Makienko conjecture

Definitions and mathematical statement

Let $f:\widehat{\mathbb C}\to\widehat{\mathbb C}$ be a rational map of degree at least two, and let $f^m$ denote its $m$-fold iterate. Its Fatou set $F(f)$ is the open set on which the family $\{f^m:m\geq0\}$ is normal in the spherical metric. Its Julia set is $J(f)=\widehat{\mathbb C}\setminus F(f)$. A Fatou component is a connected component of $F(f)$.

Suppose that some Fatou component $D$ has the entire Julia set as its boundary: \[ \partial D=J(f). \] Must $D$ be completely invariant under the second iterate, in the precise sense that \[ (f^2)^{-1}(D)=D? \] The preimage is the full set of points of the sphere whose second forward iterate belongs to $D$, not a chosen inverse branch. The assertion therefore requires both that points of $D$ remain in $D$ after two steps and that no point outside $D$ enters it after two steps.

No initial periodicity or simple connectivity assumption is made on $D$, and no local connectedness assumption is made on $J(f)$. The use of the second iterate permits a two-cycle of Fatou components. The question is whether the strong boundary equality alone forces this complete invariance for every rational map of every degree. A counterexample must have exactly the stated boundary equality, rather than only a boundary containing a substantial subset of the Julia set.

Short English statement

If one Fatou component has the whole Julia set as its boundary, must its full two-step preimage equal itself?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700005 (AMR-036-0005), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eremenko - My favorite unsolved problems, Completely invariant domains, Problem 1. Original source indexed by AMR-036-0005. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Two questions on completely invariant domains, Question 1. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/conjectures.pdf.