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\title{On the Normal Behaviour of Partially
Elliptic Lower Dimensional Tori of Hamiltonian Systems}
\author{\`Angel Jorba and Jordi Villanueva}
\date{October 16th, 1996}
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\begin{document}
\maketitle
\thispagestyle{empty}
\vspace*{-10mm}
\begin{center}
Departament de Matem\`atica Aplicada I \\
Universitat Polit\`ecnica de Catalunya \\
Diagonal 647, 08028 Barcelona, Spain. \\
E-mails: {\tt angel@tere.upc.es}, {\tt jordi@tere.upc.es}
\end{center}
\bigskip
\begin{abstract}
The purpose of this paper is to study the dynamics near a reducible
lower dimensional invariant tori of a finite-dimensional autonomous
Hamiltonian system with $\ell$ degrees of freedom. We will focus in
the case in which the torus has (some) elliptic directions.
First, let us assume that the torus is totally elliptic. In this
case, it is shown that the diffusion time (the time to move away from
the torus) is exponentially big with the initial distance to the
torus. The result is valid, in particular, when the torus is of
maximal dimension and when it is of dimension 0 (elliptic point). In
the maximal dimension case, our results coincide with previous
ones. In the zero dimension case, our results improve the existing
bounds in the literature.
Let us assume now that the torus (of dimension $r$, $0\le r<\ell$) is
partially elliptic (let us call $m_e$ to the number of these
directions). In this case we show that, given a fixed number of
elliptic directions (let us call $m_1\le m_e$ to this number), there
exist a Cantor family of invariant tori of dimension $r+m_1$, that
generalize the linear oscillations corresponding to these elliptic
directions. Moreover, the Lebesgue measure of the complementary of
this Cantor set (in the frequency space $\RR^{r+m_1}$) is proven to
be exponentially small with the distance to the initial torus. This
is a sort of ``Cantorian central manifold'' theorem, in which the
central manifold is completely filled up by invariant tori and it is
uniquely defined.
The proof of these results is based on the construction of suitable
normal forms around the initial torus.
\end{abstract}
\newpage
\tableofcontents
\newpage
%%%%
\markboth{Normal Behaviour of Lower Dimensional Tori}
{A. Jorba and J. Villanueva}
%%%%
\section{Introduction}
The study of the solutions close to an invariant object is a classical
subject in Dynamical Systems. Here we will address the problem of
describing the phase space near an invariant torus of a Hamiltonian
system. To fix the notation, let us call $H$ to a real analytic
Hamiltonian with $\ell$ degrees of freedom, and let us assume it has
an invariant $r$-dimensional torus, $0\le r\le\ell$. Note that we are
including the two limit cases, that is, when it is an equilibrium
point and when it is a maximal dimensional torus.
To start the discussion, let us assume that the torus has some
elliptic directions, this is, that the linearized normal flow contains
some harmonic oscillators. A natural question is if these oscillations
persist when the nonlinear part of the Hamiltonian is added. If the
torus is totally elliptic, another natural problem is the (nonlinear)
stability around this torus.
There are known answers to these questions in some concrete cases. If
$r=0$ (the torus is an equilibrium point) and it is totally elliptic,
KAM theory says that there is plenty of maximal dimension invariant
tori around the point (see \cite{DG}): the complementary of the set of
invariant tori has measure exponentially small with the distance to
the point. It is well known that if $\ell=2$, the maximal dimensional
tori split the energy levels $H=h$ in disconnected components. This
is the basis to prove the nonlinear stability of the point.
Unfortunately, if $\ell>2$, the invariant tori do not separate the
energy levels. In this case it is generally believed that some
diffusion can take place in the phase space (see
\cite{A64}). Nevertheless, it is still possible to give lower bounds
on the diffusion time, that are exponentially big with the distance to
the point (they follow immediately from \cite{DG}).
If $r=\ell$ (the torus has maximal dimension) we can not speak about
normal behaviour since there are not ``available directions''. The
nonlinear stability has been studied in \cite{PW} and \cite{MG94}
(among others), where it is shown that the diffusion time is also
bounded by an exponentially big (with the distance to the initial
torus) quantity. In \cite{MG94} it is also related this exponentially
big stability time with the density of invariant maximal dimensional
tori around the initial one, by showing that the total measure of the
gaps between the invariant tori nearby is not bigger than an
exponentially small quantity with respect to the distance to the
initial one. In fact, in \cite{MG94} it is proved that under an extra
steepness condition the diffusion time is, at least,
superexponential. This condition corresponds to the classical
quasi-convexity hypothesis used to obtain ``global'' and exponentially
big stability time of a perturbed integrable Hamiltonian system with
respect to the size of the perturbation (see \cite{DG95} and
references therein).
In this work we will consider these problems, without any steepness
condition, for a lower dimensional torus. The two limit cases
mentioned above are included, and the results obtained can be
summarized as follows: for a totally elliptic torus, we have obtained
lower bounds for the diffusion time. They agree with the bounds of
\cite{MG94} in the case $r=\ell$ but, for the case $r=0$, they are
better than the ones directly derived from \cite{DG}. Moreover, we
show the existence of quasiperiodic solutions that generalize the
linear oscillations of the normal flow to the complete system. If the
torus has normal behaviour of the kind ``some centres''$\times$``some
saddles'' we obtain, for any combination of centres, a Cantor family
of invant tori around the initial one, by adding to the initial set of
frequencies new ones that come from the nonlinear oscillations
associated to the chosen centres. Those invariant tori have the same
normal behaviour as the initial one (of course, skipping the centres
that give rise to the family). This result is a sort of ``Cantorian
central manifold'' theorem, in which we obtain an invariant manifold
parametrized on a Cantor set and completely filled up by invariant
tori. We note that we obtain a Cantorian central ``submanifold'' for
each combination of centres, and that it is uniquely defined.
The proofs are based on the construction of suitable normal forms.
The estimates on the difussion time are obtained bounding the
remainder of this normal form, while the existence of families of
lower dimensional tori is proved by applying a KAM scheme to this
remainder.
The paper has been organized in the following way:
Section~\ref{sec:summary} summarizes the main ideas and results
contained in the work. Section~\ref{sec:nf} contains the details
concerning the normal form and the bounds on the diffusion time.
Section~\ref{sec:efldt} is devoted to the existence of families
of tori near the initial one and, finally, in Section~\ref{sec:bl},
we have included some basic lemmas used along the paper.
\section{Summary}\label{sec:summary}
Here we have included a technical description of the problem, the
methodology used in the proofs and the results obtained. We have
ommitted the technical details of the proofs in order to simplify the
reading.
\subsection{Notation and formulation of the problem}
Let $H$ be a Hamiltonian system of $\ell$ degrees of freedom defined
on $\RR^{2\ell}$, having an invariant $r$-dimensional isotropic torus
(that is, the canonical 2-form of $\RR^{2\ell}$ restricted to the
tangent bundle of the torus vanish), $0\le r\le\ell$, with a
quasiperiodic flow given by the vector of basic frequencies
$\hat{\omega}^{(0)}\in\RR^r$. We assume, from the isotropic character
of the torus, that we can introduce (with a canonical change of
coordinates) $r$ angular variables $\hat{\theta}$ describing the
initial torus. Hence, the Hamiltonian in these coordinates takes the
form
$$
H(\hat{\theta},x,\hat{I},y)=\hat{\omega}^{(0)\top}\hat{I}+
\frac{1}{2}z^{\top}{\cal B}(\hat{\theta})z+
H_1(\hat{\theta},x,\hat{I},y),
$$
where $z^{\top}=(x^{\top},y^{\top})$. Here, $x$, $y$ are
$m$-dimensional real vectors, and $\hat{\theta}$, $\hat{I}$ belong to
$\RR^r$, $r+m=\ell$. Of course, $\hat{\theta}$, $x$ are the positions
and $\hat{I}$, $y$ the respective conjugate momenta. As $\hat{\theta}$
is an angular variable, we assume that $H$ depends on it in a
$2\pi$-periodic way. Moreover, we will use $u^{\top}v$ to denote the
scalar product of two vectors.
We also suppose that the Hamiltonian $H$ can be extended to a real
analytic function defined on the set ${\cal D}_{r,m}(\rho_0,R_0)$
given by
\begin{equation}
{\cal D}_{r,m}(\rho_0,R_0)=\{(\hat{\theta},x,\hat{I},y)
\in\CC^r\times\CC^m\times\CC^r\times\CC^m:\;
|\Im \hat{\theta}|\leq \rho_0,\;|z|\leq R_0,\;|\hat{I}|\leq R_0^2\},
\label{eq:domini}
\end{equation}
where $|.|$ denotes the infinity norm of a complex vector (we will
use the same notation for the matrix norm induced). The different
scaling for the variables $z$ and $\hat{I}$ in
${\cal D}_{r,m}(\rho_0,R_0)$ is motivated by
the definition of degree for a monomial of the Taylor expansion (with
respect to $z$ and $\hat{I}$, see (\ref{eq:Taylor})) used along the
paper:
\begin{equation}
\mbox{\rm deg}\left(h_{l,s}(\hat{\theta})z^l\hat{I}^s\right)
=|l|_1+2|s|_1,
\label{eq:grau}
\end{equation}
with $l\in\NN^{2m}$, $s\in\NN^r$, and where $|k|_1$ is defined as
$\sum_j|k_j|$.
The reason for counting twice the exponent $s$ will be clear later (it
is motivated, basically, by the properties of the Poisson bracket).
We assume the initial invariant torus is given by $z=0$ and
$\hat{I}=0$. Hence, we can take ${\cal B}(\hat{\theta})$ as a
symmetric $2m$-dimensional matrix, with real coefficients that depend
on $\hat{\theta}$ in analytic and $2\pi$-periodic way. Moreover,
the Taylor expansion of $H_1$ around $z=0$, $\hat{I}=0$ begins with
terms of degree at least three.
\subsubsection{Reducibility}
We will assume that the normal variational flow around this torus
(given by the matrix $J_m {\cal B}(\hat{\omega}^{(0)}t)$, where $J_m$
is the canonical $2$-form of $\CC^{2m}$) can be reduced to constant
coefficients with a real linear change of variables that depends
quasiperiodically on $\hat{\theta}$, having $\hat{\omega}^{(0)}$ as a
vector of basic frequencies (quasiperiodic Floquet
reduction).\footnote{If the torus is reducible except by an small
remainder, it is still possible to derive similar results (by adding a
perturbative parameter). See \cite{JRV} and \cite{JV96} for the main
ideas and related results.} The hypothesis does not seem to be very
restrictive in our context, since all the partially elliptic tori
obtained by KAM techniques have reducible normal flow (see, for
instance, \cite{El88}, \cite{P89}, \cite{JLZ}, \cite{JV96}). This
property allows to construct a canonical change of coordinates that
transforms the matrix ${\cal B}(\hat{\theta})$ to constant
coefficients. Hence, we will assume that ${\cal B}$ is a real
symmetric matrix, independent from $\hat{\theta}$, and that the initial
Hamiltonian in those Floquet variables looks like:
\begin{equation}
H(\hat{\theta},x,\hat{I},y)=\hat{\omega}^{(0)\top}\hat{I}+
\frac{1}{2}z^{\top}{\cal B}z+
H_2(\hat{\theta},x,\hat{I},y),
\label{eq:rediniham}
\end{equation}
where $H_2$ begins with terms of degree at least three.
\subsubsection{Linear normal behaviour of the torus}\label{sec:lnbt}
We also assume that the matrix $J_m {\cal B}$ has different
eigenvalues, given by the complex vector $\lambda\in\CC^{2m}$, that
takes the form $\lambda^{\top}=
(\lambda_1,\ldots,\lambda_m,-\lambda_1,\ldots,-\lambda_m)$ (this
structure comes from the canonical character of the system). We note
that in this case, different eigenvalues also means nonzero
eigenvalues. We will refer to those eigenvalues as the normal
eigenvalues of the torus. We remark that if $\lambda_j=i\beta$ (with
$\beta\in\RR\setminus\{0\}$ and $i=\sqrt{-1}$) is an eigenvalue,
then $\lambda_{j+m}=-i\beta$. The vectors of $\RR^{2m}$
that are combination of eigenvectors corresponding to (couples of)
eigenvalues of this form are called the elliptic directions of the
torus.
The study of the behaviour of the initial torus in those directions is
the main issue in this paper. Moreover, there may be other
eigenvalues with real part different from zero, that define the
hyperbolic directions of the torus. They can be grouped in one of
these two following forms:
\begin{enumerate}
\item if $\lambda_j=\alpha\in\RR\setminus\{0\}$, then
$\lambda_{j+m}=-\alpha$,
\item if $\lambda_j=\alpha+i\beta$ (with
$\alpha,\beta\in\RR\setminus\{0\}$), then, from the real character
of the matrix ${\cal B}$, we can take $\lambda_{j+1}=\alpha-i\beta$,
and hence, $\lambda_{j+m}=-\alpha-i\beta$ and
$\lambda_{j+m+1}=-\alpha+i\beta$.
\end{enumerate}
The imaginary parts of the eigenvalues are usually called normal
frequencies of the torus.
For reasons that will be clear later, it is very convenient to put the
matrix $J_m{\cal B}$ in diagonal form. This is possible with a complex
canonical change of basis, that transforms the initial real
Hamiltonian system into a complex one. Thus, the complexified
Hamiltonian has some symmetries because it comes from a real one. As
this symmetries are preserved by the transformations used along the
proofs, the final Hamiltonian can be realified. In fact,
complexification is not necessary, but it simplifies the
proofs. Nevertheless, in the proofs we have not written explicitly the
preservation of those symmetries. This is because the details are very
tedious and cumbersome and, on the other hand, the interested reader
should not have problems in writting them (it is a very standard
methodology). For further uses, we denote by
$Z^{\top}=(X^{\top},Y^{\top})$ those complex (canonical) variables,
and by ${\cal B}^*$ the complex symmetric matrix such that $J_m{\cal
B}^*=\diag(\lambda)$.
\subsubsection{Seminormal form: formal description}\label{sec:fnf}
Now we take a subbundle of $\RR^{2m}$, ${\cal G}\subset\RR^{2m}$,
invariant by the action of the matrix $J_m{\cal B}$, and such that it
only contains eigenvectors of elliptic type. We put $2m_1=\dim({\cal
G})$ (we recall that this dimension is always even) and we call
$\tilde{\omega}^{(0)}\in\RR^{m_1}$ to the vector of normal frequencies
associated to this subbundle. As ${\cal G}$ will be fixed along the
paper, we introduce some notation related to it. First, we assume that
the first $m_1$ eigenvalues of $\lambda$ are the ones associated to
${\cal G}$, that is, $\lambda_j=i\tilde{\omega}^{(0)}_j$,
$j=1,\ldots,m_1$. We also denote by $\hat{\lambda}\in\CC^{2(m-m_1)}$
the vector obtained skipping from $\lambda$ the $2m_1$ eigenvalues
associated to ${\cal G}$. This introduces in a natural way the
decomposition $X^{\top}=(\tilde{X}^{\top},\hat{X}^{\top})$,
$Y^{\top}=(\tilde{Y}^{\top},\hat{Y}^{\top})$, obtained taking apart
the first $m_1$ components from the last $m-m_1$. Moreover, we define
$\tilde{Z}^{\top}=(\tilde{X}^{\top},\tilde{Y}^{\top})$ and
$\hat{Z}^{\top}=(\hat{X}^{\top},\hat{Y}^{\top})$. A similar notation
can be used for any vector $l\in\NN^{2m}$, splitting
$l^{\top}=(l_X^{\top},l_Y^{\top})$, where $l_X$ and $l_Y$ are the
exponents of $X$ and $Y$ in the monomial $Z^l$ ($Z^l=X^{l_X}Y^{l_Y}$).
Then, we introduce $\omega^{(0)}\in\RR^{r+m_1}$ as
$\omega^{(0)\top}=(\hat{\omega}^{(0)\top},\tilde{\omega}^{(0)\top})$,
and we ask for a Diophantine condition of the following
form,\footnote{The Diophantine condition can be relaxed when $|l|_1=2$
and $l^{\top}\hat{\lambda}$ only involves hyperbolic eigenvalues. In
this case, the results are proved using a combined method based on a
fixed point scheme for the hyperbolic directions and a Newton method
for the remaining ones. This technique allows to have multiple
hyperbolic eigenvalues.}
\begin{equation}
|ik^{\top}\omega^{(0)}+l^{\top}\hat{\lambda}|\geq
\frac{\mu}{|k|_1^{\gamma}},
\;\;k\in{\ZZ}^{r+m_1}\setminus\{0\},\;\;l\in\NN^{2(m-m_1)},
\;\;0\le|l|_1\leq 2,
\label{eq:dc}
\end{equation}
being $\mu>0$ and $\gamma>r+m_1$. This nonresonance condition allows
to construct (formally) a seminormal form related to the chosen ${\cal
G}$. If we express the Hamiltonian in terms of the variables $Z$, this
seminormal form is done by removing from $H$ the monomials of the
following form (see (\ref{eq:Fourier}) for the notations):
\begin{equation}
h_{l,s,k}\exp{(ik^{\top}\hat{\theta})}Z^l\hat{I}^s,\;\;
l\in\NN^{2m},\;\; s\in\NN^r,\;\; k\in\ZZ^r,\;\;
|k|_1+|l_X-l_Y|_1\neq 0,\;\; |\hat{l}|_1\leq 2,
\label{eq:km}
\end{equation}
where $\hat{l}$ is the part of $l$ that corresponds to $\hat{\lambda}$.
After this normal form process, using the preservation of the
symmetries that come from the complexification, we can rewrite this
(formal) seminormal form, in terms of suitable real variables, in the
following form:
\begin{equation}
H(\hat{\theta},x,\hat{I},y)
=\omega^{(0)\top}I+\frac{1}{2}\hat{z}^{\top}\hat{\cal B}\hat{z}+
{\cal F}(I)+\frac{1}{2}\hat{z}^{\top}{\cal Q}(I)\hat{z}+O_3(\hat{z}),
\label{eq:fnf}
\end{equation}
where, for simplicity, we do not change the name of the Hamiltonian,
and where we extend the decomposition introduced
above to the variables $(x,y)$. Here, the matrix $\hat{\cal B}$ is a
real symmetric matrix obtained by projecting ${\cal B}$ on the
directions given by the eigenvalues corresponding to the eigenvectors
$\hat{\lambda}$. $I$ is a compact notation for
$I^{\top}=(\hat{I}^{\top},\tilde{I}^{\top})$, where the actions
$\tilde{I}$ can be taken as $\tilde{I}_j=\frac{1}{2}(x_j^2+y_j^2)$,
$j=1,\ldots,m_1$, if we choose the real normal form variables $(x,y)$
associated to the considered elliptic directions in adequate (and
standard) way (see (\ref{eq:complex}) in the proof of
Theorem~\ref{teo:dif}). Of course, ${\cal F}=O_2(I)$ and
${\cal Q}=O_1(I)$.
