Are there examples which are weak-mixing but not mixing. Let $T: X \to X$ (with measure $\mu$ and events $\mathcal{B}$). There is mixing
mixing means events become independent (eventually) $$ \lim_{n \to \infty} \mu(A \cap T^{-n}B) = \mu(A) \mu(B) $$
and also weak-mixing
weak-mixing $$ \lim_{n \to \infty} \frac{1}{n}\sum_{k=0}^{n-1}\left| \mu(A \cap T^{-n}B)- \mu(A)\mu(B)\right| = 0 $$
Do there exist dynamical systems that are weak-mixing but not mixing? Bonus points if you use geometry.
Actually, "most" transformations are weak mixing but not mixing.
More precisely, if you consider a probability measure $\mu$ and the set of all invertible $\mu$-invariant measurable transformations with the "weak" topology, the subset of weak mixing transformations is of the second category, while the set of mixing transformations is of the first category.
To give an explicit example is much more delicate:
A possibility is to consider interval exchange transformations associated to irreducible permutations $\sigma$, in the sense that $\sigma\{1,\ldots,k\}\ne\{1,\ldots,k\}$ for any $k$. Katok showed that no such transformation is mixing. The most general result on weak mixing seems to be a result by Ávila and Forni showing that if $\sigma$ is not a rotation, then almost every interval exchange transformation is weak mixing ("almost every" with respect to the lengths of the intervals using Lebesgue measure).
The first example of a transformation that is weak mixing but not mixing seems to be due to Kakutani using some combinatorial arguments, followed by Maruyama using Gaussian processes.