# Conical limit sets and continued fractions

Short, Dr. Ian and Crane, Dr. Edward (2007) Conical limit sets and continued fractions.

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## 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 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 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.

Keywords: Conical limit set, continued fraction, hyperbolic geometry, quasiconformal mapping, Diophantine approximation Science & Engineering > Mathematics 721 Ian Short 29 Nov 2007 Yes http://front.math.ucdavis.edu/0708.1730

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