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%% (Proceeding for the NATO ASI, S'Agaro, June 1995
BODY
\documentstyle[nato]{crckapb}
\def\btt#1{{\tt$\backslash$#1}}
\begin{opening}
\title{A POSSIBLE MECHANISM FOR THE KAM TORI BREAKDOWN}
\date{today}
\author{Guido Gentile}
\institute{IHES, 35 Route de Chartres, 91440 Bures sur Yvette, France}
\author{Vieri Mastropietro}
\institute{Dipartimento di Matematica, $II^a$ Universit\`a di Roma,
00133 Roma, Italia}

\end{opening}
\runningtitle{TORI BREAKDOWN}
\begin{document}
\begin{abstract}
{\it Inspired by the quantum field theory, a resummation is performed
in the Lindstedt series defining the KAM tori, and a possible mechanism
for universality of the tori breakdown is discussed.
This work is mostly based on a joint paper with G. Gallavotti.}
\end{abstract}
% 
%%%%%GRECO%%%%%%%%%
\let\a=\alpha \let\b=\beta  \let\g=\gamma    \let\d=\delta \let\e=\varepsilon
\let\z=\zeta  \let\h=\eta   \let\th=\vartheta\let\k=\kappa \let\l=\lambda
\let\m=\mu    \let\n=\nu    \let\x=\xi       \let\p=\pi    \let\r=\rho
\let\s=\sigma \let\t=\tau   \let\iu=\upsilon \let\f=\varphi\let\ch=\chi
\let\ps=\psi  \let\o=\omega \let\y=\upsilon
\let\G=\Gamma \let\D=\Delta  \let\Th=\Theta  \let\L=\Lambda\let\X=\Xi
\let\P=\Pi    \let\Si=\Sigma \let\F=\Phi     \let\Ps=\Psi  \let\O=\Omega
\let\U=\Upsilon
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DEFINIZIONI VARIE
\def\V#1{\vec#1}\let\dpr=\partial\let\io=\infty\let\ig=\int
\def\fra#1#2{{#1\over#2}}\def\media#1{\langle{#1}\rangle}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%LATINORUM
\def\etc{\hbox{\it etc}}\def\eg{\hbox{\it e.g.\ }}
\def\ap{\hbox{\it a priori\ }}\def\aps{\hbox{\it a posteriori\ }}
\def\ie{\hbox{\it i.e.\ }}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%DEFINIZIONI LOCALI
\def\AA{{\V A}}\def\aa{{\V\a}}\def\nn{{\V\n}}\def\oo{{\V\o}}
\def\mm{{\V m}}
%\def\nn{{\V\n}}
\def\lis#1{{\overline #1}}
\def\NN{{\cal N}}\def\FF{{\cal F}}\def\Im{{\rm\,Im\,}}\def\Re{{\rm\,Re\,}}
\def\={{ \; \equiv \; }}\def\Dpr{{\V \dpr}\,}
\def\sign{{\rm sign\,}}\def\atan{{\,\rm arctg\,}}
\def\hh{\V h}\def\pps{\V \psi}
\let\0=\noindent\def\*{\vskip0.3truecm}
\let\Eq=\eqno\def\equu(#1){{Eq.$\,$(#1)}}
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\def\\{\noindent}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\\{\bf 1. Introduction.} The parametric equations of KAM tori
for a $\ell$ degrees of freedom quasi integrable system can 
be shown to be one point--Schwinger functions of 
a suitable euclidean quantum field theory (QFT), \cite{G}, \cite{GGM}. 
This shows that it is very natural to use the powerful techniques 
developed in QFT (see for instance \cite{BG}) to study the convergence
of the perturbative series for the KAM tori ({\it Lindstedt series}), 
and in fact this has been done in recent proofs of the KAM theorem,
(indeed in the first of such proofs, due to Eliasson, \cite{E}, 
this is only implicit but it appears more and more clear in following
simplified proofs, \cite{G}, \cite{GM3}, \cite{GM1}, \cite{GM2}).
In QFT it is often necessary to perform a resummation in the
perturbative expansion for the Schwinger functions, so that the original
series in the perturbative parameter $\e$ is replaced by a different
power series in suitable functions of $\e$, called {\it form factors}:
inspired by the analogy with QFT, we perform a resummation of the
Lindstedt series, (see \S 3), and obtain a new expression for the
sum of the series, which we call the {\it resummed series}.
By the non resummed series, one estimates the tori
analyticity domain $D$ around $\e=0$, with boundary $\e=\e_D(\theta)$,
$\theta\in [0,2\pi)$, with a disk of radius $R$ not larger than
the largest disk contained in $D$, \ie $R\le\min_{\theta}
|\e_D(\theta)|$.
Hence, if for some system $\e(0),\e(\pi)\gg \min_{\theta} |\e_D(\theta)|$,
then, although the convergence radius of the non resummed series can be
well estimated by $R$, the invariant tori exist for real $\e$ bigger
than $R$, \ie $R$ is a bad estimate of the value $\e_c$ of $\e$ at which
there is the tori breakdown.
We will see that the resummed series, at least in principle,
could allow us to estimate the domain $D$ with a domain less simple
than a disk and this might allow to obtain better estimates of the
value of $\e_c$.
Finally the resummation leads to a conjecture about the mechanism of
the phenomenon of the tori breakdown, (see \S 4). 

