Research Problems in Function Theory — Problem 2.86

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. Put $f_\lambda(z)=\lambda(e^z-1)$, $|\lambda|=1$. A Siegel domain $S$ containing $0$ is an invariant Fatou component on which $f_\lambda$ is conformally conjugate to an irrational rotation of a disk. The unique finite asymptotic value of $f_\lambda$ is $-\lambda$. The source update attributes affirmative answers to both parts to Rempe (the first using a method of Ghys).

Statement recorded in the supplied source.

(a) Is there a parameter $\lambda$ for which a Siegel domain $S$ containing $0$ is bounded? (b) Whenever such a domain $S$ is unbounded, must $-\lambda$ lie on $\partial S$? [S2]

Short English statement

Can this exponential family have a bounded Siegel domain, and must its unique asymptotic value lie on the boundary of every unbounded Siegel domain?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302086 (AMR-022-2086). 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 2.86 and Update 2.86. https://arxiv.org/abs/1809.07200.