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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VOLUME = {134},
YEAR = {1991},
NUMBER = {3},
PAGES = {545--607},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {22E40 (58F11 58F17)},
MRNUMBER = {1135878},
MRREVIEWER = {S. G. Dani},
DOI = {10.2307/2944357},
ZBLNUMBER = {0763.28012},
} -
[Ra2]
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@article {Ra2, MRKEY = {1106945},
AUTHOR = {Ratner, Marina},
TITLE = {Raghunathan's topological conjecture and distributions of unipotent flows},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {63},
YEAR = {1991},
NUMBER = {1},
PAGES = {235--280},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {22E40 (22D40 28D10)},
MRNUMBER = {1106945},
MRREVIEWER = {Gopal Prasad},
DOI = {10.1215/S0012-7094-91-06311-8},
ZBLNUMBER = {0733.22007},
} -
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@article {Ro49, MRKEY = {0030584},
AUTHOR = {Rokhlin, V. A.},
TITLE = {On the fundamental ideas of measure theory},
JOURNAL = {Mat. Sbornik N.S.},
VOLUME = {25(67)},
YEAR = {1949},
PAGES = {107--150},
MRCLASS = {27.2X},
MRNUMBER = {0030584},
MRREVIEWER = {P. R. Halmos},
} -
[Ro67]
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@article{Ro67,
author = {Rokhlin, V. A.},
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JOURNAL={Russian Math. Surveys},
VOLUME={22},
YEAR={1967},
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MRNUMBER = {0217258},
ZBLNUMBER = {0174.45501},
DOI = {10.1070/RM1967v022n05ABEH001224},
} -
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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.},
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PUBLISHER = {Tata Inst. Fund. Res.},
ADDRESS = {Bombay},
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@book {Zim, MRKEY = {776417},
AUTHOR = {Zimmer, Robert J.},
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PAGES = {x+209},
ISBN = {3-7643-3184-4},
MRCLASS = {22E40 (22D40 28D15)},
MRNUMBER = {0776417},
MRREVIEWER = {S. G. Dani},
ZBLNUMBER = {0571.58015},
}