# Every diffeomorphism is a total renormalization of a close to identity map

Pierre Berger, Nicolaz Gourmelon, Mathieu Helfter IMJ-PRG, CNRS, Sorbonne
University, Paris University, partially supported by the ERC project 818737
Emergence of wild differentiable dynamical systems. IMJ-PRG, CNRS, Sorbonne
University, Paris University, partially supported by the ERC project 818737
Emergence of wild differentiable dynamical systems.

###### Abstract

For any $1\leq r\leq\infty$, we show that every diffeomorphism of a manifold
of the form $\mathbb{R}/\mathbb{Z}\times M$ is a total renormalization of a
$C^{r}$-close to identity map. In other words, for every diffeomorphism $f$ of
$\mathbb{R}/\mathbb{Z}\times M$, there exists a map $g$ arbitrarily close to
identity such that the first return map of $g$ to a domain is conjugate to $f$
and moreover the orbit of this domain is equal to $\mathbb{R}/\mathbb{Z}\times
M$. This enables us to localize nearby the identity the existence of many
properties in dynamical systems, such as being Bernoulli for a smooth volume
form.

###### Contents

  1. 0.1 Statements of the main theorems
  2. 0.2 Applications and open problems
  3. 0.3 Sketch of proof
  4. 1 Plugins and Pluggable dynamics
    1. 1.1 Plugins
    2. 1.2 Proof of the corollaries of the main E
    3. 1.3 Pluggable dynamics
  5. 2 Topological group structure on $\mathrm{Plug}$
    1. 2.1 Group structure on $\mathrm{Plug}$
    2. 2.2 Another subgroup included in $\mathrm{Plug}$
    3. 2.3 Closure of the group $\mathrm{Plug}$
    4. 2.4 Vector fields whose flow is pluggable
  6. 3 Construction of Pluggable flows
    1. 3.1 Proof of main F and F′
    2. 3.2 Eigenvectors of the adjoint representation
    3. 3.3 Decomposition of vector fields
  7. A Smoothness of outputs
  8. B Lie algebras associated to closed subgroups

## Introduction

###  0.1 Statements of the main theorems

Let $\mathbb{B}^{n}$ be the unit closed ball of $\mathbb{R}^{n}$.

######  Definition 0.1.

A renormalization $G$ of a diffeomorphism $g\in\mathrm{Diff}(\mathbb{B}^{n})$
is a rescaling of an iteration of $g$. In other words, there exists $N\geq 2$
and an embedding $\psi:\mathbb{B}^{n}\hookrightarrow\mathbb{B}^{n}$ such that:

$G=\psi^{-1}\circ g^{N}\circ\psi.$

A long standing open problem of dynamical systems theory is:

######  Problem 0.2 (1971).

Which dynamics can be reached by renormalization of close to identity maps?

This problem was first studied by Ruelle and Takens in [RT71]. Motivated by
the study of turbulence, they proved that for any integer $n\geq 2$, any
dynamic on the $n$-dimensional torus is the renormalization of a $C^{n}$-close
to identity map. This enabled them to construct perturbations of the identity
map of the torus with a strange attractor. Based on this, they conjectured
that this appears as well in fluid dynamics and could be used as a
mathematical definition of the notion of turbulence [Lor63].

The main mathematical issue with this result is that the regularity is limited
by the dimension of the torus. However when considering flows, this problem
was solved by Newhouse, Ruelle and Takens in [NRT78]: given any vector filed
$X$ equal to a rotation on the torus $\mathbb{T}^{n}$, $n\geq 3$ and any map
$F_{0}\in\mathrm{Diff}^{\infty}(\mathbb{T}^{n-1})$ homotopic to the identity,
they perturbed $X$ to $\widetilde{X}$ so that its first return map to a global
transverse section is $F_{0}$. Yet $\mathrm{Diff}^{\infty}(\mathbb{T}^{n})$ is
“much larger” than $\mathrm{Diff}^{\infty}(\mathbb{T}^{n-1})$ and so the
mathematical 0.2 remains unsolved.

A breakthrough was then performed in the seminal work of Turaev who proved
that a $C^{r}$-dense subset of $C^{r}$-orientation preserving embeddings of
$\mathbb{B}^{n}$ could be obtained after renormalization of an arbitrarily
close to identity map, for every $0\leq r\leq\infty$.

A first main result is a solution to 0.2, where we improve Turaev’s theorem to
obtain, via a self-contained and new proof, any $C^{r}$-orientation preserving
map of $\mathbb{B}^{n}$ (instead of maps among a dense subset):

######  Theorem A.

For any $1\leq r\leq\infty$ and any orientation preserving
$G\in\mathrm{Diff}^{r}(\mathbb{B}^{n})$, in any neighborhood
${\mathcal{N}}\subset\mathrm{Diff}^{r}(\mathbb{B}^{n})$ of the identity, there
exists $g\in{\mathcal{N}}$ such that a renormalization of $g$ is equal to $G$.
Moreover the rescaling map of this renormalization can be chosen affine.

A natural open problem is whether $g$ can be obtained conservative or
symplectic when $G$ is conservative or symplectic. In this direction let us
mention the work of Gonchenko-Shilnikov-Turaev [GST07] who proved that, for
every $0\leq r\leq\infty$, a $C^{r}$\- dense subset of volume preserving
embeddings of $\mathbb{B}^{2}$ could be obtained after renormalization of an
arbitrarily close to identity volume preserving map. Recently Fayad and
Saprikina in [FS21] showed that any conservative map of the $n$-dimensional
ball can be realized by renormalized iteration of a conservative
$C^{n}$-perturbation of the identity.

If all these theorems indicate the richness of the possible dynamical
behaviors nearby the identity, one can object the following. In the setting of
Definition 0.1, the orbit of $\bigcup_{k=0}^{N-1}\psi(\mathbb{B}^{n})$ of the
renormalization domain might be extremely small and so experimentally not
observable. This objection is lifted completely when the renormalization
domain intersects every orbit. This leads us to strengthen the notion of
renormalization by the following:

######  Definition 0.3.

Let $r\in\\{1,\dots,\infty\\}\cup\\{\omega\\}$ and let $V$ be a manifold (with
boundary). A map $g\in\mathrm{Diff}^{r}(V)$ is renormalizable if there exists
a strict submanifold with corners $\Delta\subsetneq V$ such that:

  * •

there exists a bijective, local diffeomorphism $H:\Delta\to V$, called the
rescaling map of the renormalization domain $\Delta$,

  * •

the first return time $\tau:\Delta\to\mathbb{N}^{*}$ into $\Delta$ by $g$ is
bounded and the renormalization $G=H\circ g^{\tau}\circ H^{-1}$ belongs to
$\mathrm{Diff}^{r}(V)$.

The map $g$ is totally renormalizable if the forward orbit of $\Delta$ covers
$V$, i.e. $\bigcup_{n\geq 0}g^{n}(\Delta)=V$. The map $G$ is then a total
renormalization of $g$.

######  Example 0.4.

If $V=\mathbb{B}^{n}$, any renormalizable map in the sense of Definition 0.1
satisfies Definition 0.3 with $\Delta=\psi(\mathbb{B}^{n})$ and $H=\psi^{-1}$.
However these maps are not totally renormalizable.

######  Example 0.5.

When $V$ is the circle $\mathbb{T}$, a diffeomorphism $g$ is totally
renormalizable iff it does not fix a point. Indeed in this case, take any
point $0\in\mathbb{T}$ and consider the interval $\Delta=[0,g(0))$. Then we
glue the two endpoints of $\Delta$ using $g$ to obtain a circle and we
uniformize it to obtain $\mathbb{T}$. This defines a map $H$. For this setting
one easily shows that the mapping $g$ is renormalizable. This construction was
intensively used by Yoccoz [Yoc95b].

Let $V$ be a compact manifold (possibly with corners) and $1\leq r\leq\infty$.
We recall that the support $\mathrm{supp}\>f$ of $f\in\mathrm{Diff}^{r}(V)$ is
the closure of the set of points such that $f(x)\neq x$.

######  Definition 0.6.

Let $\mathrm{Diff}^{r}_{0}(V)$ be the component of the identity in
$\mathrm{Diff}^{r}(V)$. Let $\mathrm{Diff}^{r}_{c}(V)$ be the subset of
$\mathrm{Diff}^{r}_{0}(V)$ formed by maps isotopic to $\mathrm{id}$ through
isotopies $(f_{t})_{t\in[0,1]}$ whose support
$\bigcup_{t\in[0,1]}\mathrm{supp}\>f_{t}$ is a compact subset of
$V\setminus\partial V$.

Observe that when $V$ is boundaryless, it holds
$\mathrm{Diff}^{r}_{0}(V)=\mathrm{Diff}^{r}_{c}(V)$. A natural question is:

######  Question 0.7.

For which manifold $V$, any map $F\in\mathrm{Diff}^{r}_{c}(V)$ is a total
renormalization of a close to identity map?

So far no example of such a manifold $V$ was known. In this work we give a
full class of examples:

######  Theorem B.

Let $0\leq r\leq\infty$, let $M$ be a compact manifold of dimension $\geq 1$
and put $V:=\mathbb{T}\times M$. Let
${\mathcal{N}}\subset\mathrm{Diff}^{r}(V)$ be a neighborhood of the identity.
Then any $G\in\mathrm{Diff}^{r}_{c}(V)$ is a total renormalization of some
$g\in{\mathcal{N}}$.

If A implies that every local dynamical phenomenon can be found nearby the
identity, B implies that every global dynamical phenomenon can be found nearby
the identity. A new improvement brought by the latter result is that the
renormalization domain is larger than in all of the previous extensions of
Ruelle-Takens theorems: its orbit coincides with the whole domain of the
dynamic. In E, we will give a precise formula defining the renormalization
domain and rescaling map involved in B. This will enable new applications such
as the proof of existence of maps preserving smooth SRB nearby the identity
(see Corollary C) or universal maps whose renormalization domains decrease as
slow as we want (see Corollary D).

In Proposition 0.10, we show that B is wrong when $\mathbb{T}\times
M\approx\mathbb{T}$, hence the set of dimensions of the manifold is optimal.
On the other hand, a natural open problem communicated to us by Turaev is:

######  Problem 0.8.

Show that a dense subset of $\mathrm{Diff}^{\omega}(\mathbb{B}^{n})$ is equal
to the renormalization of a close to identity map in
$\mathrm{Diff}^{\omega}(\mathbb{B}^{n})$?

Another natural question is:

######  Question 0.9.

Is B correct in the area preserving or symplectic categories?

An extension of B regards the $C^{r}$-families
$f_{\mathscr{P}}=(f_{p})_{p\in{\mathscr{P}}}$ of maps
$f_{p}\in\mathrm{Diff}^{r}(V)$ and indexed by a manifold ${\mathscr{P}}$. A
family $(f_{p})_{p\in{\mathscr{P}}}$ is of class $C^{r}$ if the following is
in $\mathrm{Diff}^{r}(V\times{\mathscr{P}})$:

(0.1) $\widehat{f_{\mathscr{P}}}:=(x,p)\mapsto(f_{p}(x),p)\;.$

We denote by $\mathrm{Diff}^{r}(V)_{\mathscr{P}}$ the space of such families
endowed with the topology induced by
$\mathrm{Diff}^{r}(V\times{\mathscr{P}})$. Let
$\mathrm{Diff}_{c}^{r}(V)_{\mathscr{P}}$ be the component of the identity in
$\mathrm{Diff}^{r}(V)_{\mathscr{P}}$ for homotopies
$(f_{p,t})_{(p,t)\in{\mathscr{P}}\times[0,1]}\in\mathrm{Diff}^{r}(V)_{{\mathscr{P}}\times[0,1]}$
whose support
$\bigcup_{(t,p)\in[0,1]\times{\mathscr{P}}}\mathrm{supp}\>f_{p,t}\times\\{p\\}$
is a compact subset of the interior of $V\times{\mathscr{P}}$. Observe that
$\widehat{f_{\mathscr{P}}}\in\mathrm{Diff}^{r}_{c}(V\times{\mathscr{P}})$.

######  Theorem B′.

Let $0\leq r\leq\infty$, let $M$ and ${\mathscr{P}}$ be compact manifolds of
dim $\geq 1$ and set $V:=\mathbb{T}\times M$. Let
${\mathcal{N}}\subset\mathrm{Diff}^{r}(V)_{\mathscr{P}}$ be a neighborhood of
$(id)_{p\in{\mathscr{P}}}$. Then for any
$(G_{p})_{p\in{\mathscr{P}}}\in\mathrm{Diff}^{r}_{c}(V)_{\mathscr{P}}$, there
exist $(g_{p})_{p\in{\mathscr{P}}}\in{\mathcal{N}}$ and a rescaling map
(independent of $p\in{\mathscr{P}}$) of a total renormalization domain which
renormalizes each $g_{p}$ to $G_{p}$.

This theorem implies that any bifurcation in $\mathrm{Diff}^{r}_{c}(V)$ occurs
at small unfolding of the identity. In Proposition 1.5, we will state the main
general E which implies Theorems A and B and also its parametric counterpart
E′ which implies B′. In Section 0.2 we will give several applications of them.
Now let us discuss the optimality of B.

######  Proposition 0.10.

When $r\geq 2$, B is wrong if $V\approx\mathbb{T}$ (and so $\dim M=0$).

###### Proof.

Indeed if $F$ is a renormalization of a close to identity map $f$ for a
renormalization domain $\Delta$, then we have necessarily
$|\tau(\theta)-\tau(\theta^{\prime})|\leq 1$ for any
$\theta,\theta^{\prime}\in\Delta$. Let
$N:=\min\\{\tau(\theta),\tau(\theta^{\prime})\\}$. We compute the derivative
of the $N-1$ first iterates $(\theta_{i})_{i}$ and $(\theta^{\prime}_{i})_{i}$
of $\theta$ and $\theta^{\prime}$:

$\log\left|\frac{D_{\theta}F}{D_{\theta^{\prime}}F}\right|=\log\left|\frac{D_{\theta}f^{N}}{D_{\theta^{\prime}}f^{N}}\right|+o(1)\quad\text{when
}f\to id$
$=\sum_{i=0}^{N-1}\log\left|\frac{D_{\theta_{i}}f}{D_{\theta^{\prime}_{i}}f}\right|+o(1)\leq\|\log|Df|\|_{C^{1}}\cdot\sum_{i=0}^{N-1}|\theta_{i}-\theta_{i}^{\prime}|+o(1)=o(1),$

where the latter inequality uses that $\|\log|Df|\|_{C^{1}}$ is small while
the segment $[\theta_{i},\theta_{i}^{\prime}]$ are disjoint and so the union
of their length is at most $1$. Hence this proves that the derivative of $F$
is constant and so that $F$ must be a rotation. ∎

Also we cannot change $\mathrm{Diff}_{c}^{r}(\mathbb{T}\times M)$ by
$\mathrm{Diff}_{0}^{r}(\mathbb{T}\times M)$ in B. Indeed the latter
proposition applied to the boundary of $[0,1]$ implies immediately:

######  Corollary 0.11.

There are $G\in\mathrm{Diff}_{0}^{\infty}(\mathbb{T}\times[0,1])$ which are
not total renormalization of $C^{2}$-close to identity map.

Yet in view of 0.7, B seems to be generalizable for a manifold $V$ on which
$\mathbb{T}$ acts properly discontinuously without any fixed point.

###  0.2 Applications and open problems

##### Smooth SRB nearby the identity

In Section 1.1 we will state E which will imply together with Katok’s theorem
[Kat79], an answer to an open question of Thouvenot:

######  Corollary C.

In any neighborhood ${\mathcal{N}}$ of
$\mathrm{id}\in\mathrm{Diff}^{\infty}(\mathbb{T}^{2})$ there is a map
$g\in{\mathcal{N}}$ which leaves invariant an ergodic smooth volume form and
displays positive Lyapunov exponent at Lebesgue a.e. point.

This corollary will be proved in Section 1.2 from Proposition 1.7, and
generalized to higher dimension using [DP02].

##### Universal mappings

A map $f\in\mathrm{Diff}^{r}(V)$ is said to be universal if there exists a
dense subset of $\mathrm{Diff}_{0}^{r}(V)$ such that each of its element is a
renormalization of $f$. Bonatti and Diaz in [BD03] have shown that universal
maps are locally $C^{1}$-generic on $\mathbb{B}^{3}$. Turaev in [Tur15] has
shown that universal maps are locally $C^{\infty}$-generic on
$\mathbb{B}^{2}$. Yet mathematicians wondered whether we can “see” the
universality of such mapping. While there are infinitely many renormalization
domains, the above proofs lead to a very small volume for the union of their
orbits.

######  Corollary D.

Let $1\leq r\leq\infty$ and $n\geq 2$. For any sequence $(F_{i})_{i\geq 0}$ of
maps $F_{i}\in\mathrm{Diff}_{0}^{r}(\mathbb{T}\times[0,1])$ and any sequence
of positive numbers $(\ell_{i})_{i\geq 0}$ s.t. $\sum\ell_{i}<1$, there exists
a $C^{r}$-arbitrary close to identity map
$f\in\mathrm{Diff}_{0}^{r}(\mathbb{T}\times[0,1])$ which displays a family of
renormalization domains $(\Delta_{i})_{i\geq 0}$ such that:

  1. 1.

a renormalization of $f$ associated to $\Delta_{i}$ is $F_{i}$ for every
$i\geq 0$,

  2. 2.

the orbit $\tilde{\Delta}_{i}:=\bigcup_{n\geq 0}f^{n}(\Delta_{i})$ has volume
equal to $\ell_{i}$,

  3. 3.

the sets $\tilde{\Delta}_{i}$ and $\tilde{\Delta}_{j}$ are disjoint for $i\neq
j$.

This corollary will be proved in Section 1.2.

The proof of the main theorem is constructive and it seems to us that, in the
case where $M$ is boundaryless, the map $g$ of B depends smoothly on $G$ in a
neighborhood of the identity. This leads us to propose:

######  Conjecture 0.12.

For every compact boundaryless manifold $M$ of dimension $\geq 1$, there
exists a neighborhood ${\mathcal{N}}_{0}$ of
$id\in\mathrm{Diff}^{\infty}(\mathbb{T}\times M)$ such that for every
neighborhood ${\mathcal{N}}$ of $id\in\mathrm{Diff}^{\infty}(\mathbb{T}\times
M)$, there is a smooth (tame) injective map ${\cal
I}:G\in{\mathcal{N}}_{0}\mapsto g\in{\mathcal{N}}$ such that $G$ is a total
renormalization of $g={{\mathcal{I}}}(G)$ for every $G\in{\mathcal{N}}_{0}$.

Roughly speaking, this conjecture asserts that modulo total renormalization, a
fixed neighborhood ${\mathcal{N}}_{0}$ of
$id\in\mathrm{Diff}^{\infty}(\mathbb{T}\times M)$ can be smoothly embedded
into any smaller neighborhood. This defines infinitely many inverse branches
of the renormalization operator with image converging to the identity.

###  0.3 Sketch of proof

##### Plugins and pluggable dynamics:

The framework of the proof of the main theorem relies on a new object called
plugin and the notion of pluggable map. A plugin is a renormalizable map of a
special form, so that it has a canonical renormalization called its output.
See Def. 1.1 and 1.3 below and Figs. 1 and 2. We will say that a map is
pluggable if it is the output of an arbitrarily close to identity plugin and
likewise for its inverse. In particular a pluggable map is a total
renormalization of a close to identity map. Most of this work will be
dedicated to show F stating that:

any map of $\mathrm{Diff}^{\infty}_{c}(\mathbb{T}\times M)$ is pluggable.

