Lower Dimensional Invariant Tori in the Regions of Instability for Nearly Integrable Hamiltonian Systems

Consider a Hamiltonian system of KAM type, H(p,q)=N(p)+P(p,q), with n degrees of freedom (n>2), where the Hessian of N is nondegenerate. For one resonance condition <I,Np>=0, \ (I∈ℤn), there is an immersed (n−1) dimensional submanifold ? in action variable space, where almost every point corresponds to a resonant torus for the unperturbed system, which is foliated by (n−1) dimensional ergodic components. It is shown in this paper that there is a subset of ? with positive (n−1)-dim Lebesgue measure, such that for each resonant torus corresponding to a point in this set at least two (n−1)-dimensional tori can survive perturbations. Generically, one is hyperbolic and the other one is elliptic.