Boundedness, compactness, and invariant norms for banach cocycles over hyperbolic systems

We consider group-valued cocycles over dynamical systems with hyperbolic behavior. The base system is either a hyperbolic diffeomorphism or a mixing subshift of finite type. The cocycle A takes values in the group of invertible bounded linear operators on a Banach space and is Hölder continuous. 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 uniformly quasiconformal or bounded or contained in a compact set, then so is the cocycle. Moreover, in the latter case the cocycle is isometric with respect to a Hölder continuous family of norms. We also obtain a general result on existence of a measurable family of norms invariant under a cocycle.