integrable Hamiltonian systems

Professor: Marcel Guardia

Dates: September 17 - 26, 2013

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.