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Recurrence for the wind-tree model. (English) Zbl 1436.37006

The Ehrenfest wind-tree model consists of the motion of a single particle in a periodic array of rectangular scatterers in the plane, with elastic collisions. Introduced by Paul and Tatiana Ehrenfest in the early 20th century, it has reemerged as an object of study in recent years with many advances using the theory of {translation surfaces}.
The key insight is that the billiard flow can be viewed as the flow on a \(\mathbb Z^2\)-cover of a genus two translation surface \(\Sigma_{a,b}\) (here \(a, b\) are the side lengths of the rectangle). Properties of the linear flows on these translation surfaces can be then translated to the properties of the wind-tree. For example, the rate of drift in the wind-tree model is related to the deviation of ergodic averages for flows on \(\Sigma_{a,b}\).
In this paper, the question of recurrence of this flow (and more generally, recurrence of flows on \(\mathbb Z^d\)-covers of translation surfaces) is studied. A sufficient condition for recurrence on covers is given, generalizing Masur’s criterion for unique ergodicity of linear flows on translation surfaces. As a consequence, the authors obtain that for almost every \((a, b)\), and almost every initial direction, the wind-tree is recurrent.

MSC:

37A30 Ergodic theorems, spectral theory, Markov operators
37C83 Dynamical systems with singularities (billiards, etc.)
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
37B10 Symbolic dynamics
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