KAM theory for properly-degenerate systems: remarks on the measure of the invariant set

The thesis extends the properly-degenerate KAM theory proposed by L. Chierchia and G. Pinzari to a very general setting. The original theorem assume that the averaged Hamiltonian function possesses the first order Birkhoff invariants which are non resonant up to order 4. And this thesis generalizes this assumption to arbitary order. Following the original proof by L. Chierchia and G. Pinzari, the measure of the set of invariant tori in the new setting is also generalized