The orphan theorem: degenerate perpendicular orbits in rational right triangles

Introduction

The question

A billiard trajectory that meets a side of a polygon perpendicularly reverses and retraces its path. Fixing a side and launching perpendicularly from each of its points therefore organises the dynamics: two launch points behave the same way exactly when their trajectories strike the same sequence of sides, and the side decomposes into intervals — branches — on which that sequence is constant.

Concretely, place the right triangle $T$ with vertices $O = (0, 0)$, $R = (\cot\alpha, 0)$ and $A = (\cot\alpha, 1)$, so that $L_1 = RA$ is the vertical launch leg, $L_2 = OR$ the horizontal leg and $H = OA$ the hypotenuse, and the acute angle at $O$ is $\alpha$. Launch the beam from $(\cot\alpha,s)$, $s \in(0,1)$, with velocity $(-1,0)$. Write

$n(P/Q) :=$ the number of branches the perpendicular beam cuts on $L_1$, when the triangle is rational, $2\alpha/\pi= P/Q$ with $\gcd(P,Q)=1$ and $0<P<Q$. We call the rational number $P/Q$ — equivalently the pair $(P,Q)$, equivalently the triangle itself — the centre, and the word is used in that sense throughout. The generic branch is a strip of parallel periodic trajectories. What we study is the exception.

Why the exception matters. The partition at $P/Q$ is one stage of a hierarchy: moving $\alpha$ toward the next continued-fraction approximant changes the partition, and the change is not uniform. A mirror pair’s perpendicular return ends on a copy of the launch leg, and that copy keeps its direction as $\alpha$ moves (Corollary 4.14: the pair turns at $R$, where the two leg reflections compose to $-1$ for every $\alpha$). The orphan’s turnaround lies on a wall copy carried a nonzero number of rhombus steps (Definition 4.4), whose direction moves with $\alpha$. We therefore expect a pair to persist under small detuning and the orphan to break into new branches, so that the orphan, not the generic branch, carries the growth of $n(P/Q)$ along the approximants of an irrational angle, and its existence and uniqueness are where that growth starts. We prove nothing about detuning here.

The orphan, and the main theorem

Unfolding keeps the beam on a fixed horizontal line, and the copy of $L_1$ on which it returns perpendicularly is, like the launch copy, a vertical segment. The first perpendicular-return map $J$ to the launch leg is therefore an isometry of the launch height, affine with $|\text{slope}|=1$, and an involution (Theorem 2.2 below). This forces a dichotomy on each branch, and the dichotomy is the subject of the paper:

• a branch is carried isometrically onto a distinct branch of equal width — generic branches occur in mirror pairs, and $J$ reverses orientation on them, with slope $-1$ (Lemma 4.9); or

• a branch is fixed pointwise: every trajectory launched in it meets a side head-on and retraces to its own launch height. We call such a branch an orphan, since it has no partner.

Theorem A says when an orphan can exist, that there is never more than one, and where there is one:

Every orphan orbit turns around at a single head-on collision, on a side of type $H$ when $Q$ is odd and of type $L_2$ when $Q$ is even. Consequently $n(P/Q)$ is even for even $P$, and for odd $P$ it is odd exactly when the orphan exists — in particular whenever $Q$ is a prime power, or twice or four times a prime.

We conjecture that (c) holds without the restriction on $Q$ (Conjecture 4.12); the companion paper [21] proves it when $\min(P,Q-P) \ge2$ and either $\gcd(P+2,Q)=\gcd(P-2,Q)=1$ (its Theorem 2) or $Q$ has at most one odd prime factor (its Theorem 3); Theorem 4.18 below covers $\min(P,Q-P)=1$. The three parts stand on different footings, and we keep them apart throughout.

Non-existence and the turnaround type are elementary and hold at every $Q$. A head-on hit on a side requires the beam direction to equal that side’s normal, and along a perpendicular orbit the direction takes only the $Q$ values $k\pi/Q$ modulo $\pi$ (Lemma 4.1). Coprimality makes the required direction unattainable when $P$ is even, and when $P$ is odd attainable on exactly one side type — the hypotenuse at odd $Q$, the far leg at even $Q$ — which types the turnaround.

