Sur les points fixes et les cycles répulsifs au voisinage d'une singularité essentielle isolée à l'instar de la méthode de renormalisation de Zalcman (Q763697)

Let \(f\) be a holomorphic function in a region of the complex plane. In [Contemp. Math. 382, 55--63 (2005; Zbl 1089.30028)], \textit{W. Bergweiler} proved that for all \(n>1\), any transcendental entire function has a sequence of repelling periodic points of period \(n\) with diverging multipliers. In the paper under review, the author proves a related result. In Théorème 3.1 he proves that given \(v\in\mathbb{C}\), \(W\subset\mathbb{C}\) a closed neighborhood of \(v\), and \(g: W\setminus\{v\}\to \mathbb{C}\) a holomorphic function with an essential singularity at \(v\), if there is a complex value \(\alpha\) omitted by \(g\) on \(W\setminus\{v\}\), then there exists a sequence of repelling fixed points of \(g\) converging to \(v\) and with diverging multipliers. Moreover, in Théorème 3.2 the author proves that under the same hypotheses, if \(v\) is not an exceptional value for \(g\) in the sense of Picard, then \(v\) can be approached by a sequence of periodic points of \(g\) of period \(2\), which are repelling (and with diverging multipliers) if \(v\) is not a completely ramified value. The proofs of these results strongly use Zalcman's renormalization method.