Pseudo-anosov homeomorphisms with quadratic expansion

We show that if f : M → M is a pseudo-Anosov homeomorphism on an orientable surface with oriented unstable manifolds and a quadratic expanding factor, then there is a hyperbolic toral automorphism on T² and a map h : M → T² such that h is a semi-conjugacy and (M, h) is a branched covering space of T². We also give another characterization of pseudo-Anosov homeomorphisms with quadratic expansion in terms of the kinds of Euclidean foliations they admit which are compatible with the affine structure associated to f.