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# Preliminaries

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