Anosov diffeomorphism

If M is a compact smooth manifoldMathworldPlanetmath, a diffeomorphism f:MM (or a flow ϕ:×MM) such that the whole space M is an hyperbolic set for f (or ϕ) is called an Anosov diffeomorphismMathworldPlanetmath (or flow).

Anosov diffeomorphisms were introduced by D.V. Anosov, who proved that they are 𝒞1-structurally stableMathworldPlanetmath.

Not every manifold admits an Anosov diffeomorphism; for example, there are no such diffeomorphisms on the sphere Sn. The simplest examples of compact manifolds admiting them are the tori 𝕋n: they admit the so called linear Anosov diffeomorphisms, which are isomorphismsPlanetmathPlanetmathPlanetmath of 𝕋n having no eigenvalue of modulus 1. It was proved that any other Anosov diffeomorphism in 𝕋n is topologically conjugateMathworldPlanetmath to one of this kind.

It is not known which manifolds support Anosov diffeomorphisms. The only known examples of are nilmanifolds and infranilmanifolds, and it is conjectured that these are the only ones. Anosov flows are more abundant; for example, if M is a Riemannian manifoldMathworldPlanetmath of negative sectional curvatureMathworldPlanetmath, then its geodesic flow is an Anosov flow.

Another famous conjecture is that the nonwandering set of any Anosov diffeomorphism is the whole manifold M. This is known to be true for linear Anosov diffeomorphisms (and hence for any Anosov diffeomorphism in a torus). For Anosov flows, there are examples where the nonwandering set is a proper subsetMathworldPlanetmathPlanetmath of M.

Title Anosov diffeomorphism
Canonical name AnosovDiffeomorphism
Date of creation 2013-03-22 13:47:43
Last modified on 2013-03-22 13:47:43
Owner Koro (127)
Last modified by Koro (127)
Numerical id 9
Author Koro (127)
Entry type Definition
Classification msc 37D20
Defines Anosov flow