Pointwise cubic average for two commuting transformations

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, A mu, S, T) with commuting transformations S and T, the average converges a.e. as N goes to infinity for any f (0), f (1), f (2) a L (a)(A mu). converges a.e. as N goes to infinity for any f(0), f(1), f(2) is an element of L-infinity(mu)