Ergodic theorems for actions of hyperbolic groups
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- by Mark Pollicott and Richard Sharp PDF
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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.References
- Roy Adler and Leopold Flatto, Geodesic flows, interval maps, and symbolic dynamics, Bull. Amer. Math. Soc. (N.S.) 25 (1991), no. 2, 229–334. MR 1085823, DOI 10.1090/S0273-0979-1991-16076-3
- Claire Anantharaman-Delaroche, On ergodic theorems for free group actions on noncommutative spaces, Probab. Theory Related Fields 135 (2006), no. 4, 520–546. MR 2240699, DOI 10.1007/s00440-005-0456-1
- M. Bourdon, Actions quasi-convexes d’un groupe hyperbolique, flot géodésique, Thesis, Orsay, 1993.
- Alexander Bufetov, Markov averaging and ergodic theorems for several operators, Topology, ergodic theory, real algebraic geometry, Amer. Math. Soc. Transl. Ser. 2, vol. 202, Amer. Math. Soc., Providence, RI, 2001, pp. 39–50. MR 1819180, DOI 10.1090/trans2/202/05
- Alexander I. Bufetov, Convergence of spherical averages for actions of free groups, Ann. of Math. (2) 155 (2002), no. 3, 929–944. MR 1923970, DOI 10.2307/3062137
- 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.
- Alexander I. Bufetov and Caroline Series, A pointwise ergodic theorem for Fuchsian groups, Math. Proc. Cambridge Philos. Soc. 151 (2011), no. 1, 145–159. MR 2801319, DOI 10.1017/S0305004111000247
- Danny Calegari and Koji Fujiwara, Combable functions, quasimorphisms, and the central limit theorem, Ergodic Theory Dynam. Systems 30 (2010), no. 5, 1343–1369. MR 2718897, DOI 10.1017/S0143385709000662
- James W. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984), no. 2, 123–148. MR 758901, DOI 10.1007/BF00146825
- Michel Coornaert, Mesures de Patterson-Sullivan sur le bord d’un espace hyperbolique au sens de Gromov, Pacific J. Math. 159 (1993), no. 2, 241–270 (French, with French summary). MR 1214072, DOI 10.2140/pjm.1993.159.241
- B. Farkas, T. Eisner, M. Hasse and R. Nagel, Ergodic Theory - An operator-theoretic approach., 12th International Internet Seminar.
- Koji Fujiwara and Amos Nevo, Maximal and pointwise ergodic theorems for word-hyperbolic groups, Ergodic Theory Dynam. Systems 18 (1998), no. 4, 843–858. MR 1645314, DOI 10.1017/S0143385798117443
- É. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1990 (French). Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648, DOI 10.1007/978-1-4684-9167-8
- R. Grigorchuk, Pointwise ergodic theorems for actions of free groups, Proc. Tambov Workshop in the Theory of Functions (1986).
- R. I. Grigorchuk, An ergodic theorem for actions of a free semigroup, Tr. Mat. Inst. Steklova 231 (2000), no. Din. Sist., Avtom. i Beskon. Gruppy, 119–133 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 4(231) (2000), 113–127. MR 1841754
- Yves Guivarc’h, Généralisation d’un théorème de von Neumann, C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A1020–A1023 (French). MR 251191
- Roger L. Jones, James Olsen, and Máté Wierdl, Subsequence ergodic theorems for $L^p$ contractions, Trans. Amer. Math. Soc. 331 (1992), no. 2, 837–850. MR 1043860, DOI 10.1090/S0002-9947-1992-1043860-6
- Bruce P. Kitchens, Symbolic dynamics, Universitext, Springer-Verlag, Berlin, 1998. One-sided, two-sided and countable state Markov shifts. MR 1484730, DOI 10.1007/978-3-642-58822-8
- Ulrich Krengel, Ergodic theorems, De Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. MR 797411, DOI 10.1515/9783110844641
- François Ledrappier, Distribution des orbites des réseaux sur le plan réel, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 1, 61–64 (French, with English and French summaries). MR 1703338, DOI 10.1016/S0764-4442(99)80462-5
- F. Ledrappier and M. Pollicott, Ergodic properties of linear actions of $(2\times 2)$-matrices, Duke Math. J. 116 (2003), no. 2, 353–388. MR 1953296, DOI 10.1215/S0012-7094-03-11626-9
- Amos Nevo and Elias M. Stein, A generalization of Birkhoff’s pointwise ergodic theorem, Acta Math. 173 (1994), no. 1, 135–154. MR 1294672, DOI 10.1007/BF02392571
- Murray Rosenblatt, Markov processes. Structure and asymptotic behavior, Die Grundlehren der mathematischen Wissenschaften, Band 184, Springer-Verlag, New York-Heidelberg, 1971. MR 0329037, DOI 10.1007/978-3-642-65238-7
- E. Seneta, Non-negative matrices and Markov chains, Springer Series in Statistics, Springer, New York, 2006. Revised reprint of the second (1981) edition [Springer-Verlag, New York; MR0719544]. MR 2209438
- Caroline Series, Geometrical Markov coding of geodesics on surfaces of constant negative curvature, Ergodic Theory Dynam. Systems 6 (1986), no. 4, 601–625. MR 873435, DOI 10.1017/S0143385700003722
- Arkady Tempelman, Ergodic theorems for group actions, Mathematics and its Applications, vol. 78, Kluwer Academic Publishers Group, Dordrecht, 1992. Informational and thermodynamical aspects; Translated and revised from the 1986 Russian original. MR 1172319, DOI 10.1007/978-94-017-1460-0
Additional Information
- Mark Pollicott
- Affiliation: Department of Mathematics, University of Warwick, Coventry, CV4 7AL, United Kingdom
- MR Author ID: 140805
- Email: mpollic@maths.warwick.ac.uk
- 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
- MR Author ID: 317352
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 1749-1757
- MSC (2010): Primary 28D15, 37A15, 37A30, 60J05
- DOI: https://doi.org/10.1090/S0002-9939-2012-11447-9
- MathSciNet review: 3020860