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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
##### Keywords:
normal family; iterates; set of normality; wandering domain
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