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\centerline{\titlefont Fixed point theorems in the Arnol'd model about 
instability}

\centerline{\titlefont of the action-variables in phase-space}
\vskip2.0truecm
\centerline{\bf Paolo Perfetti}
\centerline{Dipartimento di matematica II Universit{\accent"12 a} di Roma}

\centerline{via della Ricerca Scientifica 00133 Roma, Italy}
\vskip1.0truecm
\noindent
{\bf Abstract}\ {\ninerm 
We consider the hamiltonian $H={1\over2}(I_1^2+I_2^2)+\varepsilon(\cos\varphi_1-1)
(1+\mu(\sin\varphi_2+\cos t))$ $I\in{\Bbb R}^2$ 
(\lq\lq Arnol'd model about diffusion"); by means of 
fixed point theorems, 
the existence of the stable and unstable manifolds 
{\it (whiskers)} of invariant, \lq\lq a priori unstable
tori", for any vector-frequency $(\omega,1)\in{\Bbb R}^2$ is proven. 
Our aim is to provide detailed proofs which are missing in Arnol'd's paper, 
namely prove  
the content of the {\tt Assertion B} pag.583 of [A]. Our proofs are based on   
technical tools suggested by Arnol'd i.e.  
the contraction mapping method togheter 
with the \lq\lq conical metric" ( see the footnote 
** of pag. 583 of [A]). }
\vskip0.5truecm
\noindent
{\bf \mathhexbox2780.Introduction}\  
By means of contraction techniques we prove 
the existence of {\it whiskered 
tori} (i.e. invariant tori that possess stable and unstable manifolds) 
for the dynamical system whose hamiltonian is
(\lq\lq Arnol'd model")
$$
H(I_1,I_2,\varphi_1,\varphi_2,t)={1\over2}(I_1^2+I_2^2)+\varepsilon
(\cos\varphi_1-1)+
\varepsilon\mu(\cos\varphi_1-1)(\sin\varphi_2+\cos t).
$$
\noindent
The hamiltonian $H$ 
is defined over the \lq\lq extended phase-space" $\tilde{\Cal M}={\Cal M}\times{\Bbb T}=
{\Bbb R}^2\times{\Bbb T}^3;$ 
$(I_1,I_2)\in{\Bbb R}^2,$ $(\varphi_1,\varphi_2,t)\in{\Bbb T}^3$ 
$({\Bbb T}={\Bbb R}/2\pi{\Bbb Z}),$ $\varepsilon$ and $\mu$ are real 
parameters. Physically $H$ represents a pendulum 
$({1\over2}I_1^2+\varepsilon(\cos\varphi_1-1))$ coupled to a rotator 
$({1\over2}I_2^2)$ through a force periodically depending on 
$\varphi_1,$ $\varphi_2$ and $t.$ The canonical equations of motion are 
$$
\left\{
\eqalign{
&\dot\theta=1,\qquad\qquad\dot\varphi_1=I_1,\qquad\qquad\dot\varphi_2=I_2\cr
&\dot I_1=\varepsilon\sin\varphi_1(1+\mu(\sin\varphi_2+\cos \theta))
\qquad\qquad\qquad
\dot I_2=-\varepsilon\mu\cos\varphi_2(\cos\varphi_1-1)\cr
}\right.
\eqno(0.0)
$$
\noindent
and we denote by $\phi^\mu_t\colon\tilde M\to\tilde M$ the associated 
hamiltonian flow.

\noindent
For $\varepsilon=0$ the system is integrable. 
For $0<\varepsilon\le\varepsilon_o$ ($\varepsilon_o$ 
small enough), as well known by KAM theorem, most 
(in the sense of Lebesgue measure) of the tori  are continued. The 
measure of the initial data excluded by the theorem is 
$O(\sqrt\varepsilon)$ (although they are dense). 
 
Now for $\varepsilon\ne0$ and $\mu=0,$ following [A], we 
consider the motions that \lq\lq develop along the separatrix of the 
pendulum". 
We therefore write the equations 
$$
H^1(I_1,\varphi_1)=
{1\over2}I_1^2+\varepsilon(\cos\varphi_1-1)=0,\qquad\quad 
H^2(I_2,\varphi_2)=
{1\over2}I_2^2={1\over2}\omega^2
$$ 
and consider their orbits. A standard calculation gives 
$$
\eqalign{
&\theta(t)=\theta^o+t,\qquad\qquad 
I_2(t)\equiv\omega,\qquad\quad I_1(t)\doteq
\pm\beta(t)=\pm{2\sqrt\varepsilon \over
\cosh(\sqrt\varepsilon t)}\cr
&\varphi_2(t)=\varphi_2^o+\omega t,\qquad
\varphi_1(t)\doteq\pm\alpha(t)=\pi\pm2\arctan\sinh(\sqrt\varepsilon t).
\cr
}
\eqno(0.1)
$$
$\phi_t^0(\pm2\sqrt\varepsilon,\pi,\omega,\varphi_2^o,\theta^o):\tilde{\Cal M}\to \tilde{\Cal M}$ 
is the flow generated by the 
hamiltonian $H^0=H^1+H^2$ 
when the initial datum, for 
what concerns the variables $(I_1,\varphi_1),$ is $(\pm2\sqrt\varepsilon,\pi).$ 

\noindent
The invariant set ${\Bbb T}_\omega=\{(I_1,\varphi_1,I_2,\varphi_2,\theta)
\vert
(I_1=0, \varphi_1=0, I_2=\omega, (\varphi_2,\theta)\in{\Bbb T}^2\}$ 
is geometrically an unstable torus (from which the 
definition \lq\lq a priori unstable" used in [CG]) and it is run linearly: 
$\theta^o\to \theta^o+t,$ $\varphi^o_2\to\varphi^o_2+\omega t.$   
Adopting the definitions of [A] we call $Y^-_\omega$ 
the set of points such that, 
$dist(\phi_t^0(\pm2\sqrt\varepsilon,\pi,\omega,\varphi_2^o,\theta^o),
{\Bbb T}_\omega)\to0$ 
as 
$t\to\infty$ (it is called the {\it stable manifold} of 
${\Bbb T}_\omega$). It is a three dimensional manifold in a five dimensional 
phase-space.
The {\it unstable manifold} at ${\Bbb T}_\omega$, $Y^+_\omega,$ 
is defined as the set of points such that 
$dist(\phi_t^0(\pm2\sqrt\varepsilon,\pi,\omega,\varphi_2^o,\theta^o),
{\Bbb T}_\omega)\to0$ as 
$t\to-\infty;$ because of the degeneracy of the model we have 
$Y^-_\omega\equiv Y^+_\omega$ and 
$Y^+_\omega\cap Y^-_{\omega^\prime}=\emptyset$ 
$\forall \omega\ne{\omega^\prime}.$ This last relation is connected to 
the impossibility of the trajectories (0.1) to make a \lq\lq large 
variation" in the $(I_1,I_2)$ variables; in fact $I_2$ is a constant while 
$\vert I_1\vert\le2\sqrt\varepsilon.$

\vskip0.7truecm
\noindent
Now set $\varepsilon\ne0$ and $\mu\ne0.$ The torus ${\Bbb T}_\omega$ continues 
to be invariant for any value of $\mu$ and in [A] ({\tt Theorem 3} pag.582) the 
following theorem (\lq\lq Arnol'd diffusion") is stated
\vskip0.5truecm
\noindent
{\sl Assume $0<A<B.$ For any $\varepsilon>0$ 
there exists a $\mu_o(\varepsilon,A,B)$  
such that if $0<\mu<\mu_o$ then 
$\vert I_2(t^o)-I_2(0)\vert\ge(B-A)$ where $t^o=t^o(\mu)$
}\footnote{$\mu_o\sim e^{-1/\sqrt\varepsilon}.$ Moreover 
it is easy to verify that $\mu_o\to0$ as $B\to\infty.$  
}
\vskip0.5truecm\noindent
The proof is based on two fundamental steps:

1) {\tt Assertion B} of [A]: \lq\lq {\sl ${\Bbb T}_\omega$ 
is a whiskered transition torus for $\mu$ small enough}"

2) $Y^+_\omega\cap Y^-_{\omega^\prime}\ne\emptyset$ for $\omega^\prime$ 
close to $\omega.$
\vskip0.3truecm
\noindent
The proof of Part 1) is the object of this paper. 
Part 2) is proven in [A].

\noindent
In order to prove 1) we shall need:

i) the existence (continuation) for $\mu\ne0$ of the {\it stable manifold}, $Y^-_\omega,$ 
and {\it unstable manifold}, $Y^+_\omega,$ at 
${\Bbb T}_\omega$ for any real $\omega$ (proved in {\bf \mathhexbox2782}; 
we have retained for the manifolds the same notation of the case $\mu=0$). 
This is what in the step 1) is represented by the word 
\lq\lq ${\Bbb T}_\omega$ is a whiskered torus for $\mu$ small 
enough" 
 
ii) \lq\lq The transition property" of ${\Bbb T}_\omega$ with 
$\omega\in{\Bbb R}\backslash{\Bbb Q}$ so that 
\lq\lq ${\Bbb T}_\omega$ is a whiskered transition torus" 
(see {\bf \mathhexbox2784}).

