Measurable rigidity of the cohomological equation for linear cocycles over hyperbolic systems

TL;DR

This paper proves that measurable solutions to the cohomological equation for H"older linear cocycles over hyperbolic systems are almost everywhere H"older, with applications to conformal structures and characterizations of negatively curved manifolds.

Contribution

It establishes the regularity of measurable solutions and conformal structures for linear cocycles over hyperbolic systems, extending previous results and providing new geometric characterizations.

Findings

Measurable solutions coincide with H"older solutions almost everywhere.

Invariant conformal structures are almost everywhere continuous.

Characterization of negatively curved manifolds via cocycle irreducibility.

Abstract

We show that any measurable solution of the cohomological equation for a H\"older linear cocycle over a hyperbolic system coincides almost everywhere with a H\"older solution. More generally, we show that every measurable invariant conformal structure for a H\"older linear cocycle over a hyperbolic system coincides almost everywhere with a continuous invariant conformal structure. We also use the main theorem to show that a linear cocycle is conformal if none of its iterates preserve a measurable family of proper subspaces of $\mathbb{R}^{d}$. We use this to characterize closed negatively curved Riemannian manifolds of constant negative curvature by irreducibility of the action of the geodesic flow on the unstable bundle.