  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.
