Outer billiards

When P is a polygon, some points might not have well-defined orbits, on account of the potential ambiguity of choosing the midpoint of the relevant tangent line.

Bernhard Neumann informally posed the question as to whether or not one can have unbounded orbits in an outer billiards system, and Moser put it in writing in 1973.

This question, originally posed for shapes in the Euclidean plane and solved only recently, has been a guiding problem in the field.

theory, that outer billiards relative to a 6-times-differentiable shape of positive curvature has all orbits bounded.

In 2007, Richard Schwartz showed that outer billiards has some unbounded orbits when defined relative to the Penrose Kite, thus answering the original Moser-Neumann question in the affirmative.

Subsequently, Schwartz showed that outer billiards has unbounded orbits when defined relative to any irrational kite.

In 2008, Dmitry Dolgopyat and Bassam Fayad showed that outer billiards defined relative to the semidisk has unbounded orbits.

The proof of Dolgopyat-Fayad is robust, and also works for regions obtained by cutting a disk nearly in half, when the word nearly is suitably interpreted.

In 2003, Filiz Doǧru and Sergei Tabachnikov showed that all orbits are unbounded for a certain class of convex polygons in the hyperbolic plane.

Filiz Doǧru and Samuel Otten then extended this work in 2011 by specifying the conditions under which a regular polygonal table in the hyperbolic plane have all orbits unbounded, that is, are large.

As mentioned above, all the orbits are periodic when the system is defined relative to a convex rational polygon in the Euclidean plane.