Mesures stationnaires et fermés invariants des espaces homogénes

Yves Benoist; Jean-François Quint

doi:http://dx.doi.org/10.4007/annals.2011.174.2.8

Abstract

Soient $G$ un groupe de Lie réel simple, $\Lambda$ un réseau de $G$ et $\Gamma$ un sous-semi-groupe Zariski dense de $G$. On montre que toute partie infinie $\Gamma$-invariante dans le quotient $X=G/\Lambda$ est dense.
Soit $\mu$ une probabilité sur $G$ dont le support est compact et engendre un sous-groupe Zariski dense de $G$. On montre que toute probabilité $\mu$-stationnaire sans atome sur $X$ est $G$-invariante.On montre aussi des énoncés analogues pour le tore $X=\mathbb{T}^d$.

Let $G$ be the group of real points of a real simple Lie group, $\Lambda$ be a lattice of $G$ and $\Gamma$ be a Zariski dense subsemigroup of $G$. We prove that every infinite $\Gamma$-invariant subset in the quotient $X=G/\Lambda$ is dense. Let $\mu$ be a probability measure on $G$ whose support is compact and spans a Zariski dense subgroup of $G$. We prove that every atom free $\mu$-stationary probability measure on $X$ is $G$-invariant. We also prove similar results for the torus $X=\mathbb{T}^d$.

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@article{Ro67, author = {Rokhlin, V. A.}, TITLE = {Lectures on the entropy theory of measure-preserving transformations}, JOURNAL={Russian Math. Surveys}, VOLUME={22}, YEAR={1967}, PAGES={1--52}, MRNUMBER = {0217258}, ZBLNUMBER = {0174.45501}, DOI = {10.1070/RM1967v022n05ABEH001224}, }
[Sh] N. A. Shah, "Invariant measures and orbit closures on homogeneous spaces for actions of subgroups generated by unipotent elements," in Lie Groups and Ergodic Theory, Bombay, 1998, pp. 229-271.

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@inproceedings {Sh, MRKEY = {1699367}, AUTHOR = {Shah, Nimish A.}, TITLE = {Invariant measures and orbit closures on homogeneous spaces for actions of subgroups generated by unipotent elements}, BOOKTITLE = {Lie Groups and Ergodic Theory}, VENUE={{M}umbai, 1996}, SERIES = {Tata Inst. Fund. Res. Stud. Math.}, VOLUME = {14}, PAGES = {229--271}, PUBLISHER = {Tata Inst. Fund. Res.}, ADDRESS = {Bombay}, YEAR = {1998}, MRCLASS = {22E40 (22D30 22F30 43A85)}, MRNUMBER = {1699367}, MRREVIEWER = {Dave Witte Morris}, ZBLNUMBER = {0951.22006}, }
[Zim] R. J. Zimmer, Ergodic Theory and Semisimple Groups, Basel: Birkhäuser-Verlag, 1984, vol. 81.

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@book {Zim, MRKEY = {776417}, AUTHOR = {Zimmer, Robert J.}, TITLE = {Ergodic Theory and Semisimple Groups}, SERIES = {Monogr. Math.}, VOLUME = {81}, PUBLISHER = {Birkhäuser-Verlag}, ADDRESS = {Basel}, YEAR = {1984}, PAGES = {x+209}, ISBN = {3-7643-3184-4}, MRCLASS = {22E40 (22D40 28D15)}, MRNUMBER = {0776417}, MRREVIEWER = {S. G. Dani}, ZBLNUMBER = {0571.58015}, }

Mesures stationnaires et fermés invariants des espaces homogénes

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