Equidistribution of divergent geodesics in negative curvature

TL;DR

This paper proves that divergent geodesics in negatively curved manifolds and tree quotients become uniformly distributed with respect to natural invariant measures as their complexity increases.

Contribution

It establishes the equidistribution of divergent geodesics towards measures of maximal entropy in both negatively curved manifolds and tree quotients, extending prior results to new settings.

Findings

Divergent geodesics equidistribute towards the measure of maximal entropy.

Results hold for noncompact finite volume negatively curved manifolds.

Analogous equidistribution established for geometrically finite tree quotients.

Abstract

In the unit tangent bundle of noncompact finite volume negatively curved Riemannian manifolds, we prove the equidistribution towards the measure of maximal entropy for the geodesic flow of the Lebesgue measure along the divergent geodesic flow orbits, as their complexity tends to infinity. We prove the analogous result for geometrically finite tree quotients, where the equidistribution takes place in the quotient space of geodesic lines towards the Bowen-Margulis measure.