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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Annular itineraries for entire functions


Authors: P. J. Rippon and G. M. Stallard
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 37F10; Secondary 30D05
Published electronically: June 26, 2014
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Abstract: In order to analyse the way in which the size of the iterates of a transcendental entire function $ f$ can behave, we introduce the concept of the annular itinerary of a point $ z$. This is the sequence of non-negative integers $ s_0s_1\ldots $ defined by

$\displaystyle f^n(z)\in A_{s_n}(R),\;\;$$\displaystyle \text {for }n\ge 0, $

where $ A_0(R)=\{z:\vert z\vert<R\}$ and

$\displaystyle A_n(R)=\{z:M^{n-1}(R)\le \vert z\vert<M^n(R)\},\;\;n\ge 1. $

Here $ M(r)$ is the maximum modulus of $ f$ on $ \{z:\vert z\vert=r\}$ and $ R>0$ is so large that $ M(r)>r$, for $ r\ge R$.

We consider the different types of annular itineraries that can occur for any transcendental entire function $ f$ and show that it is always possible to find points with various types of prescribed annular itineraries. The proofs use two new annuli covering results that are of wider interest.


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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: http://dx.doi.org/10.1090/S0002-9947-2014-06354-X
PII: S 0002-9947(2014)06354-X
Received by editor(s): January 24, 2013
Published electronically: June 26, 2014
Additional Notes: Both authors were supported by the EPSRC grant EP/H006591/1
Article copyright: © Copyright 2014 American Mathematical Society