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On the packing dimension of the Julia set and the escaping set of an entire function

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Abstract

Let f be a transcendental entire function. We give conditions which imply that the Julia set and the escaping set of f have packing dimension 2. For example, this holds if there exists a positive constant c less than 1 such that the minimum modulus L(r, f) and the maximum modulus M(r, f) satisfy log L(r, f) ≤ c logM(r, f) for large r. The conditions are also satisfied if logM(2r, f) ≥ d logM(r, f) for some constant d greater than 1 and all large r.

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Authors and Affiliations

  1. Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, D-24098, Kiel, Germany

    Walter Bergweiler

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  1. Walter Bergweiler
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Correspondence to Walter Bergweiler.

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Supported by a Chinese Academy of Sciences Visiting Professorship for Senior International Scientists, Grant No. 2010 TIJ10. Also supported by the Deutsche Forschungsgemeinschaft, Grant Be 1508/7-1, the EU Research Training Network CODY and the ESF Networking Programme HCAA.

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Bergweiler, W. On the packing dimension of the Julia set and the escaping set of an entire function. Isr. J. Math. 192, 449–472 (2012). https://doi.org/10.1007/s11856-012-0033-0

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  • DOI: https://doi.org/10.1007/s11856-012-0033-0

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