Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows

Abstract

In several contexts the defining invariant structures of a hyperbolic dynamical system are smooth only in systems of algebraic origin (smooth rigidity), and we prove new results of this type for a class of flows. For a compact Riemannian manifold and a uniformly quasiconformal transversely symplectic Anosov flow we define the longitudinal KAM-cocycle and use it to prove a rigidity result: The joint stable/unstable subbundle is Zygmund-regular, and higher regularity implies vanishing of the longitudinal KAM-cocycle, which in turn implies that the subbundle is Lipschitz-continuous and indeed that the flow is smoothly conjugate to an algebraic one. To establish the latter, we prove results for algebraic Anosov systems that imply smoothness and a special structure for any Lipschitz-continuous invariant 1-form. Several features of the reasoning are interesting: The use of exterior calculus for Lipschitz-continuous forms, that the arguments for geodesic flows and infranilmanifoldautomorphisms are quite different, and the need for mixing as opposed to ergodicity in the latter case.