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\title{On the Persistence of Lower Dimensional Invariant Tori under
Quasiperiodic Perturbations}
\author{\`Angel Jorba\thanks{{\tt jorba@ma1.upc.es}}\mbox{\ } and
Jordi Villanueva\thanks{{\tt jordi@tere.upc.es}} \\
Departament de Matem\`atica Aplicada I \\
Universitat Polit\`ecnica de Catalunya \\
Diagonal 647, 08028 Barcelona, Spain.}
\date{ }
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\begin{document}
\maketitle
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\begin{abstract}
In this work we consider time dependent quasiperiodic perturbations
of autonomous Hamiltonian systems. We focus on the effect that this
kind of perturbations has on lower dimensional invariant tori. Our
results show that, under standard conditions of analyticity,
nondegeneracy and
nonresonance, most of these tori survive, adding the frequencies of
the perturbation to the ones they already have.
The paper also contains estimates on the amount of surviving tori. The
worst situation happens when the initial tori are normally elliptic.
In this case, a torus (identified by the vector of intrinsic
frequencies)
can be continued with respect to a perturbative
parameter $\epsilon\in[0,\epsilon_0]$, except for a set of
$\epsilon$ of measure exponentially small with $\epsilon_0$.
In case that $\epsilon$ is fixed (and sufficiently small), we prove the
existence of invariant tori for every vector of
frequencies close to the one of the initial torus, except for a set of
frequencies of measure exponentially small with the distance to the
unperturbed torus. As a particular case, if the perturbation is
autonomous, these results also give the same kind of estimates
on the measure of destroyed tori.
Finally, these results are applied to some problems of
celestial mechanics, in order to help in the description of the
phase space of some concrete models.
\end{abstract}
\newpage
\tableofcontents
\newpage
%%%%
\markboth{Persistence of Lower Dimensional Tori}
{A. Jorba and J. Villanueva}
%%%%
\section{Introduction}
Let $H$ be an autonomous Hamiltonian system with $\ell$ degrees
of freedom, having the origin as an elliptic equilibrium point.
If we take the linearization at this point as a first approximation
to the dynamics,
we see that all the solutions are quasiperiodic and can be described
as the product of $\ell$ linear oscillators. The solutions of each
oscillator can be parametrized by the amplitude of the orbits.
When the nonlinear part is
added, each oscillator becomes a one parametric family of
periodic orbits (usually called Lyapounov orbits), that can
be still parametrized by the amplitude, at least near the origin
(see \cite{SM}).
Generically, the frequency of these orbits varies with their amplitude.
The effect that the nonlinear part of the Hamiltonian has on
the quasiperiodic solutions is more complex. Without going into
the details, KAM theorem states that under generic
conditions of nonresonance on the frequencies of the oscillators
and generic conditions of nondegeneracy on the nonlinear part
of the Hamiltonian, most of these solutions still survive. Their
frequencies now vary with the amplitude, and the measure of the
destroyed tori is exponentially small with the distance to the
origin (see \cite{DG}).
Usually these results are proved putting first the Hamiltonian into
the more general form
$$
H=H_0(I)+H_1(\theta,I),\;\;\;\;
I=(I_1,\ldots,I_{\ell}),\;\;
\theta=(\theta_1,\ldots,\theta_{\ell}),
$$
where $H_1$ is small near the origin. This can be achieved,
for instance, applying some steps of the process to put the Hamiltonian
in (Birkhoff) normal form. If we neglect $H_1$, each
quasiperiodic solution takes place on a torus $I=I^{*}$ with
frequencies given by $\nabla H_0(I^{*})$. Here the question
is if these invariant tori are preserved when the perturbing
term $H_1$ is added. The usual hypothesis are, essentially, two:
\begin{enumerate}
\item Nonresonance. The frequencies of the torus must satisfy
a Diophantine condition:
$$
|k^{\top}\nabla H_0(I^{*})| \geq \frac{\cd}{|k|_1^{\gamma}},\;\;\;\;
\gamma > \ell-1,
$$
where $k^{\top}\nabla H_0(I^{*})$ denotes the scalar product of
$k$ with the gradient of $H_0$, and $|k|_1=|k_1|+\cdots+|k_{\ell}|$.
\item Nondegeneracy. The frequencies must depend on the actions:
$$
\det\left(
\frac{\partial^2 H_0}{\partial I^2}(I^{*})
\right)\ne 0.
$$
\end{enumerate}
The necessity of the first hypothesis comes from the fact that,
during the proof, we obtain the divisors $k^{\top}\nabla H_0(I^{*})$.
Hence, if they are too small it is not possible to prove
the convergence of the series that appear in the proof (see
\cite{A63} for the details).
An interesting case is when $k^{\top}\nabla H_0(I^{*})$ is exactly
zero, for some $k$. This implies that, as the frequencies are
rationally dependent, the flow on the torus $I=I^{*}$ is not dense.
More precisely,
if one has $\ell_i$ independent frequencies, the torus $I=I^{*}$
contains an ($\ell-\ell_i$)-family of $\ell_i$-dimensional invariant
tori, and each of these tori is densely filled up by the flow. Here
the natural problem is also to study the persistence of these
lower dimensional invariant tori when the nonintegrable part
$H_1$ is taken into account. Generically, some of these tori survive
but their normal behaviour can be either elliptic or hyperbolic
(see \cite{T}, \cite{LW}, \cite{El94}, \cite{LW} and \cite{JLZ}).
The invariant manifolds associated to the hyperbolic directions
of these tori (usually called ``whiskers'') seem to be the
skeleton that organizes the diffusion (see \cite{A64}).
Moreover, there are other families of lower dimensional tori that come
from the Hamiltonian in normal form $H_0$. They can be obtained
combining some of the elliptic directions associated to the fixed
point, that is, they come from the
product of some of the oscillators of the linearization. These tori
are generically nonresonant, and some of them also survive when we add
the nonitegrable part $H_1$ (see \cite{El88}).
They are the generalitzation of the periodic
Lyapounov orbits to higher dimensional tori and hence, we will call
them Lyapounov tori.
In this paper we will focus on every kind of nondegenerate low
dimensional torus, in the sense that its normal behaviour only contains
elliptic or hyperbolic directions but not degenerate ones (see
\cite{G} and \cite{Z} for results in the hyperbolic case and
\cite{M}, \cite{El88} and \cite{P} for previous results in the general
case).
This implies that the torus is not contained in a (resonant) higher
dimensional invariant torus.
We will develop a perturbation theory for these tori, focussing
on the case in which the perturbation also depends on time in a
quasiperiodic way. The Hamiltonian is of the form
\begin{equation}
H(\theta,x,I,y)=\tilde{\omega}^{(0)\top}\tilde{I}+
H_0(\hat{\theta},x,\hat{I},y) +
H_1(\hat{\theta},\tilde{\theta},x,\hat{I},y),
\label{eq:hini}
\end{equation}
with respect to the symplectic form
$d\hat{\theta}\wedge d\hat{I}+d\tilde{\theta}\wedge d\tilde{I}+
dx\wedge dy$.
Here, $\hat{\theta}$ are the angular variables that describe an initial
$r$-dimensional torus of $H_0$,
$x$ and $y$ are the normal directions to the torus,
$\tilde{\theta}$ are the angular variables that denotes the time,
$\tilde{I}$ are the corresponding momenta (that has only been
added to put the Hamiltonian in autonomous form) and
$\tilde{\omega}^{(0)}$ is the frequency associated to time.
These kind of Hamiltonians appear in several problems of celestial
mechanics: for instance, to study the dynamics of a small
particle (an asteroid or spacecraft) near the equilateral libration
points (\cite{S}) of the Earth--Moon system, one can take the
Earth--Moon system as a restricted three body problem (that can be
written as an autonomous Hamiltonian) plus perturbations coming for the
real motion of Earth and Moon and the presence of the Sun. As these
perturbations can be very well approximated by quasiperiodic functions
(at least for moderate time spans),
it is usual to do so. Hence, one ends up with an autonomous model
perturbed with a function that depends on time in a quasiperiodic
way. Details on these models and their applications can be found
in \cite{DJS}, \cite{GJMS91A}, \cite{GJMS91B} and \cite{GJMS93B}.
For more theoretical results, see \cite{JS92}, \cite{JS94B} and
\cite{JRV}.
The problem of the preservation of maximal dimension tori
of Hamiltonians like (\ref{eq:hini}) has already been considered
in \cite{JS94B}. There it is proved that most (in the usual
measure sense) of the tori of the unperturbed system survive
to the perturbation, but adding the perturbing frequencies
to the ones they already have. Here we will consider the problem
of the preservation of lower dimensional invariant tori,
under the same kind of perturbations. We will show that,
under some hypothesis of nondegeneracy and nonresonance
(to be precised later) some of the (lower dimensional) tori
are not destroyed but only deformed by the perturbation,
adding the perturbing frequencies to the ones they previously
had.
One of the main contributions of this paper are the estimates
on the measure of the destroyed tori. We have taken two approaches
to that point. In the first one we study the persistence of a
single invariant torus of the initial Hamiltonian, under a
quasiperiodic
time-dependent perturbation, using as a parameter the size ($\epsilon$)
of this perturbation. Our results show that this torus can be
continued for a Cantor set of values of $\epsilon$, adding the
perturbing frequencies to the ones it already have. Moreover,
if $\epsilon\in [0,\epsilon_0]$,
the measure of the complementary of that Cantor set is exponentially
small with $\epsilon_0$. If the perturbation
is autonomous this result is already contained in \cite{JLZ} but for
4-D symplectic maps.
The second approach is to fix the size of the pertubation to
a given (and small enough) value. Then it is possible that the
latter result can not be applied because $\epsilon$ can be
in the complementary of the above-defined Cantor set. In this case,
it is still possible to prove the existence of invariant tori
with frequencies the ones of the perturbation plus frequencies close
to the ones of the unperturbed torus. These tori are a Cantor family
parametrized (for instance) by the frequencies of the unperturbed
problem. Again, the measure of the complementary of this Cantor
set is exponentially small with the distance to the frequencies
of the initial torus.
It is interesting to note the implications of this last assertion
when the perturbation is autonomous and the size of the perturbation
is fixed: in this case we are proving, for the perturbed Hamiltonian,
the existence of a Cantor family of invariant tori near the
initial one (see \cite{El88} and \cite{Se}). Moreover, the measure of
the complementary of this set is exponentially small with the
distance to the initial tori.
The most difficult case is when the normal behaviour of the
torus contain some elliptic directions, because
the (small) divisors obtained contain
combinations of the intrinsic frequencies with the normal ones.
As we will see, it is not difficult to control the value of
the intrinsic frequencies but then we have no control (in principle)
on the corresponding normal ones. This is equivalent to say
that we can not select a torus with given both intrinsic and normal
frequencies, because there are not enough available parameters
(see \cite{M}, \cite{BHT} and \cite{Se}). The
main trick in the proofs is to assume that the normal frequencies move
as a function of $\epsilon$ (then we derive the existence of the torus
for a Cantor set of $\epsilon$) or as a function of the
intrinsic frequencies (then we obtain the existence of
the above-mentioned family of tori, close to the initial one).
When the initial torus is normally hyperbolic we do not need to
control the eigenvalues in the normal direction and, hence,
we do not have to deal with the lack of parameters. Of course,
in this case the results are much better and the proofs can be
seen as simplifications of the ones contained here. Hence,
this case is not explicity considered.
Finally, we have also included examples where the application
of these results helps to understand the dynamics of
concrete problems.
The paper has been organized as follows: section~\ref{sec:ideas}
contains the main ideas used to derive these results.
Section~\ref{sec:results} contains the rigorous statement
of the results. The applications of these results to some
concrete problems can be found in section~\ref{sec:applic}
and, finally, section~\ref{sec:proofs} contains the technical
details of the proofs.
\section{Main ideas}\label{sec:ideas}
Let ${\cal H}$ be a Hamiltonian system of $\ell$ degrees
of freedom in $\CC^{2\ell}$ having an invariant
$r$-dimensional torus, $0\le r\le\ell$, with a quasiperiodic flow
given by the vector of basic frequencies $\hat{\omega}^{(0)}\in\RR^r$.
Let us consider the (perturbed) Hamiltonian system
$H={\cal H}+\epsilon\hat{\cal H}$.
As it has been mentioned before, we do not restrict ourselves to
the case of autonomous perturbations, but we will assume that
$\hat{\cal H}$ depends on time in a quasiperiodic way, with
vector of basic frequencies given by $\tilde{\omega}^{(0)}\in\RR^s$.
\subsection{Reducibility}\label{sec:red}
Let us consider the variational flow around one of the quasiperiodic
orbits of the initial $r$-dimensional invariant torus of
${\cal H}$. The variational equations are a linear system with
quasiperiodic time dependence, with vector of basic frequencies
$\hat{\omega}^{(0)}$. When the torus is a periodic orbit, the
well know Floquet theorem states that we can reduce this periodic
system to constant coefficients with a linear periodic change of
variables (with the same period of the system). This change can be
selected to be canonical if the equations are Hamiltonian.
So, the reduced matrix has a pair of zero eigenvalues (associated
to the tangent vector to the periodic orbit)
plus eigenvalues that describes the linear normal behaviour around the
torus. We will assume that these eigenvalues are all different (this
condition implies, from the canonical character of the system, that
they are also non-zero). This implies that the periodic orbit is not
contained in a (resonant) higher dimensional torus.
Usually, the imaginary parts of these eigenvalues are called
normal frequencies, and $\hat{\omega}^{(0)}$ is called the vector
of intrinsic frequencies of the torus.
The quasiperiodic case ($r>1$) is more complex,
because we can not guarantee in general the reducibility to
constant coefficients of the variational equations with a linear
quasiperiodic change of variables with the same basic frequencies
as the initial system. The question of reducibility of
linear quasiperiodic systems (proved in some cases, see \cite{JS},
\cite{C}, \cite{El92}, \cite{JS92}, \cite{JS94B}, \cite{JLZ},
and \cite{K}, among others)
remains open in the general case. However, we can say that if this
reduction is possible, we have $2r$ zero eigenvalues (related to
the $r$ tangent vectors to the torus).
Here we will assume that such reduction is possible for the initial
torus. We want to remark that if this initial torus comes from
an autonomous perturbation of a resonant torus of an integrable
Hamiltonian, this hypothesis is not very strong. To justify this
assertion, we mention the following fact: let us write the
Hamiltonian as
$H={\cal H}(I)+\epsilon\hat{\cal H}(\theta,I)$,
and let ${\cal T}_{0}$ be a low dimensional invariant tori of
the integrable Hamiltonian ${\cal H}(I)$ that survives to the
perturbation $\epsilon\hat{\cal H}(\theta,I)$.
Then, under generic hypothesis of nondegeneracy and nonresonance,
this low dimensional torus exists and its normal flow is also reducible
for a Cantor set of values of $\epsilon$. The Lebesgue measure of the
complementary of this set in $[0,\epsilon_0]$ is exponentially small
with $\epsilon_0$. This fact is proved for symplectic diffeomorphisms
of $\RR^4$ in \cite{JLZ}, but it is immediate to extend to other
cases.
Moreover, let us assume that we can introduce (with a canonical
change of coordinates) $r$ angular variables $\hat{\theta}$
describing the initial torus. Hence, the Hamiltonian takes the form
$$
{\cal H}(\hat{\theta},x,\hat{I},y)=\hat{\omega}^{(0)\top}\hat{I}+
\frac{1}{2} z^{\top}{\cal B}z+{\cal H}_*(\hat{\theta},x,\hat{I},y),
$$
where $z^{\top}=(x^{\top},y^{\top})$,
being $z$, $\hat{\theta}$ and $\hat{I}$ complex vectors, $x$ and $y$
elements of $\CC^r$ and $\hat{\theta}$ and $\hat{I}$ elements of
$\CC^s$, with $r+m=\ell$. Here, $\hat{\theta}$ and $x$ are the
positions and $\hat{I}$ and $y$ are the conjugate momenta. In this
notation ${\cal B}$ is a symmetric $2m$-dimensional matrix (with
complex coefficients). Moreover, ${\cal H}_*$ is an analytic function
(with respect to all its arguments) with $2\pi$-periodic dependence on
$\hat{\theta}$. More concretely, we will assume that it is analytic
on a neighbourhood of $z=0$, $\hat{I}=0$, and on a complex strip of
positive width $\rho$ for the variable $\hat{\theta}$, that is, if
$|\Im \hat{\theta}_j|\le\rho$, for all $j=1,\ldots, r$.
Then, if we assume that ${\cal H}$ has an invariant $r$-dimensional
torus with vector of basic frequencies $\hat{\omega}^{(0)}$, given by
$\hat{I}=0$ and $z=0$, this
implies that the Taylor expansion of ${\cal H}_*$ must begin with
terms of second order in the variables $\hat{I}$ and $z$. If we have
that the normal variational flow around this torus can be reduced to
constant coefficients, we can assume that the quadratic terms of
${\cal H}_*$ in the $z$ variables vanish. Hence, the normal
variational equations are given by the matrix $J_m {\cal B}$, where
$J_m$ is the canonical $2$-form of $\CC^{2m}$. We also assume that
the matrix $J_m {\cal B}$ is in diagonal form with different
eigenvalues $\lambda^{\top}=(\lambda_1,\ldots,\lambda_m,
-\lambda_1,\ldots,-\lambda_m)$.
Let us give some remarks on these coordinates. First, note that we
have assumed that the initial torus is isotropic (this is, the
canonical 2-form of $\CC^{2\ell}$ restricted to the tangent bundle
of the torus vanishes everywhere).
This fact (that is always true for a periodic orbit)
is not a strong assumption for a torus, because all the tori
obtained by applying KAM techniques to near-integrable Hamiltonian
systems are isotropic.
Another point worth to comment is the real or complex
character of the matrix ${\cal B}$.
In this paper we work,
in principle, with complex analytic Hamiltonian
systems, but the most interesting case happens when we deal with real
analytic ones, and when the initial torus is also real.
In this case, to guarantee that the perturbative scheme
preserves the real character of the tori, we want that the initial
reduced matrix ${\cal B}$ comes from a real matrix.
We note that this is equivalent to assume that
if $\lambda$ is an eigenvalue
of $J_m{\cal B}$, then $\bar{\lambda}$ is also an eigenvalue.
This assumption is not true in general for every reducible
torus of a real analytic Hamiltonian system,
but it holds for most of the tori one can obtain near
an initial torus with a normal flow reducible over $\RR$.
Note that in the case of a periodic orbit one can always assume that
${\cal B}$ is real (doubling the period if necessary).
The fact that ${\cal B}$ is real guarantees that all the
tori obtained are also real. To see it, we note that
we can use the same proof but putting $J_m{\cal B}$ in real normal
form instead of diagonal form, and this makes that all the
steps of the proof are also real. However, the technical
details in this case are a little more tedious and, hence,
we have prefered to work with a diagonal $J_m{\cal B}$.
\subsection{Normal form around the initial torus}
The first step is to rearrange the initial Hamiltonian
${\cal H}^{(0)}:={\cal H}$ in a suitable form to apply an inductive
procedure.
In what follows, we will define degree of a monomial $z^l \hat{I}^j$
as $|l|_1 + 2|j|_1$. This definition is motivated below.
Let us expand ${\cal H}^{(0)}_{*}$ in power series with respect
to $z$ and $\hat{I}$ around the origin:
$$
{\cal H}^{(0)}_{*}=\sum_{d\ge 2} {\cal H}^{(0)}_d,
$$
where ${\cal H}^{(0)}_d$ are homogeneous polynomials of degree $d$,
that is,
$$
{\cal H}^{(0)}_d=\sum_{\begin{array}{c}
\scriptstyle l \in \NN^{2m},\; j \in \NN^{r} \\
\scriptstyle |l|_1 + 2|j|_1 =d
\end{array}}
h_{l,j}^{(0)}(\hat{\theta}) z^l \hat{I}^j.
$$
We also expand the (periodic) coefficients in Fourier series:
\begin{equation}
h_{l,j}^{(0)}(\hat{\theta})=
\sum_{k\in\ZZ^r} h_{l,j,k}^{(0)}\exp{(ik^{\top}\hat{\theta})},
\label{eq:inihf}
\end{equation}
being $i=\sqrt{-1}$.
The definition of degree for a monomial $z^l \hat{I}^j$ counting
twice the contribution of the variable $\hat{I}$ is
motivated by the definition of the Poisson bracket of two functions
depending on $(\hat{\theta},x,\hat{I},y)$:
$$
\{f,g\}= \frac{\partial f}{\partial \hat{\theta}}
\left(\frac{\partial g}{\partial \hat{I}}\right)^{\top} -
\frac{\partial f}{\partial \hat{I}}
\left(\frac{\partial g}{\partial \hat{\theta}}\right)^{\top} +
\frac{\partial f}{\partial z} J_m
\left( \frac{\partial g}{\partial z} \right)^{\top}.
$$
Note that, if $f$ is an homogeneous polynomial of degree $d_1$
and $g$ is an homogeneous polynomial of degree $d_2$ then
$\{f,g\}$ is an homogeneous polynomial of degree
$d_1+d_2-2$.
This property shows that if we try to construct canonical changes using
the Lie series method, the adequate form to put ${\cal H}^{(0)}$ in
normal form is to remove in an increasing order the terms of degree
$3$, $4$, $\ldots$, with a suitable generating function.
To introduce some of the parameters
(see section~\ref{sec:Op}), it is very convenient
that the initial Hamiltonian has the following properties:
\begin{description}
\item[{\bf P1}] The coefficients of the monomials $(z,\hat{I})$
(degree 3) and $(z,\hat{I},\hat{I})$ (degree 5) are zero.
