We prove pointwise convergence of spherical averages for a measure-preserving action of a Fuchsian group. The proof is based on a new variant of the Bowen–Series symbolic coding for Fuchsian groups that, developing a method introduced by Wroten, simultaneously encodes all possible shortest paths representing a given group element. The resulting coding is self-inverse, giving a reversible Markov chain to which methods previously introduced by the first author for the case of free groups may be applied.
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Alexander I. Bufetov, Alexey Klimenko, Caroline Series, Convergence of spherical averages for actions of Fuchsian groups. Comment. Math. Helv. 98 (2023), no. 1, pp. 41–134DOI 10.4171/CMH/548