hyperbolic set

Let $M$ be a compact smooth manifold, and let $f:M\to M$ be a diffeomorphism. An $f$-invariant subset $\mathrm{\Lambda }$ 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 $\mathrm{\Lambda }$ 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 and $c>0$ such that

1. $T_{\mathrm{\Lambda }}M=E^s\oplus E^u$;

2. $Df(x)E_x^s=E_{f(x)}^s$ and $Df(x)E_x^u=E_{f(x)}^u$ for each $x\in \mathrm{\Lambda }$;

3. for each $v\in E^s$ and $n>0$;

4. for each $v\in E^u$ and $n>0$.

using some Riemannian metric on $M$.

If $\mathrm{\Lambda }$ 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