Now, we proceed to describe the normal behaviour of the torus derived
from this seminormal form. It is not difficult to check that we have
the following (formal) quasiperiodic solutions for the canonical
equations of (\ref{eq:fnf}):
\begin{eqnarray}
\hat{\theta}(t) & = & \left(\hat{\omega}^{(0)}+
\frac{\partial {\cal F}}{\partial \hat{I}}(I(0))\right)t+
\hat{\theta}(0), \nonumber \\
\hat{I}(t) & = & \hat{I}(0), \nonumber \\
\tilde{x}_j(t) & = & \sqrt{2\tilde{I}_j(0)}\sin{\left(\left(
\tilde{\omega}^{(0)}+
\frac{\partial {\cal F}}{\partial \tilde{I}_j}(I(0))\right)t+
\tilde{\theta}_j(0) \right)}, \label{eq:nfsol}\\
\tilde{y}_j(t) & = & \sqrt{2\tilde{I}_j(0)}\cos{\left(\left(
\tilde{\omega}^{(0)}+
\frac{\partial {\cal F}}{\partial \tilde{I}_j}(I(0))\right)t+
\tilde{\theta}_j(0) \right)}, \nonumber\\
\hat{z}(t)& = & 0.\nonumber
\end{eqnarray}
That is, we obtain a $2(r+m_1)$-dimensional invariant manifold
($\hat{z}=0$) foliated by a continuous $(r+m_1)$-dimensional family of
$(r+m_1)$-dimensional invariant reducible tori, parametrized by
$I(0)$. The selection of the parameter $I(0)$ is natural, as
$I_1,\ldots,I_{r+m_1}$ are first integrals of the Hamiltonian
(\ref{eq:fnf}) restricted to the invariant manifold $\hat{z}=0$. We
remark that the tori of the family collapse to lower dimensional ones
when any of the $\tilde{I}_j(0)$ become zero. In particular, if we
take $I(0)=0$ we recover the initial $r$-dimensional one. In fact, for
every $0\leq m_2\leq m_1$ we have, for this seminormal form, $m_1
\choose m_2$ different $(r+m_2)$-dimensional families of
$(r+m_2)$-dimensional invariant tori. They are associated to every
invariant real subbundle contained in ${\cal G}$. The skeleton of
these families comes from the natural $r$-dimensional family of
$r$-dimensional tori containing the initial one. This family is
associated to the neutral directions of the torus (the neutral
directions are conjugated to the tangent ones), an it
is obtained taking ${\cal G}=0$ in our notation. Moreover, we also
remark that in (\ref{eq:nfsol}) we only have real tori when all the
$\tilde{I}_j\geq 0$. This comes directly from the definition of
$\tilde{I}$ as a function of the real normal form variables. To
explain this fact let us give the classical example of a 1-dimensional
pendulum near the elliptic equilibrium point,
$\ddot{x}+\sin(x)=0$. The linear (normal) frequency at the equilibrium
point is $1$. Moving the energy level in the real phase space we
obtain periodic orbits with frequency smaller than 1. If one wants
periodic orbits with frequency bigger than 1, one is forced to extend
the phase space from $\RR^2$ to $\CC^2$, keeping the time in $\RR$.
This same phenomenon happens when we study the normal
elliptic directions of a torus. It is important to note that,
for us, a $s$-dimensional complex torus is a map from $\TT^s$
to $\CC^{2\ell}$. Hence, we will use the word
``dimension'' to refer to the real dimension.
\subsection{Results and main ideas}
A basic result in this paper is the quantitative version of the
seminormal form, if we only kill the monomials like (\ref{eq:km}) up
to some finite order. From the estimates on this seminormal form, we
deduce (under certain nondegeneracy conditions) that the normal
behaviour of the initial torus described in Section~\ref{sec:fnf} is
``correct'' in the sense of the classical KAM ideas: the ``majority''
of these tori really exist (but slightly deformed) in the initial
Hamiltonian system. Moreover, we also deduce the long time effective
stability of any real trajectory close to a totally elliptic torus. In
the following sections we present the explicit description of those
results, and we explain the main ideas used in the proofs.
\subsubsection{Seminormal form: bounds on the remainder}\label{sec:qenf}
We start with the Hamiltonian (\ref{eq:rediniham}), where the normal
flow is reduced to constant coefficients. Then, we perform a finite
number of (semi)normal form steps, by using suitable canonical
transformations that remove the monomials (\ref{eq:km}) up to a finite
degree. This allows to show the convergence of the process on the set
${\cal D}_{r,m}(\rho_1,R)$, where $\rho_1$ is independent from $R$ and
$R$ is small enough.
By selecting the order up to which the seminormal
form is done as a suitable function of $R$, it is possible to obtain a
remainder for the seminormal form which is exponentially small with
$R$. This is contained in Theorem~\ref{teo:nf}.
\subsubsection{Elliptic tori are very sticky}
Now let us assume that the inital torus has all the normal directions
of elliptic type. In this case we can take ${\cal G}=\RR^{2m}$, the
whole set of normal directions.
Then, using the normal form explained above, one can write the
initial Hamiltonian as an integrable one plus an exponentially small
perturbation. Hence, it is very natural to obtain exponentially big
estimates for the diffusion time: the time needed for a real trajectory
to go away from the set ${\cal D}_{r,m}(\rho,R)$ (for a precise
definition of ``going away'' see Theorem~\ref{teo:dif}) is bigger than
\begin{equation}
T(R)=const.
\exp{\left(const.\left(\frac{1}{R}\right)^{\frac{2}{\gamma+1}}\right)},
\label{eq:temps-difusio}
\end{equation}
being the constants on the definition of $T(R)$ independent from $R$.
As usual, we call the exponent $\frac{2}{\gamma+1}$ the stability
exponent.
Let us compare this result with previous ones. In the case in which
the initial torus is of maximal dimension, note that the normal
variables $z^\top=(x^\top,y^\top)$ are missing everywhere. So, the set
${\cal D}_{r,m}(\rho,R)$ (see (\ref{eq:domini})) reads
$$
{\cal D}_{\ell,0}(\rho,R)=\{(\hat{\theta},\hat{I})\in\CC^{\ell}
\times\CC^{\ell}:\;|\Im \hat{\theta}|\leq \rho,\;,\;|\hat{I}|\leq R^2\}.
$$
To compare with \cite{MG94} we must redefine $R^2$ as
$R$, in order to have the same units. Then, the stability exponent in
(\ref{eq:temps-difusio}) coincides with \cite{MG94}.
If the initial torus is an equilibrium point, the variables
$\hat{\theta}$ and $\hat{I}$ are the ones that are missing. Hence,
${\cal D}_{r,m}(\rho,R)$ becomes
$$
{\cal D}_{0,\ell}(0,R)=\{(x,y)\in\CC^{\ell}\times\CC^{\ell}:\;
|(x,y)|\leq R\}.
$$
Hence, no rescaling is necessary to compare the diffusion time of
(\ref{eq:temps-difusio}) with the one derived from \cite{DG}: the
improvement is that the exponent $\frac{1}{\gamma+1}$ in \cite{DG}
here becomes $\frac{2}{\gamma+1}$. We note that this improvement is
not only on the diffusion time, but also on the measure of the
destroyed tori (see Remark~\ref{rem:mes-pfe}).
\subsubsection{Cantor families of invariant tori}\label{sec:manada}
It is clear from Section~\ref{sec:fnf} that computing the seminormal
formal form associated to ${\cal G}$ around the initial torus, up to
finite order, and skipping the non-integrable remainder, those
elliptic directions define a unique $(r+m_1)$-dimensional family of
$(r+m_1)$-dimensional tori around the initial $r$-dimensional one.
When we approach the inital torus, the intrinsic frequencies of the
tori of the family can be selected such that they tend to
$\omega^{(0)}$.
In this case we will show that when we add the remainder of the
seminormal form most of these tori still persist in the complete system
$H$, having also reducible normal flow. The normal eigenvalues of these
tori are close to the eigenvalues $\hat{\lambda}_j$ (that are the ones
not related with ${\cal G}$). Of
course, due to the different small divisors involved in the problem,
we can not prove the persistence of all the invariant tori predicted
by the normal form.
The hypotheses needed are usual in KAM methods. The first one is a
non-resonance condition involving the frequencies $\hat{\omega}^{(0)}$
and the normal ones $\lambda$, that depends on the concrete selection
of ${\cal G}$ and it is explicitly given in (\ref{eq:dc}). The second
hypothesis is a nondegeneracy condition, asking that all the
frequencies vary with the actions. Note that, in general, we have more
frequencies ($r+m$) than actions ($r+m_1$, $m_1\le m$). This
introduces the classical lack-of-parameters problem when working with
lower dimensional torus, that needs a special treatement (for related
results, see \cite{BHT}, \cite{Se} and \cite{JV96}). The idea that we
have used here is to choose a suitable $(r+m_1)$-dimensional set of
parameters, and to ask for the existence of lower dimensional tori
associated to some of the values of these parameters. Here, the
natural parameter is the vector of intrinsic frequencies
$\omega\in\RR^{r+m_1}$ of the invariant tori. To use this
parametrization we need a typical nondegeneracy condition on the
frequency map from $I$ to $\omega$, this is, that this map be a
(local) diffeomorphism around $I=0$. This condition can be explicitly
formulated computing the normal form of Section~\ref{sec:fnf} up to
degree $4$ and it is given in (\ref{eq:detC}). The control of the
remaining $m-m_1$ normal frequencies (normal to the
$(r+m_1)$-dimensional family of tori) is more difficult, since there
are no free parameters to control them. Note that those frequencies
are functions of the intrinsic ones. Then, the idea is to eliminate all
the frequencies for which the Diophantine conditions needed to construct
invariant tori are not satisfied. This will lead us to eliminate
values of $\omega$ to: a) control the intrinsic frequencies $\omega$
and b) control the normal ones as a function of the intrinsic ones. To
control the measure of the set of intrisic frequencies for which the
associated normal ones are close to resonance, we use the same kind of
method of \cite{JV96}: we ask for a extra set of nondegeneracy
conditions for the dependence of these normal frequencies with respect
to the intrinsic ones. Those conditions are given
in (\ref{eq:eigndc}). They have already been considered in \cite{Me}
and \cite{El88}.
With the formulation given above, the result is that the measure of
the complementary of the preserved tori is exponentially small:
we introduce
\begin{equation}
{\cal U}(A)=\left\{\omega\in\RR^{r+m_1}:\;|\omega-\omega^{(0)}|\leq A
\right\}, \;\;\; A>0,
\label{eq:calU}
\end{equation}
and let us define ${\cal A}(A)$ as the set of frequencies of ${\cal
U}(A)$ for which we have reducible invariant
tori. Then, if $A$ is small enough, we have
$$
\frac{\mes({\cal U}(A)\setminus{\cal A}(A))}{\mes({\cal U}(A))}
\leq const.
\exp{\left(-const.\left(\frac{1}{A}\right)^{\frac{1}{\gamma+1}}\right)},
$$
where $\mes(\cdot)$ denotes the Lebesgue measure of $\RR^{r+m_1}$,
$\gamma$ is the exponent of the Diophantine condition (\ref{eq:dc}),
and the constants that appear in this bound are positive and independent
from $A$. This result is formulated in Theorem~\ref{teo:cfldt}.
Nevertheless, as we have noted in Section~\ref{sec:fnf}, some of the
frequencies of ${\cal A}(A)$ give rise to complex tori. If one wants
to ensure that the obtained tori are real tori, one can look at the
formulation of Theorem~\ref{teo:fldt}.
Let us describe how those result are proved. For this purpose, we
start from the seminormal form provided by Theorem~\ref{teo:nf}, and
we assume that the reader is familiar with the standard KAM
techniques (see \cite{AKN88} and references therein).
Initially, we have the seminormal form tori parametrized by the
vector of ``actions'' $I\in\RR^{r+m_1}$ (see (\ref{eq:nfsol})). By
using the nondegeneracy condition of (\ref{eq:detC}), we can replace
this parameter by the $(r+m_1)$-dimensional vector of frequencies (see
Lemma~\ref{lem:invF} for the details).
The main issue is to kill, for a given frequency, the part of the
remainder that obstructs the existence of the corresponding invariant
torus in the complete Hamiltonian. This will be done by a standard
iterative Newton method. As usual, we need to have some control on
the combinations of intrinsic frequencies and normal eigenvalues
that appear in the divisors of the series used to keep them satisfying
a suitable Diophantine condition (like (\ref{eq:dc})). This control can
be done using the nondegeneracy conditions of (\ref{eq:eigndc}). As we
will start these iterations from an integrable Hamiltonian (at least
in the direction ${\cal G}$) with an exponentially small perturbation,
we can take the $\mu$ in (\ref{eq:dc}) of the same order. This produces
convergence except for a set of ``bad frequencies'' with exponentially
small measure.
Now, we use Poincar\'e variables (see (\ref{eq:Poicvar}) and
(\ref{eq:complPoicvar})) to introduce extra $m_1$ angular variables to
describe the invariant $(r+m_1)$-dimensional tori of the seminormal
form. When we introduce those variables, there is also another source
for degeneracy that, essentially, is due to the fact that the family
of $(r+m_1)$-dimensional tori comes from an $r$-dimensional one. It
causes the Poincar\'e variables to become singular when some of the
$\tilde{I}_j$ are zero.\footnote{This is the same problem that appears
when we put action-angle variables around an elliptic equilibrium
point of a one degree of freedom Hamiltonian system. A neighbourhood
of the origin has to be excluded since the change of variables is
singular there.} We remark that this degeneracy corresponds, in the
seminormal form (\ref{eq:fnf}), to the families of invariant tori of
dimension between $r$ and $r+m_1-1$ (assuming $m_1\geq 1$). If we ask
for real invariant tori, this degeneracy also corresponds to the
transition manifold from real to complex tori. We remark
that we have an exact knowledge of the manifold of degenerate
frequencies for the seminormal form, but the exponentially small
remainder makes that we only know the set of degenerate frequencies up
to an exponentially small error. This is the main reason that forces to
refine the seminormal form as we approach to the initial
torus. Moreover, the same remarks apply when we look for real tori: we
know the boundary of the set of frequencies that are candidate to give
a real torus in the complete Hamiltonian, with an exponentially small
error. To remove the degeneracy, we will take out a neighbourhood of
the frequencies corresponding to the transition manifold. As the terms
of the remainder are exponentially small with $R$, this neighbourhood
can be selected with exponentially small measure with respect to $R$.
Finally, we note that the application of the results mentioned above
show that, around the initial torus, there exists (Cantorian) families
of tori of dimensions between $r$ and $m_e$ (we recall $m_e$ is the
number of elliptic directions of the initial torus), under generic
conditions of non-resonance and nondegeneracy.
\section{Normal form and effective stability}\label{sec:nf}
This section contains the technical details of the seminormal
form process with rigorous bounds on the remainder, as well as
bounds on the diffusion time around an elliptic torus.
\subsection{Notation}
First, let us introduce some notation. We will consider analytic
functions
$h(\hat{\theta},x,\hat{I},y)$ defined on ${\cal D}_{r,m}(\rho,R)$, for
some $\rho>0$ and $R>0$, and $2\pi$-periodic with respect to
$\hat{\theta}$. We denote the Taylor series of $h$ as
\begin{equation}
h=\sum_{(l,s)\in\NN^{2m}\times\NN^r} h_{l,s}(\hat{\theta})z^l\hat{I}^s.
\label{eq:Taylor}
\end{equation}
Moreover, the coefficients $h_{l,s}$ will be expanded in Fourier series,
\begin{equation}
h_{l,s}(\hat{\theta})=
\sum_{k\in\ZZ^m} h_{l,s,k}\exp{(ik^{\top}\hat{\theta})}.
\label{eq:Fourier}
\end{equation}
We will denote by $\bar{h}_{l,s}=h_{l,s,0}$ the average of
$h_{l,s}(\hat{\theta})$, and let us define
$\tilde{h}_{l,s}(\hat{\theta})=h_{l,s}(\hat{\theta})-\bar{h}_{l,s}$.
Then, we use the expressions (\ref{eq:Taylor}) and (\ref{eq:Fourier}) to
introduce the following norms:
\begin{eqnarray}
|h_{l,s}|_{\rho} & = & \sum_{k\in\ZZ^m}|h_{l,s,k}|\exp{(|k|_1\rho)},
\label{eq:Fournorm} \\
|h|_{\rho,R} & = &
\sum_{(l,s)\in\NN^{2m}\times\NN^r}|h_{l,s}|_{\rho}R^{|l|_1+2|s|_1}.
\label{eq:Taylornorm}
\end{eqnarray}
Some basic properties of these norms are given in
Section~\ref{sec:bl}. Here we only note that, if those norms are
convergent, they are bounds for the supremum norms of
$h_{l,s}(\hat{\theta})$ (on the complex strip of width $\rho>0$) and
of $h$ (on ${\cal D}_{r,m}(\rho,R)$).
Let us recall the definition of Poisson bracket of two functions
depending on $(\hat{\theta},x,\hat{I},y)$:
$$
\{f,g\}= \frac{\partial f}{\partial \hat{\theta}}
\left(\frac{\partial g}{\partial \hat{I}}\right)^{\top} -
\frac{\partial f}{\partial \hat{I}}
\left(\frac{\partial g}{\partial \hat{\theta}}\right)^{\top} +
\frac{\partial f}{\partial z} J_m
\left( \frac{\partial g}{\partial z} \right)^{\top}.
$$
We use a similar definition when $f$ and $g$ depend on
$(\hat{\theta},X,\hat{I},Y)$. Note that, with our definition
of degree (see (\ref{eq:grau})), if $f$ and $g$ are homogeneous
polynomials and $\{f,g\}\ne 0$, one has
\begin{equation}
\deg\left(\{f,g\}\right)=\deg(f)+\deg(g)-2.
\label{eq:homog}
\end{equation}
To introduce more notation, let us define ${\cal N}=
\{(l,s)\in\NN^{2m}\times\NN^{r} :\; |l|_1+2|s|_1\geq 3 \}$, and let
${\cal S}$ be a subset of ${\cal N}$. We will say that $h\in{\cal
M}({\cal S})$ if $h_{l,s}=0$ when $(l,s)\notin{\cal S}$. We will also
use the following decomposition: given $h\in{\cal M}({\cal N})$, we
write $h={\cal S}(h)+({\cal N}\setminus{\cal S})(h)$, where ${\cal
S}(h)\in{\cal M}({\cal S})$ and $({\cal N}\setminus{\cal S})(h)
\in{\cal M}({\cal N}\setminus{\cal S})$.
Let us split $l^{\top}=(l_x^{\top},l_y^{\top})$, with
$l_x,l_y\in\NN^m$. Now, given $h\in{\cal M}({\cal S})$, we say that
$h\in\overline{\cal M}({\cal S})$ when $h_{l,s}=0$ for all
$(l,s)\in{\cal S}$ such that $l_x\neq l_y$, and
$h_{l,s}=\bar{h}_{l,s}$ if $l_x=l_y$. We say that $h\in\widetilde{\cal
M}({\cal S})$ if $\bar{h}_{l,s}=0$ for all $(l,s)\in{\cal S}$ such
that $l_x=l_y$. Note that, for any $h\in{\cal M}({\cal S})$, we have
$h=\overline{\cal S}(h)+\widetilde{\cal S}(h)$, with $\overline{\cal
S}(h)\in\overline{\cal M}({\cal S})$ and $\widetilde{\cal
S}(h)\in\widetilde{\cal M}({\cal S})$. We remark that the functions in
$\overline{\cal M}({\cal S})$ only depend on $\hat{I}$ and on the
products $x_jy_j$, $j=1,\ldots,m$.
\subsection{Bounding the remainder of the normal form}
We introduce ${\cal S}\subset{\cal N}$ in the following form: we recall
that the first $m_1$ components of $\lambda$ are eigenvalues associated
to ${\cal G}$, and then, we put ${\cal S}={\cal S}_1\cup{\cal S}_2$,
with
\begin{eqnarray}
{\cal S}_1 & = & \{(l,s)\in{\cal N}:\; |l_{m_1+1}|+\ldots+|l_{m}|+
|l_{m+m_1+1}|+\ldots+|l_{2m}|\leq 1\},
\label{eq:S1}\\
{\cal S}_2 & = & \{(l,s)\in{\cal N}:\; |l_{m_1+1}|+\ldots+|l_{m}|+
|l_{m+m_1+1}|+\ldots+|l_{2m}|=2\}.
