Journal of Modern Dynamics (JMD)

Divergent trajectories in the periodic wind-tree model

Pages: 1 - 29, Issue 1, March 2013      doi:10.3934/jmd.2013.7.1

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Vincent Delecroix - Université Paris 7, Département de Mathématiques, Bâtiment Sophie Germain, 8 Place FM/13, 75013 Paris, France (email)

Abstract: The periodic wind-tree model is a family $T(a,b)$ of billiards in the plane in which identical rectangular scatterers of size $a \times b$ are disposed periodically at each integer point. In that model, the recurrence is generic with respect to the parameters $a$, $b$, and the angle $\theta$ of initial direction of the particule. In contrast, we prove that for some parameters $(a,b)$ the set of angles $\theta$ for which the billiard flow is divergent has Hausdorff dimension greater than one half.

Keywords:  Wind-tree model, rational billiard, interval-exchange transformations.
Mathematics Subject Classification:  Primary: 37B05, 37B10.

Received: April 2012;      Available Online: May 2013.