Singularities and wandering domains in iteration of meromorphic functions
Let \(f\) be a transcendental meromorphic function and let \(U\) be a wandering domain of \(f\). If \(b\) denotes a finite point of the Julia set of \(f\) and \(\lambda(z)\) the Poincaré density on the Fatou set \(F\) of \(f\), we write \(C(a)= \inf\lambda(z) d(z,b)\), where the inf is over \(z\) in \(F\). It is shown that if \(b\) is a finite limit function of a subsequence of \(\{f^n\}\) in \(U\) and \(C(b)> 0\) then \(b\) is in the derived set of the forward orbits of the singularities of the inverse of \(f\). If \(U\) and the components of \(F\) into which it is mapped are simply-connected then \(C(b)> 0\) holds. This generalizes a result proved by \textit{W. Bergweiler}, \textit{M. Haruta}, \textit{H. Kriete}, \textit{H.-G. Meier} and \textit{N. Terglane} [Ann. Acad. Sci. Fenn., Ser. A I, Math. 18, No. 2, 369-375 (1993; Zbl 0793.30022)] for entire functions. The class \(B\) denotes those meromorphic functions for which the set of singular values of the inverse is bounded. The author shows that for such functions there is no point of \(F\) for which \(f^n(z)\to \infty\). [This was proved independently by \textit{P. J. Rippon} and \textit{G. Stallard}, Proc. Am. Math. Soc. 127, No. 11, 3251-3258 (1999; Zbl 0931.30017).] Finally, some conditions are given under which, for any wandering domain \(U\) of \(f\), there is some subsequence of iterates of \(f\) which tends to infinity in \(U\).
scientific article; zbMATH DE number 1525077