\noindent
Point ii) means: given an arbitrary point $p$ of the {\it stable manifold} at ${\Bbb T}_\omega$ 
let us consider an arbitrary 
neighborhood $U$ of $p$ and set $\Omega=\cup_{t\ge0}\phi_t(U)$ 
($\phi_t$ is the flow of the system (0.0)). Now 
consider the {\it unstable manifold}, an arbitrary point $q$ of it 
and an arbitrary manifold ${\Cal N}$ 
that crosses $Y^+_\omega$ at $q.$ Moreover the manifold ${\Cal N}$ 
is such that 
$T_q{\Cal N}\oplus T_qY^+_\omega=T_q M$ (i.e.$T_q{\Cal N}$ and $T_qY^+_\omega$ 
span the whole tangent space; 
it is said that ${\Cal N}$ {\it complements} $Y^+_\omega$ at $q$); 
then $\Omega$ intersects ${\Cal N}.$ 

\noindent
We do not prove ii) as it has been done in [Ma] and extended in [Cr]
(they consider the hamiltonian functions previously considered in [Gr] and 
[Tr] respectively). The 
idea of the proof is based on a suitable generalization 
of the $\lambda-$lemma by Palis (see [PM] pag.80, [Wi] pag. 324, [Ru] pag.88)

As said above we make use of contraction arguments in order to prove the 
existence 
of {\it stable} and {\it unstable} manifolds. We will work in the analytic  
class as the Cauchy theorem makes the 
estimates very easy. Frequently other less regular 
function spaces are introduced in connection with fixed-point theorems but 
they imply more involved estimates adding little to the comprehension of 
the problem.

In recent years there have been many papers about instability 
of quasi-integrable dynamical system 
([Xia1], [CG], [Xia2], [Ga2], [Xia3], [Ge1], [Ge2], [C1], [Moe],  
[RW1], [RW2], [C2], [Ni]). 
We just point out that the above papers use 
KAM theorem to prove the existence of the invariant tori or 
their stable and ustable manifolds. 
This forces to 
exclude, as usual, a dense set of initial data of small Lebesgue measure. 
The presence of holes in the space of the actions 
complicates a lot the construction of the transition chains (see [CG]).

>From a non-perturbative point of view we cite the work of Bessi 
[Be]. Bessi uses variational techniques in order to prove the 
existence of trajectories connecting far away regions of phase-space 
and finds good estimates about the times of transition.
\vskip0.3truecm
\noindent
The plan of the paper is the following: in {\bf\mathhexbox2781} we construct a 
suitable Poincar\'e map, say $M,$ for the system (0.0); in 
 {\bf\mathhexbox2782} we construct the stable and unstable manifold, called 
${\Cal Y}^-$ and ${\Cal Y}^+$ respectively, for $M$; in  {\bf\mathhexbox2783} using the 
equations of ${\Cal Y}^-$ and ${\Cal Y}^+$ we linearize the motion on ${\Cal Y}^\pm$; 
finally in {\bf\mathhexbox2784} we show how [Ma] applies to our setting in 
order to prove the transition property (see ii) above).
\vskip0.5truecm
\noindent
{\bf Acknowledgments}\ I thank Luigi Chierchia for many careful comments,
advises and suggestions.
\vskip0.6truecm
\noindent
{\bf \mathhexbox2781.Setup}\qquad 
In this section we do a suitable Poincar\'e map on the system (0.0). 
Therefore the relevant quantities that we shall consider: tori, stable 
and unstable manifolds are all relative to the map. 

With the transformation over the variables 
$(I_1,\varphi_1,I_2,\varphi_2,\theta)=U(q,p,I_2,\varphi_2,\theta)$ 
made of two canonical maps plus a rescaling the equations (0.0) become 
(more details are in Appendix 1)
$$
\left\{
\eqalign{
&\dot\theta=1,\qquad\quad\dot\varphi_2=I_2,\qquad\quad
\dot I_2=-\varepsilon\mu\cos\varphi_2(\cos\tilde\varphi_1
(q,p)-1)\cr
&\dot q=\sqrt\varepsilon q(1+F_J)-{\sqrt\varepsilon\over\sqrt2}\mu
\sin\tilde\varphi_1(q,p)(\sin\varphi_2+\cos \theta)
\cr
&\dot p=-\sqrt\varepsilon p(1+F_J)+{\sqrt\varepsilon\over\sqrt2}\mu\sin\tilde
\varphi_1
(q,p)(\sin\varphi_2+
\cos \theta)\cr
}\right.
\eqno(1.0)
$$
$F(J)=O(J^2)$ and $\tilde\varphi_1$ 
are analytic functions of $q$ and $p$ in the region 
$\vert q\vert +\vert p\vert\le O(1)$;  
$\tilde\varphi_1(0,0)=0.$ 

\noindent
>From the equations (1.0) one immediately realizes that: 

\noindent
1) ${\Bbb T}_\omega=\{(q,p,I_2,\varphi_2,\theta)\vert q=0, 
p=0, I_2=\omega, (\varphi_2,\theta)\in{\Bbb T}^2\}$ 
is an invariant set for any value of the parameter $\mu$ 
and is linearly i.e. $\varphi_2(t)=\varphi_2^o+
\omega t,$ $\theta(t)=\theta^o+t.$

\noindent
2) if $\mu=0$ the $q$ and $p$ axis are invariant. $F(J)$ is a first  
integral (actually any function of $J$ is a prime integral) 
and for $\vert q\vert+\vert p\vert$ small 
enough it is $1+F_J>0.$ Then the points of the $p$ axis are exponentially 
shrunk to 
$0$ while the points of the $q$ axis are exponentially 
pushed away from the origin. 
In the new variables $(q,p,I_2,\varphi_2,\theta)$ the {\it stable 
manifold}, $Y^-_\omega$ is written as 
$\{(q,p,I_2,\varphi_2,\theta)\vert
q=0,p\in(-\sigma,\sigma),I_2=\omega,(\varphi_2,\theta)\in{\Bbb T}^2\}$ 
while the {\it unstable manifold}  
$Y^+_\omega$ as 
$\{(q,p,I_2,\varphi_2,\theta)\vert q\in(-\sigma,\sigma),
p=0,I_2=\omega,(\varphi_2,\theta)\in{\Bbb T}^2\}$ 
$(\sigma=O(1)).$ 
\vskip0.4truecm
\noindent
Now we do the following Poincar\'e-map: let 
$\phi^\mu_t(q^o,p^o,I^o_2,\varphi^o_2,0)\colon 
\tilde{\Cal M}\to \tilde{\Cal M}$ 
be the flow of the 
system (1.0) ($0$ stands for $\theta^o=0$ and we omit it for brevity). 
Denoting $r^o=(q^o,p^o,I^o_2,\varphi^o_2),$ for any 
integer $n$ we define 
$\phi^\mu_{2n\pi}(r^o,0)\vert_{{\Cal M}}\doteq\hat\phi^\mu_{2n\pi}(r^o)
\colon B^2_\sigma\times{\Bbb R}\times{\Bbb T}\to B^2_\sigma\times{\Bbb R}
\times{\Bbb T}$  ($B^2_\sigma$ is a 
two-dimensional sphere of radius $2\sigma^2$ in the $(q,p)$ space).
The map $\hat\phi^\mu_{2\pi}$ is written symbolically as 
$(r^o)^\prime\doteq(x^\prime, y^\prime, I^\prime,\varphi^\prime)=
\hat\phi^\mu_{2\pi}(x,y,I,\varphi)\doteq\hat\phi^\mu_{2\pi}(r^o)$ 
$$
\left\{
\eqalign{
&\theta^\prime=\theta=0\cr
&y^\prime=y e^{-2\pi\sqrt\varepsilon(1+F_J)}
+\mu \sqrt\varepsilon f_1(x,y,I,\varphi,\mu)
\cr
&x^\prime=x e^{2\pi\sqrt\varepsilon(1+F_J)}
-\mu \sqrt\varepsilon f_2(x,y,I,\varphi,\mu)
\cr
&\varphi^\prime=\varphi+2\pi I+\mu\varepsilon f_3(x,y,I,\varphi,\mu)
\cr
&I^\prime=I-\mu\varepsilon f_4(x,y,I,\varphi,\mu)
\cr
}\right.
\eqno(1.1)
$$
where 
$f_1,\ f_2=O(\vert x\vert+\vert y\vert)\qquad f_3,\ f_4=
O((\vert x\vert+\vert y\vert)^2).$ In fact we have 
$$
\left\{
\eqalign{
&q(2\pi;r^o)=q^o e^{2\pi\sqrt\varepsilon(1+F_J)}
-\mu{\sqrt\varepsilon\over\sqrt2}\int_0^{2\pi}ds\sin\tilde\varphi_1
(q,p)(\sin\varphi_2+\cos s)\cr
&\qquad\qquad\qquad\qquad
+\sqrt\varepsilon
\int_0^{2\pi}ds\Bigl[\bigl(q(1+F_J)\bigr)(s;r^o,\mu)-
\bigl(q(1+F_J)\bigr)(s;r^o,0)\Bigr]\cr
&p(2\pi;r^o)=p^o e^{-2\pi\sqrt\varepsilon(1+F_J)}
-\mu{\sqrt\varepsilon\over\sqrt2}
\int_0^{2\pi}ds\sin\tilde\varphi_1
(q,p)
(\sin\varphi_2+\cos s)-\cr
&\qquad\qquad\qquad\qquad
-\sqrt\varepsilon
\int_0^{2\pi}ds
\Bigl[\bigl(p(1+F_J)\bigr)(s;r^o,\mu)-
\bigl(p(1+F_J)\bigr)(s;r^o,0)\Bigr]
\cr
&\varphi_2(2\pi;r^o)=\varphi_2^o+2\pi I^o_2+\mu\varepsilon
\int_0^{2\pi}
dt \int_0^t ds\cos\varphi_2(\cos\tilde\varphi_1(q,p)-1)\cr
&I_2(2\pi;r^o)=I_2^o-\mu\varepsilon\int_0^{2\pi}ds
\cos\varphi_2(\cos\tilde\varphi_1(q,p)-1)\cr
}\right.
\eqno(1.2)
$$
In analogy with the differential equations (0.0) we define: 
${\Cal Y}^-_\omega=\{r^o\in B^2_\sigma\times{\Bbb R}\times{\Bbb T}\vert
dist(\hat\phi_{2\pi n}^\mu(r^o),{\Cal T}_\omega)\to0$ as $n\to+\infty$  
(the {\it stable manifold} at ${\Cal T}_\omega$) and 
${\Cal Y}^+_\omega=\{r^o\in B^2_\sigma\times{\Bbb R}\times{\Bbb T}\vert
dist(\hat\phi_{-2\pi n}^\mu(r^o),{\Cal T}_\omega)\to0$ as $n\to\infty$  
(the {\it unstable manifold} at ${\Cal T}_\omega$). 
A different value of $\theta^o$ would generate a different Poincar\'e map. If we 
call ${\Cal Y}^-_\omega(\theta^o)$ the stable manifold,  
${\Cal Y}^+_\omega(\theta^o)$ the unstable one and ${\Cal T}_\omega(\theta^o)$ the 
invariant torus of frequency $\omega$ for the Poincar\'e map at $\theta^o$ i.e. 
taking as surface map 
$\{(p,q)\in B^2_\sigma, I\in{\Bbb R}, (\varphi,\theta)\in{\Bbb T}^2
\vert \theta=\theta^o\},$ then 
we have $Y^-_\omega=\cup_{0\le \theta^o\le2\pi}({\Cal Y}^-_\omega)(\theta^o),$  
$Y^+_\omega=\cup_{0\le \theta^o\le2\pi}({\Cal Y}^+_\omega)(\theta^o)$ and 
${\Bbb T}_\omega=\cup_{0\le \theta^o\le2\pi}{\Cal T}_\omega(\theta^o)$ (see Appendix 1).  
We then reduce to the study of the existence of 
${\Cal Y}^-_\omega$ and ${\Cal Y}^+_\omega;$ the existence of 
${\Cal T}_\omega$ is trivial matter due to the peculiarity of the model.

