arXiv subjects
Publications and source records attributed to Axel Álvarez.
We prove, modulo almost one-to-one extensions, that the topological rational Kronecker factors of a minimal $\mathbb{Z}^k$-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.
We propose and develop an approach to study nilsystems and their proximal extensions using cube structures associated with the universal minimal system. We provide alternative proofs for results regarding saturation properties of factor maps to maximal nilfactors in cubes, as well as new results and applications of independent interest to the structural theory of topological systems. In particular, we give a new proof that $\mathbf{RP}^{[d]}$ is an equivalence relation, building upon the distal case, by establishing a description of this relation in algebraic terms. This is new even for d=1.
We study several aspects of higher-order regionally proximal relations for group actions. First, we develop an algebraic approach to study higher-order regionally proximal relations. To this end, we introduce a new topology on a subgroup of the universal minimal system, which can be seen as a higher-order analogue of the classical $τ$-topology. Using this topology, we obtain an algebraic characterization of the relation $\mathbf{RP}^{[d]}$ for abelian actions. Then, we study higher-order regionally proximal relations via recurrence sets, extending results of Huang, Shao, and Ye for $\mathbb{Z}$-actions to more general group actions under suitable assumptions. We then study topological characteristic factors and prove, modulo almost one-to-one factors, that the maximal factor of order $d-1$ is the topological characteristic factor of order d for cubic configurations for arbitrary group actions, and for arithmetic progressions for finitely generated abelian group actions. As a consequence, we show that $\mathbf{RP}^{[d]}$ and $\mathbf{AP}^{[d]}$ coincide on minimal points for finitely generated abelian group actions, and we apply this to obtain results on independence along arithmetic progressions.