Abstract:
Consider a compact oriented surface without boundary endowed by a flat structure with trivial holonomy. Motion with unit speed in a given direction yields a globally defined translation flow on our surface.
Dynamical properties of such flows were apparently first investigated by A. G. Mayer in Nizhnii Novgorod in early 1940's. They have been an object of intense study since the 1960's, in particular, in a recent cycle of papers of M. Kontsevich and A. Zorich.
In this talk we will be interested in the asymptotic behaviour of ergodic integrals of translation flows. By the Masur–Veech Theorem (1982), for a generic abelian differential the corresponding flow is uniquely ergodic. The first main result of the talk, which extends earlier work of A. Zorich and G. Forni, is an asymptotic formula for ergodic integrals. The main object is a special finite-dimensional space of Hölder cocycles over flow trajectories. The asymptotic expansion implies limit theorems for these flows; limit distributions have compact support.
The proof is based on a symbolic representation of translation flows as suspension flows over Vershik's automorphisms, a construction similar to one given by S. Ito.
The main results of the talk are exposed in the preprint: http://arxiv.org/abs/0804.3970v3.