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hyperbolic set
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(Definition)
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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 $Df|_{E^s}$ is a contraction and $Df|_{E^u}$ is an expansion. This means that there are constants $0<\lambda<1$ and $c>0$ such that
- $T_\Lambda M = E^s\oplus E^u$ ;
- $Df(x)E^s_x = E^s_{f(x)}$ and $Df(x)E^u_x = E^u_{f(x)}$ for each $x\in \Lambda$ ;
- $\|Df^nv\| < c\lambda^n\|v\|$ for each $v\in E^s$ and $n> 0$ ;
- $\|Df^{-n}v\| < c\lambda^n \|v\|$ 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, i.e. one such that $c=1$ .
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"hyperbolic set" is owned by Koro.
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Cross-references: adapted, Riemannian metric, contraction, restriction, subbundles, sum, 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 6081 times total.
Classification:
| AMS MSC: | 37D20 (Dynamical systems and ergodic theory :: Dynamical systems with hyperbolic behavior :: Uniformly hyperbolic systems ) |
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Pending Errata and Addenda
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