Beurling’s theorem

Let R, R’ be hyperbolic Riemann surfaces and \phi be an analytic mapping of R into R’. Let K_{0} be a closed disk in R and let R_{0}=R-K_{0} . Let \acute{C} be the Kuramochi capacity on R_{0}\cup\Delta_{N} and \Delta_{1} be the set of all minimal Kuramochi boundary points of R. For a metrizable compactification R^{\prime*} of R’, we denote by \mathscr{F}(\phi) the set of all points in \Delta_{1} at which \phi has a fine limit in R^{\prime*} . There are two typical extensions of Beurling’s theorem [1] to analytic mappings of a Riemann surface to another one, i . e. , Z. Kuramochi’s [5, 6, 7] and C. Constantinescu and A. Cornea’s theorems [3, 4] . The former result states that if \phi is an almost finitely sheeted mapping and R^{\prime*} is H. D. separative, then \tilde{C}(\Delta_{1}-\mathscr{F}(\phi))=0 . The latter one states that if \phi is a Dirichlet mapping and R^{\prime*} is a quotient space of the Royden compactification of R’, then \overline{C}(\Delta_{1}-^{\Gamma j}(\phi))=0 . The present author [9] proved that these two results are independent. In this paper we shall give an another extension of Beurling’s theorem such that it contains the above two results: If \phi is a Dirichlet mapping and R^{\prime*} is H. D. separative, then Beurling’s theorem is valid. Notation and terminology Let R be a hyperbolic Riemann surface. For a subset A of R, we denote by \partial A and A^{i} the (relative) boundary and the interior of A respectively. We call a closed or open subset A of R is regular if \partial A is nonempty and consists of at most a countable number of analytic arcs clustering nowhere in R. We fix a closed disk K_{0} in R once for all and let R_{0}=