Continuity of Measurable Invariant Conformal Structures for Linear Cocycles over Hyperbolic Systems

We show that every measurable invariant conformal structure for a Hölder continuous linear cocycle over a subshift of finite type coincides almost everywhere with a continuous invariant conformal structure. We use this result to establish Hölder continuity of a measurable conjugacy between Hölder continuous cocycles where one of the cocycles is assumed to be uniformly quasiconformal. As a special case we derive that if a Hölder linear cocycle is a measurable coboundary, then it is a Hölder coboundary. We also use the main theorem to show that a linear cocycle is conformal if none of its iterates preserve a measurable family of proper subspaces of Rd. We use this to characterize closed negatively curved Riemannian manifolds of constant negative curvature by irreducibility of the action of the geodesic flow on the unstable bundle.