Équidistribution des orbites toriques sur les espaces homogènes [d'après M. Einsiedler, E. Lindenstrauss, Ph. Michel, A. Venkatesh]

Breuillard, Emmanuel

Séminaire Bourbaki : volume 2008/2009 exposés 997-1011 - Avec table par noms d'auteurs de 1848/49 à 2008/09, Astérisque, no. 332 (2010), Exposé no. 1008, 35 p.

Zbl | 1 citation dans Numdam

@incollection{AST_2010__332__305_0,
     author = {Breuillard, Emmanuel},
     title = {\'Equidistribution des orbites toriques sur les espaces homog\`enes [d'apr\`es {M.} {Einsiedler,} {E.} {Lindenstrauss,} {Ph.} {Michel,} {A.} {Venkatesh]}},
     booktitle = {S\'eminaire Bourbaki : volume 2008/2009 expos\'es 997-1011  - Avec table par noms d'auteurs de 1848/49 \`a 2008/09},
     series = {Ast\'erisque},
     note = {talk:1008},
     pages = {305--339},
     year = {2010},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {332},
     zbl = {1209.22005},
     language = {fr},
     url = {https://www.numdam.org/item/AST_2010__332__305_0/}
}

TY  - CHAP
AU  - Breuillard, Emmanuel
TI  - Équidistribution des orbites toriques sur les espaces homogènes [d'après M. Einsiedler, E. Lindenstrauss, Ph. Michel, A. Venkatesh]
BT  - Séminaire Bourbaki : volume 2008/2009 exposés 997-1011  - Avec table par noms d'auteurs de 1848/49 à 2008/09
AU  - Collectif
T3  - Astérisque
N1  - talk:1008
PY  - 2010
SP  - 305
EP  - 339
IS  - 332
PB  - Société mathématique de France
UR  - https://www.numdam.org/item/AST_2010__332__305_0/
LA  - fr
ID  - AST_2010__332__305_0
ER  -

%0 Book Section
%A Breuillard, Emmanuel
%T Équidistribution des orbites toriques sur les espaces homogènes [d'après M. Einsiedler, E. Lindenstrauss, Ph. Michel, A. Venkatesh]
%B Séminaire Bourbaki : volume 2008/2009 exposés 997-1011  - Avec table par noms d'auteurs de 1848/49 à 2008/09
%A Collectif
%S Astérisque
%Z talk:1008
%D 2010
%P 305-339
%N 332
%I Société mathématique de France
%U https://www.numdam.org/item/AST_2010__332__305_0/
%G fr
%F AST_2010__332__305_0

Breuillard, Emmanuel. Équidistribution des orbites toriques sur les espaces homogènes [d'après M. Einsiedler, E. Lindenstrauss, Ph. Michel, A. Venkatesh], dans Séminaire Bourbaki : volume 2008/2009 exposés 997-1011  - Avec table par noms d'auteurs de 1848/49 à 2008/09, Astérisque, no. 332 (2010), Exposé no. 1008, 35 p.. https://www.numdam.org/item/AST_2010__332__305_0/

Bibliographie
Cité par

[1] Y. Benoist & H. Oh - Equidistribution of rational matrices in their conjugacy classes, Geom. Fund. Anal. 17 (2007), p. 1-32. | Zbl | DOI

[2] Y. Benoist & H. Oh, Effective equidistribution of $S$ -integral points on symmetric varieties, prépublication arXiv:0706.1621. | Zbl | Numdam | DOI

[3] N. Bergeron - Spectre des surfaces hyperboliques, livre en préparation. | Zbl

[4] V. Blomer, G. Harcos & P. Michel - Bounds for modular $L$ -functions in the level aspect, Ann. Sci. École Norm. Sup. 40 (2007), p. 697-740. | Zbl | EuDML | DOI

[5] A. Borel - Introduction aux groupes arithmétiques, Publications de l'Institut de Mathématique de l'Université de Strasbourg, XV. Actualités Scientifiques et Industrielles, No. 1341, Hermann, 1969. | Zbl

[6] R. Bowen - Periodic orbits for hyperbolic flows, Amer. J. Math. 94 (1972), p. 1-30. | Zbl | DOI

[7] D. A. Burgess - On character sums and $L$ -series. II, Proc. London Math. Soc. 13 (1963), p. 524-536. | Zbl | DOI

