Parametrizing Unstable and Very Unstable Manifolds
Joseph B. Hubbard · Moscow Mathematical Journal · 2005
Abstract. Existence and uniqueness theorems for unstable manifolds are well-known. Here we prove certain refinements. Let f:(C n, 0) → C n 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 k → C n when f is defined on all of C n, 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. Key words and phrases. Invariant manifold, resonance. The origin of this paper lies in an attempt to numerically compute unstable manifolds for Hénon mappings H: C 2 → C 2. Suppose H(p) =p and that DH(p) has eigenvalues λ, µ with |λ |> 1> |µ|. Let v be an eigenvector for the eigenvalue λ. Our parametrization of the unstable manifold at p is given by the following theorem. Theorem 1. The limit Φ(z) = lim Hm x +