Nonuniform measure rigidity

Boris Kalinin; Anatole Katok; Federico Rodriguez Hertz

doi:http://doi.org/10.4007/annals.2011.174.1.10

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.

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@misc{DK-II, author={Damjanovic, D. and Katok, A.}, TITLE={Local Rigidity of Partially Hyperbolic Actions. {II}. The geometric method and restrictions of Weyl chamber flows on ${SL}(n, {R})/{\Gamma}$ }, URL={www.math.psu.edu/katok$\_$a/papers.html}, }
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Nonuniform measure rigidity

Abstract

Authors