# Properties of mixing BV vector fields

Stefano Bianchini, Martina Zizza

###### Abstract

We consider the density properties of divergence-free vector fields $b\in
L^{1}([0,1],\operatorname{BV}([0,1]^{2}))$ 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. 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. 2.

ergodic vector fields $b$ are a residual $G_{\delta}$-set in $\mathcal{U}$;

  3. 3.

weakly mixing vector fields $b$ are a residual $G_{\delta}$-set in
$\mathcal{U}$;

  4. 4.

strongly mixing vector fields $b$ are a first category set in $\mathcal{U}$;

  5. 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.

Preprint SISSA 19/2021/MATE

Key words: ergodicity, mixing, Baire Category Theorem, divergence-free vector
fields, Regular Lagrangian Flows, rates of mixing.

MSC2020: 26A21, 35Q35, 37A25.

###### Contents

  1. 1 Introduction
    1. 1.1 A discussion of the key points and results leading to the main theorem
      1. 1.1.1 Approximations of flows by permutations
      2. 1.1.2 From permutations of subsquares to ergodic/exponential mixing
    2. 1.2 Plan of the paper
  2. 2 Preliminaries and notations
    1. 2.1 BV functions
    2. 2.2 Regular Lagrangian Flows
  3. 3 Ergodic Theory
    1. 3.1 The neighbourhood topology as a convergence in measure.
    2. 3.2 Genericity of weakly mixing
    3. 3.3 Markov Shifts
  4. 4 Density of Strongly Mixing vector fields
    1. 4.1 Cyclic permutations of squares
    2. 4.2 Density of ergodic vector fields
    3. 4.3 Density of strongly mixing vector fields
    4. 4.4 Proof of the density of strongly mixing vector fields
  5. 5 Permutation Flow
    1. 5.1 Affine approximations of smooth flows
      1. 5.1.1 The d-dimensional case
    2. 5.2 BV estimates of perturbations
    3. 5.3 BV estimates for rotations
    4. 5.4 Main approximation theorem
  6. 6 Appendix

##  1 Introduction

Consider a divergence free vector field
$b:\mathbb{R}^{+}\times\mathbb{T}^{d}\to\mathbb{R}^{d}$ and the continuity
equation

$\partial_{t}\rho_{t}+D\cdot(\rho_{t}b_{t})=0,\quad\rho_{t=0}=\rho_{0}.$ (1.1)

In recent year the following question has been addressed: is the solution
$\rho_{t}$ approaching weakly to a constant as $t\to\infty$? The meaning of
”approaching to a constant” is usually formalized as

$\rho_{t}\underset{t\to\infty}{\to}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int\rho_{0}\mathcal{L}^{d}\quad\text{weakly in
$L^{2}$},$ (1.2)

since $\|\rho_{t}\|_{L^{p}}$ is constant (at least for positive solutions and
sufficiently regular vector fields) and this is referred to as _functional
mixing_ (another notion of mixing is the _geometric mixing_ introduced in [1],
but for our purposes the functional mixing above is the most suitable, since
it is related to Ergodic Theory.
Without any functional constraint on the space derivative $Db_{t}$, it is
quite easy to obtain mixing in finite time: a well known example is [16]. A
similar idea, used in a nolinear setting, can be found in [2]. The problem is
usually formulated as follows: assume that $b\in L^{\infty}_{t,x}\cap
L^{\infty}_{t}W^{s,p}_{x}$, what is the maximal speed of convergence in (1.2)?

This question has been addressed in several papers. In [3] the 2d-case has
been throughly analyzed, and the main results are the explicit construction of
mixing vector fields when the initial data is fixed: the authors are able to
achieve the optimal exponential mixing rate for the case $W^{1,p}$, $p>1$, and
study also the case $s<1$ (mixing in finite time) and $s>1$ (mixing at a
polynomial rate). Recall that for $s=1,p>1$ the mixing is at most exponential
[7], while the same estimate in $W^{1,1}$ (or equivalently
$\operatorname{BV}$) is still open [1]. In [9] the authors construct a vector
field which mixes at an exponential rate every initial data, and it belongs to
$L^{\infty}_{t}W^{s,p}_{x}$ for $s<\frac{1+\sqrt{5}}{2}$,
$p\in[1,\frac{2}{2s+1-\sqrt{5}})$. The autonomous 2d vector field is special,
having an Hamiltonian structure: indeed in [5] the authors show that the
mixing is polynomial with rate $t^{-1}$ when $b\in\operatorname{BV}$.

In this paper we consider the different problem: how many vector fields are
mixing? More precisely, we study the mixing properties of flows generated in
the unit square $K=[0,1]^{2}$ by divergence-free vector fields $b:[0,1]\times
K\rightarrow\mathbb{R}^{2}$ belonging to the space
$L^{\infty}([0,1],\operatorname{BV}(K))$: in order to avoid problems at the
boundary, we assume that the vector field $b$ is divergence-free and
$\operatorname{BV}$ when extended to whole $\mathbb{R}^{2}$. In order to
shorten the notation, we will sometimes write $\operatorname{BV}(K)$,
$K=[0,1]^{2}$ as the space
$\operatorname{BV}(\mathbb{R}^{2})\cap\\{\operatorname{supp}b\subset K\\}$.
All the results stated here can be extended to the case $x\in\mathbb{T}^{2}$
with minor modifications; our choice is in the spirit of [22].
In this setting, there exists a unique flow $t\rightarrow X_{t}\in
C([0,1],L^{1}(K;K))$ (called Regular Lagrangian Flow (RLF)) of the ODE

$\begin{cases}\frac{d}{dt}X_{t}(y)=b(t,X_{t}(y)),\\\ X_{t=0}(y)=y,\end{cases}$

which is measure-preserving and stable, see [13, 12] and Section 2.2. Our idea
is to consider the $\mathcal{L}^{2}$-a.e. invertible measure preserving map
$X_{t=1}:K\rightarrow K$ as an _automorphism_ of the measure space
$(K,\mathcal{B}(K),\mathcal{L}^{2}\llcorner_{K})$ and apply the tools of
Ergodic Theory. Here and in the following $\mathcal{L}^{2}\llcorner_{K}$ is
the Lebesgue measure on $K$ and $\mathcal{B}(K)$ are the Borel subsets of $K$.
We call $G(K)$ the group of automorphisms of $K$. We underline that the
additional difficulty is to retain that the maps under consideration are
generated by a divergence-free vector field in
$L^{\infty}_{t}\operatorname{BV}_{x}$.

There is a rich literature in Ergodic Theory that has deeply investigated the
genericity properties of mixing for invertible and measure-preserving maps.
These results are due mostly to Oxtoby and Ulam [17], Halmos [18, 19], Katok
and Stepin [11] and Alpern [21]. They proved that the set of ergodic
transformation is a residual $G_{\delta}$-set (i) in the set $G(K)$ with the
_neighbourhood topology_ 111The neighbourhood topology is indeed the
convergence in measure, see Subsection 3.1. [19], (ii) in the set of measure-
preserving homeomorphisms of a connected manifold with the strong topology222A
sequence of maps $T_{n}\rightarrow T$ in the strong topology if
$T_{n}\rightarrow T$ and $T_{n}^{-1}\rightarrow T^{-1}$ uniformly on $K$. [17,
11]. Moreover, the transformations satisfying a stronger condition known as
_weak mixing_ , that is

$\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\left[\mu(T^{-j}(A)\cap
B)-\mu(A)\mu(B)\right]^{2}=0,$

for every $A,B$ measurable sets, are still a residual $G_{\delta}$ set [19,
11]. In 1976 Alpern showed that these problems are indeed connected by using
the Annulus Theorem [21]. A different result holds for strongly mixing maps,
i.e such that

$\lim_{n\to\infty}\mu(T^{-n}(A)\cap B)=\mu(A)\mu(B).$

It was shown firstly by Rokhlin in [20] (see [24] for an exposition of
Rokhlin’s work) and then by D. Ornstein [10] that (strongly) mixing maps are a
first category set in the neighborhood topology.

In these settings, the genericity properties of measure-preserving (weakly)
mixing or ergodic maps are fairly understood; to our knowledge a similar
analysis has not been done for flows generated by vector fields with
additional regularity requirements (e.g. $b\in
L^{\infty}_{t}\operatorname{BV}_{x}$). The aim of our work is to extend the
above genericity results to divergence-free vector fields whose Regular
Lagrangian Flow is ergodic and weakly mixing (in dimension $d=2$, but see the
discussion below on the extension to every dimension $d\geq 3$).

Let $b\in L^{\infty}([0,1],\operatorname{BV}(K))$ be a divergence-free vector
field.

######  Definition 1.1.

We say that $b$ is _ergodic (weakly mixing, strongly mixing)_ if its unique
Regular Lagrangian Flow evaluated at time $t=1$ is ergodic (respectively
weakly mixing, strongly mixing).

In the original setting (1.1), if $b_{t+1}=b_{t}$ (i.e. it is time periodic of
period $1$), then it is fairly easy to see that if $b$ is strongly mixing as
in the above definition then (1.2) holds, while for weakly mixing vector
fields $b$ it holds the weaker limit

$\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\bigg{(}\int_{K}\bigg{(}\rho_{t}-\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int\rho_{0}\mathcal{L}^{2}\bigg{)}\phi\mathcal{L}^{2}\bigg{)}^{2}dt=0,\quad\forall\phi\in
L^{2}(K).$

Our main result is the following.

######  Theorem 1.2.

There exists a $G_{\delta}$-subset $\mathcal{U}\subset
L^{1}([0,1];L^{1}(K))\cap\\{b:D\cdot b=0\\}$ containing all divergence-free
vector fields in $L^{\infty}([0,1];\operatorname{BV}(K))$ with the following
properties:

  1. 1.

the map $\Phi$ associating $b$ with its RLF $X_{t}$,

$\Phi:\\{b\in L^{\infty}_{t}\operatorname{BV}_{x}:D\cdot b_{t}=0\\}\rightarrow
C([0,1],L^{1}(K)),$

can be extended as a continuous function to the $G_{\delta}$-set
$\mathcal{U}$;

  2. 2.

ergodic vector fields $b$ are a residual $G_{\delta}$-set in $\mathcal{U}$;

  3. 3.

weakly mixing vector fields $b$ are a residual $G_{\delta}$-set in
$\mathcal{U}$;

  4. 4.

strongly mixing vector fields $b$ are a first category set in $\mathcal{U}$;

  5. 5.

exponentially (fast) mixing vector fields are a dense subset of $\mathcal{U}$.

We will reasonably call the flow $X_{t}=\Phi(b)$, $b\in\mathcal{U}$, as the
Regular Lagrangian Flow of $b$, even if we are outside the setting where RLF
are known to be unique: however $X_{t}$ is the unique flow which can be
approximated by RLF of smoother vector fields $b^{n}$ as $b^{n}\to b$ in
$L^{1}$. The existence of such a set $\mathcal{U}$ is due to purely
topological properties of metric spaces (Proposition 2.3).

Our proof adapts some ideas from [19] to our setting: we give an outline of
Halmos’ analysis. First of all, both ergodic automorphisms and weakly mixing
automorphisms are a $G_{\delta}$-set [18, 19]. Next, it is shown that the
mixing properties are invariant for conjugation, i.e. if $T:[0,1]\to[0,1]$ is
weakly/strongly mixing and $R:[0,1]\to[0,1]$ is an automorphism, then $R\circ
T\circ R^{-1}$ is weakly/strongly mixing too. It remains to be proved that
weakly mixing maps are dense: define a _permutation_ as an automorphism of
$[0,1]$ sending dyadic intervals (subintervals of $[0,1]$ with dyadic
endpoints) into dyadic intervals by translation. Cyclic permutations (i.e.
permutations made by a unique cycle) of the same intervals are clearly
conjugate. One of the key ingredients of Halmos’ proof is that, for every non
periodic automorphism (i.e. $T^{n}x\not=x$ for all $n$ in a conegligible set
of points $x$), there exists a cyclic permutation close to it in the
neighbourhood topology, and by the previous observation about conjugation of
permutations one deduces that if $T$ is non-periodic then the maps of the form
$R\circ T\circ R^{-1}$ form a dense set. In particular, the weakly mixing maps
are a $G_{\delta}$-set containing a non periodic map, hence it is residual.

In our setting, the fact that ergodic/weakly mixing vector fields form a
$G_{\delta}$-subset of $\mathcal{U}$ is an easy consequence of the Stability
Theorem for Regular Lagrangian Flows and the definition of the map $\Phi$ (see
Point (1) and Proposition 3.9). Indeed, since both ergodic automorphisms and
weakly mixing automorphisms are a $G_{\delta}$-set [18, 19], then by the
continuity of the map $\Phi$ associating $b$ with the RLF $X_{t}$, also
ergodic and weakly mixing vector fields are a $G_{\delta}$-set.
Unfortunately, we cannot use conjugation of a RLF $X_{t}$ with an automorphism
$R$ of $K$, since in general $R\circ X_{t=1}\circ R^{-1}$ is not a RLF
generated by $b\in L^{\infty}_{t}BV_{x}$ (or even $b\in\mathcal{U}$). However,
we are able to prove the density in $\mathcal{U}$ of vector fields $b\in
L^{\infty}_{t}\operatorname{BV}_{x}$ whose RLF is a cyclic permutation of
subsquares of $K$, which is the natural extension of the permutation of
intervals used in [19]. More precisely, the map $T=X_{t=1}$ sends by a rigid
translation subsquares of some rational grid
$\mathbb{N}\times\mathbb{N}\frac{1}{D}$, where $D\in\mathbb{N}$, into
subsquares of the same grid (it will be clear later that being dyadic as in
[19] is not relevant, see Lemma 5.1 and Remark 5.3), and as a permutation of
subsquares it is made by a single cycle.
The precise statement is the following.

######  Theorem 1.3.

Let $b\in L^{\infty}([0,1],\operatorname{BV}(K))$ be a divergence-free vector
field. Then for every $\epsilon>0$ there exist $1\ll D\in\mathbb{N}$, two
positive constants $C_{1},C_{2}$ and a divergence-free vector field $b^{c}\in
L^{\infty}([0,1],\operatorname{BV}(K))$ such that

$||b-b^{c}||_{L^{1}(L^{1})}\leq\epsilon,\quad||\text{\rm
Tot.Var.}(b^{c})(K)||_{\infty}\leq C_{1}||\text{\rm
Tot.Var.}(b)(K)||_{\infty}+C_{2}$ (1.3)

and the map $X^{c}_{t=1}:K\rightarrow K$, where $X^{c}_{t}:[0,1]\times
K\rightarrow K$ is the flow associated with $b^{c}$, is a ${D}^{2}$-cycle of
subsquares of size $\frac{1}{{D}}$.

The above approximation is the most technical part of the paper, and it is the
point which forces to state the theorem in $\mathcal{U}$ and not in the
original space $b\in L^{\infty}_{t}\operatorname{BV}_{x}$: indeed, while
achieving the density in the $L^{1}_{t,x}$-topology, the total variation
increases because of the constants $C_{1},C_{2}$ in (1.3). (It is possible to
improve the first estimate of (1.3) to
$\|b-b^{c}\|_{L^{\infty}L^{1}}\leq\epsilon$, see Remark 4.7, but to avoid
additional technicalities we concentrate on the simplest results leading to
Theorem 1.2.)

We remark that the above approximation result is sufficient to prove that
strongly mixing vector fields are a set of first category (Proposition 3.9):
indeed, Theorem 1.3 shows the density in $\mathcal{U}$ of divergence free
vector fields whose flow is made of periodic trajectories with the same period
$D^{2}$. This observation is the key to obtain Point (4) of Theorem 1.2.

Looking at cyclic permutations of subsquares is an important step to obtain
ergodic (and then weakly mixing and strongly mixing) vector fields: indeed,
instead of studying the map $X^{c}_{t=1}$ (the RLF generated by $b^{c}$ of
Theorem 1.3 above) in the unit square $K$ with the Lebesgue measure
$\mathcal{L}^{2}$, it is sufficient to work in the finite space made of the
centers of the subsquares

$\Omega=\bigg{\\{}x=\bigg{(}\frac{k_{1}-1/2}{D},\frac{k_{2}-1/2}{D}\bigg{)},k_{1},k_{2}=1,\dots,D\bigg{\\}},$
(1.4)

where the measure-preserving transformation $X^{c}_{t=1}$ reduces to a cyclic
permutations. In particular in $\Omega$ it is already ergodic.

Since we cannot use the conjugation argument as we observed above, the final
steps of the proof of Theorem 1.2 differ from the ones of [19]. Indeed we give
a general procedure to perturb vector fields $b^{c}\in
L^{\infty}_{t}\operatorname{BV}_{x}$ (whose RLF $X_{t}^{c}$ at $t=1$ is a
cyclic permutation of subsquares) into ergodic vector fields $b^{e}$ (strongly
mixing vector fields $b^{s}$) still belonging to
$L^{\infty}_{t}\operatorname{BV}_{x}$: here the explicit form of $X^{c}$ plays
a major role, allowing us to construct _explicitly_ the perturbations to
$b^{c}$ (Subsections 4.2,4.3).
The key idea is to apply the (rescaled) _universal mixer_ vector field
(introduced in [9]) whose RLF at time $t=1$ is the _Folded Baker’s map_

$U=U\llcorner_{t=1}=\begin{cases}\left(-2x+1,-\frac{y}{2}+\frac{1}{2}\right)&x\in\left[0,\frac{1}{2}\right),\\\
\left(2x-1,\frac{y}{2}+\frac{1}{2}\right)&x\in\left(\frac{1}{2},1\right],\end{cases}\qquad
y\in[0,1],$ (1.5)

to the subsquares of the grid $\mathbb{N}\times\mathbb{N}\frac{1}{D}$ given by
Theorem 1.3.
In order to achieve ergodicity, it is sufficient to apply the universal mixer
$U$ inside a single subsquare, because the action of $X^{c}_{t=1}$ is already
ergodic in $\Omega$ being a cyclic permutation. Together with the fact that
ergodic vector fields are a $G_{\delta}$-set, this gives the proof of Point
(2) of Theorem 1.2.
The perturbation to achieve exponential mixing is more complicated, since we
need to transfer mass across different subsquares. The idea is to apply the
universal mixer $U$ to adjacent couples of subsquares, a procedure which
assures that the mass of $\rho$ is eventually equidistributed among all
subsquares. The exponential mixing is a consequence of the finiteness of
$\Omega$ and the properties of $U$ (see Proposition 4.10). This concludes the
proof of Point (5) of Theorem 1.2, and since strongly mixing vector fields are
a subset of weakly mixing vector fields we obtain also Point (3), concluding
the proof of the theorem.

A completely analogous result can be obtained in any dimension by adapting the
above steps, at the cost of additional heavy technicalities. In this work we
decided to sketch the proof of the key estimates (i.e. the ones requiring new
ideas) in the general case (see Section 5.1.1 and the comments in next
section).

###  1.1 A discussion of the key points and results leading to the main
theorem

The sketch of the proof given above includes steps which are somehow standard
either in the theory of linear transport or in ergodic analysis. Other points
of the proof instead require to introduce new ideas or at least to
significantly develop tools present in the literature. This section is devoted
to expand these critical parts in order to help the reader to understand the
novelties contained in this work.
In most proofs we tried to get the best possible result in term of the $L^{1}$
and $\operatorname{BV}$ norms, with the hope to prove a similar statement
inside the compact subset $\\{b:\|b_{t}\|_{L^{1}},\text{\rm
Tot.Var.}(b_{t})\leq C\\}\subset L^{1}_{t,x}$. However there are some delicate
arguments where we have to increase the total variation of $b$ of a fixed
amount: we will point out these points here below.

The most technical part of this paper is the proof of the approximation
theorem through cyclic permutations, Theorem 1.3. It is based on two results:
the first one, whose proof is the content of Section 5, is an approximation
through vector fields whose RLF $X_{t}$ at time $t=1$ is a permutation of
subsquares. The second one exploits the classical result that every
permutation is a product of disjoint cycles (Proposition 4.5), in order to
merge these cycles into a single one.