The finite regularity counterpart of F is stated as E in Section 1.1 and will
be deduced from F in Section 2.3. We will deduce Theorems A and B and
Corollaries C and D from E in Section 1.2.

In Section 1.3 we precise the topologies of the involved spaces. Also we will
show that the following group is formed by pluggable maps, see Proposition
1.16:

$\mathbb{G}_{1}:=\\{(\theta,y)\mapsto(\theta+\Xi(y),y):\Xi\in
C^{\infty}_{c}(M,\mathbb{R})\\}\;.$

##### Topological group structure on $\mathrm{Plug}$.

It is easier to work with pluggable maps rather than directly the set of total
renormalizations of close to identity maps. Indeed, we will show in
Proposition 2.4 that the set $\mathrm{Plug}$ of pluggable maps endowed with
the composition rule $\circ$ is a group. To prove this, we will define in
Section 2.1 a binary operation $\star$ on compatible plugins $g_{1},g_{2}$
such that the output of $g_{1}\star g_{2}$ is the composition of the outputs
of $g_{1}$ and $g_{2}$. See Fig. 5. In Section 2.2 we will show that the
following group is formed by pluggable maps, see Proposition 2.7:

$\mathbb{G}_{2}:=\left\\{(\theta,y)\in\mathbb{T}\times
M\mapsto(\theta,F(y)):F\in\mathrm{Diff}_{c}^{\infty}(M)\right\\}\;.$

Then in Section 2.3 we will show that $\mathrm{Plug}$ is closed in
$\mathrm{Diff}^{\infty}_{c}(\mathbb{T}\times M)$, see Proposition 2.9. To
prove this, we will construct plugins whose dynamics enlarge an iterate of the
renormalization domain and will perform a perturbation therein. Note that the
elements of $\mathrm{Diff}_{c}^{\infty}(\mathbb{T}\times M)$ generated by
compositions of elements of $\mathbb{G}_{1}$ and $\mathbb{G}_{2}$ have
constant derivatives w.r.t. $\theta$. Consequently we will need to construct
other diffeomorphisms in $\mathrm{Plug}$ to prove F. To do so we will study
the space ${\mathfrak{plug}}$ of vector fields whose flow is pluggable:

${\mathfrak{plug}}:=\\{X\in\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times
M):\mathrm{Fl}^{t}_{X}\in\mathrm{Plug},\quad\forall t\in\mathbb{R}\\}\;,$

where $\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times M)$ denotes the space of
compactly supported smooth vector fields on $\mathbb{T}\times M$ and
$\mathrm{Fl}^{t}_{X}$ denotes the flow of $X$ at time $t$. Using that
$\mathrm{Plug}$ is a closed subgroup, we will deduce in Section 2.4 that
${\mathfrak{plug}}$ is a closed sub-Lie algebra of
$\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times M)$, see Proposition 2.16.

Note that the following Lie algebras are formed by fields whose flows are in
$\mathbb{G}_{1}$ or $\mathbb{G}_{2}$:

${\mathfrak{g}}_{1}=\\{X_{1}:(\theta,y)\in\mathbb{T}\times
M\mapsto(\xi(y),0)\in\mathbb{R}\times TM\ :\ \xi\in
C^{\infty}(M,\mathbb{R})\\}\subset{\mathfrak{plug}}\;,$
${\mathfrak{g}}_{2}=\\{X_{2}:(\theta,y)\in\mathbb{T}\times
M\mapsto(0,f(y))\in\mathbb{R}\times TM\ :\
f\in\mathfrak{diff}_{c}^{\infty}(M)\\}\subset{\mathfrak{plug}}\;.$

##### Construction of pluggable flows.

In Section 3.1, using the connectedness of
$\mathrm{Diff}_{c}^{\infty}(\mathbb{T}\times M)$ and that $\mathrm{Plug}$ is a
closed group, we will show that, to prove F, it suffices to show:

${\mathfrak{plug}}=\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times M)\;.$

This equality will be stated in Proposition 3.3. Its proof relies on two
phenomena. The first one is stated in Proposition 3.7 as:

$\\{Y\in\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times M):\exists
X\in{\mathfrak{plug}}\text{ such that }Y=[X,Y]\\}\subset{\mathfrak{plug}}\;,$

where $[\cdot,\cdot]$ denotes the Lie brackets. This will be proved in Section
3.2 by noting that for any $Y=[X,Y]$, it holds
$\mathrm{Fl}_{Y}^{t}=\mathrm{Fl}_{X}^{s}\circ\mathrm{Fl}_{e^{-s}Y}^{t}\circ\mathrm{Fl}_{X}^{-s}$.
Then by using the same technique as for the proof of the closure of
$\mathrm{Plug}$, we will show that arbitrarily close to identity there exists
a plugin with output $\mathrm{Fl}_{e^{-s}Y}^{t}$ for any $s$ large enough. We
will conclude the proof by doing two $\star$ products of the latter with
plugins with outputs $\mathrm{Fl}_{X}^{s}$ and $\mathrm{Fl}_{X}^{-s}$.

The second phenomenon, stated in Proposition 3.9, is that for any vector field
$T\in\mathfrak{diff}_{c}^{\infty}(M)$, there exist finite families
$(X_{i})_{i},(Y_{i})_{i},(Z_{i})_{i}$ of vector fields in
$\mathfrak{diff}_{c}^{\infty}(M)$ such that:

$T=\sum_{i}[Y_{i},Z_{i}]\quad\text{and}\quad Y_{i}=[X_{i},Y_{i}]\;.$

From this we will deduce the same statement for vector fields in
${\mathfrak{g}}_{2}$. To show Proposition 3.9, we will remark that when
$M=\mathbb{R}$, $X=1$ and $Y(\theta,y)=y$, it holds $Y=[X,Y]$ and for any
$T\in\mathfrak{diff}^{\infty}_{c}(\mathbb{R})$, with
$Z=\int_{-\infty}^{y}T(t)dt$, it also holds $T=[Y,Z]$. Then we will deduce a
compactly supported and parametric version of this property which will enable
to prove Proposition 3.9 in the case $M=\mathbb{R}^{n}$. Finally, we will use
a partition of unity to deduce the proposition for any manifold.

These phenomena will enable to prove Proposition 3.1 stating that for any
$T\in{\mathfrak{g}}_{2}$ and $\phi\in C_{c}^{\infty}(\mathbb{T}\times M)$
depending only on $\theta$, the field $\phi\cdot T$ is in ${\mathfrak{plug}}$.
Indeed, by the second phenomenon, there are
$X_{i},Y_{i},Z_{i}\in{\mathfrak{g}}_{2}$ s.t.:

$\phi\cdot T=\sum_{i}\phi\cdot[Y_{i},Z_{i}]=\sum_{i}[\phi\cdot
Y_{i},Z_{i}]\quad\text{and}\quad Y_{i}=[X_{i},Y_{i}]\;.$

Also a simple computation shows that $[\phi\cdot Y_{i},X_{i}]=\phi\cdot
Y_{i}$. As $X_{i}$ is in ${\mathfrak{g}}_{2}\subset{\mathfrak{plug}}$, we
deduce by the first phenomenon that $\phi\cdot Y_{i}$ is in
${\mathfrak{plug}}$. This gives that $\phi\cdot T\in{\mathfrak{plug}}$ as
stated in Proposition 3.1. In Section 3.1, we will use Fourier’s Theorem and
the closure of ${\mathfrak{plug}}$ to deduce Proposition 3.1 that any vector
field of the form $(0,Y(\theta,y))$ is in ${\mathfrak{plug}}$. Finally using
this with a Lie bracket with an element of ${\mathfrak{g}}_{1}$ will enable to
obtain that any vector field has a pluggable flow
(${\mathfrak{plug}}=\mathfrak{diff}^{\infty}_{c}(\mathbb{T}\times M)$) as
stated by Proposition 3.3.

##### Parametric counterparts.

At the end of each subsection, we will prove a parametric generalization of
the aforementioned statements. This will enable to show the parametric
counterpart F′ of F. It will imply the parametric counterparts E′ of E and B′
of B.

##  1 Plugins and Pluggable dynamics

###  1.1 Plugins

For the rest of this article, we fix $1\leq r\leq\infty$ and compact connected
manifolds $M$ and ${\mathscr{P}}$ of $\dim\geq 1$.

For $\sigma>0$, define the rotation:

$R_{\sigma}:(\theta,y)\in\mathbb{T}\times
M\to(\theta+\sigma,y)\in\mathbb{T}\times M$

We are now ready to introduce:

######  Definition 1.1.

A plugin with step $\sigma\in\\{2^{-k}:k\geq 1\\}$ is a map
$g\in\mathrm{Diff}^{r}(\mathbb{T}\times M)$ satisfying the following
assertions:

  1. $(i)$

$g$ restricted to $\Delta_{\sigma}:=[0,\sigma)\times M$ is equal to
$R_{\sigma}$,

  2. $(ii)$

the first return time in $\Delta_{\sigma}$ of $g$ is a well defined and
bounded function $\tau:\Delta_{\sigma}\to\mathbb{N}^{*}$,

  3. $(iii)$

the union of the iterates $\bigcup_{k\geq 0}g^{k}(\Delta_{\sigma})$ equals
$\mathbb{T}\times M$.

######  Remark 1.2.

One can show by compactness of $M$ that, under condition $(i)$, condition
$(ii)$ is equivalent to $(iii)$ and that in $(ii)$ the return time is
necessarily bounded.

$\mathbb{T}\times M\colon$ $\theta$$y$ $g$$\Delta$$\sigma$ $g$

Figure 1: Plugin $g$ of step $\sigma$.

Let
$H_{\sigma}:=(\theta,y)\in\Delta_{\sigma}\mapsto(\theta/\sigma,y)\in\mathbb{T}\times
M$. It is a bijective local diffeomorphism.

$H\colon$

Figure 2: Rescaling map $H_{\sigma}\colon\Delta_{\sigma}\to\mathbb{T}\times
M$.

######  Definition 1.3.

The output of a plugin $g$ of step $\sigma$ is the following rescaling of the
first return map $g^{\tau}\colon\Delta_{\sigma}\to\Delta_{\sigma}$:

$G:=H_{\sigma}\circ g^{\tau}\circ H^{-1}_{\sigma}\colon\mathbb{T}\times
M\to\mathbb{T}\times M.$

######  Example 1.4.

For every $k\geq 0$, the map $g_{k}:(\theta,y)\mapsto(\theta+2^{-k},y)$ is a
plugin of step $2^{-k}$, iteration $2^{k}$ and output the identity.

Actually, we can show that the output of a plugin is always smooth:

######  Proposition 1.5.

Let $1\leq r\leq\infty$. The output of a plugin
$g\in\mathrm{Diff}^{r}(\mathbb{T}\times M)$ is in
$\mathrm{Diff}^{r}(\mathbb{T}\times M)$ and depends continuously on $g$. In
particular, the output is a total renormalization of the plugin.

This proposition is a consequence of a classical, yet beautiful, argument
which will be recalled in Appendix A. We are now ready to state the general
result:

######  Theorem E (Main).

Let $1\leq r\leq\infty$ and a compact manifold $M$ of dimension $\geq 1$, and
let ${\mathcal{N}}\subset\mathrm{Diff}^{r}(\mathbb{T}\times M)$ be a
neighborhood of the identity. Then any
$G\in\mathrm{Diff}^{r}_{c}(\mathbb{T}\times M)$ is the output of some plugin
$g_{G}\in{\mathcal{N}}$.

And here is its parametric counterpart:

######  Theorem E′.

Let $1\leq r\leq\infty$ and a compact manifold $M$ of dimension $\geq 1$, fix
a compact manifold ${\mathscr{P}}$ and a compactly supported family
$(G_{p})_{p\in{\mathscr{P}}}$ in $\mathrm{Diff}_{c}^{r}(\mathbb{T}\times
M)_{\mathscr{P}}$. Let ${\mathcal{N}}\subset\mathrm{Diff}^{r}(\mathbb{T}\times
M)_{\mathscr{P}}$ be a neighborhood of $(id)_{p\in{\mathscr{P}}}$. Then there
exists a $C^{r}$-family of plugins
$(g_{p})_{p\in{\mathscr{P}}}\in{\mathcal{N}}$ such that $G_{p}$ is the output
of $g_{p}$ for every $p\in{\mathscr{P}}$.

###  1.2 Proof of the corollaries of the main E

Observe that E implies immediately B and that E′ implies immediately B′. In
this subsection we show that E implies furthermore A and Corollaries C and D.
To this end, the following will be useful:

######  Fact 1.6.

If the output of a plugin restricted to $\\{0\\}\times M$ is the identity,
then the return time $\tau$ of the plugin is constant on the renormalization
domain $\Delta$.

###### Proof.

As $f$ preserves the orientation it suffices to show that $\tau$ is constant
on the interior of $\Delta$. As $\tau$ is integer valued and $M$ is connected,
it suffices to show that $\tau$ is continuous on $int\,\Delta$ to conclude the
proof. We start by showing that $\tau$ is lower semi-continuous on $\Delta$.
Suppose that it is not the case. Then there exist $N\geq 2$, a point
$x\in\Delta$ and a point $x^{\prime}\in\Delta$ arbitrarily close to $x$ such
that $N=\tau(x^{\prime})<\tau(x)$. By continuity of $g$, it holds
$g^{N}(x)\in\mathrm{cl}(\Delta)$. By assumption, we have
$g^{N}(x)\notin\Delta$. Thus $g^{N}(x)\in\\{\sigma\\}\times M$ and since $g$
is the translation by $\sigma$ on $\Delta$ it holds then that
$g^{N-1}(x)\in\Delta$ and consequently $\tau(x)<N$, which contradicts the
assumption.

We now show that $\tau$ is upper semi-continuous on $int\,\Delta$. Consider
$x\in int\,\Delta$. As $g^{\tau}$ is a bijection of $\Delta$ which leaves
invariant $\\{0\\}\times M$, it comes that $g^{\tau}(x)$ is in the interior of
$\Delta$. Then for every $x^{\prime}$ close to $x$, the iterate
$g^{\tau(x)}(x^{\prime})$ belongs to $\Delta$ and so
$\tau(x^{\prime})\leq\tau(x)$. ∎

###### Proof that E implies A .

Let $G$ be in
$\mathrm{Diff}^{r}(\mathbb{B}^{n})=\mathrm{Diff}^{r}_{0}(\mathbb{B}^{n})$. We
observe111 By connectedness, there exists an isotopy $(h_{t})_{t\in[0,1]}$
between $G|\partial\mathbb{B}^{n}$ and $id_{\partial\mathbb{B}^{n}}$ that can
be chosen $C^{r}$-smooth, made of diffeomorphisms and and flat at the
endpoints. Note that we can extend $G$ on
$2\cdot\mathbb{B}^{n}\setminus\mathbb{B}^{n}$ by
$h_{\|z\|-1}\left(\frac{z}{\|z\|}\right)$ to construct and element of
$\mathrm{Diff}^{r}_{0}(2\cdot\mathbb{B}^{n})$. that $G$ is the restriction to
$\mathbb{B}_{n}$ of a diffeomorphism $\bar{G}$ in
$\mathrm{Diff}^{r}_{c}(2\cdot\mathbb{B}^{n})$. As $2\cdot id$ conjugates
$\bar{G}$ to a map in $\mathrm{Diff}^{r}_{c}(\mathbb{B}^{n})$, without any
loss of generality we can assume that $G$ belongs to
$\mathrm{Diff}^{r}_{c}(\mathbb{B}^{n})$.

Let $M:=\mathbb{B}^{n-1}$ and embed
$i:\mathbb{B}^{n}\hookrightarrow\mathbb{T}\times M$ so that the embedded ball
$i(\mathbb{B}^{n})$ does not meet $\\{0\\}\times M$ nor the boundary of
$\mathbb{T}\times M$. Extend then $G$ by $\mathrm{id}$ to a diffeomorphism
$\tilde{G}\in\mathrm{Diff}_{0}^{r}(\mathbb{T}\times M)$. By E, $\tilde{G}$ is
the output of a plugin $\tilde{g}\in\mathrm{Diff}^{r}(\mathbb{T}\times M)$
arbitrarily close to $\mathrm{id}$. Since $\tilde{G}$ leaves $\\{0\\}\times M$
invariant it holds by 1.6 that the first return time $\tau$ of the plugin
$\tilde{g}$ is a constant $N$. In particular, the restriction
$\tilde{g}^{N}|i(\mathbb{B}^{n})$ is conjugate to $G$.

$\tilde{G}\colon\\!\\!\raisebox{-20.66666pt}{
\begin{picture}(1.0,1.0)\put(0.0,0.0){\includegraphics[width=55.0pt]{PluginTakensQuestion1.eps}}\put(3.44241,0.08037){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{}}}\put(0.75722,0.48161){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{$\mathrm{id}$}}}\put(0.43611,0.12339){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{$G$}}}\put(0.27493,-0.16654){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{$\mathbb{T}\times
M$}}}\end{picture}}\mbox{ is}\underset{\mbox{$\tilde{g}$ close to
$\mathrm{id}$}}{\mbox{ the output of
}\tilde{g}}\colon\raisebox{-20.66666pt}{\begin{picture}(1.0,1.0)\put(0.0,0.0){\includegraphics[width=55.0pt]{PluginTakensQuestion2.eps}}\put(1.55104,0.07414){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{}}}\put(0.25475,-0.17278){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{$\mathbb{T}\times
M$}}}\end{picture}}\lhook\joinrel\xrightarrow{\begin{array}[]{c}\mbox{embed}\\\
\mbox{in
}\mathbb{B}^{d}\end{array}}\raisebox{-20.66666pt}{\begin{picture}(1.0,1.0)\put(0.0,0.0){\includegraphics[width=55.0pt]{PluginTakensQuestion3ter.eps}}\put(0.43332,0.07414){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{}}}\put(0.38041,-0.17278){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{$\mathbb{B}^{n}$}}}\end{picture}}\begin{array}[]{c}\mbox{Extend
to }\\\ g\in\mathrm{Diff}(\mathbb{B}^{d})\end{array}$ Figure 3: Proof of A.

To conclude it suffices to embed $\mathbb{T}\times M$ into the interior of
$\mathbb{B}^{n}$ and extend $\tilde{g}$ then the diffeomorphism $\tilde{g}$ to
a diffeomorphism $g$ of $\mathbb{B}^{n}$ that is close to $\mathrm{id}$. ∎

The following enables to localize nearby the identity the existence of some
ergodic properties:

######  Proposition 1.7.

Let $g$ be a plugin of step $\sigma$ and $G$ its output. Then there is a
canonical bijection $\mu_{g}\to\mu_{G}$ between $G$-invariant probability
measures and $g$-invariant probability measures:

$\mu_{g}\mapsto\mu_{G}:=H_{\sigma*}\frac{\mu_{g}|\Delta}{\mu_{g}(\Delta)}\quad\text{and}\quad\mu_{G}\mapsto\mu_{g}:=\sum_{N\geq
0}g^{*N}H_{\sigma}^{*}\mu_{G}|\\{\tau>N\\}\;.$

Moreover:

  1. 1.

$\mu_{g}$ is ergodic iff $\mu_{G}$ is ergodic,

  2. 2.

$\mu_{g}$ is hyperbolic iff $\mu_{G}$ is hyperbolic.

  3. 3.

