Complete regularity of linear cocycles and the Baire category of the set of Lyapunov-Perron regular points

Given a continuous linear cocycle A over a homeomorphism f of a compact metric space X , we investigate its set R of Lyapunov-Perron regular points, that is, the collection of trajectories of f that obey the conclusions of the Multiplicative Ergodic Theorem. We obtain results roughly saying that the set R is of first Baire category (i.e., meager) in X , unless some rigid structure is present. In some settings, this rigid structure forces the Lyapunov exponents to be defined everywhere and to be independent of the point; that is what we call complete regularity.