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.

Write $L$ for the perimeter of $\gamma$, so $s\in\mathbb R/L\mathbb Z$, and let $\alpha\in(0,\pi)$ be the angle with the positively oriented tangent. The phase cylinder is $(\mathbb R/L\mathbb Z)\times(0,\pi)$. The ordinary billiard map follows a straight chord and reflects with equal incidence and reflection angles. Invariance means $T(C)=C$. Integrability asks for a foliation by invariant curves; KAM curves are invariant curves carrying quasiperiodic motion. Ergodicity is with respect to the invariant symplectic area measure.

Statement recorded in the supplied source.

Let $\gamma$ be a smooth closed strictly convex plane curve, parametrized by arc length $s$, and let $d>0$. Form a sphere-like surface by gluing the two ends of the height-$d$ cylinder over $\gamma$ to two copies of its enclosed domain. Its geodesic return map on the phase cylinder is $$T(s,\alpha)=(s_1+d\cot\alpha_1,\alpha_1),$$ where $(s_1,\alpha_1)$ is the ordinary billiard image of $(s,\alpha)$. (1) Does $T$ have invariant curves, for example KAM curves near the boundary? (2) Other than circles, for which $\gamma$ is $T$ integrable? (3) Can $T$ be ergodic? [S2]

Short English statement

For the cylinder-and-two-caps surface, ask whether its geodesic return map has invariant curves, including KAM curves near grazing directions, which shapes besides circles give integrability, and whether it can be ergodic.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5200001 (AMR-051-0001). 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 1, pp. 1--2. https://arxiv.org/pdf/2110.10750v2.