On the configuration of Herman rings of meromorphic functions
Let \(f\) be a transcendental meromorphic map. There is a well known classification of the connected components of the Fatou set \(\mathcal{F}\) of \(f\). One of the possible types are Herman rings. A Herman ring is a doubly connected \(p\)-periodic Fatou component \(A\) such that there exists \(\theta\in[0,1]\setminus\mathbb{Q}\) (the rotation number of \(A\)), \(0<r<1\) and a conformal change of variables \(L:\{z\in\mathbb{C}:r<|z|<1\}\to A\) which conjugates \(f^p\) (the \(p\)-th iterate of \(f\)) to the rotation \(z\mapsto e^{2\pi i\theta}z\).NEWLINENEWLINE\textit{M. Shishikura} showed that for any \(p>0\) and any \(\theta\) of Brjuno type there exists a rational function with a \(p\)-cycle of Herman rings of rotation number \(\theta\) [Ann. Sci. Éc. Norm. Supér. (4) 20, No. 1, 1--29 (1987; Zbl 0621.58030)]. The main contribution of the paper in review is to show that every configuration of Herman rings which is possible for rational maps is also possible for transcendental meromorphic maps. Thereby, a configuration is an equivalence class which provides the relative position, orientation and dynamics of a cylce of Herman rings.NEWLINENEWLINEFurthermore, the authors prove that every bounded component of \(\mathbb{C}\setminus\bigcup_{i=1}^{n}A_i\) contains a pole, where \(A_1,\dots,A_n\) are the invariant Herman rings of a transcendental meromorphic map. It is also proven that one pole is sufficient to obtain cycles of Herman rings of arbitrary period.NEWLINENEWLINEThe main technique used in the paper is quasiconformal surgery.
scientific article; zbMATH DE number 6062834
scientific article