On regularity of conjugacy between linear cocycles over partially hyperbolic systems

TL;DR

This paper investigates the regularity of conjugacies between linear cocycles over partially hyperbolic systems, proving continuity results for measurable conjugacies and solutions to cohomological equations, with implications for invariant structures.

Contribution

It establishes continuity of measurable conjugacies for linear cocycles over partially hyperbolic systems, including perturbations of constant cocycles, and extends regularity results for invariant structures.

Findings

Continuity of measurable conjugacy between constant and perturbed cocycles.

Continuity of solutions to twisted cohomological equations.

Generalized regularity results for invariant subbundles and structures.

Abstract

We consider H\"older continuous $GL(d,\mathbb R)$-valued cocycles, and more generally linear cocycles, over an accessible volume-preserving center-bunched partially hyperbolic diffeomorphism. We study the regularity of a conjugacy between two cocycles. We establish continuity of a measurable conjugacy between {\em any} constant $GL(d,\mathbb R)$-valued cocycle and its perturbation. We deduce this from our main technical result on continuity of a measurable conjugacy between a fiber bunched linear cocycle and a cocycle with a certain block-triangular structure. The latter class covers constant cocycles with one Lyapunov exponent. We also establish a result of independent interest on continuity of measurable solutions for twisted vector-valued cohomological equations over partially hyperbolic systems. In addition, we give more general versions of earlier results on regularity of invariant subbudles, Riemannian metrics, and conformal structures.