# Hamiltonian normal forms

Mauricio Garay and Duco van Straten

###### Abstract.

We study a new type of normal form at a critical point of an analytic
Hamiltonian. Under a Bruno condition on the frequency, we prove a convergence
statement. Using this result, we deduce the existence of a positive measure
set of invariant tori near the critical point.

To the memory of J.-C. Yoccoz.

## Introduction

Investigations into normal forms of Hamiltonian systems can be traced back the
earliest beginnings of celestial mechanics and perturbation theory in the
works of Euler, Laplace, Delaunay, and others. The Birkhoff normal form
provides a practical way to extend the classical theory of action-angle
coordinates to critical points of Hamiltonians. As a general rule, the map
which reduces the Hamiltonian to normal form is divergent, while under
conditions of integrability it is convergent for analytic Hamiltonians ([4, 9,
24, 39, 29, 32, 34, 35, 36, 37]).
On the other hand KAM theory always provides, under non-degeneracy conditions,
the existence of invariant tori. Since the appearance of Kolmogorov’s original
paper, the non-degeneracy conditions have been weakened and even in case the
system is degenerate, invariant tori are known to exist ([1, 3, 5, 6, 17, 21,
27, 31, 33, 34]).

Here we present an iteration scheme that leads to a different type of normal
form that we call the Hamiltonian normal form. It appears to be suited for the
application of KAM theory near critical points of a Hamiltonian. We explain
its relation to the classical Birkhoff normal form. From a formal perspective,
the Hamiltonian normal form seems to be a rather trivial variant of the usual
one, but it has some important technical advantages. The first advantage is
that the Cantor set over which we work is fixed during the iteration, whereas
in the usual proofs this set is constructed step by step and changes along the
transformations of the iteration. A second advantage of the new iteration
scheme is the control over what we call the frequency space. As a consequence,
non-degeneracy conditions will be fulfilled automatically when passing to the
limit. For the readers acquainted with Arnold’s proof of the KAM theorem, this
corresponds to the fact that the UV-cutoff can be chosen so that it involves
only the terms that appear in the linear approximation.

The structure of the paper is as follows: In $\S 1$, after setting up some
notations and a quick review of the classical Birkhoff normal form, we
introduce the frequency space associated to a non-resonant critical point of a
Hamiltonian. It is the space defined by the linear relations between the
components of the gradient of the Birkhoff normal form. This space will play
an important role and takes over the role of the usual non-degeneracy
conditions of KAM theory.
In $\S 2$ we describe in some detail the formal aspects of an iteration that
leads to our Hamiltonian normal form. The first important point is that the
normalising maps in the iteration preserve, in a precise sense, the frequency
space.
In $\S 3$ we state and prove, under a Bruno condition on the frequency, the
convergence of the iteration procedure over a Cantor-like set in the
complement of the resonance fractal. We introduce the relevant function spaces
and rewrite the iteration in this context. The estimates we need are all
simple applications of lemmas that are collected in an appendix.
In $\S 4$ we give an application to invariant tori near an elliptic fixed
point. By our control of the frequency space, we can apply the arithmetic
density theorem from [15] and obtain a measure result for the preimage under
the frequency map. A conjecture formulated by M. Herman [18] is an easy
corollary. The setup chosen here represents a significant improvement and
simplification of the original arguments used in [11, 13, 14].

At the background of our investigations is a more general theory of normal
forms based on a functional analytic theory of Banach space valued functors
that we are currently developing. The reader interested in these lines of
thought may take a look in the preliminary [8, 10, 13, 16]. From that
perspective there appeared to be a certain incongruency between the usual
Birkhoff normal form and the general theory of normal forms. The Hamiltonian
normal form presented here seems to be the more natural one: the proof is from
an abstract point of view in complete congruence with that of the ordinary
Kolmogorov theorem. Nevertheless, for the convenience of the reader, we
decided to keep the paper self-contained and provide complete proofs of all
results used, with the exception of the arithmetical density result, which is
taken from [15].

Acknowledgement. When we started to develop an abstract version of KAM theory,
J.-C. Yoccoz was among the few enthusiastic dynamicists, eager to bridge the
frontiers between algebra, topology and analysis. After two months of numerous
exchanges, he was forced to stop due to health problems. Yoccoz made several
influential and motivating remarks, and for this reason we dedicate this
research to his memory.

##  1\. The Birkhoff normal form and the frequency space

We will be concerned with the structure of an analytic Hamiltonian system with
$d$ degrees of freedom near a critical point of the form

$H=\sum_{i=1}^{d}\alpha_{i}p_{i}q_{i}+O(3).$

We assume that the frequency vector:

$\alpha:=(\alpha_{1},\alpha_{2},\dots,\alpha_{d})\in\mathbb{C}^{d}$

is non-resonant, i.e., its components $\alpha_{i}$ are $\mathbb{Q}$-linearly
independent. We can consider the Hamiltonian $H$ as an element of the formal
power series ring

$P:=\mathbb{C}[[q,p]]:=\mathbb{C}[[q_{1},\dots,q_{d},p_{1},\dots,p_{d}]].$

The Poisson bracket of $f,g\in P$, defined by

$\\{f,g\\}=\sum_{i=1}^{d}\partial_{q_{i}}f\partial_{p_{i}}g-\partial_{p_{i}}f\partial_{q_{i}}g,$

makes $P$ into a Poisson-algebra. An element $h\in P$ is a power series that
can be written as

$h:=\sum_{a,b}C_{a,b}p^{a}q^{b},\;\;\;C_{a,b}\in\mathbb{C},$

where we use the usual multi-index notation, so that

$p^{a}q^{b}:=p_{1}^{a_{1}}p_{2}^{a_{2}}\ldots
p_{d}^{a_{d}}q_{1}^{b_{1}}q_{2}^{b_{2}}\ldots q_{d}^{b_{d}},$

and so on. We will assign weight $1$ to each of the variables, so that the
monomial $p^{a}q^{b}$ has weight $|a|+|b|$. We write $h=O(k)$ if $h$ only
contains monomials of degree $\geq k$, and say that $h$ has order $k$. If $h$
is analytic, it is represented by a convergent series, and our usage of the
$O$ corresponds to its usual meaning.

###  1.1. Birkhoff normal form

A derivation $v\in Der(P)$ that preserves the Poisson-bracket:

$v(\\{f,g\\})=\\{v(f),g\\}+\\{f,v(g)\\}$

is called a Poisson-derivation and we denote by $\Theta(P)$ the vector space
of all Poisson-derivations or Poisson vector fields. The map

$P\longrightarrow\Theta(P),\;\;\;h\mapsto\\{-,h\\}$

associates to $h$ the corresponding Poisson-derivation, usually called the
Hamiltonian vector field of $h$. If $h=O(k)$ and $f=O(l)$, then clearly
$\\{f,h\\}=O(k+l-2)$, so the vector field $v:=\\{-,h\\}$ is said to be of
order $k-2$, although the coefficients of the vector field $v$ are $O(k-1)$.
In particular, if $h=O(3)$, then one can exponentiate $v$ and obtain a Poisson
automorphism of the ring $P$:

$e^{v}=Id+\\{-,h\\}+\frac{1}{2!}\\{\\{-,h\\},h\\}+\ldots\in Aut(P).$

If we let

$h_{0}:=\sum_{i=1}^{d}\alpha_{i}p_{i}q_{i},$

then

$\\{h_{0},p^{a}q^{b}\\}=(\alpha,a-b)p^{a}q^{b},$

where $(-,-)$ denotes the standard euclidean scalar product. So if $\alpha$ is
non-resonant, then each monomial $p^{a}q^{b}$ with $a\neq b$ appearing in
$H=h_{0}+O(3)$ can be removed by an application of the derivation

$v=j(p^{a}q^{b}):=\\{-,\frac{1}{(\alpha,a-b)}p^{a}q^{b}\\}.$

Hence, we can construct a sequence of automorphisms

$\varphi_{0}:=e^{-v_{0}},\;\;\;\varphi_{1}:=e^{-v_{1}},\;\;\;\varphi_{2}:=e^{-v_{2}},\ldots,\in
Aut(P)$

that remove successively all monomials $p^{a}q^{b}$, $a\neq b$ of weight $k+3$
from $H_{k}$, defined recursively by

$H_{1}:=\varphi_{0}(H),\ H_{2}=\varphi_{1}(H_{1}),\ \ldots,$

so that the automorphism

$\Phi_{k}:=\varphi_{k-1}\ldots\varphi_{1}\varphi_{0}$

maps $H$ to $H_{k}$. The infinite composition

$\Phi:=\ldots\varphi_{k}\varphi_{k-1}\ldots\varphi_{1}\varphi_{0}\in Aut(P)$

is a formal symplectic coordinate transformation that removes all monomials
$p^{a}q^{b}$, $a\neq b$ from our Hamiltonian $H$, so that

$\Phi(H)=B_{H},$

where $B_{H}$ is a series of the form

$B_{H}:=\sum_{a\in\mathbb{N}^{d}}C_{a}p^{a}q^{a}.$

The series $B_{H}$ is called the Birkhoff normal form of $H$. There exist
several variants of this algorithm, which differ in details and notation. For
example, it is possible to remove certain terms at the same time. These may
lead to different normalising transformations $\Phi$, but it is known that
these different choices lead to the same series $B_{H}$. As in the iteration
process one has to divide by the quantity $(\alpha,a-b)$ to remove the
$p^{a}q^{b}$-term, small denominators appear, which lead to convergence
problems for the formal series involved.
C. L. Siegel has shown that for a generic critical point of a real analytic
Hamiltonian $H$ with $d\geq 2$ degrees of freedom, the series of the
transformation $\Phi$ is in fact divergent [35]. It is expected that in
general the series $B_{H}$ itself is also divergent, but no published proof is
known to us (see [29] for more details).

###  1.2.  The Moser Extension

As the monomials $p_{i}q_{i}$ ($i=1,2,\ldots,d$) Poisson commute with the
Birkhoff normal form $B_{H}$, Birkhoff normalisation implies that any non-
resonant Hamiltonian $H$ is formally completely integrable. To express this
fact more clearly, it is useful to enlarge the ring $P$ and consider

$Q:=\mathbb{C}[[\tau,q,p]]=\mathbb{C}[[\tau_{1},\ldots,\tau_{d},q_{1},\ldots,q_{d},p_{1},\ldots,p_{d}]]$

with the extra $\tau$-variables, introduced by Moser. With the same definition
of the Poisson-bracket as before, $Q$ becomes a Poisson algebra with Poisson
centre $Q_{0}:=\mathbb{C}[[\tau]]$. We will assign weight $=2$ to the
variables $\tau_{i}$, so that the $d$ elements

$f_{i}:=p_{i}q_{i}-\tau_{i}\in Q$

are homogeneous of degree two. These elements Poisson commute,
$\\{f_{i},f_{j}\\}=0$, and we obtain a Poisson commuting sub-algebra

$\mathbb{C}[[\tau,f]]=\mathbb{C}[[\tau,f_{1},f_{2},\ldots,f_{d}]]=\mathbb{C}[[\tau,p_{1}q_{1},\ldots,p_{d}q_{d}]]$

containing $Q_{0}$. The $f_{1},f_{2},\ldots,f_{d}$ also generate an ideal

$I=(f_{1},f_{2},\ldots,f_{d})\subset Q=\mathbb{C}[[\tau,q,p]]$

and clearly, the canonical map

$\mathbb{C}[[\tau,p,q]]{\longrightarrow}\mathbb{C}[[p,q]],\;\;\;p_{i}\mapsto
p_{i},\;q_{i}\mapsto q_{i},\;\tau_{i}\mapsto p_{i}q_{i}.$

induces an isomorphism of the factor ring $Q/I$ with our original ring $P$:

$Q/I\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}P.$

Although $f_{i}$ maps to zero under this map, the derivation $\\{-,f_{i}\\}$
induces the non-zero derivation $\\{-,p_{i}q_{i}\\}$ on $P$, so the map
$Q\longrightarrow P$ is not a Poisson-morphism. The ideal $I^{2}\subset Q$ is
the square of the ideal $I$, i.e. generated by the elements $f_{i}f_{j}$,
$1\leq i,j\leq d$, and plays a very distinguished role in dynamics. The reason
is that if $T\in I^{2}$, then $\\{h,T\\}\subset I$. As a consequence, $H$ and
$H+T$ determine the same Hamiltonian vector field on $Q/I=P$.

Extending the multi-index notation in an obvious way, we can write

$p^{a}q^{a}=(\tau+f)^{a}=\tau^{a}+\sum_{i=1}^{d}\partial_{\tau_{i}}\tau^{a}f_{i}+I^{2}.$

The term $\tau^{a}$ is in the centre of $Q$, whereas the above remark implies
that $p^{a}q^{a}$ and $\sum_{i=1}^{d}\partial_{\tau_{i}}\tau^{a}f_{i}$ define
the same derivation on the ring $P=Q/I$.

We can consider the Birhoff normal form series $B(pq)=B_{H}(pq)$ as an element
of $Q$. When we write $pq=\tau+f$, then we find:

$B(\tau+f)=B(\tau)+\sum_{i=1}^{d}b_{i}(\tau)f_{i}\ {\rm mod\,}I^{2}.$

The first term $B(\tau)$ belongs to the Poisson centre $Q_{0}$ and is
dynamically trivial, but gets mapped to the non-trivial element $B_{H}\in P$.
The second term $\sum_{i=1}^{d}b_{i}(\tau)f_{i}$ carries the dynamical
information in $Q$, but is mapped by the canonical map $Q\longrightarrow P$ to
zero.

The formal power series $b_{1},\dots,b_{d}\in\mathbb{C}[[\tau]]$ are obtained
as partial derivatives of $B$, considered as a series in the
$\tau_{i}$-variables:

$b=(b_{1},\dots,b_{d})=\nabla B(\tau).$

One has $b(0)=\alpha$, and the higher order terms describe how the frequencies
change with $\tau$ and for this reason we call it the (formal) frequency map.
If the system happens to be integrable, then the series are convergent and the
vector $b(\tau)=(b_{1}(\tau),b_{2}(\tau),\dots,b_{d}(\tau))$ is the frequency
of motion on the corresponding manifold defined by $f_{i}(\tau,q,p)=0$,
$i=1,2,\ldots,d$.

For a multi-index $a\in\mathbb{N}^{d}$, we consider the vector

$\nabla^{a}b(0):=\frac{\partial^{a}b}{\partial\tau^{a}}(0)\in\mathbb{C}^{d}.$

######  Definition 1.1.

The frequency space of $H$ is the vector space $F(H)\subset\mathbb{C}^{d}$
generated by the vectors $\nabla^{a}b(0)$, $|a|\geq 1$.

This vector space controls an important aspect of the non-degeneracy
conditions in KAM theory. The space $F(H)=\\{0\\}$ if there are no terms of
degree $>2$ in the Birkhoff normal form. This is the case considered by
Rüßmann [32], who proved, under arithmetic conditions on the frequency vector
$\alpha$, that in this situation the transformation $\Phi$ actually is
convergent, so $H$ is analytically completely integrable. The classical non-
degeneracy condition from KAM-theory lead to the opposite case
$F(H)=\mathbb{C}^{d}$. Our main interest lies in the intermediate cases. In
$\S 4$ we will define in the real domain a $C^{\infty}$-map

$\beta:{\mathcal{W}}\longrightarrow\alpha+{\mathcal{F}}(H)\subset\mathbb{R}^{d},$

defined in a neighbourhood ${\mathcal{W}}\subset\mathbb{R}^{d}$,
${\mathcal{F}}(H):=F(H)\cap\mathbb{R}^{d}$ and whose Taylor series coincides
with the above formal power series $b$.

##  2\. The Hamiltonian normal form

We will now describe a variant of the Birkhoff normal form algorithm that
allows for a better control of the invariant tori. For this we have to
introduce $d$ further additional variables

$\omega_{1},\omega_{2},\ldots,\omega_{d},$

to which we assign weight zero. The iteration will take place in the formal
power series ring with $4d$ variables

$R:=\mathbb{C}[[\omega,\tau,q,p]]=\mathbb{C}[[\omega_{1},\omega_{2},\ldots,\omega_{d},\tau_{1},\ldots,\tau_{d},q_{1},\ldots,q_{d},p_{1},\ldots,p_{d}]].$

Again we retain the standard Poisson bracket, so now the Poisson centre is the
subring $R_{0}:=\mathbb{C}[[\omega,\tau]]$. The following sub-algebra is of
importance for our discussion:

######  Definition 2.1.

We let

$M:=\mathbb{C}[[\omega,\tau,f]]=\mathbb{C}[[\omega,\tau,pq]]\subset R,$

and call it the Moser-algebra of $R$.

We remark that $M$ is a Poisson commutative sub-algebra of $R$ and that all
monomials of $M$ have even degree. We will also make use of the projection

$\pi:R\longrightarrow M$

that maps all monomials not in $M$ to zero. Clearly, $\pi\circ i=Id_{M}$,
where $i:M\longrightarrow R$ is the inclusion.

###  2.1. The homological equation

As before, we denote the vector space of Poisson derivations of $R$ by
$\Theta(R)$, which has the structure of a module over the Poisson centre
$R_{0}$. These derivations decompose into Hamiltonian and non-exact parts:

$\Theta(R)={\rm Ham\,}(R)\oplus{\rm Der\,}(R_{0}),$

that is, an element of $\Theta(R)$ is of the form

$v=\\{-,h\\}+w$

with

$w=\sum_{i=1}^{d}A_{i}\frac{\partial}{\partial\omega_{i}}+B_{i}\frac{\partial}{\partial\tau_{i}},\;\;\;A_{i},B_{i}\in
R_{0}.$

######  Definition 2.2.

Let $F,m\in R$ and $v\in\Theta(R)$. If

$v(F)=m,$

we say that $v$ solves the homological equation for $m$ on $F$.

We will now solve the homological equation for certain special elements.

The following function will play an important role in the paper.

######  Definition 2.3.

The formal unfolding of $h_{0}=\sum_{i=1}^{d}\alpha_{i}p_{i}q_{i}$ is the
element

$A_{0}:=\sum_{i=1}^{d}(\alpha_{i}+\omega_{i})p_{i}q_{i}\in R.$

So $A_{0}$ is obtained from $h_{0}$ by detuning the frequencies in the most
general way.

