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Journal of the London Mathematical Society Advance Access originally published online on September 30, 2009
Journal of the London Mathematical Society 2009 80(3):680-698; doi:10.1112/jlms/jdp042
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© 2009 London Mathematical Society

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

Walter Bergweiler

Mathematisches Seminar
Christian–Albrechts–Universität zu Kiel
Ludewig–Meyn–Str. 4
D–24098 Kiel
Germany
bergweiler@math.uni-kiel.de

Boguslawa Karpinska

Faculty of Mathematics and Information Science
Warsaw University of Technology
Pl. Politechniki 1
00-661 Warszawa
Poland
bkarpin@mini.pw.edu.pl

Gwyneth M. Stallard

Department of Mathematics and Statistics
The Open University
Walton Hall
Milton Keynes
MK7 6AA
United Kingdom

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.

2000 Mathematics Subject Classification 37F10 (primary), 30D05, 30D15 (secondary).

All three authors were supported by the EU Research Training Network CODY. The first author was also supported by the G.I.F., the German–Israeli Foundation for Scientific Research and Development, Grant G-809-234.6/2003 and the ESF Research Networking Programme HCAA. The second author was also supported by Polish MNiSW Grant N N201 0234 33 and PW Grant 504G 1120 0011 000. The latter grant supported a visit of the first and third authors to Warsaw, during which most of the work for this paper was carried out.

Received July 15, 2008; revised April 14, 2009; published online September 30, 2009.


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