Convex billiard tables

Good starting points in the literature of the billiard problem are [23] [4] [24] or [21]. In the case, when the domain is a strictly convex planar region, the return map to the boundary T defines a map of the annulus . The circle parametrizes the table with impact point s at the boundary T. The angle of impact and s determine the next reflection point with angle . The reader can see some orbits of the map in Figures 3 and 4.

Noncontractible invariant curves in A can define caustics in the interior of the table T (see Figure 2). Caustics are curves with the property that a ball, once tangent to stays tangent after every reflection. We will discuss the precise relation between the invariant curve and the caustic in the next section. Examples of caustics are drawn in Figures 6,7 and Figure 11a.

Invariant curves of are interesting for the dynamics because they divide the phase space A into invariant regions and therefore prevent ergodicity. They often occur near the boundary of a smooth, strictly convex table and define 'Lazutkin whisper galleries` [8]. Most convex billiards have no invariant curves [11]. Also, if the curvature of the table is unbounded at some point, then there are no invariant curves in a neighborhood of the boundary of A [15]. Strict convexity is necessary for the existence of caustics [17]. The billiard in Figure 1 for example has no caustics.

We will focus in this article especially on billiard tables of constant width. They all leave the curve invariant. Figures 3 and 4 show typical trajectories in such tables. The flatness of the invariant curve will allow us to make statements about the regularity of the corresponding caustics depending on the regularity of the table. First however, in the next section, we will clarify the relation between the invariant curve in the phase space and the caustic in the interior of the table.