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Iterates of meromorphic functions on escaping Fatou components

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

Published online by Cambridge University Press:  22 November 2022

Jian-Hua Zheng  and
Cheng-Fa Wu Open the ORCID record for Cheng-Fa Wu [Opens in a new window]
Jian-Hua Zheng
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P. R. China (zheng-jh@mail.tsinghua.edu.cn)
Cheng-Fa Wu*
Affiliation:
Institute for Advanced Study, Shenzhen University, Shenzhen 518060, P. R. China College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, P. R. China (cfwu@szu.edu.cn)
*
*Corresponding author.
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Abstract

In this paper, we prove that the ratio of the modulus of the iterates of two points in an escaping Fatou component could be bounded even if the orbit of the component contains a sequence of annuli whose moduli tend to infinity, and this cannot happen when the maximal modulus of the meromorphic function is uniformly large enough. In this way we extend certain related results for entire functions to meromorphic functions with infinitely many poles.

Keywords

Escaping setsmeromorphic functionsFatou setJulia set

MSC classification

Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions
Secondary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 6 , December 2023 , pp. 1906 - 1928
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Baker, I. N.. The iteration of polynomials and transcendental entire functions. J. Aust. Math. Soc. (Ser. A) 30 (1981), 483–495.CrossRefGoogle Scholar
Baker, I. N.. Multiply connected domains of normality in iteration theory. Math. Z. 81 (1963), 206–214.CrossRefGoogle Scholar
Baker, I. N.. The domains of normality of an entire functions. Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), 277–283.CrossRefGoogle Scholar
Baker, I. N.. An entire function which has wandering domains. J. Aust. Math. Soc. (Ser. A) 22 (1976), 173–176.CrossRefGoogle Scholar
Baker, I. N.. Wandering domains in the iteration of entire functions. Proc. Lond. Math. Soc. (3) 49 (1984), 563–576.CrossRefGoogle Scholar
Baker, I. N., Kotus, J. and Lü, Y.. Iterates of meromorphic functions II: examples of wandering domains. J. Lond. Math. Soc. (2) 42 (1990), 267–278.CrossRefGoogle Scholar
Bergweiler, W.. Iteration of meromorphic functions. Bull. Am. Math. Soc. 29 (1993), 151–188.CrossRefGoogle Scholar
Bergweiler, W.. An entire function with simply and multiply connected wandering domains. Pure Appl. Math. Q. (2) 7 (2011), 107–120.CrossRefGoogle Scholar
Bergweiler, W., Rippon, P. J. and Stallard, G. M.. Multiply connected wandering domains of entire functions. Proc. Lond. Math. Soc. (3) 107 (2013), 1261–1301.CrossRefGoogle Scholar
Bergweiler, W. and Zheng, J.. On the uniform perfectness of the boundary of multiply connected wandering domains. J. Aust. Math. Soc. 91 (2011), 289–311.CrossRefGoogle Scholar
Bishop, C. J.. Constructing entire functions by quasiconformal folding. Acta Math. 214 (2015), 1–60.CrossRefGoogle Scholar
Domínguez, P.. Dynamics of transcendental meromorphic functions. Ann. Acad. Sci. Fenn. Math. 23 (1998), 225–250.Google Scholar
Eremenko, A. E. and Lyubich, M. Y.. Examples of entire functions with pathological dynamics. J. Lond. Math. Soc. (2) 36 (1987), 458–468.CrossRefGoogle Scholar
Fagella, N., Godillon, S. and Jarque, X.. Wandering domains for composition of entire functions. J. Math. Anal. Appl. 429 (2015), 478–496.CrossRefGoogle Scholar
Hayman, W. K.. Meromorphic functions (Oxford: Oxford University Press, 1964).Google Scholar
Herman, M.. Exemples de fractions rationnelles ayant une orbite dense sur la sphére de Riemann. Bull. Soc. Math. France 112 (1984), 93–142.CrossRefGoogle Scholar
Kisaka, M. and Shishikura, M., On multiply connected wandering domains of entire functions, Transcendental Dynamics and Complex Analysis (LMS Lecture Note Series vol. 348) eds P. J. Rippon and G. M. Stallard (Cambridge University Press, 2008) pp. 217–250.CrossRefGoogle Scholar
Lazebnik, K.. Several constructions in the Eremenko–Lyubich class. J. Math. Anal. Appl. 448 (2017), 611–632.CrossRefGoogle Scholar
Rippon, P. J.. Baker domains of meromorphic functions. Ergodic Theory Dyn. Syst. 26 (2006), 1225–1233.CrossRefGoogle Scholar
Rippon, P. J., Baker domains, transcendental dynamics and complex analysis (LMS Lecture Note Series vol. 348) eds P. J. Rippon and G. M. Stallard (Cambridge University Press, 2008) pp. 371–395.CrossRefGoogle Scholar
Rippon, P. J. and Stallard, G. M.. On multiply connected wandering domains of meromorphic functions. J. Lond. Math. Soc. (2) 77 (2008), 405–423.CrossRefGoogle Scholar
Rudin, W.. Real and complex analysis (Singapore: McGraw-Hill, 1986).Google Scholar
Sixsmith, D. J.. Simply connected fast escaping Fatou components. Pure Appl. Math. Q. (4) 8 (2012), 1029–1046.CrossRefGoogle Scholar
Töpfer, H.. Über die Iteration der ganzen transzendenten Funktinen, insbesondere von $\sin z$ und $\cos z$. Math. Ann. 117 (1939), 65–84.CrossRefGoogle Scholar
Zheng, J. H.. Value distribution of meromorphic functions (Beijing: Tsinghua University Press; Heidelberg: Springer, 2010).Google Scholar
Zheng, J. H.. On multiply-connected Fatou components in iteration of meromorphic functions. J. Math. Anal. Appl. 313 (2006), 24–37.CrossRefGoogle Scholar
Zheng, J. H.. On non-existence of unbounded domains of normality of meromorphic functions. J. Math. Anal. Appl. 264 (2001), 479–494.Google Scholar
Zheng, J. H.. Hyperbolic metric and multiply connected wandering domains of meromorphic functions. Proc. Edinburg Math. Soc. 60 (2017), 787–810.CrossRefGoogle Scholar