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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Analytic functions with prescribed cluster sets

Author: L. W. Brinn
Journal: Trans. Amer. Math. Soc. 300 (1987), 681-693
MSC: Primary 30D40; Secondary 30E10
MathSciNet review: 876472
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Abstract: Suppose that $ 0 < R \leq + \infty $. A monotonic boundary path (mb-path) in $ \left\vert z \right\vert < R$ is a simple continuous curve $ z = z(s)$, $ 0 \leq s < 1$, in $ \left\vert z \right\vert < R$ such that $ \left\vert {z(s)} \right\vert \to R$ strictly monotonically as $ s \to 1$. Suppose that $ f$ is a complex valued function, defined in $ \left\vert z \right\vert < R$, and that $ t$ is any mb-path in $ \left\vert z \right\vert < R$. The cluster set of $ f$ on $ t$ is the set of those points $ w$ on the Riemann sphere for which there exists a sequence $ \{ {z_n}\} $ of points of $ t$ with $ {\operatorname{lim}_{n \to \infty }}\left\vert {{z_n}} \right\vert = R$ and $ {\operatorname{lim}_{n \to \infty }}f({z_n}) = w$. The cluster set is denoted by $ {C_t}(f)$. If the cluster set is a single point set, that point is called the asymptotic value of $ f$ on $ t$. If the function $ f$ is continuous, then $ {C_t}(f)$ is a continuum on the Riemann sphere.

It is a conjecture of F. Bagemihl and W. Seidel that if $ \mathcal{T}$ is a family of mb-paths in $ \left\vert z \right\vert < R$ satisfying certain conditions, and if $ \mathcal{K}$ is an analytic set of continua on the Riemann sphere, then there exists a function $ f$, analytic in $ \left\vert z \right\vert < R$, such that $ \left\{ {{C_t}(f)\vert t \in \mathcal{T}} \right\} = \mathcal{K}$. 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 $ \mathcal{K}$ is the collection of all continua on the Riemann sphere and $ \mathcal{T}$ is a tress of a certain type. The results are generalized in several directions. In particular, our technique for constructing the analytic function $ f$ extends immediately to the case in which $ \mathcal{K}$ is any closed set of continua on the sphere. Specific examples of closed sets lead to several corollaries.

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