Non-convergence of some non-commuting double ergodic averages
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: \[\frac{1}{n}\sum_{i=0}^{n-1} \int f(S^ix)f(T^ix)\ d\mu(x) \qquad (n\ge 1).\] We provide a new construction of diverse examples for which these integrals do not …