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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Inspired by questions of convergence in continued fraction theory, Erdos, Piranian and Thron studied the possible sets of divergence for arbitrary sequences of Möbius maps acting on the Riemann sphere, . By identifying with the boundary of three-dimensional hyperbolic space, , we show that these sets of divergence are precisely the sets that arise as conical limit sets of subsets of . Using hyperbolic geometry, we give simple geometric proofs of the theorems of Erdos, 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

Email:
Edward.Crane@gmail.com

**Ian Short**

Affiliation:
Logic House, National University of Ireland, Maynooth, Maynooth, County Kildare, Ireland

Email:
Ian.Short@nuim.ie

DOI:
https://doi.org/10.1090/S1088-4173-07-00169-5

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.