Lattice actions on the plane revisited

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.