  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.
