Mesures stationnaires et fermés invariants des espaces homogénes

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 $X=\mathbb{T}^d$.

  • [Blan] Go to document F. Blanchard, "$K$-flots et théorème de renouvellement," Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, vol. 36, iss. 4, pp. 345-358, 1976.
    @article {Blan, MRKEY = {0417386},
      AUTHOR = {Blanchard, Fran{ç}ois},
      TITLE = {{$K$}-flots et théorème de renouvellement},
      JOURNAL = {Z. Wahrscheinlichkeitstheorie und Verw. Gebiete},
      VOLUME = {36},
      YEAR = {1976},
      NUMBER = {4},
      PAGES = {345--358},
      MRCLASS = {28A65 (60K05)},
      MRNUMBER = {0417386},
      MRREVIEWER = {Isaac Meilijson},
      DOI = {10.1007/BF00532699},
      ZBLNUMBER = {0328.60036},
      }
  • [BL] P. Bougerol and J. Lacroix, Products of Random Matrices with Applications to Schrödinger Operators, Boston, MA: Birkhäuser, 1985, vol. 8.
    @book {BL, MRKEY = {0886674},
      AUTHOR = {Bougerol, Philippe and Lacroix, Jean},
      TITLE = {Products of Random Matrices with Applications to {S}chrödinger Operators},
      SERIES = {Progress in Probability and Statistics},
      VOLUME = {8},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {1985},
      PAGES = {xii+283},
      ISBN = {0-8176-3324-3},
      MRCLASS = {60B15 (47F05 58G40 60H25 82A42)},
      MRNUMBER = {0886674},
      MRREVIEWER = {Shinichi Kotani},
      ZBLNUMBER = {0572.60001},
     }
  • [BFLM] Go to document J. Bourgain, A. Furman, E. Lindenstrauss, and S. Mozes, "Invariant measures and stiffness for non-abelian groups of toral automorphisms," C. R. Math. Acad. Sci. Paris, vol. 344, iss. 12, pp. 737-742, 2007.
    @article {BFLM, MRKEY = {2340439},
      AUTHOR = {Bourgain, Jean and Furman, Alex and Lindenstrauss, Elon and Mozes, Shahar},
      TITLE = {Invariant measures and stiffness for non-abelian groups of toral automorphisms},
      JOURNAL = {C. R. Math. Acad. Sci. Paris},
      FJOURNAL = {Comptes Rendus Mathématique. Académie des Sciences. Paris},
      VOLUME = {344},
      YEAR = {2007},
      NUMBER = {12},
      PAGES = {737--742},
      ISSN = {1631-073X},
      MRCLASS = {37A15 (22F10 28D15)},
      MRNUMBER = {2340439},
      MRREVIEWER = {Bryna Kra},
      DOI = {10.1016/j.crma.2007.04.017},
      ZBLNUMBER = {1119.43001},
      }
  • [Buf] Go to document A. I. Bufetov, "Convergence of spherical averages for actions of free groups," Ann. of Math., vol. 155, iss. 3, pp. 929-944, 2002.
    @article {Buf, MRKEY = {1923970},
      AUTHOR = {Bufetov, Alexander I.},
      TITLE = {Convergence of spherical averages for actions of free groups},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {155},
      YEAR = {2002},
      NUMBER = {3},
      PAGES = {929--944},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {37A15 (28D05 28D15 47A35)},
      MRNUMBER = {1923970},
      MRREVIEWER = {A. H. Dooley},
      DOI = {10.2307/3062137},
      ZBLNUMBER = {1028.37001},
      }
  • [ChOr] Go to document R. V. Chacon and D. S. Ornstein, "A general ergodic theorem," Illinois J. Math., vol. 4, pp. 153-160, 1960.
    @article {ChOr, MRKEY = {0110954},
      AUTHOR = {Chacon, R. V. and Ornstein, D. S.},
      TITLE = {A general ergodic theorem},
      JOURNAL = {Illinois J. Math.},
      FJOURNAL = {Illinois Journal of Mathematics},
      VOLUME = {4},
      YEAR = {1960},
      PAGES = {153--160},
      ISSN = {0019-2082},
      MRCLASS = {46.00},
      MRNUMBER = {0110954},
      MRREVIEWER = {J. T. Schwartz},
      URL = {http://projecteuclid.org/getRecord?id=euclid.ijm/1255455860},
      ZBLNUMBER = {0134.12102},
      }
  • [COU] Go to document L. Clozel, H. Oh, and E. Ullmo, "Hecke operators and equidistribution of Hecke points," Invent. Math., vol. 144, iss. 2, pp. 327-351, 2001.
