NON-CONVERGENCE OF SOME NON-COMMUTING DOUBLE ERGODIC AVERAGES

Let S and T be measure-preserving transformations of a probability space (X, B, mu). Let f be a bounded measurable function, and consider the integrals of the corresponding 'double' ergodic averages: 1 n n & yuml; i"0 <acute accent>1 & zdot;f(Six)f(Tix)d mu (x) (n % 1). We construct examples for which these integrals do not converge as n-> oo. 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.