[8] J. W. S. Cassels - The product of $n$ inhomogeneous linear forms, J. London Math. Soc. (1952). | Zbl

[9] J. W. S. Cassels & H. P. F. Swinnerton-Dyer - On the product of three homogeneous linear forms and the indefinite ternary quadratic forms, Philos. Trans. Roy. Soc. London. Ser. A. 248 (1955), p. 73-96. | Zbl | DOI

[10] L. Clozel & E. Ullmo - Équidistribution de mesures algébriques, Compos. Math. 141 (2005), p. 1255-1309. | Zbl | DOI

[11] L. Clozel & E. Ullmo, Équidistribution de sous-variétés spéciales, Ann. of Math. 161 (2005), p. 1571-1588. | Zbl | DOI

[12] P. B. Cohen - Hyperbolic equidistribution problems on Siegel $3$ -folds and Hilbert modular varieties, Duke Math. J. 129 (2005), p. 87-127. | Zbl | DOI

[13] S. G. Dani & G. A. Margulis - Limit distributions of orbits of unipotent flows and values of quadratic forms, in I.M. Gel'fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., 1993, p. 91-137. | Zbl | DOI

[14] W. Duke - Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math. 92 (1988), p. 73-90. | Zbl | EuDML | DOI

[15] W. Duke, J. B. Friedlander & H. Iwaniec - Bounds for automorphic $L$ -functions, Invent Math. 112 (1993), p. 1-8. | Zbl | EuDML | DOI

[16] W. Duke, J. B. Friedlander & H. Iwaniec, The subconvexity problem for Artin $L$ -functions, Invent. Math. 149 (2002), p. 489-577. | Zbl | DOI

[17] M. Einsiedler - A simplified proof of Ratner's measure classification for the action of subgroups isomorphic to $S L (2, ℝ)$ , prépublication, 2008.

[18] M. Einsiedler & A. Katok - Invariant measures on $G / Γ$ for split simple Lie groups $G$ , Comm. Pure Appl. Math. 56 (2003), p. 1184-1221. | Zbl | DOI

[19] M. Einsiedler & A. Katok, Rigidity of measures-the high entropy case and non-commuting foliations, Israel J. Math. 148 (2005), p. 169-238. | Zbl | DOI

[20] M. Einsiedler, A. Katok & E. Lindenstrauss - Invariant measures and the set of exceptions to Littlewood's conjecture, Ann. of Math. 164 (2006), p. 513-560. | MR | Zbl | DOI

[21] M. Einsiedler & E. Lindenstrauss - Diagonalizable flows on locally homogeneous spaces and number theory, in Proceedings of the International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zurich, 2006, p. 1731-1759. | MR | Zbl

[22] M. Einsiedler, E. Lindenstrauss P. Michel & A. Venkatesh, - Distribution of periodic torus orbits on homogeneous spaces I, Duke Math. J. 148 (2009), p. 119-174. | MR | Zbl | DOI

[23] M. Einsiedler, E. Lindenstrauss P. Michel & A. Venkatesh, Distribution of periodic torus orbits on homogeneous spaces II : Duke's theorem for quadratic fields, en préparation. | MR | Zbl

[24] M. Einsiedler, E. Lindenstrauss P. Michel & A. Venkatesh, Distribution of periodic torus orbits on homogeneous spaces III : Duke's theorem for cubic fields, preprint, arXiv:math.0708.1113. | MR | Zbl

[25] M. Einsiedler & T. Ward - Ergodic theory with a view towards number theory, livre à paraître dans la collection Springer Graduate Texts in Math. | MR | Zbl

[26] A. Eskin - Counting problems and semisimple groups, in Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, p. 539-552. | MR | Zbl | EuDML

[27] A. Eskin & C. Mcmullen - Mixing, counting, and equidistribution in Lie groups, Duke Math. J. 71 (1993), p. 181-209. | MR | Zbl | DOI

[28] A. Eskin, S. Mozes & N. A. Shah - Unipotent flows and counting lattice points on homogeneous varieties, Ann. of Math. 143 (1996), p. 253-299. | MR | Zbl | DOI

