The Parameterization Method for Invariant Manifolds I: Manifolds Associated to Non-resonant Subspaces

We introduce a method to prove existence of invariant manifolds and, at the same time to find simple polynomial maps which are conjugated to the dynamics on them. As a first application, we consider the dynamical system given by a Cr map F in a Banach space X close to a fixed point: F(x) = Ax + N(x), A linear, N(0) = 0, DN(0) = 0. We show that if X1 is an invariant subspace of A and A satisfies certain spectral properties, then there exists a unique Cr manifold which is invariant under F and tangent to X1. When X1 corresponds to spectral subspaces associated to sets of the spectrum contained in disks around the origin or their complement, we recover the classical (strong) (un)stable manifold theorems. Our theorems, however, apply to other invariant spaces. Indeed, we do not require X1 to be a spectral subspace or even to have a complement invariant under A.