Non-convergence of some non-commuting double ergodic averages

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: \[\frac{1}{n}\sum_{i=0}^{n-1} \int f(S^ix)f(T^ix)\ dμ(x) \qquad (n\ge 1).\] 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.

arXiv comment6 pages. [v2:] Main ideas re-arranged for clarity, and some remarks added based on correspondence about v1. [v3:] Minor corrections following referee report, will appear in Proc. Amer. Math. Soc