Research Problems in Function Theory — Problem 2.89

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 $\mathcal R_d$ be the space of rational maps of degree $d\geq2$, with its usual coefficient topology (pairs of degree-at-most-$d$ homogeneous polynomials, modulo a common nonzero scalar, with nonzero resultant). An invariant Herman ring is a doubly connected Fatou component on which the map is conformally conjugate to $w\mapsto e^{2\pi i\alpha}w$ on an annulus $1<|w|<R$. A real irrational $\alpha$ is Diophantine if some $C>0$ and $\tau>0$ satisfy $|\alpha-p/q|\geq Cq^{-(2+\tau)}$ for all integers $p$ and $q\geq1$. Fix $f_0\in\mathcal R_d$ and an invariant Herman ring $A_0$ of Diophantine rotation number $\alpha$. Let $\mathcal H_d(\alpha;f_0,A_0)$ consist of endpoints $f_1$ of continuous paths $f_t\in\mathcal R_d$, $0\leq t\leq1$, along which there are invariant Herman rings $A_t$ of rotation number $\alpha$, starting with $A_0$, that vary continuously in the Carathéodory kernel sense (compact subsets of a limiting domain eventually belong to the corresponding domains, with the component determined by a moving interior base point).

Statement recorded in the supplied source.

Is $\mathcal H_d(\alpha;f_0,A_0)$ locally closed in $\mathcal R_d$? Is its boundary relative to its closure in $\mathcal R_d$, namely $\partial_{\overline{\mathcal H_d(\alpha;f_0,A_0)}}\mathcal H_d(\alpha;f_0,A_0)$, a topological manifold? [S2]

Short English statement

For a fixed Diophantine Herman-ring rotation, is the path-connected parameter locus locally closed, and is its boundary within its closure a manifold?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302089 (AMR-022-2089). 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 2.89. https://arxiv.org/abs/1809.07200.