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Conical limit sets and continued fractions

Crane, Edward and Short, Ian (2007). Conical limit sets and continued fractions. Conformal Geometry and Dynamics, 11 pp. 224–249.

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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, S2. By identifying S2 with the boundary of three-dimensional hyperbolic space, H3, we show that these sets of divergence are precisely the sets that arise as conical limit sets of subsets of H3. 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.

Item Type: Journal Article
Copyright Holders: 2007 American Mathematical Society
ISSN: 1088-4173
Academic Unit/Department: Mathematics, Computing and Technology > Mathematics and Statistics
Mathematics, Computing and Technology
Item ID: 22457
Depositing User: Ian Short
Date Deposited: 29 Jul 2010 11:31
Last Modified: 24 Feb 2016 06:01
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