    @article {COU, MRKEY = {1827734},
      AUTHOR = {Clozel, Laurent and Oh, Hee and Ullmo, Emmanuel},
      TITLE = {Hecke operators and equidistribution of {H}ecke points},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {144},
      YEAR = {2001},
      NUMBER = {2},
      PAGES = {327--351},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11F72},
      MRNUMBER = {1827734},
      MRREVIEWER = {Stefan K{ü}hnlein},
      DOI = {10.1007/s002220100126},
      ZBLNUMBER = {1144.11301},
      }
  • [Co] J. Conze, "Extensions de systèmes dynamiques par des endomorphismes de groupes compacts," Ann. Inst. H. Poincaré Sect. B, vol. 8, pp. 33-66, 1972.
    @article {Co, MRKEY = {0310194},
      AUTHOR = {Conze, Jean-Pierre},
      TITLE = {Extensions de systèmes dynamiques par des endomorphismes de groupes compacts},
      JOURNAL = {Ann. Inst. H. Poincaré Sect. B},
      VOLUME = {8},
      YEAR = {1972},
      PAGES = {33--66},
      MRCLASS = {28A65 (58F99)},
      MRNUMBER = {0310194},
      MRREVIEWER = {Robert Azencott},
      ZBLNUMBER = {0274.46035},
      }
  • [EKL] Go to document M. Einsiedler, A. Katok, and E. Lindenstrauss, "Invariant measures and the set of exceptions to Littlewood’s conjecture," Ann. of Math., vol. 164, iss. 2, pp. 513-560, 2006.
    @article {EKL, MRKEY = {2247967},
      AUTHOR = {Einsiedler, Manfred and Katok, Anatole and Lindenstrauss, Elon},
      TITLE = {Invariant measures and the set of exceptions to {L}ittlewood's conjecture},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {164},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {513--560},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {22F10 (11J83 28A80 28D15 37A15)},
      MRNUMBER = {2247967},
      MRREVIEWER = {Thomas Ward},
      DOI = {10.4007/annals.2006.164.513},
      ZBLNUMBER = {1109.22004},
      }
  • [EM] A. Eskin and G. Margulis, "Recurrence properties of random walks on finite volume homogeneous manifolds," in Random Walks and Geometry, Walter de Gruyter GmbH & Co. KG, Berlin, 2004, pp. 431-444.
    @incollection {EM, MRKEY = {2087794},
      AUTHOR = {Eskin, Alex and Margulis, Gregory},
      TITLE = {Recurrence properties of random walks on finite volume homogeneous manifolds},
      BOOKTITLE = {Random Walks and Geometry},
      PAGES = {431--444},
      PUBLISHER = {Walter de Gruyter GmbH \& Co. KG, Berlin},
      YEAR = {2004},
      MRCLASS = {22F30 (37A99 60G50)},
      MRNUMBER = {2087794},
      MRREVIEWER = {Dmitry Y. Kleinbock},
      ZBLNUMBER = {1064.60092},
      }
  • [EMS] Go to document A. Eskin, S. Mozes, and N. Shah, "Unipotent flows and counting lattice points on homogeneous varieties," Ann. of Math., vol. 143, iss. 2, pp. 253-299, 1996.
    @article {EMS, MRKEY = {1381987},
      AUTHOR = {Eskin, Alex and Mozes, Shahar and Shah, Nimish},
      TITLE = {Unipotent flows and counting lattice points on homogeneous varieties},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {143},
      YEAR = {1996},
      NUMBER = {2},
      PAGES = {253--299},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {22E40 (11E57 11P21)},
      MRNUMBER = {1381987},
      MRREVIEWER = {Alexander Starkov},
      DOI = {10.2307/2118644},
      ZBLNUMBER = {0852.11054},
      }
  • [EO1] Go to document A. Eskin and H. Oh, "Ergodic theoretic proof of equidistribution of Hecke points," Ergodic Theory Dynam. Systems, vol. 26, iss. 1, pp. 163-167, 2006.
