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Slow escaping points of meromorphic functions


Authors: P. J. Rippon and G. M. Stallard
Journal: Trans. Amer. Math. Soc. 363 (2011), 4171-4201
MSC (2010): Primary 37F10; Secondary 30D05
DOI: https://doi.org/10.1090/S0002-9947-2011-05158-5
Published electronically: March 15, 2011
MathSciNet review: 2792984
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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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Additional Information

P. J. Rippon
Affiliation: Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom
Email: p.j.rippon@open.ac.uk

G. M. Stallard
Affiliation: Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom
Email: g.m.stallard@open.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-2011-05158-5
Received by editor(s): September 5, 2008
Received by editor(s) in revised form: June 23, 2009
Published electronically: March 15, 2011
Article copyright: © Copyright 2011 American Mathematical Society

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