A truncated transitivity status sentence and a Fourier mixing criterion

Definitions and mathematical statement

This extract is a sentence fragment referring to Conjecture 3, not a separately stated conjecture. It says that "it is not even known whether an Anosov automorphism is topologically transitive in general" and ends with the incomplete phrase "the conjugation map in". The supplied verification confirms the printed word "automorphism"; it must not be silently changed.

For the algebraic toral interpretation, let $d\ge1$, $A\in\mathrm{GL}(d,{\mathbb{Z}})$, $\mathbb T^d={\mathbb{R}}^d/{\mathbb{Z}}^d$, and $T_A(x)=Ax\pmod{{\mathbb{Z}}^d}$. Topological transitivity means that for every pair of nonempty open sets $U,V$, some $n\ge0$ satisfies $T_A^n(U)\cap V\ne\varnothing$. Topological mixing requires this for all sufficiently large $n$. Haar mixing means \[ \int_{\mathbb T^d} f(T_A^n x)g(x)\,{\,\mathrm{d}} m(x) \longrightarrow \left(\int f\,{\,\mathrm{d}} m\right)\left(\int g\,{\,\mathrm{d}} m\right) \] for all $f,g\in L^2(m)$, where $m$ is normalized Haar measure. The supplied aid reports that, for toral automorphisms, these three properties are equivalent to $A$ having no root-of-unity eigenvalue. In particular, a hyperbolic toral automorphism, meaning $|\lambda|\ne1$ for every eigenvalue, is mixing; its transitivity is not an open question.

The aid's finite-Fourier formulation is as follows. For finite $K,L\subset{\mathbb{Z}}^d$ and no root-of-unity eigenvalue, the set \[ \{n\ge0:(A^{\mathsf T})^n k=-\ell \text{ for some }k\in K,\ \ell\in L,\ (k,\ell)\ne(0,0)\} \] is finite. Thus, for trigonometric polynomials $p,q$ with supports in $K,L$, their above correlation equals the product of their means exactly for all sufficiently large $n$. The zero-frequency pair is excluded from the collision set because it supplies the constant term. The aid also reports that an affine hyperbolic toral map $x\mapsto Ax+b$ is translation-conjugate to $T_A$ and inherits mixing.

A different, unresolved reading concerns arbitrary Anosov diffeomorphisms of closed connected manifolds. A smooth diffeomorphism $f:M\to M$ is Anosov if there is a continuous $Df$-invariant splitting $TM=E^s\oplus E^u$ and constants $C>0$, $0<\lambda<1$ such that for every $n\ge0$, \[ \|Df^n v\|\le C\lambda^n\|v\|\quad(v\in E^s),\qquad \|Df^{-n}w\|\le C\lambda^n\|w\|\quad(w\in E^u). \] The plausible contextual question is whether every such $f$ is topologically transitive. The aid reports this general assertion as still open in the primary literature it checked, but the extract does not itself supply a complete statement with regularity, compactness, or connectedness hypotheses. These conventional hypotheses clarify this possible reading rather than repair the source by fiat.

Short English statement

The fragment's printed "Anosov automorphism" has two readings. Hyperbolic toral automorphisms are mixing by the reported Fourier criterion; transitivity of arbitrary Anosov diffeomorphisms is the distinct general question reported as open. The fragment is not a complete new conjecture.

Sources

• American Institute of Mathematics, Emerging applications of measure rigidity, fragment referring to Conjecture 3 and source bibliography labels [61, 14, 69, 90], https://aimath.org/WWN/measrigid/measrigid.pdf. The corresponding bibliography entries are not supplied.
• The attached research aid identifies partial-result versions arXiv:2410.15740v3, revised 9 December 2025, and arXiv:2410.12958v5, revised 6 May 2026; no titles or authors are supplied here.
• Local UnsolvedMath registry, permanent ID $20003266$, source code catalogue entry; canonical location aim-other-notes.json, notes[72], dated 14 August 2026. Research attempt 2 is labelled context only and "not a problem" at run, with proved rigor and low novelty confidence; it is machine-generated, not peer reviewed, and requires independent verification. The registry's open label is retained as metadata, not a newly verified status.