Non-convergence of some non-commuting double ergodic averages

Let S and T be measure-preserving transformations of a probability space (X, B, ). Let f be a bounded measurable functions, and consider the integrals of the corresponding `double' ergodic averages: \1n₈=₀^n-1 f (Sⁱx) f (Tⁱx) \ d (x) (n 1). \ We provide a new construction of diverse examples for which these integrals do not converge as n. These include examples in which S and T are rigid, and hence have entropy zero, answering a question of Frantzikinakis and Host. We begin with a corresponding construction of pairs of unitary operators on a complex Hilbert space, and then construct transformations of a Gaussian measure space from them.