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

Soient G un groupe semi-simple algébrique sur ℝ, H un sous-groupe algébrique sur ℝ, et Γ 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/Γ est minimale alors elle est uniquement ergodique. Notre preuve repose sur la classification de Marina Ratner des mesures sur G/Γ invariantes sous l’action des éléments unipotents, et l’analyse des “tubes” dans G/Γ.

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

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     title = {Minimality and unique ergodicity for subgroup actions},
     journal = {Annales de l'Institut Fourier},
     pages = {1533--1541},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {48},
     number = {5},
     year = {1998},
     doi = {10.5802/aif.1665},
     mrnumber = {99j:22007},
     zbl = {0910.43010},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1665/}
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Mozes, Shahar; Weiss, Barak. Minimality and unique ergodicity for subgroup actions. Annales de l'Institut Fourier, Tome 48 (1998) no. 5, pp. 1533-1541. doi : 10.5802/aif.1665. http://www.numdam.org/articles/10.5802/aif.1665/

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