research/AMR-005-0002.md

metadata

id: AMR-005-0002
classification: PARTIAL-PROGRESS
wording_corrected: 'no'

AMR-005-0002 — Coexistence of one-parameter families of p- and q-periodic billiard trajectories

Problem (corrected statement if needed)

Source: Serge Tabachnikov, "A Baker's Dozen of Problems", Arnold Mathematical Journal 1(1) (2015), 59–67, DOI 10.1007/s40598-014-0001-3 (verified via Crossref), §2, Problem 1. The dataset transcription matches the published wording essentially verbatim:

Problem 1. Are there smooth convex curves, other than ellipses, simultaneously admitting one-parameter families of $p$- and $q$-periodic billiard trajectories (for $p\neq q$)?

Context given in the source: a curve of constant width admits a one-parameter family of 2-periodic (back-and-forth) trajectories; for every $p\ge 3$ there exist non-elliptic billiard tables admitting a one-parameter family of $p$-periodic trajectories (Baryshnikov–Zharnitsky, Math. Res. Lett. 13 (2006), 587–598, DOI 10.4310/MRL.2006.v13.n4.a8, verified via the Crossref reference list of the source article). The simplest case: does any curve of constant width, other than the circle, admit a one-parameter family of 3-periodic trajectories? The source then adds: "A similar question can be asked about outer billiards." No correction to the transcription is needed.

Two clarifying remarks (standard, and consistent with the source's intent):

• A one-parameter family of $p$-periodic trajectories is an invariant circle $\Gamma$ of the billiard map $T$ in the phase cylinder, with rotation number $k/p$ ($\gcd(k,p)=1$), on which $T^p=\mathrm{id}$. For $p\ge 3$ this is an integrable rational caustic; for $p=2$ it is a circle of fixed points of $T$.
• In an ellipse, the 2-periodic orbits (the two axes) are isolated — they do not form a family. So ellipses have families for every $p\ge 3$ (Poncelet porism) but not for $p=2$; the circle additionally has a 2-periodic family (it has constant width). The problem is therefore interesting already for $(p,q)=(2,3)$, where the conjectured answer "no non-circular constant-width curve has a 3-periodic family" characterizes the circle, not arbitrary ellipses.

Status / Literature

The problem in full generality is open. It is a weakening of the Birkhoff–Poritsky conjecture (integrable convex billiards are ellipses), itself still open in general. Verified relevant literature:

• Bialy, M., "Convex billiards and a theorem by E. Hopf", Math. Z. 214(1) (1993), 147–154, DOI 10.1007/BF02572397 (bibliographic data verified from the publisher-asserted reference lists of two Crossref-verified papers below). If the whole phase cylinder is foliated by non-contractible invariant circles, the table is a disk. This settles the extreme case "families of all periods" but not two isolated periods.
• Avila, A., De Simoi, J., Kaloshin, V., "An integrable deformation of an ellipse of small eccentricity is an ellipse", Ann. of Math. 184(2) (2016), 527–558, DOI 10.4007/annals.2016.184.2.5 (verified via the Crossref-verified reference list of Glutsyuk–Shustin below). Infinitesimal/one-parameter-deformation rigidity of ellipses of small eccentricity under preservation of caustics near the boundary.
• Kaloshin, V., Sorrentino, A., "On the local Birkhoff conjecture for convex billiards", Ann. of Math. 188(1) (2018), 315–380, DOI 10.4007/annals.2018.188.1.6 (verified via Crossref). Any $C^\infty$ billiard sufficiently close to a given ellipse that admits an integrable rational caustic of rotation number $1/q$, $q\ge 3$ (i.e., a one-parameter family of $q$-periodic orbits), is an ellipse. Hence locally near ellipses even one family of period $\ge 3$ already forces ellipticity — a much stronger local statement than the two-periods question.
• Kaloshin, V., Koudjinan, C. E., "Non co-preservation of the $1/2$ & $1/(2l+1)$-rational caustics along deformations of circles", arXiv:2107.03499 (verified via the arXiv API). Every deformation of a circle preserving both the $1/2$- and the $1/(2l+1)$-rational caustics is trivial (similarities only). This is exactly the deformational (infinitesimal) version of the $(2,,\text{odd})$ case of Problem 1, including the "simplest case" $(2,3)$ highlighted by Tabachnikov.
• Bialy, M., Mironov, A. E., "The Birkhoff–Poritsky conjecture for centrally-symmetric billiard tables", Ann. of Math. 196(1) (2022), 389–413, DOI 10.4007/annals.2022.196.1.2 (verified via Crossref; note the article number is .1.2). For $C^2$ centrally symmetric tables, a 1/4-rotation-number invariant circle (a family of 4-periodic orbits) together with a $C^0$-foliation of the region between it and the boundary by invariant curves forces an ellipse. Global (not local), but the hypothesis is stronger than two isolated periodic families.
• Koval, I., "Local strong Birkhoff conjecture and local spectral rigidity of almost every ellipse", arXiv:2111.12171 (verified via the arXiv API). Local rigidity of almost every ellipse under the stronger hypothesis of integrability near the boundary (rational caustics of all rotation numbers $p/q\le 1/q_0$).

