The Boca-Cobeli-Zaharescu Map Analogue for the Hecke Triangle Groups $G_q$

The Farey sequence $\mathcal{F}(Q)$ at level $Q$ is the sequence of irreducible fractions in $[0, 1]$ with denominators not exceeding $Q$, arranged in increasing order of magnitude. A simple ``next-term'' algorithm exists for generating the elements of $\mathcal{F}(Q)$ in increasing or decreasing order. That algorithm, along with a number of other properties of the Farey sequence, was encoded by F. Boca, C. Cobeli, and A. Zaharescu into what is now known as the Boca-Cobeli-Zaharescu (BCZ) map, and used to attack several problems that can be described using the statistics of subsets of the Farey sequence. In this paper, we derive the Boca-Cobeli-Zaharescu map analogue for the discrete orbits $\Lambda_q = G_q(1, 0)^T$ of the linear action of the Hecke triangle groups $G_q$ on the plane $\mathbb{R}^2$ starting with a Stern-Brocot tree analogue for the said orbits. We derive the next-term algorithm for generating the elements of $\Lambda_q$ in vertical strips in increasing order of slope, and present a number of applications to the statistics of $\Lambda_q$.