The quasi-ergodic hypothesis and
Arnol'd diffusion in nearly
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
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
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.