Hölder cocycles and ergodic integrals for translation flows on flat surfaces
Alexander I. Bufetov
Electronic Research Announcements 2010, 17(0): 34-42 doi: 10.3934/era.2010.17.34
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: limit theorems Hölder cocycles Translation flows Forni's invariant distributions Teichmueller flow abelian differentials Vershik's automorphisms
Infinite determinantal measures
Alexander I. Bufetov
Electronic Research Announcements 2013, 20(0): 12-30 doi: 10.3934/era.2013.20.12
Infinite determinantal measures introduced in this note are inductive limits of determinantal measures on an exhausting family of subsets of the phase space. Alternatively, an infinite determinantal measure can be described as a product of a determinantal point process and a convergent, but not integrable, multiplicative functional.
    Theorem 4.1, the main result announced in this note, gives an explicit description for the ergodic decomposition of infinite Pickrell measures on the spaces of infinite complex matrices in terms of infinite determinantal measures obtained by finite-rank perturbations of Bessel point processes.
keywords: Determinantal processes infinite-dimensional groups. ergodic decomposition

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