Skip to main content
SpringerLink
Log in

A Haar component for quantum limits on locally symmetric spaces

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We prove lower bounds for the entropy of limit measures associated to non-degenerate sequences of eigenfunctions on locally symmetric spaces of non-positive curvature. In the case of certain compact quotients of the space of positive definite n × n matrices (any quotient for n = 3, quotients associated to inner forms in general), measure classification results then show that the limit measures must have a Haar component. This is consistent with the conjecture that the limit measures are absolutely continuous.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. Anantharaman, Entropy and the localization of eigenfunctions, Annals of Mathematics, Second Series 168 (2008), 435–475.

    Article  MathSciNet  MATH  Google Scholar 

  2. N. Anantharaman, Exponential decay for products of Fourier integral operators, Methods and Applications of Analysis 18 (2011), 165–181.

    Article  MathSciNet  Google Scholar 

  3. N. Anantharaman, S. Nonnenmacher, Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold Université de Grenoble, Annales de l’Institut Fourier 57 (2007), 2465–2523.

    Article  MathSciNet  MATH  Google Scholar 

  4. N. Anantharaman, H. Koch and S. Nonnenmacher, Entropy of eigenfunctions, in New Trends of Mathematical Physics, selected contributions of the 15th International Congress on Mathematical Physics, Springer, Berlin, 2009, 1–22.

    Chapter  Google Scholar 

  5. J. Bourgain and E. Lindenstrauss, Entropy of quantum limits, Communications in Mathematical Physics 233 (2003), 153–171.

    Article  MathSciNet  MATH  Google Scholar 

  6. U. Bunke and M. Olbrich, On quantum ergodicity for vector bundles, Acta Applicandae Mathematicae 90 (2006), 19–41.

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Dimassi, J. Sjöstrand, Spectral Asymptotics in the Semi-classical Limit, London Mathematical Society Lecture Note Series 268, Cambridge University Press, Cambridge, 1999.

    Book  MATH  Google Scholar 

  8. M. Einsiedler and A. Katok, Invariant measures on G/γ for split simple Lie groups G, Dedicated to the memory of Jürgen K. Moser, Communications on Pure and Applied Mathematics 56 (2003), 1184–1221.

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Einsiedler, A. Katok and E. Lindenstrauss, Invariant measures and the set of exceptions to Littlewood’s conjecture, Annals of Mathematics, Second Series 164 (2006), 513–560.

    Article  MathSciNet  MATH  Google Scholar 

  10. R. Godement, Introduction à la théorie des groupes de Lie, Tome 2, Publications Mathématiques de l’Université Paris VII 12, Université de Paris VII, U.E.R. de Mathématiques, Paris, 1982.

    Google Scholar 

  11. S. Helgason, Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions, Pure and Applied Mathematics 113, Academic Press Inc., Orlando, 1984.

    MATH  Google Scholar 

  12. S. Helgason, Geometric Analysis on Symmetric Spaces, Second edition, Mathematical Surveys and Monographs 39, American Mathematical Society, Providence, RI, 2008.

    Google Scholar 

  13. J. Hilgert An ergodic Arnold-Liouville theorem for locally symmetric spaces, in Twenty years of Bialowieza: a Mathematical Anthology, World Scientific Monograph Series in Mathematics 8, World Sci. Publ., Hackensack, NJ, 2005, pp. 163–184.

    Chapter  Google Scholar 

  14. A. Knapp, Lie Groups Beyond an Introduction, Progress in Mathematics 140, Birkhäuser Boston, Inc., Boston, MA, 1996.

    Book  MATH  Google Scholar 

  15. A. Knapp, Representation Theory of Semisimple Groups, An overview based on examples, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1986.

    Google Scholar 

  16. E. Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity, Annals of Mathematics 163 (2006), 165–219.

    Article  MathSciNet  MATH  Google Scholar 

  17. E. Lindenstrauss and B. Weiss, On sets invariant under the action of the diagonal group. Ergodic Theory and Dynamical Systems 21 (2001), 1481–1500.

    Article  MathSciNet  MATH  Google Scholar 

  18. V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Translated from the 1991 Russian original by Rachel Rowen, Pure and Applied Mathematics 139, Academic Press, Inc., Boston, MA, 1994, xii+614 pp.

    Book  MATH  Google Scholar 

  19. G. Rivière, Entropy of semiclassical measures in dimension 2, Duke Mathematical Journal 155 (2010), 271–336

    Article  MathSciNet  MATH  Google Scholar 

  20. G. Rivière, Entropy of semiclassical measures for nonpositively curved surfaces, Annales Henri Pincaré. A Journal of Theoretical and Mathematical Physics 11 (2010), 1085–1116.

    MATH  Google Scholar 

  21. Z. Rudnick and P. Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Communications in Mathematical Physics 161 (1994), 195–213.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Schröder, Patterson-Sullivan distributions for symmetric spaces of the noncompact type, PhD dissertation, Universität Paderborn, 2010.

  23. L. Silberman and A. Venkatesh, On quantum unique ergodicity for locally symmetric spaces, Geometric and Functional Analysis 17 (2007), 960–998.

    Article  MathSciNet  MATH  Google Scholar 

  24. L. Silberman and A. Venkatesh, Entropy bounds for Hecke eigenfunctions on division algebras, Geometric and Functional Analysis, to appear.

  25. K. Soundararajan, Quantum Unique Ergodicity for SL2(Z)\H, Annals of Mathematics, Second Series 172 (2010), 1529–1538.

    MathSciNet  MATH  Google Scholar 

  26. G. Tomanov, Actions of maximal tori on homogeneous spaces, in Rigidity in Dynamics and Geometry (Cambridge, 2000), Springer, Berlin, 2002, pp. 407–424.

    Chapter  Google Scholar 

  27. S. A. Wolpert, The modulus of continuity for γ0(m)\ℍ semi-classical limits, Communications in Mathematical Physics 216 (2001), 313–323.

    Article  MathSciNet  MATH  Google Scholar 

  28. S. Zelditch, Pseudodifferential analysis on hyperbolic surfaces, Journal of Functional Analysis 68 (1986), 72–105.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mathématique, Université d’Orsay Paris XI, 91405, Orsay Cedex, France

    Nalini Anantharaman

  2. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Lior Silberman

Authors
  1. Nalini Anantharaman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lior Silberman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nalini Anantharaman.

Additional information

N.A. was partly supported by the grant ANR-05-JCJC-0107-01. She is also grateful to the Miller Institute for Basic Research in Science at the University of California Berkeley, for supporting her work in the spring of 2009.

L.S. was partly supported by an NSERC Discovery Grant.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Anantharaman, N., Silberman, L. A Haar component for quantum limits on locally symmetric spaces. Isr. J. Math. 195, 393–447 (2013). https://doi.org/10.1007/s11856-012-0133-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-012-0133-x

Keywords

Search

Navigation