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The Hausdorff dimension of Julia sets of hyperbolic meromorphic functions

Published online by Cambridge University Press:  01 September 1999

GWYNETH M. STALLARD
GWYNETH M. STALLARD
Affiliation:
Department of Pure Mathematics, The Open University, Walton Hall, Milton Keynes, MK7 6AA
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Abstract

Let f be a hyperbolic transcendental meromorphic function such that the finite singularities of f−1 are in a bounded set. We show that there exists 0<s(f)[les ]2 such that

formula here

for each point a in the Julia set of f, where

formula here

We then show that s(f)[les ]dimJ(f), the Hausdorff dimension of the Julia set, and give examples of such functions for which dimJ(f)>s(f). This contrasts with the situation for a hyperbolic rational function f where it is known that dimJ(f) = s(f).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 127 , Issue 2 , September 1999 , pp. 271 - 288
Copyright
The Cambridge Philosophical Society 1999

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