Iterates of meromorphic functions. IV: Critically finite functions
[For part III see the authors in Ergod. Theory Dyn. Syst. 11, 603-618 (1991; reviewed above).] The stable set or Fatou set \(N(f)\) of a meromorphic function \(f\) is the largest subset of the sphere \(\hat{\mathbf C}\) where the iterates \(f^ n\) of \(f\) may be defined meromorphically and form a normal family. The complement of \(N(f)\) is the Julia set. Given a component \(N_ 0\) of \(N(f)\), there exist components \(N_ n\) of \(N(f)\) such that \(f^ n(N_ 0)\subset N_ n\). The component \(N_ 0\) is called wandering, if \(N_ n\cap N_ m\neq 0\) for \(n\neq m\). One of the most important results in the iteration theory of rational functions is \textit{D. Sullivan}'s theorem [Ann. Math., II. Ser. 122, 401-418 (1985; Zbl 0589.30022)] that rational functions do not have wandering domains. The main result of the paper under review is that Sullivan's result remains valid for the class of critically finite meromorphic functions. Here a meromorphic function is called critically finite, if there exists a finite set \(W\) such that \(f:{\mathbf C}\backslash f^{-1}(W)\to\hat{\mathbf C}\backslash W\) is an unbranched covering. The result that critically finite entire functions do not have wandering domains was proved by \textit{A. Eh. Eremenko} and \textit{M. Yu. Lyubich} [Sov. Math. Dokl. 30, 592-594 (1984; Zbl 0588.30027)] and \textit{L. R. Goldberg} and \textit{L. Keen} [Ergod. Theory Dyn. Syst. 6, 183-192 (1986; Zbl 0657.58011)]. The proof that critically finite meromorphic functions do not have simply-connected wandering domains uses quasiconformal mappings, a tool introduced by Sullivan into the subject. Multiply-connected wandering domains are ruled out by a more elementary (and very clever) method. It is noteworthy that this method also works for rational functions and thus yields a proof that rational functions do not have multiply-connected wandering domains which is much simpler than Sullivan's original argument.
scientific article; zbMATH DE number 147374
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