Properties of Mixing BV vector fields

We consider the density properties of divergence-free vector fields $b \in L^1([0,1],\textit{BV}([0,1]^2))$ which are ergodic/weakly mixing/strongly mixing: this means that their Regular Lagrangian Flow $X_t$ is an ergodic/weakly mixing/strongly mixing measure preserving map when evaluated at $t=1$. Our main result is that there exists a $G_\delta$-set $\mathcal U \subset L^1_{t,x}([0,1]^3)$ made of divergence-free vector fields such that $1)$ the map $\Phi$ associating $b$ with its RLF $X_t$ can be extended as a continuous function to the $G_\delta$-set $\mathcal{U}$; $2)$ ergodic vector fields $b$ are a residual $G_\delta$-set in $\mathcal{U}$; $3)$ weakly mixing vector fields $b$ are a residual $G_\delta$-set in $\mathcal{U}$; $4)$ strongly mixing vector fields $b$ are a first category set in $\mathcal{U}$; $5)$ exponentially (fast) mixing vector fields are a dense subset of $\mathcal{U}$. The proof of these results is based on the density of BV vector fields such that $X_{t=1}$ is a permutation of subsquares, and suitable perturbations of this flow to achieve the desired ergodic/mixing behavior. These approximation results have an interest of their own. A discussion on the extension of these results to $d \geq 3$ is also presented.