02-16 X. Cabre, E. Fontich, R. de la Llave

The parameterization method for invariant manifolds I: manifolds associated to non-resonant subspaces (215K, LaTeX) Jan 10, 02

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Abstract. We introduce a method to prove existence of invariant manifolds and, at the same time study their dynamics. 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 $X_1$ 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 $X_1$. The method of proof also provides information about the dynamics on the manifold. In addition, we prove regularity results on dependence on parameters. When $X_1$ 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 $X_1$ to be an spectral subspace or even to have a complement invariant under $A$.

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