Limit Theorems for Translation Flows

Alexander Bufetov, Steklov Institute of Mathematics and Rice University
Fine Hall 314

The talk is devoted to limit theorems for translation flows on flat surfaces. Consider a compact oriented surface of genus at least two endowed with a holomorphic one-form. The real and the imaginary parts of the one-form define two foliations on the surface, and each foliation defines an area-preserving translation flow. By a Theorem of H.Masur and W.Veech, for a generic surface these flows are uniquely ergodic. 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. Next, given an Lipschitz function of average zero on the surface, we consider its time integrals as random variables and study the limit behaviour of their probability distributions. Informally, the main result 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