Lattice actions on the plane revisited

François Maucourant; Barak Weiss

doi:10.1007/s10711-011-9596-x

Lattice actions on the plane revisited

François Maucourant, Barak Weiss

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

17 Scopus citations

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 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.

Original language	English
Pages (from-to)	1-21
Number of pages	21
Journal	Geometriae Dedicata
Volume	157
Issue number	1
DOIs	https://doi.org/10.1007/s10711-011-9596-x
State	Published - 1 Apr 2012

Keywords

Equidistribution
Homogeneous
Infinite measure
Lattice actions
Lie groups
Plane

ASJC Scopus subject areas

Geometry and Topology

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10.1007/s10711-011-9596-x

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title = "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 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, Str{\"o}mbergsson, Forni and Flaminio.",

keywords = "Equidistribution, Homogeneous, Infinite measure, Lattice actions, Lie groups, Plane",

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note = "Funding Information: Acknowledgments We thank Alex Gorodnik, S{\'e}bastien Gouez{\"e}l and Amos Nevo for useful discussions. The first author would like to thank the Center for Advanced Studies in Mathematics at Ben Gurion University for its hospitality during a stay where this work was first conceived. The work of the second author was supported by the Israel Science Foundation.",

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N1 - Funding Information: Acknowledgments We thank Alex Gorodnik, Sébastien Gouezël and Amos Nevo for useful discussions. The first author would like to thank the Center for Advanced Studies in Mathematics at Ben Gurion University for its hospitality during a stay where this work was first conceived. The work of the second author was supported by the Israel Science Foundation.

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N2 - 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.

AB - 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.

KW - Equidistribution

KW - Homogeneous

KW - Infinite measure

KW - Lattice actions

KW - Lie groups

KW - Plane

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Lattice actions on the plane revisited

Abstract

Keywords

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