Any counterexample to Makienko's conjecture is an indecomposable continuum

TL;DR

This paper proves Makienko's conjecture for rational functions with decomposable Julia sets, a broad class, while the case of indecomposable Julia sets remains unresolved.

Contribution

It establishes Makienko's conjecture for a wide class of Julia sets, specifically those that are decomposable continua.

Findings

Makienko's conjecture holds for decomposable Julia sets

It remains open whether indecomposable Julia sets can be counterexamples

The paper advances understanding of Julia set structures in complex dynamics

Abstract

Makienko's conjecture, a proposed addition to Sullivan's dictionary, can be stated as follows: The Julia set of a rational function R 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 functions whose Julia set is an indecomposable continuum.