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  20130322 13:40:21 
Last modified on  20130322 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 