Coboundaries of nonconventional ergodic averages

TL;DR

This paper establishes a sufficient and necessary condition for the product of bounded functions to be a coboundary in the context of nonconventional ergodic averages, advancing understanding of their structure in ergodic theory.

Contribution

It provides a precise criterion for when a product of bounded functions forms a coboundary, enhancing the theoretical framework of nonconventional ergodic averages.

Findings

Identifies a sufficient condition for bounded functions to form a coboundary.

Shows the condition is also necessary for bounded coboundaries.

Contributes to the characterization of nonconventional ergodic averages.

Abstract

Let $(X,\mathcal{A}, \mu)$ be a probability measure space and let $T_i,$ $1\leq i\leq H,$ be invertible bi measurable measure preserving transformations on this measure space. We give a sufficient condition for the product of $H$ bounded functions $f_1, f_2, ..., f_H$ to be a coboundary. This condition turns out to be also necessary when one seeks bounded coboundaries.