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A pointwise ergodic theorem for Fuchsian groups

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Bufetov, Alexander I., 1979- and Series, Caroline. (2011) A pointwise ergodic theorem for Fuchsian groups. Mathematical Proceedings of the Cambridge Philosophical Society, Vol.151 (No.1). pp. 145-159. ISSN 0305-0041

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Official URL: http://dx.doi.org/10.1017/S0305004111000247

Abstract

We use Series' Markovian coding for words in Fuchsian groups and the Bowen-Series coding of limit sets to prove an ergodic theorem for Cesaro averages of spherical averages in a Fuchsian group.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Fuchsian groups, Ergodic theory
Journal or Publication Title:	Mathematical Proceedings of the Cambridge Philosophical Society
Publisher:	Cambridge University Press
ISSN:	0305-0041
Date:	July 2011
Volume:	Vol.151
Number:	No.1
Page Range:	pp. 145-159
Identification Number:	10.1017/S0305004111000247
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	Alfred P. Sloan Foundation, Russia (Federation), Rossiĭskai︠a︡ akademii︠a︡ nauk [Russian Academy of Sciences] (RAN), Russia (Federation). Ministerstvo obrazovanii︠a︡ [Ministry of Education], Rice University, National Science Foundation (U.S.) (NSF), Rossiĭskiĭ fond fundamentalŉykh issledovaniĭ [Russian Foundation for Basic Research] (RFFI), Centre national de la recherche scientifique (France) (CNRS)
Grant number:	MK-4893.2010.1 (Russia), 2.1.1/5328 (MO), DMS 0604386 (NSF), 10-01-93115 (RFFI/CNRS)
References:	[1] C.ANANTHARAMAN ET AL. Th´eor`emes ergodiques pour les actions de groupes. Monographie 41 de L’ Enseignment Math´ematique (Gen`eve, 2010). [2] A. BEARDON. An introduction to hyperbolic geometry. Ergodic Theory and Symbolic Dynamics in Hyperbolic Spaces, T. Bedford, M. Keane and C. Series eds. (Oxford University Press, 1991). [3] J.BIRMAN and C. SERIES. Dehn’s algorithm revisited, with application to simple curves on surfaces. Combinatorial Group Theory and Topology, S. Gersten and J. Stallings eds. Ann. of Math. Studies III (Princeton University Press, 1987), 451–478. [4] L. BOWEN. Invariant measures on the space of horofunctions of a word hyperbolic group. Ergodic Theory Dynam. Syst. 30 (2010), 97–129. [5] R. BOWEN and C. SERIES. Markov maps associated with Fuchsian groups. Inst. Hautes E´tudes Sci. Pubi. Math. 50 (1979), 153–170. [6] A. I. BUFETOV. Convergence of spherical averages for actions of free groups. Ann. of Math. (2), 155 (2002), 929–944. [7] A. I. BUFETOV. Markov averaging and ergodic theorems for several operators. Topology, ergodic theory, real algebraic geometry, 39–50, Amer. Math. Soc. Transl. Ser. 2 202 (Amer. Math. Soc., 2001). [8] K. FUJIWARA and A. NEVO. Maximal and pointwise ergodic theorems for word-hyperbolic groups. Ergodic Theory Dynam. Syst. 18 (1998), 843–858. [9] A.GORODNIK and A. NEVO. The ergodic theory of lattice subgroups. Ann. of Math. Stud. 172 (Princeton University Press, 2010). [10] R. I.GRIGORCHUK. Ergodic theorems for the actions of a free group and a free semigroup. (Russian) Mat. Zametki 65 (1999), 779–783 (English trans: Math. Notes 65 (1999), 654–657). [11] R. I. GRIGORCHUK. An ergodic theorem for actions of a free semigroup. (Russian) Tr. Mat. Inst. Steklova 231 (2000), Din. Sist., Avtom. i Beskon. Gruppy, 119–133 (English trans: Proc. Steklov Inst. Math. 2000 231, 113–127). [12] G. A. MARGULIS, A. NEVO and E. M. STEIN. Analogs ofWiener’s ergodic theorems for semisimple Lie groups. II. Duke Math. J. 103 (2000), 233–259. [13] A. NEVO. Harmonic analysis and pointwise ergodic theorems for noncommuting transformations. J. Amer. Math. Soc. 7 (1994), 875–902. [14] A. NEVO. Pointwise ergodic theorems for actions of groups. Handbook of dynamical systems. Vol. 1B, 871–982 (Elsevier, Amsterdam, 2006). [15] A. NEVO and E. M. STEIN. A generalization of Birkhoff’s pointwise ergodic theorem. Acta Math. 173 (1994), 135–154. [16] A.NEVO and E. M. STEIN. Analogs of Wiener’s ergodic theorems for semisimple groups. I. Ann. of Math. 2 145 (1997), 565–595. [17] C.SERIES. The infinite word problem and limit sets in Fuchsian groups. Ergodic Theory Dynam. Syst. 1 (1981), 337–360. [18] C. SERIES. Martin boundaries of random walks on Fuchsian groups. Israel J. Math. 44 (1983), 221– 240. [19] C. SERIES. Geometrical Markov coding of geodesics on surfaces of constant negative curvature. Ergodic Theory Dynam. Syst. 6 (1986), 601–625. [20] C. SERIES. Geometrical methods of symbolic coding. Ergodic Theory and Symbolic Dynamics in Hyperbolic Spaces, T. Bedford, M. Keane and C. Series eds. (Oxford University Press, 1991).
URI:	http://wrap.warwick.ac.uk/id/eprint/38825