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On transcendental meromorphic functions which are geometrically finite

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

Published online by Cambridge University Press:  09 April 2009

Jian-Hua Zheng
Jian-Hua Zheng
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P. R. China e-mail: jzheng@math.tsinghua.edu.cn
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Abstract

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In this paper we give the definition of a meromorphic function which is geometrically finite and investigate some properties of geometrically finite meromorphic functions and the Lebesgue measure of their Julia sets.

Keywords

meromorphic functiongeometrically finiteJulia setLebesgue measure

MSC classification

Secondary: 37F10: Polynomials; rational maps; entire and meromorphic functions 37F50: Small divisors, rotation domains and linearization; Fatou and Julia sets 30D05: Functional equations in the complex domain, iteration and composition of analytic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 1 , February 2002 , pp. 93 - 108
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Baker, I. N., Kotus, J. and Lu, Y., ‘Iterates of meromorphic functions I’, Ergodic Theory Dynam. Systems 11 (1991), 241–248.CrossRefGoogle Scholar
[2]Beardon, A. F. and Pommerenke, Ch., ‘The Poincaré metric of plane domains’, J. London Math. Soc. (2) 18 (1978), 475–483.CrossRefGoogle Scholar
[3]Bergweiler, W., ‘Iteration of meromorphic functions’, Bull. Amer. Math. Soc. (New Series) 29 (1993), 151–188.CrossRefGoogle Scholar
[4]Devaney, R. L. and Keen, L., ‘Dynamics of meromorphic maps with polynomial Schwarzian derivative’, Ann. Sci. Ecole Norm. Sup. 22 (1989), 55–81.CrossRefGoogle Scholar
[5]Dominguez, P., ‘Dynamics of transcendental meromorphic functions’, Ann. Acad. Sci. Fenn. Ser A. I. Math. 23 (1998), 225–250.Google Scholar
[6]Eremenko, A. E., ‘On the iteration of entire functions’, in: Dynamical systems and ergodic theory (Warsaw, 1986) Banach Center Publ. 23 (PWN, Warsaw, 1989) pp. 339–345.Google Scholar
[7]Eremenko, A. E. and Lyubich, M. Yu., ‘Dynamical properties of some classes of entire functions’, Ann. Inst. Fourier 42 (1992), 989–1020.CrossRefGoogle Scholar
[8]Jankowski, M., ‘Newton's method for solutions of quasi-Bassel differential equations’, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 22 (1997), 187–204.Google Scholar
[9]Lyubich, M. Yu., ‘The dynamics of rational transforms: the topological picture’, Russian Math. Surveys 41 (1986), 43–117.CrossRefGoogle Scholar
[10]Lyubich, M. Yu., ‘An analysis of the stability of the dynamics of rational functions’, Selecta Math. Soviet. 9 (1990), 69–90.Google Scholar
[11]Marco, R. Perez, ‘Sur une question de Dulac et Fatou’, C. R. Acad. Sci., Paris, Ser. I 321 (1995), 1045–1048.Google Scholar
[12]McMullen, C., ‘Area and Hausdorff dimension of Julia sets of entire functions’, Trans. Amer. Math. Soc. 300 (1987), 329–342.CrossRefGoogle Scholar
[13]Misiurewicz, M., ‘On iterates of ez’, Ergodic Theory Dynam. Systems 1 (1981), 103–106.CrossRefGoogle Scholar
[14]Rippon, P. J. and Stallard, G. M., ‘Iteration of a class of hyperbolic meromorphic functions’, Proc. Amer. Math. Soc. 207 (1999), 3251–3258.CrossRefGoogle Scholar
[15]Stallard, G. M., ‘Entire functions with Julia sets of zero measure’, Math. Proc. Cambridge Philos. Soc. 108 (1990), 551–557.CrossRefGoogle Scholar
[16]Steinmetz, N., Rational iteration (Walter de Gruyter, Berlin, 1993).CrossRefGoogle Scholar
[17]Zheng, J. H., ‘Singularities and limit functions in iteration of meromorphic functions’, preprint, Tsinghua University.Google Scholar
[18]Zheng, J. H., ‘Singularities and wandering domains in iteration of meromorphic functions’, Illinois J. Math. 44 (2000), 520–530.CrossRefGoogle Scholar
[19]Zheng, J. H., ‘Uniformly perfect sets and distortion of holomorphic functions’, Nagoya Math. J., to appear.Google Scholar