The dynamics of Super-Apollonian continued fractions Journal Article uri icon

Overview

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.

publication date

  • May 23, 2019

Date in CU Experts

  • February 3, 2019 8:27 AM

Full Author List

  • Chaubey S; Fuchs E; Hines R; Stange K

author count

  • 4

Other Profiles

International Standard Serial Number (ISSN)

  • 0002-9947

Electronic International Standard Serial Number (EISSN)

  • 1088-6850

Additional Document Info

start page

  • 2287

end page

  • 2334

volume

  • 372

issue

  • 4