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. For a direction $u$, let $I_u$ exchange the two endpoints of each parallel chord, with tangency points fixed. Then $F=I_v\circ I_u$ for the chosen ordered pair of directions. For a pencil through $P$, define the analogous chord involution $I_P$, wherever the construction is defined, and set $F=I_Q\circ I_P$. Conjugacy means $h\circ F\circ h^{-1}$ is the stated circle map for a change of coordinate $h$; a Möbius transformation is a projective fractional-linear map on $\mathbb RP^1$.

Statement recorded in the supplied source.

For a planar oval $\gamma$, alternately follow chords in two fixed directions to obtain a circle map $F:\gamma\to\gamma$. If $F$ is conjugate to a rotation for every pair of directions, must $\gamma$ be an ellipse? In the projective version, use pencils through points $P,Q$; if for every $P,Q$ with $PQ$ meeting $\gamma$ the resulting $F$ is conjugate to a Möbius transformation, must $\gamma$ be an ellipse? [S2]

Short English statement

Does rotation conjugacy for every pair of parallel directions characterize ellipses? Does projective conjugacy for every pair of pencils whose joining line meets the oval also characterize ellipses?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5200014 (AMR-051-0014). 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 2, pp. 10--11. https://arxiv.org/pdf/2110.10750v2.