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 ${\mathbb{Z}}^d$.