Renormalization Basins of Quasiperiodic Schrödinger Cocycles

Definitions and mathematical statement

Let $\mathbb T={\mathbb{R}}/{\mathbb{Z}}$ and let $T_\alpha(x)=x+\alpha\pmod 1$. In the customary irrational-frequency interpretation of a one-frequency quasiperiodic cocycle, $\alpha\in{\mathbb{R}}\setminus{\mathbb{Q}}$. A matrix cocycle $(\alpha,A)$ acts by \[ (x,u)\longmapsto (T_\alpha x,A(x)u). \] For reference, the standard real Schrödinger convention uses a real-analytic potential $v:\mathbb T\to{\mathbb{R}}$, an energy $E\in{\mathbb{R}}$, and \[ A_{E,v}(x)=\begin{pmatrix}E-v(x)&-1\\1&0\end{pmatrix}\in\operatorname{SL}(2,{\mathbb{R}}). \] This convention explains the terminology; the supplied record does not explicitly specify the potential and energy domains, analytic extension requirements, or additional arithmetic restrictions on $\alpha$.

To make a basin question precise, one must specify a renormalization map $\mathcal R:D\to\mathcal X$, where $D\subseteq\mathcal X$, the ambient space $\mathcal X$, its topology, and an attractor $\mathcal K\subseteq\mathcal X$. If that topology is specified by a metric $d$ and $\mathcal K$ is nonempty, the corresponding basin can be defined by \[ \mathcal B_{\mathcal R}(\mathcal K) =\left\{z\in\mathcal X: \mathcal R^n z\text{ is defined for every }n\geq0, \quad\lim_{n\to\infty}\inf_{w\in\mathcal K}d(\mathcal R^n z,w)=0 \right\}. \] This is a conditional definition, not a specification of the renormalization intended by the source.

Source question. For a one-frequency analytic quasiperiodic Schrödinger cocycle $(\alpha,A)$ over $T_\alpha$, characterize the basin of the renormalization attractor: which admissible cocycles have renormalization iterates converging to that attractor in the intended topology?

Unresolved qualifications. The supplied record does not identify $\mathcal R$, its domain and normalization, the attractor $\mathcal K$, the ambient topology, or whether renormalization acts on cocycles themselves or on equivalence classes. It also does not specify whether frequencies are fixed or allowed to vary under renormalization, or which energies and potentials are admissible. Consequently, the displayed basin definition cannot be instantiated as a unique mathematical target from this record alone. These missing choices require source clarification; no characterization of the basin is asserted.

Short English statement

The question asks which analytic one-frequency quasiperiodic Schrödinger cocycles approach a specified attractor under repeated renormalization. The supplied record leaves the renormalization, attractor, topology, and admissible class unspecified, so the draft gives a conditional formulation and records the missing information. The supplied status is open; this status has not been independently verified.

Sources

• Local UnsolvedMath registry: supplied complete source record, permanent ID $30002023$, title Renormalization Basins of Quasiperiodic Schrödinger Cocycles, exact source problem code catalogue entry, collection Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2012.
• Supplied report citation: Arbeitsgemeinschaft: Quasiperiodic Schrödinger Operators (2013), Oberwolfach Reports, pp. 1035--1106, DOI 10.4171/owr/2012/17. https://doi.org/10.4171/owr/2012/17. The background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions. The citation's parenthetical year 2013 differs from the supplied proposed year and year tag, both 2012; the discrepancy is unresolved here.
• Supplied literature source, labeled Global theory I: https://arxiv.org/abs/0905.3902.
• Supplied literature source, labeled Avila ICM survey: https://www.imj-prg.fr/wp-content/uploads/2020/prix/avila2010.pdf.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that renormalization and rigidity theory provides major classification results for one-frequency cocycles, while the basin in the stated generality remains unresolved. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official report, with notation, context, and question form normalized without changing the source-level claim. This is a supplied verification assertion; no source was visited or independently checked for this draft.