The infinitesimal action

$\Theta(R)\longrightarrow R,\;\;v\mapsto v(A_{0})$

on $A_{0}$ takes a simple form in the monomial basis:

$\displaystyle\\{A_{0},p^{a}q^{b}\\}$
$\displaystyle=(\alpha+\omega,a-b)p^{a}q^{b},$
$\displaystyle\partial_{\omega_{k}}A_{0}$ $\displaystyle=p_{k}q_{k},$
$\displaystyle\partial_{\tau_{k}}A_{0}$ $\displaystyle=0.$

######  Definition 2.4.

We define a $\mathbb{C}[[\omega,\tau]]$-linear map

$L:R\longrightarrow\Theta(R)={\rm Ham\,}(R)\oplus{\rm Der\,}(R_{0}),m\mapsto
Lm$

by setting for $a\neq b$:

$Lp^{a}q^{b}:=\\{-,\frac{1}{(\alpha+\omega,a-b)}p^{a}q^{b}\\}.$

For $a=b$, or more generally for a series

$m=g(p_{1}q_{1},p_{2}q_{2}\,\ldots,p_{d}q_{d})=g(pq)$

we set

$Lm:=\sum_{i=1}^{d}\frac{\partial
g(\tau)}{\partial\tau_{i}}\partial_{\omega_{i}}.$

We note that the first case of the definition applies to the monomials in the
kernel of the projection map $\pi$, whereas the second part of the definition
applies to the elements of the Moser sub-algebra $M$.

######  Definition 2.5.

For $A=A_{0}+T,\ T\in I^{2}$ we define a linear map

$j_{A}:R\longrightarrow\Theta(R)$

in terms of $L$ by the formula

$j_{A}:m\mapsto Lm-L(Lm(T))=L(m-Lm(T))$

######  Proposition 2.6.

For any $A=A_{0}+T\in A_{0}+I^{2}$, there exists $t\in R_{0}+I^{2}$ such that

$j_{A}(m)(A)=m+t.$

###### Proof.

First, for $A=A_{0}$ we have $j_{A_{0}}=L$. For $m=p^{a}q^{b}$ with $a\neq b$
we have

$j_{A_{0}}(m)(A_{0})=\\{A_{0},\frac{1}{(\alpha+\omega,a-b)}p^{a}q^{b}\\}=p^{a}q^{b}=m$

and for $m=g(pq)$ we have, with $g_{i}=\partial_{\tau_{i}}g$,

$\displaystyle j_{A_{0}}(m)(A_{0})$
$\displaystyle=\sum_{i=1}^{d}g_{i}(\tau)\frac{\partial
A_{0}}{\partial\omega_{i}}=\sum_{i=1}^{n}g_{i}(\tau)p_{i}q_{i}$
$\displaystyle=\sum_{i=1}^{d}g_{i}(\tau)f_{i}\ {\rm mod\,}R_{0}=g(pq)\ {\rm
mod\,}R_{0}+I^{2},$

where we used the Taylor expansion

$g(pq)=g(\tau+f)=g(\tau)+\sum_{i=1}^{d}g_{i}(\tau)f_{i}\ {\rm mod\,}I^{2}.$

This shows the correctness for $T=0$. For the general case $A=A_{0}+T$, we get

$\displaystyle j_{A}(m)(A_{0}+T)$
$\displaystyle=Lm(A_{0})+Lm(T)-L(Lm(T))A_{0}-L(Lm(T))(T)$
$\displaystyle=m+Lm(T)-Lm(T)-L(Lm(T))(T)\ {\rm mod\,}R_{0}+I^{2}$
$\displaystyle=m+L(Lm(T))(T)\ {\rm mod\,}R_{0}+I^{2}.$

Because $T\in I^{2}$, it follows that $Lm(T)\in I$. Furthermore, for any $g\in
I$, we have $Lg(T)\in I^{2}$. This can be seen by writing $g$ as
$\mathbb{C}[[\omega,\tau]]$-linear combination of terms of the form
$p^{a}q^{b}f_{i}$. If $a\neq b$, $\\{T,p^{a}q^{b}f_{i}\\}\in I^{2}$, whereas
for $a=b$, we obtain a combination of terms $\partial_{\omega_{i}}T$, which is
in $I^{2}$, as the generators $f_{i}=p_{i}q_{i}-\tau_{i}$ are independent of
$\omega_{i}$. ∎

###  2.2. Hamiltonian normal form iteration

Our Hamiltonian normal form is obtained from the following basic iteration.
Starting from a Hamiltonian

$H=\sum_{i=0}^{d}\alpha_{i}p_{i}q_{i}+O(3),$

we first form

$F_{0}:=H+\sum_{i=1}^{d}\omega_{i}p_{i}q_{i}=A_{0}+O(3).$

Solving the homological equation on $A_{0}$ for the degree $3$ part of $F_{0}$
determines a Poisson derivation $v_{0}$. The application of $e^{-v_{0}}$ to
$F_{0}$ produces $F_{1}$, where this term is removed and we put $A_{1}=A_{0}$.
The solution of the homological equation for the degree $4$ and $5$ part of
$F_{1}$ on $A_{1}$ defines $v_{1}$. Then the application of $e^{-v_{1}}$ to
$F_{1}$ produces $F_{2}$, where now the terms of degree $4$ and $5$ are
removed, modulo terms in $(R_{0}+I^{2})\cap M$. These terms we add to
$A_{1}=A_{0}$ and obtain $A_{2}$. Then we solve the homological equation for
the terms of degree $6,7,8,9$ of $F_{2}$, but now on $A_{2}$, etcetera. Thus
we obtain a by iteration a sequence of triples

$(F_{n},A_{n},v_{n}),\;\;\;n=0,1,2,\ldots$

that we call the Hamiltonian normal form iteration and which we will describe
now in some detail. It is convenient to write

$\left[h\right]_{i}^{j}$

for the sum of terms of $h$ of weight $\geq i$ and $<j$, so that
$\left[h\right]_{i}^{i+1}$ represents the part of $h$ of pure weight $i$. When
$j=+\infty$ we omit the letter $j$, when $i=0$ we omit the letter $i$. In a
similar way we can truncate a vector field by truncating its coefficients, but
taking the shift of grading by $1$ into account.

We begin with the initialisation step

$\displaystyle F_{0}$
$\displaystyle=H+\sum_{i=1}^{d}\omega_{i}p_{i}q_{i}=A_{0}+O(3)$ $\displaystyle
A_{0}$ $\displaystyle=\sum_{i=1}^{d}(\alpha_{i}+\omega_{i})p_{i}q_{i}$
$\displaystyle v_{0}$
$\displaystyle=[j_{A_{0}}\left([F_{0}]_{3}^{4}\right)]^{2}.$

The next terms are determined by the iteration step:

From $F_{n},A_{n}$ we first compute

$v_{n}=[j_{A_{n}}(\left[F_{n}\right]_{2^{n}+2}^{2^{n+1}+2})]^{2^{n+1}}.$

and then obtain

$\displaystyle F_{n+1}$ $\displaystyle=e^{-v_{n}}F_{n},$ $\displaystyle
A_{n+1}$
$\displaystyle=A_{n}+\left[F_{n}-v_{n}(F_{n})\right]_{2^{n}+2}^{2^{n+1}}.$

Then compute

$v_{n+1}:=[j_{A_{n+1}}(\left[F_{n+1}\right]_{2^{n+1}+2}^{2^{n+2}+2})]^{2^{n+2}}$

etcetera. The extra truncation of $v_{n}$ is not needed for the arguments
here, but it will play a role in 2.6.
It is useful to define the increments

$S_{n+1}:=\left[F_{n}-v_{n}(F_{n})\right]_{2^{n}+2}^{2^{n+1}+2},$

so that:

$A_{n+1}=A_{n}+S_{n+1}.$

There are a few simple but important points to notice:

######  Proposition 2.7.

(i) The derivation $v_{n}$ has order $2^{n}$, i.e.
$v_{n}=[v_{n}]_{2^{n}}=[v_{n}]_{2^{n}}^{2^{n+1}}$.
(ii) $F_{n}=A_{n}+O(2^{n}+2)$.

(iii) $S_{n}\in(R_{0}+I^{2})\cap M$.

###### Proof.

(i) From the recursive definition we see that $v_{n}$ is obtained by solving
the homological equation with the terms of degrees ${2^{n}+2}$ up to
${2^{n+1}+2}$ from $F_{n}$. Taking Poisson-bracket with a term of degree
$2^{n}+2$ shifts degrees by $2^{n}$, and similarly for the non-exact part of
$v_{n}$. So indeed $v_{n}$ has order $2^{n}$.
(ii) This follows from an easy induction on $n$. By definition, the statement
holds for $n=0$. Let us assume that

$F_{n}=A_{n}+O(2^{n}+2)$

From the definition of $F_{n+1}$ we have

$F_{n+1}=e^{-v_{n}}F_{n}=F_{n}-v_{n}(F_{n})+\frac{1}{2}v_{n}^{2}(F_{n})-\ldots$

and as $v_{n}$ has order $2^{n}$, it follows that

$v_{n}^{2}(F_{n})=O(2+2^{n}+2^{n})=O(2^{n+1}+2).$

So we have

$F_{n+1}=A_{n}+[F_{n}-v_{n}(F_{n})]_{2^{n}+2}^{2^{n+1}+2}+O(2^{n+1}+2)=A_{n+1}+O(2^{n+1}+2).$

(iii) We use induction on $n$ and assume that $S_{n}\in(R_{0}+I^{2})\cap M$.
From (ii) we have

$F_{n}=A_{n}+O(2^{n}+2).$

The derivation
$v_{n}=[j_{A_{n}}([F_{n}]_{2^{n}+2}^{2^{n+1}+2})]_{2^{n}}^{2^{n+1}}$ is
constructed to solve the homological equation up to terms of high order:

$v_{n}(A_{n})=[F_{n}]_{2^{n}+2}^{2^{n+1}+2}+t+O(2^{n+1}+2),\;\;\;t\in(R_{0}+I^{2})\cap
M$

As we have

$v_{n}(F_{n})=v_{n}(A_{n}+O(2^{n}+2))=v_{n}(A_{n})+O(2^{n+1}+2),$

we see that the increment

$[F_{n}-v_{n}(F_{n})]_{2^{n}+2}^{2^{n+1}+2}\in(R_{0}+I^{2})\cap M,$

hence also $S_{n+1}\in(R_{0}+I^{2})\cap M$. ∎

###  2.3. Quadratic and non-quadratic nature of the iteration

The Hamiltonian normal form iteration produces a sequence
$(F_{n},A_{n},v_{n})$: the series

$F_{0}=H+\sum_{i=1}^{d}\omega_{i}p_{i}q_{i}=A_{0}+O(3)$

is transformed by

$\Phi_{n}:=e^{-v_{n-1}}\dots e^{-v_{0}}$

to a series of the form

$F_{n}=A_{n}+O(2^{n}+2).$

If we let $n$ go to $\infty$, we obtain a formal Poisson automorphism

$\Phi_{\infty}:=\ldots e^{-v_{n}}\dots e^{-v_{0}}\in Aut(R),$

and obtain

$F_{\infty}:=\Phi_{\infty}(F_{0})=A_{\infty},\;\;\;A_{\infty}\in
A_{0}+(R_{0}+I^{2})\cap M$

The automorphism $\Phi_{\infty}$ transforms the perturbation
$F_{0}=A_{0}+O(3)$ back to the normal form $A_{0}$, plus terms that have no
effect on the dynamics.

For convenience of the reader we include the following diagram that indicates
the degrees of the quantities that appear in the iteration.

$\begin{array}[]{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline\cr&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18\\\
\hline\cr\hline\cr
F_{0}&\bullet&\times&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare\\\
\hline\cr
F_{1}&\bullet&\circ&\times&\times&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare\\\
\hline\cr
F_{2}&\bullet&\circ&\bullet&\circ&\times&\times&\times&\times&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare\\\
\hline\cr
F_{3}&\bullet&\circ&\bullet&\circ&\bullet&\circ&\bullet&\circ&\times&\times&\times&\times&\times&\times&\times&\times&\blacksquare\\\
\hline\cr
F_{4}&\bullet&\circ&\bullet&\circ&\bullet&\circ&\bullet&\circ&\bullet&\circ&\bullet&\circ&\bullet&\circ&\bullet&\circ&\times\\\
\hline\cr\end{array}$

The bullets $\bullet$ and circles $\circ$ represent terms of $A_{n}$. They
belong to the $R_{0}+I^{2}$ part of the Moser-algebra : the $\bullet$ terms
are constant in columns, the circles $\circ$ are zero, as the Moser-algebra
only has terms of even degree.
So $\bullet$ and $\circ$ represents the normal form range, consisting of terms
of $F_{n}$ of degree

$2\leq degree<2^{n}+2$

The crosses $\times$ represent the terms of $F_{n}$ that determine the
derivations $v_{n}$. These make up what we call the active range of degrees:

$2^{n}+2\leq degree<2^{n+1}+2.$

The black squares $\blacksquare$ represent the terms of $F_{n}$ that of degree
higher than $2^{n+1}+2$ that do not directly influence the next iteration
step, but of course must be carried along.

We now rewrite the iteration in a form where this trichotomy in degrees is
manifest. Consider the decomposition

$F_{n}:=A_{n}+M_{n}+U_{n}=\bullet+\times+\blacksquare,$

where

$A_{n}:=[F_{n}]^{2^{n}+2},\;\;\;M_{n}:=[F_{n}]_{2^{n}+2}^{2^{n+1}+2},\;\;\;U_{n}:=[F_{n}]_{2^{n+1}+2},$

are the lower, middle and upper part of $F_{n}$. The middle term decomposes,
by the construction of $v_{n}$, as sum of two terms

$(1)\quad M_{n}=S_{n+1}+[v_{n}(A_{n})]_{2^{n}+2}^{2^{n+1}+2},$

where

$S_{n+1}=[F_{n}-v_{n}(F_{n})]_{2^{n}+2}^{2^{n+1}+2}\in(R_{0}+I^{2})\cap M,$

where we also used

$[v_{n}(A_{n})]_{2^{n}+2}^{2^{n+1}+2}=[v_{n}(F_{n})]_{2^{n}+2}^{2^{n+1}+2}$

The quantity $v_{n}(A_{n})$ has only terms of middle and high degree:

$(2)\quad v_{n}(A_{n})=[v_{n}(A_{n})]_{2^{n}+2}^{2^{n+1}+2}+R_{n},$

where

$R_{n}:=[v_{n}(A_{n})]_{2^{n+1}+2}.$

represents a remainder part. We then have

$\displaystyle F_{n+1}$ $\displaystyle=e^{-v_{n}}F_{n}$
$\displaystyle=e^{-v_{n}}(A_{n}+M_{n}+U_{n})$
$\displaystyle\stackrel{{\scriptstyle
1}}{{=}}e^{-v_{n}}(A_{n}+S_{n+1}+[v_{n}(A_{n})]_{2^{n}+2}^{2^{n+1}+2}+U_{n})$
$\displaystyle\stackrel{{\scriptstyle
2}}{{=}}e^{-v_{n}}(A_{n}+S_{n+1}+v_{n}(A_{n}))+e^{-v_{n}}(U_{n}-R_{n})$
$\displaystyle=A_{n}+S_{n+1}+(e^{-v_{n}}-{\rm
Id\,})(A_{n}+S_{n+1})+e^{-v_{n}}(v_{n}(A_{n}))+e^{-v_{n}}(U_{n}-R_{n})$
$\displaystyle=A_{n}+S_{n+1}+(e^{-v_{n}}({\rm Id\,}+v_{n})-{\rm
Id\,})A_{n}+(e^{-v_{n}}-{\rm Id\,})(S_{n+1})+e^{-v_{n}}(U_{n}-R_{n})$

where we indicated the equations we used to establish the equalities. The
third and fourth equalities are just obtained by rearrangement of the
different terms.

Now, when we set

$B_{n}:=M_{n}+U_{n}=[F_{n}]_{2^{n}+2}=\times+\blacksquare,$

we can formulate the KAM-form of the iteration in the formal setting:

$\begin{array}[]{| ll |}\hline\cr&\\\
S_{n+1}&=[B_{n}-v_{n}(A_{n})]_{2^{n}+2}^{2^{n+1}+2},\\\ &\\\
A_{n+1}&=A_{n}+S_{n+1},\\\ &\\\
B_{n+1}&=\psi(v_{n})S_{n+1}+\phi(v_{n})A_{n}+e^{-v_{n}}([B_{n}-v_{n}(A_{n})]_{2^{n+1}+2})\\\
&\\\ \hline\cr\end{array}$

with

$\psi(z)=e^{-z}-1,\;\;\;\phi(z)=e^{-z}(1+z)-1$

and where we used the abbreviation:

$v_{n}=j_{n}B_{n}=[j_{A_{n}}(B_{n})]_{2^{n}}^{2^{n+1}}.$

Note that the increments $S_{n+1}$ and the vector fields $v_{n}$ are auxiliary
quantities, which depend linearly on $B_{n}$. We see that the terms involving
$\phi$ and $\psi$ are quadratic in $B_{n}$, whereas the third term is not. But
that term only depends on the higher order terms of the series.

In the classical Kolmogorov scheme, the convergence is ensured by the
quadraticity of the iteration, while in the Arnold-Nash-Moser case there does
appear such a remainder term from the classical UV cutoff technique. However,
one may hope that these terms will remain small, as they involve terms of
sufficiently high order.
As we will see, unlike the cases considered by these authors, in our situation
it is important to be very precise about the degrees in the truncation, as
otherwise we would loose control over the fields $v_{n}$ and the convergence
properties of the iteration could be spoiled.

###  2.4. Some further remarks on the iteration

Let us denote by

$R^{an}:=\mathbb{C}\\{\omega,\tau,p,q\\}$

the Poisson sub-algebra of $R$ consisting of convergent power series. Its
Poisson centre is $R_{0}^{an}:=\mathbb{C}\\{\omega,\tau\\}$. If
$v\in\Theta(R^{an})$ is an analytic Poisson-derivation of order $\geq 1$, then
the exponential series $e^{v}$ converges as an element of $Aut(R^{an})$. As a
result, if we start with an analytic Hamiltonian, the terms of iteration
$(F_{k},S_{k},v_{k})$ are are all analytic, but of course the limit objects a
priori are given by formal series.
The variable $\omega$ plays a very special role. From the definition of the
operation $L$, it follows that the iteration can be formulated in a much
smaller ring. We define the ring $SD_{\alpha}$ of small denominators at
$\alpha$ as the subring of the field $\mathbb{C}(\omega)$ of rational
functions, defined by localisation of $\mathbb{C}[\omega]$ with respect with
the multiplicative subset $S$ generated by all linear polynomials
$(\alpha+\omega,J),J\in\mathbb{Z}^{n}\setminus\\{0\\}$:

$SD_{\alpha}:=\mathbb{C}[\omega]_{S}:=\mathbb{C}[\omega,\frac{1}{(\alpha+\omega,J)},J\in\mathbb{Z}^{n}\setminus\\{0\\}]\subset\mathbb{C}(\omega).$

Then the iteration makes sense in the Poisson-algebra

$R^{an}\cap SD_{\alpha}[[\tau,p,q]],$

but we will not make use of this fact in this paper.

In section 3 we will formulate a version of this iteration in appropriate
function spaces and study its convergence properties.

