On regularity of conjugacy between linear cocycles over partially hyperbolic systems

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.