On reflections in Jordan curves

Abstract

A purely geometric method for constructing reflections in Jordan curves on the Riemann sphere based on hyperbolic geodesics is introduced. It is then possible to investigate the relations between the geometry of a Jordan domain D D and the properties of the reflection by studying properties of hyperbolic geodesics. This idea is used to characterize unbounded Jordan John domains in terms of reflections satisfying a kind of Lipschitz condition.