\label{eq:S2}
\end{eqnarray}
This splitting of ${\cal S}$ will be used during the proof of
Lemma~\ref{lem:lnf}, to identify in a precise form the contribution to
${\cal M}(\cal S)$ from the Poisson brackets involving monomials of
${\cal M}(\cal S)$ (see the bounds (\ref{eq:cotS1}) and
(\ref{eq:cotS2})). This is essential to obtain the estimates of
Lemma~\ref{lem:lnfgs}.
We take the Hamiltonian $H(\hat{\theta},x,\hat{I},y)$ of
(\ref{eq:rediniham}), we write it in the variables $Z$ (introduced at
the end of Section~\ref{sec:lnbt}) and we decompose it in the
following form:
\begin{equation}
H(\hat{\theta},X,\hat{I},Y)=\hat{\omega}^{(0)\top}\hat{I}+
\frac{1}{2}Z^{\top}{\cal B}^*Z+
N(X,\hat{I},Y)+S(\hat{\theta},X,\hat{I},Y)+T(\hat{\theta},X,\hat{I},Y),
\label{eq:nfham}
\end{equation}
with $N\in\overline{\cal M}({\cal S})$, $S\in\widetilde{\cal M}({\cal
S})$ and $T\in{\cal M}({\cal N}\setminus{\cal S})$. We also define
$S_1:={\cal S}_1(S)$ and $S_2:={\cal S}_2(S)$. With this formulation,
we say that $N$ is in normal form with respect to ${\cal S}$, that $S$
contains the terms of $H$ that are not in normal form with respect to
${\cal S}$, and that $T$ contains the terms of the Taylor expansion of
$H$ not associated to ${\cal S}$. We will show that, assuming the
Diophantine conditions of (\ref{eq:dc}), we can put $H$ in normal form
with respect to ${\cal S}$, with a canonical transformation defined
around the initial $r$-dimensional torus, $Z=0$ and $\hat{I}=0$,
leaving a small remainder of non-resonant terms. This remainder will
be exponentially small with respect to $R$ on the set ${\cal
D}_{r,m}(\rho_1,R)$, provided that $R$ be small enough, and for
certain $\rho_1>0$ independent from $R$. This is done with a finite
iterative scheme, with general step described in the following lemma:
\begin{lemma}\label{lem:lnf}
We consider the Hamiltonian $H$ given in \mbox{\rm (\ref{eq:nfham})}.
We assume that it is defined on ${\cal D}_{r,m}(\rho,R)$, with
$0<\rho<1$ and $00$ and
$\gamma>r+m_1$ such that
$$
|ik^{\top}\hat{\omega}^{(0)}+l^{\top}\lambda|\geq
\frac{\mu_0}{(|k|_1+|l_x-l_y|_1)^{\gamma}}
\;\;
\forall (l,s)\in{\cal S}\;\forall k\in\ZZ^r\;
\mbox{\rm with } |l_x-l_y|_1+|k|_1\neq 0,
$$
and, given $\delta>0$, let us introduce $\rho_j=\rho-j\delta$ and
$R_j=R\exp{(-j\delta})$. Then, we can construct an analytic function
$G(\hat{\theta},X,\hat{I},Y)\in\widetilde{\cal M}({\cal S})$, such
that for any $0<\delta\le\rho/8$ we have the following properties :
\begin{enumerate}
\item $G$ is defined on
${\cal D}_{r,m}(\rho_1,R_1)$,
and if we decompose $G=G_1+G_2$, being
$G_1=\widetilde{\cal S}_1(G)$
and $G_2=\widetilde{\cal S}_2(G)$, the bounds for
$|G|_{\rho_1,R_1}$,
$|G_1|_{\rho_1,R_1}$ and
$|G_2|_{\rho_1,R_1}$ are given in
\mbox{\rm (\ref{eq:cotG})}.
\item Let us denote by $\Psi^{G}_t$ the flow at time $t$ of the
Hamiltonian system $G$. Then, if
\begin{equation}
\Delta\frac{|S|_{\rho,R}}{\delta^{\gamma+2}R^2}\leq 1,
\label{eq:Delta}
\end{equation}
where $\Delta$ depends only on $\gamma$ and $\mu_0$, we have
$$
\Psi^{G}_1, \Psi^{G}_{-1}:
{\cal D}_{r,m}(\rho_4,R_4)
\longrightarrow {\cal D}_{r,m}(\rho_3,R_3).
$$
\item If we take $(\hat{\theta},X,\hat{I},Y)\in
{\cal D}_{r,m}(\rho_4,R_4)$, and we put
$(\hat{\theta}^*,X^*,\hat{I}^*,Y^*)=
\Psi^{G}_1(\hat{\theta},X,\hat{I},Y)$,
then we have $|\hat{\theta}^*-\hat{\theta}|\leq\delta$,
$|Z^*-Z|\leq R\delta\exp{(-1/2)}/2$,
$|\hat{I}^*-\hat{I}|\leq R^2\delta\exp{(-1)}$. The same
bounds also hold for $\Psi^{G}_{-1}$.
\item $\Psi^{G}_1$ transforms
\begin{equation}
H^{(1)}:=H\circ\Psi^G_1=\hat{\omega}^{(0)\top}\hat{I}+
\frac{1}{2}Z^{\top}{\cal B}^*Z+N^{(1)}+S^{(1)}+T^{(1)},
\label{eq:H1}
\end{equation}
decomposition analogous to \mbox{\rm (\ref{eq:nfham})}, with the
bounds \mbox{\rm (\ref{eq:cotH*})}--\mbox{\rm(\ref{eq:cotT})}.
\end{enumerate}
\end{lemma}
\begin{remark}
In the Diophantine condition of the statement of the lemma, we remark
that if we write
${\lambda}^{\top}=(\varlam^{\top},-\varlam^{\top})$,
then, for any $l\in\NN^{2m}$, we have
$l^{\top}\lambda=(l_x-l_y)^{\top}\varlam$. Hence, this condition
is equivalent to the one formulated in \mbox{\rm(\ref{eq:dc})}, and one
can take $\mu_0$ as the minimum of $\mu$ and
$\min{\{|\hat{l}^{\top}\hat{\lambda}|\}}$, where this last expression is
taken on the $\hat{l}\in\ZZ^{2(m-m_1)}$ with $0<|\hat{l}|_1\leq 2$ and
$\hat{l}_x-\hat{l}_y\neq 0$. Moreover, we remark that if we
take $\gamma>r+m_1$, the set of vectors $\hat{\omega}^{(0)}$ and
$\lambda$ for which any Diophantine condition of this kind is not
satisfied, has zero measure.
\end{remark}
\begin{remark}\label{rem:choose}
The canonical transformation generated by $G$ has been chosen to
remove the term $S$ in the decomposition \mbox{\rm(\ref{eq:nfham})},
formulating the homological equation in terms of the monomials of
degree $2$ of the Hamiltonian. This is, in fact, a classical (and
linearly convergent) normal form scheme.
\end{remark}
\begin{remark}
The bounds on $H^{(1)}$ given by Lemma~\mbox{\rm \ref{lem:lnf}} are
not very concrete. This is because we will use this lemma in iterative
form, but the estimates used in the first steps will be different from
the ones used in a general step of the iterative process. A
description of a general step is given in Lemma~\mbox{\rm
\ref{lem:lnfgs}}.
\end{remark}
\prova
We look for a generating function
$G\in\widetilde{\cal M}({\cal S})$, such that
$$
S+\{\hat{\omega}^{(0)\top}\hat{I}+\frac{1}{2}Z^{\top}{\cal B}^*Z,G\}=0.
$$
From the definition of the Poisson bracket, we have
$$
S+\left(-\frac{\partial G}{\partial \hat{\theta}}\hat{\omega}^{(0)}+
Z^{\top}{\cal B}^*J_m\left(\frac{\partial G}{\partial Z}
\right)^{\top}\right)=0.
$$
Expanding $G$ and $S$, we obtain
\begin{equation}
G_{l,s,k}=\frac{S_{l,s,k}}{ik^{\top}\hat{\omega}^{(0)}+
(l_x-l_y)^{\top}\varlam},
\label{eq:coefG}
\end{equation}
for the admissible scripts $(l,s,k)$ in the expansion of $S$
(otherwise $G_{l,s,k}$ is defined as 0). Then, from the definition of
$G$, we have
\begin{eqnarray}
H^* & := &
H\circ\Psi^{G}_{1}-(\hat{\omega}^{(0)\top}\hat{I}+
\frac{1}{2}Z^{\top}{\cal B}^*Z+N+T+\{N+T,G\}) = \nonumber
\\ & = &
\int_0^1 \frac{d}{dt}\left(tH+(1-t)(\hat{\omega}^{(0)\top}\hat{I}+
\frac{1}{2}Z^{\top}{\cal B}^*Z+N+T+\{N+T,G\})\right)\circ\Psi^{G}_t dt =
\nonumber
\\ & = &
\int_0^1\{tH+(1-t)(\hat{\omega}^{(0)\top}\hat{I}+
\frac{1}{2}Z^{\top}{\cal B}^*Z+N+T+\{N+T,G\}),G\}\circ\Psi^{G}_tdt+
\nonumber
\\ & & +
\int_0^1(H-\hat{\omega}^{(0)\top}\hat{I}-
\frac{1}{2}Z^{\top}{\cal B}^*Z-N-T-\{N+T,G\})\circ\Psi^{G}_tdt=\nonumber
\\ & = &
\int_0^1\{tS+(1-t)\{N+T,G\},G\}\circ\Psi^{G}_tdt.
\label{eq:transfham}
\end{eqnarray}
Hence, we have $N^{(1)}=N+\overline{\cal S}(\{N+T,G\}+H^*)$,
$S_1^{(1)}=\widetilde{\cal S}_1(\{N,G_1\}+H^*)$,
$S_2^{(1)}=\widetilde{\cal S}_2(\{N,G\}+\{T,G_1\}+H^*)$ and
$T^{(1)}=T+({\cal N}\setminus{\cal S})(\{N+T,G\}+H^*)$. To give the
expressions of $S_1^{(1)}$ and $S_2^{(1)}$, we remark that
$\{T,G_1\}\in{\cal M}({\cal N}\setminus{\cal S}_1)$,
$\{T,G_2\}\in{\cal M}({\cal N}\setminus{\cal S})$ and
$\{N,G_2\}\in{\cal M}({\cal N}\setminus{\cal S}_1)$. Those facts are
consequence of the definition of ${\cal S}_1$, ${\cal S}_2$, the
structure of $N\in\overline{\cal M}({\cal S})$, and the properties of
the Poisson bracket.
We proceed to describe the effect of the transformation $\Psi_1^{G}$
and to bound the transformed Hamiltonian $H\circ\Psi_1^{G}$. For this
purpose, we take a fixed value of $\delta$, $0<\delta\leq\rho/8$. Then,
for any
$(l,s)\in{\cal S}$, $k\in\ZZ^r$, with $|l_x-l_y|_1+|k|_1\neq 0$, we have
from (\ref{eq:coefG})
\begin{eqnarray}
\lefteqn{
\left|\frac{S_{l,s,k}Z^l \exp{(ik^{\top}\hat{\theta})}}
{ik^{\top}\hat{\omega}^{(0)}+(l_x-l_y)^{\top}\varlam}
\right|_{\rho_1,R_1}
\leq} \nonumber
\\ & & \leq
\frac{(|k|_1+|l_x-l_y|_1)^{\gamma}}{\mu_0}
\exp{(-\delta |k|_1-\delta |l|_1)}
|S_{l,s,k}| R^{|l|_1}\exp{(|k|_1\rho)} \leq \nonumber
\\ & & \leq
\sup_{\alpha\geq 1}\left\{\alpha^{\gamma}\exp{(-\delta\alpha)}\right\}
\frac{|S_{l,s,k}|}{\mu_0} R^{|l|_1}\exp{(|k|_1\rho)}.
\label{eq:cotcoefG}
\end{eqnarray}
Now, using that for any $\gamma>0$ and $\delta>0$,
\begin{equation}
\sup_{\alpha\geq 1}\left\{\alpha^{\gamma}\exp{(-\delta\alpha)}\right\}
\leq \left(\frac{\gamma}{\delta\exp{(1)}}\right)^{\gamma},
\label{eq:cotpd}
\end{equation}
we deduce from (\ref{eq:cotcoefG}),
\begin{equation}
\left|G\right|_{\rho_1,R_1}
\leq \left(\frac{\gamma}{\delta\exp{(1)}}\right)^{\gamma}
\frac{|S|_{\rho,R}}{\mu_0}.
\label{eq:cotG}
\end{equation}
Moreover, the same bounds hold for $G_1={\cal S}_1(G)$ and $G_2={\cal
S}_2(G)$, if one adds the subscripts ``1'' or ``2'' to $G$ and $S$ in
(\ref{eq:cotG}).
Hence, using Lemma~\ref{lem:normt}, we have \begin{equation}
\left|\frac{\partial G}{\partial\hat{I}}\right|_{\rho_2,R_2} \leq
\frac{|G|_{\rho_1,R_1}}{R^2\exp{(-2\delta)}(1-\exp{(-2\delta)})}\leq
\frac{|G|_{\rho_1,R_1}}{R^2\delta\exp{(-2\delta)}},
\label{eq:cotG1}
\end{equation}
\begin{equation}
\left|\frac{\partial G}{\partial Z}\right|_{\rho_2,R_2}\leq
\frac{|G|_{\rho_1,R_1}}{R\exp{(-\delta)}(1-\exp{(-\delta)})}\leq
\frac{2|G|_{\rho_1,R_1}}{R\delta\exp{(-\delta)}},
\label{eq:cotG2}
\end{equation}
$$
\left|\frac{\partial G}{\partial\hat{\theta}}\right|_{\rho_2,R_2} \leq
\frac{|G|_{\rho_1,R_1}}{\delta\exp{(1)}},
$$
where in (\ref{eq:cotG1}) and (\ref{eq:cotG2}) we have used that,
if $0<\alpha\leq 1$, then
\begin{equation}
\frac{\alpha}{2}\leq 1-\exp{(-\alpha)}.
\label{eq:exp}
\end{equation}
Now, to check the bounds
$$
\left|\frac{\partial G}{\partial\hat{I}}\right|_{\rho_2,R_2}\leq\delta,
\;\;\;\;
\left|\frac{\partial G}{\partial Z}\right|_{\rho_2,R_2}
\leq\frac{R\delta\exp{(-1/2)}}{2},\;\;\;\;
\left|\frac{\partial G}{\partial\hat{\theta}}\right|_{\rho_2,R_2}
\leq R^2\delta\exp{(-1)},
$$
we use (\ref{eq:Delta}) with the following $\Delta$:
$$
\Delta(\gamma,\mu_0)=\left(\frac{\gamma}{\exp{(1)}}\right)^{\gamma}
\frac{4\exp{(1)}}{\mu_0}.
$$
If we use the notation $\Psi_t^G-Id=(\hat{\Theta}_t^G,{\cal X}_t^G,
\hat{\cal I}_t^G,{\cal Y}_t^G)$ (see Lemma~\ref{lem:canon}),
then we obtain,
\begin{equation}
\begin{array}{rclrcl}
|\hat{\cal I}_t^G|_{\rho_2,R_2} & \leq & R^2\delta\exp{(-1)}, &
|\hat{\Theta}_t^G|_{\rho_2,R_2} & \leq & \delta, \\
|{\cal Z}_t^G|_{\rho_2,R_2} & \leq & R\delta\exp{(-1/2)}/2, & & &
\end{array}
\label{eq:canonlnf}
\end{equation}
for any $-1\leq t\leq 1$, being ${\cal Z}_t^G=({\cal X}_t^G,{\cal
Y}_t^G)$. From the bounds of (\ref{eq:canonlnf}), and using the
inequality (\ref{eq:exp}), we can also deduce that the transformations
$\Psi_{1}^G$ and $\Psi_{-1}^G$ act as we describe in the statement.
Moreover, (\ref{eq:canonlnf}) and Lemma~\ref{lem:comp} allows to bound
(\ref{eq:transfham}) as
\begin{eqnarray}
|H^*|_{\rho_4,R_4} & \leq & |\{S,G\}|_{\rho_2,R_2}+
|\{\{T,G\},G\}|_{\rho_3,R_3} + |\{\{N,G\},G\}|_{\rho_3,R_3}.
\label{eq:cotH*}
\end{eqnarray}
Finally, the same arguments hold to bound the terms
of $H^{(1)}$ in (\ref{eq:H1}) by
\begin{eqnarray}
|N^{(1)}-N|_{\rho_4,R_4} & \leq & |\{N,G\}|_{\rho_2,R_2}+
|\{T,G\}|_{\rho_2,R_2}+|H^*|_{\rho_4,R_4}, \label{eq:cotN} \\
|S_1^{(1)}|_{\rho_4,R_4} & \leq & |\{N,G_1\}|_{\rho_2,R_2}+
|H^*|_{\rho_4,R_4}, \label{eq:cotS1} \\
|S_2^{(1)}|_{\rho_4,R_4} & \leq & |\{N,G\}|_{\rho_2,R_2}+
|\widetilde{\cal S}_2(\{T,G_1\})|_{\rho_2,R_2}+
|H^*|_{\rho_4,R_4}, \label{eq:cotS2} \\
|T^{(1)}-T|_{\rho_4,R_4} & \leq & |\{N,G\}|_{\rho_2,R_2}+
|\{T,G\}|_{\rho_2,R_2}+|H^*|_{\rho_4,R_4}. \label{eq:cotT}
\end{eqnarray}
\fiprova
Before giving more concrete estimates on the bounds of
Lemma~\ref{lem:lnf}, we assume that $H$ is in normal form up to
certain order $p$, to be determined later (the reduction of $H$ to
this finite normal form will be described in the proof of
Theorem~\ref{teo:nf}). Then, taking advantage of this fact, the bounds
of Lemma~\ref{lem:lnf} produce better estimates on the different
steps of the normal form process (this is done in
Lemma~\ref{lem:lnfgs}). This allows to produce a very accurate bound
on the final remainder. We want to stress that these bounds are not so
good if the initial Hamiltonian is not in normal form up to degree
$p$.
Let us introduce now the following notation: we break the Hamiltonian
(\ref{eq:nfham}) as
\begin{equation}
N=N_4+N^*,\;\;\; T=T_3+T^*,
\label{eq:breaks}
\end{equation}
where $N_4$ contains the monomials of $N$ of degree $4$ and $T_3$
contains the monomials of degree $3$ of $T$. Then, We assume that, for
$R$ small enough, we have the bounds
\begin{equation}
\begin{array}{rclrclrlc}
|S_1|_{\rho,R} & \leq & \hat{S}R^{p+1}, &
|S_2|_{\rho,R} & \leq & \hat{S}R^{p}, &
|S|_{\rho,R} & \leq & \hat{S}R^p, \\
|N_4|_{\rho,R} & \leq & \hat{N}_4R^4, &
|N^*|_{\rho,R} & \leq & \hat{N}^*R^6, &
& & \\
|T_3|_{\rho,R} & \leq & \hat{T}_3R^3, &
|T^*|_{\rho,R} & \leq & \hat{T}^*R^4, &
& &
\end{array}
\label{eq:cotbreak}
\end{equation}
being $\hat{S}$, $\hat{N}_4$, $\hat{N}^*$, $\hat{T}_3$ and $\hat{T}^*$
positive constants. Here, $p\in\NN$, $p\geq 6$, is the order of the
previous normal form and will be chosen later.
\begin{lemma}\label{lem:lnfgs}
Let us consider the Hamiltonian $H$ of \mbox{\rm (\ref{eq:nfham})},
with the same hypotheses as in Lemma~\mbox{\rm \ref{lem:lnf}}. We use
the notations \mbox{\rm (\ref{eq:breaks})}, and we assume
\mbox{\rm (\ref{eq:cotbreak})}. We also assume that
$\hat{S}\leq\hat{S}^*$, $\hat{N}^*\leq\hat{N}^{**}$ and
$\hat{T}^*\leq\hat{T}^{**}$, for some $\hat{S}^{*}$, $\hat{N}^{**}$
and $\hat{T}^{**}$. Let $G$ be the generating function obtained in
Lemma~\mbox{\rm \ref{lem:lnf}}, and let $\delta$,
$0<\delta\leq\rho/8$, be such that
$$
\Delta\frac{\hat{S}R^{p-2}}{\delta^{\gamma+2}}\leq 1,
$$
($\Delta$ is given by Lemma~\mbox{\rm \ref{lem:lnf}}).
