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Exponential mixing of frame flows for convex cocompact hyperbolic manifolds

Part of: Smooth dynamical systems: general theory Lie groups Ergodic theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  08 November 2021

Pratyush Sarkar  and
Dale Winter
Pratyush Sarkar
Affiliation:
Department of Mathematics, Yale University, New Haven, CT06511, USApratyush.sarkar@yale.edu
Dale Winter
Affiliation:
Winchester, MA01890, USAdale.alan.winter@gmail.com
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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.

Keywords

convex cocompact subgroupZariski dense subgrouphyperbolic manifoldframe flowmixingexponential mixingtransfer operatorDolgopyat operatorasymptotic formulamatrix coefficientequidistributionholonomyBowen–Margulis–Sullivan measure

MSC classification

Primary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 37A25: Ergodicity, mixing, rates of mixing 37C30: Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems 22E40: Discrete subgroups of Lie groups
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 12 , December 2021 , pp. 2585 - 2634
Copyright
© 2021 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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