$\mu_{g}$ is a smooth volume form iff $\mu_{G}$ is a smooth volume form,

###### Proof.

We recall that $H_{\sigma}$ conjugates $G$ with the first return time
$g^{\tau}$ of $g$ in $\Delta$. Then it is sufficient to prove the analogous
properties for the following classical bijection between $g^{\tau}$-invariant
probability measures and $g$-invariant probability measures:

$\mu_{g^{\tau}}\mapsto\mu_{g}:=\sum_{N\geq
0}g^{*N}\mu_{g^{\tau}}|\\{\tau>N\\}\quad\text{and}\quad\mu_{g}\mapsto\mu_{g^{\tau}}:=\frac{\mu_{g}|\Delta}{\mu_{g}(\Delta)}$

Indeed it is well known that $\mu_{g^{\tau}}$ is ergodic iff $\mu_{g}$ is
ergodic (see for instance [EW13, Lemma 2.43]), this implies 1. Also it well
known that the Lyapunov exponent of $\mu_{G}$ are equal to the mean of $\tau$
times the Lyapunov exponent of $\mu_{g}$, from which we deduce 2.

Let us prove 3. If $\mu_{G}$ is smooth, then $\mu_{g}|\Delta=\mu_{g^{\tau}}$
is smooth on $\Delta$ and $\mu_{g}|\Delta\cup
g(\Delta)=\mu_{g^{\tau}}+g^{*}\mu_{g^{\tau}}$ is smooth on $\Delta\cup
g(\Delta)$. Now observe that for every $x\in\mathbb{T}\times M$ there exists a
neighborhood $U$ and $N\geq 0$ such that $g^{-N}(U)\subset\Delta\cup
g(\Delta)$. Then by invariance, the density of $\mu_{g}$ on $U$ is as smooth
as $\mu_{g}|\Delta\cup g(\Delta)$. Conversely, if $\mu_{g}$ is smooth, then
$\mu_{g^{\tau}}:=\frac{\mu_{g}|\Delta}{\mu_{g}(\Delta)}$ must be smooth. Since
$g$ is a plugin, the density $\mu_{g}$ at the left hand side of $\Delta$ is
equal to the translation by $(\sigma,0)$ of the density of $\mu_{g}$ at the
right hand side of $cl(\Delta)$. Thus $\mu_{g^{\tau}}$ is pushed forward by
$H_{\sigma}$ to a smooth volume form $\mu_{G}$ on $\mathbb{T}\times M$. ∎

###### Proof of Corollary C.

Katok in [Kat79][Thm B] showed the existence of Bernoulli diffeomorphisms
homotopic to the identity and preserving a given smooth measure on any
surface. Hence there exists $G\in\mathrm{Diff}^{\infty}_{0}(\mathbb{T}^{2})$
preserving a smooth Bernouilli volume form $\mu_{G}$. Equivalently by the
Pesin Theorem the measure $\mu_{G}$ is a volume form which is ergodic and
hyperbolic. By E, arbitrarily close to the identity, there exists $g$ whose
output is $G$. Then by Proposition 1.7, the volume form $\mu_{G}$ induces an
ergodic and hyperbolic volume form $\mu_{g}$ for $g$. ∎

We now proceed to:

###### Proof of Corollary D.

We will construct a sequence $(\Delta_{i})_{i\geq 1}$ of domains with disjoint
closures and of the form
$\Delta_{i}:=[0,\sigma_{i})\times(s_{i},s_{i}+l_{i})\subset\mathbb{T}\times[0,1]$
with $0<\sigma_{i}<1$ and $0<s_{i}<1-l_{i}$. Then we will be able to construct
a map $f$ that renormalizes to $F_{i}$ on each $\Delta_{i}$ and such that the
orbit of the renormalization domain $\Delta_{i}$ is exactly
$\mathbb{T}\times(s_{i},s_{i}+l_{i})$. Moreover we will be able to take $f$
arbitrarily close to identity.

To do so, consider a sequence $(l^{\prime}_{i})_{i\geq 0}$ such that
$\sum_{i}l^{\prime}_{i}<1$ and $l_{i}^{\prime}>l_{i}$ for any $i\geq 0$. For
$i\geq 0$ denote $s_{i}:=\sum_{j<i}l^{\prime}_{j}$ and
$B_{i}:=\mathbb{T}\times[s_{i},s_{i}+l_{i}]$. Note that $B_{i}$ has volume
$l_{i}$ and that $\mathrm{cl}(B_{i})\cap\mathrm{cl}(B_{j})=\emptyset$ for any
$i\neq j\geq 0$. Let us define the map
$\psi_{i}:(\theta,y)\in\mathbb{T}\times[0,1]\mapsto(\theta,s_{i}+l_{i}\cdot
y)$ that sends $\mathbb{T}\times[0,1]$ to $\mathrm{cl}(B_{i})$. Now by E, for
any $i\geq 0$, there exists a plugin
$\tilde{f}_{i}:\mathbb{T}\times[0,1]\to\mathbb{T}\times[0,1]$ arbitrarily
close to identity with renormalization domain with output $F_{i}$. Let step
$\sigma_{i}$ be its step. The map
$f_{i}:=\psi_{i}\circ\tilde{f}_{i}\circ\psi_{i}^{-1}:B_{i}\to B_{i}$ is close
to identity. Also a renormalization of $f_{i}$ associated to
$\Delta_{i}:=[0,\sigma_{i})\times[s_{i},s_{i}+l_{i}]$ is $F_{i}$. Since the
supports $\mathrm{supp}\>f_{i}=\mathrm{cl}(B_{j})$ are disjoint and each
$f_{i}$ is close to identity, there exists a close to identity map
$f:\mathbb{T}\times[0,1]\to\mathbb{T}\times[0,1]$ that coincides with $f_{i}$
on $B_{i}$. Such a map verifies the conditions of the Corollary D. ∎

###  1.3 Pluggable dynamics

We will first work in the $C^{\infty}$-topology. Indeed, the proof relies on
the fact that $\mathrm{Diff}^{\infty}(V)$ is a Fréchet Lie group, which is not
the case of $\mathrm{Diff}^{r}(V)$ for $r<\infty$. The $C^{r}$ case will be
deduced from the $C^{\infty}$ case in Section 2.3.

####  1.3.1 Topologies on spaces of smooth maps and parameter families

We endow $V$ with a Riemannian metric. Let $\mathfrak{diff}^{\infty}(V)$ be
the space of smooth vector fields on $V$ that are tangent to the boundary (if
any). We endow $\mathrm{Diff}^{\infty}(V)$ and the space
$\mathfrak{diff}^{\infty}(V)$ with the following distances:

$d_{C^{\infty}}(f,g)=\max_{x\in V}\sum_{k\geq
1}2^{-k}\min(1,d(D_{x}^{k}f,D^{k}_{x}g))$

and

$d_{C^{\infty}}(X,Y)=\max_{x\in V}\sum_{k\geq
1}2^{-k}\min(1,\|D_{x}^{k}X-D^{k}_{x}Y\|_{C^{r}})\;.$

For these distances $\mathrm{Diff}^{\infty}(V)$ and
$\mathfrak{diff}^{\infty}(V)$ are complete. Actually
$\mathrm{Diff}^{\infty}(V)$ is a Lie group with algebra the Fréchet space
$\mathfrak{diff}^{\infty}(V)$. We endow the connected component
$\mathrm{Diff}^{\infty}_{0}(V)$ of the identity with the topology induced by
$\mathrm{Diff}^{\infty}(V)$.

On the other hand, we endow the space $\mathrm{Diff}^{\infty}_{c}(V)$ and the
space of compactly supported smooth vector fields
$\mathfrak{diff}^{\infty}_{c}(V)$ with the finer Whitney topology. A basis of
open sets of these respective topologies is:

$U_{\eta,f,m}:=\\{g\in\mathrm{Diff}^{\infty}_{c}(V):d(D_{x}^{k}f,D^{k}_{x}g)\leq\eta(x),\forall
k\leq m\\}$

and

$U_{\eta,X,m}:=\\{Y\in\mathfrak{diff}^{\infty}_{c}(V):\|D_{x}^{k}X-D^{k}_{x}Y\|\leq\eta(x),\forall
k\leq m\\}\;,$

among $m\in\mathbb{N}$, $f\in\mathrm{Diff}^{\infty}_{c}(V)$,
$X\in\mathfrak{diff}^{\infty}_{c}(V)$, and positive continuous functions
$\eta:V\setminus\partial V\to(0,\infty)$. A well known theorem asserts that
$f_{n}\to f$ in $\mathrm{Diff}^{\infty}_{c}(V)$ if and only if there exists a
compact subset $K\subset V\setminus\partial V$ such that the supports of $f$
and $f_{n}$ are included in $K$ for $n$ and for every $r\geq 1$, we have
$f_{n}\to f$ in the uniform $C^{r}$-topology when $n\to\infty$. The analogous
property holds true for $\mathfrak{diff}^{\infty}_{c}(V)$.

We endow $\mathrm{Diff}^{r}(V)_{\mathscr{P}}$ and
$\mathrm{Diff}_{c}^{r}(V)_{{\mathscr{P}}}$ with the topologies induced by
$\mathrm{Diff}^{r}(V\times{\mathscr{P}})$ and
$\mathrm{Diff}_{c}^{r}(V\times{\mathscr{P}})$ with the inclusions:

$\mathrm{Diff}^{r}(V)_{{\mathscr{P}}}\hookrightarrow\mathrm{Diff}^{r}(V\times{\mathscr{P}})\quad\text{and}\quad\mathrm{Diff}_{c}^{r}(V)_{{\mathscr{P}}}\hookrightarrow\mathrm{Diff}_{c}^{r}(V\times{\mathscr{P}})$

via
$f_{\mathscr{P}}=(f_{p})_{p\in{\mathscr{P}}}\mapsto\widehat{f_{\mathscr{P}}}$
where:

$\widehat{f_{\mathscr{P}}}:=(v,p)\in
V\times{\mathscr{P}}\mapsto(f_{p}(v),p)\;.$

The spaces $\mathrm{Diff}^{r}(V)_{\mathscr{P}}$ and
$\mathrm{Diff}_{c}^{r}(V)_{\mathscr{P}}$ endowed with the composition law
$f_{\mathscr{P}}\circ g_{\mathscr{P}}:=(f_{p}\circ g_{p})_{p\in{\mathscr{P}}}$
are groups.

Note that we have:

######  Fact 1.8.

The following inclusion is a morphism of topological group:

$f_{\mathscr{P}}=(f_{p})_{p\in{\mathscr{P}}}\in\mathrm{Diff}^{\infty}_{0}(V)_{\mathscr{P}}\mapsto\widehat{f_{\mathscr{P}}}\in\mathrm{Diff}^{\infty}_{0}(V\times{\mathscr{P}})\;.$

The latter fact will be used among some proofs of the parametric counterparts.

From now on we will work with $V=\mathbb{T}\times M$.

####  1.3.2 The space $\mathrm{Plug}$ of Pluggable maps

Every time, but when explicitly stated, we will focus on the case $r=\infty$:
plugins will be of class $C^{\infty}$ and their outputs as well. Recall that
we endow $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ with the Euclidean Riemannian
metric, $M$ and ${\mathscr{P}}$ with their Riemannian metric. The product
spaces $\mathbb{T}\times M$, $\mathbb{T}\times M\times{\mathscr{P}}$ and
$M\times{\mathscr{P}}$ are endowed with their product Riemannian metric.

######  Definition 1.9.

A map $G\in\mathrm{Diff}^{\infty}_{c}(\mathbb{T}\times M)$ is semi-pluggable
if there is a sequence $(g_{k})_{k\geq 1}$ of plugins
$g_{k}\in\mathrm{Diff}^{\infty}(\mathbb{T}\times M)$ with step $2^{-k}$, so
that for every large integer $k$ the output of $g_{k}$ is $G$ and
$g_{k}\to\mathrm{id}$ for the $C^{\infty}$-topology when $k\to\infty$. The map
$G$ is pluggable if $G$ and $G^{-1}$ are semi-pluggable. Let:

$\mathrm{Plug}:=\\{G\in\mathrm{Diff}^{\infty}_{c}(\mathbb{T}\times M):G\text{
is pluggable}\\}\;.$

######  Example 1.10.

The identity $\mathrm{id}$ of $\mathbb{T}\times M$ is pluggable with the
family $\left((\theta,y)\in\mathbb{T}\times
M\mapsto(\theta+2^{-k},y)\right)_{k\geq 1}$ as in Example 1.4.

We give in Section 1.3.3 more sophisticated examples of pluggable dynamics.
Note that in Definition 1.9, we are only interested by plugins whose output is
compactly supported (outside of the boundary). This is because not every
mapping of $\mathrm{Diff}^{\infty}_{0}(\mathbb{T}\times M)$ is pluggable by
0.11, while we will show that every map in
$\mathrm{Diff}^{\infty}_{c}(\mathbb{T}\times M)$ is pluggable:

######  Theorem F.

We have $\mathrm{Plug}=\mathrm{Diff}^{\infty}_{c}(\mathbb{T}\times M)$.

Observe that the first assertion of E in the case $r=\infty$ is an immediate
consequence of F.

##### Parametric counterpart.

The proof of F is (basically) constructive and depends smoothly on the output.
For the sake of completeness, we will verify this by giving the parametric
counterpart of each statement. Some of the proofs will be designed to be
verbatim the same. For a first reading of the proofs, we advise the reader to
skip the all parametric counterpart of the arguments. Here is the parametric
counterpart of the notion of plugin:

######  Definition 1.11.

A family
$g_{\mathscr{P}}=(g_{p})_{p\in{\mathscr{P}}}\in\mathrm{Diff}^{\infty}(\mathbb{T}\times
M)_{\mathscr{P}}$ defines a ${\mathscr{P}}$-plugin if the diffeomorphism:

$\widehat{g_{\mathscr{P}}}:(z,p)\in M\times{\mathscr{P}}\mapsto(g_{p}(x),p)$

in $\mathrm{Diff}^{\infty}(\mathbb{T}\times M\times{\mathscr{P}})$ is a
plugin.

Similarly we have the parametric counterpart of the output of a
${\mathscr{P}}$-plugin:

######  Definition 1.12.

The output of a ${\mathscr{P}}$-plugin
$g_{\mathscr{P}}\in\mathrm{Diff}^{\infty}(\mathbb{T}\times M)_{\mathscr{P}}$
of step $\sigma$ is the family $G_{\mathscr{P}}$ such that for each
$p\in{\mathscr{P}}$, the map $G_{p}$ is the rescaling of the return map
$g_{p}^{\tau_{p}}\colon\Delta_{\sigma}\to\Delta_{\sigma}$:

$G_{p}:=H_{\sigma}\circ g_{p}^{\tau_{p}}\circ
H^{-1}_{\sigma}\colon\mathbb{T}\times M\to\mathbb{T}\times M.$

By Proposition 1.5 the output $\widehat{G_{\mathscr{P}}}$ of the plugin
$\widehat{g_{\mathscr{P}}}$ on $\mathbb{T}\times M\times{\mathscr{P}}$ is
smooth and so:

######  Fact 1.13.

The output of a ${\mathscr{P}}$-plugin of $\mathbb{T}\times M$ lies in
$\mathrm{Diff}_{0}^{r}(\mathbb{T}\times V)_{{\mathscr{P}}}$.

In the parametric setting, Definition 1.9 becomes:

######  Definition 1.14.

A family $G_{\mathscr{P}}\in\mathrm{Diff}_{c}^{\infty}(\mathbb{T}\times
M)_{\mathscr{P}}$ is ${\mathscr{P}}$-semi-pluggable if there is a sequence
$\left(g_{k{\mathscr{P}}}\right)_{k\geq 1}$ of ${\mathscr{P}}$-plugins with
step $2^{-k}$, so that for every large $k$, the output of $g_{k{\mathscr{P}}}$
is $G_{\mathscr{P}}$ and
$\widehat{g_{k{\mathscr{P}}}}\to\mathrm{id}_{\mathbb{T}\times
M\times{\mathscr{P}}}$ for the $C^{\infty}$-topology. The family of maps
$G_{\mathscr{P}}$ is ${\mathscr{P}}$-semi-pluggable if $G_{\mathscr{P}}$ and
$G_{\mathscr{P}}^{-1}:=(G^{-1}_{p})_{p\in{\mathscr{P}}}$ are
${\mathscr{P}}$-semi-pluggable. Let:

$\mathrm{Plug}_{\mathscr{P}}:=\\{G_{\mathscr{P}}\in\mathrm{Diff}^{\infty}_{c}(\mathbb{T}\times
M)_{\mathscr{P}}:G_{\mathscr{P}}\text{ is ${\mathscr{P}}$-pluggable}\\}\;.$

To obtain E′ in the case $r=\infty$, it suffices to prove:

######  Theorem F′.

We have
$\mathrm{Plug}_{\mathscr{P}}=\mathrm{Diff}^{\infty}_{c}(\mathbb{T}\times
M)_{{\mathscr{P}}}$.

####  1.3.3 Examples of pluggable dynamics

Consider the following subgroup of
$\mathrm{Diff}_{c}^{\infty}(\mathbb{T}\times M)$:

$\mathbb{G}_{1}:=\\{(\theta,y)\in\mathbb{T}\times
M\mapsto(\theta+\nu(y),y):\nu\in C^{\infty}_{c}(M,\mathbb{R})\\}\;.$

This gives a first example of a subgroup of pluggable maps:

$\mathbb{G}_{1}\subset{\mathfrak{plug}}\;.$

######  Proposition 1.15.

The group is included in $\mathrm{Plug}$.

The subgroup $\mathbb{G}_{1}$ was first studied in [BT22] into a set of
generators of symplectomorphisms.

$y$$2^{-k}\nu(y)$$0$$\Delta$$T_{1}\Delta$ Figure 4: Dynamics of an element of
$\mathbb{G}_{1}$.

###### Proof.

Let $\rho\in C^{\infty}(\mathbb{T},\mathbb{R}^{+})$ be a function with support
contained in $[\tfrac{1}{3},\tfrac{2}{3}]\subset\mathbb{T}$ and integral $1$.
For $\epsilon\in[-2^{-k-2},2^{-k-2}]$, we define the smooth vector field:

(1.1) $X_{\epsilon}:\theta\in\mathbb{T}\mapsto
2^{-k}/(1+\epsilon\cdot\rho(\theta))\;.$

Let $\phi^{t}_{\epsilon}$ be the flow of $X_{\epsilon}$. The time taken to go
all around the circle equals to:

(1.2)
$\tau(\epsilon)=\int_{\mathbb{T}}\frac{1}{X_{\epsilon}(\theta)}d\theta=\int_{\mathbb{T}}2^{k}\cdot(1+\epsilon\cdot\rho(\theta))d\theta=2^{k}\cdot(1+\epsilon)\;.$

If we stop at time $2^{k}$, then the lacking or exceeding time for a complete
lap is $2^{k}\epsilon\in[-\tfrac{1}{4},\tfrac{1}{4}]$. As nearby $0$ the
vector field $X_{\epsilon}$ equal $2^{-k}$, this implies that the image of $0$
by $\phi^{2^{k}}_{\epsilon}$ is equal to $\epsilon$. So for every $\nu\in
C^{\infty}_{c}(M,\mathbb{R})$, for every $k$ large, the map:

(1.3) $g_{k}:=(\theta,y)\mapsto(\phi^{1}_{2^{-k}\nu(y)}(\theta),y)$

coincides with $R_{2^{-k}}$ on a set which contains the complement of
$S=[\tfrac{1}{4},\tfrac{3}{4}]\times\mathrm{supp}\>\nu$. Thus $g_{k}$ is a
plugin with step $2^{-k}$. Furthermore, by the above discussion, its output
equals to:

(1.4) $G:(\theta,y)\mapsto\left(\theta+\nu(y),y\right)\text{ with }\nu\in
C^{\infty}_{c}(M,\mathbb{R})\;.$

Finally observe that $g_{k}$ is $C^{\infty}$-close to identity. Thus $G$ is
semi-pluggable. Hence $\mathbb{G}_{1}$ is formed by semi-pluggable maps, and
as $\mathbb{G}_{1}$ is a group, it is formed by pluggable maps. ∎

##### Parametric counterpart.

Let $\mathbb{G}_{1{\mathscr{P}}}$ be the subset of
$\mathrm{Diff}_{c}^{\infty}(\mathbb{T}\times M)_{\mathscr{P}}$ formed by
families $(G_{p})_{p\in{\mathscr{P}}}$ such that $G_{p}\in\mathbb{G}_{1}$ for
every $p\in{\mathscr{P}}$.

