Let be a closed convex and differentiable curve.
We denote by the space of rays intersecting T.
The space is a closed cylinder
with coordinates , where is the arc-length
parameter on T and where
is the natural angle. The cylinder
is the phase space for the billiard map
of the billiard table T (compare Figure 1).
If T is and strictly convex then
is a twist diffeomorphism of class
which preserves the area-form . The
time-reversal involution conjugates
Let be a closed convex curve with perimeeter
. For any string length , the string construction
produces a closed convex curve
encircling . As varies from to , the curves
form a continuous foliation of the exterior of
in (see the Figures 2,4).
We denote by the
billiard map of the table . The set
is a one-parameter family of area-preserving
twist maps of the phase cylinder .
A homotopically nontrivial topological circle in
the phase space of a twist map is called an invariant circle
The restriction of to is a Lipschitz homeomorphism,
isotopic to the identity.
We denote by its rotation number.
By construction, has a canonical invariant circle,
. It is formed by the rays supporting ,
with orientation induced by the positive orientation
of (Figures 3,5). The opposite choice of the orientation yields the
invariant circle . Thus, is the caustic
corresponding to the invariant circle ,
for any and the family
billiard tables with the same caustic .
We denote by the induced homeomorphism. We choose a reference direction in and parameterize by the directions of the supporting rays (Figure 5). With this parameterization, becomes a family of Lipschitz homeomorphisms of the circle . We use the notation for the homeomorphisms and write for the derivatives with respect to .
If is a convex n-gon, then . In particular, if is an interval with endpoints A, B, then . The curves are then the ellipses with foci A, B, see Figure 3 and circles if A=B. Considering individual caustics , we will assume . Then .
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