####  1.1.1 Approximations of flows by permutations

The approximation through divergence free vector fields $b$ whose flow at
$t=1$ is a permutation of squares has been already studied in [22] in the
context of generalized flows for incompressible fluids. Indeed the starting
point of Section 5 is Lemma 5.1, whose statement is almost identical to Lemma
4.3 of [22]: it says that if $T$ is a smooth map sufficiently close to
identity, there exists an arbitrarily close flow $\sigma_{t}$, $t\in[0,1]$,
such that $\sigma_{t=0}=T$ and $\sigma_{t=1}$ maps affinely rectangles whose
edges are on a dyadic grid into rectangles belonging to the same grid. Even if
the ideas of the proof are completely similar to the original ones, we choose
to make them more explicit (see also Remark 5.3 for some comments on the
original proof). The extension to general dimension seems non-trivial (at
least to us): we decided to sketch the main ideas in Section 5.1.1, leaving
the details to the interested reader (we observe however that our approach is
somehow different from the $2d$ case, differently from what it is stated in
[22]).

At this point the proof diverges from [22], due to the fact that in his case
one has to control the $L^{2}$-norm of the vector field while here we need to
build a perturbation of a vector field (not of a map) and to estimate its
$\operatorname{BV}_{x}$-norm. In Lemma 5.9 we prove that the perturbation
$\sigma_{t}$ constructed in the above paragraph (i.e. in Lemma 5.1) can be
encapsulated inside the flow $X_{t}$ so that the resulting vector field is
close in $L^{\infty}_{t}L^{1}_{x}$ and remains in
$L^{\infty}_{t}\operatorname{BV}_{x}$ if the grid is sufficiently small,
always under the assumption that $X_{t}$ is close to identity.

We finally arrive to the approximation theorem through permutations (Theorem
5.14), which we think that can have an independent interest:

######  Theorem 1.4.

Let $b\in L^{\infty}([0,1];BV(K))$ be a divergence-free vector field. Then for
every $\epsilon>0$ there exist $\delta^{\prime},C_{1},C_{2}>0$ positive
constants, $D\in\mathbb{N}$ arbitrarily large and a divergence-free vector
field $b^{\epsilon}\in L^{\infty}([0,1];BV(K))$ such that

  1. 1.

$\operatorname{supp}b^{\epsilon}_{t}\subset\subset
K^{\delta^{\prime}}=[\delta^{\prime},1-\delta^{\prime}]^{2}$,

  2. 2.

it holds

$\|b-b^{\epsilon}\|_{L^{\infty}(L^{1})}\leq\epsilon,\quad||\text{\rm
Tot.Var.}(b^{\epsilon})(K)||_{\infty}\leq C_{1}||\text{\rm
Tot.Var.}(b)(K)||_{\infty}+C_{2},$ (1.6)

  3. 3.

the map $X^{\epsilon}\llcorner_{t=1}$ generated by $b^{\epsilon}$ at time
$t=1$ translates each subsquare of the grid
$\mathbb{N}\times\mathbb{N}\frac{1}{D}$ into a subsquare of the same grid,
i.e. it is a permutation of squares.

We remark that in the statement of Theorem 5.14 it is also assumed that there
exists $\delta>0$ such that for $\mathcal{L}^{1}$-a.e. $t\in[0,1]$,
$\operatorname{supp}b_{t}\subset\subset K^{\delta}$. This is for technical
reason, as standard approximation methods allows to deduce the Theorem 1.4
above from Theorem 5.14.

The starting point of its proof is to divide the time interval $[0,1]$ into
subintervals $[t_{i},t_{i+1}]$ and apply the previous perturbations (Lemma
5.1) to $b_{t}$, $t\in[t_{i},t_{i-1}]$. We however need an additional
mechanism in order to obtain a permutation of subsquares and not a piecewise
affine map at $t=1$, as it would be the case if we only use the perturbations
above.

The introduction of this new perturbation is done in Section 5.3: the idea is
that if a measure preserving map $T$ is diagonal with rational eigenvalues,
then there exists a subgrid and a map $R$ made by two rotations such that
$T\circ R$ maps subsquares of the new grid into subsquares instead of
rectangles (Lemma 5.12). The key point is that the total variation of the new
map is bounded independently on the grid size, while the $L^{1}$-norm
converges to $0$ as the grid becomes smaller and smaller. This gives better BV
estimates than the construction of [22].
In the proof of the theorem, this rotation mechanism has to act differently in
each subrectangle where the map is affine at all times $t_{i}$ of the time
partition: this means that the rotation mechanism is calibrated on the single
subrectangle, depending on its evolution, and it differs from one to the
other. More precisely, whenever a rectangle is distorted of a fixed amount, we
let the rotation to act on a subgrid in order to obtain a permutation of
subsquares (see Figure 1). Without entering into technicalities, we just
notice that this has to be done during the time evolution, and the grid has to
be chosen sufficiently small so that it is the same for all rotations.

$t=0$
---
$t=t_{1}$
---
$t=t_{2}$
---
Figure 1: A graphic explanation of the action of rotations: the three top
frames above shows the evolution of curves at different times $t_{i}$, where
the perturbation of Lemma 5.1 takes care of transforming the green square
affinely into the green rectangles. The three bottom frames is the action of
the rotation on a finer grid: the red rectangle is chosen so that the action
of the affine map $X_{t=t_{1}}$ coincides with a rotation by $\pi/2$ (as a
set, but the black grid (image through an affine map) is not the image of the
red grid after a rotation), and then the red grid is mapped into itself when
composed with a rotation of $\pi/2$ inside the red rectangle. At the next
step, one chooses again a finer grid (the light blue one) to perform the same
transformation, so that the blue grid is mapped into itself.

The interesting part of the above theorem is the form of the estimate for the
Total Variation in (1.6). The constant $C_{1}$ is easily explained: in the
first two paragraph above we explained how to perturb a smooth map/flow into a
piecewise affine flow on subrectangles. The key argument is that we need to
contrast rotations of the original vector field $b_{t}$, and then this action
needs in principle a total variation of the order of $\text{\rm
Tot.Var.}(b_{t})$: we believe that this constant $C_{1}$ can be optimized, but
it is not necessary here, because the hard term is the one leading to $C_{2}$.
Indeed, the counter-rotation is of the order of the area of the region where
it is acting (see Lemma 2.6), if it is not too much distorted. The partition
into times where these counter-rotations act is then related to their
distorsion, which is related to the BV norm spent by the vector field.
However, it is in general necessary to use a final counter-rotation, and this
is the main reason for the constant $C_{2}$.

####  1.1.2 From permutations of subsquares to ergodic/exponential mixing

The advantage of having a flow $X_{t}$ such that $X_{t=1}$ is a permutations
of subsquares is that its action is sufficiently simple to perturb in order to
achieve a desired property. Nevertheless it requires some smart constructions,
since in any case we need to control the $L^{1}$-distance and the BV norm.

The first step is to perturb a permutation of subsquares into a cyclic
permutation of subsquares, i.e. a permutation made of a single cycle: this is
clearly a necessary condition for ergodicity. Roughly speaking, the idea is to
exchange two adjacent subsquares belonging to different cycles in order to
merge them. We do this operation in two steps.
In Lemma 4.4 it is shown that one can arbitrarily refine the grid
$\mathbb{N}\times\mathbb{N}\frac{1}{D}$ into
$\mathbb{N}\times\mathbb{N}\frac{1}{DM}$ so that each cycle of length $k$ in
the original grid becomes a cycle of length $kM^{2}$ in the new one. Moreover
the perturbation is going to $0$ in $L^{\infty}_{t}L^{1}_{x}$ as $M\to\infty$
and its $L^{\infty}\operatorname{BV}_{x}$ is arbitrarily small when $D$ is
large.
The above result allows now to exchange sets of size $(DM)^{-1}$ when merging
cycles: this is done in Proposition 4.5. This proposition faces a new problem:
in the previous case the exchange of subsquares of size $(DM)^{-1}$ occurs
within the same subsquare of size $D^{-1}$: the latter is only deformed during
the evolution and hence the merging can be done in the whole time interval
$[0,1]$. In the case of Proposition 4.5, instead, we are exchanging subsquares
of size $(DM)^{-1}$ which are then shifted away during the flow, since they
belong to different subquares of the grid $D^{-1}$. This requires to do the
exchange sufficiently fast (i.e. during the time where they share a common
boundary, Remark 4.7), or to freeze the evolution for an interval of time
$[0,\delta]$ and perform the exchanges here and then let the flow permuting
the subsquares to evolve during the remaining time interval $[\delta,1]$. We
choose for simplicity this second line, being easier and not changing the
final result: notice however that now the constant $M$ plays the role of
controlling the constant $\delta^{-1}$, appearing because the exchange action
occurs in the time interval $[0,\delta]$.

Once we have a cyclic permutation of subsquares, the perturbation to get an
ergodic vector field is straighforward: indeed as a map of the quotient space
$\Omega$ (defined as in (1.4) for the grid
$\mathbb{N}\times\mathbb{N}\frac{1}{DM}$) it is already ergodic, hence one has
only to add an ergodic vector field inside each subsquare: this is done in
Proposition 4.9, where the universal mixer $U$ given by 1.5 is used inside a
single subsquare. In particular, we can spread its action on the whole time
interval $[0,1]$, and the smallness estimates are optimal.

To achieve the exponential mixing, instead, we need to transfer mass across
different subsquares, and hence we face again the problem of Proposition 4.5
above: we let the mixing action occurs in an interval of time where the
evolution is frozen, and then let the cyclic permutation to act in the time
interval $[\delta,1]$ (see also Remark 4.11). The idea is again to use the
universal mixer (1.5) to exchange mass across to nearby subsquares. The
additional difficulty here is that in order to avoid resonant phenomena we mix
all squares with 2 neighboring ones, so that by simple computations the Markov
Shift obtained through this map is exponentially mixing, Proposition 4.10.

To collect all the above results into a proof of Point (5) of Theorem 1.2 is
not difficult at this point, and we devote a section (Section 4.4 and
Corollary 3.10) to shows how to merge these result and get the desired
statement.

###  1.2 Plan of the paper

The paper is organized as follows.

In Section 2, after listing some of the notation used in the paper, we give a
short overview on BV functions (Section 2.1) and Regular Lagrangian Flows
(Section 2.2), proving the extension of the continuous dependence to a
complete set $\mathcal{U}$ in Proposition 2.3 (providing the proof of Point
(1) of Theorem 1.2) and stating some technical estimates on composition of
maps (Theorem 2.4 and (2.2), (2.3)) and on the vector field (2.5) generating a
rotation (Lemma 2.6).

In Section 3 we collect some classical results in Ergodic Theory which are
needed for Theorem 1.2, and also give the proof of the $G_{\delta}$-properties
of the set of ergodic/weakly mixing vector fields of Theorem 1.2. First we
introduce the basic definitions, then in Section 3.1 we clarify the relation
with the neighborhood topology and the $L^{1}$-topology used in Theorem 1.2.
In Section 3.2 we restate in our setting the well known fact that weakly
mixing are a $G_{\delta}$-set, as well as the first category property of
strongly mixing vector fields (Proposition 3.9). The proof of the remaining
parts of Theorem 1.2 is a corollary of the previous statement (Corollary
3.10), if we know that the strongly mixing vector fields are dense.
The construction of exponential mixing vector fields is based on the analysis
of Markov Shift: in Section 3.3 we give the results which are linked to our
construction.

In Section 4 we present the proof of the the density of exponentially mixing
vector fields, under the assumption that permutation flows are dense in
$L^{1}_{t}\operatorname{BV}_{x}$ w.r.t. the $L^{1}_{t,x}$-norm. We decide to
put first this construction because it is in some sense independent on the
proof of the density of permutation flows: the idea is that different
functional settings can be studied by changing this last part (i.e. the
density of permutation flow), while keeping the construction of approximation
by permutations more or less the same.
The first statement is Lemma 4.4 which allows to partition the subsquares of a
given cycle into smaller subquares still belonging to the same cycle: as we
already said, its proof is based on a doubling scale approach, merging larger
and larger cycles. The usefullness of this estimate is shown in Proposition
5.4, where we need to exchange mass only on an area which is of order
$M^{-2}$, and hence obtaining that the perturbation is small in $L^{1}_{t,x}$
and $L^{\infty}_{t}BV_{x}$ (Proposition 4.8 of Remark 4.7 addresses the
problem of exchanging two subquares during the evolution, a refinement not
needed for the proof of Theorem 1.2).
The last two subsection address the density of ergodic vector fields
(Proposition 4.9) and of exponentially mixing vector fields (Proposition
4.10): the basic idea is the same (i.e. perturb the cyclic permutation) but
the strongly mixing case requires more effort and uses the Markov chains.
Section 4.4 shows at this point how the assumptions of Corollary 3.10 are
verified, concluding the proof of Theorem 1.2 under the assumption of the
density of vector fields whose flow is a permutation of subsquares.

The last section, Section 5, proves the cornerstone approximation result, i.e.
the density of vector fields whose flow at $t=1$ is a permutation of
subsquares, Theorem 5.14 (whose statement is the same of Theorem 1.6).
In Section 5.1 we approximate a smooth flow close to identity with a BV flow
which is locally affine in subrectangles: Lemma 5.1 considers the 2d-case as
in [22], while the needed variations for the $d$-dimensional case are in
Section 5.1.1.
The BV estimates for such perturbed flow are studied in Section 5.2. A
preliminary result (Lemma 5.7) takes care of the conditions that the area of
the subsquares has to be a dyadic rational, while the key estimates are in
Lemma 5.9: an important fact is that as the grid becomes finer the
perturbation becomes smaller.
An ingredient for obtaining a flow which is a permutation of subsquares is the
use of rotations: in Section 5.3 we study these elementary transformations.
The main approximation theorem, Theorem 5.14, is stated and proved in Section
5.4. Its proof uses all the ingredients of the previous sections, and an
additional argument on how to encapsulate rotations in order to control the
total variation.

##  2 Preliminaries and notations

First, a list of standard notations used throughout this paper.

  * •

$\Omega\subset\mathbb{R}^{n}$ denotes in general an open set;
$\mathcal{B}(\Omega)$ denotes the $\sigma$-algebra of Borel sets of $\Omega$;

  * •

$\text{dist}(x,A)$ is the distance of $x$ from the set $A\subset\Omega$,
defined as the infimum of $|x-y|$ as $y$ varies in $A$;

  * •

$\forall A\subset\Omega$, $\mathring{A}$ denotes the interior of $A$ and
$\partial A$ its boundary, moreover, if $\epsilon>0$, then $A^{\epsilon}$ is
the $\epsilon$-neighbourhood of $A$, that is

$A^{\epsilon}=\\{x\in\Omega:\text{dist}(x,\partial A)\leq\epsilon\\};$

  * •

$\mathcal{M}_{b}(\Omega)$ bounded Radon measures;

  * •

if $\nu\in\mathcal{M}_{b}(\Omega)$ then $\|\nu\|$ denotes its total variation;

  * •

$\operatorname{BV}(\Omega)$ is the set of functions with bounded variation,
and if $u\in\operatorname{BV}(\Omega)$ we will use instead $\text{\rm
Tot.Var.}(u)$ to denote $\|Du\|$;

  * •

$\mathcal{L}^{d}$ denotes the $d$-dimensional Lebesgue measure on
$\mathbb{R}^{d}$, and $\mathcal{H}^{k}$ the $k$-dimensional Hausdorff measure;

  * •

$K=[0,1]^{2}$ is the unit square;

  * •

$\mathcal{L}^{2}\llcorner_{K}$ denotes the normalized Lebesgue measure on $K$;

  * •

let $b:[0,1]\times\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$, and let
$t,s\in[0,1]$ then we denote by $X(t,s,x)$ a solution of

$\begin{cases}\dot{x}(t)=b(t,x(t))\\\ x(s)=x,\end{cases}$

moreover we will use $X(t)(x)$ or (alternatively $X_{t}(x)$) for $X(t,0,x)$
(in our setting as a flow the function $X(t,s,x)$ is unique a.e.);

  * •

$(S,\Sigma,\mu)$ denotes a locally compact separable metric space where $\mu$
is a normalized complete measure;

  * •

$G(S)$ denotes the space of automorphisms of $S$.

###  2.1 BV functions

In this subsection we recall some results concerning functions of bounded
variation. For a complete presentation of the topic, see [4]. Let
$u\in\operatorname{BV}(\Omega;\mathbb{R}^{m})$ and
$Du\in\mathcal{M}_{b}(\Omega)^{n\times m}$ the $n\times m$-valued measure
representing its distributional derivative. We recall the decomposition of the
measure $Du$

$Du=D^{\textrm{cont}}u+D^{\textrm{jump}}u=D^{\textrm{a.c.}}u+D^{\mathrm{cantor}}u+D^{\mathrm{jump}}u,$

where
$D^{\mathrm{cont}}u,D^{\mathrm{a.c.}}u,D^{\mathrm{cantor}}u,D^{\mathrm{jump}}u$
are respectively the continuous part, the absolutely continuous part, the
Cantor part and the jump part of the measure. We also recall that for $u\in
BV(\Omega)$ the following estimate on the translation holds: for every
$C\subset\Omega$ compact and $z\in\mathbb{R}^{n}$ such that
$|z|\leq\operatorname{dist}(C,\partial\Omega)$

$\int_{C}|u(x+z)-u(x)|dx\leq\left|\sum_{i=1}^{n}z_{i}D_{i}u\right|(C^{|z|}).$
(2.1)

###  2.2 Regular Lagrangian Flows

Throughout the paper we will consider divergence-free vector fields
$b:[0,1]\times K\rightarrow\mathbb{R}^{2}$ in the space
$L^{\infty}([0,1];\operatorname{BV}(K))$ (in short $b\in
L^{\infty}_{t}\operatorname{BV}_{x}$) such that
$\operatorname{supp}(b_{t})\subset\subset\mathring{K}$ for
$\mathcal{L}^{1}$-a.e. $t\in[0,1]$: it is standard to extend the analysis to
divergence-free BV-vector fields in $\mathbb{R}^{2}$ with support in $K$. When
the velocity field $b$ is Lipschitz, then its _flow_ is well-defined in the
classical sense, indeed it is the map $X:[0,1]\times K\rightarrow K$
satisfying

$\begin{cases}\frac{d}{dt}X_{t}(x)=b(t,X_{t}(x));&\\\ X_{0}(x)=x.\end{cases}$

But when we allow the velocity fields to be discontinuous (as in our case
$\operatorname{BV}$ regular in space) we can still give a notion of a flow
(namely the _Regular Lagrangian Flow_). These flows have the advantage to
allow rigid _cut and paste_ motions, since they do not preserve the property
of a set to be connected. More in detail, we give the following

######  Definition 2.1.

Let $b\in L^{1}([0,1]\times\mathbb{R}^{2};\mathbb{R}^{2})$. A map
$X:[0,1]\times\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ is a _Regular
Lagrangian Flow_ (RLF) for the vector field $b$ if

  1. 1.

for a.e. $x\in\mathbb{R}^{2}$ the map $t\rightarrow X_{t}(x)$ is an absolutely
continuous integral solution of

$\begin{cases}\frac{d}{dt}x(t)=b(t,x(t));&\\\ x(0)=x.\end{cases}$

  2. 2.

there exists a positive constant $C$ independent of $t$ such that

$\mathcal{L}^{2}(X_{t}^{-1}(A))\leq C\mathcal{L}^{2}(A),\quad\forall
A\in\mathcal{B}(\mathbb{R}^{2}).$

DiPerna and Lions proved existence, uniqueness and stability for Sobolev
vector fields with bounded divergence [8], while the extension to the case of
$\operatorname{BV}$ vector fields with divergence in $L^{1}$ has been done by
Ambrosio in [13]. When dealing with divergence-free vector fields $b$ the
unique Regular Lagrangian Flow $t\rightarrow X_{t}$ associated with $b$ is a
flow of measure-preserving maps, main objects of investigations in Ergodic
Theory. In the sequel we will build flows of measure-preserving maps
originating from divergence-free vector fields; more precisely, if a flow
$X:[0,1]\times K\rightarrow K$ is invertible, measure-preserving for
$\mathcal{L}^{1}$-a.e. $t$ and the map $t\rightarrow X_{t}$ is differentiable
for $\mathcal{L}^{1}$-a.e. $t$ and $\dot{X}_{t}\in L^{1}(K)$, then the _vector
field associated with $X_{t}$_ is the divergence-free vector field defined by

$b_{t}(x)=b(t,x)=\dot{X}_{t}(X^{-1}_{t}(x)).$

######  Theorem 2.2 (Stability, Theorem 6.3 [12]).

Let $b_{n},b\in L^{\infty}([0,1],\operatorname{BV}(K))$ be divergence-free
vector fields and let $X^{n},X$ be the corresponding Regular Lagrangian Flows.
Assume that

$||b_{n}-b||_{L^{1}_{t,x}}\to 0\quad\text{as }n\to\infty,$

then

$\lim_{n\to\infty}\int_{K}\sup_{t\in[0,1]}|X^{n}_{t}(x)-X_{t}(x)|dx=0.$

In this setting we can extend the family of vector fields we consider to a
Polish subspace of $L^{1}_{t,x}$ in which we still have a notion of
uniqueness. This extension allows us to apply Baire Category Theorem for the
results of genericity that we will give for weakly mixing vector fields.

