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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The dynamics of Super-Apollonian continued fractions


Authors: Sneha Chaubey, Elena Fuchs, Robert Hines and Katherine E. Stange
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11A55, 11J70, 11E20, 37A45, 37F30, 52C26
DOI: https://doi.org/10.1090/tran/7372
Published electronically: May 23, 2019
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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.


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Additional Information

Sneha Chaubey
Affiliation: Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois 61801
Address at time of publication: Indraprastha Institute of Information Technology, Delhi Okhla Industrial Estate, Phase III, New Delhi, India, 110020
Email: sneha@iiitd.ac.in

Elena Fuchs
Affiliation: Department of Mathematics, University of California Davis, One Shields Avenue, Davis, California 95616
Email: efuchs@math.ucdavis.edu

Robert Hines
Affiliation: Department of Mathematics, University of Colorado, Campus Box 395, Boulder, Colorado 80309-0395
Email: robert.hines@colorado.edu

Katherine E. Stange
Affiliation: Department of Mathematics, University of Colorado, Campus Box 395, Boulder, Colorado 80309-0395
Email: kstange@math.colorado.edu

DOI: https://doi.org/10.1090/tran/7372
Keywords: Bianchi group, dynamical system, ergodic theory, geodesic flow, invertible extension, invariant measure, Pythagorean triple, Lorentz quadruple, Descartes quadruple, orthogonal group, continued fractions, projective linear group, M\"obius transformation, hyperbolic isometry, quadratic form, Apollonian circle packing
Received by editor(s): May 2, 2017
Received by editor(s) in revised form: August 9, 2017
Published electronically: May 23, 2019
Additional Notes: The work of the second author was supported by NSF DMS-1501970 and a Sloan Research Fellowship.
The work of the fourth author was sponsored by the National Security Agency under Grant Number H98230-14-1-0106 and NSF DMS-1643552. The United States government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein.
Article copyright: © Copyright 2019 American Mathematical Society