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 line field $L$ is smooth on the smooth boundary pieces and is transverse to their tangent lines. A harmonic quadruple in the pencil through $p$ means that the nonidentity projective involution fixing $L(p)$ and $T_p\partial\Omega$ exchanges the incident and reflected lines (equivalently their cross-ratio is $-1$ in this ordering). The reflected ray is oriented back into the domain. A $k$-periodic point returns after $k$ applications of the billiard map; $k$-reflective means that such points include a nonempty open subset.

Statement recorded in the supplied source.

A planar projective billiard is a bounded domain $\Omega$ whose piecewise-smooth boundary carries a transverse line field $L$. At $p\in\partial\Omega$, an incident line $\ell$ reflects to $\ell'$ when $\ell,\ell',L(p),T_p\partial\Omega$ form a harmonic quadruple. Call the billiard $k$-reflective if its billiard map has an open set of $k$-periodic points. (1) Construct a $k$-reflective projective billiard for some odd $k\ge5$. (2) For fixed $k\ge4$, classify $k$-reflective projective billiards within natural boundary-smoothness classes (polygonal, piecewise algebraic, analytic, and so on)—the projective Ivrii conjecture. [S2]

Short English statement

Find an odd-period reflective projective billiard with period at least five; for each period at least four, determine the examples within specified boundary regularity classes.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5200006 (AMR-051-0006). 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, Corentin Fierobe, Problem 1, pp. 4--5. https://arxiv.org/pdf/2110.10750v2.