The dichotomy of harmonic measures of compact hyperbolic laminations

TL;DR

This paper explores the properties of harmonic measures on compact hyperbolic laminations, revealing a dichotomy in ergodic harmonic measures when all leaves are hyperbolic, and analyzing associated harmonic functions on universal covers.

Contribution

It introduces a classification of ergodic harmonic measures into two distinct types for hyperbolic laminations with all leaves hyperbolic, based on properties of harmonic functions on universal covers.

Findings

Ergodic harmonic measures are divided into two classes.

Positive harmonic functions are defined on universal covers of leaves.

Properties of harmonic functions relate to the structure of harmonic measures.

Abstract

Given a harmonic measure of a hyperbolic lamination on a compact metric space, a positive harmonic function is defined on the universal cover of a typical leaves. We discuss some properties of this function. Especially if all the leaves are hyperbolic, ergodic harmonic measures are divided into two classes.