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Hyperbolic Metric and Multiply Connected Wandering Domains of Meromorphic Functions

Part of: Complex dynamical systems Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  30 January 2017

Jian-Hua Zheng
Jian-Hua Zheng*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People's Republic of China (jzheng@math.tsinghua.edu.cn)
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Abstract

In this paper, in terms of the hyperbolic metric, we give a condition under which the image of a hyperbolic domain of an analytic function contains a round annulus centred at the origin. From this, we establish results on the multiply connected wandering domains of a meromorphic function that contain large round annuli centred at the origin. We thereby successfully extend the results of transcendental meromorphic functions with finitely many poles to those with infinitely many poles.

Keywords

wandering domainhyperbolic metricannulus domainsFatou setJulia set

MSC classification

Secondary: 37F10: Polynomials; rational maps; entire and meromorphic functions 30D05: Functional equations in the complex domain, iteration and composition of analytic functions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 3 , August 2017 , pp. 787 - 810
Copyright
Copyright © Edinburgh Mathematical Society 2017 

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References

1. Baker, I. N., The domains of normality of an entire function, Annales Acad. Sci. Fenn. Math. 1 (1975), 277–283.Google Scholar
2. Baker, I. N., Kotus, J. and Lü, Y., Iterates of meromorphic functions III: examples of wandering domains, J. Lond. Math. Soc. 42(2) (1990), 267–278.Google Scholar
3. Beardon, A. F., Iteration of rational functions (Springer, 1991).CrossRefGoogle Scholar
4. Beardon, A. F. and Pommerenke, Ch., The Poincaré metric of plane domains, J. Lond. Math. Soc. 18(2) (1978), 475–483.Google Scholar
5. Bergweiler, W., Iteration of meromorphic functions, Bull. Am. Math. Soc. 29 (1993), 151–188.Google Scholar
6. Bergweiler, W. and Terglane, N., Weakly repelling fixpoints and the connectivity of wandering domains, Trans. Am. Math. Soc. 348 (1996), 1–12.Google Scholar
7. Bergweiler, W., Rippon, P. J. and Stallard, G. M., Multiply connected wandering domains of entire functions, Proc. Lond. Math. Soc. 107 (2013), 1261–1301.Google Scholar
8. Carleson, L. and Gamelin, T. W., Complex dynamics (Springer, 1993).CrossRefGoogle Scholar
9. Dominguez, P., Dynamics of transcendental meromorphic functions, Annales Acad. Sci. Fenn. Math. 23 (1998), 225–250.Google Scholar
10. Milnor, J., Dynamics in one complex variable: introductory lectures, Stony Brook Institute for Mathematical Sciences, Preprint (arXiv:math/9201272 [math.DS]; 1990).Google Scholar
11. Rippon, P. J. and Stallard, G. M., Slow escaping points of meromorphic functions, Trans. Am. Math. Soc. 363 (2011), 4171–4201.Google Scholar
12. Rudin, W., Real and complex analysis (McGraw-Hill, 1986).Google Scholar
13. Whittington, J. E., On the fixpoints of entire functions, Proc. Lond. Math. Soc. 17 (1967), 530–546.Google Scholar
14. Zheng, J.-H., On non-existence of unbounded domains of normality of meromorphic functions, J. Math. Analysis Applic. 264 (2001), 479–494.Google Scholar
15. Zheng, J.-H., Uniformly perfect sets and distortion of holomorphic functions, Nagoya Math. J. 164 (2001), 17–33.Google Scholar
16. Zheng, J.-H., Dynamics of transcendental meromorphic functions (in Chinese) (Tsinghua University Press, 2004).Google Scholar
17. Zheng, J.-H., On multiply-connected Fatou components in iteration of meromorphic functions, J. Math. Analysis Applic. 313 (2006), 24–37.CrossRefGoogle Scholar
18. Zheng, J.-H., Value distribution of meromorphic functions (Tsinghua University Press/Springer, 2010).Google Scholar
19. Zheng, J.-H., Domain constants and their applications in dynamics of meromorphic functions, J. Jiangxi Normal Univ. (Nat. Sci. Edn) 34(5) (2010), 1–7.Google Scholar