Non-convergence of some non-commuting double ergodic averages

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: $$\frac{1}{n}\sum _{i=0}^{n-1}\int f(S^{i}x)f(T^{i}x)d\mu (x)(n\ge 1)\mathrm{.}$$ We construct examples for which these integrals do not converge as $n\to \mathrm{\infty }$. 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.