###  2.5. Relation to the Birkhoff normal form

When we start from an analytic Hamiltonian $H(p,q)$, we first formed

$F_{0}(\omega,p,q)=H(p,q)+\sum_{i=1}^{d}\omega_{i}p_{i}q_{i},$

and the Hamiltonian normal form iteration produces a sequence of Poisson
derivations:

$v_{n},\;\;\;n=0,1,2,\ldots$

These exponentiate to Poisson-automorphisms $\varphi_{n}:=e^{-v_{n}}$ and the
compositions

$\Phi_{n}=\varphi_{n-1}\varphi_{n-1}\ldots\varphi_{1}\varphi_{0},\;\;\;\varphi_{n}:=e^{-v_{n}},$

preserve the ring $R^{an}\subset R$ of convergent series. We have

$\Phi_{n}(F_{0})=F_{n}\in\mathbb{C}\\{\omega,\tau,p,q\\},$

and the infinite composition is a Poisson automorphism of the formal power
series ring $R$:

$\Phi_{\infty}=\ldots\varphi_{n-1}\varphi_{n-1}\ldots\varphi_{1}\varphi_{0}\in
Aut(R),$

and correspondingly

$\Phi_{\infty}(F_{0})=F_{\infty}\in\mathbb{C}[[\omega,\tau,p,q]].$

Note that

$F_{\infty}=A_{\infty}=A_{0}+T_{\infty},\;\;\;T_{\infty}\in R_{0}+I^{2}.$

From the construction of the vector fields $v_{n}$ we see that

$\Phi_{n}(\tau_{i})=\tau_{i},\;\;\Phi_{\infty}(\tau_{i})=\tau_{i}.$

As any Poisson-automorphism, $\Phi_{n}$ and $\Phi_{\infty}$ preserve the
Poisson centre, so we have

$\Phi_{n}(\omega_{i})\in\mathbb{C}\\{\omega,\tau\\},\;\;\Phi_{\infty}(\omega_{i})\in\mathbb{C}[[\omega,\tau]].$

As these element will play an important role in the sequel, will give the them
a fixed name:

######  Definition 2.8.

We set

$R_{n,i}:=\Phi_{n}(\omega_{i})\in\mathbb{C}\\{\omega,\tau\\},$
$R_{\infty,i}:=\Phi_{\infty}(\omega_{i})\in\mathbb{C}[[\omega,\tau]].$

To see what happens to the original Hamiltonian $H(p,q)$ during the iteration,
we remark that

$F_{0}(\omega=0,\tau,p,q)=H(p,q).$

But during the iteration, the condition $\omega_{i}=0$ is transformed into the
condition $R_{n,i}(\omega,\tau)=0$:

$\Phi_{n}(F_{0})=F_{0}(R_{n}(\omega,\tau),\tau,P_{n}(\omega,\tau,p,q),Q_{n}(\omega,\tau,p,q))=F_{n}(\omega,\tau,p,q).$

So if we want to follow $H(p,q)$ during these transformations, we will need to
solve for $\omega$ the equations

$R_{n,i}(\omega,\tau)=0,\;\;i=1,2,\ldots,d,$

and

$R_{\infty,i}(\omega,\tau)=0,\;\;i=1,2,\ldots,d.$

From the fact that vector fields $v_{n}$ have order $2^{n}$, we readily see
that

$R_{0,i}=\omega_{i},\;\;\;R_{n,i}=R_{n-1,i}+O(2^{n}),$

so the equations $R_{n,i}(\omega,\tau)=0$ can be solved for the $\omega_{i}$
and we obtain convergent power series

$\omega_{n,i}(\tau)\in\mathbb{C}\\{\tau\\},$

such that

$R_{n,i}(\omega_{n}(\tau),\tau)=0,$

and similarly for $n=\infty$, we find formal power series

$\omega_{\infty,i}(\tau)\in\mathbb{C}[[\tau]]$

solving

$R_{\infty,i}(\omega_{\infty}(\tau),\tau)=0.$

So we then have

$\Phi_{n}(H(p,q))=F_{n}(\omega_{n}(\tau),\tau,p,q)=:H_{n}(\tau,p,q).$

and is thus of the form

$H_{n}=h_{n}(\tau)+\sum_{i=1}^{d}(\alpha_{i}+\omega_{n,i}(\tau))f_{i}+O(2^{n}+2)\
{\rm mod\,}I^{2}\cap M$

with $f_{i}=p_{i}q_{i}-\tau_{i}$.

Comparing with the Taylor expansion of 1.2:

$B(qp)=B(\tau+f)=B(\tau)+\sum_{i=1}^{d}b_{i}(\tau)f_{i}\ {\rm mod\,}I^{2},$

we deduce that the coefficients $b_{i}(\tau)$ of the frequency map are related
to the $\omega_{n,i}(\tau)$ by the congruence

$\alpha_{i}+\omega_{n,i}(\tau)=b_{i}(\tau)+O(2^{n}+2),$

and that the constant term gives back the Birkhoff normal form of $H$:

$h_{n}(\tau)=B(\tau)+O(2^{n}+2).$

One may be tempted to go one step further and use the projection
$Q\longrightarrow P$, where we eliminate the $\tau$-variables using the
relations $f_{i}=0$, $i=1,2,\ldots,d$, in order to create an iteration taking
place in the ring $P$. This certainly can be done, but the resulting formulas
are much more complicated and less revealing. The great advantage of working
with the Hamiltonian normal form iteration is that we avoid solving these
implicit relations $R_{n,i}(\omega,\tau)=0$ and $f_{i}=0$. This leads to a
great simplification of the setup.

###  2.6. Behaviour of the frequency space

We will now see that the precise truncations that we made for the derivations
$v_{n}$ allow us to control some non degeneracy properties from the formal
iteration.
We can find open subsets $U_{n}\subset\mathbb{C}^{2d}$ in the $2d$-dimensional
$\omega,\tau$ space, on which the power series $R_{n,i}(\omega,\tau),\
\omega_{k,i}(\tau)$, $i=1,2,\ldots,n$ converge. They define $d$-dimensional
complex analytic manifolds

$X_{n}:=\\{(\tau,\omega)\in
U_{n}\;|\;R_{n,1}(\tau,\omega)=\dots=R_{n,d}(\tau,\omega)=0\\},$

which are the graphs of the map

$\omega_{n}=(\omega_{n,1}(\tau),\dots,\omega_{n,d}(\tau)).$

######  Proposition 2.9.

The manifolds $X_{n}$ are contained in $F(H)\times\mathbb{C}^{d}$, where
$F(H)$ denotes the frequency space of $H$.

###### Proof.

We prove the statement by induction on $n$. Clearly, this is true for $X_{0}$,
which reduces to the $d$-dimensional plane

$\omega_{1}=\dots=\omega_{d}=0.$

As $X_{n+1}$ is the image of $X_{n}$ under the automorphism by
$\varphi_{n}=e^{-v_{n}}$, it is sufficient show that the restriction of the
vector field $v_{n}$ to $X_{n}$ is contained in the frequency space $F(H)$.

To see this, decompose the field $v_{n}$ into Hamiltonian and non-Hamiltonian
part

$v_{n}=\sum_{i=1}^{d}v_{i,n}(\omega,\tau)\partial_{\omega_{i}}+w_{n},\;\;\;w_{n}\in{\rm
Ham\,}(Q).$

Only the non-Hamiltonian part enters the computation of

$\varphi_{n}(\omega_{i})=\omega_{i}-v_{n,i}(\omega,\tau)+O(2^{k}+2).$

As the $R_{n,i}(\omega,0)=\omega_{i}+O(2)$, we may apply Weierstrass division
and divide the coefficients $v_{n,i}$ successively by the $R_{n,j}$’s and get
an expression of the form:

$v_{n,i}=a_{n,i}(\tau)+\sum_{j=1}^{d}R_{n,j}(\omega,\tau)b_{n,j}(\omega,\tau).$

By definition, the coefficients $R_{n,j}(\tau,\omega)$ vanish on $X_{n}$. The
Hamiltonian $H_{n+1}=(F_{n+1})_{|X_{n+1}}$ is in Birkhoff normal form up to
order $2^{n+1}+1$ and moreover a direct computation shows that:

$\displaystyle H_{n+1}$
$\displaystyle=(A_{n}+v_{n}(A_{n}))_{|X_{n+1}}+O(2^{n+1}+2)$
$\displaystyle=(A_{0})_{|X_{n+1}}+\sum_{i=1}^{d}a_{n,i}(\tau)p_{i}q_{i}+O(2^{n+1}+2)\
{\rm mod\,}R_{0}+I^{2}$

Therefore by identification with the Birkhoff normal form, we get that

$a_{n,i}(\tau)=[b_{n,i}(\tau)]^{2^{n+1}}_{2^{n}}.$

This shows that the vector field $v_{n}$ is tangent to the frequency space
along $X_{n}$. Thus its time one flow $e^{-v_{n}}$ maps the analytic manifold
$X_{n}\subset F(H)\times\mathbb{C}^{d}$ to an analytic submanifold
$X_{n+1}\subset F(H)\times\mathbb{C}^{d}$. This proves the proposition. ∎

This simple proposition will turn out to be a fundamental property of the
formal Hamiltonian normal form iteration.

##  3\. Convergence of the Hamiltonian normal form

We now proceed to the convergence properties of the iteration scheme of the
Hamiltonian normal form. Our strategy will be to transfer the Hamiltonian
normal form from power series rings to function spaces and “replace” the ring
$R$ by a sequence of Banach spaces $E_{n}$, where the $E_{n}$ are certain
spaces of functions on a shrinking sequence of sets

$W_{0}\supset W_{1}\supset W_{2}\supset\ldots\supset W_{\infty},$

so that there will be natural restriction mappings

$E_{0}\xhookrightarrow{}E_{1}\xhookrightarrow{}E_{2}\xhookrightarrow{}\ldots\xhookrightarrow{}E_{\infty}$

of norm $\leq 1$. The iteration step will construct from $A_{n},B_{n}\in
E_{n}$ elements $A_{n+1},B_{n+1}\in E_{n+1}$, using certain operators that
emulate the operators that appear in the iteration scheme of $\S 2$.

It is important to have a certain flexibility in the possible choices of the
spaces $E_{n}$. The maps occurring in the iteration will always satisfy
precise estimates. The resulting technical difficulties are just of notational
nature and largely irrelevant, as we shall see.

###  3.1. The resonance fractal

The $\omega$-variables are crucial: we can control the estimates only if we
restrict the $\omega$-domain over a sequence of decreasing sets. In this way
we obtain convergence over the remaining Cantor-like set.

Let us denote by

$\Xi_{v}:=\\{\beta\in\mathbb{C}^{d}\;\;|\;\;(\beta,v)=0\\}\subset\mathbb{C}^{d}$

the hyperplane orthogonal to vector $v\in\mathbb{C}^{d},v\neq 0$, with respect
to the standard scalar product

$(\beta,v):=\sum_{i=1}^{d}\beta_{i}v_{i}.$

A vector $\beta\in\mathbb{C}^{d}$ is non-resonant precisely if for all
$J\in\mathbb{Z}^{d}\setminus\\{0\\}$

$(\beta,J)\neq 0.$

######  Definition 3.1.

The complex resonance fractal is the set

$\Xi:=\bigcup_{J\in\mathbb{Z}^{d}\setminus\\{0\\}}\Xi_{J},$

The real resonance fractal is the set

${\mathcal{X}}:=\bigcup_{J\in\mathbb{Z}^{d}\setminus\\{0\\}}{\mathcal{X}}_{J},$

where ${\mathcal{X}}_{J}=\Xi_{v}\cap\mathbb{R}^{d}$.

######  Proposition 3.2.

  1. i)

The set $\Xi\subset\mathbb{C}^{d}$ is dense for $d>2$.

  2. ii)

The set ${\mathcal{X}}\subset\mathbb{R}^{d}$ is dense for $d>1$.

###### Proof.

We only prove the first assertion. The second one is similar, but easier. As
$\mathbb{Q}$ is dense in $\mathbb{R}$, it follows that $\Xi$ is dense in the
set

$S:=\\{\beta\in\mathbb{C}^{d}:\exists\;\;v\in\mathbb{R}^{d}\setminus\\{0\\},(\beta,v)=0\\},$

consisting of all points the belong to the complexification of a real
hyperplane. Let us show that for $d>2$, we have $S=\mathbb{C}^{d}$. Let
$z\in\mathbb{C}^{d}=\mathbb{R}^{d}+i\mathbb{R}^{d}$ and write

$z=x+iy,\ x,y\in\mathbb{R}^{d}.$

As $d>2$, there exists a real hyperplane $V\subset\mathbb{R}^{d}$ containing
the vectors $x,y\in\mathbb{R}^{d}$, so

$z=x+iy\in V\otimes\mathbb{C}.$

This proves the statement. ∎

###  3.2. Arithmetical classes

In general, if $\beta$ is non-resonant, the scalar product $(\beta,J)$,
$J\in\mathbb{Z}^{d}\setminus\\{0\\}$ is non-zero, but can become arbitrary
small. To quantify this, we consider the following.

######  Definition 3.3.

The arithmetic sequence $\sigma(\beta)=(\sigma(\beta)_{k})$ of a vector
$\beta\in\mathbb{C}^{d}$ is defined as

$\sigma(\beta)_{k}:=\min\\{|(\beta,J)|:J\in\mathbb{Z}^{d}\setminus\\{0\\},\|J\|\leq
2^{k}\\}.$

The precise nature of the falling sequence $\sigma(\beta)$ encodes important
arithmetical properties of the vector $\beta$.

######  Definition 3.4.

Let $a=(a_{k})$ be any real positive decreasing sequence. The complex
arithmetic class associated to $a$ is the set

${\mathcal{C}}(a)_{\infty}:=\cap_{m=0}^{\infty}{\mathcal{C}}(a)_{m},\;\textup{where}\;\;{\mathcal{C}}(a)_{m}:=\\{\beta\in\mathbb{C}^{n}:\forall
k\leq m\;\;\sigma(\beta)_{k}\geq a_{k}\\}.$

Similarly, the real arithmetic class associated to $a$ is the set

${\mathcal{R}}(a)_{\infty}:=\cap_{m=0}^{\infty}{\mathcal{R}}(a)_{m}\;\;\textup{where}\;\;{\mathcal{R}}(a)_{m}:=\\{\beta\in\mathbb{R}^{d}:\forall
k\leq m\;\;\sigma(\beta)_{k}\geq a_{k}\\}.$

Note that

${\mathcal{R}}(a)_{m}={\mathcal{C}}(a)_{m}\cap\mathbb{R}^{d},\ \forall
m\in\mathbb{N}\cup\\{\infty\\}.$

If a vector $\beta$ belongs to ${\mathcal{C}}(a)_{\infty}$, then
$|(\beta,J)|\geq a_{k}$ for all lattice vectors $0\neq\|J\|\leq 2^{k}$, and
thus small denominators are controlled by the sequence $a$.

As we are dealing with a descending chain of closed sets
${\mathcal{C}}(a)_{m}$ and ${\mathcal{R}}(a)_{m}$, it follows that
${\mathcal{C}}(a)_{\infty}$ and ${\mathcal{R}}(a)_{\infty}$ are closed
subsets. From the above propositions we can conclude immediately:

######  Corollary 3.5.

  1. i)

The set ${\mathcal{C}}(a)_{\infty}$ has an empty interior for $d>2$.

  2. ii)

The set ${\mathcal{R}}(a)_{\infty}$ has an empty interior for $d>1$.

So the case of two degrees of freedom is special. In this case, the complex
arithmetic classes ${\mathcal{C}}(a)_{\infty}\subset\mathbb{C}^{2}$ have a
non-empty interior. More precisely, if the frequency ratio is non-real,

$\beta_{2}/\beta_{1}\notin\mathbb{R},$

then it is an interior point of $\mathbb{C}^{2}\setminus\Xi$. This ensures a
convergent Birkhoff normalisation in two degrees of freedom. This peculiar
fact was already observed by Moser back in 1958 [25] (see also [28]).

Although we tautologically have

$\beta\in{\mathcal{R}}(\sigma(\beta))_{\infty},$

it might be an isolated point of that set.

Note however, that $a\geq a^{\prime}$ then
${\mathcal{C}}(a)_{\infty}\subset{\mathcal{C}}(a^{\prime})_{\infty}$ and
similarly ${\mathcal{R}}(a)_{\infty}\subset{\mathcal{R}}(a^{\prime})$. The
following elementary but fundamental result shows that although arithmetic
classes may have an empty interior, after replacing $a$ by a slightly smaller
sequence $\nu a$, they are big in the sense of measure theory.

######  Proposition 3.6 ([15]).

Let $a=(a_{n})$ be a positive decreasing sequence and
$\beta\in{\mathcal{R}}(a)_{\infty}$. Let $\nu=(\nu_{n})$ be another positive
decreasing sequence with $\nu_{i}\leq 1$ and
$\sum_{i=1}^{\infty}\nu_{i}<\infty$. Then the density of ${\mathcal{R}}(\nu
a)_{\infty}$ at $\beta$ is equal to $1$:

$\lim_{\varepsilon\longrightarrow
0}\frac{\mu(B(\beta,\varepsilon)\cap{\mathcal{R}}(\nu
a)_{\infty})}{\mu(B(\beta,\varepsilon))}=1$

Here $\nu a$ is the sequence with terms $\nu_{n}a_{n}$ and $\mu$ denotes the
Lebesgue measure, $B(\beta,\varepsilon)$ the ball with radius $\varepsilon$,
centred at $\beta$.

######  Definition 3.7.

For a given vector $\beta\in\mathbb{R}^{d}$, we say that a falling sequence
$a=(a_{n})$ is $\beta$-dense, if the set ${\mathcal{R}}(a)_{\infty}$ has
density $1$ at the point $\beta$:

$\lim_{\varepsilon\longrightarrow
0}\frac{\mu(B(\beta,\varepsilon)\cap{\mathcal{R}}(a)_{\infty})}{\mu(B(\beta,\varepsilon))}=1$

So the above proposition says we can always find $a<\sigma(\beta)$ slightly
smaller, such that $a$ is $\beta$-dense. Similar statements holds in the
complex case be we will not use it, as our range of applications is in the
real domain.

###  3.3. The sets $Z_{n}$ and $W_{n}$

We will have to study the set ${\mathcal{C}}(a)_{\infty}$ in the neighbourhood
of a fixed frequency vector $\alpha$. For this we consider a decreasing
sequence $s=(s_{n})$ and converging to a positive limit $s_{\infty}$. We
denote by $B(r)\subset\mathbb{C}^{d}$ the ball of radius $r$ centred at the
origin.

