Non-convergence of some non-commuting double ergodic averages

Abstract

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 converge as $n\to\infty$. 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.

Locations

• arXiv (Cornell University) - View - PDF