Research projects

1 Billiards and related systems 1.1 Polygonal outer billiards in the hyperbolic plane The outer billiard about a convex polygon P in the plane R 2 is a piece-wise isometry, T , of the exterior of P defined as follows: given a point x outside of P , find the support line to P through x having P on the left, and define T (x) to be the reflection of x in the support vertex. See [10, 35]. Figure 1: The outer billiard map in the plane C. Culter proved (Penn State REU 2004) that every polygon in the plane admits periodic outer billiard orbits, see [36]. Outer billiard can be defined on the sphere and in the hyperbolic plane. On the sphere, there exist polygons without periodic outer billiard orbits. Conjecture: every polygonal outer bil-liard in the hyperbolic plane has periodic orbits. These orbits may lie on the circle at infinity. Figure 2: Outer billiards on equilateral triangles in the hyperbolic plane 1