Positive measure of KAM tori for finitely differentiable Hamiltonians

Abstract

Consider an integer $n\ge 2$ and real numbers $\tau >n-1$ and $l>2(\tau +1)$. Using ideas of Moser, Salamon proved that individual Diophantine tori persist for Hamiltonian systems which are of class $C^l$. Under the stronger assumption that the system is a $C^{l+\tau }$ perturbation of an analytic integrable system, P{ö}schel proved the persistence of a set of positive measure of Diophantine tori. We improve the last result by showing it is sufficient for the perturbation to be of class $C^l$ and the integrable part to be of class $C^{l+2}$.