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Equidistribution in S-arithmetic and adelic spaces
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 23 (2014) no. 5, pp. 1023-1048.

We give an introduction to adelic mixing and its applications for mathematicians knowing about the mixing of the geodesic flow on hyperbolic surfaces. We focus on the example of the Hecke trees in the modular surface.

Cet article présente une introduction au mélange adélique et ses applications. La présentation faite est pensée pour les mathématiciens connaissant le mélange du flot géodésique sur les surfaces hyperboliques. L’accent est principalement mis sur l’exemple des arbres de Hecke dans la surface modulaire.

Published online:
DOI: 10.5802/afst.1434
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     title = {Equidistribution in $S$-arithmetic and adelic spaces},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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Antonin Guilloux. Equidistribution in $S$-arithmetic and adelic spaces. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 23 (2014) no. 5, pp. 1023-1048. doi : 10.5802/afst.1434. https://afst.centre-mersenne.org/articles/10.5802/afst.1434/

[1] Aka (M.) and Shapira (U.).— On the evolution of continued fractions in a fixed quadratic field. arXiv preprint arXiv:1201.1280 (2012).

[2] Benoist (Y.) and Quint (J.-F.).— Mesures stationnaires et fermés invariants des espaces homognes. Ann. of Math. (2), 174(2), p. 1111-1162 (2011). | MR: 2831114 | Zbl: 1241.22007

[3] Clozel (L.).— Démonstration de la conjecture τ. Invent. Math., 151(2), p. 297-328 (2003). | MR: 1953260 | Zbl: 1025.11012

[4] Clozel (L.), Oh (H.), and Ullmo (E.).— Hecke operators and equidistribution of Hecke points. Invent. Math., 144(2), p. 327-351 (2001). | MR: 1827734 | Zbl: 1144.11301

[5] Duke (W.), Rudnick (Z.), and Sarnak (P.).— Density of integer points on affine homogeneous varieties. Duke Math. J., 71(1), p. 143-179 (1993). | MR: 1230289 | Zbl: 0798.11024

[6] Dani (S. G.) and Smillie (J.).— Uniform distribution of horocycle orbits for Fuchsian groups. Duke Math. J., 51(1), p. 185-194 (1984). | MR: 744294 | Zbl: 0547.20042

[7] Einsiedler (M.), Katok (A.), and Lindenstrauss (E.).— Invariant measures and the set of exceptions to LittlewoodÕs conjecture. Ann. of Math. (2), 164(2), p. 513-560 (2006). | MR: 2247967 | Zbl: 1109.22004

[8] Einsiedler (M.), Lindenstrauss (E.), Michel (P.), and Venkatesh (A.).— Distribution of periodic torus orbits and DukeÕs theorem for cubic fields. Ann. of Math. (2), 173(2), p. 815-885 (2011). | MR: 2776363 | Zbl: 1248.37009

[9] Eskin (A.) and McMullen (C.).— Mixing, counting, and equidistribution in Lie groups. Duke Math. J., 71(1), p. 181-209 (1993). | MR: 1230290 | Zbl: 0798.11025

[10] Eskin (A.), Mozes (S.), and Shah (N.).— Unipotent flows and counting lattice points on homogeneous varieties. Ann. of Math. (2), 143(2), p. 253-299 (1996). | MR: 1381987 | Zbl: 0852.11054

[11] Eskin (A.) and Oh (H.).— Ergodic theoretic proof of equidistribution of Hecke points. Ergodic Theory Dynam. Systems, 26(1), p. 163-167 (2006). | MR: 2201942 | Zbl: 1092.11023

[12] Goldstein (D.) and Mayer (A.).— On the equidistribution of Hecke points. Forum Math., 15(2), p. 165-189 (2003). | MR: 1956962 | Zbl: 1048.11057

[13] Gorodnik (A.), Maucourant (F.), and Oh (H.).— Manin’s and Peyre’s conjectures on rational points and adelic mixing. Ann. Sci. Éc. Norm. Supér. (4), 41(3), p. 383-435 (2008). | Numdam | MR: 2482443 | Zbl: 1161.14015

