Counting problem on wind-tree models

Pardo, Angel

doi:10.2140/gt.2018.22.1483

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Counting problem on wind-tree models

Angel Pardo

Geometry & Topology 22 (2018) 1483–1536

DOI: 10.2140/gt.2018.22.1483

Abstract

We study periodic wind-tree models, that is, billiards in the plane endowed with $ℤ^{2}$ –periodically located identical connected symmetric right-angled obstacles. We give asymptotic formulas for the number of (isotopy classes of) closed billiard trajectories (up to $ℤ^{2}$ –translations) on the wind-tree billiard. We also explicitly compute the associated Siegel–Veech constant for generic wind-tree billiards depending on the number of corners on the obstacle.

Keywords

billiards, translations surfaces, periodic orbits, counting problem, Siegel–Veech constants

Mathematical Subject Classification 2010

Primary: 37D50, 37C35

Secondary: 30F30, 37A40, 37D40

References

Publication

Received: 23 May 2016

Revised: 23 January 2017

Accepted: 15 May 2017

Published: 16 March 2018

Proposed: Danny Calegari

Seconded: Anna Wienhard, Leonid Polterovich

Authors

Angel Pardo
Institut de Mathématiques de Marseille Aix-Marseille Université Marseille France

Volume 22, issue 3 (2018)