Limit theorem for generic interval exchange transformations

The talk is focused on the spatial limit theorem for interval exchange transformations (IETs). Namely, let us consider an ergodic sum at the point x as the random variable. That is, x is assumed to be uniformly distributed on the segment (note that this distribution is the only one that is invariant under the dynamics). We show that the asymptotical behavior of the appropriately normalized ergodic sums is the same as for the ergodic integrals for the translation flows on flat surfaces. The latter case was analyzed by A.I. Bufetov, who proved that these distributions has no limit but has some limit points. These limit poins were described in terms of the Teichmueller flow in the space of translation flows. The case of interval exchange transformations brings some additional difficulties since in this case the renormalizing dynamics, the Rauzy induction, change the time differently on different trajectories.