\item[{\bf P2}] The coefficients of the monomials $(z,z,\hat{I})$
(degree 4) and $(\hat{I},\hat{I})$ (degree 4) do not depend on
$\hat{\theta}$ and, in the case of $(z,z,\hat{I})$, they vanish except
for the coefficients of the trivial resonant terms.
\end{description}
Here, we have used the following notation: for instance, by the terms
of order $(z,z,\hat{I})$ we denote the monomials
$z^l\hat{I}^j$, with $|l|_1=2$ and $|j|_1=1$, with the corresponding
coefficients.
We will apply three steps of a normal form procedure in
order to achieve these conditions.
Each step is done using a generating function of the following type:
$$
S^{(n)}(\hat{\theta},x,\hat{I},y) = \sum_{\begin{array}{c}
\scriptstyle l \in \NN^{2m},\; j \in \NN^{r} \\
\scriptstyle |l|_1 + 2|j|_1 =n
\end{array}}
s^{(n)}_{l,j}(\hat{\theta}) z^l \hat{I}^j,
$$
for $n=3$, $4$ and $5$. Then, if we put $\Psi^{S^{(n)}}$ for the flow
at time one of the Hamiltonian system associated to $S^{(n)}$, we
transform the initial Hamiltonian into
\begin{eqnarray*}
{\cal H}^{(n-2)} & = &
{\cal H}^{(n-3)}\circ\Psi^{S^{(n)}}= \\
& = & {\cal H}^{(n-3)}+
\{{\cal H}^{(n-3)},S^{(n)}\}+
\frac{1}{2!}\{\{{\cal H}^{(n-3)},S^{(n)}\},S^{(n)}\}+O_{n+1} =
\\ & = & \hat{\omega}^{(0)\top}\hat{I}+\frac{1}{2}z^{\top}{\cal B}z+
{\cal H}_*^{(n-3)}+\{\hat{\omega}^{(0)\top}\hat{I}+
\frac{1}{2}z^{\top}{\cal B}z, S^{(n)}\}+ O_{n+1},
\end{eqnarray*}
for $n=3,4,5$. In each step, we take $S^{(n)}$ such that
${\cal H}^{(n-3)}_{n}+\{\hat{\omega}^{(0)\top}\hat{I}+
\frac{1}{2}z^{\top}{\cal B}z,{S}^{(n)}\}$ satisfies conditions {\bf P1}
and {\bf P2} for the monomials of degree $n$ ($n=3,4,5$).
To compute ${S}^{(n)}$ we expand ${\cal H}^{(n-3)}_{n}$ and we find
(formally) an expansion for $S^{(n)}$:
$$
s_{l,j,k}^{(n)}=\frac{h_{l,j,k}^{(n-3)}}{ik^{\top}\hat{\omega}^{(0)} +
l^{\top}\lambda},
$$
where the indices have the same meaning as in (\ref{eq:inihf}). If we
split $l=(l_x,l_y)$ ($z^l=x^{l_x}y^{l_y}$), the exactly resonant terms
correspond to $k=0$ and $l_x=l_y$ (we recall that $\lambda^{\top}=
(\lambda_1,\ldots,\lambda_m, -\lambda_1,\ldots,-\lambda_m)$). Hence,
it would be possible to formally compute
a normal form depending only on $\hat{I}$ and the products $x_jy_j$,
$j=1,\ldots,m$. As it has been mentioned before, our purpose is much
more modest. To kill the monomials mentioned above (in conditions
{\bf P1} and {\bf P2}) with a convergent change of variables,
one needs a condition on the smallness of $|ik^{\top}\hat{\omega}^{(0)}
+l^{\top}\lambda|$, $k\in\ZZ^r\setminus\{0\}$, $l\in\NN^{2m}$
and $|l|_1\le 2$. We have used the usual one
$$
|ik^{\top}\hat{\omega}^{(0)}+l^{\top}\lambda|\ge
\frac{\cd_0}{|k|_1^{\gamma}},
$$
that we will assume true in the statement of the results. We notice
that with these conditions we can construct convergent expressions
for the different generating functions $S^{(n)}$, $n=3,4,5$, to achieve
conditions {\bf P1} and {\bf P2}. We can call this process a seminormal
form construction.
Then, the final form for the Hamiltonian is
\begin{equation}
{\cal H}=\hat{\omega}^{(0)\top}\hat{I}+\frac{1}{2}z^{\top}{\cal B}z+
\frac{1}{2}\hat{I}^{\top}{\cal C}\hat{I}+
H_*(\hat{\theta},x,\hat{I},y),
\label{eq:fn3}
\end{equation}
for which conditions {\bf P1} and {\bf P2} holds. Here ${\cal C}$ is a
symmetric constant matrix and we will assume
$\det{\cal C}\neq 0$ (this is one of the nondegeneracy hypothesis).
Now let us introduce the quasiperiodic time-dependent perturbation.
To simplify the notation, we write this perturbation in the normal form
variables, and we add this perturbation to (\ref{eq:fn3}). We call $H$
to the new Hamiltonian:
\begin{equation}
H(\theta,x,I,y,\epsilon)={\omega}^{(0)\top}{I}+
\frac{1}{2}z^{\top}{\cal B}z+
\frac{1}{2}\hat{I}^{\top}{\cal C}\hat{I}+
H_*(\hat{\theta},x,\hat{I},y)+
\epsilon \hat{\cal H}(\theta,x,\hat{I},y,\epsilon),
\label{eq:hp}
\end{equation}
for a fixed $\omega^{(0)\top}=
(\hat{\omega}^{(0)\top},\tilde{\omega}^{(0)\top})$,
${\omega}^{(0)}\in\RR^{r+s}$, where
$\theta^{\top}=(\hat{\theta}^{\top},\tilde{\theta}^{\top})$,
$I^{\top}=(\hat{I}^{\top},\tilde{I}^{\top})$ and
$z^{\top}=(x^{\top},y^{\top})$,
being $\tilde{\theta}$, $\tilde{I}$ ($s$-dimensional complex vectors)
the new positions and momenta added to put in autonomous form the
quasiperiodic perturbation. Hence, $H$ is $2\pi$-periodic in $\theta$.
Moreover, $\epsilon$ is a small positive parameter. This is the
Hamiltonian that we consider in the formulation of the results.
\subsection{The iterative scheme}\label{sec:its}
Before the explicit formulation of the results, let us describe a
generic step of the iterative method used in the proof. So, let us
consider a Hamiltonian of the form:
\begin{equation}
H(\theta,x,I,y)={\omega}^{(0)\top}{I}+\frac{1}{2}z^{\top}{\cal B}z+
\frac{1}{2}\hat{I}^{\top}{\cal C}({\theta})\hat{I}+
H_*({\theta},x,\hat{I},y)+
\epsilon \hat{H} (\theta,x,\hat{I},y),
\label{eq:hits}
\end{equation}
with the same notations of (\ref{eq:hp}), where we assume that skipping
the term $\epsilon\hat H$, we have that $z=0$, $\hat{I}=0$ is a
reducible ($r+s$)-dimensional torus with vector of basic frequencies
$\omega^{(0)}$,
such that the variational normal flow is given by
$J_m {\cal B}=
\mbox{diag}(\lambda_1,\ldots,\lambda_m,-\lambda_1,\ldots,-\lambda_m)$,
and that $\det \bar{\cal C}\neq 0$, where $\bar{\cal C}$ means the
average of ${\cal C}$ with respect to its angular variables
(although initially ${\cal C}$ does not depend on $\theta$,
during the iterative scheme it will).
Moreover, we suppose that in $H_*$ the terms of order $(\hat{I},z)$
vanish (that is, we suppose that the ``central'' and ``normal''
directions of the unperturbed torus have been uncoupled up to
first order). Here we only use the parameter $\epsilon$
to show that the perturbation $\epsilon \hat H$ is of $O(\epsilon)$.
We expand $\hat{H}$ in power series around $\hat{I}=0$, $z=0$ and we
add these terms to the previous expansion of the unperturbed
Hamiltonian. This makes that the initial torus is not longer invariant.
Hence, the expression of the Hamiltonian must be (without writting
explicity the dependence on $\epsilon$):
\begin{equation}
H(\theta,x,I,y)=\tilde{\omega}^{(0)\top}\tilde{I}+
H^{*}(\theta,x,\hat{I},y),
\label{eq:ha}
\end{equation}
where
$$
H^{*}= a(\theta) +
b(\theta)^{\top} z +
c(\theta)^{\top} \hat{I} +
\frac{1}{2} z^{\top} B(\theta) z +
\hat{I}^{\top} E(\theta) z +
\frac{1}{2}\hat{I}^{\top}C(\theta)\hat{I}+
\Omega(\theta,x,\hat{I},y),
$$
being $\Omega$ the remainder of the expansion.
Looking at this expression, we introduce the notation
$[H^{*}]_{(z,z)}=B$, $[H^{*}]_{(\hat{I},\hat{I})}=C$,
$[H^{*}]_{(\hat{I},z)}=E$ and $=H^*-\Omega$.
We have that $\tilde{a}$, $b$, $c-\hat{\omega}^{(0)}$, $B-{\cal B}$,
$C-{\cal C}$ and $E$ are $O(\epsilon)$, where if $f(\theta)$ is a
periodic function on $\theta$, $\tilde{f}=f-\bar{f}$.
Note that if we are able to kill the terms $\tilde{a}$, $b$ and
$c-\hat{\omega}^{(0)}$ we obtain a lower dimensional
invariant torus with intrinsic frequency $\omega^{(0)}$. We will try to
do that by using a quadratically convergent scheme. As it is usual in
this kind of Newton methods, it is very convenient to kill something
more. Before continuing, let us introduce the following notation:
if $A$ is a $n\times n$ matrix, $\mbox{dp}(A)$ denotes the
diagonal part of $A$, that is, $\mbox{dp}(A)=
\mbox{diag}(a_{1,1},\ldots,a_{n,n})^{\top}$,
where $a_{i,i}$ are the diagonal entries of $A$.
Here, we want that the new matrix $B$ verifies
$B={\cal J}_m(B)$,
where we define ${\cal J}_m(B)=-J_m \mbox{dp}(J_m \bar{B})$ (this is,
we ask the normal flow to the torus to be reducible and given by a
diagonal matrix like for the unperturbed torus) and to eliminate
$E$ (to uncouple the ``central'' and the normal directions of the
torus up to first order in $\epsilon$). Hence, the torus we will
obtain has also these two properties. This is a very usual technique
(see \cite{El88}, \cite{JLZ}).
At each step of the iterative procedure, we use a canonical change of
variables similar to the ones used in \cite{BGGS} to prove the
Kolmogorov theorem. The generating function is of the form
\begin{equation}
S(\theta,x,\hat{I},y)=\xi^{\top} \hat{\theta} +
d(\theta) +
e(\theta)^{\top} z +
f(\theta)^{\top} \hat{I} +
\frac{1}{2} z^{\top} G(\theta) z +
\hat{I}^{\top} F(\theta) z,
\label{eq:gf2}
\end{equation}
where $\xi\in\CC^{r}$, $\bar{d}=0$, $\bar{f}=0$ and G is a symmetric
matrix with ${\cal J}_m(G)=0$. Keeping the same name for the new
variables, the transformed Hamiltonian is
$$
H^{(1)}=H\circ\Psi^{S}=\tilde{\omega}^{(0)\top}\tilde{I}+
H^{(1)*}(\theta,x,\hat{I},y),
$$
being
\begin{eqnarray*}
H^{(1)*}(\theta,x,\hat{I},y) & = & a^{(1)}(\theta) +
b^{(1)}(\theta)^{\top} z +
c^{(1)}(\theta)^{\top} \hat{I} +
\frac{1}{2} z^{\top} B^{(1)}(\theta) z +\\
& & +\hat{I}^{\top} E^{(1)}(\theta) z +
\frac{1}{2}\hat{I}^{\top}C^{(1)}(\theta)\hat{I}+
\Omega^{(1)}(\theta,x,\hat{I},y).
\end{eqnarray*}
We want $\tilde{a}^{(1)}=0$, $b^{(1)}=0$,
$c^{(1)}-\hat{\omega}^{(0)}=0$, $E^{(1)}=0$ and $J_m B^{(1)}$
to be a constant diagonal matrix. We will show
that this can be achieved up to first order in $\epsilon$.
So, we write those conditions in terms of the initial Hamiltonian and
the generating function. Skipping terms of $O_2(\epsilon)$, we obtain:
\begin{itemize}
\item[$(eq_1)$]
$\tilde{a}-\frac{\partial d}{\partial\theta}\omega^{(0)}=0$,
\item[$(eq_2)$]
$b-\frac{\partial e}{\partial\theta}\omega^{(0)}+{\cal B}J_me=0$,
\item[$(eq_3)$]
$c-\hat{\omega}^{(0)}-\frac{\partial f}{\partial\theta}\omega^{(0)}
-{\cal C}\left(\xi+\left(\frac{\partial d}{\partial\hat{\theta}}
\right)^{\top}\right)=0$,
\item[$(eq_4)$]
$B^*-{\cal J}_m(B^*)-\frac{\partial G}{\partial\theta}\omega^{(0)}+
{\cal B}J_m G-GJ_m{\cal B}=0$,
\item[$(eq_5)$]
$E^*-\frac{\partial F}{\partial\theta}\omega^{(0)}-FJ_m{\cal B}=0$,
\end{itemize}
being
\begin{equation}
B^*=B-\left[\frac{\partial H_*}{\partial\hat{I}}
\left(\xi+\left(\frac{\partial d}{\partial\hat{\theta}}
\right)^{\top}\right)-\frac{\partial H_*}{\partial z}J_me\right]_{(z,z)}
\label{eq:Bstar}
\end{equation}
and
\begin{equation}
E^*=E-{\cal C}\left(\frac{\partial e}{\partial\hat{\theta}}
\right)^{\top}-\left[\frac{\partial H_*}{\partial\hat{I}}
\left(\xi+\left(\frac{\partial d}{\partial\hat{\theta}}\right)^{\top}
\right)-\frac{\partial H_*}{\partial z}J_m e\right]_{(\hat{I},z)}.
\label{eq:Estar}
\end{equation}
Here we denote by $\frac{\partial}{\partial q}$ the matrix of partial
derivatives with respect to the variables $q$ and, for instance,
$\frac{\partial G}{\partial\theta}\omega^{(0)}$ means
$\sum_{j=1}^{r+s} \frac{\partial G}{\partial\theta_j}\omega^{(0)}_j$.
These equations are solved formally by expanding in Fourier series
and equating the corresponding coefficients. This lead us to the
following expressions for $S$:
\begin{itemize}
\item[$(eq_1)$]
$$
d(\theta)=\sum_{k \in {\ZZ}^{r+s} \backslash \{0\}}
\frac{a_k}{ik^{\top} \omega^{(0)}} \exp(i k^{\top} \theta).
$$
\item[$(eq_2)$]
If we put $e^{\top}=(e_1,\ldots,e_{2m})$,
$$
e_{j}(\theta)=\sum_{k \in {\ZZ}^{r+s}}
\frac{b_{j,k}}{ik^{\top}\omega^{(0)}+\lambda_j}\exp(i k^{\top}\theta).
$$
\item[$(eq_3)$]
$$
\xi=(\bar{\cal C})^{-1} \left(\bar{c}-\hat{\omega}^{(0)}-\overline{
{\cal C}\left(\frac{\partial d}{\partial\hat{\theta}}\right)^{\top}}
\right),
$$
and if we define
$$
c^*=\tilde{c}-\tilde{\cal C}\xi-
{\cal C}\left(\frac{\partial d}{\partial\hat{\theta}}\right)^{\top} +
\overline{{\cal C}\left(\frac{\partial d}{\partial\hat{\theta}}
\right)^{\top}},
$$
we have for $f^{\top}=(f_1,\ldots,f_r)$:
$$
f_j(\theta)=\sum_{k \in {\ZZ}^{r+s} \backslash\{0\}}
\frac{c^{*}_{j,k}}{ik^{\top}\omega^{(0)}} \exp(i k^{\top} \theta).
$$
\item[$(eq_4)$]
If we define
\begin{equation}
B^{**}=B^*-{\cal J}_m(B^*),
\label{eq:Bsatar2}
\end{equation}
then we have for
$G=(G_{j,l})$, $1\leq j,l \leq 2m$,
$$
G_{j,l}(\theta)=\sum_{k \in {\ZZ}^{r+s}}
\frac{B^{**}_{j,l,k}}{ik^{\top}\omega^{(0)}+{\lambda}_{j}+{\lambda}_{l}}
\exp(i k^{\top} \theta),\;\;j,l=1,\ldots,2m.
$$
In the definition of $G_{j,l}$, we notice that we have trivial zero
divisors when $|j-l|=m$ and $k=0$, but from the expression of $B^{**}$,
in these cases the coefficient $B^{**}_{j,l,0}$ is $0$. Moreover, the
matrix $G$ is symmetric.
\item[$(eq_5)$]
If $F=(F_{j,l})$, $j=1,\ldots,r$ and $l=1,\ldots,2m$, then
$$
F_{j,l}(\theta)=\sum_{k\in {\ZZ}^{r+s}}
\frac{E^{*}_{j,l,k}}{ik^{\top}\omega^{(0)}+{\lambda}_l}
\exp(ik^{\top}\theta).
$$
\end{itemize}
Note that if we have Diophantine hypothesis on the small divisors of
these expressions,
\begin{equation}
|ik^{\top}\omega^{(0)}+l^{\top}\lambda|\geq\frac{\cd_0}{|k|_1^{\gamma}},
\;\;k\in{\ZZ}^{r+s}\backslash\{0\},\;\;l\in\NN^{2m},
\;\;|l|_1\leq 2,\;\;\gamma>r+s-1,
\label{eq:dc0}
\end{equation}
we can guarantee the convergence of the expansion of $S$. We assume
that they hold in the first step, and we want to have similar
conditions after each step of the process, to be able to iterate.
As the frequencies $\tilde{\omega}^{(0)}$ are fixed in all the
process and $\hat{\omega}^{(0)}$ can be preserved by the
nondegeneracy and the kind of generating function we are using (this is
done by the $\xi$ term), we will be able to recover the Diophantine
properties on them. The main problem are the eigenvalues $\lambda$,
because, in principle, we can not preserve their value. Hence, we will
control the way they vary, to try to ensure they are still satisfying a
good Diophantine condition. Our first approach is to consider $\lambda$
as a function of $\epsilon$ (the size of the perturbation).
This leads us to eliminate a Cantor
set of values of these parameters in order to have all the time
good (in a Diophantine sense) values of $\lambda$.
Another possibility is to consider $\lambda$ as a function of the
(frequencies of the) torus. This leads us to eliminate
a Cantor set of those tori.
\subsection{Estimates on the measure of preserved tori}\label{sec:empt}
The technique we are going to apply to produce exponentially small
estimates has already been used in \cite{JS94B}.
It is based on working at every step $n$ of the iterative procedure
with values of $\epsilon$ for which we have
Diophantine conditions of the type
\begin{equation}
|ik^{\top}\omega^{(0)}+{\lambda}_l^{(n)}(\epsilon)|\ge
\frac{\cd_n}{|k|_1^{\gamma}}
\exp{(-\delta_n |k|_1)},\;\;\;\;
k\in {\ZZ}^{r+s} \backslash\{0\},
\label{eq:codi}
\end{equation}
where ${\lambda}_l^{(n)}(\epsilon)$ denotes the eigevalues of
$J_m {\cal B}^{(n)}(\epsilon)$, being ${\cal B}^{(n)}(\epsilon)$ the
matrix that replaces ${\cal B}$ after $n$ steps of the iterative
process.
Of course, we ask for the same condition for the sum of eigenvalues
$\lambda_j^{(n)}+\lambda_l^{(n)}$.
We will see that, if we take a suitable sequence of
$\delta_n$, the exponential term in (\ref{eq:codi}) is not an
obstruction to the convergence of the scheme. This condition will
be used to obtain exponentially small estimates for the
measure of the values of $\epsilon$ for which we do not have invariant
tori of frequency $\omega^{(0)}$ in the perturbed system.
The key idea can be described as follows: for the values of
$\epsilon$ for which we can prove convergence, we obviously have that,
if $\epsilon$ is small enough,
$|\lambda_l^{(n)}(\epsilon)-\lambda_l|\leq a\epsilon$, at every step
$n$. Now, if we assume that $\cd_n\leq\cd_0/2$,
from the Diophantine bounds on $ik^{\top}\omega^{(0)}+\lambda_l$ in
(\ref{eq:dc0}), we only need to worry about the resonances
corresponding to values of $k$ such that
$$
|k|_1\geq\left(\frac{\cd_0}{2a\epsilon}\right)^{1/\gamma}:=
K(\epsilon).
$$
This is equivalent to say that we do not have low order resonances
nearby, hence we only have to eliminate higher order ones.
When we eliminate the values of $\epsilon$ for which the Diophantine
condition is not fullfiled for some $k$, we only need to worry about
controlling the measure of the ``resonant'' sets associated to
$|k|_1\geq K(\epsilon)$. From that, and from the exponential
in $|k|_1$ for the admissible small divisors, we obtain exponentially
small estimates for the set of values of $\epsilon$ for which we can
not prove the existence of invariant tori. If
$\epsilon\in]0,\epsilon_0]$ this measre is of order
$\exp(-1/\epsilon_0^{c})$, for any $00$ and $\gamma>r+s-1$, the following
Diophantine conditions hold:
$$
|ik^{\top}{\omega}^{(0)}+l^{\top}\lambda|\ge
\frac{\cd_0}{|k|_1^{\gamma}},
\;\; k\in\ZZ^{r+s}\setminus\{0\}, \;\;l\in\NN^{2m},\;\;|l|_1\leq 2.
$$
\end{itemize}
Then, under certain generic nondegeneracy conditions
for the Hamiltonian $H$ (that are given explicitly in {\bf NDC} at the
end of section~\mbox{\rm \ref{sec:ivfp})}, the following assertions
hold:
\begin{itemize}
\item[$(a)$] There exists a Cantor set ${\cal I}_*\subset{\cal I}_0$,
such that for every $\epsilon\in{\cal I}_*$ the Hamiltonian $H$ has
a reducible ($r+s$)-dimensional invariant torus with vector of basic
frequencies $\omega^{(0)}$. Moreover, for every $0 < \sigma < 1$:
$$
\mbox{mes} ([0,\bar{\epsilon}] \; \backslash \; \bar{\cal I}_* ) \leq
\exp{(-(1/\bar{\epsilon})^{\frac{\sigma}{\gamma}})},
$$
if $\bar{\epsilon}$ is small enough (depending on $\sigma$) where,
for every $\bar{\epsilon}$,
$\bar{\cal I}_*=\bar{\cal I}_*(\bar{\epsilon}):=
[0,\bar{\epsilon}]\cap {\cal I}_*$.
