Geodesic
Nonuniform measure rigidity
Boris Kalinin1 ; Anatole Katok2 ; Federico Rodriguez Hertz3
1 Department of Mathematics and Statistics<br/>University of South Alabama<br/> Mobile, AL 36688
2 Mathematics Department<br/>The Pennsylvania State University<br/>University Park<br/> State College, PA 16802
3 IMERL<br/> Facultad de Ingeniería<br/>Universidad de la República<br/> 11300 Montevideo<br/> Uruguay
Annals of mathematics, Tome 174 (2011) no. 1, pp. 361-400.

Voir la notice de l'article provenant de la source Annals of Mathematics website

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.
DOI : 10.4007/annals.2011.174.1.10

Boris Kalinin 1 ; Anatole Katok 2 ; Federico Rodriguez Hertz 3

1 Department of Mathematics and Statistics<br/>University of South Alabama<br/> Mobile, AL 36688
2 Mathematics Department<br/>The Pennsylvania State University<br/>University Park<br/> State College, PA 16802
3 IMERL<br/> Facultad de Ingeniería<br/>Universidad de la República<br/> 11300 Montevideo<br/> Uruguay 
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Boris Kalinin; Anatole Katok; Federico Rodriguez Hertz. Nonuniform measure rigidity. Annals of mathematics, Tome 174 (2011) no. 1, pp. 361-400. doi : 10.4007/annals.2011.174.1.10. http://geodesic.mathdoc.fr/articles/10.4007/annals.2011.174.1.10/

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