Context around Anosov classification: transitivity and smooth-conjugacy obstructions

Definitions and mathematical statement

This record is contextual prose, not a standalone statement of Conjecture 3. The aid reconstructs its opening as "the conjugation map in Conjecture 3 is not necessarily smooth." The actual conjecture and preceding sentence are absent. Its concluding fragment, "2.2. One can also state an analog of", belongs to the next discussion and supplies no complete additional assertion.

The historical context reports Anosov diffeomorphisms on manifolds homeomorphic but not diffeomorphic to a torus [54], non-topologically-transitive Anosov flows [63], and a classification conjecture in the codimension-one stable or unstable case [195, 70]. Here Anosov means an invariant tangent splitting with uniformly contracting and expanding directions, together with the flow direction for a flow. These reports distinguish topological conjugacy, smooth conjugacy and orbit equivalence; none may be interchanged merely from hyperbolicity.

The aid's first independent audit is the following exact statement. Let $T:X\to X$ be a homeomorphism of a compact metric space and $r:X\to(0,\infty)$ continuous. Form the suspension \[ X^r=\{(x,s):0\leq s\leq r(x)\}/((x,r(x))\sim(Tx,0)). \] The suspension flow has a dense full two-sided orbit if and only if $T$ has a dense full ${\mathbb{Z}}$-orbit. Positivity and compactness bound the roof away from zero and make the full flow well defined. This uses two-sided transitivity, without silently substituting a different forward-orbit convention.

The second audit concerns differentiable conjugacy. If a $C^1$ diffeomorphism $h$ conjugates smooth maps $f,g$ and $x$ has period $k$, then \[ Dg^k_{h(x)}=Dh_x\,Df^k_x\,(Dh_x)^{-1}. \] Thus periodic derivative spectra and determinants must agree. For a concrete version of the aid example, take the cat map on ${\mathbb{R}}^2/{\mathbb{Z}}^2$ induced by $A=\begin{pmatrix}2&1\\1&1\end{pmatrix}$, whose fixed point is unique. A sufficiently small smooth perturbation $f_\varepsilon=\psi_\varepsilon\circ A$, supported in local coordinates near that fixed point, may fix it and have $D\psi_\varepsilon=\operatorname{diag}(1+\varepsilon,1)$ there. For small nonzero $\varepsilon$, the aid invokes Anosov structural stability to retain topological conjugacy, while \[ \det Df_\varepsilon=1+\varepsilon\ne1=\det A \] at the corresponding fixed points excludes a $C^1$ diffeomorphic conjugacy. This example is on the standard torus and is separate from the exotic-torus historical claim.

The aid additionally reports a May 2026 primary preprint claiming four-dimensional codimension-one Anosov flows not orbit equivalent to suspensions, described there as refuting the broad Verjovsky suspension conjecture. No authors, title, identifier or full hypotheses are supplied. This is a dated unverified preprint report, not independently confirmed literature status and not the missing statement of Conjecture 3. The paired audits and historical qualifications do not constitute a classification of general Anosov flows.

Short English statement

The source entry is a truncated context paragraph about Anosov classification. The supplied aid separates a suspension-transitivity criterion from a periodic-derivative obstruction to smooth conjugacy and records a newer preprint claim with its missing references, rather than inventing a standalone conjecture.

Sources

• Local UnsolvedMath registry, permanent ID $20003267$, source code catalogue entry; canonical aim-other-notes.json, notes[73].
• AIM workshop Emerging applications of measure rigidity: https://aimath.org/WWN/measrigid/measrigid.pdf. The excerpt cites [54] for exotic-torus Anosov maps, [63] for nontransitive flows and [195, 70] for codimension-one classification context. Full bibliographic entries are not supplied; original remarks and literature field are empty.
• The aid reports a May 2026 primary preprint on four-dimensional nonsuspension codimension-one Anosov flows without an authoritative complete citation or independently verified hypotheses.
• Research attempt 2: context_only, problem status at run not_a_problem, current registry status open, classification OPEN-TRIAGE, rigor proved, novelty confidence low. These are supplied labels. The machine-generated aid is not peer reviewed and requires independent verification before citation.