Analytic functions with prescribed cluster sets

Abstract:

Suppose that $0 < R \leq + \infty$. A monotonic boundary path (mb-path) in $\left | z \right | < R$ is a simple continuous curve $z = z(s)$, $0 \leq s < 1$, in $\left | z \right | < R$ such that $\left | {z(s)} \right | \to R$ strictly monotonically as $s \to 1$. Suppose that $f$ is a complex valued function, defined in $\left | z \right | < R$, and that $t$ is any mb-path in $\left | z \right | < R$. 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 ${\operatorname {lim}_{n \to \infty }}\left | {{z_n}} \right | = R$ 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 $\left | z \right | < R$ 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 $\left | z \right | < R$, such that $\left \{ {{C_t}(f)|t \in \mathcal {T}} \right \} = \mathcal {K}$. 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.