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Complexity of nilsystems and systems lacking nilfactors

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Abstract

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.

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Authors and Affiliations

  1. Laboratoire d’analyse et de mathématiques appliquées, Université Paris-Est Marne la Vallée & CNRS UMR 8050, 5 Bd. descartes, Champs sur Marne, 77454, Marne la Vallée Cedex 2, France

    Bernard Host

  2. Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL, 60208-2730, USA

    Bryna Kra

  3. Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, UMI 2071 UCHILE-CNRS Universidad de Chile, Casilla 170/3 correo 3, Santiago, Chile

    Alejandro Maass

Authors
  1. Bernard Host
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  2. Bryna Kra
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  3. Alejandro Maass
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Correspondence to Bernard Host.

Additional information

Partially supported by NSF grant 1200971

Partially supported by Fondap 15090007 and CMM-Basal grants.

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Host, B., Kra, B. & Maass, A. Complexity of nilsystems and systems lacking nilfactors. JAMA 124, 261–295 (2014). https://doi.org/10.1007/s11854-014-0032-7

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  • DOI: https://doi.org/10.1007/s11854-014-0032-7

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