The parameterization method for invariant manifolds I: Manifolds associated to non-resonant subspaces

作者：Xavier Cabré, Ernest Fontich, Rafael de la Llave · 发表于：Indiana University Mathematics Journal · 年份：2003 · 被引用次数：300 · 研究领域：Advanced Differential Equations and Dynamical Systems、Mathematical Dynamics and Fractals、Quantum chaos and dynamical systems

ABSTRACT. 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 C r 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 C r 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. 1.