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

Valery V. Ryzhikov

paper

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

arXiv:2407.13741

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 $∣ N_{m + 1} ∣/∣ N_{m} ∣ \to \infty$ such that $T^{i} A \cap S^{i} A = ϕ$ for $i \in [N_{4 k}, N_{4 k + 1}]$ and $T^{i} A = S^{i} A$ for $i \in [N_{4 k + 2}, N_{2 k + 3}]$ . For the corresponding deterministic Gaussian and Poisson suspensions $S_{\circ}, T_{\circ}$ over $S, T$ for some $f \in L_{\infty}$ there is no limit of $\frac{1}{N} \sum_{n = 1}^{N} T_{\circ}^{n} f S_{\circ}^{n} f$ in $L_{2}$ .

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