Uniqueness is not elementary. Counting vertical directions does not bound the vertical copies a single return passage meets, since the return word can be far longer than one turn of the direction cycle. What settles uniqueness is the unfolded surface $S_\alpha$ of § 6 and its genus-zero quotient of § 7. Each orphan branch lies in a cylinder that the deck involution moves to a different cylinder, and an Euler count on the quotient, which allows minimal components, leaves room for at most one exchanged pair. § 4.3 states the theorem and § 7.6 proves it.

Existence is not elementary either. Folding the development by its symmetries of the horizontal direction gives a flat disc on which every horizontal leaf from a non-singular point of the launch leg ends either back on the leg or on the one vertical copy of the far side (at odd $P$), and an orphan is a leaf of the second kind. If there is none, Gauss–Bonnet sorts the leaves into fans, each turning about one image of the right-angle vertex; the fans tile the leg, so 1 is a signed sum of twice the heights of those images. When $Q$ is a power of a prime $p$ every doubled height lies in the prime of $\mathbb{Q}(e^{i\pi/Q})$ over $p$, and 1 does not (Theorem 4.11). When $Q$ has two distinct prime factors that membership no longer excludes 1 (Remark 4.13), and § 4.5 uses more of the configuration. Modulo 2 the identity becomes linear over $\mathbb{F}_2$, indexed by the images whose horizontal rays reach the leg, and near the two acute corners some of those rays are forced onto the leg and others onto the far wall. When enough are forced the identity has no solution (Theorem 4.18); when fewer are it has one (Proposition 4.19). This settles the edges of every row, and every odd $P$ at $Q=2p$ (Corollary 4.20); at $Q=4p$ three more rays, followed by hand, cover the row (Theorem 4.24).

Theorem A fixes the parity of $n(P/Q)$, not its value, which is arithmetically irregular (Remark 5.2) and is the subject of the companion paper [22].

The orphan is a property of the triangle, not of the launch leg. Each of the three sides carries its own perpendicular beam and its own retracing criterion — $L_1$ requires $P$ odd, $L_2$ requires $P+Q$ odd, $H$ requires $Q$ odd — so at most two of the three sides carry a retracing cell, a cell of that side’s own perpendicular partition whose orbits retrace. When they do, the two cells are the two ends of one band of double normals, billiard segments meeting the boundary perpendicularly at both of their ends. § 5 states this precisely and proves it (Theorem 5.3).

The surface reading

The second half of the paper reads Theorem A on the unfolded surface, and proves its uniqueness half there. The Katok–Zemlyakov unfolding $S_\alpha$ of $T$ is a closed translation surface glued from $4Q$ copies of $T$, of genus $\lfloor Q/2\rfloor$ (Theorem 6.1), and its rotation through $\pi$ exhibits it as the holonomy double cover of the genus-zero quadratic differential $Q(P-2,Q-P-2,-1^Q)$ on $\mathbb{CP}^1$ (Proposition 7.1), as Apisa observes [1 §1.2, §7]. The deck involution $\tau$ is then the hyperelliptic involution, and its fixed points — the Weierstrass points — are the preimages of the odd-order singularities of the base: the $Q$ points over the right angle, which are regular points of the translation surface but branch points of the cover, together with $O$ when $P$ is odd and $A$ when $Q-P$ is odd, $2g+2$ in all (Proposition 7.2). The points over the right angle are the ones the orphan problem uses. $\tau$ does not preserve the launch leg, so $J$ is not a restriction of $\tau$; instead the two copies of the leg in each direction meet at a point over $R$ and form one geodesic segment, $\tau$ reflects that segment about the point, and $J$ is the first return to it read in the quotient (§ 7.4). The same surface carries the proof of uniqueness: each orphan uses up a $\tau$-swapped pair of cylinders, and an Euler count on the genus-zero base allows at most one such pair (§ 7.6). That $O$ is a Weierstrass point exactly when $P$ is odd, the parity condition of Theorem A, is a match of conditions with a one-line explanation in the reflection group, not a map from the orphan to that point (Remark 7.3).

