Nonuniform measure rigidity

Abstract

We consider an ergodic invariant measure µ for a smooth actionof 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 µ is hyperbolic with the Lyapunov hyperplanes in general position and if one element in Z k has positive entropy, then µ 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.