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.
Spiders' webs in the punctured plane
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abstract
Many authors have studied sets, associated with the dynamics of a transcendental entire function, which have the topological property of being a spider's web. In this paper we adapt the definition of a spider's web to the punctured plane. We give several characterisations of this topological structure, and study the connection with the usual spider's web in $\mathbb{C}$. We show that there are many transcendental self-maps of $\mathbb{C}^*$ for which the Julia set is such a spider's web, and we construct a transcendental self-map of $\mathbb{C}^*$ for which the escaping set $I(f)$ has this structure and hence is connected. By way of contrast with transcendental entire functions, we conjecture that there is no transcendental self-map of $\mathbb{C}^*$ for which the fast escaping set $A(f)$ is such a spider's web.
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math.DS 1years
2019 1verdicts
CONDITIONAL 1representative citing papers
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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.