######  Proposition 2.3 (Extension).

Let

$\Phi:\\{b\in L^{\infty}_{t}\operatorname{BV}_{x}:D\cdot
b_{t}=0\\}\subset\\{b\in L^{1}([0,1],L^{1}(K)),D\cdot b_{t}=0\\}\rightarrow
C([0,1],L^{1}(K))$

the map that associates $b$ with its unique Regular Lagrangian Flow $X_{t}$.
Then $\Phi$ can be extended as a continuous function to a $G_{\delta}$-set
$\mathcal{U}$ containing $\\{b\in L^{\infty}_{t}\operatorname{BV}_{x}:D\cdot
b_{t}=0\\}$.

This proposition proves Point (1) of Theorem 1.2.

###### Proof.

We recall that for every $f:A\rightarrow Z$ continuous where $A\subset W$ is
metrizable and $Z$ is a complete metric space, there exists a $G_{\delta}$-set
$A\subset G$ and a continuous extension $\tilde{f}:G\rightarrow Z$
(Proposition 2.2.3, [14]). Thus we have to prove the continuity of the map
$\Phi$ which follows by

$\begin{split}\|\Phi(b^{n})-\Phi(b)\|_{C_{t}L^{1}_{x}}&=\sup_{t\in[0,1]}\int_{K}\big{|}X_{t}^{n}(x)-X_{t}(x)\big{|}dx\\\
&\leq\int_{K}\sup_{t\in[0,1]}\big{|}X_{t}^{n}(x)-X_{t}(x)\big{|}dx.\end{split}$

This concludes the proof. ∎

We will also use the following tools to prove the main approximation theorems
of the paper. The first one gives a rule to compute the total variation of the
composition of vector fields, while the second one is a direct computation of
the cost, in terms of the total variation of the vector field whose flow
rotates rectangles.

######  Theorem 2.4 (Change of variables, Theorem 3.16, [4]).

Let $\Omega,\Omega^{\prime}$ two open subsets of $\mathbb{R}^{n}$ and let
$\phi:\Omega\rightarrow\Omega^{\prime}$ invertible with Lipschitz inverse,
then $\forall u\in BV(\Omega^{\prime})$ the function $v=u\circ\phi$ belongs to
$BV(\Omega)$ and

$\text{\rm Tot.Var.}(v)\leq\operatorname{Lip}(\phi^{-1})^{n-1}\text{\rm
Tot.Var.}(u).$

######  Corollary 2.5.

Let $\Omega,\Omega^{\prime}\subset\mathbb{R}^{n}$ open sets where
$\partial\Omega$ is Lipschitz and let
$\phi:\bar{\Omega}\rightarrow\bar{\Omega}^{\prime}$ invertible with Lipschitz
inverse, then $\forall u\in BV(\mathbb{R}^{n})$ the function

$v=\begin{cases}u\circ\phi&x\in\Omega,\\\ 0&\text{otherwise},\end{cases}$

belongs to $BV(\mathbb{R}^{n})$ and

$\begin{split}\text{\rm
Tot.Var.}(v)(\mathbb{R}^{n})&\leq\operatorname{Lip}(\phi^{-1})^{n-1}\big{(}\text{\rm
Tot.Var.}(u)(\Omega^{\prime})+\|\mathrm{Tr}(u,\partial\Omega^{\prime})\|_{L^{1}(\mathcal{H}^{n-1}\llcorner_{\partial\Omega^{\prime}})}\big{)}\\\
&=\operatorname{Lip}(\phi^{-1})^{n-1}\text{\rm
Tot.Var.}(u)(\mathbb{R}^{n}).\end{split}$

In the following, we have often to study the properties of the vector field
$b_{3}$ associated with the composition $Y_{3}(t)$ of two smooth measure
preserving flows $t\mapsto Y_{i}(t)$, $i=1,2$, with associated vector fields
$b_{1},b_{2}$. By direct computation

$b_{3}(t,Y_{3}(t,y))=\partial_{t}Y_{1}(t,Y_{2}(t,y))=b_{1}(t,Y_{3}(t,y))+\nabla
Y_{1}(t,Y_{2}(t,y))b_{2}(t,Y_{2}(t,y)),$
$\begin{split}b_{3}(t,x)&=b_{1}(t,x)+\nabla
Y_{1}(t,Y_{2}(t,Y_{3}^{-1}(t,x)))b_{2}(t,Y_{2}(t,Y_{3}^{-1}(t,x)))\\\
&=b_{1}(t,x)+\nabla
Y_{1}(t,Y_{1}^{-1}(t,x))b_{2}(t,Y_{1}^{-1}(t,x)).\end{split}$ (2.2)

Hence using Theorem 2.4 we conclude that (being $Y_{1}\circ Y_{2}$ measure
preserving too)

$\begin{split}\text{\rm Tot.Var.}(b_{3})&\leq\text{\rm
Tot.Var.}(b_{1})+\operatorname{Lip}(Y_{1})^{n-1}\text{\rm
Tot.Var.}(DY_{1}(t)b_{2})\\\ &\leq\text{\rm Tot.Var.}(b_{1})+\|\nabla
Y_{1}\|^{n}_{\infty}\text{\rm Tot.Var.}(b_{2})+\|\nabla
Y_{1}\|^{n-1}_{\infty}\|b_{2}\|_{\infty}\text{\rm
Tot.Var.}(DY(t)).\end{split}$ (2.3)

Throughout the paper we will extensively use a flow rotating rectangles and
the vector field associated with it. More precisely we define the _rotation
flow_ $r_{t}:K\rightarrow K$ for $t\in[0,1]$ in the following way: call

$V(x)=\max\left\\{\left|x_{1}-\frac{1}{2}\right|,\left|x_{2}-\frac{1}{2}\right|\right\\}^{2},\quad(x_{1},x_{2})\in
K.$

Then the _rotation field_ is $r:K\rightarrow\mathbb{R}^{2}$

${r}(x)=\nabla V^{\perp}(x),$ (2.4)

where $\nabla^{\perp}=(-\partial_{x_{2}},\partial_{x_{1}})$ is the orthogonal
gradient. Finally the rotation flow $r_{t}$ is the flow of the vector field
$r$, i.e. the unique solution to the following ODE system:

$\begin{cases}\dot{r}_{t}(x)={r}(r_{t}(x)),\\\ r_{0}(x)=x.\end{cases}$ (2.5)

This flow rotates the cube counterclockwise of an angle $\frac{\pi}{2}$ in a
unit interval of time.

######  Lemma 2.6.

Let $R\subset\mathbb{R}^{2}$ a rectangle of sides $a,b>0$. Consider the
rotating flow

$R_{t}=\chi^{-1}\circ r_{t}\circ\chi,$

where $\chi:R\rightarrow K$ is the affine map sending $R$ into the unit cube
and $r_{t}$ is the rotation flow defined in (2.5). Let $b_{t}^{R}$ the
divergence-free vector field associated with $R_{t}$. Then

$\text{\rm Tot.Var.}(b^{R}_{t})(\mathbb{R}^{2})=4a^{2}+4b^{2},\quad\forall
t\in[0,1].$

###### Proof.

The potential $V$ generating the rotation of $\pi/2$ in this case is the
function

$V(x)=\max\bigg{\\{}\frac{b}{a}\bigg{(}x_{1}-\frac{a}{2}\bigg{)}^{2},\frac{a}{b}\bigg{(}x_{2}-\frac{b}{2}\bigg{)}^{2}\bigg{\\}},$

where we assume that $R=[0,a]\times[0,b]$, so that the vector field is given
by

$r(x)=\nabla^{\perp}V=\begin{cases}\big{(}0,\frac{2b}{a}\big{(}x_{1}-\frac{a}{2}\big{)}\big{)}&|x_{1}|\geq\frac{b}{a}|x_{2}|,0\leq
x_{1}\leq a,\\\
\big{(}-\frac{2a}{b}\big{(}x_{2}-\frac{b}{2}\big{)},0\big{)}&|x_{1}|<\frac{b}{a}|x_{2}|,0\leq
x_{2}\leq b.\end{cases}$

Hence by elementary computations

$\|D^{\mathrm{cont}}r\|=a^{2}+b^{2},\quad\|D^{\mathrm{jump}}r\|=3a^{2}+3b^{2},$

and then we conclude. ∎

##  3 Ergodic Theory

We will consider flows of divergence-free vector fields from the point of view
of Ergodic Theory. Even if we apply the results to the case
$(K,\mathcal{B}(K),\mathcal{L}^{2}\llcorner_{K})$ in this section we will give
the notions of ergodicity and mixing in more general spaces [6, Chapter 1].
More precisely, let $(\Omega,\Sigma,\mu)$ be a locally compact separable
metric space where $\mu$ is complete and normalized, that is $\mu(\Omega)=1$.

######  Definition 3.1.

An _automorphism_ of the measure space $(\Omega,\Sigma,\mu)$ is a one-to-one
map $T:\Omega\rightarrow\Omega$ bi-measurable and measure-preserving, that is

$\mu(A)=\mu(T(A))=\mu(T^{-1}(A)),\quad\forall A\in\Sigma.$

We call $G(\Omega)$ the group of automorphisms of the measure space
$(\Omega,\Sigma,\mu)$.

######  Definition 3.2.

A _flow_ $\\{X_{t}\\}$, $t\in\mathbb{R}$, is a one-parameter group of
automorphisms of $(\Omega,\Sigma,\mu)$ such that for every
$f:\Omega\rightarrow\mathbb{R}$ measurable, the function $f(X_{t}(x))$ is
measurable on $\Omega\times\mathbb{R}$.

######  Definition 3.3.

Let $T:\Omega\rightarrow\Omega$ an automorphism. Then

  * •

$T$ is _ergodic_ if for every $A\in\Sigma$

$T(A)=A\quad\Rightarrow\quad\mu(A)=0\textrm{ or }\mu(A)=1;$ (3.1)

  * •

$T$ is _weakly mixing_ if $\forall A,B\in\Sigma$

$\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\left[\mu(T^{-j}(A)\cap
B)-\mu(A)\mu(B)\right]^{2}=0;$ (3.2)

  * •

$T$ is _(strongly) mixing_ if $\forall A,B\in\Sigma$

$\lim_{n\to\infty}\mu(T^{-n}(A)\cap B)=\mu(A)\mu(B).$ (3.3)

######  Remark 3.4.

It is a well-known and quite elementary fact that strongly mixing
$\Rightarrow$ weakly mixing $\Rightarrow$ ergodic.

We can give the analogous definitions for the flow:

######  Definition 3.5.

Let $\\{X_{t}\\}$ a flow of automorphisms. Then

  * •

$\\{X_{t}\\}$ is _ergodic_ if for every $A\in\Sigma$

$X_{t}(A)=A\quad\Rightarrow\quad\mu(A)=0\textrm{ or }\mu(A)=1;$ (3.4)

  * •

$\\{X_{t}\\}$ is _weakly mixing_ if $\forall A,B\in\Sigma$

$\lim_{t\to\infty}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{0}^{t}\left[\int_{\Omega}\chi_{A}(X_{-s}(x))\chi_{B}(x)d\mu-\mu(A)\mu(B)\right]^{2}ds=0;$
(3.5)

  * •

$\\{X_{t}\\}$ is _(strongly) mixing_ if $\forall A,B\in\Sigma$

$\lim_{t\to\infty}\int_{\Omega}\chi_{A}(X_{-t}(x))\chi_{B}(x)d\mu=\mu(A)\mu(B).$
(3.6)

###  3.1 The neighbourhood topology as a convergence in measure.

To get a genericity result it is necessary to identify the correct topology on
$G(\Omega)$. Following the work of Halmos [19] we define the _neighbourhood
topology_ as the topology generated by the following base of open sets: let
$T\in G(\Omega)$ then

$N(T)=\\{S\in G(\Omega):|T(A_{i})\triangle S(A_{i})|<\epsilon,\quad
i=1,\dots,n\\},$

where $\epsilon>0$ and $A_{i}\in\Sigma$ are measurable sets.

Since for our purposes we will consider the $L^{1}$ topology on $G(\Omega)$,
we recall the following

######  Proposition 3.6.

Let $\\{{T_{n}}\\},T\subset G(\Omega)$ and assume that $T_{n}\rightarrow T$ in
measure. Then $T_{n}\rightarrow T$ in the neighbourhood topology. Conversely,
if $T_{n}\rightarrow T$ in the neighbourhood topology, then $T_{n}$ converges
to $T$ in measure.

Since in our case $\Omega$ is a compact set, then the convergence in measure
is equivalent to the convergence in $L^{1}$: hence we will use the $L^{1}$
topology for maps as in Proposition 2.3.

We will be concerned with flows of vector fields extended periodically to the
real line, that is $b(t+1)=b(t)$. Even if $X_{t}$ is not a flow of
automoprhisms, the quantities in the r.h.s. of (3.5),(3.6) can be computed and
are related to the mixing properties of $T=X_{t=1}$. Also the ergodic
properties of $T=X_{1}$ are equivalent to an ergodic property of $X_{t}$.

Let $\\{X_{s}\\}_{s\in[0,1]}$ be a family of autmorphisms of $\Omega$ such
that $s\rightarrow X_{s}$ is continuous (hence uniformly continuous) with
respect to the neighborhood topology of $G(\Omega)$. Let $T=X_{t=1}$ and
define

$X_{t}=X_{s}\circ T^{n}=T^{n}\circ X_{s},\quad t=n+s,s\in[0,1).$

######  Lemma 3.7.

The following holds

  1. 1.

if $T$ is ergodic then for every set $A\in\Sigma$

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{0}^{t}\chi_{X_{s}(A)}ds\to_{L^{1}}|A|;$

  2. 2.

$T$ is weakly mixing iff for every $A,B\in\Sigma$

$\lim_{t\to\infty}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{0}^{t}\big{[}|X_{s}(A)\cap
B|-|A||B|\big{]}^{2}ds=0;$

  3. 3.

$T$ is mixing iff $\forall A,B\in\Sigma$

$\lim_{t\to\infty}\big{|}X_{t}(A)\cap B\big{|}=|A||B|.$

The proof of this lemma is given in Appendix 6, since we believe it is
standard and not strictly related to our results.

######  Definition 3.8.

Let $b\in L^{\infty}([0,1],\operatorname{BV}(\mathbb{R}^{2}))$,
$\operatorname{supp}b_{t}\subset K$, be a divergence-free vector field. We
will say that $b$ is ergodic (weakly mixing, strongly mixing) if its unique
RLF $X_{t}$ evaluated at $t=1$ is ergodic (respectively weakly mixing,
strongly mixing).

###  3.2 Genericity of weakly mixing

Let $\mathcal{U}$ be the $G_{\delta}$-subset of $L^{1}_{t,x}$ where the
Regular Lagrangian Flow can be uniquely extended by continuity (Proposition
2.3). The first statement has the same proof of [Theorem 2, [19]] and [Page
77,[10]]:

######  Proposition 3.9.

The set of ergodic/weakly-mixing vector fields is a $G_{\delta}$-set in
$\mathcal{U}$, the set of strongly mixing is a first category set.

We repeat the proof for convenience only for weakly/strongly mixing, the case
for ergodic vector fields is completely analogous [18].

###### Proof.

Since the map $\tilde{\Phi}(b)(t=1)=T(b)$ defined in Proposition 2.3 is
continuous from $\mathcal{U}$ into $L^{1}(K,K)$, it is enough to prove that
the set of weakly mixing maps is a $G_{\delta}$. For simplicity we define a
new topology on $G(K)$ that coincides with the neighbourhood topology known as
_Von Neumann strong neighbourhood topology_. So take $T\in G(K)$ then it
defines a linear operator $T:L^{2}(K,\mathbb{C})\rightarrow
L^{2}(K,\mathbb{C})$

$(Tf)(x)=f(Tx)\quad\forall f\in L^{2}(K,\mathbb{C})$

such that $||Tf||_{L^{2}}=||f||_{L^{2}}$ (being $T$ measure-preserving).
Consider $f_{i}$ a countable dense subset in $L^{2}$: a base of open sets in
the strong neighbourhood topology is given by

$N(T)=\\{S\in G(K):||Tf_{i}-Sf_{i}||_{2}\leq\epsilon,\quad i=1,\dots,n\\}.$

Then we define

$E(i,j,m,n)=\big{\\{}T\in
G(K):\big{|}(T^{n}f_{i},f_{j})-(f_{i},1)(1,f_{j})\big{|}<2^{-m}\big{\\}},$

where $(\cdot,\cdot)$ denotes the scalar product in $L^{2}$. Simply observing
that $T\rightarrow(Tf,g)$ is continuous in the strong neighbourhood topology
then by Proposition 3.6 it follows that $E(i,j,n,m)$ is open in $L^{1}(K,K)$,
and then

$G=\bigcap_{i,j,m}\bigcup_{n}E(i,j,m,n)$

is a $G_{\delta}$-set. By the Mixing Theorem [Theorem 2, page 29, [6]] $G$
coincides with the set of weakly mixing maps in $L^{1}(K,K)$. Indeed if $T$ is
not mixing, then there exists $f\not=0$ and a complex eigenvalue
$\lambda\in\\{|z|=1,z\not=0,1\\}$ such that $Tf=\lambda f$. We can assume that
$f$ is orthogonal to the eigenvector $1$, that is $(f,1)=0$, and also that
$||f||_{2}=1$. Now choose $i$ such that $||f-f_{i}||\leq\epsilon$ for some
$\epsilon$ to be chosen later and take $f_{j}=f_{i}$. Then

$\displaystyle 1$ $\displaystyle=|(T^{n}f,f)-(f,1)(f,1)|$
$\displaystyle\leq|(T^{n}f,f)-(T^{n}f,f_{i})|+|(T^{n}f,f_{i})-(T^{n}f_{i},f_{i})|+|(T^{n}f_{i},f_{i})-(f_{i},1)(1,f_{i})|$
$\displaystyle\quad+|(f_{i},1)(1,f_{i})-(f,1)(f_{i},1)|+|(f,1)(1,f_{i})-(f,1)(f,1)|$
$\displaystyle\leq
2||f-f_{i}||_{2}+2||f_{i}||_{2}||f-f_{i}||_{2}+|(T^{n}f_{i},f_{i})-(f_{i},1)(1,f_{i})|,$

so since $||f_{i}||_{2}\leq 1+\epsilon$ we get that

$1\leq
2\epsilon+2(1+\epsilon)\epsilon+|(T^{n}f_{i},f_{i})-(f_{i},1)(1,f_{i})|.$

With the choice of $\epsilon>0$ small enough we get that
$\frac{1}{2}\leq|(T^{n}f_{i},f_{i})-(f_{i},1)(1,f_{i})|$, that is $T\not\in
G$. This concludes the proof of the first part of the statement.

