Weak mixing

Remarks

Example: every mixing map is weakly mixing, since the averages of a sequence that tends to $0$ also tend to $0$. So the doubling map $x \mapsto 2x \bmod 1$ with Lebesgue measure is weakly mixing.

Non-example: let $T(x) = x + \alpha \bmod 1$ with $\alpha$ irrational, and let $f(x) = e^{2\pi i x}$. Then $f \circ T = e^{2\pi i \alpha} f$, so $f$ is a nonconstant eigenfunction of the Koopman operator. By form 2 below, $T$ is not weakly mixing, although it is ergodic.

Equivalent forms, for $T$ as above:

1. The product map $T \times T$ is ergodic for the product measure $\mu \times \mu$.
2. Every $f \in L^2(\mu)$ with $f \circ T = \lambda f$ for some $\lambda \in \C$ is constant almost everywhere.
3. For all $A, B \in \mathcal{B}$ there is a set $J \subseteq \N$ of density zero such that $\mu(T^{-n} A \cap B) \to \mu(A)\mu(B)$ as $n \to \infty$ with $n \notin J$.

There are weakly mixing maps that are not mixing; Chacon's transformation is a classical example.