Lower dimensional invariant tori in the regions of instability for nearly integrable Hamiltonian systems (Q1302023)

The author considers Hamiltonian systems of more than two degrees of freedom, consisting of a non-degenerate integrable part and a perturbation. The frequencies of the unperturbed part satisfy a simple resonant condition on an \((n-1)\)-dimensional submanifold \(\mathbb{M}\) in the action space. Almost every point of this manifold corresponds to a resonant \(n\)-dimensional torus of the unperturbed system, which is foliated by \((n-1)\)-dimensional ergodic components. It is proved that for each resonant \(n\)-dimensional torus of a subset of \(\mathbb{M}\) with positive measure, at least two \((n-1)\)-dimensional tori are continued under the perturbation, one being hyperbolic and the other elliptic. This important result fills a gap between the Poincaré-Birkhoff fixed point theory on the continuation of isolated periodic orbits from the non-isolated ones of the fully resonant tori of the integrable part, and the KAM theory, which guarantees the conservation under the perturbation of the non-resonant tori with frequencies which satisfy certain diophantine condition.