Integrability breaking in extended Hamiltonian systems

In low dimensional Hamiltonian systems, several classical results such as the KAM theorem or Nekhoroshev estimates guarantee that the dynamics remains close to integrable in the vicinity of an integrable point. In statistical physics and thermodynamics, one needs to consider extensive systems at positive temperature. In this case, the common belief is that integrability is completely lost as soon as one leaves the integrable limit. Nevertheless, as we will see in this talk, the dynamics on some intermediate time scales may be strongly affected by integrable effects. The understanding of the dynamics on such timescales is directly relevant to evaluate the thermal or electrical conductivity.
The talk will consist of two parts: First, I will introduce the topic, describe the phenomenology and state a few mathematical results that we obtained in the last years (works with W. De Roeck). Second, I will discuss the Green-Kubo formula for the conductivity and I will present some open problems related to our results.