hyperbolic set

Let M be a compact smooth manifoldMathworldPlanetmath, and let f:MM be a diffeomorphism. An f-invariant subset Λ of M is said to be hyperbolic (or to have an hyperbolic structure) if there is a splitting of the tangent bundle of M restricted to Λ into a (Whitney) sum of two Df-invariant subbundles, Es and Eu such that the restriction of Df|Es is a contraction and Df|Eu is an expansion. This means that there are constants 0<λ<1 and c>0 such that

  1. 1.


  2. 2.

    Df(x)Exs=Ef(x)s and Df(x)Exu=Ef(x)u for each xΛ;

  3. 3.

    Dfnv<cλnv for each vEs and n>0;

  4. 4.

    Df-nv<cλnv for each vEu and n>0.

using some Riemannian metricMathworldPlanetmath on M.

If Λ is hyperbolic, then there exists an adapted Riemannian metric, i.e. one such that c=1.

Title hyperbolic set
Canonical name HyperbolicSet
Date of creation 2013-03-22 13:40:21
Last modified on 2013-03-22 13:40:21
Owner Koro (127)
Last modified by Koro (127)
Numerical id 5
Author Koro (127)
Entry type Definition
Classification msc 37D20
Synonym hyperbolic structure
Synonym uniformly hyperbolic
Related topic HyperbolicFixedPoint