Diffusion for a chaotic plane sections of 3-periodic surfaces

Studying of asymptotic behavior of plane sections of 3-periodic surfaces was initiated by S. P. Novikov in 1982. Now all interesting remaining questions in this context are devoted to so called chaotic sections. In the talk, we discuss some applications of the dynamical tools to this topological problem. Namely, we use a natural generalization of interval exchange transformations – systems of isometries – and introduce Lyapunov exponents for a suspension flow of this kind of systems. For a very particular family of chaotic sections we express a diffusion rate of the trajectory in a terms of our Lyapunov exponents and prove that it is always strictly less than 1. By the last we answer on a question suggested us by A. Zorich. Joint work with Artur Avila and Sasha Skripchenko.