The purpose of this paper is to study the dynamics near a reducible lower dimensional invariant tori of a nite-dimensional autonomous Hamiltonian system withd egrees 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 a way 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 0 (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, 0 r <`) is partially elliptic (let us call m e to the number of these directions). In this case we show that, given a xed numberof elliptic directions (let us call m 1 m e to this number), there exist a Cantor family of invariant tori of dimension r+m 1 , 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 R r+m 1) is proven to beexponentially 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 i n variant 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.

In this work w e c o n s i d er time d ependent quasiperiodic perturbations of autonomous Hamiltonian systems. We focus on the e ect that t his kind o f p e r t urbations has on lower dimensional invariant t ori. Our results s h ow t hat, under standard conditions of analyticity, nondegeneracy and nonresonance, most of these tori survive, adding the frequencies of the p e r t urbation to t he o n es they already have.The paper also contains estimates on the amount of surviving t ori. The w orst situation happens when the initial tori are normally elliptic. In this case, a torus (identi ed by t he v ector of intrinsic frequencies) can be continued with respect to a p e r t urbative p a r a m eter " 2 0 " 0 ], except for a set of " of measure exponentially small with " 0 . I n c a s e t hat " is xed (and su ciently small), we prove t he existence of invariant t ori for every vector of frequencies close to t he o n e o f t he initial torus, except for a set of frequencies of measure exponentially small with t he d i s t ance to the u nperturbed torus. As a particular case, if the p e r t urbation is autonomous, these results also give t he s a m e k i n d o f e s t imates on the m easure of destroyed tori.Finally, t hese results are applied to s o m e problems of celestial mechanics, in order to h elp in the d escripti o n o f t he p h ase space of some concrete m o d els.jorba@ma1.upc.es y jordi@tere.upc.es 2

In this paper we present a numerical method to compute Diophantine rotation numbers of circle maps with high accuracy. We mainly focus on analytic circle diffeomorphisms, but the method also works in the case of (enough) finite differentiability. The keystone of the method is that, under these conditions, the map is conjugate to a rigid rotation of the circle. Moreover, albeit it is not fully justified by our construction, the method turns to be quite efficient for computing rational rotation numbers. We discuss the method through several numerical examples.

We perform a bifurcation analysis of normal-internal resonances in parametrized families of quasi-periodically forced Hamiltonian oscillators, for small forcing. The unforced system is a one degree of freedom oscillator, called the 'backbone' system; forced, the system is a skew-product flow with a quasiperiodic driving with n basic frequencies. The dynamics of the forced system are simplified by averaging over the orbits of a linearization of the unforced system. The averaged system turns out to have the same structure as in the well-known case of periodic forcing (n = 1); for a real analytic system, the nonintegrable part can even be made exponentially small in the forcing strength. We investigate the persistence and the bifurcations of quasi-periodic n-dimensional tori in the averaged system, filling normal-internal resonance 'gaps' that had been excluded in previous analyses. However, these gaps cannot completely be filled up: secondary resonance gaps appear, to which the averaging analysis can be applied again. This phenomenon of 'gaps within gaps' makes the quasiperiodic case more complicated than the periodic case.

In this paper we introduce a general methodology for computing (numerically) the normal form around a periodic orbit of an autonomous analytic Hamiltonian system. The process follows two steps. First, we expand the Hamiltonian in suitable coordinates around the orbit and second, we perform a standard normal form scheme, based on the Lie series method. This scheme is carried out up to some nite order and, neglecting the remainder, we obtain an accurate description of the dynamics in a (small enough) neighbourhood of the orbit. In particular, we obtain the invariant tori that generalize the elliptic directions of the periodic orbit. On the other hand, bounding the remainder one obtains lower estimates for the di usion time around the orbit.This procedure is applied to an elliptic periodic orbit of the spatial Restricted Three Body Problem. The selected orbit belongs to the Lyapunov family associated to the vertical oscillation of the equilibrium point L 5 . The mass parameter has been chosen such that L 5 is unstable but the periodicorbit is still stable. This allows to show the existence of regions of e ective stability near L 5 for values of bigger that the Routh critical value. The computations have been done using formal expansions with numerical coe cients.2