2012/03/16 by Bernard Host, Host, Bernard, Bryna Kra +3
Computer Science · Mathematics · #Cellular Automata and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals
paper · doi:10.48550/arxiv.1203.3778
openalex publication_date 2012/03/16 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
Nilsystems are a natural generalization of rotations and arise in various contexts, including in the study of multiple ergodic averages in ergodic theory, in the structural analysis of topological dynamical systems, and in asymptotics for patterns in certain subsets of the integers. We show, however, that many natural classes in both measure preserving systems and topological dynamical systems contain no higher order nilsystems as factors, meaning that the only nilsystems they contain as factors are rotations. In the ergodic setting, we show that there are spectral obstructions that give rise to this behavior. In the topological setting, nilsystems have a particular type of complexity of polynomial growth, where the polynomial (with explicit degree) is an asymptotic both from below and above. We also deduce several ergodic and topological applications of these results.