Conical limit sets and continued fractions

Inspired by questions of convergence in continued fraction theory, Erdos, Piranian and Thron studied the possible sets of divergence for arbitary sequences of Mobius maps acting on the Riemann sphere. By identifying the Riemann sphere with the boundary of three-dimensional hyperbolic space, we show that thesse sets of divergence are precisely the sets that arise as conical limit sets of subsets of hyperbolic space. 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 sets that arise as conical limit sets, for example that it is closed under locally quasisymmetric homeomorphisms. Applications are given to continued fractions.