Identifying points of a pseudo-Anosov homeomorphism

Volume 180 / 2003

Gavin Band Fundamenta Mathematicae 180 (2003), 185-198 MSC: 37B05, 37D10, 37D20, 37E30. DOI: 10.4064/fm180-2-4


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.


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

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