Abstract
We describe a new approach to proving the continuity of asymptotic entropy as a function of a transition measure under a finite first moment condition. It is based on using conditional random walks and amounts to checking uniformity in the strip criterion for the identification of the Poisson boundary. It is applicable to word hyperbolic groups and in several other situations when the Poisson boundary can be identified with an appropriate geometric boundary.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. A. Kaimanovich, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 123 (1983), 167–184.
V. A. Kaimanovich, Dokl. AN SSSR, 280:5 (1985), 1051–1054; English transl.: Soviet Math. Dokl., 31 (1985), 193–197.
A. M. Vershik, Uspekhi Mat. Nauk, 55:4(334) (2000), 59–128; English transl.: Russian Math. Surveys, 55:4 (2000), 667–733.
G. Amir, O. Angel, and B. Virág, http://arxiv.org/abs/0905.2007.
A. Avez, C. R. Acad. Sci. Paris Sér. A-B, 275 (1972), A1363–A1366.
Y. Derriennic, in: Conference on Random Walks (Kleebach, 1979), Astérisque, vol. 74, Soc. Math. France, Paris, 1980, 183–201, 4.
A. Erschler, in: Random Walks, Boundaries and Spectra. Proceedings of the Workshop on Boundaries, Graz, Austria, June 29–July 3, 2009 and the Alp-workshop, Sankt Kathrein, Austria, July 4–5, 2009, Progress in Probability, vol. 64 (eds. Lenz, Daniel et al.), Birkhäuser, Basel, 2011, 55–64.
H. Furstenberg, Trans. Amer. Math. Soc., 108 (1963), 377–428.
Sur les groupes hyperboliques d’après Mikhael Gromov, Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988, Progress in Mathematics, vol. 83 (eds. É. Ghys, P. de la Harpe), Birkhäuser, Boston, MA, 1990.
M. Gromov, “Hyperbolic groups,” in: Essays in Group Theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer-Verlag, New York, 1987, 75–263.
V. A. Kaimanovich, C. R. Acad. Sci. Paris Sér. I Math., 318:1 (1994), 59–64.
V. A. Kaimanovich, Ann. of Math. (2), 152:3 (2000), 659–692.
V. A. Kaimanovich and A. M. Vershik, Ann. Probab., 11:3 (1983), 457–490.
F. Ledrappier, in: Topics in Probability and Lie Groups: Boundary Theory, CRM Proc. Lecture Notes, vol. 28, Amer. Math. Soc., Providence, RI, 2001, 117–152.
F. Ledrappier, http://arxiv.org/abs/1110.3156.
F. Ledrappier, Groups Geom. Dyn., 6:2 (2012), 317–333.
P. Mathieu, http://arxiv.org/abs/1209.2020.
J. Mairesse and F. Mathéus, J. Lond. Math. Soc. (2), 75:1 (2007), 47–66.
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Funktsional’ nyi Analiz i Ego Prilozheniya, Vol. 47, No. 2, pp. 84–89, 2013
Original Russian Text Copyright © by A. Erschler and V. A. Kaimanovich
The work was supported by European Research Council under European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement 257110-RAWG, by the ANR (France) program “DiscGroup: facettes des groupes discrets,” by the Canada Research Chairs program, and by NSERC (Canada).
Rights and permissions
About this article
Cite this article
Erschler, A., Kaimanovich, V.A. Continuity of asymptotic characteristics for random walks on hyperbolic groups. Funct Anal Its Appl 47, 152–156 (2013). https://doi.org/10.1007/s10688-013-0020-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10688-013-0020-1