Lattice actions on the plane revisited

TL;DR

This paper investigates the behavior of lattice actions in SL(2,R) on the plane, providing formulas for orbit visits to various sets and discussing the shrinking target problem, with results applicable to specific initial points.

Contribution

It introduces explicit formulas for orbit visits under lattice actions and extends understanding of the shrinking target problem for certain initial vectors.

Findings

Derived formulas for orbit visits to fixed and expanding/contracting sets.

Established results for nonzero vectors in cocompact lattices.

Extended analysis to vectors satisfying Diophantine conditions in SL(2,Z).

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.