We show that the Birkhoff normal form near a positive definite KAM torus is given by the function of Mather. This observation is due to Siburg [Si2], [Si1] in dimension 2. It clarifies the link between the Birkhoff invariants and the action spectrum near the torus. Our extension to high dimension is made possible by a simplification of the proof given in [Si2].
Nondegeneracy conditions need to be imposed in K.A.M. theorems to insure that the set of diophantine tori has a large measure. Although they are usually expressed in action coordinates, it is possible to give a geometrical formulation using the notion of regular completely integrable systems defined by a fibration of a symplectic manifold by lagrangian tori together with a Hamiltonian function constant on the fibers. In this paper, we give a geometrical definition of different nondegeneracy conditions,...