    @article {EO1, MRKEY = {2201942},
      AUTHOR = {Eskin, Alex and Oh, Hee},
      TITLE = {Ergodic theoretic proof of equidistribution of {H}ecke points},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {26},
      YEAR = {2006},
      NUMBER = {1},
      PAGES = {163--167},
      ISSN = {0143-3857},
      MRCLASS = {11F70 (22E40 37A15 37A45)},
      MRNUMBER = {2201942},
      MRREVIEWER = {Alexander Gorodnik},
      DOI = {10.1017/S0143385705000428},
      ZBLNUMBER = {1092.11023},
      }
  • [Fur] Go to document A. Furman, "Random walks on groups and random transformations," in Handbook of Dynamical Systems, Vol. 1 A, Amsterdam: North-Holland, 2002, pp. 931-1014.
    @incollection {Fur, MRKEY = {1928529},
      AUTHOR = {Furman, Alex},
      TITLE = {Random walks on groups and random transformations},
      BOOKTITLE = {Handbook of Dynamical Systems, {V}ol. 1{{\rm A}}},
      PAGES = {931--1014},
      PUBLISHER = {North-Holland},
      ADDRESS = {Amsterdam},
      YEAR = {2002},
      MRCLASS = {60G50 (37A50 37H15)},
      MRNUMBER = {1928529},
      MRREVIEWER = {Gernot Greschonig},
      DOI = {10.1016/S1874-575X(02)80014-5},
      ZBLNUMBER = {1053.60045},
      }
  • [Fu63] Go to document H. Furstenberg, "Noncommuting random products," Trans. Amer. Math. Soc., vol. 108, pp. 377-428, 1963.
    @article {Fu63, MRKEY = {0163345},
      AUTHOR = {Furstenberg, H.},
      TITLE = {Noncommuting random products},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {108},
      YEAR = {1963},
      PAGES = {377--428},
      ISSN = {0002-9947},
      MRCLASS = {60.08 (60.66)},
      MRNUMBER = {0163345},
      MRREVIEWER = {G.-C. Rota},
      DOI = {10.2307/1993589},
      ZBLNUMBER = {0203.19102},
      }
  • [Fu98] H. Furstenberg, "Stiffness of group actions," in Lie Groups and Ergodic Theory, Bombay, 1998, pp. 105-117.
    @inproceedings {Fu98, MRKEY = {1699360},
      AUTHOR = {Furstenberg, H.},
      TITLE = {Stiffness of group actions},
      BOOKTITLE = {Lie Groups and Ergodic Theory},
      VENUE={{M}umbai, 1996},
      SERIES = {Tata Inst. Fund. Res. Stud. Math.},
      VOLUME = {14},
      PAGES = {105--117},
      PUBLISHER = {Tata Inst. Fund. Res.},
      ADDRESS = {Bombay},
      YEAR = {1998},
      MRCLASS = {22D99 (28D15)},
      MRNUMBER = {1699360},
      MRREVIEWER = {Adolf R. Mirotin},
      ZBLNUMBER = {0942.22006},
      }
  • [FuK] Go to document H. Furstenberg and H. Kesten, "Products of random matrices," Ann. Math. Statist., vol. 31, pp. 457-469, 1960.
    @article {FuK, MRKEY = {0121828},
      AUTHOR = {Furstenberg, H. and Kesten, H.},
      TITLE = {Products of random matrices},
      JOURNAL = {Ann. Math. Statist.},
      FJOURNAL = {Annals of Mathematical Statistics},
      VOLUME = {31},
      YEAR = {1960},
      PAGES = {457--469},
      ISSN = {0003-4851},
      MRCLASS = {60.00},
      MRNUMBER = {0121828},
      MRREVIEWER = {R. E. Bellman},
      DOI = {10.1214/aoms/1177705909},
      ZBLNUMBER = {0137.35501},
      }
  • [GoMa] Go to document Y. I. Gol$’$dsheid and G. A. Margulis, "Lyapunov exponents of a product of random matrices," Uspekhi Mat. Nauk, vol. 44, iss. 5(269), pp. 13-60, 1989.
