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 oval is a smooth closed strictly convex plane curve. The phase points are oriented chords $(x,y)$ and the billiard map sends $(x,y)$ to $(y,z)$ according to the parallel-tangent rule. For a polygon, use the line containing the edge at a regular impact point; trajectories hitting a vertex or an undefined reflection are excluded from regular orbits. A periodic orbit returns to its initial chord after finitely many steps. A stadium consists of two equal semicircular caps joined by parallel straight segments. The source asks broadly about chaos without selecting a particular numerical criterion.

Statement recorded in the supplied source.

In a planar symplectic billiard on an oval, the chord $xy$ reflects to $yz$ when the tangent at $y$ is parallel to $xz$; define polygonal symplectic billiards similarly. (1) Classify polygons for which every symplectic billiard orbit is periodic. (2) Does every polygon have a periodic orbit? (3) Is the symplectic billiard in a stadium chaotic? [S2]

Short English statement

Classify polygons with all regular symplectic billiard orbits periodic, decide whether every polygon has some periodic orbit, and establish whether the symplectic stadium dynamics is chaotic.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5200015 (AMR-051-0015). 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, Serge Tabachnikov, Problem 3, pp. 11--12. https://arxiv.org/pdf/2110.10750v2.