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

TL;DR

This paper constructs examples of non-commuting transformations with non-converging 2-fold ergodic averages, demonstrating complex behavior in ergodic theory related to automorphisms and their suspensions.

Contribution

It provides explicit examples of automorphisms with non-converging ergodic averages, extending previous results and exploring non-commuting transformations in ergodic theory.

Findings

Existence of automorphisms with non-converging 2-fold averages

Construction of transformations with specific intersection properties

Demonstration of non-convergence in Gaussian and Poisson suspensions

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 $T^iA\cap S^iA=\phi$ for $i\in [N_{4k}, N_{4k+1}]$ and $T^iA=S^iA$ for $i\in [N_{4k+2}, N_{2k+3}]$. 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^nf\,S_\circ^nf$ in $ L_2$.