Pointwise convergence along cubes for measure preserving systems

الأصل · EN

Let (X, B, μ) be a probability measure space and T₁, T₂, T₃ three not necessarily commuting measure preserving transformations on (X, B, μ). We prove that for all bounded functions f₁, f₂, f₃ the averages 1/N²∑ₙ, ₘ ₌₁ⁿ f₁(T₁ⁿx)f₂(T₂ᵐx)f₃(T₃ⁿ⁺ᵐx) converges a.e. Generalizations to averages of 2ᵏ -1 functions are also given for not necessarily commuting weakly mixing systems.