Lattice actions on the plane revisited

François Maucourant, Barak Weiss

Research output: Contribution to journalArticlepeer-review

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 u ∈ R 2 in the case of a cocompact lattice, and all u satisfying certain diophantine conditions in case Γ = SL(2, ℤ). The proofs combine the method of Ledrappier with effective equidistribution results for the horocycle flow on Γ\G due to Burger, Strömbergsson, Forni and Flaminio.

Original languageEnglish
Pages (from-to)1-21
Number of pages21
JournalGeometriae Dedicata
Volume157
Issue number1
DOIs
StatePublished - Apr 2012
Externally publishedYes

Keywords

  • Equidistribution
  • Homogeneous
  • Infinite measure
  • Lattice actions
  • Lie groups
  • Plane

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