Research Problems in Function Theory — Problem 2.90

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.

The circle is $\mathbb R/\mathbb Z$. For an orientation-preserving circle homeomorphism $f$ with lift $F(x+1)=F(x)+1$, its rotation number is $\lim_{n\to\infty}(F^{\circ n}(x)-x)/n$ modulo $1$. An irrational $\alpha$ is non-Diophantine if no $C,\tau>0$ satisfy $|\alpha-p/q|\geq Cq^{-(2+\tau)}$ for every $p\in\mathbb Z$, $q\geq1$. An analytic conjugacy is a real-analytic circle diffeomorphism $h$ such that $h\circ f\circ h^{-1}(x)=x+\alpha\pmod1$.

Statement recorded in the supplied source.

Does there exist a non-Diophantine irrational $\alpha$ for which every orientation-preserving real-analytic circle diffeomorphism of rotation number $\alpha$ admits an analytic conjugacy to the rotation by $\alpha$? [S2]

Short English statement

Is there a non-Diophantine rotation number for which every analytic circle diffeomorphism with that number can be analytically linearized?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302090 (AMR-022-2090). 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 2.90. https://arxiv.org/abs/1809.07200.