Global Rigidity of Diophantine Pseudo-Rotations

Definitions and mathematical statement

The supplied question is: "Are Diophantine pseudo-rotations always globally conjugate to a rotation?"

A global conjugacy between maps $f\colon M\to M$ and $R\colon M\to M$ is a bijection $h\colon M\to M$, with whatever regularity the intended category requires, satisfying \[ h\circ f=R\circ h \qquad\text{on all of }M. \] The record does not specify the phase space $M$, the definition or regularity of a pseudo-rotation, the rotation data to which the Diophantine condition applies, or the required regularity of $h$. Consequently, it does not determine a unique precise conjecture.

A conditional formalization can be given once these missing choices are supplied. Let $\mathcal D(M)$ denote the intended class of Diophantine pseudo-rotations on $M$, let $\mathcal R(M)$ denote the intended family of rotations, and let $\mathcal H(M)$ denote the permitted global changes of coordinates. The question then has the form \[ \forall f\in\mathcal D(M)\quad \exists R\in\mathcal R(M)\quad \exists h\in\mathcal H(M)\quad h\circ f=R\circ h. \] Here these three classes are placeholders for missing source specifications, not definitions recovered from the record. In particular, this formula does not select an underlying manifold, an arithmetic condition, a preservation hypothesis, or a conjugacy category. Whether the target rotation must have prescribed rotation data also remains unspecified.

The supplied dated curation labels the problem partially solved and reports global conjugacy in perturbative and other restricted regimes, while describing the general question as open. Those regimes and their hypotheses are not stated in the record, so this assessment supplies no additional precise theorem or restriction for the displayed question.

Short English statement

The question asks whether every Diophantine pseudo-rotation admits a change of coordinates on its entire phase space that turns it into a rotation. The phase space, meaning of pseudo-rotation, Diophantine condition, and regularity of the conjugacy are unspecified, so the displayed statement is a conditional template requiring human mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete source record for permanent ID $30002375$, source problem code catalogue entry, title "Global Rigidity of Diophantine Pseudo-Rotations." The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2013; no individual proposer is supplied.
• Supplied source citation: Dynamische Systeme (2014), Oberwolfach Reports, pp. 1975--2033, DOI 10.4171/owr/2013/34. https://doi.org/10.4171/owr/2013/34. The literature list labels this source "Oberwolfach Report 34/2013." The differing year designations are retained as supplied and require bibliographic clarification.
• Supplied literature source: "Herman's last geometric theorem." https://www.numdam.org/articles/10.24033/asens.2093/.
• Supplied literature source: "Questions around quasi-periodic dynamics." https://webusers.imj-prg.fr/~bassam.fayad/ICM_FayadKrikorian-cor.pdf.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved, reports conjugacy results in perturbative and other restricted regimes, and describes the general global-conjugacy question as open. This is supplied dated curation derived from the Oberwolfach material, not an independently verified literature assessment.
• The registry marks the statement as exact and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized. This verification claim is supplied provenance and has not been independently checked for this draft.