Spiders' webs in the punctured plane

Many authors have studied sets associated with the dynamics of transcendental entire functions which have the topological property of being a spider's web. Roughly speaking, a subset of the complex plane is a spiders' web if it is connected and contains an increasing sequence of loops that tends to infinity. Such sets can also be characterised by saying that they separate every point of the plane from infinity. We will consider the analogue of this structure in the punctured plane, and study the connection with the usual spiders' webs in the plane. We say that $f$ is a transcendental self-map of the punctured plane if $f:\mathbb{C}^*\to \mathbb{C}^*$ is a holomorphic function, $\mathbb{C}^*=\mathbb{C}\setminus\{0\}$, and both $0$ and $\infty$ are essential singularities of $f$. The escaping set of such maps consists of the points whose orbit accumulates to a subset of $\{0,\infty\}$. We use the spider's web structure to give the first example of a transcendental self-map of $\C^*$ for which the escaping set is connected. This is a joint work with Vasiliki Evdoridou (Open University) and Dave Sixsmith (University of Liverpool).