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Decomposition of multicorrelation sequences and joint ergodicity
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 44 / Issue 2 / February 2024
- Published online by Cambridge University Press:
- 04 May 2023, pp. 432-480
- Print publication:
- February 2024
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- Article
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We show that, under finitely many ergodicity assumptions, any multicorrelation sequence defined by invertible measure-preserving
$\mathbb {Z}^d$-actions with multivariable integer polynomial iterates is the sum of a nilsequence and a nullsequence, extending a recent result of the second author. To this end, we develop a new seminorm bound estimate for multiple averages by improving the results in a previous work of the first, third, and fourth authors. We also use this approach to obtain new criteria for joint ergodicity of multiple averages with multivariable polynomial iterates on 
${\mathbb Z}^{d}$-systems.