What is proved, what is classical, and what is cited

Every numbered statement is proved here except Conjecture 4.12, which is labelled as one (it is proved on large families in [21]; § 4.4), and none rests on computation: the exact branch counts the paper quotes (Figure 3, Remark 5.2, the census behind Conjecture 4.12) carry the completeness certificate of Appendix A and illustrate proved statements, their limits or the conjecture. Where a remark quotes a value proved elsewhere it cites the proof, and no proof depends on it. Three proofs are completed away from their statements: Theorem 4.7’s at § 7.6, which also carries the one-cell clauses of Theorem 5.3, and Lemmas 4.22 and 4.23, whose trajectories are followed in Appendix B. No proof takes anything from the companion paper [22], which imports results of this one; Remark 5.2 quotes one of its counts for comparison. The external inputs are classical: Boshernitzan’s theorem that a perpendicular orbit in a rational polygon avoiding the vertices is periodic [6 Prop. 1.8] (also [12 §5]), the first step of Proposition 3.4; the Katok–Zemlyakov unfolding [33] and its cone-angle bookkeeping; the decomposition of a direction on a flat surface into cylinders and minimal components [Str84] §11.4, [MT02], used in Lemma 7.8; the Gauss–Bonnet formula for a flat disc with cone points [28], used in §4.4; the classical fact that the fixed points of the hyperelliptic involution are the Weierstrass points [11] (Ch. III, §7); and the arithmetic of cyclotomic fields [32]. Apisa’s identification of the genus-zero base [1], the description of the branch points of its double cover in [3], the classification of the lattice locus [1, 17] and McMullen’s regular-polygon models [20] are cited for comparison and are not inputs: Proposition 7.1 derives what we use. No proof in §§2–5 uses surface theory except §7.6’s, on which Theorem 4.7 and the one-cell clauses of Theorem 5.3 rest; a reader can stop after §5 with non-existence, the turnaround type and existence (Proposition 4.3, Theorems 4.6, 4.11, 4.18 and 4.24, Corollary 4.20) and their proofs intact.

Related work

The perpendicular problem. The perpendicular class, and the partition of a side by it, were studied numerically before they were studied as theorems. Ruijgrok [24] singles out “all orbits starting in a direction perpendicular to a side”, presents evidence that “almost all orbits of this kind are either periodic or end in a corner”, and observes that “the starting points with the same period form intervals” — the partition of this paper, for arbitrary and mainly irrational triangles; he attributes the original observation, for a right triangle and a hypotenuse launch, to Rabouw and Ruijgrok [RR81]. What he offers is numerical evidence and, in his own words, “plausibility arguments”; the statements he draws from it for a right triangle are the elementary ones, and they are consistent with the partition as we describe it. Cipra, Hanson and Kolan [7] proved that in a right triangle almost every trajectory launched perpendicular to a leg returns to that leg in the same direction — the form in which [26] Theorem 7.5 states it — which is what makes the branch partition well-defined in the irrational case. Troubetzkoy [29] treated irrational right triangles whose smaller angle lies in $(\pi/6,\pi/4)$: every perpendicular periodic beam is symmetric (Theorem 1, whose rational counterpart is Corollary 4.14), there is at most one non-singular escape orbit in the perpendicular direction (Theorem 2), and every other non-singular orbit on the invariant surface of that direction is periodic (Theorem 3); the sequel [30] proves periodic orbits dense in the phase space of every irrational right triangle. All of these concern measure, density or symmetry, and outside the rational case except where [7] recovers the known rational result. Our setting is the rational one, where the partition is finite and we determine its degenerate element exactly. Galperin and Zvonkin [13] treat the same trajectories by elementary means — the rhombus code word, and (§4B) a proof that every perpendicular periodic trajectory in a right triangle is unstable, the number of its links never being divisible by 4 as [12]’s criterion requires of a stable one — and report the conjecture of [7]. (Their mirror trajectories, after [12], are a different object from our mirror pairs.) In [7] that conjecture is a speculation drawn from computation: in every triangle the authors examined, the launch heights resolved into periodic beams but for a single “mystery” interval whose periods exceeded their computation, which inclines them to “speculate that there may be at most one nonperiodic trajectory left over”. The qualifier per direction is [13]’s and the qualifier nonsingular is [29]’s; neither is in [7]. The orphan is not that object: it is periodic, and by Lemma 2.1 its turnaround lies in the relative interior of a side, so it is nonsingular in the sense [13] gives the word (a trajectory that meets a vertex; the right angle is exempted there as a removable singularity). The conjecture is the irrational question, and [29] Theorems 2 and 3 together prove it for irrational right triangles whose smaller angle lies in $(\pi/6,\pi/4)$. Hooper [16] extended the instability to every periodic billiard path in a right triangle.

