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.

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