Slow escaping points of meromorphic functions

Abstract

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