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

DOI:
https://doi.org/10.1090/S0002-9939-2011-11032-3

Published electronically:
August 15, 2011

MathSciNet review:
2869121

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Abstract | References | Similar Articles | Additional Information

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:
https://doi.org/10.1090/S0002-9939-2011-11032-3

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.