On the connectivity of the escaping set in the punctured plane

We consider the dynamics of transcendental self-maps of the punctured plane, $\mathbb{C}^*=\mathbb{C}\setminus \{0\}$. We prove that the escaping set $I(f)$ is either connected, or has infinitely many components. We also show that $I(f)\cup \{0,\infty\}$ is either connected, or has exactly two components, one containing $0$ and the other $\infty$. This gives a trichotomy regarding the connectivity of the sets $I(f)$ and $I(f)\cup \{0,\infty\}$, and we give examples of functions for which each case arises. Finally, whereas Baker domains of transcendental entire functions are simply connected, we show that Baker domains can be doubly connected in $\mathbb{C}^*$ by constructing the first such example. We also prove that if $f$ has a doubly connected Baker domain, then its closure contains both $0$ and $\infty$, and hence $I(f)\cup\{0,\infty\}$ is connected.