We next prove that the set of strongly mixing vector fields is a first
category set. Let $A\subset K$ be a measurable set such that
$|A|=\frac{1}{2}$. Then define the $F_{\sigma}$-set

$F=\bigcup_{n}\bigcap_{k>n}\left\\{T\in G(K):\left||(T^{-k}(A)\cap
A)|-\frac{1}{4}\right|\leq\frac{1}{5}\right\\}.$

Clearly strongly mixing maps are contained in $F$ by definition and therefore
strongly mixing vector fields are contained in
$\tilde{F}=\tilde{\Phi}^{-1}(t=1)(F)$. This $\tilde{F}$ is a set of first
category: indeed consider the set

$\bigcup_{k>n}\tilde{\Phi}^{-1}(t=1)\left(\left\\{T\in
G(K):\left||(T^{-k}(A)\cap
A)|-\frac{1}{4}\right|\leq\frac{1}{5}\right\\}\right)^{c}{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color<EMAIL_ADDRESS>(3.7)

By our main result (Theorem 5.14) $\forall b\in\mathcal{U}$ for all
$n\in\mathbb{N}$ there exists $k>n$ and $b^{p}\in
L^{\infty}_{t}(\operatorname{BV}_{x})$ such that the RLF $X^{p}_{t=1}$
associated with $b^{p}$ evaluated at $t=1$ is a permutation of subsquares of
period $k$. Hence

$\bigcup_{k>n}\\{b\in L^{\infty}_{t}(\operatorname{BV}_{x})\text{ permutation
of period }k\\}$

is dense and contained in (3.7), so that we conclude that (3.7) is open and
dense for all $n$, i.e. $F$ is of first category. ∎

######  Corollary 3.10.

Assume that the set

$SM=\big{\\{}b\in\mathcal{U}:b\ \text{is strongly mixing}\big{\\}}$

is dense in $\mathcal{U}$. Then the set of weakly mixing vector fields is
residual.

###### Proof.

Elementary. ∎

Our aim will be to prove the assumption of the above corollary, which together
with Proposition 3.9 will conclude the proof of Theorem 1.2 once we show that
the dense set of strongly mixing vector fields are actually exponentially
mixing.

######  Remark 3.11.

The above situation, namely

  * •

$b$ strongly mixing is dense in $\mathcal{U}$,

  * •

$b$ weakly mixing is second category in $\mathcal{U}$,

is in some sense the best situation we can hope in $\mathcal{U}$. Indeed, the
strongly mixing vector fields are a set of first category and then it is not a
”fat” set. On the other hand, the weakly mixing vector fields would be a ”fat”
set once we know their density, which one deduces from the density of the
strongly mixing vector fields.

###  3.3 Markov Shifts

When dealing with finite spaces $X=\\{1,\dots,n\\}$ and processes whose
outcome at time $k$ depends only on their outcome at time $k-1$ it is easier
to determine some statistical properties of the dynamical system, as
ergodicity and mixing (see for a reference [15],[23]). More precisely let
$B(n)=\\{\theta:\mathbb{Z}\rightarrow X\\}$ the space of sequences and define
a cylinder

$C(m,k_{1},\dots,k_{r})=\Big{\\{}\theta\in B(n):\theta(m+i)=k_{i+1},\
i=0,\dots,r-1\Big{\\}}$

where $m\in\mathbb{Z}$ and $k_{i}\in X$. Therefore, since the Borel
$\sigma$-algebra on $B(n)$ is generated by disjoint union of cylinders, we can
define a probability measure $\mu$ on $B(n)$ simply determining its value on
cylinders. A _Markov measure_ $\mu$ is a probability measure on $B(n)$ for
which there exist $p_{i}>0,P_{ij}\geq 0$, $i,j=1,\dots n$, with

$\sum_{i}p_{i}=\sum_{j}P_{ij}=1,\quad\sum_{i}p_{i}P_{ij}=p_{j},$

such that

$\mu(C(m,k_{1},\dots,k_{r}))=p_{k_{1}}P_{k_{1}k_{2}}\dots P_{k_{r-1}k_{r}}$

for every cylinder $C(m,k_{1},\dots,k_{r})$. The $P_{ij}$ are called
_transition probabilities_ and $P=(P_{ij})$ is the _transition matrix_. The
transition matrix is a stochastic matrix, that is $\sum_{j}P_{ij}=1$ for every
$i$. Now define $P^{(m)}_{ij}$ the coefficients of the matrix $P^{m}$.

######  Definition 3.12.

A matrix $P$ with positive coefficients is _irreducible_ if $\forall i,j$
there exists $m$ such that $P^{(m)}_{ij}>0$.

######  Definition 3.13.

A matrix $P$ with positive coefficients is _aperiodic_ if there exists $m$
such that $P^{(m)}_{ij}>0$ $\forall i,j$.

A _Markov shift_ is a map $\sigma:(B(n),\mu)\rightarrow(B(n),\mu)$ such that

$\sigma(\theta)(i)=\theta(i+1),\quad\forall\theta\in B(n).$

Then it can be proved that $\sigma_{\sharp}\mu=\mu$. We conclude this
subsection with the following results on ergodicity and mixing properties of
Markov shifts.

######  Proposition 3.14 (Ergodicity).

The following are equivalent:

  1. 1.

$\sigma:(B(n),\mu)\rightarrow(B(n),\mu)$ is ergodic;

  2. 2.

$P$ is irreducible;

  3. 3.

$\displaystyle{\lim_{m\to\infty}\frac{1}{m}\sum_{k=0}^{m-1}P^{(k)}_{ij}=p_{j}}$.

######  Proposition 3.15 (Mixing).

The following are equivalent:

  1. 1.

$\sigma:(B(n),\mu)\rightarrow(B(n),\mu)$ is strongly mixing;

  2. 2.

$P$ is aperiodic;

  3. 3.

$\displaystyle{\lim_{m\to\infty}P^{(m)}_{ij}=p_{j}}$.

##  4 Density of Strongly Mixing vector fields

In Section 5 we prove that _permutation vector fields_ (i.e. vector fields
whose RLF $X_{t}$, when evaluated at $t=1$, is a permutation of subsquares,
Point (3) of Theorem 5.14) are dense in $\mathcal{U}$. In this section, we
show that each permutation vector field can be approximated by a vector field
whose RLF evaluated at time $t=1$ is a unique cycle. This approximation result
will be used to get first the density of ergodic vector fields, then the
density of strongly mixing vector fields.

###  4.1 Cyclic permutations of squares

We start by recalling some basic facts about permutations. Denote by $S_{n}$
the set of permutations of the elements $\\{1,\dots,n\\}$.

######  Definition 4.1.

Let $\sigma\in S_{n}$ be a permutation and $k\leq n\in\mathbb{N}$. We say that
$\sigma$ is a _$k$ -cycle_ $c$ (or simply a _cycle_) if there exist $k$
distinct elements $a_{1},\dots,a_{k}\in\\{1,\dots,n\\}$ such that

$\sigma(a_{i})=a_{i+1},\quad\sigma(a_{k})=a_{1},\quad\sigma(x)=x\quad\forall
x\not=a_{1},\dots,a_{k}.$

We identify the permutation with the ordered set $c=(a_{1}a_{2}\dots a_{k})$.
The number $k$ is the _length_ of the cycle. We say that $c$ is _cyclic_ if
$k=n$. We call _transpositions_ the $2$-cycles.

######  Definition 4.2.

Let $c_{1},c_{2}$ be the cycles $c_{1}=(a_{1}\dots a_{t})$ and
$c_{2}=(b_{1}\dots b_{s})$. We say that $c_{1},c_{2}$ are _disjoint_ cycles if
$a_{i}\not=b_{j}$ for every $i=1,\dots,t$, $j=1,\dots,s$.

Recall the following result.

######  Theorem 4.3.

Every permutation $\sigma\in S_{n}$ is the product of disjoint cycles.

From now on we will address flows $X_{t}$ of divergence-free vector fields
such that $X_{t=1}$ is a permutation of squares of size $\frac{1}{D}$.

Let us fix the size $D\in\mathbb{N}$ of the grid in the unit square $K$. We
enumerate the $D^{2}$ subsquares of the grid and we consider $S_{D^{2}}$ the
set of the permutations of $\\{\kappa_{1},\dots,\kappa_{D^{2}}\\}$. We say
that two squares (ore more in general two rectangles) are _adjacent_ if they
have a common side. We will use also the word _adjacent_ for cycles: two
disjoint cycles of squares $c_{1},c_{2}$ are adjacent if there exist
$\kappa_{1}\in c_{1}$, $\kappa_{2}\in c_{2}$ adjacent subsquares. Two
_adjacent_ squares can be _connected_ by a transposition, which can be defined
simply as an exchange between the two squares: let $\kappa_{i},\kappa_{j}$ two
adjacent squares of size $\frac{1}{D}$ and let $R=\kappa_{i}\cup\kappa_{j}$,
then the _transposition flow_ between $\kappa_{i},\kappa_{j}$ is
$T_{t}(\kappa_{i},\kappa_{j}):[0,1]\times K\rightarrow K$ defined as

$T_{t}(\kappa_{i},\kappa_{j})=\begin{cases}\chi^{-1}\circ
r_{4t}\circ\chi&x\in\mathring{R},\ t\in\left[0,\frac{1}{2}\right],\\\
\chi_{i}^{-1}\circ r_{4t}\circ\chi_{i}&x\in\mathring{\kappa}_{i},\
t\in\left[\frac{1}{2},1\right],\\\ \chi_{j}^{-1}\circ
r_{4t}\circ\chi_{j}&x\in\mathring{\kappa}_{j},\
t\in\left[\frac{1}{2},1\right],\\\ x&\text{otherwise},\end{cases}$ (4.1)

where the map $\chi:R\rightarrow K$ is the affine map sending the rectangle
$R$ into the unit square $K$, $\chi_{i},\chi_{j}$ are the affine maps sending
$\kappa_{i},\kappa_{j}$ into the unit square $K$ and $r$ is the rotation flow
(2.5). This invertible measure-preserving flow has the property to exchange
the two subsquares in the unit time interval (Figure 2).

Figure 2: The action of the _transposition flow_ $T_{t}$.

Moreover, by the computations done in Lemma 2.6, we can say that

$\text{\rm
Tot.Var.}(\dot{T}_{t}(\kappa_{i},\kappa_{j}))(\bar{R})\leq\frac{20}{D^{2}}.$
(4.2)

######  Lemma 4.4.

Let $b\in L^{\infty}_{t}(\operatorname{BV}_{x})$ be a divergence-free vector
field and assume that its flow at time $t=1$, namely $X\llcorner_{t=1}$, is a
$k$-cycle of squares of the grid $\mathbb{N}\times\mathbb{N}\frac{1}{D}$ where
$k,D\in\mathbb{N}$. Then for every $M=2^{p}$, there exists $b^{c}\in
L^{\infty}_{t}(\operatorname{BV}_{x})$ divergence-free vector field such that

$||b-b^{c}||_{L^{\infty}(L^{1})}\leq\mathcal{O}\left(\frac{1}{D^{3}M}\right),$
$||\text{\rm
Tot.Var.}(b^{c}-b)(K)||_{\infty}\leq\mathcal{O}\left(\frac{1}{D^{2}}\right),$

and the map $X^{c}_{t=1}:K\rightarrow K$ is a $kM^{2}$-cycle of squares of
size $\frac{1}{DM}$, where $X^{c}_{t}:[0,1]\times K\rightarrow K$ is the flow
associated with $b^{c}$.

Here and in the following we will write $T(\kappa_{i})=\kappa_{j}$ meaning
that $T$ is a rigid translation of $\kappa_{i}$ to $\kappa_{j}$. This to avoid
cumbersome notation.

###### Proof.

Let us call $T\doteq X_{t=1}$: being a cycle, there exist
$\\{\kappa_{1},\dots,\kappa_{k}\\}\subset\\{1,\dots,D^{2}\\}$ such that

$T(\kappa_{i})=\kappa_{i+1},\quad T(\kappa_{n})=\kappa_{1},\quad
T(x)=x\quad\text{otherwise}.$

Now fix some $M=2^{p}$ and divide each subsquare $\kappa_{i}$ into $M^{2}$
subsquares $\kappa^{j}_{i}$ with $j=1,\dots,M^{2}$. Since $T$ is a translation
of subsquares and choosing cleverly the labelling $j\to\kappa^{j}_{i}$, then
we have also $T(\kappa^{j}_{i})=\kappa^{j}_{i+1}$ so that $T$ is a permutation
of subsquares $\kappa^{j}_{i}$. More precisely, it is the product of $M^{2}$
disjoint cycles of length $k$. The idea is to connect these cycles with
transpositions in order to have a unique cycle of length $kM^{2}$: we will
need a parturbation inside $\kappa_{1}$.

Divide the $M^{2}$ subsquares of $\kappa_{1}$ into $\frac{M^{2}}{2}$ couples
$R^{2}_{h}=\kappa^{j}_{1}\cup\kappa^{j^{\prime}}_{1}$ with
$h=1,\dots,\frac{M^{2}}{2}$ and $\kappa^{j}_{1},\kappa^{j^{\prime}}_{1}$ are
adjacent squares. In the time interval $\left[0,\frac{1}{2}\right]$ perform
$\frac{M^{2}}{2}$ transpositions, one in each $R^{2}_{h}$, that is

$X^{c}_{t}(x)=X_{t}\circ T^{2}_{t}(x),\quad t\in\left[0,\frac{1}{2}\right],$

where the flow $T^{2}:\left[0,\frac{1}{2}\right]\times K\rightarrow K$

$T^{2}_{t}\llcorner_{R^{2}_{h}}\doteq
T_{2t}(\kappa^{j}_{1},\kappa^{j^{\prime}}_{1})\quad\text{and}\quad
T^{2}_{t}(x)=x\ \text{otherwise},$

is the transposition flow (4.1) between $\kappa_{1}^{j}$ and
$\kappa_{1}^{j^{\prime}}$ as defined in (4.1) above. Then for
$t\in\left[0,\frac{1}{2}\right]$ fixed,

$\displaystyle\text{\rm
Tot.Var.}(b^{c}_{t}-b_{t})({\kappa}_{1})\leq\mathcal{O}(1)2\frac{M^{2}}{2}\frac{20}{M^{2}D^{2}},$

where we have used (4.2). We observe that at this time step we have obtained
$\frac{M^{2}}{2}$ disjoint $2k$-cycles.

In the time interval $\left[\frac{1}{2},\frac{3}{4}\right]$ we divide the unit
square into squares $R^{4}_{h}=R^{2}_{j}\cup R^{2}_{j^{\prime}}$ with
$h=1,\dots,\frac{M^{2}}{4}$ where $R^{2}_{j},R^{2}_{j^{\prime}}$ are adjacent
(in particular there exist $\kappa_{1}^{j}\subset
R^{2}_{j},\kappa_{1}^{j^{\prime}}\subset R^{2}_{j^{\prime}}$ adjacent
squares). Now we perform $\frac{M^{2}}{4}$ transpositions of squares
connecting the two rectangles $R^{2}_{j},R^{2}_{j^{\prime}}$ as in Figure 3.
More precisely we define for $t\in\left[\frac{1}{2},\frac{3}{4}\right]$

$X^{c}_{t}(x)=X_{t}\circ T^{4}_{t}(x),\quad
t\in\left[\frac{1}{2},\frac{3}{4}\right],$

where the flow $T^{4}:\left[\frac{1}{2},\frac{3}{4}\right]\times K\rightarrow
K$

$T^{4}_{t}\llcorner_{R^{4}_{h}}\doteq
T_{4t-2}(\kappa^{j}_{1},\kappa^{j^{\prime}}_{1})\quad\text{and}\quad
T^{4}_{t}(x)=x\ \text{otherwise},$

is the transposition flow (4.1) between $\kappa_{1}^{j}$ and
$\kappa_{1}^{j^{\prime}}$. Again,

$\displaystyle\text{\rm
Tot.Var.}(b^{c}_{t}-b_{t})({\kappa}_{1})\leq\mathcal{O}(1)4\frac{M^{2}}{4}\frac{20}{M^{2}D^{2}}.$

Figure 3: Subdivision of the initial square $\kappa_{1}$ into
subrectangles/subsquares where transpositions (the bars) occurr between
subsquares $\kappa^{j}_{i}\subset R^{2i}_{j},\kappa^{j^{\prime}}_{i}\subset
R^{2i}_{j^{\prime}}$ of side $\frac{1}{DM}$ (dotted lines). Notice that at the
first/third step the initial square $\kappa_{1}$ is divided into rectangles
(see Point (1) of the procedure), while at the second step it is divided into
squares (see Point (2)).

Repeating the procedure (see Figure 3),

  1. 1.

at the $2i-1$-th step we divide our initial square $\kappa_{1}$ into
$\frac{M^{2}}{2^{2i-1}}$ rectangles (made of two squares of obtained at the
step $2(i-1)$) so that we perform ${2^{2p-i}}$ transpositions of subsquares
$\kappa_{1}^{j}$ in the time interval
$\left[\sum_{j=1}^{2i-2}\frac{1}{2^{j}},\sum_{j=1}^{2i-1}\frac{1}{2^{j}}\right]$;

  2. 2.

at the $2i$-th step, we divide our initial square $\kappa_{1}$ into
$\frac{M^{2}}{2^{2i}}$ squares (made of 2 rectangles of the previous step) so
that we perform ${2^{2p-i}}$ transpositions of subsquares $\kappa_{1}^{j}$ in
the time interval
$\left[\sum_{j=1}^{2i-1}\frac{1}{2^{j}},\sum_{j=1}^{2i}\frac{1}{2^{j}}\right]$.

In both cases we find in the interval
$\left[\sum_{j=1}^{i-1}\frac{1}{2^{j}},\sum_{j=1}^{i}\frac{1}{2^{j}}\right]$
that

$\text{\rm
Tot.Var.}(b^{c}_{t}-b_{t})({\kappa}_{1})\leq\mathcal{O}(1)2^{i}\frac{M^{2}}{2^{i}}\frac{20}{M^{2}D^{2}}.$

Call $t_{i}=\sum_{j=1}^{i}2^{-j}$. We will prove that the map $X(1,t_{i})\circ
X^{c}\llcorner_{t=t_{i}}$ is a permutation given by the product of
$\frac{M^{2}}{2^{i}}$ disjoint $2^{i}k$-cycles simply by induction on $i$.

The case $i=1$ is immediate from the definition. So let us assume that the
property is valid for $i$ and call $c_{1},c_{2},\dots,c_{\frac{M^{2}}{2^{i}}}$
the disjoint $2^{i}k$-cycles made of rectangles of subsquares as in Figure 3,
where we have ordered them in such a way that $c_{2h-1},c_{2h}$ with
$h=1,\dots,\frac{M^{2}}{2^{i+1}}$ are adjacent along the long side. Then fix a
couple of adjacent cycles, for simplicity $c_{1},c_{2}$. Then

$\begin{split}&c_{1}=(\kappa^{1}_{1}\quad\dots\quad\kappa^{1}_{2^{i}k}),\\\
&c_{2}=(\kappa^{2}_{1}\quad\dots\quad\kappa^{2}_{2^{i}k}),\end{split}$

and assume that there exist $j,j^{\prime}$ such that
$\kappa^{1}_{j},\kappa^{2}_{j^{\prime}}$ are the adjacent subsquares in which
we perform the transposition. By simply observing that

$X^{c}\llcorner_{t=t_{i+1}}(x)=X^{c}\llcorner_{t=t_{i}}(x)+\int_{t_{i}}^{t_{i+1}}b^{c}(s,X^{c}_{s}(x))ds,$

we deduce that, when restricted to $c_{1}\cup c_{2}$, the map
$X(1,t_{i+1})\circ X^{c}\llcorner_{t_{i+1}}$ is the following permutation

$\left(\begin{array}[]{lcllcllcllcl}\kappa^{1}_{1}&\dots&\kappa^{1}_{j-1}&\kappa^{1}_{j}&\dots&\kappa^{1}_{2^{i}k}&\kappa^{2}_{1}&\dots&\kappa^{2}_{j^{\prime}-1}&\kappa^{2}_{j^{\prime}}&\dots&\kappa^{2}_{2^{i}k}\\\
&&&&&&&&&&&\\\
\kappa^{1}_{2}&\dots&\kappa^{2}_{j^{\prime}}&\kappa^{1}_{j+1}&\dots&\kappa^{1}_{1}&\kappa^{2}_{2}&\dots&\kappa^{1}_{j}&\kappa^{2}_{j^{\prime}+1}&\dots&\kappa^{2}_{1}\end{array}\right).$

Clearly this is a single cycle of length $2^{(i+1)k}$, and it is supported on
a rectangle.

