Completely invariant Fatou components

Definitions and mathematical statement

Let $f:\mathbb C\to\mathbb C$ be a transcendental entire function, meaning an entire function that is not a polynomial. Its Fatou set $F(f)$ consists of those points with a neighborhood on which the iterates $\{f^m:m\geq0\}$ form a normal family in the spherical metric. A Fatou component is a maximal connected open subset of $F(f)$.

A Fatou component $U$ is completely invariant if \[ f^{-1}(U)=U. \] Thus $f(U)\subseteq U$, and every point whose image lies in $U$ already belongs to $U$. This full-preimage condition is stronger than mere forward invariance of the component. Surjectivity onto values omitted by $f$ is not required.

What is the largest possible number of completely invariant Fatou components of a transcendental entire function? In particular, is that number always at most one, or can there exist two disjoint Fatou components $U,V$ with $f^{-1}(U)=U$ and $f^{-1}(V)=V$? The target covers all transcendental entire functions, without assuming finite order, finitely many singular values, or any special topology of their Fatou components.

Any negative example to the at-most-one assertion must use actual components of the dynamical Fatou set. Merely finding disjoint plane domains whose preimages are connected does not give such an example. Conversely, an upper bound must apply uniformly to the full class of entire functions.

Short English statement

Can a transcendental entire function have two completely invariant Fatou components?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700006 (AMR-036-0006), 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 2. Original source indexed by AMR-036-0006. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Two questions on completely invariant domains, Question 2. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/conjectures.pdf.