    @article {GoMa, MRKEY = {1040268},
      AUTHOR = {Gol{$'$}dsheid, I. Ya. and Margulis, G. A.},
      TITLE = {Lyapunov exponents of a product of random matrices},
      JOURNAL = {Uspekhi Mat. Nauk},
      FJOURNAL = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
      NOTE={in Russian; translated in {\it Russian Math. Surveys} {\bf 44} (1989), 11--71},
      VOLUME = {44},
      YEAR = {1989},
      NUMBER = {5(269)},
      PAGES = {13--60},
      ISSN = {0042-1316},
      MRCLASS = {60B15 (22E30 47E05 60H25 82B44)},
      MRNUMBER = {1040268},
      MRREVIEWER = {Philippe Bougerol},
      DOI = {10.1070/RM1989v044n05ABEH002214},
      ZBLNUMBER = {0687.60008},
     }
  • [GR] Go to document Y. Guivarc$’$h and A. Raugi, "Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence," Z. Wahrsch. Verw. Gebiete, vol. 69, iss. 2, pp. 187-242, 1985.
    @article {GR, MRKEY = {0779457},
      AUTHOR = {Guivarc$'$h, Yves and Raugi, Albert},
      TITLE = {Frontière de {F}urstenberg, propriétés de contraction et théorèmes de convergence},
      JOURNAL = {Z. Wahrsch. Verw. Gebiete},
      FJOURNAL = {Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete},
      VOLUME = {69},
      YEAR = {1985},
      NUMBER = {2},
      PAGES = {187--242},
      ISSN = {0044-3719},
      CODEN = {ZWVGAA},
      MRCLASS = {60J15 (22E30 60B15)},
      MRNUMBER = {0779457},
      MRREVIEWER = {Ren{é} Schott},
      DOI = {10.1007/BF02450281},
      ZBLNUMBER = {0558.60009},
      }
  • [GR07] Go to document Y. Guivarc$’$h and A. Raugi, "Actions of large semigroups and random walks on isometric extensions of boundaries," Ann. Sci. École Norm. Sup., vol. 40, iss. 2, pp. 209-249, 2007.
    @article {GR07, MRKEY = {2339285},
      AUTHOR = {Guivarc$'$h, Yves and Raugi, Albert},
      TITLE = {Actions of large semigroups and random walks on isometric extensions of boundaries},
      JOURNAL = {Ann. Sci. École Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {40},
      YEAR = {2007},
      NUMBER = {2},
      PAGES = {209--249},
      ISSN = {0012-9593},
      CODEN = {ASENAH},
      MRCLASS = {60B15 (22F30 28D15)},
      MRNUMBER = {2339285},
      MRREVIEWER = {C. R. E. Raja},
      DOI = {10.1016/j.ansens.2007.01.003},
      ZBLNUMBER = {1166.60301},
      }
  • [GS] Go to document A. Guivarc$’$h and A. N. Starkov, "Orbits of linear group actions, random walks on homogeneous spaces and toral automorphisms," Ergodic Theory Dynam. Systems, vol. 24, iss. 3, pp. 767-802, 2004.
    @article {GS, MRKEY = {2060998},
      AUTHOR = {Guivarc$'$h, Albert and Starkov, A. N.},
      TITLE = {Orbits of linear group actions, random walks on homogeneous spaces and toral automorphisms},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {24},
      YEAR = {2004},
      NUMBER = {3},
      PAGES = {767--802},
      ISSN = {0143-3857},
      MRCLASS = {37C85 (22E40 37A15 60B11 60B15 60G50)},
      MRNUMBER = {2060998},
      MRREVIEWER = {Mikhail S. Kulikov},
      DOI = {10.1017/S0143385703000440},
      ZBLNUMBER = {1050.37012},
      }
  • [KS] Go to document A. Katok and R. J. Spatzier, "Invariant measures for higher-rank hyperbolic abelian actions," Ergodic Theory Dynam. Systems, vol. 16, iss. 4, pp. 751-778, 1996.
