The structure and distribution of prime ends

Summary In this paper, we study the prime ends of a simply connected, plane domain. We use the metric concept of the separation of subsets of a domain to define the lateral points of a prims end (Section 3). The idea of right and left side-chains of rankn then leads to the concept of right and left lateral points of rankn (n = 1, 2, ..., ) (Section 5). Thus we achieve an analysis which goes beyond Carathéodory's classification (sea Section 2) of the points of a prime end into principal and subsidiary points (Hauptpunkte andNebenpunkte). Each subsidiary point of a prime endP is a lateral point ofP; but even a principal point ofP can be a right and left lateral point of infinite rank ofP.The need for the new concepts and for the extended analysis arises from the following fact: There exist very simple domains with boundary points that are principal points of a prime end but which, at the same time, possess some of the properties of subsidiary points, relative to the same prime end. In other words, a boundary point may not only belong to two or more prime ends; but in each of the prime ends, it may play a multiple role.It is known [2, p. 349] that, in the space of the prime ends of a fixed domain, the set of prime ends having subsidiary points is a set of first category. In Section 3, we show that the analogue of this proposition for lateral points is false; all the prime ends of certain highly laminar domains have lateral points.A theorem of Lindelöf and others ([4, p. 28]; see [2, p. 347] for further references) asserts that if the function maps the unit disk ¦ ¦ < 1 conformally onto a simply connected domain, andL is a Stolz path approaching the point = 1, then the closure of the set (L) contains no subsidiary points of the prime end corresponding to the point = 1. In Section 4, we formulate and prove an analogous theorem for lateral points.In Section 6, we consider a special phenomenon which at first sight might be expected to re exceptional, but which we show can affect a residual set of prime ends. Section 7 deals with a theorem of Frankl.G. Piranian's contribution to this paper was made under Contract DA 20-018-ORD-13585, Office of Ordnance Research, U. S. Army.