Crane, Edward and Short, Ian
(2007).

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URL:  http://www.ams.org/journals/ecgd/20071116/S1088... 

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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 threedimensional 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.
Item Type:  Journal Article 

Copyright Holders:  2007 American Mathematical Society 
ISSN:  10884173 
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:  15 Jan 2016 21:29 
URI:  http://oro.open.ac.uk/id/eprint/22457 
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