\item[$(b)$] Given $R_0>0$ small enough and a fixed
$0\leq\epsilon\leq R_0^{\frac{\gamma}{\gamma+1}}$, there exists a
Cantor set ${\cal W}_*({\epsilon},R_0)\subset\{\hat{\omega}\in\RR^r:\;
|\hat{\omega}-\hat{\omega}^{(0)}|\leq R_0\}:={\cal V}(R_0)$, such that
for every $\hat{\omega}\in{\cal W}_*({\epsilon},R_0)$ the Hamiltonian
$H$ corresponding to this fixed value of $\epsilon$, has a reducible
($r+s$)-dimensional invariant torus with vector of basic frequencies
$\omega$, ${\omega}^{\top}=
(\hat{\omega}^{\top},\tilde{\omega}^{(0)\top})$. Moreover, for every
$0<\sigma<1$, if $R_0$ is small enough (depending on $\sigma$):
$$
\mbox{mes} ( {\cal V}(R_0) \backslash {\cal W}_*({\epsilon},R_0)) \leq
\exp{(-(1/R_0)^{\frac{\sigma}{\gamma+1}})}.
$$
\end{itemize}
Here, $\mbox{mes} (A)$ denotes the Lebesgue measure of the set $A$.
\end{theorem}
\subsection{Remarks}\label{sec:rem}
The result $(b)$ has special interest if we take ${\epsilon}=0$.
It shows that for the unperturbed system, around the initial
$r$-dimensional reducible torus there exist an $r$-dimensional family
(with Cantor structure) of $r$-dimensional reducible tori
parametrized by $\hat{\omega}\in{\cal W}_*(0,R_0)$, with relative
measure for the complementary of the Cantor set exponentially small
with $R_0$, for values of $\hat{\omega}$ $R_0$-close to
$\hat{\omega}^{(0)}$. There are previous results on the
existence of these lower dimensional tori (see the references), but the
estimates on the measure of preserved tori close to a given one are
not so good as the ones presented here.
Moreover, we have the same result around every ($r+s$)-dimensional
torus that we can obtain for the perturbed system for some
$\epsilon\neq 0$ small enough, if we assume that their intrinsic and
normal frequencies verify the same kind of Diophantine bounds as
the frequencies of the unperturbed torus. In this case, for every
$R_0$ small enough we have a (Cantor) family of ($r+s$)-dimensional
reducible tori parametrized by
$\hat{\omega}\in{\cal W}_*(\epsilon,R_0)$, with the same kind of
exponentially small measure with respect to $R_0$ on the complementary
of this set. To prove it, we remark that we can reduce to the case
$\epsilon=0$ if we note that it is easy to see that Theorem \ref{teo}
also holds if the
unperturbed Hamiltonian depends on $\theta$ and not only on
$\hat{\theta}$ (that is, if the initial torus is ($r+s$)-dimensional).
If the initial torus is normally hyperbolic, the problem is
easier. For instance, it is possible to prove the existence
of invariant tori without using reducibility conditions. Then,
in case $(a)$, one obtains an open set of values
of $\epsilon$ for which the torus exists, although its normal
flow could not be reducible. The reason is that
the intrinsic frequencies of the torus are fixed with respect to
$\epsilon$ and the normal eigenvalues (that depend on $\epsilon$)
do not produce extra small divisors if we consider only equations
$(eq_1)-(eq_3)$ of the iterative scheme described in
section~\ref{sec:its} (we take $G=0$ and $F=0$ in equation
(\ref{eq:gf2})).
Note that now we can solve $(eq_2)$ using a fixed point method,
because the matrices $J_m{\cal B}^{(n)}$ are $\epsilon$-close to
the initial hyperbolic matrix $J_m{\cal B}$ (that is supposed to be
reducible). This makes unnecessary to consider $(eq_4)$ and
$(eq_5)$. Of course, the tori produced in this way are not
necessarily reducible. If one wants to ensure
reducibility, it is necessary to use the normal eigenvalues
and this can produce (depending on some conditions on those
eigenvalues, see \cite{JS94B}) a Cantor set of $\epsilon$ of the same
measure as the one in $(a)$. If we consider the case $(b)$
when the normal behaviour is hyperbolic, the results do not change
with respect to the normally elliptic case. As we are ``moving'' the
intrinsic frequencies, we have to take out the corresponding
resonances. The order (when $R_0$ goes to zero) of the measure of
these resonances is still exponentially small with $R_0$ (see
lemma~\ref{lem:meas2}).
Finally, let us recall that the Diophantine condition $(iv)$
is satisfied for all the frequencies $\omega^{(0)}$ and
eigenvalues $\lambda$, except for a set of zero measure.
\section{Applications}\label{sec:applic}
In this section we are going to illustrate the possible
applications of these results to some concrete problems of
celestial mechanics.
\subsection{The bicircular model near $L_{4,5}$}
The bicircular problem is a first approximation to study the motion
of a small particle in the Earth--Moon system, including perturbations
coming for the Sun. In this model it is assumed that Earth and Moon
revolve in a circular orbits around their centre of masses, and that
this centre of masses moves in circular orbit around the Sun.
Usually, in order to simplify the equations, the units
of lenght, time and mass are chosen such that the angular velocity
of rotation of Earth and Moon (around their centre of masses),
the sum of masses of Earth and Moon and the gravitational constant
are all equal to one. With these normalized units, the Earth--Moon
distance is also one. The system of reference is defined as follows:
the origin is taken at the centre of mass of the Earth--Moon system,
the $X$ axis is given by the line that goes from Moon to Earth,
the $Z$ axis has the direction of the angular momentum of Earth
and Moon and the $Y$ axis is taken such that the system is
orthogonal and positive-oriented. Note that, in this (non-inertial)
frame, called synodic system, Earth and Moon have fixed positions and
the Sun is rotating around the barycentre of the Earth-Moon system.
If we define momenta $P_{X}=\dot{X}-Y$, $P_{Y}=\dot{Y}+X$ and
$P_{Z}=\dot{Z}$, in these coordinates,
the motion of a infinitessimal particle moving under the gravitational
attraction of Earth, Moon and Sun is given by the Hamiltonian
$$
H=\frac{1}{2}(P_{X}^{2}+P_{Y}^{2}+P_{Z}^{2})+YP_{X}-XP_{Y}
-\frac{1-\mu}{r_{PE}}-\frac{\mu}{r_{PM}}-\frac{m_{s}}{r_{PS}}
-\frac{m_{s}}{a_{s}^{2}}(Y\sin \theta - X\cos \theta),
$$
where $\theta =w_{S}t$, being $w_{S}$ the mean angular velocity of
the Sun in synodic coordinates, $\mu$ the mass parameter for the
Earth--Moon system, $a_s$ the semimajor axis of the Sun, $m_{s}$ the
Sun mass, and $r_{PE}$, $r_{PM}$, $r_{PS}$ are defined in the following
form:
\begin{eqnarray*}
r_{PE}^{2}&=&(X-\mu)^{2}+Y^{2}+Z^{2}, \\
r_{PM}^{2}&=&(X-\mu+1)^{2}+Y^{2}+Z^{2}, \\
r_{PS}^{2}&=&(X-X_{s})^{2}+(Y-Y_{s})^{2}+Z^{2},
\end{eqnarray*}
where $X_{s}=a_{s}\cos \theta$ and $Y_{s}=-a_{s}\sin \theta$.
Note that one can look at this model as a time-periodic perturbation
of an autonomous system, the Restricted Three Body Problem (usually
called RTBP, see \cite{S} for definition and basic properties). Hence,
the Hamiltonian is of the form
$$
H=H_0(x,y)+\epsilon H_1(x,y,t),
$$
where $\epsilon$ is a parameter such that $\epsilon=0$ corresponds
to the unperturbed RTBP and $\epsilon=1$ to the bicircular model with
the actual values for the perturbation.
Note that the bicircular model is not dynamically consistent, because
the motion of Earth, Moon and Sun does not follow a true orbit of the
system (we are not taking into account the interaction between the Sun
and the Earth--Moon system). Nevertheless, numerical simulation shows
that, in some regions of the phase space, this model gives the same
qualitative behaviour as the real system and this makes it worth to
study (see \cite{SGJM}).
We are going to focus in the dynamics near the equilateral points
$L_{4,5}$ of the Earth--Moon system. These points are linearly stable
for the unperturbed problem ($\epsilon=0$), so we can associate three
families of periodic (Lyapounov) orbits to them: the short period
family, the long period family and the vertical family of periodic
orbits. Classical results about these families can be found in
\cite{S}.
When the perturbation is added the points $L_{4,5}$ become (stable)
periodic orbits with the same period as the perturbation.
These orbits become
unstable for the actual value of the perturbation ($\epsilon=1$ in
the notation above). In this last case, numerical simulation shows
the existence of a region of stability not very close to the orbit
and outside of the plane of motion of Earth and Moon.
This region seems to be centered around some of the (Lyapounov)
periodic orbits of the vertical family. See \cite{GJMS93B} or
\cite{SGJM} for more details.
Let us consider the dynamics near $L_{4,5}$ for $\epsilon$ small.
In this case, the equilibrium point has been replaced by a small
periodic orbit. Our results imply that the three families of Lyapounov
periodic orbits become three cantorian families of 2-D invariant
tori, adding the perturbing frequency to the one of the periodic
orbit. Moreover, the Lyapounov tori (the 2-D invariant tori of the
unperturbed problem
that are obtained by ``product'' of two families of periodic
orbits) become 3-D invariant tori, provided they are nonresonant
with the perturbation. Finally, the maximal dimension (3-D) invariant
tori of the unperturbed problem become 4-D tori, adding the frequency
of the Sun to the ones they already had (this last result is already
contained in \cite{JS94B}).
Now let us consider $\epsilon=1$. This value of $\epsilon$ is too
big to apply these results. In particular, $\epsilon$ is big
enough to cause a change of stability in the periodic orbit
that replaces the equilibrium point. Hence, if one wants to
apply the results of this paper to this case, it is necessary
to start by putting the Hamiltonian in a suitable form.
To describe the dynamics near the unstable periodic orbit that
replaces the equilibrium point, we can perform some steps of
a normal form procedure to write the Hamiltonian as an autonomous
(and integrable) Hamiltonian plus a small time dependent periodic
perturbation (see \cite{GJMS93B}, \cite{JS94A} or \cite{SGJM} for more
details about these kind of computations). Then, if we are close enough
to the periodic orbit, Theorem~\ref{teo} applies and we have invariant
tori of dimensions 1, 2 and 3. They are in the ``central''
directions of the periodic orbit.
The application to the stable region that is in the vertical direction
is more difficult. A possiblity is to compute (numerically) an
approximation to a 2-D invariant torus of the vertical family
(note that its existence has not already been proved rigorously)
and to perform some steps of a normal form procedure, in order
to write the problem as an integrable autonomous Hamiltonian
plus a time dependent periodic perturbation. Then, if the (approximate)
torus satisfies the equations within a small enough error,
it should be possible to show the existence of a torus nearby,
and to establish that it is stable and surrounded by invariant
tori of dimensions 1 to 4. Numerical experiments suggest (see
\cite{GJMS93B} or \cite{SGJM}) that this is what happens in this case.
\subsubsection{Extensions}
In fact, the bicircular model is only the first step in the study
of the dynamics near the libration points of Earth-Moon system.
One can construct better models taking into account the
non-circular motion of Earth and Moon (see \cite{DJS}, \cite{GJMS91B},
\cite{GJMS93B}).
Our results can be applied to these models in the same way it has
been done in the bicircular case. The main difference is that now
the equilibrium point is replaced by a quasiperiodic solution
that, due to the resonances, does not exist for all values of $\epsilon$
but only for a Cantor set of them (see \cite{JS94B}).
\subsection{Halo orbits}
Let us consider the Earth and Sun as a RTBP, and let us focus
in the dynamics near the equilibrium point that it is in between
(the so called $L_1$ point). It is well known the existence of a
family of periodic orbits (called Halo orbits, see \cite{R}) such
that, when one looks at them from the Earth, they seem to describe an
halo around the solar disc. These orbits are a very suitable
place to put a spacecraft to study the Sun: from that place,
the Sun is always visible and it is always possible to send data
back to Earth (because the probe does not cross the solar disc,
otherwise the noise coming from the Sun would make communications
impossible). These orbits have been used by missions ISEE-C (from
1978 to 1982) and SOHO (launched in 1995).
In the RTBP, Halo orbits are a one parameter family of periodic
orbits with a normal behaviour of the type centre$\times$saddle.
Unfortunately, the RTBP is too simple to produce good
approximations to the dynamics. If one wants to have a cheap
station keeping it is necessary to compute the nominal orbit
with a very accurate model (see \cite{GJMS91C}, \cite{GJMS91B},
\cite{GJMS93A} and \cite{GJMS93B}).
The usual analytic models for this problem are written as an autonomous
Hamiltonian (the RTBP) plus the effect coming from the real motion
of Earth and Moon, the effect of Venus, etc. All these effects can be
modelled very accurately using quasiperiodic functions that depend
on time in a quasiperiodic way. Hence, we end up with an autonomous
Hamiltonian plus a quasiperiodic time dependent perturbation with
$r>0$ frequencies. As usual, we add a parameter $\epsilon$ in front of
this perturbation.
Then, Theorem~\ref{teo} implies that, if $\epsilon$ is small enough,
the Halo orbits become a cantorian family of ($r+1$)-D invariant tori.
The normal behaviour of these tori is also of the type
centre$\times$saddle.
To study the case $\epsilon=1$ we refer to the remarks for the
case of the bicircular problem.
\section{Proofs}\label{sec:proofs}
This section contains the proof of Theorem~\ref{teo}.
It has been split in several parts to simplify the reading.
Section~\ref{sec:notac} introduces the basic notation used along
the proof. In section~\ref{sec:basic} we give the
basic lemmas needed during the proof. Section~\ref{sec:itl} gives
quantitative estimates on one step of the iterative scheme and
section~\ref{sec:pt} contains the technical details of the proof.
\subsection{Notations}\label{sec:notac}
Here we introduce some of the notations used to prove the
different results.
\subsubsection{Norms and Lipschitz constants}\label{sec:nlc}
As usual we denote by $|v|$ the absolute value of $v\in\CC$,
and we use the same notation to refer to the (maximum) vectorial or
matrix norm on $\CC^n$ or $\MM_{n_1,n_2}(\CC)$.
Let us denote by $f$ an analytic function defined on a complex
strip of width $\rho>0$, having $r$ arguments
and being $2\pi$-periodic in all of them. The range of this
function can be in $\CC$, $\CC^{n}$ or $\MM_{n_1,n_2}(\CC)$.
If we write its Fourier expansion as
$$
f(\theta) = \sum_{k \in \ZZ^{r}} f_{k} \exp{(i k^{\top} \theta)},
$$
we can introduce the norm
$$
|f|_{\rho}=\sum_{k \in \ZZ^{r}} |f_{k}| \exp{(|k|_{1} \rho)}.
$$
Let $f(\theta,q)$ be a $2\pi$-periodic function on $\theta$, and
analytic on the domain
$$
{\cal U}_{\rho,R}^{r,m}=\{(\theta,q)\in\CC^{r}\times\CC^{m}:\;
|\Im \theta|\leq\rho,\;|q|\leq R\}.
$$
If we write its Taylor expansion around $q=0$ as:
$$
f(\theta,q)=\sum_{l\in\NN^{m}}f_{l}(\theta)q^l,
$$
then, from this expansion we define the norm:
$$
|f|_{\rho,R}=\sum_{l\in\NN^{m}}|f_{l}|_{\rho}R^{|l|_1}.
$$
If $f$ takes values in $\CC$, we put $\nabla f$ to denote the gradient
of $f$ with respect to $(\theta,q)$.
Now, we introduce the kind of Lipschitz dependence considered.
Assume that $f(\varphi)$ is a function defined for
$\varphi\in {\cal E}$, ${\cal E}\subset\RR^j$ for some $j$,
and with values in $\CC$, $\CC^{n}$ or ${\MM}_{n_1,n_2}(\CC)$.
We call $f$ a Lipschitz function with respect to $\varphi$ on the set
${\cal E}$ if:
$$
{\cal L}_{\cal E}\{f\}= \sup_{
\begin{array}{c}
\scriptstyle \varphi_1,\varphi_2 \in {\cal E} \\
\scriptstyle \varphi_1 \neq \varphi_2 \end{array}}
\frac{|f(\varphi_2)-f(\varphi_1)|}{|\varphi_2-\varphi_1|} <+\infty.
$$
The value ${\cal L}_{\cal E}\{f\}$ is called the Lipschitz constant of
$f$ on ${\cal E}$. For these kind of functions we define
$\|f\|_{\cal E}=\sup_{\varphi \in {\cal E}} |f(\varphi)|$.
Similarly, if $f(\theta,\varphi)$ is a $2\pi$-periodic analytic
function on $\theta$ for every $\varphi\in{\cal E}$, we denote:
$$
{\cal L}_{{\cal E},\rho}\{f\}= \sup_{
\begin{array}{c}
\scriptstyle \varphi_1,\varphi_2 \in {\cal E} \\
\scriptstyle \varphi_1 \neq \varphi_2 \end{array} }
\frac{|f(.,\varphi_2)-f(.,\varphi_1)|_{\rho}}{|\varphi_2-\varphi_1|}.
$$
In the same way we can introduce
${\cal L}_{{\cal E},\rho,R}\{f\}$, if we work with
$f(\theta,q,\varphi)$ and the norm $|.|_{\rho,R}$.
We can also extend $\|.\|_{\cal E}$ to both cases to define
$\|.\|_{{\cal E},\rho}$ and $\|.\|_{{\cal E},\rho,R}$.
\subsubsection{Canonical transformations}\label{sec:canon}
The changes of variables are performed by means of a Lie
series method, with a suitable generating function. For the sake of
clarity, we will use here the same notations for the different
variables as in the formulation of the results.
We want to keep the quasiperiodic time dependence (after each
transformation) with the same vector of basic frequencies
$\tilde{\omega}^{(0)}$ as the initial one. This is achieved when the
generating function does not depend on $\tilde{I}$.
Let us consider a generating function $S(\theta,x,\hat{I},y)$ such that
$\nabla S$ depends analytically on
$(\theta,x,\hat{I},y)$ and it is $2\pi$-periodic in $\theta$.
The equations related to the Hamiltonian function $S$ are
$$
\dot{\hat{\theta}}=
\left(\frac{\partial S}{\partial\hat{I}}\right)^{\top},\;
\dot{\tilde{\theta}}=
\left(\frac{\partial S}{\partial\tilde{I}}\right)^{\top}=0,\;
\dot{\hat{I}}=
-\left(\frac{\partial S}{\partial\hat{\theta}}\right)^{\top},\;
\dot{\tilde{I}}=
-\left(\frac{\partial S}{\partial\tilde{\theta}}\right)^{\top},\;
\dot{z}=
J_m\left(\frac{\partial S}{\partial z}\right)^{\top}.
$$
We denote by $\Psi_{t}^{S}(\theta,x,I,y)$ the flow at time $t$
of $S$ with initial conditions $(\theta,x,I,y)$ when $t=0$.
We note that $\Psi_t^{S}$ is (for a fixed t) a canonical change
of variables that acts in a trivial way on $\tilde{\theta}$.
If we put
$(\theta(t),x(t),I(t),y(t))={\Psi}_{t}^{S}(\theta(0),x(0),I(0),y(0))$,
we can express the change as :
$$
\hat{\theta}(t)=\hat{\theta}(0)+
\int_{0}^{t}\left(\frac{\partial S}{\partial\hat{I}}
(\theta(\tau),x(\tau),\hat{I}(\tau),y(\tau)) \right)^{\top} d \tau,
$$
$$
I(t)=I(0) - \int_{0}^{t} \left( \frac{ \partial S}{\partial \theta}
(\theta(\tau),x(\tau),\hat{I}(\tau),y(\tau)) \right)^{\top} d \tau,
$$
$$
z(t)= z(0) + J_m \int_{0}^{t} \left( \frac{\partial S}{\partial z}
(\theta(\tau),x(\tau),\hat{I}(\tau),y(\tau)) \right)^{\top} d \tau,
$$
and $\tilde{\theta}(t)=\tilde{\theta}(0)$. We note that the function
$\Psi_{t}^{S}-Id$ does not depend on the auxiliar variables $\tilde{I}$.
Then, we put $\theta(0)=\theta$, $\hat{I}(0)=\hat{I}$ and $z(0)=z$
to introduce the transformations $\hat{\Psi}_{t}^{S}$ and
$\hat{\Phi}_{t}^{S}$, defined as
$\hat{\Psi}_{t}^{S}(\theta,x,\hat{I},y)=
(\theta(t),x(t),\hat{I}(t),y(t))$
and $\hat{\Phi}_{t}^{S}=\hat{\Psi}_{t}^{S}-Id$. It is not difficult
to check that $\hat{\Phi}_{t}^{S}(\theta,x,\hat{I},y)$ is (for a
fixed $t$) $2\pi$-periodic in $\theta$.
