Invariant Manifolds Associated to Non-Resonant Spectral Subspaces.

. We show that, if the linearization of a map at a fixed point leaves invariant a spectral subspace which satisfies certain non-resonance conditions, the map leaves invariant a smooth manifold tangent to this subspace. This manifold is as smooth as the map, but is unique among C L invariant manifolds, where L depends only on the spectrum of the linearization. We show that if the non-resonance conditions are not satisfied, a smooth invariant manifold need not exist and also establish smooth dependence on parameters. We also discuss some applications of these invariant manifolds and briefly survey related work. 1. Introduction Besides their intrinsic appeal, invariant manifold theorems are interesting in Dynamics because they provide landmarks which organize the long--time behavior. ?From this point of view, having more invariant manifolds is quite desirable, since it means having more tools for the analysis of the dynamical system. In particular, it is often the case, that associate..