Complexity of nilsystems and systems lacking nilfactors

Bernard Host*, Bryna Kra, Alejandro Maass

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


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. Our main result is that 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.

Original languageEnglish (US)
Pages (from-to)261-295
Number of pages35
JournalJournal d'Analyse Mathematique
Issue number1
StatePublished - Oct 2014

ASJC Scopus subject areas

  • Analysis
  • General Mathematics


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