Cohomology of GL(2,ℝ)-valued cocycles over hyperbolic systems

We consider Hölder continuous GL(2,ℝ)-valued cocycles over a transitive Anosov diffeomorphism. We give a complete classification up to Hölder cohomology of cocycles with one Lyapunov exponent and of cocycles that preserve two transverse Hölder continuous sub-bundles. We prove that a measurable cohomology between two such cocycles is Hölder continuous. We also show that conjugacy of periodic data for two such cocycles does not always imply cohomology, but a slightly stronger assumption does. We describe examples that indicate that our main results do not extend to general GL(2,ℝ)-valued cocycles.