---
id: AMR-005-0004
classification: OPEN-TRIAGE
wording_corrected: no
---

# AMR-005-0004 — Periodic orbits of polygonal outer billiards in the hyperbolic plane

## 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, Section 4 ("Polygonal Outer Billiards in the Hyperbolic Plane"), Conjecture 2. The published wording, verified verbatim against the journal HTML:

> **Conjecture 2.** Every polygonal outer billiard in the hyperbolic plane has periodic orbits. These orbits may lie on the circle at infinity.

The dataset transcription ("Does every polygonal outer billiard in the hyperbolic plane have periodic orbits, possibly lying on the circle at infinity?") is a faithful question-form restatement; no correction was needed.

Context from the same section: the outer billiard map about a convex polygon $P$ reflects a point $x\notin P$ in the support vertex of the tangent line through $x$ having $P$ on the left. C. Culter proved that every polygon in the Euclidean (affine) plane admits periodic outer billiard orbits (Tabachnikov 2007). On the sphere there exist polygons without any periodic outer billiard orbits. A companion problem in the same section: describe the hyperbolic polygonal tables for which *all* orbits are periodic (right-angled regular $n$-gons, $n\ge5$, have this property by Dogru–Tabachnikov 2003).

## Status / Literature

All citations below verified via Crossref metadata or the arXiv API (abstracts/journal refs seen verbatim).

- **F. Dogru, S. Tabachnikov, "On polygonal dual billiard in the hyperbolic plane", *Regul. Chaotic Dyn.* 8 (2003), 67–82, DOI 10.1070/RD2003v008n01ABEH000226** (Crossref metadata verified: authors, journal, volume, year, first page 67; full text not accessed, but its main theorems are restated verbatim in the two papers below). Establishes: (i) the outer billiard map extends continuously to a circle homeomorphism $f$ on the circle at infinity, with a well-defined Poincaré rotation number $\rho$; (ii) a class of "large" $n$-gons — those for which $\rho(f)=1/n$ and $f$ has a (hyperbolic, i.e. attracting) $n$-periodic orbit at infinity; for a triangle, large $\iff H>1$ where $H=\sinh h_i\sinh a_i=\sin\alpha_i\sinh a_{i+1}\sinh a_{i+2}=\dots$ (explicit hyperbolic-trigonometric quantity; $\rho=1/3$ iff $H\ge1$, with $H=1$ giving a unique 3-periodic orbit at infinity); (iii) **if $C$ is a large polygon then all orbits of the dual billiard map escape to infinity** — so for large polygons the periodic orbits exist precisely on the circle at infinity; (iv) for right-angled regular $n$-gons ($n\ge5$), every orbit is periodic, with $\rho(f)=\bigl(n-\sqrt{n(n-4)}\bigr)/(2n)$ (irrational).
- **S. Tabachnikov, "A proof of Culter's theorem on the existence of periodic orbits in polygonal outer billiards", *Geom. Dedicata* 129 (2007), 83–87, DOI 10.1007/s10711-007-9196-y** (Crossref verified). The Euclidean analogue of the conjecture: every polygon in the affine plane admits periodic outer billiard orbits.
- **F. Dogru, E. M. Fischer, C. M. Munteanu, "Outer Billiards and Tilings of the Hyperbolic Plane", *Involve* 8 (2015), 637–651, DOI 10.2140/involve.2015.8.637, arXiv:1311.1930** (arXiv API verified; journal ref seen verbatim; full text read). Extends the all-orbits-periodic result to tables that are tiles of regular two-piece $(M,N)$-tilings of $\mathbb{H}^2$ (four tiles per vertex, $1/M+1/N<1/2$): for $(3,N)$, $N\ge7$, and for $M,N\ge4$, the map preserves the rank of each tile, hence every orbit is periodic; explicit formulas for the number of tiles of each rank and for $\rho(f)$ are given. (Full text read — the paper does not address arbitrary polygons.)
- **T. Noda, S. Yasutomi, "Billiards in a circle with trajectories circumscribing a triangle", arXiv:2111.04495 (2021, preprint; no journal ref listed in the arXiv record as of 2026-08)** (abstract and full text read via arXiv/ar5iv). Reproves and Euclidean-izes the Dogru–Tabachnikov largeness criterion for triangles in the Klein–Beltrami model: a triangle is large iff a certain altitude-type quantity exceeds $\Delta(P,Q)=\log\coth(d(P,Q)/2)$; equivalently iff there exist two triangles inscribed in the circle at infinity and circumscribing it (these are the 3-periodic orbits of $f$). Restates DT2003's Theorems 1.1–1.2 verbatim (used above).
- **T. Noda, S. Yasutomi, M. Yoshida, "Star-shaped trajectories of certain billiards around a triangle", arXiv:2304.08148 (2023, preprint; no journal ref listed as of 2026-08)** (abstract seen verbatim via arXiv API). Studies triangle outer billiards at infinity with rotation number $2/5$: gives a sufficient condition for $\rho=2/5$ (and necessity for large isosceles triangles), i.e. further families with 5-periodic orbits at infinity; ends with a conjecture.
- Background on the rotation-number calculus used by both preprints: $\rho$ is monotone under inclusion of tables (DT2003 Lemma 1: $C_1\subset C_2\Rightarrow \rho(C_1)\ge\rho(C_2)$) and continuous in the table, so rational values of $\rho$ — hence periodic orbits at infinity — persist on open regions of table space near any large polygon.

