Equidistribution of continuous low-lying pairs of horocycles via Ratner’s theorem

Claire Burrin

doi:10.5802/crmath.693

Research article - Dynamical systems

Equidistribution of continuous low-lying pairs of horocycles via Ratner’s theorem

Claire Burrin ¹

¹ Institute of Mathematics, University of Zurich, Switzerland

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1-5.

Abstract (Original language)
Résumé

We record an alternative proof of a recent joint equidistribution result of Blomer and Michel, based on Ratner’s topological rigidity theorem. This approach has the advantage of extending to non-uniform lattices.

Nous présentons une preuve alternative d’un résultat récent d’équidistribution jointe de Blomer et Michel, basée sur le théorème de rigidité topologique de Ratner. Cette approche a l’avantage de s’étendre aux réseaux non uniformes.

Received: 2024-08-01
Revised: 2024-09-23
Accepted: 2024-09-28
Published online: 2025-03-06

DOI: 10.5802/crmath.693

Classification: 37A10, 37A44, 22F30
Keywords: Joint equidistribution, measure classification, commensurator, Hecke operators, matrix coefficients
Mots-clés : Équidistribution jointe, classification des mesures, commensurateur, opérateurs de Hecke, coefficients matriciels

Author's affiliations:

Claire Burrin ¹

¹ Institute of Mathematics, University of Zurich, Switzerland

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMATH_2025__363_G1_1_0,
     author = {Claire Burrin},
     title = {Equidistribution of continuous low-lying pairs of horocycles via {Ratner{\textquoteright}s} theorem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1--5},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.693},
     language = {en},
}

TY  - JOUR
AU  - Claire Burrin
TI  - Equidistribution of continuous low-lying pairs of horocycles via Ratner’s theorem
JO  - Comptes Rendus. Mathématique
PY  - 2025
SP  - 1
EP  - 5
VL  - 363
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.693
LA  - en
ID  - CRMATH_2025__363_G1_1_0
ER  -

%0 Journal Article
%A Claire Burrin
%T Equidistribution of continuous low-lying pairs of horocycles via Ratner’s theorem
%J Comptes Rendus. Mathématique
%D 2025
%P 1-5
%V 363
%I Académie des sciences, Paris
%R 10.5802/crmath.693
%G en
%F CRMATH_2025__363_G1_1_0

Claire Burrin. Equidistribution of continuous low-lying pairs of horocycles via Ratner’s theorem. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1-5. doi : 10.5802/crmath.693. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.693/

References
Cited by

[1] Valentin Blomer; Philippe Gabriel Michel The unipotent mixing conjecture, J. Anal. Math., Volume 151 (2023), pp. 25-57 | DOI | MR | Zbl

[2] Claire Burrin; Uri Shapira; Shucheng Yu Translates of rational points along expanding closed horocycles on the modular surface, Math. Ann., Volume 382 (2022) no. 1, pp. 655-717 | DOI | MR | Zbl

[3] Laurent Clozel; Hee Oh; Emmanuel Ullmo Hecke operators and equidistribution of Hecke points, Invent. Math., Volume 144 (2001) no. 2, pp. 327-380 | DOI | MR | Zbl

[4] Manfred Leopold Einsiedler; Elon Lindenstrauss Joinings of higher rank torus actions on homogeneous spaces, Publ. Math., Inst. Hautes Étud. Sci., Volume 129 (2019), pp. 83-127 | DOI | MR | Zbl

[5] Dennis A. Hejhal On value distribution properties of automorphic functions along closed horocycles, XVIth Rolf Nevanlinna Colloquium (Joensuu, 1995), Walter de Gruyter, 1996, pp. 39-52 | MR | Zbl

[6] Grigoriĭ Aleksandrovich Margulis Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 17, Springer, 1991, x+388 pages | DOI | MR

[7] Jens Marklof Spectral theta series of operators with periodic bicharacteristic flow, Ann. Inst. Fourier, Volume 57 (2007), pp. 2401-2427 | DOI | Numdam | MR | Zbl

[8] Jens Marklof Horospheres and Farey fractions, Dynamical numbers—Interplay between dynamical systems and number theory (Contemporary Mathematics), Volume 532, American Mathematical Society, 2010, pp. 97-106 | DOI | MR | Zbl

[9] Jens Marklof; Andreas Strömbergsson Equidistribution of Kronecker sequences along closed horocycles, Geom. Funct. Anal., Volume 13 (2003) no. 6, pp. 1239-1280 | DOI | MR | Zbl

[10] Toshitsune Miyake Modular forms, Springer, 1989, x+335 pages | DOI | MR

[11] Marina Ratner Raghunathan’s topological conjecture and distributions of unipotent flows, Duke Math. J., Volume 63 (1991), pp. 235-281 | MR | Zbl

[12] Peter Clive Sarnak; Adrián Ubis The horocycle flow at prime times, J. Math. Pures Appl., Volume 103 (2015) no. 2, pp. 575-618 | DOI | MR | Zbl

[13] Nimish A. Shah Limit distribution of expanding translates of certain orbits on homogeneous spaces, Proc. Indian Acad. Sci., Math. Sci., Volume 106 (1996), pp. 105-125 | DOI | MR | Zbl

[14] Andreas Strömbergsson On the uniform equidistribution of long closed horocycles, Duke Math. J., Volume 123 (2004), pp. 507-547 | MR | Zbl

Cited by Sources:

Comments - Policy