research/AMR-005-0005.md

metadata

id: AMR-005-0005
classification: OPEN-TRIAGE
wording_corrected: 'no'

AMR-005-0005 — Completely periodic 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), DOI 10.1007/s40598-014-0001-3, Section 4 ("Polygonal Outer Billiards in the Hyperbolic Plane"), the second unnumbered problem. The original wording, verified verbatim against the published article (both the journal HTML page and the Springer PDF):

Another problem is to describe polygonal outer billiard tables in the hyperbolic plane for which all orbits are periodic. For example, right-angled regular $n$-gons (with $n \geq 5$) have this property (Dogru and Tabachnikov [2003]). In the affine plane, every outer billiard orbit about a lattice polygon is periodic.

The outer billiard map $T$ about a convex polygon $P$ is the piecewise isometry of the exterior of $P$ defined by reflecting the point $x$ in the support vertex of $P$ (the support line through $x$ having $P$ on the left). The dataset transcription ("Describe the polygonal outer billiard tables in the hyperbolic plane for which every orbit is periodic") is a faithful paraphrase; no correction needed. Note that Section 4 of the same source also contains Conjecture 2: every polygonal outer billiard in the hyperbolic plane has periodic orbits (possibly on the circle at infinity) — the existence counterpart to this classification problem, also open as far as I could verify.

Status / Literature

References verified via Crossref metadata and the arXiv API.

• F. Dogru, S. Tabachnikov, "On polygonal dual billiard in the hyperbolic plane", Regul. Chaotic Dyn. 8 (2003), 67–82, DOI 10.1070/RD2003v008n01ABEH000226. (Existence verified: this DOI, first page 67, appears as reference CR7 in the Crossref record of the authors' Math. Intelligencer paper; I could not obtain the full text.) This is the foundational paper on polygonal outer billiards in $\mathbb{H}^2$. Per the Baker's Dozen itself, it proves that right-angled regular $n$-gons ($n\ge 5$) have all orbits periodic — the mechanism being that such an $n$-gon tiles $\mathbb{H}^2$ by reflections, and the second iterate $T^2$ is compatible with the tiling group. Per the secondary literature (the ICERM REU problem list and the introduction of Dogru–Fischer–Munteanu below), the same paper introduces a class of "large" polygons (roughly, polygons whose side-extending geodesics are pairwise ultraparallel) for which every orbit escapes to infinity, so such tables have no periodic orbits in $\mathbb{H}^2$ at all — the basic obstruction to complete periodicity. I did not re-read DT03 itself, so these content attributions are via the sources cited.
• F. Dogru, S. Tabachnikov, "Dual billiards", Math. Intelligencer 27(4) (2005), 18–25, DOI 10.1007/BF02985854 (Crossref-verified). Survey containing the state of the art as of 2005.
• S. Tabachnikov, "Dual billiards in the hyperbolic plane", Nonlinearity 15 (2002), 1051–1072, DOI 10.1088/0951-7715/15/4/305 (Crossref-verified). Smooth dual billiards in $\mathbb{H}^2$; background for the induced map on the circle at infinity.
• F. Dogru, E. M. Fischer, C. M. Munteanu, "Outer billiards and tilings of the hyperbolic plane", Involve 8 (2015), 637–651, arXiv:1311.1930, DOI 10.2140/involve.2015.8.637 (arXiv API verified, including journal ref). Abstract (seen verbatim): "we present new results regarding the periodicity of outer billiards in the hyperbolic plane around polygonal tables which are tiles in regular two-piece tilings of the hyperbolic plane." This enlarges the known stock of completely periodic tables beyond right-angled regular polygons to tiles of "regular two-piece tilings" of $\mathbb{H}^2$.
• S. Tabachnikov, "A proof of Culter's theorem on the existence of periodic orbits in polygonal outer billiards", Geom. Dedicata 129 (2007), 83–87 (cited in the source article; not independently re-verified). Euclidean counterpart: every Euclidean polygon admits periodic outer billiard orbits, and every orbit about a lattice polygon is periodic — the contrast motivating the problem.

I found no published work (searches through 2026) that gives a complete classification or resolves either the classification problem or Conjecture 2. The problem is open.

