Research Problems in Function Theory — Problem 2.20

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 $n$".

Statement recorded in the supplied source.

For every transcendental entire $f$ and every integer $n\ge2$, must there exist a point of exact period $n$? [S2] Update 2.20 records Bergweiler's affirmative result, in the stronger form of infinitely many points of every exact period $n\ge2$. [S2]

Short English statement

Does every transcendental entire function have a periodic point of each exact period at least two?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302020 (AMR-022-2020). 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 2.20 and Update 2.20. https://arxiv.org/abs/1809.07200.