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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Hausdorff dimension of sets of divergence arising from continued fractions


Author: Ian Short
Journal: Proc. Amer. Math. Soc. 140 (2012), 1371-1385
MSC (2010): Primary 37F35, 40A15; Secondary 30B70, 30F45, 51B10
Published electronically: August 15, 2011
MathSciNet review: 2869121
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Abstract: 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
Email: i.short@open.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11032-3
PII: S 0002-9939(2011)11032-3
Keywords: Continued fractions, convergence, divergence, Hausdorff dimension, Möbius transformations, sets of divergence.
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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.