Diffusion in periodic billiards and for Novikov"s problem

I will describe the behavior of chaotic trajectories in Novikov"s problem (joint work with A. Avila and A. Skripchenko) in a very special situation first studied by Dynnikov.I will explain how to define a natural measure on the set of chaotic directions in Novikov"s problem.This set is a fractal set of zero measure, thus this is a non trivial problem. Using some results on Lyapunov exponents and following some ideas from Zorich and Forni, I will describe the asymptotic behavior of a generic chaotic trajectory. I will also give some results on the ergodic properties of the associated foliations.