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arXiv · 0901.3014

On the Hausdorff dimension of the escaping set of certain meromorphic functions

Walter Bergweiler·Janina Kotus
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Abstract

The escaping set I(f) of a transcendental meromorphic function f consists of all points which tend to infinity under iteration. The Eremenko-Lyubich class B consists of all transcendental meromorphic functions for which the set of finite critical and asymptotic values of f is bounded. It is shown that if f is in B and of finite order of growth, if infinity is not an asymptotic value of f and if the multiplicities of the poles of f are uniformly bounded, then the Hausdorff dimension of I(f) is strictly smaller than 2. In fact, we give a sharp bound for the Hausdorff dimension of I(f) in terms of the order of f and the bound for the multiplicities of the poles. If f satisfies the above hypotheses but is of infinite order, then the area of I(f) is zero. This result does not hold without a restriction on the multiplicities of the poles.

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Walter Bergweiler, Janina Kotus. 2009-01-20. On the Hausdorff dimension of the escaping set of certain meromorphic functions. https://doi.org/10.1090/s0002-9947-2012-05514-0

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