×

A theorem of Picard type. (English) Zbl 0604.30037

The author extends some results of Picard type due to E. F. Collingwood and M. L. Cartwright [Acta Math. 87, 83-146 (1952; Zbl 0046.084)] to the case of Riemann surfaces of Parreau-Widom type. Let R be a regular Riemann surface of Parreau-Widom type [cf., for instance, the reviewer’s notes ”Hardy classes on infinitely connected Riemann surfaces” (1983; Zbl 0523.30028)]. Let \(R^*\) be the Martin compactification of R and set \(\Delta =R^*-R\). Then, for almost all \(p\in \Delta\) (with respect to any fixed harmonic measure on \(\Delta)\), we can draw (in R) Stolz domains of arbitrary opening \((<\pi)\) with vertex p. S(p) denotes the totality of such Stolz domains with vertex p [cf. Duke. Math. J. 43, 731-746 (1976; Zbl 0339.30025)]. Let f(z) be a meromorphic function on R. f is said to have angular limit at \(p\in \Delta\) if f tends uniformly to some limit inside any \(S\in S(p)\). We denote by \(F=F(f)\) (resp. \(P=P(f))\) the set of \(p\in \Delta\) such that f has angular limit at p (resp. f takes every value in the extended complex plane infinitely often in every neighborhood of p with two possible exceptions).
The main theorem: Let f be meromorphic on R and let \(p\in \Delta\). Assume the condition (*): every neighborhood, in \(R^*\), of p intersects the boundary \(\Delta\) at a set of positive harmonic measure. Then either \(p\in P\) or \(p\in F'\) \((=the\) derived set of F). Corollary: If the condition (*) holds everywhere on \(\Delta\) and P is empty, then F is everywhere dense in \(\Delta\).
Reviewer: M.Hasumi

MSC:

30D40 Cluster sets, prime ends, boundary behavior
30F25 Ideal boundary theory for Riemann surfaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] E. F. Collingwood and M. L. Cartwright, Boundary theorems for a function meromorphic in the unit circle,Acta Math.,87 (1952), 83–146. · Zbl 0046.08402
[2] M. Hasumi, Invariant subspaces on open Riemann surfaces,Ann. Inst. Fourier, Grenoble,24 (1974), 241–286. · Zbl 0287.46066
[3] M. Hasumi, Invariant subspaces on open Riemann surfaces II,Ann. Inst. Fourier, Grenoble,26 (1976), 273–299. · Zbl 0322.46058
[4] M. Parreau, Théorème de Fatou et problème de Dirichlet pour les lignes de Green de certaines surfaces de Riemann,Ann. Acad. Sci. Fenn. Ser. A. I. no. 250/25 (1958), 8pp. · Zbl 0086.08503
[5] L. Sario and M. Nakai,Classification theory of Riemann surfaces, Springer-Verlag (Berlin-Heidelberg-New York, 1970). · Zbl 0199.40603
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.