We have similarly:

######  Proposition 1.16.

The group $\mathbb{G}_{1{\mathscr{P}}}$ is included in
$\mathrm{Plug}_{\mathscr{P}}$.

###### Proof.

An element $(G_{p})_{p\in{\mathscr{P}}}\in\mathbb{G}_{1{\mathscr{P}}}$ is
formed by mapping of the form $G_{p}:(\theta,y)\mapsto(\theta+\nu_{p}(y),y)$
with $(\nu_{p})_{p\in{\mathscr{P}}}$ smooth. Thus the vector field
$Z_{p}:(\theta,y)\mapsto(X_{2^{-k}\nu_{p}}(y),0)$ depends smoothly on $p$.
Hence the time one maps $g_{p}$ form a family of plugins
$(g_{p})_{p\in{\mathscr{P}}}$ (with output $G_{p}$ and step $2^{-k}$) which
depends smoothly on $p\in{\mathscr{P}}$. ∎

##  2 Topological group structure on $\mathrm{Plug}$

###  2.1 Group structure on $\mathrm{Plug}$

In this section we show that $\mathrm{Plug}$ endowed with the composition rule
is a group. To this end, let $\pi:\bar{\mathbb{T}}\times
M:=\mathbb{R}/2\mathbb{Z}\times M\to\mathbb{T}\times
M=\mathbb{R}/\mathbb{Z}\times M$ be the canonical 2-sheeted covering map and
denote by $\psi:(\theta,y)\in\bar{\mathbb{T}}\times
M\mapsto(\theta/2,y)\in\mathbb{T}\times M$ the canonical diffeomorphism. The
following defines a binary operation $\star$ on the space of plugins:

######  Definition 2.1.

Let $g_{1}$ and $g_{0}$ be two plugins with same step $\sigma=2^{-k}$. Let
$\bar{g}_{1}$ and
$\bar{g}_{0}\in\mathrm{Diff}^{\infty}(\widehat{\mathbb{T}}\times M)$ be the
lifts of $g_{1}$ and $g_{0}$ such that
$\bar{g}_{1}(0,y)=\bar{g}_{0}(0,y)=(\sigma,y)$ for every $y\in M$. Let
$\overline{g_{1}\star g_{0}}$ be equal to the lift $\bar{g}_{0}$ on the first
half of $\bar{\mathbb{T}}\times M$ and be equal to the lift of $\bar{g}_{1}$
on second half of $\bar{\mathbb{T}}\times M$:

$\overline{g_{1}\star g_{0}}:(\theta,y)\in\bar{\mathbb{T}}\times
M\mapsto\left\\{\begin{array}[]{cl}\bar{g}_{0}(\theta,y)&\text{if
}\theta\in[0,1)+2\mathbb{Z}\;,\\\ \bar{g}_{1}(\theta,y),&\text{if
}\theta\in[1,2)+2\mathbb{Z}\;,\end{array}\right.\text{ and put }g_{1}\star
g_{0}:=\psi^{-1}\circ\overline{g_{1}\star g_{0}}\circ\psi\;.$

######  Remark 2.2.

Given a neighborhood $V$ of $\mathrm{id}$ in
$\mathrm{Diff}^{r}_{0}(\mathbb{T}\times M)$, there exists a neighborhood $W$
of $\mathrm{id}$ such that for any pair of plugins $f,g\in W$ of same steps,
we have $f\star g\in V$.

$\raisebox{-6.88889pt}{\begin{picture}(1.0,1.0)\put(0.0,0.0){\includegraphics[width=30.0pt]{Concatenation1.eps}}\put(2.17198,0.08424){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{}}}\put(3.9566,0.26885){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{}}}\put(8.23352,-0.43884){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{}}}\put(-0.34322,-0.44397){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{$\begin{array}[]{c}\mbox{\footnotesize{plugins
}}g_{i}\\\ \mbox{\footnotesize{of step
}}2^{-k}\end{array}$}}}\put(0.20275,1.13039){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{$\mathbb{T}\times
M$}}}\end{picture}}\;\xrightarrow{\begin{array}[]{c}\mbox{$2$-sheeted}\\\
\mbox{lift}\end{array}}\;\raisebox{-17.22221pt}{\begin{picture}(1.0,1.0)\put(0.0,0.0){\includegraphics[width=50.0pt]{Concatenation2.eps}}\put(0.03032,0.38218){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{}}}\put(1.2262,0.50589){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{}}}\put(0.11371,1.08027){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{$\nicefrac{{\mathbb{R}}}{{2\mathbb{Z}}}\times
M$}}}\put(0.34389,-0.19956){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{$\bar{g}_{i}$}}}\put(4.09218,0.03166){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{}}}\end{picture}}\quad\xrightarrow{\begin{array}[]{c}\mbox{cut
\&}\\\
\mbox{paste}\end{array}}\quad\raisebox{-17.22221pt}{\begin{picture}(1.0,1.0)\put(0.0,0.0){\includegraphics[width=50.0pt]{Concatenation3.eps}}\put(-2.34843,0.39123){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{}}}\put(-1.15255,0.51494){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{}}}\put(0.12204,1.09276){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{$\nicefrac{{\mathbb{R}}}{{2\mathbb{Z}}}\times
M$}}}\put(1.71343,0.04071){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{}}}\put(0.03755,0.45245){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{$\bar{g}_{1}$}}}\put(0.80832,0.45515){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{$\bar{g}_{0}$}}}\end{picture}}\quad\xrightarrow{\theta/2}\quad\raisebox{-6.88889pt}{\begin{picture}(1.0,1.0)\put(0.0,0.0){\includegraphics[width=30.0pt]{Concatenation4.eps}}\put(-6.22511,0.09152){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{}}}\put(-4.44049,0.27614){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{}}}\put(0.16753,1.13843){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{$\mathbb{T}\times
M$}}}\put(-0.39128,-0.45559){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{$\begin{array}[]{c}\\!\\!\\!\\!\\!g_{1}\star
g_{0}\\\ \mbox{\footnotesize{of step
}}2^{-k-1}\end{array}$}}}\put(-0.16357,-0.43156){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lb]{\smash{}}}\end{picture}}$
Figure 5: Concatenation of two plugins.

The $\star$-product associates to a pair of plugins of the same step a plugin
of half that step and whose output is the composition of the outputs:

######  Proposition 2.3.

If $g_{0}$ and $g_{1}$ are plugins with a same step $\sigma$ and outputs
$G_{0}$ and $G_{1}$ then $g_{1}\star g_{0}$ is a plugin with step $\sigma/2$
and output $G_{1}\circ G_{0}$.

###### Proof.

The set $\Delta=([0,\sigma)+\mathbb{Z})\times M$ lifts into the union of the
two sets:

$\bar{\Delta}_{0}=([0,\sigma)+2\mathbb{Z})\times
M\quad\text{and}\quad\bar{\Delta}_{1}=([1,1+\sigma)+2\mathbb{Z})\times M.$

Let $\pi_{i}$ be the restriction of $\pi$ to each $\bar{\Delta}_{i}$.

Note that $\bar{g}_{0}$ and $\bar{g}_{1}$ coincide with the translation by
$(\sigma,0)$ on $\bar{\Delta}_{0}\cup\bar{\Delta}_{1}$. Hence the map
$\overline{g_{0}\star g_{1}}$ is a smooth diffeomorphism that also coincides
with the translation by $(\sigma,0)$ on
$\bar{\Delta}_{0}\cup\bar{\Delta}_{1}$. Then $\overline{g_{1}\star
g_{0}}^{\tau_{1}\circ\pi}$ sends $\bar{\Delta}_{0}$ onto $\bar{\Delta}_{1}$ by
$\bar{g}_{0}^{\tau_{0}\circ\pi}$ and sends $\bar{\Delta}_{1}$ onto
$\bar{\Delta}_{0}$ by $\bar{g}_{1}^{\tau_{1}\circ\pi}$. Thus we have:

${\bar{g}_{0}^{\tau_{0}\circ\pi}}{|\bar{\Delta}_{0}}=\pi_{1}^{-1}\circ
H_{\sigma}^{-1}\circ G_{0}\circ
H_{\sigma}\circ\pi_{0}\quad\text{and}\quad\bar{g}_{1}^{\tau_{1}\circ\pi}|\bar{\Delta}_{1}=\pi_{0}^{-1}\circ
H_{\sigma}^{-1}\circ G_{1}\circ H_{\sigma}\circ\pi_{1}\;.$

Therefore the return time of $\overline{g_{1}\star g_{0}}$ into
$\bar{\Delta}_{0}$ is defined on $\bar{\Delta}_{0}$ and the first return map
is:

$\pi_{0}^{-1}\circ H_{\sigma}^{-1}\circ G_{1}\circ G_{0}\circ
H_{\sigma}\circ\pi_{0}.$

This implies that $g_{1}\star g_{0}$ is a plugin with output $G_{1}\circ
G_{0}$ and step $\sigma/2$. ∎

######  Proposition 2.4.

The set $\mathrm{Plug}$ is a subgroup of
$\mathrm{Diff}^{\infty}_{c}(\mathbb{T}\times M)$ endowed with the composition
rule.

###### Proof.

By Example 1.10, $\mathrm{Plug}$ contains the identity and by definition it is
stable by inversion. Thus it remains to show that semi-pluggability is stable
by composition. Let $G_{1},G_{0}$ be semi-pluggable. Then for every $k$ large,
$C^{\infty}$-close to identity, there are plugins $g_{1}$ and $g_{0}$ with
step $2^{-k}$ and with outputs $G_{1}$ and $G_{0}$ respectively. Then by
Remark 2.2 , the map $g_{1}\star g_{0}$ is close to identity when $k$ is
large. By Proposition 2.3, the output of $g_{1}\star g_{0}$ is $G_{1}\circ
G_{0}$ and its step is $2^{-k-1}$. Thus $G_{1}\circ G_{0}$ is semi-pluggable.
The second assertion is proved similarly. ∎

##### Parametric counterpart.

The proofs of the two latter propositions imply immediately:

######  Proposition 2.5.

If $f_{\mathscr{P}},g_{\mathscr{P}}$ are two ${\mathscr{P}}$-plugins with same
step $\sigma$ and output $F_{\mathscr{P}}$ and $G_{\mathscr{P}}$, then
$(f_{p}\star g_{p})_{p\in{\mathscr{P}}}$ is a ${\mathscr{P}}$-plugin with step
$\sigma/2$ and output $(F_{p}\circ G_{p})_{p\in{\mathscr{P}}}$.

Thus we deduce:

######  Proposition 2.6.

The set $\mathrm{Plug}_{\mathscr{P}}$ is a subgroup of
$\mathrm{Diff}^{\infty}_{c}(\mathbb{T}\times M)_{\mathscr{P}}$ endowed with
operation:

$(F_{p})_{p\in{\mathscr{P}}}\circ(G_{p})_{p\in{\mathscr{P}}}=(F_{p}\circ
G_{p})_{p\in{\mathscr{P}}}\;.$

###  2.2 Another subgroup included in $\mathrm{Plug}$

Consider the following sub-group of
$\mathrm{Diff}^{\infty}_{c}(\mathbb{T}\times M)$:

$\mathbb{G}_{2}:=\left\\{(\theta,y)\in\mathbb{T}\times
M\mapsto(\theta,F(y)):F\in\mathrm{Diff}_{c}^{\infty}(M)\right\\}$

######  Proposition 2.7.

The group $\mathbb{G}_{2}$ is included in $\mathrm{Plug}$.

$y$$\Delta$$T_{1}\Delta$$f(y)$ Figure 6: Dynamics of a plugin with output in
$\mathbb{G}_{2}$.

###### Proof.

As $\mathbb{G}_{2}$ is connected and $\mathrm{Plug}$ is a group by Proposition
2.4, it suffices to show that a neighborhood $W$ of $\mathrm{id}$ in
$\mathbb{G}_{2}$ is included in $\mathrm{Plug}$. Indeed by [Wil84, Prop 3.18],
any such neighborhood $W$ generates $\mathbb{G}_{2}$. Up to replacing $W$ by
$W\cap W^{-1}$, it suffices to show that any element of $W$ is semi-pluggable.

In order to do so, we develop an idea which appears in [NRT78]. Take $W$
sufficiently small so that for every $G:(\theta,y)\mapsto(\theta,F(y))\in W$,
the map $F$ is sufficiently close to $\mathrm{id}$ to be isotopic to it via a
smooth path. In other words, there exists a $C^{\infty}$-family
$(F_{t})_{t\in[0,1]}$ of maps $F_{t}\in\mathrm{Diff}^{\infty}_{c}(M)$ such
that $F_{0}=\mathrm{id}$ and $F_{1}=F$. Such a family can be obtained using
the exponential map $\mathrm{exp}$ of the Riemannian metric, via the formula
$F_{t}:=y\mapsto\mathrm{exp}_{y}(t\cdot\mathrm{exp}_{y}^{-1}F(y))$. Define:

(2.1) $Y(t,y):=\partial_{t}F_{t}\circ F_{t}^{-1}(y)\;,$

and observe that $F$ is the time one map of the (compactly supported) non-
autonomous vector field $Y$. Let $\tau:\mathbb{T}\to[0,1]$ be a smooth map
which is smooth on $\mathbb{T}\setminus\\{0\\}$ and such that nearby $0^{+}$
it equals $0$ and nearby $0^{-}$ is equal $1$. For $k$ large, let:

(2.2) $X_{k}:(\theta,y)\in\mathbb{T}\times
M\mapsto(2^{-k},2^{-k}\cdot\partial_{\theta}\tau(\theta)\cdot
Y(\tau(\theta),y))\;.$

Let $g_{k}$ be the time one map of this vector field. Observe that for $k$
large enough, $g_{k}$ is a plugin with step $\sigma=2^{-k}$ and return time
$\sigma^{-1}$. Furthermore, its output is $G$. Indeed the second coordinate of
the output is the time $\sigma^{-1}$ map of the flow of
$\partial_{\theta}\tau(\theta)\cdot Y(\tau(\theta),y)$ which is the time one
map $F$ of the flow of $Y$. Thus we have:

(2.3) $g^{1/\sigma}_{k}\circ H_{\sigma}(x)=H_{\sigma}\circ G(x),\quad\forall
x\in\Delta_{\sigma}\;.$

Furthermore, when $k$ is large, the plugin $g_{k}$ is close to identity. Thus
$G$ is semi-pluggable. ∎

##### Parametric counterpart.

Let $\mathbb{G}_{2{\mathscr{P}}}$ be the subset of
$\mathrm{Diff}_{c}^{\infty}(\mathbb{T}\times M)_{\mathscr{P}}$ formed by
families $(G_{p})_{p\in{\mathscr{P}}}$ such that $G_{p}\in\mathbb{G}_{2}$ for
every $p\in{\mathscr{P}}$. The following is a counterpart of Proposition 2.7:

######  Proposition 2.8.

The group $\mathbb{G}_{2{\mathscr{P}}}$ is included in
$\mathrm{Plug}_{\mathscr{P}}$.

###### Proof.

Let $W$ be as defined in the proof of Proposition 2.7. We define
$W_{\mathscr{P}}$ as the subset of
$\mathrm{Diff}_{c}^{\infty}(\mathbb{T}\times M)_{\mathscr{P}}$ formed by
families $(G_{p})_{p\in{\mathscr{P}}}$ such that $G_{p}\in W$ for every
$p\in{\mathscr{P}}$. For the same reasons it suffices to show that any element
of $W_{\mathscr{P}}$ is ${\mathscr{P}}$-semi-pluggable. Similarly, for any
family $(G_{p})_{p\in{\mathscr{P}}}=:(id_{\mathbb{T}}\times
F_{p})_{p\in{\mathscr{P}}}\in W_{{\mathscr{P}}}$ , the family
$(F_{p})_{p\in{\mathscr{P}}}$ is isotopic to the identity via a smooth path
$\left((F_{pt})_{p\in{\mathscr{P}}}\right)_{t\in[0,1]}$ where
$F_{pt}:=y\mapsto\mathrm{exp}_{y}(t\cdot\mathrm{exp}_{y}^{-1}F_{p}(y))$. We
define:

(2.4) $Y_{p}(t,y):=\partial_{t}F_{pt}\circ F_{pt}^{-1}(y)\;.$

Note that the family of vector fields $(Y_{p})_{p\in{\mathscr{P}}}$ is smooth.
Define then the family of vector fields:

(2.5) $X_{kp}:(\theta,y)\in\mathbb{T}\times
M\mapsto(2^{-k},2^{-k}\cdot\partial_{\theta}\tau(\theta)\cdot
Y_{p}(\tau(\theta),y))\;,$

where $\tau$ is the function defined in Proposition 2.7. For $k$ large enough,
the family of time one maps $(g_{kp})_{p\in{\mathscr{P}}}$ is a
${\mathscr{P}}$-plugin with output the family $(G_{p})_{p\in{\mathscr{P}}}$.
Moreover, when $k\to\infty$, the ${\mathscr{P}}$-plugin
$(g_{kp})_{p\in{\mathscr{P}}}$ tends to the identity
$\mathrm{id}\in\mathrm{Diff}^{\infty}_{c}(M)_{\mathscr{P}}$. ∎

###  2.3 Closure of the group $\mathrm{Plug}$

In this section we prove that $\mathrm{Plug}$ is closed in
$\mathrm{Diff}^{\infty}_{c}(\mathbb{T}\times M)$.

######  Proposition 2.9.

The subgroup $\mathrm{Plug}\subset\mathrm{Diff}^{\infty}_{c}(\mathbb{T}\times
M)$ is closed.

This proposition uses the following lemma proved below:

######  Lemma 2.10.

For any $1\leq r\leq\infty$, for any neighborhood ${\mathcal{N}}$ of
$\mathrm{id}\in\mathrm{Diff}^{r}(\mathbb{T}\times M)$, there exist $N\geq 1$
and a neighborhood ${\mathcal{N}}_{c}$ of
$\mathrm{id}\in\mathrm{Diff}_{c}^{r}(\mathbb{T}\times M)$ such that for all
$G\in{\mathcal{N}}_{c}$ and $k\geq N$, there is a $C^{r}$-plugin
$g\in{\mathcal{N}}$ with output $G$ and step $2^{-k}$.

Note that the latter lemma is redacted for any regularity $1\leq r\leq\infty$.
It will allow to deduce E from F.

###### Proof of Proposition 2.9.

It is suffices to show that the set of semi-pluggable maps is closed. Indeed,
the continuity of the involution $G\mapsto G^{-1}$ implies that the set of
maps with semi-pluggable inverse is closed; and so it comes that the
intersection $\mathrm{Plug}$ of these two sets is closed.

Let $(G_{j})_{j\geq 0}$ be a sequence semi-pluggable maps converging in the
$C^{\infty}$-topology to a diffeomorphism
$G\in\mathrm{Diff}_{c}^{\infty}(\mathbb{T}\times M)$. Let us show that $G$ is
semi-pluggable. In other words, let us show that for every neighborhood
${\mathcal{V}}$ of $\mathrm{id}\in\mathrm{Diff}^{\infty}(\mathbb{T}\times M)$,
for every $k$ large enough, there exists a plugin $g\in{\mathcal{V}}$ with
output $G$ and step $2^{-k}$.

