$\rho_{n}a_{n+1}^{{}^{\prime\prime\prime\prime}}\sigma_{n+1}^{-l}\leq\rho_{n+2}^{1/2}\leq\rho_{n+1}^{1/2}.$

Now using Conditions 3), we deduce that

$4e|j_{n+1}|\sigma_{n+1}^{-1}\rho_{n+1}^{1/2}\leq\rho_{n+2},$

and by 5) the right hand side is smaller than $1$. This shows $N_{2}$.

Let us now check $N_{1}$ at rank $n+1$:

$(N_{1})\qquad|\delta_{j}|\leq\frac{1}{2^{j+1}},\ j=1\dots,n.$

Using the estimate $E_{1}$, we deduce that

$|\delta_{n+1}|\leq\frac{a_{n}^{\prime}}{\sigma_{n}^{k/2}}\rho_{n+1}^{1/2},$

and using conditions 4):

$\frac{a_{n}^{\prime}}{\sigma_{n}^{k+l}}\rho_{n+1}^{1/2}\leq\rho_{n+2}^{1/4}$

Finally by 5), we get that the right hand side is at most $1/2^{n+2}$.
Therefore assumption $N_{1}$ is satisfied at rank $n+1$. Thus the iteration is
well defined and converges. This proves the theorem. ∎

###  5.6. The involutivity statement

The Nash-Moser requires an inverse in a whole neighbourhood which is an
exceptional situation while in our versal deformation theorem we only need an
inverse along the transversal. One may ask if the existence of an inverse at
the initial point implies the existence along the transversal. In practise
this requires additional assumptions such as involutivity properties, which
formally spells out as follows:

######  Proposition 5.55.

Let ${\bf E}\longrightarrow{\bf B}$ be an $A1$-space and $({\bf E},{\bf
f+M},{\bf\mathfrak{g}})$ be an Arnold action, ${\bf T}\subset{\bf M}$ a
transversal with the choice with projector $\pi\in{\mathcal{L}}({\bf M},{\bf
M})$. Assume that:

  1. i)

there exists a map $L\in{\mathcal{L}}({\bf M},\mathfrak{g}))$ such that

$\kappa_{0}:=(\iota-\pi)(\iota-L)\in{\mathcal{K}}({\bf M},{\bf M})$

is a cutoff operator,

  2. ii)

for any $x\in M$, $L(x)\in\mathfrak{g}$ maps the transversal ${\bf T}$ to
itself

then the map $j=L\circ(\iota-L):\delta\mapsto[r\mapsto L(r-Lr(\delta))]$ is a
quasi-right inverse to the infinitesimal action.

###### Proof.

$\displaystyle j(\tau)(r)(x+\delta)$ $\displaystyle=L(r-L(r)\delta)(x+\delta)$
$\displaystyle=L(r)x+L(r)\delta-L(L(r)\delta)x-L(L(r)\delta)\delta$
$\displaystyle=r+\kappa_{0}(r)+L(r)\delta-L(r)\delta+\kappa_{0}(L(r)\delta)-L(L(r)\delta)\delta$
$\displaystyle=r+\kappa_{0}(r+L(r)\delta)-L(L(r)\delta)\delta$

and by assumption $L(L(r)\delta)\delta\in{\bf T}$. This proves the
proposition. ∎

###  5.7. The abstract versal deformation theorem for vector fields

Our versal deformation theorem contains in essence all classical theorems on
normal forms and versal deformations. The interested reader might consult to
see how to deduce them [16]. But it contains also more recent and new results.

We now state an abstract versal deformation theorem for vector fields which
might be considered as an analog of Tyurina’s theorem for hypersurface
singularities [41] (see also [19, 34]). The statement might look obscure to
non specialists while specialists will easily recognise the existence of an
analytic versal deformation over a Cantor set. More details on this theorem
will be given in a forthcoming work [18].

We say that a Kolmogorov space

$\mathfrak{g}\longrightarrow B$

is a Kolmogorov-Lie-algebra if there is a bilinear bracket $[-,-]$ which is
local in each argument.

An Arnold space

${\bf\mathfrak{g}\longrightarrow B}$

is an Arnold-Lie algebra if its fibres are Kolmogorov-Lie algebras and the
norm sequence of the bracket is a Bruno sequence.

Here is a special case of the versal deformation theorem.

######  Theorem 5.56.

Let $\mathfrak{g}$ be an Arnold-Lie algebra, $X\in\mathfrak{g}$ and
$T=\mathfrak{h}\subset\mathfrak{g}$ a sub-Arnold-Lie algebra which is a direct
summand, that is, the projection on $\mathfrak{h}$ is a local operator whose
norm is a Bruno sequence. Assume that

  1. 1)

there exists a a local operator $L\in{\mathcal{L}}(\mathfrak{g},{\bf M})$ and
a cutoff operator $\kappa\in{\mathcal{K}}(M,M)$ solving approximatively the
homological equation modulo $\mathfrak{h}$, in the sense that:

$[L(Y),X]=Y+\kappa(Y)\ {\rm mod\,}\mathfrak{h}$

  2. 2)

the norm sequence $|L|,\ |\kappa|$ are Bruno sequences.

Then there exists $r\in\mathbb{R}_{>0}$ and $k$ such that for any $Y\in
rt^{k}(B_{\mathfrak{g}})_{t}$ there exists
$v\in\Gamma(\mathbb{N},\mathfrak{g})$ satisfying

$e^{v}(X+rY)=\iota X\ {\rm mod\,}\mathfrak{h}$

######  Example 5.57.

Consider again the example of the cotangent space to $\mathbb{C}^{d}$, the
open set

$U\longrightarrow\mathbb{R}_{>0}$

with fibres

$U_{t}=\\{z=(q,p)\in\mathbb{C}^{2d}:|z_{i}|<t,\ i=1,\dots,n\\}$

We extend the symplectic structure into a Poisson structure on
$\mathbb{C}^{2d}\times\mathbb{C}^{2d}$ with coordinates $z,\omega,\tau$.
Consider the Hamiltonian vector field

$x_{0}=\\{-,\sum_{i=1}^{d}(\nu_{i}+\omega_{i})p_{i}q_{i}\\}$

and $\mathfrak{h}$ is the Arnold space defined as the module of symplectic
vector fields vanishing on the manifold given by the equations
$p_{i}q_{i}-\tau_{i}=0$. We consider the weights $1,1,2,0$ for the variables
$p_{i},q_{i},\tau_{i},\omega_{i}$. Then the map $L$ is defined as an
approximate solution of the homological equation by taking terms of degree up
to $2^{n}$ and $\kappa$ is the remainder term. The theorem then says that any
perturbed vector field can be linearized modulo a vector field of
$\mathfrak{h}$. Again for full details see [17].

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