Minimality and unique ergodicity for subgroup actions
Annales de l'Institut Fourier, Volume 48 (1998) no. 5, p. 1533-1541

Let $G$ be an $ℝ$-algebraic semisimple group, $H$ an algebraic $ℝ$-subgroup, and $\Gamma$ a lattice in $G$. Partially answering a question posed by Hillel Furstenberg in 1972, we prove that if the action of $H$ on $G/\Gamma$ is minimal, then it is uniquely ergodic. Our proof uses in an essential way Marina Ratner’s classification of probability measures on $G/\Gamma$ invariant under unipotent elements, and the study of “tubes” in $G/\Gamma$.

Soient $G$ un groupe semi-simple algébrique sur $ℝ$, $H$ un sous-groupe algébrique sur $ℝ$, et $\Gamma$ un réseau dans $G$. Répondant partiellement à une question de Hillel Furstenberg remontant à 1972, nous prouvons que si l’action de $H$ sur $G/\Gamma$ est minimale alors elle est uniquement ergodique. Notre preuve repose sur la classification de Marina Ratner des mesures sur $G/\Gamma$ invariantes sous l’action des éléments unipotents, et l’analyse des “tubes” dans $G/\Gamma$.

@article{AIF_1998__48_5_1533_0,
author = {Mozes, Shahar and Weiss, Barak},
title = {Minimality and unique ergodicity for subgroup actions},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {48},
number = {5},
year = {1998},
pages = {1533-1541},
doi = {10.5802/aif.1665},
zbl = {0910.43010},
mrnumber = {99j:22007},
language = {en},
url = {http://www.numdam.org/item/AIF_1998__48_5_1533_0}
}

Mozes, Shahar; Weiss, Barak. Minimality and unique ergodicity for subgroup actions. Annales de l'Institut Fourier, Volume 48 (1998) no. 5, pp. 1533-1541. doi : 10.5802/aif.1665. http://www.numdam.org/item/AIF_1998__48_5_1533_0/

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