Introduced by Diana Davis and her collaborators, tiling billiards are the class of dynamical systems that model the propagation of the refracted light in a heterogeneous material. To every tiling of a plane is associated a billiard. There are very few tilings for which corresponding billiards are well understood.
I will give an introduction to this domain with a focus on the simplest example — billiards in periodic triangular tilings. We will see that their dynamics is reduced to that of a family of interval exchange transformations (with flips!) and can be understood via renormalization. Lately, my motivation to study such billiards was reinforced by the discovery of their connection to a topological question from the eighties : how do foliations defined by the restriction of a constant 1-form on a level surface of a 3-periodic function ?
The talk is based on collaborations with Pascal Hubert, Paul Mercat, Ivan Dynnikov and Alexandra Skripchenko, as well as on my personal work.