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Dimensions of Julia sets of meromorphic functions

Rippon, P.J. and Stallard, G.M. (2005). Dimensions of Julia sets of meromorphic functions. Journal of the London Mathematical Society, 71(3) pp. 669–683.

DOI (Digital Object Identifier) Link: http://dx.doi.org/10.1112/S0024610705006393
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Abstract

We show that for any meromorphic function $f$ the Julia set $J(f)$ has constant local upper and lower box dimensions, $\overline{d}(J(f))$ and $\underline{d}(J(f))$, respectively, near all points of $J(f)$ with at most two
exceptions. Further, the packing dimension of the Julia set is equal to $\overline{d}(J(f))$. Using this result we show that, for any transcendental entire function $f$ in the class $B$ (that is, the class of functions such that the singularities of the inverse function are bounded), both the local upper box dimension and packing dimension of $J(f)$ are equal to 2. Our approach is to show that the subset of the Julia set containing those points that escape to infinity as quickly as possible has local upper box dimension equal to 2.

Item Type: Journal Article
ISSN: 1469-7750
Extra Information: Some of the symbols may not have transferred correctly into this bibliographic record and/or abstract.
Keywords: meromorphic function; Julia set; packing dimension; box dimension
Academic Unit/Department: Mathematics, Computing and Technology > Mathematics and Statistics
Item ID: 7590
Depositing User: Philip Rippon
Date Deposited: 01 May 2007
Last Modified: 02 Dec 2010 19:59
URI: http://oro.open.ac.uk/id/eprint/7590
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