Hostname: page-component-cb9f654ff-lqqdg Total loading time: 0 Render date: 2025-08-28T23:47:11.566Z Has data issue: false hasContentIssue false
  • English
  • Français

Any counterexample to Makienko’s conjecture is an indecomposable continuum

Published online by Cambridge University Press:  01 June 2009

CLINTON P. CURRY ,
JOHN C. MAYER ,
JONATHAN MEDDAUGH  and
JAMES T. ROGERS Jr
CLINTON P. CURRY
Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294-1170, USA (email: clintonc@uab.edu, mayer@math.uab.edu)
JOHN C. MAYER
Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294-1170, USA (email: clintonc@uab.edu, mayer@math.uab.edu)
JONATHAN MEDDAUGH
Affiliation:
Department of Mathematics, Tulane University, New Orleans, LA 70118, USA (email: jmeddaugh@math.tulane.edu, jim@math.tulane.edu)
JAMES T. ROGERS Jr
Affiliation:
Department of Mathematics, Tulane University, New Orleans, LA 70118, USA (email: jmeddaugh@math.tulane.edu, jim@math.tulane.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

Makienko’s conjecture, a proposed addition to Sullivan’s dictionary, can be stated as follows: the Julia set of a rational function R:ℂ∞→ℂ∞ has buried points if and only if no component of the Fatou set is completely invariant under the second iterate of R. We prove Makienko’s conjecture for rational functions with Julia sets that are decomposable continua. This is a very broad collection of Julia sets; it is not known if there exists a rational function whose Julia set is an indecomposable continuum.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 3 , June 2009 , pp. 875 - 883
Copyright
Copyright © Cambridge University Press 2009

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.)

Article purchase

Temporarily unavailable

References

[1]Beardon, A. F.. Iteration of Rational Functions. Springer, Berlin, 1991.CrossRefGoogle Scholar
[2]Childers, D. K., Mayer, J. C. and Rogers Jr, J. T.. Indecomposable continua and the Julia sets of polynomials. II. Topology Appl. 153(10) (2006), 1593–1602.CrossRefGoogle Scholar
[3]Childers, D. K., Mayer, J. C., Murat Tuncali, H. and Tymchatyn, E. D.. Indecomposable continua and the Julia sets of rational maps. Complex Dynamics (Contemporary Mathematics, 396). American Mathematical Society, Providence, RI, 2006, pp. 1–20.Google Scholar
[4]Domínguez, P. and Fagella, N.. Residual Julia sets of rational and transcendental functions. Transcendental Dynamics and Complex Analysis (London Mathematical Society Lecture Note Series, 348). Cambridge University Press, Cambridge, 2007.Google Scholar
[5]Devaney, R. L., Look, D. M. and Uminsky, D.. The escape trichotomy for singularly perturbed rational maps. Indiana Univ. Math. J. 54(6) (2005), 1621–1634.CrossRefGoogle Scholar
[6]Eremenko, A. E. and Lyubich, M. Yu. The dynamics of analytic transformations. Leningrad Math. J. 1(3) (1990), 562–634.Google Scholar
[7]Kuratowski, C.. Sur la structure des frontières communes à deux régions. Fund. Math. 12(1) (1928), 21–42.CrossRefGoogle Scholar
[8]Mayer, J. C. and Rogers Jr, J. T.. Indecomposable continua and the Julia sets of polynomials. Proc. Amer. Math. Soc. 117(3) (1993), 795–802.CrossRefGoogle Scholar
[9]McMullen, C.. Automorphism of rational maps. Holomorphic Functions and Moduli, I (Mathematical Sciences Research Institute Publications, 10). Springer, New York, 1988,pp. 31–60.CrossRefGoogle Scholar
[10]Milnor, J. and Lei, T.. A “Sierpiński carpet” as Julia set. Appendix F in Geometry and dynamics of quadratic rational maps. Experiment. Math. 2(1) (1993), 37–83.CrossRefGoogle Scholar
[11]Morosawa, S.. On the residual Julia sets of rational functions. Ergod. Th. & Dynam. Sys. 17(1) (1997), 205–210.CrossRefGoogle Scholar
[12]Morosawa, S.. Julia sets of subhyperbolic rational functions. Complex Var. Theory Appl. 41(2) (2000), 151–162.Google Scholar
[13]Ng, T. W., Zheng, J. H. and Choi, Y. Y.. Residual Julia sets of meromorphic functions. Math. Proc. Cambridge Philos. Soc. 141(1) (2006), 113–126.CrossRefGoogle Scholar
[14]Rogers Jr, J. T.. Diophantine conditions imply critical points on the boundaries of Siegel disks of polynomials. Comm. Math. Phys. 195(1) (1998), 175–193.CrossRefGoogle Scholar
[15]Qiao, J.. Topological complexity of Julia sets. Sci. China Ser. A 40(11) (1997), 1158–1165.CrossRefGoogle Scholar
[16]Sun, Y. and Yang, C.C.. Buried points and Lakes of Wada continua. Discrete Contin. Dyn. Syst. 9(2) (2003), 379–382.Google Scholar