Dynamics of meromorphic functions with direct or logarithmic singularities

Details Dynamics of meromorphic functions with direct or logarithmic singularities Dynamics of meromorphic functions with direct or logarithmic singularities

2007-04-20

arxiv.org/abs/0704.2712v2

Proc. London. Math. Soc. 97 (2008), 368-400.

Mathematics

Complex Variables

29 pages, 2 figures; v2: some overall revision, with comments and references added; to appear in Proc. London Math. Soc

Scientific paper

10.1112/plms/pdn007

We show that if a meromorphic function has a direct singularity over infinity, then the escaping set has an unbounded component and the intersection of the escaping set with the Julia set contains continua. This intersection has an unbounded component if and only if the function has no Baker wandering domains. We also give estimates of the Hausdorff dimension and the upper box dimension of the Julia set of a meromorphic function with a logarithmic singularity over infinity. 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. The method is also applied to complex differential equations.

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