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ON THE ITERATIONS AND THE ARGUMENT DISTRIBUTION OF MEROMORPHIC FUNCTIONS

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

Published online by Cambridge University Press:  15 February 2024

JIE DING  and
JIANHUA ZHENG
JIE DING*
Affiliation:
School of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, PR China
JIANHUA ZHENG
Affiliation:
Department of Mathematical Science, Tsinghua University, Beijing 100084, PR China e-mail: zheng-jh@mail.tsinghua.edu.cn
*
e-mail: dingjie@tyut.edu.cn
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Abstract

This paper consists of two parts. The first is to study the existence of a point a at the intersection of the Julia set and the escaping set such that a goes to infinity under iterates along Julia directions or Borel directions. Additionally, we find such points that approximate all Borel directions to escape if the meromorphic functions have positive lower order. We confirm the existence of such slowly escaping points under a weaker growth condition. The second is to study the connection between the Fatou set and argument distribution. In view of the filling disks, we show nonexistence of multiply connected Fatou components if an entire function satisfies a weaker growth condition. We prove that the absence of singular directions implies the nonexistence of large annuli in the Fatou set.

Keywords

meromorphic functionFatou setJulia setBorel directionfilling disk

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
Journal of the Australian Mathematical Society , Volume 116 , Issue 2 , April 2024 , pp. 145 - 170
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Milena Radnovic

The first author was supported by Grant No. 202103021224069 of Shanxi Basic Research Program.

The second author was supported by Grant No. 12071239 of the NSF of China.