######  Definition 3.8.

For fixed decreasing sequences $a$ and $s$ we define the closed set

$Z_{n}:=Z_{n}(\alpha,a,s):=\\{\omega\in B(s_{n}):\forall k\leq
n,\sigma(\alpha+\omega)_{k}\geq a_{k}(s_{0}-s_{n})\\}$

Note that $Z_{0}=B(s_{0})$. As the sequence $(s_{n})$ is decreasing we have

$a_{k}(s_{0}-s_{n+1})\geq a_{k}(s_{0}-s_{n}),$

so the sets $(Z_{n})$ form a descending chain and can be considered as a local
variant of the chain $({\mathcal{C}}(a)_{n})$. Moreover if $s_{0}<1$ then
$a_{k}>a_{k}(s_{0}-s_{n})$ for all $n$’s and therefore

$B(s_{n})\cap{\mathcal{C}}(a)_{n}\subset Z_{n}$

In the iteration, we will have to control the shrinking of the sets $Z_{n}$.
To do so, given two open sets $V\subset U\subset\mathbb{C}^{d}$, we denote by
$\delta(U,V)$ the supremum of the real numbers $\rho$ for which

$V+\rho B\subset U.$

where $B$ denotes the unit ball.

######  Lemma 3.9.

The sequence $\left(\delta(Z_{n},Z_{n+1})\right)_{n\in\mathbb{N}}$ satisfies
the estimate

$\delta(Z_{n},Z_{n+1})\geq 2^{-n}a_{n}(s_{n}-s_{n+1}).$

###### Proof.

The proof is straightforward. Assume that $\omega\in Z_{n+1}$ and take
$x\in\mathbb{C}^{d}$ satisfying

$\|x\|\leq 2^{-n}a_{n}(s_{n}-s_{n+1}).$

For $k\leq n$ and $\|J\|<2^{k}$, we have:

$\displaystyle|(\alpha+\omega+x,J)|$
$\displaystyle\geq|(\alpha+\omega,J)|-|(x,J)|$ $\displaystyle\geq
a_{k}(s_{0}-s_{n+1})-\|x\|\,\|J\|$ $\displaystyle\geq
a_{k}(s_{0}-s_{n+1})-a_{n}(s_{n}-s_{n+1})$ $\displaystyle\geq
a_{k}(s_{0}-s_{n+1})-a_{k}(s_{n}-s_{n+1})=a_{k}(s_{0}-s_{n}).$

This shows that $\omega+x\in Z_{n}$ and thus proves the lemma. ∎

The sets $W_{n}$ we will be working with, are defined as follows.

######  Definition 3.10.

For fixed decreasing sequences $a$ and $s$ we set

$W_{n}:=Z_{n}\times D_{s_{n}}^{d}\times
D_{s_{n}}^{2d}\subset\mathbb{C}^{d}\times\mathbb{C}^{d}\times\mathbb{C}^{2d}$

where $D_{t}\subset\mathbb{C}$ denotes the closed disc of radius $t$.
Furthermore, we put

$V_{n}:=Z_{n}\times D_{s_{n}}^{d}\subset\mathbb{C}^{d}\times\mathbb{C}^{d}$

The coordinates on $\mathbb{C}^{d}\times\mathbb{C}^{d}\times\mathbb{C}^{2d}$
are

$\omega_{1},\ldots,\omega_{d},\;\tau_{1},\ldots,\tau_{d},\;q_{1},\ldots,q_{d},\;p_{1},\ldots,p_{d},$

and there are projection maps

$W_{n}\longrightarrow V_{n},\;\;\;(\omega,\tau,q,p)\mapsto(\omega,\tau).$

###  3.4. The Banach spaces

The Banach spaces that we will use, consist of functions that are holomorphic
on the interior of the closed sets $W_{n}$ defined above. They come in
different flavours: we may require that they extend continuously to the
boundary, or more generally be Whitney $C^{k}$, or be square integrable.

For a closed subset $X\subset\mathbb{R}^{d}$, we denote by

$C^{k}(X,\mathbb{R})$

the vector space of $C^{k}$-Whitney differentiable functions,
$k\in\mathbb{N}$, for which the norm

$\|f\|=\max_{|I|\leq k}\sup_{x\in X}|\partial^{I}f(x)|$

is finite and therefore defines a Banach space structure on it111Recall that a
function $f:X\longrightarrow\mathbb{R}$ defined on a closed subset $X$ of
Euclidean space is called Whitney differentiable at $x\in X$, if there exists
functions $D^{I}f(x)$ called the Whitney derivatives of $f$ at $x$ such that:
$f(y)=\sum_{|I|\leq m}\frac{D^{I}f(x)}{|I|!}(y-x)^{I}+o(\|y-x\|^{m})$ The sole
difference that the Whitney definition bears is the uniformity of the limit in
the $x,y$ variables..

There is a natural definition of a holomorphic function at a point $x$ of a
closed subset $X\subset\mathbb{C}^{k}$. Let us say that a vector $v$ at $x\in
X$ is interior if:

$\exists\delta,|t|<\delta\implies x+tv\in X.$

Then $f$ is holomorphic at $x\in X$ if for any interior vector $v$ with the
property the limit

$\lim_{t\mapsto 0}\frac{f(x+tv)-f(x)}{t}$

exists for complex values of $t$.

If $x$ is an interior point this is equivalent to the standard definition. In
the general case it allows us to control analyticity of the function on the
boundary of a set and also to avoid defining our spaces as topological tensor
products.

If $f$ is defined in a closed polydisc $D_{1}\times D_{2}$, to be holomorphic
means that

  1. (1)

$f$ is holomorphic in the interior of the polydisc.

  2. (2)

$f(x_{1},-)$ and $f(-,x_{2})$ define holomorphic functions for any $x_{1}$ and
any $x_{2}$.

It follows from Morera’s theorem that if $f$ is continuous then the first
condition implies the second one. On the other extreme if $f$ is defined in
$X=\\{0\\}\times D_{2}$, then to be holomorphic means that $f(0,-)$ is
holomorphic. In this case, the set $X$ has an empty interior. In our iteration
the limit set over which the functions are defined, namely $W_{\infty}$ has
indeed an empty interior while for finite $n$, the sets $W_{n}$ are the
closures of their interior.

######  Definition 3.11.

Given a closed subset $X\subset\mathbb{C}^{d}$, we denote by
${\mathcal{O}}^{k}(X)$ the Banach space of (bounded) $C^{k}$-Whitney function
which are holomorphic in $X$:

${\mathcal{O}}^{k}(X)=C^{k}(X,\mathbb{C})\cap{\mathcal{O}}(X)$

For $k=0$, we use the notation ${\mathcal{O}}^{c}$ instead of
${\mathcal{O}}^{k}$.
We define similarly ${\mathcal{O}}^{h}(X)$ as the Hilbert space of square
integrable functions on $X$, holomorphic on the interior.

For our iteration scheme, we will use the following Banach spaces:

######  Definition 3.12.

$E_{n}^{b}:={\mathcal{O}}^{b}(W_{n}),\ b=h,c\text{ or }k,\
n\in\mathbb{N}\cup\\{\infty\\}.$

Note that $E_{\infty}^{b}$ is the subspace of ${\mathcal{O}}^{b}(W_{\infty})$
consisting of functions for which $f(\omega,-)$ is holomorphic for any
$\omega\in Z_{\infty}$:

$E_{\infty}^{k}=\\{f\in C^{k}(W_{\infty}):\forall\omega\in Z_{n},\
f(\omega,-)\in{\mathcal{O}}^{k}(D_{s_{n}}^{3d})\\}$

That these subspaces are closed follows from Morera’s theorem.

Also the spaces of functions

$F_{n}^{b}:={\mathcal{O}}^{b}(V_{n})$

do play a role. Note that
${\mathcal{O}}^{b}(V_{n})\subset{\mathcal{O}}^{b}(W_{n})$, so $F_{n}^{k}$ can
be identified with the closed subspaces of $E_{n}^{k}$, consisting of those
functions that do not depend on $q,p$.

The restriction mappings induce injective mappings

$(E_{n}^{b},|\cdot|_{n})\longrightarrow(E_{m}^{b},|\cdot|_{m}),\ m>n,$

and this holds also for $m=\infty$, since the image consists of Whitney
$C^{\infty}$ functions having analytic Taylor expansion. As the set $W_{n}$ is
connected, this expansion defines uniquely a holomorphic function in this
neighbourhood.

Usually there is no ambiguity to drop the $n$ in $|x|_{n}$; for $x\in E_{n}$
the notation $|x|$ just means $|x|_{n}$. Furthermore, if $(x_{n})$ is a
sequence with $x_{n}\in E_{n}$, then the notation $|x|$ can be used for the
sequence $|x_{n}|$ and be called the norm sequence of $x$, etc.

We say that a sequence $(x_{n}),\ x_{n}\in E_{n}$ is convergent, if the image
sequence in $E_{\infty}$ converges and that it converges quadratically, if
there is an estimate of the form

$|x-x_{n}|_{\infty}\leq Cq^{2^{n}},$

for some $q\in]0,1[$ and $C>0$.

###  3.5. Bruno sequences

We will make use of Bruno sequences in two different ways. First, we will have
to impose a Bruno-condition on the frequency vector $\alpha$ and second, we
will use a Bruno sequence $\rho$ to determine the sequence $s$ of polyradii.
This leads to a simple way to control the small denominator estimates for the
operators that will be defined in the next section.

######  Definition 3.13 ([2]).

A strictly monotone positive sequence $a$ is called a Bruno sequence if the
infinite product

$\prod_{k=0}^{\infty}a_{k}^{1/2^{k}}$

converges to a strictly positive number or equivalently if

$\sum_{k\geq 0}\left|\frac{\log a_{k}}{2^{k}}\right|<+\infty.$

Since their introduction in [2], these sequences have played a key role in
KAM-theory. We denote respectively by ${\mathcal{B}}^{+}$ and
${\mathcal{B}}^{-}$ the set of increasing and decreasing Bruno sequences. The
set of Bruno sequences some obvious multiplicative properties:

  1. i)

Taking the multiplicative inverse interchanges ${\mathcal{B}}^{+}$ and
${\mathcal{B}}^{-}$,

  2. ii)

The product of two elements in ${\mathcal{B}}^{\pm}$ is again in
${\mathcal{B}}^{\pm}$.

  3. iii)

An element of ${\mathcal{B}}^{\pm}$ raised to a positive power remains in
${\mathcal{B}}^{\pm}$.

Note also that any geometrical sequence $a_{n}=q^{n},\;\;q\neq 1$ is a Bruno
sequence, belonging to ${\mathcal{B}}^{-}$ if $q<1$, to ${\mathcal{B}}^{+}$ if
$q>1$. The sequence $a_{n}=e^{\pm\alpha^{n}}$ belongs to ${\mathcal{B}}^{\pm}$
if and only if $1<\alpha<2$.

######  Definition 3.14.

A vector $\alpha\in\mathbb{C}^{d}$ is said to satisfy Bruno’s arithmetical
condition if $\sigma(\alpha)\in{\mathcal{B}}^{-}$.

If $(a_{n})\in{\mathcal{B}}^{+},a_{0}>1$, then from the convergence of the sum
$\sum\frac{\log a_{n}}{2^{n}}$ we see that for any $\varepsilon>0$, there is
an $N$ such that for $n\geq N$ we have

$\log a_{n}\leq 2^{n}\varepsilon,\;\;a_{n}\leq(e^{\varepsilon})^{2^{n}},$

so the sequence can not increase faster than a quadratic iteration

$u_{n+1}=u_{n}^{2},\ u_{0}>1.$

Similarly, for $(a_{n})\in{\mathcal{B}}^{-},a_{0}<1$ we find an estimate

$a_{n}\geq(e^{-\varepsilon})^{2^{n}}\;\;\textup{for}\;\;n\geq N,$

so that the sequence can not decrease too quickly.

### Sequences of type ${\mathcal{P}}(a)$.

We need to make a special choice of our sequence $s$ that determines the size
of the sets $W_{n}$ in terms of an auxiliary Bruno sequence
$\rho=(\rho_{n})\in{\mathcal{B}}^{-}$.

We will always assume that $\rho_{0}<1$ and that $\rho$ converges to $0$. For
a given $s_{0}>0$, we define $s_{n}$ recursively by

$s_{n+1}=(\rho_{n})^{1/2^{n}}s_{n}.$

Because of the Bruno property of $\rho$, the sequence $s=(s_{n})$ converges to
a positive limit $s_{\infty}$.

The sequence $\rho$ will be taken small enough to counteract certain small
denominators of the form

$a_{n}^{l}(s_{n}-s_{n+1})^{m}=a_{n}^{l}(1-\rho^{1/2^{n}})^{m}s_{n}^{m}$

that appear norm estimates.Therefore we define

######  Definition 3.15.

Given a sequence $a=(a_{n})\in{\mathcal{B}}^{-}$, we denote by
${\mathcal{P}}={\mathcal{P}}(a)$ (resp. ${\mathcal{P}}^{+}$) the set of
sequences depending on parameters $\rho_{n}>0,\ n~\in\mathbb{N}$, whose terms
are majorated by $C\rho_{n}^{k}a_{n}^{-l}(s_{n}-s_{n+1})^{-m}$ for some
$C,k,l,m\geq 0$ (resp. $k>0$):

$u_{n}\leq\frac{C\rho_{n}^{k}}{a_{n}^{l}(s_{n}-s_{n+1})^{m}}$

The following lemma shows that an appropriate choice of $\rho$ absorbs the
small denominators.

######  Proposition 3.16.

For any falling Bruno sequence $b\in{\mathcal{B}}^{-}$ and any
$u\in{\mathcal{P}}^{+}(a)$, there exists values of the sequence
$\rho=\rho(a,b)$ and a constant $K\in\mathbb{R}_{+}$ so that $u<Kb$ and
$\rho\in{\mathcal{B}}^{-}$.

###### Proof.

We let

$u_{n}<C\rho_{n}^{k}a_{n}^{-l}(s_{n}-s_{n+1})^{-m},$

and show that the falling Bruno sequence

$\rho_{n}=2^{(-n-1)m/k}M^{-1/k}a_{n}^{l/k}b_{n}^{1/k}\text{ with
}M=\max(C,2^{k}a_{0}^{l}b_{0})$

satisfies the condition.

For this choice, we have:

$u_{n}<2^{(-n-1)m}b_{n}(s_{n}-s_{n+1})^{-m}$

By our choice of the sequence $s=(s_{n})$ we have:

$s_{n}-s_{n+1}=(1-\rho_{n}^{1/2^{n}})s_{n}\geq(1-\rho_{n}^{1/2^{n}})s_{\infty}.$

The constant $M$ is defined so that the falling sequence $\rho$ is bounded by
$1/2$ and hence:

$(1-\rho_{n}^{1/2^{n}})\geq(1-2^{-1/2^{n}})>2^{(-n-1)}.$

Thus we have the estimates

$u_{n}<2^{(-n-1)m}b_{n}(s_{n}-s_{n+1})^{-m}<b_{n}s_{\infty}^{-m}=Kb_{n}.$

This proves the proposition. ∎

So in the iteration process, small denominators $a_{n}^{l}(s_{n}-s_{n+1})^{m}$
can be absorbed by the presence of such a sequence $\rho$ in the numerator,
and in fact be pushed below any pre-given Bruno sequence.

###  3.6. The operators

The basic step in the iteration of $\S 2$ that determines $A_{n+1},B_{n+1}$
form $A_{n},B_{n}$ depended on the truncation operators

$[-]_{2^{n}+1}^{2^{n+1}+1},\;\;\;[-]_{2^{n+1}+2}$

and the map $j_{n}$, which came from a map $j_{A}:R\longrightarrow\Theta(R)$,
which was defined as a composition of several other maps. If we want to lift
these operator in the Banach space context, we often need to shrink the domain
of definition slightly. For this reason it is convenient to define Banach
spaces $E_{n+\varepsilon}$ between $E_{n}$ and $E_{n+1}$, and set up things in
such a way that the iteration step brings us from $E_{n}$ to $E_{n+1}$, using
one or more intermediate spaces $E_{n+\varepsilon}$, where $0<\varepsilon<1$.
We implement this idea in the following simple way: we interpolate the
sequence $s=(s_{n})$ by setting:

$s_{n+\varepsilon}=(\rho_{n})^{\varepsilon/2^{n}}s_{n},\ \varepsilon\in]0,1[,$

and then set

$W_{n+\varepsilon}:=Z_{n+\varepsilon}\times D_{s_{n+\varepsilon}}^{d}\times
D_{s_{n+\varepsilon}}^{2d},$
$E_{n+\varepsilon}^{b}:={\mathcal{O}}^{b}(W_{n+\varepsilon}),$

where we put

$Z_{n+\varepsilon}=\\{\omega\in\mathbb{C}^{d}\;|\;\forall k\leq
n,\sigma(\alpha+\omega)_{k}\geq a_{k}(s_{0}-s_{n+\varepsilon}\\}.$

One then has:

$\delta(W_{n},W_{n+\varepsilon})\geq\frac{a_{n}}{2^{n}}(s_{n}-s_{n+\varepsilon})\geq\frac{a_{n}}{2^{n}}(1-\rho_{n}^{\varepsilon/2^{n}})s_{\infty}.$

The definition of ${\mathcal{P}}$ and ${\mathcal{P}}^{+}$ extend for a fixed
$\varepsilon$ as well as the absorption principle.

### Restriction maps

First, there are the restriction operators $E^{b}_{n}\longrightarrow
E^{b}_{m}$, $n\geq m$. These have norm $\leq 1$ and thus are completely
harmless. We will not mention or even give names to these maps. A different
matter is that we also have to use ’flavour changing’ restriction maps, like
the canonical map

$r_{n,m}:E_{n}^{h}\longrightarrow E_{m}^{c},(n<m),$

which arise by remarking that a square integrable holomorphic function on
$W_{n}$ is continuous on the smaller set $W_{m}$. From the local equivalence
lemma (Lemma 5.2), it follows that the norm sequence of $r_{n,n+\varepsilon}$
belongs to ${\mathcal{P}}$. There is also a canonical map
$i_{n}:E_{n}^{c}\longrightarrow E_{n}^{h}$, whose norm is bounded by the
volume of $W_{n}$ and as these volumes form a falling sequence, this norm
sequence is also in ${\mathcal{P}}$.

### Truncation maps

For the power series ring we used the truncation operations

$[-]_{2^{n}+2}^{2^{n+1}+2},\;\;\;[-]_{2^{n+1}+2},$

that pick out some range of monomials from a series. Inside the Hilbert spaces
$E^{h}_{n}$ the monomials form an orthogonal set on polydiscs, so these
truncations are just special orthogonal projectors. In this way we obtain
corresponding maps

$\tau_{n,m}^{h}:E_{n}^{h}\longrightarrow E^{h}_{m},\;\;\;(n<m),$

and

$\sigma_{n,m}^{h}:E_{n}^{h}\longrightarrow E^{h}_{m},\;\;\;(n<m).$

Corresponding truncation operators on $E^{c}$-spaces can then be defined as
compositions

$\tau_{n,m}^{c}:E^{c}_{n}\stackrel{{\scriptstyle
i_{n}}}{{\longrightarrow}}E^{h}_{n}\stackrel{{\scriptstyle\tau_{n,m^{\prime}}^{h}}}{{\longrightarrow}}E^{h}_{m^{\prime}}\stackrel{{\scriptstyle
r_{m^{\prime},m}}}{{\longrightarrow}}E_{m}^{c},$

where $n<m^{\prime}<m$. We can make the specific but arbitrary choice
$m^{\prime}=(m+n)/2$.

