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 function, 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.