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Makienko's conjecture, a proposed addition to Sullivan's dictionary, can bestated as follows: The Julia set of a rational function R has buried points ifand only if no component of the Fatou set is completely invariant under thesecond iterate of R. We prove Makienko's conjecture for rational functions withJulia sets that are decomposable continua. This is a very broad collection ofJulia sets; it is not known if there exists a rational functions whose Juliaset is an indecomposable continuum.

Any counterexample to Makienko’s conjecture is an indecomposable continuum