Higher-rank rigidity context and a finite-phase circle normal form

Definitions and mathematical statement

The source is a boundary-damaged context paragraph, not the statement of Conjecture 4. According to the supplied aid, its opening continues "Some partial results on Conjecture 4". It attributes such partial results to [93, 142], F. Rodriguez-Hertz [162], and B. Kalinin and R. Spatzier. It then proposes broader classification directions:

• "irreducible" partially hyperbolic actions of ${\mathbb{Z}}^k$ and ${\mathbb{R}}^k$, $k\ge2$;
• higher-rank actions of commuting expanding maps;
• analogous actions of higher-rank connected semisimple Lie groups and their lattices under unspecified hyperbolicity assumptions, referring to [85, 131].

An action of ${\mathbb{Z}}^k$ is given by $k$ commuting invertible maps, whereas commuting expanding coverings generally define semigroup actions and need not be invertible. The paragraph does not define irreducibility, the precise partially hyperbolic or higher-rank hypotheses, the ambient manifolds, or the equivalence relation for classification. It therefore does not determine a single fully quantified classification conjecture.

The attached research aid reports this precise circle model. Write $\mathbb T={\mathbb{R}}/{\mathbb{Z}}$. Let $f:\mathbb T\to\mathbb T$ be an orientation-preserving $C^1$ expanding covering of degree $m\ge2$, where expansion means that $|(f^j)'(x)|\ge C\lambda^j$ for all $x$ and $j\ge1$, for some $C>0$ and $\lambda>1$. If $g:\mathbb T\to\mathbb T$ is any continuous map of degree $n\in{\mathbb{Z}}$ with $fg=gf$, there are a homeomorphism $h:\mathbb T\to\mathbb T$ and $c\in\mathbb T$ such that \[ hfh^{-1}(x)=mx,\qquad hgh^{-1}(x)=nx+c, \qquad (m-1)c=0\quad\text{in }\mathbb T. \] Thus the residual translation phase lies in the finite group \[ \left\{\frac{j}{m-1}\pmod1:0\le j<m-1\right\}. \] The conclusion is a joint topological normal form; expansion of $g$ is not assumed, and smoothness of $h$ is not asserted.

The aid reports substantial progress by Spatzier and Yang for genuinely higher-rank smooth semigroup actions, without supplying their complete theorem or citation. It also cautions that general partially hyperbolic classification depends on hypotheses and that a blanket $C^1$ smooth-rigidity assertion is false. The circle model neither supplies those missing hypotheses nor classifies all the actions mentioned in the paragraph.

Short English statement

This fragment lists higher-rank classification directions rather than a complete conjecture. Its supplied concrete model jointly conjugates an expanding circle covering and any commuting continuous circle map to $x\mapsto mx$ and $x\mapsto nx+c$, with the exact finite constraint $(m-1)c=0$.

Sources

• American Institute of Mathematics, Emerging applications of measure rigidity, context after Conjecture 4 and the transition to section 2.3, https://aimath.org/WWN/measrigid/measrigid.pdf. Supplied attributions: [93, 142], F. Rodriguez-Hertz [162], B. Kalinin, R. Spatzier, and [85, 131]; the bibliography entries are not included in this record.
• The attached research summary attributes progress on genuinely higher-rank expanding semigroup actions to Spatzier and Yang but supplies no full bibliographic reference.
• Local UnsolvedMath registry, permanent ID $20003270$, source code catalogue entry, canonical location aim-other-notes.json, notes[76], dated 14 August 2026. Research attempt 1 is labelled context only, "not a problem" at run, proved rigor and low novelty confidence; it is a machine-generated, non-peer-reviewed aid requiring independent verification.