We consider a family of conformally symplectic maps, which are characterized by the property that they transform a symplectic form into a multiple of itself. We assume that the conformal factor depends on a parameter, such that we recover the symplectic case when the parameter goes to zero. We study the perturbative expansions and the domains of analyticity in the symplectic limit of the parameterization of the quasi--periodic orbits with Diophantine frequency. Our main result is to prove that the tori are analytic in a domain in the complex parameter plane, obtained by taking from a ball centered at zero a sequence of smaller balls with center along smooth lines going through the origin. The proof is based on developing a theorem in an "a-posteriori" format, that is used to validate (under certain conditions) the formal asymptotic expanions. The rigorous results match very well recent numerical explorations that, in turn, suggest new conjectures. Joint work with R. Calleja and R. de la Llave

One distinguishing feature of Hamiltonian dynamical systems is that such systems, with very few exceptions, tend to have numerous periodic orbits and these orbits carry a lot of information about the dynamics of the system. In 1984 Conley conjectured that a Hamiltonian diffeomorphism (i.e., the time-one map of a Hamiltonian flow) of a torus has infinitely many periodic points. This conjecture was proved by Hingston some twenty years later, in 2004. Similar results for Hamiltonian diffeomorphisms of surfaces of positive genus were also established by Franks and Handel. Of course, one can expect the Conley conjecture to hold for a much broader class of closed symplectic manifolds and this is indeed the case as has been proved by Gurel, Hein and the speaker. However, the conjecture is known to fail for some, even very simple, phase spaces such as the sphere. These spaces admit Hamiltonian diffeomorphisms with finitely many periodic orbits -- the so-called pseudo-rotations -- which are of particular interest in dynamics. In this talk, we will discuss the role of periodic orbits in Hamiltonian dynamics and the methods used to prove their existence, and examine the situations where the Conley conjecture does not hold.

In this talk we present some results on the existence and regularity of invariant manifolds (or whiskers) for either equilibrium points or invariant tori (including periodic orbits) whose normal directions are neutral in the sense that the corresponding eigenvalues are one. We call these objects parabolic. The sufficient conditions for their existence come from higher order terms in the Taylor expansion of the normal dynamics. We consider both the case of maps and the case of differential equations with special attention for the maps case. Some of the analytical results are presented in \textit{a posteriori} format, meaning that if we have a sufficiently good approximation of the manifolds together with some non-degeneracy conditions, then there is a true manifold nearby. A methodology to find this approximated manifold is also provided by using the celebrated parametrization method. Some examples showing the differences between the \textit{non degenerate} (hyperbolic) case and our case are also provided along this talk. When the stable invariant manifold associated to the eigenvalue one, is one dimensional, we prove the Gevrey character of this manifolds. We apply our result to prove that the set of parabolic orbits in the Sitnikov problem and in the planar restricted three body problem is an invariant manifold which is $1/3$-Gevrey. The ``whiskers'' (stable and unstable manifolds) of Diophantine parabolic tori are present in several problems of celestial mechanics for open sets of the mass parameters. More concretely, we will show that one can find them in the restricted planar $n$-body problem and in the full planar $n$-body problem, for any $n\ge 1$, at least for values of the masses corresponding to the planetary problem (that is, where one mass is much larger than the rest). The whiskers of these parabolic tori have an important role in the dynamics of the $n$-body problem and may be used to obtain oscillatory orbits (solutions where one of the masses goes closer and closer to infinity but always returning to a fixed neighborhood of the rest) or diffusion orbits (solutions where the semiaxis of the orbits change along time). This is a joint work with E. Fontich and P.Martin.

The problem of the existence and the qualitative properties of parabolic and other selected trajectories (periodic, collision) for the N-body problem (from the classical celestial mechanics point of view to more recent advances in molecular and quantum models) has been extensively studied over the decades, and, more recently, new tools and approaches have given a significant boost to the field. We shall review some old an new results on the existence and classication of selected trajectories of the classical N-centre and N-body problem, with an emphasis on new analytical and geometrical techniques.