Slow escaping points of meromorphic functions

Rippon, P.; Stallard, G.

doi:10.1090/S0002-9947-2011-05158-5

Slow escaping points of meromorphic functions
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by P. J. Rippon and G. M. Stallard PDF

Trans. Amer. Math. Soc. 363 (2011), 4171-4201 Request permission

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 $f^n(z)$ 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 $f^n(z)$ tends to $\infty$ 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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