Weakly mixing measure-preserving transformation

A measure-preserving transformation is weakly mixing when its correlations converge to products in Cesaro mean. Equivalently, its Cartesian square is ergodic, or its product with every ergodic measure-preserving system is ergodic.

If $T$ is weakly mixing, then $T\times S$ is ergodic exactly when $S$ is ergodic. In particular, ergodicity of $T\times T$ is equivalent to weak mixing of $T$.

A weakly mixing system has no nonconstant Koopman operator eigenfunction. If $U_{T}f=\lambda f$, then $f(x)\overline{f(y)}$ is invariant under $T\times T$, contradicting the product characterization of weak mixing unless $f$ is constant.

A system is weakly mixing exactly when, for every positive-measure $A,B,C$, some $n\ge 1$ makes both $T^{-n}A\cap B$ and $T^{-n}A\cap C$ have positive measure. This is the finite simultaneous-hitting form of the density-one correlation characterization.