Lattice actions on the plane revisited

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.