To this end, let us fix a small neighborhood of ${\mathcal{N}}$ of
$\mathrm{id}\in\mathrm{Diff}^{\infty}(\mathbb{T}\times M)$ and $j$ large so
that $G_{j}^{-1}\circ G$ belongs to the open set ${\mathcal{N}}_{c}$ given by
Lemma 2.10. Hence for every $k\geq 0$ large enough, there exists a plugin
$g_{1}\in{\mathcal{N}}$ with output $G_{j}^{-1}\circ G$ and step $2^{-k}$. As
$G_{j}$ is pluggable, for every $k$ large enough, there exists a plugin
$g_{0}\in{\mathcal{N}}$ with output $G_{j}$ and step $2^{-k}$. Now we merge
the plugins $g_{1}$ and $g_{0}$ to obtain a plugin $g=g_{1}\star g_{0}$ of $G$
of step $2^{-k-1}$. By Remark 2.2, when ${\mathcal{N}}$ is small, the map $g$
is close to identity and so in ${\mathcal{V}}$. ∎

The idea of the proof of Lemma 2.10 is to find, for each fixed small
$\delta>0$, a sequence of close to identity plugins $g_{k}$ with step $2^{-k}$
and output the identity such that some iterates of each $g_{k}$ stretch the
$2^{-k}$-thin fundamental domain $\Delta_{2^{-k}}$ onto a wider fundamental
domain, isometric to $[0,2\delta[\times M$. The iteration by $g_{k}$ will
produce a horizontal zooming effect on $\Delta_{2^{-k}}$. Then we will be able
to perturb the plugin on this stretched fundamental domain, to obtain an open
set of outputs independent of $k$.

The following produces the sequence $(g_{k})_{k}$:

######  Sub-Lemma 2.11.

For every neighborhood ${\mathcal{N}}$ of
$\mathrm{id}\in\mathrm{Diff}^{\infty}(\mathbb{T}\times M)$, for every
$\delta>0$ small, there exists $N\geq 1$ and a sequence $(g_{k})_{k\geq N}$ of
plugins in ${\mathcal{N}}$ with step $2^{-k}$, output $\mathrm{id}$ such that:

  * •

for all $y\in M$ and $\theta\in[\tfrac{1}{2},\tfrac{2}{3}]$; we have
$g_{k}(\theta,y)=(\theta+\delta,y),$

  * •

$g_{k}$ is of the form $g_{k}:(\theta,y)\in\mathbb{T}\times
M\mapsto(\phi_{k}(\theta),y)$.

###### Proof.

Let $A=[\tfrac{1}{2},\tfrac{2}{3}]$ and $B=[\tfrac{1}{4},\tfrac{1}{3}]$. Let
$\psi_{A}$, $\psi_{B}\in C^{\infty}(\mathbb{T},[0,1])$ be two non-negative
functions with disjoint supports, vanishing at a neighborhood of $0$ and such
that:

(2.6) $\psi_{A}|A=1\quad\text{and}\quad\psi_{B}|B=1\;.$

For $\beta\geq 0$ we define the following vector field on the circle
$\mathbb{T}$:

(2.7)
$X_{\beta,k}:=\delta\cdot\psi_{A}+\beta\cdot\psi_{B}+(1-\psi_{A}-\psi_{B})\cdot
2^{-k}.$

Let $\tau_{\beta,k}$ be the time needed to make one turn around the circle
along the flow of $X_{\beta}$. This number is large since $\delta$ is small.
Note that $\tau_{\beta,k}$ depends smoothly on $\beta>0$. Also
$\tau_{\beta,k}\to\infty$ when $\beta\to 0$ and
$\partial_{\beta}\tau_{\beta,k}<0$. Let $N\geq 1$ be so that $\delta>2^{-N}$.
Let $k\geq N$. We have $\delta>2^{-k}$. If $\beta=2^{-k}$, then the time
$\tau_{\beta,k}$ is smaller than $2^{k}$. Thus by the mean value theorem,
there exists a unique $\beta=\beta(k,\alpha)$ close to $2^{-k}$ such that
$\tau_{\delta,\beta(k,\delta),k}=2^{k}$.

Then the time $1$ map $g_{k}$ of the flow of $(X_{\beta(k),k},0)$ satisfies
the sought properties.∎

We have now the tools to prove the following restricted version of Lemma 2.10:

######  Sub-Lemma 2.12.

For any $1\leq r\leq\infty$, for any neighborhood ${\mathcal{V}}$ of
$\mathrm{id}\in\mathrm{Diff}^{r}(\mathbb{T}\times M)$, there exist $N\geq 1$
and a neighborhood ${\mathcal{N}}_{c}$ of
$\mathrm{id}\in\mathrm{Diff}_{c}^{r}(\mathbb{T}\times M)$ such that for any
$k\geq N$ and every $G\in{\mathcal{N}}_{c}$ whose restriction to a
neighborhood of $\\{0\\}\times M$ or a neighborhood of
$\\{\tfrac{1}{2}\\}\times M$ is the identity, there is a $C^{r}$-plugin
$g\in{\mathcal{V}}$ with output $G$ and step $2^{-k}$.

###### Proof of 2.12 .

Let ${\mathcal{N}}$ be a neighborhood of
$\mathrm{id}\in\mathrm{Diff}^{r}(\mathbb{T}\times M)$ such that222in the sense
that the distance between ${\mathcal{N}}$ and the complement of
${\mathcal{V}}$ is positive. ${\mathcal{N}}\Subset{\mathcal{V}}$. We apply
2.11 which provides $\delta>0$, $N\geq 1$ and a sequence $(g_{k})_{k\geq N}$
of plugins $g_{k}:(\theta,y)\mapsto(\phi_{k}(\theta),y)$ with output
$\mathrm{id}$ and step $2^{-k}$. By 2.11, there exists $n_{k}>0$ minimal such
that $\theta_{k}:=\phi^{n_{k}}_{k}(0)\in[\tfrac{1}{2},1]$. Taking
${\mathcal{N}}$ small, we have that $\theta_{k}$ is smaller than
$\tfrac{2}{3}-\tfrac{3}{2}\delta$, and so $g_{k}$ equals the translation by
$\delta$ on $(\theta_{k},\theta_{k}+\tfrac{3}{2}\delta)$. Let
${\mathcal{N}}_{c}$ be a small neighborhood of
$\mathrm{id}\in\mathrm{Diff}_{c}^{r}(\mathbb{T}\times M)$. Let
$G\in{\mathcal{N}}_{c}$ be equal to the identity nearby $\\{0\\}\times M$ or
$\\{\tfrac{1}{2}\\}\times M$. We would like to $C^{r}$-perturb $g_{k}$ so that
its output is $G$.

Case 1: If $G\in{\mathcal{N}}_{c}$ coincides with the identity nearby
$\\{0\\}\times M$, then we perform a perturbation of $g_{k}$ supported by
$(\theta_{k},\theta_{k}+\delta)\times M$ and therein equals to:

(2.8) $\begin{array}[]{rcl}\tilde{g}_{k}:[\theta_{k},\theta_{k}+\delta]\times
M&\to&[\theta_{k}+\delta,\theta_{k}+2\delta]\times M\\\
(\theta_{k}+x,y)&\mapsto&(\theta_{k}+\delta,0)+\delta\cdot\tilde{G}(\delta^{-1}x,y)\end{array}$

where $\tilde{G}:\mathbb{R}\times M\to\mathbb{R}\times M$ is a lifting of $G$
which fixes $\\{0\\}\times M$. Note that $\tilde{g}_{k}$ is a $C^{r}$-plugin
with output $G$ and step $2^{-k}$. Furthermore, if ${\mathcal{N}}_{c}$ is
small enough (at $\delta$ fixed), then for every $k\geq N$, the map
$\tilde{g}_{k}$ is uniformly close to $g_{k}\in{\mathcal{N}}$ in the
$C^{r}$-topology and so $\tilde{g}_{k}$ belongs to ${\mathcal{V}}$.

Case 2: If $G\in{\mathcal{N}}_{c}$ coincides with the identity nearby
$\\{\tfrac{1}{2}\\}\times M$, then we perform a perturbation of $g_{k}$
supported by
$(\theta_{k}+\tfrac{\delta}{2},\theta_{k}+\tfrac{3}{2}\delta)\times M$ and
therein equals to:

(2.9)
$\begin{array}[]{rcl}\tilde{g}_{k}:[\theta_{k}+\tfrac{\delta}{2},\theta_{k}+\tfrac{3}{2}\delta]\times
M&\to&[\theta_{k}+\tfrac{3}{2}\delta,\theta_{k}+\tfrac{5}{2}\delta]\times M\\\
(\theta_{k}+\tfrac{\delta}{2}+x,y)&\mapsto&(\theta_{k}+\tfrac{\delta}{2}+\delta,0)+\delta\cdot\tilde{G}(\delta^{-1}x,y)\;.\end{array}$

Similarly, this is a $C^{r}$-plugin of output $G$ and step $2^{-k}$, which is
in ${\mathcal{V}}$ for every $k$ provided that ${\mathcal{N}}_{c}$ is small
enough. ∎

###### Proof that 2.12 implies Lemma 2.10 .

First let us ‘fragment’ any $C^{r}$-close to identity map
$G\in{\mathcal{N}}_{c}$ into the composition of two $C^{r}$-maps $G_{1}\circ
G_{0}$ such that $G_{0}$ coincides with the identity nearby $\\{0\\}\times M$
and $G_{1}$ coincides with the identity nearby $\\{\tfrac{1}{2}\\}\times M$.

To this end, we use the exponential map $\mathrm{exp}$ associated to the
geodesic flow of $\mathbb{T}\times M$ and a function $\rho\in
C^{\infty}(\mathbb{T}\times M,[0,1])$ such that $\rho|\\{0\\}\times M=1$ and
$\rho|\\{\tfrac{1}{2}\\}\times M=0$. Let ${\mathcal{N}}_{c}$ be a sufficiently
small neighborhood of
$\mathrm{id}\in\mathrm{Diff}^{\infty}_{c}(\mathbb{T}\times M)$ such that for
every $G\in{\mathcal{N}}_{c}$ the following is a smooth diffeomorphism:

(2.10) $G_{0}:=x\mapsto\mathrm{exp}(\rho(x)\cdot\mathrm{exp}_{x}^{-1}G(x))\;.$

Let $G_{1}:=G\circ G_{0}^{-1}$ and note that $G=G_{1}\circ G_{0}$.

2.12 states that there are $C^{r}$-plugins $g_{1}$ and $g_{0}$ close to
identity with step $2^{-k}$ and outputs $G_{1}$ and $G_{0}$ for every $k$
large enough. Then by Remark 2.2 and Proposition 2.3, the $C^{r}$-plugin
$g_{1}\star g_{0}$ of step $2^{-k-1}$ is close to identity and has output
$G_{1}\circ G_{0}=G$. ∎

###### Proof that F implies E .

When $r=\infty$, the result of E corresponds to the one F. Consider now
$r<\infty$. Let $G\in\mathrm{Diff}_{c}^{r}(\mathbb{T}\times M)$, and
${\mathcal{N}}\subset{\mathcal{V}}$ two neighborhoods of
$\mathrm{id}\in\mathrm{Diff}_{c}^{r}(\mathbb{T}\times M)$. We smooth the map
$G$ into a map $\tilde{G}\in\mathrm{Diff}_{c}^{\infty}(\mathbb{T}\times M)$
such that the map $\tilde{G}^{-1}\circ G$ belongs to the neighborhood
${\mathcal{N}}_{c}$ given by Lemma 2.10. Then for every $k\geq 0$ large
enough, there exists a plugin $g_{0}\in{\mathcal{N}}$ with output
$\tilde{G}^{-1}\circ G$ and step $2^{-k}$. By F, for $k$ large enough, there
exists a plugin $g_{1}\in{\mathcal{N}}$ with output $\tilde{G}$. We merge
$g_{0}$ and $g_{1}$ to get a plugin $g=g_{1}\star g_{0}$ with output $G$. By
Remark 2.2, when ${\mathcal{N}}$ is small, the map $g$ is in ${\mathcal{V}}$.
∎

##### Parametric counterpart.

Here is the parametric counterpart of Proposition 2.9:

######  Proposition 2.13.

The set $\mathrm{Plug}_{\mathscr{P}}$ is closed.

To show this proposition we will use the following counterpart of Lemma 2.10
proved below:

######  Lemma 2.14.

For any $1\leq r\leq\infty$, for any neighborhood ${\mathcal{N}}$ of
$\mathrm{id}\in\mathrm{Diff}^{r}(\mathbb{T}\times M)$, there exists $N\geq 1$
and a neighborhood ${\mathcal{N}}_{c}$ of
$\mathrm{id}\in\mathrm{Diff}_{c}^{r}(\mathbb{T}\times M)_{\mathscr{P}}$ such
that for all $(G_{p})_{p\in{\mathscr{P}}}\in{\mathcal{N}}_{c}$ and $k\geq N$,
there is a ${\mathscr{P}}$-$C^{r}$-plugin
$(g_{p})_{p\in{\mathscr{P}}}\in{\mathcal{N}}$ with output
$(G_{p})_{p\in{\mathscr{P}}}$ and step $2^{-k}$.

###### Proof of Proposition 2.13.

We proceed literally as in the proof Proposition 2.9, by applying Lemma 2.14
instead of Lemma 2.10, and considering families instead of single maps. Note
that the continuity of $\star$ given by Remark 2.2 is also valid for families
since $(f_{p})_{p}\in\mathrm{Diff}^{\infty}(\mathbb{T}\times
M)_{\mathscr{P}}\mapsto f_{\mathscr{P}}\in\mathrm{Diff}^{r}(\mathbb{T}\times
M\times{\mathscr{P}})$ is an embedding which commutes with the
$\star$-product. ∎

###### Proof of Lemma 2.14.

The explicit construction of 2.12 gives directly a parametric counterpart of
this lemma. Indeed note that the maps $\tilde{g}_{k}$ defined in its proof
depends smoothly on $G$. Moreover the maps $G_{0}$ and $G_{1}$ obtained in the
proof of Lemma 2.10 by the fragmentation formula depends smoothly on the
involved diffeomorphism. ∎

Similarly, we show:

###### Proof that F′ implies E′.

This goes exactly the same as for the proof that F implies E′. It suffices to
replace maps in $\mathrm{Diff}^{r}_{c}(\mathbb{T}\times M)$ by families of
maps in $\mathrm{Diff}^{r}_{c}(\mathbb{T}\times M)_{\mathscr{P}}$, plugins by
${\mathscr{P}}$-plugins and Lemma 2.10 by Lemma 2.14.

∎

###  2.4 Vector fields whose flow is pluggable

In this section $V$ denotes a manifold. We recall that
$\mathrm{Diff}_{c}^{\infty}(V)$ endowed with the composition rule $\circ$ is a
Lie group. The space $\mathfrak{diff}_{c}^{\infty}(V)$ of $C^{\infty}$-vector
fields $X$ on $V$ whose support is a compact subset of $V\setminus\partial V$
is a Lie algebra endowed with the Lie bracket:

$[X,Y]:=DY(X)-DX(Y)\;,\quad\forall X,Y\in\mathfrak{diff}^{\infty}_{c}(V).\;$

We will work with the Lie algebra counterpart of the considered subgroups. We
recall that a subset of $\mathfrak{diff}_{c}^{\infty}(V)$ is a Lie subalgebra
if it is a vector space stable by Lie Bracket. To define the counterpart, we
will use the flow $(\mathrm{Fl}_{X}^{t})_{t}$ of vector fields
$X\in\mathfrak{diff}^{\infty}_{c}(V)$.

######  Definition 2.15.

We denote ${\mathfrak{plug}}$ the set of vector fields whose flow is
pluggable. In short:

${\mathfrak{plug}}:=\\{X\in\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times
M):\mathrm{Fl}_{X}^{t}\in\mathrm{Plug},\ \forall t\in\mathbb{R}\\}\;.$

Using that $\mathrm{Plug}$ is a closed subgroup, the following is an immediate
consequence of Proposition B.1 of Appendix B:

######  Proposition 2.16.

The space ${\mathfrak{plug}}$ is a closed Lie subalgebra of
$\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times M)$.

The following are closed Lie algebras of
$\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times M)$:

${\mathfrak{g}}_{1}={\mathfrak{g}}_{1}(M):=\\{X\in\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times
M):\mathrm{Fl}_{X}^{t}\in\mathbb{G}_{1},\ \forall
t\in\mathbb{R}\\}=\\{X:(\theta,y)\mapsto(v(y),0):v\in
C^{\infty}(M,\mathbb{R})\\}\;.$
${\mathfrak{g}}_{2}={\mathfrak{g}}_{2}(M):=\\{X\in\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times
M):\mathrm{Fl}_{X}^{t}\in\mathbb{G}_{2},\ \forall
t\in\mathbb{R}\\}=\\{X:(\theta,y)\mapsto(0,f(y)):f\in\mathfrak{diff}_{c}^{\infty}(M)\\}\;.$

The subgroups $\mathbb{G}_{1}$ and $\mathbb{G}_{2}$ are in $\mathrm{Plug}$,
so:

(2.11)
${\mathfrak{g}}_{1}\subset{\mathfrak{plug}}\quad\text{and}\quad{\mathfrak{g}}_{2}\subset{\mathfrak{plug}}\;.$

##### Parametric counterpart.

We denote $\mathfrak{diff}^{\infty}_{c}(V)_{\mathscr{P}}$ the subspace of
families $X_{\mathscr{P}}=(X_{p})_{p\in{\mathscr{P}}}$ in
$\mathfrak{diff}^{\infty}_{c}(V)$ such that

(2.12) $\widehat{X_{\mathscr{P}}}\colon(x,p)\mapsto(X_{p}(x),0)$

is smooth and compactly supported, that is, such that
$\widehat{X_{\mathscr{P}}}\in\mathfrak{diff}^{\infty}_{c}(V\times{\mathscr{P}})$.
By 1.8, the space $\mathfrak{diff}_{c}^{\infty}(V)_{\mathscr{P}}$ is a Lie
algebra endowed with the Lie bracket
$[X_{\mathscr{P}},Y_{\mathscr{P}}]:=([X_{p},Y_{p}])_{p\in{\mathscr{P}}}$.

######  Definition 2.17.

The ${\mathscr{P}}$-families of vector fields whose flow is
${\mathscr{P}}$-pluggable is denoted:

${\mathfrak{plug}}_{\mathscr{P}}:=\left\\{(X_{p})_{p\in{\mathscr{P}}}\in\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times
M)_{{\mathscr{P}}}:(\mathrm{Fl}_{X_{p}}^{t})_{p\in{\mathscr{P}}}\in\mathrm{Plug}_{\mathscr{P}},\
\forall t\in\mathbb{R}\right\\}\;.$

Using that $\mathrm{Plug}_{\mathscr{P}}$ is a closed subgroup, the following
is an immediate consequence of B.2 of Appendix B:

######  Proposition 2.18.

The space ${\mathfrak{plug}}_{\mathscr{P}}$ is a closed Lie subalgebra of
$\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times M)_{\mathscr{P}}$.

