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Conformal Geometry and Dynamics

ISSN 1088-4173



Conical limit sets and continued fractions

Authors: Edward Crane and Ian Short
Journal: Conform. Geom. Dyn. 11 (2007), 224-249
MSC (2000): Primary 51B10; Secondary 40A15
Published electronically: October 31, 2007
MathSciNet review: 2354097
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Abstract: Inspired by questions of convergence in continued fraction theory, Erdős, Piranian and Thron studied the possible sets of divergence for arbitrary sequences of Möbius maps acting on the Riemann sphere, $S^2$. By identifying $S^2$ with the boundary of three-dimensional hyperbolic space, $H^3$, we show that these sets of divergence are precisely the sets that arise as conical limit sets of subsets of $H^3$. Using hyperbolic geometry, we give simple geometric proofs of the theorems of Erdős, Piranian and Thron that generalise to arbitrary dimensions. New results are also obtained about the class of conical limit sets; for example, it is closed under locally quasisymmetric homeomorphisms. Applications are given to continued fractions.

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Additional Information

Edward Crane
Affiliation: Department of Mathematics, University Walk, Bristol, BS8 1TW, United Kingdom

Ian Short
Affiliation: Logic House, National University of Ireland, Maynooth, Maynooth, County Kildare, Ireland
MR Author ID: 791601
ORCID: 0000-0002-7360-4089

Keywords: Conical limit set, continued fraction, hyperbolic geometry, quasiconformal mapping, Diophantine approximation
Received by editor(s): January 3, 2007
Published electronically: October 31, 2007
Additional Notes: The first author was supported by a junior research fellowship at Merton College, Oxford, and by the University of Bristol.
The second author was supported by Science Foundation Ireland grant 05/RFP/MAT0003.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.