  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.
