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Dynamics of meromorphic functions with direct or logarithmic singularities

Bergweiler, W.; Rippon, Philip J. and Stallard, Gwyneth M. (2008). Dynamics of meromorphic functions with direct or logarithmic singularities. Proceedings of the London Mathematical Society, 97(2) pp. 368–400.

DOI (Digital Object Identifier) Link: http://dx.doi.org/10.1112/plms/pdn007
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Abstract

Let f be a transcendental meromorphic function and denote by J(f) the Julia set and by I(f) the escaping set. We show that if f has a direct singularity over infinity, then I(f) has an unbounded component and I(f)∩J(f) contains continua. Moreover, under this hypothesis I(f)∩J(f) has an unbounded component if and only if f has no Baker wandering domain. If f has a logarithmic singularity over infinity, then the upper box dimension of I(f)∩J(f) is 2 and the Hausdorff dimension of J(f) is strictly greater than 1. The above theorems are deduced from more general results concerning functions which have ‘direct or logarithmic tracts’, but which need not be meromorphic in the plane. These results are obtained by using a generalization of Wiman–Valiron theory. This method is also applied to complex differential equations.

Item Type: Journal Article
ISSN: 1460-244X
Project Funding Details:
Funded Project NameProject IDFunding Body
Not SetNot SetLondon Mathematical Society
Not SetNot SetGerman–Israeli Foundation for Scientific Research and Development [grant number G-809-234.6/2003]
Not SetNot SetEU Research Training Network
Not SetNot SetESF Research Networking Programme
Academic Unit/Department: Mathematics, Computing and Technology > Mathematics and Statistics
Item ID: 15753
Depositing User: Colin Smith
Date Deposited: 14 Apr 2009 20:29
Last Modified: 02 Dec 2010 20:26
URI: http://oro.open.ac.uk/id/eprint/15753
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