Geometriae Dedicata

, Volume 181, Issue 1, pp 293–306 | Cite as

Mean convergence of Markovian spherical averages for measure-preserving actions of the free group

Original Paper
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Abstract

Mean convergence of Markovian spherical averages is established for a measure-preserving action of a finitely-generated free group on a probability space. We endow the set of generators with a generalized Markov chain and establish the mean convergence of resulting spherical averages in this case under mild nondegeneracy assumptions on the stochastic matrix \(\varPi \) defining our Markov chain. Equivalently, we establish the triviality of the tail sigma-algebra of the corresponding Markov operator. This convergence was previously known only for symmetric Markov chains, while the conditions ensuring convergence in our paper are inequalities rather than equalities, so mean convergence of spherical averages is established for a much larger class of Markov chains.

Keywords

Ergodic theorems Markov operators Spherical averages Free groups 

Mathematics Subject Classification (2010)

37A30 

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Lewis Bowen
    • 1
  • Alexander Bufetov
    • 2
    • 3
    • 4
    • 5
  • Olga Romaskevich
    • 5
    • 6
  1. 1.University of Texas at AustinAustinUSA
  2. 2.CNRS, Centrale Marseille, I2M, UMR 7373Aix-Marseille UniversitéMarseilleFrance
  3. 3.The Steklov Institute of MathematicsMoscowRussia
  4. 4.The Institute for Information Transmission ProblemsMoscowRussia
  5. 5.National Research University Higher School of EconomicsMoscowRussia
  6. 6.École Normale Supérieure de LyonLyonFrance

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