If we consider the Hamiltonian function $H$ of (\ref{eq:ha}), and we put
\begin{equation}
H^{**}=\{H^{*},S\}-
\frac{\partial S}{\partial\tilde{\theta}}\tilde{\omega}^{(0)},
\label{eq:Hstar2}
\end{equation}
$\Psi_{t}^{S}$ transforms the Hamiltonian $H$ into
$$
H \circ {\Psi}_{t}^{S}(\theta,x,I,y)=\tilde{\omega}^{(0)\top}\tilde{I}+
H^{*}(\theta,x,\hat{I},y)+tH^{**}(\theta,x,\hat{I},y) +
\Sigma_{t}(H^{**},S)(\theta,x,\hat{I},y),
$$
where
\begin{equation}
\Sigma_t(H^{**},S)=\sum_{j\geq 2}\frac{t^{j}}{j!}L_{S}^{j-1}(H^{**}),
\label{eq:sigma}
\end{equation}
with $L_{S}^{0}(H^{**})=H^{**}$ and
$L_{S}^j(H^{**})=\{L_{S}^{j-1}(H^{**}),S\}$, for $j\geq 1$.
We remark that if we transform a Hamiltonian function $H$ by the
canonical change of variables $\Psi_{t}^{S}$, we only need to control
the transformation $\hat{\Phi}_{t}^{S}$ and to see that the new
Hamiltonian, $H\circ\Psi_{t}^{S}$, is well defined on a suitable domain.
Finally, as the change of variables is selected as the flow at time one
of a Hamiltonian $S$, in what follows we will omit the subscript $t$
and we will assume that it means $t=1$.
\subsection{Basic lemmas}\label{sec:basic}
\subsubsection{Lemmas on norms and Lipschitz constants}
\label{sec:lanlc}
In this section we give some bounds used when working with the
norms and Lipschitz constants introduced in section~\ref{sec:nlc}.
We follow here the same notations of section~\ref{sec:nlc} for the
different analytic functions used in the lemmas.
\begin{lemma}\label{lem:normf}
Let $f(\theta)$ and $g(\theta)$ be analytic functions on a strip of
width $\rho>0$, $2\pi$-periodic in $\theta$ and taking values in $\CC$.
Let us denote by $f_k$ the Fourier coefficients of $f$,
$f(\theta)=\sum_{k\in\ZZ^{r}}f_k\exp{(ik^{\top}\theta)}$.
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 $0<\rho_0<\rho$
$$
\left|\frac{\partial f}{\partial \theta_j}\right|_{\rho-\rho_0} \leq
\frac{ |f|_{\rho}}{\rho_0 \exp(1)}, \; j=1, \ldots,r.
$$
\item[$(iv)$] Let $\{d_k\}_{k\in{\ZZ}^{r}\backslash\{0\}}\subset\CC$,
with the following bounds:
$$
| d_k| \geq \frac{\cd}{|k|_1^{\gamma}} \exp{(-\delta |k|_1)},
$$
for some $\cd>0$, $\gamma\geq 0$, $0\leq\delta<\rho$. If we assume
that $\bar{f}=0$, then the function $g$ defined as
$$
g(\theta) = \sum_{k \in \ZZ^{r}\backslash \{0\}}
\frac{f_k}{d_k} \exp{(i k^{\top} \theta)},
$$
satisfies the bound
$$
|g|_{\rho-\rho_0}\leq
\left(\frac{\gamma}{(\rho_0-\delta)\exp(1)}\right)^{\gamma}
\frac{|f|_{\rho} }{\cd},
$$
for every $\rho_0\in ]\delta,\rho[$.
\end{itemize}
All these bounds can be extended to the case when $f$ and $g$ take
values in $\CC^n$ or $\MM_{n_1,n_2}(\CC)$. Of course, in the
matrix case, in $(ii)$ it is
necessary that the product $fg$ be well defined.
\end{lemma}
\prova Items $(i)$ and $(ii)$ are easily verified.
Proofs of $(iii)$ and $(iv)$ are essentially contained in
\cite{JS94B}, but working with the supremum norm.
\begin{lemma}\label{lem:normt}
Let $f(\theta,q)$ and $g(\theta,q)$ be analytic functions
on a domain ${\cal U}^{r,m}_{\rho,R}$ and
$2\pi$-periodic in $\theta$. Then we have:
\begin{itemize}
\item[$(i)$] If we expand $f(\theta,q)=\sum_{l\in\NN^{m}}
f_l(\theta)q^l$,
then $|f_l|_{\rho}\leq\frac{|f|_{\rho,R}}{R^{|l|_1}}$.
\item[$(ii)$] $|f g|_{\rho,R} \leq |f|_{\rho,R} |g|_{\rho,R}$.
\item[$(iii)$] For every $0<\rho_0<\rho$ and $00$, with Lipschitz dependence with respect to $\varphi$.
Let us expand $f(\theta,\varphi)=
\sum_{k\in\ZZ^{r}}f_k(\varphi)\exp{(ik^{\top}\theta)}$.
Then, we have:
\begin{itemize}
\item[$(i)$] ${\cal L}_{\cal E}\{f_k\}\leq
{\cal L}_{\cal E,\rho}\{f\}\exp{(-|k|_1\rho)}$.
\item[$(ii)$] For every $0<\rho_0<\rho$
$$
{\cal L}_{{\cal E},\rho-\rho_0}
\left\{\frac{\partial f}{\partial\theta_j}\right\}\leq
\frac{{\cal L}_{\cal E,\rho}\{f\}}{\rho_0\exp{(1)}},\; j=1,\ldots,r.
$$
\item[$(iii)$] Let $\{d_k(\varphi)\}_{k\in{\ZZ}^{r}\backslash\{0\}}$
be a set of complex-valued functions defined for $\varphi\in{\cal E}$,
with the following bounds:
$$
|d_k(\varphi)|\geq\frac{\cd}{|k|_1^{\gamma}}\exp{(-\delta |k|_1)},
$$
and
$$
{\cal L}_{\cal E}\{d_k\}\leq A+B|k|_1,
$$
for some $\cd>0$, $\gamma\geq 0$, $0\leq 2\delta<\rho$, $A\geq 0$ and
$B\geq 0$. As in lemma~\ref{lem:normf} we assume $\bar{f}=0$
for every $\varphi\in{\cal E}$. If
$$
g(\theta,\varphi) = \sum_{k \in \ZZ^{r}\backslash \{0\}}
\frac{f_k(\varphi)}{d_k(\varphi)} \exp{(i k^{\top} \theta)},
$$
then, for every $\rho_0$, $2\delta< \rho_0 <\rho$, we have:
\begin{eqnarray*}
{\cal L}_{{\cal E},\rho-\delta_0} \{g\} & \leq &
\left(\frac{\gamma}{(\delta_0-\delta)\exp(1)}\right)^{\gamma}
\frac{{\cal L}_{\cal E,\rho} \{f\}}{\cd} +
\left(\frac{2\gamma+1}{(\delta_0-2\delta)\exp(1)}\right)^{2\gamma+1}
\frac{\|f\|_{{\cal E},\rho }} {\cd^2}B +
\\ & & +
\left(\frac{2\gamma}{(\delta_0 -2\delta)\exp(1)} \right)^{2\gamma}
\frac{\|f\|_{{\cal E},\rho }}{\cd^2}A.
\end{eqnarray*}
\end{itemize}
\end{lemma}
\prova It is analogous to lemma~\ref{lem:normf}, using also
the results of lemma~\ref{lem:tli}.
\begin{lemma}\label{lem:lipnormt}
We assume that $f(\theta,q,\varphi)$ is, for every
$\varphi\in{\cal E}$, an analytic function on
${\cal U}^{r,m}_{\rho,R}$ and $2\pi$-periodic in $\theta$. Then we
have:
\begin{itemize}
\item[$(i)$] If we write $f(\theta,q,\varphi)=
\sum_{l\in\NN^{m}} f_{l}(\theta,\varphi) q^{l}$, then
${\cal L}_{{\cal E},\rho} \{f_l \} \leq
\frac{{\cal L}_{{\cal E},\rho,R} \{ f \}}{R^{|l|_1}}$.
\item[$(ii)$] For every $0<\rho_0<\rho$ and $00$, $b\geq0$, $c>0$ and $1 < \varrho < 2$. Then:
$$
K_{n+1} \leq \frac{1}{a} \left( \left( \frac{5}{3} \right)^b a K_1
\exp{\left(\frac{c \varrho}{2-\varrho} \right)} \right)^{2^n}.
$$
\end{lemma}
\prova The proof is a direct combination of results contained in
\cite{JS92} and \cite{JS94B}.
\subsubsection{Lemmas on the control of the measure}\label{sec:lacm}
In the following lemmas, we consider a fixed
$\omega^{(0)\top}=(\hat{\omega}^{(0)\top},\tilde{\omega}^{(0)\top})$,
with $\hat{\omega}^{(0)}\in\RR^r$ and $\tilde{\omega}^{(0)}\in\RR^s$.
Let $\lambda(\varphi)$ be a function defined on
${\cal E}\subset\RR^{r+1}$ with range in $\CC$, where
$\varphi^{\top}=(\hat{\omega}^{\top},\epsilon)$,
with $\hat{\omega}\in\RR^{r}$ and $\epsilon\in\RR$.
We assume that $\lambda$ takes the form:
$$
\lambda(\varphi)=
\lambda_0+iu\epsilon+iv^{\top}(\hat{\omega}-\hat{\omega}^{(0)})
+ \tilde{\lambda}(\varphi),
$$
where $\lambda_0$, $u\in\CC$, $v\in\CC^r$ and, if we denote by
$\bar{\cal E}= \bar{\cal E}(\bar{\vartheta}):= \left\{\varphi\in{\cal E}
\;:|\varphi-\varphi^{(0)}|\leq\bar{\vartheta}\right\}$,
$\varphi^{(0)\top}=(\hat{\omega}^{(0)\top},0)$, then we have that
${\cal L}_{\bar{\cal E}}\{\tilde{\lambda}\}\leq L \bar{\vartheta}$ for
certain $L\geq0$, for all $0\leq\bar{\vartheta}\leq {\vartheta}_0$.
We also assume that
$|\lambda(\varphi)-\lambda_0|\leq M|\varphi-\varphi^{(0)}|$ for all
$\varphi\in\bar{\cal E}(\vartheta_0)$. We remark that the Lipschitz
bound for $\tilde{\lambda}$ formulated on a
sufficiently smooth function, means that $\tilde{\lambda}$ is of
$O_2(\varphi-\varphi^{(0)})$.
Now, we take $\cd>0$, $\gamma>r+s-1$ and $0<\delta\leq 1$
to define from $\lambda$ and ${\cal E}$ the following
``resonant'' sets:
\begin{eqnarray*}
{\cal R}(\epsilon_0,R_0) & = & \left\{ \hat{\omega} \in \RR^{r} : \;
|\hat{\omega}-\hat{\omega}^{(0)}|\leq R_0,\;
(\hat{\omega}^{\top},\epsilon_0)^{\top}=\varphi\in{\cal E}
\mbox{ and }\right.
\\ & & \left.
\; \exists k\in\ZZ^{r+s} \backslash \{0\}
\mbox{ such that } |i k^{\top} \omega + \lambda(\varphi)| <
\frac{\cd}{|k|_1^{\gamma}}\exp{(-\delta |k|_1)}
\right\},
\end{eqnarray*}
for every $\epsilon_0 \geq 0$ and $R_0 \geq 0$, and
\begin{eqnarray*}
{\cal A}(\epsilon_0,\hat{\omega}) & = &
\left\{ \epsilon \in [0,\epsilon_0] : \;
(\hat{\omega}^{\top},\epsilon)^{\top}=\varphi\in{\cal E}
\mbox{ and } \right.
\\ & & \left.
\; \exists k \in \ZZ^{r+s} \backslash \{0\}
\mbox{ such that } |i k^{\top} \omega + \lambda(\varphi)| <
\frac{\cd}{|k|_1^{\gamma}}\exp{(-\delta |k|_1)}
\right\},
\end{eqnarray*}
for every $\hat{\omega}\in\RR^r$ and $\epsilon_0>0$, where in both cases
$\omega\in\RR^{r+s}$ is defined from
$\varphi^{\top}=(\hat{\omega}^{\top},\epsilon)$ as
$\omega^{\top}=(\hat{\omega}^{\top},\tilde{\omega}^{(0)\top})$.
Note that these sets depend on $\delta$ and $\cd$.
As the purpose of this section is to deal with the measure
of these resonant sets, we will always assume we are in the worst
case: $\Re\lambda_0 = 0$. When this is not true (this is, when there
are no resonances) it is not difficult to see that the sets
${\cal R}$ and ${\cal A}$ are empty if we are close enough to
$\varphi^{(0)}$ (the value of the parameter for the unperturbed
system). We want to remark that we are not making any assumption
on the values $\Im u$ and $\Im v$. According to the size of
the resonant sets the worst case happens when $\Im u=0$ and/or
$\Im v=0$. Hence, the proof will be valid in this case,
although it is possible to improve the measure estimates
assuming that $\Im u\ne 0$ and $\Im v\ne 0$.
\begin{lemma}\label{lem:meas1}
If we assume that
$|ik^{\top}\omega^{(0)}+\lambda_0|\geq\frac{\cd_0}{|k|_1^{\gamma}}$,
$\gamma>r+s-1$,
for all $k\in\ZZ^{r+s}\backslash\{0\}$, for certain $\cd_0\geq 2\cd$,
then, if $K$ is the only positive solution of
$\frac{\cd_0}{2 K^{\gamma}} = M \max{\{R_0,\epsilon_0\}} + K R_0 $,
we have:
\begin{itemize}
\item[$(i)$]
If $\Re v \notin \ZZ^{r}$, and $\epsilon_0$, $R_0$ are small enough
(condition that depends only on $v$, $L$, $\vartheta_0$, $\gamma$,
$\cd_0$ and $M$), then,
$$
\mbox{mes}({\cal R}(\epsilon_0,R_0))\leq
16\cd(2\sqrt{r}R_0)^{r-1}(r+s)\hat{K}(v)
K^{r+s-1-\gamma}\frac{\exp{(-\delta K)} }{\delta},
$$
where $\hat{K}(v):=\sup_{\hat{k}\in\ZZ^r}
{\left\{\frac{1}{|\hat{k}+\Re v|_2}
\right\}}$, being $|.|_2$ the Euclidean norm of $\RR^r$.
\item[$(ii)$]
If $u \neq 0$, and $\epsilon_0$,
$R_0:=|\hat{\omega}-\hat{\omega}^{(0)}|$ are small enough (condition
that depends only on $u$, $L$, $\vartheta_0$, $\gamma$, $\cd_0$ and
$M$), then
$$
\mbox{mes}({\cal A}(\epsilon_0,\hat{\omega}))\leq
\frac{16\cd}{|u|}(r+s)K^{r+s-1-\gamma}\frac{\exp{(-\delta K)} }{\delta}.
$$
\end{itemize}
\end{lemma}
\prova We prove here the part $(i)$. Similar ideas can be used to
prove $(ii)$. To study the measure of ${\cal R}(\epsilon_0,R_0)$, we
consider the following decomposition:
$$
{\cal R}(\epsilon_0,R_0)=
\bigcup_{k \in \ZZ^{r+s} \backslash \{0\}} {\cal R}_k(\epsilon_0,R_0),
$$
where ${\cal R}_k (\epsilon_0,R_0)$ is defined as:
\begin{eqnarray*}
{\cal R}_k(\epsilon_0,R_0) & = & \left\{ \hat{\omega} \in \RR^{r} : \;
|\hat{\omega}-\hat{\omega}^{(0)}| \leq R_0, \;
(\hat{\omega}^{\top},\epsilon_0)^{\top}=\varphi\in{\cal E} \right.
\\ & & \left.
\mbox{ and } |ik^{\top}\omega+\lambda(\varphi)|<
\frac{\cd}{|k|_1^{\gamma}}\exp{(-\delta |k|_1)} \right\}.
\end{eqnarray*}
To compute the measure of these sets, we take
$\hat{\omega}^{(1)}$, $\hat{\omega}^{(2)}\in{\cal R}_k(\epsilon_0,R_0)$,
and we put $\varphi^{(j)\top}=(\hat{\omega}^{(j)\top},\epsilon_0)$ and
$\omega^{(j)\top}=(\hat{\omega}^{(j)\top},\tilde{\omega}^{(0)\top})$.
Then, from $|ik^{\top}\omega^{(j)}+\lambda(\varphi^{(j)})| <
\frac{\cd}{|k|_1^{\gamma}}\exp{(-\delta |k|_1)}$, we clearly have that
$|i \hat{k}^{\top}(\hat{\omega}^{(1)}-\hat{\omega}^{(2)})+
\lambda(\varphi^{(1)})-\lambda(\varphi^{(2)})|<
\frac{2\cd}{|k|_1^{\gamma}}\exp{(-\delta |k|_1)}$, where we have
split $k^{\top}=(\hat{k}^{\top},\tilde{k}^{\top})$, with
$\hat{k}\in\ZZ^r$
and $\tilde{k}\in\ZZ^s$. From that, and using the definition of
$\lambda(\varphi)$ one obtains:
$$
|i ( \hat{k} + v )^{\top} (\hat{\omega}^{(1)} - \hat{\omega}^{(2)}) +
\tilde{\lambda} (\varphi^{(1)}) - \tilde{\lambda}(\varphi^{(2)}) | <
\frac{2\cd}{|k|_1^{\gamma}}\exp{(-\delta |k|_1)}.
$$
Note that the set ${\cal R}_k$ is a slice of the set of
$\hat{\omega}$ such that
$|\hat{\omega}-\hat{\omega}^{(0)}|\le R_0$. To estimate its
measure, we are going to take the values $\hat{\omega}^{(1)}$ and
$\hat{\omega}^{(2)}$ such that $\hat{\omega}^{(1)}-\hat{\omega}^{(2)}$
is (approximately) perpendicular to the slice, that is,
parallel to the vector $\hat{k} + \Re v$. Then,
$\mbox{mes}({\cal R}_k)$ can be bounded by the product of
a bound of the value $|\hat{\omega}^{(1)}-\hat{\omega}^{(2)}|$ by
(a bound of) the measure of the worst (biggest) section of an
hyperplane (of codimension 1) with the set
$|\hat{\omega}-\hat{\omega}^{(0)}|\le R_0$.
Hence, assuming now that $\hat{\omega}^{(1)}-\hat{\omega}^{(2)}$ is
parallel to the vector $\hat{k} + \Re v$, we have:
\begin{eqnarray*}
| \hat{\omega}^{(1)} - \hat{\omega}^{(2)} |_2 & = &
\frac{|(\hat{k}+\Re v)^{\top}(\hat{\omega}^{(1)}-\hat{\omega}^{(2)})|}
{| \hat{k} + \Re v |_2} \leq
\frac{|(\hat{k}+v)^{\top}(\hat{\omega}^{(1)} - \hat{\omega}^{(2)}) |}
{| \hat{k} + \Re v |_2} \leq
\\ & \leq &
\frac{1}{| \hat{k} + \Re v |_2}
\left(L\max{\{R_0,\epsilon_0\}}|\hat{\omega}^{(1)}-\hat{\omega}^{(2)}|+
\frac{2\cd}{|k|_1^{\gamma}}\exp{(-\delta |k|_1)}\right).
\end{eqnarray*}
In consequence:
$$
\left(1-\frac{L\max{\{R_0,\epsilon_0\}}}{|\hat{k}+\Re v|_2}\right)
|\hat{\omega}^{(1)}-\hat{\omega}^{(2)}|_2\leq
\frac{2\cd}{|k|_1^{\gamma}}\exp{(-\delta |k|_1)})
\frac{1}{|\hat{k}+\Re v|_2}.
$$
So, if $\epsilon_0$ and $R_0$ are small enough (independent on $k$) we
can bound:
$$
|\hat{\omega}^{(1)}-\hat{\omega}^{(2)}|_2\leq\frac{4\cd}{|k|_1^{\gamma}}
\exp{(-\delta |k|_1)} \hat{K},
$$
where we put $\hat{K}=\hat{K}(v)$. From that:
$$
\mbox{mes}({\cal R}_k(\epsilon_0,R_0))\leq\frac{4\cd}{|k|_1^{\gamma}}
\exp{(-\delta |k|_1)}(2\sqrt{r}R_0)^{r-1}\hat{K},
$$
where $2 \sqrt{r} R_0$ is a bound for the diameter of the set
$\{\hat{\omega}\in\RR^r:\;|\hat{\omega}-\hat{\omega}^{(0)}|\leq R_0\}$.
Then, we have:
\begin{equation}
\mbox{mes}({\cal R}(\epsilon_0,R_0))\leq
\sum_{k\in\ZZ^{r+s}\backslash \{0\}}\frac{4\cd}{|k|_1^{\gamma}}
\exp{(-\delta|k|_1)}(2\sqrt{r} R_0)^{r-1}\hat{K}.
\label{eq:sumeas}
\end{equation}
In fact, in this sum we only need to consider
$k\in\ZZ^{r+s}\backslash\{0\}$ such that
${\cal R}_k(\epsilon_0,R_0)\neq\emptyset$. Now, let us see that
${\cal R}_k(\epsilon_0,R_0)$ is empty if $|k|_1$ is less than some
critical value $K$.