[14] Guilloux (A.).— Existence et équidistribution des matrices de dénominateur n dans les groupes unitaires et orthogonaux. Ann. Inst. Fourier (Grenoble), 58(4), p. 1185-1212 (2008). | Numdam | MR: 2427958 | Zbl: 1149.11017

[15] Guilloux (A.).— A brief remark on orbits of SL(2,) in the Euclidean plane. Ergodic Theory Dynam. Systems, 30(4), p. 1101-1109 (2010). | MR: 2669411 | Zbl: 1203.37010

[16] Guilloux (A.).— Polynomial dynamic and lattice orbits in S-arithmetic homogeneous spaces. Confluentes Math., 2(1), p. 1-35 (2010). | MR: 2649235 | Zbl: 1193.37007

[17] Gorodnik (A.) and Weiss (B.).— Distribution of lattice orbits on homogeneous varieties. Geom. Funct. Anal., 17(1), p. 58-115 (2007). | MR: 2306653 | Zbl: 1112.37001

[18] Ledrappier (F.).— Distribution des orbites des réseaux sur le plan réel. C. R. Acad. Sci. Paris Sér. I Math., 329(1), p. 61-64 (1999). | MR: 1703338 | Zbl: 0928.22012

[19] Linnik (Yu. V.).— Ergodic properties of algebraic fields. Translated from the Russian by M. S. Keane. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 45. Springer-Verlag New York Inc., New York (1968). | MR: 238801 | Zbl: 0162.06801

[20] Lindenstrauss (E.).— Invariant measures and arithmetic quantum unique ergodicity. Ann. of Math. (2), 163(1), p. 165-219 (2006). | MR: 2195133 | Zbl: 1104.22015

[21] Margulis (G. A.).— Certain applications of ergodic theory to the investigation of manifolds of negative curvature. Func. Anal. Appl., 4, p. 333-335 (1969). | MR: 257933 | Zbl: 0207.20305

[22] Maucourant (F.).— Homogeneous asymptotic limits of Haar measures of semisimple linear groups and their lattices. Duke Math. J., 136(2), p. 357-399 (2007). | MR: 2286635 | Zbl: 1117.22006

[23] Margulis (G. A.) and Tomanov (G. M.).— Invariant measures for actions of unipotent groups over local fields on homogeneous spaces. Invent. Math., 116(1-3), p. 347-392 (1994). | EuDML: 144192 | MR: 1253197 | Zbl: 0816.22004

[24] Maucourant (F.) and Weiss (B.).— Lattice actions on the plane revisited. Geom. Dedicata, 157, p. 1-21 (2012). | MR: 2893477 | Zbl: 1252.37004

[25] Nogueira (A.).— Orbit distribution on R 2 under the natural action of SL(2; Z ). Indag. Math. (N.S.), 13(1), p. 103-124 (2002). | MR: 2014978 | Zbl: 1016.37003

[26] Platonov (V.) and Rapinchuk (A.).— Algebraic groups and number theory, volume 139 of Pure and Applied Mathematics. Academic Press Inc., Boston, MA (1994). Translated from the 1991 Russian original by Rachel Rowen. | MR: 1278263 | Zbl: 0841.20046

[27] Ratner (M.).— Invariant measures and orbit closures for unipotent actions on homogeneous spaces. Geom. Funct. Anal., 4(2), p. 236-257 (1994). | EuDML: 58162 | MR: 1262705 | Zbl: 0801.22008

[28] Sarnak (P. C.).— Diophantine problems and linear groups. In Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), pages 459-471, Tokyo (1991). Math. Soc. Japan. | MR: 1159234 | Zbl: 0743.11018

[29] Serre (J.-P.).— A course in arithmetic. Springer-Verlag, New York (1973). Translated from the French, Graduate Texts in Mathematics, No. 7. | MR: 344216 | Zbl: 0256.12001

[30] Serre (J.-P.).— Trees. Springer Monographs in Mathematics. Springer-Verlag, Berlin (2003). Translated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation. | MR: 607504 | Zbl: 1013.20001

[31] Skubenko (B. F.).— The asymptotic distribution of integers on a hyperboloid of one sheet and ergodic theorems. Izv. Akad. Nauk SSSR Ser. Mat., 26, p. 721-752 (1962). | MR: 151436 | Zbl: 0107.04201

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