Hyperbolic set

In dynamical systems theory, a subset Λ of a smooth manifold M is said to have a hyperbolic structure with respect to a smooth map f if its tangent bundle may be split into two invariant subbundles, one of which is contracting and the other is expanding under f, with respect to some Riemannian metric on M. An analogous definition applies to the case of flows.

In the special case when the entire manifold M is hyperbolic, the map f is called an Anosov diffeomorphism; the dynamics of f on a hyperbolic set, or hyperbolic dynamics, exhibits features of local structural stability and has been much studied, cf. Axiom A.

Definition

Let M be a compact smooth manifold, f: MM a diffeomorphism, and Df: TMTM the differential of f. An f-invariant subset Λ of M is said to be hyperbolic, or to have a hyperbolic structure, if the restriction to Λ of the tangent bundle of M admits a splitting into a Whitney sum of two Df-invariant subbundles, called the stable bundle and the unstable bundle and denoted Es and Eu. With respect to some Riemannian metric on M, the restriction of Df to Es must be a contraction and the restriction of Df to Eu must be an expansion. Thus, there exist constants 0<λ<1 and c>0 such that

$T_{\Lambda }M=E^{s}\oplus E^{u}$ and

$(Df)_{x}E_{x}^{s}=E_{f(x)}^{s}$ and $(Df)_{x}E_{x}^{u}=E_{f(x)}^{u}$ for all $x\in \Lambda$ and

$\|Df^{n}v\|\leq c\lambda ^{n}\|v\|$ for all $v\in E^{s}$ and $n>0$ and

$\|Df^{-n}v\|\leq c\lambda ^{n}\|v\|$ for all $v\in E^{u}$ and $n>0$ .

If Λ is hyperbolic then there exists a Riemannian metric for which c = 1 — such a metric is called adapted.