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Julia sets of rational functions are uniformly perfect

Published online by Cambridge University Press:  24 October 2008

A. Hinkkanen
A. Hinkkanen
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, U.S.A.
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Abstract

Let f be a rational function of degree at least two. We shall prove that the Julia set J(f) of f is uniformly perfect. This means that there is a constant c∈(0, 1) depending on f only such that whenever z∈J(f) and 0 < r < diam J(f) then J(f) intersects the annulus .

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 113 , Issue 3 , May 1993 , pp. 543 - 559
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Baker, I. N.. An entire function which has wandering domains. J. Austral. Math. Soc. (A) 22 (1976), 173–176.CrossRefGoogle Scholar
[2]Baker, I. N., Kotus, J. and Yinian, Lü. Iterates of meromorphic functions. III: Preperiodic domains. Ergodic Theory of Dynamical Systems 11 (1991), 603–618.CrossRefGoogle Scholar
[3]Beardon, A. F.. Iteration of Rational Functions (Springer-Verlag, 1991).CrossRefGoogle Scholar
[4]Beardon, A. F. and Pommerenke, Ch.. The Poincaré metric of plane domains. J. London Math. Soc. (2) 18 (1978), 475–483.CrossRefGoogle Scholar
[5]Blanchard, P.. Complex analytic dynamics on the Riemann sphere. Bull. Amer. Math. Soc. 11 (1984), 85–141.CrossRefGoogle Scholar
[6]Brolin, H.. Invariant sets under iteration of rational functions. Ark. Math. 6 (1965), 103–144.CrossRefGoogle Scholar
[7]Doob, J. L.. Classical Potential Theory and its Probabilistic Counterpart (Springer-Verlag. 1984).CrossRefGoogle Scholar
[8]Fatou, P.. Sur les équations fonctionnelles. Bull. Soc. Math. France 47 (1919), 161–271.CrossRefGoogle Scholar
[9]Fatou, P.. Sur les équations fonctionnelles. Bull. Soc. Math. France 48 (1920), 33–94, 208–314.CrossRefGoogle Scholar
[10]Fatou, P.. Sur l'itération des fonctions transcendantes entières. Acta Math. 47 (1926). 337–370.CrossRefGoogle Scholar
[11]Hayman, W. K. and Kennedy, P. B.. Subharmonic Functions, vol. 1 (Academic Press, 1976).Google Scholar
[12]Hinkkanen, A.. Julia sets of polynomials are uniformly perfect. Preprint.Google Scholar
[13]Julia, G.. Mémoire sur l'itération des fonctions rationnelles. J. Math. Pures Appl. 8 (1918), 47–245.Google Scholar
[14]Lehto, O. and Viktanen, K. I.. Quasiconformal Mappings in the Plane (Springer-Verlag, 1973).CrossRefGoogle Scholar
[15]Miller, T. L. and Olin, R.. Analytic curves. Amer. Math. Monthly 91 (1984), 127–130.CrossRefGoogle Scholar
[16]Pommerenke, C.. Uniformly perfects sets and the Poincaré metric. Arch. Math. (Basel) 32 (1979), 192–199.CrossRefGoogle Scholar
[17]Pommerenke, C.. On uniformly perfect sets and Fuchsian groups. Analysis 4 (1984), 299–321.CrossRefGoogle Scholar
[18]Shishikura, M.. On the quasiconformal surgery of rational functions. Ann. Sci. École Norm. Sup. (4) 20 (1987), 1–29.CrossRefGoogle Scholar
[19]Sullivan, D.. Quasiconformal homeomorphisms and dynamics. I: Solution of the Fatou–Julia problem on wandering domains. Ann. of Math. 122 (1985), 401–418.CrossRefGoogle Scholar
[20]Wolff, J.. Sur une généralisation d'un théorème de Schwarz. C.R. Acad. Sci. Paris 182 (1926), 918–920.Google Scholar