Identifying points of a pseudo-Anosov homeomorphism

Gavin Band¹

¹ Department of Mathematics University of Warwick Coventry CV4 7AL United Kingdom and Institute of Mathematics Polish Academy of Sciences P.O. Box 21 00-956 Warszawa, Poland

Fundamenta Mathematicae, Tome 180 (2003) no. 2, pp. 185-198

Résumé

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.

Détail

DOI : 10.4064/fm180-2-4

Keywords: investigate question due smale whether hyperbolic automorphism n dimensional torus have compact invariant subset homeomorphic compact manifold positive dimension other finite union subtori simplest manifold would closed surface result fathi says sometimes have invariant subset which finite to one image closed surface under continuous map which locally injective except possibly finite number points these being singularities invariant foliations suitable pseudo anosov homeomorphism class pseudo anosov homeomorphisms whose invariant foliations particularly simple type map never locally injective singularities proof involves finding pairs points having lifts universal abelian cover whose orbits similar whole pairs horseshoes worth points