Skip to main content
Log in

Tori Approximation of Families of Diagonally Invariant Measures

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

Let A be the full diagonal group in \({\text {SL}}_{n}(\mathbb {R})\). We study possible limits of Haar measures on periodic A-orbits in the space of unimodular lattices \(X_n\). We prove the existence of non-ergodic measures which are also weak limits of these compactly supported A-invariant measures. In fact, given any countably many A-invariant ergodic measures, we show that there exists a sequence of Haar measures on periodic A-orbits such that the ergodic decomposition of its weak limit has these measures as factors with positive weight. In particular, we prove that any compactly supported A-invariant and ergodic measure is the weak limit of the restriction of different compactly supported periodic measures to a fixed proportion of the time. In addition, for any \(c\in (0,1]\) we find a sequence of Haar measures on periodic A orbits that converges weakly to \(cm_{X_n}\) where \(m_{X_n}\) denotes the Haar measure on \(X_n\). In particular, we prove the existence of partial escape of mass for Haar measures on periodic A orbits. These results give affirmative answers to questions posed by Shapira in [Sha16]. Our proofs are based on a modification of Shapira’s proof in [Sha16] and on a generalization of a construction of Cassels, as well as on effective equidistribution estimates of Hecke neighbors by Clozel, Oh and Ullmo in [COU01].

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1

Similar content being viewed by others

References

  1. Menny Aka and Uri Shapira. On the evolution of continued fractions in a fixed quadratic field. J. Anal. Math., 134(1):335–397, 2018.

    Article  MathSciNet  MATH  Google Scholar 

  2. Yves Benoist and Hee Oh. Equidistribution of rational matrices in their conjugacy classes. GAFA Geometric And Functional Analysis, 17(1):1–32, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  3. J. W. S. Cassels. An introduction to the geometry of numbers. Springer Science & Business Media, 2012.

  4. JWS Cassels. The product of n inhomogeneous linear forms in n variables. Journal of the London Mathematical Society, 1(4):485–492, 1952.

    Article  MathSciNet  Google Scholar 

  5. Laurent Clozel, Hee Oh, and Emmanuel Ullmo. Hecke operators and equidistribution of Hecke points. Invent. Math., 144(2):327–351, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  6. Laurent Clozel and Emmanuel Ullmo. Équidistribution des points de hecke. Contributions to automorphic forms, geometry, and number theory, 193–254, 2004.

  7. Keith Conrad. Hensel’s lemma. Unpublished notes, 2015.

  8. Ofir David and Uri Shapira. Dirichlet shapes of unit lattices and escape of mass. International Mathematics Research Notices, 2018(9):2810–2843, 2018.

    MathSciNet  MATH  Google Scholar 

  9. W. Duke. Hyperbolic distribution problems and half-integral weight maass forms. Inventiones mathematicae, 92(1):73–90, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  10. Manfred Einsiedler, Elon Lindenstrauss, Philippe Michel, and Akshay Venkatesh. Distribution of periodic torus orbits on homogeneous spaces. Duke Mathematical Journal, 148(1):119 – 174, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  11. Manfred Einsiedler, Elon Lindenstrauss, Philippe Michel, and Akshay Venkatesh. (2011) Distribution of periodic torus orbits and duke’s theorem for cubic fields. Annals of mathematics, 815–885, .

  12. Alex Eskin and Hee Oh. Ergodic theoretic proof of equidistribution of Hecke points. Ergodic Theory Dynam. Systems, 26(1):163–167, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  13. Wee Teck Gan and Hee Oh. Equidistribution of integer points on a family of homogeneous varieties: a problem of Linnik. Compositio Math., 136(3):323–352, 2003.

  14. M. Kline. Calculus: an intuitive and physical approach. Courier Corporation, 1998.

  15. ON Solan L. Liao, R. Shi, N. Tamam Hausdorff. dimension of weighted singular vectors in \(\mathbb{R}^{2}\). JEMS, 22(3):833–875,2019.

  16. C. McMullen. Minkowski’s conjecture, well-rounded lattices and topological dimension. Journal of the American Mathematical Society, 18(3):711–734, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  17. Mark Pinsky. Convergence of probability measures (patrick billingsley). SIAM Review, 11(3):414, 1969.

    Article  Google Scholar 

  18. Uri Shapira. Full Escape of Mass for the Diagonal Group. International Mathematics Research Notices, 2017(15):4704–4731 2016.

    MathSciNet  MATH  Google Scholar 

  19. Uri Shapira and Cheng Zheng. Translates of s-arithmetic orbits and applications. arXiv preprintarXiv:2107.05017, 2021.

  20. G. Tomanov and B. Weiss. Closed orbits for actions of maximal tori on homogeneous spaces. Duke Mathematical Journal, 119(2):367–392, 2003.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The second author would like to express his deep gratitude to Uri Shapira for his support and encouragement. Without them, this paper would not exist. We thank Uri Shapira for bringing the main questions answered in this paper to our attention and for many intriguing discussions with him which contributed a lot to this paper. Moreover, the first author thanks Andreas Wieser and Elon Lindenstrauss for many fruitful discussions. The first author thanks the generous donation of Dr. Arthur A. Kaselemas. The second author acknowledges the support of ISF Grants Number 871/17. This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 754475). This work is part of the first author’s Ph.D. thesis.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yuval Yifrach.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Solan, O.N., Yifrach, Y. Tori Approximation of Families of Diagonally Invariant Measures. Geom. Funct. Anal. 33, 1354–1378 (2023). https://doi.org/10.1007/s00039-023-00646-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-023-00646-7

Navigation