Hyperbolic set

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

[edit] Definition

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

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

and

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

and

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

and

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

using some Riemannian metric on M. If Λ 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.

[edit] See also

[edit] References

  • Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin/Cummings Publishing, Reading Mass. ISBN 0-8053-0102-X

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