Uniformly quasiconformal partially hyperbolic systems

TL;DR

This paper characterizes when smooth volume-preserving perturbations of geodesic flows on negatively curved manifolds are smoothly orbit equivalent to the original flow, based on Lyapunov exponents and quasiconformality conditions.

Contribution

It provides a classification of volume-preserving, partially hyperbolic systems with uniform quasiconformality, extending to perturbations of geodesic and Anosov flows.

Findings

Equal extremal Lyapunov exponents imply smooth orbit equivalence.

Classification of systems with compact center foliation or as perturbations of Anosov flows.

Techniques applicable to a broad class of partially hyperbolic diffeomorphisms.

Abstract

We study smooth volume-preserving perturbations of the time-1 map of the geodesic flow $\psi_{t}$ of a closed Riemannian manifold of dimension at least three with constant negative curvature. We show that such a perturbation has equal extremal Lyapunov exponents with respect to volume within both the stable and unstable bundles if and only if it embeds as the time-1 map of a smooth volume-preserving flow that is smoothly orbit equivalent to $\psi_{t}$. Our techniques apply more generally to give an essentially complete classification of smooth, volume-preserving, dynamically coherent partially hyperbolic diffeomorphisms which satisfy a uniform quasiconformality condition on their stable and unstable bundles and have either compact center foliation with trivial holonomy or are obtained as perturbations of the time-1 map of an Anosov flow.