Dynamics of meromorphic functions with direct or logarithmic singularities

Bergweiler, W.; Rippon, Philip J. and Stallard, Gwyneth M. (2008). Dynamics of meromorphic functions with direct or logarithmic singularities. Proceedings of the London Mathematical Society, 97(2) pp. 368–400.

DOI: https://doi.org/10.1112/plms/pdn007

Abstract

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.

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• Item ORO ID
• 15753
• Item Type
• Journal Item
• ISSN
• 1460-244X
• 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

• Academic Unit or School
• Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
  Faculty of Science, Technology, Engineering and Mathematics (STEM)
• Depositing User
• Colin Smith