Also note that the space ${\mathfrak{plug}}_{\mathscr{P}}$ contains:

${\mathfrak{g}}_{1{\mathscr{P}}}:=\\{(X_{p})_{p\in{\mathscr{P}}}\in\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times
M)_{\mathscr{P}}:X_{p}\in{\mathfrak{g}}_{1},\ \forall p\in{\mathscr{P}}\\}\;.$

and

${\mathfrak{g}}_{2{\mathscr{P}}}:=\\{(X_{p})_{p\in{\mathscr{P}}}\in\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times
M)_{\mathscr{P}}:X_{p}\in{\mathfrak{g}}_{2},\ \forall p\in{\mathscr{P}}\\}\;.$

We define also:

$\widehat{{\mathfrak{g}}_{1{\mathscr{P}}}}:=\\{\widehat{X_{\mathscr{P}}}:X_{\mathscr{P}}\in{\mathfrak{g}}_{1{\mathscr{P}}}\\}\quad\text{and}\quad\widehat{{\mathfrak{g}}_{2{\mathscr{P}}}}:=\\{\widehat{X_{\mathscr{P}}}:X_{\mathscr{P}}\in{\mathfrak{g}}_{2{\mathscr{P}}}\\}\;.$

Observe that:

$\widehat{{\mathfrak{g}}_{1{\mathscr{P}}}}=\\{(\theta,y,p)\mapsto(\nu(y,p),0,0):\nu\in
C^{\infty}_{c}(M\times{\mathscr{P}},\mathbb{R})\\}\;$

and

$\widehat{{\mathfrak{g}}_{2{\mathscr{P}}}}=\\{(\theta,y,p)\mapsto(0,f(y,p),0):(f,0)\in\mathfrak{diff}^{\infty}_{c}(M\times{\mathscr{P}})\\}\;.$

Thus we have
$\widehat{{\mathfrak{g}}_{1{\mathscr{P}}}}={\mathfrak{g}}_{1}(M\times{\mathscr{P}})$
and
$\widehat{{\mathfrak{g}}_{2{\mathscr{P}}}}\subsetneq{\mathfrak{g}}_{2}(M\times{\mathscr{P}})$.

##  3 Construction of Pluggable flows

In this section we show F by proving that any vector field has a pluggable
flow. In order to do so we will show that the following subspace of
$\mathfrak{diff}^{\infty}_{c}(\mathbb{T}\times M)$ is in ${\mathfrak{plug}}$:

${\mathfrak{g}}_{3}:={\mathfrak{g}}_{3}(M)=\\{X\in:\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times
M):X\text{ has null $\mathbb{T}$-coordinate}\\}\;.$

The non-trivial remaining part of the proof is to show the following:

######  Proposition 3.1.

Any vector field of ${\mathfrak{g}}_{3}$ has pluggable flows:

${\mathfrak{g}}_{3}\subset{\mathfrak{plug}}\;.$

The proof of this proposition will occupy Sections 3.2 and 3.3. In the next
section we show that it implies main F. The proof of several propositions will
involve the following notation. If ${\mathfrak{g}}$ and ${\mathfrak{h}}$ are
two sub-Lie algebras, we denote $[{\mathfrak{g}},{\mathfrak{h}}]$ the vector
space spanned by Lie brackets of elements of ${\mathfrak{g}}$ and
${\mathfrak{h}}$:

$[{\mathfrak{g}},{\mathfrak{h}}]:=\left\\{\sum_{\text{i:finite}}[X_{i},Y_{i}]:X_{i}\in{\mathfrak{g}},\
Y_{i}\in{\mathfrak{h}}\right\\}\;.$

##### Parametric counterpart.

Similarly we set:

${\mathfrak{g}}_{3{\mathscr{P}}}:=\\{(X_{p})_{p\in{\mathscr{P}}}\in\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times
M)_{\mathscr{P}}:X_{p}\in{\mathfrak{g}}_{3},\ \forall p\in{\mathscr{P}}\\}\;,$

and

$\widehat{{\mathfrak{g}}_{3{\mathscr{P}}}}:=\\{\widehat{X_{\mathscr{P}}}:X_{\mathscr{P}}\in{\mathfrak{g}}_{3{\mathscr{P}}}\\}\subsetneq{\mathfrak{g}}_{3}(M\times{\mathscr{P}})\;.$

We will prove the following parametric counterpart of Proposition 3.1 in
Sections 3.2 and 3.3:

######  Proposition 3.2.

Every smooth family of vector fields in ${\mathfrak{g}}_{3}$ has a
${\mathscr{P}}$-pluggable flow:

${\mathfrak{g}}_{3{\mathscr{P}}}\subset{\mathfrak{plug}}_{\mathscr{P}}\;.$

###  3.1 Proof of main F and F′

The first step of the proof is the following:

######  Proposition 3.3.

Any vector field has a pluggable flow:

${\mathfrak{plug}}=\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times M)\;.$

###### Proof.

As ${\mathfrak{plug}}$ is a Lie algebra by Proposition 2.16 and since
${\mathfrak{g}}_{1}$ and ${\mathfrak{g}}_{3}$ are in ${\mathfrak{plug}}$ by
Eq. 2.11 and Proposition 3.1, it suffices to show that the Lie algebra
generated by ${\mathfrak{g}}_{1}$ and ${\mathfrak{g}}_{3}$ equals
$\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times M)$. We first prove the
statement of the proposition for $M=\mathbb{R}^{n}$. Let
$X\in\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times M)$ and $f\in
C^{\infty}_{c}(\mathbb{T}\times\mathbb{R}^{n},\mathbb{R})$ be its
$\mathbb{T}$-coordinate. Let $W\in{\mathfrak{g}}_{3}$ be:

(3.1) $W:=(0,f,0,\ldots,0)\;.$

Let $\rho\in C^{\infty}(\mathbb{T}\times M,\mathbb{R})$ be a compactly
supported function which is equal to $1$ nearby the support of $f$. Let
$Y\in{\mathfrak{g}}_{1}$ be:

(3.2) $Y(\theta,y_{1},\ldots,y_{n})=(\rho(\theta,y)\cdot y_{1},0,\ldots,0)\;.$

A computation gives a Lie bracket of the form:

(3.3) $[W,Y](\theta,y)=(f(\theta,y),-\partial_{\theta}f(\theta,y)\cdot
y_{1},0,\ldots,0)\;.$

Thus $X-[W,Y]$ is in ${\mathfrak{g}}3$ which gives the sought result for
$M=\mathbb{R}^{n}$.

When $M$ is another manifold, we fix a locally finite covering $(U_{i})_{i}$
by balls $U_{i}$. Using a partition of the unity, every
$X\in\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times M)$ can be written as a
sum:

(3.4) $X=\sum_{i}X_{i}\;.$

where each $X_{i}$ is supported by $U_{i}$. As $X$ is compactly supported, the
$X_{i}$ are almost all null and thus the above sum is finite. As each $U_{i}$
is diffeomorphic to $\mathbb{R}^{n}$, we can apply the case $M=\mathbb{R}^{n}$
which gives $Y_{i},Z_{i}\in{\mathfrak{g}}_{3}$ and
$W_{i}\in{\mathfrak{g}}_{1}$ all supported by $\mathbb{T}\times U_{i}$ such
that $[W_{i},Y_{i}]+Z_{i}=X_{i}$. We conclude by summing over $i$. ∎

######  Remark 3.4.

We proved that
$\left[{\mathfrak{g}}_{3},{\mathfrak{g}}_{1}\right]+{\mathfrak{g}}_{3}=\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times
M)$.

We now have the tools to show:

###### Proof of F.

Let $G\in\mathrm{Diff}_{c}^{\infty}(\mathbb{T}\times M)$ and let
$(G_{t})_{t\in[0,1]}$ be a compactly supported smooth path from
$G_{0}=\mathrm{id}$ to $G_{1}=G$ in
$\mathrm{Diff}_{c}^{\infty}(\mathbb{T}\times M)$. Derivatives
$X_{t}:=\partial_{t}G_{t}$ define a smooth family $X=(X_{t})_{t}$ of vector
fields all supported in a compact subset $K\subset V\setminus\partial V$. In
particular, the time $1/N$-map $F_{i}$ of the vector field $X_{i/N}$ is
supported by $K$, and likewise for $F=F_{N-1}\circ\cdots\circ F_{0}$. By
definition of the Whitney topology, it suffices to show that for every $r\geq
1$, when $N$ is large, the map $F$ is $C^{r}$-close to $G$ to obtain that $F$
is close to $G$ in $\mathrm{Diff}_{c}^{\infty}(\mathbb{T}\times M)$.

Note that indeed each $F_{i}$ is $O_{C^{r}}(1/N^{2})$-close to
$\tilde{F}_{i}:=G_{(i+1)/N}\circ G_{i/N}^{-1}$ and so it holds:

(3.5) $G=\tilde{F}_{N-1}\circ\cdots\circ\tilde{F}_{0}=F_{N-1}\circ\cdots\circ
F_{0}+O_{C^{r}}(1/N)=F+O_{C^{r}}(1/N)\;.$

As each $F_{i}$ belongs to $\mathrm{Plug}$, and since $\mathrm{Plug}$ is a
closed group by Proposition 2.13, it comes that $F$ belongs to $\mathrm{Plug}$
and its limit $G$ when $N\to\infty$ as well. ∎

##### Parametric counterpart.

To prove F′ we will use the following parametric counterpart of Proposition
3.3:

######  Proposition 3.5.

The flow of every ${\mathscr{P}}$-family of vector fields is
${\mathscr{P}}$-pluggable:

(3.6)
${\mathfrak{plug}}_{\mathscr{P}}=\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times
M)_{\mathscr{P}}\;.$

###### Proof.

As in the proof of Proposition 3.3, it suffices to show that the Lie algebra
generated by ${\mathfrak{g}}_{1{\mathscr{P}}}$ and
${\mathfrak{g}}_{3{\mathscr{P}}}$ equals ${\mathfrak{plug}}_{\mathscr{P}}$. We
start with the case $M=\mathbb{R}^{n}$ and ${\mathscr{P}}=\mathbb{R}^{d}$. Let
$X_{\mathscr{P}}=(X_{p})_{p\in{\mathscr{P}}}$ be a ${\mathscr{P}}$-family in
$\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times M)_{\mathscr{P}}$ and
$f_{p}\in\mathfrak{diff}_{c}(\mathbb{T}\times M)$ be the
$\mathbb{T}$-coordinate of $X_{p}$ for each $p\in{\mathscr{P}}$. Let
$W_{\mathscr{P}}\in{\mathfrak{g}}_{3{\mathscr{P}}}$ be:

$W_{\mathscr{P}}:=((0,f_{p},0,\dots,0))_{p\in{\mathscr{P}}}\;.$

With $Y$ as defined in Eq. 3.2, it holds by Eq. 3.3:

$[W_{p},Y](\theta,y)=(f_{p}(\theta,y),-\partial_{\theta}f_{p}(\theta,y)\cdot
y_{1},0,\dots,0))_{p\in{\mathscr{P}}}\;.$

We set
$Y_{\mathscr{P}}:=(Y)_{p\in{\mathscr{P}}}{\mathfrak{g}}_{1{\mathscr{P}}}$. The
latter computation gives:

(3.7)
$[W_{\mathscr{P}},Y_{\mathscr{P}}](\theta,y)=((f_{p}(\theta,y),-\partial_{\theta}f_{p}(\theta,y)\cdot
y_{1},0,\dots,0))_{p\in{\mathscr{P}}}\;.$

Thus $X_{\mathscr{P}}-[W_{\mathscr{P}},Y_{\mathscr{P}}]$ is in
${\mathfrak{g}}_{3{\mathscr{P}}}$. Now for the general case we conclude by
using a partition of unity as before. ∎

It allows to conclude:

###### Proof of F′.

Consider $G_{\mathscr{P}}\in\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times
M)_{\mathscr{P}}$. As a corollary of F, for any $r\geq 0$, the map
$\widehat{G_{\mathscr{P}}}\in\mathrm{Diff}_{c}^{\infty}(\mathbb{T}\times
M\times{\mathscr{P}})$ can be approximated by a composition of flows of vector
fields with a zero ${\mathscr{P}}$-coordinate. Thus the family
$G_{\mathscr{P}}$ can be approximated arbitrarily close by a composition of
compactly supported families of flows of vector fields in the $C^{r}$-norm for
any $r\geq 0$. By Proposition 3.3, each of the families of flows belongs to
$\mathrm{Plug}_{\mathscr{P}}$. Since $\mathrm{Plug}_{\mathscr{P}}$ is a group
by Proposition 2.13 the above composition belongs to
$\mathrm{Plug}_{\mathscr{P}}$ as well and it comes that
$(G_{p})_{p\in{\mathscr{P}}}$ is in $\mathrm{Plug}_{\mathscr{P}}$ by closure
of $\mathrm{Plug}_{\mathscr{P}}$. ∎

###  3.2 Eigenvectors of the adjoint representation

Observe that the Lie algebra generated by $\mathfrak{g}_{1}$ and
$\mathfrak{g}_{2}$ contains the vector fields that do not depend on the
$\mathbb{T}$-coordinate. To obtain more elements in ${\mathfrak{plug}}$ we
study the eigenvectors of operators of the form:

$\mathrm{ad}_{X}:Y\in\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times
M)\mapsto[X,Y]\in\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times M)\;,$

for $X\in\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times M)$.

In this subsection, we show that if $Y$ is an eigenvector of $\mathrm{ad}_{X}$
for $X\in{\mathfrak{plug}}$ then $Y$ is in ${\mathfrak{plug}}$ (see
Proposition 3.7). This will enable to prove Proposition 3.1 in Section 3.3.

######  Definition 3.6.

Let $V$ be a manifold and let ${\mathfrak{g}}$ be a subspace of
$\mathfrak{diff}_{c}^{\infty}(V)$. We denote:

${\mathfrak{Eig}}(\mathrm{ad}_{\mathfrak{g}}):=\\{Y\in\mathfrak{diff}_{c}^{\infty}(V):\exists
X\in{\mathfrak{g}}\text{ such that }Y=[X,Y]\\}\;.$

The following is the key proposition enabling to construct new examples of
pluggable flows:

######  Proposition 3.7.

If $Y\in\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times M)$ satisfies
$\mathrm{ad}_{X}Y=Y$ for $X\in{\mathfrak{plug}}$ then $Y$ is in
${\mathfrak{plug}}$:

${\mathfrak{Eig}}(\mathrm{ad}_{\mathfrak{plug}})\subset{\mathfrak{plug}}\;.$

###### Proof.

We recall that the adjoint of $X$ by $f\in\mathrm{Diff}_{c}^{\infty}(V)$ is
$\mathrm{Ad}_{f}X:=Df\circ X\circ f^{-1}$. The following contains a key idea
for the proof of the main theorem:

######  Lemma 3.8.

If $Y$ satisfies $[X,Y]=Y$, then for every $s,t\in\mathbb{R}$ it holds:

(3.8) $\mathrm{Fl}^{t}_{Y}=\mathrm{Fl}_{X}^{-s}\circ\mathrm{Fl}_{Y}^{t\cdot
e^{-s}}\circ\mathrm{Fl}_{X}^{s}\;.$

###### Proof.

Let $Y_{s}:=\mathrm{Ad}_{\mathrm{Fl}_{X}^{s}}(Y):=D\mathrm{Fl}_{X}^{s}\circ
Y\circ\mathrm{Fl}_{X}^{-s}$. Observe that
$X=\mathrm{Ad}_{\mathrm{Fl}_{X}^{s}}(X)$. Thus since $Y=[X,Y]$, it holds:

(3.9)
$[X,Y_{s}]=[\mathrm{Ad}_{\mathrm{Fl}_{X}^{s}}(X),\mathrm{Ad}_{\mathrm{Fl}_{X}^{s}}(Y)]=\mathrm{Ad}_{\mathrm{Fl}_{X}^{s}}([X,Y])=\mathrm{Ad}_{\mathrm{Fl}_{X}^{s}}(Y)=Y_{s}\;.$

Also for every $s\in\mathbb{R}$, we have:

(3.10)
$\partial_{s}Y_{s}=\partial_{t}(Y_{s+t})_{|t=0}=\partial_{t}\mathrm{Ad}_{\mathrm{Fl}_{X}^{t}}(Y_{s})_{|t=0}=\partial_{t}(D\mathrm{Fl}_{X}^{t}\circ
Y_{s}\circ\mathrm{Fl}_{X}^{-t})_{|t=0}$

and thus:

(3.11) $\partial_{s}Y_{s}=DX\circ Y_{s}-DY_{s}\circ X=-[X,Y_{s}]=-Y_{s}\;.$

Consequently $\partial_{s}Y_{s}=-Y_{s}$ and thus $Y_{s}=e^{-s}\cdot Y$. It
follows that $e^{s}\cdot Y=\mathrm{Ad}_{\mathrm{Fl}_{X}^{s}}(Y)$. After
integration between $0$ and $e^{-s}\cdot t$, we obtain the sought equality. ∎

We can now prove Proposition 3.7. Fix a neighborhood ${\mathcal{N}}$ of
$\mathrm{id}\in\mathrm{Diff}^{\infty}(\mathbb{T}\times M)$ and let us show the
existence of a plugin in ${\mathcal{N}}$ whose output is
$\mathrm{Fl}^{t}_{Y}$. Let ${\mathcal{N}}_{c}$ be the neighborhood of
$\mathrm{id}\in\mathrm{Diff}_{c}^{\infty}(\mathbb{T}\times M)$ given by Lemma
2.10. Let $s\in\mathbb{R}$ be sufficiently large such that
$\mathrm{Fl}^{e^{-s}\cdot t}_{Y}$ belongs to the neighborhood
${\mathcal{N}}_{c}$ of the identity. Then for any $k\geq 1$ large enough
$\mathrm{Fl}^{e^{-s}\cdot t}_{Y}$ is the output of a plugin
$g\in{\mathcal{N}}$ of step $2^{-k}$. Since $X\in{\mathfrak{plug}}$, it holds
$\mathrm{Fl}_{X}^{s}\in\mathrm{Plug}$ so for any $k\geq 1$ large enough
$\mathrm{Fl}_{X}^{s}$ and $\mathrm{Fl}_{X}^{-s}$ are the outputs of plugins
$h$ and $f$ in ${\mathcal{N}}$ of respective steps $2^{-k}$ and $2^{-k-1}$. So
for any large $k$ the map $\mathrm{Fl}^{t}_{Y}$ is the output of the plugin
$(h\star g)\star f$ with step $2^{-k-2}$. Since this holds for any
neighbourhood ${\mathcal{N}}$ of the identity the plugin $(h\star g)\star f$
can be taken arbitrary close to identity. Therefore for every $t$, there
exists a plugin with output $\mathrm{Fl}_{X}^{t}$ arbitrarily close to
identity for any small enough step. Hence $X$ is in ${\mathfrak{plug}}$. ∎

A second main ingredient of the proof of the main theorem is the following of
independent interest:

######  Proposition 3.9.

For every $T\in\mathfrak{diff}^{\infty}_{c}(V)$, there exist finite families
$(X_{i})_{i},(Y_{i})_{i},(Z_{i})_{i}$ of vector fields in
$\mathfrak{diff}^{\infty}_{c}(V)$ such that:

$T=\sum_{i}[Y_{i},Z_{i}]\quad\text{and}\quad Y_{i}=[X_{i},Y_{i}]\;.$

The proof of this proposition and its parametric counterpart will occupy the
full LABEL:{proof_of_deceig}. Now note that the following is an isomorphism of
Lie algebras:

${\mathfrak{i}}_{2}:X\in\mathfrak{diff}_{c}^{\infty}(M)\mapsto(0,X)\in{\mathfrak{g}}_{2}\;.$

Thus, by applying this isomorphism to the image of sets involved in the
statement of Proposition 3.9 for every $T\in{\mathfrak{g}}_{2}$, there exist
finite families $(X_{i})_{i},(Y_{i})_{i},(Z_{i})_{i}$ of vector fields in
${\mathfrak{g}}_{2}$ such that:

$T=\sum_{i}[Y_{i},Z_{i}]\quad\text{and}\quad Y_{i}=[X_{i},Y_{i}]\;.$

In other words, we proved:

######  Corollary 3.10.

Any element of ${\mathfrak{g}}_{2}$ can be written as a sum of Lie brackets of
elements of ${\mathfrak{g}}_{2}$ with elements in
${\mathfrak{Eig}}(\mathrm{ad}_{{\mathfrak{g}}_{2}})\cap{\mathfrak{g}}_{2}$.