    @article {KS, MRKEY = {1406432},
      AUTHOR = {Katok, A. and Spatzier, R. J.},
      TITLE = {Invariant measures for higher-rank hyperbolic abelian actions},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {16},
      YEAR = {1996},
      NUMBER = {4},
      PAGES = {751--778},
      ISSN = {0143-3857},
      MRCLASS = {58F11},
      MRNUMBER = {1406432},
      MRREVIEWER = {Scot Adams},
      DOI = {10.1017/S0143385700009081},
      ZBLNUMBER = {0859.58021},
      }
  • [Kec] Go to document A. S. Kechris, "Countable sections for locally compact group actions," Ergodic Theory Dynam. Systems, vol. 12, iss. 2, pp. 283-295, 1992.
    @article {Kec, MRKEY = {1176624},
      AUTHOR = {Kechris, Alexander S.},
      TITLE = {Countable sections for locally compact group actions},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {12},
      YEAR = {1992},
      NUMBER = {2},
      PAGES = {283--295},
      ISSN = {0143-3857},
      MRCLASS = {22D40 (03E15 54H11)},
      MRNUMBER = {1176624},
      MRREVIEWER = {Ivan L. Reilly},
      DOI = {10.1017/S0143385700006751},
      ZBLNUMBER = {0761.28014},
      }
  • [Lall] Go to document S. P. Lalley, "Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits," Acta Math., vol. 163, iss. 1-2, pp. 1-55, 1989.
    @article {Lall, MRKEY = {1007619},
      AUTHOR = {Lalley, Steven P.},
      TITLE = {Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits},
      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {163},
      YEAR = {1989},
      NUMBER = {1-2},
      PAGES = {1--55},
      ISSN = {0001-5962},
      CODEN = {ACMAA8},
      MRCLASS = {58F20 (20H10 22E40 30F40 54F50 54H20 58F03 58F17)},
      MRNUMBER = {1007619},
      MRREVIEWER = {Caroline Series},
      DOI = {10.1007/BF02392732},
      ZBLNUMBER = {0701.58021},
     }
  • [LY1] Go to document F. Ledrappier and L. -S. Young, "The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula," Ann. of Math., vol. 122, iss. 3, pp. 509-539, 1985.
    @article {LY1, MRKEY = {0819556},
      AUTHOR = {Ledrappier, F. and Young, L.-S.},
      TITLE = {The metric entropy of diffeomorphisms. {I}. {C}haracterization of measures satisfying {P}esin's entropy formula},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {122},
      YEAR = {1985},
      NUMBER = {3},
      PAGES = {509--539},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {58F11 (58F15)},
      MRNUMBER = {0819556},
      MRREVIEWER = {D. Newton},
      DOI = {10.2307/1971328},
      ZBLNUMBER = {0605.58028},
     }
  • [LY2] Go to document F. Ledrappier and L. -S. Young, "The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension," Ann. of Math., vol. 122, iss. 3, pp. 540-574, 1985.
    @article {LY2, MRKEY = {0819557},
      AUTHOR = {Ledrappier, F. and Young, L.-S.},
      TITLE = {The metric entropy of diffeomorphisms. {II}. {R}elations between entropy, exponents and dimension},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {122},
      YEAR = {1985},
      NUMBER = {3},
      PAGES = {540--574},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {58F11 (58F15)},
      MRNUMBER = {0819557},
      MRREVIEWER = {D. Newton},
      DOI = {10.2307/1971329},
      ZBLNUMBER = {0605.58028},
     }
  • [LS] Go to document F. Ledrappier and J. Strelcyn, "A proof of the estimation from below in Pesin’s entropy formula," Ergodic Theory Dynam. Systems, vol. 2, iss. 2, pp. 203-219 (1983), 1982.
    @article {LS, MRKEY = {0693976},
      AUTHOR = {Ledrappier, Fran{ç}ois and Strelcyn, Jean-Marie},
      TITLE = {A proof of the estimation from below in {P}esin's entropy formula},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {2},
      YEAR = {1982},
      NUMBER = {2},
      PAGES = {203--219 (1983)},
      ISSN = {0143-3857},
      MRCLASS = {58F11 (28D20 34C35)},
      MRNUMBER = {0693976},
      MRREVIEWER = {M. I. Brin},
      DOI = {10.1017/S0143385700001528},
      ZBLNUMBER = {0533.58022},
      }
  • [LP] Go to document &. Le Page, "Théorèmes limites pour les produits de matrices aléatoires," in Probability Measures on Groups, New York, 1982, pp. 258-303.
