Research Problems in Function Theory — Problem 2.76

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. For a set $E\subset\mathbb C$, $\dim_H E$ denotes its Hausdorff dimension, defined by the critical exponent of the Hausdorff measures $\mathcal H^s(E)$. The source update describes counterexamples of Bishop: boundaries of dimension $1$ need not be circles or lines.

Statement recorded in the supplied source.

For every entire function $f$ and every Fatou component $\Omega$, is it true that \[\dim_H\partial\Omega>1\quad\text{or}\quad\partial\Omega\text{ is a Euclidean circle or a straight line}?\] [S2]

Short English statement

Must the boundary of an entire-function Fatou component have Hausdorff dimension greater than one unless it is a circle or a line?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302076 (AMR-022-2076). 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 2.76 and Update 2.76. https://arxiv.org/abs/1809.07200.