math.DS — Dynamical Systems

The necklace of a rational right triangle: exact counting and complete periodicity

Contributed by Joshua Boorstein

Let $T$ be a right triangle with acute angle $\alpha$, $2\alpha/\pi = P/Q$ in lowest terms. The Katok--Zemlyakov unfolding of $T$, quotiented by its holonomy involution, is a necklace of $Q$ kites: an explicit genus-zero half-translation surface in the stratum $Q(P-2, Q-P-2, -1^Q)$, whose kites glue in steps of $P$ and whose interfaces have explicit transverse measures. We use it to count the cells into which the vertices of $T$ cut the beam perpendicular to each side. For a completely periodic direction class the separatrix diagram has exactly $Q$ edges, and sorting them by the involution gives the identity $Q = h + f + 2p$ ($h$ sides of $T$ in the class, $f$ extra cells of the perpendicular beams, $p$ involution-swapped edge pairs). When $p = 0$ it is a covering law: the perpendicular beams of a class cut $Q - 1$ or $Q + 1$ cells in total at odd $Q$, according to the class, and $Q$ at even $Q$; with no hypothesis the shortfall is even, non-negative, and zero only on a completely periodic class. Complete periodicity of a class — that almost every trajectory in it meets a side at a right angle — is equivalent to $p = 0$ and to the covering law (on two-beam classes off the Veech locus, one direction assumes an orphan branch, which companion papers prove at broad classes of centres); it fails at $8/15$. For the beam perpendicular to a leg the count $n(P/Q)$ is at most $Q - 1$ at every centre and is known exactly on the Veech locus $\min(P, Q-P) \leq 2$; off it we prove $n(3/Q) \leq 2\cdot\text{round}(Q/3) - 1$ for $Q \geq 5$, with equality exactly under complete periodicity, and $n(5/(5m+1)) = 5$ for every $m \geq 3$. Finally, every periodic billiard path of a rational right triangle in a mirror direction meets a side at a right angle.

Author comments: First version. Companion papers: "The orphan theorem" (hexagon:2611.00191); "No arrivals" and "Non-perpendicular periodic orbits in mirror directions of right triangles" in preparation. Code: https://doi.org/10.5281/zenodo.24037744