Research Problems in Function Theory — Problem 2.23

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.

Let $f:\mathbb C\to\mathbb C$ be entire and set $f^{\circ1}=f$, $f^{\circ(n+1)}=f\circ f^{\circ n}$. A point $z_0$ has exact period $n$ if $f^{\circ n}(z_0)=z_0$ and $f^{\circ k}(z_0)\ne z_0$ for $1\le k<n$. The source calls such a point a "fixed point of exact order $n$". A straight line means a set $\ell=\{a+tb:t\in\mathbb R\}$ with $b\ne0$.

Statement recorded in the supplied source.

For every transcendental entire function $f$ and every straight line $\ell$, must there be a periodic point of exact period one or two outside $\ell$? Equivalently, can all points of period at most two lie on one line? [S2] Update 2.23 gives a negative answer to this concentration question: for each $n\ge2$ and each line there are infinitely many fixed points of $f^{\circ n}$ off that line, with the later result giving repelling ones. [S2]

Short English statement

Can all fixed points and all points of exact period two of a transcendental entire function lie on a single straight line?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302023 (AMR-022-2023). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter 2, Problem 2.23 and Update 2.23. https://arxiv.org/abs/1809.07200.