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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3

References

  1. 1.

    Arnold, V.I., Krylov, A.L.: Equidistribution of points on a sphere and ergodic properties of solutions of ordinary differential equations in a complex domain. Dokl. Akad. Nauk SSSR 148, 9–12 (1963)

    MathSciNet  Google Scholar 

  2. 2.

    Birman, J., Series, C.: Dehns algorithm revisited, with application to simple curves on surfaces. In: Gersten, S., Stallings, J., (eds.), Combinatorial Group Theory and Topology. Ann. of Math. Studies III, Princeton U.P., pp. 451–478 (1987)

  3. 3.

    Bowen, L.: Invariant measures on the space of horofunctions of a word hyperbolic group. Ergod. Theory Dyn. Syst. 30(1), 97–129 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Bowen, L., Nevo, A.: Geometric covering arguments and ergodic theorems for free groups. LEnseignement Mathématique 59(1/2), 133–164 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Bowen, L., Nevo, A.: Amenable equivalence relations and the construction of ergodic averages for group actions. J. Anal. Math. 126, 359–388 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Bowen, L., Nevo, A.: von-Neumann and Birkhoff ergodic theorems for negatively curved groups, arXiv:1303.4109 [math.DS]

  7. 7.

    Bowen, L., Nevo, A.: A horospherical ratio ergodic theorem for actions of free groups. Groups Geom. Dyn. 8(2), 331–353 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Bowen, R., Series, C.: Markov maps associated with Fuchsian groups. IHES Publ. 50, 153–170 (1979)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Bufetov, A.I.: Ergodic theorems for actions of several mappings. (Russian) Uspekhi Mat. Nauk 54(4 (328)), 159–160 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Bufetov, A.I.: Translation in Russian math. Surveys 54(4), 835–836 (1999)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Bufetov, A.I.: Operator ergodic theorems for actions of free semigroups and groups. Funct. Anal. Appl. 34, 239–251 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Bufetov, A.I.: Markov averaging and ergodic theorems for several operators. Topol. Ergod. Theory Algebraic Geom. AMS Transl. 202, 39–50 (2001)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Bufetov, A.I.: Convergence of spherical averages for actions of free groups. Ann. Math. 155, 929–944 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Bufetov, A.I., Khristoforov, M., Klimenko, A.: Cesàro convergence of spherical averages for measure-preserving actions of Markov semigroups and groups. Int. Math. Res. Not. IMRN 2012(21), 4797–4829 (2012)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Bufetov, A.I., Series, C.: A pointwise ergodic theorem for Fuchsian groups, arXiv:1010.3362v1 [math.DS]

  16. 16.

    Calegari, D., Fujiwara, K.: Combable functions, quasimorphisms, and the central limit theorem. Ergod. Theory Dyn. Syst. 30(5), 1343–1369 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Cannon, J.: The combinatorial structure of cocompact discrete hyperbolic groups. Geom. Dedic. 16(2), 123–148 (1984)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Fujiwara, K., Nevo, A.: Maximal and pointwise ergodic theorems for word-hyperbolic groups. Ergod. Theory Dyn. Syst. 18, 843–858 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Sur les groupes hyperboliques d’après Mikhael Gromov. Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. In: É. Ghys and P. de la Harpe (eds.) Progress in Mathematics, 83. Birkhäuser Boston, Inc., Boston, MA (1990)

  20. 20.

    Grigorchuk, R.I.: Pointwise ergodic theorems for actions of free groups. Proceedings of Tambov Workshop in the Theory of Functions (1986)

  21. 21.

    Grigorchuk, R.I.: Ergodic theorems for actions of free semigroups and groups. Math. Notes 65, 654–657 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Grigorchuk, R.I.: An ergodic theorem for actions of a free semigroup. (Russian) Tr. Mat. Inst. Steklova 231, 119–133 (2000). Din. Sist., Avtom. i Beskon. Gruppy

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Grigorchuk, R.I.: Translation in Proc. Steklov Inst. Math. no. 4 (231), 113–127 (2000)

  24. 24.

    M. Gromov, Hyperbolic groups. In: Essays in Group Theory, MSRI Publ. 8: 75–263. Springer-Verlag, New York (1987)

  25. 25.

    Guivarc’h, Y.: Généralisation d’un théorème de von Neumann, C. R. Acad. Sci. Paris Sér. A-B 268, 1020–1023 (1969)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Kakutani, S.: Random ergodic theorems and Markoff processes with a stable distribution, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley and Los Angeles, pp. 247–261 (1951)

  27. 27.

    Nevo, A.: Harmonic analysis and pointwise ergodic theorems for noncommuting transformations. J. Am. Math. Soc. 7(4), 875–902 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Nevo, A.: Pointwise ergodic theorems for actions of groups, in Handbook of dynamical systems, Vol. 1B, pp. 871–982, Elsevier B. V., Amsterdam, (2006)

  29. 29.

    Nevo, A., Stein, E.M.: A generalization of Birkhoff’s pointwise ergodic theorem. Acta Math. 173, 135–154 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Ornstein, D.: On the pointwise behavior of iterates of a self-adjoint operator. J. Math. Mech 18, 473–477 (1968/1969)

  31. 31.

    Oseledets, V.I.: Markov chains, skew-products, and ergodic theorems for general dynamical systems. Theory Probab. Appl. 10, 551–557 (1965)

    Article  Google Scholar 

  32. 32.

    Rota, G.-C.: An “Alternierende Verfahren” for general positive operators. Bull. Am. Math. Soc. 68, 95–102 (1962)

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Pollicott, M., Sharp, R.: Ergodic theorems for actions of hyperbolic groups. Proc. Am. Math. Soc. 141, 1749–1757 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Series, C.: Geometrical methods of symbolic coding. In: Bedford, T., Keane, M., Series, C. (eds.) Ergodic Theory and Symbolic Dynamics in Hyperbolic Spaces. Oxford Univ. Press, Oxford (1991)

    Google Scholar 

Download references

Acknowledgments

The authors are deeply grateful to Vadim Kaimanovich for useful discussions. Lewis Bowen is supported in part by NSF grant DMS-0968762, NSF CAREER Award DMS-0954606 and BSF grant 2008274. Alexander Bufetov’s research is carried out thanks to the support of the A*MIDEX project (no. ANR-11-IDEX-0001-02) funded by the programme “Investissements d’Avenir ” of the Government of the French Republic, managed by the French National Research Agency (ANR). Bufetov is also supported in part by the Grant MD-2859.2014.1 of the President of the Russian Federation, by the Programme “Dynamical systems and mathematical control theory” of the Presidium of the Russian Academy of Sciences, and by the RFBR grant 13-01-12449. Olga Romaskevich is supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). She is as well supported by RFBR project 13-01-00969-a and Poncelet Laboratory.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Olga Romaskevich.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bowen, L., Bufetov, A. & Romaskevich, O. Mean convergence of Markovian spherical averages for measure-preserving actions of the free group. Geom Dedicata 181, 293–306 (2016). https://doi.org/10.1007/s10711-015-0124-2

Download citation

Keywords

  • Ergodic theorems
  • Markov operators
  • Spherical averages
  • Free groups

Mathematics Subject Classification (2010)

  • 37A30