\*

\\{\bf 2. Formalism.} We consider for simplicity $\ell$ 
rotators with inertia momenta $J$, angular momenta 
$\vec A=(A_1,\ldots,A_{\ell})\in {\bf R}^{\ell}$, and angular positions 
$\vec a=(\a_1,\ldots,\a_{\ell})\in {\bf T}^{\ell}$, described by the 
Hamiltonian $ {\cal H} = {1\over 2 J} \vec A \cdot \vec A +
\e f(\vec\a)$, where $\vec A\in {\bf R}^{\ell}$, $\vec\a\in {\bf T}^{\ell}$
and $f(\vec\a)$ is analytic and even in its argument.
Similar considerations holds for much more general hamiltonians. 
Let $\oo_0=J^{-1}\vec A_0$ be a rotation vector
verifying for $C_0, \t>0$ the {\sl diophantine property}
$C_0|\oo_0\cdot\vec\nu|>|\vec\nu|^{-\t}$, $\vec 0 \neq
\vec\nu \in {\bf Z}^{\ell} $.
The KAM theorem states the existence of a one parameter family
$\e\to T_\e$ of tori with parametric equations
$ \vec A=\vec A_0+\vec H(\vec\psi)$,
$\vec a=\vec \psi+\vec h(\vec\psi)$, $\vec \psi\in{\bf T}^{\ell}$,
where $\vec H(\vec \psi)$ and $\vec h(\vec \psi)$ are analytic functions
of $\e$, $\psi_j$, $j=1,\ldots,\ell$, divisible by $\e$,
defined for $|\e|$, $|\hbox{Im}\psi_j|$ small enough.
Writing $\vec H(\pps)=\sum_{k=1}^\io\sum_\nn e^{i\nn\cdot\pps}
\e^k\vec H^{(k)}_\nn$, $\vec h(\pps)=\sum_{k=1}^\io \sum_\nn
e^{i\nn\cdot\pps}\e^k\vec h^{(k)}_\nn$, it is easy to define
a graphical representation for $\vec h_{\nn}^{(k)}$
and $\vec H_{\nn}^{(k)}$ in the following way.
Consider $k$ oriented lines, labeled from $1$ to $k$: the final extreme
$v'$ of a line is called the root and the other extreme
$v$ the vertex, and the line is denoted $v'\leftarrow v$.
The lines are arranged on a plane by attaching in all possible ways
the vertices of some lines to the roots of others, to form a
connected tree: each tree, following a therminology mutuated from QFT,
can be called a {\it Feynman graph}.
In this way only one root $r$ remains unmatched and it will be called
the {\it root} of the graph, whose lines will be called {\it branches}
and whose vertices other than the root will be called {\it nodes}.
Between the nodes there is an ordering relation $\le$;
we write $w<v$ is $v$ is closer to the root.
Each node $v$ is given a {\it mode} label $\nn_v$ which is one of the
Fourier modes $\nn$'s such that $f_\nn\neq 0$.
We define the {\it momentum} flowing through the branch going from
$v$ to $v'$ as $\nn(v)=\sum_{w\le v}\nn_w$: the parity properties of
the model impliy $\nn(v)\neq\V0$, for any $v$.
To each tree we associate a {\it value} obtained by assigning to a
branch $v'\leftarrow v$, the {\it line factor}
$\frac{-i\vec\nu_{v'}\cdot iJ^{-1}\vec\nu_{v}}
{(i\oo_0\cdot\vec\nu(v))^2}$
for all the branches distinct from the one containing the root;
to the root branch we associate, instead, the
quantity $\frac{-iJ^{-1}\vec\nu_v}{(i\oo_0\cdot\vec\nu(v))^2}$.
We multiply all the above operators and multiply the resulting factor
times the function $\prod_v f_{\nn_v}$.  
This defines the {\it Feynman rules}: the $\vec h^{(k)}_\nn$ is 
given by $k!^{-1}$ times the sum of all the
values of all the $k$ branches trees with momentum $\nn$
flowing through the branch leading to the root, and 
$\vec H^{(k)}_\nn=(i\oo_0\cdot\nn(v)) \vec h_\nn^{(k)}$.
The scaling properties of the factor ${1\over (i\oo\cdot\nn(v))^2}$
({\it propagator}) suggests decomposing it into components
relative to various {\it scales}. We can write
%
$$ \frac{1}{(i\oo_0\cdot\nn(v))^2}=
\frac{\chi_1(\oo_0\cdot\nn(v))}{(i\oo_0\cdot\nn(v))^2}+\sum_{n=-\io}^0
\frac{\chi(2^{-n}\oo_0\cdot\nn(v))}{(i\oo_0\cdot\nn(v))^2} \; , $$
%
where $\chi_1(x)$ and $\chi(x)$ are the characteristic functions
of the sets $|x|\geq 1$ and, respectively, $1/2\le |x|<1$;
correspondingly we can break each Feynamn graph into a sum of
many terms by developing the sums. This can be simply
represented by assigning to each branch $\l$ an extra label $n_\l$
and multiplying the factor associated to such a line times
$\chi(2^{-n_\l}\oo_0\cdot\nn)$:
the value of $\vec H^{(k)}_\nn,\vec h^{(k)}_\nn$ will be the sum
over all possible new graphs which once deprived of the new scale
labels would become ``old" graphs contributing to
$\vec H^{(k)}_{\vec\nu},\vec h^{(k)}_\nn$  respectively.
The branches of the new graphs are naturally collected into connected
{\it clusters} ``of fixed scale": a cluster of scale $n$
consists in a maximal connected set of branches
with scale label $\ge n$, containing at least one line of scale $n$.
The lines which are not contained in a cluster,
but have an extreme inside the clusters will be called the external
lines of the cluster: if the extreme
inside the cluster is the root, they will be {\it incoming},
while if the extreme is the vertex they will be {\it outgoing}.
Examining the convergence of the perturbation series it becomes clear
that if one considers the sum of the contributions to $\vec H^{(k)}_{\nn}$,
$\vec h^{(k)}_{\nn}$ by all the graphs that
{\it do not contain clusters with
just one incoming and one outgoing branch which, furthermore, have
the same momentum}, then the series so generated converge for
$\e$ small, \cite{E}, \cite{G}.
Therefore the clusters with one incoming and one
outgoing equal momentum branches are called {\it resonances} and the
KAM theory can be interpreted as an analysis of the reason why the
resonances do not destroy the analyticity in $\e$ at $\e$ small, \ie
of the cancellations that make the resonances give a contribution much
smaller than one could fear ($\e^k C^k$ instead of $\e^k k!$).

