Conical limit sets and continued fractions

TL;DR

This paper links divergence sets in continued fraction theory to conical limit sets in hyperbolic geometry, providing new geometric proofs and results, with applications to continued fractions.

Contribution

It establishes a geometric characterization of divergence sets as conical limit sets and extends classical theorems to higher dimensions with new properties of these sets.

Findings

Divergence sets are exactly conical limit sets of hyperbolic space subsets.

The class of conical limit sets is closed under locally quasisymmetric homeomorphisms.

New geometric proofs generalize previous theorems to arbitrary dimensions.

Abstract

Inspired by questions of convergence in continued fraction theory, Erd\H{o}s, Piranian and Thron studied the possible sets of divergence for arbitrary sequences of M\"obius 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 Erd\H{o}s, 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.