We have to keep in mind that after the integrations with respect 
to time, there remain functions only of the initial data $r^o.$ 
For any $I^o\in{\Bbb R}$ and for any $\mu$ 
the set ${\Cal T}_{I^o}=\{q=0, p=0, I=I^o, \varphi\in{\Bbb T}\}$ is invariant. 
Geometrically it is a circle contained in the hyperplane $\theta=0;$ on it  
the motion is linear $\varphi_2^{(n)}=\varphi_2^{(0)}+2\pi I^on,$ 
$n\in{\Bbb N};$ it is periodic if $I^o\in{\Bbb Q}$ and quasi-periodic if 
$I^o\in{\Bbb R}\backslash{\Bbb Q}.$ 
Of course we have 
${\Bbb T}_{I^o}=\cup_{0\le \theta^o\le2\pi}{\Cal T}_{I^o}(\theta^o).$

When $\mu=0$ the following sets are also invariant: 
${\Cal Y}^-_{I^o}
=\{q=0, p\in(-\sigma,+\sigma), I=I^o, \varphi\in{\Bbb T}\}$ 
(the stable manifold) and 
${\Cal Y}^+_{I^o}
=\{q\in(-\sigma,+\sigma), p=0, I=I^o, \varphi\in{\Bbb T}\}$ 
(the unstable manifold); they are two dimensional in the 
four dimensional space ${\Bbb R}^3\times{\Bbb T}.$

\vskip0.5truecm
\noindent
{\bf \mathhexbox2782.Existence of perturbed stable and unstable manifolds}
\vskip0.3truecm
\noindent
Now we investigate the point i) of the introduction. 
>From now on we shall consider only the map (1.1). 

What follows 
is closely related to the Hadamard's theorem about the existence of stable 
and unstable manifolds of fully hyperbolic systems and in general to the 
theory of 
invariant sets for flows and maps defined on a manifold (possibly ${\Bbb R}^N$). 
There is an abundant literature on this 
argument and we mention only few references [Ke], [Fe1], [Fe2], [Fe3], 
[La2], [La1], [Ga3], [Ze], [HPS], [Wi], [PM], [Ir], [Ru].  

\noindent
Following Hadamard's approach and using a suggestion by Arnol'd (see  
footnote ** of pag.583 of [A]) 
we look for the stable manifold at ${\Cal T}_\omega$ 
as ${\Cal Y}^-_\omega=\{(x,y,I,\varphi)\vert x=\mu yA^s(y,\varphi;\omega,\mu), 
y\in(-\sigma,+\sigma), I=\omega+\mu\sqrt\varepsilon 
y{\Cal I}^s(y,\varphi;\omega,\mu),
\varphi\in{\Bbb T}\}$ while 
${\Cal Y}^+_\omega=\{(x,y,I,\varphi)\vert x\in(-\sigma,+\sigma), 
y=\mu xA^u(x,\varphi;\omega,\mu), I=\omega+
\mu\sqrt\varepsilon y{\Cal I}^u(x,\varphi;\omega,\mu),
\varphi\in{\Bbb T}\}.$ In order to simplify the estimates (leaving all the 
essential difficulties) we shall consider the space of the holomorphic 
functions ${\Cal A}(\Delta_{y,\varphi}^{\sigma,\xi})$ (for ${\Cal Y}^-_\omega$)
given by the 
analytic functions over 
$(-\sigma,\sigma)\times{\Bbb T}$ that admit an 
holomorphic extension to 
the complex domain $\Delta_{y,\varphi}^{\sigma,\xi}
=\{(y,\varphi)\in{\Bbb C}^2\vert\ \vert y\vert<\sigma, 
\vert Im\varphi\vert<\xi<1\}$ and are continuous on 
$\overline\Delta_{y,\varphi}^{\sigma,\xi}.$ With the norm 
$\Vert g\Vert=
\sup_{\overline\Delta_{y\varphi}^{\sigma,\xi}}\vert g(y,\varphi)\vert,$ 
${\Cal A}(\Delta_{y,\varphi}^{\sigma,\xi})$ is a Banach space. 
Of course for ${\Cal Y}^+_\omega$ the 
definition of the space of functions is exactly the same except that there is 
$x$ in place of $y$ and $u$ in place of $s.$ Now we prove the following
\vskip0.5truecm
\noindent
{\bf Theorem 2.0}{\sl\  For any $\omega\in{\Bbb R}\backslash\{0\}$ and for any 
$\vert\mu\vert\le\mu_o$ 
there exist two unique functions
$(A^s,{\Cal I}^s)\colon[-{\sigma\over2},{\sigma\over2}]\times{\Bbb T}
\to{\Bbb R}\times{\Bbb R}$ 
such that 
the set 
${\Cal Y}^-_\omega=\{(x,y,I,\varphi)=(\mu yA^s(y,\varphi;\omega,\mu,\varepsilon),y,
\omega+\mu\sqrt\varepsilon y{\Cal I}^s(y,\varphi;\omega,\mu,\varepsilon)
,\varphi)\vert y\in[-{\sigma\over2},{\sigma\over2}],\ \varphi\in{\Bbb T}\}$ 
is invariant respect to the Poincar\'e map (1.1). $A^s$ and ${\Cal I}^s$ are 
analytic in $(y,\varphi),$ in $\omega\in{\Bbb R}\backslash\{0\}$ and $\mu.$ 
 
}
\vskip0.4truecm
\noindent
Remarks\quad
i) in the functions $A^s$ and ${\Cal I}^s,$ $\varphi$ and $y$ are the variables  
while the presence of $\omega,$ $\mu$ and $\varepsilon$ is of parametric 
type\quad
ii) clearly the theorem could be formulated choosing 
less regular function 
spaces (see e.g. [La1]) \quad iii) 
$\sigma=O(1);$ $\mu_o$ is a 
function of $\sigma,$ $\varepsilon,$ $\xi,$ and of the Poincar\'e map; 
moreover $\mu_o\to0$ as $\sigma\to0,$ as $\varepsilon\to0$ and 
$\varepsilon\to\infty$
\qquad iv) in the next section we 
will prove that ${\Cal Y}^-_\omega$ is the stable manifold
\vskip0.3truecm
\noindent
{\bf Proof.}\qquad Let's consider the closed set 
$\bar{\Cal A}\subset{\Cal A}(\Delta_{y\varphi}^{\sigma,\xi})$ whose functions have the 
norm bounded by a constatnt ${\Cal A}$ 
$\Vert g\Vert_{\sigma,\xi}=
\sup_{\Delta_{y,\varphi}^{\sigma,\xi}}\vert g(y,\varphi)\vert\le A_o.$ Moreover 
we will consider the functions from 
$\Delta_{x,y,I,\varphi}^{\sigma,\xi,\rho}$ into ${\Bbb C}$ where 
$\Delta_{x,y,I,\varphi}^{\sigma,\xi,\rho}=
\{(x,y,I,\varphi)\in{\Bbb C}^4\vert\ 
\vert x\vert<\sigma, \vert y\vert<\sigma, \vert I-\omega\vert
<\rho, \vert Im\varphi\vert<\xi<1\}.$ Analogously to what happens in 
${\Cal A}(\Delta_{y\varphi}^{\sigma,\xi})$ we define 
$\Vert h\Vert_{\sigma,\xi,\rho}=
\sup_{\overline\Delta_{x,y,I,\varphi}^{\sigma,\xi,\rho}}
\vert h(x,y,I,\varphi)\vert$ for functions defined and analytic on 
$\Delta_{x,y,I,\varphi}^{\sigma,\xi,\rho}$ and continuous on 
$\overline\Delta_{x,y,I,\varphi}^{\sigma,\xi,\rho}.$

The function
$(A^s,{\Cal I}^s)$ will be found as fixed point of a contraction map from 
$\bar{\Cal A}\times\bar{\Cal A}$ into itself. Then the uniqueness and analiticity 
immediately follow.

