Higher-rank conjugacy regularity: scope, obstruction, and a toral affine criterion

Definitions and mathematical statement

This source record is a contextual paragraph between Conjectures 3 and 4, not the statement of either conjecture. It mentions Anosov ${\mathbb{Z}}^k$- and ${\mathbb{R}}^k$-actions, $k\geq2$, says continuous conjugacy is often smooth in the intended situations [85,93,97], and introduces a following conjecture communicated by R. Spatzier. That conjecture and the necessary genuine higher-rank hypotheses are absent; no replacement conjecture is invented here.

The aid's negative scope test concerns parameter rank alone. It reports rank-one Anosov diffeomorphisms $F,G$ with a topological conjugacy but no $C^1$-diffeomorphic conjugacy, obstructed by periodic derivative data. For a nonzero homomorphism $\ell:{\mathbb{Z}}^k\to{\mathbb{Z}}$, the actions \[ \alpha(n)=F^{\ell(n)},\qquad\beta(n)=G^{\ell(n)} \] retain that topological conjugacy and obstruction. A $C^1$-diffeomorphic conjugacy would identify the corresponding periodic derivatives by linear conjugation, which the reported example violates. These inflated actions factor through rank one and generally have a kernel. They do not refute a theorem requiring effective, irreducible, or genuinely higher-rank actions. The explicit diffeomorphisms and periodic matrices are not supplied. No ${\mathbb{R}}^k$ counterexample is asserted from this discrete construction.

The positive supplied criterion is for linear toral actions, including different source and target dimensions. Let \[ A:{\mathbb{Z}}^k\to\operatorname{GL}(d,{\mathbb{Z}}),\qquad B:{\mathbb{Z}}^k\to\operatorname{GL}(e,{\mathbb{Z}}), \] and let $f:\mathbb T^d\to\mathbb T^e$, $\mathbb T^j={\mathbb{R}}^j/{\mathbb{Z}}^j$, be continuous with \[ f(A(n)x)=B(n)f(x)\qquad(n\in{\mathbb{Z}}^k). \] Define the target bounded joint subspace \[ V_b(B)=\{v\in{\mathbb{R}}^e:\sup_{n\in{\mathbb{Z}}^k}\|B(n)v\|<\infty\}. \] The aid states that the hypotheses \[ V_b(B)=0,\qquad\det(I-B(a))\ne0\text{ for some }a\in{\mathbb{Z}}^k \] force $f$ to be affine: \[ f(x)=Cx+b\pmod{{\mathbb{Z}}^e},\qquad C\in\operatorname{Mat}_{e\times d}({\mathbb{Z}}), \] with $CA(n)=B(n)C$ and $B(n)b=b$ in $\mathbb T^e$ for all $n$. In particular the map is smooth; when it is a homeomorphic conjugacy, the compatible equal-dimensional linear part is unimodular. The target condition is joint boundedness under all group parameters, not boundedness for one selected iterate.

If the target matrices are simultaneously diagonalizable over ${\mathbb{C}}$, write their joint eigencharacters as $\chi_j:{\mathbb{Z}}^k\to{\mathbb{C}}^\times$ and their Lyapunov characters as $L_j(n)=\log|\chi_j(n)|$. The aid gives the explicit test that every $L_j$ be nonzero. In that setting it rules out bounded joint vectors and permits choosing one $a$ off the finitely many zero-character hyperplanes, yielding the stated invertibility condition. This is a criterion for continuous equivariant maps of linear toral actions, not a smoothing theorem for arbitrary nonlinear Anosov actions. The broad contextual observation and both missing conjectures remain outside this precise special case.

Short English statement

The record is explanatory context with missing conjecture statements. Its supplied scope test shows that parameter rank alone does not imply smoothing, while a separate target-character criterion forces equivariant maps of linear toral actions to be affine. Rank-one inflation is kept distinct from genuine higher-rank rigidity.

Sources

• Local UnsolvedMath registry, permanent ID $20003268$, exact source code catalogue entry; canonical aim-other-notes.json, notes[74], AIM domain Other, registry category Miscellaneous, source tag conjecture.
• AIM workshop "Emerging applications of measure rigidity": https://aimath.org/WWN/measrigid/measrigid.pdf. The aid identifies the stored OCR fragment as the paragraph between Conjectures 3 and 4 rather than an autonomous conjecture. The registry's exact label is retained as source metadata. No PDF was fetched.
• Source-local labels [85], [93], and [97] support the contextual smoothing observation. R. Spatzier is credited only with the following missing conjecture. No full bibliographic entries, actual rank-inflation examples, or authoritative citation for the affine criterion are supplied.
• Research attempt 1: result context_only, problem status at run not_a_problem, rigor label proved, novelty confidence low; classification OPEN-TRIAGE, registry status open, suggested difficulty L2. Created and updated 14 August 2026. These are supplied assessments, not a resolution of an unreproduced conjecture.
• Attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. The machine-generated research aid is not peer reviewed and requires independent verification before citation.