The norm of these maps can be estimated directly from the Arnold-Moser lemma
(Lemma 5.6) and the conclusion is that the norm sequences for
$\tau_{n,n+\varepsilon}^{b}$ and $\sigma_{n,n+\varepsilon}^{b}$
($b\in\\{h,c\\}$ ) belong to ${\mathcal{P}}^{+}$ for $b=h$. Due to local
equivalence (Lemma 5.2), it also belongs to ${\mathcal{P}}^{+}$ for $b=c$.

More generally, one may define projectors for any subset
$S\subset\mathbb{N}^{3d}$ of monomials $p^{a}q^{b}\tau^{c}$. Such a subset
generates a $\mathbb{C}[[\omega]]$-module $M_{S}$ in the power series ring
$R$. By the same method as above one can define the projection maps for
$b\in\\{c,h\\}$

$\pi_{S,n,m}^{b}:E^{b}_{n}\longrightarrow E_{m}^{b},(n<m),$

on the space $M_{S}\cap E_{m}^{b}$ ’spanned’ by the monomials in $S$. The
corresponding norm sequence $(|\pi_{S,n,n+\varepsilon}^{b}|)$ belongs to
${\mathcal{P}}$.

### Poisson derivations

We introduce the space of Poisson derivations

$\Theta_{n,m}^{c}=\Theta(R)\cap L(E_{n}^{c},E_{m}^{c}).$

The quantity

$\|u\|=\sup_{n<p<q<m}\\{(s_{p}-s_{q})|u(x)|_{q}/(e|x|_{p})\\}$

is well-defined and defines a Banach space structure on $\Theta_{n.m}^{c}$.
(The normalisation constant $e\approx 2.718$ is purely conventional and serves
to simplify the estimates of the Borel lemma 5.4.)

The map $H_{n,m}:E^{c}_{n}\longrightarrow\Theta_{n,m}^{c}$ ($n<m$), which
associates to a function $f$ its hamiltonian field $\\{-,f\\}$ is well-
defined, and the norm sequence for $H_{n,n+1}$ belongs to ${\mathcal{P}}$,
again by Cauchy-Nagumo (Lemma 5.1).

Truncation maps for derivations factor:

$\Theta_{n,n^{\prime}}^{c}\stackrel{{\scriptstyle
ev}}{{\longrightarrow}}(E^{h}_{n^{\prime}})^{4d}\longrightarrow(E^{h}_{n^{\prime}})^{4d}\longrightarrow\Theta_{m^{\prime},m}^{c},\
n<n^{\prime}<m^{\prime}<m$

where $ev$ is the map which evaluates derivations on the coordinates
$\tau,\omega,q,p$ and the middle map of the diagram corresponds to the
truncation

$x\mapsto[x]_{2^{n}+1}^{2^{n+1}+1}$

Therefore the associated norm sequence for any choice of $m=n+\varepsilon$,
$n^{\prime}$, $m^{\prime}$ depending on $n$ belongs to ${\mathcal{P}}^{+}$. We
will denote this map also by $\tau_{n,m}$ when $n,n^{\prime},m^{\prime},m$ are
at equal mutual distances.

### The map $L^{c}$

Recall that in iteration scheme for the Hamiltonian normal form of $\S 2$ used
a specific $\mathbb{C}[[\omega,\tau]]$-linear map

$L:R\longrightarrow\Theta(R)={\rm Ham\,}(R)\oplus{\rm Der\,}(R_{0}),m\mapsto
Lm$

that was defined by setting for $a\neq b$:

$Lp^{a}q^{b}:=\\{-,\frac{1}{(\alpha+\omega,a-b)}p^{a}q^{b}\\},\ a\neq b.$

and

$Lg(qp):=\sum_{i=1}^{d}\frac{\partial
g(\tau)}{\partial\tau_{i}}\partial_{\omega_{i}}.$

We have to realise this on the level of Banach spaces.

We will define for $n<m^{\prime}<m^{\prime\prime}<m$ a map

$L_{n,m^{\prime},m^{\prime\prime},m}^{c}:E^{c}_{n}\longrightarrow\Theta_{m^{\prime},m}$

as a composition of five basic maps.

Inclusion step:

$i_{n}:E_{n}^{c}\longrightarrow E_{n}^{h}.$

Clearly, the norm sequence belongs to ${\mathcal{P}}$.

Truncation step:

$\tau_{n,m^{\prime}}:E^{h}_{n}\longrightarrow E^{h}_{m^{\prime}}.$

As remarked before, its norm sequence belongs to ${\mathcal{P}}^{+}$.

Division step: We now use the map

$div_{m^{\prime}}\longrightarrow div_{m^{\prime}},$

where the map $div_{n}:E_{n}^{h}\longrightarrow E_{n}^{h}$ is defined as by

$p^{I}q^{J}\mapsto\frac{1}{(\alpha+\omega,I-J)}p^{I}q^{J}$

on $(M_{n}^{h})^{\perp}$ and equal to the identity on $M_{n}^{h}$. As
$\omega\in Z_{n}$, we have

$|\frac{1}{(\alpha+\omega,I-J)}p^{I}q^{J}|_{n}\leq\frac{1}{a_{n}}|p^{I}q^{J}|_{n}.$

As the monomials $p^{I}q^{J}$ are orthogonal, we deduce that the norm sequence
of $div_{n}$ has its norm bounded by $a^{-1}\in{\mathcal{P}}$.

Restriction step: After that, we use the restriction map

$r_{m^{\prime},m^{\prime\prime}}:E^{h}_{m^{\prime}}\longrightarrow
E^{c}_{m^{\prime\prime}}$

As remarked above, the norm sequence of $r_{m+\varepsilon,m+2\varepsilon}$
belongs to ${\mathcal{P}}$.

Poisson step: We use the map $D_{m^{\prime\prime},m}$, where the map

$D_{n,m}:E^{c}_{n}\longrightarrow\Theta_{n,m}^{c}$

is defined on the level of monomials by

$\left\\{\begin{matrix}Dp^{I}q^{J}=\\{-,p^{I}q^{J}\\},\;\;\;\ I\neq J,\\\
Dg(pq)=\sum_{i=1}^{d}\frac{\partial
g(\tau)}{\partial\tau_{i}}\partial_{\omega_{i}}.\end{matrix}\right.$

As we are dealing with a projection and partial differential operators, its
norm is again in ${\mathcal{P}}$.

So we define

$L_{n,m^{\prime},m^{\prime\prime},m}^{c}:=D_{m^{\prime\prime},m}\circ
r_{m^{\prime},m^{\prime\prime}}\circ
div_{m^{\prime}}\circ\tau_{m^{\prime},n}\circ i_{n}$

as the composition of these five maps:

$E_{n}^{c}\stackrel{{\scriptstyle
i_{n}}}{{\xhookrightarrow{}}}E_{n}^{h}\stackrel{{\scriptstyle\tau_{n,m^{\prime}}^{h}}}{{\longrightarrow}}E_{m^{\prime}}^{h}\stackrel{{\scriptstyle
div_{m^{\prime}}}}{{\longrightarrow}}E_{m^{\prime}}^{h}\stackrel{{\scriptstyle
r_{m^{\prime},m^{\prime\prime}}}}{{\longrightarrow}}E_{m^{\prime\prime}}^{c}\stackrel{{\scriptstyle
D_{m^{\prime\prime},m}}}{{\longrightarrow}}\Theta^{c}_{m^{\prime\prime},m}$

We will make specific, but arbitrary choices for the intermediate points by
setting

$m^{\prime}:=n+\varepsilon,\;\;m^{\prime\prime}:=n+2\varepsilon,\;\;m=n+3\varepsilon,\;\;\varepsilon:=(m-n)/3$

and use the choices to define

$L_{n,m}^{c}:=L^{c}_{n,m^{\prime},m^{\prime\prime},m}$

In our iteration we will use specifically $L^{c}_{n,n+1/4}$ and
$L^{c}_{n+1/6,n+1/3}$.

######  Lemma 3.17.

The norm sequences $|L^{c}_{n,n+1/4}|$ and $|L^{c}_{n+1/6,n+1/3}|$ belong to
${\mathcal{P}}^{+}$

###### Proof.

The norm of the composition is at most the product of the norms, which are all
in ${\mathcal{P}}$; the norm of truncation is in ${\mathcal{P}}^{+}$. Hence
the result follows. ∎

### The map $j$

We proceed to the definition of the map $j$ inside our functional spaces.
Recall that in $\S 2$ we defined it in term of $L$:

$j_{A}:R\longrightarrow\Theta(R),\;\;\;m\mapsto L(m-Lm(T)).$

It is quite simple to realise this on the level of Banach spaces; as we need
to apply it to only to specific elements

$A_{n}=A_{0}+T_{n},\;\;\;T_{n}=\sum_{i=1}^{n}S_{n},$

we consider the map

$M_{n}:E_{n}^{c}\longrightarrow E_{n+1/4}^{c},\ m\mapsto
m-L^{c}_{n,n+1/4}(m)T_{n}.$

Note that the map

$L^{c}_{n,n+1/4}:E^{c}_{n}\longrightarrow\Theta^{c}_{n+1/6,n+1/4},$

so that $L^{c}_{n,n+1/4}(m)\in\Theta^{c}_{n+1/6,n+1/4}$ indeed can be applied
to $T_{n}\in E_{n}^{c}$. We can now define a map

$j_{n}^{c}:E_{n}^{c}\longrightarrow\Theta_{n+1/3,n+1/2}^{c},\ x\mapsto
L^{c}_{n+1/4,n+1/2}\circ M_{n}(x)$

that emulates the map

$j_{n}:x\mapsto[j_{A_{n}}([x]_{2^{n+1}+2}^{2^{n+2}+2})]_{2^{n}}^{2^{n+1}}=[j_{A_{n}}([x]_{2^{n+1}+2}^{2^{n+2}+2})]^{2^{n+1}}.$

defined in the formal KAM-iteration.

Note that by our implementation of $L$, we incorporated the truncation step,
which leads to a norm sequence in ${\mathcal{P}}^{+}$.

######  Lemma 3.18.

We have $\frac{|j^{c}|}{1+|T|}\in{\mathcal{P}}^{+}.$

###### Proof.

By definition one has:

$|j_{n}^{c}|\leq|L^{c}_{n,n+1/4}|\,|L^{c}_{n+1/4,n+1/2}|(1+|T_{n}|)\in(1+|T|){\mathcal{P}}^{+}.$

∎

###  3.7. The iteration in the Banach spaces $E^{c}_{n}$

We may now formulate the Hamiltonian normal form iteration in the spaces
$E_{n}^{c}$ for appropriate choices of $\rho$ and $s_{0}$.

######  Definition 3.19.

We will simplify notation slightly and write:

$\tau_{n}:=\tau_{n+1/2,n+1}^{c}:E^{c}_{n+1/2}\longrightarrow E_{n+1}^{c}$

for the maps that emulates the truncation $[-]_{2^{n}+1}^{2^{n+1}+1}$, and
similarly

$\sigma_{n}:=\sigma_{n,n+1/2}^{c}:E^{c}_{n}\longrightarrow E_{n+1/2}^{c}$

for the map the emulates the tail-truncation $[-]_{2^{n+1}+2}$ and $j_{n}$ for
$j_{n}^{c}$.

We are given an arithmetic class ${\mathcal{C}}(a)_{\infty}$ depending on a
sequence $a=(a_{n})$ and, without loss of generality, we may assume that
$a_{0}\leq 1$. We also fix an arbitrary $\alpha\in]3/2,2[$.

We now list all the estimates which appear in the proof of the convergence.
The precise form of these estimates is irrelevant for the proof of the
convergence, since the absorption principle implies that these are fulfilled
for sufficiently small $\rho$. More precisely, according to Proposition 3.16,
as the norm sequences $|j|/(1+|T|),\ |\tau|,\ |\sigma|$ belong to
${\mathcal{P}}^{+}$, we may find $\rho$ such that for some constant $R\geq 1$
we have:

(1) $\displaystyle|j_{0}|$ $\displaystyle\leq\frac{Ra_{0}(s_{1/2}-s_{1})}{2},$
(2) $\displaystyle\frac{|j_{n}|\,|\tau_{n}|}{1+|T_{n}|}$
$\displaystyle\leq\frac{Ra_{n}(s_{n+1/4}-s_{n+1/2})}{2e(n+1)},$ (3)
$\displaystyle|\tau_{n}|$ $\displaystyle\leq\frac{R}{2},$ (4)
$\displaystyle\frac{|j_{n}|}{1+|T_{n}|}$
$\displaystyle\leq\frac{Ra_{n}(s_{n+1/2}-s_{n+1})}{8(n+1)},$ (5)
$\displaystyle|\sigma_{n}|$ $\displaystyle\leq\frac{R^{2}e^{-\alpha^{n}}}{8},$
(6) $\displaystyle|\sigma_{n}|\,|j_{n}|$
$\displaystyle\leq\frac{R^{2}e^{-\alpha^{n}}}{8(n+1)}.$

Note that (1) follows form the $n=0$ case of (4) and that some estimates are
redundant. As $A_{0},B_{0}$ vanish at the origin, we may choose $s_{0}$ small
enough so that

$\displaystyle(a)$ $\displaystyle\qquad|A_{0}|\leq 1,$ $\displaystyle(b)$
$\displaystyle\qquad|B_{0}|\leq R^{-1},$ $\displaystyle(c)$
$\displaystyle\qquad|B_{0}|\leq R^{-2}e^{1/(2-\alpha)}.$

Clearly $(c)$ is in general stronger than $(b)$! The estimates used in the
proof will be numbered by letters and numbers; for instance $(1b)$ means that
we use the estimates $(1)$ and $(b)$.

######  Theorem 3.20 ([14]).

Consider an analytic Hamiltonian of the form

$H=\sum_{i=1}^{n}\alpha_{i}p_{i}q_{i}+O(3)\in\mathbb{C}\\{p,q\\}$

and put

$F_{0}:=H+\sum_{i=1}^{d}\omega_{i}p_{i}q_{i}=A_{0}+B_{0}$

Assume that the frequency vector $\alpha\in{\mathcal{C}}(a)_{\infty}$, where
$a$ is a Bruno sequence. Under the above assumptions on the sequence $\rho$
and $s_{0}$, the iteration

$\displaystyle A_{n+1}$ $\displaystyle=A_{n}+S_{n+1}$ $\displaystyle B_{n+1}$
$\displaystyle=\phi(v_{n})A_{n}+\psi(v_{n})S_{n+1}+e^{-v_{n}}(\sigma_{n}(B_{n}-v_{n}(A_{n})))$

with

$S_{n+1}:=\tau_{n}(B_{n}-v_{n}(A_{n})),\;\;\;v_{n}:=j_{n}(B_{n})=j_{n}B_{n}$

and

$\phi(z)=e^{-z}(1+z)-1,\;\;\;\psi(z)=e^{-z}-1$

is well-defined in the Banach space $E_{n}^{c}$, i.e. for all $n\in\mathbb{N}$
we have:

  1. i)

$A_{n},B_{n},S_{n}\in E_{n}^{c}$.

  2. ii)

The sequence $(B_{n})$ converges quadratically to zero.

  3. iii)

The vector fields $v_{n}$ exponentiate to elements

$\varphi_{n}=e^{-v_{n}}\in L(E_{n},E_{n+1})$

Moreover the composition

$\Phi_{n}=\varphi_{n-1}\varphi_{n-2}\ldots\phi_{1}\phi_{0}$

converges to a Poisson morphism

$\Phi_{\infty}\in L(E_{0}^{c},E_{\infty}^{c}),$

which reduces $F_{0}$ to its Hamiltonian normal form.

Remark. It could happen that the real part of the set $Z_{\infty}$ reduces to
the origin as the set ${\mathcal{R}}(a)$ might have density $0$ at the point
$\alpha\in\mathbb{R}^{d}$. In this case, our statement would simply be empty.
As we already explained, this difficulty however can always be circumvented by
an appropriate choice of the sequence $a$. According to Proposition 3.6, given
a frequency vector $\alpha\in\mathbb{R}^{d}$ satisfying the Bruno condition,
we can always find a Bruno sequence $a<\sigma(\alpha)$, for which
${\mathcal{R}}(a)_{\infty}$ has density $1$ at the point $\alpha$. So the
above pathology can be avoided easily by an appropriate choice of the Bruno
sequence $a=(a_{n})$. This is a fundamental difference between KAM theory and
Diophantine analysis, where one is concerned with optimal bounds [15, 19, 20].

###### Proof.

We start with

$F_{0}=A_{0}+B_{0},$

and consider $A_{0}$ and $B_{0}$ as elements of $E_{0}^{c}$. We have:

$v_{0}=j_{0}(B_{0})\in\Theta_{1/4,1/2}^{c}$

and, as $T_{0}=0$, we get the estimate:

$(1b)\qquad|v_{0}|\leq|j_{0}||B_{0}|\leq\frac{a_{0}(s_{1/2}-s_{1})}{2}.$

According to the Borel lemma (Lemma 5.4), the linear map

$e^{-v_{0}}:E_{1/2}^{c}\longrightarrow E_{1}^{c}$

is well-defined and moreover, as $e^{z}$ is the Borel transform of $1/(1-z)$,
we deduce from the previous estimate that:

$|e^{-v_{0}}|\leq\frac{1}{1-1/2}=2.$

In the next step we get

$F_{1}=e^{-v_{0}}F_{0}\in E_{1}^{c},$

and $A_{1}=A_{0}$, $S_{1}=0$.

