Boundedness and invariant metrics for diffeomorphism cocycles over hyperbolic systems

Let $A$ be a H\"older 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 Diff$^{\,q}(M)$, $q>1$, then the set of values of the cocycle is bounded in Diff$^{\,r}(M)$ for each $r<q$. Moreover, such a cocycle is isometric with respect to a H\"older continuous family of Riemannian metrics on $M$.