Short, Dr. Ian and Crane, Dr. Edward
(2007)
Conical limit sets and continued fractions.
[Preprint]
Abstract
Inspired by questions of convergence in continued fraction theory,
Erdos, Piranian and Thron studied the possible sets of divergence for arbitrary sequences of Moebius 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 Erdos, Piranian and Thron that generalise to arbitrary dimensions. New results are also obtained about the class of conical limit sets, for example, that it is closed under locally quasisymmetric homeomorphisms. Applications are given to continued fractions.
Item Type: 
Preprint

Keywords: 
Conical limit set, continued fraction, hyperbolic geometry, quasiconformal mapping, Diophantine approximation 
Subjects: 
Science & Engineering > Mathematics 
Item ID: 
721 
Depositing User: 
Ian Short

Date Deposited: 
29 Nov 2007 
Refereed: 
Yes 
URI: 

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