Decomposition of multicorrelation sequences and joint ergodicity

TL;DR

This paper extends the decomposition of multicorrelation sequences into nilsequences and null sequences for $bZ^d$-actions with polynomial iterates, providing new criteria for joint ergodicity under ergodicity assumptions.

Contribution

It introduces a new seminorm bound for multiple averages and extends the decomposition result to multivariable polynomial iterates, advancing ergodic theory.

Findings

Decomposition of multicorrelation sequences into nilsequences and null sequences.

New seminorm bounds for multiple averages.

Criteria for joint ergodicity with polynomial iterates.

Abstract

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 null sequence, 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.