Lattice actions on the plane revisited

Details Lattice actions on the plane revisited Lattice actions on the plane revisited

2010-01-27

arxiv.org/abs/1001.4924v1

Mathematics

Dynamical Systems

Scientific paper

We study the action of a lattice in the group SL(2,R) on the plane. We obtain a formula which simultaneously describes visits of an orbit 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 nonzero vectors in the case of a cocompact lattice, and all vectors satisfying certain diophantine conditions in case SL(2,Z). The proofs combine the method of Ledrappier with effective equidistribution results for the horocycle flow due to Burger, Strombergsson, Forni and Flaminio.

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