Exponential mixing of frame flows for three dimensional manifolds of quarter-pinched negative curvature

TL;DR

This paper proves exponential mixing of the frame flow on certain negatively curved 3D manifolds, extending to torus extensions of Anosov flows, using a Young tower model and Dolgopyat estimates.

Contribution

It establishes exponential mixing for frame flows on 3D quarter-pinched negatively curved manifolds and extends results to specific torus extensions of Anosov flows.

Findings

Exponential mixing of frame flow on the specified manifolds.

Extension of mixing results to certain torus extensions of Anosov flows.

Application of Young tower models and Dolgopyat estimates to prove mixing.

Abstract

For a compact three-dimensional smooth Riemannian manifold of strictly 1/4-pinched negative sectional curvature, we establish exponential mixing of the frame flow with respect to the normalized volume. More generally this result extends to a class of torus extensions of Anosov flows, subject to assumptions on the Brin transitivity group and the smoothness of the stable subbundle. Our approach is based on a simplified dynamical model for studying the extension flow, constructed via a Young tower of the underlying Anosov flow. Exponential mixing is then obtained through a strengthened Dolgopyat type estimate on the corresponding transfer operators.