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Let be the equilateral triangle with sidelength 1/3.
We construct tables T obtained from by
the so called string construction (see i.e. [11, 12]).
One takes an unstretchable string having length ,
wraps it around , pulls it tight at a point, M,
and drags it around . The point M then traces the table.
Varying the parameter , we get a oneparameter family
of billiard tables . Each of these tables is composed of
piecewise elliptic arcs. (See Fig. 1.).

Fig. 1. Four examples of tables given by the string construction at
an equilateral triangle.

The corresponding billiard maps, ,
form a natural oneparameter family of
twist maps. We investigate this family in the
present work.
We denote by the space of unit vectors, with
foot points in , directed inwards.
The space is a closed cylinder with natural coordinates
, where is the normalized
arclength parameter on T, and where
measures the height in . The cylinder
is the phase space for the billiard map, ,
of the billiard table T (Fig. 2.).

Fig. 2. The billiard map and the phase space .

By construction, each billiard map
has a canonical invariant circle,
. It is formed by the rays
supporting . Their orientation is induced by the positive orientation
of . The opposite choice of orientation yields an other
invariant circle .
The curve is the caustic
corresponding to the invariant circle ,
for any [2].

Fig. 3. A family of billiard tables obtained by the string
construction. ( ). The invariant circles are
moving up in the phase space as increases.

Thus the family
, consists of
billiard tables with the same caustic .

Fig. 4. Some orbits for the parameter .


Fig. 5. Some orbits for the parameter .


Fig. 6. Some orbits for the parameter .

By our choice of ,
the rotation number of satisfies
.
Next: A Family of Circle
Up: Billiards that share a
Previous: Billiards that share a
Oliver Knill, Jul 8, 1998