The growth rate of an entire function and the Hausdorff dimension of its Julia set

Bergweiler, Walter; Karpińska, Boguslawa and Stallard, Gwyneth M. (2009). The growth rate of an entire function and the Hausdorff dimension of its Julia set. Journal of the London Mathematical Society, 80(3) pp. 680–698.

DOI: https://doi.org/10.1112/jlms/jdp042

Abstract

Let f be a transcendental entire function in the Eremenko-Lyubich class B. We give a lower bound for the Hausdorff dimension of the Julia set of f that depends on the growth of f. This estimate is best possible and is obtained by proving a more general result concerning the size of the escaping set of a function with a logarithmic tract.

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