Uniformly quasiconformal partially hyperbolic systems

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.

42 pages. Final arXiv version. To appear in Ann. Sci. École Norm. Sup