### Limit theorems for translation flows

Alexander I. Bufetov

Abstract.
The talk is devoted to limit theorems for translation flows on flat surfaces.
The first result of the talk, which extends earlier work of
A.Zorich and G.Forni, is an asymptotic formula for time
integrals of Lipschitz functions.
One of the main objects of the talk is the space of
finitely-additive Hoelder transverse invariant measures for our
foliations. These measures are classified and related to G.
Forni's invariant distributions of Sobolev regularity -1 for
translation flows. Time integrals of Lipschitz functions are
then shown to admit an asymptotic expansion in terms of the
finitely-additive measures.
The limit theorem then states that the probability
distributions of time integrals converge to an orbit of an
ergodic dynamical system in the space of random variables with
compactly supported distributions.
The argument relies on a symbolic representation of translation
flows as suspension flows over Vershik's automorphisms, a
construction developing one proposed by S.Ito.
The talk is based on the preprint
http://arxiv.org/abs/0804.3970