Non-commuting transformations with non-converging 2-fold ergodic averages

2024/07/18 by Valery V. Ryzhikov, Ryzhikov, Valery V. · 1 citation
Mathematics · #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2407.13741

openalex publication_date 2024/07/18 · openalex created_date 2024/08/29 · openalex updated_date 2026/07/28

Abstract

In connection with the results of Tim Austin, and Wen Huang, Song Shao, Xiangdong Ye we present the following assertion: there are infinite automorphisms S,T, some set A of positive finite measure and a sequence N_m with |N_m+1|/|N_m| →∞ such that TⁱA∩ SⁱA=ϕ for i∈ [N_4k, N_4k+1] and TⁱA=SⁱA for i∈ [N_4k+2, N_2k+3]. For the corresponding deterministic Gaussian and Poisson suspensions S_∘,T_∘ over S,T for some f∈ L_∞ there is no limit of \frac 1 N ∑_n=1^N T_∘ⁿf S_∘ⁿf in L₂.

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