Tohoku Mathematical Journal

The dichotomy of harmonic measures of compact hyperbolic laminations

Shigenori Matsumoto

Full-text: Open access

Abstract

Given a harmonic measure $m$ of a hyperbolic lamination $\mathcal L$ on a compact metric space $M$, a positive harmonic function $h$ on the universal cover of a typical leaf is defined in such a way that the measure $m$ is described in terms of these functions $h$ on various leaves. We discuss some properties of the function $h$. We show that if $m$ is ergodic and not completely invariant, then $h$ is typically unbounded and is induced by a probability $\mu$ of the sphere at infinity which is singular to the Lebesgue measure. A harmonic measure is called Type I (resp. Type II) if for any typical leaf, the measure $\mu$ is a point mass (resp. of full support). We show that any ergodic harmonic measure is either of type I or type II.

Article information

Source
Tohoku Math. J. (2), Volume 64, Number 4 (2012), 569-592.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1356038979

Digital Object Identifier
doi:10.2748/tmj/1356038979

Mathematical Reviews number (MathSciNet)
MR3008239

Zentralblatt MATH identifier
1317.53039

Subjects
Primary: 53C12: Foliations (differential geometric aspects) [See also 57R30, 57R32]
Secondary: 37C85: Dynamics of group actions other than Z and R, and foliations [See mainly 22Fxx, and also 57R30, 57Sxx]

Keywords
Lamination foliation harmonic measure ergodicity

Citation

Matsumoto, Shigenori. The dichotomy of harmonic measures of compact hyperbolic laminations. Tohoku Math. J. (2) 64 (2012), no. 4, 569--592. doi:10.2748/tmj/1356038979. https://projecteuclid.org/euclid.tmj/1356038979


Export citation

References

  • C. Anantharaman-Delaroche and J. Renault, Amenable groupoids, Monogr. Enseign. Math. 36 l'Enseignement Mathématique, Geneva, 2000.
  • M. T. Anderson and R. Shoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. Math. 121 (1985), 429–461.
  • Y. Bakhtin and M. Martinez, A characterization of harmonic measures on laminations by hyperbolic Riemann surfaces, Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008), 1078–1089.
  • A. Candel, The harmonic measures of Lucy Garnett, Adv. Math. 176 (2003), 187–247.
  • A. Candel and L. Conlon, Foliations II, Grad. Stud. Math. 60, Amer. Math. Soc. Providence, R.I., 2003.
  • S. Y. Cheng, P. Li and S. T. Yau, On the upper estimate of the heat kernel of a complete Riemannian manifold, Amer. J. Math. 103 (1980), 1021–1063.
  • B. Deroin and V. Kleptsyn, Random conformal dynamical systems, Geom. Funct. Anal. 17 (2007), 1043–1105.
  • A. Furman, Random walks on groups and random transformations, Handbook of dynamical systems, Vol. 1A, 931–1014, Norht-Holland, Amsterdam, 2002.
  • L. Garnett, Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal. 51 (1983), 285–311.
  • V. A. Kaimanovich, Brownian motion on foliations: entropy, invariant measures, mixing, Funct. Anal. Appl. 22 (1988), 326–328.
  • V. A. Kaimanovich, Boundary amenability of hyperbolic spaces, Discrete geometric analysis, 83–111, Contemp. Math. 347, Amer. Math. Soc. Providence, R.I., 2004.
  • A. S. Kechris, Classical descriptive set theory, Graduate Texts in Math. 156, Springer Verlag, New York, 1995.
  • M. Martinez, Measures on hyperbolic surface laminations, Ergodic Theory Dynam. Systems 26 (2006), 847–867.
  • M. Martinez and A. Verjovsky, Hedlund's theorem for compact minimal laminations, Preprint, arXiv:0711.2307v2.
  • L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa 13 (1959), 115–162.
  • B. Øksendal, Stochastic differential equations, Sixth Edition, Universitext, Springer Verlag, Berlin, 2007.
  • R. Sauer, $L^2$-invariants of groups and discrete measured groupoids, Dissertation, Universität Münster, 2003.