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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[Cassels-Swinnerton-Dyer]
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@article{Cassels-Swinnerton-Dyer, mrkey = {0070653},
author = {Cassels, J. W. S. and Swinnerton-Dyer, H. P. F.},
title = {On the product of three homogeneous linear forms and the indefinite ternary quadratic forms},
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mrclass = {10.0X},
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zblnumber = {0065.27905},
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author = {Dani, S. G. and Margulis, G. A.},
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[Einsiedler-Katok]
M. Einsiedler and A. Katok, "Invariant measures on $G/\Gamma$ for split simple Lie groups $G$," Comm. Pure Appl. Math., vol. 56, iss. 8, pp. 1184-1221, 2003.
@article{Einsiedler-Katok, mrkey = {1989231},
author = {Einsiedler, Manfred and Katok, Anatole},
title = {Invariant measures on {$G/\Gamma$} for split simple {L}ie groups {$G$}},
note = {Dedicated to the memory of J{ü}rgen K. Moser},
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mrnumber = {1989231},
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[EinsiedlerKatokNonsplit]
M. Einsiedler and A. Katok, "Rigidity of measures—the high entropy case and non-commuting foliations," Israel J. Math., vol. 148, pp. 169-238, 2005.
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author = {Einsiedler, Manfred and Katok, Anatole},
title = {Rigidity of measures---the high entropy case and non-commuting foliations},
note = {Probability in mathematics},
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mrclass = {37C85 (37A15 37C40 37D30)},
mrnumber = {2191228},
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doi = {10.1007/BF02775436},
zblnumber = {1097.37017},
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[Einsiedler-Katok-Lindenstrauss]
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.
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[Einsiedler-Lindenstrauss-joining]
M. Einsiedler and E. Lindenstrauss, "Joinings of higher-rank diagonalizable actions on locally homogeneous spaces," Duke Math. J., vol. 138, iss. 2, pp. 203-232, 2007.
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author = {Einsiedler, Manfred and Lindenstrauss, Elon},
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volume = {138},
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pages = {203--232},
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coden = {DUMJAO},
mrclass = {37A17 (22F30 28D15)},
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author = {Einsiedler, Manfred and Lindenstrauss, Elon},
title = {Diagonalizable flows on locally homogeneous spaces and number theory},
booktitle = {International {C}ongress of {M}athematicians. {V}ol. {II}},
pages = {1731--1759},
publisher = {Eur. Math. Soc., Zürich},
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[Low-entropy] M. Einsiedler and E. Lindenstrauss, "On measures invariant under diagonalizable actions: the rank-one case and the general low-entropy method," J. Mod. Dyn., vol. 2, iss. 1, pp. 83-128, 2008.
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volume = {2},
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mrnumber = {2366231},
mrreviewer = {Dmitry Y. Kleinbock},
zblnumber = {1146.37006},
} -
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@incollection{Pisa-notes, mrkey = {2648695},
author = {Einsiedler, Manfred and Lindenstrauss, Elon},
title = {Diagonal actions on locally homogeneous spaces},
booktitle = {Homogeneous Flows, Moduli Spaces and Arithmetic},
series = {Clay Math. Proc.},
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publisher = {Amer. Math. Soc.},
address = {Providence, RI},
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} -
[ELMV1]
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[ELMV3]
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zblnumber = {1248.37009},
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[Furstenberg-disjointness-1967]
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[Johnson-invariant-measures]
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[KatokSpatzier96]
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.
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[new16]
A. Katok and R. J. Spatzier, "Corrections to: “Invariant measures for higher-rank hyperbolic abelian actions”," Ergodic Theory Dynam. Systems, vol. 18, iss. 2, pp. 503-507, 1998.
@article {new16, MRKEY={1619571},
AUTHOR = {Katok, A. and Spatzier, R. J.},
TITLE = {Corrections to: ``{I}nvariant measures for higher-rank hyperbolic abelian actions''},
JOURNAL = {Ergodic Theory Dynam. Systems},
FJOURNAL = {Ergodic Theory and Dynamical Systems},
VOLUME = {18},
YEAR = {1998},
NUMBER = {2},
PAGES = {503--507},
ISSN = {0143-3857},
MRCLASS = {58F11},
MRNUMBER = {1619571},
DOI = {10.1017/S0143385798110969},
} -
[Kleinbock-nondense-orbits]
D. Y. Kleinbock, "Nondense orbits of flows on homogeneous spaces," Ergodic Theory Dynam. Systems, vol. 18, iss. 2, pp. 373-396, 1998.
