Bounded orbit of a wandering domain

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.

An entire function is a holomorphic map $f:\mathbb C\to\mathbb C$. Its iterates are $f^0=\mathrm{id}$ and $f^{n+1}=f\circ f^n$. Its Fatou set is the open set where the family of iterates is normal (in the spherical metric), and its Julia set $J(f)$ is the complement. The exponential and trigonometric parameter families here are \[E_\lambda(z)=\lambda e^z,\quad S_\lambda(z)=\lambda\sin z,\quad C_\lambda(z)=\lambda\cos z,\qquad \lambda\in\mathbb C\setminus\{0\}.\]A Fatou component $D$ is wandering when $f^m(D)\cap f^n(D)=\varnothing$ for all integers $m>n\geq0$. Its orbit is bounded when $\bigcup_{n\geq0}f^n(D)$ is a bounded subset of $\mathbb C$.

Statement recorded in the supplied source.

Does there exist an entire function $f$ with a wandering Fatou component $D$ whose orbit is bounded? [S2]

Short English statement

Can an entire function have a wandering Fatou component all of whose forward images stay in one bounded region?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300067 (AMR-052-0067). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ben Bielefeld, Mikhail Lyubich and contributors, Problems in Holomorphic Dynamics, arXiv:math/9205209v1 (1992), Alexandre Eremenko and Mikhail Lyubich, "Wandering Domains for Holomorphic Maps, Question 1. https://arxiv.org/abs/math/9205209v1.