Rapid mixing for compact group extensions of hyperbolic flows

TL;DR

This paper establishes explicit conditions under which compact group extensions of hyperbolic flows, including geodesic flows on negatively curved manifolds, exhibit rapid mixing with quantifiable decay rates of correlations.

Contribution

It provides the first explicit criteria for rapid mixing in compact group extensions of hyperbolic flows, including error estimates for equidistribution of holonomy.

Findings

Mixing rate faster than any polynomial for certain flows

Explicit error bounds for holonomy equidistribution

Applicability to frame flows on negatively curved manifolds

Abstract

In this article, we give explicit conditions for compact group extensions of hyperbolic flows (including geodesic flows on negatively curved manifolds) to exhibit quantifiable rates of mixing (or decay of correlations) with respect to the natural probability measures, which are locally the product of a Gibbs measure for a H\"older potential and the Haar measure. More precisely, we show that the mixing rate with respect to H\"older functions will be faster than any given polynomial (i.e., rapid mixing). We also give error estimates on the equidistribution of the holonomy around closed orbits. In particular, these results apply to some frame flows for manifolds with negative sectional curvatures.