Elliptical billiard systems and chaotic micro-lasers

This paper will discuss billiard systems, especially elliptical ones, with the main focus on finding the mean minimal action function $\alpha$ of these systems. This function plays a crucial role in understanding different rigidity phenomena that appear in the study of convex billiards and is also widely used in Aubrey-Mather theory. Billiard systems in general will be discussed before deriving the $\alpha$ function. These results will then be illustrated using an example of circular billiards. Then the elliptical billiard systems will be discussed in detail, the mean minimal action $\alpha$ will be derived and results will be compared with the circular billiards. Furthermore, chaotic micro-lasers will be covered, what they are, and how they are related to billiard systems.