Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps

TL;DR

This paper proves exponential mixing of geodesic flows on certain hyperbolic manifolds with cusps and uses this to derive results about the Laplacian's resolvent, employing coding and spectral estimates.

Contribution

It establishes exponential mixing for geodesic flows on geometrically finite hyperbolic manifolds with cusps, a significant advancement in understanding their dynamical properties.

Findings

Proves exponential mixing of geodesic flow with respect to Bowen-Margulis-Sullivan measure.

Derives a resonance free region for the Laplacian's resolvent.

Develops a coding and spectral estimate approach for the analysis.

Abstract

Let $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{SO}(d+1,1)^{\circ}$ with parabolic elements. We establish exponential mixing of the geodesic flow on the unit tangent bundle $\operatorname{T}^1(\Gamma\backslash \mathbb{H}^{d+1})$ with respect to the Bowen-Margulis-Sullivan measure, which is the unique probability measure on $\operatorname{T}^1(\Gamma\backslash \mathbb{H}^{d+1})$ with maximal entropy. As an application, we obtain a resonance free region for the resolvent of the Laplacian on $\Gamma\backslash \mathbb{H}^{d+1}$. Our approach is to construct a coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator.