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

TL;DR

This paper investigates Banach cocycles over hyperbolic systems, establishing conditions under which cocycles are bounded, compact, or isometric, and demonstrating the existence of invariant norms.

Contribution

It provides new criteria linking periodic data properties to cocycle boundedness, compactness, and invariance of norms in hyperbolic dynamical systems.

Findings

Periodic data being uniformly quasiconformal implies cocycle boundedness.

Cocycles with bounded periodic data are shown to be isometric under a H"older continuous norm.

Existence of a measurable invariant norm family for cocycles is established.

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 $A$ takes values in the group of invertible bounded linear operators on a Banach space and is H\"older 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 H\"older continuous family of norms. We also obtain a general result on existence of a measurable family of norms invariant under a cocycle.