Convergence of spherical averages for actions of Fuchsian groups

Date(s) : 19/10/2018   iCal
9 h 30 min - 10 h 30 min

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.

The talk is based on the joint work with A. Bufetov and C. Series (arXiv:1805.11743).

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