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 | References | Similar Articles | Additional Information
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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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1987-0876472-3
Article copyright:
© Copyright 1987
American Mathematical Society