Research Problems in Function Theory — Problem 2.22

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 $n$". Let $J(f)$ be the set of points with no neighbourhood on which the family $(f^{\circ n})_{n\ge1}$ is normal, where normality means that every sequence has a subsequence converging locally uniformly in the spherical metric. The source denotes this nonnormality set by $\mathcal F(f)$.

Statement recorded in the supplied source.

Is there an entire function $f$ for which $J(f)=\mathbb C$? In particular, is $J(e^z)=\mathbb C$, equivalently, are the periodic points of $z\mapsto e^z$ dense in $\mathbb C$? [S2] Update 2.22 records Baker's example and Misiurewicz's proof for $f(z)=e^z$, answering both questions affirmatively. [S2]

Short English statement

Can the iterates of an entire function fail to form a normal family everywhere, and does the exponential function already have this property?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302022 (AMR-022-2022). 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 2.22 and Update 2.22. https://arxiv.org/abs/1809.07200.