On the Normal Behaviour of Partially Elliptic Lower Dimensional Tori of Hamiltonian Systems

The purpose of this paper is to study the dynamics near a reducible lower dimen sional invariant tori of a nite dimensional autonomous Hamiltonian system with degrees of freedom We will focus in the case in which the torus has some elliptic directions First let us assume that the torus is totally elliptic In this case it is shown that the di usion time the time to move away from the torus is exponentially big with the initial distance to the torus The result is valid in particular when the torus is of maximal dimension and when it is of dimension elliptic point In the maximal dimension case our results coincide with previous ones In the zero dimension case our results improve the existing bounds in the literature Let us assume now that the torus of dimension r r is partially elliptic let us call me to the number of these directions In this case we show that given a xed number of elliptic directions let us call m me to this number there exist a Cantor family of invariant tori of dimension r m that generalize the linear oscillations corresponding to these elliptic directions Moreover the Lebesgue measure of the complementary of this Cantor set in the frequency space Rr m is proven to be exponentially small with the distance to the initial torus This is a sort of Cantorian central manifold theorem in which the central manifold is completely lled up by invariant tori and it is uniquely de ned The proof of these results is based on the construction of suitable normal forms around the initial torus