Non-convergence of some non-commuting double ergodic averages

Let S S and T T be measure-preserving transformations of a probability space ( X , B , μ ) (X,B,μ ) . Let f f be a bounded measurable function, and consider the integrals of the corresponding ‘double’ ergodic averages: \[ 1 n ∑ i = 0 n − 1 ∫ f ( S i x ) f ( T i x ) d μ ( x ) ( n ≥ 1 ) . 1/n∑ ᵢ₌₀ⁿ⁻¹ ∫ f(S^ix)f(T^ix)\,dμ (x) (n≥ 1). \] We construct examples for which these integrals do not converge as n → ∞ n→ ∞ . These include examples in which S S and T 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.