Superexponential Stability of Quasi-Periodic Motion in Hamiltonian Systems

We prove that generically, both in a topological and measure-theoretical sense, an invariant Lagrangian Diophantine torus of a Hamiltonian system is superexponentially stable in the sense that nearby solutions remain close to the torus for an interval of time which dominate any time that is exponentially large with respect to the inverse of the distance to the torus. More specifically, we prove stability over times that are doubly exponentially large with respect to the inverse of the distance to the torus. We also prove that for an arbitrary small perturbation of a generic integrable Hamiltonian system, there exists a set of almost full positive Lebesgue measure of KAM tori which are superexponentially stable with the previous estimates. Our results hold true for real-analytic but more generally for Gevrey smooth systems.