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

Le Prince, Vincent

doi:10.1090/S0002-9947-07-04108-6

Dimensional properties of the harmonic measure for a random walk on a hyperbolic group
HTML articles powered by AMS MathViewer

by Vincent Le Prince PDF

Trans. Amer. Math. Soc. 359 (2007), 2881-2898 Request permission

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.

References

André Avez, Entropie des groupes de type fini, C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A1363–A1366 (French). MR 324741
S. D. Chatterji, Masse, die von regelmässigen Kettenbrüchen induziert sind, Math. Ann. 164 (1966), 113–117 (German). MR 193079, DOI 10.1007/BF01429048
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
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
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, DOI 10.1007/BFb0084913
Yves Derriennic, Quelques applications du théorème ergodique sous-additif, Conference on Random Walks (Kleebach, 1979) Astérisque, vol. 74, Soc. Math. France, Paris, 1980, pp. 183–201, 4 (French, with English summary). MR 588163
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
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
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
M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
Y. Guivarc’h, Sur la loi des grands nombres et le rayon spectral d’une marche aléatoire, Conference on Random Walks (Kleebach, 1979) Astérisque, vol. 74, Soc. Math. France, Paris, 1980, pp. 47–98, 3 (French, with English summary). MR 588157
Vadim A. Kaimanovich, Hausdorff dimension of the harmonic measure on trees, Ergodic Theory Dynam. Systems 18 (1998), no. 3, 631–660. MR 1631732, DOI 10.1017/S0143385798108180
Vadim A. Kaimanovich, The Poisson formula for groups with hyperbolic properties, Ann. of Math. (2) 152 (2000), no. 3, 659–692. MR 1815698, DOI 10.2307/2661351
V. A. Kaĭmanovich and A. M. Vershik, Random walks on discrete groups: boundary and entropy, Ann. Probab. 11 (1983), no. 3, 457–490. MR 704539, DOI 10.1214/aop/1176993497
Yuri Kifer and François Ledrappier, Hausdorff dimension of harmonic measures on negatively curved manifolds, Trans. Amer. Math. Soc. 318 (1990), no. 2, 685–704. MR 951889, DOI 10.1090/S0002-9947-1990-0951889-7
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, DOI 10.1007/BF02772607
F. Ledrappier, Quelques propriétés des exposants caractéristiques, École d’été de probabilités de Saint-Flour, XII—1982, Lecture Notes in Math., vol. 1097, Springer, Berlin, 1984, pp. 305–396 (French). MR 876081, DOI 10.1007/BFb0099434
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, DOI 10.1090/crmp/028/05
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
J. Mairesse, F. Mathéus, Random walks on free products of cyclic groups and on Artin groups with two generators, preprint.
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, DOI 10.7208/chicago/9780226662237.001.0001
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, DOI 10.2307/1971475
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, DOI 10.1070/rm2000v055n04ABEH000314