This corollary allows to deduce:

###### Proof of Proposition 3.1.

Let $p_{\theta}:\mathbb{T}\times M\to\mathbb{T}$ be the projection on the
$\mathbb{T}$-coordinate.

First note that by Fourier’s theorem, the space ${\mathfrak{g}}_{3}$ is the
closure of the vector space spanned by elements of the form:

$\phi\circ p_{\theta}\cdot Y\quad\text{for }\phi\in
C^{\infty}(\mathbb{T},\mathbb{R})\quad\text{and}\quad
Y\in{\mathfrak{g}}_{2}\;.$

Since ${\mathfrak{plug}}$ is a closed vector space by Proposition 2.16, it
suffices to show any such $\phi\circ p_{\theta}\cdot Y$ is in
${\mathfrak{plug}}$. To do so, we first start with the case where there exists
$X\in{\mathfrak{g}}_{2}$ such that $Y=[X,Y]$, i.e. we assume that
$Y\in{\mathfrak{Eig}}(\mathrm{ad}_{{\mathfrak{g}}_{2}})$. Since the
$\mathbb{T}$-coordinate $p_{\theta}\circ X$ of $X$ is zero, it follows:

(3.12) $[X,\phi\circ p_{\theta}\cdot Y]=\phi\circ p_{\theta}\cdot
DY(X)-DX(\phi\circ p_{\theta}\cdot Y)=\phi\circ p_{\theta}\cdot[X,Y]=\phi\circ
p_{\theta}\cdot Y\;.$

Thus by Proposition 3.7 we have $\phi\circ p_{\theta}\cdot
Y\in{\mathfrak{plug}}$ since $X\in{\mathfrak{g}}_{2}\subset{\mathfrak{plug}}$.
Now in the general case, for every $Y\in{\mathfrak{g}}_{2}$, by 3.10, there
exist an integer $N\geq 1$, $Z_{i}\in{\mathfrak{g}}_{2}$ and
$Y_{i}\in{\mathfrak{Eig}}(\mathrm{ad}_{{\mathfrak{g}}_{2}})\cap{\mathfrak{g}}_{2}$
for any $1\leq i\leq N$, such that:

(3.13) $Y=\sum_{1\leq N}[Y_{i},Z_{i}]\;.$

By the first case for any $1\leq i\leq N$ we have $\phi\circ p_{\theta}\cdot
Y_{i}\in{\mathfrak{plug}}$. Since ${\mathfrak{plug}}$ is a Lie-algebra and
each $Z_{i}$ is in ${\mathfrak{g}}_{2}\subset{\mathfrak{plug}}$, it
immediately follows:

(3.14) $\phi\circ p_{\theta}\cdot Y=\sum_{1\leq i\leq N}\phi\circ
p_{\theta}\cdot[Y_{i},Z_{i}]=\sum_{1\leq i\leq N}[\phi\circ p_{\theta}\cdot
Y_{i},Z_{i}]\in{\mathfrak{plug}}\;.$

∎

##### Parametric counterpart.

The following is the parametric counterpart of Proposition 3.7.

######  Proposition 3.11.

If $Y_{\mathscr{P}}\in\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times
M)_{\mathscr{P}}$ satisfies
$[X_{\mathscr{P}},Y_{\mathscr{P}}]=Y_{\mathscr{P}}$ for
$X_{\mathscr{P}}\in{\mathfrak{plug}}_{\mathscr{P}}$, then
$Y_{\mathscr{P}}\in{\mathfrak{plug}}_{\mathscr{P}}$.

###### Proof.

Let $X_{\mathscr{P}}=:(X_{p})_{p\in{\mathscr{P}}}$ and
$Y_{\mathscr{P}}=:(Y_{p})_{p\in{\mathscr{P}}}$. By Lemma 3.8, it holds:

(3.15)
$\mathrm{Fl}_{Y_{p}}^{t}=\mathrm{Fl}_{X_{p}}^{-s}\circ\mathrm{Fl}_{Y_{p}}^{t\cdot
e^{-s}}\circ\mathrm{Fl}_{X_{p}}^{s}\;,$

for any $t,s\in\mathbb{R}$. For any neighborhood ${\mathcal{N}}$ of
$\mathrm{id}\in\mathrm{Diff}^{\infty}(\mathbb{T}\times M)_{\mathscr{P}}$, we
denote ${\mathcal{N}}_{c}$ the neighborhood of
$\mathrm{id}\in\mathrm{Diff}_{c}^{\infty}(\mathbb{T}\times M)_{\mathscr{P}}$
given by Lemma 2.14. For all $t\in\mathbb{R}$ and for $s$ large enough
$(\mathrm{Fl}_{Y_{p}}^{t\cdot
e^{-s}})_{p\in{\mathscr{P}}}\subset{\mathcal{N}}_{c}$ and thus it is the
output of a ${\mathscr{P}}$-plugin in ${\mathcal{N}}$ of any small step. We
now regard the $\star$ product of the latter plugins with
${\mathscr{P}}$-plugins of $(\mathrm{Fl}^{s}_{X_{p}})_{p\in{\mathscr{P}}}$ and
$(\mathrm{Fl}^{-s}_{X_{p}})_{p\in{\mathscr{P}}}$ to obtain a
${\mathscr{P}}$-plugin with output
$(\mathrm{Fl}_{Y_{p}}^{t})_{p\in{\mathscr{P}}}$ of any small step. Moreover,
by construction, this ${\mathscr{P}}$-plugin can be taken arbitrarily close to
identity family, which ends the proof. ∎

For
${\mathfrak{g}}_{\mathscr{P}}\subset\mathfrak{diff}_{c}^{\infty}(V)_{\mathscr{P}}$,
we denote:

${\mathfrak{Eig}}(\mathrm{ad}_{{\mathfrak{g}}_{\mathscr{P}}}):=\\{Y_{\mathscr{P}}\in\mathfrak{diff}_{c}^{\infty}(V)_{\mathscr{P}}:\exists
X_{\mathscr{P}}\in{\mathfrak{g}}_{\mathscr{P}}\text{ such that
}Y_{\mathscr{P}}=[X_{\mathscr{P}},Y_{\mathscr{P}}]\\}\;.$

The following parametric counterpart of Proposition 3.9 holds:

######  Proposition 3.12.

For every $T_{\mathscr{P}}\in\mathfrak{diff}^{\infty}_{c}(V)_{\mathscr{P}}$,
there exist finite families
$(X_{i{\mathscr{P}}})_{i},(Y_{i{\mathscr{P}}})_{i},(Z_{i{\mathscr{P}}})_{i}$
of vector fields in $\mathfrak{diff}^{\infty}_{c}(V)_{\mathscr{P}}$ such that:

$T_{\mathscr{P}}=\sum_{i}[Y_{i{\mathscr{P}}},Z_{i{\mathscr{P}}}]\quad\text{and}\quad
Y_{i{\mathscr{P}}}=[X_{i{\mathscr{P}}},Y_{i{\mathscr{P}}}]\;.$

Using the isomorphism:

$X_{\mathscr{P}}\in\mathfrak{diff}_{c}^{\infty}(M)_{\mathscr{P}}\mapsto(0,X_{p})_{p\in{\mathscr{P}}}\in{\mathfrak{g}}_{2{\mathscr{P}}}$

leads as before to:

######  Corollary 3.13.

Any element of ${\mathfrak{g}}_{2{\mathscr{P}}}$ can be written as a sum of
Lie brackets of elements of ${\mathfrak{g}}_{2{\mathscr{P}}}$ with elements in
${\mathfrak{Eig}}(\mathrm{ad}_{{\mathfrak{g}}_{2{\mathscr{P}}}})\cap{\mathfrak{g}}_{2{\mathscr{P}}}$.

The latter allows to deduce:

###### Proof of Proposition 3.2.

By the Fourier theorem, the space ${\mathfrak{g}}_{3{\mathscr{P}}}$ is the
closure in $\mathfrak{diff}_{c}^{\infty}(\mathbb{T}\times M)$ of the vector
space spanned by vector fields of the form $(\phi\circ p_{\theta}\cdot
Y_{p})_{p\in{\mathscr{P}}}$ with $\phi\in C^{\infty}(\mathbb{T},\mathbb{R})$
and $(Y_{p})_{p\in{\mathscr{P}}}\in{\mathfrak{g}}_{2{\mathscr{P}}}$. Since
${\mathfrak{plug}}_{\mathscr{P}}$ is a closed vector space by Proposition
2.18, it suffices then to show that $(\phi\circ p_{\theta}\cdot
Y_{p})_{p\in{\mathscr{P}}}$ is in ${\mathfrak{plug}}_{\mathscr{P}}$. To do so,
we first start with the case where there exists
$X_{\mathscr{P}}\in{\mathfrak{g}}_{2{\mathscr{P}}}$ such that
$Y_{\mathscr{P}}=[X_{\mathscr{P}},Y_{\mathscr{P}}]$. By Eqs. 3.12 and 3.14 of
the proof of Proposition 3.1, for each $p\in{\mathscr{P}}$, it holds
$\phi\circ p_{\theta}\cdot Y_{p}=[X_{p},\phi\circ p_{\theta}\cdot Y_{p}]$. And
thus the family $(\phi\circ p_{\theta}\cdot Y_{p})_{p\in{\mathscr{P}}}$ is in
${\mathfrak{plug}}_{\mathscr{P}}$ by Proposition 3.11.

Now in the general, by the latter 3.13 we can decompose:

(3.16) $Y_{\mathscr{P}}=\sum[Y_{i{\mathscr{P}}},Z_{i{\mathscr{P}}}]\;,$

with
$Y_{i{\mathscr{P}}}\in{\mathfrak{Eig}}({\mathfrak{g}}_{2{\mathscr{P}}})\cap{\mathfrak{g}}_{2{\mathscr{P}}}$
and $Z_{i{\mathscr{P}}}\in{\mathfrak{g}}_{2{\mathscr{P}}}$. Thus Eq. 3.14
applied for each $p\in{\mathscr{P}}$ leads to:

(3.17) $(\phi\circ p_{\theta}\cdot Y_{p})_{p\in{\mathscr{P}}}=\sum[(\phi\circ
p_{\theta}\cdot Y_{ip})_{p\in{\mathscr{P}}},Z_{i{\mathscr{P}}}]\;,$

where $(Y_{ip})_{p\in{\mathscr{P}}}=Y_{i{\mathscr{P}}}$. Since
$Z_{i{\mathscr{P}}}\in{\mathfrak{g}}_{2{\mathscr{P}}}\subset{\mathfrak{plug}}_{\mathscr{P}}$
and $(\phi\circ p_{\theta}\cdot
Y_{ip})_{p\in{\mathscr{P}}}\in{\mathfrak{plug}}_{\mathscr{P}}$ by the first
case, it follows that $(\phi\circ p_{\theta}\cdot Y_{p})_{p\in{\mathscr{P}}}$
lies in the sub Lie algebra ${\mathfrak{plug}}_{\mathscr{P}}$ as well. ∎

###  3.3 Decomposition of vector fields

To prove the F, it remains only to show Proposition 3.9. We first prove this
proposition in the case $M=\mathbb{R}^{n}$, which we will deduce from the case
$M=\mathbb{R}$ and a parametric version thereof. This whole section is
dedicated to this proof. For a manifold $M$, let us denote:

${\mathfrak{Eig}}(M):=\\{Y\in\mathfrak{diff}_{c}^{\infty}(M):\exists
X\in\mathfrak{diff}_{c}^{\infty}(M)\text{ such that
}\mathrm{ad}_{X}Y=[X,Y]=Y\\}\;.$

We can rephrase Proposition 3.9 as:

######  Proposition 3.14.

Every vector field in $T\in\mathfrak{diff}^{\infty}_{c}(V)$ is a finite sum of
vector fields of the form $[Y_{i},Z_{i}]$ with $Y_{i}\in{\mathfrak{Eig}}(M)$.
In other words:

$[{\mathfrak{Eig}}(M),\mathfrak{diff}^{\infty}_{c}(M)]=\mathfrak{diff}^{\infty}_{c}(M).$

Two key ingredients of the proof of this proposition are the following
observations:

######  Fact 3.15.

The vector field $Y:y\mapsto 1$ on $\mathbb{R}$ satisfies $[X,Y]=Y$ with
$X:y\in\mathbb{R}\mapsto-y$.

######  Fact 3.16.

For any diffeomorphism $\psi\colon W\to V$ between manifolds and any vector
fields $X,Y\in\mathfrak{diff}_{c}^{\infty}(V)$ satisfying $[X,Y]=Y$, it holds
$[\psi^{*}X,\psi^{*}Y]=\psi^{*}Y$.

An important consequence of the latter fact is:

######  Fact 3.17.

For any diffeomorphism $\psi\colon W\to V$ from a manifold $W$ into $V$, if
$X\in[{\mathfrak{Eig}}(V),\mathfrak{diff}^{\infty}_{c}(V)]$, then
$\psi^{*}X\in[{\mathfrak{Eig}}(W),\mathfrak{diff}^{\infty}_{c}(W)]$.

Let ${\mathcal{Q}}$ be a manifold. We define:

${\mathfrak{Eig}}(\mathbb{R})_{\mathcal{Q}}=\Bigl{\\{}Y_{\mathcal{Q}}\in\mathfrak{diff}^{\infty}_{c}(\mathbb{R})_{\mathcal{Q}}:\text{
there exists a family
}X_{\mathcal{Q}}\in\mathfrak{diff}^{\infty}_{c}(\mathbb{R})_{\mathcal{Q}}\text{
such that }[X_{\mathcal{Q}},Y_{\mathcal{Q}}]=Y_{\mathcal{Q}}\Bigr{\\}}.$

We can prove Proposition 3.14 in the case $M=\mathbb{R}$:

######  Lemma 3.18.

We have:

$[{\mathfrak{Eig}}(\mathbb{R}),\mathfrak{diff}_{c}^{\infty}(\mathbb{R})]=\mathfrak{diff}_{c}^{\infty}(\mathbb{R})\;.$

Moreover for any manifold ${\mathcal{Q}}$, every
$T_{\mathcal{Q}}\in\mathfrak{diff}_{c}^{\infty}(\mathbb{R})_{{\mathcal{Q}}}$
satisfies $T_{\mathcal{Q}}=[Y_{\mathcal{Q}},Z_{\mathcal{Q}}]$ for some
$Y_{\mathcal{Q}}\in{\mathfrak{Eig}}(\mathbb{R})_{{\mathcal{Q}}}$ and
$Z_{\mathcal{Q}}\in\mathfrak{diff}_{c}^{\infty}(\mathbb{R})_{{\mathcal{Q}}}$.

###### Proof.

We first give an intuitive idea of the proof using 3.15. First note that for
every $T\in\mathfrak{diff}_{c}^{\infty}(\mathbb{R})$, there exist
$X,Y,Z\in\mathfrak{diff}^{\infty}(\mathbb{R})$ such that $Y=[X,Y]$ and
$T=[Y,Z]$. Indeed it suffices to take $X\colon y\mapsto-y$, $Y\colon y\mapsto
1$ and $Z\colon y\mapsto\int_{-\infty}^{y}T(t)\,dt$. This almost proves the
first assertion of the Lemma. To have exactly the sought result we shall
modify $X$, $Y$ and $Z$ to make them compactly supported. Let
$\widetilde{T}\in\mathfrak{diff}^{\infty}_{c}(\mathbb{R})$. Take intervals
$[-A,A]\subset(-a,a)$ containing its support. Consider the map $\psi\colon
y\mapsto y\cdot e^{e^{(a^{2}-y^{2})^{-1}}}$ from $(-a,a)$ to $\mathbb{R}$. We
compute:

(3.18) $\partial_{y}\psi(y)=\phi(y)\cdot
e^{e^{(a^{2}-y^{2})^{-1}}}\;,\quad\text{with
}\phi(y)=1+2y^{2}\cdot(a^{2}-y^{2})^{-2}e^{(a^{2}-y^{2})^{-1}}\;.$

Thus the map $\psi$ is a diffeomorphism. This allows to consider the push-
forward:

(3.19) $T=\psi_{*}\widetilde{T}=D\psi\circ\widetilde{T}\circ\psi^{-1}\;.$

Now we define as above:

(3.20) $X\colon y\mapsto-y\,,\quad Y\colon y\mapsto 1\quad\text{and}\quad
Z\colon y\mapsto\int_{-\infty}^{y}T(t)\,dt\,.$

Note that $\tilde{T}=\psi^{*}T$ and put $\widetilde{X}=\psi^{*}X$ and
$\widetilde{Y}=\psi^{*}Y$. Then by 3.16, it holds:

(3.21)
$\widetilde{Y}=[\widetilde{X},\widetilde{Y}]\quad\text{and}\quad[\widetilde{Y},\widetilde{Z}]=\widetilde{T}\;.$

Let us show that these pulled-back vector fields on $(-a,a)$ extend smoothly
by $0$ to $\mathbb{R}$. For any vector field
$S\in\mathfrak{diff}_{c}^{\infty}(\mathbb{R})$, we have
$\psi^{*}S=(\partial_{y}\psi)^{-1}\cdot
S\circ\psi=\tfrac{S\circ\psi}{\partial_{y}\psi}$. It follows:

(3.22)
$\widetilde{X}(y)=\psi^{*}X(y)=\tfrac{-\psi(y)}{\partial_{y}\psi(y)}=\tfrac{-y}{\phi(y)}\quad\text{and}\quad\widetilde{Y}(y)=\psi^{*}Y(y)=\tfrac{1}{\partial_{y}\psi(y)},$

Observe that $\phi(y)$ grows exponentially fast to $+\infty$ when $|y|\to a$.
Thus $\widetilde{X}$ and $\widetilde{Y}$ have all their derivatives tending to
$0$ as $y$ tends to $\pm a$. So they extend smoothly by $0$ to vector fields
in $\mathfrak{diff}_{c}^{\infty}(\mathbb{R})$. Also note that $Z(y)=0$ for
$y\leq\psi(-A)$ and $Z(y)=\int_{\mathbb{R}}T(t)\,dt$ for $y\geq\psi(A)$. Thus:

(3.23) $\widetilde{Z}(y)=\psi^{*}Z(y)=0\quad\mbox{ for
}-a<y\leq-A\quad\text{and}\quad\widetilde{Z}(y)=\frac{1}{\partial_{y}\psi(y)}\cdot\int_{\mathbb{R}}T(t)\cdot
dt\quad\mbox{ for }A\leq y<a.$

Likewise $\widetilde{Z}$ extends smoothly by $0$ to form a vector field in
$\mathfrak{diff}_{c}^{\infty}(\mathbb{R})$. As
$\widetilde{Y}=[\widetilde{X},\widetilde{Y}]$ and
$[\widetilde{Y},\widetilde{Z}]=\widetilde{T}$, this proves the first assertion
of the lemma.

