Analysis Seminar

Speaker: Terry Adams, US DoD
Authors: Terry Adams and Cesar Silva

Abstract: Suppose $T$ is an invertible finite measure preserving transformation. If $T\times T$ is ergodic, then $T^p \times T^q$ is ergodic for all nonzero integers $p$ and $q$. The transformation $T$ is weakly mixing, and $S\times T$ is ergodic for any ergodic finite measure preserving transformation $S$. The situation is different for infinite measure preserving transformations. See Kakutani/Parry (1963) for early examples demonstrating a difference. In the case of product powers $T^p\times T^q$, we show anything goes. There exists a class of rank-one infinite measure preserving transformations such that given any subset $R\subset \mathbb{Q} \cap (0,1)$, there exists $T$ in the class such that $T^p\times T^q$ is ergodic if and only if ${p} / {q} \in R$. Also, the same class is rich in conservative product powers. We make a connection with recent work of Johnson/Sahin on $\mathbb{Z}^d$-directional recurrence