An example of a weakly mixing BV vector field which is not strongly mixing

Abstract

We give an example of a weakly mixing vector field $b\in L^{\infty }([0,1],\text{BV}({\mathbb{T}}^2))$ which is not strongly mixing. The example is based on a work of Chacon who constructed a weakly mixing automorphism which is not strongly mixing on $([0,1],\mathcal{B}([0,1]),|\cdot |)$, where $\mathcal{B}([0,1])$ are the Borel subsets of $[0,1]$ and $|\cdot |$ is the one-dimensional Lebesgue measure.