A saturation theorem for distinct-degree polynomials and an application to joint transitivity

Abstract: We prove, modulo almost one-to-one extensions, that the topological rational Kronecker factors of a minimal $\mathbb{Z}<sup>k$-system are the topological characteristic factors for commuting transformations along polynomials with distinct degrees. As an application, we obtain necessary and sufficient conditions for the joint transitivity of these polynomial iterates.