Let $\varphi \in {\cal R}_k (\epsilon_0,R_0)$, then we can write:
\begin{eqnarray*}
\frac{\cd_0}{|k|_1^{\gamma}}& \leq & |ik^{\top}\omega^{(0)}+\lambda_0|
\leq |ik^{\top}\omega+\lambda(\varphi)|+|\lambda(\varphi)-\lambda_0|+
|\hat{k}^{\top}(\hat{\omega}-\hat{\omega}^{(0)})| \leq
\\ & \leq &
\frac{\cd}{|k|_1^{\gamma}}\exp{(-\delta|k|_1)}+M\max{\{R_0,\epsilon_0\}}
+|\hat{k}|_1|\hat{\omega}-\hat{\omega}^{(0)}|\leq
\\ & \leq & \frac{\cd}{|k|_1^{\gamma}}+M\max{\{R_0,\epsilon_0\}}+
|\hat{k}|_1 R_0,
\end{eqnarray*}
and then:
$$
\frac{\cd_0}{2|k|_1^{\gamma}}\leq M\max{\{R_0,\epsilon_0\}}+|k|_1 R_0.
$$
So, in the sum (\ref{eq:sumeas}), we only need to consider
$k\in\ZZ^{r+s}\backslash\{0\}$ such that $|k|_1\geq K$, where $K$
(that depends on $R_0$ and $\epsilon_0$) is defined in the
statement of the lemma. We assume $R_0$ and $\epsilon_0$ small enough
such that $K \geq 1$. Now, using that
$\#\{k\in\ZZ^{r+s}:\;|k|_1 =j\}\leq 2(r+s)j^{r+s-1}$ and
that $\gamma > r+s-1$, we have:
\begin{eqnarray*}
\mbox{mes}({\cal R}(\epsilon_0,R_0)) & \leq &
4\cd(2\sqrt{r}R_0)^{r-1}\hat{K} \sum_{ \begin{array}{c}
\scriptstyle k \in \ZZ^{r+s} \backslash \{0\} \\
\scriptstyle |k|_1 \geq K
\end{array} }
\frac{\exp{(-\delta |k|_1)}}{|k|_1^{\gamma}} \leq
\\ & \leq &
4 \cd (2 \sqrt{r} R_0)^{r-1} \hat{K} \sum_{j \geq K} 2(r+s) j^{r+s-1}
\frac{\exp{(-\delta j)}}{j^{\gamma}} \leq
\\ & \leq &
8\cd(2\sqrt{r}R_0)^{r-1}(r+s)\hat{K}K^{r+s-1-\gamma}
\sum_{j \geq K}{\exp{(-\delta j)}}=
\\ & = &
8\cd(2\sqrt{r}R_0)^{r-1}(r+s)\hat{K}K^{r+s-1-\gamma}
\frac{\exp{(-\delta K)}}{1 - \exp{(-\delta)}} \leq
\\ & \leq &
16\cd(2\sqrt{r}R_0)^{r-1}(r+s)\hat{K}K^{r+s-1-\gamma}
\frac{ \exp{(-\delta K)} }{\delta},
\end{eqnarray*}
where we used that $\frac{1}{1-\exp{(-\delta)}}\leq\frac{2}{\delta}$, if
$0<\delta\leq 1$.
\begin{lemma}\label{lem:meas2}
With the previous notations, we introduce the set
\begin{eqnarray*}
{\cal D}(R_0) & = & \left\{ \hat{\omega} \in \RR^{r} : \;
|\hat{\omega}-\hat{\omega}^{(0)}| \leq R_0 \mbox{ and }\right.
\\ & & \left.\;\exists k\in\ZZ^{r+s}\backslash\{0\} \mbox{ such that }
|k^{\top}\omega|<
\frac{\cd}{|k|_1^{\gamma}}\exp{(-\delta |k|_1)}\right\}.
\end{eqnarray*}
Let us assume $|k^{\top}\omega^{(0)}|\geq\frac{\cd_0}{|k|_1^{\gamma}}$
for all $k\in\ZZ^{r+s}\backslash\{0\}$, for certain $\cd_0\geq 2\cd$.
Then, if $R_0$ is small enough (depending only on $\gamma$ and
$\cd_0$), one has:
$$
\mbox{mes}({\cal D}(R_0))\leq 8\cd(2\sqrt{r} R_0)^{r-1}(r+s)
K^{r+s-1-\gamma}\frac{\exp{(-\delta K)}}{\delta},
$$
being $K=\left( \frac{\cd_0}{2R_0} \right)^{\frac{1}{\gamma+1}}$.
\end{lemma}
\prova It is similar to the one of lemma~\ref{lem:meas1}.
\begin{lemma}\label{lem:meas3}
Let $\cd_0>0$, $\rho>0$ and $1<\varrho<2$ fixed. We put
$\sigma_n=\frac{6}{\pi^2 n^2}$, $\delta_n=\frac{\sigma_n\rho}{18}$ and
$\cd_n=\cd_0\exp{(-\varrho^n)}$, for all $n\geq 1$. Then, for every
$0<\sigma<1$, we have that, if $K$ is big enough (depending only on
$\varrho$, $\rho$, $\cd_0$ and $\sigma$),
$$
\sum_{n\geq 1}\cd_n\frac{\exp{(-\delta_n K)}}{\delta_n}\leq
\exp{(-K^{\sigma})}.
$$
\end{lemma}
\prova Let $n_*(K)=\frac{\ln{K}}{\ln{\varrho}}$. We remark that
$n_*(K)\rightarrow +\infty$ as $K\rightarrow +\infty$. Then, if $K$ is
big enough one has that, for all $n\geq n_*(K)$,
$$
\frac{(n+1)^2\exp{\left(-\varrho^{n+1}\right)}}
{n^2\exp{\left(-\varrho^{n}\right)}} =
\left(\frac{n+1}{n}\right)^2\exp{\left(-\varrho^{n}(\varrho-1)\right)}
\leq
\exp{\left( -\frac{\varrho -1 }{2} \right)},
$$
that allows to bound
$$
\sum_{n\geq 1} n^2\exp{\left(-\varrho^{n}\right)}\leq
\frac{n_*^2\exp{\left(-\varrho^{n}\right)}}
{1-\exp{\left( -\frac{\varrho -1 }{2}\right)}}.
$$
Hence,
\begin{eqnarray*}
\lefteqn{
\sum_{n \geq 1} \cd_n \frac{\exp{(-\delta_n K)}}{\delta_n} =
\sum_{n \geq 1} \frac{3 \cd_0\pi^2}{\rho} n^2
\exp{ \left( -\varrho^n - \frac{K \rho}{3 \pi^2 n^2} \right) } = }
\\ & = &
\frac{3\cd_0\pi^2}{\rho}\left(\sum_{1\leq n=0$ (see section~\ref{sec:its} for the definition)
for all $\varphi$.
This implies that, skipping the term $\hat{H}$ (this is the small
perturbation), we have for every $\varphi\in{\cal E}$ an invariant
($r+s$)-dimensional reducible torus with basic frequencies $\omega$.
Moreover, we also assume that ${\cal B}$ and ${\cal C}$ are symmetric
matrices with ${\cal J}_m({\cal B})={\cal B}$ and
$\det\bar{\cal C}\neq 0$. Hence, $J_m{\cal B}$ is a diagonal matrix,
with eigenvalues
$\lambda(\varphi)^{\top}=(\lambda_1(\varphi),\ldots,\lambda_m(\varphi),
-\lambda_1(\varphi),\ldots,-\lambda_m(\varphi))$, that we asume
all different, that gives the normal
behaviour around the unpertubed invariant torus.
More concretely, let us assume that the following bounds hold:
for the unperturbed part, for every $j,l=1,\ldots,2m$ with $j\neq l$,
we have $0<\alpha_1\leq |\lambda_j(\varphi)-\lambda_l(\varphi)|$,
$\alpha_1/2 \leq|\lambda_j(\varphi)|\leq \alpha_2/2$
for all $\varphi\in{\cal E}$,
and that ${\cal L}_{\cal E}\{\lambda_j\}\leq\beta_2/2$.
Moreover, $\|(\bar{\cal C})^{-1}\|_{\cal E}\leq\bar{m}$ and,
for certain $\rho>0$ and $R>0$, $\|{\cal C}\|_{\cal E,\rho}\leq\hat{m}$,
${\cal L}_{{\cal E},\rho} \{{\cal C} \} \leq \tilde{m}$,
$\|H_*\|_{{\cal E},\rho,R}\leq\hat{\nu}$ and
${\cal L}_{{\cal E},\rho,R} \{H_* \} \leq \tilde{\nu}$.
Finally, we bound the size of the perturbation $\hat{H}$ by
$\|\hat{H}\|_{{\cal E},\rho,R}\leq M$ and
${\cal L}_{{\cal E},\rho,R} \{H_* \} \leq L$.
To simplify the bounds, we will assume that $M\le L$.
\begin{lemma} \mbox{\bf (Iterative lemma)}\label{lem:itl}
Let us consider a Hamiltonian $H$ as the one we have just described
above. We assume that we can bound $\rho$, $R$, $\alpha_2$, $\beta_2$,
$\bar{m}$, $\hat{m}$, $\tilde{m}$, $\hat{\nu}$, $\tilde{\nu}$, $M$ and
$L$ by certain fixed absolute constants $\rho_0$, $R_0$,
$\alpha_2^*$,
$\beta_2^*$, $\bar{m}^*$, $\hat{m}^*$, $\tilde{m}^*$, $\hat{\nu}^*$,
$\tilde{\nu}^*$, $M_0$ and $L_0$, and that for some fixed $R^*>0$ and
$\alpha_1^*>0$ we have $R^* \leq R$ and $\alpha_1^*\leq\alpha_1$.
We assume that for every $\varphi\in{\cal E}$, the corresponding
$\omega$ verifies $|\omega|\leq\cotw^*$, for some fixed
$\cotw^*>0$. Finally, we also consider fixed $\tilde{\delta}_0>0$,
$\gamma >r+s-1$ and $\cd_0>0$.
In these conditions, there exist a constant $\hat{N}$, only depending
on the fixed constants showed above plus $r$, $m$ and $s$, such that
for every $\delta>0$, $\hat{\delta}>0$ and $\cd>0$
for which the following three conditions hold,
\begin{itemize}
\item[{\rm a)}] $0<9\delta<\rho$, $0<9\hat{\delta}<\rho$ and
$\delta/\hat{\delta}\leq\tilde{\delta}_0$,
\item[{\rm b)}] for every $\varphi\in{\cal E}$,
\begin{equation}
|ik^{\top}\omega+l^{\top}{\lambda}(\varphi)|\ge
\frac{\cd}{|k|_1^{\gamma}}\exp{(-\delta |k|_1)},\;\;
k\in {\ZZ}^{r+s}\backslash\{0\},\;\; l\in\NN^{2m}, |l|_1\leq 2,
\label{eq:dcil}
\end{equation}
being $0<\cd\leq\cd_0$,
\item[{\rm c)}]
$\Theta:=\hat{N}\frac{M}{\delta^{2\gamma+3}\cd^2}\leq 1/2$,
\end{itemize}
we have that there exist a function $S(\theta,x,\hat{I},y,\varphi)$,
defined for every $\varphi\in{\cal E}$, with $\nabla S$ an analytic
function with respect to $(\theta,x,\hat{I},y)$ on
${\cal U}_{\rho-8\delta,R-8\hat{\delta}}^{r+s,2m+r}$, $2\pi$-periodic
on $\theta$ and with Lipschitz dependence on
$\varphi\in{\cal E}$, such that
$\|{\nabla} S\|_{{\cal E},\rho-8\delta,R-8\hat{\delta}}\leq
\min\{\delta,\hat{\delta}\}$. Moreover, following the notations of
section~\ref{sec:canon}, the canonical change of variables
$\Psi^S$ is well defined for every $\varphi\in{\cal E}$,
\begin{equation}
\hat{\Psi}^S : {\cal U}_{\rho-9\delta,R-9\hat{\delta}}^{r+s,2m+r}
\longrightarrow
{\cal U}_{\rho-8\delta,R-8\hat{\delta}}^{r+s,2m+r},
\label{eq:Psiacts}
\end{equation}
and transforms $H$ into
$$
H^{(1)}(\theta,x,I,y,\varphi):=H \circ \Psi^{S}(\theta,x,I,y),
$$
where
$$
H^{(1)}=\ti^{(1)}(\varphi)+{\omega}^{\top}I+
\frac{1}{2}z^{\top}{\cal B}^{(1)}(\varphi)z+
\frac{1}{2}\hat{I}^{\top}{\cal C}^{(1)}(\theta,\varphi)\hat{I}+
H_*^{(1)}+\hat{H}^{(1)},
$$
with $=0$, and where ${\cal B}^{(1)}$ and ${\cal C}^{(1)}$
are symmetric matrices with
${\cal J}_m({\cal B}^{(1)})={\cal B}^{(1)}$. Moreover if we put
$R^{(1)}=R-9\hat{\delta}$ and $\rho^{(1)}=R-9\delta$, we have the
following bounds:
$$
\begin{array}{rclrcl}
\|{\nabla} S\|_{{\cal E},\rho-8\delta,R-8\hat{\delta}} & \leq &
\hat{N} \frac{M}{ \delta^{2+2\gamma} \cd^{2}}, &
\|\ti^{(1)}-\ti\|_{\cal E} & \leq &
\hat{N}\frac{M}{\delta^{1+\gamma}\cd},\\
\|{\cal B}^{(1)}-{\cal B}\|_{\cal E} & \leq &
\hat{N}\frac{M}{\delta^{1+\gamma}\cd}, &
{\cal L}_{{\cal E}}\{{\cal B}^{(1)}-{\cal B}\} & \leq &
\hat{N}\frac{L}{ \delta^{2+2\gamma} \cd^{2}},\\
\|{\cal C}^{(1)}-{\cal C}\|_{{\cal E},\rho^{(1)}} & \leq &
\hat{N}\frac{M}{\delta^{2+2\gamma}\cd^{2}}, &
{\cal L}_{{\cal E},\rho^{(1)}}\{{\cal C}^{(1)}-{\cal C}\} & \leq &
\hat{N}\frac{L}{\delta^{3+3\gamma}\cd^{3}},\\
\|H_*^{(1)}-H_*\|_{{\cal E},\rho^{(1)},R^{(1)}} & \leq &
\hat{N}\frac{M}{\delta^{3+2\gamma}\cd^{2}}, &
{\cal L}_{{\cal E},\rho^{(1)},R^{(1)}}\{H_*^{(1)}-H_* \} & \leq &
\hat{N}\frac{L}{\delta^{4+3\gamma}\cd^{3}},\\
\|\hat{H}^{(1)}\|_{{\cal E},\rho^{(1)},R^{(1)}} & \leq &
\hat{N}\frac{M^2}{\delta^{6 + 4\gamma} \cd^{4}}, &
{\cal L}_{{\cal E},\rho^{(1)},R^{(1)}}\{\hat{H}^{(1)}\} & \leq &
\hat{N} \frac{ M L}{ \delta^{7+5\gamma} \cd^{5}}.
\end{array}
$$
\end{lemma}
\prova The idea is to use the scheme described in section~\ref{sec:its},
to remove the perturbative terms that are an obstruction for the
existence of a (reducible) torus with vector of basic frequencies
$\omega$ up to first order in the size of the perturbation.
Hence, as we described in section~\ref{sec:its}
we expand $\hat{H}$ in power series around $\hat{I}=0$, $z=0$ to
obtain $H=\tilde{\omega}^{(0)}\tilde{I}+ H^*$, being
$$
H^{*}= a(\theta) +
b(\theta)^{\top} z +
c(\theta)^{\top} \hat{I} +
\frac{1}{2} z^{\top} B(\theta) z +
\hat{I}^{\top} E(\theta) z +
\frac{1}{2}\hat{I}^{\top}C(\theta)\hat{I}+
\Omega(\theta,x,\hat{I},y),
$$
with $<\Omega>=0$, where we have not written explicitly the dependence
on $\varphi$. We look for a generating function $S$,
$$
S(\theta,x,\hat{I},y)=\xi^{\top} \hat{\theta} +
d(\theta) +
e(\theta)^{\top} z +
f(\theta)^{\top} \hat{I} +
\frac{1}{2} z^{\top} G(\theta) z +
\hat{I}^{\top} F(\theta) z,
$$
with the same properties as the one given in (\ref{eq:gf2}).
If we want to obtain the transformed Hamiltonian $H^{(1)}$
we need to compute (see section~\ref{sec:canon}):
$$
H^{**}=\{H^{*},S\}-
\frac{\partial S}{\partial\tilde{\theta}}\tilde{\omega}^{(0)}.
$$
We introduce the decomposition $H^{**}=H^{**}_1+H^{**}_2$, with
$$
H^{**}_1=\{\omega^{\top}I+\frac{1}{2}z^{\top}{\cal B}z+
\frac{1}{2}\hat{I}^{\top}{\cal C}\hat{I}+ H_*,S \}
-\frac{\partial S}{\partial\tilde{\theta}}\tilde{\omega}^{(0)},
$$
and $H^{**}_2=\{\hat{H},S\}$. Then, we want to select $S$ such that
$H+H^{**}_1$ takes the form
\begin{equation}
H+H^{**}_1=\ti^{(1)}(\varphi)+{\omega}^{\top}I+
\frac{1}{2}z^{\top}{\cal B}^{(1)}(\varphi)z+
\frac{1}{2}\hat{I}^{\top}{\cal C}^{(1)}(\theta,\varphi)\hat{I}+
H_*^{(1)}(\theta,x,\hat{I},y,\varphi),
\label{eq:elt}
\end{equation}
and hence, $\hat{H}^{(1)}=H^{**}_2+\Sigma(H^{**},S)$.
We can explicitly compute $H^{**}_1$:
\begin{eqnarray}
H^{**}_1 & = &
\left(
\frac{1}{2} \frac{\partial}{\partial\hat{\theta}}
\left(\hat{I}^{\top}{\cal C}\hat{I} \right)+
\frac{\partial H_*}{\partial\hat{\theta}} \right)\left(f+Fz\right) -
\left(\hat{I}^{\top}{\cal C}+\frac{\partial H_*}{\partial\hat{I}}\right)
\left(\frac{\partial S}{\partial\hat{\theta}}\right)^{\top}+
\nonumber \\ & & +
\left(z^{\top}{\cal B}+\frac{\partial H_*}{\partial z} \right)
J_m ( e + G z + F^{\top} \hat{I} ) -
\frac{\partial S}{\partial {\theta}} {\omega}.
\label{eq:H1star2}
\end{eqnarray}
Then, it is not difficult to see that equation (\ref{eq:elt}) leads to
equations $(eq_1)-(eq_5)$ given in section~\ref{sec:its}, replacing
$\omega^{(0)}$ by $\omega$, and that
\begin{equation}
\ti^{(1)}=\ti-\hat{\omega}^{\top} \xi,
\label{eq:ti1}
\end{equation}
\begin{equation}
{\cal B}^{(1)}={\cal J}_m(B^*),
\label{eq:calB1}
\end{equation}
\begin{equation}
{\cal C}^{(1)}={\cal C}+\left[\hat{I}^{\top} \left(
\frac{1}{2}\frac{\partial {\cal C}}{\partial \hat{\theta}} f -
{\cal C}
\left(\frac{\partial f}{\partial\hat{\theta}}\right)^{\top}\right)
\hat{I}- \frac{\partial H_*}{\partial\hat{I}}\left(\xi +
\left(\frac{\partial d}{\partial\hat{\theta}} \right)^{\top} \right) +
\frac{\partial H_*}{\partial z}J_me\right]_{(\hat{I},\hat{I})},
\label{eq:calC1}
\end{equation}
\begin{equation}
H^{(1)}_*=\Omega+H_1^{**}-.
\label{eq:Hstar1}
\end{equation}
We will prove that, from the Diophantine bounds of (\ref{eq:dcil}),
it is possible to construct a convergent expression for $S$,
and to obtain suitable bounds for the transformed Hamiltonian.
The first step is to bound the solutions of $(eq_1)-(eq_5)$,
using lemmas of section~\ref{sec:lanlc}.
In what follows, $\hat{N}$ denotes a constant that bounds all the
expressions depending on all the absolute fixed constants of the
statement of the lemma, and its value is redefined several
times during the proof in order to simplify the notation.
Moreover, sometimes we do not write explicitly the dependence on
$\varphi$, but all the bounds hold for all $\varphi\in{\cal E}$.
First, we remark that using the bounds on $\hat{H}$, and from lemmas
\ref{lem:normt} and \ref{lem:lipnormt} we can bound
$\|.\|_{{\cal E},\rho}$ and ${\cal L}_{{\cal E},\rho}\{.\}$ of
$a-\ti$, $b$, $c-\hat{\omega}$, $B-{\cal B}$, $E$ and $C-{\cal C}$ by
$\hat{N} M$ and $\hat{N} L$ respectively, with an $\hat{N}$ that only
depends on $R^*$, $r$ and $m$.
