Identifying points of a pseudo-Anosov homeomorphism
We investigate the question, due to S.~Smale, of whether a hyperbolic automorphism $T$ of the $n$-dimensional torus can have a compact invariant subset homeomorphic to a compact manifold of positive dimension, other than a finite union of subtori. In the simplest case such a manifold would be a closed surface. A result of Fathi says that $T$ can sometimes have an invariant subset which is a finite-to-one image of a closed surface under a continuous map which is locally injective except possibly at a finite number of points, these being the singularities of the invariant foliations of a suitable pseudo-Anosov homeomorphism. For a class of pseudo-Anosov homeomorphisms whose invariant foliations are of a particularly simple type, we show that this map is never locally injective at the singularities. The proof involves finding pairs of points having lifts in the universal abelian cover whose orbits are similar, and in fact we find whole pairs of horseshoes worth of such points.