Hostname: page-component-7c8c6479df-995ml Total loading time: 0 Render date: 2024-03-28T15:54:17.048Z Has data issue: false hasContentIssue false

Entire functions with Julia sets of zero measure

Published online by Cambridge University Press:  24 October 2008

Gwyneth M. Stallard
Affiliation:
Department of Mathematics, Imperial College of Science and Technology, London SW7 2BZ

Abstract

We extend results of McMullen about the dynamics of entire functions for which the orbits of the critical values stay away from the Julia set. In particular we show that such functions are expanding on their Julia sets which have self-similarity properties. Under suitable further conditions the Julia sets have plane measure zero.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1990

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.. Limit functions and sets of non-normality in iteration theory. Ann. A cad. Ser. Fenn. Serm. A I Math. 467 (1970), 117.Google Scholar
[2]Douady, A.. Systèmes dynamiques holomorphes. Asterisque 105 (1983), 3963.Google Scholar
[3]Duren, P. L.. Univalent Functions (Springer-Verlag, 1953).Google Scholar
[4]Eremenko, A. and Lyubich, M.. Iterations of entire functions. Dokl. Acad. Nauk. SSSR 279 (1984), 2537.Google Scholar
[5]Fatou, P.. Sur l'itération des fonctions transcendantes entières. Acta Math. 47 (1926), 337370.CrossRefGoogle Scholar
[6]Hayman, W. K.. Meromorphic Functions (Oxford University Press, 1964).Google Scholar
[7]McMullen, C.. Area and Hausdorff dimension of Julia sets of entire functions. Trans. Amer. Math. Soc. 300 (1987), 329342.CrossRefGoogle Scholar
[8]Sullivan, D.. Conformal dynamical systems. In Geometric Dynamics, Lecture Notes in Math. vol. 1007 (Springer-Verlag, 1983), pp. 725752.CrossRefGoogle Scholar
[9]Sullivan, D.. Quasiconformal homeomorphisms and dynamics 1. Ann. of Math. (2) 122 (1985), 401418.CrossRefGoogle Scholar
[10]Tsuji, M.. Potential Theory in Modern Function Theory (Maruzen Company Ltd., 1959).Google Scholar