Work done

• Retrieved the original Section 4 wording from the published AMJ article (HTML and PDF versions) and confirmed the dataset transcription.
• Crossref verification of the Tabachnikov 2002 (Nonlinearity) and Dogru–Tabachnikov 2005 (Math. Intelligencer) records; the Dogru–Tabachnikov 2003 DOI was confirmed via the verified reference list of the latter (a direct Crossref lookup of the neighbouring DOI ...000227 returned a mismatched record, so I report ...000226 as the correct one on that evidence).
• arXiv API verification of Dogru–Fischer–Munteanu (arXiv:1311.1930), including its Involve journal reference and DOI.
• Searches for post-2015 progress on the classification problem and on Conjecture 2 (existence of periodic orbits for arbitrary hyperbolic polygonal tables): nothing beyond the tilings paper above. An attempt to fetch the Involve PDF returned binary content, so precise theorem statements of that paper were taken from its arXiv abstract only.
• No computation performed (per constraints); the remarks in "Result" are pure reasoning.

Result

The literature state can be synthesized as follows.

Known completely periodic tables. (i) Right-angled regular $n$-gons, $n\ge 5$ (DT03); more generally (ii) polygonal tiles of "regular two-piece tilings" of $\mathbb{H}^2$ (Dogru–Fischer–Munteanu 2015). In both cases the proof strategy is tiling-based: the table is a fundamental domain (or a union of two tiles) of a discrete reflection group, and compatibility of $T^2$ with the group confines every orbit to a compact set of tiles on which the piecewise isometry has uniformly finite order.

Known obstruction. "Large" polygons in the sense of DT03 (side-geodesics pairwise ultraparallel): every orbit escapes to the circle at infinity, so no complete periodicity — indeed no periodic orbits in $\mathbb{H}^2$ whatsoever.

A necessary condition from the dynamics at infinity (my synthesis, not a published theorem). Write $R_i$ for the half-turn (elliptic involution) about vertex $v_i$, and $A_i$ for the exterior region on which $T = R_i$. For distinct $i, j$ the product $R_iR_j$ is loxodromic: a hyperbolic translation by $2,d(v_i,v_j)$ along the geodesic through the two vertices, with two fixed points on $\partial\mathbb{H}^2$ and none in $\mathbb{H}^2$. Consequently, if $x$ is a periodic point of $T$ with itinerary word $w = R_{i_1}\cdots R_{i_m}$, then $w(x)=x$, and since loxodromic (and parabolic) isometries fix no point of $\mathbb{H}^2$, the word $w$ must be elliptic or trivial. Hence:

A polygonal table is completely periodic only if every admissible itinerary that is realized by a periodic orbit has an elliptic product of vertex half-turns; and every admissible infinite itinerary whose word-growth produces loxodromic products with an attracting basin covering the realizing region forces escape to infinity.

This is exactly the DT03 mechanism for large polygons, and it explains why all known completely periodic examples come from reflection tilings: for tiling polygons the relevant words lie in a discrete reflection group and the admissible itineraries are forced to be elliptic. A full classification would require showing that, conversely, any polygon whose side-geodesics intersect (a "small" polygon) with all admissible periodic itineraries elliptic is necessarily of tiling type — or exhibiting a counterexample. Neither direction is presently known; even the existence of a single aperiodic orbit for some small polygon (which would kill the hope that all small polygons are completely periodic) is not established in the literature I could verify, and the weaker Conjecture 2 (existence of one periodic orbit for every table) is open.

What remains

• The full classification: no necessary-and-sufficient geometric condition on $P$ is known. Open even for specific simple shapes, e.g. arbitrary (non-right-angled) regular $n$-gons, or right-angled irregular pentagons/hexagons.
• Conjecture 2 of the source (every polygonal table has at least one periodic orbit, possibly at infinity) is open; on the sphere there are polygons with no periodic outer billiard orbits, so the hyperbolic case cannot be settled by uniform arguments.
• Precise delineation of the "small/large" dichotomy of DT03: whether every small polygon has a periodic orbit, and whether completely periodic tables must be "quasirational"/tiling-type in a suitable hyperbolic sense.
• Whether bounded but aperiodic orbits (the hyperbolic analogue of the Euclidean irrational-polygon phenomenon, cf. Schwartz's resolution of the Moser–Neumann question) can occur for polygonal tables in $\mathbb{H}^2$.
• Next concrete steps: read DT03 (Regul. Chaotic Dyn. 8 (2003), 67–82) and Dogru–Fischer–Munteanu in full to extract exact definitions ("large", "regular two-piece tiling") and check whether the map on the circle at infinity for small polygons must always have an attracting periodic point — a plausible route to showing that complete periodicity is equivalent to the tiling property.