Perpendicular families by elementary construction. Holt [15] constructs periodic paths in an arbitrary right triangle — no rationality assumed — by two partial tilings: the rhombus of four copies about the right angle, and a fan of $2N+2$ copies reflected about the smallest-angle vertex, where $N=\lceil\pi/2\beta\rceil-1$ for $\beta$ the smallest angle; with $\beta=\alpha$ and $\alpha/\pi=P/2Q$ one has $N+1=\lceil Q/P\rceil$. His paths are perpendicular and retracing in the sense of § 1.1 — each “leaves either the hypotenuse or the longer leg at a right angle . . . and retraces the directed segments in the opposite direction” — so they are members of the perpendicular class this paper partitions, reached from the other two sides. What he proves is existence, one family at a time: for each $k=1,\ldots,N-1$ every interior point of the shorter leg lies on a path of $4k+2$ reflections per cycle, and for $k=N$ the points of a subsegment do. That is $N$ families each sweeping the leg, not a partition of one beam, and the terminal subsegment is where the last family survives rather than a cell of a partition; neither the count $n(P/Q)$ nor the orphan is visible from it. A second result of his is a genuine negative, and of a kind we do not reach: there are no periodic paths of 3, 5 or 7 reflections per cycle in any right triangle, so Birkhoff’s theorem for smooth convex bodies has no polygonal analogue.

Counting, the lattice locus, and genus-zero strata. The counting literature for rational billiards — Veech [31], Eskin–Masur [10], Apisa [1], Gutkin–Rams [14] — counts orbits in all directions up to length $L$, and the work on the stratum $Q(P-2,Q-P-2,-1^{Q})$ our base inhabits — Athreya–Eskin–Zorich [3], Boissy [Boi09], Delecroix–Goujard–Zograf–Zorich [8] — is asymptotic, statistical or generic over the stratum. Our statements are about one direction, at every admissible $(P,Q)$, and the only count this paper fixes is the parity of $n(P/Q)$; the comparison with that literature belongs to the count itself and is made in [22].

Organisation

$§ 1.4. Kenyon–Smillie’s criterion [17] for a lattice triangle, “smallest angle $\pi/n$”, is $\min(P,Q-P) \in\{1,2\}$ for a right triangle; on that locus McMullen [20] shows that the per-direction combinatorics of the regular polygons is controlled by arithmetic, and where his results meet ours — the Weierstrass points — the comparison is $§ 7.5. We do not present the genus-zero construction itself as new.

${xref}`\S2` sets up the partition and proves {xref}`Theorem 2.2`.$§ 3 gives the unfolding dictionary and the fixed development, and counts the vertex images to prove the partition finite (Proposition 3.4). $§ 4 proves Theorem A’s non-existence half and turnaround type (Proposition 4.3, Theorem 4.6), states its uniqueness half (Theorem 4.7), proves existence when $Q$ is a prime power ($§ 4.4, Theorem 4.11) — reading off from the same disc that mirror pairs have slope $-1$ and are adjacent (Lemma 4.9, Corollary 4.14) — and proves it at the edges of every row, at $Q=2p$ and at $Q=4p$ (${xref}`\S4.5`).$§ 5 collects the consequences: the parity, what it leaves undetermined, and the three-sided refinement. $§ 6 computes the surface data of $S_\alpha$; § 7 derives the genus-zero base, identifies the Weierstrass points, proves the fold (§ 7.4) and, at ${xref}`\S7.6`, proves {xref}`Theorem 4.7`.$§ 8 states what is not settled. Appendix § 9 describes the exact computation behind every count the paper quotes, and Appendix § 10 follows the two trajectories of $§ 4.5 by hand.

The perpendicular partition

${xref}`\S2.2` uses the unfolding apparatus of$§ 3.1 — the developed copies and the isometries placing them (Lemma 3.1) — which is a construction and uses nothing from $§ 2.

