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Uniformly perfect Julia sets of meromorphic functions

Published online by Cambridge University Press:  17 April 2009

Sheng Wang  and
Liang-Wen Liao
Sheng Wang
Affiliation:
College of Economics and Management, South China Agricultural University, Guangzhou 510642, Guangdong, China, e-mail: [email protected]
Liang-Wen Liao
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, China, e-mail: [email protected]
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Julia sets of meromorphic functions are uniformly perfect under some suitable conditions. So are Julia sets of the skew product associated with finitely generated semigroup of rational functions.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 71 , Issue 3 , June 2005 , pp. 387 - 397
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Baker, I.N., ‘Some entire functions with multiply-connected wandering domains’, Ergodic Theory Dynam. Systems 5 (1985), 163–169.CrossRefGoogle Scholar
[2]Baker, I.N., Kotus, J. and Lü, Y.N., ‘Iterates of meromorphic functions IV: Critically finite functions’, Results Math. 22 (1992), 651–656.CrossRefGoogle Scholar
[3]Baker, I.N., Domínguez, P. and Herring, M.E., ‘Dynamics of functions meromorphic outside a small set’, Ergodic Theory Dynam. Systems 21 (2001), 647–672.CrossRefGoogle Scholar
[4]Beardon, A.F., Iteration of rational functions (Springer-Verlag, Berlin, 1991).CrossRefGoogle Scholar
[5]Beardon, A.F. and Pommerenke, Ch., ‘The Poincaré metric of plane domains’, J. London Math. Soc. 41 (1979), 475–483.Google Scholar
[6]Carleson, L. and Gamelin, W., Complex dynamics (Springer-Verlag, New York, 1993).CrossRefGoogle Scholar
[7]Fatou, P., ‘Sur l'itération des fonctions transcendentes entiéres’, Acta Math. 47 (1926), 337–370.CrossRefGoogle Scholar
[8]Hinkkanen, A., ‘Julia sets of rational functions are uniformly perfect’, Math. Proc. Cambridge Philos. Soc. 113 (1993), 543–559.CrossRefGoogle Scholar
[9]Hinkkanen, A. and Martin, G.J., ‘Julia sets of rational functions’, Math. Z. 222 (1996), 161–169.CrossRefGoogle Scholar
[10]Hua, X.H. and Yang, C.C., Dynamics of transcendental functions, Asian Mathematics Series 1 (Gordon and Breach Science Publishers, Amsterdam, 1998).Google Scholar
[11]Māné, R. and Da Rocha, L.F., ‘Julia sets are uniformly perfect’, Proc. Amer. Math. Soc. 62 (1992), 251–257.CrossRefGoogle Scholar
[12]Pommerenke, Ch., ‘Uniformly perfect sets and Poincaré metric’, Arch Math. 32 (1979), 192–199.CrossRefGoogle Scholar
[13] Ch. Pommerenke, ‘On uniformly perfect sets and Fuchsian groups’, Analysis 4 (1984), 299–321.Google Scholar
[14]Stallard, G.M., ‘The Hausdorff dimension of Julia sets of meromorphic functions II’, J. London Math. Soc. 2 60 (1999), 847–859.CrossRefGoogle Scholar
[15]Sumi, H., ‘Skew product maps related to finitely generated rational semigroups’, Nonlinearity 13 (2000), 995–1019.CrossRefGoogle Scholar
[16]Wang, S., On some properties of Fatou and Julia sets of meromorphic functions, (in Chinese, Ph.D. Disssertation) (Tsinghua University, Beijing, 2002).Google Scholar
[17]Zheng, J.H., ‘On uniformly perfect boundaries of stable domains in iteration of meromorphic functions’, Bull. London Math. Soc. 32 (2000), 439–446.Google Scholar
[18]Zheng, J.H., ‘Uniformly perfect sets and distortion of holomorphic functions’, Nagoya Math. J. 164 (2001), 17–33.CrossRefGoogle Scholar
[19]Zheng, J.H., ‘On uniformly perfect boundaries of stable domains in iteration of meromorphic functions II’, Math. Proc. Cambridge Philos. Soc. 132 (2002), 531–544.Google Scholar
[20]Zheng, J.H., ‘Singularities and limit functions in iteration of meromorphic functions’, J. London Math. Soc. 67 (2003), 195–207.CrossRefGoogle Scholar