On invariant holonomies between centers

Abstract

Abstract We prove that for $C^{1+\theta }$ , $\theta $ -bunched, dynamically coherent partially hyperbolic diffeomorphisms, the stable and unstable holonomies between center leaves are $C^1$ , and the derivative depends continuously on the points and on the map. Also for $C^{1+\theta }$ , $\theta $ -bunched partially hyperbolic diffeomorphisms, the derivative cocycle restricted to the center bundle has invariant continuous holonomies which depend continuously on the map. This generalizes previous results by Pugh, Shub, and Wilkinson; Burns and Wilkinson; Brown; Obata; Avila, Santamaria, and Viana; and Marin.