    @inproceedings {LP, MRKEY = {0669072},
      AUTHOR = {Le Page, {É}mile},
      TITLE = {Théorèmes limites pour les produits de matrices aléatoires},
      BOOKTITLE = {Probability Measures on Groups},
      VENUE={{O}berwolfach, 1981},
      SERIES = {Lecture Notes in Math.},
      VOLUME = {928},
      PAGES = {258--303},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1982},
      MRCLASS = {60B15 (60F05 60F10)},
      MRNUMBER = {0669072},
      MRREVIEWER = {Albert Raugi},
      ZBLNUMBER = {0506.60019},
      DOI = {10.1007/BFb0093229},
     }
  • [M0] G. Margulis, "Problems and conjectures in rigidity theory," in Mathematics: Frontiers and Perspectives, Providence, RI: Amer. Math. Soc., 2000, pp. 161-174.
    @incollection {M0, MRKEY = {1754775},
      AUTHOR = {Margulis, Gregory},
      TITLE = {Problems and conjectures in rigidity theory},
      BOOKTITLE = {Mathematics: Frontiers and Perspectives},
      PAGES = {161--174},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2000},
      MRCLASS = {22E40 (37C85 37D20 53C24)},
      MRNUMBER = {1754775},
      MRREVIEWER = {A. I. Danilenko},
      ZBLNUMBER = {0952.22005},
      }
  • [MT] Go to document G. Margulis and G. M. Tomanov, "Invariant measures for actions of unipotent groups over local fields on homogeneous spaces," Invent. Math., vol. 116, iss. 1-3, pp. 347-392, 1994.
    @article {MT, MRKEY = {1253197},
      AUTHOR = {Margulis, Gregory and Tomanov, G. M.},
      TITLE = {Invariant measures for actions of unipotent groups over local fields on homogeneous spaces},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {116},
      YEAR = {1994},
      NUMBER = {1-3},
      PAGES = {347--392},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {22E40 (11E57 20G25 22D40)},
      MRNUMBER = {1253197},
      MRREVIEWER = {Nimish A. Shah},
      DOI = {10.1007/BF01231565},
      ZBLNUMBER = {0816.22004},
      }
  • [Mu] R. Muchnik, "Semigroup actions on $\Bbb T^n$," Geom. Dedicata, vol. 110, pp. 1-47, 2005.
    @article {Mu, MRKEY = {2136018},
      AUTHOR = {Muchnik, Roman},
      TITLE = {Semigroup actions on {$\Bbb T\sp n$}},
      JOURNAL = {Geom. Dedicata},
      FJOURNAL = {Geometriae Dedicata},
      VOLUME = {110},
      YEAR = {2005},
      PAGES = {1--47},
      ISSN = {0046-5755},
      CODEN = {GEMDAT},
      MRCLASS = {37B05 (20M20)},
      MRNUMBER = {2136018},
      MRREVIEWER = {Daniel Berend},
      ZBLNUMBER = {1071.37008},
     }
  • [Ra1] Go to document M. Ratner, "On Raghunathan’s measure conjecture," Ann. of Math., vol. 134, iss. 3, pp. 545-607, 1991.
    @article {Ra1, MRKEY = {1135878},
      AUTHOR = {Ratner, Marina},
      TITLE = {On {R}aghunathan's measure conjecture},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      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] Go to document M. Ratner, "Raghunathan’s topological conjecture and distributions of unipotent flows," Duke Math. J., vol. 63, iss. 1, pp. 235-280, 1991.
    @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},
      }
  • [Ro49] V. A. Rokhlin, "On the fundamental ideas of measure theory," Mat. Sbornik N.S., vol. 25(67), pp. 107-150, 1949.
    @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] Go to document V. A. Rokhlin, "Lectures on the entropy theory of measure-preserving transformations," Russian Math. Surveys, vol. 22, pp. 1-52, 1967.
    @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.
    @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.
    @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},
      }

Authors

Yves Benoist

CNRS -- Université; Paris-Sud
91405 Orsay
France

Jean-François Quint

CNRS -- Université; Paris-Nord
93 430 Villetaneuse
France