an:05937007
Zbl 1233.37025
Hubert, Pascal; LeliÃ¨vre, Samuel; Troubetzkoy, Serge
The Ehrenfest wind-tree model: periodic directions, recurrence, diffusion
EN
J. Reine Angew. Math. 656, 223-244 (2011).
0075-4102 1435-5345
2011
j
37D50 37A60 37D35 37C27 37D05 37C75 37B20
Ehrenfest wind-free model; planar billiards; periodic obstacles; periodic directions; recurrence; billiard flow; diffusion; compact translation surfaces; Veech surfaces; square-tiled surfaces; Veech groups
Results of previous papers by the last author are extended by pursuing the article by \textit{W. A. Veech} [Invent. Math. 97, No. 3, 553--583 (1989; Zbl 0676.32006)] on compact translation surfaces. Periodic wind-free models are investigated with an emphasis on unbounded planar billiards with periodically located rectangular obstacles. Previous observations by \textit{J. Hardy} and \textit{J. Weber} [J. Math. Phys. 21, No. 7, 1802--1808 (1980)] on recurrence and abnormal diffusion of the billiard flow are addressed anew to demonstrate that, depending on the range of rational parameters, the existence of completely periodic directions and recurrences can be proved. In another parameter range, there are escape directions for all trajectories; rates of escape for almost all directions are proved.
Piotr Garbaczewski (Opole)
0676.32006