Characterization of trivial topological rational Kronecker factors for transitive systems

Determine whether, for a transitive dynamical system (X,T), the topological rational Kronecker factor X_rat is trivial if and only if T is totally transitive, meaning that every nonzero power T^n is transitive.

Background

For minimal systems, the paper recalls that the topological rational Kronecker factor is trivial exactly when the system is totally minimal. It asks whether the analogous equivalence holds for merely transitive systems, with total minimality replaced by total transitivity.

The paper proves that the implication from total transitivity to triviality of X_rat is valid. It also gives a counterexample to the converse: the one-point compactification of the integer shift has a transitive transformation with trivial topological rational Kronecker factor, although its square is not transitive. Thus, the question is explicitly raised but is resolved negatively within the paper; under the user’s requirement to include only unresolved questions, it is excluded from the extracted open-problem list.

References

For a minimal system $(X,T)$, the definition in implies that $X_{\operatorname{rat}$ is trivial if and only if $(X,T)$ is totally minimal, meaning that $(X,Tn)$ is minimal for every $n\inZ\setminus{0}$. It is therefore natural to ask whether the analogous statement holds for transitive systems: is $X_{\operatorname{rat}$ trivial if and only if $T$ is totally transitive, meaning that $(X,Tn)$ is transitive for every $n\inZ\setminus{0}$? The implication from triviality of $X_{\operatorname{rat}$ to total transitivity is false.

— A saturation theorem for distinct-degree polynomials and an application to joint transitivity  (2608.26570 - Álvarez, 27 Aug 2026) in Section 4, subsection 'The topological rational Kronecker factor'