math.DS — Dynamical Systems

The orphan theorem: degenerate perpendicular orbits in rational right triangles

Contributed by Joshua Boorstein

Let $T$ be a right triangle with acute angle $\alpha$, legs $L_1$, $L_2$ and hypotenuse $H$, and suppose $T$ is rational: $2\alpha/\pi = P/Q$ in lowest terms. Shoot a beam perpendicular to the leg $L_1$ from each of its points. The leg is partitioned into branches — maximal intervals of launch heights sharing a combinatorial type — and we prove that the partition is finite, with at most $2Q + 1$ branches, which cover the leg but for finitely many points. The first-return map $J$ of the beam to $L_1$ is an involution, and on each branch it does one of two things: it carries the branch onto a distinct branch of equal width by a reflection, or it fixes the branch pointwise. A pointwise-fixed branch — a whole interval of trajectories that hit a side head-on and retrace — is an orphan. Theorem A. There is no orphan branch when $P$ is even and at most one when $P$ is odd, with no hypothesis on the direction; when $P$ is odd and $Q$ is a prime power, or twice or four times a prime, the orphan exists. Non-existence is a parity count on the velocity set of the unfolded beam. Uniqueness is an Euler count on the genus-zero quotient of the unfolded surface, which admits minimal components. Existence is arithmetic: on a folded disc, an absent orphan forces $1$ to be a signed sum of doubled heights of images of the right-angle vertex, which is impossible at the centres named. Running the direction count and the Euler count for each side shows that at most two of the three sides carry a retracing cell, at most one each, the two ends of one band of double normals. On the unfolded surface — the holonomy double cover of a genus-zero quadratic differential $Q(P-2, Q-P-2, -1^Q)$ — the images of the right-angle vertex are Weierstrass points, the leg is the quotient by the hyperelliptic involution of a geodesic segment through one of them, and the first-return map is read in that quotient.

Author comments: First version. Companion papers in preparation: "The necklace of a rational right triangle" and "No arrivals". Code: https://doi.org/10.5281/zenodo.24037744