Hölder cocycles and ergodic integrals for translation flows on flat surfaces
Alexander I. Bufetov - Department of Mathematics, Rice University, Texas, United States (email)
Abstract: 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 .
Keywords: Translation flows, limit theorems, Vershik's
automorphisms, abelian differentials, Hölder cocycles, Forni's
invariant distributions, Teichmueller flow
Received: March 2010; Revised: April 2010; Available Online: June 2010.