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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
DOI: https://doi.org/10.1090/S0002-9947-1987-0876472-3
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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  • [1] N. U. Arakeljan, Uniform and tangential approximation by holomorphic functions, Izv. Akad. Nauk. Armjan SSR Ser. Mat. 3 (1968), 273-286. MR 0274770 (43:530)
  • [2] F. Bagemihl and J. E. McMillan, Uniform approach to cluster sets of arbitrary functions in a disk, Acta Math. Acad. Sci. Hungar. 17 (1966), 411-418. MR 0203032 (34:2891)
  • [3] F. Bagemihl and W. Seidel, A problem concerning the cluster sets of analytic functions, Math. Z. 62 (1955), 99-110. MR 0072220 (17:249d)
  • [4] -, Some boundary properties of analytic functions, Math. Z. 61 (1954), 186-199. MR 0065643 (16:460d)
  • [5] -, Spiral and other asymptotic paths, and paths of complete indetermination of analytic and meromorphic functions, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 1251-1258. MR 0059352 (15:515c)
  • [6] J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 0193606 (33:1824)
  • [7] P. M. Gauthier and W. Seidel, Some applications of Arakelian's approximation theorems to the theory of cluster sets, Izv. Akad. Nauk Armjan SSR Ser. Mat. 6 (1971), 458-464. MR 0302919 (46:2062)
  • [8] S. Kierst and E. Szpilrajn, Sur certaines singularités des fonctions analytiques uniformes, Fund. Math. 21 (1933), 276-294.
  • [9] K. Kuratowski, Topology. I, Academic Press, New York, 1966. MR 0217751 (36:840)
  • [10] -, Topology. II, Academic Press, New York, 1968.
  • [11] S. N. Mergelyan, Uniform approximation to functions of a complex variable, Uspekhi Mat. Nauk (N.S.) 7 (1952), 33-122; Amer. Math. Soc. Transl. 101 (1954).
  • [12] A. Roth, Appoximationseigenschaften und strahlengrenzewerte Meromorpher und ganzer Funktionen, Comment. Math. Helv. 11 (1938), 77-125. MR 1509593
  • [13] J. L. Stebbins, A construction of meromorphic functions with prescribed boundary behavior, Nagoya Math. J. 41 (1971), 75-87. MR 0291468 (45:559)
  • [14] J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, Amer. Math. Soc. Colloq. Publ., vol. 20, Amer. Math. Soc., Providence, R.I., 1960. MR 0218587 (36:1672a)

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DOI: https://doi.org/10.1090/S0002-9947-1987-0876472-3
Article copyright: © Copyright 1987 American Mathematical Society

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