Cocycles measurably conjugate to unipotent over hyperbolic systems

TL;DR

This paper proves that certain measurable conjugacies to unipotent cocycles over hyperbolic systems are actually H"older continuous, extending to a broader class called Zimmer blocks, which include unipotent and compact groups.

Contribution

It establishes H"older regularity of conjugacies for cocycles measurably conjugate to unipotent or Zimmer block-valued cocycles over hyperbolic systems.

Findings

Measurable conjugacy implies H"older conjugacy for unipotent cocycles.

Introduces Zimmer blocks as a general class of matrices with similar properties.

Extends regularity results to a broader class of matrix groups.

Abstract

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