The procedure stops at $t=\sum_{j=1}^{2p}\frac{1}{2^{j}}$ when we have
obtained a unique $M^{2}k$-cycle. Summing up, for $t$ fixed

$\text{\rm Tot.Var.}(b^{c}_{t}-b)(K)\leq\mathcal{O}(1)\frac{20}{D^{2}},$

that is

$||\text{\rm
Tot.Var.}(b^{c}_{t}-b_{t})(K)||_{\infty}\leq\mathcal{O}\left(\frac{1}{D^{2}}\right).$

We conclude with the $L^{\infty}_{t}L^{1}_{x}$ estimate of the vector field:
to do this computation it is necessary to observe that $b_{t}$ and $b^{c}_{t}$
differ only in the couples of adjacent squares in which we perform the
transpositions. Using (2.2) and simple estimates on the rotation (2.4) we
obtain, for $t\in[t_{i-1},t_{i}]$ fixed,

$\|b^{c}_{t}-b_{t}\|_{1}\leq\mathcal{O}(1)\frac{M^{2}}{2^{i}}2^{i}\frac{2}{DM}\frac{2}{D^{2}M^{2}}\leq\mathcal{O}\left(\frac{1}{D^{3}M}\right),$

which concludes the proof. ∎

We state now the approximation result by vector fields whose flow at time
$t=1$ is a unique cycle.

######  Proposition 4.5.

Let $b\in L^{\infty}_{t}(\operatorname{BV}_{x})$ be a divergence-free vector
field and assume that $b_{t}=0$ for $t\in[0,\delta]$, $\delta>0$, and its flow
at time $t=1$, namely $X\llcorner_{t=1}$, is a permutation of squares of the
grid $\mathbb{N}\times\mathbb{N}\frac{1}{D}$ where $D\in\mathbb{N}$. Then for
every $M=2^{p}\gg 1$ there exists a divergence-free vector field $b^{c}\in
L^{\infty}_{t}(\operatorname{BV}_{x})$ such that

$||b-b^{c}||_{L^{1}(L^{1})}\leq\mathcal{O}\left(\frac{1}{DM^{3}}\right),\quad||\text{\rm
Tot.Var.}(b^{c}_{t}-b_{t})(K)||_{\infty}\leq\mathcal{O}\left(\frac{1}{\delta
M^{2}}\right)$

and the map $X^{c}_{1}:K\rightarrow K$, being $X^{c}_{t}:[0,1]\times
K\rightarrow K$ is the flow associated with $b^{c}$, is a $M^{2}D^{2}$-cycle
of subsquares of size $\frac{1}{DM}$.

###### Proof of Proposition 4.5..

Let us fix $\epsilon>0$ and consider $M=2^{p}$ to be chosen later. Let
$C\doteq X\llcorner_{t=1}$ be a permutation, which we write by Theorem (4.3)

$C=(\kappa^{1}_{1}\dots\kappa^{1}_{k_{1}})(\kappa^{2}_{1}\dots\kappa^{2}_{k_{2}})\dots(\kappa^{n}_{1}\dots\kappa^{n}_{k_{n}})=c_{1}\dots
c_{n},$

where $\sum_{i=1}^{n}k_{i}\leq D^{2}$. Define $c_{n+1},\dots c_{N}$,
$N=D^{2}-\sum_{i}k_{i}+n$, the $1$-cycles representing the subsquares that are
sent into themselves. By the previous lemma we can also assume that
$C\llcorner_{c_{i}}$ $i=1,\dots,N$ is a cyclic permutation of subsquares
$a^{i}_{jk}$, $j=1,\dots,M^{2}$, of the grid
$\mathbb{N}\times\mathbb{N}\frac{1}{MD}$. To find a $D^{2}M^{2}$ cycle we
should consider all the couples of adjacent subsquares (of size
$\frac{1}{MD}$), and then we should connect them by transpositions in a
precise way.

Fix $c_{1}$ and consider

$C^{1}=\\{c_{h}\not=c_{1}\ \text{ s.t. }c_{h}\text{ adjacent to
}c_{1}\\}=\\{c^{1}_{1},\dots,c^{1}_{|C^{1}|}\\}.$

Now for every $c^{1}_{j}\in C^{1}$ define by induction the disjoint families
of cycles

$C^{2}_{j}=\\{c_{h}\notin\\{c_{1}\\}\cup C^{1}\cup C^{2}_{1}\cup\dots\cup
C^{2}_{j-1}\text{ s.t. }c_{h}\text{ is adjacent to }c^{1}_{j}\\},$

and call

$C^{2}=C^{2}_{1}\cup\dots\cup
C^{2}_{|C^{1}|}=\\{c^{2}_{1},\dots,c^{2}_{|C^{2}|}\\}.$

At the $i-1$-th step we have

$C^{i-1}=\\{c^{i-1}_{1},\dots,c^{i-1}_{|C^{i}|}\\}.$

and, for every $c^{i-1}_{j}\in C^{i-1}$,

$C^{i}_{j}=\\{c_{h}\notin\\{c_{1}\\}\cup C^{1}\cup C^{2}\cup\dots\cup
C^{i-1}\cup C^{i}_{1}\cup\dots C^{i}_{j-1}\text{ s.t. }c_{h}\text{ is adjacent
to }c^{i-1}_{j}\\}.$

The procedure ends when we have arranged all $c_{i}$ into sets $C^{i}$, and
hence for some $K\in\mathbb{N}$ we obtain $C^{K+1}=\emptyset$ (see Figure 4).
Indeed, by contradiction assume that $|\\{c_{1}\\}\cup C^{1}\cup
C^{2}\cup\dots\cup C^{K}|<N$. Then this set has a boundary, i.e. there exists
a cycle $c\not\in\\{c_{1}\\}\cup C^{1}\cup C^{2}\cup\dots\cup C^{K}$ adjacent
to a cycle of $\\{c_{1}\\}\cup C^{1}\cup C^{2}\cup\dots\cup C^{K}$, which is a
contradiction by definition.

The partition

$C(c_{1})=\\{c_{1}\\}\cup C^{2}\cup\dots C^{K}$

has the natural structure of a directed tree: indeed every two cycles
$c_{i}\in C^{i}$, $c_{j}\in C^{j}$ are connected by a unique sequence of
cycles: the direction of each edge is given by the construction
$c^{i-1}_{j}\to c_{h}$ whenever $c_{h}\in C^{i}_{j}$. This tree-structure
gives us a selection of the $N-1$ couples of subsquares $\kappa_{i}^{j}$ of
disjoint cycles $c_{i}$ in which we can perform a transposition among the
subsquares $a^{i}_{jk}$ to connect all of them in a unique $D^{2}M^{2}$-cycle.
More precisely, for every connected couple $c^{i-1}_{j},c_{h}$ such that
$c_{h}\in C^{i}_{j}$, there exist cubes $\kappa\in
c^{i-1}_{j},\kappa^{\prime}\in c_{h}$, and hence there are adjacent subsquares
$a\subset\kappa,a^{\prime}\subset\kappa^{\prime}$ of size $1/(MD)$: assuming
$M\geq 4$, we can take $a,a^{\prime}$ not being on the corners of
$\kappa,\kappa^{\prime}$, respectively.

Let $T_{t}:[0,\delta]\times K\rightarrow K$ be the transposition flow (4.1)
acting in the selected $N-1$ couples of subsquares $a,a^{\prime}$
reparametrized on the time interval $[0,\delta]$ and define (being
$X_{t}=\mathrm{id}$ for $t\in[0,\delta]$)

$X^{c}_{t}(x)=\begin{cases}T_{t}(x)&t\in[0,\delta],\\\ X_{t}\circ
T_{\delta}(x)&t\in[\delta,1].\end{cases}$ (4.3)

The transposition is well defined: indeed it can happen that
$c_{i},c_{j},c_{k}$ are adjacent cycles and the couples of adjacent squares
(of size $\frac{1}{D}$) are $\kappa_{i},\kappa_{j}$ and
$\kappa_{i},\kappa_{k}$ (where $\kappa_{i}\in c_{i},\kappa_{j}\in c_{j}$ and
$\kappa_{k}\in c_{k}$), that is: $\kappa_{i}$ is in common. But since the
transposition occurs between subsquares of size $\frac{1}{DM}$ nor belonging
to the corners, it is always guaranteed that the transpositions act on
disjoint subsquares. By using the explicit formula (2.4) we get that for
$t\in[0,\delta]$

$\begin{split}||b^{c}_{t}-b_{t}||_{1}&\leq\frac{\mathcal{O}(1)}{\delta}\left(\frac{N-1}{D^{3}M^{3}}\right)\leq\frac{\mathcal{O}(1)}{\delta}\left(\frac{1}{DM^{3}}\right),\end{split}$

while for $t\in[\delta,1]$ it clearly holds $b^{c}_{t}=b_{t}$. Coupling these
last two estimates we get the $L^{1}_{t}L^{1}_{x}$ estimate:

$||b-b^{c}||_{L^{1}(L^{1})}\leq\mathcal{O}\left(\frac{1}{DM^{3}}\right),$

for $\delta<<1$ and $M$ sufficiently large.

Next we compute the total variation for $t\in[0,\delta]$: by using (4.2), we
get

$\displaystyle\text{\rm Tot.Var.}(b^{c}_{t})(K)$
$\displaystyle\leq\frac{N-1}{\delta}\frac{20}{M^{2}D^{2}}\leq\frac{1}{\delta}\frac{20}{M^{2}},$

while for $t\in[\delta,1-\delta]$ we find

$\text{\rm Tot.Var.}(b^{c}_{t})(K)=\text{\rm Tot.Var.}(b_{t})(K),$

therefore

$||\text{\rm
Tot.Var.}(b^{c}_{t}-b_{t})(K)||_{\infty}\leq\mathcal{O}\left(\frac{1}{\delta
M^{2}}\right).$

To conclude we have to prove that $X^{c}_{1}$ is a unique cycle, which follows
by the tree-structure of the selection of adjacent cycles. The end points of
the tree are clearly cycles. By recurrence, assume that $c^{i-1}_{j}$ is
connected to cycles $\gamma_{h}$, each one made of all squares belonging to
$c_{h}\in C^{i}_{j}$ and all subsequent cycles to $c_{h}$. It is fairly easy
to see that the transposition merging $c^{i-1}_{j}$ to each $c_{h}\in
C^{i}_{j}$ generates a unique cycles $\gamma^{i}_{j}$, made of the cubes of
$c^{i-1}_{j}$ and all $\gamma_{h}$. We thus conclude that the map $X^{c}_{1}$
is a cycle of size $M^{2}D^{2}$. ∎

$\kappa^{1}_{1}$
---
$\kappa^{3}_{1}$
---
$\kappa^{1}_{2}$
---
$\kappa^{1}_{4}$
---
$\kappa^{3}_{2}$
---
$\kappa^{3}_{3}$
---
$\kappa^{1}_{5}$
---
$\kappa^{1}_{3}$
---
$\kappa^{2}_{7}$
---
$\kappa^{2}_{1}$
---
$\kappa^{2}_{2}$
---
$\kappa^{2}_{4}$
---
$\kappa^{2}_{3}$
---
$\kappa^{4}_{1}$
---
$\kappa^{5}_{2}$
---
$\kappa^{5}_{11}$
---
$\kappa^{3}_{4}$
---
$\kappa^{3}_{8}$
---
$\kappa^{5}_{9}$
---
$\kappa^{5}_{1}$
---
$\kappa^{5}_{7}$
---
$\kappa^{5}_{8}$
---
$\kappa^{4}_{6}$
---
$\kappa^{5}_{6}$
---
$\kappa^{5}_{10}$
---
$\kappa^{4}_{4}$
---
$\kappa^{3}_{5}$
---
$\kappa^{8}_{2}$
---
$\kappa^{8}_{1}$
---
$\kappa^{5}_{13}$
---
$\kappa^{4}_{5}$
---
$\kappa^{4}_{2}$
---
$\kappa^{8}_{7}$
---
$\kappa^{10}_{1}$
---
$\kappa^{8}_{6}$
---
$\kappa^{8}_{5}$
---
$\kappa^{6}_{9}$
---
$\kappa^{8}_{9}$
---
$\kappa^{8}_{3}$
---
$\kappa^{3}_{6}$
---
$\kappa^{6}_{1}$
---
$\kappa^{8}_{10}$
---
$\kappa^{5}_{3}$
---
$\kappa^{6}_{11}$
---
$\kappa^{6}_{8}$
---
$\kappa^{3}_{7}$
---
$\kappa^{8}_{8}$
---
$\kappa^{6}_{7}$
---
$\kappa^{6}_{12}$
---
$\kappa^{9}_{1}$
---
$\kappa^{6}_{5}$
---
$\kappa^{8}_{4}$
---
$\kappa^{5}_{4}$
---
$\kappa^{2}_{5}$
---
$\kappa^{4}_{3}$
---
$\kappa^{6}_{9}$
---
$\kappa^{5}_{12}$
---
$\kappa^{5}_{5}$
---
$\kappa^{2}_{6}$
---
$\kappa^{6}_{2}$
---
$\kappa^{6}_{10}$
---
$\kappa^{7}_{1}$
---
$\kappa^{6}_{3}$
---
$\kappa^{6}_{4}$
---
Figure 4: Concatenation of cycles in a specific example, Remark 4.6. The
orange subrectangles are the couples of $a^{i}_{jk}$ on which the
transposition $T_{t}$ of (4.3) acts.

######  Remark 4.6.

An example of how the proof works is in Figure 4: the decomposition in cycles
is

$C={\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}(\kappa_{1}^{1}\dots\kappa^{1}_{5})}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\kappa^{2}_{1}\dots\kappa^{2}_{7})}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(\kappa^{3}_{1}\dots\kappa^{3}_{8})}{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\kappa^{4}_{1}\dots\kappa^{4}_{6})}{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}(\kappa^{5}_{1}\dots\kappa^{5}_{13})}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}(\kappa^{6}_{1}\dots\kappa^{2}_{12})}(\kappa^{7}_{1}){\color[rgb]{1,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,0}\pgfsys@color@cmyk@stroke{0}{0}{1}{0}\pgfsys@color@cmyk@fill{0}{0}{1}{0}(\kappa^{8}_{1}\dots\kappa^{8}_{10})}(\kappa^{9}_{1})(\kappa^{10}_{1}).$

The black arrow indicates the adjacent subsquares where the exchanges are
performed: the tree of concatenation is then

${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}(\kappa_{1}^{1}\dots\kappa^{1}_{5})}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\kappa^{2}_{1}\dots\kappa^{2}_{7})}$${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}(\kappa^{6}_{1}\dots\kappa^{2}_{12})}$$(\kappa^{7}_{1})$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(\kappa^{3}_{1}\dots\kappa^{3}_{8})}$${\color[rgb]{1,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,0}\pgfsys@color@cmyk@stroke{0}{0}{1}{0}\pgfsys@color@cmyk@fill{0}{0}{1}{0}(\kappa^{8}_{1}\dots\kappa^{8}_{10})}$${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\kappa^{4}_{1}\dots\kappa^{4}_{6})}$$(\kappa^{9}_{1})$${\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}(\kappa^{5}_{1}\dots\kappa^{5}_{13})}$

Note that in the subsquares
$(\kappa^{1}_{2},\kappa^{2}_{1}),(\kappa^{1}_{2},\kappa^{3}_{1})$ and
$(\kappa^{1}_{5},\kappa^{4}_{1}),(\kappa^{1}_{5},\kappa^{5}_{1})$ the exchange
occurs actually in the subsquares
$(a^{1}_{2j},a^{2}_{1\ell}),(a^{1}_{2j^{\prime}},a^{3}_{1\ell^{\prime}})$ and
$(a^{1}_{5k},a^{4}_{1\ell^{\prime\prime}}),(a^{1}_{5k^{\prime}},a^{5}_{1,\ell^{\prime\prime\prime}})$,
so that it is always acting on different couples of subsquares.

######  Remark 4.7.

The construction of the cyclic flow (4.3) gives us only the
$L^{1}_{t}L^{1}_{x}$ estimate on the vector fields, which is what we need for
our genericity result. We can get the more refined estimate in
$L^{\infty}_{t}L^{1}_{x}$ allowing for mass flowing (when performing the
transposition) during the time evolution of the flow $X_{t}$ (see Figure 5).

Figure 5: The two adjacent cycles $c_{1}$ (light green) and $c_{2}$ (blue)
touch in $\kappa_{1}$ and $\kappa_{2}$, which exchange their mass during the
time evolution.

In this case, the time spent by the squares of size $(MD)^{-1}$ to transfer
the mass is of order $(MD)^{-1}$, so that the vector field moving it should be
of the order

$\frac{\text{length}}{\text{time}}=\mathcal{O}(1),\quad\text{acting on a
region of area}\ \frac{N-1}{(MD)^{2}}\leq M^{-2}.$ (4.4)

Hence the $L^{\infty}_{t}L^{1}_{x}$ estimate can be obtained by (2.2) as

$\|b^{c}_{t}-b_{t}\|_{1}\leq\mathcal{O}(1)M^{-2},$

while the total variation estimate becomes

$\text{\rm Tot.Var.}(b^{c}_{t}-b_{t})=\mathcal{O}(1)\frac{D}{M}.$

The statement one can prove is then the following.

######  Proposition 4.8.

Let $b\in L^{\infty}_{t}(\operatorname{BV}_{x})$ be a divergence-free vector
field and assume that its flow at time $t=1$, namely $X\llcorner_{t=1}$, is a
permutation of squares of the grid $\mathbb{N}\times\mathbb{N}\frac{1}{D}$
where $D\in\mathbb{N}$. Then for every $\epsilon>0$ there exist $M=2^{p}$ and
$b^{c}\in L^{\infty}_{t}(\operatorname{BV}_{x})$ a divergence-free vector
field such that

$||b^{c}_{t}-b_{t}||_{L^{1}}\leq\mathcal{O}(M^{-2})\leq\epsilon,\quad\text{\rm
Tot.Var.}(b^{c}_{t}-b_{t})(K)\leq\mathcal{O}(D/M),$

and the map $X^{c}_{t=1}:K\rightarrow K$, being $X^{c}_{t}:[0,1]\times
K\rightarrow K$ the flow associated with $b^{c}$, is a $M^{2}D^{2}$-cycle of
subsquares of size $\frac{1}{DM}$.

###  4.2 Density of ergodic vector fields

Starting from the cyclic permutation we have built in the previous section, we
construct an ergodic vector field arbitrarily close to a given vector field in
$L^{\infty}_{t}\operatorname{BV}_{x}$. The density of ergodic vector fields is
not strictly relevant for the genericity result of weakly mixing vector
fields, but it can be considered as a simple case study for the construction
of strongly mixing vector fields. Moreover it will give a direct proof of
Point (2) of Theorem 1.2.

We will use the _universal mixer_ that has been constructed in [9]: it is the
time periodic divergence-free vector field $u\in
L^{\infty}_{t}([0,1],BV_{x}(\mathbb{R}^{2}))$ whose flow $U_{t}:[0,1]\times
K\rightarrow K$ of measure-preserving maps realizes at time $t=1$ the _folded
Baker’s map_ , that is

$U=U\llcorner_{t=1}=\begin{cases}\left(-2x+1,-\frac{y}{2}+\frac{1}{2}\right)&x\in\left[0,\frac{1}{2}\right),\\\
\left(2x-1,\frac{y}{2}+\frac{1}{2}\right)&x\in\left(\frac{1}{2},1\right],\end{cases}\qquad
y\in[0,1],$ (4.5)

(see Theorem $1$, [9]).

