For transcendental self-maps of the punctured plane, the escaping set is either connected or has infinitely many components, and the first doubly connected Baker domain is constructed.
Bergweiler, Iteration of meromorphic functions, Bull
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On the connectivity of the escaping set in the punctured plane
For transcendental self-maps of the punctured plane, the escaping set is either connected or has infinitely many components, and the first doubly connected Baker domain is constructed.