A pointwise cubic average for two commuting transformations

Abstract

Huang, Shao and Ye recently studied pointwise multiple averages by using suitable topological models. Using a notion of dynamical cubes introduced by the authors, the Huang-Shao-Ye technique and the Host machinery of magic systems, we prove that for a system $(X,\mu ,S,T)$ with commuting transformations $S$ and $T$, the average \[\frac{1}{N^2} \sum_{i,j=0}^{N-1} f_0(S^i x)f_1(T^j x)f_2(S^i T^j x)\] converges a.e. as $N$ goes to infinity for any $f_1,f_2,f_3\in L^{\infty }(\mu )$.

References in corpus (3)

• Pointwise convergence of multiple ergodic averages and strictly ergodic models
• Strictly ergodic models and the convergence of non-conventional pointwise ergodic averages
• Dynamical cubes and a criteria for systems having products extensions

Cited by in corpus (3)

• Pointwise convergence of some multiple ergodic averages
• Pointwise multiple averages for systems with two commuting transformations
• Uniformly rigid models for rigid actions