For the parametric assertion, observe that $\psi$, $\widetilde{X}$ and
$\widetilde{Y}$ depends only on the segment $[-a,a]$. Hence given
$\widetilde{T}_{\mathcal{Q}}=(\widetilde{T}_{q})_{q\in{\mathcal{Q}}}\in\mathfrak{diff}_{c}^{\infty}(\mathbb{R})_{{\mathcal{Q}}}$,
we set $[-A,A]\subset(-a,a)$ containing the supports of all
$\widetilde{T}_{q}$ and define $\psi$, $\widetilde{X}$ and $\widetilde{Y}$ as
above. Then we observe that $\widetilde{Z}_{q}=\psi^{*}\int T_{q}$ depends
smoothly on $q$ and define a family
$\widetilde{Z}_{\mathcal{Q}}\in\mathfrak{diff}_{c}^{\infty}(\mathbb{R})_{\mathcal{Q}}$
which satisfies the sough equalities with $\widetilde{T}_{\mathcal{Q}}$ and
the constant families $(\widetilde{X})_{q\in{\mathcal{Q}}}$ and
$(\widetilde{Y})_{q\in{\mathcal{Q}}}$. ∎

We are going to use the parametric assertion of the latter lemma to obtain:

######  Lemma 3.19.

We have
$[{\mathfrak{Eig}}(\mathbb{R}^{n}),\mathfrak{diff}^{\infty}_{c}(\mathbb{R}^{n})]=\mathfrak{diff}^{\infty}_{c}(\mathbb{R}^{n})$.

###### Proof.

Lemma 3.18 corresponds to the case $n=1$. For $n\geq 2$, given
$T\in\mathfrak{diff}^{\infty}_{c}(\mathbb{R}^{n})$, we write its components as
$T=(T_{1},\dots,T_{n})$. By linearity of the condition, it suffices to show
that each vector field $(0,\dots,0,T_{i},0,\dots,0)$ is in
$[{\mathfrak{Eig}}(\mathbb{R}^{n}),\mathfrak{diff}^{\infty}_{c}(\mathbb{R}^{n})]$.
Using an adjunction by a permutation of the coordinates and 3.17, its suffices
to show that each $(T_{i},0,\dots,0)$ is in
$[{\mathfrak{Eig}}(\mathbb{R}^{n}),\mathfrak{diff}^{\infty}_{c}(\mathbb{R}^{n})]$.
In other words, it suffices to prove that the following subalgebra
${\mathfrak{h}}(\mathbb{R}^{n})$ of
$\mathfrak{diff}^{\infty}_{c}(\mathbb{R}^{n})$ is included in
$[{\mathfrak{Eig}}(\mathbb{R}^{n}),\mathfrak{diff}^{\infty}_{c}(\mathbb{R}^{n})]$
:

${\mathfrak{h}}(\mathbb{R}^{n}):=\\{y\in\mathbb{R}^{n}\mapsto(h(y),0,\dots,0):h\in
C^{\infty}_{c}(\mathbb{R}^{n},\mathbb{R})\\}\;.$

To this end, note that
$X_{\mathbb{R}^{n-1}}\in\mathfrak{diff}^{\infty}_{c}(\mathbb{R})_{\mathbb{R}^{n-1}}\mapsto\widehat{X_{\mathbb{R}^{n-1}}}\in{\mathfrak{h}}(\mathbb{R}^{n})$
is an isomorphism of Lie algebras. By Lemma 3.18 with
${\mathcal{Q}}=\mathbb{R}^{n-1}$, we have:

${\mathfrak{h}}(\mathbb{R}^{n})=[\widehat{{\mathfrak{Eig}}(\mathbb{R})_{{\mathcal{Q}}}},{\mathfrak{h}}(\mathbb{R}^{n})]\;.$

Finally we note that $\widehat{{\mathfrak{Eig}}(\mathbb{R})_{{\mathcal{Q}}}}$
is formed by vector fields of the form $\widehat{Y_{\mathcal{Q}}}$ such that
$Y_{\mathcal{Q}}=[X_{\mathcal{Q}},Y_{\mathcal{Q}}]$ for
$X_{\mathcal{Q}}\in\mathfrak{diff}_{c}^{\infty}(\mathbb{R})_{\mathcal{Q}}$.
Thus
$\widehat{Y_{\mathcal{Q}}}=[\widehat{X_{\mathcal{Q}}},\widehat{Y_{\mathcal{Q}}}]$,
this proves that
${\mathfrak{h}}(\mathbb{R}^{n})\subset[{\mathfrak{Eig}}(\mathbb{R}^{n}),{\mathfrak{h}}(\mathbb{R}^{n})]$.

∎

We can now treat the general case:

###### Proof of Proposition 3.14.

Let $T\in\mathfrak{diff}^{\infty}_{c}(M)$. Then it decomposes in a finite sum
$T=\sum_{i}T_{i}$ where each $T_{i}$ is compactly supported in an open set
$U_{i}$ which is diffeomorphic to $\mathbb{R}^{n}$ via a map $\psi_{i}\colon
U_{i}\to\mathbb{R}^{n}$. Consider the push forward
$\psi_{i*}T_{i}\in\mathfrak{diff}^{\infty}_{c}(\mathbb{R}^{n})$. By Lemma 3.19
the field $\psi_{i*}T_{i}|U_{i}$ belongs to
$[{\mathfrak{Eig}}(\mathbb{R}^{n}),\mathfrak{diff}^{\infty}_{c}(\mathbb{R}^{n})]$.
Thus by Lemma 3.18, it holds
$T_{i}|U_{i}\in[{\mathfrak{Eig}}(U_{i}),\mathfrak{diff}^{\infty}_{c}(U_{i})]$.
This means that $T_{i}=\sum_{\text{Finite}}[Y_{j},Z_{j}]$ with
$Y_{j}=[X_{j},Y_{j}]$ for some
$X_{j},Y_{j},Z_{j}\in\mathfrak{diff}^{\infty}_{c}(\mathbb{R}^{n})$. Extending
all these vector fields by $0$, we obtain that $T_{i}$ belongs to
$[{\mathfrak{Eig}}(M),\mathfrak{diff}^{\infty}_{c}(M)]$. So does
$T=\sum_{i}T_{i}$. ∎

##### Parametric counterpart.

We set:

${\mathfrak{Eig}}(M)_{\mathscr{P}}:=\\{Y_{\mathscr{P}}\in\mathfrak{diff}_{c}^{\infty}(M)_{\mathscr{P}}:Y_{\mathscr{P}}=[X_{\mathscr{P}},Y_{\mathscr{P}}]\text{
with }X_{\mathscr{P}}\in\mathfrak{diff}_{c}^{\infty}(M)_{\mathscr{P}}\\}\;.$

We shall prove Proposition 3.12 that we rephrase as:

######  Proposition 3.20.

It holds:

$[{\mathfrak{Eig}}(M)_{\mathscr{P}},\mathfrak{diff}^{\infty}_{c}(M)_{\mathscr{P}}]=\mathfrak{diff}^{\infty}_{c}(M)_{\mathscr{P}}.$

Similarly to the proof of Proposition 3.14, Proposition 3.20 is an easy
consequence of the following:

######  Lemma 3.21.

We have
$[{\mathfrak{Eig}}(\mathbb{R}^{n})_{\mathscr{P}},\mathfrak{diff}^{\infty}_{c}(\mathbb{R}^{n})_{\mathscr{P}}]=\mathfrak{diff}^{\infty}_{c}(\mathbb{R}^{n})_{\mathscr{P}}$.

###### Proof.

As in the proof of Proposition 3.14, we only need to prove that the following
Lie subalgebra ${\mathfrak{h}}(\mathbb{R}^{n})_{\mathscr{P}}$ of
$\mathfrak{diff}^{\infty}_{c}(\mathbb{R}^{n})_{\mathscr{P}}$ is included in
$[{\mathfrak{Eig}}(\mathbb{R}^{n})_{\mathscr{P}},\mathfrak{diff}^{\infty}_{c}(\mathbb{R}^{n})_{\mathscr{P}}]$
:

${\mathfrak{h}}(\mathbb{R}^{n})_{\mathscr{P}}:=\\{((h_{p},0,\dots,0))_{p\in{\mathscr{P}}}:h_{\mathscr{P}}=(h_{p})_{p\in{\mathscr{P}}}\in
C^{\infty}_{c}(\mathbb{R}^{n},\mathbb{R})_{\mathscr{P}}\\}\;.$

To this end we proceed as in Lemma 3.19, by using the isomorphism of Lie
algebra:

$X_{\mathbb{R}^{n-1}\times{\mathscr{P}}}\in\mathfrak{diff}^{\infty}_{c}(\mathbb{R})_{\mathbb{R}^{n-1}\times{\mathscr{P}}}\mapsto
Y_{\mathscr{P}}\in{\mathfrak{h}}(\mathbb{R}^{n})_{\mathscr{P}}\text{ such that
}\widehat{X_{\mathbb{R}^{n-1}\times{\mathscr{P}}}}=\widehat{Y_{\mathscr{P}}}\;,$

and using Lemma 3.18. ∎

##  Appendix A Smoothness of outputs

The proof of Proposition 1.5 stating that the output of a plugin is
necessarily smooth is similar to the classical renormalization performed by
Douady-Ghys [Dou87], Yoccoz [Yoc95a] and Shilnikov-Turaev [ST00].

###### Proof of Proposition 1.5.

Let $g$ be a plugin with step $\sigma$. Let $\pi:\tilde{\mathbb{T}}\times
M:=\mathbb{R}\times M\to\mathbb{T}\times M=$ be the canonical cover. Let
$\tilde{g}$ be a lifting of $g$ such that $\tilde{g}(0,y)=(\sigma,y)$ for
every $y\in M$.

######  Fact A.1.

The action $\phi:(k,z)\in\mathbb{Z}\times\tilde{\mathbb{T}}\times
M\mapsto\tilde{g}^{k}(z)\in\tilde{\mathbb{T}}\times M$ is free, proper and
discontinuous.

###### Proof.

The action is free since no point of $\Delta+\mathbb{Z}$ is fixed by
$\tilde{g}$ nor in its complement (every point must come back to
$\Delta+\mathbb{Z}$). It is discontinuous since any $x\in\mathbb{R}\times M$
has its orbit which equals the one of a certain $z\in[k,k+\sigma)\times M$ for
some $k\in\mathbb{Z}$ by Definition 1.1.(iii), and the orbit of $z$ is
discrete by Definition 1.1.(i). Finally the action is proper since $\tau$ is
bounded by some $N\geq 1$ by Definition 1.1.(ii), and so any
$\tilde{g}^{N}(\theta,y)$ has its $\mathbb{R}$-coordinate greater than
$\theta+1$. ∎

Thus the quotient $C:=\tilde{\mathbb{T}}\times M/\phi$ is a manifold. As
$\tilde{g}$ sends the left hand side of $\Delta$ to its right hand side, the
image of $\Delta$ by the group action is both open and close, hence equal to
the connected set $C$. Therefore, $\Delta_{\sigma}=[0,\sigma)\times M$ is a
fundamental domain of this group action. Also the rescaling map
$H_{\sigma}:\Delta_{\sigma}\to\mathbb{T}\times M$ induces a diffeomorphism
between $C$ and $\mathbb{T}\times M$.

$g\colon$$\tilde{g}\colon$$T_{-1}$

Now observe that $\tilde{T}:(\theta,y)\mapsto(\theta+1,y)$ and $\tilde{g}$
commutes: $\tilde{T}\circ\tilde{g}=\tilde{g}\circ\tilde{T}$. Thus $\tilde{T}$
defines a smooth diffeomorphism $T$ on $C$. To determine $T$, we (abusively)
identify $\Delta_{\sigma}$ to a subset of both $\tilde{\mathbb{T}}\times M$
and $\mathbb{T}\times M$. Given $x\in\Delta_{\sigma}$, the point
$\tilde{T}(x)\in[1,1+\sigma)\times M$ is equivalent to the point
$y\in\Delta_{\sigma}$ such that there exists $k\geq 0$ satisfying
$\tilde{g}^{k}(y)=\tilde{T}(x)$. Note that $k=\tau(y)$ and so
$\tilde{T}(x)=\tilde{g}^{\tau}(y)$. Composing by $\pi$, we obtain that
$x=g^{\tau}(y)$. Hence $T$ is equal to the inverse of the action of $g^{\tau}$
on $C$. Therefore $G$ is a diffeomorphism. Observe that this construction
depends continuously on $g$ and so the output depends continuously on the
plugin. As the space of plugins is connected, it comes that the space of
output is connected to $\mathrm{id}$ by Example 1.4.

∎

##  Appendix B Lie algebras associated to closed subgroups

This section is dedicated to show Proposition 2.16 which states that
${\mathfrak{plug}}$ is a closed Lie algebra. We prove this using general
arguments on closed subgroups of $\mathrm{Diff}_{c}^{\infty}(V)$ where $V$ is
a manifold.

######  Proposition B.1.

For every closed subgroup $\mathbb{G}\subset\mathrm{Diff}_{c}^{\infty}(V)$,
the following is a closed Lie subalgebra of $\mathfrak{diff}_{c}^{\infty}(V)$:

${\mathfrak{g}}:=\\{X\in\mathfrak{diff}_{c}^{\infty}(V):\mathrm{Fl}_{X}^{t}\in\mathbb{G},\;\forall
t\in\mathbb{R}\\}.$

We immediately deduce the result of Proposition 2.16 by applying the latter
proposition with $V=\mathbb{T}\times M$ and $\mathbb{G}=\mathrm{Plug}$.

###### Proof of Proposition B.1.

${\mathfrak{g}}$ is a vector space. First note that if $X\in\mathbb{G}$ and
$\lambda\in\mathbb{R}$, it holds $\lambda\cdot X\in{\mathfrak{g}}$. Now for
$X,Y\in{\mathfrak{g}}$, for any large integer $N$ and $r\geq 1$, observe that
$\mathrm{Fl}_{X+Y}^{1/N}=\mathrm{Fl}_{X}^{1/N}\circ\mathrm{Fl}_{Y}^{1/N}+O(N^{-2})$
for the $C^{r}$-norm. Thus we have:

$Fl_{X+Y}^{1}=\left(Fl_{X}^{1/N}\circ\mathrm{Fl}_{Y}^{1/N}\right)^{N}+O_{C^{r}}(N^{-1}),$

and the supports of $\left(Fl_{X}^{1/N}\circ\mathrm{Fl}_{Y}^{1/N}\right)^{N}$
are included in the union of those of $X$ and $Y$. Thus $Fl_{X+Y}^{1}$ is the
limit of $\left(Fl_{X}^{1/N}\circ\mathrm{Fl}_{Y}^{1/N}\right)^{N}$ when
$N\to\infty$ in the topology of $\mathrm{Diff}^{\infty}_{c}(V)$. As
$\mathbb{G}$ is a group, the map
$(Fl_{X}^{1/N}\circ\mathrm{Fl}_{Y}^{1/N})^{N}$ belongs to $\mathbb{G}$, and
since $\mathbb{G}$ is closed the map $Fl_{X+Y}^{1}$ also belongs to
$\mathbb{G}$. Also for every $t\in\mathbb{R}$, by replacing $X,Y$ by
$(tX,tY)$, we obtain that $Fl_{X+Y}^{t}=Fl_{tX+tY}^{1}$ belongs to
$\mathbb{G}$.

${\mathfrak{g}}$ is a Lie algebra. For $X,Y\in{\mathfrak{g}}$ and $r\geq 1$,
we have for the $C^{r}$-norm:

$\mathrm{Fl}_{[X,Y]}^{\tau^{2}}=[\mathrm{Fl}_{X}^{\tau},\mathrm{Fl}_{Y}^{\tau}]+O(\tau^{3})$

Thus by taking $\tau^{2}=1/N$ small we have:

$\mathrm{Fl}_{[X,Y]}^{1}=\left([\mathrm{Fl}_{X}^{\sqrt{N}},\mathrm{Fl}_{Y}^{\sqrt{N}}]\right)^{N}+O({\sqrt{N}}^{-1})\;.$

So $\mathrm{Fl}_{[X,Y]}^{1}$ is in $\mathbb{G}$. Similarly, we have for any
$t$ that $\mathrm{Fl}_{[tX,Y]}^{1}\in\mathbb{G}$, and so
$\mathrm{Fl}_{[X,Y]}^{t}=\mathrm{Fl}_{[tX,Y]}^{1}\in\mathbb{G}$.

${\mathfrak{g}}$ is closed. As for every $t\geq 0$, the map
$\mathrm{Fl}^{t}:X\in\mathrm{Diff}_{c}^{\infty}(V)\mapsto\mathrm{Fl}_{X}^{t}$
is continuous and $\mathbb{G}$ is closed, the set
$\\{X\in\mathfrak{diff}_{c}^{\infty}(V):\mathrm{Fl}_{X}^{t}\in\mathbb{G}\\}$
is closed. Thus the intersection ${\mathfrak{g}}$ of the latter sets for all
$t$ is closed. ∎

To state the parameteric counterpart of the latter proposition, given a
manifold ${\mathscr{P}}$, we define:

$\mathrm{Fl}^{t}:X_{\mathscr{P}}=(X_{p})_{p\in{\mathscr{P}}}\in\mathfrak{diff}^{\infty}_{c}(V)_{\mathscr{P}}\mapsto(\mathrm{Fl}^{t}_{X_{p}})_{p\in{\mathscr{P}}}\in\mathrm{Diff}^{\infty}_{c}(V)_{\mathscr{P}}\;.$

Note that the following diagram commutes:

$\begin{array}[]{rcccl}&&\mathfrak{inc}&&\\\
&\mathfrak{diff}^{\infty}_{c}(V)_{\mathscr{P}}&\hookrightarrow&\mathfrak{diff}^{\infty}_{c}(V\times{\mathscr{P}})&\\\
\mathrm{Fl}^{t}&\downarrow&&\downarrow&\mathrm{Fl}^{t}\\\
&\mathrm{Diff}^{\infty}_{c}(V)_{\mathscr{P}}&\hookrightarrow&\mathrm{Diff}^{\infty}_{c}(V\times{\mathscr{P}})&\\\
&&{inc}&&\\\ \end{array}$

where $\mathfrak{inc}(X_{\mathscr{P}}):=\widehat{X_{\mathscr{P}}}$ and
$inc(f_{\mathscr{P}}):=\widehat{f_{\mathscr{P}}}$.

######  Corollary B.2.

For every closed subgroup
$\mathbb{G}_{\mathscr{P}}\subset\mathrm{Diff}_{c}^{\infty}(V)_{\mathscr{P}}$,
the following is a closed Lie subalgebra of
$\mathfrak{diff}_{c}^{\infty}(V)_{\mathscr{P}}$:

${\mathfrak{g}}_{\mathscr{P}}:=\\{X_{\mathscr{P}}\in\mathfrak{diff}_{c}^{\infty}(V)_{\mathscr{P}}:\mathrm{Fl}_{X}^{t}\in\mathbb{G},\;\forall
t\in\mathbb{R}\\}.$

###### Proof.

First note that $inc(\mathbb{G}_{\mathscr{P}})$ is a closed Lie sub-group of
$\mathrm{Diff}^{\infty}_{c}(V\times{\mathscr{P}})$. Hence it define via
Proposition B.1 a closed Lie algebra $\widehat{{\mathfrak{g}}_{\mathscr{P}}}$.
By commutativity of the diagram, we have:

$\mathfrak{inc}({\mathfrak{g}}_{\mathscr{P}}):=(\widehat{{\mathfrak{g}}_{\mathscr{P}}})\;.$

Hence ${\mathfrak{g}}_{\mathscr{P}}$ is a closed Lie subalgebra of
$\mathfrak{diff}_{c}^{\infty}(V)_{\mathscr{P}}$. ∎

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