Maximal inequality and ergodic theorems for Markov groups

  • A. I. Bufetov
  • A. V. Klimenko


The paper shows that for actions of Markov semigroups, in particular, of finitely generated word hyperbolic groups, the Cesàro means of spherical averages converge almost everywhere for any function from the class L p , p > 1.


Probability Space STEKLOV Institute Directed Graph Ergodic Theorem Nonnegative Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. Bowen, “Invariant Measures on the Space of Horofunctions of a Word Hyperbolic Group,” Ergodic Theory Dyn. Syst. 30(1), 97–129 (2010).MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    L. Bowen and A. Nevo, “Geometric Covering Arguments and Ergodic Theorems for Free Groups,” arXiv: 0912.4953v2 [math.DS].Google Scholar
  3. 3.
    A. I. Bufetov, “Ergodic Theorems for Actions of Several Maps,” Usp. Mat. Nauk 54(4), 159–160 (1999) [Russ. Math. Surv. 54, 835–836 (1999)].MathSciNetCrossRefGoogle Scholar
  4. 4.
    A. I. Bufetov, “Operator Ergodic Theorems for Actions of Free Semigroups and Groups,” Funkts. Anal. Prilozh. 34(4), 1–17 (2000) [Funct. Anal. Appl. 34, 239–251 (2000)].MathSciNetCrossRefGoogle Scholar
  5. 5.
    A. I. Bufetov, “Markov Averaging and Ergodic Theorems for Several Operators,” in Topology, Ergodic Theory, Real Algebraic Geometry: Rokhlin’s Memorial (Am. Math. Soc., Providence, RI, 2001), AMS Transl., Ser. 2, 202, pp. 39–50.Google Scholar
  6. 6.
    A. I. Bufetov, “Convergence of Spherical Averages for Actions of Free Groups,” Ann. Math., Ser. 2,155, 929–944 (2002).MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. I. Bufetov, M. Khristoforov, and A. Klimenko, “Cesàro Convergence of Spherical Averages for Measure-Preserving Actions of Markov Semigroups and Groups,” Int. Math. Res. Not., doi: 10.1093/imrn/rnr181 (2011).Google Scholar
  8. 8.
    A. I. Bufetov and C. Series, “A Pointwise Ergodic Theorem for Fuchsian Groups,” Math. Proc. Cambridge Philos. Soc. 151, 145–159 (2011).MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    D. Calegari and K. Fujiwara, “Combable Functions, Quasimorphisms, and the Central Limit Theorem,” Ergodic Theory Dyn. Syst. 30(5), 1343–1369 (2010).MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    J. W. Cannon, “The Combinatorial Structure of Cocompact Discrete Hyperbolic Groups,” Geom. Dedicata 16(2), 123–148 (1984).MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory (Interscience, New York, 1958), Applied Mathematics 7.Google Scholar
  12. 12.
    K. Fujiwara and A. Nevo, “Maximal and Pointwise Ergodic Theorems for Word-Hyperbolic Groups,” Ergodic Theory Dyn. Syst. 18(4), 843–858 (1998).MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Sur les groupes hyperboliques d’après Mikhael Gromov, Ed. by É. Ghys and P. de la Harpe (Birkhäuser, Boston, 1990), Prog. Math. 83.zbMATHGoogle Scholar
  14. 14.
    R. I. Grigorchuk, “An Individual Ergodic Theorem for Actions of Free Groups,” in XII School on Operator Theory in Function Spaces (Tambov, 1987), Part 1, p. 57 [in Russian].Google Scholar
  15. 15.
    R. I. Grigorchuk, “Ergodic Theorems for Actions of Free Groups and Free Semigroups,” Mat. Zametki 65(5), 779–783 (1999) [Math. Notes 65, 654–657 (1999)].MathSciNetCrossRefGoogle Scholar
  16. 16.
    R. I. Grigorchuk, “An Ergodic Theorem for the Action of a Free Semigroup,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 231, 119–133 (2000) [Proc. Steklov Inst. Math. 231, 113–127 (2000)].MathSciNetGoogle Scholar
  17. 17.
    M. Gromov, “Hyperbolic Groups,” in Essays in Group Theory (Springer, New York, 1987), Math. Sci. Res. Inst. Publ. 8, pp. 75–263.CrossRefGoogle Scholar
  18. 18.
    A. Nevo, “Harmonic Analysis and Pointwise Ergodic Theorems for Noncommuting Transformations,” J. Am. Math. Soc. 7(4), 875–902 (1994).MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    A. Nevo, “Pointwise Ergodic Theorems for Actions of Groups,” in Handbook of Dynamical Systems (Elsevier, Amsterdam, 2006), Vol. 1B, pp. 871–982.CrossRefGoogle Scholar
  20. 20.
    A. Nevo and E. M. Stein, “A Generalization of Birkhoff’s Pointwise Ergodic Theorem,” Acta Math. 173, 135–154 (1994).MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    M. Pollicott and R. Sharp, “Ergodic Theorems for Actions of Hyperbolic Groups,” Proc. Am. Math. Soc. (in press).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  • A. I. Bufetov
    • 1
    • 2
    • 3
  • A. V. Klimenko
    • 1
    • 3
  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.Institute for Information Transmission Problems (Kharkevich Institute)Russian Academy of SciencesMoscowRussia
  3. 3.National Research University Higher School of EconomicsMoscowRussia

Personalised recommendations