# The Dynamics of Super-Apollonian Continued Fractions

## TL;DR

This paper introduces new dynamical systems based on Super-Apollonian groups that generate complex and real continued fractions, providing insights into Gaussian rational approximations, reduction algorithms, and invariant measures.

## Contribution

It develops invertible extensions and invariant measures for these systems, and explores their ergodic properties and statistical behavior, extending classical continued fraction theory.

## Key findings

- Constructed invertible extensions of the dynamical systems.
- Identified an invariant measure conjectured to be ergodic.
- Analyzed continued fraction statistics and real line restrictions.

## 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.