First of all we impose the invariance. A brief computation gives 
$$
\eqalign{
&A={\sqrt\varepsilon\over ay}f_2(\mu yA,y,\omega+\mu\sqrt\varepsilon y{\Cal I},\varphi)+
{1\over a}\bigl(
{1\over a}+{\mu\sqrt\varepsilon\over y}
f_1(\mu yA,y,\omega+\mu\sqrt\varepsilon y{\Cal I},\varphi)\bigl)
A({y\over a}+\cr
&+\mu\sqrt\varepsilon f_1(\mu yA,y,\omega+\mu\sqrt\varepsilon y{\Cal I},\varphi),\cr
&\qquad\qquad\qquad
,\varphi+2\pi\omega+
2\pi\mu\sqrt\varepsilon y{\Cal I}+
\varepsilon\mu f_3(\mu yA,y,\omega+\mu\sqrt\varepsilon y{\Cal I},\varphi))\cr
&{\Cal I}={\sqrt\varepsilon\over y}
f_4(\mu yA,y,\omega+\mu\sqrt\varepsilon y{\Cal I},\varphi)+\bigl(
{1\over a}+{\mu\sqrt\varepsilon\over y} 
f_1(\mu yA,y,\omega+\mu\sqrt\varepsilon y{\Cal I},\varphi)\bigl)
{\Cal I}({y\over a}+\cr
&+\mu\sqrt\varepsilon f_1(\mu yA,y,\omega+\mu\sqrt\varepsilon y{\Cal I},\varphi),\cr
&\qquad\qquad\qquad
,\varphi+2\pi\omega+
2\pi\mu\sqrt\varepsilon y{\Cal I}+\mu\varepsilon 
f_3(\mu yA,y,\omega+\mu\sqrt\varepsilon y{\Cal I},\varphi))\cr
}
\eqno(2.0)
$$
and $a=e^{2\pi\sqrt\varepsilon(1+F_J)}$. 

We indicate (2.0) as $M(A,{\Cal I})(y,\varphi)=\bigl(M_1(A,{\Cal I}),M_2(A,{\Cal I})\bigl)
(y,\varphi)$ and try to solve it by means of contraction techniques: we look for  
two functions $(A^s,{\Cal I}^s)$ such that $M_1(A^s,{\Cal I}^s)=A^s, M_2(A^s,{\Cal I}^s)={\Cal I}^s.$ 
We show that there exists a real number $0<q<1$   
such that 
$$
\Vert M(A,{\Cal I})-M(A^\prime,{\Cal I}^\prime)\Vert_{\sigma^\prime,\xi^\prime}
\le q\Vert (A,{\Cal I})-(A^\prime,{\Cal I}^\prime)\Vert_{\sigma^\prime,\xi^\prime}
\qquad \sigma^\prime={\sigma\over2}\quad \xi^\prime={\xi\over4}
$$
and $\Vert M(A,{\Cal I})\Vert=\Vert M_1(A,{\Cal I})\Vert+\Vert M_2(A,{\Cal I})\Vert.$ 
 
\noindent
More precisely 
$$
\Vert M(A,{\Cal I})-M(A^\prime,{\Cal I}^\prime)\Vert_{\sigma^\prime,\xi^\prime}
\le q\bigl(\Vert A-A^\prime)\Vert_{\sigma^\prime,\xi^\prime}+
\Vert{\Cal I}-{\Cal I}^\prime\Vert_{\sigma^\prime,\xi^\prime}\bigl)
\eqno(2.1)
$$
and $q\le e^{-\pi\sqrt\varepsilon}+
\mu{P(\varepsilon)A_o^2\over\sigma\xi},$ 
$(P(\varepsilon)=1+87\sqrt\varepsilon+1521\varepsilon).$ $q<1$ implies 
$\mu{A_o^2 P(\varepsilon)\over\sigma\xi}<1-e^{-\pi\sqrt\varepsilon}$ 
and a sufficient 
condition is $\mu\le\mu_o(\sigma,\xi,A_o,\varepsilon)=
{\sigma\xi\over 2P(\varepsilon) A^2_o}(1-e^{-\pi\sqrt\varepsilon});$
\footnote{The estimate of the size of $\mu$ looks a 
little complicated. We can bound $\mu_o<{\sigma\xi\over2A_o^2}
\pi\sqrt\varepsilon$ which is the asymptotic formula when $\varepsilon$ goes 
to zero.} for 
$\mu=\mu_o$ it is $q={1\over2}(1-e^{-\pi\sqrt\varepsilon})<1$

\vskip0.3truecm\noindent
About the proof we use the Cauchy estimates on the holomorphic functions 
(see the Appendix 1). We take 
$\mu\le{1\over2A_o}\min\Bigl\{\sigma,{\xi\over5\pi\sqrt\varepsilon}\Bigr\}$, 
$\sigma\le1, \sqrt\varepsilon\le\rho$, 
$\sup_{\vert J\vert\le\sigma^2}\vert F_J\vert\le{1\over2}$, 
$A_o\ge\max\Bigr\{1,{2\tilde F\over\pi}\Bigr\},$ 
$e^{\pi\sqrt\varepsilon}\le a\le e^{{3\over2}\pi\sqrt\varepsilon}$
$$
\eqalign{
&\max_{j=1,2}\max_{s=0,1}
\{\Vert (x^2+y^2)^{-s/2}f_j\Vert_{\sigma,\xi,\rho}=
\sup_{\Delta_{x,y,I,\varphi}^{\sigma,\xi,\rho}}
\vert (x^2+y^2)^{-s/2}f_j(x,y,I,\varphi)\vert\}=F\cr
&\max_{s=0,1,2}\{\Vert (x^2+y^2)^{-s/2}f_4\Vert_{\sigma,\xi,\rho}=
\sup_{\Delta_{x,y,I,\varphi}^{\sigma,\xi,\rho}}
\vert (x^2+y^2)^{-s/2}f_4(x,y,I,\varphi)\vert\}=
F_1\cr
&\max_{s=0,1,2}\{\Vert (x^2+y^2)^{-s/2}\rho f_3\Vert_{\sigma,\xi,\rho}=
\sup_{\Delta_{x,y,I,\varphi}^{\sigma,\xi,\rho}}
\vert (x^2+y^2)^{-s/2}\rho f_3(x,y,I,\varphi)\vert\}=
F_2\cr
}
$$
and $\tilde F=\max\{F, F_1, F_2\};$ then obtain the estimates
$$
\eqalign{
\Vert M_1(A,{\Cal I})-M_1(A^\prime,{\Cal I}^\prime)\Vert_{\sigma^\prime,\xi^\prime}\le
&\Vert A-A^\prime\Vert_{\sigma^\prime,\xi^\prime}(e^{-2\pi\sqrt\varepsilon}
+(1+39\sqrt\varepsilon)^2{\mu\over3\sigma\xi}A_o^2)+\cr
&+\Vert I-I^\prime\Vert_{\sigma^\prime,\xi^\prime}
2(1+\sqrt\varepsilon){\mu\sqrt\varepsilon\over\xi}A_o^2\cr
}
\eqno(2.2)
$$
$$
\eqalign{
\Vert M_2(A,{\Cal I})-M_2(A^\prime,{\Cal I}^\prime)\Vert_{\sigma^\prime,\xi^\prime}\le
&\Vert I-I^\prime\Vert_{\sigma^\prime,\xi^\prime}(e^{-\pi\sqrt\varepsilon}+
(1+6\pi\sqrt\varepsilon)^2{\mu\over\sigma\xi}A_o)+\cr
&+\Vert A-A^\prime\Vert_{\sigma^\prime,\xi^\prime}
{20\over3}({4\over3}+{\pi\sqrt\varepsilon\over4})
{\mu\sqrt\varepsilon\over\sigma\xi}A_o^2\cr
}
\eqno(2.3)
$$
Reuniting (2.2) and (2.3) we get (2.1) \vru

