On Infinite Transformations with Maximal Control of Ergodic Two-fold Product Powers

We study the rich behavior of ergodicity and conservativity of Cartesian products of infinite measure preserving transformations. A class of transformations is constructed such that for any subset $R\subset \mathbb Q\cap (0,1)$ there exists $T$ in this class such that $T^p\times T^q$ is ergodic if and only if $\frac{p}{q} \in R$. This contrasts with the finite measure preserving case where $T^p\times T^q$ is ergodic for all nonzero $p$ and $q$ if and only if $T\times T$ is ergodic. We also show that our class is rich in the behavior of conservative products. For each positive integer $k$, a family of rank-one infinite measure preserving transformations is constructed which have ergodic index $k$, but infinite conservative index.