Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups

We prove absolute continuity of 'high-entropy'' hyperbolic invariant measuresfor smooth actions of higher-rank abelian groups assuming that there are noproportional Lyapunov exponents. For actions on tori and infranilmanifoldsthe existence of an absolutely continuous invariant measure of this kind isobtained for actions whose elements are homotopic to those of an action byhyperbolic automorphisms with no multiple or proportional Lyapunov exponents.In the latter case a form of rigidity is proved for certain natural classes ofcocycles over the action.