Outer billiards (the analogous question is also open in full generality):

• Tabachnikov, S., "On algebraically integrable outer billiards", Pacific J. Math. 235(1) (2008), 101–104, DOI 10.2140/pjm.2008.235.89 (verified via Crossref reference lists). If the outer billiard map admits a non-constant algebraic first integral (in a real-analytic sense near the curve), the curve is an ellipse.
• Glutsyuk, A., Shustin, E., "On polynomially integrable planar outer billiards and curves with symmetry property", Math. Ann. 372(3–4) (2018), 1481–1501, DOI 10.1007/s00208-018-1726-4 (verified via Crossref). Every polynomially integrable planar outer billiard is elliptic — the solution of the polynomial/algebraic version of the outer-billiard integrability problem.
• Bialy, M., "Integrable outer billiards and rigidity", arXiv:2306.12494 (verified via the arXiv API; journal version announced 2024). If the vicinity of a smooth convex plane curve $\gamma$ of positive curvature is foliated by continuous curves invariant under the outer billiard map, then $\gamma$ is an ellipse (outer-billiard analogue of Bialy's 1993 Hopf-type rigidity, via a new generating function and the Blaschke–Santaló inequality).

I found no published result that settles either the inner or the outer two-periods question as stated; all known results either are local (near an ellipse/circle), deformational, or assume a full foliation / full integrability, which is strictly stronger than two isolated periodic families.

Work done

• Located and read the source article (AMJ open HTML) and confirmed the dataset wording matches published Problem 1 (§2) — wording_corrected: no.
• Ran targeted web searches for (i) direct attacks on the two-periods problem, (ii) the constant-width/3-periodic sub-case, (iii) outer-billiard analogues.
• Verified every citation above against Crossref (api.crossref.org/works/<DOI>) or the arXiv API; the Avila–De Simoi–Kaloshin, Bialy 1993, and Tabachnikov 2008 entries were cross-verified through publisher-asserted reference lists inside Crossref-verified records. One initially guessed DOI for Bialy–Mironov (.196.1.5) returned 404 and was corrected to .196.1.2 — only the verified DOI is cited.
• Mathematical analysis (no computation used):

Reformulation and reduction of the $p=2$ case. A one-parameter family of 2-periodic orbits is a circle of fixed points of the billiard map projecting onto the whole boundary. Through every boundary point there is then a chord orthogonal to the boundary at both endpoints, and the involution swapping its endpoints is the antipodal map; equality of the two support-line distances along every direction forces the curve to have constant width. Conversely every constant-width curve has such a family (all diameters are double normals). This classical reduction is exactly the premise stated by Tabachnikov.

