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.



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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