Analytic functions with prescribed cluster sets

Author:
L. W. Brinn

Journal:
Trans. Amer. Math. Soc. **300** (1987), 681-693

MSC:
Primary 30D40; Secondary 30E10

DOI:
https://doi.org/10.1090/S0002-9947-1987-0876472-3

MathSciNet review:
876472

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Abstract: Suppose that . A monotonic boundary path (mb-path) in is a simple continuous curve , , in such that strictly monotonically as . Suppose that is a complex valued function, defined in , and that is any mb-path in . The cluster set of on is the set of those points on the Riemann sphere for which there exists a sequence of points of with and . The cluster set is denoted by . If the cluster set is a single point set, that point is called the asymptotic value of on . If the function is continuous, then is a continuum on the Riemann sphere.

It is a conjecture of F. Bagemihl and W. Seidel that if is a family of mb-paths in satisfying certain conditions, and if is an analytic set of continua on the Riemann sphere, then there exists a function , analytic in , such that . A restricted form of the conjecture is mentioned in [**3**, p. 100].

Our principal results show the correctness of the conjecture in the case that is the collection of all continua on the Riemann sphere and is a tress of a certain type. The results are generalized in several directions. In particular, our technique for constructing the analytic function extends immediately to the case in which is any closed set of continua on the sphere. Specific examples of closed sets lead to several corollaries.

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DOI:
https://doi.org/10.1090/S0002-9947-1987-0876472-3

Article copyright:
© Copyright 1987
American Mathematical Society