Any counterexample to Makienko’s conjecture is an indecomposable continuum

Abstract

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.