Hyperbolic pseudo-Anosov maps almost everywhere embed into a toral automorphism

Abstract

Fathi and Franks showed that a pseudo-Anosov diffeomorphism $f$ with orientable foliations and dilation coefficient $\lambda$ with no conjugates (over ${\mathbb Q}$) in the unit circle factors onto a (homologically non-trivial) invariant subset of a hyperbolic toral automorphism. After recounting this result, we show that the factor map is either almost everywhere one-to-one or almost everywhere $m$-to-one for some $m>1$ and the pseudo-Anosov map $f$ is an $m$-to-one ramified covering of another pseudo-Anosov (or Anosov) map on a surface of smaller genus. As a corollary, any pseudo-Anosov diffeomorphism with orientable foliations and hyperbolic action on the first homologies almost everywhere embeds into a hyperbolic toral automorphism.