hyperbolic set

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 $D f$ -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

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.

$\|Df^{n}v\|<c\lambda^{n}\|v\|$ for each $v\in E^{s}$ and $n>0$ ;
4.

$\|Df^{-n}v\|<c\lambda^{n}\|v\|$ 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 $c=1$ .

Title	hyperbolic set
Canonical name	HyperbolicSet
Date of creation	2013-03-22 13:40:21
Last modified on	2013-03-22 13:40:21
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	5
Author	Koro (127)
Entry type	Definition
Classification	msc 37D20
Synonym	hyperbolic structure
Synonym	uniformly hyperbolic
Related topic	HyperbolicFixedPoint