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Wandering domains for maps of the punctered plane. (English) Zbl 0606.30029
Denote \({\mathbb{C}}\setminus \{0\}\) by \({\mathbb{C}}_ *\). Let f denote an analytic (non-Möbius) map of \({\mathbb{C}}_ *\) to itself and N(f) the set of points in whose neighbourhood \(f^ n\) is a normal family. The following results are proved: I. The components of N(f) are simply or doubly-connected. There is at most one doubly-connected component, except when \(f(z)=kz^ n\), \(k\neq 0\), \(n\in {\mathbb{Z}}\), \(n\neq 0,\pm 1\). II. If \(0<\alpha <\) and \(f(z)=\exp \{\alpha (z-)\}\), then N(f) consists of a single multiply-connected component, which has 0, \(\infty\) on its boundary. III. There is a function f which maps \({\mathbb{C}}_ *\) to \({\mathbb{C}}_ *\) and for which N(f) has a doubly-connected component in which f is analytically conjugate to a rotation \(z\to e^{i\alpha \pi}z\), \(\alpha\) \(\in {\mathbb{R}}\setminus {\mathbb{Q}}\). IV. There is an example of a function f such that N(f) has a wandering component.

MSC:
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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