Lower dimensional tori

Livia Corsi

Título: Lower dimensional tori

Resumen:

Consider a Liouville-integrable Hamiltonian system: then the phases space is foliated in invariant

tori on which the dynamics is linear. The celebrated KAM theorem states that all tori with Diophantine

frequency survive any perturbation smooth and small enough.

On the other hand, for each resonant frequency there is a family of ``lower dimensional tori'' which are

invariant for the unperturbed dynamics. When a perturbation is added, typically such families do not

survive,but a finite number of elements of the family might. All classical results concernig the existence

of such tori require some non-degeneracy condition on the perturbation.

Very recently an existence result has been estabilished for every perturbation, but only when the lower

dimensional tori have co-dimension 1 and the unperturbed Hamiltonian is of ``hyperbolic type'', or for

a class quasi-periodically forced equations. In higher co-dimension, the non-degeneracy condition can

be replaced by a parity condition for a class of quasi-periodically forced systems.

I will discuss the known results and I will try to illustrate the difficulties that one meets in the general case.