On Iversen's property and the existence of bounded analytic functions (Q798804)

Let R be a Riemann surface and f be a meromorphic function on R. If for any domain V of \({\bar {\mathbb{C}}}\) every component of \(f^{-1}(V)\) covers V up to a set of capacity zero then f is said to have the C.I. property. The author denotes by \(P_ I\) the class of Riemann surfaces R such that every meromorphic function on R possesses the C.I. property and by \(P_{HI}\) the class of Riemann surfaces R such that every holomorphic function on R possesses the C.I. property. Riemann surfaces belonging to \(P_{HI}\backslash P_ I\) and \(P_ I\backslash O_ G\) are constructed. The Martin and the Kuramochi ideal boundaries play an important part in these constructions.