A pointwise cubic average for two commuting transformations

الأصل · EN

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,μ,S,T) with commuting transformations S and T, the average 1/N² ∑ᵢ,ⱼ₌₀ⁿ⁻¹ f₀(Sⁱ x)f₁(Tʲ x)f₂(Sⁱ Tʲ x) converges a.e. as N goes to infinity for any f₁,f₂,f₃∈ L∞(μ).