Group Actions on 1-Manifolds — Question 27

Definitions and mathematical statement

Let $S^1={\mathbb{R}}/{\mathbb{Z}}$, and let $f:S^1\to S^1$ be a $C^2$ circle diffeomorphism with irrational Poincaré rotation number $\rho\in{\mathbb{R}}/{\mathbb{Z}}$. Here the usual rotation-number convention implicitly requires $f$ to preserve orientation. For a lift $F:{\mathbb{R}}\to{\mathbb{R}}$ satisfying $F(x+1)=F(x)+1$, this number is \[ \rho=\left[\lim_{n\to\infty}\frac{F^n(x)-x}{n}\right]\in{\mathbb{R}}/{\mathbb{Z}}, \] independently of $x$ and of the choice of lift after passage to ${\mathbb{R}}/{\mathbb{Z}}$. Irrationality means that $\rho$ has no rational representative.

For each $\varepsilon>0$, define the mapping torus \[ M_{\varepsilon}=\left(S^1\times[0,\varepsilon]\right)/\!\sim, \qquad (x,0)\sim(f(x),\varepsilon). \] The supplied source then proposes a parameter $\rho(\varepsilon)\in{\mathbb{R}}/{\mathbb{Z}}$ through the asserted correspondence \[ M_{\varepsilon}\ \text{corresponds to}\ {\mathbb{C}}/\left({\mathbb{Z}}+i\rho(\varepsilon){\mathbb{Z}}\right), \] and asks whether, for every such $f$, \[ \lim_{\varepsilon\to0^+}\rho(\varepsilon)=\rho \quad\text{in }{\mathbb{R}}/{\mathbb{Z}}. \]

Unresolved formulation qualifications. The displayed elliptic-curve expression is retained exactly as supplied, but it does not define $\rho(\varepsilon)$ as written. Multiplication of an element of ${\mathbb{R}}/{\mathbb{Z}}$ by $i$ does not specify a complex lattice generator. Choosing a real representative $r$ would give ${\mathbb{Z}}+ir{\mathbb{Z}}$, which depends on that choice and is not a rank-two lattice when $r=0$. The mapping-torus construction specifies a topological surface, but the supplied record does not explain the conformal structure, the meaning of "corresponds," or the marking and normalization used to extract a parameter. In the usual presentation ${\mathbb{C}}/({\mathbb{Z}}+\tau{\mathbb{Z}})$, one takes a complex modulus $\tau$ with $\operatorname{Im}\tau>0$, with additional choices needed to select a representative. No replacement of the source's parameter or lattice formula is assumed here. Consequently, the proposed convergence question requires clarification of these data before it is a fully determined mathematical statement.

Short English statement

The source asks whether an elliptic-curve parameter associated with increasingly thin mapping tori of a $C^2$ circle diffeomorphism converges to its irrational rotation number. Its stated parameter and lattice formula do not give a well-defined construction, so the intended parameter, conformal structure, and normalization require clarification. This is a statement-only draft.

Sources

• Local UnsolvedMath registry, permanent ID $800310$, source problem code catalogue entry, title "Group Actions on 1-Manifolds --- Question 27." The supplied record attributes the problem to Navas (2017) and labels it open; that label is supplied registry metadata, not an independently verified assessment.
• Navas, Group Actions on 1-Manifolds: A List of Very Concrete Open Questions (2017), source item recorded as "Question 27 (#26)." Supplied source URL: https://arxiv.org/abs/1712.06462. The background records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not a claim of independent access or verification in this draft.