Conical limit sets and continued fractions
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- by Edward Crane and Ian Short
- Conform. Geom. Dyn. 11 (2007), 224-249
- DOI: https://doi.org/10.1090/S1088-4173-07-00169-5
- Published electronically: October 31, 2007
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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.References
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Bibliographic Information
- Edward Crane
- Affiliation: Department of Mathematics, University Walk, Bristol, BS8 1TW, United Kingdom
- Email: Edward.Crane@gmail.com
- Ian Short
- Affiliation: Logic House, National University of Ireland, Maynooth, Maynooth, County Kildare, Ireland
- MR Author ID: 791601
- ORCID: 0000-0002-7360-4089
- Email: Ian.Short@nuim.ie
- 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. - © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Conform. Geom. Dyn. 11 (2007), 224-249
- MSC (2000): Primary 51B10; Secondary 40A15
- DOI: https://doi.org/10.1090/S1088-4173-07-00169-5
- MathSciNet review: 2354097