Boundedness and invariant metrics for diffeomorphism cocycles over hyperbolic systems

TL;DR

This paper investigates the boundedness and invariance properties of diffeomorphism cocycles over hyperbolic systems, showing that bounded periodic data implies boundedness of the cocycle and existence of an invariant metric.

Contribution

It establishes that bounded periodic data in a diffeomorphism cocycle ensures boundedness of the entire cocycle and the existence of a H"older continuous invariant metric.

Findings

Bounded periodic data implies boundedness of the cocycle in higher regularity spaces.

The cocycle is isometric with respect to a H"older continuous family of Riemannian metrics.

Results connect periodic data boundedness with geometric invariance properties.

Abstract

Let $A$ be a H\"older continuous cocycle over a hyperbolic dynamical system with values in the group of diffeomorphisms of a compact manifold $M$. 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 bounded in Diff$^{\,q}(M)$, $q>1$, then the set of values of the cocycle is bounded in Diff$^{\,r}(M)$ for each $r<q$. Moreover, such a cocycle is isometric with respect to a H\"older continuous family of Riemannian metrics on $M$.