Nonuniform measure rigidity

Abstract

We consider an ergodic invariant measure $\mu$ for a smooth action $\alpha$ of $\mathbb{Z}^k$, $k\ge 2$, on a $(k+1)$-dimensional manifold or for a locally free smooth action of $\mathbb{R}^k$, $k\ge 2$, on a $(2k+1)$-dimensional manifold. We prove that if $\mu$ is hyperbolic with the Lyapunov hyperplanes in general position and if one element in $\mathbb{Z}^k$ has positive entropy, then $\mu$ 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.