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Mathematics  2014 

A pointwise cubic average for two commuting transformations

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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 \[\frac{1}{N^2} \sum_{i,j=0}^{N-1} f_1(S^i x)f_2(T^j x)f_3(S^i T^j x)\] converges a.e. as $N$ goes to infinity for any $f_1,f_2,f_3\in L^{\infty}(\mu)$.

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