Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T00:23:44.116Z Has data issue: false hasContentIssue false
  • English
  • Français

Hausdorff spectrum of harmonic measure

Published online by Cambridge University Press:  21 July 2015

RYOKICHI TANAKA
RYOKICHI TANAKA*
Affiliation:
Tohoku University, 2-1-1 Katahira, Aoba-ku, 980-8577 Sendai, Japan email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

For every non-elementary hyperbolic group, we show that for every random walk with finitely supported admissible step distribution, the associated entropy equals the drift times the logarithmic volume growth if and only if the corresponding harmonic measure is comparable with Hausdorff measure on the boundary. Moreover, we introduce one parameter family of probability measures which interpolates a Patterson–Sullivan measure and the harmonic measure, and establish a formula of Hausdorff spectrum (multifractal spectrum) of the harmonic measure. We also give some finitary versions of dimensional properties of the harmonic measure.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 1 , February 2017 , pp. 277 - 307
Copyright
© Cambridge University Press, 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ancona, A.. Positive harmonic functions and hyperbolicity. Potential Theory (Prague, 1987) (Lecture Notes in Mathematics, 1344) . Springer, Berlin, 1988, pp. 1–23.Google Scholar
Avez, A.. Entropie des groupes de type fini. C. R. Acad. Sci. Paris Sér. A 275 (1972), 1363–1366.Google Scholar
Barreira, L. and Schmeling, J.. Sets of ‘non-typical’ points have full topological entropy and full Hausdorff dimension. Israel J. Math. 116 (2000), 29–70.Google Scholar
Blachère, S., Haissinsky, P. and Mathieu, P.. Asymptotic entropy and Green speed for random walks on countable groups. Ann. Probab. 36(3) (2008), 1134–1152.CrossRefGoogle Scholar
Blachère, S., Haissinsky, P. and Mathieu, P.. Harmonic measures versus quasiconformal measures for hyperbolic groups. Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), 683–721.Google Scholar
Bourgain, J.. Finitely supported measures on SL2(ℝ) which are absolutely continuous at infinity. Geometric Aspect of Functional Analysis (Lecture Notes in Mathematics, 2050) . Ed. Klartag, B. et al. . Springer, Berlin, 2012, pp. 133–141.Google Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470) . Revised 2nd edn. Springer, Berlin, 2008.Google Scholar
Brieussel, J. and Tanaka, R.. Discrete random walks on the group Sol. Israel J. Math. to appear, 2013, arXiv:1306.6180v1 [math.PR].Google Scholar
Calegari, D.. The ergodic theory of hyperbolic groups. Geometry and Topology Down Under (Contemporary Mathematics, 597) . American Mathematical Society, Providence, RI, 2013, pp. 15–52.Google Scholar
Calegari, D. and Fujiwara, K.. Combable functions, quasimorphisms, and the central limit theorem. Ergod. Th. & Dynam. Sys. 30 (2009), 1343–1369.Google Scholar
Cannon, J. W.. The combinatorial structure of cocompact discrete hyperbolic groups. Geom. Dedicata 16 (1984), 123–148.Google Scholar
Connell, C. and Muchnik, R.. Harmonicity of quasiconformal measures and Poisson boundaries of hyperbolic spaces. Geom. Funct. Anal. 17 (2007), 707–769.Google Scholar
Connell, C. and Muchnik, R.. Harmonicity of Gibbs measures. Duke Math. J. 137(3) (2007), 461–509.Google Scholar
Coornaert, M.. Mesures de Patterson–Sullivan sur le bord d’un espace hyperbolique an sens de Gromov. Pacific J. Math. 159(2) (1993), 241–270.Google Scholar
Derriennic, Y.. Quelques applications du théorème ergodique sous-additif. Conference on Random Walks (Kleebach, 1979) (Astérisque, 74) . Société Mathématique de France, Paris, 1980, pp. 183–201.Google Scholar
Falconer, K.. Fractal Geometry, Mathematical Foundations and Applications, 3rd edn. John Wiley & Sons, Chichester, 2014.Google Scholar
Feng, D.-J.. Gibbs properties of self-conformal measures and the multifractal formalism. Ergod. Th. & Dynam. Sys. 27(3) (2007), 787–812.Google Scholar
Ghys, E. and de la Harpe, P.. Sur les groupes hyperboliques d’après Mikhael Gromov (Progress in Mathematics, 83) . Birkhäuser, Boston, 1990.Google Scholar
