Research Problems in Function Theory — Problem 2.67

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. For a Fatou component $D$, let $L_D\subset\widehat{\mathbb C}$ be the set of constants $a$ for which some subsequence $f^{\circ n_j}$, $n_j\to\infty$, converges locally uniformly in the spherical metric on $D$ to $a$. A bounded subset here means a subset of $\mathbb C$ contained in some finite Euclidean disk.

Statement recorded in the supplied source.

Do there exist an entire function $f$ and a Fatou component $D$ such that the complete set $L_D$ of constant limit functions is infinite and bounded? [S2]

Short English statement

Can the complete set of constant subsequential limits of entire-function iterates on one Fatou component be both infinite and bounded?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302067 (AMR-022-2067). 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 2.67. https://arxiv.org/abs/1809.07200.