The authors continue their investigations on bounded analytic functions on two sheeted discs [Pac. J. Math. 134, No. 2, 261-273 (1988; Zbl 0627.30031); Constructive Approximation 6, No. 2, 195-223 (1990; Zbl 0698.30036); Trans. Am. Math. Soc. 333, No. 2, 799-819 (1992; Zbl 0759.30019)].
Let \(D\subset\Omega\) be a subdomain of the unit disc \(\Delta\) and let \(\widetilde D\) be a two sheeted unlimited covering surface of \(D\) with projection map \(\pi: \widetilde D\to D\). The authors investigate the Myrberg phenomenon i.e. the validity of \[ H^ \infty(\widetilde D)= H^ \infty(D)\circ \pi\tag{1} \] for certain coverings \((\widetilde D,D,\pi)\). Here \(H^ \infty(D)\) is the space of bounded analytic functions on \(D\). For \(D=\Delta\) and \(\{z_ n\}\) the set of branch points of \(\widetilde D\), Selberg proved (1937) that (1) is true iff \[ \sum_{n\geq 1} (1- | z_ n|)= \infty.\tag{2} \] Let now be \(\{z_ n\}\subset\Delta\) be a set without accumulation point in \(\Delta\) which satisfies (2) (admissible set). A sequence \(\{\Delta_ n\}\) of closed discs centered at \(\{z_ n\}\) is called an admissible sequence if the \(\{\Delta_ n\}\) are disjoint and contained in \(\Delta\). If \((\widetilde\Delta,\Delta,\pi)\) is the two sheeted covering surface with is branched over \(\{z_ n\}\), let \(D:= \Delta\backslash\bigcap_{n\geq 1}\Delta_ n\) and \(\widetilde D:= \pi^{-1}(D)\). The main theorem (in qualitative form):
For any admissible set \(\{z_ n\}\) there exists a sequence \(\{\bar r_ n\}\) of positive numbers such that any associated admissible sequence \(\{\Delta_ n\}\) of discs with radii \(0< r_ n\leq \bar r_ n\) leads to a covering surface \((\widetilde D,D,\pi)\) with property (1).
The very interesting quantitative form of the theorem is too complicated to be stated here.