Exterior Billiards

It is an unsolved problem, whether there is a convex table in the plane such that the exterior billiard dynamical systems has an unbounded orbit. The problem had been suggested in 1960 by B.H. Neumann. It appeared listed as an unsolved problem in Mosers book "stable and unstable motion". Serge Tabatchnikov suggested to look at the semi circle. Numerical experiments show there, that an orbit escapes to infinity. But this is not proven. Here are some numerically computed orbits.

Update since posting this page in the year 2005: there are polygons known now for which the map is unbounded: see Outer billiards on kites by Richard Evan Schwartz. Update 2009: there is now a book on the proof. Here is the first page of that book, which contains some milestones on the Moser-Neumann problem:
Source: Outer Billiards on Kites (AM-171) by Richard Evan Schwartz, 2009.

Exterior Billiard


Click on the thumbnail to download a 800x800 picture. Many orbits of the exterior billiard at the semi circle. Visible are stable islands around periodic points. Many orbits starting between these islands seem to escape to infinity. A typical orbit. A stable orbit The distance to the origin for an orbit of length 106 (1 million iterations).

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