Nonuniform measure rigidity

We consider an ergodic invariant measure for a smooth action of Z k , k 2, on a (k + 1)-dimensional manifold or for a locally free smooth action of R k , k 2, on a (2k +1)-dimensional manifold. We prove that if is hyperbolic with the Lyapunov hyperplanes in general position and if one element in Z k has positive entropy, then is absolutely continuous. The main ingredient is absolute continuity of conditional measures on Lyapunov foliations which holds for a more general class of smooth actions of higher rank abelian groups.