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In mathematics, a subset of a manifold is said to have hyperbolic structure when its tangent bundle may be split into two invariant subbundles, one of which is contracting, and the other expanding.

[edit] Definition

Let M be a compact smooth manifold, and let <math>f:M\to M</math> be a diffeomorphism. An f-invariant subset <math>\Lambda</math> of M is said to be hyperbolic (or to have a hyperbolic structure) if there is a splitting of the tangent bundle of M restricted to <math>\Lambda</math> into a Whitney sum of two <math>Df</math>-invariant subbundles, <math>E^s</math> and <math>E^u</math>, the stable manifold and the unstable manifold. The splitting is such that the restriction of <math>Df|_{E^s}</math> is a contraction and <math>Df|_{E^u}</math> is an expansion. This means that there are constants <math>0<\lambda<1</math> and <math>c>0</math> such that

<math>T_\Lambda M = E^s\oplus E^u</math>

and

<math>Df(x)E^s_x = E^s_{f(x)}</math> and <math>Df(x)E^u_x = E^u_{f(x)}</math> for each <math>x\in \Lambda</math>

and

<math>\|Df^nv\| < c\lambda^n\|v\|</math> for each <math>v\in E^s</math> and <math>n> 0</math>

and

<math>\|Df^{-n}v\| < c\lambda^n \|v\|</math> for each <math>v\in E^u</math> and <math>n>0</math>.

using some Riemannian metric on M. If <math>\Lambda</math> is hyperbolic, then there exists an adapted Riemannian metric, that is, one such that c=1.

When the subset Λ is the entire manifold M, then the diffeomorphism f is called an Anosov diffeomorphism.

This article incorporates material from Hyperbolic Set on PlanetMath, which is licensed under the GFDL.