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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 
with 
.
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
if 
.
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
consists of
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 .
The following Lemma summarizes the elementary properties of the family
of the circle homeomorphisms, determined by a
caustic .
If
is a closed convex curve, and
is the
corresponding family of invariant circles, we set
and
sometimes suppress the subscript .
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 
.
Next: Devil's staircase
Up: Billiards that share a
Previous: Introduction
Oliver Knill
Wed Jul 8 11:57:32 CDT 1998
