Slow Escaping Points of Meromorphic Functions

We show that for any transcendental meromorphic function f there is a point z in the Julia set of f such that the iterates fn(z) escape, that is, tend to ∞, arbitrarily slowly. The proof uses new covering results for analytic functions. We also introduce several slow escaping sets, in each of which fn(z) tends to ∞ at a bounded rate, and establish the connections between these sets and the Julia set of f . To do this, we show that the iterates of f satisfy a strong distortion estimate in all types of escaping Fatou components except one, which we call a plane-filling wandering domain. We give examples to show how varied the structures of these slow escaping sets can be.