Mini Course - 5 classes

The quasi-ergodic hypothesis, proposed by Ehrenfest and Birkhoff, says that a typical Hamiltonian system on a typical energy surface has a dense orbit. This question is wide open. In the early sixties, V. Arnold constructed a nearly integrable Hamiltonian system presenting instabilities and he conjectured that such instabilities existed in typical nearly integrable Hamiltonian systems. A proof of Arnold's conjecture in two and half degrees of freedom was announced by J. Mather in 2003. In these lectures I will explain a recent proof of Arnol'd conjecture for two and a half degrees of freedom systems based on two works, which use a different approach. One by V. Kaloshin, P. Bernard and K. Zhang, and another by V. Kaloshin and K. Zhang. Their approach is based on constructing a net of normally hyperbolic invariant cylinders and a version of Mather variational method. In these lectures I will also explain a more recent work by myself and V. Kaloshin. In this work, using also this approach, we prove a weak form of the quasi-ergodic hypothesis. We prove that for a dense set of non-autonomous perturbations of two degrees of freedom Hamiltonian systems there exist unstable orbits which accumulate in a set of positive measure containing KAM tori.