Nonuniform measure rigidity

Details Nonuniform measure rigidity Nonuniform measure rigidity

2008-03-20

arxiv.org/abs/0803.3094v2

Mathematics

Dynamical Systems

To appear in Annals of Mathematics

Scientific paper

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

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