KAM theory

The trajectories of an integrable Hamiltonian system are, in general of a simple nature : in some special coordinates they become just straight lines on an abelian manifold, or on a torus if the situation is real. Such motions are called quasi-periodic. The KAM theorem states that under reasonable assumptions on the initial Hamiltonian, most quasi-periodic motions persist under perturbation, where most as to be understood in the sense of Lebesgue measure. In his thesis, Arnold was able to prove that, in the planar three- body problem, the neighbourhood of an elliptic point contains invariant tori with quasi-periodic motions. In the 80's, Eliasson proved that most of the singular locus of the singular abelian varieties are also stable under pertubation. Finally, in 1998, Herman made the conjecture that such a critical point always contain a set of positive measure of invariant tori for ANY analytic hamiltonian (it is remarkable that it is important for Herman's that the Hamiltonian is analytic and not only $C^\infty$). My purpose will be to explain how to generalise the standard theory of Lie group and Lie algebra actions on manifolds to the infinite dimensional case for spaces which are neither Banach nor Fréchet. Applying these general notion to the KAM situation, we get a generalised KAM theorem , which in particular contains Herman's conjecture as a particular case.