Pointwise multiple averages for systems with two commuting transformations

الأصل · EN

We show that if (X,X,μ,S,T) is an ergodic measure preserving system with commuting transformations S and T, then the average 1/N³ ∑ᵢ,ⱼ,ₖ₌₀ⁿ⁻¹ f₀(Sʲ Tᵏ x) f₁ (Sⁱ⁺ʲ Tᵏ x) f₂ (Sʲ Tⁱ⁺ᵏ x) converges for μ-a.e. x∈ X as N→ ∞ for f₀,f₁, f₂∈ L∞(μ). We also show that if (X,X,μ,S,T) is a measurable distal system, the average 1/N∑ᵢ₌₀ⁿ⁻¹ f₁ (Sⁱ x) f₂ (Tⁱ x) converges for μ-a.e. x∈ X as N→ ∞ for f₁,f₂∈ L∞(μ).