Pointwise multiple averages for systems with two commuting transformations

2015/09/30 by Sebastián Donoso, Wenbo Sun, Donoso, Sebastian +1
Mathematics · #37A05 #37A30 #37B05 #37B99 #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1509.09310

openalex publication_date 2015/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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=0^N-1 f₀(S^j T^k x) f₁ (S^i+j T^k x) f₂ (S^j T^i+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=0^N-1 f₁ (Sⁱ x) f₂ (Tⁱ x) converges for μ-a.e. x∈ X as N→ ∞ for f₁,f₂∈ L^∞(μ).

Citations

• Dynamical cubes and a criteria for systems having products extensions

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