Gouëzel, S.. Local limit theorem for symmetric random walks in Gromov-hyperbolic groups. J. Amer. Math. Soc. 27(3) (2014), 893–928.Google Scholar
Gouëzel, S.. Martin boundary of random walks with infinite range in hyperbolic groups. Annal. Probab. to appear. Preprint, 2013, arXiv:1302.5388v1 [math.PR].Google Scholar
Gouëzel, S.. Private communication, 2014.Google Scholar
Gouëzel, S. and Lalley, S. P.. Random walks on co-compact Fuchsian groups. Ann. Sci. Éc. Norm. Supér., (4) 46(1) (2013), 129–173.Google Scholar
Gouëzel, S., Mathéus, F. and Maucourant, F.. Sharp lower bounds for the asymptotic entropy of symmetric random walks. Preprint, 2014, arXiv:1209.3378v3 [math.PR].Google Scholar
Gouëzel, S., Mathéus, F. and Maucourant, F.. Entropy and drift in word hyperbolic groups. Preprint, 2015, arXiv:1501.05082v1 [math.PR].Google Scholar
Guivarc’h, Y.. Sur la loi des grands nombres et le rayon spectral d’une marche aléatoire. Conference on Random Walks (Kleebach, 1979) (Astérisque, 74) . Société Mathématique de France, Paris, 1980, pp. 47–98.Google Scholar
Haissinsky, P.. Marches aléatoires sur les groupes hyperboliques. Géométrie Ergodique (Monographie de L’Enseignement Mathématique, 43) . Ed. Dal’Bo-Milonet. L’Enseignement Mathématique, Généve, 2013, pp. 199–265.Google Scholar
Haissinsky, P., Mathieu, P. and Müller, S.. Renewal theory for random walks on surface groups. Preprint, 2013, arXiv:1304.7625v1 [math.PR].Google Scholar
Heinonen, J.. Lectures on Analysis on Metric Spaces. Springer, New York, 2001.Google Scholar
Izumi, M., Neshveyev, S. and Okayasu, R.. The ratio set of the harmonic measure of a random walk on a hyperbolic group. Israel J. Math. 163 (2008), 285–316.Google Scholar
Kaimanovich, V. A.. Hausdorff dimension of the harmonic measure on trees. Ergod. Th. & Dynam. Sys. 18 (1998), 631–660.Google Scholar
Kaimanovich, V. A.. The Poisson formula for groups with hyperbolic properties. Ann. of Math. (2) 152(3) (2000), 659–692.Google Scholar
Kaimanovich, V. A. and Le Prince, V.. Matrix random products with singular harmonic measure. Geom. Dedicata 150 (2011), 257–279.Google Scholar
Kaimanovich, V. A. and Vershik, A. M.. Random walks on discrete groups: boundary and entropy. Ann. Probab. 11(3) (1983), 457–490.Google Scholar
Ledrappier, F.. Some asymptotic properties of random walks on free groups. Topics in Probability and Lie Groups: Boundary Theory (CRM Proceedings & Lecture Notes, 28) . American Mathematical Society, Providence, RI, 2001, pp. 117–152.Google Scholar
Le Prince, V.. Dimensional properties of the harmonic measure for a random walk on a hyperbolic group. Trans. Amer. Math. Soc. 359(6) (2007), 2881–2898.Google Scholar
Lyons, R.. Equivalence of boundary measures on covering trees of finite graphs. Ergod. Th. & Dynam. Sys. 14 (1994), 575–597.Google Scholar
Lyons, R. and Peres, Y.. Probability on Trees and Networks. Cambridge University Press, in preparation. Current version available at http://mypage.iu.edu/∼rdlyons/.Google Scholar
Lyons, R., Pemantle, R. and Peres, Y.. Ergodic theory on Galton–Watson trees: speed of random walk and dimension of harmonic measure. Ergod. Th. & Dynam. Sys. 15 (1995), 593–619.Google Scholar
Parry, W. and Pollicott, M.. Zeta functions and the periodic orbit structure of hyperbolic dynamics (Astérisque) . Société Mathématique de France, 1990, pp. 187–188.Google Scholar
Patterson, S. J.. The limit set of a Fuchsian group. Acta Math. 136 (1976), 241–273.Google Scholar
Pesin, Y. B.. Dimension Theory in Dynamical Systems. The University of Chicago Press, Chicago and London, 1997.Google Scholar
Przytycki, F. and Urbański, M.. Conformal Fractals: Ergodic Theory Methods (London Mathematical Society Lecture Note Series, 371) . Cambridge University Press, Cambridge, 2010.Google Scholar
Vershik, A. M.. Dynamic theory of growth in groups: Entropy, boundaries, examples. Russian Math. Surveys 55(4) (2000), 667–733.Google Scholar
Woess, W.. Random Walks on Infinite Graphs and Groups (Cambridge Tracts in Mathematics, 138) . Cambridge University Press, Cambridge, 2000.Google Scholar