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Ergodic theorems for actions of hyperbolic groups

Authors: Mark Pollicott and Richard Sharp
Journal: Proc. Amer. Math. Soc. 141 (2013), 1749-1757
MSC (2010): Primary 28D15, 37A15, 37A30, 60J05
Published electronically: November 30, 2012
MathSciNet review: 3020860
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Abstract | References | Similar Articles | Additional Information

Abstract: In this note we give a short proof of a pointwise ergodic theorem for measure-preserving actions of word hyperbolic groups, also obtained recently by Bufetov, Khristoforov and Klimenko. Our approach also applies to infinite measure spaces, and one application is to linear actions of discrete groups on the plane.

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  • 1. R. Adler and L. Flatto, Geodesic flows, interval maps, and symbolic dynamics, Bull. Amer. Math. Soc. 25 (1991), 229-334. MR 1085823 (92b:58172)
  • 2. C. Anantharaman-Delaroche, On ergodic theorems for free group actions on noncommutative spaces, Probability Theory and Related Fields 135 (2006), 520-546. MR 2240699 (2007m:46104)
  • 3. M. Bourdon, Actions quasi-convexes d'un groupe hyperbolique, flot géodésique, Thesis, Orsay, 1993.
  • 4. A. Bufetov, Markov averaging and ergodic theorems for several operators, Topology, ergodic theory, real algebraic geometry, Amer. Math. Soc. Transl. Ser. 2, 202, Amer. Math. Soc., Providence, RI, 2001, pp. 39-50. MR 1819180 (2002b:37006)
  • 5. A. Bufetov, Convergence of spherical averages for actions of free groups, Ann. of Math (2) 155 (2002), 929-944. MR 1923970 (2003f:37008)
  • 6. A. Bufetov, M. Khristoforov and A. Klimenko, Cesàro convergence of spherical averages for measure-preserving actions of Markov semigroups and groups, to appear, Internat. Math. Res. Notices.
  • 7. A. Bufetov and C. Series, A pointwise ergodic theorem for Fuchsian groups, Math. Proc. Camb. Phil. Soc. 151 (2011), 145-159. MR 2801319 (2012d:37011)
  • 8. D. Calegari and K. Fujiwara, Combable functions, quasimorphisms, and the central limit theorem, Ergodic Theory Dynam. Systems 30 (2010), 1343-1369. MR 2718897 (2011k:20088)
  • 9. J. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984), 123-148. MR 758901 (86j:20032)
  • 10. M. Coornaert, Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov, Pac. J. Math. 159 (1993), 241-270. MR 1214072 (94m:57075)
  • 11. B. Farkas, T. Eisner, M. Hasse and R. Nagel, Ergodic Theory - An operator-theoretic approach., 12th International Internet Seminar.
  • 12. K. Fujiwara and A. Nevo, Maximal and pointwise ergodic theorems for word-hyperbolic groups, Ergod. Th. Dynam. Sys. 18 (1998), 843-858. MR 1645314 (99j:22006)
  • 13. E. Ghys and P. de la Harpe, Sur les groupes hyperboliques d'après Mikhael Gromov, Progr. Math., 83, Birkhäuser, Boston, 1990. MR 1086648 (92f:53050)
  • 14. R. Grigorchuk, Pointwise ergodic theorems for actions of free groups, Proc. Tambov Workshop in the Theory of Functions (1986).
  • 15. R. Grigorchuk, An ergodic theorem for actions of a free semigroup, Proc. Steklov Inst. Math. 231 (2000), 113-127. MR 1841754
  • 16. Y. Guivarc'h, Généralisation d'un théorème de von Neumann, C. R. Acad. Sci. Paris Ser. A 268 (1969), 1010-1013. MR 0251191 (40:4422)
  • 17. R. Jones, J. Olsen and M. Wierdl, Subsequence ergodic theorems for $ L^{p}$ contractions, Trans. Amer. Math. Soc. 331 (1992), 837-850. MR 1043860 (92h:47011)
  • 18. B. Kitchens, Symbolic dynamics. One-sided, two-sided and countable state Markov shifts, Universitext, Springer-Verlag, Berlin, 1998. MR 1484730 (98k:58079)
  • 19. U. Krengel, Ergodic Theorems, de Gruyter Studies in Mathematics, 6, Walter de Gruyter, Berlin, 1985. MR 797411 (87i:28001)
  • 20. F. Ledrappier, Distribution des orbites des réseaux sur le plan réel, C. R. Acad. Sci. Paris Ser. I Math. 329 (1999), 61-64. MR 1703338 (2000c:22009)
  • 21. F. Ledrappier and M. Pollicott, Ergodic properties of linear actions of $ (2 \times 2)$-matrices, Duke Math. J. 116 (2003), 353-388. MR 1953296 (2003j:37041)
  • 22. A. Nevo and E. Stein, A generalization of Birkhoff's pointwise ergodic theorem, Acta Math. 173 (1994), 135-154. MR 1294672 (95m:28025)
  • 23. M. Rosenblatt, Markov processes: structure and asymptotic behavior, Die Grundlehren der mathematischen Wissenschaften, Band 184, Springer-Verlag, New York, 1971. MR 0329037 (48:7379)
  • 24. E. Seneta, Non-negative matrices and Markov chains, Revised reprint of the second (1981) edition. Springer Series in Statistics, Springer, New York, 2006. MR 2209438
  • 25. C. Series, Geometrical Markov coding of geodesics on surfaces of constant negative curvature, Ergod. Th. Dynam. Sys. 6 (1986), 601-625. MR 873435 (88k:58130)
  • 26. A. Templeman, Ergodic theorems for group actions, Kluwer Acad. Publ., Dordrecht, 1992. MR 1172319 (94f:22007)

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Additional Information

Mark Pollicott
Affiliation: Department of Mathematics, University of Warwick, Coventry, CV4 7AL, United Kingdom

Richard Sharp
Affiliation: School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
Address at time of publication: Department of Mathematics, University of Warwick, Coventry CV4 7AL, United Kingdom

Received by editor(s): July 28, 2011
Received by editor(s) in revised form: September 14, 2011
Published electronically: November 30, 2012
Communicated by: Bryna Kra
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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