Absence of wandering domains for some real entire functions with bounded singular sets (Q383598)

The authors consider \(f: \mathbb{C} \to \mathbb{C}\) an entire function, \(S(f)\) the set of singular values of \(f\), \(B=\{ f \text{ transcendental entire function: } S(f) \text{ is bounded }\}\), \(B_{\mathrm{real}} = \{f \in B : f \text{ is real }\}\), \(B^{*}_{\mathrm{real}} = \{ f \in B_{\mathrm{real}}: f\) has real singular values\(\}\) and a geometric condition, defined in the article as the ``sector condition''. The authors give a first result which is stated as follows: Let \(f \in B\) be a function which \(f^n|_{S(f)} \to \infty\) uniformly. Let \(A \subset \mathbb{C}\) be a closed set with \((S(f) \cap f(A)) \subset A\) such that all conneceted components of \(A\) are unbounded. Suppose that there exists \(\epsilon >0\) and \(c \in (0,1)\) with the following property: if \(z \in A\) is sufficiently large and \(w \in \mathbb{C}\) satisfies \(|w - z| < c| z|\), then \(\mathrm{dist}(f(w), S(f)) >\epsilon\). Then \(f\) has no wandering domains. Other important result of the paper is that if \(f \in B^{*}_{\mathrm{real}}\) satisfies the (real) sector condition, then \(f\) has no wandering domains. This result includes all maps of the form \(z \to \frac {\lambda\sinh(z)}{z} + a\), with \(\lambda, a \in \mathbb{R}\), \(\lambda \neq 0\). As special case of the results above the authors give a short, elementary and non-technical proof of \(J(e^z) = \mathbb{C}\). Furthermore, the authors extend a result of \textit{W. Bergweiler} [Math. Proc. Camb. Philos. Soc. 117, No. 3, 525--532 (1995; Zbl 0836.30016)] concerning Baker domains of entire functions and the relation to the postsingular set to the case of meromorphic functions.