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. 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. We also consider actions on the torus TN with induced action on the first homology corresponding to a finite index subgroup of a maximal semisimple abelian subgroup of SL(N, R). Such an action has a unique invariant measure, called large measure, which projects to the Lebesgue measure under the semiconjugacy with the linear action and this measure is absolutely continuous. Finally, we consider cocycles over an action on the torus with Cartan homotopy data. Every cocycle which is Hölder with respect to a Lyapunov Riemannian metric a.e. for the large invariant measure is cohomologous to a constant cocycle via a Lyapunov-Hölder transfer function.
|Original language||English (US)|
|Number of pages||14|
|Journal||Electronic Research Announcements of the American Mathematical Society|
|State||Published - Dec 1 2008|
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