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We consider group-valued cocycles over a partially hyperbolic diffeomorphismwhich is accessible volume-preserving and center bunched. We study cocycleswith values in the group of invertible continuous linear operators on a Banachspace. We describe properties of holonomies for fiber bunched cocycles andestablish their Holder regularity. We also study cohomology of cocycles and itsconnection with holonomies. We obtain a result on regularity of a measurableconjugacy, as well as a necessary and sufficient condition for existence of acontionuous conjugacy between two cocycles.

Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms