Description:
We give a general necessary condition for the extremal (largest and smallest) Lyapunov exponents of a Ho͏̈lder continuous cocycle over a volume preserving partially hyperbolic diffeomorphism to coincide. This condition applies to smooth cocycles, with linear and projective cocycles as special cases. It is based on an abstract rigidity result for fiber bundle sections that are holonomy-invariant, or even just continuous, over the strong-stable leaves and the strong-unstable leaves of the diffeomorphism. As an application, we prove that the subset of Ho͏̈lder continuous linear cocycles for which the extremal Lyapunov exponents do coincide is meager and even has infinite codimension.