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Seminorms for multiple averages along polynomials and applications to joint ergodicity

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Abstract

Exploiting the recent work of Tao and Ziegler on the concatenation theorem on factors, we find explicit characteristic factors for multiple averages along polynomials on systems with commuting transformations, and use them to study criteria of joint ergodicity for sequences of the form \((T_{1}^{p_{1,j}(n)}\cdots T_{d}^{p_{d,j}(n)})_{n\in\mathbb{Z}}\), 1 ≤ j ≤ k, where T1, …, Td are commuting measure preserving transformations on a probability measure space and pi, j are integer polynomials. To be more precise, we provide a sufficient condition for such sequences to be jointly ergodic, giving also a characterization for sequences of the form \((T_{i}^{p(n)})_{n\in\mathbb{Z}}\), 1 ≤ i ≤ d to be jointly ergodic, answering a question due to Bergelson.

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Authors and Affiliations

  1. Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, CNRS IRL 2807, Universidad de Chile, Beauchef, 851, Santiago, Chile

    Sebastián Donoso

  2. Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki, 54124, Greece

    Andreas Koutsogiannis

  3. Department of Mathematics, Virginia Tech, 902 Prices Fork RD, Blacksburg, VA, 24061, USA

    Wenbo Sun

Authors
  1. Sebastián Donoso
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  2. Andreas Koutsogiannis
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  3. Wenbo Sun
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Corresponding author

Correspondence to Sebastián Donoso.

Additional information

The first author is supported by ANID/Fondecyt/1200897 and Centro de Modelamiento Matemático (CMM), ACE210010 and FB210005, BASAL funds for centers of excellence from ANID-Chile.

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Donoso, S., Koutsogiannis, A. & Sun, W. Seminorms for multiple averages along polynomials and applications to joint ergodicity. JAMA 146, 1–64 (2022). https://doi.org/10.1007/s11854-021-0186-z

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  • DOI: https://doi.org/10.1007/s11854-021-0186-z

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