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A symmetric Markov coding and the ergodic theorem for actions of Fuchsian Groups
Alexander I. Bufetov1; Alexey Klimenko2; Caroline Series3
1 Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France; and Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina str. 8, 119991, Moscow, Russia
2 Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina str. 8, 119991, Moscow, Russia; and National Research University Higher School of Economics, Usacheva str. 6, 119048, Moscow, Russia
3 Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
Mathematics Research Reports, Volume 1 (2020), pp. 5-14.

The main result of this note is the pointwise convergence of spherical averages for measure-preserving actions of Fuchsian groups. The proof relies on a new self-inverse Markovian symbolic coding for Fuchsian groups and the method of Markov operators.

Received:
Accepted:
Published online:
DOI: 10.5802/mrr.3
Classification: 20H10, 22D40, 37A30
Keywords: Ergodic theorem, Fuchsian group, Markov coding, Markov operator, spherical averages.
Alexander I. Bufetov 1; Alexey Klimenko 2; Caroline Series 3

1 Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France; and Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina str. 8, 119991, Moscow, Russia
2 Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina str. 8, 119991, Moscow, Russia; and National Research University Higher School of Economics, Usacheva str. 6, 119048, Moscow, Russia
3 Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK 
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Alexander I.  Bufetov; Alexey Klimenko; Caroline Series. A symmetric Markov coding and the ergodic theorem for actions of Fuchsian Groups. Mathematics Research Reports, Volume 1 (2020), pp. 5-14. doi : 10.5802/mrr.3. https://mrr.centre-mersenne.org/articles/10.5802/mrr.3/

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