Boundedness and invariant metrics for diffeomorphism cocycles over hyperbolic systems

Let A be a Hölder continuous cocycle over a hyperbolic dynamical system with values in the group of diffeomorphisms of a compact manifold M. We consider the periodic data of A, i.e., the set of its return values along the periodic orbits in the base. We show that if the periodic data of A is bounded in Diffq(M), q> 1 , then the set of values of the cocycle is bounded in Diffr(M) for each r< q. Moreover, such a cocycle is isometric with respect to a Hölder continuous family of Riemannian metrics on M.