We characterize inverse limits of nilsystems in topological dynamics, via a structure theorem for topological dynamical systems that is an analog of the structure theorem for measure preserving systems. We provide two applications of the structure. The first is to nilsequences, which have played an important role in recent developments in ergodic theory and additive combinatorics; we give a characterization that detects if a given sequence is a nilsequence by only testing properties locally, meaning on finite intervals. The second application is the construction of the maximal nilfactor of any order in a distal minimal topological dynamical system. We show that this factor can be defined via a certain generalization of the regionally proximal relation that is used to produce the maximal equicontinuous factor and corresponds to the case of order 1. © 2009 Elsevier Inc. All rights reserved.
Host, B., Kra, B., & Maass, A. (2010). Nilsequences and a structure theorem for topological dynamical systems. Advances in Mathematics, 224(1), 103–129. https://doi.org/10.1016/j.aim.2009.11.009