Rippon, Philip and Stallard, Gwyneth
(2011).
*Transactions of the American Mathematical Society*, 363(8) pp. 4171–4201.

DOI (Digital Object Identifier) Link: | http://dx.doi.org/10.1090/S0002-9947-2011-05158-5 |
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Google Scholar: | Look up in Google Scholar |

## 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

*∞*, arbitrarily slowly. The proof uses new covering results for analytic functions. We also introduce several slow escaping sets, in each of which

*f*tends to

^{n}(z)*∞*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.

Item Type: | Journal Article |
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Copyright Holders: | 2011 American Mathematical Society |

ISSN: | 1088-6850 |

Academic Unit/Department: | Mathematics, Computing and Technology > Mathematics and Statistics Mathematics, Computing and Technology |

Related URLs: | |

Item ID: | 30391 |

Depositing User: | Philip Rippon |

Date Deposited: | 08 Dec 2011 09:44 |

Last Modified: | 18 Jan 2016 11:36 |

URI: | http://oro.open.ac.uk/id/eprint/30391 |

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