Then, there exists a constant $\Pi$, depending only on $r$, $m$,
$\gamma$, $\mu_0$, $\hat{N}_4$, $\hat{N}^{**}$, $\hat{S}^{*}$,
$\hat{T}_3$ and $\hat{T}^{**}$, such that the following bounds hold
for the transformed Hamiltonian $H\circ\Psi^G_1$,
\begin{eqnarray*}
|N^{(1)}-N|_{\rho_4,R_4} & \leq & \Pi\hat{S}\left(
\frac{R^{p+1}}{\delta^{\gamma+2}}+\frac{R^{2p-1}}{\delta^{2(\gamma+2)}}
\right), \\
|S_1^{(1)}|_{\rho_4,R_4} & \leq & \Pi\hat{S}R^{p+1}\left(
\frac{R^{2}}{\delta^{\gamma+1}}+\frac{R^{4}}{\delta^{\gamma+2}}+
\frac{R^{p-3}}{\delta^{\gamma+2}}+\frac{R^{p-2}}{\delta^{2(\gamma+2)}}
\right), \\
|S_2^{(1)}|_{\rho_4,R_4} & \leq & \Pi\hat{S}R^p\left(
\frac{R^{2}}{\delta^{\gamma+1}}+\frac{R^{3}}{\delta^{\gamma+2}}+
\frac{R^{p-1}}{\delta^{2(\gamma+2)}}
\right), \\
|T^{(1)}-T|_{\rho_4,R_4} & \leq & \Pi\hat{S}\left(
\frac{R^{p+1}}{\delta^{\gamma+2}}+\frac{R^{2p-1}}{\delta^{2(\gamma+2)}}
\right).
\end{eqnarray*}
\end{lemma}
\begin{remark}\label{rem:ordres}
(A very important one)
If $p$ is big enough and $\delta>R$, the dominant term in the bounds of
$S_1^{(1)}$ and $S_2^{(1)}$ is given by the factor
$R^{2}/\delta^{\gamma+1}$.
This will be the factor of decreasing of those terms during the normal
form process and it allows to take $\delta$ of order
$R^{2/(\gamma+1)}$, that will produce the exponent $2/(\gamma+1)$
in \mbox{\rm (\ref{eq-angel:cota-reste})}. As we have $2/(\gamma+1)<1$,
we can deduce that an adequate selection for $p$ is $p=8$. This allows
to keep bounds like
\mbox{\rm (\ref{eq:cotbreak})} during all the iterative process.
If we start with a ``raw'' Hamiltonian (without any previous step of
normal form) the decreasing factor obtained is of order
$R/\delta^{\gamma+1}$, that forces us to select $\delta$ of order
$R^{1/(\gamma+1)}$. This produces a worse exponent $1/(\gamma+1)$ in
\mbox{\rm (\ref{eq-angel:cota-reste})}. For instance, let us assume
that the normal form has been done around an elliptic equilibrium
point. Here the important issue is to note that the bounds obtained
when killing degree $3$ are much worse than the bounds obtained for
the other degrees (this has been observed numerically in \mbox{\rm
\cite{S89}}). Hence, to apply the same bounds to all the degrees
results in poor estimates.
\end{remark}
\begin{remark}
The exponent $2/(\gamma+1)$ in Remark~\mbox{\rm\ref{rem:ordres}} can
be improved in some very degenerate cases. For instance, let us
consider a totally elliptic torus, and we take ${\cal
G}=\RR^{2m}$. Let $q$ be the lowest degree of the monomials of $N$
corresponding to the (formal) normal form of $H$ around the torus (of
course, $q\geq 4$). Then, $\delta$ can be taken of order
$R^{(q-2)/(\gamma+1)}$, that produces the exponent $(q-2)/(\gamma+1)$
in \mbox{\rm (\ref{eq-angel:cota-reste})}.
\end{remark}
\prova
During this proof we will use different constants $\Pi_j$, $j\geq 0$,
that will depend only on the same parameters as the final constant
$\Pi$ of the statement of the lemma. First, from the bound
(\ref{eq:cotG}) of Lemma~\ref{lem:lnf}, we have that
$$
|G_1|_{\rho_1,R_1} \leq
\Pi_0\frac{\hat{S}R^{p+1}}{\delta^{\gamma}},\;\;\;\;
|G_2|_{\rho_1,R_1} \leq
\Pi_0 \frac{\hat{S}R^{p}}{\delta^{\gamma}},\;\;\;\;
|G|_{\rho_1,R_1} \leq \Pi_0 \frac{\hat{S}R^{p}}{\delta^{\gamma}},
$$
where, as in Lemma~\ref{lem:lnf}, $\rho_j=\rho-j\delta$ and
$R_j=R\exp{(-j\delta})$. Then, to obtain the bounds for the different
terms of the transformed Hamiltonian, we only need to bound the
Poisson brackets that appear in
(\ref{eq:cotH*})--(\ref{eq:cotT}).
To obtain precise estimates, we will look carefully into the critical
bounds of the different partial derivatives involved, that is, the
ones associated to $N_4$ and $T_3$. So, we estimate, separately, the
contribution of $N_4$, $N^*$, $T_3$ and $T^*$, taking into account
that $N$ does not depend on $\hat{\theta}$, $N_4$ is a polynomial of
degree $4$, and $T_3$ only contains terms of degree $3$. Moreover, to
bound $\widetilde{\cal S}_2(\{T,G_1\})$ we note that (from the
definition of ${\cal S}_1$ and ${\cal S}_2$) it only contains terms
corresponding to $\frac{\partial}{\partial \hat{Z}}$, and not to
$\frac{\partial}{\partial \hat{\theta}}$ or $\frac{\partial}{\partial
\hat{I}}$. Thus, using the bounds on the Poisson bracket provided by
Lemma~\ref{lem:bPb} (see Remark~\ref{rem:fte} for the case in which
one of the terms has finite degree), we have
\begin{eqnarray*}
|\widetilde{\cal S}_2(\{T,G_1\})|_{\rho_2,R_2} & \leq & \Pi_1\hat{S}R^p
\left( \frac{R^2}{\delta^{\gamma+1}}+\frac{R^3}{\delta^{\gamma+2}}
\right), \\
|\{T,G\}|_{\rho_2,R_2} & \leq &
\Pi_2\hat{S}\frac{R^{p+1}}{\delta^{\gamma+2}}, \\
|\{N,G_1\}|_{\rho_2,R_2} & \leq & \Pi_3\hat{S}R^{p+1}
\left(\frac{R^2}{\delta^{\gamma+1}}+ \frac{R^4}{\delta^{\gamma+2}}
\right), \\
|\{N,G\}|_{\rho_2,R_2} & \leq & \Pi_4\hat{S}R^{p}
\left( \frac{R^2}{\delta^{\gamma+1}}+ \frac{R^4}{\delta^{\gamma+2}}
\right), \\
|\{S,G\}|_{\rho_2,R_2} & \leq & \Pi_5
\hat{S}\frac{R^{2p-2}}{\delta^{\gamma+2}}, \\
|\{\{T,G\},G\}|_{\rho_3,R_3} & \leq &
\Pi_6\hat{S}\frac{R^{2p-1}}{\delta^{2(\gamma+2)}}, \\
|\{\{N,G\},G\}|_{\rho_3,R_3} & \leq &
\Pi_7\hat{S} \left(
\frac{R^{2p}}{\delta^{2\gamma+3}}+
\frac{R^{2p+2}}{\delta^{2(\gamma+2)}}
\right),
\end{eqnarray*}
and finally
$$
|H^*|_{\rho_4,R_4}\leq \Pi_8\hat{S}\left(
\frac{R^{2p-2}}{\delta^{\gamma+2}}+
\frac{R^{2p-1}}{\delta^{2(\gamma+2)}} \right).
$$
From that, with a suitable definition of $\Pi$ as a function of
$\Pi_0$--$\Pi_8$, the bounds of the statement of the lemma are clear, if
we recall that we have taken $p\geq 6$.
\fiprova
Now, we are in conditions to formulate a quantitative result about
``partial reduction to seminormal form'' of the initial Hamiltonian.
For this purpose, we
consider the Hamiltonian $H$ of (\ref{eq:rediniham}), written as in
(\ref{eq:nfham}) in terms of the $Z$ variables. We assume that $H$ is
defined on ${\cal D}_{r,m}(\rho_0,R_0)$, for some $0<\rho_0<1$ and
$00$ and $\gamma>r+m_1$ such that
$$
|ik^{\top}\hat{\omega}^{(0)}+l^{\top}\lambda|\geq
\frac{\mu_0}{(|k|_1+|l_x-l_y|_1)^{\gamma}}
\;\;
\forall (l,s)\in{\cal S}\;\forall k\in\ZZ^r\;
\mbox{\rm with } |l_x-l_y|_1+|k|_1\neq 0.
$$
Then, for any $R>0$ small enough (this condition on $R$
depends only on $r$, $m$, $\gamma$, $\mu_0$, $\rho_0$, $R_0$,
$\hat{N}$, $\hat{S}$ and $\hat{T}$), there exists an analytical
canonical transformation $\Psi^R$ such that
\begin{enumerate}
\item $\Psi^R-Id$ and $(\Psi^R)^{-1}-Id$ are $2\pi$-periodic on
$\hat{\theta}$.
\item
$$
\Psi^R:
{\cal D}_{r,m}(3\rho_0/4,R\exp{(-\rho_0/4)}) \longrightarrow
{\cal D}_{r,m}(\rho_0,R),
$$
and
$$
(\Psi^R)^{-1}:
{\cal D}_{r,m}(11\rho_0/16,R\exp{(-5\rho_0/16)}) \longrightarrow
{\cal D}_{r,m}(\rho_0,R).
$$
\item If we take $(\hat{\theta},X,\hat{I},Y)\in
{\cal D}_{r,m}(3\rho_0/4,R\exp{(-\rho_0/4)})$ and we define
$(\hat{\theta}^*,X^*,\hat{I}^*,Y^*)=\Psi^R(\hat{\theta},X,\hat{I},Y)$,
then $|\hat{\theta}^*-\hat{\theta}|\leq\rho_0/16$,
$|Z^*-Z|\leq R\rho_0\exp{(-1/2)}/32$,
$|\hat{I}^*-\hat{I}|\leq R^2\rho_0\exp{(-1)}/16$.
Moreover, the same bounds hold for $(\Psi^R)^{-1}$
if $(\hat{\theta},X,\hat{I},Y)\in
{\cal D}_{r,m}(11\rho_0/16,R\exp{(-5\rho_0/16)})$.
\item $\Psi^R$ transforms
$$
H^{R}:=H\circ\Psi^R=\hat{\omega}^{(0)\top}\hat{I}+
\frac{1}{2}Z^{\top}{\cal B}^*Z+N^{R}+S^{R}+T^{R},
$$
decomposition analogous to \mbox{\rm (\ref{eq:nfham})},
with the bounds:
$|N^R-N_4|_{3\rho_0/4,R\exp{(-\rho_0/4)}}\leq const.R^6$,
$|T^R-T_3|_{3\rho_0/4,R\exp{(-\rho_0/4)}}\leq const.R^4$, where $N_4$
and $T_3$ were introduced in \mbox{\rm (\ref{eq:breaks})}, and can be
computed with a normal form with respect to ${\cal S}$ up to degree
$4$, and
\begin{equation}
|S^{R}|_{3\rho_0/4,R\exp{(-\rho_0/4)}}\leq const. \exp{\left(
-const.\left(\frac{1}{R}\right)^{\frac{2}{\gamma+1}} \right)} R^8,
\label{eq-angel:cota-reste}
\end{equation}
being the constats that appear in the bounds of $N^{R}$, $T^{R}$ and
$S^{R}$, positive and independent from $R$.
Moreover, for any $R$ for which the result holds, $H^{R}$
is in normal form with respect to ${\cal S}$, at least up to degree $8$.
\end{enumerate}
\end{theorem}
\begin{remark}
The dependence of $\Psi^R$ on $R$ is not continuous but piecewise
analytic.
\end{remark}
\begin{remark}
From the bounds provided by Lemma~\mbox{\rm\ref{lem:lnfgs}} for the
iterative
normal form procedure described in Lemma~\mbox{\rm\ref{lem:lnf}}, this
exponentially small bound seems to be the best that one can obtains
by using this linearly convergent scheme.
\end{remark}
\prova
The proof is done simultaneously for any $00$. We also assume that all the
eigenvalues of $J_m{\cal B}$ are of elliptic type, and that there
exists $\mu_0>0$ and $\gamma>\ell$ such that
$$
|ik^{\top}\hat{\omega}^{(0)}+l^{\top}\lambda|\geq
\frac{\mu_0}{(|k|_1+|l_x-l_y|_1)^{\gamma}}
\;\;
\forall l\in\NN^{2m}\,\forall k\in\ZZ^r\;
\mbox{\rm with } |l_x-l_y|_1+|k|_1\neq 0.
$$
Let $R\in(0,R_0)$, and let us take real
initial conditions at $t=0$ contained in ${\cal D}_{r,m}(0,R)$.
Then, we can define $\alpha>2$ such that, if $R$ is small enough, the
corresponding trajectories belong to ${\cal D}_{r,m}(0,\alpha R)$ for
any time $0\leq t\leq T(R)$, with
$$
T(R)=const.
\exp{\left(const.\left(\frac{1}{R}\right)^{\frac{2}{\gamma+1}}
\right)},
$$
being the constants in the definition of $T(R)$ independent from $R$.
\end{theorem}
\begin{remark}
In the proof, and only for technical reasons,
$\alpha$ depends
on $\rho_0$. Nevertheless, we can take $\alpha$ as close as we want to
$2$ (by taking an initial $\rho_0$ small enough, see the proof for
details), but this implies a reduction on the set of allowed $R$,
and on the constants of the stability time.
The reason that forces to take $\alpha>2$ is the norm used for the
normal variables. If one takes the Euclidean norm instead of the
supremum norm, the condition $\alpha>2$ is replaced by $\alpha>1$.
\end{remark}
\prova
In order to simplify the proof, we assume that the initial real
variables $(x,y)$ of (\ref{eq:rediniham}) correspond to the ones that
put ${\cal B}$ in canonical real form, that is,
$$
z^{\top}{\cal B}z=\sum_{j=1}^m \alpha_j(x_j^2+y_j^2),
$$
with $\lambda_j=i\alpha_j$, $j=1,\ldots,m$. Moreover, we assume that
$R_0<1$. We introduce $(X,Y)$ to denote the complexified variables
\begin{equation}
x_j=\frac{X_j+iY_j}{\sqrt{2}},\;\;\;\;
y_j=\frac{iX_j+Y_j}{\sqrt{2}},\;\;\;\;
j=1,\ldots,m,
\label{eq:complex}
\end{equation}
that put the matrix $J_m{\cal B}$ in the diagonal form
$J_m{\cal B}^*$. Then, we can write the Hamiltonian in these variables
as
$$
{\cal H}=\hat{\omega}^{(0)\top}\hat{I}+\frac{1}{2}Z^{\top}{\cal B^*}Z+
N(X,\hat{I},Y)+S(\hat{\theta},X,\hat{I},Y),
$$
where $N$ can be rewritten as a function of $I$,
$I^{\top}=(\hat{I}^{\top},\tilde{I}^{\top})$, with
$\tilde{I}_j=iX_jY_j=\frac{1}{2}(x_j^2+y_j^2)$, and $S$ verifies
$\overline{\cal N}(S)=0$. This corresponds to the decomposition
(\ref{eq:nfham}) if one puts ${\cal S}={\cal N}$. ${\cal H}$ is defined
on ${\cal D}_{r,m}(\rho_0,R_0/\sqrt{2})$, with bounds
of the following form: $|N|_{\rho_0,R}\leq\hat{N}R^4$
and $|S|_{\rho_0,R}\leq\hat{S}R^3$, for any $00$, we can define
$\alpha=R_1/R_2$.
\fiprova
\section{Estimates on the families of lower dimensional tori}
\label{sec:efldt}
Let us consider the real analytic reduced Hamiltonian $H$ of
(\ref{eq:rediniham}) and a fixed subbundle ${\cal G}$ of
elliptic directions of $J_m{\cal B}$. In Theorem~\ref{teo:nf} we have
proved that, under standard Diophantine conditions, one can put $H$
in normal form with respect to the set ${\cal S}$ (see (\ref{eq:S1})
and (\ref{eq:S2}) for the definition), with an exponentially small
remainder. If we write this seminormal form in terms of the complexified
variables $Z$, and without changing the name of the Hamiltonian, one has
\begin{equation}
H=\omega^{(0)\top}I+\frac{1}{2}\hat{Z}^{\top}\hat{\cal B}^*\hat{Z}+
{\cal F}(I)+\frac{1}{2}\hat{Z}^{\top}{\cal Q}(I)\hat{Z}+
{\cal T}(\hat{\theta},X,\hat{I},Y)+{\cal R}(\hat{\theta},X,\hat{I},Y).
\label{eq:expnfham}
\end{equation}
To explain the notation used, let us recall that the different resonant
terms depend only on $\hat{I}$ and on the products $X_jY_j$,
$j=1,\ldots,m$, but, from the structure of ${\cal S}$, not all
the possible combinations of those monomials take place in
$\overline{\cal M}({\cal S})$. Then, we introduce
$I^{\top}=(\hat{I}^{\top},\tilde{I}^{\top})$, with
$\tilde{I}_j=iX_jY_j$, $j=1,\ldots,m_1$,
and with this definition (\ref{eq:expnfham}) can be described as
follows: the symmetric matrix $\hat{\cal B}^*$ is
defined from ${\cal B}^*$ skipping the $2m_1$ eigenvalues associated
to ${\cal G}$, $J_{m-m_1}\hat{\cal B}^*=\mbox{diag}(\hat{\lambda})$.
${\cal F}$ and ${\cal Q}$ correspond to the normal
form with respect to ${\cal S}$, with the expansion of
${\cal F}$ starting at second order with respect to $I$,
and with ${\cal Q}(0)=0$. It
is not difficult to check that by choosing the variables $\tilde{Z}$ in
suitable form (as it has been done in the proof of
Theorem~\ref{teo:dif}), ${\cal F}$ is real analytic. Moreover,
${\cal Q}$ is a symmetric matrix such that $J_{m-m_1}{\cal Q}$ is
diagonal, ${\cal T}\in{\cal M}({\cal N}\setminus{\cal S})$ (so
${\cal T}\equiv O_3(\hat{Z})$) and
${\cal R}\in\widetilde{\cal M}({\cal S})$.