######  Proposition 4.9.

Let $b\in L^{\infty}_{t}(\operatorname{BV}_{x})$ and let $X_{t}$ be its RLF,
and assume that $X_{t=1}$ is a cyclic permutation of squares of the grid
$\mathbb{N}\times\mathbb{N}\frac{1}{D}$. Then there exists $b^{e}\in
L^{\infty}_{t}(\operatorname{BV}_{x})$ divergence-free ergodic vector field
such that

$\begin{split}&||b-b^{e}||_{L^{\infty}(L^{1})}\leq\mathcal{O}\left(\frac{1}{D^{2}}\right),\\\
&||\text{\rm Tot.Var.}(b^{e})(K)||_{\infty}\leq||\text{\rm
Tot.Var.}(b)(K)||_{\infty}+\mathcal{O}\left(\frac{1}{D^{2}}\right).\end{split}$
(4.6)

###### Proof.

Let us call $T=X\llcorner_{t=1}$ and $\kappa_{1},\dots,\kappa_{D^{2}}$ the
subsquares of the grid where the numbering is chosen such that

$T(\kappa_{i})=\kappa_{i+1},\quad T(\kappa_{n})=\kappa_{1}.$

Let us define

$X^{e}_{t}=\begin{cases}X_{t}\circ U^{1}_{t}&x\in\kappa_{1},\\\
X_{t}&\text{otherwise},\end{cases}$

where the flow $U^{1}_{t}=\theta^{-1}\circ U_{t}\circ\theta$ and $\theta$ is
the affine map from $\kappa_{1}$ to $K$, i.e. $\theta(x,y)=(Dx,Dy)$.

We first prove the ergodicity of $T^{e}=X^{e}\llcorner_{t=1}$. Assume by
contradiction that $T^{e}$ is not ergodic, then there exists a measurable set
$B$ such that $T^{e}(B)=B$ and $0<|B|<1$. We claim that $|B\cap\kappa_{1}|>0$.
Indeed, since $|B|>0$ there exists $i$ such that $|B\cap\kappa_{i}|>0$. If
$i=1$ we have nothing to prove, if not, since $T^{e}$ is measure-preserving,
then $|T^{e}(B\cap\kappa_{i})|>0$. But

$0<|T^{e}(B\cap\kappa_{i})|=|T^{e}(B)\cap
T^{e}(\kappa_{i})|=|B\cap\kappa_{i+1}|$

(we have used that the set $B$ is invariant) and re-applying the map $T^{e}$
sufficiently many times we have the claim. Moreover,
$|B\cap\kappa_{1}|<\frac{1}{D^{2}}$. If not, that is
$|B\cap\kappa_{1}|=\frac{1}{D^{2}}$, then $|B\cap\kappa_{i}|=\frac{1}{D^{2}}$
for every $i=1,\dots,D^{2}$, again by using the fact that $B$ is invariant and
that $T^{e}(\kappa_{i})=\kappa_{i+1}$ and $T^{e}(\kappa_{D^{2}})=\kappa_{1}$.
But now

$|B|=\sum_{i=1}^{D^{2}}|B\cap\kappa_{i}|=\sum_{i=1}^{D^{2}}\frac{1}{D^{2}}=1,$

which is a contradiction, since $|B|<1$. Now, the fact that
$0<|B\cap\kappa_{1}|<\frac{1}{D^{2}}$ implies that
$U^{1}_{1}(B\cap\kappa_{1})\not=B\cap\kappa_{1}$ because $U^{1}_{1}$ is mixing
(and thus ergodic). But this is a contradiction because
$T^{e}(B\cap\kappa_{1})=B\cap\kappa_{2}$ and applying to both of them
$T^{D^{2}-1}$ we find that

$U^{1}_{1}(B\cap\kappa_{1})=T^{D^{2}-1}(T^{e}(B\cap\kappa_{1}))=T^{D^{2}-1}(B\cap\kappa_{2})=(T^{e})^{D^{2}-1}(B\cap\kappa_{2})=B\cap\kappa_{1},$

where we have used that $T^{D^{2}}=Id$. To prove the estimates (4.6) we have
to observe first that $U_{t}$ acts only on $\kappa_{1}$, then that it is the
composition of two rotations (see [Figure 1, [9]]), that is $\text{\rm
Tot.Var.}(\dot{U}_{t}(U^{-1}_{t}))(\bar{\kappa}_{1})\leq\mathcal{O}\left(\frac{1}{D^{2}}\right)$
(see again Lemma 2.6). ∎

###  4.3 Density of strongly mixing vector fields

As in the previous section, we use the density of cyclic permutations to show
that the vector fields whose flow is strongly mixing are dense in
$\mathcal{U}$ with the $L^{1}_{t,x}$-topology. Again we use the universal
mixer constructed in [9]. The main result here is the following

######  Proposition 4.10.

Let $b\in L^{\infty}_{t}(\operatorname{BV}_{x})$ and let $X_{t}$ be its RLF,
and assume that $b_{t}=0$ for $t\in[0,2\delta]$ and $X_{t=1}$ is a cyclic
permutation of squares of the grid $\mathbb{N}\times\mathbb{N}\frac{1}{D}$,
$D=2^{p}$. Then there exists $b^{s}\in L^{\infty}_{t}(\operatorname{BV}_{x})$
divergence-free strongly mixing vector field such that

$\begin{split}&||b-b^{s}||_{L^{1}(L^{1})}\leq\mathcal{O}\left(\frac{1}{\delta
D}\right),\\\ &||\text{\rm Tot.Var.}(b^{s})(K)||_{\infty}\leq||\text{\rm
Tot.Var.}(b)(K)||_{\infty}+\mathcal{O}\left(\delta^{-1}\right).\end{split}$
(4.7)

In the proof it is shown that the mixing is actually exponential, in the sense
that for every set in a countable family of sets $\\{B_{i}\\}_{i}$ generating
the Borel $\sigma$-algebra it holds

$\big{|}T^{q}(B_{i})\cap
B_{j}\big{|}-|B_{i}||B_{j}|=\mathcal{O}(1)c_{ij}^{q},\quad c_{ij}<1.$

###### Proof.

Let us call $T=X\llcorner_{t=1}$ and $\kappa_{1},\dots,\kappa_{D^{2}}$ the
subsquares of the grid where the numbering is chosen such that

$T(\kappa_{i})=\kappa_{i+1},\quad T(\kappa_{D^{2}})=\kappa_{1}.$

If $\\{1,\dots,D^{2}\\}\ni\ell\mapsto j(\ell)\in\\{1,\dots,D^{2}\\}$ is an
enumeration of $\kappa_{i}$ such that $\kappa_{j(\ell)},\kappa_{j(\ell+1)}$
are adjacent, consider the rescaled universal mixer $U^{\ell,\ell+1}_{t}$
acting on $\kappa_{\ell},\kappa_{\ell+1}$ in the time interval $[0,\delta]$,
whose generating vector field $b^{U^{\ell,\ell+1}}$ satisfies the estimates

$\|b^{U^{\ell,\ell+1}}_{t}\|_{L^{1}}=\mathcal{O}(1)\frac{1}{\delta}\frac{1}{D^{3}},\quad\text{\rm
Tot.Var.}(b^{U^{\ell,\ell+1}}_{t})=\mathcal{O}(1)\frac{1}{\delta}\frac{1}{D^{2}}.$

The idea is to define the a new vector field as in (4.3)

$X^{s}_{t}(x)=\begin{cases}M_{t}(x)&t\in[0,2\delta],\\\ X_{t}\circ
M_{2\delta}(x)&t\in[2\delta,1],\end{cases}$

where the map $M_{t}$, $t\in[0,2\delta]$, is defined as follows:

$M_{t}(x)=\begin{cases}U^{\ell,\ell+1}_{t}(x)&t\in[0,\delta],\ell\
\text{even},\\\ U^{\ell,\ell+1}_{t}(x)&t\in[\delta,2\delta],\ell\
\text{odd}.\end{cases}$

The estimates (4.7) follows as in Proposition 4.9, so we are left with the
proof that $T^{s}=X^{s}_{1}$ is strongly mixing.

The map $T^{s}$ is the composition of $3$ maps $T_{3}\circ T_{2}\circ T_{1}$
acting as follows (all indexes should be indended modulus $D^{2}$):

  1. 1.

$T_{1}$ is the folded Baker’s map $U$ acting on the couples $\ell,\ell+1$,
$\ell=0,2,\dots$ even;

  2. 2.

$T_{2}$ is the folded Baker’s map $U$ acting on the couples $\ell,\ell+1$,
$\ell=1,3,\dots$ odd;

  3. 3.

$T_{3}$ is a cyclic permutation $\ell\to j^{-1}(j(\ell)+1)$.

We first compute the evolution of a rectangle $a$ of the form

$a=2^{-p}[k,k+1]\times
2^{-p^{\prime}}[k^{\prime},k^{\prime}+1]\frac{1}{D},\quad k=0,\dots,2^{p}D-1,\
k^{\prime}=0,\dots,2^{p^{\prime}}D-1,\ p,p^{\prime}\in\mathbb{N}.$

By definition of $U$ (4.5) we obtain that if $p\geq 1$ then the map $T_{1}$
does not split $a$ into disjoint rectangles, i.e.

$T_{1}a=2^{1-p}[\tilde{k},\tilde{k}+1]\times
2^{-p^{\prime}-1}[\tilde{k}^{\prime},\tilde{k}^{\prime}+1]\frac{1}{D},\quad\tilde{k}=0,\dots,2^{p-1}D-1,\tilde{k}^{\prime}=0,\dots,2^{p^{\prime}+1}D-1,$

and the same happens for $T_{2}$:

$T_{2}a=2^{1-p}[\hat{k},\hat{k}+1]\times
2^{-p^{\prime}-1}[\hat{k}^{\prime},\hat{k}^{\prime}+1]\frac{1}{D},\quad\hat{k}=0,\dots,2^{p-1}D-1,\hat{k}^{\prime}=0,\dots,2^{p^{\prime}+1}D-1.$

Hence if

$a=2^{-2p}[k,k+1]\times
2^{-2p^{\prime}}[k^{\prime},k^{\prime}+1]\frac{1}{D},\quad
k=0,\dots,2^{2p}D-1,\ k^{\prime}=0,\dots,2^{2p^{\prime}}D-1,\
p,p^{\prime}\in\mathbb{N},$ (4.8)

then

$T_{2}\circ T_{1}a=2^{2(1-p)}[\check{k},\check{k}+1]\times
2^{-2(p^{\prime}+1)}[\check{k}^{\prime},\check{k}^{\prime}+1]\frac{1}{D},\quad\check{k}=0,\dots,2^{2(p-1)}D-1,\check{k}^{\prime}=0,\dots,2^{2(p^{\prime}+1)}D-1,$

and being the action of $T_{3}$ just a permutation, the final form
$T^{s}a=T_{3}\circ T_{2}\circ T_{1}a$ is again a rectangle.
When $p=0$, instead the rectangle $a$ is mapped into two rectangles belonging
to two different subsquares $\kappa,\kappa^{\prime}$

$T_{1}a=[\tilde{k}_{1},\tilde{k}_{1}+1]\times
2^{-p^{\prime}-1}[\tilde{k}_{1}^{\prime},\tilde{k}_{1}^{\prime}+1]\frac{1}{D}\cup[\tilde{k}_{2},\tilde{k}_{2}+1]\times
2^{-p^{\prime}-1}[\tilde{k}_{2}^{\prime},\tilde{k}_{2}^{\prime}+1]\frac{1}{D},$

and the action of $T_{2}$ divides $T_{1}a$ into 4 rectangles of horizontal
length $1/D$ belonging to 4 different subsquares. As before, $T_{3}$ just
shuffles them into new locations.
The same happens when considering $(T^{s})^{-1}$: if $p^{\prime}\leq 1$ and
$a$ is given by (4.8) then $(T^{s})^{-1}a$ is still a rectangle of side
$2^{-2(p+1)}\times 2^{-2(p^{\prime}-1)}\frac{1}{D}$, while for $p^{\prime}=0$
it is split into 4 rectangles with vertical size equal to $1/D$.

In particular, starting from two squares $a,a^{\prime}$ of side
$(2^{-2p}D)^{-2}$, for $q\geq p$ the set $(T^{s})^{q}a$ is made of disjoint
rectangles whose horizontal side is $D^{-1}$, and $(T^{s})^{-q}a^{\prime}$ is
made of disjoint rectangles whose vertical side is $D^{-1}$. Hence if the
masses of $(T^{s})^{q}a$, $(T^{s})^{-q^{\prime}}a^{\prime}$ inside
$\kappa_{i}$ are $m_{i}(q)$, $m_{i}^{\prime}(-q^{\prime})$, then by Fubini

$\mathcal{L}^{2}\big{(}(T^{s})^{q}a\cap(T^{s})^{-q^{\prime}}a^{\prime}\big{)}=\sum_{i=1}^{D^{2}}D^{2}m_{i}(q)m^{\prime}_{i}(-q^{\prime}).$

In order to prove the strong mixing it is enough to show that

$m_{i}(q)\to\frac{\mathcal{L}^{2}(a)}{D^{2}},\
m^{\prime}_{i}(-q^{\prime})\to\frac{\mathcal{L}^{2}(a^{\prime})}{D^{2}}\quad
q,q^{\prime}\to\infty.$

Actually, we will show that the above convergence is exponential, which
implies that the mixing is exponential. We prove the above exponential
convergence for $m_{i}(q)$, the other being completely similar.

Once $(T^{s})^{q}a$ has become a rectangle of horizontal side $1/D$, the
distribution of mass by $T^{s}$ is computed by the action of the following
matrices on the vector $(m_{i})_{i}$:

  1. 1.

the matrix $A_{1}$ corresponding to the map $T_{1}$,

$(A_{1})_{\ell^{\prime}\ell}=\frac{1}{2}\begin{cases}\delta_{\ell^{\prime}\ell}+\delta_{\ell^{\prime}(\ell-1)}&\ell^{\prime}=0,2,\dots,\\\
\delta_{\ell^{\prime}(\ell+1)}+\delta_{\ell^{\prime}\ell}&\ell^{\prime}=1,3,\dots;\end{cases}$

  2. 2.

the matrix $A_{2}$ corresponding to the map $T_{2}$,

$(A_{2})_{\ell^{\prime}\ell}=\frac{1}{2}\begin{cases}\delta_{\ell^{\prime}\ell}+\delta_{\ell^{\prime}(\ell+1)}&\ell^{\prime}=0,2,\dots,\\\
\delta_{\ell^{\prime}(\ell-1)}+\delta_{\ell^{\prime}\ell}&\ell^{\prime}=1,3,\dots;\end{cases}$

  3. 3.

the permutation matrix $A_{3}$ corresponding to $T_{3}$.

Being the Markov process generated by the matrix $P=A_{3}A_{2}A_{1}$ finite
dimensional, exponential mixing is equal to strong mixing, and we prove
directly that $P$ has a simple eigenvalue of modulus $1$ whose eigenvector is
necessarily the uniform distribution $(1/D^{2},1/D^{2},\dots)$: in particular
this gives that $P$ is aperiodic (Definition 3.13 and Proposition 3.15).
Indeed, for $v\in\mathbb{C}^{D}$ one considers the functional $|v|$, and by
simple computations it holds $|A_{3}v|=|v|$ and

$|A_{1}v|=|v|\quad\text{iff}\quad v_{\ell}=v_{\ell+1}\ \text{for}\
\ell=0,2,\dots,$ $|A_{2}v|=|v|\quad\text{iff}\quad v_{\ell}=v_{\ell+1}\
\text{for}\ \ell=1,3,\dots.$

Hence the unique $v$ such that $|Av|=|v|$ is $v=(1/D^{2},1/D^{2},\dots)$, and
$1$ is a simple eigenvector. ∎

######  Remark 4.11.

As in Remark 4.7, one could let the Bakers map to act during the time
evolution of $X_{t}$, but in this case the distance in $L^{\infty}L^{1}$ would
be of order $1$. The problem is that the maps $T_{1},T_{2}$ are acting on the
whole set $K=[0,1]^{2}$, and the vector field $b^{s}_{t}-b_{t}$ is of order
$1$ as in (4.4).

###  4.4 Proof of the density of strongly mixing vector fields

We are now ready to prove the density of strongly mixing vector fields in
$\mathcal{U}$, which implies the statement by Corollary 3.10. It will be
obtained through the following steps.

  1. 1.

Let $b\in\mathcal{U}$: by the very construction of the set $\mathcal{U}$
(Proposition 2.3), we can assume that $b\in
L^{\infty}_{t}\operatorname{BV}_{x}$. Fix $\epsilon>0$.

  2. 2.

By the continuity of translation in $L^{1}$, we can take $0<\delta\ll 1$ such
that defining

$b^{\delta}=\begin{cases}0&t\in[0,3\delta),\\\
\frac{1}{1-3\delta}b_{(t-3\delta)/(1-3\delta)}&t\in[3\delta,1],\end{cases}$

it holds

$\|b^{\delta}-b\|_{L^{1}_{t,x}}<\frac{\epsilon}{4}.$

Since

$\|\text{\rm Tot.Var.}(b^{\delta})\|_{\infty}=\frac{1}{1-3\delta}\|\text{\rm
Tot.Var.}(b)\|_{\infty}$

then $b^{\delta}\in\mathcal{U}$. Clearly we can also assume that $b^{\delta}$
is compactly supported in $K$.

  3. 3.

Use Theorem 5.14 to approximate $b^{\delta}$ in $[3\delta,1]$ with a vector
field $b^{\epsilon\delta}\in
L^{\infty}_{t}\operatorname{BV}_{x}\subset\mathcal{U}$ such that

$\|b^{\delta}-b^{\epsilon\delta}\|_{L^{1}_{t,x}}<\frac{\epsilon}{4},$

and such that its RLF is a permutation of squares of size $D^{-1}$. We can
assume that

$D\gg\frac{1}{\epsilon\delta}.$ (4.9)

  4. 4.

Apply Lemma 4.4 together with Proposition 4.5 to $b^{\epsilon\delta}$ for
$t\in[2\delta,1]$ obtaining a new vector field $b^{\epsilon\delta c}\in
L^{\infty}_{t}\operatorname{BV}_{x}\subset\mathcal{U}$ such that

$\|b^{\epsilon\delta}-b^{\epsilon\delta
c}\|_{L^{1}_{t,x}}\leq\mathcal{O}\bigg{(}\frac{1}{DM}\bigg{)}<\frac{\epsilon}{4}$

for $M=2^{p^{\prime}}\gg 1$, and such that its RLF is a single cycle of
squares of size $(DM)^{-1}$.

  5. 5.

Finally, apply Proposition 4.10 to $b^{\epsilon\delta c}$ in $t\in[0,1]$
obtaining a strongly (exponentially) mixing vector field $b^{\epsilon\delta
cs}\in L^{\infty}_{t}\operatorname{BV}_{x}\subset\mathcal{U}$ such that

$\|b^{\epsilon\delta c}-b^{\epsilon\delta
cs}\|_{L^{1}_{t,x}}\leq\mathcal{O}\bigg{(}\frac{1}{\delta
D}\bigg{)}<\frac{\epsilon}{4}$

by using (4.9).

We thus conclude that for every $b\in L^{\infty}_{t}\operatorname{BV}_{x}$ and
$\epsilon>0$ there is a vector field $b^{s}\in
L^{\infty}_{t}\operatorname{BV}_{x}$ exponentially mixing such that

$\|b-b^{s}\|_{L^{1}_{t,x}}<\epsilon,$

which is our aim.

##  5 Permutation Flow

In this section we prove the key tool of this paper, namely the approximation
in $L^{1}$ of any BV vector field with another BV vector field such that its
flow at $t=1$ is a permutation of subsquares, i.e. it is a rigid translation
of subsquares of a grid partition of $K=[0,1]^{2}$. The approach is inspired
by [22], with the additional difficulty that we need to control the BV norm of
the approximating vector field. We will address also the $d$-dimensional case,
explaining the additional technicalities needed to prove the same
approximation result in the general case.

This section is divided into two parts: in the first one we collect some
preliminary estimates which will be used as building blocks in the proof of
the main theorem, while in the second part we state the main approximation
theorem and give its proof.

###  5.1 Affine approximations of smooth flows

The next lemma is almost the same of [22, Lemma 4.3]. In order to follow the
original Shnirelman’s Lemma we require the subrectangles in the next lemma to
be dyadic (i.e. their corners belong to a dyadic partition, see Remark 5.3
however), but we notice that the proof of the main theorem works in the same
way just asking subrectangles with rational coordinates to be mapped affinely
onto subrectangles with rational coordinates. At the end this section we will
address the same lemma in the general case $d>2$, which in the original paper
is not proved.

