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Conformal Geometry and Dynamics

Published by the American Mathematical Society since 1997, the purpose of this electronic-only journal is to provide a forum for mathematical work in related fields broadly described as conformal geometry and dynamics. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4173

The 2024 MCQ for Conformal Geometry and Dynamics is 0.49.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Serendipitous decompositions of higher-dimensional continued fractions
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by Anton Lukyanenko and Joseph Vandehey;
Conform. Geom. Dyn. 29 (2025), 57-89
DOI: https://doi.org/10.1090/ecgd/396
Published electronically: January 21, 2025

Abstract:

We prove a suite of dynamical results, including exactness of the transformation and piecewise-analyticity of the invariant measure, for a family of continued fraction systems, including specific examples over reals, complex numbers, quaternions, octonions, and in $\mathbb {R}^3$. Our methods expand on the work of Nakada and Hensley, and in particular fill some gaps in Hensley’s analysis of Hurwitz complex continued fractions. We further introduce a new “serendipity” condition for a continued fraction algorithm, which controls the long-term behavior of the boundary of the fundamental domain under iteration of the continued fraction map, and which is under reasonable conditions equivalent to the finite range property. We also show that the finite range condition is extremely delicate: perturbations of serendipitous systems by non-quadratic irrationals do not remain serendipitous, and experimental evidence suggests that serendipity may fail even for some rational perturbations.
References
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Bibliographic Information
  • Anton Lukyanenko
  • Affiliation: Department of Mathematics, George Mason University, Fairfax, Virginia 22030
  • MR Author ID: 995877
  • ORCID: 0000-0002-2252-7407
  • Email: alukyane@gmu.edu
  • Joseph Vandehey
  • Affiliation: Department of Mathematics, University of Texas at Tyler, Tyler, Texas 75799
  • MR Author ID: 999182
  • Email: jvandehey@uttyler.edu
  • Received by editor(s): September 26, 2023
  • Received by editor(s) in revised form: July 12, 2024, and September 5, 2024
  • Published electronically: January 21, 2025
  • © Copyright 2025 by the authors
  • Journal: Conform. Geom. Dyn. 29 (2025), 57-89
  • MSC (2020): Primary 11K50, 37A44; Secondary 11R52
  • DOI: https://doi.org/10.1090/ecgd/396