[29] A. Eskin & H. Oh - Representations of integers by an invariant polynomial and unipotent flows, Duke Math. J. 135 (2006), p. 481-506. | MR | Zbl | DOI

[30] W. T. Gan & H. Oh - Equidistribution of integer points on a family of homogeneous varieties : a problem of Linnik, Compositio Math. 136 (2003), p. 323-352. | MR | Zbl | DOI

[31] É. Ghys - Dynamique des flots unipotents sur les espaces homogènes, Séminaire Bourbaki, vol. 1991/92, exposé n° 747, Astérisque 206 (1992), p. 93-136. | MR | Zbl | EuDML | Numdam

[32] A. Gorodnik, F. Maucourant & H. Oh - Manin's and Peyre's conjectures on rational points and adelic mixing, Ann. Sci. Éc. Norm. Super. 41 (2008), p. 383-435. | MR | Zbl | EuDML | Numdam

[33] A. Gorodnik & H. Oh - Rational points on homogeneous varieties and equidistribution of adelic periods, prépublication arXiv:0803. 1996. | MR

[34] G. Harcos - Equidistribution and $L$ -functions, Lecture Notes Clay Math. School, Pisa 2007.

[35] G. Harcos & P. Michel - The subconvexity problem for Rankin-Selberg $L$ -functions and equidistribution of Heegner points. II, Invent. Math. 163 (2006), p. 581-655. | MR | Zbl | DOI

[36] B. Host - Nombres normaux, entropie, translations, Israel J. Math. 91 (1995), p. 419-428. | MR | Zbl | DOI

[37] A. Katok & R. J. Spatzier - Invariant measures for higher-rank hyperbolic abelian actions, Ergodic Theory Dynam. Systems 16 (1996), p. 751-778. | MR | Zbl | DOI

[38] B. Klingler & A. Yafaev - On the André-Oort conjecture, prépublication, 2006. | Zbl

[39] S. Lang - Algebraic number theory, 2e éd., Graduate Texts in Math., vol. 110, Springer, 1994. | MR | Zbl

[40] F. Ledrappier & L.-S. Young - The metric entropy of diffeomorphisms. I and II, Ann. of Math. 122 (1985), p. 509-574. | MR | Zbl

[41] E. Lindenstrauss - Rigidity of multiparameter actions, Israel J. Math. 149 (2005), p. 199-226. | MR | Zbl | DOI

[42] E. Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity, Ann. of Math. 163 (2006), p. 165-219. | MR | Zbl | DOI

[43] E. Lindenstrauss & B. Weiss - On sets invariant under the action of the diagonal group, Ergodic Theory Dynam. Systems 21 (2001), p. 1481-1500. | MR | Zbl | DOI

[44] Y. Linnik - Ergodic properties of number fields, Springer, 1970. | MR

[45] G. A. Margulis - Oppenheim conjecture, in Fields Medallists' lectures, World Sci. Ser. 20th Century Math., vol. 5, World Sci. Publ., River Edge, NJ, 1997, p. 272-327. | MR | DOI

[46] G. A. Margulis, Problems and conjectures in rigidity theory, in Mathematics : frontiers and perspectives, Amer. Math. Soc., 2000, p. 161-174. | MR | Zbl

[47] G. A. Margulis, On some aspects of the theory of Anosov systems, Springer Monographs in Math., Springer, 2004. | MR | Zbl

[48] G. A. Margulis & G. M. Tomanov - Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math. 116 (1994), p. 347-392. | MR | Zbl | EuDML | DOI

[49] F. Maucourant - A non-homogeneous orbit closure of a diagonal subgroup, prépublication arXiv:0707.2920, à paraître dans Annals of Math. | MR | Zbl

[50] P. Michel & A. Venkatesh - Equidistribution, $L$ -functions and ergodic theory : on some problems of Yu. Linnik, in Proceedings of the International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, p. 421-457. | MR | Zbl

[51] P. Michel & A. Venkatesh, Subconvexity for $G L_{2}$ , prépublication, 2009.

