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

2018/10/25 by Taha, Diaaeldin · 2 citations
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1810.10668

Abstract

The Farey sequence 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 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 Λ_q = G_q(1, 0)^T of the linear action of the Hecke triangle groups G_q on the plane ℝ² starting with a Stern-Brocot tree analogue for the said orbits. We derive the next-term algorithm for generating the elements of Λ_q in vertical strips in increasing order of slope, and present a number of applications to the statistics of Λ_q.

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