Research Problems in Function Theory — Problem 2.21

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 $n$". An exact-period-$n$ point is repelling if $|(f^{\circ n})'(z_0)|>1$.

Statement recorded in the supplied source.

Does every transcendental entire function have a repelling periodic point, that is, do there exist $n\ge1$ and $z_0$ of exact period $n$ with $|(f^{\circ n})'(z_0)|>1$? [S2] Here "fixed points" has the source's all-periods meaning; the question does not assert the existence of a repelling point of period one. Update 2.21 records the affirmative result first proved by Baker. [S2]

Short English statement

Must a transcendental entire function have a periodic point that repels nearby points?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302021 (AMR-022-2021). 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 2.21 and Update 2.21. https://arxiv.org/abs/1809.07200.