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On prime ends and plane continua. (English) Zbl 1060.30013

This paper contains several interesting results on the boundary behavior of conformal mappings in relation with the theory of continua in the plane. Among other things, it investigates the relation between the impression and the principal point set of a prime end, and it generalizes various results for prime ends to non-locally connected continua. We will describe in a precise manner only one of its six theorems. Let \(f\) be a conformal map of the unit disk \(\mathbb D\) onto a domain \(G\). For \(\zeta\in\partial \mathbb D\), we define \(I(\zeta)=\{w\in \partial G:\exists z_n\in\mathbb D \text{ with } z_n\to \zeta,\; f(z_n)\to w\}\) (the impression of the prime end \(f(\zeta)\)). \(\Pi(\zeta)=\{w\in \partial G:\exists r_n\in (0,1) \text{ with } r_n\to 1,\;f(r_n\zeta)\to w\}\) (the principal point set of the prime end \(f(\zeta)\)). \[ \begin{aligned} I^+(\zeta)&=\{w\in \partial G:\exists z_n\in\mathbb D \text{ with } \arg z_n>\arg \zeta,\;z_n\to \zeta,\;f(z_n)\to w\}.\\ I^-(\zeta)&=\{w\in \partial G:\exists z_n\in\mathbb D \text{ with } \arg z_n<\arg \zeta,\;z_n\to \zeta,\;f(z_n)\to w\}.\end{aligned} \] The prime end \(f(\zeta)\) is called symmetric if \(I^+(\zeta)=I^-(\zeta)=I(\zeta)\). Theorem 1 asserts that if a continuum \(E \subset G^c\) contains the principal point set of a symmetric prime end, then \(E \cap \partial G\) either contains its impression or is contained in its impression.

MSC:

30C35 General theory of conformal mappings
30D40 Cluster sets, prime ends, boundary behavior
54F15 Continua and generalizations
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