Ergodic Behaviour of Nonconventional Ergodic Averages for Commuting Transformations

Abstract

Based on T. Tao's celebrated result on the norm convergence of multiple ergodic averages for commuting transformations, we find that there is a subsequence which converges almost everywhere. Meanwhile, we obtain the ergodic behaviour of diagonal measures, which indicates the time average equals the space average. According to the classification of transformations, we also give several different results. Additionally, on the torus $\mathbb{T}^d$ with special rotation, we prove the pointwise convergence in $\mathbb{T}^d$ , and get a result for ergodic behaviour.