Upper Hausdorff dimension estimates for invariant sets of evolutionary systems on Hilbert manifolds

Abstract

We prove a generalization of the Douady-Oesterl\'{e} theorem on the upper bound of the Hausdorff dimension of an invariant set of a smooth map on an infinite dimensional manifold. It is assumed that the linearization of this map is a noncompact linear operator. A similar estimate is given forthe Hausdorff dimension of an invariant set of a dynamical system generated by a differential equation on a Hilbert manifold.