Let $T$ be a measure-preserving diffeomorphism
$T:[0,1]^{2}\rightarrow[0,1]^{2}$ of class $C^{3}$ and such that
$T=\mathrm{id}$ in a neighborhood of $\partial[0,1]^{2}$. Assume that it is
close to the identity, i.e. there exists $\delta>0$ sufficiently small such
that $||T-\mathrm{id}||_{\mathcal{C}^{1}}\leq\delta$.

######  Lemma 5.1.

There exists $N\in\mathbb{N}$, $N=2^{p}$, and a path of measure-preserving
invertible maps $t\rightarrow\sigma_{t}$ piecewise smooth w.r.t. the time
variable $t$ such that $\sigma_{0}=T$ and $\sigma_{1}$ maps arbitrarily small
dyadic rectangles $P_{ij}\in\mathbb{N}\times\mathbb{N}\frac{1}{N}=K_{N}$
(meaning that their boundaries are in the net $K_{N}$) affinely onto dyadic
rectangles $\tilde{P}_{ij}\in K_{N}$.

Moreover, the map $\sigma$ is of the form

$\sigma_{t}=T\circ\xi_{3t}1\\!\\!\mathrm{I}_{[0,1/3]}(t)+\zeta_{3t-1}\circ
T\circ\xi_{1}1\\!\\!\mathrm{I}_{[1/3,2/3]}(t)+\eta_{3t-2}\circ\zeta_{1}\circ
T\circ\xi_{1}1\\!\\!\mathrm{I}_{[2/3,1]}(t).$ (5.1)

where $\xi,\eta:[0,1]\times[0,1]^{2}\to[0,1]^{2}$ are piecewise smooth and
$\zeta:[0,1]\times[0,1]^{2}\to[0,1]^{2}$ is smooth, so that for every
$t\in[0,1]$, the map $\sigma_{t}$ is piecewise smooth on each subrectangle
$\kappa$ and it extends continuosly on $\bar{\kappa}$.
Finally, the space differential $D\sigma_{1}$ of $\sigma_{t=1}$ is a constant
diagonal matrix in each subrectangle.

The number $N$ is used in the next results in order to have that the
perturbation is arbitrarily small in $L^{1}_{t,x}$.

###### Proof.

The proof is given in $3$ steps:

  1. 1.

first by an arbitrarily small perturbation of the final configuration we make
sure the area of the regions which will be mapped into rectangles is dyadic;

  2. 2.

secondly we perturb along horizontal slabs in order to have that vertical
sections of the slabs are mapped into vertical segments;

  3. 3.

finally we perturb vertical slabs so that the image of particular rectangles
are rectangles and vertical segments remains vertical segments.

The composition of all 3 maps with $T$ as in (5.1) will be the movement
$\sigma_{t}$. We will use the notation

$[0,1]^{2}\ni(x_{1},x_{2})\mapsto T(x_{1},x_{2})=(z_{1},z_{2})\in[0,1]^{2}$

to avoid confusion between the final coordinates and the initial ones. When
piecing together maps which are defined in closed sets with piecewise regular
boundaries, we will neglect the negligible superposition of boundaries for
simplicity: this slight inaccuracy should not generate confusion.

Step 0: initial grid and perturbation. For $N_{0}=2^{p_{0}}\gg 1$ define the
horizontal and vertical slabs

$H_{j}=[0,1]\times 2^{-p_{0}}[j-1,j],\quad
V_{i}=2^{-p_{0}}[i-1,i]\times[0,1],\quad i,j=1,\dots,2^{p_{0}}.$

The image of the horizontal lines

$x_{1}\mapsto T(x_{1},x_{2})$

can be written as graphs of functions

$z_{1}\mapsto g(z_{1},x_{2}),$

and divides every vertical slab $V_{i}$ into $N_{0}=2^{p_{0}}$ parts

$\tilde{\omega}_{ij}=\Big{\\{}(i-1)2^{-p_{0}}\leq z_{1}\leq
i2^{-p_{0}},g(z_{1},(j-1)2^{-p_{0}})\leq z_{2}\leq
g(z_{1},j2^{-p_{0}})\Big{\\}}.$

Let $\zeta_{t}:[0,1]^{2}\to[0,1]^{2}$ be a measure preserving flow, moving
mass across the boundary of $\tilde{\omega}_{ij}$: we can assume w.l.o.g that
the mass flow $\phi_{ij,i^{\prime}j^{\prime}}$ across the boundary from
$\tilde{\omega}_{ij}$ to $\tilde{\omega}_{i^{\prime}j^{\prime}}$ occurs in the
relative interior of
$\partial\tilde{\omega}_{ij}\cap\partial\tilde{\omega}_{i^{\prime}j^{\prime}}$.
The measure preserving condition requires that

$\phi_{ij,(i-1)j}+\phi_{ij,(i+1)j}+\phi_{ij,i(j+1)}+\phi_{ij,i(j-1)}=0.$

Set $T^{\prime}=\zeta_{1}\circ T$ and consider the new curves

$z_{1}\mapsto
g^{\prime}(z_{1},x_{2}),\quad\operatorname{Graph}g^{\prime}=T^{\prime}([0,1]\times\\{x_{2}\\}).$

Let $\tilde{\omega}^{\prime}_{ij}$ be the new regions

$\tilde{\omega}^{\prime}_{ij}=\Big{\\{}(i-1)2^{-p_{0}}\leq z_{1}\leq
i2^{-p_{0}},g^{\prime}(z_{1},(j-1)2^{-p_{0}})\leq z_{2}\leq
g^{\prime}(z_{1},j2^{-p_{0}})\Big{\\}},$

whose new area is

$\mathcal{L}^{2}(\tilde{\omega}^{\prime}_{ij})=\phi_{ij,i(j-1)}+\phi_{ij,i(j+1)}.$

Starting with $\tilde{\omega}^{\prime}_{11}$, we move a mass
$\phi_{11,12}<2^{-p_{0}-p_{1}}\ll 1$ so that

$\mathcal{L}^{2}(\tilde{\omega}^{\prime}_{11})=2^{-p_{0}-p_{1}}n_{11}\in
2^{-p_{0}-p_{1}}\mathbb{N}.$

Hence a mass $-\phi_{11,21}$ is flowing to the region $\tilde{\omega}_{21}$.
Assuming that we have

$\mathcal{L}^{2}(\tilde{\omega}^{\prime}_{i1})=2^{-p_{0}-p_{1}}n_{i1}\in
2^{-p_{0}-p_{1}}\mathbb{N},$

and that the mass flowing by $\phi_{i1,i2},\phi_{i1,(i+1)1}$ is $<2^{-p_{1}}$,
we consider two cases:

  1. 1.

if $\phi_{i1,(i+1)1}\in 2^{-p_{0}-p_{1}}[0,1)$, then we flow a mass
$\phi_{(i+1)1,(i+1)2}\in 2^{-p_{0}-p_{1}}[0,1)$ so that

$\mathcal{L}^{2}(\tilde{\omega}^{\prime}_{(i+1)1})=2^{-p_{0}-p_{1}}n_{(i+1)1}\in
2^{-p_{1}}\mathbb{N},$

and the flow to the right is then

$\phi_{(i+1)1,(i+2)i}=\phi_{i1,(i+1)i}-\phi_{(i+1)1,(i+1)2}\in(-1,1)2^{-p_{0}-p_{1}},$

by the balance and because they have different sign;

  2. 2.

if $\phi_{i1,(i+1)1}\in 2^{-p_{0}-p_{1}}(-1,0)$, then we flow a mass
$\phi_{(i+1)1,(i+1)2}\in 2^{-p_{0}-p_{1}}(-1,0)$ and obtain the same estimate.

The last term $\tilde{\omega}^{\prime}_{N_{0}1}$ is computed by conservation:
indeed

$\sum_{i}\phi_{i1,i2}=0,$

and then

$\mathcal{L}^{2}(T^{\prime}([0,1]\times[0,2^{-p_{0}}])=2^{-p_{0}}=\sum\mathcal{L}^{2}(\tilde{\omega}^{\prime}_{i1})=2^{-p_{0}-p_{1}}\sum_{i=0}^{2^{p_{0}}-1}n_{i1}+\mathcal{L}^{2}(\tilde{\omega}^{\prime}_{N_{0}1}),$

so that

$\mathcal{L}^{2}(\tilde{\omega}^{\prime}_{N_{0}1})=2^{-p_{0}-p_{1}}\bigg{(}2^{p_{1}}-\sum_{i=0}^{2^{p_{0}}-1}n_{i1}\bigg{)}\in
2^{-p_{0}-p_{1}}\mathbb{N}.$

The estimate of $\phi_{N_{0}1,N_{0}2}$ is automatic from the flow
$\phi_{(N_{0}-1)1,N_{0}1}$.

The above procedure is then repeated for each region

$T([0,1]\times 2^{-p_{0}}[j-1,j])=\bigcup_{i=1}^{N_{0}}\tilde{\omega}_{ij},$

and the flow across each boundary is $\leq 2^{-p_{0}-p_{1}}$: the conservation
of the measure of $T([0,1]\times[0,j]2^{-p_{0}})$ yields that the last element
$\tilde{\omega}_{N_{0}j}$ is again dyadic.

From now on we work with the map $T^{\prime}=\zeta_{1}\circ T$.

Step 1: perturbation along horizontal slabs. Consider the curves

$z_{2}\mapsto(T^{\prime})^{-1}(z_{1},z_{2}),$

which can be parameterized as

$x_{2}\mapsto f^{\prime}(z_{1},x_{2})$

being $T^{\prime}$ close to the identity in $C^{1}$. In each $H_{j}$ we can
determine uniquely the value

$x_{1,j}(z_{1})=\fint_{(j-1)2^{-p_{0}}}^{j2^{-p_{0}}}f^{\prime}(z_{1},x_{2})dx_{2},$
(5.2)

and since $T^{\prime}$ is close to identity, again every map $z_{1}\mapsto
x_{1,j}(z_{1})$ is invertible: denote its inverse by $z_{1,j}(x_{1})$.

In particular, we consider the values

$x_{1,ij}=x_{1,j}(i2^{-p_{0}}).$ (5.3)

By (5.2) it follows that

$\big{(}x_{1,ij}-x_{1,(i-1)j}\big{)}2^{-p_{0}}=\mathcal{L}^{2}\big{(}(T^{\prime})^{-1}(\tilde{\omega}^{\prime}_{ij})\big{)}\in
2^{-p_{0}-p_{1}}\mathbb{N},$ (5.4)

so that we deduce that the elements $x_{1,ij}$ are dyadic, i.e. $x_{1,ij}\in
2^{-p_{1}}\mathbb{N}$ (being $x_{1,0j}=0$).

Consider the family of ordered curves parametrized by $x_{1}\in[0,1]$

$[0,1]\times[j-1,j]2^{-p_{0}}\ni t,x_{2}\mapsto
f^{\prime}_{j,t}(x_{1},x_{2})=(1-t)x_{1}+tf^{\prime}(z_{1,j}(x_{1}),x_{2}),$

and let
$\xi_{j,t}:[0,1]\times[j-1,j]2^{-p_{0}}\to[0,1]\times[j-1,j]2^{-p_{0}}$ be the
unique measure preserving map mapping each segment
$\\{x_{1}\\}\times[j-1,j]2^{-p_{0}}$ into the image of
$(f^{\prime}_{j,t}(x_{1},x_{2}),x_{2})$, $x_{2}\in[j-1,j]2^{-p_{0}}$. This map
is uniquely defined by the balance of mass, which reads as

$\int_{(j-1)2^{-p_{0}}}^{(\xi_{j,t})_{2}(x_{1},x_{2})}\partial_{x_{1}}f^{\prime}_{j,t}(x_{1},w)dw=x_{2}-(j-1)2^{-p_{0}}.$
(5.5)

Being $f^{\prime}_{j,t}$ close to the identity, $\xi_{j,t}$ is smooth and
close to the identity.

Let $\xi_{t}:[0,1]^{2}\to[0,1]^{2}$ be the measure preserving map obtained by
piecing together the maps $\xi_{j,t}$. By construction the map
$T^{\prime\prime}=T^{\prime}\circ\xi_{1}$ maps each vertical segment
$\\{x_{1}\\}\times[j-1,j]2^{-p_{0}}$ into the vertical segment

$\\{z_{1,j}(x_{1})\\}\times\big{[}g^{\prime}(z_{1,j}(x_{1}),(j-1)2^{-p_{0}}),g^{\prime}(z_{1,j}(x_{1}),j2^{-p_{0}})\big{]}.$

Step 2: construction of the affine maps. The next step is to rectify the
pieces of curves

$[i-1,i]2^{-p_{0}}\ni z_{1}\mapsto
g_{j}(z_{1})=g^{\prime}(z_{1},j2^{-p_{0}}),$ (5.6)

which are the horizontal slab of the sets $\tilde{\omega}^{\prime}_{ij}$.
Fixing a vertical slide $v_{i}$, one considers the unique measure preserving
map $\eta_{i,t}:[i-1,i]\times[0,1]\to[i,i-1]\times[0,1]$ such that the
segments (5.6) are mapped into vertical segments and such that maps the curve
$g_{j}([i-1,i]2^{-p_{0}})$ into the curve

$\begin{split}g^{\prime}_{t,ij}(z_{1})&=(1-t)g^{\prime}(z_{1},j2^{-p_{0}})+t\fint_{(i-1)2^{-p_{0}}}^{i2^{-p_{0}}}g^{\prime}(w,j2^{-p_{0}})dw\\\
&=(1-t)g^{\prime}(z_{1},j2^{-p_{0}})+tz_{2,ij}.\end{split}$ (5.7)

In each $\tilde{\omega}^{\prime}_{ij}$ this map is uniquely determined by the
balance

$\int_{(i-1)2^{-p_{0}}}^{(\eta_{i,t})_{2}(z_{1},z_{2})}\big{(}g^{\prime}_{t,ij}(w)-g^{\prime}_{t,i(j-1)}(w)\big{)}dw=\mathrm{constant},$

while the vertical coordinate is affine in each vertical segment.

Let $\eta_{t}:[0,1]^{2}\to[0,1]^{2}$ be the measure preserving map obtained by
piecing together the maps $\eta_{i,t}$.

Conclusion. Up to a time scaling, the map we are looking for is

$\sigma_{t}=T\circ\xi_{t}1\\!\\!\mathrm{I}_{[0,1]}(t)+\zeta_{t-1}\circ
T\circ\xi_{1}1\\!\\!\mathrm{I}_{[1,2]}(t)+\eta_{t-2}\circ\zeta_{1}\circ
T\circ\xi_{1}1\\!\\!\mathrm{I}_{[2,3]}(t).$

It is clearly measure preserving and at $t=3$ it maps affinely the rectangles
with dyadic coordinates

$P_{ij}=\big{[}x_{1,(i-1)j},x_{1,ij}\big{]}\times[j-1,j]2^{-p_{0}}$

into the rectangles with dyadic coordinates

$\tilde{P}_{ij}=[i-1,i]2^{-p_{0}}\times\big{[}z_{2,i(j-1)},z_{2,ij}\big{]}.$

The values $x_{1,ij}$, $z_{2,ij}$ are given by (5.3), (5.7) and belong to
$2^{-p_{1}}\mathbb{N}$. Thus $N_{1}=N$ is the number of the statement.

The fact that $\sigma_{t}$ is piecewise smooth and it extends continuously to
the boundary of each $P_{ij}$ are immediate from the construction, and its
smallness follows by observing that as $p_{0},p_{1}$ diverge the maps
$\xi,\zeta,\eta$ converge to the identity. ∎

Figure 6: The action of $\sigma_{t}$ in Lemma 5.1: first the map $\xi_{t}$
moves the mass in $H_{j}$ in order to map the vertical green segments into the
counterimages of the vertical red segments; then the map $T$ acts and the
horizontal black boundaries of $H_{j}$ becomes the black curves, but vertical
segments remain vertical; finally the action of $\eta_{t}$ rectifies the
horizontal boundaries, while keeping vertical segments vertical.

Being the rectangles dyadic in the grid
$\mathbb{N}\times\mathbb{N}\frac{1}{N}$, then we have the following

######  Corollary 5.2.

Every subsquare of the grid $\mathbb{N}\times\mathbb{N}\frac{1}{N}$ is sent by
$\sigma_{1}$ affinely by a diagonal matrix onto a subrectangle with rational
coordinates.

######  Remark 5.3.

The previous lemma also tells us that the map $\sigma_{1}$ is piecewise
affine, in particular there exists $N=2^{p_{1}}\in\mathbb{N}$ refinement of
the grid such that $\sigma_{1}$ maps each $\kappa$ subsquare of the grid
$\mathbb{N}\times\mathbb{N}\frac{1}{N}$ affinely by a diagonal matrix onto a
subrectangle $q$ with rational coordinates. It’s false, in general, that $q$
has dyadic rational coordinates as stated in [22, Lemma 4.3]. More precisely,
the previous lemma states that each

$P_{ij}=[x_{1,(i-1)j},x_{1,ij}]\times[j-1,j]2^{-p_{0}}$

is sent into

$\tilde{P}_{ij}=[i-1,i]2^{-p_{0}}\times[z_{2,i(j-1)},z_{2,ij}],$

where $x_{1,(i-1)j},x_{1,ij},z_{2,i(j-1)},z_{2,ij}$ are dyadic. Call $\Delta
x=x_{1,ij}-x_{1,(i-1)j}$ and $\Delta z=z_{2,ij}-z_{2,i(j-1)}$. Then by (5.4)
$\Delta x=2^{-p_{1}}n_{ij}$ with $n_{ij}\in\mathbb{N}$. Up to translation the
perturbed map $\sigma_{1}$ (5.1) can be written as

$\sigma_{1}\llcorner_{P_{ij}}=\left(\begin{matrix}\frac{2^{-p_{0}}}{\Delta
x}&0\\\ 0&2^{p_{0}}(\Delta z)\end{matrix}\right).$

Take a subsquare $\kappa=[h-1,h]2^{-p_{1}}\times[k-1,k]2^{-p_{1}}\subset
P_{ij}$, then
$q=\sigma_{1}\llcorner_{P_{ij}}(\kappa)=\left[\frac{2^{-p_{0}}}{n_{ij}},2^{p_{0}-p_{1}}(\Delta
z)\right]$, which is dyadic only with further requirements on $n_{ij}$. For a
more detailed analysis consider $H_{j}$ and call $\Delta
x_{i}=x_{1,ij}-x_{1,(i-1)j}=2^{-p_{1}}n_{ij}$, where $i=1,\dots 2^{p_{0}}$. If
we assume that every subsquare of the grid
$\mathbb{N}\times\mathbb{N}\frac{1}{N}$ is sent into a dyadic rectangle then
we find the conditions

$n_{ij}=2^{m_{ij}},\quad\forall i,j.$

This condition tells us that, being measure-preserving,

$\sigma_{1}\llcorner_{P_{ij}}=\left(\begin{matrix}2^{-p_{0}+p_{1}-m_{ij}}&0\\\
0&2^{p_{0}-p_{1}+m_{ij}}\end{matrix}\right),$

that is, all possible matrices are of the form

$\left(\begin{matrix}1&0\\\
0&1\end{matrix}\right)\quad\left(\begin{matrix}\frac{1}{2}&0\\\
0&2\end{matrix}\right)\quad\left(\begin{matrix}2&0\\\
0&\frac{1}{2}\end{matrix}\right)\quad\left(\begin{matrix}\frac{1}{4}&0\\\
0&4\end{matrix}\right)\quad\dots\quad.$

This condition is not compatible with the fact that $\sigma_{1}$ is an
approximation of the original map $T$, which has been chosen to be close to
the identity.

######  Remark 5.4.

From the previous lemma it easily follows that, if $N$ is the size of the
grid, then every rectangle contained in the unit square $K$ is sent by the
perturbed flow into a union of rectangles.

######  Remark 5.5.

To use Theorem 2.4 we observe that in our case the change of variables $\phi$
is given by the flow $X_{t}$. In particular, since $X_{t}$ is close to the
identity with all its derivatives, the costant $C_{X_{t}}$ given by the
previous theorem, is $C_{X_{t}}\leq(1+\delta)^{d-1}$ ($d=2$ here).

####  5.1.1 The d-dimensional case

The analysis of the general case can be done as follows.

The starting point is the following approximation assumption in
$d-1$-dimension.

