Diffeomorphism cocycles over partially hyperbolic systems

TL;DR

This paper studies cocycles over partially hyperbolic systems with diffeomorphism values, establishing conditions for invariant metrics, conjugacy continuity, and relations between holonomies and conjugacies, with minimal regularity loss.

Contribution

It provides new results on the existence and regularity of invariant metrics and conjugacies for diffeomorphism cocycles over partially hyperbolic systems.

Findings

Existence of continuous invariant Riemannian metrics for bounded cocycles.

Continuity of measurable conjugacies under bunching and holonomy conditions.

Conditions for the existence of continuous conjugacies based on cycle weights.

Abstract

We consider H\"older continuous cocycles over an accessible partially hyperbolic system with values in the group of diffeomorphisms of a compact manifold $M$. We obtain several results for this setting. If a cocycle is bounded in $C^{1+\gamma}$, we show that it has a continuous invariant family of $\gamma$-H\"older Riemannian metrics on $M$. We establish continuity of a measurable conjugacy between two cocycles assuming bunching or existence of holonomies for both and pre-compactness in $C^0$ for one of them. We give conditions for existence of a continuous conjugacy between two cocycles in terms of their cycle weights. We also study the relation between the conjugacy and holonomies of the cocycles. Our results give arbitrarily small loss of regularity of the conjugacy along the fiber compared to that of the holonomies and of the cocycle.