Foliations and Diffeomorphism Groups — Question 50

Definitions and mathematical statement

Let $S^1={\mathbb{R}}/{\mathbb{Z}}$ with its standard orientation, and let $G=\mathrm{Diff}_+^\infty(S^1)$ be the group, under composition, of orientation-preserving smooth diffeomorphisms of $S^1$ with smooth inverses. For $g,h\in G$, define their commutator by \[ [g,h]=g\circ h\circ g^{-1}\circ h^{-1}. \] The question is whether \[ \forall f\in G\quad \exists g,h\in G\quad f=[g,h]. \] Thus the requested factors must belong to the same smooth, orientation-preserving circle diffeomorphism group.

For the supplied background, write $R_\alpha(x)=x+\alpha$ on ${\mathbb{R}}/{\mathbb{Z}}$. The rotation number of $f\in G$ is \[ \rho(f)=\left(\lim_{n\to\infty}\frac{F^n(x)-x}{n}\right)\bmod{\mathbb{Z}}, \] where $F:{\mathbb{R}}\to{\mathbb{R}}$ is an increasing lift satisfying $F(x+1)=F(x)+1$; the resulting class is independent of the lift and of $x$. Here a rotation number is called Diophantine if a real representative $\alpha$ satisfies $\operatorname{dist}(q\alpha,{\mathbb{Z}})\geq Cq^{-\tau}$ for every integer $q\geq1$, for some constants $C>0$ and $\tau>0$.

The supplied source attributes to Michel Herman the theorem that every element of $G$ with Diophantine rotation number is smoothly conjugate to a rotation. It further states that continuity of rotation number yields a factorization of every element of $G$ as a rotation times an element smoothly conjugate to a rotation, and that every element is consequently a product of two commutators in $G$. These are supplied background assertions; the question asks whether one commutator always suffices. The source record labels this question open, and this status is not independently verified here.

Short English statement

Does every orientation-preserving smooth diffeomorphism of the circle have the form $g\circ h\circ g^{-1}\circ h^{-1}$, where $g$ and $h$ are also orientation-preserving smooth circle diffeomorphisms? The supplied source states that two commutators suffice and asks whether one always suffices.

Sources

• Local UnsolvedMath registry: permanent ID $800109$, exact source code catalogue entry, supplied complete source record. The record attributes the problem to Nariman and Yazdi, with proposed year 2025, and labels it open.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 50. Supplied source URL: https://arxiv.org/abs/2503.18274. The record reports access on 2026-10-16 and extraction from arXiv LaTeX; these are supplied provenance details, not a claim of independent access or verification.
• The supplied statement attributes the Diophantine rotation-number conjugacy theorem to Michel Herman. No separate bibliographic reference for that theorem is supplied.