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Dimensional properties of the harmonic measure for a random walk on a hyperbolic group

Author: Vincent Le Prince
Journal: Trans. Amer. Math. Soc. 359 (2007), 2881-2898
MSC (2000): Primary 60G50, 20F67, 28D20, 28A78
Published electronically: January 26, 2007
MathSciNet review: 2286061
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Abstract: This paper deals with random walks on isometry groups of Gromov hyperbolic spaces, and more precisely with the dimension of the harmonic measure $ \nu$ associated with such a random walk. We first establish a link of the form $ \dim \nu\leq h/l$ between the dimension of the harmonic measure, the asymptotic entropy $ h$ of the random walk and its rate of escape $ l$. Then we use this inequality to show that the dimension of this measure can be made arbitrarily small and deduce a result on the type of the harmonic measure.

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  • 1. André Avez, Entropie des groupes de type fini, C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A1363–A1366 (French). MR 0324741
  • 2. S. D. Chatterji, Masse, die von regelmässigen Kettenbrüchen induziert sind, Math. Ann. 164 (1966), 113–117 (German). MR 0193079,
  • 3. Michel Coornaert, Sur les groupes proprement discontinus d’isométries des espaces hyperboliques au sens de Gromov, Publication de l’Institut de Recherche Mathématique Avancée [Publication of the Institute of Advanced Mathematical Research], vol. 444, Université Louis Pasteur, Département de Mathématique, Institut de Recherche Mathématique Avancée, Strasbourg, 1990 (French). Dissertation, Université Louis Pasteur, Strasbourg, 1990. MR 1116319
  • 4. 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
  • 5. M. Coornaert, T. Delzant, and A. Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990 (French). Les groupes hyperboliques de Gromov. [Gromov hyperbolic groups]; With an English summary. MR 1075994
  • 6. Y. Derriennic, Quelques applications du théorème ergodique sous-additif, Astérisque 74 (1980), pp. 183-201. MR 0588163 (82e:60013)
  • 7. E. B. Dynkin and M. B. Maljutov, Random walk on groups with a finite number of generators, Dokl. Akad. Nauk SSSR 137 (1961), 1042–1045 (Russian). MR 0131904
  • 8. Harry Furstenberg, Boundary theory and stochastic processes on homogeneous spaces, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 193–229. MR 0352328
  • 9. 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
  • 10. M. Gromov, Hyperbolic groups, Essays in Group Theory (S.M. Gersten, ed.), MSRI Publ., vol. 8, Springer, New York, 1987, pp. 75-263. MR 0919829 (89e:20070)
  • 11. Y. Guivarc'h, Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire, Astérisque 74 (1980), pp. 47-78. MR 0588157 (82g:60016)
  • 12. Vadim A. Kaimanovich, Hausdorff dimension of the harmonic measure on trees, Ergodic Theory Dynam. Systems 18 (1998), no. 3, 631–660. MR 1631732,
  • 13. Vadim A. Kaimanovich, The Poisson formula for groups with hyperbolic properties, Ann. of Math. (2) 152 (2000), no. 3, 659–692. MR 1815698,
  • 14. V. A. Kaimanovich, A. M. Vershik, Random walks on discrete groups : boundary and entropy, Ann. Prob. 11 (1983), pp. 457-490. MR 0704539 (85d:60024)
  • 15. Y. Kifer, F. Ledrappier, Hausdorff dimension of the harmonic measure on negatively curved manifolds, Trans. Amer. Math. Soc. 318 (1990), pp. 685-704. MR 0951889 (91a:58205)
  • 16. Yuri Kifer, Yuval Peres, and Benjamin Weiss, A dimension gap for continued fractions with independent digits, Israel J. Math. 124 (2001), 61–76. MR 1856504,
  • 17. F. Ledrappier, Quelques propriétés des exposants caractéristiques, Lecture Notes in Math., vol. 1097, Springer, Berlin, 1982. MR 0876081 (88b:58081)
  • 18. François Ledrappier, Some asymptotic properties of random walks on free groups, Topics in probability and Lie groups: boundary theory, CRM Proc. Lecture Notes, vol. 28, Amer. Math. Soc., Providence, RI, 2001, pp. 117–152. MR 1832436
  • 19. B. Ja. Levit and S. A. Molčanov, Invariant chains on a free group with a finite number of generators, Vestnik Moskov. Univ. Ser. I Mat. Meh. 26 (1971), no. 4, 80–88 (Russian, with English summary). MR 0298721
  • 20. J. Mairesse, F. Mathéus, Random walks on free products of cyclic groups and on Artin groups with two generators, preprint.
  • 21. Yakov B. Pesin, Dimension theory in dynamical systems, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. Contemporary views and applications. MR 1489237
  • 22. Feliks Przytycki, Mariusz Urbański, and Anna Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps. I, Ann. of Math. (2) 130 (1989), no. 1, 1–40. MR 1005606,
  • 23. A. M. Vershik, Dynamic theory of growth in groups: entropy, boundaries, examples, Uspekhi Mat. Nauk 55 (2000), no. 4(334), 59–128 (Russian, with Russian summary); English transl., Russian Math. Surveys 55 (2000), no. 4, 667–733. MR 1786730,

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Additional Information

Vincent Le Prince
Affiliation: IRMAR, Université de Rennes 1, Campus de Beaulieu, 35 042 Rennes cedex, France

Keywords: Ergodic theory, random walk, hyperbolic group, harmonic measure, entropy
Received by editor(s): December 16, 2004
Received by editor(s) in revised form: June 17, 2005
Published electronically: January 26, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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