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Geometriae Dedicata

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

Lattice actions on the plane revisited

  • François Maucourant
  • Barak WeissEmail author
Original Paper

Abstract

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.

Keywords

Lattice actions Plane Homogeneous Equidistribution Lie groups Infinite measure 

Mathematics Subject Classification (2000)

37A17 22F30 

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References

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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

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