Hausdorff measure of escaping sets of certain meromorphic functions

Baranski and Schubert showed that for a meromorphic function f in Eremenko-Lyubich class B of finite order and for which infinity is an asymptotic value, the Hausdorff dimension of its escaping set, denoted by dim I(f), is two. Assume that for f in B the value infinity is not asymptotic and that there exists M such that the multiplicity of all poles, except finitely many, is at most M. Bergweiler and Kotus proved that for such a function, the Hausdorff dimension of its escaping set is bounded by 2Mp/(2+Mp), where p is the order of f. If f is chosen as above except that the order is infinite, then the area of I(f) is zero, while an example of such a function f with dim I(f)=2 was given. In 1987 McMullen proved that the Julia set of the exponential function has Hausdorff dimension two but, in the presence of an attracting periodic cycle, its area is zero. He further remarked that the Hausdorff measure is infinity induced by a certain gauge function. Peter gave in his dissertation a fairly precise description of the gauge functions for which the Hausdorff measure is infinity. We aim in this thesis to find a gauge function for which the Hausdorff measure of I(f) is 0 or infty for certain meromorphic functions in B of infinite order. As results we find a gauge function, for which the exponent term is depending on the growth rate of the functions that we study.


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