[52] S. Mozes - Actions of Cartan subgroups, Israel J. Math. 90 (1995), p. 253-294. | MR | Zbl | DOI

[53] J. Neukirch - Algebraic number theory, Grund. Math. Wiss., vol. 322, Springer, 1999. | MR | Zbl

[54] H. Oh - Finiteness of compact maximal flats of bounded volume, Ergodic Theory Dynam. Systems 24 (2004), p. 217-225. | MR | Zbl | DOI

[55] H. Oh, Orbital counting via mixing and unipotent flows, in Scuola Normale Superiore Summer school, Pisa 2007, Clay Math. Proceedings, vol. 8, 2008. | MR | Zbl

[56] V. Platonov & A. Rapinchuk - Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press Inc., 1994. | MR | Zbl

[57] A. A. Popa - Central values of Rankin $L$ -series over real quadratic fields, Compos. Math. 142 (2006), p. 811-866. | MR | Zbl | DOI

[58] M. S. Raghunathan - Discrete subgroups of Lie groups, Ergebnisse Math. Grenzg., vol. 68, Springer, 1972. | MR | Zbl

[59] M. Ratner - Factors of horocycle flows, Ergodic Theory Dynam. Systems 2 (1982), p. 465-489. | MR | Zbl | DOI

[60] M. Ratner, Rigidity of horocycle flows, Ann. of Math. 115 (1982), p. 597-614. | MR | Zbl | DOI

[61] M. Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math. 118 (1983), p. 277-313. | MR | Zbl | DOI

[62] M. Ratner, Distribution rigidity for unipotent actions on homogeneous spaces, Bull. Amer. Math. Soc. 24 (1991), p. 321-325. | MR | Zbl | DOI

[63] M. Ratner, On Raghunathan's measure conjecture, Ann. of Math. 134 (1991), p. 545-607. | MR | Zbl | DOI

[64] M. Ratner, Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J. 63 (1991), p. 235-280. | MR | Zbl

[65] D. J. Rudolph - $\times 2$ and $\times 3$ invariant measures and entropy, Ergodic Theory Dynam. Systems 10 (1990), p. 395-406. | MR | Zbl | DOI

[66] P. C. Sarnak - Diophantine problems and linear groups, in Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, 1991, p. 459-471. | MR | Zbl

[67] C. Series - On coding geodesics with continued fractions, in Ergodic theory (Sem., Les Plans-sur-Bex, 1980), Monograph. Enseign. Math., vol. 29, Univ. Genève, 1981, p. 67-76. | MR | Zbl

[68] N. A. Shah - Uniformly distributed orbits of certain flows on homogeneous spaces, Math. Ann. 289 (1991), p. 315-334. | MR | Zbl | EuDML | DOI

[69] U. Shapira - A solution to a problem of Cassels and Diophantine properties of cubic fields, prépublication, 2008. | Zbl

[70] G. M. Tomanov - Values of decomposable forms at $S$ -integral points and orbits of tori on homogeneous spaces, Duke Math. J. 138 (2007), p. 533-562. | MR | Zbl | DOI

[71] G. M. Tomanov & B. Weiss - Closed orbits for actions of maximal tori on homogeneous spaces, Duke Math. J. 119 (2003), p. 367-392. | MR | Zbl | DOI

[72] E. Ullmo & A. Yafaev - Galois orbits and equidistribution of special subvarieties : towards the André-Oort conjecture, prépublication, 2006.

[73] A. Venkatesh - The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture, Bulletin of the A.M.S. 45 (2008), p. 117-134(électronique). | MR | Zbl | DOI

[74] A. Venkatesh, Sparse equidistribution problems, period bounds, and subconvexity, pré-publication arXiv:math.NT/0506224. | MR | Zbl | DOI

[75] J.-L. Waldspurger - Sur les valeurs de certaines fonctions $L$ automorphes en leur centre de symétrie, Compositio Math. 54 (1985), p. 173-242. | MR | Zbl | EuDML | Numdam

[76] F. Wlelonsky - Séries d'Eisenstein, intégrales toroïdales et une formule de Hecke, Enseign. Math. 31 (1985), p. 93-135. | MR | Zbl

[77] S. Zelditch - Selberg trace formulae and equidistribution theorems for closed geodesies and Laplace eigenfunctions : finite area surfaces, Mem. Amer. Math. Soc. 96 (1992). | MR | Zbl

[78] S.-W. Zhang - Equidistribution of $C M$ -points on quaternion Shimura varieties, Int. Math. Res. Not. 2005 (2005), p. 3657-3689. | MR | Zbl | DOI