Évènement

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 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 in the self-similar case (when a relevant compact translation surface is periodic under renormalisation), all closures of trajectories have non-maximal Hausdorff dimension.

This is achieved by constructing explicit invariant functions for the flow, and studying their levels sets using a symbolic coding for Interval Exchange Transformations.