There exists an outer billiards system with an unbounded orbit

Analysis, Geometry and Topology 1 Minute

You need to know: Euclidean plane {\mathbb R}^2, convex set in {\mathbb R}^2, bounded set or sequence in {\mathbb R}^2, the notion of (Lebesgue) almost every point in {\mathbb R}^2.

Background: Given a bounded convex set S \subset {\mathbb R}^2, and a point x_0 \in {\mathbb R}^2 outside S, let x_1 \in {\mathbb R}^2 be the point such that the segment x_0x_1 is tangent to S at its midpoint and a person walking from x_0 to x_1 would see S on the right. Such x_1 exists and unique for almost every x_0 \in {\mathbb R}^2, and the map x_0 \to x_1 is called the outer billiards map. Iterating this maps starting from x_0, we get an infinite sequence x_0 \to x_1 \to x_2 \to \dots, which is called an outer billiards orbit of x_0.

The Theorem: On 3rd February 2007, Richard Schwartz submitted to arxiv a paper in which he proved the existence of set S \subset {\mathbb R}^2 and x_0 \in {\mathbb R}^2 such that the outer billiards orbit of x_0 is well-defined and unbounded.

Short context: The question whether an outer billiards orbit can be unbounded was asked by Neumann in the 1950s, and since then has been a guiding question in the field. The Theorem answers this question affirmatively. In fact, Schwartz provides an explicit example of set S in the Theorem: this is the convex quadrilateral known as the Penrose kite.

Links: Free arxiv version of the original paper is here, journal version is here.

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Published by Bogdan Grechuk

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