JournalscmhVol. 98, No. 1pp. 41–134

Convergence of spherical averages for actions of Fuchsian groups

  • Alexander I. Bufetov

    Aix–Marseille Université, CNRS, France; and Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

  • Alexey Klimenko

    Steklov Mathematical Institute of Russian Academy of Sciences; and National Research University Higher School of Economics, Moscow, Russia

  • Caroline Series

    University of Warwick, Coventry, UK
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Abstract

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.

Cite this article

Alexander I. Bufetov, Alexey Klimenko, Caroline Series, Convergence of spherical averages for actions of Fuchsian groups. Comment. Math. Helv. 98 (2023), no. 1, pp. 41–134

DOI 10.4171/CMH/548