A counterexample on multiple convergence without commutativity

2024/07/15 by Huang, Wen, Shao, Song, Ye, Xiangdong · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2407.10728

Abstract

It is shown that there exist a probability space (X,\mathcal X,μ), two ergodic measure preserving transformations T,S acting on (X,\mathcal X,μ) with h_μ(X,T)=h_μ(X,S)=0, and f, g ∈ L^∞(X,μ) such that the limit lim_N→∞(1)/(N)∑_n=0^N-1 f(Tⁿx)g(Sⁿx) does not exist in L²(X,μ).

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• Counterexamples to double recurrence for non-commuting deterministic transformations

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