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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)
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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.

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

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