A counterexample on multiple convergence without commutativity

TL;DR

This paper constructs a specific example demonstrating that multiple ergodic averages can fail to converge in $L^2$ when the involved transformations are not assumed to commute, challenging previous assumptions in ergodic theory.

Contribution

It provides the first known counterexample showing non-convergence of multiple ergodic averages without the commutativity condition.

Findings

Existence of non-convergent multiple averages in non-commuting systems

Counterexample with zero entropy transformations

Failure of $L^2$ convergence in specific ergodic setups

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 $L^2(X,\mu)$.