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A counterexample on multiple convergence without commutativity
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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 $L^2(X,\mu)$.
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Cited by 1 Pith paper
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Counterexamples to double recurrence for non-commuting deterministic transformations
Non-commuting, mixing, zero-entropy transformations can fail double recurrence along p1(n), p2(n) whenever the two polynomials are both linear or both have degree at least two.
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