Bounded wandering domains of entire functions

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a nonlinear entire function $f:\mathbb C\to\mathbb C$, its Fatou set is the open set on which the iterates $\{f^n:n\ge0\}$ form a normal family for locally uniform convergence in the spherical metric. Let $D$ be a connected component of that set. A constant limit function is a value $c\in\widehat{\mathbb C}$ for which some subsequence $f^{n_k}$ tends locally uniformly on $D$ to the constant $c$, with $n_k\to\infty$. Suppose every subsequential limit function on $D$ is constant, and put \[ \mathcal L(D)=\{c\in\widehat{\mathbb C}: f^{n_k}|_D\longrightarrow c\text{ locally uniformly for some }n_k\to\infty\}. \] A component is wandering if the Fatou components containing $f^n(D)$ are pairwise distinct. Boundedness of $\mathcal L(D)$ means $\mathcal L(D)\subset\mathbb C$ and $\sup_{c\in\mathcal L(D)}|c|<\infty$, so infinity is excluded.

Statement recorded in the supplied source.

Can $\mathcal L(D)$ be both infinite and bounded? Equivalently in the author's formulation, can there be a nonempty open subdomain $U\subset D$ of a wandering component whose forward images remain in a bounded subset of $\mathbb C$?

The September 2026 paper of Drach, Pardo-Simón and Učakar states a negative answer in Theorem B: for every point $z$ of a wandering component of a transcendental meromorphic function, $\limsup_{n\to\infty}|f^n(z)|=\infty$. Its historical discussion explicitly identifies the infinite bounded constant-limit-set question. [S2] [S3]

Short English statement

Can an entire function have a Fatou component with infinitely many constant subsequential limits, all in a bounded part of the plane? The cited 2026 theorem reports that bounded wandering orbits do not occur.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3700004 (AMR-036-0004). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ben Bielefeld and Mikhail Lyubich (editors); Alexandre Eremenko and Mikhail Lyubich (problem authors), Problems in Holomorphic Dynamics (1992), Eremenko–Lyubich, Wandering Domains for Holomorphic Maps, Question 1, pp.39–40. https://arxiv.org/abs/math/9205209.
• S3 Kostiantyn Drach, Leticia Pardo-Simón and Beno Učakar, Bounded-orbit wandering domains do not exist (2026), Theorems A,B; Remark 1.1; §1.1, pp.1–3. https://arxiv.org/abs/2609.30103.