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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, Anton Zorich of the Université Paris 7 Jussieu shows how to apply this technique to description of the diffusion of billiard trajectories in the plane with periodic polygonal obstacles, June 24, 2014 at the Bill Thurston Legacy Conference.
The conference, "What's Next? The mathematical legacy of Bill Thurston," held at Cornell June 23-27, 2014, brought together mathematicians from a broad spectrum of areas to describe recent advances and explore future directions motivated by Thurston's transformative ideas.