A counterexample on polynomial multiple convergence without commutativity

TL;DR

This paper constructs a counterexample showing that polynomial multiple convergence can fail without commutativity of transformations, even in systems with zero entropy, answering a question in ergodic theory.

Contribution

It provides the first known counterexample demonstrating failure of polynomial multiple convergence in non-commuting ergodic systems with zero entropy.

Findings

Counterexample exists for polynomials of degree at least 5

Convergence fails in L^2 space for certain ergodic systems

Addresses a question posed by Frantzikinakis and Host

Abstract

It is shown that for polynomials $p_1, p_2 \in {\mathbb Z}[t]$ with ${\rm deg}\ p_1, {\rm deg}\ p_2\ge 5$ 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.