Pseudo-Anosov homeomorphisms with quadratic expansion

We show that if $f: M \rightarrow M$ 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 $\mathbb{T}^2$ and a map $h: M \rightarrow \mathbb{T}^2$ such that $h$ is a semi-conjugacy and $(M, h)$ is a branched covering space of $\mathbb{T}^2$. 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 $f$.