Cube structures of the universal minimal system, nilsystems and applications
Axel Álvarez, Sebastián Donoso
TL;DR
The paper develops a cube-structure framework for studying nilsystems and proximal extensions within topological dynamics, employing Host-Kra cube groups and the universal minimal system to obtain algebraic characterizations of dynamical cubes and regionally proximal relations. It provides a new algebraic description of $\boldsymbol{RP}^{[d]}$, proves that it is an equivalence relation, and demonstrates lifting and saturation properties for factor maps along cubes, along with a topological cubic characteristic-factor theory. A structure theorem is established showing that the dynamical cubes $\bm{Q}^{[d]}(X)$ share the same structural architecture as the base system, including PI/AI/I-type towers and distal factors, and the maximal distal/infinite order factors of cubes are identified as $\bm{Q}^{[d]}(X_{dis})$ and $\bm{Q}^{[d]}(X/\boldsymbol{RP}^{[\infty]}(X))$, respectively. Collectively, these results advance the algebraic understanding of nilsystems, distal factors, and cubic dynamics, with implications for ergodic theory, additive combinatorics, and the structural theory of minimal systems.
Abstract
We propose and develop an approach to studying 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.