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A Family of Billiard Maps

Let tex2html_wrap_inline402 be the equilateral triangle with sidelength 1/3. We construct tables T obtained from tex2html_wrap_inline402 by the so called string construction (see i.e. [11, 12]). One takes an unstretchable string having length tex2html_wrap_inline424 , wraps it around tex2html_wrap_inline402 , pulls it tight at a point, M, and drags it around tex2html_wrap_inline402 . The point M then traces the table. Varying the parameter tex2html_wrap_inline434 , we get a one-parameter family of billiard tables tex2html_wrap_inline400 . Each of these tables is composed of piecewise elliptic arcs. (See Fig. 1.).

tex2html_wrap512 Fig. 1. Four examples of tables given by the string construction at an equilateral triangle.
The corresponding billiard maps, tex2html_wrap_inline438 , form a natural one-parameter family of twist maps. We investigate this family in the present work.

We denote by tex2html_wrap_inline440 the space of unit vectors, with foot points in tex2html_wrap_inline400 , directed inwards. The space tex2html_wrap_inline440 is a closed cylinder with natural coordinates tex2html_wrap_inline446 , where tex2html_wrap_inline448 is the normalized arc-length parameter on T, and where tex2html_wrap_inline452 measures the height in tex2html_wrap_inline440 . The cylinder tex2html_wrap_inline440 is the phase space for the billiard map, tex2html_wrap_inline458 , of the billiard table T (Fig. 2.).

tex2html_wrap512 Fig. 2. The billiard map tex2html_wrap_inline462 and the phase space tex2html_wrap_inline440 .
By construction, each billiard map tex2html_wrap_inline466 has a canonical invariant circle, tex2html_wrap_inline468 . It is formed by the rays supporting tex2html_wrap_inline402 . Their orientation is induced by the positive orientation of tex2html_wrap_inline402 . The opposite choice of orientation yields an other invariant circle tex2html_wrap_inline474 . The curve tex2html_wrap_inline402 is the caustic corresponding to the invariant circle tex2html_wrap_inline410 , for any tex2html_wrap_inline406 [2].
tex2html_wrap514 Fig. 3. A family tex2html_wrap_inline400 of billiard tables obtained by the string construction. ( tex2html_wrap_inline484 ). The invariant circles tex2html_wrap_inline410 are moving up in the phase space tex2html_wrap_inline440 as tex2html_wrap_inline406 increases.
Thus the family tex2html_wrap_inline492 , consists of billiard tables with the same caustic tex2html_wrap_inline402 .
tex2html_wrap516 Fig. 4. Some orbits for the parameter tex2html_wrap_inline496 .
tex2html_wrap518 Fig. 5. Some orbits for the parameter tex2html_wrap_inline498 .
tex2html_wrap520 Fig. 6. Some orbits for the parameter tex2html_wrap_inline500 .
By our choice of tex2html_wrap_inline410 , the rotation number tex2html_wrap_inline504 of tex2html_wrap_inline410 satisfies tex2html_wrap_inline508 .

Next: A Family of Circle Up: Billiards that share a Previous: Billiards that share a

Oliver Knill, Jul 8, 1998