Constant perimeter lemma. For any smooth one-parameter family $x(t)=(x_0(t),\dots,x_{p-1}(t))$ of $p$-periodic billiard trajectories, the perimeter $L(t)=\sum_i |x_{i+1}(t)-x_i(t)|$ is constant. Proof: writing $h(x,y)=|x-y|$ for the generating function, the billiard reflection law gives $\partial_2 h(x_{i-1},x_i)=-s_i$ and $\partial_1 h(x_i,x_{i+1})=s_i$ (the outgoing/incoming "momenta"), so $dL=\sum_i(\partial_1 h(x_i,x_{i+1})+\partial_2 h(x_{i-1},x_i)),dx_i=\sum_i (s_i-s_i),dx_i=0$. (This is the classical reason Poncelet families have constant perimeter.) Hence the data of Problem 1 include two marked constants $L_p, L_q$ (plus the width $w$ when $p=2$, with $L_2=2w$), and the associated invariant circles are Lagrangian circles of rational rotation number in the phase cylinder.

Dynamical consequence. Between the two invariant circles $\Gamma_{1/p}$, $\Gamma_{1/q}$ the billiard map is a Birkhoff twist map; Aubry–Mather theory yields Birkhoff periodic orbits of every intermediate rotation number and Mather sets for irrational ones. So the two-family hypothesis generates rich structure "in between" — but no contradiction, and the circles need not belong to a foliation: the gap to the Bialy/Bialy–Mironov-type hypotheses is exactly the missing foliation.

Result

The problem is not solved, and I could not solve it; the honest classification is partial progress via reformulation plus a precise map of how close the literature comes:

1. Near ellipses the answer is "no" in a strong sense (Kaloshin–Sorrentino 2018): a single one-parameter family of $q$-periodic orbits, $q\ge 3$, already characterizes ellipses locally among $C^\infty$ tables. So any counterexample to Problem 1 must be far (in a $C^\infty$ sense) from every ellipse.
2. The simplest case $(2,3)$ is deformationally settled near the circle (Kaloshin–Koudjinan 2021): no non-trivial deformation of the circle preserves both the 2-periodic family (constant width to first order) and the 3-periodic family; the same holds for $(2, 2l+1)$. Thus a non-circular constant-width curve with a 3-periodic family, if it exists, is isolated from the circle in a deformation sense.
3. Global results all need strictly stronger hypotheses: full foliation of the phase cylinder (Bialy 1993 $\Rightarrow$ disk), or a 1/4-caustic plus foliation below it with central symmetry (Bialy–Mironov 2022 $\Rightarrow$ ellipse). Two isolated rational invariant circles are not known to force a foliation — this is precisely the open gap.
4. Outer billiards: the algebraic/polynomial integrability versions are solved (Tabachnikov 2008; Glutsyuk–Shustin 2018: only ellipses), and full integrability near the curve is solved (Bialy 2023: only ellipses); the exact two-periods question remains open there as well.
5. Elementary but useful contributions recorded above: the $p=2$ $\Leftrightarrow$ constant-width reduction, and the constant-perimeter lemma for any one-parameter family of periodic orbits, which packages the hypothesis into two rotation numbers and two marked action constants.

What remains

• The $(2,3)$ case globally: does a (smooth, strictly convex) constant-width curve other than the circle admit a one-parameter family of 3-periodic trajectories? Open. Natural approaches: (a) extend the Bialy–Mironov integral-geometry/Hopf-rigidity machinery from the 1/4-caustic to the pair (1/2-, 1/3-caustics), exploiting that constant width gives an explicit circle of fixed points; (b) Fourier/collision-operator analysis near constant-width curves generalizing the Kaloshin–Koudjinan deformation computation beyond the circle.
• The general $(p,q)$ case for $p,q\ge 3$ globally, without proximity to an ellipse: open. Key obstacle: two rational invariant circles do not imply a foliation of the annulus between them; Birkhoff zones of instability may a priori occur there.
• Outer-billiard two-periods question: open; even a deformational analogue of Kaloshin–Koudjinan for outer billiards seems to be missing, and Bialy's new generating function for outer billiards (arXiv:2306.12494) is a plausible tool.
• A related open direction suggested by the literature: whether the two-period hypothesis implies rational integrability near the boundary (then Koval's local strong Birkhoff result would apply, settling the problem near almost every ellipse under any finite number of periods).