Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-25T00:29:49.383Z Has data issue: false hasContentIssue false
  • English
  • Français

Rational maps whose Fatou components are Jordan domains

Published online by Cambridge University Press:  14 October 2010

Kevin M. Pilgrim
Kevin M. Pilgrim
Affiliation:
Mathematics Department, White Hall, Cornell University, Ithaca, NY 14853, USA, (e-mail: pilgrim@math.Cornell.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove: If f(z) is a critically finite rational map which has exactly two critical points and which is not conjugate to a polynomial, then the boundary of every Fatou component of f is a Jordan curve. If f(z) is a hyperbolic critically finite rational map all of whose postcritical points are periodic, then there exists a cycle of Fatou components whose boundaries are Jordan curves. We give examples of critically finite hyperbolic rational maps f with a Fatou component ω satisfying f(ω) = ω and f|∂ω not topologically conjugate to the dynamics of any polynomial on its Julia set.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 6 , December 1996 , pp. 1323 - 1343
Copyright
Copyright © Cambridge University Press 1996

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

[Ahm] Ahmadi, D.. Dynamics of certain rational maps of degree two. PhD thesis, University of Liverpool, 1991.Google Scholar
[Bea] Beardon, A. F.. Iteration of Rational Functions (Graduate Texts in Mathematics 132). Springer, 1991.CrossRefGoogle Scholar
[Car] Carathéodory, C.. Theory of Functions of a Complex Variable. Chelsea, 1960.Google Scholar
[DH1] Douady, A. and Hubbard, J.. Etude Dynamique des Polynomes Complexes, I, II. Publications Mathematique Orsay, 1984/1985.Google Scholar
[DH2] Douady, A. and Hubbard, J.. A proof of Thurston's topological characterization of rational functions. Ada. Math. 171 (1993), 263–297.Google Scholar
[Mill] Milnor, J.. Dynamics in one complex variable: introductory lectures. Preprint, SUNY Stonybrook Institute for Mathematical Sciences, 1990/1995.Google Scholar
[Mil2] Milnor, J.. Remarks on quadratic rational maps. Preprint, SUNY Stonybrook Institute for Mathematical Sciences, 1992/1914.Google Scholar
[PT] Pilgrim, K. M. and Lei, T.. Blowing up an arc in a postcritically finite rational map. Ergod. Th. & Dynam. Sys. To appear.Google Scholar
[Ree] Rees, M.. A partial description of the parameter space of rational maps of degree two: part I. Ada Math. 168 (1992), 11–87.Google Scholar
[Sul] Sullivan, D.. Quasiconformal homeomorphisms and dynamics I: solution to the Fatou-Julia problem on wandering domains. Ann. Math. 122 (1985), 401–418.CrossRefGoogle Scholar
[Thu] Thurston, W. P.. On the combinatorics and dynamics of iterated rational maps. Preprint, Princeton University and IAS, 1985.Google Scholar