\*

\\{\bf 3. Lindstedt series resummation.}
If we multiply each graph value by the appropriate power of $\e$
(equal to the number of branches of the graph) we see that the values of
$\vec H$ and $\vec h$ can be computed 
by considering all the graphs {\it without
resonances}  but modifing the line factor of scale $n$ as:
%
$$ \frac{\chi(2^{-n}\oo_0\cdot\vec\nu(v))}
{(i\oo_0\cdot\vec\nu(v))^2} \; (-i\vec\nu_{v'}\cdot iJ^{-1}\vec\nu_v) \to
\frac{ \chi(2^{-n}\oo_0\cdot\vec\nu(v))} {(i\oo_0\cdot\vec\nu(v))^2}
(-i\vec\nu_{v'}\cdot {1\over 1-\s_{n,\e}(\oo_0\cdot\vec\nu(v))}iJ^{-1}
\vec\nu_v) \; , $$
%
where $\sigma_{n,\e}(\oo_0\cdot\nn(v))$ is a suitable matrix
representing the sum of all the possible insertions of resonances
on the line connecting $v'$ with $v$.
The function $\s_{n,\e}(\oo_0\cdot\nn(v)) \equiv\s_{n,\e}(2^{n}x)$
does not vanish only for $x\in (1/2,1]$.
It is possible to prove that (1) the matrix $\s_{n,\e}(2^{n}x)$ is
analytic in $\e$ for $\e$ small, independently on $n$, and
(2) there is a constant ${\cal C}$ such that
$||\s_{n,\e}(2^{n}x)||<{\cal C}|\e|$, \cite{GGM}.
Furthermore the limit $\lim_{n\to-\io} \s_{n,\e}(2^n x)=\s_\e$
exists and is a $x$--independent function of $\e$, analytic for
$\e$ small enough and divisible by $\e$, \cite{GGM}.
The constants $\s_\e$ will be called the {\it form
factors}. It is reasonable to think that the singularities of
$\V H$, $\V h$, depending on the form factors $\sigma_{n,\e}$, are
the same as those of the series $\V H^*$, $\V h^*$
obtained replacing $\sigma_{n,\e}$ by $\sigma_\e$.
Thus the series for $\V H^*$, $\V h^*$ depend only on the variables
$\h=\e(1-\s_{\e})^{-1}$, and the possibility arises that a
singularity is reached at a value $\e^*$ at which $\s_{\e^*}$ is
still finite, while $\h$ crosses the boundary of the analyticity
domain for $\V H^*$, $\V h^*$.

