Non-convergence of some non-commuting double ergodic averages

TL;DR

This paper constructs examples showing that certain double ergodic averages for measure-preserving transformations can fail to converge, even when transformations are rigid and have zero entropy, answering a previously open question.

Contribution

It provides the first known examples of non-convergence of double ergodic averages for non-commuting transformations with zero entropy.

Findings

Examples of non-converging double ergodic averages

Transformations can be rigid with zero entropy

Answers an open question by Frantzikinakis and Host

Abstract

Let $S$ and $T$ be measure-preserving transformations of a probability space $(X,{\mathcal B},\mu)$. Let $f$ be a bounded measurable functions, and consider the integrals of the corresponding `double' ergodic averages: \[\frac{1}{n}\sum_{i=0}^{n-1} \int f(S^ix)f(T^ix)\ d\mu(x) \qquad (n\ge 1).\] We construct examples for which these integrals do not converge as $n\to\infty$. These include examples in which $S$ and $T$ are rigid, and hence have entropy zero, answering a question of Frantzikinakis and Host. Our proof begins with a corresponding construction for orthogonal operators on a Hilbert space, and then obtains transformations of a Gaussian measure space from them.