# Nonuniform measure rigidity | Annals of Mathematics

Abstract We consider an ergodic invariant measure μ for a smooth action α of ℤk, k≥2, on a (k+1)-dimensional manifold or for a locally free smooth action of ℝk, 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 ℤk 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.

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### 数学年刊（Annals of Mathematics）

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