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

Kalinin, Boris; Sadovskaya, Victoria

doi:10.1090/proc/13720

Boundedness, compactness, and invariant norms for Banach cocycles over hyperbolic systems
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by Boris Kalinin and Victoria Sadovskaya PDF

Proc. Amer. Math. Soc. 146 (2018), 3801-3812 Request permission

Abstract:

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 $\mathcal {A}$ takes values in the group of invertible bounded linear operators on a Banach space and is Hölder continuous. We consider the periodic data of $\mathcal {A}$, i.e., the set of its return values along the periodic orbits in the base. We show that if the periodic data of $\mathcal {A}$ 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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