Universal system of weak mixing transformations

Question

Suppose that $T: X \to X$ is an invertible measure preserving transformation on a probability space. The task is to find a probability space $Y$ and an invertible measure preserving map $S: Y \to Y$ such that $T$ is weakly mixing iff $T \times S$ is ergodic. My guess is that $Y$ is the circle and $S$ is some map with discrete spectrum since most ergodic transformations are conjugate to some rotation on a compact abelian group but I'm not really sure how to start. Also, it should be noted that we are looking for a fixed $S$ and $Y$ that works for every mixing $T$, not just one that depends on $T$.

Answer 1

Here is one such example: $Y= b\mathbb Z$, the Bohr compactification of $\mathbb Z$, $\nu = $ Haar probability measure on $Y$, and $S:Y\to Y$ is defined by $Sy = y+1$ (where we consider $\mathbb Z$ as a dense subgroup of $b\mathbb Z$). Now $S$ is, in a sense, the universal ergodic group rotation: it is an ergodic group rotation, and every ergodic group rotation is a factor of $S$. The important feature, for this example, is that every $\lambda\in \mathbb C$ with $|\lambda|=1$ appears as an eigenvalue of $S$.

The fact that $T\times S$ is ergodic whenever $T$ is weak mixing is a consequence of the definition (or rather, one of the several equivalent definitions) of weak mixing.

The fact that $T$ is weak mixing whenever $T\times S$ is ergodic follows from the fact that every $\lambda$ of unit modulus is an eigenvalue $S$, combined with a standard characterization of ergodicity of products: $T\times S$ is ergodic if and only if $T$ and $S$ are both ergodic and have no nontrivial eigenvalues in common. This characterization follows from Lemma 4.17 on page 91 of Furstenberg's book:

Furstenberg, H., Recurrence in ergodic theory and combinatorial number theory, M. B. Porter Lectures, Rice University, Department of Mathematics, 1978. Princeton, New Jersey: Princeton University Press. XI, 203 p. (1981). ZBL0459.28023.