## Tohoku Mathematical Journal

### The dichotomy of harmonic measures of compact hyperbolic laminations

Shigenori Matsumoto

#### 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

https://projecteuclid.org/euclid.tmj/1356038979

Digital Object Identifier
doi:10.2748/tmj/1356038979

Mathematical Reviews number (MathSciNet)
MR3008239

Zentralblatt MATH identifier
1317.53039

#### 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

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