\noindent
A theorem analogous to 2.0 could be formulated in order to find 
the {\it unstable manifold} ${\Cal Y}^+_\omega.$ We write only its statement
\vskip0.5truecm
\noindent
{\bf Theorem 2.1}{\sl\  For any $\omega\in{\Bbb R}\backslash\{0\},$ 
$\vert\mu\vert\le\mu_o$  
there exist two unique functions
$(A^u,{\Cal I}^u)\colon[-{\sigma\over2},{\sigma\over2}]\times{\Bbb T}
\to{\Bbb R}\times{\Bbb R}$ 
such that the set 
${\Cal Y}^+_\omega=\{(x,y,I,\varphi)=(x,\mu xA^u(x,\varphi;\omega,\mu,\varepsilon),
\omega+
\mu\sqrt\varepsilon x{\Cal I}^u(x,\varphi;\omega,\mu,\varepsilon)
,\varphi)\vert x\in[-{\sigma\over2},{\sigma\over2}],\ \varphi\in{\Bbb T}\}$ 
is invariant respect to the Poincar\'e map (1.1). The functions $A^u$ and 
${\Cal I}^u$ are analytic in $(x,\varphi),$ in $\omega\in{\Bbb R}\backslash\{0\}$ and in 
$\mu$.
} 
\vskip0.5truecm
\noindent
{\bf \mathhexbox2783.Linearizations}\qquad 
In this section we write the linearizations of the motion. 
Roughly speaking we find new variables and 
functions, 
say $p$ and $\Gamma(p),$ such that if we indicate the Poincar\'e map 
with $x^\prime=F(x)$ and with $x=\Gamma(p)$ the change of coordinates, we then 
have 
$p^\prime=\Gamma^{-1}\circ F\circ \Gamma(p)=\alpha p $ and $\alpha$ a 
constant. Unfortunately using the contaction 
arguments we can linearize all the variables only \lq\lq on the 
whiskers" while near them we should use the KAM theorem. In [Ma] and [Cr] 
the authors prove that this is enough for an extension 
of the $\lambda-$lemma to this type of problem.
\vskip0.3truecm
\noindent
We make the following, regular, change of variables 

\noindent
$(\xi,\eta,\bar I,\bar\varphi)=U(x,y,I,\varphi)\colon
\Delta_{x,y,I,\varphi}^{\sigma_o,\xi/4}\to
\Delta_{\xi,\eta,\bar I,\bar\varphi}^{\sigma_1,\xi/4}$ where 

\noindent
$U(x,y,I,\varphi)=(x-\mu y A^s(y,\varphi;\omega,\mu), y-\mu x A^u(x,\varphi;
\omega,\mu),I,\varphi)$  
$(\sigma_o\doteq{\sigma\over2}-{\sigma\over2}\mu_oA_o$ 
$(\sigma_1={\sigma\over2}-{3\over2}\sigma\mu_oA_o$ and being $\mu_o A_o$ 
very small the map $U$  
is onto and invertible obtaining
$$
x=X(\xi,\eta,\bar\varphi;\omega,\mu,\varepsilon)=\xi+O(\mu),\quad 
y=Y(\xi,\eta,\bar\varphi;\omega,\mu,\varepsilon)=\eta+O(\mu),\quad 
I=\bar I,\quad \varphi=\bar\varphi.
$$ 
We have of course \footnote{In almost all the function we 
meet or define there is a parametric $\omega,$ $\mu$ and 
$\varepsilon$ dependences 
but sometimes we do not write them for brevity.
}
$X(0,0,\varphi)=Y(0,0,\varphi)\equiv0$ 

\noindent
$X(0,\eta,\varphi)=
\mu Y(0,\eta,\varphi)A^s\bigl(
Y(0,\eta,\varphi),\bar\varphi\bigl)$ $Y(\xi,0,\varphi)=\mu X(\xi,0,\varphi)
A^u\bigl(
X(\xi,0,\varphi),\bar\varphi\bigl)$

\noindent  
The Poincar\'e map relative to 
the new variables $(\xi,\eta,\bar I,\bar\varphi),$ is 
$$
\eqalign{
&\xi^\prime=\xi a(XY)+\mu g(X(\xi,\eta,\varphi),Y(\xi,\eta,\varphi),I,\varphi)\cr
&\eta^\prime={\eta\over a(XY)}+\mu h(X(\xi,\eta,\varphi),
Y(\xi,\eta,\varphi),I,\varphi)\cr
&\varphi^\prime=\varphi+2\pi I+\mu \varepsilon f_3
(X(\xi,\eta,\varphi),Y(\xi,\eta,\varphi),I,\varphi)\cr
&I^\prime=I-\mu\varepsilon f_4(X(\xi,\eta,\varphi),Y(\xi,\eta,\varphi),I,\varphi)\cr
}
\eqno(3.0)
$$
where
$$
\eqalign{
&g(X(\xi,\eta,\varphi),Y(\xi,\eta,\varphi),I,\varphi)=a(XY)YA^s\bigl(Y,\varphi)-
\sqrt\varepsilon f_2(X,Y,I,\varphi)-
{Y\over a(XY)}A^s({Y\over a}+\cr 
&+\sqrt\varepsilon\mu f_1(X,Y,I,\varphi),\varphi+2\pi I+
\mu\varepsilon f_3(X,Y,I,\varphi)\bigl)-\sqrt\varepsilon
\mu f_1(X,Y,I,\varphi)
A^s\bigl({Y\over a}+\cr 
&\qquad\qquad\qquad +\sqrt\varepsilon\mu f_1(X,Y,I,\varphi),\varphi+2\pi I+\mu
\varepsilon f_3(X,Y,I,\varphi)\bigl),\cr
}
$$
$$
\eqalign{
&h(X(\xi,\eta,\varphi),Y(\xi,\eta,\varphi),I,\varphi)={X\over a(XY)}A^u(X,\varphi)+ 
\sqrt\varepsilon f_1(X,Y,I,\varphi)-Xa(XY)\cdot\cr
&\cdot A^u\bigl(a(XY)X-\mu\sqrt\varepsilon f_2(X,Y,I,\varphi),\varphi+2\pi I+
\mu \varepsilon f_3
(X,Y,I,\varphi)\bigl)-
\mu\sqrt\varepsilon f_2(X,Y,I,\varphi)\cdot\cr 
&\cdot A^u\bigl(a(XY)X-\mu\sqrt\varepsilon 
f_2(X,Y,I,\varphi),\varphi+2\pi I+\mu\varepsilon f_3(X,Y,I,\varphi)\bigl).\cr
}
$$
The theorems 2.0 and 2.1 of course imply that 
$$
\eqalign{
&g(X(0,\eta,\varphi)
,Y(0,\eta,\varphi),\omega+\mu\sqrt\varepsilon Y(0,\eta,\varphi)
{\Cal I}^s(Y(0,\eta,\varphi),\varphi)\equiv0\cr
&h(X(\xi,0,\varphi),Y(\xi,0,\varphi),\omega+\mu\sqrt\varepsilon X(\xi,0,\varphi)
{\Cal I}^u(X(\xi,0,\varphi),\varphi)\equiv0.\cr
}
$$
\noindent
Now let's define 
$$
\eqalign{
&a(XY)\equiv b(\xi,\eta,\varphi),\quad 
b(\xi,\eta,\varphi)\vert_{\mu=0}=a(\xi\eta)=e^{2\pi\sqrt\varepsilon(1+F_J(J))},
\quad b(0,0,\varphi)\equiv 
e^{2\pi\sqrt\varepsilon}\cr
&\omega+\mu \sqrt\varepsilon Y(0,\eta,\varphi){\Cal I}^s(Y(0,\eta,\varphi),\varphi)\doteq
\omega+\mu\sqrt\varepsilon{\Cal I}^s_\omega(\eta,\varphi)\cr
&\omega+\mu\sqrt\varepsilon X(\xi,0,\varphi){\Cal I}^u(X(\xi,0,\varphi),\varphi)\doteq
\omega+\mu\sqrt\varepsilon{\Cal I}^u_\omega(\xi,\varphi)\cr
}
\eqno(3.1)
$$
and write the Poincar\'e map restricted respectively on the {\it stable manifold} and {\it unstable manifold} 
(which are invariant sets  
of course)
$$
\left\{
\eqalign{
&\eta^\prime={\eta\over b(\eta,\varphi)}+\mu h(\eta,\varphi)\qquad\qquad\qquad
\xi=0,\qquad I=\omega+\mu\sqrt\varepsilon{\Cal I}^s_\omega(\eta,\varphi)
\cr
&\varphi^\prime=\varphi+2\pi\omega+2\pi\mu\sqrt\varepsilon{\Cal I}^s_\omega(\eta,\varphi)+
\mu\varepsilon 
F_3(\eta,\varphi)\cr
}\right.
\eqno(3.2)
$$
$$
\left\{
\eqalign{
&\xi^\prime=\xi b(\xi,\varphi)+\mu g(\xi,\varphi)\qquad\qquad
\eta=0,\qquad I=\omega+\mu\sqrt\varepsilon{\Cal I}^u_\omega(\xi,\varphi)
\cr
&\varphi^\prime=\varphi+2\pi\omega+2\pi\mu\sqrt\varepsilon{\Cal I}^u_\omega(\xi,\varphi)+\mu
\varepsilon F_3^\prime(\xi,\varphi)\cr
}\right.
\eqno(3.3)
$$
where
$$
\eqalign{
&h(\mu Y(0,\eta,\varphi)
A^s(Y(0,\eta,\varphi),\varphi),Y(0,\eta,\varphi),\omega+\mu
\sqrt\varepsilon{\Cal I}^s_\omega(\eta,\varphi),\varphi)
\doteq h(\eta,\varphi)\cr
&g(X(\xi,0,\varphi),\mu X(\xi,0,\varphi)
A^u(X(\xi,0,\varphi),\varphi),\omega+\mu\sqrt\varepsilon
{\Cal I}^u_\omega(\xi,\varphi),\varphi)\doteq g(\xi,\varphi)
\cr
&f_3(\mu Y(0,\eta,\varphi)
A^s(Y(0,\eta,\varphi),\varphi),Y(0,\eta,\varphi),\omega+\mu\sqrt\varepsilon
{\Cal I}^s_\omega(\eta,\varphi),\varphi,\mu)\doteq 
F_3(\eta,\varphi)\cr
&f_4(X(\xi,0,\varphi),\mu X(\xi,0,\varphi)
A^u(X(\xi,0,\varphi),\varphi),\omega+\mu\sqrt\varepsilon
{\Cal I}^u_\omega(\xi,\varphi),\varphi,\mu)\doteq 
F_3^\prime(\xi,\varphi)\cr
&a(\mu Y^2(0,\eta,\varphi)A^s(Y(0,\eta,\varphi),\varphi))=
b(\eta,\varphi), \qquad{\Cal I}^s_\omega(0,\varphi)\equiv0\cr
&a(\mu X^2(\xi,0,\varphi)A^u(X(\xi,0,\varphi),\varphi))=b(\xi,\varphi),\qquad
{\Cal I}^u_\omega(0,\varphi)\equiv0
\cr
}
\eqno(3.4)
$$
Actually we have $\lim_{\eta\to0}\eta^{-2}{\Cal I}^s_\omega(\eta,\varphi)\equiv0$ and 
$\lim_{\xi\to0}\xi^{-2}{\Cal I}^u_\omega(\xi,\varphi)\equiv0.$ 
In fact we observe that $Y(0,\eta,\varphi)\to0$ as $\eta\to0$ and 
$X(\xi,0,\varphi)\to0$ as $\xi\to0$ and being $X,Y$ analytic they make finite 
the following limit 
$\lim_{\eta\to0}\eta^{-1}Y(0,\eta,\varphi)$ as well as 
$\lim_{\xi\to0}\xi^{-1}X(\xi,0,\varphi).$ So we are left 
with the relations $\lim_{\eta\to0}\eta^{-1}{\Cal I}^s(Y(0,\eta,\varphi)\varphi)\equiv0$ 
and $\lim_{\xi\to0}\xi^{-1}{\Cal I}^u(X(\xi,0,\varphi)\varphi)\equiv0$. 
Now we set $y=0$ in the second of (2.0) and obtain ${\Cal I}(\varphi)=({1\over a}+
\mu\sqrt\varepsilon g(\omega,\varphi)){\Cal I}(\varphi+2\pi\omega)$ (where 
$g(\omega,\varphi)=
\lim_{y\to0}{1\over y}f_1(\mu yA,y,\omega+\mu y{\Cal I},\varphi)$ while 
$\lim_{y\to0}{1\over y}f_4(\mu yA,y,\omega+\mu y{\Cal I},\varphi)\equiv0$. 
Using again the contraction argument we find the solution of the last  
functional equation. Being ${\Cal I}\equiv0$ a solution, 
by uniqueness , we can say that the only solution is the identically zero one. 

