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DIMENSIONS OF JULIA SETS OF MEROMORPHIC FUNCTIONS

Published online by Cambridge University Press:  24 May 2005

P. J. RIPPON
Affiliation:
Department of Pure Mathematics, Open University, Walton Hall, Milton Keynes MK7 6AA, United [email protected], [email protected]
G. M. STALLARD
Affiliation:
Department of Pure Mathematics, Open University, Walton Hall, Milton Keynes MK7 6AA, United [email protected], [email protected]
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Abstract

It is shown 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 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.

Type
Notes and Papers
Copyright
The London Mathematical Society 2005

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