hyperbolic set
Let be a compact smooth manifold, and let be a diffeomorphism.An -invariant subset of is said to be hyperbolic (or to have an hyperbolic structure) if there is a splitting of the tangent bundle of restricted to into a (Whitney) sum of two -invariant subbundles, and such that the restriction of is a contraction and is an expansion. This means that there are constants and such that
- 1.
;
- 2.
and for each ;
- 3.
for each and ;
- 4.
for each and .
using some Riemannian metric on .
If is hyperbolic, then there exists an adapted Riemannian metric, i.e. one such that .