Slow escaping points of meromorphic functions

Rippon, Philip and Stallard, Gwyneth (2011). Slow escaping points of meromorphic functions. Transactions of the American Mathematical Society, 363(8) pp. 4171–4201.

DOI: https://doi.org/10.1090/S0002-9947-2011-05158-5

Abstract

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 quasi-nested wandering domain. We give examples to show how varied the structures of these slow escaping sets can be.

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