Positive metric entropy in nearly integrable Hamiltonian systems

The celebrated Kolmogorov-Arnold-Moser (KAM) theorem asserts that a small perturbation of a non-degenerate integrable Hamiltonian system preserves the quasi-periodicity of trajectories and invariant tori on a set of large measure. The question remains: how chaotic the system's behavior can be on the remaining “small” set? I will speak on a recent result of Dima Burago, Dong Chen and myself saying that every integrable system can be perturbed so that the resulting Hamiltonian system has positive measure-theoretic entropy.