We assume that this normal form has been done for a given (and small
enough) $R$, as in the formulation of Theorem~\ref{teo:nf}. We only
consider the $R$-dependence when we give the bounds of the different
terms of (\ref{eq:expnfham}). To obtain these bounds, let us
define $\rho_1=3\rho_0/4$, where we recall that $\rho_0$ is the width of
the strip of analiticity, with respect to $\hat{\theta}$, for the
initial Hamiltonian. Then, Theorem~\ref{teo:nf} implies that,
for any $R$ small enough, we have
\begin{equation}
\begin{array}{rclrcl}
|{\cal F}|_{0,R} & \leq & \hat{\cal F}R^4, &
|{\cal F}_3|_{0,R} & \leq & \hat{\cal F}_3R^6, \\
|{\cal Q}|_{0,R} & \leq & \hat{\cal Q}R^2, &
|{\cal Q}_2|_{0,R} & \leq & \hat{\cal Q}_2R^4, \\
|{\cal T}|_{\rho_1,R} & \leq & \hat{\cal T}R^3, &
|{\cal R}|_{\rho_1,R} & \leq & const.\exp{\left(
-const.\left(\frac{1}{R}\right)^{\frac{2}{\gamma+1}}
\right)} R^8,
\end{array}
\label{eq:expnfcot}
\end{equation}
To derive these bounds on ${\cal D}_{r,m}(0,R)$, we have considered
the functions that depend on $\tilde{I}$ as functions of
$\tilde{Z}$. Here, we have split ${\cal F}={\cal F}_2+{\cal F}_3$ and
${\cal Q}={\cal Q}_1+{\cal Q}_2$. ${\cal F}_2$ and the components of
${\cal Q}_1$ are polynomials on $I$ of degrees $2$ and $1$
respectively. ${\cal F}_3$ and ${\cal Q}_2$ contain the remaining
terms. We note that the definition of ${\cal F}_2$ and ${\cal Q}_1$
does not depend on the order of the seminormal form.
This seminormal form has been formally explained in
Section~\ref{sec:fnf}, and we will use the notation related to
(\ref{eq:nfsol}) to represent the normal form tori.
The main purpose of this section is to study the persistence of those
tori when we add the remainder ${\cal R}$. We note that, as $|{\cal
R}|$ is exponentially small with $R$, we can expect that the tori of
(\ref{eq:nfsol}) will survive, except the ones corresponding to a set
of parameters ($I(0)$) of exponentially small measure with respect to
$R$. We will show that this assertion holds, assuming certain
standard nondegeneracy conditions on this family of tori, that have
been explained in Section~\ref{sec:manada} (conditions that, as we
will see, can be checked by computing a normal form up to degree $4$,
that is, from ${\cal F}_2$ and ${\cal Q}_1$). As it is a more natural
parameter, the results will be formulated in terms of frequencies
instead of actions.
\subsection{Nondegeneracy conditions}
Before the rigorous formulation of the results, let us give in explicit
form these nondegeneracy conditions.
\subsubsection{Nondegeneracy of the intrinsic frequencies}
The first one is a standard nondegeneracy condition on the dependence
of the frequencies with respect to the actions: we require
\begin{equation}
\det {\cal C}\ne 0,\;\;\;\;
{\cal C}=\frac{\partial^2 {\cal F}_2}{\partial I^2}(0).
\label{eq:detC}
\end{equation}
This allows to parametrize the tori of the family by their vector of
intrinsic frequencies (instead of $I(0)$). Of course, we have to be
close enough to the initial $r$-dimensional torus. This assertion is
justified by the following lemma:
\begin{lemma}\label{lem:invF}
Let us assume that $\det {\cal C}\neq 0$.
Then, if $R$ is small enough, there exists a real analytic vectorial
function ${\cal I}(\omega)$, defined on the set
\begin{equation}
\left\{\omega\in\CC^{r+m_1}:\; |\omega-\omega^{(0)}|\leq
\frac{1}{8}(|{\cal C}^{-1}|)^{-1}R^2 \right\},
\label{eq:setI}
\end{equation}
such that
$$
\frac{\partial {\cal F}}{\partial I}({\cal I}(\omega))=\omega
-\omega^{(0)},
$$
with ${\cal I}(\omega^{(0)})=0$.
Moreover, we have $|{\cal I}(\omega)|\leq \frac{1}{4}R^2$ for any
$\omega$ in the set \mbox{\rm (\ref{eq:setI})}, and if
$\omega^{(1)}$, $\omega^{(2)}$ belong in
\mbox{\rm (\ref{eq:setI})}, then
$$
|{\cal I}(\omega^{(1)})-{\cal I}(\omega^{(2)})|\leq
2|{\cal C}^{-1}||\omega^{(1)}-\omega^{(2)}|.
$$
Of course, we are still using the notation of
Section~\mbox{\rm\ref{sec:efldt}}.
\end{lemma}
\prova
We have ${\cal F}(I)=\frac{1}{2}I^{\top}{\cal C}I+{\cal F}_3$. Then,
we take a fixed $\omega$ in the set (\ref{eq:setI}), and we want
to solve the equation:
\begin{equation}
{\cal I}(\omega)={\cal C}^{-1}\left(\omega-\omega^{(0)}-
\frac{\partial {\cal F}_3}{\partial I}({\cal I}(\omega))\right).
\label{eq:its}
\end{equation}
Putting the superscripts ``$(k+1)$'' and ``$(k)$'' to
${\cal I}(\omega)$ in (\ref{eq:its}), we can consider this expression as
an iterative
procedure, using ${\cal I}^{(0)}(\omega)=0$ as the seed. If we assume
$|{\cal I}^{(k)}(\omega)|\leq \frac{1}{4}R^2$, then, using
Cauchy inequalities, we have for $R$ small enough,
$$
|{\cal I}^{(k+1)}(\omega)|\leq |{\cal C}^{-1}|\left(
\frac{1}{8}(|{\cal C}^{-1}|)^{-1}R^2+
\frac{\hat{\cal F}_3R^6}{\frac{3}{4}R^2}\right)\leq\frac{1}{4}R^2,
$$
where we have used the bounds of (\ref{eq:expnfcot}) for ${\cal F}_3$,
remarking that $|{\cal F}_3|_{0,R}$ is a bound for the supremum norm
of ${\cal F}_3(I)$ if $|I|\leq R^2$. Moreover, to ensure convergence,
we remark that using the main value theorem one has,
$$
|{\cal I}^{(k+1)}(\omega)-{\cal I}^{(k)}(\omega)|\leq
(r+m_1)\frac{\hat{\cal F}_3R^6}{\left(\frac{3}{8}\right)^2R^4}
|{\cal I}^{(k)}(\omega)-{\cal I}^{(k-1)}(\omega)|\leq
\frac{1}{2}|{\cal I}^{(k)}(\omega)-{\cal I}^{(k-1)}(\omega)|,
$$
if $R$ is small enough. Clearly, the limit function is analytic with
respect to $\omega$, and from the real analytic character of ${\cal
F}$, ${\cal I}$ is in fact real analytic. Taking $\omega^{(1)}$,
$\omega^{(2)}$ in the set (\ref{eq:setI}), one has
$$
{\cal I}(\omega^{(1)})-{\cal I}(\omega^{(2)})=
{\cal C}^{-1}(\omega^{(1)}-\omega^{(2)})+
{\cal C}^{-1}\left(
\frac{\partial {\cal F}_3}{\partial I}({\cal I}(\omega^{(2)}))-
\frac{\partial {\cal F}_3}{\partial I}({\cal I}(\omega^{(1)})) \right),
$$
and with the same arguments previously used, we obtain for $R$ small
enough
$$
|{\cal I}(\omega^{(1)})-{\cal I}(\omega^{(2)})|\leq
2|{\cal C}^{-1}||\omega^{(1)}-\omega^{(2)}|.
$$
\fiprova
\subsubsection{Nondegeneracy of the normal frequencies}
The other nondegeneracy condition considered refers to the normal
eigenvalues. Skipping again the remainder ${\cal R}$, for every
invariant tori parametrized by $I(0)$ in (\ref{eq:nfsol}), the
corresponding normal eigenvalues are the ones of the diagonal matrix
$J_{m-m_1}(\hat{\cal B}^*+{\cal Q}(I(0)))$. Using the parametrization
$I\equiv{\cal I}(\omega)$ provided by Lemma~\ref{lem:invF}, we can
consider those eigenvalues as functions of $\omega$ instead of functions
of $I(0)$.
\begin{equation}
J_{m-m_1}(\hat{\cal B}^*+{\cal Q}({\cal I}(\omega)))\equiv
\mbox{diag}(\hat{\lambda}^{(0)}(\omega)),
\label{eq:nfeig}
\end{equation}
we ask for the condition
\begin{equation}
\Im\left(\frac{\partial}{\partial\omega}(l^{\top}\hat{\lambda}^{(0)})
(\omega^{(0)})\right)\notin\ZZ^{r+m_1},\;\;
l\in\ZZ^{2(m-m1)},\;
0<|l|_1\leq 2,\;\; l_{\hat{X}}\neq l_{\hat{Y}},
\label{eq:eigndc}
\end{equation}
where we have used the notation
$l^{\top}=(l^{\top}_{\hat{X}},l^{\top}_{\hat{Y}})$. To check this
condition, we only need to know the first order approximation to
${\cal Q}$, remarking that from (\ref{eq:nfeig}) one has
\begin{equation}
\hat{\lambda}^{(0)}_j(\omega)=\hat{\lambda}_j+
\left(\frac{\partial}{\partial I}{\cal Q}_{1,j,j+m-m_1}(0)\right)
{\cal C}^{-1}(\omega-\omega^{(0)})+O_2(\omega-\omega^{(0)}),
\label{eq:foeig}
\end{equation}
$j=1,\ldots,m-m_1$, and that
$\hat{\lambda}^{(0)}_{j+m-m_1}=-\hat{\lambda}^{(0)}_{j}$.
With those assumptions and notations, we can formulate the following
results.
\subsection{Main theorems}
\begin{theorem}\label{teo:cfldt}
We consider the real analytic Hamiltonian $H$ of
\mbox{\rm(\ref{eq:rediniham})}, defined on ${\cal
D}_{r,m}(\rho_0,R_0)$ (for some $0<\rho_0<1$ and $R_0>0$), and such
that the first $m_1$ components ($0\leq m_1\leq m$) of the vector
$\lambda$ of eigenvalues of $J_m{\cal B}$ are of elliptic type. Let
${\cal S}\subset {\cal N}$ be the set introduced in
\mbox{\rm(\ref{eq:S1})} and \mbox{\rm(\ref{eq:S2})}. Then, we also
assume that there exists $\mu_0>0$ and $\gamma>r+m_1$ such that
$$
|ik^{\top}\hat{\omega}^{(0)}+l^{\top}\lambda|\geq
\frac{\mu_0}{(|k|_1+|l_x-l_y|_1)^{\gamma}}
\;\;
\forall (l,s)\in{\cal S}\,\forall k\in\ZZ^r\;
\mbox{\rm with } |l_x-l_y|_1+|k|_1\neq 0.
$$
This allows to put $H$ in seminormal form with respect to ${\cal S}$ up
to finite degree. We assume that this seminormal form up to degree
$4$ is nondegenerate, in the sense that the two nondegeneracy conditions
given in \mbox{\rm (\ref{eq:detC})} and \mbox{\rm (\ref{eq:eigndc})}
hold.
Then, there exist a Cantor subset ${\cal A}\subset\RR^{r+m_1}$ such
that, for any $\omega\in{\cal A}$, the Hamiltonian system $H$ has an
invariant $(r+m_1)$-dimensional (complex) torus with $\omega$ as a
vector of basic frequencies, with reducible normal flow. Moreover, if
${\cal A}(A)={\cal U}(A)\cap{\cal A}$ (being ${\cal U}(A)$ the set
defined in \mbox{\rm(\ref{eq:calU})}), then,
$$
\mes({\cal U}(A)\setminus{\cal A}(A))\leq const.
\exp{\left(-const.\left(\frac{1}{A}\right)^{\frac{1}{\gamma+1}}\right)},
$$
where the constants in this bound are independent from $A$.
\end{theorem}
The key to prove Theorem~\ref{teo:cfldt} is the parametrization
${\cal I}(\omega)$ of the invariant tori of the normal form given by
Lemma~\ref{lem:invF}. To construct this function, we take $R$ of
order $\sqrt{A}$ in Theorem~\ref{teo:nf}, and we obtain a Hamiltonian in
normal form with respect to ${\cal S}$ (as the one of
(\ref{eq:expnfham})), with exponentially small bounds for ${\cal R}$ as
a function of $A$, of the same order of the measure of destroyed tori in
Theorem~\ref{teo:cfldt}. Using Lemma~\ref{lem:invF} on
the function ${\cal F}$ of (\ref{eq:expnfham}), we can construct for
any frequency $A$-close to $\omega^{(0)}$ the corresponding action,
$I\equiv{\cal I}(\omega)$, that gives in (\ref{eq:nfsol}) the
invariant torus of the seminormal form having this concrete vector of
intrinsic frequencies. Nevertheless, as the action $I$ can have some of
the $\tilde{I}_j<0$,
the corresponding torus in (\ref{eq:nfsol}) can be complex. This is
not an obstruction to construct an invariant (and complex) torus for
the complete Hamiltonian (\ref{eq:expnfham}), but the final torus and
its reduced variational normal flow can be in $\CC$. If we want to have
real tori in (\ref{eq:nfsol}), we need to take $\omega\in{\cal W}(A)$,
\begin{equation}
{\cal W}(A)=\left\{\omega\in{\cal U}(A):\;\omega=\omega^{(0)}+
\frac{\partial{\cal F}}{\partial I}(I),\mbox{ with } \tilde{I}_j\geq 0,
\; j=1,\ldots,m_1 \right\}.
\label{eq:calW}
\end{equation}
Note that the degenerate (transition) tori have frequencies whose
corresponding actions satisfy $\tilde{I}_j=0$, for some $j$. Then, we
are forced to remove actions in a tiny slice around the hyperplanes
$\tilde{I}_j=0$, that implies to take out in ${\cal W}(A)$ the
corresponding frequencies. Unfortunately, ${\cal F}$ changes with $A$
as $A\rightarrow 0$ by increasing the order up to which this
seminormal form is done. This is necessary because the successive
approximations to ${\cal F}$ given by Theorem~\ref{teo:nf} do not
converge in general, and hence, as we want to eliminate only an
exponentially small set of frequencies, we need to know this map with
an exponentially small precision.
In this context, we have the following result about the
existence of real invariant tori:
\begin{theorem}\label{teo:fldt}
With the same hypoteses as in Theorem~\mbox{\rm\ref{teo:cfldt}}, there
exist a Cantor subset ${\cal A}\subset\RR^{r+m_1}$ such that, for any
$\omega\in{\cal A}$, the Hamiltonian system $H$ has an invariant
$(r+m_1)$-dimensional real torus with vector of basic frequencies
given by $\omega$. The normal flow of this torus can be reduced to
constant coefficients by means of a real change of variables.
${\cal A}$ can be caracterized in the following form: for any $R>0$
small enough, there exists a convergent (partial) seminormal form with
respect to ${\cal S}$ (it takes the form \mbox{\rm(\ref{eq:expnfham})})
defined on ${\cal D}_{r,m}(\rho_1,R)$ (being $\rho_1$ independent from
$R$), and such that if we put
${\cal A}(R^2)={\cal W}(R^2)\cap{\cal A}$ (see \mbox{\rm(\ref{eq:calW})}
for the definition of ${\cal W}(A)$), then, we have
$$
mes({\cal W}(R^2)\setminus{\cal A}(R^2))\leq const.
\exp{\left(-const.\left(\frac{1}{R}\right)^{\frac{2}{\gamma+1}}\right)},
$$
being the constants in this bound independent from $R$.
\end{theorem}
\begin{remark}
Let us explain how the transition set (in the frequency space) from
real to complex tori can be constructed independently from the
seminormal form. For any tori of dimension $s$, $r\leq s0$, such that
$0<\alpha_1^{(0)}\leq
|\hat{\lambda}_j^{(0)}(\omega)-\hat{\lambda}^{(0)}_l(\omega)|$,
$\alpha_1^{(0)}/2\leq|\hat{\lambda}^{(0)}_j(\omega)|\leq
\alpha_2^{(0)}/2$, for any $\omega\in{\cal E}^{(0)}$, $j\neq l$, and
${\cal L}_{{\cal E}^{(0)}}\{\hat{\lambda}^{(0)}_j\}\leq\beta^{(0)}_1$.
We do not give here explicit bounds on the $\breve{\lambda}^{(0)}_j$,
as those functions do not appear in the iterative process, but we
remark that one has that
${\cal L}_{{\cal E}^{(0)}}\{\breve{\lambda}^{(0)}_j\}$ is of order $R$.
This will be used in Section~\ref{sec:bm}. Moreover, if one uses bounds
like (\ref{eq:cotcalF}) for ${\cal F}$ and ${\cal Q}$, and the ones of
(\ref{eq:cotTR}) and (\ref{eq:lipT}) for ${\cal T}^*$, it is not
difficult to check that for certain positive $R$-independent constants
$\hat{\nu}^{(0)}$ and $\tilde{\nu}^{(0)}$, one has
$\|H_*^{(0)}\|_{{\cal E}^{(0)},\rho^{(0)},R^{(0)}}\leq\hat{\nu}^{(0)}$
and ${\cal L}_{{\cal E}^{(0)},\rho^{(0)},R^{(0)}}\{H_*^{(0)}\}
\leq\tilde{\nu}^{(0)}$. Finally, using the bounds of
(\ref{eq:cotTR}) and (\ref{eq:lipR}) for ${\cal R}^*$, one
can bound the size of the
perturbative term $\hat{H}^{(0)}$ by $\|\hat{H}^{(0)}\|_{{\cal
E}^{(0)},\rho^{(0)},R^{(0)}}\leq M$ and ${\cal L}_{{\cal
E}^{(0)},\rho^{(0)},R^{(0)}}\{\hat{H}^{(0)}\} \leq M^{1-\alpha}$. Some
of these bounds are far from optimal, but they suffice for our
purposes.
\subsubsection{The iterative scheme}
Now, we can describe the iterative procedure used to construct
invariant $(r+m_1)$-dimensional tori. This process is given by a
sequence of canonical changes of variables, constructed as the time
one flow of a suitable generating function $S_\omega$. The changes are
constructed to kill the terms that obstructs the existence of an
invariant reduced torus with vector of basic frequencies given by
$\omega$. As usual (to overcome the effect of the small divisors), the
changes are chosen to produce a quadratically convergent scheme,
instead of the linear one of Lemma~\ref{lem:lnf}.
First, we describe a generic step of this iterative process. For this
purpose, we expand the Hamiltonian $H^{(0)}$ in the following form
\begin{equation}
H^{(0)}=a(\theta)+b(\theta)^{\top}\hat{Z} + c(\theta)^{\top} I +
\frac{1}{2}\hat{Z}^{\top}B(\theta)\hat{Z}+I^{\top} E(\theta)\hat{Z} +
\frac{1}{2}I^{\top}C(\theta)I+\Omega(\theta,\hat{X},I,\hat{Y}),
\label{eq:expH0}
\end{equation}
where we do not write explicitly the $\omega$-dependence and where we
have skipped the superscript ``$(0)$'' in the different parts of the
Hamiltonian. From this expansion, we introduce the following
notations: $[H^{(0)}]_{(\hat{Z},\hat{Z})}=B$,
$[H^{(0)}]_{(I,\hat{Z})}=E$ and $=H^{(0)}-\Omega$. From the
bounds on the terms of the decomposition (\ref{eq:h0}), we have that
$\tilde{a}$, $b$, $c-\omega$, $B-\hat{\cal B}^{(0)*}$, $C-{\cal
C}^{(0)}$ and $E$ are all $O(\hat{H}^{(0)})$. Note that if we are able
to kill the terms $\tilde{a}$, $b$ and $c-\omega$, we will obtain an
invariant torus with intrinsic frequency $\omega$. Nevertheless, as
we want to have simple equations at every step of the iterative scheme
(this is, linear equations with constant coefficients), we are forced
to kill something more. Then, we ask the final torus to have reducible
normal flow given by a diagonal matrix. This is, we want that the new
matrix $B$ verifies $B={\cal J}_{m-m_1}(B)$ where, for a
$(2s)$-dimensional matrix $A(\theta)$ depending $2\pi$-periodically on
$\theta$, we define ${\cal J}_s(A)=-J_s \mbox{dp}(J_{s}\bar{A})$.
