Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms

TL;DR

This paper investigates the properties of holonomies and cohomology for group-valued cocycles over partially hyperbolic diffeomorphisms, establishing regularity results and conditions for conjugacy.

Contribution

It introduces new regularity results for holonomies and provides criteria for the existence of continuous conjugacies between cocycles.

Findings

Holonomies are shown to be Holder continuous.

A necessary and sufficient condition for continuous conjugacy is established.

Connections between cohomology and holonomies are clarified.

Abstract

We consider group-valued cocycles over a partially hyperbolic diffeomorphism which is accessible volume-preserving and center bunched. We study cocycles with values in the group of invertible continuous linear operators on a Banach space. We describe properties of holonomies for fiber bunched cocycles and establish their Holder regularity. We also study cohomology of cocycles and its connection with holonomies. We obtain a result on regularity of a measurable conjugacy, as well as a necessary and sufficient condition for existence of a contionuous conjugacy between two cocycles.