A counterexample on multiple convergence without commutativity

Abstract

It is shown that there exist a probability space $(X,\mathcal{X},\mu )$, two ergodic measure preserving transformations $T,S$ acting on $(X,\mathcal{X},\mu )$ with $h_{\mu }(X,T)=h_{\mu }(X,S)=0$, and $f,g\in L^{\infty }(X,\mu )$ such that the limit \begin{equation*} \lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1} f(T^{n}x)g(S^{n}x) \end{equation*} does not exist in $L^2(X,\mu )$.

10 pages. arXiv admin note: substantial text overlap with arXiv:2301.12409