Effective equidistribution of horospherical flows in infinite volume rank one homogeneous spaces

TL;DR

This paper establishes effective equidistribution of horospherical flows in certain infinite volume hyperbolic spaces, leveraging exponential mixing properties and properties of Patterson-Sullivan measures.

Contribution

It proves effective equidistribution results for horospherical flows in geometrically finite hyperbolic manifolds with exponential mixing, extending understanding of flow dynamics in infinite volume settings.

Findings

Effective equidistribution of horospherical flows in infinite volume spaces.

Exponential mixing of frame flows under specific geometric conditions.

Patterson-Sullivan measure satisfies friendly-like properties in geometrically finite cases.

Abstract

We prove effective equidistribution of horospherical flows in $\operatorname{SO}(n,1)^\circ / \Gamma$ when $\Gamma$ is geometrically finite and the frame flow is exponentially mixing for the Bowen-Margulis-Sullivan measure. We also discuss settings in which such an exponential mixing result is known to hold. As part of the proof, we show that the Patterson-Sullivan measure satisfies some friendly-like properties when $\Gamma$ is geometrically finite.