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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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  • [1] N. U. Arakeljan, Uniform and tangential approximations by analytic functions, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 3 (1968), no. 4-5, 273–286 (Russian, with Armenian and English summaries). MR 0274770
  • [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,
  • [3] F. Bagemihl and W. Seidel, A problem concerning cluster sets of analytic functions, Math. Z. 62 (1955), 99–110. MR 0072220,
  • [4] F. Bagemihl and W. Seidel, Some boundary properties of analytic functions, Math. Z. 61 (1954), 186–199. MR 0065643,
  • [5] F. Bagemihl and W. Seidel, 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
  • [6] James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
  • [7] P. Gauthier and W. Seidel, Some applications of Arakélian’s approximation theorems to the theory of cluster sets, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 6 (1971), no. 6, 458–464 (English, with Armenian and Russian summaries). MR 0302919
  • [8] S. Kierst and E. Szpilrajn, Sur certaines singularités des fonctions analytiques uniformes, Fund. Math. 21 (1933), 276-294.
  • [9] K. Kuratowski, Topology. Vol. I, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. MR 0217751
  • [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] Alice Roth, Approximationseigenschaften und Strahlengrenzwerte meromorpher und ganzer Funktionen, Comment. Math. Helv. 11 (1938), no. 1, 77–125 (German). MR 1509593,
  • [13] J. L. Stebbins, A construction of meromorphic functions with prescribed asymptotic behavior, Nagoya Math. J. 41 (1971), 75–87. MR 0291468
  • [14] J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, Third edition. American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I., 1960. MR 0218587

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