On the Existence and Stability of an Infinite-Dimensional Invariant Torus

We consider an annular set of the form K = B x T-infinity, where B is a closed ball of the Banach space E, T-infinity is the infinite-dimensional torus (the direct product of a countable number of circles with the topology of coordinatewise uniform convergence). For a certain class of smooth maps Pi: K -> K, we establish sufficient conditions for the existence and stability of an invariant toroidal manifold of the form A = {(v,phi) epsilon K : v = h(phi) epsilon E, phi epsilon T-infinity}, where h(phi) is a continuous function of the argument phi epsilon T-infinity. We also study the question of the Cm-smoothness of this manifold for any natural m.