Pointwise convergence along cubes for measure preserving systems

TL;DR

This paper proves pointwise convergence of certain multiple ergodic averages along cubes for measure-preserving systems, extending results to non-commuting and weakly mixing transformations.

Contribution

It establishes almost everywhere convergence of cubic averages for non-commuting measure-preserving transformations, including generalizations to higher dimensions and weakly mixing systems.

Findings

Almost everywhere convergence of cubic averages for non-commuting transformations

Extension to averages involving 2^k - 1 functions in weakly mixing systems

Generalization to higher-dimensional cube averages

Abstract

Let $(X, \mathcal{B}, \mu)$ be a probability measure space and $T_1$, $T_2$, $T_3$ three not necessarily commuting measure preserving transformations on $(X, \mathcal{B}, \mu)$. We prove that for all bounded functions $f_1$, $f_2$, $f_3$ the averages $$\frac{1}{N^2}\sum_{n, m =1}^N f_1(T_1^nx)f_2(T_2^mx)f_3(T_3^{n+m}x)$$ converges a.e. Generalizations to averages of $2^k -1$ functions are also given for not necessarily commuting weakly mixing systems.