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Let tex2html_wrap_inline1408 be a closed convex and differentiable curve. We denote by tex2html_wrap_inline1684 the space of rays intersecting T. The space tex2html_wrap_inline1422 is a closed cylinder with coordinates tex2html_wrap_inline1690 , where tex2html_wrap_inline1692 is the arc-length parameter on T and where tex2html_wrap_inline1696 is the natural angle. The cylinder tex2html_wrap_inline1698 is the phase space for the billiard map tex2html_wrap_inline1700 of the billiard table T (compare Figure 1). If T is tex2html_wrap_inline1706 and strictly convex then tex2html_wrap_inline1700 is a twist diffeomorphism of class tex2html_wrap_inline1710 which preserves the area-form tex2html_wrap_inline1712 . The time-reversal involution tex2html_wrap_inline1714 conjugates tex2html_wrap_inline1414 with tex2html_wrap_inline1718 .

Let tex2html_wrap_inline1460 be a closed convex curve with perimeeter tex2html_wrap_inline1462 . For any string length tex2html_wrap_inline1724 , the string construction produces a closed convex curve tex2html_wrap_inline1726 encircling tex2html_wrap_inline1372 . As tex2html_wrap_inline1378 varies from tex2html_wrap_inline1462 to tex2html_wrap_inline1664 , the curves tex2html_wrap_inline1726 form a continuous foliation of the exterior of tex2html_wrap_inline1372 in tex2html_wrap_inline1740 (see the Figures 2,4).

We denote by tex2html_wrap_inline1742 the billiard map of the table tex2html_wrap_inline1744 . The set tex2html_wrap_inline1374 is a one-parameter family of area-preserving twist maps of the phase cylinder tex2html_wrap_inline1422 .

A homotopically nontrivial topological circle tex2html_wrap_inline1436 in the phase space of a twist map tex2html_wrap_inline1414 is called an invariant circle if tex2html_wrap_inline1754 . The restriction of tex2html_wrap_inline1414 to tex2html_wrap_inline1436 is a Lipschitz homeomorphism, isotopic to the identity. We denote by tex2html_wrap_inline1760 its rotation number.

By construction, tex2html_wrap_inline1374 has a canonical invariant circle, tex2html_wrap_inline1764 . It is formed by the rays supporting tex2html_wrap_inline1372 , with orientation induced by the positive orientation of tex2html_wrap_inline1372 (Figures 3,5). The opposite choice of the orientation yields the invariant circle tex2html_wrap_inline1770 . Thus, tex2html_wrap_inline1372 is the caustic corresponding to the invariant circle tex2html_wrap_inline1384 , for any tex2html_wrap_inline1378 and the family tex2html_wrap_inline1778 consists of billiard tables with the same caustic tex2html_wrap_inline1372 .

We denote by tex2html_wrap_inline1782 the induced homeomorphism. We choose a reference direction in tex2html_wrap_inline1740 and parameterize tex2html_wrap_inline1384 by the directions tex2html_wrap_inline1788 of the supporting rays (Figure 5). With this parameterization, tex2html_wrap_inline1380 becomes a family of Lipschitz homeomorphisms of the circle tex2html_wrap_inline1792 . We use the notation tex2html_wrap_inline1794 for the homeomorphisms and write tex2html_wrap_inline1796 for the derivatives with respect to tex2html_wrap_inline1788 .

tex2html_wrap1914 tex2html_wrap1916
The following Lemma summarizes the elementary properties of the family tex2html_wrap_inline1804 of the circle homeomorphisms, determined by a caustic tex2html_wrap_inline1372 .



tex2html_wrap1918 tex2html_wrap1920
tex2html_wrap1922 tex2html_wrap1924
If tex2html_wrap_inline1372 is a closed convex curve, and tex2html_wrap_inline1856 is the corresponding family of invariant circles, we set tex2html_wrap_inline1858 and sometimes suppress the subscript tex2html_wrap_inline1372 .



If tex2html_wrap_inline1372 is a convex n-gon, then tex2html_wrap_inline1894 . In particular, if tex2html_wrap_inline1372 is an interval with endpoints A, B, then tex2html_wrap_inline1900 . The curves tex2html_wrap_inline1376 are then the ellipses with foci A, B, see Figure 3 and circles if A=B. Considering individual caustics tex2html_wrap_inline1372 , we will assume tex2html_wrap_inline1910 . Then tex2html_wrap_inline1912 .

Next: Devil's staircase Up: Billiards that share a Previous: Introduction

Oliver Knill
Wed Jul 8 11:57:32 CDT 1998