research/AMR-005-0001.md

metadata

id: AMR-005-0001
classification: SOLVED-IN-LITERATURE
wording_corrected: 'yes'

AMR-005-0001 — Commuting billiard ball maps

Problem (corrected statement if needed)

Source: S. Tabachnikov, "A Baker's Dozen of Problems", Arnold Mathematical Journal 1 (2015), 59–67, DOI 10.1007/s40598-014-0001-3 (Problem 1 in the list). The original wording, verified against the published article:

Consider two nested convex domains. Then one has two billiard ball maps, $T_1$ and $T_2$, acting on the oriented lines that intersect both domains. If the domains are bounded by confocal ellipses, then the respective billiard ball maps commute. Assume that the two maps commute: $T_1 \circ T_2 = T_2 \circ T_1$. Conjecture. The two domains are bounded by confocal ellipses. For outer (a.k.a. dual) billiards, an analogous fact is proved in Tabachnikov (1994). For piece-wise analytic billiards, this conjecture was proved by Glutsyuk (2014). Of course, this problem has a multi-dimensional version, open both for inner and outer billiards.

Correction made to the garbled transcription: the list version merged the 2-dimensional conjecture with the multidimensional question into a single imperative sentence and omitted the status remarks (planar dual-billiard case already solved in 1994; piecewise-analytic case solved by Glutsyuk in 2014). The transcription's mathematical content is otherwise faithful.

Status / Literature

All references below were verified via Crossref, the arXiv API, and publisher pages (abstracts seen verbatim).

• Planar inner billiards — SOLVED. A. Glutsyuk, "On 4-reflective complex analytic planar billiards", J. Geom. Anal. 27 (2017), 183–238 (online 2016), DOI 10.1007/s12220-016-9679-x, arXiv:1405.5990. The published abstract states that the paper provides "solutions of Tabachnikov's Commuting Billiard Conjecture ... in two dimensions; the boundary is required to be piecewise $C^4$-smooth."
• Higher-dimensional inner billiards — SOLVED. A. Glutsyuk, "On commuting billiards in higher-dimensional spaces of constant curvature", Pacific J. Math. 305 (2020), 577–595, DOI 10.2140/pjm.2020.305.577, arXiv:1807.10567. Abstract (seen verbatim): "We consider two nested billiards in $\mathbb{R}^d$, $d\ge 3$, with $C^2$-smooth strictly convex boundaries. We prove that if the corresponding actions by reflections on the space of oriented lines commute, then the billiards are confocal ellipsoids. This together with the previous analogous result of the author in two dimensions solves completely the Commuting Billiard Conjecture due to Sergei Tabachnikov." The higher-dimensional case is deduced from Marcel Berger's classical theorem that in dimension $\ge 3$ only quadrics may have caustics; the paper also proves versions of Berger's theorem and the commuting result in space forms (constant curvature).
• Planar outer (dual) billiards — SOLVED already in 1994. S. Tabachnikov, "Commuting dual billiard maps", Geom. Dedicata 53 (1994), 57–68, DOI 10.1007/BF01264044. Abstract (seen verbatim): "...We prove that if two curves are given, such that the corresponding dual billiard transformations commute, then the curves are concentric homothetic ellipses." (Note the dual-billiard answer is concentric homothetic ellipses, not confocal — dual billiard maps are affinely covariant.)
• Higher-dimensional outer billiards — apparently OPEN. Multidimensional dual billiards exist (symplectic setting in $\mathbb{R}^{2n}$, Tabachnikov, "On the dual billiard problem", Adv. Math. 115 (1995), 221–249), but I found no published resolution of the commuting question there; a Crossref/arXiv search (2015–present) for commuting higher-dimensional dual/outer billiard maps returned nothing relevant.

Work done

• Retrieved the original statement from the published AMJ article (link.springer.com/article/10.1007/s40598-014-0001-3) and corrected the garbled list wording.
• Verified every citation above against Crossref metadata and, where possible, publisher abstracts (Springer page for Tabachnikov 1994; arXiv abstracts for Glutsyuk 1405.5990 and 1807.10567, including journal references).
• Searched for post-2015 work on the multidimensional outer-billiard commuting question via the arXiv API ("outer billiard" AND commuting: 0 hits) and Crossref (no relevant result).

Result

The problem is solved in the literature, with one sub-case apparently still open:

1. Planar inner case: commuting billiard ball maps of two nested convex domains with piecewise $C^4$-smooth boundaries $\Rightarrow$ confocal ellipses (Glutsyuk 2017). The proof goes through complexified billiards: commuting forces a 4-reflective complex analytic pseudo-billiard structure near the curves, and the classification of 4-reflective germs forces the curves to be confocal conics.
2. Higher-dimensional inner case ($d\ge3$): commuting actions by reflections for nested strictly convex $C^2$ billiards $\Rightarrow$ confocal ellipsoids (Glutsyuk 2020), via Berger's theorem (in dimension $\ge 3$ only quadrics admit caustics); also extended to spaces of constant curvature.
3. Planar outer case: commuting dual billiard maps $\Rightarrow$ concentric homothetic ellipses (Tabachnikov 1994) — predates the list.
4. Higher-dimensional outer case: no resolution found; appears to remain open.

What remains

• The multidimensional commuting question for outer/dual billiards (symplectic dual billiard maps in $\mathbb{R}^{2n}$) seems unresolved; nothing in the literature post-2015 addresses it as far as I could verify. A natural conjecture would be: commuting dual billiard maps of nested strictly convex hypersurfaces $\Rightarrow$ concentric homothetic ellipsoids.
• In the planar inner case the published solution assumes piecewise $C^4$ regularity; whether $C^2$ (or lower) smoothness suffices in dimension 2 is a residual regularity question (dimension $\ge 3$ needs only $C^2$, thanks to Berger-type rigidity).