Now (3.2)-(3.3) and the Theorems 2.0-2.1 respectively 
give the next two theorems about the 
linearization of the motion when we are set \lq\lq on" the whiskers.
\vskip0.5truecm
\noindent
{\bf Theorem 3.0}\ {\sl\ For any $\omega\in{\Bbb R}\backslash\{0\}$ and for any 
$\vert\mu\vert\le\mu_1$  there exist two analytic 
functions $(R^s,\Phi^s)\colon [-\sigma_2,\sigma_2]\times{\Bbb T}\to{\Bbb R}\times
{\Bbb T}$ such that on the stable manifold (see system (3.2)), 
in the variables $(r,\theta)\in{\Bbb R}\times{\Bbb T}$ given by 
$\eta=r+\mu R^s(r,\theta),$ 
$\varphi=\theta+\mu r^2\Phi^s(r,\theta),$ the motion is 
linear i.e. $r^\prime=re^{-2\pi\sqrt\varepsilon},$ 
$\theta^\prime=\theta+2\pi\omega.$
} 
\vskip0.3truecm
\noindent
Remarks\qquad i) The proof implies $R^s(0,\theta)\equiv0$\qquad 
ii) $\sigma_2={\sigma_1\over4},$ 

\noindent
$\mu_1={1\over2} 
(1-e^{-2\pi\sqrt\varepsilon})e^{-3\pi\sqrt\varepsilon}\mu_o
{\sigma^2\xi\over130\pi A_o}$
\vskip0.2truecm
\noindent 
The proof is obtained by means of the contraction method again and more 
details are given in Appendix 2
\vskip0.3truecm
\noindent
Of course we have the unstable counterpart of Theorem 3.0. By the 
implicit function theorem invert the system 
(3.3) 
$$
\xi=\xi^\prime e^{-2\pi\sqrt\varepsilon}+\mu\xi^\prime 
A(\xi^\prime,\varphi^\prime),\qquad\qquad
\varphi=\varphi^\prime-2\pi\omega+\mu B(\xi^\prime,\varphi^\prime)
$$
with suitable functions $A$ and $B.$ Then by the following change of 
variables $\xi^\prime=q+\mu R^u(q,\theta),$ $\varphi^\prime=
\theta+\mu q^2\Phi^u(q,\theta)$ we can linearize the inverse of 
system (3.3).
Of course we have $q\to qe^{-2\pi\sqrt\varepsilon}$ and $\theta\to\theta-
2\pi\omega.$ About the domains we loose something because of 
the inversion of the system and the Cauchy estimates of the derivatives 
needed in the contraction theorem; 
\vskip1.0truecm
\noindent
{\bf\mathhexbox2784.About the obstruction mechanism}\qquad 
Here we want just to show how the proof of the obstruction mechanism 
of [Ma] and [Cr] can be adapted to our construction. Before we write a 
very simple argument which implies that the we can deal with the 
obstruction mechanism for the 
Poincar\'e map (${\Cal T}_\omega$) instead of the flow (${\Bbb T}_\omega$).
\vskip0.2truecm
\noindent
{\sl Suppose to have proved the obstruction for 
${\Cal T}_\omega(\theta^o)$ for any $\theta^o.$ 
Then the same is true for ${\Bbb T}_\omega.$}
\vskip0.3truecm
\noindent
It is a simple 
corollary of the result of the Lemma 1 in Appendix 1. Suppose in fact that the 
obstruction mechanism is not verified for ${\Bbb T}_\omega.$ This means that 
there exists a point $p\in Y^+_\omega,$ a manifold ${\Cal N}$ that complements 
$Y^+_\omega$ in $p,$ a neighbourhhood $U\ni p,$ a point $q\in Y^-_\omega,$ 
a neighbourood $V\ni q,$ such that $\phi_t(\bar q)\not\in U$ for any 
$\bar q\in V$ and for any time $t.$ If we take a section at 
$\theta=\theta^o$ mod.$2\pi$ of the previous quantities we are considering the 
projection of the evolution on the surface-map at $\theta^o$ 
i.e. a Poincar\'e map. But a fortiori the projected torus 
${\Cal T}_\omega(\theta^o)$ 
will not verify the obstruction mechanism falling in contradiction.\vru
\vskip0.3truecm
\noindent

Then we change the variables $\xi\to\xi,$ $\eta\to\eta,$ 
$\varphi\to\varphi,$ $I\to z=I-\omega-\mu\sqrt\varepsilon
Y(0,\eta,\varphi){\Cal I}^s(Y(0,\eta,\varphi),\varphi)-
\mu\sqrt\varepsilon X(\xi,0,\varphi){\Cal I}^u(X(\xi,0,\varphi),\varphi)=
I-\omega-Z$ 
which permits 
to write (3.0) as  
$$
\eqalign{
&\xi^\prime=\xi e^{2\pi\sqrt\varepsilon}+
\xi e^{2\pi\sqrt\varepsilon}(e^{2\pi\sqrt\varepsilon F_J(\xi\eta)}-1)+
\mu(\xi g_1+g)\cr
&\eta^\prime=\eta e^{-2\pi\sqrt\varepsilon} +
\eta e^{-2\pi\sqrt\varepsilon}(e^{-2\pi\sqrt\varepsilon F_J(\xi\eta)}-1)+
\mu(\eta h_1+h)
\cr
&\varphi^\prime=\varphi+2\pi z+2\pi\omega+\mu \varepsilon f_3
(X(\xi,\eta,\varphi),Y(\xi,\eta,\varphi),z+\omega+Z(\xi,\eta,\varphi),\varphi)\cr
&z^\prime=z-\mu\varepsilon 
\tilde f_4(X(\xi,\eta,\varphi),Y(\xi,\eta,\varphi),z+\omega+
Z(\xi,\eta,\varphi),\varphi)\cr
}
\eqno(4.0)
$$
where $g=g(X(\xi,\eta,\varphi),Y(\xi,\eta,\varphi),z+\omega+Z(\xi,\eta,\varphi)
,\varphi)=O(\vert\xi\vert+\vert\eta\vert),$ 

\noindent
$h=h(X(\xi,\eta,\varphi),Y(\xi,\eta,\varphi),z+\omega+Z(\xi,\eta,\varphi)
,\varphi)=O(\vert\xi\vert+\vert\eta\vert),$ 

