Israel Journal of Mathematics

, Volume 210, Issue 1, pp 245–295 | Cite as

Escape of mass and entropy for diagonal flows in real rank one situations



Let G be a connected semisimple Lie group of real rank 1 with finite center, let Γ be a non-uniform lattice in G and a any diagonalizable element in G. We investigate the relation between the metric entropy of a acting on the homogeneous space Γ\G and escape of mass. Moreover, we provide bounds on the escaping mass and, as an application, we show that the Hausdorff dimension of the set of orbits (under iteration of a) which miss a fixed open set is not full.


Symmetric Space Homogeneous Space Haar Measure Fundamental Domain Height Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BK83]
    M. Brin and A. Katok, On local entropy, in Geometric Dynamics (Rio de Janeiro, 1981), Lecture Notes in Mathematics, Vol. 1007, Springer, Berlin, 1983, pp. 30–38.CrossRefGoogle Scholar
  2. [CDKR91]
    M. Cowling, A. Dooley, A. Korányi and F. Ricci, H-type groups and Iwasawa decompositions, Advances in Mathematics 87 (1991), 1–41.MathSciNetCrossRefMATHGoogle Scholar
  3. [CDKR98]
    M. Cowling, A. Dooley, A. Korányi and F. Ricci, An approach to symmetric spaces of rank one via groups of Heisenberg type, Journal of Geometric Analysis 8 (1998), 199–237.MathSciNetCrossRefMATHGoogle Scholar
  4. [Dan84]
    S. G. Dani, On orbits of unipotent flows on homogeneous spaces, Ergodic Theory and Dynamical Systems 4 (1984), 25–34.MathSciNetCrossRefMATHGoogle Scholar
  5. [EK12]
    M. Einsiedler and S. Kadyrov, Entropy and escape of mass for SL3(ℤ)\SL3(ℝ), Israel Journal of Mathematics 190 (2012), 253–288.MathSciNetCrossRefMATHGoogle Scholar
  6. [EL10]
    M. Einsiedler and E. Lindenstrauss, Diagonal actions on locally homogeneous spaces, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Mathematics Proceedings, Vol. 10, American Mathematical Society, Providence, RI, 2010, pp. 155–241.Google Scholar
  7. [ELMV12]
    M. Einsiedler, E. Lindenstrauss, Ph. Michel and A. Venkatesh, The distribution of closed geodesics on the modular surface, and Duke’s theorem, L’Enseignement Mathématique 58 (2012), 249–313.MathSciNetCrossRefMATHGoogle Scholar
  8. [ELW]
    M. Einsiedler, E. Lindenstrauss and T. Ward, Entropy in ergodic theory and homogeneous dynamics, book project,
  9. [GR70]
    H. Garland and M. S. Raghunathan, Fundamental domains for lattices in ℝ-rank 1 semisimple Lie groups, Annals of Mathematics 92 (1970), 279–326.MathSciNetCrossRefMATHGoogle Scholar
  10. [Hel00]
    S. Helgason, Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions., Mathematical Surveys and Monographas, Vol. 83, American Mathematical Society, Providence, RI, 2000.Google Scholar
  11. [Hel01]
    S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Graduate Studies in Mathematics, Vol. 34, American Mathematical Society, Providence, RI, 2001.MATHGoogle Scholar
  12. [HW13]
    P. Hubert and B. Weiss, Ergodicity for infinite periodic translation surfaces, Compositio Mathematica 149 (2013), 1364–1380.MathSciNetCrossRefMATHGoogle Scholar
  13. [Kad12]
    S. Kadyrov, Entropy and escape of mass for Hilbert modular spaces, Journal of Lie Theory 22 (2012), no. 3, 701–722.MathSciNetMATHGoogle Scholar
  14. [KKLM]
    S. Kadyrov, D. Y. Kleinbock, E. Lindenstrauss and G. A. Margulis Singular systems of linear forms and nonescape of mass in the space of lattices, Journal d’Analyse Mathematique, to appear.Google Scholar
  15. [KP]
    S. Kadyrov and A. Pohl, Amount of failure of upper-semicontinuity of entropy in noncompact rank one situations, and Hausdorff dimension, Scholar
  16. [MT94]
    G. A. Margulis and G. M. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Inventiones Mathematicae 116 (1994), 347–392.MathSciNetCrossRefMATHGoogle Scholar
  17. [Poh10]
    A. Pohl, Ford fundamental domains in symmetric spaces of rank one, Geometriae Dedicata 147 (2010), 219–276.MathSciNetCrossRefMATHGoogle Scholar
  18. [Shi12]
    R. Shi, Convergence of measures under diagonal actions on homogeneous spaces, Advances in Mathematics 229 (2012), 1417–1434.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2015

Authors and Affiliations

  1. 1.Departement MathematikETH ZürichZürichSwitzerland
  2. 2.School of MathematicsUniversity of BristolBristolUK
  3. 3.Mathematisches InstitutGeorg-August-Universität GöttingenGöttingenGermany

Personalised recommendations