On periodic billiard trajectories in obtuse triangles (Q2706417)

This clear and interesting paper (even for non-specialists) deals with the existence of periodic billiard trajectories in a plane domain bounded by an obtuse triangle, i.e. a triangle such that one of its three angles is obtuse. Such trajectories are generated by reflections on the domain boundary. After reviewing results obtained on a class of obtuse triangles, for which the existence of particular periodic paths was proved, the authors extend the study to arbitrary obtuse triangle. NEWLINENEWLINENEWLINEThe former results are improved by using new techniques of analysis of periodic paths. This is made parametrizing the set of obtuse triangles, equipped with the topology and measure inherited from the Euclidean topology and the Lebesgue measure on the parameter plane. Each periodic trajectory is associated with an unique element (called its ``code'') by considering the number of the sides in the order in which they are visited by the reflecting path. The problem is to describe how one can determine whether for a given triangle, defined by its two acute angles, a periodic trajectory, associated with a given code, exists. To this end the notion of ``generators of stable trajectories'' is introduced as an infinite subset made up of the concatenation of copies of words constituted by the elements ``1, 2, 3'', resulting from the numbering of the triangle sides. Considering the plane of the two acute angles, the parameter regions related to a given generator are defined. About 50 regions for code words of length less or equal to 25 are given. The authors hope to detect enough generators to fill the whole parameter set.