Polynomial ergodic averages of measure-preserving systems acted by $\mathbb{Z}^{d}$

Abstract

In this paper, we reduce pointwise convergence of polynomial ergodic averages of general measure-preserving system acted by $\mathbb{Z}^{d}$ to the case of measure-preserving system acted by $\mathbb{Z}^{d}$ with zero entropy. As an application, we can build pointwise convergence of polynomial ergodic averages for $K$-system acted by $\mathbb{Z}^{d}$.