Research Problems in Function Theory — Problem 2.87

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 an entire function $f$, write $f^{\circ0}(z)=z$ and $f^{\circ(n+1)}=f\circ f^{\circ n}$. Its Fatou set is the set on which the iterates form a normal family: every sequence has a subsequence converging locally uniformly in the spherical metric to a meromorphic function or to $\infty$. A Fatou component is a connected component of this set. A wandering domain $W$ has pairwise disjoint forward images $g^{\circ n}(W)$, $n\geq0$.

Statement recorded in the supplied source.

Can a nonlinear entire function $g$ have a wandering Fatou component $W$ for which $\bigcup_{n\geq0}g^{\circ n}(W)$ is bounded? In this setting, is it impossible that $g^{\circ n}\to p$ locally uniformly on $W$ for a finite fixed point $p$ of $g$? [S2]

Short English statement

Can an entire wandering domain have a uniformly bounded forward orbit, and can all its iterates converge to a finite fixed point?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302087 (AMR-022-2087). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, arXiv:1809.07200v2 (2018), Problem 2.87. https://arxiv.org/abs/1809.07200.