Cocycles with one exponent over partially hyperbolic systems

We consider Hölder 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 the quasiconformal distortion from the periodic data.