Rigidity of Maximal Rank Abelian Actions

Definitions and mathematical statement

The supplied record does not determine a self-contained conjecture. It concerns abelian actions described as having maximal rank and Lyapunov exponents in general position, but does not specify the acting group, the manifold, the invariant measure, the regularity, or the meanings of these two hypotheses.

For terminology only, an action of an abelian group $G$ on a manifold $M$ is a homomorphism $\alpha:G\to\operatorname{Diff}(M)$, when a differentiable action is intended. Write $d=\dim M$. If a probability measure $\mu$ on $M$ is invariant under every $\alpha(g)$, then $h_\mu(\alpha(g))$ denotes the measure-theoretic entropy of the transformation $\alpha(g)$. When the derivative and integrability hypotheses needed for Lyapunov theory hold, Lyapunov exponents describe asymptotic exponential growth rates of tangent vectors under derivatives of the action. These conventions explain possible terminology; the source does not supply the hypotheses needed to adopt this framework as its intended setting. It also does not specify whether its entropies are measure-theoretic or topological.

The retained source formulation asks, for maximal-rank abelian actions with Lyapunov exponents in general position:

1. Which values can the entropies and Lyapunov exponents take?
2. Are these quantities logarithms of algebraic numbers, with algebraic degrees related to $d$? The algebraic numbers, the meaning of the degree relation, and the normalization of the quantities are unspecified.
3. Are there finitely many ergodic components? The invariant measure and whether this concerns the full action or individual transformations are unspecified.
4. Are the actions measurably conjugate to algebraic actions? For specified probability-preserving $G$-actions $\alpha$ and $\beta$, measurable conjugacy means an isomorphism of their probability spaces modulo null sets that intertwines $\alpha(g)$ and $\beta(g)$ almost everywhere for every $g\in G$. The source does not specify the algebraic models or their measures.

These questions are retained as an incomplete source formulation, not asserted as a universal conjecture. The supplied verification field says that the report immediately states a theorem resolving the questions in its particular setting, and marks the statement as unrecoverable, with no separate clean statement promoted. The supplied open label and literature assessment do not resolve this qualification. Recovering the theorem's exact hypotheses and conclusions is necessary before a precise independent problem can be stated.

Short English statement

The record asks about possible entropy and Lyapunov values, algebraicity, finitely many ergodic components, and measurable algebraic models for maximal-rank abelian actions. Essential hypotheses are missing. Although the supplied status is open, the supplied verification says that a theorem immediately resolves the questions in the report's setting. This is therefore an incomplete statement-only draft requiring mathematical review.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30001834$, exact source problem code catalogue entry, title: Rigidity of Maximal Rank Abelian Actions. Collection: Oberwolfach Reports Open Problems; category: Dynamical Systems. No individual proposer is supplied; the proposed year is 2011.
• Supplied report citation: Dynamische Systeme (2012), Oberwolfach Reports, DOI 10.4171/owr/2011/34, pp. 1913--1970. Source URL and supplied literature-source link, labelled "original Oberwolfach report": https://doi.org/10.4171/owr/2011/34. The background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature assessment, checked 2026-08-21: status open; the questions about algebraicity of logarithms, finiteness, and measurable conjugacy are described as extracted without sufficient system data and as having no general resolution in the cited report. This is supplied dated curation, not independently verified. Its time-sensitive status and cited claims require independent verification.
• Supplied statement verification: the official report introduces the rigidity questions and immediately states a theorem resolving them in the setting considered. The statement status is unrecoverable; no separate clean statement is promoted, and the original record is retained publicly for provenance. This qualification must be reconciled with the supplied open label and literature assessment.