## Ergodic theorems for actions of hyperbolic groups.(English)Zbl 1266.28008

The authors give a short proof of the almost everywhere convergence for the Cesàro averages of the spherical averages for $${L_\infty}$$ functions with respect to measure-preserving actions of word hyperbolic groups. This result recently is also obtained by A. I. Bufetov, A. V. Klimenko and M. I. Khristoforov [Russ. Math. Surv. 66, No. 3, 633–634 (2011); translation from Usp. Mat. Nauk. 66, No. 3, 203–204 (2011; Zbl 1244.37006)].
The proof is based on an association to strongly Markov groups (word hyperbolic groups are the particular case of such groups) of some directed graph. This allows to reduce the problem to the ergodic theorem for the Markov matrix that encodes a word hyperbolic group. The structure of such a matrix observed by D. Calegari and K. Fujiwara [Ergodic Theory Dyn. Syst. 30, No. 5, 1343–1369 (2010; Zbl 1217.37025)] is used in the proof.
At the end of the paper, an application to linear actions of discrete groups on the plane is considered.

### MSC:

 28D15 General groups of measure-preserving transformations 37A15 General groups of measure-preserving transformations and dynamical systems 37A30 Ergodic theorems, spectral theory, Markov operators 60J05 Discrete-time Markov processes on general state spaces

### Citations:

Zbl 1244.37006; Zbl 1217.37025
Full Text:

### References:

 [1] Roy Adler and Leopold Flatto, Geodesic flows, interval maps, and symbolic dynamics, Bull. Amer. Math. Soc. (N.S.) 25 (1991), no. 2, 229 – 334. · Zbl 0802.58037 [2] Claire Anantharaman-Delaroche, On ergodic theorems for free group actions on noncommutative spaces, Probab. Theory Related Fields 135 (2006), no. 4, 520 – 546. · Zbl 1106.46047 [3] M. Bourdon, Actions quasi-convexes d’un groupe hyperbolique, flot géodésique, Thesis, Orsay, 1993. [4] 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. · Zbl 1002.47004 [5] Alexander I. Bufetov, Convergence of spherical averages for actions of free groups, Ann. of Math. (2) 155 (2002), no. 3, 929 – 944. · Zbl 1028.37001 [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. · Zbl 1267.22003 [7] Alexander I. Bufetov and Caroline Series, A pointwise ergodic theorem for Fuchsian groups, Math. Proc. Cambridge Philos. Soc. 151 (2011), no. 1, 145 – 159. · Zbl 1219.22008 [8] Danny Calegari and Koji Fujiwara, Combable functions, quasimorphisms, and the central limit theorem, Ergodic Theory Dynam. Systems 30 (2010), no. 5, 1343 – 1369. · Zbl 1217.37025 [9] James W. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984), no. 2, 123 – 148. · Zbl 0606.57003 [10] 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). · Zbl 0797.20029 [11] B. Farkas, T. Eisner, M. Hasse and R. Nagel, Ergodic Theory - An operator-theoretic approach., 12th International Internet Seminar. [12] Koji Fujiwara and Amos Nevo, Maximal and pointwise ergodic theorems for word-hyperbolic groups, Ergodic Theory Dynam. Systems 18 (1998), no. 4, 843 – 858. · Zbl 0919.22002 [13] É. Ghys and P. de la Harpe , 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. · Zbl 0731.20025 [14] R. Grigorchuk, Pointwise ergodic theorems for actions of free groups, Proc. Tambov Workshop in the Theory of Functions (1986). [15] 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. [16] Yves Guivarc’h, Généralisation d’un théorème de von Neumann, C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A1020 – A1023 (French). · Zbl 0176.11703 [17] Roger L. Jones, James Olsen, and Máté Wierdl, Subsequence ergodic theorems for \?^{\?} contractions, Trans. Amer. Math. Soc. 331 (1992), no. 2, 837 – 850. · Zbl 0786.47005 [18] Bruce P. Kitchens, Symbolic dynamics, Universitext, Springer-Verlag, Berlin, 1998. One-sided, two-sided and countable state Markov shifts. · Zbl 0892.58020 [19] Ulrich Krengel, Ergodic theorems, De Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. · Zbl 0575.28009 [20] 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). · Zbl 0928.22012 [21] F. Ledrappier and M. Pollicott, Ergodic properties of linear actions of (2\times 2)-matrices, Duke Math. J. 116 (2003), no. 2, 353 – 388. · Zbl 1020.37009 [22] Amos Nevo and Elias M. Stein, A generalization of Birkhoff’s pointwise ergodic theorem, Acta Math. 173 (1994), no. 1, 135 – 154. · Zbl 0837.22003 [23] Murray Rosenblatt, Markov processes. Structure and asymptotic behavior, Springer-Verlag, New York-Heidelberg, 1971. Die Grundlehren der mathematischen Wissenschaften, Band 184. · Zbl 0236.60002 [24] 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]. · Zbl 0471.60001 [25] Caroline Series, Geometrical Markov coding of geodesics on surfaces of constant negative curvature, Ergodic Theory Dynam. Systems 6 (1986), no. 4, 601 – 625. · Zbl 0593.58033 [26] 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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.