Nonuniform measure rigidity


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.


Boris Kalinin

Department of Mathematics and Statistics
University of South Alabama
Mobile, AL 36688

Anatole Katok

Mathematics Department
The Pennsylvania State University
University Park
State College, PA 16802

Federico Rodriguez Hertz

Facultad de Ingeniería
Universidad de la República
11300 Montevideo