Tiling billiards and interval exchange transformations with flips

Let us consider a tiling of the Euclidean plane by polygons. We play the billiard on this tiling in the following way. A trajectory goes in straight line in each tile. When it reaches a boundary between two tiles, it crosses the boundary and is refracted in the new tile. We get a zigzaging trajectory in the plane. Our goal is to understand what behaviour it can have: Is it periodic or not? bounded or not? If not, how does it go to infinity?

I will first present this dynamical system, and explain how to study it with interval exchange transformations with flips, which are piecewise isometries of the circle.

This will allows me to state a result of my PhD thesis about deviations from asymptotic direction of some tiling billiards trajectories and to explain its proof.