Global Linearization of Rank-One Rotation Domains

Definitions and mathematical statement

A domain $D\subset{\mathbb{C}}^2$ is a nonempty connected open set. A biholomorphic conjugacy between a self-map $f|_\Omega$ and a linear map $L$ on an invariant domain $D$ means a biholomorphism $h:\Omega\to D$ such that \[ L(D)=D,\qquad h\circ(f|_\Omega)=L\circ h. \] In particular, such a conjugacy requires $f|_\Omega$ to be a biholomorphic self-map of $\Omega$.

Question, with unresolved source hypotheses. For every pair $(f,\Omega)$ for which $\Omega$ is a rank-one rotation domain in the sense intended by the source, do there exist a domain $D\subset{\mathbb{C}}^2$, integers $p,q\in{\mathbb{Z}}$ with $\gcd(p,q)=1$, a complex number $\alpha$, and a biholomorphism $h:\Omega\to D$ such that \[ |\alpha|=1,\qquad \alpha^n\ne1\quad\text{for every integer }n\ge1, \] \[ L(z_1,z_2)=(\alpha^p z_1,\alpha^q z_2),\qquad L(D)=D, \] and \[ h\left(f(z)\right)=L\left(h(z)\right)\qquad\text{for every }z\in\Omega? \] The source imposes no further explicit sign or nonvanishing conditions on $p,q$, and does not require $0\in D$.

Unresolved scope. The supplied record does not define "rank-one rotation domain," specify the ambient complex manifold or the class of maps $f$, or state whether the terminology includes additional dynamical hypotheses. Thus the quantifier over $(f,\Omega)$ above retains the source's intended but incompletely specified class. A fully self-contained formulation requires those hypotheses from the report; they cannot be recovered from this record alone.

Supplied qualification. The dated curation describes this as a question about a global $(p,q)$-linear model. It reports that the report proves linearization when a fixed point exists, while leaving the fixed-point-free case unsettled. This is a supplied assessment, not an independently verified theorem or current literature determination; the hypotheses of the reported fixed-point result are not reproduced in the record.

Short English statement

Does every rank-one rotation domain admit a global biholomorphic change of coordinates to an invariant domain in ${\mathbb{C}}^2$, on which the dynamics is $(z_1,z_2)\mapsto(\alpha^p z_1,\alpha^q z_2)$ for coprime integers $p,q$ and a unit complex number $\alpha$ of infinite order? The precise dynamical setting and definition of rank one are missing. The supplied curation labels the question open and reports linearization in the presence of a fixed point, with the fixed-point-free case unsettled.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30003923$, titled Global Linearization of Rank-One Rotation Domains, in the Oberwolfach Reports Open Problems collection. Exact source problem code: catalogue entry. No individual proposer is supplied; the proposed year is recorded as 2018.
• Supplied report citation: Geometric Methods of Complex Analysis (2019), Oberwolfach Reports, pp. 2253--2302, DOI 10.4171/owr/2018/37. Supplied source URL: https://doi.org/10.4171/owr/2018/37. The background describes Oberwolfach Reports as reports of mathematical workshops and their open-problem discussions.
• Supplied literature source: "Oberwolfach Report 37/2018," https://doi.org/10.4171/owr/2018/37. Literature checked at: 2026-08-21. This exact date records supplied dated curation, not independent source verification. That curation labels the problem open, reports linearization when a fixed point exists, and says that the fixed-point-free case is not settled by the report. Its own qualification requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, explaining that mathematical content was preserved while notation, typography, and sentence structure were repaired. This supplied editorial label does not establish independent mathematical verification of this draft.