On the convergence of multiple ergodic means

2022/07/30 by G. A. Karagulyan, Michael T. Lacey, Karagulyan, Grigori A. +3
Mathematics · #37A30 #37A46 #42B25 #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Equations Stability Results #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2208.00215

openalex publication_date 2022/07/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given sequence of measure preserving transformations \U_k: k=1,2,…, n\ on a measurable space (X,μ). We prove a.e. convergence of the ergodic means \frac1s₁⋯ s_n∑_j1=0^s₁-1⋯∑_jn=0^s_n-1f(U₁^j₁⋯ U_n^j_n x ) as min_j s_j→∞ , for any function f∈ Llog^d-1(X), where d≤ n is the rank of the transformations. The result gives a generalization of a theorem by N. Dunford and A. Zygmund, claiming the convergence of the means in a narrower class of functions Llog^n-1(X).

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