Abstract
For a polynomial P it is well known that its Julia set \({\cal J_P}\) is connected if and only if the orbits of the finite critical points are bounded. But there is no such simple criteria for the connectedness of the Julia set of a rational function. Indeed, up to the very nice result of Shishikura that any rational function which has one repelling fixed point only has a connected Julia set almost nothing is known on the connectivity. In the first part of the paper we give constructive sufficient conditions for a basin of attraction to be completely invariant and the Julia set to be connected. Then it is shown that the connectedness of a basin of attraction depends heavily on the fact whether the critical points from the basin tend to the attracting fixed point z 0 via a preimage of z 0 or not. As a consequence we obtain for instance that rational functions with a finite postcritical set or with a Fatou set which contains no Herman rings and each component of which contains at most one critical point, counted without multiplicity, have a connected Julia set.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
A. F. Beardon, Iteration of Rational Functions, Graduate Texts in Mathematics 132, Springer, New York, 1991.
L. Carleson and T. W. Gamelin, Complex Dynamics, Springer, New York, 1993.
C. Inninger and F. Peherstorfer, A new simple class of rational functions whose Julia set is the whole Riemann sphere, J. London Math. Soc. 65 (2002), 453–463.
C. Inninger and F. Peherstorfer, Explicite rational functions whose Julia set is a Jordan arc, submitted.
J. Milnor, Geometry and dynamics of quadratic rational maps, Exp. Math. 2 (1993), 37–83.
J. Milnor, Dynamics in One Complex Variable, Introductory Lectures, Vieweg, Braunschweig, 1999.
F. Peherstorfer and C. Stroh, Julia and Mandelbrot Sets of Chebyshev Families, Int. J. Bifur. Chaos Appl. Sci. Engrg. 11 (2001), 2463–2481.
F. Przytycki, Remarks on the simple connectedness of basins of sinks for iterations of rational maps, Dynamical systems and ergodic theory, 29th Sem. St. Banach Int. Math. Cent., Warsaw/pol. 1986 (K. Krzyzzewski, ed.), Banach Ent. Publ., vol. 23, 1989, 229–235.
F. Przytycki, Iterations of rational functions: which hyperbolic components contain polynomials?, Fundamenta Math. 149 (1996), 95–118.
M. Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. École Norm. Sup. (4) 20 (1987), 1–29.
M. Shishikura, The connectivity of the Julia set of rational maps and fixed points, preprint, Inst. Hautes Études Sci., Bures-sur-Yvette, 1990.
N. Steinmetz, Rational Iteration, de Gruyter Studies in Mathematics 16, de Gruyter, Berlin, 1993.
Young Chen Yin, On the Julia sets of quadratic rational maps, Complex Variables Theory Appl. 18 (1992), 141–147.
Author information
Authors and Affiliations
Corresponding author
Additional information
The second author has been supported by the Austrian Science Fund (FWF) under the project P12985-TEC.
Rights and permissions
About this article
Cite this article
Peherstorfer, F., Stroh, C. Connectedness of Julia Sets of Rational Functions. Comput. Methods Funct. Theory 1, 61–79 (2001). https://doi.org/10.1007/BF03320977
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03320977