Convergence of spherical averages for actions of Fuchsian groups
Date(s) : 19/10/2018 iCal
9h30 - 10h30
Consider a measure-preserving action of a Fuchsian group G on a Lebesgue probability space X. Given a fundamental domain R, we obtain a symmetric generating set consisting of all group elements that map R to adjacent domains. This generating set endows the group G with the norm, and for a function f on X, we define its spherical average of order n as the average with equal weights of f shifted by all elements in G with the norm n.
Assume now that R has even corners, that is, that for the tessellation of the hyperbolic plane by images of R the boundaries between domains comprise of complete geodesic lines. Our result now says that if the even corners condition holds, then for any L^p-function f, p>1, its spherical averages of even orders converge almost surely.
The main ingredient of the proof is the construction of the new Markov coding for a Fuchsian group with the even corners condition. The key property of our coding is the following symmetry condition: the sequence of states generating an element g^{-1} is obtained from the sequence for g as follows: we reverse the order of its terms and apply an involution on the state space to each of these terms.
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The talk is based on the joint work with A. Bufetov and C. Series (arXiv:1805.11743).
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http://www.hse.ru/en/org/persons/36852818
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