Research Problems in Function Theory — Problem 8.26

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 $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. Let $A$ be the boundary restrictions to $\mathbb T$ of the disc algebra: functions continuous on $\overline{\mathbb D}$ and analytic in $\mathbb D$. Let $\operatorname{Re}A=\{\operatorname{Re}f:f\in A\}\subset C(\mathbb T;\mathbb R)$, and for a circle homeomorphism $\psi$ put $(\operatorname{Re}A)\circ\psi=\{u\circ\psi:u\in\operatorname{Re}A\}$.

Statement recorded in the supplied source.

Characterize all homeomorphisms $\psi:\mathbb T\to\mathbb T$ satisfying \[\operatorname{Re}A=(\operatorname{Re}A)\circ\psi.\] The source reports absolute continuity as necessary and $C^{1+\varepsilon}$ regularity as sufficient. [S2]

Short English statement

Characterize the circle homeomorphisms that preserve the real parts of disc-algebra functions as a class.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2308026 (AMR-022-8026). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200. Problem 8.26, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.