Averages along cubes for not necessarily commuting measure preserving transformations

TL;DR

This paper proves the almost everywhere convergence of certain weighted averages along cubes in measure-preserving systems that are not necessarily commuting, extending classical results to more general non-commutative settings.

Contribution

It establishes pointwise convergence of cube averages in non-commuting measure-preserving systems, a significant generalization of previous commuting cases.

Findings

Almost everywhere convergence of cube averages in non-commuting systems

Extension of classical ergodic theorems to non-commutative settings

Convergence holds for bounded functions in finite measure spaces

Abstract

We study the pointwise convergence of some weighted averages linked to averages along cubes. We show that if $(X,\mathcal{B},\mu, T_i)$ are not necessarily commuting measure preserving systems on the same finite measure space and if $f_i,$ $1\leq i\leq 6$ are bounded functions then the averages $$\frac{1}{N^3}\sum_{n, m, p=1}^N f_1(T_1^nx) f_2(T_2^mx) f_3(T_3^px) f_4(T_4^{n+m}x) f_5(T_5^{n+p}x) f_6(T_6^{m+p}x)$$ converge almost everywhere.