Smooth Rigidity of Diophantine Step-Two Affine Maps

Definitions and mathematical statement

The intended question concerns an affine map $a$ described in the supplied record as Diophantine and step-two. The record does not specify the underlying space, the definition of step-two, the Diophantine condition, or the meaning and required smallness of a perturbation. These qualifications remain unresolved; the following expresses the conjugacy question conditionally on their specification.

Let $M$ denote the underlying smooth manifold, which must be specified, and write $\operatorname{Diff}^{r}(M)$ for the diffeomorphisms of $M$ whose maps and inverses are of class $C^{r}$. Interpret smooth as $C^{\infty}$. For every admissible Diophantine step-two affine diffeomorphism $a$ of $M$, and every $f\in\operatorname{Diff}^{\infty}(M)$ satisfying the intended perturbation hypothesis relative to $a$, is the following implication true? \[ \left(\exists h\in\operatorname{Diff}^{1}(M)\;:\;h\circ f=a\circ h\right) \quad\Longrightarrow\quad \left(\exists H\in\operatorname{Diff}^{\infty}(M)\;:\;H\circ f=a\circ H\right). \] The conclusion asks for the existence of a smooth conjugacy; it does not require the particular given conjugacy $h$ to be smooth. This is a conditional formulation of the supplied question, not a fully specified assertion until the missing geometric, arithmetic, and perturbation hypotheses are supplied.

Short English statement

Must every smooth perturbation of a Diophantine step-two affine map that admits a $C^{1}$ conjugacy also admit a smooth conjugacy? The underlying space, arithmetic condition, step-two condition, and perturbation hypothesis are unspecified, so the formulation requires human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004737$, exact source problem code catalogue entry, title "Smooth Rigidity of Diophantine Step-Two Affine Maps," in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2021$ and names no proposer.
• Supplied report citation: Dynamische Systeme (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/33, pp. 1735--1803. https://doi.org/10.4171/owr/2021/33. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions. The report contents have not been independently consulted here.
• Supplied literature reference, labeled "parabolic KAM rigidity": https://doi.org/10.1007/s00222-026-01415-7.
• Supplied literature reference, labeled "higher-rank local rigidity": https://arxiv.org/abs/2106.01585.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and states that the specific one-map $C^{1}$-to-smooth conjugacy question is not settled in full generality. It further attributes to 2026 work a locked--unlocked dichotomy and tame KAM rigidity for almost every translation in parabolic $\mathbb{Z}^{2}$ actions with a step-two generator, and states that higher-step cases remain open. This is supplied dated curation derived from the OWR collection, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification; the higher-rank results are not asserted here to settle the one-map question.