Dynamics of meromorphic functions with direct or logarithmic singularities

By Walter Bergweiler, P. Rippon, G. Stallard. 2007

97 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0f80e8bf59543f8fed8abb1be588f30f80159d13

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In Applied Mathematics

1. Walter Bergweiler
2. G. Stallard
3. P. Rippon
4. A. Zdunik
5. Krzysztof Bara'nski
6. B. Karpinska
7. Lasse Rempe

In Geometry and Topology

1. Walter Bergweiler
2. G. Stallard
3. P. Rippon
4. Krzysztof Bara'nski
5. B. Karpinska
6. A. Zdunik
7. Lasse Rempe

In Mathematical Physics

1. Walter Bergweiler
2. G. Stallard
3. P. Rippon
4. Krzysztof Bara'nski
5. B. Karpinska
6. A. Zdunik
7. Lasse Rempe

Citations in the corpus

• Slow escaping points of meromorphic functions (2008, unknown): “Next we describe a family of transcendental meromorphic functions which have many dynamical properties in common with transcendental entire functions; see [12].”
• Multiply connected wandering domains of entire functions (2011, methodology): “We remark that [22] gives a special case of this result but Eremenko’s technique allows for the flexibility described below; see [44] for more details and also [15] for a version of Wiman–Valiron theory that allows this technique to be applied to meromorphic functions with a direct tract.”
• Hyperbolic Dimension of Julia Sets of Meromorphic Maps with Logarithmic Tracts (2007, methodology): “The result was extended by Rippon and Stallard in [6] to meromorphic maps with finitely many poles, and by Bergweiler, Rippon and Stallard in [2] to maps with logarithmic tracts.”