Properties of Mixing BV Vector Fields

Abstract

We consider the density properties of divergence-free vector fields $b \in L^1([0,1],{{\,\textrm{BV}\,}}([0,1]^2))$ 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 ${\mathcal {U}} \subset L^1_{t,x}([0,1]^3)$ containing all 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 $d \ge 3$ is also presented.

Appendix

Proof of Lemma 3.7

By the Ergodic Theorem, $T = X_{t=1}$ is ergodic iff

$$\begin{aligned} \frac{1}{n} \sum _{i=0}^{n-1} \chi _{T^i(A)} \rightarrow _{L^1} |A| \end{aligned}$$

In particular, if T is ergodic, then by writing

$$\begin{aligned} \frac{1}{n} \int _0^n \chi _{X_t(A)} dt = \int _0^1 \frac{n-1}{n} \bigg ( \frac{1}{n-1} \sum _{i=0}^{n-1} \chi _{T^i(X_s(A))} \bigg ) ds \end{aligned}$$

we see that

It is immediate to find a counterexample to the converse implication: just consider rotation of the unit circle with period 1.

The proof of the implication $\Rightarrow$ in the second point is analogous. For the converse, let $A,B \in \Sigma$ such that

$$\begin{aligned} \frac{1}{n} \sum _{i=0}^n \big [ |T^i(A) \cap B| - |A||B| \big ]^2 > \epsilon . \end{aligned}$$

By the continuity of $s \mapsto X_s$ in the neighborhood topology we have that there exists ${\bar{s}}$ such that for $0 \le s \le {\bar{s}}$ it holds

$$\begin{aligned} \big | X_s(B) \triangle B \big | = \big | B \triangle (X_s)^{-1}(B) \big | < \frac{\epsilon }{2}. \end{aligned}$$

Hence we can write

for $n \gg 1$. Hence

Finally, if T is strongly mixing, the continuity of $s \mapsto X_s$ in the neighborhood topology gives that $s \mapsto X_s^n = X_s \circ T^n$ is a family of equicontinuous functions, and since for all s fixed

$$\begin{aligned} \lim _{n \rightarrow \infty } |X_s(T^n(A)) \cap B| = |A| |B| \end{aligned}$$

we conclude that $X_s^n$ converges to 0 uniformly in s. The opposite implication is trivial.

$\square$