Branches

Fix the triangle and the beam as in $§ 1.1: $T$ has vertices $O=(0,0)$, $R=(\cot\alpha,0)$ and $A=(\cot\alpha,1)$, with $L_1=RA$, $L_2=OR$, $H=OA$, and the beam leaves $(\cot\alpha,s)$, $s \in(0,1)$, with velocity $(-1,0)$. Let $\gamma_s$ denote the resulting billiard trajectory.

Follow $\gamma_s$ until it next meets $L_1$ perpendicularly, and record the sequence of sides it strikes in between. Call that finite sequence the word $W(s) \in\{L_1,L_2,H\}^{*}$ of the launch height, and call a maximal open interval of heights on which the word is constant a branch. Write $n(P/Q)$ for the number of branches. The map that sends a launch height to the height of that first perpendicular return is the first-return map $J$; it is defined at every $s$ whose trajectory avoids the vertices, that is, at every $s$ interior to a branch.

Two heights lie in the same branch exactly when their trajectories are combinatorially identical, so a branch is an equivalence class of the dynamics rather than an artefact of how the count is organised. Everything in $§§§ 2–§ 5 is a statement about this partition.

That the partition is well defined. Three statements are easily run together here. The first two hold unconditionally and are ours to use; the third is stronger than both and is false in general. We separate them once, and keep them separate throughout.

• The number of branches is finite, and the branches cover $L_1$ but for finitely many points, so their widths sum to 1. This is Proposition 3.4, proved in §3.3 with $n(P/Q) \le2Q + 1$, and it is what rationality buys: a perpendicular launch reaching no vertex is periodic by Boshernitzan’s theorem [6 Prop. 1.8], and the launches that do reach a vertex number at most $2Q$, by a count of the horizontal rays arriving at vertex images.

• The branches cover $L_1$ in the sense of measure, in every right triangle, rational or not — almost every trajectory launched perpendicular to a leg returns to it in the same direction [7], in the form of [26] Theorem 7.5. On the rational locus this is weaker than Proposition 3.4, and this paper does not rest on it.

• The perpendicular direction on the unfolded surface is completely periodic — the surface is, in that direction, a finite union of closed cylinders, so that every leaf is closed or a saddle connection. This is strictly stronger than either of the above, and it is false in general. It is part of Boshernitzan’s Conjecture 2.2(e) [6]; a remark at the end of [29] §3 states that its method verifies it on an infinite family of rational right triangles and the companion paper [21] on larger ones ([22] supplies a proof of Troubetzkoy’s remark), and on the lattice locus it follows from Veech’s dichotomy [31 Propositions (2.10)–(2.11)]. But [9] certify a minimal component in a direction perpendicular to a side of the triangle of 8/15. It is a property of a direction: each centre carries two perpendicular-beam direction classes, which decompose $S_\alpha$ independently — at odd $Q$ the $L_1$ beam and the $L_2$ beam lie in different classes, at even $Q$ in the same one, the other being the class of the $H$ beam — and every direction perpendicular to a side lies in one of them, so at 8/15 one of the two classes is not completely periodic. No statement of this paper assumes complete periodicity.

The third deserves a word, because the gap between it and the first two is not an artefact of what we have managed to prove. At 8/15 one of the two perpendicular-beam directions carries a minimal component of positive area, and the branch partition of the leg it is perpendicular to is nonetheless finite, complete, and of total width 1. There is no contradiction, and Proposition 3.4 — which holds for either leg, relabelling the legs replacing $P/Q$ by $(Q-P)/Q$ — says something sharper than an appeal to measure would: the leg misses the minimal component entirely. A launch interior to a minimal component is not periodic, so by Proposition 3.4 it is one of the finitely many singular launches; but the leg meets an open subsurface in a relatively open set, and an open subset of the leg is either empty or infinite. Hence it is empty, and the leg can meet the component’s closure in at most finitely many points, on its boundary saddle connections. Complete periodicity is a statement about the whole surface; finiteness and covering are statements about the leg, and the leg can miss what the surface contains.

The involution

The first-return map is rigid, and this is the structural fact the paper turns on. Before the involution itself we isolate the criterion that decides which branches retrace. It is elementary — time-reversibility and uniqueness of billiard orbits — and it is used in the proof of Theorem 2.2 below and again in §4.