\noindent
$h_1={1\over\mu}\int_0^1d\lambda {d\over d\lambda}
{1\over a(XY;\lambda\mu)},$ 
$g_1={1\over\mu}\int_0^1d\lambda {d\over d\lambda}
a(XY;\lambda\mu),$ see (3.1)   
$$
\eqalign{
&\tilde f_4=\bigl\{f_4(X,Y,z+\omega+Z,\varphi)-
f_4(X,Y,z+\omega+Z,\varphi)\vert_{\xi=0}-
f_4(X,Y,z+\omega+\cr
&+Z,\varphi)\vert_{\eta=0}\bigr\}+
\bigl\{{\Cal I}^s_\omega(\eta^\prime,\varphi^\prime)-
{\Cal I}^s_\omega(\eta^\prime,\varphi^\prime)\vert_{\xi=0}\bigr\}+
\bigl\{{\Cal I}^u_\omega(\xi^\prime,\varphi^\prime)-
{\Cal I}^u_\omega(\xi^\prime,\varphi^\prime)\vert_{\eta=0}\bigr\}.\cr
}
$$
Of course $\tilde f_4(X(0,\eta,\varphi),Y(0,\eta,\varphi),\omega+Z,\varphi)\equiv
\tilde f_4(X(\xi,0,\varphi),Y(\xi,0,\varphi),\omega+Z,\varphi)\equiv0$ as now 
${\Cal Y}^+_\omega=\{(\xi,\eta,z,\varphi)\vert \eta=0, z=0\}$ while 
${\Cal Y}^-_\omega=\{(\xi,\eta,z,\varphi)\vert \xi=0, z=0\}.$ 
Now all the conditions of the Lemme of pag.11 of [Ma]
 are verified \footnote
{The reader could note that $\xi g_1+g$ and $\eta h_1+h$ are of order 
one in $\xi, \eta, z$ and $f_3$ 
and $\tilde f_4$ are of order 2 while all the analogous terms in the 
Lemme of pag.11 of [Ma] have order 2. We think that this causes no problem in 
the proof because the order two is necessary to write the map as a sum of 
two addends, the second one much smaller that the first. In our case, for 
what concerns the first two equations the presence of $\mu$ helps us.
}
and we can apply to 
(4.0) the Condition (OS) of pag.7  which is the obstruction mechanism
when $\omega\in{\Bbb R}\backslash{\Bbb Q}$
\vskip0.3truecm\noindent
{\bf Condition (OS)} {\sl For any submanifold $\Delta$ that intersects 
transversaly ${\Cal Y}^-_\omega,$ for any point $b\in{\Cal Y}^+_\omega,$ for any 
neighborhood $B$ of $b$  there exists an integer $k$ such that 
$\hat\phi_{2\pi k}(\Delta)\cap B\ne\emptyset$} 

\vskip0.5truecm\noindent 
{\bf Appendix 1}\qquad The transformation $U$  
consists of a composition of three transformations (the last of them is 
actually the composition of infinite canonical transformations). 
The first one (canonical) 
is defined in the set $\{(I_1,\varphi_1)\vert\ \vert\varphi_1\vert<\pi,\ 
\left\vert I_1-\vert\beta(t)\vert\right\vert<{1\over2}\sqrt\varepsilon\},$ 
(see (0.1)) and it is given by 

\centerline{
$x={1\over\sqrt2}(\varepsilon^{-{1\over4}}I_1-\varepsilon^{{1\over4}}
\varphi_1)\qquad 
y={1\over\sqrt2}(\varepsilon^{-{1\over4}}I_1+\varepsilon^{{1\over4}}
\varphi_1)$}
\noindent  
so $(x,y)$ are defined in a neighborhood of radius $O(\varepsilon^{1/4})$ of the 
origin. 
In the second one we rescale by a constant factor 
$x^\prime=x\varepsilon^{-{1\over4}},$ $y^\prime=y\varepsilon^{-{1\over4}}$ 
The equations (0.0) become 
$$
\eqalign{
&\dot \theta=1,\qquad\quad\dot\varphi_2=I_2,\qquad\quad
\dot I_2=-\varepsilon\mu\cos\varphi_2(\cos\varphi_1
(x^\prime,y^\prime)-1)\cr
&\dot x^\prime=\sqrt\varepsilon 
x^\prime+{\sqrt\varepsilon\over\sqrt2}{\partial f\over\partial\varphi_1}
(\varphi_1(x^\prime,y^\prime))-{\sqrt\varepsilon\over\sqrt2}\mu
\sin\varphi_1(x^\prime,y^\prime)(\sin\varphi_2+\cos \theta)\cr
&\dot y^\prime=-\sqrt\varepsilon 
y^\prime+{\sqrt\varepsilon\over\sqrt2}{\partial f\over\partial\varphi_1}
(\varphi_1(x^\prime,y^\prime))-{\sqrt\varepsilon\over\sqrt2}\mu
\sin\varphi_1(x^\prime,y^\prime)(\sin\varphi_2+\cos \theta)\cr
\cr
}
$$
where $f(x^\prime,y^\prime)=\cos\varphi_1(x^\prime,y^\prime)-1+
{1\over2}\varphi_1^2(x^\prime,y^\prime).$ The previous system is non-canonical 
for two reasons: the first one (less important) is the presence of 
$\dot\theta$ while the second and more important one is due to the rescaling 
only on the $(x,y)$ variables. Nevertheless the following equations 
are canonical with hamiltonian 
$\tilde H=\sqrt\varepsilon x^\prime y^\prime+
\sqrt\varepsilon f(\varphi_1(x^\prime,y^\prime))$
$$
\eqalign{
&\dot x^\prime=\sqrt\varepsilon 
x^\prime+{\sqrt\varepsilon\over\sqrt2}{\partial f\over\partial\varphi_1}
(\varphi_1(x^\prime,y^\prime))
\qquad\qquad\dot y^\prime=-\sqrt\varepsilon 
y^\prime+{\sqrt\varepsilon\over\sqrt2}{\partial f\over\partial\varphi_1}
(\varphi_1(x^\prime,y^\prime))
}
$$
At this point we do a third transformation only on the  
$(x^\prime,y^\prime)$ introducing the normal coordinates 
$(q,p)$ in such a way that 
$\tilde H(x^\prime(q,p),y^\prime
(q,p))=K(qp)=qp+F(qp)$ and 
$F(qp)=O((qp)^2)$
(the dependence in only through the product $qp).$ 
The variables $(q,p)$ are analytic 
functions of $(x^\prime,y^\prime)$ and the series (that 
we do not write) begin with a linear term. This suitable linear dependence 
and the linearity of each other transformation we 
have already introduced imply that $(q,p)$ are defined in 
a neighborhood of radius $O(1)$ around $(0,0)\in{\Bbb R}^2.$ 
The reference [CG] (lemma 0 
of \mathhexbox2785) is a good one where to read (and possibly learn) the 
construction of this normal form.

\noindent
$\tilde\varphi_1(q,p)=
\varphi_1(x^\prime(q,p),y^\prime(q,p)),
\quad\varphi_1(x^\prime,y^\prime)={y^\prime-x^\prime\over\sqrt2}=
\varepsilon^{-1\backslash4}{y-x\over\sqrt2},$ 
$J=qp,\ F_J={\partial F\over
\partial J}$
\vskip0.5truecm
\noindent
{\bf Lemma 1}
Let us write the system of differential equations (0.0) as 
$$
\left\{
\eqalign{
&\dot x=f(x,\theta),
\qquad\qquad\dot\theta=1\cr
&x(0)=p,\ \ \theta(0)=\theta^o\qquad\qquad\qquad\qquad\qquad
f(x,\theta+2\pi)=f(x,\theta)
\cr
}\right.
$$ 
where $x\in{\Cal M},$ $\theta\in{\Bbb T}$ and let be 
$\phi_t\colon\tilde{\Cal M}\to\tilde{\Cal M}$ 
($(p,\theta^o)\to\phi_t(p,\theta^o)$) 
the flow of the 
system. Let $\hat\phi_{2k\pi}(p)\colon {\Cal M}\to {\Cal M},$ 
$k\in{\Bbb Z}$ be the Poincar\'e map at 
time $\theta^o$ and suppose that $M^s_{\theta^o}$ is its stable manifold. 
Of course the Poincar\'e map is defined taking the projection on
${\Cal M}$ of the following equations  
\footnote{The Poincar\'e map written down is equal to (1.2) with $s+\theta^o$ 
in place of $s$ inside the integrals} 
$$
\left\{
\eqalign{
&x(2k\pi;p,\theta^o)=p+
\int_0^{2k\pi}d\tau f\bigl(x(\tau;p,\theta^o),\theta^o+\tau\bigr)\cr
&\theta(2k\pi;\theta^o)-\theta(0;\theta^o)=2\pi=0\ (mod.2\pi).\cr
}
\right.
\eqno(4.1)
$$ 
We show that $M^s\doteq\cup_{0\le \theta^o\le2\pi}M^s_{\theta^o}$ 
is the stable manifold of the flow $\phi_t.$
\vskip0.3truecm
\noindent
$\Rightarrow:$ $(p,\theta^o)\in M^s$ and then 
$\lim_{t\to\infty}\phi_t(p,\theta^o)\in{\Bbb T}_\omega$ is the hypotheses. 
Being trivially 
${\Bbb T}_\omega=\cup_{0\le \theta^o\le2\pi}{\Cal T}_\omega(\theta^o)$ we get 
$\lim_{t\to\infty}\phi_t(p,\theta^o)\vert_{{\Cal M}}
\in{\Cal T}_\omega(\theta^o)$ so 
that $p\in M^s_{\theta^o}.$
\vskip0.3truecm
\noindent
$\Leftarrow:$ Now suppose that $p\in M^s_{\theta^o}$
and consider  
$$
\eqalign{
&\lim_{k\to\infty}p+\int_0^{\theta^o+2\pi k}d\tau
f(x(\tau;p,0),\tau)=\cr
&=\lim_{k\to\infty}p+
\int_0^{2\pi k}d\tau f(x(\tau;p,0),\tau)+
\int_0^{\theta^o}d\tau f(x(\tau+2\pi k;p,0),\tau)=\cr
&=\hat\phi_{\theta^o}\circ\lim_{k\to\infty}\phi_{2\pi k}(p,0).\cr
}
$$
Being $\lim_{k\to\infty}\hat\phi_{2\pi k}(p,0)\in{\Cal T}_\omega(0)$ by 
hypotheses, we have 
$\lim_{k\to\infty}\hat\phi_{\theta^o}\circ\phi_{2\pi k}(p,0)\in{\Cal
T}_\omega
(\theta^o)\subset{\Bbb T}_\omega$ and then $(p,\theta^o)\in M^s.$\vru 
\vskip0.5truecm
\noindent
Here we write the basic (elementary) tool (a \lq\lq Cauchy estimate") which 
is fundamental if we would write down explicitely the calculations leading to 
(2.1) and the other contraction estimates.