**Open status.** I found no publication solving the conjecture for arbitrary convex polygons in $\mathbb{H}^2$. An arXiv API search ("outer billiard" AND "hyperbolic", 10 hits) and a web search turned up only the partial results above; recent activity (2021–2023 preprints, a 2024–2025 line of work on outer billiards in higher-rank/complex hyperbolic spaces by Godoy–Harrison–Salvai, arXiv:2110.01679 and arXiv:2503.06865) treats special classes or different settings, not the general conjecture. As of this review the conjecture appears open.

## Work done

- Retrieved the original statement from the published AMJ article (publisher HTML) and confirmed the dataset wording is faithful (question form of Conjecture 2).
- Verified Dogru–Tabachnikov 2003 (DOI 10.1070/RD2003v008n01ABEH000226) and Tabachnikov 2007 (DOI 10.1007/s10711-007-9196-y) against Crossref records.
- Read the full text of Dogru–Fischer–Munteanu (arXiv:1311.1930) and Noda–Yasutomi (arXiv:2111.04495), and the abstracts of arXiv:2304.08148, arXiv:2110.01679, arXiv:2503.06865 via the arXiv API, to map exactly which cases are settled.
- Searched for post-2015 resolutions (arXiv API: "outer billiard" AND "hyperbolic"; web search on the conjecture). Nothing claims a general solution.
- Reasoned about the structure of the problem (below) but did not find a new proof; the general case appears genuinely hard (its Euclidean inner-billiard analogue — periodic orbits in every triangle — is a famous open problem despite intensive work).

## Result

Synthesis of the rigorous state of the art, with a structural reformulation.

1. **Reformulation.** In $\mathbb{H}^2$ the reflection of $x$ in a support vertex $v$ is the half-turn $H_v$ about $v$ (an orientation-preserving isometry). Hence the outer billiard map $T$ is a piecewise orientation-preserving isometry, and an orbit with periodic itinerary through vertices $v_1,\dots,v_k$ closes iff the composition $H_{v_k}\circ\cdots\circ H_{v_1}$ has a fixed point realizing that itinerary — i.e. iff this composition is *elliptic* (a rotation) with fixed point in the appropriate continuity cell, or *parabolic/hyperbolic* with an (attracting) fixed point on the circle at infinity. The conjecture thus asks: for every convex polygon, does some periodic itinerary produce a non-hyperbolic composition (or a hyperbolic one with fixed points at infinity)? This is the hyperbolic analogue of the "elliptic composition" mechanism behind Culter's Euclidean theorem.

2. **Settled cases.**
   - *Large polygons* (in particular all triangles with $H>1$): all interior orbits escape to infinity, and $f$ has an attracting $n$-periodic orbit on the circle at infinity — the conjecture holds, with the periodic orbits at infinity exactly as the conjecture allows (Dogru–Tabachnikov 2003; quantitative triangle criterion reproved by Noda–Yasutomi 2021).
   - *Right-angled regular $n$-gons* ($n\ge5$) and *tables of two-piece regular $(M,N)$-tilings*: **every** orbit is periodic (interior orbits; the web coincides with the tiling's grid lines, rank is preserved, finitely many tiles per rank, so some iterate is the identity on each tile) — Dogru–Tabachnikov 2003; Dogru–Fischer–Munteanu 2015.
   - *Triangle tables at infinity with $\rho=p/q$ rational*: periodic orbits at infinity exist; families realizing $\rho=1/3$ (DT2003) and $\rho=2/5$ (Noda–Yasutomi–Yoshida 2023) are explicitly characterized.

3. **The gap.** For a "small" generic polygon (one not contained in any tiling and failing the largeness conditions), the map at infinity typically has irrational rotation number (so no periodic orbits at infinity), and interior orbits are bounded but aperiodic in general. Nothing in the literature produces even a single periodic orbit for an arbitrary such table; the tiling-based proofs rely essentially on the global grid structure, and the large-polygon arguments force escape to infinity, leaving no interior periodic orbits. The two known mechanisms are complementary and each covers a measure-zero-ish/structured part of the space of polygons.

## What remains

- The full conjecture for arbitrary convex polygons, especially "small" ones with bounded, non-tiling dynamics: no periodic-orbit existence result is known. Even the case of an arbitrary (non-large, non-right-angled) triangle seems unproved.
- Decide whether $\rho(f)$ rational can occur at all without a periodic orbit at infinity being realizable, and conversely classify tables with $\rho(f)$ irrational but possessing interior periodic orbits (the tiling examples show this happens).
- Characterize all "totally periodic" tables (companion problem stated by Tabachnikov): known examples are the right-angled regular $n$-gons and the $(M,N)$-tiling tables; are there others not coming from tilings?
- Natural next steps: (a) perturbative arguments near tiling tables, using continuity of $\rho$ and stability of hyperbolic/attracting periodic orbits at infinity; (b) an extremal/variational approach à la Culter–Tabachnikov (maximize perimeter or area over candidate $k$-periodic inscribed configurations) adapted to $\mathbb{H}^2$, where compactness must come from the boundedness of orbits for small polygons; (c) computational search for periodic cells of the web for small triangles to guide conjectures (outside the scope of this review).
- Caveat: the two preprints arXiv:2111.04495 and arXiv:2304.08148 had no journal reference in the arXiv record at the time of review; their restatements of DT2003's theorems are internally consistent with Dogru–Fischer–Munteanu's, but the original 2003 text itself was not read (journal full text not freely accessible).