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Variations on topological recurrence

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Abstract

Recurrence properties of systems and associated sets of integers that suffice for recurrence are classical objects in topological dynamics. We describe relations between recurrence in different sorts of systems, study ways to formulate finite versions of recurrence, and describe connections to combinatorial problems. In particular, we show that sets of Bohr recurrence (meaning sets of recurrence for rotations) suffice for recurrence in nilsystems. Additionally, we prove an extension of this property for multiple recurrence in affine systems.

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Notes

  1. After this paper was submitted, Wenbo Sun adapted the methods we use to generalize this theorem, showing that a set of \(s\)-recurrence for \(s\)-step nilsystems is also a set of \(t\)-recurrence for all \(t\ge s\).

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Authors and Affiliations

  1. Université de Paris-Est Marne-la-Vallée, Laboratoire d’analyse et de mathématiques appliquées 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. Department of Mathematical Engineering and Center for Mathematical Modeling, UMI 2071 UCHILE-CNRS, University of Chile, Beauchef 851, 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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Corresponding author

Correspondence to Bryna Kra.

Additional information

Communicated by H. Bruin.

The second author was partially supported by NSF Grant DMS-\(1200971\) and the third author was partially supported by the Bézout Chair of the Université Paris-Est Marne-la-Vallée.

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Host, B., Kra, B. & Maass, A. Variations on topological recurrence. Monatsh Math 179, 57–89 (2016). https://doi.org/10.1007/s00605-015-0765-0

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  • DOI: https://doi.org/10.1007/s00605-015-0765-0

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