A section-boundary fragment and a periodic-data obstruction

Definitions and mathematical statement

This record is connective prose, not an autonomous conjecture. The supplied source inspection says it ends at the colon immediately before separately numbered Conjecture 3 in the 2004 AIM report. It first comments on a preceding Conjecture 2, then surveys local rigidity, then introduces global Anosov rigidity. Neither full conjecture is included, so their missing statements are not reconstructed here.

The source reports that [97] proves the preceding Conjecture 2 when $A$ is the full split Cartan subgroup, and attributes to A. Katok the suggestion to use the method of [39]. It describes the higher-rank semisimple-group/lattice local-rigidity program of R. Zimmer [209], mentioning Anosov-action results [85,94,95,97], weaker-hyperbolicity results [131], and results without hyperbolicity [60]. Its next paragraph motivates a folklore conjecture using infranilmanifold Anosov examples and topological conjugacy to hyperbolic automorphisms [62,127], then cites [130] for the following conjecture. These are historical context and references with no bibliography supplied here. The phrase about the "only known examples" must not be promoted to an updated literal classification of smooth diffeomorphisms; the currency symbol in its "100,000" phrase is extraction-damaged.

The aid develops a separate background distinction between topological and $C^1$ conjugacy. Let $A\in GL(d,{\mathbb{Z}})$ be hyperbolic, meaning it has no eigenvalue of modulus one, and let $L_A:T^d\to T^d$ be the induced toral automorphism. For all sufficiently small nonzero real $\varepsilon$, choose a smooth diffeomorphism $\psi_\varepsilon$ supported in a coordinate ball at $0$, with \[ \psi_\varepsilon(0)=0,\qquad D\psi_\varepsilon|_0=(1+\varepsilon)I, \qquad\psi_\varepsilon\longrightarrow\mathrm{id}\text{ in }C^1. \] For example a compactly supported coordinate bump with value one near zero provides this family. Put $f_\varepsilon=\psi_\varepsilon\circ L_A$. The aid invokes openness and structural stability of Anosov diffeomorphisms: $f_\varepsilon$ is smooth Anosov, arbitrarily $C^1$-close to $L_A$, and topologically conjugate to it.

It nevertheless has no conjugacy to $L_A$ by a $C^1$ diffeomorphism. Any such $h$ would carry its fixed origin to a fixed point of $L_A$ and give the similarity relation \[ Dh|_0\,Df_\varepsilon|_0=A\,Dh|_0. \] But $Df_\varepsilon|_0=(1+\varepsilon)A$, so its determinant is $(1+\varepsilon)^d\det A\ne\det A$ for the chosen small nonzero $\varepsilon$. This violates similarity. The obstruction requires the conjugating map to be a $C^1$ diffeomorphism with invertible derivative, not merely a possibly singular differentiable homeomorphism.

This single-diffeomorphism perturbation is contextual mathematics. It neither supplies the missing folklore conjecture nor refutes the higher-rank action-rigidity assertions, which impose additional action relations and hypotheses.

Short English statement

The extracted entry joins historical rigidity discussion and stops before the actual next conjecture. Its attached aid illustrates why topological conjugacy is weaker than $C^1$ diffeomorphic conjugacy: an arbitrarily small Anosov perturbation of a toral automorphism changes a fixed-point derivative determinant, obstructing a $C^1$ conjugacy while structural stability preserves a topological one.

Sources

• Local UnsolvedMath registry, permanent ID $20003264$, source code catalogue entry; canonical location aim-other-notes.json, notes[70].
• AIM workshop report Emerging Applications of Measure Rigidity, 2004, https://aimath.org/WWN/measrigid/measrigid.pdf. The supplied source assessment identifies this as connective prose preceding Conjecture 3, despite the local conjecture/open tags and reconstructed/unverified label.
• Source labels [97] (full split-Cartan case), [39] (method suggested by A. Katok), [209] (R. Zimmer program), [85,94,95,97] (Anosov actions), [131] (weaker hyperbolicity), [60] (without hyperbolicity), [62,127] (infranilmanifold conjugacy), and [130] (following folklore conjecture). No full entries are supplied, so none is guessed.
• Research attempt 1 is marked context-only/not-a-problem, proved, and low novelty confidence, with an explicit machine-generated, non-peer-reviewed research-aid caveat requiring independent verification before citation. Structural stability and the derivative obstruction are its developed background, not the missing source question.