Periodic data conjugacy for diffeomorphism cocycles

We consider Hölder continuous cocycles over a hyperbolic dynamical system with values in the group of diffeomorphisms of a compact manifold. We focus on establishing conjugacy between two cocycles from conjugacy of their periodic data. We obtain such a result under bunching, or domination, assumption on the cocycles and a mild regularity assumption in C⁰ topology on the maps conjugating the periodic data. We also obtain conjugacy to a constant cocycle assuming boundedness in C¹ or pre-compactness in C⁰ for the constant cocycle and the conjugating maps.