Here, $\mbox{dp}(A)$ denotes the diagonal matrix obtained taking the
diagonal entries of $A$. Moreover, we have to eliminate $E$ to
uncouple the ``neutral'' and the normal directions of the torus up to
first order. Thus, for each step of the iterative process, we use a
canonical change of variables, given by a generating function of the
form
$$
S(\theta,\hat{X},I,\hat{Y})=\xi^{\top}\theta+ d(\theta) +
e(\theta)^{\top}\hat{Z}+f(\theta)^{\top}I +
\frac{1}{2}\hat{Z}^{\top}G(\theta)\hat{Z}+I^{\top}F(\theta)\hat{Z},
$$
where $\xi\in\CC^{r+m_1}$, $\bar{d}=0$, $\bar{f}=0$ and $G$ is a
symmetric matrix, with ${\cal J}_{m-m_1}(G)=0$. The transformed
Hamiltonian is $H^{(1)}=H^{(0)}\circ\Psi^{S}_1$. We expand $H^{(1)}$
in the same way as $H^{(0)}$ in (\ref{eq:expH0}), keeping the same
name for the new variables, but adding the superscript ``$(1)$'' to
$a$, $b$, $c$, $B$, $C$, $E$ and $\Omega$. Then, we ask
$\tilde{a}^{(1)}=0$, $b^{(1)}=0$, $c^{(1)}-\omega=0$, $E^{(1)}=0$ and
$B^{(1)}={\cal J}_{m-m_1}(B^{(1)})$. We will show that this can be
achieved up to first order in the size of $\hat{H}^{(0)}$. For this
purpose, we write those conditions in terms of the initial Hamiltonian
and the generating function, and then, we obtain the following
equations:
\begin{itemize}
\item[$(eq_1)$] $\tilde{a}-\frac{\partial d}{\partial\theta}\omega=0$,
\item[$(eq_2)$] $b-\frac{\partial e}{\partial\theta}\omega+
\hat{\cal B}^{(0)*}J_{m-m_1}e=0$,
\item[$(eq_3)$] $c-\omega-\frac{\partial f}{\partial\theta}\omega
-{\cal C}^{(0)}\left(\xi+\left(\frac{\partial d}{\partial{\theta}}
\right)^{\top}\right)=0$,
\item[$(eq_4)$] $B^*-{\cal J}_{m-m_1}(B^*)-
\frac{\partial G}{\partial\theta}\omega+
\hat{\cal B}^{(0)*}J_{m-m_1} G-GJ_{m-m_1}\hat{\cal B}^{(0)*}=0$,
\item[$(eq_5)$] $E^*-\frac{\partial F}{\partial\theta}\omega-
FJ_{m-m_1}\hat{\cal B}^{(0)*}=0$,
\end{itemize}
being
\begin{eqnarray*}
B^* & = & B-\left[\frac{\partial H_*^{(0)}}{\partial I}
\left(\xi+\left(\frac{\partial d}{\partial \theta} \right)^{\top}\right)
-\frac{\partial H_*^{(0)}}{\partial\hat{Z}}J_{m-m_1}e
\right]_{(\hat{Z},\hat{Z})}, \\
E^* & = & E-{\cal C}^{(0)}\left(\frac{\partial e}{\partial \theta}
\right)^{\top}-\left[\frac{\partial H_*^{(0)}}{\partial I}
\left(\xi+\left(\frac{\partial d}{\partial\theta}\right)^{\top}
\right)-\frac{\partial H_*^{(0)}}{\partial\hat{Z}}J_{m-m_1}e
\right]_{(I,\hat{Z})}.
\end{eqnarray*}
To solve those homological equations, we expand them in Fourier
series and we equate the corresponding coefficients, obtaining the
formal solutions. The next step is to derive bounds on those
solutions. As we will use these bounds in iterative form, we want
to make clear which expressions change from one
step to another, and which ones can be bounded
independently from the step. For this purpose, we take fixed positive
constants $\bar{m}$, $\hat{m}$, $\tilde{m}$, $\alpha_2$, $\beta_1$,
$\hat{\nu}$, $\tilde{\nu}$ defined as twice the corresponding
inital values $\bar{m}^{(0)}$, $\hat{m}^{(0)}$, $\tilde{m}^{(0)}$,
$\alpha_2^{(0)}$, $\beta_1^{(0)}$, $\hat{\nu}^{(0)}$,
$\tilde{\nu}^{(0)}$ and a fixed $\alpha_1$,
$0<\alpha_1<\alpha_1^{(0)}$. In what follows, $\hat{N}$ will denote an
expression depending only on $\bar{m}$, $\hat{m}$, $\alpha_1$,
$\alpha_2$, $\hat{\nu}$, the different dimensions $r$, $m$, $m_1$,
plus $\gamma$ and $\rho_0$. $\hat{N}$ will be redefined during the
description of the iterative scheme to meet a finite number of
conditions. The idea is to perform the bounds on the iterative scheme
putting the superscript ``$(0)$'' on the terms that change at every
iteration. Hence, we write the bounds on $\hat{H}^{(0)}$ as
$\|\hat{H}^{(0)}\|_{{\cal E}^{(0)},\rho^{(0)},R^{(0)}}\leq M^{(0)}$
and ${\cal L}_{{\cal E}^{(0)},\rho^{(0)},R^{(0)}}\{\hat{H}^{(0)}\}\leq
L^{(0)}$, with $M^{(0)}(R)\equiv M(R)$ and
$L^{(0)}(R)\equiv (M(R))^{1-\alpha}$. Hence, using
Lemma~\ref{lem:normt},
\begin{equation}
\begin{array}{rclrcl}
\|a-\phi^{(0)}\|_{{\cal E}^{(0)},\rho^{(0)}} & \leq & M^{(0)}, &
\|E\|_{{\cal E}^{(0)},\rho^{(0)}} &
\leq & \frac{2(m-m_1)M^{(0)}}{(R^{(0)})^3}, \\
\|c-\omega\|_{{\cal E}^{(0)},\rho^{(0)}} & \leq &
\frac{M^{(0)}}{(R^{(0)})^2}, &
\|B-\hat{\cal B}^{(0)*}\|_{{\cal E}^{(0)},\rho^{(0)}} &
\leq & \frac{(2(m-m_1)+1)M^{(0)}}{(R^{(0)})^2}, \\
\|b\|_{{\cal E}^{(0)},\rho^{(0)}} & \leq & \frac{M^{(0)}}{R^{(0)}}, &
\|C-{\cal C}^{(0)}\|_{{\cal E}^{(0)},\rho^{(0)}} &
\leq & \frac{(2(r+m_1)+1)M^{(0)}}{(R^{(0)})^4}, \\
\|\Omega\|_{{\cal E}^{(0)},\rho^{(0)},R^{(0)}} &
\leq & \hat{\nu}^{(0)}+M^{(0)}. & & &
\end{array}
\label{eq:itbo}
\end{equation}
Moreover, we can use Lemma~\ref{lem:lip} to deduce that the same
bounds hold for their Lipschitz constants on ${\cal E}^{(0)}$,
replacing $M^{(0)}$ by $L^{(0)}$, and $\hat{\nu}^{(0)}$ by
$\tilde{\nu}^{(0)}$. Then, to prove the convergence of the expansion of
$S$, we need some kind of control on the different small divisors
involved. For this purpose, we restrict the parameter $\omega$ to the
subset ${\cal E}^{(1)}(R)\subset{\cal E}^{(0)}(R)$ for which the
following Diophantine estimates hold: we say that $\omega\in{\cal
E}^{(1)}$, if $\omega\in{\cal E}^{(0)}$, and
\begin{equation}
|ik^{\top}\omega+l^{\top}\hat{\lambda}^{(0)}(\omega)|\geq
\frac{\mu^{(0)}(R)}{|k|_1^{\gamma}},
\;\;k\in{\ZZ}^{r+m_1}\setminus\{0\},\;\;l\in\NN^{2(m-m_1)},
\;\;0<|l|_1\leq 2,
\label{eq:dc0}
\end{equation}
for certain $\mu^{(0)}>0$. We expect the measure of ${\cal E}^{(0)}
\setminus{\cal E}^{(1)}$ to be of order $\mu^{(0)}$ and, hence, as we
want to have exponentially small bounds for this measure, we take
$\mu^{(0)}\equiv(M^{(0)})^{\alpha}$. Then, we proceed to bound the
solutions of the different homological equations. For this purpose, we
use Lemma~\ref{lem:normf}. More precisely, we define
$\delta^{(0)}=(M^{(0)})^{\alpha}$, and we take $\delta^{(0)}$ as a
value for $\delta$ to use the different estimates provided by this
lemma. In order to simplify the proofs, we assume
$\rho^{(0)}-N\delta^{(0)}\geq\rho_0/4$, where $N\in\NN$ will be a
fixed integer that will be determined before the description of the
iterative scheme. Moreover, we also assume that
$(M^{(0)})^{\alpha}\leq R^{(0)}\leq 1$. Then, one can solve
$(eq_1)-(eq_5)$ as follows:
\begin{itemize}
\item[$(eq_1)$] For $d$, we have
$$
d(\theta)=\sum_{k\in{\ZZ}^{r+m_1} \setminus \{0\}}
\frac{a_k}{ik^{\top} \omega} \exp(i k^{\top} \theta),
$$
that implies,
$$
\|d\|_{{\cal E}^{(1)},\rho^{(0)}-\delta^{(0)}}\leq
\left(\frac{\gamma}{\delta^{(0)}\exp{(1)}}\right)^{\gamma}
\frac{\|\tilde{a}\|_{{\cal E}^{(1)},\rho^{(0)}}}{\mu^{(0)}}
\leq \hat{N}(M^{(0)})^{1-\alpha-\alpha\gamma}.
$$
\item[$(eq_2)$] For any $j$, $1\leq j\leq 2(m-m_1)$, we have
$$
e_{j}(\theta)=\sum_{k \in {\ZZ}^{r+m_1}}
\frac{b_{j,k}}{ik^{\top}\omega+\hat{\lambda}^{(0)}_j}
\exp(i k^{\top}\theta),
$$
and hence,
$$
\|e\|_{{\cal E}^{(1)},\rho^{(0)}-\delta^{(0)}} \leq
\left( \frac{2}{\alpha_1}+\left(
\frac{\gamma}{\delta^{(0)}\exp{(1)}}\right)^{\gamma}
\frac{1}{\mu^{(0)}}\right)\|b\|_{{\cal E}^{(1)},\rho^{(0)}}\leq
\hat{N}(M^{(0)})^{1-2\alpha-\alpha\gamma}.
$$
\item[$(eq_3)$] Taking average with respect to $\theta$, we obtain
$$
\xi=(\bar{\cal C}^{(0)})^{-1} \left(\bar{c}-\omega-\overline{
{\cal C}^{(0)}\left(\frac{\partial d}{\partial \theta}\right)^{\top}}
\right).
$$
Thus,
\begin{eqnarray*}
\|\xi\|_{{\cal E}^{(1)}} & = &
\|(\bar{\cal C}^{(0)})^{-1}\bar{\cal C}^{(0)}\xi\|_{{\cal E}^{(1)}}\leq
\|(\bar{\cal C}^{(0)})^{-1}\|_{{\cal E}^{(1)}}
\|\bar{\cal C}^{(0)}\xi\|_{{\cal E}^{(1)}} \leq \\
& \leq & \bar{m}\left(\|\bar{c}-\omega\|_{{\cal E}^{(1)},0} +
\left\|{\cal C}^{(0)}\left(\frac{\partial d}{\partial\theta}
\right)^{\top} \right\|_{{\cal E}^{(1)},0}\right) \leq \\
& \leq & \bar{m} \left(\|c-\omega\|_{{\cal E}^{(1)},\rho^{(0)}}+
\hat{m}\frac{\|d\|_{{\cal E}^{(1)},\rho^{(0)}-\delta^{(0)}}}
{(\rho^{(0)}-\delta^{(0)})\exp{(1)}}\right)\leq
\hat{N}(M^{(0)})^{1-\alpha-\alpha\gamma}.
\end{eqnarray*}
To solve the equation for $f$, we define
$$
c^*=\tilde{c}-\tilde{\cal C}^{(0)}\xi-
{\cal C}^{(0)}\left(\frac{\partial d}{\partial \theta}\right)^{\top} +
\overline{{\cal C}^{(0)}\left(\frac{\partial d}{\partial\theta}
\right)^{\top}},
$$
and then, for any $1\leq j\leq r+m_1$, we have
$$
f_j(\theta)=\sum_{k \in {\ZZ}^{r+m_1} \setminus\{0\}}
\frac{c^{*}_{j,k}}{ik^{\top}\omega} \exp(i k^{\top} \theta).
$$
To bound $f$, first we have that
\begin{eqnarray*}
\|c^*\|_{{\cal E}^{(1)},\rho^{(0)}-2\delta^{(0)}} & \leq &
\|\tilde{c}\|_{{\cal E}^{(1)},\rho^{(0)}}+
\|{\cal C}^{(0)}\|_{{\cal E}^{(1)},\rho^{(0)}}
\left(\|\xi\|_{{\cal E}^{(1)}}+
\frac{\|d\|_{{\cal E}^{(1)},\rho^{(0)}-\delta^{(0)}}}
{\delta^{(0)}\exp{(1)}}\right)\leq \\ & \leq &
\hat{N}(M^{(0)})^{1-2\alpha-\alpha\gamma},
\end{eqnarray*}
and from here
$$
\|f\|_{{\cal E}^{(1)},\rho^{(0)}-3\delta^{(0)}}\leq
\left(\frac{\gamma}{\delta^{(0)}\exp{(1)}}\right)^{\gamma}
\frac{\|c^*\|_{{\cal E}^{(1)},\rho^{(0)}-2\delta^{(0)}}}{\mu^{(0)}}
\leq\hat{N}(M^{(0)})^{1-3\alpha-2\alpha\gamma}.
$$
\item[$(eq_4)$] We define $B^{**}=B^*-{\cal J}_{m-m_1}(B^*)$,
and then, if $G=(G_{j,l})$, $1\leq j,l \leq 2(m-m_1)$, we have
$$
G_{j,l}(\theta)=\sum_{k \in {\ZZ}^{r+m_1}}
\frac{B^{**}_{j,l,k}}{ik^{\top}\omega+\hat{\lambda}^{(0)}_{j}+
\hat{\lambda}^{(0)}_{l}}\exp(ik^{\top} \theta).
$$
In this sum we have to avoid the indices $(j,l,k)$ for which
$|j-l|=m-m_1$ and $k=0$. In these cases we have trivial zero divisors,
but also the coefficient $B^{**}_{j,l,0}$ is $0$. Moreover, we remark
that the matrix $G$ is symmetric. Then, to bound $G$, we have to bound
$B^{**}$. First, we have
\begin{eqnarray*}
\lefteqn{
\|B^*-\hat{\cal B}^{(0)*}\|_{{\cal E}^{(1)},\rho^{(0)}-2\delta^{(0)}}
\leq\|B-\hat{\cal B}^{(0)*}\|_{{\cal E}^{(1)},\rho^{(0)}-2\delta^{(0)}}
+} \\ && + (2(m-m_1)+1)(r+m_1)
\frac{\|H_*^{(0)}\|_{{\cal E}^{(1)},\rho^{(0)},R^{(0)}}}{(R^{(0)})^4}
\left(\|\xi\|_{{\cal E}^{(1)}}+
\frac{\|d\|_{{\cal E}^{(1)},\rho^{(0)}-\delta^{(0)}}}
{\delta^{(0)}\exp{(1)}}\right)+\\ & & + 24(m-m_1)^2
\frac{\|H_*^{(0)}\|_{{\cal E}^{(1)},\rho^{(0)},R^{(0)}}}{(R^{(0)})^3}
\|e\|_{{\cal E}^{(1)},\rho^{(0)}-\delta^{(0)}} \leq \hat{N}
(M^{(0)})^{1-6\alpha-\alpha\gamma},
\end{eqnarray*}
and from the definition of $B^{**}$ and the norm used, the same bound
holds for $B^{**}$. Then,
\begin{eqnarray*}
\|G\|_{{\cal E}^{(1)},\rho^{(0)}-3\delta^{(0)}} & \leq &
\left(\frac{1}{\alpha_1}+\left(
\frac{\gamma}{\delta^{(0)}\exp{(1)}}\right)^{\gamma}
\frac{1}{\mu^{(0)}}\right) 2(m-m_1)
\left\|B^{**}\right\|_{{\cal E}^{(1)},\rho^{(0)}-2\delta^{(0)}}
\leq \\ & \leq & \hat{N} (M^{(0)})^{1-7\alpha-2\alpha\gamma}.
\end{eqnarray*}
\item[$(eq_5)$]
The different components of $F$ are given by
$$
F_{j,l}(\theta)=\sum_{k\in {\ZZ}^{r+m_1}}
\frac{E^{*}_{j,l,k}}{ik^{\top}\omega+\hat{\lambda}^{(0)}_l}
\exp(ik^{\top}\theta),
$$
for $j=1,\ldots,r+m_1$ and $l=1,\ldots,2(m-m_1)$. Thus,
\begin{eqnarray*}
\lefteqn{\left\|E^*\right\|_{{\cal E}^{(1)},\rho^{(0)}-2\delta^{(0)}}\le
\|E\|_{{\cal E}^{(1)},\rho^{(0)}} +
2(m-m_1)\|{\cal C}^{(0)}\|_{{\cal E}^{(1)},\rho^{(0)}}
\frac{\|e\|_{{\cal E}^{(1)},\rho^{(0)}-\delta^{(0)}}}
{\delta^{(0)}\exp{(1)}}+} \\
& & + 4(m-m_1)(r+m_1)
\frac{\|H_*^{(0)}\|_{{\cal E}^{(1)},\rho^{(0)},R^{(0)}}}{(R^{(0)})^5}
\left(\|\xi\|_{{\cal E}^{(1)}}+
\frac{\|d\|_{{\cal E}^{(1)},\rho^{(0)}-\delta^{(0)}}}
{\delta^{(0)}\exp{(1)}} \right) + \\ & & +
8(m-m_1)^2 \frac{\|H_*^{(0)}\|_{{\cal E}^{(1)},\rho^{(0)},R^{(0)}}}
{(R^{(0)})^4} \|e\|_{{\cal E}^{(1)},\rho^{(0)}-\delta^{(0)}}\leq
\hat{N}(M^{(0)})^{1-7\alpha-\alpha\gamma},
\end{eqnarray*}
and, hence,
\begin{eqnarray*}
\|F\|_{{\cal E}^{(1)},\rho^{(0)}-3\delta^{(0)}} & \leq &
\left(\frac{2}{\alpha_1}+\left(
\frac{\gamma}{\delta^{(0)}\exp{(1)}}\right)^{\gamma}
\frac{1}{\mu^{(0)}}\right) 2(m-m_1)
\left\|E^{*}\right\|_{{\cal E}^{(1)},\rho^{(0)}-2\delta^{(0)}}
\leq \\ & \leq & \hat{N}(M^{(0)})^{1-8\alpha-2\alpha\gamma}.
