Approximate invariant manifolds up to exponentially small terms

This paper is devoted to analytic vector fields near an equilibrium for which the linearized system is split in two invariant subspaces E0 ( ), E1 ( ). Under light Diophantine conditions on the linear part, we prove that there is a polynomial change of coordinate in E1 allowing to eliminate, in the E1 component of the vector field, all terms depending only on the coordinate u0 E0, up to an exponentially small remainder. This main result enables to prove the existence of analytic center manifolds up to exponentially small terms and extends to infinite-dimensional vector fields. In the elliptic case, our results also proves, with very light assumptions on the linear part in E1, that for initial data very close to a certain analytic manifold, the solution stays very close to this manifold for a very long time, which means that the modes in E1 stay very small.