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Dimensions of Julia Sets of Meromorphic Functions
Author(s) -
Rippon P. J.,
Stallard G. M.
Publication year - 2005
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610705006393
Subject(s) - meromorphic function , julia set , mathematics , dimension (graph theory) , infinity , gravitational singularity , bounded function , combinatorics , inverse , transcendental number , class (philosophy) , function (biology) , packing dimension , pure mathematics , mathematical analysis , minkowski–bouligand dimension , geometry , fractal dimension , computer science , artificial intelligence , evolutionary biology , fractal , biology
It is shown that for any meromorphic function f the Julia set J(f) has constant local upper and lower box dimensions, d (J(f)) and 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 d (J(f)) . Using this result it is shown 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. The 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.