Lyapunov exponents of the Hodge bundle and diffusion in periodic billiards

Asymptotic behavior of leaves of a measured foliation on a Riemann surface is governed by the mean monodromy of the Hodge bundle along the associated trajectory of the Teichmüller geodesic flow in the moduli space. As a consequence, recent progress in the study of the Teichmüller flow (inspired by the fundamental work of A. Eskin and M. Mirzakhani) and in the study of the Lyapunov exponents of the Hodge bundle along this flow leads to new results on measured foliations on surfaces.

Following ideas of V. Delecroix, P. Hubert, and S. Lelièvre I will show how to apply this technique to description of the diffusion of billiard trajectories in the plane with periodic polygonal obstacles. The results presented in the talk are obtained in collaboration with J. Athreya, V. Delecroix, A. Eskin, and M. Kontsevich.