Ergodic Theory and Dynamical Systems

Research Article

Any counterexample to Makienko’s conjecture is an indecomposable continuum

CLINTON P. CURRYa1, JOHN C. MAYERa1, JONATHAN MEDDAUGHa2 and JAMES T. ROGERS Jra2

a1 Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294-1170, USA (email: clintonc@uab.edu, mayer@math.uab.edu)

a2 Department of Mathematics, Tulane University, New Orleans, LA 70118, USA (email: jmeddaugh@math.tulane.edu, jim@math.tulane.edu)

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.

(Received June 01 2008)

(Revised July 16 2008)

Footnotes

Dedicated to Bob Devaney on the occasion of his 60th birthday