Ergodic behaviour of nonconventional ergodic averages for commuting transformations

TL;DR

This paper extends Tao's work on multiple ergodic averages by establishing almost everywhere convergence along subsequences and analyzing ergodic behavior for diagonal measures, with special results on the torus.

Contribution

It provides new almost everywhere convergence results for nonconventional ergodic averages and classifies ergodic behavior based on transformation types, including special cases on the torus.

Findings

Existence of subsequences with almost everywhere convergence.

Ergodic behavior of diagonal measures varies with transformation classification.

Convergence results on the torus with special rotations.

Abstract

Based on T.Tao's result of norm convergence of multiple ergodic averages for commut-ing transformation, we obtain there is a subsequence which converges almost everywhere. Meanwhile, the ergodic behaviour, which the time average is equal to the space average, of diagonal measures is obtained and we give different result according to the classification of transformations. Additionally, on the torus with special rotation. we can not only get the convergence in T.Tao's paper for every point in Td, but also get a beautiful result for ergodic behaviour.