Group Actions on 1-Manifolds — Question 30

Definitions and mathematical statement

Let $S^1={\mathbb{R}}/{\mathbb{Z}}$, and write $R_\alpha(x)=x+\alpha\pmod{{\mathbb{Z}}}$. For an orientation-preserving homeomorphism $f:S^1\to S^1$, choose a lift $F:{\mathbb{R}}\to{\mathbb{R}}$ satisfying $F(x+1)=F(x)+1$. Its rotation number is \[ \rho(f)=\left[\lim_{n\to\infty}\frac{F^n(x)-x}{n}\right]\in{\mathbb{R}}/{\mathbb{Z}}, \] independent of $x$ and of the chosen lift modulo ${\mathbb{Z}}$. The rotation number is irrational if it has a representative $\alpha\in{\mathbb{R}}\setminus{\mathbb{Q}}$.

The usual irrational-rotation conclusion of Denjoy's theorem is that there exists a homeomorphism $h:S^1\to S^1$ such that \[ h\circ f\circ h^{-1}=R_\alpha, \qquad [\alpha]=\rho(f). \] Using this interpretation of the reference to Denjoy's theorem, the supplied question asks: does every orientation-preserving circle homeomorphism $f$ with irrational rotation number and real-analytic graph \[ \Gamma_f=\{(x,f(x)):x\in S^1\}\subset S^1\times S^1 \] satisfy this conjugacy conclusion?

The source does not explicitly state the orientation and irrational-rotation hypotheses; these are made explicit here to identify the usual Denjoy conclusion. It also does not define "real-analytic graph." In particular, it does not specify whether $\Gamma_f$ must be a real-analytic embedded one-dimensional submanifold of the standard real-analytic torus, or may be a real-analytic subset with singularities. Neither interpretation is silently selected, and real analyticity of the map $f$ is not substituted for the stated graph condition. The intended graph class therefore requires clarification before this can be regarded as a fully specified mathematical statement.

Short English statement

Navas asks whether a circle homeomorphism with a real-analytic graph satisfies Denjoy's conclusion: under the usual orientation-preserving and irrational-rotation hypotheses, it is topologically conjugate to the corresponding rigid rotation. The source leaves the precise meaning of a real-analytic graph unspecified.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800313$, source problem code catalogue entry, title Group Actions on 1-Manifolds --- Question 30. The record attributes the question to Navas (2017) and labels it open; this is a supplied status label, not an independently verified assessment.
• Navas, Group Actions on 1-Manifolds: A List of Very Concrete Open Questions (2017), https://arxiv.org/abs/1712.06462. The supplied provenance identifies the item as "Question 30 (#29)," records access on 2026-10-16, and gives the extraction method as arxiv_latex. These provenance details are supplied by the registry and were not independently verified here.