Invariant tori for Gevrey-smooth Hamiltonian systems under Rüssmann’s non-degeneracy condition

We consider Gevrey-smooth perturbation of a completely integrable Gevrey-smooth Hamiltonian under Rüssmann’s non-degeneracy condition, and obtain a family of invariant tori using an improved KAM iteration; they are Gevrey smooth in the sense of Whitney, with a Gevrey index depending on the Gevrey class of the Hamiltonian and on the exponent in the Diophantine condition.