CF-Nil systems and convergence of two-dimensional ergodic averages

2025/10/20 by Kangbo Ouyang, Ouyang, Kangbo, Qinqi Wu +1
Computer Science · Mathematics · #Cellular Automata and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2510.17267

openalex publication_date 2025/10/20 · openalex created_date 2025/10/22 · openalex updated_date 2026/07/28

Abstract

A topological dynamical system (X,T) is called CF-Nil(k) if it is strictly ergodic and the maximal measurable and maximal topological k-step pro-nilfactors coincide as measure preserving systems. Through constructing specific ``CF-Nil'' models, we prove that for any ergodic system (X,X,μ,T), any nilsequence \ψ(m,n)\_m,n∈ℤ and any f₁,…,f_d∈ L^∞(μ), the averages \dfrac1N² ∑_m,n=0^N-1 ψ(m,n)∏_j=1^df_j(T^m+jnx) converge pointwise as N goes to infinity. Moreover, we show the L²-convergence of a certain two-dimensional averages for non-commuting transformations without zero entropy condition.

Citations

• Polynomial Furstenberg joinings and its applications
• Joint ergodicity for commuting transformations and applications to polynomial sequences
• Multiple recurrence and convergence without commutativity
• Maximal pronilfactors and a topological Wiener-Wintner theorem
• Pro-nilfactors of the space of arithmetic progressions in topological dynamical systems
• Topological characteristic factors and nilsystems
• Nilspace factors for general uniformity seminorms, cubic exchangeability and limits
• Higher order regionally proximal equivalence relations for general minimal group actions
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• Infinite-step nilsystems, independence and complexity
• Regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence
• Notes on Austin's multiple ergodic theorem
• Nilsequences and a structure theorem for topological dynamical systems
• Mean-L-stable systems

Related

• Maximal pronilfactors and a topological Wiener-Wintner theorem
• Pointwise convergence of some multiple ergodic averages
• Multidimensional local limit theorem in deterministic systems and an application to non-convergence of polynomial multiple averages
• On the pointwise convergence of multiple ergodic averages
• On the uniform convergence of ergodic averages for C^*-dynamical systems