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 functions whose Julia set is an indecomposable continuum.

Buried points in Julia sets

“…to Theorem 1.3, but for a special class of maps (those with exactly two critical points). Improving on these results, Curry et al [10] recently proved the following.…”

“…In fact, it is not known if the Julia set of a rational function can be as complicated as required. We restate here the questions which appear in [9][10][11]25,29,31]. Theorems in [9,11] indicate that the answers to these two questions are the same for polynomial Julia sets and for rational functions whose Julia set contains no buried points.…”

The maximal entropy measure of Fatou boundaries

“…The Makienko conjecture can be stated as saying that, if J 0 = ∅, then there is a Fatou component that is completely invariant under R 2 . A discussion of conditions under which Makienko's conjecture has been proved can be found in [2]. Of particular use here is the following result of [11]:…”

Buried points in complex dynamical systems and glassy transitions

“…1991 年, Beardon [6] 数淹没点存在性的最为完备的结果. 虽然围绕淹没点的几何和拓扑性质还有一些深入的研究, 但 关于淹没点的本质特性, 特别是动力系统在其上的性态还远远没有被揭示 [8,12,16] .…”