The dynamics of Super-Apollonian continued fractions

Abstract

We examine a pair of dynamical systems on the plane induced by a pair of spanning trees in the Cayley graph of the Super-Apollonian group of Graham, Lagarias, Mallows, Wilks and Yan. The dynamical systems compute Gaussian rational approximations to complex numbers and are "reflective" versions of the complex continued fractions of A. L. Schmidt. They also describe a reduction algorithm for Lorentz quadruples, in analogy to work of Romik on Pythagorean triples. For these dynamical systems, we produce an invertible extension and an invariant measure, which we conjecture is ergodic. We consider some statistics of the related continued fraction expansions, and we also examine the restriction of these systems to the real line, which gives a reflective version of the usual continued fraction algorithm. Finally, we briefly consider an alternate setup corresponding to a tree of Lorentz quadruples ordered by arithmetic complexity. Quadruples and the Super-Apollonian Group2.1. Simple continued fractions. We begin with a brief overview of simple continued fractions on the real line, described from the perspective and in the language we plan to use for complex continued fractions in the remainder of this paper. The purpose is to provide an explicit analogy for much of what follows.Over the integers, the Euclidean algorithm is the iteration of the division algorithm, which, for a, b ∈ Z, returns q, r ∈ Z so that a = bq + r, 0 ≤ r < |b|. Replacing the pair (a, b) with (b, r), we

Number theoretical properties of Romik's dynamical system

“…To summarize, we proved that each primitive Pythagorean triple (a, b, c) (except for (3,4,5) • any vertex coming out of (a, b, c) has its x 3 -coordinate greater than c, while the vertex going into (a, b, c) has a smaller x 3 -coordinate than c. We will say that a vertex (a, b, c) is a funnel vertex, if it satisfies the above three properties. The idea behind this terminology is this.…”

“…Next, suppose v is a vertex not equal to (3,4,5) and (4,3,5). We argue that v = (a, b, c) has indegree 1, or, equivalently,…”

“…For the exceptional case v = (3, 4, 5) and (4, 3, 5), we have U j Hv = (1, 0, 1) and (0, 1, 1), which are not (technically) primitive Pythagorean triples. So, (3,4,5) and (4, 3, 5) are the only vertices of indegree 0.…”

“…Then the repeated actions of M 1 , M 2 , M 3 via left-multiplication on (the column vectors) (3,4,5) and (4,3,5) generate all primitive Pythagorean triples and each primitive Pythagorean triple shows up in the trees exactly once, as pictured in Figure 1. As far as we are aware, the oldest literature containing this theorem is the paper [4] by Berggren.…”

“…In the same paper, Romik initiates the investigation of a dynamical system T : Q −→ Q, where Q is the (closed) unit quarter circle Q = {(x, y) ∈ R 2 | x ≥ 0, y ≥ 0 and x 2 + y 2 = 1}, (3,4,5) ( 15,8,17) (21, 20, 29) (5,12,13) . .…”