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Proceedings of the London Mathematical Society Advance Access published online on March 21, 2008
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdn007
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© 2008 London Mathematical Society

Dynamics of meromorphic functions with direct or logarithmic singularities

Walter Bergweiler

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

Philip J. Rippon and Gwyneth M. Stallard

Department of Mathematics
The Open University
Walton Hall
Milton Keynes MK7 6AA
United Kingdom
g.m.stallard@open.ac.uk

Received 18 April 2007. Revision received 21 January 2008.

Let f be a transcendental meromorphic function and denote by J(f) the Julia set and by I(f) the escaping set. We show that if f has a direct singularity over infinity, then I(f) has an unbounded component and I(f){cap}J(f) contains continua. Moreover, under this hypothesis I(f){cap}J(f) has an unbounded component if and only if f has no Baker wandering domain. If f has a logarithmic singularity over infinity, then the upper box dimension of I(f){cap}J(f) is 2 and the Hausdorff dimension of J(f) is strictly greater than 1. The above theorems are deduced from more general results concerning functions which have ‘direct or logarithmic tracts’, but which need not be meromorphic in the plane. These results are obtained by using a generalization of Wiman–Valiron theory. This method is also applied to complex differential equations.

The authors were supported by a London Mathematical Society Scheme 1 grant. The first author was also supported by the GIF, the German–Israeli Foundation for Scientific Research and Development, Grant G-809-234.6/2003, the EU Research Training Network CODY and the ESF Research Networking Programme HCAA.

2000 Mathematics Subject Classification 37F10 (primary), 30D05, 30D20, 30D30, 34M05 (secondary).


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