Dynamical billiards

Definition

Let $Q \subset \R^2$ be a compact connected set whose boundary $\partial Q$ is a finite union of smooth curves, called the table. A billiard trajectory in $Q$ is a path $t \mapsto q(t)$ in $Q$ with the following properties:

1. Between two times at which $q(t) \in \partial Q$, the path is a straight segment run at unit speed.
2. At a time when $q(t)$ is a smooth point of $\partial Q$, the velocity after the collision is the mirror image of the velocity before it. The component tangent to $\partial Q$ is kept and the normal component changes sign.

The billiard flow is the flow on the unit tangent vectors $(q, v)$, $q \in Q$, $\abs{v} = 1$, that moves $(q, v)$ along its trajectory. The billiard map acts on the pairs $(s, \theta)$. Here $s$ is the arc-length parameter of a point of $\partial Q$, and $\theta \in (0, \pi)$ is the angle between an inward unit vector and the boundary. The map sends $(s, \theta)$ to the next collision point and its outgoing angle. These two dynamical systems are called dynamical billiards.