On invariant holonomies between centers

TL;DR

This paper proves that for certain smooth, dynamically coherent partially hyperbolic diffeomorphisms, the stable and unstable holonomies between center leaves are continuously differentiable and depend smoothly on the map, extending previous results.

Contribution

It establishes the $C^1$ regularity and continuous dependence of holonomies and their derivatives in a broad class of partially hyperbolic systems, generalizing earlier work.

Findings

Holonomies between center leaves are $C^1$ for the specified class.

Derivative cocycle on the center bundle has invariant continuous holonomies.

Results extend previous theorems by multiple researchers.

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 diffeomorphism, 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-Wilkinson, Burns-Wilkinson, Brown, Obata, Avila-Santamaria-Viana, Marin.