A KAM theorem for finitely differentiable Hamiltonian systems

Abstract

Given l>2ν>2d≥4, we prove the persistence of a Cantor–family of KAM tori of measure O(ε1/2−ν/l) for any non–degenerate nearly integrable Hamiltonian system of class Cl(D×Td), where ν−1 is the Diophantine power of the frequencies of the persistent KAM tori and D⊂Rd is a bounded domain, provided that the size ε of the perturbation is sufficiently small. This extends a result by D. Salamon in [25] according to which we do have the persistence of a single KAM torus in the same framework. Moreover, as for the persistence of a single torus, the regularity assumption is essentially optimal.