Slow escaping points of meromorphic functions

By P. Rippon, G. Stallard. 2008

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We show that for any transcendental meromorphic function f there is a point z in the Julia set of f such that the iterates f n (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 f n (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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/744494aa9d46483a5edf16682f0fa03ccd86ae8c