Research Problems in Function Theory — Problem 2.77

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.

Statement recorded in the supplied source.

Do there exist an entire function $f$, a Fatou component $\Omega$, and a single constant $R<\infty$ such that \[|f^{\circ n}(z)|\leq R\quad(z\in\Omega,\ n\geq0),\qquad f^{\circ n}(\Omega)\cap f^{\circ m}(\Omega)=\varnothing\quad(n\ne m)?\] [S2]

Short English statement

Can a wandering Fatou component have all its forward images inside one fixed bounded disk?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302077 (AMR-022-2077). 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 2.77. https://arxiv.org/abs/1809.07200.