DYNAMICAL OUTER BILLIARDS

'Outer billiards' is an emerging topic within dynamical systems that is related to inner billiards. However, while billiard theory is concerned with the continuous motion of a mass-point (or billiard ball) inside a Riemannian manifold with a piecewise smooth boundary (or billiard table), outer billiards moves the ball to the outside of the table and works in discrete time.
The outer (or dual) billiard transformation is defined as follows: Given a table ''P'' and a ball ''x'' in the exterior of ''P'', there are two points on ''P'' that form support lines through ''x''. The outer billiard transformation ''F'', which has either a clockwise or counterclockwise orientation, maps ''x'' to its reflection through the support point in the given direction. The set of all iterations of a point around a table is called an orbit, and if the point maps back onto itself it is called periodic.
Outer billiards was introduced in 1945 by M. Day and has attracted attention from Moser and others concerning the stability problem. It has been studied in the Euclidean plane as well as the hyperbolic plane and higher dimensions. Bernhard Neumann posed the question of whether or not orbits of the outer billiard map can escape to infinity. This question has been answered in the hyperbolic plane where certain tables, classified as “large”, have all of their orbits escape to infinity. In the Euclidean plane, certain maps about the Penrose kite have recently been found to have unbounded orbits. Much more is known and many more problems remain open.

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References

★ On Polygonal Dual Billiards in the Hyperbolic Plane, F. Dogru, S. Tabachnikov, , , Regular Chaotic Dynamics, 2003

★ Is the Solar System Stable?, J. Moser, , , Mathematical Intelligencer, 1978

★ R.E. Schwartz "Unbounded Orbits for Outer Billiards" Penn State 2007

★ Billiards, S. Tabachnikov, , , SMF Panoramas et Syntheses, 1995, ISBN 2-85629-030-2

★ Dual Billiards in the Hyperbolic Plane, S. Tabachnikov, , , Nonlinearity, 2002