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hyperbolic set (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 an 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$ such that the restriction of $ Df\vert _{E^s}$ is a contraction and $ Df\vert _{E^u}$ is an expansion. This means that there are constants $ 0<\lambda<1$ and $ c>0$ such that

  1. $ T_\Lambda M = E^s\oplus E^u$;
  2. $ Df(x)E^s_x = E^s_{f(x)}$ and $ Df(x)E^u_x = E^u_{f(x)}$ for each $ x\in \Lambda$;
  3. $ \Vert Df^nv\Vert < c\lambda^n\Vert v\Vert$ for each $ v\in E^s$ and $ n> 0$;
  4. $ \Vert Df^{-n}v\Vert < c\lambda^n \Vert v\Vert$ 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, i.e. one such that $ c=1$.



"hyperbolic set" is owned by Koro.
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See Also: hyperbolic fixed point

Other names:  hyperbolic structure, uniformly hyperbolic
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Cross-references: adapted, Riemannian metric, contraction, restriction, subbundles, sum, restricted, tangent bundle, subset, diffeomorphism, smooth manifold, compact
There are 5 references to this entry.

This is version 2 of hyperbolic set, born on 2003-06-11, modified 2006-06-08.
Object id is 4338, canonical name is HyperbolicSet.
Accessed 5132 times total.

Classification:
AMS MSC: 37D20 (Dynamical systems and ergodic theory :: Dynamical systems with hyperbolic behavior :: Uniformly hyperbolic systems )
Pending Errata and Addenda
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anoteher PM article on same topic by Linas on 2006-06-09 12:33:08
It appears that PM has another article on the same topic

http://planetmath.org/?op=getobj&from=objects&id=3315

perhaps these should be merged??
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