Suppose that the analycity domain in $\e\equiv |\e|e^{i\theta}$ of 
$\vec h$, $\vec H$ is given by the complex domain $D$ whose 
boundary is $\e_D(\theta)$. By the study of the non resummed
Lindstedt series one obtains a circular estimate for the
analicity domain which will be {\it not larger that the biggest circle
contained in $D$}, so that in general it is a bad estimate for the
value of $\e_c$.
 From the resummed series, on the other hand, one finds, if $\|\cdot\|$
is the matrix norm, $\|{\e\over 1-\sigma_\e}\|\le R^*$, if $R^*$ is
the convergence radius in $\h$ of the series for $\V H^*$, $\V h^*$,
and the $\e$'s which verify the above inequality define 
a domain $D'$ different from a disk and possibly with a very
complicated form.
Then the problem is reduced to the study of the form factors $\s_{\e}$
appearing in the resummed series: one can expect that they produce
an estimate for $\e_D(0)$, $\e_D(\p)$ better than $R$,
(see \S 1 for the notation).
For instance, if one could truncate $\s_\e$ at the first non trivial
order (the second one), one would find ${|\e|^2 \over A+B |\e|
\cos(2\theta)}\le R^*$, for some constants $A,B$, so that the
estimated analyticity domain for the tori would be an ellipse.
In general such a truncation is not justified at all,
because, for $\e$ near $\e^*$, $\s_\e$ is not well
approximated by the second order expansion: on the contrary
the full functional dependence on $\e$ would be required.
Nevertheless one can envisage the following scenario:
although the convergence radii for the tori $\vec h^*$, $\vec H^*$,
and for the form factors $\s_\e$ are probably the same,
there is the possibility that $\s_\e$  remains small {\it at real
$\e^*$ on the boundary of $D'$}, so that a second order truncation
for $\s_\e$ represents a good approximation for $\s_\e$.
We have not been able to prove if such a phenomenon
really occurs: perhaps one could proceed as in QFT, 
where one defines a recursive relation ({\it beta function},
see \cite{BG}) for the ``form factors on scale $n$''
$\s_{n,\e}(2^nx)$, so obtaining useful informations about $\s_{\e}$,
and in this way one to could try to really perform the above analysis.
One difficulty in this direction could be that the {\it relevant part}
of the beta function (\ie the part of the beta function which one
expects to describe the important features of the model,
by dimensional arguments) seems to be identically vanishing:
anyway a numerical study would be highly desirable.

