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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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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Additional Information
  • P. J. Rippon
  • Affiliation: Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom
  • MR Author ID: 190595
  • 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
  • MR Author ID: 292621
  • Email: g.m.stallard@open.ac.uk
  • Received by editor(s): September 5, 2008
  • Received by editor(s) in revised form: June 23, 2009
  • Published electronically: March 15, 2011
  • © Copyright 2011 American Mathematical Society
  • 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
  • MathSciNet review: 2792984