Quasiperiodic Poincaré Persistence at High Degeneracy

For Hamiltonian systems with high-order degenerate perturbation, we study the persistence of resonant invariant tori, where the resonant tori might be elliptic, hyperbolic or mixed types. As a consequence, we prove a quasiperiodic Poincaré theorem at high degeneracy. This answers a long standing conjecture on the persistence of resonant invariant tori in general situations.