Mathematical Proceedings of the Cambridge Philosophical Society

The Hausdorff dimension of Julia sets of hyperbolic meromorphic functions

a1 Department of Pure Mathematics, The Open University, Walton Hall, Milton Keynes, MK7 6AA


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)[less-than-or-eq, slant]2 such that

formula here

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

formula here

We then show that s(f)[less-than-or-eq, slant]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).

(Received February 23 1998)
(Revised September 16 1998)