Tohoku Mathematical Journal

The dichotomy of harmonic measures of compact hyperbolic laminations

Shigenori Matsumoto

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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

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

First available in Project Euclid: 20 December 2012

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Zentralblatt MATH identifier

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]

Lamination foliation harmonic measure ergodicity


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.

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