A closed nonplanar totally integrable wire billiard in three-space
Let $\gamma : \mathbb{R}/\ell\mathbb{Z} \to \mathbb{R}^3$ be a smooth embedding, parametrized by arclength, with nowhere-vanishing curvature, and let $L(s,t) = |\gamma(t) - \gamma(s)|$. Assume that $\partial_s \partial_t L(s,t) > 0$ for all distinct parameters $s, t$. The wire billiard map acts on the annulus of ordered pairs of distinct parameters by $T(s,t) = (t,r)$, where $r \ne t$ is the unique parameter with $$\partial_2 L(s,t) + \partial_1 L(t,r) = 0,$$ that is, the chords $[\gamma(s), \gamma(t)]$ and $[\gamma(t), \gamma(r)]$ make equal angles with the tangent line to $\gamma$ at $\gamma(t)$.
Bialy, Mironov and Tabachnikov introduced wire billiards and proved that, for the curves they call nice, $T$ is an area-preserving twist map. They showed that closed curves in $\mathbb{R}^4 = \mathbb{C}^2$ such as $t \mapsto (e^{it}, \varepsilon e^{imt})$, with $m \ge 2$ an integer and $\varepsilon$ small, which are orbits of one-parameter subgroups of $SO(4)$, give wire billiards whose phase cylinder is foliated by invariant curves, and they conjectured that every wire with this property is an orbit of a one-parameter subgroup of the orthogonal group. In $\mathbb{R}^3$ such closed orbits are circles in a plane, and integrable round helices are not closed (Bialy and Mironov). For a planar convex billiard table, Bialy proved that a foliation of the whole phase cylinder by continuous invariant curves occurs only for the circle.
Problem. Is there such a curve $\gamma$, with image not contained in a plane, for which the entire annulus is foliated by continuous $T$-invariant simple curves, each winding once around it?
Source: Bialy, Mironov and Tabachnikov (2020), §8.3, whose rigidity conjecture would rule out such a wire; asked by Bialy and Mironov (2026), §9.7, Question (1). Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DS — Dynamical Systems
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, integrable billiards, twist maps, wire billiards
References
@article{bialy2020wire,
author = {Bialy, Misha and Mironov, Andrey E. and Tabachnikov, Serge},
title = {Wire billiards, the first steps},
journal = {Adv. Math.},
volume = {368},
year = {2020},
pages = {107154},
doi = {10.1016/j.aim.2020.107154},
eprint = {1905.13617},
archivePrefix = {arXiv},
}
@incollection{bialy2026icm,
author = {Bialy, Misha and Mironov, Andrey E.},
title = {Integrable billiards and related topics},
booktitle = {Proceedings of the International Congress of Mathematicians 2026, Volume 5: Invited Lectures (Sections 9--11)},
volume = {5},
year = {2026},
pages = {42--58},
doi = {10.1137/25M1804078},
eprint = {2510.03790},
archivePrefix = {arXiv},
}
@article{bialy1993,
author = {Bialy, Misha},
title = {Convex billiards and a theorem by {E}. {H}opf},
journal = {Math. Z.},
volume = {214},
number = {1},
year = {1993},
pages = {147--154},
doi = {10.1007/BF02572397},
}
@misc{aim-nonplanar-integrable-wire-billiard,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A closed nonplanar totally integrable wire billiard in three-space}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/264-nonplanar-integrable-wire-billiard.md},
note = {Entry 264 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00540, "A closed nonplanar totally integrable wire billiard in three-space", clopen.solutions.