Cocycles measurably conjugate to unipotent over hyperbolic systems

Abstract

We show that if a Hölder continuous linear cocycle over a hyperbolic system is measurably conjugate to a cocycle taking values in a unipotent group, then the cocycle is Hölder continuously conjugate to a cocycle taking values in a unipotent group. More generally, we introduce some natural classes of matrices contained in $\mathrm{GL}(d,\mathbb{R})$, which we call Zimmer blocks. Examples of Zimmer blocks are unipotent and compact subgroups. We show that the same conclusion holds for Zimmer blocks.

26 pages