Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-09T14:23:13.270Z Has data issue: false hasContentIssue false

A NEW SIMPLE CLASS OF RATIONAL FUNCTIONS WHOSE JULIA SET IS THE WHOLE RIEMANN SPHERE

Published online by Cambridge University Press:  24 March 2003

CLEMENS INNINGER
Affiliation:
Institut für Analysis, Universität Linz, Altenbergerstrasse 69, 4040 Linz, Austria; [email protected], [email protected]
FRANZ PEHERSTORFER
Affiliation:
Institut für Analysis, Universität Linz, Altenbergerstrasse 69, 4040 Linz, Austria; [email protected], [email protected]
Get access

Abstract

The paper first gives sufficient conditions on the critical points and the Schwarzian derivative of a real rational function $R$ such that the Julia set of $R$ is $\bar{{\bb C}}$ . Further, it is shown that under mild conditions on another real rational function $\tilde{R}$ with possibly non-empty Fatou set, the Julia set of $\tilde{R} \circ R$ is the whole Riemann sphere again. Then families of rational functions are given whose Julia set is $\bar{{\bb C}}$ and whose critical points are not necessarily preperiodic. Concrete examples were previously available only for the preperiodic case. Finally, it is demonstrated that the methods presented also apply to the construction of polynomials whose Julia sets are dendrites and whose critical points in the Julia set are not necessarily preperiodic.

Type
Research Article
Copyright
The London Mathematical Society, 2002

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.)

Footnotes

This work was supported by the Austrian Science Fund FWF, project number P12985-TEC.