Hyperbolic set

Summary

A hyperbolic set of a diffeomorphism is a compact invariant set with a splitting of the tangent space at each of its points. One part is uniformly contracted and the other is uniformly expanded. It extends the notion of a hyperbolic fixed point from one point to a whole set, possibly a Cantor set or the whole manifold. Hyperbolic sets have stable and unstable manifolds, the shadowing property and robustness under perturbation. They are the setting of uniformly hyperbolic dynamics, which includes Anosov diffeomorphisms and the Smale horseshoe.