The dichotomy of harmonic measures of compact hyperbolic laminations

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.