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

Full-text PDF

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.

**1.**B. Aebischer, The limiting behavior of sequences of Möbius transformations, Math. Z.**205**(1990), no. 1, 49-59. MR**1069484 (91i:51007)****2.**A. F. Beardon,*The geometry of discrete groups*, Springer, New York, 1983. MR**698777 (85d:22026)****3.**A. F. Beardon, Continued fractions, discrete groups and complex dynamics, Comput. Methods Funct. Theory**1**(2001), no. 2, 535-594. MR**1941142 (2003m:30010)****4.**A. F. Beardon and I. Short, The Seidel, Stern, Stolz and Van Vleck Theorems on continued fractions, Bull. London Math. Soc.**42**(2010), no. 3, 457-466. MR**2651941****5.**E. Crane and I. Short, Conical limit sets and continued fractions, Conform. Geom. Dyn.**11**(2007), 224-249 (electronic). MR**2354097 (2008i:30002)****6.**P. Erdős and G. Piranian, Sequences of linear fractional transformations, Michigan Math. J**6**(1959), 205-209. MR**0109227 (22:114)****7.**K. Falconer,*Fractal geometry*, Second edition, Wiley, Hoboken, NJ, 2003. MR**2118797 (2006b:28001)****8.**W. K. Hayman and P. B. Kennedy,*Subharmonic functions. Vol. I*, Academic Press, London, 1976. MR**0460672 (57:665)****9.**L. Jacobsen, General convergence of continued fractions, Trans. Amer. Math. Soc.**294**(1986), no. 2, 477-485. MR**825716 (87j:40004)****10.**W. B. Jones and W. J. Thron,*Continued fractions*, Addison-Wesley Publishing Co., Reading, Mass., 1980. MR**595864 (82c:30001)****11.**L. Lorentzen and H. Waadeland,*Continued fractions. Vol. 1*, Second edition, Atlantis Press, Paris, 2008. MR**2433845 (2009b:30005)****12.**R. D. Mauldin and M. Urbański, Conformal iterated function systems with applications to the geometry of continued fractions, Trans. Amer. Math. Soc.**351**(1999), no. 12, 4995-5025. MR**1487636 (2000c:28016)****13.**P. J. Nicholls,*The ergodic theory of discrete groups*, Cambridge Univ. Press, Cambridge, 1989. MR**1041575 (91i:58104)****14.**G. Piranian and W. J. Thron, Convergence properties of sequences of linear fractional transformations, Michigan Math. J.**4**(1957), 129-135. MR**0093578 (20:102)****15.**J. G. Ratcliffe,*Foundations of hyperbolic manifolds*, Springer, New York, 1994. MR**1299730 (95j:57011)****16.**C. A. Rogers,*Hausdorff measures*, reprint of the 1970 original, Cambridge Univ. Press, Cambridge, 1998. MR**1692618 (2000b:28009)****17.**B.-W. Wang and J. Wu, Hausdorff dimension of certain sets arising in continued fraction expansions, Adv. Math.**218**(2008), no. 5, 1319-1339. MR**2419924 (2009d:11115)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
37F35,
40A15,
30B70,
30F45,
51B10

Retrieve articles in all journals with MSC (2010): 37F35, 40A15, 30B70, 30F45, 51B10

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.