The parameterization method for invariant manifolds III: overview and applications

We describe a method to establish existence and regularity of invariant manifolds and, at the same time to find simple maps which are conjugated to the dynamics on them. The method establishes several invariant manifold theorems. For instance, it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in Banach spaces. We also present several other applications of the method.