Pointwise multiple averages for systems with two commuting transformations

Abstract

We show that if $(X,\mathcal{X},\mu,S,T)$ is an ergodic measure preserving system with commuting transformations $S$ and $T$, then the average \[\frac{1}{N^3} \sum_{i,j,k=0}^{N-1} f_0(S^j T^k x) f_1 (S^{i+j} T^k x) f_2 (S^j T^{i+k} x)\] converges for $\mu$-a.e. $x\in X$ as $N\to \infty$ for $f_0,f_1, f_2\in L^\infty(\mu)$. We also show that if $(X,\mathcal{X},\mu,S,T)$ is a measurable distal system, the average \[ \frac{1}{N}\sum_{i=0}^{N-1} f_1 (S^i x) f_2 (T^i x) \] converges for $\mu$-a.e. $x\in X$ as $N\to \infty$ for $f_1,f_2\in L^{\infty}(\mu)$.