphism on a compact smooth orientable manifold M of dimension 2, and let μψ be an equilibrium measure for a Hölder-continuous potential ψ: M → ℝ. We show that if μψ has positive measure-theoretic entropy, then f is measure-theoretically isomorphic mod μψ to the product of a Bernoulli scheme and a finite rotation.