Exponential mixing of frame flows for convex cocompact hyperbolic manifolds

TL;DR

This paper proves exponential mixing of frame flows in convex cocompact hyperbolic manifolds, leading to precise asymptotics for matrix coefficients and equidistribution of geodesic holonomy, using spectral bounds and Dolgopyat's method.

Contribution

It establishes the exponential mixing property for frame flows in higher-dimensional convex cocompact hyperbolic manifolds, extending previous results.

Findings

Exponential decay of correlations for frame flows.

Asymptotic formulas for matrix coefficients with exponential error.

Exponential equidistribution of holonomy of closed geodesics.

Abstract

The aim of this paper is to establish exponential mixing of frame flows for convex cocompact hyperbolic manifolds of arbitrary dimension with respect to the Bowen-Margulis-Sullivan measure. Some immediate applications include an asymptotic formula for matrix coefficients with an exponential error term as well as the exponential equidistribution of holonomy of closed geodesics. The main technical result is a spectral bound on transfer operators twisted by holonomy, which we obtain by building on Dolgopyat's method.