We show by induction the following estimates:

$\displaystyle|S_{n+1}|$ $\displaystyle\leq R|B_{n}|,$
$\displaystyle|B_{n+1}|$
$\displaystyle\leq\frac{R^{2}}{2}|B_{n}|^{2}+\frac{R^{2}}{2}e^{-\alpha^{n}}|B_{n}|,$
$\displaystyle|B_{n+1}|$ $\displaystyle<R^{-1}.$

Assuming the validity of these estimates up to index $n$, we may conclude that

$(b)\qquad|T_{n}|=|\sum_{k=1}^{n}S_{k}|\leq\sum_{k=1}^{n}|S_{k}|\leq
R\sum_{k=1}^{n}|B_{k}|\leq n.$

In particular, we may simplify the estimates in which $1+|T_{n}|$ is involved.
Note also that:

$(a)\qquad|A_{n}|\leq|A_{0}|+|T_{n}|\leq n+1.$

As formal power series, the term $S_{n+1}$ is the difference of two terms

$S_{n+1}=\tau_{n}(B_{n}-j_{n}(B_{n})(A_{n}))=\tau_{n}(B_{n})-\tau_{n}(j_{n}(B_{n})(A_{n})).$

Inside our Banach spaces, the second term

$J_{n}(B_{n}):=\tau_{n}(j_{n}(B_{n})(A_{n}))$

can be estimated as follows. The map $J_{n}$ is obtained as composition

$E_{n}^{c}\stackrel{{\scriptstyle
j_{n}}}{{\longrightarrow}}\Theta^{c}_{n+1/4,n+1/2}\stackrel{{\scriptstyle
ev_{n}}}{{\longrightarrow}}E_{n+1/2}^{c}\stackrel{{\scriptstyle\tau_{n}}}{{\longrightarrow}}E_{n+1}^{c},$

where the evaluation map $ev_{n}$ is defined by

$ev_{n}(v)=v(A_{n}).$

Note that by definition of the norm in $\Theta^{c}_{n+1/4,n+1/2}$, we have:

$|v(A_{n})|\leq\frac{e|A_{n}|}{a_{n}(s_{n+1/4}-s_{n+1/2})}.$

Consequently:

$\displaystyle(2)$
$\displaystyle\qquad|J_{n}|\leq|j_{n}|\,|\tau_{n}|\frac{e|A_{n}|}{a_{n}(s_{n+1/4}-s_{n+1/2})}\leq\frac{R}{2},$
$\displaystyle(3)$
$\displaystyle\qquad|S_{n+1}|\leq(|\tau_{n}|+|J_{n}|)|B_{n}|\leq R|B_{n}|.$

This proves the first step of the induction.

We can now form

$A_{n+1}=A_{n}+S_{n+1}\in E_{n+1}^{c},$

and set as usual:

$v_{n}:=j_{n}(B_{n}).$

Now:

$(4b)\qquad|v_{n}|\leq|j_{n}||B_{n}|\leq\frac{a_{n}(s_{n+1/2}-s_{n+1})}{8}<\frac{a_{n}(s_{n+1/2}-s_{n+1})}{2},$

and therefore by the Borel lemma (Lemma 5.4), the linear map

$e^{-v_{n}}:E_{n+1/2}^{c}\longrightarrow E_{n+1}^{c}$

is well-defined and, as $e^{-z}$ is the Borel transform of $1/(1+z)$, we also
get that:

$|e^{-v_{n}}|\leq\frac{1}{1-1/2}=2.$

So we can form $F_{n+1}=e^{-v_{n}}F_{n}\in E_{n+1}^{c}$ and so indeed

$B_{n+1}=F_{n+1}-A_{n+1}\in E_{n+1}^{c}.$

Let us now prove the announced central estimate of $|B_{n+1}|$. The power-
series

$e^{-z}(1+z)-1\in\mathbb{C}\\{z\\}$

is the Borel transform of

$-\frac{z^{2}}{(1+z)^{2}}\in z^{2}\mathbb{C}\\{z\\},$

which has radius of convergence equal to $1$ and, choosing $r\leq 1/2$, we get
that:

$\left|\frac{1}{(1+z)^{2}}\right|\leq\frac{1}{(1-r)^{2}}\leq 4.$

We apply the Borel lemma (Lemma 5.4) and obtain:

$(4)\qquad|\phi(j_{n}B_{n})|\leq
4\left(\frac{|j_{n}|\,|B_{n}|}{a_{n}(s_{n+1/2}-s_{n+1})}\right)^{2}\leq\frac{R^{2}|B_{n}|^{2}}{4}.$

Similarly the series $\psi(z)$ is the Borel transform of $z/(1-z)$ and, as

$|S_{n+1}|<|B_{n}|\text{ and }\frac{1}{1-1/2}=2,$

the Borel estimate gives

$(4)\qquad|\psi(j_{n}(B_{n}))S_{n+1}|\leq
2\frac{|j_{n}|\,|B_{n}|\,|S_{n+1}|}{a_{n}(s_{n+1/2}-s_{n+1})}\leq\frac{R^{2}|B_{n}|^{2}}{4}.$

Finally we look at the remainder term
$e^{-v_{n}}(\sigma_{n}(B_{n}-j_{n}(B_{n})(A_{n}))$:

$\displaystyle(5)$ $\displaystyle\qquad|e^{-v_{n}}\sigma_{n}(B_{n})|\leq
2|\sigma_{n}||B_{n}|\leq\frac{1}{4}e^{-\alpha^{n}}R^{2}|B_{n}|,$
$\displaystyle(6)$
$\displaystyle\qquad|e^{-v_{n}}(\sigma_{n}(j_{n}(B_{n})(A_{n}))\leq
2|\sigma_{n}|\,|j_{n}|\,|A_{n}|\,|B_{n}|\leq\frac{R^{2}e^{-\alpha^{n}}}{4}|B_{n}|.$

This proves the second step of the induction

$|B_{n+1}|\leq\frac{R^{2}}{2}|B_{n}|^{2}+\frac{R^{2}}{2}e^{-\alpha^{n}}|B_{n}|.$

The ’norm map’:

$E^{c}:=\bigsqcup_{n}E_{n}^{c}\longrightarrow\mathbb{R}_{+},\
\eta\mapsto|\eta|$

reduces the issue of convergence to the analysis of some elementary iterations
of positive numbers.

So let us put $x_{n}:=|B_{n}|$ so that

$x_{n+1}\leq\frac{R^{2}}{2}x_{n}^{2}+\frac{R^{2}}{2}e^{-\alpha^{n}}x_{n},$

where we have $x_{0}\leq R^{-2}e^{-1/(2-\alpha)}$.

We show that $(x_{n})$ converges quadratically to zero. To see this, consider
the real sequence $(y_{n})$ defined by

$y_{0}=R^{-2}e^{-1/(2-\alpha)},\
y_{n+1}=\frac{R^{2}}{2}\left(e^{\alpha^{n}}y_{n}^{2}+e^{-\alpha^{n}}y_{n}\right),$

which clearly majorates the sequence $(x_{n})$. It follows with an easy
induction that one has the inequality

$y_{n}\geq e^{-2\alpha^{n}}.$

Indeed, assuming the truth for $y_{n}$, we get

$y_{n+1}\geq\frac{R^{2}}{2}\left(e^{-3\alpha^{n}}+e^{-3\alpha^{n}}\right)=R^{2}e^{-3\alpha^{n}}\geq
e^{-2\alpha^{n+1}},$

as $3/2<\alpha<2$ and $R>1$. The inequality can also be written as:

$e^{-\alpha^{n}}y_{n}<e^{\alpha^{n}}y_{n}^{2},$

and therefore:

$y_{n+1}=\frac{R^{2}}{2}\left(e^{\alpha^{n}}y_{n}^{2}+e^{-\alpha^{n}}y_{n}\right)<\frac{R^{2}}{2}\left(e^{\alpha^{n}}y_{n}^{2}+e^{\alpha^{n}}y_{n}^{2}\right)=R^{2}e^{\alpha^{n}}y_{n}^{2}.$

This shows that the sequence

$z_{0}=R^{-2}e^{-1/(2-\alpha)},\ z_{n+1}=R^{2}e^{\alpha^{n}}z_{n}^{2}$

majorates both $(y_{n})$ and $(x_{n})$. This sequence is easily integrated

$z_{n}=R^{2^{n+1}-2}e^{\beta_{n}}z_{0}^{2^{n}}.$

with

$\beta_{n}:=2^{n-1}\sum_{k=0}^{n-1}\left(\frac{\alpha}{2}\right)^{k}=2^{n-1}\frac{1-(\alpha/2)^{n}}{1-(\alpha/2)}\sim
2^{n}\frac{1}{2-\alpha}.$

Writing $(z_{n})$ in the form

$z_{n}=R^{-2}(R^{2}e^{\gamma_{n}}z_{0})^{2^{n}},\
\gamma_{n}:=\frac{1-(\alpha/2)^{n}}{2-\alpha}.$

The sequence $\gamma_{n}$ increases and is bounded by $1/(2-\alpha)$ thus:

$(c)\qquad R^{2}e^{\gamma_{n}}z_{0}\leq R^{2}e^{1/(2-\alpha)}z_{0}<1,$

thus the sequence $(z_{n})$ is decreasing and converges quadratically to zero.

This shows the quadratic convergence of $(B_{n})$ and concludes the proof of
(b). The quadratic convergence for $v_{n}=j_{n}(B_{n})$ follows from this and
the Composition Lemma (Lemma 5.3) implies (c). ∎

###  3.8. Regularity of the normal form

We formulated the iteration scheme in terms of the sequence of Banach spaces
$E_{n}^{c}$. Without much difficulty one can formulate a version of the
iteration in the spaces $E^{k}_{n}$. Without going into all details, we state
the following

######  Proposition 3.21 ([14]).

Under the assumptions of Theorem 3.20, the iteration is well-defined in
$E_{n}^{k}$ and the sequences $(B_{n}^{k})$ and $(v_{n}^{k})$ converge
quadratically to zero. Furthermore, the Poisson morphism $\Phi_{\infty}$ maps
$E_{0}^{k}\subset E_{0}^{c}$ to $E_{\infty}^{k}\subset E_{0}^{c}$.

###### Proof.

This is a direct consequence of Pöschel’s regularity lemma (5.8). Indeed, the
norms $\nu=(\nu_{n})$ of the maps

$E_{n}^{c}\longrightarrow E_{n+1}^{k}$

are bounded by a positive Bruno sequence. But the sequences of the iteration
converge to zero quadratically, so the multiplication by $\nu$ has no effect
on the convergence. ∎

We remark that the Local Equivalence Lemma (Lemma 5.2 ) implies that we can
also formulate and prove quadratic convergence for the iteration in
$(E_{n}^{h})$.

##  4\. Application to invariant tori

We describe now an application of our theorem to the analysis of invariant
tori near elliptic critical points of analytic Hamiltonians, whose frequency
satisfies a Bruno condition. For this we have to consider the appropriate real
form of $H$ and restrict to the real domain.

###  4.1. Hyperbolic and Elliptic fixed points

The dynamics of the harmonic oscillator

$H_{e}=\frac{1}{2}\sum_{i=1}^{d}\beta_{i}(p_{i}^{2}+q_{i}^{2})$

describes quasi-periodic motions with frequency vector $\beta$. All orbits are
bounded and the phase space is filled out by a $d$-parameter family of
invariant tori $p_{i}^{2}+q_{i}^{2}=t_{i}$, on which the solutions spiral
around. The geometry of the situation is well-known: the fibres of the map

$\mathbb{R}^{2d}\longrightarrow\mathbb{R}_{>0}^{d},\;\;\;(q,p)\mapsto
p^{2}+q^{2}:=(p_{1}^{2}+q_{1}^{2},\ldots,p_{d}^{2}+q_{d}^{2})$

are tori, which are of real dimension $d$ over the strictly positive orthant
$\mathbb{R}_{>0}^{d}$.

In the real domain there is a big difference in the dynamical behaviour
between $H_{e}$ and its hyperbolic cousin

$H_{h}=\sum_{i=1}^{d}\alpha_{i}p_{i}q_{i},$

for which all orbits are unbounded and there exist no invariant tori.

Yet when considered over $\mathbb{C}$, the canonical coordinate transformation
$\phi$

$p_{j}\mapsto\frac{1}{\sqrt{2}}(p_{j}+iq_{i}),\;\;\;q_{j}\mapsto\frac{1}{\sqrt{2}}(q_{j}+ip_{j})$

maps $H_{h}$ to $H_{e}$, when we put

$\beta=i\alpha.$

Another way of expressing the relation between $H_{h}$ and $H_{e}$ is by
saying that the evolution for $H_{h}$ in purely imaginary time is equivalent
to the real time evolution of $H_{e}$ and vice versa. As a consequence of this
relation, we can immediately translate results about $H_{h}$ into results
about $H_{e}$.

###  4.2. The coordinate transformation

Consider an analytic hamiltonian of the form

$H=\frac{1}{2}\sum_{i=1}^{d}\alpha_{i}(p_{i}^{2}+q_{i}^{2})+O(3)\in\mathbb{R}\\{p,q\\}.$

and assume that the frequency vector $\alpha\in{\mathcal{R}}(a)_{\infty}$,
where $a$ is a Bruno sequence. We can apply our theorem 3.20, so for
appropriate choice of the sequence $\rho$ and radius $s_{0}$, we find sets

$W_{0}=Z_{0}\times D_{0}\times
D_{0}^{2},\;\;\;Z_{0}=B(s_{0}),\;\;D_{0}:=D_{s_{0}}^{d}$

and

$W_{\infty}=Z_{\infty}\times D_{\infty}\times
D_{\infty}^{2},\;\;D_{\infty}:=D_{s_{\infty}}^{d},$

such the sequence

$A_{0},\;\;A_{1},\;\;A_{2},\ldots$

which converges in the Banach space ${\mathcal{O}}^{c}(W_{\infty})$ to an
element

$A_{\infty}\in{\mathcal{O}}^{c}(W_{\infty})$

and the sequence

$\Phi_{0}=e^{-v_{0}},\;\;\Phi_{1}=e^{-v_{1}}e^{-v_{0}},\;\ldots,\Phi_{n}=\prod_{i=0}^{n}e^{-v_{k}},$

converges in the operator norm to $\Phi_{\infty}\in
L({\mathcal{O}}^{c}(W_{0}),{\mathcal{O}}^{c}(W_{\infty}))$. This
transformation maps, for any $k$, the subspace ${\mathcal{O}}^{k}(W_{0})$ to
${\mathcal{O}}^{k}(W_{\infty})$ (Proposition 3.21). In particular, if the
closed set $W_{0}$ is chosen inside the holomorphy domain of $A_{0}$, then it
is in particular $C^{\infty}$ on $W_{0}$ and therefore belongs to
${\mathcal{O}}^{k}(W_{0})$ for any $k$. The function $A_{\infty}$ is then
$C^{\infty}$ on $W_{\infty}$ and for fixed $\omega\in Z_{\infty}$ it is
holomorphic.

Of course, in a sense we get a ’half-way theorem’, as we start with a real
Hamiltonian, but obtain a statement about its behaviour in the complexified
domain. But it is clear from the explicit form of the description of the
iteration that, starting from a real Hamiltonian, the algorithm produces real
vector fields $v_{n}$, which exponentiate to real analytic coordinate
transformations $\varphi_{n}=e^{-v_{n}}$, etc. As a consequence the limit
transformation $\Phi_{\infty}$ is ’real’. Furthermore, we remark that it
follows from the construction of the vector fields $v_{n}$ that the
transformation $\varphi_{n}$ maps the subspace
$F_{n}^{c}={\mathcal{O}}^{c}(V_{n})$, ($V_{n}=W_{n}\times D_{n}$) to
$F_{n+1}^{c}$, so that $\Phi_{\infty}$ maps $F_{0}^{c}$ to $F_{\infty}^{c}$
and by the regularity property $F_{0}^{k}$ to $F_{\infty}^{k}$.

The coordinate functions $\omega,\tau,q,p$ can be considered as elements of
the space ${\mathcal{O}}^{c}(W_{0})$ and we write

$\omega^{\prime}=\Phi_{\infty}(\omega),\;\;\tau^{\prime}=\Phi_{\infty}(\tau),\;\;q^{\prime}=\Phi_{\infty}(q),\;\;p^{\prime}=\Phi_{\infty}(p).$

Note that $\tau^{\prime}=\tau$ and

$\omega^{\prime}\in
F_{\infty}^{k}={\mathcal{O}}^{k}(V_{\infty}),\;\;\;\textup{for
all}\;\;k\in\mathbb{N},$

so is independent of $q,p$. These functions define a $C^{\infty}$-map

$\phi^{\prime}:W_{\infty}\longrightarrow
W_{0},\;\;x\mapsto(\omega^{\prime}(x),\tau^{\prime}(x),q^{\prime}(x),p^{\prime}(x)),\
x=(\omega,\tau,q,p)$

and for each $g\in{\mathcal{O}}^{c}(W_{0})$ we have the relation

$g(\phi^{\prime}(x))=\Phi_{\infty}(g)(x).$

The reality of $\Phi_{\infty}$ implies that the map $\phi^{\prime}$ maps the
real part

${\mathcal{W}}_{\infty}:=W_{\infty}\cap\mathbb{R}^{d}\textup{ to
}{\mathcal{W}}_{0}:=W_{0}\cap\mathbb{R}^{d}.$

Thus we obtain a real $C^{\infty}$-map

$\varphi^{\prime}:{\mathcal{W}}_{\infty}\longrightarrow{\mathcal{W}}_{0}.$

As the $\varphi^{\prime}$ sends the $(\omega,\tau)$-space to itself, the map
is fibred over the $(\omega,\tau)$-space and we obtain a commutative diagram:

$\textstyle{{\mathcal{W}}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{\prime}}$$\textstyle{{\mathcal{W}}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{V}}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{\prime}}$$\textstyle{{\mathcal{V}}_{0}}$

with $\psi^{\prime}=(\omega^{\prime},\tau^{\prime})$ and vertical maps in the
diagram forget the coordinates $q,p$.

Recall the following Whitney extension theorem:

######  Theorem 4.1 ([38]).

Let $X\subset\mathbb{R}^{d}$ be a compact subset. Any function $f\in
C^{\infty}(X,\mathbb{R})$ is the restriction of a $C^{\infty}$ function
defined on $\mathbb{R}^{d}$.

Invoking this theorem to the component functions
$\omega^{\prime},\tau^{\prime},q^{\prime},p^{\prime}$ of $\varphi^{\prime}$,
we obtain a $C^{\infty}$-maps

$\psi:\mathbb{R}^{2d}\longrightarrow\mathbb{R}^{2d},\;\;\;\varphi:\mathbb{R}^{4d}\longrightarrow\mathbb{R}^{4d}.$

We restrict $\psi$ and $\varphi$ to the preimages

${\mathcal{V}}_{e}:=\psi^{-1}({\mathcal{V}}_{0})\supset{\mathcal{V}}_{\infty},\;\;\;{\mathcal{W}}_{e}:=\varphi^{-1}({\mathcal{W}}_{0})\supset{\mathcal{W}}_{\infty},$

and we arrive at a diagram

$\textstyle{{\mathcal{W}}_{e}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{{\mathcal{W}}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{V}}_{e}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{{\mathcal{V}}_{0}}$

that extends the previous diagram. Obviously, the maps $\varphi$ and $\psi$
are not unique, but its restrictions to ${\mathcal{W}}_{\infty}$ and
${\mathcal{V}}_{\infty}$ are.

As the Taylor series of $\varphi$ is given by the series $\Phi_{\infty}$,
which is $Id+O(2)$, $\varphi$ is a diffeomorphism near the origin.
Consequently, by restriction to smaller polydiscs, we may and will assume that

  1. (1)

$\psi$ is a diffeomorphism between ${\mathcal{V}}_{e}$ and
${\mathcal{V}}_{0}$,

  2. (2)

$\varphi$ is a diffeomorphism between ${\mathcal{W}}_{e}$ and
${\mathcal{W}}_{0}$.

