KAM Theorem for Gevrey Hamiltonians

Abstract

We consider Gevrey perturbations $H$ of a completely integrable Gevrey Hamiltonian $H_0$. Given a Cantor set ${\Omega }_{\kappa }$ defined by a Diophantine condition, we find a family of KAM invariant tori of $H$ with frequencies $\omega \in {\Omega }_{\kappa }$ which is Gevrey smooth in a Whitney sense. Moreover, we obtain a symplectic Gevrey normal form of the Hamiltonian in a neighborhood of the union $\Lambda$ of the invariant tori. This leads to effective stability of the quasiperiodic motion near $\Lambda$.