Identifying points of a pseudo-Anosov homeomorphism

• Autorzy: Gavin Band
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
• 37D10: Invariant manifold theory
• 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces
• 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2003
• Tom: 180
• Numer: 2
• Strony: 185-198

Daty

wydano

2003

Twórcy

autor

Gavin Band

• Department 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

Identyfikatory

DOI

10.4064/fm180-2-4