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 $f_1,f_2\in L^{\infty }(\mu )$.

Some proofs clarified, following the referee's suggestions

References in corpus (3)

• Pointwise convergence of multiple ergodic averages and strictly ergodic models
• Dynamical cubes and a criteria for systems having products extensions
• A pointwise cubic average for two commuting transformations

Cited by in corpus (4)

• Norm-variation of ergodic averages with respect to two commuting transformations
• Pointwise convergence of some multiple ergodic averages
• Almost sure convergence of the multiple ergodic average for certain weakly mixing systems
• Uniformly rigid models for rigid actions