Minimality and unique ergodicity for subgroup actions

Shahar Mozes; Barak Weiss

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 5, page 1533-1541
  • ISSN: 0373-0956

Abstract

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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 .

How to cite

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Mozes, Shahar, and Weiss, Barak. "Minimality and unique ergodicity for subgroup actions." Annales de l'institut Fourier 48.5 (1998): 1533-1541. <http://eudml.org/doc/75329>.

@article{Mozes1998,
abstract = {Let $G$ be an $\{\Bbb R\}$-algebraic semisimple group, $H$ an algebraic $\{\Bbb R\}$-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 $.},
author = {Mozes, Shahar, Weiss, Barak},
journal = {Annales de l'institut Fourier},
keywords = {minimal action; uniquely ergodic; homogeneous space; reductive algebraic -group; lattice},
language = {eng},
number = {5},
pages = {1533-1541},
publisher = {Association des Annales de l'Institut Fourier},
title = {Minimality and unique ergodicity for subgroup actions},
url = {http://eudml.org/doc/75329},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Mozes, Shahar
AU - Weiss, Barak
TI - Minimality and unique ergodicity for subgroup actions
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 5
SP - 1533
EP - 1541
AB - Let $G$ be an ${\Bbb R}$-algebraic semisimple group, $H$ an algebraic ${\Bbb R}$-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 $.
LA - eng
KW - minimal action; uniquely ergodic; homogeneous space; reductive algebraic -group; lattice
UR - http://eudml.org/doc/75329
ER -

References

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  3. [DMa] S.G. DANI and G.A. MARGULIS, Limit Distributions of Orbits of Unipotent Flows and Values of Quadratic Forms, Advances in Soviet Mathematics, 16, Part 1, (1993), 91-137. Zbl0814.22003MR95b:22024
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  6. [MaT] G.A. MARGULIS and G.M. TOMANOV, Measure Rigidity for Almost Linear Groups and its Applications, J. d'Analyse Math., 69 (1996), 25-54. Zbl0864.22005MR98i:22016
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  8. [Mo] S. MOZES, Epimorphic Subgroups and Invariant Measures, Ergodic Theory and Dynamical Systems, Vol. 15, Part 6 (1995), 1207-1210. Zbl0843.28008MR96m:58143
  9. [PR] G. PRASAD and M.S. RAGHUNATHAN, Cartan Subgroups and Lattices in Semisimple Groups, Annals of Math., 96 (1972), 296-317. Zbl0245.22013MR46 #1965
  10. [R] M. RATNER, Invariant Measures and Orbit Closures for Unipotent Actions on Homogeneous Spaces, Geometric and Functional Analysis, 4 (1994), 236-257. Zbl0801.22008MR95c:22018
  11. [St] A.N. STARKOV, Minimal Sets of Homogeneous Flows, Ergodic Theory and Dynamical Systems, 15 (1995), 361-377. Zbl1008.22004MR96k:22023
  12. [V] W.A. VEECH, Unique Ergodicity of Horospherical Flows, American Journal of Mathematics, Vol. 99, 4, 827-859. Zbl0365.28012MR56 #5788
  13. [W] B. WEISS, Finite Dimensional Representations and Subgroup Actions on Homogeneous Spaces, to appear in Israel J. of Math. Zbl0909.22022

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