Parametrizing Unstable and Very Unstable Manifolds

Existence and uniqueness theorems for unstable manifolds are well-known. Here we prove certain refinements. Let f : (C, 0) → C be a germ of an analytic diffeomorphism, whose derivative Df(0) has eigenvalues λ1, . . . , λn such that |λ1| ≥ · · · ≥ |λk| > |λk+1| ≥ · · · ≥ |λn|, with |λk| > 1. Then there is a unique k-dimensional invariant submanifold whose tangent space is spanned by the generalized eigenvectors associated to the eigenvalues λ1, . . . , λk, and it depends analytically on f . Further, there is a natural parametrization of this “very unstable manifold,” which can be extended to an analytic map C → C when f is defined on all of C, and is an injective immersion if f is a global diffeomorphism. We also give the corresponding statements for stable manifolds, which are analogous locally but quite different globally. 2000 Math. Subj. Class. Primary 37D10; Secondary 37F15, 37G05.