"Translations surfaces and their geodesics" by Jean-Christophe Yoccoz

In a rectangular billiard with no friction, orbits are either periodic or equidistributed. The study of such billiards is very related to the study of rotations on a circle. In general, the study of the dynamics of a polygonal billiard table whose angles are commensurable to \pi leads to fascinating dynamical objects (interval exchanges) and geometrical objects (translation surfaces). We shall try to give an overview of this topic.