The escaping set of transcendental self-maps of the punctured plane
We study the different rates of escape of points under iteration by holomorphic self-maps of $\mathbb{C}^{\ast }=\mathbb{C}\setminus \{0\}$ for which both zero and infinity are essential singularities. Using annular covering lemmas we construct different types of orbits, including fast escaping and arbitrarily slowly escaping orbits to either zero, infinity or both. We also prove several properties about the set of fast escaping points for this class of functions. In particular, we show that there is an uncountable collection of disjoint sets of fast escaping points, each of which has $J(f)$ as its boundary.