smoothness of invariant manifolds

Question

Suppose M be compact manifold and f be a diffeomorphism on M.and A be hyperbolic set respect to f.How can we proof that the global stable and unstable manifolds of A are embedded manifolds?

Answer 1

The full proof you can find in any standard book on dynamical systems, say, Katok and Hasselblatt.

The idea is that first you show that local (un)stable manifolds have the same smoothness as the map and are embedded (it is called stable manifold theorem sometimes). Than you just iterate these local manifolds applying powers of the diffeo (and its inverse) to get a maximal invariant set, containing them. As diffeo is smooth, you will get a smooth manifold as a result.