Nearly integrable systems with orbits accumulating to KAM tori
The quasi-ergodic hypothesis, proposed by Ehrenfest and Birkhoff, says that
a typical Hamiltonian system of n degrees of freedom on a typical energy
surface has a dense orbit.
This question is wide open. In this talk I will explain a recent result by
V. Kaloshin and myself which can be seen as a weak form of the
quasi-ergodic hypothesis. We prove that a dense set of perturbations of
integrable Hamiltonian systems of two and a half degrees of freedom
possess orbits which accumulate in sets of positive measure. In
particular, they accumulate in prescribed sets of KAM tori.