Electronic Research Announcements in Mathematical Sciences (ERA-MS)

Hölder cocycles and ergodic integrals for translation flows on flat surfaces

Pages: 34 - 42, January 2010      doi:10.3934/era.2010.17.34

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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 [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].

Keywords:  Translation flows, limit theorems, Vershik's automorphisms, abelian differentials, Hölder cocycles, Forni's invariant distributions, Teichmueller flow
Mathematics Subject Classification:  Primary: 37A50; Secondary: 60F99.

Received: March 2010;      Revised: April 2010;      Available Online: June 2010.