\end{eqnarray*}
\end{itemize}
We use these estimates to bound the transformed Hamiltonian $H^{(1)}$.
For this purpose, we define
$H^{(0)*}:=\{H^{(0)},S\}=H^{(0)*}_1+H^{(0)*}_2$, with
$$
H^{(0)*}_1=\left\{\omega^{\top}I+
\frac{1}{2}\hat{Z}^{\top}\hat{\cal B}^{(0)*}\hat{Z}+
\frac{1}{2}I^{\top}{\cal C}^{(0)}I+H_*^{(0)},S\right\},
$$
and $H^{(0)*}_2=\{\hat{H}^{(0)},S\}$. Note that we are splitting the
contributions that are $O_1(\hat{H}^{(0)})$ and
$O_2(\hat{H}^{(0)})$. Then, by construction of $S$, one has
$$
H^{(0)}+H^{(0)*}_1=\phi^{(1)}+{\omega}^{\top}I+
\frac{1}{2}\hat{Z}^{\top}\hat{\cal B}^{(1)*}\hat{Z}+
\frac{1}{2}I^{\top}{\cal C}^{(1)}(\theta)I+ H_*^{(1)},
$$
with $\hat{\cal B}^{(1)*}={\cal J}_{m-m_1}(\hat{\cal B}^{(1)*})$
and $=0$. Hence, $H^{(1)}$ takes the same
form as $H^{(0)}$ in (\ref{eq:h0}) if we define
\begin{equation}
\hat{H}^{(1)}=H^{(0)}\circ\Psi^{S}_{1}-H^{(0)}-H^{(0)*}_1=\int_0^1
\left(H^{(0)*}_2+(1-t)\{H^{(0)*}_1,S\}\right)\circ\Psi^{S}_tdt.
\label{eq:barretH1}
\end{equation}
To bound the different terms of $H^{(1)}$, we use Lemma~\ref{lem:bPb}
to bound the Poisson brackets involved in the previous expressions:
\begin{eqnarray*}
\|H^{(0)*}_1\|_{{\cal E}^{(1)},\rho^{(0)}-4\delta^{(0)},R^{(0)}
\exp{(-\delta^{(0)})}} & \leq &
\hat{N}(M^{(0)})^{1-12\alpha-2\alpha\gamma},
\label{eq:H1*}\\
\|\{H^{(0)*}_1,S\}\|_{{\cal E}^{(1)},\rho^{(0)}-5\delta^{(0)},
R^{(0)}\exp{(-2\delta^{(0)})}} & \leq & \hat{N}(M^{(0)})^{2-24\alpha-
4\alpha\gamma},\label{eq:H1*S}\\
\|H_2^{(0)*}\|_{{\cal E}^{(1)},\rho^{(0)}-4\delta^{(0)},
R^{(0)}\exp{(-\delta^{(0)})}} & \leq & \hat{N}(M^{(0)})^{2-12\alpha-
2\alpha\gamma}.
\end{eqnarray*}
Hence, to bound $\hat{H}^{(1)}$ one only needs to control the effect
of $\Psi^{S}_{t}$. To this end, we remark that from the bounds on the
solutions of $(eq_1)-(eq_5)$, one has
\begin{equation}
\|\nabla S\|_{{\cal E}^{(1)},\rho^{(0)}-4\delta^{(0)},R^{(0)}}\leq
\hat{N}(M^{(0)})^{1-9\alpha-2\alpha\gamma},
\label{eq:nablaS}
\end{equation}
where $\nabla S$ is taken with respect to
$(\theta,\hat{X},I,\hat{Y})$. If we assume that
\begin{equation}
\|\nabla S\|_{{\cal E}^{(1)},\rho^{(0)}-4\delta^{(0)},R^{(0)}}\leq
(R^{(0)})^2\delta^{(0)}\exp{(-1)}/2,
\label{eq:nablaS1}
\end{equation}
then,
$\Psi^{S}_{t}$ is well defined from ${\cal D}_{r+m_1,m-m_1}(
\rho^{(0)}-5\delta^{(0)},R^{(0)}\exp{(-\delta^{(0)})})$ to ${\cal
D}_{r+m_1,m-m_1}(\rho^{(0)}-4\delta^{(0)},R^{(0)})$, for any $-1\leq
t\leq 1$, and for any $\omega\in{\cal E}^{(1)}$
(this follows from Lemma~\ref{lem:canon} and (\ref{eq:exp})).
More precisely, we
have that
\begin{equation}
\|\Psi^{S}_{t}-Id\|_{{\cal E}^{(1)},\rho^{(0)}-5\delta^{(0)},
R^{(0)}\exp{(-\delta^{(0)})}} \leq\|\nabla S\|_{{\cal
E}^{(1)},\rho^{(0)}-4\delta^{(0)},R^{(0)}},
\label{eq:Psi}
\end{equation}
for any $-1\leq t\leq 1$. From (\ref{eq:nablaS}) we have
that (\ref{eq:nablaS1}) holds if
$\hat{N}(M^{(0)})^{1-12\alpha-2\alpha\gamma}\leq 1$, condition that
will follow immediately from the inductive restrictions.
Applying the bounds (\ref{eq:H1*}), (\ref{eq:H1*S}) and (\ref{eq:Psi})
to (\ref{eq:barretH1}) and using Lemma~\ref{lem:comp}, we deduce
\begin{equation}
\|\hat{H}^{(1)}\|_{{\cal E}^{(1)},\rho^{(0)}-6\delta^{(0)},
R^{(0)}\exp{(-3\delta^{(0)})}}\leq\hat{N}(M^{(0)})^{2-24\alpha-
4\alpha\gamma}.
\label{eq:hatH1}
\end{equation}
Moreover, the bound on $H_1^{(0)*}$ produces
\begin{equation}
\left.
\begin{array}{rcl}
\|\phi^{(1)}-\phi^{(0)}\|_{{\cal E}^{(1)}} & \leq &
\hat{N}(M^{(0)})^{1-12\alpha-2\alpha\gamma}, \\
\|\hat{\cal B}^{(1)*}-\hat{\cal B}^{(0)*}\|_{{\cal E}^{(1)}} & \leq &
\hat{N}(M^{(0)})^{1-14\alpha-2\alpha\gamma}, \\
\|{\cal C}^{(1)}-{\cal C}^{(0)}\|_{{\cal E}^{(1)},
\rho^{(0)}-4\delta^{(0)}} & \leq &
\hat{N}(M^{(0)})^{1-16\alpha-2\alpha\gamma}, \\
\|H_*^{(1)}-H_*^{(0)}\|_{{\cal E}^{(1)},\rho^{(0)}-4\delta^{(0)},
R^{(0)}\exp{(-\delta^{(0)})}} & \leq &
\hat{N}(M^{(0)})^{1-12\alpha-2\alpha\gamma}.
\end{array}
\right.
\label{eq:cot1}
\end{equation}
We take $N\geq 6$, and we define
$\rho^{(1)}=\rho^{(0)}-N\delta^{(0)}$, and
$R^{(1)}=R^{(0)}\exp{(-(N-3)\delta^{(0)})}$. Then, it is not difficult
to rewrite the bounds on $H^{(1)}$ as the ones on $H^{(0)}$, but now
on ${\cal D}_{r+m_1,m-m_1}(\rho^{(1)},R^{(1)})$. To iterate this
scheme, we only need to check that the bounds assumed on $H^{(0)}$ to
define $\hat{N}$ still hold on $H^{(1)}$. This is done in the next
section.
\subsubsection{Convergence of the iterative scheme}
Looking at the bounds of the previous section, we take $\alpha>0$
small enough such that, for $s=2(1-16\alpha-2\alpha\gamma)$, we have
$s>1$. Then, assuming $\hat{N}\geq 1$, we define
$M^{(1)}=(\hat{N}M^{(0)})^s$ (note that this is a bound for the norm
of $\hat{H}^{(1)}$ in (\ref{eq:hatH1})). If the
hypotheses needed to iterate hold, we obtain recursively
$M^{(n)}=(\hat{N}M^{(0)})^{s^n}$, and hence, for $R$ small enough, we
have $\lim_{n\rightarrow\infty}M^{(n)}=0$. Let us define
${\cal E}^*(R)$ as the set of parameters $\omega$ for which all the steps
are well defined. We assume that, for any $\omega\in{\cal E}^*(R)$,
the composition of canonical transformations
$\Psi^{*}=\Psi_1^{S^{(0)}}\circ\Psi_1^{S^{(1)}}\circ\ldots$ (being
$S^{(n)}$ the generating function used at the $n$-step of the
iterative procedure) is convergent. Then, the limit
Hamiltonian $H^{*}=H^{(0)}\circ\Psi^{*}$ takes the form:
$$
H^{*}=\phi^{*}(\omega)+{\omega}^{\top}I+
\frac{1}{2}\hat{Z}^{\top}\hat{\cal B}^{**}(\omega)\hat{Z}+
\frac{1}{2}I^{\top}{\cal C}^{*}(\theta,\omega)I+
H_*^{*}(\theta,\hat{X},I,\hat{Y},\omega),
$$
with $=0$. This is, we obtain for any $\omega\in{\cal E}^*$ a
Hamiltonian with an $(r+m_1)$-dimensional reducible torus, with linear
quasiperiodic flow given by $\omega$.
Let us prove that the inductive bounds hold. First, we check that we
can define, recursively, constants $\bar{m}^{(n)}$, $\hat{m}^{(n)}$,
$\alpha_1^{(n)}$, $\alpha_2^{(n)}$ and $\hat{\nu}^{(n)}$, replacing
the initial super-``$(0)$'' ones, such that they are also bounded by
$\bar{m}$, $\hat{m}$, $\alpha_1$, $\alpha_2$ and $\hat{\nu}$,
respectively. To prove that, we note that the expressions
in the right-hand side of (\ref{eq:cot1}) can be bounded by
$(\hat{N}M^{(0)})^{s/2}$ (we remark that the same bound holds for
(\ref{eq:nablaS})). Hence, iterating this bounds, we only need to use
that the sum
\begin{equation}
\sum_{n\geq 0}\left(\hat{N}M^{(0)}\right)^{\frac{s^{n+1}}{2}},
\label{eq:convsum}
\end{equation}
is convergent for $R$ small enough (and in fact, that it goes to zero
when $R$ does), to justify these $n$-independent bounds. The same
arguments can be used to prove that $\|\phi^{*}\|_{{\cal E}^*}<+\infty$.
Here, we only check the bound $\bar{m}^{(n)}\leq\bar{m}$, because is the
only one that does not follow directly: note that one can define
$$
\bar{m}^{(1)}=
\frac{\bar{m}^{(0)}}{1-\bar{m}^{(0)}(\hat{N}M^{(0)})^{s/2}},
$$
and then, taking $R$ small enough, we have $\bar{m}^{(1)}\leq\bar{m}$.
Hence, iterating this definition and assuming
$\bar{m}^{(n)}\leq\bar{m}$ by induction, we have
$$
\bar{m}^{(n)}\leq\bar{m}^{(0)}\prod_{j=0}^{n-1}
\frac{1}{1-\bar{m}(\hat{N}M^{(0)})^{s^{n+1}/2}}.
$$
Under this inductive hypotesis, one can bound $\bar{m}^{(n)}$ by an
infinite product that it is convergent because (\ref{eq:convsum})
does. From here, the bound $\bar{m}^{(n)}\leq\bar{m}$ follows
immediately for $R$ small enough. Finally, with the inductive
definitions $\rho^{(n+1)}=\rho^{(n)}-N\delta^{(n)}$ and
$R^{(n+1)}=R^{(n)}\exp{(-(N-3)\delta^{(n)})}$, $n\geq 0$, we need to
check that $\rho^{(n)}\geq\rho_0/4$ and $R^{(n)}\geq M^{(n)}$. We
remark that, as we take $\delta^{(n)}=(M^{(n)})^{\alpha}$, we have,
\begin{equation}
\sum_{n\geq 0}\delta^{(n)}\leq (M^{(0)})^{\alpha}+\sum_{n\geq 1}
(\hat{N}M^{(0)})^{s^n\alpha}\leq 2(M^{(0)})^{\alpha},
\label{eq:cotdeltan}
\end{equation}
at least for $R$ small enough. Then, as $N$ will be a fixed number,
the bound on $\rho^{(n)}$ is clear, taking $R$ small enough. Moreover,
we also have $R^{(n)}\geq
R^{(0)}\exp{(-\rho_0/4)}>R^{(0)}/2=M^{(0)}\geq M^{(n)}$. To justify
this last inequality, we only need to take $R$ small enough such that
$M^{(1)}\leq M^{(0)}$. Under this assumption, the sequence
$\{M^{(n)}\}_{n\geq 0}$ is clearly decreasing.
Finally, to prove the well defined character of the limit Hamiltonian,
it only remains to check the convergence of $\Psi^*$. To do that we
write, for simplicity, $\Psi^{(n)}=\Psi^{S^{(n)}}_1$ and we define
$\breve{\Psi}^{(n)}=\Psi^{(0)}\circ\ldots\circ\Psi^{(n)}$, for $n\geq
0$. We also put $\rho'_{n}=\rho^{(n)}-\rho_0/8$ and
$R'_{n}=R^{(n)}\exp{(-\rho_0/8)}$, $n\geq 1$. Then, using in inductive
form the bounds (\ref{eq:Psi}), (\ref{eq:nablaS}) and
(\ref{eq:nablaS1}),
it is not difficult to check that from Lemma~\ref{lem:mvt} we have
\begin{eqnarray*}
\lefteqn{
\|\breve{\Psi}^{(n+1)}-\breve{\Psi}^{(n)}\|_{{\cal E}^*,\rho'_{n+2},
R'_{n+2}}\leq} \\
& & \le (1+\hat{\Delta}(\hat{N}M^{(0)})^{\frac{s}{2}-2\alpha})
\|\Psi^{(1)}\circ\ldots\circ\Psi^{(n+1)}-
\Psi^{(1)}\circ\ldots\circ\Psi^{(n)}\|_{{\cal E}^{*},\rho'_{n+2},
R'_{n+2}},
\end{eqnarray*}
where $\hat{\Delta}$ only depends on $r$, $m$, $m_1$, $\rho_0$ and
$\hat{N}$. Iterating this bound and taking $\alpha$ small enough, one
obtains for $R$ small enough
$$
\|\breve{\Psi}^{(n+1)}-\breve{\Psi}^{(n)}\|_{{\cal E}^*,\rho'_{n+2},
R'_{n+2}}\leq \prod_{j=0}^{n}
(1+\hat{\Delta}(\hat{N}M^{(0)})^{\frac{s^{j+1}}{2}-2\alpha})
(\hat{N}M^{(0)})^{\frac{s^{n+2}}{2}}\leq
2(\hat{N}M^{(0)})^{\frac{s^{n+2}}{2}},
$$
where we have used again the convergent character of the sum
(\ref{eq:convsum}). From this bound, it is clear that if $p>q\geq 0$,
then
$$
\|\breve{\Psi}^{(p)}-\breve{\Psi}^{(q)}\|_{{\cal E}^*,\rho_0/8,
R^{(0)}\exp{(-3\rho_0/8)}}\leq\sum_{j\geq q}
2(\hat{N}M^{(0)})^{\frac{s^{n+2}}{2}},
$$
bound that goes to zero as $p,q\rightarrow +\infty$. This allows
to check that the limit canonical transformation $\Psi^{*}$ goes
from ${\cal D}_{r+m_1,m-m_1}(\rho_0/8,R^{(0)}\exp{(-3\rho_0/8)})$
to ${\cal D}_{r+m_1,m-m_1}(\rho^{(0)},R^{(0)})$.
\subsubsection{Bounds on the measure}\label{sec:bm}
Then, we have shown the existence of real invariant reducible tori for
a set of parameters $\omega\in{\cal E}^*$. It only remains to bound
the measure of ${\cal E}^*$ or, equivalently, the measure of the
complementary set. To do that, we start recalling how ${\cal E}^*$ is
constructed. Iterating the definition of ${\cal E}^{(1)}$ from ${\cal
E}^{(0)}$, we define ${\cal E}^{(n+1)}$ from ${\cal E}^{(n)}$ in the
same way as it has been done in (\ref{eq:dc0}), replacing
$\mu^{(0)}\equiv (M^{(0)})^{\alpha}$ by $\mu^{(n)}\equiv
(M^{(n)})^{\alpha}$. Then, we have ${\cal E}^*=\cap_{n\geq 1}{\cal
E}^{(n)}$. This is, ${\cal E}^*$ is constructed by taking out, in
recursive form, the set of parameters $\omega$ for which the
Diophantine conditions (\ref{eq:dc0}), formulated on the eigenvalues
of the previous step and depending on the size of the remaining
perturbative terms, do not hold. Then, the set of removed parameters
can be obtained as union of sets for which one of those conditions is
not satisfied at some step of the iterative process.
To estimate the size of the removed sets, we will use a Lipschitz
condition with respect to $\omega$ for the different eigenvalues
$\hat{\lambda}_j^{(n)}$ of ${\cal B}^{(n)*}$, for $n\geq 0$. To this
end, we will prove that this kind of regularity holds for the
successive transformed Hamiltonians. As this condition holds for the
initial one, we have to check, by induction, that the canonical
transformations used preserve this kind of dependence. The key point
is to bound the Lipschitz constants of the different solutions of
$(eq_1)-(eq_5)$. To do it, we recall that we have bounds like the ones
of (\ref{eq:itbo}) for the Lipschitz constants of the different terms
of the decomposition (\ref{eq:h0}) of $H^{(0)}$. Then, we only have
to prove that those bounds for the Lipschitz constants, can be
iterated in the same way as the bounds on the norms. To see that, we
can use the different results given in item $(a)$ of
Lemma~\ref{lem:lip} to bound the Lipschitz constants of the solutions
of $(eq_1)-(eq_5)$. We remark that, for the denominators that appear
solving these equations, we have
$$
{\cal L}_{{\cal E}^{(0)}}\{ik^{\top}\omega+
l^{\top}\hat{\lambda}^{(0)}\}\leq |k|_1+\beta_1^{(0)}|l|_1.
$$
Then, combining Lemma~\ref{lem:lip} with standard inequalities to bound
the Lipschitz constants of sums and products, it is not difficult to
check that one can iterate bounds of the following form:
\begin{eqnarray*}
{\cal L}_{{\cal E}^{(1)},\rho^{(1)},R^{(1)}}\{\hat{H}^{(1)}\} & \leq &
\tilde{N}(M^{(0)})^{2s_1},\\
{\cal L}_{{\cal E}^{(1)},\rho^{(1)},R^{(1)}}\{\hat{\cal B}^{(1)*}-
\hat{\cal B}^{(0)*}\}\ & \leq & \tilde{N}(M^{(0)})^{s_1},\\
{\cal L}_{{\cal E}^{(1)},\rho^{(1)},R^{(1)}}\{{\cal C}^{(1)}-
{\cal C}^{(0)}\} & \leq & \tilde{N}(M^{(0)})^{s_1},\\
{\cal L}_{{\cal E}^{(1)},\rho^{(1)},R^{(1)}}\{H_*^{(1)}-H_*^{(0)}\}
& \leq & \tilde{N}(M^{(0)})^{s_1},
\end{eqnarray*}
that are analogous to the ones of (\ref{eq:hatH1}) and
(\ref{eq:cot1}). $\tilde{N}\geq 1$ depends on the same parameters as
$\hat{N}$, plus $\tilde{m}$, $\tilde{\nu}$ and $\beta_1$. Moreover,
taking $\alpha$ small enough, we have $2s_1>1$. Here, the selection of
$N$ (used to define $\rho^{(1)}$ and $R^{(1)}$) is done depending on
the number of times that we need to use Cauchy estimates to bound the
different norms and Lipschitz constants. Iterating those expressions,
it is not difficult to check (by induction) that we can define
inductively $\tilde{m}^{(n)}$, $\tilde{\nu}^{(n)}$ and $\beta_1^{(n)}$
for which the assumed $n$-independent bounds hold. The deduction of
those Lipschitz bounds is tedious but it only involves simple
inequalities. Full details in a very similar context can be found in
\cite{JS94} or \cite{JV96}.