\*

\\{\bf 4. Tori breakdown and universality.}
The possibility that the singularities of
the resummed series, as functions
both of $\e$ and $\vec\psi$, have a {\it universal nature} is
related to the behaviour of the large order coefficients, which
are likely to be very mildly dependent on the actual
values of the Fourier components $f_\nn$.
This can indeed be seen to happen when only the contributions
to the coefficients arising from simple classes of trees are
taken into account.
The simplest class of graphs not giving a trivial contribution,
\ie contribution which is an entire function of $\e$, to the invariant
tori is given by the set of trees of the form of a linear chain
$k\leftarrow\ldots\leftarrow 2 \leftarrow 1$, see \cite{GGM}.
For concreteness we fix $\ell=2$, $\oo=(r,1)$ with $r={\sqrt{5}-1\over
2}=$ {\it golden mean}, and the perturbation to be an even function of
$\V\a$ only as $f(\V\a)=a \cos\a_1+ b\cos (\a_1-\a_2)$,
(``Escande--Doveil pendulum'', \cite{ED}).
Let us call ``resonant line" the line ortogonal to $\oo$,
and  let $(p_n,q_n)$ be the convergents for continued fraction for $r$. 
Any integer $s\ge1$ can be written $s=q_n+\sigma_{n-2}q_{n-2}+\ldots
+\sigma_1 q_1$, if $q_n\le s< q_{n+1}$ and
$\sigma_1,\sigma_2,\ldots,\sigma_{n-2}=0,1$,
with the constraint $\s_j\s_{j+1}=0$, $j=1,\ldots,n-3$.
Let $\L_{q_n}$ be the family of self--avoiding walks on the integer
lattice ${\bf Z}^2$ starting at $(0,0)$, ending at $(q_{n},-p_n)$ and
contained in the strip $0<x\leq q_n$, except for the left extreme
points. 
Then a self--avoiding walk joining $(0,0)$ to $(s,s')$ with 
$=q_n+\sigma_{n-2} q_{n-2}+...+\s_1q_1$
and $s'=p_n+\sigma_{n-2}p_{n-2}+\ldots+\sigma_1 p_1$
can be obtained by simply joining a walk in $\Lambda_{q_n}$, one in
$\Lambda_{q_{n-2}}$ if $\sigma_{n-2}=1,\ldots,$ one in $\Lambda_1$ if
$\sigma_1=1$: the latter self--avoiding walks will define the class
$\Lambda_s$ of walks. 
It is clear by the construction that the class $\Lambda_s$ of
self--avoiding walks contains many of the ones which
have the largest products $\prod_j {1\over (i\oo\cdot\nn(j))^2}$ of
small divisors. Therefore we define:
%
$$\vec Z(\Lambda_{q_n})=\sum_{\rm walks\ in \ \Lambda_{q_n}} (-i
\h J^{-1}\nn_{{1}}) { f_{\vec
\n_{{1}}}\over(i\oo\cdot\vec\n(1))^2}
\prod_{j=2}^{k}{f_{\vec\n_{v_j}}(\vec \n_{j-1}\cdot \h
J^{-1}\vec\n_{{j}})\over
(\oo\cdot\vec\n(j))^2}e^{i(q_n\psi_1-p_n\psi_2)} \; , $$
%
(with $\vec\nu(1)=(q_n,-p_n)$ which can be realized as a sum
of copies of the vectors $\nn_1=(1,0)$ and $\nn_2=(1,-1)$).
We expect:
%
$$\vec Z(\Lambda_{q_n})=\vec\z\
{C(\h,f)^{q_n}\over q_n^\d} \, e^{i(q_n\psi_1-p_n\psi_2)}(1+O(
q_n^{-1}))\equiv\vec\z Z(\L_{q_n})
= Z_n\,e^{i(q_n\psi_1-p_n\psi_2)} \; , $$
%
where $\vec\z$ is a suitable unit vector, $C(\h,f)$ is a suitable function
of $\h$ and $f$, and $\d$ is a ``critical exponent''
characteristic of the golden mean.
Let $\e_n=(r p_n- q_n)$; then $Z(\L_s)$ is given by
$Z(\L_s)\simeq Z(\Lambda_{q_n})
Z(\Lambda_{q_{n-2}})^{\sigma_{n-2}}\ldots Z(\Lambda_{q_1})^{\sigma_1}$,
if $s<q_{n+1}$, while $Z(\Lambda_{q_{n+1}})\simeq
Z(\Lambda_{q_n})Z(\Lambda_{q_{n-1}})
\Big({\e_{n-1}\over\e_{n+1}}\Big)^2$,
if $s=q_{n+1}$, (we can define, for consistency,
$Z(\L_{q_0})=\e_0^{-2}=r^{-2}$ and $Z(\L_{q_{-1}})=(\e_0/\e_{-1})^2=r^2$).
Then the contribution to $\V h^*$ due to the above classes of trees
and walks can be written, approximately, if $r_n={p_n\over q_n}$, as
%
$$ \vec\z\sum_{n=1}^\i { [C(\h,f)e^{i(\psi_1-r_n\psi_2)}]^{q_n}\over
q_n^\d}=\vec\z \sum_{n=1}^\i {[C(\h,f) e^{i(\psi_1-r\psi_2)}]^{q_n}
\over q_n^\d}e^{-i\psi_2 O({1\over q_n})} \; . $$
%
We expect that {\it the singularities of the resummed series,
as $\e$ grows, are the same as those of $\vec H,\vec h$}, (see \S 3), and
furthermore we expect the above considered contributions to the
parametric equations of the
resummed tori to be the {\it most singular}.
Hence we interpret the above equation as saying that we
should expect $\vec h,\vec H$ to be, at the breakdown of the invariant torus
which corresponds to $|C(\h,f)|=1$, singular as functions of
$\psi_1,\psi_2$ and of $\h$ (hence of $\e$).
Furthermore (1) the set $|C(\h,f)|=1$ is in the $\h$--plane a natural
boundary for the above approximations to the
functions $\V h^*,\V H^*$ as functions of $\h$, and (2) if
$\h={\e\over 1-\sigma_\e}$ is supposed smooth in $\e$ or at least
Lipshitz continuous, (a possibility mentioned above),
when $\e\to\e_c^-$, one has $C(\h,f)\simeq (1-\g(\e_c-\e))$,
so that the singularity of $\vec
h(\psi_1,0)$ or $\vec H(\psi_1,0)$ in $\e,\psi_1$ is described by the
singularity of a single function $\x(z)$, or, respectively,
$\x'(z)$, of the single
variable $z=e^{-\g(\e_c-\e)} e^{i\psi}$,
$\x(z)=\sum_{k=1}^\io \frac{z^{q_k}}{q_k^\d}$, or
$\x'(z)=\sum_{k=1}^\io \frac{z^{q_k}}{q_k^{\d+1}}$.
%
This would mean that the {\it critical torus} has a Lipshitz continuous
regularity with {\it any} exponent $\d'<\d$ in the $\psi_1$--variable
and $\d$ in the $\e-\e_c$ variable, \cite{K}.
For instance if we fix $\psi_2$, \eg as $\psi_2=0$ (which can be
regarded as a special Poincar\'e section of the invariant torus), then
$\V h^*$ is a $C^{\d'}$ function and $\vec H^*$, which is obtained from $\V
h$ by applying the operator $\vec\omega\cdot\partial_{\vec\psi}$, 
is a $C^{1+\d}$
function, \cite{K}.  Therefore, based on
the hypothesis that the singularity of $\vec H,\vec h$ and of
$\vec H^*,\vec h^*$ are the same and on the above heuristic
discussion, the following conjecture emerges.

