Nonuniform measure rigidity

Abstract

We consider an ergodic invariant measure µ for a smooth action a of Zk, k ≥ 2, on a (k + 1)-dimensional manifold or for a locally free smooth action of Rk, 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 Zk 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.