As the map $\varphi$ arose from the transformation $\Phi_{\infty}$, it has the
property that after restriction to ${\mathcal{W}}_{\infty}$, it transforms

$F_{0}=A_{0}+H=\frac{1}{2}\sum_{i=1}^{d}\omega_{i}(p_{i}^{2}+q_{i}^{2})+H(p,q)$

to $F_{\infty}=A_{\infty}$. Furthermore

$A_{\infty}=A_{0}+T_{\infty},\;\;\;T_{\infty}\in(R_{0}+I^{2})\cap{\mathcal{O}}^{k}(W_{\infty}).$

This means that for $\omega\in Z_{\infty}$ one has

$F_{0}\circ\varphi(\omega,\tau,q,p)=A_{\infty}(\omega,\tau,q,p),$

and moreover, for such a value, the map $\varphi(\omega,-)$ is an analytic
Poisson morphism in the variables $(\tau,q,p)$.

###  4.3. Frequency maps

In this section we return to the complex situation and start to analyse the
limit $n\longrightarrow\infty$ of the frequency manifolds $X_{n}$ that were
considered in section 2. We need a more careful use of the Whitney extension
theorem.

We consider the extension diagram of neighbourhoods of the previous section,
but in the complex setting:

$\textstyle{W_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{e}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{W_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{e}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\textstyle{V_{0}}$

We consider

$X_{n}=\\{(\omega,\tau)\in
V_{n}\;|\;\;R_{n,1}(\omega,\tau)=\dots=R_{n,d}(\omega,\tau)=0\\},$

where $R_{n,i}:=\Phi_{n}(\omega_{i})\in{\mathcal{O}}^{k}(V_{n})$. We will have
to take the degree of differentiability $k\geq 1$. For $n=\infty$ we have the
functions

$R_{\infty,i}=\Phi_{\infty}(\omega_{i})\in
F^{k}_{\infty}={\mathcal{O}}^{k}(V_{\infty}),$

which are analytic in $\tau$ and $C^{k}$ in $\omega$.

We can consider the limit set

$X_{\infty}:=\\{(\omega,\tau)\in
V_{\infty}\;|\;\;R_{\infty,1}(\omega,\tau)=\dots=R_{\infty,d}(\omega,\tau)=0\\}$

and invoke the following theorem due to Fefferman :

######  Theorem 4.2 ([7]).

Let $X\subset\mathbb{C}^{N}$ be a compact subset and $k\in\mathbb{N}$. There
exists a bounded linear operator

$T:C^{k}(X,\mathbb{C})\longrightarrow C^{k}(\mathbb{C}^{N},\mathbb{C}),$

which is right inverse to the restriction mapping.

We consider the inclusion

$V_{\infty}\subset V_{e}$

with $k=1$ and get an extension operator $T$. We consider the functions

$R^{\prime}_{n,i}:=R_{n,i}|V_{\infty},\ n\in\mathbb{N}$

and set

$r_{n,i}:=T(R^{\prime}_{n,i}),\ r_{n}:=(r_{n,1},\dots,\ r_{n,d}).$

As $T$ is a bounded operator, we can conclude the existence of a limit

$r_{\infty}=\lim_{n\longrightarrow\infty}r_{n}$

By the implicit function theorem each manifold

$X_{n,e}:=\\{(\omega,\tau)\in V_{e}:r_{n}(\omega,\tau)=0\\},\
n\in\mathbb{N}\cup\\{\infty\\}$

is, near the origin, the graph of a function

$\\{\omega=f_{n}(\tau)\\}.$

These functions can be constructed using Picard iteration, whose convergence
is controlled by a condition on the derivative $dr_{n}$. Since $(r_{n})$
converge in the $C^{1}$-topology, we may find a common compact neighbourhood
of the origin for all these maps

$f_{n}:U\longrightarrow U^{\prime}.$

So by possible shrinking of $V_{e}$ to the set $U\times
U^{\prime}\subset\mathbb{C}^{d}\times\mathbb{C}^{d}$ we can express the
manifolds $X_{n,e}$ has graphs over a common neighbourhood.

Note that these functions converge pointwise to $f_{\infty}$. Indeed, take
$\tau\in U$, as $U^{\prime}$ is compact $(f_{n}(\tau))$ admits converging
subsequences. Let $\omega$ be a limit value of such a subsequence
$(f_{n_{k}}(\tau))$. Passing to the limit in the equality

$r_{n_{k}}(f_{n_{k}}(\tau),\tau)=0$

we get that the point $(\tau,\omega)$ belongs to the manifold $X_{\infty,e}$
and therefore $\omega=f_{\infty}(\tau)$. We will not use this fact.

The map

$U\longrightarrow\mathbb{C}^{d},\;\;\tau\mapsto\alpha+f_{\infty}(\tau).$

will called a frequency map of our Hamiltonian system.

According to 2.5, the Taylor series of this map at the origin at order $k$
coincides with that of the formal frequency map $b$. Of course, the process of
extension is not unique, but the restriction to the preimage of
${\mathcal{C}}(a)_{\infty}$ is the same for any choice. The construction can
done so that the resulting frequency map restricts to a real map

$\beta:{\mathcal{U}}\longrightarrow\mathbb{R}^{d},\;\;\tau\mapsto\alpha+f_{\infty}(\tau).$

###  4.4. The non-degeneracy condition

We now wish to construct a frequency map whose image lies inside
$\alpha+F(H)$. Later, this condition will guarantee positive measure for the
set of invariant tori. This property depends on our choice of Whitney
extension, but there is an intrinsic underlying statement: Proposition 2.9
stating that all $X_{n}$’s are contained in $F(H)\times\mathbb{C}^{d}$ extends
to $n=\infty$.

######  Proposition 4.3.

The limit set $X_{\infty}$ satisfies the same non-degeneracy condition as the
manifolds $X_{n}$:

$X_{\infty}\subset F(H)\times\mathbb{C}^{d}.$

###### Proof.

Consider a unit vector $n\in\mathbb{C}^{d}$ normal to the frequency space
$F(H)$ induces, via the Euclidean scalar product, a linear form

$u:\mathbb{C}^{2d}\longrightarrow\mathbb{C},x\mapsto(x,n),\ F(H)\subset{\rm
Ker\,}u$

Assume that there exists a point $x_{0}=(\omega_{0},\tau_{0})\in X_{\infty}$
which does not lie inside the frequency space:

$u(x_{0})\neq 0\text{ and }R_{\infty}(x_{0})=0.$

The point $x_{0}=(\omega_{0},\tau_{0})$ remains at distance $K=|u(x_{0})|$
from the hyperplane ${\rm Ker\,}u$. By Proposition 2.9, the manifolds $X_{n}$
are contained inside the frequency space therefore the point $(\omega,\tau)$
remains also at distance $K$ from all the manifolds

$X_{n}=\\{(\omega,\tau)\in V_{e}:R_{n}(\omega,\tau)=0\\}.$

We consider the Whitney extensions:

$X_{n,e}=\\{(\omega,\tau)\in V_{e}:r_{n}(\omega,\tau)=0\\}$

and as both manifolds intersected with ${\mathcal{C}}(a)\times\mathbb{C}^{d}$
are the same the points of

$\left(X_{n,e}\cap\\{\omega=\omega_{0}\\}\right)=\left(X_{n}\cap\\{\omega=\omega_{0}\\}\right)$

are also at distance at least $K$ from the point $x_{0}\in X_{\infty}$.

Now let the parameter $\tau$ vary and define the functions

$h_{n,i}(\tau)=r_{n,i}(\omega_{0},\tau),\ n\in\mathbb{N}\cup\\{+\infty\\}.$

By definition of our functional spaces, as $\omega_{0}\in{\mathcal{C}}(a)$,
these functions are holomorphic in $\tau$ even for $n=+\infty$.

At least one of the functions $h_{\infty,i}$ is not constant for some
$i=1,\dots,d$, otherwise this common constant will necessarily be zero and
$x_{0}$ will be the origin, contradicting the assumption $x_{0}\notin
F(H)\times\mathbb{C}^{d}$.

Now choose a complex line $L\subset\mathbb{C}^{d}$ containing $x_{0}$ along
which the function $h_{\infty,i}$ is not constant. By restriction to this line
we get a sequence of one variable holomorphic functions

$g_{n}:L\cap V_{e}\longrightarrow\mathbb{C},z\mapsto h_{n,i}(z)$

which has the following properties:

  1. 1)

$g_{\infty}(x_{0})=0$.

  2. 2)

$\|x-x_{0}\|<K\implies g_{n}(x)\neq 0$ for any $n\in\mathbb{N}$.

  3. 3)

the sequence $(g_{n})$ converges to $g_{\infty}$ in the $C^{1}$-norm.

Condition 2) 3) are in obvious contradiction with 1). Indeed The number of
zeroes $Z_{n,i}$ counted with multiplicities contained inside the disk
$D\subset L$ centred at $x_{0}$ of radius $K/2$ is given by the integral
formula

$Z_{n,i}=\frac{1}{2i\pi}\int_{\partial
D}\frac{g_{n,i}^{\prime}(z)}{g_{n,i}(z)}dz$

for any $n\in\mathbb{N}\cup\\{\infty\\}$. We know that for $n\in\mathbb{N}$,
$Z_{n,i}=0$, thus passing to the limit gives $Z_{\infty,i}=0$. This
contradicts 1) and concludes the proof of the proposition.∎

As we know that $X_{\infty}$ is contained inside $F(H)\times\mathbb{C}^{d}$,
we can improve our previous construction of a frequency map. First we restrict
the functions $R_{\infty,i}$ to

$V_{\infty}\cap\left(F(H)\times\mathbb{C}^{d}\right)$

Now we use Whitney $C^{\infty}$-extension to and obtain functions

$s_{\infty,1},\dots,s_{\infty,d}\in C^{\infty}(\mathbb{C}^{2d},\mathbb{C})$

whose common zero set, when restricted to $F(H)\times U$, can be written as a
graph of a map:

$b:U\longrightarrow\alpha+F(H)\subset\mathbb{C}^{d},$

and similarly in the real case

$\beta:{\mathcal{U}}\longrightarrow\alpha+{\mathcal{F}}(H)\subset\mathbb{R}^{d}.$

In this way, we construct frequency maps $b$ satisfy the non degeneracy
condition which will be needed to prove positive measure of invariant tori:
the partial derivatives of the frequency map evaluated at the origin generate
the frequency space. Moreover as the condition is open, we may assume, up to a
possible shrinking of ${\mathcal{U}}$, that the condition holds not only at
the origin, but at any point of ${\mathcal{U}}$.

###  4.5. The elliptic normal form Theorem

Using the coordinate transformation and the frequency map, we may sum up our
results in the following way:

######  Theorem 4.4 ([11, 12]).

Let $a=(a_{n})$ be a sequence satisfying the Bruno condition and
$\alpha\in{\mathcal{R}}(a)_{\infty}$. Let $H\in\mathbb{R}\\{q,p\\}$ be a real
analytic function with an elliptic fixed point:

$H=\frac{1}{2}\sum_{i=1}^{d}\alpha_{i}(p_{i}^{2}+q_{i}^{2})+O(3).$

Then there exists an open neighbourhood of the origin
$U\subset\mathbb{C}^{d}$, $V\subset\mathbb{C}^{2d}$ with real parts
${\mathcal{U}},{\mathcal{V}}$ and $C^{\infty}$-maps

$\beta:{\mathcal{U}}\longrightarrow\alpha+{\mathcal{F}}(H)\subset\mathbb{R}^{d},$
$\textstyle{{\mathcal{U}}\times{\mathcal{V}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\textstyle{{\mathcal{U}}\times\mathbb{R}^{2d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{U}}$

such that for any $\tau\in\beta^{-1}({\mathcal{R}}(a)_{\infty})$, one has:

  1. i)

The Taylor series expansion of $\beta$ at the origin is equal to $\nabla
B(H)$.

  2. ii)

The map $\Psi$ is a fibred diffeomorphism over its image.

  3. iii)

The map $\Psi(\tau,-)$ is an analytic symplectomorphism.

  4. iv)

$H\circ\Psi(\tau,q,p)=\frac{1}{2}\sum_{i=1}^{n}\beta_{i}(\tau)(p_{i}^{2}+q_{i}^{2})+T_{\infty}(\tau,q,p)$

  5. v)

$T_{\infty}(\tau,-)\in I^{2}+\mathbb{C}$, where $I\subset{\mathcal{O}}^{c}(V)$
is the ideal generated by the $p_{i}^{2}+q_{i}^{2}-\tau_{i}$’s.

The map $\Psi$ of the theorem is defined in terms of the map $\varphi$ and the
map $\tau\mapsto\omega(\tau)$ of the previous section by the relation

$\Psi(\tau,q,p)=\varphi(\omega(\tau),\tau,q,p).$

We note that in the extremal case where ${\mathcal{F}}(H)=\\{0\\}$, the
frequency map $\beta$ is constant, and the condition
$\beta(\tau)\in{\mathcal{R}}(a)_{\infty}$ is always satisfied. In this case
the map $\Psi$ is therefore analytic, because $\varphi$ is analytic in the
$\tau$-variables. So the theorem implies that $H$ is integrable, and thus we
recover a classical result of Rüßmann [32].

In the general case, our iteration produces a $C^{\infty}$ function $\beta$,
whose Taylor expansion at the origin is the formal frequency map given by the
Birkhoff normal form. In a similar way, our construction shows that the
sequence $(h_{n})$ of 2.5 converges to a limit $h_{\infty}$. This limit
function being the constant term in the expression

$T_{\infty}(\tau,q,p)=h_{\infty}(\tau)+\sum_{i,j}t_{ij}(q,p)f_{i}f_{j}$

with $\tau\in\beta^{-1}({\mathcal{R}}(a)_{\infty})$.

The Taylor expansion of $h_{\infty}$ at the origin is the Birkhoff normal form
and the map $\beta$ can then be chosen to be the gradient of $h_{\infty}$. In
analogy with Pöschel terminology [30], we might say that the situation is
similar to that of the Birkhoff normal form but over a Cantor set. But we will
not use this fact in the sequel.

Eliasson posed the question whether the frequency map $\beta$ is analytic or
not [4, 29]. We do not have an answer, but we remark that the frequency map
$\beta$ is constructed out of the frequency manifolds $X_{n}$ and more
precisely by the Malgrange-Mather division theorem [22, 23]:

$R_{\infty}(\omega,\tau)=A(\omega,\tau)(\omega-\beta(\tau)).$

We only know that the function $R_{\infty}$ is analytic in $\tau$ and
Eliasson’s question concerns the map $\beta$. So the mystery remains…

###  4.6. A big set of invariant tori

A direct corollary of the elliptic normal form theorem is the following.

######  Corollary 4.5.

For $\tau\in\beta^{-1}({\mathcal{R}}(a)_{\infty})$, the image under
$\Psi(\tau,-)$ of the torus

$T_{\tau}:p_{1}^{2}+q_{1}^{2}=\tau_{1},\dots,p_{1}^{2}+q_{1}^{2}=\tau_{n}$

is invariant under the Hamiltonian flow of $H$. The motion on this torus is
quasi-periodic with frequency $\beta(\tau)$.

So we get a collection of invariant tori in our hamiltonian system,
parametrised by the inverse image of ${\mathcal{R}}(a)_{\infty}$ by the
frequency map

$\beta:{\mathcal{U}}\longrightarrow\alpha+{\mathcal{F}}(H)\subset\mathbb{R}^{d}.$

As pointed out in remark 3.7, we may and will suppose that $a$ is chosen so
that the set ${\mathcal{R}}(a)_{\infty}$ is $\alpha$-dense. But without
further precautions, the inverse image under $\beta$ might still be a very
small set, maybe reduced to an half line. We know however that our frequency
map $\beta$ is non-degenerate in the frequency space
$\alpha+{\mathcal{F}}(H)$. The following arithmetic density theorem then can
be used to control the density of the inverse image of
${\mathcal{R}}(a)_{\infty}$ with $n=d$.

######  Theorem 4.6 ([15]).

Consider a real positive decreasing sequence $\sigma=(\sigma_{k})$ and let
$\nu=(\nu_{k})$ be a real positive sequence such that the sequence

$(2^{kn}\nu_{k}^{1/dl})$

is summable and $\nu_{k}<1$ for all $k$’s. Consider a mapping

$f=(f_{1},\dots,f_{d}):\mathbb{R}^{d}\supset U\longrightarrow\mathbb{R}^{n}$

such that $f(U)$ is contained in an affine space spanned by the partial
derivatives of $f$ up to order $l$. Then the density of the set
$f^{-1}({\mathcal{R}}(\nu\sigma)_{\infty})$ at the origin is equal to $1$.

So the logic of our argument is the following: we fix a vector
$\alpha\in\mathbb{R}^{d}$ and assume that
$\sigma(\alpha)\in{\mathcal{B}}^{-}$. We consider the sequences

$\nu_{k}=(2^{-(k+1)d^{2}l}),\ a=\nu\sigma(\alpha).$

Note that

$\nu\in{\mathcal{B}}^{-}\text{ and }\sigma(\alpha)\in{\mathcal{B}}^{-}\implies
a=v\sigma(\alpha)\in{\mathcal{B}}^{-}.$

Now the elliptic normal form theorem 4.4 applies. By Proposition 4.3, the
frequency mapping $\beta$ we construct satisfies the assumption of the
arithmetic density theorem. Consequently the set
$\beta^{-1}({\mathcal{R}}(a)_{\infty})$ has density one at the origin and is,
in particular, a set of positive measure.

We will now see how these tori fit together in a neighbourhood of the origin
of our original Hamiltonian $H(p,q)$. The map

$\Psi:{\mathcal{U}}\times{\mathcal{V}}\longrightarrow{\mathcal{U}}\times\mathbb{R}^{2d}$

of the previous theorem has an inverse $\Gamma=\Psi^{-1}$ over a sufficiently
small neighbourhood of the origin of the form
${\mathcal{U}}\times{\mathcal{B}}$, ${\mathcal{B}}\subset\mathbb{R}^{2d}$:

$\Gamma:{\mathcal{U}}\times{\mathcal{B}}\longrightarrow{\mathcal{U}}\times{\mathcal{V}};(\tau,p,q)\mapsto(\tau,P(\tau,p,q),Q(\tau,p,q))$

So we have the relation

$H(p,q)=A_{0}(\omega(\tau),P(\tau,p,q),Q(\tau,p,q))+T_{\infty}\circ\Gamma(\tau,p,q).$

We can, in principle, eliminate the variables
$\tau_{1},\tau_{2},\ldots,\tau_{d}$ from the right hand side by solving the
implicit equations

$P_{i}(\tau,p,q)^{2}+Q_{i}(\tau,p,q)^{2}=\tau_{i},\;\;i=1,2,\ldots,d,$

which produces a map

$T:{\mathcal{B}}\longrightarrow{\mathcal{U}};\;\;(p,q)\mapsto(\tau_{1}(p,q),\ldots,\tau_{d}(p,q)).$

As one has

$\tau_{i}(p,q)=p_{i}^{2}+q_{i}^{2}+O(3),$

the map $T$ is generically a submersion. In fact, it is a submersion on
${\mathcal{B}}\setminus C$, $C:=T^{-1}(\Delta)$, where
$\Delta\subset{\mathcal{U}}$ is the set of critical values of $T$.

