Open Problems on Billiards and Geometric Optics

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

An outer billiard point is outside the domain enclosed by $\gamma$. Choose one of the two tangent branches consistently with the orientation of $\gamma$; reflect the point in its point of tangency. This defines $T$ near $\gamma$. An invariant curve is a curve preserved by $T$; integrability here concerns a foliation by such curves. A homothetic ellipse is a scaled copy with the same center.

Statement recorded in the supplied source.

Let $\gamma$ be a smooth closed strictly convex plane curve. The outer billiard map $T$ sends a point $A$ near $\gamma$ to the point $T(A)$ for which $[A,T(A)]$ is tangent to $\gamma$ at its midpoint. Outer billiards about ellipses are integrable, with phase space foliated by homothetic invariant ellipses. Are there other integrable outer billiards? [S2]

Short English statement

Determine whether a smooth strictly convex outer billiard other than an ellipse can be integrable.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5200002 (AMR-051-0002). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 M. Bialy, C. Fierobe, A. Glutsyuk, M. Levi, A. Plakhov and S. Tabachnikov, Open problems on billiards and geometric optics (2021), arXiv:2110.10750v2, Misha Bialy, Problem 2, pp. 2--3. https://arxiv.org/pdf/2110.10750v2.