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volume = {18},
year = {1998},
number = {2},
pages = {373--396},
issn = {0143-3857},
mrclass = {58F11 (22E40)},
mrnumber = {1619563},
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doi = {10.1017/S0143385798100408},
zblnumber = {0915.58078},
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[Lindenstrauss-03]
E. Lindenstrauss, "Invariant measures and arithmetic quantum unique ergodicity," Ann. of Math., vol. 163, iss. 1, pp. 165-219, 2006.
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author = {Lindenstrauss, Elon},
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mrnumber = {2195133},
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zblnumber = {1104.22015},
} -
[Lyons-2-and-3]
R. Lyons, "On measures simultaneously $2$- and $3$-invariant," Israel J. Math., vol. 61, iss. 2, pp. 219-224, 1988.
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title = {Oppenheim conjecture},
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[Margulis-Tomanov]
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[Margulis-Tomanov-almost-linear]
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[Ratner-factors]
M. Ratner, "Factors of horocycle flows," Ergodic Theory Dynam. Systems, vol. 2, iss. 3-4, pp. 465-489 (1983), 1982.
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number = {3-4},
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doi = {10.2307/2007014},
zblnumber = {0506.58030},
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zblnumber = {0745.28010},
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[Ratner-Annals]
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volume = {134},
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number = {3},
pages = {545--607},
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} -
[Ratner-Duke]
M. Ratner, "Raghunathan’s topological conjecture and distributions of unipotent flows," Duke Math. J., vol. 63, iss. 1, pp. 235-280, 1991.
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author = {Ratner, Marina},
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fjournal = {Duke Mathematical Journal},
volume = {63},
year = {1991},
number = {1},
pages = {235--280},
issn = {0012-7094},
coden = {DUMJAO},
mrclass = {22E40 (22D40 28D10)},
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doi = {10.1215/S0012-7094-91-06311-8},
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[Ratner-padic]
M. Ratner, "Raghunathan’s conjectures for Cartesian products of real and $p$-adic Lie groups," Duke Math. J., vol. 77, iss. 2, pp. 275-382, 1995.
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[Rees-example] M. Rees, Some ${R}^2$-Anosov flows, 1982.
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[Rudolph-2-and-3]
D. J. Rudolph, "$\times 2$ and $\times 3$ invariant measures and entropy," Ergodic Theory Dynam. Systems, vol. 10, iss. 2, pp. 395-406, 1990.
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series = {Progr. Math.},
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edition = {Second},
publisher = {Birkhäuser},
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[Tomanov-orbits] G. Tomanov, "Orbits on homogeneous spaces of arithmetic origin and approximations," in Analysis on Homogeneous Spaces and Representation Theory of Lie Groups, Tokyo: Math. Soc. Japan, 2000, vol. 26, pp. 265-297.
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title = {Orbits on homogeneous spaces of arithmetic origin and approximations},
booktitle = {Analysis on Homogeneous Spaces and Representation Theory of {L}ie Groups},
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[Tomanov] G. Tomanov, "Actions of maximal tori on homogeneous spaces," in Rigidity in Dynamics and Geometry, New York: Springer-Verlag, 2002, pp. 407-424.
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author = {Tomanov, George},
title = {Actions of maximal tori on homogeneous spaces},
booktitle = {Rigidity in Dynamics and Geometry},
venue = {{C}ambridge, 2000},
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publisher = {Springer-Verlag},
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address = {New York},
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[Waldschmidt]
M. Waldschmidt, "Open Diophantine problems," Mosc. Math. J., vol. 4, iss. 1, pp. 245-305, 312, 2004.
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title = {Open {D}iophantine problems},
journal = {Mosc. Math. J.},
fjournal = {Moscow Mathematical Journal},
volume = {4},
year = {2004},
number = {1},
pages = {245--305, 312},
issn = {1609-3321},
mrclass = {11Jxx (11Dxx 11G50 14G05)},
mrnumber = {2074991},
mrreviewer = {Damien Roy},
zblnumber = {1066.11030},
url = {http://www.ams.org/distribution/mmj/vol4-1-2004/abst4-1-2004.html#waldschmidt_abstract},