References

Baker, I. N., ‘The domain of normality of an entire function’, Ann. Acad. Sci. Fenn. Ser. Math. 1 (1975), 277–283.CrossRefGoogle Scholar
Baker, I. N., Kotus, J. and Lü, Y., ‘Iterates of meromorphic functions I’, Ergodic Theory Dynam. Systems 11 (1991), 241–248.CrossRefGoogle Scholar
Beardon, A. F., Iteration of Rational Functions (Springer, Berlin, 1991).CrossRefGoogle Scholar
Bergweiler, W., ‘Iteration of meromorphic functions’, Bull. Amer. Math. Soc. (N.S.) 29 (1993), 151–188.CrossRefGoogle Scholar
Bergweiler, W., ‘On the packing dimension of the Julia set and the escaping set of an entire function’, Israel J. Math. 192 (2012), 449–472.CrossRefGoogle Scholar
Bergweiler, W. and Chyzhykov, I., ‘Lebesgue measure of escaping sets of entire functions of completely regular growth’, J. Lond. Math. Soc. (2) 94 (2016), 639–661.CrossRefGoogle Scholar
Bergweiler, W. and Hinkenna, A., ‘On semiconjugation of entire functions’, Math. Proc. Cambridge Philos. Soc. 126 (1999), 565–574.CrossRefGoogle Scholar
Bergweiler, W. and Karpinska, B., ‘On the Hausdorff dimension of the Julia set of a regularly growing entire function’, Math. Proc. Cambridge Philos. Soc. 148 (2010), 531–551.CrossRefGoogle Scholar
Bergweiler, W., Rippon, P. and Stallard, G., ‘Multiply connected wandering domains of entire functions’, Proc. Lond. Math. Soc. (3) 107 (2013), 1261–1301.CrossRefGoogle Scholar
Bergweiler, W. and Zheng, J. H., ‘On the uniform perfectness of the boundary of multiply connected wandering domains’, J. Aust. Math. Soc. 91 (2011), 289–311.CrossRefGoogle Scholar
Devaney, R. L. and Krych, M., ‘Dynamics of $\exp \left({z}\right)$ ’, Ergodic Theory Dynam. Systems 4 (1984), 35–52.CrossRefGoogle Scholar
Devaney, R. L. and Tangerman, F., ‘Dynamics of entire functions near the essential singularity’, Ergodic Theory Dynam. Systems 6 (1986), 489–503.CrossRefGoogle Scholar
Dominguez, P., ‘Dynamics of transcendental meromorphic functions’, Ann. Acad. Sci. Fenn. Math. 23 (1998), 225–250.Google Scholar
Eremenko, A. E., ‘On the iteration of entire functions’, in: Dynamical Systems and Ergodic Theory, Banach Center Publications, 23 (ed. Krzyżewski, K.) (Polish Scientific Publishers, Warsaw, 1989), 339–345.Google Scholar
Goldberg, A. A., Levin, B. Y. and Ostrovskii, I. V., ‘Entire and meromorphic functions’, in: Complex Anaysis. I, Encyclopaedia of Mathematical Sciences, 85 (eds. Gonchar, A. A., Havin, V. P. and Nikolski, N. K.) (Springer-Verlag, Berlin–Heidelberg, 1997), 1–193, 254–261.Google Scholar
Goldberg, A. A. and Ostrovskii, I. V., Value Distribution of Meromorphic Functions, Translations of Mathematical Monographs, 236 (American Mathematical Society, Providence, RI, 2008).CrossRefGoogle Scholar
Hayman, W. K., Meromorphic Functions (Clarendon Press, Oxford, UK, 1964).Google Scholar
Milloux, H., ‘Le Théorème de Picard, suites de fonctions holomorphes; fonctions holomorphes et fonctions entières’, J. Math. Pures Appl. (9) 3 (1924), 345–401.Google Scholar
Milnor, J., Dynamics in One Complex Variable (Friedr. Vieweg $\&$ Sohn, Braunschweig, 1999).Google Scholar
Petrenko, V. P., ‘On the growth of meromorphic functions with finite order’, Isv. Akad. Nauk. SSSR 33 (1969), 414–454 (in Russian).Google Scholar
Qiao, J. Y., ‘Borel directions and iterated orbits of meromorphic functions’, Bull. Aust. Math. Soc. 61(1) (2000), 1–9.CrossRefGoogle Scholar
Rempe, L., ‘On a question of Eremenko concerning escaping components of entire functions’, Bull. Lond. Math. Soc. 39 (2007), 661–666.CrossRefGoogle Scholar
Rempe, L., ‘Rigidity of escaping dynamics for transcendental entire functions’, Acta Math. 203 (2009), 235–267.CrossRefGoogle Scholar
Rempe, L., Rippon, P. J. and Stallard, G. M., ‘Are Devaney hairs fast escaping?’, J. Difference Equ. Appl. 6 (2010), 739–762.CrossRefGoogle Scholar
Rippon, P. J. and Stallard, G. M., ‘On sets where iterates of a meromorphic function zip towards infinity’, Bull. Lond. Math. Soc. 32 (2000), 528–536.CrossRefGoogle Scholar
Rippon, P. J. and Stallard, G. M., ‘Slow escaping points of meromorphic functions’, Trans. Amer. Math. Soc. 363 (2011), 4171–4201.CrossRefGoogle Scholar
Rippon, P. J. and Stallard, G. M., ‘Fast escaping points of entire functions’, Proc. Lond. Math. Soc. (3) 105 (2012), 787–820.CrossRefGoogle Scholar
Rippon, P. J. and Stallard, G. M., ‘Eremenko points and the structure of the escaping set’, Trans. Amer. Math. Soc. 372(5) (2019), 3083–3111.CrossRefGoogle Scholar
Sixsmith, D. J., ‘Maximally and non-maximally fast escaping points of transcendental entire functions’, Math. Proc. Cambridge Philos. Soc. 158 (2015), 365–383.CrossRefGoogle Scholar
Yang, L., Value Distribution Theory (Springer-Verlag, Berlin–Heidelberg–New York; Science Press, Beijing, 1993).Google Scholar
Zheng, J. H., Dynamics of Transcendental Meromorphic Functions (Tsinghua University Press, Beijing, 2006) (in Chinese).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., Value Distribution of Meromorphic Functions (Tsinghua University Press, Beijing; Springer-Verlag, Berlin, 2010).Google Scholar
Zheng, J. H., ‘Hyperbolic metric and multiply connected wandering domains of meromorphic functions’, Proc. Edinb. Math. Soc. (2) 60 (2017), 787–810.CrossRefGoogle Scholar