On measures invariant under tori on quotients of semisimple groups

Abstract

We classify invariant and ergodic probability measures on arithmetic homogeneous quotients of semisimple $S$-algebraic groups invariant under a maximal split torus in at least one simple local factor and show that the algebraic support of such a measure splits into the product of four homogeneous spaces: a torus, a homogeneous space on which the measure is (up to finite index) the Haar measure, a product of homogeneous spaces on each of which the action degenerates to a rank one action, and a homogeneous space in which every element of the action acts with zero entropy.

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      mrreviewer = {Caroline Series},
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      }
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    @article{Ratner-isomorphisms, mrkey = {0657240},
      author = {Ratner, Marina},
      title = {Rigidity of horocycle flows},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {115},
      year = {1982},
      number = {3},
      pages = {597--614},
      issn = {0003-486X},
      coden = {ANMAAH},
      mrclass = {58F25 (22E40 28D10 58F11)},
      mrnumber = {0657240},
      mrreviewer = {S. G. Dani},
      doi = {10.2307/2007014},
      zblnumber = {0506.58030},
      }
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    @article{Ratner-products, mrkey = {0717825},
      author = {Ratner, Marina},
      title = {Horocycle flows, joinings and rigidity of products},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {118},
      year = {1983},
      number = {2},
      pages = {277--313},
      issn = {0003-486X},
      coden = {ANMAAH},
      mrclass = {58F17 (22E40)},
      mrnumber = {0717825},
      mrreviewer = {S. G. Dani},
      doi = {10.2307/2007030},
      zblnumber = {0556.28020},
      }
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    @article{Ratner-Acta, mrkey = {1075042},
      author = {Ratner, Marina},
      title = {On measure rigidity of unipotent subgroups of semisimple groups},
      journal = {Acta Math.},
      fjournal = {Acta Mathematica},
      volume = {165},
      year = {1990},
      number = {3-4},
      pages = {229--309},
      issn = {0001-5962},
      coden = {ACMAA8},
      mrclass = {57S20 (22E40 58F25)},
      mrnumber = {1075042},
      mrreviewer = {M. Rees},
      doi = {10.1007/BF02391906},
      zblnumber = {0745.28010},
      }
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    @article{Ratner-Annals, 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},
      }
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    @article{Ratner-Duke, 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{Ratner-padic, mrkey = {1321062},
      author = {Ratner, Marina},
      title = {Raghunathan's conjectures for {C}artesian products of real and {$p$}-adic {L}ie groups},
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      mrclass = {22E40 (22E50)},
      mrnumber = {1321062},
      mrreviewer = {Dave Witte Morris},
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      zblnumber = {0914.22016},
      }
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      author = {Rees, M.},
      title = {Some ${R}^2$-{A}nosov flows},
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      }
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      mrclass = {20G15 (14L10)},
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      title = {Actions of maximal tori on homogeneous spaces},
      booktitle = {Rigidity in Dynamics and Geometry},
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      mrclass = {22E40 (11H46 37C85)},
      mrnumber = {1919414},
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      mrclass = {11Jxx (11Dxx 11G50 14G05)},
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Authors

Manfred Einsiedler

ETH Zürich, Zürich, Switzerland

Elon Lindenstrauss

Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel