A KAM Theorem for finitely differentiable Hamiltonian systems

Abstract

Given $l>2\nu >2d\ge 4$, we prove the persistence of a Cantor--family of KAM tori of measure $O({\epsilon }^{1/2-\nu /l})$ for any non--degenerate nearly integrable Hamiltonian system of class $C^l(\mathcal{D}\times {\mathbb{T}}^d)$, where $\mathcal{D}\subset {\mathbb{R}}^d$ is a bounded domain, provided that the size $\epsilon$ of the perturbation is sufficiently small. This extends a result by D. Salamon in \cite{salamon2004kolmogorov} according to which we do have the persistence of a single KAM torus in the same framework. Moreover, it is well--known that, for the persistence of a single torus, the regularity assumption can not be improved.