Minimal invariant tori in the resonant regions for nearly integrable Hamiltonian systems

Consider a real analytical Hamiltonian system of KAM type that has degrees of freedom () and is positive definite in . Let . In this paper we show that for most rotation vectors in , in the sense of ()-dimensional Lebesgue measure, there is at least one ()-dimensional invariant torus. These tori are the support of corresponding minimal measures. The Lebesgue measure estimate on this set is uniformly valid for any perturbation.