######  Assumption 5.6.

If the $\mathcal{L}^{d-1}$-measure-preserving diffeomorphism
$T:[0,1]^{d-1}\rightarrow[0,1]^{d-1}$ is sufficiently close to the identity
and equal to $\mathrm{id}$ in a neighborhood of $\partial[0,1]^{d-1}$, then
there exists $N\in\mathbb{N}$, $N=2^{p}$, and a measure-preserving piecewise
smooth invertible map $\sigma$ close to $T$ such that $T\circ\sigma$ maps
dyadic rectangles $P_{ij}\in\frac{\mathbb{N}^{d-1}}{N}$ onto dyadic rectangles
$\tilde{P}_{ij}\in\frac{\mathbb{N}^{d-1}}{N}$ by a diagonal linear map (up to
a translation).

The above assumption is true for $d=3$: indeed if $\sigma_{t}$ is the map of
Lemma 5.1, then $\sigma=T^{-1}\circ\sigma_{t=1}$ does the job.

Now let $T:[0,1]^{d}\to[0,1]^{d}$ be a diffeomorphism sufficiently close to
the identity and equal to the identity near $\partial[0,1]^{d}$ (Figure 7). We
will not address the perturbation $\zeta$ used to obtain dyadic
parallelepipeds (Step 0 of the proof above), being the idea completely similar
to the $2d$-case. We will also neglect the time dependence (i.e. how to split
$t\in[0,1]$ into time intervals where the different maps are acting), because
it is a fairly easy extension of the $2d$ case.

Step 1. Consider the curves

$z_{d}\mapsto T^{-1}(z_{1},\dots,z_{d}).$

The first step is to perturb $T$ to a map $T^{\prime}$ in order to have that
the above curves are segments along the $x_{d}$-direction in each slab
$x_{d}\in[k_{d},k_{d}+1]/N$ (Figure 8).
Being $T$ close to the identity, the surface
$T^{-1}(\\{z_{d-1}=\text{const.}\\})$ is parameterized by
$x_{1},\dots,x_{d-2},x_{d}$, and then in each strip

$(x_{1},\dots,x_{d-2})=\text{const.},\quad x_{d-1}\in[0,1],\
x_{d}\in[k_{d},k_{d}+1]\frac{1}{N}$

one can use the same measure preserving map $\xi_{1}$ defined in Step 1 of the
proof of Lemma 5.1 above to obtain a perturbation $\hat{T}=T\circ\xi$ such
that

$\hat{T}^{-1}(\\{z_{d-1}=\text{const.}\\})\cap\big{\\{}x_{d}\in[k_{d},k_{d}+1]/N\big{\\}}$

is independent of $x_{d}$, in the sense that it is the graph of a function
depending only on $x_{1},\dots,x_{d-2}$ times the segment
$x_{d}\in[k_{d},k_{d}+1]/N$.
Disintegrate the Lebesgue measure $\mathcal{L}^{d}$ as

$\mathcal{L}^{d}\llcorner_{\\{x_{d}\in[k_{d},k_{d}+1]/N\\}}=\int\Big{[}a(x_{1},\dots,x_{d-2},z_{d-1})dx_{1}dx_{d-2}dx_{d}\Big{]}dz_{d-1},$

according to the partition $\hat{T}^{-1}(z_{d-1}=\text{const})$ (the density
$a$ does not depend on $x_{d}$ because the surfaces contains the segments
along $x_{d}$), and consider the $2$-dimensional surfaces

$\hat{T}^{-1}(z_{d-1}=\text{const})\cap\\{x_{1},\dots,x_{d-3}=\text{const}\\}.$

We use the same map $\xi_{1}$ of Step 1 of the proof above to rectify the
curves

$\begin{split}z_{d}\mapsto
f(x_{1},\dots,x_{d-3},z_{d-2},z_{d-1};z_{d})&=(\hat{T})^{-1}(z_{d-2},z_{d-1}=\text{const})\\\
&\quad\cap\\{x_{1},\dots,x_{d-3}=\text{const}\\}\cap\\{x_{d}\in[k_{d},k_{d}+1]/N\\}.\end{split}$

The main difference w.r.t. the maps (5.2), (5.5) is that instead of the
Lebesgue measure we use the density $a(x_{1},\dots,x_{d-2},z_{d-1})$.
Eventually, the composition of the two maps above gives a new map $\check{T}$
such that $(\check{T})^{-1}(z_{d-2},z_{d-1}=\text{const})$ is a
$(d-2)$-dimensional surface made of the graph of a function depending on
$x_{1},\dots,x_{d-3}$ times the segment $x_{d}\in k_{d},k_{d}+1]/N$.
The argument is then repeated in the $d-2$-regions
$(\check{T})^{-1}(z_{d-2},z_{d-1}=\text{const})$ (i.e. disintegrate the
Lebesgue measure and shift along the $x_{d-3}$ direction to rectify
$(\check{T})^{-1}(z_{d-3},\dots,z_{d-1}=\text{const}$), and so on until we
obtain that a new map $T^{\prime}$ such that

$(T^{\prime})^{-1}(\\{z_{1},\dots,z_{d-1}=\text{const.}\\})\cap\big{\\{}x_{d}\in[k_{d},k_{d}+1]/N\big{\\}}$

is independent on $x_{d}$. This means that lines along the $z_{d}$ are mapped
back into $N$ segments of length $1/N$ along $x_{d}$.

$T$
---
$z_{2}=\mathrm{const}$
---
$T^{-1}(z_{1},z_{2})$
---
$x_{3}\in[k_{3},k_{3}+1]/N$
---
$z_{1}=\mathrm{const}$
---
Figure 7: Starting point: the map $T:[0,1]^{3}\to[0,1]^{3}$ maps the slap
$x_{3}\in[k_{3},k_{3}+1]/N$ into a $3d$-set with purple intersections, and
$T^{-1}(z_{1},z_{2})$ is the red curve at the left.
$x_{3}\in[k_{3},k_{3}+1]/N$
---
$x_{1}=\text{const}$
---
$T^{-1}(z_{2})$
---
$T^{-1}(z_{1},z_{2})$
---
Figure 8: First move the mass in the yellow $2d$-rectangle
$x_{1}=\text{const}$ so that its intersection with $T^{-1}(z_{2})$ is vertical
in $x_{3}\in[k_{3},k_{3}+1]/N$, next move the mass along $T^{-1}(z_{2})$ so
that $T^{-1}(z_{1},z_{2})$ is vertical in $x_{3}\in[k_{3},k_{3}+1]/N$.
$(T^{\prime})^{-1}\circ\sigma$
---
$(T^{\prime})^{-1}(z_{1})$
---
$(T^{\prime})^{-1}(z_{2})$
---
Figure 9: The recurrence assumption yields a map $\sigma$ which maps affinely
subsquares into rectangles: in the picture it is shows how it acts before the
composition with $T^{\prime}$ (see also Figure 6). $z_{2}=\text{const}$
---
$z_{1}=\text{const}$
---
$G$
---
Figure 10: The last step is to map the subcubes deformed in the direction
$x_{3}$ into parallelepipeds such that the Lebesgue measure is preserved and
the map $G$ is of triangular form: these conditions imply that
$\tilde{T}=G\circ\bar{T}$ is affine.

Step 2. Differently from the $2d$ case, it is not enough to perturb the
vertical slab as in Step 2, since the sets
$(T^{\prime})^{-1}(\\{z_{k}=\text{const.}\\})$ are not of (piecewise) the form
$\\{x_{k}=\text{const.}\\}$. Observe however that in each slab
$\\{x_{d}\in[k_{d},k_{d}+1]/N\\}$ the map

$(x_{1},\dots,x_{d-1})=(T^{\prime}_{k_{d}})^{-1}(z_{1},\dots,z_{d-1})$

is well defined, where $(T^{\prime})^{-1}_{k_{d}}$ denotes the first
$(d-1)$-components of $(T^{\prime})^{-1}$ restricted to
$\\{x_{d}\in[k_{d},k_{d}+1]/N\\}$: we have used the property that segments
along $z_{d}$ are mapped back into segments along $x_{d}$.

We use Assumption 5.6 to get a map $\sigma_{k_{d}}:[0,1]^{d-1}\to[0,1]^{d-1}$
such that $T^{\prime\prime}_{k_{d}}=T^{\prime}_{k_{d}}\circ\sigma_{k_{d}}$
maps affinely parallepipeds of a grid $\mathbb{N}^{d-1}/(NN_{1})$,
$N_{1}=2^{p_{1}}$, into cubes of the same grid: we can take $N_{1}\gg 1$ in
order to be independent of $k$ (Figure 9).
Hence the map $T^{\prime\prime}_{k_{d}}$ maps parallelepipeds of the form

$\prod_{i=1}^{d-1}\frac{[k_{i},k_{i}+1]}{NN_{1}}\times\frac{[k_{d},k_{d}+1]}{N}$

into regions for the form

$\bigg{\\{}z_{d}\in\big{[}g(x_{1},\dots,x_{d-1},k_{d}/N),g(x_{1},\dots,x_{d-1},(k_{d}+1)/N)\big{]},z_{i}\in\prod_{i=1}^{d-1}\frac{[k^{\prime}_{i},k^{\prime}_{i}+1]}{NN_{1}}\bigg{\\}},$

and up to a translation it is a linear diagonal map in the first $d-1$
coordinates and segments along $x_{d}$ remains along $x_{d}$.

Step 3. Piecing together the maps $T^{\prime\prime}_{k_{d}}$, we obtain a
measure preserving map $\bar{T}$ close to $T$ with the properties listed at
the end of the previous step. We will use the fact that it is affine in the
first $(d-1)$ coordinates to use a map similar $\eta_{1}$ of Step 2 of the
proof of Lemma 5.1 to rectify the set

$\bar{T}(\\{x_{d}\in[k_{d},k_{d}+1]/N\\})\cap\big{\\{}z_{j}\in[k^{\prime}_{j},k^{\prime}_{j}+1]/(NN_{1}),j=1,\dots,d-1\big{\\}}.$

It is defined as the unique measure preserving map $G(z_{1},\dots,z_{d-1})$ of
the form

$G(z_{1},\dots,z_{d-1})=\Big{(}G_{1}(z_{1},\dots,z_{d-1}),G_{2}(z_{2},\dots,z_{d-1}),\dots,G_{d-1}(z_{d-1}),G(z_{1},\dots,z_{d})\Big{)}.$

Note that since $z_{d}$ enters in the last component, segments along $z_{d}$
are mapped into segments along $z_{d}$, and the triangular form of the map
assures its uniqueness (Figure 10).

The last part of the analysis is to deduce that if a measure preserving
transformation $\tilde{T}=G\circ\bar{T}:[0,1]^{d}\to[0,1]^{d}$ is such that
$\tilde{T}$ is of triangular form then it is the identity: we have rescaled
every rectangle to a cube by linear scaling.
If the map has this triangular form, we conclude that the measure preserving
condition reads as

$\prod_{i=1}^{d}\partial_{i}\tilde{T}_{i}(x_{i},\dots,x_{d})=1,$

which together with

$\int\partial_{i}\tilde{T}_{i}(x_{i},\dots,x_{d})dx_{i}=1$

gives $\partial_{i}T_{i}(x_{i},\dots,x_{d})=1$, i.e. that the map is the
identity.

###  5.2 BV estimates of perturbations

Let $X_{t}:K\rightarrow K$ be a smooth flow of measure-preserving
diffeomorphisms and assume

$\|X_{t}-\mathrm{id}\|_{C^{3}},\|X^{-1}_{t}-\mathrm{id}\|_{C^{3}}\leq\wp,$
(5.8)

with $\wp\ll 1$. Call $T(x)=X_{t=1}(x)$, and let $N=2^{p_{0}}\in\mathbb{N}$ be
the dimension of the grid given by Lemma 5.1. In this section we compute the
BV norms of the perturbations of the form (5.1) constructed in Lemma 5.1.

We first address the action of the map $\zeta_{t}$ on $X(t)$. Define the
perturbed flow

$t\mapsto X^{\prime}_{t}(x)=\zeta_{t}\circ X_{t}(x).$

######  Lemma 5.7.

There exists a perturbation $\zeta_{t}$ as required by Step 0 of Lemma 5.1
such that if $v$ is its associated vector field then

$\|\zeta_{t}-\mathrm{id}\|_{C_{0}}+2^{-p_{0}}\|\nabla\zeta_{t}-\mathrm{id}\|_{C_{0}}+2^{-2p_{0}}\|\nabla^{2}\zeta_{t}\|_{C_{0}}\leq\mathcal{O}(1)2^{-p_{1}},$
$\|v\|_{C_{0}}+2^{-p_{0}}\|\nabla
v\|_{C_{0}}+2^{-2p_{0}}\|\nabla^{2}v\|_{C^{0}}\leq\mathcal{O}(1)2^{-p_{1}}.$

for $p_{1}$ sufficiently large.

###### Proof.

The request of Step 0 of Lemma 5.1 are that the flow across the each region
$\tilde{\omega}_{ij}$ is $0$ (plus the dyadic condition on the new region
$\tilde{\omega}_{ij}$). Hence the problem reduces in finding a suitable
incompressible flow with a given boundary flux: we will construct a flow
generated by a vector field constant in time.

Consider a function $v_{n}$ on $\partial\omega_{ij}$ such that

  * •

its support is at distance $2^{-p_{0}-2}$ from the corners of
$\tilde{\omega}_{ij}$,

  * •

the integral on each of the regular sides is the required flux
$\phi_{ij,i^{\prime}j^{\prime}}$,

  * •

$\|v_{n}\|_{\infty},2^{-p_{0}}\|v_{n}^{\prime}\|,2^{-2p_{0}}\|v_{n}^{\prime\prime}\|\leq\mathcal{O}(1)2^{-p_{1}}$,
where $v^{\prime}_{n}$ is the derivative of $v_{n}$.

Its existence follows from the fact that $\omega_{ij}$ is close to a square of
side $2^{-p_{0}}$, being $X$ close to the identity. The last point is a
consequence of the fact that $|\phi_{ij,i^{\prime}j^{\prime}}|\leq
2^{-p_{0}-p_{1}}$.

The integral of $v$ on $\partial\tilde{\omega}_{ij}$ is a potential function
$p$, which is constant in the $2^{-p_{0}-2}$-neighborhood of every corner and
such that

$\|p\|_{C_{0}},2^{-p_{0}}\|p^{\prime}\|_{C_{0}},2^{-2p_{0}}\|p^{\prime\prime}\|_{C_{0}},2^{-3p_{0}}\|p^{\prime\prime\prime}\|_{C_{0}}\leq\mathcal{O}(1)2^{-p_{0}-p_{1}},$

where $p^{\prime},p^{\prime\prime}$ are its first and second derivative.

Extend $p$ to a $C^{2}$-function inside $\tilde{\omega}_{ij}$: since this
extension can be required to vary in a region of size $2^{-p_{0}-2}$, we get

$\|p\|_{C_{0}}\leq\mathcal{O}(1)2^{-p_{0}-p_{1}},\quad\|\nabla
p\|_{C_{0}}\leq\mathcal{O}(1)\frac{2^{-p_{0}-p_{1}}}{2^{-p_{0}-2}}=\mathcal{O}(1)2^{-p_{1}},$
$\|\nabla^{2}p\|_{C_{0}}\leq\mathcal{O}(1)\frac{2^{-p_{0}-p_{1}}}{2^{-2p_{0}-4}}=\mathcal{O}(1)2^{p_{0}-p_{1}}.$
$\|\nabla^{3}p\|_{C_{0}}\leq\mathcal{O}(1)\frac{2^{-p_{0}-p_{1}}}{2^{-3p_{0}-6}}=\mathcal{O}(1)2^{2p_{0}-p_{1}}.$

In particular the vector field $v=\nabla^{\perp}p$ satisfies the statement,
and if $\zeta_{t}$ is the flow generated by $v$ then the same holds by the
estimates

$\|\zeta_{t}-\mathrm{id}\|_{C^{0}}\leq\|v\|_{C_{0}}t,\quad\|\nabla\zeta_{t}-\mathrm{id}\|_{C^{0}}\leq
e^{\|\nabla v\|_{C_{0}}t}-1,$ $\|\nabla^{2}\zeta(t)\|_{C^{0}}\leq e^{\|\nabla
v\|_{C_{0}}t}\|\nabla^{2}v\|_{C_{0}}\|\nabla\zeta\|_{C_{0}}^{2},$

with $p_{1}\gg 1$. ∎

######  Corollary 5.8.

If $b^{\prime}$ is the vector field associated with
$X^{\prime}_{t}=\zeta_{t}\circ X_{t}$, then

$\|b^{\prime}_{t}-b_{t}\|_{C_{1}}\leq\mathcal{O}(1)2^{2p_{0}-p_{1}}.$

###### Proof.

From the formula (2.2)

$b^{\prime}(t,x)-b(t,x)=v(x)+\big{(}\nabla\zeta_{t}(t,\zeta_{t}^{-1}(x))-\mathrm{id}\big{)}b(t,x),$

where $v(x)$ is the time independent vector field associated with $\zeta_{t}$.
Hence from the previous lemma

$\begin{split}\|b^{\prime}(t)-b(t)\|_{C^{1}}&\leq\|v\|_{C^{1}}+\mathcal{O}(1)\|\zeta_{t}-\mathrm{id}\|_{C^{2}}\|b\|_{C^{1}}\\\
&\leq\mathcal{O}(1)2^{2p_{0}-p_{1}}.\qed\end{split}$

Define the perturbed flow $t\rightarrow\tilde{X}_{t}$ as

$\tilde{X}_{t}(x)=\begin{cases}X^{\prime}(t,0,\xi_{t}(x))&t\in\left[0,\frac{1}{2}\right],\\\
X^{\prime}(t,1,\eta(t,1,w(x)))&t\in\left[\frac{1}{2},1\right],\end{cases}$

where $\xi_{t}$ and $\eta_{t}$ are given by formula (5.1) of Lemma 5.1 (here
since the map $\zeta$ is not needed we rescale $\xi_{t},\eta_{t}$ with
$t\in[0,1/2]$) and

$w(x)=\eta_{1}\circ T^{\prime}\circ\xi_{\frac{1}{2}}(x),\quad
T^{\prime}=\zeta_{1}\circ T.$

Call $\tilde{b}_{t}$ the vector field associated with $\tilde{X}_{t}$.

######  Lemma 5.9 (BV estimates).

There exists a positive constant $C=C(\wp)$ such that

$\|b-\tilde{b}\|_{L^{\infty}({L^{1}})}\leq\frac{C\wp}{N},\quad\|\text{\rm
Tot.Var.}({\tilde{b}})\|_{\infty}\leq C\|\text{\rm
Tot.Var.}(b)\|_{\infty}+\frac{C\wp}{N}.$ (5.9)

###### Proof.

From Corollary 5.8, we have that (for $p_{1}\gg 1$)

$\|b-b^{\prime}\|_{C^{1}}\leq\mathcal{O}(1)2^{2p_{0}-p_{1}}\ll\frac{\wp}{N},$

so that we are left to prove (5.9) with $b^{\prime}$ in place of $b$:

$\|b^{\prime}-\tilde{b}\|_{L^{\infty}({L^{1}})}\leq\frac{C\wp}{N},\quad\|\text{\rm
Tot.Var.}({\tilde{b}_{t}})\|_{\infty}\leq C\wp.$

We will prove the above estimates for $t\in[0,1/2]$, i.e. only for
$X^{\prime}_{t}\circ\xi_{t}$, being the analysis of
$X^{\prime}(t,1,\eta(t,1,w(x)))$ completely analogous.

We start by observing that there exists a constant $C>0$ such that

$\|\dot{\xi_{t}}\|_{\infty}\leq\frac{C\wp}{N}.$

Indeed, the map $\xi^{i}_{t}=\xi_{t}\llcorner_{H_{i}}$ is given by the
formulas (see Step 1 of the proof of Lemma 5.1)

$\xi^{i}_{1,t}(x_{1},x_{2})=f^{\prime}_{i,2t}(x_{1},\xi^{i}_{t}(x_{1},x_{2}))=(1-2t)x_{1}+2tf^{\prime}(z_{1,i}(x_{1}),\xi^{i}_{2,t}(x_{1},x_{2})),$