Abstract:

Let $f$ be a holomorphic self-map of $mathbb C^*=mathbb Csetminus {0}$ with $0$ and $infty$ essential singularities. A point is called escaping if its orbit accumulates to $0$, $infty$ or both of them under iteration by $f$. We introduce the concept of fast escaping point for this class of maps using the iterates of the maximum and minimum modulus functions. As in the entire case, we are able to show that this set is always non-empty and its connected components are unbounded. We also construct points with different rates of escape, including arbitrarily slow escaping points and points with periodic itineraries.