Invariant sets in the wind-tree model

The wind-tree model is a billiard model of gas diffusion introduced by Ehrenfest and Ehrenfest in the early 20th century. A small particle (the "wind") bounces between periodically placed rectangles (the "trees") on the plane. A lot of progress has been made in the past 15 years by studying the wind-tree model from the point of view of translation surfaces.

In particular, Fraçzek and Ulcigrai proved in 2014 that in typical directions the billiard flow on the wind-tree model has no dense trajectories. This opens the natural question of trying to understand what the closures of the trajectories look like.

The new result that I will present in this talk is that when the wind-tree model has some self-similarity, all closures of trajectories have non-maximal Hausdorff dimension.