| NONUNIFORM MEASURE RIGIDITY (2009) | |||||||||||||||
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| Abstract. We consider an ergodic invariant measure µ for a smooth action α of Z k, k ≥ 2, on a (k +1)-dimensional manifold or for a locally free smooth action of R k, 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 Z k 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. 1. | |||||||||||||||
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