Tiling billiard in the wind-tree model

In this talk, I will present the meeting of different dynamical systems: tiling billiards, the wind-tree model and Eaton lenses. The three of them are motivated by physics.

In the beginning of the 2000's, physicists have conceived metamaterials with negative index of refraction. Tilling billiards' trajectories consist of light rays moving in a arrangement of metamaterials with opposite indew of refraction. The wind-tree model was instoduced by Paul and Tatyana Ehrenfest to study a gaz: a particule is moving in a plane where obstacles are periodically placed, on which the particule bounces. The Eaton lenses are a periodic array of lenses in the plane, in which we consider a light ray that is reflected each time it crosses a lens.

After having introduced these dynamical systems, I will consider a mix of them: an arrangement of rectangles in the plane, like in the wind-tree model, but made of metamaterials, like for tiling billiards. I study the trajectories of light in this plane. They are refracted each time they cross a rectangle. I show that these trajectories are traped in a strip, for almost every parameter. This behavior is similar to the one of Eaton lenses.