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

TL;DR

This paper reduces the problem of pointwise convergence of polynomial ergodic averages in measure-preserving systems acted by  to the zero entropy case, enabling convergence results for  systems with higher complexity.

Contribution

It establishes a reduction technique for polynomial ergodic averages, extending convergence results to  systems with positive entropy, especially  K-systems.

Findings

Reduction of convergence problem to zero entropy systems

Pointwise convergence established for  K-systems

Framework applicable to general  measure-preserving actions

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