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# A Family of Billiard Maps

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 one-parameter 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 one-parameter 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 arc-length 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 .

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Oliver Knill, Jul 8, 1998