Bergweiler, W.; Rippon, Philip J. and Stallard, Gwyneth M.
Dynamics of meromorphic functions with direct or logarithmic singularities.
Proceedings of the London Mathematical Society, 97(2) pp. 368–400.
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)∩J(f) contains continua. Moreover, under this hypothesis I(f)∩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)∩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.
|External Project Funding Details:
|Funded Project Name||Project ID||Funding Body|
|Not Set||Not Set||London Mathematical Society|
|Not Set||Not Set||German–Israeli Foundation for Scientific Research and Development [grant number G-809-234.6/2003]|
|Not Set||Not Set||EU Research Training Network|
|Not Set||Not Set||ESF Research Networking Programme|
||Mathematics, Computing and Technology > Mathematics and Statistics
||14 Apr 2009 20:29
||02 Dec 2010 20:26
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