Oseledets Regularity Functions for Anosov Flows

Oseledets regularity functions quantify the deviation between the growth associated with a dynamical system along its Lyapunov bundles and the corresponding uniform exponential growth. Precise degree of regularity of these functions is unknown. We show that for every invariant Lyapunov bundle of a volume preserving Anosov flow on a closed smooth Riemannian manifold, the corresponding Oseledets regularity functions are in L(m), for some p > 0, where m is the probability measure defined by the volume form. We prove an analogous result for essentially bounded cocycles over volume preserving Anosov flows.