Geometriae Dedicata

, Volume 157, Issue 1, pp 1–21 | Cite as

Lattice actions on the plane revisited

  • François Maucourant
  • Barak WeissEmail author
Original Paper


We study the action of a lattice Γ in the group G = SL(2, R) on the plane. We obtain a formula which simultaneously describes visits of an orbit Γu to either a fixed ball, or an expanding or contracting family of annuli. We also discuss the ‘shrinking target problem’. Our results are valid for an explicitly described set of initial points: all \({{\bf u} \in {\bf R}^2}\) in the case of a cocompact lattice, and all u satisfying certain diophantine conditions in case \({\Gamma = {\rm SL}(2, \mathbb {Z})}\) . The proofs combine the method of Ledrappier with effective equidistribution results for the horocycle flow on \({\Gamma {\backslash} G}\) due to Burger, Strömbergsson, Forni and Flaminio.


Lattice actions Plane Homogeneous Equidistribution Lie groups Infinite measure 

Mathematics Subject Classification (2000)

37A17 22F30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Burger M.: Horocycle flow on geometrically finite surfaces. Duke Math. J. 61(3), 779–803 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Flaminio L., Forni G.: Invariant distributions and time averages for horocycle flows. Duke Math. J. 119(3), 465–526 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Gorodnik A., Weiss B.: Distribution of lattice orbits on homogeneous varieties. Geom. Funct. An. 17, 58–115 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn.Google Scholar
  5. 5.
    Laurent, M., Nogueira, A.: Approximation to points in the plane by \({{\rm SL}(2,\mathbb {Z})}\) -orbits, preprint, available at
  6. 6.
    Ledrappier F.: Distribution des orbites des réseaux sur le plan réel. C.R. Acad. Sci. Paris Sr. I Math. 329(1), 61–64 (1999)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Maucourant F.: Homogeneous asymptotic limits of Haar measures of semisimple linear groups and their lattices. Duke Math. J. 136(2), 357–399 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Nogueira A.: Orbit distribution on \({\mathbb{R}^2}\) under the natural action of \({{\rm SL}(2,\mathbb{Z})}\) . Indag. Math. (N.S.) 13(1), 103–124 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Nogueira, A.: Lattice orbit distribution on \({\mathbb{R}^2}\) , to appear in Ergodic Theory and Dynamical SystemsGoogle Scholar
  10. 10.
    Strömbergsson, A.: On the deviation of ergodic averages for horocycle flows. Preprint, available at

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Université Rennes I, IRMARRennes cedexFrance
  2. 2.Ben Gurion UniversityBe’er ShevaIsrael

Personalised recommendations