Hyperbolic set

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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.

Definition

Let M be a compact smooth manifold, and let [f:M\to M] be a diffeomorphism. An f-invariant subset [\Lambda] 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 [\Lambda] into a Whitney sum of two [Df]-invariant subbundles, [E^s] and [E^u], the stable manifold and the unstable manifold. The splitting is such that the restriction of [Df|_] is a contraction and [Df|_] is an expansion. This means that there are constants [0<\lambda<1] and [c>0] such that

[T_\Lambda M = E^s\oplus E^u]

and

[Df(x)E^s_x = E^s_] and [Df(x)E^u_x = E^u_] for each [x\in \Lambda]

and

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

and

[\|Df^v\| < c\lambda^n \|v\|] for each [v\in E^u] and [n>0].

using some Riemannian metric on M. If [\Lambda] 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 on PlanetMath, which is licensed under the [Text of the GNU Free Documentation LicenseGFDL].

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