We recall that from the expressions of section~\ref{sec:its} the
solutions of $(eq_1)-(eq_5)$ are unique, and for them we have
(working here for a fixed $\varphi\in{\cal E}$):
\begin{itemize}
\item[$(eq_1)$] From the expression of $d$ as a function of the
coefficients of the Fourier expansion of $a$, it is clear that if we
use the bounds of the denominators given by the Diophantine conditions
of (\ref{eq:dcil}) and lemma \ref{lem:normf} we can see that:
$$
|d|_{\rho-\chi}\leq
\left(\frac{\gamma }{(\chi-\delta)\exp{(1)}}\right)^{\gamma}
\frac{|\tilde{a}|_{\rho}}{\cd},
$$
if $\rho > \chi > \delta$, and then using that
$|\tilde{a}|_{\rho}\leq|a-\ti|_{\rho}$, we can write:
$$
|d|_{\rho - \chi} \leq \hat{N} \frac{M}{(\chi-\delta)^{\gamma} \cd}.
$$
\item[$(eq_2)$] We have for $e$
$$
|e|_{\rho -\chi} \leq \left( \frac{2}{\alpha^*_1} + \left(
\frac{\gamma}{(\chi-\delta)\exp{(1)}}\right)^{\gamma}\frac{1}{\cd}
\right) |b|_{\rho},
$$
for all $\rho > \chi > \delta$. Consequently:
$$
|e|_{\rho -\chi} \leq \hat{N}\frac{M}{(\chi -\delta)^{\gamma} \cd}.
$$
\item[$(eq_3)$] First we bound $\xi$:
\begin{eqnarray*}
|\xi| & = & |(\bar{\cal C})^{-1}\bar{\cal C}\xi|\leq
|(\bar{\cal C})^{-1}||\bar{\cal C}\xi|\leq
\bar{m}^*\left|\bar{c}-\hat{\omega}-\overline{{\cal C}\left(
\frac{\partial d}{\partial\hat{\theta}}\right)^{\top}}
\right|_{\rho-\chi} \leq
\\ & \leq & \bar{m}^* \left( |c -\hat{\omega}|_{\rho} +
\left|{\cal C}\left(\frac{\partial d}{\partial \hat{\theta}}
\right)^{\top}
\right|_{\rho-\chi} \right) \leq \bar{m}^* \left( \hat{N} M +
|{\cal C}|_{\rho} \frac{2 |d|_{\rho-\chi/2}}{\chi \exp{(1)}} \right),
\end{eqnarray*}
where $\rho\geq\chi>2\delta$. Hence,
$$
|\xi| \leq \hat{N} \frac{M}{(\chi-2\delta)^{\gamma+1} \cd},
$$
for all $\rho\geq\chi>2\delta$. Then, for $c^*$ we have:
\begin{eqnarray*}
|c^*|_{\rho-\chi} & \leq & \left|\tilde{c}-\tilde{\cal C}\xi -
{\cal C}\left( \frac{\partial d}{\partial \hat{\theta}} \right)^{\top}+
\overline{{\cal C}\left(\frac{\partial d}{\partial\hat{\theta}}
\right)^{\top}}
\right|_{\rho-\chi}\leq \\
& \leq & | \tilde{c}|_{\rho}+|{\cal C}|_{\rho}\left(|\xi|+
\left|\left(\frac{\partial d}{\partial \hat{\theta}}\right)^{\top}
\right|_{\rho-\chi} \right) \leq
\\ & \leq &
|c -\hat{\omega}|_{\rho} + \hat{m}^* \left( |\xi| +
\frac{2 |d|_{\rho-\chi/2}}{\chi \exp{(1)}} \right) \leq \hat{N}
\frac{M}{(\chi - 2\delta)^{\gamma+1} \cd}.
\end{eqnarray*}
Hence, if $\rho>\chi>3\delta$,
$$
|f|_{\rho-\chi}\leq
\left(\frac{3\gamma}{(\chi-3\delta)\exp{(1)}}\right)^{\gamma}
\frac{|c^*|_{\rho -2\chi/3}}{\cd}
\leq \hat{N} \frac{M}{(\chi-3\delta)^{2\gamma+1}\cd^2}.
$$
\item[$(eq_4)$] From the definition of $B^*$ given in (\ref{eq:Bstar}),
we have
\begin{eqnarray*}
|B^*-{\cal B}|_{\rho-\chi} & \leq & |B-{\cal B}|_{\rho-\chi} + \\
& & + \left|\left[\frac{\partial H_*}{\partial \hat{I}}
\left(\xi+\left(\frac{\partial d}{\partial\hat{\theta}}\right)^{\top}
\right)\right]_{(z,z)}\right|_{\rho-\chi}+
\left|\left[\frac{\partial H_*}{\partial z}
J_me\right]_{(z,z)}\right|_{\rho-\chi} \leq
\\ & \leq &
\hat{N} M + (2m+1)r\frac{|H_*|_{\rho,R}}{(R^*)^3}
\left( | \xi | + \frac{2 |d|_{\rho-\chi/2}}{\chi \exp{(1)}} \right) +
24m^2\frac{|H_*|_{\rho,R}}{ (R^*)^3} |e|_{\rho - \chi},
\end{eqnarray*}
and then
$$
|B^*-{\cal B}|_{\rho-\chi}\leq
\hat{N}\frac{M}{(\chi-2\delta)^{\gamma+1}\cd},
$$
if $\rho>\chi>2\delta$, and the same bound holds for
$|B^{**}|_{\rho-\chi}$ (see (\ref{eq:Bsatar2})). Lemma~\ref{lem:normf}
allows to bound
$$
|G|_{\rho-\chi}\leq
\left(\frac{1}{\alpha_1^*}+ \left(
\frac{3\gamma}{(\chi-3\delta)\exp{(1)}}\right)^{\gamma}
\frac{1}{\cd}\right) 2m \left|B^{**}\right|_{\rho - 2\chi/3},
$$
with $\rho > \chi > 3\delta$. Hence,
$$
|G|_{\rho - \chi}\leq
\hat{N} \frac{M}{ (\chi - 3\delta)^{2\gamma+1} \cd^2}.
$$
\item[$(eq_5)$] If $\rho>\chi>2\delta$, we have for $E^*$ defined in
(\ref{eq:Estar}):
\begin{eqnarray*}
\left| E^* \right|_{\rho - \chi} & \leq & |E|_{\rho -\chi} +
\left|{\cal C}
\left(\frac{\partial e}{\partial\hat{\theta}}\right)^{\top}
\right|_{\rho-\chi}+\left| \left[\frac{\partial H_*}{\partial \hat{I}}
\left(\xi+
\left(\frac{\partial d}{\partial\hat{\theta}}\right)^{\top}\right)
\right]_{( \hat{I}, z)} \right|_{\rho - \chi} +
\\ & & +
\left| \left[\frac{\partial H_*}{\partial z} J_m e
\right]_{( \hat{I}, z)} \right|_{\rho - \chi} \leq
\hat{N}M+|{\cal C}|_{\rho} 2m \frac{2|e|_{\rho-\chi/2}}{\chi\exp{(1)}}+
\\ & & +
4mr \frac{|H_*|_{\rho,R}}{\left( R^* \right)^3}
\left( |\xi| + \frac{2 |d|_{\rho-\chi/2}}{\chi \exp{(1)}} \right) +
8m^2 \frac{|H_*|_{\rho,R}}{\left( R^* \right)^3} |e|_{\rho - \chi}.
\end{eqnarray*}
Then,
$$
\left|E^*\right|_{\rho-\chi}\leq
\hat{N}\frac{M}{(\chi-2\delta)^{\gamma+1} \cd}.
$$
Now, if $\rho > \chi > 3\delta$,
$$
\left|F \right|_{\rho-\chi}\leq 2m \left( \frac{2}{\alpha_1^*} +
\left( \frac{3\gamma}{(\chi - 3\delta) \exp{(1)}}\right)^{\gamma}
\frac{1}{\cd} \right) \left|E^{*} \right|_{\rho - 2\chi/3},
$$
that implies
$$
\left| F \right|_{\rho - \chi}
\leq \hat{N} \frac{M}{ (\chi -3\delta)^{2\gamma+1} \cd^2}.
$$
\end{itemize}
Now, we repeat the same process to bound the Lipschitz constants for
the solutions of these equations. For that purpose, we will also need
the results of lemmas \ref{lem:lipnormf} and \ref{lem:lipnormt}
to work with the different Lipschitz dependences. We remark that,
for the different denominators, we can bound:
$$
{\cal L}_{\cal E}\{ik^{\top}\omega+l^{\top}\lambda\}\leq
|k|_1+\frac{\beta_2^*}{2}|l|_1,
$$
for every $k\in\ZZ^{r+s}$, $l\in\NN^{2m}$, $|l|_1\leq 2$. Moreover, we
will also use the hypothesis $M\leq L$ to simplify the bounds. Then we
have:
\begin{itemize}
\item[$(eq_1)$] We need to take into account the $\varphi$ dependence
for all the functions, and so for $d$ we have
$$
d(\theta,\varphi)=\sum_{k \in {\ZZ}^{r+s} \backslash \{0\}}
\frac{a_k(\varphi)}{ik^{\top} \omega} \exp(i k^{\top} \theta).
$$
Then, using lemma~\ref{lem:lipnormf} and
${\cal L}_{{\cal E},\rho}\{\tilde{a}\}\leq
{\cal L}_{{\cal E},\rho}\{a-\ti\}$, one obtains
\begin{eqnarray*}
{\cal L}_{{\cal E},\rho-\chi}\{d\} & \leq &
\left( \frac{\gamma}{(\chi-\delta)\exp{(1)}} \right)^{\gamma}
\frac{{\cal L}_{{\cal E},\rho}\{\tilde{a}\}}{\cd}+
\left( \frac{2\gamma +1}{(\chi-2\delta)\exp{(1)}} \right)^{2\gamma+1}
\frac{\|\tilde{a}\|_{{\cal E},\rho}}{\cd^2} \leq
\\ & \leq &
\hat{N} \frac{L}{(\chi-2\delta)^{2\gamma+1} \cd^2},
\end{eqnarray*}
for every $\rho > \chi >2 \delta$.
\item[$(eq_2)$]
\begin{eqnarray*}
{\cal L}_{{\cal E},\rho-\chi} \{e\} & \leq &
\left( \frac{\gamma}{(\chi-\delta)\exp{(1)}} \right)^{\gamma}
\frac{{\cal L}_{{\cal E},\rho}\{b\}}{\cd}+
\left( \frac{2\gamma+1}{(\chi-2\delta)\exp{(1)}} \right)^{2\gamma+1}
\frac{\|b\|_{{\cal E},\rho}}{\cd^2} + \\ & & +
\left( \frac{2\gamma}{(\chi-2\delta)\exp{(1)}} \right)^{2\gamma}
\frac{\|b\|_{{\cal E},\rho}}{\cd^2} \frac{\beta_2^*}{2} +
\frac{2}{\alpha_1^*} {\cal L}_{{\cal E},\rho} \{b\} +
\frac{4}{(\alpha_1^*)^2} \| b \|_{{\cal E},\rho}\frac{\beta^*_2}{2} \leq
\\ & \leq &
\hat{N} \frac{L}{(\chi-2\delta)^{2\gamma+1} \cd^2},
\end{eqnarray*}
if $\rho > \chi > 2\delta$.
\item[$(eq_3)$] If $\rho\geq\chi>3\delta$, we have:
\begin{eqnarray*}
{\cal L}_{\cal E}\{\xi\} & \leq &
{\cal L}_{\cal E}\left\{(\bar{\cal C})^{-1} \right\}
\left\|\bar{c}-\hat{\omega}-\overline{{\cal C}\left(
\frac{\partial d}{\partial\hat{\theta}}\right)^{\top}}
\right\|_{{\cal E},\rho-\chi}+
\\ & & +
\|(\bar{\cal C})^{-1}\|_{\cal E}
{\cal L}_{{\cal E},\rho-\chi} \left\{ \bar{c} -\hat{\omega} -
\overline{{\cal C}\left(
\frac{\partial d}{\partial\hat{\theta}}\right)^{\top}}\right\}\leq
%\\ & \leq &
\hat{N} \frac{L}{(\chi-3\delta)^{2\gamma+2} \cd^2},
\end{eqnarray*}
where we have used that, from lemma \ref{lem:lbim},
$$
{\cal L}_{\cal E} \{ (\bar{\cal C})^{-1} \}
\leq\|(\bar{\cal C})^{-1}\|^2_{\cal E}{\cal L}_{\cal E}\{\bar{\cal C}\}
\leq (\bar{m}^*)^2 {\cal L}_{{\cal E},\rho} \{{\cal C} \},
$$
and also that
$$
{\cal L}_{{\cal E},\rho-\chi}
\left\{\left(\frac{\partial d}{\partial\hat{\theta}}\right)^{\top}
\right\}\leq\frac{3}{\chi \exp{(1)}}
{\cal L}_{{\cal E},\rho-2\chi/3} \{d\},
$$
and
$
{\cal L}_{{\cal E},\rho-\chi}\{\bar{c} -\hat{\omega}\}\leq
{\cal L}_{{\cal E},\rho}\{c-\hat{\omega}\}.
$
Then, if $\rho > \chi > 3\delta$, using that
$
{\cal L}_{{\cal E},\rho-\chi} \{\tilde{c} \} \leq
{\cal L}_{{\cal E},\rho-\chi} \{c -\hat{\omega} \},
$
one has
$$
{\cal L}_{{\cal E},\rho-\chi}\{c^* \} \leq
{\cal L}_{{\cal E},\rho-\chi} \{\tilde{c} \} +
{\cal L}_{{\cal E},\rho-\chi} \left\{\tilde{\cal C} \xi \right\} +
{\cal L}_{{\cal E},\rho-\chi} \left\{
{\cal C}\left(\frac{\partial d}{\partial\hat{\theta}}\right)^{\top}
\right\}
\leq\hat{N}\frac{L}{(\chi-3\delta)^{2\gamma+2} \cd^2}.
$$
Hence,
\begin{eqnarray*}
{\cal L}_{{\cal E},\rho-\chi}\{f\} & \leq &
\left( \frac{3\gamma}{(\chi-3\delta)\exp{(1)}} \right)^{\gamma}
\frac{{\cal L}_{{\cal E},\rho-2\chi/3}\{c^*\}}{\cd}+
\\ & & +
\left( \frac{2(2\gamma+1)}{(\chi-6\delta)\exp{(1)}} \right)^{2\gamma+1}
\frac{\|c^*\|_{{\cal E},\rho-2\chi/3}}{\cd^2} \leq
%\\ & \leq &
\hat{N} \frac{L}{(\chi-6\delta)^{3\gamma+2}\cd^3},
\end{eqnarray*}
if $\rho > \chi > 6\delta$.
\item[$(eq_4)$] We first bound:
\begin{eqnarray*}
{\cal L}_{{\cal E},\rho-\chi} \{ B^* -{\cal B} \} & \leq &
{\cal L}_{{\cal E},\rho-\chi} \{ B- {\cal B} \} +
{\cal L}_{{\cal E},\rho-\chi}\left\{
\left[\frac{\partial H_*}{\partial\hat{I}}
\left(\xi+\left(\frac{\partial d}{\partial\hat{\theta}}\right)^{\top}
\right) \right]_{(z,z)}
\right\} +
\\ & & +{\cal L}_{{\cal E},\rho-\chi} \left\{
\left[\frac{\partial H_*}{\partial z}J_me\right]_{(z,z)} \right\} \leq
\hat{N} \frac{L}{(\chi-3\delta)^{2\gamma+2} \cd^2},
\end{eqnarray*}
if $\rho>\chi>3\delta$, and the same bound holds for
${\cal L}_{{\cal E},\rho-\chi} \{ B^{**} \}$. This implies
\begin{eqnarray*}
{\cal L}_{{\cal E},\rho-\chi}\left\{ G \right\} & \leq &
(2m-1)\frac{1}{\alpha_1^*} {\cal L}_{{\cal E},\rho-\chi} \{B^{**}\} +
(2m-1)\frac{1}{(\alpha_1^*)^2}\|B^{**}\|_{{\cal E},\rho-\chi}\beta^*_2 +
\\ & & +
2m \left( \frac{3\gamma}{(\chi-3\delta)\exp{(1)}} \right)^{\gamma}
\frac{{\cal L}_{{\cal E},\rho-2\chi/3}\{B^{**}\}}{\cd} +
\\ & & +
2m\left( \frac{3(2\gamma+1)}{(\chi-6\delta)\exp{(1)}}\right)^{2\gamma+1}
\frac{\|B^{**}\|_{{\cal E},\rho-2\chi/3}}{\cd^2} +
\\ & & +
2m\left( \frac{6\gamma}{(\chi-6\delta)\exp{(1)}} \right)^{2\gamma}
\frac{\|B^{**}\|_{{\cal E},\rho-2\chi/3}}{\cd^2} \beta_2^*
\leq \\ & \leq & \hat{N} \frac{L}{(\chi-6\delta)^{3\gamma+2}\cd^3},
\end{eqnarray*}
if $\rho > \chi>6\delta$.
\item[$(eq_5)$] From the definition of $E^*$,
\begin{eqnarray*}
{\cal L}_{{\cal E},\rho-\chi} \left\{E^* \right\} & \leq &
{\cal L}_{{\cal E},\rho-\chi}\left\{E\right\} +
%{\cal L}_{{\cal E},\rho-\chi} \left\{{\cal C}
%\left(\frac{\partial e}{\partial\hat{\theta}}\right)^{\top}\right\} +
{\cal L}_{{\cal E},\rho-\chi}
\left\{ \left[ \frac{\partial H_*}{\partial \hat{I}}
\left(\xi+\left(\frac{\partial d}{\partial\hat{\theta}}\right)^{\top}
\right) \right]_{( \hat{I}, z)} \right\} +
\\ & & +
{\cal L}_{{\cal E},\rho-\chi} \left\{{\cal C}
\left(\frac{\partial e}{\partial\hat{\theta}}\right)^{\top}\right\} +
{\cal L}_{{\cal E},\rho-\chi}\left\{\left[
\frac{\partial H_*}{\partial z}J_me\right]_{(\hat{I},z)}\right\}\leq
\\ & \leq &
\hat{N} \frac{L}{(\chi-4\delta)^{2\gamma+2}\cd^2},
\end{eqnarray*}
if $\rho>\chi>4\delta$. Hence, if now $\rho>\chi>6\delta$,
we can bound:
\begin{eqnarray*}
{\cal L}_{{\cal E},\rho-\chi} \{F\} & \leq &
2m\frac{2}{\alpha_1^*} {\cal L}_{{\cal E},\rho-\chi} \{E^{*}\} +
2m\frac{4}{(\alpha_1^*)^2}\|E^*\|_{{\cal E},\rho-\chi}
\frac{\beta^*_2}{2} +
\\ & & +
2m \left( \frac{3\gamma}{(\chi-3\delta)\exp{(1)}} \right)^{\gamma}
\frac{{\cal L}_{{\cal E},\rho-2\chi/3}\{E^{*}\}}{\cd}+
\\ & & +
2m\left(\frac{3(2\gamma+1)}{(\chi-6\delta)\exp{(1)}} \right)^{2\gamma+1}
\frac{\|E^{*}\|_{{\cal E},\rho-2\chi/3}}{\cd^2} +
\\ & & +
2m\left( \frac{6\gamma}{(\chi-6\delta)\exp{(1)}} \right)^{2\gamma}
\frac{\|E^{*}\|_{{\cal E},\rho-2\chi/3}}{\cd^2} \frac{\beta_2^*}{2} \leq
\\ & & +
\\ & \leq & \hat{N} \frac{L}{(\chi-6\delta)^{3\gamma+2}\cd^3}.
\end{eqnarray*}
\end{itemize}
Before bounding the transformed Hamiltonian, let us check that the
change given by the generating function $S$ is well defined.
First, we have that:
\begin{equation}
\|\nabla S\|_{{\cal E},\rho-\chi,R} \leq
\hat{N} \frac{M}{(\chi -4\delta)^{2\gamma+2}\cd^2},
\label{eq:nablaS}
\end{equation}
and that
$$
{\cal L}_{{\cal E},\rho-\chi,R} \left\{ \nabla S \right\} \leq \hat{N}
\frac{L}{(\chi -7\delta)^{3\gamma+3}\cd^3},
$$
provided that $\rho>\chi>7\delta$. If we select $\chi=8\delta$, and if
we consider (\ref{eq:nablaS}), we have a bound of the type:
$$
\|\nabla S \|_{{\cal E},\rho - 8\delta, R - 8\hat{\delta}} \leq
\hat{N} \frac{M}{ \delta^{2\gamma+2} \cd^{2}}.
$$
Before continuing, let us ask to the quantity
\begin{equation}
\left(r+(r+2m)\exp{(1)}\max{\{1,\tilde{\delta}_0\}}\right)\hat{N}
\frac{M}{\delta^{2\gamma+3} \cd^2},
\label{eq:ask}
\end{equation}
to be bounded by $1/2$ (this will be used in (\ref{eq:con1S}) and
(\ref{eq:con2S})). We can bound expression (\ref{eq:ask}) by
$\Theta:=\hat{N}\frac{M}{\delta^{2\gamma+3} \cd^2}$ with a
redefinition of $\hat{N}$, and hence, this condition (on the size of
$M$) can be reduced to $\Theta\leq 1/2$. We note that $\hat{N}$ only
depends on the absolute constants given in the hypothesis of the lemma,
and this is, in fact, the assumption on the size of $\Theta$ that
appears in those hypothesis.
In what follows, we will redefine the value of $\hat{N}$ in order
to meet a few more conditions, but this redefinition will not
change the fact that the final $\hat{N}$ is an absolute
constant.
From this last bound one obtains,
\begin{equation}
\| {\nabla} S \|_{{\cal E},\rho-8\delta,R-8\hat{\delta}} \leq
\frac{\Theta \delta}{\max{\{1,\tilde{\delta}_0\}}} \leq
\min{\{\delta,\delta/\tilde{\delta}_0\}} \leq
\min{\{ \delta, \hat{\delta} \}},
\label{eq:con1S}
\end{equation}
and
\begin{equation}
\Delta_{\delta,\hat{\delta}}
\exp{(1)}\|\nabla S\|_{{\cal E},\rho-8\delta,R-8\hat{\delta}}\leq\Theta,
\label{eq:con2S}
\end{equation}
where we use the definition of $\Delta_{\delta,\hat{\delta}}$ given
in (\ref{eq:Delta}).
%%%%$$
%%%%\Delta_{\delta,\hat{\delta}}=
%%%%\left(\frac{r}{\delta\exp{(1)}}+\frac{r+2m}{\hat{\delta}}\right).
