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

1. 1.

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

2. 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. 3.

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

4. 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 $c=1$.