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Annular itineraries for entire functions

Rippon, P. J. and Stallard, G. M. (2015). Annular itineraries for entire functions. Transactions of the American Mathematical Society, 367(1) pp. 377–399.

DOI (Digital Object Identifier) Link: https://doi.org/10.1090/S0002-9947-2014-06354-X
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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 s0s1 . . . defined by

fn(z)Asn(R), for n ≥ 0,

where A0(R) = {z : |z| < R} and

An(R) = {z : Mn−1(R) ≤ |z|<Mn(R)}, n≥ 1.

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

Item Type: Journal Item
Copyright Holders: 2014 American Mathematical Society
ISSN: 1088-6850
Project Funding Details:
Funded Project NameProject IDFunding Body
Baker's conjecture and Eremenko's conjecture: a unified approach.EP/H006591/1EPSRC (Engineering and Physical Sciences Research Council)
Academic Unit/School: Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM)
Item ID: 40671
Depositing User: Philip Rippon
Date Deposited: 06 Aug 2014 10:02
Last Modified: 23 May 2019 14:24
URI: http://oro.open.ac.uk/id/eprint/40671
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