Weak form of equidistribution theorem for harmonic measures of foliations by hyperbolic surfaces

TL;DR

This paper investigates the limitations of a known equidistribution theorem for foliations by hyperbolic surfaces and proposes a weaker, more generally applicable version.

Contribution

It demonstrates the failure of the original theorem in general cases and develops a weaker form that holds more broadly.

Findings

Original equidistribution theorem does not hold universally.

A weaker form of the theorem is proposed for general foliations.

The new form extends applicability to a wider class of foliations.

Abstract

We show that the equidistribution theorem of C. Bonatti and X. G\'omez-Mont for a special kind of foliations by hyperbolic surfaces does not hold in general, and seek for a weaker form valid for general foliations by hyperbolic surfaces.