Cocycles with one exponent over partially hyperbolic systems

TL;DR

This paper studies the regularity and structure of linear cocycles over partially hyperbolic systems, establishing continuity of invariant structures and extending Zimmer's theorem, with additional growth estimates in hyperbolic cases.

Contribution

It proves continuity of invariant conformal structures and sub-bundles for fiber bunched cocycles with one Lyapunov exponent, and extends Zimmer's Amenable Reduction Theorem to this setting.

Findings

Continuity of invariant conformal structures and sub-bundles.

Extension of Zimmer's Amenable Reduction Theorem.

Polynomial growth estimates for cocycles over hyperbolic systems.

Abstract

We consider Holder continuous linear cocycles over partially hyperbolic diffeomorphisms. For fiber bunched cocycles with one Lyapunov exponent we show continuity of measurable invariant conformal structures and sub-bundles. Further, we establish a continuous version of Zimmer's Amenable Reduction Theorem. For cocycles over hyperbolic systems we also obtain polynomial growth estimates for the norm and quasiconformal distortion from the periodic data.