Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-25T02:31:35.405Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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