Let $f$ be a conformal map of the unit disk ${\bb D}$ onto the domain $G \subset \hat{\bb C} = {\bb C} \cup \{\infty\}$. We shall always use the spherical metric in $\hat{\bb C}$.

Carathéodory [3] introduced the concept of a prime end of $G$ in order to describe the boundary behaviour of $f$ in geometric terms; see for example [6, Chapter 9] or [12, Section 2.4]. There is a bijective map $\hat{f}$ of ${\bb T} = \partial {\bb D}$ onto the set of prime ends of $G$.