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.

The intended higher-dimensional setting is $d\ge3$. A billiard trajectory travels along lines in the enclosed domain and reflects at $S$ in its tangent hyperplane. The phase space $\mathbb A$ consists of oriented lines crossing the interior; a smooth hypersurface $\Sigma$ has codimension one in this phase space and is invariant when $T(\Sigma)=\Sigma$. Such an invariant hypersurface is not required to consist of lines tangent to a spatial caustic.

Statement recorded in the supplied source.

For $d\ge3$, find a closed smooth strictly convex nonellipsoidal hypersurface $S\subset\mathbb R^d$ whose billiard map preserves a smooth hypersurface $\Sigma\subset\mathbb A$. Determine the geometric and dynamical properties of such invariant hypersurfaces. [S2]

Short English statement

In dimension at least three, construct a nonellipsoidal convex billiard with a smooth invariant hypersurface in phase space, and describe the geometry and dynamics of these hypersurfaces.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5200004 (AMR-051-0004). 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 4, p. 3. https://arxiv.org/pdf/2110.10750v2.