\noindent
{\bf Lemma 2}\quad{\sl Let $g$ be an analytic function from ${\Cal D}
\subset 
{\Bbb C}^N\to{\Bbb C}$ ($\partial{\Cal D}$ smooth). 
Then for any subdomain ${\Cal D}'\subset {\Cal D}$ 
with $\delta\equiv$ dist$({\Cal D}',\partial {\Cal D})>0$ 
and any $n\in {\Bbb N}^N$, one has:
$$
\Vert\partial_z^n g\Vert_{{\Cal D}'}\equiv\sup_{{\Cal D}'}
\vert{\partial^{\vert n\vert} g\over 
\partial_{z_1}^{n_1}\cdot\cdot\cdot\partial_{z_N}^{n_N}}\vert\le 
\vert n\vert!\ \delta^{-\vert n\vert} \ 
\Vert g\Vert_{\Cal D}
$$

\noindent
The proof is a straightforward exercise based upon Cauchy's integral 
formula using the contour $\vert z^\prime_h-z_h\vert=\delta$, 
$h=1,...,N$ ($z\in {\Cal D}'$ fixed 
and $z^\prime\in {\Cal D}$ variable of integration)}.
\vskip0.5truecm
\noindent
{\bf Appendix 2}\qquad Sketch of the proof of theorem 3.0. 
We rewrite the system (3.2) as 
$$
\left\{
\eqalign{
&\eta^\prime=\eta e^{-2\pi\sqrt\varepsilon}
+\mu\eta\bigl(h_1+h_2\bigr)(\eta,\varphi)\cr
&\varphi^\prime=\varphi+2\pi\omega+2\pi\mu\sqrt\varepsilon{\Cal I}^s_\omega(\eta,\varphi)+
\mu\varepsilon F_3(\eta,\varphi)\cr
}\right.
\eqno(4.2)
$$
where 
$$
\eqalign{
&h_1={1\over\eta\mu}\int_0^1d\lambda {d\over d\lambda}
{1\over b(\eta,\varphi;\lambda\mu)}={1\over\eta}
\int_0^1-d\lambda {1\over b^2}{d\over d\zeta}
a[\zeta Y^2(\eta,\varphi;\zeta)A^s\Bigl(
Y(\eta,\varphi;\zeta),\varphi\Bigr)]\vert_{\zeta=\lambda\mu}\cr
}
$$
$\vert\eta\vert\le\sigma_1,$ $\vert Im\varphi\vert\le{\xi\over4};$ for the 
definition of $a[\cdot]$ see (3.4). Now 
$$
{d\over d\zeta}a[\zeta Y^2(\eta,\varphi;\zeta)A^s\bigl(
Y(\eta,\varphi;\zeta),\varphi\bigr)]=
\Bigl(Y^2A^s+2\zeta Y A^s{\partial Y\over\partial\zeta}+
\zeta Y^2{\partial Y\over\partial\zeta}
{\partial A^s\over\partial Y}\Bigr){da\over dJ}
$$
and in the domain 
$\vert\eta\vert\le\sigma_1,$ $\vert\mu\vert\le{1\over2}\mu_o,$ 
$\vert Im\varphi\vert\le{\xi\over4}$ we have 
$\Vert{1\over\eta}{d\over d\zeta}a[\cdot]\Vert
\le{A_o e^{3\pi\sqrt\varepsilon}\over\sigma\mu_o}$ and then 
$\vert-
{1\over\eta\mu}
\int_0^1d\lambda b^{-2}(\eta,\varphi;\lambda\mu){d\over d\lambda}
b(\eta,\varphi;\lambda\mu)\vert=\vert
h_1(\varphi;\lambda\mu)\vert\le{4A_o\over\mu_o}
e^{\pi\sqrt\varepsilon}.$ 

For what concerns $h_2$ we observe that $Y(\eta,\varphi)\to0$ as $\eta\to0$ 
and then $h(\eta,\varphi)\to0$ as $\eta\to0$ (see (3.4)). Using the 
maximum principle for holomorphic functions in the form of Schwarz's lemma 
we bound $\Vert{h\over\eta}\Vert_{\sigma_1,{\xi\over4}}\le
{\pi A_o\over\sigma_1}\sqrt\varepsilon e^{3\pi\sqrt\varepsilon}.$

Now substitute 
$\eta=r+\mu R(r,\theta)$ and $\varphi=\theta+\mu r^2\Phi(r,\theta)$ into 
the system (4.2) and write a fixed point equation 
$$
\eqalign{
&R(re^{-2\pi\sqrt\varepsilon},\theta+2\pi\omega)=e^{-2\pi\sqrt\varepsilon}
R(r,\theta)+\bigl(r+\mu R(r,\theta)\bigr)\cdot\cr
&\cdot\bigl(h_1+h_2\bigr)
(r+\mu R(r,\theta),\theta+\mu r^2\Phi(r,\theta))\doteq V_1(R,\Phi)\cr
}
$$
$$
\eqalign{
&\Phi(r,\theta)=e^{-4\pi\sqrt\varepsilon}
\Phi(re^{-2\pi\sqrt\varepsilon},\theta+2\pi\omega)-
2\pi{\sqrt\varepsilon\over r^2}{\Cal I}^s_\omega(\eta(r,\theta),\varphi(r\theta))-\cr
&-{\varepsilon\over r^2}F_3(\eta(r,\theta),\varphi(r\theta))
\doteq V_2(R,\Phi).\cr
}
$$
As in the previous cases 
the Cauchy estimates enable us to say that in the domain 
$\Delta^{\sigma_2,{\xi\over16}}_{\eta,\theta}$ and for 
$\vert\mu\vert\le\mu_1$ there exists a unique 
solution (fixed point) of the previous equations that we call 
$(R^s,\Phi^s).$ About their norm we have taken 
$\Vert R^s\Vert_{\sigma_2,{\xi\over16}}\ge{20A_o\sigma_1\over\mu_o\sigma_o}
e^{3\pi\sqrt\varepsilon}(1-e^{-2\pi\sqrt\varepsilon})^{-1},$ 
$\Vert\Phi^s\Vert_{\sigma_2,{\xi\over16}}\ge
{7A_o\sigma_o\sqrt\varepsilon\over\sigma_2^2}
(1-e^{-4\pi\sqrt\varepsilon})^{-1}.$

\noindent
It is important to note that $R^s(0,\varphi)\equiv0$ and being $R^s$ an 
analytic function it can be written as $r\hat R^s.$ Moreover for writing 
${{\Cal I}^s_\omega\over r^2}$ meaningful we need to keep in mind the content 
of what written immediately before the Theorem 3.0.
\vskip0.5truecm
\centerline{\bf References}
\vskip0.2truecm
\noindent
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\noindent
[Be] U.BESSI {\sl An Approach to Arnold Diffusion through the 
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\noindent
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\noindent
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\noindent
and 

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\noindent
[Fe3] N.FENICHEL {\sl Asymptotic stability with rate conditions, II}, 
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\noindent
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[Ni] L.NIEDERMAN {\sl Dynamic Around a Chain of Simple Resonant Tori in 
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\noindent
Math\'ematiques B\^ atiment 425 91405 Orsay France 

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[Ru] D.RUELLE, \lq\lq Elements of Differentiable Dynamics 
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mail to the archive) 

\noindent
[RW2] M.RUDNEV, S.WIGGINS {\sl KAM theory multiplicity one resonant surfaces 
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[Xia3] Z.XIA {\sl Arnol'd Diffusion and Oscillatory Solutions in the 
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[Ze] E.ZEHNDER {\sl Stability and Instability in Celestial Mechanics}, 
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\vskip2.0truecm
\noindent
E-mail: perfetti{\rm \char 64}mat.utovrm.it
\end