One now obtains a diagram

$\textstyle{{\mathcal{B}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\scriptstyle{\gamma}$$\textstyle{(\alpha+{\mathcal{F}}(H))\times{\mathcal{V}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{{\mathcal{U}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{\alpha+{\mathcal{F}}(H)}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\alpha+{\mathcal{F}}(H))\cap{\mathcal{R}}(a)_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

related to our normal forms as follows.

On the right hand side we have the standard Hamiltonian

$A_{0}=\frac{1}{2}\sum_{i=1}^{d}(\alpha_{i}+\omega_{i})(p_{i}^{2}+q_{i}^{2}),$

defined on $(\alpha+{\mathcal{F}}(H))\times{\mathcal{V}}$, where the map

$\pi:(\omega,q,p)\mapsto\alpha+\omega$

gives the frequency of motion.

On the left hand side we have a neighbourhood ${\mathcal{B}}$, on which the
original Hamiltonian $H(p,q)$ is defined. The vertical map on the left is the
$\tau$-map $T(p,q)$ defined above.

The horizontal map $\gamma$ stems from coordinate transformation $\Gamma$:

$\displaystyle\gamma:{\mathcal{B}}$
$\displaystyle\longrightarrow(\alpha+{\mathcal{F}}(H))\times{\mathcal{V}}$
$\displaystyle(q,p)$
$\displaystyle\mapsto(\beta(T(p,q)),P(T(p,q),p,q),Q(T(p,q),p,q))$

The horizontal map in the middle is the frequency map $\beta$, which is non-
degenerate in the affine sub-space $\alpha+{\mathcal{F}}(H)$. The inverse
image $S:=\beta^{-1}({\mathcal{R}}(a)_{\infty})$ under $\beta$ parametrises
invariant tori for $H$ in the neighbourhood ${\mathcal{B}}$. As $T$ is a
submersion outside $\Delta$, which by Sard’s theorem has measure zero, the set
$T^{-1}(S\setminus\Delta)\subset{\mathcal{B}}$ yields a set of positive
measure consisting of invariant tori in the neighbourhood ${\mathcal{B}}$ of
elliptic critical point, as conjectured by Herman [18].

##  5\. Appendix: Lemmas from the hall of fame

In this appendix we collect some fundamental lemmas of great use in the type
of analysis we are pursuing here. We include proofs, as these are usually
elementary.

Given two open sets $V\subset U\subset\mathbb{C}^{n}$, we denote by $r=d(U,V)$
the supremum of the real numbers $\rho$ for which

$V+\rho D\subset U.$

where $D$ denotes the unit polydisc.

###  5.1. Cauchy-Nagumo lemma

Let $U$ be an open set in $\mathbb{C}^{n}$ and $V\subset U$ with $d(U,V)=r>0.$
For a differential operator

$P=\sum_{|J|\leq k}a_{J}\partial^{J}\in
L({\mathcal{O}}^{c}(U),{\mathcal{O}}^{c}(V))$

of order $k$ we have:

$\|P\|\leq C\frac{k!}{r^{k}}$

where $C=\sup_{|J|\leq a_{k}}|a_{J}|$.

###### Proof.

If $z\in V$ and $f\in{\mathcal{O}}^{c}(U)$, then one has

$f(z)=\frac{1}{(2\pi
i)^{d}}\int_{\gamma_{z}}\frac{f(\xi)}{\prod_{i=1}^{d}(\xi_{i}-z_{i})}d\xi_{1}\wedge\dots\wedge
d\xi_{d},$

where $\gamma_{z}$ denotes the cycle defined by $|\xi_{i}-z_{i}|=r$. We write
it symbolically as

$f(z)=\frac{1}{(2\pi i)^{d}}\int_{\gamma_{z}}\frac{f(\xi)}{(\xi-z)}d\xi,$

Differentiation under the integral sign leads to

$\partial^{I}f(z)=\frac{I!}{(2\pi
i)^{d}}\int_{\gamma_{z}}\frac{f(\xi)}{(\xi-z)^{1+I}}d\xi.$

We parametrise $\gamma_{z}$ by:

$\theta\mapsto\xi(\theta):=z+re^{\langle 2\pi i,\theta\rangle}$

and thus

$d\xi=(2\pi ir)^{d}e^{\langle 2\pi i,\theta\rangle}d\theta,$

so that

$\partial^{I}f(z)=\frac{I!}{r^{|I|}}\int_{0}^{1}\frac{f(\xi(\theta))}{e^{2\pi
i\theta}}d\theta,$

so finally

$|\partial^{I}f|\leq\frac{I!}{r^{|I|}}|f|.$

From this the lemma follows. ∎

For the special case of Hamiltonian derivations, we deduce that the norm of
the map

${\mathcal{O}}^{c}(U)\longrightarrow
L({\mathcal{O}}^{c}(U),{\mathcal{O}}^{c}(V)),\ h\mapsto\\{h,-\\}$

is bounded by $d/r^{2}$.

###  5.2. Local equivalence lemma

Let $V\subset U\subset\mathbb{C}^{d}$ be such that $d(V,U)=r>0.$ Then the
restriction mapping

$\rho^{hc}:{\mathcal{O}}^{h}(U)\longrightarrow{\mathcal{O}}^{c}(V)$

has norm smaller than $\pi^{-d/2}r^{-d}$ Furthermore, the canonical map

$\rho^{ch}:{\mathcal{O}}^{c}(U)\longrightarrow{\mathcal{O}}^{h}(U)$

has its norm bounded by ${\rm Vol}(U)$.

###### Proof.

Let $f\in{\mathcal{O}}^{h}(U)$ and $w\in V$. The Taylor expansion of $f$ at a
point $w$

$f(z)=\sum_{J\in\mathbb{N}^{d}}a_{J}(z-w)^{J},\ a_{J}\in\mathbb{C}.$

The polydisc $D_{w}$ centred at $w$ with radius $r$ is contained in $U$. We
have

$\int_{D_{w}}|f|^{2}=\sum_{J\in\mathbb{N}^{d}}C(J)|a_{J}|^{2}r^{2|J|+2n},\;\;\;C(I)=\prod_{k=1}^{d}\frac{\pi}{j_{k}+1}.$

So we obtain

$C(0)|a_{0}|^{2}r^{2n}\leq\int_{D_{w}}|f|^{2}\leq\int_{U}|f|^{2}=|f|^{2}.$

This shows that

$|f(w)|=|a_{0}|\leq\frac{c}{r^{d}}|f|,\;\;c:=\sqrt{\frac{1}{C(0)}}=\pi^{-d/2}.$

As the point $w\in V$ was general the result follows. ∎

###  5.3. Composition lemma

Consider the family of polydiscs
$U_{t}=\\{z\in\mathbb{C}^{n}\;|\;|z_{i}|<t\\}$ and the Banach spaces
${\mathcal{O}}^{c}(U_{t})$. Often linear operators require an arbitrary small
shrinking of domain and we end up with linear maps

$u(t,s)\in L({\mathcal{O}}^{c}(U_{t}),{\mathcal{O}}^{c}(U_{s}))$

defined only for $s<t$. If such a collection $u=(u(t,s),t>s)$ of linear
operators is compatible with the restrictions
${\mathcal{O}}^{c}(U_{t^{\prime}})\longrightarrow{\mathcal{O}}^{c}(U_{t})$
($t^{\prime}\geq t$) and
${\mathcal{O}}^{c}(U_{s})\longrightarrow{\mathcal{O}}^{c}(U_{s^{\prime}})$
($s\geq s^{\prime}$), we say that $u$ is a horizontal section. The composition
lemma says:

The composition of horizontal sections $u$ and $v$ defined by

$uv(t,s)=u(t,\frac{t+s}{2})v(\frac{t+s}{2},s)$

is again horizontal and satisfies

$\|uv(t,s)\|\leq\|u(t,\frac{t+s}{2})\|\,\|v(\frac{t+s}{2},s)\|$

The proof is immediate. Often one has estimates for $u(t,s)$ and $v(t,s)$
involving only the difference $(t-s)$. The fact that we take the midpoint then
leads to powers of $2$ in the estimate of the composition. However, to define
the composition one could take any point between $t$ and $s$. For example, the
the $m$-fold composition of horizontal sections $u_{1},u_{2},\ldots,u_{m}$
could be defined by repeated composition of two factors, but is more
conveniently defined as

$u_{1}u_{2}\cdots u_{m}(t,s):=u_{1}\left(t,\frac{(m-1)t+s}{m}\right)\cdots
u_{m}\left(\frac{t+(m-1)s}{m},s\right),$

which often leads to powers of $m$ in estimates, as can be observed for
example in the next lemma.

###  5.4. Borel lemma

If for $t>s$ we have linear operators $u(t,s)\in
L({\mathcal{O}}^{c}(U_{t}),{\mathcal{O}}^{c}(U_{s}))$ that define a horizontal
section $u=(u(t,s),t>s)$ in the sense described above. The sections for which
the quantity

$\|u\|=\sup_{s<q<p<t}\\{(p-q)|u(x)|_{q}/(e|x|_{p})\\}$

is well-defined and finite are called $1$-local. They form a Banach space. The
normalising constant $e\approx 2.718$ is purely conventional.

By composition we can form horizontal sections $u^{2},u^{3},u^{4},\ldots$. So
one may try to develop a functional calculus for $1$-local operators. For this
to work one has to invoke a Borel transform.

If $f=\sum_{n\geq 0}a_{n}z^{n}\in\mathbb{C}\\{z\\}$ is an analytic series, we
define its Borel transform as

${\mathcal{B}}f:=\sum_{n\geq 0}\frac{a_{n}}{n!}z^{n},$

and absolute value as

$|f|=\sum_{n\geq 0}|a_{n}|z^{n}.$

Let $f=\sum_{n\geq 0}a_{n}z^{n}\in\mathbb{C}\\{z\\}$ a power series with $R$
as radius of convergence and $u=(u(t,s),t>s)$ a $1$-local horizontal section.

If $\|u\|<R(t-s)$, then the series ${\mathcal{B}}f(u)(t,s)$ converges in the
operator norm and one has the estimate

$\|{\mathcal{B}}f(u(t,s))\|\leq|f|\left(\frac{\|u\|}{t-s}\right).$

###### Proof.

From the composition lemma we find

$\|u^{n}(t,s)\|\leq{\|u\|^{n}e^{-n}n^{n}}{(t-s)^{n}}\leq{(\|u\|^{n}n!}{(t-s)^{n}},$

where we used the standard inequality $n^{n}\leq e^{n}n!$ and therefore

$\|\sum_{n\geq 0}\frac{a_{n}}{n!}u^{n}(t,s)\|\leq\sum_{n\geq
0}|a_{n}|\left(\frac{\|u\|}{(t-s)}\right)^{n}.$

This proves the lemma. ∎

###  5.5. Product lemma

Let $E_{n}$ be an increasing Banach scale and $(t:=t_{0},t_{1},t_{2},\ldots)$
a decreasing sequence converging to $s>0$. For any sequence $(u_{n})$ of
$1$-local operators $u_{n}\in L(E_{n},E_{n+1})$, such that

  1. i)

$\|u_{n}\|<t_{n}-t_{n+1},$

  2. ii)

$\sigma:=\sum_{n\geq 0}||u_{n}||/(t_{n}-t_{n+1})<+\infty,$

the sequence

$g_{0}=e^{u_{0}},\;\;g_{1}=e^{u_{1}}e^{u_{0}},$
$g_{n}:=e^{u_{n}}e^{u_{n-1}}\cdots e^{u_{0}}$

converges to an element $g$ in the Banach space $L(E_{t_{0}},E_{s})$ (with
operator norm). Furthermore, we have the estimate:

$|g|<\frac{1}{1-\sigma/(t-s)}.$

###### Proof.

We have seen that $e^{u_{i}}$ exists as a section and thus defines elements of
the Banach space $L(E_{t_{i}},E_{t_{i+1}})$ as long as $\|u_{i}\|\leq
t_{i}-t_{i+1}$, which holds by the first assumption. As a consequence the
compositions

$e^{u_{0}},\;\;e^{u_{1}}e^{u_{0}},\;\;\ldots,e^{u_{n}}\ldots
e^{u_{1}}e^{u_{0}},\ldots$

are well defined. Furthermore, the Borel estimate gives

$|e^{u_{i}}|\leq\frac{1}{1-\nu_{i}},\ \nu_{i}:=\|u_{i}\|/(t_{i}-t_{i+1}).$

As

$\frac{1}{1-x}\times\frac{1}{1-y}<\frac{1}{1-(x+y)}$

for $x,y\in]0,1[$, we get for the composition $e^{u_{i+1}}e^{u_{i}}$

$|e^{u_{i+1}}e^{u_{i}}|\leq\frac{1}{1-(\nu_{i}+\nu_{i+1})}.$

By a straighforward induction (and the fact that restrictions have norm $\leq
1$), we obtain the estimate

$|g_{n}|\leq\frac{1}{1-(\sum_{i=0}^{n}\nu_{i})}.$

Therefore

$|g_{n+1}-g_{n}|\leq\frac{|e^{u_{n+1}}-{\rm
Id\,}|}{1-(\sum_{i=0}^{n}\nu_{i})}$

Using again the Borel estimate

$|e^{u_{n+1}}-1|\leq\frac{\nu_{n+1}}{1-\nu_{n+1}}$

we get

$|g_{n+1}-g_{n}|\leq\frac{\nu_{n+1}}{1-(\sum_{i=0}^{n+1}\nu_{i})}.$

From this it follows that the sequence $g_{n}$ converges in the Banach space
$L(E_{t},E_{s})$ with operator norm. ∎

###  5.6. Arnold-Moser lemma

Let, as before, $U_{t}$ denote the open polydisc of radius $t$ and let

$\rho(t,s):{\mathcal{O}}^{h}(U_{t})\longrightarrow{\mathcal{O}}^{h}(U_{s})$

be the restriction mappings.
The following simple result appears in [1] and [26] and is of great use:

Let $f\in{\mathcal{O}}^{h}(U_{t})$ be such that its Taylor series expansions
starts at order $N$:

$f(z):=\sum_{|I|\geq N}a_{I}z^{I}.$

then:

$|\rho(t,s)f|\leq\left(\frac{s}{t}\right)^{d+N}|f|.$

###### Proof.

The monomials $z^{I}$ form an orthogonal basis of ${\mathcal{O}}^{h}(U_{s})$
with norms

$|z^{I}|=C(I)^{1/2}s^{d+|I|},\
C(I):=\frac{\pi^{d}}{\prod_{k=1}^{d}(1+i_{k})}.$

By the Pythagorean theorem, for $f\in{\mathcal{O}}^{h}(U_{t})$, we have:

$\displaystyle|\rho(t,s)f|^{2}$ $\displaystyle=\sum_{|I|\geq
N}|a_{I}|^{2}C(I)s^{2d+2|I|}$ $\displaystyle=\sum_{|I|\geq
N}|a_{I}|^{2}C(I)\frac{s^{2d+2|I|}}{t^{2d+2|I|}}t^{2d+2|I|}$
$\displaystyle\leq\frac{s^{2d+2N}}{t^{2d+2N}}|f|^{2}.$

∎

###  5.7. Approximation lemma

A simple consequence of the Arnold-Moser lemma is the the following: Let a
sequence $(f_{N})$ converge to $f$ in ${\mathcal{O}}^{h}(U_{t})$. Consider the
polynomials $p_{N}$ obtained by truncating $f_{N}$ at degree $N$ as analytic
functions in $U_{t}$. Then the sequence of polynomial $(p_{N})$ converges on
any smaller polydisc to the the same limit as the restriction of $f$.

###### Proof.

$\displaystyle\|\rho(t,s)(p_{N}-f)\|$
$\displaystyle\leq\|\rho(t,s)(p_{N}-f_{N})\|+\|\rho(t,s)(f_{N}-f)\|$
$\displaystyle\leq\left(\frac{s}{t}\right)^{n+N}\|f_{N}\|+\|\rho(t,s)(f_{N}-f)\|\xrightarrow[N\longrightarrow+\infty]{}0$

∎

###  5.8. Pöschel’s lemma

For an increasing Banach scale indexed by $n\in\mathbb{N}\cup\\{+\infty\\}$

$E_{1}\xhookrightarrow{}E_{2}\xhookrightarrow{}\ldots
E_{n}\longrightarrow\ldots\xhookrightarrow{}E_{\infty},$

we say that a sequence $(f_{n}),\ f_{n}\in E_{n}$ is convergent, if it maps to
a converging sequence in the Banach space $E_{\infty}$. A decreasing sequences
of open sets

$U_{1}\supset U_{2}\supset\ldots\supset U_{n}\supset\ldots\supset
U_{\infty}:=\cap_{n}U_{n}$

gives such a Banach scale with

$E_{n}:={\mathcal{O}}^{k}(U_{n}),$

where ${\mathcal{O}}^{k}(U)$ is the Banach space function of complex valued
$C^{k}$-function on $\overline{U}$, with bounded $C^{k}$ norm which are
holomorphic on the interior of $U$. Consider a second such decreasing sequence

$V_{1}\supset V_{2}\supset\ldots\supset V_{n}\supset\ldots\supset
V_{\infty}:=\cap_{n}V_{n}$

with $V_{n}\subset U_{n}$. The following result goes back to [30]:

Let $d(U_{n},V_{n})>r_{n}$. If a sequence $(f_{n}),\
f_{n}\in{\mathcal{O}}^{c}(U_{n})$ converges to a limit $f$ in
${\mathcal{O}}^{c}(U_{\infty})$ faster than $(r_{n}^{k})$, then the sequence
of restrictions $\rho_{n}(f_{n})\in{\mathcal{O}}^{c}(V_{n})$ converge to the
restriction of $f$ in ${\mathcal{O}}^{k}(V_{\infty})$

###### Proof.

By the Cauchy-Nagumo lemma, the restriction maps

$\rho_{n}:{\mathcal{O}}^{c}(U_{n})\longrightarrow{\mathcal{O}}^{k}(V_{n})$

satisfy the estimate

$\|\rho_{n}\|\leq\frac{k!}{r_{n}^{k}}.$

Therefore

$\|\rho_{\infty}(f_{n}-f)\|\leq\frac{k!\|f_{n}-f\|}{r_{n}^{k}}.$

Consequently the condition (for fixed $k$):

$\|f_{n}-f\|=o(r_{n}^{k})$

implies the convergence of $\rho_{n}(f_{n})$. ∎

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