Let us particularize those bounds on the eigenvalues of
${\cal B}^{(n)*}$. If we expand $\hat{\lambda}_j^{(n)}$,
$j=1,\ldots,2(m-m_1)$, $n\geq 0$, as in (\ref{eq:eig0}), replacing only
the superscript ``${(0)}$'' by ``${(n)}$'', we have that
${\cal L}_{{\cal E}^{(n)}}\{\breve{\lambda}^{(n)}_j\}\leq \breve{N}R$,
being $\breve{N}$ a positive constant independent from $R$, $j$ and $n$.
To justify this assertion, we note that it holds for $n=0$, and that
the contributions that come from the next steps are exponentially
small with $R$.
Those bounds on the Lipschitz constants of $\lambda_j^{(n)}$ plus the
nondegeneracy conditions (\ref{eq:eig0}) are the key to
control the measure of ${\cal E}^{(n)}\setminus{\cal E}^{(n+1)}$. We
consider the decomposition
$$
{\cal E}^{(n)}\setminus{\cal E}^{(n+1)}=
\bigcup_{\begin{array}{c} l\in\ZZ^{2(m-m_1)} \\ 0<|l|_1\leq 2 \\
l_{\hat{X}}\neq l_{\hat{Y}} \end{array}}
\bigcup_{k\in\ZZ^{r+m_1}\setminus\{0\}} {\cal R}_{l,k}^{(n)},
$$
with
$$
{\cal R}_{l,k}^{(n)}(R)=\left\{\omega\in{\cal E}^{(n)}(R):\;
|ik^{\top}\omega+l^{\top}\hat{\lambda}^{(n)}(\omega)|<
\frac{\mu^{(n)}(R)}{|k|_1^{\gamma}}\right\}.
$$
To estimate the measure of ${\cal R}_{l,k}^{(n)}$, we take
$\omega^{(1)}$ and $\omega^{(2)}$ in this set and then, we have
$$
|ik^{\top}(\omega^{(1)}-\omega^{(2)})+
l^{\top}(\hat{\lambda}^{(n)}(\omega^{(1)})-
\hat{\lambda}^{(n)}(\omega^{(2)}))|<
\frac{2\mu^{(n)}}{|k|_1^{\gamma}}.
$$
Let us start with the case $|l|_1=1$. Then,
$l^{\top}\hat{\lambda}^{(n)}=\hat{\lambda}^{(n)}_j$ for some
$j=1,\ldots,2(m-m_1)$. Hence, the previous expression can be
rewritten as
$$
|i(k+v_j)^{\top}(\omega^{(1)}-\omega^{(2)})+
\breve{\lambda}^{(n)}_j(\omega^{(1)})-
\breve{\lambda}^{(n)}_j(\omega^{(2)})|<
\frac{2\mu^{(n)}}{|k|_1^{\gamma}}.
$$
Assuming that $\omega^{(1)}-\omega^{(2)}$ is parallel to $k+\Re(v_j)$,
we have
\begin{eqnarray*}
|{\omega}^{(1)}-{\omega}^{(2)}|_2 & = &
\frac{|(k+\Re(v_j))^{\top}({\omega}^{(1)}-{\omega}^{(2)})|}
{|k+\Re(v_j)|_2} \leq
\frac{|(k+v_j)^{\top}({\omega}^{(1)}-{\omega}^{(2)})|}{|k+\Re(v_j)|_2}
\leq \\ & \leq & \frac{1}{|k+\Re(v_j)|_2}\left(|
\breve{\lambda}^{(n)}_j(\omega^{(1)})-
\breve{\lambda}^{(n)}_j(\omega^{(2)})|+\frac{2\mu^{(n)}}{|k|_1^{\gamma}}
\right)\leq \\ & \leq &
\frac{1}{|k+\Re(v_j)|_2}\left(\breve{N}R|{\omega}^{(1)}-{\omega}^{(2)}|
+\frac{2\mu^{(n)}}{|k|_1^{\gamma}}\right).
\end{eqnarray*}
being $|.|_2$ the Euclidean norm of a real vector. Using that
$\Re(v_j)\neq 0$ (see (\ref{eq:eig0})), we obtain that there exists a
positive constant $\Pi_1$, independent from $j$, $k$ and $n$, such that
$$
|{\omega}^{(1)}-{\omega}^{(2)}|_2\leq
\Pi_1\frac{\mu^{(n)}}{|k|_1^{\gamma}},
$$
for $R$ small enough.
In fact, this bound can be extended to the case $|l|_1=2$,
$l_x\neq l_y$, using that $\Re(v_{j_1,j_2})\neq 0$ if $j_1\neq j_2$.
This is a bound for the width of a section of ${\cal R}^{(n)}_{l,k}$
by a line in the direction $k+\Re(v_j)$. Then, the measure of
${\cal R}^{(n)}_{l,k}$ can be bounded by
$$
\mbox{mes}({\cal R}_{l,k})\leq \Pi_1\frac{\mu^{(n)}}{|k|_1^{\gamma}}
\left(\sqrt{r+m_1}\frac{1}{4}(|{\cal C}^{-1}|)^{-1}R^2
\right)^{r+m_1-1},
$$
where $2\sqrt{r+m_1}\frac{1}{8}(|{\cal C}^{-1}|)^{-1}R^2$ is a bound
for the diameter of ${\cal E}^{(0)}(R)$. Then, we have
$$
\mbox{mes}({\cal E}^{(n)}\setminus{\cal E}^{(n+1)})\leq
\Pi_2 R^{2(r+m_1-1)}\mu^{(n)}\sum_{k\in\ZZ^{r+m_1}\setminus\{0\}}
\frac{1}{|k|_1^{\gamma}},
$$
where $\Pi_2$ does not depend on $n$ and $R$. Using that
$\#\{k\in\ZZ^{r+m_1}:\;|k|_1 =j\}\leq 2(r+m_1)j^{r+m_1-1}$ and that
$\gamma>r+m_1$ we obtain
$$
\mbox{mes}({\cal E}^{(n)}\setminus{\cal E}^{(n+1)})\leq
\Pi_2 R^{2(r+m_1-1)}\mu^{(n)}\sum_{j\geq 1}2(r+m_1)j^{r+m_1-1-\gamma}
\leq\Pi_3R^{2(r+m_1-1)}\mu^{(n)},
$$
being $\Pi_3$ also independent from $n$ and $R$. As
$\mu^{(n)}=(M^{(n)})^{\alpha}$, we deduce, using (\ref{eq:cotdeltan}),
that for $R\leq 1$ small enough,
$$
\mbox{mes}({\cal E}^{(0)}\setminus{\cal E}^{*})\leq
\Pi_3R^{2(r+m_1-1)}\left((M^{(0)})^{\alpha}+
\sum_{n\geq 1}(\hat{N}M^{(0)})^{s^{n}\alpha}\right)\leq
2\Pi_3(M^{(0)})^{\alpha}.
$$
Taking into account the bound on the measure of ${\cal
W}\left(\frac{1}{8}(|{\cal C}^{-1}|)^{-1}R^2\right)\setminus {\cal
E}^{(0)}$ (we have shown, from (\ref{eq:meswt}), that it is of order
$(M^{(0)})^{2\alpha}$), one obtains the exponentially small bounds on
the measure of destroyed tori. To finish the proof, we define ${\cal
A}$ as $\cup_{00$, $2\pi$-periodic on
$\theta$, and taking values in $\CC$. Let us denote by $f_k$ the Fourier
coefficients of $f$, $f=\sum_{k\in\ZZ^{r}}f_k\exp{(ik^{\top}\theta)}$.
Then, we have:
\begin{itemize}
\item[$(i)$] $|f_k|\leq |f|_{\rho}\exp{(-|k|_1 \rho)}$.
\item[$(ii)$] $|f g|_{\rho} \leq |f|_{\rho} |g|_{\rho}$.
\item[$(iii)$] For every $0<\delta<\rho$,
$$
\left|\frac{\partial f}{\partial\theta_j}\right|_{\rho-\delta} \leq
\frac{|f|_{\rho}}{\delta\exp(1)}, \; j=1, \ldots,r.
$$
\item[$(iv)$] Let $\{d_k\}_{k\in{\ZZ}^{r}\setminus\{0\}}\subset\CC$,
with $|d_k|\geq\frac{\mu}{|k|_1^{\gamma}}$, for some $\mu>0$ and
$\gamma\geq 0$. If we assume that $\bar{f}=0$, then, for any
$0<\delta<\rho$, we have that the function $g$
defined as
$$
g(\theta) = \sum_{k \in \ZZ^{r}\setminus \{0\}}
\frac{f_k}{d_k} \exp{(i k^{\top} \theta)},
$$
satisfies the bound
$$
|g|_{\rho-\delta}\leq\left(\frac{\gamma}{\delta\exp(1)}\right)^{\gamma}
\frac{|f|_{\rho}}{\mu}.
$$
\end{itemize}
All these bounds can be extended to the case in which $f$ and $g$ take
values in $\CC^{n_1}$ or $\MM_{n_1,n_2}(\CC)$.
\end{lemma}
\prova
Items {\em (i)} and {\em (ii)} are easily verified. Proofs of
{\em (iii)} and {\em (iv)} follows immediately using (\ref{eq:cotpd}).
\fiprova
\begin{lemma}\label{lem:normt}
Let $f(\theta,x,I,y)$ and $g(\theta,x,I,y)$ be analytic functions
on ${\cal D}_{r,m}(\rho,R)$,
and $2\pi$-periodic on $\theta$. Then,
\begin{itemize}
\item[$(i)$] If
$f=\sum_{(l,s)\in\NN^{2m}\times\NN^r} f_{l,s}(\theta)z^l\hat{I}^s$,
we have
$|f_{l,s}|_{\rho}\leq\frac{|f|_{\rho,R}}{R^{|l|_1+2|s|_1}}$.
\item[$(ii)$] $|fg|_{\rho,R}\leq |f|_{\rho,R} |g|_{\rho,R}$.
\item[$(iii)$] For every $0<\delta<\rho$ and $0<\chi<1$, we have for
$j=1,\ldots,r$ and $k=1,\ldots,2m$:
$$
\left|\frac{\partial f}{\partial\theta_j}\right|_{\rho-\delta,R}\leq
\frac{|f|_{\rho,R}}{\delta\exp(1)},\;\;\;\;
\left|\frac{\partial f}{\partial I_j}\right|_{\rho,R\chi}\leq
\frac{|f|_{\rho,R}}{(1-\chi^2)R^2}, \;\;\;\;
\left|\frac{\partial f}{\partial z_k}\right|_{\rho,R\chi}\leq
\frac{|f|_{\rho,R}}{(1-\chi)R}.
$$
\end{itemize}
As in Lemma~\mbox{\rm\ref{lem:normf}}, all the bounds hold if $f$ and
$g$ take values in $\CC^{n_1}$ or $\MM_{n_1,n_2}(\CC)$.
\end{lemma}
\prova
The proof of {\em (i)} and {\em (ii)} is straightforward.
{\em (iii)} is proved using item {\em (iii)} of
Lemma~\mbox{\rm\ref{lem:normf}} and applying Cauchy estimates
to the function
$\sum_{(l,s)\in\NN^{2m}\times\NN^r}|f_{l,s}|_{\rho}z^l\hat{I}^s$.
\fiprova
\begin{lemma}\label{lem:bPb}
Let us consider $f(\theta,x,I,y)$ and $g(\theta,x,I,y)$ complex-valued
functions, such that $f$ and $\nabla g$ are analytic functions defined
on ${\cal D}_{r,m}(\rho,R)$, $2\pi$-periodic on $\theta$.
Then, for every $0<\delta<\rho$ and $0<\chi<1$, we have:
\begin{eqnarray*}
|\{f,g\}|_{\rho-\delta,R\chi} & \leq &
\frac{r|f|_{\rho,R}}{\delta\exp{(1)}}
\left|\frac{\partial g}{\partial I}\right|_{\rho-\delta,R\chi}+
\frac{r|f|_{\rho,R}}{R^2(1-\chi^2)}
\left|\frac{\partial g}{\partial\theta}
\right|_{\rho-\delta,R\chi}
+\frac{2m|f|_{\rho,R}}{R(1-\chi)}
\left|\frac{\partial g}{\partial z}\right|_{\rho-\delta,R\chi}.
\end{eqnarray*}
\end{lemma}
\begin{remark}\label{rem:fte}
If $f$ has a finite Taylor expansion with respect to $(I,z)$, the
expressions in the bound of $|\{f,g\}|_{\rho-\delta,R\chi}$ that come
from the Cauchy estimates on the derivatives of $f$ with respect to
$I$ or $z$, can be replaced by bounds on the degree of the different
Taylor expansions. Moreover, if $f$ does not depend on $\theta$, the
first term on the bound can be eliminated. Similar comments can be
extended to $\nabla g$. This remark has been used in the proof of
Lemma~\mbox{\rm\ref{lem:lnfgs}}.
\end{remark}
\prova
It follows from Lemma~\ref{lem:normt}.
\fiprova
\begin{lemma}\label{lem:comp}
Let us take $0<\rho_0<\rho$ and $00$ and $R>0$, being
$\nabla S$ analytic on ${\cal D}_{r,m}(\rho,R)$ and
$2\pi$-periodic on $\theta$. If we assume that
$$
\left|\frac{\partial S}{\partial\theta}\right|_{\rho,R} \leq
R^2(1-\chi^2), \;\;\;\;
\left|\frac{\partial S}{\partial I}\right|_{\rho,R}\leq\delta, \;\;\;\;
\left|\frac{\partial S}{\partial z}\right|_{\rho,R}\leq R(1-\chi),
$$
for certain $0<\chi<1$ and $0<\delta<\rho$, then one has
\begin{itemize}
\item[$(a)$] $\Psi^S_t:{\cal D}_{r,m}(\rho-\delta,R\chi)\longrightarrow
{\cal D}_{r,m}(\rho,R)$, for every $-1\leq t\leq 1$, where $\Psi^S_t$
is the flow time $t$ of the Hamiltonian system given by $S$.
\item[$(b)$] If one writes $\Psi^S_t-Id=(\Theta_t^S,{\cal X}^S_t,
{\cal I}^S_t,{\cal Y}^S_t)$, then, for every $-1\leq t\leq 1$, we have
that $\Theta_t^S$, ${\cal Y}^S_t$ and
${\cal Z}^S_t=({\cal X}^S_t,{\cal Y}^S_t)$ are analytic functions on
${\cal D}_{r,m}(\rho-\delta,R\chi)$,
$2\pi$-periodic on $\theta$. Moreover, the following bounds hold:
$$
\left|\Theta_t^S\right|_{\rho-\delta,R\chi}\leq
\left|\frac{\partial S}{\partial I}\right|_{\rho,R},\;\;\;\;
\left|{\cal I}^S_t\right|_{\rho-\delta,R\chi} \leq
\left|\frac{\partial S}{\partial\theta}\right|_{\rho,R},\;\;\;\;
\left|{\cal Z}^S_t\right|_{\rho-\delta,R\chi}\leq
\left|\frac{\partial S}{\partial z}\right|_{\rho,R}.
$$
\end{itemize}
\end{lemma}
\prova
A similar result can be found in \cite{DG95}, where it is proved
working with the supremum norm. The ideas are basically the same, but
here we use Lemma~\ref{lem:comp} to bound the composition of functions.
\fiprova
\begin{lemma}\label{lem:Poincv}
Let $I^{(0)},I^{(1)}\in\RR$, $L^2\leq I^{(0)},I^{(1)}$, with $L>0$, and
let us consider the functions $f_{I^{(j)}}(I)=\sqrt{I^{(j)}+I}$,
$j=0,1$. Then, for every $00$, $\gamma\geq 0$, $A\geq 0$ and $B\geq 0$. We
assume $\bar{f}=0$ for every $\varphi\in{\cal E}$, and we consider the
function $g(\theta,\varphi)$ defined from $f$ and $\{d_k(\varphi)\}$ as
in the item $(iv)$ of Lemma~\mbox{\rm\ref{lem:normf}}. Then, for any
$0<\delta<\rho$, we have:
\begin{eqnarray*}
{\cal L}_{{\cal E},\rho-\delta}\{g\} & \leq &
\left(\frac{\gamma}{\delta\exp(1)}\right)^{\gamma}
\frac{{\cal L}_{\cal E,\rho}\{f\}}{\mu} +
\left(\frac{2\gamma+1}{\delta\exp(1)}\right)^{2\gamma+1}
\frac{\|f\|_{{\cal E},\rho }} {\mu^2}B +
\\ & & +
\left(\frac{2\gamma}{\delta\exp(1)} \right)^{2\gamma}
\frac{\|f\|_{{\cal E},\rho }}{\mu^2}A.
\end{eqnarray*}
\end{itemize}
\item[$(b)$]
\begin{itemize}
\item[$(i)$] If
$g=\sum_{(l,s)\in\NN^{2m}\times\NN^r}g_{l,s}(\theta,\varphi)z^lI^s$,
then ${\cal L}_{{\cal E},\rho} \{g_{l,s}\}\leq \frac{{\cal L}_{{\cal
E},\rho,R}\{g\}}{R^{|l|_1+2|s|_1}}$.
\item[$(ii)$] For every $0<\delta<\rho$ and $0<\chi<1$, we have for
$j=1,\ldots,r$ and $k=1,\ldots,2m$:
\begin{eqnarray*}
{\cal L}_{{\cal E},\rho-\delta}
\left\{\frac{\partial g}{\partial\theta_j}\right\} & \leq &
\frac{{\cal L}_{{\cal E},\rho,R}\{g\}}{\delta\exp{(1)}},\;\;\;\;
{\cal L}_{{\cal E},\rho,R\chi}
\left\{\frac{\partial g}{\partial I_j}\right\}\leq
\frac{{\cal L}_{{\cal E},\rho,R}\{g\}}{(1-\chi^2)R^2},\\
{\cal L}_{{\cal E},\rho,R\chi}
\left\{\frac{\partial g}{\partial z_k}\right\} & \leq &
\frac{{\cal L}_{{\cal E},\rho,R}\{g\}}{(1-\chi)R}.
\end{eqnarray*}
\end{itemize}
\end{itemize}
\end{lemma}
\prova
It can be immediately verified, using the same ideas as in
Lemmas~\ref{lem:normf} and \ref{lem:normt}, plus standard inequalities
for the Lipschitz dependence.
\fiprova
\section{Acknowledgements}
The authors want to thank interesting discussions with R. de la Llave,
and remarks from A. Delshams and C. Sim\'o.
The research of A. Jorba has been supported by the Spanish grant
DGICYT PB94--0215, the EC grant ER\-BCHRXCT\-940460, and the Catalan
grant CIRIT GRQ93--1135. The research of J. Villanueva has been
supported by the UPC grant PR9409.
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\end{document}