The main results announced in this note are an asymptotic expansion for ergodic integrals of
translation flows on flat surfaces of higher genus (Theorem 1)
and a limit theorem for such flows (Theorem 2).
Given an abelian differential on a compact oriented surface,
consider the space $\mathfrak B^+$ of Hölder cocycles over the corresponding vertical flow that are
invariant under holonomy by the horizontal flow.
Cocycles in $\mathfrak B^+$ are closely related to G.Forni's invariant distributions for
translation flows . Theorem 1 states that ergodic integrals of Lipschitz functions are approximated
by cocycles in $\mathfrak B^+$ up to an error that grows more slowly than any power of time. Theorem 2 is obtained using the renormalizing action of the Teichmüller flow on the space $\mathfrak B^+$.
A symbolic representation of translation flows as suspension flows over Vershik's automorphisms allows one to construct cocycles in $\mathfrak B^+$ explicitly.
Proofs of Theorems 1, 2 are given in .