Hostname: page-component-7479d7b7d-m9pkr Total loading time: 0 Render date: 2024-07-13T22:22:12.083Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on the Julia set of a rational function

Published online by Cambridge University Press:  24 October 2008

S. D. Letherman  and
R. M. W. Wood
S. D. Letherman
Affiliation:
Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL
R. M. W. Wood
Affiliation:
Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL
Get access Rights & Permissions [Opens in a new window]

Extract

The purpose of this note is to present a few facts about the Julia set of a rational function that are well known to the experts in the subject of complex dynamics but whose documented exposition in the literature seems to need a little clarification. For example, under conditions set out in Theorem 1, the Julia set of a rational function can be expressed as the limit of a sequence of finite sets. In particular, for certain choices of a point α, the Julia set is the limit as n increases of the inverse image sets R−n(α). This formulation is widely exploited in the backwards iteration algorithm to produce computer illustrations of Julia sets (see for example Section 5·4 of [14]).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 118 , Issue 3 , November 1995 , pp. 477 - 485
Copyright
Copyright © Cambridge Philosophical Society 1995

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]Baribeau, L. and Ransford, T. J.. Meromorphic multifunctions in complex dynamics. Ergod. Th. Dynam. Sys. 12 (1992), 39–52.CrossRefGoogle Scholar
[2]Barnsley, M. F.. Fractals everywhere (Academic Press, 1988).Google Scholar
[3]Beardon, A. F.. Iteration of rational functions (Springer-Verlag, 1991).CrossRefGoogle Scholar
[4]Branner, B.. The Mandelbrot set: chaos and fractals. Proc. Symp. App. Math. 39 (1980).Google Scholar
[5]Brolin, H.. Invariant sets under iteration of rational functions. Ark. Mat. 6 (1965), 103–144.CrossRefGoogle Scholar
[6]Douady, A.. The parabolic implosion. Proc. A M.S. Conference (Cincinnati, January 1994).Google Scholar
[7]Herman, M.. Are there critical points on the boundary of singular domains? Comm. Math. Phys. 99 (1985), 593–612.CrossRefGoogle Scholar
[8]Kriete, H.. On the stability of Julia sets. Preprint, Ruhr-Universität Bochum (1992).Google Scholar
[9]Lavours, P.. Systèmes dynamiques holomorphes: explosion de points périodiques paraboliques. Ph.D. Thesis, Université de Paris-Sud (1989).Google Scholar
[10]Letherman, S. D.. Fractals and the continuity of the Julia set. Masters Thesis, University of Manchester (1993).Google Scholar
[11]Peitgen, H.-O. and Richter, P. H.. The beauty of fractals (Springer-Verlag, 1986).CrossRefGoogle Scholar
[12]Steinmetz, N.. Rational iteration (de Gruyter, 1993).CrossRefGoogle Scholar
[13]Sullivan, D. P.. Quasiconformal homeomorphisms and dynamics III - topological conjugacy of analytic endomorphisms (preprint, 1985).Google Scholar
[14]Wagon, S.. Mathematica in action (W. H. Freeman, 1991).Google Scholar