%%%%$$
From (\ref{eq:con1S}) and lemma~\ref{lem:lemcan} we have that $\Psi^S$
is well defined (for every $\varphi\in{\cal E}$), according to
(\ref{eq:Psiacts}). From (\ref{eq:con2S}) and lemma~\ref{lem:bsigma}
we can bound the expression of $\Sigma (H^{**},S )$ that appear in the
transformed Hamiltonian,
$$
\|\Sigma ( H^{**},S )\|_{{\cal E},\rho^{(1)},R^{(1)}} \leq
\left(\sum_{j\geq 1}\frac{1}{j+1}\left(\frac{1}{2}\right)^{j-1}\right)
\Theta \|H^{**}\|_{{\cal E},\rho-8\delta,R-8\hat{\delta}}.
$$
and, similarly, for the Lipschitz constant we can use lemma
\ref{lem:lipsigma} to produce
\begin{eqnarray*}
{\cal L}_{{\cal E},\rho^{(1)},R^{(1)}}\left\{\Sigma(H^{**},S) \right\}
& \leq & \sum_{j\geq 1}\left( \frac{1}{j+1}
\left(\Delta_{\delta,\hat{\delta}}\exp{(1)}\right)^{j}
\hat{F}_2^{j-1}(j\hat{L}_2\hat{F}_1+\hat{L}_1\hat{F}_2)\right),
\end{eqnarray*}
with
$\hat{F}_1=\|H^{**}\|_{{\cal E},\rho-8\delta,R-8\hat{\delta}}$,
$\hat{F}_2=\|\nabla S\|_{{\cal E},\rho-8\delta,R-8\hat{\delta}}$,
$\hat{L}_1={\cal L}_{{\cal E},\rho-8\delta,R-8\hat{\delta}}
\left\{H^{**}\right\}$ and
$\hat{L}_2={\cal L}_{{\cal E},\rho-8\delta,R-8\hat{\delta}}
\left\{\nabla S \right\}$.
Then
\begin{eqnarray*}
{\cal L}_{{\cal E},\rho^{(1)},R^{(1)}}\left\{\Sigma(H^{**},S)\right\}
& \leq &
\left(\sum_{j\geq 1}\frac{j}{j+1}\left(\frac{1}{2}\right)^{j-1}\right)
\Delta_{\delta,\hat{\delta}}\exp{(1)}\hat{L}_2\hat{F}_1 + \\ & & +
\left(\sum_{j\geq 1}\frac{1}{j+1}\left(\frac{1}{2}\right)^{j-1}\right)
\Delta_{\delta,\hat{\delta}}\exp{(1)}\hat{L}_1\hat{F}_2.
\end{eqnarray*}
With those expressions, to bound $\Sigma(H^{**},S)$ is reduced to
bound $H^{**}$, with the only remark that the sums
$\sum_{j\geq 1}\frac{1}{j+1}\nu^{j-1}=-\frac{\ln{(1-\nu)}+\nu}{\nu^2}$
and $\sum_{j\geq 1}\frac{j}{j+1}\nu^{j-1}=
\frac{(1-\nu)\ln{(1-\nu)}+ \nu}{\nu^2(1-\nu)}$ are well defined for
$\nu=1/2$.
Now, we can bound the transformed Hamiltonian. From the bounds that come
from the solutions of $(eq_1)-(eq_5)$ we have:
$$
\|H_1^{**}\|_{{\cal E},\rho - \chi,R - \eta} \leq
\hat{N}\frac{M}{(\chi -4\delta)^{2\gamma+3}\cd^2}\max{\{1,\chi/\eta\}},
$$
and
$$
{\cal L}_{{\cal E},\rho-\chi,R-\eta}\left\{ H_1^{**} \right\} \leq
\hat{N} \frac{L}{(\chi-7\delta)^{3\gamma+4}\cd^3} \max{\{1,\chi/\eta\}}.
$$
To obtain these bounds, we use the explicit expression of $H_1^{**}$
given in (\ref{eq:H1star2}),
%%%%%combined with all the previous bounds
%%%obtained in $(eq_1)-(eq_5)$
and lemmas \ref{lem:normf},
\ref{lem:normt}, \ref{lem:lipnormf} and \ref{lem:lipnormt} to bound the
different partial derivatives.
We remark that here we need to use that $|\omega|\leq\cotw^*$ for any
$\varphi\in{\cal E}$. Moreover, from the bound for the Poisson brackets
given in lemmas \ref{lem:bPb} and \ref{lem:lipPb} we have, for
$H_2^{**}$,
$$
\| H_2^{**}\|_{{\cal E},\rho - \chi,R-\eta}\leq
\hat{N}\frac{M^2}{(\chi-4\delta)^{2\gamma+3}\cd^2}\max{\{1,\chi/\eta\}},
$$
and
$$
{\cal L}_{{\cal E},\rho-\chi,R-\eta}\left\{ H_2^{**} \right\} \leq
\hat{N}\frac{LM}{(\chi-7\delta)^{3\gamma+4}\cd^3}
\max{\{1,\chi/\eta\}}.
$$
The techniques that we use to control the reduction in the different
domains when we use Cauchy estimates, are analogous to the ones used in
all the previous bounds. Hence, it is clear that we can estimate
$H^{**}$ with an analogous bounds as the ones for $H_1^{**}$.
Finally, using all those bounds and from the explicit expressions of
$\ti^{(1)}$,
${\cal B}^{(1)}$, ${\cal C}^{(1)}$, $H_1^{(1)}$ and $\hat{H}^{(1)}$
in (\ref{eq:ti1})--(\ref{eq:Hstar1}) it is not
difficult to obtain the final $\hat{N}$ such that all the bounds in the
statements of the lemma hold.
\subsection{Proof of the theorem}\label{sec:pt}
We split the proof of the theorem in several parts: in the first one we
use one step of the iterative method described in section~\ref{sec:its}
as a linear scheme to reduce the size of the perturbation.
Then, we introduce $\hat{\omega}$ as a new parameter
to describe the family of lower dimensional tori near the initial one.
The next step is to apply the bounds of the iterative scheme given by
lemma~\ref{lem:itl}, and we prove the
convergence of this scheme for a suitable set of parameters.
Finally, we obtain the different estimates on the measure of this set.
\subsubsection{Linear scheme with respect to $\epsilon$}
We consider the initial Hamiltonian given in the formulation of
Theorem~\ref{teo}, and we apply one step of the iterative method
described in section~\ref{sec:its}. We remark that
from the Diophantine bounds in the statements of the theorem, we can
guarantee that this step is possible for small enough values of
$\epsilon$, and that it keeps the initial $C^2$ differentiability with
respect to $\epsilon$ on the transformed
Hamiltonian. We put $H^{(0)}$ for this Hamiltonian that, if we skip
the constant term, looks like:
\begin{equation}
H^{(0)}=\omega^{(0)\top}I+
\frac{1}{2} z^{\top} {\cal B}^{(0)}(\epsilon) z +
\frac{1}{2}\hat{I}^{\top}{\cal C}^{(0)}(\theta,\epsilon)\hat{I} +
H_*^{(0)}({\theta},x,\hat{I},y,\epsilon) +
\epsilon^2 \hat{\cal H}^{(0)}(\theta,x,\hat{I},y,\epsilon),
\label{eq:hls}
\end{equation}
with the same kind of analytic properties with respect to
$(\theta,x,\hat{I},y)$ as the initial one,
in a new domain that is independent on
$\epsilon$ (small enough). We remark that the new matrices
${\cal B}^{(0)}$ and ${\cal C}^{(0)}$ depend on $\epsilon$, and that
${\cal C}^{(0)}$ depends also on $\theta$.
Moreover, for $H_*$ we do not have the semi-normal form conditions
given in {\bf P1} and {\bf P2}. As this step comes from a perturbative
(linear) method, we have that ${\cal B}^{(0)}-{\cal B}$,
${\cal C}^{(0)}-{\cal C}$ and $H_*^{(0)}-H_*$ are of $O(\epsilon)$.
Our aim is to repeat the same iterative scheme. We remark that in the
next step and in the ones that follows, we can not guarantee good
Diophantine properties for the new eigenvalues of $J_m{\cal B}^{(0)}$
because this matrix changes at each step of the process. This is
the reason that forces us to use parameters to control these
eigenvalues. So, we can only work in the set of
parameters for which certain Diophantine bounds hold.
But before that, we want to introduce a new parameter.
\subsubsection{Introduction of the vector of frequencies as a parameter}
\label{sec:ivfp}
Here we add a new parameter to introduce a family of
Hamiltonians $H^{(0)}$. We consider values of $\hat{\omega}\in\RR^r$
close to $\hat{\omega}^{(0)}$, and for any of these
values we perform the change given in (\ref{eq:trans}).
So, following the notation introduced in section~\ref{sec:lacm},
we put $\varphi^{\top}=(\hat{\omega}^{\top},\epsilon)$
and then we write this family of Hamiltonians in the following form:
\begin{eqnarray*}
H^{(1)}(\theta,x,I,y,\varphi) & = &
\tilde{\omega}^{(0)\top}\tilde{I} +
\hat{\omega}^{(0)\top}
(\hat{I}+{\cal C}^{-1}(\hat{\omega}-\hat{\omega}^{(0)}))+
\frac{1}{2} z^{\top} {\cal B}^{(0)}(\epsilon) z +
\\ & & +
\frac{1}{2}
(\hat{I}+{\cal C}^{-1}(\hat{\omega}-\hat{\omega}^{(0)}))^{\top}
{\cal C}^{(0)}(\theta,\epsilon)
(\hat{I}+{\cal C}^{-1}(\hat{\omega}-\hat{\omega}^{(0)})) +
\\ & & +
H_*^{(0)}(\theta,x,\hat{I}+
{\cal C}^{-1}(\hat{\omega}-\hat{\omega}^{(0)}),y,\epsilon)+
\\ & & + \epsilon^2{\cal H}^{(0)}(\theta,x,
\hat{I}+{\cal C}^{-1}(\hat{\omega}-\hat{\omega}^{(0)}),y,\epsilon),
\end{eqnarray*}
that can be expanded as
$$
H^{(1)}=\ti^{(1)}(\varphi) + {\omega}^{\top}I+
\frac{1}{2}z^{\top}{\cal B}^{(1)}(\varphi)z+
\frac{1}{2}\hat{I}^{\top}{\cal C}^{(1)}({\theta},\varphi)\hat{I}+
H_*^{(1)}(\theta,x,\hat{I},y,\varphi)+
\hat{H}^{(1)}(\theta,x,\hat{I},y,\varphi),
$$
(we recall
$\omega^{\top}=(\hat{\omega}^{\top},\tilde{\omega}^{(0)\top})$)
with analogous properties for the different terms as in (\ref{eq:hls}),
where we can take $\hat{H}^{(1)}$ of $O_2(\varphi - \varphi^{(0)})$,
$\varphi^{(0)\top}=(\hat{\omega}^{(0)\top},0)$. This comes
from the semi-normal form structure that we have for $H_*$, and the
fact that $H_*^{(0)}$ is $\epsilon$-close to $H_*$. We also remark
that we have differentiable dependence of this Hamiltonian with
respect to $\varphi$ (in fact it is analytic with respect to
$\hat{\omega}$), but since we will work in the following
steps on Cantor sets, we can not keep this kind of dependence.
So, we replace the differentiable dependence by a Lipschitz one, in
the sense given in section~\ref{sec:nlc}.
To quantify all these facts, we take
$0<\rho\leq 1$, $00$. To simplify the following
bounds we assume, without loss of generality, that $N_1\geq 1$.
Finally, we finish this part with an explicit formulation of the
nondegeneracy hypothesis of the normal eigenvalues with respect
to the parameters. Let us consider
${\cal B}^{(1)}$. By construction, we have that
$J_m{\cal B}^{(1)}$ is a diagonal matrix with the same kind of
eigenvalues as the matrix $J_m{\cal B}$ that appears in the statement
of the theorem. Then, using the $C^2$ differentiability with respect
to $\varphi$, we can write its eigenvalues as:
\begin{equation}
\lambda_j^{(1)}(\varphi)=\lambda_j+i u_j \epsilon +
iv_j^{\top}(\hat{\omega}-\hat{\omega}^{(0)})+
\tilde{\lambda}_j^{(1)}(\varphi),
\label{eq:ndc}
\end{equation}
for $j=1,\ldots,2m$, with $u_j\in\CC$ and $v_j\in\CC^r$, and where
the Lipschitz constant of $\tilde{\lambda}_j^{(1)}$ on
$\bar{\cal E}^{(1)}$ is of $O(\bar{\vartheta})$. Then, those generic
nondegeneracy conditions are:
\begin{description}
\item[{\bf NDC}] For any $j$ such that $\Re\lambda_j=0$, we have
$u_j\neq 0$ and $Re(v_{j})\notin \ZZ^r$. Moreover, if we define
$u_{j,l}=u_{j}-u_{l}$ and $v_{j,l}=v_{j}-v_{l}$, we have these same
conditions for $u_{j,l}$ and $v_{j,l}$ for any $j\neq l$ such that
$\Re(\lambda_j-\lambda_l)=0$.
\end{description}
Note that we have used the $C^2$ dependence on $\varphi$ to
ensure that the Lipschitz constant of $\tilde{\lambda}_j^{(1)}$ on
$\bar{\cal E}^{(1)}$ is $O(\bar{\vartheta})$. If the dependence
is $C^1$ we can only say that this constant is $o(\bar{\vartheta})$.
Nevertheless, it is still possible in this case to derive the same
results as in the $C^2$ case, but the details are more tedious.
The nondegeneracy conditions with respect to $\epsilon$ are the same
ones used in
\cite{JS94B} to study the quasiperiodic perturbations of elliptic fixed
points, and the nondegeneracy conditions with respect to the
$\hat{\omega}$-dependence are analogous to the ones appeared in
\cite{Me} and \cite{El88}, but in those cases they were formulated for
an unperturbed system having an $r$-dimensional analytic family of
$r$-dimensional reducible elliptic tori.
\subsubsection{Inductive part}
We want to apply here the iterative lemma in an inductive form. For this
purpose, we define $\sigma_n=\frac{6}{\pi^2n^2}$ for every $n\geq1$,
and we note that $\sum_{n\geq 1}\sigma_n=1$. From this definition,
we put $\delta_n=\frac{\sigma_n\rho}{18}$,
$\hat{\delta}_n=\frac{\sigma_n R}{18}$
and we introduce $\rho^{(n+1)}=\rho^{(n)}-9\delta_n$ and
$R^{(n+1)}=R^{(n)}-9\hat{\delta}_n$ for every $n\geq 1$.
We also consider a fixed $1<\varrho<2$, to define
$\cd_n=\exp{\left(-\varrho^n\right)}\cd_0$.
We suppose that, at step $n$, we have a Hamiltonian $H^{(n)}$ like
$H^{(1)}$ defined for $\varphi$ in a set
${\cal E}^{(n)}\subset{\cal E}^{(1)}$, with analogous bounds as
$H^{(1)}$, replacing the superscript $(1)$ by $(n)$ in the unperturbed
part, and with bounds for the perturbation given by
$\bar{M}_n=M_n(\bar{\vartheta})=N_n \bar{\vartheta}^{2^n}$ and
$\bar{L}_n=L_n(\bar{\vartheta})=N_n \bar{\vartheta}^{2^n-1}$, in
every set of the form $\bar{\cal E}^{(n)}(\bar{\vartheta})$,
for all $0\leq\bar{\vartheta}\leq\vartheta_1$, being $N_n$ independent
on $\bar{\vartheta}$. We will show that this is
possible if $\vartheta_1$ is small enough, with conditions on
$\vartheta_1$ that are independent on the actual step.
At this point, we define the new set ${\cal E}^{(n+1)}$ of good
parameters from ${\cal E}^{(n)}$ looking at the new Diophantine
conditions. We have that $\varphi\in{\cal E}^{(n+1)}$ if
$\varphi\in{\cal E}^{(n)}$ and the following conditions hold:
\begin{equation}
|k^{\top}\omega+l^{\top}\lambda^{(n)}(\varphi)|
\geq \frac{\cd_n}{|k|_1^{\gamma}}\exp{(-\delta_n |k|_1)},
\label{eq:esdb}
\end{equation}
for all $k\in{\ZZ}^{r+s}\backslash\{0\}$, $l\in\NN^{2m}$, $|l|_1\leq 2$.
Now, we use the iterative lemma for $\varphi\in{\cal E}^{(n+1)}$.
First we remark that at every step we have
$\rho^{(n)}\leq\rho$,
$R/2\leq R^{(n)}\leq R$, $\delta_n/\hat{\delta}_n=\rho/R$,
$\cd_n \leq \cd_0$ and, as $\vartheta_1\leq 1$, we have
that for every $\varphi\in{\cal E}^{(1)}$,
$|\omega|\leq\max{\{|\tilde{\omega}^{(0)}|,|\hat{\omega}^{(0)}|+1\}}$.
Moreover, we assume that we can bound
$\alpha_1^{(1)}/2\leq\alpha_1^{(n)}$,
$\alpha_2^{(n)}\le 2\alpha_2^{(1)}$, $\beta_2^{(n)}\le 2\beta_2^{(1)}$,
$\bar{m}^{(n)}\leq 2\bar{m}^{(1)}$, $\hat{m}^{(n)}\leq 2\hat{m}^{(1)}$,
$\tilde{m}^{(n)}\leq 2\tilde{m}^{(1)}$,
$\hat{\nu}^{(n)}\leq 2\hat{\nu}^{(1)}$
$\tilde{\nu}^{(n)}\leq 2\tilde{\nu}^{(1)}$ and
$N_n{\vartheta}_1^{2^n-2}\leq N_1$.
We remark that all those bounds hold for $n=1$.
Then we consider the constant $\hat{N}$, given by the iterative
lemma, corresponding to these bounds.
If we assume that in the actual step we have for $\Theta_n:=
\hat{N}\frac{N_n\vartheta_1^{2^n}}{\delta_n^{2\gamma+3}\cd_n^2}$,
$\Theta_n\leq 1/2$, then we can apply the iterative lemma to obtain
the generating function $S^{(n)}(\theta,x,\hat{I},y,\varphi)$,
with $\|\nabla S^{(n)}\|_{{\cal E}^{(n+1)},\rho^{(n)}-8\delta_n,
R^{(n)}-8\hat{\delta}_n}\leq\min{\{\delta_n,\hat{\delta}_n\}}$.
So, in this case we have for $\Psi^{S^{(n)}}$
$$
\hat{\Psi}^{S^{(n)}} :
{\cal U}_{\rho^{(n+1)},R^{(n+1)}}^{r+s,2m+r} \longrightarrow
{\cal U}_{\rho^{(n)}-8\delta_n,R^{(n)}-8\hat{\delta}_{n}}^{r+s,2m+r}.
$$
The next step is to bound the transformed Hamiltonian
$H^{(n+1)}= H^{(n)} \circ \Psi^{S^{(n)}}$. We work in a set of
the form $\bar{\cal E}^{(n+1)}$, for all
$0<\bar{\vartheta}\leq\vartheta_1$.
From the bounds of the iterative lemma, and the explicit expressions of
$\sigma_n$, $\delta_n$ and $\cd_n$, we can deduce that there exists
$\tilde{N}$ (we can assume $\tilde{N}\geq 1$) depending on the same
constants as $\hat{N}$, such that
\begin{eqnarray*}
\|\nabla S^{(n)}\|_{\bar{\cal E}^{(n+1)},\rho^{(n)}-8\delta_n,
R^{(n)}-8\hat{\delta}_n} & \leq &
\tilde{N}n^{4+4\gamma}(\exp{(\varrho^n)})^2N_n\bar{\vartheta}^{2^n},\\
\|\ti^{(n+1)}-\ti^{(n)}\|_{\bar{\cal E}^{(n+1)}} & \leq &
\tilde{N}n^{2+2\gamma}\exp{(\varrho^n)}N_n\bar{\vartheta}^{2^n}, \\
\|{\cal B}^{(n+1)}-{\cal B}^{(n)}\|_{\bar{\cal E}^{(n+1)}} & \leq &
\tilde{N}n^{2+2\gamma}\exp{(\varrho^n)}N_n\bar{\vartheta}^{2^n}, \\
{\cal L}_{\bar{\cal E}^{(n+1)}}\{{\cal B}^{(n+1)}-
{\cal B}^{(n)}\} & \leq &
\tilde{N}n^{4+4\gamma}(\exp{(\varrho^n)})^2N_n\bar{\vartheta}^{2^n-1},\\
\|{\cal C}^{(n+1)}-{\cal C}^{(n)}\|_{\bar{\cal E}^{(n+1)},
\rho^{(n+1)}} & \leq &
\tilde{N}n^{4+4\gamma}(\exp{(\varrho^n)})^2N_n\bar{\vartheta}^{2^n},\\
{\cal L}_{\bar{\cal E}^{(n+1)},\rho^{(n+1)}}
\{{\cal C}^{(n+1)}-{\cal C}^{(n)}\} & \leq &
\tilde{N}n^{6+6\gamma}(\exp{(\varrho^n)})^3N_n\bar{\vartheta}^{2^n-1},\\
\|H_*^{(n+1)}-H_*^{(n)}\|_{\bar{\cal E}^{(n+1)},
\rho^{(n+1)},R^{(n+1)}} & \leq &
\tilde{N}n^{6+4\gamma}(\exp{(\varrho^n)})^2N_n\bar{\vartheta}^{2^n},\\
{\cal L}_{\bar{\cal E}{(n+1)},\rho^{(n+1)},R^{(n+1)}}
\{H_*^{(n+1)}-H_*^{(n)} \} & \leq &
\tilde{N}n^{8+6\gamma}(\exp{(\varrho^n)})^3N_n\bar{\vartheta}^{2^n-1},\\
\|\hat{H}^{(n+1)}\|_{\bar{\cal E}^{(n+1)},
\rho^{(n+1)},R^{(n+1)}} & \leq &
\tilde{N}n^{12+8\gamma}(\exp{(\varrho^n)})^4N_n^2
\bar{\vartheta}^{2^{n+1}},\\
{\cal L}_{\bar{\cal E}^{(n+1)},\rho^{(n+1)},R^{(n+1)}}
\{\hat{H}^{(1)}\} & \leq &
\tilde{N}n^{14+10\gamma}(\exp{(\varrho^n)})^5N_n^2
\bar{\vartheta}^{2^{n+1}-1}.
