A counterexample on polynomial multiple convergence without commutativity

Abstract

It is shown that for polynomials $p_1,p_2\in \mathbb{Z}[t]$ with $\mathrm{deg}{p}_1,\mathrm{deg}{p}_2\ge 5$ 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^{p_1(n)}x)g(S^{p_2(n)}x) \end{equation*} does not exist in $L^2(X,\mu )$, which in some sense answers a question by Frantzikinakis and Host.

arXiv commentrevised version according to referee's suggestion