Elise Goujard (Nantes Université) and Magali Jay (Max Planck Institute Leipzig)

Abstract: I will review some recent results about the dynamics of polygonal billiards, focusing mostly on rational polygons. In this case the dynamical properties of the billiard are related to dynamical and geometric properties of corresponding flat surfaces and their moduli space.

Abstract: In this talk, I will present the meeting of different dynamical systems: tiling billiards, the wind-tree model and the 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 an arrangement of metamaterials with opposite index of refraction. The wind-tree model was introduced by Paul and Tatyana Ehrenfest to study a gas: a particle is moving in a plane where obstacles are periodically placed, on which the particle 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. Using dynamics on half-translation surfaces and their moduli space, I show that these trajectories are trapped in a strip, for almost every parameter. This behavior is similar to the one of the Eaton lenses.