\end{eqnarray*}
Moreover, we assume that we can bound
$\Theta_n\leq\tilde{N}n^{6+4\gamma}(\exp{(\varrho^n)})^2N_n
\vartheta_1^{2^n}$, with the same constant $\tilde{N}$.
Then, we use all these expressions as a motivation to define
$N_{n+1}=\tilde{N}n^{14+10\gamma}(\exp{(\varrho^n)})^5N_n^2$,
for $n \geq 1$. To bound how fast $N_{n+1}$ grows with $n$ and $N_1$
we use lemma~\ref{lem:laqc}:
$$
N_n\le\frac{1}{\tilde{N}}\left(\left(\frac{5}{3}\right)^{14+10\gamma}
\tilde{N}N_1\exp{\left(\frac{5\varrho}{2-\varrho}\right)}
\right)^{2^{n-1}},
$$
if $n \geq 1$. If we also define
$\tilde{N}_{n+1}=\tilde{N}n^{8+6\gamma}(\exp{(\varrho^n)})^5N_n$, for
$n\geq 1$, we clearly have, using that $N_1\geq 1$ and
$\tilde{N}\geq 1$, that $\tilde{N}_n\leq N_n$ for $n\geq 2$.
Now, we have to justify that we can use the iterative lemma in this
inductive form when $n\geq 2$. To this end we need to see that the
bounds that we have assumed at the step $n$ (to define $\hat{N}$ and to
use the iterative lemma) hold at every step if $\vartheta_1$ is small
enough. So, we note that if $\vartheta_1$ is small enough, the
following sum:
\begin{equation}
\sum_{n \geq 1} {N}_{n+1} \vartheta_1^{2^{n}-2},
\label{eq:sum}
\end{equation}
is bounded by $\hat{N}^*$ that depends on $\varrho$ and the same
constants as $\hat{N}$.
This bound is not difficult to obtain if we look at how fast
$\tilde{N}_n$ grows. Moreover the same ideas can be used to prove that
$N_n\vartheta_1^{2^{n}-2}\leq N_1$, if $n \geq 1$ and $\vartheta_1$ is
small enough.
Then, we can define
$\alpha_1^{(n+1)}=\alpha_1^{(n)}-2{N}_{n+1}\vartheta_1^{2^n}$,
$\alpha_2^{(n+1)}=\alpha_2^{(n)}+2{N}_{n+1}\vartheta_1^{2^n}$,
$\beta_2^{(n+1)}=\beta_2^{(n)}+{N}_{n+1}\vartheta_1^{2^n-1}$,
$\hat{m}^{(n+1)}=\hat{m}^{(n)} +{N}_{n+1}\vartheta_1^{2^n}$,
$\tilde{m}^{(n+1)}=\tilde{m}^{(n)}+{N}_{n+1}\vartheta_1^{2^n-1}$,
$\hat{\nu}^{(n+1)}=\hat{\nu}^{(n)}+{N}_{n+1}\vartheta_1^{2^n}$ and
$\tilde{\nu}^{(n+1)}=\tilde{\nu}^{(n)}+{N}_{n+1}
\vartheta_1^{2^n -1}$, that from the convergence of (\ref{eq:sum})
allows to apply another step of the iterative scheme, at least for
sufficiently small values of $\vartheta_1$. Moreover, it is clear that
$\Theta_n\le{N}_{n+1}\vartheta_1^{2^n}\le\hat{N}^*\vartheta_1^2\le
1/2$ taken $\vartheta_1$ small enough.
Then, it only reamains to bound $\bar{m}^{(n+1)}$. For that purpose we
first consider the bound $\|\bar{\cal C}^{(n+1)}-
\bar{\cal C}^{(n)}\|_{{\cal E}^{(n+1)},\rho^{(n+1)}}\leq
{N}_{n+1}\vartheta_1^{2^n}$, and then, if we work with a fixed
value of $\varphi\in\bar{\cal E}^{(n+1)}(\vartheta_1)$, we have for
any $W\in\CC^r$:
$$
|\bar{C}^{(n+1)} W| \geq |\bar{C}^{(n)} W|-
\left|\left(\bar{C}^{(n+1)}-\bar{C}^{(n)}\right) W\right|\geq\left(
\left(\bar{m}^{(n)}\right)^{-1}-{N}_{n+1}
\vartheta_1^{2^n} \right) |W|.
$$
We note that, from the equivalence
$|\left(\bar{C}^{(n)}\right)^{-1}|\leq \bar{m}^{(n)}$ $\iff$
$|\bar{C}^{(n)}W|\geq\left(\bar{m}^{(n)}\right)^{-1}|W|$, for any
$W\in\CC^r$, we can take $\bar{m}^{(n+1)}=
\frac{\bar{m}^{(n)}}{1-\bar{m}^{(n)}{N}_{n+1}\vartheta_1^{2^n}}$,
provided that $\bar{m}^{(n)}{N}_{n+1}\vartheta_1^{2^n}<1$.
Then, using this expression we can see that
$\bar{m}^{(n)}\leq 2\bar{m}^{(1)}$ for any $n\geq 1$,
if $\vartheta_1$ is small enough: if we assume that it holds for $n$,
when we compute $\bar{m}^{(n+1)}$ we have that
$$
\bar{m}^{(n)}{N}_{n+1} \vartheta_1^{2^n} \leq
2\bar{m}^{(1)}{N}_{n+1}\vartheta_1^{2^n}\leq \frac{1}{2},
$$
if $ \vartheta_1$ is small enough. Moreover, we have by induction that
$$
\bar{m}^{(n+1)} \leq \bar{m}^{(1)} \prod_{j=1}^{n}
\frac{1}{1-2\bar{m}^{(1)}{N}_{j+1}\vartheta_1^{2^j}}.
$$
So, it is clear that, if $\vartheta_1$ is small enough,
$$
\sum_{j\geq 1}2\bar{m}^{(1)}{N}_{j+1}\vartheta_1^{2^j}
\leq 2\bar{m}^{(1)}\hat{N}^*\vartheta_1^{2}\leq\frac{1}{2}
\ln{(2)},
$$
and hence, if we note that when $0 \leq X \leq 1/2$,
$$
\ln{\left(\frac{1}{1-X}\right)}=\ln{\left(1+\frac{X}{1-X}\right)}\leq
\frac{X}{1-X} \leq 2 X,
$$
we can bound
$\ln{\left(\bar{m}^{(n+1)}\right)}\leq\ln{(\bar{m}^{(1)})}+\ln{(2)}$,
that proves $\bar{m}^{(n+1)}\leq 2\bar{m}^{(1)}$.
\subsubsection{Convergence of the changes of variables}
Now, we are going to prove the convergence of the composition of changes
of variables. Let ${\cal E}^*=\cap_{n \geq 1}{\cal E}^{(n)}$ be the set
of $\varphi$ where everything is well defined for all the steps. We
consider a fixed $\varphi\in{\cal E}^*$, but in fact, the
results will hold in the whole set ${\cal E}^*$ provided that
$\vartheta_1$ is small enough.
We put $\breve{\Psi}^{(n)}=
\hat{\Psi}^{(1)}\circ\ldots\circ\hat{\Psi}^{(n)}$
for $n \geq 1$, that goes from
${\cal U}_{\rho^{(n+1)},R^{(n+1)}}^{r+s,2m+r}$ to
${\cal U}_{\rho,R}^{r+s,2m+r}$, where $\hat{\Psi}^{(n)}$ means
$\hat{\Psi}^{S^{(n)}}$. Then, if $p > q \geq 1$, we have
$$
\breve{\Psi}^{(p)} - \breve{\Psi}^{(q)} = \sum_{j=q}^{p-1} \left(
\breve{\Psi}^{(j+1)} - \breve{\Psi}^{(j)} \right).
$$
To bound $\breve{\Psi}^{(j+1)}-\breve{\Psi}^{(j)}$, we define
$\rho'_{j}=\rho^{(j)}-\rho/4$ and $R'_{j}=R^{(j)}-R/4$, and
we put $\hat{\Delta}_{\rho,R}=\frac{1}{\exp{(1)}\rho}+
\frac{r+2m}{R}$.
%%$\hat{\Delta}_{\rho,R}=1/(\exp{(1)}\rho)+(r+2m)/R$.
Now, let us see that
\begin{eqnarray}
& &
|\breve{\Psi}^{(j+1)}-\breve{\Psi}^{(j)}|_{\rho'_{j+2},R'_{j+2}}=
|\hat{\Psi}^{(1)}\circ\ldots\circ\hat{\Psi}^{(j+1)}-
\hat{\Psi}^{(1)}\circ\ldots\circ\hat{\Psi}^{(j)}
|_{\rho'_{j+2},R'_{j+2}} \leq
\nonumber \\& \leq &
\left(1+4\hat{\Delta}_{\rho,R}|\hat{\Phi}^{(1)}|_{\rho^{(2)},R^{(2)}}
\right)|\hat{\Psi}^{(2)}\circ\ldots\circ\hat{\Psi}^{(j+1)}-
\hat{\Psi}^{(2)}\circ\ldots\circ\hat{\Psi}^{(j)}
|_{\rho'_{j+2},R'_{j+2}},
\label{eq:difcomp}
\end{eqnarray}
where we note
$\hat{\Psi}^{(n)}-Id=\hat{\Phi}^{S^{(n)}}:=\hat{\Phi}^{(n)}$,
if $n\geq 1$. To prove it, we write
$ \hat{\Psi}^{(1)} \circ \ldots \circ \hat{\Psi}^{(j)} =
\hat{\Psi}^{(2)} \circ \ldots \circ \hat{\Psi}^{(j)} +
\hat{\Phi}^{(1)}\circ\hat{\Psi}^{(2)}
\circ\ldots\circ\hat{\Psi}^{(j)}$,
for every $j\geq 1$, and then we note that can bound
$|\hat{\Psi}^{(2)}\circ\ldots\circ \hat{\Psi}^{(j+1)}-Id
|_{\rho'_{j+2},R'_{j+2}}$ and
$|\hat{\Psi}^{(2)}\circ\ldots\circ
\hat{\Psi}^{(j)}-Id|_{\rho'_{j+2},R'_{j+2}}$ by
$\min{\{\rho^{(2)}-\rho'_{j+2}-\rho/4,R^{(2)}-R'_{j+2}- R/4\}}$.
We prove the first bound, the second is analogous. To prove this, we
have:
\begin{eqnarray*}
& &
|\hat{\Psi}^{(2)}\circ\ldots\circ \hat{\Psi}^{(j+1)} -
Id |_{\rho'_{j+2},R'_{j+2}}\leq
|\hat{\Psi}^{(2)}\circ\ldots\circ \hat{\Psi}^{(j+1)} -
Id |_{\rho^{(j+2)},R^{(j+2)}}\leq
\\ & & \leq \sum_{l=2}^{j}
|\hat{\Psi}^{(l)}\circ\ldots\circ\hat{\Psi}^{(j+1)} -
\hat{\Psi}^{(l+1)}\circ\ldots\circ\hat{\Psi}^{(j+1)}
|_{\rho^{(j+2)},R^{(j+2)}} +
\\ & & +
|\hat{\Psi}^{(j+1)}-Id|_{\rho^{(j+2)},R^{(j+2)}} \leq
\sum_{l=2}^{j+1}|\hat{\Phi}^{(l)}|_{\rho^{(l+1)},R^{(l+1)}}
= \sum_{l=2}^{j+1} \min{\{\delta_l,\tilde{\delta}_{l}}\} =
\\ & = &
\min{\{\rho^{(2)}-\rho^{(j+2)},R^{(2)}-R^{(j+2)}\}} =
\min{\{\rho^{(2)}-\rho'_{j+2}-\rho/4,R^{(2)}-R'_{j+2}-R/4 \}},
\end{eqnarray*}
where we have used lemma \ref{lem:comp} to bound the norms of the
compositions. From that, we can prove (\ref{eq:difcomp}) from
lemma \ref{lem:mvt}.
Now, if we iterate (\ref{eq:difcomp}) using similar ideas
at every step, we produce the bound
$$
|\breve{\Psi}^{(j+1)}-\breve{\Psi}^{(j)}|_{\rho'_{j+2},R'_{j+2}}
\leq \prod_{l=1}^{j}\left(1+4\hat{\Delta}_{\rho,R}
|\hat{\Phi}^{(l)}|_{\rho^{(l+1)},R^{(l+1)}} \right)
|\hat{\Phi}^{(j+1)}|_{\rho^{(j+2)},R^{(j+2)}}.
$$
So, from lemma~\ref{lem:lemcan},
$$
|\hat{\Phi}^{(n)}|_{\rho^{(n+1)},R^{(n+1)} } \leq
|\nabla S^{(n)}|_{\rho^{(n)}-8\delta_n,R^{(n)}-8\hat{\delta}_n}\leq
{N}_{n+1}\vartheta_1^{2^n},
$$
and if we assume
$$
4\hat{\Delta}_{\rho,R}\sum_{l\geq 1}{N}_{l+1}\vartheta_1^{2^l}\leq
4\hat{\Delta}_{\rho,R}\hat{N}^*\vartheta_1^{2}\leq\ln{2},
$$
we have that:
$$
\prod_{l=1}^{j}\left(1+4\hat{\Delta}_{\rho,R}
{N}_{l+1}\vartheta_1^{2^l}\right)\leq 2,\mbox{ for every }
j\geq 1.
$$
Hence, using the convergent character of the sum (\ref{eq:sum}), one
obtains
$$
|\breve{\Psi}^{(p)}-\breve{\Psi}^{(q)}|_{\rho/4,R/4} \leq
\sum_{j \geq q}2{N}_{j+2}\vartheta_1^{2^{j+1}}
\rightarrow 0,\; \mbox{as} \; p,q \rightarrow +\infty.
$$
This fact allows to define
$$
\hat{\Psi}^{(*)} := \lim_{n \rightarrow +\infty} \breve{\Psi}^{(n)},
$$
that maps
${\cal U}_{\rho/4,R/4}^{r+s,2m+r}$ into ${\cal U}_{\rho,R}^{r+s,2m+r}$.
So, as we remark in section~\ref{sec:canon} for this kind of canonical
transformations that keeps the quasiperiodic time dependence of
the Hamiltonian at every step,
we only need to show that the final Hamiltonian is well
defined to obtain the convergence of the final canonical change
${\Psi}^{(*)}$, defined as the composition of all the
${\Psi}^{S^{(n)}}$. This follows immediately from the different bound
for the terms of $H^{(n)}$.
Hence, the limit Hamiltonian $H^{(*)}$ takes the form:
$$
H^{(*)}(\theta,x,I,y,\varphi)=\ti^{(*)}(\varphi)+\omega I +
\frac{1}{2} z^{\top} {\cal B}^{(*)}(\varphi) z+
\frac{1}{2} z^{\top} {\cal C}^{(*)}(\theta,\varphi) z+
H_*^{(*)}(\theta,x,\hat{I},y,\varphi),
$$
with $=0$, that is, for every $\varphi\in{\cal E}^{*}$,
we have a ($r+s$)-dimensional reducible torus.
\subsubsection{Control of the measure}
To prove the assumptions $(a)$ and $(b)$ of Theorem~\ref{teo}, we only
need to control the measure of the set of parameters for which we can
prove convergence of the scheme or, in an equivalent form, which is
the measure of the different sets that we remove at each step of the
iterative method: the key idea is to study the characteritzation of
these sets given by the Diophantine conditions of (\ref{eq:esdb}).
Hence, we only need to look at the eigenvalues of ${\cal B}^{(n)}$.
From the bounds of the inductive scheme, we have that
$\|\lambda_j^{(n)}-\lambda_j^{(1)}\|_{\bar{\cal E}^{(n)}}=
O_2(\bar{\vartheta})$ and ${\cal L}_{\bar{\cal E}^{(n)}}
\left\{\lambda_j^{(n)}-\lambda_j^{(1)}\right\}
=O(\bar{\vartheta})$ for every $j=1,\ldots,2m$ and $n\geq 2$, provided
that $0<\bar{\vartheta}\leq\vartheta_1$, where the constants that give
the different $O_2(\bar{\vartheta})$ and $O(\bar{\vartheta})$ are
independent on $n$ and $j$. Then, from expression (\ref{eq:ndc})
we can write:
$$
\lambda_j^{(n)}(\varphi)=\lambda_j + i u_j \epsilon +
i v_j^{\top} (\hat{\omega} - \hat{\omega}^{(0)}) +
\tilde{\lambda}_j^{(n)}(\varphi),
$$
with
${\cal L}_{\bar{\cal E}^{(n)}}\{\tilde{\lambda}_j^{(n)}\}
\leq L\bar{\vartheta}$
and
$|\lambda_j^{(n)}(\varphi)-\lambda_j|\leq M|\varphi-\varphi^{(0)}|$,
for certain $L$ and $M$ positive. Then, if we use the nondegeneracy
conditions of {\bf NDC} plus the Diophantine assumptions for the
frequencies and eigenvalues of
the initial torus, the results of $(a)$ and $(b)$ are consequence of
lemmas of section~\ref{sec:lacm}. Here we skip any kind of
``hyperbolicity'' and we assume that we are always in the worst case,
that is, we assume all the normal directions to be of elliptic type.
\begin{itemize}
\item[$(a)$]
From the bound for the measure of the set ${\cal A}$ in
lemma~\ref{lem:meas1}, we cleary have that if we put in this lemma
$\lambda \equiv \lambda_j^{(n)}$ or $\lambda \equiv \lambda_j^{(n)} -
\lambda_l^{(n)}$, $j\neq l$, in the set ${\cal E}\equiv{\cal E}^{(n)}$
with $\vartheta_0\equiv\vartheta_1$, $\cd\equiv\cd_n$ and
$\delta\equiv\delta_n$, we can bound
$$
\mbox{mes}(\bar{\cal I}^{(n)}\backslash \; \bar{\cal I}^{(n+1)} ) =
O\left(K^{r+s-1-\gamma}\cd_n\frac{\exp{(-\delta_n K)}}{\delta_n}\right),
$$
with $\bar{\cal I}^{(n)}=\{\epsilon\in[0,\bar{\epsilon}]:\;
(\hat{\omega}^{(0)\top},\epsilon)^{\top}=\varphi\in{\cal E}^{(n)}\}$
($\bar{\epsilon}>0$ small enough) and we can take $K$ such
that $\frac{\cd_0}{2 K^{\gamma}}=M \bar{\epsilon}$, that is,
$K:=K(\bar{\epsilon})=
\left(\frac{\cd_0}{2M\bar{\epsilon}}\right)^{\frac{1}{\gamma}}$.
Then, if we put $\bar{\cal I}_* =\cap_{n\geq 1}\bar{\cal I}^{(n)}$ we
have from the bounds of lemma \ref{lem:meas3} that for every
$0<\sigma<1$, if $K(\bar{\epsilon})$ is big enough (that is equivalent
to take $\bar{\epsilon}$ small enough) depending on $\sigma$:
$$
\mbox{mes}([0,\bar{\epsilon}] \; \backslash \; \bar{\cal I}_* ) \leq
\exp{(-(1/\bar{\epsilon})^{\frac{\sigma}{\gamma}})}.
$$
\item[$(b)$]
Now we can use the result $(i)$ of lemma \ref{lem:meas1} plus lemma
\ref{lem:meas2}, working on sets of the form
${\cal W}^{(n)}({\epsilon}_0,R_0)=\{\hat{\omega}\in\RR^r:\;
(\hat{\omega}^{\top},\epsilon_0)^{\top}=\varphi\in{\cal E}^{(n)}\}$.
We have
$$
\mbox{mes} ( {\cal W}^{(n)} \backslash \; {\cal W}^{(n+1)}) =
O\left(R_0^{r-1}K^{r+s-1-\gamma}\cd_n
\frac{\exp{(-\delta_n K)}}{\delta_n}\right),
$$
where for $K$ we can take (depending on $\epsilon_0$ and $R_0$),
$K=\min{\{K_1,K_2\}}$, with
$\frac{\cd_0}{2K_1^{\gamma}}=M\max{\{R_0,\epsilon_0\}}+K_1 R_0$,
condition that comes from lemma \ref{lem:meas1}, and
$K_2=\left(\frac{\cd_0}{2 R_0}\right)^{\frac{1}{\gamma+1}}$,
that comes from lemma \ref{lem:meas2}. Then, if we take a fixed
$0\leq\epsilon_0\leq R_0^{\frac{\gamma}{\gamma+1}}$
(we recall $R_0 \leq 1$) we can obtain a lower bound for
$K$ of $O\left(R_0^{-\frac{1}{\gamma+1}}\right)$, where the
constant that give this order depends only on $\cd_0$, $\gamma$ and $M$.
So, if we use lemma~\ref{lem:meas3} we have the desired bound for the
measure of the set ${\cal W}_*({\epsilon}_0,R_0)=
\cap_{n\geq 1}{\cal W}^{(n)}({\epsilon }_0,R_0)$:
$$
\mbox{mes}( {\cal V}(R_0) \backslash {\cal W}_*({\epsilon},R_0)) \leq
\exp{(-(1/R_0)^{\frac{\sigma}{\gamma+1}})},
$$
for every $0<\sigma<1$, if $R_0$ is small enough depending on $\sigma$.
\end{itemize}
\section{Acknowledgements}
The authors want to thank Rafael del la Llave and Carles Sim\'o for
interesting discussions.
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}