Polynomial dynamic and lattice orbits in S-arithmetic homogeneous spaces
Confluentes Mathematici, Volume 2 (2010) no. 1, pp. 1-35.

Consider a homogeneous space under a locally compact group G and a lattice Γ in G. Then the lattice naturally acts on the homogeneous space. Looking at a dense orbit, one may wonder how to describe its repartition. One then adopts a dynamical point of view and compare the asymptotic distribution of points in the orbits with the natural measure on the space. In the setting of Lie groups and their homogeneous spaces, several results show an equidistribution of points in the orbits.

We address here this problem in the setting of p-adic and S-arithmetic groups.

Published online:
DOI: 10.1142/S1793744210000120

Antonin Guilloux 1

1
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Antonin Guilloux. Polynomial dynamic and lattice orbits in $S$-arithmetic homogeneous spaces. Confluentes Mathematici, Volume 2 (2010) no. 1, pp. 1-35. doi : 10.1142/S1793744210000120. https://cml.centre-mersenne.org/articles/10.1142/S1793744210000120/

[1] Y. Benoist and H. Oh, Effective equidistribution of S-integral points on symmetric varieties , arXiv:0706.1621 .

[2] A. Borel , Linear Algebraic Groups , 2nd edn. , Graduate Texts in Mathematics 126 ( Springer-Verlag , 1991 ) .

[3] S. G. Dani and S. Raghavan, Israel J. Math. 36, 300 (1980), DOI: 10.1007/BF02762053 .

[4] S. R. Dani, Ergod. Th. Dynam. Syst. 2, 139 (1982).

[5] J. Denef, Séminaire de Théorie des Nombres, Paris 1983-84, Progr. Math. 59 (Birkhäuser, 1985) pp. 25-47.

[6] J. S. Ellenberg and A. Venkatesh, Inven. Math. 171, 257 (2008), DOI: 10.1007/s00222-007-0077-7 .

[7] A. Eskin, S. Mozes and N. Shah, Geom. Funct. Anal. 7, 48 (1997), DOI: 10.1007/PL00001616 .

[8] A. Gorodnik, Duke Math. J. 122, 549 (2004), DOI: 10.1215/S0012-7094-04-12234-1 .

[9] A. Gorodnik and H. Oh, Rational points on homogeneous varieties and equidistribution of adelic periods, preprint .

[10] A. Gorodnik and B. Weiss, Distribution of lattice orbits on homogeneous varieties, Geom. Funct. Anal. to appear .

[11] A. Guilloux, Équidistribution dans les espaces homogènes, PhD thesis, Université Paris 11 Orsay, 2007 .

[12] D. Kleinbock and G. Tomanov, Comment. Math. Helv. 82, 519 (2007).

[13] F. Ledrappier, C. R. Acad. Sci. 329, 61 (1999).

[14] F. Ledrappier and M. Pollicott, Bull. Braz. Math. Soc. 36, 143 (2005).

[15] G. A. Margulis , Discrete Subgroups of Semisimple Lie Groups ( Springer , 1991 ) .

[16] F. Maucourant, Homogeneous asymptotic limits of Haar measures of semisimple Lie groups and their lattices, Duke Math. J. to appear .

[17] A. Nogueira, Indag. Math. 13, 103 (2002), DOI: 10.1016/S0019-3577(02)90009-1 .

[18] V. Platonov and A. Rapinchuk , Algebraic Groups and Number Theory ( Academic Press , 1994 ) .

[19] N. A. Shah, Proc. Indian Acad. Sci. (Math. Sci.) 106, 105 (1996), DOI: 10.1007/BF02837164 .

[20] J. Tits, Proc. Symp. Pure Math. 33 (1979) pp. 20-70.

[21] G. Tomanov, Adv. Stud. Pure Math. 26, 265 (2000).

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