Cluster Sets

For the first systematic investigations of the theory of cluster sets of analytic functions, we are indebted to IVERSEN [1-3J and GROSS [1-3J about forty years ago. Subsequent important contributions before 1940 were made by SEIDEL [1-2J, DOOE [1-4J, CARTWRIGHT [1-3J and BEURLING [1]. The investigations of SEIDEL and BEURLING gave great impetus and interest to Japanese mathematicians; beginning about 1940 some contributions were made to the theory by KUNUGUI [1-3J, IRIE [IJ, TOKI [IJ, TUMURA [1-2J, KAMETANI [1-4J, TsuJI [4J and NOSHIRO [1-4J. Recently, many noteworthy advances have been made by BAGEMIHL, SEIDEL, COLLINGWOOD, CARTWRIGHT, HERVE, LEHTO, LOHWATER, MEIER, OHTSUKA and many other mathematicians. The main purpose of this small book is to give a systematic account on the theory of cluster sets. Chapter I is devoted to some definitions and preliminary discussions. In Chapter II, we treat extensions of classical results on cluster sets to the case of single-valued analytic functions in a general plane domain whose boundary contains a compact set of essential singularities of capacity zero; it is well-known that HALLSTROM [2J and TsuJI [7J extended independently Nevanlinna's theory of meromorphic functions to the case of a compact set of essential singUlarities of logarithmic capacity zero. Here, Ahlfors' theory of covering surfaces plays a funda­ mental role. Chapter III "is concerned with functions meromorphic in the unit circle.

I. Definitions and preliminary discussions.- § 1. Definitions of cluster sets.- § 2. Some classical theorems.- II. Single-valued analytic functions in general domains.- § 1. Compact set of capacity zero and Evans-Selberg’s theorem.- § 2. Meromorphic functions with a compact set of essential singularities of capacity zero.- § 3. Extension of Iversen’s theorem on asymptotic values.- § 4. Extension of Iversen-Gross-Seidel-Beurling’s theorem.- § 5. Hervé’s theorems.- III. Functions meromorphic in the unit circle.- §1. Functions of class (U) in Seidel’s sense.- § 2. Boundary theorems of Collingwood and Cartwright.- § 3. Baire category and cluster sets.- § 4. Boundary behaviour of meromorphic functions.- § 5. Meromorphic functions of bounded type and normal meromorphic functions.- IV. Conformal mapping of Riemann surfaces.- § 1. Gross’ property of covering surfaces.- § 2. Iversen’s property of covering surfaces.- § 3. Boundary theorems on open Riemann surfaces.- Appendix: Cluster sets of pseudo-analytic functions.