Orphans in Forests of Linear Fractional Transformations

Sandie Han,Johann Thiel,Satyanand Singh,Ariane M Masuda

doi:10.37236/5684

Sandie Han, Johann Thiel + Show 2 more

Open Access

https://doi.org/10.37236/5684

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Abstract

A positive linear fractional transformation (PLFT) is a function of the form $f(z)=\frac{az+b}{cz+d}$ where $a,b,c$ and $d$ are nonnegative integers with determinant $ad-bc\neq 0$. Nathanson generalized the notion of the Calkin-Wilf tree to PLFTs and used it to partition the set of PLFTs into an infinite forest of rooted trees. The roots of these PLFT Calkin-Wilf trees are called orphans. In this paper, we provide a combinatorial formula for the number of orphans with fixed determinant $D$. In addition, we derive a method for determining the orphan ancestor of a given PLFT. Lastly, taking $z$ to be a complex number, we show that every positive complex number has finitely many ancestors in the forest of complex $(u,v)$-Calkin-Wilf trees.

Highlights

In [5], Calkin and Wilf introduced a rooted infinite binary tree where every vertex is labeled by a positive rational number according to the following rules: (A) the root is labeled 1/1, (B) the left child of a vertex a/b is labeled a/(a + b), and (C) the right child of a vertex a/b is labeled (a + b)/b
By a positive linear fractional transformation (PLFT), we mean a function of the form f (z) =
From [15], we find that the set of PLFTs is partitioned into an infinite forest of PLFT CW-trees

Summary

Introduction

In [5], Calkin and Wilf introduced a rooted infinite binary tree where every vertex is labeled by a positive rational number according to the following rules:. As noted by several authors [10, 15], replacing a/b in (B) and (C) above by the variable z shows that the vertex labels of the Calkin-Wilf tree are generated by applying one of two transformations. For any vertex labeled z in the Calkin-Wilf tree, the left child of z z and the right child of z is R(z) := z + 1. The isomorphism between the two sets shows that we can compute the vertices of a PLFT CW-tree via matrix multiplication by the matrices. Using [2, Theorem 3.3] and [2, Theorem 3.4], we see that the contribution from the rightmost sum in (6) is O(x2 log x)

Now let

Positive linear fractional transformations and continued fractions