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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Hausdorff dimension of sets of divergence arising from continued fractions
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by Ian Short PDF
Proc. Amer. Math. Soc. 140 (2012), 1371-1385 Request permission


A complex continued fraction can be represented by a sequence of Möbius transformations in such a way that the continued fraction converges if and only if the sequence converges at the origin. The set of divergence of the sequence of Möbius transformations is equivalent to the conical limit set from Kleinian group theory, and it is closely related to the Julia set from complex dynamics. We determine the Hausdorff dimensions of sets of divergence for sequences of Möbius transformations corresponding to certain important classes of continued fractions.
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Additional Information
  • Ian Short
  • Affiliation: Department of Mathematics and Statistics, The Open University, Milton Keynes MK7 6AA, United Kingdom
  • MR Author ID: 791601
  • ORCID: 0000-0002-7360-4089
  • Email:
  • Received by editor(s): July 27, 2010
  • Received by editor(s) in revised form: January 5, 2011
  • Published electronically: August 15, 2011
  • Additional Notes: The author thanks the referee for useful remarks—in particular, for suggestions which strengthened Theorem 3.4.
  • Communicated by: Mario Bonk
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 140 (2012), 1371-1385
  • MSC (2010): Primary 37F35, 40A15; Secondary 30B70, 30F45, 51B10
  • DOI:
  • MathSciNet review: 2869121