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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.
** Next:** Caustics of convex billiards
**Up:** On nonconvex caustics of
** Previous:** Introduction

*Oliver Knill *

Fri Jun 12 13:34:37 CDT 1998