2010, 17: 34-42. doi: 10.3934/era.2010.17.34

Hölder cocycles and ergodic integrals for translation flows on flat surfaces

1. 

Department of Mathematics, Rice University, Texas, United States

Received  March 2010 Revised  April 2010 Published  June 2010

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].
Citation: Alexander I. Bufetov. Hölder cocycles and ergodic integrals for translation flows on flat surfaces. Electronic Research Announcements, 2010, 17: 34-42. doi: 10.3934/era.2010.17.34
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