\*
  
\\{\bf 5. Conjecture.} {\it Consider the conjugacy
$\aa \leftrightarrow \pps$ to a pure rotation
of the motion generated by the Poincar\'e map on a circle on the
critical torus. There is $\d>0$ such that
the coniugacy is described by two functions $\V h,\V H$ and written as
$\V\a=(\psi,0)+(h_1(\psi),h_2(\psi))$ and $\vec A=(H_1(\psi),H_2(\psi))$
with $\vec h$ H\"older continuous with exponent $\d'<\d$ and $\vec H$ of
class $C^{1+\d'}$. Furthermore the above conjugacy has a H\"older
continuous regularity $\d'<\d$ in the $\e-\e_c$ variable.}

In particular  $\vec H$ is ``once more differentiable'' than $\vec h$,
for $\e=\e_c$; this is quite surprising and, perhaps, quite unexpected as 
$\V H$ is obtained from $\V h$, by applying to it 
$\oo\cdot\partial_{\pps}$. The idea of studying the
universality of the tori breakdown from a tree expansion is a refined
version of an important idea in \cite{PV}: except that we have
{\it not} made here the simplifying assumption of
absence of resonances (\ie we {\it allow} for non zero Fourier
components of opposite wave label $\pm\nn$,
and find resummations that in some sense eliminate them).
If one accepts that the above pendulum system has the same critical
exponents for the golden mean torus in the standard map then it follows
that $\d=0.7120834$ by the scaling argument on p. 207 of \cite{M}.\

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%
\bibitem{E}
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Absolutely convergent series expansions
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%
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%
\bibitem{G}
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%
\bibitem{GGM}
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%
\bibitem{GM3}
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%
\bibitem{GM1}
{Gentile, G., Mastropietro, V. (1995)
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%
\bibitem{GM2} 
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Methods for the analysis of the Lindstedt series for KAM tori
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%
\bibitem{K}
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%
\bibitem{M}
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%
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%
\end{thebibliography}

\end{document}