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Dimensions of Julia sets of hyperbolic entire functions

Stallard, Gwyneth M. (2004). Dimensions of Julia sets of hyperbolic entire functions. Bulletin of the London Mathematical Society, 36(2) pp. 263–270.

URL: http://journals.cambridge.org/action/displayAbstra...
DOI (Digital Object Identifier) Link: http://dx.doi.org/10.1112/S0024609303002698
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Abstract

It is known that, if $f$ is a hyperbolic rational function, then the Hausdorff, packing and box dimensions of the Julia set $J(f)$ are equal. It is also known that there is a family of hyperbolic transcendental meromorphic functions with infinitely many poles for which this result fails to be true. In this paper, new methods are used to show that there is a family of hyperbolic transcendental entire functions $f_K$, $K \in \N$, such that the box and packing dimensions of $J(f_K)$ are equal to two, even though as $K \to \infty$ the Hausdorff dimension of $J(f_K)$ tends to one, the lowest possible value for the Hausdorff dimension of the Julia set of a transcendental entire function.

Item Type: Journal Article
ISSN: 1469-2120
Academic Unit/Department: Mathematics, Computing and Technology > Mathematics and Statistics
Item ID: 3839
Depositing User: Gwyneth Stallard
Date Deposited: 30 Jun 2006
Last Modified: 02 Dec 2010 19:50
URI: http://oro.open.ac.uk/id/eprint/3839
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