Lattice actions on the plane revisited
Dynamical Systems
2010-01-28 v1 Group Theory
Abstract
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.
Keywords
Cite
@article{arxiv.1001.4924,
title = {Lattice actions on the plane revisited},
author = {Francois Maucourant and Barak Weiss},
journal= {arXiv preprint arXiv:1001.4924},
year = {2010}
}