Crane, Edward and Short, Ian
(2007).
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| URL: | http://www.ams.org/journals/ecgd/2007-11-16/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, 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 |
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| Copyright Holders: | 2007 American Mathematical Society |
| ISSN: | 1088-4173 |
| Academic Unit/Department: | Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics Faculty of Science, Technology, Engineering and Mathematics (STEM) |
| Item ID: | 22457 |
| Depositing User: | Ian Short |
| Date Deposited: | 29 Jul 2010 11:31 |
| Last Modified: | 02 Aug 2016 16:10 |
| URI: | http://oro.open.ac.uk/id/eprint/22457 |
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