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Hölder cocycles and ergodic integrals for translation flows on flat surfaces
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 [10]. 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 [5].